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Title:
Data storage, processing and visualisation for the ATCA |
Abstract: We present three Virtual Observatory tools developed at the ATNF for the
storage, processing and visualisation of ATCA data. These are the Australia
Telescope Online Archive, a prototype data reduction pipeline, and the Remote
Visualisation System. These tools were developed in the context of the Virtual
Observatory and were intended to be both useful for astronomers and technology
demonstrators. We discuss the design and implementation of these tools, as well
as issues that should be considered when developing similar systems for future
telescopes.
| https://export.arxiv.org/pdf/astro-ph/0601354 |
\twocolumn[
\begin{changemargin}{.8cm}{.5cm}
\begin{minipage}{.9\textwidth}
\vspace{-1cm}
\small{\bf Abstract:} We present three Virtual Observatory tools developed at the ATNF for
the storage, processing and visualisation of ATCA data. These are the Australia
Telescope Online Archive, a prototype data reduction pipeline, and the Remote Visualisation
System. These tools were developed in the context of the Virtual Observatory and were
intended to be both useful for astronomers and technology demonstrators.
We discuss the design and implementation of these tools, as well as issues that should be
considered when developing similar systems for future telescopes.
\medskip{\bf Keywords:} astronomical data bases: miscellaneous --- methods: data analysis
\medskip
\medskip
\end{minipage}
\end{changemargin}
]
\small
\section{Introduction}
The so-called data explosion in astronomy promises exciting new scientific developments,
but brings with it many technical challenges, in collecting, storing, transporting,
processing and visualising data.
Virtual Observatories (VO) have developed to meet some of these technical challenges.
Falling under the broad area of e-science (which incorporates other scientific domains
facing similar challenges, such as genetics and particle physics) the aim of Virtual
Observatory research is to provide the tools necessary for dealing with this data.
The Australian Virtual Observatory (Aus-VO)\footnote{{\tt http://www.aus-vo.org}}
was started in 2003 with the aim of both contributing to the international VO effort,
and developing tools of use to Australian astronomers.
Australia has many areas of expertise (for example radio astronomy) and it makes sense
to focus our efforts on providing modern tools for working in these areas.
In this context the ATNF decided to develop a range of tools for storing, processing
and visualising data from the Australia Telescope Compact Array.
The aim was that these tools would be useful to astronomers now, and at the same time
let us explore the technology that would be necessary for developing software for
future telescopes such as the Square Kilometre Array (SKA).
This paper is based on a talk given at the ASA Annual Meeting in Sydney in July 2005.
After giving some background about the ATCA and the Virtual Observatory, we discuss three
tools developed at the ATNF over the last few years.
Firstly the Australia Telescope Online Archive which contains all of the data collected
so far by the ATCA; secondly a prototype data reduction pipeline for ATCA data; and
finally the Remote Visualisation System for viewing large datasets.
\section{The ATCA}
The Australia Telescope Compact Array (ATCA) is an east-west earth-rotation synthesis
interferometer, with six 22 m antennas on a 6 km baseline.
It has been in operation at Narrabri since 1990.
The telescope can observe at 6 bands with wavelengths 20 cm, 12 cm, 6 cm, 3 cm, 1 cm
and 3 mm.
Each antenna observes two frequencies simultaneously.
There are six bandwidths available on Frequency 1 (128, 64, 32, 16, 8 and 4 MHz) and two
bandwidths on Frequency 2 (128 and 64 MHz).
The telescope produces $\sim 0.5$ GB of raw data per day, and this is likely to increase
significantly with future telescope upgrades.
In the rest of this section we outline the existing systems for archiving, processing
and visualising ATCA data.
\subsection{Data archiving}\label{s_da}
Since the commencement of operation of the ATCA in June 1990, a complete record of all
data observed from the telescope has been maintained offline at the telescope site,
mostly recorded on CD.
In conjunction with this, the ATNF maintained a record of the project proposals for
observations on the telescope --– the {\it Projects database} --- and a short form of the
observation parameters for each day’s observing --– the {\it Positions database}.
After a proprietary period of 18 months, in which the observing team has sole access to
the data obtained in an observation, the data is made publicly available.
Astronomers can search for observations on the ATNF webpage, submit the details
of the observation data required via e-mail and have a CD containing the data prepared
for them at nominal cost.
\subsection{Data processing}
ATCA Data processing (reduction) is generally performed with one of the standard radio data
reduction packages; most commonly Miriad \citep{sault95}, but also
AIPS\footnote{Astronomical Image Processing System (AIPS), http://www.cv.nrao.edu/aips.}
and AIPS++\footnote{Astronomical Image Processing System (AIPS++), http://aips2.nrao.edu.}.
After loading, editing and calibrating the data, the resulting product is an intensity map
referred to as the dirty image.
At this stage a deconvolution algorithm, usually a variant of {\sc clean}
\citep{hogbom74}, is required to produce the final image.
At each stage in the process there are a range of parameters that can be set to control
the type of processing performed.
Both general parameters (such as calibration strategy, {\sc clean} method and type of
data editing) as well as fine-grained parameters (such as calibration solution interval,
number of {\sc clean} iterations and median filter size) need to be modified to obtain
the best results.
Hence data processing is typically a highly interactive process.
There is an existing system operating at the telescope {\tt CAONIS} designed for on-the-fly
imaging of ATCA data.
However the design of this system makes it difficult to port to current Linux systems.
\subsection{Data visualisation}
The final images from the ATCA are usually visualised using tools such as Miriad or
{\tt kvis} \citep{gooch96}.
These are well established tools that cover many of the visualisation requirements of
ATCA observers.
As mentioned in the previous section, the {\tt CAONIS} system which runs at Narrabri also
allows basic visualisation of images.
\section{The Virtual Observatory}
The umbrella organisation for Virtual Observatory work is the International Virtual
Observatory Alliance (IVOA).
The IVOA was formed in June 2002 with a mission to
\begin{quotation}{\it
\noindent facilitate the international coordination
and collaboration necessary for the development and deployment of the tools, systems and
organizational structures necessary to enable the international utilization of astronomical
archives as an integrated and interoperating virtual observatory.}
\end{quotation}
The IVOA is a collaboration between over 15 member countries including Australia.
The focus so far has been developing the standards required for interoperability between
software developed and data produced in all areas of astronomy.
Another significant aim is to develop the infrastructure required (networks and
organisations) for the large scale storage and distribution of astronomical data.
The IVOA working groups address a range of issues such as grid and web services,
data modelling and standards for the data access.
There are also four interest groups
\begin{itemize}
\item Applications IG
\item Astronomy Grid IG
\item Data Curation IG
\item Theory IG
\end{itemize}
which focus on the requirements of particular application domains.
The aim of the Australian Virtual Observatory (Aus-VO) is to provide distributed, uniform
interfaces to the data archives of Australia's major observatories and the archives of
simulation data.
Aus-VO is a collaboration between many Australian institutions, including the Universities of
Melbourne, Sydney, New South Wales and Queensland, Monash University,
Swinburne University of Technology, the Australian National University and Mount Stromlo
Observatory, the Victorian Partnership for Advanced Computing, the ATNF and the AAO.
There are a range of VO projects underway in Australia, including the development of
data archives and software for HIPASS \citep{meyer04}, RAVE \citep{siebert04},
2QZ \citep{croom04} and SUMSS \citep{bock99}.
The initial focus of most of these projects has been to make data from Australian projects
widely available within the international community, in a VO compliant format.
In addition there are several projects investigating novel methods for astronomical
data mining and data analysis, for example \citet{rohde05} apply machine learning
techniques to catalogue crossmatching.
The Melbourne University group has also been setting up infrastructure such as a registry
for Australian web services and data archives.
\section{The Australia Telescope \\ Online Archive}
In June 2003, a joint project between the ATNF and the CSIRO ICT Centre was commenced to
make the ATNF archive data available online as the
Australia Telescope Online Archive (ATOA).
This was planned as a new data resource for astronomers, as well as the foundation for the
development of online data processing systems to make the raw data more accessible to
non-expert users (see Section \ref{s_pipe}).
The construction of the ATOA required the copying of the offline archive (at the time,
$\sim 2700$ CDs, totalling $\sim 1.7$ TB) from the telescope site to Canberra where
the online archive was to be developed, creating a meta-database describing the data, and
making a web front-end to search and download the data.
The database consists of two parts.
The first is the raw data from the telescope (RPFITS files) which is stored as normal files
on the host system.
In addition there is a relational database which stores all the metadata
(discussed in Section \ref{s_meta}).
The `vital statistics' of the ATOA are shown in Table \ref{t_atoa}.
The current rate of growth of the archive is $\sim 0.5$ Gb/day.
However this is likely to increase significantly in the future as new instruments
come online.
To maintain an growing archive (rather than a static one) it is necessary to ensure the
RPFITS files are stored in a readily accessible way (currently on a RAID system) that is
easily distributed over a number of drives.
Also, that the database itself is easy to update in a robust manner.
The ATOA was made publicly available in December 2004 and
can be accessed from {\tt http://atoa.atnf.csiro.au}.
\begin{table}[ht]
\begin{center}
\caption{ATOA Statistics}\label{t_atoa}
\begin{tabular}{lr}
\hline
Projects & 2261 \\
Files & 57147 \\
Sources & 128111 \\
Metadata size & $\sim 4$ Gb \\
RPFITS data size & $\sim 2$ Tb \\
Growth rate & $\sim 0.5$ Gb/day \\
\hline
\end{tabular}
\end{center}
\medskip
\end{table}
\subsection{Metadata}\label{s_meta}
Metadata is simply data which describes other data, for example the project code or the
name of the primary calibrator.
The meta-database for the ATOA consists of three main parts; the contents of the original
ATNF online Projects database, metadata describing the observation that is extracted directly
from the raw data files produced by the telescope’s software, and metadata inferred from all
of the available data sources to assist in the automation of reducing the telescope’s raw data
to images.
The types of metadata used in the ATOA are summarised in Table \ref{t_meta}.
\begin{table}[ht]
\begin{center}
\caption{Metadata in the ATOA}\label{t_meta}
\begin{tabular}{ll}
\hline Metadata source & Examples \\ \hline
Projects database & proposal; observer name\\
& country; institution \\
Positions database$^*$ & source names \& positions \\
& observing band; receivers \\
RPFITS files & scans; polarisations \\
& array configuration \\
Inferred & calibrator names \& roles\\
& calibrator--target matches \\
\hline
\end{tabular}
\end{center}
$^*$ The positions database is included in our data model, and some of the metadata
is used to reconstruct the observation metadata. However, it is not actually loaded
into the ATOA.
\medskip\\
\end{table}
Most of the metadata available in the ATNF Positions
database is also available from the metadata in the raw data files, and is finer-grained,
since the Positions data is a daily summary, while the file metadata is available for each
telescope pointing.
The inferred metadata in the ATOA is `value added' information that is automatically
determined from the existing metadata, for example the calibration role of each source
(primary calibrator, secondary calibrator, target, etc).
This is discussed further in Section \ref{s_roles}.
\subsection{A data model for the ATCA}\label{s_dm}
A data model is a comprehensive scheme describing how data is to be represented, for
manipulation by humans or computer programs.
Data models are critical for planning how data will be organised within a database as they
describe all the relationships between the different entities.
A section of our data model for the ATCA is shown in Figure \ref{f_dm}.
We now briefly explain the UML (Unified Modelling Language) notation used in the data model.
Each box contains an entity (e.g. {\tt Scan}) that has been identified in
the metadata.
Each entity has attributes (e.g. {\tt restFreq}), each of which are of a specified
data type (e.g. {\tt float}).
Associated entities are connected to each other with lines, which also specify the
cardinality of the relationship. For example \\
\begin{minipage}{8cm}
\begin{center}
\resizebox{7cm}{!}{\includegraphics{uml.eps}}
\end{center}
\end{minipage}
should be read as {\bf ``A scan has 0 or more spectral windows. A spectral windows has 1 or
more scans.''}
The development of a data model that covers the whole of astronomy is an ongoing project
within the international VO community.
We have contributed this data model to the IVOA Data Model WG as an example of a data
model for radio astronomy.
For more information on this topic, see the IVOA Data Modelling
website\footnote{{\tt http://www.ivoa.net/twiki/bin/view/IVOA/IvoaDataModel}}.
The ATOA archive database structure is created directly from the
definitions in the ATOA data model. Parts of the data model contain
information for specific database implementations so that all of the
implementation-specific parts of the database creation are handled in
this process.
The data model in Figure \ref{f_dm} corresponds to the part of the data model
that describes the metadata contained directly in the archive RPFITS
files. The data model for the inferred data is available from the ATOA
web pages\footnote{{\tt http://www.atnf.csiro.au/computing/web/atoa/implementation.html}}.
\subsection{Implementation}
The ATOA web interface was implemented as a Java
(ver. 1.4.2)\footnote{{\tt http://java.sun.com}}
application and is hosted using the Apache Tomcat
(ver. 5.0.28)\footnote{{\tt http://tomcat.apache.org}} web container.
Relational database services are provided by an Oracle 9i instance
running on the same machine.
A web based interface was chosen so as to maximise interoperability and provide
easy access to users.
For example, RPFITS files may simply be downloaded by
constructing a suitable URL for the ATOA file server.
This allows files on the server to be downloaded by a Web browser, by
command-line programs that allow users to fetch the data referred to by
a URL, or by application programs using libraries that allow a URL to be
opened in a similar way to a file on a local file system.
The user interface centres around two main web pages: the query
page which allows users to specify criteria for selecting RPFITS
files from the archive, and a results page which provides the means
for users to inspect the metadata of matching files and download
particular files if desired.
The results page initially presents the
user with a broad, global view of the query results in tabular form
listing details such as file name, file size, principal investigator
and array configuration.
The user may also interactively `drill-down'
for a more detailed view of any file in the list.
RPFITS files can be downloaded individually or in batches.
As mentioned in Section \ref{s_da} ATCA data has a proprietary period of 18 months,
in which it is only accessible to members of the project team.
Authorised access is supported for data within the proprietary access
period.
If a user wishes to access proprietary data they must first go
through a manually verified authentication process after which a password is issued to the
Principal Investigator for that project.
In the future we plan to replace this authentication and
authorisation method with a streamlined system that links the new ATNF
proposal system, OPAL\footnote{{\tt http://opal.atnf.csiro.au/}}, and its authentication
database to the ATOA.
Users will then be given access to proprietary data by using their
OPAL credentials based on the projects they are associated with in
the OPAL system
The ATOA web server and database are hosted on a Dell PowerEdge 750
running Debian Linux 3.0. The host has a 2.8GHz Pentium 4 processor,
2GB of RAM and is attached to a 3 terabyte Apple Xserve RAID for
archive storage.
\section{A data processing pipeline framework}\label{s_pipe}
The data products in radio astronomy are often less accessible to the non-expert than
those in other domains such as optical astronomy.
It requires a reasonably high level of domain expertise to process the raw data and
produce an image.
Obviously for carrying out detailed scientific analysis it would be necessary to
develop this expertise, or collaborate with a radio astronomer.
However in an era of multiwavelength astronomy, astronomers expect to download and
compare data from a variety of telescopes, at a variety of wavelengths.
With this in mind we have developed an automatic pipeline for people who want to
quickly inspect the data in the ATOA, to see if it was suitable for further processing.
One of the aims of this project was to test the viability of `driving' the pipeline
using the metadata discussed in Section \ref{s_meta}. In other words the pipeline should
make decisions about what kind of processing to do --- both on a general
(e.g. continuum/spectral line) and specific level (e.g. number of {\sc clean} iterations).
In this section we discuss the development of extra metadata required to driving the pipeline,
in particular the calibration process.
We then outline our prototype pipeline which can process
single pointing continuum data from the ATOA and is available for testing at
{\tt http://atoa.atnf.csiro.au/test}.
\subsection{Metadata for automatic \\ processing}\label{s_pmeta}
The metadata in the Project and Position databases, while providing information about
which astronomical sources have been recorded in an observing session, does not (in general)
provide any information about the role that the observer intended the source to play in the
observation (eg. primary calibrator, target source).
This would be relatively easy to record in a new system, but as we are dealing with
existing data we had to infer the roles of sources.
Another problem for automatic processing is the grouping of data into valid `observations'.
An expert would typically choose an appropriate subset of files from the archive to image.
However, a non-radio astronomer may choose an subset that contains files that should be imaged
separately, or files that contain data that should be ignored entirely.
Although it is impossible to deal with all cases, our aim was to have the pipeline
group together the selected data in such a way that an image could be made in at
least $80\%$ of cases.
A wide range of observation types can be recognised and characterised using the meta-database
but are not yet processed by the prototype pipeline (e.g. millimetre and spectral line
observations).
In the following section we discuss how we assign the source roles within an observation, and
the algorithm we used to match target sources with the appropriate calibrators.
\subsection{Determining source roles}\label{s_roles}
While matching target sources with their calibrators would be straightforward for an
astronomer it is a challenge for an automatic system.
In a typical (simple) observing session the primary calibrator is recorded for a short period
at the start or end of the observing session; and alternating pointings are made to the
secondary, and the source of interest, or target.
However there is a great variety of different ways that the observer can choose to structure
their observations.
If an observation contains more than one target,
the targets may share, or have distinct, secondary calibrators, depending on their
separation in the sky.
There may be several secondary calibrators for each target, and the same source may be
used for primary and secondary calibration.
In addition, some observers use secondary calibrators that are not in the list of recommended
calibrators, and that list has itself changed over time.
In order to classify the sources in an observing session the following metadata is used
\begin{itemize}
\item The locations and names of sources extracted from the raw telescope data
\item The times and durations of the source pointings
\item The names and locations of the four primary calibrators commonly used at the ATCA
\item A recent ATCA catalogue of recommended secondary calibrators
\item Names of sources extracted from project titles
\item A pre-assembled list of possible calibrator sources
\end{itemize}
Once the source roles have been determined, the proximity in the sky and the proximity
in observation time of the targets and their secondary calibrators are used to match targets
with their respective calibrator(s). For each target pointing, a weight is calculated for
each secondary calibration pointing made within two hours of observation of the target pointing:
\begin{displaymath}
w_{t,s} = \Sigma_{P} \Sigma_{S} e^\frac{-(3a/a_{max})^2}{2}
e^\frac{-(3\Delta t / \Delta t_{max})^2}{2}
\end{displaymath}
\begin{displaymath}
\Delta t_{t,s} < \Delta t_{max}
\end{displaymath}
where $S$ is the set of candidate secondary calibrators, $a$ is the angular separation
between the target and the secondary calibrator, $a_{max}$ is the maximum desirable
separation between the target and the secondary calibrator (and is a function of the
observing frequency band).
$\Delta t$ is the separation of the time midpoints of the target and calibrator
pointings and $\Delta t_{max}$ is the maximum desirable time separation (two hours).
The summation is over all pointings at a target ($\Sigma_P$) and all secondary
calibrators within two hours of a target pointing ($\Sigma_S$).
The $w_{t,s}$ are used to select suitable secondary calibrators for the respective
targets from the calibrators whose $w_{t,s}$ weights dominate for a particular target.
This procedure constructs the metadata required for continuum imaging at centimetre
wavelengths.
The algorithm works well in general, but there are some problematic cases, for example
where the target is a source from the secondary calibrator catalogue.
\subsection{Implementation}
The underlying processing of the ATCA data is carried
out using the Glish scripting language in AIPS++.
The ATOA imaging Web Services interface was constructed using the Apache Axis tools
(ver. 1.1)\footnote{{\tt http://ws.apache.org/axis}}, and
interfaces to the processing scripts through a Perl
(ver. 5.4.8)\footnote{{\tt http://www.perl.com}} script that deals with the control of
the execution of the Glish scripts.
The pipeline client is written using Python (ver. 2.3)\footnote{{\tt http://python.org}},
and the SOAPpy web services tools
(ver. 0.11.3)\footnote{{\tt http://pywebsvcs.sourceforge.net}}.
There were some minor, but difficult to find, problems in interoperation between the SOAPpy
tools and Apache Axis; the data structures used in the web services calls are possibly more
complicated than had been previously used between the two web services implementations.
Documentation in both was not as informative or complete as it might have been.
The pipeline web services can be configured to run directly on the server host, or be
directed to run on other machines through a batch queuing system, since some stages
in the pipeline can run for several CPU minutes.
We used the OpenPBS Batch Queuing System (ver. 2.3)\footnote{{\tt http://www.openpbs.org}}
for queue management, but unfortunately it has no mechanism for
reporting job completion to another program.
After processing for a web service completes, the batch job doing the
processing sends a completion message to the program invoked by the web
service that controls the execution of the processing for the service.
However, at this point, the batch processing system has not yet
transferred the job's output data back to the pipeline server. The
control program then polls the PBS batch queue at five second intervals
to ensure that the batch job has completed.
The raw data from the ATOA, all the intermediate files from the data processing, the log files,
and the resulting images are stored temporarily on the pipeline server.
The first web service call made by a pipeline client reserves a private location for storage,
and requests a lifetime for the storage. The pipeline server has a configurable
maximum lifetime, and the stored data will be deleted after this time expires.
Only clients who have the name of the storage area (a randomly generated string) can
access it.
There is no quota on the storage use of any individual temporary storage
area.
However, a quota may be imposed on the total amount of storage available to
all active storage areas.
The ATOA and pipeline web services
return a URL for the generated images to the end-user's system.
This allows the URL to be passed on to the Remote Visualisation System (see Section
\ref{s_rvs}) image viewing system so that
the image can be viewed online while it is still resident on the
pipeline server.
Figure \ref{f_arch} show the overall system architecture, in particular how the ATOA,
pipeline and RVS interact.
\section{The Remote Visualisation \\ System}\label{s_rvs}
The Remote Visualisation System (RVS) was designed to enable visualisation of
and interaction with large astronomical images in the context of the VO.
As opposed to other VO image displays, such as CDS Aladin \citep{fernique04},
RVS does not require the user to download the data to the client machine.
Furthermore it provides rendering of image cubes, such as spectral-line cubes
created from ATCA data.
The RVS server accepts FITS images - which can be compressed - through local
{\tt file://} URLs and remote http or ftp access.
The data should be co-located with or at least be available to the server on
high bandwidth connection, while it places no such requirements on the client.
Only minimal data transfer to the client is necessary and this is independent
of the size of the source data set.
The server-side architecture is distributed to enable workload sharing and
extensibility.
RVS makes use of several software components: CORBA to make it distributed,
AIPS++ as the image rendering component and Java for the web services
and client.
The architecture of the RVS system is shown in Figure \ref{rvs_arch}.
RVS is exposed through a web service interface using the standard Web Service
Description Language (WSDL ver. 1.1)\footnote{{\tt http://www.w3.org/TR/wsdl}}.
This can easily be integrated into custom applications.
Several client applications make use of the web service interface; the
RVSViewer - a traditional image viewer, a thumbnail service - providing
preview images and a session viewer.
The session viewer connects to an existing RVSViewer via a key.
Multiple instances can be run at the same time, making it a possible to use
it as a conferencing tool where people can observe and interact with the data.
The ATOA pipeline re-uses the existing RVSViewer client by passing it the file
location of the output image.
RVS is not specific to the ATOA or prototype pipeline and there are plans to
use it for all ATCA archives.
It has been successfully tested on images and data cubes from various surveys and has
good performance on large datasets.
For example a 1.5 Gb data cube from the Galactic All-Sky Survey (GASS) \citep{mcclure05}
takes about one minute to load.
Compare this with downloading the full cube from say the U.S. to Australia which could
take $\sim 1-2$ hours.
For more information and direct access to RVS, see {\tt http://www.atnf.csiro.au/vo/rvs/}.
\section{Discussion}
The ATOA has been public since December 2004.
We hope that it will encourage the reuse of ATCA data for projects other than
those it was originally intended for.
The framework used for the ATOA could easily be extended to include data from other
telescopes and can be updated as additional metadata is required.
The most significant improvement of the ATOA over existing online archives (such
as NRAO\footnote{{\tt http://archive.nrao.edu/archive/e2earchive.jsp}} and
MAST\footnote{{\tt http://archive.stsci.edu/}}) is the data delivery mechanism.
Most existing archives do not support on demand delivery
of data over the web, instead requiring the user to submit a form
requesting files that then have to be transfered to a publicly
accessible ftp site or to other media (such as CD) for physical
delivery.
In the ATOA, the batch downloading of multiple files is handled by a
streaming TAR or ZIP archiving algorithm that performs dynamic archiving as
files are streamed over the web, requiring no additional disk space on the
server for these operations.
In developing the ATCA data model and considering the type of metadata required for automatic
processing we identified several new metadata types that would be useful to store in the
RPFITS files.
As a result the following fields have been added to the RPFITS files and will be available in
all future ATCA data:
\begin{itemize}
\item four calibrator codes
\begin{description}
\item[C] (standard phase calibrator)
\item[F] (primary flux calibrator)
\item[B] (bandpass calibrator)
\item[P] (pointing calibrator)
\end{description}
\item Pointing offsets
\item Weather data: added rain gauge and phase rms and difference
\item Attenuator settings at start of scan
\item Subreflector position
\item Correlator configuration
\item Scan type
\item Coordinate type
\item Line mode
\item CACAL counter
\end{itemize}
These will help both automatic processing systems and astronomers assess the data quality
in the observations they are interested in.
A full e-logbook system will be used in the future as currently the logs are all stored on
paper at the telescope and hence are not easily accessible to ATOA users.
We have developed a prototype pipeline for processing of raw data for single-pointing
continuum images.
This is attached to the ATOA to provide an improved service for users of the ATOA.
At this stage the image quality is suitable for previewing the data in archive to see
if it is of interest.
Further manual processing would then be required to obtain images of scientific quality.
A significant challenge in developing the ATOA and the prototype pipeline were integrating
pre-existing software with modern software tools.
For example, the Glish scripting language has no web service libraries and so an extra
layer had to be developed between the data processing level and the web services.
If re-implementing from scratch, a language such as Python would be a better alternative
for developing the pipeline.
In developing these tools we have started to explore the techniques necessary for
astronomical software development in the VO era.
This is essential for future telescopes and surveys that Australia will produce.
Making access to existing Australian data as easy as possible will maximise
its use in the international community.
\section*{Acknowledgments}
The authors would like to acknowledge the software development done
on the RVS project, primarily by Anil Chandra and also by Praveena Tokachichu.
The ATNF side of the prototype pipeline and ATOA development was managed by
Neil Killeen and Jessica Chapman.
Vince McIntyre contributed extensively to all three projects, in particular
in setting up the hardware required.
A number of ATNF staff put in significant effort to get the ATOA set up,
in particular Robin Wark, Bob Sault and Mark Wieringa.
Warwick Wilson and Mark Wieringa implemented the changes to add extra
metadata to the RPFITS files.
From the ICT Centre, Robert Power made the initial data model designs,
the ATOA data loader software and ATOA query front end. Geoff Squire and
Bella Robinson made significant contributions to the prototype pipeline.
|
Title:
The Araucaria Project. The Distance to the Local Group Galaxy IC 1613 from Near-Infrared Photometry of Cepheid Variables |
Abstract: We have measured accurate near-infrared magnitudes in the J and K bands of 39
Cepheid variables in IC 1613 with well-determined periods and optical VI light
curves. Using the template light curve approach of Soszy{\'n}ski, Gieren and
Pietrzy{\'n}ski, accurate mean magnitudes were obtained from these data which
allowed to determine the distance to IC 1613 relative to the LMC from a
multiwavelength period-luminosity solution in the optical VI and near-IR JK
bands, with an unprecedented accuracy. Our result for the IC 1613 distance is
$(m-M)_{0} = 24.291 \pm 0.014$ (random error) mag, with an additional
systematic uncertainty smaller than 2%. From our multiwavelength approach, we
find for the total (average) reddening to the IC 1613 Cepheids $E(B-V) = 0.090
\pm 0.007$ mag,which is significantly higher than the foreground reddening of
about 0.03 mag,showing the presence of appreciable dust extinction inside the
galaxy. Our data suggest that the extinction law in IC 1613 is very similar to
the galactic one.Our distance result agrees, within the uncertainties, with two
earlier infrared Cepheid studies in this galaxy of Macri et al. (from HST data
on 4 Cepheids), and McAlary et al. (from ground-based H-band photometry of 10
Cepheids), but our result has reduced the total uncertainty on the distance to
IC 1613 (relative to the LMC) to less than 3%. With distances to nearby
galaxies from Cepheid infrared photometry at this level of accuracy, which are
currently being obtained in our Araucaria Project, it seems possible to
significantly reduce the systematic uncertainty of the Hubble constant as
derived from the HST Key Project approach, by improving the calibration of the
metallicity effect on PL relation zero points, and by improving the distance
determination to the LMC.
| https://export.arxiv.org/pdf/astro-ph/0601309 |
\newcommand{\up}[1]{\ifmmode^{\rm #1}\else$^{\rm #1}$\fi}
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\title{The Araucaria Project. The Distance to the Local Group
Galaxy IC 1613 from
Near-Infrared Photometry of Cepheid Variables.
\footnote{Based on observations obtained with the NNT telescope
at ESO/La Silla for programs 074.D-0318(B) and 074.D-0505(B)
}
}
\author{Grzegorz Pietrzy{\'n}ski}
\affil{Universidad de Concepci{\'o}n, Departamento de Fisica, Astronomy
Group,
Casilla 160-C,
Concepci{\'o}n, Chile}
\affil{Warsaw University Observatory, Al. Ujazdowskie 4, 00-478, Warsaw,
Poland}
\authoremail{pietrzyn@hubble.cfm.udec.cl}
\author{Wolfgang Gieren}
\affil{Universidad de Concepci{\'o}n, Departamento de Fisica, Astronomy Group,
Casilla 160-C, Concepci{\'o}n, Chile}
\authoremail{wgieren@astro-udec.cl}
\author{Igor Soszy{\'n}ski}
\affil{Universidad de Concepci{\'o}n, Departamento de Fisica, Astronomy Group,
Casilla 160-C, Concepci{\'o}n, Chile}
\affil{Warsaw University Observatory, Al. Ujazdowskie 4, 00-478, Warsaw,
Poland}
\authoremail{soszynsk@astro-udec.cl}
\author{Fabio Bresolin}
\affil{Institute for Astronomy, University of Hawaii at Manoa, 2680 Woodlawn
Drive,
Honolulu HI 96822, USA}
\authoremail{bresolin@ifa.hawaii.edu}
\author{Rolf-Peter Kudritzki}
\affil{Institute for Astronomy, University of Hawaii at Manoa, 2680 Woodlawn
Drive,
Honolulu HI 96822, USA}
\authoremail{kud@ifa.hawaii.edu}
\author{Massimo Dall'Ora}
\affil{INAF-Osservatorio Astronomico di Capodimonte, via Moiariello 16, I-80131 Naples, Italy}
\authoremail{dallora@na.astro.it}
\author{Jesper Storm}
\affil{Astrophysikalisches Institut Potsdam, An der Sternwarte 16, D-14482 Potsdam,
Germany}
\authoremail{jstorm@aip.de}
\author{Giuseppe Bono}
\affil{INAF-Osservatorio Astronomico di Roma, via Frascati 33, 00040 Monte Porzio Catone,
Italy}
\authoremail{bono@mporzio.astro.it}
\keywords{distance scale - galaxies: distances and redshifts - galaxies:
individual(IC 1613) - stars: Cepheids - infrared photometry}
\section{Introduction}
Cepheid variables are the most important standard candles to calibrate
the first rungs of the extragalactic distance ladder, out to some 30 Mpc. As young
stars, Cepheids tend to lie in dusty regions in their spiral or irregular parent galaxies.
As a consequence, Cepheid distances derived from the period-luminosity (PL) relation
in optical photometric bands are quite sensitive to a precise knowledge of the total
reddening, foreground and intrinsic, of their parent galaxy. While the galactic
foreground reddening towards any
direction in the sky is usually well established, particularly in directions far away
from the galactic equator, the correct assessment of the reddening produced by
dust extinction {\it intrinsic to the host galaxy} is usually a difficult task, and in most work
on Cepheid distances based on optical data such an intrinsic contribution to the
reddening has simply been ignored. Just for this one
particular reason, it is clear that more accurate Cepheid distances to galaxies
can be derived in near-infrared passbands, where dust absorption is small as compared
to visual wavelengths, and the distance results become increasingly independent of errors
in the assumed total reddenings. Efforts along these lines have started in the early eighties
with the pioneering work of McGonegal et al. (1982), and Welch et al. (1985).
Yet, an important obstacle to carry out accurate Cepheid distance work
in the infrared has been, until very recently, the lack of well-calibrated fiducial
PL relations in the near-infrared JHK bands. This problem has now been solved
by the work of Persson et al. (2004) who provided such well-calibrated relations for the LMC.
Very recently, Gieren et al. (2005a) have also provided
well-calibrated PL relations in the JHK bands for Milky Way Cepheids, which
agree with the corresponding LMC relations when an improved version of their infrared surface
brightness technique (Gieren, Fouqu{\'e} \& G{\'o}mez, 1997, 1998) is used.
In the {\it Araucaria Project}, started by our group some time ago (Gieren et al. 2005c),
we have conducted surveys for Cepheid variables in a number of galaxies in the Local Group,
and in the more distant Sculptor Group in order to investigate the effect of environmental properties on
the PL relation, and to improve the accuracy of Cepheids as distance indicators. While we
are discovering Cepheids in optical photometric bands, where these stars are rather easy to detect
due to their relatively large amplitudes and typical light curve shapes (e.g. Pietrzynski et al. 2002, 2004),
the main goal of the program is
to undertake near-infrared follow-up imaging of selected subsamples of Cepheids
in our target galaxies to obtain accurate reddening information, and thus to obtain more
accurate distances than what is possible from optical (VI) data alone. Near-infrared PL relations
from such Cepheids with existing information on their periods, and V and/or I light curves
can be obtained very economically because it is possible to obtain accurate mean JHK magnitudes for these stars
from just one single-phase observation from the template
light curve approach of Soszynski, Gieren and Pietrzynski (2005). The success
of this approach was recently demonstrated in the case of the Sculptor galaxy NGC 300 (Gieren et al. 2005b).
For this galaxy, a combination of the PL relations obtained in the optical VI and
infrared JK bands has allowed to determine a distance which is practically unaffected
by any remaining uncertainty on reddening. It has also been shown in that paper that
from the combined optical/near-infrared approach
a total uncertainty as small as 3 \% can be obtained for the Cepheid distance (as measured
relative to the LMC) for such a relatively nearby (2 Mpc) galaxy.
In the present paper, we are applying the same approach to the Local Group dwarf irregular galaxy IC 1613.
IC 1613 is a very important galaxy in the Araucaria Project because of the very low metallicity of its young stellar
population close to -1.0 dex (Skillman et al. 2003), making it the lowest-metallicity galaxy
in our sample. It is therefore a key object
in our effort to determine the effect of metallicity on the Cepheid PL relation, and on other
stellar distance indicators, like blue supergiant stars (Kudritzki et al. 2003).
A first survey for Cepheid variables in IC 1613 was carried out by Sandage (1971) who used
photographic images previously obtained by Baade. Modern work
on the Cepheid PL relation in IC 1613, in the optical V and I bands, has been carried out
by the OGLE Project (Udalski
et al. 2001) who has discovered many new, previously unknown Cepheids in this irregular galaxy.
More recently, Antonello et al. (2006) have extended this work to the B and R bands. From the
work of the OGLE group, it could be established that the slope of the PL relation in optical bands is
identical to the slope observed for the more metal-rich LMC Cepheids, arguing for a metallicity-
independent slope of the PL relation. In the near-infrared, a pioneering paper on the
distance of IC 1613 from H-band photometry of 10 Cepheids was published by McAlary et al. (1984)
already 20 years ago;
however, the uncertainty on this distance result was rather large due to the technical difficulties
to obtain accurate IR photometry at those times, and the lack of an accurate calibrating PL relation.
Much more recently, Macri et al. (2001) determined a near-infrared Cepheid distance to IC 1613 from
H-band photometry of four variables obtained with HST/NICMOS. The accuracy of this determination
suffers, however, from the
very small number of stars used in the PL solution. A main goal of the present study was
to derive {\it truly accurate near-infrared PL relations for IC 1613}, based on a large number
of well-observed and well-selected Cepheids (see section 3.2.), and this way reduce the current uncertainty
on the distance to IC 1613 to the very small level of 3-5\% we have achieved in our previous study
of NGC 300.
We have organized this paper in the following way: in section 2,
we describe the observations, reductions and calibration of our data; in section 3, we present
the calibrated infrared mean magnitudes of the Cepheids in our selected fields in IC 1613, and
determine the distance and reddening; in section 4, we discuss our results; and in section 5, we
summarize the main results of this work, and present some conclusions.
\section{Observations, Data Reduction and Calibration}
\subsection{Optical data}
Our infrared observations of IC 1613 (see next section) were obtained
about four years after the OGLE-II optical observations (Udalski et al.
2001) of this galaxy. This long gap in time made it necessary to improve the periods
of the Cepheids in order to calculate accurate $<K>$ and $<J>$
mean magnitudes from single-phase infrared observations with the method of Soszynski et al. (2005).
For this purpose, three new V-band observations
of IC 1613 with the 1.3 m Warsaw telescope located at Las Campanas
Observatory were secured in September 2005. This telescope is equipped with a mosaic 8k x 8k detector,
with a field of view of 35 x 35 arcmin and a scale
of 0.25 arcsec/pixel. Preliminary data reductions (i.e. debiasing
and flatfielding) were done with the IRAF \footnote{IRAF is distributed by the
National Optical Astronomy Observatories, which are operated by the
Association of Universities for Research in Astronomy, Inc., under cooperative
agreement with the NSF.} package. The point-spread
function photometry was obtained with the DAOPHOT and ALLSTAR programs,
in an identical way as described in Pietrzy{\'n}ski et al. (2002).
Our photometry was then transformed to the standard system using the
OGLE-II list of carefully calibrated stars in this galaxy (Udalski et
al. 2001).
\subsection{Infrared data}
The near-infrared data presented in this paper were collected with the ESO
NTT telescope on La Silla, equipped with the
SOFI infrared camera. In the setup we used (Large Field), the field
of view was 4.9 x 4.9 arcmin, with a scale of 0.288 arcsec / pixel.
The gain and readout noise were 5.4 e / ADU and 0.4 e, respectively.
The data were obtained under two observational programs:
074.D-0318(B), 074.D-0505(B) (PI: Pietrzy{\'n}ski) as part
of the Araucaria Project.
Alltogether, six different, slightly overlapping fields were
observed through J and Ks filters. Their location is shown
in Figure 1, and the equatorial coordinates of their centers
are given in Table 1.
Single deep J and Ks observations of our six fields
were obtained under excellent seeing conditions during three different
photometric nights. On these nights, we also observed a large number of photometric standard
stars from the UKIRT system (Hawarden et al. 2001). In order to account
for the rapid sky level variations in the infrared domain, the
observations were performed with a dithering technique.
In the Ks filter, we obtained six consecutive 10 s integrations (DITs)
at a given sky position and then moved the telescope by about 20 arcsec
to a different random position. Integrations obtained at 65 different
dithering positions resulted in a total net exposure time of 65 minutes in this
filter, for a given field. In the case of the J filter, in which the sky level variations
are less pronounced than in K, two consecutive 20 s exposures were
obtained at each of 25 dithering positions, which corresponded to
a total net exposure of about 17 minutes, for any given field.
Sky subtraction was performed by using a two-step process implying the
masking of stars with the XDIMSUM IRAF package in an analogous manner as described in
Pietrzy{\'n}ski and Gieren (2002). Then the single images were flatfielded
and stacked into the final images. PSF photometry was obtained
using DAOPHOT and ALLSTAR, following
the procedure described in Pietrzy{\'n}ski, Gieren, and Udalski (2002).
In order to derive the aperture corrections for each frame, about 7-10
relatively isolated and bright stars were selected, and all
neighbouring stars were removed using an iterative procedure. Finally,
we measured the aperture magnitudes for the selected stars with the
DAOPHOT program using apertures of 16 pixels. The median from the
differences between the aperture magnitudes obtained this way, and the corresponding PSF
magnitudes, averaged over all selected stars was finally adopted as
the aperture correction for a given frame. The rms scatter from
all measurements was always smaller than 0.02 mag.
In order to accurately transform our data to the standard system
a large number (between 8 and 15) of standard stars from the UKIRT
system (Hawarden et al. 2001) was observed under photometric conditions
at a variety of airmasses, together with our six science fields.
The standard stars were chosen to have colors bracketting the colors of the
Cepheids in IC 1613. The aperture photometry for our standard stars
was performed with DAOPHOT using the same aperture as for the
calculation of the aperture corrections. Given the relatively
large number of standard stars we observed on each night, the transformation coefficients
were derived for each night. The accuracy of the zero points of our photometry was
determined to be about 0.02 mag.
Since our six fields overlap (see Fig. 1), we were able to perform an
internal check of our photometry comparing the magnitudes of stars located
in the common regions. In every case, the independently calibrated
magnitudes agree within 0.02 - 0.03 mag in both, K and J filters.
Unfortunately, we are not aware of any other deep near-infrared JK images
obtained for IC 1613, so an external check of our photometry
is not possible. However, the magnitudes of the bright stars
in our fields can be compared with the 2MASS photometry. Fig. 2
presents the difference between our K magnitudes and J-K colors, and the corresponding
2MASS data for common bright stars. Before calculating these differences, we transformed
our photometry, which
had been calibrated onto the UKIRT system, to the 2MASS
system using the equations provided by Carpenter (2001). In spite of the
relatively large uncertainties of the 2MASS data for the fainter stars in Fig. 2,
it is appreciated that there is
no evident zero point offset either in K or in J-K, supporting the conclusion
that both datasets are well calibrated, within 0.02-0.03 mag.
The pixel positions of the stars were transformed to the equatorial
coordinate system using Digital Sky Survey (DSS) images. For this purpose, we used
the algorithm developed and used in the
OGLE project (Udalski et al. 1998). The accuracy of our
astrometric transformations is better than 0.3 arcsec.
\section{Results}
\subsection{The Cepheid mean K and J band magnitudes}
In the six NTT/SOFI fields observed in this project, 39 objects from
the Cepheid list presented by Udalski et at. (2001) were identified.
It is worth noticing that most of the (few) long-period OGLE-II Cepheids in IC 1613
are located in our fields. Thanks to the depth of our infrared photometry, we were able
to detect Cepheids with periods down to about 2 days.
In Table 2, we present the journal of individual observations of these 39
Cepheid variables. Most of them were observed only once. However, several
objects located in the overlapping areas were observed twice.
Before deriving the mean K and J magnitudes of the Cepheids from these data,
we tried to improve the periods given by Udalski et al. (2001)
using our new three-epoch optical observations, which are given in Table 3.
For the long-period Cepheids (P larger than about 5 days),
we could indeed improve the periods with these new data. However, for the variables
with shorter periods
the time elapsed between two sets of observations
was too large in order to unambiguously count the number of elapsed
cycles. For these Cepheids, we adopted the OGLE II periods from Udalski et al. (2001).
The mean magnitudes were obtained from the template light curve method of Soszy{\'n}ski, Gieren and
Pietrzy{\'n}ski (2005) which uses the V-band phases of the individual near-IR observations,
and the light curves amplitudes in V and I to calculate the differences of the individual
single-phase magnitudes to the mean magnitudes in J and K.
For a detailed description of this technique, the reader is referred to
this paper. It has been demonstrated by
these authors that the mean K and J magnitudes of Cepheids can be derived from just
one random-phase observation with an accuracy of 0.02-0.03 mag provided that
high-quality optical and infrared data, and periods are available for the stars.
In Table 4, we present the final intensity mean JK magnitudes of the 39 Cepheids in our fields
with their estimated uncertainties and their adopted periods. The last
column contains remarks on some of the variables. V2, V6, etc.,
correspond to the numbering system introduced by Sandage (1971).
\subsection{Selection of the final sample}
In Fig. 3, we show the optical V vs. V-I color-magnitude diagram of IC 1613
obtained from the OGLE II
data (Udalski et al. 2001) on which the locations of the Cepheids observed
in the present study are marked. In Fig. 4, we display the
PL relations in the K and J bands which we obtain from the data of all the 39
Cepheids in Table 4. While the data define tight PL relations in both bands,
there are some objects which clearly deviate from the bulk of the Cepheids in these diagrams,
and which need individual discussion. These stars are indicated with open
circles in Fig. 4.
Star 13682 is most probably not a Cepheid variable (Sandage 1971,
Antonello et al. 1999). Udalski et al. (2001) supported this conclusion
from the position of this star on the V, V-I CMD (see Fig. 3), and
its abnormal location on the optical PL relations on which
13682 appears much brighter than other Cepheids with similar periods.
This proves also true for its near-IR magnitudes in Fig. 4. We therefore exclude this
object for the distance determination.
Besides star 13682, the variables 13709, 11743, 12068,
8173 and 2771 are also very significantly brighter than other Cepheids with
similar periods. The two latter Cepheids with their very short periods
are almost certainly first overtone pulsators. Due to the detection limit
in our present near-infrared photometry
we would not see fundamental mode pulsators at this very short period
of about 1.3 days. The three other over-bright Cepheids are
probably blended by relatively bright stars. These Cepheids are also located above the Cepheid
PL relations in the V and I bands (Udalski et al. 2001).
In Fig. 3, variable 11743 appears close to the red edge of the
instability strip, while the heavily blended Cepheid 13709 lies outside the strip,
supporting the blending hypothesis.
For star 12068, unfortunately no V-I color is available.
The remaining two clearly deviating stars, 10421 and 17473, were already
classified as Population II Cepheids by Udalski et al. (2001).
Indeed, these stars are located about 2 magnitudes below the
IR Cepheid PL relations (see Fig. 4), which fully supports the conclusion
about the Pop. II nature of these objects.
In the light of the arguments presented above, we decided to reject
all these eight objects from the final sample of Cepheids used for the
distance determination.
Finally, we would like to comment on the Cepheid designated as 7647.
Udalski et al. (2001) suspected this star to be a heavily blended
Cepheid. Indeed, as can be seen in Fig. 3, this star is located bluewards to
the instability strip,
suggesting that this variable is blended with a very bright blue star.
In the infrared, Cepheid 7647 appears with normal flux and colors.
This finding is consistent with the presence of a blue, unresolved companion star,
which does contaminate the optical, but not the near-infrared photometry.
We therefore retain this Cepheid
in our final list of stars for the distance solution in the infrared. We remark that
an omission of
this star from the final sample would not significantly alter the results we will
present below.
The errors of the mean K-band magnitudes for Cepheids with
log P (days) $<$ 0.5 become large due to a) the relatively low accuracy
of the K-band photometry for such a faint stars ( $K > 20.5$ mag), and b)
the increasingly uncertain mean magnitude corrections for these stars,
caused by their relatively noisy optical
light curves, and less accurate periods. We obtained linear regressions to the
PL relations in the J and K bands for the whole sample, including the faintest stars, and
for the subsamples limited to the Cepheids with log P (days) $>$ 0.5,
finding very good agreement (to better than 1 $\sigma$) between the corresponding solutions.
However, since the inclusion of the shortest-period Cepheids in the solutions does increase the noise
significantly, we decided to adopt log P (days) = 0.5 as a lower cut-off period
for our solutions. This way, our final samples in J and K still comprise some
20 Cepheids with excellent photometry, which is sufficient for a very accurate
determination of the distance to IC 1613.
\subsection{Determination of the distance and reddening}
The least-squares fits to the mean magnitudes of the Cepheids from our
carefully selected final list yield the following
slopes of the PL relations: $ -3.117 \pm 0.044$ in J and $-3.148
\pm 0.053 $ in K. The stated errors are $1 \sigma$ uncertainties.
These values agree very well with the slopes of the PL relations
for the LMC Cepheids in these bands derived by Persson et al.
(2004) (-3.153 and -3.261 in J and K, respectively), and are consistent
with the LMC values within the combined uncertainties.
We therefore calulated the zero points of the Cepheid J and K band PL relations
in IC 1613 by adopting the corresponding slopes from Persson et al.
(2004). This yields the following results: \\
J = -3.153 log P + (22.187 $\pm$ 0.040) \\
K = -3.261 log P + (21.827 $\pm$ 0.045) \\
The adopted linear regressions to our K and J Cepheid data are shown in Fig. 5.
Before calculating, from the determined zero points, the relative distance of IC 1613
with respect to the LMC, we need
to convert our PL relation zero point magnitudes calibrated onto the UKIRT system
(Hawarden et al. 2001) to the NICMOS system on which the corresponding
LMC zero points were calibrated (Persson et al. 2004). According to
Hawarden et al. (2001) there are just zero point offsets between
the UKIRT and NICMOS systems (e.g. no color dependence) in the J and K
filters, which amount to 0.034 and 0.015 mag, respectively.
After adding these offsets, and assuming an LMC true distance modulus
of 18.5 mag (see next section for discussion on this assumption), we derived
the following distance moduli for IC 1613: 24.385 (J),
and 24.306 mag (K) . The corresponding distance moduli in the optical
V (24.572 mag) and I (24.488 mag) bands had been previously calculated from the
OGLE-II data by Udalski et al. (2001) adopting the linear LMC Cepheid P-L
relations (Udalski et al. 1999, Udalski 2000).
With the values of the distance moduli of IC 1613 derived in four different bands,
providing the large coverage in wavelength from 0.5-2.2 microns, we can compute
the reddening, and true distance modulus
of the galaxy very accurately.
Adopting the extinction law of Schlegel et al. (1998), and following the approach
we have developed in the study of NGC 300 (Gieren et al. 2005b), we fit
a straight line to the relation $(m-M)_{0} =
(m-M)_{\lambda} - A_{\lambda} = (m-M)_{\lambda} - E(B-V) * R_{\lambda} $.
The best least squares fit to this relation yields: \\
$(m-M)_{0} = 24.291 \pm 0.014$
$ E(B-V) = 0.090 \pm 0.007$
From Fig. 6, it is appreciated that the true distance modulus, and the total reddening
of IC 1613 are indeed very well determined from the available distance moduli in
the different photometric bands.
\section{Discussion}
In the following, we will discuss the various assumptions we made, and
possible systematic errors which could affect our distance
determination of IC 1613.
Almost certainly, the largest contribution to our total error budget
comes from the current uncertainty of the distance to the LMC. Since
this problem has been extensively discussed in the recent literature
(e.g. Benedict et al. 2002; Walker 2003), we will not focus on this discussion here.
The value of 18.50 mag for the true LMC distance modulus adopted in this
paper assures to have our distance results on the same scale as the results
from the HST Key Project team (Freedman et al. 2001)
and our own previous distance studies in the course of the Araucaria Project
(Pietrzynski et al. 2004, Gieren et al. 2004, Gieren et al. 2005b).
The adopted fiducial slopes of the J- and K-band Cepheid PL relations from
the work of Persson et al. (2004) are based on about 100 LMC Cepheids
with periods bracketting those of the IC 1613 Cepheids used in our present study.
The Persson et al. infrared PL relations clearly represent the most accurate
determination of these relations currently available in the literature. From
the recent work of Gieren et al. (2005a), there is evidence that the infrared
Cepheid PL relations in the Milky Way agree with the corresponding LMC relations,
within the combined $1 \sigma$ uncertainties. In Gieren et al. (2005b), we found
that the slopes of the Persson et al. PL relations in J and K do also provide excellent
fits to the Cepheid near-IR data in NGC 300, with its slightly more
metal-rich young population than the one in the calibrating LMC (Urbaneja et al. 2005).
From the present study of IC 1613, we now see that the slopes of the LMC near-IR PL relations
give an excellent fit to the metal-poor population of Cepheids in IC 1613, too. This
indicates that on the one hand, use of the Persson et al. LMC PL relations does not
introduce any significant systematic error in our current determination of the IC 1613
distance; on the other hand, it strongly suggests that in the near-infrared domain, as
in the optical domain, the slope of the Cepheid PL relation is independent of metallicity
in the wide range from about -1.0 dex up to solar abundance. This empirical finding is
in good agreement with the model predictions of Bono et al. (1999) who have found that
both, the zero-point and the slope of the K-band PL relation depend only marginally on
metal abundance. They found that the predicted slope in K is 3.19 $\pm$ 0.09 for the LMC,
and 3.27 $\pm$ 0.09 in the SMC, compatible with both the empirical value determined by Persson
et al. (2004) for the LMC, and with a zero change in the slope of the K-band PL relation when going
from LMC (-0.3 dex) metallicity to the SMC (-0.7 dex) metal abundance.
Finally, it is worthwhile to notice that the adoption of the slightly non-linear P-L
relations for Cepheids in the LMC, as advocated by Tammann and Reindl 2002, and more recently Ngeow et
al. 2005, would practically have no influence on the results presented in
this paper. Indeed, as has already been stated in Ngeow et al. (2005), such an effect
would introduce a change less than 3 percent for the derived
distance modulus, which is in the order of the one $\sigma$ error of our current determination.
In order to check this out more carefully, we used the Ngeow et al. P-L relations in both optical
and infrared bands for LMC Cepheids with periods longer
than 10 days as fiducial relations and re-calculated the distance moduli in the VIJK bands.
This exercise resulted in revised distance moduli to
IC 1613 which in all bands were consistent within one $\sigma$ with our original
results obtained by using the Cepheid P-L relations of Udalski (2000) and
Persson et al. (2004) for the LMC. The possible non-linearity of the LMC P-L relation, and
the associated slight change of its slope for the long-period Cepheids, is therefore not
a significant problem in the context of our current distance work. Yet, it will be very important
to improve on the slope for the long-period LMC Cepheid P-L relation by using very
accurate and homogeneous new data. We are currently involved in a project to obtain
such new data in the V and I bands.
The sample of Cepheids used for our present distance determination to IC 1613
is relatively large, making our distance result invulnerable to the problem of an inhomogeneous
filling of the instability strip which is ideally required in such studies.
We suspect that the main reason for the difference of 0.14 mag between
our current distance result for IC 1613
and the one obtained by Macri et al. (2001) is the small number of Cepheids available
for their study (4), which does not guarantee a homogeneous filling of the instability strip
and can cause a relatively large systematic offset of the derived distance modulus from the
true value. Therefore, we consider our present result as consistent with the HST-based result
of Macri et al. (2001).
The location of the Cepheids of our final sample in the CMD in Fig. 3 shows that they do indeed cover
the instability strip quite homogeneously. Moreover, the period range for the PL solution
is very wide and rather uniformly covered with stars-we chose our IR fields in such a way as to optimize
the period distribution of the Cepheids in these fields.
Applying different cut-off periods
to our sample (e.g. log P = 0.5, 0.7, 1), we always reproduce the zero point results
to within one $\sigma$. From this we conclude that our choice for the cut-off period
does not affect our final results in any significant way.
The most important source of uncertainty while using the optical data {\it alone}
is the interstellar reddening. Our present study shows that most of the reddening
to the IC 1613 Cepheids is actually contributed in the galaxy itself, which explains
the overestimation of the distance to IC 1613 in previous studies from optical
data which had only used the very small foreground extinction to IC 1613 to
make the reddening correction.
Using infrared data, and in
particular K-band photometry where the reddening is by an order of
magnitude smaller than in the optical bands, the error due to reddening is minimized
to a practically insignificant level of about 0.01 mag.
From the fact that our new value of E(B-V) yields very consistent
distances from the PL relations in all optical and infrared bands, we can also
conclude that the extinction law in IC 1613 is not significantly different
from the Milky Way law of Schlegel et al. (1998). This is the same conclusion
we had already reached in the case of NGC 300 (Gieren et al. 2005b).
Another contribution to the error budget comes from the effect
of unresolved companions on the Cepheid magnitudes. The few strongly
blended Cepheids in our sample were easily detected from their positions on the
multiband PL relations, and on the CMD, and were discarded from our
further analysis (see section 3.2.). As we extensively discussed
in our previous papers (e.g. Gieren et al. 2004, 2005b; Bresolin et al. 2005,
Pietrzynski et al. 2004) the blending effect was found to be very small in the cases of
NGC 300, and of NGC 6822. In the paper of Bresolin et al. (2005), we were able
to demonstrate from HST/ACS images that those Cepheids in NGC 300 which we had
identified as strongly blended in the ground-based photometry were indeed the ones
with the brightest nearby companions. In that paper, it was shown that the
effect of unresolved companion stars on the Cepheids which constituted the
final sample was less than 2 percent. Given that IC 1613 is located at less than
half the distance of NGC 300, and has a much smaller stellar density, it is
reasonable to assume that the effect of blending due to unresolved companion
stars on its distance is even smaller than in the case of NGC 300, and
does not contribute in a significant way to the systematic uncertainty of
our present result.
While it now seems well established that the slopes of the
Cepheid P-L relations in optical and near-infared bands
do not depend, within our current detection sensitivity, on metallicity over a very broad range of
this parameter ( -1 $<$ [Fe/H] $<$ 0 ; see previous discussion),
a possible metallicity dependence
of their {\it zero points} is still under discussion (Sakai et al.
2004; Storm et al. 2004; Pietrzynski et al. 2004; Pietrzynski and Gieren 2005).
In particular, due to the fact that up to now very few galaxies have been {\it exhaustively}
surveyed for Cepheids in the infrared,
it is currently not possible to draw any firm conclusion about the
potential variation of the infrared PL relation zero points with
metallicity. Soon, once the data for all target galaxies
observed in the course of the Araucaria Project will be analyzed,
we should be able to put tighter constraints on this open question,
and if needed calibrate the metallicity dependence of PL relation zero
points in both optical and infrared domains with high precision.
\section{Summary and conclusions}
We have measured accurate NIR magnitudes in the J and K bands for 39 Cepheid variables
in the Local Group galaxy IC 1613 with well-determined periods and optical (VI)
light curves. Mean magnitudes in J and K were derived for these variables using
the single-phase approach of Soszy{\'n}ski et al. (2005). After carefully cleaning
the Cepheid list from blended objects, Population II variables and overtone pulsators,
we have determined accurate PL relations. Fits to these observed relations were made
using the slopes of the LMC relations determined by Persson et al. (2004), which gave an excellent
representation of the IC 1613 data, providing for the first time solid evidence
that the slope of the Cepheid PL relation is independent of metallicity down to
the low metallicity of -1.0 dex of the IC 1613 young population in the near-infrared
domain, too. This is in agreement with the theoretical predictions of Bono et al. (1999).
By combining the zero points of the J and K band PL relations in our
study with the ones derived by Persson et al. for the LMC, we derive relative distance
moduli of IC 1613 with respect to the LMC in both bands. Combining these infrared
moduli with the distance moduli previously derived by Udalski et al. (2001) in V
and I, we determine the total (average) reddening of the Cepheids in IC 1613, and
the true distance modulus of this galaxy with an unprecedented accuracy. For the
reddening, we find E(B-V) = 0.090 $\pm$ 0.007 mag, and for the true distance modulus
of IC 1613 from our multiwavelength approach we obtain 24.291 $\pm$ 0.014 mag (random error).
As in the case of our study of NGC 300 with the same method, we find evidence that there
is a significant contribution to the total reddening from dust absorption {\it intrinsic}
to IC 1613, which had been neglected in the previous Cepheid distance work on
this galaxy. The excellent fit of the distance moduli to the assumed galactic extinction law
suggests that the interstellar extinction in this small irregular galaxy follows closely
the galactic law.
We show that our derived Cepheid distance is very insensitive to systematic uncertainties
caused by the fiducial PL relations used in our fits, possible inhomogeneous filling
of the instability strip by our Cepheid sample, and problems with blending of the variables.
Any remaining influence of the uncertainty of reddening on our distance result is
negligible. All these possible sources of error contribute less to the total systematic
uncertainty of our result than the two dominant sources of error, which are the zero
points of our JK photometry ($\pm$ 0.03 mag), and the distance to the LMC, which we have
{\it adopted} as 18.50 mag, and whose current uncertainty seems in the order of $\pm$ 0.10 mag.
Our distance determination for IC 1613 is in reasonable agreement with the previous determination
of Macri et al. (2001) from HST H-band photometry of four Cepheids, 24.43 $\pm$ 0.08 mag.
We attribute the 0.14 mag difference mainly to the small number of stars available to Macri et al.
in their study.
Our new distance determination is also in very good agreement with the very early infrared
work of McAlary et al. (1984); these authors had obtained a distance modulus of 24.31 $\pm$ 0.12
from H-band data of 10 Cepheids in IC 1613. A change of their assumed reddening of 0.03 mag
to our larger value found in this study still produces excellent agreement of their result
with ours.
The distance to IC 1613 was also determined by Dolphin et al. (2001) using HST data, and employing
a number of distance indicators (TRGB, red clump stars, RR Lyrae stars, Cepheids). We note
that their Cepheid sample was very small, which can clearly lead to spurious results.
Udalski et al. 2001 have observed an order of magnitude larger sample of
Cepheids in this galaxy and showed that all these different distance indicators yield
consistent distances to this galaxy. Those distance measurements are all in very good
agreement with the distance of IC 1613 obtained in this study if the revised reddening of 0.09 mag
found in this study, and a LMC true distance modulus of 18.5 mag are assumed.
As a final conclusion, we have produced in this work a determination of the Cepheid distance
to IC 1613 whose random error is of the order of 1\%, and estimated systematic error (excluding
the uncertainty of the adopted LMC distance) is in the order of 2\%. This accuracy, when
combined with distance determinations of similar accuracy we pretend to obtain for most of the other
galaxies of the Araucaria Project, should enable us to pin down the metallicity dependence
of the PL relation zero points in the different optical and near-infrared photometric bands with
the 1-2\% accuracy needed to produce a significant improvement in the determination of
the Hubble constant from distance determinations to galaxies in the nearby field from
their Cepheid populations, which is the approach used in the HST Key Project of Freedman et al. (2001).
Our work in the Araucaria Project should therefore strongly contribute, in the very near future,
to make best use of the past work of the Key Project team.
\acknowledgments
We would like to thank the anonymous referee for his interesting suggestions
which helped to improved this paper.
We gratefully acknowledge the generous allocation of observing time
by ESO to our distance scale projects. We also appreciate the excellent staff
support at the telescope at ESO/La Silla where these data were obtained.
WG and GP gratefully acknowledge financial support for this
work from the Chilean Center for Astrophysics FONDAP 15010003.
Support from the Polish KBN grant No 2P03D02123 and BST grant for
Warsaw University Observatory is also acknowledged.
\clearpage
\begin{deluxetable}{c c c}
\tablewidth{0pc}
\tablecaption{Coordinates of the Centers of the Observed SOFI/NTT Fields in IC 1613}
\tablehead{ Field & \colhead{RA} & \colhead{DEC} }
\startdata
C1 & 01:04:58.9 & 02:05:48.5 \nl
C2 & 01:04:56.6 & 02:04:13.2 \nl
C3 & 01:04:59.6 & 02:09:23.9 \nl
C4 & 01:04:37.2 & 02:05:58.8 \nl
C5 & 01:04:35.1 & 02:09:29.4 \nl
C6 & 01:05:03.0 & 02:09:44.4 \nl
\enddata
\end{deluxetable}
\clearpage
\begin{deluxetable}{ccccccc}
\tablewidth{0pc}
\tablecaption{Journal of the Individual J and K Observations of IC 1613
Cepheids}
\tablehead{ \colhead{ID} & HJD (J) & J & $\sigma_{\rm J}$ & HJD (K) & K & $\sigma_{\rm K}$}
\startdata
11446 & 53215.86015 & 16.997 & 0.009 & 53215.80516 & 16.492 & 0.009 \\
11446 & 53315.64411 & 17.400 & 0.016 & 53315.58668 & 16.822 & 0.016 \\
10421 & 53315.64411 & 19.551 & 0.060 & 53315.58668 & 18.877 & 0.071 \\
1987 & 53370.57443 & 17.938 & 0.019 & 53370.52660 & 17.593 & 0.024 \\
736 & 53315.72831 & 17.833 & 0.018 & 53315.67116 & 17.260 & 0.018 \\
7647 & 53370.57443 & 18.144 & 0.020 & 53370.52660 & 17.716 & 0.027 \\
13738 & 99999.99999 & 99.999 & 9.999 & 53215.88146 & 18.165 & 0.032 \\
13738 & 53315.56228 & 18.763 & 0.043 & 53315.50158 & 18.249 & 0.046 \\
13682 & 99999.99999 & 99.999 & 9.999 & 53215.88146 & 15.763 & 0.007 \\
13682 & 53315.56228 & 16.760 & 0.010 & 53315.50158 & 15.751 & 0.007 \\
\enddata
\end{deluxetable}
\clearpage
\begin{deluxetable}{cccc}
\tablewidth{0pc}
\tablecaption{Journal of the Individual V band Observations of IC 1613
Cepheids}
\tablehead{ \colhead{ID} & HJD & V & $\sigma_{\rm V}$}
\startdata
11446 & 53620.76797 & 18.385 & 0.009 \nl
11446 & 53621.75534 & 18.422 & 0.008 \nl
11446 & 53621.82741 & 18.412 & 0.008 \nl
10421 & 53620.76797 & 21.632 & 0.079 \nl
10421 & 53621.75534 & 21.648 & 0.069 \nl
1987 & 53620.76797 & 19.764 & 0.019 \nl
1987 & 53621.75534 & 19.815 & 0.018 \nl
1987 & 53621.82741 & 19.829 & 0.019 \nl
736 & 53620.76797 & 19.593 & 0.017 \nl
736 & 53621.75534 & 19.671 & 0.018 \nl
736 & 53621.82741 & 19.675 & 0.023 \nl
7647 & 53620.76797 & 19.110 & 0.014 \nl
7647 & 53621.75534 & 19.080 & 0.012 \nl
7647 & 53621.82741 & 19.100 & 0.012 \nl
13738 & 53620.76797 & 20.035 & 0.020 \nl
13738 & 53621.75534 & 20.115 & 0.021 \nl
13738 & 53621.82741 & 20.109 & 0.023 \nl
\enddata
\end{deluxetable}
\clearpage
\begin{deluxetable}{cccccccccccc}
\tablewidth{0pc}
\tablecaption{Final Intensity Mean J and K Magnitudes of IC 1613 Cepheids}
\tablehead{ \colhead{OGLE ID} & P & $\log{P}$ & $\langle{J}\rangle$ & $\sigma_J$ & $\langle{K}\rangle$ &
$\sigma_K$& remarks}
\startdata
11446 & 41.87 & 1.62194 & 17.114 & 0.009 & 16.605 & 0.009& V20\\
10421 & 29.19 & 1.46529 & 19.694 & 0.060 & 19.029 & 0.071&PII, V47\\
1987 & 25.398 & 1.40480 & 17.715 & 0.019 & 17.405 & 0.024&V11\\
736 & 23.469 & 1.37049 & 17.745 & 0.018 & 17.256 & 0.018&V2\\
7647 & 16.488 & 1.21716 & 18.056 & 0.020 & 17.688 & 0.027&blend\\
13738 & 16.420 & 1.21537 & 18.590 & 0.043 & 18.066 & 0.028&V18\\
13682 & 14.317 & 1.15585 & 16.815 & 0.010 & 15.818 & 0.005¬ Cepheid, V39\\
17473 & 13.154 & 1.11906 & 99.999 & 9.999 & 20.669 & 0.251 &PII\\
7664 & 10.4390 & 1.01866 & 18.996 & 0.030 & 18.555 & 0.048&V16\\
926 & 9.4286 & 0.97445 & 19.109 & 0.028 & 18.736 & 0.036&V6\\
879 & 9.2130 & 0.96440 & 19.156 & 0.046 & 18.689 & 0.054&V25\\
13808 & 7.572 & 0.87921 & 19.591 & 0.092 & 19.160 & 0.058&\\
13759 & 7.3403 & 0.86571 & 19.272 & 0.074 & 18.832 & 0.112&V7\\
13709 & 6.741 & 0.82872 & 18.480 & 0.041 & 17.739 & 0.020&blend\\
5037 & 6.3175 & 0.80055 & 19.824 & 0.065 & 19.484 & 0.127&\\
11604 & 5.8191 & 0.76486 & 19.685 & 0.051 & 19.133 & 0.097&\\
13780 & 5.5771 & 0.74641 & 19.973 & 0.137 & 19.256 & 0.069&V9\\
11831 & 5.0269 & 0.70130 & 19.902 & 0.063 & 19.532 & 0.111&\\
8146 & 4.5630 & 0.65925 & 20.306 & 0.075 & 19.730 & 0.122&\\
14287 & 4.365 & 0.63998 & 99.999 & 9.999 & 19.951 & 0.142 &\\
12109 & 4.1364 & 0.61662 & 20.128 & 0.079 & 19.344 & 0.093&\\
13784 & 4.0657 & 0.60914 & 99.999 & 9.999 & 19.459 & 0.096 &V10\\
11743 & 3.8953 & 0.59054 & 19.348 & 0.035 & 18.795 & 0.047&blend, V53\\
8127 & 3.8444 & 0.58483 & 20.636 & 0.097 & 20.267 & 0.159&\\
2240 & 3.0733 & 0.48760 & 20.941 & 0.138 & 20.221 & 0.147&V35\\
18349 & 2.8700 & 0.45788 & 99.999 & 9.999 & 19.639 & 0.102 &V29\\
19024 & 2.8418 & 0.45359 & 99.999 & 9.999 & 19.859 & 0.154 &\\
12068 & 2.781 & 0.44420 & 20.013 & 0.060 & 19.339 & 0.091&blend\\
2760 & 2.7123 & 0.43334 & 21.101 & 0.127 & 20.651 & 0.203&\\
10804 & 2.6629 & 0.42535 & 21.041 & 0.137 & 20.404 & 0.197&V48\\
12526 & 2.6310 & 0.42012 & 99.999 & 9.999 & 20.982 & 0.358 &\\
7322 & 2.3378 & 0.36881 & 21.125 & 0.141 & 21.145 & 0.328&\\
6128 & 2.2578 & 0.35369 & 20.712 & 0.114 & 20.140 & 0.174&\\
8782 & 2.0930 & 0.32077 & 21.013 & 0.137 & 20.673 & 0.255&\\
5996 & 2.0682 & 0.31559 & 20.888 & 0.183 & 20.508 & 0.284& V60\\
2389 & 2.0286 & 0.30720 & 20.837 & 0.115 & 20.363 & 0.175&\\
13481 & 1.678 & 0.22479 & 99.999 & 9.999 & 21.226 & 0.392 &\\
2771 & 1.3290 & 0.12352 & 99.999 & 9.999 & 20.193 & 0.150 &FO\\
8173 & 1.3103 & 0.11737 & 20.587 & 0.099 & 20.020 & 0.130&FO\\
\enddata
\end{deluxetable}
\clearpage
\begin{deluxetable}{cccccc}
\tablewidth{0pc}
\tablecaption{Reddened and Extinction-Corrected Distance Moduli for IC 1613 in Optical and Near-Infrared Bands}
\tablehead{ \colhead{Band} & $V$ & $I$ & $J$ & $K$ & $E(B-V)$ }
\startdata
$m-M$ & 24.572 & 24.488 & 24.385 & 24.306 & -- \nl
${\rm R}_{\lambda}$ & 3.24 & 1.96 & 0.902 & 0.367 & -- \nl
$(m-M)_{0}$ & 24.277 & 24.309 & 24.302 & 24.273 & 0.090 \nl
\enddata
\end{deluxetable}
|
Title:
The broadband afterglow of GRB 030328 |
Abstract: We here report on the photometric, spectroscopic and polarimetric monitoring
of the optical afterglow of the Gamma-Ray Burst (GRB) 030328 detected by
HETE-2. We found that a smoothly broken power-law decay provides the best fit
of the optical light curves, with indices alpha_1 = 0.76 +/- 0.03, alpha_2 =
1.50 +/- 0.07, and a break at t_b = 0.48 +/- 0.03 d after the GRB. Polarization
is detected in the optical V-band, with P = (2.4 +/- 0.6)% and theta = (170 +/-
7) deg. Optical spectroscopy shows the presence of two absorption systems at z
= 1.5216 +/- 0.0006 and at z = 1.295 +/- 0.001, the former likely associated
with the GRB host galaxy. The X-ray-to-optical spectral flux distribution
obtained 0.78 days after the GRB was best fitted using a broken power-law, with
spectral slopes beta_opt = 0.47 +/- 0.15 and beta_X = 1.0 +/- 0.2. The
discussion of these results in the context of the "fireball model" shows that
the preferred scenario is a fixed opening angle collimated expansion in a
homogeneous medium.
| https://export.arxiv.org/pdf/astro-ph/0601293 |
\title{The broadband afterglow of GRB 030328}
\classification{95.75.De; 95.75.Fg; 95.75.Hi; 95.85.Kr; 98.70.Rz}
\keywords {gamma-ray bursts; astronomical observations: visible;
photometry; spectroscopy; polarimetry}
\author{E. Maiorano}{
address={INAF-IASF, Bologna, Italy}
}
\author{N. Masetti}{
address={INAF-IASF, Bologna, Italy}
}
\author{E. Palazzi}{
address={INAF-IASF, Bologna, Italy}
}
\author{S. Savaglio}{
address={The Johns Hopkins Univ., Baltimore, USA}
}
\author{E. Rol}{
address={Univ. of Leicester, UK}
}
\author{P.M. Vreeswijk}{
address={ESO-Santiago, Chile}
}
\author{E. Pian}{
address={INAF-Oss. Astron. Trieste, Italy}
}
\author{P.A. Price}{
address={Univ. of Hawaii, Honolulu, USA}
}
\author{B.A. Peterson}{
address={Australian National Univ., Weston, Australia}
}
\author{M. Jel\'{i}nek}{
address={IAA-CSIC, Granada, Spain}
}
\author{S.B. Pandey}{
address={IAA-CSIC, Granada, Spain}
}
\author{M.I. Andersen}{
address={AIP, Potsdam, Germany}
}
\author{A.A. Henden}{
address={US Naval Observatory, Flagstaff, USA}
}
\section{Introduction}
GRB 030328 was a long, bright GRB detected on 2003 Mar 28.4729 UT, by the
FREGATE, WXM, and SXC instruments onboard {\it HETE-2}, and rapidly
localized with sub-arcminute accuracy (Villasenor et al. 2003).
About $\sim$1 hour after the GRB, its optical afterglow has been detected
by the 40-inch Siding Spring Observatory (SSO) telescope (Peterson \&
Price 2003). A study of the X--ray afterglow of GRB 030328 was performed
by Butler et al. (2005) using {\it Chandra} data.
We report here on the study of the optical afterglow emission of GRB
030328 made, within the GRACE\footnote{GRB Afterglow Collaboration at ESO;
see {\tt http://www.gammaraybursts.org/grace/}} collaboration, performed
with 7 different optical telescopes.
A more detailed presentation of these data will appear in Maiorano et
al. (2005).
\section{Observations}
Optical $UBVRI$ data of the GRB 030328 Optical Transient (OT), for a total
of 130 photometry points, were acquired at the 40-inch SSO (Australia), 1m
ARIES (India), 2.5m NOT (Spain), 1.54m ESO Danish, 2.2m ESO/MPG, ESO
VLT-$Antu$ (Chile) and 1m USNO-FS (USA) telescopes.
A series of six 10-min optical spectra was obtained at ESO-Paranal with
VLT-{\it Antu} starting 0.59 d after the GRB. The Grism 300V was used with
a nominal spectral coverage of 3600--8000 \AA~and a spectral dispersion of
2$\farcs$7 \AA/pixel.
Linear polarimetry $V$-band observations were acquired between 0.66 and
0.88 d after the GRB at VLT-{\it Antu}. Five complete imaging polarimetry
cycles were performed.
\section{Results}
\subsection{Photometry}
In Fig. 1 (left) we plot our photometric measurements together with
those reported in the GCN circulars\footnote{{\tt
http://gcn.gsfc.nasa.gov/gcn/gcn3\_archive.html}}. For the cases in which
no error was reported, a 0.3 mag uncertainty was assumed. The $UBVRI$
zero-point calibration was performed using the photometry by Henden (2003).
The optical data of Fig. 1 (left) were corrected for the Galactic
foreground reddening assuming $E(B-V)$ = 0.047 mag (Schlegel et al. 1998).
The GRB 030328 host galaxy emission in the $BVRI$ bands was computed from
the data of Gorosabel et al. (2005) and subtracted from our optical data
set.
The best fit of the $R$-band data (in Fig. 1, left)
is obtained using a smoothly broken powerlaw (Beuermann et al. 1999), with
temporal indices $\alpha_1 = 0.76 \pm 0.03$ and $\alpha_2 = 1.50 \pm 0.07$
before and after a break occurring at $t_b = 0.48 \pm 0.03$ d from the GRB
trigger, and with $s = 4.0 \pm 1.5$ the parameter modeling the slope
change rapidity. This best-fit curve describes well the data in the other
bands also (see Fig. 1, left). This means that the decay of the OT can be
considered as achromatic.
From the measured jet break time we can compute the jet opening angle
value for GRB 030328 which is, following Sari et al. (1999), $\theta_{jet}
\sim 3\fdg2$.
\subsection{Broadband analysis}
By using the available information, we have constructed the
optical-to-X--ray spectral flux distribution of the GRB 030328
afterglow at the epoch 0.78 days after the GRB, that is, the time with
best broadband photometric coverage.
As Fig. 1 (right) shows, the best descriptions are a single powerlaw
(dashed line) with a spectral index $\beta_{\rm X-opt}$ = 0.83$\pm$0.01,
or a broken powerlaw with $\beta_{\rm opt}$ = 0.47$\pm$0.15 and assuming
$\beta_{\rm X}$ = 1.0$\pm$0.2 from Butler et al. (2005).
However, the single powerlaw description of the broadband afterglow does
not fit any of the synchrotron fireball scenarios (Sari et al. 1998,
1999). Instead, in the broken powerlaw description, which means that the
synchrotron cooling frequency $\nu_{\rm c}$ lies between the optical and
X--ray bands, we obtain that the GRB 030328 afterglow broadband evolution
is consistent with a jet-collimated expansion in a homogeneous medium with
fixed opening angle (M\'esz\'aros \& Rees 1999) and with an electron
distribution index $p$ = 2.
Assuming a negligible host absorption and using the optical and X--ray
spectral slopes above, we obtain for $\nu_{\rm c}$ the value $5.9 \times
10^{15}$ Hz, which places this frequency in the ultraviolet band.
\subsection{Spectroscopy}
Figure 2 shows the spectrum of the GRB 030328 OT. Most of the
significant features can be identified with Fe {\sc ii}, Mg {\sc ii},
Al {\sc ii} and C {\sc iv} absorption lines in a
system at a redshift of $z$ = 1.5216 $\pm$ 0.0006. These lines are
associated with the circumburst gas or interstellar medium in the
GRB host galaxy. A lower redshift absorption system at $z$ = 1.295
$\pm$ 0.001 is also found: for it, only two lines can be identified
(Fe {\sc ii} $\lambda$2600 and the unresolved Mg {\sc ii}
$\lambda$$\lambda$2796, 2803 doublet). Its detection indicates the
presence of a foreground absorber.
\subsection{Polarimetry}
After correcting for spurious field polarization, we found $Q_{\rm OT} =
0.029 \pm 0.008$ and $U_{\rm OT} = -0.004 \pm 0.008$. The fit of the data
with the relation of Di Serego Alighieri (1997) yielded for the OT a
linear polarization $P = (2.4 \pm 0.6)$ \% and a polarization angle
$\theta = 170^{\circ} \pm 7^{\circ}$, corrected for the polarization bias
(Wardle \& Kronberg 1974).
In order to check whether variations of $P$ and $\theta$ occurred during
the polarimetric run, we also separately considered each of the 5 single
polarimetry cycles. Although with lower S/N, $P$ and $\theta$ are
consistent with being constant across the whole polarimetric observation
run.
|
Title:
A genetic algorithm for the non-parametric inversion of strong lensing systems |
Abstract: We present a non-parametric technique to infer the projected-mass
distribution of a gravitational lens system with multiple strong-lensed images.
The technique involves a dynamic grid in the lens plane on which the mass
distribution of the lens is approximated by a sum of basis functions, one per
grid cell. We used the projected mass densities of Plummer spheres as basis
functions. A genetic algorithm then determines the mass distribution of the
lens by forcing images of a single source, projected back onto the source
plane, to coincide as well as possible. Averaging several tens of solutions
removes the random fluctuations that are introduced by the reproduction process
of genomes in the genetic algorithm and highlights those features common to all
solutions. Given the positions of the images and the redshifts of the sources
and the lens, we show that the mass of a gravitational lens can be retrieved
with an accuracy of a few percent and that, if the sources sufficiently cover
the caustics, the mass distribution of the gravitational lens can also be
reliably retrieved. A major advantage of the algorithm is that it makes full
use of the information contained in the radial images, unlike methods that
minimise the residuals of the lens equation, and is thus able to accurately
reconstruct also the inner parts of the lens.
| https://export.arxiv.org/pdf/astro-ph/0601124 |
\date{} %
\pagerange{\pageref{firstpage}--\pageref{lastpage}} \pubyear{2005}
\label{firstpage}
\begin{keywords}
gravitational lensing -- methods:~data analysis -- dark matter --
galaxies:~clusters:~general
\end{keywords}
\section{Introduction}
The deflection of light caused by a gravitational lens and the
amplifying and distorting effect thereof on the images of background
sources, provide us with a means to measure the total mass of the
lensing object. If only, of course, that it is possible to ``invert''
such a lensing system, i.e. to infer the mass distribution of the lens
given the positions and shapes of a set of lensed images of background
sources and the redshifts of the lens and the sources. The inversion
of gravitational lensing systems is interesting in its own right,
since it puts constraints on the spatial dark matter distribution of
the lensing objects and thus helps constrain dark matter physics
\citep{n04,d04}, but it may also contribute to cosmology. Good
reconstructions of lensing systems may help to constrain the density
parameter and the redshift evolution of the dark energy
\citep{y01,s04,m05}.
The use of the gravitational lensing effect to measure masses of
point-mass lenses (or ``stars'') was envisaged by \citet{r64} and
\citet{l64}. The idea of using simple parametric models for extended
lenses, such as galaxies or clusters of galaxies, can be traced back
to \citet{dr80}, who use a King model to estimate the mass
distribution of the galaxy that produces the two images of
QSO~0957+561A,~B. Since then a host of parametric inversion methods
has been developed and applied to observed strong lensing systems,
such as the ring cycle method of \citet{k89} or the maximum entropy
method of \citet{w94}. \citet{kn93} fitted a bimodal lensing
potential, constructed from two elliptical pseudo-isothermal
potentials, images of the cluster A370, obtained from the ground under
excellent seeing conditions. More elaborate parametric models for
lensing clusters associate a simple mass distribution, e.g. a
power-law of radius, to each galaxy and to the cluster as a whole. The
many parameters that define the lens model are then determined by a
$\chi^2$ fit to the observed images \citep{tkd98}. \citet{b05} suggest
using a gravitational lens as a cosmic magnifying glass to recover
structural details about distant sources. These authors assume a
parametric form for the lens mass distribution and use a genetic
algorithm to non-parametrically reconstruct the surface brightness
distribution of the source. If only a few sources are being lensed or
if the sources do not sufficiently cover the caustics, parametric
methods are clearly the preferred approach.
However, the multitude of arcs and distorted images visible in massive
clusters (e.g. \citealt{s87,lh88}), which are observed routinely now
at high spatial resolution with the Hubble Space Telescope
(e.g. \citealt{br05}), contain a wealth of information and call for
more flexible and model independent inversion methods. Using the
pixelation method \citep{sw97,a98}, the lens plane is divided into a
static grid. The mass in each grid cell and the source positions are
estimated so as to construct a solution that best reproduces the
observed image positions, subject to regularizing constraints that
ensure a smooth mass distribution for the lens that stays close to the
luminosity distribution. \citet{t00} perform a multipole-Taylor
expansion of the two-dimensional lensing potential, the coefficients
of which are determined by a $\chi^2$-fit to the observed images. In
\citet{d05}, a non-parametric inversion technique, called {\sc slap},
is presented and applied to an HST image of the cluster Abell 1689
\citep{d05b,k02} that, like the pixelation method, makes use of a grid
division of the mass distribution of the lens. This time, however, the
grid is dynamic:~it is refined iteratively where the mass density is
large. Recently, \citet{d05c} extended {\sc slap} to {\sc wslap} in
order to also take into account information in the weak lensing
regime. Thus, {\sc slap} and {\sc wslap} are fast and versatile tools
for inverting observed lens systems. However, both methods assume the
background galaxies to be point sources, which may lead to an
over-estimation of the central mass density of the lens in order to
focus the images into very compact sources and to physically
implausible regions with negative mass density in the lens plane.
Adding weak-lensing information alleviates the dependence of the
solution on this minimization threshold. {\sc wslap} can make use of
quadratic programming to avoid unphysical negative mass densities for
the lens. However, this limits the analysis to observables that are
linear functions of the lens mass density, such as image positions,
and does not allow to incorporate e.g. surface brightness information,
which depends non-linearly on the lens mass density. \citet{bra05}
also used weak and strong lensing data to invert the X-ray cluster
RX~J1347.5$-$1145. Their method evaluates the gravitational potential
on a non-dynamic grid. The best fitting gravitational potential is
then constructed non-parametrically by minimising a $\chi^2$ function,
starting from a parametric priorsolution.
The ideal non-parametric lens inversion algorithm {\em (i)} should be
free of any assumptions regarding the mass distribution of the lens or
the luminosity distributions of the sources, {\em (ii)} should not
depend on any prior on the lens mass distribution or any
regularisation scheme that could bias the solution, {\em (iii)} should
not produce unphysical, i.e. negative, mass densities {\em (iv)}
should be free of any uncontrollable parameters, {\em (v)} should be
easily extendible to any kind of data, both in the strong and weak
lensing regimes, without having to change the inner workings of the
algorithm or having to worry about features like continuity or
differentiability of the objective function that is extremised. These
conceptual issues are the main motivation for this paper, rather than
computational speed.
Genetic algorithms do an excellent job at fulfilling all these
constraints. In this paper, we describe and test a new non-parametric
lens inversion technique. The technique makes use of a dynamic grid on
which the mass distribution of the lens is approximated by a weighted
sum of basis functions and of a genetic algorithm to determine the
unknown weights. In the strong lensing regime, where multiple images
of each source are available, the following data are offered to the
algorithm:~the redshifts of the sources and the lens, and the
observed positions of the images. The genetic algorithm generates
solutions that satisfy only one minimal constraint:~the
back-projected images of a single source should overlap as well as
possible in the source plane.
We briefly describe the relevant background to gravitational lens
systems and genetic algorithms in Section \ref{sec:back}. The details
of the inversion method are given in Sect. \ref{sec:invert}. We
discuss a number of tests to which we subjected the method in
Sect. \ref{sec:sim}. Finally, our conclusions are summarized in
Sect. \ref{sec:conc}.
\section{Background} \label{sec:back}
\subsection{The lens equation}
In the thin lens approximation, the lens equation relates viewing
directions $\Vec{\theta}$, that are defined in the lens plane, to
positions $\Vec{\beta}$ in the source plane:~ \begin{equation}
\Vec{\beta}(\Vec{\theta}) = \Vec{\theta} -
\frac{D_{ds}}{D_s}\;\Vec{\hat{\alpha}}(\Vec{\theta}) \mcm
\label{eq_lenseqn} \end{equation} with $\Vec{\hat{\alpha}}$ the
deflection angle, $D_s$ the distance between the observer and the
source, and $D_{ds}$ the distance between the lens and the source. The
gravitational bending of light rays, described by the deflection angle
$\Vec{\hat{\alpha}}$, depends on the viewing direction $\Vec{\theta}$,
the mass distribution of the lens and the distance between the lens
and the observer. Here and in the following, distances should be
interpreted as angular-diameter distances. For simplicity, we will
adopt a standard CDM cosmology, with a matter density $\Omega=1$ and a
Hubble parameter $H_0 = 70$~km~s$^{-1}$~Mpc$^{-1}$, in which the
angular-diameter distance between an observer at redshift $z_1$ and a
source at redshift $z_2$ is given by \begin{equation} D(z_1,z_2) =
\frac{2c}{H_0} \frac{1}{1+z_2} \left(\frac{1}{\sqrt{1+z_1}}-
\frac{1}{\sqrt{1+z_2}} \right) \mpt \end{equation} We will always
assume that the redshifts of the lens and of the source(s) are known
to the observer. The lens equation projects the images back onto their
respective sources in the source plane. When a gravitational lens
produces multiple images of a single source, one can use the lens
equation to find, for each image, the corresponding region in the
source plane. Since all images correspond to a single source, all
back-projected images have to coincide.
\subsection{The Plummer lens}
We first describe the gravitational lens effect caused by a Plummer
sphere \citep{p11} at a distance $D_d$ from the observer. The
projected density distribution of a Plummer sphere with total mass $M$
and angular scale-length $\theta_P$ as a function of angular distance
$\theta$ is given by \begin{equation} \Sigma(\theta) = \frac{M}{\pi
D_d^2}\frac{\theta_P^2}{(\theta^2+\theta_P^2)^2}.
\label{eq_sigma_plummer} \end{equation} This mass
distribution leads to the following lens equation: \begin{equation}
\Vec{\beta}(\Vec{\theta}) = \Vec{\theta}-\frac{D_{ds}}{D_s D_d}\frac{4
G M}{c^2}\frac{\Vec{\theta}}{\theta^2+\theta_P^2} \end{equation} if
the coordinate system in the lens plane is centered on the Plummer
sphere.
As a first step towards inverting a given lens system, we write the
(unknown) projected mass distribution of the lens as a sum of Plummer
mass distributions, of the form given by
eq. (\ref{eq_sigma_plummer}). We chose the Plummer mass distribution
as basis function because it is well-behaved at all radii and yields a
finite total mass. The deflection angle is then simply the sum of the
deflection angles caused by each individual Plummer distribution. For
$N$ individual Plummer lenses, this yields the following lens
equation: \begin{equation} \Vec{\beta}(\Vec{\theta}) = \Vec{\theta} -
\frac{D_{ds}}{D_s D_d} \frac{4 G}{c^2} \sum_{i = 1}^N
\frac{\Vec{\theta}-\Vec{\theta}_{s,i}}{|\Vec{\theta}-\Vec{\theta}_{s,i}|^2+\theta_{P,i}^2}
M_i \mcm \label{eq_lenseqn_mplum} \end{equation} with
$\Vec{\theta}_{s,i}$ the position of the centre of a Plummer
distribution in the lens plane, $M_i$ its mass, and $\theta_{P,i}$ its
angular scale-length.
A given set of $R$ points in the lens plane, $\Vec{\theta}_k,\,k = 1
\ldots R$, is related to a corresponding set of $R$ points in the
source plane by a matrix equation \citep{d05}. Indeed, let $\Theta$ be
a vector of length $2R$, containing the coordinates of the points in
the image plane, in which $x$ and $y$ components alternate. Similarly,
$B$ is a vector of length $2R$ which will contain the coordinates of
the corresponding points in the source plane. The masses $M_i$ of the
Plummer distributions that make up the mass distribution of the lens
are stored in an $N$ dimensional column vector $M$. The lens equation
can then be rewritten as
\begin{equation} B = \Theta - \gamma M \mcm \end{equation} with
$\gamma$ a $2R \times N$ matrix whose components are given by:
\begin{eqnarray} \gamma_{2k-1,l} &=& \frac{D_{ds}}{D_d D_s} \frac{4
G}{c^2}
\frac{(\Vec{\theta}_k-\Vec{\theta}_{s,l})_x}{|\Vec{\theta}_k-\Vec{\theta}_{s,l}|^2+\theta_{P,l}^2}
\nonumber \\ \gamma_{2k,l} &=& \frac{D_{ds}}{D_d D_s} \frac{4 G}{c^2}
\frac{(\Vec{\theta}_k-\Vec{\theta}_{s,l})_y}{|\Vec{\theta}_k-\Vec{\theta}_{s,l}|^2+\theta_{P,l}^2}
\mpt \end{eqnarray} The problem of inverting a gravitational lens
system is thus transformed into the problem of finding the vector $M$,
given the matrices $\Theta$ and $\gamma$.
\subsection{Genetic algorithms} \label{genalg}
With genetic algorithms, one tries to breed good solutions to a given
problem. A central concept is the genome, which is an encoded
representation of a possible solution. Usually, the genome will encode
the parameters of a specific model. For a particular genome, there has
to be some kind of measure of how adequate it fits the data. This
value is usually called the fitness of the genome. The algorithm
starts with a random set of genomes: the population. From this
population, a new one will be created using the following procedure:
\begin{itemize} \item For each genome, the fitness is calculated.
\item A new set of genomes is created by combining and copying genomes
of the current population. Selection of genomes in this reproduction
step should favor genomes with a better fitness.
\item Finally, mutations are introduced in the new population to
ensure genetic variety. \end{itemize} When creating the new
population, the best genome is often copied without mutations. This
approach is often referred to as elitism and ensures that the best
member of the new population will perform at least as well as the
fittest member of the old population. Thus, generation after
generation, one tries to breed increasingly better solutions to a
problem. A complete overview of genetic programming techniques can be
found in \cite{koza:book}.
\section{The inversion method}\label{sec:invert}
In the following, we discuss two key features of our inversion
method: the use of a dynamic grid in the lens plane on which the
Plummer lenses are positioned (which defines the matrix $\gamma$) and
the genetic algorithm employed to breed the best approximation to the
projected mass distribution of the lens (i.e., the vector $M$).
\subsection{The dynamic grid}
The procedure starts with a square grid, large enough to encompass the
projected mass density of the lens. At first, this area is uniformly
subdivided in square grid cells. At the centre of each grid cell, a
Plummer mass distribution is positioned. The width of each Plummer
distribution is set proportional to the side of its grid cell. We
tested which proportionality factor allows to best reproduce a wide
range of mass densities and found that a value of 1.7 yields a good
trade-off between smoothness and dynamic range. The same scale factor
was subsequently used in our lens inversion simulations. The genetic
algorithm (see subsection \ref{sub:gen}) then breeds, for this given
grid, the best solution $M$. Given this first approximation of the
total mass density, a new grid is constructed by further subdividing
grid cells that contain a large fraction of the total mass or that
reside in areas with large density gradients. This way, the new grid
will allow a better approximation of the mass density, without wasting
resources on areas which contain little mass or detail. With each cell
of this new grid a Plummer distribution is associated and the
individual masses are determined by the genetic algorithm, as before.
In our implementation, this procedure of refining the grid is repeated
unless the number of grid cells exceeds one thousand. Fig. \ref{fig1}
illustrates the procedure. At first, a uniform grid is used. With this
grid, a first estimate of the distribution is found and this is used
to create a new grid. The figure shows a few additional mass density
estimates on which new grids are based.
\subsection{The genetic algorithm} \label{sub:gen}
\subsubsection{Genome and fitness}
The goal of the genetic algorithm is to determine good values for the
masses of the individual Plummer distributions which are laid out
according to a specific grid. Therefore, the genome in our genetic
algorithm will encode the masses of these Plummer distributions.
For a specific set of Plummer masses, we need to define a way to
evaluate how good the corresponding solution is. Since we are working
in the strong lensing regime, it is assumed that the gravitational
lens system produces multiple images of one or more sources. If one
would project these images back to the source plane using the exact
lens equation for the lens under study, one would find that
back-projected images of the same source will overlap perfectly. For
this reason, the degree in which back-projected images of the same
source overlap will be used to determine how good the suggested
solution actually is.
The way this is implemented is as follows. For a given solution of the
mass distribution of the lens, the images of a single source are
projected back to the source plane. The areas occupied by each image
are surrounded by rectangles: two examples are shown in Fig.
\ref{fig2}. Corresponding corners of the rectangles are connected with
imaginary springs. Consider two rectangles, each enclosing a
backprojected image. For corresponding corners, the distance is
calculated in absolute units, for example in units of arcminutes or
arcseconds. In a previous step, a length scale was calculated as the
average of the lengths of the sides of all the rectangles belonging to
a specific source. The distance between two corresponding corners is
then divided by this length, yielding a dimensionless distance
$d$. The ``potential energy'' for this pair of corners is then simply
$d^2$. Repeating this for the other three corners and adding together
the energies then gives the potential energy of these two
rectangles. For a specific source, this procedure is then done for all
pairs of backprojected images and the sum of these potential energies
is then the potential energy contribution of this source. The fitness
value of a given lens solution is the sum of the potential energies of
all sources. It is important to take into account the scaling of the
rectangles when calculating the potential energy values. Comparing the
left and right parts of Fig. \ref{fig2}, it is clear that the left
situation definitely corresponds to a better overlap, while on an
absolute scale the potential energy of the right situation will be the
lower one. For this reason, we express distances between corners of
rectangles relative to the size of the rectangles, or, in other words,
relative to the size of the source.
As was mentioned above, the genome represents the masses of the
individual Plummer distributions. To be more precise, the genome only
represents the relative contribution of each Plummer distribution:
each Plummer mass is represented by a dimensionless, integer number
between 0 and 1000. These numbers are stored in the vector $M$ and the
matrix product \begin{equation} \Theta' = \gamma M \end{equation} is
calculated. For the dimensionless masses to be converted into real
masses, the vector $M$ needs to be multiplied with a factor $\mu$,
bringing the lens equation in the form \begin{equation} B = \Theta -
\mu \Theta'. \end{equation} Since $\Theta$ and $\Theta'$ are constant
column matrices, it is an easy and computationally inexpensive task to
find, for a given $M$, the factor $\mu$ that maximises the fitness,
or, in other words, for which the back-projected images of the sources
coincide best. The value of the fitness of that particular situation
is then considered to be the fitness of the genome.
\subsubsection{Reproduction and mutation}\label{sub:rep}
In our implementation, a population of 250 genomes is used. Based on
the many simulations we did (see below), 250 genomes has always led to
good solutions within an acceptable amount of time. To obtain a new
population, some genomes are copied from the original population while
others are obtained by merging two genomes. The procedure of merging
two genomes consists of a few steps which are illustrated in
Fig. \ref{fig3}. At first, the values between 0 and 1000 of each
genome are multiplied with their best $\mu$ value to obtain the true
Plummer masses they represent. Then, for each Plummer distribution,
the procedure selects at random the mass from one of the two
genomes. Finally, these values are rescaled to integer numbers in such
a way that the largest number is 500.
When the new population is complete, mutations are introduced in some
genomes. In early generations, some values are simply changed to a
random number between 0 and 1000. It is for this reason that the
previous step rescaled the Plummer masses to a maximum value of
500. This way, a random change of the value will also allow a
considerable increase in mass for that Plummer distribution.
When the best fitness values of successive generations start to
converge, a new mutation rule is adopted. In this case, random integer
numbers in the interval $[-200,200]$ are generated and added to some
of the genomes' values. Resulting values which are negative or larger
than $1000$, are set to zero or $1000$ respectively. The first
mutation rule makes sure that a large range of mass densities can be
inspected. When the algorithm starts to converge near a good solution,
the second mutation rule assures that the algorithm can more closely
approach the best solution.
\subsubsection{Stopping criterion}
The algorithm can be stopped if the fitness of the best genome ceases
to improve significantly. We use the following stopping criterion : if
the fitness of the last generation, denoted by {\tt new\_fit} fulfills
the constraint $|${\tt new\_fit}$-${\tt old\_fit}$| <$ {\tt
new\_fit}/50, with {\tt old\_fit} the fitness of 150 generations ago,
the algorithm is stopped. We inspected that raising the factor of 50,
so that the code runs longer, does not significantly change the
solution, i.e. that the solution has converged. Just for safety, we
implemented an upper limit of 15000 on the number of generations but
all inversions we tried so far converged after less than 5000
generations.
\subsection{Averaging multiple solutions}
The genetic algorithm uses a random initial population, selects
genomes at random and introduces random mutations. Because of this,
multiple applications of the inversion procedure for a specific set of
images will in general yield slightly different solutions. These
solutions are all equally acceptable: in all cases, the back-projected
images of a single source coincide very well with each other and with
the true position of the source. Given this variety of possible
solutions, it is interesting to calculate the average of a set of
solutions. This averaging procedure will enhance the common
characteristics of all the individual lens solutions while suppressing
random fluctuations. One can also calculate the standard deviation of
these individual solutions. This will identify the regions in which
the solutions agree as well as the regions in which there is a lot of
uncertainty about the mass density. Averaging the solutions will also
increase the smoothness of the retrieved mass density.
What determines the convergence of the fitness value (see section
\ref{genalg}) is the amplitude of the mutations. Once the difference
between the best possible lens solution and the best genome becomes
comparable to or smaller than the mutation amplitude, the best genomes
of subsequent generations merely scatter around some lowest achieved
fitness value. Lowering the mutation amplitude when the fitness starts
to converge and averaging a few tens of independent solutions both
help to get as close as possible to the best possible solution.
Being able to create an averaged solution is an attractive feature of
our approach, but it would be of little use if the resulting mass
density would not be a good solution of the inversion problem (or a
worse solution than the individual solutions). Using simulations (see
Sect. \ref{sec:sim}), we found that the averaged solution is indeed
also a good solution, with a very high fitness, and in many cases even
does a better job than many of the individual solutions. This is
because the random mutations that occur during the reproduction
process of the genomes, cause the best solution to oscillate around
the ``true'' solution. Since averaging a set of solutions suppresses
these random fluctuations, the averaged solution can be a more faithful
realisation of the true solution than any of the individual
solutions. Also, the inversion of a gravitational lens is clearly an
ill-posed problem so it's no great supprise that multiple solutions
exist. For these reasons, the averaged solution is definitely a very
acceptable one.
\section{Simulations}\label{sec:sim}
We conducted many simulations in order to test the validity of our
approach. Having full knowledge of the original lens as well as of the
original sources, we can easily check the accuracy with which lenses
can be reconstructed in ideal circumstances, i.e. when the redshifts
of the lens and the sources are known exactly. The mass distributions
of the lenses in these simulations were created by randomly adding a
number of Plummer distributions. The number of sources, their
positions and redshifts were also chosen at random. A wide variety of
these gravitational lens systems were used to test the algorithm. In
the example that we present below, a lens with mass of the order of
$10^{15}\,M_\odot$ was positioned at $z = 0.45$ while the redshifts of
the sources were sampled from a uniform distribution in the interval
$[1.2,4.0]$.
In Fig. \ref{fig4}, we show the mass distribution and the positions
and shapes of the sources. The total mass of the lens within a radius
of $1.5${\arcmin}, which is slightly further out than the position of
the outermost image, is $0.95\times 10^{15} \,M_\odot$ and the number
of sources in this simulation is $15$. This configuration was used to
generate the images shown in the left panel of Fig. \ref{fig5}, which
in turn serves as input for the inversion algorithm. The resolution of
this image is $1024 \times 1024$ pixels. Critical lines and caustics
for a source at redshift $z=2.5$ are presented in the right panel of
Fig. \ref{fig5}, in which the source positions are also indicated. The
genetic algorithm then constructs a lens solution that projects images
of a single source onto overlapping regions in the source plane. For
this particular simulation, the fitness converged for a grid
containing about 400 Plummer mass distributions. As explained before,
multiple applications of the inversion algorithm yield different
solutions. Still, each solution manages to produce overlapping
back-projected images and, while this is in no way enforced by the
algorithm, the positions of the back-projected images are very close
to the true source positions.
After applying the inversion routine $25$ times and averaging the
individual solutions, we obtained the final solution presented in the
left panel of Fig. \ref{fig6}. This figure shows a striking
resemblance to the left panel of Fig. \ref{fig4}. Clearly,
the mass distribution of the lens is retrieved with very high
accuracy. The fitness values of the 25 individual solutions and of the
averaged solution are shown in the left panel of
Fig. \ref{fig7}. Since averaging the individual solutions
suppresses random generation-to-generation fluctuations, which can
even prevent the solutions from further lowering the fitness value,
and enhances their common traits, the averaged solution outperforms
each individual solution. When the images of Fig. \ref{fig5}
are projected back onto the source plane, we obtain the situation
shown in the right panel of Fig. \ref{fig6}. The
back-projected images overlap very well and are close to the true
source positions. The critical lines and caustics of the averaged
solution for a source at redshift $z=2.5$ are shown in the right panel
of Fig. \ref{fig7}, which can be compared with the right
panel of Fig. \ref{fig5}. Again, the resemblance is
striking. In the left panel of Fig. \ref{fig8}, we show the
absolute value of the difference between the mass distributions of the
input lens and of the averaged solution. In the right panel of
Fig. \ref{fig8}, the standard deviation of the 25 individual
solutions is presented. The first quantity is a measure for the
quality of the fit, the second measures the disagreement between the
individual solutions.
For a circularly symmetric lens, only the total mass enclosed within
the radius of the outermost image can be determined. The lens employed
in the simulation is not spherically symmetric but one can still
surmise that we do not have a very good handle on the mass outside the
outermost image. In Fig. \ref{fig9}, we show the circularly averaged
density profiles of the input lens and of the averaged solution. As
expected, both agree excellently with each other within the inner
$\sim 1.5${\arcmin}, which is about the position of the outermost
image. Outside that radius, the density is no longer well constrained
by the data and the profile of the best lens solution drops below that
of the input lens. For the averaged solution, the mass enclosed within
a radius of 1.5{\arcmin} is $0.96 \times 10^{15}\,M_\odot$ which can
be compared with the input lens, which comprises a mass of $0.95
\times 10^{15}\,M_\odot$ within the same radius.
From this simulation and the many others we have ran, we conclude that
given enough observational constraints, our method succeeds in
inferring the mass distribution of a lens given the redshifts of the
lens and the sources and the positions and shapes of the images. If
the sources sample the caustics sufficiently well, the density profile
of the lens can be reconstructed with great accuracy out to the
outermost image. A very important feature of our method is that it
does not minimise residuals of the lens equation, like e.g. {\sc
slap}. Because tangential images are larger than radial ones, this
residual is dominated by the tangential arcs and, as a consequence,
methods that make use of it are less sensitive to the information
contained in the radial images. This is clear from e.g. \citet{d05b},
where the non-parametrically reconstructed density profile of the
cluster A1689 shows a central decline which the authors contribute to
this effect. Our method is insensitive to the size of the images and
thus makes full use of the information contained in the radial
images. This is reflected in Fig. \ref{fig9}, which shows that the
central density of the input lens is very accurately retrieved.
\section{Discussion and conclusion}\label{sec:conc}
The procedure described and illustrated above is a non-parametric
method for inverting gravitational lenses, making no a priori
assumptions regarding the shape of the lens. We only impose the
condition that an acceptable solution must be able to map images of
the same source onto overlapping regions in the source plane. The
procedure only requires that one can identify which images correspond
to the same source. In particular, no information about the sizes of
the sources needs to be provided. The size of the grid on which the
algorithm will determine the mass distribution of the lens needs to be
specified. However, because a single solution can be obtained
relatively fast, it is an easy task to try a variety of sizes until
one is obtained which generates a lens with a good fitness value. A
multi-resolution grid with a few hundred cells is usually sufficient
to represent any plausible lens mass distribution. The implementation
used for the simulation in this paper employs a population of $250$
genomes. Based on a suite of simulations, this value proved to be
sufficiently large to to yield good solutions within an acceptable
amount of time. The calculations were done in a distibuted manner,
using sixteen Intel \textregistered{} Xeon \texttrademark{} 2.4 GHz
processors of a computer cluster. Depending on the number of sources,
creating a single solution may require several hours. To give a
specific example, the $25$ solutions used in the simulation were
created in four days.
The simulation discussed in this article, together with the many
others we performed, indicate that our inversion technique
successfully solves the lens inversion problem. The reconstructed
sources lie close to their true positions and their shapes are
retrieved quite accurately as well. The procedure determines the mass
of the lens very accurately. The averaging procedure guarantees the
removal of random fluctuations and yields a smooth mass density very
close to the true solution. Note that because the masses of the
individual Plummer distributions are always represented by positive
numbers in the genome, no negative mass densities will be produced.
Of course, the quality of the reconstruction depends on the quality of
the information at hand. The algorithm depends on the availability of
multiply imaged sources, which identifies the relevant area in our
procedure as the area within the outermost caustic. When this area is
sampled well by the sources, we can expect a good reconstruction of
the mass density, as indicated by the simulation described
previously. A major advantage of the algorithm is that it makes full
use of the information contained in the radial images, unlike methods
that minimise the residuals of the lens equation, and is thus able to
accurately reconstruct also the inner parts of the lens.
Another advantage of a genetic algorithm is the ease with which the
fitness criterion can be specified. One simply has to devise a way to
associate a fitness value with a specific genome, without worrying
about features like continuity or differentiability. E.g., to test
whether information about the amplifying effect of the gravitational
lens can improve the lens reconstruction, we added a contribution to
the fitness value:~the maximum brightness values in the back-projected
images of a single source should lie as close as possible to each
other. Several simulations indicated that augmenting the procedure in
this fashion does not improve the final result since the
$\Vec{\beta}(\Vec{\theta})$ mapping was already very well approximated
anyway. Not requiring surface brightness information not only reduces
the computational cost, but also makes the method less sensitive to
noise in the images. Incorporating information about the shear field
at larger radii outside the gravitational lens is straightforward. For
each genome the shear components $\gamma_1(\Vec{\theta})$ and
$\gamma_2(\Vec{\theta})$ can be calculated and compared with the
observed values. The expression for the fitness can be changed so as
to penalise genomes with large differences between the measured and
the model shear. Further improvements of the genetic algorithm, such
as making the mutation amplitude depend automatically on the
convergence of the fitness, are easily implemented and are left as
future research. An application to real data will be presented in a
subsequent paper.
\section*{Acknowledgments}
We would like to thank Prof. Philippe Bekaert and Tom Van Laerhoven of
the Expertise Centre for Digital Media for granting us access to the
computer cluster and for taking care of the related practical
issues. We also would like to thank the anonymous referee for the
valuable remarks and suggestions. They very much improved the content
and presentation of this paper.
\bsp
\label{lastpage} |
Title:
Percolation Galaxy Groups and Clusters in the SDSS Redshift Survey: Identification, Catalogs, and the Multiplicity Function |
Abstract: We identify galaxy groups and clusters in volume-limited samples of the SDSS
redshift survey, using a redshift-space friends-of-friends algorithm. We
optimize the friends-of-friends linking lengths to recover galaxy systems that
occupy the same dark matter halos, using a set of mock catalogs created by
populating halos of N-body simulations with galaxies. Extensive tests with
these mock catalogs show that no combination of perpendicular and line-of-sight
linking lengths is able to yield groups and clusters that simultaneously
recover the true halo multiplicity function, projected size distribution, and
velocity dispersion. We adopt a linking length combination that yields, for
galaxy groups with ten or more members: a group multiplicity function that is
unbiased with respect to the true halo multiplicity function; an unbiased
median relation between the multiplicities of groups and their associated
halos; a spurious group fraction of less than ~1%; a halo completeness of more
than ~97%; the correct projected size distribution as a function of
multiplicity; and a velocity dispersion distribution that is ~20% too low at
all multiplicities. These results hold over a range of mock catalogs that use
different input recipes of populating halos with galaxies. We apply our
group-finding algorithm to the SDSS data and obtain three group and cluster
catalogs for three volume-limited samples that cover 3495.1 square degrees on
the sky. We correct for incompleteness caused by fiber collisions and survey
edges, and obtain measurements of the group multiplicity function, with errors
calculated from realistic mock catalogs. These multiplicity function
measurements provide a key constraint on the relation between galaxy
populations and dark matter halos.
| https://export.arxiv.org/pdf/astro-ph/0601346 |
\title{Percolation Galaxy Groups and Clusters in the SDSS Redshift Survey: Identification, Catalogs, and the Multiplicity Function}
\author{
Andreas A. Berlind, \altaffilmark{1,2}
Joshua Frieman, \altaffilmark{2}
David H. Weinberg, \altaffilmark{3}
Michael R. Blanton, \altaffilmark{1}
Michael S. Warren, \altaffilmark{4}
Kevork Abazajian, \altaffilmark{4}
Ryan Scranton, \altaffilmark{5}
David W. Hogg, \altaffilmark{1}
Roman Scoccimarro, \altaffilmark{1}
Neta A. Bahcall, \altaffilmark{6}
J. Brinkmann, \altaffilmark{7}
J. Richard Gott III, \altaffilmark{6}
S.J. Kleinman, \altaffilmark{7}
J. Krzesinski, \altaffilmark{8,9}
Brian C. Lee, \altaffilmark{10}
Christopher J. Miller, \altaffilmark{11}
Atsuko Nitta, \altaffilmark{7}
Donald P. Schneider, \altaffilmark{12}
Douglas L. Tucker, \altaffilmark{13}
Idit Zehavi, \altaffilmark{14,15}
for the SDSS collaboration
}
\keywords{cosmology: large-scale structure of universe --- galaxies: clusters}
\altaffiltext{1}{Center for Cosmology and Particle Physics, New York
University, New York, NY 10003, USA; aberlind@cosmo.nyu.edu}
\altaffiltext{2}{Center for Cosmological Physics and Department of Astronomy
and Astrophysics, University of Chicago, Chicago, IL 60637; frieman@fnal.gov}
\altaffiltext{3}{Department of Astronomy, The Ohio State University, Columbus,
OH 43210; dhw@astronomy.ohio-state.edu}
\altaffiltext{4}{Theoretical Division, Los Alamos National Laboratory, Los Alamos,
NM 87545}
\altaffiltext{5}{Physics and Astronomy Department, University of Pittsburgh,
Pittsburgh PA, 15260}
\altaffiltext{6}{Department of Astrophysical Sciences, Princeton University,
Princeton NJ, 08544}
\altaffiltext{7}{Subaru Telescope, 650 N A'ohoku Pl., Hilo, HI 96720}
\altaffiltext{8}{Apache Point Observatory, P.O. Box 59, Sunspot, NM 88349}
\altaffiltext{9}{Mt. Suhora Observatory, Cracow Pedagogical University, ul.
Podchorazych 2, 30-084 Cracow, Poland}
\altaffiltext{10}{Lawrence Berkeley National Lab, Berkeley CA 94720}
\altaffiltext{11}{Cerro-Tololo Inter-American Observatory, NOAO, Casilla 603, La Serena,
Chile}
\altaffiltext{12}{Department of Astronomy and Astrophysics, Pennsylvania State University,
University Park, PA 16802}
\altaffiltext{13}{Fermi National Accelerator Laboratory, MS 127, PO Box 500,
Batavia, IL 60510}
\altaffiltext{14}{Steward Observatory, University of Arizona, 933 N. Cherry Ave.,
Tucson AZ 85721}
\altaffiltext{15}{Deptartment of Astronomy, Case Western Reserve University,
Cleveland, OH 44106}
\section{Introduction} \label{intro}
Galaxies are gregarious by nature. Bright galaxies typically reside in groups
or clusters, surrounded by less luminous neighbors. Interactions within the
group or cluster environment may have important effects on the star formation
history, morphology, dynamics, and other properties of member galaxies.
Characterizing the relation between galaxy properties and their group environment
is thus a key step in understanding galaxy formation and evolution.
At the density thresholds often used to identify groups, most members should
belong to the same, gravitationally bound dark matter (DM) halo.\footnote{Throughout
this paper, we use the term ``halo'' to refer to a gravitationally bound structure
with overdensity $\rho/\bar{\rho}\sim200$, so an occupied halo may host a single
luminous galaxy, a group of galaxies, or a cluster. Higher overdensity concentrations
around individual galaxies of a group or cluster constitute, in this terminology,
halo substructure, or ``sub-halos''.} Recent approaches to describing the relation
between galaxies and DM focus on galaxy populations of DM halos as a function of
halo mass. Specifically, the bias of a particular class of galaxies can be
characterized by its Halo Occupation Distribution (HOD), which specifies the
probability distribution $P(N|M)$ that a halo of mass $M$ contains $N$ such galaxies,
together with relations describing the relative spatial and velocity distributions
of galaxies and dark matter within halos (\citealt{berlind_weinberg_02} and references
therein). A well defined group catalog with well understood properties can play
a central role in the empirical determination of this relation.
This paper presents a group and cluster catalog defined from the Sloan Digital Sky
Survey (SDSS, \citealt{york_etal_00}). While this catalog is useful for many purposes,
our overriding objective is to obtain a well understood measurement of the group
multiplicity function (the space density of groups as a function of richness), with the
goal of determining the HOD in the high mass regime \citep{peacock_smith_00,berlind_weinberg_02,marinoni_hudson_02,kochanek_etal_03,lin_etal_04}.
With this objective in mind, we have adopted a simple group-finding algorithm,
friends-of-friends in redshift space \citep{huchra_geller_82}, and carried out
extensive tests on realistic mock catalogs in order to assess its performance and
optimize parameter choices. We apply the group-finding algorithm to volume-limited
samples of galaxies so that the resulting group statistics characterize the clustering
of well defined populations of galaxies.
Galaxy clusters have been the focus of study since they were first seen on optical
photographic plates \citep{shapley_ames_26}. \citet{zwicky_37} pioneered the study
of clusters as dynamical objects by using imaging and spectroscopy of the Coma cluster
to estimate its mass. However, the most influential pioneering work on clusters was
done by \citet{abell_58}, who assembled the first large sample of galaxy clusters. The
Abell catalog of rich galaxy clusters \citep{abell_58,abell_etal_89} was created by
eyeball identification in the Palomar Observatory Sky Survey and it spawned numerous
follow-up studies. \citet{devaucouleurs_71} shifted focus to poorer systems by
studying nearby groups of galaxies. \citet{gott_turner_77b} made the first measurement
of the group multiplicity function using the \citep{turner_gott_76} catalog of groups
selected based on the projected surface density of galaxies.
With the advent of large redshift surveys, group identification became three dimensional
and thus less subject to projection effects. Group-finding in redshift space was
pioneered by \citet{huchra_geller_82} and \citet{geller_huchra_83}, using the Center for
Astrophysics (CfA) redshift survey. Subsequent versions of the CfA redshift survey were
used to identify groups by various authors
\citep{nolthenius_white_87,ramella_etal_89,moore_etal_93,ramella_etal_97}.
Other redshift surveys that spawned group catalogs were the Nearby Galaxies Catalog
\citep{tully_87}, the ESO Slice Project \citep{ramella_etal_99}, the Las Campanas
Redshift Survey (LCRS) \citep{tucker_etal_00}, the Nearby Optical Galaxy Sample (NOG)
\citep{giuricin_etal_00}, the Southern Sky Redshift Survey (SSRS)
\citep{ramella_etal_02}, the 2dF redshift survey
\citep{merchan_zandivarez_02,eke_etal_04a,yang_etal_05}, and even the high redshift DEEP2
survey \citep{gerke_etal_05}.
There have been several efforts to detect clusters in the SDSS to date, most of them
using the photometric data rather than the redshift data. \citet{annis_etal_99}
developed the maxBCG technique, where Brightest Cluster Galaxy (BCG) candidates
are identified based on their colors and magnitudes and other cluster members are
selected from nearby galaxies that have the colors of the E/S0 ridgeline.
\citet{kim_etal_02} developed a hybrid matched filter (HMF) technique that assumes
a radial profile for clusters and convolves the data with that filter.
\citet{goto_etal_02} developed the cut-and-enhance (CE) method, which selects
overdensities of galaxies that have similar colors. All these techniques were applied
to the early SDSS commissioning data \citep{bahcall_etal_03,goto_etal_02}.
\citet{lee_etal_04} identified compact groups by looking for small and isolated
concentrations of galaxies in the SDSS Early Data Release (EDR;
\citealt{stoughton_etal_02}). Cluster searches in the SDSS redshift survey have also
been carried out. \citet{goto_etal_05} used a friends-of-friends algorithm (though
with linking lengths that do not scale with the changing number density of galaxies
due to the flux limit) to identify clusters in the SDSS Data Release 2 (DR2;
\citealt{abazajian_etal_04}). \citet{merchan_zandivarez_05} used a friends-of-friends
algorithm to identify groups in the SDSS Data Release 3 (DR3;
\citealt{abazajian_etal_05}). \citet{weinmann_etal_05} used the \citet{yang_etal_05}
algorithm to identify groups in SDSS DR2. \citet{miller_etal_05} developed the C4
algorithm for finding clusters in redshift space and also applied it to the SDSS DR2.
The C4 algorithm looks for concentrations of galaxies in a seven-dimensional position
and color space. It takes advantage of the color similarity of cluster member galaxies
and thus minimizes contamination due to projection. However, some correlations are built
into the method, and modeling it in order to understand the properties of the resulting
cluster catalog requires a complete model of the galaxy population (including colors
and luminosities). Our method complements the C4 catalog by applying a simple and
easily modeled algorithm to volume-limited samples with homogeneous properties.
In \S~\ref{data} we describe the SDSS data that we use. In \S~\ref{mocks} we describe
the mock catalogs that we use to optimize our group-finder and to estimate uncertainties
for our measured group statistics. In \S~\ref{groupfinder} we outline our group-finding
algorithm and choice of parameters. We present a detailed discussion of tests with mock
catalogs in the Appendix, with the key points summarized in the main text.
We discuss incompleteness in our group catalogs due to fiber collisions and survey edges
in \S~\ref{incompleteness}. The group catalogs are published in electronic tables
and their contents are described in \S~\ref{catalog}. Finally, in \S~\ref{multiplicity},
we present our measured group multiplicity function. We will use this to constrain
the HOD in future work. We summarize our results in \S~\ref{summary}.
\section{Data} \label{data}
\subsection{SDSS}
The SDSS is a large imaging and spectroscopic survey that is mapping two-fifths of the
Northern Galactic sky and a smaller area of the Southern Galactic sky, using a
dedicated 2.5 meter telescope \citep{gunn_etal_06} at Apache Point, New Mexico.
The survey uses a photometric camera \citep{gunn_etal_98} to scan the sky
simultaneously in five photometric bandpasses \citep{fukugita_etal_96,smith_etal_02}
down to a limiting $r$-band magnitude of $\sim22.5$. The imaging data are processed
by automatic software that does astrometry \citep{pier_etal_03}, source identification,
deblending and photometry \citep{lupton_etal_01,lupton_05}, photometric calibration
\citep{hogg_etal_01,smith_etal_02,tucker_etal_05}, and data quality assessment
\citep{ivezic_etal_04}. Algorithms are applied to select spectroscopic targets for
the main galaxy sample \citep{strauss_etal_02}, the luminous red galaxy sample
\citep{eisenstein_etal_01}, and the quasar sample \citep{richards_etal_02}.
The main galaxy sample is approximately complete down to an apparent $r$-band
Petrosian magnitude limit of $<17.77$. Targets are assigned to spectroscopic plates
using an adaptive tiling algorithm \citep{blanton_etal_03a}. Finally,
spectroscopic data reduction pipelines produce galaxy spectra and redshifts.
We use the large-scale structure sample \texttt{sample14} from the NYU Value
Added Galaxy Catalog (NYU-VAGC; \citealt{blanton_etal_04a}) as our primary galaxy
sample. Galaxy magnitudes are corrected for Galactic extinction
\citep{schlegel_etal_98} and absolute magnitudes are k-corrected
\citep{blanton_etal_03b} and corrected for passive evolution \citep{blanton_etal_03c}
to rest-frame magnitudes at redshift $z=0.1$. A significant fraction of
the sample that we use was made publicly available with the SDSS Data Release~3
\citep{abazajian_etal_05}.
\begin{table}
\begin{center}
\centerline{\small Table~1. Volume-limited Sample Parameters}
\begin{tabular}[t]{lccccc}
\tableline
\tableline
Name & $\zmin$ & $\zmax$ & $<\Mr$ & $\Ng$ & $\ng$ \\
\tableline
$Mr20$ & 0.015 & 0.100 & -19.9 & 57138 & 0.00673 \\
$Mr19$ & 0.015 & 0.068 & -19.0 & 37820 & 0.01396 \\
$Mr18$ & 0.015 & 0.045 & -18.0 & 18895 & 0.02434 \\
\tableline
\label{tab:samples}
\end{tabular}
\end{center}
Note---Absolute magnitude thresholds listed are for $\zmax$. $\ng$ is in
units of $\hden$.
\end{table}
The galaxy redshift sample has an incompleteness due to the mechanical restriction
that spectroscopic fibers cannot be placed closer to each other than their own
thickness. This fiber collision constraint makes it impossible to obtain redshifts
for both galaxies in pairs that are closer than $55''$ on the sky. In the case of a
conflict, the target selection algorithm randomly chooses which galaxy gets a fiber
\citep{strauss_etal_02}.\footnote{In cases where a target galaxy fiber collides with a
target quasar fiber, priority is always given to the quasar, but such collisions
only constitute $\sim 5\%$ of all cases.} Spectroscopic plate overlaps alleviate this
problem to some extent, but fiber collisions still account for a $\sim 6\%$
incompleteness in the main galaxy sample. Since this incompleteness is most severe in
regions of high galaxy density, it is necessary to correct for it in
studies of groups and clusters. We correct for fiber collisions by giving each
collided galaxy the redshift of its nearest neighbor on the sky (usually the galaxy
it collided with), and we show in \S~\ref{incompleteness} that this procedure is
adequate for our purposes.
Putting collided galaxies at the redshifts of their nearest neighbors will
cause some nearby galaxies to be placed at high redshift, artificially making
their estimated luminosities very high. Since the abundance of highly luminous
galaxies is low, this contamination can become a significant fraction of all highly
luminous galaxies. For this reason, we also give collided galaxies the magnitudes
(in addition to the redshifts) of their nearest neighbors. The resulting luminosity
distribution is thus unbiased.
There is some additional incompleteness due to bright foreground stars
blocking background galaxies, but this is at the $\sim 1\%$ level. In order
to limit the effects of incompleteness on our group identification, we
restrict our sample to regions of the sky where the completeness (ratio of
obtained redshifts to spectroscopic targets) is greater than $90\%$. Our final
sample covers 3495.1 square degrees on the sky and contains 298729 galaxies.
\subsection{Volume-limited Samples}
In this and subsequent papers, we are primarily interested in using galaxy
groups to constrain the properties of galaxies as a function of their
underlying dark matter halo mass. It is therefore important that the population
of galaxies constituting the groups is homogeneous within the sample volume.
For this reason, we construct volume-limited subsamples of the full SDSS redshift
sample that are each complete in a specified redshift range down to a limiting
$r$-band absolute magnitude threshold. We construct each sample by choosing
redshift limits $\zmin$ and $\zmax$, and only keeping galaxies whose
evolved, redshifted spectra would still make the redshift survey's apparent
magnitude and surface brightness cuts at the limiting redshifts of the sample.
Since the apparent magnitude limit of the redshift sample varied across the sky in
the commissioning phases of the survey, we cut the $r$-band magnitude limit from
$\sim17.77$ back to 17.5. This more conservative limit is uniform across the sky.
We construct three such volume-limited samples. Figure~\ref{fig:vollim} shows
these samples in the luminosity-redshift plane. Each dot in the figure shows
a galaxy in the SDSS redshift survey. The sharp cutoff curve along the lower-right
part of the plot shows our $r=17.5$ apparent magnitude limit.
We select three redshift ranges for our volume-limited samples: $0.015-0.1$,
$0.015-0.068$, and $0.015-0.045$. These samples are complete down to absolute
$r$-band magnitudes of $\Mr<-19.9$, $-19$, and $-18$, respectively.\footnote{All
absolute magnitudes are quoted for $\Omegam=0.3$, $\Omegal=0.7$, and a value of the
Hubble constant $h \equiv H_0 / 100\Hunits = 1$. For other values of $H_0$, one
should add $5\log h$ to the quoted absolute magnitudes.} We refer to these samples
as $Mr20$, $Mr19$, and $Mr18$, henceforth. Regions of the plot that make it into
these three samples are shown in blue, green, and red, respectively. The limiting
absolute magnitude of each sample changes slightly with redshift due to the passive
evolution corrections applied to galaxy luminosities: as a galaxy is moved to
the outer edge of a given volume-limited sample, its luminosity increases
somewhat, allowing lower redshift galaxies to make it into the sample at lower
luminosities than they do at higher redshifts. We choose the first limiting
redshift of $\zmax=0.1$ because this yields the largest possible volume-limited
sample (largest number of galaxies). We choose lower redshift samples in
order to probe galaxy populations less luminous than $\Lstar$. We use a lower
redshift limit of $0.015$ for all three samples to alleviate some of the problems
associated with obtaining accurate photometry of nearby highly extended galaxies.
The redshift limits, luminosity thresholds at $\zmax$, number of galaxies, and space
densities of these samples are listed in Table~1.
Figure~\ref{fig:skyvollim} shows a Hammer (equal area) projection (in equatorial
coordinates) of sample $Mr20$. Points represent galaxies in the sample. The curve
shows the location of the Galactic plane. The figure illustrates the patchy and
non-uniform nature of the sample footprint on the sky, which has irregular edges,
as well as multiple holes. This irregularity exacerbates systematic errors due to edge
effects. We deal with incompleteness due to edge effects in \S~\ref{incompleteness}.
Figure~\ref{fig:slicevollim} shows an equatorial slice through sample $Mr20$. The
slice is $4^\circ$ thick and each point shows the RA and redshift of a galaxy in
the sample. Prominent in this projection of the data is the the giant supercluster
at $z\sim0.08$ at the left end of the Sloan Great Wall of Galaxies, which extends from
longitude 132 degrees (at $z\sim0.05$) to longitude 210 degrees (at $z\sim0.08$) (See
\citealt{gott_etal_05}).
\section{Mock Catalogs} \label{mocks}
Our main scientific motivation for constructing group catalogs from the SDSS
data requires that identified groups most closely resemble systems of galaxies
that occupy a common dark matter halo. Moreover, it is important that we
statistically quantify the degree to which our groups do not satisfy this
criterion. For both these reasons, it is imperative that we use mock galaxy
catalogs that are constructed by populating dark matter halos in N-body
simulations with mock galaxies. The N-body simulations must satisfy two
basic conditions: they must contain a large enough volume to fit our largest
volume-limited sample, $Mr20$, and they must resolve the smallest mass halos
that can host a galaxy in our least luminous volume-limited sample, $Mr18$.
HOD fits to the SDSS two-point correlation function of galaxies suggest that
the minimum dark matter halo mass that can host a galaxy of luminosity
$\Mr\sim -18$ is approximately $2\times 10^{11}\hMsun$
\citep{zehavi_etal_05,tinker_etal_05}. Requiring that a halo contain at least
forty dark matter particles to be resolved means that we need N-body simulations
with particle masses less than $5\times 10^{9}\hMsun$.
We use a series of N-body simulations of a $\Lambda$CDM cosmological model,
with $\Omegam=0.3$, $\Omegal=0.7$, $\Omegab=0.04$,
$h\equiv H_0/(100~\mathrm{km~s}^{-1}~\mathrm{Mpc}^{-1})=0.7$, $n_s=1.0$, and
$\sigma_8=0.9$. This model is in good agreement with a wide variety of
cosmological observations (see, e.g., \citealt{spergel_etal_03,tegmark_etal_04b,
abazajian_etal_05b}).
Initial conditions were set up using the transfer function calculated
for this cosmological model by CMBFAST \citep{seljak_zaldarriaga_96}. The
simulations were run at Los Alamos National Laboratory (LANL) using the
Hashed-Oct-Tree (HOT) code \citep{warren_Salmon_93}. We use a total of
six independent simulations of varying size and resolution, which we refer
to as \texttt{LANL1-6}. The size of box $\Lbox$,
number of particles
$N_\mathrm{p}$, and resulting particle mass $m_\mathrm{p}$ for each
simulation are listed in Table~2. The gravitational force
softening is $\epsilon_{\rm grav}=12\hkpc$ (Plummer equivalent).
\begin{table*}
\begin{center}
\centerline{\small Table~2. Mock Catalog Parameters}
\begin{tabular}{l|cccc|cccc}
\tableline
\tableline
&
\multicolumn{4}{c|}{N-body} &
\multicolumn{4}{c}{HOD} \\
Mock & Name & $\Lbox$ & $N_\mathrm{p}$ & $m_\mathrm{p}$ & $\Mmin$ & $\Mcut$ & $M_1$ & $\alpha$ \\
& & ($\hmpc$) & & ($10^9\hMsun$) & ($10^{11}\hMsun$) & ($10^{13}\hMsun$) & ($10^{12}\hMsun$) & \\
\tableline
\texttt{LANL1.Mr20} & \texttt{LANL1} & $384$ & $1024^3$ & 4.39 & 10.0 & --- & 25.0 & 1.1 \\
\texttt{LANL1.Mr20b} & & & & & 9.08 & 1.14 & 12.3 & 0.9 \\
\texttt{LANL1.Mr19} & & & & & 3.7 & --- & 8.2 & 1.0 \\
\texttt{LANL1.Mr18} & & & & & 1.9 & --- & 3.4 & 0.9 \\
\tableline
\texttt{LANL2.Mr20} & \texttt{LANL2} & $384$ & $1024^3$ & 4.39 & 10.0 & --- & 25.0 & 1.1 \\
\texttt{LANL2.Mr20b} & & & & & 9.08 & 1.14 & 12.3 & 0.9 \\
\texttt{LANL2.Mr19} & & & & & 3.7 & --- & 8.2 & 1.0 \\
\texttt{LANL2.Mr18} & & & & & 1.9 & --- & 3.4 & 0.9 \\
\tableline
\texttt{LANL3.Mr20} & \texttt{LANL3} & $384$ & $1024^3$ & 4.39 & 10.0 & --- & 25.0 & 1.1 \\
\texttt{LANL3.Mr20b} & & & & & 9.08 & 1.14 & 12.3 & 0.9 \\
\texttt{LANL3.Mr19} & & & & & 3.7 & --- & 8.2 & 1.0 \\
\texttt{LANL3.Mr18} & & & & & 1.9 & --- & 3.4 & 0.9 \\
\tableline
\texttt{LANL4.Mr20} & \texttt{LANL4} & $400$ & $1280^3$ & 2.54 & 10.0 & --- & 25.0 & 1.1 \\
\texttt{LANL4.Mr20b} & & & & & 9.08 & 1.14 & 12.3 & 0.9 \\
\texttt{LANL4.Mr19} & & & & & 3.7 & --- & 8.2 & 1.0 \\
\texttt{LANL4.Mr18} & & & & & 1.9 & --- & 3.4 & 0.9 \\
\tableline
\texttt{LANL5.Mr20} & \texttt{LANL5} & $543$ & $1024^3$ & 12.4 & 10.0 & --- & 25.0 & 1.1 \\
\texttt{LANL5.Mr20b} & & & & & 9.08 & 1.14 & 12.3 & 0.9 \\
\tableline
\texttt{LANL6.Mr20} & \texttt{LANL6} & $768$ & $1024^3$ & 35.1 & 10.0 & --- & 25.0 & 1.1 \\
\tableline
\label{tab:mocks}
\end{tabular}
\end{center}
\end{table*}
We identify halos in the dark matter particle distributions using a
friends-of-friends algorithm with a linking length equal to $0.2$ times the
mean interparticle separation. We then populate these halos with galaxies
using a simple model for the HOD of galaxies more luminous than a luminosity
threshold. Every halo with a mass $M$ greater than a minimum mass $\Mmin$ gets a
central galaxy that is placed at the halo center of mass and is given the mean halo
velocity. A number of satellite galaxies is then drawn from a Poisson distribution
with mean $\Nsat = ((M-\Mmin)/M_1)^\alpha$, for $M\geq\Mmin$. These satellite
galaxies are assigned the positions and velocities of randomly selected dark matter
particles within the halo. In order to construct mock catalogs for each of our
three volume-limited samples $Mr20$,
$Mr19$, and $Mr18$, we select sets of
values for the parameters $\Mmin$, $M_1$, and $\alpha$ that yield
the observed \citet{zehavi_etal_05} galaxy-galaxy correlation functions for
these samples. These HOD parameter values are similar to the best-fit values
given by \citet{zehavi_etal_05} (they are slightly different because the
model for $\Nsat$ was different in that paper). We refer to these sets of
mock catalogs with the suffixes \texttt{.Mr20}, \texttt{.Mr19}, and \texttt{.Mr18}.
In addition to these mock catalogs, we construct a set of
catalogs for the $Mr20$ sample using an alternative HOD model, where the mean
number of satellites in a halo of mass $M$ is
$\Nsat = \mathrm{exp}[-\Mcut/(M-\Mmin)] (M/M_1)^\alpha$, for $M>\Mmin$
(also used by \citealt{tinker_etal_05}). We fix the value of the slope $\alpha$
to 0.9, which is lower than that for the \texttt{.Mr20} mocks, and we choose
values for the remaining HOD parameters that yield the observed
\citet{zehavi_etal_05} correlation function of $\Mr<-20$ galaxies. We refer
to these sets of mock catalogs with the suffix \texttt{.Mr20b}. The values
for all mock HOD parameters are listed in Table~2.
We construct ten realizations of each mock catalog listed in Table~2
by using different random number generator seeds when we (a) draw a number of
satellite galaxies for each halo from a Poisson distribution of mean $\Nsat$,
and (b) select random dark matter halo particles to give their positions and
velocities to these satellite galaxies. The dispersion among the ten
realizations for one mock catalog therefore represents the scatter among
possible observed states for a given halo distribution and HOD model.
We now have a set of mock catalogs containing galaxies in real space and in
the cubical geometry of the N-body simulations. We refer to these as our
``real-space cube mocks''. We create a redshift-space version of these catalogs
by assuming the distant observer approximation and aligning the line-of-sight
along one of the axes of the simulation cubes. We use the mock galaxies'
peculiar velocities to move them along the line-of-sight into redshift space.
We refer to the resulting mock catalogs as our ``redshift-space cube mocks''.
We use these real-space and redshift-space cube mocks to determine optimal
parameters for our group-finding algorithm. We summarize this determination in
\S\ref{groupfinder} and discuss details in the Appendix.
For the purpose of studying the effects of SDSS incompleteness on our measured
groups, as well as for obtaining estimates of the uncertainty in our measured
group multiplicity function, we also require mock catalogs that have the same
geometry as our SDSS volume-limited samples. The total volume of our largest
sample, $Mr20$, is approximately $210^3\hvol$, which is more than six times
smaller than any of our mock cubes. However, the SDSS geometry is highly
irregular (as seen in Fig.~\ref{fig:skyvollim}) and can only be fully embedded in
a cube of much larger volume than the survey itself. The $Mr20$ sample, for
example, has a maximum extent of $\sim600\hmpc$ when both the North and
South Galactic portions are included. In order to carve this sample geometry
out of our mock catalogs, we create mock cubes with eight times larger volume by
tiling each mock cube $2\times2\times2$. Since the N-body simulations used
to construct the mocks were run with periodic boundary conditions, we can tile
the cubes without having density discontinuities at the boundaries. We set the
center of this tiled cube to be the origin and put galaxies into redshift space
using the line-of-sight component of their peculiar velocities. We then
compute RA, DEC, and redshift coordinates for every mock galaxy in the tiled
cube. Finally, we only keep galaxies whose coordinates on the sky would place
them in regions of the SDSS survey that have completeness greater than $90\%$,
and whose redshifts lie within the redshift limits of the specific volume-limited
sample we are constructing mock catalogs for.
Since the volume of each simulation cube is at least six times larger than
our largest volume-limited sample $Mr20$, we try to carve out as many independent
volumes with the $Mr20$ geometry as possible without too much overlap. We do
this by performing many sets of three rotations (one around each Cartesian axis)
and testing how much overlap the resulting catalogs have with each other (i.e.,
how many common mock galaxies do they share). With the right combination
of rotation angles, we can carve out two $Mr20$ mock catalogs that share fewer than
$3\%$ of their galaxies with each other, but we cannot obtain more without
significant overlap. We create two such independent mock
catalogs, with the correct SDSS geometry, from every one of the ten HOD
realizations of the mock cubes listed in Table~2, except for the
\texttt{LANL6.Mr20} mock. This procedure yields 200 mock catalogs for
the $Mr20$ sample (5 N-body simulations $\times$ 2 HOD models $\times$ 10 HOD
realizations $\times$ 2 mocks per simulation cube), and 80 mock catalogs each
for the $Mr19$ and $Mr18$ samples (4 N-body simulations $\times$ 1 HOD model
$\times$ 10 HOD realizations $\times$ 2 mocks per simulation cube).
The final step in creating mock SDSS catalogs is to incorporate the fiber
collision constraint. We use a friends-of-friends algorithm to identify groups
of mock galaxies that are linked together by the $55''$ minimum angular
separation of fibers. We then select ``collided'' mock galaxies (whose redshifts
will be unknown) in each such collision group in a way that minimizes the number of
such galaxies. For example, if a collision group contains three galaxies in a row,
where the first is closer than $55''$ from the second and the second is closer than
$55''$ from the third, but the first is more than $55''$ from the third, we will
always select the middle galaxy to be the collided one. In cases where multiple
choices yield the same number of collided galaxies, we select randomly (e.g., in
collision groups with only two galaxies). This procedure is designed to mimic
the tiling code that assigns spectroscopic fibers to SDSS target galaxies
\citep{blanton_etal_03a}. If we perform this operation on the \texttt{.Mr20} catalogs
we end up with only $\sim 3\%$ of mock galaxies being tagged as collided. This
is about half the fraction of SDSS galaxies in our $Mr20$ sample that don't have
measured redshifts due to fiber collisions. The reason for this discrepancy is
that galaxies in the $Mr20$ volume-limited sample do not only collide with each
other; they also collide with galaxies more luminous than $\Mr\sim-20$ at redshifts
higher than the sample limit $z=0.1$ and galaxies less luminous than $\Mr\sim-20$ at
lower redshifts. Most of these additional galaxies that can collide with a given
galaxy in $Mr20$ are uncorrelated background or foreground galaxies. It is
therefore sufficient to model them as a background screen of galaxies on the sky
that have an angular correlation function equal to the mean for all SDSS galaxies.
For this purpose, we use the very large volume \texttt{LANL6.Mr20} cube mock.
We use \texttt{LANL6.Mr20} to construct a ``screen'' catalog with the correct SDSS
angular geometry and a variable outer redshift limit, and superpose it onto each
of our \texttt{.Mr20}, \texttt{.Mr19}, and \texttt{.Mr18} mock catalogs. We then
allow all galaxies to collide with each other and keep track of collided mock galaxies.
We set the outer redshift limit of the screen catalog to the value that results in
$\sim 6\%$ of mock galaxies being tagged as collided. We find that we need
approximately seven times more galaxies in the screen catalog than in the mocks
in order to achieve this collided fraction.
Using this approach we construct three versions of every mock catalog described
above: a version with no fiber collisions applied (``true'' version), a version
where collided galaxies have no redshifts and are dropped out of the mock catalog
altogether (``uncorrected'' version), and a version where collided galaxies
are assigned the redshift of the galaxy they collided with (``corrected'' version).
These mock catalogs allow us to test the effects of fiber collisions on our
measured group multiplicity function (discussed in \S~\ref{incompleteness}.)
\section{Group-Finding Algorithm} \label{groupfinder}
We wish to identify galaxy groups primarily in order to measure the group
multiplicity function and use it to constrain the HOD of galaxies as a function
of galaxy properties.
This goal places a number of demands on the group-finding algorithm:
(1) It should identify galaxy systems that occupy the same dark matter halos with
the least possible merging of different halos into the same group and the least
possible splitting of individual halos into multiple groups. (2) It should
produce a group multiplicity function that is unbiased with respect to the
halo multiplicity function. (3) It should be simple and well-defined so that the
statistical and systematic uncertainty in the measured group multiplicity function
can be accurately characterized. (4) It should use only the spatial positions
of galaxies in redshift space to identify groups, and not galaxy properties
such as color or luminosity. These requirements point to an algorithm that
uniquely identifies density enhancements in redshift space.
We adopt the simple and well understood friends-of-friends (FoF) algorithm, where
galaxies are recursively linked to other galaxies within a specified linking volume
around each galaxy. The FoF algorithm has several attractive features. First, for a
given linking volume (usually specified by one linking length in real space and two
linking lengths in redshift space), FoF produces a unique group catalog. Second, it
does not assume or enforce any particular geometry for groups (e.g., spherical), but
rather identifies structures that are approximately enclosed by an isodensity surface
whose density is monotonically related to the linking lengths. Third, the algorithm
satisfies a nesting condition: all the members of a group identified with one set of
linking lengths are also members of the same group identified using larger linking
lengths.
The FoF algorithm has been used extensively to identify dark matter halos in N-body
simulations (e.g., \citealt{davis_etal_85}) and has been shown to produce halo
catalogs with mass functions that are close to universal (within $\sim 20\%$) for a wide
range of epochs and cosmological models \citep{jenkins_etal_01}. FoF has also been the
most used algorithm for identifying galaxy groups in redshift surveys
\citep{huchra_geller_82,geller_huchra_83,nolthenius_white_87,ramella_etal_89,
moore_etal_93,ramella_etal_97,ramella_etal_99,tucker_etal_00,giuricin_etal_00,
ramella_etal_02,merchan_zandivarez_02,eke_etal_04a}, though alternative methods have
also been used (see e.g., \citealt{tully_87,marinoni_etal_02,gerke_etal_05,
yang_etal_05}). These FoF studies all used the same basic algorithm, but differed in
their choices for linking lengths and in their methods for dealing with the varying
density of galaxies inherent in flux-limited surveys.
We use the basic \citet{huchra_geller_82} algorithm, where two galaxies are linked to
each other if both their transverse and line-of-sight separations are smaller
than a given pair of projected and line-of-sight linking lengths, respectively.
Specifically, two galaxies $i$ and $j$ with angular separation $\theta_{ij}$ and
redshifts $z_i$ and $z_j$, have a projected separation $D_{\perp,ij}$ and a
line-of-sight separation $D_{\parallel,ij}$ (both in $\hmpc$) given by
\footnote{We use these simple equations, rather than the exact formulae for the
redshift-distance and angular diameter-distance relations because, at $z=0.1$ (the
outer limit of our sample), the difference between these formulae is less than $1\%$.}
\begin{eqnarray}
D_{\perp,ij} & = & (c/H_0)(z_i+z_j)~\mathrm{sin}(\theta_{ij}/2), \\
D_{\parallel,ij} & = & (c/H_0)|z_i-z_j|.
\end{eqnarray}
The two galaxies are then linked to each other if
\begin{equation}
D_{\perp,ij} \leq \bperp~\ng^{-1/3}
\end{equation}
and
\begin{equation}
D_{\parallel,ij} \leq \bpar~\ng^{-1/3},
\end{equation}
where $\ng$ is the mean number density of galaxies, and $\bperp$ and $\bpar$ are
the projected and line-of-sight linking lengths in units of the mean intergalaxy
separation. Since we use volume-limited samples of SDSS galaxies, $\ng$ is
constant throughout the sample volumes, and thus the linking lengths are also
constant.
The resulting linking volume around each galaxy is very similar to a cylinder,
oriented along the line-of-sight, whose radius is equal to the projected linking
length and whose height is equal to twice the line-of-sight linking length. It
is not a perfect cylinder because its radius increases with redshift, making it
slightly wider at the far end than at the near end, and its bases are slightly
curved. However, for the small linking lengths considered here, a cylinder is
a good approximation. The FoF algorithm works recursively, whereby a galaxy is
linked to all its ``friends'', which are in turn linked to their ``friends'',
etc., to yield a unique group of galaxies.
\subsection{Choice of Linking Lengths}
The most important ingredient of our group-finding algorithm is our choice
for the linking lengths $\bperp$ and $\bpar$. If the linking lengths are
too small, then the group-finder will break up single halos into multiple
groups. If the linking lengths are too large, then different halos will be
fused together into single groups. There are no values for the linking lengths
that will work perfectly for every halo, even in real space. In redshift space
this problem becomes substantially worse, since redshift-space distortions both
move halos and elongate them along the line-of-sight, often causing them to
overlap with each other. The right choice of linking lengths depends on the
purpose for which groups are being identified. If we require a group catalog that
is highly inclusive and groups together every galaxy inhabiting the same halo, then
we will use larger linking lengths than if our goal is to minimize contamination by
galaxies that come from different halos. For our purposes, we wish to obtain a
balance between being inclusive and reducing contamination, while producing groups
that have an unbiased multiplicity function.
In order to find the right combination of linking lengths, we use the mock galaxy
catalogs described in \S~\ref{mocks}. Specifically, we use the real-
and redshift-space cube mocks, which are constructed by applying simple HOD
models to the \texttt{LANL1} and \texttt{LANL4} N-body simulations. Since we
know which mock galaxies occupy the same dark matter halos, we can evaluate
how well a particular choice of linking lengths recovers features of the halo
population. The mocks that we use here have a cubical geometry, and we assume the
distant observer approximation when we put mock galaxies into redshift space.
We use the full cubical mocks rather than those with the correct SDSS geometry
because the full mocks have a much larger volume and thus better statistics.
Moreover, our goal is to find the best linking lengths for any redshift survey, and
we will deal with systematic effects specific to our SDSS sample geometry separately.
The FoF algorithm that we use is therefore slightly different from the one outlined
above, in that the linking volume is a perfect cylinder (i.e., $D_{\perp,ij}$ is
simply the projected distance between two mock galaxies).
We run the FoF group-finder on the mock catalogs for a grid of linking length
values, and we study the properties of the resulting group catalogs. Specifically,
we investigate four features of the recovered group distribution: (1) the group
multiplicity function compared to the ``true'' halo multiplicity function;
(2) The relation between the number of galaxies in a halo $\Ntrue$ and the
number of galaxies in its associated group $\Nobs$; (3) The distribution of projected
group sizes as a function of group richness compared to the ``true'' distribution
of projected halo sizes as a function of halo multiplicity; (4) The distribution of
group velocity dispersions as a function of group richness compared to the ``true''
distribution of halo velocity dispersions as a function of halo multiplicity.
We check how each set of linking lengths performs in the above four tests, for
each of the four HOD model mock cubes (\texttt{.Mr20, .Mr20b, .Mr19, .Mr18}).
In the case of each HOD model, we average results over the 10 HOD realizations
described in \S~\ref{mocks} and over the \texttt{LANL1} and \texttt{LANL4} N-body
simulations. We do this procedure for groups that are identified in both real space
(for which there is only one linking length), and redshift space. These tests are
described in detail in the Appendix. Here we summarize the main results.
In real space, a linking length choice of $b=0.2$ yields galaxy groups with ten or
more members that pass all four tests listed above. Groups with $N<10$ show
systematic deviations in abundance, multiplicity, projected sizes, and velocity
dispersions from the corresponding halos with $N<10$. The choice of $b=0.2$ is not
surprising, given that the same linking length was used to identify halos in the
N-body simulations. It is also not surprising that the group-finding fails the
tests for small groups, where adding or losing a couple of galaxies makes a large
fractional difference to the group size. The threshold of $N\sim 10$ is independent
of the underlying dark matter halo mass. This means that we can push the regime in
which the groups are reliable to lower mass systems by using a lower luminosity
sample (where each halo will contain more galaxies). Of course, the change of
luminosity threshold comes at the expense of statistical power, since low luminosity
samples have smaller volumes than high luminosity samples. The number of groups in a
volume-limited sample scales roughly with the number of galaxies, and a luminosity
threshold near the characteristic luminosity $L_*$ maximizes this number.
In redshift space the situation is more complicated. No set of transverse and
line-of-sight linking lengths is able to produce groups that pass all four tests
listed above, even for large size groups. Figure~\ref{fig:linkinglengths.hod20}
summarizes our tests for the \texttt{.Mr20} HOD model mocks. Results for the
other HOD models are similar and are shown in the Appendix. The figure shows
regions (shaded) of the two-dimensional linking length space ($\bpar$ vs. $\bperp$)
that pass each of our four tests.
\subsubsection{Multiplicity Function}
The dark and thin shaded region in Figure~\ref{fig:linkinglengths.hod20}, labeled
$n(N)$, shows linking lengths that pass the group multiplicity function test. In
other words, these linking lengths yield mock group catalogs whose multiplicity
functions are unbiased relative to the ``true'' input halo multiplicity function,
in the regime $N\geq 10$. In this case, ``unbiased'' means that the shape of the
multiplicity function is on average the same as the ``true'' shape and its amplitude
is within $10\%$ of the ``true'' amplitude. Linking length values that lie along
the upper boundary of the shaded region (e.g, the values $\bperp=0.11$, $\bpar=1.5$)
yield multiplicity functions that are $10\%$ too high in amplitude, whereas values
that lie along the lower boundary yield multiplicity functions whose amplitudes are
$10\%$ too low. These results show that an increase in either linking length
generally leads to an increase in the multiplicity function for $N\geq 10$. This
increase is compensated for by a corresponding decrease in the abundance of isolated
(i.e., $N=1$) and low $N$ groups. The shaded region appears to be close to horizontal
only because the vertical axis is highly compressed with respect to the horizontal
axis.
\subsubsection{$\Ntrue$ vs. $\Nobs$}
The group multiplicity function is an average statistic showing the abundance
of all groups as a function of $N$. It is therefore possible, in principle, for it
to be unbiased relative to the halo multiplicity function, without the relation
between individual halo multiplicities and their recovered group multiplicities
being correct. For this reason, we also require that the group-finder yield an
unbiased relation between the multiplicity of individual halos, $\Ntrue$, and their
recovered groups, $\Nobs$. In order to check this, we must match input halos to
recovered groups in a one-to-one way. There are many ways to do this matching, and no
one way is more correct than another. For example, a halo can be associated with the
group that contains most of its galaxies, or the group that contains its central
galaxy, or the group whose centroid is closest to the halo center. We associate
each halo to the group that contains its central galaxy. When two or more halos are
matched to the same group, we choose the halo that shares the largest number of
common galaxies with the group. Halos that are not associated with any group are
considered ``undetected,'' and groups that are not associated with any halo (because
they don't contain any halo central galaxies) are considered ``spurious''.
The light (and green) shaded region in Figure~\ref{fig:linkinglengths.hod20} that
roughly tracks and is slightly wider than the $n(N)$ region shows linking lengths that
pass the $\Ntrue$ vs. $\Nobs$ test. In other words, these linking lengths yield mock
group catalogs with an unbiased median relation between $\Ntrue$ and $\Nobs$ for
associated halos and groups, in the regime $N\geq 10$. We consider the relation to
be unbiased if its slope is within $10\%$ of unity. Linking length values that lie
along the upper boundary of the shaded region yield associated halos and groups with
a median relation $\Ntrue=1.1\Nobs$, whereas values that lie along the lower boundary
yield the relation $\Ntrue=0.9\Nobs$. As expected, most linking lengths that pass
the multiplicity function test also pass the $\Ntrue$ vs. $\Nobs$ test. This breaks
down, however, for values of $\bperp$ greater than 0.16-0.17.
\subsubsection{Projected Sizes}
The (blue) shaded region in Figure~\ref{fig:linkinglengths.hod20}, labeled
``Projected sizes'', shows linking lengths that pass the projected sizes test. These
linking lengths yield mock groups with an unbiased median relation between rms
projected size and group multiplicity $N$, in the regime $N\geq 10$. We consider the
relation to be unbiased if it is within $10\%$ of the ``true'' relation between
median rms projected halo size and halo multiplicity. This shaded region is roughly
vertically oriented because the projected linking length $\bperp$ affects the
projected sizes of groups much more than the line-of-sight linking length $\bpar$.
Clearly, increasing $\bperp$ leads to galaxy groups with larger projected sizes.
The shaded region is not completely vertical, however, because increasing $\bpar$
also leads to larger projected size groups, albeit in a much less sensitive way.
\subsubsection{Velocity Dispersions}
The (red) shaded region in Figure~\ref{fig:linkinglengths.hod20}, labeled
``Velocity dispersions'', shows linking lengths that pass the velocity dispersion
test. These linking lengths yield mock groups with an unbiased median relation
between group velocity dispersion and group multiplicity $N$, in the regime $N\geq 10$.
We consider the relation to be unbiased if it is within $10\%$ of the ``true''
relation between median halo velocity dispersion and halo multiplicity. This shaded
region is roughly horizontally oriented because the line-of-sight linking length
$\bpar$ affects the velocity dispersions of groups much more than $\bperp$.
Clearly, increasing $\bpar$ leads to galaxy groups with larger velocity dispersions.
The shaded region is not completely horizontal, because changing $\bperp$ also
affects the velocity dispersions of groups, though not consistently in the same
sense.
\subsubsection{Our Adopted Linking Lengths}
It is obvious from Figure~\ref{fig:linkinglengths.hod20} that no combination of
FoF linking lengths passes all four tests listed above. We can choose linking
lengths that successfully recover the abundance and projected sizes, or the abundance
and velocity dispersions of groups as a function of multiplicity, but not all three
simultaneously. We can also choose linking lengths that successfully recover
both the projected sizes and velocity dispersions of groups as a function of
multiplicity, but since the multiplicity function of such groups is incorrect, the
overall size and velocity dispersion distributions will also be incorrect.
This failure to recover all features of groups in redshift space is a fundamental
shortcoming of the FoF group-finder when applied to redshift space. Given that
most redshift-space group-finding algorithms operate on very similar principles,
i.e., they identify overdense regions that are elongated along the line-of-sight,
it is likely that this shortcoming is shared by other group-finders as well. To
our knowledge, no group-finder has been shown to pass all four of the tests
considered here for a single choice of parameters.
Figure~\ref{fig:linkinglengths.hod20} shows that in order to recover groups with
unbiased velocity dispersions, the line-of-sight linking length must be substantially
larger than the mean intergalaxy separation. With $\bpar$ that large, groups are
bound to be linked together along the line-of-sight.
The only way to then obtain groups with the correct multiplicity function is to have
a transverse linking length small enough that galaxies in the outer parts of halos are
not included in the recovered groups. The resulting groups bear little physical
resemblance to their parent halos. If, on the other hand, we recover groups with
unbiased projected sizes, then the groups will be missing some of their fastest moving
galaxies and this decrease in multiplicity will be compensated by including as group
members a few galaxies in the infall regions of halos. These groups are much more
physically similar to their parent halos. For this reason, we choose to sacrifice
velocity dispersions, rather than projected sizes, when selecting values for the FoF
linking lengths.
Figure~\ref{fig:linkinglengths.hod20} shows the linking length values that we
adopt and use in this paper (yellow star). These values are
\begin{equation}
\bperp=0.14, \qquad \bpar=0.75 ~.
\end{equation}
Our mock catalog tests show that the FoF algorithm with these linking lengths finds
galaxy groups with $N\geq 10$ that have: (1) an unbiased multiplicity function;
(2) an unbiased median relation between the multiplicities of groups and their
associated halos; (3) a spurious group fraction of less than $\sim 1\%$; (4) a halo
completeness (fraction of halos that are associated one-to-one with groups) of more
than $\sim 97\%$; (5) the correct projected size distribution as a function of
multiplicity; (6) a velocity dispersion distribution that is $\sim 20\%$ too low
at all multiplicities. These results hold for all of the mock catalogs that we
have used (see results for other HOD models in the Appendix) and are thus not very
sensitive to the HOD model assumed or to the specific realization of the underlying
density field. We note that our adopted group-finder only has the above properties
when dark matter halos are defined using a FoF algorithm with a linking length of
0.2 times the mean interparticle separation, since that was the definition used to
construct our mock catalogs. A different halo definition (such as FoF using a
different linking length, or a spherical overdensity halo-finder) will result in a
different optimal group-finder.
Previous FoF group analyses have used different linking lengths. For example,
\citet{eke_etal_04a} adopt $\bperp=0.13, \bpar=1.43$ in their analysis of groups in the
2dF Galaxy Redshift Survey (2dFGRS; \citealt{colless_etal_01}). With a similar
transverse linking length but much larger line-of-sight linking length than used here,
this parameter combination yields unbiased projected sizes and velocity dispersions,
but it overpredicts the abundances of halos by $20-30\%$ at large multiplicities (see
Figure~\ref{fig:linkinglengths.hod20}). These groups are thus poorly suited to our
primary objective of using group abundances as a cosmological test.
\citet{yang_etal_05} and \citet{weinmann_etal_05} use a group-finder that assumes
a mass, radius, and velocity dispersion for each preliminary group and then includes or
discards galaxies from the group based on these assumed properties (similar to a
matched filter technique). This method might, in principle, be able to simultaneously
recover groups with unbiased abundances, projected sizes, and velocity dispersions -
at the expense of model independence - but this remains to be tested.
\section{Incompleteness} \label{incompleteness}
There are two main sources of incompleteness that will affect the richnesses of
groups, and hence the multiplicity function, in our SDSS group catalogs: fiber
collisions and survey edges. Both these effects will prevent galaxies from
being included in some groups, and thus cause the richness of these groups to be
underestimated. These sources of incompleteness and their effects on the measured
group multiplicity function must be accounted for.
\subsection{Fiber Collisions} \label{fibcols}
Fiber collisions cause an incompleteness that grows with the surface density of
galaxies and is thus especially important in group and cluster studies. Moreover,
the surface density in groups is likely a function of group richness. The mean
surface density of a group of richness $N$, mass $M$, and radius $R$ scales like
$\Sigma \sim N/R^2 \sim N/M^{2/3}$. For a power-law relation between mean richness
and halo mass $N\sim M^\alpha$, the surface density is $\Sigma \sim N^{1-2/3\alpha}$.
This scaling relation is clearly a crude approximation, but it illustrates that
the incompleteness due to fiber collisions likely varies with group richness and
can thus affect both the amplitude and slope of the multiplicity function.
We use the 100 \texttt{LANL1-5.Mr20} mock catalogs (5 N-body simulations $\times$
10 HOD realizations $\times$ 2 mocks per simulation cube) to assess the impact of
fiber collisions on the group multiplicity function. We apply the group-finder
described in \S~\ref{groupfinder} to the ``uncorrected'' and ``true'' versions of
these mock catalogs and measure the resulting multiplicity functions.
Figure~\ref{fig:nbodyfibcol} shows these multiplicity functions averaged over all
the mock catalogs. The figure shows that dropping collided galaxies from
the sample lowers the amplitude of the multiplicity function by more than 10\% and
also slightly changes its slope. The amplitude drops because some groups in each
richness bin lose galaxies and are thus shifted to lower $N$ bins. There are also
some groups from higher $N$ bins that are shifted into these bins, but their number
is smaller than the number of groups lost because the abundance of groups drops
steeply with increasing $N$.
\citet{zehavi_etal_05} show that the effect of fiber collisions on the galaxy
two-point correlation function can be successfully corrected for by including
each collided galaxy at the redshift of its nearest neighbor. We apply the same
correction to our mock catalogs to produce a set of ``corrected'' mocks.
Figure~\ref{fig:nbodyfibcol} shows that this correction works very well in the regime
$N\geq 10$, and we therefore adopt it for our group identification.
\subsection{Survey Edges} \label{edges}
Groups that are identified near the edges of a given sample could be missing
galaxies that are located just outside the sample. Similar to fiber collisions,
edge effects always shift groups from higher to lower richness. Moreover, large and
extended groups have a higher probability of being affected by edges than do small
and compact groups because they can straddle an edge while being further away from it.
Edge effects are most severe when the ratio of a sample's surface area to its enclosed
volume is high. Figure~\ref{fig:skyvollim} shows that the SDSS sample has a highly
irregular footprint on the sky, which implies a high surface-to-volume ratio. Edge
effects are, therefore, potentially severe in our samples. When the SDSS survey is
complete and the gap in the North Galactic cap is filled in, edge effects will be
much less important.
We can measure the effects of edges using our mock catalogs, since we know what
galaxies lie on the other side of edges. For every group identified in our
\texttt{LANL1-5.Mr20} mock catalogs, we determine how many galaxies are missing due to
edges. An edge can lie either in the perpendicular direction, or along the
line-of-sight due to a sample's redshift limits.
The solid curve in the right panel of Figure~\ref{fig:nbodyedgestats} shows the fraction
of mock groups that are missing one or more galaxies due to edges, as a function of
group richness $N$. The affected fraction climbs from 10\% to 40\% as $N$ goes from
5 to 50. Edges clearly affect a large fraction of high richness groups in our
sample, but counting a group as affected if it loses only a single galaxy is a very
conservative test. It makes more sense to calculate the fraction of groups that lose
a fixed fraction of their galaxies, rather than just a single galaxy. The dashed curve
in the same panel shows the fraction of groups that lose 25\% or more of their
galaxies. The affected fraction defined this way is $\sim10\%$, roughly independent
of richness. Figure~\ref{fig:nbodyedge} shows the effect of edges on the multiplicity
function (blue curve). The effect of edges on the abundance of mock groups grows from
zero at $N=2$ to approximately 20\% at $N=50$. It is, therefore, very important to
correct for edges, since they systematically change the shape of the multiplicity
function and, hence, the derived HOD.
We measure the shortest distance of every galaxy from the survey edges by laying down
points around each galaxy at successively larger radii and checking if they also lie
within our sample volume. The smallest radius at which points fall outside the sample
volume is the distance of the galaxy from the edge. Any group that contains at least
one galaxy within a linking length from the edge, whether it is a projected
linking length in the tangential direction or a line-of-sight linking length in the
redshift direction, is potentially affected, since there could be galaxies on the other
side that would be linked to the same group. One possible way to deal with edges is
to throw out all such groups. This is a very conservative solution, since it ensures
that all groups in our final sample are uncontaminated by edges. However, it is tricky
to estimate the new effective volume of the sample, which is necessary for measuring the
multiplicity function. Moreover, the effective volume for large groups will be smaller
than that for small groups. Another possibility is to keep all groups, but somehow
correct the multiplicities of those that are potentially affected by edges. This
solution has the advantage that no groups are lost, but it is once again difficult to
estimate the effective volume of the sample, even if all multiplicity corrections are
exactly right. A third possibility is to reject all groups whose centers
lie less than a minimum distance from the edge. This correction has the advantage that
it produces an unbiased sample and it is simple to estimate the new effective volume.
However, it is important to use the correct minimum distance. If it is too small, then
the correction will not work for the largest groups; if it is too big, then we will
unnecessarily reduce our sample size.
The left panel of Figure~\ref{fig:nbodyedgestats} shows the fraction of mock groups that
are missing one or more galaxies due to edges, as a function of the distance from the
group centroid to the edge. The fraction drops from 20\% at 100 Kpc to 5\% at 500 Kpc
and less than 1\% at 1 Mpc. It does not go to zero at larger distance because there are
groups with high velocity dispersion that can be far from the edge and still have
galaxies within a linking length of the outer or lower redshift limit of our sample.
This figure suggests that if we set the minimum distance to 500 Kpc in the tangential
direction and 500 km/s in the redshift direction, we should eliminate most groups that
are affected by edges. We make this correction on our mock group catalogs, and the
number of groups in the resulting catalog is reduced by $\sim 22\%$ on average. We
estimate the new effective volume of each group catalog by scaling the original volume
by the fraction of groups that survive the edge cut. This estimate, though not exactly
accurate, is simple to make and adequate for our purposes. Figure \ref{fig:nbodyedge}
shows that this correction results in a multiplicity function that is unbiased due to
edges (dashed red curve).
Our mock catalog tests show that we can deal with survey edges effectively if we measure
the multiplicity function after eliminating all groups whose centers (estimated as the
centroids of their member galaxy positions) lie less than 500 Kpc from an edge in the
tangential direction or less than 500 km/s from an edge in the radial direction.
Applying this edge cut to the $Mr20$, $Mr19$, and $Mr18$ SDSS group catalogs reduces the
numbers of groups by 22.0\%, 30.2\%, and 41.1\%, respectively. Our measurement of the
multiplicity function for these samples includes this correction, though the group
catalogs that we present include all groups.
\section{Group and Cluster Catalog} \label{catalog}
We apply our group-finding algorithm to the three volume-limited samples described
in \S~\ref{data} and get three group catalogs. The fractions of ungrouped, isolated
galaxies are 43.7\%, 41.2\%, and 39.8\% for the $Mr20$, $Mr19$, and $Mr18$ samples,
respectively. The fractions of galaxies grouped in pairs are 19.1\%, 18.3\%, and
17.9\%. The remaining 37.2\%, 40.6\%, and 42.3\% of galaxies are in groups of three or
more members. Samples $Mr20$, $Mr19$, and $Mr18$ contain a total of 4107, 2684, and
1357 groups with richness $N\geq3$, respectively.
Figure~\ref{fig:groupslice} shows an equatorial slice with groups identified from
sample $Mr20$. The slice is $4^\circ$ thick and each point shows the RA and
redshift of a group with $N\geq3$. A comparison of this figure to
Figure~\ref{fig:slicevollim} shows that groups and clusters trace the large-scale
structure of galaxies, as expected. Larger groups are preferentially located
in higher density regions, whereas smaller groups are more uniformly distributed.
It is striking that the majority of very large groups reside within the large
supercluster at $z=0.08$. Figure~\ref{fig:groupslice2} shows the same slice, but
with points representing the positions of member galaxies in $N\geq3$ groups.
A visual inspection of the figure shows that group velocity dispersions, which
are responsible for the finger-of-God effect, are largest in the most luminous
groups.
For each group, we compute an unweighted group centroid, which consists of a group
right ascension, declination, and mean redshift. We compute a total group
luminosity that is the sum of luminosities of its member galaxies. Since we are
dealing with volume-limited samples, the luminosity of a given group in samples
$Mr20$, $Mr19$, $Mr18$, only counts galaxies with absolute magnitudes brighter
than -19.9, -19, -18, respectively. For example, for the $Mr20$ sample, the total
group absolute magnitude is
\begin{equation}
\Mrtot = -2.5 \mathrm{log}\left( \sum_{i=1}^{N} 10^{-0.4M_{\band{0.1}{r},i}} \right),
\end{equation}
and it is equivalent to integrating the galaxy luminosity function within the
group from $\Mr=-19.9$ to $-\infty$. Note that we compute these group absolute
magnitudes using the altered absolute magnitudes for galaxies that do not have
measured redshifts due to fiber collisions (see \S~\ref{data}).
We also compute a total group color, which is simply defined as
$\grgrp = \Mgtot - \Mrtot$. We compute a group one-dimensional velocity
dispersion given by
\begin{equation}
\sigv = \frac{1}{1+\bar{z}}\sqrt{\frac{1}{N-1}\sum_{i=1}^{N} (cz_i - c\bar{z})^2},
\end{equation}
and an rms projected group radius given by
\begin{equation}
\Rproj = \sqrt{\frac{1}{N}\sum_{i=1}^{N} r_i^2},
\end{equation}
where $r_i$ is the projected distance between each member galaxy and the group
centroid.
In the three portions of Table~3, we present the groups and clusters with $N\geq3$,
selected from samples $Mr20$, $Mr19$, and $Mr18$. For each group, we list a group
ID (column 1); the (J2000) right ascension and declination of the group centroid
(columns 2, 3); the mean redshift of the cluster (column 4); the group richness $N$
(column 5); the total $r$-band absolute magnitude of the group, $\Mrtot$ (column 6);
the total color of the group, $\grgrp$ (column 7); the line-of-sight velocity
dispersion of the group, $\sigv$ (column 8); the projected rms radius of the
group $\Rproj$ (column 9); the perpendicular distance of the group center from
the survey edge $\redge$ (column 10). The groups in each portion of Table~3 are
ranked in decreasing order of richness $N$. We show the first few rows of each
portion of the table in the text and make the entire table available in the electronic
version of the journal, as well as at \texttt{http://cosmo.nyu.edu/aberlind/Groups}.
In Table~4, we present the member galaxies of the groups listed in Table~3. For
each galaxy we list the ID of the group to which it belongs (column 1); the (J2000)
right ascension and declination (columns 2, 3); the redshift (column 4); the
$r$-band absolute magnitude $\Mr$\footnote{Galaxies without measured redshifts due
to fiber collisions are assigned the absolute magnitude of their nearest neighbor,
as described in \S~\ref{data}.} (column 5); the $\gr$ color (column 6); a fiber
collision flag that is equal to 0 if the galaxy has its own measured redshift and 1
if it has been given the redshift of its nearest neighbor (column 7); the perpendicular
distance of the galaxy from the survey edge $\redge$ (column 8). As before, we show
the first few rows of each portion of Table~4 in the text and make the entire table
available in the electronic version of the journal, as well as at
\texttt{http://cosmo.nyu.edu/aberlind/Groups}.
\begin{table*}
\begin{center}
\centerline{\small Table~3. Group and Cluster Catalogs for Samples $Mr20$, $Mr19$, and $Mr18$}
\smallskip
\begin{tabular}{rrrccccccr}
\tableline
\tableline
ID & RA & DEC & $\bar{z}$ & $N$ & $\Mrtot$ & $\grgrp$ & $\sigv$ & $\Rproj$ & $\redge$ \\
& (deg) & (deg) & & & & & (km/s) & ($\hmpc$) & ($\hmpc$) \\
\tableline
\cutinhead{$Mr20$}
33974 & 239.580740 & 27.312343 & 0.08797 & 132 & -25.920 & 0.946 & 723.7 & 1.371 & 17.7 \\
16089 & 247.172589 & 40.164633 & 0.03057 & 97 & -25.468 & 0.891 & 661.1 & 1.318 & 89.3 \\
8817 & 358.535971 & -10.372017 & 0.07405 & 61 & -25.190 & 0.921 & 736.0 & 0.734 & 17.9 \\
14552 & 183.450292 & 59.266666 & 0.09386 & 51 & -24.861 & 0.808 & 338.3 & 1.079 & 22.9 \\
12289 & 159.824898 & 4.987457 & 0.06815 & 51 & -24.859 & 0.899 & 661.4 & 1.161 & 47.1 \\
3025 & 195.700154 & -2.627141 & 0.08183 & 49 & -24.805 & 0.911 & 377.1 & 1.247 & 57.9 \\
20593 & 169.362355 & 54.469262 & 0.06907 & 49 & -24.831 & 0.906 & 426.4 & 1.202 & 35.5 \\
\cutinhead{$Mr19$}
9501 & 246.963120 & 40.182569 & 0.03009 & 197 & -25.839 & 0.886 & 588.7 & 1.317 & 88.2 \\
4915 & 10.447791 & -9.381301 & 0.05543 & 95 & -25.068 & 0.927 & 572.4 & 0.981 & 38.8 \\
4634 & 329.333792 & -7.765802 & 0.05727 & 86 & -25.016 & 0.724 & 564.0 & 0.677 & 52.5 \\
10986 & 14.231949 & -0.655097 & 0.04378 & 86 & -24.944 & 0.935 & 385.4 & 1.076 & 5.2 \\
5585 & 351.303515 & 14.909898 & 0.04113 & 83 & -24.622 & 0.871 & 496.8 & 1.045 & 53.2 \\
3709 & 214.187113 & 1.962572 & 0.05333 & 81 & -24.902 & 0.887 & 368.3 & 1.160 & 42.9 \\
11585 & 18.686704 & 0.254973 & 0.04442 & 68 & -24.704 & 0.903 & 386.8 & 0.744 & 27.0 \\
\cutinhead{$Mr18$}
4792 & 247.062059 & 40.107520 & 0.03011 & 311 & -25.934 & 0.865 & 584.2 & 1.300 & 90.5 \\
2748 & 351.183638 & 14.580962 & 0.04128 & 152 & -25.057 & 0.903 & 446.6 & 1.014 & 72.3 \\
6984 & 173.640705 & 49.042739 & 0.03270 & 65 & -24.086 & 0.918 & 526.2 & 0.533 & 45.7 \\
1968 & 220.146510 & 3.491413 & 0.02680 & 54 & -23.853 & 0.946 & 274.1 & 0.506 & 23.6 \\
5607 & 14.274495 & -0.247149 & 0.04303 & 52 & -24.066 & 0.915 & 309.0 & 0.760 & 13.0 \\
5948 & 18.760997 & 0.307893 & 0.04326 & 49 & -24.108 & 0.876 & 264.9 & 0.659 & 26.5 \\
5692 & 51.279369 & -0.496506 & 0.03664 & 48 & -23.871 & 0.870 & 246.1 & 0.802 & 44.6 \\
\tableline
\label{tab:groups}
\end{tabular}
\end{center}
Note---The rest of the table can be found in the electronic version of the ApJ, or at
\texttt{http://cosmo.nyu.edu/aberlind/Groups}
\end{table*}
\begin{table*}
\begin{center}
\centerline{\small Table~4. Member Galaxies of Groups and Clusters for Samples $Mr20$, $Mr19$, and $Mr18$}
\smallskip
\begin{tabular}{lcrccccc}
\tableline
\tableline
groupID & RA & DEC & $z$ & $\Mr$ & $\gr$ & fibcol & $\redge$ \\
& (deg) & (deg) & & & & & ($\hmpc$) \\
\tableline
\cutinhead{$Mr20$}
14 & 196.769894 & -0.039161 & 0.08086 & -20.168 & 0.945 & 1 & 72.3 \\
14 & 196.799107 & -0.024688 & 0.08051 & -20.498 & 0.918 & 0 & 72.3 \\
14 & 196.788454 & -0.029741 & 0.08086 & -20.168 & 0.945 & 1 & 72.3 \\
14 & 196.779246 & -0.038656 & 0.08086 & -20.168 & 0.945 & 0 & 72.3 \\
15 & 197.264020 & -0.053520 & 0.07962 & -20.302 & 0.457 & 0 & 72.4 \\
15 & 197.207327 & 0.047123 & 0.07987 & -19.950 & 0.895 & 0 & 72.4 \\
15 & 197.165432 & 0.102322 & 0.08016 & -20.467 & 0.872 & 0 & 72.4 \\
\cutinhead{$Mr19$}
1 & 169.180550 & -0.213320 & 0.03917 & -19.355 & 0.752 & 0 & 13.5 \\
1 & 169.195964 & -0.100215 & 0.03898 & -19.315 & 0.584 & 0 & 13.5 \\
1 & 169.387065 & -0.187503 & 0.03999 & -20.762 & 0.967 & 0 & 13.5 \\
5 & 199.555960 & -0.148218 & 0.04825 & -19.267 & 0.321 & 0 & 65.9 \\
5 & 199.656619 & -0.226944 & 0.04731 & -19.705 & 0.960 & 0 & 65.9 \\
5 & 199.665084 & -0.175183 & 0.04708 & -20.975 & 0.976 & 1 & 65.9 \\
5 & 199.679052 & -0.178932 & 0.04708 & -20.975 & 0.976 & 0 & 65.9 \\
5 & 199.671638 & -0.173772 & 0.04708 & -20.975 & 0.976 & 1 & 65.9 \\
\cutinhead{$Mr18$}
1 & 194.342587 & -0.630508 & 0.02247 & -18.821 & 0.744 & 1 & 57.7 \\
1 & 194.353591 & -0.622488 & 0.02247 & -18.821 & 0.744 & 0 & 57.7 \\
1 & 194.313130 & -0.657646 & 0.02295 & -18.837 & 0.894 & 0 & 57.7 \\
2 & 169.180550 & -0.213320 & 0.03917 & -19.355 & 0.752 & 0 & 13.4 \\
2 & 169.195964 & -0.100215 & 0.03898 & -19.315 & 0.584 & 0 & 13.4 \\
2 & 169.387065 & -0.187503 & 0.03999 & -20.762 & 0.967 & 0 & 13.4 \\
2 & 169.300864 & -0.189302 & 0.03972 & -18.203 & 0.819 & 0 & 13.4 \\
\tableline
\label{tab:members}
\end{tabular}
\end{center}
Note---The rest of the table can be found in the electronic version of the ApJ, or at
\texttt{http://cosmo.nyu.edu/aberlind/Groups}
\end{table*}
\clearpage
\section{Multiplicity Function} \label{multiplicity}
With group catalogs in hand, we can now measure the group multiplicity function.
The differential group multiplicity function, $\ngrpN$, is defined as the
number density of groups in bins of richness $N$, where richness bins can have
a width of unity or more. Before computing $\ngrpN$, we must make the
corrections for incompleteness described in \S~\ref{incompleteness}.
Though the catalogs presented in \S~\ref{catalog} already include the fiber
collision correction, we also compute the multiplicity function from an
alternate $Mr20$ group catalog that does not include this correction in order
to see the magnitude of the correction. Figure~\ref{fig:groupmultfibedge}
shows this uncorrected multiplicity function, as well as the multiplicity
function that includes the fiber collision correction. The figure shows that
applying the correction boosts the amplitude of the multiplicity function, just
as it did in our mock tests in \S~\ref{incompleteness}.
Figure~\ref{fig:groupmultfibedge} also shows the effect on the multiplicity
function of applying the edge correction described in \S~\ref{incompleteness}.
This effect is small, typically less than 5\%, though it is larger in individual bins
at high $N$, where the number of groups is small.
\begin{table}[t]
\begin{center}
\centerline{\small Table~5. Group Multiplicity Function for $Mr20$ Sample}
\begin{tabular}[t]{lccc}
\tableline
\tableline
$\Nmin$--$\Nmax$ & $\ngrpN$ & $\signgrpN$ & $\signgrpN$ (Poisson) \\
\tableline
3--3 & $2.290\times 10^{-4}$ & $1.110\times 10^{-5}$ & $5.881\times 10^{-6}$ \\
4--4 & $1.054\times 10^{-4}$ & $4.890\times 10^{-6}$ & $3.990\times 10^{-6}$ \\
5--5 & $4.909\times 10^{-5}$ & $4.181\times 10^{-6}$ & $2.723\times 10^{-6}$ \\
6--6 & $3.263\times 10^{-5}$ & $4.465\times 10^{-6}$ & $2.220\times 10^{-6}$ \\
7--7 & $1.962\times 10^{-5}$ & $1.979\times 10^{-6}$ & $1.722\times 10^{-6}$ \\
8--8 & $1.496\times 10^{-5}$ & $2.250\times 10^{-6}$ & $1.503\times 10^{-6}$ \\
9--9 & $1.118\times 10^{-5}$ & $2.398\times 10^{-6}$ & $1.299\times 10^{-6}$ \\
10--10 & $8.906\times 10^{-6}$ & $1.502\times 10^{-6}$ & $1.160\times 10^{-6}$ \\
11--11 & $5.139\times 10^{-6}$ & $1.292\times 10^{-6}$ & $8.810\times 10^{-7}$ \\
12--12 & $4.223\times 10^{-6}$ & $8.632\times 10^{-7}$ & $7.986\times 10^{-7}$ \\
13--13 & $3.780\times 10^{-6}$ & $7.200\times 10^{-7}$ & $7.555\times 10^{-7}$ \\
14--14 & $2.565\times 10^{-6}$ & $1.283\times 10^{-6}$ & $6.224\times 10^{-7}$ \\
15--15 & $2.873\times 10^{-6}$ & $9.335\times 10^{-7}$ & $6.587\times 10^{-7}$ \\
16--16 & $2.868\times 10^{-6}$ & $1.165\times 10^{-6}$ & $6.581\times 10^{-7}$ \\
17--17 & $1.361\times 10^{-6}$ & $6.868\times 10^{-7}$ & $4.533\times 10^{-7}$ \\
18--18 & $1.358\times 10^{-6}$ & $4.131\times 10^{-7}$ & $4.530\times 10^{-7}$ \\
19--19 & $1.209\times 10^{-6}$ & $5.133\times 10^{-7}$ & $4.273\times 10^{-7}$ \\
20--21 & $9.817\times 10^{-7}$ & $3.079\times 10^{-7}$ & $3.851\times 10^{-7}$ \\
22--24 & $6.039\times 10^{-7}$ & $2.253\times 10^{-7}$ & $3.020\times 10^{-7}$ \\
25--28 & $3.401\times 10^{-7}$ & $9.522\times 10^{-8}$ & $2.266\times 10^{-7}$ \\
29--30 & $9.061\times 10^{-7}$ & $4.483\times 10^{-7}$ & $3.699\times 10^{-7}$ \\
31--34 & $3.398\times 10^{-7}$ & $7.501\times 10^{-8}$ & $2.265\times 10^{-7}$ \\
35--42 & $1.699\times 10^{-7}$ & $6.455\times 10^{-8}$ & $1.602\times 10^{-7}$ \\
43--61 & $6.360\times 10^{-8}$ & $2.982\times 10^{-8}$ & $9.801\times 10^{-8}$ \\
\tableline
\label{tab:mult20}
\end{tabular}
\end{center}
Note---$\ngrp$ and $\signgrpN$ are in units of $\hden$.
\end{table}
We must calculate errorbars for the multiplicity function in order to use it to
constrain the HOD. We use our mock catalogs for this purpose. Specifically,
we compute fractional errors from the dispersion among 10 independent mock catalogs for
the $Mr20$ sample (\texttt{LANL1-5.Mr20} mocks $\times$ 1 HOD realization $\times$
2 mocks per simulation cube), and 8 mock catalogs for each of the $Mr19$ and $Mr18$
samples (\texttt{LANL1-4.Mr19}/\texttt{LANL1-4.Mr18} mocks $\times$ 1 HOD
realization $\times$ 2 mocks per simulation cube). Note that we do not use multiple
HOD realizations because the underlying halo populations themselves would not be
independent. Before computing errors, we correct each mock catalog for fiber
collisions and edge effects in the same way as in the data. The computed errors
thus implicitly include any contribution from these correction procedures.
The SDSS multiplicity function shown in Figure~\ref{fig:groupmultfibedge} becomes
very noisy at high richness because the abundance of groups drops with $N$ and
the figure uses richness bins with a width of unity. It makes more sense to
increase the bin width with $N$ so as to beat down the noise. Moreover, since
we calculate errorbars for the multiplicity function using our mock catalogs,
each richness bin must contain enough mock groups so that an errorbar can be
reliably estimated. We choose richness bins for each group catalog so that each
bin contains at least eight SDSS groups and twenty mock groups (among all mock
catalogs used). At low multiplicities, the bin width is always unity because
there are many groups with low $N$. At higher multiplicities, however, the
richness bins grow wider in order to satisfy these criteria. The bin widths
for samples $Mr20$, $Mr19$, and $Mr18$, are listed in the first columns of Tables~5,
6, and~7, respectively. Once a richness bin is defined, the abundance of groups in
that bin, $\ngrpN$, is simply the number of groups having richnesses within the bin,
divided by the sample volume and divided by the bin width. The values of $\ngrpN$ are
listed in the second columns of Tables~5, 6, and~7. We use the same richness bins
to compute the abundance of mock groups for each independent mock catalog, and we
compute errors, $\signgrpN$, in the SDSS multiplicity function by measuring the
dispersion among the mock multiplicity functions. These errors are listed in the
third columns of Tables~5, 6, and~7. Finally, we also compute Poisson errors for
the SDSS $\ngrpN$, which we list in the fourth columns of Tables~5, 6, and~7.
In some of the highest multiplicity bins, the Poisson errors are larger than the mock
errors. In these cases, the mock errors are likely underestimated and it is
best to use the Poisson errors in their place.
\begin{table}[t]
\begin{center}
\centerline{\small Table~6. Group Multiplicity Function for $Mr19$ Sample}
\begin{tabular}[t]{lccc}
\tableline
\tableline
$\Nmin$--$\Nmax$ & $\ngrpN$ & $\signgrpN$ & $\signgrpN$ (Poisson) \\
\tableline
3--3 & $4.514\times 10^{-4}$ & $2.872\times 10^{-5}$ & $1.545\times 10^{-5}$ \\
4--4 & $1.889\times 10^{-4}$ & $1.201\times 10^{-5}$ & $9.996\times 10^{-6}$ \\
5--5 & $1.085\times 10^{-4}$ & $9.323\times 10^{-6}$ & $7.575\times 10^{-6}$ \\
6--6 & $6.292\times 10^{-5}$ & $8.977\times 10^{-6}$ & $5.769\times 10^{-6}$ \\
7--7 & $5.027\times 10^{-5}$ & $5.465\times 10^{-6}$ & $5.157\times 10^{-6}$ \\
8--8 & $2.856\times 10^{-5}$ & $2.434\times 10^{-6}$ & $3.887\times 10^{-6}$ \\
9--9 & $1.853\times 10^{-5}$ & $2.832\times 10^{-6}$ & $3.131\times 10^{-6}$ \\
10--10 & $1.534\times 10^{-5}$ & $2.799\times 10^{-6}$ & $2.849\times 10^{-6}$ \\
11--11 & $1.534\times 10^{-5}$ & $2.577\times 10^{-6}$ & $2.849\times 10^{-6}$ \\
12--12 & $1.164\times 10^{-5}$ & $2.236\times 10^{-6}$ & $2.482\times 10^{-6}$ \\
13--13 & $8.994\times 10^{-6}$ & $2.135\times 10^{-6}$ & $2.181\times 10^{-6}$ \\
14--14 & $7.936\times 10^{-6}$ & $2.105\times 10^{-6}$ & $2.049\times 10^{-6}$ \\
15--15 & $5.819\times 10^{-6}$ & $1.186\times 10^{-6}$ & $1.755\times 10^{-6}$ \\
16--16 & $5.819\times 10^{-6}$ & $1.718\times 10^{-6}$ & $1.755\times 10^{-6}$ \\
17--18 & $5.819\times 10^{-6}$ & $1.318\times 10^{-6}$ & $1.755\times 10^{-6}$ \\
19--20 & $2.380\times 10^{-6}$ & $5.168\times 10^{-7}$ & $1.122\times 10^{-6}$ \\
21--23 & $2.292\times 10^{-6}$ & $5.243\times 10^{-7}$ & $1.101\times 10^{-6}$ \\
24--26 & $1.587\times 10^{-6}$ & $4.621\times 10^{-7}$ & $9.164\times 10^{-7}$ \\
27--32 & $7.054\times 10^{-7}$ & $2.228\times 10^{-7}$ & $6.109\times 10^{-7}$ \\
33--38 & $7.054\times 10^{-7}$ & $3.069\times 10^{-7}$ & $6.109\times 10^{-7}$ \\
39--51 & $3.256\times 10^{-7}$ & $4.634\times 10^{-8}$ & $4.151\times 10^{-7}$ \\
52--86 & $1.209\times 10^{-7}$ & $3.602\times 10^{-8}$ & $2.529\times 10^{-7}$ \\
\tableline
\label{tab:mult19}
\end{tabular}
\end{center}
Note---Same units as Table~5.
\end{table}
\begin{table}[t]
\begin{center}
\centerline{\small Table~7. Group Multiplicity Function for $Mr18$ Sample}
\begin{tabular}[t]{lccc}
\tableline
\tableline
$\Nmin$--$\Nmax$ & $\ngrpN$ & $\signgrpN$ & $\signgrpN$ (Poisson) \\
\tableline
3--3 & $7.311\times 10^{-4}$ & $6.909\times 10^{-5}$ & $4.000\times 10^{-5}$ \\
4--4 & $3.436\times 10^{-4}$ & $3.325\times 10^{-5}$ & $2.742\times 10^{-5}$ \\
5--5 & $1.948\times 10^{-4}$ & $2.200\times 10^{-5}$ & $2.065\times 10^{-5}$ \\
6--6 & $1.248\times 10^{-4}$ & $1.629\times 10^{-5}$ & $1.652\times 10^{-5}$ \\
7--7 & $1.182\times 10^{-4}$ & $1.546\times 10^{-5}$ & $1.608\times 10^{-5}$ \\
8--8 & $5.686\times 10^{-5}$ & $9.917\times 10^{-6}$ & $1.116\times 10^{-5}$ \\
9--9 & $3.284\times 10^{-5}$ & $5.340\times 10^{-6}$ & $8.477\times 10^{-6}$ \\
10--10 & $3.066\times 10^{-5}$ & $5.777\times 10^{-6}$ & $8.191\times 10^{-6}$ \\
11--11 & $2.626\times 10^{-5}$ & $8.403\times 10^{-6}$ & $7.581\times 10^{-6}$ \\
12--13 & $1.423\times 10^{-5}$ & $1.629\times 10^{-6}$ & $5.580\times 10^{-6}$ \\
14--15 & $8.756\times 10^{-6}$ & $1.443\times 10^{-6}$ & $4.378\times 10^{-6}$ \\
16--17 & $1.203\times 10^{-5}$ & $1.761\times 10^{-6}$ & $5.132\times 10^{-6}$ \\
18--23 & $3.647\times 10^{-6}$ & $7.402\times 10^{-7}$ & $2.825\times 10^{-6}$ \\
24--31 & $2.188\times 10^{-6}$ & $6.091\times 10^{-7}$ & $2.188\times 10^{-6}$ \\
32--152 & $1.447\times 10^{-7}$ & $1.673\times 10^{-8}$ & $5.627\times 10^{-7}$ \\
\tableline
\label{tab:mult18}
\end{tabular}
\end{center}
Note---Same units as Table~5.
\end{table}
Figure~\ref{fig:groupmult} shows the SDSS multiplicity functions for the three
volume-limited samples, along with the mock errorbars for the $Mr20$ sample.
Though we measure and show the multiplicity function down to a multiplicity of
$N=3$, our tests with mock catalogs have shown that it is only unbiased with
respect to the true halo multiplicity function for $N\geq 10$.
When using this measured multiplicity function to constrain the HOD, we must
either only use bins with $N\geq 10$, or attempt to calibrate the relation between
the measured group multiplicity function and the true halo multiplicity function
at lower values of $N$. The central curve of Figure~\ref{fig:nbodymultbxybz},
discussed in the Appendix, effectively provides this calibration for $Mr20$ and the
cosmology adopted in our mock catalogs.
The multiplicity functions shown in Figure~\ref{fig:groupmult} appear to be close
to power-law relations. In order to test this, we perform a simple power-law fit
to each multiplicity function in the regime $N\geq 10$. We use only the diagonal
errors of the full covariance matrix (i.e., the errors listed in Tables~5, 6,
and ~7). We find that all three multiplicity functions are well-fit by power-law
relations, with best-fit slopes of $-2.72\pm0.16$, $-2.48\pm0.14$, and $-2.49\pm0.28$
for the $Mr20$, $Mr19$, and $Mr18$ samples, respectively.
\section{Summary and Discussion} \label{summary}
We have used a simple friends-of-friends algorithm to identify galaxy groups in
volume-limited samples of the SDSS redshift survey. We have selected FoF
linking lengths that are best at grouping together galaxies that occupy the same
dark matter halos. We based this choice on extensive tests with mock galaxy
catalogs, which we constructed by populating halos in N-body simulations with
galaxies. The result of our mock tests is that no combination of perpendicular
and line-of-sight linking lengths can yield groups that successfully recover
all aspects of the parent halo distribution, even for large richness systems.
Specifically, FoF cannot identify groups that simultaneously have unbiased
abundances, projected sizes, and velocity dispersions. The ideal group-finding
parameters for a given study depend on its scientific objectives. Given our
objective of using the multiplicity function to constrain the HOD, it makes
sense to sacrifice velocity dispersions and obtain groups with unbiased abundances
and projected sizes. Our choice of linking lengths results in a group catalog
that, for groups of ten or more members, has an unbiased multiplicity function,
an unbiased median relation between the multiplicities of groups and their parent
halos, an unbiased projected size distribution as a function of multiplicity, and
a velocity dispersion distribution that is $\sim 20\%$ too low for all
multiplicities. We correct for fiber collisions and survey edge effects and present
three SDSS group catalogs (for three different volume-limited samples) and their
measured multiplicity functions.
It is important to recognize that our adopted group finder has the above properties
only for halos defined using FoF with a linking length of 0.2 times the mean
interparticle separation, since this is how halos were identified in our mock
catalogs. A different halo definition (such as FoF with a different linking
length, or spherical overdensity halos) would require a different set of optimal
group-finding parameters. This is not a problem as long as the same halo
definition is used consistently. For example, an HOD measured from these
group catalogs will hold for this halo definition, and any theoretical model should
use the same halo definition to compare its predictions to the measured HOD.
We chose this particular halo finder because it has been widely used and tested,
and the properties of the resulting halo distribution (e.g., mass function) are
well understood.
The groups and clusters that we present here are intended to be systems of galaxies
that belong to the same virialized dark matter halo. We can test whether these
systems are virialized by computing crossing times for the groups and checking
if they are sufficiently less than the Hubble time. We define the crossing time
divided by the hubble time as
\begin{equation}
\frac{\tcross}{\tH} = \frac{(\Rrms/\hmpc)}{(\sigv/100\kms)},
\end{equation}
where $\Rrms$ is the one-dimensional group radius, which is equal to the projected
(two-dimensional) radius, $\Rproj$, divided by the square root of two. We correct
for the velocity dispersion bias revealed in our mock tests by applying a 20\% upward
correction to all group velocity dispersions, and we compute $\tcross/\tH$ for all
groups. We find that, for all three group catalogs, the median value of
$\tcross/\tH$ is $\sim0.15$, and 80\% of all groups have values less than $\sim0.29$.
These numbers can be interpreted in terms of the spherical infall model
\citep{gunn_gott_72,gott_turner_77a}, or other analytic or numerical models.
However, at a first glance, the numbers are encouraging and suggest that most of our
groups are likely virialized systems.
The group and cluster catalogs presented here are well-suited for testing many of the
predictions and assumptions made by galaxy formation models regarding the relationship
between galaxies and their underlying dark matter halos. We will investigate several
of these issues in subsequent papers.
\acknowledgments
We thank Zheng Zheng and Jeremy Tinker for their help with choosing HOD parameters for
constructing mock catalogs and Luis Teodoro for his help with making the mock catalogs.
AAB acknowledges support by NSF grant AST-0079251 and the NSF Center for Cosmological
Physics, while at the University of Chicago, and by NASA grant NAG5-11669, NSF grant
PHY-0101738, and a grant from NASA administered by the American Astronomical Society,
while at New York University. AAB also acknowledges the hospitality of the Aspen
Center for Physics, where some of this work was completed. MRB and DWH acknowledge
support by NSF grant AST-0428465. DHW acknowledges support by NSF grant AST-0407125.
JRG acknowledges support by NSF grant AST-0406713.
Portions of this work were performed under the auspices of the U.S. Dept. of Energy,
and supported by its contract \#W-7405-ENG-36 to Los Alamos National Laboratory.
Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, Cambridge University, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPA), the Max-Planck-Institute for Astrophysics (MPIA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.
\section*{Appendix}
In this appendix, we describe the mock catalog tests that help us choose optimal
FoF parameters. Since our primary goal for identifying groups is to measure the
group multiplicity function and use it to constrain the HOD, we clearly require
our FoF algorithm to produce groups that have an unbiased multiplicity function
with respect to the true halo multiplicity function. In addition, we require an
unbiased relation between the multiplicities of groups and their associated halos.
Finally, we would like our groups to have unbiased projected size and velocity
dispersion distributions as a function of multiplicity. We create a grid of
FoF linking lengths and check how each set of linking lengths performs in the
above tests, for each of the four HOD model mock cubes
(\texttt{.Mr20, .Mr20b, .Mr19, .Mr18}). In the case of each HOD model, we average
results over the 10 HOD realizations described in \S~\ref{mocks} and over the
\texttt{LANL1} and \texttt{LANL4} N-body simulations.
Before focusing on redshift space, we briefly examine how well FoF recovers
the true multiplicity function in real space, since this represents the best
possible case (any group finder will almost certainly perform worse in redshift
space). We apply FoF to the real-space cube mocks using a single linking length
(the linking volume around each mock galaxy is a sphere), and investigate how
the recovered multiplicity function varies with the value of this linking length.
In particular, we compare the mock group multiplicity functions to the input
halo multiplicity functions that were used to construct the mock catalogs.
Figure~\ref{fig:nbodymultbxy} shows this comparison for the \texttt{.Mr20} mocks.
The bottom panel of the figure shows the logarithm of the ratio of group to halo
multiplicity function, and the horizontal solid line therefore denotes the
``unbiased'' case. The figure reveals that, at large $N$, the group multiplicity
function has an unbiased shape that is independent of the choice of linking length
(at least for the range of linking lengths shown). The amplitude, however, is
dependent on the linking length used, with larger linking lengths leading to a
higher abundance of groups at large $N$. A linking length of $b=0.2$ (in units of
the mean intergalaxy separation) yields a group multiplicity function with an unbiased
amplitude at large $N$. This is not surprising given that the same value was used
to identify dark matter halos in the N-body simulations while constructing mock
catalogs.
At low $N$, the multiplicity function is highly biased, both in shape and amplitude.
The abundance of groups relative to halos at a given multiplicity $N$ decreases when
FoF splits these halos into smaller groups or merges them to form larger groups.
This decrease is countered by an increase due to the merging of smaller halos or the
splitting of larger halos. The balance between these competing effects determines
whether the multiplicity function is biased or not. For linking lengths near $b=0.2$,
merging dominates over splitting, which means that group abundances at a given
multiplicity are mainly determined by a balance between halos at that $N$ merging to
yield larger groups and smaller halos merging to replenish the lost groups. However,
this balance breaks at $N=1$ because, while FoF merges $N=1$ halos (i.e., isolated
galaxies) to form larger groups, there are no smaller halos that can merge to
replenish $N=1$ groups. The abundance of $N=1$ groups is therefore necessarily
less than that of $N=1$ halos (it can only be more if the linking length is so small
- approximately $b\sim0.1$ - that single galaxy groups splinter off in large numbers
from larger halos). Since most galaxies live in $N=1$ halos ($\sim70\%$ in these mock
catalogs), merging a small fraction of them to form larger groups will fractionally
increase the abundance of larger $N=2, 3, 4$, etc. groups significantly. This is seen
in Figure~\ref{fig:nbodymultbxy}: the abundance of $N=1$ groups is lower than that of
halos by $\sim20\%$ for $b=0.2$, causing the abundance of $N=2$ and $N=3$ groups to
be $\sim50\%$ higher. Only for $N>10$ does the group abundance settle down and become
unbiased. This behavior is a fundamental limitation of the FoF algorithm, and it
has the consequence that group abundances can only be trusted for large multiplicity
groups.
In redshift space, group finding is much more challenging because finger-of-god
distortions stretch groups along the line-of-sight, making it more likely that
single halos will be split into multiple groups and that neighboring halos will
be merged into the same groups. Figure~\ref{fig:nbodyslice} illustrates these
effects by showing the performance of FoF in a small slice through a single mock
catalog (one HOD realization of the \texttt{LANL4.Mr20} mock catalog). The top-left
panel shows the mock galaxies in real space, with each $N>4$ halo denoted by a unique
color. The bottom-left panel shows the same galaxies in redshift space, where the
line-of-sight is oriented along the $z$-axis of the mock cube. Large open circles have
radii equal to the halo virial radii and are centered at the halo centers in real space,
and the galaxy centroids in redshift space. We run our adopted FoF group-finder
(described in \S~\ref{groupfinder}) on the redshift-space mock and denote each
resulting $N>4$ group with a unique color in the bottom-right panel. Finally, we
show the group galaxies' real-space positions in the top-right panel. Large dotted
circles are centered at the group centroids and have virial radii that are estimated
by assuming a halo mass function and a monotonic relation between group multiplicity
and mass. A visual comparison of the real- and redshift-space panels reveals many of
the failure modes of FoF group-finding in redshift space. The halo denoted by green
in the left-side panels is fairly well recovered by FoF as the group denoted by green in
the right-side panels. However, a couple of halo galaxies are missed in group finding,
such as the one whose velocity moved it the furthest away from the center of the halo.
Most of the galaxies in the halo denoted by blue are linked together in the same group,
also denoted by blue. However, many galaxies that do not belong to the ``blue'' halo
are also linked to the same group. This is seen clearly in the top-right panel, where
seven of the ``blue'' group galaxies' real-space positions place them well outside the
halo. A similar thing occurs to the halos and corresponding groups denoted by magenta
and cyan. Most of the galaxies in the large ``red'' halo are recovered correctly into
the ``red'' group, but there are some galaxies added to this group that do not belong
to the ``red'' halo, as well as a few galaxies that do belong to that halo, but have
splintered off into a different group (denoted by dark green). Despite these
imperfections, there is clearly a substantial correspondence between the groups
identified by FoF and the true population of halos in this slice.
We now examine the relative multiplicity functions of groups and halos when the groups
are identified in redshift space. If we use the same linking length in transverse and
line-of-sight directions, finger-of-god distortions will cause halos to be split into
multiple small groups along the line-of-sight. This is demonstrated by the dashed
curve in Figure~\ref{fig:nbodymultbxybz}, which shows the multiplicity function of groups
identified with a single linking length of $b=0.2$. The abundance of groups is
vastly underestimated for $N\gtrsim 5$, and the effect grows with $N$ because richer
halos have higher velocity dispersions. We therefore need to use different linking
lengths in the line-of-sight and perpendicular directions. We apply FoF to our
redshift-space cube mocks for a grid of perpendicular and line-of-sight linking lengths
and find that we can recover an unbiased multiplicity function at large $N$ for the right
combinations of linking lengths. Figure~\ref{fig:nbodymultbxybz} shows one such
combination ($\bperp=0.14$, $b_z=0.75$) and demonstrates how the group multiplicity
function changes with the line-of-sight linking length $b_z$. Generally,
larger linking lengths in either direction lead to a higher abundance of groups at
large $N$. We record all linking length combinations that yield unbiased multiplicity
functions in the large $N$ regime and show the successful parameter space in
Figure~\ref{fig:linkinglengths.hod20}, as discussed in \S~\ref{groupfinder}.
Recovering an unbiased multiplicity function does not guarantee that the one-to-one
relation between the multiplicities of halos and their recovered groups is also
unbiased. We therefore also investigate this relation. As described in
\S~\ref{groupfinder}, we associate each halo to the recovered group that contains
the halo's central galaxy. Groups that contain central galaxies from more than one
halo are associated with the halo with which they share the largest number of galaxies.
Halos that end up not being associated with any group are considered ``undetected,''
and groups that are not associated with any halo (i.e., they contain no halo central
galaxies) are considered ``spurious''. Once we have associated mock groups one-to-one
with their parent halos, we can look at the relation between the halo and group
multiplicities (i.e., $\Ntrue$ vs. $\Nobs$). In addition, we can look at the fraction
of halos that are detected and the fraction of groups that are spurious.
Figure~\ref{fig:nbodycompbxybz} shows how these relations depend on the line-of-sight
linking length. The bottom panel of the figure shows one set of linking lengths
($\bperp=0.14$, $b_z=0.70$) that yields an unbiased median relation between $\Ntrue$
and $\Nobs$, but the scatter around this relation is large and quite asymmetric.
90\% of groups at a given $\Nobs$ are associated with halos that have up to 40\%
higher and 60\% lower $\Ntrue$. Increasing the line-of-sight linking length causes
groups to grow and thus biases the median $\Ntrue$ vs. $\Nobs$ relation by tilting it
toward larger $\Nobs$. As before, we record all linking length combinations that
yield unbiased median relations between group and halo multiplicities, and we show the
successful parameter space in Figure~\ref{fig:linkinglengths.hod20}.
The top panel of Figure~\ref{fig:nbodycompbxybz} shows the completeness (fraction of
halos that are associated one-to-one with groups) as a function of halo multiplicity
$\Ntrue$, and the middle panel shows the spurious group fraction as a function of
group multiplicity $\Nobs$. Over a wide range of FoF linking lengths, the completeness
for halos with $N\gtrsim 5$ is over 95\%, and the spurious fraction for groups with
$N\gtrsim 5$ is less than 5\%. Increasing the line-of-sight linking length causes
a drop in the halo completeness and a corresponding drop in the spurious group fraction,
since more halos get linked to the same groups. For the final linking lengths that
we use (see \S~\ref{groupfinder}), the halo completeness is greater than 97\% and
the spurious group fraction less than 1\% for $N\gtrsim 10$. The high completeness
and low spurious fraction are a result of how we associate groups to halos. Since we
only require a group to have a halo's central galaxy in order to be associated with it,
most groups and halos have one-to-one associations. If we used a more stringent
criterion for group-halo association, for example by requiring that a group contain
some minimum fraction of a halo's galaxies, then the halo completeness would be lower
and the spurious group fraction higher, but the scatter in $\Ntrue$ vs. $\Nobs$ would
be reduced. The three panels of Figure~\ref{fig:nbodycompbxybz}, put together,
characterize the errors in the FoF group finder. Changing the definition for how groups
are associated to halos does not change the errors in group-finding; it merely
redistributes the errors among the three panels.
In addition to requiring that our groups have unbiased abundances and multiplicities,
we would also like them to have unbiased size distributions. For every group in our
redshift-space cube mocks, we measure the projected rms radius and the line-of-sight
velocity dispersion of galaxies. We compare these to the projected rms radii and
actual velocity dispersions of halo galaxies. Figure~\ref{fig:nbodygrpstatsbxybz}
shows the median, 10th, and 90th percentile projected size and velocity dispersion as a
function of multiplicity for halos, compared to that for groups identified with two
different line-of-sight linking lengths. Increasing the line-of-sight linking length
produces groups with higher velocity dispersions, but it has less impact on
the projected size distributions. The opposite is naturally true when we increase the
perpendicular linking length. Linking length combinations that yield groups with
unbiased abundances and projected sizes tend to yield velocity dispersions that are
biased low. This is illustrated in Figure~\ref{fig:nbodygrpstatsbxybz}, which
shows that the linking length combination $\bperp=0.14$, $b_z=0.7$ yields groups with
velocity dispersions that are $\sim 20\%$ too low relative to halos. The line-of-sight
linking length must be more than doubled to repair this bias, but then the abundances
of groups would be too high.
Figure~\ref{fig:linkinglengths.hod20} shows the linking length parameter space that
satisfies each of the above tests. As discussed in \S~\ref{groupfinder}, there
is no combination of perpendicular and line-of-sight linking lengths that yields
groups with unbiased abundances, projected sizes, and velocity dispersions, even at
high multiplicity. We choose to sacrifice velocity dispersions and adopt the
parameters $\bperp=0.14$, $b_z=0.75$. All the above tests and resulting choice
of linking lengths were done using the \texttt{.Mr20} mock catalogs. Since we plan
to use our group catalog to constrain the HOD, it is vital that our choice of
linking lengths does not depend sensitively on the input HOD assumed when constructing
the mocks. For this reason, we repeat all the above tests with the \texttt{.Mr20b}
mock catalogs, which use a different input HOD to model the same $Mr20$ sample of
SDSS galaxies. The results are shown in Figure~\ref{fig:linkinglengths.hod20b}.
It is clear that our adopted group finder performs equally well in both sets of mock
catalogs, demonstrating that our choice of linking lengths is insensitive to the
underlying HOD.
It is also important to show how well our linking lengths work on lower luminosity
galaxy samples, since we apply them to the SDSS $Mr19$ and $Mr18$ samples. We thus
repeat our mock tests with the \texttt{.Mr19} and \texttt{.Mr18} mock catalogs and show
the results in Figures~\ref{fig:linkinglengths.hod19} and~\ref{fig:linkinglengths.hod18},
respectively. The figures show that lower luminosity (higher density) samples
require slightly higher line-of-sight linking lengths in order to retain unbiased
multiplicity functions. However, this effect is small. When applied to the
\texttt{.Mr18} mock catalogs, our adopted linking lengths yield a multiplicity function
that is 10\% too low in amplitude. Overall, Figures~\ref{fig:linkinglengths.hod20},
\ref{fig:linkinglengths.hod20b}, \ref{fig:linkinglengths.hod19},
and~\ref{fig:linkinglengths.hod18} demonstrate that our choice of linking lengths
is fairly robust.
\def\baselinestretch{1}
\bibliographystyle{apj}
\bibliography{}
|
Title:
Resolving the Stellar Outskirts of M31 and M33 |
Abstract: Many clues about the galaxy assembly process lurk in the faint outer regions
of galaxies. The low surface brightnesses of these parts pose a significant
challenge for studies of diffuse light, and few robust constraints on galaxy
formation models have been derived to date from this technique. Our group has
pioneered the use of extremely wide-area star counts to quantitatively address
the large-scale structure and stellar content of external galaxies at very
faint light levels. We highlight here some results from our imaging and
spectroscopic surveys of M31 and M33.
| https://export.arxiv.org/pdf/astro-ph/0601121 |
\articletitle[Resolving the Stellar Outskirts of M31 and M33]{Resolving the Stellar Outskirts of M31 and M33}
\author{Annette Ferguson\altaffilmark{1}, Mike Irwin\altaffilmark{2}, Scott Chapman\altaffilmark{3},
Rodrigo Ibata\altaffilmark{4}, Geraint Lewis\altaffilmark{5}, Nial Tanvir\altaffilmark{6}}
\affil{\altaffiltext{1}{Institute for Astronomy, University of Edinburgh, UK}
\altaffiltext{2}{Institute of Astronomy, University of Cambridge, UK}
\altaffiltext{3}{California Institute for Technology, Pasadena, USA}
\altaffiltext{3}{Observatoire de Strasbourg, Strasbourg, France}
\altaffiltext{3}{Institute of Astronomy, University of Sydney, Australia}
\altaffiltext{3}{Centre for Astrophysics Research, University of Hertfordshire, UK}}
\section{Introduction}
The study of galaxy outskirts has become increasingly important in
recent years. From a theoretical perspective, it has been realised
that many important clues about the galaxy assembly process should lie
buried in these parts. Cosmological simulations of disk galaxy
formation have now been carried out by several groups and have led to
testable predictions for the large-scale structure and stellar content
at large radii -- for example, the abundance and nature of stellar
substructure (e.g. Bullock \& Johnston 2005, Font et
al. 2005), the ubiquity, structure and content of stellar halos and
thick disks (e.g. Abadi et al. 2005, Governato et al. 2004, Brook et
al. 2005) and the age distribution of stars in the outer regions of thin disks
(e.g. Abadi et al. 2003).
Bullock \& Johnston (2005) find that Milky Way-like galaxies will have
accreted 100-200 luminous satellites during the last 12~Gyr and that
the signatures of this process should be readily visible at surface
brightnesses of V$\sim 30$ magnitudes per square arcsec and lower. Although
traditional surface photometry at such levels (roughly 9 magnitudes
below sky) remains prohibitive, star count analyses of nearby galaxies
have the potential to reach these effective depths (e.g. Pritchet \&
van den Bergh 1994). The requirement of a large survey area (to
provide a comprehensive view of the galaxy) and moderate-depth imagery
can now be achieved in a relatively straightforward manner using
wide-field imaging cameras attached to medium-sized telescopes.
\section{The INT WFC Surveys of M31 and M33}
In 2000, we began a program to map the outer regions of our nearest
large neighbour, M31, with the Wide-Field Camera equipped to the INT
2.5m. The success of this program led us to extend our survey to M33 in
the fall of 2002. To date, more than 45 and 7 square degrees have been
mapped around these galaxies respectively. Our imagery reaches to
V$\sim$24.5 and {\sl i}$\sim$23.5 and thus probes the top 3 magnitudes
of the red giant branch (RGB) in each system. The raw data are
pipeline-processed in Cambridge and source catalogues are produced
containing positions, magnitudes and shape parameters. The M31 survey
currently contains more than 7 million sources, and the M33 survey more than 1
million. Magnitude and colour cuts are applied to point-like
sources in order to isolate distinct stellar populations
and generate surface density maps (see Figure 1). The faint
structures visible by eye in Figure 1 have effective V-band surface
brightnesses in the range 29-30 magnitudes per square arcsec. Early
versions of our M31 maps have been discussed in Ibata et al. (2001),
Ferguson et al. (2002) and Irwin et al. (2005).
\subsection{Results for M31}
The left-hand panel of Figure 1 shows the distribution of blue
(i.e. presumably more metal-poor) RGB stars in and around M31. A
great deal of substructure can be seen including the giant stream in
the south-east, various overdensities near both ends of the major axes, a
diffuse extended structure in the north-east and a loop of stars projected near
NGC~205.
{\sl Origin of the Substructure:} Do the substructures in M31
represent debris from one or more satellite accretions, or are they
simply the result of a warped and/or disturbed outer disk? We are
addressing these issues with deep ground-based imagery from
the INT and CFHT, Keck-10m spectroscopy and deep HST/ACS
colour-magnitude diagrams (CMDs). Our findings to date can be
summarized as follows:
\begin{itemize}
\item{M31 has at least 12 satellites lying within a
projected radius of 200~kpc. The bulk of these systems,
the low-luminosity dwarf spheroidals, are unlikely to be associated with the
stellar overdensities since their RGB stars are much bluer than
those of the substructure (Ferguson et al. 2002).}
\item{The combination of line-of-sight distances and radial velocities
for stars at various locations along the giant stellar stream
constrains the progenitor orbit (e.g. McConnachie et al. 2003,
Ibata et al. 2004). Currently-favoured orbits do not connect the
more luminous inner satellites (e.g. M32, NGC~205) to the stream
in any simple way however this finding leaves some remarkable
coincidences (e.g. the projected alignment on the sky, similar
metallicities) yet unexplained.}
\item{Deep HST/ACS CMDs reaching well below the horizontal branch
reveal different morphologies between most substructures in
the outskirts of M31 (Ferguson et al. 2005, see Figure 2). These
variations reflect differences in the mean age and/or metallicity
of the constituent stellar populations. Analysis is underway to
determine whether multiple satellite accretions are required, or
whether consistency can be attained with a single object which has
experienced bursts of star formation as it has orbited M31. The
giant stream is linked to another stellar overdensity, the NE
shelf, on the basis of nearly identical CMD morphologies and RGB
luminosity functions; indeed, this coupling seems likely in view
of progenitor orbit calculations (e.g. Ibata et al.
2004).}
\end{itemize}
{\sl Smooth Structure:} The INT/WFC survey provides the first
opportunity to investigate the smooth underlying structure of M31 to
unprecedented surface brightnesses. We have used the dataset to
map the minor axis profile from the innermost regions to $\gtrsim 55$~kpc
(Irwin et al. 2005). Figure 3 shows how the combination of
inner diffuse light photometry and outer star count data can be used to
trace the effective {\sl i}-band surface brightness profile
to $\sim 30$~magnitudes per square arcsec. The profile
shows an unexpected flattening (relative to the inner R$^{1/4}$ decline)
at large radius, consistent with the presence of an additional shallow power-law
stellar component (index $\approx -2.3$) in these parts. This component may
extend out as far as 150~kpc (Guhathakurta, these proceedings). Taken
together with our knowledge of the Milky Way halo, this finding
supports the ubiquitous presence of power-law stellar halos around
bright disk galaxies (see also Zibetti et al. 2004, Zibetti \& Ferguson 2004).
The kinematics of M31's outer regions are being probed with Keck
DEIMOS spectroscopy (e.g. Ibata et al. 2005). Two surprising results
have emerged from our program so far. Firstly, there is a high degree
of rotational support at large radius, extending well beyond the
extent of M31's bright optical disk. Secondly, the overall coherence
of this kinematic component is in striking contrast to its clumpy substructured
appearance in the star count maps. Further work is underway
to understand the nature and origin of this rotating component.
\subsection{Results for M33}
The right-hand panel of Figure 1 shows the RGB map of the low mass
system, M33. Although it has the same limiting absolute depth as the
M31 map, the stellar density distribution is extremely smooth and
regular (Ferguson et al. 2006, in preparation). To a limiting depth
of $\sim 30$~magnitudes per square arcsec (readily visible by eye
here), the outer regions of M33 display no evidence for stellar
substructure (c.f. the simulations of Bullock \& Johnston 2005).
Equally surprising, our analysis of the isophote shape as a function of
radius indicates no evidence for any twisting or asymmetries.
M33 appears to be a galaxy which has evolved in relative
isolation.
The radial {\sl i}-band profile of M33 has been quantified via azimuthally-averaged
photometry in elliptical annuli of fixed PA and inclination (Figure 3).
The inner parts of the profile are constructed
from diffuse light photometry, whereas the outer regions are derived
from RGB star counts. The luminosity profile displays an exponential
decline out to $\sim 8$~kpc (roughly 4.5 scalelengths) beyond which it
significantly steepens. This behaviour is reminiscent of the ``disk
truncations'' first pointed out by van der Kruit in the 80's, but until
now not seen directly with resolved star counts. The steep
outer component dominates the M33 radial light profile out to at least 14~kpc
and limits the contribution of any shallow power-law stellar halo
component in M33 to be no more than a few percent of the disk luminosity
(Ferguson et al. 2006, in preparation).
\section{Future Work}
Quantitative study of the faint outskirts of galaxies provides
important insight into the galaxy assembly process. The outskirts
of our nearest spiral galaxies, M31 and M33, exhibit intriguing
differences in their large-scale structure and stellar content. While
M31 appears to have formed in the expected hierarchical fashion, M33
shows no obvious signatures of recent accretions. Observations of
the outer regions of additional galaxies are required to determine
which of these behaviours is most typical of the general disk
population.
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\end{chapthebibliography} |
Title:
A radial velocity survey of low Galactic latitude structures: III. The Monoceros Ring in front of the Carina and Andromeda galaxies |
Abstract: As part of our radial velocity survey of low Galactic latitude structures
that surround the Galactic disc, we report the detection of the so called
Monoceros Ring in the foreground of the Carina dwarf galaxy at Galactic
coordinates (l,b)=(260,-22) based on VLT/FLAMES observations of the dwarf
galaxy. At this location, 20 degrees in longitude greater than previous
detections, the Ring has a mean radial velocity of 145+/-5 km/s and a velocity
dispersion of only 17+/-5 km/s. Based on Keck/DEIMOS observations, we also
determine that the Ring has a mean radial velocity of -75+/-4 km/s in the
foreground of the Andromeda galaxy at (l,b)\sim(122,-22), along with a velocity
dispersion of 26+/-3 km/s. These two kinematic detections are both highly
compatible with known characteristics of the structure and, along with previous
detections provide radial velocity values of the Ring over the 120<l<260 range.
This should add strong constraints on numerical models of the accretion of the
dwarf galaxy that is believed to be the progenitor of the Ring.
| https://export.arxiv.org/pdf/astro-ph/0601176 |
\begin{keywords} Galaxy: structure -- galaxies: interactions -- Galaxy: formation
\end{keywords}
\section{Introduction}
With the public release of all sky surveys, many details have been gained in the structure of the outer parts of the
Galactic disc. In particular, the Sloan Digital Sky Survey (SDSS) revealed the existence of a stellar structure
in the anticentre direction, near the Galactic plane and slightly over the edge of the disc \citep{newberg02}. Visible
as a clear main sequence at a Galactocentric distance of $18\kpc$ for $180\deg<l<225\deg$ and $|b|<30\deg$, this
structure is unlike what is expected for the Galactic disc. \citet{ibata03} used the INT Wide Field Camera to show
this structure is in fact present in the second and third Galactic quadrants, circling the disc in a ring-like fashion.
Radial velocity measurements also revealed a kinematically cold population with a velocity dispersion of only
$15-25\kms$, once more unexpected for a Galactic structure, leading to the conclusion that this so-called
Ring\footnote{This structure has been called many names including the Monoceros Ring, the Galactic Anticentre Stellar
Structure or GASS and the Ring. Given its extent in Galactic longitude and its presence in numerous constellations, we
prefer to call it simply the Ring.} is produced by the accretion of a dwarf galaxy in the Galactic plane whose
tidal arms are wrapped around the Milky Way.
Subsequently, much work has been invested in trying to determine the true extent of this Ring and where possible
determine its kinematics to constrain the orbit of its progenitor. \citet{rocha-pinto03} used the 2MASS catalogue to probe the
distribution of M giant stars and found that the Ring may extend over the $120\deg\simlt l\simlt270\deg$ range in the
Northern hemisphere and be present in the Southern hemisphere, with $|b|<35\deg$. \citet{conn05a}
extended the \citet{ibata03} INT/WFC survey to show the Ring is present within 30 degrees of the Galactic plane
throughout the whole second quadrant but seems to disappear at $l\sim90\deg$. More constraints on the structure
were gained by the \citet{crane03} spectroscopic survey of putative Ring M-giant stars in the anticentre direction
($150\deg<l<230\deg$, $|b|<40\deg$) which confirmed that most of these stars are indeed linked to this structure. Using
the simple model of a population orbiting the Milky Way in a prograde, circular orbit, they determined their sample was
best fitted by a population at a Galactocentric distance of $18\kpc$ and with a rotational velocity of $220\kms$. The
resulting velocity dispersion of $20\pm4\kms$ around this model is in good agreement with the velocity dispersion
measured by the SDSS team \citep{yanny03}.
While tracking the Ring, two other structures were discovered that do not seem to be directly related to the Monoceros
Ring. Using the 2MASS catalogue, \citet{rocha-pinto04} reported the existence of a diffuse population of M giants in the
direction of the Triangulum and Andromeda constellations, behind known detections of the Ring. Also using the 2MASS
catalogue, our group presented evidence of a dwarf galaxy located closer than the Ring, just at the edge of the Galactic
disc, in the Canis Major constellation \citep{martin04,bellazzini04}. In particular, we argued that the accretion of the
Canis Major galaxy onto the Galactic disc would naturally reproduce similar features as the one observed for the Ring.
However, a link between these three structures remains putative at the moment, even though current models show such
a scenario is highly plausible \citep{penarrubia05,martin05,dinescu05}.
To gain more insight into the nature of these structures, we have started a radial velocity survey of regions at low
Galactic latitude where the Ring structure may be located. Using the AAT/2dF multi-fibre spectrograph, we first
targeted the Canis Major dwarf \citep{martin05}. This survey also revealed the presence of the Ring behind the dwarf,
with a Heliocentric radial velocity of $133\pm1\kms$ and dispersion of $23\pm2\kms$ at a Galactocentric distance of
$19\kpc$, highly compatible with previous detections \citep{conn05b}. In this third paper of the series, we report
the presence of the Ring in the foreground of the Carina dwarf galaxy from VLT/FLAMES observations of the dwarf at
$(l,b)=(260\deg,-22\deg)$. Using Keck2/DEIMOS observations of regions around M31, we also determine the radial velocity
of the Ring at $(l,b)\sim(122\deg,-22\deg)$ where the Ring is known to exist but where its radial velocity has not yet
been measured. Section 2 presents the Ring detection in front of Carina and Section 3 deals with the detection in
front of the Andromeda galaxy. Section 4 concludes this letter.
In the following, all the magnitudes have been corrected for extinction using the maps from
\citet{schlegel98}. We also assume that the Solar radius is $R_\odot = 8\kpc$, that the LSR circular velocity is
$220\kms$, and that the peculiar motion of the Sun is ($U_0=10.00\kms, V_0=5.25\kms, W_0=7.17\kms$; \citealt{dehnen98}).
Except when stated otherwise, the radial velocities, $v_{r}$, are Heliocentric radial velocities, not corrected for the
motion of the Sun.
\section{The Ring in front of the Carina dwarf galaxy}
FLAMES observations in the Carina fields were taken from the public access ESO raw data
archive\footnote{http://www.eso.org/archive} and included 14 setups centred on various fields designed to study the
Carina dwarf galaxy. The total integration time of each setup was typically $15000$ seconds giving excellent
signal-to-noise ($>$20:1 per 0.2\AA\ sampling element) for the Galactic foreground stars in these directions. The
Carina data plus calibration sequences were downloaded and processed through the standard ESO FLAMES low resolution
pipeline.
At the time of using the pipeline, sky subtraction was not part of the pipeline procedure so the remaining processing
steps used the FLAMES software developed for the DART project (see e.g. \citealt{tolstoy04}). Briefly, this
software stacks the individual repeat spectra on the same target field; combines all the sky observations to form a
master sky spectrum; and then optimally scales, shifts (if necessary) and resolution-matches the sky to the object
spectrum prior to sky subtraction. The sky-subtracted spectra are then searched for CaII near infrared triplet lines
and a model template CaII spectrum is used to extract velocity information.
The direct imaging used here also came from the ESO archive and comprised 4 $V,I$ sequences of ESO/WFI data covering
$\approx 1 \times 1$ square degrees centred on the Carina dwarf. This was processed through the standard Cambridge
pipeline \citep{irwin01} to produce catalogues of magnitudes, colours and object morphological classifications.
The velocity distribution of all the stars targeted with FLAMES is displayed on Figure~1. The most prominent feature is
of course the peak of stars at $v_{r}\sim220\kms$ produced by Red Giant Branch stars belonging to the Carina dwarf
galaxy (the primary targets of these FLAMES observations). Stars with $v_{r}<180\kms$ are expected to be Galactic stars
belonging to the thin disc, thick disc and/or stellar halo. However, they seem to follow a bimodal Gaussian distribution
produced by populations with central velocities and dispersions of $(\mu_A,\sigma_A)=(49\pm7\kms,33\pm2\kms)$ and
$(\mu_B,\sigma_B)=(145\pm5\kms,17\pm5\kms)$ and a 9 to 1 ratio according to a maximum likelihood fit of a two component
Gaussian model. Although population A has the expected characteristics of a disc-like population in the foreground, the
low dispersion of population B is more puzzling.
The Colour-Magnitude Diagram (CMD) of Figure~2 displays the location of stars belonging to these two populations with
hollow circles for population A stars ($v_{r}<110\kms$) and filled circles for population B stars ($110<v_{r}<180\kms$).
The two populations show drastically different colour-magnitude distributions, pointing at a genuine difference between
them. Population A is widely spread in colour and follows, once more, the expected distribution of foreground disc
stars. On the other hand population B is confined on the bluer part of our sample, with almost all stars having
$(V-I)_0<1.3$. Of the three stars with $(V-I)_0>1.3$, the one with the highest $(V-I)_0$ is only just a population B
star with $v_{r}=112\kms$ (the two others have radial velocities of 143 and $149\kms$).
Such a sharp colour cut makes it very unlikely that population B is produced by disc stars. Among the Galactic
components, only the stellar halo is expected to have such behaviour (see e.g. \citealt{conn05a} for a comparison of the
disc and halo Galactic component CMDs as they appear in the Besan\c{c}on model of \citealt{robin03}). However, the
velocity dispersion of the halo is $\sim100\kms$ \citep[e.g.][]{gould03}, at odds with the $17\kms$ found for population
B. Since this low dispersion could be an artifact of the radial velocity cuts we used, we investigate the number of
stars that would be expected for a halo population at $(l,b)=(260\deg,-22\deg)$. According to the synthetic Besan\c{c}on
model of Galactic stellar populations \citep{robin03}, there should be less than 225 halo stars per square degree in the
CMD region highlighted on Figure~2 and that contains most of our population B stars, independently of radial velocity.
Since our FLAMES targets roughly cover 0.25 square degrees, we would expect our sample to contain less than 60 halo
stars if complete. Assuming the Milky Way is surrounded by a non-rotating stellar halo with a velocity dispersion of
$100\kms$, our sample should contain only $\sim15$ halo stars within $110\kms<v_{r}<180\kms$, once again, if it were
complete. Since within the selection box of Figure~2, the completeness is under 5\%, it is highly unlikely that the 56
population B stars belong to the halo. Therefore, population B is most likely a non-Galactic population of stars that
lie in front of the Carina dwarf galaxy, with a mean radial velocity of $145\pm5\kms$ and a velocity dispersion of
$17\pm5\kms$.
A direct comparison with the velocity distribution of all the stars from the model that fall in the same region of the
CMD would of course be more suited for our purpose. However the model greatly overpredicts the number of thick disc
stars in this region of the sky when compared to the observations. This is probably due to an overestimate of the thick
disc flare in the model. Although population A is very well reproduced by the thin disc population of the model, the
modeled thick disc population is twice as numerous, centred on $v_{r}\sim80\kms$ and with a high dispersion of
$\sim50\kms$. Such a population is clearly not present in our data. Since the sharp $(V-I)_0<1.3$ colour limit of
population B is not expected for a disc population, we prefer comparing population B with only the halo population of
the model.
With a detection of the Ring behind the Canis Major galaxy under the Galactic disc only 20 degrees away in Galactic
longitude from the feature we detect in front of Carina, this population could be another detection of the Ring.
Previous detections of the Ring have mainly relied on CMDs and especially on the main sequence of
this population compared to the disc population at brighter magnitudes \citep{newberg02,ibata03,conn05a}. Unfortunately,
for the Carina CMD, the red clump of the dwarf galaxy at $(V-I)_0\sim0.7$ and $V_0\sim20.5$ lies in the region of
interest for detecting the Ring main sequence. However, comparison with fiducials shows the location of population B
stars is not incompatible with turn-off stars from the old metal-rich population at a Galactocentric distance of
$\sim20\kpc$ that is usually assumed for the Ring; even though the Carina-driven selection criteria applied to select
the stars in the sample prevents a reliable comparison.
The radial velocity characteristics of this population in front of Carina further supports a connection with the Ring.
Indeed, the determined velocity dispersion of $17\pm5\kms$ is within the $15-25\kms$ range of the SDSS and
\citet{crane03} detections and close to the $24\pm2\kms$ of the detection in the background of Canis Major. For this
latter case, the higher uncertainties on the 2dF radial velocity value of each star ($\sim10\kms$) compared to those of
the FLAMES derived velocities ($\sim3\kms$) may also artificially increase the dispersion. When corrected from the Solar
motion, the mean radial velocity of population B, $v_{\mathrm{gsr,B}}=-65\kms$, is very close to the
$v_{\mathrm{gsr,bCMa}}=-67\kms$ found only 20 degrees away behind the Canis Major dwarf galaxy.
Therefore, we conclude that the non-Galactic population we have uncovered in front of the Carina dwarf galaxy is most
likely part of the Ring.
\section{Kinematics of the Ring in front of the Andromeda galaxy}
The presence of the Ring in front of the Andromeda galaxy was first reported by \citet{ibata03} from the analysis of
INT/WFC Colour-Magnitude diagrams. The CMD of their `M31-N' field is shown on the left panel of Figure~3 and the Ring is
clearly visible as a main sequence that extends from $(V-i,V)_{0}\sim(1.2,22)$ to $(V-i,V)_{0}\sim(0.6,20.0)$. During
our Keck/DEIMOS survey of M31 outer disc and halo substructures \citep[e.g.][]{ibata05}, we took the opportunity to
target foreground Ring stars that fortuitously fall in the targeted regions. The data were reduced as in \citet{ibata05}
and the CMD of all the stars with a radial velocity uncertainty lower than $10\kms$ is presented on the right panel of
Figure~3. To select only probable members of the Ring structure, we construct a selection box around the Ring main
sequence from the INT CMD (box C in Figure~3). Among the stars in the DEIMOS sample, 86 fall within this Ring selection
box and the radial velocity distribution of these stars is displayed on Figure~4. Aside from Galactic halo and M31 disc
stars at $v_{r}<-200\kms$, a peak is apparent at $\sim-70\kms$. Applying a maximum likelihood algorithm to fit with a
Gaussian model those stars with $-140<v_{r}<0\kms$ that produce the peak, reveals that this population has a mean
velocity of $-75\pm4\kms$ and an intrinsic dispersion of $26\pm3\kms$, corrected for the uncertainties on each measured
radial velocity.
A direct comparison with the radial velocity distribution of stars from the Besan\c{c}on model within the same sky
region ($119\deg<l<124\deg$ and $-24\deg<b<-19\deg$) and that fall in the same CMD selection box reveals the Ring
sub-sample is not incompatible with the model. Indeed, a Kilmogorov-Smirnov test yields a probability of 10 percent that
the two populations are identical. However, it is not unexpected that disc stars and Ring stars should show similar
behaviour since both populations are believed to orbit the Milky Way on nearly circular orbits and the small shift in
distance between them does not translate into a significant difference in radial velocity. Given that our
selection box is constructed to contain Ring stars, it would be surprising that all the stars of the sample belong to the
Galactic disc. On the contrary, we find it more likely that we are observing mainly Ring stars, as is suggested by the
relatively low velocity dispersion of $26\pm3\kms$ in the sample, at odds with the $\sim50\kms$ found in the
Besan\c{c}on model within the selection box. This low dispersion is also compatible with previous detections, especially
since some disc and/or halo stars certainly fall in the same radial velocity range and increase the dispersion. The mean
Heliocentric velocity of $-75\pm4\kms$ which converts to a Galactocentric standard of rest radial velocity of
$v_{\mathrm{gsr}}=94\pm4\kms$ is also only slightly higher than the simple circular \citet{crane03} model. As for the
detection in the foreground of the Carina dwarf, the radial velocity similarities between the previous detections of the
Ring and the population in front of M31 strengthen the Ring nature of our detection.
\section{Summary and Conclusion}
We have presented the detection of two groups of stars that lie in front of the Carina dwarf galaxy at
$(l,b)=(260\deg,-22\deg)$ and in front of the Andromeda galaxy at $(l,b)\sim(122\deg,-22\deg)$ and that cannot be
satisfyingly explained by known Galactic components. The proximity with known detections of the Ring that surrounds the
Galactic disc makes it highly probable that they belong to the same structure. Both detections have a low velocity
dispersion ($17\pm5\kms$ and $26\pm3\kms$ respectively), a characteristic value encountered in all previous detections of
the Ring.
With the Ring detection behind the Canis Major dwarf galaxy reported by \citet{conn05b}, the radial velocity of the Ring
population is now sampled throughout the $120\deg<l<260\deg$ range, which should provide important constraints on
N-body models. However, it can be directly seen in Figure~5 that currently known radial velocity values for the Ring are
not exactly reproducible by the \citet{crane03} simple circular model. In fact, trying to fit all the detections in a
single orbit of the progenitor proves unsatisfactory, whether the orbit is forced to be circular or allowed to be
slightly elliptical. It would therefore seem that models where the Ring completely surrounds the Galactic disc with
multiple tidal arms are following the right track (see e.g. \citealt{penarrubia05} and \citealt{martin05}). In
addition to the new radial velocities we report in this letter, such models would highly benefit from a similar survey
as the one presented in \citet{conn05a}, but this time to higher longitudes to study in more detail the morphology of
the Ring in these regions and especially to add a distance constraint to the detection in front of the Carina dwarf.
\section*{acknowledgements}
NFM is grateful to the IoA for the kind hospitality during the months at Cambridge in which this work was mainly performed.
NFM acknowledges support from a Marie Curie Stage Research Training Fellowship under contract MEST-CT-2004-504604.
GFL acknowledges support from ARC DP 0343508 and is grateful to the Australian Academy of Science for financially
supporting a collaboratory visit to Strasbourg Observatory.
\newcommand{\mnras}{MNRAS}
\newcommand{\pasa}{PASA}
\newcommand{\nat}{Nature}
\newcommand{\araa}{ARAA}
\newcommand{\aj}{AJ}
\newcommand{\apj}{ApJ}
\newcommand{\apjl}{ApJ}
\newcommand{\apjs}{ApJSupp}
\newcommand{\aap}{A\&A}
\newcommand{\aaps}{A\&ASupp}
\newcommand{\pasp}{PASP}
|
Title:
Multifrequency observations of the jets in the radio galaxy NGC 315 |
Abstract: We present images of the jets in the nearby radio galaxy NGC 315 made with
the VLA at five frequencies between 1.365 and 5 GHz with resolutions between
1.5 and 45 arcsec FWHM. Within 15 arcsec of the nucleus, the spectral index of
the jets is 0.61. Further from the nucleus, the spectrum is flatter, with
significant transverse structure. Between 15 and 70 arcsec from the nucleus,
the spectral index varies from 0.55 on-axis to 0.44 at the edge. This spectral
structure suggests a change of dominant particle acceleration mechanism with
distance from the nucleus and the transverse gradient may be associated with
shear in the jet velocity field. Further from the nucleus, the spectral index
has a constant value of 0.47. We derive the distribution of Faraday rotation
over the inner +/-400 arcsec of the radio source and show that it has three
components: a constant term, a linear gradient (both probably due to our
Galaxy) and residual fluctuations at the level of 1 - 2 rad/m^2. These residual
fluctuations are smaller in the brighter (approaching) jet, consistent with the
idea that they are produced by magnetic fields in a halo of hot plasma that
surrounds the radio source. We model this halo, deriving a core radius of
approximately 225 arcsec and constraining its central density and
magnetic-field strength. We also image the apparent magnetic-field structure
over the first +/-200 arcsec from the nucleus.
| https://export.arxiv.org/pdf/astro-ph/0601660 |
\label{firstpage}
\begin{keywords}
galaxies: jets -- radio continuum:galaxies -- galaxies: individual: NGC\,315 --
magnetic fields -- polarization -- MHD
\end{keywords}
\section{Introduction}
\label{intro}
The giant FR\,I \citep{FR74} radio source NGC\,315 was first imaged by
\citet{Brid76}, who showed that it has an angular size of nearly 1$^\circ$. The
extended radio structure is described in more detail by \citet{Brid79},
\citet{Fom80}, \citet{Willis81}, \citet{Jaegers}, \citet{Venturi93},
\citet{Mack97} and \citet{Mack98}. The main jet has also been imaged extensively
on parsec scales \citep{Linfield81,Venturi93,Cotton99,Xu00}. X-ray emission from
the first 10\,arcsec of the main jet was detected by \citet{WBH}, but no optical
emission from this region has yet been reported.
The source is associated with a giant elliptical galaxy at a redshift of 0.01648
\citep{Trager}, giving a scale of 0.335\,kpc/arcsec for our adopted cosmology
(Hubble constant $H_0$ = 70\,$\rm{km\,s^{-1}\,Mpc^{-1}}$, $\Omega_\Lambda =
0.7$ and $\Omega_M = 0.3$). NGC\,315 is a member of a group or poor cluster of
galaxies \citep{Nolthenius,Miller02} located in one of the filaments of the
Pisces-Perseus supercluster \citep{Ensslin01,Huchra}. HST images
\citep{Verdoes99} show a 2.5-arcsec diameter dust lane and a nuclear
point source. The dust lane is associated with a disk of ionized gas which is
probably in ordered rotation \citep{Noel-Storr}. CO emission, also with a line
profile indicating rotation, was detected by \citet{Leon}. The inferred mass of
molecular hydrogen is $(3.0 \pm 0.3) \times 10^8$\,M$_\odot$ and the cold gas is
likely to be cospatial with the dust. HI absorption against the
nucleus was detected by \citet{vanG89}. There is evidence for a weak, polarized
broad H$\alpha$ line in the nuclear spectrum \citep{Ho97,Barth99,Noel-Storr}.
Hot gas associated with the galaxy has been imaged using {\sl ROSAT} and {\sl
Chandra} \citep{WB,WBH}.
Within $\approx$\,90 arcsec of the nucleus, the jets in NGC\,315 are initially
narrow, then expand rapidly (``flare'') and re-collimate \citep{Brid82,CLBC}. We
have modelled the inner $\pm$70\,arcsec of this {\em flaring region} as a
two-sided, symmetrical, relativistic flow, fitting to deep, high-resolution VLA
observations at 5\,GHz in order to derive the three-dimensional distributions of
velocity, proper emissivity and magnetic-field structure \citep{CLBC}. Our main
conclusions are as follows.
\begin{enumerate}
\item The jets are inclined by
$38^\circ \pm 2^\circ$ to the line of sight.
\item Where they first brighten, their on-axis velocity is $\beta = v/c \approx
0.9$. They decelerate to $\beta \approx 0.4$ between 8 and 18\,kpc from the
nucleus (15 -- 33\,arcsec in projection) and the velocity thereafter remains
constant.
\item The ratio of the speed at the edge of the jet to its value on-axis ranges
from $\approx$0.8 close to the nucleus to $\approx$0.6 further out.
\item The longitudinal profile of proper emissivity
is split into three power-law regions separated by shorter transition zones and
the emission is intrinsically centre-brightened.
\item To a first approximation, the magnetic field evolves from a mixture of
longitudinal and toroidal components to predominantly toroidal by 26\,kpc
(48\,arcsec in projection).
\item Simple adiabatic models fail to fit the emissivity
variations.
\end{enumerate}
In the present paper, we investigate the energy spectrum of the relativistic
particles in the jets of NGC\,315 in the context of the models developed by
\citet{CLBC}. We use VLA observations at frequencies between 1.365 and
5\,GHz\footnote{The 5-GHz observations are those discussed by \citet{CLBC}} to
derive the spectrum of the jets at resolutions of 5.5 and 1.5\,arcsec and relate
the observed spectral gradients to velocity, emissivity and field structure. A
separate paper (Worrall et al., in preparation) will describe the radio structure of
the main jet at high resolution and its relation to new {\sl Chandra} images.
We also determine the variations of Faraday rotation over the jets and test the
hypothesis that these result from magnetic-field irregularities in hot, X-ray
emitting plasma associated with the surrounding group of galaxies. Finally, we
determine the apparent magnetic-field structure of the jets on scales larger
than those covered by \citet{CLBC}.
In Section~\ref{obs}, we describe the observations and their reduction. The
total-intensity images are presented in Section~\ref{Images} and we use them to
derive distributions of spectral index in Section~\ref{Spectra}. We then discuss
the distributions of Faraday rotation (Section~\ref{Faraday}) and apparent
magnetic-field structure (Section~\ref{Field}) derived from observations of
linear polarization. Section~\ref{Summary} summarizes our main results.
\section{Observations and images}
\label{obs}
\subsection{Observations}
VLA data were obtained at 4.985\,GHz in the B, C/D and A/D configurations as
described by \citet{Venturi93} and \citet{Cotton99}. These were supplemented by
additional observations in the A and C configurations with a centre frequency of
4.860\,GHz to give complete coverage of the spatial scales accessible to the VLA
in a single pointing. In order to map Faraday rotation, we observed at 1.365,
1.413, 1.485 and 1.665\,GHz in the B and C configurations of the VLA using a
lower bandwidth. We also extracted observations in A configuration at 1.413\,GHz
from the VLA archive.\footnote{In addition, we re-analysed the C-configuration
dataset at 8.4\,GHz from \citet{Venturi93}, but poor weather during the
observations precluded accurate absolute flux and polarization calibration, so
we do not discuss them here.} A journal of observations is given in
Table~\ref{record}.
\begin{center}
\begin{table}
\caption{Record of VLA observations. $\nu$ and $\Delta\nu$ are the centre
frequency and bandwidth, respectively, and t is the on-source integration
time.}
\begin{center}
\begin{tabular}{clllc}
\hline
Config- & Date & $\nu$ & $\Delta\nu$ & t \\
uration & & (MHz) & (MHz) & (min) \\
\hline
B & 1989 Apr 13 & 4985.1 & 50 & 279 \\
B & 1995 Oct 25 & 4985.1 & 50 & 396 \\
C/D & 1996 May 10 & 4985.1 & 50 & 417 \\
A/D & 1996 Oct 07 & 4985.1 & 50 & 428 \\
A & 1996 Nov 2 & 4860.1 & 100 & 586 \\
C & 1997 Jul 12 & 4860.1 & 100 & 283 \\
A & 1980 Dec 21 & 1413.0 & 25 & 473 \\
B & 2001 Mar 19 & 1365.0 & 12.5 & 108 \\
B & 2001 Mar 19 & 1413.0 & 12.5 & 108 \\
B & 2001 Mar 19 & 1485.0 & 12.5 & 109 \\
B & 2001 Mar 19 & 1665.0 & 12.5 & 107 \\
C & 2001 Jul 17 & 1365.0 & 12.5 & 61 \\
C & 2001 Jul 17 & 1413.0 & 12.5 & 61 \\
C & 2001 Jul 17 & 1485.0 & 12.5 & 65 \\
C & 2001 Jul 17 & 1665.0 & 12.5 & 57 \\
\hline
\end{tabular}
\end{center}
\label{record}
\end{table}
\end{center}
\subsection{Data reduction}
\label{reduction}
All of the data reduction was done in the {\sc aips} package. Initial amplitude
and phase calibration were applied using standard methods and the flux-density
scales were set using observations of 3C\,48 or 3C\,286. Standard instrumental
polarization calibration was applied and the zero-points of ${\bf E}$-vector
position angle were determined using observations of 3C\,138 or 3C\,286. The
data for each configuration were first adjusted to a common phase centre in
J2000 coordinates, imaged and self-calibrated separately. They were then
concatenated in turn, starting with the widest configuration. The slight
difference in centre frequency between the datasets at 4.86 and 4.985\,GHz was
ignored (we show in Section~\ref{RM} that this has a negligible effect on the
analysis of polarization) and we will refer to the combination as the ``5\,GHz
dataset''. At this frequency, the core showed significant variability between
observations (cf.\ \citealt{Lazio01}) and the flux density of the unresolved
component in the larger-configuration dataset was adjusted to match that
observed with the smaller configuration when both were imaged at matched
resolution. No core variability was detected at lower frequencies. A further
iteration of phase self-calibration was done after each combination. Our final
datasets are listed in Table~\ref{Datasets}, together with the minimum and
maximum spatial scales they sample.
\begin{center}
\begin{table}
\caption{Final uv datasets. The columns are: (1) centre frequency, (2) array
configurations used, (3) minimum and (4) maximum spatial scales.
\label{Datasets}}
\begin{tabular}{llll}
\hline
$\nu$ & Configurations &\multicolumn{2}{c}{Scales (arcsec)}\\
(MHz) & & Min & Max \\
1365 & BC & 4 & 900\\
1413 & ABC & 1.2 & 900\\
1485 & BC & 4 & 900\\
1665 & BC & 4 & 900\\
4985/4860 & ABCD & 0.4 & 300\\
\hline
\end{tabular}
\end{table}
\end{center}
Our observations were designed to give the maximum sensitivity for the inner
jets of NGC\,315 and were therefore taken with the pointing centre on or near
the nucleus. As can be seen from Table~\ref{Datasets}, the maximum spatial scale
sampled adequately at 5\,GHz is $\approx$300\,arcsec. Even at L-band, the lobe
associated with the counter-jet (\citealt{Mack97} and Fig.~\ref{fig:ilow}a) is
severely attenuated by the primary-beam response of the VLA and is not visible
on our images. We did not recover the total flux density of the source at any
frequency, so we estimated appropriate zero-spacing flux densities from
the shortest-spacing visibility amplitudes. We made images at five resolutions:
45, 5.5, 2.35, 1.5 and 0.4 arcsec FWHM, using similar baseline ranges at all
frequencies and weighting the data in the uv plane as required. After imaging,
we made both {\sc clean} and maximum-entropy deconvolutions. Although the latter
algorithm gave slightly smoother images, it introduced a significant
large-scale ripple parallel to the jet axis, whereas {\sc clean} gave a flat
background. We therefore show the {\sc clean} images, although we quote
quantitative results only where the two deconvolution methods agree. We also
compared the $I$ images made with and without zero-spacing flux densities and
before and after subtraction of a local zero-level. None of these differences
led to significant changes in spectral index or degree of polarization compared
with the errors quoted below. After deconvolution, all of the images were
corrected for primary beam attenuation. We then took averages of the $I$ images
at 1.365 -- 1.665\,MHz (``mean L-band images'').
Data in Stokes $Q$ and $U$ were imaged without zero-spacing flux densities and
{\sc clean}ed. A first-order correction for Ricean bias \citep{WK} was applied
to the images of polarized intensity $P = (Q^2+U^2)^{1/2}$ used to derive the
degree of polarization $p = P/I$.
The off-source noise levels at the centre of the field for the final images are
given in Table~\ref{noise} (the 0.4-arcsec image at 5\,GHz is discussed in
detail by Worrall et al., in preparation, and is therefore not considered
further here). Note that the wide-field L-band images at a resolution of
5.5\,arcsec are significantly affected by bandwidth smearing in their outer
regions, images of point sources being broadened by a factor of 2 in the radial
direction at a distance of 22\,arcmin from the phase centre \citep{ObsSS}. This
limitation needs to be taken into account only in the discussion of the source
morphology on large scales (Section~\ref{Images}). Measurements of spectra are
restricted to the inner 200\,arcsec of the field, where the effects of bandwidth
smearing are $<$3\% in peak intensity or image size for any of our
frequency/resolution combinations. Our estimates of Faraday rotation, which
extend to larger scales, should not be affected systematically by bandwidth
smearing.
\begin{center}
\begin{table}
\caption{Image resolutions and noise levels. $\sigma_I$ is the
off-source noise level on the $I$ image; $\sigma_P$ the average of
the noise levels for $Q$ and $U$. The noise levels were evaluated before
correction for the primary beam response and apply only at the centre of the
field for corrected images. Asterisks denote the images used by \citet{CLBC}
\label{noise}}
\begin{tabular}{llcc}
\hline
$\nu$ & FWHM &\multicolumn{2}{c|}{rms noise level} \\
(GHz) & (arcsec) &\multicolumn{2}{c|}{($\mu$Jy / beam area)} \\
& &$\sigma_I$&$\sigma_P$ \\
Mean L & 45 & 200 &$-$ \\
1.365 & 5.5 &41 & 35 \\
1.413 & 5.5 &38 & 34 \\
1.485 & 5.5 &37 & 33 \\
1.665 & 5.5 &37 & 30 \\
Mean L & 5.5 & 17 &$-$ \\
5 & 5.5 &15 & 12 \\
5*& 2.35 & 10 & 8 \\
1.413 & 1.5 & 36 & 36 \\
5 & 1.5 & 10 & 10 \\
5*& 0.40 & 13 & 7 \\
\hline
\end{tabular}
\end{table}
\end{center}
\section{Total intensity}
\label{Images}
A grey-scale of the large-scale radio structure of NGC\,315 at 327\,MHz
\citep{Mack97} is given in Fig.~\ref{fig:ilow}(a). Our observations are
sensitive only to emission from the region indicated by the box in this figure
and our mean L-band image of this region at a resolution of 45\,arcsec FWHM is
displayed in Fig.~\ref{fig:ilow}(b). A higher-resolution (5.5\,arcsec FWHM)
image of the same area is shown in Fig.~\ref{fig:i5.5full} and a detail of the
sharp bend in the main jet $\approx$20\,arcmin from the nucleus is plotted at
the same resolution but on a larger scale in Fig.~\ref{fig:ibend}. Finally, the
5-GHz emission from the inner 4\,arcmin of the jets is shown in
Fig.~\ref{fig:i2.35} at a resolution of 2.35\,arcsec FWHM.
We refer to the NW and SE jets as the {\em main} and {\em counter-}jets, as the
former is brighter at most distances from the nucleus. A striking feature of
the main jet is its almost constant width between $\approx$100 and
$\approx$400~arcsec from the nucleus (Fig.~\ref{fig:i5.5full}). This is the {\em
collimation shoulder} identified by \citet{Willis81}; a similar feature is
visible in the counter-jet, but cannot be traced out as far. The lack of
expansion over such an extended region is surprising if the jets are confined
solely by thermal plasma associated with the surrounding galaxy group, as a
significant pressure gradient would be expected on scales of a few hundred
arcsec (we argue in Section~\ref{RMorigin} that the core radius of the
group-scale plasma is $\approx$225~arcsec). An alternative possibility is that
the jets also respond to the ${\bf J \times B}$ forces of their own toroidal
fields on scales $\ga$100\,arcsec. In Section~\ref{Field}, we show that the
observed polarization structure is consistent with a dominant toroidal component
(see also \citealt{CLBC}), but we cannot tell from the high-frequency
synchrotron emission alone whether this component is vector-ordered or has many
reversals (evidence from Faraday rotation is also inconclusive; see
Section~\ref{RMorigin}). The possibility that both pressure confinement and
magnetic confinement could act together to produce a collimation shoulder was
discussed by \citet{BCH}, but they assumed rather different physical parameters
from those we now consider appropriate for NGC\,315. Our observations show that
the collimation shoulder in the main jet ends at a bright feature with a sharp
edge on the side towards the nucleus, inclined by $\approx$60$^\circ$ to the jet
axis. At this point (marked ``Deflection'' in Fig.~\ref{fig:i5.5full}), the flow
changes direction by $\approx$8$^\circ$ and re-expands with an opening angle
$\approx$10$^\circ$ (defined in terms of the jet FWHM and the angular distance
from the nucleus; \citealt{Willis81}). At a similar distance, the counter-jet
does not deflect significantly, its surface brightness decreases monotonically
away from the nucleus and it expands less rapidly than the main jet
\citep[Fig.~\ref{fig:ilow}b]{Willis81}.
The brightness distributions in both jets show large-scale ``banding'' --
repeated, but irregular alternation of bright and faint regions with surface
brightnesses differing by factors of 1.5 to 2 -- along their lengths on
arcminute scales (Fig.~\ref{fig:i5.5full}). The brightness bands extend across
both jets but their variations are slower than those in the flaring region or at
the edges of the jets. These variations could, in principle, result either from
periods of enhanced activity in the nucleus or from interactions between the
jets and their surroundings. If they were due to fluctuations in activity in
the nucleus that propagated outwards at constant velocity $\beta c$, then they
would appear at projected distances $D_{\rm j}$ and $D_{\rm cj}$ in the main and
counter-jets, respectively, where $D_{\rm j}/D_{\rm cj} =
(1+\beta\cos\theta)/(1-\beta\cos\theta)$ and $\theta \approx$ 38$^\circ$
\citep{CLBC}. Any transverse velocity gradients or deceleration will complicate
this expression and the former effect should distort the bands into arcs that
are concave towards the nucleus. We see no obvious relation between the
distances of the bands in the two jets for any plausible value of $\beta$ and no
evidence for systematic concave curvature of the bands beyond the flaring
region. Furthermore, the most prominent banding appears to be associated with
regions where the jets deflect or change their collimation properties. It
therefore seems more likely that the banding is associated with ongoing
interactions between the jets and their surroundings, although we cannot rule
out a contribution to large-scale brightness fluctuations from slow variations
in the jet output.
The remarkable 180$^\circ$ bend in the main jet at the West end of the source is
well known from earlier observations. Our data (Fig.~\ref{fig:ilow}b) show the
emission after the bend at a resolution comparable to the 610-MHz WSRT image
presented by \citet{Mack97}. The brightness distribution at the first part of
the bend (where the jet deflects by $\approx$100$^\circ$) shows complex
structure at 5.5-arcsec resolution. A bright ridge runs along the outside edge,
with a lane of reduced emission next to it (both features are labelled on
Fig.~\ref{fig:ibend}). Note that the suggestion that the flow is re-energised by
an intergalactic shock \citep{Ensslin01} applies to emission further downstream,
after the second bend. The compact knot of emission at the SW edge of the jet
just before the bend (Fig.~\ref{fig:ibend}) coincides in location and shape with
the optical emission from the flattened background galaxy FGC 0110 (z =
0.021965) and appears not to be physically related to NGC\,315.
The source at RA 00 57 38.710, Dec. 30 22 44.99 (J2000; labelled ``Background
source'' in Fig.~\ref{fig:i2.35}a) is unresolved and has a flat spectrum. The
polarized flux density and ${\bf E}$-vector position angle vary smoothly across
this position, consistent with addition of an unpolarized point source to the
jet emission. There is a faint optical counterpart on the Digital Sky Survey
and an X-ray point source is detected at a consistent position with the {\em
ROSAT} PSPC (\citealt{WB}; fig.~2). The source is likely to be a background
quasar, despite its location on the projected jet axis. There is a strong,
diffuse component of emission on the axis of the main jet (``On-axis
enhancement'' in Fig.~\ref{fig:i2.35}a) at $\approx$225\,arcsec from the
nucleus, with no obvious counterpart in the counter-jet.
The 2.35-arcsec resolution images illustrate the initial flaring of the jets
(discussed in detail by \citealt{CLBC} and Worrall et al., in preparation)
followed by recollimation to an almost uniform diameter
\citep{Willis81,Brid82}. Although similar behaviour is observed in other sources
\citep{LB02a,CL}, the physical scale on which NGC\,315 flares and recollimates
is unusually large. This is probably a consequence of the low external density
\citep{WBH}, and we will explore this idea quantitatively elsewhere, using the
conservation-law approach developed by \citet{LB02b}. The jets in NGC\,315 bend
slightly as they recollimate, from a position angle of $-48.5^\circ$ close to
the nucleus to $-52.8^\circ$ at distances $\ga$100\,arcsec from the nucleus.
The outer isophotes of the main and counter-jets are very similar before the
bend, but the counter-jet is slightly wider at larger distances. The main jet is
brighter than the counter-jet {\em on-axis} at all distances from the nucleus,
but the counter-jet is brighter at the {\em edge} between $\approx$100 and
200\,arcsec. In the flaring region, by contrast, the main jet is significantly
brighter than the counter-jet both on-axis and at the edge of the jet
\citep{CLBC}.
We successfully fit the structure of the inner $\pm$70\,arcsec of the jets in
NGC\,315 using an intrinsically symmetrical model in which all apparent
differences between the main and counter-jets result from relativistic
aberration and beaming \citep{CLBC}. Such models clearly cannot be continued to
indefinitely large scales, as FR\,I jets almost always show evidence (e.g.\
bends and disruption) for asymmetric interaction with the external medium. The
largest-scale structure of NGC\,315 shown in Fig.~\ref{fig:ilow}(a) is an
excellent example, with jets of unequal length terminating in entirely different
ways. This raises the question: how large is the region over which symmetrical,
relativistic models can be applied? At 400~arcsec from the nucleus, there are
clear asymmetries in both the deflection and collimation properties; these must
be intrinsic, although the overall brightness asymmetry persists. There is also
good evidence for interaction with the external medium at distances of 70 --
100\,arcsec on both sides of the nucleus, where both jets bend (symmetrically,
this time). It seems likely that intrinsic and relativistic effects become
comparable between 100 and 200~arcsec, where the sidedness difference on-axis is
the same as on small scales, but the edge value is reversed. Asymmetries in
apparent magnetic-field structure, which also imply that the flow remains mildly
relativistic in this region, are discussed in Section~\ref{Field}. Our working
hypothesis is therefore that relativistic effects dominate the observed
differences between the two jets only before the first bends, but that
environmental effects become first comparable and then dominant at larger
distances, although slightly relativistic bulk flow probably continues to the
largest scales and may remain responsible for the generally brighter appearance
of the NW jet far from the nucleus.
\section{Spectra}
\label{Spectra}
\subsection{Accuracy}
We define spectral index $\alpha$ in the sense $S(\nu) \propto \nu^{-\alpha}$.
We estimated spectral indices both for individual pixels and by integration of
flux density over well-defined regions. Values at 5.5-arcsec resolution were
determined from weighted power-law fits to data at all five frequencies between
1.365 and 5\,GHz. At 1.5-arcsec resolution, spectral indices were calculated
between 1.413 and 5\,GHz.
There are three main sources of error in the estimate of $\alpha$, as
follows.
\begin{enumerate}
\item The transfer of the amplitude scale from the primary calibrator. The
errors for the four L-band frequencies are likely to be tightly correlated,
since they were observed during the same periods, data being taken
simultaneously at 1365 and 1413\,MHz, and at 1485 and 1665\,MHz. Consequently,
the principal effect of flux-density scale transfer errors is a constant
offset in spectral index.
\item Residual deconvolution effects, typically on scales of 5 --
20\,arcsec. These are approximately proportional to surface brightness.
\item Thermal noise.
\end{enumerate}
We model the error from (ii) as 0.03 times the flux density and that from (iii)
as the noise level estimated off-source (from Table~\ref{noise}), appropriately
integrated. These two contributions are added in quadrature. In addition, we
estimate the rms spectral-index offset due to transfer errors in the
flux-density scale to be 0.02. This should be taken in addition to the errors
quoted below.
\subsection{Spectral-index images and tomography}
\label{tomography}
Figs~\ref{fig:spec}(a) and (b) show the spectral indices, $\alpha$, determined
from a weighted power-law fit to data at all five frequencies between 1.365 and
5\,GHz at 5.5\,arcsec resolution and between 1.413 and 5\,GHz at a resolution of
1.5\,arcsec, respectively.
It is clear from Fig.~\ref{fig:spec}(a) that there are transverse gradients in
spectral index where the jets are expanding rapidly. These gradients can only be
see clearly on spectral-index images where the errors are small, and for this
reason both panels of Fig.~\ref{fig:spec} are blanked where the rms error in
$\alpha$ is $>0.05$. In order to search for transverse variations over a larger
area, we need to average along the jets. The gradients between 34.5 and
69\,arcsec from the nucleus are best displayed by averaging along radii from the
nucleus and plotting the results against angle from the local jet axis, as is
shown for the main and counter-jets in
Fig.~\ref{fig:transspec_radial}(a). Further from the nucleus, where the jets
recollimate, we have averaged along the local jet axis to derive transverse
spectral-index profiles. The results are shown for two regions in each of the
main and counter-jets in Fig.~\ref{fig:transspec_long}. The fluctuations in
these regions are dominated by quasi-periodic deconvolution errors: this problem
is particularly acute for the counter-jet at $\sim$100\,arcsec from the nucleus
(Fig.~\ref{fig:transspec_long}a). The spectral index is everywhere consistent
with the mean value of $\langle \alpha \rangle = 0.47$ between 70 and
160\,arcsec.
Another method of displaying spatial variations of the spectrum is ``spectral
tomography'' \citep{K-SR,KSetal}. This involves the generation of a set of
images $I_{\rm t}({\bf r}) = I({\bf r},\nu_1) - (\nu_2/\nu_1)^{\alpha_{\rm t}}
I({\bf r},\nu_2)$ for a range of values of $\alpha_{\rm t}$, where $I({\bf
r},\nu)$ is the brightness at position ${\bf r}$ and frequency $\nu$. If the
brightness distribution can be represented as the sum of two components with
different spectral indices $I({\bf r},\nu) = S_{\rm a}({\bf r})\nu^{-\alpha_{\rm
a}} + S_{\rm b}({\bf r})\nu^{-\alpha_{\rm b}}$, then the ``a'' component will
disappear from the image $I_{\rm t}$ when $\alpha_{\rm t} = \alpha_{\rm a}$. We
made a set of images of $I_{\rm t}$ with $\nu_1 = 1.365$\,GHz, $\nu_2 = 5$\,GHz
and $\alpha_{\rm t}$ from 0.4 -- 0.7 in steps of 0.01. For $\alpha_{\rm t}
\approx 0.44$, the outer edges of both jets disappear at distances from the
nucleus between $\approx$22\,arcsec and $\approx$80\,arcsec. The image of
$I_{\rm t}$ for $\alpha_{\rm t} = 0.44$ is shown in
Fig.~\ref{fig:spec_tomo}. Further from the nucleus, $I_{\rm t}$ is close to
zero across the whole width of both jets for $\alpha_{\rm t} \approx
0.47$. There is no single value of $\alpha_{\rm t}$ for which the steep-spectrum
component vanishes completely in an image of $I_{\rm t}$.
The main features of the spectral-index distribution are as follows.
\begin{enumerate}
\item Variations in the spectral index are subtle ($0.4 \la \alpha \la
0.65$ everywhere).
\item The spectral index is slightly, but significantly steeper in the jet bases
than elsewhere. Between 7.5 and 22.5\,arcsec from the nucleus the mean values
at 5.5-arcsec resolution are 0.63 and 0.58 in the main and counter-jets,
respectively (the difference between them is not significant).
\item At 1.5 arcsec resolution (Fig.~\ref{fig:spec}b), the spectral index of the
main jet is essentially constant for the first $\approx$15\,arcsec, with a
mean spectral index $\langle\alpha\rangle = 0.61$, consistent with the value
determined at lower resolution.
\item Between $\approx$15 and 70 arcsec from the nucleus, the spectral index is
steeper on-axis ($\alpha \approx 0.5$) than at the edges ($\alpha \approx
0.44$) in both jets. This is illustrated by the transverse profiles averaged
between 34.5 and 69\,arcsec from the nucleus
(Fig.~\ref{fig:transspec_radial}a).
\item At 1.5-arcsec resolution, the on-axis spectral index is slightly higher
between 15 and 60\,arcsec from the nucleus ($\langle\alpha\rangle = 0.55$;
Fig.~\ref{fig:spec}b) than at smaller distances. $\alpha$ cannot be determined
to adequate accuracy at the edges of the jet for distances $\ga$15\,arcsec at
this resolution.
\item The tomographic analysis shows the spectral gradient in a different way:
if we subtract off a component with $\alpha_{\rm t} = 0.44$
(Fig.~\ref{fig:spec_tomo}), the emission at the edges of the jet and at large
distances from the nucleus essentially vanishes. What remains (positive in
Fig.~\ref{fig:spec_tomo}) corresponds to the jet bases and to a ridge of
steeper-spectrum emission at larger distances. The latter is clearly visible
in both jets.
\item The flatter-spectrum edge first becomes detectable at $\approx$15\,arcsec
from the nucleus and widens thereafter. It occupies the entire width of the jet
from $\approx$70\,arcsec outwards. The transition between steeper
and flatter spectrum on-axis is poorly defined.
\item There is no evidence for any transverse spectral gradient at larger
distances, after the jets recollimate, although the data are noisy and do not
cover quite the full width of the jets (Fig.~\ref{fig:transspec_long}).
\item This is confirmed by the tomographic analysis: $I_{\rm t}$ for the outer parts
of the region vanishes for $\alpha_{\rm t} = 0.47$, the mean spectral index,
confirming that $\alpha$ is constant within our errors.
\end{enumerate}
\subsection{Deprojection of the spectral-index distribution}
\citet{KSetal} and \citet{K-SR} suggested that the spectral index of an {\em
on-axis} component in a jet is the value of $\alpha_{\rm t}$ at which the
component appears to vanish against the background of the surrounding emission
(exactly as for an {\em edge} component such as that in the NGC\,315 jets;
Section~\ref{tomography}) and can therefore be derived simply from a tomographic
analysis. This requires an additional assumption which may not be correct,
namely that when the the on-axis component is subtracted, the remaining emission
has a smooth brightness distribution (this is {\em not} true for the model
proposed below). For NGC\,315, our three-dimensional model of the emissivity
\citep{CLBC} gives a good fit to the observed emission at 5\,GHz, so we isolated
the on-axis component and measured its spectral index, as follows.
\begin{enumerate}
\item We first used the tomography image with $\alpha_{\rm t} = 0.44$ as a template for
the on-axis component. To a reasonable approximation, this defines a cone with a
half-opening angle of 13.6$^\circ$ projected on the sky. Assuming an angle to
the line of sight of $\theta = 38^\circ$ \citep{CLBC},
the half-opening angle in the jet frame is $8.3^\circ$ .
\item We then made a model 5-GHz image, as in \citet{CLBC} but with the
emissivity set to zero within this cone, convolved it to a resolution of 5.5
arcsec and subtracted it from the observed 5-GHz $I$ image. The residual
corresponds to the observed emission from within the central cone.
\item We scaled the model to 1.365\,GHz assuming a spectral index of 0.44, as
appropriate for the edge emission and then subtracted it from the observed
1.365-GHz image.
\item Both residual images showed little emission towards the edges of the jets,
implying that the model subtraction was reasonably accurate.
\item Finally, we derived a spectral-index image for the on-axis component
alone. This is shown in Fig.~\ref{fig:spinespec}.
\end{enumerate}
Fig.~\ref{fig:spinespec} shows that the spectrum of the on-axis component
flattens slightly with distance from the nucleus. The mean values of $\alpha$
between 7.5 and 22.5\,arcsec are 0.60 and 0.61 for the main and counter-jets,
respectively. The corresponding values for distances between 22.5 and 66.5
arcsec are 0.56 and 0.55. For the main jet, these values are consistent with
the measurements at 1.5-arcsec resolution without subtraction. At 5.5-arcsec
resolution in both jets, they are slightly higher than the values measured
on-axis before subtraction, which include a contribution from the
flatter-spectrum, off-axis component. We conclude that the structure observed
in Fig.~\ref{fig:spec}(a) does not result entirely from the superposition of two
components with constant, but different spectral indices. The sketch in
Fig.~\ref{fig:specsketch} summarizes our results on the distribution of spectral
index in the jets.
\subsection{Comparison with other sources}
Typical spectral indices for the flaring regions of other FR\,I radio jets are
in the range 0.5 -- 0.6 (\citealt{Young} and references therein). Indeed,
\citet{Young} suggested that FR\,I jets have a ``canonical'' low-frequency
spectral index of $\alpha = 0.55$, but this conclusion is based primarily on
lower-resolution data than we consider here. Their canonical value is
intermediate between that for the jet bases in NGC\,315 and the significantly
flatter spectra seen at larger distances.
There are very few published studies of spectral {\em variations} in the flaring
regions of FR\,I jets. Short regions of slightly steeper spectrum than the
average have been detected in the bases of three other FR\,I jets: 3C\,449
\citep{K-SR}, PKS\,1333$-$33 \citep{KBE} and 3C\,31 (Laing et al., in
preparation). The measurement of $\alpha = 0.7 \pm 0.2$ in 3C\,449 applies to
the faint inner jets (corresponding to the innermost 5\,arcsec in NGC\,315), but
differs marginally from that of the brighter nearby emission, given the large
error. In PKS\,1333$-$33, the spectrum flattens from $\alpha \approx 1.0$ to
$\alpha \approx 0.6$ between 10 and 35\,arcsec from the nucleus (2.4 -- 8.4\,kpc
in projection); this includes both the faint inner jets and the bright part of
the flaring region, as in NGC\,315. In 3C\,31, there is a steeper-spectrum
region extending to $\approx$6\,arcsec in the main jet (plausibly also in the
counter-jet).
There is a flatter-spectrum edge on one side of the main jet in 3C\,31 (Laing et
al., in preparation). Transverse variations of spectral index in the sense that
the spectrum is {\em steeper} on-axis have not been reported in any other
sources, but their flaring regions are smaller in both linear and angular size
and are difficult to resolve. Gradients at the level of $\Delta\alpha \approx$
0.05 -- 0.1 are also tricky to detect without data at more than two
frequencies.
The tendency for the jet spectrum to be flatter at the edge is in the opposite
sense to the spectral gradients found on large scales in tailed radio sources
\citep[Laing et al., in preparation]{KSetal,K-SR} but the latter effect occurs
in completely different regions of the jets, where they merge into the tails.
\subsection{Acceleration mechanisms}
The X-ray emission detected by \citet{WBH} in NGC\,315 coincides with the
steeper-spectrum ($\alpha =0.61$) region at the base of the main jet. The form
of the synchrotron spectrum in those FR\,I jet bases which emit X-rays is now
well established \citep{Hard01,Hard02,Parma03,Hard05,PW05}. It can be
characterised as a broken power law with spectral indices of 0.5 -- 0.6 at radio
wavelengths and 1.2 -- 1.6 in X-rays. The spectrum of NGC\,315 is consistent
with this pattern, although its high-frequency slope is poorly constrained
\citep{WBH}. This spectral shape is consistent with synchrotron emission from a
single electron population; ongoing particle acceleration is therefore required
(see also Worrall et al., in preparation). The magnetic-field strengths
calculated for the on-axis emissivity model of \citet{CLBC}, assuming a
minimum-pressure condition, range from 3.3\,nT at 6\,arcsec in projection to
0.44\,nT at 69\,arcsec. For electrons with Lorentz factor $\gamma$ radiating at
the synchrotron critical frequency $\nu_{\rm c}$ in a magnetic field $B$, we
have $\nu_{\rm c}$/Hz = 41.99 $\gamma^2$ ($B$/nT) \citep{Longair}, so at 5\,GHz, $6
\times 10^3 \la \gamma \la 1.6 \times 10^4$ and in the X-ray band at 1\,keV, $4
\times 10^7 \la \gamma \la 1.1 \times 10^8$.
The flatter-spectrum edges occur where \citet{CLBC} infer substantial velocity
shear across the jets. If the jets are relativistic and faster on-axis than at
their edges, the approaching jet always appears more centre-brightened than the
receding one. This difference is reflected in the sidedness-ratio image. The
average transverse sidedness profile between 34.5 and 69\,arcsec from the
nucleus is shown in Fig.~\ref{fig:transspec_radial}(b) for comparison with the
spectral-index profile over the same region. The velocity profile is modelled as
a truncated Gaussian function with $\beta = 0.38$ on-axis and 0.22 at the edge,
although \citet{CLBC} note that the on-axis velocity may be larger ($\beta
\approx$ 0.5) if the shear occurs over a narrow range of radii in the jet: this
might give a better fit to the sidedness profile.
The flatter-spectrum edge first becomes visible $\approx$15\,arcsec from the
nucleus. This coincides to within the errors with:
\begin{enumerate}
\item the {\em start} of rapid deceleration, as inferred by \citet{CLBC}, 14\,arcsec in
projection from the nucleus;
\item the {\em end} of the region of enhanced radio and (in the main jet only) X-ray
emissivity \citep[Worrall et al., in preparation]{CLBC}.
\item The first point at which the observed jet/counter-jet sidedness image
gives any evidence for transverse velocity gradients, 16\,arcsec from the
nucleus \citep{CLBC}.
\end{enumerate}
The deceleration and enhanced emission regions are marked on
Fig.~\ref{fig:specsketch}. Note that the detection of transverse gradients in
sidedness and spectral index may be limited by resolution.
The changes of spectral index observed in NGC\,315 suggest that (at least) two different
electron acceleration mechanisms are required, as follows.
\begin{enumerate}
\item The first mechanism dominates at the base of the flaring region (the
initial 15\,arcsec in NGC\,315) and may continue at a lower level on the axis
of the jet to $\approx$70\,arcsec. It generates emission from radio to X-ray
wavelengths and has a characteristic spectral index $\alpha \approx 0.6$ in
the former band. Three pieces of evidence suggest that this mechanism is
dominant where the jet is fast ($\beta \approx 0.9$). Firstly, its
characteristic spectral index is observed across the whole of the jet width in
NGC\,315 until the start of rapid deceleration. Secondly, in both 3C\,31
\citep{LB04} and NGC\,315 \citep{WBH}, the bright X-ray emission occurs
upstream of the deceleration region. Finally, \citet{LB04} show from radio
data alone that significant injection of fresh relativistic particles is
required before the start of deceleration in 3C\,31 to counter-balance
adiabatic losses.
\item The second mechanism causes the flattening of the spectrum towards the
edges of the jets observed from 15\,arcsec outwards, but eventually spreading
over the entire jet width. It produces spectral indices in the range $0.44 \la
\alpha \la 0.5$ for electrons emitting at radio wavelengths in NGC\,315 and
appears to be associated with velocity shear across the jets. A possible
candidate for the this mechanism is the shear acceleration process described
by \shortcite{RD1,RD2}, but their estimates of the acceleration timescale for
electrons in conditions appropriate to FR\,I jets are very long (at least if
the mean free path $\sim$ gyro-radius), and it is unclear whether the process
is efficient enough to influence the spectrum.
\end{enumerate}
\section{Faraday rotation and depolarization}
\label{Faraday}
\subsection{Faraday rotation}
\label{RM}
In order to investigate the variations of Faraday rotation along the jets of
NGC\,315, we made images of rotation measure (RM) at a resolution of 5.5\,arcsec
by least-squares fitting to the relation $\chi(\lambda^2) = \chi(0) + {\rm
RM}\lambda^2$ (where $\chi$ is the {\bf E}-vector position angle) for all 5
frequencies between 1.365 and 5\,GHz. The fits were weighted by errors in $\chi$
derived from Table~\ref{noise}. We excluded a small region around the core which
was affected by residual instrumental polarization and included only points
where the rms error in position angle was $<$15$^\circ$ at all frequencies. The
resulting RM image is shown in Fig.~\ref{fig:RMimages}(a). The fit to a
$\lambda^2$ law is very good everywhere: two examples are shown in
Fig.~\ref{fig:rmegs}. The extreme value of RM $\approx -90$\,rad\,m$^{-2}$
results in rotations of $\approx1^\circ$ between the two centre frequencies
combined in our 5\,GHz dataset and $\approx$0.4$^\circ$ and $\approx$5$^\circ$
across the bands at 5 and 1.4\,GHz, respectively. The worst of these effects,
rotation across the band at the lowest frequencies, results in a spurious
depolarization $<$0.1\%, which is negligible compared with errors due
to noise. The images of $Q$ and $U$ show little power on large spatial scales
and our estimates of position angle should be reliable over the full range of
distances from the nucleus shown in Fig.~\ref{fig:RMimages}(a). The area over
which the RM can be determined accurately is limited primarily by the primary
beam at 5\,GHz (540\,arcsec FWHM).
The mean RM is $-$75.7\,rad\,m$^{-2}$. The most obvious feature of the RM image
is a nearly linear gradient along the jets, as shown by the profile in
Fig.~\ref{fig:RMprofiles}(a). In order to reveal smaller-scale structure in the
RM, we initially fitted and subtracted a function ${\rm RM} = {\rm RM}_0 + a_x
x$, where $x$ is measured along the axis from the nucleus ($a_x$ and ${\rm
RM}_0$ are constants). We used an unweighted least-squares fit, as any attempt
to weight by the estimated errors caused the brightest part of the main jet to
be fitted at the expense of other regions. The gradient is $a_x = $
0.025\,rad\,m$^{-2}$\,arcsec$^{-1}$ and the residual image is shown in
Fig.~\ref{fig:RMprofiles}(b). This indicates that variations in RM across the
jet are also significant, as can be seen more clearly in averaged profiles,
particularly between 69 and 112.5\,arcsec from the nucleus
(Fig.~\ref{fig:RMtrans}b). Closer to the nucleus, the gradient is barely visible
(Fig.~\ref{fig:RMtrans}a), but the jets are narrower there and the profiles are
consistent with the gradient measured at larger distances. The transverse
variation also appears to be linear, so we fitted and subtracted a function
${\rm RM} = {\rm RM}_0 + a_x x + a_y y$, where $y$ is a coordinate transverse to
the jet and $a_y$ is a constant. The gradients along and transverse to the jet
axis become $a_x$ = 0.018\,rad\,m$^{-2}$\,arcsec$^{-1}$ and $a_y$ =
0.051\,rad\,m$^{-2}$\,arcsec$^{-1}$, respectively. Taken at face value, the
best-fitting gradient is 0.054\,rad\,m$^{-2}$\,arcsec$^{-1}$ at an angle of
72$^\circ$ to the mean jet axis. Note, however, that the transverse gradient is
essentially determined by a subset of the data from the widest parts of the main
and counter-jets (Fig.~\ref{fig:RMtrans}b).
Removal of the large-scale gradient leaves fluctuations in the local mean RM
which appear significantly larger on the counter-jet side
(Fig.~\ref{fig:RMimages}c). By definition, the signal-to-noise ratio in $I$ is
lower in the counter-jet. Although this is partially offset by a higher average
degree of polarization, the errors in RM are still larger than in the main
jet. To evaluate the significance of the fluctuations, we considered only points
with fitting errors $\leq$2.5\,rad\,m$^{-2}$ and calculated the expected errors
in the means for boxes of length 30\,arcsec along the jet axis containing more
than 50 such points (the errors are corrected for oversampling). The results
are shown in Fig.~\ref{fig:RMprofiles}(b). The fluctuations are significant, and
form an ordered pattern with a typical scale $\sim$100\,arcsec. They are larger
by a factor of $\approx$2 in the counter-jet and the first bin of the main jet
(within 30 arcsec of the nucleus) compared with the rest of the main jet.
The residual fluctuations within the boxes (i.e.\ after subtracting the local
mean) are comparable with the errors in RM except in the brightest regions close
to the nucleus. We therefore made a first-order correction to the rms RM,
$\sigma_{\rm RM raw}$, by subtracting the fitting error $\sigma_{\rm fit}$ in
quadrature to give $\sigma_{\rm RM} = (\sigma_{\rm RM raw}^2 - \sigma_{\rm
fit}^2)^{1/2}$. The profiles of $\sigma_{\rm RM raw}$ and $\sigma_{\rm RM}$ are
both plotted in Fig.~\ref{fig:RMprofiles}(c), for the same selection of points
as in Fig.~\ref{fig:RMprofiles}(b). The corrected profile is very uncertain,
but suggests that $\sigma_{\rm RM}$ has a maximum of 2\,rad\,m$^{-2}$ close to
the nucleus on the counter-jet side and may be slightly asymmetric in the sense
that the rms RM is lower in the main jet than the counter-jet at the same
distance from the nucleus.
We can image these smaller-scale fluctuations directly only at the bright base
of the main jet. There, the RM at a resolution of 1.5\,arcsec FWHM can be
derived accurately from the difference between 1.413 and 5\,GHz position-angle
images, using the lower-resolution data to resolve the $n\pi$ ambiguity.
Fig.~\ref{fig:RMimages}(d) shows the RM at this resolution. Data are
plotted only where the rms error in the fitted RM
$<$2\,rad\,m$^{-2}$. Fluctuations are clearly detected: the rms is $\sigma_{\rm
RM raw}$ = 2.1\,rad\,m$^{-2}$, giving $\sigma_{\rm RM} =$ 1.6\,rad\,m$^{-2}$
after making a first-order correction for fitting error, as above. This is in
good agreement with the value for the innermost bin of the profile of rms RM for
the main jet at lower resolution (Fig.~\ref{fig:RMprofiles}c).
\subsection{Depolarization}
\label{Depol}
The variation of $p$ with wavelength at low resolution potentially measures
fluctuations of RM across the observing beam which cannot
be imaged directly with adequate sensitivity. This
variation is small, and is best quantified by fitting to the first-order
approximation $p(\lambda^2) \approx p(0) +p^\prime(0)\lambda^2$ where
$p^\prime(\lambda^2) = dp/d(\lambda^2)$. We expect $p^\prime(0) < 0$
(depolarization) under most circumstances. The quantity $p^\prime(0)/p(0)$ is
directly related to the commonly quoted depolarization ratio, but is biased in
the present case, since both the gradient and the degree of polarization depend
directly on a single high-frequency measurement (at 5\,GHz), so deviations in $p(0)$
and $p^\prime(0)$ are anticorrelated. We therefore tested for the presence of
depolarization using the gradient $p^\prime(0)$ alone. We derived $p^\prime(0)$
by weighted least-squares fitting to images of $p$ at the 5 frequencies between
1.365 and 5\,GHz at a resolution of 5.5\,arcsec FWHM. Two sets of $p$ images
were used: in the first, an estimate of the local zero-level was subtracted from
the $I$ images before calculating $p = P/I$; in the second, the original $I$
images were used (any differences indicate systematic errors in the estimation
of large-scale structure). The fitting weights were the inverse squares of
errors in $p$ derived from the values in Table~\ref{noise} and points were only
included if the errors were $<$0.3 at all frequencies. The resulting
polarization gradients are very small, and there are no obvious variations. In
order to determine the significance of the gradients, we measured their mean
values over the main and counter-jets. The maximum scale of structure imaged
accurately in total intensity is $\approx$300\,arcsec (Table~\ref{Datasets}), so
we calculated the means for points between 9 and 150\,arcsec from the nucleus
along the jet axis (excluding the small region around the core to avoid spurious
instrumental polarization, as in Section~\ref{RM}). The mean values for the
main and counter-jets derived from the zero-level corrected images were $\langle
p^\prime(0)\rangle = 0.03 \pm 0.08$ and $0.16 \pm 0.16$, respectively. Without
zero-level correction, the values become $0.05 \pm 0.08$ and $0.22 \pm 0.16$.
We conclude that the increase of $p$ with wavelength for $\lambda \leq 0.22$\,m
($\nu \geq 1.365$\,GHz), which is in any case opposite to the expected effects
of Faraday rotation, is not significant.
\subsection{The origin of the rotation measure}
\label{RMorigin}
NGC\,315 has Galactic coordinates $l = 124.^\circ6$, $b = -32.^\circ5$. This is
on the outskirts of Region A of \citet{SK}, where the majority of sources have
large negative RM's ($\sim -100$\,rad\,m$^{-2}$ at the centre of the
region). \citet{DK05} derived spherical-harmonic models of the Galactic RM
distribution by fitting to RM's of large numbers of extragalactic sources. These
models predict a Galactic contribution $\approx
-47$\,rad\,m$^{-2}$ at the position of NGC\,315. The bulk of the mean RM of
$-$75.7\,rad\,m$^{-2}$ is therefore likely to be Galactic in origin.
\citet{SC86} determined RM variations, which they argued to be primarily
Galactic, across a number of sources within Region A (their Region 1). Their
plot of squared RM difference $\Delta{\rm RM}^2$ against separation suggests
$\langle \Delta{\rm RM}^2\rangle^{1/2} \sim$ 10 -- 30\,rad\,m$^{-2}$ on a scale
of 700\,arcsec, but with large uncertainties. The observed $\Delta{\rm RM}$
($\approx 15$\,rad\,m$^{-2}$ on the same scale along the jets of NGC\,315;
Fig.~\ref{fig:RMprofiles}a) is again consistent with a Galactic origin. The
maximum change in RM across the jet ($\approx 4$\,rad\,m$^{-2}$ over 80\,arcsec;
Fig.~\ref{fig:RMtrans}b) is within the range of the upper limits plotted by
\citet{SC86}. We note, however, that NGC\,315 is located outside the core of
Region A, so it may be that smaller values of $\langle \Delta{\rm
RM}^2\rangle^{1/2}$ are appropriate.
The linearity of the RM gradient along the jet and the fact that the maximum
gradient is aligned neither with the jet axis nor with the minor axis of the
galaxy both imply that little of the Faraday rotating medium is associated with
the jets or with the host galaxy. In either case we would expect some
non-linear variation with distance from the nucleus.
We conclude that most of the mean RM and its linear gradient are likely to be
Galactic in origin, but a significant contribution from material local to
NGC\,315 is not ruled out. In particular, we cannot exclude the hypothesis that
some of the RM gradient transverse to the jet results from an ordered toroidal
field within or just outside the jet \citep{L81,Blandford93}. The associated
position angle rotation between 1.365 and 5\,GHz is at most a few degrees and
significant departures from $\lambda^2$ rotation would not be detectable even if
the thermal plasma responsible for the Faraday rotation is mixed with the
synchrotron-emitting material \citep{Burn66}. If local toroidal fields are
solely responsible for the transverse RM gradients, then their vector directions
must be the same in the main and counter-jets.
The {\em residual} RM fluctuations are qualitatively very similar to those in
3C\,31 (Laing et al., in preparation) but have amplitudes that are 10 times
smaller on similar angular scales. The larger-scale ($\sim$100\,arcsec)
fluctuations are systematically lower on the main (approaching) jet side. The
distribution of fluctuations on smaller scales is also consistent with such an
asymmetry, but is not well determined. As with the transverse gradients
discussed earlier, the observed position-angle rotations are too small to be
sure that the RM fluctuations are due to foreground plasma, but the asymmetry
between approaching and receding jets suggests an origin in a distributed
magnetoionic medium surrounding the host galaxy, a possibility we now explore.
The thermal plasma associated with NGC\,315 and observed using {\em Chandra} can
be described by a beta model with a core radius of 1.55\,arcsec
\citep{WBH}. This cannot be responsible for the RM fluctuations, which occur on
far larger scales. The most likely hot plasma component to be responsible for
the Faraday rotation would be associated with the poor group of galaxies
surrounding NGC\,315 \citep{Nolthenius,Miller02}, but has not yet been detected
in X-ray observations. Since RM fluctuations are seen in both jets, a spherical
distribution is plausible. In order to make a rough estimate of the parameters
of the putative group component, we took a simple model for the field structure
in which cells of fixed size $l$ at radius $r$ contain randomly orientated
fields $B(r)$ \citep{LD82,Felten96}. The density distribution was taken to be a
beta model: $n(r) = n_0 (1+r^2/r^2_c)^{-3\beta_{\rm atm}/2}$ with $B(r) \propto
n(r)^N$. We derived the variation of $\sigma^2_{\rm RM}$ along the projection
of the jet axis by numerical integration, assuming that the jets have $\theta =
37.9^\circ$ everywhere. This is a similar approach to the calculation of
depolarization asymmetry by \citet{GC91} and \citet{Tribble92}; our code also
reproduces the analytical results of \citet{Felten96} for $\theta = 90^\circ$
and $N = 0$ or 0.5. Following \shortcite{Dolag01,Dolag06}, we assumed that
$B(r) \propto n(r)$ or $N = 1$ (an empirical result derived from RM's of radio
sources in and behind cluster and rich groups). Our detection of RM variations
on a range of scales is qualitatively consistent with the idea that the power
spectrum of the magnetic-field fluctuations is a power law
\citep{Tribble91,EV03,Murgia,VE05}, but our data are too noisy and poorly
sampled to constrain its functional form. \citet{Murgia} show that a
single-scale model gives a very similar relation between the RM variance
$\sigma^2_{\rm RM}$ and radius $r$ to that derived for the more realistic case
of a power-law power spectrum provided that $l$ is interpreted as the
correlation length of the magnetic field. Finally, we fitted the resulting
$\sigma_{\rm RM}$ curves by eye to the profiles of RM fluctuations on different
scales in Fig.~\ref{fig:RMprofiles}(b) and (c). We fixed the value of
$\beta_{\rm atm} = 0.5$ and adjusted the core radius to give a reasonable fit to
the profile. For both plots, we found that $r_c \approx 225$\,arcsec gave an
adequate fit. Strictly speaking, $\sigma^2_{\rm RM}$ is the RM variance
evaluated over a window much larger than the maximum fluctuation scale, whereas
the profiles in Fig.~\ref{fig:RMprofiles} describe fluctuations over two
different ranges of scale. We have therefore added the two model variances to
give a rough estimate of the total (the curves shown in
Fig.~\ref{fig:RMprofiles}b and c actually have the same amplitude). Note that
the process of removing a linear trend from the RM profile will have suppressed
some power in large-scale fluctuations, particularly transverse to the jet. The
amplitude of the model variance profile is related to the central density and
field, and to the correlation length \citep{Felten96}: $(n_0/{\rm m}^{-3})^2
(B_0/{\rm nT})^2 (l/{\rm kpc}) \approx 700$. This is a very rough estimate, but
is enough to establish that a low-density group-scale gas component with a low
magnetic field can generate the observed Faraday rotation. Plausible parameters
might be $n_0 \approx$ 600\,m$^{-3}$, $B_0 \approx$ 0.015\,nT (0.15\,$\mu$Gauss)
and $l \approx$ 10\,kpc. We note that the recollimation of the jets may also
require a large-scale hot gas component.
\section{Magnetic-field structure}
\label{Field}
Vectors whose magnitudes are proportional to $p$ at zero wavelength and whose directions
are those of the apparent magnetic field inferred from the rotation-measure fit
of Section~\ref{RM} are plotted in Fig.~\ref{fig:ivec5.5}. At higher resolution,
we derived the apparent field direction by interpolating the RM image onto a
finer grid and using it to correct the observed 5-GHz position angles
(Fig.~\ref{fig:ivec2.35}). The apparent field structure in the flaring region
(up to $\approx$70 arcsec from the nucleus) is
discussed extensively by \citet{CLBC}. Here, we concentrate on larger scales,
after the jets recollimate.
The signal-to-noise ratio for individual points, particularly at the jet edges,
is often $<$3 in linear polarization, causing them to be blanked in
Figs~\ref{fig:ivec5.5} and \ref{fig:ivec2.35}. In particular, it is impossible
to see whether the parallel-field edge continues to large distances in the main
jet. We therefore adopted the following procedure to derive the average degree of
polarization.
\begin{enumerate}
\item We first corrected the observed 5-GHz, 2.35-arcsec position angles for
Faraday rotation using the linear model derived in Section~\ref{RM}, which is
defined everywhere in the field, unlike the RM image.
\item We then changed the origin of position angle to be along the jet axis, so
that apparent field along or orthogonal to the jet appears entirely in the $Q$
Stokes parameter (also verifying that there is very little signal in $U$).
\item We then integrated $Q$ and $I$ along the jet axis from 69 -- 113 and from 113.5
-- 157.5 arcsec from the nucleus (the same areas used for the profiles of
spectral index in Fig.~\ref{fig:transspec_long}). Lack of short spacings at
5\,GHz precludes extension of this analysis to larger distances.
\item Finally, we divided the results to give transverse profiles of
$Q/I$. Provided that the apparent field is either along or orthogonal to the
jet axis, $p = |Q/I|$. We have chosen the sign convention so that $Q > 0$ for
transverse apparent field and $< 0$ for longitudinal field.
\end{enumerate}
The resulting profiles are shown in Fig.~\ref{fig:QIprof}.
Figs~\ref{fig:ivec5.5} -- \ref{fig:QIprof} show that the apparent field
configuration found in the flaring region -- transverse on-axis and longitudinal
at the edges -- persists until at least 160\,arcsec in both jets. In particular,
the longitudinal-field edge of the main jet is easily detected in the profiles,
even though it is not clearly visible on the images. The main difference from
the corresponding profiles for the flaring region (fig.~9 of \citealt{CLBC}) is
that the on-axis polarization is higher in both jets at larger distances. This
is a continuation of the trend in the longitudinal profile shown by \citet[their
fig.~8]{CLBC}. As in the flaring region, the on-axis (perpendicular)
polarization is always higher in the counter-jet, reaching levels close to the
theoretical maximum of $p_0 = 0.69$ for the observed spectral index. In the main
jet, $p \approx 0.4$ on-axis. Both jets are very highly polarized at their
edges.
\citet{CLBC} modelled the three-dimensional structure of the field in the outer
parts of the flaring region as a mixture of toroidal and longitudinal components
of roughly equal magnitude on-axis but with the former dominant at the edge of
the jet. The differences between the two jets are attributed to relativistic
aberration, so the fact that they persist after the jets recollimate suggests
that there is little further deceleration in this region, despite the reversal
in sidedness at the edges of the jets (Section~\ref{Images}). A decrease in the
on-axis longitudinal field component, bringing the configuration closer to a
purely toroidal one, would result in a polarization distribution consistent with
that observed. Some longitudinal component must remain, however, otherwise $p$
would be close to $p_0$ on the axis for both jets. The very high degree of
polarization at the jet edge requires the radial field component to be very
small, as inferred for the flaring region by \cite{CLBC}.
\section{Summary}
\label{Summary}
We have imaged the jets in the nearby FR\,I radio galaxy NGC\,315 with the VLA
at five frequencies in the range 1.365 -- 5\,GHz and at resolutions ranging from
45 -- 1.5\,arcsec FWHM. Our total intensity observations reveal new details of
the structure, particularly around the sharp bend in the main jet.
The flaring regions of both jets, where they initially expand rapidly and then
recollimate, show a complex and previously unknown spectral structure. Within
15\,arcsec of the nucleus, the spectral index has a uniform value of $\alpha =$
0.61 in both jets. This region is associated with strong X-ray emission in the
main jet, high radio emissivity, complex filamentary structure, and fast flow
with $\beta \approx 0.9$ \citep{CLBC,WBH}. Between 15 and 70\,arcsec, the
spectrum is steeper on-axis than at the edges of the jet. We have developed a
novel deprojection technique which allows us to isolate two spectral
components. The first (on-axis) forms a continuation of the jet base, its
spectral index flattening gradually from 0.61 to 0.55. The second (at the edge
of the jet) has $\alpha \approx$ 0.44 and is associated with a region where
strong shear is inferred \citep{CLBC}. We speculate that two different
acceleration mechanisms are involved, one associated with fast flow, dominant
close to the nucleus and capable of accelerating electrons to the very high
energies required to produce X-ray emission ($\gamma \sim 10^8$), the other
being driven by shear and generating the flatter spectral indices seen at the
edges of the jet. Both mechanisms must efficiently generate the electrons with
$\gamma \sim 10^4$ which radiate at cm wavelengths. At distances
$\ga$70\,arcsec, the spectral index is consistent with $\alpha \approx 0.47$
everywhere.
We have imaged the variations of Faraday rotation over the jets. All of the
rotation is resolved and must originate mostly in foreground material. There is
no detectable depolarization. The largest contributions -- a constant term and a
linear gradient -- are probably Galactic in origin. We have also detected
residual fluctuations of $\approx$1 -- 2 rad\,m$^{-2}$ rms on scales $\sim$ 5 --
100\,arcsec. The amplitude of fluctuations on scales $\ga$30\,arcsec is larger
by a factor $\approx$2 for the counter-jet, consistent with an origin in
magnetoionic material around the source, but not in the known X-ray-emitting
halo, whose core radius is too small. We model the Faraday-rotating medium as a
spherical halo with a core radius $\approx$225\,arcsec and derive an approximate
value for the product $(n_0/{\rm m}^{-3})^2 (B_0/{\rm nT})^2 (l/{\rm kpc})
\approx 700$, where $n_0$ is the central density, $B_0$ the central magnetic
field and $l$ is the magnetic-field correlation length. Our analysis is
therefore consistent with models of Faraday rotation proposed for rich clusters
(e.g.\ \citealt{CT}), but requires much lower densities and field strengths. We
predict that a tenuous, group-scale halo should be detectable in sensitive X-ray
observations; measurement of its density will allow us to estimate the
magnetic-field strength.
We have derived the apparent magnetic field direction (corrected for Faraday
rotation) and degree of polarization at distances between 70 and 160\,arcsec
from the nucleus. The structure is qualitatively similar to that seen in the
flaring region, with transverse field on-axis and longitudinal field at the
edges of both jets, but the degree of polarization on-axis is larger. The
difference in polarization structure between the main and counter-jets observed
in the flaring region by \cite{CLBC} persists at larger distances. This can be
explained fully as an effect of differential aberration on radiation from
intrinsically identical jets, as long as their velocities remain significantly
relativistic on the relevant scales. The asymmetry in RM fluctuation amplitude is
consistent with the jet orientation required by this analysis and the presence
of a tenuous, magnetized group halo.
The large angular size of the flaring region in NGC\,315 and our use of deep
observations at several frequencies has allowed us to image spectral variations
at a level of detail not yet achieved in any other jet. Taken together with
X-ray imaging and modelling of the jet velocity field, this has given important
insights into the particle acceleration mechanisms. It will be interesting to see
whether our results apply to other FR\,I jets and to study the spectral
variations in NGC\,315 over a wider frequency range.
\section*{Acknowledgments}
JRC acknowledges a research studentship from the UK Particle Physics and
Astronomy Research Council (PPARC). The National Radio Astronomy Observatory is
a facility of the National Science Foundation operated under cooperative
agreement by Associated Universities, Inc. We thank Greg Taylor for the use of
his rotation-measure code, Karl-Heinz Mack for providing the 327-MHz WSRT image,
Frank Rieger for discussions on shear acceleration and the referee for helpful
comments. We also acknowledge the use of the {\sc healpix} package ({\tt
http://healpix.jpl.nasa.gov}) and the provision of the models of \citet{DK05} in
{\sc healpix} format.
\label{lastpage} |
Title:
A VLT-UVES spectrscopic analysis of C-rich Fe-poor stars |
Abstract: Large surveys of very metal-poor stars have revealed in recent years that a
large fraction of these objects were carbon-rich, analogous to the more
metal-rich CH-stars. The abundance peculiarities of CH-stars are commonly
explained by mass-transfer from a more evolved companion. In an effort to
better understand the origin and importance for Galactic evolution of Fe-poor,
C-rich stars, we present abundances determined from high-resolution and high
signal-to-noise spectra obtained with the UVES instrument attached to the
ESO/VLT. Our analysis of carbon-enhanced objects includes both CH stars and
more metal-poor objects, and we explore the link between the two classes. We
also present preliminary results of our ongoing radial velocity monitoring.
| https://export.arxiv.org/pdf/astro-ph/0601253 |
\title{A VLT/UVES spectroscopic analysis of C-rich Fe-poor stars}
\author{T. Masseron\inst{1,2}, B. Plez\inst{2}, F. Primas\inst{1},
S. Van Eck\inst{3}, \and A. Jorissen\inst{3}}
\institute{ESO, Garching, Germany,
\and GRAAL, Universit\'e Montpellier II, France,
\and Institut d'Astronomie et d'Astrophysique,
Universit\'e Libre de Bruxelles, Belgium}
\section{Introduction}
A subgroup of carbon stars, the CH stars, were first
distinguished
from other carbon stars 60 years ago (\cite{keenan42}).
Their chief characteristics
are : strong CH absorption lines, strong Swan bands
of C$_2$, strong resonance lines of Ba~II and Sr~II
(later shown to reflect genuine overabundances of
s-process elements),
weak lines of the iron-group elements and high proper motion.
Since it was shown that all field CH stars
were spectroscopic binaries (\cite{mcclure90}),
the binary scenario was accepted: CH stars have been
polluted by a nearby
companion, formerly on the AGB, now a defunct white dwarf.
But some recent results have cast doubts on this
general picture.
Conflicting evidences include the radial velocity
monitoring of targets from the HK survey (\cite{beers92})
which show no radial velocity variations (\cite{preston01}),
and the discovery of carbon but not s-process enhanced stars
(e.g. \cite{sneden96})
. Furthermore, the unexpectedly high fraction of
carbon-enhanced stars
among metal-poor stars discovered in the HK survey
(about 25\% in the metallicity range [Fe/H] $\le$ -2.5 compared
to 1-2 \% among stars of higher metal abundances), make
them crucial for the study of the early stages of Galaxy evolution.
\begin{table}
\caption{The sample and the atmospheric parameters.}
\label{table}
\begin{center}
\leavevmode
\footnotesize
\begin{tabular}{l|c|c|c|c}
\hline
star &T$_{\rm eff}$ & log~g & [Fe/H] & $^{12}$C/$^{13}$C \\
\hline
CS 22891-171 & 5100 & 1.8 & -2.31 & 5 \\
CS 22942-019 & 5100 & 2.5 & -2.48 & 7 \\
CS 22945-017* & 6400 & 4.3 & -2.60 & 4 \\
CS 22953-003 & 4600 & 1.0 & -3.27 & $>$5 \\
CS 22956-028* & 6700 & 3.5 & -2.38 & 3 \\
CS 30322-023 & 4000 & 0.0 & -3.78 & 5 \\
HD 26* & 5200 & 2.6 & -0.71 & ? \\
HD 5424* & 5000 & 3.0 & -0.41 & 4 \\
HD 24035* & 5000 & 3.7 & 0.16 & 4 \\
HD 168214 & 5200 & 3.5 & -0.03 & ? \\
HD 187861* & 4800 & 1.8 & -2.21 & 10 \\
HD 196944* & 5250 & 1.7 & -2.21 & ? \\
HD 206983 & 4550 & 1.6 & -0.93 & 4 \\
HD 207585* & 5800 & 4.0 & -0.35 & 10 \\
HD 211173 & 4800 & 2.5 & -0.17 & 13 \\
HD 218875 & 4600 & 1.5 & -0.63 & 100 \\
HD 219116 & 4800 & 1.8 & -0.45 & 7 \\
HD 224959* & 5000 & 2.0 & -2.08 & 7 \\
HE 1419-1324 & 4900 & 1.8 & -3.28 & 4 \\
HE 1410+0213* & 4550 & 1.0 & -2.38 & 4 \\
HE 1001-0243* & 5000 & 2.3 & -3.12 & 30 \\
\hline
\multicolumn{5}{l}{* probably binary, from radial velocity monitoring}\\
\end{tabular}
\end{center}
\end{table}
\section{Observations and abundance determinations}
We have started a new extensive high-resolution analysis of
carbon-enhanced stars, based on a sample including at the
moment 86 stars, aimed at investigating the C-enrichment
phenomenon over a wide range of metallicities, chosen in the
Barktevicius catalog (1996), in the HK survey and in the
Hamburg/ESO survey (\cite{christlieb01}). Observations
were made with the VLT/UVES echelle spectrograph
at high S/N and spectral resolution, as well as with the
ESO 1m52/FEROS spectrograph. A few radial velocity observations were made
with the OHP 1m93 ELODIE instrument.\\
In Table~\ref{table}, we present a subset of our sample for
which we have derived C, N (from CN and CH bands), O
(from [O~I] 630nm, when possible),
and Ba and Eu abundances, from VLT observations.\\
Effective temperatures were initially estimated from photometry
available in the literature, using the
calibration of \cite{alonso99}. However, due to strong \
absorption by molecular CH and CN bands impacting the photometry
by an unknown amount, T$_{\rm eff}$ were instead
derived by forcing the abundance determined
from individual Fe I lines to show no dependence on excitation
potential. The gravity was determined
from the ionization equilibrium of Fe~I and Fe~II. The microturbulent
velocity was set by requiring
no trend of Fe~I abundance with equivalent width.
The observed spectra are compared to synthetic ones, computed
with the "turbospectrum" package
(\cite{alvaplez98}). This program uses OSMARCS atmosphere
models, initially developed by \cite{gustafsson75} and
later improved by \cite{plez92}; see \cite{gustafsson03}
for details on recent improvements. \\
The line lists are the same as used by \cite{hill02} for
atoms and for CH, C$_2$ and CN, and their various isotopes.\\
\section{Results}
Figures~\ref{figcarbon}, \ref{figcarbonnitrogen}, \ref{fignitrogen},
and \ref{figeuba}
present the derived abundances.
Carbon is indeed enhanced in the atmospheres of most of our
metal-poor targets (Fig.~\ref{figcarbon}).
Excluding the 2 stars around [Fe/H] $= -3.5$ that do not show
a large C enhancement, the average carbon abundance is almost
constant from the CH stars ([Fe/H] $> -2.0$)
to the very metal-poor carbon-enhanced stars
(from [C/H]$\approx 0$ at solar Fe/H to [C/H]$\approx -0.5$
at [Fe/H]$\approx -3$).
The lowest metallicity star HE 0107-5240 ([Fe/H] $=-5.3$, recently
discovered by \cite{christlieb02}), shows a carbon abundance
close to the solar value. This extreme star follows the trend of
Fig.~\ref{figcarbon}.
The nitrogen abundance, when combined to the carbon abundance
and the carbon isotopic ratio, provides additional information.
Figure~\ref{figcarbonnitrogen} shows the sum of C and N overabundance
as a function of Fe/H. Some of the stars that did not show a large C/Fe
do show a large (C+N)/Fe, as the most metal-poor star of our sample.
The $^{12}$C/$^{13}$C ratio is generally low (see Table~\ref{table}),
often close to the
CN-cycle equilibrium value.
There is a general anticorrelation between the N overabundance and
$^{12}$C/$^{13}$C (Fig.~\ref{fignitrogen}), characteristic
of the operation of the CN cycle.
Note that the stars of our sample with low $^{12}$C/$^{13}$C
and still on the main
sequence (cf. log~g values in Table~\ref{table}) are binaries,
supporting the mass-transfer scenario.
The carbon and nitrogen enhancements are large, and whether
the CN cycling happened in the stars we observe when they evolved
to the giant stage, or in the companions that polluted them,
remains to be determined. The combined (C+N)/Fe overabundance is
much larger at lower metallicity, pointing towards a primary origin of
these elements.
\cite{asplund03} warns for 3D effects that may
lead to overestimates of abundances derived from molecular lines
in metal-poor stars, but we doubt that the trend could be
erased by NLTE and 3D effects.\\
Ba is an s-process element and, as for carbon, its overabundance
is explained by
mass-transfer from a now extinct AGB star. The r-process element Eu is
believed to originate from
supernovae, although the r-process site(s) is still debated.
Figure~\ref{figeuba} shows [Eu/Fe] vs. [Ba/Fe] for the stars of
Table~\ref{table}.
In addition
to a few stars that are very
enriched in both r- and s-process elements, two groups emerge:
one enriched in Eu and not in
Ba, and a large group of stars with [Ba/Fe] around +1, and no
or little Eu enhancement.
This latter group encompasses the solar metallicity Ba stars
present in our sample.\\
The carbon-rich, very iron-poor stars are of various origins:
(i) mass-transfer binaries polluted by
an AGB or a more massive star,
(ii) single stars, some enriched in s-process elements, other
in r-process elements, some maybe in both.
We are pursuing our
analysis of more stars
in our sample, and of more chemical elements, in order to
provide useful constraints
on their origin, and on the early chemical evolution of the Galaxy.
|
Title:
Proper Motions of Dwarf Spheroidal Galaxies from Hubble Space Telescope Imaging. IV: Measurement for Sculptor |
Abstract: This article presents a measurement of the proper motion of the Sculptor
dwarf spheroidal galaxy determined from images taken with the Hubble Space
Telescope using the Space Telescope Imaging Spectrograph in the imaging mode.
| https://export.arxiv.org/pdf/astro-ph/0601547 |
\setcounter{figure}{0}
\title{Proper Motions of Dwarf Spheroidal
Galaxies from \textit{Hubble Space Telescope} Imaging. IV:
Measurement for Sculptor.\footnote{Based on observations with NASA/ESA
\textit{Hubble Space Telescope}, obtained at the Space Telescope
Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract NAS
5-26555.}}
\author{Slawomir Piatek} \affil{Dept. of Physics, New Jersey Institute
of Technology,
Newark, NJ 07102 \\ E-mail address: piatek@physics.rutgers.edu}
\author{Carlton Pryor}
\affil{Dept. of Physics and Astronomy, Rutgers, the State University
of New Jersey, 136~Frelinghuysen Rd., Piscataway, NJ 08854--8019 \\
E-mail address: pryor@physics.rutgers.edu}
\author{Paul Bristow}
\affil{Space Telescope European Co-ordinating Facility,
Karl-Schwarzschild-Str. 2, D-85748, Garching bei Munchen, Germany \\
E-mail address: bristowp@eso.org }
\author{Edward W.\ Olszewski}
\affil{Steward Observatory, The University of Arizona,
Tucson, AZ 85721 \\ E-mail address: eolszewski@as.arizona.edu}
\author{Hugh C.\ Harris}
\affil{US Naval Observatory, Flagstaff Station, P. O. Box 1149,
Flagstaff, AZ 86002-1149 \\ E-mail address: hch@nofs.navy.mil}
\author{Mario Mateo} \affil{Dept. of Astronomy, University of
Michigan, 830 Denninson Building, Ann Arbor, MI 48109-1090 \\
E-mail address: mateo@astro.lsa.umich.edu}
\author{Dante Minniti}
\affil{Universidad Catolica de Chile, Department of Astronomy and
Astrophysics, Casilla 306, Santiago 22, Chile \\
E-mail address: dante@astro.puc.cl}
\author{Christopher G.\ Tinney}
\affil{Anglo-Australian Observatory, PO Box 296, Epping, 1710,
Australia \\ E-mail address: cgt@aaoepp.aao.gov.au}
\keywords{galaxies: dwarf spheroidal --- galaxies: individual (Sculptor) ---
astrometry: proper motion}
\section{Introduction}
\label{sec:intro}
Shapley (1938) discovered the Sculptor dwarf spheroidal (dSph)
galaxy --- the first example of this type of galaxy in the vicinity of
the Milky Way --- on a plate with a 3~hour exposure time taken with the
Bruce telescope. Shapely notes ``... that systems such as the Sculptor
cluster may not be uncommon; their luminosity characteristics would
enable them to escape easy discovery.'' Since the detection of
Sculptor, astronomers have identified eight other dSphs.
Sculptor is at a celestial location of $(\alpha,\delta) =
(01^{\mbox{h}}00^{\mbox{m}}09^{\mbox{s}}, -33^{\circ}
42^{\prime}30^{\prime\prime})$ (J2000.0; Mateo 1998), which corresponds
to Galactic coordinates of $(\ell, b)=(287\fdg 5,-83\fdg 2)$. Thus,
Sculptor lies nearly at the South Galactic Pole.
Kaluzny \etal\ (1995) searched for variable stars in a
$15^{\prime} \times 15^{\prime}$ field centered approximately on the
dSph by taking $V-$ and $I-$band images with the 1-m Swope telescope
at Las Campanas Observatory over a period of more than two months.
The search resulted in the identification of 226 RR~Lyr stars. The
average $V-band$ magnitude of the RR~Lyr stars gives a distance
modulus of $(m-M)_{V}=19.71$, which corresponds to a heliocentric
distance of 87~kpc. This estimate is practically the same as that
obtained by Hodge (1965); it is consistent with the estimate of Baade
\& Hubble (1939), but somewhat larger than the estimate of Kunkel
\& Demers (1977). This study adopts the estimate of Kaluzny \etal\
(1995) for the distance to Sculptor.
Irwin \& Hatzidimitriou (1995) derives the most comprehensive
set of structural parameters for Sculptor --- and seven other dSphs ---
using star counts from UK Schmidt telescope plates. With a luminosity
of $(1.4\pm 0.6) \times 10^{6}$~L$_{\odot}$, Sculptor is among the most
luminous dSphs. Its major-axis core and limiting radii are $5.8 \pm
1.6$~arcmin and $76.5 \pm 5.0$~arcmin, respectively, which are in good
agreement with the values derived by Demers, Kunkel, \& Krautter
(1980). However, they differ from the values derived by Eskridge
(1988a), who also uses star counts from UK Schmidt telescope plates.
The discrepancy is likely due to an underestimation of the central
density in the latter study for reasons that are discussed in Irwin \&
Hatzidimitriou (1995). The isopleth map of Sculptor (Panel (f) of
Figure~1 in Irwin \& Hatzidimitriou 1995) shows that the ellipticity of
the isodensity contours increases with increasing projected radius from
the center of the dSph: the ellipticity is consistent with 0 in the
inner 10~arcmin and smoothly increases to a value of 0.32 in the
outermost region. The study observes that Sculptor ``... looks
remarkably similar to numerical simulations of dSph galaxies that are
tidally distorted.'' The position angle of the major axis is $99 \pm
1$~degrees. Eskridge (1988b) finds asymmetric ``structure'' in
Sculptor in an isopleth map of the difference between the stellar
surface density and a fitted 2-D model. In contrast, Irwin \&
Hatzidimitriou (1995) finds only the increase of the ellipticity with
projected radius in a similar map. The left panel in
Figure~\ref{fig:fields} shows a 30~arcmin $\times$ 30~arcmin region of
the sky centered on Sculptor. The dashed ellipse is the boundary of
the core.
Walcher \etal\ (2003) studies the structure of Sculptor ---
together with those of Carina and Fornax --- using $V$-band images
taken with the MPG/ESO 2.2-m telescope at La Silla. The images have an
areal coverage of 16.25~square degrees and reach a limiting magnitude of $V
\approx 23.5$. The study derives major-axis core and limiting radii of
$7.56 \pm 0.7$~arcmin and $40 \pm 4$~arcmin, respectively, using a
King (1962) model, as did Irwin \& Hatzidimitriou (1995). The
1-$\sigma$ disagreement between the two derived core radii is perhaps
larger than expected from two data sets that have many stars in
common. The apparently more serious disagreement between the two
limiting radii is most likely due only to a larger true uncertainty in
the limiting radius caused by uncertainties in the background surface
density and the poor fit of the model to the outer part of the surface
density profile. Walcher \etal\ (2003) confirms that the ellipticity
of the surface-density contours increases with increasing projected
radius; the contours also suggest ``extensions'' from the ends of the
major axis that are interpreted as tidal tails. The radial projected
density profile shows a ``break'' --- a departure from the fitted King
model --- at around 30~arcmin, which the study interprets as evidence
for the existence of an extended stellar component. Walcher \etal\
(2003) uses the relation between the King-model tidal radius, the
mass, and the perigalacticon of a Galactic satellite developed by Oh,
Lin, \& Aarseth (1992) to deduce that Sculptor has a perigalacticon of
28~kpc.
A recent article by Coleman \etal\ (2005) does not confirm the
finding in Walcher \etal\ (2003) that Sculptor has tidal tails and an
extended stellar component. Instead, the analysis of photometric data
in the $V$ and $I$ bands for a $3.1^{\circ}\times3.1^{\circ}$ field
shows that a King model with a limiting radius of $72.5\pm4.0$~arcmin
is a satisfactory fit to the radial profile of stars that lie on the
giant branch. The limiting magnitudes of the photometry are $V=20$ and
$I=19$, respectively. The study notes that oversubtracting the field
population in its data produces a radial profile with the smaller
fitted limiting radius and significant extratidal structure found in
Walcher \etal\ (2003). Using additional information from the
spectroscopy of 723 stars selected from the red giant branch, Coleman
\etal\ (2005) derives an upper limit of $2.3\% \pm 0.6\%$ for the
contribution from stars beyond the tidal boundary to the total mass of
Sculptor. The study does not find any conclusive evidence for tidal
interaction between the Milky Way and Sculptor.
In contrast, Westfall \etal\ (2005) does find evidence. This
study uses imaging in the $M$, $T_{2}$, and $DDO51$ bands to separate
member giants from foreground dwarfs in a 7.82 degrees$^{2}$ area that
covers the eastward side of Sculptor including the central region.
Candidate members are also selected from the region of the blue
horizontal branch in the color-magnitude diagram. The selection of
members is checked with spectroscopy for 147 candidates. The study
finds members up to 150 arcmin from the center of Sculptor --- the
spatial extent of the survey and beyond the tidal boundary if that is
identified with the measured King limiting radius of 80~arcmin.
Several of these stars are spectroscopically-confirmed members. The
radial surface brightness profile shows a break to a shallower slope
at a radius of about 60~arcmin, which resembles the radial profiles
seen in simulations of satellites interacting with the Milky Way
(Johnston \etal\ 1999). Thus, Westfall \etal\ (2005) argues in favor
of a significant tidal interaction between the Milky Way and Sculptor.
It is beyond the scope of this work to resolve the apparent conflict
between Coleman \etal\ (2005) and Westfall \etal\ (2005) by judging
the merits of the analyses presented in both articles. Needless to
say, a disagreement exists about the effect of the Galactic tidal
field on the structure of Sculptor; measuring the proper motion of the
dSph may allow us to impose constraints on this effect.
Armandroff \& Da Costa (1986) measured the radial velocities
of 16 giants in Sculptor which average to produce a systemic
heliocentric velocity of $107.4 \pm 2.0$~km~s$^{-1}$. This
measurement alleviated the large uncertainty in this quantity, which
existed due to mutually contradictory estimates from Hartwick \&
Sargent (1978) and Richter \& Westerlund (1983). More recently,
Queloz, Dubath, \& Pasquini (1995) measured the radial velocities of
23 giant stars. The sample includes 15 stars observed previously by
Armandroff \& Da Costa (1986). The implied systemic heliocentric
velocity of $109.9 \pm 1.4$~km~s$^{-1}$ (after excluding two stars
that are likely binaries) agrees within the quoted uncertainties with
the measurement of Armandroff \& Da Costa (1986). Our article adopts
a mean velocity of $109.9 \pm 1.4$~km~s$^{-1}$ for calculating the
space velocity of Sculptor. Queloz, Dubath, \& Pasquini (1995) finds
no apparent rotation of the dSph around its minor axis. Tolstoy
\etal\ (2004) measured radial velocities for 308 potential members of
the dSph and find a systemic velocity of 110~km~s$^{-1}$. There is no
discussion of rotation, but Figure~4 in that article shows that the
velocity dispersion does not increase with radius, as would be
expected if there were a net rotation larger than the central
dispersion. Interestingly, the study finds that the red horizontal
branch stars have a more compact spatial distribution and a smaller
velocity dispersion than the older and more metal-poor blue horizontal
branch stars. The two most recent photometric and spectroscopic
surveys by Coleman \etal\ (2005) and Westfall \etal\ (2005) confirm the
greater central concentration of the more metal-rich stars. The two
studies also find no evidence for rotation.
The mass-to-light ratio, $(M/L)$, of Sculptor is larger than a
typical value for a Galactic globular cluster; it is, however, smaller
than the $M/L$s for some other Galactic dSphs. Armandroff \& Da Costa
(1986) derives a central $M/L_{V}$ of $6.0 \pm 3.1$ and Queloz, Dubath,
\& Pasquini (1995) determines the somewhat larger value of $13 \pm 6$,
in solar units. Both studies note that the measured $M/L$ does not
imply unequivocal support for dark matter in Sculptor.
The stars of Sculptor are old. Fitting isochrones to the
principal sequences in the color-magnitude diagram, Da Costa (1984)
finds that the majority of the stars are younger by ``2--3~Gyr than
Galactic globular clusters of similar metal abundance provided the
helium abundances and the CNO/Fe ratios are also similar.'' Da Costa
(1984) also detects ``blue stragglers'' and estimates their age to be
about 5~Gyr under the assumption that they are ``normal'' main-sequence
stars, i.e., stars which did not acquire mass from a companion. No
stars younger than 5~Gyr exist in Sculptor, indicating an absence of
ongoing or recent star formation. However, Sculptor contains HI gas.
Carignan \etal\ (1998) and later Bouchard \etal\ (2003) detect
two distinct clouds of HI that are diametrically opposite to each other
almost along the minor axis and 20 -- 30~arcmin from the center of the
dSph. These clouds are within the tidal radius. The HI gas is very
likely to be associated with Sculptor because its mean heliocentric
velocity is similar to that of the dSph.
Carignan \etal\ (1998) discusses mechanisms that might account
for the existence of the clouds. Removing the gas from the dSph by a
time-dependent tidal force due the Milky Way is one possibility.
Carignan \etal\ (1998) suggests that the alignment between the proper
motion vector from Schweitzer \etal\ (1995) and a line passing through
the two clouds supports this hypothesis for the origin of the clouds.
However, if the tidal force affects the HI, it should also affect the
stars and the possible signatures of tides in the stellar component of
Sculptor are either absent or inconsistent with the direction of the
Schweitzer \etal\ (1995) proper motion vector. For example, the
increasing ellipticity of isodensity contours with increasing
projected radius could be due to tides (e.g., Johnston, Spergel, \&
Hernquist 1995), but then the major axis should be along the proper
motion vector. The possible ``tidal extensions'' reported by Walcher
\etal\ (2003) are at the ends of the major axis. If these extensions
are a continuation of the increasing ellipticity, they also argue for
an orbital plane parallel to the major axis. Walcher \etal\ (2003)
claims that the eastern extension bends to the south, i.e., parallel to
the minor axis and so argues that the orbital plane is aligned in the
north-south direction. However, this alignment is inconsistent with
the increasing ellipticity being due to the tidal force since, given
the large distance of Sculptor, the Sun is nearly in the orbital plane
and so the increasing ellipticity should be aligned with the tidal
extensions.
Schweitzer \etal\ (1995) reports the first measurement of the
proper motion for Sculptor: $(\mu_{\alpha},\mu_{\delta})=(72 \pm 22,
-6 \pm 25)$~mas~century$^{-1}$. This value includes contributions
from the motions of the Sun and LSR; this article refers to this
quantity as the ``measured proper motion." The measurement derives
from 26 photographic plates imaged with a variety of ground-based
telescopes using either a ``blue'', B, or V filter. The earliest
epoch is 1938 and the latest is 1991. The study estimates, among
other quantities, the perigalacticon of the implied orbit. The best
estimate ranges from 60~kpc for the ``infinite halo'' potential of the
Milky Way to 78~kpc for the ``point mass'' potential. If the
perigalacticon is no smaller than 60~kpc, then the Galactic tidal
force has not played a significant role in the evolution of Sculptor.
Numerical simulations of Piatek \& Pryor (1995) or Oh, Lin, \& Aarseth
(1995) show that for a typical dSph, even with a $M/L_V$ as low as 3,
a perigalacticon of 60~kpc is too large for tides to have an important
effect.
Motivated by the idea that some of the Galactic dSphs and
globular clusters may be pieces of a tidally-disrupted progenitor
satellite galaxy, several studies propose that they form ``streams''
in the Galactic halo. Lynden-Bell (1982) hypothesizes that Fornax,
Leo I, Leo II, and Sculptor are members of the ``FLS stream.''
Majewski (1994) adds the newly-discovered Sextans to the FLS stream
and recalculates its common plane --- naming it the ``FL$^{2}$S$^{2}$
plane.'' The FLS and the FL$^{2}$S$^{2}$ planes differ only slightly.
In a more extensive study, Lynden-Bell \& Lynden-Bell (1995) infers
that Sculptor may belong to one of three possible streams (see their
Table~2). Stream \# 2 contains the LMC, SMC, Draco, Ursa Minor,
and, possibly, Sculptor and Carina; stream \# 4a contains Sextans,
Sculptor, Pal 3, and, possibly, Fornax; finally, stream \# 4b
contains Sextans, Sculptor, and, possibly, Fornax. For each stream,
Lynden-Bell \& Lynden-Bell (1995) calculates the expected proper
motion of Sculptor.
Kroupa \etal\ (2004) notes that the 11 dwarf galaxies nearest
to the Milky Way form a disk with a thickness to radius ratio of
$\leq$~0.15. The article argues that the distribution expected for
such nearby substructure in a cold-dark-matter universe is spherical,
that the observed distribution is not, and, thus, that these objects
are the tidal debris from the disruption of a larger satellite galaxy.
In contrast, Kang \etal\ (2005) and Zentner \etal\ (2005) find that a
planar distribution of nearby Galactic satellites is actually common
in numerical simulations of galaxy formation. A direct comparison
between the results of the simulations and the distribution of nearby
satellites finds that they are consistent.
A test of the reality of streams or planar alignments is to
measure the space motions of the satellites. Piatek \etal\ (2005;
P05) reports a proper motion for Ursa Minor. The implied orbit for
Ursa Minor is not in the plane defined by Kroupa \etal\ (2004). The
proper motion also rules out membership in the stream proposed by
Lynden-Bell \& Lynden-Bell (1995). The measured proper motion for
Carina (Piatek \etal\ 2003; P03) does not agree well with the
predictions of Lynden-Bell \& Lynden-Bell (1995), but is not precise
enough to rule out membership in a stream. Piatek \etal\ (2002; P02)
finds that a preliminary proper motion for Fornax is inconsistent with
the predictions of Lynden-Bell \& Lynden-Bell (1995) and that its
direction is also inconsistent with an orbit in the FL$^{2}$S$^{2}$
plane. Dinescu \etal\ (2004) reports an independent measurement of
the proper motion of Fornax. This motion is consistent, within its
uncertainty, with the predictions of Lynden-Bell \& Lynden-Bell (1995)
and the direction of this motion is along the great circle defined by
the FL$^{2}$S$^{2}$ plane. Dinescu \etal\ (2004) notes that the
proper motion for Sculptor in Schweitzer \etal\ (1995) is inconsistent
with the FL$^{2}$S$^{2}$ plane.
This article reports a second independent measurement of the
proper motion for Sculptor and discusses the implications of the
derived space motion on the dSph-Galaxy interaction.
Section~\ref{sec:data} describes observations and the data. The
following section describes the analysis of the data leading to the
derivation of the proper motion. Section~\ref{sec:pmm} compares the
proper motion from Schweitzer \etal\ (1995) with the one reported in
this article. The next section, Section~\ref{sec:orbit}, integrates
and describes the orbit of Sculptor. Section~\ref{sec:disc} discusses
the implications of the orbit for the importance of the Galactic tidal
force on the structure and internal kinematics of Sculptor, for the
star formation history, and for the membership of Sculptor in the
proposed streams of galaxies and globular clusters in the Galactic
halo. The final section is a summary of the main results and
conclusions.
\section{Observations and Data}
\label{sec:data}
The Hubble Space Telescope (HST, hereafter) imaged two
distinct fields in Sculptor using the Space Telescope Imaging
Spectrograph (STIS, hereafter) in imaging mode with no filter (50CCD).
Each field contains a known quasi-stellar object (QSO, hereafter),
which serves as a reference point. The left panel of
Figure~\ref{fig:fields} depicts the locations of the two fields on the
sky: two small squares --- one inside and the other outside of the
core. The name of the field inside the core is SCL~$J0100-3341$,
which derives from the IAU designation of the QSO in this field.
Tinney \etal\ (1997) confirms the identity of this QSO: it is at
$(\alpha, \delta) = (01^{\mbox{h}}00^{\mbox{m}}25\fs 3, -33^{\circ}
41^{\prime}07^{\prime\prime})$ (J2000.0), has a redshift $z=0.602 \pm
0.001$, and has a magnitude $B=20.4$. The observations of the
SCL~$J0100-3341$ field occurred on September 24, 2000 and on September
26, 2002. At each epoch, there are three exposures at each of the
eight dither pointings for the total of 24 images. The
``ORIENTAT'' angle --- the position angle of the Y axis of the CCD
measured eastward from north --- is the same to within one-tenth of a
degree for all of the exposures and equal to -67.5 degrees. The
top-right panel in Figure~\ref{fig:fields} shows the SCL~$J0100-3341$
field. The QSO is in the cross-hair.
The name of the field outside of the core is SCL~$J0100-3338$.
The QSO in this field, also confirmed by Tinney \etal\ (1997), is at
$(\alpha, \delta) = (01^{\mbox{h}}00^{\mbox{m}}32\fs 6, -33^{\circ}
38^{\prime}32^{\prime\prime})$ (J2000.0), has a redshift $z=0.728 \pm
0.001$, and has a magnitude $B=20.4$. $HST$ observed this field on
September 13, 1999; September 28, 2000; and on September 28, 2002. At
each of the three epochs, there are three exposures at each of the
eight dither pointings for a total of 24 images. The ``ORIENTAT''
angle is the same to within one-tenth of a degree and equal to -69.3
degrees for all of the exposures for this field. The bottom-right
panel of Figure~\ref{fig:fields} shows the SCL~$J0100-3338$ field.
The QSO is in the cross-hair. Owing to its greater distance from the
center of the dSph, the SCL~$J0100-3338$ field contains fewer stars
than does the SCL~$J0100-3341$ field.
Bristow (2004) and P05 discuss the effect of the decreasing
charge transfer efficiency of the STIS CCD on astrometric
measurements. If not accounted for, the decreasing charge transfer
efficiency may introduce a spurious contribution to a measured proper
motion. Bristow \& Alexov (2002) developed computer software which
approximately restores an image taken with STIS to its pre-readout
condition. All of the results that this article reports are based on
images restored using the program of Bristow \& Alexov (2002).
\section{Analysis}
\label{sec:analysis}
P02 describes our method of deriving a proper motion from
images taken with \textit{HST} and containing at least one QSO.
Fundamental to the method is the concept of an effective point-spread
function (ePSF, hereafter), which Anderson \& King (2000) describes in
detail. The subsequent two articles in this series, P03 and P05,
expand and improve upon the basic method. The analysis reported here
incorporates only minor new features into the method; thus, the reader
should consult those earlier articles for the details. Instead, this
study mentions the major elements of the method alongside figures
depicting key diagnostics of the performance of the method and briefly
describes the new features.
\subsection{Flux Residuals}
\label{sec:rf}
Equation 22 in P02 defines a ``flux residual'' diagnostic,
$\cal{RF}$. It is the measure of how the shape of the constructed
ePSF matches the shape of an image of an object. In the case of a
perfect match, ${\cal RF} =0$; if the ePSF is narrower, ${\cal RF} >
0$; otherwise, ${\cal RF} < 0$.
Several factors affect the shape of the PSF for an object. 1.
Type of an object. A PSF for a galaxy is generally wider than that
for a star, all else being equal. 2. Color of an object. Because of
diffraction and aberrations, the width of the PSF is color-dependent.
3. Tilt or curvature of the focal plane. The PSF varies with location
because the CCD surface and focal plane do not coincide everywhere.
4. Thermal expansion. Because the \textit{HST} moves in and out of
the Earth's shadow, its temperature is continuously changing. These
changes cause the telescope to expand or contract, affecting its focal
length. 5. Charge traps in the CCD. As the packets of charge
representing an object move along the $Y$ axis (the direction of
readout for STIS), those on its leading side fill partially each trap
encountered, so that there are fewer traps available to remove charge
from subsequent packets (Bristow \& Alexov 2002). This non-uniform
loss of charge across the object changes its PSF.
Given the aforementioned factors affecting the shape of the
PSF, a plot of ${\cal RF}$ \textit{versus} the $X$- or $Y$-coordinate
of an object will, in the best case, show that the points scatter
around ${\cal RF}=0$. In a less desirable case, the points may show
trends with $X$ or $Y$ or both. These trends signal that the true PSF
varies with location.
Because of the scarcity of stars in the observed fields, our
method constructs a single and constant ePSF for a given field and
epoch. A constant ePSF is one that does not vary with either $X$ or
$Y$. Figures \ref{fig:rf-scl1} and \ref{fig:rf-scl2} show plots of
${\cal RF}$ \textit{versus} $X$ (panels in the left-hand column) and
${\cal RF}$ \textit{versus} $Y$ (panels in the right-hand column) for
the SCL~$J0100-3341$ and SCL~$J0100-3338$ fields, respectively. The
rows of panels from top to bottom are each one epoch, arranged in
chronological order. The filled squares in a plot correspond to the
QSO. Note that the number of ${\cal RF}$ values for a given object
may be equal to the number of exposures --- individual images --- at a
given epoch, or be less if the object is not measured in one or more
exposures.
No panel in Figure~\ref{fig:rf-scl1}, except for the top-left
one, shows a trend between ${\cal RF}$ and $X$ or ${\cal RF}$ and $Y$.
The top-left panel shows that the mean ${\cal RF}$ decreases linearly
with $X$, implying that the shape of the true PSF becomes
progressively narrower and more peaked than that of the constructed
ePSF with increasing $X$. We are unable to trace the origin of this
dependence. The values of ${\cal RF}$ for the QSO are larger than
those for other objects at both epochs and are all positive, implying
that the true PSF for the QSO is wider than the constructed ePSF and
than that for a star.
No panel in Figure~\ref{fig:rf-scl2} shows a trend as
conspicuous as the one in the upper-left panel of
Figure~\ref{fig:rf-scl1}. Nevertheless, the left-hand panel in the
middle row does show a hint of variability of the true PSF with
location. The PSF of the QSO in the SCL~$J0100-3338$ is similar to
that of a star. The values of ${\cal RF}$, though still biased towards
positive values, are comparable to those for bright stars.
There are two reasons why the PSF of a QSO can be different
from that of a star. 1. The underlying galaxy can broaden the image
of a QSO. 2. The color of a typical QSO is bluer than that of a
typical star. So, particularly for the unfiltered STIS imaging, the
true PSF is narrower for a bluer object. Thus, depending on the
interplay between the distance to a QSO and its color, the values of
${\cal RF}$ for the QSO can average more positive than, more negative
than, or the same as those for a bright star. Visual inspection of
the QSO in the SCL~$J0100-3341$ field shows what appears to be a
single spiral arm or tidal feature extending from its image,
suggesting that the underlying galaxy is indeed the cause of the large
positive values of ${\cal RF}$ for this QSO.
Experience with the data for other dSphs (P02, P03, and P05)
has shown that trends in the ${\cal RF}$ values with position do not
necessarily produce systematic errors in the positions of objects. The
next section searches for such systematic errors in the position.
\subsection{Position Residuals}
\label{sec:rx-ry}
Fitting an ePSF to the science data array of an object (the
$5\times5$ array of pixels representing an object; see P02 for more
detail on this array and our procedures) determines its centroid.
With 24 images per field and epoch, there can be up to 24 measurements
of the centroid. The actual number will be smaller than 24 if an
object is flagged out from one or more images because its array is
corrupted by cosmic rays or hot pixels. The dithering, rotation, and
change of scale (e.g., due to ``breathing'' of the \textit{HST})
between any two images cause the centroid of an object measured in
these two images to differ. Therefore, at each epoch, every field has
a fiducial coordinate system that coincides with the coordinate system
of the first image in chronological order. The adopted transformation
from the coordinate system of each subsequent image to the fiducial
system contains a linear translation, rigid rotation, and a uniform
scale change. Let $(X_{0,j}^{i,k}, Y_{0,j}^{i,k})$ be the centroid of
object $i$ at epoch $j$ in image $k$ transformed to the fiducial
coordinate system and the mean centroid of object $i$ in the fiducial
coordinate system of epoch $j$ be $(<X_{0,j}>^{i},<Y_{0,j}>^{i})$.
Define position residuals, ${\cal RX}_{j}^{i,k}$ and ${\cal
RY}_{j}^{i,k}$, for an object $i$ as ${\cal
RX}_{j}^{i,k}=<X_{0,j}>^{i} - X_{0,j}^{i,k}$ and ${\cal
RY}_{j}^{i,k}=<Y_{0,j}>^{i}- Y_{0,j}^{i,k}$. Ideally, ${\cal
RX}_{j}^{i,k} = {\cal RY}_{j}^{i,k} = 0$ for all $j$ and $k$. Random
noise causes ${\cal RX}_{j}^{i,k}$ and ${\cal RY}_{j}^{i,k}$ to differ
from zero, but it does not cause any trends with respect to other
quantities. However, systematic errors can cause such trends.
Anderson \& King (2000) demonstrates that a mismatch between the true
PSF and the ePSF causes ${\cal RX}_{j}^{i,k}$ and ${\cal
RY}_{j}^{i,k}$ to depend on the location of a centroid within a pixel
--- the pixel phase $\Phi_{x}$ or $\Phi_{y}$. By definition,
$\Phi_{x,j}^{i,k}
\equiv X_{0,j}^{i,k}-Int(X_{0,j}^{i,k})$ and $\Phi_{y,j}^{i,k} \equiv
Y_{0,j}^{i,k}-Int(Y_{0,j}^{i,k})$, where the function $Int(x)$ returns
the integer part of the variable $x$.
Figure~\ref{fig:rxry-scl1} plots ${\cal RX}$ and ${\cal RY}$
\textit{versus} $\Phi_{x}$ or $\Phi_{y}$ for the SCL~$J0100-3341$
field. The plots in the panel \ref{rx-ry-9-scl1} are for the 2000
epoch and those in the panel \ref{rx-ry-10-scl1} are for the 2002
epoch. The filled squares correspond to the QSO and the dots to stars
with a $S/N$ greater than 30.
The plots of ${\cal RX}$ \textit{versus} $\Phi_{x}$ and ${\cal
RY}$ \textit{versus} $\Phi_{y}$ in Figures~\ref{rx-ry-9-scl1} and
\ref{rx-ry-10-scl1} show trends between these quantities for the QSO.
Values of ${\cal RX}$ and ${\cal RY}$ tend to be negative for
$\Phi_{x}$ and $\Phi_{y}$ less than about 0.5~pixel, and they tend to
be positive for $\Phi_{x}$ and $\Phi_{y}$ greater than about
0.5~pixel. The points corresponding to the stars do not show these
trends. The plots of the cross terms, ${\cal RX}$ \textit{versus}
$\Phi_{y}$ and ${\cal RY}$ \textit{versus} $\Phi_{x}$, do not show any
trends for the QSO or for the stars. These trends indicate a mismatch
between the ePSF and the true PSF (Anderson \& King 2000). Both stars
with $S/N > 15$ and the QSO contribute to the construction of the
ePSF. Therefore, the more-extended true PSF of the QSO causes the
ePSF to be wider than an ePSF constructed using only stars; in other
words, the ePSF is a ``compromise'' between that of the stars and that
of the QSO. An ePSF constructed using objects with $S/N > 100$
diminishes the trends in the values of ${\cal RX}$ and ${\cal RY}$ for
the QSO because the shape of the ePSF is more akin to the shape of the
true PSF of the QSO. However, increasing the $S/N$ threshold to 100
or more in the construction of the ePSF is undesirable because the
resulting ePSF is poorly sampled because there are only a few stars
with $S/N$ greater than this limit. Instead, we choose to allow the
errors in the position of the QSO to remain and be reflected in a
greater uncertainty for the measured proper motion for this field.
Figure \ref{fig:rxry-scl2} plots ${\cal RX}$ and ${\cal RY}$
\textit{versus} $\Phi_{x}$ or $\Phi_{y}$ for the SCL~$J0100-3338$
field. Figures~\ref{rx-ry-8-scl2}, \ref{rx-ry-9-scl2}, and
\ref{rx-ry-10-scl2} are for the 1999, 2000, and 2002 epochs,
respectively. Only objects with a $S/N$ greater than 15 are shown.
No plot shows clear evidence for trends between ${\cal RX}$ or ${\cal
RY}$ and $\Phi_{x}$ or $\Phi_{y}$ for the QSO or for the stars. In this
field, the true PSF of the QSO resembles that for a star, which is
confirmed by Figure~\ref{fig:rf-scl2}, where the values of $\cal{RF}$
for the QSO are indistinguishable from those for stars.
\section{Proper Motion of Sculptor}
\label{sec:pm}
At this point, there are two lists of fiducial coordinates, one
for each epoch, for the SCL~$J0100-3341$ field, and three for the
SCL~$J0100-3338$ field. Define the standard coordinate system to be
that which moves uniformly together with the stars of Sculptor. Thus,
transforming the fiducial coordinates of a star of Sculptor from
different epochs into the standard coordinate system produces the same
value within the measurement uncertainties. In contrast, the
transformed coordinates of the QSO or any other object that is not a
member of Sculptor will show uniform motion. The proper motion of
Sculptor derives from the motion of the QSO in the standard coordinate
system.
P05 describes a procedure for deriving the motion of the QSO,
and any other object that is not a member of the dSph, in the standard
coordinate system from lists of fiducial coordinates at three epochs.
The procedure includes a linear motion in the fitted transformations
between the fiducial coordinate systems and the standard coordinate
system for those objects whose $\chi^2$ calculated with zero motion is
above a threshold. The SCL~$J0100-3341$ field has only two epochs, so
we have modified the procedure for this case by excluding those
objects with $\chi^2$ values above a threshold from the calculation of
the transformations between the coordinate systems. The motion of the
QSO is just the difference of the two transformed coordinates. The
following two sections describe the results from applying these
procedures to the two fields.
\subsection{Motion of the QSO in the SCL~$J0100-3341$ field}
\label{sec:pm1}
The number of objects with a measured centroid is 567 and 516
in epochs 2000 and 2002, respectively. Among these, 470 are common to
the two epochs. The choice for the individual $\chi^2$ that triggers
fitting for uniform linear motion is 15. The multiplicative constant
that ensures a $\chi^2$ of one per degree of freedom is 1.151 (see
P05 for a discussion of these parameters).
The transformation of the measured centroids to the standard coordinate
system used in this article is
\begin{eqnarray}
x_{j}^{\prime\, i} &=& x_{off} + c_{1} + c_{2}(x_{j}^{i} - x_{off}) +
c_{3}(y_{j}^{i} - y_{off})
\label{eq:tranx} \\
y_{j}^{\prime\, i} &=& y_{off} + c_{4} + c_{5}(x_{j}^{i} -
x_{off}) + c_{6}(y_{j}^{i} - y_{off})
\label{eq:trany} \\
\sigma_{xj}^{\prime\, i}&=&
\sqrt{(c_{2}\sigma_{xj}^{i})^{2}
+ (c_{3}\sigma_{yj}^{i})^{2}} \\
\sigma_{yj}^{\prime\, i}&=&
\sqrt{(c_{5}\sigma_{xj}^{i})
+ (c_{6}\sigma_{yj}^{i})^{2}}.
\label{eq:tranuy}
\end{eqnarray}
The above represents a modification of the method described in P05,
afforded here because of the greater number of stars. In the
equations, $c_{1}$ through $c_{6}$ are the free parameters, $(x_{off},
y_{off}) =(512, 512)$~pixel defines the reference point for the
transformation, and $(x^{i}_{j},y^{i}_{j})$ is a measured centroid of
the $i$th object at the $j$th epoch which is transformed to $(x^{\prime
i}_{j},y^{\prime i}_{j})$ in the standard coordinate system.
Equations 10 and 11 in P05 define position residuals
$RX_{j-1}^{i}$ and $RY_{j-1}^{i}$ for an object $i$ transformed to the
standard coordinate system from the fiducial coordinate system of the
$j$th epoch. For an ideal case, $RX_{j-1}^{i} = RY_{j-1}^{i} =0$.
Figure~\ref{fig:rx-ry-scl1} shows $RX$ \textit{versus} $X$ and $RY$
\textit{versus} $Y$ for the SCL~$J0100-3341$ field.
The most prominent feature is a ``step'' in $RX_{1-1}$ \textit{versus}
$X$ at $X \simeq 320$~pixel. The values of $RX_{1-1}$ tend to be
negative for $X$ below the step, indicating the presence of a
systematic error in the $X$ coordinates whose source we are unable to
trace. The values of $RX_{2-1}$ tend to be positive for $X \lesssim
320$~pixel, which is forced by the fitting procedure.
An \textit{ad hoc} approach for removing the ``steps'' is to
replace $x_{j}^{i}$ with $x_{j}^{i} + c_{7}$ in the
Equations~\ref{eq:tranx} through \ref{eq:tranuy} when $x_{j}^{i} \leq
320$~pixels and to fit for the additional free parameter $c_{7}$.
Applying this remedy removes the ``steps,'' as is shown by
Figure~\ref{fig:rx-ry-scl1-c} which plots the same quantities as
Figure~\ref{fig:rx-ry-scl1}. In this corrected fitting procedure, the
value of the multiplicative constant that ensures $\chi^2$ of one per
degree of freedom decreased to 1.123 because of the smaller residuals.
The fitted value of $c_7$ is 0.019~pixel. The proper motion for this
field derives from this fit. Figure~\ref{fig:rx-ry-scl1-cb} is the
same as Figure~\ref{fig:rx-ry-scl1-c} except that the points are the
weighted mean residuals in ten equal-length bins in $X$ or $Y$. Note
the different vertical scale. The points are plotted at the mean of
the coordinate values in the bin. The average residuals show no
systematic trends above a level of 0.001~pixel.
Figure~\ref{fig:xq-yq-scl1} shows the location of the QSO as a
function of time in the standard coordinate system. The top panel
shows the variation of the $X$ coordinate and the bottom panel does
the same for the $Y$ coordinate. The motion of the QSO is
$(\mu_{x},\mu_{y})=(0.0032 \pm 0.0032, 0.0005 \pm
0.0035)$~pixel~yr$^{-1}$. The contribution to the total $\chi^2$ from
the QSO, and from any other object whose motion was fit for, is 0
because a line always passes exactly through two points.
\subsection{Motion of the QSO in the SCL~$J0100-3338$ field}
\label{sec:pm2}
The number of objects with measured centroids is 343, 326, and
314 in epochs 1999, 2000, and 2002, respectively. Among these, 257
are common to the three epochs. The choice for the individual
$\chi^2$ that triggers fitting for uniform linear motion is 15. The
multiplicative constant that ensures a $\chi^2$ of one per degree of
freedom is 1.176.
Figures~\ref{fig:rx-ry-scl2} and \ref{fig:rx-ry-scl2-b} show
position residuals, $RX$ and
$RY$, as a function of position in the standard coordinate system for
the SCL~$J0100-3338$ field. They are analogous to
Figures~\ref{fig:rx-ry-scl1} and \ref{fig:rx-ry-scl1-cb}. From top to bottom, the rows of panels
are for epochs 1999, 2000, and 2002. No panel shows unambiguous
trends between $RX$ and $X$ or $RY$ and $Y$. The largest deviations of
the average residuals are $RY \simeq 0.004$~pixel for $Y < 100$~pixel. Any
systematic trends at the location of the QSO are on the order of
0.001~pixel. Although not shown in
the figures, the plots of the cross-terms do not show trends either.
Figure~\ref{fig:xq-yq-scl2} is analogous to
Figure~\ref{fig:xq-yq-scl1} for the SCL~$J0100-3338$ field. Note that
the slopes in the corresponding plots in Figures~\ref{fig:xq-yq-scl2}\
and \ref{fig:xq-yq-scl1}\ need not be the same because the two fields
are rotated with respect to each other --- though for the fields in
Sculptor the rotation
is only a few degrees. The uncertainties shown for the points in
Figure~\ref{fig:xq-yq-scl2} are those calculated from the
scatter of the measurements about the mean for an individual epoch
increased by a multiplicative factor. The introduction of this factor
reduces the contribution to the total $\chi^2$ from the QSO. Without it,
the contribution was 9.52. The contribution to
the $\chi^2$ has approximately two degrees of freedom, which implies a
0.9\% probability of a $\chi^2$ larger than 9.52 by
chance. Such a small probability likely indicates the presence of
unaccounted-for systematic errors. We choose to increase the uncertainty
in our fitted proper motion by multiplying the uncertainties of the
mean positions at each epoch by the same numerical factor so that contribution
to the total $\chi^2$ is about one per degree of freedom. Our fitting
procedure calculates a value for the factor for all objects whose
contribution to the total $\chi^2$ exceeds 4.6, which is expected
10\% of the time by chance. The value of the factor is 2.2 for the
QSO and the uncertainty in the fitted motion of the QSO increases by
essentially the same amount. The motion of the QSO is
$(\mu_{x},\mu_{y})=(-0.0043\pm 0.0050, 0.0034\pm 0.0038)$~pixel~yr$^{-1}$.
\subsection{Measured Proper Motion}
\label{sec:pmm}
Table~1 gives the measured proper motion for each field in the
equatorial coordinate system and their weighted mean. Table~2
tabulates the proper motions for those objects in the SCL~$J0100-3341$
field for which it was measured. Table~3 does the same for the
SCL~$J0100-3338$ field. The first line of Table~2 and Table~3
corresponds to the QSO and subsequent objects are listed in order of
decreasing $S/N$. The ID number of an object is in column~1, the $X$
and $Y$ coordinates of an object in the earliest image of the first
epoch (o65q09010 for SCL~$J0100-3341$ and o5bl02010 for
SCL~$J0100-3338$) are in columns 2 and 3, and the $S/N$ of the object
at the first epoch is in column 4. The components of the measured
proper motion, expressed in the equatorial coordinate system, are in
columns 5 and 6. Each value is the measured proper motion in the
standard coordinate system corrected by adding the weighted mean
proper motion of Sculptor given in the bottom line of Table~1. To
indicate that this correction has been made, the proper motion of the
QSO is given as zero. The listed uncertainty of each proper motion is
the uncertainty of the measured proper motion, calculated in the same
way as for the QSO, added in quadrature to that of the average proper
motion of the dSph. The contribution of the object to the total
$\chi^2$ is in column~7. Although column 7 is in Table 2 for the sake
of symmetry with Table 3, the $\chi^2$ contributions are not
meaningful.
Schweitzer \etal\ (1995) reports the first measurement of the
proper motion for Sculptor; this study reports an additional two
independent measurements. Figure~\ref{fig:pm} compares the three
independent measurements, each represented by a rectangle. A dot at
the center of a rectangle is the best estimate of the proper motion.
The sides of a rectangle are offset from the center by the 1-$\sigma$
uncertainties. Rectangles 1, 2, and 3 represent the measurements by
Schweitzer \etal (1995), this study (field SCL~$J0100-3341$), and this
study (field SCL~$J0100-3338$), respectively.
The $\alpha$ components of our measurements 2 and 3 agree
almost exactly and their $\delta$ components differ by only
1.4$\times$ the uncertainty of their difference. While the
$\delta$ component of measurement 1 agrees with the $\delta$
components of measurements 2 and 3, the $\alpha$ component does not
agree with either one. The $\alpha$ components of measurements 1 and
2 differ by 2.3$\times$ the uncertainty of their difference and those
for measurements 1 and 3 differ by 2.2$\times$. Because of the large
difference in the $\alpha$ components of the proper motion between the
measurement from Schweitzer \etal\ (1995) and from our two fields, we
choose to use the weighted average proper motion from Table~1 to
determine the space velocity of Sculptor.
\subsection{Galactic Rest Frame Proper Motion}
\label{sec:pmgrf}
The measured proper motion of the dSph contains contributions
from the motion of the LSR and the peculiar motion of the Sun. The
magnitude of the contributions depend on the Galactic longitude and
latitude of the dSph. Removing them yields the Galactic-rest-frame
proper motion --- the proper motion measured by a hypothetical
observer at the location of the Sun but at rest with respect to the
Galactic center. Columns (2) and (3) of Table~4 give the equatorial
components, $(\mu_{\alpha}^{\mbox{\tiny{Grf}}},
\mu_{\delta}^{\mbox{\tiny{Grf}}})$, of the Galactic-rest-frame proper
motion. Their derivation assumes: 220~km~s$^{-1}$ for the circular
velocity of the LSR; 8.5~kpc for the distance of the Sun from the
Galactic center; and $(u_\odot, v_\odot, w_\odot) = (-10.00 \pm 0.36,
5.25 \pm 0.62 , 7.17 \pm 0.38)$~km~s$^{-1}$ (Dehnen \& Binney 1998)
for the peculiar velocity of the Sun, where the components are
positive if $u_{\odot}$ points radially away from the Galactic center,
$v_{\odot}$ is in the direction of rotation of the Galactic disk, and
$w_\odot$ points in the direction of the North Galactic Pole. Columns
(4) and (5) give the Galactic-rest-frame proper motion in the Galactic
coordinate system,
$(\mu_{l}^{\mbox{\tiny{Grf}}},\mu_{b}^{\mbox{\tiny{Grf}}})$. The next
three columns give the $\Pi$, $\Theta$, and $Z$ components of the
space velocity in a cylindrical coordinate system centered on the
dSph. The components are positive if $\Pi$ points radially away from
the Galactic axis of rotation, $\Theta$ points in the direction of
rotation of the Galactic disk, and $Z$ points in the direction of the
North Galactic Pole. The derivation of these components assumes
87~kpc (Kaluzny
\etal\ 1995) for the heliocentric distance to and $109.9 \pm
1.4$~km~s$^{-1}$ (Queloz, Dubath, \& Pasquini 1995) for the
heliocentric radial velocity of Sculptor. The last two columns give
the radial and tangential components of space velocity for an observer
at rest at the Galactic center. The component $V_{r}$ is positive if
it points radially away from the Galactic center. Thus, at present,
Sculptor is moving away from the Milky Way.
\section{Orbit and Orbital Elements of Sculptor}
\label{sec:orbit}
Knowing the space velocity of a dSph permits a determination
of its orbit for a given form of the Galactic potential. This study
adopts a Galactic potential that has a contribution from a disk of the
form (Miyamoto \& Nagai 1975)
\begin{equation}
\label{diskpot}
\Psi_{\mbox{\small{disk}}}=-\frac{G
M_{\mbox{\small{disk}}}}{\sqrt{R^{2}+(a+\sqrt{Z^{2}+b^{2}})^{2}}},
\end{equation}
from a spheroid of the form (Hernquist 1990)
\begin{equation}
\label{spherpot}
\Psi_{\mbox{\small{spher}}}=-\frac{GM_{\mbox{\small{spher}}}}
{R_{\mbox{\small{GC}}}+c},
\end{equation}
and from a halo of the form
\begin{equation}
\label{logpot}
\Psi_{\mbox{\small{halo}}}=v^{2}_{\mbox{\small{halo}}}\ln
(R^{2}_{\mbox{\small{GC}}}+d^{2}).
\end{equation}
In the above equations, $R_{\mbox{\small GC}}$ is the Galactocentric
distance, $R$ is the projection of $R_{\mbox{\small GC}}$ onto the
plane of the Galactic disk, and $Z$ is the distance from the plane of
the disk. All other quantities in the equations are adjustable
parameters and their values are the same as those adopted by Johnston,
Sigurdsson, \& Hernquist (1999):
$M_{\mbox{disk}}=1.0\times10^{11}$~M$_{\odot}$,
$M_{\mbox{spher}}=3.4\times10^{10}$~M$_{\odot}$,
$v_{\mbox{halo}}=128$~km~s$^{-1}$, $a=6.5$~kpc, $b=0.26$~kpc,
$c=0.7$~kpc, and $d=12.0$~kpc.
Figure~\ref{fig:orbit} shows the projections of the orbit of
Sculptor onto the $X-Y$ (top-left panel), $X-Z$ (bottom-left panel),
and $Y-Z$ (bottom-right panel) Cartesian planes. The orbit results
from an integration of the motion in the Galactic potential given by
Equations~\ref{diskpot}, \ref{spherpot}, and \ref{logpot}. The
integration extends for 3~Gyr backwards in time and begins at the
current location of Sculptor with the negative of the space velocity
components given in the bottom line of columns (6), (7), and (8) of
Table~4. The filled square marks the current location of the dSph, the
filled star indicates the center of the Galaxy, and the two small
circles mark the points on the orbit where $Z=0$ or, in other words,
where the orbit crosses the plane of the Galactic disk. The large
circle is for reference: it has a radius of 30~kpc. In the
right-handed coordinate system of Figure~\ref{fig:orbit}, the current
location of the Sun is on the positive $X$-axis. The figure shows that
Sculptor is moving away from the Milky Way, is closer to perigalacticon
than apogalacticon, and that it has a nearly polar orbit with a modest
eccentricity.
Table~5 tabulates the elements of the orbit of Sculptor. The
value of the quantity is in column (4) and its $95\%$ confidence
interval is in column (5). The latter comes from 1000 Monte Carlo
experiments, where an experiment integrates the orbit using an initial
velocity that is generated by randomly choosing the line-of-sight
velocity and the two components of the measured proper motion from
Gaussian distributions whose mean and standard deviation are the best
estimate of the quantity and its quoted uncertainty, respectively. The
eccentricity of the orbit is defined as
\begin{equation}
\label{eccentricity}
e = \frac{(R_{a} - R_{p})}{(R_{a} + R_{p})}.
\end{equation}
The most likely orbit has about a 2:1 ratio of apogalacticon to
perigalacticon, though the 95\% confidence interval for the
eccentricity allows ratios approximately between 1.7:1 and 4:1. The
orbital period of Sculptor, 2.2~Gyr, is about 50\% longer than those of
Carina (1.4~Gyr; P03) and Ursa~Minor (1.5~Gyr; P05).
\section{Discussion}
\label{sec:disc}
Knowing the orbit can help answer several questions about
Sculptor, or, at least, increase the level of our understanding of
this galaxy. These questions are: 1. Is Sculptor a member of a stream
of galaxies? 2. Is its star formation history correlated with the
orbit? 3. What is the origin of the HI clouds detected in close
proximity to the dSph? 4. Does Sculptor contain dark matter?
\subsection{Is Sculptor a Member of a Stream?}
\label{stream}
Lynden-Bell \& Lynden-Bell (1995) proposes that Sculptor may be
a member of one of three possible streams: stream No.~2 (together with
the LMC, SMC, Draco, Ursa Minor, and Carina); No.~4a (together with
Sextans, Pal 3, and Fornax); or No.~4b (together with Sextans and
Fornax). Columns (2) and (3) in Table~6 give the predicted
heliocentric (i.e., ``measured'' in our terminology) proper motion in
the equatorial coordinate system for Sculptor if it indeed belongs to
any of the three streams. The magnitude of the proper motion vector,
$\vert \vec{\mu} \vert = \sqrt {\mu_{\alpha}^{2} + \mu_{\delta}^{2}}$,
and its position angle are in columns (4) and (5). For easy
comparison, the corresponding quantities from our study are in the
bottom line of the table. Comparing the entries shows that the
predictions for streams 2 and 4a disagree significantly with our
measurement. However, the prediction for stream No. 4b is closer: the
magnitudes differ by 1.6$\times$ the uncertainty in their difference,
while the position angles differ by 1.6$\times$. Differences of this
size should occur by chance 1\% of the time. The measured proper
motion based on only the three-epoch data in the SCL~$J0100-3338$ field
improves the agreement with the prediction for stream 4b. Thus, while
we rule out the possibility that Sculptor is a member of stream No.~2
or 4a, its membership in stream~4b is possible.
Stream 4b contains both Sculptor and Fornax. The Dinescu
\etal\ (2004) proper motion for Fornax is $(\mu_\alpha, \mu_\delta) =
(59 \pm 16, -15 \pm 16)$~mas~cent$^{-1}$. The magnitude and position
angle of the proper motion are $61 \pm 16$~mas~cent$^{-1}$ and $104\pm
15$~degrees. The prediction for stream 4b from Lynden-Bell \&
Lynden-Bell (1995) is 20~mas~cent$^{-1}$ and 162~degrees. The
difference between the measured and predicted proper motions would be
this large or larger by chance only 0.4\% of the time. Thus, the
physical reality of stream~4b is doubtful.
Dinescu \etal\ (2004) argues that Fornax and Sculptor are
members of the same stream that also includes Leo I, Leo II, and
Sextans. Together the galaxies define the FL$^{2}$S$^{2}$ plane. If
they do form a stream, their Galactic-rest-frame proper motion vectors
should be aligned with the great circle passing through the galaxies.
The position angle of the great circle passing through Sculptor and
Fornax is about 99~degrees at the location of Sculptor and 95~degrees
at Fornax. The position angle of the Galactic-rest-frame proper
motion for Fornax reported by Dinescu \etal\ (2004) is $79 \pm
25$~degrees, which differs by 0.64$\times$ its uncertainty from the
position angle of the great circle. If Sculptor and Fornax form a
stream, then they should move in the same direction along the great
circle connecting them. Thus, the proper motion of Fornax from Dinescu
\etal\ (2000) implies that the position angle of the
Galactic-rest-frame proper motion of Sculptor should be 99~degrees.
The position angle for the proper motion of Sculptor reported here is
$333 \pm 15$~degrees, which differs from the
prediction by 8.4$\times$ its uncertainty. Discounting the
proper motion from this study and instead using the position angle of
$40 \pm 24$~degrees implied by the proper motion measured by
Schweitzer \etal\ (1995) does not remove the disagreement: the
position angle differs by $2.5\times$ its uncertainty from that of the
great circle. We conclude that Sculptor and Fornax do not belong to
the same stream.
Kroupa \etal\ (2004) shows that the 11 dwarf
galaxies nearest to the Milky Way are nearly on a plane, whose two
poles are at $(\ell,b) = (168,-16)$~degrees and $(348,+16)$~degrees.
Adopting the direction of the angular momentum vector as the pole of
the orbit, then the location of the pole is
\begin{equation}
(\ell,b) = (\Omega+90^{\circ},\Phi-90^{\circ}).
\end{equation}
Because of the left-handed nature of the Galactic rotation, prograde
orbits have $b < 0$ and retrograde orbits have $b > 0$. Thus, the pole
of our orbit for Sculptor is $(\ell,b) = (5 \pm 16, -4 \pm
1.6)$~degrees, where the uncertainties are 1-$\sigma$ values from the
Monte Carlo simulations. The galactic longitudes of the poles of the
plane and orbit agree within the uncertainty, but the galactic
latitudes do not. They differ by 20~degrees, which is more than
12$\times$ the uncertainty in the location of the pole of the orbit.
However, there is also some uncertainty in the orientation of the plane
passing through the dwarf galaxies near the Milky Way. We conclude the
plane of the orbit of Sculptor is similar to the plane defined by the
nearby dwarf galaxies.
\subsection{The Effect of the Galactic Tidal Force on the Structure of
Sculptor}
\label{sec:tides}
The measured ellipticity of the isodensity contours increases
with projected distance from the center of Sculptor (see Figure~1 in
Irwin \& Hatzidimitriou 1995), akin to surface density contours of a
model dSph in the numerical simulations of Johnston, Spergel, \&
Hernquist (1995; e.g., see Figure~4). If the Galactic tidal force
deformed Sculptor from an initial nearly-spherical shape to its
present elongated shape in the outer regions then, from our vantage
point nearly in the orbit plane, the position angle of its projected
major axis should be similar to --- or differ by 180 degrees from ---
the position angle of the Galactic-rest-frame proper motion vector, as
predicted by the numerical simulations of Oh, Lin, \& Aarseth (1995),
Piatek \& Pryor (1995), or Johnston, Spergel, \& Hernquist (1995).
The position angle of the projected major axis is $99 \pm 1$~degrees
and the position angle of our measured Galactic-rest-frame proper
motion vector is $333 \pm 15$~degrees. Allowing for the
180-degree degeneracy, the difference between the two position angles
is 3.6$\times$ the uncertainty of their difference, which
suggests that the Galactic tidal force has not elongated Sculptor.
\subsection{Does Star Formation History Correlate with the Orbital
Motion of Sculptor?}
\label{sec:sfh}
Da Costa (1984) imaged a 3~arcmin $\times$ 5~arcmin field
located just outside of the core radius of Sculptor in three bands:
$B$, $V$, and $R$. The photometry reaches the main-sequence turn off.
Comparing theoretical isochrones with the distribution of stars in the
color-magnitude diagram, the study finds that the majority of stars is
about 2-3 Gyr younger than the galactic globular clusters of comparable
metal abundance. An earlier study by Kunkel \& Demers (1977) based on
$B$ and $V$ photometry extending to 0.4 magnitudes below the horizontal
branch reaches a similar conclusion. The color-magnitude diagram also
shows a population of ``blue stragglers,'' which might be indicative of
an extended period of star formation. Da Costa (1984) concludes,
however, that, if an intermediate-age stellar population exists in
Sculptor, it is ``infinitesimal compared to that of the Carina system.''
Deep \textit{HST} imaging by Monkiewicz \etal\ (1999) in a
single field, reaching 3 magnitudes below the main-sequence turn-off,
confirms the basic picture of Sculptor uncovered by Kunkel \& Demers
and Da Costa (1984). This color-magnitude diagram also reveals the
presence of ``blue stragglers'' and implies an age comparable to that
of the galactic globular clusters. The small number of stars in the
small field made a search for an intermediate-age stellar population
inconclusive.
Majewski \etal\ (1999), Hurley-Keller \etal\ (1999), and
Harbeck \etal\ (2001) use wide-field imaging to show the presence of
two stellar populations with distinctly different metallicities
([Fe/H] = --2.3 and --1.5; Majewski \etal\ 1999). The more metal rich
population is more centrally concentrated in the galaxy. Most
recently, Tolstoy \etal\ (2004) confirms the above picture using
wide-field imaging and spectroscopy. Spectroscopically-determined
metallicities range from --2.8 to --0.9. Stars more metal-rich than
--1.7 are more centrally concentrated and have a smaller velocity
dispersion than the rest of the sample. However, both stellar
populations are older than 10~Gyrs.
The aforementioned studies show that there were at least two
episodes of star formation at times more than 10~Gyr ago. Because
10~Gyrs is much longer than the orbital period of approximately
2.2~Gyr, there is no clear connection between the stimulation of
star formation and processes such as the Galaxy-Sculptor tidal
interaction or the effects of ram pressure. The lack of correlation
could be due to the loss of all of the gas in Sculptor about 10~Gyr
ago. But, surprisingly, the observations indicate that Sculptor has
HI today.
\subsection{HI Gas in Sculptor}
\label{sec:gas}
Unlike most other Galactic dSphs, Sculptor contains a
detectable amount of HI. Knapp \etal\ (1978) detects three clouds of
HI in the vicinity of Sculptor and speculates that one of them, with a
radial velocity of 120~km~s$^{-1}$, may be associated with Sculptor,
whose radial velocity at the time was uncertain. Carignan
\etal\ (1998) confirms and refines this detection and puts a lower
limit on the mass of HI of $3.0 \times 10^{4}$~M$_{\odot}$. Bouchard
\etal\ (2003) repeats the observations over a wider field with the
Parkes single-dish telescope and, over a smaller region, at higher
angular resolution with the Australia Telescope Compact Array. The
better data show that the HI is not associated with a background
galaxy and the probability of a chance superposition of a galactic
high velocity cloud is less than 2\%. These arguments, together with
the agreement within 4~km~s$^{-1}$ of the radial velocity of the HI
and the radial velocity of Sculptor make a strong case for the
physical association of the clouds with the dSph.
The gas is in two clouds. Figure~\ref{fig:gas}\ shows the
distribution of HI on the sky in the direction of Sculptor based on
the Australia Telescope Compact Array data from Bouchard \etal\
(2003). The asterisk marks the optical center of Sculptor and the two
clouds are about 20--30~arcmin from the center, one to the northeast
and one to the southwest. The masses of the clouds are $(4.1 \pm 0.2)
\times 10^4$~M$_\odot$ and $(1.93 \pm 0.02) \times 10^5$~M$_\odot$,
respectively. The two clouds lie nearly along the minor axis of the
dSph; the orientation of Sculptor is shown in Figure~\ref{fig:gas}\ by
the ellipse representing the optical boundary.
Although the association between the gas and the dSph seems
well-established, why Sculptor still has gas when most of the other
galactic dSphs do not and the cause of the observed configuration of
the gas are debated. Mayer \etal\ (2005) studies the loss of gas from
dwarf galaxies in the Local Group using numerical simulations that
include ram pressure stripping. It quantifies the expected result that
a galaxy with a deeper potential well or with a larger perigalacticon
is more likely to retain its gas. The orbit of Sculptor is similar to
those of Carina and Ursa Minor, which suggests that differences in the
degree of tidal shocking or ram pressure stripping are not the reason
for the difference in gas retention. However, note that the 95\%
confidence intervals for the perigalacticons of all three orbits are
still too large to make a conclusive statement. The larger mass of
Sculptor compared to Carina and Ursa Minor is the most likely reason
why it was able to retain gas.
Mechanisms that could affect the distribution of HI within the
dSph are tidal interaction, ram pressure, and forces from supernovae
or winds from young stars. Tidal interaction and ram pressure tend to
spread the gas in the plane of the orbit. The two arrows in
Figure~\ref{fig:gas}\ represent the Galactic-rest-frame proper motions
as measured by Schweitzer \etal\ (1995; dashed) and this study
(solid). The ratio of their lengths is the same as the ratio of the
magnitudes of the proper motions. The dotted line is a section of a
great circle that passes through Sculptor and Fornax; its position
angle is 99~degrees.
The Schweitzer \etal\ (1995) proper motion (dashed arrow) is
nearly aligned with the line that connects the centers of the clouds.
Carignan \etal\ (1998) notes this alignment and suggests that it may
indicate a ``tidal'' origin for the clouds: presumably, the Galactic
tidal force stretches the gas, initially centered within the dSph, into
its observed distribution. The most serious problem with such a
picture is that the tidal force would stretch the distribution of both
the stars and the gas, which then aligns the major axes of the gaseous
and stellar distributions from our perspective as an observer nearly in
the plane of the orbit of Sculptor. They are not aligned, perhaps
because the motion of the gas is governed by both gravitational forces
and pressure gradients. As was noted in Section~\ref{sec:tides}, if
the observed ellipticity of Sculptor is due to the tidal force, then
the Galactic-rest-frame proper motion should be aligned with the major
axis. Figure~\ref{fig:gas}\ shows that the solid arrow, our
measurement, is closer to such an alignment than the dashed arrow.
Because the tidal force is zero at the center of a dSph, it cannot by
itself separate a single cloud centered on the dSph into two clouds.
Thus, within the context of a picture in which tides have had an
important effect on Sculptor, our proper motion is more plausible than
that of Schweitzer \etal\ (1995). However, is our proper motion
consistent with the geometry of the two clouds?
We think yes. First, the two clouds are elongated in the
direction of our proper motion (see particularly Figure~1 in Bouchard
\etal\ 2003). The observed elongation could be due to ram pressure
from the motion through a gaseous Galactic halo. Second and more
speculatively, the two clouds could be due to the Rayleigh-Taylor
instability when the HI in Sculptor moves through the hot and
low-density gaseous halo. Or the gas could have been squeezed out
perpendicular to the direction of motion by the compressive tidal
shock when Sculptor crosses the Galactic disk. Expansion combined
with infall of the gas forms a ring that looks like two clouds in
projection.
\subsection{Is there dark matter in Sculptor?}
\label{sec:dm}
Estimates of the $M/L_V$ for Sculptor range from $6 - 13$ and
estimates of the limiting radius range from about $40 - 80$~arcmin.
The estimates of $M/L$ assume that mass follows light. If this is
true, then the implied mass of Sculptor must be large enough, given our
orbit, to produce a tidal radius that is at least as large as the
observed limiting radius. Equating the tidal radius and the limiting
radius predicts a value for $M/L$, which should agree with the measured
value. Also, the dSph must have a mass and, hence, $M/L$ large enough
for it to have survived destruction by the Galactic tidal force on our
orbit.
In lieu of numerical simulations, an approximate analytical approach is
to calculate the tidal radius, $r_t$, beyond which a star becomes
unbound from the dSph. For a logarithmic Galactic potential, $r_t$ is
given by (King 1962; Oh, Lin, \& Aarseth 1992)
\begin {equation}
r_t = \left(\frac{(1-e)^2}{[(1+e)^2/2e]\ln[(1+e)/(1-e)] +1} \,
\frac{M}{M_G}\right)^{1/3} a.
\label{eq:rtidal}
\end {equation}
Here $e$ is eccentricity of the orbit, $a$ is the semi-major axis ($a
\equiv (R_{a}+R_{p})/2$), $M$ is the mass of the dSph, and $M_G$ is the
mass of the Galaxy within $a$. Equating $r_t$ with the observed
limiting radius derived by fitting a King (1966) model, $r_k$, gives a
value for $M/L_V$ for a given orbit. If $r_k = 40$~arcmin, then 28\%
of the orbits in Monte Carlo simulations have $M/L_V > 6$ and
10\% of the orbits have $M/L_{V} > 13$. If $r_k = 80$~arcmin, then
100\% of the orbits have $M/L_V > 13$. These
results show that the global $M/L$ of Sculptor is probably larger than
the measured $M/L$, if the larger of the measured limiting radii is
identified with the
tidal radius. The $M/L$ calculated assuming that mass follows light
underestimates the true global $M/L$ if Sculptor contains dark matter
that is more spatially extended than the luminous matter (e.g., Pryor
\& Kormendy 1990). However, equation~\ref{eq:rtidal}\ shows that $M
\propto r_t^3$, so the values for $M/L$ derived using this equation
are sensitive to the measured value of the limiting radius. Until
kinematic measurements definitively identify the tidal radius, an
$M/L$ derived with the above argument should be treated with caution.
The average of the measured values of $M/L_V$ for Galactic globular
clusters is 2.3 (Pryor \& Meylan 1993). Could the true $M/L_V$ of
Sculptor be similar to this average? Numerical simulations by Oh, Lin,
\& Aarseth (1995) and Piatek \& Pryor (1995) show that the ratio of the
limiting radius derived by fitting a theoretical King model (King
1966), $r_k$, to the tidal radius defined by Equation~(\ref{eq:rtidal})
is a useful indicator of the importance of the Galactic tidal force on
the structure of a dSph. These simulations show that: if $r_{k}/r_t
\lesssim 1.0$, the Galactic tidal force has little effect on the
structure of the dSph; at $r_k/r_t \approx 2.0$, the effect of the
force increases rapidly with increasing $r_k/r_t$; and, for $r_k/r_t
\approx 3.0$, the dSph disintegrates in a few orbits. Assuming that
$M/L_V = 2.3$ and $r_k = 40$~arcmin, $r_k/r_t > 2.0$ for 6\%
of the orbits generated in Monte Carlo simulations. If $r_k = 80$~arcmin,
the fraction is 100\%. Thus, it is possible
that Sculptor could have survived for a Hubble time on its current
orbit if it only contains luminous matter. We conclude that measured
orbit of Sculptor does not require it to contain dark matter.
\subsection{A Lower Limit for the Mass of the Milky Way}
\label{sec:massofg}
Sculptor is bound gravitationally to the Milky Way. The
Galactocentric space velocity of the dSph imposes a lower limit on the
mass of the Milky Way within the present Galactocentric radius of the
dSph, $R$. Assuming a spherically symmetric mass distribution and
zero for the total energy of the dSph, the lower limit for the mass of
the Milky Way is given by
\begin{equation}
\label{mwmass}
M=\frac{R\left (V_{r}^{2} + V_{t}^{2} \right )}{2G}.
\end{equation}
Setting $R=87$~kpc and using the values from Table~4 for $V_{r}$ and
$V_{t}$, $M = (4.6\pm 2.0) \times 10^{11}\ M_{\odot}$.
This lower limit is consistent with other recent estimates of the mass
of the Milky Way, such as the mass of $5.4^{+0.1}_{-0.4} \times
10^{11}\ M_{\odot}$ within $R=50$~kpc found by Sakamoto, Chiba, \&
Beers (2003). The Milky Way potential adopted in
Section~\ref{sec:orbit} has a mass of $7.8\times 10^{11}\ M_{\odot}$
out to $R=87$~kpc.
\section{Summary}
\label{sec:sum}
This article presents a measurement of the proper motion of
Sculptor using data taken with \textit{HST} and STIS in imaging mode.
Using this measurement, it derives the orbit and discusses: membership
in proposed streams, tidal interaction with the Milky Way, the relation
between the orbit and the star-formation history, the HI gas associated
with the dSph, the dark matter content, and a lower limit on the mass
of the Milky Way. The list below enumerates our findings.
1. Two independent measurements of the proper motion give a
weighted mean value of $(\mu_\alpha,\mu_\delta)=(9\pm 13, 2 \pm
13)$~mas~cent$^{-1}$ in the equatorial coordinate system for a
heliocentric observer.
2. Removing the contributions to the measured proper motion
from the motions of the Sun and of the LSR gives a Galactic-rest-frame
proper motion of $(\mu_{\alpha}^{\mbox{\tiny{Grf}}},
\mu_{\delta}^{\mbox{\tiny{Grf}}})=(-23\pm 13, 45 \pm 13)$~mas~cent$^{-1}$
in the equatorial coordinate system for an
observer at the location of the Sun but at rest with respect to the
Galactic center. In the Galactic coordinate system this motion is
$(\mu_{l}^{\mbox{\tiny{Grf}}},\mu_{b}^{\mbox{\tiny{Grf}}}) = (11
\pm 13, -50 \pm 14)$~mas~cent$^{-1}$.
3. The radial and tangential components of the space velocity
are $V_{r}=79 \pm 6$~km~s$^{-1}$ and $V_{t}=198 \pm 50$~km~s$^{-1}$, respectively, as measured by a Galactocentric
observer at rest.
4. The best estimate of the orbit shows that Sculptor is
approaching its apogalacticon of $R_{a}=122$~kpc on a polar
orbit with eccentricity $e=0.29$. The perigalacticon of the orbit is
$R_{p}=68$~kpc and the orbital period is $T=2.2$~Gyr.
5. Sculptor is not a member of streams~2 and 4a proposed by
Lynden-Bell \& Lynden-Bell (1995). It could be a member of stream~4b,
though the proper motion of Fornax measured by Dinescu \etal\ (2004)
makes the physical reality of this stream doubtful.
6. The proper motion vectors of Sculptor and Fornax show that
they cannot be members of the same stream.
7. The pole of the orbit of Sculptor is 26~degrees from the
pole of the plane of the Galactic dSphs noted by Kroupa \etal\ (2004).
This difference is much larger than the uncertainty in the pole of the
orbit, but is probably within the uncertainty of the definition of the
plane of the dSphs.
8. A comparison of the orbit of Sculptor to those of other
dSphs does not provide a clear reason for why Sculptor contains HI
while the others do not. The origin and distribution of HI remain
puzzling. This article proposes that, while the line connecting the
two clouds of HI is nearly perpendicular to our Galactic-rest-frame
proper motion, some combination of ram pressure, tidal interaction, and
Rayleigh-Taylor instability could produce this geometry.
\acknowledgments
We thank the anonymous referee for comments that helped to improve the
presentation of our work. We also thank Sergei Maschenko for pointing
out to us that our method for propagating uncertainties from the
measured proper motions to the space velocity was incorrect. CP and SP
acknowledge the financial support of the Space Telescope Science
Institute through the grants HST-GO-07341.03-A and HST-GO-08286.03-A
and from the National Science Foundation through the grant
AST-0098650. EWO acknowledges support from the Space Telescope Science
Institute through the grants HST-GO-07341.01-A and HST-GO-08286.01-A
and from the National Science Foundation through the grants AST-9619524
and AST-0098518. MM acknowledges support from the Space Telescope
Science Institute through the grants HST-GO-07341.02-A and
HST-GO-08286.02-A and from the National Science Foundation through the
grant AST-0098661. DM is supported by FONDAP Center for Astrophysics
15010003.
\clearpage
\setcounter{figure}{0}
\clearpage
\clearpage
\clearpage
\clearpage
\clearpage
\clearpage
\clearpage
\clearpage
\clearpage
\clearpage
\clearpage
\clearpage
\clearpage
\setcounter{table}{0}
\newdimen\digitwidth\setbox0=\hbox{\rm 0}\digitwidth=\wd0
\catcode`@=\active\def@{\kern\digitwidth}
\begin{deluxetable}{lrr}
\tablecolumns{3}
\tablewidth{3.5truein}
\tablecaption{Measured Proper Motion of Sculptor}
\tablehead{
&\colhead{$\mu_{\alpha}$}&\colhead{$\mu_{\delta}$}\\
\colhead{Field}&\multicolumn{2}{c}{(mas cent$^{-1}$)}\\
\colhead{(1)}&\colhead{(2)}&\colhead{(3)}}
\startdata
SCL@J$0100-3341$ &$9\pm17$&$14\pm16$ \\
\noalign{\vspace{1pt}}
SCL@J$0100-3338$ &$8\pm20$&$-27\pm25$ \\
\hline
\noalign{\vspace{1pt}}
Weighted mean&$9\pm13$&$2\pm13$\\
\enddata
\end{deluxetable}
\begin{deluxetable}{ccccccc}
\tablecolumns{7}
\tablewidth{5.0truein}
\tablecaption{Measured Proper Motions For Objects in
the SCL~$J0100-3341$ Field}
\tablehead{ &X&Y& &$\mu_{\alpha}$&$\mu_{\delta}$ & \\
\colhead{ID}&\colhead{(pixels)}&\colhead{(pixels)}&\colhead{$S/N$}&\colhead{(mas
@cent$^{-1}$)}&
\colhead{(mas@cent$^{-1}$)} & \colhead{$\chi^2$} \\
\colhead{(1)}&\colhead{(2)}&\colhead{(3)}&\colhead{(4)}&\colhead{(5)}&
\colhead{(6)} & \colhead{(7)}
}
\startdata
1& 459& 352& 125& $ 0 \pm 21 $& $ 0 \pm 21 $& \nodata \\
2& 327& 740& 164& $ 2766 \pm 17 $& $ 1048 \pm 18 $& \nodata \\
3& 588& 330& 112& $ -812 \pm 16 $& $-1714 \pm 17 $& \nodata \\
4& 137& 627& 99& $ 180 \pm 17 $& $ -127 \pm 21 $& \nodata \\
5& 496& 398& 10& $ -66 \pm 46 $& $ 178 \pm 72 $& \nodata \\
6& 574& 710& 6& $ 36 \pm 93 $& $ 258 \pm 63 $& \nodata \\
\enddata
\end{deluxetable}
\begin{deluxetable}{ccccccc}
\tablecolumns{7}
\tablewidth{5.0truein}
\tablecaption{Measured Proper Motions For Objects in
the SCL~$J0100-3338$ Field}
\tablehead{ &X&Y& &$\mu_{\alpha}$&$\mu_{\delta}$ & \\
\colhead{ID}&\colhead{(pixels)}&\colhead{(pixels)}&\colhead{$S/N$}&\colhead{(mas
@cent$^{-1}$)}&
\colhead{(mas@cent$^{-1}$)} & \colhead{$\chi^2$} \\
\colhead{(1)}&\colhead{(2)}&\colhead{(3)}&\colhead{(4)}&\colhead{(5)}&
\colhead{(6)} & \colhead{(7)}
}
\startdata
1& 523& 490& 180& $ 0 \pm 24 $& $ 0 \pm 28 $& 2.10 \\
2& 414& 177& 42& $ -188 \pm 22 $& $ -299 \pm 20 $& 0.61 \\
3& 950& 832& 20& $ 89 \pm 31 $& $ -562 \pm 27 $& 4.25 \\
\enddata
\end{deluxetable}
\hoffset=-0.5truein
\begin{deluxetable}{lrrrrrrrrr}
\tablecolumns{10}
\tablewidth{8.8truein}
\tablecaption{Galactic-Rest-Frame Proper Motion and Space Velocity of Sculptor}
\tablehead{&\colhead{$\mu_{\alpha}^{\mbox{\tiny{Grf}}}$}&
\colhead{$\mu_{\delta}^{\mbox{\tiny{Grf}}}$}&
\colhead{$\mu_{l}^{\mbox{\tiny{Grf}}}$}&\colhead{$\mu_{b}^{\mbox{\tiny{Grf}}}$}&
\colhead{$\Pi$}&\colhead{$\Theta$}&\colhead{$Z$}&\colhead{$V_{r}$}&
\colhead{$V_{t}$}\\
\colhead{Field}&\multicolumn{2}{c}{(mas cent$^{-1}$)}
&\multicolumn{2}{c}{(mas cent$^{-1}$)}&\colhead{(km s$^{-1}$)}&\colhead{(km s$^{-1}$)}&\colhead{(km s$^{-1}$)}
&\colhead{(km s$^{-1}$)}&\colhead{(km s$^{-1}$)}\\
\colhead{(1)}&\colhead{(2)}&\colhead{(3)}&\colhead{(4)}
&\colhead{(5)}&\colhead{(6)}&\colhead{(7)}&\colhead{(8)}&\colhead{(9)}
&\colhead{(10)}}
\startdata
SCL@J$0100-3341$&$-23\pm17$&$57\pm16$& $6\pm17$&$-62\pm17$& $-186\pm69$&$159\pm68$&$-107\pm8$& $82\pm7$&$254\pm66$\\
\noalign{\vspace{5pt}}
SCL@J$0100-3338$&$-24\pm20$&$17\pm25$& $18\pm21$&$-23\pm25$& $-113\pm88$&$10\pm100$&$-88\pm12$& $73\pm9$&$124\pm76$\\
\noalign{\vspace{3pt}}
\hline
\noalign{\vspace{5pt}}
Weighted mean&$-23\pm13$&$45\pm13$& $11\pm13$&$-50\pm14$& $-158\pm54$&$112\pm56$&$-101\pm7$& $79\pm6$&$198\pm50$\\
\enddata
\end{deluxetable}
\begin{deluxetable}{lcccc}
\tablecolumns{5}
\tablewidth{4.75truein}
\tablecaption{Orbital elements of Sculptor}
\tablehead{Quantity&Symbol&Unit&Value&95\% Conf. Interv.\\
\colhead{(1)}&\colhead{(2)}&\colhead{(3)}&\colhead{(4)}&\colhead{(5)}}
\startdata
Perigalacticon&$R_{p}$&kpc&$68$&$(31,83)$ \\
\noalign{\vspace{1pt}}
Apogalacticon&$R_{a}$&kpc&$122$&$(97,313)$\\
\noalign{\vspace{1pt}}
Eccentricity &$e$&&$0.29$&$(0.26,0.60)$\\
\noalign{\vspace{1pt}}
Period&$T$&Gyr&$2.2$&$(1.5,4.9)$\\
\noalign{\vspace{1pt}}
Inclination&$\Phi$°&86&$(83,90)$ \\
\noalign{\vspace{1pt}}
Longitude&$\Omega$°&275&$(243,306)$ \\
\enddata
\end{deluxetable}
\begin{deluxetable}{lcccc}
\tablecolumns{5}
\tablewidth{4.5truein}
\tablecaption{Predicted Proper Motion of Sculptor}
\tablehead{
&
\colhead{$\mu_{\alpha}$}&
\colhead{$\mu_{\delta}$}&
\colhead{$\vert \vec{\mu}\vert$}&
\colhead{PA}\\
\colhead{Stream No.}&\multicolumn{3}{c}{(mas
cent$^{-1}$)}&\colhead{(degrees)}\\
\colhead{(1)}&\colhead{(2)}&\colhead{(3)}&\colhead{(4)}&\colhead{(5)}}
\startdata
2 & 51 & --80 & 95 & 147 \\
\noalign{\vspace{1pt}}
4a & 80 & --61 & 101 & 127 \\
\noalign{\vspace{1pt}}
4b & -13 & -27 & 30 & 205\\
\noalign{\vspace{1pt}}
\hline
\noalign{\vspace{1pt}}
Our Result&$9\pm13$&$2\pm13$&$9\pm13$&$77\pm81$ \\
\enddata
\end{deluxetable} |
Title:
Exploring the surface properties of Transneptunian Objects and Centaurs with polarimetric FORS1/VLT observations |
Abstract: Polarization is a powerful remote-sensing method to investigate solar system
bodies. It is an especially sensitive diagnostic tool to reveal physical
properties of the bodies whose observational characteristics are governed by
small scatterers (dust, regolith surfaces). For these objects, at small phase
angles, a negative polarization is observed, i.e., the electric vector E
oscillates predominantly in the scattering plane, contrary to what is typical
for rather smooth homogeneous surfaces. The behavior of negative polarization
with phase angle depends on the size, composition and packing of the
scatterers. These characteristics can be unveiled by modelling the light
scattering by the dust or regolith in terms of the coherent backscattering
mechanism.
We have investigated the surface properties of TNOs and Centaurs by means of
polarimetric observations with FORS1 of the ESO VLT.
TNOs Ixion and Quaoar, and Centaur Chiron show a negative polarization surge.
The Centaur Chiron has the deepest polarization minimum (-1.5 - 1.4%). The two
TNOs show differing polarization curves: for Ixion, the negative polarization
increases rapidly with phase; for Quaoar, the polarization is relatively small
(~ -0.6%), and nearly constant at the observed phase angles. For all three
objects, modelling results suggest that the surface contains an areal mixture
of at least two components with different single-scatterer albedos and photon
mean-free paths.
| https://export.arxiv.org/pdf/astro-ph/0601414 |
\title{Exploring the surface properties of Transneptunian Objects and
Centaurs with polarimetric FORS1/VLT observations
\thanks{Based on observations made with ESO Telescopes at the
Paranal Observatory under programme ID 69.C-0133
and 073.C-0561 (PI: H.\ Boehnhardt)}}
\author{
S.~Bagnulo \inst{1}
\and
H.~Boehnhardt \inst{2}
\and
K.~Muinonen \inst{3}
\and
L.~Kolokolova \inst{4}
\and
I.~Belskaya \inst{5}
\and
M.A.~Barucci \inst{6}
}
\institute{European Southern Observatory,
Alonso de Cordova 3107, Vitacura,
Santiago, Chile.
\email{sbagnulo@eso.org}
\and
Max-Planck-Institut f\"{u}r Sonnensystemforschung,
Max-Planck-Strasse 2,
37191 Katlenburg-Lindau,
Germany.\\
\email{hboehnha@linmpi.mpg.de}
\and
Observatory, PO Box 14, 00014 University of Helsinki, Finland.
\email{muinonen@cc.helsinki.fi}
\and
University of Maryland, College Park, MD, USA.
\email{ludmilla@astro.umd.edu}
\and
Astronomical observatory of Kharkiv National University,
35 Sumska str., 61022 Kharkiv, Ukraine.
\email{irina@astron.kharkov.ua}
\and
LESIA, Observatoire de Paris, 5,
pl.~Jules Janssen, FR-92195 Meudon cedex, France.
\email{antonella.barucci@obspm.fr}
}
\date{Received: 15 November 2005 / Accepted: 3 January 2006}
\abstract{
Polarization is a powerful remote-sensing method to investigate solar
system bodies. It is an especially sensitive diagnostic tool to reveal
physical properties of the bodies whose observational characteristics
are governed by small scatterers (dust, regolith surfaces). For these
objects, at small phase angles, a negative polarization is observed,
i.e., the electric vector $\vec{E}$ oscillates predominantly in the
scattering plane, contrary to what is typical for rather smooth
homogeneous surfaces. The behavior of negative polarization with
phase angle depends on the size, composition and packing of the
scatterers. These characteristics can be unveiled by modelling the
light scattering by the dust or regolith in terms of the coherent
backscattering mechanism.
}
{We investigate the surface properties of TNOs and Centaurs by means
of polarimetric observations with FORS1 of the ESO VLT.
}
{
We have obtained new broadband polarimetric measurements over
a range of phase angles for a TNO, 50000\,Quaoar (in the $R$
Bessel filter), and a Centaur, 2060\,Chiron (in the $BVR$
Bessel filters). Simultaneously to the polarimetry, we have obtained
$R$ broadband photometry for both objects. We have modelled these new
observations of Quaoar and Chiron, and revised the modelling of
previous observations of the TNO 28978\,Ixion using an
improved value of its geometric albedo.
}
{
TNOs Ixion and Quaoar, and Centaur Chiron show a negative polarization
surge. The Centaur Chiron has the deepest polarization minimum
(-1.5 -- 1.4\,\%). The two TNOs show differing polarization curves: for
Ixion, the negative polarization increases rapidly with phase; for
Quaoar, the polarization is relatively small ($\simeq -0.6$\,\%), and
nearly constant at the observed phase angles. For all three objects,
modelling results suggest that the surface contains an areal mixture
of at least two components with different single-scatterer albedos and
photon mean-free paths.
}
{}
\keywords{Kuiper Belt -- polarization}
\titlerunning{Polarimetry of Kuiper belt TNO objects and Centaurs}
\authorrunning{S.~Bagnulo et al.\ }
\section{Introduction}\label{Sect_Introduction}
Transneptunian objects (TNOs) in the Kuiper Belt are considered to
represent one of the oldest and possibly most original population of
solar system bodies that can be observed from Earth. Centaurs are
escapees from the Kuiper Belt through gravitational interaction with
Neptune and the other giant planets. They may eventually become members
of the Jupiter family of comets, or may be ejected from the planet
region due to close encounters with the giant planets.
The intense study of physical properties of TNOs and Centaurs was
triggered by the advent of large telescopes on the ground: besides a
large set of photometric colours, also visible and near-IR spectra of
a number of objects are available now. Polarimetric observations are
more scarce: except for Pluto/Charon system (that was observed
unresolved, e.g., by Kelsey \& Fix \cite{KelFix73}), it was only
recently that broadband polarized radiation of a TNO, the Plutino
28978\,Ixion, has been observed and modelled (Boehnhardt et
al.~\cite{Boeetal04}).
Polarimetry is a powerful tool to investigate the physical properties
of atmosphereless bodies. At small ($\le 30\degr$) phase angles (the
phase angle is the angle between the Sun and the observer as seen from
the object), these objects exhibit a phenomenon of \textit{negative
polarization}: the observed flux perpendicular to the plane
Sun-Object-Observer (the scattering plane) minus the observed flux
perpendicular to that plane, divided by the sum of the two fluxes,
turns to be a negative quantity. This phenomenon, first discovered
through lunar observations by Lyot (\cite{Lyot29}), escapes from
common sense interpretation, since elementary physics tells that
reflected electric vector $\vec{E}$ oscillates predominantly in the
plane perpendicular to the scattering plane rather than in the
scattering plane. Solar-system objects show two types of angular
dependence of negative polarization: either a smooth phase-angle
change that has the minimum at $\sim 10\degr$ (S-, C- asteroids, Moon)
or a sharp surge with the minimum at $\sim 1-2\degr$ (Saturn rings,
Europa, E-asteroids) (see, e.g., Rosenbush et
al.~\cite{Rosetal02}). Both types of negative polarization, which also
were observed in powdered laboratory samples, are currently
interpreted in terms of enhanced backscattering of multiply scattered
rays (Shkuratov \cite{Shkuratov89}; Muinonen \cite{Muinonen90}).
Observations of negative polarization and simultaneous photometry of
main-belt asteroids and other solar system bodies (see, e.g., Belskaya et
al. \cite{Beletal05}; Rosenbush et al. \cite{Rosetal05}) can be
modelled to infer the properties of the surface texture of these
objects. Faintness of the targets was the main obstacle hampering the
same kind of study in TNOs and Centaurs\footnote{Another difference in
the observing and modelling techniques is that, due to the larger
distance, the observed phase angle range is much smaller for TNOs and
Centaurs than for main-belt asteroids}. Thanks to the advent of the large
telescopes and instruments equipped with polarimetric capabilities,
observations of TNOs and Centaurs are nowadays possible with signal to
noise ratio comparable to that commonly reached for main-belt asteroid
observations with small and middle-size telescopes.
After our first polarimetric study of 28978\,Ixion (Boehnhardt et
al. \cite{Boeetal04}), in this paper we present new polarimetric and
photometric measurements obtained with FORS1 at the ESO Very Large
Telescope (VLT) for a TNO, 50000\,Quaoar, and a Centaur,
2060\,Chiron. We also present a revised modelling of the observed
polarization and photometry of 28978\,Ixion based on a determination
of the geometric albedo that has been recently obtained, and that was
not available at the time of our first modelling effort.
\section{Target summary}\label{Sect_Targets}
Criteria used for target selection are that the targets are bright
enough to allow us to measure the polarization with an error bar
smaller than 0.05\,\% in less than two hours telescope time. With
FORS1 at the ESO VLT, this sets the $R$ magnitude limit to about
20. Another constraint is the possibility to observe the largest
possible phase angle range. Complementary information on the geometric
albedo and surface composition is essential for the modelling
part. Moreover, it is desirable to study members of the various
dynamical and taxonomic groups identified among the TNO population
(Plutinos, Centaurs, classical and scattered-disk Objects). We finally
selected three objects with known physical parameters
(geometric albedos, colours, spectral slopes, and surface composition):
Ixion, Quaoar, and Chiron.
\subsection{28978\,Ixion}
28978\,Ixion, discovered in 2001, belongs to the dynamical class of
Plutinos, and it is one of the largest known TNOs (400--550\,km
according to Stansberry et al. \cite{Stansbe05}). The visible
spectrum by Marchi et al. (\cite{Maretal03}) is featureless with a
gradient $S'$ of 19.8\,\%/100\,nm. Optical and near-IR (Licandro et
al. \cite{Licetal02}) spectra have been interpreted by Boehnhardt et
al. (\cite{Boeetal04}) using an areal mixture of Titan tholin,
amorphous carbon, water ice, and ice tholin. The same authors also
present a surface model of Ixion based upon their $R$ filter
polarimetry and simultaneous $R$ band photometry of the objects
spanning the phase angle range 0.25\degr\ -- 1.34\degr. In that work,
an $R$-band geometric albedo of 0.1 was assumed for the
modelling. Here we repeat the analysis for the higher $R$-band
geometric albedo now available from Spitzer observations (0.23;
Stansberry, priv.\ comm.).
Further details about the properties of this object are given by
Boehnhardt et al.~(\cite{Boeetal04}).
\subsection{50000\,Quaoar}
50000\,Quaoar is a classical disk object in the Kuiper Belt. Orbital
elements and red visible colours (Fornasier et al.~\cite{Foretal04})
suggest that the object could be a member of the ``dynamically hot''
population that is supposed to have migrated to the classical disk
only after formation closer to the Sun (Gomes \cite{Gomes03}). Apart
from Pluto/Charon, 50000\,Quaoar is the only TNO so far for that
disk-resolved photometry could be performed: HST measurements
allowed to determine the overall size and geometric albedo of the
object to be $1260 \pm 190$\,km, and about 0.1, respectively (Brown \&
Trujillo \cite{BroTru04}). The photometric lightcurve of the object
seems to be double-peaked with a period of about 17.6\,h and an
amplitude of 0.13\,mag suggesting an aspherical shape of the body
and/or geometric albedo variations of the surface (Ortiz et
al.\ \cite{Ortetal03}).
Quaoar visible spectra were obtained by Marchi et al.\
(\cite{Maretal03}) and by Fornasier et al.\ (\cite{Foretal04}). The
reflectivity gradients $S'$ obtained in the two papers are not fully
consistent, and their mean value is $27.6 \pm
0.3$\,\%/100\,nm. Visible spectrum appears to be featureless.
Quaoar has been observed also in the near-infrared by Jewitt and Luu
(\cite{JewLuu04}) at the Subaru 8\,m telescope. The complete spectrum
shows a positive-slope continuum from 0.4 up to 1.3\,$\mu$m, that is
considered typical for the presence of organic materials on its
surface. The spectrum shows strong absorption bands at 1.5 and
2.0\,$\mu$m due to H$_2$O ice with the band at 1.65\,$\mu$m typical
for the crystalline structure in the ice. A small presence of ammonia
hydrate has also been supposed on the basis of the presence of faint
features at 2.2\,$\mu$m (detected also by Pinilla-Alonso et al.\
\cite{Pinetal04}). This was the first time that the quality of a TNO
spectrum was good enough to distinguish between crystalline and
amorphous ice. The detection of crystalline ice indicates that the
temperature has reached at least 110\,K (critical temperature
necessary for crystallization). This object is large enough to be
cryovolcanically active, and crystalline ice and ammonia hydrate might
be products of this type of activity. Jewitt and Luu
(\cite{JewLuu04}) suggested that Quaoar has been recently resurfaced
either by impacts or by cryovolcanic outgassing or by a combination
of these two processes.
\subsection{2060\,Chiron}
2060\,Chiron is the first discovered (1977) and best observed
Centaur. The intermediate character of the object between TNOs and
comets is apparent from photometric observations that show recurrent
episodes of coma activity (gas and dust) and of stellar appearance
(see for example Meech \& Belton \cite{MeeBel90}, Luu \& Jewitt
\cite{LuuJew90}, Bus et al.\ \cite{Busetal91}, Duffard et al.\
\cite{Dufetal02}).
Chiron was observed spectroscopically in the visible region by many
authors (see Barucci et al.\ \cite{Baretal03}) showing a flat spectrum
with no absorption features. The reflectivity gradient $S'$, which
ranges between $-0.2$ up to $2.3 \pm 0.1$\,\%/100\,nm, seems more
similar to that of C-type asteroids rather than to the mean
reflectance slope of cometary nuclei. The small variation on the
optical reflectivity gradient could be due to dust production
variation connected to episodes of recurrent cometary activity.
Several spectra obtained in the near-infrared did not show any
features. Only Foster et al.~(\cite{Fosetal99}) and Luu et
al.~(\cite{Luuetal00}) detected a 2\,$\mu$m absorption band suggesting
the presence of H$_2$O ice on Chiron's surface. Later on, Romon-Martin
et al. (\cite{Rometal03}) observed Chiron again during high activity
in the visible and NIR showing a flat behaviour without any spectral
features finding that is compatible with the hypothesis made by Luu et
al.\ (\cite{Luuetal00}) that the detection of water ice in Chiron
spectra would be correlated with its cometary activity level. Such
activity could cause a rain of cometary debris on its surface changing
the surface mantle. The ice present on the surface, is probably mixed
with dark impurities which mask the spectral bands.
Nucleus properties using multi-wavelength information has revealed an
about 70\,km nucleus of relatively bright geometric albedo (0.17) and
moderate axis ratio (1.16) or surface albedo variations (Groussin et
al.\ \cite{Groetal04}).
From the large number of publications on Chiron (more than 150
to-date) several interesting properties of 2060\,Chiron have been
worked out. However, a synoptic picture of the nucleus and its surface
properties has not yet evolved.
\section{New observations with FORS1}\label{Sect_New_Observations}
Observations of Chiron and Quaoar have been obtained at the ESO VLT
with the FORS1 instrument in service mode during the observing period
from April to September 2004. Until June 1, 2004, FORS1 was attached
at the VLT Unit Telescope 1 (Antu). After that date, FORS1 was moved
to the VLT Unit Telescope 2 (Kueyen).
FORS1 is a multi-mode instrument for imaging and (multi-object)
spectroscopy equipped with polarimetric optics. For the present study,
FORS1 has been used to measure the broadband polarization of Chiron at
six different epochs in the Bessel $BVR$ filters, and to measure the
broadband polarization of Quaoar at five different epochs in the
Bessel $R$ filter (Sect.~\ref{Sect_Polarimetry}). In fact, one series
of Chiron measurements was started on night 2004-08-05/06 and aborted
after the observations in the $R$ filter because the seeing conditions
were too good ($\la 0.6''$) with consequent risk of CCD saturation.
The series was repeated the following night. From target acquisition
images, a by-product of polarimetric observations, we could also
obtain photometry in the Bessel $R$ filter (Sect.~\ref{Sect_Photometry}).
Differential tracking was used for all our observations, so that
long exposure time images of Chiron, obtained in polarimetric mode, could
be combined altogether to search for coma activity
(Sect.~\ref{Sect_Coma_Searching}).
Taking advantage of the flexibility offered by the `VLT service
observing mode', we distributed the observations along a few months as
to obtain data points approximately equally spread over the phase
angle ranges of the targets. We set precise time intervals for the
execution of the observations. In presence of the Moon, the
sky-background is highly polarized, hence we generally tried to avoid
observations with the target close to the Moon, and with a too large
fraction of lunar illumination. However, \textit{a posteriori} we
found that the presence of the Moon did not jeopardize our
observations. The log of the observations can be inferred from Tables
\ref{Tab_Pol_Chi} to \ref{Tab_Pho_Qua}.
\subsection{Polarimetry}\label{Sect_Polarimetry}
To perform linear polarization measurements, a $\lambda/2$ retarder
waveplate and a Wollaston prism are inserted in the FORS1 optical path
(see Appenzeller \cite{App67}). The $\lambda/2$ retarder waveplate can
be rotated in 22.5\degr\ steps. Stokes~$Q$ and $U$ parameters (defined
as in Shurcliff \cite{Shu62}) are measured by combining the photon
counts (background subtracted) of ordinary and extra-ordinary beams
(\fo\ and \fe, respectively) observed at various retarder waveplate
positions $\alpha$, where $\alpha$ indicates the angle between the
acceptance axis of the ordinary beam of the Wollaston prism and the
fast axis of the retarder waveplate.
In the following, we will always work with the ratios $Q/I$ and $U/I$,
and will adopt the notation:
\[
\pq = \frac{Q}{I}\ \ {\rm and}\ \ \pu = \frac{U}{I}
\]
In the ideal case, \pq\ is
obtained measuring the quantity
\[
r = (-1)^k\ \frac{\fo - \fe}{\fo +\fe}
\]
at any retarder waveplate position $\alpha=k\,45\degr$,
and \pu\ is obtained measuring the
ratio $r$ at any position $\alpha=k\,45\degr + 22.5\degr$
($k=0,\,1,\,2,\,\ldots,\,7$). The validity of
this assertion can be verified e.g.\ with the help of Eq.~(1.33) of
Landi Degl'Innocenti \& Landolfi (\cite{LanLan04}). In practice, there
are several deviations from the ideal case. For instance, the actual
retardance value of the retarder waveplate may deviate from the
nominal $\pi$ value; the transmission of the ordinary and
extraordinary beam are not identical, even after flat fielding
correction. The effect of these (and other) sources of
\textit{instrumental polarization} can be largely reduced at the first
order by measuring
\begin{equation}
\begin{array}{rcl}
\pq^{ij} &=&
\frac{1}{2} \Bigg\{
\left(\frac{\fo - \fe}{\fo + \fe}\right)_{\alpha = 2(i-1) \times 45^\circ} -
\left(\frac{\fo - \fe}{\fo + \fe}\right)_{\alpha = (2j-1) \times 45^\circ}
\Bigg\} \\
\pu^{ij} &=&
\frac{1}{2} \Bigg\{
\left(\frac{\fo - \fe}{\fo + \fe}\right)_{\alpha = 2(i-1) \times 45^\circ + 22.5^\circ} -
\left(\frac{\fo - \fe}{\fo + \fe}\right)_{\alpha = (2j-1) \times 45^\circ + 22.5^\circ}
\Bigg\} \\
\end{array}
\label{Eq_Stokes_QU}
\end{equation}
where $i$ and $j$ are integers numbers\footnote{We note that in a
similar formula that we reported in Boehnhardt et al.\ (\cite{Boeetal04}),
the factor 2 in the indices that denote the angles is missing because of
a typo.}. In the simplest case, linear polarization can be measured from
the observations obtained at four angles of the retarder waveplate:
\[
\begin{array}{rcl}
\pq & = &
\frac{1}{2} \Bigg\{
\left(\frac{\fo - \fe}{\fo + \fe}\right)_{\alpha= 0\degr} -
\left(\frac{\fo - \fe}{\fo + \fe}\right)_{\alpha=45\degr}
\Bigg\} \\
\pu & = &
\frac{1}{2} \Bigg\{
\left(\frac{\fo - \fe}{\fo + \fe}\right)_{\alpha=22.5\degr} -
\left(\frac{\fo - \fe}{\fo + \fe}\right)_{\alpha=67.5\degr}
\Bigg\} \\
\end{array}
\]
It is convenient (and recommended in the FORS1/2 user manual) to
obtain Stokes $Q$ and $U$ adding up observations obtained with
the retarder waveplate at various positions:
\begin{equation}
\begin{array}{rcl}
\pq&=&\frac{1}{m}\, \sum_{l=1}^{m} \pq^{ll} \\ [3mm]
\pu&=&\frac{1}{m}\, \sum_{l=1}^{m} \pu^{ll} \;,\\
\end{array}
\label{Eq_Sumpol}
\end{equation}
where $m$ represents the number of pairs of observations for each
Stokes parameter, and $\pq^{ll}$ and $\pu^{ll}$ are obtained from
Eq.~(\ref{Eq_Stokes_QU}) setting $i=j=l$. We performed simple
numerical simulations to study the impact on the precision of the
polarimetric measurements of a deviation of the waveplate retardance
from its nominal value (180\degr). We found that using
Eq.~(\ref{Eq_Sumpol}) with $m=2$, a deviation from the nominal value
of the waveplate retardance as large as 5\degr, for the polarization
value observed in our targets, would introduce a spurious contribution
$\ll 0.01$\,\%. Figure~4.1 of FORS1/2 user manual shows
that actual deviation of the retarder waveplate from the nominal value
is well within 5\degr. We conclude that in our data, the effect of
instrumental polarization due to the chromathism of the retarder
waveplate is negligible.
It should be noted that FORS1 is affected by a problem with
spurious linear instrumental polarization that cannot be eliminated
even by using the reduction technique explained above. This spurious
polarization, due to the presence of rather curved lenses in the
collimator, combined with the non complete removal of reflections by
the coatings, is axially symmetric, and smoothly increases from less
than 0.03\,\% on the optical axis to 0.7\,\% at an axis distance of 3
arcmin (in the $V$ band). This problem has been discovered and
investigated by Patat \& Romaniello (\cite{PatRom05}). In our case,
since our targets are always in the center of the field of view, the
problem of the instrumental polarization can be safely ignored.
The error-bar due to photon-noise on the \pq\ or \pu\ measured from a pair of
observations is
\begin{equation}
\begin{array}{rcl}
\sigma^2_{X^{ij}} & = &
\left(\left(\frac{\fe}{(\Den)^2}\right)^2 \sigma^2_{\fo} +
\left(\frac{\fo}{(\Den)^2}\right)^2 \sigma^2_{\fe}\right)_{\alpha=2(i-1) \times 45^\circ + \phi_0 } + \\
& &
\left(\left(\frac{\fe}{(\Den)^2}\right)^2 \sigma^2_{\fo} +
\left(\frac{\fo}{(\Den)^2}\right)^2 \sigma^2_{\fe}\right)_{\alpha=(2j-1) \times 45^\circ + \phi_0} \;, \\
\end{array}
\label{Eq_Sigma_QU}
\end{equation}
where $\phi_0 = 0$ in case $X$ represents \pq\ and $\phi_0=22.5\degr$
in case $X$ represents \pu.
When \pq\ and \pu\ are obtained adding up $m$ pairs as in
Eq.~(\ref{Eq_Sumpol}), the error is given by
\[
\sigma^2_{X} = \frac{1}{m} \sum_{l=1}^{m} \sigma^2_{X^{ll}}
\]
For the sake of simplicity, assuming $\fo
= \fe = \cal{N}$ and also assuming $\cal{N}$ to be the same for all
positions of the retarder waveplate, we obtain
\begin{equation}
\sigma_{X} = \frac{1}{2} \frac{1}{\sqrt{m\cal{N}}}
\label{Eq_Photon_Noise}
\end{equation}
where $\sqrt{\cal{N}}$ represents the signal to noise ratio (SNR)
measured in the individual beam for each of the $m$ exposures (in
other words, $\sqrt{m\cal{N}}$ is the cumulative SNR in each beam).
It appears for instance that to get an error bar on the Stokes
parameter \pq\ or \pu\ of about 0.1\,\%, one should take a pair of
exposures with SNR of about 500 each (integrated over the
individual point-spread function area of each beam of each exposure).
When multiple pairs of exposures are taken, it is useful to study the
distribution of the $\pq^{ij}$ ($\pu^{ij}$) values obtained
substituting $i,j = 1, 2, \ldots m$. In particular, a
$\sigma$-clipping algorithm can be applied to the $\pq^{ij}$ and
$\pu^{ij}$ distributions in order to ``clean'' the data, by rejecting
those values that deviate more than a certain distance from the median
(see Boehnhardt et al.\ \cite{Boeetal04}, and Bagnulo et al.\
\cite{Bagetal05}).
Stokes~$Q$ and $U$ are usually measured with the instrument position
angle = 0\degr, i.e. to have the acceptance axis of the
ordinary beam of the Wollaston prism aligned to the North Celestial
Meridian (and the acceptance axis of the extra-ordinary beam perpendicular
to it). We then transform the Stokes parameters according to
\begin{equation}
\begin{array}{rcl}
\pq' &=& \phantom{-}\cos(2(\Phi+\pi/2))\, \pq + \sin(2(\Phi+\pi/2))\, \pu\\
\pu' &=& - \sin(2(\Phi+\pi/2))\, \pq + \cos(2(\Phi+\pi/2))\, \pu\\
\end{array}
\label{Eq_QU_Transform}
\end{equation}
where $\Phi$ is the angle between the direction Object-North Pole
and the direction Object-Sun. This angle can be calculated applying
the four parts formula to the spherical triangle defined by the object
(with coordinates \alphaobj,\deltaobj), the Sun (with coordinates
(\alphasun,\deltasun) and the North celestial pole:
\[
\sin \deltaobj \cos(\alphasun - \alphaobj) =
\cos(\deltaobj) \tan(\deltasun) - \sin(\alphasun - \alphaobj) \frac{1}{\tan(\Phi)}\;.
\]
This way $\pq'$ \textit{represents the flux perpendicular to the
plane Sun-Object-Earth (the scattering plane) minus the flux parallel
to that plane, divided by the sum of the two fluxes}.
The angle of maximum polarization is obtained as
\begin{equation}
\Thetar = \frac{1}{2} \arctan \left(\frac{\pu'}{\pq'}\right) + \Theta_0'
\end{equation}
where
\[
\Theta_0' = \cases {0 &{\rm if} $\pq' > 0$ and $\pu'\ge0$ \cr
\pi &{\rm if} $\pq' > 0$ and $\pu'<0$ \cr
\pi/2 &{\rm if} $\pq' < 0$ \cr } \;.
\]
or
\[
\Thetar = \cases{\pi/4 &{\rm if} $\pq' = 0$ and $\pu' > 0$ \cr
3\pi/4 &{\rm if} $\pq' = 0$ and $\pu' < 0$ \cr
}
\]
(see Landi Degl'Innocenti \& Landolfi \cite{LanLan04}.) Incidentally,
it should be noted that the $\Theta_0'$ term is occasionally incorrectly
neglected.
Observations of Chiron were performed with the retarder waveplate at
all positions between 0 and 157.5\degr\ (at 22.5\degr\ steps), i.e.,
setting $m=2$ in Eq.~(\ref{Eq_Sumpol}), using the broadband Bessel
$B$, $V$, and $R$ filters. For Quaoar we used all positions of the
retarder waveplate from 0\degr\ to 337.5\degr, i.e., setting $m=4$ in
Eq.~(\ref{Eq_Sumpol}), using the Bessel $R$ filter. For each frame,
exposure times $t$ were as follows. For Chiron we set $t=460, 140$,
and 110\,s in the $B$, $V$, and $R$ filter, respectively. For Quaoar
($R$ filter only), we set $t = 250$\,s. Note that, for each Stokes
parameter we obtained 2 pairs of exposures for Chiron, and 4 pairs of
exposures for Quaoar. Therefore the total shutter time for Chiron
observations was of about 60, 20, and 15\,min in the $B$, $V$, and $R$
filter, respectively, and of about 67\,min for Quaoar ($R$ filter only).
For each observation of Chiron, the typical SNR accumulated in each
beam was about 1700, 1600, and 1600, in the $B$, $V$, and $R$ filters
respectively. For each observation of Quaoar, in the $R$ filter, we
obtained a SNR accumulated in each beam of about 1400.
The photon counts \fo\ and \fe\ were measured via simple aperture
photometry performed on the images, obtained after bias subtraction
and flatfield correction (master flat field was obtained from sky
images obtained during twilight with no polarimetric optics in). More
sophisticated methods based on point spread function (PSF)
fitting are difficult to apply because of the star trailing due to
differential tracking on the moving targets. Sky background was
measured in an annulus with 5\,\arcsec\ inner radius, centered around
the source, of 2\arcsec\ to 6\arcsec\ width. The errors on \fo\ and
\fe\ were estimated as explained in the Sect.~3.3.5.8 of Davis
(\cite{Dav87}). \fo, \fe, and their error estimates were measured for
aperture values ranging from 0.6\arcsec to 4\arcsec.
\pq\ and \pu\ values were found to be slightly dependent on the
aperture adopted to measure \fo\ and \fe. This effect was more
critical when the target was in a crowded field and/or in presence of
strong background polarization by the Moon. The final value was
selected as the one for which the error on \pq\ and \pu\ was minimum,
i.e., usually for aperture values between 1.2\arcsec\ and 2.5\,\arcsec.
\begin{table*}
\caption{The observed polarization of 2060\,Chiron. The meaning of
the various columns is given in the text}
\label{Tab_Pol_Chi}
\centering
\begin{tabular}{cccclcccrrr}
\hline \hline
Date &
Time (UT) &
\multicolumn{1}{c}{Moon} &
\multicolumn{1}{c}{} &
\multicolumn{1}{l}{Sky} &
Phase angle &
Filter &
\multicolumn{1}{c}{$\pq'$} &
\multicolumn{1}{c}{$\pu'$} &
\multicolumn{1}{c}{$\Thetar$} \\
\multicolumn{1}{c}{(yyyy mm dd)} &
\multicolumn{1}{c}{(hh:mm)} &
\multicolumn{1}{c}{dist.} &
\multicolumn{1}{c}{FLI} &
\multicolumn{1}{l}{transp.} &
\multicolumn{1}{c}{(DEG)} &
&
\multicolumn{1}{c}{(\%)} &
\multicolumn{1}{c}{(\%)} &
\multicolumn{1}{c}{(DEG)} \\
\hline
2004 05 27 & 06:37 &139\degr&0.3&CLR & 3.469 &$R$&$-1.14 \pm 0.04$&$ 0.02 \pm 0.04$&$89.4 \pm 1.0$\\
2004 05 27 & 07:03 & & & & 3.468 &$V$&$-1.20 \pm 0.04$&$ 0.13 \pm 0.04$&$86.9 \pm 0.9$\\
2004 05 27 & 07:52 & & & & 3.467 &$B$&$-1.24 \pm 0.04$&$-0.04 \pm 0.05$&$90.9 \pm 1.1$\\[2mm]
2004 06 29 & 03:19 & 64\degr&0.4&CLR? & 1.410 &$R$&$-1.40 \pm 0.05$&$ 0.05 \pm 0.04$&$89.1 \pm 0.8$\\
2004 06 29 & 03:44 & & & & 1.409 &$V$&$-1.35 \pm 0.05$&$-0.08 \pm 0.04$&$91.6 \pm 0.9$\\
2004 06 29 & 04:33 & & & & 1.407 &$B$&$-1.32 \pm 0.04$&$ 0.08 \pm 0.05$&$88.2 \pm 1.2$\\[2mm]
2004 08 06 & 05:53 & 94\degr&0.7&PHO & 1.766 &$R$&$-1.40 \pm 0.04$&$ 0.00 \pm 0.04$&$89.9 \pm 0.8$\\[2mm]
2004 08 07 & 03:46 &105\degr&0.7&THN (c)& 1.828 &$R$&$-1.33 \pm 0.04$&$-0.01 \pm 0.04$&$90.2 \pm 0.8$\\
2004 08 07 & 04:13 & & & & 1.829 &$V$&$-1.46 \pm 0.03$&$-0.14 \pm 0.03$&$92.7 \pm 0.7$\\
2004 08 07 & 05:00 & & & & 1.832 &$B$&$-1.52 \pm 0.04$&$-0.09 \pm 0.05$&$91.8 \pm 0.9$\\[2mm]
2004 08 13 & 00:17 &167\degr&0.9&PHO (c)& 2.219 &$R$&$-1.33 \pm 0.04$&$ 0.08 \pm 0.04$&$88.3 \pm 1.0$\\
2004 08 13 & 00:42 & & & & 2.221 &$V$&$-1.34 \pm 0.04$&$-0.08 \pm 0.04$&$91.6 \pm 0.9$\\
2004 08 13 & 01:31 & & & & 2.223 &$B$&$-1.45 \pm 0.03$&$-0.19 \pm 0.05$&$93.8 \pm 1.1$\\[2mm]
2004 09 01 & 00:56 & 74\degr&0.5&THN? (c)& 3.327&$R$&$-1.12 \pm 0.04$&$ 0.05 \pm 0.04$&$88.7 \pm 1.1$\\
2004 09 01 & 01:21 & & & & 3.328 &$V$&$-1.09 \pm 0.05$&$-0.08 \pm 0.04$&$92.1 \pm 1.2$\\
2004 09 01 & 02:10 & & & & 3.330 &$B$&$-1.15 \pm 0.05$&$-0.24 \pm 0.06$&$95.9 \pm 1.6$\\[2mm]
2004 09 27 & 01:24 & 56\degr&0.4&CLR? & 4.232 &$R$&$-0.89 \pm 0.07$&$ 0.03 \pm 0.05$&$89.1 \pm 1.8$\\
2004 09 27 & 01:49 & & & & 4.232 &$V$&$-1.05 \pm 0.05$&$-0.20 \pm 0.05$&$95.4 \pm 1.7$\\
2004 09 27 & 02:38 & & & & 4.233 &$B$&$-0.87 \pm 0.09$&$-0.10 \pm 0.08$&$93.3 \pm 3.0$\\[2mm]
\hline
\end{tabular}
\end{table*}
\begin{table*}
\caption{The observed polarization of 50000\,Quaoar. The meaning of
the various columns is given in the text}
\label{Tab_Pol_Qua}
\centering
\begin{tabular}{cccclcccrrr}
\hline \hline
Date &
Time (UT) &
\multicolumn{1}{c}{Moon} &
\multicolumn{1}{c}{} &
\multicolumn{1}{l}{Sky} &
Phase angle &
Filter &
\multicolumn{1}{c}{$\pq'$} &
\multicolumn{1}{c}{$\pu'$} &
\multicolumn{1}{c}{$\Thetar$} \\
\multicolumn{1}{c}{(yyyy mm dd)} &
\multicolumn{1}{c}{(hh:mm)} &
\multicolumn{1}{c}{dist.} &
\multicolumn{1}{c}{FLI} &
\multicolumn{1}{l}{transp.} &
\multicolumn{1}{c}{(DEG)} &
&
\multicolumn{1}{c}{(\%)} &
\multicolumn{1}{c}{(\%)} &
\multicolumn{1}{c}{(DEG)} \\
\hline
2004 04 18 & 04:43 & 47\degr&1.0&CLR & 0.952 &$R$&$-0.64 \pm 0.05$&$ 0.14 \pm 0.08$&$83.9 \pm 3.9$\\
2004 05 13 & 07:38 & 93\degr&0.8&CLR? & 0.496 &$R$&$-0.50 \pm 0.06$&$-0.14 \pm 0.05$&$97.7 \pm 3.6$\\
2004 05 26 & 03:37 &107\degr&0.2&PHO & 0.252 &$R$&$-0.49 \pm 0.06$&$ 0.10 \pm 0.05$&$84.3 \pm 3.5$\\
2004 07 10 & 01:44 &118\degr&0.7&PHO (c) & 0.797 &$R$&$-0.53 \pm 0.04$&$ 0.09 \pm 0.05$&$85.3 \pm 3.1$\\
2004 08 11 & 02:34 & 75\degr&0.8&CLR? & 1.231 &$R$&$-0.65 \pm 0.04$&$-0.01 \pm 0.06$&$90.4 \pm 2.6$\\
\hline
\end{tabular}
\end{table*}
The results of the polarimetric measurements are given in
Tables~\ref{Tab_Pol_Chi} and \ref{Tab_Pol_Qua}, that are organized as
follows. Columns 1 and 2 give the epoch of the observations (date and
UT time).
Columns~3 and 4 give Moon angular distance and fraction of lunar
illumination (FLI), respectively.
Column 5 gives an estimate of the night time sky conditions: THN =
thin cirrus, CLR= clear, PHO = photometric. The classification given
in these Tables is based on our inspection of reduction products and
of the atmospheric monitors of the observatory at \\ {\tt
http://www.eso.org/gen-fac/pubs/astclim/}\\ \ \ {\tt
forecast/meteo/CIRA/images/repository/lossam/} \\ and is somewhat
arbitrary. Those nights when no photometric standard stars were
observed, or when the zeropoints were not considered stable for the
entire night, are indicated with a question mark. Sky transparency
does not affect the precision of the polarization measurements, but it
does affect the precision of the photometry (see
Tables~\ref{Tab_Pho_Chi} and \ref{Tab_Pho_Qua}). Note that the
precision of the measurements (both polarimetric and photometric)
depends also on how the field of view close to the target is crowded
with background objects. This situation of course changes from epoch
to epoch. Observations labelled with (c) are hampered to some extent
by a crowded background. In the case of Chiron observations on
2004-09-01 the situation was probably especially critical.
Column 6 gives the Sun-Target-Observer angle, i.e., the
target's apparent phase angle as seen at observer's location,
expressed in degrees. This phase angle was obtained from the
object ephemeris calculated at the JPL's
solar system dynamics WWW site at {\tt http://ssd.jpl.nasa.gov}.
Column~7 specifies the broadband filter used for the observations.
Columns~8, 9, and 10 give \pq', \pu', and the angle of maximum
polarization $\Thetar$ , respectively, after the transformation of
Eq.~(\ref{Eq_QU_Transform}). It should be noted that for both Chiron
and Quaoar, the measured $\pu'$ is generally consistent with zero
(equivalent to the fact that $\Thetar$ is generally consistent with
90\degr). This means that the principal axes of the polarization
ellipse are aligned with the coordinate axes perpendicular and
parallel to the scattering plane. Furthermore, the observed $\pq'$,
i.e., the flux perpendicular to the scattering plane minus the flux
parallel to that plane, divided by the sum of the two fluxes, is
always a \textit{negative} quantity. This means that the direction of
the polarization is included in the scattering plane (this case is
normally referred to as ``negative polarization''), in contrast to
what is expected for a dielectric medium, i.e., that the direction of
polarization be perpendicular to the scattering plane (this case is
normally referred to as ``positive polarization'').
Polarimetric measurements of Quaoar and Chiron are plotted against
object phase angle in Figs.~\ref{Fig_Pol_Qua} and \ref{Fig_Pol_Chi}.
As far as Chiron observations are concerned, it appears that the
observed polarization does not depend on the filter used, at least
when considering the error bars of the observations. The absolute
value of the polarization increases as the phase angle decreases,
perhaps reaching a minimum around phase 1.5-2.0 deg. Observations of
this object at smaller phase angles were unfortunately not taken
although originally scheduled in our service observing campaign.
\subsection{Photometry}\label{Sect_Photometry}
The object magnitudes were measured in the acquisition images
obtained in the $R$ Bessel filter with no polarimetric optics in the
light path. Exposure times were 5\,s for Chiron, and 30\,s for Quaoar.
Since our program was aimed at polarization measurements, we did not
obtain a number of observations of photometric standard stars
sufficient to estimate precise zeropoints and extinction coefficients
for the various nights. In most of the cases, the zeropoints were
obtained from only one frame obtained during the night when our
observations were executed (and not necessarily close to them in
time). In some cases, no photometric standard stars were observed at
all (see Tables~\ref{Tab_Pol_Chi} and \ref{Tab_Pol_Qua}). In these
cases we adopted the values measured in the nearest night with similar
sky conditions. For the extinction coefficient $K_R$ and the colour
term $k_{VR}$ we used the values estimated for the P73 ESO observing
period published at\\
{\tt http://www.eso.org/observing/dfo/quality/FORS1/}\\ \ \ {\tt
qc/photcoeff/photcoeffs\_fors1.html}\\ i.e.,
\[
\begin{array}{rcl}
K_R &=& 0.045 \pm 0.019 \\
k_{VR} &=& 0.0090 \pm 0.0017 \\
\end{array}
\]
For the Quaoar colour indices we adopted $V-R = 0.64 \pm 0.01$ (Tegler
et al. \cite{Tegetal03}), and for Chiron $V-R = 0.37 \pm 0.03$ (Davies
et al. \cite{Davetal98}). Finally, reduced magnitude $R$ of the
target was obtained by taking into account the distance Sun-Object
$r$ and the distance Earth-Object $\Delta$ at the epoch of the
observations, and calculating the reduced magnitude
\begin{equation}
R = m_R -5\,\log (r)- 5\,\log (\Delta)
\label{Eq_Reduced_Mag}
\end{equation}
where $m_R$ is the observed $R$-band magnitude.
The results of our photometry are given in Tables~\ref{Tab_Pho_Chi}
and \ref{Tab_Pho_Qua}.
From a linear least-square fit of the data, we found that for Chiron
the $H_R$ absolute magnitude at zero phase angle is $5.52 \pm
0.07$\,mag, and the slope of the opposition surge is $0.045 \pm
0.023$\,mag/deg. Previous numerous photometric observations of Chiron
(for summary see Groussin et al. \cite{Groetal04}) have shown
considerable variations of absolute magnitude with heliocentric
distance attributed to its sporadic cometary behavior. The absolute
magnitude of Chiron in 2004 is brighter as compared to the
observations in 1988-1990 made at the same heliocentric distances (Bus
et al.\ \cite{Busetal91}). It may indicate that we observed Chiron
closely after an activity period which may have caused partial
resurfacing of the object. This is why we paid a special attention
to search for a coma around Chiron (see
Sect.~\ref{Sect_Coma_Searching}).
For Quaoar, the $H_R$ absolute magnitude at zero phase angle, obtained
from a \textit{linear fit} is $2.16 \pm 0.05$\,mag, and the linear
slope is $0.16 \pm 0.06$\,mag/deg. The determined value of absolute
magnitude coincides with that used for Quaoar for geometric albedo
determination (Brown \& Trujillo \cite{BroTru04}).
Note that $H_R$ absolute magnitudes at zero phase angles and linear
slopes will be re-estimated based on our modelling technique in
Sect.~\ref{Sect_Modelling}.
\begin{table}
\caption{The observed $R$ photometry of 2060\,Chiron}
\label{Tab_Pho_Chi}
\centering
\begin{tabular}{ccccc}
\hline \hline
Date &
Time (UT) &
Phase angle &
\multicolumn{1}{c}{$m_R$} &
\multicolumn{1}{c}{$R$} \\
\multicolumn{1}{c}{(yyyy mm dd)} &
\multicolumn{1}{c}{(hh:mm)} &
\multicolumn{1}{c}{(DEG)} &
(mag) &
(mag) \\
\hline
2004 05 27 & 06:23 & 3.470 & $16.64 \pm 0.04$& 5.68 \\
2004 06 29 & 03:04 & 1.411 & $16.48 \pm 0.05$& 5.56 \\
2004 08 06 & 05:40 & 1.766 & $16.49 \pm 0.04$& 5.54 \\
2004 08 07 & 03:33 & 1.828 & $16.65 \pm 0.04$& 5.69 \\
2004 08 13 & 00:02 & 2.219 & $16.55 \pm 0.05$& 5.58 \\
2004 09 01 & 00:39 & 3.326 & $16.65 \pm 0.04$& 5.63 \\
2004 09 27 & 01:10 & 4.232 & $16.84 \pm 0.04$& 5.74 \\
\hline
\end{tabular}
\end{table}
\begin{table}
\caption{The observed $R$ photometry of 50000\,Quaoar}
\label{Tab_Pho_Qua}
\centering
\begin{tabular}{ccccc}
\hline \hline
Date &
Time (UT) &
Phase angle &
\multicolumn{1}{c}{$m_R$} &
\multicolumn{1}{c}{$R$} \\
\multicolumn{1}{c}{(yyyy mm dd)} &
\multicolumn{1}{c}{(hh:mm)} &
\multicolumn{1}{c}{(DEG)} &
(mag) &
(mag) \\
\hline
2004 04 18 & 03:59 & 0.952 & $18.60 \pm 0.05$& 2.27 \\
2004 05 13 & 06:54 & 0.497 & $18.60 \pm 0.04$& 2.27 \\
2004 05 26 & 02:52 & 0.252 & $18.51 \pm 0.04$& 2.19 \\
2004 07 10 & 00:59 & 0.796 & $18.57 \pm 0.04$& 2.24 \\
2004 08 11 & 01:49 & 1.231 & $18.73 \pm 0.04$& 2.38 \\
\hline
\end{tabular}
\end{table}
\subsection{Search for a coma around 2060\,Chiron}\label{Sect_Coma_Searching}
Visual inspection and measurements of the full-width-at-half-maximum
of the Chiron and neighbouring star images did not reveal any
indications for the presence of a coma around the Centaur. For a more
thorough analysis we calculated the radial flux profile measured in
concentric rings around the object, and compared it with the radial
profile across the star trails, averaged along the trail direction and
for both sides of the trail. The resulting object and star profiles
are normalized to unity brightness at 'one pixel' central distance and
to zero at background distance 30\,pixels from the
center. Figure~\ref{Fig_Coma} shows the results for three observing
dates (for other dates the presence of star trails close to the object
image jeopardized the accuracy of this analysis). Since the radial
profile of Chiron (solid line) is basically identical to that of the
comparison stars (dotted line) or falls at slightly lower flux levels,
we conclude that a coma around Chiron is not present or it is well
beyond our detection limit (order of 31\,mag/arcsec) - if present at
all. Hence, we assume that the polarimetry of the object is not
contaminated by dust around the object.
\section{Modelling of the observed polarization}\label{Sect_Modelling}
\begin{table}
\caption{
The best fit coherent-backscattering model parameters for Ixion,
Quaoar, and Chiron. We give the single-scattering albedos
$\tilde{\omega}$ and dimensionless mean free paths $k\ell$ for the
dark (subscript $d$) and bright components ($b$), the weight of the
dark component $w_d$, the rms values of the polarimetric fits, as well
as the $R$-band absolute magnitudes $H_R$ and slope parameters $k_R$.
}
\label{Table_Results}
\begin{tabular}{llll}
\hline\hline
& Ixion & Quaoar & Chiron \\
\hline
$\tilde{\omega}_d$ & 0.45 & 0.35 & 0.15 \\
$\ell_d$ & 250 & 300 & 120 \\
$\tilde{\omega}_b$ & 0.80 & 0.50 & 0.60 \\
$\ell_b$ & 20 & 10 & 500 \\
$w_d$ & 0.74 & 0.46 & 0.14 \\
rms & 0.029\,\% & 0.069\,\% & 0.067\,\% \\
$H_R$ & 3.25 & 2.15 & 5.41 \\
$k_R$ & 0.12\,deg$^{-1}$& 0.11\,deg$^{-1}$& 0.041\,deg$^{-1}$\\
\hline
\end{tabular}
\end{table}
We interpret the polarimetric and photometric phase curves of Ixion,
Quaoar, and Chiron through extensive numerical simulations of coherent
backscattering by Rayleigh scatterers. Note that we avoid making
an assumption that the fundamental scatterers responsible for the
coherent backscattering contribution would be the single particles in
the regolith. With the phenomenological modeling currently including
the first multipole contribution of an electric dipole, we allow for
the possibility that the fundamental scatterers can be the volume and
surface inhomogeneities within and on the particles, respectively.
Shadowing among the regolith particles is known to contribute to the
opposition effect but not to the negative polarization surge. Here, as
in Boehnhardt et al. (\cite{Boeetal04}), the shadowing effect
manifests itself in the residual slope parameter resulting from the
combined coherent backscattering modeling on the polarimetry and
photometry.
The coherent-backscattering mechanism is a multiple-scattering
mechanism for scattering orders higher than the first one. For a
recent review of the coherent-backscattering mechanism and its
relevance in asteroid studies, see Muinonen et al. (\cite{Muietal02})
and Muinonen (\cite{Muinonen04}). Note that the computational
technique for coherent backscattering accounts for all orders of
scattering contributing in a non-negligible way to the backscattering
peaks and polarization surges.
In the following we limit ourselves to illustrate the mechanism at the
second order of scattering. Let us consider a semi-infinite random
medium of discrete scatterers, constrained by a plane-parallel
boundary with free space, and an electromagnetic plane wave incident
on the random medium from the free space. Let us assume that the
incident wave interacts with one of the scatterers, giving rise to
first-order scattering, and that, subsequently, the first-order
scattered field interacts with another scatterer, giving rise to
second-order scattering. Let us assume that the field scattered by the
second scatterer escapes the medium and is detected by the observer in
the free space. Now there is the reciprocal sequence of scatterings
in the opposite direction involving the very same two scatterers. In
the exact backward scattering direction, due to the identical lengths
of the propagation paths, the two reciprocal wave components interfere
constructively whereas, in other directions, the interference
characteristics vary. After configurational averaging, a
backscattering peak results in the proximity of the backward
scattering direction. Focusing in on the polarization characteristics,
the interference is selective and tends to invert the linear
polarization characteristics in single scattering: for positively
polarizing single scattering, coherent backscattering tends to result
in negative polarization. The illustration can be readily
generalized to higher orders of scattering. In the modeling that
follows, all relevant orders of scattering are taken into account.
For the modelling of Ixion, Quaoar, and Chiron, we carried out
coherent backscattering computations for a total of 360 spherical
media of Rayleigh scatterers, as follows: we assumed 18 different
single-scattering albedos
\[
\tilde{\omega}=0.05, 0.10, \ldots, 0.90
\]
and 20 different dimensionless mean free paths
\[
\begin{array}{rcl}
k\ell = 2\pi \ell/\lambda &=& 10, 20, 30,\ \ldots,\ 100, 120, 140,\ \ldots, \\
& & 200, 250, 300,\ \ldots,\ 400, 500 \\
\end{array}
\]
where $k$ and $\lambda$ are the wave number and wavelength, respectively.
As to the $R$-band geometric albedos $p_R$, we adopted $p_R=0.23$ for
Ixion (Stansberry, priv.\ comm.), 0.11 for Quaoar
(calculated using the diameter from Brown \& Trujillo \cite{BroTru04}
and our determination of the absolute magnitude), and 0.17 for Chiron
(Barucci et al. \cite{Baretal04}). In our previous interpretation of
Ixion polarimetric data (Boehnhardt et al. \cite{Boeetal04}), we had
assumed $p_R = 0.1$.
We compared the polarimetric observations against the spherical media
composed of monodisperse Rayleigh scatterers with given
single-scattering albedo and mean free path (two parameters). The fits
were poor: for a fixed geometric albedo, the monodisperse
Rayleigh-scattering model tends to result in polarizations that are
substantially pronounced as compared to the polarizations observed.
As in Boehnhardt et al. (\cite{Boeetal04}), we then studied a
two-component Rayleigh-scattering model consisting of dark and bright
scatterers. There are five parameters in such a model: two
single-scattering albedos and two mean free paths, and the weight
factor for the dark component (one minus the weight factor of the
bright component). Fixing the geometric albedo fixes the weight
factor, reducing the number of free parameters to four. After a
systematic study of physically realistic combinations of the two kinds
of scatterers, satisfactory fits were obtained for the polarizations
of all three objects. For Ixion, the data point at phase angle
0.43\degr\ was omitted as an outlier. The model parameters and rms
values of the fits are summarized in Table~\ref{Table_Results} and the
actual fits are depicted in Figs.~\ref{Fig_Pol_Ixi} --
\ref{Fig_Pol_Chi}. After the polarization fits, approximate brightness
fits were obtained by varying the absolute magnitude $H_R$ and slope
parameter $k_R$ of a linear phase dependence multiplying the coherent
backscattering contribution (see Fig.~\ref{Fig_Pho_Ixi} --
\ref{Fig_Pho_Chi}). For Chiron, the observation at 1.8\degr\ phase
angle was omitted as outlier.
For all three objects, the dark component shows a mean free path
substantially longer than the wavelength. For Chiron, the mean free
path of the bright component is also considerably longer than the
wavelength. The dark components of Ixion and Quaoar resemble one
another and the bright component of Chiron: this is concluded from the
similarity of the parameters of the corresponding scatterers on the
surfaces of the three objects. The phase curves of Ixion and Chiron
resemble each other to an extent where their combined polarimetric and
photometric data could be explained using a single two-component
scattering model.
The polarimetric observations suggest that Ixion and Chiron could have
similar surface structure. It is notable that the geometric albedo
ranges of Ixion and Chiron overlap, whereas Quaoar stands out as a
darker object. More observational and theoretical work is required for
further conclusions.
\section{Discussion and Conclusions}\label{Sect_Discussion}
We have presented the results of the first polarimetric observations
for a Centaur and a classical disk object in the Kuiper Belt. Together
with polarimetric observations of Ixion (Boehnhardt et al.,
\cite{Boeetal04}), a representative of the Plutino population in the
belt, they give us a first idea about the behavior of linear
polarization and intensity of the light reflected by surfaces of
Kuiper belt objects and Centaurs. Two Kuiper Belt objects, Ixion and
Quaoar, were observed practically in the same range of phase angles
(0.25\degr--1.3\degr). They have shown completely different
polarization-phase behaviour. For Ixion the negative polarization
rapidly increases with phase angle ($-1.02 \pm 0.025$\,\%/deg), while
for Quaoar the polarization degree is changed very slowly ($-0.17 \pm
0.10$\,\%/deg).
Our polarimetric observations of Chiron were made for larger phase
angles, 1.4\degr--4.2\degr. They are characterized by a pronounced
branch of negative polarization with a minimum of 1.4--1.5\,\% at
phase angles of 1.5\degr--2.0\degr. Such polarimetric characteristics
are unique among Solar System bodies. For the majority of Solar System
bodies the polarization at the phase angle 1\degr\ is characterized by
values within 0.1--0.5\,\%. The largest values of the polarization at 1\degr\
measured for non-TNO objects were 0.83\,\% for the dark side of
Iapetus (albedo = 0.05) and 0.73\,\% for Saturn Ring A (albedo = 0.75)
(Rosenbush et al.\ \cite{Rosetal02}) that is noticeably smaller than the values
obtained at our observations of Ixion and Chiron. Polarimetric
observations of Chiron at smaller phase angles are urgently needed for
a better modelling, and to compare Chiron data with the polarimetric
curve of Ixion and Quaoar. Further observations would be also desirable
to identify the inversion angle.
Our modelling has shown that the possible way to explain observed
polarization properties of KBOs and Centaur is to assume two-component
surface media consisting of dark and bright scatterers. Such a model
succeeds in fitting all observed polarimetric and photometric
characteristics of the three objects and the two-component modelling
is realistic. A more thorough theoretical study is beyond the scope of
the present article and should be carried out in the nearest future.
\begin{acknowledgements}
The authors wish to thank D.\ Rabinowitz for sharing his unpublished
photometric data of Quaoar which have helped us to verify the
calibration of our data, E.~Landi Degl'Innocenti and M.\ Landolfi for
their help to write Sect.~3, and O.~Hainaut for his help to identify
adequate observing periods for our targets (i.e., with no or little
risk of background star confusion).
\end{acknowledgements}
|
Title:
The Offline Software Framework of the Pierre Auger Observatory |
Abstract: The Pierre Auger Observatory is designed to unveil the nature and the origins
of the highest energy cosmic rays. The large and geographically dispersed
collaboration of physicists and the wide-ranging collection of simulation and
reconstruction tasks pose some special challenges for the offline analysis
software. We have designed and implemented a general purpose framework which
allows collaborators to contribute algorithms and sequencing instructions to
build up the variety of applications they require. The framework includes
machinery to manage these user codes, to organize the abundance of
user-contributed configuration files, to facilitate multi-format file handling,
and to provide access to event and time-dependent detector information which
can reside in various data sources. A number of utilities are also provided,
including a novel geometry package which allows manipulation of abstract
geometrical objects independent of coordinate system choice. The framework is
implemented in C++, and takes advantage of object oriented design and common
open source tools, while keeping the user side simple enough for C++ novices to
learn in a reasonable time. The distribution system incorporates unit and
acceptance testing in order to support rapid development of both the core
framework and contributed user code.
| https://export.arxiv.org/pdf/astro-ph/0601016 |
\title{The Offline Software Framework of the Pierre Auger Observatory}
\author{S. Argir{\`o},
S.L.C. Barroso,
J. Gonzalez,
L. Nellen,
T. Paul,
T.A. Porter,\\
L. Prado Jr.,
M. Roth,
R. Ulrich and
D. Veberi{\v c}
\thanks{The work of J. Gonzalez and T. Paul was funded in
part by the National Science Foundation. The work of T.A. Porter
was funded in part by the US Department of Energy.
}%
\thanks{S. Argir{\'o} is with the INFN and University of Torino,
S.L.C. Barroso is with the Centro Brasileiro de Pesquisas F{\'i}sicas,
J.Gonzalez and T. Paul are with Northeastern University,
L. Nellen is with the Universidad Nacional Aut{\'o}noma de M{\'e}xico,
T. Porter is with Louisiana State University,
L. Prado Jr. is with the University of Campinas,
M. Roth and R. Ulrich are with the Forschungzentrum Karlsruhe,
and D. Veberi{\v c} is with Nova Gorica Polytechnic.
}}
\begin{keywords}
offline software, framework, object oriented, simulation, cosmic rays
\end{keywords}
\section{Introduction}
\PARstart{T}{he} offline software framework of the
Pierre Auger Observatory~\cite{Abraham:2004dt}
provides an infrastructure to support a variety of distinct
computational tasks
necessary to analyze data gathered by the observatory.
The observatory itself is designed to measure the
extensive air showers produced by the highest
energy cosmic rays ($> 10^{19}$~eV) with the goal
of discovering their origins and composition.
Two different techniques are used to detect
air showers. Firstly, a collection of telescopes
is used to sense the fluorescence light produced
by excitation of nitrogen induced by the cascade
of particles in the atmosphere.
This method can be used only when the sky is moonless and dark,
and thus has roughly a 10\% duty cycle.
The second method uses an array of
detectors on the ground to sample particle
densities as the air shower arrives at the Earth's surface.
Each surface detector consists of a tank containing 12 tons of
purified water instrumented with photomultiplier tubes to detect
the Cherenkov light produced when particles pass through
it. The surface detector has a 100\% duty cycle.
A subsample of air showers detected by both instruments,
dubbed hybrid events, are very precisely measured and
provide an invaluable energy calibration tool.
In order to cover the full sky, the observatory
will consist of two sites, one in the southern
hemisphere and one in the north. The southern site
is located in Mendoza, Argentina, and construction
there is nearing completion, at which time the observatory
will comprise 24 fluorescence telescopes overlooking
1600 surface detectors spaced 1.5~km apart on a
hexagonal grid. Colorado has been selected
as the location for the northern site.
The requirements of this project place rather strong demands
on the software framework underlying data analysis. Most importantly,
the framework must be flexible and robust
enough to support the collaborative effort of a large
number of physicists developing a variety of applications over
the projected 20 year lifetime of the experiment.
Specifically, the software must
support simulation and reconstruction of events using
surface, fluorescence and hybrid methods, as well as
simulation of calibration techniques
and other ancillary tasks such as data preprocessing.
Further, as the experimental
run will be long, it is desirable that the software be
extensible in case of future upgrades to the observatory
instrumentation. The offline framework
must also handle a number of data formats in order to deal with
event and monitoring information from a variety of instruments,
as well as the output
of air shower simulation codes.
Additionally, it is desirable that all physics code
be ``exposed'' in the sense that any collaboration member must be able to
replace existing algorithms with his own in a straightforward
manner. Finally, while the underlying framework itself may
exploit the full power of C++ and object-oriented design,
the portions of the code directly used by physicists
should not assume a particularly detailed knowledge of
these topics.
The offline framework was designed with these principles
in mind. Implementation has taken place over the last
three years, and the first physics results based upon this
code were recently presented at the $29^{\rm th}$ International Cosmic
Ray Conference~\cite{icrc2005}.
\section{Overview}
The offline framework comprises three principal parts:
a collection of processing {\em modules} which can
be assembled and sequenced through instructions
provided in an XML file, an {\em event} structure
through which modules can relay data to one another
and which accumulates all simulation and reconstruction
information, and a {\em detector description} which provides
a gateway to data describing the configuration and
performance of the observatory as well as atmospheric
conditions as a function of time. These principal
ingredients are depicted in figure~\ref{f:general}.
These components are complimented by
a set of foundation classes and utilities for
error logging, physics and mathematical manipulation,
as well as a unique package supporting
abstract manipulation of geometrical objects.
Each of these aspects of the framework is described in more detail
below.
\section{User code, Modules and Configuration} \label{sec:config}
Experience has shown that most tasks of interest of the Pierre Auger
Collaboration can be reasonably factorized into sequences of well-defined
processing steps. Physicists prepare such processing algorithms
in so-called {\em modules}, which they can insert into the framework
via a registration macro in their code. This modular design allows
collaborators to easily exchange code, compare algorithms and
build up a wide variety of applications by combining modules in various
sequences.
Run-time control over module sequences is afforded through
a {\em run controller} which invokes the various
processing steps within the modules according to a set of externally
provided instructions. As most of our applications
do not require extremely sophisticated sequencing control
at the module level, we have chosen to construct
a very simple XML~\cite{xml}-based language for
specifying sequencing instructions. This provides
users with a tool which can be learned
very quickly, but which is still rich enough to still support most common
applications. Figure~\ref{f:xml} shows a simple example
of the structure of a sequencing file.
Parameters, cuts and configuration instructions used by
modules or by the framework itself are also stored in
XML files. A central directory points modules to their configuration
file(s) using a local filename or URI and creates parsers to assist in
reading information from these files. This directory is
constructed from a {\em bootstrap} file whose name is passed
on the command line at run time.
The configuration
mechanism can also concatenate and write a log file with
all configuration data accessed during a run. The log
file format identical to the {\em bootstrap} format, so
the logs can be subsequently read in
to reproduce a run with an identical configuration. This
configuration logging mechanism may also be used to record the versions
of modules and external libraries which are used during a run.
To check configuration files for errors, we employ XML Schema~\cite{schema}
validation throughout. This has proved successful in saving coding time
for developers and users alike, and facilitates much more detailed
error checking than most users are likely to implement on their own.
All XML handling is based upon the Xerces~\cite{xerces} validating parser,
augmented by a wrapper to simplify use and compliment functionality
with features such as unit handling, expression evaluation, and
casting of data in XML files to atomic types or STL containers.
\section{Data Access}
The offline framework provides two parallel hierarchies for accessing
data: the {\em event} for reading and writing information that changes per event,
and the read-only {\em detector description} for retrieving static or slowly
varying information such as detector geometry, calibration constants,
and atmospheric conditions.
\subsection{Event}
The event data structure contains all raw, calibrated,
reconstructed and Monte Carlo data and acts as the
principal backbone for communication between modules.
As it is a communication backbone, reference semantics are used
throughout to access data structures in the event and constructors are
kept private to prevent accidental copying of event components.
The event structure is built up dynamically as needed, and is
instrumented with a simple protocol allowing modules to interrogate the
event at any point to discover its current constituents.
The event representation in memory is decoupled from the
representation on disk. Serialization is currently implemented
using the ROOT~\cite{root} toolkit, though the design is intended
to allow for relatively straightforward changes of
serialization machinery.
A set of simple-to-use input/output modules are provided to
allow users to transfer all or part of the event from memory
to a file at any stage in the processing, and to reload the event and
continue processing from that point onward. These modules are
are built upon a a set of utilities to support the multi-format
reading and writing required to deal with different raw event and
monitoring formats as well as the different formats used by the
AIRES~\cite{aires}, CORSIKA~\cite{corsika} and CONEX~\cite{conex}
air shower simulation packages.
\subsection{Detector Description} \label{sec:detector}
The {\em detector description} provides a unified interface
from which module authors can retrieve information about
the detector configuration and performance at a particular time.
The interface is organized following the hierarchy normally
associated with the observatory instruments. Requests for data
are passed by this interface to a registry
of so-called {\em managers}, each of which is capable of
extracting a particular sort of information from a
particular data source. Data retrieved from the manager
are cached in the interface for future use.
In this approach, the user deals with a
single interface even though the data sought may reside in
any number of different sources. Generally we choose to store
static detector information in XML files, and
time-varying monitoring and calibration data in MySQL~\cite{mysql}
databases. The structure of the detector description
machinery is illustrated in figure~\ref{f:detector}.
Note that it is possible to implement more than one manager
for a particular sort of data. In this way, a special
manager can override data from a general manager. For
example, a user can decide to use a database for the
majority of the description of the detector, but override some data
by writing them in an XML file which is interpreted by
the special manager. The specification of which data
sources are accessed by the manager registry and in
what order they are queried is detailed in a configuration
file. The configuration of the manager registry is transparent
to the user code.
The detector description is also equipped to support a
set of plug-in functions, called {\em models} which can be used for
additional processing of data retrieved through the detector.
These are used primarily to interpret atmospheric
monitoring data. As an example, users can invoke
a model designed to evaluate attenuation due to aerosols
between two points in the atmosphere. This model
interrogates the detector interface to find the
atmospheric conditions at the specified time, and
computes the attenuation. Models can also be
placed under command of a {\em super-model} which
can attempt various methods of computing the desired
result, depending on what data are available at the
specified time.
\section{Utilities}
The offline framework is built on a collection of
utilities, including a XERCES-based XML parser, an
error logger, various mathematics and physics services,
testing utilities and a set of foundation classes to represent
objects such as signal traces, tabulated functions and
particles. The utilities collection also includes
a novel geometry package, which we describe in more
detail here.
As discussed previously, the Pierre Auger Observatory
comprises many instruments spread over a large area and, in
the case of the fluorescence telescopes, oriented in different
directions. Consequently there is no natural
preferred coordinate system for the observatory;
indeed each detector component
has its own preferred system, as do components
of the event such as the air shower itself.
Furthermore, since the detector spans some 40~km from side to side,
the curvature of the earth cannot generally be neglected.
In such a circumstance, the necessity of
keeping track of all the
the required transformations when performing
geometrical computations is tedious and
error prone.
This problem is alleviated in
the offline geometry package by
providing geometrical objects such as points
and vectors which keep track of the coordinate
system in which they are represented.
Operations on these objects can then be
written in an abstract way,
independent of any particular coordinate system choice.
There is no need for pre-defined coordinate
system conventions, or coordinate system
conversions at module boundaries.
Coordinate systems themselves are defined
in terms of transformations of other coordinate
systems, with an ultimate root coordinate system
as the foundation. In order to avoid
reliance on this root coordinate system by all
of the client code, a registry of pre-defined
coordinate systems is provided.
Furthermore,
various specialized coordinate systems,
such as the coordinate system of the shower
or one of the telescopes,
can be retrieved from different parts of the
event and detector description.
Locations of detector components are provided
by survey teams in UTM (Universal Transverse Mercator)
coordinates, which are convenient for navigation, but less so
for data analysis.
The geometry package therefore includes support for
transformations between geodetic and Cartesian
coordinates.
\section{Build System and Quality Control}
To help ensure code maintainability and stability in
the face of a large number of contributors and
a decades long experimental run, unit and acceptance testing
are integrated into the offline framework build and
distribution system. This sort of quality assurance
mechanism is crucial for any software which must continue
to develop over a timescale of many years.
Our build system is based on the GNU autotools~\cite{autotools},
which provide hooks for integrating tests with the
build and distribution system. A substantial collection
of unit tests has been developed, each of which
is designed to comprehensively test a single framework
component. These unit tests are run at regular
intervals, and in particular prior to releasing a new version
of the software. We have employed the CppUnit~\cite{cppunit}
testing framework as an aid in implementing these unit tests.
We are currently in the process of developing more involved acceptance
tests which will be used to verify that modules and
framework components working in concert continue to function
properly during ongoing development.
\section{External packages}
The choice of external packages upon which to build the offline
framework was dictated not only by package features, support
and the requirement of being open-source, but also by our
best assessment of prospects for longevity. At the same time,
we attempted to avoid locking the design to any single-provider
solution. To help achieve this, the functionality of
external libraries is often provided
to the client code
through wrappers or fa{\c c}ades, as in the case of XML
parsing described in section~\ref{sec:config}, or through
a bridge, as in the case of the detector description described
in section~\ref{sec:detector}. The collection of external
libraries currently employed includes ROOT~\cite{root} for
serialization, Xerces~\cite{xerces} for XML parsing and validation,
CLHEP~\cite{clhep} for expression evaluation and geometry foundations,
Boost~\cite{boost} for its many valuable C++ extensions, and optionally
Geant4~\cite{g4} for detailed detector simulations.
\section{Conclusions}
We have implemented an offline software framework for the Pierre Auger Observatory.
It provides machinery to help collaborators work together
on data analysis problems, compare results, and carry out
production runs of large quantities of simulated or real data.
The framework is configurable enough to adapt to a diverse
set of applications, while the user side remains simple enough
for C++ non-experts to learn in a reasonable time. The modular
design allows straightforward swapping of algorithms for quick comparisons
of different approaches to a problem. The interfaces to detector
and event information free the users from having to deal individually
with multiple data formats and data sources. This software, while still
undergoing vigorous development and improvement, has been
used in production of the first physics results from the observatory.
\section*{Acknowledgment}
The authors would like to thank the fearless
early users of the offline framework.
|
Title:
XMM-Newton View of the z>0 Warm-Hot Intergalactic Medium Toward Markarian 421 |
Abstract: The recent detection with Chandra of two warm-hot intergalactic medium (WHIM)
filaments toward Mrk 421 by Nicastro et al. provides a measurement of the bulk
of the "missing baryons" in the nearby universe. Since Mrk 421 is a bright
X-ray source, it is also frequently observed by the XMM-Newton Reflection
Grating Spectrometer (RGS) for calibration purposes. Using all available
archived XMM observations of this source with small pointing offsets (<15"), we
construct the highest-quality XMM grating spectrum of Mrk 421 to date with a
net exposure time (excluding periods of high background flux) of 437 ks and
\~15000 counts per resolution element at 21.6A, more than twice that of the
Chandra spectrum. Despite the long exposure time neither of the two intervening
absorption systems is seen, though the upper limits derived are consistent with
the Chandra equivalent width measurements. This appears to result from (1) the
larger number of narrow instrumental features caused by bad detector columns,
(2) the degraded resolution of XMM/RGS as compared to the Chandra/LETG, and (3)
fixed pattern noise at \lambda > 29A. The non-detection of the WHIM absorbers
by XMM is thus fully consistent with the Chandra measurement.
| https://export.arxiv.org/pdf/astro-ph/0601620 | command.
\def\hi{\ifmmode {\rm H}\,{\sc i}~ \else H\,{\sc i}~\fi}
\def\kms{\rm\,km\,s^{-1}}
\def\hubunits{\rm\,km\,s^{-1}\,Mpc^{-1}}
\def\K{\,{\rm K}}
\def\cm{{\rm cm}}
\def\chandra {{\it Chandra}}
\def\xmm {{\it XMM}}
\def\xmmnewton {{\it XMM--Newton}}
\def\fuse {{\it FUSE}}
\def\cvi {\ion{C}{6}}
\def\nvi {\ion{N}{6}}
\def\neix {\ion{Ne}{9}}
\def\nvii {\ion{N}{7}}
\def\nvi {\ion{N}{6}}
\def\ovi {\ion{O}{6}}
\def\ovii {\ion{O}{7}}
\def\oviii {\ion{O}{8}}
\def\ovihv {\ion{O}{6}$_{\rm HV}$}
\def\novi {N_{\rm OVI}}
\def\novii {N_{\rm OVII}}
\def\noviii {N_{\rm OVIII}}
\def\kms {~km~s$^{-1}$}
\shorttitle{XMM Observations of Mrk 421}
\shortauthors{Williams et al.}
\begin{document}
\title{\boldmath{\it XMM--Newton} View of the $z>0$ Warm--Hot
Intergalactic Medium Toward Markarian 421}
\author{Rik J. Williams\altaffilmark{1},
Smita Mathur\altaffilmark{1},
Fabrizio Nicastro\altaffilmark{2,3,4},
Martin Elvis\altaffilmark{2}}
\altaffiltext{1}{Department of Astronomy, The Ohio State University,
140 West 18th Avenue, Columbus OH 43210, USA}
\altaffiltext{2}{Harvard--Smithsonian Center for Astrophysics, Cambridge,
MA, 01238, USA}
\altaffiltext{3}{Instituto de Astronom\'ia, Universidad Aut\'onomica de
M\'exico, Apartado Postal 70-264, Ciudad Universitaria,
M\'exico, D.F., CP 04510, M\'exico}
\altaffiltext{4}{Osservatorio Astronomico di Roma, Istituto Nazionale di
Astrofisica, Italy}
\email{williams,smita@astronomy.ohio-state.edu}
\keywords{ intergalactic medium --- X-rays: general --- cosmology:
observations }
\section{Introduction}
Most of the baryons that comprise 4\% of the mass--energy budget
of the universe are found in the intergalactic medium (IGM), primarily
appearing as the Lyman--alpha ``forest'' in high--redshift
quasar spectra \citep{weinberg97}. At more recent times ($z\la 2$)
the process of structure formation has
shock--heated the IGM to temperatures of $\sim 10^{5-7}$\,K, thus
rendering the hydrogen nearly fully ionized and producing (at most)
broad, extremely weak Lyman--alpha absorption
\citep[e.g.][]{sembach04,richter04}. Known as the warm--hot intergalactic
medium (WHIM), this phase is predicted to contain roughly half of the
baryonic matter at low redshifts \citep{cen99,dave01}. Its extremely low
density (typically $\delta\sim 10-100$)
precludes the detection of WHIM thermal or line emission with current
facilities, so the only way to directly measure these ``missing'' baryons
is through far--ultraviolet and X-ray spectroscopic measurements of
absorption lines from highly ionized heavy elements
\citep{perna98,hellsten98,fangb02}.
Several early attempts to detect these intervening WHIM absorbers in X-rays
\citep[e.g.][]{fang02,mathur03,mckernan03} and more recent surveys
\citep{fang05} yielded only tentative detections
at best. Intervening Lyman--alpha \citep{shull96,sembach04} and \ion{O}{6}
\citep[e.g.][]{savage02} absorbers had also been seen in \fuse\ and HST
quasar spectra, but their ionization states and possible galactic halo origins
\citep[e.g.][]{tumlinson05} are quite uncertain. These uncertainties
are largely mitigated with the recent detection by
\citet{nicastro05a,nicastro05b}
of two X-ray absorption systems at $z=0.011$ and $z=0.027$ along the
line of sight to the blazar Mrk 421. These filaments account for
a critical density fraction of
$\Omega_{\rm WHIM}=0.032^{+0.042}_{-0.021}$, fully consistent with the mass
of the missing baryons in the local universe (albeit with large
uncertainties). While future proposed missions such as \emph{Constellation--X},
\emph{XEUS}, or \emph{Pharos} (Nicastro et al., in preparation) will be able to
measure $\Omega_{\rm WHIM}$ to far greater precision with
detections of numerous weaker X-ray forest lines, the
\citet[][hereafter N05]{nicastro05a} results present a key
early confirmation of numerical predictions using observational
capabilities that are \emph{currently} within our grasp --- hence any
test of their correctness is of great importance.
Although each of these two absorption systems was detected with high confidence
through multiple redshifted X-ray absorption lines, the \emph{individual}
absorption lines were generally quite weakly detected, mostly at the
$2-4\sigma$ level. Moreover, while they employed high--quality \chandra\
and \fuse\ data taken during exceptionally bright outbursts of Mrk 421,
the many archived \xmmnewton\ observations of this source
were not included in the analysis. With roughly twice the effective
area of \chandra/LETG, \xmm/RGS is in principle superior for X-ray
grating spectroscopy
between $\sim 10-40$\,\AA; however, its slightly worse resolution
(approximately 60\,m\AA\ FWHM, versus 50\,m\AA\ for \chandra/LETG), higher
susceptibility to background flares, and multitude of narrow instrumental
features can present serious complications for WHIM searches.
Independent confirmation of the \chandra\ results with a separate
instrument like \xmm\ is thus important. While some groups have searched
for WHIM features in a limited number of \xmm\ Mrk 421 spectra
\citep[e.g.,][]{ravasio05}, a complete and systematic analysis has yet
to be performed. Here we present a search for $z>0$ WHIM features employing
all ``good'' observations of Mrk 421 available in the \xmm\
archive, and a comparison of these results to those presented by N05.
\section{Data Reduction and Measurements}
We searched the \xmm\ archive for all Mrk 421 Reflection Grating
Spectrometer (RGS) data. Although
31 separate observations were available, 16 had pointing offsets
$\Delta\theta \ga 60$\arcsec\ while the rest were offset by less than
15\arcsec. Since spectral resolution and calibration quality can degrade at
large offsets, we only included
those with $\Delta\theta < 15$\arcsec. One extremely short observation
(0158971101, with $t_{\rm exp}=237$\,s) was also excluded to simplify the
data reduction process. Using the standard \xmm\ Science Analysis System
version 6.5.0
routines\footnote{See \url{http://xmm.vilspa.esa.es/sas}}, RGS1
light curves were built for the remaining
14 ``good'' observations (see Table~\ref{tab_log}), and the spectra were
reprocessed to exclude periods of high background levels and coadded.
These combined, filtered
RGS1 and RGS2 spectra have effective exposure times of $\sim 440$\,ks and
over $9\times 10^6$
combined RGS1$+$RGS2 first--order counts between $10-36$\,\AA\ with
$\sim 15000$ counts per 0.06\,\AA\ resolution element in RGS1 near 21\,\AA,
over twice that in the N05 Mrk~421 \chandra\ spectrum.
We first used the spectral fitting program
\emph{Sherpa}\footnote{\url{http://cxc.harvard.edu/sherpa/}} to fit
a simple power law plus Galactic foreground absorption model to the
RGS1 and RGS2 data; however, at such high spectral quality the RGS
response model does not appear to be well--determined, and large residuals
remained. For line measurements, we thus only considered
$\sim 2$\,\AA\ windows around each wavelength of interest, using a power
law to independently model
the RGS1 and RGS2 continua within each window
and excluding the strongest narrow detector features (with typical
widths of 70\,m\AA\ or less). None of the intervening absorption
lines were apparent through a visual inspection of the \xmm\ spectrum, though
several of the $z=0$ lines reported by \citet{williams05} could be seen
clearly.
A narrow Gaussian absorption line (FWHM$=5$\,m\AA)
was included in the model for each
line measurement or upper limit reported by N05. When convolved with
the RGS instrumental response these absorption lines appeared broadened
to the RGS line spread function (LSF) widths \citep[typically
FWHM$=60-70$\,m\AA;][]{denherder01}. The $2\sigma$
upper limits on all equivalent widths were then calculated (allowing the
central line wavelengths to vary within the $1\sigma$ errors reported
by N05). Since the shapes of the RGS1 and RGS2 instrumental responses
are quite different, these limits were calculated using both a joint
fit to the RGS1$+$RGS2 spectra as well as the individual RGS1 and RGS2
spectra. It should be noted that wherever one RGS unit is unusable,
the total response is effectively halved, at which point it has a similar
effective area to \chandra/LETG. The resulting equivalent width limits
are listed in Table~\ref{tab_ew}.
\section{Discussion}
Figure~\ref{fig_xmmspec} shows the spectral windows used to determine
upper limits on the N05 measured lines, with the data (black), continuum fit
(blue), \chandra\ measurements and limits (N05; red solid and dotted lines
respectively), and \xmm\ limits (green) overplotted.
In all cases, the N05 measurements (or $3\sigma$ upper
limits) appear to be consistent with the
$2\sigma$ upper limits we have derived directly from the \xmm\ data,
as shown in the figure and listed in Table~\ref{tab_ew}.
The \ovii\ line at $z=0.027$
looks as though it might be visible in the spectrum, but this is most
likely due to the weak instrumental feature at $\sim 22.1$\,\AA.
For two lines (\nvii\ and \nvi\ at
$z=0.027$) the \xmm\ $2\sigma$ upper limits are approximately equal to
the N05 best--fit measurements, but since the N05 values are quite uncertain
this result is still consistent.
Why, then, with $2-4$ times the counts per resolution element, was
\xmm\ unable to detect the intervening absorption systems seen by
\chandra? Several factors appear to have been involved in this
non--detection, primarily: (1) narrow instrumental features caused by
bad detector columns, (2) the broader LSF as compared
to \chandra/LETG, and (3) fixed--pattern noise at long wavelengths:
\begin{enumerate}
\item{While broad instrumental features can be taken into
account by modifications to the continuum model (as in N05), it is
difficult or impossible
to distinguish narrow features from true absorption lines; thus, any line
falling near one of the detector features shown in Figure~\ref{fig_rgsmod}
can easily be lost\footnote{These response file data can be found at
\url{http://www.astronomy.ohio-state.edu/$\sim$smita/xmmrsp/}}.
This was responsible for the non-detection of the $z=0.011$ \ovii\ K$\alpha$
line. Although it was the strongest line reported by N05,
its wavelength falls directly on a narrow RGS1 feature and within
the non--functional CCD4 region on RGS2,
thereby preventing this line from being detectable with either RGS. Since
18\% of the wavelength space for studying redshifted \ovii\
($\lambda>21.6$\,\AA) toward Mrk 421 is directly blocked by these narrow
features
(with this number climbing to about 60\% if resolution elements immediately
adjacent to bad columns are included), these bad columns present the
single greatest hindrance to \xmm/RGS studies of the WHIM.}
\item{Even for lines where both RGS1 and RGS2 data are available and the
instrumental response appears to be relatively smooth, the lower resolution
of \xmm\ contributes to the nondetectability of the
weaker $z>0$ absorption lines. Figure~\ref{fig_lsfplot} shows the
LSFs for both \xmm/RGS1 (solid) and \chandra/LETG assuming an unresolved
line with $W_\lambda=10$\,m\AA\ at 21.6\,\AA. While the core of the RGS1
response is
$\sim 20$\% broader than that of the LETG, the RGS1 LSF
has extremely broad wings: only 68\% of the line flux is contained within
the central 0.1\AA\ of the RGS1 LSF, as opposed to 96\% for the LETG.
This reduces the apparent depth of absorption lines by about a factor of
two as compared to \chandra/LETG, severely decreasing the line detectability.}
\item{At long wavelengths ($\lambda\ga 29$\,\AA)
strong fixed--pattern noise is apparent as a sawtooth pattern in the
instrumental response, strongly impeding the detection of species
such as \nvi\ and \cvi. Indeed, in these wavelength regimes
(the lower two panels of Figure~\ref{fig_xmmspec}), the \nvi\ and \cvi\
absorption lines are nearly indistinguishable from the continuum.}
\end{enumerate}
\section{Conclusion}
We have presented the highest signal--to--noise coadded \xmm\ grating
spectrum of
Mrk 421 to date, incorporating all available archival data. This
spectrum serves as an independent check on the recent detection of
two $z>0$ WHIM filaments by N05. While none of the \chandra--detected
absorption lines are seen in the \xmm\ spectrum, the upper limits derived
from the \xmm\ data are consistent with the equivalent widths
reported by N05 (even though the \xmm\ data contain a larger number
of counts), and hence do not jeopardize the validity of the \chandra\
measurement. The non--detections can be attributed primarily to narrow
instrumental features in RGS1 and RGS2, as well as the inferior spectral
resolution of \xmm\ and fixed--pattern noise at longer wavelengths.
This underscores the extreme difficulty
of detecting the WHIM, illustrates how the aforementioned (apparently small)
effects can greatly affect the delicate measurement
of weak absorption lines, and re--emphasizes the importance of high resolution
and a smooth instrumental response function for current and future WHIM
absorption line studies.
\acknowledgments
We thank the \xmm\ team for their efforts on this excellent mission,
the helpdesk staff for their assistance with the data reduction, and
the anonymous referee for helpful comments.
This research is based on archival data obtained with XMM-Newton, an ESA
science mission with instruments and contributions directly funded by ESA
Member States and NASA. RJW is supported by an Ohio State University
Presidential Fellowship, and FN acknowledges the support of NASA Long--Term
Space Astrophysics Grant NNG04GD49G.
\clearpage
\begin{deluxetable}{lcccc}
\tabletypesize{\footnotesize}
\tablecolumns{5}
\tablewidth{240pt}
\tablecaption{\xmmnewton\ observation log \label{tab_log}}
\tablehead{
\colhead{ID} &
\colhead{Date} &
\colhead{$t_{\rm exp}$\tablenotemark{a}} &
\colhead{$t_{\rm filt}$\tablenotemark{b}} &
\colhead{Rate\tablenotemark{c}}
\\
\colhead{} &
\colhead{} &
\colhead{ks} &
\colhead{ks} &
\colhead{s$^{-1}$}
}
\startdata
0099280101 &2000 May 25 &63.8 &21.2 &15.7\\
0099280201 &2000 Nov 01 &40.1 &34.1 &5.4\\
0099280301 &2000 Nov 13 &49.8 &46.6 &15.3\\
0099280501 &2000 Nov 13 &21.2 &17.8 &17.2\\
0136540101 &2001 May 08 &38.8 &36.1 &11.7\\
0136540301 &2002 Nov 04 &23.9 &20.5 &11.7\\
0136540401 &2002 Nov 04 &23.9 &20.1 &13.6\\
0136540701 &2002 Nov 14 &71.5 &62.8 &16.4\\
0136541001 &2002 Dec 01 &70.0 &58.1 &8.3\\
0158970101 &2003 Jun 01 &43.0 &25.3 &9.0\\
0158970201 &2003 Jun 02 &9.0 &6.6 &9.7\\
0158970701 &2003 Jun 07 &48.9 &29.9 &5.4\\
0158971201 &2004 May 06 &65.7 &40.5 &19.5\\
0162960101 &2003 Dec 10 &30.0 &17.5 &9.8\\
\hline\hline
TOTAL & &572.3 &437.1 &12.2\\
\enddata
\tablenotetext{a}{Total observation duration.}
\tablenotetext{b}{Effective RGS1 exposure time after filtering for periods
of high background levels.}
\tablenotetext{c}{Average count rate in the filtered RGS1 first--order source
spectral extraction region.}
\end{deluxetable}
\clearpage
\begin{deluxetable}{lccccccc}
\tabletypesize{\footnotesize}
\tablecolumns{8}
\tablecaption{Absorption line equivalent width measurements \label{tab_ew}}
\tablehead{
\colhead{Line} &
\colhead{$\lambda$\tablenotemark{a}} &
\colhead{$z$\tablenotemark{a}} &
\colhead{$W_{\lambda, {\rm N05a}}$\tablenotemark{a}} &
\colhead{$W_{\lambda, {\rm R1}}$\tablenotemark{b}} &
\colhead{$W_{\lambda, {\rm R2}}$\tablenotemark{b}} &
\colhead{$W_{\lambda, {\rm R1+R2}}$\tablenotemark{b}} &
\colhead{Note}
\\
\colhead{} &
\colhead{\AA} &
\colhead{} &
\colhead{m\AA} &
\colhead{m\AA} &
\colhead{m\AA} &
\colhead{m\AA} &
\colhead{}
}
\startdata
\ion{Ne}{9}$_{K\alpha}$ &$13.80\pm 0.02$ &$0.026\pm 0.001$ &$<1.5$ &$<5.2$ &$<1.9$ &$<2.9$ &1\\
\ion{O}{7}$_{K\beta}$ &$19.11\pm 0.02$ &$0.026\pm 0.001$ &$<1.8$ &$<2.5$ &$<2.1$ &$<1.5$ &\\
\ion{O}{8}$_{K\alpha}$ &$19.18\pm 0.02$ &$0.011\pm 0.001$ &$<4.1$ &$<7.6$ &$<5.8$ &$<4.1$ &\\
\ion{O}{8}$_{K\alpha}$ &$19.48\pm 0.02$ &$0.027\pm 0.001$ &$<1.8$ &\nodata &$<3.9$ &\nodata &2\\
\ion{O}{7}$_{K\alpha}$ &$21.85\pm 0.02$ &$0.011\pm 0.001$ &$3.0^{+0.9}_{-0.8}$ &\nodata &\nodata &\nodata &2,3\\
\ion{O}{7}$_{K\alpha}$ &$22.20\pm 0.02$ &$0.028\pm 0.011$ &$2.2\pm 0.8$ &$<3.9$ &\nodata &\nodata &3\\
\ion{N}{7}$_{K\alpha}$ &$25.04\pm 0.02$ &$0.010\pm 0.001$ &$1.8\pm 0.9$ &$<3.0$ &$<6.0$ &$<4.4$ &\\
\ion{N}{7}$_{K\alpha}$ &$25.44\pm 0.02$ &$0.027\pm 0.001$ &$3.4\pm 1.1$ &$<4.3$ &$<4.2$ &$<3.5$\\
\ion{N}{6}$_{K\alpha}$ &$29.54\pm 0.02$ &$0.026\pm 0.001$ &$3.6\pm 1.2$ &$<3.8$ &$<8.7$ &$<3.4$ &\\
\ion{C}{6}$_{K\alpha}$ &$34.69\pm 0.02$ &$0.028\pm 0.001$ &$2.4\pm 1.3$ &$<5.5$ &$<5.2$ &$<4.2$ &\\
\enddata
\tablenotetext{a}{\ Line wavelength, redshift, and equivalent width measurements
(or $3\sigma$ upper limits) from \citet{nicastro05a}.}
\tablenotetext{b}{$2\sigma$ equivalent width upper limits measured from the
RGS1 only (R1), RGS2 only (R2), and joint (R1+R2) fits to the XMM--Newton
spectrum, when available.}
\tablecomments{
(1) A nearby chip gap in RGS1 renders this measurement unreliable, so only
the RGS2 measurement was used in Figure~\ref{fig_xmmspec};
(2) Line was unmeasurable in RGS1 because of a detector feature;
(3) Line was unmeasurable in RGS2 because of a detector feature.
}
\end{deluxetable}
|
Title:
{Interstellar Plasma Weather Effects in Long-term Multi-frequency Timing of Pulsar B1937+21 |
Abstract: We report here on variable propagation effects in over twenty years of
multi-frequency timing analysis of pulsar PSR B1937+21 that determine
small-scale properties of the intervening plasma as it drifts through the sight
line. The phase structure function derived from the dispersion measure
variations is in remarkable agreement with that expected from the Kolmogorov
spectrum, with a power law index of $3.66\pm 0.04$, valid over an inferred
scale range of 0.2--50 A.U. The observed flux variation time scale and the
modulation index, along with their frequency dependence, are discrepant with
the values expected from a Kolmogorov spectrum with infinitismally small inner
scale cutoff, suggesting a caustic-dominated regime of interstellar optics.
This implies an inner scale cutoff to the spectrum of $\sim 1.3\times 10^9$
meters. Our timing solutions indicate a transverse velocity of 9 km sec$^{-1}$
with respect to the solar system barycenter, and 80 km sec$^{-1}$ with respect
to the pulsar's LSR. We interpret the frequency dependent variations of DM as a
result of the apparent angular broadening of the source, which is a sensitive
function of frequency ($\propto\nu^{-2.2}$). The error introduced by this in
timing this pulsar is $\sim$2.2 $\mu$s at 1 GHz. The timing error introduced by
``image wandering'' from the slow, nominally refractive scintillation effects
is about 125 nanosec at 1 GHz. The error accumulated due to positional error
(due to image wandering) in solar system barycentric corrections is about 85
nanosec at 1 GHz.
| https://export.arxiv.org/pdf/astro-ph/0601242 |
\title{Interstellar Plasma Weather Effects in Long-term
Multi-frequency \\
Timing of Pulsar B1937+21}
\author{R. Ramachandran, P. Demorest, D. C. Backer}
\affil{Department of Astronomy and Radio Astronomy Laboratory,
University of California, Berkeley, CA 94720-3411, USA; \\ e-mail:
ramach, demorest, dbacker@astro.berkeley.edu}
\author{I. Cognard}
\affil{Laboratoire de Physique et Chimie
de l'Environnement, CNRS, 3A avenue de la Recherche
Scientifique, F-45071 Orleans, France}
\author{A. Lommen}
\affil{Department of Physics and Astronomy, Franklin and Marshall College,
P.O.Box 3003, Lancaster, PA 17604, USA}
\keywords{ISM: general --- pulsars: general --- radio continuum:
general --- scattering --- turbulence}
\section{Introduction}
The dispersion measure (DM) of a pulsar probes
the column density of free electrons along the line of sight (LOS).
Observed DM variations over time scales of several weeks to years
sample structures in the electron plasma over length scales
of $10^{10}$ m to $10^{12}$ m. Diffraction of pulsar signals is the
result of scattering by
structures on scales below the Fresnel radius, $10^{8}$ m or so.
The DM as well as the scattering measure (SM) variability along the LOS
to the Crab pulsar was first reported by Rankin \& Isaacman (1977),
who reported that the DM variability poorly correlated with the SM
variability. Helfand et al. (1980) inferred an upper limit for DM
variations of a few parts in a thousand for several pulsars. In an
earlier study of PSR B1937+21 Cordes et al. (1990) measured a DM
change of $\Delta DM\sim 0.003$ pc cm$^{-3}$ over a period of a
thousand days. The work of Phillips \& Wolszczan (1991) presented the
variations of DM observed along the LOS to a few pulsars. They
connected these variations to those on diffractive scales,
and derived an electron density fluctuation
spectrum slope of $3.85\pm0.04$ over a scale range of $10^7-10^{13}$
meters. Backer et al. (1993) report on further DM variability and
show that the amplitude of the variations known at that time are
consistent with a scaling by the square root of DM. Another important
investigation by Kaspi et al. (1994) studied DM variations of the
millisecond pulsars PSR B1937+21 and B1855+09 over a time interval of
calendar years 1984 to 1993. In addition to establishing a secular
variation in DM over this time interval, they also show that the
underlying density power spectrum has an index of 3.874$\pm$0.011,
which is close to what we would expect if the density fluctuations are
described by Kolmogorov spectrum. An anomalous dispersion event
towards the Crab pulsar was reported by Backer et al. (2000), where
they report a DM ``jump'' as large as 0.1 pc cm$^{-3}$.
In this work, we present results of various
long term monitoring programs on PSR
B1937+21. Our data, which includes that of Kaspi, Taylor \& Ryba
(1994), spans calendar years from 1983 to 2004. These
data sets have been taken with five different telescopes, the
NRAO\footnote{The National Radio Astronomy Observatory (NRAO)
is owned and operated by Associated Universities, Inc under
contract with the US National Science Foundation.} Green
Bank 42-m (140-ft) and 26-m (85-ft) telescopes, the
NAIC\footnote{The National Astronomy and Ionosphere Center is operated
by Cornell University under contract with the US National Science
Foundation.} Arecibo telescope and the \nancay ~telescope, at
frequency bands of 327, 610, 800, 1400 and 2200 MHz.
After giving the details of our observations in \S\ref{sec-obsvn},
we describe our analysis methods in \S\ref{sec-anal}. This is
followed in \S\ref{sec-sightline} by a discussion on the distribution of
scattering material along the LOS. As we describe, the knowledge
of temporal and angular broadening of the source, proper motion, and
scintillation based velocity estimates enables us to at least qualitatively
study the distribution of scattering matter as well as properties
of its wave-number spectrum.
We have measured some of the basic refractive scintillation parameters from
our observations, and these are discussed in \S\ref{sec-diffref}.
The frequency dependence of the refractive scintillation
time scale and the modulation index indicate a caustic-dominated
regime that results from a large inner scale in the spectrum.
We have detected DM variations as a function of time and frequency.
We determine the phase structure function of the medium with the
knowledge of the time dependent DM variations, which is consistent
with a Kolmogorov distribution of density
fluctuations between scale sizes of about 1 to 100 A.U. These are
summarized in \S\ref{sec-dmvar} and \S\ref{sec-dmfreq}.
\begin{table}
\begin{center}
\caption[]{Template fit parameters at various frequencies.}
\begin{tabular}{lcccccccc} \hline
{\bf $\nu$ (MHz)} & backend & ~~$w_1$~~ & ~~$l_2$~~
& ~~$w_2$~~ & ~~$h_2$~~ & ~~$l_3$~~ & ~~$w_3$~~ & ~~$h_3$~~ \\ \hline\hline
327 & GBPP & 8.7 & 0 & 0 & 0 & 186.9 & 12.0 & 0.51 \\
430 & ABPP & 10.3 & 7.0 & 2.5 & 0.19 & 186.8 & 12.4 & 0.53 \\
610 & GBPP & 9.4 & 7.1 & 2.9 & 0.34 & 187.3 & 10.8 & 0.60 \\
863 & EBPP & 8.9 & 7.7 & 5.2 & 0.38 & 187.7 & 10.9 & 0.55 \\
1000 & GBPP & 9.2 & 8.6 & 3.9 & 0.56 & 187.9 & 11.3 & 0.54 \\
1419 & ABPP & 8.5 & 8.5 & 3.7 & 0.37 & 187.5 & 10.1 & 0.45 \\
1689 & EBPP & 9.1 & 9.2 & 3.9 & 0.37 & 187.4 & 10.5 & 0.40 \\
2200 & GBPP & 9.4 & 9.8 & 3.2 & 0.29 & 187.6 & 10.4 & 0.36 \\
2379 & ABPP & 10.0 & 9.8 & 3.0 & 0.29 & 187.1 & 10.8 & 0.33 \\
\hline
\end{tabular}
\label{tab:fitpars}
\end{center}
\end{table}
PSR B1937+21 is known for its short term timing stability. However,
the achievable long term timing accuracy is suspected to be
seriously limited by the interstellar scattering properties. With
our sensitive measurements, we are in a position to quantify these
errors. In \S\ref{sec-timingerror}, we describe in detail various
sources of these errors and quantify them.
\section{Observations }
\label{sec-obsvn}
We have used five different primary data sets for this analysis.
The first set is the 1984--1992 Arecibo pulse timing and
dispersion measurements obtained by Kaspi et al. (1994; hereafter KTR94).
Their observations were performed with their Mark~II backend (Rawley
1986; Rawley, Taylor \& Davis 1988) and later their Mark~III backend
(Stinebring et al. 1992) at two different radio frequency
bands, 1420 MHz and 2200 MHz. Their analysis methods are described in
KTR94.
The second data set is from 800-MHz and 1400-MHz observations
at the NRAO 140-ft telescope in Green Bank, WV. The Spectral
Processor backend, a hardware FFT device, was used. Details about
the observations and analysis are contained in an earlier report
on dispersion measure variability (Backer et al. 1993).
The third data set consists of observations at 327 MHz and 610 MHz
using the 26m (85-ft) pulsar monitoring telescope at NRAO's Green
Bank, WV site. Room temperature (uncooled) receivers at the two bands
are mounted off-axis. At 327 MHz the total bandwidth used was 5.5 MHz,
and 16 MHz was used at 610 MHz. The two orthogonally polarized signals
were split into 32 frequency channels in a hybrid analog/digital
filter bank in the GBPP (Green Bank--Berkeley Pulsar Processor).
Dispersion effects were removed in the GBPP in real-time with a
coherent (voltage) deconvolution algorithm. At the end of the
real-time processing folded pulse profiles were recorded for each
frequency channel and polarization. Further details of the backend
and analysis can be found in Backer, Wong \& Valanju (2000). PSR
B1937+21 was observed for about two hours per day starting in
mid-1995.
The fourth data set comes from a bi-monthly precision timing program
that includes B1937+21 at the Arecibo Observatory which we started in
1999 after the telescope upgrade. Signals at 1420 MHz and 2200 MHz
were recorded using the ABPP backend (Arecibo--Berkeley Pulsar
Processor), which is identical to the GBPP. Our typical observing
sessions at 1420 MHz and 2200 MHz had bandwidths of 45 MHz and 56 MHz,
respectively, and integration times of approximately 10 minutes per
session.
\begin{table}
\tt
\begin{center}
\caption[]{Parameters of PSR B1937+21.}
\begin{tabular}{ll} \hline
{\bf Parameter} & {\bf value} \\ \hline\hline
PSR & 1937+21 \\
RAJ (hh:mm:ss) & 19:39:38.561 (1) \\
DECJ (dd:mm:ss) & 21:34:59.136 (6) \\
PMRA (mas yr$^{-1}$) & 0.04 (20) \\
PMDEC (mas yr$^{-1}$) & -0.45 (6) \\
$f$ (Hz) & 641.9282626021 (1) \\
$\dot{f}_{-15}$ (Hz s$^{-1}$) & -43.3170 (6) \\
$\ddot{f}_{-26}$ (Hz s$^{-2}$) & 1.5 (3) \\
PEPOCH (MJD) & 47500.000000 \\
START & 45985.943 \\
FINISH & 52795.286 \\
EPHEM & DE405 \\
CLK & UTC (NIST) \\ \hline
\end{tabular}
\label{tab:1937par}
\end{center}
\end{table}
The fifth data set is from a pulsar timing program that has been
ongoing since 1989 October with the large decimetric radio telescope
located at \nancay, France. The Nan\c cay telescope has a surface
area of 7000 m$^2$, which provides a telescope gain of 1.6 K
Jy$^{-1}$. Observations are performed with dual-linear feeds at
frequencies 1280, 1680 and 1700 MHz. Then the signal is dedispersed by
using a swept frequency oscillator (at 80 MHz) in the receiver IF
chain. The pulse spectra are produced by a digital autocorrelator
with a frequency resolution of 6.25 kHz. Cognard et al. (1995)
describe in detail the backend setup and the analysis procedure.
A small amount of additional data from the Effelsberg telescope was
used in our profile analysis. At Effelsberg the EBPP backend, a copy
of the GBPP/ABPP, was used.
\section{Basic analysis }
\label{sec-anal}
We first present several results from the analysis of these data
sets: a description of the frequency-dependent profile template used
for timing; spin and astrometric timing parameters from high frequency
data; pulse broadening, flux densities and dispersion measure
as functions of time. In \S\ref{sec-sightline} we proceed to interpret
these results and return to finer details regarding dispersion measure
variations in \S\ref{sec-dmvar}.
Our basic data set consists of average pulse profiles obtained
approximately every 5 minutes in each of the radio frequency bands --
327, 610, 800, 1420 and 2200 MHz. Figure \ref{fig:obs_summary} provides
a graphical summary of observation epochs vs date.
For data sets corresponding to all
frequencies except 327 MHz, {\it Times of Arrival} (TOAs) were
computed by cross correlating these average profiles with a template
profile. The template profile at a given frequency was made by using
multiple Gaussian fits to very high signal to noise ratio average profiles
at that frequency; the interactive program {\it bfit}, which is
based on M. Kramer's original program {\it fit} was used.
These fit parameters are listed in Table 1.
Col.~1 in the Table gives the radio frequency
and the backend name is in col.~2. Col.~3 gives the width
of component 1 ($w_1$; its location is taken to be 0 degrees and its
amplitude is set to 1.0); cols.~4-6 and cols.~7-9 give the location ($l$),
width ($w$) and amplitude ($h$) values for components 2 and 3,
respectively. The location and width are given in units of
longitudinal degrees, where 360$^{\circ}$ indicates one full rotation
cycle. The results of this analysis can be compared with that of
Foster et al. (1991) which are given on the line at 1000 MHz
\footnote{The widths $w_1$ and $w_3$ are inverted in Table 4 of Foster
et al.}. There is reasonable agreement for all values except $h_2$
which must have been erroneously entered in Table 4 of Foster et al.
In our analysis templates corresponding to arbitrary frequencies are
produced by spline-interpolation of the component parameters.
We used the Arecibo (1420 MHz and 2200 MHz) TOAs, and the GBT 140-ft
(800 MHz and 1420 MHz) TOAs to fit for pulsar spin (rotation frequency
($f$), first time derivative ($\dot{f}$), and second time derivative
($\ddot{f}$)) and astrometric (position ({\tt RAJ}, {\tt DECJ}),
proper motion ({\tt PMRA}, $\mu_{\alpha}$, along right acension, and
{\tt PMDEC}, $\mu_{\delta}$, along declination) parameters. All TOAs
were referred to the UTC time scale kept by the National Institute of
Standards and Technology (NIST) via GPS satellite comparison. We
removed the effects of variable dispersion from this fitting procedure
with weekly estimation of DMs and subsequent extrapolation of the dual
frequency data to infinite frequency prior to parameter
estimation. The nature of achromatic timing noise makes it
particularly difficult to determine a precise timing model. As one
adds additional higher derivatives of rotation frequency (e.g., a
third derivative), the best fit parameters change by amounts much
larger than the nominal errors reported by the package that we used,
TEMPO. The results are listed in Table 2. The errors
presented in the table incorporates the range of variation of each
parameter, as additional derivative terms are included. In comparison
to Kaspi, Taylor \& Ryba (1994), the derived proper motion values are
marginally different. We attribute this difference to the variable
influence of timing noise. An important point that needs to be
stressed here is that there is no reason for us to assume that the
higher derivative terms of rotation period (e.g., $\ddot{f}$ or
higher) has anything to do with the radiative braking index. They are
most likely dominated by some intrinsic instabilities of the star
itself, or some other perturbation on the star.
Extension of dispersion measurement to 327 MHz requires removal of the
time-variable broadening of the intrinsic pulse profile owing to
multipath propagation in the interstellar medium. We deconvolved the
effect of interstellar scattering following precepts first introduced
by Rankin et al. (1970). We assume that the interstellar temporal
broadening is quantified in terms of convolution of a Gaussian
function and a truncated exponential function. If there is only one
scattering screen along the LOS, the assumption of a truncated
exponential function will suffice to represent the scatter
broadening. However, since the scattering may arise from material
distributed all along the LOS, a more realistic representation is
approximated by a truncated exponential function ``smoothed''
(convolved) with a Gaussian function. The intrinsic pulse profile was
estimated by extrapolation of parameters from the higher frequency
profiles. In the deconvolution procedure, we minimized the normalized
$\chi^2$ value by varying the width of the Gaussian $w_g$ and the
decay time scale of the truncated exponential $\tau_e$, while keeping
the intrinsic pulse profile fixed. The pulse scatter broadening is
quantified as $\tausc\; = \; (w_g^2 + \tau_e^2)^{1/2}$. We repeated this
for average profiles obtained at every epoch to obtain the $\tausc$
measurement. In our
fits, the average value of $w_g$ came to about 74 $\mu$sec,
whereas the corresponding value for $\tau_e$ was about 85 $\mu$sec.
The measurement of $\tausc$ versus time at 327~MHz is plotted in the
Figure \ref{fig:tausc}. This quantity has a mean value of 120~$\mu$s,
an RMS variation of 20~$\mu$s, and a fluctuation timescale of
$\sim60$~days. We explain these variations as the result of refractive
modulation of this inherently diffractive parameter in discussion
below. The estimated RMS variation at the next higher frequency in
our data set, 610 MHz, is $\sim$2.5 $\mu$s, using a frequency
dependence of $\tausc\propto\nu^{-4.4}$. This is too small to allow
fitting at this frequency band.
In the strong scintillation regime, time dependent variations in the
observed flux occur in two distinct regimes --- {\it diffractive}
and {\it refractive}. The diffractive effects are dominated by
structures smaller than the Fresnel scale, and appear on short time
scales and over narrow bandwidths. In our observations diffractive
modulations are strongly suppressed. On the other hand refractive
effects occur on days time scales and are correlated over wide
bandwidths. We have analyzed our best data sets -- the densely sampled
data at 327 MHz and 610 MHz from Green Bank and at 1410 MHz from Nan\c
cay -- for flux density variations as a function of time. The data
are presented in Figure \ref{fig:fluxplot}.
In analyzing the low frequency flux data from Green Bank, we have not
adopted a rigorous flux calibration procedure. While there is a pulsed
calibration noise source installed in this system, equipment changes and
the nature of the automated observing have led to large gaps in the
calibration record. Rather than dealing with a mix of calibrated and
uncalibrated data, or lose a large fraction of the data, we decided not
to apply any calibration. Instead, we normalize our data by assuming
the system temperature is constant. In order to see what effect this
has on our results, we did two tests.
First, we analyzed observations of PSR~B1641$-$45, taken with the same
system, over a similar time range. This pulsar is known to have a very
long refractive timescale, $\tref >1800$~days (Kaspi \& Stinebring,
1992), so it can be used as a flux calibrator. In our data, we
find it to have a modulation index, $m=0.10$. This immediately puts a
upper limit of 10\% on any systematic gain and/or system
temperature variations. Since modulation adds in
quadrature, and we observe modulation indeces of $m\sim0.4$ for
PSR~B1937+21, gain fluctuations represent at most a small fraction
of the observed modulation.
We also considered the possibility that gain variations could influence
our measurement of $\tref $. This might happen if they occur with a
characteristic timescale longer than 1~day. In order to test this, we
analyzed observations of the Crab pulsar, PSR~B0531+21, again taken with
the same system over the same time range. The refractive parameters of
this pulsar were studied in detail by Rickett \& Lyne (1990). It makes
a good comparison since it has modulation index of $m=0.4$ at 610~MHz,
very similar to PSR~B1937+21. Applying the structure function analysis
(see \S\ref{sec-diffref}) to this data gives $\tref =11$~days at
610~MHz, and $\tref =63$~days at 327~MHz, consistent with the previously
published results and a scaling law of $\tref \propto\nu^{-2.2}$.
The procedure that we have adopted to calibrate our data set from
\nancay~ telescope is described in detail in Cognard et al. (1995).
\section{Distribution of scattering material along the line of sight}
\label{sec-sightline}
Several authors have shown how the scattering parameters of a pulsar
can be used to assess the distribution of scattering material along
the LOS (Gwinn et al. 1993; Deshpande \& Ramachandran 1998; Cordes \&
Rickett 1998). This results from the varied dependences of the
scattering parameters on the fractional distance of scattering
material along the LOS. PSR B1937+21 is viewed through the local
spiral arm as well as the Sagittarius arm which are both potential
sites of strong scattering. The parameters employed in this analysis
are: the temporal pulse broadening by scattering ($\tausc$; or its
conjugate parameter $\Delta\nu$, the diffractive scintillation
bandwidth), the diffractive scintillation time scale ($\tdiff$), the
angular broadening from scattering ($\theta_H$), the proper motion of
the pulsar ($\mu_{\alpha}$, $\mu_{\delta}$), and a distance estimate
of the pulsar ($D$).
Let us first compare $\theta_H$ and $\tausc$ that are the result of
multiple scattering along the LOS, and express them as
(Blandford \& Narayan 1985)
\begin{eqnarray}
\tausc &=& \frac{1}{2cD}\int_0^D x(D-x)\;\psi(x)\; {\rm d}x \\
\theta_H^2 &=& \frac{4\ln2}{D^2}\int_0^D x^2\psi(x)\; {\rm d}x.
\label{eq:tautheta}
\end{eqnarray}
In these equations, $x$ is the coordinate along the LOS, with the
pulsar at $x=0$ and the observer at $x=D$. $\psi(x)$ is the mean
scattering rate. If the scattering material is uniformly distributed
along the LOS, then the relation between the two quantities can be
expressed as $\theta_H^2=16\ln2\left(c\tausc/D\right)$. With the
distance to the pulsar of 3.6~kpc to the pulsar (according to the
distance model of Cordes \& Lazio 2002), and the average pulse
broadening time scale of 120 $\mu$s (from the present work), we obtain
an estimate of the angular broadening, $\theta_{\tau}$, of 12
mas. This is in modest agreement with the measured value of 14.6$\pm
1.8$ mas (Gwinn et al. 1993), given the uncertainty in the distance to the pulsar and the
simple assumption that the scattering material is uniformly
distributed along the LOS.
Next, we formulate two approaches to estimation of the velocity of the
LOS with respect to the scattering medium, and use these approaches to
assess the location and extent of the medium. The transverse velocity
of the pulsar based on the measured proper motion
(Table 2) an assumed distance of $D=3.6$ kpc (Cordes \&
Lazio 2002) is 9 km s$^{-1}$. This value is the velocity of the pulsar
with respect to the solar system barycenter. With the assumed ``Flat
Rotation Curve'' linear velocity of the Galaxy of 225 km s$^{-1}$, and
the Sun's peculiar velocity of 15.6 km s$^{-1}$ in the Galactic
coordinate direction of $(l,b) = (48.8^{\circ}, 26.3^{\circ})$
(Murray 1986), the
transverse velocity of the pulsar in its LSR ($V_p$) is 80 km
s$^{-1}$.
The {\it scintillation velocity}
($\viss$), which is an estimate of the velocity of the {\it
diffraction pattern} at the location of the Earth, is
estimated from the decorrelation bandwidth ($\Delta\nu$) and the
diffractive scintillation time scale ($\tdiff$).
Gupta et al. (1994) conclude that
\begin{equation}
\viss\;=\;3.85\times 10^4\; \sqrt{\frac{D \; z \; \Delta
\nu}{(1-z)}} \; \frac{1}{\tdiff\;\nu_{\rm GHz}} \;\;\; {\rm km\;s^{-1}}
\end{equation}
where $\nu_{\rm GHz}$ is the observing frequency in GHz, $D$ is in
kpc, $\Delta\nu$ is in MHz, and $\tdiff$ is in seconds. The variable
$z$ gives the fractional distance to the scattering screen, where
$z=0$ gives the observer's position, and $z=1$ gives the pulsar's
position. The value of decorrelation bandwidth is computed by the
relation $\Delta \nu \; =\; 1 / (2 \pi \tausc)$. When the effective
scattering screen is midway along the LOS ($z=0.5$), $\viss\;=\;V_p$,
and when the screen is at the location of the pulsar ($z=1.0$), $\viss
= \infty$.
While doing this, an important assumption is that the pulsar proper
motion is dominant over contributions from differential Galactic
motion, solar peculiar velocity, and the Earth's annual orbital
modulation. In the case of PSR B1937+21, this assumption is not
justified. The effective
scattering screen, which is located somewhere along the LOS, has a
Galactic motion whose component along the LOS direction is different
from that of the pulsar or the Sun. In order to correct for this
effect, let us calculate the LOS velocity across the effective
scattering screen at a fractional distance $z$ from the observer:
\begin{equation}
\vlos\;=\; 3.85\times 10^4\; \frac{\sqrt{D \; z \; (1-z)\; \Delta
\nu}}{\tdiff\;\nu_{\rm GHz}} \;\;\; {\rm km\;s^{-1}}
\label{eq:vlos}
\end{equation}
Then, let us assume that the scattering along the LOS can be
adequately expressed by having a thin screen alone, at a distance of
$D_s=zD$ from the observer. Then, Equation \ref{eq:tautheta} can be
expressed as
\begin{eqnarray}
\tausc &=& \frac{\psi_{\circ}}{2c}\; D\;z \; (1-z) \\
\theta_H^2 &=& 4\;\ln 2\; (1-z)^2 \psi_{\circ}
\end{eqnarray}
Here, $\psi_{\circ}$ gives the mean scattering rate corresponding
to the effective thin screen. Then, let us express independently the
transverse velocity of the LOS across the scattering screen as
\begin{eqnarray}
\vec{V}_{\perp}' &=& (1-z)\;\vec{V}_e\; + \; z\vec{V}_p \;-\; \vec{V}_G
(zD\hat{n}) \nonumber\\
&=& \vec{V}_E + zD\vec{\mu} - \vec{V}_G (zD\hat{n}),
\label{eq:vlos2}
\end{eqnarray}
where $V_e$ is Earth's orbital velocity, $V_p$ is the pulsar transverse
velocity in its LSR, and $V_G$ is the transverse velocity contribution from
the Galactic differential motion to the screen. $V_E$ gives the
contribution of the Earth's motion on the LOS velocity across the screen.
Equations \ref{eq:vlos} and \ref{eq:vlos2} give two independent
estimates of the line of sight velocity across the effective
scattering screen and therefore allow us to solve for the value of $z$ given
$D$. With $D=3.6$ kpc (Cordes \& Lazio 2002),
we find $z=0.7$. The LOS velocity is 51 km s$^{-1}$.
The assumed value of $\tdiff$ is
78 seconds at 327 MHz (scaled from 444 seconds at 1400 MHz of Cordes
et al. 1991), and the value of $\Delta\nu$ is 0.0013 MHz calculated
from $\tausc=120$ $\mu$s.
To summarize, the measured value of $\theta_H=14.6\pm 1.8$ mas and the
estimated value of $\theta_{\tau}$ are consistent with each other,
suggesting a uniformly distributed scattering medium.
On the other hand, comparison of velocity components, $V_p$,
$\vlos$ and $V_{\rm los}^\prime$ suggest a thin-screen at $z\sim
2/3$. As Deshpande \& Ramachandran (1998) demonstrate, this solution
is equivalent to having a uniformly distributed scattering medium!
Therefore, we conclude that the line of sight to PSR B1937+21 can be
described adequately by a uniformly distributed scattering matter.
The Earth's orbital velocity around the Sun will modulate the observed
scintillation speed, and therefore the diffractive scintillation time
scale, with a one-year time scale. The amplitude of this modulation
will depend on the effective $z$ of the diffracting material, and so
monitoring could provide an estimate of the effective screen location.
If the effective screen is close to the Earth, then the modulation is
strong, and if it is located close to the pulsar, then it is
negligible. Figure \ref{fig:annualmod} demonstrates this effect. The
ordinate and abscissa give the LOS velocity across the effective
scattering screen along the galactic longitude and latitude,
respectively. For an assumed distance of 3.6 kpc, the straight line
shows the expected centroid velocity of $\vlos^\prime$. The left most
end of the line (origin of the plot) corresponds to $z=0$, and the
right most end corresponds to $z=1$. The annual modulation of
$\vlos^\prime$, shown as the two ellipses, correspond to z=0.5 and
$z=2/3$. We have no way of identifying this annual modulation in our
data, as we are insensitive to diffractive effects in our data set.
Another measurement that could help us is the direct measurement of
distance to this object by parallax measurements. Chatterjee et
al. (2005, private communication), from their preliminary Very Long Baseline Array (VLBA)
based parallax measurements, report that the distance to PSR B1937+21
is $2.3^{+0.8}_{-0.5}$ kpc, if they force the proper motion value to
be the same as that of our timing based measurements (Table 2).
In the coming year, accuracy of their measurements
will improve with further sensitive observations.
\section{Refractive scintillation }
\label{sec-diffref}
\subsection{Parameter estimation}
\label{sec-diffrefparm}
We determine refractive scintillation parameters from the data
presented in Figure \ref{fig:fluxplot} following the structure
function approach in previous studies (Stinebring et al 2000; Kaspi \&
Stinebring; Rickett \& Lyne 1990). We define the structure function
$D_F$ for flux time series $F(t)$ as
\begin{equation}
D_F(\delta t) = \frac{\langle [F(t)
- F(t+\delta t)]^2\rangle}{\langle F(t) \rangle^2},
\label{eq:structflux}
\end{equation}
where $\delta t$ is a time delay. Since our flux measurments occur at
discrete and unevenly spaced time
interals, we compute the flux difference for all possible lags, then
average results into logarithmically spaced bins.
The flux structure function typically has a form described by Kaspi \&
Stinebring et al. (1992) - a flat, noise dominated section at small
lags, then a power-law increase which finally saturates at a value
$D_s$ at large lags. In practice, the saturation regime may have
large ripples in it, an effect of the finite length of any data set.
In addition, the measured flux structure function is offset from the
``true'' flux structure function due to a contribution from
uncorrelated measurement errors. At 327~MHz and 610~MHz, we estimate
this noise term from the short-lag (noise regime) values. At 1410~MHz
(from \nancay), we use the individual flux error bars to get the noise
level. After subtracting the noise value, we fit the result to a
function of the form
\begin{equation}
D_F(\delta t) =
\left\{ \begin{array}{ll}
D_s(\delta t/T_s)^\alpha, & [0<\delta t<T_s] \\
D_s, & [\delta t > T_s]
\end{array} \right.
\label{eqn:sffit}
\end{equation}
In this fit, the power law slope $\alpha$, the saturation timescale
$T_s$, and the saturation value $D_s$ are all free parameters. The
flux structure function data and fits are shown in Figure
\ref{fig:fluxstruct}.
As shown in Rickett \& Lyne (1990), the refractive parameters can be
measured from the flux structure function using the following
relationships: The modulation index $m$ is given by $m =
\sqrt{D_s/2}$, and the refractive scintillation timescale $\tref$ is
given by $D_F(\tref) = D_s/2$. All the measured parameters, including
those measured by earlier investigators are summarized in Table 3.
Based on a propagation model through a simple power-law density fluctuation
spectrum, we expect to see refractive variations in the flux measurements
on a timescale $\tref\sim0.5\theta_H D/\vlos$, where $\vlos$ is the line of
sight velocity across the effective scattering screen.
For the argument sake, if we assume an effective scattering screen at
$z=0.5$, then $\vlos\sim 40$ km sec$^{-1}$. With $\theta_H = 14.6$
mas, the expected refractive scintillation time scale is $\sim$3 years
at 327 MHz. This is more than an order of magnitude in excess of the
measured value. Furthermore, if the density fluctuations in the
medium are distributed according to the Kolmogorov power law
distribution, then the expected frequency scaling law is
$\tref\propto\lambda^{2.2}$. Our measured values indicate a
significantly different scaling. Although it is consistent with
$\tref\propto\lambda^{2.2}$ between 610 and 1420 MHz, it is not so
between 327 and 610 MHz, where it is consistent with being directly
proportional to $\lambda$. Our observed modulation index ($m$) values
are also considerably larger than predicted, and show a ``flatter''
wavelength dependence, as listed in Table 3. We
will address this issue in detail in the following section.
\subsection{Nature of the spectrum -- inner scale cutoff}
\label{sec-cutoff}
The three disagreements with a simple model
summarized in \S\ref{sec-diffrefparm} force us to
explore a few aspects of the electron density power spectrum that may
possibly explain what we observe. The effects of {\it caustics} on the
observed scintillations have been explored by several earlier
investigators, most notably Goodman et al. (1987) and Blandford \&
Narayan (1985). In particular, if the power law scale distribution in
the medium is truncated at an inner scale that is considerably larger
than the diffractive scale, as they show, the observed flux variations
are dominated by caustics. This is of great interest to us, as this
seems to explain all the discrepancies that we note in our observed
refractive parameters. For instance, as Goodman et al. (1987) show
that if the inner scale cutoff is a considerable fraction of the
Fresnel scale, then the observed fluctuation spectrum of flux is
dominated by fluctuation frequencies that are lower than the
diffractive frequencies, but significantly higher than that expected
from refractive scintillation. This is what we observe. Moreover, as
they note, the observed wavelength dependence of the refractive time
scale, as well as the modulation index is expected to be ``shallower''
than the expected values of $\lambda^{2.2}$ and $\lambda^{-0.57}$,
respectively.
A ``shallow'' frequency dependence of the modulation index has been
reported by others (Coles et al. 1987; Kaspi \& Stinebring 1992; Gupta
et al. 1993; Stinebring et al. 2000).
While Kaspi \& Stinebring (1992) find that the observed refractive
quantities matched well with the predicted values for five objects,
three other objects, especially PSR B0833--45, has a significantly
shorter measured $\tref$ and greater modulation index than
expected. This is very similar to our situation here with PSR
B1937+21.
Stinebring et al. (2000) concluded that the 21 objects that they
analysed fell into two groups. The first group followed the frequency
dependence predicted by a Kolmogorov spectrum with the inner cutoff
scale far less than the diffractive scales ({\it Kolmogorov-consistent
group}). The second group, which is the {\it super-Kolmogorov group},
is consistent with a Kolmogorov spectrum with an inner scale cutoff at
$\sim 10^{8}$ meters. The observed modulation indices were
consistently greater than that of the Kolmogorov predictions, as we
have seen in our measurements of PSR B1937+21. This group includes
pulsars like PSRs B0833--45 (Vela), B0531+21 (Crab), B0835--41,
B1911--04 and B1933+16. An important physical property that binds them
all is that, excepting one object, all objects have a strong {\it
thin-screen} scatterer somewhere along the LOS. This is either a
supernova remnant (or a plerion) like in the case of Vela and Crab
pulsars, a HII region as in the case of B1942--03 and B1642--03, or a
Wolf-Reyet star as in the case of B1933+16 (see Prentice \& ter Haar
1969; Smith 1968). Although our measurements show that pulsar PSR
B1937+21 is consistent with the characteristics of the {\it
super-Kolmogorov} group, as we describe in \S\ref{sec-sightline}, we
find no compelling evidence for the presence of any dominant scatterer
somewhere along the LOS.
To summarize, while some investigators have reported agreement of
the measured refractive properties with the theoretical expectations
from a Kolmogorov spectrum with an infinitismally small inner scale,
there are a considerable number of cases where
the observed properties are significantly different from that predicted
by the simple Kolomogorov spectrum.
These other cases can be explained by invoking
spectrum with a large inner scale cutoff, including the case where
the cutoff approaches the Fresnel radius and leads to
a caustic-dominated regime.
From Gupta et al. (1993) and Stinebring et al. (2000), the modulation
index can be specified as a function of the inner cutoff scale as
\begin{equation}
m\;=\;0.85\; \left( \frac{\Delta\nu}{\nu} \right)^{0.108}\; \left(
\frac{r_i}{10^8{\rm m}} \right)^{0.167} D_{\rm kpc}^{-0.0294}.
\end{equation}
With the known value of $\Delta\nu$ at 327 MHz of 1.33 kHz, the
distance to the pulsar of 3.6 kpc, and the observed modulation index
of 0.39, the inner scale cutoff, $r_i$, comes to $1.3\times 10^9$
meters.
\section{DM variations}
\label{sec-dmvar}
We turn now to the dispersion measure variations presented in Figure
\ref{fig:dmvstime} that sample density variations on transverse scales
much larger than those involved with diffractive and refractive
effects. The most striking feature in Figure \ref{fig:dmvstime} is
the large secular decline from 71.040 pc cm$^{-3}$ in 1985 to 71.033
pc cm$^{-3}$ in 1991 and then to 71.022 pc cm$^{-3}$ by late
2004. These long-term secular variations are many times greater than
the RMS fluctuations of $\sim 10^{-3}$ pc cm$^{-3}$ on short time
scales. An important question that arises is whether these variations
are the result of a spectrum of electron-density turbulence, or
whether there might be a contribution from the smooth gradient of a
cloud, or clouds along the LOS. We look at this question from two
angles. First we present a phase structure function analysis of the
dispersion measure data and estimate a power-law index of the electron
density spectrum. Then we estimate the probability that such a
spectrum would produce a 22-y realization that was so strongly
dominated by the large, monotonic changes mentioned above.
We write the power spectrum of electron-density fluctuations as
\begin{equation}
P(q)\;=\; C_n^2\;q^{-\beta},\;\;\;\;\;\;\;\;\;\;\;\;[\qo<q<\qi]
\end{equation}
\noindent
where $\beta$ is the power law index, $\qo$ and $\qi$ are the spatial
frequencies corresponding to the outer and the inner boundary scale,
within which this power law description is valid. $C_n^2$ is the
amplitude, or strength, of the fluctuations. A quantity that is
closely related to the density spectrum which can be quantified by
observable variables is the phase structure function, $D_{\phi}(b)$,
with $b=2\pi/q$. This is defined as the mean square geometric phase
between two straight line paths to the observer, with a separating
distance of $b$ between them in the plane normal to the observer's
sight line. The phase structure function and the density power
spectrum are related by a transform (Rickett 1990; Armstrong,
Rickett, Spangler 1995),
\begin{eqnarray}
D_{\phi}(b)\;=\;\int_0^{\infty} 8\pi^2\lambda^2 r_e^2\;dz'\;
\int_0^{\infty}q\;[1-J_0(bqz'/z)]\;dq \nonumber \\
\times P(q=0)
\end{eqnarray}
Here, $r_e$ is the classical electron radius (2.82$\times$10$^{-15}$
meters), $J_{\circ}$ is the Bessel function. Under the conditions
that we have assumed, $D_{\phi}(b)$ is also a
power law (Rickett 1990; Armstrong, Rickett \& Spangler 1995), given
by
\begin{equation}
D_{\phi}(b)\;=\;\left(\frac{b}{b_{\rm coh}}\right)^{\beta-2}
\end{equation}
\noindent
where $b_{\rm coh}$ is the coherence spatial scale. Dispersion measure
can be written as
\begin{equation}
DM\;=2.410\times 10^{-16}\, \left[\frac{(\nu_1^2-\nu_2^2)}{ \nu_1^2
\nu_2^2}\right]\; \left(\frac{\Delta\phi}{f}\right)\; {\rm pc\;
cm^{-3}},
\end{equation}
\noindent
where $\Delta\phi$ is the difference in the arrival phases
$(\phi_2-\phi_1)$ of the pulse at the two barycentric radio
frequencies (Hz) $\nu_1$ and $\nu_2$, with $f$ being the barycentric
rotation frequency (Hz) of the pulsar. With this linear relation between DM
and geometric phase difference, then structure function can be written
as (KTR94)
\begin{eqnarray}
D_{\phi}(b_{\circ})&=&\left( \frac{2\pi}{\nu} \frac{{\rm Hz}}{ 2.410
\times 10^{-16}\;{\rm pc\;cm^{-3}}}\right)^2 \nonumber\\
&& \times \langle [DM(b+b_{\circ}) - DM(b)]^2 \rangle.
\label{eqn:struct}
\end{eqnarray}
Here, the angular brackets indicate ensemble averaging. The
transformation between the spatial coordinate $b$ (and the spatial
delay $b_{\circ}$) and the time coordinate $t$ (or time delay $\tau$)
is simply given by $b=V_\perp t$, where $V_\perp$ is the transverse
velocity of the LOS across the effective scattering screen.
\begin{table}
\label{tab:scintparams}
\begin{center}
\caption[]{Measured and expected parameters.}
\begin{tabular}{c|c|c|c|c|c|c|c} \hline
$\tausc$ & $\theta_H$ & $\tdiff$ & \multicolumn{2}{|c|}{$T_{\rm ref}$}
& \multicolumn{2}{|c|}{$m$} & {\bf $\nu$} \\
($\mu$s) & (mas) &
(s) & observed & expected & observed & expected & (MHz) \\
\hline\hline
120$^{\dagger}$ & 14.6$^{b}$ & -- & 73 days & 3
y$^f$ & 0.33 & 0.14$^c$ & 327 \\
38$^e$ & -- & 100$^e$ & -- & -- & -- & -- & 430 \\
-- & -- & -- & 43.9 days & 6 mon$^g$ & 0.39 & 0.2 & 610 \\
-- & -- & 260$^a$ & 3 days$^d$ &
45 days$^g$ & 0.45 & 0.33$^c$ & 1400 \\
0.17$^e$ & -- & 444$^e$ & -- & -- & -- & -- & 1400 \\
\hline
\end{tabular}
\end{center}
$^{\dagger}$Has a time dependent RMS variation of 20$\mu$s\\
$^a$Cordes et al. (1986) \\
$^b$Gwinn et al. (1993) \\
$^c$Romani et al. (1986); Kaspi \& Stinebring (1992) \\
$^d$Lestrade et al. (1998) give the value as 13 days\\
$^e$Cordes et al. 1990 \\
$^f$Calculated with $\tref\sim\theta_HD/2\vlos$ \\
$^g$Extrapolated with $T_{\rm ref}\propto\lambda^{2.2}$
\end{table}
With the understanding that any difference in DM that we compute for a
time baseline from Figure \ref{fig:dmvstime} corresponds to a point in
the phase structure function, we can derive the phase structure
function on the basis of Equation \ref{eqn:struct}. This is given in
Figure \ref{fig:struct}.
There are several important points in Figure \ref{fig:struct}. The
solid line gives the best fit line for the data in the time interval
of 5 days to 2000 days. The derived values of the intercept and the
power law index ($\beta$) are,
\begin{eqnarray}
{\rm intercept} &=& 4.46\pm 0.09 \nonumber\\
\beta &=& 3.66\pm 0.04
\label{eq:fitpars}
\end{eqnarray}
The value of $\beta$ is remarkably close to the value expected from a
Kolmogorov power law distribution ($\beta=11/3$). We are using the
terminology ``intercept" only to indicate the value of
$\log[D_{\phi}(\tau)]$ when $\log[time lag (days)]$ is zero. Here, a
cautionary remark is warranted. Given the finite time span of our data
set, and the fact that the low spatial frequencies dominate the long
term variations in DM, we do not have a stationary sample of noise
spectrum. We have estimated the error in each bin of the structure
function as
\[
\sigma_s = \frac{\sigma_{D}}{\sqrt{N_i}},
\]
where $\sigma_{D}$ is the root mean square deviation with respect to
the mean phase structure function value in a bin, $D_{\phi}(\tau)$,
and $N_i$ is the number of ``independent'' samples in the bin. This is
estimated as the smaller of $(T/\tau)$ and the actual number of samples
that have gone into the estimation of $D_{\phi}(\tau)$.
Here, $T$ is the time span of the data.
By assuming that the transverse speed of the sightline across the
effective scattering screen is $\sim$40 km sec$^{-1}$ (half of
pulsar's velocity in its LSR), we can translate the delay range
between which this slope is valid, to 0.2 to 50 A.U.
The time delay value corresponding to the phase structure function
value of unity is, by definition, the coherent diffractive time scale
($\tdiff$) at the corresponding radio frequency, with the assumption
that the scattering material is uniformly distributed along the
LOS. From the fit parameters given in Equation~\ref{eq:fitpars},
this delay is 180 seconds. This should
be compared with the measured $\tdiff$ value of 444$ \pm $28 seconds
tabulated in Table 3. If we interpret the inner
scale cutoff value of $r_i \sim 1.3\times 10^9$ meters as the scale
size below which the slope ($\beta$) of the density fluctuation
spectrum changes to a value greater than that given in Equation
\ref{eq:fitpars}, then the fact that the measured $\tdiff$ value of
444 seconds being significantly greater than 180 seconds is
understandable. In the limiting case, where the slope of the
density irregularity power spectrum changes to the
critical value of $\beta = 4$ below the inner scale cutoff value, the
expected $\tdiff$ value is about 1100 seconds. This makes it very
important to measure the exact frequency dependence of the diffractive
parameters like temporal scatter broadening and diffractive
scintillation time scale. To the best of our knowledge, Cordes et
al. (1990) show the most complete multi-frequency measurements of the
diffractive scintillation parameters of this pulsars. Their
measurements are not accurate enough to distinguish between such small
variations in slope.
While our analysis of DM variability suggests a Kolmogorov spectrum at
AU scales, we are struck by the long term monotonic decrease of DM and
wonder if we might be seeing the effects of smooth gradients in large
scale galactic structures that are not part of a turbulent cascade.
We performed a Monte Carlo simulation to investigate this. In each
realization of the simulation, we generated with a different random
number seed, a screen of density fluctuations. We assumed that the
random fluctuations corresponding to a given spatial frequency are
described by a Gaussian function, but the total power as a function of
spatial frequencies is described by a single power law of index
--11/3. Assuming that the screen is located at the mid point along
the sight line, we let the pulsar drift with its transverse velocity,
and measured the implied column density (DM) as a function of time.
We developed a procedure similar to that of Deshpande (2000) to
compare the observed $\dmt$ curve with the simulated ones. From the
observed $\dmt$ curve, we computed the parameter $\Delta DM\;=\;[DM(t)
- DM(t-\tau_{\circ})]$, where $\tau_{\circ}$ is the time delay. Our
aim is to compare the distribution of this parameter in very short
delays and very large delays. As we can see in Figure
\ref{fig:struct}, the structure function describes a well defined
slope between the delay range of $\sim$30 days to $\sim$2000 days. We
defined two delay bins, 30--60 and 1300--2000 days, within which we
monitored the distribution function of the quantity $\Delta DM$. From
this, we could infer that the distribution at the bin of 1300--2000
days had a span of $\sim 20 \sigma_s$, where $\sigma_s$ is the RMS
deviation of the distribution at the delay range of 30--60 days. That
is, the $\Delta DM$ values that we see at largest delays is as high as
20 times that of the typical deviations at short delays. We performed
the same procedure on the simulated set of data to quantify the
likelihood of such deviations. We simulated 1024 number phase
screens. Out of these 1024 screens, we found that such large
deviations were possible $\sim$7\% of the times. This is perhaps not
surprising, as with such a steep spectrum, it is obvious that most of
the power is in large scales (smaller spatial frequencies), and hence
they tend to dominate our $\Delta DM$ measurements. We conclude that
while monotonic changes of this magnitude are rare, it is consistent
with a turbulent cascade spectrum of density fluctutations.
\section{Frequency dependence of DM}
\label{sec-dmfreq}
Dispersion depends on the column density of electrons. In a
uniform medium radio wave propagation senses the average density
in a tube whose beam waist is set by the Fresnel radius
$\sqrt{z(1-z)\lambda D}$. In a
turbulent medium frequency-dependent multipath propagation can expand
this tube considerably. With refraction, the center of the tube
wanders from the geometric LOS. Indeed there may be a number of wave
propagation tubes, each with their independent relative gain. The
consequence is that DM and related effects will show frequency
dependence:
\begin{enumerate}
\item the effective DM depends on frequency.
\item the DM variations at lower frequencies will be
much ``smoother'' than that at higher frequencies, as the apparent
angular size of the source acts as a {\it smoothing function} on the
measured DM variations.
\item since the apparent size of the source is larger at low
frequencies, some features of the ISM that are visible at lower
frequencies may be invisible at higher frequencies!
\end{enumerate}
We can explore these effects by assuming that the timing residuals at
327 MHz and 610 MHz, which are relative to the timing model derived at
higher frequencies that included removal of DM variations, are due to
DM variations.
The smoothing effect of scattering could be revealed by a spectral
analysis. The slow variations were removed to pre-whiten the spectrum
that would otherwise be severely contaminated. The 327 MHz data was
fit to a fourth order polynomial and the result was subtracted from
both data sets. The two right side panels give the residual DM values
after subtracting the best fit curve from the actual DM curve. The
resulting spectral comparison fails to have sufficient signal to
clearly demonstrate increased smoothing at 327 MHz relative to 610
MHz. Higher signal-to-noise ratio is required. The DM variations
relative to long term trends in the right-hand panels of Figure
\ref{fig:dmindiv} are different.
An important source of systematic error that can affect our analysis
here is the effect of scattering on the derived DM as a function of
time at a given frequency. At 327 MHz, as we described in
\S\ref{sec-anal}, we perform an elaborate procedure to fit for the
scatter broadening of the pulse profile, in order to compute the
``true'' TOA of the profile. However, we do not follow this procedure
at 610 MHz (or any other higher frequency). The error due to this can
be quantified easily from Figure \ref{fig:tausc}. The temporal scatter
broadening value varies by an RMS value of some 19.6 $\mu$s. With the
wavelength dependence of $\tausc\propto\lambda^{4.4}$, the expected
RMS variation at 610 MHz is 1.3 $\mu$s. The equivalent DM perturbation
at 610 MHz with respect to infinite frequency is $\sim
10^{-4}$~pc~cm$^{-3}$.
\section{Achievable timing accuracy}
\label{sec-timingerror}
In this section, we will estimate quantitatively errors introduced by
various scintillation related effects. For PSR B1937+21, although a
typical observation with highly sensitive telescopes like Arecibo
telescope helps us achieve a TOA accuracy of a few tens of
nanoseconds, the ultimate long term accuracy seems to be far greater
than this. In general, it is a combination of frequency independent
``intrinsic timing nose" from the pulsar itself, and the frequency
dependent effects, such as what we are addressing here. With some 18
years of data at 800, 1400 and 2200 MHz, Lommen (2002) quantifies the
timing timing residual, after fitting for position, proper motion,
rotation frequency and its time derivative (see also Kaspi et
al. 1994). A large fraction of the left over residuals is presumably
the intrinsic timing noise. As we have mentioned before, we have
absorbed a good part of this by fitting for the second time derivative
of the rotation frequency, $\ddot{f}$ (see
Table~2). In this section, our aim is to quantify
possible timing errors from various ``chromatic'' effects related to
interstellar scintillation.
\subsection{Fluctuation of apparent angular size}
The temporal variability of pulse broadening, $\tausc$,
(as shown in Figure \ref{fig:tausc}) means that even the apparent
angular broadening of the source, $\theta_H$, is also changing as a
function of time. Since $\tausc \propto \theta_H^2$, with the RMS
variation in $\tausc$ of 19.6 $\mu$s at the radio frequency of 327
MHz, the corresponding variation in $\theta_H$ comes out to be
$\sim$8\% of the mean value. This change occurs with typical time
scales of $\sim$67 days, which is the time scale with which $\tausc$
changes. Since we have only one epoch of $\theta_H$ measurement, we
have no way of observationally verifying the mean value or the time
scale of its variation.
\subsection{Image wandering and the associated timing error}
Due to non-diffractive scintillation that ``steers'' the direction of
rays (``refractive focussing''), the position of the pulsar is
expected to change as a function of time. This is an important and
significant effect, as it introduces a TOA residual as a function of
time, depending on the instantaneous position of source on the
sky. Several authors have investigated this effect in the past
(Cordes et al. 1986; Romani et al. 1986; Rickett \&
Coles 1988; Fey \& Mutel 93; Lazio \& Fey 2001). For a Kolmogorov
spectrum of irregularities ($\beta = 11/3$) with infinitismally small
inner scale cutoff, Cordes et al. (1986) predict the value of
RMS image wandering as
\begin{equation}
\langle\delta\theta^2\rangle^{1/2} = 0.18\; {\rm mas} \; \left(
\frac{D_{\rm kpc}}{\lambda_{\rm cm}} \right)^{-1/6} \theta_H^{2/3}
\end{equation}
For an assumed distance to PSR B1937+21 of 3.6 kpc, this comes to 2
mas at 327 MHz (wavelength, $\lambda = 92$ cm). The value of 2 mas is
still significantly less than the apparent angular size of the source,
14.6 mas, measured by Gwinn et al. (1993). However, for a spectrum
with a steeper slope or with a significantly larger inner scale cutoff
(as in our case), the value of $\langle\delta\theta^2\rangle^{1/2}$ is
expected to be much larger, perhaps comparable to the value of
$\theta_H$.
In order to estimate the timing errors introduced by this image
wandering, we need an estimate of scattering measure ($SM$) and $C_n^2$
along the LOS to this pulsar. Following Cordes et al. (1991),
\begin{eqnarray}
SM &=& \int_0^D C_n^2(x)\;\;{\rm d}x \nonumber\\ &=&
\left(\frac{\theta_H}{128\;{\rm mas}} \right)^{5/3} \nu_{\rm
GHz}^{11/3} \nonumber\\ &=& 292\; \left( \frac{\tausc}{D_{\rm kpc}}
\right)^{5/6} \nu_{\rm GHz}^{11/3}.
\end{eqnarray}
Here, $\tausc$ is specified in seconds. $SM$ is specified in units of
kpc m$^{-20/3}$. Assuming a distance of 3.6 kpc, $\tausc$ = 120 $
\mu $s, $\nu$=0.327 GHz, the value of $SM$ comes to $\sim 8.8\times
10^{-4}$ kpc m$^{-20/3}$. Assuming that the scattering material is
uniformly distributed along the LOS, $C_n^2\sim 2.4\times 10^{-4}$
m$^{-20/3}$.
Then, for a Kolmogorov spectrum, the RMS timing residual due to the
image wandering can be written as (Cordes et al. 1986)
\begin{equation}
\sigma_{\delta t_{\theta}} = 26.5 \;\; {\rm ns} \;\; \nu^{-49/15}
D^{2/3} \left( \frac{C_n^2}{10^{-4} m^{-20/3}} \right)^{4/5}.
\end{equation}
With the computed value of $C_n^2$ and a distance of 3.6 kpc, this
amounts to 4.8 $\mu$s at 327 MHz. Given the frequency dependence, this
effect can be minimized by timing the pulsar at higher frequencies.
For instance, at frequencies of 1 GHz and 2.2 GHz, this error
translates to 125 and 2 nanosec, respectively. However, given the
significantly large value of the inner scale cutoff, the RMS timing
error that we have computed may well be a lower limit, and it is
likely to be higher.
Given the fact that the exact source position due to this effect is
unknown at any given time, it is very difficult to compensate for this
effect.
\subsection{Positional errors in solar system barycentric corrections}
As we saw above, due to image wandering, the apparent position of the
source wanders in the sky. This introduces yet another timing
error. While translating the TOA at the observatory to the solar
system barycenter, we assume a position which is away from the actual
apparent position at the time of observation. This introduces an
error, which can be quantified as (Foster \& Cordes 1990)
\begin{equation}
\Delta t_{\rm bary}\;=\; \frac{1}{c}\; (\vec{r_e}\cdot\hat{n}) \;
(1-z) \Delta\theta_r(\lambda).
\end{equation}
Here, $c$ is the velocity of light, the dot product term gives the
projected extra path length travelled by the ray due to Earth's annual
cycle around the Sun, and $\Delta\theta_r(\lambda)$ is the positional
error due to image wandering. Of course, this term is a function of
frequency, and hence the error accumulated is different at different
frequencies.
At 327 MHz, with an RMS image wandering angle of 2 mas, for an object
at the ecliptic plane, $\Delta t_{\rm bary}\sim 2$ $\mu$s. For PSR
B1937+21, this error amounts to $\sim$0.8 $\mu$s. At frequencies 1
GHz and 2.2 GHz, this error translates to 85 and 17 nanosec, respectively.
\subsection{Timing error due to DM variation as a function of frequency}
\label{sec-accuracy}
An important issue that arises due to the frequency dependent DM
variation is the timing accuracy. Pulsars like PSR B1937+21 are known
for the accuracy to which one can compute the pulse TOA. Given this,
one wishes to eliminate any error that is incurred due to systematic
effects like what we have here. Between 327 and 610 MHz (the two
curves in Figure \ref{fig:dmindiv}), the typical relative fluctuation
of DM that we see is about $5\times 10^{-4}$ pc cm$^{-3}$. As we
discussed before, this is purely due to the fact that the effective
interstellar column length sampled at one frequency is different from
that at another frequency, due to the scatter broadening of the
source. At 610 MHz, this relative DM fluctuation corresponds to some 6
$\mu$s at 610 MHz. That is, at 610 MHz, typically an unaccounted
residual of 6 $\mu$s is incurred due to just effective DM errors. Even
if the behavior of the pulse emission is extremely stable, at low
frequencies, interstellar scattering limits our timing capabilities.
Due to the fact that dispersion delay goes as $\nu^{-2}$, although the
above mentioned effect seems significant, one should be able to reduce
it by going to higher frequencies. For instance, at 2.2 GHz, the
DM-limited TOA error for PSR B1937+21 will be $\sim$0.5$\mu$s. This
is not necessarily encouraging news, as a timing residual error of
0.5$\mu$s is large when compared to the accuracy that we can achieve
in quantifying the TOAs (a few tens of ns) for this pulsar, given
our observations with sensitive telescopes like Arecibo.
To summarize, although one takes into account time dependent DM
changes while analysing the data, in order to achieve high accuracy
timing, it is important to correct for a frequency dependent DM
term. This introduces another dimension of correction in the timing
analysis.
\section{Concluding remarks}
We have presented in this paper a summary of over twenty years of
timing of PSR B1937+21. These observations have been done over
frequencies ranging from 327 MHz to 2.2 GHz with three different
telescopes.
Given the agreement between the measured apparent angular broadening
and that estimated by the temporal broadening, and the measured
proper motion velocity and that estimated by the knowledge of
scintillation parameters, we conclude that the scattering material
is uniformly distributed along the sightline.
There are three significant discrepancies between the expected values
and the measured refractive parameters. These are,
\begin{enumerate}
\item The measured flux variation time scale is about an order of
magnitude shorter than what is expected from the knowledge of the
observed apparent angular broadening.
\item The flux variation time scale is observed to be directly
proportional to the wavelength, whereas it is expected to vary as
proportional to $\lambda^{2.2}$ (for a Kolmogorov spectrum).
\item The flux modulation index is observed to have a wavelength
dependence that is much ``shallower" than the expected value.
\end{enumerate}
These three discrepancies consistently imply that the optics is
``caustic-dominated''. This would mean that the density irregularity
spectrum has a large inner scale cutoff, $1.3\times 10^9$ m. Our
extrapolation of the phase structure function from the regime sampled
by DM variations to the diffractive regime seems to indicate that the
expected $\tdiff$ value is considerably shorter than the measured
value. This is in favor of the above conclusion. Accurate measurements
of frequency dependence of diffractive parameters is much needed.
In general, Millisecond pulsars are known for their timing stability.
Potentially, we may achieve adequate accuracy in timing some of these
pulsars to understand some of the most important questions related to
the gravitational background radiation, or the internal structure of
these neutron stars. However, our analysis here shows that
interstellar scattering could be an important and significant source
of timing error. As we have shown, although PSR B1937+21 is known to
produce short term TOA errors as low as 10--20 nanosec with sensitive
observations, the long term error is far larger than this. After
fitting for $\ddot{f}$ (which absorbes most of the achromatic timing
noise), the best accuracy that we can achieve for this pulsar is
$0.9\;\mu$sec at 1.4 GHz, and about 0.5 $\mu$sec at 2.2 GHz (by one of
the authors, Andrea Lommen). It appears that almost all of this error
can be accounted for by various effects that we have discussed in
\S\ref{sec-timingerror}. In general, for millisecond pulsars with
substantial DM, even if achromatic timing noise is small, interstellar
medium may be a major source of timing noise.
\acknowledgements We thank M. Kramer for sharing the data from
the Effelsberg-Berkeley Pulsar Processor (EBPP), and S. Chatterjee and
W. Brisken for sharing their VLBA based proper motion and parallax
results of PSR B1937+21 prior to the publication. This work was in part
supported by the NSF grant, AST--9820662.
|
Title:
Spectra of the spreading layers on the neutron star surface and constraints on the neutron star equation of state |
Abstract: Spectra of the spreading layers on the neutron star surface are calculated on
the basis of the Inogamov-Sunyaev model taking into account general relativity
correction to the surface gravity and considering various chemical composition
of the accreting matter. Local (at a given latitude) spectra are similar to the
X-ray burst spectra and are described by a diluted black body. Total spreading
layer spectra are integrated accounting for the light bending, gravitational
redshift, and the relativistic Doppler effect and aberration. They depend
slightly on the inclination angle and on the luminosity. These spectra also can
be fitted by a diluted black body with the color temperature depending mainly
on a neutron star compactness. Owing to the fact that the flux from the
spreading layer is close to the critical Eddington, we can put constraints on a
neutron star radius without the need to know precisely the emitting region area
or the distance to the source. The boundary layer spectra observed in the
luminous low-mass X-ray binaries, and described by a black body of color
temperature Tc=2.4+-0.1 keV, restrict the neutron star radii to R=14.8+- 1.5 km
(for a 1.4-Msun star and solar composition of the accreting matter), which
corresponds to the hard equation of state.
| https://export.arxiv.org/pdf/astro-ph/0601689 |
\date{Accepted, Received}
\pagerange{\pageref{firstpage}--\pageref{lastpage}} \pubyear{2005}
\label{firstpage}
\begin{keywords}
{accretion, accretion discs -- radiative transfer -- X-rays: binaries -- stars: neutron }
\end{keywords}
\section{Introduction}
Matter accreting on to a weakly magnetized neutron star (NS) in low mass X-ray
binaries (LMXRBs) can form an accretion disc which extend down to the
NS surface. A boundary layer (BL)
is formed between the accretion disc and the NS surface,
where a rapidly rotating matter of the disc is decelerated down to the
NS angular velocity. The amount of the energy, which is
generated during this process, is comparable with the energy generated in
the accretion disc \citep{SS86,SS98}.
There is no generally accepted theory of the BL. Two different approaches to
the BL description are considered. First of them, which we will call
a `classical model', considers the BL between a central star (a white
dwarf or a NS) as a part of the accretion disc
\citep{P77,PS79,T81,SS88,BK94,PN95,PS01}.
In this model the component of velocity normal to the accretion disc plane
is zero. The half-thickness of the BL is determined by the same relation,
as for the accretion disc:
\be \label{eq:Hbl}
\Hbl\sim \frac{c_{\rm s}}{v_{\rm K}} R,
\ee
where $c_{\rm s}$ is a sound speed in the BL and $v_{\rm K}$ is the
Keplerian velocity close to the NS surface of radius $R$.
The radial extension
of the BL is determined by the relation \citep{P77}
\be \label{eq:hbl}
\hbl\sim \frac{c^2_{\rm s}}{v^2_{\rm K}} R
\sim \Hbl\frac{\Hbl}{R}.
\ee
In this classical model the accreting matter in the BL
is decelerated in the accretion disc plane, along radial coordinate only,
due to the viscosity operating within the differentially rotating BL,
similarly to the accretion disc. From the observational point
of view, the classical BL is a bright equatorial belt close to the
NS surface. The effective temperature of the BL is higher than
the maximum accretion disc effective temperature, because the BL is
smaller than the accretion disc, while their luminosities are comparable.
Another approach was suggested by \citet[][ hereafter IS99]{IS99}.
The BL is considered as a spreading layer (SL)
on the NS surface. The accreting matter diffusing along
the radial direction in the accretion disc and reaching the neutron
star surface gains the velocity component normal to the accretion disc
plane due to the ram pressure from the accretion disc. Then the
matter spirals along the NS surface towards the
poles. Rotating at the NS surface, the matter is decelerated due
to a turbulent friction between the rapidly rotating matter and a slowly
rotating NS surface. The kinetic energy of the accreting gas
is mostly deposited in
two bright belts at some latitude above and below the NS equator.
The width and the latitude of the belts depend on the mass accretion rate. The larger
the accretion rate, the wider the belts are and the closer they are to the NS
poles. At the accretion rate close to the Eddington limit ($L_{\rm BL}
\sim \Ledd$) the bright belts expand all over the NS surface.
The observational difference between two BL models is not significant.
At low accretion rates ($\Lbl\sim 0.01 \Ledd$)
the latitude of bright belts of the SL is small ($\sim$ few
degrees) and the vertical extension of the SL is comparable
to the classical BL thickness. At high accretion
rates ($\Lbl\sim \Ledd$) the classical BL thickness is
comparable to the NS radius \citep[see][]{PS01}.
Therefore, the effective temperatures of these BL models are of the
same order.
In the approach by IS99, the NS radius is assumed to be larger that $3\Rs$
(where $\Rs=2GM/c^2$ is the Schwarzschild radius of a NS of mass $M$),
but the accretion disc structure is not significantly changed up to the NS
surface, and the radial velocity is always subsonic.
If the radial velocity of accreting gas is supersonic at the surface
(see e.g. \citealt*{PN92}, for a possibility of the supersonic radial velocity
in classical BL, and \citealt*{KW91}, for the ``gap accretion'' when
the NS is within the innermost stable circular orbit),
some fraction of the kinetic energy (associated with
a small radial velocity component) should be dissipated in an oblique shock, but
most of it still remains stored in the kinetic energy of the gas rotating at the surface
to be dissipated later in the SL.
The gap accretion model of \citet{KW91} is rather similar to the SL, but they
did not consider the fate of the accreting material and its spread over the surface.
At very low accretion rate, both models
should produce hard Comptonization spectra extending to $\sim100$ keV.
The aim of this work is the calculation of the radiation spectra of the
SLs and their comparison to the observed
X-ray spectra of the BLs in the luminous LMXRBs.
\section{Spreading layer model}
The theory of the SL was developed by IS99
under the assumption of Newtonian gravity. They
considered accretion of the pure hydrogen plasma. Here we
repeat the IS99 theory for plasmas of arbitrary chemical composition
taking into account general relativity (GR) corrections using the
pseudo-Newtonian potential. These corrections may be important, because
the maximum effective temperature of the SL, which should be
smaller than the local Eddington effective temperature $\Tcr$,
depends on the opacity and the gravity. The critical
temperature is determined by the balance between the surface gravity and
the radiative acceleration:
\be \label{eq:tc}
\frac{G \Mns}{\Rns^2 \sqrt{1-\Rs/\Rns}} = \frac{\sigmasb \Tcr^4}{c}
\sigmae,
\ee
where $\sigmae = 0.2 (1+X)$ cm$^2$~g$^{-1}$ is the electron scattering opacity, $X$ is the hydrogen mass fraction, and $\sigmasb$ is the Stefan-Boltzmann constant.
It is clear, that solar (or larger) helium
abundance together with the GR correction will lead to higher $\Tcr$,
and, therefore, to a higher maximum effective temperature of the SL.
This could have important consequences for the determination of the neutron
star parameters from observations.
We use the pseudo-Newtonian potential in the form:
\be \label{eq:fi}
\Psi (r) = -c^2 \left(1 - \sqrt{1-\Rs/r}\right).
\ee
This potential gives the correct GR surface gravity
\be \label{eq:g0}
g_{\rm 0}(\Rns)= \frac{G \Mns}{\Rns^2\sqrt{1-\Rs/\Rns}},
\ee
but gives the Keplerian velocity at
the NS surface
\be \label{eq:vk}
v^2_{\rm K}(\Rns) = \frac{G \Mns}{\Rns\sqrt{1-\Rs/\Rns}},
\ee
which is smaller than the correct GR value.
\subsection{Main equations}
Below we rewrite the IS99 equations for the SL for the pseudo-Newtonian
potential (\ref{eq:fi}) and considering arbitrary abundances.
We consider the dynamics of the SL on the NS
surface (see Fig.~\ref{fig1}). The full hydrodynamic equations, which
describe this process are as follows \citep[see for example][]{M78}. The
continuity equation is
\be \label{eq:cont}
\frac{\partial \rho}{\partial t} + {\bf \nabla \cdot} (\rho \vecv) =0,
\ee
where $\rho$ is the plasma density, $\vecv$ is the vector of the
gas velocity in the SL with components $\vphi$, $\vtheta$
and
$v_{r}$, which are velocities of the SL along longitude, latitude and
radius correspondingly.
Conservation of momentum for each gas element is described by the
vector Euler equation
\be \label{eq:euler}
\rho \frac{\partial \vecv}{\partial t} +
\rho \vecv \cdot \nabla \vecv = - \nabla P + \vecf,
\ee
where $P=P_{\rm rad}+P_{\rm g}$ is the total pressure which is a sum of the radiation
and gas pressures, and $\vecf$ is a force density. The energy equation
for the gas in the SL is
\begin{eqnarray} \label{eq:energy}
\frac{\partial}{\partial t} \left( \frac{1}{2} \rho v^2
+\varepsilon\right) + \nabla \cdot
\left[ \left(\frac{1}{2} \rho v^2 + \varepsilon + P\right){\vecv}\right] =
\\ \nonumber \vecf \cdot \vecv - \nabla \cdot \vecq +Q^+.
\end{eqnarray}
Here $\varepsilon = \varepsilon_{\rm rad}+ \varepsilon_{\rm g}$ is the total
density of internal energy, where $\varepsilon_{\rm rad}=aT^4$ is the
radiation energy density and $\varepsilon_{\rm g} = (3/2) P_{\rm g}$ is the density
of the internal gas energy. The first term on the right hand side
is the power produces by the force density, the second is
the energy, which is lost by radiation ($\vecq$ is a vector of the
radiation flux), and the third is the heat, which is generated within a
unit volume of the SL.
Following IS99 we consider the steady state SL model in the
spherical coordinate system $(r,\theta,\varphi)$, where $\theta$
is the latitude and $\varphi$ is the azimuthal angle (see Fig.~\ref{fig1}).
We also assume that the SL has a
small thickness (in comparison with the NS radius $\Rns$, therefore the
radial coordinate $r=\Rns$), the radial velocity component
is zero $v_r=0$, and it is axially symmetric
(therefore, all of the derivatives $\partial / \partial \varphi$ equal to
zero). In this case
equations (\ref{eq:cont})--(\ref{eq:energy}) take the following form.
The continuity equation is
\be \label{eq:cont1}
\frac {1}{R \cos \theta} \frac{\partial}{\partial \theta} (\cos \theta\ \rho
\vtheta) = 0 ,
\ee
the
three components of the Euler equation are
\be \label{eq:e1}
-\rho\left(\frac{\vtheta^2 +\vphi^2}{R}\right) =
-\frac{\partial P}{\partial r} +\fr,
\ee
\be \label{eq:e2}
\rho \frac{\vtheta}{R} \frac{\partial \vtheta}{\partial
\theta} + \rho \frac{\vphi^2}{R} \tan \theta =
-\frac{1}{R}\frac{\partial P}{\partial \theta} +\ftheta,
\ee
\be \label{eq:e3}
\rho \frac{\vtheta}{R} \frac{\partial \vphi}{\partial
\theta} - \rho \frac{\vphi \vtheta}{R} \tan \theta =
\fphi,
\ee
and the energy equation is
\begin{eqnarray} \label{eq:energy1}
\frac{1}{R \cos \theta} \frac{\partial}{\partial \theta}
\left[ \cos\theta \ \vtheta
\left(\frac{1}{2} \rho v_0^2 + \varepsilon +P\right)\right] =
\\ \nonumber
\ftheta\vtheta + \fphi\vphi - \frac{\partial q}{\partial r} + Q^+,
\end{eqnarray}
where
\be \label{v0}
v_0^2= \vphi^2 + \vtheta^2.
\ee
Here the radiation flux has only one (radial) non-zero
component and its divergence is computed in the plane-parallel approximation
which is a consequence of our assumption of small height of the SL.
A small azimuthal component of the radiation flux arises due to the aberration,
which we neglect here.
It is clear that equation~(\ref{eq:e1}) can be solved independently on equations
(\ref{eq:e2})--(\ref{eq:e3}) and we can consider some averaging over the
layer's height. Therefore, we arrive at a one-dimensional problem.
In this case instead of equations (\ref{eq:cont1})--(\ref{eq:energy1})
we obtain
\be \label{eq:cont2}
\frac {1}{R \cos \theta} \frac{\partial}{\partial \theta} \left(
\cos \theta \int \rho \vtheta \d r\right) = 0 ,
\ee
\be \label{eq:e22}
\int \rho \vtheta \frac{\partial \vtheta}{\partial \theta} \d r + \tan \theta \int \rho \vphi^2 \d r
= -\frac{\partial}{\partial \theta} \int P \d r + R \int \ftheta \d r ,
\ee
\be \label{eq:e32}
\int \rho \vtheta
\frac{\partial \vphi}{\partial \theta} \d r - \tan \theta
\int \rho \vphi \vtheta \d r = R \int \fphi \d r ,
\ee
\begin{eqnarray} \label{eq:energy2}
\frac{1}{R \cos \theta} \frac{\partial}{\partial \theta}
\left[ \cos\theta \int \vtheta
\left(\frac{1}{2} \rho v_0^2 + \varepsilon +P\right) \d r \right] =
\\ \nonumber
\int \ftheta\vtheta \d r + \int \fphi\vphi \d r - q + \int Q^+ \d r.
\end{eqnarray}
Here the integration over radius is from $R$ to $R+\hs$,
where $\hs$ is the local SL thickness. We define the corresponding
force densities in the next section.
\subsection{Vertical averaging}
\label{sec:avehei}
The one-dimensional equations for the SL structure are derived using
the averaging along the height at a given latitude. It means that we have
to calculate all the integrals in equations
(\ref{eq:cont2})--(\ref{eq:energy2}) for some model of the SL vertical structure.
The simplest way is just to consider the variables averaged over the height.
IS99 used a
more complicated model of averaging. They constructed a simple model
of the SL using assumptions that velocities $\vtheta$, $\vphi$ and the radiation flux
do not depend on the height $q(r)=\mbox{const}=\sigmasb \Teff^4$.
The latter suggestion means that all of the thermal
energy in the SL is generated at the bottom. This model is described by the
hydrostatic equilibrium equation (\ref{eq:e1}) taken in the form
\be \label{eq:he}
\frac{\d P}{\d m} = \geff \equiv g_0 - \frac{\vphi^2+ \vtheta^2}{R},
\ee
and the radiation transfer
equation in the diffusion approximation
\be \label{eq:re}
\frac{\d\varepsilon_{\rm rad}}{\d m} = \frac{3q}{c} \sigmae.
\ee
Here and below we use a new independent variable: a column mass
$m$ and a new geometrical coordinate $z$, which are defined as
\be
\d m = \rho \d z = -\rho \d r. \nonumber
\ee
Coordinate $z$ has an opposite direction relative to $r$ and $z$=0 at
$r=R+\hs$. We also defined the $r$ component of the force density
\be
\fr = -g_0 \rho. \nonumber
\ee
Equations (\ref{eq:he})--(\ref{eq:re}) have to be supplemented by the equation
of state
\be \label{eq:se}
P = \frac{\rho kT}{\mu m_{\rm p}} +
\frac{\varepsilon_{\rm rad}}{3},
\ee
where $\mu=4/(3+5X)$ is the mean molecular
weight and $m_{\rm p}$ is the proton mass.
Equations (\ref{eq:he})--(\ref{eq:se}) can be solved with the simple
boundary conditions $P(m=0)=0$, $T(m=0)=0$:
\be
P=\geff m,
\ee
\be
\varepsilon_{\rm rad}= \frac{3q}{c} m \sigmae,
\ee
\be
\rho = \mu m_{\rm p} \frac{\gwr}{k} \left( \frac{a c}{3q
\sigmae}m^3 \right)^{1/4},
\ee
\be
T=\left( \frac{3q}{ac} m \sigmae \right)^{1/4}= \Teff
\left( \frac{3}{4} \taue\right)^{1/4},
\ee
where $\taue= m \sigmae$ is the electron scattering optical
depth of the layer, and
\be
\gwr \equiv \geff - \grad = \geff - \frac{q}{c}
\sigmae.
\ee
The column density $m$ is related to the geometrical depth~$z$
\be
m = \frac{(\mu m_{\rm p} \gwr)^4}{4^4 \sigmae k^4}
\frac{ac}{3q} z^4,
\ee
which gives the following dependence of temperature on height
\be
T=\frac{\mu m_{\rm p} \gwr}{4k} z.
\ee
Following IS99, we consider the values of temperature and density at the
bottom of the SL $\TS$ and $\rho_{\rm S}$ as parameters.
In this case the local SL thickness $\hs$ is:
\be \label{eq:height}
\hs =
\frac{4k\TS}{\mu m_{\rm p} \gwr}.
\ee
We can also calculate all of the integrals in
equations~(\ref{eq:cont2})--(\ref{eq:energy2}): the total surface density
\be
\int_{0}^{\hs} \rho \d z = m(z=\hs) = \Sigmas,
\ee
the pressure surface density
\be
\int^{\hs}_0 P \d z = \frac{1}{5}
\geff \hs \Sigmas= \frac{4}{5} \frac{\geff}{\mu
m_{\rm p}\gwr} \Sigmas k \TS,
\ee
the surface density of the internal energy
\begin{eqnarray}
E_{\rm int} = \int^{\hs}_0 \left(\varepsilon_{\rm rad} +
\frac{3}{2}
P_{\rm g}\right) \d z =
\frac{3}{2} \left(\geff+\grad\right)
\frac{\Sigmas \hs}{5},
\end{eqnarray}
the local flux
\be
q=\sigmasb \Teff^4 = \frac{ac}{3\sigmae}
\frac{\TS^4}{\Sigmas},
\ee
and the enthalpy flux
\be
H= E_{\rm int} + \int^{\hs}_0 P \d z =
\left( \frac{5}{2} \geff + \frac{3}{2} \grad\right)
\frac{\Sigmas \hs}{5} .
\ee
If we take $X=1$, all these relations will be the same, as
derived by IS99 with one exception: there is no
potential energy of the SL in our energy equation.
Thus our expression for $H$ contains a factor
$5/2$ instead of $7/2$ as in IS99. Below we will show that this
produces only a small quantitative differences between our
and IS99 models.
In the IS99 model there are two forces, which give contribution to
the force density in equations (\ref{eq:cont2})--(\ref{eq:energy2}). These
are the gravity force, which has only the radial component (see above) and the
force arising due to the friction between the NS surface and
the SL. This force is directed along the NS surface and is
expressed in the IS99 model through stress $\tau$ and its azimuthal and
meridional components
$\tauphi$ and $\tautheta$.
IS99 have parameterized it in the form:
\begin{eqnarray}
\tauphi&=& -\int_0^{\hs} \fphi\d z = \alphab
\rhos\vphi v_0,\\ \nonumber
\tautheta &=& -\int_0^{\hs} \ftheta\d z = \alphab
\rhos\vtheta v_0,
\end{eqnarray}
where $\alphab=v_*^2/v_0^2$ is the parameter of the stress parametrization,
$v_*$ is the velocity of turbulent pulsations.
We should note that $\alphab$ is not the same $\alpha$ that is
used in the accretion disc theory. In accretion discs, $\alpha$
(in the first approximation) is the square of the ratio of the
turbulent velocity to the sound speed $\alpha = v_*^2/c_{\rm s}^2$ and
can be quite high, up to 0.1--1.
In the SL, the plasma velocity $v_0$ is close to the Keplerian velocity
at the NS surface and is orders of magnitude larger than the sound speed.
The velocity of turbulent pulsation is also limited by the radiation
viscosity at the SL bottom. IS99 carefully investigated this matter
and estimated $\alphab \sim 10^{-3}$.
We used this value in most of the paper.
IS99 have ignored the mechanical work between the SL and the NS (which
accelerates the stellar rotation). In our work we use the same approximation.
A fraction of the kinetic energy of the accreting gas that goes to
increase the rotational energy of the NS is approximately $2 \Omegans/\Omegak$,
where $\Omegans$ is the NS angular velocity and $\Omegak$ is the Keplerian
angular velocity at the NS surface.
As we consider a non-rotating NS and the characteristic time to increase
its angular velocity is orders of magnitude larger than the
characteristic time of the SL $t= \Rns/\vtheta=10\ \mbox{km}/ 10^3\ \mbox{km s}^{-1} = 0.01$ s,
ignoring the mechanical work on the NS is a reasonable approximation.
In this approximation, therefore, all the work due to friction between the SL and the
NS transforms to heat:
\be \label{eq:heat}
\int_0^{\hs} Q ^+ \d z = - (\tauphi\vphi +
\tautheta \vtheta) = -\tau v_0.
\ee
\subsection{One-dimensional model of the spreading layer}
\label{sec:structure}
Using relations (\ref{eq:height})--(\ref{eq:heat}) we can rewrite equations,
which describe the one-dimensional SL structure. The
continuity equation can be rewritten via the accretion rate as
\be \label{eq:cn}
\dot{M} = 4 \pi R \ \cos\theta\ \vtheta \Sigmas,
\ee
Therefore, the product $\cos\theta \ \vtheta \Sigmas =const$. The Euler
equations are
\begin{eqnarray} \label{eq:fine1}
\cos\theta\ \vtheta \Sigmas \vtheta' + \frac{4}{5} \cos\theta\
\left( \frac{\geff}{\gwr}\frac{k\TS}{\mu m_{\rm p}}
\Sigmas\right)' = \\ \nonumber
-R\cos\theta\ \tautheta - \sin\theta\ \Sigmas \vphi^2 ,
\end{eqnarray}
\be \label{eq:fine2}
\Sigmas\vtheta(\cos\theta\ \vphi)' = -R \cos\theta \tauphi.
\ee
Here the prime means the derivative over $\theta$.
The second term in the left hand side of equation~(\ref{eq:fine1}) is the
lateral force gradient, and the terms in the right hand side are the
components of the stress force and the centrifugal force. We
see from equation~(\ref{eq:fine2}) that the momentum along $\varphi$
coordinate is changed due to the friction with the NS surface
only. The system of equations is closed by the energy equation
\be \label{eq:en}
\Sigmas\vtheta \left( \frac{v_0^2}{2} + \frac{2}{5}
\frac{k\TS}{\mu m_{\rm p}}\frac{5\geff+3\grad}{\gwr}
\right)' = -R q .
\ee
The system of equations~(\ref{eq:cn})--(\ref{eq:en}) can be transformed to the three
dimensionless equations for $\vphi(\theta)$,
$\vtheta(\theta)$ and $\TS (\theta)$ as was done by IS99.
These equations are solved with the boundary conditions at
the transition zone between the accretion disc and the SL: the initial
latitude, where the SL starts, is close to the
NS equator $\theta_{\rm 0} \sim $ 10$^{-2}$;
the initial relative
deviation $\delta$ of $\vphi$ from the Keplerian velocity
$\vphi=v_{\rm K}(1-\delta)$; and the initial ratio of the
kinetic energy of the SL along $\theta$ coordinate and
it's thermal energy $\Theta \equiv (\mu m_{\rm p} v^2_{\theta_0})/k\TS $.
As was demonstrated by IS99, the solution of equations~(\ref{eq:cn})--(\ref{eq:en})
depends very little on $\theta_0$ (if $\theta_0$ sufficiently
small $<0.1$, see below) and $\delta$, but strongly depend on
parameter $\Theta$. We choose the solutions which are closest to the critical
solution (in this solution $\vtheta$ is equal to the sound speed at the
maximum latitude of the spreading layer), but slightly subsonic. The
necessary critical value of parameter $\Theta$ is found by the bisection
method.
The distributions of $\vtheta$, the effective
temperature $\Teff$, and the surface density $\Sigmas$ along the
latitude $\theta$ for four models with the same accretion rate,
corresponding to first model luminosity are shown in Fig.~\ref{fig2}.
The first model (shown by the solid
curves) is our model with the pseudo-Newtonian potential and solar hydrogen
abundance $X=0.7$. The dashed curves are for our model
with GR correction, but with pure hydrogen $X=1$; the dotted curves correspond to
our model without GR corrections
($\Rs=0$ in the equations) with solar hydrogen abundance ($X=0.7$), while the
dot-dashed curves are for the IS99 model with GR corrections and
solar hydrogen abundance. It is clear
that the solar abundance lead to a narrower spreading
layer with a smaller surface density. A higher helium abundance as well as the
the GR corrections lead to a higher effective temperature and a larger
latitudinal velocity.
Our model gives a slightly wider SL with slightly smaller latitudinal
velocity but same effective temperature and surface density. Most
calculations below were performed for our model with the GR
correction and solar abundances. The surface density and the
effective temperature distributions along the latitude for models with 0.1,
0.2, 0.4 and 0.8 of the Eddington luminosity are presented in Fig.~\ref{fig3}.
Variations of parameter $\alphab$ lead to some changes in the SL structure.
The SL column density is inversely proportional to $\alphab$,
while the resulting effective temperature decreases only by about 1 per cent with
decreasing of $\alphab$ by an order of magnitude.
The lower boundary of the SL was taken very close
to the equator, $\theta_{\rm 0} \approx 0.01-0.001$, in IS99. Formally, the
accretion disc thickness is close to zero at the inner boundary, if we take
the usual inner boundary condition for the component of the stress tensor
$W_{r\varphi}(R_{\rm in})=0$. In the case of the accretion disc
around a NS this condition is not correct, and the disc
thickness at the NS surface is considerable. The
luminous accretion disc half-thickness can be evaluated from the
balance of the radiation force and $z$-component of gravity:
\be \label{z0}
z_0 = \frac{3 \sigmae}{8 \pi c} \dot{M}.
\ee
We calculated the SL models with the initial latitudes
$\theta_{01} = \arcsin (z_0/\Rns)$ and
$\theta_{02} = \arcsin (0.5 z_0/\Rns)$.
The surface density and the effective temperature
distributions along the latitude for the second case are shown in Fig.~\ref{fig4} .
The qualitative behavior of these distributions is close to the case of
small $\theta_0$ with some shift along the latitude. The main
difference is the maximum possible luminosity of the SL. At this luminosity
the SL reaches the NS poles.
For $\theta_{02}$, the maximum possible luminosity is about 0.4 $\Ledd$,
while for $\theta_{01}$ it is about 0.25 $\Ledd$.
\subsection{Vertical structure of the spreading layer}
\label{sec:vertical}
The IS99 SL model was constructed under an assumption that the
local layer velocity and the radiation flux along height is constant. It
means that the SL is decelerated and energy is liberated in
the infinitely thin layer at the NS surface. This is an
approximation only and the velocity distribution should not be uniform and
the energy should be generated at all heights. Therefore, we need a more detailed
vertical model of the SL for calculating its radiation spectrum.
For evaluation of the viscosity parameter $\alphab$,
IS99 used classical theory of
hydrodynamic BLs \citep{LL59}
with the logarithmic velocity and the energy generation distribution along the height.
In this case,
both the energy generation rate and the velocity gradient are inversely
proportional to the distance from the NS surface $z$
\be
\label{eq:lgr}
\frac{\d q}{\d z} \propto \frac{\d v}{\d z} \propto
\frac{1}{z}.
\ee
It is clear that these dependencies cannot be correct
in a SL. The SL has a finite thickness with a low
density at the surface. However according to equation~(\ref{eq:lgr}) some
amount of energy has to be generated in the surface layers.
At present time, a theory of the radiation-dominated turbulent boundary
layer does not exist. Thus we here can only make similar assumptions about
the energy generation and velocity gradient along the
height. We assume that these values are inversely proportional to the
surface density measured from the NS surface:
\be \label{eq:qgr}
\frac{\d q}{\d m} = -A~ \frac{q_{\rm 0}}{\Sigmas-m},
\ee
\be \label{eq:vgr}
\frac{\d v}{\d m} = -A~ \frac{v_0}{\Sigmas-m},
\ee
where $A= 2.5 \alphab^{1/2}$, $q_{\rm 0}$ and $v_0$ are the
local radiation flux and the average layer velocity at a given latitude
obtained from the one-dimensional model. Equations (\ref{eq:qgr}) and (\ref{eq:vgr})
are very close to the IS99 SL
model assumptions (the layer is decelerated and the energy is
generated at the bottom of the layer).
Integration of these equations yields
\be \label{eq:q}
q(m)=q_{\rm 0} \left[1+A~ \ln\left(1-\frac{m}{\Sigmas}\right)\right],
\ee
\be \label{eq:v}
v(m)=v_0 \left[ 1+A~ \ln\left(1-\frac{m}{\Sigmas}\right)\right].
\ee
These equations can be used up to some critical column density
\be \label{eq:mstr}
m_*=\Sigmas\left(1-\exp[-A^{-1}]\right),
\ee
which is very close to the local surface density.
The hydrostatic equilibrium equation then
reads
\be \label{eq:hyd}
\frac{\d P_{\rm g}}{\d m} = g_{\rm 0} - \frac{v^2(m)}{\Rns}
-\frac{q(m)}{c}\sigmae
\ee
and the radiation transfer equation is
\be \label{eq:rad}
\frac{1}{3} \frac{\d\varepsilon}{\d m} = \frac{q(m)}{c}\sigmae.
\ee
The temperature and the gas pressure distributions along the
height are:
\begin{eqnarray} \label{eq:tm}
T(m) &=& \Teff~ \left[ \frac{3}{4}m \sigmae \
\left(1-A\left[1+ \frac{\Sigmas-m}{m} \right. \right. \right.
\\ \nonumber
& \times& \left. \left. \left. \ln\left(1-\frac{m}{\Sigmas} \right) \right] \right) +\frac{1}{2}\right] ^{1/4}
\end{eqnarray}
\begin{eqnarray} \label{eq:pg}
P_{\rm g}(m) & = & g_0 m - \frac{v^2_0}{\Rns} m\left(1-2A+2A^2\right) \\ \nonumber
& +& \frac{v^2_0}{\Rns}A\left(\Sigmas-m\right) \ln\left(1-\frac{m}{\Sigmas}\right)\\ \nonumber
&\times& \left[A~\ln\left(1-\frac{m}{\Sigmas}\right)-2A+2\right] \\
\nonumber
&- & \frac{q_0\sigmae}{c}\left[m-A\left(m+\left(\Sigmas-m\right)
\ln\left(1-\frac{m}{\Sigmas}\right)\right)\right].
\end{eqnarray}
These solutions are obtained
using the boundary conditions at the surface $\varepsilon(m=0)=2q_0/c$
and $P_{\rm g}(m=0)=0$.
At the same time, there is a disagreement between this vertically explicit
model and one-dimensional model, because the velocity and the flux vertical
profiles are different. We suggest, that the model can be made
self-consistent, if we find a new value of the surface density $\Sigmas'$
at a given latitude, which conserves the mass flux
\be \label{eq:msfl}
v_0 \Sigmas= \int_0^{\Sigmas'} v(m) \d m = v_0
(1-A) \Sigmas'.
\ee
Therefore, the new value of the surface density $\Sigmas' =
\Sigmas/(1-A)$. For $\alphab = 10^{-3}$, this gives
$\Sigmas' = 1.086 \Sigmas$ and we take these values
below for our calculations. There are
similar disagreements for other integrals over the height in equations
(\ref{eq:cont2})--(\ref{eq:energy2}). For example:
\be \label{eq:ke}
v_0^2 \Sigmas= \int_0^{\Sigmas'} v^2(m) \d m =
v_0 (1-2A+A^2) \Sigmas'.
\ee
In this case, we have to take a new value of the surface density
$\Sigmas' = \Sigmas/(1-2A+A^2)$, which gives
$\Sigmas' = 1.171 \Sigmas$ if $\alphab = 10^{-3}$.
Our vertically explicit models disagree
with the one-dimensional ones by about 10 per cent.
Fortunately, the emitted local spectra
depend very little on the surface density of the SL.
\section{Spectrum of the spreading layer}
For calculation of the SL spectra we divide it into a
number of rings over the latitude which have different effective
temperatures $\Teff$, matter velocities $v_0$ and surface densities
$\Sigmas$.
We then calculate the vertically explicit model for
each ring, solve the radiative transfer equation and obtain
the local SL spectrum. Then we integrate
local spectra from the SL surface accounting for the general and
special relativity effects.
\subsection{Local spectra}
To calculate a vertically explicit hydrodynamical model with
the radiation transfer we use standard methods
for stellar atmospheres modelling \citep{M78}. Our models are
obtained in the hydrostatic and the plane-parallel approximations. The
effective temperatures of the considered SL models are rather high
($\sim$ 2 keV) and these models are similar to the atmospheres of
bursting NSs, where Compton scattering have to be taken
into account.
The vertically explicit local SL model is described by the following
equations: the equation of
hydrostatic equilibrium (\ref{eq:hyd}), the
energy generation law (\ref{eq:q}), the
velocity law (\ref{eq:v}), the RTE accounting for the Compton
effect using the \citet{K57} operator:
\begin{eqnarray} \label{eq:rtr}
\frac{\partial^2 ( f_{\nu} J_{\nu})}{\partial \tau_{\nu}^2} =
\frac{k_{\nu}}{k_{\nu}+\sigmae} \left(J_{\nu} - B_{\nu}\right) -
\frac{\sigmae}{k_{\nu}+\sigmae} \frac{kT}{\me c^2}
\times \\ \nonumber
x \frac{\partial}{\partial x} \left(x \frac{\partial J_{\nu}}{\partial x} -
3J_{\nu} + \frac{\Teff}{T} x J_{\nu} \left[ 1 + \frac{CJ_{\nu}}{x^3} \right] \right),
\end{eqnarray}
where $x=h \nu /k\Teff$ is dimensionless frequency,
$f_{\nu}(\tau_{\nu}) \approx 1/3$ is the variable Eddington factor, $J_{\nu}$
is the mean intensity of radiation, $B_{\nu}$ is the black body (Planck)
intensity, $k_{\nu}$ is the opacity due to the free-free and bound-free
transitions, $\sigmae$ is the electron (Thomson) opacity, $T$ is the local
electron temperature, $\Teff$ is the effective temperature of SL at a
given latitude, and $C=c^2 h^2/2(k\Teff)^3$. The optical
depth $\tau_{\nu}$ is defined as
\be
\d \tau_{\nu} = (k_{\nu}+\sigmae) \d m.
\ee
These equations have to be completed by the energy balance equation
\begin{eqnarray} \label{eq:econs}
\int_0^{\infty} k_{\nu}\left(J_{\nu} - B_{\nu}\right) \d\nu -
\frac{1}{4\pi}\frac{\d q}{\d m} -
\sigmae \frac{kT}{\me c^2} \times \\ \nonumber
\left[ 4 \int_0^{\infty} J_{\nu}
\d\nu - \frac{\Teff}{T} \int_0^{\infty} x J_{\nu}
\left( 1+\frac{CJ_{\nu}}{x^3}\right) \d\nu \right]=0
\end{eqnarray}
and by the ideal gas law
\be \label{eq:gstat}
P_{\rm g} = N_{\rm tot} kT,
\ee
where $N_{\rm tot}$ is the number density of all particles, as well as
by the particle and charge conservation laws. We assume local
thermodynamical equilibrium (LTE) in our calculations, so the number
densities of all ionization and excitation states of all elements have been
calculated using Boltzmann and Saha equations.
For solving these equations and computing the local SL model we used
the Kurucz's code {\sc ATLAS} \citep{K70,K93} modified for high temperature.
All ionization states of the 15 most abundant elements are
taken into consideration. The photoionization cross-sections from the
ground states of all ions are calculated using {\sc phfit2} code \citep{V96}.
For details see \citet{SGS02} and \citet{I03}.
The code was also modified to account for Compton scattering.
The scheme of calculation is the following.
First, the input parameters of the
local SL model are defined from the total one-dimensional SL model (see
Sect. \ref{sec:structure}): the effective temperature $\Teff$, the surface gravity $g_0$,
the surface density $\Sigmas$, and the local average layer velocity $v_0$.
Then the analytical vertically explicit model (\ref{eq:tm}--\ref{eq:msfl})
are calculated together with the new value of surface density
$\Sigmas'=\Sigmas/(1-A)$. The calculations are
performed for the set of 98 column densities
$m$, distributed logarithmically with equal steps from $m=
10^{-5}$ g cm$^{-2}$ to $0.99 m_{*}$. The gas
pressure, which is found from equation~(\ref{eq:pg}),
is not varied during the iterations.
For this starting model, all number densities and
the opacities at all depth points and all the frequencies (we use 300
logarithmically equidistant frequency points) are calculated. The
RTE (\ref{eq:rtr}) is solved
by the Feautrier method \citep{M78,ZS91,PSZ91,GS02}
iteratively, because it is non-linear.
Between the iterations we
calculate the variable Eddington factors $f_{\nu}$ and $h_{\nu}$, using
the formal solution of the RTE for three angles.
Usually 5--6 iterations are sufficient to achieve convergence.
We used the usual condition at the outer boundary
\be
\frac{\partial ( f_{\rm \nu} J_{\nu}) }{\partial \tau_{\nu}} = h_{\nu} J_{\nu},
\ee
where $h_{\nu}$ is the surface variable Eddington factor,
and the inner boundary condition
\be
\frac{\partial J_{\nu}}{\partial \tau_{\nu}} =
\frac{\partial B_{\nu}}{\partial \tau_{\nu}}.
\ee
The outer boundary condition is found from the lack of the incoming
radiation at the SL surface, and the inner boundary condition is obtained
from the diffusion approximation $J_{\nu} \approx B_{\nu}$ and $q_{\nu}
\approx 4\pi/3 \times \partial B_{\nu}/\partial \tau_{\nu}$. This condition
is satisfied for any SL optical thickness, because the SL bottom is the
NS surface.
The boundary conditions along the frequency axis are
\be \label{eq:lbc}
J_{\nu} = B_{\nu}
\ee
at the lower frequency boundary, $\nu=\nu_{\rm min}=10^{14}$ Hz
($h\nu_{\rm min} \approx$ 0.03 eV $\ll k\Teff$) and
\be \label{eq:hbc}
x \frac{\partial J_{\nu}}{\partial x} - 3J_{\nu} + \frac{\Teff}{T} x
J_{\nu} \left( 1 + \frac{CJ_{\nu}}{x^3} \right)=0
\ee
at the higher frequency
boundary $\nu=\nu_{\rm max}=3\ 10^{19}$~Hz ($h \nu_{\rm max}
\approx$ 100 keV $\gg k\Teff$). Condition (\ref{eq:lbc})
means that at the lowest energies the true opacity dominates the
scattering $k_{\nu} \gg \sigmae$, and therefore $J_{\nu} \approx
B_{\nu}$. Condition (\ref{eq:hbc}) means that there is no photon flux
along the frequency axis at the highest energy.
The solution of the RTE (\ref{eq:rtr}) should also satisfy
the energy balance equation (\ref{eq:econs}) and the surface
flux condition
\be
\int_0^{\infty} q_{\nu} (m=0) \d\nu = \sigmasb \Teff^4.
\ee
We calculated the relative flux error along the depth
\be
\varepsilon_{q}(m) = 1 - \frac{q(m)}{\int_0^{\infty} q_{\nu} (m) \d\nu},
\ee
where $q(m)$ is found from the energy generation law (\ref{eq:q}), and $q_{\nu}
(m)$ is radiation flux at a given depth obtained from the first
moment of the RTE
\be
4\pi \frac{\partial (f_{\nu} J_{\nu})}{\partial \tau_{\nu}} = q_{\nu}.
\ee
Then the temperature corrections were evaluated using three different
procedures. The first procedure is the
integral $\Lambda$-iteration method based on the energy
balance equation (\ref{eq:econs}) which was modified to account for
Compton scattering. It works well in the upper layers. The
second one is the modified Avrett-Krook flux correction, which uses the
relative flux error and is good in deep layers. And the third one
is the surface correction, which is based on the emergent flux error.
See \citet{K70} for the detailed description of the methods.
The iteration procedure is repeated until the relative flux error is
smaller than 1 per cent, and the relative flux derivative error is smaller
than 0.01 per cent. As a result we obtain the self-consistent local SL model
together with the emergent spectrum of radiation.
Our method of calculation was checked on the atmosphere model of bursting
NS. The equations which describe the bursting
atmosphere are simpler, because there is no velocity field along the
surface ($v_0=0$) and the integral flux is constant along depth
($\d q/\d m=0$).
We compared our model
atmospheres with the most recent models of \citet*{MJR04}.
The radiation spectra and the temperature structure for
some models with $\Teff=2\ 10^7$~K, solar H/He abundances, and
various surface gravities are shown in Fig.~\ref{fig5}.
These results are in a perfect
agreement with the results of \citet{MJR04}. The
emergent spectra and the temperature structure for the models with the
solar abundance of heavy elements are shown in Fig.~\ref{fig6}.
In the surface layers, local cooling is small
because of the low density, and the temperature equals the Compton temperature of
radiation which is slightly higher than the effective temperature.
In dipper layers, at $\taue\sim 0.1$,
the cooling due to thermal emission (free-free and bound-free) becomes
important (as thermal emissivity per gram is proportional to density)
and the temperature decreases.
At large optical depth the temperature rises again and follows the
$\taue^{1/4}$ relation, typical for a grey atmosphere.
At higher surface gravity (at fixed $\Teff$), the plasma density is higher,
resulting in a more significant temperature dip.
We see that heavy elements
have rather minimal influence on the models close to the Eddington limit
(lower~$g$).
The comparison between the bursting NS models and different local SL models for
the same $\Teff$ and effective $\log g$ is shown
in Fig.~\ref{fig7}.
For this SL model we use the vertical structure model, which
is described in Section~\ref{sec:vertical}. We also investigated, whether the model for the
vertical structure is important for the emergent spectra of local SL models. We also
calculated the SL model with the constant velocity and flux derivatives:
\be
\frac{\d v}{\d m} = -\frac{v_0}{\Sigmas}
\ee
and
\be
\frac{\d q}{\d m} = -\frac{q_0}{\Sigmas}.
\ee
In this case the mass flux conservation requirement
(\ref{eq:msfl}) leads to $\Sigmas'= 2 \Sigmas$.
The spectrum and the temperature structure of this model are shown in
Fig.~\ref{fig7} by squares. The surface temperatures of the local SL
models are higher than the bursting NS model surface temperature. The
reason is the non-zero flux derivative in the energy conservation equation
(\ref{eq:econs}). This means that a part of the energy is released in the upper
atmosphere and is heating it additionally. The smaller the surface density
(i.e. the larger the flux derivative),
the higher the surface temperature. But the
differences in the temperature structure have very small influence on the
emergent spectra. Therefore we conclude, that details of the vertical structure
have negligible influence on the emergent spectrum for the
optically thick models ($\Sigmas\ge 100\ \mbox{g cm}^{-2}$).
It is well known that the model spectra of bursting NS close to the
Eddington limit
are well described by a diluted Planck spectrum with the color temperature
$\Tc = \fc \Teff$ with the hardness factor $\fc$
varying in the interval 1.6--1.9 and the dilution factor $D=\fc^{-4}$.
\citet{PSZ91} have derived an analytical formula for the
hardness factor, which successfully describes high
luminosity ($L \approx \Ledd$) burst spectra:
\be \label{eq:fc}
\fc = \left( 0.15 \ \ln C_1+0.59 \right)^{-4/5} C_1^{2/15}
\ell^{3/20},
\ee
where $C_1=(3+5X)/(1-\ell)$ and $\ell=L/\Ledd=\grad/\geff$.
Equation (\ref{eq:fc}) works well also for models with heavy elements.
The local spectra of the optically thick SL (with $L > 0.2 \Ledd$)
are very similar to the burst spectra with corresponding parameters (see
Fig. ~\ref{fig7}). The local SL are very close to the Eddington limit
\be \label{eq:grgeff}
\grad = \frac{q_0}{c}
\sigmae \approx g_0 - \frac{v_0^2}{\Rns} = \geff.
\ee
For example, the distributions of the ratios $\grad/g_0$ and
$\grad/\geff$ along the latitude for SL models with three different
luminosities are shown in Fig.~\ref{fig8}a. Corresponding hardness
factor distributions are shown in Fig.~\ref{fig8}b.
The comparison of the two local SL spectra
(close to equator and at higher latitude) with the diluted
Planck spectra and hardness factors given by equation (\ref{eq:fc})
are shown in Fig.~\ref{fig9}a.
Closer to the equator, effective gravity is low as centrifugal force
is large. The gas is levitating above the NS and $\ell$ is close to
unity. The energy dissipation and the effective temperature are low.
Thus, $\fc$ is large and the spectrum is close to the diluted Planck.
At higher latitude, the layer is decelerated, while the energy dissipation and $\Teff$
grow. However, the effective gravity grows faster reducing $\ell$ and the color correction
$\fc$. The spectrum shows deviations from the diluted Planck spectrum at low energies.
At high energies, the Wien part of both spectra can
well be described by the diluted Planck.
\subsection{Integral spectra}
Now we can compute the integral
total model spectrum of the SL, which is seen by a distant observer
accounting for the relativistic effects such as gravitational
redshift, light bending, relativistic Doppler effect and aberration.
We take into account only half of the SL
because another half is hidden by the accretion disc and
divide the SL surface on 10 latitude
rings and on 100 angles in azimuth.
In a spherical coordinate system, where the accretion disc coincides with the
$\theta = 0\degr$ plane, the spectrum of the SL is
\citep{PG03}
\be \label{eq:totsp}
F_{\rm E} = \frac{\Rns^2}{D^2} \int\limits_0^{\theta_{\rm SL}}
\int\limits_0^{2 \pi} \eta^3 \delta^3
I(E',\cos \alpha', \theta) \cos \alpha'
\cos \theta\ \d\theta\ \d\varphi.
\ee
Here the observed and the emitted photon energies are connected by the
relation $ E=E' \ \eta\ \delta$, where $\eta = \sqrt{1-\Rs/\Rns}$,
the Doppler factor $\delta = 1/\gamma (1-\beta \cos\xi)$,
$ \beta=\vphi(\theta)/c$ (here we neglected low latitudinal velocity),
the Lorentz factor $\gamma =1/\sqrt{1-\beta^2}$, and
$ \cos\xi = - \sin\alpha\ \sin i\ \sin\varphi/\sin\psi$.
The light bending is accounted for by the relation \citep{B02}
\be
\cos\alpha = \frac{\Rs}{\Rns} + \eta^2 \cos\psi,
\ee
where $\cos\psi = \cos i \sin\theta + \sin i \cos\theta \cos\varphi$,
and the relativistic aberration gives $ \cos\alpha' = \delta\cos\alpha$
\citep{PG03}.
Here $i$ is the inclination angle of the NS polar
axis to the line of sight, $D$ is the distance to the observer, and
$\theta_{\rm SL}$ is the SL boundary. Only visible surface elements
with $\cos\alpha > 0$ give contribution to the total spectrum.
The emitted specific intensity $I(E',\cos\alpha', \theta)$ is taken
from the computed local SL flux assuming angular dependence for the
electron scattering atmosphere
\be \label{eq:ang}
I(E',\cos\alpha', \theta) =
\frac{q_{\rm E'}(\theta)}{\pi} (0.4215+0.86775 \cos\alpha').
\ee
This formula gives a good approximation to the specific intensity
of the emergent radiation (see Fig.~\ref{fig10}).
The total spectra of the SL model for two inclination
angles, $i=0{\degr}$ and $90{\degr}$, are shown in Fig.~\ref{fig9}b.
The spectra computed using the local
diluted Planck spectra are shown also for comparison.
The difference is very small in the high energy part ($E> 10$ keV) and more
significant at lower energies ($E < 7$ keV).
Dependence of spectral shape on the inclination angle is not significant (see
also Fig.~\ref{fig11}a). Differences between the SL spectra, which are seen at
different inclinations are comparable to the differences due to change in
the SL luminosities (see Fig.~\ref{fig11}b). It is interesting, that the
total spectra can also be well described by the diluted Planck spectrum.
The color temperature depends slightly on the assumed turbulence parameter $\alphab$.
Decreasing $\alphab$ by an order of magnitude increases $\Tc$ by 0.1 keV.
\section{Comparison with observations}
\label{sec:obs}
In LMXRBs a weakly magnetized NS is surrounded by the accretion disc which
transforms to the boundary/SL close to the NS surface.
At present, about 100 LMXRBs are known.
They can be divided into two different classes. The Z-sources
are very luminous ($L \sim 0.1 - 1 \Ledd$) and have relatively soft,
two-component spectra. Both components are close to the black body with color
temperatures of about 1 keV and 2--2.5 keV.
The atoll sources are less luminous ($L \sim 0.01 - 0.05 \Ledd$) and are
observed in two states, the high/soft and the low/hard.
In the soft state, the radiation spectra are similar to those of the Z-sources,
while in the hard state they are close to the spectra of the Galactic
black hole sources in the hard states \citep[e.g. Cyg X-1, see e.g.][]{P98,B00}.
These hard spectra are well
described by unsaturated Comptonization of soft photons in the hot ($kT
\sim$ 30 -- 100 keV) optically thin ($\taue\sim $ 1) plasma.
The soft component can be associated with the radiation from the accretion
disc, while the hard one with the boundary/SL \citep{M84} or possibly with
a corona or hot optically thin inner accretion flow \citep[see discussion in][]{DG03}
in case of low-luminosity atoll sources. At high luminosities, the BL
is optically thick and its effective temperature is higher than that of the accretion disc,
because the BL is smaller than the accretion disc, while their luminosities are
comparable. The hard component is also more variable than the soft component
at the timescales from millisecond to 1000 seconds \citep*{M84,GRM03}.
The Fourier-frequency resolved spectroscopy confirms that a component variable
at high frequencies (and sometimes showing quasi-periodic oscillations, see
\citealt{vdK00}) has a blackbody-like
spectrum with the color temperature $\Tc= 2.4\pm 0.1$ keV \citep{GRM03,RG06}
which is very similar for the five investigated sources.
On the other hand, the variability of the soft component is very similar to the variability of the
soft component of black hole sources in their soft states, which is associated with the accretion disc.
Based on these arguments, we associate the hard blackbody-like component with the BL
and compare our theoretical SL spectra with it.
Spectra computed for one SL model together with the observed
BL spectra obtained by the Fourier-frequency resolved spectroscopy
\citep{GRM03,RG06} are shown in Fig.~\ref{fig12}.
We see a very good agreement between theoretical spectra and the
spectrum of GX 340+0 at the normal brach (at high accretion rates).
The spectra of five Z- and atoll-sources (open circles) are
similar to our SL spectra at high energies, but have a soft excess.
This excess may be related to the emission of the classical BL,
the inner part of the accretion disc.
The observed spectral similarity gives us a confidence to try
to determine NS parameters from the observed spectra.
As we have shown above the spectrum of the SL can be
represented by a diluted blackbody. The effective temperature of radiation
is determined by the critical temperature from equation (\ref{eq:tc}),
where the left-hand side is multiplied by $\ell$, the ratio of the local flux to the
critical Eddington one (reduced due to the action of the centrifugal force).
The observed color temperature is $\Tc=\fc \sqrt{1-\Rs/\Rns}\ \Tcr$,
where corrections are made for spectral hardening and gravitational redshift.
For the known color correction and $\ell$, the NS radius as a function of
compactness $\Mns/\Rns$ can then be found from
\be \label{eq:eos}
\Rns= \frac{ \ell \fc^4 c^3} { 2 \sigmasb \Tc^4 \sigmae}
\frac{\Rs}{\Rns} \left( 1- \frac{\Rs}{\Rns} \right) ^{3/2} .
\ee
Assuming $\fc=1.6-1.8$ and $\ell=0.8$, \citet{RG06} obtained
constraints on the NS mass-radius relation (shown in Fig. \ref{fig13} by dotted curves).
The maximum NS radius is reached for $\Rs/\Rns=2/5$:
\be \label{eq:rmax}
\Rns_{\max}= \frac{24.6}{1+X} \frac{ \ell}{0.8} \left( \frac{ \fc}{1.7} \right)^4
\left( \frac{ \Tc}{2.4\ \mbox{keV}} \right)^{-4} \ \mbox{km} .
\ee
Here instead we calculate exactly a grid of the SL model spectra, where the main input
parameters are the NS mass $\Mns$ and radius $\Rns$, and
the SL luminosity. The NS mass is varied from 1 to 2 $\msun$
with a step of 0.2 $\msun$, and the NS radius is varied from 10 to 24 km with
a step 1 km. Only the models with $\Rns > 3 \Rs$ are considered.
We take $\alphab=10^{-3}$ and luminosity of $0.4 \Ledd$, and compute
spectra for four inclination angles 0, 30, 60 and 90 degrees and for three
chemical compositions: pure hydrogen ($X=1$), solar
abundance ($X=0.7$), and pure helium ($X=0, Y=1$).
The spectra are fitted by the black body and the corresponding
color temperatures are found.
Models with higher He abundance have a smaller hardness factor as can be
seen from equation (\ref{eq:fc}). However, the local effective temperature of the SL is
higher for larger He abundance (see eq. \ref{eq:tc} and Fig. \ref{fig2}c).
The higher $\Teff$ leads to a higher color temperature of the integral SL spectrum.
For example, at NS radius of 13 km and mass $1.4\msun$
pure hydrogen models give color temperature of about 2.5 keV, while
pure helium models produce harder spectra with $\Tc\approx 3$ keV.
Contours corresponding to the color temperature equal 2.3 (right), 2.4
(central) and 2.5 keV (left) are shown on the $\Mns-\Rns$
plane (Fig.~\ref{fig13}) together with the NS models for
various equations of state. These iso-temperature curves are
shown for the inclination angle $i=45{\degr}$.
Comparison of the observed spectra to the theoretical spectra
of the SL constrains the NS radius at $13.5 \pm 1.5$ km (for pure
hydrogen $X=1$ model), $14.8 \pm 1.5$ km (solar composition $X=0.7$)
and $19 \pm 1.5$ km (pure helium $X=0, Y=1$) assuming the NS mass of
1.4 solar mass. For pure hydrogen and solar abundance, the permitted
radii are consistent with the hard equation of state of the NS matter.
If the composition is solar, but the heavier elements are able to sink,
the emitted spectra would correspond to a pure hydrogen atmosphere
requiring thus smaller radii.
Increasing the inclination to $90{\degr}$ increases the deduced NS radii by about 10 per cent,
while assuming $i=0{\degr}$, gives a 15 per cent reduction on $R$.
The uncertainty in the luminosity increases the width of possible NS radii by about 50 per cent.
Another source of uncertainty comes from the turbulence parameter $\alphab$.
With $\alphab$ decreasing by an order of magnitude the spectrum hardens by 0.1 keV.
This results in about 15 per cent decrease of the NS radius that is required
to produce the observed spectra. Thus $\alphab\sim 10^{-5}$ is needed
to reconcile the derived NS radii with the soft equations of state (assuming solar
composition). Such a small $\alphab$ at the same time yields a very large column
density of the SL and a rather long life-time of the accreting gas in the layer
(of the order of 1 s, instead of 10 ms as in the model of IS99).
Finally, we would like to emphasize that our method of determination of the
NS radius from the SL spectrum is based on the
observed color temperature of radiation alone,
because the SL radiates locally at almost Eddington flux.
The color temperature can be related to the effective temperature which
is a function of the stellar compactness (and chemical composition) as given
by equation~(\ref{eq:tc}).
This method is identical to that used for the radius-expansion X-ray bursts which are
believed to reach Eddington luminosity \citep*[see e.g.][]{LvPT93}.
In contrast to the standard methods based on the modeling of the thermal emission
from the NS surface \citep[see for example][]{vpL87,T05},
there is no need to know precisely either the area
of the emitting region, or the distance to the source.
As the standard method gives the apparent stellar radius at infinity,
which is related to the NS parameters through
\be \label{eq:rmrinf}
R_{\infty} = \Rns \left( 1- \frac{\Rs}{\Rns} \right)^{-1/2} ,
\ee
the allowed band of $\Rns$ and $\Mns$ is nearly orthogonal to that obtained
from the color temperature and equation (\ref {eq:eos})
(see the almost vertical dashed curve
in Fig. \ref{fig13}). Thus for a NS, where both the thermal emission from the surface
(e.g. during the quiescence) and the BL emission (during the accretion phase)
are observed, it would be possible to determine $\Rns$ and $\Mns$ independently.
Interestingly our constraints on the NS radius are very similar
to those obtained by \citet{HR05} for the thermally emitting
quiescent NS X7 in the globular cluster 47 Tucanae $\Rns=14.5^{+1.8}_{-1.6}$ km.
They are also consistent with the lower limit $\Rns>14$ km obtained by \citet{T05}
for the isolated NS RX~J1856-3754.
\section{Conclusions}
We have derived the one-dimensional equations describing the SL model on a spherical
NS surface from the usual hydrodynamic equations. The obtained equations are
similar to those in IS99, except for the energy conservation law where we neglected
the surface density of the gravitational potential energy which is of the second order in $H/R$.
This difference, however, leads only to small quantative changes.
We have also implemented a pseudo-Newtonian potential
to account for the main general relativity corrections and considered
various chemical compositions of the accreting matter.
We have studied the vertical (radial) structure of the SL with different
assumptions about the vertical distributions of the radiation flux and azimuthal velocity.
The temperature structure and the emergent radiation spectra of the SL
are computed accounting for the effect of Compton scattering. We showed that
the local (at a given latitude) emergent spectra depend very little on details
of the SL vertical structure in optically thick cases with
$\Sigmas\gtrsim 100\ \mbox{g\ cm}^{-2}$ ($L \gtrsim 0.1 \Ledd$).
These spectra can be described by the diluted Planck spectrum and are similar
to the spectra of X-ray bursts with the same effective temperature and the
effective surface gravity.
The integral SL spectra were computed accounting for relativistic effects such as
the gravitational redshift and light bending, the relativistic Doppler
effect and aberration. These spectra slightly
depend on the inclination angle to the line of sight and on the SL luminosity.
The local effective temperature increases with latitude, while the hardness factor
$\fc$ decreases. This leads to only slight variation of the color temperature
on latitude. As a result, the integral spectra can also be well
described by a single-temperature diluted Planck spectrum.
We compared our theoretical integral SL spectra
with the observed spectra of the LMXRBs BLs.
The observed color temperature of 2.4 $\pm$ 0.1 keV \citep{GRM03,RG06}
can be reproduced for hard equations of state of NS material.
Our model constrains radii of NSs in LMXRBs to 13--16 km for a
1.4 solar mass star. Soft equations of state (smaller NS radii)
can be reconciled with the observed spectra only for very low viscosity
$\alphab\sim10^{-5}$.
Calculation of $\alphab$ from the first principles is a challenging
problem that deserves further attention.
\section*{Acknowledgments}
This work was supported by
the Academy of Finland grants 107943 and 102181,
the Jenny and Antti Wihuri Foundation,
RFBR grant 05-02-17744, and the
Russian President program for support of the leading
science school (grant Nsh - 784.2006.2).
We are grateful to M. Revnivtsev for providing us with the
spectral data, and to D. G. Yakovlev and P. Haensel for
the theoretical mass-radius relations for neutron and strange stars.
We thank the referee for useful comments.
\label{lastpage}
|
Title:
Spitzer Reveals Hidden Quasar Nuclei in Some Powerful FR II Radio Galaxies |
Abstract: We present a Spitzer mid-infrared survey of 42 Fanaroff-Riley class II radio
galaxies and quasars from the 3CRR catalog at redshift z<1. All of the quasars
and 45+/-12% of the narrow-line radio galaxies have a mid-IR luminosity of
nuLnu(15 micron) > 8E43 erg/s, indicating strong thermal emission from hot dust
in the active galactic nucleus. Our results demonstrate the power of Spitzer to
unveil dust-obscured quasars. The ratio of mid-IR luminous narrow-line radio
galaxies to quasars indicates a mean dust covering fraction of 0.56+/-0.15,
assuming relatively isotropic emission. We analyze Spitzer spectra of the 14
mid-IR luminous narrow-line radio galaxies thought to host hidden quasar
nuclei. Dust temperatures of 210-660 K are estimated from single-temperature
blackbody fits to the low and high-frequency ends of the mid-IR bump. Most of
the mid-IR luminous radio galaxies have a 9.7 micron silicate absorption trough
with optical depth <0.2, attributed to dust in a molecular torus. Forbidden
emission lines from high-ionization oxygen, neon, and sulfur indicate a source
of far-UV photons in the hidden nucleus. However, we find that the other
55+/-13% of narrow-line FR II radio galaxies are weak at 15 micron, contrary to
single-population unification schemes. Most of these galaxies are also weak at
30 micron. Mid-IR weak radio galaxies may constitute a separate population of
nonthermal, jet-dominated sources with low accretion power
| https://export.arxiv.org/pdf/astro-ph/0601485 |
\title{Spitzer Reveals Hidden Quasar Nuclei in Some Powerful FR II Radio Galaxies}
\author{Patrick Ogle}
\affil{Spitzer Science Center, California Institute of Technology,
Mail Code 220-6, Pasadena, CA 91125}
\email{ogle@ipac.caltech.edu}
\author{David Whysong\altaffilmark{1} \& Robert Antonucci}
\affil{Physics Dept., University of California, Santa Barbara, CA 93106}
\altaffiltext{1}{now at NRAO, Array Operations Center, P. O. Box O, 1003 Lopezville Rd., Socorro, NM 87801-0387}
\shorttitle{Hidden Quasar Nuclei}
\shortauthors{Ogle et al.}
\keywords{galaxies: active, galaxies: quasars, galaxies: jets, infrared: galaxies}
\section{Unification of Quasars and Radio Galaxies}
The nature of the energy source in active galactic nuclei (AGNs) is a
fundamental problem. The basic model attributes the large luminosity of these
systems to gravitational energy release in an accretion disk around a
supermassive black hole. A jet may be driven by magnetic fields threading
the disk \citep{bp82}. The black hole spin energy may also be tapped and
converted into electromagnetic Poynting flux and particles in a relativistic
jet \citep{bz77,pc90,m99,d04}.
Extragalactic radio sources are categorized by their morphology as either of two
types \citep{fr74}. Fanaroff-Riley (FR) type I sources are edge-darkened, while FR IIs
are edge-brightened. The different morphology of FR Is indicates that they are not
related to FR IIs by orientation. FR Is also have lower radio luminosities than FR IIs
for a given host galaxy luminosity \citep{ol94,b96} and most have low-ionization nuclear
emission region (LINER) spectra \citep{hl79}. However, not all FR Is can be characterized by low
accretion power \citep{wa04,cr04}. The present paper focuses on FR IIs, which contain powerful
jets with bright terminal hot spots and lobes. Furthermore, we count broad-line radio galaxies
(BLRGs) as low-luminosity quasars.
Quasars and narrow-line radio galaxies (NLRGs) may be unified by orientation-dependent obscuration.
Radio galaxies are thought to host quasar nuclei that are obscured by circumnuclear dusty tori
aligned with the radio jets \citep{ant84} . Unification of radio galaxies and quasars can therefore
explain the lack of quasars viewed at large angles to the radio axis \citep{b89}. The percentage
of high-redshift radio galaxies (60\% of the 3CRR FR II sample at $z>0.5$) would then indicate a torus
covering fraction of $\sim 0.6$.
However, there appears to be a discrepancy between the redshift distributions of quasars and
radio galaxies at $z<0.5$, with a factor of $\sim 4$ more narrow-line radio galaxies
than quasars \citep{s93}. Furthermore, the median projected linear size of
these 'excess' radio galaxies is smaller than expected for quasars seen
in the sky plane \citep{s93,w05}. The unification hypothesis may be
modified to include a second population of lower luminosity, low-excitation
FR II radio galaxies \citep{wj97,grw04}. Alternatively, it has been argued that the
torus covering fraction may increase with decreasing radio luminosity \citep{l91}.
The unification hypothesis has been qualitatively confirmed by
spectropolarimetry of radio galaxies, many of which have been shown
to have highly polarized broad emission lines and blue continuum, scattered from material
which has a direct view of the active galactic nucleus \citep{cdb97, cot99}. Of particular note
are the original discovery of highly polarized broad H$\alpha$ from the hidden quasar nucleus in
3C 234 \citep{ant84}, and the discovery of highly polarized broad H$\alpha$ in the
spectrum of the powerful radio galaxy Cygnus A \citep{ocm97}. However, this method of detecting
hidden quasars relies on an appropriately placed scattering region to view the otherwise
hidden nucleus. Such a region is not guaranteed to exist for all radio
galaxies, and thus spectropolarimetry can easily yield false negatives.
Polarimetry is also ineffective at determining the luminosity of the hidden
nucleus, since the scattering efficiency is usually unknown.
Another way to search for hidden quasar nuclei is to observe radio
galaxies in the mid-IR. If the unification hypothesis is correct, the dusty torus should serve
as a crude calorimeter of the central engine \citep{mhm01,sfk04,hmb04,wa04}.
Optical, UV, and X-ray photons from the quasar nucleus are absorbed by dust in the
torus and the energy is re-emitted in the thermal infrared. This explains why blue, UV color-selected
quasars emit 10-50\% of their luminosity in the IR \citep{spn89,hmc00}. There appears
to be no connection between the bulk of this IR emission and nonthermal radio emission,
except in core-dominated radio sources such as blazars. Observations of matched 3CR quasars and
radio galaxies by ISO indicate similar IR luminosities, consistent with the unification
picture \citep{mhm01,hmb04}. However, differences in 24 $\mu$m/ 70 $\mu$m color
may indicate that mid-IR emission from the torus is anisotropic by a factor of $\le 3$
\citep{srh05}.
We present {\it Spitzer} observations of a sample of 42 FR II radio galaxies and quasars selected
from the 3CRR survey. The goals are to search for mid-IR emission from hidden quasar nuclei and test
the ubiquity of the unification hypothesis. The {\it Spitzer} Infrared Spectrograph (IRS) combines
the advantages of unprecedented sensitivity from 5-36.5 $\mu$m to measure the mid-IR continuum and
spectral resolution to measure high ionization emission lines powered by hidden AGNs.
In the current paper, we present evidence for hidden quasar nuclei based on mid-IR photometry extracted from
the IRS spectra. We examine in detail the spectra of the subset of 14 mid-IR luminous radio galaxies which
appear to contain hidden quasar nuclei. Spectra of the quasars and mid-IR weak radio
galaxies and a statistical study of the complete sample will be presented in separate papers.
\section{Sample}
We begin by selecting a well-defined, radio flux-limited and redshift-limited sample of 55
radio galaxies and quasars from the 3CRR catalog \citep{lrl83}. We include all 3CRR sources with
FR II radio morphology, a flux of $S_{178}>16.4$ Jy\footnote{The radio flux limit is 15 Jy using
\cite{lrl83} flux values and 16.4 Jy on the standard \cite{b77} scale.} at 178 MHz, and a redshift of
$z<1$. The original 3CRR catalog has a flux limit of 10 Jy at 178 MHz, is restricted to northern
declinations ($\delta >10 \arcdeg$), and has galactic latitude $|b|>10\arcdeg$. It is the canonical
low-frequency selected catalog of bright radio sources, has optical identifications and redshifts
for all entries, and has been extensively observed in most wavebands.
We select only sources with FR II radio morphology. We verify or update the FR classification of
all sources by inspection of the latest published radio maps. Compact, steep-spectrum sources
(CSSs: 3C 48, 138, 147, 286, and 309.1) with radio major axis $D<10$ kpc \citep{ffp85} are excluded
from the sample because they may constitute a class of young or frustrated radio sources. Here and
throughout this paper, we assume a cosmology with $H_0=70$ km s$^{-1}$ Mpc$^{-1}$,
$\Omega_\mathrm{m}=0.3$, and $\Omega_\Lambda=0.7$. Size and morphology indicate that CSSs are not
related to FR IIs by orientation.
It is essential for our unification studies that we select a sample based on isotropic
radio lobe flux, and {\it not} on optical or IR properties, so that it is unbiased by
orientation-dependent selection effects. In particular, our sample includes no blazars. No sources
make the flux limit only because of beamed emission from the core of the radio jet. Our sample
includes quasars as well as radio galaxies, and we use the quasar subsample as a control. We aim to
determine whether and which narrow-line FR II radio galaxies have mid-IR power comparable to
quasars or broad-line radio galaxies of similar radio lobe flux and redshift.
The 42/55 sources in our sample which we have observed with {\it Spitzer}, or which have {\it Spitzer} data in the
public archive are listed in Tables 1 and 2. The 25 {\it mid-IR luminous} sources with
$\nu L_\nu(15\mu\mathrm{m}) >8\times 10^{43}$ erg s$^{-1}$ (14 NLRGs and 11 quasars or BLRGs) are listed
in Table 1, and the 17 {\it mid-IR weak} galaxies with $\nu L_\nu(15\mu\mathrm{m}) <8\times 10^{43}$ erg s$^{-1}$
are listed in Table 2. The reason for this particular division is explained below.
Optical source classifications are based on emission line properties.
Type 1 sources have directly visible broad emission lines (quasars and BLRGs), and type 2 sources
(NLRGs) do not. The NLRGs are further classified using their forbidden emission lines
\citep{jr97,wrb99}\footnote{Updated optical classifications are available at
http://www-astro.physics.ox.ac.uk/$\sim$cjw/3crr/3crr.html.}.
High-excitation galaxies (HEGs) are defined to have [O {\sc iii}] $\lambda$5007
equivalent widths of $>10$ \AA~ and [O {\sc iii}] $\lambda$5007/[O {\sc ii}] $\lambda$3727 $>1$. The
sources which do not meet these criteria are classified as low-excitation galaxies (LEGs).
The equivalent width criterion ensures that [O {\sc iii}] is measurable in moderate S/N spectra. However,
it remains to be seen whether some sources with low [O {\sc iii}] equivalent width might have
[O {\sc iii}] $\lambda$5007/[O {\sc ii}] $\lambda$3727 $>1$. In addition, we caution that [O {\sc ii}]
and [O {\sc iii}] may be subject to differing amounts of extinction.
\section{Observations}
We observed the sources in our sample with the Infrared Spectrograph (IRS) on the
{\it Spitzer} Space Telescope \citep{h04,w04}. We used the low-resolution ($R\sim 64-128$) modules
Short-Low (SL) and Long-Low (LL) for accurate spectrophotometry over the wavelength
range 5-36.5 $\mu$m. Wavelengths 36.5-40 $\mu$m are unusable because of
low-S/N and 2nd-order bleed-through caused by filter delamination in LL 1st order (LL1). The
absolute and relative flux accuracies of IRS are generally better than 10\% and 4\%, respectively, as
judged from observations of bright standard stars. However, additional low-level
instrumental artifacts may become important for faint sources.
We used IRS in standard flux-staring mode, for 2 cycles at 2 nod positions in each
of the modules SL1, SL2 (SL 1st and 2nd order), and LL2 (LL 2nd order). We executed
1 cycle at 2 nod positions for LL1, which covers the 20-36.5 $\mu$m range. A typical
observation includes 240 s of on-source exposure time in each of SL1, SL2, and LL1, and
480 s in LL2, for a total of 2000 s per target (including overhead).
Archived IRS data are used for 14 sources which were observed partly or in full by other
investigators (with similar or longer exposure times).
Nod or off-slit observations were subtracted to remove foreground emission from the
telescope, zodiacal light, and interstellar medium. Spectra were then extracted from the
Basic-Calibrated Datasets (BCDs), using the {\it Spitzer} IRS Custom Extraction
(SPICE\footnote{ http://ssc.spitzer.caltech.edu/postbcd/spice.html} version 1.1) software
and standard tapered extraction windows. The extraction window full-widths are proportional to
wavelength in each order to match the diffraction-limited telescope point-spread function
(SL2: $7\farcs2$ at 6 $\mu$m, SL1: $14\farcs 4$ at 12 $\mu$m, LL2: $21\farcs7$ at 16 $\mu$m,
LL1: $36\farcs6$ at 27 $\mu$m). We rebinned portions of the spectra of 3C 55, 172, 220.1, 244.1, 263.1,
280, and 330 by factors of 4-8 in order to improve the S/N at short wavelengths. Spectral
orders were trimmed at the edges and merged to produce final spectra.
The SL and LL slits have widths of $3\farcs7$ and $10\farcs6$, respectively. Standard point-source flux
calibrations (version 12.0) were applied to correct for slit and aperture losses and convert the spectra from
electron s$^{-1}$ to Jy. In most cases, fluxes match to $<15\%$ across order boundaries, consistent with a
point source that is well-centered in all of the slits. However, in 5 cases (3C 192, 216, 220.1, 380, and
381) SL2 fluxes are larger by 17-35\% relative to the other orders. Assuming that these mismatches owe
to variable slit-loss caused by pointing errors, the orders with low flux are adjusted upward to match
the orders with high flux. Order mismatches may alternatively be an indication of extended mid-IR
emission.
The results for a few sources with nearby neighbors in the slit should be viewed with caution.
In the case of 3C 310, a nearby companion galaxy (to the east) may contribute a significant fraction of the
flux ($<50\%$) in the LL1 slit. Similarly, a nearby source may potentially contribute to the LL spectra
of 3C 438 (which is, however undetected at 15 $\mu$m). The SL2 spectrum of 3C 388 may be weakly affected by flux
from a nearby star on the slit ($<20\%$). The northern component of the double nucleus in 3C 401 falls
outside of the SL slits, but falls inside the LL slit used to measure the 15 $\mu$m flux.
\subsection{Mid-Infrared and Radio Luminosities}
We measure the mean $6.5-7.5$ $\mu$m and $13.0-17.0$ $\mu$m flux densities $F_\nu(7$ and 15 $\mu$m,
rest) of each target (Tables 1 \& 2). All {\it Spitzer} flux densities in this paper are in observed units at
a constant rest-frame wavelength defined by $\lambda_\mathrm{rest}=\lambda_\mathrm{obs}/(1+z)$, where
$z$ is the redshift measured from optical emission lines and cataloged in the NASA Extragalactic
Database (NED\footnote{ http://nedwww.ipac.caltech.edu}). This avoids any complication from potentially
large cosmological K-corrections that could otherwise be introduced by a steep IR continuum slope or
redshifted silicate absorption features.
We choose to measure the mid-IR flux at 7 and 15 $\mu$m to avoid the 9.7 $\mu$m trough and the deepest part of the
18 $\mu$m silicate absorption trough. We exclude the 14.0-14.5 $\mu$m and 15.3-15.8 $\mu$m wavelength regions from
our photometry, to avoid emission from Ne {\sc v} and Ne {\sc iii}. The 7 and 15 $\mu$m bands are within the
Spitzer IRS bandpass for redshifts $z<1.28$. However, the archival LL data for two quasars (3C 254 and 275.1) are
not yet public. We extrapolate their SL spectra to obtain $F_\nu($15 $\mu$m, rest) using $F_\nu($7 $\mu$m, rest) and
the observed (relatively line-free) 5-7 $\mu$m spectral index.
Radio luminosities $\nu L_\nu($178 MHz, rest) are estimated from the observed 178 MHz fluxes
and K-corrected using the 178-750 MHz radio spectral index \citep{lrl83}. The sources in our sample
display a large range of nearly 3 orders of magnitude in mid-IR to radio luminosity:
$\nu L_\nu(15$ $\mu\mathrm{m})/\nu L_\nu(178$ MHz$)=0.8-680$ (Fig.1). This quantity is thought to reflect the
relative importance of accretion luminosity and jet kinetic power dissipation. However, different
size and time scales are probed by the radio (10 kpc-1 Mpc) and mid-IR (0.1-100 pc), and the radio
power may be sensitive to differences in environmental conditions.
For the purpose of studying quasar and radio galaxy unification, it is natural to divide the sample into
the {\it mid-IR luminous} NLRGs which emit as powerfully as quasars or BLRGs, and the {\it mid-IR weak}
NLRGs that do not. We adopt an empirical dividing line of $\nu L_\nu(15$ $\mu\mathrm{m})> 8 \times 10^{43}$
erg s$^{-1}$ to separate hidden quasars from mid-IR weak radio galaxies. The cutoff is set at 1/2 the
luminosity of the mid-IR weakest BLRG (3C 219) to allow for some degree of anisotropy at 15 $\mu$m.
Fourteen NLRGs satisfy our criterion and are thus likely to contain hidden quasar or BLRG
nuclei (Table 1). Notably, all of these NLRGs are optically classified as HEGs.
The 17 mid-IR weak NLRGs with $\nu L_\nu(15$ $\mu\mathrm{m})<8 \times 10^{43}$ erg s$^{-1}$ (Table 2)
have mixed optical classifications, including both HEGs and LEGs. These sources have lower S/N mid-IR
spectra, which will be considered in detail in a later paper. Six mid-IR weak NLRGs (including 2 HEGs
and 4 LEGs) are undetected by {\it Spitzer} at 15 $\mu$m, and one is also undetected at 7 $\mu$m.
\subsection{Hidden Quasar Spectra}
\subsubsection{Continuum Emission}
We now present {\it Spitzer} spectra of the 14 mid-IR luminous NLRGs that ostensibly contain
hidden quasar nuclei (Figs. 2-4). We also plot the spectral energy distributions (SEDs) of the sources
with published near-IR photometry (Fig. 5). The collected photometric data were measured in the J, H,
K, L$^\prime$, and M wavelength bands from the ground \citep{ll84,llm85,srl99,sww00}.
The photometric apertures range in size from $3-11\arcsec$, with preference given to the apertures
that most closely match the {\it Spitzer} SL slit width. Where available, the ground-based L$^\prime$
and M-band photometry agrees with {\it Spitzer} spectrophotometry remarkably well. There is no
indication of variability over the time span of 20 yr.
Four of the low-redshift NLRGs (3C 33, 234, 381, and 452) have broad peaks in their $\nu L_\nu$ spectra (and
SEDs) at $1.5-2.5\times 10^{13}$ Hz (12-20 $\mu$m). A maximum and spectral curvature near 20 $\mu$m are
also suggestive of broad peaks in the {\it Spitzer} spectra of 3C 55, 244.1, 265, and 330.
The large amplitude ($\sim 0.5-1.0$ dex) of the mid-IR bump (Fig. 5) excludes a large contribution of
synchrotron emission to the mid-IR continuum of most sources. This is not surprising if the equatorial
plane of the dusty torus is roughly perpendicular to the radio jet, such that jet emission is beamed
away. The high redshifts of the NLRGs 3C 172, 220.1, 263.1, 268.1, and 280 preclude
the identification of a mid-IR bump in the SEDs of these sources. The unusually flat, blue SED of 3C 433 may
indicate a quasar viewed at {\it low} inclination (Section 3.2.2).
We attribute the mid-IR continuum bump visible in most sources to thermal emission from warm or hot
dust. Fitting the mid-IR peak with a single-temperature blackbody model indicates
dust with a temperature of $210-225\pm 0.5$ K (Fig. 5). While this temperature characterizes the peak
of the mid-IR SED, hotter dust must also be present. At frequencies greater than the peak of the SED
($2.0-7.5\times 10^{13}$ Hz), the continuum emission of most sources can be characterized using a power law
with spectral index $\alpha = 1.1-2.1$ (Table 1 \& Fig. 6). This emission likely comes from a continuous
distribution of dust temperature. We measure the spectral index between 7 and 15
$\mu$m, avoiding the 9.7 $\mu$m and 18 $\mu$m silicate absorption troughs. The most blue
and apparently hottest mid-IR luminous NLRG is 3C 265, while the most red and coolest are 3C 55 and 3C 268.1
(Fig. 6). In comparison, some mid-IR weak sources such as 3C 310 and 3C 388 are quite blue
($\alpha \sim -0.1- +0.7$), indicating a large contribution of starlight from the host galaxy to the
7 $\mu$m continuum.
The near-IR continuum shifts into the {\it Spitzer} IRS passband for the highest redshift ($z>0.7$) sources.
The spectra of the NLRGs 3C 55 and 3C 265 steepen above $7.5\times10^{13}$ Hz (below 4 $\mu$m). Fitting these
spectral turnovers with single-temperature blackbodies, we find emission from hot dust with temperatures of
$520\pm 10$ K and $660\pm 10$ K, respectively. Altogether, the mid-IR luminous radio galaxies in our sample show
emission from dust with temperatures distributed in the range 210-660 K. Hotter temperature dust (up to the
sublimation temperature) may be present but not visible for radio galaxy tori viewed at high inclination \citep{pk92}.
Extinction by cold foreground dust in the host galaxy may also affect the spectral index.
For Galactic-type dust, $A(7,15,35~\mu\mathrm{m})/A(\mathrm{V})=(0.020,0.015,0.004)$ \citep{m00}. An
extinction of $A(\mathrm{V})=100$ would steepen the 7-15 $\mu$m spectral index by $\delta \alpha= 0.6$
(Fig. 6a). The observed range in spectral index for the mid-IR luminous NLRGs is $\delta \alpha=1.0$,
corresponding to $A(\mathrm{V}, 7, 15, 35~\mu\mathrm{m})=(167, 3.3, 2.5, 0.6)$ mag. Thus if reddening
by a cold foreground dust screen accounted entirely for the range in mid-IR slope, the mid-IR emission
could be anisotropic by factors of $f_\mathrm{A}(7,15,35~\mu\mathrm{m})\sim (22,10,1.3)$. However,
these are upper limits since variations in the spectral index are also controlled by the physical
temperature distribution of the visible dust.
The SEDs of several sources (3C 33, 55, 172, 265, and 452) have upturns at short wavelengths, which we
attribute to stellar emission from the host galaxy (Fig. 5). The wavelength of the upturn (1-5 $\mu$m)
is an indicator of the relative strength of the mid-IR bump seen from our direction vs. host galaxy
light, occurring at shorter wavelength for sources with a stronger mid-IR bump. This may have important
consequences for understanding the K-z Hubble diagram for 3C radio galaxies, for which it has been
argued that AGNs contribute a negligible fraction of the K-band flux \citep[e.g.,][]{sww00}. This may
be incorrect for a few of the most luminous mid-IR sources in our sample, including 3C 234 and 3C 280
where there appears to be much emission from hot dust in the K band.
Detailed spectral modeling, combined with radio orientation indicators, promises to further
characterize the temperature distribution, optical depth, and inclination of the dusty torus that is
thought to produce most of the mid-IR emission from hidden quasar nuclei. Such an analysis is, however,
outside the scope of the present paper.
\subsubsection{Silicate Absorption}
The silicate absorption trough at 9.7 $\mu$m is detected in 12/14 of the mid-IR luminous
NLRG spectra (Table 3 \& Figs. 2-4). The equivalent width EW$_{9.7}$ and apparent optical depth
$\tau_\mathrm{9.7}$ are measured relative to a local continuum fit to either side of the trough,
indicated in Figures 2-4. The optical depth is averaged over the trough bottom (rest 9.2-10.2 $\mu$m)
to improve the S/N. It should be kept in mind that the apparent $\tau_\mathrm{9.7}$ is just a convenient
parameterization of (and lower limit to) the total optical depth since there must also be broad-band
silicate absorption of the adjacent continuum.
The apparent silicate optical depths are small ($\tau_\mathrm{9.7}=0.02-0.2$), for all but 3C 55 and
3C 433. If attributed to foreground dust screens, this would indicate optical extinction of only
$A_\mathrm{V}=0.2-5.1$ mag (Fig. 6b), assuming a Galactic extinction law with
$A_\mathrm{V}/\tau_\mathrm{9.7}=12.3-25.6$ mag \citep{rl85,dl84}. The extinction values are clearly
underestimated since they imply that the hidden nuclei in 3C 234, 265, 381, and 452, which have
$\tau_\mathrm{9.7}\le 0.1$, should be reddened but directly visible at H$\alpha$. The same discrepancy
between $\tau_\mathrm{9.7}$ and estimates of extinction at shorter wavelengths is seen for the hidden
quasar nucleus in Cygnus A, and attributed to a radial gradient in torus dust temperature \citep{iu00}.
The observed range of $\tau_\mathrm{9.7}$ may correspond to a range of equatorial silicate dust column
densities in the torus, or alternatively a range of viewing angles. In this regard, more detailed
modeling of the torus, including its geometric and temperature structure is clearly called for.
Filling-in of the silicate troughs by silicate {\it emission} from the torus or
narrow-line region (NLR) may also reduce the apparent silicate optical depths in some sources.
This is predicted for an optically thick torus viewed at an intermediate or face-on inclination
\citep{pk92}. Recently, strong silicate emission features were detected by {\it Spitzer} in several
radio-loud (3C) and radio quiet (PG) quasars \citep{shk05,hss05}. The failure of previous attempts to
observe this feature inspired torus models with large dust grain size \citep{ld93} or a spatial
distribution of optically thick clumps \citep{nie02}. However, it appears that past non-detections owe
to inadequate wavelength coverage to determine the underlying continuum.
The NLRGs 3C 55 and 3C 433 have significantly deeper silicate troughs than other NLRGs, with
$\tau_\mathrm{9.7}=0.9$ and 0.7, respectively (Fig. 6b). We suggest that their active nuclei and tori are
absorbed by an additional (kpc-scale) cold dust screen in the host galaxy. As noted above, the NLRG 3C 433
is unusual in having a flat, blue continuum (similar to some of the quasars in our sample). A blue mid-IR
spectrum is not necessarily at odds with deep silicate absorption features. It can be understood if this
is a quasar viewed at low inclination to the jet and torus axes, but through an
($A_\mathrm{V}\sim 10$) cold dust screen. This amount of extinction would result in very little
reddening at 7-15 $\mu$m ($\delta \alpha=0.05-0.14$), but would be sufficient to create the deep 9.7 $\mu$m
trough (Fig. 6b) and would obscure any optical broad lines.
The NLRG 3C 433 is also unique in having the only unambiguously detected 18 $\mu$m silicate trough,
with equivalent width $EW_{18}=0.42 \pm 0.01$ $\mu$m and apparent optical depth
$\tau_\mathrm{18}=0.07 \pm 0.03$ (averaged over 17-19 $\mu$m). The ratio of $\tau_\mathrm{18}$ to
$\tau_\mathrm{9.7}$ apparent silicate trough depths is $0.10 \pm 0.04$, consistent with a $0.11$ ratio
for Galactic-type silicate dust \citep{dl84}. We do not see the full 18 $\mu$m silicate trough in the
spectrum of 3C 55 because of inadequate rest-wavelength coverage.
\subsubsection{Forbidden Emission Lines}
All of the mid-IR luminous NLRGs with high S/N spectra have forbidden emission lines from
highly ionized metals, including [O {\sc iv}] $\lambda 25.89$ $\mu$m, [Ne {\sc ii}] $\lambda 12.81$,
[Ne {\sc iii}] $\lambda 15.55$, [Ne {\sc v}] $\lambda 14.3$, [Ne {\sc v}] $\lambda 24.31$,
[Ne {\sc vi}] $\lambda 7.65$, [S {\sc iii}] $\lambda 18.71$, [S {\sc iii}] $\lambda 33.48$, and
[S {\sc iv}] $\lambda 10.51$ (Figs. 2-4). We measure the line flux and rest equivalent
width of each emission line relative to the local continuum level (Table 4). Formal
uncertainties are computed from the noise in the continuum to either side of the line. Upper
limits are estimated for undetected emission lines, assuming they are unresolved.
The large range of ionization states (especially high-ionization Ne {\sc v}, Ne {\sc vi}, and
S {\sc iv}) indicates photoionization by a hidden source of far-UV photons \citep{asl99,slv02,acs04},
e.g. a quasar nucleus. Low critical densities in the range $10^3-10^6$ cm$^{-3}$ \citep{asl99} indicate
that the forbidden lines arise in the NLR. There could plausibly be contributions
from starburst emission to the lower-ionization emission lines such as [Ne {\sc ii}].
For 3C 55 and 3C 433, the [S {\sc iv}] $\lambda 10.51$ line has a relatively large flux even
though it sits at the bottom of a deep silicate trough. This line must then arise from a region
not heavily obscured by dust, such as the extended NLR. The resolving power of IRS is
insufficient to measure the intrinsic emission line widths, which are therefore $<4700$ km s$^{-1}$.
In order to assess the ionization level of the emission line regions, we compute several emission
line ratios (Fig. 7). In particular, the [O {\sc iv}]/[Ne {\sc ii}] and [Ne {\sc v}]/[Ne {\sc ii}] ratios
can be used as diagnostics of the relative contributions of AGN and starburst emission to the mid-IR emission
line spectra of galaxies \citep{g98,slv02}. In the sources where O {\sc iv} , Ne {\sc v}, and Ne {\sc ii}
emission are all detected (3C 33, 234, 381, and 433), we find
[O {\sc iv}] $\lambda 25.89$ $\mu$m / [Ne {\sc ii}] $\lambda 12.81$ $\mu$m $>1.0$ (Fig. 7a) and
[Ne {\sc v}] $\lambda 14.3$/ [Ne {\sc ii}] $\lambda 12.81$ $\mu$m $> 0.5$ (Fig. 7c), indicating a $>50\%$ AGN
contribution to the emission line spectrum. The [Ne {\sc ii}] $\lambda 12.81$ $\mu$m line is relatively weak or
undetected in the $z>0.2$ sources, making it difficult to apply these diagnostics. However, the large EWs of the
[Ne {\sc vi}] or [S {\sc iv}] lines in 3C 55, 265, and 330 indicate that the emission line spectra of these sources
are also AGN-dominated.
\subsubsection{Molecular Emission}
Polycyclic aromatic hydrocarbon (PAH) emission features are commonly seen in star-forming regions,
starbursts, and starburst-dominated ULIRGs \citep[e.g.,][]{g98,acs04}. The only PAH feature we detect
is the weak 11.3 $\mu$m line in the spectrum of 3C 33, with a flux of $2.3\pm 0.3 \times 10^{-14}$ erg s$^{-1}$
cm$^{-2}$ and equivalent width of $0.009\pm 0.003$ $\mu$m (Fig. 2). We do not detect a 6.2 $\mu$m PAH feature in
3C 33 (EW$<0.8$ $\mu$m) or any of the other mid-IR luminous NLRGs, though this spectral region is generally
noisier. If present, we could not cleanly resolve the 7.7 $\mu$m PAH feature from the adjacent [Ne {\sc vi}]
line, nor the 12.7 $\mu$m PAH feature from the adjacent [Ne {\sc ii}] line. Regardless, we do not see any hint
of these PAH features in any source, suggesting EW$<<0.1$ $\mu$m. Therefore in most cases, neither the 7.7 or 11.3
$\mu$m PAH features can have a significant impact on the measurement of the silicate trough.
The general lack of PAH features is a strong indication that the primary power source in mid-IR luminous
radio galaxies is accretion power, not hot stars. It is likely that PAHs are destroyed in the torus, which is
exposed to intense X-ray emission from the AGN \citep{v91}. The weak PAH emission that is present in 3C 33 may
arise in star-forming regions shielded from the AGN.
We detect the H$_2$ 0-0 S(3) 9.67 $\mu$m and H$_2$ 0-0 S(1) 17.03 $\mu$m pure rotational emission lines of
molecular hydrogen (at the $3\sigma$ level) only in the spectrum of the NLRG 3C 433 (Fig. 2). The line fluxes are
0.6 $\pm 0.2 \times 10^{-14}$ and 1.4 $\pm 0.4 \times 10^{-14}$ erg s$^{-1}$ cm$^{-2}$ respectively (EW $=$ 0.008
and 0.013 $\mu$m). The location of the 9.67 $\mu$m line at the bottom the deep 9.7 $\mu$m silicate trough may
indicate that the H$_2$ emission region is exterior to the obscuring dust screen. The H$_2$ emission lines can be
produced in warm molecular gas heated either by shocks or X-ray photons from the AGN \citep{rkl02}. The ratio of
S(3)/S(1) line fluxes is $0.5 \pm 0.2$, which indicates warm H$_2$ with an excitation temperature of $300\pm30$ K.
We estimate a warm H$_2$ mass of roughly $2\times 10^8 M_\odot$, assuming a Boltzmann distribution of rotational
level populations, an unresolved source, and negligible mid-IR extinction.
\section{Discussion}
\subsection{Hidden Quasar Nuclei}
The high mid-IR luminosities $\nu L_\nu(15$ $\mu\mathrm{m})= 10^{44}-10^{46}$ erg s$^{-1}$ of
$45 \pm 12\%$ (14/31) of the FR II NLRGs in our sample are consistent with hidden quasar or BLRG nuclei. In
fact, such copious hot dust emission directly requires a hidden source of quasar-like luminosity to power it.
Including the 11 quasars and BLRGs seen directly, we find that at least $60 \pm 12\%$ (25/42) of 3CRR FR II
sources at $z<1$ with $S_{178}>16.4$ Jy contain powerful AGNs.
The percentage of {\it mid-IR luminous} AGNs obscured by dust and therefore classified as NLRGs is
$56\pm 15\%$ (14/25), corresponding to a mean torus covering fraction of 0.56 and mean torus opening
half-angle of $55\pm 11 \arcdeg$. (If the 8 mid-IR weak HEGs are counted as highly obscured quasars,
then the mean torus covering fraction increases to $0.67 \pm 0.14$.) Both of these numbers are
consistent with the estimated 60\% mean torus covering fraction required to unify $z=0.5-1.0$ radio
galaxies and quasars \citep{b89}.
The receding torus model \citep{l91} predicts a larger torus covering fraction for low luminosity sources.
If so, we would expect a larger mean mid-IR/radio ratio and a smaller type 1 fraction for low-redshift than for
high-redshift sources. The fraction of type 1 mid-IR luminous sources is 3/9 ($0.3\pm 0.2$) at $z<0.5$ vs. 8/16
($0.5 \pm 0.2$) at $z>0.5$. Clearly, a larger sample is needed to put meaningful constraints on the variation
of torus covering fraction with redshift or luminosity.
For the galaxies that contain a powerful mid-IR source, we find that
there are other indications of a hidden AGN. The high-ionization, mid-IR
forbidden lines such as [Ne {\sc v}], even more-so than the strong optical [O {\sc iii}] lines,
are telltale signatures of a non-stellar source of FUV photons in the radio galaxies that show
them. Because of the lower extinction in the mid-IR relative to the optical, it is likely that we
can see these lines closer to the nucleus than optical narrow lines such as [O {\sc iii}] \citep{hss5}.
At least three of the mid-IR luminous NLRGs in our sample are known to have highly polarized
broad emission lines. The NLRG 3C 234 has quasar-like mid-IR luminosity and a highly polarized broad
H$\alpha$ line \citep{ant84}. The NLRG 3C 265 has quasar luminosity, highly polarized UV flux, and a
highly polarized broad Mg {\sc ii} line \citep{tco98}. The low-redshift NLRG 3C 33 has
mid-IR luminosity comparable to the BLRG 3C 219, and highly polarized broad H$\alpha$ and
H$\beta$ \citep{cot99}. We have an ongoing program to obtain optical spectropolarimetry
of the rest of the sources in our sample. While the radio galaxies with highly polarized broad lines
are known to contain hidden type-1 AGNs, there were previously no reliable measurements
of the hidden AGN luminosities. Our mid-IR flux measurements yield rough calorimetric estimates
of the hidden AGN luminosities, subject to uncertainties in SED, torus covering fraction, and
any mid-IR anisotropy.
\subsection{Mid-IR Weak Radio Galaxies}
The majority of radio galaxies in our sample (17/31 or $55\pm 13\%$) are relatively weak mid-IR
sources. It is possible that some of the AGNs are highly obscured even at 15 $\mu$m because they are
viewed through a very large dust column. A mid-IR source obscured by a nearly Compton-thick
($\tau_\mathrm{e}=0.7$) disk with Galactic dust/gas ratio would have an equatorial extinction of
$\sim 5$ mag (factor of 100) at 15 $\mu$m. Even in this case, any mid-IR emission above the disk might not
be obscured. For example, there may be a contribution from dust in the narrow-line region (NLR), above the
hole in the torus \citep[e.g. NGC 1068,][]{gpa05,bnm00}, tending to make the mid-IR emission more isotropic.
For galaxies with redshift $z \le 0.22$, {\it Spitzer} can measure the flux at $\lambda = 30$ $\mu$m (rest),
which should be more isotropic and less subject to extinction than the 15 $\mu$m flux.
Nevertheless, 9/11 mid-IR weak galaxies in this redshift range are also weak at 30 $\mu$m, with
$\nu L_\nu(30$ $\mu\mathrm{m}) <8 \times 10^{43}$ erg s$^{-1}$. The two exceptions are
3C 61.1 and 3C 123, with $\nu L_\nu(30$ $\mu\mathrm{m})=1.27 \pm 0.08$ and $1.4\pm 0.2\times 10^{44}$ erg s$^{-1}$,
respectively. In comparison 3C 452, the weakest mid-IR luminous NLRG in this redshift range, has
$\nu L_\nu(30$ $\mu\mathrm{m})=8.63 \pm 0.09\times 10^{43}$ erg s$^{-1}$. Therefore, reclassifying the NLRGs by
their luminosity at 30 $\mu$m would only gain us an additional 2 mid-IR luminous sources. This leads us to believe
that most of the mid-IR weak radio galaxies truly lack a powerful accretion disk. Relatively low accretion
power suggests, but does not prove, that some FR II jets may be driven by radiatively inefficient
accretion flows or black hole spin-energy \citep{bbr84, m99}.
As we mentioned above, roughly half (9/17) of the mid-IR weak NLRGs are are classified as LEGs with weak optical
[O {\sc iii}] emission. The [O {\sc iii}] emission in these sources may be weak because there is no
strong source of UV photons to power the NLR. Qualitatively similar conclusions have been drawn by other
investigators \citep[e.g.,][]{hl79, ccc00,grw04}. Alternatively, the NLR may be partly or completely obscured in
LEGs \citep{hss5}. It will be important to make a more quantitative assessment of the optical and IR emission line
strengths, to evaluate the extinction and determine what UV luminosity is necessary to power the NLR in mid-IR weak
NLRGs.
\subsection{Radio Properties and Unification}
One of the major motivations for the radio galaxy and quasar unification theory is the deficit of lobe-dominant
vs. core-dominant FR II quasars \citep{b89}. Relativistic beaming models predict that there should be relatively
more sources where the radio jet is beamed away and the high frequency radio core is weak. Core fluxes at 5 GHz
are tabulated for the 3CRR catalog by
\cite{lrl03}\footnote{The online 3CRR catalog is available at http://www.3crr.dyndns.org.}. We identify the
mid-IR luminous NLRGs in our sample with the missing lobe-dominant quasar population (Fig. 8). Their median core
to lobe ratio is $R_\mathrm{c}=\nu L_\nu($core, 5 GHz$)/ \nu L_\nu(178$ MHz$)=5$, while the
median $R_\mathrm{c}=180$ for the quasars. Conversely, all of the mid-IR luminous sources with
$R_\mathrm{c}>100$ are classified as quasars. We will perform a more detailed statistical analysis when the
Spitzer observations of our full sample are complete.
For most mid-IR weak NLRGs, we reject the possibility that the AGN and radio core are viewed in an 'off'
state while the radio lobes are still active. A 5 GHz core is detected in 13/17 of the mid-IR weak radio galaxies
(Fig. 8) and 13/14 of the mid-IR luminous galaxies. Furthermore, mid-IR weak and mid-IR luminous NLRGs have
comparable core to total luminosity ratios of $R_\mathrm{c} = 1-100$. Deeper 5 GHz observations of the 5
non-detected cores (in 3C 28, 153, 172, 315, and 319) will be necessary to determine whether or not they are in
an off state (e.g., $R_\mathrm{c}<0.1$).
We find that FR II radio morphology is not a reliable predictor of nuclear mid-IR luminosity for
radio galaxies. {\it Contrary to the simple unification paradigm, not all narrow-line
FR II galaxies host nuclei as powerful as quasars with matched radio lobe luminosity}.
Unification theories must be modified to account for an additional population of mid-IR weak radio galaxies.
Both intrinsic jet power and interaction with the interstellar and intergalactic medium are likely to
be important for determining radio morphology. The break luminosity between FR I and FR II radio
sources is found to increase with host galaxy optical luminosity \citep{ol94,b96}. The existence of
radio sources with hybrid FR I/II morphology also points to the importance of environmental effects in
determining radio morphology \citep{gkw00,gmk05}. Furthermore, most FR I radio jets are one-sided,
relativistic, and narrowly collimated on sub-parsec scales, just like FR IIs, and decollimate
only on kpc scales \citep[e.g. M87,][]{jbl99}.
Contrary to a common misconception, not all FR I sources have radiatively inefficient nuclei. For
example, Centaurus A is persuasively argued to have a powerful hidden AGN \citep{wa04}, and the BLRG
3C 120 is a well-known FR I source. Deep VLA observations of the optically luminous, 'radio-quiet'
quasar E1821+643 demonstrate that it has an FR I radio morphology \citep{br01}. These and other cases
\citep{ant01} demonstrate that many powerful AGNs are FR Is.
The observed variation in radio morphology and a wide range in AGN radio-loudness \citep{kss89} do not
necessarily require a weak coupling between jet power and accretion power, but may demonstrate that
multiple factors are at work. At least five parameters may be necessary to theoretically unify all AGN
types: black hole mass, black hole spin, accretion rate, radiative efficiency, and viewing angle. There
is still much work ahead before we completely understand how basic physical parameters regulate the
activity of supermassive black hole systems. Models that tie jet production to accretion onto a
spinning black hole are particularly promising \citep[e.g.,][]{m99,hk06}.
Much progress has been made in understanding the aspect-dependent
appearance of AGN disks and jets, as a consequence of relativistic beaming and obscuration \citep{up95}.
Our {\it Spitzer} observations confirm that many FR II radio galaxies would appear as powerful quasars if
viewed from an unobscured direction (e.g. along the radio axis). However, just as many FR II radio galaxies
would not. Our {\it Spitzer} observations also demonstrate that powerful radio jets may be produced even by
mid-IR weak AGN. A powerful, luminous accretion disk is not always necessary to produce a highly collimated,
relativistic jet.
\section{Conclusions}
(1.) We report on a large {\it Spitzer} spectroscopic survey of 3CRR FR II radio sources with $S_{178}>16.4$ Jy
at $z<1$. We find strong mid-IR emission from $45 \pm 12\%$ (14/31) of NLRGs, which have luminosities comparable
to matched BLRGs and quasars. Other indicators including high-ionization mid-IR lines and highly polarized broad
emission lines confirm that some of these sources contain hidden quasar or BLRG nuclei. This demonstrates the power
of {\it Spitzer} IRS for unveiling hidden quasars and estimating their luminosities.
(2.) We present {\it Spitzer} spectra of the 14 mid-IR luminous radio galaxies. In most cases, the mid-IR
continuum bump from 3-30 $\mu$m can be produced by a distribution of hot dust with temperatures in the range
210-$660$ K. These high temperatures are most likely maintained by hidden AGNs. The silicate
absorption trough at 9.7 $\mu$m has an apparent optical depth of $\tau=0-0.2$ in most cases,
consistent with dust temperatures decreasing outward from the center of a dusty torus. Two
sources, 3C 55 and 433, have deeper silicate troughs which may be produced by additional cool dust
in the host galaxy.
(3.) However, not all FR II radio galaxies emit strongly in the mid-IR. Contrary to single-population
unification schemes, the majority of narrow-line radio galaxies in our sample (17/31 or $55 \pm 13\%$)
have weak or undetected mid-IR emission compared to matched quasars and BLRGs, with
$\nu L_\nu(15$ $\mu\mathrm{m}) < 8 \times 10^{43}$ erg s$^{-1}$.
For a few sources, this may possibly be the result of anisotropic torus emission viewed
through a large column density of dust. However, it is likely that most of the weakest sources do not
contain a powerful accretion disk. These may be truly nonthermal, jet-dominated AGNs, where the jet is powered by
a radiatively inefficient accretion flow or black hole spin-energy rather than energy extracted from an
accretion disk.
\acknowledgements
This work is based on observations made with the {\it Spitzer} Space Telescope, which is operated
by the Jet Propulsion Laboratory, California Institute of Technology under NASA contract 1407.
We have also made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion
Laboratory, California Institute of Technology, under contract with NASA. Support for this research was
provided by NASA through an award issued by JPL/Caltech. We thank Dave Meier, Lee Armus, Bill Reach, and the
anonymous referee for their helpful input and comments on the manuscript.
\begin{deluxetable}{cccccccc}
\tablecaption{Mid-IR Luminous Sources}
\tablewidth{0pt}
\tablehead{
\colhead{3C} & \colhead{type\tablenotemark{a}}
& \colhead{z}
& \colhead{$F_7$\tablenotemark{b}}
& \colhead{$F_{15}$\tablenotemark{c}}
& \colhead{log $\nu L_{15}$ \tablenotemark{d }}
& \colhead{$\alpha$ \tablenotemark{e}}
&\colhead{S178\tablenotemark{f}}
}
\startdata
175 & QSR & 0.7700 & 6.96 $\pm$ 0.07 & 21.6 $\pm$ 0.7 & 45.83 &1.49 $\pm$ 0.04 & 19.2 \\
196 & QSR & 0.871 & 8.0 $\pm$ 0.1 & 22.9 $\pm$ 0.6 & 45.96 &1.38 $\pm$ 0.04 & 74.3 \\
216 & QSR & 0.6703 & 9.8 $\pm$ 0.1 & 28.7 $\pm$ 0.6 & 45.83 &1.41 $\pm$ 0.03 & 22.0 \\
219 & BLRG & 0.1744 & 3.6 $\pm$ 0.1 & 11.2 $\pm$ 0.4 & 44.21 &1.50 $\pm$ 0.06 & 44.9 \\
254 & QSR & 0.7361 & 6.02 $\pm$ 0.08 & \nodata & 45.56:&\nodata & 21.7 \\
263 & QSR & 0.646 & 13.6 $\pm$ 0.1 & 29.8 $\pm$ 0.8 & 45.81 &1.03 $\pm$ 0.04 & 16.6 \\
275.1 & QSR & 0.5551 & 3.04 $\pm$ 0.07 & \nodata & 44.76:&\nodata & 19.9 \\
325 & QSR & 0.8600 & 1.17 $\pm$ 0.07 & 4.1 $\pm$ 0.2 & 45.20 &1.6 $\pm$ 0.1 & 17.0 \\
380 & QSR & 0.6920 & 15.1 $\pm$ 0.1 & 40.4 $\pm$ 1.2 & 46.00 &1.29 $\pm$ 0.04 & 64.7 \\
382 & BLRG & 0.0579 & 86.1 $\pm$ 0.3 &114. $\pm$ 2. & 44.24 &0.37 $\pm$ 0.02 & 21.7 \\
390.3 & BLRG & 0.0561 & 56.8 $\pm$ 0.5 &164. $\pm$ 4. & 44.37 &1.39 $\pm$ 0.03 & 51.8 \\
\hline
33 & HEG & 0.0597 & 19.8 $\pm$ 0.3 & 75. $\pm$ 2. & 44.08 &1.75 $\pm$ 0.04 & 59.3 \\
55 & HEG & 0.7348 & 4.7 $\pm$ 0.1 & 23.3 $\pm$ 0.7 & 45.82 &2.11 $\pm$ 0.06 & 23.4 \\
172 & HEG & 0.5191 & 0.40 $\pm$ 0.04 & 1.5 $\pm$ 0.2 & 44.31 &1.7 $\pm$ 0.2 & 16.5 \\
220.1 & HEG & 0.610 & 0.77 $\pm$ 0.05 & 2.4 $\pm$ 0.1 & 44.67 &1.5 $\pm$ 0.1 & 17.2 \\
234 & HEG\tablenotemark{g} & 0.1848 & 86. $\pm$ 2. & 239. $\pm$ 3. & 45.59 &1.35 $\pm$ 0.03 & 34.2 \\
244.1 & HEG & 0.4280 & 3.50 $\pm$ 0.09 & 14.4 $\pm$ 0.3 & 45.13 &1.86 $\pm$ 0.04 & 22.1 \\
263.1 & HEG & 0.8240 & 0.63 $\pm$ 0.07 & 2.7 $\pm$ 0.1 & 44.98 &1.9 $\pm$ 0.2 & 19.8 \\
265 & HEG & 0.8110 & 9.0 $\pm$ 0.1 & 21.1 $\pm$ 0.5 & 45.86 &1.12 $\pm$ 0.04 & 21.2 \\
268.1 & HEG & 0.970 & $<0.5$ & 3.0 $\pm$ 0.2 & 45.17 &2.1 $\pm$ 0.2\tablenotemark{h} & 23.3 \\
280 & HEG & 0.996 & 4.58 $\pm$ 0.09 & 13.2 $\pm$ 0.4 & 45.83 &1.39 $\pm$ 0.05 & 25.8 \\
330 & HEG & 0.550 & 1.72 $\pm$ 0.06 & 6.4 $\pm$ 0.2 & 45.00 &1.72 $\pm$ 0.06 & 30.3 \\
381 & HEG & 0.1605 & 11.1 $\pm$ 0.2 & 36.4 $\pm$ 0.8 & 44.65 &1.56 $\pm$ 0.05 & 18.1 \\
433 & HEG & 0.1016 & 40.0 $\pm$ 0.4 & 98. $\pm$ 1. & 44.67 &1.18 $\pm$ 0.02 & 61.3 \\
452 & HEG & 0.0811 & 11.3 $\pm$ 0.2 & 45. $\pm$ 1. & 44.13 &1.80 $\pm$ 0.04 & 59.3 \\
\enddata
\tablenotetext{a}{Optical spectral type \citep{lrh99,jr97}.}
\tablenotetext{b,c}{Flux densities (mJy) and $3\sigma$ upper limits at 7 $\mu$m (rest) and
15 $\mu$m (rest) from {\it Spitzer} IRS.}
\tablenotetext{d}{~Logarithm of luminosity (erg s$^{-1}$) at 15 $\mu$m (rest). Values for 3C 254
and 3C 275.1 are extrapolated from 7 $\mu$m because the LL data are unavailable.}
\tablenotetext{e}{Spectral power law index for $F_\nu\sim \nu^{-\alpha}$ from
7-15 $\mu$m (rest).}
\tablenotetext{f}{Radio flux density (Jy) at 178 MHz (observed) \citep{lrl83}, multiplied by
a factor of 1.09 to convert to the \cite{b77} standard flux scale.}
\tablenotetext{g}{The broad H$\alpha$ line visible in total flux is entirely scattered \citep{ant84}.}
\tablenotetext{h}{Spectral power law index for 3C 268.1 measured from 8-15 $\mu$m (rest).}
\end{deluxetable}
\begin{deluxetable}{cccccccc}
\tablecaption{Mid-IR Weak Sources}
\tablewidth{0pt}
\tablehead{
\colhead{3C} & \colhead{type\tablenotemark{a}}
& \colhead{z}
& \colhead{$F_7$\tablenotemark{b}}
& \colhead{$F_{15}$\tablenotemark{c}}
& \colhead{log $\nu L_{15}$ \tablenotemark{d }}
& \colhead{$\alpha$ \tablenotemark{e}}
&\colhead{S178\tablenotemark{f}}
}
\startdata
61.1 & HEG & 0.1878 & 0.64 $\pm$ 0.06 & 3.0 $\pm$ 0.2 & 43.70 &2.0 $\pm$ 0.2 & 34.0 \\
192 & HEG & 0.0597 & 1.28 $\pm$ 0.07 & 3.2 $\pm$ 0.2 & 42.71 &1.2 $\pm$ 0.1 & 23.0 \\
274.1 & HEG & 0.4220 & 0.20 $\pm$ 0.05 & $<$0.9 &$<$43.91 &\nodata & 18.0 \\
300 & HEG & 0.270 & 0.26 $\pm$ 0.04 & 0.7 $\pm$ 0.2 & 43.40 &1.3 $\pm$ 0.4 & 19.5 \\
315 & HEG & 0.1083 & 0.97 $\pm$ 0.08 & 1.9 $\pm$ 0.2 & 43.01 &0.9 $\pm$ 0.2 & 19.4 \\
388 & HEG & 0.0917 & 0.98 $\pm$ 0.06 & 0.84 $\pm$ 0.09& 42.66 &0.4 $\pm$ 0.2 & 26.8 \\
436 & HEG & 0.2145 & 0.65 $\pm$ 0.05 & 1.5 $\pm$ 0.2 & 43.52 &1.1 $\pm$ 0.2 & 19.4 \\
438 & HEG & 0.290 & 0.16 $\pm$ 0.04 &$<$0.45 &$<$43.27 &\nodata & 48.7 \\
\hline
28 & LEG & 0.1953 & 0.45 $\pm$ 0.06 & $<$0.30 &$<$42.74 &\nodata & 17.8 \\
123 & LEG & 0.2177 & 1.07 $\pm$ 0.08 & 2.8 $\pm$ 0.4 & 43.81 &0.7 $\pm$ 0.2 &206.0 \\
153 & LEG & 0.2769 & 0.29 $\pm$ 0.05 & 1.0 $\pm$ 0.2 & 43.59 &1.7 $\pm$ 0.3 & 16.7 \\
173.1 & LEG & 0.2921 & 0.38 $\pm$ 0.04 & 0.6 $\pm$ 0.1 & 43.40 &0.6 $\pm$ 0.3 & 16.8 \\
288 & LEG & 0.2460 & 0.40 $\pm$ 0.05 &$<$0.60 &$<$43.25 &\nodata & 20.6 \\
310 & LEG & 0.0538 & 0.81 $\pm$ 0.07 & 0.73$\pm$ 0.1 & 41.98 &-0.1 $\pm$ 0.2 & 60.1 \\
319 & LEG & 0.1920 & $<$0.15 &$<$0.27 &$<$42.68 &\nodata & 16.7 \\
326 & LEG & 0.0895 & 0.67 $\pm$ 0.09 &$<$0.39 &$<$42.16 &\nodata & 22.2 \\
401 & LEG & 0.2011 & 0.25 $\pm$ 0.05 & 0.8$\pm$ 0.2 & 43.17 &1.5 $\pm$ 0.4 & 22.8 \\
\enddata
\tablenotetext{a-f}{See Table 1.}
\end{deluxetable}
\begin{deluxetable}{ccc}
\tablecaption{Silicate Trough}
\tablewidth{0pt}
\tablehead{
\colhead{3C} & \colhead{EW$_{9.7}$ \tablenotemark{a}} & \colhead{$\tau_{9.7}$ \tablenotemark{b}}
}
\startdata
33 & -0.222 $\pm$ 0.007 & 0.14 $\pm$ 0.02 \\
55 & -1.51 $\pm$ 0.03 & 0.9 $\pm$ 0.1 \\
172 & $>-0.4$ & \nodata \\
220.1 & $>-0.3$ & \nodata \\
234 & -0.028 $\pm$ 0.006 & 0.019 $\pm$ 0.008 \\
244.1 & -0.23 $\pm$ 0.03 & 0.18 $\pm$ 0.08 \\
263.1 & -1.3 $\pm$ 0.2 & \nodata \\
265 & -0.11 $\pm$ 0.03 & 0.04 $\pm$ 0.06 \\
268.1 & -0.4 $\pm$ 0.1 & 0.2 $\pm$ 0.5: \\
280 & -0.23 $\pm$ 0.02 & 0.16 $\pm$ 0.09 \\
330 & -0.41 $\pm$ 0.06 & 0.1 $\pm$ 0.1 \\
381 & -0.19 $\pm$ 0.02 & 0.07 $\pm$ 0.02 \\
433 & -1.30 $\pm$ 0.01 & 0.71 $\pm$ 0.07 \\
452 & -0.196 $\pm$ 0.009 & 0.10 $\pm$ 0.02 \\
\enddata
\tablenotetext{a}{The 9.7 $\mu$m silicate trough (rest) equivalent width in $\mu$m.}
\tablenotetext{b}{Apparent 9.7 $\mu$m silicate optical depth, averaged over 9.2-10.2 $\mu$m (rest).}
\end{deluxetable}
\begin{deluxetable}{ccccccccc}
\tablecaption{Emission Lines\tablenotemark{a}}
\tablewidth{0pt}
\tablehead{
\colhead{3C} & \colhead{[Ne {\sc vi}]} & \colhead{[S {\sc iv}]} & \colhead{[Ne {\sc ii}]}
& \colhead{[Ne {\sc v}]} & \colhead{[Ne {\sc iii}]} & \colhead{[S {\sc iii}]}
& \colhead{[Ne {\sc v}]} & \colhead{[O {\sc iv}]} \\
\colhead{ } & \colhead{$\lambda$ 7.65} & \colhead{10.51} & \colhead{12.81}
& \colhead{14.3} & \colhead{15.55} & \colhead{18.71}
& \colhead{24.31} & \colhead{25.89}
}
\startdata
33 & 2.8(0.2)& 1.2(0.1)& 3.9(0.2)& 2.0(0.3)& 5.3(0.2)& 2.5(0.4)& 1.6(0.2)& 8.1(0.2)\\
& 0.022 & 0.012 & 0.037 & 0.019 & 0.051 & 0.028 & 0.029 & 0.159 \\
55 &1.82(.06)& 2.2(0.3)& $<0.5$ & 1.1(0.2)& 2.0(0.3)& $<0.4$ & \nodata & \nodata \\
& 0.062 & 0.17 & $<0.01$ & 0.036 & 0.068 & $<0.01$ & \nodata & \nodata \\
172 & $<0.1$ & $<0.2$ & $<0.3$ & $<0.6$ & $<0.6$ & $<0.8$ & \nodata & \nodata \\
& $<0.08$ & $<0.1$ & $<0.08$ & $<0.2$ & $<0.2$ & $<0.3$ & \nodata & \nodata \\
220.1 & $<0.2$ & 0.5(0.2)& $<0.2$ & $<0.5$ &0.35(.09)& $<0.2$ & \nodata & \nodata \\
& $<0.06$ & 0.21 & $<0.05$ & $<0.1$ & 0.11 & $<0.05$ & \nodata & \nodata \\
234 & 1.7(0.1)& 3.2(0.3)& 0.8(0.2)& 3.3(0.7)& 8.2(0.7)& $<3.$ & 3.9(0.7)& 7.5(1.0)\\
& 0.0034 & 0.0077 & 0.0022 & 0.010 & 0.027 & $<0.01$ & 0.026 & 0.056 \\
244.1 & 0.7(0.2)& 0.9(0.2)& 1.4(0.2)& 0.6(0.4)& 0.3(0.2)&0.68(.06)& \nodata & \nodata \\
& 0.033 & 0.041 & 0.067 & 0.033 & 0.02 & 0.043 & \nodata & \nodata \\
263.1 & $<0.6$ & $<0.8$ & 0.6(0.2)& $<0.6$ & 0.6(0.2)& \nodata & \nodata & \nodata \\
& $<0.1$ & $<0.3$ & 0.19 & $<0.2$ & 0.23 & \nodata & \nodata & \nodata \\
265 & 0.6(0.2)& 1.6(0.4)& $<0.8$ & $<0.6$ & $<0.5$ & \nodata & \nodata & \nodata \\
& 0.012 & 0.040 & $<0.02$ & $<0.02$ & $<0.02$ & \nodata & \nodata & \nodata \\
268.1 &0.49(.08)& $<0.4$ & $<0.4$ & $<0.8$ & $<0.2$ & \nodata & \nodata & \nodata \\
& 0.15 & $<0.1$ & $<0.1$ & $<0.4$ & $<0.04$ & \nodata & \nodata & \nodata \\
280 & $<0.3$ & $<0.5$ & 0.3(0.1)& $<0.4$ & 0.9(0.2)& \nodata & \nodata & \nodata \\
& $<0.01$ & $<0.02$ & 0.014 & $<0.02$ & 0.050 & \nodata & \nodata & \nodata \\
330 & 0.4(0.2)&0.64(.08)& $<0.5$ & 0.4(0.1)& 0.7(0.2)&0.19(.08)& \nodata & \nodata \\
& 0.03 & 0.076 & $<0.05$ & 0.04 & 0.09 & 0.02 & \nodata & \nodata \\
381 & 1.3(0.1)&0.62(.07)& 0.6(0.2)& 0.7(0.3)& 1.2(.3) &1.49(.08)&0.56(.08)& 3.9(0.2)\\
& 0.019 & 0.012 & 0.011 & 0.014 & 0.025 & 0.040 & 0.026 & 0.196 \\
433 & 3.1(0.3)& 2.2(0.2)& 1.9(0.3)& 2.7(0.3)& 5.2(0.3)& 1.2(0.8)& 2.3(0.5)& 7.9(0.4)\\
& 0.014 & 0.023 & 0.012 & 0.020 & 0.042 & 0.013 & 0.028 & 0.105 \\
452 & $<0.5$ & 0.7(0.1)&2.11(.07)& $<0.3$ & 1.9(0.2)& 1.4(0.4)& $<0.3$ & 1.3(0.2)\\
& $<.008$ & 0.012 & 0.034 & $<.005$ & 0.032 & 0.026 & $<0.01$ & 0.057 \\
\enddata
\tablenotetext{a}{Notes. For each source, emission line fluxes ($10^{-14}$ erg s$^{-1}$ cm$^{-2}$) or
2$\sigma$ upper limits are on the first line and rest equivalent widths ($\mu$m)
are on the second line. Emission line rest wavelengths ($\mu$m) are at the top of
each column.}
\end{deluxetable}
|
Title:
Dispersion relation for electromagnetic wave propagation in a strongly magnetized plasma |
Abstract: A dispersion relation for electromagnetic wave propagation in a strongly
magnetized cold plasma is deduced, taking photon-photon scattering into
account. It is shown that the combined plasma and quantum electrodynamic effect
is important for understanding the mode-structures in magnetar and pulsar
atmospheres. The implications of our results are discussed.
| https://export.arxiv.org/pdf/astro-ph/0601311 |
\title[Dispersion relation in strongly a magnetized plasma]{Dispersion relation for electromagnetic wave propagation in a strongly magnetized plasma}
\author{G. Brodin M. Marklund, L. Stenflo and P.K. Shukla}
\address{Department of Physics, Ume{\aa} University, SE--901 87 Ume{\aa},
Sweden}
\pacs{52.25.Xz (Magnetized plasmas), 52.35.Mw (Nonlinear phenomena),
52.27.Ep (Electron-positron plasmas)}
\section{Introduction}
The quantum electrodynamical (QED) phenomenon of elastic photon--photon
scattering, due to the interaction of photons with virtual
electron--positron pairs, has recently received increased attention
\cite{Ding1992,Moulin1999,BMS2001,Eriksson2004,Shen-etal,Shen-Yu,Bulanov-etal,JPP,%
Marklund-Brodin-Stenflo,MSSBS2005,RMP}.
Several papers are motivated by the desire to detect
photon--photon scattering in laboratories \cite
{Ding1992,Moulin1999,BMS2001,Eriksson2004}, whereas others \cite
{Shen-etal,Shen-Yu,Bulanov-etal} concern phenomena that might be relevant
when the laser power is further increased to produce electric fields
strengths close to the Schwinger field $\sim 10^{18}\, \mathrm{V\,m^{-1}}$ \cite{Bulanov-etal}%
. Up to now, however, observable effects of photon--photon scattering are
likely to occur only for astrophysical systems \cite
{Baring-Harding,Shaviv-etal,Marklund-Brodin-Stenflo,magnetar,Bialynicki-Birula1970,%
Adler,MSSBS2005}, where the
large magnetic field strengths in pulsar and magnetar environments \cite{magnetar,Duncan-Thompson,Palmer-etal}
open up
for QED processes to play an important role, leading to phenomena such as
frequency down-shifting \cite{Adler,Bialynicki-Birula1970} and lensing
\cite{Shaviv-etal}. \ The
frequency down-shifting \ is a result of so called photon splitting \cite
{Adler,Bialynicki-Birula1970}, which is one of the consequences of elastic
photon-photon scattering, and the process may even be responsible for the
radio silence of magnetars \cite{Baring-Harding}. Another QED-process of interest
in pulsar and magnetar environments is pair-production \cite{Beskin-book}
due to the strong field interactions, which lead to the presence of an
electron-positron pair plasma in the pulsar and magnetar atmospheres.
However, with a few exceptions (e.g.\ \cite{MSSBS2005,JPP}), we note that
most papers considering photon--photon scattering have omitted plasma effects
when considering electromagnetic wave propagation under these conditions.
In the present paper we will consider electromagnetic wave propagation at an
arbitrary angle to a strong external magnetic field $\mathbf{B}_{0}$, and
include the QED-effects associated with that field, as well as the influence
of an electron-positron pair plasma. The former effect is described within
the framework of the Heisenberg--Euler Lagrangian, which constitutes an
effective theory of photon--photon scattering \cite
{Heisenberg-Euler,Schwinger}, and the latter contribution follows from
elementary plasma theory. A comparatively general dispersion relation will
be derived. It reduces to previous results in a number of limiting cases
\cite{Chen-book,Bialynicki-Birula1970,MSSBS2005}. In order to determine the
contribution from the pair-plasma on the propagation properties in pulsar
and magnetar atmospheres, we adopt the Goldreich-Julian expression for the
plasma density \cite{Beskin-book}, and evaluate the dispersion relation for
field strengths in the pulsar and magnetar range, $B_{0}\sim 10^{8}-10^{10}%
\,\mathrm{T}$. In the radio-wave regime it then turns out that for one of the
EM-wave polarizations, the plasma effects are typically negligible as
compared to the QED-effects, whereas for the other polarization, the
opposite is true in most cases. Noting that important processes in pulsar
and magnetar environments, e.g.\ photon splitting, typically involve both
EM-wave polarizations, we will conclude that QED and plasma effects should
be simultaneously included when studying radio wave propagation in such
environments.
\section{Derivations}
If vacuum fluctuations are taken into account, such as under highly
energetic conditions (e.g.\ pulsar plasmas and the next generation of
laser-plasma systems), Maxwell's equation will be altered by the quantum
vacuum self-interaction through the polarization
\begin{equation}
\mathbf{P}=2\kappa \epsilon _{0}^{2}\left[ 2(E^{2}-c^{2}B^{2})\mathbf{E}%
+7c^{2}(\mathbf{E}\cdot \mathbf{B})\mathbf{B}\right]
\end{equation}
and magnetization
\begin{equation}
\mathbf{M}=2\kappa \epsilon _{0}^{2}c^{2}\left[ -2(E^{2}-c^{2}B^{2})\mathbf{B%
}+7(\mathbf{E}\cdot \mathbf{B})\mathbf{E}\right]
\end{equation}
respectively, see e.g.\ \cite{BMS2001}. Here $\kappa =(\alpha /90\pi
)(1/\epsilon _{0}E_{\mathrm{crit}}^{2})$ gives the strength of the quantum
vacuum nonlinearity, $\alpha \approx 1/137$ is the fine-structure constant, $%
E_{\mathrm{crit}}=m^{2}c^{3}/e\hbar \sim 10^{18}\,\mathrm{V\,m^{-1}}$ is the
Schwinger critical field, $m$ is the electron rest mass, $c$ is the speed of
light in vacuum, $e$ is the magnitude of the electron charge, and $\hbar $
is Planck's constant divided by $2\pi $. These corrections to Maxwell's
vacuum equations are valid as long as $|\mathbf{E}|\ll E_{\mathrm{crit}}$ and $%
\omega \ll \omega _{e}=mc^{2}/\hbar $, where $\omega _{e}\approx 8\times
10^{20}\,\mathrm{rad\,s^{-1}}$ is the Compton frequency.
Next we Fourier decompose the electromagnetic perturbations, which have
frequencies $\omega $ and wavevectors $\mathbf{k}$. Maxwell's equations
together with the plasma equations of motion then yield
\begin{equation}
\Delta ^{ab}\delta E_{b}=0.
\end{equation}
using index notation. Here the matrix $\Delta ^{ab}=n^{a}n^{b}-n^{2}\delta
^{ab}+\epsilon ^{ab}$, where $n^{a}=k^{a}c/\omega $, $n=kc/\omega $ is the
plasma refractive index, where $k=|\mathbf{k}|$, $\epsilon ^{ab}=\epsilon _{%
\mathrm{classical}}^{ab}+\epsilon _{\mathrm{QED}}^{ab}$ is the dielectric
tensor,
\begin{equation}
\epsilon _{\mathrm{classical}}^{ab}=\delta ^{ab}+i\omega \sum_{s}\left( \frac{%
\omega _{\mathrm{p}s}}{\omega }\right) ^{2}\sigma _{s}^{ab},
\label{eq:dielectric-classical}
\end{equation}
\begin{equation}
\epsilon _{\mathrm{QED}}^{ab}=-4\xi \left[ \delta ^{ab}+n^{a}n^{b}-n^{2}\delta
^{ab}-\frac{7}{2}b^{a}b^{b}-2(\eta ^{aij}n_{i}b_{j})(\eta ^{bkl}n_{k}b_{l})%
\right] ,
\end{equation}
$s$ denotes the plasma particle species, $\omega
_{\mathrm{p}s}=(q_{s}^{2}n_{s}/\epsilon _{0}m_{s})^{1/2}$ is the plasma frequency for
species $s$, $\ \xi =\kappa \epsilon _{0}c^{2}B_{0}^{2}=(\alpha /90\pi
)(cB_{0}/E_{\mathrm{crit}})^{2}$ is the dimensionless QED parameter, $%
b^{a}=B_{0}^{a}/B_{0}$, and
\begin{equation}
(\sigma _{s}^{ab})^{-1}=-i\omega \delta ^{ab}+\omega _{\mathrm{c}s}\eta ^{abj}b_{j},
\label{eq:sigmainv}
\end{equation}
with the cyclotron frequency $\omega _{\mathrm{c}s}=q_{s}B_{0}/m_{s}$ for species $s,$
$\delta ^{ab}$ is the Kronecker dela and $\eta _{abc}$ is the totally
anti-symmetric unit tensor. From the definition (\ref{eq:sigmainv}) we
obtain
\begin{equation}
\sigma _{s}^{ab}=\frac{i\omega }{\omega ^{2}-\omega _{\mathrm{c}s}^{2}}(\delta
^{ab}-b^{a}b^{b})-\frac{\omega _{\mathrm{c}s}}{\omega ^{2}-\omega _{\mathrm{c}s}^{2}}\eta
^{abj}b_{j}+\frac{i}{\omega }b^{a}b^{b}, \label{sigma-QED}
\end{equation}
and the dielectric tensor (\ref{eq:dielectric-classical}) is thus
\begin{equation}
\fl \epsilon _{\mathrm{classical}}^{ab}=\delta ^{ab}-\sum_{s}\left[ \frac{\omega
_{\mathrm{p}s}^{2}}{\omega ^{2}-\omega _{\mathrm{c}s}^{2}}(\delta ^{ab}-b^{a}b^{b})+\frac{%
i\omega _{\mathrm{p}s}^{2}\omega _{\mathrm{c}s}}{\omega (\omega ^{2}-\omega _{\mathrm{c}s}^{2})}\eta
^{abj}b_{j}+\left( \frac{\omega _{\mathrm{p}s}}{\omega }\right) ^{2}b^{a}b^{b}\right]
, \label{Eps-classic}
\end{equation}
We note that the full dielectric tensor depends on the wavevector through
the QED contribution $\epsilon _{\mathrm{QED}}^{ab}$. Freely propagating waves
are characterized by the vanishing of the dispersion relation $D(\omega ,%
\mathbf{k})=\mathrm{det}(\Delta ^{ab})$. Writing the QED-tensor $\epsilon _{%
\mathrm{QED}}^{ab}$in matrix form, letting the $\mathbf{k}$-vector lie in
the $xz$-plane, we then obtain
\begin{equation}
\epsilon^{ab}_{\mathrm{QED}}=-4\xi \left(
\begin{array}{ccc}
1-n_{\Vert }^{2} & 0 & n_{\bot }n_{\Vert } \\
0 & 1-n^{2}-2n_{\bot }^{2} & 0 \\
n_{\bot }n_{\Vert } & 0 & -\frac{5}{2}-n_{\bot }^{2}
\end{array}
\right) \label{QED-matrix}
\end{equation}
where $n_{\Vert }=k_{\Vert }c/\omega $, $n_{\bot }=k_{\bot }c/\omega $ and
the $\mathbf{k}$-vector is written as $\mathbf{k}=k_{\bot }\widehat{\mathbf{x%
}}+k_{\Vert }\widehat{\mathbf{z}}$. From (\ref{Eps-classic}) the classical
contributions to $\Delta ^{ab}$ is
\begin{equation}
\!\!\!\!\!\!\! \left(
\begin{array}{ccc}
1- \displaystyle{\sum\limits_{s}\frac{\omega _{\mathrm{p}s}^{2}}{\omega ^{2}-\omega _{\mathrm{c}s}^{2}} }%
-n_{\Vert }^{2} & \displaystyle{i\sum\limits_{s}\frac{\omega _{\mathrm{p}s}^{2}\omega _{\mathrm{c}s}}{\omega
(\omega ^{2}-\omega _{\mathrm{c}s}^{2})}} & n_{\bot }n_{\Vert } \\
\displaystyle{-i\sum\limits_{s}\frac{\omega _{\mathrm{p}s}^{2}\omega _{\mathrm{c}s}}{\omega (\omega
^{2}-\omega _{\mathrm{c}s}^{2})}} & 1-\displaystyle{\sum\limits_{s}\frac{\omega _{\mathrm{p}s}^{2}}{\omega
^{2}-\omega _{\mathrm{c}s}^{2}}}-n^{2} & 0 \\
n_{\bot }n_{\Vert } & 0 & 1-\displaystyle{\sum\limits_{s}\frac{\omega _{\mathrm{p}s}^{2}}{\omega
^{2}}}-n_{\bot }^{2}
\end{array}
\right) \label{Classical-matrix}
\end{equation}
The determinant of the sum of the matrixes (\ref{QED-matrix}) and (\ref
{Classical-matrix}) is then evaluated to give the dispersion relation
\begin{eqnarray}
\fl
0 =\left( (1-n^{2})(1-4\xi )-\sum\limits_{s}\frac{\omega _{\mathrm{p}s}^{2}}{\omega
^{2}-\omega _{\mathrm{c}s}^{2}}+8\xi n_{\bot }^{2}\right) \times
\nonumber \\ \fl
\left[\! \left(\!\! (1-n_{\Vert }^{2})(1-4\xi )- \!\! \sum\limits_{s}\frac{\omega
_{\mathrm{p}s}^{2}}{\omega ^{2}-\omega _{\mathrm{c}s}^{2}}\! \right) \!\! \left(\!\! 1+10\xi
- \!\! \sum\limits_{s}\frac{\omega _{\mathrm{p}s}^{2}}{\omega ^{2}}-n_{\bot }^{2}\left(
1-4\xi \right)\!\! \right)\!\! -n_{\bot }^{2}n_{\Vert }^{2}\left( 1-4\xi \right) \!
\right] -
\nonumber \\ \fl
\left( \sum\limits_{s}\frac{\omega _{\mathrm{p}s}^{2}\omega _{\mathrm{c}s}}{\omega (\omega
^{2}-\omega _{\mathrm{c}s}^{2})}\right) ^{2}\left( 1+10\xi -\sum\limits_{s}\frac{%
\omega _{\mathrm{p}s}^{2}}{\omega ^{2}}-n_{\bot }^{2}\left( 1-4\xi \right) \right)
\label{Full-DR}
\end{eqnarray}
The dispersion relation (\ref{Full-DR}) is the main result of the present
paper. It describes wave propagation at any angle to the external magnetic
field in a multi-component plasma, and it includes the QED effects
associated with the external magnetic field.
Thus, it applies to high frequency electromagnetic waves of any polarization, as well as electrostatic oscillations and low frequency waves, such as Alfv\'en waves. As a specific example of how the plasma dispersion relation is affected by the QED effects we consider the case of an electron--positron plasma with $\omega \sim \omega_{\mathrm{p}} \ll |\omega_{\mathrm{c}}|$. The dispersion relation relation for the ordinary mode propagating perpendicular to the background magnetic field, with strength $\sim 10^{10}\,\mathrm{T}$ is depicted in figure 1.
A number of limiting cases of (%
\ref{Full-DR}) have previously appeared in the literature. First, neglecting
the QED-effects (i.e.\ letting $\xi \rightarrow 0$), we immediately
obtain the standard dispersion relation for a cold multi-component plasma
(see e.g.\ \cite{Chen-book}). Alternatively, letting $\omega
_{\mathrm{p}}\rightarrow 0$, we note that the dispersion relation depends on the
propagation angle relative to the magnetic field. Furthermore, we note that
the indicies of refraction depend on the polarization even without a plasma.
These QED-effects due to a strong external magnetic field are wellknown
(often referred to as ''birefringence of vacuum''). Our dispersion relation
in the limit $\omega _{\mathrm{p}}\rightarrow 0$ agrees with those of previous
works, see e.g.\ \cite{Adler,Bialynicki-Birula1970}. The combined contribution
from the QED-effects due to a strong magnetic field and a non-zero plasma
density have previously been considered \cite{MSSBS2005} in the limit of
parallel propagation and allowing for large amplitudes. Taking the limit of
a small wave amplitudes in the dispersion relation (11) of reference \cite
{MSSBS2005}, and letting $n_{\bot }\rightarrow 0$ in (\ref{Full-DR})
we obtain agreement with \cite{MSSBS2005}.
\section{Conclusion}
QED-effects associated with the external magnetic field are likely to be of
importance in environments with extreme magnetic fields, in particular in
the vicinity of astrophysical objects like pulsars and magnetars. For
example, the radio silence of magnetars is assumed to be connected with
QED-effects associated with the magnetar fields \cite{Baring-Harding}, which
could reach $10^{10} - 10^{11}\,\mathrm{T}$ $\ $\ close to the surface. However, in
the same environments, we also expect the presence of an electron-positron
plasma \cite{Beskin-book}. Thus we evaluate (\ref{Full-DR}) with $%
\sum_{s}=\sum_{e,p}$, where $e$ and $p$ denotes electrons and positrons,
respectively. Considering propagation at an arbitrary angle to the magnetic
field in an electron-positron plasma, letting $\omega _{\mathrm{p}e,\mathrm{p}p}\!\sim\! \omega\! \ll \!
\left| \omega _{\mathrm{c}e,\mathrm{c}p}\right| $ , using $\xi\! \ll\! 1$ and noting that the
factor $[\sum_{e,p}\omega _{\mathrm{p}s}^{2}\omega _{\mathrm{c}s}/\omega (\omega ^{2}-\omega
_{\mathrm{c}s}^{2})]^{2}$ then becomes negligibly small (due to the approximate
cancellation of the electron and positron contributions), we find from (%
\ref{Full-DR}) that the dispersion relation separates in two modes that can
be approximated by
\begin{equation}
1-n^{2}+8\xi n_{\bot }^{2}+\frac{\omega _{\mathrm{p}}^{2}}{\omega _{\mathrm{c}}^{2}(1-4\xi )}%
\approx 0 \label{Magnetosonic-DR}
\end{equation}
and
\begin{equation}
(1-n^{2})(1-4\xi ) - \left( -14\xi +\frac{\omega _{\mathrm{p}}^{2}}{\omega ^{2}}%
\right) (1-n_{\Vert }^{2}) \approx 0 ,
\label{O-mode-DR}
\end{equation}
where $\omega _{\mathrm{p}}=(\omega _{\mathrm{p}e}^{2}+\omega _{\mathrm{p}p}^{2})^{1/2}$ is the total
plasma frequency, and $\omega _{\mathrm{c}}$ $=eB_{0}/m$ is the magnitude of the
electron (or positron) cyclotron frequency. In the vicinity of pulsars or
magnetars where $\omega _{\mathrm{c}}\sim 10^{19}\! -\! 10^{21}\,\mathrm{rad\,s^{-1},}$ the last term of (\ref{Magnetosonic-DR}) is negligible unless the plasma density is
extremely high. Omitting that term, the dispersion relation (\ref
{Magnetosonic-DR}) is then the same as that used in \cite
{Bialynicki-Birula1970} for the high phase velocity mode when considering
photon-splitting. Similarly the ordinary mode described by (\ref
{O-mode-DR}), reduces to the mode with the lower phase velocity of reference \cite
{Bialynicki-Birula1970} when the plasma is removed. However, for the latter
dispersion relation we note that a relatively modest plasma density is
enough to significantly affect the propagation properties in the radio wave
regime. To make a concrete estimate, we adopt the Goldreich--Julian density
\begin{equation}
n_{\mathrm{GJ}}=7\times 10^{15}\left(\frac{0.1}{\tau }\right)\left(\frac{B_{\mathrm{pulsar}}}{10^{8}}\right)\,\mathrm{m}^{-3}
\label{Julian-Goldreich}
\end{equation}
where $\tau $ is the pulsar period time (in seconds) and $B_{\mathrm{pulsar}}$ the pulsar
magnetic field (in tesla). The pair plasma density is expected to satisfy $%
n_{e}=n_{p}=Mn_{\mathrm{GJ}}$, where $M$ is the multiplicity \cite
{Beskin-book,Luo-etal}. Moderate estimates then give $M=10$ \cite{Luo-etal}.
Choosing this value and letting $\tau =1\,\mathrm{s}$, we note that for
magnetar field strengths, $B_{\mathrm{pulsar}}=10^{10}\,\mathrm{T}$, the term due to the
plasma $\propto \omega _{\mathrm{p}}^{2}/\omega ^{2}$ in (\ref{O-mode-DR})
dominates over the term due to QED $\propto 14\xi $ for frequencies up to $%
\omega \sim 10^{14} - 10^{15}\,\mathrm{rad\,s^{-1},}$ i.e.\ in the infrared regime and
below. Furthermore, we note that photon splitting \cite
{Adler,Bialynicki-Birula1970} as described by standard QED (i.e.\ with zero plasma
density) requires that the phase velocity of the dispersion relation in (%
\ref{Magnetosonic-DR}) is higher than that of (\ref{O-mode-DR}). While
this is always true in the absence of a plasma, we note that for wave
frequencies in the infra-red regime and below, the Goldreich--Julian density
given by (\ref{Julian-Goldreich}) is enough to increase the phase velocity
of the mode in (\ref{O-mode-DR}) above that of (\ref{Magnetosonic-DR}),
unless we choose the period time $\tau $ extremely low. \ Thus we conclude
that photon--photon splitting as described by vacuum theories is not likely
to apply to magnetar atmospheres, unless the pair-production \cite
{Beskin-book} responsible for the Goldreich-Julian expression is effectively
suppressed. Wave cascade processes as a mechanism to explain the radio
silence of magnetars \cite{Baring-Harding} could still be possible, but for
densities of the order of (\ref{Julian-Goldreich}), plasma nonlinearities
are likely to dominate over the pure QED effects.
\section*{References}
|
Title:
Introduction: Paleoheliosphere versus PaleoLISM |
Abstract: Speculations that encounters with interstellar clouds modify the terrestrial
climate have appeared in the scientific literature for over 85 years. This
article introduces a series of articles that seek to give substance to these
speculations by examining the exact mechanisms that link the pressure and
composition of the interstellar medium surrounding the Sun to the physical
properties of the inner heliosphere at the Earth.
| https://export.arxiv.org/pdf/astro-ph/0601356 |
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\articletitle[Introduction: Paleoheliosphere versus PaleoLISM]{Introduction: \\ Paleoheliosphere versus PaleoLISM}
\chaptitlerunninghead{Paleoheliosphere versus PaleoLISM}
\author{Priscilla C. Frisch}
\affil{University of Chicago}
\email{frisch@oddjob.uchicago.edu}
\begin{keywords}
Heliosphere, interstellar clouds, interstellar medium, cosmic rays,
magnetosphere, atmosphere, climate, solar wind, paleoclimate
\end{keywords}
\section{The Underlying Query}
If the solar galactic environment is to have a discernible effect on
events on the surface of the Earth, it must be through a subtle and
indirect influence on the terrestrial climate. The scientific and
philosophical literature of the 18th, 19th and 20th centuries all
include discussions of possible cosmic influences on the terrestrial
climate, including the effect of cometary impacts on Earth
(\nolinebreak \cite{Halley:1694}), and the diminished solar radiation
from sunspots, which Herschel attributed to ``holes'' in the luminous
fluid on the surface of the Sun\footnote{In this same paper Herschel
commented that ``Whatever fanciful poets might say, in making the sun
the abode of blessed spirits, or angry moralists devise, in pointing
it out as a fit place for the punishment of the wicked, it does not
appear that they had any other foundation for their assertions than
mere opinion and vague surmise; but now I think myself authorized,
\emph{upon astronomical principles,} to propose the sun as an
inhabitable world, and am persuaded that the foregoing observations,
with the conclusions I have drawn from them, are fully sufficient to
answer every objection that may be made against it. '' These
comments show that valuable data are not always interpreted
correctly.} (\nolinebreak \cite{Herschel:1795}). The discovery of interstellar
material in the 20th century led to speculations that encounters with
dense clouds initiated the ice ages (\nolinebreak
\cite{Shapley:1921}), and many papers appeared that explored the
implications of such encounters, including the influence of
interstellar material (ISM) on the interplanetary medium and planetary
atmospheres
(e.g. \cite{Fahr:1968,BegelmanRees:1976,McKayThomas:1978,Thomas:1978,McCrea:1975,TalbotNewman:1977,Willis:1978,ButlerNewmanTalbot:1978}).
The ISM-modulated heliosphere was also believed to affect climate
stability and astrospheres
(e. g. \cite{Frisch:1993a,Frisch:1997,ZankFrisch:1999}). Recent
advances in our understanding of the solar wind and heliosphere
(e. g. \cite{WangRichardson:2005,Fahr:2004}) justify a new look at
this age-old issue. This book addresses the underlying question:
\begin{verse}
\emph{How does the heliospheric interaction} \emph{with the interstellar medium
affect the heliosphere, interplanetary medium, and Earth?}
\end{verse}
The heliosphere is the cavity in the interstellar medium created by
the dynamic ram pressure of the radially expanding solar wind, a halo
of plasma around the Sun and planets, dancing like a candle in the
wind and regulating the flux of cosmic rays and interstellar material
at the Earth. Neutral interstellar gas and large interstellar dust
grains penetrate the heliosphere, but the solar wind acts as a buffer
between the Earth and most other interstellar material and low energy
galactic cosmic rays (GCR). Together the solar wind and interstellar
medium determine the properties of the heliosphere. In the present
epoch the densities of the solar wind and interstellar neutrals are
approximately equal outside of the Jupiter orbit. Solar activity
levels drive the heliosphere from within, and the physical properties
of the surrounding interstellar cloud constrain the heliosphere from
without, so that the boundary conditions of the heliosphere are set by
interstellar material. Figure \ref{fig:1} shows the Sun and
heliosphere in the setting of the Milky Way Galaxy.
The answer to the question posed above lies in an interdisciplinary
study of the coupling between the interstellar medium and the solar
wind, and the effects that ISM variations have on the 1 AU environment
of the Earth through this coupling. The articles in this book explore
different viewpoints, including \emph{gedanken} experiments, as well
as data-rich summaries of variations in the solar environment and
paleoclimate data on cosmic ray flux variations at Earth.
The book begins with the development of theoretical models of the
heliosphere that demonstrate the sensitivity of the heliosphere to the
variations in boundary conditions caused by the passage of the Sun
through interstellar clouds (\cite{Zanketal:2006jos,PogorelovZank:2006jos}). A series of \emph{gedanken} experiments
then yield the response of planetary magnetospheres to encounters with
denser ISM (\cite{Parker:2006jos}). Variations in the galactic environment of the Sun, caused
by the motions of the Sun and clouds through the Galaxy, are shown to
occur for both long and short timescales (\cite{Shaviv:2006jos,FrischSlavin:2006jos}).
The heliosphere acts as a buffer between the Earth and interstellar
medium, so that dust and particle populations inside of the
heliosphere, which have an interstellar origin, vary as the Sun
traverses interstellar clouds. These buffering mechanisms determine
the interplanetary medium\footnote{The buffering processes convert
interstellar neutrals into low energy ions, which are convected
outwards with the solar wind and accelerated to low cosmic ray
energies that have an anomalous composition, including abundant
elements with FIP$>$13.6 eV. The high energy galactic cosmic ray
population incident on the heliosphere is also modulated.}. The
properties of these buffering interactions are evaluated for
heliosphere models that have been developed using boundary conditions
appropriate for when the Sun traverses different types of interstellar
clouds (\cite{Landgraf:2006jos,Moebiusetal:2006jos,FlorinskiZank:2006jos,Fahretal:2006jos}).
The consequences of Sun-cloud encounters are then discussed in terms
of the accretion of ISM onto the terrestrial atmosphere for dense
cloud encounters, and the possibly extreme variations expected for
cosmic ray modulation when interstellar densities vary
substantially (\cite{Fahretal:2006jos,YeghikyanFahr:2006jos}). Radioisotope records on Earth extending backwards in
time for over $\sim$0.5 Myrs, together with paleoclimate data, suggest
that cosmic ray fluxes are related to climate. The galactic
environment of the Sun must have left an imprint on the geological
record through variations in the concentrations of radioactive
isotopes (\cite{KirkbyCarslaw:2006jos}).
The selection of topics in this book is based partly on scientific
areas that have already been discussed in the literature. The authors
who were invited to contribute chapters have previously studied the
heliosphere or terrestrial response to variable ISM conditions or cosmic rays.
Figure \ref{fig:1} shows the heliosphere in our setting of the Milky
Way Galaxy. A postscript at the end of this chapter lists basic useful
information. I introduce the term ``paleoheliosphere'' to represent the
heliosphere in the past, when the boundary conditions set by the local
interstellar material (LISM) may have differed substantially from the boundary
conditions for the present-day heliosphere. The ``paleolism'' is the
local ISM that once surrounded the heliosphere.
\section[Addressing the Query: The Heliosphere for Different Interstellar Environments]{Addressing the Query: The Heliosphere and Particle Populations
for Different Interstellar Environments}
The solar wind drives the heliosphere from the inside, with the
properties of the solar wind varying with ecliptic latitude and the
phase of the 11-year solar activity cycle. The global heliosphere is
the volume of space occupied by the supersonic and subsonic solar
wind. Interstellar material forms the boundary conditions of the
heliosphere, and the windward side of the heliosphere, or the ``upwind
direction'', is defined by the interstellar velocity vector
with respect to the Sun. The leeward side of the heliosphere
is the ``downwind direction''. Figure \ref{fig:1} shows a cartoon of
the present-day heliosphere, with labels for the major landmarks such
as the termination shock, heliopause, and bow shock.
In the present-day heliosphere, the transition from solar wind to
interstellar plasma occurs at a contact discontinuity known as the
``heliopause'', which is formed where the total solar wind and
interstellar pressures equilibrate (\nolinebreak \cite{Holzer:1989}).
For a non-zero interstellar cloud velocity in the solar rest frame,
the solar wind turns around at the heliopause and flows around the
flanks of the heliosphere and into the downwind heliotail. Before
reaching the heliopause, the supersonic solar wind slows to subsonic
velocities at the ``termination shock'', where kinetic energy is
converted to thermal energy.
The subsonic solar wind region between the termination shock and
heliopause is called the inner ``heliosheath''. The outer heliosheath lies
just beyond the heliopause, where the pristine ISM is distorted by the
ram pressure of the heliosphere. A bow shock, where the interstellar
gas becomes subsonic, is expected to form ahead of the present-day
heliosphere in the observed upwind direction of the ISM flow through
the solar system.
Large interstellar dust grains and interstellar atoms that remain
neutral inside of the orbit of Earth, such as He, are gravitationally
focused in the downwind direction. This ``focusing cone'' is traversed
by the Earth every year in early December, and extends many AU from the Sun
in the leeward direction
(e.g. \cite{Landgraf:2000,Moebiusetal:2004,Frisch:2000amsci}). The
heliotail itself extends $>10^3$ AU from the Sun in the downwind
direction, forming a cosmic wake for the solar system.
Of significance when considering the interaction of the heliosphere
with an interstellar cloud is that neutral particles enter the
heliosphere relatively unimpeded, after which they are ionized and
convected outwards with the solar wind. Ions and small charged dust
grains are magnetically deflected in the heliosheath around the flanks
of the heliosphere (see Figure \ref{fig:1}).
Space and astronomical data now confirm the basic milestones of the
outer heliosphere. Voyager 1 crossed the termination shock at 94 AU
on 16 December, 2004 (UT), and observed the signature of the
termination shock on low-energy particle populations, the solar wind
magnetic field, low-energy electrons and protons, and Langmuir radio
emission (\nolinebreak
\cite{Stoneetal:2005,Burlagaetal:2005,Gurnett:2005,Deckeretal:2005}).
The present-day termination shock appears to be weak, with a solar
wind velocity jump ratio (the ratio of upstream to downstream values)
of $\sim$2.6 and a magnetic field compression ratio of $\sim$3. The
magnetic wall that is predicted for the heliosphere (\nolinebreak
\cite{Linde:1998,Ratkiewicz:1998}, Chapter 3 by Pogorelov and Zank)
appears to have been detected through observations of magnetically
aligned dust grains (\nolinebreak
\cite{Frisch:2005L}), and the offset between upwind directions of
interstellar \HI\ and \HeI\ (\nolinebreak \cite{Lallementetal:2005}).
The compressed and heated \HI\ in the hydrogen wall region of the
outer heliosheath has now been detected around a number of stars
(\nolinebreak \cite{Woodetal:2005}).
The present-day solar wind is the baseline for evaluating the
heliosphere response to ISM variations in the following articles, so a
short review of the solar wind is first presented. The remaining part
of $\S$1.2 introduces the topics in the following articles in terms of
the underlying query of the book.
\subsection{The Present Day Solar Wind }
The solar wind originates in the million degree solar corona that
expands radially outwards, with a density $\sim 1/R^2_\mathrm{ S }$
where $R_\mathrm{ S }$ is the distance to the Sun, and contains both
features that corotate with the Sun, and transient structures (\nolinebreak
e.~g.~\cite{Gosling:1996}). The properties of the solar wind vary
with the phase of the solar magnetic activity cycle and with ecliptic
latitude. The best historical indicator of solar magnetic activity
levels is the number of sunspots, first detected by Galileo in 1610,
which are magnetic storms in the convective zone of the Sun. Sunspot
numbers indicate that the magnetic activity levels fluctuate with a
$\sim$11 year cycle, or the ``solar cycle'', and solar maximum/minimum
corresponds to the maximum/minimum of sunspot numbers. The magnetic
polarity of the Sun varies with a $\sim$22 year cycle. During solar
maximum, a low-speed wind, with velocity $\sim$300--600 \kms\ and
density $\sim 6-10$ particles \cc\ at 1 AU, extends over most of the
solar disk. Open magnetic field lines\footnote{Open magnetic field lines
are formed in coronal holes that reconnect in the outer heliosphere
and contain low density and very high speed, $\sim$700 \kms, solar wind.}
are limited to solar pole regions. A neutral current sheet $\sim$0.4 AU
thick forms between the solar wind containing negative magnetic polarity
fields and the solar wind that contains positive magnetic polarity fields.
The neutral current sheet reaches its largest
inclination ($\ge 70^\circ$) during solar maximum.
During the conditions of solar minimum, a high speed
wind with velocity $\sim$600--800 \kms\ and density $\sim$5 \cc\ is
accelerated in the open magnetic flux lines in coronal holes. During
mininum, the
high speed wind and open field lines extend from the polar regions
down to latitudes of $\leq 40$\deeg
(\nolinebreak \cite{Smithetal:2003,Richardson:1995}). The higher solar wind
momentum flux associated with solar minimum conditions produces an
upwind termination shock that is $\sim$5--40 AU more distant in the
upwind direction than during solar maximum conditions
(e.g. \cite{SchererFahr:2003,ZankMueller:2003,Whang:2004}).
The alignment and strength of the solar magnetic multipoles depend on the phase of the solar cycle
(\cite{Bravoetal:1998}). During solar minimum conditions, the magnetic field
is dominated by the dipole and hexapole moments, and the dipole moment
is generally aligned with the solar rotation axis.
Sunspots migrate from high to low heliographic latitudes.
The magnetic poles follow the coronal holes to the solar equator as solar
activity increases. During the solar maximum period, the galactic cosmic
rays undergo their maximum modulation, the dipole component of the
magnetic field is minimized, the quadrapole moment dominates, and the polarity of the solar magnetic field
reverses (\cite{LockwoodWebber:2005}, Figure
\ref{fig:2}). Over historic times, the cosmic ray modulation by
the heliosphere correlates better with the open magnetic flux line
coverage than with sunspot numbers (\cite{McCrackenetal:2004}).
Variable cosmic ray modulation produced by a variable heliosphere may
be a primary factor in both solar and ISM forcing of the terrestrial
climate. The heliosphere modulation of cosmic rays is well
established. John Simpson, to whom this book is dedicated, initiated
a program 5 solar cycles ago in 1951 to monitor cosmic ray fluxes on
Earth using high-altitude neutron detectors (\cite{Simpson:2001}).
The results show a pronounced anticorrelation between cosmic ray flux
levels and solar sunspot numbers, which trace the 11-year Schwabe
magnetic activity cycle, and which also show that the polarity of the solar
magnetic field affects cosmic ray modulation (see Figure \ref{fig:2}).
The articles in this book show convincingly that the ISM also modulates the
heliosphere, and the effect of the solar wind on the heliosphere
must be differentiated from the influence of interstellar matter.
Variations in solar activity levels are also seen over $\sim 100-200$
year timescales, such as the absence of sunspots during the Maunder
Minimum in the 17th century. Modern climate records show that the
Maunder Minimum corresponded to extremely cold weather, and
radioisotope records show that the flux of cosmic rays
was unusually high at this time (see Kirkby and Carslaw, Chapter 12).
Similar effects will occur from the modulation of galactic cosmic rays
by the passage of the Sun through an interstellar cloud.
These temporal and latitudinal variations in the solar wind momentum
flux produce an asymmetric heliosphere, which varies with time. Any possible
historical signature of the ISM on the heliosphere must first be
distinguished from variations driven by the solar wind itself.
\subsection{Present Day Heliosphere and Sensitivity to ISM}
The ISM forms the boundary conditions of the heliosphere, so that
encounters with interstellar clouds will affect the global
heliosphere, the interplanetary medium, and the inner heliosphere
region where the Earth is located. Today an interstellar wind passes
through the solar system at --26.3 \kms\ (\cite{Witte:2004}). An
entering parcel of ISM takes about 20 years to reach the inner
heliosphere, so that ISM near the Earth is constantly replenished with
new inflowing material. This warm gas is low density and partially
ionized, with temperature $T \sim$6,300 K, and densities of neutral
and ionized matter of \nHI$\sim 0.2$ \cc, and
\nHII$\sim 0.1$ \cc.
An elementary perspective of the response of the heliosphere to
interstellar pressures is given by an analytical expression for the
heliopause distance based on the locus of positions where the solar
wind ram pressure, $P_\mathrm{SW}$, and the total interstellar
pressure equilibrate (\cite{Holzer:1989}). The solar wind density
$\rho$ falls off as $\sim 1/R^2$, where $R$ is the distance to the
Sun, while the velocity $v$ is relatively constant. At 1 AU the solar
wind ram pressure is $P_\mathrm{SW,1AU} \sim \rho~v^2 $ so the
heliosphere distance, $R_\mathrm{HP} $, is given by:
\begin{eqnarray*}
P_\mathrm{SW,1AU}/R_\mathrm{HP}^2 \sim P_\mathrm{B} +P_\mathrm{Ions, thermal} +
P_\mathrm{Ions, ram} + P_\mathrm{Dust} + P_\mathrm{CR}
\end{eqnarray*}
The interstellar pressure terms include the magnetic pressure
$P_\mathrm{B}$, the thermal, $P_\mathrm{Ions, thermal}$, and the ram,
$P_\mathrm{Ions, ram}$, pressures of the charged gas, and the
pressures of dust grains, $P_\mathrm{Dust}$, and cosmic rays,
$P_\mathrm{CR}$, which are excluded by heliosphere magnetic fields and
plasma. Some interstellar neutrals convert to ions through charge
exchange with compressed interstellar proton gas in heliosheath
regions, adding to the confining pressure. An important response
characteristic is that, for many clouds, the encounter will be
ram-pressure dominated, where $P_\mathrm{ram} \sim m v^2$ for
interstellar cloud mass density $m$ and relative Sun-cloud velocity
$v$, so that variations in the cloud velocity perturb the heliosphere
even if the thermal pressures remain constant.
The multifluid, magnetohydrodynamic (MHD), hydrodynamic and hybrid
approaches used in the following chapters provide much more
substantial models for the heliosphere, and include the coupling
between neutrals and plasma, and field-particle interactions. These
sophisticated models predict variations in the global heliosphere in
the face of changing interstellar boundary conditions, and for a range
of different cloud types. Although impossible to model a solar
encounter with every type of interstellar cloud, the following
articles include discussions of many of the extremes of the
interstellar parameter space, including low density gas with a range
of velocities, very tenuous plasma, high velocity clouds, dense ISM,
and magnetized material for a range of field orientations and
strengths. The discussions in these chapters extrapolate from our
best theoretical understanding of the heliosphere boundary conditions
today to values that differ, in some cases dramatically, from the
boundary conditions that prevailed at the beginning of the third
millennium in the Gregorian calendar.
The Sun has been, and will be, subjected to many different
physical environments over its lifetime. Theoretical heliosphere
models yield the properties
of the solar wind-ISM interaction for these different environments,
which in turn determine the nature and properties of interstellar
populations inside of the heliosphere for a range of galactic environments.
These models form the foundation for understanding the significance
of our galactic environment for the Earth.
The interstellar parameter space is explored
by Zank et al. (Chapter 2), where 28 sets of
boundary conditions are evaluated with computationally efficient multifluid
models. Moebius et al. (Chapter 8), Fahr et al. (Chapter 9),
Florinski and Zank (Chapter 10), and Yeghikyan and Fahr (Chapter 11) also develop heliosphere models
for a range of interstellar conditions. Together these models
evaluate the heliosphere response to interstellar density, temperature,
and velocity variations of factors of $\sim 10^{9}$, $\sim 10^{5}$, and $\sim 10^{2}$,
respectively.
The interstellar magnetic field introduces an asymmetric pressure
on the heliosphere, affecting the heliosphere current sheet
and cosmic ray modulation.
Pogorelov and Zank (Chapter 3) use MHD models to probe the heliosphere
response to the interstellar magnetic field,
including charge exchange between
the neutrals and solar wind. The resulting asymmetry provides a
test of the magnetic field direction, and shows strong
differences between cases where the interstellar flow is parallel,
instead of perpendicular, to the interstellar magnetic field direction.
Since the random component of the interstellar magnetic field
is stronger, on the average, than the ordered component, particularly
in spiral arm regions where active star formation occurs,
a range of interstellar magnetic field strengths and
orientations are expected over the solar lifetime
(Shaviv, Chapter 5, and Frisch and Slavin, Chapter 6).
\subsection{Planetary Magnetospheres}
The Earth's magnetosphere acts as a buffer between the solar wind and
atmosphere, and as such is an ingredient in understanding the effect
of our galactic environment on the Earth. The decreasing solar wind
density in the outer heliosphere results in an interplanetary medium
around outer planets that is more sensitive to ISM variations than for
inner planets, with implications for the magnetospheres of Jupiter,
Neptune, and Uranus. Most topics in this book are already considered
in the scientific literature, but questions about magnetosphere
variations from an ISM-modulated heliosphere have received scant
attention. In a quintessential \emph{gedanken} experiment, Parker
explores the interaction between magnetospheres and the solar wind for
variations in the interstellar density, and for inner versus outer
planets (Chapter 4).
\subsection{Short and Long Term Variations in the Galactic Environment}
There is every reason to expect that the galactic environment of the
Sun varies over geological timescales. The Sun moves through space at
a velocity of 13--20 \kms, and interstellar clouds have velocities
ranging up to hundreds of \kms. The Arecibo Millennium survey showed
that $\sim$25\% of the mass contained in interstellar \HI, including
both warm and cold ISM, is in clouds traveling with velocities $\ge$10
\kms\ through the local standard of rest (\nolinebreak \cite{HTI}).
Thus Sun-cloud encounters with relative velocities exceeding 25 \kms\
are quite likely, and for a typical cloud length of $\sim$1 pc the
cloud transit time would be $\sim$40,000 years. The many types of ISM
traversed by the Sun during the past several million years have
affected the heliosphere, the inner solar system, and the flux of
anomalous and galactic cosmic rays at Earth
(\cite{FrischYork:1986,Frisch:1997,Frisch:1998}).
For the past $\sim$3 Myrs the Sun has been in a nearly empty region of
space, the ``Local Bubble'', with very low densities of $<$10$^{-26}$
gr \cc. Within the past 44,000--140,000 years the Sun entered a
flow of tenuous, partly neutral ISM, nick-named the ``Local Fluff'', with density
$\sim$60 times higher (Chapter 6). This transition was accompanied by
the appearance of interstellar dust and neutrals in the heliosphere,
along with the pickup ion and anomalous cosmic ray populations.
Galactic cosmic ray modulation was affected, providing a possible link
between our galactic environment and climate. Intriguingly, the
averaged cosmic ray flux at Earth, as traced by \Beten\ records, was
lower in the past $\sim$135 kyrs than for earlier times (Chapter 12).
Was the decrease in the galactic cosmic ray flux $\sim$135 kyrs ago
caused by an increase in modulation as the Sun entered the Local
Fluff?
The galactic environment of the Sun also varies quite dramatically over
long time scales, as discussed by Shaviv (Chapter 5). Over its 4.5 billion year lifetime,
the Sun traverses spiral arm and interarm regions, with atomic
densities varying from less than $10^{-26.1}$ g \cc\ to over
$10^{-20.1}$ g \cc, and temperatures ranging over 7 orders of
magnitude, 10--10$^7$ K. The Sun is now in low density space between
the Perseus and Sagittarius spiral arms, and on the inner edge of what
is known as the Orion Spur on the Local Arm. The Local Arm is not
shown in Figure \ref{fig:1}, as is consistent with the usual Galaxy
depictions. The Local Arm does not appear to be a grand design spiral shock (Bochkarev,
1984\nocite{Bochkarev:1984}). The Sun has a systematic motion of
13--20 km s$^{-1}$ with respect to the nearest stars, corresponding to
$\sim$3--4 AU per year. The Local Interstellar Cloud (LIC) now
surrounding the Sun traverses the heliosphere at $\sim$5.5 AU per
year. The Sun oscillates vertically through the galactic plane once
every $\sim$34 Myrs, and orbits the center of the Milky Way Galaxy
once per $\sim$220 Myrs.
Shaviv evaluates variations in the galactic environment of the Sun
over long timescales. This bold discussion compares
various geologic records of cosmic ray flux variations, based on
radioisotope data that sample timescales of $\sim 10^8$ years, with
models of the Milky Way Galaxy spiral arm pattern to reconstruct the
timing of the Sun's passage through spiral arms. The chapter concludes that star
formation in spiral arms leaves a signature on the radioisotope
records of the solar system.
Frisch and Slavin (Chapter 6) reconstruct short-term variations of the
galactic environment of the Sun using observations of interstellar
matter towards nearby stars and inside of the solar system. Radiative
transfer models of the LIC show that ionization varies across
this low density cloud, so that the heliosphere boundary conditions vary
from radiative transfer considerations alone as the Sun traverses the LIC.
Cloud transitions are predicted during the past $\sim$3 Myrs, including the
departure of the Sun from the Local Bubble interior 44,000--140,000
years ago, and entry into the surrounding cloud 1000--40,000 years ago.
\subsection{Interstellar Dust}
The particle populations formed by the interactions between the solar
wind and interstellar dust, gas, and cosmic rays are emissaries
between the cosmos and inner heliosphere, varying as the Sun moves
through clouds.
About $\sim$1\% of the mass of the cloud surrounding the Sun is
contained in interstellar dust grains. The largest of these charged
grains, mass $> 10^{-13} $ g, have large magnetic Larmor radii of
$>$500 AU at the heliopause for an interstellar field of $\sim$1.5
$\mu$G, and flow into the solar system. The Earth passes through the
gravitational focusing cone formed by these grains early each
December. The smallest charged grains, mass$< 10^{-14.5} $ g and
radii$< 0.01 ~\mu$m, have Larmor radii of $\sim$20 AU, depending on
the magnetic field strength and radiation field, and are deflected
around the heliosheath (\cite{Frischetal:1999}). Interstellar
dust grains are measured in
the inner heliosphere within $\sim$5 AU of the Sun, and over the solar
poles, by satellites such as Ulysses, Galileo and Cassini. Landgraf
(Chapter 7) reviews the properties of the interaction between
interstellar dust and the solar wind, and speculates on the changes
that might be expected from an encounter with a dense interstellar
cloud.
Should it some day be possible to
compare the ratio of large to small interstellar dust grains on the
surfaces of the inner versus outer planets, it would become possible to
disentangle cloud encounters from solar activity effects.
At the very large end of the dust population mass spectrum we find
interstellar micrometeorites, with masses $\sim$3 $\times$ 10$^{-7}$
g, open orbits, and inflow velocities greater than the 42 \kms\ escape
velocity from the solar system at 1 AU. These interstellar objects,
detected by radar as they impact the atmosphere, evidently originate
in circumstellar disks such as that around $\beta$ Pictoris, and in the
interior of the Local Bubble (\cite{Baggaley:2000,Meiseletal:2002}).
These objects do not collisionally couple to the interstellar gas
(\cite{GruenLandgraf:2000}), and should not vary with the type of ISM
surrounding the Sun.
\subsection{Particle Populations in the Inner and Outer Heliosphere}
Presently, low energy interstellar neutrals, high energy galactic
cosmic rays, and interstellar dust all enter the heliosphere.
The characteristics of each of these populations and their secondary
products are modified as the Sun transits the ISM, or the cloud
ionization changes.
The first ionization potential (FIP) of \HI\ is 13.6 eV. Neutral
interstellar atoms with FIP$<$13.6 eV are ionized in nearly all
interstellar clouds because the main source of interstellar opacity is
\HI. Interstellar ions are deflected around the heliosheath, so
the result is that only interstellar atoms with FIP$>$13.6 eV
enter the heliosphere where they are then destroyed, primarily by charge
exchange with solar wind ions.
The density of interstellar neutrals in the inner heliosphere
depends on the density and ionization of the surrounding cloud, the
ionization (or ``filtration'') of those neutrals by the heliosheath,
and the subsequent interactions with the solar wind inside of the heliosphere.
Secondary products produced by solar wind interactions
with interstellar neutrals inside of the heliosphere
include pickup ions
\footnote{The pickup ions are interstellar neutrals formed by charge
exchange with the solar wind. Energetic neutral atoms are formed by
energetic ions that capture an electron from a low energy neutral by
charge exchange. The gravitational focusing cone contains heavy
elements (mainly He) that are predominantly ionized inside of 1 AU and
therefore gravitationally focused downwind of the Sun (Chapter 8).
Large interstellar dust grains are also gravitationally focused (Chapter
7). The anomalous cosmic ray population is formed from pickup ions
accelerated to low cosmic ray energies, $<$\nolinebreak 1 GeV, in the
solar wind and at the termination shock, and then subjected to the
same modulation and propagation processes as galactic cosmic rays
(\cite{Jokipii:2004}).}, energetic neutral atoms, the gravitational
focusing cone formed by helium (also seen in dust), and the anomalous cosmic ray
population with energies $<$\nolinebreak 1 GeV. Interstellar neutrals
inside of the heliosphere, and the heliosphere itself, form a coupled
system that together respond to variations in the heliosphere boundary
conditions.
Moebius et al. (Chapter 8) model the heliosphere for several
different conditions, and then probe the response of the inner
heliosphere to the density of interstellar neutrals flowing
into this ISM-modified heliosphere. At 1 AU, the neutral densities, particle
populations derived from interstellar neutrals, and
characteristics of the helium focusing cone all respond to
variations in the interstellar boundary conditions.
For some cases, increased neutral fluxes fall on the atmosphere
of Earth (also see Yeghikyan and Fahr, Chapter 11).
The velocity structure of the ISM appears to vary on subparsec scale
lengths (Frisch and Slavin, Chapter 6), and these variations may in
some cases result in significant modifications of the inner heliosphere,
particularly the gravitational focusing cone, when all other interstellar parameters
such as thermal pressure are invariant (Zank et al., Chapter 2, Moebius
et al., Chapter 8).
The most readily available diagnostics of the paleoheliosphere are
radioisotopes, formed by cosmic ray spallation on the atmosphere,
interplanetary and interstellar dust, and meteorites. Thus,
the evaluation of cosmic ray modulation for various types of
interstellar cloud boundary conditions is a key part of understanding
the paleoclimate records that might trace the solar journey
through the Milky Way Galaxy. Fahr et
al. (Chapter 9) and Florinski and Zank (Chapter 10) use our
understanding of galactic cosmic ray modulation in the modern-day
heliosphere as a basis for making detailed calculations of the
response of the paleoheliosphere, or the heliosphere as it once was,
to the paleolism, or the local interstellar medium that once
surrounded the Sun. The predictions of these calculations are quite
intriguing. Both the termination shock compression ratio and the
solar wind turbulence spectrum may vary dramatically with different
environments, as mass-loading by pickup ions and the heliosphere
properties vary. The problem of galactic cosmic ray modulation in an
ISM-forced heliosphere is extremely important to understanding the
paleoheliosphere signature in the terrestrial isotope record.
Today, galactic cosmic rays (GCR) with energies $\ge 0.25$ GeV
penetrate the solar system, and anomalous cosmic rays
(energies $<1$ GeV) are formed from accelerated pickup ions.
The cosmic ray flux at Earth is sampled by geological radioisotope records,
as reviewed Kirkby and Carslaw (Chapter 12, also see Florinski and Zank, Chapter 10).
Astronomical data indicates that the Sun has emerged from a region of space with
virtually no neutral ISM within the past $\sim$0.4--1.5 10$^5$ years,
and entered the Local Fluff (Chapter 6). The GCR modulation
discontinuity that accompanied this transition may be in the
geologic record, which show lower cosmic ray fluxes at Earth,
on the average, for the past 135 kyrs years
than the 135 kyrs before that (\cite{Christl:2004}).
\subsection{Atmosphere Accretion from Dense Cloud Encounters}
Harlow Shapley (1921) suggested \nocite{Shapley:1921} that an
encounter between the Sun and giant dust clouds in Orion may have
perturbed the terrestrial climate and caused ice ages. The discovery
of interstellar \HI\ and \HeI\ inside the heliosphere was soon
followed by studies of the ISM influence on the atmosphere for dense
cloud conditions
(\cite{Fahr:1968,BegelmanRees:1976,McKayThomas:1978,Thomas:1978,McCrea:1975,TalbotNewman:1977,Willis:1978,ButlerNewmanTalbot:1978}).
Yeghikyan and Fahr (Chapter 11), evaluate the density of ISM at the
Earth based on models describing the heliosphere inside of an dense cloud,
and the interactions between the solar wind and ISM for these dense
cloud conditions (also see Chapter 9, by Fahr et al.). These models
then yield the concentration of interstellar hydrogen at the Earth,
and the flow of water downward towards the Earth's
surface, as a function of the dense cloud density.
Significant atmosphere modifications are predicted in some cases.
Enhanced neutral populations at 1 AU for a somewhat lower interstellar cloud density regime
are discussed in Chapter 8, by Moebius et al.
\subsection{Possible Effects of Cosmic Rays}
Both solar activity cycles (Figure \ref{fig:2}) and ISM variations
modulate the cosmic ray flux in the heliosphere, and Kirkby and
Carslaw (Chapter 12) compare galactic cosmic ray records with
paleoclimate archives. They examine sources of climate forcing such
as solar irradiance and cosmic ray fluxes, and conclude that arguments
in favor of cosmic ray climate forcing are strong although the
mechanism is uncertain. This relation between cosmic ray flux levels
and the climate is shown by radioisotope records and climate archives,
such as ice cores, stalagmites, and ice-rafted debris, and for modern
times, by historical records. Paleoclimate archives include
terrestrial records of cosmic ray spallation in the atmosphere, as
traced by radioisotopes with short half-lives (\tauhalf), e.~g.~
\Cfourteen\ (\tauhalf=5,730 yrs) and \Beten\ (\tauhalf=1.6 Myrs).
Possible mechanisms linking the cosmic ray flux at 1 AU and the
climate include cloud nucleation by cosmic rays, and the global
electrical circuit (see Chapter 12 and \cite{RobleHaysII:1979}). The discussion in
Chapter 12 provides persuasive evidence linking the
surface temperature to cosmic ray fluxes at Earth. The
anticorrelation between sunspot number and cosmic ray fluxes in Figure
\ref{fig:2} shows the heliosphere role in cosmic ray modulation;
this mechanism must have also been a prominent mechanism for relating
the ISM-modulated heliosphere with the climate. Fortunately this
hypothesis is also verifiable by comparing paleoclimate data with
astronomical data on the timing of cloud transitions.
The radioisotope records also indicate that cosmic ray fluctuations
have occurred over longer timescales of many $10 ^8$ years.
Shaviv compares the \Clthirtysix\ (\tauhalf $\sim$0.3 Myrs) and
\Kforty\ (\tauhalf $\sim$1.3 Gyr) cosmic ray exposure records in iron
meteorites (Chapter 5), but in this case to obtain cosmic ray flux
increases due to the Sun's location in spiral arms where active star
formation occurs.
A number of studies, none convincing, have invoked the geological
\Beten\ record, as a proxy for cosmic ray fluxes at Earth, to infer
historical encounters with interstellar clouds. As a way of dating
the Loop I supernova remnant, it was suggested that the relative
constancy of \Beten\ in sea sediments precluded a strong nearby X-ray
source within the past $\sim$2 Myrs (\cite{Frisch:1981}). Sonett
(1992) \nocite{Sonett:1992} suggested that peaks in \Beten\ layers
35,000 and 65,000 years ago resulted from a compressed heliosphere
caused by the passage of a high-velocity interstellar shock. This
extreme heliosphere compression expected for a rapidly moving cloud is
supported by heliosphere models (Chapter 2). Structure in the \Beten\
peaks has also been related to spatial structure in the local ISM
(\cite{Frisch:1997}), and solar wind turbulence caused by mass-loading
of interstellar neutrals may supply the required mechanism.
Global geomagnetic excursions such as the events
$\sim$32 kyr and $\sim$40 kyr ago also affect the \Beten\ record, and
can not be ignored (\nolinebreak \cite{Christl:2004}). Indeed, Figure
\ref{fig:2} shows the sensitivity of galactic cosmic ray fluxes on
Earth to geomagnetic latitude.
\section{Closing Comments}
This brief summary of the scientific question motivating this
book does not relay the full significance of the galactic environment
of the Sun to the heliosphere and Earth; the following chapters
provide deeper insights into this question.
Historical and paleoclimate data show a correspondence between high
cosmic ray flux levels and cool temperatures on Earth
(\cite{Parker:1996}). The disappearance of sunspots for extended
periods of time, such as the Maunder Minimum in the years 1645 to
1715, shows up in terrestrial radioisotope records such as \Beten\ in
ice cores (Chapter 12). The solar magnetic activity cycle was present
during this period, and cosmic ray modulation by the heliosphere was
still evident (\cite{McCrackenetal:2004}). The \Beten\ record now
extends to $\sim$10$^5$ years before present, raising the hope that
encounters between the Sun and interstellar clouds can be separated
from solar activity effects, and from the global signature of
geomagnetic pole wandering.
Sunspots have long been controversial as an influence on the
terrestrial climate. Sir William Herschel carefully observed them, and
postulated that diminished solar radiation at Earth during sunspot
maximum affected the terrestrial climate
(1801). \nocite{Herschel:1801} Prof. Langley (1876)
\nocite{Langley:1876} measured the radiative heat from sunspot umbral
and penumbral regions, and concluded the $<$0.1\% solar radiation
decrease associated with sunspots was inadequate to affect the
climate. Climate records show that the Maunder Minimum and other
periods of low solar activity levels have been exceptionally cold,
which implicates high cosmic ray fluxes with cold climate conditions.
Solar activity levels have returned to historic highs in the past few
decades (\cite{CaballeroMcCracken:2004}), and the historic
correlations indicate these high levels also yield warm climate
conditions. Unfortunately, these scientific conclusions also impact
the politically loaded issue of global warming.
The possibility that the cosmos has affected the terrestrial climate
is a longtime source of speculation, with many of the first
discussions focused on explaining the ``Universal Deluge". In 1694
Edmond Halley presented his thoughts to the Royal Society as to
whether the "casual Shock of a Comet, or other transient Body" might
instantly alter the axis orientation or diurnal rotation of the Earth,
thus disturbing sea levels, or whether the impact of a comet could
explain the presence of "vast Quantities of Earth and high Cliffs upon
Beds of Shells, which once were the Bottom of the Sea"
(\cite{Halley:1694}). Halley's speculation has resurfaced in the
hypothesis that the impact of a comet led to the extinction of
dinosaurs 65 Myrs ago at the Cretaceous-Tertiary boundary
(\nolinebreak \cite{Alvarez:1982}). The common sense disclaimer that accompanied
Halley's discussion is timeless: \emph{ ``... the Almighty generally
making us of Natural Means to bring about his Will, I thought it not
amiss to give this Honourable Society an Account of some Thoughts that
occurr'd to me on this Subject; wherein, if I err, I shall find myself
in very good Company.''}
The articles in this volume show firmly that the interaction between
the heliosphere and ISM depends on the detailed boundary conditions
set for the heliosphere by each type of interstellar cloud encountered
by the Sun, and that the galactic environment of the Sun changes over
both geologically short time scales of $< 10^5$ years, and long
time scales of $> 10 ^7$ years. This interaction, in turn, affects
the flux of gas, dust, and energetic particles in the inner
heliosphere.
The discussions in this book also apply to the study of astrospheres
around cool stars, which are expected to have similar properties as
the heliosphere. Is the historical astrosphere of a star a factor in
climate stability for planetary systems? I think so
(\cite{FrischYork:1986}). If so, then the sample of $\sim$100
detected extrasolar planetary systems can be narrowed to those that
are the most likely to harbor technological civilizations by
evaluating the astrosphere characteristics suitable to the space
trajectory of each star (\nolinebreak \cite{Frisch:1993a}).
Astrospheres have now been detected towards $\sim$60\% of the observed cool
stars within 10 pc (\nolinebreak \cite{Woodetal:2005}), and extensive
efforts to detect Earth-sized exoplanets are underway. Perhaps some
day these questions will be answered.
\vspace*{0.1in}
\emph{Acknowledgments:} The author thanks Dr. Clifford Lopate,
of the University of New Hampshire, for providing Figure \ref{fig:2},
and Dr. Lopate thanks NSF Grant 03-39527 for supporting the research
displayed in this figure. The author thanks NASA for supporting her
research, including grants NAG5-13107 and NNG05GD36G. Additional support
has been provided by grants NAG 5-13558 and NAG5-11999. This article
will appear in the book ''Solar Journey: The Significance of Our Galactic Environment
for the Heliosphere and Earth'',
Springer, in press (2006), editor P. C. Frisch.
\begin{table}[h]
\caption[Commonly Used Terms and Acronyms.]{Commonly Used Terms and Acronyms }
\begin{tabular}{ll}
Object & Description \\
\hline
\emph{Interstellar:}& \\
Interstellar Material, ISM & Atoms in the space between stars \\
Local Fluff or CLIC & ISM within $\sim$30 pc, density $10^{-24.3}$ g \cc \\
& CLIC=Cluster of Local Interstellar Clouds \\
Local Interstellar Cloud, LIC & The cloud feeding ISM into the solar system \\
Local Bubble, LB & Nearby ISM with density $< 10 ^{-26.1}$ g \cc\\
& \\
\emph{Heliosphere:}& \\
Solar Wind, SW & Solar plasma expanding to form heliosphere \\
& Density$\sim$5 ions \cc, velocity $\sim$450 km s$^{-1}$ \\
& at Earth \\
Neutral Current Sheet & Thin neutral region separating SW \\
& with opposite magnetic polarities \\
Heliosphere, HS & Region of space containing the solar wind \\
Termination Shock, TS & Shock where solar wind becomes subsonic \\
& TS at $\sim$94 AU on 16 December, 2004 \\
Heliosheath & Subsonic solar wind, outside TS \\
Heliosphere Bow Shock & Shock where LIC becomes subsonic \\
Focusing Cone & Gravitationally focused ISM dust \\
& and helium gas downwind of the Sun\\
& \\
\multicolumn{2}{l}{\emph{Interstellar Products in the Heliosphere:}} \\
Pickup Ions, PUI & Ions from SW-ISM charge exchange \\
Energetic Neutral Atoms & ENAs, Energetic atoms formed by \\
& charge exchange with ions \\
Cosmic Rays: & \\
\hspace{3mm}Anomalous, ACR & Accelerated pickup ions, energy $<$1 GeV \\
\hspace{3mm}Galactic, GCR & From supernova, energy $>$1 GeV at Earth\\
\hline
\end{tabular}
\end{table}
\section*{Postscript: Definitions}
The nine planets of the solar system (including Pluto as a planet)
extend out to 39 AU, compared to the distance of the solar wind
termination shock in the upwind direction of 94 AU.
The Earth is 8.3 light minutes from the Sun, versus the $\sim$0.5 light day
distance to the upwind termination shock of the solar wind. The
ecliptic and galactic planes are tilted with respect to each other by
$\sim$60$^\circ$, and the north ecliptic pole points towards the
galactic coordinates $\ell$=96.4$^\circ$ and $b$=+29.8$^\circ$. This
tilt allows the separation of large scale ecliptic and large scale
galactic phenomena by geometric considerations.
Acronyms are used throughout this
book, and some of these are listed in Table 1.
For those new to this subject, an astronomical unit, AU, is the
distance between the Earth and Sun. A parsec, pc, is 206,000 AU, 3.3
light years (ly), or 3.1 $\times$ 10$^{18}$ cm. For comparison, the
Earth is 8.3 light minutes from the Sun, and the nearest star,
$\alpha$ Cen, is 1.3 pc from the Sun. The planet Pluto is at 39 AU,
versus the 94 AU termination shock distance.
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\emph{Submitted 15 November 2005; accepted 17 November 2005} |
Title:
Two-Current-Sheet Reconnection Model of Interdependent Flare and Coronal Mass Ejection |
Abstract: Time-dependent resistive magnetohydrodynamic simulations are carried out to
study a flux rope eruption caused by magnetic reconnection with implication in
coexistent flare-CME (coronal mass ejection) events. An early result obtained
in a recent analysis of double catastrophe of a flux rope system is used as the
initial condition, in which an isolated flux rope coexists with two current
sheets: a vertical one below and a transverse one above the flux rope. The flux
rope erupts when reconnection takes place in the current sheets, and the flux
rope dynamics depends on the reconnection sequence in the two current sheets.
Three cases are discussed: reconnection occurs (1) simultaneously in the two
current sheets, (2) first in the transverse one and then in the vertical, and
(3) in an order opposite to case 2. Such a two-current-sheet reconnection
exhibits characteristics of both magnetic breakout for CME initiation and
standard flare model. We argue that both breakout-like and tether-cutting
reconnections may be important for CME eruptions and associated surface
activities.
| https://export.arxiv.org/pdf/astro-ph/0601231 |
\title{TWO-CURRENT-SHEET RECONNECTION MODEL OF INTERDEPENDENT FLARE
AND CORONAL MASS EJECTION}
\author{Y. Z. ZHANG\altaffilmark{1}, J. X. WANG\altaffilmark{1} AND Y. Q. HU\altaffilmark{2}
}
\altaffiltext{1}{National Astronomical Observatories, Chinese
Academy of Sciences, Beijing 100012, China}
\altaffiltext{2}{School of Earth and Space Sciences, University of
Science and Technology of China, Hefei 230026, China}
\keywords{Sun: corona $-$ Sun: coronal mass ejections (CMEs) $-$
Sun: flares $-$ Sun: magnetic fields}
\section{INTRODUCTION}
A number of coronal mass ejections (CMEs) showed structures
consistent with the ejection of a magnetic flux rope as it has
been reported by Chen et al. (1997), Wood et al. (1999) and Dere
et al. (1999). Therefore, magnetic flux ropes have been presumed
to be typical structures in the solar corona, and their eruptions
might be closely related to solar flares and CMEs (Forbes, 2000;
Low, 2001). A lot of studies, both analytical and numerical, tried
to explain such eruptive phenomena (Anzer, 1978; Priest, 1988;
Forbes \& Isenberg, 1991; Isenberg et al., 1993; Mikic \& Linker,
1994; Forbes \& Priest, 1995; Low, 1996; Wu et al., 1997;
Antiochos et al., 1999; Chen \& Shibata, 2000; Hu \& Liu, 2000;
Lin \& Forbes, 2000; Amari et al., 2000; Lin et al., 2001, Cheng,
et al., 2005, T\"{o}r\"{o}k \& Kliem, 2005). Most of them were
associated with a bipolar magnetic configuration and assumed that
reconnection in the current sheet below the flux rope triggers the
eruption by the so-called tether cutting of the field lines.
However, observations always show complicated magnetic
configuration and global coupling of different flux systems (see
an example described by Wang et al. (2005) and a statistic
analysis by Zhou et al. (2005)).
Two types of models are popular in the investigation of solar
eruptive phenomena: the standard flare model and the magnetic
breakout model. The standard flare model for magnetic explosions
in eruptive flares was first proposed by Sturrock (1966), and
advanced by a lot of latter studies ( Hirayama, 1974; Heyvaerts et
al., 1977; Sturrock et al., 1984; Shibata et al., 1995; Tsuneta,
1997; Shibata, 1999; Chen \& Shibata, 2000; Moore et al., 2001).
Recently, Chen \& Shibata (2000) proposed an emerging flux trigger
mechanism for CMEs, in which reconnection in the current sheet
below the rope leads to an eruption of the CME and a cusp-shaped
solar flare. All of these studies showed that a cusp structure and
a two ribbon flare occur in the lower corona, and that the
reconnection is tether-cutting at the internal current sheet.
Another type of models is the breakout model (Antiochos et
al.,1999) that involves multipolar topology and requires external
magnetic reconnection to occur on the top of the sheared arcade.
In their model the background field has a spherically symmetric
quadrupolar configuration, rather than a simple bipolar one.
Many observations have shown that CMEs and flares are often two
aspects of the same eruptive event. In a recent study (Zhang, et
al. 2005) we found that a double catastrophe exists for an
isolated flux rope embedded in a quadrupolar background field.
After the first catastrophe, the flux rope levitates in the solar
corona and two current sheets coexist with the rope, a transverse
one above and a vertical one below the rope. As a product of
interaction between the central and overlying arcades, the
transverse current sheet represents the large-scale nature of the
flux system. On the other hand, the vertical current sheet is
limited to the interior of the central arcade and comes from a
local interaction between small-scale bipoles. The coexistence of
the two current sheets differentiates the present magnetic
configuration with either the configuration of magnetic breakout
model, or that of standard flare model. In the absence of
reconnection, the flux rope may levitate in the corona in
equilibrium. The resulting magnetic configuration provides a
pre-eruption magnetic topology for a potential CME and its
associated surface magnetic activity, and meets the requirements
of magnetic breakout and standard flare models. Once reconnection
sets in across one of the two current sheets or both, an eruption
of the flux rope is inevitable, which is presumably responsible
for concurrence of CMEs and flares. To explore this possibility,
we take one of the force-free field solutions obtained by Zhang et
al. (2005) as the initial state, which is located right after the
first catastrophic point, introduce resistive dissipation in the
current sheets, and examine the dynamic evolution of the flux rope
system. The numerical results will show both breakout and
tether-cutting.
We describe the time-dependent, resistive magnetohydrodynamic
(MHD) equations and the solution procedures in section 2. We
discuss the evolution of the flux rope system in section 3, and
conclude our work in section 4.
\section{BASIC EQUATIONS AND SOLUTION PROCEDURES}
We use time-dependent resistive MHD simulations to study the
dynamic evolution of a flux rope system in the presence of
resistance. For 2.5-dimensional (2.5-D) MHD problems in spherical
coordinates (r,$\theta,\varphi$), one may introduce a magnetic
flux function $\psi(t,r,\theta)$ related to the magnetic field by
$$
{\mathbf B} = \bigtriangledown\times \left (
\frac{\psi}{r\sin\theta}\hat{\varphi} \right ) + {\mathbf
B}_\varphi, \ \ \ \ {\mathbf B}_\varphi = B_\varphi \hat{\varphi}
, \eqno (1)
$$
where $B_\varphi$ is the azimuthal component of the magnetic
field. Then the 2.5-D resistive MHD equations are cast in the
following form
$$
\frac{\partial\rho}{\partial t} + \bigtriangledown\cdot(\rho
{\mathbf v})=0, \eqno (2)
$$
$$
\frac{\partial{\mathbf v}}{\partial t} + {\mathbf
v}\cdot\bigtriangledown {\mathbf v} +
\frac{1}{\rho}\bigtriangledown p + \frac{1}{\mu\rho}
[L\psi\bigtriangledown\psi + {\mathbf
B}_\varphi\times(\bigtriangledown \times{\mathbf B}_\varphi )]
$$
$$
+\frac{1}{\mu\rho r\sin\theta}\bigtriangledown\psi
\cdot(\bigtriangledown\times{\mathbf B}_\varphi )\hat{\varphi} +
\frac{GM_{\odot}}{r^2}\hat{r}=0, \eqno (3)
$$
$$
\frac{\partial\psi}{\partial t}+ {\mathbf
v}\cdot\bigtriangledown\psi - {1 \over \mu}\eta r^{2}
\sin^{2}\theta L \psi = 0, \eqno (4)
$$
$$
\frac{\partial B_{\varphi}}{\partial t} +
r\sin\theta\bigtriangledown \cdot \left ( \frac{B_\varphi{\mathbf
v}}{r\sin\theta} \right ) + \left [ \bigtriangledown \psi\times
\bigtriangledown \left ( \frac{v_\varphi}{r\sin\theta} \right )
\right ]_\varphi-\frac{1}{r sin\theta}\bigtriangledown\eta \cdot
\bigtriangledown(\mu r \sin\theta B_{\varphi})
$$
$$
-{1\over \mu}\eta r \sin \theta L(r B_{\varphi} \sin \theta)= 0,
\eqno (5)
$$
$$
\frac{\partial T}{\partial t}+{\mathbf v}\cdot\bigtriangledown T +
(\gamma-1)T\bigtriangledown\cdot{\mathbf
v}-\frac{\gamma-1}{\rho}\eta {\mathbf j}^2 = 0, \eqno (6)
$$
where
$$
L\psi\equiv\frac{1}{r^{2}\sin^{2}\theta} \left (
\frac{\partial^{2}\psi}{\partial
r^{2}}+\frac{1}{r^{2}}\frac{\partial^{2}\psi}{\partial\theta^{2}}-
\frac{\cot\theta}{r^{2}}\frac{\partial\psi}{\partial\theta} \right
) , \eqno (7)
$$
$$
{\mathbf j} = {1\over \mu}\bigtriangledown \times {\mathbf B} =
-{1\over \mu}r\sin\theta L\psi \hat{\varphi} +{1\over
\mu}\bigtriangledown \times (B_\varphi \hat{\varphi}), \eqno (8)
$$
$\rho$ is the density, ${\mathbf v}$ is the flow velocity, $\mu$
is the vacuum magnetic permeability, $G$ is the gravitational
constant, $M_\odot$ is the mass of the Sun, $T$ is the
temperature, $\gamma$ (= 1.05) is the polytropic index, $\eta$ is
the resistivity, and ${\mathbf j}$ is the current density.
The computational domain is taken to be $1 \leq r \leq 30$ in the
unit of $R_\odot$ ($R_\odot$ is the solar radius),
$0\leq\theta\leq \pi/2$, discretized into $130 \times 90$ grid
points. The grid spacing increases according to a geometrical
series of common ratio 1.03 from 0.02 at the base ($r$ = 1) to
0.86 at the top ($r$ = 30), whereas a uniform mesh is adopted in
the $\theta$-direction. The multistep implicit scheme (Hu 1989) is
used to solve equations (2)-(6). As for the boundary conditions,
we use appropriate symmetrical conditions at the pole and equator,
and calculate the quantities at the top in terms of equivalent
extrapolation except for $B_\varphi$ and $\psi$. The magnetic
field is potential above the transverse current sheet that is
below the top boundary. Therefore, $B_\varphi$ is set to be zero
and $\psi$ is calculated from $j_\varphi$ = $-r\sin\theta L\psi$ =
0 at the top (see Hu et al., 2003; Hu, 2004; Zhang et al., 2005).
The initial corona is assumed to be isothermal and static with
$T=T_0=2\times 10^6$ K and $\rho = \rho_0 = 1.67\times 10^{-13}$
kg$\cdot$m$^{-3}$ at the coronal base, where $T_0$ and $\rho_0$
are taken to be the units for temperature and density,
respectively. Taking a characteristic value of 0.01 for $\beta$,
the ratio of gas pressure to magnetic pressure, leads to a
characteristic value of $\psi_0$ = $(2\mu \rho_0 R T_0 R_\odot^4 /
\beta )^{1/2}$ = 5.69$\times 10^{14}$ Wb, taken to be the unit of
$\psi$. Other units of interest are $B_0$ = $\psi_0/R_\odot^2$ =
1.18$\times 10^{-3}$ T for field strength, $v_A$ =
$B_0/(\mu\rho_0)^{1/2}$ = 2570 km$\cdot$s$^{-1}$ for velocity,
$\tau_A$ = $R_\cdot /v_A$ = 271 s for time, and $j_0 = B_0/(\mu
R_\odot )$ = 1.35$\times 10^{-6}$ A$\cdot$m$^{-2}$ for electric
current density.
We choose a force-free field solution as the initial magnetic
field. This solution was obtained by Zhang et al. (2005) right
after the first catastrophic point, characterized by an isolated
flux rope levitating in the corona and accompanied by two current
sheets, a transverse one above and a vertical one below the rope.
The annular magnetic flux per radian is 0.6 in the unit of
$\psi_0$, and the axial magnetic flux is 0.0416 in the unit of
$\psi_0$ for the flux rope, and both of them are conserved during
subsequent dynamic evolutions of the flux rope system. The
magnetic energy of the initial field is 1.71, which is still
larger than the energy of the associated partially open field,
1.662, by 2.9\% (see Zhang, et al.,2005). The excess energy is
obviously in favor of high-speed CMEs.
The initial field chosen above is in equilibrium in the ideal MHD
regime, but will certainly evolve into a dynamic state once
reconnection sets in across the current sheets. The temporal
evolution of the whole system depends on how reconnection occurs
in the two current sheets. Three cases will be treated, labelled
A, B, and C hereinafter, and they differ in the sequence of
reconnection. Reconnection starts simultaneously in the two
current sheets in case A, first in the transverse current sheet
and later on in the vertical one in case B, and in the opposite
order in case C. To control the sequence of reconnection, we
introduce a critical current density for each current sheet,
denoted by $j_t$ for the transverse current sheet and $j_v$ for
the vertical one. When the current density nearby the transverse
current sheet exceeds $j_t$ or that nearby the vertical current
sheet exceeds $j_v$, the resistivity of $\eta$ is set to be 0.01,
and $\eta$ is set to be 0 elsewhere. Consequently, we may simply
set $j_t$ larger than the initial peak current density in the
transverse current sheet to delay reconnection or smaller than the
initial peak current density to start reconnection across the
sheet. Notice that a larger value of $j_t$ just causes a delay of
reconnection, rather than prohibits it. As a mater of fact, the
current density in the transverse current sheet grows with time
during the rope eruption, so it may eventually exceed $j_t$
somewhere, leading to a delayed onset of reconnection in the
sheet. The same is the case for the vertical current sheet. Such
an expedient measure is somewhat artificial but satisfies our
purpose. Through tentative calculations, we find that the initial
peak current density is 5.3 in the transverse current sheet and
22.1 in the vertical current sheet. Consequently, we choose
($j_t$, $j_v$) = (5, 20) for case A, (5, 40) for case B, and (10,
20) for case C.
\section{SIMULATION RESULTS}
As mentioned in the previous section, we intend to discuss three
cases, a simultaneous reconnection in the transverse and vertical
current sheets for case A, a first reconnection in the transverse
current sheet followed by a second in the vertical current sheet
for case B, and a first reconnection in the vertical current sheet
followed by a second in the transverse current sheet for case C.
In each case, we use the height of the rope axis relative to the
solar surface, $h_a$, to mark the position of the flux rope. For
the initial state, we have $h_a$ = 1.70.
In case A, reconnection occurs simultaneously in the transverse
and vertical current sheets. Figures 1a-1c show the magnetic
configuration at three separate times, along with the temperature
distribution in color. Figure 1a corresponds to the initial state,
and resistive dissipation is switched on in both current sheets at
$t$ = 0. Since then, high temperature appears in the current sheet
regions because of reconnection, and the flux rope erupts upward,
as shown in Figures 1b and 1c. The rope is immediately accelerated
without an initial slow rising phase as shown in Figure 2 (solid),
and it gains its maximum eruption speed of 595 km$\cdot$s$^{-1}$
at about $t$ = 5 $\tau_A$, when $h_a$ reaches 2.31 (Figure 1c).
Meanwhile, a cusp-shaped structure with high temperature is
clearly seen in Figure 1c, a typical feature of flares. Also, a
high temperature structure appears in the corona right above the
cusp structure at 1.5 in height.
\placefigure{fig1}
\placefigure{fig2}
In case B, reconnection occurs first in the transverse current
sheet, and then with the growth of the current in the vertical
current sheet, reconnection follows over there. Figures 3a-3c show
the magnetic configuration and temperature distribution at several
separate times. At $t$ = 1 $\tau_A$ when reconnection is initiated
in the transverse current sheet, the temperature along the sheet
rises. As shown by dashed line in Figure 2, the flux rope's speed
increases with time very slowly until reconnection sets in across
the vertical current sheet at about $t$ = 7 $\tau_A$ (Figure 3b).
Then the flux rope undergoes a slight deceleration of short
duration (about 1 $\tau_A$), followed by a quick acceleration. The
rope gains its maximum speed of 670 km$\cdot$s$^{-1}$ at about $t$
= 12 $\tau_A$, when $h_a$ reaches 2.34 (Figure 3c). Similarly, a
cusp-shaped structure with high temperature and a coronal high
temperature structure also appear in this case.
\placefigure{fig3}
In case C, reconnection occurs first in the vertical current
sheet, and then with the growth of the current in the transverse
current sheet, reconnection follows over there. Figures 4a-4c show
the magnetic configuration and temperature distribution at several
separate times. At $t$ = 1 $\tau_A$ when reconnection occurs only
in the vertical current sheet, the temperature along the sheet
rises, as shown in Figure 4a. It can be seen from the dash-dotted
profile in Figure 2 that the flux rope is accelerated before that
time, slightly decelerated afterwards about 2 $\tau_A$ in
duration, and then accelerated again with a much larger
acceleration. The flux rope gains a maximum speed of 568
km$\cdot$s$^{-1}$ at about $t$ = 10 $\tau_A$, when $h_a$ reaches
3.0 (Figure 4c). This case differs from case B in that the
cusp-shaped structure is formed much earlier: it becomes clear as
early as $t$ = 4.7 $\tau_A$ (Figure 4b). And at that time the
reconnection initiates in the transverse current sheet.
\placefigure{fig4}
In summary, magnetic reconnection causes an eruption of the flux
rope and the formation of a cusp-shaped structure of high
temperature in all three cases. The former is presumably a
manifestation of CMEs whereas the latter characterizes a
two-ribbon flare. The reconnection sequence plays a critical role
in the motion of the erupting flux rope and the formation of the
cusp-shaped structure. The reconnection in the transverse current
sheet is apt to produce a gradual acceleration of the flux rope
but a higher peak speed and has little bearing on the formation of
the cusp-shaped structure. On the other hand, the reconnection in
the vertical current sheet is directly responsible for the
formation of the cusp-shaped structure and leads to an immediate
acceleration of the flux rope. It is interesting to note that a
short term deceleration occurs before the rapid acceleration
caused by reconnection across the vertical current sheet, as seen
in cases B and C. Presently we do not know exactly why the flux
rope has such a behavior. A possible reason might be that the
magnetic pressure decreases right beneath the flux rope when
reconnection starts in the vertical current sheet. High resolution
observations at both optical and radio bands show indications that
flux systems shrink first during the impulsive phases of flares,
and then explode later in the main phases of flares (Ji et al.,
2004, Li \& Gan, 2005). This seems to be consistent with the
simulation results of reconnection occurring in the vertical
current sheet in cases B and C. More careful work needs to be done
in order to judge whether this is a common behavior of flux rope
dynamics in the flare impulsive phase. Incidentally, since we have
not considered the background solar wind, the flux rope's speed
decreases after they obtain a peak speed in all three cases.
\section{Concluding Remarks}
Using time-dependent resistive MHD simulations, we find solutions
associated with an isolated coronal flux rope embedded in a
quadrupolar background field and accompanied by a transverse
current sheet above and a vertical current sheet below the rope.
Reconnection may occur in the current sheets either simultaneously
or one after another. The present model agrees with the breakout
model (Antiochos, 1999; Lynch, et al. 2004) if reconnection is
initiated in the transversal current sheet, and it returns to the
standard flare model (Chen \& Shibata, 2000) if reconnection is
initiated in the vertical current sheet. Nevertheless, we argue
that both breakout-like external reconnections and tether-cutting
internal reconnections are essential to the magnetic eruption in
general. Williams et al. (2005) showed observational evidence for
the presence of both tether-cutting and breakout in eruptive
events. Our simulations just combine the two models together,
which is probably more relevant to observations that many eruptive
events occur in background fields of quadrupolar magnetic
configuration (Sterling \& Moore, 2004; Sterling \& Moore, 2004;
Gary \& Moore, 2004).
The present magnetic configuration and the dynamical evolution
shed new light on understanding the relationship between CMEs and
flares, which is a topic with great interest and hot debates. More
and more investigations prefer a closer and rather intrinsic
association between CMEs and surface activities (see Zhang et
al.a,b; Zhou et al. 2003).
Zhang et al. (2001a) reported that the kinematic evolution of CMEs
can be described in a three-phase scenario: the initiation phase,
the impulsive acceleration phase, and the propagation phase.
Furthermore, they found that following the initiation phase, the
CME displays an impulsive acceleration phase, which starts almost
simultaneously with the flare onset time. After the acceleration
phase the CME undergoes a propagation phase. And Zhang et al.
(2001b) found a halo CME that moved slowly in the initial phase,
and was later on accelerated and erupted. This is consistent with
our case B, in which reconnection starts first in the transversal
current sheet, leading to a slow upward motion of the CME, and
subsequently, because of reconnection onset in the vertical
current sheet, the CME acceleration is quickened until it reaches
the maximum speed. In other words, the breakout first occurs and
the tether-cutting follows. However, this is just one possibility,
the other two cases we work out would appear in different
circumstances. Zhou et al.(2003) gave a statistic result that
$59\%$ of the selected 197 halo CMEs initiate earlier than the
flare onset and $41\%$ are preceded by flare onsets. The latter
samples may relate to our case C. Furthermore, Zhang et al.
(2001a) also found one CME that did not show an initiation phase,
but was immediately accelerated to the maximum speed. This example
is very similar to our case A in which reconnection occurs
simultaneously in the two current sheets.
Another point is worthy of mentioning as to the effect of the
reconnection sequence on the maximum speed of CMEs. The flux rope,
identified as the CME here, has the largest speed when
reconnection starts first in the transverse current sheet. On the
other hand, the maximum speed is the lowest when reconnection
starts first in the vertical current sheet. This implies that the
reconnection sequence may affect the maximum speed of CMEs.
\acknowledgments
The authors are greatly indebted to the anonymous referee for
helpful comments and valuable suggestions on the manuscript. One
of the authors (YZZ) thanks J.Y. Ding for kind assistance in
coding and P.F. Chen for helpful discussions. The work is
supported by the National Natural Science Foundation of China
(10233050, 40274049) and the National Key Basic Science Foundation
(TG2000078404).
\clearpage
\clearpage
|
Title:
Radio Linear and Circular Polarization from M81* |
Abstract: We present results from archival Very Large Array (VLA) data and new VLA
observations to investigate the long term behavior of the circular polarization
of M81*, the nuclear radio source in the nearby galaxy M81. We also used the
Berkeley-Illinois-Maryland Association (BIMA) array to observe M81* at 86 and
230 GHz. M81* is unpolarized in the linear sense at a frequency as high as 86
GHz and shows variable circular polarization at a frequency as high as 15 GHz.
The spectrum of the fractional circular polarization is inverted in most of our
observations. The sign of circular polarization is constant over frequency and
time. The absence of linear polarization sets a lower limit to the accretion
rate of $10^{-7} M_\odot y^{-1}$. The polarization properties are strikingly
similar to the properties of Sgr A*, the central radio source in the Milky Way.
This supports the hypothesis that M81* is a scaled up version of Sgr A*. On the
other hand, the broad band total intensity spectrum declines towards milimeter
wavelengths which differs from previous observations of M81* and also from Sgr
A*.
| https://export.arxiv.org/pdf/astro-ph/0601474 |
\newcommand\degd{\ifmmode^{\circ}\!\!\!.\,\else$^{\circ}\!\!\!.\,$\fi}
\newcommand{\rdm}{{\rm\ rad\ m^{-2}}}
\title{Radio Linear and Circular Polarization from M81*}
\author{Andreas Brunthaler\inst{1,2},
Geoffrey C. Bower\inst{3},
Heino Falcke\inst{4,5}
}
\institute{Joint Institute for VLBI in Europe, Postbus 2, 7990 AA
Dwingeloo, The Netherlands
\and
Max-Planck-Institut f\"ur Radioastronomie, Auf dem H\"ugel 69,
53121 Bonn, Germany
\and
Radio Astronomy Laboratory, University of California,
Berkeley, CA 94720, USA
\and
ASTRON, Postbus 2, 7990 AA Dwingeloo, The Netherlands
\and
Department of Astrophysics, Radboud Universiteit
Nijmegen, Postbus 9010, 6500 GL Nijmegen, The Netherlands}
\offprints{brunthal@mpifr-bonn.mpg.de}
\date{Received 30 December 2005 / Accepted 19 January 2006}
\abstract{We present results from archival Very Large Array (VLA) data
and new VLA observations to investigate the long term behavior of
the circular polarization of M81*, the nuclear radio source
in the nearby galaxy M81. We also used the
Berkeley-Illinois-Maryland Association (BIMA) array to observe
M81* at 86 and 230 GHz. M81* is unpolarized in the
linear sense at a frequency as high as 86 GHz
and shows variable circular polarization at a frequency as
high as 15 GHz. The spectrum of the fractional circular
polarization is inverted in most of our observations. The
sign of circular polarization is constant over frequency and time.
The absence of linear polarization sets a lower limit to
the accretion rate of $10^{-7} M_\odot y^{-1}$.
The polarization properties are strikingly similar to the
properties of Sgr A*, the central radio source in the Milky Way.
This supports the hypothesis that M81* is a scaled up version of
Sgr A*. On the other hand, the broad band total intensity spectrum
declines towards milimeter wavelengths which differs from previous
observations of M81* and also from Sgr A*.
\keywords{galaxies: active, galaxies: individual: Messier Number:
M81, polarization
}
}
\section{Introduction}
The nearby spiral galaxy M81 (NGC\,3031) shares many properties with
the Milky Way. It is similar in type, size and mass and it also
contains a nuclear radio source, M81*, that is most likely associated
with a supermassive black hole. M81* has been studied extensively using
Very Long Baseline Interferometry (VLBI) in the
past. \citeN{BietenholzBartelRupen2000} resolved M81* into a stationary
core with a one sided jet. Multi-wavelength
(\citeNP{HoFilippenkoSargent1996}) and sub-millimeter observations
(\citeNP{ReuterLesch1996}) showed many similarities between M81* and
Sgr A*, the central radio source in our Milky Way (\citeNP{MeliaFalcke2001}).
A jet model of Sgr A* has been applied to M81*, where it can reproduce the
radio flux density and the size of the radio core by changing the accretion
rate (\citeNP{Falcke1996}). The sizes of both radio sources show a $\sim 1/\nu$
dependency on the frequency (e.g. \citeNP{BietenholzBartelRupen2004}
for M81*, and \citeNP{BowerFalckeHerrnstein2004} and
\citeNP{ShenLoLiang2005} for Sgr A*).
M81* is an apparent transitional object between Sgr A* and high
luminosity AGN. As the brightest of the nearby low
luminosity AGN (LLAGN), it is 5 orders of magnitude brighter than Sgr A* at
radio wavelengths. M81* is substantially underluminous at X-ray
wavelengths ($L\sim 10^{-5} L_{edd}$), yet not as much as Sgr A*
($L\sim
10^{-10} L_{edd}$). Still, it is the faintest LLAGN we can study.
Furthermore, the polarization properties of M81* and Sgr A* are very similar.
Sgr A* shows circular polarization in absence of linear polarization
(\citeNP{BowerFalckeBacker1999}, \citeNP{BowerBackerZhao1999},
\citeyearNP{BowerWrightBacker1999}) and we detected the same behaviour in M81*
(\citeNP{BrunthalerBowerFalcke2001}.)
The polarization properties of Sgr A* and M81* are in contrast to the
properties of most radio jets in active galactic nuclei where linear
polarization often exceeds circular polarization by a large factor
(e.g.~\citeNP{WardleHomanOjha1998}; \citeNP{RaynerNorrisSault2000}).
The absence of linear polarized emission in Sgr A* can be explained as a
consequence of the accretion flow.
\citeN{BowerFalckeSault2002} investigated the long term behavior of the
circular polarization in Sgr A* from archival VLA
data and showed that {\it i)} the circular polarization is variable on
timescales of days to months, {\it ii)} the sign of the circular polarization
stayed constant over the entire time range of almost 20 years, and {\it iii)}
the average spectrum of circular polarization is inverted.
After the discovery of circular polarization in M81* we used the VLA to
investigate the variability of the circular polarization on short timescales.
We used additional archival VLA data to investigate the long term
behavior of the circular polarization of M81*.
\section{Observations}
\subsection{Archival VLA Data}
M81* and the supernova SN1993J in M81 were observed many
times with the Very Long Baseline Array (VLBA) and the phased Very
Large Array (VLA) over the last decade
(e.g. \citeNP{BartelBietenholzRupen1994};
\citeNP{BietenholzBartelRupen2000}). The observations were typically 16
hours in duration, were made at different frequencies and involved rapid
switching between M81*, SN1993J and scans roughly every hour on the
extragalactic background source 0954+658 for calibration purposes. We used
the VLA data from the observations on 5 November 1993 ($\nu=$ 8.4 and 15 GHz),
16 December 1993 (8.4 and 15 GHz), 29 January 1994 (4.8 and 8.4 GHz),
21 April 1994 (4.8 GHz), 29 August 1994 (4.8 and 8.4 GHz), 31 October 1994
(8.4 GHz), 23 December 1994 (8.4 GHz), and 7 April 1996 (8.4 GHz). The VLA
data had 50 MHz of bandwidth in two sidebands in right (RCP) and left (LCP)
circular polarization modes.
Data reduction was performed with the Astronomical Image Processing
System (AIPS). 3C\,48 was used as primary amplitude calibrator. Then
amplitude and phase self-calibration was performed on 0954+658. This
forces 0954+658 to have zero circular polarization. The amplitude
calibration was transfered to M81* and SN1993J before we performed phase
self-calibration on M81* and SN1993J. Finally, we mapped all three sources
in Stokes I and V. Flux densities were determined by fitting an
elliptical Gaussian component to the sources.
\subsection{New VLA observations}
In addition to the archival VLA data, we used the VLA to observe M81* on 02 March 2001 and 9
dates between 15 June and 30 August 2001. During the latter period,
the minimum and maximum separations between observing dates were 2 and
15 days, respectively. The VLA was in B configuration for the first
observation and C array for the remaining observations. Observations
were made at 5.0, 8.4, 15 and 22 GHz with 50 MHz of bandwidth in two
sidebands in RCP and LCP modes.
Each observation was between two and four hours in duration. Observing
and analysis was performed with AIPS and followed the procedures outlined in
\citeN{BrunthalerBowerFalcke2001}. The extragalactic point source
J1044+719 was used as a phase, amplitude and polarization calibrator as
well as reference pointing source. The extragalactic point source
J1053+704 was used to check for polarization calibration errors . 3C 286 was
used as an absolute amplitude calibration source. Finally, we mapped
all three sources in Stokes I, U, Q, and V. Flux densities were
determined by fitting an elliptical Gaussian component to the sources.
We also used the VLA to observe M81* on 09 August 2003 at 15 GHz, 22 GHz, and
43 GHz in polarimetric mode. The VLA was in A configuration with a resolution
of about 50 milliarcseconds at 43 GHz. The total integration time on M81* was
8, 48, and 52 minutes at 15, 22 and 43 GHz respectively and spread over a
time range of 11 hours. Phase and amplitude calibration were
performed using the nearby compact source, J1056+714. Phase self-calibration
was also performed on M81* to eliminate the effects of atmospheric
decorrelation. Polarization leakage calibration was performed using
simultaneous full track observations of J1056+714 and J1048+701.
\subsection{BIMA observations}
We used the Berkeley-Illinois-Maryland Association (BIMA) array to
observe M81* at 3\,mm wavelength (\citeNP{WelchThorntonPlambeck1996}).
Observations were made in the multiplexed
polarimetric mode described in \citeN{BowerWrightBacker1999}. The receivers
were tuned to sky frequencies of 82.8 GHz (lower sideband) and
86.2 GHz (upper sideband). The compact source 0954+658 was
used for phase calibration. Leakage calibration was determined
from observations of 3C 279 on 11 November 2003. Observations of M81* were
made on 6 dates in September and October 2003 (Table~\ref{High}).
BIMA observations at 230 GHz on M81* were also performed in November 2003 (Table~\ref{High}).
These data were also obtained in polarimetric mode with similar observing
parameters to the 3\,mm observations. Each observation was a 8 hour track.
\subsection{Error Analysis}
The Stokes parameter V is measured as the difference between the left- and
right-handed parallel polarization correlated visibilities. Errors
in circular polarization measurements with the VLA have numerous origins:
thermal noise, gain errors, beam squint, second-order leakage corrections,
unknown calibrator polarization, background noise and radio frequency
interference. The errors caused by amplitude calibration errors, beam
squint, and polarization leakage scale with the source strength, and
therefore the fractional circular polarization is a more relevant
indicator for the detection of circular polarization.
A detailed discussion of these errors is given in
\citeN{BowerFalckeBacker1999} and \citeN{BowerFalckeSault2002}.
We calculated the systematic errors based on the model for the VLA for
circular polarization from \citeN{BowerFalckeSault2002}. For M81*,
SN1993J, and J1053+704 the errors on the fractional circular
polarization in Tables~\ref{all-c} -- \ref{all-u} are separated into
statistical and systematic terms, while for the calibrator sources
0954+658 and J1044+719 only the statistical error is given. The calibrator
sources do not have a systematic error, since their circular polarization was
assumed to be zero during the calibration.
\section{Results}
\subsection{Circular and Linear Polarization}
The results for the archival data at 4.8, 8.4, and 15 GHz are shown in
Tables~\ref{all-c}, \ref{all-x}, and \ref{all-u} respectively. We consider
circular polarization as detected if the measured flux density exceeds the
combined statistical and systematic errors (added in quadrature) by a factor
of three. 0954+658 showed no circular polarization as
expected. M81* showed circular polarization in all observations except
one (16 December 1993 at 15 GHz). SN1993J showed
circular polarization only in one epoch (29 August 1994 at 4.8 GHz)
and no circular polarization in all other observations. The upper
limits on the circular polarization of SN1993J are not very meaningful
at 15 GHz due to the low flux density and unexpected high noise.
The results for the new observations at 4.8, 8.4, and 15 GHz are shown
in Tables~\ref{all-c}, \ref{all-x}, and \ref{all-u} respectively. The
22 GHz data and the high frequency observations on 9 August 2003 gave no useful
limits on the circular polarization, mainly
because the short integration time and higher systematic errors. At
4.8. 8.4, and 15 GHz, 1044+719 showed no circular polarization as
expected. M81* showed circular polarization in three epochs at 4.8 GHz,
all except one epoch at 8.4 GHz and four epochs at 15 GHz. The check
source 1053+704 showed only circular polarization in two epochs at 15
GHz.
The three {\it detections} of circular polarization in the check sources
SN1993J and 1053+704 are caused by either remaining amplitude
calibration errors or by a small level of circular polarization in the
sources. Since both sources show no circular polarization at 8.4 GHz,
the frequency band with the highest sensitivity, it is most likely that
the {\it detected} circular polarization in SN1993J and 1053+704 comes
from residual amplitude calibration errors. The fractional circular
polarization in M81* at 15 GHz is higher by a factor of 2 and 4 than in
1053+704 in the observations on 15 June 2001 and 4 July 2001
respectively. In these two cases, the measured circular polarization of M81*
is probably only partly caused by the amplitude calibration errors.
In the BIMA observations at 86 and 230 GHz neither linear nor circular
polarization is detected for M81* in any individual epoch. Mean linear
polarization at 86 GHz is $1.2$ mJy, or 1.6\% ($3\sigma$). Mean circular
polarization at 86 GHz is $2.8 \pm 0.4$ mJy, or $3.9 \pm 0.5$\%.
Although this is formally a detection, it is not clear
whether systematic errors are significant. Due to the low flux density, limits
on the polarized flux density are not significant at 230 GHz.
The 8.4 GHz data seems to be the most reliable data since M81* showed
circular polarization in all epochs except one while the check sources
SN1993J and 1053+704 never showed circular polarization. The light-curve of
total intensity and fractional circular polarization is shown in
Fig.~\ref{x-light}. The fractional circular polarization shows significant
variability on timescales of a few weeks which is not correlated with the
variability in the total intensity. Between 4 August 2001 and 16 August 2001,
the fractional circular polarization at 8.4 GHz dropped from 0.78\% to less
than 0.35\% while the total intensity showed no significant change.
However it is remarkable that, despite
the strong variability, the sign of the circular polarization is always
positive. At the other two frequencies, the sign is also positive
when circular polarization is detected in M81*.
The spectrum of the fractional circular polarization is inverted
($\alpha > 0$ for $m_{c} \propto \nu^\alpha$) with values between 0.4 and 2
in most epochs when it was detected at more than one frequency. The mean
spectral index between 4.8 and 8.4 GHz is 0.51, while the mean spectral
index between 8.4 and 15 GHz is 1.52. Only the
observation on 18 August 2001 shows a steep spectrum of $\alpha$=-0.77
between 4.8 and 8.4 GHz..
Linear polarization was not detected in the VLA observations on 9 August 2003
and the BIMA observations at 86 GHz (Fig.~\ref{lp}).
The archival data was not searched for linear polarization.
\subsection{Total intensity}
The high frequency VLA data taken on 9 August 2003 give a simultaneous total
intensity spectrum of M81* that shows the flux density decreasing from 15 to
43 GHz with a spectral index of $\sim -0.6$. In the BIMA observations M81* is
clearly detected at 86 GHz in total intensity and is strongly variable on a
time scale $\sim 10$ days. The mean total intensity is 71 mJy. Due to low
sensitivity, M81* is only marginally detected at 230 GHz in total intensity.
The mean flux density for M81* from this observation is $31 \pm 4 \pm 15$ mJy,
where the first error is the statistical error and the second error is the
systematic error due to decorrelation. Atmospheric decorrelation may be
serious and these results may significantly underestimate the total flux
density of M81* (Table~\ref{High}).
Fig.~\ref{bb-spec} shows a non simultaneous broad-band spectrum of M81*. The
15 -- 43 GHz data points are from the VLA observations on 9 August 2003. The
4.8 and 8.4 GHz data points are from the observation on 19 July 2001, where
the 15 GHz flux was comparable to the 15 GHz flux in the 9 August 2003
observation. The 86 GHz and 230 GHz data points are from the BIMA observations
on 7 September 2003 and November 2003 respectively. Also shown are the maximal
and minimal measured values at each frequency in our observations. Although
the spectrum is not simultaneous it is clear that the spectrum declines
towards higher frequencies.
M81* underwent a flare in total intensity during the new VLA observations
between June and August 2001. The peak was reached at 8.4 GHz before 15 June
2001, while the flux density continued to rise at 4.8 until 4 July 2001. At 15
GHz, the peak was reached in 30 June 2001. Fig.~\ref{flare-spec} shows the
spectral indices between 4.8 and 8.4 GHz, and bewteen 8.4 and 15 GHz during
this flare. The spectrum at the lower frequencies shows a smooth transition
from an inverted ($\alpha\sim$ +0.7) to a steep ($\alpha\sim$~-~0.4) spectrum.
At higher frequencies, the spectral index does not follow a trend and is
scattered between -0.4 and +0.5. The fast change in spectral index between
4.8 and 8.4 GHz could be caused by a drop in the turnover frequency of a
syncrotron self-absorpted jet from above 8.4 to below 4.8 GHz and
should be accompanied by a fast expansion of the jet. This
behaviour is known in other active galactic nuclei (e.g. III~Zw~2:
\citeNP{BrunthalerFalckeBower2000}, \citeNP{BrunthalerFalckeBower2005}).
The scatter of the spectral index between 8.4 and 15 GHz could be caused by
multiple sub-flares that occur at 15 GHz.
\section{Discussion}
The origin of circular polarization in AGN is still not known.
Several mechanisms have been proposed in the literature.
Interstellar propagation effects predict a very steep spectrum
(\citeNP{MacquartMelrose2000}) which is not consistent with our observations.
One possible mechanism could be Faraday conversion
(\citeNP{Pacholczyk1977}; \citeNP{JonesODell1977}) of linear polarization to
circular polarization caused by the lowest energy relativistic electrons.
\citeN{BowerFalckeBacker1999} proposed a simple model for Sgr~A* in which
low-energy electrons reduce linear polarization through Faraday de-polarization
and convert linear polarization into circular polarization. Faraday conversion
can also affect the spectral properties of circular polarization and may
lead to a variety of spectral indices, including inverted spectra
(\citeNP{JonesODell1977}). In inhomogeneous sources, conversion can
produce relatively high fractional circular polarization (\citeNP{Jones1988}).
Gyro-synchrotron emission, can also lead to high circular polarization
with an inverted spectrum and low linear polarization (\citeNP{Ramaty1969}).
However, this mechanism is to some degree related and also requires that
M81* and Sgr A* both contain a rather large number of low-energy
electrons. Faraday conversion is also favored by \citeN{BeckertFalcke2002} and
\citeN{RuszkowskiBegelman2002}.
The long-term stability of the sign of circular polarization suggests that the
Faraday conversion is connected to fundamental properties of the source and the
material which is responsible for the conversion. It requires uniformity in the
magnetic pole and accretion conditions over the observed timescales. One
possible scenario is described by \citeN{ensslin2003} where the sign of the
circular polarization is connected to the sense of rotation of the central
engine. In this scenario M81* is expected to rotate counter-clockwise.
The size of the radio emission of M81* at 8.4 GHz is $\sim 0.45$ mas, or
$\sim$ 1800 AU at 4 Mpc (\citeNP{BietenholzBartelRupen1996}). The fact that
M81* is depolarized at at level of $<$ 0.1$\%$ at the same frequency requires
that the depolarizing material is also present at a scale of $\sim$ 1800 AU.
While the polarization properties of M81* and Sgr A* are strikingly similar
the total intensity spectrum seems to be different. In Sgr A*, the radio flux
density rises towards the sub-mm regime (e.g.
\citeNP{ZylkaMezgerWard-Thompson1995}, \citeNP{FalckeMarkoff2000}). Our
measurements indicate a different trend for M81*. Although the spectrum in
Fig.~\ref{bb-spec} is not simultaneous and M81* shows strong variability
our measured flux densities at 86 and 230 GHz are lower than the typical flux
densities at centimeter wavelengths. This is different from the results in
\citeN{ReuterLesch1996} who find an inverted spectrum up to $\sim$ 100 GHz.
We can not tell whether we observed M81* in a phase with unusual low millimeter
emission or the \citeN{ReuterLesch1996} observations were made during an
outburst at millimeter wavelengths. A simultaneous monitoring project from
centimeter to millimeter wavelengths would be needed to decide this question.
Linear polarization is not detected for M81* at any wavelength
longward of 3.6 mm. The presence of a jet at VLBI resolution,
radio synchrotron emission, and circular polarization suggest
that M81* is intrinsically linearly polarized but depolarized
during propagation through a magnetized plasma. The case
is similar to that of Sgr A*, which is detected in linear polarization
only at wavelengths shortward of 3.6 mm. For Sgr A*, the detection
of linear polarization at short wavelengths provides an upper limit
to the rotation measure of a few times $10^6 {\rm\ rad\ m^{-2}}$.
This provides an upper limit to the accretion rate of $\sim 10^{-7}
M_\odot y^{-1}$. For M81*, we find a lower limit to the rotation
measure of $\sim 10^4 \rdm$ for the case of beam depolarization
and $\sim 4\times 10^5 \rdm$ for the case of bandwidth depolarization,
under the assumption that the intrinsic source is polarized. The
lesser value could originate in the dense interstellar medium but
the larger value exceeds that seen anywhere in the ISM. ADAF
and Bondi-Hoyle accretion models will depolarize the source unless
the accretion rate falls below $10^{-9} M_\odot y^{-1}$
(\citeNP{QuataertGruzinov2000}). However, for radiatively
inefficient accretion flows, the larger of the RM limits implies
a lower limit to the accretion rate of $10^{-7} M_\odot y^{-1}$.
The accretion rate necessary for the X-ray luminosity is $10^{-5}
M_\odot y^{-1}$. Since bandwidth depolarization effects decrease
as $\lambda^3$, measurement of the linear polarization at
a wavelength of 0.8 mm would increase the accuracy of the accretion
rate constraint by nearly two orders of magnitude.
\section{Summary \& Conclusion}
We have presented VLA observations of M81* from 1994 until 2002
that show that circular polarization is present at 4.8, 8.4, and 15
GHz in absence of linear polarization. The fractional circular
polarization is variable on timescales of days and months and not
correlated with the total flux density of the source. The sign of the
circular polarization was, if detected, at all frequencies and times always
positive. The polarization properties are strikingly similar to the
properties of Sgr A*, the central radio source in the Milky Way. This
supports the hypothesis that M81* is a scaled up version of Sgr A*.
\citeN{AitkenGreavesChrysostomou2000} and \citeN{BowerWrightFalcke2003}
detected linear polarization at 230 GHz and higher frequencies that also shows
variability (\citeNP{BowerFalckeWright2005}; \citeNP{MarroneMoranZhao2006}).
Given the similarity between M81* and Sgr A* we expect to see also linear
polarization in M81* at higher frequencies.
\begin{acknowledgements}
This research was partially supported by the DFG Priority Programme 1177. The
National Radio Astronomy Observatory is operated by Associated Universities,
Inc., under a cooperative agreement with the National Science Foundation.
\end{acknowledgements}
\bibliography{brunthal_refs}
\bibliographystyle{aa}
\appendix{}
\section{Tables}
\begin{table*}
\begin{center}
\caption{Circularly polarized flux at 4.8 GHz for M81* and calibrators. The errors on the fractional circular polarization are separated into statistical and systematic terms for the target and check source. For the calibrator source only the statistical error is given.\label{all-c}}
\begin{tabular}{rrrrrr}
\hline\hline
Date & Source & I & $P_{c}$ & rms & $m_{c}$\\
& & [mJy] &[mJy] &[mJy] & $[\%]$\\
\hline
28 Jan. 1994 & 0954+658 & 622.2 & $<$ 0.43 &0.11 & $<$ 0.07 $\pm$ 0.02\\
& M81* & 95.0 & 0.36 &0.06 & 0.38 $\pm$ 0.06
$\pm$ 0.04\\
& SN1993J & 77.1 & $<$ 0.17 &0.05 & $<$ 0.22 $\pm$ 0.06
$\pm$ 0.03\\
\hline
21 Apr. 1994 & 0954+658 & 534.7 & $<$ 0.21 &0.07 & $<$ 0.04 $\pm$ 0.01\\
& M81* & 114.1 & 0.52 &0.05 & 0.46 $\pm$ 0.04
$\pm$ 0.03\\
& SN1993J & 62.1 & $<$ 0.22 &0.04 & $<$ 0.35 $\pm$ 0.06
$\pm$ 0.03\\
\hline
29 Aug. 1994 & 0954+658 & 597.2 & $<$ 0.18 &0.06 & $<$ 0.03 $\pm$ 0.01\\
& M81* & 109.0 & 0.21 &0.04 & 0.19 $\pm$ 0.04
$\pm$ 0.03\\
& SN1993J & 54.6 & 0.18 &0.03 & 0.33 $\pm$ 0.05
$\pm$ 0.03\\
\hline
\hline
02 Mar. 2001 & 1044+719 & 1568.2 & 0.11 & 0.11 & $<$ 0.01 $\pm$ 0.01\\
& M81* & 130.5 & 0.44 & 0.10 & 0.34 $\pm$ 0.08 $\pm$0.05\\
& 1053+704 & 388.2 & 0.03 & 0.07 & $<$ 0.01 $\pm$ 0.02 $\pm$0.06\\
\hline
15 Jun. 2001 & 1044+719 & 1710.9 & 0.33 & 0.35 & $<$ 0.02 $\pm$ 0.02 \\
& M81* & 136.7 & 0.45 & 0.11 & 0.33 $\pm$ 0.08 $\pm$0.05\\
& 1053+704 & 490.1 & -0.58 & 0.21 & $<$ 0.12 $\pm$ 0.04 $\pm$0.06\\
\hline
30 Jun. 2001 & 1044+719 & 1645.4 & 0.4 & 0.32 & $<$ 0.02 $\pm$ 0.02\\
& M81* & 140.9 & 0.3 & 0.12 & $<$ 0.21 $\pm$ 0.09 $\pm$0.05\\
& 1053+704 & 497.5 & -1.41 & 0.33 & $<$ 0.28 $\pm$ 0.07 $\pm$0.07\\
\hline
04 Jul. 2001 & 1044+719 & 1687.5 & 0.01 & 0.01 & $<$ 0.01 $\pm$ 0.01\\
& M81* & 142.8 & 0.12 & 0.05 & $<$ 0.09 $\pm$ 0.04 $\pm$0.04\\
& 1053+704 & 484 & 0.01 & 0.01 & $<$ 0.01 $\pm$ 0.01 $\pm$0.05\\
\hline
19 Jul. 2001 & 1044+719 & 1645.3 & 0.08 & 0.24 & $<$ 0.01 $\pm$ 0.05\\
& M81* & 134.9 & 0.46 & 0.19 & $<$ 0.34 $\pm$ 0.14 $\pm$0.07\\
& 1053+704 & 516.8 & 0.12 & 0.12 & $<$ 0.02 $\pm$ 0.02 $\pm$0.10\\
\hline
04 Aug. 2001 & 1044+719 & 1641.8 & 0.1 & 0.31 & $<$ 0.01 $\pm$ 0.02\\
& M81* & 128.6 & 0.07 & 0.07 & $<$ 0.05 $\pm$ 0.05 $\pm$0.07\\
& 1053+704 & 564.3 & 0.14 & 0.14 & $<$ 0.02 $\pm$ 0.02 $\pm$0.10\\
\hline
16 Aug. 2001 & 1044+719 & 1636.5 & 0.06 & 0.23 & $<$ 0.01 $\pm$0.01 \\
& M81* & 122.7 & 0.44 & 0.18 & $<$ 0.36 $\pm$ 0.15 $\pm$0.07\\
& 1053+704 & 524.3 & -0.38 & 0.23 & $<$ 0.07 $\pm$0.04 $\pm$0.10 \\
\hline
18 Aug. 2001 & 1044+719 & 1607.5 & 0.01 & 0.01 & $<$ 0.01 $\pm$ 0.01\\
& M81* & 118.2 & 0.74 & 0.16 & 0.63 $\pm$ 0.14 $\pm$ 0.07\\
& 1053+704 & 532.6 & 0.14 & 0.14 & $<$ 0.03 $\pm$ 0.03 $\pm$ 0.10\\
\hline
25 Aug. 2001 & 1044+719 & 1602.7 & 0.02 & 0.15 & $<$ 0.01 $\pm$ 0.01\\
& M81* & 112.7 & 0.37 & 0.16 & $<$ 0.33 $\pm$ 0.14 $\pm$ 0.07\\
& 1053+704 & 546 & 0.13 & 0.13 & $<$ 0.02 $\pm$ 0.02 $\pm$ 0.10\\
\hline
30 Aug. 2001 & 1044+719 & 1655.7 & 0.13 & 0.13 & $<$ 0.01 $\pm$ 0.01\\
& M81* & 113.9 & 0.23 & 0.11 & $<$ 0.2 $\pm$ 0.10 $\pm$ 0.07\\
& 1053+704 & 548 & -0.08 & 0.14 & $<$ 0.01 $\pm$ 0.03 $\pm$ 0.10\\
\hline
\end{tabular}
\end{center}
\end{table*}
\begin{table*}
\begin{center}
\caption{Circularly polarized flux at 8.4 GHz for M81* and calibrators. The errors on the fractional circular polarization are separated into statistical and systematic terms for the target and check source. For the calibrator source only the statistical error is given.\label{all-x}}
\begin{tabular}{rrrrrr}
\hline\hline
Date & Source & I & $P_{c}$ & rms & $m_{c}$\\
& & [mJy] &[mJy] &[mJy] & $[\%]$\\
\hline
05 Nov. 1993 & 0954+658 & 663.5 & $<$ 0.27 &0.04& $<$ 0.04 $\pm$ 0.01\\
& M81* & 110.4 & 0.59 &0.03& 0.53 $\pm$ 0.03
$\pm$ 0.04\\
& SN1993J & 62.2 & $<$ 0.07 &0.02& $<$ 0.11 $\pm$ 0.03
$\pm$ 0.03\\
\hline
16 Dec. 1993 & 0954+658 & 655.5 & $<$ 0.12 &0.04& $<$ 0.02 $\pm$ 0.01\\
& M81* & 85.7 & 0.23 &0.03& 0.27 $\pm$ 0.04
$\pm$ 0.03\\
& SN1993J & 54.7 & $<$ 0.06 &0.02& $<$ 0.11 $\pm$ 0.04
$\pm$ 0.03\\
\hline
28 Jan. 1994 & 0954+658 & 600.6 & $<$ 0.24 &0.08& $<$ 0.04 $\pm$ 0.01 \\
& M81* & 111.2 & 0.79 &0.04& 0.71 $\pm$ 0.04
$\pm$ 0.04 \\
& SN1993J & 48.9 & $<$ 0.14 &0.03& $<$ 0.29 $\pm$ 0.06
$\pm$ 0.03\\
\hline
29 Aug. 1994 & 0954+658 & 648.1 & $<$ 0.11 &0.04& $<$ 0.02 $\pm$ 0.01\\
& M81* & 102.0 & 0.35 &0.03& 0.34 $\pm$ 0.03
$\pm$ 0.04\\
& SN1993J & 34.5 & $<$ 0.08 &0.03& $<$ 0.23 $\pm$ 0.09
$\pm$ 0.03\\
\hline
31 Oct. 1994 & 0954+658 & 670.9 & $<$ 0.12 &0.04& $<$ 0.02 $\pm$ 0.01\\
& M81* & 117.2 & 0.33 &0.03& 0.28 $\pm$ 0.03
$\pm$ 0.04\\
& SN1993J & 33.5 & $<$ 0.08 &0.03& $<$ 0.24 $\pm$ 0.09
$\pm$ 0.03\\
\hline
23 Dec. 1994 & 0954+658 & 693.7 & $<$ 0.43 &0.14& $<$ 0.06 $\pm$ 0.02\\
& M81* & 76.7 & 0.52 &0.09& 0.68 $\pm$ 0.12
$\pm$ 0.07\\
& SN1993J & 30.4 & $<$ 0.22 &0.07& $<$ 0.72 $\pm$ 0.23
$\pm$ 0.05\\
\hline
07 Apr. 1996 & 0954+658 & 786.3 & $<$ 0.11 &0.04& $<$ 0.01 $\pm$ 0.01\\
& M81* & 165.8 & 1.15 &0.03& 0.69 $\pm$ 0.02
$\pm$ 0.04\\
& SN1993J & 20.5 & $<$ 0.15 &0.05& $<$ 0.73 $\pm$ 0.24
$\pm$ 0.04\\
\hline
\hline
02 Mar. 2001 & 1044+719 & 1478.7 & 0.13 & 0.13 & $<$ 0.01 $\pm$ 0.01\\
& M81* & 143.8 & 0.62& 0.07 & 0.43 $\pm$ 0.05 $\pm$ 0.05\\
& 1053+704 & 524.8 & 0.35 & 0.10 & $<$ 0.07 $\pm$ 0.02 $\pm$ 0.06\\
\hline
15 Jun. 2001 & 1044+719 & 2136.1 & 0.11 & 0.11 & $<$ 0.01 $\pm$ 0.01\\
& M81* & 201.7 & 1.08 & 0.10 & 0.53 $\pm$ 0.05 $\pm$ 0.05\\
& 1053+704 & 1006.1 & 0.17 & 0.17 & $<$ 0.02 $\pm$ 0.02 $\pm$ 0.06\\
\hline
30 Jun. 2001 & 1044+719 & 1488.6 & 0.01 & 0.01 & $<$ 0.01 $\pm$ 0.01\\
& M81* & 193.9 & 0.75 & 0.11 & 0.39 $\pm$ 0.06 $\pm$ 0.05\\
& 1053+704 & 763.0 & 0.34 & 0.19 & $<$ 0.04 $\pm$ 0.02 $\pm$ 0.07\\
\hline
04 Jul. 2001 & 1044+719 & 1525.8 & 0.57 & 0.57 & $<$ 0.04 $\pm$ 0.04\\
& M81* & 175.7 & 0.93 & 0.09 & 0.53 $\pm$ 0.05 $\pm$ 0.05\\
& 1053+704 & 753.3 & 0.13 & 0.13 & $<$ 0.02 $\pm$ 0.02 $\pm$ 0.06\\
\hline
19 Jul. 2001 & 1044+719 & 1474.8 & 0.10 & 0.10 & $<$ 0.01 $\pm$ 0.01\\
& M81* & 135.1 & 0.60 & 0.09 & 0.45 $\pm$ 0.07 $\pm$ 0.05\\
& 1053+704 & 769.2 & 0.27 & 0.13 & $<$ 0.04 $\pm$ 0.02 $\pm$ 0.07\\
\hline
04 Aug. 2001 & 1044+719 & 1485.4 & 0.12 & 0.12 & $<$ 0.01 $\pm$ 0.01\\
& M81* & 107.0 & 0.84 & 0.17 & 0.78 $\pm$ 0.16 $\pm$ 0.05\\
& 1053+704 & 785.1 & 0.57 & 0.26 & $<$ 0.07 $\pm$ 0.03 $\pm$ 0.07\\
\hline
16 Aug. 2001 & 1044+719 & 1469.1 & 0.27 & 0.27 & $<$ 0.02 $\pm$ 0.02\\
& M81* & 101.4 & 0.36 & 0.16 & $<$ 0.35 $\pm$ 0.16 $\pm$ 0.07\\
& 1053+704 & 758.3 & 1.20 & 0.27 & $<$ 0.16 $\pm$ 0.04 $\pm$ 0.10\\
\hline
18 Aug. 2001 & 1044+719 & 1476.8 & 0.41 & 0.65 & $<$ 0.03 $\pm$ 0.03\\
& M81* & 98.0 & 0.40 & 0.09 & 0.41 $\pm$ 0.09 $\pm$ 0.07\\
& 1053+704 & 775.2 & 0.41 & 0.30 & $<$ 0.05 $\pm$ 0.04 $\pm$ 0.10\\
\hline
25 Aug. 2001 & 1044+719 & 1406.9 & 0.11 & 0.11 & $<$ 0.01 $\pm$ 0.01\\
& M81* & 89.2 & 0.32 & 0.09 & 0.36 $\pm$ 0.10 $\pm$ 0.05\\
& 1053+704 & 746.0 & 1.30 & 0.24 & $<$ 0.17 $\pm$ 0.03 $\pm$ 0.07\\
\hline
30 Aug. 2001 & 1044+719 & 1520 & 0.17 & 0.17 & $<$ 0.01 $\pm$ 0.01\\
& M81* & 92.3 & 0.42 & 0.16 & 0.45 $\pm$ 0.17 $\pm$ 0.07\\
& 1053+704 & 789.2 & 0.30 & 0.30 & $<$ 0.04 $\pm$ 0.04 $\pm$ 0.10\\
\hline
\end{tabular}
\end{center}
\end{table*}
\begin{table*}
\begin{center}
\caption{Circularly polarized flux at 15 GHz for M81* and calibrators. The errors on the fractional circular polarization are separated into statistical and systematic terms for the target and check source. For the calibrator source only the statistical error is given.\label{all-u}}
\begin{tabular}{rrrrrr}
\hline\hline
Date & Source & I & $P_{c}$ & rms & $m_{c}$\\
& & [mJy] &[mJy] &[mJy] & $[\%]$\\
\hline
05 Nov. 1993 & 0954+658 & 664.5 & $<$ 0.26 &0.09& $<$ 0.04 $\pm$ 0.01\\
& M81* & 108.2 & 1.14 &0.18& 1.05 $\pm$ 0.17
$\pm$ 0.05\\
& SN1993J & 42.3 & $<$ 4.05 &1.35& $<$ 9.57 $\pm$ 3.19
$\pm$ 0.03\\
\hline
16 Dec. 1993 & 0954+658 & 606.4 & $<$ 0.28 &0.09& $<$ 0.05 $\pm$ 0.01\\
& M81* & 87.2 & $<$ 0.06 &0.19& $<$ 0.07 $\pm$ 0.22
$\pm$ 0.05\\
& SN1993J & 39.0 & $<$ 9.0 &3.32& $<$ 23.0 $\pm$ 8.51
$\pm$ 0.03\\
\hline
\hline
02 Mar. 2001 & 1044+719 & 999.8 & 0.03 & 0.12 & $<$ 0.01 $\pm$ 0.01\\
& M81* & 114.2 & 0.81 & 0.19 & 0.71 $\pm$ 0.17 $\pm$ 0.05\\
& 1053+704 & 485.2 & 0.25 & 0.18 & $<$ 0.05 $\pm$ 0.04 $\pm$ 0.07\\
\hline
15 Jun. 2001 & 1044+719 & 2275 & 0.36 & 0.36 & $<$ 0.02 $\pm$ 0.02\\
& M81* & 188.1 & 2.16 & 0.38 & 1.15 $\pm$ 0.20 $\pm$ 0.06\\
& 1053+704 & 1354.7 & 8.28 & 0.75 & 0.61 $\pm$ 0.06 $\pm$ 0.07\\
\hline
30 Jun. 2001 & 1044+719 & 1669.9 & 0.43 & 0.43 & $<$ 0.03 $\pm$ 0.03\\
& M81* & 257.4 & 3.14 & 0.4 & 1.22 $\pm$ 0.16 $\pm$ 0.07\\
& 1053+704 & 1051.5 & 2.61 & 0.66 & $<$ 0.25 $\pm$ 0.06 $\pm$ 0.09\\
\hline
04 Jul. 2001 & 1044+719 & 1955.8 & 1.67 & 1.67 & $<$ 0.09 $\pm$ 0.09 \\
& M81* & 221.2 & 3.66 & 0.46 & 1.66 $\pm$ 0.21 $\pm$ 0.06\\
& 1053+704 & 1162.1 & 4.9 & 0.88 & 0.42 $\pm$ 0.08 $\pm$ 0.07\\
\hline
19 Jul. 2001 & 1044+719 & 1663.3 & 0.58 & 0.58 & $<$ 0.03 $\pm$ 0.03\\
& M81* & 123.3 & 1.65 & 0.68 & $<$ 1.34 $\pm$ 0.55 $\pm$ 0.09\\
& 1053+704 & 1011.9 & 1.76 & 1.05 & $<$ 0.17 $\pm$ 0.10 $\pm$ 0.12\\
\hline
04 Aug. 2001 & 1044+719 & 1663 & 0.29 & 1.14 & $<$ 0.02 $\pm$ 0.07\\
& M81* & 113.4 & 0.36 & 0.36 & $<$ 0.31 $\pm$ 0.31 $\pm$ 0.09\\
& 1053+704 & 973.4 & 6.42 & 1.92 & $<$ 0.66 $\pm$ 0.20 $\pm$ 0.12\\
\hline
16 Aug. 2001 & 1044+719 & 1876 & 0.51 & 0.51 & $<$ 0.03 $\pm$ 0.03\\
& M81* & 107 & 0.38 & 0.42 & $<$ 0.35 $\pm$ 0.39 $\pm$ 0.09\\
& 1053+704 & 1043.9 & 3.42 & 1.37 & $<$ 0.33 $\pm$ 0.13 $\pm$ 0.12\\
\hline
18 Aug. 2001 & 1044+719 & 1496.6 & 0.46 & 0.98 & $<$ 0.03 $\pm$ 0.07\\
& M81* & 79.3 & 0.2 & 0.2 & $<$ 0.25 $\pm$ 0.25 $\pm$ 0.09\\
& 1053+704 & 830.4 & 2.44 & 0.95 & $<$ 0.29 $\pm$ 0.11 $\pm$ 0.12\\
\hline
25 Aug. 2001 & 1044+719 & 1378 & 0.49 & 0.49 & $<$ 0.04 $\pm$ 0.04\\
& M81* & 82.9 & 0.47 & 0.76 & $<$ 0.56 $\pm$ 0.56 $\pm$ 0.09\\
& 1053+704 & 756.7 & 0.82 & 1.65 & $<$ 0.11 $\pm$ 0.22 $\pm$ 0.12\\
\hline
30 Aug. 2001 & 1044+719 & 1846.3 & 0.76 & 0.76 & $<$ 0.04 $\pm$ 0.04\\
& M81* & 104.7 & 0.54 & 0.53 & $<$ 0.52 $\pm$ 0.52 $\pm$ 0.09\\
& 1053+704 & 988.9 & 0.41 & 0.41 & $<$ 0.04 $\pm$ 0.04 $\pm$ 0.12\\
\hline
\end{tabular}
\end{center}
\end{table*}
\begin{table*}
\begin{center}
\caption{Polarized and total flux density of M81* at high frequencies unsing the VLA (15, 22, and 43 GHz) and BIMA (83, 86, and 230 GHz).~\label{High}}
\begin{tabular}{rccrrrrr}
\hline\hline
Date & Frequency & Sideband & I & Q & U & V & m$_p$\\
&[GHz]&&[mJy] & [mJy] &[mJy] &[mJy] &$[\%]$\\
\hline
09 Aug. 2003 & 15 & & 125.9 $\pm$ 1.5 & $<$ 1.4 & $<$ 1.4 & $<$ 1.4 & $<$ 1.0 \\
& 22 & & 118.2 $\pm$ 1.3 & $<$ 1.1 & $<$ 1.1 & $<$ 1.3 & $<$ 1.0 \\
& 43 & & 66.8 $\pm$ 2.0 & $<$ 3.3 & $<$ 3.3 & $<$ 4.2 & $<$ 4.9 \\
\hline
\hline
07 Sep. 2003 & 83 & lsb & 44.0 $\pm$ 2.6 & -2.0 $\pm$ 2.6 & 5.6 $\pm$ 2.6 & 0.3 $\pm$ 2.6 & \\
& 86 & usb & 41.8 $\pm$ 2.6 & -1.8 $\pm$ 2.6 & -3.4 $\pm$ 2.6 & 3.2 $\pm$ 2.6 & \\
& & avg & 42.9 $\pm$ 1.8 & -1.8 $\pm$ 1.8 & 1.1 $\pm$ 1.8 & 1.8 $\pm$ 1.8 & 5.1 $\pm$ 4.2 \\
\hline
12 Sep. 2003 & 83 & lsb & 89.4 $\pm$ 1.8 & -1.3 $\pm$ 1.8 & -2.1 $\pm$ 1.8 & 5.7 $\pm$ 1.8 & \\
& 86 & usb & 92.4 $\pm$ 1.8 & -0.8 $\pm$ 1.8 & -1.1 $\pm$ 1.8 & 2.5 $\pm$ 1.8 & \\
& & avg & 90.9 $\pm$ 1.3 & -1.1 $\pm$ 1.3 & -1.6 $\pm$ 1.3 & 4.1 $\pm$ 1.3 & 2.1 $\pm$ 1.4 \\
\hline
21 Sep. 2003 & 83 & lsb & 86.4 $\pm$ 1.5 & -0.2 $\pm$ 1.5 & -0.7 $\pm$ 1.5 & 3.8 $\pm$ 1.5 & \\
& 86 & usb & 86.9 $\pm$ 1.5 & -1.3 $\pm$ 1.5 & -0.2 $\pm$ 1.5 & 3.4 $\pm$ 1.5 & \\
& & avg & 86.7 $\pm$ 1.1 & -0.2 $\pm$ 1.1 & -0.5 $\pm$ 1.1 & 3.6 $\pm$ 1.1 & 0.6 $\pm$ 1.3 \\
\hline
06 Oct. 2003 & 83 & lsb & 70.4 $\pm$ 2.0 & 3.5 $\pm$ 2.0 & 0.7 $\pm$ 2.0 & 0.2 $\pm$ 2.0 & \\
& 86 & usb & 72.7 $\pm$ 2.0 & -0.3 $\pm$ 2.0 & 0.3 $\pm$ 2.0 & 3.5 $\pm$ 2.0 & \\
& & avg & 71.6 $\pm$ 1.4 & 1.6 $\pm$ 1.4 & 0.5 $\pm$ 1.4 & 1.8 $\pm$ 1.4 & 2.3 $\pm$ 2.0 \\
\hline
09 Oct. 2003 & 83 & lsb & 46.1 $\pm$ 1.8 & 5.1 $\pm$ 1.8 & -2.1 $\pm$ 1.8 & 1.0 $\pm$ 1.8 & \\
& 86 & usb & 45.1 $\pm$ 1.8 & -3.2 $\pm$ 1.8 & 2.7 $\pm$ 1.8 & 1.5 $\pm$ 1.8 & \\
& & avg & 45.6 $\pm$ 1.3 & 1.0 $\pm$ 1.3 & 0.3 $\pm$ 1.3 & 1.2 $\pm$ 1.3 & 2.2 $\pm$ 2.9 \\
\hline
12 Oct. 2003 & 83 & lsb & 43.9 $\pm$ 1.5 & -2.4 $\pm$ 1.5 & -1.0 $\pm$ 1.5 & 3.8 $\pm$ 1.5 & \\
& 86 & usb & 34.5 $\pm$ 1.5 & 6.1 $\pm$ 1.5 & 4.9 $\pm$ 1.5 &-1.0 $\pm$ 1.5 & \\
& & avg & 39.2 $\pm$ 1.1 & 1.9 $\pm$ 1.1 & 2.0 $\pm$ 1.1 & 1.4 $\pm$ 1.1 & 7.0 $\pm$ 2.8 \\
\hline
\hline
01 Nov. 2003 & 230 & lsb & 33.7 $\pm$ 6 & -6.6 $\pm$ 6 & 1.2 $\pm$ 6 & -4.5 $\pm$ 6 & \\
& & usb & 27.5 $\pm$ 6 & 2.7 $\pm$ 6 & -19.9 $\pm$ 6 & 7.3 $\pm$ 6 & \\
& & avg & 30.6 $\pm$ 4 & -2.0 $\pm$ 4 & -9.4 $\pm$ 4 & 1.4 $\pm$ 4 & 31.4 $\pm$ 13\\
\hline
\end{tabular}
\end{center}
\end{table*}
|
Title:
Theoretical foundations for on-ground tests of LISA PathFinder thermal diagnostics |
Abstract: This paper reports on the methods and results of a theoretical analysis to
design an insulator which must provide a thermally quiet environment to test on
ground delicate temperature sensors and associated electronics. These will fly
on board ESA's LISA PathFinder (LPF) mission as part of the thermal diagnostics
subsystem of the LISA Test-flight Package (LTP). We evaluate the heat transfer
function (in frequency domain) of a central body of good thermal conductivity
surrounded by a layer of a very poorly conducting substrate. This is applied to
assess the materials and dimensions necessary to meet temperature stability
requirements in the metal core, where sensors will be implanted for test. The
analysis is extended to evaluate the losses caused by heat leakage through
connecting wires, linking the sensors with the electronics in a box outside the
insulator. The results indicate that, in spite of the very demanding stability
conditions, a sphere of outer diameter of the order one metre is sufficient.
| https://export.arxiv.org/pdf/gr-qc/0601096 |
\jl{6}
\title[Theoretical foundations for\ldots]{Theoretical foundations for on-ground
tests of \textsl{LISA PathFinder} thermal diagnostics}
\author{A Lobo$^{1,2}$\footnote[3]{To whom correspondence should be
addressed.}, M Nofrarias$^2$, J Ramos-Castro$^3$ and J Sanju\'an$^2$}
\address{$^1$ Institut de Ci\`encies de l'Espai, {\sl CSIC}}
\address{$^2$ Institut d'Estudis Espacials de Catalunya ({\sl IEEC\/}), Edifici
{\sl Nexus}, Gran Capit\`a~2--4, 08034 Barcelona, Spain}
\address{$^3$ Departament d'Enginyeria Electr\`onica, {\sl UPC},
Campus Nord, Edif.\ C4, Jordi Girona 1--3, 08034 Barcelona, Spain
\ead{lobo@ieec.fcr.es}}
\date{\today}
\pacs{04.80.Nn, 95.55.Ym, 04.30.Nk}
\submitto{\CQG}
\section{Introduction
\label{sec.1}}
\lisa Pathfinder (\lpf) is an \esa mission, whose main objective is
to put to test critical parts of \lisa (Laser Interferometer Space
Antenna), the first space borne gravitational wave (GW) observatory
\cite{bender}. The science module on board \lpf is the \lisa Test-flight
Package (\ltp)~\cite{lpfall}, which basically consists in two test masses
in nominally perfect geodesic motion (free fall), and a laser metrology
system, which reports on \emph{residual deviations} of the test masses'
actual motion from the ideal free fall, to a given level of accuracy
\cite{gerhar}.
In order to ensure that the test masses are not deviated from their
geodesic trajectories by external (non-gravitational) agents, a so
called Gravitational Reference System (GRS) is used~\cite{rita}. This
consists in position sensors for the masses which send signals to a set
of micro-thrusters; the latter take care of correcting as necessary the
spacecraft trajectory, so that at least one of the test masses remains
centred relative to the spacecraft at all times. The combination of the
GRS plus the actuators is known as \emph{drag-free} subsystem\footnote{
The term \emph{drag-free} dates back to the early days of space
navigation, when it was used to name a trajectory correction system
designed to compensate for the effect of atmospheric drag on satellites
in low altitude orbits.}.
The \emph{drag-free} is of course a central component of \lisa, and
needs to be operated at extremely demanding levels of accuracy. The
laser metrology system should then be sufficiently precise to measure
relative test mass deviations. The overall level of noise acceptable
for \lisa is defined in terms of rms acceleration spectral density,
and has been set to
\begin{equation}
S_{a,{\rm LISA}}^{1/2}(\omega)\leq 3\!\times\!10^{-15}\,\left[
1 + \left(\frac{\omega/2\pi}{3\ {\rm mHz}}\right)^{\!\!2}\right]\,
{\rm m}\,{\rm s}^{-2}/\sqrt{\rm Hz}
\label{eq.1}
\end{equation}
in the frequency range
$\quad 10^{-4}\,{\rm Hz}\leq\omega/2\pi\leq 10^{-3}\,{\rm Hz}$.
This is equivalent to $S_h^{1/2}$\,$\sim$\,4$\times$10$^{-21}$
Hz$^{-1/2}$, with the same frequency dependence.
Because \lpf is a \emph{technological mission}, aimed to assess the
feasibility of \lisa, its ultimate goal has been relaxed to~\cite{toplev}
\begin{equation}
S_{a,{\rm LPF}}^{1/2}(\omega)\leq 3\!\times\!10^{-14}\,\left[
1 + \left(\frac{\omega/2\pi}{3\ {\rm mHz}}\right)^{\!\!2}\right]\,
{\rm m}\,{\rm s}^{-2}/\sqrt{\rm Hz}
\label{eq.2}
\end{equation}
in the frequency range $1\,{\rm mHz}\leq\omega/2\pi\leq 30\,{\rm mHz}$,
i.e., one order of magnitude less demanding, both in noise amplitude and
in frequency band.
Equation (\ref{eq.2}) gives the \emph{global} noise budget. This is
naturally made up of contributions from different perturbative agents,
such as temperature and magnetic field fluctuations, GRS and interferometer
noise, etc. As a general rule, a requirement on the magnitude of each of
the various perturbing factors is set at a 10\,\% fraction of the total.
In the case of temperature fluctuations, this is equivalent to
\begin{equation}
S_{T}^{1/2}(\omega)\leq 10^{-4}\,{\rm K}/\sqrt{\rm Hz}\ ,
\quad 1\,{\rm mHz}\leq \omega/2\pi \leq 30\,{\rm mHz}
\label{eq.3}
\end{equation}
Because temperature stability is important, a decision has been taken to
place high precision thermometers in several strategic spots across the
\ltp ---as part of what is called \emph{Diagnostics Subsystem}~\cite{lobo}
\footnote{
The Diagnostics Subsystem of the \ltp also includes magnetometric
measurements and a charged particle flux detector.}. Such high precision
temperature measurements will be useful to identify the fraction of the
total system noise which is due to thermal fluctuations only, and this
will in turn provide important debugging information to assess the
performance of the \ltp.
\subsection{Temperature measurements
\label{sec.1-1}}
If the temperature gauges are to be sensitive to fluctuations at the
level given by (\ref{eq.3}) then clearly the entire measuring device
should be less noisy, typically by a factor of~10. This means that
such device, which includes both the sensors \emph{and} the associated
electronics, can generate a maximum level of noise of
\begin{equation}
S_{T, {\rm sensor}}^{1/2}(\omega)\leq 10^{-5}\,{\rm K}/\sqrt{\rm Hz}\ ,
\quad 1\,{\rm mHz}\leq \omega/2\pi \leq 30\,{\rm mHz}
\label{eq.4}
\end{equation}
Research work is currently being conducted at \ieec (Barcelona, Spain)
to identify the appropriate sensors and design the better suited front
end electronics. But the prototype system needs of course to be tested
for compliance with equation~(\ref{eq.4}). Thus, in order to do a
meaningful test, the system must be sufficiently thermally isolated
that the observed fluctuations in the readout data can be attributed
\emph{solely} to sensor noise, rather than to a combination of it with
real ambient temperature fluctuations. This means temperature fluctiations
in the thermomters' placements should again be at least one order of
magnitude below the target sensitivity, equation~(\ref{eq.4}), or
\begin{equation}
S_{T, {\rm testbed}}^{1/2}(\omega)\leq 10^{-6}\,{\rm K}/\sqrt{\rm Hz}\ ,
\quad 1\,{\rm mHz}\leq \omega/2\pi \leq 30\,{\rm mHz}
\label{eq.5}
\end{equation}
It turns out that 10$^{-6}\,{\rm K}/\sqrt{\rm Hz}$ is a truly demanding
temperature stability, orders of magnitude beyond the capabilities of
normal thermally regulated rooms. We thus need to design a specific
thermal insulator to shield the sensors from ambient temperature
fluctuations during the test process.
In the ensuing pages we describe in detail the insulator design. It is
extremely important to stress at this point that the performance of the
insulator, i.e., its ability to screen out ambient temperature fluctuations,
\emph{cannot be checked} experimentally, at least under working thermal
conditions in the laboratory. This is because, by definition, the insulator
is the tool to check the sensing instruments, \emph{not viceversa}: we need
to rely on the results of theoretical argumentation to make a decission on
which is the appropriate thermal insulator for our purposes. An experimental
verification of the model is only thinkable under much more extreme
conditions, where external temperature fluctuations are orders of magnitude
higher than the ones which will be met during the test.
\section{Thermal insulator design concept
\label{sec.2}}
The idea of the insulator design is displayed in figure~\ref{fig.1}:
an interior metal core of good thermal conductivity is surrounded by
a thick layer of a poorly conductive material. The inner block ensures
thermal stability of the sensors attached to it, while the surrounding
substrate efficiently shields it from external temperature fluctuations
in the laboratory ambient. We propose a spherical shape for the sake
of simplicity of the mathematical analysis, even though this will be
eventually changed to cubic in the actual experimental device due to
practical feasibility issues.
\subsection{Mathematical model
\label{sec.2.1}}
The basic assumption of the mathematical analysis we shall present is
that heat flows from the interior of the insulator to the air outside,
and from the latter to the interior of the insulator, only by thermal
\emph{conduction}. This is a very realistic hypothesis in the context
of the experiment, as radiation mechanisms are certainly negligible
and convection should not play any significant role, either, since
the entire body is solid, and temperature fluctuations will be small
at all times anyway.
Let then $T({\bf x},t)$ be the temperature at time $t\/$ of a point
positioned at vector {\bf x} relative to the centre of the sphere.
$T({\bf x},t)$ thus satisfies Fourier's partial differential equation
\cite{carslaw}
\begin{equation}
\rho c_{\rm p}\,\frac{\partial}{\partial t} T({\bf x},t) =
\nabla\cdot\left[\kappa\nabla T({\bf x},t)\right]
\label{eq.6}
\end{equation}
where $\rho$, $c_{\rm p}$ and $\kappa$ are the density, specific heat
and thermal conductivity, respectively, of the substrate. We shall
assume these are uniform values within each of the two materials making
up the insulating body, with abrupt changes in the interface. We can
thus represent them as discontinuous functions of the radial coordinate,
as follows:
\begin{equation}
\rho, c_{\rm p}, \kappa({\bf x}) = \left\{\begin{array}{ll}
\rho_1, c_{{\rm p}1}, \kappa_1 \quad & {\rm if}\ \ 0\leq r < a_1 \\
\rho_2, c_{{\rm p}2}, \kappa_2 \quad & {\rm if}\ \ a_1\leq r < a_2
\end{array}\right.
\label{eq.7}
\end{equation}
with $r\/$\,$\equiv$\,$|{\bf x}|$. Initial and boundary conditions are
the following:
\begin{equation}
T({\bf x},t=0) = 0\ ,\quad T(r=a_2,t) = T_0(\theta,\varphi;t)
\label{eq.8}
\end{equation}
where $\theta\/$ and $\varphi\/$ are spherical angles which define positions
on the sphere's surface. The boundary temperature can be expediently
expressed as a multipole expansion:
\begin{equation}
T_0(\theta,\varphi;t) = \sum_{lm}\,b_{lm}(t)\,Y_{lm}(\theta,\varphi)
\label{eq.9}
\end{equation}
where $Y_{lm}(\theta,\varphi)$ are spherical harmonics, and $b_{lm}(t)$
are boundary multipole temperature components.
In practice, the boundary temperature will be \emph{randomly fluctuating},
therefore $b_{lm}(t)$ will be considered \emph{stochastic} functions of
time. We shall also reasonably assume them to be \emph{stationary Gaussian}
noise processes with known spectral densities, $S_{lm}(\omega)$.
As shown in the appendix, the frequency analysis of this problem leads to
a \emph{transfer function} expression of the temperature inside the body:
\begin{equation}
\tilde T({\bf x},\omega) =
\sum_{lm}\,H_{lm}({\bf x},\omega)\,\tilde b_{lm}(\omega)
\label{eq.10}
\end{equation}
where \emph{tildes} (\,$\tilde{}$\,) stand for Fourier transforms, e.g.,
\begin{equation}
\tilde T({\bf x},\omega)\equiv\int_{-\infty}^\infty\,
T({\bf x},t)\,e^{-i\omega t}\,dt
\label{eq.11}
\end{equation}
etc. If we make the further assumption that different multipole temperature
fluctuations at the boundary are \emph{uncorrelated}, i.e.,
\begin{equation}
\langle\tilde b^*_{l'm'}(\omega)\,\tilde b_{lm}(\omega)\rangle
= S_{lm}(\omega)\,\delta_{l'l}\,\delta_{m'm}
\label{eq.12b}
\end{equation}
then the spectral density of fluctuations at any given point inside the
insulating body is given by
\begin{equation}
S_T({\bf x},\omega) =
\sum_{lm}\,\left|H_{lm}({\bf x},\omega)\right|^2\,
S_{lm}(\omega)
\label{eq.12}
\end{equation}
It is ultimately the spectral density $S_T({\bf x},\omega)$ which has to
comply with the requirement expressed by equation~(\ref{eq.4}). Based on
knowledge (by direct measurement) of ambient laboratory temperature
fluctuations, equation~(\ref{eq.12}) will provide the guidelines, as
regards materials and dimensions, for the actual design of a suitable
insulator jig.
\section{Homogeneous boundary conditions
\label{sec.3}}
Thermal conditions in the laboratory are rather \emph{homogeneous}. This
means that the boundary temperature fluctuations will be in practice
essentially independent of the angles $\theta\/$ and $\varphi$, i.e.,
\begin{equation}
T_0(\theta,\varphi;t) = B(t)
\label{eq.13}
\end{equation}
and consequently the generic expansion equation~(\ref{eq.9}) includes
only the \emph{monopole} term, hence
\begin{equation}
b_{00}(t) = \sqrt{4\pi}\,B(t)
\label{eq.14}
\end{equation}
The temperature $T({\bf x},\omega)$ in this case will only depend on
radial depth, $r$, therefore,
\begin{equation}
\tilde T(r,\omega) = H(r,\omega)\,\tilde B(\omega)
\label{eq.15}
\end{equation}
with $H(r,\omega)$\,$\equiv$\,$\sqrt{4\pi}\,H_{00}({\bf x},\omega)$.
According to equation~(\ref{eq.a13}) of the Appendix, this is
\begin{equation}
\hspace*{-0.6 cm}
H(r,\omega) = \left\{\begin{array}{ll}
\xi_0(\omega)\,j_0(\gamma_1r)\ , &
0\leq r \leq a_1 \\[1 em]
\eta_0(\omega)\,j_0(\gamma_2r) +
\zeta_0(\omega)\,y_0(\gamma_2r)\ , &
a_1\leq r \leq a_2 \end{array}\right.
\label{eq.16}
\end{equation}
This is a low-pass filter transfer function ---even though the cumbersome
frequency dependencies involved in the expressions above do not make it
immediately obvious. A plot of the square modulus of $H(r,\omega)$ is
shown in figure~\ref{fig.2} for $r\/$\,=\,0 (red curve). The figure also
shows a low-pass filter of the first order with the same frequency cut-off,
$|H_{\rm 1st\ order}(\omega)|^2$\,=\,$(1+\omega^2\tau^2)^{-1}$, for conceptual
comparison (blue curve).
The most salient feature emerging out of the plot is the stronger drop
in $H(r,\omega)$ at the high frequency tails. The latter can be easily
assessed in quantitative detail, and the result is
\begin{equation}
|H(0,\omega)|\sim\omega\tau\,e^{-\sqrt{\omega\tau}}
\label{eq.19}
\end{equation}
where $\tau$ is the filter's time constant ---a complicated function of
the insulator's physical and geometric properties, to be discussed below.
As already mentioned in the Introduction section, to test the temperature
sensors and electronics we need a very strong noise suppression factor in
the \ltp frequency band. A look at figure~\ref{fig.2} readily shows that
high damping factors require such frequency band to lie in the filter's
tails. The thermal insulator should therefore be designed in such a way
that its time constant $\tau\/$ be sufficiently large to ensure that the
\ltp frequencies are high enough compared to 1/$\tau\/$. The exponential
drop in the transfer function shown by equation~(\ref{eq.19}) makes the
filter actually feasible with reasonable dimensions.
\section{Numerical analysis
\label{sec.4}}
In this section we consider the application of the above formalism to
obtain practically useful numbers for the actual implementation of a
real insulator device which complies with the needs of our experiment.
First of all, a selection of an \emph{aluminum} core surrounded by a
layer of \emph{polyurethane} was made. Aluminum is a good heat conductor
and is easy to work with in the laboratory; polyurethane is a good
insulator and is also convenient to handle, as it can be foamed to any
desired shape from canned liquid. Other alternatives are certainly
possible, but this appears sufficiently good and we shall therefore
only make reference to this specific one.
The relevant physical properties of aluminium and polyurethane are
specified in table~\ref{tab.1}.
\begin{table}[h!]
\begin{center}
\begin{tabular}{lccc}
& Density
& Specific heat
& Thermal conductivity \\
& $\rho$ (kg\,m$^{-3}$)
& $c_{\rm p}$ (J\,kg$^{-1}$\,K$^{-1}$)
& $\kappa$ (W\,m$^{-1}$\,K$^{-1}$) \\[1ex]
\hline \\[-1.3ex]
{\sf Aluminum} & 2700 & 900 & 250 \\
{\sf Polyurethane} & 35 & 1000 & 0.04
\end{tabular}
\caption{Density, specific heat and thermal conductivity of aluminium
and polyurethane. Units are given in the International System.
\label{tab.1}}
\end{center}
\end{table}
Figure \ref{fig.3} plots the \emph{amplitude damping coefficient} of
the insulator block, $|H(r,\omega)|$, at the lower end of the \ltp
frequency band, i.e., 1 mHz, and at the interface position,
$r\/$\,=\,$a_1$. Each of the curves corresponds to a fixed value of
the latter, and is represented as a function of the outer radius of
the insulator. This choice is useful because the sensors are implanted
for test on the surface of the aluminium core, and also because at
higher frequencies thermal damping is stronger. So in practice the
actual damping power of the device will be the one plotted, and better
at the higher frequencies in the measuring bandwidth. The figure clearly
shows that the assymptotic regime of equation~(\ref{eq.19}) is quite
early established.
The choice of dimensions for the insulating body must of course ensure
that the minimum requirement, equation~(\ref{eq.5}) is met. For this,
a primary consideration is the size of the ambient temperature
fluctuations in the site where the experiment is made. Dedicated
measurements in our laboratory showed that
\begin{equation}
S_{T, {\rm ambient}}^{1/2}(\omega)\sim 10^{-1}\,{\rm K}/\sqrt{\rm Hz}\ ,
\quad 1\,{\rm mHz}\leq \omega/2\pi \leq 30\,{\rm mHz}
\label{eq.20}
\end{equation}
We therefore need to implement a device such that
$|H(a_1,\omega)|$\,$\leq$\,10$^{-5}$ throughout the measuring bandwidth
(MBW). Suitable dimensions can then be readily read off figure~\ref{fig.3},
and various alternatives are possible, as seen. Before making a decission,
however, we need to make an additional estimate of the heat leakage
down the electric wires which connect the temperature sensors with
the elctronics, which lies of course outside the insulator. We come
to this next.
\subsection{Heat leakage through connecting wires}
We use a simple model, consisting in assuming the connecting wires
behave as straight metallic rods which connect the central aluminum
core with the electronics, placed in the external laboratory ambient.
Because the polyurethane provides a very stable insulation, we can
neglect the lateral flux, hence only a unidimensional heat flow
needs to be considered. For this, the following equation relates
the heat flux to the temperature difference between the two wires'
edges:
\begin{equation}
\dot{Q}(t) = \kappa_{\rm wire}\,\frac{\pi R_{\rm wire}^2}{\ell_{\rm wire}}\;
[T(a_2,t)-T(a_1,t)]
\label{eq.21}
\end{equation}
where $\kappa_{\rm wire}$ is the thermal conductivity of the wire,
$R_{\rm wire}$ its transverse radius, and $\ell_{\rm wire}$ its length
\emph{inside} the polyurethane layer.
On the other hand, the heat flux results in temperature variations in
the metal core, given by
\begin{equation}
\dot{Q}(t) = \rho_1c_{{\rm p}1}V_1\,
\frac{\partial T}{\partial t}(a_1,t)
\label{eq.22}
\end{equation}
where $V_1$\,=\,$4\pi a_1^3/3$ is the volume of the metal core. Equating
the above expressions we find
\begin{equation}
\kappa_{\rm wire}\,\frac{\pi R_{\rm wire}^2}{\ell_{\rm wire}}\;
[T(a_2,t)-T(a_1,t)] =
\rho_1c_{{\rm p}1}V_1\, \frac{\partial T}{\partial t}(a_1,t)
\label{eq.23}
\end{equation}
For fluctuating temperatures, we can now obtain the relationship between
the spectral density at the aluminium core and the ambient, due to heat
conduction along the wire:
\begin{equation}
S_{T,{\rm wire}}^{1/2}(a_1,\omega) = |H_{\rm wire}(\omega)|\,
S_{T,{\rm ambient}}^{1/2}(\omega)
\label{eq.24}
\end{equation}
where
\begin{equation}
|H_{\rm wire}(\omega)|\simeq\frac{\pi}{\omega}\,
\frac{\kappa_{\rm wire}\,R_{\rm wire}^2}
{\rho_1c_{{\rm p}1}V_1\,\ell_{\rm wire}}
\label{eq.25}
\end{equation}
and where the approximation has been made that the temperature fluctuations
at the inner end of the wire are much smaller than those at the outer
end, due to the presence of the polyurethane layer.
In practice, there will be several sensors for test inside the insulator.
Under the hypothesis made that no lateral heat flux is relevant, the transfer
function for a bundle of $N\/$ of wires is, at most, $N\/$ times that of a
single wire. Thus,
\begin{equation}
|H_{N{\rm wires}}(\omega)| = \frac{3N}{\omega/2\pi}\,
\frac{\kappa_{\rm wire}\,R_{\rm wire}^2}
{8\pi\rho_1c_{{\rm p}1}a_1^3\,\ell_{\rm wire}}
\label{eq.26}
\end{equation}
Let us consider numerical values in this expression. We use thin
copper wires ($\kappa_{\rm Cu}$\,=\,401\,Wm$^{-1}$K$^{-1}$) of
radius $R_{\rm wire}$\,=\,0.1\,mm, and assume some fiducial
parameters for the size of the aluminium core, $a_1$, the wire
length, $\ell_{\rm wire}$, the number of connecting wires, $N$,
and the frequency, $\omega/2\pi$. The following obtains:
\begin{equation}
\hspace*{-1.2 cm}
|H_{N{\rm wires}}(\omega)| = 1.1\times 10^{-5}\,
\left(\frac{N}{30}\right)\!
\left(\frac{a_1}{13\ {\rm cm}}\right)^{\!-3}\!
\left(\frac{\ell_{\rm wire}}{25\ {\rm cm}}\right)^{\!-1}\!
\left(\frac{\omega/2\pi}{1\ {\rm mHz}}\right)^{\!-1}
\label{eq.27}
\end{equation}
This result indicates that, for laboratory fluctuations in the level of
equation~(\ref{eq.20}), leakage through wiring causes fluctuations in
the temperature sensors of about 10$^{-6}$\,K/$\sqrt{\rm Hz}$,
equation~\eref{eq.24}, which is compliant with the requirement of
stability of equation~\eref{eq.5}. The most sensitive parameter in
the above expression is the size of the metal core, and this determines
the need to make it somewhat large. The length of the wires has been
taken to be 25~cm, but this does not necessarily mean we need
$a_2$\,=\,38~cm (assuming the radius of the aluminum core is
$a_1$\,=\,13~cm), because the wires can be partly wound inside the
polyurethane layer to further protect the system against leakage.
In fact, this wire lengthening is an easy way to improve attenuation.
As regards frequency dependence, compliance is guaranteed in the entire
MBW if it is at its lower end: indeed, not only $|H_{\rm wire}(\omega)|$
decreases as $\omega^{-1}$, also ambient noise fluctuations drop below
10$^{-1}$\,K/$\sqrt{\rm Hz}$ at higher frequencies.
\section{Conclusions}
Temperature fluctuation measurement is very demanding in the \ltp, and
subsequently \lisa, as reflected by equation~\eref{eq.4}. Accordingly,
very delicate sensor and associated electronics must be designed, and
of course tested in ground before boarding.
However, even the best laboratory conditions are orders of magnitude
worse than the above requirement, so meaningful tests of the temperature
sensing system cannot be tested without suitably screening the sensors
from ambient temperature fluctuations. We have addressed how this can
be accomplished by means of an insulating system consisting of a central
metallic core surrounded by a thick layer of a very poorly conducting
material. The latter provides good thermal insulation, while the central
core, having a large thermal inertia, ensures stability of the sensors'
environemnt. The choice of materials is flexible, so aluminium and
polyurethane, which are easily available in the market, has been
adopted. Thereafter, the dimensions need to be fixed.
The appropriate sensors for the needs are temperature sensitive resistors,
more specifically thermistors ---also known as NTCs. It appears that,
because these sensors need to be wired to external electronics, heat leakage
through such wires is an effect which needs to be quantitatively assessed
to prevent losses. We have analysed this problem, and concluded that it
strongly depends on the central metallic core size, and imposes that it
be somewhat large.
Laboratory ambient temperature fluctuations, determined by dedicated
\emph{in situ} measurements, are of the order of 10$^{-1}$\,K/$\sqrt{\rm Hz}$
at 1~mHz, and dropping at higher frequencies within the MBW. The required
stability conditions at the sensors, attached at the core's surface, thus
need an attenuation factor of 10$^{-5}$, or better. Our analysis determines
that a central aluminium core of 13~cm of radius, surrounded by a concentric
layer of polyurethane 15--20~cm thick, comfortably provides the needed
thermal screening which guarantees a meaningful test of the sensors'
performance.
The results of this paper are based on modelling. Because our aim is to
produce a very stable thermal environment for the temperature sensors,
we cannot check \emph{experimentally} the correctness of our conclusions.
We must instead rely on the validity of the hypotheses made ---essentially
that heat only flows by thermal conduction--- and on the underlying physical
laws which govern heat conduction. Even though there is good reason to
believe that both are sufficiently accurate, unexpected behaviour e.g. at
the interface between the metal core and the insulator, may partly distort
the results. Direct measurements with very large temperature gradients
applied across the insulating device are envisaged, and will be reported
elsewhere as an auxiliary independent test of the model.
\ack
We want to thank Albert Tom\`as, from {\sl NTE}, for discussions on
the subject of this paper. Support for this work came from Project
ESP2004-01647 of Plan Nacional del Espacio of the Spanish Ministry
of Education and Science (MEC). MN acknowledges a grant from Generalitat
de Catalunya, and JS a grant from MEC.
\appendix
\section{Thermal insulator frequency response functions
\label{sec.a1}}
Here we present some mathematical details of the solution to the Fourier
problem, equations~(\ref{eq.6})-(\ref{eq.9}). We first of all Fourier
transform equations~(\ref{eq.6}) and (\ref{eq.9}):
\begin{equation}
i\omega\,\rho c_{\rm p}\,\tilde T({\bf x},\omega) =
\nabla\cdot\left[\kappa\nabla\tilde T({\bf x},\omega)\right]
\label{eq.a1}
\end{equation}
\begin{equation}
\tilde T_0(\theta,\varphi;\omega) =
\sum_{l=0}^\infty\sum_{m=-l}^l\,\tilde b_{lm}(\omega)\,Y_{lm}(\theta,\varphi)
\label{eq.a2}
\end{equation}
Equation (\ref{eq.a1}) can be recast in the form
\begin{equation}
\left(\nabla^2 + \gamma_1^2\right)\,\tilde T({\bf x},\omega) = 0\ ,
\quad 0\leq r\leq a_1
\label{eq.a3a}
\end{equation}
\begin{equation}
\left(\nabla^2 + \gamma_2^2\right)\,\tilde T({\bf x},\omega) = 0\ ,
\quad a_1\leq r\leq a_2
\label{eq.a3b}
\end{equation}
where $r\/$\,$\equiv$\,$|{\bf x}|$, and
\begin{equation}
\gamma_1^2\equiv -i\omega\,\frac{\rho_1 c_{\rm p,1}}{\kappa_1}\ ,\quad
\gamma_2^2\equiv -i\omega\,\frac{\rho_2 c_{\rm p,2}}{\kappa_2}
\label{eq.a4}
\end{equation}
To these, matching conditions at the interface\footnote{
The temperature and the \emph{heat flux} should be continuous across
the interface.}
and boundary conditions must be added:
\begin{equation}
\tilde T(r=a_1-0,\omega) = \tilde T(r=a_1+0,\omega)
\label{eq.a7a}
\end{equation}
\begin{equation}
\kappa_1\,\frac{\partial \tilde T}{\partial r}(r=a_1-0,\omega) =
\kappa_2\,\frac{\partial \tilde T}{\partial r}(r=a_1+0,\omega)
\label{eq.a7b}
\end{equation}
\begin{equation}
\tilde T(r=a_2,\omega) = \tilde T_0(\theta,\varphi;\omega)
\label{eq.a7c}
\end{equation}
Equations (\ref{eq.a3a}) and (\ref{eq.a3b}) are of the Helmholtz kind. Their
solutions are thus respectively given by
\begin{equation}
\hspace*{-2.25 cm}
\tilde T({\bf x},\omega) = \left\{\begin{array}{ll}
\displaystyle
\sum_{lm}\,A_{lm}(\omega)\,j_l(\gamma_1r)\,Y_{lm}(\theta,\varphi)\ , &
0\leq r \leq a_1 \\[1.7 em]
\displaystyle
\sum_{lm}\,\left[C_{lm}(\omega)\,j_l(\gamma_2r) +
D_{lm}(\omega)\,y_l(\gamma_2r)\,\right]\,
Y_{lm}(\theta,\varphi)\ , &
a_1\leq r \leq a_2 \end{array}\right.
\label{eq.a5}
\end{equation}
where $j_l\/$ and $y_l\/$ are spherical Bessel functions \cite{as72},
\begin{equation}
\hspace*{-0.8 cm}
j_l(z) = z^l\,\left(-\frac{1}{z}\,\frac{d}{dz}\right)^{\!\!l}\,
\frac{\sin z}{z}\ ,\quad
y_l(z) = -z^l\,\left(-\frac{1}{z}\,\frac{d}{dz}\right)^{\!\!l}\,
\frac{\cos z}{z}
\label{eq.a6}
\end{equation}
and the coefficients $A_{lm}(\omega)$, $C_{lm}(\omega)$ and $D_{lm}(\omega)$
are to be determined by equations~(\ref{eq.a7a})--(\ref{eq.a7c}). These can
be expanded as follows, respectively:
\begin{eqnarray}
\sum_{lm}\,A_{lm}(\omega)\,j_l(\gamma_1a_1)\,Y_{lm}(\theta,\varphi)\ =
& & \nonumber \\
\ \ =\ \sum_{lm}\,\left[C_{lm}(\omega)\,j_l(\gamma_2a_1) +
D_{lm}(\omega)\,y_l(\gamma_2a_1)\,\right]\,
Y_{lm}(\theta,\varphi) & &
\label{eq.a8a}
\end{eqnarray}
\begin{eqnarray}
\kappa_1\gamma_1\,
\sum_{lm}\,A_{lm}(\omega)\,j'_l(\gamma_1a_1)\,Y_{lm}(\theta,\varphi)\ =
& & \nonumber \\
\ \ =\ \kappa_2\gamma_2\,
\sum_{lm}\,\left[C_{lm}(\omega)\,j'_l(\gamma_2a_1) +
D_{lm}(\omega)\,y'_l(\gamma_2a_1)\,\right]\,
Y_{lm}(\theta,\varphi)
\label{eq.a8b}
\end{eqnarray}
\begin{eqnarray}
\sum_{lm}\,\left[C_{lm}(\omega)\,j_l(\gamma_2a_2) +
D_{lm}(\omega)\,y_l(\gamma_2a_2)\,\right]\,
Y_{lm}(\theta,\varphi)\ =
& & \nonumber \\
\ \ =\ \sum_{lm}\,\tilde b_{lm}(\omega)\,Y_{lm}(\theta,\varphi)
\label{eq.a8c}
\end{eqnarray}
Because of the completeness property of the spherical harmonics, the
above equations completely determine the coefficients $A_{lm}(\omega)$,
$C_{lm}(\omega)$ and $D_{lm}(\omega)$. The result is
\begin{equation}
\hspace*{-2 cm}
A_{lm}(\omega) = \xi_l(\omega)\,\tilde b_{lm}(\omega)\ ,\ \
C_{lm}(\omega) = \eta_l(\omega)\,\tilde b_{lm}(\omega)\ ,\ \
D_{lm}(\omega) = \zeta_l(\omega)\,\tilde b_{lm}(\omega)
\label{eq.a9}
\end{equation}
with
\begin{equation}
\hspace*{-1 cm}
\xi_l(\omega) = \frac{1}{\Delta_l(\omega)}\,\left[
\kappa_2\gamma_2\,j_l(\gamma_2a_1)\,y'_l(\gamma_2a_1) -
\kappa_2\gamma_2\,j'_l(\gamma_2a_1)\,y_l(\gamma_2a_1)\right]
\label{eq.a9a}
\end{equation}
\begin{equation}
\hspace*{-1 cm}
\eta_l(\omega) = \frac{1}{\Delta_l(\omega)}\,\left[
\kappa_2\gamma_2\,j_l(\gamma_1a_1)\,y'_l(\gamma_2a_1) -
\kappa_1\gamma_1\,j'_l(\gamma_1a_1)\,y_l(\gamma_2a_1)\right]
\label{eq.a9b}
\end{equation}
\begin{equation}
\hspace*{-1. cm}
\zeta_l(\omega) = \frac{1}{\Delta_l(\omega)}\,\left[
\kappa_1\gamma_1\,j_l(\gamma_2a_1)\,j'_l(\gamma_1a_1) -
\kappa_2\gamma_2\,j'_l(\gamma_2a_1)\,j_l(\gamma_1a_1)\right]
\label{eq.a9c}
\end{equation}
and
\begin{eqnarray}
\hspace*{-1 cm}
\Delta_l(\omega) & = &
\ \kappa_1\gamma_1\,j'_l(\gamma_1a_1)\,\left[
j_l(\gamma_2a_1)\,y_l(\gamma_2a_2) -
j_l(\gamma_2a_2)\,y_l(\gamma_2a_1)\right]\ + \nonumber \\
& + &
\ \kappa_2\gamma_2\,j_l(\gamma_1a_1)\,\left[
j_l(\gamma_2a_2)\,y'_l(\gamma_2a_1) -
j'_l(\gamma_2a_1)\,y_l(\gamma_2a_2)\right]
\label{eq.a10}
\end{eqnarray}
When the above results, equations~(\ref{eq.a9a}) through (\ref{eq.a10}),
are inserted back into equation~(\ref{eq.a5}) the result stated in
equation~(\ref{eq.10}) in the main text obtains, i.e.,
\begin{equation}
\tilde T({\bf x},\omega) =
\sum_{lm}\,H_{lm}({\bf x},\omega)\,\tilde b_{lm}(\omega)
\label{eq.a11}
\end{equation}
where
\begin{equation}
\hspace*{-1.8 cm}
H_{lm}({\bf x},\omega) = \left\{\begin{array}{ll}
\xi_l(\omega)\,j_l(\gamma_1r)\,Y_{lm}(\theta,\varphi)\ , &
0\leq r \leq a_1 \\[1 em]
\left[\eta_l(\omega)\,j_l(\gamma_2r) +
\zeta_l(\omega)\,y_l(\gamma_2r)\,\right]\,
Y_{lm}(\theta,\varphi)\ , &
a_1\leq r \leq a_2 \end{array}\right.
\label{eq.a12}
\end{equation}
For monopole only boundary conditions, equation~(\ref{eq.15}), the
transfer function is
\begin{equation}
\hspace*{-0.6 cm}
H(r,\omega) = \left\{\begin{array}{ll}
\xi_0(\omega)\,j_0(\gamma_1r)\ , &
0\leq r \leq a_1 \\[1 em]
\eta_0(\omega)\,j_0(\gamma_2r) +
\zeta_0(\omega)\,y_0(\gamma_2r)\ , &
a_1\leq r \leq a_2 \end{array}\right.
\label{eq.a13}
\end{equation}
|
Title:
The X-ray properties of young radio-loud AGN |
Abstract: We present XMM-Newton observations of a complete sample of five archetypal
young radio-loud AGN, also known Gigahertz Peaked Spectrum (GPS) sources. They
are among the brightest and best studied GPS/CSO sources in the sky, with radio
powers in the range L_{5GHz}=10^{43-44} erg/s and with 4 sources having
measured kinematic ages of 570 to 3000 yrs. All sources are detected, and have
2-10 keV luminosities from 0.5 to 4.8x10^{44} erg/s. In comparison with the
general population of radio galaxies, we find that: 1) GPS galaxies show a a
range in absorption column densities similar to other radio galaxies. We
therefore find no evidence that GPS galaxies reside in significantly more dense
circumnuclear environment, such that they could be hampered in their expansion.
2) The ratio of radio to X-ray luminosity is significantly higher than for
classical radio sources. This is consistent with an evolution scenario in which
young radio sources are more efficient radio emitters than large extended
objects at a constant accretion power. 3) Taking the X-ray luminosity of radio
sources as a measure of ionisation power, we find that GPS galaxies are
significantly underluminous in their [OIII]_{5007 Angstrom}, including a weak
trend with age. This is consistent with the fact that the Stroemgren sphere
should still be expanding in these young objects. This would mean that here we
are witnessing the birth of the narrow line region of radio-loud AGN.
| https://export.arxiv.org/pdf/astro-ph/0601141 |
\date{}
\pagerange{\pageref{firstpage}--\pageref{lastpage}} \pubyear{2002}
\label{firstpage}
\newcommand{\apj}{{ApJ}}
\newcommand{\apjs}{{ApJS}}
\newcommand{\apjl}{{ApJ}}
\newcommand{\aj}{{AJ}}
\newcommand{\aap}{{A\&A}}
\newcommand{\aaps}{{A\&AS}}
\newcommand{\nat}{{Nat}}
\newcommand{\jetp}{{JETP}}
\newcommand{\mnras}{{MNRAS}}
\newcommand{\phrvl}{{PhRvL}}
\newcommand{\phrc}{{PhRvC}}
\newcommand{\prc}{{PhRvC}}
\newcommand{\araa}{{ARA\&A}}
\newcommand{\pasj}{{PASJ}}
\newcommand{\pasp}{{PASP}}
\newcommand{\npa}{{NuPhA}}
\newcommand{\iaucirc}{{IAU circ.}}
\newcommand{\aplett}{{Astrophysical Letters}}
\newcommand{\rvmp}{{\it Rev. Mod. Physics}}
\newcommand{\xmm}{{\it XMM-Newton}}
\newcommand{\chandra}{{\it Chandra}}
\newcommand{\asca}{{\it ASCA}}
\newcommand{\rosat}{{\it ROSAT}}
\newcommand{\einstein}{{\it Einstein}}
\newcommand{\cangeroo}{{\it CANGEROO}}
\newcommand{\whipple}{{\it Whipple}}
\newcommand{\hegra}{{\it HEGRA}}
\newcommand{\hess}{{\it HESS}}
\newcommand{\smm}{{\it SMM}}
\newcommand{\sax}{{\it BeppoSAX}}
\newcommand{\rxte}{{\it RXTE}}
\newcommand{\osse}{{\it OSSE}}
\newcommand{\egret}{{\it CGRO-EGRET}}
\newcommand{\integr}{{\it INTEGRAL}}
\newcommand{\glast}{{\it GLAST}}
\newcommand{\comptel}{{\it COMPTEL}}
\newcommand{\cgro}{{\it CGRO}}
\newcommand{\xspec}{{\it xspec}}
\newcommand{\oiii}{\hbox{[O\,III]}}
\newcommand{\msun}{{$M_{\odot}$}}
\newcommand{\nh}{{$N_{\rm H}$}}
\newcommand{\ep}{{e$^+$e$^-$}}
\newcommand{\fluxunit}{{ph\,cm$^{-2}$s$^{-1}$}}
\newcommand{\kms}{{km\,s$^{-1}$}}
\newcommand{\ndot}{{\dot{N}_{UV}}}
\newcommand{\NH}{{N_{\rm H}}}
\newcommand{\loiii}{{L_{\rm [O III]}}}
\begin{keywords}
galaxies: active -- X-ray: galaxies
\end{keywords}
\begin{table*}
\centering
\begin{minipage}{\textwidth}
\caption{The sample of GPS/CSO radio sources with $|b|>20$\degr\ from
the \citet{pearson88} catalogue.
Indicated are, (column 1) the coordinates,
(column 2) redshift, $z$,
(column 3) radio flux density at 5~GHz, $S_{5 \rm GHz}$ (erg s$^{-1}$),
(column 4) radio luminosity at 5~GHz, $L_{5\rm GHz}$\ (erg s$^{-1}$),
(column 5) the 5007~\AA \oiii\ emission line luminosity,
$L_{\rm [O III]}$\ (erg s$^{-1}$) from \citet{lawrence96},
and (column 6) the kinematic age of the radio hot spots
\citep[][except for B1358+624]{polatidis03}
\label{tab-sample}}
\begin{tabular}{@{}lllcccc@{}}\hline
Source & Position & $z$ & $S_{5 \rm GHz}$\footnote{Obtained from the NASA/IPAC Extragalactic Database (NED)}& $\log L_{5\rm GHz}$ &
$\log L_{\rm [O III]}$ & Kinematic Age\\
& (J2000) & & Jy & & & yr\\
\hline
B0108+388 & $01^h11^m37.3^s$\ +39\degr 06\arcmin 28\arcsec & 0.668 & 1.6 & 44.0 &40.8 & $570\pm50$\\
B0710+439 & $07^h13^m38.1^s$\ +43\degr 49\arcmin 17\arcsec & 0.518 & 1.6 & 43.7 & 42.4 & $930\pm100$\\
B1031+567 & $10^h35^m07.0^s$\ +56\degr 28\arcmin 47\arcsec & 0.45 & 1.3 & 43.4 & 41.6 & $1800\pm600$\\
B1358+624 & $14^h00^m28.6^s$\ +62\degr 10\arcmin 39\arcsec & 0.431 & 1.8 & 43.5 & 41.8 & $2400\pm1000$
\footnote{
The kinematic age of B1358+624 has not been directly measured, but is based on
on its size and the average size-age
relation of GPS/CSO sources in \citet{polatidis03}.}
\\
B2352+495 & $23^h55^m09.4^s$\ +49\degr 50\arcmin 08\arcsec & 0.238 & 1.5 & 43.0 & 41.3 & $3000\pm750$\\
\hline
\end{tabular}
\end{minipage}
\end{table*}
\section{Introduction}
Ever since their discovery, it has been speculated that those compact radio
sources that show convex-shaped radio spectra at cm wavelengths,
may be young objects \citep{shklovsky65,blake70}. As a class, they were
named Gigahertz Peaked Spectrum (GPS) sources after their characteristic
radio spectrum \citep[see][for a review]{odea98},
most likely caused by synchrotron self absorption \citep[e.g.]{fanti90,snellen00a}.
High resolution Very Long Baseline Interferometry
(VLBI) observations have shown that these sources are
typically up to a few hundred parsec in size, often
exhibiting jet and/or lobe structures on two opposite sides from their central
core $-$ the reason why they are also called Compact Symmetric Objects
\citep[CSO; eg.][]{Wilkinson94}. The most compelling evidence that GPS/CSO
are indeed young radio sources comes from VLBI monitoring observations, showing
that the bright archetypal objects in this class have
hot-spot propagation velocities of $\sim$0.1-0.2c
\citep{owsianik98a,owsianik98b,tschager00},
indicating kinematic ages of $\sim$10$^{3}$ years.
This is in contrast to speculations that
GPS/CSO galaxies are small due to confinement by a particularly dense and
clumpy interstellar medium (ISM) that impedes the outward propagation of
the jets \citep{vanbreugel84,odea91}.
Note that a large fraction of GPS sources, in particular those showing a
convex radio spectrum peaking at higher frequencies than a few GHz,
turn out to be identified with high redshift quasars (z$\sim$2-3).
Their connection with the population of GPS/CSO galaxies at lower redshift
is not clear, and they may well be a completely separate class of object
which just also happen to exhibit a convex shaped spectrum \citep{snellen99}.
By no means it has been established that the GPS quasars
may also represent a young stage of radio source evolution.
Here we present \xmm\ X-ray observations of a small
but complete sample of all five
GPS sources from the \citet{pearson88}
sample with Galactic latitude $|b| > 20$\degr\ (Table~\ref{tab-sample}).
These are among the brightest GPS/CSO galaxies in the sky.
All these sources appear to be young radio loud AGN with
four out of five sources having measured
kinematic ages of their hot spots, indicating ages of up to $\sim3000$~yr.
No kinematic age measurement for B1358+624 exists, but based on its size and the
age-size measurements of GPS/CSO sources \citep{polatidis03}
we estimate its age to be 2400 yr.
Although the sample is limited in size,
it allows us to study the correlations between their
X-ray, optical and radio properties, and to assess how they
compare to those of mature radio galaxies.
One expects that the X-ray luminosity is largely a manifestation of the
instantaneous accretion power of the central black hole,
whereas the radio luminosity is expected to evolve
substantially over the life time of the source \citep[e.g.]{readhead96,snellen00a}.
Moreover, X-ray absorption measurements are an excellent means to
probe the circum nuclear density of the GPS galaxies.
Several papers on X-ray observations of GPS sources have appeared in the
literature, but one should be cautious to interpret these results in
terms of X-ray properties of young radio-loud AGN.
A first success was obtained by \citep{odea00} with ASCA.
Although they did not detect B2352+495, they obtained a firm detection of
GPS galaxy B1345+125 (PKS1345+125).
Furthermore, \citet{guainazzi04} detected B1404+288
(Mkn 668).\footnote{Prior to the submission of this publication we learned that
Guainazzi et al. have detected a number of other GPS/CSO galaxies in X-rays,
which do not overlap with our sample \citep{guainazzi05}}
Both are low redshift GPS galaxies exhibiting strong optical
line emission and are powerful infrared emitters, and may be not representative
to the class of young radio-loud AGN (yet no reliable ages have been measured
for these sources). The combined X-ray and radio observations of the
GPS {\em quasar}
B0738+313 presented by \citet{siemiginowska03b} show that, with its
prominent kpc-scale X-ray/radio jet,
it is certainly not a young radio-loud AGN.
In order to easily compare our results for GPS/CSO galaxies with the X-ray
properties
of a large sample of AGN by \citet{sambruna99} we adapt here a cosmology with
$H_0 = 75$~km s$^{-1}$ Mpc$^{-1}$ and $q_0 = 0.5$.
\begin{table}
\centering
\caption{Log of the \xmm\ observations. The MOS exposure time
and event rates
refer to the average value for MOS1 and MOS2.\label{tab-obs}}
{%
\begin{tabular}{@{}lcccc@{}}\hline
Source & Observation ID & Start Date & Exposures & Event rates \\
& & & (MOS/PN) & (MOS/PN)\\
& & d/m/y & ks & ct s$^{-1}$\\
\hline
B0108+388 & 0202520101 & 09/01/2004 & 16.4/12.0 & 1.6/13.6\\
B0710+439 & 0202520201 & 22/01/2004 & 14.2/11.2 & 3.2/26.6\\
B1031+567 & 0202520301 & 21/10/2004 & 22.2/12.8 & 34.4/93.0\\
B1358+624 & 0202520401 & 14/04/2004 & 12.4/12.0 & 23.2/120.6\\
B2352+495 & 0202520501 & 25/12/2004 & 15.8/12.8 & 3.7/28.5\\
\hline
\end{tabular}
}
NOTE -- The event rates refer to the total detector count rates,
not the source count rates.
\end{table}
\begin{table*}
\begin{center}
\begin{minipage}{\textwidth}
\caption{Observational properties and parameters obtained by modeling the observed X-ray spectra in the range 0.5-10~keV.\label{tab-res}}
\begin{tabular}{@{}llllll@{}}\hline\noalign{\smallskip}
& B0108+388 & B0710+439 & B1031+567 & B1358+624 &B2352+495 \\
\noalign{\smallskip}\hline
\noalign{\smallskip}
PN 0.5-10~keV source count rate ($10^{-3}$~cts\,s$^{-1}$) & $4.7\pm0.9$ & $89.5\pm3.0$ &$12.6\pm2.0$ & $44.3\pm2.7$ & $8.6\pm1.2$\\
MOS1+2 0.5-10~keV source count rate ($10^{-3}$~cts\,s$^{-1}$) & $0.71\pm0.25$ & $33.0\pm1.1$ & $4.3\pm0.5$ & $15.8\pm0.9$ & $3.4\pm0.4$\\
\noalign{\smallskip}
Galactic \nh\
($10^{20}$cm$^{-2}$) & 5.80 & 8.11 & 0.56 & 1.96 &12.4\\
\noalign{\smallskip}
Normalization \footnote{Statistical errors correspond to $\Delta C=1$ (68\% confidence limits).}
($10^{-5}$ph s$^{-1}$keV$^{-1}$ cm$^{-2}$@ 1 keV) & $3.3\pm1.1$\footnote{
The 3$\sigma$ lower limit is $1.1\times10^{-5}$ph s$^{-1}$keV$^{-1}$ cm$^{-2}$.
The source is detected at the $7\sigma$ level.}
&$8.1\pm0.6$ & $1.2\pm0.2$ & $6.1\pm1.6$ & $1.1\pm0.2$\\
Power law slope
\footnote{Brackets indicate that the power
law slope was fixed to this value.}
($\Gamma$) & (1.75) & $1.59\pm0.06$ & (1.75) & $1.24\pm0.17$ & (1.75) \\
Intrinsic \nh\ ($10^{22}$cm$^{-2} $) & $57\pm20$\footnote{
The 3$\sigma$ lower limit is $1.8\times10^{23}$~cm$^{-2}$.}
& $0.44\pm 0.08$ & $0.50\pm0.18$ & $3.0\pm0.7$ & $0.66\pm0.27$\\
\noalign{\smallskip}
Flux (2-10 keV) \footnote{Including absorption.}
($10^{-13}$~erg\, s$^{-1}$cm$^{-2}$) & 0.50 & 4.0 & 0.51 & 4.8 & 0.41\\
Luminosity (2-10 keV)\footnote{Calculated for rest frame energies, ignoring absorption.}
($10^{44}$~erg\, s$^{-1}$) & 1.18 & 2.16 & 0.22 & 1.67 & 0.046\\
C-statistic/bins &137.5/99 & 497.4/401 & 104.2/97 & 122.9/99 & 126.3/99 \\
\noalign{\smallskip}
\hline
\end{tabular}
\end{minipage}
\end{center}
\end{table*}
\section{Observations, spectral analysis and results}
\xmm\ \citep{jansen01} observed the five GPS/CSO sources as part of its guest
observation program from January to December 2004 (Table~\ref{tab-obs}).
All observations were made with the ``Thin1'' optical blocking filter.
For the data reduction we used the standard \xmm\ software package SAS v6.0.0.
Unfortunately several observations were plagued by a high particle background
(see Table~\ref{tab-obs}).
In the case of {B0108+388}, {B0710+439}, {B2325+495}
we removed time intervals with a high background
count rate, using cut off rates of 15.5~ct\,s$^{-1}$ and 2.5~ct\,s$^{-1}$
for resp. PN and MOS.
For {B1031+567} and {B1358+624}
the high background persisted throughout the
observation, and we simply used all available data.
The observation {B2325+495} had an intermediate background activity,
so we selected time intervals with $< 40$~ct\,s$^{-1}$ and $< 5$~ct\,s$^{-1}$
for PN and MOS.
For spectral extraction we used circular extraction regions
with radii of 15\arcsec\ for {B1031+567} and {B1358+624}, and
25\arcsec\ for the other three sources.
Background spectra were obtained from rectangular regions near the source
position, but excluding regions around 35\arcsec\ of the source.
We extracted spectra for the two MOS CCD cameras \citep{turner01}, and the PN
camera \citep{strueder01}. For each source we combined the spectra of MOS1 and
MOS2, into one spectrum, which we analysed using averaged
instrumental response matrices.
The potential systematic
error introduced is small compared to the statistical
errors, given the fact that the MOS1 and MOS2 are virtually
identical instruments with similar instrumental response functions.
All the five sources of the sample are detected.
For the spectral analysis we
employed a simple model consisting of a power law continuum
and two absorption components: One represents the
Galactic absorption, with an absorption column, \nh, fixed
to the Galactic value of
\citet{dickey90}\footnote{We extracted the absorption
columns from the online ``\nh-tool'', \\
\url{http://heasarc.gsfc.nasa.gov/Tools}}.
The other
absorption component corresponds to the absorption column
intrinsic to the host galaxy. The redshift of this component was fixed
to that of the galaxy, but the absorption column density was a free parameter.
The spectral analysis was done with the spectral fitting program
{\it xspec} \citep{xspec}, using the absorption models of
\citet{wilms00} (called {\it tbabs} and {\it ztbabs} in \xspec).
The slope of the power law continuum was a free parameter for
the high signal to noise spectra of {B0710+439} and {B1358+624},
but fixed to 1.75 for the other three sources whose spectra are
statistically more limited.
The best fit parameters, together with the inferred intrinsic
X-ray luminosities between 2-10~keV are listed in
Table~\ref{tab-res}.
The spectra and best fit models are shown in Fig.~\ref{spectra1}.
\section{Interpretation}
As we are interested in whether GPS/CSO galaxies are different from
other radio-loud AGN we compare their X-ray properties
to those of the sample of radio-loud AGN whose X-ray properties were
determined by \citet{sambruna99} from ASCA observations.
We use all the sources of their broad line radio galaxies (BLRG),
narror line radio galaxies (NLRG) and radio galaxies (RG) subsamples.
Since \citet{sambruna99} do not quote upper limits for the non-detected
intrinsic X-ray asborption components, we estimate upper limits proportional
to the observed flux taking into account the errors on the detected instrinsic
absorption components.
In Fig.~\ref{correlations} we have set out the radio and \oiii\
luminosities of the galaxies in our sample to their X-ray luminosities,
and we compare them to the \citet{sambruna99} sample.
In Fig.~\ref{fig-nhd} we show the
intrinsic absorption column density distribution.
These figures reveal three important properties of our sample of
GPS/CSO galaxies:
1) for their X-ray emission GPS/CSO galaxies are relatively radio-loud;
2) their
\oiii\ emission is relatively low; 3) the column density distribution is
similar to those of radio-loud AGN classified by \citet{sambruna99} as
narrow line radio galaxies (NLRGs) and radio galaxies (RGs),
but the absorption is on average higher than those of broad line
radio galaxies.
As we discuss below these three properties support the
hypothesis that GPS/CSO galaxies are indeed young radio-galaxies. {%
Note that there are strong indications that
GPS radio galaxies have relatively
low [OIII] lumininosities with respect to their radio luminosities as compared
to compact steep spectrum (CSS) sources \citep{odea98}.
This is probably the same trend
between radio, [OIII], and X-ray luminosity that we report here,
but without the X-ray luminosity as intermediary quantity.
As noted in the
introduction, the X-ray luminosity is the quantity that
is probably the best indicator for the intrinsic power of the AGN.
}
\begin{table}
\caption{A comparison of the X-ray derived absorption column $\NH$
and radio absorption column measurements $N_{\rm HI}$ \citep{pihlstroem03}.
\label{tab-nh}}
\centering
\begin{tabular}{@{}lcr@{}}\hline
&$\log \NH$ &$\log N_{\rm HI}$ \\\hline
B0108+388 & 23.8 & 21.9 \\
B1031+567 & 21.7 & $<20.1$ \\
B1358+624 & 22.5 & 20.3\\
B2352+495 & 21.8& 20.5 \\\hline
\end{tabular}
\end{table}
\subsection{The absorbing column density}
\label{sec-nh}
An alternative explanation for the small extent of the radio
jets in GPS/CSO galaxies that is
still often considered in the literature \citep{odea98} is that the
radio jets are quenched by a high density in the vicinity of the nucleus,
in other words they are ``frustrated radio sources''.
It is clear from Table~\ref{tab-res} and Fig.~\ref{fig-nhd}
that this is unlikely to be the case, as the intrinsic X-ray absorption
is similar to other radio-loud AGN, with column densities
ranging from a relatively
modest $4\times10^{21}$~cm$^{-2}$\ to a considerably large
$6\times10^{23}$~cm$^{-2}$.
A similar conclusion was drawn by \citet{pihlstroem03} based on HI
radio absorption observations of a large sample of compact radio sources,
and by \citep{odea05} from upper limits to the molecular gas content in
GPS sources.
Note that the X-ray absorption gives more stringent constraints on the actual
gas column densities than HI and molecular absorption densities,
as X-ray absorption depends on the total column
toward the central source, whereas radio observations only
probe the neutral fraction of the gas.
This is quite an important distinction since AGN are expected to
create an extended ionised region, as we discuss in section~\ref{sec-oiii}.
Furthermore, X-ray absorption
probes the gas toward the accretion disk, whereas the radio absorption probes
the neutral gas toward the radio source, which is situated at larger radii.
It is therefore not surprising that for the four galaxies in our sample
for which also HI absorption measurements have been made the HI
column is always one to two orders of magnitude lower than the X-ray
absorption column (Table~\ref{tab-nh} and Fig.~\ref{fig-nh}).
Attributing the difference solely to ionisation effects would mean ionisation
fractions of 90\% to 99\%.
However, the ionisation fractions are likely to be lower,
because a substantial part of the X-ray absorption may
occur inside the central 100~pc, which is not probed by
absorption toward the radio hot spots.
\citet{pihlstroem03} found a strong anti-correlation between the linear
size of the radio emission and the HI column density, which they use to probe
the average density profile of the interstellar medium.
They do not consider ionisation effects, whereas this could be
an additional cause for the observed anti-correlation:
small jets are associated with young AGN, which do therefore not yet have an
extended narrow line emission region of ionised gas (see section~\ref{sec-oiii}).
If this is the case it makes it less straightforward
to derive an average interstellar medium density profile from the
relation between $N_{\rm HI}$\ and linear size of the radio emission,
since the growth of the emission line region also depends on the
density and UV luminosity of the AGN.
\subsection{The optical line emission:
the case for an expanding emission line region}
\label{sec-oiii}
Usually a high [O III] luminosity is taken as an indication for the
presence of a powerful AGN, but Fig.~\ref{correlations} indicates
that GPS/CSO galaxies are relatively underluminous in [O III] compared to
their radio or X-ray flux.
This may not be too surprising if one takes into account that
these are young radio galaxies, in which the AGN has
switched on only a few thousand years ago.
The reason is that it takes time to establish a large emission line region
by photo-ionisation.
This is best illustrated by a simple calculation, for which we
assume an average interstellar medium
density of 1~cm$^{-3}$\ and a typical luminosities of
$\log L_{\rm \oiii} = 42$, and $\log L_{X} = 44$.
The number rate of ionising photons (i.e. $> 13.6$ eV) is
$\ndot \sim 10^{54}$ ph s$^{-1}$\ for a power law spectrum with photon
index -1.75.
Comparing this to the total number of hydrogen atoms within a
typical region of 5~kpc radius \citep{baum89b}, one finds that
$\sim 10^{67}$ atoms have to be ionised.
In other words the central source has to shine for at least
$\sim 10^{67}/10^{54} = 10^{13}$~s, or $\sim$300,000 yr before it
has completely ionized a region with a radius of 5~kpc.
This means that if an AGN has become only recently active
it must be surrounded by a small, but rapidly expanding ionisation nebula
\citep[see][for a calculation concerning the first generation of quasars]{white03}.
Hence, this would imply that in GPS/CSO galaxies we
see the birth of the narrow line region of radio-loud AGN.
In order to see whether this is the reason that
GPS/CSO galaxies are relatively underluminous
in [O III] we consider a simplified model for the
evolution of a Str\"omgen sphere.
For an old ionisation nebula in equilibrium, the number rate
of ionising photons ($\ndot$)
should equal the number
of recombinations, from which follows the
equilibrium radius, $R_i$, of a Str\"omgen sphere:
\begin{equation}
R_i^3 = \frac{3 \ndot}{4\pi n_{\rm H} n_{\rm e} \alpha_{\rm H}},
\end{equation}
with $\alpha_{\rm H}$\ the hydrogen recombination coefficient.
The initial rapid expansion of the ionisation nebula is described
by e.g. \citet{spitzer68}
\begin{equation}
r_i^3 = R_i^3\{1- \exp( -n_{\rm e} \alpha_{\rm H} t)\} ,\label{ifront}
\end{equation}
with $r_i$\ the radius of the ionisation front and
$t$\ the time since the central source switched on.
For small $n_{\rm e} \alpha_{\rm H} t$ we can approximate this by
\begin{equation}
r_i^3 \approx n_{\rm e} \alpha_{\rm H} t R^3 =
\frac{3 \ndot}{4\pi n_{\rm H} } t
\end{equation}
The \oiii\ line emissivity is then given by
\begin{eqnarray}
\dot{N}_{\rm \oiii} = \frac{4\pi}{3} r_i^3 n_{\rm e} n_{\rm OIV} f_{5007 \rm\AA}\alpha_{\rm O III} =\nonumber\\
n_{\rm e} \frac{n_{\rm OIV}}{ n_{\rm H}}f_{5007 \rm\AA}\ \alpha_{\rm O III} \ndot t,
\end{eqnarray}
with $f_{5007 \rm\AA}$ the probability of emitting a photon at $5007$~\AA\ after recombination.
Assuming the interstellar medium densities in the GPS/CSO galaxies are more or
less similar, we can expect the following
correlation between the \oiii\ luminosity and the number rate
of ionising photons for young GPS/CSO sources of kinematic age $\tau$, provided
that the age of the radio galaxy coincides with the
birth of the ionisation nebula:
\begin{equation}
L_{\rm \oiii} \propto \tau \ndot \label{eq-scaling}.
\end{equation}
As a first approximation we consider whether there is a relation
between the observables $L_{\rm \oiii}$, and $\tau$\ and $L_X$.
However, Fig.~\ref{fig-opt-age} illustrates that the five sources
in our sample do not support a simple
scaling of $L_{\rm \oiii} \propto \tau L_X$.
There may be various reasons why this is not the case:
e.g.
the interstellar medium density varies from galaxy to galaxy,
their is no simple proportionality between $L_X$ and $\ndot$, due
to different spectral energy distributions (different spectral slopes,
or spectral breaks), and shocks induced
by jet cloud interactions may provide an additional source of
ionisation.\footnote{Although we do not have kinematic age estimates
of {PKS1345+125} and Mkn668 \citep{guainazzi04}, reasonable values
for the age in fact show these galaxies to be too bright in \oiii\
compared to the galaxies in our sample. In this case the reason is very likely
that a large part of the \oiii\ emission is not related to the activity of the
central nucleus, as both galaxies show evidence of recent merger activity,
and are bright infrared sources.} Moreover, one of the outliers is
B1358+624, for which we only have an approximate age.
However, a hint of what may the prime reason for deviations
from Eq.~\ref{eq-scaling} is provided by the very low \oiii\ luminosity
of {B0108+388}, because this is also the source
with the highest absorption column (Table~\ref{tab-res}). So the most
likely reason that the optical emission is lower than expected is that
the ionising UV flux is blocked by absorbing material close
to the nucleus. The absorbing material is probably not the result
of neutral hydrogen and helium, as, being close to the nucleus
it would be ionised almost immediately,
but dust grains, which
may survive the extreme conditions close to the nucleus for
1000~yr to $10^6$~yr, depending on the destruction mechanisms
and dust particle sizes \citep[e.g.][]{villar-martin01}. For a young source
like {B0108+388} this means that an appreciable amount of dust may still
enshroud the nucleus, frustrating the formation of a narrow emission
line region.
Note that this may also explain the relatively high neutral hydrogen column
density of {B0108+388} \citep[][Fig.~\ref{fig-nh}]{pihlstroem03};
the hydrogen ionisation fraction of the inner stellar medium is likely to be
low, which would also support the idea that photo-ionisation
is the dominant source of ionisation. If ionisation is dominated
by shocks generated by the jet, one would expect that {B0108+388}
would be relatively bright in \oiii\ as its interstellar medium density is
apparently high. B1358+624 deviates less from the expected relation in the
right hand panel of Fig.~\ref{fig-opt-age} due
to its relatively flat X-ray spectrum, which makes that the number
flux of UV photons is relatively small with respect to the X-ray luminosity.
However, since we do not know the broad band spectral shape, we do not
want to overemphasise this.
Let us now consider the possible implications of dust absorption.
The optical depth of dust particles depends on
dust particles cross sections ($\sigma_d = \pi r^2$ with $r$
the physical size of the particles, $r\sim0.1~\mu$m)
and the column density of dust particles $N_d$.
To obtain an order of magnitude estimate we assume that most
dust particles consist of silicates, and that all silicon is depleted into
dust. As $N_{\rm Si} \approx 4\times10^{-4} \NH$, and
a typical dust particle density is $\rho = 3.5$~g cm$^{-3}$, we have
$N_d \approx 10^{-13} \NH$, and the optical depth should be around
$\sigma_d N_d = 3\times10^{-23} N_{\rm H}$. This means that dust particles absorb
an appreciable amount of UV flux if the hydrogen column density is comparable
to, or exceeds, $10^{23}$cm$^{-2}$, which is only the case for
{B0108+388}.
We have therefore extrapolated from the observed $L_X$\ the ionising
photon luminosity $\ndot$\ using the observed spectral properties.
Plotting now $\loiii$\ as a function of $\tau\ndot$ we see less scatter,
certainly if we allow for dust absorption (Fig.~\ref{fig-opt-age}).
In order to bring {B0108+388} on the expected relation
we need a conversion of $\NH$ to dust optical depth that is 16\% of the above
order magnitude estimate. Note that in a log-log plot the absorption
enters linearly, since
$\ln(\dot{N}_{UV-abs}) = \ln(\ndot) - \sigma_d N_d$.
Hence, only {B0108+388}
is likely to be significantly affected by dust absorption.
The large uncertainties in extrapolating from the observed $L_X$\ to
an ionising UV flux makes that we cannot use Fig.~\ref{fig-opt-age}
(right panel)
to prove that Eq.~\ref{eq-scaling} is an accurate description,
but it makes it at least
plausible that for GPS/CSO galaxies the \oiii\ emission is relatively
low due to an underdeveloped narrow emission line region. This is
consistent with the idea that GPS-galaxies have AGN that
switched on around the same time that the radio jets were formed.
Further support for the idea that GPS/CSO galaxies are in the process of
creating an extended narrow line region comes from
the fact that neutral hydrogen apparently extends close
to the compact radio jets,%
given the fact that there is strong anti-correlation between
the neutral hydrogen column density and jet-size \citep{pihlstroem03}.
Note that GPS/CSO galaxies have sizes of the order of a few 100~pc, whereas
narrow emission line regions can extend up to 10~kpc.
\subsection{The radio luminosity versus the X-ray luminosity}
The X-ray emission from AGN is thought to come predominantly from
the immediate vicinity of the central black hole,
i.e. thermal emission
from the accretion disk reprocessed by
the hot plasma in its vicinity.
It is unlikely that synchrotron radiation
from the jet makes a dominant contribution to the X-ray
band. The reason is that, given the typical magnetic fields
infered from radio luminosities \citep[$>1$~mG, ][]{odea98},
the synchrotron cooling
time relevant for X-ray synchrotron radiation is in the order of
only 2~yr for an electron energy of 10~erg. This means that
a very small fraction of the total radio jet volume could
produce X-ray synchrotron emission.{
Another potential source of X-ray emission from outside the
central region could be inverse Compton emission by the
relativistic electron population in the jets \citep[c.f.][]{belsole05}.
Although we cannot totally exclude a significant inverse Compton
contribution, it seems unlikely to be the case for our sample.
The reason is that the X-ray
emission should in that case come from the same locations as the
radio emission (the bright regions in the jets). However, we would then
expect that the measured X-ray absorption columns would be more consistent
with radio absorption measurements toward the jets, which is not the case
(section~\ref{sec-nh}).
}
It is therefore reasonable to assume
that, compared to the radio
and \oiii\ luminosity, the X-ray luminosity is more directly related to
the accretion power of the AGN.
Nevertheless, the radio and optical luminosities are still indirectly
related to the accretion power, given the
correlations between radio, optical and X-ray
luminosities \citep[e.g.][]{sambruna99}.
It is, therefore, interesting that GPS/CSO galaxies seem on average radio
bright compared to the X-ray luminosity (Fig.~\ref{correlations}).
Given the absorption column distribution and low \oiii\
emission, we can ignore the interpretation
that the radio emission is relatively bright because the
interstellar medium is dense in GPS/CSO galaxies, as argued by advocates
of the ``frustrated radio source'' scenario. It is, therefore,
very probable that GPS/CSO galaxies are radio bright because they are young.
However, within our sample no relation between age and radio over X-ray ratio
can be seen (Fig.~\ref{fig-radio-age}), nor is there a correlation
with column density.
Nevertheless, the fact that for the sample as a whole the radio to X-ray
luminosity is brighter than for other radio-loud AGN
is at least qualitatively in agreement with several radio evolution models,
such as \citet{fanti95,readhead96,kaiser97,alexander00} and
\citet{snellen00a}.
These models describe the evolution of radio jets,
with radio brightnesses depending on the radial
density distribution of the interstellar medium. The radio jets
are relatively bright as long as the they plow through the dense
regions of the galaxies, but decline as soon as they propagate
outside the core of the galaxy.
The difference between the various models
is that some assume a density distribution
described by a King profile \citep{snellen00a,alexander00},
whereas others assume a density profile
falling of as power law of radius \citep{kaiser97}.
As a result, the non-power law models predict that galaxies in the
GPS-phase are still increasing in radio luminosity in time, until the jet
has reached the core radius of $\sim 1$~kpc, after which the luminosity
declines. In this case the relatively brightest phase of
radio galaxies would be represented by the so-called compact steep spectrum
sources (CSS), which have more extended radio jets than GPS/CSO sources.
The power law density models predict that right after the
jet emerges from the core region the radio emission starts to decline.
Our results are inconclusive regarding the details of the early evolution
of the radio luminosity. However, future X-ray observations of CSS galaxies
could help to clarify the brightness evolution further,
as models with a King profile predict that
CSS galaxies should, on average, have a higher radio to
X-ray luminosity ratio than GPS/CSO galaxies, whereas models that assume
power law density profiles predict that CSS galaxies should have a smaller
radio to X-ray luminosity.
\section{Conclusions}
We have presented \xmm\ observations of a sample of all the
GPS/CSO galaxies with $|b|>20$\degr\
from the \citet{pearson88} catalog, four of which
have measured kinematic time scales for the jet expansion.
All five of the sources are detected by \xmm\
thereby increasing the number of X-ray detected GPS/CSO galaxies
from 2 to 7.
These detections allow us to compare the X-ray properties of GPS/CSO
galaxies with those of other radio loud AGN.
The results presented here support the hypothesis that GPS/CSO galaxies
represent the young phases in the evolution of radio-loud AGN.
The alternative explanation that the radio sources are compact due
to confinement by an exceptionally high density of the interstellar medium in
those galaxies, seems extremely unlikely in view of the low intrinsic
X-ray absorption column densities,
which ranges from
$\NH = 4\times10^{21}$~cm$^{-2}$\ to $6\times10^{23}$~cm$^{-2}$,
and has a distribution similar to other radio loud AGN.
{%
After submission of our manuscript, a preprint by \citet{guainazzi05}
arrived at apparently different conclusions based on a sample
of five different GPS galaxies observed by \chandra\ and \xmm.
Only one of the five GPS galaxies in their sample
has \nh $< 10^{22}$~cm$^{-2}$,
whereas this is 75\%$\pm$26\% for their control sample. Note, however,
that in our sample three out of five have \nh $< 10^{22}$~cm$^{-2}$.
Taking both samples together, this means that four out of ten GPS galaxies,
or 40\%$\pm$20\% have \nh $< 10^{22}$~cm$^{-2}$\ consistent with the
control sample.
The fact that the absorption columns toward GPS galaxies
are consistent with those of other radio galaxies
suggests that the interstellar medium densities in
the cores of GPS/CSO galaxies are similar to those of other radio loud AGN.}
A similar conclusion was reached by \citet{pihlstroem03} based on HI radio
absorption observations, but the X-ray data provide stronger constraints,
as the total column density
contributes to the absorption, including ionized regions
of the interstellar medium.
This may contribute to the fact that in all cases the X-ray column densities
are higher than the HI column densities. A difference
between the radio column densities and the X-ray column densities is also that
the X-ray column is measured toward the central source, whereas the HI column
density is toward the jets, which extend outside the central region.
If the difference between radio and X-ray column density is dominated
by those geometrical effects, this would be additional evidence against the
confinement scenario, since the jets have apparently been able to pierce
through the dense local regions that contribute most to the X-ray absorption
column.
Although the $\NH$\ distribution of our sample cannot be distinguished from
other radio-loud AGN, the ratio of radio to X-ray luminosity shows that GPS
galaxies have a strong tendency to be relatively radio bright. This supports
the view that for the same thrust of the jets
younger radio sources are relatively bright \citep{kaiser97,snellen00a}.
However, the data is inconclusive concerning whether GPS-galaxies
represent the most radio-luminous phases in the
lifes of radio-loud AGN, as would be the case if the source develops in a
density profile
that drops of as power law with distance \citep{kaiser97,pihlstroem03},
or whether they are still
in their brightening phase. This would be the case
if the interstellar medium is best described by a King profile with
a relative uniform density within $\sim$1~kpc of the center and then dropping
of as a power law \citep[e.g.][]{snellen00a}.
In the latter case the brightest evolutionary phase of radio-loud AGN would
be represented by the compact steep spectrum sources (CSS), which have more
extended radio emission than GPS/CSO galaxies. A similar study to this one
concerning CSS-galaxies can clarify this issue.
Finally, we find that GPS/CSO galaxies are relatively weak in \oiii\ line
emission. This is again in
support of the idea that GPS/CSO galaxies represent the very earliest stages
of
the evolution of radio-loud AGN, since narrow line regions need time to build
up to their equilibrium size, and young narrow line regions are therefore not
as bright as fully developed ones.
The narrow line region is powered by the UV flux of the central source,
but also shocks
induced by the expanding jets are likely to contribute to their formation.
For those very
young radio-loud AGN we advocate here that their
emission line regions are powered by
the UV flux from the central sources.
A case in point is that the relatively weakest \oiii\
source, {B0108+388}, has also the highest X-ray column density,
which suggests that a dusty torus is partially blocking the UV light.
The low \oiii\ luminosity therefore shows that the birth of the narrow
emission line region must coincide more or less with the birth of the radio
jet. {%
However, we caution that we only have a limited knowledge
of the nature of the ionization mechanism for the [O III] line emission.
i.e. both shocks from the jets, as the UV radiation from the
AGN may contribute to the ionization. Moreover, compact steep spectrum (CSS)
sources,
probably representing a more advanced evolutionary state of radio galaxies
than GPS galaxies, show that the forbidden line emission tends to be
aligned with the radio jet \citep{devries99}.
This phenomenon is not well understood, but it should be accounted
for if one wants to build a more detailed model of the evolution of
emission line nebulae in radio galaxies.}
In summary, the findings from this X-ray study lends further support
to the theory that GPS/CSO galaxies represent the early phases of radio-loud
AGN,
in which also the narrow line emission nebula is still in the early
phases of its evolution. Future X-ray studies may help to further clarify
the relation between age or jet-size, the extent or brightness of the emission
line region and the power of the X-ray emission. It is important
to include also compact steep spectrum (CSS) sources in such a study, as they
are likely to represent the next phase in the evolution of radio-loud AGN.
Comparing their X-ray to radio luminosity may help clarify
whether the radio emission declines already during the GPS-phase,
or first increases, then peaks around the CSS-phase and from then on weakens.
\section*{Acknowledgments}
We thank for Elisa Costantini for helpful discussions on dust
grain properties.
The Space Research Organization of the Netherlands is
supported financially by NWO,
the Netherlands Organization for Scientific Research.
This research has made use of the NASA/IPAC Extragalactic Database (NED)
which is operated by the Jet Propulsion Laboratory, California Institute
of Technology, under contract with the National Aeronautics and Space
Administration.
\xmm\ is an ESA science mission, with instruments and
contributions directly funded by the ESA member states and the USA (NASA).
|
Title:
The Anomalous Early Afterglow of GRB 050801 |
Abstract: The ROTSE-IIIc telescope at the H.E.S.S. site, Namibia, obtained the earliest
detection of optical emission from a Gamma-Ray Burst (GRB), beginning only 21.8
s from the onset of Swift GRB 050801. The optical lightcurve does not fade or
brighten significantly over the first ~250 s, after which there is an
achromatic break and the lightcurve declines in typical power-law fashion. The
Swift/XRT also obtained early observations starting at 69 s after the burst
onset. The X-ray lightcurve shows the same features as the optical lightcurve.
These correlated variations in the early optical and X-ray emission imply a
common origin in space and time. This behavior is difficult to reconcile with
the standard models of early afterglow emission.
| https://export.arxiv.org/pdf/astro-ph/0601350 |
\title{The Anomalous Early Afterglow of GRB 050801}
\author{
E.~S.~Rykoff,\altaffilmark{1},
V.~Mangano,\altaffilmark{2},
S.~A.~Yost\altaffilmark{1},
R.~Sari\altaffilmark{3},
F.~Aharonian\altaffilmark{4},
C.~W.~Akerlof\altaffilmark{1},
M.~C.~B.~Ashley\altaffilmark{5},
S.~D.~Barthelmy\altaffilmark{6},
D.~N.~Burrows\altaffilmark{7},
N.~Gehrels\altaffilmark{6},
E.~G\"{o}\v{g}\"{u}\c{s}\altaffilmark{8},
D.~Horns\altaffilmark{4},
\"{U}.~K{\i}z{\i}lo\v{g}lu\altaffilmark{10},
H.~A.~Krimm\altaffilmark{6,11},
T.~A.~McKay\altaffilmark{1},
M.~\"{O}zel\altaffilmark{12},
A.~Phillips\altaffilmark{5},
R.~M.~Quimby\altaffilmark{13},
G.~Rowell\altaffilmark{4},
W.~Rujopakarn\altaffilmark{1},
B.~E.~Schaefer\altaffilmark{14},
D.~A.~Smith\altaffilmark{15},
H.~F.~Swan\altaffilmark{1},
W.~T.~Vestrand\altaffilmark{16},
J.~C.~Wheeler\altaffilmark{13},
J.~Wren\altaffilmark{16},
F.~Yuan\altaffilmark{1},
}
\altaffiltext{1}{University of Michigan, 2477 Randall Laboratory, 450 Church
St., Ann Arbor, MI, 48109, erykoff@umich.edu}
\altaffiltext{2}{INAF-IASF, Palermo, Italy} %
\altaffiltext{3}{California Institute of Technology, Pasadena, CA, 91125, USA}
\altaffiltext{4}{Max-Planck-Institut f\"{u}r Kernphysik, Saupfercheckweg 1,
69117 Heidelberg, Germany}
\altaffiltext{5}{School of Physics, Department of Astrophysics and Optics,
University of New South Wales, Sydney, NSW 2052, Australia}
\altaffiltext{6}{NASA Goddard Space Flight Center, Laboratory for High Energy
Astrophysics, Greenbelt, MD 20771}
\altaffiltext{7}{Pennsylvania State University, University Park, PA, 16802,
USA}
\altaffiltext{8}{Sabanc{\i} University, Istanbul, Turkey}
\altaffiltext{9}{Istanbul University Science Faculty, Department of Astronomy
and Space Sciences, 34119, University-Istanbul, Turkey}
\altaffiltext{10}{Middle East Technical University, 06531 Ankara, Turkey}
\altaffiltext{11}{Universities Space Research Association, 10227 Wincopin
Circle, Suite 212, Columbia, MD 21044}
\altaffiltext{12}{\c{C}anakkale Onsekiz Mart \"{U}niversitesi, Terzio\v{g}lu
17020, \c{C}anakkale, Turkey}
\altaffiltext{13}{Department of Astronomy, University of Texas, Austin, TX
78712}
\altaffiltext{14}{Department of Physics and Astronomy, Louisiana State
University, Baton Rouge, LA 70803}
\altaffiltext{15}{Guilford College, Greensboro, NC, 27410, USA}
\altaffiltext{16}{Los Alamos National Laboratory, NIS-2 MS D436, Los Alamos, NM
87545}
\keywords{gamma rays:bursts}
\section{Introduction}
Gamma-ray bursts (GRBs) are the most luminous explosions in the universe, but
the origin of their emission remains elusive. With the launch of the
\emph{Swift} $\gamma$-ray Burst Explorer~\citep{gcgmn04} in late 2004, great
progress has been made in the study of the early afterglow phase of GRBs.
However, only a small number of bursts have been imaged simultaneously in both
the optical and X-ray bands in the first minutes after the
burst~\citep{nkgpg05,qryaa05,rykaa05,bbbbc05}.
In this letter, we report on the earliest detection of optical emission,
starting at 21.8 seconds after the onset of GRB~050801 with the ROTSE-IIIc
(Robotic Optical Transient Search Experiment) telescope located at the
H.E.S.S. site in Namibia. This is the most densely sampled early lightcurve
yet obtained. It does not fade or brighten significantly over the first
$\sim250$ seconds, after which there is a break and the lightcurve declines in
a typical power-law fashion. The \emph{Swift}/XRT also obtained early
observations starting at 69 seconds after the burst onset. The X-ray
lightcurve shows the same features as the optical lightcurve. These correlated
variations in the early optical and X-ray emission imply a common origin in
space and time. This behavior differs from that seen in
GRB~050319~\citep{qryaa05}, GRB~050401~\citep{rykaa05}, and
GRB~050525a~\citep{bbbbc05}. It is difficult to explain this behavior with
standard models of early afterglow emission without assuming there is
continuous late time injection of energy into the afterglow.
\section{Observations and Analysis}
\label{sec:observations}
The ROTSE-III array is a worldwide network of 0.45~m robotic, automated
telescopes, built for fast ($\sim 6$ s) responses to GRB triggers from
satellites such as HETE-2 and \emph{Swift}. They have wide ($1\fdg85 \times
1\fdg85$) fields of view imaged onto Marconi $2048\times2048$ back-illuminated
thinned CCDs, and operate without filters. The ROTSE-III systems are described
in detail in \citet{akmrs03}.
On 2005 August 01, \emph{Swift}/BAT detected GRB~050801 (\emph{Swift} trigger
148522) at 18:28:02.1 UT. The position was distributed as a Gamma-ray Burst
Coordinates Network (GCN) notice at 18:28:16 UT, with a $4\arcmin$ radius
$3\sigma$ error circle. The burst had a $T_{90}$ duration of
$20\pm3\,\mathrm{s}$ in the 15-350 keV band, and consisted of two peaks
separated by around 3 seconds. The position was released during the tail end
of the $\gamma$-ray emission~\citep{smbbc05}. The \emph{Swift} satellite
immediately slewed to the target, with the XRT beginning observations in
windowed timing mode at 69 s after the start of the burst and switching to
photon counting mode at 89.3 s after the trigger.
ROTSE-IIIc, at the H.E.S.S. site in Namibia, responded automatically to the GCN
notice, beginning its first exposure in less than 8 s, at 18:28:23.9 UT. The
automated burst response included a set of ten 5-s exposures, ten 20-s
exposures, and 134 60-s exposures before the burst position dropped below our
elevation limit. The first set of ten exposures were taken with subframe
readout mode to allow rapid sampling (3-s readout between each 5-s exposure).
Near real-time analysis of the ROTSE-III images detected a $15^{th}$ magnitude
source at $\alpha=13^h36^m35\fs4$, $\delta=-21\arcdeg55\arcmin42\farcs0$
(J2000.0) that was not visible on the Digitized Sky Survey red plates, which we
reported via the GCN Circular e-mail exploder within 7 minutes of the
burst~\citep{ryr05}. No spectroscopic redshift has been reported for
this GRB, although the \emph{Swift}/UVOT detected the afterglow in all filters
including the $UVW2$ filter at $188\,\mathrm{nm}$~\citep{bbhgc05}, which implies
that the redshift is $\lesssim1.2$. In addition, the afterglow was dimmer than
23 mag with no evidence for a bright host galaxy~\citep{fjhww05b}.
The X-ray photometry is shown in Table~\ref{tab:xray}. Time bin midpoints and
durations are listed in seconds, relative to the \emph{Swift} trigger time,
18:28:02 UT. The count rate is in counts/s and the flux is in
$10^{-11}\,\mathrm{erg}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}$, for the energy
range 0.2-10 keV. The X-ray data has been corrected for a hot CCD column
crossing the source as well as a nearby source $30''$ away. Photon counting
data from the first orbit have been corrected for pile-up. We chose a time
binning that ensures a detection of at least $3.5\sigma$ for each time bin
before corrections were applied. The gaps in the data are caused by earth
occultation. There is no spectral variation across the lightcurve, and the
$N_H$ value is consistent with the Galactic value
($7\times10^{20}\,\mathrm{cm}^{-2}$). The best-fit spectrum (with $N_H$ fixed
to $7\times10^{20}\,\mathrm{cm}^{-2}$) is a power law with photon index
$1.87\pm0.15$ (90\% confidence level). The relative errors for the fluxes are
slightly larger than those for the count rate due to the additional systematic
error from the conversion.
\begin{deluxetable}{cccc}
\tablewidth{0pt}
\tablecaption{\emph{Swift}/XRT observations of the afterglow
of GRB~050801.\label{tab:xray}}
\tablehead{
\colhead{T-mid (s)} & \colhead{Duration (s)} & \colhead{Count Rate
($\mathrm{cts}\,\mathrm{s}^{-1}$)} &
\colhead{Flux ($10^{-11}\,\mathrm{erg}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}$)}
}
\startdata
74.1 & 10.0 & $3.46\pm0.81$ & $19.2\pm 5.7$ \\
84.1 & 10.0 & $1.99\pm0.70$ & $11.1\pm 4.4$ \\
111.8 & 45.0 & $1.62\pm0.39$ & $ 9.0\pm 2.7$ \\
164.3 & 60.0 & $1.40\pm0.31$ & $ 7.8\pm 2.2$ \\
214.3 & 40.0 & $1.83\pm0.43$ & $10.1\pm 3.0$ \\
254.3 & 40.0 & $1.83\pm0.43$ & $10.1\pm 3.0$ \\
291.8 & 35.0 & $2.25\pm0.51$ & $12.5\pm 3.6$ \\
334.3 & 50.0 & $1.52\pm0.35$ & $ 8.4\pm 2.5$ \\
396.8 & 75.0 & $1.01\pm0.24$ & $ 5.6\pm 1.6$ \\
471.8 & 75.0 & $1.05\pm0.24$ & $ 5.8\pm 1.7$ \\
561.8 & 105.0 & $0.69\pm0.167$ & $ 3.8\pm 1.1$ \\
686.8 & 145.0 & $0.50\pm0.12$ & $ 2.76\pm 0.83$ \\
859.3 & 200.0 & $0.31\pm0.08$ & $ 1.71\pm 0.55$ \\
4346.7 & 320.0 & $0.069\pm0.021$ & $ 0.38\pm 0.14$ \\
4856.7 & 700.0 & $0.060\pm0.013$ & $ 0.333\pm 0.092$ \\
5715.3 & 510.0 & $0.043\pm0.014$ & $ 0.241\pm 0.086$ \\
6357.8 & 775.0 & $0.027\pm0.009$ & $ 0.149\pm 0.055$ \\
11249.6 & 2560.0 & $0.012\pm0.003$ & $ 0.065\pm 0.021$ \\
17040.6 & 2550.0 & $0.0095\pm0.0028$ & $ 0.053\pm 0.018$ \\
22814.0 & 2575.0 & $0.0069\pm0.0025$ & $ 0.038\pm 0.015$ \\
31556.0 & 8237.1 & $0.0049\pm0.0015$ & $ 0.0275\pm 0.0096$ \\
47812.8 & 17585.3 & $0.0033\pm0.0010$ & $ 0.0182\pm 0.0063$ \\
366425.9 & 515983.6 & $< 0.0004$ & $< 0.00222$ \\
\enddata
\tablecomments{Time bin midpoints and durations are relative
to the \emph{Swift} trigger time, 18:28:02 UT.}
\end{deluxetable}
The optical photometry is shown in Table~\ref{tab:opt}. The ROTSE-IIIa images
were bias-subtracted and flat-fielded by our automated pipeline. The
flat-field image was generated from 30 twilight images. We used
SExtractor~\citep{ba96} to perform the initial object detection and to
determine the centroid positions of the stars. The images were processed with
our custom RPHOT photometry program based on the DAOPHOT PSF-fitting photometry
package~\citep{qryaa05}. The unfiltered thinned ROTSE-III CCDs have a peak
response similar to an $R$-band filter. The magnitude zero-point for was
calculated from the median offset of the fiducial reference stars to the USNO
B1.0 $R$-band measurements to produce $C_R$ magnitudes. After the first 30
images, frames were co-added in logarithmic time bins to maintain roughly
constant signal-to-noise.
\begin{deluxetable}{ccc}
\tablewidth{0pt}
\tablecaption{ROTSE-IIIc Optical Photometry of GRB~050801.\label{tab:opt}}
\tablehead{
\colhead{$t_{\mathrm{start}}$} &
\colhead{$t_{\mathrm{end}}$} &
\colhead{$C_R$}
}
\startdata
21.8 & 26.8 & $14.93\pm 0.05$\\
29.9 & 34.9 & $14.79\pm 0.05$\\
38.0 & 43.0 & $14.80\pm 0.04$\\
46.1 & 51.1 & $14.91\pm 0.06$\\
54.2 & 59.2 & $14.83\pm 0.05$\\
62.4 & 67.4 & $14.91\pm 0.04$\\
70.5 & 75.5 & $14.75\pm 0.04$\\
78.6 & 83.6 & $14.87\pm 0.05$\\
86.7 & 91.7 & $14.88\pm 0.05$\\
94.8 & 99.8 & $14.93\pm 0.05$\\
113.5 & 133.5 & $14.98\pm 0.03$\\
143.3 & 163.3 & $15.09\pm 0.03$\\
172.7 & 192.7 & $15.12\pm 0.03$\\
203.0 & 223.0 & $15.06\pm 0.03$\\
232.5 & 252.5 & $15.13\pm 0.04$\\
262.3 & 282.3 & $15.21\pm 0.04$\\
291.8 & 311.8 & $15.35\pm 0.04$\\
321.0 & 341.0 & $15.47\pm 0.04$\\
350.8 & 370.8 & $15.59\pm 0.03$\\
380.3 & 400.3 & $15.70\pm 0.04$\\
409.9 & 469.9 & $15.89\pm 0.04$\\
479.8 & 539.8 & $16.12\pm 0.03$\\
549.0 & 609.0 & $16.29\pm 0.04$\\
618.2 & 678.2 & $16.31\pm 0.05$\\
688.1 & 748.1 & $16.63\pm 0.06$\\
757.2 & 817.2 & $16.59\pm 0.06$\\
826.6 & 886.6 & $16.66\pm 0.07$\\
896.3 & 956.3 & $16.75\pm 0.06$\\
965.5 & 1025.5 & $16.93\pm 0.07$\\
1034.9 & 1094.9 & $16.92\pm 0.09$\\
1104.7 & 1233.9 & $16.99\pm 0.06$\\
1243.6 & 1442.0 & $17.10\pm 0.05$\\
1451.4 & 1650.3 & $17.39\pm 0.07$\\
1659.7 & 1858.6 & $17.48\pm 0.07$\\
1867.9 & 2136.8 & $17.60\pm 0.06$\\
2146.5 & 2485.3 & $17.78\pm 0.07$\\
2495.2 & 2832.6 & $17.88\pm 0.07$\\
2841.9 & 3249.7 & $18.26\pm 0.11$\\
3259.7 & 3736.8 & $18.24\pm 0.09$\\
3745.9 & 4332.1 & $18.71\pm 0.20$\\
4341.4 & 4956.6 & $18.49\pm 0.09$\\
4966.5 & 5721.7 & $18.88\pm 0.12$\\
5731.0 & 6554.7 & $18.99\pm 0.15$\\
6564.4 & 7527.4 & $18.83\pm 0.13$\\
7536.7 & 8619.8 & $19.63\pm 0.22$\\
8629.6 & 10357.0 & $19.49\pm 0.16$\\
\enddata
\tablecomments{Start and end times are relative to the \emph{Swift} trigger
time, 18:28:02 UT.}
\end{deluxetable}
\section{Results}
With a detection only 21.8 s after the start of the burst, this is the earliest
detection of an optical counterpart of a GRB, as well as the most densely
sampled early afterglow. Only four GRBs have had optical counterparts detected
within the first minute, and none of these had more than two detections in the
first minute. The first 250 s of the optical afterglow shows short timescale
variability relative to an overall flat lightcurve. This is in stark contrast
to the prompt counterpart of GRB~990123~\citep{abbbb99}, which had a very bright
$9^{th}$ mag peak at 60~s after the burst onset, generally interpreted as the
signature of reverse shock emission~\citep{sp99b}. This afterglow shows no
evidence for reverse shock emission.
Figure~\ref{fig:optandxraylc} shows a comparison of the early optical and X-ray
lightcurves of GRB~050801, combined with the prompt $\gamma$-ray emission. The
prompt BAT $\gamma$-ray flux densities have been extrapolated to the X-ray band
[0.2-10 keV]. This extrapolation was performed with the best-fit photon index
of $2.0\pm0.2$ for the time-averaged $\gamma$-ray spectrum from 20-150 keV, as
in \citet{tgcmc05}. The statistical errors scaled from the BAT count rate are
shown; the gray region denotes the uncertainties from the extrapolation to the
X-ray regime. The X-ray flux values have been converted to flux density (Jy)
using an effective frequency of $<\nu> = 6.89\times10^{17}\,\mathrm{Hz}$, the
flux weighted average in the 0.2-10 keV range with the best-fit photon index
$\Gamma = 1.87$. The ROTSE-III optical magnitudes have been converted to flux
density assuming the unfiltered ROTSE-III images are equivalent to $R_c$, and
have been approximately adjusted for Galactic extinction by 0.24
mag~\citep{sfd98}. The de-extinction does not have a significant effect on the
derived spectral indices. After the break at $\sim250$ seconds, the optical
lightcurve decays as $t^{-1.31\pm0.11}$, followed by a brief but significant
plateau at $\sim800$ seconds. The top panel shows the ratio of optical flux to
X-ray count rate for the first 7000 s, scaled to the average ratio value. The
X-ray count rate rather than the X-ray flux was used to avoid the systematic
error introduced when converting from count rate to flux, and is made possible
by the lack of X-ray spectral evolution. The ROTSE-III observations have been
co-added to match the times of the XRT integrations as closely as possible.
The flux ratio is consistent with a constant value (dashed-line) with a
$\chi^2$ of 15.9 (16 degrees of freedom). The break at $\sim250\,\mathrm{s}$
has no systematic change in the optical to X-ray flux ratio, and is therefore
achromatic.
Only three $\gamma$-ray bursts have had prompt optical detections
contemporaneous with the $\gamma$-ray emission. The prompt optical counterpart
of GRB~041219a~\citep{vwwfs05} was correlated with the $\gamma$-ray emission,
implying a common origin. However, both GRB~990123~\citep{abbbb99} and
GRB~050401~\citep{rykaa05} demonstrated a different origin for the $\gamma$-rays
and the optical radiation. Although we do not have a prompt optical detection
in the case of GRB~050801, we can interpolate between the high energy prompt
lightcurve scaled to the X-ray band (gray band in Figure~\ref{fig:optandxraylc})
and the first X-ray detection. During this interval the high energy emission
falls by a factor of $\gtrsim100$ while the optical emission is unchanged.
This suggests a different origin for the prompt $\gamma$-ray emission and the
early optical emission. However, the X-ray and optical afterglow of GRB~050801
do appear to arise from a similar origin after $\sim80$~s. The two lightcurves
are plotted in the main panel of Figure~\ref{fig:optandxraylc}. Each lightcurve
shows similar flat behavior at the early time, with a break around 250~s.
\section{Discussion}
In the standard fireball model of GRB afterglow emission, the spectral energy
distribution of GRB afterglows can be fit by a broken power-law with spectral
segments $F_\nu \propto \nu^\beta$ (for a review, see \citet{p05}). The
spectral index obtained by comparing the de-extincted optical (see
Figure~\ref{fig:optandxraylc}) to X-ray flux density during the second XRT
integration is $\beta_{\mathrm{opt-X}} = -0.92\pm0.05$, consistent with the
X-ray only spectral index of $\beta_{\mathrm{X}} = -0.87\pm0.15$ [0.2-10 keV].
To test for evolution in the broadband spectral index, we have compared the
optical and X-ray lightcurves during the first 7000 s (top panel of
Figure~\ref{fig:optandxraylc}). The optical to X-ray flux ratio is consistent
with a constant value ($\chi^2=15.9$ with 16 degrees of freedom). Across the
break at 250 s, both $\beta_{\mathrm{opt-X}}$ and $\beta_{\mathrm{X}}$ are
unchanged, and therefore the break is achromatic. Furthermore, there is no
evidence of a spectral change in the UVOT images~\citep{bbhgc05}, although the
time resolution is insufficient to constrain the time of the break. Many X-ray
lightcurves have been seen to steepen around 1000 s - 5000 s post-burst with no
change in the X-ray spectral index~\citep{nkgpg05}, For the few bursts with
sufficient early optical and X-ray coverage~\citep{qryaa05, bbbbc05}, this
behavior has not been mirrored in the optical band.
The tight correlation between the optical and X-ray emission suggests that they
share the same origin in space and time. The standard fireball model of GRB
afterglows can explain the behavior of the optical and X-ray lightcurve after
250~s. The observed spectral parameters and decay indices are most consistent
with a fireball expanding adiabatically into a constant density medium, with
the typical synchrotron frequency $\nu_\mathrm{m}$ below the optical band, and
the cooling frequency $\nu_\mathrm{c}$ above the X-ray band. For example, this
can be produced by the following parameters: the electron energy index $p =
2.8$; the isotropic equivalent energy $E \sim 10^{53}\,\mathrm{erg}$ at a
redshift of $z \sim 0.5$; the circumburst density $n \sim
0.7\,\mathrm{cm}^{-3}$; the energy fraction in the electrons $\epsilon_e \sim
0.07$; and the energy fraction in the magnetic field $\epsilon_B \sim
0.0002$. These values of the electrons and magnetic energy are consistent with
those deduced for other bursts albeit on the lower side. If the ejecta were
expanding into a $1/r^2$ density profile (a so-called ``wind'' medium), the
fireball model predicts a relationship between the spectral and temporal
behavior that is inconsistent at the $4\sigma$ level with the observations
after 250~s.
We now investigate the possible explanations of the flat early lightcurve and
the origin of the break at 250~s. First, any spectral transition (e.g. $\nu_m$
crossing the optical band) would fail to explain the achromatic nature of the
break. Achromatic breaks observed in other afterglows have been interpreted as
geometric, when the edge of a conical jet becomes visible to the observer and
the jet starts to spread~\citep{hbfsk99,sgkpt99}. At 250 s, this would be the
earliest such ``jet break'' detected. In the fireball model, the post jet
break afterglow is expected to decay as $t^{-p}$, where $p$ is the electron
energy index with $N_e \propto E^{-p}$, provided that $p>2$~\citep{sph99}. A
hard electron index of $p<2$ predicts a post-jet decay even steeper than
$t^{-p}$~\citep{dc01}. Therefore, the observed post-break temporal decay
implies $p \le 1.3$ which predicts a significant pre-break decay~\citep{dc01}
that is inconsistent with the observed pre-break flatness as well as the
observed spectral index $\beta$. Therefore, the achromatic evolution of
GRB~050801 cannot be explained with a jet break.
We have investigated whether the early afterglow is consistent with the
predictions of a structured jet viewed off axis~\citep{gk03}. In this case, it
is difficult to create a sharp early break; under such conditions, the
post-break evolution should track closely with the electron energy index $p$,
which is inconsistent with observations as described above.
Such an early break at 250~s, can perhaps be explained as the onset time of the
afterglow. If the reverse shock is non relativistic (as indicated by the
relatively short duration of the burst, see \citet{sari97}) then self
similar expansion starts once the mass collected from the environment is a
factor $\gamma$ smaller than that in the ejecta:
\begin{equation}
t_{{\rm afterglow}}=100\,{\rm s}\,(1+z)
\left( \frac{E}{10^{53}\,{\rm erg}} \right)^{1/3} \left( \frac{n}{1\,{\rm
cm^{-3}}} \right)
\left(\frac{\gamma}{100} \right)^{-8/3}.
\end{equation}
A value of the initial Lorentz factor $\gamma$ just below a hundred would
therefore be consistent with an onset time of 250~s. However, it is difficult
to reconcile the flat part before 250~s as the rise of the afterglow. During
the onset, since the fireball is coasting with a constant Lorentz factor, the
bolometric luminosity is given by $L_\mathrm{B} \propto t^2 n$, the surface
area times the density. For a constant density a sharp rise $\propto t^2$ is
therefore expected. For a wind-like decreasing density, the lightcurve should
be flat as observed. However, as stated before, a wind density profile seems
inconsistent with the behavior after 250~s.
Continuous energy injection has been suggested as a source of early X-ray
light\-curve flat\-tening~\citep{nkgpg05}. This injection could be observed if
the initial fireball ejecta had a range of Lorentz factors, with the slower
shells catching up with the decelerating afterglow~\citep{rm98,sm00}. However,
we require a very steady injection of energy to produce the observed
lightcurve, flat for more than a decade in time. If we adopt this explanation,
the afterglow must start before our first optical observation, implying an
initial Lorentz factor of more than 200, and energy injection rate which is
roughly constant over a decade in time, and which shuts off suddenly at 250~s.
Flat or very slowly decaying optical lightcurves have been seen in a
number of other early afterglows (eg, GRB~030418~\citep{rspaa04},
GRB~050319~\citep{qryaa05}, and GRB~041006~\citep{msmy04,ysr04}). Early
X-ray lightcurves detected by \emph{Swift} are typically more complex,
with rapidly fading sections and short timescale flares~\citep{nkgpg05}.
The early afterglow of GRB~050801, flat in both optical and X-rays, is,
so far, unique. It is inconsistent with the standard fireball model for
early afterglow emission, unless continuous energy injection is
involved. Further \emph{Swift} prompt GRB detections, combined with
rapid follow-up by \emph{Swift} and ground-based telescopes, will
provide further opportunities to explore the origin of this type of early
afterglow behavior.
\acknowledgements
This work has been supported by NASA grants NNG-04WC41G and NGT5-135, NSF
grants AST-0407061, the Australian Research Council, the University of New
South Wales, and the University of Michigan. Work performed at LANL is
supported through internal LDRD funding. The Palermo work is supported at INAF
by funding from ASI on grant number I/R/039/04. Special thanks to Toni Hanke
at the H.E.S.S. site.
\newcommand{\noopsort}[1]{} \newcommand{\printfirst}[2]{#1}
\newcommand{\singleletter}[1]{#1} \newcommand{\switchargs}[2]{#2#1}
|
Title:
Radio continuum and molecular line observations of four bright-rimmed clouds |
Abstract: We present the results of radio continuum and molecular line observations
conducted using the Mopra millimetre-wave telescope and Australia Telescope
Compact Array. These observations reveal the presence of a dense core embedded
within each cloud, and the presence of a layer of hot ionised gas coincided
with their bright-rims. The ionised gas has electron densities significantly
higher than the critical density above which an ionised boundary layer can form
and be maintained, strongly supporting the hypothesis that these clouds are
being photoionised by the nearby OB star(s). From an evaluation of the pressure
balance between the ionised and molecular gas, SFO 58 and SFO 68 are identified
as being in a post-pressure balance state, while SFO 75 and SFO 76 are more
likely to be in a pre-pressure balance state. We find secondary evidence for
the presence of ongoing star formation within SFO 58 and SFO 68, such as
molecular outflows, OH, H$_2$O and methanol masers, and identify a potential
embedded UC HII region, but find no evidence for any ongoing star formation
within SFO 75 and SFO 76. Our results are consistent with the star formation
within SFO 58 and SFO 68 having been triggered by the radiatively driven
implosion of these clouds.
| https://export.arxiv.org/pdf/astro-ph/0601718 |
\title{Radio continuum and molecular line observations of four bright-rimmed clouds}
\author{J. S. Urquhart\inst{1}, M. A. Thompson\inst{2}, L. K. Morgan\inst{3,4} \& Glenn J. White\inst{4,5}}
\offprints{J. S. Urquhart: jsu@ast.leeds.ac.uk}
\institute{
Department of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, UK
\and
School of Physics Astronomy \& Maths, University of Hertfordshire, College Lane, Hatfield, AL10 9AB, UK
\and
Centre for Astrophysics and Planetary Science, School of Physical Sciences, University of
Kent, Canterbury, CT2 7NR, UK
\and
Green Bank Telescope, P.O. Box 2, Green Bank, WV 24944, USA
\and
Dept. of Physics \& Astronomy, The Open University, Walton Hall, Milton
Keynes, MK7 6AA, UK
\and
Space Physics Division, Space Science \& Technology Division, CCLRC
Rutherford Appleton Laboratory, Chilton, Didcot, Oxfordshire, OX11 0QX,
UK
}
\date{}
\abstract{}{To search for evidence of triggered star formation within four bright-rimmed clouds, SFO~58, SFO~68, SFO~75 and
SFO~76.}{We present the results of radio continuum and molecular line observations conducted using the Mopra millimetre-wave telescope and Australia Telescope Compact Array. We use the \mbox{$J$=1--0} transitions of $^{12}$CO, $^{13}$CO and C$^{18}$O to trace the distribution of molecular material and to study its kinematics.}{These observations reveal the presence of a dense core ($n_{\rm{H}_2}>10^4$~cm$^{-3}$) embedded within each cloud, and the presence of a layer of hot ionised gas coincided with their bright-rims. The ionised gas has electron densities significantly higher than the critical density ($>$~25 cm$^{-3}$) above which an ionised boundary layer can form and be maintained, strongly supporting the hypothesis that these clouds are being photoionised by the nearby OB star(s). Using a simple pressure-based argument, photoionisation is shown to have a profound effect on the stability of these cores, leaving SFO~58 and SFO~68 on the edge of gravitational stability, and is also likely to have rendered SFO~75 and SFO~76 unstable to gravitational collapse. From an evaluation of the pressure balance between the ionised and molecular gas, SFO~58 and SFO~68 are identified as being in a post-pressure balance state, while SFO~75 and SFO~76 are more likely to be in a pre-pressure balance state. We find secondary evidence for the presence of ongoing star formation within SFO~58 and SFO~68, such as molecular outflows, OH, H$_2$O and methanol masers, and identify a potential embedded UC HII region, but find no evidence for any ongoing star formation within SFO~75 and SFO~76.}{Our results are consistent with the star formation within SFO~58 and SFO~68 having been triggered by the radiatively driven implosion of these clouds.}
\keywords{Stars: formation -- ISM: clouds -- ISM: HII regions -- ISM: individual object: bright-rimmed clouds: SFO~58, SFO~68, SFO~75 and SFO~76 -- ISM: molecules -- Radio continuum: ISM}
\authorrunning{J. S. Urquhart et al.}
\titlerunning{Radio observations of four bright-rimmed clouds}
\section{Introduction}
From the moment they turn on OB stars begin to drive an ionisation front into the surrounding molecular material, photo-evaporating and dissipating the molecular cloud from which they have formed. The rapidly expanding HII region leads to the formation of a dense shell of neutral gas, swept up in front of the ionisation front. These dense shells, and dense neutral clumps of material, surrounding evolved HII regions have long been suspected to be regions where star formation could have been triggered \mbox{(\citealt{elmegreen1977,sandford1982})}. Bright-Rimmed Clouds (BRCs) are small molecular clouds located on the edges of evolved HII regions and are considered, due to their relatively simple geometry and isolation within HII regions, to be ideal laboratories in which to study the physical processes involved in triggered star formation.
The photoionisation of the BRCs surface layers by UV photons from nearby OB stars leads to the formation of a layer of hot ionised gas, known as an \emph{Ionised Boundary Layer} (IBL), which surrounds the rim of the molecular cloud. The hot ionised gas streams off the surface of the cloud into the low density HII regions, resulting in a continuous mass loss by the cloud, known as a \emph{photo-evaporative flow} (\citealt{megeath1997}). Within the IBL the incoming ionising photon flux is balanced by recombination, with only a small fraction of ionising photons penetrating the IBL to ionise new material (\citealt{lefloch1994}), which replenishes the ionised material within the IBL lost to the photo-evaporative flow. If the IBL is over-pressured with respect to the molecular gas within the BRC, shocks are driven into the molecular gas, resulting in the compression of the cloud, and can lead to the formation of dense cores which are then triggered to collapse by the same (or a subsequent) shock front (\citealt{elmegreen1992}). The propagating shock front may also serve to trigger the collapse of pre-existing dense cores, thus leading to the creation of a new generation of stars. This method of triggered star formation is known as \emph{Radiative--Driven Implosion} (RDI) and may be responsible for the production of hundreds of stars in each HII region (\citealt{ogura2002}), and perhaps even contributing up to $\sim 15$ \% of the low-to-intermediate mass IMF (\citealt{sugitani2000}).
Shocks continue to be driven into the cloud, compressing the molecular material until the internal density and
pressure is balanced with the pressure of the IBL; after which the shock fronts dissipate and the cloud is
considered to be in a quasi-steady state known as the cometary stage (\citealt{bertoldi1990,lefloch1994}). Once
equilibrium is reached the ionisation front is unable to have any further influence on the internal dynamics of the
cloud, but continues to propagate into the cloud photo-evaporating the molecular gas, which streams into the HII
region, eroding the cloud and accelerating it radially away from the ionising star via the \emph{rocket effect}
(\citealt{oort1955}). The mass loss resulting from photoionisation ultimately leads to the destruction of the
cloud on a timescale of several million years.
A search of the IRAS point source catalogue, correlated with the Sharpless HII region catalogue (\citealt{sharpless1959}) and the ESO(R) Southern Hemisphere Atlas resulted in a total of 89 BRCs being identified with an associated IRAS point source, 44 in the northern and 45 in the southern sky (\citealt{sugitani1991, sugitani1994}; collectively known as the SFO catalogue). The association of an IRAS point source located within these BRCs suggests that these clouds might contain embedded protostars. Several comprehensive studies of individual BRCs have been reported (e.g. \citealt{lefloch1997, megeath1997,yamaguchi1999,codella2001, devries2002,dobashi2002,thompson2004a}), all of which have confirmed their association with protostellar cores. However, the question of star formation occurring more widely in these objects still remains unclear, and evidence for triggered star formation within the small number of BRCs so far observed remains circumstantial and inconclusive. To address these issues we are currently conducting a complete census of the SFO catalogue; here we report the results of a detailed study of four southern BRCs.
In an earlier paper (Thompson, Urquhart \& White, 2004b; hereafter Paper~I) we reported the results of a relatively
low angular resolution radio
continuum survey of the 45 southern BRCs taken from the SFO catalogue. In that survey we detected radio continuum
emission toward eighteen BRCs. In each case the radio emission was found to correlate extremely well with both the
morphology of the optical bright rim seen in the DSS~(R band) image and the Photo-Dominated Region (PDR) traced by
the Midcourse Space eXperiment (MSX)\footnote{Available from the NASA/IPAC Infrared Processing and Analysis Center
and NASA/IPAC Infrared Space Archive both held at http://www.ipac.caltech.edu.} 8~$\mu$m image
(\citealt{price2001}), consistent with the hypothesis that these eighteen BRCs are being photoionised by the
nearby OB star(s).
In this paper we present high resolution molecular line and radio continuum observations toward four clouds (i.e. SFO~58, SFO~68, SFO~75 and SFO~76) from the eighteen photoionised clouds identified in Paper~I which displayed the best correlation between the optical, MIR and radio emission and appeared to possess the simplest geometry. These clouds are thus considered ideal candidates for further investigation into the applicability of the RDI star-formation mechanism. From these observations we investigate the internal and external structure of these clouds and calculate their physical parameters, which when combined with archival data will provide a comprehensive picture of star formation within these BRCs and allow us to determine whether or not it could have been triggered.
The structure of this paper is as follows: in Section~2 we briefly describe the morphologies of the BRCs and the HII regions in which they are located, including a summary of any previous observations that have been reported toward them. In Section~3 we describe our observation strategy and the molecular line and radio continuum observations, followed in Section~4 by the observational results and analyses. In Section~5 we discuss the impact the ionisation front has had on the stability, morphology and future evolution of these clouds and try to evaluate the current state of star formation within each cloud and whether it could have been triggered. We present a summary and our conclusions in Section~6.
\section{Description of individual BRCs}
\label{sect:summary_clouds}
In Figure~\ref{fig:dss_brc_images} we present a large scale DSS image of each BRC and the HII region in which they
are situated. In these we have indicated the position of the ionising star(s) as identified by
\citet{yamaguchi1999} and the IRAS point source with a
cross and an ellipse respectively. In these images it would appear that these BRCs are dense condensations of
material that are connected to an extended shell of molecular gas which surrounds the HII region that are beginning
to protrude into the HII region as the ionisation front ionises the less dense material around them. An arrow has
been added from the centre of each bright rim which points directly toward the ionising star(s). These images
clearly show the cloud morphologies to be curved in the general direction of the ionising stars; this is especially
evident for SFO~58 and SFO~68. The physical parameters relating to each HII region are presented in
Table~\ref{tbl:HII_regions}; the distances and spectral types of the ionising star(s) have been taken from
\citet{yamaguchi1999}, the age of each HII region has been estimated by calculating their expansion timescale as
described by \citet{thompson2004a}.
\begin{table*}
\begin{center}
\caption[HII regions containing selected BRCs]{HII regions containing selected bright-rimmed clouds.}
\begin{tabular}{lccccc}
\hline
\hline
HII & Distance & Associated & Ionising & Spectral & HII region \\
region & (kpc)& BRC & star(s)& type & age (Myr)\\
\hline
RCW 32 & 0.7 & SFO~58 & HD 74804 & B0 V & 0.26 \\
RCW 62 & 1.7 & SFO~68 & HD 101131 & O6.5 N & 1.20 \\
& & & HD 101205 & O6.5 \\
& & & HD 101436 & O7.5 \\
RCW 98 & 2.8 & SFO~75 & LSS3423 & O9.5 IV &0.31 \\
RCW 105 & 1.8 & SFO~76 & HD 144918 & O7 & 0.73 \\
\hline
\end{tabular}
\label{tbl:HII_regions}
\end{center}
\end{table*}
\begin{table*}[!ht]
\begin{center}
\caption[Parameters of the embedded IRAS point sources]{Parameters of the embedded IRAS point sources. (Upper limits are indicated by parenthesis.)}
\begin{tabular}{cccccc}
\hline
\hline
Cloud id. &IRAS id. &IRAS colour type & Luminosity (\lsun) & Spectral type & Mass (\msun) \\
\hline
SFO~58 & 08435--4105 & hot cirrus & 140 & B7 V & 4.2 \\
SFO~68 & 11332--6258 & class 0/UC HII & (3400) & (B1--B2) &(10.6) \\
SFO~75 & 15519--5430 & hot cirrus & 34000 & O9.5 & 20.6 \\
SFO~76 & 16069--4858 & class 0/UC HII & 5600 & B1 & 12.2 \\
\hline
\end{tabular}
\label{tbl:IRAS_sources}
\end{center}
\end{table*}
The IRAS point sources can be clearly seen to lie within the cloud, slightly behind the rim with respect to the
direction of ionisation. The parameters of the IRAS point sources associated with each BRC are presented in
Table~\ref{tbl:IRAS_sources}; the luminosities and classifications of the IRAS colours have been taken from
\citet{sugitani1994}, the spectral types of the embedded IRAS sources have been estimated using the tables of
\citet{panagia1973} and \citet{de_jager1987} assuming that the IRAS infrared luminosity is due to the presence of a
single Zero Age Main Sequence (ZAMS) star, the masses have been estimated using the \mbox{L$_{\rm{star}}$ $\simeq$
M$_{\rm{star}}^{3.45}$} relationship (\citealt{allen1973}). Although the assumption that the infrared luminosity
of each IRAS point source is due to a single embedded star is a rather crude approximation, it does enable an upper
limit to the spectral type of the embedded star to be estimated.
There are a couple of interesting points to note from Table~\ref{tbl:IRAS_sources}. Firstly, three of the four BRCs could possibly harbour Massive Young Stellar Objects (MYSOs), and secondly, two BRCs have IRAS colours consistent with the presence of UC HII regions.
\subsection{SFO~58}
The BRC SFO~58 is situated on the edge of the HII region RCW~32, which is located at a heliocentric distance of $\sim$~700 pc (\citealt{georgelin1973}). The ionising star of RCW~32 has been identified as HD 74804, a B0 V--B4 II star (\citealt{yamaguchi1999}, and references therein), which is located at a projected distance of 1.53~pc from the bright rim of SFO~58.
\subsection{SFO~68}
SFO~68 is located on the northwestern edge of RCW~62, a bright HII region located approximately \mbox{1.7 kpc} from the Sun (\citealt{yamaguchi1999}).
The HII region is driven by three O stars, HD~101131, HD~101205 and HD~101436, which have the spectral types of O6.5, O6.5 and O7.5 respectively (\citealt{yamaguchi1999}). At \mbox{1.7 kpc} the projected distances between SFO~68 and the three ionising stars range between 7.4--14.9 pc, with HD 101131 providing the vast majority of the ionising flux ($>90$ \%) impinging upon the surface of SFO~68.
SFO~68 has IRAS colours consistent with that of an UC~HII region, which has led to SFO~68 being included in several surveys of high-mass star forming regions, such as maser surveys (\citealt{braz1989,macleod1992,caswell1995}), and molecular line surveys (\citealt{zinchenko1995,bronfman1996}). H$_2$O, OH (\citealt{braz1989}) and 6.7~GHz methanol masers (\citealt{macleod1992,caswell1995}) have all been detected toward SFO~68, suggesting the presence of ongoing high-mass star formation within SFO~68. This is supported by the FIR luminosity, from which the presence of a B1--B2~ZAMS star embedded within the cloud can be inferred. The \vlsr~of the detected maser emission range between $-$12 and $-$17~\kms, which is in good agreement with the \vlsr~obtained from our CO observations ($-$16.7~\kms; see Section~4.1).
In addition to the detected masers, a survey of candidate UC HII regions conducted by \citet{bronfman1996} detected CS(\emph{J}=2--1) emission toward the IRAS point source embedded within SFO~68. CS is a high density tracer with a critical excitation density threshold between 10$^4$--10$^5$~cm$^{-3}$, and therefore confirms the presence of dense gas coincident with the position of the IRAS point source. This CS emission has a \vlsr~of $-$15.4 \kms; similar to our CO velocity, and the velocities measured from the masers, confirms they are dynamically associated with this cloud. \citet{zinchenko1995} mapped SFO~68 using the \mbox{CS(\emph{J}=2--1)} transition as well as making single pointing observations toward the molecular peak, identified from the CS map, in the C$^{34}$S(\emph{J}=2--1) and $^{12}$CO(\emph{J}=1--0) molecular transition lines. \citet{zinchenko1995} estimated the main beam temperature and mass of the molecular cloud to be $\sim$~24~K and 425~\msun~respectively.
\subsection{SFO~75}
The bright-rimmed cloud SFO~75 is located on the southeastern edge of the HII region RCW~98. Embedded within SFO~75 is the IRAS point source 15519--5430, which is the most luminous in the SFO catalogue with an FIR luminosity, \emph{L}$_{\rm{FIR}}$ $\sim$ $3.4\times10^{4}$~L$_\odot$. The surface exposed to the HII region is being ionised by LSS 3423, an O9.5~IV star located 0.61~pc to the northwest. RCW~98 lies at a heliocentric distance of 2.8~kpc (\citealt{yamaguchi1999}).
\subsection{SFO~76}
The exciting star of RCW~105 is HD~144918, an O6 star located at a projected distance of 1.79~pc to the northwest of SFO~76 assuming a heliocentric distance of 1.8~kpc (\citealt{yamaguchi1999}). The IRAS point source 16069--4858 is located very close to the rim of the BRC and has an FIR luminosity of 5600~\lsun, and colours consistent with the presence of an UC HII region, which has led to this source being included in many of the same surveys of high-mass star forming regions as SFO~68. However, unlike SFO~68, searches for H$_2$O, OH (\citealt{braz1989,caswell1995}) and 6.7~GHz methanol masers (\citealt{walsh1997}) resulted in non-detections. CS emission has been detected toward SFO~76 (\citealt{bronfman1996}), confirming the presence of dense molecular gas coincident with the position of the IRAS point source. The FIR luminosity is consistent with the presence of a single B1 ZAMS star, supporting the identification of this cloud as a high-mass star forming region. However, the non-detection of maser emission suggests that either SFO~76 is at an earlier stage of development than SFO~68, or that it is simply in the process of forming a cluster of intermediate-mass stars.
\section{Observations and data reduction}
\subsection{Survey strategy}
Theoretical RDI models \citep{bertoldi1989,bertoldi1990,lefloch1994} suggest that the pressure balance between the IBL and the molecular gas can be used to identify clouds in which star formation may have been triggered, or is likely to be triggered in the future. A comparison of the internal and external pressures can result in three possible scenarios depending on whether the cloud is (1) over-pressured, (2) under-pressured or (3) in approximate pressure balance with the respect to the IBL. The implications of each of these scenarios are as follows:
\begin{enumerate}
\item Over-pressured clouds: the ionisation front is likely to have stalled at the surface of these clouds where it
will remain, unable to overcome the internal pressure of the cloud until the pressure in the IBL increases to match
that of the molecular cloud. The ionisation front has no dynamical effect on
these clouds and therefore is unlikely to have influenced any current, or future, star formation within these
clouds.
\item Under-pressured clouds: these clouds are thought to have only recently been exposed to the ionisation front, and although it is highly likely that shocks are currently being driven into these clouds, these shocks have not yet led to the equalisation of the internal and external pressures, leaving the ionisation front stalled at the surface of these clouds. These clouds are thought to be in a \emph{pre-pressure balance state}. Any current star formation present in these clouds is unlikely to have been triggered and is more likely to be pre-existing.
\item Approximate pressure balance: these clouds are thought to have been initially under-pressured with respect to the ionisation front, however, shocks have propagated through the surface layers compressing the molecular gas, leading to an equalisation of the internal and external pressures, and leaving a dense core in its wake as it continues toward the rear of the cloud. These clouds are said to be in a \emph{post-pressure balance state}. Models suggest that the star formation within these clouds may have been triggered (\citealt{lefloch1994}).
\end{enumerate}
In order to evaluate the state of the pressure balance the physical properties of the ionised and molecular gas need to be measured. A combination of radio continuum and molecular line observations have previously been successfully used to determine the current state of several clouds (e.g. \citealt{lefloch1997,white1999,lefloch2002,thompson2004a}). To build on the models, and these previous observational studies, we have made high resolution molecular line and radio continuum observations of four BRCs.
\subsection{CO observations}
\label{sect:co_observations}
Observations of the four BRCs were made during June 2003 in the $J$=1--0 rotational lines of $^{12}$CO, $^{13}$CO and C$^{18}$O using the Mopra millimetre-wave telescope. Mopra is a 22 metre telescope located near Coonabarabran, New South Wales, Australia.\footnote{Mopra is operated by the Australia Telescope National Facility, CSIRO and the University of New South Wales.} The telescope is situated at an elevation of 866 metres above sea level, and at a latitude of 31 degrees south.
The receiver is a cryogenically cooled \mbox{($\sim$ 4 K)}, low-noise, Superconductor-Insulator-Superconductor (SIS) junction mixer with a frequency range between 85--116 GHz, corresponding to a half-power beam-width of \mbox{36--33 $\pm$ 2\arcsec} (Mopra Technical Summary version 10).\footnote{Available at http://www.narrabri.atnf.csiro.au/mopra/.} The receiver can be tuned to either single or double side-band mode. The incoming signal is separated into two channels, using a polarisation splitter, each of which can be tuned separately allowing two channels to be observed simultaneously. The receiver backend is a digital autocorrelator capable of providing two simultaneous outputs with an instantaneous bandwidth between 4--256 MHz.
For these observations a bandwidth of 64 MHz with a 1024-channel digital autocorrelator was used, giving a frequency resolution of \mbox{62.5 kHz} and a
velocity resolution of \mbox{0.16--0.17 km s$^{-1}$} over the \mbox{109--115 GHz} frequency range. For the $^{12}$CO and $^{13}$CO observations the second channel was tuned to \mbox{86.2 GHz} (SiO maser frequency) to allow pointing corrections to be performed during the observations. However, both bands were tuned to \mbox{109.782 GHz} for the C$^{18}$O observations in order to optimise the signal-to-noise ratio. System temperatures were between $\sim 500$--$600$ K for $^{12}$CO and $\sim 250$--$350$~K for both $^{13}$CO and C$^{18}$O depending on weather conditions and telescope elevation, but were found to be stable over the short time periods required to complete each map, varying by no more than approximately 10 \%.\footnote{With the exception of the $^{12}$CO emission observed toward SFO~58 which suffered from large system temperature variation due to poor weather rendering the $^{12}$CO map unreliable. (Only the $^{13}$CO map will be presented for this source.)} Position-switching was used to subtract sky emission. Antenna pointing checks every two hours showed that the average pointing accuracy was better than 10$^{\prime\prime}$ r.m.s..
\begin{table}[!ht]
\begin{center}
\caption[Summary of Mopra CO observations]{Summary of Mopra CO observations.}
\label{tbl:co_observations}
\begin{tabular}{lcccc}
\hline
\hline
Isotope & Frequency & Velocity res. & Grid & Integration \\
(\emph{J}=1--0) & (GHz) & (km s$^{-1}$) & size& time (s)\\
\hline
$^{12}$CO & 115.271 & 0.162 & $9\times9$ & 30 \\
$^{13}$CO& 110.201 & 0.170 &$9\times9$ & 30\\
C$^{18}$O& 109.782 & 0.170 &$3\times3$ & 120 \\
\hline
\end{tabular}
\label{tbl:co_line}
\end{center}
\end{table}
The $^{12}$CO and $^{13}$CO observations consisted of spectra taken of a $9\times9$ pixel grid centred on the cometary head of each cloud, using a grid spacing of 15\arcsec. Each grid position was observed for 30 seconds, interleaved with observations at an off-source reference position for 90 seconds after each row of 9 points. The C$^{18}$O maps consist of a smaller grid of $3\times3$ points with the same spacing, and were centred on the molecular peaks identified from the $^{13}$CO maps. For each of the C$^{18}$O grid positions a total integration time of 2 minutes was used. A summary of the observational parameters is presented in Table~\ref{tbl:co_line} with the grid centres and off-source reference positions presented in Table~\ref{tbl:positions}.
\begin{table*}
\begin{center}
\caption[Pointing centres and off-source reference positions]{Pointing centres and off-source reference positions for all four BRCs.}
\label{tbl:positions}
\begin{tabular}{cccccc}
\hline
\hline
Cloud & IRAS &\multicolumn{2}{c}{Pointing centre} & \multicolumn{2}{c}{Reference position} \\
id.& id.& $\alpha$(2000) & $\delta$(2000) & $\alpha$(2000) & $\delta$(2000) \\
\hline
SFO~58 & 08435-4105& 08:45:25.4 & $-$41:16:02 & 08:45:26.1 & $-$41:15:10 \\
SFO~68 & 11332-6258 & 11:35:31.9 & $-$63:14:51 & 11:24:20.5 & $-$64:09:56 \\
SFO~75 & 15519-5430 & 15:55:50.4 & $-$54:38:58 & 16:02:17.6 & $-$55:19:12 \\
SFO~76 & 16069-4858 & 16:10:38.6 & $-$49:05:52 & 16:04:09.3 & $-$48:48:24 \\
\hline
\end{tabular}
\end{center}
\end{table*}
The measured antenna temperatures, $T_A^*$, were corrected for atmospheric absorption, ohmic losses and rearward
spillover, by taking measurements of an ambient load (assumed to be at 290 K) placed in front of the receiver
following the method of \citet{kutner1981}. To correct for forward spillover and scattering, these data are
converted to the corrected receiver temperature scale, T$_R^*$, by taking account of the main beam efficiency,
\emph{B$_{eff}/F_{eff}$} $\sim0.42\pm0.02.$\footnote{Main beam efficiencies have only been accurately determined at
86, 100 and 115 GHz which have efficiencies of 0.49$\pm0.03$, 0.44$\pm0.03$ and 0.42$\pm0.02$ respectively
(\citealt{ladd2005}). Interpolating from these efficiencies it is easy to show that, although the efficiency is
probably not the same at 110 GHz (i.e. $^{13}$CO and C$^{18}$O ) to that at 115 GHz, any difference is smaller
that the errors involved. We have therefore adopted the 115 GHz efficiency for all three CO lines.} All of the BRCs
have angular diameters larger than $\sim$ 80$^{\prime\prime}$, and therefore to take account of the contribution
made to the measured intensity from extended material that couples to the error beam, we have used the
extended beam efficiency (i.e. $\eta_{xb}\sim0.55$) to correct the $^{12}$CO measurements. Absolute calibration was
performed by comparing measured line temperatures of Orion~KL and M17SW to standard values. We estimate the
combined calibration uncertainties to be no more than 10 \%.
The ATNF data reduction package, SPC, was used to process the individual spectra. Sky-subtracted spectra were obtained by subtracting emission from the off-source reference position from the on-source data. A correction was made to account for the change in the shape of the dish as a function of elevation. The data have been Hanning-smoothed to improve the signal-to-noise ratio, reducing the velocity resolution to 0.32--0.34 \kms.
\subsection{Radio observations}
Centimetre-wave continuum observations were carried out with the Australia Telescope Compact Array (ATCA)\footnote{The Australia Telescope Compact Array is funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO.} between June 2002 and September 2003. ATCA is located at the Paul Wild Observatory, Narrabri, New South Wales, Australia. The ATCA consists of 6$\times$22 metre antennas, 5 of which lie on a 3 km east-west railway track with the sixth antenna located 3 km farther west. Each antenna is fitted with a dual feedhorn system allowing simultaneous measurements of two wavelengths, either 20 \& 13 cm or 6 \& 3.6 cm. Additionally, during a recent upgrade, 12 and 3 mm receivers were installed allowing observations at $\sim$ 20 and $\sim$ 95 GHz respectively. The 6/3.6 cm receiver system was used for the observations of SFO~58, SFO~68 and SFO~76, while observations of SFO~75 were carried out using the new 12 mm receivers. A summary of the observational parameters is presented in Table~\ref{tbl:radio_observations}.
\begin{table*}
\begin{center}
\caption[ATCA radio observational parameters]{Observational parameters for the ATCA radio observations.}
\begin{tabular}{ccccccc}
\hline
\hline
Cloud& Wavelength&\multicolumn{2}{c}{Phase centre} & Array & Integration& Phase\\
id.&(cm)& $\alpha$(J2000) & $\delta$(J2000)& configurations& time (hrs) & calibrators\\
\hline
SFO~58\dotfill&6/3.6& 08:45:25.4 & $-$41:16:02& 750/352/214 &36& 0826--373\\
SFO~68\dotfill&6/3.6& 11:35:31.9 & $-$63:14:51& 750/352/367&18& 1129--58\\
SFO~76\dotfill&6/3.6& 16:10:38.6 & $-$49:05:52& 750/352/367&18& 1613--586 \\
\hline
SFO~75\dotfill&1.3& 15:55:50.4 & $-$54:38:58& 352&12& 1613-586 \\
\hline
\label{tbl:radio_observations}
\end{tabular}
\end{center}
\end{table*}
The 6/3.6 cm observations of SFO~58, SFO~68 and SFO~76 were made at two different frequency bands centred at 4800 and 8309~MHz using bandwidths of 128 and 8~MHz respectively. The second frequency was observed in spectral band mode (using a total of 512 channels, giving a frequency resolution of 15.6~kHz) in order to observe the H92$\alpha$ radio recombination line, however, this proved too faint to be detected toward all three clouds. Each source was observed using three separate configurations over a twelve hour period for SFO~58, and six hours each for SFO~68 and SFO~76 (split into 8$\times$40 minute observations spread over a 12~hour period to optimise \emph{u-v} coverage).
Observations of SFO~75 were made using the 12 mm receivers centred at 23569 MHz ($\sim1.3$~cm) and used a bandwidth of 128 MHz. SFO~75 was observed for a total of 12 hours in a single configuration.
To correct these data for fluctuations in the phase and amplitude caused by atmospheric and instrumental effects, a phase calibrator was observed for two minutes after approximately every 40 minutes of on-source integration ($\sim$ 15 minutes for SFO~75 due to the atmosphere being less stable at higher frequencies). The primary flux calibrator, 1934--638, was observed once during each set of observations to allow for the absolute calibration of the flux density. To calibrate the bandpass the bright point source 1921--293 was also observed once during each set of observations. The phase centres, array configurations, total integration time and phase calibrators are tabulated in Table~\ref{tbl:radio_observations}.
The calibration and reduction of these data were performed using the MIRIAD reduction package
(\citealt{sault1995}) following standard ATCA procedures. The data were CLEANed using a robust weighting of 0.5 to obtain the same sensitivity as natural weighting, but with a much improved beam-shape and lower sidelobe contamination, with the exception of the 3.6 cm data for SFO~58 and SFO~68. Using a robust weighting of 0.5 for these two clouds resulted in poorer imaging of the large scale structure of the ionised gas surrounding their bright rims. (The smaller bandwidth used for the 3.6 cm observations results in these images being a factor of three less sensitive than the 6 cm observations.) For the 3.6 cm data for SFO~58 and SFO~68 a robust weighting of 1 was used, resulting in a slight loss of resolution but improved sensitivity. The data obtained from baselines which included the 6th antenna were found to distort the processed images (due to the large gap in \emph{u-v} coverage at intermediate baselines) and so were excluded from the final images. There are three calibration errors that need to be considered: absolute flux calibration, r.m.s pointing errors and the lack of short baselines. The uncertainties introduced by the first two of these are no more than a few percent each. The shortest baseline for all of these radio observations was 31 metres, and therefore flux loss due to short baselines is not considered to be significant. The combined uncertainty is estimated to be no more than $\sim$ 10 \%.
\section{Results and analysis}
In Figures \ref{fig:co_sfo58}--\ref{fig:co_sfo76} we present plots of the integrated $^{13}$CO (\emph{upper left
panel}), $^{12}$CO (\emph{lower left panel}), radio continuum emission (\emph{upper right panel}) contoured over a
DSS image of the BRC and surrounding region. These images reveal a strong spatial correlation between the
distribution of both the molecular and ionised gas with the optical morphology of the bright rims.
The molecular gas traced by the $^{12}$CO and $^{13}$CO contours displays a similar morphology, both tightly
correlated within the rim of the cloud and with a steep intensity gradient decreasing toward the HII region.
The steep intensity gradient is possibly a consequence of shock compression of the molecular gas which is swept up
in front of the expanding ionisation front and accelerated into the cloud, consistent with the predictions of RDI.
The $^{13}$CO emission reveals the presence of a dense molecular core embedded within every BRC, set back slightly
from the bright rim of the cloud with respect to the direction of the ionising star(s). All of the molecular cores
appear to be centrally condensed, which suggests they may be gravitationally bound, or have a gravitationally bound
object embedded within them, such as a protostar.
There is an interesting difference between the spatial distribution of $^{12}$CO and $^{13}$CO emission toward SFO~75. The peak of the integrated $^{12}$CO emission detected toward SFO~75 is correlated with the position of the bright rim of the cloud, and is slightly elongated in a direction parallel to the morphology of the rim, whereas the integrated $^{13}$CO emission peaks farther back within the molecular cloud and is elongated in a direction perpendicular to the rim. There are a few possible explanations that should be considered: heating of the surface layers of the cloud by the FUV radiation field, the ionisation front is preceeded by a PDR, which could sharpen the $^{12}$CO emission, or that the cloud is angled to the line of sight and that we are seeing the body of the cloud through the bright rim.
The morphology of the radio and optical emission seen toward all four clouds shows the rims to be curving directly away from the ionising star(s), starting to form the typical cometary structure seen toward more evolved BRCs (e.g. the Eagle nebula). The presence of radio emission and its tight correlation with the morphology of the rim strongly supports the hypothesis that an IBL is present between the ionising star(s) and the molecular material within these BRCs. In addition to the IBL, the radio emission image of SFO~58, also reveals the presence of a compact radio source within the optical boundary of the cloud, which is coincident with the position of the molecular core identified in $^{13}$CO image, possibly indicating the presence of an UC HII region within this cloud.
The radio images were analysed using the visualisation package \emph{kvis} (part of the \emph{karma} image analysis
suite (\citealt{gooch1996}). The image parameters and measurements of the peak and source integrated emission are
summarised in Table~\ref{tbl:image5}.
\begin{table*}
\caption[Summary of physical parameters derived from radio observations]{Summary of physical parameters derived from radio observation images.}
\begin{center}
\small
\begin{tabular}{cccccccc}
\hline
\hline
Cloud & & Restoring &Position &Peak & Integrated & Source-averaged & Image \\
id.& $\lambda$ & beam &angle & emission & emission & emission& r.m.s. noise\\
& (cm)& (arcsec)& (degrees)&(mJy/beam) & (mJy) & (mJy/beam)&(mJy/beam) \\
\hline
SFO~58 & 3.6 &$27.6\times17.6$ & 2.8 &2.66 &23.1& 1.33 & 0.11\\
& 6 & $21.6\times12.6$ & 1.7 & 2.38 &38.5& 1.44 & 0.10 \\
\hline
SFO~68 & 3.6 &$20.9\times15.6$ & 7.5 & 5.22 & 69.3 & 1.63 & 0.17 \\
& 6 & $21.6\times15.6$ & 10.42 & 6.36 & 151.1 & 2.97 & 0.16\\
\hline
SFO~76 & 3.6 & $13.3\times10.9$ & 50.1 & 9.84 & 195.4 & 1.74 & 0.25\\
& 6 & $20.6\times17.3$ & 35.4 & 20.82 & 255.7 & 6.99 & 0.30\\
\hline
SFO~75 & 1.3 & $11.5\times4.9$ & $-$9.2 & 3.39 & 30.3 & 0.53 & 0.20 \\
\hline
\end{tabular}
\label{tbl:image5}
\end{center}
\end{table*}
The IRAS point sources within SFO~68, SFO~75 and SFO~76 are located slightly behind the ionisation front, with respect to the ionising star(s), and toward the centre of the bright rim, where one would expect photoionisation induced shocks to focus molecular material, and where the RDI models predict the cores to form (\citealt{lefloch1994}). The positions of the $^{13}$CO peak emission and the IRAS point source seen toward SFO~58 (Figure~\ref{fig:co_sfo58}) are not so well correlated; the IRAS point source is located approximately 1\arcmin~to the south of the molecular peak. It is possible that the IRAS point source is unrelated to the molecular core detected, but is associated with another core which has formed on the edge of the molecular cloud, however, considering the positional correlation of the $^{13}$CO core with the compact radio source and taking account of the pointing errors, size of the IRAS beam ($\sim2^{\prime}$ at 100 $\mu$m), and the positional inaccuracy of the IRAS point source, we do not consider the displacement between the IRAS point source and the centre of the $^{13}$CO core to be significant.\footnote{Comparing the positions of dense cores identified from ammonia maps of the Orion and Cepheus clouds with the positions of their associated IRAS sources \citet{harju1993} found that they could be offset from each other by up to 80$^{\prime\prime}$.}
\subsection{Physical parameters of the cores}
\label{sect:co_analysis}
The angular size of each of the four molecular cores was estimated from the FWHM of azimuthally averaged $^{13}$CO intensity maps. Averaging all spectra within the derived angular size of each core, a source-averaged spectrum was produced for each of the three molecular lines; these spectra are presented in the \emph{lower right panels} of Figures~\ref{fig:co_sfo58}--\ref{fig:co_sfo76}. We note that the baseline for SFO~76 is poor due to the fact that the reference position was found to contain emission at a nearby velocity (i.e. $-$26.5 km s$^{-1}$) to the main line.
Gaussian profiles were fitted to the core-averaged spectra of each core to determine the emission peak, FWHM line width and V$_{\rm{LSR}}$ for each spectral line. These values are listed in Table~\ref{tbl:co_data5} along with the central position of each core derived from a 2D Gaussian fit to the $^{13}$CO emission map. The measured \vlsr~of each source was compared to those reported by \citet{yamaguchi1999} and found to agree to better than 2~\kms, with the exception of SFO~76. For this source \citet{yamaguchi1999} reported a V$_{\rm{LSR}}$ of $\sim$ $-$37 \kms~compared to our measurement of $-$23~\kms. The \vlsr~obtained from our CO observations compares well with that reported by \citet{bronfman1996} from CS observations toward SFO~76 (i.e. $-$22.2~\kms). It is therefore unclear why there is such a large disagreement between the \vlsr~reported by \citet{yamaguchi1999} and that reported in this survey for SFO~76.
Comparisons between the $^{12}$CO lines and other isotopomers for each core reveal no significant variations in the kinematic velocities of the emission peaks for either SFO~75 or SFO~76. However, inspection of the SFO~58 source-averaged $^{12}$CO spectrum shows evidence of a blue wing component not present in either the $^{13}$CO or C$^{18}$O spectra (this spectrum was best fitted by two Gaussian components). Comparing the source-averaged $^{12}$CO spectrum of SFO~68 to the $^{13}$CO and C$^{18}$O spectra reveals a small shift in velocity, which suggests the presence of a broad blue wing. There are several physical phenomenon that could give rise to these observed line wings such as: another cloud on the line of sight, a signature of shock compressions in the surface layers, or the presence of a protostellar outflow.
In order to try to determine the source of these blue wings and to look for kinematic signatures of shocks we produced channel maps and position-velocity diagrams of the $^{12}$CO emission for each cloud. Only the position-velocity diagram produced for SFO~68 shows any evidence of a large scale velocity gradient, revealing the presence of moderate velocity wing components with a FWHM $\sim$ 8~\kms (see Figure \ref{fig:pv_sfo68}); these can either be attributed to a protostellar outflow, or could be tracing the compression/expansion motions of the surface layers of a collapsing cloud. The spatially localised wings seen in the position-velocity diagram are more suggestive of a bipolar outflow, however, our observations do not have either the mapping coverage or the angular resolution necessary to be able to spatially resolve the red and blue outflow lobes that would confirm the presence of an outflow.
The nature of the blue wings is at present unclear, however, taking into account the presence of a UC HII region within SFO~58 (see Section \ref{sect:uchii_region}) and the association of SFO~68 with methanol, OH and H$_2$O masers -- both of which are strong indications of ongoing star formation -- we consider the protostellar outflow hypothesis to be the most likely. We must stress that these are only tentative detections and further observations are required to confirm the presence of protostellar outflows in these two sources.
\begin{table*}
\caption[Physical values derived from CO spectra]{Results of Gaussian fitting of the CO spectra observed toward the molecular cores within BRCs.}
\begin{center}
\begin{tabular}{ccccccc}
\hline
\hline
Cloud id.&\multicolumn{2}{c}{Core position}&Molecular & V$_{\rm{LSR}}$ & Peak T$_R^*$ &FWHM \\
& $\alpha$ (J2000) & $\delta$ (J2000) &line & (km s$^{-1}$) & (K) & (km s$^{-1}$)\\
\hline
SFO~58 &08:45:26 & $-$41:15:10 &$^{12}$CO& 2.1& 11.9& 1.3\\
&& && 3.6& 33.7& 1.51\\
&&&$^{13}$CO & 3.4& 17.3 & 1.5 \\
&&&C$^{18}$O & 3.4 & 2.4& 1.2 \\
\hline
SFO~68 & 11:35:31 & $-$63:14:31&$^{12}$CO& $-$17.1& 23.0& 4.0\\
&&&$^{13}$CO & $-$16.7 & 10.8& 2.5\\
&&&C$^{18}$O & $-$16.4 & 1.8&2.3\\
\hline
SFO~75 & 15:55:49 & $-$54:39:13&$^{12}$CO& $-$37.5 & 29.4& 3.5\\
&&&$^{13}$CO & $-$37.5 & 16.1& 2.6 \\
&&&C$^{18}$O & $-$37.5 & 2.7& 2.3\\
\hline
SFO~76 & 16:10:40 & $-$49:06:17&$^{12}$CO& $-$23.3 & 26.4& 2.7\\
&&&$^{13}$CO & $-$23.4 & 10.1& 2.1\\
&&&C$^{18}$O & $-$23.0 & 1.2& 1.5\\
\hline
\end{tabular}
\label{tbl:co_data5}
\end{center}
\end{table*}
The optically thin transitions of C$^{18}$O, and the moderately optically thick ($\tau$ $<$ 1) $^{13}$CO, were used
to determine the optical depth, gas excitation temperature and C$^{18}$O column density following the procedures
described by \citet{urquhart2004} using the following equations
\begin{equation}
\frac{T_{13}}{T_{18}}=\frac{1-{\rm{e}}^{-\tau_{13}}}{1-{\rm{e}}^{-\tau_{18}}}
\end{equation}
\noindent where $T_{13}$ and $T_{18}$ can be either the corrected antenna or receiver temperatures, and $\tau_{13}$ and $\tau_{18}$ are the optical depths of the $^{13}$CO and C$^{18}$O transitions respectively. The optical depths are related to each other by their abundance ratio
such that $\tau_{13}$ = X$\tau_{18}$, where X is the $^{13}$CO/C$^{18}$O abundance ratio. To estimate the $^{13}$CO/C$^{18}$O abundances we first need to estimate $^{12}$C/$^{13}$C abundances for all four clouds. The galactocentric distances for each source lie between 6.5 and 8.5 kpc, which were compared to the $^{12}$C/$^{13}$C gradient measured over the Galactic disk by \citet{langer1990}. This gives a $^{12}$C/$^{13}$C ratio range of between $\sim$ 45-55, and therefore a value of 50 was adopted for the $^{12}$C/$^{13}$C ratio. Assuming the abundance of $^{16}$O/$^{18}$O in all of the sources to be similar to solar system abundances (i.e. $\sim$ 500; \citealt{zinner1996}), gives a $^{13}$CO/C$^{18}$O abundance ratio of 10.
The gas excitation temperature was estimated using the optically thin C$^{18}$O line in the following
equation,
\begin{equation}
T_R^*\simeq [T_{\rm{ex}}- T_{\rm{bg}}] \tau_{18}
\end{equation}
\noindent where $T_R^*$ is the corrected receiver temperature and $T_{\rm{bg}}$ is the background temperature assumed to be $\simeq$ \mbox{2.7 K}. We have assumed the cores are in Local Thermodynamic Equilibrium (LTE) and can therefore be described by a single temperature (i.e. \mbox{$T_{\rm{ex}} = T_{\rm{Kin}} = T$}). The derived core temperature and optical depth are related to the column densities, $N$ (cm$^{-2}$), through the following equation,
\begin{equation}\label{eq:column_density}
N({\rm{C}}^{18}{\rm{O}})=2.42 \times 10^{14}\tau_{18} \left[ \frac{\Delta v ~T}{1-{\rm{exp}}(-5.27/T)} \right]
\end{equation}
\noindent where $\Delta v$ is the FWHM of the C$^{18}$O line, $\tau_{18}$ and \emph{T} are as previously defined. To convert the C$^{18}$O column densities to H$_2$ column densities a fractional abundance of (C$^{18}$O/H$_2$) = $1.7\times 10^{-7}$ (\citealt{goldsmith1996}) was assumed. The H$_2$ number density was calculated assuming the cores to be spherical, and that contamination effects from emission along the line of sight can be neglected. The total mass (M$_\odot$) was estimated by multiplying the volume densities of each core by the total volume using,
\begin{equation}
M_{\rm{core}}=6.187\times10^{25}R^3n_{\rm{H_2}}\mu m_{\rm{H}}
\label{eq:mass}
\end{equation}
\noindent where \emph{R} is the radius of the core (pc), \emph{n$_{\rm{H_2}}$} is the molecular hydrogen number density (cm$^{-3}$), $\mu$~is the mean molecular weight (taken to be 2.3, assuming a 25 \% abundance of helium by mass), and \emph{m$_{\rm{H}}$} is the mass of a hydrogen atom.
The physical parameters for the molecular cores calculated from the $^{13}$CO and C$^{18}$O data are summarised in Table~\ref{tbl:co_summary5}. We estimate the uncertainties involved in the estimate of the column density to be no more than 50~\%. In estimating the density we have considered the additional uncertainties in the distance to the cores and in the assumptions that the cores are spherical. Taking these additional uncertainties into account we estimate the mass and density calculated to be accurate to within a factor of two.
\begin{table*}
\caption[Summary of parameters derived from $^{13}$CO and C$^{18}$O data]{Summary of parameters derived from $^{13}$CO and C$^{18}$O data.}
\begin{center}
\begin{tabular}{ccccccccc}
\hline
\hline
Cloud & \vlsr &$\tau_{18}$ & \emph{T} & Log(N$_{\rm{H_2}}$) & Log(n$_{\rm{H_2}}$) & Angular & Physical & Mass \\
id. & (\kms) & & (K) & (cm$^{-2}$) & (cm$^{-3}$) & size (\arcsec) & diameter (pc) & (M$_{\odot}$) \\
\hline
SFO~58 &3.4 & 0.07 & 37.0 & 22.53 & 4.89 & 47 & 0.16 & 9.5 \\
SFO~68 & $-$16.7&0.06 & 29.4 & 22.51 & 4.36 & 68 & 0.46 & 66.4 \\
SFO~75 & $-$37.5&0.13 & 23.5 & 22.75 & 4.53 & 41 & 0.56 & 177.3\\
SFO~76 &$-$23.3 &0.05 & 26.7 & 22.26 & 4.06 & 59 & 0.52 & 48.1 \\
\hline
\end{tabular}
\label{tbl:co_summary5}
\end{center}
\end{table*}
The physical parameters of the cores are similar, with the densities ranging between $\sim$ 10$^4$--10$^5$~cm$^{-3}$, the physical diameters varying between $\sim$ 0.2--0.6~pc, kinetic temperatures ranging from $\sim$~24 to 37~K, and masses from $\sim$~10--180~M$_{\odot}$. The temperature of all of the cores are significantly higher than would be expected for starless cores \mbox{(\emph{T} $\sim$ 10 K}; \citealt{evans1999}), which suggests these cores possess an internal heating mechanism, possibly a YSO, or an UC HII region (as indicated by the IRAS colours; see Table~\ref{tbl:IRAS_sources}). It is possible these cores are being heated by the surrounding FUV radiation field, however, if that were the case, we would expect to find a temperature gradient that peaked at the position of the bright rim and decreased with distance into the cloud, but this is not observed.
\subsection{Physical parameters of the ionised boundary layers}
\label{sect:IBL}
\label{sect:radio_analysis}
Using the radio emission detected toward the rims of SFO~58, SFO~68, SFO~75 and SFO~76 we can quantify the ionising photon flux impinging upon them. Making the assumption that all of the ionising photon flux is absorbed within the IBL we can determine the photon flux, $\Phi$ (cm$^{-2}$ s$^{-1}$), and the electron density, n$_{e}$ (cm$^{-3}$), using the following equations which have been modified from Equations~2 and 6 of \citet{lefloch1997} (see Paper~I for details):
\begin{equation}
\Phi=1.24\times10^{10}S_{\nu}T_e^{0.35}\nu^{0.1}\theta^{-2}
\end{equation}
\begin{equation}
n_{e}=122.21\times\sqrt{\frac{S_{\nu}T_e^{0.35}\nu^{0.1}}{\eta R\theta^2}}
\end{equation}
\noindent where \emph{$S_{\nu}$} is the integrated flux density in mJy, $\nu$ is the frequency at which the integrated flux density is evaluated in GHz, $\theta$ is the angular diameter over which the flux density is integrated in arc-seconds, $\eta$R is the shell thickness in pc, and \emph{T$_{e}$} is the electron temperature in K. Note, an average HII~region electron temperature of \mbox{$\sim$ 10$^{4}$ K} and \mbox{$\eta$ = 0.2} (\citealt{bertoldi1989}) have been assumed.
The values calculated for the ionising photon fluxes, and electron densities within the IBLs of the four clouds, are presented in Table~\ref{tbl:physical_parameters}. The main uncertainties in these values are due to flux calibration ($\sim10$~\%), the approximation of the electron temperature, $T_e\simeq10^4$~K (e.g. a difference in the electron temperature of 2000~K corresponds to an uncertainly in the photon flux of $\sim$~25~\%), and the $\eta=0.2$. The uncertainties in flux calibration and electron temperature combine to give a total uncertainty in the calculated photon fluxes and electron densities of no more than 30~\%. However, approximating $\eta=0.2$ leads to the the electron density being underestimated by at most a factor of $\sqrt{2}$, and since the uncertainty in $\eta=0.2$ dominates the others it effectively sets a lower limit for the electron density.
\begin{table*}[!hbt]
\caption[Summary of derived physical parameters of the IBL]{Summary of derived physical parameters of the IBL.}
\begin{center}
\begin{tabular}{cccccc}
\hline
\hline
Cloud & \multicolumn{3}{c}{Photon fluxes (10$^8$ cm$^{-2}$~s$^{-1}$)} & \multicolumn{2}{c}{Electron densities $n_e$(cm$^{-3}$)}\\
id.& Predicted $\Phi_P$ & Peak $\Phi$ & Mean $\Phi$ & Peak & Mean \\
\hline
SFO~58 & 20.4 & 31.9 & 19.3 & 338& 262 \\
SFO~68& 126.0 & 68.5 & 32.0 & 341 & 233 \\
SFO~75 & 448.0 & 257 & 40.0 & 839 & 332 \\
SFO~76 & 441 & 263.1& 46.5 & 1242& 526 \\
\hline
\end{tabular}
\label{tbl:physical_parameters}
\end{center}
\end{table*}
Predicted ionising fluxes were calculated from the Lyman flux of the candidate ionising stars and their projected
distances to the clouds following the method described in \mbox{Paper I}. Comparing the predicted fluxes to the
measured fluxes, and taking into account any attenuation between the OB star(s) and the clouds, we find good
agreement (to within a factor of two). In the majority of cases the measured fluxes at the surface of the BRCs are
lower than the predicted flux, which is to be expected, as the predicted flux is a strict upper limit. The one
notable exception to this is SFO~58 where the measured flux is a factor of one and half times greater than the
predicted upper limit. There are two possible explanations for this discrepancy: the spectral type of the ionising
star is incorrect, or the distance to the HII region is incorrect. We favour the former as a misclassification of
the ionising star's spectral type by even half a spectral class can alter its predicted Lyman flux by up to a factor of two,
which would more than account for the difference in fluxes reported here.
The high resolution radio observations result in a much tighter correlation between the predicted and measured
fluxes than were found with the low angular resolution data presented in Paper I. The variation between measured and
predicted fluxes for the low resolution data for these four clouds ranged from a factor of a few to more than ten
(in the case of SFO~58). Moreover, the analysis of the distribution of radio emission suggests that two sources
(SFO~68 and SFO~76) lie in the foreground relative to the locations of the ionising star(s), and thus are located
farther from the ionising star(s) than the projected distance, used to derive the predicted flux, would suggest. In
these two cases the correlation could be considerably better than the factor of two quoted above.
The main reason for the improved correlation between the predicted and measured fluxes is because the higher
resolution observations have been able to resolve the radio emission, resulting in the detected emission being much
more tightly peaked. This has allowed more accurate measurements of the flux density to be obtained, and
consequently more realistic values for the ionising fluxes and electron densities to be calculated. Values
calculated from the low resolution observations presented in Paper I suffer due to the large size of the
synthesised beams (typically $\sim$ 90\arcsec~ and $\sim$ 60\arcsec~ for the 20~cm + 6~cm and 13~cm + 3~cm
observations respectively) which dilutes the emission if the IBL is not resolved, resulting in significantly lower
flux densities being measured. This explains why, in every case, the high resolution radio observations have
resulted in an increase in the calculated ionising fluxes. Another reason is the vast improvement in the \emph{u-v}
coverage obtained by using multiple array configurations, which allows the brightness distribution of the emission
to be more accurately deconvolved from the visibility data. Therefore, although low resolution radio observations
may be useful in identifying clouds that are likely to possess an IBL and are thus subject to photoionisation from
the nearby OB star(s), their main use is to limited to the determination of global estimates for the physical
parameters of the IBLs.
The mean electron densities calculated for each cloud range between 233--526 cm$^{-3}$, considerably greater than the critical value of \emph{n}$_{\rm{e}}\sim$ 25 cm$^{-3}$ above which an IBL is able to develop around the cloud (\citealt{lefloch1994}). The excellent correlation of the radio emission with the bright-rim of the clouds strongly supports the presence of an IBL at the surface of each of these clouds, confirming their identification as potential triggered star forming regions. It is therefore clear that these clouds are being photoionised by the nearby OB star(s), however, it is not yet clear to what extent the ionisation has influenced the evolution of these clouds, and what part, if any, it has played a part in triggering star formation within these clouds.
\subsection{Evaluation of the pressure balance}
\label{sect:pressure_balance}
\label{sect:implications_pressure_balance}
In this section the results of the molecular line and radio observations will be used to evaluate the pressure balance between the hot ionised gas of the IBLs, and the cooler neutral gas within the BRCs. Following the method described in Paper~I we calculated the internal ($P_{\rm{int}}$) and external ($P_{\rm{ext}}$) pressures (N cm$^{-2}$) using,
\begin{equation}
P_{\rm{int}}\simeq\sigma^{2}\rho_{\rm{int}}
\label{eq:internal_pressure}
\end{equation}
\begin{equation}
P_{\rm{ext}}=2\rho_{\rm{ext}}c^2
\end{equation}
\noindent where $\sigma^2$ is the square of the velocity dispersion (i.e. $\sigma^2=\langle\Delta
v\rangle^2/(8\rm{ln}2$), where $\Delta v$ is the core-averaged C$^{18}$O line width (\kms)), $\rho_{\rm{int}}$ is
the core-averaged density calculated in Section~\ref{sect:co_analysis}, $\rho_{\rm{ext}}$ and $c$ are the ionised
gas density and sound speed (assumed to be $\sim$ 11.4 \kms) respectively. The external pressure term includes
contributions from both thermal and ram pressure. The electron densities calculated in Section~\ref{sect:IBL}
were used to estimate the ionised gas pressures for each cloud's IBL. The calculated internal and external
pressures are presented in Table~\ref{tbl:pressure_balance}.
\begin{table}
\caption[Summary of the pressure balance analysis]{Evaluation of the pressure balance.}
\begin{center}
\label{tbl:pressure_balance}
\begin{tabular}{ccc}
\hline
\hline
Cloud & \multicolumn{2}{c}{Pressure ($P/k_B$) ($10^6$ cm$^{-3}$ K)}\\
id.& Internal ($P_{\rm{int}}$)& External ($P_{\rm{ext}}$)\\
\hline
SFO~58 & 5.3 & 7.8\\
SFO~68 & 4.4 & 7.0 \\
SFO~75 & 9.1 & 26.5\\
SFO~76 & 1.2 & 15.7\\
\hline
\end{tabular}
\end{center}
\end{table}
The largest two uncertainties in the calculation of the molecular pressure are: the uncertainty in the observed
line temperature, and the possibility that the C$^{18}$O derived density may be affected by depletion, either onto
dust grain ice mantles, or through selective photo-dissociation. The uncertainties in the densities are thought to
be no more than a factor of two, the effects of depletion and photo-dissociation are harder to quantify. However,
all of the core temperatures are considerably larger than 10 K, where depletion is expected to be greatest, and
given that they are embedded within the clouds, away from the ionisation front, where they are shielded from much of the
ionising radiation, these effects are not thought to be significant. We estimate the molecular pressures presented
to be accurate to within a factor of two. The IBL pressures are lower limits (due to the electron densities being
lower limits) and taking account of the uncertainties are considered to be accurate to $\sim$ 30~\%.
As discussed in Section~3.1 theoretical models (\citealt{bertoldi1989,lefloch1994}) have revealed the pressure balance to be a sensitive diagnostic that can be used to determine the evolutionary state of BRCs. Comparing the internal and external pressures calculated from the molecular line and radio observations (presented in Table~\ref{tbl:pressure_balance}), reveals that all of the clouds are under pressured with respect to their IBLs, and are thus in the process of having shocks driven into them. However, taking account of the possible factor of two uncertainty in our calculations of these parameters, it is possible that two of these clouds are in approximate pressure balance (i.e. SFO~58 and SFO~68); these clouds are likely to be in a post-pressure balance state, and it is therefore possible that any current, or imminent, star formation within these clouds could have been triggered.
The remaining two clouds, SFO~75 and SFO~76, are under-pressured by factors of three and twelve respectively, with respect to their IBLs, strongly suggesting that these clouds have only recently been exposed to the HII region and that shocks are currently being driven into the surface layers of these clouds, closely followed by a D-critical ionisation front. These clouds are likely to be in a pre-pressure balance state where the shocks have not propagated very far into the surface layers; it is therefore unlikely that the molecular cores within these two clouds have been formed by RDI, but are more likely to pre-date the arrival of the ionisation front and have only recently been exposed to the HII region. Any current star formation taking place within these clouds is unlikely to have been triggered.
\subsection{Compact radio source associated with SFO~58}
\label{sect:uchii_region}
The 6 cm radio continuum image (see \emph{upper left panel} Figure~3) of SFO~58 clearly shows the presence of a radio source positionally coincident with the $^{13}$CO core embedded within this BRC, both of which lie at the focus of the BRC where the photoionisation induced shock is expected to concentrate the majority of the mass within the cloud. The radio source and embedded $^{13}$CO core are offset from the IRAS point source, but their positional correlation hints at a possible association between the two. At a distance of 700~pc the angular size of the compact radio source ($\sim$~18\arcsec) corresponds to a physical diameter of $<$~0.06~pc, which suggested it might be a compact HII region similar to those found within other BRCs reported in \mbox{Paper I} (i.e. SFO~59, SFO~62, SFO~74, SFO~79 and SFO~85; see \citealt{urquhart2004} for a detailed investigation of SFO~79).
The radio flux of the compact radio source is consistent with the presence of a single embedded B2--B3 ZAMS star. Furthermore, the correlation of the position of the radio source with that of the molecular core detected in the CO observations, both of which are located at the focus of the bright rim (see Figure~\ref{fig:co_sfo58}), offer some circumstantial support for this hypothesis. However, it is possible that the presence of the radio source is an unfortunate alignment of the cloud with an extragalactic background source. To try to determine the nature of this radio emission we attempted to calculate the spectral index ($\alpha$) of the emission using the integrated flux at both frequencies (i.e. $S_\nu \propto \nu^{\alpha}$), however, this proved inconclusive due to the poor sensitivity of the 3.6 cm map. We therefore tentatively identify this compact radio source as a possible compact HII region embedded within SFO~58.
\begin{table}
\caption[Physical parameters of CRS 4]{Derived parameters of the compact radio source associated with SFO~58.}
\begin{center}
\begin{tabular}{lc}
\hline
\hline
Position\dotfill & $\alpha$(J2000) = 08:45:26 \\
\dotfill&$\delta$(J2000) = $-$41:15:06\\
Source size\dotfill & $<$ 18\arcsec \\
Physical diameter & $<0.06$ pc \\
3.6 cm flux density\dotfill & 1.69 mJy\\
6 cm flux density\dotfill & 2.28 mJy\\
log(\emph{N}$_i$)\dotfill & 44.0 photon s$^{-1}$\\
Spectral type\dotfill & ZAMS B2--B3 \\
\hline
\end{tabular}
\label{tbl:embedded_source}
\end{center}
\end{table}
\section{Discussion}
Whilst a full hydrodynamic analysis is beyond the scope of this paper, an insight into the potential effect that exposure to the FUV radiation field has had upon the stability of the cores can be gained from a simple static analysis. Additionally, we will estimate the lifetime of these clouds in the light of the continued mass loss through photo-evaporation by the nearby OB star(s), and evaluate its effect on future star formation within these clouds.
\subsection{Gravitational stability of the cores pre- and post-exposure of the clouds to UV radiation}
\label{sect:stability}
To investigate the effect that the arrival of the ionisation front has on the stability of the cores we need to compare the stability of the cores while they were still embedded within their parental molecular cloud to that of the cores once exposed to the FUV radiation field. In the following analysis we implicitly assume that the cores pre-date the arrival of the ionisation front (as shown in the previous section, this is certainly likely for the cores embedded with SFO~75 and SFO~76), and that there was negligible external pressure from the surrounding molecular material.
The pre-exposure stability of the cores can be estimated using the standard virial equation to derive the virial mass, $M_{\rm{vir}}$ (\msun). Comparing these masses with the core masses calculated from the CO data will give an indication of their pre-exposure stability. The virial mass can be calculated using the standard equation (e.g. \citealt{evans1999}),
\begin{equation}
M_{\rm{vir}}\simeq210R_{\rm{core}} \langle\Delta v\rangle^2
\end{equation}
\noindent where \emph{R}$_{\rm{core}}$ is the core radius (pc) measured from the integrated $^{13}$CO maps, $\Delta v$ is the FWHM line width of the C$^{18}$O line (\kms). The results are presented in Table~\ref{tbl:virial_mass} with the core masses derived from the CO observations. Comparing the calculated masses of each core with their virial mass, it is clear that three cores were gravitationally stable against collapse while still embedded within their parental molecular cloud, however, taking the errors into account it is possible that SFO~75 was close to being unstable to gravitational collapse.
Now the effect of exposure to the FUV radiation field of the HII region will be examined. This will be estimated
using a modified version of the virial mass (i.e.~the Bonnor Ebert approach) that takes account of the external
pressure of the surrounding medium, in this case the pressure of the IBL. Following the notation of
\citet{thompson2004a}, this pressure-sensitive virial mass will be referred to as the \emph{pressurised virial
mass}, $M_{\rm{pv}}$ (\msun), given as, \\ \begin{equation} M_{\rm{pv}}\simeq5.8\times10^{-2}\frac{\langle \Delta
v\rangle^4}{G^{3/2}P_{\rm{ext}}^{1/2}} \end{equation} \\ \noindent where $P_{\rm{ext}}$ (N m$^{-2}$) is the
pressure of the ionised gas within the IBL, \emph{G} is the gravitational constant, and $\Delta v$ is in \kms. The
calculated values for the pressurised virial masses are presented in Table~\ref{tbl:virial_mass}.
This equation is sensitive to the accuracy of the measured line width, and taking into account the errors involved with Gaussian fits to the spectral lines, the calculated values for the pressurised virial masses are considered to be accurate to within a factor of two (\citealt{thompson2004a}). In this case it was assumed that the external pressure acts over the entire surface of the core, not just the side illuminated by the OB star, and that the cores were pre-existing cores recently uncovered by the expansion of the ionisation front. This is a rather simplistic approach but does allow the effect that exposure to the FUV radiation has on the stability of the cores to be investigated.
\begin{table}
\caption[Summary of core masses: physical, virial and pressurised virial masses]{Summary of core masses as derived from the CO data as well as the virial masses, $M_{\rm{vir}}$, and pressurised virial masses, $M_{\rm{pv}}$.}
\begin{center}
\small
\begin{tabular}{cccccc}
\hline
\hline
Cloud& $M_{\rm{co}}$ & $M_{\rm{vir}}$ & $M_{\rm{pv}}$ & $\dot{M}$ & Lifetime \\
id.& (M$_\odot$) &(M$_\odot$) &(M$_\odot$) & (M$_\odot$ Myr$^{-1}$) & (Myr)\\
\hline
SFO~58 & 9.5 & 24.2 & 10.7 & 14.9 & 0.64\\
SFO~68 & 66.4 & 255.5 & 152.2 & 58.6& 1.13\\
SFO~75 & 177.3 & 311.1 & 78.2 & 64.6& 2.75\\
SFO~76 & 48.1 & 122.8 & 18.4 & 12.6& 3.80\\
\hline
\end{tabular}
\label{tbl:virial_mass}
\end{center}
\end{table}
Comparing the values for the virial and pressurised virial masses shows that exposure to the FUV radiation field dramatically reduces the mass above which the clouds become unstable against gravitational collapse. The difference between the virial and pressurised viral masses range from a factor of $\sim$~2 (i.e. SFO~58 and SFO~68) to a factor of $\sim$ 7 for SFO~76. Taking account of the errors involved in this analysis, only differences of a factor of four or larger are significant. We are therefore unable to determine if exposure to the FUV radiation has had an impact on the stability of SFO~58 or SFO~68, however, it is clear that the exposure is likely to have had a significant impact on the stability of SFO~75 and SFO~76. Moreover, comparing the pressurised virial masses of the cores with those estimated from the CO data reveals that SFO~75 and SFO~76 are both more than a factor of two more massive than the critical pressurised viral mass. It is therefore likely that the exposure of these two cores to their respective HII regions has rendered them unstable against gravitational collapse. However, detailed hydrodynamical or radiative transfer modelling (e.g. \citealt{thompson-white2004,deVries2005}) of higher signal-to-noise molecular line data are needed to investigate the presence of collapse motions in these clouds.
\subsection{Eventual fate of the BRCs}
Once a cloud has reached the cometary stage the shocks dissipate, however, the cloud's mass continues to be slowly
eroded away as the D-type ionisation front continues to propagate into it (\citealt{lefloch1994}). In this
situation the propagation of the ionisation front leads to a constant mass loss in the form of a photo-evaporated
flow into the HII region (\citealt{megeath1997}). The amount of material within the boundary of a cloud is finite,
and thus the effect of the ionisation is to slowly erode the limited reservoir of material available for star
formation. This mass loss eventually results in the total ionisation and dispersion of the cloud, and perhaps the
disruption of ongoing star formation either by disrupting any molecular cores before the accretion phase has begun,
or by exposing the protostar to the FUV radiation field before the accretion phase has finished. Once the
protostar has been exposed much of the surrounding envelope of molecular material becomes ionised, therefore
limiting the possible size of the forming protostar (see \citealt{whitworth2004}). Therefore the mass loss rate is
an important parameter that can help determine the effect photoionisation has on current, and future star formation
within BRCs, and in estimating their lifetime.
To evaluate the mass loss we use Equation 36 from \citet{lefloch1994} which relates the mass loss (M$_\odot$~$\rm{Myr}^{-1}$) to the ionising flux illuminating the cloud ($\Phi$ photons cm$^{-2}$ s$^{-1}$), i.e.
\begin{equation}
\dot{M}=4.4\times10^{-3}\Phi^{1/2}R_{\rm{Cloud}}^{3/2}
\label{eqn:mass_loss}
\end{equation}
The globally averaged photon flux calculated in Section~\ref{sect:radio_analysis} and the cloud radii presented in Paper I were used to estimate the mass loss rate for each cloud using Equation~\ref{eqn:mass_loss}; these values are presented in Table~\ref{tbl:virial_mass} along with an estimate for the lifetime of each cloud. The BRC mass loss rates range between $\sim$ 12--59 $M_\odot$ Myr$^{-1}$, corresponding to cloud lifetimes from as little as $6\times10^5$ yr to several Myr.
The accretion phase of protostar formation is known to last for several \mbox{10$^5$ yr} (\citealt{andre2000}). Therefore any ongoing, or imminent, star formation within SFO~58, SFO~68, SFO~75 and SFO~76 will be unaffected by the ionisation and mass loss, especially SFO~68 where the star formation already appears to be well developed and unlikely to be affected by the mass loss experienced by the cloud. However, the future star formation within SFO~58 may be adversely affected as the ionisation front propagates into the cloud. Although there is no evidence of any current star formation taking place within either SFO~75 or SFO~76, we have shown these clouds are likely to be undergoing RDI as well as having sufficiently long lifetimes for RDI to be a viable method of triggered star formation.
\subsection{Star formation and the evolution of the BRCs}
Direct evidence of whether star formation within these clouds has been triggered is not readily available, however,
it is possible to investigate the probability that the star formation has been triggered by considering the
circumstantial evidence. It is interesting to note that there is strong evidence for the presence of ongoing star
formation within the two clouds (SFO~58 and SFO~68) that fall into the post-pressure balance cloud category, such
as the molecular outflows (see Section~\ref{sect:co_analysis}), association with OH, H$_2$O and methanol masers
(\citealt{braz1989,macleod1992,caswell1995}, i.e. SFO~68) and the possible association with an embedded UC HII
region (Section~\ref{sect:uchii_region}, i.e. SFO~58). Moreover, the association of the UC~HII region with SFO~58
and of H$_2$O and methanol masers with SFO~68 --- which are respectively and almost exclusively associated with
Class 0 protostars (\citealt{furuya2001}) and high-mass star formation (\citealt{minier2003}) --- lead us
to conclude that the star formation within these two clouds is relatively recent, being no more than a few 10$^5$
years old. Contrary to the evidence of recent high-mass star formation within the post-pressure balanced clouds we
find no evidence for any ongoing star formation within either of the two pre-pressure balance clouds (SFO~75 and
SFO~76), short of the presence of the embedded IRAS point source.
If, as suggested by the RDI models (e.g. \citealt{lefloch1994, vanhala1998}) and observations (e.g. \citealt{sugitani1991}), that rim morphologies represent an evolutionary sequence (see Figure~\ref{fig:rim_classification}), we should expect to find clouds at similar stages of evolution to exhibit similar physical parameters, and furthermore, the star formation within clouds at different evolutionary states to be at different stages of development. It is therefore useful to compare the observational results to the evolutionary sequence predicted by the models to investigate any differences in the star formation within clouds with different rim morphologies. However, we must point out that the boundary conditions of where the clouds meet the larger-scale molecular material are very important and may affect the following analysis.
To emphasise the morphology of each rim a black curved line has been fitted (by eye) to the radio contours following the minimum gradient of the emission (see \emph{upper right panel} of Figures~\ref{fig:co_sfo58}--\ref{fig:co_sfo76}). The four clouds separate quite nicely into two morphological groups after comparison of the rim morphologies presented in Figure~\ref{fig:rim_classification}. SFO~58 and SFO~68 are typical of a type A rim morphology, whereas SFO~75 and SFO~76 show only slight curvature intermediate between type A and type 0.
Comparing these rim morphologies with those predicted by the RDI models of \citet{lefloch1994} we find that SFO~75
and SFO~76 closely resemble the 0.036 My and SFO~58 and SFO~68 closely resemble the 0.126 My snapshots (i.e. Figure
4a and b of \citet{lefloch1994}). This comparison should be viewed with caution as the Lefloch \& Lazareff models
were calculated for a simple cloud with specific ionising fluxes and so the absolute timescales are more than
likely invalid for our ensemble of BRCs. However the comparison between individual clouds and their relative model
ages does support our conclusion that both SFO~75 and SFO~76 are in the early stages of ionisation, having only
recently been exposed to the ionising radiation of their HII region, and that SFO~58 and SFO~68 have been exposed
for a significantly longer period of time. Moreover, we find that the evolutionary age of the SFO~58 and SFO~68
suggested by the models compares well with the age of the star formation indicator mentioned in an earlier
paragraph (i.e. $\sim$ several 10$^5$ yr).
Although we have not been able to conclusively prove that the star formation has been triggered within SFO~58 and SFO~68, we have shown that is possible, if not likely. Furthermore, we have shown there are clearly significant morphological and star formation differences between the post- and pre-pressure balance clouds in this survey, which are consistent with the predictions of RDI models. From the observations of the four clouds presented here, it seems that there is reasonably good evidence to support the suggestion that the schematic presented in the Figure~\ref{fig:rim_classification} is an evolutionary picture of the changing morphology of BRCs under the influence of photoionisation.
\section{Summary and conclusions}
In this paper the results of a detailed investigation of four BRCs (SFO~58, SFO~68, SFO~75 and SFO~76) are
presented, including high resolution radio molecular line and continuum observations obtained with the Mopra
millimetre telescope and the ATCA. The main aim is to distinguish between pre- and post-pressure balance clouds,
and to evaluate to what extent the star formation within these BRCs has been influenced by the photoionisation from
the nearby OB star(s). Each of the BRCs was mapped using the $^{12}$CO, $^{13}$CO and C$^{18}$O rotational
transitions using the Mopra telescope. To complement the molecular line observations, high resolution radio
continuum maps of all four BRCs were obtained using the ATCA.
The CO observations reveal the presence of a dense molecular core within every BRC, located behind the bright rim and, in most cases, coincident with the position of the IRAS point sources. The distribution of the $^{13}$CO emission maps suggest that the cores have a spherical structure and are centrally condensed, consistent with the presence of an embedded gravitationally bound object, such as a YSO. The H$_2$ number densities and cores masses range between 3--$8\times10^4$~cm$^{-3}$ and 10--180~\msun~respectively. The core temperatures ($\sim$ 30~K) are significantly higher than expected for starless cores ($\sim$ 10~K, \citealt{clemens1991}), supporting evidence for the presence of an internal heating source, such as a protostar.
The high angular resolution radio observations have confirmed the presence of an IBL surrounding the rim of all four
clouds, with the detected emission displaying excellent correlation with the morphology of the cloud rim seen in
the optical images. The increased resolution and improved \emph{u-v} coverage of the radio observations has
resulted in significantly higher flux densities being measured, which in turn has led to higher values for the
ionising fluxes, which are now more in line with the predicted fluxes than the low angular resolution observations
presented in Paper I. The electron densities are all significantly higher than the critical density of 25~cm$^{-3}$
(\citealt{lefloch1994}) above which an IBL can form and be maintained. All these facts strongly support the
hypothesis that these clouds are being photoionised by the nearby OB star(s).
CO and radio continuum data are used to evaluate the pressure balance at the HII/molecular region interface. Comparing these values the clouds are found to fall into two categories: pre- and post-pressure balance states; SFO~75 and SFO~76 are identified as being in a pre-pressure balance state, and SFO~58 and SFO~68 are identified as being in a post-pressure balance state (taking account of the errors, see Section~\ref{sect:pressure_balance}). We draw the following conclusions from our observations:
\begin{enumerate}
\item Analysis has revealed clear morphological and evolutionary differences between the pre- and post-pressure balance clouds:
\begin{itemize}
\item The two clouds identified in this survey as being in a post-pressure balance state are also the same two identified as having a type A rim morphology and show strong evidence of ongoing high- to intermediate-mass star formation (e.g. UC HII region, masers and molecular outflows).
\item The two clouds identified as being in a pre-pressure balance state have all been classified as type 0 clouds, and show no evidence of recent or ongoing star formation.
\end{itemize}
\item The two classifications of rim morphologies, type 0 and type A, correspond to the 0.036 Myr and 0.126 Myr snapshots from the \citet{lefloch1994} RDI model, where the time indicates the exposure time of a cloud to an ionising front. This is consistent with our conclusion that SFO~75 and SFO~76 have only recently been exposed to the ionisation front and that SFO~58 and SFO~68 have been exposed for a significantly longer period of time. Moreover, the morphological age predicted by the RDI models is similar to the estimated age of the star formation within SFO~58 and SFO~68, support the possibility that the star formation has been triggered.
\item Using a simple pressure-based argument, exposure to the FUV radiation field within the HII regions is shown
to have a profound effect on the stability of these cores. All of the cores were stable whilst embedded in their
natal molecular clouds, however, in all cases exposure to the FUV radiation field of the HII region reduces the
stability of the cores by more than a factor of two (in the case of SFO~76 by almost a factor of seven). The
reduced stability leaves two clouds on the edge of being unstable to gravitational collapse, and two clouds that
have masses at least a factor of two greater than the pressurised virial masses. Analysis of the pre-exposure
stability indicates that it is possible that the core embedded within SFO~75 was unstable to gravitational collapse
prior to being exposed to the ionisation front. From this analysis we conclude that the cores within SFO~75 and
SFO~76 are both unstable to gravitational collapse.
\item The radio continuum observations toward SFO~58 reveal the presence of an embedded compact radio source within
the optical boundary of the bright rim. The radio source is offset by 1\arcmin~from the position of the IRAS point
source, but correlates extremely well with the position of the peak of the molecular core, both of which are
located at the focus of the bright rim, suggesting that the radio source is associated with the cloud. The size
($<$~0.06 pc) and integrated radio flux are all consistent with the presence of an UC HII region embedded within
the molecular core. The radio flux of this source is consistent with the presence of an UC HII region excited by a
single ZAMS B2--B3 star. Inspection of the core-averaged $^{12}$CO spectrum reveals evidence of a substantial blue
wing, possibly indicating the the presence of a molecular outflow. This is the first tentative evidence for ongoing
star formation within SFO~58.
\item The physical sizes and masses of the molecular cores are typically larger than a single star might be expected to form from. The IRAS luminosities are generally much higher than the bolometric luminosities typical for individual Class 0 and Class I stars (\citealt{andre1993,chandler2000}). Additionally, evidence that two BRCs are active high-mass star forming regions is presented in this paper, which form exclusively in clusters. It is therefore highly likely that the presence of the IRAS point source indicates the presence of multiple protostellar systems rather than a single protostar. Higher resolution molecular line observations are required to investigate the possible multiplicity of these sources.
\end{enumerate}
\begin{acknowledgements}
The authors thank the Director and staff of the Paul Wild Observatory, Narrabri, New South Wales, Australia, for their hospitality and assistance during our Compact Array and Mopra observing runs, and the Mopra support scientist Stuart Robertson for his help and advice. We would also like to thank the referee Bertrand Lefloch for his very helpful comments and suggestions. This research would not have been possible without the SIMBAD astronomical database service operated at CDS, Strasbourg, France, and the NASA Astrophysics Data System Bibliographic Services. We have made use of Digitised Sky Survey images was produced at the Space Telescope Science Institute under U.S. Government grant NAG W-2166. These images are based on photographic data obtained using the Oschin Schmidt Telescope on Palomar Mountain and the UK Schmidt Telescope. The plates were processed into the present compressed digital form with the permission of these institutions. This research has also made use of the NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology,
under contract with the National Aeronautics and Space Administration.
\end{acknowledgements}
\bibliography{3417}
\bibliographystyle{aa}
|
Title:
Far Ultraviolet Spectral Images of the Vela Supernova Remnant |
Abstract: We present far-ultraviolet (FUV) spectral-imaging observations of the Vela
supernova remnant (SNR), obtained with the Spectroscopy of Plasma Evolution
from Astrophysical Radiation (SPEAR) instrument, also known as FIMS. The Vela
SNR extends 8 degrees in the FUV and its global spectra are dominated by
shock-induced emission lines. We find that the global FUV line luminosities can
exceed the 0.1-2.5 keV soft X-ray luminosity by an order of magnitude. The
global O VI:C III ratio shows that the Vela SNR has a relatively large fraction
of slower shocks compared with the Cygnus Loop.
| https://export.arxiv.org/pdf/astro-ph/0601586 |
\title{Far Ultraviolet Spectral Images of the Vela Supernova Remnant}
\author{K. Nishikida\altaffilmark{1}, J. Edelstein\altaffilmark{1}, E. J. Korpela\altaffilmark{1},
R. Sankrit\altaffilmark{1}, W. M. Feuerstein\altaffilmark{1}, K. W. Min\altaffilmark{2},
J-H. Shinn\altaffilmark{2}, D-H. Lee\altaffilmark{2},I-S. Yuk\altaffilmark{3}, H. Jin\altaffilmark{3},
K-I. Seon\altaffilmark{3}}
\altaffiltext{1}{Space Sciences Laboratory, University of California,
Berkeley, CA 94720}
\altaffiltext{2}{Korea Advanced Institute of Science and Technology,
305-701, Daejeon, Korea}
\altaffiltext{3}{Korea Astronomy and Space Science Institute, 305-348,
Daejeon, Korea}
\keywords{ISM: individual (Vela Supernova Remnant) -- supernova remnants -- ultraviolet: ISM}
\section{Introduction}
The Vela supernova remnant (SNR) has been studied in great detail due to its proximity \citep[250pc;][]{vela_distance} and its large angular diameter of $8^{\circ}$ \citep[]{VelaROSAT}. The remnant is $\sim$10,000 years old \citep{vela_age} and its overall emission is dominated by the interaction of the SN blast wave with the interstellar medium (ISM), but also has a pulsar and plerionic nebula near its center.
Vela is one of two Galactic SNRs (the other being the Cygnus Loop) that has been extensively studied in the UV. The UV emission from SNRs arises primarily in shocks driven by the supernova blast wave into interstellar clouds. The shock velocities responsible for the UV emission lie in the range $\sim$50-300 km s$^{-1}$, and heat the gas to temperatures of ~$10^5 - 10^6$ K. The hot shocked gas can also be studied via absorption so long as there are suitable background continuum sources.
Absorption line studies of Vela using \textit{Copernicus} showed the presence of O~{\small VI} and N~{\small V}, high ionization species expected in shocked gas, and also high velocity components of lower ionization species \citep{Jenkins1976a,Jenkins1976b}. Absorption line studies have also been carried out with \textit{IUE} \citep{Jenkins,Nichols}, \emph{HST} \citep{Jenkins1995,Jenkins1998} and \textit{FUSE} \citep{Slavin}. These studies have shown the existence of fast shocks ($\gtrsim$160 km s$^{-1}$) distributed widely but inhomogeneously across the face of the remnant, and also the variations in the dynamic pressure driving these shocks.
Emission line studies of Vela have been carried out using \textit{IUE} \citep{Vela_IUE}, \textit{HUT} \citep{RaymondVela}, \textit{FUSE} \citep{ravi2,ravi} and \emph{Voyager 2} UVS \citep[henceforth BVL]{vela_voyager}. These observations, except for the ones obtained by \emph{Voyager 2}, were of regions with angular extents of order an arc-minute and probed the properties
of individual shock fronts. BVL analyzed \emph{Voyager 2} spectra of a $1.5^{\circ} \times 2.0^{\circ}$ region in the northern part of Vela, which showed strong C~{\small III} and O~{\small VI} emission features. They found variations in the flux ratios on scales of 0.2$^{\circ}$. They also concluded from the overall flux ratio between the two lines that slower shocks (those unable to produce O~{\small VI}) were more prevalent in Vela than in the Cygnus Loop. \textit{FUSE} spectra of a few regions in Vela well separated from each other have strong C~{\small III} lines and relatively weak or no O~{\small VI}, and show the presence of slower shocks ($100 - 140$ km s$^{-1}$) spread over the face of the remnant \citep{raviproc}.
We present spectral images of the Vela SNR in several FUV lines and FUV spectra of the entire remnant. The data were obtained with \emph{SPEAR} (The Spectroscopy of Plasma Emission from Astrophysical Radiation), also known as \emph{FIMS} (Far-ultraviolet Imaging Spectrograph). \emph{SPEAR}, launched Sepember 27, 2003 on the Korean satellite STSAT-1, is a dual-channel FUV imaging spectrograph (S channel 900 - 1150 \AA, L channel 1350 - 1750 \AA, $\lambda/\Delta\lambda\sim$550) with a large imaged field of view (S: 4.0$^{\circ} \times 4.6'$, L: 7.5$^{\circ} \times 4.3'$, spatial resolution $\sim10'$) optimized for the observation of diffuse emission \citep[see][for an overview of the instrument and mission.]{Instrument,Mission}. A large effective field of view can be obtained by sweeping across the sky. This combination of instrument properties yields a dataset that supplements data obtained in previous UV studies of the remnant. The data allow us to estimate the total flux from Vela in several FUV lines.
In the following sections we present the observations (\S2), discuss the spectral images (\S3) and the total spectrum (\S4). The last section (\S5) summarizes the importance of these data for the study of
SNRs.
\section{Observation and Data Analysis}
The Vela SNR region was observed between January 31 and February 4, 2004. The data were processed as described in \citet[]{Instrument,Mission} including rejection of data for which the attitude knowledge was poor ($>$30') or data contaminated by airglow, evident from an increased count rate at the end of each orbit ($\leq$10\% of the data). The resulting number of photons and exposure obtained over 16 orbits were (4.4$\times10^{6}$, 7214 s) and (1.8$\times10^{5}$, 6307 s) for the L channel and S channel, respectively. While the SNR region was entirely observed in the L channel, the S channel coverage was incomplete.
The photon's sky coordinates ($\alpha$, $\delta$) and wavelengths ($\lambda$) were binned to 0.15$^{\circ}$ (similar to the \emph{SPEAR} imaging resolution after attitude reconstruction) and to 1 \AA, respectively and combined with exposure maps to create a 3-d count-rate data cube ($\alpha$, $\delta$, $\lambda$) for each spectral channel. We identified data contaminated by bright stars in each channel as pixels in the wavelength-integrated total count rate image that exceeded 3 times the median count rate of a relatively star-free area. The L and S channels contained 20\% and 10\% of the total pixels identified as stars, respectively.
To create spectral images, the data cube was summed across the waveband of interest with star pixels removed. Star pixel ``holes'' were filled with a linear fit to adjacent pixels at the ``hole's'' declination. The undetected stars have a flux $<1.5\times 10^{-11}$ ergs s$^{-1}$ cm$^{-2}$\AA$^{-1}$ in the \emph{SPEAR} L channel. Improved stellar identification, removal and reconstruction methods are currently being developed.
Net spectral images for specific emission lines, $I_{\lambda_{Net}}$, were constructed by subtracting a continuum image, $I_{cont}$, made from a similar width spectral region \emph{adjacent} to the emission line from an image, $I_{\lambda}$, made from a spectral region \emph{including} the emission line, i.e. $I_{\lambda_{Net}}=I_{\lambda}- I_{cont}$. This approach often results in an over-subtraction of the lower intensity pixels. The continuum selected from the global spectra is not suitable for lower intensity pixels because of the non-zero slope of the adjacent continuum. Therefore we apply a variable scaling factor, $f$, to the subtraction of the continuum image: $I_{\lambda_{Net}} = I_{\lambda}-f \times I_{cont}$. For each line image, the factor $f$ was set for each image pixel associated with a bin of a five-bin intensity histogram of $I_{\lambda}$ such that 70\% (i.e. $\sim$2$\sigma$) of those pixels would have a net positive flux. The resulting values of $f$ were 1.0 at high to medium intensity bins and decreased for the lowest one to three intensity bins, depending on $\lambda$, with a typically minimum value of $f=$0.6 for the lowest intensity bin. The resulting $I_{\lambda_{Net}}$ were smoothed using a 3-pixel (0.45$^{\circ}$) square median smoothing function. Diffuse spectra were derived by totaling photons and exposure over regions of interest, excluding bright star pixels, and then smoothed by a 3 \AA\ boxcar function.
\section{Spectral Images}
We show the C {\small IV} (1543--1557 \AA) image of Vela in Fig.~\ref{c4image}. This is the most prominent emission line in the L channel, and the figure shows the overall morphology of the FUV emission from the SNR. Images of Vela in C {\small III} (974-980 \AA), O {\small VI} (1019-1033 \AA), Si {\small IV} \& O {\small IV]} (1398--1413 \AA) are shown in Fig.~\ref{lineimages2}. Also shown for comparison is the \emph{ROSAT} All-Sky Survey (RASS) 1/4 keV image of the same region. The \emph{SPEAR} maps show that FUV emission is present over most of the Vela SNR. The C~{\small III} map in particular shows the widespread presence of radiative shocks. O {\small VI} and C {\small III} trace gas at very different ionization states. O~{\small VI} production requires a shock velocity of about 150 km/s while C~{\small III} is produced even in 80 km/s shocks. There is some overall correspondence between the two, showing where the FUV producing shocks exist, but C~{\small III} is more extended. C~{\small IV} and Si~{\small IV} \& O~{\small IV]} images show a close correlation: C~{\small IV}, Si~{\small IV}, and O~{\small IV]} have comparable ionization potentials and are roughly co-extensive in the post-shock gas.
Although the FUV and X-ray emission features are not closely correlated, their extent is roughly the same. \citet{lu} attribute the X-ray emission to thermal emission from a hot, thin gas in the SNR interior. We used the plasma temperatures and emission measures in Table 1 of \citet{lu} as input for CHIANTI \citep{chianti0,chianti} and confirmed that they produce insufficient thermal FUV emission by several orders of magnitudes compared with the observed emission. The FUV emission comes from shocks that have been driven into interstellar clouds. These clouds
are large enough that they have not been destroyed by the blast wave sweeping over them, and they have a high covering factor. The total FUV luminosity of Vela exceeds the total soft X-ray luminosity (see
\S4). Thus, the overall radiation rate of the SNR is dominated by shocks in higher density regions traced by their FUV emission.
The most prominent localized feature is an intense knot of emission near ($\alpha$, $\delta$)=(8.6 h, -42.5$^{\circ}$) that appears at all FUV wavelengths, including the soft X-ray band. A detailed comparison shows that the peak emission for each FUV wavelength and the soft X-ray are non-coincident and arranged in an arc, suggesting that the feature could be a complex and intense bow shock perhaps analogous to the X-ray ``bullet shocks,'' or ``knots,'' of similar scale. \citet{VelaROSAT} identified six (labeled A through F) of these X-ray knots and suggested that they were created by high-density ejecta shock-heating the ambient medium.
The FUV images appear to be limb brightened along the north-east shell boundary and, notably, joins the X-ray knot D to the main shell in C {\small IV} and Si {\small IV}/O {\small IV]}.
FUV emission is absent from the knot B
region despite its similarity in X-ray intensity to knot D. This is consistent with FUV emission arising from shocked interstellar media because the knot D bullet is believed to be encountering a more dense ambient media than knot B. An FUV enhancement particularly notable in O {\small VI}, whose localized intensity peak is coincident with the Vela pulsar, and C {\small III}, which coincides with the Vela-X radio continuum nebula \citep[Figures 1 \& 2]{vela_480,vela_843}, is coincident with a soft X-ray ridge, suggestive of a shock structure. Curiously, there is little associated C {\small IV} or Si {\small IV} \& O {\small IV]} emission from the same area in comparison to the region of maximum intensity ($\alpha$, $\delta$)=(8.6 h, -42.5$^{\circ}$). This suggests that we are detecting at least two markedly different physical conditions in the region surrounding the Vela pulsar. A more detailed examination will be necessary to tell if these are related, or merely happen to lie along the same line of sight.
\section{FUV Spectra}
The diffuse \emph{SPEAR} FUV spectra from the entire Vela SNR region are shown in Fig.~\ref{spectra}. The spectra show strong emission lines from highly ionized atoms produced by high velocity shocks and/or hot plasmas. Detected lines include C {\small III} (977 \AA), N~{\small III} (990 \AA), O~{\small VI} (1032, 1038 \AA), O {\small IV]} and Si {\small IV} (1400, 1403 \AA\ unresolved), N~{\small IV} (1486 \AA), C {\small IV} (1548, 1550 \AA), He {\small II} (1640 \AA), and O {\small III]} (1660, 1666 \AA). The emission line-like feature near 1695 \AA\ is instrumental.
Table~\ref{luminosity} shows the observed FUV line intensities averaged over the SNR. The C {\small IV}, He {\small II}, and O {\small III]} line profiles were fitted with a Gaussian line profile and a local linear background, while the C {\small III} and O~{\small VI} 1032 \AA\ lines required an additional Gaussian line profile to fit airglow lines adjacent to C~{\small III} and O {\small VI} $\lambda$1032. The O {\small VI} doublet intensity was calculated by multiplying the 1032 \AA\ line intensity by 1.5 (assuming a 2:1 ratio between $\lambda \lambda$1032, 1038) since \emph{SPEAR} does not have the spectral resolution necessary to resolve O {\small VI} $\lambda$1038 and C {\small II}$^{*}$ $\lambda$1037. For C~{\small III} and O~{\small VI}, we assumed that the average intensity calculated from areas with sky coverage applied to the entire SNR. Since the Vela SNR appears to be faint at the edges, it is likely that the intensities are slightly overestimated.
The estimated systematic uncertainty in the \emph{SPEAR} sensitivity is $\sim25\%$ \citep{Instrument}. The \emph{Voyager 2} UVS measurements of O {\small VI} and C {\small III} line intensities at 28 \AA\ resolution (BVL) are in agreement within a factor of two below \emph{SPEAR} measurements in an area near ($\pm1^{\circ}$) UVS pointings. The global observed O~{\small VI}:C~{\small III} ratio is $\sim1$, consistent within a factor of about two (both above and below) reported by BVL.
We apply multiplicative correction factors (shown in Table~\ref{luminosity}) to the observed global FUV luminosities, presuming a diameter of $8^{\circ}$ and distance of 250 pc, to derive dereddened luminosities. These correction factors were calculated by \citet{ravi} by assuming R=3.1 for selective extinction, E(B-V)=0.1 \citep{Vela_color}, and using the extinction curve suggested by \citet{Vela_ExtinctionCurve}. The integrated C~{\small III} or O~{\small VI} line luminosity can exceed the entire 0.1 - 2.5 keV luminosity, $2.2\times 10^{35}$ ergs s$^{-1}$ \citep{lu}, by more than an order of magnitude, confirming the importance of the FUV waveband to SNR cooling.
Vela's global FUV luminosity can be compared with that of the Cygnus Loop \citep{cygnus_voyager}.
Vela is about five times less luminous than Cygnus in the X-ray and in C {\small IV}, about twice as faint in O {\small VI}, and is equally as luminous in C {\small III}. The observed O~{\small VI}:C~{\small III} ratio of Cygnus is $\sim2$, illustrating the relatively large fraction of slower shocks in Vela compared with Cygnus. For Cygnus, \citet{cygnus_rocket} showed that the O {\small VI} luminosity is at least as much as the 0.1--4 keV X-ray luminosity \citep{cygnus_einstein}; the combined O~{\small VI}, C~{\small III}, C~{\small IV} luminosity was found to be ten times as large as the X-ray luminosity \citep{cygnus_voyager}.
In both the Cygnus Loop and Vela, the supernova blast wave is expanding into inhomogeneous surroundings, and the radiative shocks in the denser interaction regions emit strongly in the FUV.
\section{Summary and Future Work}
We have presented global FUV spectral images and spectra of the Vela SNR. Our data indicate inhomogeneous shock-induced emission from the SNR surface. The images show limb brightening and knots of emission. The spectra are consistent with past FUV observations of Vela. The global FUV luminosities of emission lines can exceed the soft X-ray luminosity by an order of magnitude, providing an efficient cooling channel to the SNR.
The data will be used to compare the global FUV morphology with images in other wavebands and examine how the SNR evolves and interacts with the ambient ISM. \emph{SPEAR} spectra toward previously observed regions \citep[ex.][]{RaymondVela,ravi2} will be modeled in detail to determine the physical properties of the regions. Mapping the O~{\small VI}:C~{\small III} ratio will allow us to map the distribution of shock velocities across Vela. Furthermore, the global \emph{SPEAR} spectra can be compared with those of neighboring regions to investigate Vela's association with the Gum nebula.
\acknowledgements
\emph{SPEAR / FIMS} is a joint project of KASSI \& KAIST (Korea) and U.C., Berkeley (USA), funded by the Korea MOST and NASA Grant NAG5-5355. We used NASA's \emph{SkyView} (http://skyview.gsfc.nasa.gov) facility and CHIANTI, a collaboration of NRL (USA), RAL (UK), U. Florence
(Italy) and Cambridge (UK).
\clearpage
\begin{table}[htbp]
\caption{Observed global FUV intensities
and dereddened luminosities of the Vela SNR}
\begin{tabular}{c|cccccc}
Species & C~{\small III} & O~{\small VI} & C~{\small IV} & He~{\small II} & O~{\small III]} & X-ray \\ \hline\hline
Observed intensity ($10^5 LU$) & 2.6 & 2.7 & 2.3 & 0.5 & 0.5 & \\
Reddening correction & 4.47 & 3.73 & 2.07 & 2.03 & 2.03 & \\
Dereddened luminosity ($10^{35}$ ergs s$^{-1}$) & 26.8 & 22.7 & 8.0 & 1.3 & 1.5 & 2.2 \\
\end{tabular}
\tablecomments{Diameter of 8$^{\circ}$ and 250 pc distance assumed. The O~VI luminosity was calculated assuming a 2:1 line ratio between O~VI 1032 \AA\ and 1038 \AA\ lines. Reddening correction values are taken from \citet{ravi}. X-ray (0.1--2.5 keV) luminosity from \citet{lu}. 1 Line Unit (LU) = 1 photon s$^{-1}$ cm$^{-2}$ sr$^{-1}$ = 1.9$\times 10^{-11}$ ergs s$^{-1}$ cm$^{-2}$ sr$^{-1}$ at 1032 \AA\ and 1.3$\times 10^{-11}$ ergs s$^{-1}$ cm$^{-2}$ sr$^{-1}$ at 1550 \AA.} \label{luminosity}
\end{table}
\clearpage
\clearpage
\clearpage
|
Title:
Theoretical Isochrones with Extinction in the K Band. II. J - K versus K |
Abstract: We calculate theoretical isochrones in a consistent way for five filter pairs
near the J and K band atmospheric windows (J-K, J-K', J-Ks, F110W-F205W, and
F110W-F222M) using the Padova stellar evolutionary models of Girardi et al. We
present magnitude transformations between various K-band filters as a function
of color. Isochrones with extinction of up to 6 mag in the K band are also
presented. As found for the filter pairs composed of H & K band filters, we
find that the reddened isochrones of different filter pairs behave as if they
follow different extinction laws, and that the extinction curves of Hubble
Space Telescope NICMOS filter pairs in the color-magnitude diagram are
considerably nonlinear. Because of these problems, extinction values estimated
with NICMOS filters can be in error by up to 1.3 mag. Our calculation suggests
that the extinction law implied by the observations of Rieke et al for
wavelengths between the J and K bands is better described by a power-law
function with an exponent of 1.66 instead of 1.59, which is commonly used with
an assumption that the transmission functions of J and K filters are Dirac
delta functions.
| https://export.arxiv.org/pdf/astro-ph/0601470 |
\title{Theoretical Isochrones with Extinction in the $K$ Band. II. \\
$J$ -- $K$ versus $K$}
\author{Sungsoo S. Kim\altaffilmark{1}, Donald F. Figer\altaffilmark{2}, and
Myung Gyoon Lee\altaffilmark{3}}
\altaffiltext{1}{Department of Astronomy and Space Science, Kyung Hee
University, Yongin-shi, Kyungki-do 449-701, South Korea; sungsoo.kim@khu.ac.kr.}
\altaffiltext{2}{Space Telescope Science Institute, 3700 San Martin Drive,
Baltimore, MD 21218; figer@stsci.edu.}
\altaffiltext{3}{Astronomy Program, SEES, Seoul National University,
Seoul 151-742, South Korea; mglee@astrog.snu.ac.kr.}
\keywords{Hertzsprung-Russell diagram --- techniques: photometric ---
stars: fundamental parameters --- infrared: stars}
\section{INTRODUCTION}
\label{sec:introduction}
Kim et al. (2005, hereafter Paper I) have calculated theoretical isochrones
with extinction for some $H$ and $K$ band filters using the Padova stellar
evolutionary models by Girardi et al. (2002). In Paper I, we found that the
reddened isochrones of different filter pairs in $H$ and $K$ bands behave
as if they follow different extinction laws, and that care is needed when
applying an extinction law obtained with one filter pair to other,
similar filter pairs. For example, if the extinction law for
the Johnson-Glass $H$ and $K$ filters obtained by Rieke, Rieke, \& Paul (1989)
is directly applied to the photometry from the {\it Hubble Space
Telescope} ({\it HST}) NICMOS filters
(F160W, F205W, and F222M), estimated extinction values can be in error
by up to 0.3 mag for true extinction at $K$ of 6 mag or less. To reduce
this error, Paper I introduced an ``effective extinction slope'' for each
filter pair and isochrone model. It was also found that the extinction
behavior of isochrones in the color-magnitude diagram (CMD) for filter pair
F160W--F222M is highly nonlinear (i.e., the amount of extinction is not
proportional to color excess) because of a significant width difference
in the two filters.
These problems are certainly not limited to the isochrones for filter pairs
in the $H$ and $K$ bands. This problem will apply to any situation in
which one applies an extinction law deduced from one filter pair to other
similar filter pairs. Furthermore, the nonlinear behavior of the extinction
vector in the CMD will be problematic for filter pairs
with significant difference in width. In the present paper, we extend
the calculations performed in Paper I to the isochrones for filter pairs
in the $J$ and $K$ bands. The filters considered here are the four
ground-based filters $J$, $K$ (Johnson et al. 1966), $K'$ (Wainscoat
\& Cowie 1992), and $K_s$ ($K$-short; developed by M. Skrutskie; see the
appendix of Persson et al. 1998), and the three NICMOS filters
F110W, F205W, and F222M (transmission functions of these filters are
shown in Figure~\ref{fig:filter}). Out of these seven filters, we consider
five filter pairs: $J$--$K$, $J$--$K'$, $J$--$K_s$, F110W--F205W, and
F110W--F222M.
We adopt a Vega-based photometric system (VEGAMAG system), which uses
Vega ($\alpha$ Lyr) as the calibrating star. For photometric zero
points of NICMOS filters, we adopt $\langle f_\nu^{\rm Vega} \rangle$
values from the NICMOS Data Handbook (ver. 5.0): 1775~Jy for F110W,
703.6~Jy for F205W, and 610.4~Jy for F222M. For the spectra of
synthetic stellar atmospheres, we adopt Kurucz ATLAS9 no-overshoot
models\footnote{See NOVER files at http://kurucz.harvard.edu/grids.html.}
(Kurucz 1993) calculated by Castelli et al. (1997). The metallicities of
these models cover the values of [M/H] = $-2.5$ to $+0.5$. A microturbulent
velocity $\xi=2\,{\rm km \, s^{-1}}$ and a mixing length parameter
$\alpha =1.25$ are adopted in the present study. For the temporal
evolution of effective temperature and luminosity as functions of stellar
mass (i.e., stellar evolutionary tracks), we adopt the ``basic set" of
the Padova models\footnote{See http://pleiadi.pd.astro.it.} (Girardi et al.
2002). We consider isochrones with a metallicity $Z$ = 0.0001, 0.001, 0.019,
and 0.03. The stellar spectral library and the evolutionary tracks we
adopted assume a solar chemical ratios.
For more details on the magnitude system, stellar spectral library, and
evolutionary tracks that we adopt here, readers are referred to Paper I.
Throughout this paper, we generically refer to the atmospheric wavebands
centered near 1.25, 1.65, and 2.2~\micron, as the $J$, $H$,
and $K$ bands, whereas we refer to the Johnson-Glass filters (Johnson et al.
1966; Glass 1974) as the $J$, $H$, and $K$ filters.
\section{Isochrones}
\label{sec:isochrones}
We first prepare a table of magnitudes for all spectra in ATLAS9 models in
the $J$ and $K$ band filters, covering a large range in $T_{eff}$,
$\log g$, and [M/H], using equations (5) and (6) of Paper I.
We use this table as a set of interpolates for the $T_{eff}$, $\log g$, and
$Z$ values predicted by the stellar evolution models for a given age in order
to estimate synthetic isochrones.
Isochrones for $A_\lambda = 0$, calculated in this way, are shown in
Figures~\ref{fig:iso1}$-$\ref{fig:iso4} for four different metallicities
and four ages. The color differences between filters are more prominent
for the highest metallicity isochrones. In most cases, isochrones for
$K'$ and $K_s$ are nearly indistinguishable, and those for F205W and
F222M are quite close to each other. In general, for red giants, intrinsic
color differences between the atmospheric and NICMOS filters are 0.2--0.4~mag.
As an independent check of our procedure, in Figure~\ref{fig:padova}
we compare our $J-K$ versus $K$ isochrones to those calculated by
Girardi et al. (2002). The isochrones match nicely, except
at the extremes. The discrepancy in the bright end
is caused from the empirical M giant spectra that Girardi et al. (2002)
added to their spectral library, and that in the faint end is by the
addition of late M dwarf spectra. The discrepancies are considerable only
at the top and bottom $\sim$ 1 mag of the isochrone, where only a
small fraction of giants reside, or else stars are too faint for most
observational situations.
Magnitude transformations between $K$-band filters can be obtained from
our isochrones. We find that the magnitude difference can be well fitted
by a third-order polynomial for $K < 4$~mag, and by a separate second-order
polynomial for $K > 4$~mag. The largest residuals from the fit are
0.012~mag for the former and 0.008~mag for the latter. The coefficients
of the best-fit functions are presented in Tables~\ref{table:trans1} and
\ref{table:trans2}, along with the residuals and fitting ranges.
One useful way of using these tables would be to compare the magnitudes
of helium-burning clump giant stars, which are rather insensitive to
metallicity or age and are often used as distance indicators, observed
with different photometric systems (the clump stars show a small variation
with age, however; see Figer et al. 2004).
We present here isochrones with $K$-band extinctions of up to 6 mag, some of
which are shown in Figures~\ref{fig:red1}$-$\ref{fig:red5}. For the
extinction between the $J$ and $K$ bands, we adopt a power law,
\begin{equation}
\label{extinction}
A_\lambda = A_0 \left ( \frac{\lambda}{\lambda_0}
\right )^{-\alpha},
\end{equation}
where we choose $\lambda_0 = 2.2 \, \mu$m, and $A_0$ is the extinction at
$\lambda_0$. When assuming that the transmission functions of the $J$ and $K$
filters are Dirac delta functions centered at 1.24 and 2.21~$\mu$m,
respectively, the extinction law by Rieke et al. (1989) gives
$\alpha=1.59$.
However, as discussed below in this section, the apparent extinction behavior
of isochrones in the CMD can differ from the actual extinction law, as a
result of a nonzero width and asymmetry of the filter transmission functions.
We find that $\alpha=1.66$ makes the isochrone for the $Z = 0.019$,
age = $10^9$~yr model behave in the CMD as if it followed an extinction law
with $\alpha=1.59$. We choose this particular isochrone for calibrating
the extinction law, with the assumption that the stars used in Rieke et al.
(1989) to derive their extinction law, which are the stars in the central
parsec of our Galaxy, can be represented by the same metallicity and age.
For the sake of comparison, isochrones in
Figures~\ref{fig:red1}$-$\ref{fig:red5} have been dereddened by the amount
$A_0 ( \lambda_c / \lambda_0 )^{-1.66}$, where the central wavelength
of the filter $\lambda_c$ is defined by equation (8) of Paper I, and
given in Table~\ref{table:lambdac}.
Since we have dereddened the isochrones with the known amount of
extinction at $\lambda_c$, all the dereddened isochrones with different
extinction values in Figures~\ref{fig:red1}$-$\ref{fig:red5} should
be coincident if the filter transmission functions were Dirac delta
functions centered at $\lambda_c$. As in Paper I, the dereddened
isochrones misalign significantly, and this implies that the amount of
extinction inferred from a CMD is sensitively dependent on the shape of
the filter transmission function.
When estimating the amount of extinction from an observed CMD, one
converts an observed color excess to an extinction value, following
an assumed extinction law, which usually has the form of a power law.
When one has photometric data from a pair of two filters, $X$ and $Y$, the
amount of extinction can be estimated by
\begin{eqnarray}
\label{A_est}
A_Y^{est} & = & \frac{(m_X-m_Y)-(m_X-m_Y)_0}{A_X/A_Y-1} \cr
& = & \frac{(m_X-m_Y)-(m_X-m_Y)_0}
{(\lambda_X/\lambda_Y)^{-\alpha}-1},
\end{eqnarray}
where $m_X$, $m_Y$ and $\lambda_X$, $\lambda_Y$ are the magnitudes and
the central wavelengths of the two filters, respectively, and subscript 0
denotes the intrinsic value. For estimating extinction from our isochrones,
we first use $\alpha=1.59$. Figure~\ref{fig:adiff1} shows the difference
between the inferred extinction values, using equation~(\ref{A_est}) and
colors from our reddened isochrones, and the actual extinction values.
Here the extinction of each isochrone has been calculated using the mean
color (for $A^{est}_Y$) and magnitude (for $A_Y$) of the reddened isochrone
data points having intrinsic $K$-band magnitudes between $-$6 and 0 mag.
As the figure shows, the differences between estimated and actual extinction
values are much larger for the NICMOS filter pairs. The largest
relative difference is $\sim 24$\%, and the largest absolute difference is
1.25 mag. Note that the extinction estimates for the $Z = 0.019$
and age = $10^9$~yr model inferred from $H$ and $K$ are very close to
the actual extinction values, justifying our choice of $\alpha=1.66$
for equation~(\ref{extinction}). The error bar in the figure represents
the standard deviation of $A^{est}_Y-A_Y$ values. Some of the F110W
isochrones show quite large deviations, as pointed out in Appendix A of
Lee et al. (2001).
To reduce the problems seen in Figure~\ref{fig:adiff1}, Paper I introduced
an ``effective extinction slope'' $\alpha_{eff}$ for each filter pair and
isochrone model, which is defined such that it better describes the
extinction behavior in the CMD:
\begin{equation}
\label{alpha_eff}
\alpha_{eff} = - \frac{ \log (1+1/b) }{ \log (\lambda_X/\lambda_Y) },
\end{equation}
where $b$ is the slope of the straight line that fits the distribution of
reddened magnitudes versus reddened colors, as in Figure~\ref{fig:extlaw}.
This figure shows reddened $K$-band magnitudes and colors for the $Z = 0.019$
and age = $10^9$~yr isochrone (the figure only shows an isochrone data
point whose intrinsic $K$ magnitude is 0, as an example).
We calculate $b$ for data points of each isochrone whose intrinsic
$K$ magnitudes are between $-6$ and 0~mag, and take an average for
each isochrone model. Table~\ref{table:alpha_eff} shows the averages
and standard deviations of $\alpha_{eff}$ values for each isochrone model.
For atmospheric filters, the standard deviations of $\alpha_{eff}$ in an
isochrone is generally much smaller than the differences of average
$\alpha_{eff}$ values between different isochrones, while those for
NICMOS filters are relatively larger.
The average $\alpha_{eff}$ values range from 1.403 to 1.610,
which are 15\% to 0.02\% smaller than the original $\alpha$ value we
adopted for extinction, 1.66.
As seen in Figure~\ref{fig:adiff3}, extinction values estimated by
equation~(\ref{A_est}) with $\alpha_{eff}$ are closer to
the actual values for atmospheric filters, but still deviate significantly
from the actual values for NICMOS filters, because of the nonlinear
extinction seen in Figure~\ref{fig:extlaw}.
As pointed out in Paper I, the nonlinear extinction behavior of NICMOS
filters is due to a significant difference in relative widths of the two
filters: the width to central wavelength ratio $\Delta \lambda / \lambda_c$
is $\sim 0.5$ for F110W, while those for F205W and F222M are $\sim 0.3$
and $\sim 0.07$, respectively.
Figure~\ref{fig:nonlin} shows the effect of the filter width by comparing
the extinction behavior of six imaginary filter pairs. Filter pair $a$
represents $J$ and $K$, whose $\Delta \lambda / \lambda_c$ values are
both $\sim 0.16$, and its extinction behavior in the CMD is nearly linear.
On the other hand, filter pairs $b$ and $c$, which represent filter pairs
F110W--F205W and F110W--F222M, show considerable nonlinearity. When
the $\Delta \lambda / \lambda_c$ of the short-wavelength filter is reduced
by $\sim 60$\%, however, the extinction behaves much more linearly
($d$ and $e$). This shows that the nonlinear extinction in
filter pairs F110W--F205W and F110W--F222M is due to a relatively larger
$\Delta \lambda / \lambda_c$ value of the F110W filter. When both
filters have the same large $\Delta \lambda / \lambda_c$ values
($\sim 0.5$), the extinction becomes almost linear again ($f$).
The introduction of effective extinction slopes does not alleviate the
nonlinear extinction problem of NICMOS filter pairs.
So in Table~\ref{table:nonlin} we provide the coefficients of best-fit
third-order polynomials of the extinction curves for NICMOS
filter pairs shown in Figure~\ref{fig:adiff1} so that one can accurately
estimate the extinction value for NICMOS filter
pairs as well. Note that we assumed $\alpha = 1.59$ for all isochrone models
when estimating $A_Y^{est}$ in Figure~\ref{fig:adiff1}.
As in paper I, we find that for the filters whose extinction behavior
is relatively linear, the transformation of extinction values from filter
$Y$ to filter $Y'$ can be obtained by
\begin{equation}
\label{A_trans}
A_{Y'} = A_Y \left ( \frac{\lambda_{Y'}}{\lambda_Y} \right )^{-\alpha},
\end{equation}
if $A_Y$ is estimated with $\alpha_{eff}$, and the original $\alpha$ value
of 1.66 is used in the above equation.
\section{SUMMARY}
\label{sec:summary}
We have calculated in a consistent way five near-infrared theoretical
isochrones for filter pairs composed of $J$ and $K$ filters: $J$--$K$,
$J$--$K'$, $J$--$K_s$, F110W--F205W, and F110W--F222M. We presented
isochrones for a $Z$ of 0.0001--0.03 and an age of $10^7$--$10^{10}$~yr.
Even in the same Vega magnitude system, near-infrared colors of the same
isochrone can be different by up to $\sim 0.4$ mag at the bright end of
the isochrone for different filter pairs.
The difference in intrinsic colors for a red giant for atmospheric
filters and the {\it HST} NICMOS filters is generally 0.2--0.4 mag.
We have provided magnitude transformations between $K$-band filters
as a function of color from $J$ and $K$ band filters.
We also presented isochrones with $A_K$ of up to 6 mag.
We found that care is needed when comparing extinction
values that are estimated using different filter pairs, in particular
when comparing those of atmospheric and NICMOS filter pairs:
extinction values inferred using NICMOS filters can be in error
by up to 1.3 mag. To alleviate this problem, we introduced an
``effective extinction slope'' for each filter pair and isochrone model,
which describes the extinction-dependent behavior of isochrones in the
observed CMD. We also provided a procedure to accurately estimate
the extinction value for NICMOS filter pairs, whose extinction curves
in the CMD are highly nonlinear.
\acknowledgements
We thank Jae-Woo Lee for a helpful discussion.
S. S. K. was supported by the Astrophysical Research Center for the
Structure and Evolution of the Cosmos (ARCSEC) of the Korea Science and
Engineering Foundation through the Science Research Center (SRC) program.
M. G. L. was in part supported by the ABRL (R14-2002-058-01000-0) and the BK21
program.
\clearpage
\clearpage
\begin{deluxetable}{cclrrrrcc}
\tabletypesize{\scriptsize}
\tablecolumns{9}
\tablewidth{0pt}
\tablecaption{
\label{table:trans1}Best-Fit Coefficients for Magnitude
Differences ($K < 4$~mag)}
\tablehead{
\colhead{} &
\colhead{Magnitude} &
\colhead{} &
\colhead{} &
\colhead{} &
\colhead{} &
\colhead{} &
\colhead{Residual\tablenotemark{a}} &
\colhead{Fitting Range\tablenotemark{b}} \\
\colhead{Color} &
\colhead{Difference} &
\colhead{$Z$} &
\colhead{$c_0$} &
\colhead{$c_1$} &
\colhead{$c_2$} &
\colhead{$c_3$} &
\colhead{(mag)} &
\colhead{(mag $\sim$ mag)}
}
\startdata
$J - K$ & $K'$ $-$$K$ & 0.0001 & $-0.001$ & $ 0.028$ & $-0.064$ & $ 0.057$ & $0.004$ & $-0.236 \sim 0.720$ \\
$J - K$ & $K'$ $-$$K$ & 0.001 & $-0.001$ & $ 0.032$ & $-0.039$ & $-0.036$ & $0.007$ & $-0.162 \sim 0.859$ \\
$J - K$ & $K'$ $-$$K$ & 0.019 & $ 0.002$ & $ 0.036$ & $-0.147$ & $ 0.074$ & $0.011$ & $-0.213 \sim 1.250$ \\
$J - K$ & $K'$ $-$$K$ & 0.03 & $ 0.001$ & $ 0.042$ & $-0.163$ & $ 0.083$ & $0.007$ & $-0.129 \sim 1.237$ \\
$J - K$ & $K_s$$-$$K$ & 0.0001 & $-0.000$ & $ 0.012$ & $-0.013$ & $ 0.006$ & $0.002$ & $-0.236 \sim 0.720$ \\
$J - K$ & $K_s$$-$$K$ & 0.001 & $-0.000$ & $ 0.015$ & $-0.003$ & $-0.052$ & $0.006$ & $-0.162 \sim 0.859$ \\
$J - K$ & $K_s$$-$$K$ & 0.019 & $ 0.002$ & $ 0.017$ & $-0.103$ & $ 0.051$ & $0.009$ & $-0.213 \sim 1.250$ \\
$J - K$ & $K_s$$-$$K$ & 0.03 & $ 0.001$ & $ 0.024$ & $-0.121$ & $ 0.061$ & $0.006$ & $-0.129 \sim 1.237$ \\
$J - K$ & F205W$-$$K$ & 0.0001 & $-0.030$ & $ 0.052$ & $-0.143$ & $ 0.150$ & $0.006$ & $-0.236 \sim 0.720$ \\
$J - K$ & F205W$-$$K$ & 0.001 & $-0.030$ & $ 0.056$ & $-0.095$ & $ 0.028$ & $0.005$ & $-0.162 \sim 0.859$ \\
$J - K$ & F205W$-$$K$ & 0.019 & $-0.029$ & $ 0.060$ & $-0.147$ & $ 0.074$ & $0.007$ & $-0.213 \sim 1.250$ \\
$J - K$ & F205W$-$$K$ & 0.03 & $-0.029$ & $ 0.061$ & $-0.147$ & $ 0.077$ & $0.004$ & $-0.129 \sim 1.237$ \\
$J - K$ & F222M$-$$K$ & 0.0001 & $-0.031$ & $-0.002$ & $-0.001$ & $-0.019$ & $0.005$ & $-0.236 \sim 0.720$ \\
$J - K$ & F222M$-$$K$ & 0.001 & $-0.030$ & $ 0.001$ & $-0.010$ & $-0.046$ & $0.006$ & $-0.162 \sim 0.859$ \\
$J - K$ & F222M$-$$K$ & 0.019 & $-0.028$ & $ 0.002$ & $-0.113$ & $ 0.060$ & $0.012$ & $-0.213 \sim 1.250$ \\
$J - K$ & F222M$-$$K$ & 0.03 & $-0.028$ & $ 0.009$ & $-0.134$ & $ 0.071$ & $0.007$ & $-0.129 \sim 1.237$ \\
$J - K'$ & $K$ $-$$K'$ & 0.0001 & $ 0.001$ & $-0.029$ & $ 0.068$ & $-0.061$ & $0.004$ & $-0.224 \sim 0.715$ \\
$J - K'$ & $K$ $-$$K'$ & 0.001 & $ 0.001$ & $-0.033$ & $ 0.047$ & $ 0.024$ & $0.006$ & $-0.155 \sim 0.881$ \\
$J - K'$ & $K$ $-$$K'$ & 0.019 & $-0.001$ & $-0.037$ & $ 0.142$ & $-0.070$ & $0.011$ & $-0.203 \sim 1.290$ \\
$J - K'$ & $K$ $-$$K'$ & 0.03 & $-0.001$ & $-0.042$ & $ 0.154$ & $-0.076$ & $0.006$ & $-0.123 \sim 1.278$ \\
$J - K'$ & $K_s$$-$$K'$ & 0.0001 & $ 0.001$ & $-0.017$ & $ 0.054$ & $-0.055$ & $0.003$ & $-0.224 \sim 0.715$ \\
$J - K'$ & $K_s$$-$$K'$ & 0.001 & $ 0.001$ & $-0.018$ & $ 0.038$ & $-0.017$ & $0.002$ & $-0.155 \sim 0.881$ \\
$J - K'$ & $K_s$$-$$K'$ & 0.019 & $ 0.000$ & $-0.019$ & $ 0.043$ & $-0.022$ & $0.003$ & $-0.203 \sim 1.290$ \\
$J - K'$ & $K_s$$-$$K'$ & 0.03 & $ 0.001$ & $-0.019$ & $ 0.040$ & $-0.021$ & $0.002$ & $-0.123 \sim 1.278$ \\
$J - K'$ & F205W$-$$K'$ & 0.0001 & $-0.029$ & $ 0.025$ & $-0.085$ & $ 0.099$ & $0.003$ & $-0.224 \sim 0.715$ \\
$J - K'$ & F205W$-$$K'$ & 0.001 & $-0.029$ & $ 0.024$ & $-0.054$ & $ 0.058$ & $0.004$ & $-0.155 \sim 0.881$ \\
$J - K'$ & F205W$-$$K'$ & 0.019 & $-0.031$ & $ 0.024$ & $-0.002$ & $ 0.001$ & $0.005$ & $-0.203 \sim 1.290$ \\
$J - K'$ & F205W$-$$K'$ & 0.03 & $-0.030$ & $ 0.019$ & $ 0.012$ & $-0.004$ & $0.005$ & $-0.123 \sim 1.278$ \\
$J - K'$ & F222M$-$$K'$ & 0.0001 & $-0.030$ & $-0.032$ & $ 0.067$ & $-0.080$ & $0.005$ & $-0.224 \sim 0.715$ \\
$J - K'$ & F222M$-$$K'$ & 0.001 & $-0.029$ & $-0.032$ & $ 0.032$ & $-0.012$ & $0.003$ & $-0.155 \sim 0.881$ \\
$J - K'$ & F222M$-$$K'$ & 0.019 & $-0.030$ & $-0.035$ & $ 0.036$ & $-0.015$ & $0.004$ & $-0.203 \sim 1.290$ \\
$J - K'$ & F222M$-$$K'$ & 0.03 & $-0.029$ & $-0.035$ & $ 0.031$ & $-0.013$ & $0.003$ & $-0.123 \sim 1.278$ \\
$J - K_s$ & $K$ $-$$K_s$ & 0.0001 & $ 0.000$ & $-0.012$ & $ 0.013$ & $-0.005$ & $0.002$ & $-0.232 \sim 0.718$ \\
$J - K_s$ & $K$ $-$$K_s$ & 0.001 & $ 0.000$ & $-0.015$ & $ 0.008$ & $ 0.043$ & $0.006$ & $-0.160 \sim 0.879$ \\
$J - K_s$ & $K$ $-$$K_s$ & 0.019 & $-0.002$ & $-0.017$ & $ 0.098$ & $-0.047$ & $0.009$ & $-0.209 \sim 1.289$ \\
$J - K_s$ & $K$ $-$$K_s$ & 0.03 & $-0.001$ & $-0.023$ & $ 0.113$ & $-0.055$ & $0.006$ & $-0.127 \sim 1.278$ \\
$J - K_s$ & $K'$ $-$$K_s$ & 0.0001 & $-0.001$ & $ 0.017$ & $-0.052$ & $ 0.053$ & $0.003$ & $-0.232 \sim 0.718$ \\
$J - K_s$ & $K'$ $-$$K_s$ & 0.001 & $-0.001$ & $ 0.018$ & $-0.037$ & $ 0.017$ & $0.002$ & $-0.160 \sim 0.879$ \\
$J - K_s$ & $K'$ $-$$K_s$ & 0.019 & $-0.000$ & $ 0.019$ & $-0.042$ & $ 0.022$ & $0.003$ & $-0.209 \sim 1.289$ \\
$J - K_s$ & $K'$ $-$$K_s$ & 0.03 & $-0.001$ & $ 0.018$ & $-0.039$ & $ 0.021$ & $0.002$ & $-0.127 \sim 1.278$ \\
$J - K_s$ & F205W$-$$K_s$ & 0.0001 & $-0.029$ & $ 0.041$ & $-0.133$ & $ 0.147$ & $0.005$ & $-0.232 \sim 0.718$ \\
$J - K_s$ & F205W$-$$K_s$ & 0.001 & $-0.030$ & $ 0.042$ & $-0.090$ & $ 0.075$ & $0.004$ & $-0.160 \sim 0.879$ \\
$J - K_s$ & F205W$-$$K_s$ & 0.019 & $-0.031$ & $ 0.042$ & $-0.044$ & $ 0.022$ & $0.006$ & $-0.209 \sim 1.289$ \\
$J - K_s$ & F205W$-$$K_s$ & 0.03 & $-0.031$ & $ 0.037$ & $-0.027$ & $ 0.016$ & $0.005$ & $-0.127 \sim 1.278$ \\
$J - K_s$ & F222M$-$$K_s$ & 0.0001 & $-0.030$ & $-0.014$ & $ 0.012$ & $-0.024$ & $0.005$ & $-0.232 \sim 0.718$ \\
$J - K_s$ & F222M$-$$K_s$ & 0.001 & $-0.030$ & $-0.014$ & $-0.007$ & $ 0.006$ & $0.004$ & $-0.160 \sim 0.879$ \\
$J - K_s$ & F222M$-$$K_s$ & 0.019 & $-0.030$ & $-0.016$ & $-0.008$ & $ 0.008$ & $0.005$ & $-0.209 \sim 1.289$ \\
$J - K_s$ & F222M$-$$K_s$ & 0.03 & $-0.030$ & $-0.016$ & $-0.010$ & $ 0.008$ & $0.004$ & $-0.127 \sim 1.278$ \\
F110W$-$F205W & $K$ $-$F205W & 0.0001 & $ 0.030$ & $-0.040$ & $ 0.085$ & $-0.070$ & $0.007$ & $-0.304 \sim 0.928$ \\
F110W$-$F205W & $K$ $-$F205W & 0.001 & $ 0.030$ & $-0.043$ & $ 0.053$ & $-0.010$ & $0.005$ & $-0.210 \sim 1.122$ \\
F110W$-$F205W & $K$ $-$F205W & 0.019 & $ 0.029$ & $-0.047$ & $ 0.086$ & $-0.033$ & $0.006$ & $-0.272 \sim 1.621$ \\
F110W$-$F205W & $K$ $-$F205W & 0.03 & $ 0.029$ & $-0.047$ & $ 0.086$ & $-0.034$ & $0.004$ & $-0.166 \sim 1.615$ \\
F110W$-$F205W & $K'$ $-$F205W & 0.0001 & $ 0.029$ & $-0.018$ & $ 0.048$ & $-0.043$ & $0.003$ & $-0.304 \sim 0.928$ \\
F110W$-$F205W & $K'$ $-$F205W & 0.001 & $ 0.029$ & $-0.018$ & $ 0.035$ & $-0.030$ & $0.003$ & $-0.210 \sim 1.122$ \\
F110W$-$F205W & $K'$ $-$F205W & 0.019 & $ 0.031$ & $-0.017$ & $-0.000$ & $-0.000$ & $0.005$ & $-0.272 \sim 1.621$ \\
F110W$-$F205W & $K'$ $-$F205W & 0.03 & $ 0.030$ & $-0.013$ & $-0.010$ & $ 0.003$ & $0.005$ & $-0.166 \sim 1.615$ \\
F110W$-$F205W & $K_s$$-$F205W & 0.0001 & $ 0.029$ & $-0.031$ & $ 0.078$ & $-0.067$ & $0.005$ & $-0.304 \sim 0.928$ \\
F110W$-$F205W & $K_s$$-$F205W & 0.001 & $ 0.030$ & $-0.032$ & $ 0.055$ & $-0.036$ & $0.004$ & $-0.210 \sim 1.122$ \\
F110W$-$F205W & $K_s$$-$F205W & 0.019 & $ 0.031$ & $-0.032$ & $ 0.026$ & $-0.011$ & $0.006$ & $-0.272 \sim 1.621$ \\
F110W$-$F205W & $K_s$$-$F205W & 0.03 & $ 0.031$ & $-0.027$ & $ 0.014$ & $-0.007$ & $0.005$ & $-0.166 \sim 1.615$ \\
F110W$-$F205W & F222M$-$F205W & 0.0001 & $-0.001$ & $-0.041$ & $ 0.085$ & $-0.079$ & $0.007$ & $-0.304 \sim 0.928$ \\
F110W$-$F205W & F222M$-$F205W & 0.001 & $-0.000$ & $-0.042$ & $ 0.050$ & $-0.033$ & $0.004$ & $-0.210 \sim 1.122$ \\
F110W$-$F205W & F222M$-$F205W & 0.019 & $ 0.001$ & $-0.043$ & $ 0.018$ & $-0.006$ & $0.006$ & $-0.272 \sim 1.621$ \\
F110W$-$F205W & F222M$-$F205W & 0.03 & $ 0.001$ & $-0.037$ & $ 0.005$ & $-0.002$ & $0.004$ & $-0.166 \sim 1.615$ \\
F110W$-$F222M & $K$ $-$F222M & 0.0001 & $ 0.031$ & $ 0.001$ & $ 0.001$ & $ 0.008$ & $0.005$ & $-0.324 \sim 0.955$ \\
F110W$-$F222M & $K$ $-$F222M & 0.001 & $ 0.030$ & $-0.001$ & $ 0.004$ & $ 0.021$ & $0.005$ & $-0.222 \sim 1.152$ \\
F110W$-$F222M & $K$ $-$F222M & 0.019 & $ 0.028$ & $-0.003$ & $ 0.063$ & $-0.024$ & $0.011$ & $-0.291 \sim 1.665$ \\
F110W$-$F222M & $K$ $-$F222M & 0.03 & $ 0.028$ & $-0.009$ & $ 0.076$ & $-0.029$ & $0.006$ & $-0.176 \sim 1.669$ \\
F110W$-$F222M & $K'$ $-$F222M & 0.0001 & $ 0.030$ & $ 0.021$ & $-0.033$ & $ 0.032$ & $0.005$ & $-0.324 \sim 0.955$ \\
F110W$-$F222M & $K'$ $-$F222M & 0.001 & $ 0.029$ & $ 0.022$ & $-0.013$ & $ 0.003$ & $0.003$ & $-0.222 \sim 1.152$ \\
F110W$-$F222M & $K'$ $-$F222M & 0.019 & $ 0.030$ & $ 0.024$ & $-0.017$ & $ 0.005$ & $0.004$ & $-0.291 \sim 1.665$ \\
F110W$-$F222M & $K'$ $-$F222M & 0.03 & $ 0.029$ & $ 0.024$ & $-0.014$ & $ 0.004$ & $0.003$ & $-0.176 \sim 1.669$ \\
F110W$-$F222M & $K_s$$-$F222M & 0.0001 & $ 0.030$ & $ 0.010$ & $-0.006$ & $ 0.011$ & $0.005$ & $-0.324 \sim 0.955$ \\
F110W$-$F222M & $K_s$$-$F222M & 0.001 & $ 0.030$ & $ 0.009$ & $ 0.005$ & $-0.003$ & $0.004$ & $-0.222 \sim 1.152$ \\
F110W$-$F222M & $K_s$$-$F222M & 0.019 & $ 0.030$ & $ 0.011$ & $ 0.007$ & $-0.004$ & $0.005$ & $-0.291 \sim 1.665$ \\
F110W$-$F222M & $K_s$$-$F222M & 0.03 & $ 0.030$ & $ 0.010$ & $ 0.009$ & $-0.005$ & $0.004$ & $-0.176 \sim 1.669$ \\
F110W$-$F222M & F205W$-$F222M & 0.0001 & $ 0.001$ & $ 0.038$ & $-0.076$ & $ 0.070$ & $0.007$ & $-0.324 \sim 0.955$ \\
F110W$-$F222M & F205W$-$F222M & 0.001 & $ 0.000$ & $ 0.040$ & $-0.045$ & $ 0.030$ & $0.004$ & $-0.222 \sim 1.152$ \\
F110W$-$F222M & F205W$-$F222M & 0.019 & $-0.001$ & $ 0.041$ & $-0.017$ & $ 0.005$ & $0.006$ & $-0.291 \sim 1.665$ \\
F110W$-$F222M & F205W$-$F222M & 0.03 & $-0.001$ & $ 0.036$ & $-0.004$ & $ 0.002$ & $0.004$ & $-0.176 \sim 1.669$ \\
\enddata
\tablecomments{Magnitude differences are fitted to a function
$[{\rm Mag\, Diff}] = c_0 + c_1[{\rm Color}] + c_2[{\rm Color}]^2 +
c_3[{\rm Color}]^3$. Only the data points that have $\log T_{eff} \ge
3500$~K and $\log g \ge 0$ were considered for the fitting.}
\tablenotetext{a}{The largest absolute residual.}
\tablenotetext{b}{Color range where the fit is valid.}
\end{deluxetable}
\clearpage
\begin{deluxetable}{cclrrrcc}
\tabletypesize{\scriptsize}
\tablecolumns{8}
\tablewidth{0pt}
\tablecaption{
\label{table:trans2}Best-Fit Coefficients for Magnitude
Differences ($K > 4$~mag)}
\tablehead{
\colhead{} &
\colhead{Magnitude} &
\colhead{} &
\colhead{} &
\colhead{} &
\colhead{} &
\colhead{Residual\tablenotemark{a}} &
\colhead{Fitting Range\tablenotemark{b}} \\
\colhead{Color} &
\colhead{Difference} &
\colhead{$Z$} &
\colhead{$c_0$} &
\colhead{$c_1$} &
\colhead{$c_2$} &
\colhead{(mag)} &
\colhead{(mag $\sim$ mag)}
}
\startdata
$J - K$ & $K'$ $-$$K$ & 0.0001 & $-0.012$ & $ 0.047$ & $-0.003$ & $0.001$ & $ 0.332 \sim 0.772$ \\
$J - K$ & $K'$ $-$$K$ & 0.001 & $ 0.034$ & $-0.126$ & $ 0.127$ & $0.003$ & $ 0.338 \sim 0.893$ \\
$J - K$ & $K'$ $-$$K$ & 0.019 & $ 0.102$ & $-0.331$ & $ 0.252$ & $0.004$ & $ 0.539 \sim 0.992$ \\
$J - K$ & $K'$ $-$$K$ & 0.03 & $ 0.129$ & $-0.401$ & $ 0.291$ & $0.003$ & $ 0.559 \sim 0.987$ \\
$J - K$ & $K_s$$-$$K$ & 0.0001 & $-0.010$ & $ 0.042$ & $-0.019$ & $0.001$ & $ 0.332 \sim 0.772$ \\
$J - K$ & $K_s$$-$$K$ & 0.001 & $ 0.018$ & $-0.059$ & $ 0.055$ & $0.002$ & $ 0.338 \sim 0.893$ \\
$J - K$ & $K_s$$-$$K$ & 0.019 & $ 0.055$ & $-0.182$ & $ 0.134$ & $0.003$ & $ 0.539 \sim 0.992$ \\
$J - K$ & $K_s$$-$$K$ & 0.03 & $ 0.073$ & $-0.229$ & $ 0.160$ & $0.002$ & $ 0.559 \sim 0.987$ \\
$J - K$ & F205W$-$$K$ & 0.0001 & $-0.049$ & $ 0.082$ & $ 0.000$ & $0.001$ & $ 0.332 \sim 0.772$ \\
$J - K$ & F205W$-$$K$ & 0.001 & $ 0.026$ & $-0.203$ & $ 0.217$ & $0.004$ & $ 0.338 \sim 0.893$ \\
$J - K$ & F205W$-$$K$ & 0.019 & $ 0.127$ & $-0.490$ & $ 0.386$ & $0.007$ & $ 0.539 \sim 0.992$ \\
$J - K$ & F205W$-$$K$ & 0.03 & $ 0.173$ & $-0.601$ & $ 0.447$ & $0.004$ & $ 0.559 \sim 0.987$ \\
$J - K$ & F222M$-$$K$ & 0.0001 & $-0.026$ & $-0.023$ & $ 0.003$ & $0.001$ & $ 0.332 \sim 0.772$ \\
$J - K$ & F222M$-$$K$ & 0.001 & $-0.027$ & $-0.016$ & $-0.007$ & $0.001$ & $ 0.338 \sim 0.893$ \\
$J - K$ & F222M$-$$K$ & 0.019 & $-0.030$ & $-0.027$ & $ 0.002$ & $0.001$ & $ 0.539 \sim 0.992$ \\
$J - K$ & F222M$-$$K$ & 0.03 & $-0.035$ & $-0.020$ & $-0.002$ & $0.001$ & $ 0.559 \sim 0.987$ \\
$J - K'$ & $K$ $-$$K'$ & 0.0001 & $ 0.012$ & $-0.050$ & $ 0.004$ & $0.001$ & $ 0.328 \sim 0.750$ \\
$J - K'$ & $K$ $-$$K'$ & 0.001 & $-0.037$ & $ 0.138$ & $-0.140$ & $0.003$ & $ 0.334 \sim 0.871$ \\
$J - K'$ & $K$ $-$$K'$ & 0.019 & $-0.127$ & $ 0.401$ & $-0.300$ & $0.005$ & $ 0.543 \sim 0.975$ \\
$J - K'$ & $K$ $-$$K'$ & 0.03 & $-0.174$ & $ 0.520$ & $-0.369$ & $0.003$ & $ 0.593 \sim 0.971$ \\
$J - K'$ & $K_s$$-$$K'$ & 0.0001 & $ 0.002$ & $-0.006$ & $-0.017$ & $0.001$ & $ 0.328 \sim 0.750$ \\
$J - K'$ & $K_s$$-$$K'$ & 0.001 & $-0.018$ & $ 0.074$ & $-0.080$ & $0.003$ & $ 0.334 \sim 0.871$ \\
$J - K'$ & $K_s$$-$$K'$ & 0.019 & $-0.060$ & $ 0.184$ & $-0.143$ & $0.002$ & $ 0.543 \sim 0.975$ \\
$J - K'$ & $K_s$$-$$K'$ & 0.03 & $-0.080$ & $ 0.233$ & $-0.172$ & $0.001$ & $ 0.593 \sim 0.971$ \\
$J - K'$ & F205W$-$$K'$ & 0.0001 & $-0.038$ & $ 0.036$ & $ 0.004$ & $0.001$ & $ 0.328 \sim 0.750$ \\
$J - K'$ & F205W$-$$K'$ & 0.001 & $-0.006$ & $-0.087$ & $ 0.101$ & $0.002$ & $ 0.334 \sim 0.871$ \\
$J - K'$ & F205W$-$$K'$ & 0.019 & $ 0.040$ & $-0.202$ & $ 0.166$ & $0.003$ & $ 0.543 \sim 0.975$ \\
$J - K'$ & F205W$-$$K'$ & 0.03 & $ 0.077$ & $-0.286$ & $ 0.211$ & $0.002$ & $ 0.593 \sim 0.971$ \\
$J - K'$ & F222M$-$$K'$ & 0.0001 & $-0.013$ & $-0.074$ & $ 0.007$ & $0.001$ & $ 0.328 \sim 0.750$ \\
$J - K'$ & F222M$-$$K'$ & 0.001 & $-0.065$ & $ 0.125$ & $-0.150$ & $0.004$ & $ 0.334 \sim 0.871$ \\
$J - K'$ & F222M$-$$K'$ & 0.019 & $-0.160$ & $ 0.383$ & $-0.305$ & $0.004$ & $ 0.543 \sim 0.975$ \\
$J - K'$ & F222M$-$$K'$ & 0.03 & $-0.214$ & $ 0.513$ & $-0.380$ & $0.003$ & $ 0.593 \sim 0.971$ \\
$J - K_s$ & $K$ $-$$K_s$ & 0.0001 & $ 0.010$ & $-0.043$ & $ 0.020$ & $0.001$ & $ 0.330 \sim 0.761$ \\
$J - K_s$ & $K$ $-$$K_s$ & 0.001 & $-0.018$ & $ 0.060$ & $-0.056$ & $0.002$ & $ 0.335 \sim 0.885$ \\
$J - K_s$ & $K$ $-$$K_s$ & 0.019 & $-0.060$ & $ 0.197$ & $-0.143$ & $0.003$ & $ 0.544 \sim 0.989$ \\
$J - K_s$ & $K$ $-$$K_s$ & 0.03 & $-0.084$ & $ 0.257$ & $-0.177$ & $0.002$ & $ 0.594 \sim 0.986$ \\
$J - K_s$ & $K'$ $-$$K_s$ & 0.0001 & $-0.002$ & $ 0.006$ & $ 0.016$ & $0.001$ & $ 0.330 \sim 0.761$ \\
$J - K_s$ & $K'$ $-$$K_s$ & 0.001 & $ 0.017$ & $-0.069$ & $ 0.075$ & $0.003$ & $ 0.335 \sim 0.885$ \\
$J - K_s$ & $K'$ $-$$K_s$ & 0.019 & $ 0.054$ & $-0.165$ & $ 0.128$ & $0.002$ & $ 0.544 \sim 0.989$ \\
$J - K_s$ & $K'$ $-$$K_s$ & 0.03 & $ 0.070$ & $-0.204$ & $ 0.151$ & $0.001$ & $ 0.594 \sim 0.986$ \\
$J - K_s$ & F205W$-$$K_s$ & 0.0001 & $-0.040$ & $ 0.042$ & $ 0.019$ & $0.001$ & $ 0.330 \sim 0.761$ \\
$J - K_s$ & F205W$-$$K_s$ & 0.001 & $ 0.010$ & $-0.149$ & $ 0.168$ & $0.003$ & $ 0.335 \sim 0.885$ \\
$J - K_s$ & F205W$-$$K_s$ & 0.019 & $ 0.085$ & $-0.343$ & $ 0.276$ & $0.005$ & $ 0.544 \sim 0.989$ \\
$J - K_s$ & F205W$-$$K_s$ & 0.03 & $ 0.134$ & $-0.454$ & $ 0.337$ & $0.003$ & $ 0.594 \sim 0.986$ \\
$J - K_s$ & F222M$-$$K_s$ & 0.0001 & $-0.015$ & $-0.068$ & $ 0.024$ & $0.001$ & $ 0.330 \sim 0.761$ \\
$J - K_s$ & F222M$-$$K_s$ & 0.001 & $-0.045$ & $ 0.046$ & $-0.064$ & $0.002$ & $ 0.335 \sim 0.885$ \\
$J - K_s$ & F222M$-$$K_s$ & 0.019 & $-0.092$ & $ 0.174$ & $-0.144$ & $0.003$ & $ 0.544 \sim 0.989$ \\
$J - K_s$ & F222M$-$$K_s$ & 0.03 & $-0.122$ & $ 0.245$ & $-0.184$ & $0.002$ & $ 0.594 \sim 0.986$ \\
F110W$-$F205W & $K$ $-$F205W & 0.0001 & $ 0.054$ & $-0.071$ & $-0.000$ & $0.001$ & $ 0.454 \sim 0.961$ \\
F110W$-$F205W & $K$ $-$F205W & 0.001 & $-0.050$ & $ 0.223$ & $-0.176$ & $0.005$ & $ 0.484 \sim 1.107$ \\
F110W$-$F205W & $K$ $-$F205W & 0.019 & $-0.162$ & $ 0.460$ & $-0.278$ & $0.008$ & $ 0.724 \sim 1.252$ \\
F110W$-$F205W & $K$ $-$F205W & 0.03 & $-0.231$ & $ 0.584$ & $-0.328$ & $0.005$ & $ 0.784 \sim 1.257$ \\
F110W$-$F205W & $K'$ $-$F205W & 0.0001 & $ 0.040$ & $-0.030$ & $-0.003$ & $0.001$ & $ 0.454 \sim 0.961$ \\
F110W$-$F205W & $K'$ $-$F205W & 0.001 & $-0.002$ & $ 0.088$ & $-0.074$ & $0.002$ & $ 0.484 \sim 1.107$ \\
F110W$-$F205W & $K'$ $-$F205W & 0.019 & $-0.036$ & $ 0.150$ & $-0.097$ & $0.003$ & $ 0.724 \sim 1.252$ \\
F110W$-$F205W & $K'$ $-$F205W & 0.03 & $-0.068$ & $ 0.203$ & $-0.118$ & $0.002$ & $ 0.784 \sim 1.257$ \\
F110W$-$F205W & $K_s$$-$F205W & 0.0001 & $ 0.042$ & $-0.033$ & $-0.015$ & $0.001$ & $ 0.454 \sim 0.961$ \\
F110W$-$F205W & $K_s$$-$F205W & 0.001 & $-0.026$ & $ 0.162$ & $-0.133$ & $0.004$ & $ 0.484 \sim 1.107$ \\
F110W$-$F205W & $K_s$$-$F205W & 0.019 & $-0.095$ & $ 0.291$ & $-0.182$ & $0.005$ & $ 0.724 \sim 1.252$ \\
F110W$-$F205W & $K_s$$-$F205W & 0.03 & $-0.141$ & $ 0.370$ & $-0.214$ & $0.003$ & $ 0.784 \sim 1.257$ \\
F110W$-$F205W & F222M$-$F205W & 0.0001 & $ 0.030$ & $-0.091$ & $ 0.002$ & $0.001$ & $ 0.454 \sim 0.961$ \\
F110W$-$F205W & F222M$-$F205W & 0.001 & $-0.079$ & $ 0.218$ & $-0.186$ & $0.005$ & $ 0.484 \sim 1.107$ \\
F110W$-$F205W & F222M$-$F205W & 0.019 & $-0.190$ & $ 0.438$ & $-0.277$ & $0.008$ & $ 0.724 \sim 1.252$ \\
F110W$-$F205W & F222M$-$F205W & 0.03 & $-0.266$ & $ 0.569$ & $-0.330$ & $0.005$ & $ 0.784 \sim 1.257$ \\
F110W$-$F222M & $K$ $-$F222M & 0.0001 & $ 0.025$ & $ 0.018$ & $-0.002$ & $0.001$ & $ 0.465 \sim 1.017$ \\
F110W$-$F222M & $K$ $-$F222M & 0.001 & $ 0.027$ & $ 0.010$ & $ 0.006$ & $0.001$ & $ 0.496 \sim 1.170$ \\
F110W$-$F222M & $K$ $-$F222M & 0.019 & $ 0.025$ & $ 0.029$ & $-0.005$ & $0.001$ & $ 0.742 \sim 1.322$ \\
F110W$-$F222M & $K$ $-$F222M & 0.03 & $ 0.031$ & $ 0.023$ & $-0.002$ & $0.001$ & $ 0.806 \sim 1.327$ \\
F110W$-$F222M & $K'$ $-$F222M & 0.0001 & $ 0.011$ & $ 0.055$ & $-0.003$ & $0.001$ & $ 0.465 \sim 1.017$ \\
F110W$-$F222M & $K'$ $-$F222M & 0.001 & $ 0.069$ & $-0.103$ & $ 0.089$ & $0.004$ & $ 0.496 \sim 1.170$ \\
F110W$-$F222M & $K'$ $-$F222M & 0.019 & $ 0.122$ & $-0.208$ & $ 0.131$ & $0.004$ & $ 0.742 \sim 1.322$ \\
F110W$-$F222M & $K'$ $-$F222M & 0.03 & $ 0.152$ & $-0.259$ & $ 0.151$ & $0.003$ & $ 0.806 \sim 1.327$ \\
F110W$-$F222M & $K_s$$-$F222M & 0.0001 & $ 0.013$ & $ 0.053$ & $-0.014$ & $0.001$ & $ 0.465 \sim 1.017$ \\
F110W$-$F222M & $K_s$$-$F222M & 0.001 & $ 0.048$ & $-0.043$ & $ 0.042$ & $0.002$ & $ 0.496 \sim 1.170$ \\
F110W$-$F222M & $K_s$$-$F222M & 0.019 & $ 0.078$ & $-0.103$ & $ 0.068$ & $0.003$ & $ 0.742 \sim 1.322$ \\
F110W$-$F222M & $K_s$$-$F222M & 0.03 & $ 0.100$ & $-0.140$ & $ 0.082$ & $0.002$ & $ 0.806 \sim 1.327$ \\
F110W$-$F222M & F205W$-$F222M & 0.0001 & $-0.028$ & $ 0.083$ & $-0.001$ & $0.001$ & $ 0.465 \sim 1.017$ \\
F110W$-$F222M & F205W$-$F222M & 0.001 & $ 0.065$ & $-0.173$ & $ 0.148$ & $0.005$ & $ 0.496 \sim 1.170$ \\
F110W$-$F222M & F205W$-$F222M & 0.019 & $ 0.139$ & $-0.313$ & $ 0.200$ & $0.006$ & $ 0.742 \sim 1.322$ \\
F110W$-$F222M & F205W$-$F222M & 0.03 & $ 0.195$ & $-0.404$ & $ 0.235$ & $0.004$ & $ 0.806 \sim 1.327$ \\
\enddata
\tablecomments{Magnitude differences are fitted to a function
${\rm [Mag\, Diff]} = c_0 + c_1[{\rm Color}] + c_2[{\rm Color}]^2$.
Only the data points that have $\log T_{eff} \ge
3500$~K and $\log g \ge 0$ were considered for the fitting.}
\tablenotetext{a}{The largest absolute residual.}
\tablenotetext{b}{Color range where the fit is valid.}
\end{deluxetable}
\clearpage
\begin{deluxetable}{ccccccc}
\tablecolumns{7}
\tablewidth{0pt}
\tablecaption{
\label{table:lambdac}Central Wavelength $\lambda_c$ ($\mu$m)}
\tablehead{
\colhead{$J$} &
\colhead{$K$} &
\colhead{$K'$} &
\colhead{$K_s$} &
\colhead{F110W} &
\colhead{F205W} &
\colhead{F222M}
}
\startdata
1.237 & 2.212 & 2.114 & 2.160 & 1.140 & 2.079 & 2.219 \\
\enddata
\tablecomments{$\lambda_c$ is defined by eq. (8) of Paper I.}
\end{deluxetable}
\clearpage
\begin{deluxetable}{lcccccc}
\tabletypesize{\scriptsize}
\tablecolumns{7}
\tablewidth{0pt}
\tablecaption{
\label{table:alpha_eff}Averages and Standard Deviations of $\alpha_{eff}$
Values}
\tablehead{
\multicolumn{2}{c}{Isochrone Model} &
\colhead{} &
\colhead{} &
\colhead{} &
\colhead{} &
\colhead{} \\ \cline{1-2}
\colhead{$Z$} &
\colhead{Age} &
\colhead{$J-K$} &
\colhead{$J-K'$} &
\colhead{$J-K_s$} &
\colhead{F110W$-$F205W} &
\colhead{F110W$-$F222M}
}
\startdata
0.0001 & $10^7$ & 1.610$\pm$0.000 & 1.608$\pm$0.000 & 1.610$\pm$0.000 & 1.479$\pm$0.001 & 1.500$\pm$0.002 \\
0.0001 & $10^8$ & 1.608$\pm$0.003 & 1.605$\pm$0.004 & 1.607$\pm$0.004 & 1.467$\pm$0.011 & 1.486$\pm$0.011 \\
0.0001 & $10^9$ & 1.600$\pm$0.004 & 1.597$\pm$0.005 & 1.598$\pm$0.004 & 1.440$\pm$0.013 & 1.460$\pm$0.012 \\
0.0001 & $10^{10}$ & 1.596$\pm$0.003 & 1.593$\pm$0.003 & 1.595$\pm$0.003 & 1.429$\pm$0.008 & 1.450$\pm$0.007 \\
0.001 & $6.3 \times 10^7$ & 1.608$\pm$0.004 & 1.605$\pm$0.004 & 1.607$\pm$0.004 & 1.467$\pm$0.012 & 1.487$\pm$0.012 \\
0.001 & $10^8$ & 1.605$\pm$0.006 & 1.603$\pm$0.006 & 1.604$\pm$0.006 & 1.459$\pm$0.017 & 1.478$\pm$0.017 \\
0.001 & $10^9$ & 1.595$\pm$0.004 & 1.593$\pm$0.003 & 1.595$\pm$0.003 & 1.430$\pm$0.008 & 1.451$\pm$0.007 \\
0.001 & $10^{10}$ & 1.591$\pm$0.005 & 1.590$\pm$0.003 & 1.592$\pm$0.003 & 1.421$\pm$0.010 & 1.444$\pm$0.008 \\
0.019 & $10^7$ & 1.610$\pm$0.000 & 1.608$\pm$0.000 & 1.610$\pm$0.000 & 1.478$\pm$0.001 & 1.498$\pm$0.002 \\
0.019 & $10^8$ & 1.599$\pm$0.011 & 1.599$\pm$0.009 & 1.600$\pm$0.009 & 1.448$\pm$0.028 & 1.470$\pm$0.025 \\
0.019 & $10^9$ & 1.588$\pm$0.006 & 1.590$\pm$0.004 & 1.590$\pm$0.004 & 1.418$\pm$0.012 & 1.442$\pm$0.010 \\
0.019 & $10^{10}$ & 1.582$\pm$0.005 & 1.586$\pm$0.003 & 1.587$\pm$0.003 & 1.406$\pm$0.011 & 1.432$\pm$0.009 \\
0.03 & $6.3 \times 10^7$ & 1.606$\pm$0.007 & 1.605$\pm$0.006 & 1.606$\pm$0.006 & 1.465$\pm$0.019 & 1.485$\pm$0.018 \\
0.03 & $10^8$ & 1.599$\pm$0.012 & 1.599$\pm$0.009 & 1.600$\pm$0.010 & 1.448$\pm$0.029 & 1.470$\pm$0.026 \\
0.03 & $10^9$ & 1.586$\pm$0.006 & 1.589$\pm$0.004 & 1.590$\pm$0.004 & 1.415$\pm$0.012 & 1.440$\pm$0.010 \\
0.03 & $10^{10}$ & 1.581$\pm$0.005 & 1.585$\pm$0.003 & 1.586$\pm$0.003 & 1.403$\pm$0.011 & 1.429$\pm$0.009 \\
\enddata
\tablecomments{Data are presented in the form of average $\pm$ standard
deviation. The average and standard deviation values are calculated from
the data points of each isochrone whose intrinsic $K$ magnitudes are
between $-6$ and 0~mag.}
\end{deluxetable}
\clearpage
\begin{deluxetable}{lccrrrrrcrrrrr}
\tabletypesize{\tiny}
\tablecolumns{14}
\tablewidth{0pt}
\tablecaption{\label{table:nonlin}Extinction Behavior of {\it HST} NICMOS
Filter Pairs}
\tablehead{
\multicolumn{2}{c}{Isochrone Model} &
\colhead{} &
\multicolumn{5}{c}{F110W$-$F205W} &
\colhead{} &
\multicolumn{5}{c}{F110W$-$F222M} \\ \cline{1-2} \cline{4-8} \cline{10-14}
\colhead{$Z$} &
\colhead{Age} &
\colhead{} &
\colhead{$c_0$} &
\colhead{$c_1$} &
\colhead{$c_2$} &
\colhead{$c_3$} &
\colhead{$\sigma(A^{est})$} &
\colhead{} &
\colhead{$c_0$} &
\colhead{$c_1$} &
\colhead{$c_2$} &
\colhead{$c_3$} &
\colhead{$\sigma(A^{est})$}
}
\startdata
0.0001 & $10^7$ & & 7.07E-04 & 2.07E-01 & $-$1.08E-01 & 7.77E-03 & 0.011 & & $-$9.83E-05 & 2.67E-01 & $-$1.18E-01 & 7.01E-03 & 0.013 \nl
0.0001 & $10^8$ & & 9.78E-04 & 1.75E-01 & $-$1.05E-01 & 7.70E-03 & 0.077 & & 1.18E-04 & 2.37E-01 & $-$1.16E-01 & 7.07E-03 & 0.080 \nl
0.0001 & $10^9$ & & 1.61E-03 & 1.14E-01 & $-$1.00E-01 & 7.88E-03 & 0.082 & & 6.21E-04 & 1.84E-01 & $-$1.14E-01 & 7.41E-03 & 0.083 \nl
0.0001 & $10^{10}$ & & 1.82E-03 & 8.93E-02 & $-$9.82E-02 & 7.95E-03 & 0.049 & & 7.90E-04 & 1.62E-01 & $-$1.13E-01 & 7.53E-03 & 0.050 \nl
0.001 & $6.3 \times 10^7$ & & 9.79E-04 & 1.75E-01 & $-$1.05E-01 & 7.69E-03 & 0.082 & & 1.01E-04 & 2.38E-01 & $-$1.16E-01 & 7.05E-03 & 0.084 \nl
0.001 & $10^8$ & & 1.16E-03 & 1.58E-01 & $-$1.04E-01 & 7.77E-03 & 0.115 & & 2.30E-04 & 2.21E-01 & $-$1.16E-01 & 7.17E-03 & 0.118 \nl
0.001 & $10^9$ & & 1.75E-03 & 9.25E-02 & $-$9.86E-02 & 7.93E-03 & 0.049 & & 6.72E-04 & 1.66E-01 & $-$1.13E-01 & 7.53E-03 & 0.047 \nl
0.001 & $10^{10}$ & & 1.86E-03 & 7.52E-02 & $-$9.74E-02 & 7.98E-03 & 0.057 & & 8.48E-04 & 1.51E-01 & $-$1.12E-01 & 7.57E-03 & 0.054 \nl
0.019 & $10^7$ & & 7.77E-04 & 2.03E-01 & $-$1.08E-01 & 7.74E-03 & 0.010 & & $-$7.16E-05 & 2.64E-01 & $-$1.18E-01 & 7.01E-03 & 0.011 \nl
0.019 & $10^8$ & & 1.33E-03 & 1.36E-01 & $-$1.02E-01 & 7.82E-03 & 0.172 & & 3.57E-04 & 2.05E-01 & $-$1.15E-01 & 7.23E-03 & 0.171 \nl
0.019 & $10^9$ & & 1.93E-03 & 6.95E-02 & $-$9.71E-02 & 7.97E-03 & 0.068 & & 8.10E-04 & 1.46E-01 & $-$1.12E-01 & 7.55E-03 & 0.066 \nl
0.019 & $10^{10}$ & & 2.07E-03 & 4.13E-02 & $-$9.42E-02 & 7.93E-03 & 0.064 & & 8.75E-04 & 1.22E-01 & $-$1.09E-01 & 7.48E-03 & 0.062 \nl
0.03 & $6.3 \times 10^7$ & & 1.03E-03 & 1.72E-01 & $-$1.05E-01 & 7.73E-03 & 0.120 & & 1.43E-04 & 2.35E-01 & $-$1.16E-01 & 7.06E-03 & 0.122 \nl
0.03 & $10^8$ & & 1.30E-03 & 1.36E-01 & $-$1.02E-01 & 7.83E-03 & 0.182 & & 3.61E-04 & 2.06E-01 & $-$1.15E-01 & 7.25E-03 & 0.180 \nl
0.03 & $10^9$ & & 1.92E-03 & 6.25E-02 & $-$9.61E-02 & 7.93E-03 & 0.071 & & 8.03E-04 & 1.40E-01 & $-$1.11E-01 & 7.51E-03 & 0.068 \nl
0.03 & $10^{10}$ & & 2.04E-03 & 3.35E-02 & $-$9.28E-02 & 7.84E-03 & 0.063 & & 8.45E-04 & 1.15E-01 & $-$1.08E-01 & 7.44E-03 & 0.063 \nl
\enddata
\tablecomments{Coefficients of best-fit third-order polynomials for the
extinction curves in Figure~\ref{fig:adiff1} for {\it HST} NICMOS filter pairs.
The difference of the estimated extinction and the true extinction is fitted
to a function $[A_Y^{est}-A_Y] = c_0 + c_1[A_Y^{est}] + c_2[A_Y^{est}]^2 +
c_3[A_Y^{est}]^3$; $\sigma$($A^{est}$) is the average of the standard
deviations of $A_Y^{est}-A_Y$ values.}
\end{deluxetable}
\clearpage
\clearpage
|
Title:
The Ultra Luminous X-ray sources in the High Velocity System of NGC 1275 |
Abstract: We report the results of a study of X-ray point sources coincident with the
High Velocity System (HVS) projected in front of NGC 1275. A very deep X-ray
image of the core of the Perseus cluster made with the Chandra Observatory has
been used. We find a population of Ultra-Luminous X-ray sources (ULX; 7 sources
with LX [0.5-7 keV] > 7x10^39 erg/s). As with the ULX populations in the
Antennae and Cartwheel galaxies, those in the HVS are associated with a region
of very active star formation. Several sources have possible optical
counterparts found on HST images, although the X-ray brightest one does not.
Absorbed power-law models fit the X-ray spectra, with most having a photon
index between 2 and 3.
| https://export.arxiv.org/pdf/astro-ph/0601180 |
\label{firstpage}
\begin{keywords}
galaxies: clusters: individual: Perseus - ULX - galaxies: individual:
NGC\,1275
\end{keywords}
\section{Introduction}
The study of Ultra-Luminous X-ray sources (ULX) has been greatly
expanded by the high spatial resolution and spectral grasp of the
\emph{Chandra} and \emph{XMM-Newton} observatories, respectively. ULX
sources (Fabbiano \& White 2003; Miller \& Colbert 2004) have
2--10~keV X-ray luminosities exceeding $10^{39}\ergps$ and are found
some distance from the centres of galaxies; they are not active
galactic nuclei. Their luminosity exceeds that for a $10\Msun$ black
hole accreting at the Eddington limit which radiates isotropically and
so have created much interest in the possibility that they contain
even higher mass black holes, such as InterMediate Black Holes (IMBH)
of $\sim10^3\Msun$ (Makishima et al 2000; Miller, Fabian \& Miller 2004).
Alternatively they may appear so luminous because of beaming (Reynolds
et al 1999; King et al 2001, Zezas \& Fabbiano 2002) or due to super
Eddington accretion (Begelman 2002).
ULX are most common in starburst galaxies and in very active
star-forming regions, such as in the Antennae and the Cartwheel
galaxy, where populations of tens of them are found (Zezas et al 2002;
Gao et al 2003; Wolter \& Trinchieri 2004). In some cases variability
rules out the possibility that they are just clusters of
lower-luminosity objects. The origin of IMBH is unclear. They may form
as a result of binary interactions in dense stellar environments
(Portegies Zwart \& McMillan 2002). A comparison of IMBH ULX
candidates with a number of well known stellar-mass black holes
candidates (BHC; Miller et al 2004) demonstrates that the
ULX are more luminous but have cooler thermal disk components than
standard stellar-mass BHC. Therefore, ULX in this sample are clearly
different from the sample of stellar-mass BHC and are consistent with
being IMBH.
Here we report on the discovery of a population of 8 point X-ray
sources to the N of the nucleus of NGC\,1275, which is the central
galaxy in the Perseus cluster. All exceed $10^{39}\ergps$ in X-ray
luminosity, and 7 are formally ULX, if they are at the distance of the
cluster. The spatial region where they lie coincides with the High
Velocity System of NGC\,1275. We assume that they are part of that
system. We see no other point sources (apart from the nucleus)
over the body of NGC\,1275
(Fig.~1).
NGC\,1275 is embedded in a complex multiphase environment. Optical
imaging and spectroscopy first established the existence of two
distinct emission-line system toward NGC\,1275: a low-velocity
component associated with the galaxy itself at 5200 km $\rm{s^{-1}}$
and a high-velocity component at 8200 km $\rm{s^{-1}}$ projected
nearby on the sky (Minkowski 1955, 1957). This latter component is
associated with a small gas-rich galaxy falling into the cluster along
our line of sight (Haschick, Crane \& van der Hulst 1982). A merger
scenario has been proposed (Minkowski 1955, 1957). However,
interaction of the low and/or high-velocity system with a third
gas-rich galaxy or system of galaxies (Holtzman et al. 1992;
Conselice, Gallagher \& Wyse 2001), or influences from the surrounding
dense intracluster medium (ICM) (Sarazin 1988; Boroson 1990; Caulet et
al. 1992) have been discussed.
Deep \emph{Chandra} observations have clarified the position of the
High Velocity System (HVS). The depth of the observed X-ray
absorption (e.g. Fig.~1) is nor infilled by emission from hot gas
projected along the line-of-sight so the HVS must lie well in front of
NGC\,1275. Gillmon, Sanders \& Fabian (2004) have estimated a lower
limit on the distance of the HVS from the nucleus of 57 kpc. The low-
and high-velocity system are therefore not yet directly interacting.
The HVS $\emph{is}$ however strongly interacting with the ICM of the
Perseus cluster, which has triggered strong star formation. In this
paper we describe the detailed analysis of the X-ray spatial and
spectral properties of the discrete sources in the high velocity
system.
The paper is organized as follows: in Sect. 2 and 3 we present
reduction and results from the imaging analysis and spectral analysis,
respectively; in Sect. 4 we discuss the results; and Sect. 5
summarizes our findings.
Throughout this paper we use a redshift of 0.018 and
$H_0=70~\rm{km~s^{-1}~Mpc^{-1}}$. This gives a luminosity distance to
the cluster of 80 Mpc; 1 arcsec corresponds to a physical distance of
370~pc.
\section{Imaging analysis}
The \emph{Chandra} datasets included in this analysis are listed in
Table~\ref{tab:obs}. The total exposure time, after removing periods
containing flares, is 890~ks. To prepare the data for analysis, all of
the datasets were reprocessed to use the latest appropriate gain file
(acisD2000-01-29gain\_ctiN0003). The datasets analysed each used an
aimpoint on the ACIS-S3 CCD. The datasets were filtered using the
lightcurve in the 2.5 to 7~keV band on ACIS-S1 CCD, which is a
back-illuminated CCD like the ACIS-S3. The CIAO \textsc{lc\_clean}
tool was used to remove periods 20 per~cent away from the median count
rate for all the lightcurves. This procedure was not used for datasets
03209 and 04289 which did not include the S1 CCD, however no flares
were seen in these observations on the S3 CCD. Each of the
observations was reprojected to match the coordinate system of the
04952 observation.
\begin{table*}
\begin{tabular}{lllllll}
Obs. ID & Sequence & Observation date & Exposure (ks) & Nominal roll
(deg) & Pointing RA & Pointing Dec \\ \hline
3209 & 800209 & 2002-08-08 & 95.8 & 101.2 & 3:19:46.86 & +41:31:51.3 \\
4289 & 800209 & 2002-08-10 & 95.4 & 101.2 & 3:19:46.86 & +41:31:51.3 \\
6139 & 800397 & 2004-10-04 & 51.6 & 125.9 & 3:19:45.54 & +41:31:33.9 \\
4946 & 800397 & 2004-10-06 & 22.7 & 127.2 & 3:19:45.44 & +41:31:33.2 \\
4948 & 800398 & 2004-10-09 & 107.5 & 128.9 & 3:19:44.75 & +41:31:40.1 \\
4947 & 800397 & 2004-10-11 & 28.7 & 130.6 & 3:19:45.17 & +41:31:31.3 \\
4949 & 800398 & 2004-10-12 & 28.8 & 130.9 & 3:19:44.57 & +41:31:38.7 \\
4950 & 800399 & 2004-10-12 & 73.4 & 131.1 & 3:19:43.97 & +41:31:46.1 \\
4952 & 800400 & 2004-10-14 & 143.2 & 132.6 & 3:19:43.22 & +41:31:52.2 \\
4951 & 800399 & 2004-10-17 & 91.4 & 135.2 & 3:19:43.57 & +41:31:42.6 \\
4953 & 800400 & 2004-10-18 & 29.3 & 136.2 & 3:19:42.83 & +41:31:48.5 \\
6145 & 800397 & 2004-10-19 & 83.1 & 137.7 & 3:19:44.66 & +41:31:26.7 \\
6146 & 800398 & 2004-10-20 & 39.2 & 138.7 & 3:19:43.92 & +41:31:32.7 \\
\end{tabular}
\caption{\emph{Chandra} observations included in this analysis. The
exposure given is the time remaining after filtering the
lightcurve for flares. All observations were taken with the
aimpoint on the ACIS-S3 CCD. All positions are in J2000 coordinates.}
\label{tab:obs}
\end{table*}
The 900~ks X-ray image covering the energy range $\rm{0.3-0.8~keV}$ is
shown in Fig \ref{fig:HVS}. The bright NGC\,1275 nucleus is clearly
seen at RA $3^h 19^m 48^s$ and Dec. +$41^o$30'42" (J2000) and the
high-velocity system is seen in absorption to the north of the
nucleus.
The CIAO \textsc{celldetect} source detection routine was then used on
the reprocessed level 2 event data to produce a preliminary list of
point sources. The cell size ranges between 4 pixels to 8 pixels. This
algorithm strongly depends on the local background and the detection
cell in not adjustable to the size of the source. As the X-ray diffuse
emission of the NGC 1275 is very strong, the source list may well
include false detections in high background level regions. Therefore
problematic sources embedded in such regions have been excluded in our
analysis. Moreover, as mentioned above, we only included sources
associated with the HVS.
We have detected 8 bright sources close to the nucleus of NGC\,1275,
located in the northern inner radio lobe of 3C 84. All of these source
are embedded in the same region as the HVS (see Fig.
\ref{fig:whole_picture}). There are no sources associated with the
southern lobe (Fig. \ref{fig:whole_picture}), thus we assume these
sources are associated with the HVS.
Fig. \ref{fig:figure} (\emph{left}) shows the smoothed ACIS-S3 image in the
0.3--7.0 keV band, including numbered labels of all the detected sources
(\emph{top}), centred on source labelled N3 (\emph{centre}) and centred
on source N5 (\emph{bottom}).
All the point-like sources are listed in Table~\ref{tab:positions},
showing their positions and count rates.
\begin{table}
\begin{center}
\begin{tabular}{lcc} \hline \hline
N & Position(J2000)& Count Rate \\
& & (counts $\rm{ ks^{-1}}$) \\ \hline
1.... & 03:19:48.736 +41:30:47.25 & 0.34$\pm$0.05 \\
2.... & 03:19:48.166 +41:30:46.64 & 1.69$\pm$0.06 \\
3.... & 03:19:48.090 +41:31:01.88 & 2.60$\pm$0.08 \\
4.... & 03:19:47.994 +41:30:52.30 & 1.42$\pm$0.09\\
5.... & 03:19:47.925 +41:30:47.50 & 1.19$\pm$0.09 \\
6.... & 03:19:47.602 +41:30:47.01 & 0.74$\pm$0.06 \\
7.... & 03:19:47.422 +41:30:51.93 & 0.95$\pm$0.08\\
8.... & 03:19:47.214 +41:30:47.62 & 1.28$\pm$0.08\\ \hline
\end{tabular}
\caption{Positions of sources detected near the NGC\,1275 centre and
displayed in Fig. \ref{fig:figure} (column 2) and count rate in the
energy range between 0.5--7.0 keV (column 3).}
\label{tab:positions}
\end{center}
\end{table}
We have used archival \emph{HST} observations of NGC\,1275 in order to
search for optical counterparts. The galaxy was imaged with the WFPC2
camera on \emph{HST} using the F814W ($\sim$ I, on 2001 November 6
with an exposure time of 1200~s) and F702W ($\sim$ R, on 1994 March 31
with an exposure time of 140~s) broad-band filters. Several
coincidences between X-ray sources and optical knots of emission
(F814W) can be seen in Fig. \ref{fig:figure} ({right}), showing the
same regions as Fig. \ref{fig:figure} ({left}).
The \emph{HST} image shows many highly absorbed features. When we
compare in detail, sources N7 and N8 are located in star forming
regions, while N2 and N6 have a point-like counterpart. Sources N1,
N3, N4 and N5 have no optical identification. Therefore, we have found
a possible correlation between compact X-ray sources and regions of
vigorous star formation. The implications are discussed later.
\begin{table}
\begin{center}
\begin{tabular}{lcccc} \hline \hline
N & F${\rm _X}$/F$_{{\rm F814W}}$& F$_{\rm X}$/F$_{{\rm
F702W}}$& ${\rm M_{F814W}}$ & ${\rm M_{F702W}}$ \\
& & & &\\ \hline
1.... & $>26.5$ & $>16.2$ & $>$22.6 & $>$22.1 \\
2.... & 25.4 & 23.6 & 20.4 & 20.3 \\
3.... & $>$18800 & ... & $>$26.8 & ... \\
4.... & $>$1081 & $>$800 & $>$24.5 & $>$24.2 \\
5.... & $>$123 & $>$60.6 & $>$22.6 & $>$21.9 \\
6.... & 26.2 & 28.4 & 21.7 & 21.7 \\
7.... & 76.7 & 51.4 & 22.3 & 21.9 \\
8.... & 134 & 90.1 & 22.2 & 21.7 \\ \hline
\end{tabular}
\caption{Optical analysis. X-ray to optical ratios (columns 2 and 3) and
magnitude determinations (columns 4 and 5) for the filters F814W and F702W,
respectively, with the X-ray flux between 1.0--7.0~keV.}
\label{tab:optical}
\end{center}
\end{table}
In order to investigate the emission mechanism of these ULX, the X-ray
to optical flux ratios have been computed between the F702W and F814W
\emph{HST} broad-bands and 1.0--7.0~keV X-ray band. Preliminary
processing of the raw images including corrections for the flat
fielding was done remotely at the \emph{Space Telescope Science
Institute} through the standard pipeline. For each frame, cosmic
rays were removed by image combination, using the {\sc imcombine}
routine in IRAF. After cosmic ray removal, the frames were added using
task {\sc wmosaic} in STSDAS package. Photometric measurements were
made with {\sc phot} task, within the NOAO package. Finally the fluxes
and magnitudes have been determined using the photometric zero-point
information in the header of the calibrated image files.
These results are shown in Table \ref{tab:optical}, including the
X-ray to optical flux ratios from the F814W and F702W broad-band
filters, and the magnitude determinations from the same filters. In
the cases where an optical counterpart has not been found (N1, N3, N4
and N5), the magnitudes and fluxes are just a lower limit.
\section{Spectral analysis}
We extracted spectra for all the detected sources close to the HVS,
using extraction regions defined to include as many of the
source photons as possible, but at the same time minimizing
contamination from nearby sources and background. The background
region was either a source-free circular annulus or several circles
surrounding each source, in order to take into account the spatial
variations of the diffuse emission and to minimize effects related to
the spatial variation of the CCD response.
For each source, we extracted spectra from each of the datasets. These
spectra were summed to form a total spectrum for each source. Response
and ancillary response files were created for each source in each of
the observations using the CIAO \textsc{mkacisrmf} and \textsc{mkwarf}
tools. The responses for a particular source were summed together,
weighting according to the number of counts in each observation.
The spectra were fitted using XSPEC v.11.3.2. In order to use the
${\rm \chi^{2}}$ statistic, we grouped the data to include at least 20
counts per spectral bin, before background subtraction. In spectral
fitting we excluded any events with energies above 7.0 keV or below
0.5 keV.
\begin{table}
\begin{center}
\begin{tabular}{lccr} \hline \hline
N & N${_{\rm H}}$ & ${\rm \Gamma}$ & ${\rm \chi^{2}}$/d.o.f. \\
& (${\rm 10^{21}cm^{-2}}$) & & \\ \hline
1.... & 2.5$^{(a)}$ & 3.20$^{+0.23}_{-0.37}$ & 112.90/101 \\
2.... & 2.72$^{+1.43}_{-0.87}$ & 1.78$^{+0.30}_{-0.24}$ & 101.50/109 \\
3.... & 2.49$^{+0.40}_{-0.40}$ & 2.08$^{+0.09}_{-0.09}$ & 153.86/142 \\
4.... & 2.05$^{+0.91}_{-0.96}$ & 2.29$^{+0.44}_{-0.28}$ & 156.24/152 \\
5.... & 2.64$^{+1.23}_{-0.93}$ & 2.92$^{+1.44}_{-0.36}$ & 124.09/139 \\
6.... & 3.74$^{+1.57}_{-1.39}$ & 3.51$^{+0.48}_{-0.66}$ & 102.58/92 \\
7.... & 4.03$^{+1.78}_{-1.45}$ & 3.20$^{+1.39}_{-0.48}$ & 133.69/135 \\
8.... & 2.66$^{+1.00}_{-0.91}$ & 2.13$^{+0.52}_{-0.25}$ & 150.81/138 \\ \hline
\end{tabular}
\caption{Spectral fits. (a) The column density of source N1 has been fixed due to the low
count rate.}
\label{tab:fittings}
\end{center}
\end{table}
Table \ref{tab:fittings} summarizes our spectral results in terms of
the absorbing column density and photon index.
The sources have been modelled with an absorbed power law slope with
photon index between ${\rm \Gamma=}$[1.78-5.56] and an equivalent
column density of $\rm{N_H=[2.05-4.03]\times 10^{21} cm^{-2}}$. In all
the cases the single component power law give satisfactory fits. The
column density of source N1 has been fixed due to the low count rate.
The fitted $\rm{N_H}$ values are consistent with the intrinsic
absorption measured e.g. in the optical band; the value of
A$\rm{_V}$=0.54 corresponds to $\rm{N_H\sim 1.1\times
10^{21}cm^{-2}}$, assuming $\rm{A_V=N_H \times 5.3 \times 10^{-22}}$
for $\rm{ R_V=3.1}$ (Bohlin et al. 1978). This value should be a lower
limit to the fitted $\rm{N_H}$ value to be consistent, as is seen in
Table \ref{tab:fittings}.
As an example of our spectral fits, the brightest source, N3, has been
fitted with a power-law with spectral index of $2.08\pm0.09$ and absorption of
$\rm{N_H=2.5\pm0.4 \times 10^{21}~cm^{-2}}$ (see Fig.
\ref{fig:source1_spec}).
\begin{table}
\begin{center}
\begin{tabular}{llll} \hline \hline
N & F$\rm{_{obs}}$(0.5--7.0 keV) & F$\rm{_{corr}}$(0.5--7.0 keV) & $\rm{log~L_{X}}$ \\
& erg $\rm{cm^{-2}~s^{-1}}$ & erg $\rm{cm^{-2}~s^{-1}}$ & 0.5--7.0 keV \\ \hline
1.... & 2.09 $\times 10^{-15}$ & 4.34 $\times 10^{-15}$ &39.51 \\
2.... & 7.59 $\times 10^{-15}$ & 9.97 $\times 10^{-15}$ &39.86 \\
3.... & 1.64 $\times 10^{-14}$ & 2.28 $\times 10^{-14}$ &40.22 \\
4.... & 7.76 $\times 10^{-15}$ & 1.10 $\times 10^{-14}$ &39.91 \\
5.... & 5.36 $\times 10^{-15}$ & 1.07 $\times 10^{-14}$ &39.90 \\
6.... & 3.02 $\times 10^{-15}$ & 9.23 $\times 10^{-15}$ &39.84 \\
7.... & 4.28 $\times 10^{-15}$ & 1.18 $\times 10^{-14}$ &39.95 \\
8.... & 7.67 $\times 10^{-15}$ & 1.16 $\times 10^{-14}$ &39.93 \\ \hline
\end{tabular}
\caption{Fluxes (observed and k-corrected) and luminosities assuming a cosmological
model with $\rm{H_{0}=70~km~s^{-1} Mpc^{-1}}$ and z=0.018.}
\label{tab:luminosities}
\end{center}
\end{table}
In Table \ref{tab:luminosities} we list the 0.5-7~keV flux and
(absorption corrected) luminosities of the individual sources based on
the best-fit power law model.
The lower limit of the luminosity of point sources in the image, if at
the distance of NGC\,1275, is $\rm{L_X(0.5-7.0~keV)=3.2\times
10^{39}erg~s^{-1}}$, which is already well above the Eddington limit for
a neutron star binary ($\rm{L_X\sim 3 \times 10^{38}erg~ s^{-1}}$) and
is also above the limit of canonical ULX, i.e. $\rm{\ge 10^{39}erg ~
s^{-1}}$.
The brightest point source has a luminosity of
$\rm{L_X(0.5-7.0~keV)=1.67 \times 10^{40} erg ~ s^{-1}}$, and is one
of the brightest individual sources found in a galaxy. A ULX source
more luminous than the entire X-ray luminosity of a normal galaxy has
been found in the Cartwheel system with a luminosity of at least
$\rm{L_X \sim 2-4 \times 10^{40} erg ~ s^{-1}}$ (Gao et al. 2003;
Wolter \& Trinchieri 2004). They explain this luminosity with a
high-mass X-ray binary source (HMXB). The high X-ray luminosity
suggests either a single extremely bright source, or a very dense
collection of several high $\rm{L_X}$ sources, which would be even
more peculiar. Evidence of time variability might suggest that is a
single high $\rm{L_X}$ source.
Time variability analysis has been performed. The observations span
about two years. Two data files were observed on 2002 August 8 and 10,
and the other eleven data files were observed from 2004 October 4 to
2004 October 20, giving an almost daily coverage. The exposure times
are between 22 and 143 ks. The data characteristics allows us
determine short variation in 16 days (second period) and long-term
variability of 2 years. Because of the low count rates of the sources
in NGC\,1275 (see Table \ref{tab:positions}), it is very hard to
search for short-term variability. We extracted light-curves, using
{\sc dmextract} CIAO task for the two brightest sources (N3 and N4)
(net count rate greater than 0.98 count $\rm{s^{-1}}$) binned with bin
sizes of 500, 1000, 2500 and 5000 s. In both cases the points were
consistent with the respective mean values and variability has not
been found. Furthermore, the mean values between 2002 and 2004 are the
same, including errors bars. Therefore, evidence of time variability
has not been found during the whole set of observations.
\section{Discussion}
\emph{Chandra} has revealed significant populations of ULX in the
interacting systems of the Antennae (NGC 4038/9; Zezas, Fabbiano \&
Murray 2002) and the Cartwheel ring galaxy (Gao et al. 2003; Wolter \&
Trinchieri 2004), where dramatic events have stimulated massive star
formation. We have reported here on another example (Fig.
\ref{fig:figure} left) in the HVS of NGC\,1275 which is interacting
with the ICM of the Perseus cluster.
The sources are spatially associated with the distribution of
absorbing clouds seen in soft X-ray (Fig.~2) and optical (Fig.~3)
images. Two sources (N7 and N8) are directly linked with dust knots
and another two (N2 and N6) have an optical point-like counterpart
(Fig. \ref{fig:figure} \emph{bottom}). Similar correspondence have
been found in the Cartwheel galaxy with the outer ring (Wolter \&
Trinchieri 2004) and in the Antennae galaxies with 39 X-ray sources
within the WFPC2 field (Zezas et al. 2002). The optical brightness of
the counterparts in the HVC are too high to be individual stars and so
may be associated with young star clusters. Following the discussion
of young star clusters in NCG\,1275 given by Richer et al (1993), an
object of magnitude 22 corresponds to a cluster mass of about
$10^6\Msun$ if its age is about $10^7\yr$. The HVC system travels at
least 30~kpc in $10^7\yr$ so if a strong interaction with the core of
the Perseus galaxy cluster has triggered star cluster formation in the
HVC, then the clusters should have ages less than $\sim 10^8\yr$.
Our interpretation of the spatial correspondence with star clusters is
that the regions are especially active, indicating a real link between
ULX and star-forming regions, and meaning they are young objects.
However the optical limits on sources N3 and 4 rule out any
association with massive clusters in those cases (the limit on the
absolute magnitude is about $-8$).
In M31 and the Milky Way (Grimm, Gilfanov \& Sunyaev 2003), XRB have
luminosities consistent with the Eddington limit of a $\rm{\sim 2
M_\odot}$ accreting object. They produce luminosities $\rm{\sim 3
\times 10^{38}~ erg~s^{-1}}$, about one order of magnitude below the
limiting luminosity in our sample ($\rm{3.2 \times 10^{39}~
erg~s^{-1}}$). It is possible that our ULX consist of at least 15 (or
130, in the case of the brightest source found) `normal' XRB clustered
together, perhaps in a young star cluster. However in other objects we
know that variability requires the presence of intrinsically luminous
X-ray sources (e.g. M82; Griffiths et al. 2000, Kaaret et al. 2001).
Alternative possibilities are that black hole sources, with masses in
the range of galactic black hole binaries, are mildly beamed (Reynolds
et al. 1999 and King et al. 2001). Spectral and timing features
however rule out this possibility in some ULX (e.g. Strohmayer \&
Mushotzky 2003). We note that compact supernova remnants sometimes
have ULX luminosities (e.g. Fabian \& Terlevich 1998), but no recent
supernovae have been reported for NGC\,1275 (SN1968A was to the S of
the HVS; Capetti 2002).
Finally, we recall the IMBH model which has spectral support from some
sources (Miller et al 2004; the level of absorption in NGC\,1275 is
too high for any soft excess to be observed). They may form in dense
star clusters.
Our optical studies have clearly shown that the ULX have very high
X-Ray to optical flux ratios. X-ray selected AGN from the \emph{Rosat
all sky survey} tend to have $\rm{log(F_X/F_{opt})\sim 1}$. Thus the
ULX do not have the optical properties expected if their were simple
extensions of AGN (IMBH, as low luminosity limit). However, low mass
X-ray binaries in the Milky Way have $\rm{F_X/F_{opt}\sim 100-10000}$
(Mushotzky 2004).
The results found in our system indicate that we have a mixed group of
objects (see Table \ref{tab:optical}). At least 4 out of 8 sources
(N3, N4, N5 and N8) have high X-ray to optical flux ratios. At
least 3 out of 8 (N1, N2 and N6) have lower X-ray to optical ratios,
possibly because they lie in star clusters.
Our data are consistent with no significant variability, similar to
the result obtained on NGC\,3256 by Lira et al. (2002). Time
variability is frequently observed in ULX (e.g. IC\,342, Sugiho et al.
2001 or M51 X-1, Liu et al. 2002), arguing that most of them are
single compact objects, rather than a sum of numerous lower luminosity
objects in the same object. While most ULX vary, many show low
amplitude variability on long time scales (e.g. the Antennae galaxies,
Zezas et al. 2002), which is very different to galactic black holes.
Portegies Zwart, Dewi \& Maccarone (2004) find that a persistent
bright ULX requires a doner star exceeding $15\Msun$. The search for
characteristic frequencies is one of the most productive way of
determining the nature of the ULX.
\section{Conclusions}
We have described the detailed analysis of the spatial and spectral
properties of the discrete X-ray sources detected with a deep
\emph{Chandra} ACIS-S observation around NGC\,1275. Our results are
summarized below:
\begin{enumerate}
\item We have detected a total of 8 sources to the north of NGC\,1275
nucleus.
\item The sources are spatially coincident with the High Velocity
System and thus probably associated with it. They are therefore ULX.
\item Four of the sources have an optical counterpart in the I and R
bands (from \emph{HST} images); two of which are point-like sources and the
other two are associated with star-forming regions.
\item In all the cases a single component power law gives satisfactory
fits, with spectral index of $\rm{\Gamma=}$[1.78-3.51] and an
equivalent column density of $\rm{N_H=[2.05-4.03]\times
10^{21}cm^{-2}}$.
\item The minimum luminosity is $\rm{L_X(0.5-7.0 keV)=3.2\times
10^{39}erg~ s^{-1}}$ (source N1), which is already above the limit
of canonical ULX.
\item No variability was detected in the two brightest sources found.
\end{enumerate}
Our results add to the growing evidence that some episodes of rapid
star formation lead to the production of ULX. Young, massive, star
clusters may be involved in some, but not all of the sources.
\section*{Acknowledgements}
OGM acknowledges the financial support by the Ministerio de Educacion
y Ciencia through the program AYA2003-00128 and grant FPI
BES-2004-5044. ACF thanks the Royal Society for support.
|
Title:
The Most Metal-Rich Intervening Quasar Absorber Known |
Abstract: The metallicity in portions of high-redshift galaxies has been successfully
measured thanks to the gas observed in absorption in the spectra of quasars, in
the Damped Lyman-alpha systems (DLAs). Surprisingly, the global mean
metallicity derived from DLAs is about 1/10th solar at 0<z<4 leading to the
so-called ``missing-metals problem''. In this paper, we present high-resolution
observations of a sub-DLA system at z_abs=0.716 with super-solar metallicity
toward SDSS J1323-0021. This is the highest metallicity intervening quasar
absorber currently known, and is only the second super-solar absorber known to
date. We provide a detailed study of this unique object from VLT/UVES
spectroscopy. We derive [Zn/H]=+0.61, [Fe/H]=-0.51, [Cr/H]=<-0.53, [Mn/H] =
-0.37, and [Ti/H] = -0.61. Observations and photoionisation models using the
CLOUDY software confirm that the gas in this sub-DLA is predominantly neutral
and that the abundance pattern is probably significantly different from a Solar
pattern. Fe/Zn and Ti/Zn vary among the main velocity components by factors of
\~ 3 and ~ 35, respectively, indicating non-uniform dust depletion. Mn/Fe is
super-solar in almost all components, and varies by a factor of ~ 3 among the
dominant components. It would be interesting to observe more sub-DLA systems
and determine whether they might contribute significantly toward the cosmic
budget of metals.
| https://export.arxiv.org/pdf/astro-ph/0601079 |
\title{The Most Metal-Rich Intervening Quasar Absorber Known\thanks{Based
on the UVES observations collected during the DDT ESO prog. ID
No. 274.A-5030 at the VLT/Kueyen telescope, Paranal, Chile } }
\author{C. P\'eroux$^1$, V. P. Kulkarni$^2$, J. Meiring$^2$, R. Ferlet$^3$, P. Khare$^4$, J. T. Lauroesch$^5$, G. Vladilo$^6$, \&
D. G. York$^7$. }
\offprints{C. P\'eroux.}
\institute{$^1$ European Southern Observatory, Garching-bei-M\"unchen, Germany. \email{cperoux@eso.org}\\
$^2$ Dept. of Physics and Astronomy, Univ. of South Carolina, Columbia, USA.\\
$^3$ Institut d'Astrophysique de Paris, UMR7095 CNRS,
Universite Pierre \& Marie Curie, France.\\
$^4$ Dept. of Physics, Utkal University, Bhubaneswar, India.\\
$^5$ Dept. of Physics and Astronomy, Northwerstern University, Evanston, USA.\\
$^6$ Osservatorio di Trieste, Trieste, Italy.\\
$^7$ Dept. of Astronomy and Astrophysics, Univ. of Chicago, Chicago, USA.
}
\authorrunning{C. P\'eroux et al.}
\titlerunning{The Most Metal-Rich Intervening Quasar Absorber Known}
\date{Received August 17, 2005; accepted January 3, 2006}
\abstract{The metallicity in portions of high-redshift galaxies has
been successfully measured thanks to the gas observed in absorption in
the spectra of quasars, in the Damped Lyman-$\alpha$ systems
(DLAs). Surprisingly, the global mean metallicity derived from DLAs is
about 1/10$^{\rm th}$ solar at 0$\la$z$\la$4 leading to the so-called
``missing-metals problem''. In this paper, we present high-resolution
observations of a sub-DLA system at \zabs=$0.716$ with super-solar
metallicity toward SDSS J1323$-$0021. This is the highest metallicity
intervening quasar absorber currently known, and is only the second
super-solar absorber known to date. We provide a detailed study of
this unique object from VLT/UVES spectroscopy. We derive
[Zn/H]=$+$0.61, [Fe/H]=$-$0.51, [Cr/H]=$<-$0.53, [Mn/H] = $-$0.37, and
[Ti/H] = $-$0.61. Observations and photoionisation models using the
CLOUDY software confirm that the gas in this sub-DLA is predominantly
neutral and that the abundance pattern is probably significantly
different from a Solar pattern. Fe/Zn and Ti/Zn vary among the main
velocity components by factors of $\sim 3$ and $\sim 35$,
respectively, indicating non-uniform dust depletion. Mn/Fe is
super-solar in almost all components, and varies by a factor of $\sim
3$ among the dominant components. It would be interesting to
observe more sub-DLA systems and determine whether they might
contribute significantly toward the cosmic budget of metals.
\keywords{Galaxies: abundances -- intergalactic medium -- quasars:
absorption lines -- quasars: individual: SDSS J1323$-$0021} }
\section{Introduction}
Damped Lyman-$\alpha$ systems (DLAs) seen in absorption in the spectra
of background quasars are selected over all redshifts independent of
the intrinsic luminosities of the underlying galaxies. They have
hydrogen column densities, \loghi\ $\ga$ 20.3 and are the major
contributors to the neutral gas in the Universe at high redshifts
(Storrie-Lombardi \& Wolfe 2000; P\'eroux \e\ 2003b). But it has been
suggested that at least some of the \hi\ lies in systems with \hi\
column density below that required by the traditional DLA definition,
in the ``sub-Damped Lyman-$\alpha$ Systems (sub-DLAs)'' with $19.0$ $<$
\loghi\ $<$ 20.3. The DLAs and sub-DLAs offer direct probes
of element abundances over $ > 90 \%$ of the age of the Universe. Zn
is a good probe of the total (gas and solid phase) metallicity, in
DLAs because Zn tracks Fe in most Galactic stars with [Fe/H]$> -$3, it
is undepleted on interstellar dust grains, and the lines of the
dominant ionisation species Zn II are often unsaturated (e.g., Pettini
et al. 1999). Abundances of depleted elements such as Cr or Fe
relative to Zn probe the dust content and the relative abundances can
also yield information about the nucleosynthetic processes (e.g.,
Pettini et al. 1997; Kulkarni, Fall, \& Truran, 1997; P\'eroux et al.,
2002; Khare et al. 2004). A study of the cosmological evolution of the
\hi\ column density-weighted mean metallicity in DLAs (e.g., Kulkarni
\& Fall, 2002) shows surprising results. Contrary to most models of
cosmic chemical evolution (e.g., Malaney \& Chaboyer, 1996; Pei, Fall
\& Hauser, 1999), recent observations indicate at most a mild
evolution in DLA global metallicity with redshift for 0$\la$z$\la$ 4
(Prochaska \e\ 2003; Khare et~al. 2004; Kulkarni, et~al., 2005; and
references therein). Even theoretical models such as
Smoothed-particle-hydrodynamics simulations (Nagamine, Springel, \&
Hernquist, 2004) predict that the true DLA metallicities could be 1/3
solar at z=2.5 and higher at lower redshifts.
Even at z=2.5, making a census of the predicted and observed neutral
comoving densities of gas, $\Omega$, and metals, $\Omega_{\rm Z}$, one
finds that most of the baryons are in the Lyman-$\alpha$ forest but
its metal content is extremely low. The measured value of $\Omega_{\rm
HI}$(DLA) is only a small fraction of $\Omega_{\rm baryons}$ and the
DLA global mean metallicity is about 1/10$^{\rm th}$ solar. The
metallicity of Lyman break galaxies is still poorly constrained; but,
in any case, these objects are known to be star-forming galaxies and
may not be representative of the normal galaxy population. In total,
these three components account for no more than $\approx 10-15$\% of
what we expect to have been produced by z=2.5 (Pettini \e\ 2003;
Bouch\'e \e\ 2005). The missing metals problem in low-redshift DLAs is
even more surprising since the high global star formation rate
estimates at z$>$1.5 (e.g. Madau \e\ 1998) imply that higher
metallicities should be expected at low redshift.
It is possible that $\Omega_Z$(DLA) has been
underestimated. Metal-rich DLAs could obscure quasars due to their
possible high dust content (Fall \& Pei 1993). This may be the reason
for the apparent low metallicity in the DLAs observed in optically
selected quasars (e.g., Fall \& Pei 1993; Boiss\'e et
al. 1998). Recently Vladilo and P\'eroux (2005) have shown that the
fraction of high-redshift DLAs missed due to dust obscuration, could
be up to 50\%, which is consistent with the results of surveys of
radio selected quasars (Ellison \e\ 2001). They have estimated that at
z$\sim$2.3, the real mean metallicity of DLAs could be 5 to 6 times
higher than what is observed, which may help alleviate the missing
metals problems. Indeed, systems at lower redshift may have
significantly more dust at any given metallicity simply because the
dust in these objects has had more time to process the elements.
On the other hand, new lines of evidence are pointing toward lower
\nhi\ quasar absorbers like Lyman Limit Systems (LLS) and sub-DLAs
being more metal-rich (P\'eroux \e\ 2003a; Jenkins \e\ 2005). The
dust bias, if real, is also likely to be less severe for metal-rich
sub-DLAs as compared to the metal-rich DLAs due to the lower gas and
therefore dust content in the former, for a constant dust to gas
ratio. Thus, the obscuration bias will affect the DLAs at a lower
dust-to-gas ratio as compared to the sub-DLAs. This scenario is
consistent with the recent radio surveys (e.g., Vladilo \& P\'eroux
2005), but still needs to be further quantified observationally.
Indeed, Zn measurements exist for only two sub-DLAs at low $z$: the
marginally super-solar sub-DLA toward Q0058$+$019 with $z_{\rm
abs}$=0.61, \loghi=20.08, and [Zn/H]=$+$0.08 (Pettini \e\ 2000), and
the supersolar sub-DLA toward SDSS J1323$-$0021 with $z_{\rm
abs}$=0.72, \loghi=20.21, and [Zn/H]=$+$0.40 (Khare et al. 2004).
Although Khare et al. reported [Zn/H]=$+$0.40 for this latter
absorber, the modest resolution of their MMT data could not resolve
the Mg I+Zn II $\lambda$ 2026 and Cr II+Zn II $\lambda$ 2062
blends.The extent of line saturation on the derived column densities
was also unclear. With the goal of addressing these issues with
high-resolution data, we obtained VLT/UVES spectra of this quasar,
which are presented here. These new data are essential to confidently
determine the metallicity by minimising the problem of line
saturation. Section 2 presents the observational set-up and data
reduction process, while section 3 presents the analysis. Section 4
provides a discussion of the results.
\section{Observations and Data Reduction}
Spectra of SDSS J1323$-$0021 (z$_{\rm em}$=1.390; SDSS mag $g=18.49$)
were acquired in service mode as Director's Discretionary Time (DDT)
on 3$^{\rm rd}$ of March and 13$^{\rm th}$ of March 2005 with the
high-resolution UVES spectrograph mounted on Kueyen Unit 2 VLT
(D'Odorico \e\ 2000). Three exposures of length 4100 sec, 3500 sec and
4100 sec were obtained with standard 390$+$562 settings thus providing
a wavelength coverage of $\sim$3300\AA-4400\AA, 4700\AA-5600\AA\ and
5800\AA-6600\AA.
The data were reduced using the most recent version of the UVES
pipeline to accommodate for the new format of the raw fits file
(version: uves/2.1.0 flmidas/1.1.0). Master bias and flat images were
constructed using calibration frames taken the closest in time to the
science frames. The science frames were extracted with the ``optimal''
option. The spectrum was then corrected to vacuum heliocentric
reference. The resulting spectra were combined weighting each spectrum
with its signal-to-noise. The final spectra have resolution of 4.7 km
s$^{-1}$ at Zn II $\lambda 2026$. The spectra were divided into 100
{\AA} regions, and each region normalised using cubic spline functions
of orders 1 to 5.
\section{Analysis}
\begin{table*}
\begin{center}
\caption{Parameter fit to the $z_{\rm abs} = 0.716$ sub-DLA model. Velocities
and b are in km s$^{-1}$ and N's are in cm$^{-2}$.}
\label{t:fit}
\begin{tabular}{l r r r r r r r r r c c}
\hline\hline
&Vel & b &\ion{Mg}{i} &\ion{Mg}{ii} &\ion{Fe}{ii} &\ion{Zn}{ii} &\ion{Cr}{ii} &\ion{Mn}{ii} &\ion{Ti}{ii} &[Fe/Zn] &[Mn/Fe]\\
\hline
N(X) &$-$120.8 & 7.1 &... &5.63e12 &5.82e12 &... &... &3.75e11 &... &... & 0.79 \\
$\sigma$ &... &... &... &1.66e11 &4.08e11 &... &... &1.26e11 &... &... &... \\
N(X) &$-$96.9 &16.3 &... &3.65e12 &3.64e12 &2.32e12 &$<$2.81e12 &2.97e11 &2.09e11 &$-$2.65 &0.89 \\
$\sigma$ &... &... &... &1.21e11 &4.31e11 &6.53e11 &... &1.93e11 &1.27e11 &... &... \\
N(X) &$-$80.1 & 6.2 &3.16e10 &3.55e12 &3.92e12 &... &$<$1.69e12 &2.09e11 &1.86e11 &... &0.71 \\
$\sigma$ &... &... &1.15e10 &1.36e11 &3.69e11 &... &... &1.23e11 &8.41e10 &... &... \\
N(X) &$-$62.9 & 5.5 &1.20e11 &8.13e12 &4.25e12 &$<$3.10e11 &$<$4.77e11 &2.01e11 &1.07e11 &$-$1.71 &0.65 \\
$\sigma$ &... &... &1.23e10 &3.68e11 &4.22e11 &... &... &1.01e11 &8.00e10 &... &... \\
N(X) &$-$51.4 & 5.0 &$<$1.33e10 &$>$2.86e12 &5.74e12 &$<$2.56e11 &... &1.44e11 &$<$6.92e10 &$-$1.49 &0.38 \\
$\sigma$ &... &... &... &... &1.30e12 &... &... &1.19e11 &... &... &... \\
N(X) &$-$43.0 & 7.4 &4.80e11 &$>$6.01e13 &2.41e13 &1.07e12 &... &1.98e11 &1.32e11 &$-$1.49 &$-$0.11\\
$\sigma$ &... &... &2.29e10 &... &3.54e12 &5.18e11 &... &1.43e11 &9.85e10 &... &... \\
N(X) &$-$13.0 &12.5 &3.04e12 &$>$2.38e14 &1.97e14 &2.57e12 &... &2.40e12 &4.79e11 & $-$0.96 &0.07 \\
$\sigma$ &... &... &1.48e11 &... &2.54e13 &6.22e11 &... &3.14e11 &2.16e11 &... &... \\
N(X) &3.9 & 30.5 &2.36e12 &$>$1.48e14 &4.00e14 &... &... &5.33e12 &$<$4.17e11 &... &0.10 \\
$\sigma$ &... &... &2.03e11 &... &2.63e13 &... &... &5.28e11 &... &... &... \\
N(X) &12.0 & 5.0 &1.44e11 &... &... &$<$3.66e11 &$<$3.20e11 &... &$<$3.89e10 &... &0.01 \\
$\sigma$ &... &... &4.81e10 &... &... &... &... &... &... &... &... \\
N(X) & 28.3 &12.0 &2.06e12 &$>$7.14e13 &1.15e14 &1.53e12 &$<$2.37e12 &1.28e12 &3.80e11 &$-$0.97 &0.03 \\
$\sigma$ &... &... &1.25e11 &... &2.94e13 &7.60e11 &... &2.86e11 &1.53e11 &... &... \\
N(X) &44.5 &11.0 &4.76e12 &$>$2.30e13 &2.89e14 &7.76e12 &... &4.34e12 &4.37e11 &$-$1.27 &0.16 \\
$\sigma$ &... &... &4.37e11 &... &5.42e13 &1.48e12 &... &2.90e11 &1.22e11 &... &... \\
N(X) &56.6 &6.9 &8.70e11 &$>$4.62e13 &2.83e13 &$<$2.73e11 &... &6.30e11 &2.69e11 &$-$0.83 &0.33 \\
$\sigma$ &... &... &1.15e11 &... &1.82e13 &... &... &1.97e11 &9.48e10 &... &... \\
N(X) &71.8 &11.9 &1.93e12 &$>$5.09e14 &3.12e14 &2.49e12 &6.07e12 &3.62e12 &1.58e11 &$-$0.75 &0.04 \\
$\sigma$ &... &... &5.19e10 &... &4.29e13 &7.30e11 &3.24e12 &2.22e11 &1.05e11 &... &... \\
N(X) &92.7 &6.4 &2.42e12 &$>$2.99e14 &3.76e13 &7.35e12 &$<$1.54e12 &1.20e12 &2.09e11 &$-$2.14 &0.48 \\
$\sigma$ &... &... &8.51e10 &... &4.70e12 &1.49e12 &... &1.40e11 &7.87e10 &... &... \\
N(X) &119.3 &8.0 &7.45e10 &4.49e12 &2.33e12 &5.13e11 &6.01e12 &1.98e11 &1.02e08 &$-$2.19 &0.91 \\
$\sigma$ &... &... &1.32e10 &4.90e07 &2.85e11 &4.48e11 &3.11e12 &1.27e11 &1.42e07 &... &... \\
N(X) &194.1 &4.6 &2.44e10 &1.91e12 &... &... &... &... &... &... &... \\
$\sigma$ &... &... &1.00e10 &7.56e10 &... &... &... &... &... &... &... \\
\hline
\end{tabular}
\end{center}
\end{table*}
Several lines of Zn II, Cr II, Fe II, Mn II, and Ti II were detected
at z$_{\rm abs}$=0.716. Fe I, Zn I, and Co II were not detected. The
column densities were estimated by fitting multi-component Voigt
profiles to the observed absorption lines using the program FITS6P
(Welty, Hobbs, \& York 1991) that evolved from the code used by
Vidal-Madjar et al. (1977). FITS6P minimizes the $\chi^{2}$ between
the data and the theoretical Voigt profiles convolved with the
instrumental profile. The atomic data were adopted from Morton
(2003).
The absorption profiles show a complex velocity structure with a total
of 16 components needed. The velocity and Doppler b parameters of the
various components were estimated from the Mg I, Mg II, and Fe II
lines. The component at 12 km s$^{-1}$ is negligible in most species
except Mg I, Zn II and Cr II. The component at 194 km s$^{-1}$ is
negligible in all species but Mg II. The component at $-$121 km
s$^{-1}$ is detected in Mg II, Fe II, and Mn II, but not in Mg I, Zn
II, Cr II, or Ti II. The best-fit column densities in the individual
components and their uncertainties were estimated assuming the same
fixed b and v values for all species in a given component
(Figure~\ref{f:fit}). The results of the profile fitting analysis are
summarised in Table~\ref{t:fit}. Column densities in the few weak
components that could not be well-constrained due to noise are marked
with ``...''; their contributions to the total column densities
(listed in Table~\ref{t:ab}) are negligible. As an additional check,
we also estimated the total column densities using the apparent
optical depth (AOD; Savage \& Sembach 1991) method for the various
detected lines and obtained results consistent with those from the
profile-fitting method. The agreement was within 0.01 dex for Mg I and
Mn II, within 0.1 dex for Fe II and Zn II, and within 0.15 dex for Ti
II.
The Mg I $\lambda 2026.5$ contribution to the Zn II $\lambda 2026.1$
line was estimated using the component parameters for Mg I derived
from the Mg I $\lambda 2852$ profile. This contribution, indicated by
a dashed blue curve in the top left panel of Fig~\ref{f:fit}, was a
small fraction of the observed strength of the $\lambda 2026$
line. The remaining part of the $\lambda 2026$ line was fitted with a
15-component model for Zn II, using the same $b$ values and
velocities, but varying the column densities in the individual
components. The Zn II fit thus obtained was used to estimate the Zn II
contribution to the $\lambda 2062$ line. The remaining part of the
$\lambda 2062$ line was fitted with a 15-component model for Cr II,
using the same set of $b$ values and velocities. This was the only
available estimate for Cr II, since the Cr II $\lambda \lambda$ 2056,
2066 lines lie in noisy regions and are undetected. The relatively
large errors in the Cr II column densities arise from the noisy nature
of the $\lambda 2062$ line. The Fe II column densities in the
components at low velocities were constrained by using the Fe II
$\lambda \lambda 2260, 2374$ lines, since the stronger Fe II lines are
saturated. The weaker Fe II components at high positive and negative
velocities were constrained in column density using the $\lambda
\lambda 2374, 2382, 2600$ lines, since these components are poorly
constrained by the weaker $\lambda \lambda 2260,2374$ lines. The Mg
II $\lambda 2796, 2803$ profiles were fitted together, but provide
only a lower limit to \mg2\ owing to saturation in the central
components. The relative abundances were calculated using solar
abundances from Asplund et al. (2005), adopting the mean of
photospheric and meteoritic values for Mg, Ti, Cr, Fe, Zn, and the
meteoritic value for Mn.
One concern is that sub-DLAs may be partially ionised in H,
artificially enhancing the ratio of Zn II to H I, for instance. To
investigate the ionisation corrections, we used the CLOUDY software
package (version 94.00, Ferland 1997) and computed photoionisation
models assuming ionisation equilibrium and a solar abundance
pattern. We thus obtained the theoretical column density predictions
for any ionisation state of all observed ions as a function of the
ionisation parameter U. Our findings confirm the observations: from a
comparison of the observed and theoretical Mg II/Mg I ratios, we
deduced that the gas in this sub-DLA is predominantly neutral ($\log U
<- 5$) and the overall abundance pattern is probably not solar. This
latter point is also clear from the relative abundances listed for
each component in Table~\ref{t:fit}. It should be pointed out
that there are no third ionisation stage detected in the system under
study. Nevertheless, we do have Ti II, which has the same ionisation
potential as H I. The fact that Ti is not suppressed compared to Fe or
Mn also implies that the gas is neutral with low ionisation
parameter.
\begin{table}
\begin{center}
\caption{Summary of total abundances.}
\label{t:ab}
\begin{tabular}{l c l c }
\hline\hline
Id&$\log N_{\rm total}$&A(X/N)$_{\sun}$&[X/H]$^*$\\
\hline
\ion{Mg}{i} &$13.26\pm0.01$ &... &... \\
\ion{Mg}{ii} &$>15.15$ &$-$4.47 &$>-$0.58\\
\ion{Fe}{ii} &$15.15 \pm 0.03$ &$-$4.55 &$-$0.51 $\pm 0.20$\\
\ion{Zn}{ii} &$13.43 \pm 0.05$ &$-$7.40 &$+$0.61$\pm 0.20$\\
\ion{Cr}{ii} &$<13.33$ &$-$6.37 &$<-$0.52\\
\ion{Mn}{ii} &$13.31 \pm 0.02$&$-$6.53 &$-$0.37$\pm 0.20$\\
\ion{Ti}{ii} &$12.49 \pm 0.11$&$-$7.11 &$-$0.61$\pm 0.22$\\
\hline
\end{tabular}
\end{center}
$^*$ The error bars on [X/H] include the errors in log $N(X)$ and
\loghi.
\end{table}
\section{Discussion and Conclusions}
Table~\ref{t:ab} lists the abundances, using
$\loghi=20.21^{+0.21}_{-0.18}$ (Khare \e\ 2004) that we derived from
Voigt profile fitting of the damped Ly-$\alpha$ line in the
publicly available HST/STIS spectrum of SDSS J1323$-$0021 (program GO
9382; PI: Rao). Rao et al. (2005) obtained
$\loghi=20.54^{+0.16}_{-0.15}$ from the same data set. We use the
former value since it gives a smaller residual with respect to the
data and therefore regard this absorber as a sub-DLA. For either \nhi,
the strength of the Zn II lines detected in our UVES spectrum implies
a super-solar metallicity. Using the standard definition: $[X/H] =
\log [N(X)/N(H)]_{DLA}- \log [N(X)/N(H)]_{\odot}$, we find
[Zn/H]=$+$0.61.
In principle, if Mg I $\lambda$ 2852 were substantially saturated, the
contribution of Mg I $\lambda$ 2026 could be higher than our best-fit
estimate. However, based on our apparent optical depth measurements
and profile-fitting results, we estimate that it would take $\sim$8
times more total Mg~{\sc i} than the value derived from the $\lambda
2852$ line to contribute the entirely of the $\lambda 2026$ line. Such
a high value of Mg~{\sc i} can be ruled out by the observed profile of
the $\lambda 2852$ line. (Of course, such a scenario would also be
inconsistent with the observed strength and profile of the $\lambda
2062$ line.) To understand this issue in more detail, we estimated the
maximum Mg~{\sc i} in the dominant components that would still give
the shape of the Mg I $\lambda$2852 profile consistent with the
observed profile within the noise level in the continuum. This maximum
total Mg~{\sc i} $= 2.7e13$ is about $50 \%$ larger than the best-fit
value listed in Table 2. Putting this maximum Mg I model in the
$\lambda 2026$ line, the corresponding total Zn~{\sc ii} needed to fit
the remaining part of the $\lambda$ 2026 line would be 2.5e13, lower
by $< 10 \%$ from our best-fit value. Thus [Zn/H] is at least
$>+$0.59, indicating that our result would not be affected much by
saturation of Mg I $\lambda$ 2852. Finally, as an additional check, we
also estimated the maximum contribution of Cr II 2062 by rebinning our
spectrum by factors of 10 or 20, measuring the upper limit for Cr
II$\lambda$ 2056 in the rebinned spectrum. We then spread this upper
limit for N(Cr II) over the 2062 profile, using the velocity model
derived from the combination of the lines. We assume the same Fe/Cr
ratio in all components, taking the percentage of Cr II in each
components relative to total N(CrII) summed over all components to be
the same as the corresponding percentage of Fe II in that component.
Fitting the remaining part of the $\lambda 2062$ line with a
15-component model of Zn II, we estimated $N_{\rm Zn II} > 2.05e13$,
i.e. $[Zn/H] > 0.50$.
The abundances of Fe, Cr, Mn, and Ti lie in the range of $-$0.4 to
$-$0.6 dex, and indicate that this absorber is not only metal-rich,
but also very dusty. Using the model from Vladilo (2004), we find that
95\% of the Fe is in dust phase and the total metallicity is even
slightly higher than 0.6 dex. This sightline also shows substantial
reddening compared to the SDSS quasar composite ($\Delta (g-i) =
0.47$; Khare et al. 2004). Considering only the better-determined
components between $-$13 and 100 km s$^{-1}$, Fe/Zn varies by a factor
of $\sim3$ and Ti/Zn varies by $\sim$35. [Mn/Fe] varies by $\sim 3$,
but indicates a super-solar Mn abundance with respect to Fe in all
components. The relative abundance of Mn with respect to Fe is not
expected to exceed the solar value as Mn, unlike Fe, has an odd atomic
number. However, [Mn/Fe]$>$0 is often seen in the Galactic
interstellar gas due to the stronger depletion of Fe on to dust
grains.
To summarise, our high resolution VLT/UVES data have allowed us
to alleviate the saturation issue in Zn II and Cr II lines and
therefore unambiguously prove the super-solar metallicity of the
sub-DLA at $z_{\rm abs}$=0.716 toward SDSS J1323$-$0021. If the dust
obscuration bias for DLAs is indeed significant, as proposed by
Vladilo and P\'eroux (2005), sub-DLA systems such as the one reported
here could be better probes of dusty regions with significant past
star formation (e.g. Lauroesch \e\ 1996; York \e\ 2006), as similar
DLA systems will be missed due to dust obscuration. On the other
hand, it is also, possible to envisage a scenario where the dust
obscuration is not very significant in quasar absorbers (as is
indicated by the rising spectrum of gamma-ray burst afterglows). In
this scenario it is possible that the sub-DLA systems may indeed be
more metal-rich as compared to DLAs as indicated by observations of
P\'eroux \e\ (2003a) and by the observations of super-solar
metallicity in such systems presented here and in Pettini et al
(2003). With the large-scale spectroscopic surveys of quasars
currently underway (e.g. the SDSS, York \e\ 2000, York \e\ 2001), such
metal-rich sub-DLA systems may be found in large numbers. If future
observations indeed find such systems, sub-DLAs may contribute
significantly to the overall global metallicity.
\begin{acknowledgements}
We are grateful to ESO director, Catherine Cesarsky, for time
allocation to this DDT program, and to the VLT staff for carrying out
our observations in service mode. VPK and JM acknowledge support from
the U. S. National Science Foundation grant AST-0206197.
\end{acknowledgements}
|
Title:
The complex X-ray morphology of NGC 7618: A major group-group merger in the local Universe? |
Abstract: We present results from a short {\em Chandra}/ACIS-S observation of NGC 7618,
the dominant central galaxy of a nearby ($z$=0.017309, d=74.1 Mpc) group. We
detect a sharp surface brightness discontinuity 14.4 kpc N of the nucleus
subtending an angle of 130$^\circ$ with an X-ray tail extending $\sim$ 70 kpc
in the opposite direction. The temperature of the gas inside and outside the
discontinuity is 0.79$\pm$0.03 and 0.81$\pm$0.07 keV, respectively. There is
marginal evidence for a discontinuous change in the elemental abundance
($Z_{inner}$=0.65$\pm$0.25,$Z_{outer}$=0.17$\pm$0.21 at 90% confidence),
suggesting that this may be an `abundance' front. Fitting a two-temperature
model to the ASCA/GIS spectrum of the NGC 7618/UGC 12491 pair shows the
presence of a second, much hotter ($T$=$\sim$2.3 keV) component. We consider
several scenarios for the origin of the edge and the tail including a radio
lobe/IGM interaction, non-hydrostatic `sloshing', equal-mass merger and
collision, and ram-pressure stripping. In the last case, we consider the
possibility that NGC 7618 is falling into UGC 12491, or that both groups are
falling into a gas poor cluster potential. There are significant problems with
the first two models, however, and we conclude that the discontinuity and tail
are most likely the result of ram pressure stripping of the NGC 7618 group as
it falls into a larger dark matter potential.
| https://export.arxiv.org/pdf/astro-ph/0601378 |
\title{The complex X-ray morphology of NGC 7618: A major group-group merger in the
local Universe?}
\author{R. P. Kraft}
\affil{Harvard/Smithsonian Center for Astrophysics, 60 Garden St., MS-67, Cambridge, MA 02138}
\author{C. Jones}
\affil{Harvard/Smithsonian Center for Astrophysics, 60 Garden St., MS-2, Cambridge, MA 02138}
\author{P. E. J. Nulsen}
\affil{Harvard/Smithsonian Center for Astrophysics, 60 Garden St., MS-6, Cambridge, MA 02138}
\author{M. J. Hardcastle}
\affil{University of Hertfordshire, School of Physics, Astronomy, and Mathematics, Hatfield AL10 9AB, UK}
\keywords{galaxies: individual (NGC 7618) - X-rays: galaxies - galaxies: ISM - groups: mergers}
\section{Introduction}
The complex cluster morphology seen in X-ray images and in galaxy distributions gave support
to the hypothesis that large structures form hierarchically; that is, that small
groups of galaxies merge to form low-mass subclusters, which then merge to form a massive
rich cluster with the infalling groups aligned along large filaments.
Galaxies and groups are the building blocks of the observable Universe and
contain the bulk of the observable baryons. Groups are estimated to
contain a significant fraction, 20-30\%, of the total matter in the Universe.
Thus groups are important cosmological indicators of the distribution and properties of
the dark matter. However, because they are not as luminous as clusters,
they have received less study than their more massive cousins.
Observations of rich clusters show many examples of pending or ongoing
mergers of subclusters. In particular in X-ray cluster catalogs, about
40\% of rich clusters show substructure \citep{jon84, jon99, moh95}.
Virtually all stages of cluster mergers
have been thoroughly investigated with both the {\em Chandra} and XMM-Newton
observatories \citep{mar00,vik01,bri04,hen04}.
However, less attention has been paid to the merging of
groups and the formation of low-mass clusters due to their lower X-ray luminosity and
the paucity of examples in the local Universe.
To our knowledge, the only nearby example of the early stages of the merger of two roughly equal
mass groups is the NGC 499/NGC 507 pair \citep{kim95,kra03}.
In the hierarchical scenario, group mergers
represent a critical transitional phase in the formation of larger scale structure.
An understanding of the group merger process is thus fundamental to understanding
the growth of structure.
In this paper, we report results from analysis of a short {\em Chandra}/ACIS-S observation
of the nearby elliptical galaxy NGC 7618 ($z$=0.017309 or d$_L$=74.1 Mpc
for WMAP cosmology \citep{spe03} - 1$''$=350 pc). The X-ray luminosity of NGC 7618 is
$\sim$7$\times$10$^{42}$ ergs s$^{-1}$ in the 0.1-10 keV band, typical of groups, not
isolated elliptical galaxies, although it appears to be optically
isolated \citep{col01}.
We find a sharp surface brightness discontinuity in the X-ray emission north of
of the nucleus of NGC 7618, and an extended tail to the south.
We conclude that these features are either the result of a
major group-group merger with UGC 12491, a group
that lies 14.1$'$ on the sky from NGC 7618 at virtually identical
redshift ($z$=0.017365) \citep{ebl02}, or ram-pressure stripping due
to infall of NGC 7618 into a larger gravitational potential (which may include
UGC 12491).
This pair has been poorly studied in large part because of its relatively low galactic
latitude ($\ell=105.575$, $b=-16.909$, $A_V$=0.97, $N_H$=1.19$\times$10$^{21}$ cm$^{-2}$)).
\section{{\em Chandra} and ASCA Observations}
NGC 7618 was observed for 18.4 ks with Chandra/ACIS-S on
December 10, 1999 (OBSID 802). The lightcurve of events in the 5.0-10.0 keV
bandpass on the entire S3 chip, excluding the NGC 7618 nucleus and any point sources
visible by eye, was created using 259 s bins and examined for periods of flaring background.
Intervals where the rate was more than 3$\sigma$ above the mean rate were removed.
There was considerable background flaring during this observations, and
almost 10 ks of data were excluded. Only 8438 s of good time remained. Bad pixels,
hot columns, and columns along node boundaries were also removed.
We present data from the S2 and S3 chips in this paper.
Absorption by foreground gas in our galaxy ($N_H$=1.19$\times$10$^{21}$ cm$^{-2}$) was
included in all spectral fits.
NGC 7618 was also observed by ASCA for $\sim$54 ks on July 7, 1998.
UGC 12491 is contained within the FOV of the ASCA/GIS, and we use
these data to measure the temperature of the gas around this galaxy
and the diffuse emission between NGC 7618 and UGC 12941.
Data from the SIS was not used because of its smaller field of view.
\section{Analysis}
An adaptively smoothed ASCA/GIS image of the NGC 7618/UGC 12491 pair
(usin the CIAO program `csmooth') is shown in Figure~\ref{gisimg}.
The optical and X-ray properties of both groups are summarized in
Table~\ref{galtab}.
Their X-ray luminosities are each $\sim$6-7$\times$10$^{42}$ ergs s$^{-1}$, and their
X-ray emission extends to radii of 150-200 kpc.
The luminosities and spatial extents far exceed those of
typical isolated elliptical galaxies and are more representative of
poor clusters or fossil groups \citep{vik99}.
The central elliptical galaxies represent only a small fraction of
the gravitating mass that resides in a much larger dark matter halo.
No optical census of the galaxy populations around either NGC 7618 or UGC 12491
has been undertaken, but their recessional velocities differ by only 17 km s$^{-1}$.
The temperature of the gas around these galaxies is $\sim$0.8 keV,
typical of groups.
Based on their X-ray luminosities, X-ray extents, and spatial proximity we
conclude that these two objects are groups likely within $\sim$300 kpc of each other
and gravitationally interacting.
An adaptively smoothed, exposure corrected, background subtracted
{\em Chandra}/ACIS-S image in the 0.5-2.0 keV band
of NGC 7618 is shown in Figure~\ref{acisimgb}.
The X-ray bright active nucleus of the host galaxy is labeled at the center.
There are three unusual features to note in this image.
First, the peak of the diffuse X-ray emission (shown orange in
Figure~\ref{acisimgb}), and therefore presumably
the peak in gas density, as well as the center of the host
galaxy lie $\sim$1$'$ North of the center of the larger scale diffuse
X-ray emission (shown green).
An X-ray `tail' extends $\sim$3.3$'$ (69.3 kpc) South of the nucleus.
The statistical significance of this `tail' can be seen in the
X-ray surface brightness profiles shown in Figure~\ref{tailwedge}.
These profiles were made in two 90$^\circ$ sectors
to the North (green) and South (red) with the nucleus of NGC 7618 at
the vertex. The surface brightness in the S sector at distances larger than 70$''$ from
the nucleus is several times higher than in the North sector.
Either the dark matter has a very unusual distribution, or the gas is
not in hydrostatic equilibrium in the gravitational potential.
Second, there is a sharp surface brightness discontinuity $\sim$41$''$ (14.4 kpc)
North of the nucleus that spans $\sim$130$^\circ$. This discontinuity
is delineated by the white arrows in Figure~\ref{acisimgb}.
and is probably a contact discontinuity between two moving fluids.
Third, on smaller scales,
the X-ray emission from the gas peaks $\sim$9$''$ (3.2 kpc) to the E of the
nucleus of the host galaxy as seen in Figure~\ref{xoptovl}.
In the central 10 kpc of the host galaxy, the stars will dominate the gravitating
mass, so there clearly is an offset between the hot gas and the gravitating mass.
All of these strongly suggest that the gas has partially
separated from the gravitating matter and indicate non-hydrostatic motions.
There is an X-ray point source coincident with the optical nucleus, presumably
a low-luminosity AGN. The point source contains 48 counts in the 0.5-5.0 keV band.
Assuming a power-law spectrum with photon index 1.7 and galactic
absorption, the X-ray luminosity of the active nucleus is
4.2$\times$10$^{40}$ ergs s$^{-1}$ in the 0.1-10.0 keV band (unabsorbed).
The X-ray luminosity of the nucleus is roughly an order of magnitude larger than
that expected based on the correlation of X-ray and radio cores \citep{can99}.
This may be a considerable underestimate if the nucleus is heavily absorbed,
but is typical of that found in other ``normal'' elliptical galaxies \citep{jon05}.
The surface brightness profile in a 60$^\circ$
sector to the North of NGC 7618, shown in Figure~\ref{sbprof},
drops by approximately a factor of two
across the discontinuity. At the sharpest region of the edge to the NE of
the nucleus, the surface brightness drops by a factor of
4 over a distance of $\sim$4$''$ (1.4 kpc).
The surface brightness profile in a 30$^\circ$ sector to the NE is shown
in Figure~\ref{newedge}.
The morphology of this discontinuity is similar to that seen in {\em Chandra} observations
of cluster 'cold-fronts', although there is no evidence for a temperature discontinuity
between the two moving fluids as commonly seen in clusters of galaxies \citep{vik01,mar01,maz01}.
We fitted absorbed APEC models to the spectra in 90$^\circ$ sectors of two
annular regions, one inside the discontinuity and one outside.
The thickness of the inner and outer annuli were 33.0$''$ (11.6 kpc)
and 66.4$''$ (23.2 kpc), respectively.
The gas temperature does not change significantly
across the discontinuity ($T_{inner}$=0.785$\pm$0.025 keV,
$T_{outer}$=0.810$\pm$0.070 keV).
There is marginal evidence for a jump
in the elemental abundance ($Z_{inner}$=0.65$\pm$0.25, $Z_{outer}$=0.17$\pm$0.21 - all
uncertainties at 90\% confidence), however.
This suggests that the discontinuity could be an `abundance' front due to
a sharp discontinuity in the elemental abundance, and therefore the emissivity
of the gas as observed in NGC 507 \citep{kra03}.
The temperature and elemental abundance structure in the
gas is probably more complex than we have assumed here, but the short
exposure time and limited quality of the data prevent a more detailed analysis.
Fitting power laws to the surface brightness profiles interior to and
exterior to the discontinuity and assuming hydrostatic equilibrium, we find
a large change in the power law index across the discontinuity,
$\beta_{in}$=0.30 and $\beta_{out}$=0.64.
The change in $\beta$ is almost certainly not related to a change
in the gravitational potential and suggestive of non-hydrostatic gas motions.
We estimate the gas density on both sides of the discontinuity by
deprojecting the surface brightness profile (assuming spherical symmetry) and find
proton densities of 6.0$^{+1.2}_{-0.8}$ and 5.0$^{+2.4}_{-1.2}$
$\times$10$^{-3}$ inside and outside the discontinuity, respectively.
The upper limit of the velocity of the gas interior to the discontinuity estimated
from the maximum pressure difference is Mach 0.9 or $\sim$ 420 km s$^{-1}$ \citep{vik01}.
The X-ray morphology of the ASCA/GIS image (Figure~\ref{gisimg}) suggests that
the two groups reside in a larger-scale dark matter potential.
A third X-ray peak is seen 8.5$'$ to the east of NGC 7618, and diffuse emission
extends at least 15$'$ to the north and 10$'$ to the south.
Any gas that resides in this larger scale dark matter halo should
be hotter than the gas in the NGC 7618 group. We fitted the ASCA/GIS spectrum
in three regions, two 7$'$ radius circles centered on each galaxy and
a third region 15$'$ in radius centered between NGC 7618 and UGC 12491 excluding
the two 7$'$ radius circles centered on each galaxy.
The radius of the two circles centered on NGC 7618 and UGC 12491 corresponds
to the 90\% encircled energy radius for the ASCA/GIS in the 1-2 keV band.
Background was determined from ASCA/GIS high-latitude, blank sky
observations taken from the HEASARC and generated using the FTOOL `mkgisbgd'.
We fit absorbed, single temperature APEC models to each spectrum with $N_H$ frozen at
the Galactic value and the elemental abundance frozen at 0.6$Z_\odot$.
The results of these fits are summarized in the top half of Table~\ref{spectab}.
Single temperature models are poor fits for the regions centered on
NGC 7618 and UGC 12491, but provide
an adequate description of the diffuse emission between the galaxies.
The temperature of the diffuse gas between NGC 7618 and UGC 12491
is 2.32$^{+0.50}_{-0.34}$.
We also fit two temperature models to the two spectra extracted from
the regions centered on NGC 7618 and UGC 12491. As before, both the $N_H$ and the elemental
abundance were frozen. For NGC 7618, we also froze the temperature
of one component at 0.8 keV,
the value determined from the ACIS-S spectral fitting, to reduce the
number of free parameters. If this parameter
is allowed to freely vary, the fit value is consistent with 0.8 keV. For UGC 12491,
the temperatures of both components were allowed to freely vary.
The results of these fits are summarized in the bottom half of Table~\ref{spectab}.
The two temperature model provides acceptable fits in both cases, and
the temperature of the hotter component is $\sim$2.3 keV.
\section{Discussion}
There are at least four possible explanations for the observed X-ray
structures. First, the complex X-ray morphology could be the result
of a radio lobe/IGM interaction.
NGC 7618 is a radio source. It is detected in the NVSS with a flux
density of 20 mJy, with some evidence of extension NW/SE.
At higher frequencies (5 and 8 GHz) two 15-min archival
VLA observations only detect a point-like core coincident with the
center of the galaxy, with a flux density of 4.5 mJy at 4.9 GHz and
3.0 mJy at 8.4 GHz. However, at lower frequencies, the flux density
is much higher: 1.2 Jy in the 151-MHz 6C catalog \citep{hal93},
0.26 Jy in the 408-MHz B3 catalog \citep{fic85},
and 1.0 Jy in the 74-MHz VLA Low-Frequency Sky
Survey (VLSS: http://lwa.nrl.navy.mil/VLSS/). The high flux density
at low frequencies suggests that there may be a relic radio source,
the aged synchrotron plasma from a more energetic phase of the
active nucleus.
The X-ray morphology is very different from that seen in other examples
of radio lobe/ISM interactions, however.
There are no obvious X-ray cavities as are commonly seen in such radio
lobe/ISM interactions (e.g. \citep{mcn00,jon02,hei02}),
nor is there evidence of hot, shock-heated shell
that would be present if the radio lobes were expanding supersonically \citep{kra03}.
It is not clear whether any of these
features would be expected in the case of a relic radio source, however.
A sensitive low-frequency radio map would give us a better
understanding of the possible interaction between radio-emitting
plasma and the hot gas.
Second, it is possible that the NGC 7618 gas is oscillating, or 'sloshing', in the gravitational
potential because of a recent merger/interaction with a lower mass sub-group.
A dust lane has been detected in the optical host galaxy, perhaps
indicative of a recent merger and adding support to this hypothesis \citep{col01}.
Similar structures (on larger scales) to those presented
here have been seen in X-ray observations of clusters of galaxies \citep{mar01,tit05}.
In this scenario, the gas is oscillating, or `sloshing', in the dark matter
potential due to a recent merger with a sub-cluster. The infall of the sub-cluster
drives a shock into the gas that displaces it from the gravitating matter. The
gas oscillates around the center of mass, eventually returning to hydrostatic
equilibrium via viscous dissipation.
Such processes can contribute a significant amount of
energy to the gas and may disrupt or prevent the formation of cooling flows in clusters of
galaxies. If we naively assume the gas on both sides of the
discontinuity is in hydrostatic equilibrium, we find
an unphysical discontinuity in the gravitating mass.
The apparent mass discontinuity is the result of the
non-zero acceleration of the gas inside the discontinuity.
The distribution of the gas will reflect the reduced gravity.
That is, the acceleration term in Euler's equation is non-zero for gas
inside the discontinuity.
Assuming that the core is at maximum displacement from the center, the gravitational potential
energy of the gas is $U\sim G\Delta M r^{-1} \sim$7$\times$10$^{14}$ ergs gm$^{-1}$,
or roughly one third of the thermal energy of the gas.
There is one important difficulty with this interpretation, however. On tens of
kpc scales, the emission peak of the gas and the optical galaxy (NGC 7618) are both
offset (to the N) relative to the larger
scale X-ray isophotes (shown green in Figure~\ref{acisimgb}).
If the sloshing scenario is correct only the gas
and not the galaxy/gravitating mass, should be moving relative to the large scale dark
matter halo. The host galaxy should be resting at the center of the gas
distribution. In addition, the peak of the X-ray emission lies to the East
of the nucleus of the host galaxy, but the larger scale `sloshing' is North/South.
This suggests that there are significant gas motions along perpendicular axes, and
that the gas is coupled to the gravitating mass (the stars)
along the North/South axis, but partially decoupled along the East/West axis.
It is difficult to see how there could be the case unless the merger were off-axis. If this is the case,
the merging galaxy should be detectable by an optical census.
The third possibility is that the observed structures
are the result of a 'near-miss' flyby of the nearby UGC 12491 group.
In this scenario, the UGC 12491 group passed by NGC 7618 from the SE toward
its current position to the WNW, and
the complex X-ray morphology of NGC 7618
is the result of fluid motions in the gas induced by the
gravitational and hydrodynamical interaction.
The observed X-ray morphology is the result of the NGC 7618 core
being displaced relative to the larger scale gravitational potential, and
the surface brightness discontinuity represents a discontinuity in the elemental abundance.
X-ray observations of groups and clusters show that the elemental abundance
is strongly peaked toward the center. The displacement of the core of a group
relative to its halo could create the observed structures.
Hydrodynamic simulations of head-on collisions of equal mass clusters
show that the close approach of the cores can create strong non-hydrostatic motions in
the gas \citep{rot97}.
Elongation of the dark matter potential, and thus the gas distribution, is a natural
consequence of these collisions.
In these simulations, the gas becomes partially separated from the gravitating matter and
is not a good tracer of the dark matter
distribution. The separation between UGC 12491 and NGC 7618 was
probably (much) smaller in the past in order
to create such a large disturbance in the NGC 7618 gas.
The simulations of merging clusters show little separation of the gas from
the dark matter as the two clusters initially approach each other.
It is only as the cores collide/merge and subsequently separate
that significant gas velocities develop.
If the current positions of the two groups
are, in fact, their closest approach so far, non-hydrostatic
effects should just be starting to manifest themselves.
Therefore, the two groups must have already passed each other at least once.
There are several difficulties with this scenario, however.
If the hotter gas seen in the ASCA/GIS spectrum is shock-heated group gas,
the measured temperature ratio (0.8 to 2.3 keV) implies
an infall/approach velocity of the two groups of $\sim$1170 km s$^{-1}$ \citep{lan89},
much larger than typical peculiar velocities.
It would also be difficult to explain the large amount of hot gas ($\sim$10$^{12}$ M$_\odot$)
seen in the ASCA/GIS observation in this scenario under the assumption that
this component is entirely shock-heated group gas.
The total (thermal) energy of the hotter component is only $\sim$1.3$\times$10$^{61}$ ergs,
a reasonable value for the collison of two groups (we note that an AGN
outburst could easily supply this much energy to the gas as well \citep{mcn04}).
If the hot component is, in fact, shock-heated group gas, it is
not bound to the group potential and a transient phenomenon, escaping as
a wind.
It is likely that these two groups are gravitationally bound.
Ignoring angular momentum (which is not believed to be important
on large scales \citep{hof86}) and dissipative forces, and assuming that
the mass of each group is 10$^{13}$ M$_\odot$, the escape velocity of this
pair is $\sim$550 km s$^{-1}$. If either of these groups has a velocity
relative to the center of mass larger than this value, the pair will be
unbound. This is larger than typical peculiar velocities of galaxies/groups,
and it is reasonable to conclude that the system is bound.
Significant angular momentum will decrease this value, whereas dissipative
forces will increase it. The dynamic parameters of this pair are too
poorly known to make a definitive statement, however.
The fourth possibility is that NGC 7618 is falling into
a larger gravitational potential. In this scenario, either
UGC 12491 is at the center of the potential and NGC 7618 is
falling into it, or both NGC 7618 and UGC 12491 are falling
into a larger scale potential.
The existense of the hotter ($\sim$2.3 keV)
gas component in the ASCA/GIS observation (Figure~\ref{gisimg})
supports this scenario.
The X-ray structures seen in the gas around NGC 7618 are then
the result of ram-pressure stripping.
This is the classic `cold-front' scenario discussed by \citet{vik01}.
The effects of ram-pressure stripping on the hot gas atmospheres of
early-type galaxies falling into clusters have been studied in {\em Chandra}
observations of NGC 4472, NGC 4552 (falling into the Virgo cluster) and NGC 1404
(falling into/toward NGC 1399, the dominant member of the Fornax cluster) \citep{bil04,mac05a,mac05b}.
In these cases, a sharp surface brightness discontinuity
in the direction of infall and a diffuse tail in the opposite direction
have been observed. The gas around NGC 7618 is being stripped by
hotter, lower density gas that presumably resides in the larger/deeper dark
matter halo.
If this scenario is correct, either NGC 7618 is falling into UGC 12491 or
both groups are merging with/falling into a larger dark matter potential in
which there is no central dominant elliptical galaxy.
The high temperature of the second spectral component in the ASCA analysis and the
lack of an azimuthally symmetric X-ray halo around UGC 12491 (Figure~\ref{gisimg})
support the latter hypothesis.
This phenomenon has in fact already been observed on a smaller
scale in the Pegasus I group \citep{kra05}. Neither
of the massive ellipticals in this group, NGC 7619 or NGC 7626, lie at the
center of the extended X-ray halo.
{\em Chandra} observations of NGC 7619 show a sharp surface brightness
discontinuity to the NE (presumably the direction of infall), and an
extended, ram-pressure stripped tail in the opposite direction.
If we assume the hotter (2.3 keV) component is in hydrostatic equilibrium
with the dark matter potential and that the gas density follows
a beta-model profile with $\beta$=0.67 and $r_0$=50 kpc (typical for clusters
of galaxies),
the gravitating mass, $M_{grav}$, within a radius of 325 kpc of the midpoint between NGC 7618
and UGC 12491 is $\sim$5.6$\times$10$^{14}$ M$_\odot$.
The gas mass, $M_{gas}$, required to account for the observed ASCA/GIS flux of
the hotter component within this
radius is $\sim$7$\times$10$^{11}$ M$_\odot$ for this density profile.
Ignoring the stellar component, which is insignificant on these spatial scales,
the baryon fraction, $f=M_{gas}/M_{grav}$ is $\sim$1.5\%.
Such a low value of baryon fraction is not improbable for a 2.3 keV
cluster. We speculate that this larger dark matter halo may be a `failed' cluster
in which much of the gas was blown off during a cataclysmic event
early in its formation (either a merger or a powerful AGN remnant).
We caution, however, that the spatial resolution of the ASCA/GIS is poor and
our knowledge about the morphology of the gas limited. In addition, this gas may
not be in hydrostatic equilibrium, so the uncertainties on both (gas and
gravitating) mass estimates are large. A moderate XMM-Newton observation of this pair could measure
the temperature and morphology of the hotter component and resolve this issue.
We note that in this scenario the surface brightness discontinuity north of the NGC 7618 nucleus
cannot be the stagnation point between the hot gas in NGC 7618 and the
gas of this putative larger scale halo.
If the observed X-ray surface brightness discontinuity
is in fact the stagnation point between
these two gases, the temperature of the gas exterior to the discontinuity must be
on the order or higher than that of the halo gas (from Bernoulli's equation).
The stagnation point therefore must lie beyond the observable emission, and the
surface brightness discontinuity represents a contact discontinuity between two fluids.
Unless the infall is highly supersonic, the gas interior to the stagnation point should
not be highly disturbed and should remain in rough hydrostatic equilibrium in the
gravitational potential of NGC 7618.
A deeper {\em Chandra} observation of the central regions of
NGC 7618 is required to elucidate the hydrodynamics of the gas.
\section{Conclusions}
We have observed a sharp surface brightness discontinuity in the X-ray emission
from the hot gas in the NGC 7618 group, and an X-ray `tail' extending
70 kpc in the opposite direction in an 8 ks {\em Chandra}/ACIS-S observation.
Archival ASCA/GIS observations indicate the presence of a hotter (2.3 keV)
component, although the morphology of this gas is poorly constrained.
We conclude that there are three possible explanations for these features.
First, the NGC 7618/UGC 12491 pair underwent a recent `near-miss' flyby.
If this is the case, this pair is the nearest early-stage merger of two
roughly equal mass groups.
Second, UGC 12491 may be at the center of a cluster, and NGC 7618 is falling into
it. Third, NGC 7618 and UGC 12491 are both falling into a gas poor cluster with
no dominant central elliptical galaxy.
Whether the observed features are the result of a group-group merger,
or the infall of two groups into a larger dark matter potential, the NGC 7618/UGC 12491
pair is one of the best examples of an ongoing merger in the local Universe.
Deeper X-ray observations are required to better constrain the thermodynamic
parameters of the gas in the central regions and the larger scale halo. Radio
observations will be critical in assessing the role of radio plasma/IGM interactions.
If the observed X-ray features are the result of ram-pressure stripping or a merger
interaction between the groups, the effects
on the relic radio halo are likely to have been dramatic.
A detailed optical census including velocities of the other galaxies in the both groups will
be useful to constrain their dynamics and their relationship to
the larger dark matter potential.
\acknowledgements
This work was supported by NASA contracts NAS8-38248, NAS8-39073,
the Chandra X-ray Center, and the Smithsonian Institution.
We would like to thank the anonymous referee for comments
that improved this paper.
\clearpage
\clearpage
\clearpage
\clearpage
\clearpage
\clearpage
\clearpage
\clearpage
\begin{table}
{\small
\begin{center}
\begin{tabular}{|l|c|c|}\hline
& NGC 7618 & UGC 12491 \\ \hline
$m_B$ & 14.0 & 14.9 \\ \hline
$M_B$ & -21.3 & -20.2 \\ \hline
$z$ & 0.017309 & 0.017365 \\ \hline
Distance (Mpc) & 74.1 & 74.3 \\ \hline
$L_X$ (ergs s$^{-1}$) & 6.9$\times$10$^{42}$ & 6.2$\times$10$^{42}$ \\ \hline
X-ray Radius & $\sim$170 kpc & $\sim$200 kpc \\ \hline
\end{tabular}
\caption{Summary of the X-ray and optical properties of the
NGC 7618 and UGC 12491 galaxies.
The X-ray luminosity is in the 0.1-10 keV
bandpass (unabsorbed) within 7$'$ (146 kpc) of the nucleus. Absolute magnitudes have
been corrected for extinction.}\label{galtab}
\end{center}
}
\end{table}
\clearpage
\begin{table}
{\small
\begin{center}
\begin{tabular}{|l|c|c|c|}\hline
& NGC 7618 & UGC 12491 & Diffuse \\ \hline\hline
\multicolumn{4}{|c|}{Single Temperature Fits} \\ \hline
$k_BT$ (keV) & 1.43$^{+0.08}_{-0.14}$ & 1.32$^{+0.10}_{-0.13}$ & 2.32$^{+0.50}_{-0.34}$ \\ \hline
Flux & 3.06$\times$10$^{-12}$ & 5.37$\times$10$^{-12}$ & 2.53$\times$10$^{-12}$ \\ \hline
$\chi^2_\nu$ & 1.95 & 1.60 & 0.73 \\ \hline\hline
\multicolumn{4}{|c|}{Two Temperature Fits} \\ \hline
$k_BT_1$ (keV) & 0.80 & 0.86$^{+0.16}_{-0.09}$ & \\ \hline
Flux & 2.26$\times$10$^{-12}$ & 3.69$\times$10$^{-12}$ & \\ \hline
$k_BT_2$ (keV) & 2.21$^{+0.33}_{-0.67}$ & 2.26$^{+1.19}_{-0.35}$ & \\ \hline
Flux & 1.48$\times$10$^{-12}$ & 2.15$\times$10$^{-12}$ & \\ \hline
$\chi^2_\nu$ & 1.03 & 0.78 & \\ \hline\hline
\end{tabular}
\caption{Best-Fit Temperatures and Fluxes of the ASCA/GIS data in three regions.
All uncertainties are 90\% confidence for one parameter of interest. Units of
fluxes are ergs cm$^{-2}$ s$^{-1}$ (unabsorbed) in the 0.5-2.0 keV band.}\label{spectab}
\end{center}
}
\end{table}
|
Title:
The Local Effects of Cosmological Variations in Physical 'Constants' and Scalar Fields II. Quasi-Spherical Spacetimes |
Abstract: We investigate the conditions under which cosmological variations in physical
`constants' and scalar fields are detectable on the surface of local
gravitationally-bound systems, such as planets, in non-spherically symmetric
background spacetimes. The method of matched asymptotic expansions is used to
deal with the large range of length scales that appear in the problem. We
derive a sufficient condition for the local time variation of the scalar fields
driving variations in 'constants' to track their large-scale cosmological
variation and show that this is consistent with our earlier conjecture derived
from the spherically symmetric problem. We perform our analysis with spacetime
backgrounds that are of Szekeres-Szafron type. They are approximately
Schwarzschild in some locality and free of gravitational waves everywhere. At
large distances, we assume that the spacetime matches smoothly onto a Friedmann
background universe. We conclude that, independent of the details of the
scalar-field theory describing the varying `constant', the condition for its
cosmological variations to be measured locally is almost always satisfied in
physically realistic situations. The very small differences expected to be
observed between different scales are quantified. This strengthens the proof
given in our previous paper that local experiments see global variations by
dropping the requirement of exact spherical symmetry. It provides a rigorous
justification for using terrestrial experiments and solar system observations
to constraint or detect any cosmological time variations in the traditional
`constants' of Nature in the case where non-spherical inhomogeneities exist.
| https://export.arxiv.org/pdf/gr-qc/0601056 |
\title{The Local Effects of Cosmological Variations in Physical 'Constants'
and Scalar Fields \\
II. Quasi-Spherical Spacetimes}
\author{Douglas J. Shaw}
\affiliation{DAMTP, Centre for Mathematical Sciences, University of Cambridge,
Wilberforce Road, Cambridge CB3 0WA, UK}
\author{John D. Barrow}
\affiliation{DAMTP, Centre for Mathematical Sciences, University of Cambridge,
Wilberforce Road, Cambridge CB3 0WA, UK}
\date{\today}
\section{\protect\bigskip Introduction}
Over the past few years there has been a resurgence of observational and
theoretical interest in the possibility that some of the fundamental
`constants' of Nature might be varying over cosmological timescales \cite%
{webb}. In respect of two such `constants', the fine structure constant, $%
\alpha $, and Newton's `constant' of gravitation, $G$, the idea of such
variations is not new, and was proposed by authors such as Milne \cite{milne}%
, Dirac \cite{dirac}, and Gamow \cite{gam} as a solution to some perceived
cosmological problems of the day \cite{btip}. At first, theoretical attempts
to model such variations in constants were rather crude and equations
derived under the assumption that constants like $G$ and $\alpha $ are true
constants were simply altered by writing-in an explicit time variation. This
approach was first superseded in the case of varying $G$ by the creation of
scalar-tensor theories of gravity \cite{jordan}, culminating in the standard
form of Brans and Dicke \cite{bd} in which $G$ varies through a dynamical
scalar field which conserves energy and momentum and contributes to the
curvature of spacetime by a means of a set of generalised gravitational
field equations. More recently, such self-consistent descriptions of the
spacetime variation of other constants, like $\alpha $ \cite{bek, bsm}, the
electroweak couplings \cite{ewk}, and the electron-proton mass ratio, $\mu $%
, \cite{bm} have been formulated although most observational constraints in
the literature are imposed by simply making constants into variables in
formulae derived under the assumption that are constant.
The resurgence of interest in possible time variations in $\alpha $ and $\mu
$ has been brought about by significant progress in high-precision quasar
spectroscopy. In addition to quasar spectra, we also have available a
growing number of laboratory, geochemical, and astronomical observations
with which to constrain any local changes in the values of $\ $these
constants \cite{reviews}. Studies of the variation of other constants, such
as $G$, the electron-proton mass ratio, $\mu =m_{e}/m_{pr}$, and other
standard model couplings, are confronted with an array of other data
sources. The central question which this series of papers addresses is how
to these disparate observations, made over vastly differing scales, can be
combined to give reliable constraints on the allowed global variations of $%
\alpha $ and the other constants. If $\alpha $ varies on cosmological scales
that are gravitationally unbound and participate in the Hubble expansion of
the universe, will we see any trace of this variation in a laboratory
experiment on Earth? After all, we would not expect to find the expansion of
the universe revealed by any local expansion of the Earth. In Paper I \cite%
{shawbarrow1}, we examined this question in detail for spherically symmetric
inhomogeneous universes that model the situation of a planet or a galaxy in
an expanding Friedmann-Robertson-Walker (FRW)-like universe. In this paper
we relax the strong assumption of spherical symmetry and examine the
situation of local observations in a universe that contains non-spherically
symmetric inhomogeneity. Specifically, we use the inhomogeneous metrics
found by Szekeres to describe a non-spherically symmetric universe
containing a static star or planet. As in Paper I, we are interested in
determining the difference (if any) between variations of a supposed
'constant' or associated scalar field when observed locally, on the surface
of the planet or star, and on cosmological scales.
When a `constant', $\mathbb{C}$, is made dynamical we can allow it to vary
by making it a function of a new scalar field, $\mathbb{C}\rightarrow
\mathbb{C}(\phi )$, that depends on spacetime coordinates: $\phi =\phi (\vec{%
x},t)$. It has become general practice to combine take all observational
bounds on the allowed variations of $\mathbb{C}(\phi )$. This practice
assumes implicitly that any time variation of $\mathbb{C,}$ on or near the
Earth, is comparable to any cosmological variation that it might experience,
that is to high precision
\begin{equation}
\dot{\phi}(\vec{x},t)\approx \dot{\phi}_{c}(t), \label{wettcond}
\end{equation}%
\noindent for almost all locations $\vec{x}$, where $\phi _{c}$ is the
cosmological value of $\phi $. This assumption is always made without proof,
and there is certainly no \emph{a priori} reason why it should be valid.
Strictly, $\phi $ mediates a new or `fifth' force of Nature. If the assumed
behaviour is correct then this force is unique amongst the fundamental
forces in that its value locally reflects its cosmological variation
directly.
In this series of papers we are primarily interested in theories where the
scalar field, or `dilaton' as we shall refer to it, $\phi $, evolves
according to the conservation equation
\begin{equation*}
\square \phi =B_{,\phi }(\phi )\kappa T-V_{,\phi }(\phi ),
\end{equation*}%
where $T$ is the trace of the energy momentum tensor, $T=T_{\mu }^{\mu }$,
(with the contribution from any cosmological constant neglected). We absorb
any dilaton-to-cosmological constant coupling into the definition of $V(\phi
)$. The dilaton-to-matter coupling $B(\phi )$ and the self-interaction
potential, $V(\phi )$, are arbitrary functions of $\phi $ and units are
defined by $\kappa =8\pi G$ and $c=\hslash =1$. This covers a wide range of
theories which describe the spacetime variation of `constants' of Nature; it
includes Einstein-frame Brans-Dicke (BD) and all other, single-field,
scalar-tensor theories of gravity \cite{bd, bsm, poly, posp}. In cosmologies
that are composed of perfect fluids and a cosmological constant, it will
also contain the Bekenstein-Sandvik-Barrow-Magueijo (BSBM) theory of varying
$\alpha $, \cite{bsm}, and other single-dilaton theories which describe the
variation of standard model couplings, \cite{posp}. We considered some other
possible generalisations in \cite{shawbarrow1}. It should be noted that our
analysis and results apply equally well to any theory which involves
weakly-coupled, `light', scalar fields, and not just those that describe
variations of the standard constants of physics.
In first paper of this series, \cite{shawbarrow1}, we determined the
conditions under which condition \ref{wettcond} would hold near the surface
of a virialised over-density of matter, such as a galaxy or star, or a
planet, such as the Earth, under the assumption of spherical symmetry. We
chose to refer to this object as our `star'. In Paper I, matched asymptotic
expansions were employed to analyse the most general, \emph{%
spherically-symmetric}, dust plus cosmological constant embeddings of the
`star' into an expanding, asymptotically homogeneous and isotropic
spherically symmetric universe. We proved that, independent of the details
of the scalar-field theory describing the varying `constant', that \ref%
{wettcond} is almost always satisfied under physically realistic conditions.
The latter condition was quantified in terms of an integral over sources
that can be evaluated explicitly for any local spherical object.
In this paper we extend that analysis, and our main result, to a class of
embeddings into cosmological background universes that possess \emph{no}
Killing vectors i.e. \emph{no} symmetries. The mathematical machinery that
we use to do this is, as before, the method of matched asymptotic
expansions, employed in \cite{shawbarrow1}, where the technical machinery is
described in detail. A summary of the results obtained there can also be
found in \cite{shawbarrowlett}.
This paper is organised as follows: We shall firstly provide a very brief
summary of the method of matched asymptotic expansions used here. In section
II we will introduce the geometrical set-up that we will use. We will be
working in spacetime backgrounds of Szekeres-Szafron type \cite{szek,
szafron}. We describe theses particular solutions of Einstein's equations
briefly in section II and then in greater detail in section III. In section
IV we extend the analysis of \cite{shawbarrow1} to include non-spherically
symmetric backgrounds of Szekeres-Szafron type. In section V, we consider
the validity of the approximations used, and state the conditions under
which they should be expected to hold. In section VI we perform the matching
procedure (as outlined below), and extend the main result of \cite%
{shawbarrow1} to Szekeres-Szafron spacetimes. We consider possible
generalisations of our result in section VII before considering the
implications of the results in the section VIII.
We will employ the method of matched asymptotic expansions \cite{hinch,
Death}. We solve the dilaton conservation equations as an asymptotic series
in a small parameter, $\delta $, about a FRW background and the
Schwarzschild metric which surrounds our star. The deviations from these
metrics are introduced perturbatively. The former solution is called the
\emph{exterior expansion} of $\phi $, and the latter the \emph{interior
expansion} of $\phi $. The exterior expansion is found by assuming that the
length and time scales involved are of the order of some intrinsic exterior
length scale, $L_{E}$. Similarly in the interior expansion we assume all
length and time scales we be of the of $L_{I}$, the interior length scale.
Neither of the two different expansions will be valid in both regions. In
general, we define $\delta :=L_{I}/L_{E}\ll 1$. This means that in general
only a subset of our boundary conditions we will be enforceable for each
expansion, and as a result both the interior and exterior solutions will
feature unknown constants of integration. To remove this ambiguity, and
fully determine both expansions, we used the formal matching procedure. The
idea is to assume that both expansions are valid in some intermediate
region, where length scales go like $L_{int}=L_{I}^{\alpha }L_{E}^{1-\alpha }
$, for some $\alpha \in (0,1)$. Then by the uniqueness property of
asymptotic expansions, both solutions must be equal in that intermediate
region. This allows us to set the value of constants of integration, and
effectively apply \emph{all} the boundary conditions to both expansions. A
fuller discussion of this method, with examples, and its application in
general relativity is given in \cite{shawbarrow1}.
\section{Geometrical Set-Up}
We shall consider a similar geometrical set-up to that of Paper I. We assume
that the dilaton field is only weakly coupled to gravity, and so its energy
density has a negligible effect on the expansion of the background universe.
This allows us to consider the dilaton evolution on a fixed background
spacetime. We will require this background spacetime to have the same
properties as in Paper I, but with the requirement of spherical symmetry
removed:
\begin{itemize}
\item The metric is approximately Schwarzschild, with mass $m$, inside some
closed region of spacetime outside a surface at $r=R_{s}$. The metric for $%
r<R_{s}$ is left unspecified.
\item Asymptotically, the metric must approach FRW and the whole spacetime
should tend to the FRW metric in the limit $m\rightarrow 0$.
\item When the local inhomogeneous energy density of asymptotically FRW
spacetime tends to zero, the spacetime metric exterior to $r=R_{s}$ must
tend to a Schwarzschild metric with mass $m$ .
\end{itemize}
We will also limit ourselves to considering spacetimes in which the
background matter density satisfies a physically realistic equation of
state, specifically that of pressureless dust ($p=0$). We also allow for the
inclusion of a cosmological constant, $\Lambda $. The set of all
non-spherical spacetimes that satisfy these conditions is too large and
complicated for us to examine fully here; and such an analysis is beyond the
scope of this paper. We can simplify our analysis greatly, however, we
specify four further requirements:
\begin{enumerate}
\item The flow-lines of the background matter are geodesic and non-rotating.
This implies that the flow-lines are orthogonal to a family of spacelike
hypersurfaces, $S_{t}$.
\item Each of the surfaces $S_{t}$ is conformally flat.
\item The Ricci tensor for the hypersurfaces $S_{t}$, ${}^{(3)}R_{ab}$, has
two equal eigenvalues.
\item The shear tensor, as defined for the pressureless dust background, has
two equal eigenvalues.
\end{enumerate}
The last three of these conditions seem rather artificial; however, when the
deviations from spherical symmetry are in some sense `small' we might expect
them to hold as a result of the first condition. In the spherically
symmetric case, condition 1 implies conditions 2, 3 and 4. In the absence of
spherical symmetry, these conditions require the background spacetime to be
of Szekeres-Szafron type, containing pressureless matter and (possibly) a
cosmological constant. The conditions (1 - 4) combined with the background
matter being of perfect fluid type provide an invariant definition of the
Szekeres-Szafron class of metrics that is due to Szafron and Collins \cite%
{collins, kras}.
We have demanded that the `local' or interior region be approximately
Schwarzschild. The intrinsic length scale of the interior is defined by the
curvature invariant there:
\begin{equation}
L_{I}\equiv \left( \tfrac{1}{12}R_{abcd}R^{abcd}\right) ^{-1/4}=\frac{%
R_{s}^{3/2}}{\left( 2m\right) ^{1/2}}. \label{invar}
\end{equation}%
The exterior (or cosmological) region is approximately FRW, and so its
intrinsic length scale is proportional to the inverse square root of the
local energy density: $1/\sqrt{\kappa \varepsilon +\Lambda }$, where $%
\varepsilon $ is the matter density. In accord with current astronomical
observations, we assume that this FRW region is approximately flat, and so
we set our exterior length scale appropriate for the present epoch, $t=t_{0},
$ by the inverse Hubble parameter at that time:
\begin{equation*}
L_{E}\equiv 1/H_{0}.
\end{equation*}%
We can now define a small parameter by the ratio of the interior and
exterior length scales:
\begin{equation*}
\delta =L_{I}/L_{E}.
\end{equation*}
\section{Szekeres-Szafron Backgrounds}
In 1975 Szekeres \cite{szek} solved the Einstein equations with perfect
fluid source by assuming a metric of the form:
\begin{equation*}
\mathrm{d}s^{2}=\mathrm{d}t^{2}-e^{2\alpha }\mathrm{d}r^{2}-e^{2\beta
}\left( \mathrm{d}x^{2}+\mathrm{d}y^{2}\right) ,
\end{equation*}%
with $\alpha $ and $\beta $ being functions of $(t,r,x,y)$. The coordinates
where assumed to be comoving so that the fluid-flow vector is of the form: $%
u^{\mu }=\delta _{0}^{\mu }$; This implies $p=p(t)$ and $\ $the acceleration
$\dot{u}^{\mu }=0$. Szekeres assumed a dust source with no cosmological
constant, $p=0$, although his results were later generalised to arbitrary $%
p(t)$ by Szafron \cite{szafron} and the explicit dust plus $\Lambda $
solutions were found by Barrow and Stein-Schabes \cite{JBJSS}.\emph{\ }In
general, these metrics have \emph{no} Killing symmetries \cite{bonn}.
Spherically-symmetric solutions of this type with $\alpha (r,t)$ and $\beta
(r,t)$ were, in fact, first discussed by Lemaоtre \cite{lem} and are usually
referred to as the Tolman-Bondi models \cite{tolbondi}; much of the analysis
of Paper I assumed a Tolman-Bondi background.
The Szekeres-Szafron models can be divided into two classes: $\beta _{,r}=0$
and $\beta _{,r}\neq 0$. Both classes include all FRW models in their
homogeneous and isotropic limit; however, only the latter 'quasi-spherical'
class includes the external Schwarzschild solution. Since we want to have
some part of our spacetime look Schwarzschild we will only consider the $%
\beta _{,r}\neq 0$ quasi-spherical solutions. We will also limit ourselves
to spacetimes with a cosmological constant, \cite{JBJSS}, so in effect the
total pressure is $p=-\Lambda $. These universes contain no gravitational
radiation as can be deduced from the existence of Schwarzschild as a special
case which ensures a smooth matching to Schwarzschild, which contains no
gravitational radiation. With these restrictions, $\alpha $ and $\beta $ are
given by:
\begin{eqnarray}
e^{\beta } &=&\Phi (t,r)e^{\tilde{\nu}(r,x,y)}, \\
e^{\alpha } &=&h(r)e^{-\tilde{\nu}(r,x,y)}\left( e^{\beta }\right) _{,r}, \\
e^{-\tilde{\nu}} &=&\tilde{A}(r)(x^{2}+y^{2})+2\tilde{B}_{1}(r)x+2\tilde{B}%
_{2}(r)y+\tilde{C}(r),
\end{eqnarray}%
where $\Phi (t,r)$ satisfies:
\begin{equation*}
\Phi _{,t}^{2}=-\tilde{k}(r)+2\tilde{M}(r)/\Phi +\frac{1}{3}\Lambda \Phi
^{2}.
\end{equation*}%
The functions $\tilde{A}(r)$, $\tilde{B}_{1}(r)$, $\tilde{B}_{2}(r)$, $%
\tilde{C}(r)$, $\tilde{M}(r)$, $\tilde{k}(r)$ and $\tilde{h}(r)$ are
arbitrary up to the relations:
\begin{equation*}
\tfrac{1}{4}E(r):=\tilde{A}\tilde{C}-\tilde{B}_{1}^{2}-\tilde{B}_{2}^{2}=%
\tfrac{1}{4}\left[ \tilde{h}^{-2}(r)+\tilde{k}(r)\right] .
\end{equation*}%
The surfaces $(t,r)=const$ have constant curvature $E(r)$. We will require
that the inhomogeneous region of our spacetime is localised, so that it is
by some measure finite. This implies that the surfaces of constant curvature
must be closed; we must therefore restrict ourselves to only considering
backgrounds where $E>0$. Whenever this is the case, we always can rescale
the arbitrary functions so that $E\ $can be set equal to $1$ by the
rescalings
\begin{eqnarray}
A(r):= &&\tilde{A}(r)/\sqrt{E(r)},\;B_{1}(r):=\tilde{B}_{1}(r)/\sqrt{E(r)}%
,\;B_{2}(r):=\tilde{B}_{2}(r)/\sqrt{E(r)},\;C(r):=\tilde{C}(r)/\sqrt{E(r)}%
,e^{\nu }:=\sqrt{E}e^{\tilde{\nu}} \notag \\
k:= &&\tilde{k}(r)/E(r),\;h^{-2}:=\tilde{h}(r)^{-2}/E(r)=1-k(r),\;R(t,r):=%
\Phi (r,t)/\sqrt{E},\;M(r)=\tilde{M}(r)/E^{3/2}. \notag
\end{eqnarray}%
These transformations can be viewed as the `gauge-fixing' of arbitrary
functions. In this gauge, $R(t,r)$ is a `physical' radial coordinate, i.e.
the surfaces $(t,r)=const$ have surface area $4\pi R^{2}$ and the metric
becomes
\begin{equation*}
\mathrm{d}s^{2}=\mathrm{d}t^{2}-\frac{\left( 1+\nu _{,R}R\right)
^{2}R_{,r}^{2}\mathrm{d}r^{2}}{1-k(r)}-R^{2}e^{2\nu }\left( \mathrm{d}x^{2}+%
\mathrm{d}y^{2}\right) ,
\end{equation*}%
\noindent where $e^{-\nu }=A(r)(x^{2}+y^{2})+2B_{1}(r)x+2B_{2}(r)y+C(r)$ and
$AC-B_{1}^{2}-B_{2}^{2}=\tfrac{1}{4}$, and $\nu _{,R}:=\nu _{,r}/R_{,r}$
and:
\begin{equation*}
R_{,t}^{2}=-k(r)+2M(r)/R+\frac{1}{3}\Lambda R^{2}.
\end{equation*}%
In this quasi-spherically symmetric subcase of the Szekeres-Szafron
spacetimes the surfaces of constant curvature, $(t,r)=const$, are 2-spheres
\cite{szek2}; however, they are not necessarily concentric. In the limit $%
\nu _{,r}\rightarrow 0$, the $(t,r)=const$ spheres becomes concentric (see
fig. \ref{fig1}). We can make one further coordinate transformation so that
the metric on the surfaces of constant curvature, $(t,r)=const$, is the
canonical metric on $S^{2}$ i.e. $\mathrm{d}\theta ^{2}+\sin ^{2}\theta
\mathrm{d}\phi ^{2}$:
\begin{eqnarray}
x\rightarrow X &=&2\left( A(r)x+B_{1}(r)\right) , \notag \\
y\rightarrow Y &=&2\left( A(r)y+B_{2}(r)\right) , \notag
\end{eqnarray}%
\noindent where $X+iY=e^{i\varphi }\cot \theta /2$. This yields
\begin{equation*}
-\nu _{,r}|_{x.y}=\frac{\lambda _{z}(X^{2}+Y^{2}-1)+2\lambda _{x}X+2\lambda
_{y}Y}{X^{2}+Y^{2}+1}=\lambda _{z}(r)\cos \theta +\lambda _{x}(r)\sin \theta
\cos \varphi +\lambda _{y}(r)\sin \theta \sin \varphi ,
\end{equation*}%
\noindent where we have defined:
\begin{eqnarray}
\lambda _{z}(r):= \frac{A^{\prime }}{A}, \qquad \lambda _{x}(r):= \left( \frac{2B_{1}}{A}\right) ^{\prime }A, \qquad \lambda _{y}(r):= \left( \frac{2B_{2}}{A}\right) ^{\prime }A.
\end{eqnarray}%
With this choice of coordinates, the local energy density of the dust
separates uniquely into a spherical symmetric part, $\varepsilon _{s}$, and
and a non-spherical part, $\varepsilon _{ns}$:
\begin{equation*}
\varepsilon =\varepsilon _{s}(t,R)+\varepsilon _{ns}(t,R,\theta ,\varphi ),
\end{equation*}%
\noindent where:
\begin{eqnarray}
\kappa \varepsilon _{s} &=&\frac{2M_{,R}}{R^{2}}, \\
\kappa \varepsilon _{ns} &=&-\frac{R\nu _{,R}}{1+\nu _{,R}R}\cdot \left(
\frac{2M}{R^{3}}\right) _{,R}.
\end{eqnarray}%
We define $M_{,R}=M_{,r}/R_{,r}$. Following the conventions of our previous
paper we write define
\begin{equation*}
M:=m+Z(r),
\end{equation*}
where $m$ is the gravitational mass of our `star'.
\subsection{Exterior Expansion}
As a result of the way that the inhomogeneity is introduced in these models,
we want the FRW limit to be `natural' , that is for the $O(3)$ orbits to
become concentric in this limit; we therefore require $\nu _{r}\sim o(1)$ as
$\delta \rightarrow 0$ in the exterior. This follows from the requirement
that the whole spacetime should become homogeneous in a smooth fashion in
the limit where the mass of our `star' vanishes: $m\rightarrow 0$. Put
another way, the introduction of our star is the only thing responsible for
making the surfaces of constant curvature non-concentric. We define, as in
the previous paper, dimensionless `radial' and time coordinates appropriate
for the exterior by
\begin{equation*}
\tau =H_{0}t,\text{ \ \ }\rho =H_{0}r.
\end{equation*}
The \emph{exterior limit} is defined by $\delta \rightarrow 0$ with $\tau $
and $\rho $ fixed. In the exterior region we find asymptotic expansions in
this limit. According to our prescription, we write
\begin{equation*}
H_{0}Z(\rho )\sim \frac{1}{2}\Omega _{m}\rho ^{3}+\delta
^{p}z_{1}(r)+o(\delta ^{p}),
\end{equation*}
and
\begin{equation*}
H_{0}^{-1}\lambda _{i}\sim \delta ^{s}l_{i}(\rho )+\mathcal{o}(\delta ^{s}).
\end{equation*}
Since $H_{0}^{-2}\left( \frac{2M}{R^{3}}\right) _{,R}\sim \mathcal{O}(\delta
^{p},\delta )$, we have that: $H_{0}^{-2}\kappa \varepsilon _{ns}\sim \mathcal{O%
}(\delta ^{p+s},\delta ^{1+s})$ whereas $H_{0}^{-2}\kappa \varepsilon _{s}\sim
\mathcal{O}(\delta ^{p},\delta )$. Thus, the non-spherical perturbation to
the energy density is always of subleading order compared to the first order
in spherical perturbation. The first-order, non-spherical, metric
perturbation appears at $\mathcal{O}(\delta ^{s})$; however, since this is
equivalent to a coordinate transform on $(r,\theta ,\varphi )$ and the
dilaton field, $\phi $, is homogeneous to leading order in the exterior,
this perturbation does not make any corrections to the dilaton conservation
equation at $\mathcal{O}(\delta ^{s})$. Thus, both at leading order, and at
next-to-leading order, both the energy density and the dilaton field will
behave in the same way as in the spherically-symmetric Tolman-Bondi case -
with the possible addition of a non-spherically symmetric vacuum
perturbation to the dilaton, $\phi $, i.e. $\phi \sim \phi _{s}+\phi
_{ns}+o(\delta ^{p})$ where $\phi _{s}$ is the spherically symmetric
solution and $\square _{FRW}\phi _{ns}=0$. As in our previous paper,
however, we are not especially interested in the exterior solution for $\phi
$ beyond zeroth order, just the effect of any background variation in $\phi $
on what is measured on the surface of a local 'star'.
\subsection{Interior Expansion}
We define dimensionless coordinates for the interior in the same way as we
did for the spherically symmetric case:
\begin{equation*}
T=L_{I}^{-1}(t-t_{0})\text{ and }\xi =R/R_{s}.
\end{equation*}
We define the \emph{interior limit} to be $\delta \rightarrow 0$ with $T$
and $\xi $ fixed, and perform out interior asymptotic expansions in this
limit. To lowest order in the interior region, we write $Z\sim \delta
^{q}R_{s}\mu _{1}$, and $\lambda _{i}:=\delta _{q^{\prime }}R_{s}^{-1}b_{i}$%
, where $i=\{x,y,z\}$. The condition that $\kappa \varepsilon >0$ everywhere
requires $q^{\prime }\geq q$ and then, to next-to-leading order, the
interior expansion of $\phi $ will be the same as it was in the
spherically-symmetric Tolman-Bondi case. We can potentially include a
non-spherical vacuum component for $\phi $; however, this will be entirely
determined by a boundary condition on $R=R_{s}$ and the need that it should
vanish for large $R$. To find the leading-order behaviour of the $\phi _{,T}$
we need to know $\phi $ at next-to-leading order. The only new case we need
to consider therefore is when $q^{\prime }=q$, i.e. $\kappa \varepsilon
_{ns}\sim \kappa \varepsilon _{s}$. In the spherically symmetric case we
considered two distinct subclasses of the Tolman-Bondi models: the flat, $k=0
$, Gautreau-Tolman-Bondi spacetimes, \cite{gautreau, kras} and the non-flat,
$k\neq 0$, Tolman-Bondi models with a simultaneous initial singularity. In
Gautreau-Tolman-Bondi models the initial singularity is non-simultaneous
from the point of view of geodesic observers. The latter class is the more
realistic, since in the former the world-lines of matter particles stream
out of the surface of our star at $R=R_{s}|_{R=R_{s}}$ i.e. $R_{,t}>0$,
whereas in the simultaneous big-bang models we can demand that matter
particles fall \emph{onto} this surface i.e. $R_{,t}|_{R=R_{s}}<0$. With
this choice, and if $R_{s}=2m$, the non-flat models properly describe the
embedding of a black hole into an expanding universe, whereas the
Gautreau-Tolman-Bondi model technically describes the embedding of a
white-hole in the same universe. In this paper we shall, therefore, only
give the results explicitly for the non-flat case -- however, we can present
a simple procedure to transform our results to the flat Gautreau case.
We define \emph{\ }
\begin{equation*}
\eta =\left( \xi ^{3/2}-3T/2\right) ^{2/3};\text{ }R_{s}\eta =r+\mathcal{O}%
(\delta ^{q},\delta ^{2/3}).
\end{equation*}%
From the exact solutions we find:
\begin{equation*}
k(\eta )=\delta ^{2/3}k_{0}\left( 1+\delta ^{q}\mu _{1}(\eta )+o\left(
\delta ^{q}\right) \right) +\mathcal{O}\left( \delta ^{5/3}\right) ,
\end{equation*}%
where
\begin{equation*}
k_{0}(\delta T)=\frac{2m}{R_{s}}\left( \frac{\pi }{H_{0}t_{0}+\delta T}%
\right) ^{2/3}.
\end{equation*}%
We can remove the $\mathcal{O}(\delta ^{2/3})$ metric perturbation by a
redefinition of the $T$ coordinate, $T\rightarrow T^{\ast }$:
\begin{equation*}
\sqrt{1-\delta ^{2/3}k_{0}}T^{\ast }=T+\int^{\xi }\frac{\sqrt{\frac{2m}{%
R_{s}\xi ^{\prime }}}\left( 1-\sqrt{1-\left( \frac{\delta ^{2/3}\pi \xi
^{\prime }}{H_{0}t_{0}+\delta T}\right) }\right) }{1-\frac{2m}{R_{s}\xi
^{\prime }}}\mathrm{d}\xi ^{\prime }.
\end{equation*}%
To leading order we see that $T\sim T^{\ast }$. The interior expansion of
the metric, for $q^{\prime }=q$, is written:
\begin{equation*}
\mathrm{d}s_{int}^{2}\sim R_{s}^{2}\left( j_{ab}^{(0)}(\xi )+\delta
^{q}j_{ab}^{(1)s}(\xi ,\chi )+\delta ^{q}j_{ab}^{(1)ns}(\xi ,\chi )+o(\delta
^{q})\right) \mathrm{d}x^{a}\mathrm{d}x^{b}+o(\delta ^{q}).
\end{equation*}%
where $j_{ab}^{(0)}$ and $j_{ab}^{(1)s}$ are given by:
\begin{eqnarray}
j_{ab}^{(0)}\mathrm{d}x^{a}\mathrm{d}x^{b} &=&\frac{R_{s}}{2m}\mathrm{d}%
T^{\ast 2}-\left( \mathrm{d}\xi +\xi ^{-1/2}\mathrm{d}T^{\ast }\right)
^{2}-\xi ^{2}\{\mathrm{d}\theta ^{2}+\sin ^{2}\theta \mathrm{d}\varphi
^{2}\}, \label{j1eq.2} \\
j_{ab}^{(1)s}\mathrm{d}x^{a}\mathrm{d}x^{b} &=&-\frac{\mu _{1}(\chi )}{\xi
^{1/2}}\mathrm{d}\xi \mathrm{d}T^{\ast }-\frac{\mu _{1}(\chi )}{\xi }\mathrm{%
d}T^{\ast 2}.
\end{eqnarray}%
These are the same as in the spherically symmetric case. The non-spherically
symmetric perturbation is given by
\begin{eqnarray}
j_{ab}^{(1)ns}\mathrm{d}x^{a}\mathrm{d}y^{b} &=&2(b_{z}\cos \theta
+b_{x}\cos \varphi \sin \theta +b_{y}\sin \varphi \sin \theta )\xi \mathrm{d}%
\eta ^{2} \\
&&-2\xi ^{2}\left[ b_{z}\sin \theta +b_{x}\cos \varphi (1-\cos \theta
)+b_{y}\sin \varphi (1-\cos \theta )\right] \mathrm{d}\theta \mathrm{d}\eta
\notag \\
&&-2\xi ^{2}(1-\cos \theta )\sin \theta (b_{x}\sin \varphi -b_{y}\cos
\varphi )\mathrm{d}\varphi \mathrm{d}\eta . \notag
\end{eqnarray}%
\noindent The spherically symmetric part of the local energy density, $%
\kappa \varepsilon _{s}$ is the same as it was in the Tolman-Bondi cases:
\begin{equation*}
R_{s}^{2}\kappa \varepsilon _{s}=\delta ^{q}\frac{2m}{R_{s}}\frac{\mu _{1,\xi }%
}{\xi ^{3/2}\eta ^{1/2}}.
\end{equation*}%
The non-spherically symmetric part is:
\begin{equation*}
R_{s}^{2}\kappa \varepsilon _{ns}=-\delta ^{q}\frac{6m}{R_{s}}\frac{\left(
b_{z}\cos \theta +b_{x}\sin \theta \cos \phi +b_{y}\sin \theta \sin \phi
\right) }{\xi ^{3/2}\eta ^{1/2}}.
\end{equation*}%
and to ensure that the energy density is everywhere positive we need $\mu
_{,\eta }^{(1)}\geq 3b_{i}$.
\section{Extension to quasi-spherical situations}
\subsection{Boundary Conditions}
We demand the same boundary conditions as before: as the physical radius
tends to infinity, $R\rightarrow \infty $, we demand that the dilaton tends
to its homogeneous cosmological value: $\phi (R,t)\rightarrow \phi _{c}(t)$.
This can be applied to the exterior approximation. In the interior, we
demand that the dilaton-flux passing out from the surface of our `star' at $%
R=R_{s}$ is, at leading order, parametrised by:
\begin{equation}
-R_{s}^{2}\left( 1-\frac{2m}{R_{s}}\right) \left. \partial _{\xi }\phi
_{0}\right\vert _{\xi =R_{s}}=2mF\left( \bar{\phi}_{0}\right)
=\int_{0}^{R_{s}}\mathrm{d}R^{\prime }R^{\prime }{}^{2}B_{,\phi }(\phi _{0}(%
\hat{\xi}^{\prime }))\kappa \varepsilon (R^{\prime }), \label{phiflux}
\end{equation}%
where $\bar{\phi}_{0}=\phi _{0}(R=R_{s})$. The function $F(\phi )$ can be
found by solving the dilaton field equations to leading order in the $R<R_{s}
$ region. If the interior region is a black-hole ($R_{s}=2m$) then we must
have $F(\phi )=0$; otherwise we expect $F(\phi )\sim B_{,\phi }(\phi )$.
Without considering the sub-leading order dilaton evolution inside our
`star', i.e. at $R<R_{s}$, we cannot rigorously specify any boundary
conditions beyond leading order. Despite this, we can guess at a general
boundary condition by perturbing eq. (\ref{phiflux}):
\begin{equation}
-R_{s}^{2}\left( 1-\frac{2m}{R_{s}}\right) \left. \partial _{R}\tilde{\delta}%
(\phi )\right\vert _{\xi =R_{s}}=-\left. \tilde{\delta}\left( \sqrt{-g}%
g^{RR}\right) \partial _{R}\phi _{0}\right\vert _{R=R_{s}}+2\tilde{\delta}%
(M)F\left( \bar{\phi}_{0}\right) +2mF_{,\phi }(\bar{\phi}_{0})\tilde{\delta}%
\left( \bar{\phi}_{0}\right) +\mathrm{smaller}\;\mathrm{terms},
\label{pertbdry}
\end{equation}%
where $\tilde{\delta}(X)$ is the first sub-leading order term in the
interior expansion of $X$; $M$ is the total mass contained inside $\xi <R_{s}
$ and is found by requiring the conservation of energy; and at $t=t_{0}$ we
have $M=m$. Only $\tilde{\delta}\left( \bar{\phi}_{0}\right) $ remains
unknown; however, we shall assume it to be the same order as $\tilde{\delta}%
(\phi )$ and see that this unknown term is usually suppressed by a factor of
$2m/R_{s}$ relative to the other terms in eq. (\ref{pertbdry}).
\subsection{Interior Expansion}
In the spherically symmetric case we found that $\phi \sim \phi
_{I}^{(0)}+\delta ^{q}\phi _{I}^{(1)}+o(\delta ^{q})$. In the non-spherical
case, where $q^{\prime }=q$, we relabel $\phi _{I}^{(1)}\rightarrow \phi
_{I}^{(1)s}$ and we have additional non-spherical modes:
\begin{equation*}
\phi \sim \phi _{I}^{(0)}(\xi ,T)+\delta ^{q}\phi _{I}^{(1)s}(\xi ,T)+\delta
^{q}\phi _{I}^{(1)z}(\xi ,T)\cos \theta +\delta ^{q}\phi _{I}^{(1)x}(\xi
,T)\sin \theta \cos \varphi +\delta ^{q}\phi _{I}^{(1)y}(\xi ,T)\sin \theta
\sin \varphi +o(\delta ^{q})
\end{equation*}%
where:
\begin{eqnarray}
-\frac{2m}{R_{s}}\left( \xi ^{3/2}\phi _{I,TT}^{(1)i}+\frac{3}{2}\phi
_{I,T}^{(1)i}\right) &+&\frac{1}{\eta ^{1/2}}\left( \frac{\xi ^{5/2}}{\eta
^{1/2}}\phi _{I,\eta }^{(1)i}\right) _{,\eta }-\frac{2}{\xi ^{1/2}}\phi
_{I}^{(1)i}=\frac{6m}{R_{s}}B_{,\phi }\left( \phi _{I}^{0}\right) \frac{%
b_{i}\left( \eta \right) }{\eta ^{1/2}} \label{nsphieqn} \\
&+&\left( \frac{2m}{R_{s}}\right) \frac{1}{\eta ^{1/2}}F\left( \bar{\phi}%
_{0}\right) \left[ \left( b_{i}(\eta )\xi \left( \frac{1+\frac{2m}{R_{s}\xi }%
}{1-\frac{2m}{R_{s}\xi }}\right) \right) _{,\eta }-2b_{i}(\eta )\right] .
\notag
\end{eqnarray}%
We can solve this order by order in $2m/R_{s},$ and to lowest order we find:
\begin{eqnarray}
\phi _{I}^{(1)i} &\sim &\frac{2m}{R_{s}}B_{,\phi }\left( \phi
_{I}^{0}\right) \xi \int^{\eta }\mathrm{d}\eta ^{\prime }\frac{b_{i}(\eta
^{\prime })}{\xi ^{^{\prime }2}}-\frac{2m}{R_{s}}B_{,\phi }\left( \phi
_{I}^{0}\right) \frac{1}{\xi ^{2}}\int_{\xi =1}^{\eta }\mathrm{d}\eta
^{\prime }\xi ^{\prime }b_{i}(\eta ^{\prime }) \\
&+&\frac{2m}{R_{s}}F\left( \bar{\phi}_{0}\right) \frac{1}{\xi ^{2}}\int_{\xi
=1}^{\eta }\mathrm{d}\eta ^{\prime }b_{i}(\eta ^{\prime })\xi ^{\prime }+%
\frac{C_{i}}{\xi ^{2}}+D_{i}\xi +\mathcal{O}((2m/R_{s})^{2}) \notag
\end{eqnarray}%
Since we are interested in finding when and where the local time variation
of $\phi $ deviates from its cosmological value, we are chiefly concerned
with the case $q\leq 1$. The matching condition then requires that we fix $%
D_{i}$ so that in the intermediate limit we have $\phi _{I}^{(1)i}\sim \xi
^{n}$ with $n<1$. The value of $C_{i}$ should be set by a boundary condition
on $R=R_{s}$. We cannot specify $C_{i}$ exactly without further information
about the interior of our `star' in $R<R_{s}$. If we assume that the
prescription for the sub-leading order boundary condition given above is
correct then we find:
\begin{eqnarray}
\partial _{\xi }\phi _{I}^{(1)i}|_{\xi =1} &\sim &\frac{2m}{R_{s}}\left.
\frac{b_{i}}{\eta ^{1/2}}\right\vert _{\xi =1}F\left( \bar{\phi}_{0}\right) +%
\mathcal{O}((2m/R_{s})^{2}) \notag \\
\Rightarrow C_{i} &=&\frac{m}{R_{s}}B_{,\phi }\left( \phi _{I}^{0}\right)
\int^{\xi =1}\mathrm{d}\eta ^{\prime }\frac{b_{i}(\eta ^{\prime })}{\xi
^{^{\prime }2}}+\tfrac{1}{2}D \notag
\end{eqnarray}%
From now onwards we set $C_{i}=0,$ for simplicity; even when this is not
correct we do not expect the magnitude of $C_{i}$ or $C_{i,T}$ to be larger
than any of the other terms in $\phi _{I}^{(1)i}$ or $\phi _{I,T}^{(1)i}$,
respectively. The time-derivative of $\phi _{I}^{(1)i}$ for fixed $R$ is:
\begin{equation}
\phi _{I,T}^{(1)i}\sim \frac{4m}{R_{s}}B_{,\phi }\left( \phi _{I}^{0}\right)
\xi \int^{\eta }\mathrm{d}\xi ^{\prime }\frac{b_{i}(\eta ^{\prime })}{\xi
^{^{\prime }5/2}}+\frac{2m}{R_{s}}B_{,\phi }\left( \phi _{I}^{0}\right)
\frac{1}{\xi ^{2}}\int_{\xi =1}^{\eta }\mathrm{d}\xi ^{\prime }\frac{%
b_{i}(\eta ^{\prime })}{\xi ^{1/2}}-\frac{2m}{R_{s}}F\left( \bar{\phi}%
_{0}\right) \frac{1}{\xi ^{2}}\int_{\xi =1}^{\eta }\mathrm{d}\eta ^{\prime }%
\frac{b_{i}(\eta ^{\prime })}{\xi ^{1/2}}+D_{,T}\xi \label{nsphitev}
\end{equation}%
In the next section we shall discuss what we require of the $b_{i}$ for the
matching procedure to be valid. In section VI we will then use the matching
conditions to find $D$ and $D_{,T}$.
We could also relax the requirement that the leading-order mode in $\phi $
be spherically symmetric. At next-to-leading order these new modes would
generate extra terms in $\phi _{I}^{(1)}$. In general, an $l$-pole at
leading order becomes an $l+1$-pole at next-to-leading order. The magnitude
of the extra time-dependence that is picked up is, however, the same each
time. Hence, we restrict ourselves by taking the leading-order mode to be
spherically symmetric for the time being. Note also that we can pass from
the simultaneous big-bang case, to the spatially flat, `Gautreau', case by
setting $k=0$ and making the transform $\eta \rightarrow \chi =\left( \xi
^{3/2}+3T/2\right) ^{2/3}$. This will also mean that $\phi _{I,T}\rightarrow
-\phi _{I,T}$.
\section{Validity of Approximations}
All of the conditions found in Paper I for the matching of the spherically
symmetric parts of $\phi $ to be possible still apply here. However, we must
now satisfy some extra conditions that come from the requirement that the
non-spherical parts should also be matchable.
We assume that $b_{i}\left( \eta \right) \propto \eta ^{d_{i}}$ as $\eta
\rightarrow \infty $ for some $d_{i}>0$. At order $\delta ^{q}$, the growing
mode in the non-spherically symmetric part of the interior approximation
will then grow like $\delta ^{q}\eta ^{d_{i}+1}/\xi $. In the intermediate,
or matching, region we have that $\eta ,\xi \sim \delta ^{-\alpha }$ for
some $\alpha \in (0,1)$. We require $\phi _{I}$ to have a valid asymptotic
expansion this region. This implies that there exists some $\alpha \in (0,1)$
such that, for each $i$, we have $\alpha -q/d_{i}>0$.
In the exterior we shall write $H_{0}^{-1}\lambda _{i}\sim \delta
^{p_{i}^{\prime }}l_{i}(\rho )$, where $p_{i}^{\prime }>0$ comes from the
requirement that the 2-spheres of constant curvature become concentric in
the exterior limit. As $\rho \rightarrow 0$ we assume that $l_{i}(\rho
)\propto \rho ^{-f_{i}}$. We previously stated that $Z\sim \frac{1}{2}\Omega
_{m}\rho ^{3}+\delta ^{p}z_{1}+o(\delta ^{p})$ in the exterior. We assume
that as $\rho \rightarrow 0$, we have $z_{1}\propto \rho ^{-m}$. Although we
did not explicitly consider the exterior expansion of $\phi $ we can now
examine the behaviour of the leading-order non-spherically symmetric mode in
the intermediate limit of that exterior expansion. We noted above that there
will be no $\mathcal{O}(\delta ^{p_{i}^{\prime }})$ correction resulting
from the $l_{i}$. The leading-order mode will therefore either go like $%
\max_{i}\left( \delta ^{p+p_{i}^{\prime }}z_{1}(\rho )l_{i}(\rho )\right) $
if $p<1$ or $\max_{i}\left( \delta ^{1+p_{i}^{\prime }}(\rho )l_{i}(\rho
)\right) $ otherwise, and $\rho \sim \mathcal{O}(\delta ^{1-\alpha })$ in
the intermediate region. For the exterior expansion to be valid in the
intermediate region we therefore require
\begin{eqnarray}
\max_{i}(p_{i}^{\prime } &+&(1-\alpha )(f_{i}+m))>-p\;\mathrm{if}\;p\leq 1,
\notag \\
\max_{i}(p_{i}^{\prime } &+&(1-\alpha )f_{i})>-1\;\mathrm{if}\;p\geq 1.
\notag
\end{eqnarray}
These conditions on $\alpha $ are equivalent to the following: there exists $%
\alpha $ such that the interior expansion of $R^{2}\kappa \varepsilon _{ns}$ is
$o(1)$ as $\delta \rightarrow 0$ for all $0<\alpha ^{\prime }<\alpha $ where
$\xi ,T\sim \mathcal{O}(\delta ^{-\alpha })$, and the exterior expansion of $%
R^{2}\kappa \varepsilon _{ns}$ is also $o(1)$ as $\delta \rightarrow 0$ for all
$0<\alpha ^{\prime \prime }<\alpha $ where $\rho ,\tau -\tau _{0}\sim
\mathcal{O}(\delta ^{1-\alpha })$. This suggests that the condition for the
matching procedure to work, as far as the spherically non-symmetric modes
are concerned, is simply that
\begin{equation*}
R^{2}\kappa \varepsilon _{ns}\ll 1\;\mathrm{everywhere.}
\end{equation*}%
We can also rephrase and generalise the conditions for the matching
procedure to be possible w.r.t. the spherically symmetric modes (as found in
\cite{shawbarrow1}) in a similar fashion: for all $\alpha \in (0,1)$, and
keeping $L_{I}^{\alpha }L_{E}^{1-\alpha }(t-t_{0}),L_{I}^{\alpha
}L_{E}^{1-\alpha }R$ fixed, we have $\lim_{\delta \rightarrow 0}{R^{2}\kappa
\Delta \varepsilon _{s}}=o(1)$ and $\lim_{\delta \rightarrow 0}{2(m+Z)/R}=o(1)$%
. We can combine our two conditions by simply replacing $\Delta \varepsilon _{s}
$ by $\Delta \varepsilon $ in the above expression. Strictly speaking, since $%
\alpha \in (0,1)$ (as opposed to $[0,1)$, $(0,1]$ or $[0,1]$) we can also
replace $\Delta \varepsilon $ by just $\varepsilon $ since $R^{2}\kappa \varepsilon
_{FRW}$ is small everywhere outside the exterior region. For Szekeres
backgrounds the first of these conditions implies the second everywhere
outside the interior region. Therefore, the matching procedure is certainly
possible to zeroth order, if:
\begin{equation*}
\forall \alpha \in (0,1):\;\;\lim_{\delta \rightarrow 0}\left( R^{2}\kappa
\varepsilon (R,t)\right) =o(1)\;\mathrm{and}\;\;\lim_{\delta \rightarrow
0}\left( M(R,t)/R\right) =o(1)\;\;\mathrm{with}\;\;\{L_{I}^{\alpha
}L_{E}^{1-\alpha }(t-t_{0}),L_{I}^{\alpha }L_{E}^{1-\alpha }R\}\;\mathrm{%
fixed},
\end{equation*}%
\noindent where $M(R,t)$ is the gravitational mass inside the surface $%
(t,R)=const$. Equivalently, in \emph{any} intermediate region the background
spacetime is asymptotically Minkowski as $\delta \rightarrow 0$: everywhere
which is not in either the interior or exterior regions can be considered to
be a weak-field perturbation of Minkowski spacetime. The power of our method
is that we do \emph{not} require this to be true of the interior and
exterior regions. So long as this condition holds in the intermediate
region, we can match the zeroth-order approximations in some region and find
the circumstances under which condition (\ref{wettcond}) holds by comparing
the relative sizes of the derivatives $\phi _{c,t}$ and $\phi _{I,t}^{(1)}$.
\section{Matching}
We rewrite the expression for the $\phi _{I}^{(1)i}$ in terms of the
non-spherical part of local density:
\begin{eqnarray}
\delta ^{q}\phi _{I}^{(1)ns} &=&\delta ^{q}\left( \phi _{I,t}^{(1)z}\cos
\theta +\phi _{I,t}^{(1)x}\sin \theta \cos \varphi +\phi _{I,t}^{(1)y}\sin
\theta \sin \varphi \right) \notag \\
&\sim &-\frac{1}{3}B_{,\phi }\left( \phi _{I}^{0}\right) R\int^{r}\mathrm{d}%
r^{\prime }R_{,r}\kappa \varepsilon _{ns}(r^{\prime },t)-\frac{R}{R_{s}}\hat{D}%
(T,\theta ,\phi ) \notag \\
&&-\frac{1}{3}B_{,\phi }\left( \phi _{I}^{0}\right) \frac{1}{R^{2}}%
\int_{R=R_{s}}^{r}\mathrm{d}r^{\prime }R_{,r}R^{3}\kappa \varepsilon
_{ns}(r^{\prime },t)-\frac{1}{3}F\left( \bar{\phi}_{0}\right) \frac{1}{R^{2}}%
\int_{R=R_{s}}^{r}\mathrm{d}r^{\prime }R_{,r}R^{3}\kappa \varepsilon
_{ns}(r^{\prime },t) \notag
\end{eqnarray}%
\noindent where $\hat{D}(T,\theta ,\phi ):=D_{z}\cos \theta +D_{x}\sin
\theta \cos \varphi +D_{y}\sin \theta \sin \varphi $. By examining the
dilaton equations of motion in the FRW region, we can see there is a
component of the leading-order $(\theta ,\varphi )$-dependent term in the
exterior expansion or $\phi $ behaves like
\begin{equation*}
-\frac{1}{3}B_{,\phi }(\phi _{c})R\int_{\infty }^{r}\mathrm{d}r^{\prime
}R_{,r}\kappa \varepsilon _{ns}(r^{\prime },t)
\end{equation*}%
\noindent for $R\ll H_{0}^{-1}$ and $t$ fixed. Therefore matching requires
that we choose $\hat{D}$ such that
\begin{eqnarray}
\delta ^{q}\phi _{I}^{(1)ns} &=&\delta ^{q}\left( \phi _{I,t}^{(1)z}\cos
\theta +\phi _{I,t}^{(1)x}\sin \theta \cos \varphi +\phi _{I,t}^{(1)y}\sin
\theta \sin \varphi \right) \\
&\sim &-\frac{1}{3}B_{,\phi }\left( \phi _{I}^{0}\right) R\int_{\infty }^{r}%
\mathrm{d}r^{\prime }R_{,r}\kappa \varepsilon _{ns}(r^{\prime },t)+\frac{1}{3}%
B_{,\phi }\left( \phi _{I}^{0}\right) \frac{1}{R^{2}}\int_{R=R_{s}}^{r}%
\mathrm{d}r^{\prime }R_{,r}R^{3}\kappa \varepsilon _{ns}(r^{\prime },t) \\
&&-\frac{1}{3}F\left( \bar{\phi}_{0}\right) \frac{1}{R^{2}}\int_{R=R_{s}}^{r}%
\mathrm{d}r^{\prime }R_{,r}R^{3}\kappa \varepsilon _{ns}(r^{\prime },t). \notag
\end{eqnarray}%
The interior expansion is now fully specified to order $\mathcal{O}(\delta
^{p})$. We are interested in the behaviour of $\phi _{I,t}$ and we find
\begin{eqnarray}
\delta ^{q}\phi _{I,t}^{(1)ns} &\sim &\frac{2}{3}B_{,\phi }\left( \phi
_{I}^{0}\right) R\int_{\infty }^{r}\mathrm{d}r^{\prime }R_{,r}R_{,t}\frac{%
\kappa \varepsilon _{ns}(r^{\prime },t)}{R}+\frac{1}{3}B_{,\phi }\left( \phi
_{I}^{0}\right) \frac{1}{R^{2}}\int_{R=R_{s}}^{r}\mathrm{d}r^{\prime
}R_{,r}R_{,t}R^{2}\kappa \varepsilon _{ns}(r^{\prime },t) \\
&&-\frac{1}{3}F\left( \bar{\phi}_{0}\right) \frac{1}{R^{2}}\int_{R=R_{s}}^{r}%
\mathrm{d}r^{\prime }R_{,r}R_{,t}R^{3}\kappa \varepsilon _{ns}(r^{\prime },t)+%
\frac{1}{3}F\left( \bar{\phi}_{0}\right) RR_{,t}\kappa \varepsilon _{ns}(r,t).
\notag
\end{eqnarray}%
This expression is valid whenever $R_{s}\gg 2m$, and the requirements for
matching are satisfied. In these cases we expect $F\left( \bar{\phi}%
_{0}\right) \approx B_{,\phi }\left( \phi _{I}^{0}\right) +\mathcal{O}%
(2m/R_{s})$; so, approximately, we have
\begin{equation*}
\delta ^{q}\phi _{I,t}^{(1)ns}\sim \frac{2}{3}B_{,\phi }\left( \phi
_{c}\right) R\int_{\infty }^{r}\mathrm{d}r^{\prime }R_{,r}R_{,t}\frac{\kappa
\varepsilon _{ns}(r^{\prime },t)}{R}+\frac{1}{3}B_{,\phi }\left( \phi
_{c}\right) RR_{,t}\kappa \varepsilon _{ns}(r,t).
\end{equation*}%
In the case, where $R_{s}=2m,$ and our `star' is actually a black-hole, we
require $F\left( \bar{\phi}_{0}\right) $ to ensure that the $\phi $ is
well-defined as $R\rightarrow 2m$. Even so, in this case, equation (\ref%
{nsphitev}) will not be strictly valid, since it was derived under the
assumption of $R_{s}\gg 2m$. By inspection of the dilaton evolution equation
in the interior, eq. (\ref{nsphieqn}), however, we expect that $\delta
^{q}\phi _{I,t}^{(1)ns}$ near the black-hole horizon to be of similar
magnitude to the RHS of eq. (\ref{nsphitev}).
Combining the results of this paper with those for the spherically symmetric
case we find:
\begin{equation*}
\phi _{I,t}-\phi _{c,t}\sim B_{,\phi }\left( \phi _{c}\right) \int_{\infty
}^{r}\mathrm{d}r^{\prime }R_{,r}R_{,t}\kappa \Delta \varepsilon _{s}(r^{\prime
},t)+\frac{2}{3}B_{,\phi }\left( \phi _{c}\right) R\int_{\infty }^{r}\mathrm{%
d}r^{\prime }R_{,r}R_{,t}\frac{\kappa \varepsilon _{ns}(r^{\prime },t)}{R}+%
\frac{1}{3}B_{,\phi }\left( \phi _{c}\right) RR_{,t}\kappa \varepsilon
_{ns}(r,t).
\end{equation*}%
We require that $|(\phi _{I,t}-\phi _{c,t})/\phi _{c,t}|\ll 1$ for \ref%
{wettcond} to hold and so ensure that local observations will detect
variations of $\phi $ occurring on cosmological scales.
\section{Generalisation: a conjecture}
So far, we have found an analytic approximation to the values of $\phi $ and
$\phi _{c,t}$ in the interior. More succinctly (although less explicitly) we
can say that, to leading order in $\delta $, the values of $\phi $, $\phi
_{,t}$ and $\phi _{,r}$ can all be found everywhere outside the exterior
region from the approximation:
\begin{equation}
\phi \approx \phi _{hom}(t)+\phi _{l}(\vec{x},t), \label{phicon}
\end{equation}%
where $\phi _{l}$ is the solution to:
\begin{equation*}
\square _{sch}\phi _{l}=B_{,\phi }\kappa \Delta \varepsilon
\end{equation*}%
with $\Delta \varepsilon =\varepsilon (\vec{x},t)-\varepsilon _{c}(t)$, $\square
_{sch}$ is the wave operator in a Schwarzschild background, and $t$ is the
proper time of a comoving observer. This is solved w.r.t. the boundary
conditions $\phi _{l}\rightarrow 0$ as $R\rightarrow \infty $ (where $R=0$
is the centre of our `star') and the flux out of the `star' is as given by
equations (\ref{phiflux}) and (\ref{pertbdry}). The homogeneous term is
\begin{equation*}
\phi _{hom}(t)=\phi _{c}(t+\Delta t(\vec{x},x))
\end{equation*}%
where the \emph{lag}, $\Delta t(\vec{x},t)$, is defined by:
\begin{equation*}
\vec{\nabla}^{2}\Delta t-\vec{v}^{\ast }\cdot \vec{\nabla}(\vec{v}^{\ast
}\cdot \vec{\nabla}\Delta t)-\vec{v}^{\ast }\cdot \vec{\nabla}\Delta t(\vec{%
\nabla}\cdot \vec{v}^{\ast })=-\vec{\nabla}\cdot \Delta \vec{v},
\end{equation*}%
with $\nabla _{i}=\partial _{i}$, $i=\{1,2,3\}$, and $\Delta v=\vec{v}-H\vec{%
x}$, where $\vec{v}$ is the velocity of the dust particles relative to $%
R=\Vert \vec{x}\Vert =0$. The velocity $\vec{v}^{\ast }$ has the following
properties: $\vec{v}^{\ast }=\vec{v}$ in some region that includes all the
interior and excludes all of the exterior; $\vec{v}^{\ast }=\Delta \vec{v}$
everywhere else. In a general sense, the interior and exterior are two
disjoint regions of total spacetimes where general-relativistic effects are
non-negligible at leading order (e.g. such as when $\Vert \vec{v}\Vert
\approx 1$). The interior region should be closed, and in the exterior
region $\Vert \Delta \vec{v}\Vert $ is small. So, $\vec{v}^{\ast }$ should
be defined in such a way that it respects all the symmetries of the
spacetime and so that $\Vert \vec{v}^{ast}\Vert \ll 1$ everywhere outside
the interior region. This is required to ensure that $\Delta t$, as defined
above, is finite. It can be seen to come out of the matching procedure. When
the background spacetime satisfies the conditions given below, the precise
way in which $\vec{v}^{\ast }$ is defined does not effect the leading order
behavior of $\Delta t$. For boundary conditions, we must require the flux
out of $\Delta t$ out of the `star' to vanish, and require $\Delta
t\rightarrow 0$ as $\Delta v\rightarrow 0$, i.e. as $R\rightarrow \infty $.
This is the natural generalisation of what has been seen in the
Szekeres-Szafron backgrounds $\vec{v}^{i}=R_{,t}(R,t)\delta _{R}^{i}$. In
these cases the equation is just an ordinary differential equation in $R$
with solution:
\begin{equation*}
\Delta t=\int_{R}^{A}\mathrm{d}R^{\prime }\frac{(R_{,t}(R^{\prime
},t)-HR^{\prime }+R_{,t}(R_{s},t)+HR_{s})}{1-R_{,t}^{2}(R^{\prime },t)}%
+\int_{A}^{\infty }\mathrm{d}R^{\prime }\frac{(R_{,t}(R^{\prime
},t)-HR^{\prime }+R_{,t}(R_{s},t)+HR_{s})}{1-(R_{,t}(R^{\prime },t)-HR)^{2}},
\end{equation*}%
where $A$ is some arbitrary value of $A$ in the intermediate region, and
each $A$ represents a particular choice of definition for $\vec{v}^{\ast }$.
This expression is only valid to leading order in the interior and
intermediate regions. To this order all choices for $A$ are equivalent. Near
$R=R_{s}$, to leading order in $\delta =L_{I}/L_{E}$, this ensures that $%
\mathrm{d}(t+\Delta t)\sim \mathrm{d}v$, where $v=t_{sch}+R+2m\ln (R/2m-1)$
is the advanced time coordinate and $t_{sch}$ is the standard,
curvature-defined, Schwarzschild time-coordinate. The solution for $\phi
_{hom}$ is then, to leading order in $\delta $, just the particular one
given by Jacobson in \cite{jacobson}. We have assumed that the
generalisations of the Szekeres-Szafron result for $\phi $ hold. We have
only proved that this assumption holds for the subset of Szekeres-Szafron
spacetimes for which the matching procedure works. Nonetheless, based on
this analysis, we conjecture that \ref{phicon} provides a good numerical
approximation to the value of $\phi $, and by differentiating once, to $\phi
_{,t}$ and $\partial _{i}\phi $, $i=\{1,2,3\}$, near the surface of our
`star', for any dust plus $\Lambda $ spacetime that can be everywhere
considered to be a weak-field perturbation of either Schwarzschild,
Minkowski, or FRW spacetime; that is,
\begin{equation*}
R^{2}\kappa \Delta \varepsilon (R,t)\ll 1,\qquad 2(M(R,t)-m)/R\ll 1
\end{equation*}%
where $M(R,t)$ is the gravitational mass contained inside the surface $%
(R,t)=const$. One could seek to motivate our conjecture as some sort of
analytical continuation from the Szekeres-Szafron spacetimes to more general
backgrounds, but such arguments would, we believe, be hard to frame in any
rigorous context and are beyond the scope of the analysis in this paper.
\section{Discussion}
In this paper we have extended the analysis of \cite{shawbarrow1} to include
a class of dust-filled spacetimes without any symmetries provided by the
Szekeres-Szafron metrics. Again, we have used the method of matched
asymptotic expansions to link the evolution of the dilaton field, $\phi $,
in an approximately Schwarzschild region of spacetime to its evolution in
the cosmological background. By these methods, we have provided a rigorous
construction of what has been simply assumed about the matching procedures
in earlier studies \cite{early}. We have also analysed, more fully, the
conditions that we need the background spacetime to satisfy for the matching
procedure to be valid, and we have interpreted these conditions in terms of
their requirements on the local energy density. Finally, we have conjectured
a generalisation of our result to more general spacetime backgrounds than
those considered here.
By combining the results found here with those of the previous paper, we
conclude that, in the class of quasi-spherical Szekeres spacetimes in which
the matching procedure is valid, the local time variation of the dilaton
field will track its cosmological value whenever:
\begin{equation}
\left\vert \frac{B_{,\phi }\left( \phi _{c}\right) \int_{\infty }^{r}\mathrm{%
d}r^{\prime }R_{,r}\kappa \Delta (R_{,t}\varepsilon _{s}(r^{\prime },t))+\frac{2%
}{3}B_{,\phi }\left( \phi _{c}\right) R\int_{\infty }^{r}\mathrm{d}r^{\prime
}R_{,r}R_{,t}\frac{\kappa \varepsilon _{ns}(r^{\prime },t)}{R}+\frac{1}{3}%
B_{,\phi }\left( \phi _{c}\right) RR_{,t}\kappa \varepsilon _{ns}(r,t)}{\dot{%
\phi}_{c}}\right\vert \ll 1. \label{condition1}
\end{equation}%
When the cosmological evolution of $\phi $ is dominated by its matter
coupling: $\dot{\phi}_{c}\sim \mathcal{O}(B_{,\phi }H_{0}^{-1}\kappa
\varepsilon _{c}),$ this condition is equivalent to:
\begin{equation*}
\left\vert H_{0}\int_{\infty }^{r}\mathrm{d}r^{\prime }R_{,r}\frac{\Delta
(R_{,t}\varepsilon _{s}(r^{\prime },t))}{\varepsilon _{c}(t)}+\frac{2}{3}%
H_{0}R\int_{\infty }^{r}\mathrm{d}r^{\prime }R_{,r}R_{,t}R^{-1}\frac{%
\varepsilon _{ns}(r^{\prime },t)}{\varepsilon _{c}(t)}+\frac{1}{3}H_{0}R_{,t}\frac{%
\varepsilon _{ns}(r,t)}{\varepsilon _{c}(t)}\right\vert \ll 1.
\end{equation*}
In the other extreme, when the potential term dominates the cosmic dilaton
evolution, the left-hand side of the above condition is further suppressed
by a factor of $B_{,\phi }(\phi _{c})/V_{,\phi }(\phi _{c})\ll 1$. As in our
previous paper, \cite{shawbarrow1}, we can see that for a given evolution of
the background matter density, condition (\ref{wettcond}) is more likely to
hold (or will hold more strongly) when $\left\vert B_{,\phi }(\phi
_{c})\kappa \varepsilon _{c}/V_{,\phi }(\phi _{c})\right\vert \ll 1$. We
reiterate our previous statement that: \emph{domination by the potential
term in the cosmic evolution of the dilaton has a homogenising effect on the
time variation of} $\phi $.
The non-spherically symmetric parts of energy density enter into the
expression differently. The magnitude of the terms on the left-hand side of
eq. (\ref{condition1}) is, as in the spherically symmetric case, still $%
\left\langle H_{0}R\Delta R_{,t}\varepsilon /\varepsilon \right\rangle (R,t)$
where $\left\langle \cdot \right\rangle (R,t)$ represents some `average'
over the region outside the surface $(R,t)=const$. We should note that,
given the condition on $\kappa \varepsilon $ that has been required for
matching, the leading-order contribution to $\kappa \varepsilon _{ns}$ is
everywhere of dipole form and this is responsible for the special form of
the average over the non-spherically symmetric terms. We can also see that,
as a result of form of eqn. (\ref{condition1}), peaks in $\kappa \varepsilon
_{ns}$ that occur outside of the interior region will, in the interior,
produce a weaker contribution to the left-hand side of eqn. (\ref{condition1}%
) than a peak of similar amplitude in a spherically symmetric energy density
$\kappa \varepsilon _{s}$. This behaviour would continue if we were also to
account for higher multipole terms in $\kappa \varepsilon _{ns}$. The higher
the multipole, the more `massive' the mode, and the faster it dissipates.
If we are interested in finding a sufficient condition (as opposed to a
necessary and sufficient one) for (\ref{wettcond}) to hold locally, then in
most circumstances we will be justified in averaging over the
non-spherically symmetric modes in the same way as we average over the
spherically symmetric ones. In most cases, this will over-estimate rather
than under-estimate the magnitude of the left-hand side of our condition, (%
\ref{condition1}). This reasoning leads us to the statement that for $\dot{%
\phi}(\mathbf{x},t)\approx \dot{\phi}_{c}(t)$ to hold locally it is
sufficient that:
\begin{equation}
\mathcal{I}:=\int_{\gamma (R)}\mathrm{d}lH_{0}R^{\prime }\frac{\Delta
(v\varepsilon )}{\varepsilon _{c}}\ll 1 \label{suff}
\end{equation}%
\noindent where $\mathrm{d}l:=\mathrm{d}rR_{,r}$, $v=R_{,t}$ is the velocity
of the dust particles, $\lim_{R\rightarrow \infty }v=H_{0}R$. We make the
same generalisation that we did in Paper I by taking $\gamma (R)$ to run
from $R$ to spatial infinity along a past, radially-directed light-ray. In
this way, we incorporate the limitations imposed by causality. We should
also assume that the above expression includes some sort of average over
angular directions; to be safe we could replace $\varepsilon $ by its maximum
value for fixed $R$ and $t$. This sufficient condition, (\ref{suff}), is
precisely the generalised condition proposed in our first paper on this
issue. The inclusion of deviations from spherical symmetry, therefore, has
little effect of the qualitative nature of the conclusions that were found
in \cite{shawbarrow1}. If anything, we have seen that the non-spherical
modes dissipate faster and, as a result, will produce smaller than otherwise
expected deviations in the local time derivative of $\phi $ from the
cosmological ones.
On Earth we should expect, as before, that the leading-order deviation of $%
\dot{\phi}$ from $\dot{\phi}_{c}$ is produced by the galaxy cluster in which
we sit, and that for a dilaton evolution that is dominated by its coupling
to matter, this effect gives $\mathcal{I}\approx 6\times 10^{-3}\Omega
_{m}^{-1}h\ll 1$, where $\Omega _{m}\approx 0.27$ and $h\approx 0.71$. If
the cosmic dilaton evolution is potential dominated then $\mathcal{I}$ is
even smaller. We conclude, as before, that irrespective of the value of the
dilaton-to-matter coupling, and what dominates the cosmic dilaton evolution,
that
\begin{equation*}
\dot{\phi}(\mathbf{x},t)\approx \dot{\phi}_{c}(t)
\end{equation*}%
will hold in the solar system in general, and on Earth in particular, to a
precision determined by our calculable constant $\mathcal{I}$. We also
conclude, as before, that whenever $\mathcal{I}\ll 1$ near the horizon of a
black hole, there will be no significant gravitational memory effect for
physically reasonable values of the parameters \cite{memory, jacobson}.
Our result relies on one major assumption: the physically realistic
condition that the scalar field should be weakly coupled to matter and
gravity -- in effect, the variations of 'constants' on large scales must
occur more slowly than the universe is expanding and so their dynamics have
a negligible back-reaction on the cosmological background metric. In this
paper we have removed the previous condition of spherical symmetry at least
in as far as the spacetime background is well described by Szekeres-Szafron
solution. We have therefore extended the domain of applicability our general
proof: that \emph{terrestrial} and \emph{solar system} based observations
can legitimately be used to constrain the \emph{cosmological} time variation
of supposed `constants' of Nature and other light scalar fields.
\begin{acknowledgments}
\bigskip We thank Tim Clifton and Peter D'Eath for discussions. D. Shaw is
supported by a PPARC studentship.
\end{acknowledgments}
|
Title:
Resolving the compact dusty discs around binary post-AGB stars using N-band interferometry |
Abstract: We present the first mid-IR long baseline interferometric observations of the
circumstellar matter around binary post-AGB stars. Two objects, SX Cen and HD
52961, were observed using the VLTI/MIDI instrument during Science
Demonstration Time. Both objects are known binaries for which a stable
circumbinary disc is proposed to explain the SED characteristics. This is
corroborated by our N-band spectrum showing a crystallinity fraction of more
than 50 % for both objects, pointing to a stable environment where dust
processing can occur. Surprisingly, the dust surrounding SX Cen is not resolved
in the interferometric observations providing an upper limit of 11 mas (or 18
AU at the distance of this object) on the diameter of the dust emission. This
confirms the very compact nature of its circumstellar environment. The dust
emission around HD 52961 originates from a very small but resolved region,
estimated to be ~ 35 mas at 8 micron and ~ 55 mas at 13 micron. These results
confirm the disc interpretation of the SED of both stars. In HD 52961, the dust
is not homogeneous in its chemical composition: the crystallinity is clearly
concentrated in the hotter inner region. Whether this is a result of the
formation process of the disc, or due to annealing during the long storage time
in the disc is not clear.
| https://export.arxiv.org/pdf/astro-ph/0601169 |
\title{
Resolving the compact dusty discs around binary post-AGB stars using
N-band interferometry \thanks{Based on observations made with the Very
Large Telescope Interferometer of the European Southern Observatory
(program id 073.A-9002(A)), the 1.2\,m Flemish Mercator telescope at
Roque de los Muchachos, Spain and the 1.2\,m Swiss Euler telescope at
La Silla, Chile } }
\author{
P. Deroo\inst{1} \and H. Van Winckel\inst{1} \and M. Min\inst{2} \and
L.B.F.M. Waters\inst{1,2} \and T. Verhoelst\inst{1} \and
W. Jaffe\inst{3} \and S. Morel\inst{4} \and F. Paresce\inst{4} \and
A. Richichi\inst{4} \and P. Stee\inst{5} \and M. Wittkowski\inst{4} }
\institute{Instituut voor Sterrenkunde, K.U. Leuven, Celestijnenlaan
200B, B-3001 Leuven, Belgium \and Astronomical Institute ``Anton
Pannekoek'', University of Amsterdam, Kruislaan 403, 1098 SJ
Amsterdam, the Netherlands \and Leiden Observatory, P.B. 9513, Leiden
2300 RA, the Netherlands \and European Southern Observatory,
Karl-Scharzschild-Strasse 2, 85748 Garching, Germany \and Observatoire
de la C\^ote d' Azur, CNRS-UMR 6203, Avenue Copernic, 06130 Grasse,
France}
\date{Received <date> / Accepted <date>}
\offprints{
P. Deroo \\
\email{Pieter.Deroo@ster.kuleuven.be}
}
\abstract{
We present the first mid-IR long baseline interferometric observations
of the circumstellar matter around binary post-AGB stars. Two objects,
\object{SX Cen} and \object{HD 52961}, were observed using the
VLTI/MIDI instrument during Science Demonstration Time. Both objects
are known binaries for which a stable circumbinary disc is proposed to
explain the SED characteristics. This is corroborated by our N-band
spectrum showing a crystallinity fraction of more than 50\,\% for both
objects, pointing to a stable environment where dust processing can
occur. Surprisingly, the dust surrounding SX\,Cen is not resolved in
the interferometric observations providing an upper limit of 11 mas
(or 18 AU at the distance of this object) on the diameter of the dust
emission. This confirms the very compact nature of its circumstellar
environment. The dust emission around \object{HD\,52961} originates from a very
small but resolved region, estimated to be $\sim$35 mas at 8\,$\mu$m
and $\sim$55 mas at 13\,$\mu$m. These results confirm the disc
interpretation of the SED of both stars. In \object{HD\,52961}, the dust is not
homogeneous in its chemical composition: the crystallinity is clearly
concentrated in the hotter inner region. Whether this is a result of
the formation process of the disc, or due to annealing during the long
storage time in the disc is not clear.
\keywords{Stars: circumstellar matter -- Stars: AGB and post-AGB
-- Stars: individual: \object{HD\,52961} and SX\,Cen -- Techniques:
interferometric -- Infrared: stars
}
}
\titlerunning{
Discs around evolved objects: \object{HD\,52961} and SX\,Cen
}
\authorrunning{P. Deroo et al.}
\section{Introduction}
The fast stellar evolution connecting the Asymptotic Giant Branch
(AGB) to the Planetary Nebulae (PNe) phase is still poorly understood
\citep[e.g.][]{Vanwinckel_2003}. Many detailed studies of individual
transition objects (post-AGB stars) exist, but it is not clear how
these objects are related by evolutionary channels. Moreover, there
is general agreement that binary interactions must play a significant
role in many well studied sources. Binarity is for instance invoked in the
physical models to understand the observational characteristics of
some spectacular geometries observed in PNe. More
recently, also the geometries and kinematical structures around
resolved post-AGB stars might be linked to binarity \citep[][ and
references therein]{Balick_2002}. Since many uncertainties remain in
our understanding of the final evolution of single stars, it is no
surprise that this is even more the case when the
star is a member of a binary system.
Direct detection of the binary nature of central stars of resolved
nebulae is often difficult due to the high obscuration. Moreover, in
crossing the HR-diagram, the stars must pass the pop\,II Cepheid
instability strip, in which pulsational instabilities occur. This
makes radial velocity variations not a straightforward signature of
orbital variations.
In the sample of optically bright post-AGB stars, binaries are being
detected, however, and for an overview we refer to \citet[][ and
references therein]{Vanwinckel_2003}. One of the important
observational characteristics of those binaries is the shape of their
SED. They show a dust-excess starting near sublimation temperature,
irrespective of the effective temperature of the central object and
this despite the lack of a current dusty mass loss \citep[][ and
references therein]{Deruyter_2005, Deruyter_2006}. Moreover, when
available, the long wavelength fluxes show a black-body slope which
indicates the presence of a component of large mm-sized grains. It is
argued in \cite{Deruyter_2006} that in all the investigated objects,
gravitationally bound dust is present, likely in a Keplerian disc.
Note that only for the most famous example, the \object{Red Rectangle}, this
dust emission is resolved and it shows a clear disc structure, both in
the near-IR and in the visible \citep{Menshchikov_2002,Cohen_2004}.
Moreover, the gaseous component was spatially resolved using
mm-interferometry \citep{Bujarrabal_2003,Bujarrabal_2005}. The latter
observations clearly demonstrated the Keplerian rotation of the
disc. In all other cases, the presence of a disc is postulated. More
detailed studies of individual cases can be found: examples are
\object{89\,Her} \citep{Waters_1993}, \object{HR\,4049}
\citep{Waelkens_1991a, Dominik_2003} and \object{IRAS08544-4431}
\citep{Maas_2003}. Given the orbits detected so far, one of the
conclusions is that it is clear that most binaries cannot have evolved
along single star evolutionary tracks.
The high spatial resolution of the mid-IR instrument MIDI mounted on
the VLTI interferometer of ESO makes this an ideal instrument to probe
the circumstellar material around these binaries for two reasons: (i)
the discs are likely compact so high spatial resolution measurements
are needed to resolve the discs and (ii) the discs are shown to emit a
significant part of their total luminosity in the N-band. We therefore
carefully selected 2 binary post-AGB stars for which there is
significant indirect evidence for the presence of a stable
circumstellar dust reservoir. The data presented in this contribution
are taken during Science Demonstration Time to illustrate the
potential of MIDI coupled to the VLTI to study the compact
circumstellar environment suspected in those evolved stars.
In Sect.~\ref{sect:global_characteristics}, we introduce both objects
and refine the orbital parameters published in our previous
papers. The observational log is presented in
Sect.~\ref{sect:observations} while the reduction of the
interferometric dispersed fringes is reported in
Sect.~\ref{sect:reduction}. We discuss our findings in
Sect.~\ref{sect:discussion} and come to our conclusions in
Sect.~\ref{sect:conclusions}.
\section{\object{HD\,52961} and \object{SX\,Cen}: global
characteristics}\label{sect:global_characteristics}
\object{SX\,Cen} and \object{HD\,52961}, having both a spectral type F-G
\citep{Kholopov_1999,Shenton_1994,Waelkens_1991}, are located in the
pop\,II instability strip with \object{SX\,Cen} known as a very regular
RV\,Tauri star with a period of 32.9 days, while \object{HD\,52961} shows a
photometric periodicity of 72 days. They are members of the chemically
anomalous post-AGB stars for which the photospheric abundances of the
different elements are closely linked to their condensation
temperature \citep{Waelkens_1991, Vanwinckel_1992, Vanwinckel_1995,
Maas_2002}. Members of this class show higher photospheric abundances
for chemical elements with a lower condensation temperature. In fact,
\object{HD\,52961} is one of the most extreme examples of this class of
objects. It is a highly metal-poor object \citep[\hbox{[Fe/H] = -4.8,
}][]{Waelkens_1991} which has more zinc than iron in absolute (!)
number \citep[\hbox{[Zn/Fe] = +3.1, }][]{Vanwinckel_1992}. There is
general agreement that this abundance pattern is caused by a chemical
fractionation process caused by dust formation in the circumstellar
environment. After decoupling of the gas and dust, reaccretion of the
gas causes the observed abundance pattern. \cite{Waters_1992} proposed
a scenario in which the circumstellar dust is trapped in a stable
disc. The occurrence of such a disc likely implies binarity for post-AGB
stars. Indeed, radial velocity measurements proved that all the
extremely depleted objects are binaries \citep{Vanwinckel_1995}. In
the following we refine our previously published Spectral Energy
Distribution (SED) as well as the orbital elements of both objects.
\subsection{Spectral energy distribution}\label{sect:SED}
The SED of \object{SX\,Cen} is discussed in the literature
\citep{Goldsmith_1987,Shenton_1994,Maas_2002} where a total reddening
of \hbox{E(B-V) = 0.3\,$\pm$\,0.1} is found. They find a broad
infrared excess starting already at K which, in combination with the
confirmed binarity, is interpreted in \cite{Maas_2002} as a signature
of a dusty disc, not an outflow. The SED is reproduced in the top
panel of Fig. \ref{fig:sedBOTH} in which the ISO/SWS spectrum is
overplotted. This spectrum shows a broad silicate emission feature at
10 $\mu$m, inside the MIDI wavelength range. Due to the low flux
levels at longer wavelengths, we cannot be conclusive about the origin
of the feature around 18 $\mu$m. This is possibly an artifact,
although it is present in both scans. The distance of \object{SX\,Cen} can be
estimated comparing the intrinsic luminosity with the integrated flux
of the scaled Kurucz model. The luminosity of \object{SX\,Cen} is estimated
using the period-luminosity relation derived for the LMC RV\,Tauri
variables \citep{Alcock_1998} to be about
\hbox{600\,$\pm$\,400\,L$_{\sun}$}. The large uncertainty is a relic
of the scatter in the P-L relation. This provides a rough distance
estimate for \object{SX\,Cen} of \hbox{1.6\,$\pm$\,0.5\,kpc}.
For \object{HD\,52961}, we constructed a SED using IUE data (0.115 $\mu$m --
0.320 $\mu$m), Geneva optical photometry, near-IR JHKLM photometry
\citep{Bogaert_1994} and far-IR IRAS photometry. In addition, one SCUBA
\citep{Holland_1999} observation was made to obtain a continuum
measurement at 850 $\mu$m, providing F$_{850\mu\rm{m}} = 2.8 \pm
1.9$\,mJy with Mars as flux calibrator.
As the object suffers photometric variations over time, we constructed
the SED for photometric maximum only. The colour excess due to
interstellar and circumstellar extinction was estimated by searching
the best correspondence between the appropriate Kurucz model and the
dereddened SED in the optical and UV. The SED was dereddened using the
average interstellar extinction law of \cite{Savage_1979} and the
Kurucz model was chosen according to the stellar parameters given in
\cite{Waelkens_1991}, i.e. T$_{\rm{eff}} = 6000$\,K, log(g)$=0.5$ and
[Fe/H]$=-4.5$. The result is shown in Fig.~\ref{fig:sedBOTH}, where
the Geneva photometry at photometric minimum is overplotted using grey
crosses. While the photometry and the Kurucz model are consistent in
the UV and optical, a clear IR excess due to dust is observed at
longer wavelengths. This excess distribution, in combination with the
confirmed binarity \citep[][ and refinements in
Sect. \ref{sect:orbital_elements}]{Vanwinckel_1995} and the lack of a
current dusty mass loss, is also interpreted as evidence for a dusty
disc instead of an outflow. The luminosity of \object{HD\,52961} is estimated
as \hbox{1900\,$\pm$\,1300\,L$_{\sun}$} using the same
period-luminisoty relation as for \object{SX\,Cen}, providing a distance of
\hbox{1.4\,$\pm$\,0.5\,kpc}.
In the MIDI wavelength range \hbox{(8 -- 13 $\mu$m)}, the amount of
flux emitted by the stellar photosphere with respect to the total flux
is only 1\,\% for \object{SX\,Cen} and 5\,\% for \object{HD\,52961}. In addition, both
objects show a clear silicate resonance in emission in the N-band (see the
ISO/SWS spectrum in the top panel of Fig. \ref{fig:sedBOTH} and the
MIDI spectra in Fig. \ref{fig:totalspectrum}). Therefore, the MIDI
instrument, providing spectrally dispersed visibilities over the
N-band, is ideally suited to probe the circumstellar geometries of the
dust around both objects.
\subsection{Orbital elements}\label{sect:orbital_elements}
We refined the orbital elements which were published already in
\cite{Vanwinckel_1999} and \cite{Maas_2002} for \object{HD\,52961} and \object{SX\,Cen}
respectively. Our accumulation of data is now such that we covered
close to 3 (\object{HD\,52961}) and 2 (\object{SX\,Cen}) orbital cycles. The
heliocentric radial velocity data folded on the orbital periods are
given in Fig.~\ref{fig:orbits} and the orbital elements are listed in
Table~\ref{tab:orbitalelements}.
The data sampling of \object{HD\,52961} is not very extensive and in the
residuals no clear modulation on the pulsational period is found.
For the RV\,Tauri star \object{SX\,Cen}, the pulsational amplitude
in radial velocity is significant. After pre-whitening of the orbital
solution, the pulsational period of 16.46\,d is clearly recovered
(Fig.~\ref{fig:pulsationSXCEN}). We cleaned the original data with a
harmonic least square fit of the 16.46 days pulsation period and three
harmonics. The variance reduction of the pulsation model is 81\%.
After cleaning the original data with this pulsation model, we
redetermined the orbital elements. The eccentric orbit was found to be
significant according to the classical Lucy and Sweeney test
\citep{Lucy_1971}.
\begin{table}
\caption{The orbital elements of the program stars. All symbols have
their usual meaning. The number of measurements (N) and the
number of covered orbital cycles in our monitoring program are
also given.}\label{tab:orbitalelements}
\begin{tabular}{llll}
\hline \hline
& unit & \object{HD\,52961} & \object{SX\,Cen} \\
\hline
P & days & 1297 $\pm$ 7 & 592 $\pm$ 13 \\
T$_{o}$ & JD & 2448591 $\pm$ 38 & 2452107 $\pm$ 10 \\
K & km\,s$^{-1}$ & 13.3 $\pm$ 0.9 & 22.9 $\pm$ 0.5 \\
$\gamma$ & km\,s$^{-1}$ & 7.4 $\pm$ 0.5 & 19.1 $\pm$ 0.4 \\
e & & 0.22 $\pm$ 0.05 & 0.16 $\pm$ 0.02 \\
$a\,\sin i$ & AU & 1.54 & 1.23 \\
f(M) & M$_{\odot}$ & 0.29 & 0.70 \\
N & \# & 31 & 78 \\
cycles covered & & 2.9 & 2.0\\
\hline
\end{tabular}
\end{table}
As shown in our previous papers, both objects show a long term trend
in their photometric light-curve which is due to variable circumstellar
reddening in the line of sight towards the object. For \object{SX\,Cen} this
long term trend is periodic with a period of 615 days
\citep{Oconnell_1933,Voute_1940}, very close to the orbital period.
For \object{HD\,52961} it was not very clear whether the subtle effect is
periodic or not.
If the circumstellar material is indeed mainly stored in a disc around
the objects and assuming this disc is located in the orbital plane,
the inclination of the disc cannot be very small. Assuming an
inclination varying between edge-on (i\,=\,90$^{o}$) up to 60$^{o}$,
and a mass of the evolved component of 0.6 M$_{\odot}$, the mass of
the companion varies between 0.8\,--\,1.1 M$_{\odot}$ for \object{HD\,52961}
and 1.4\,--\,1.9 M$_{\odot}$ for \object{SX\,Cen}. The unseen companion is
probably an unevolved main sequence star, since the lack of an UV
excess and of any sign of symbiotic activity make the presence of a
massive compact object very unlikely. Moreover, the orbital
characteristics in period and eccentricity make it very unlikely that
the companion is a post red giant as well \citep{Vanwinckel_2003}.
Neither stars are filling their Roche Lobe now, but in both cases it
is clear that the actual orbit is too small to accommodate a full
grown AGB star. The stars must have suffered an evolutionary phase
with severe binary interaction when at giant dimensions.
\section{Observations}\label{sect:observations}
The VLTI/MIDI interferometer \citep{Leinert_2003} was used to combine
the light coming from the UT2 and UT3 telescopes. The observations of
the targets, \object{SX\,Cen} and \object{HD\,52961}, were performed in three nights of
Science Demonstration Time in February at a projected baseline in the
range of 40 to 50 meters. A detailed log of the observations of the
science targets is presented in Table \ref{tab:log}.
The following observing sequence was carried out, according to the
standard procedures for MIDI, and repeated for target stars and
calibrators. First, acquisiton images are obtained by both telescopes
independently (i.e. without beam combiner and prism) to ensure overlap
of the beams, which is required for interferometric combination. Then,
the MIDI beam combiner, the slit and the prism are inserted. This
produces two spectrally dispersed interferometric outputs of opposite
phase. The zero optical path difference (OPD) is searched for by
scanning a range of a few millimeters around the expected value. When
found, MIDI uses its piezo-driven mirrors to keep the fringe pattern
at a fixed position within a $\approx 200$\,$\mu$m scan length, while the
VLTI delay lines compensate for the drift in OPD position due to
sidereal motion and for the slow component of atmospheric
piston. Fringes are integrated for about 1-3 minutes. Finally,
photometric data are recorded using one telescope at a time, with the
same optical set-up but using chopping to subtract sky and background.
To correct for optical imperfections and atmospheric turbulence, a
calibrator of known diameter is measured as well. The time-lag between
the measurement of this calibrator and the science object is about 30
min. However, considering the present accuracy per single visibility
measurement of about 10\,\%, we can also use calibrators observed in
the same mode one or two hours earlier or later \citep[see
e.g.][]{Leinert_2004}. A list of the calibrator observations is given
in Table \ref{tab:log}.
\begin{table*}
\caption{A summary of the observations with the MIDI instrument of
\object{SX\,Cen} and \object{HD\,52961}. For each science target, the calibrators used
to calibrate the visibility are given (the flux calibrators are given
in Table \ref{tab:photcal}). The angular diameter in the Limb
Darkened Disc approximation is obtained from \cite{Verhoelst_2005}
(cf. http://www.ster.kuleuven.ac.be/$\sim$tijl/MIDI\_calibration/mcc.txt).
The reported flux for the calibrator sources is the IRAS 12.5\,$\mu$m
flux. Nomenclature: UT = Universal Time, PB = Projected Baseline and PA
= Projected baseline Angle}
\label{tab:log}
\begin{center}
\begin{tabular}{l c c c c c c c c c c} \hline \hline
\multicolumn{1}{c}{night} &
\multicolumn{1}{c}{science} &
\multicolumn{1}{c}{UT} &
\multicolumn{1}{c}{PB} &
\multicolumn{1}{c}{PA} &
\multicolumn{1}{c}{airmass} &
\multicolumn{1}{c}{calibrator} &
\multicolumn{1}{c}{UT} &
\multicolumn{1}{c}{spectral} &
\multicolumn{1}{c}{diameter} &
\multicolumn{1}{c}{flux} \\
\multicolumn{1}{c}{yyyy/mm/dd} &
\multicolumn{1}{c}{target} &
\multicolumn{1}{c}{hh mm ss} &
\multicolumn{1}{c}{(m)} &
\multicolumn{1}{c}{(\degr )} &
\multicolumn{1}{c}{}&
\multicolumn{1}{c}{target} &
\multicolumn{1}{c}{hh mm ss} &
\multicolumn{1}{c}{type} &
\multicolumn{1}{c}{(mas)} &
\multicolumn{1}{c}{(Jy)} \\
\hline
2004/02/09 &
\object{HD\,52961} & 02 20 02 & 39.7 & 45 & 1.2
& HD\,49161 & 02 45 52 & K4\:III & 2.44 $\pm$ 0.01 & 10.35\\ &
\object{SX\,Cen} & 07 37 52 & 44.6 & 41 & 1.1
& HD\,67582 & 06 24 15 & K3\:III & 2.30 $\pm$ 0.01 & 9.33 \\ & & & & &
& HD\,67582 & 07 13 39 & K3\:III & 2.30 $\pm$ 0.01 & 9.33 \\ & & & & &
& HD\,107446 & 08 08 53 & K3.5\:III & 4.43 $\pm$ 0.02 & 32.42 \\
2004/02/10 &
\object{HD\,52961} & 04 13 04 & 46.1 & 46 & 1.4
& HD\,67582 & 03 46 12 & K3\:III & 2.30 $\pm$ 0.01 & 9.33 \\ &
\object{SX\,Cen} & 07 37 52 & 44.6 & 41 & 1.1
& HD\,107446 & 08 16 44 & K3.5\:III & 4.43 $\pm$ 0.02 & 32.42 \\
2004/02/11 &
\object{SX\,Cen} & 08 03 25 & 43.6 & 46 & 1.1
& HD\,120404 & 08 27 38 & K7\,III & 2.96 $\pm$ 0.02 & 13.28 \\
\hline
\end{tabular}
\end{center}
\end{table*}
\section{Reduction}\label{sect:reduction}
\subsection{incoherent vs coherent analysis}
We used two different methods for the MIDI data reduction. The first
method is based on power spectrum analysis (hereafter called
incoherent analysis), while the second method reduces all frames to
the same OPD and adds them coherently (hereafter called coherent
analysis). For the incoherent analysis of the data, we used the MIA
package (MIDI Interactive Analysis,
http://www.mpia-hd.mpg.de/MIDISOFT/) developed at the Max-Planck
Institut f\"ur Astronomie in Heidelberg, while for the coherent
analysis we used the EWS package (Expert Work Station) developed by
Walter Jaffe at the Leiden observatory \citep{Jaffe_2004}.
During the incoherent analysis, we separated the different scans in
those with and without fringes, where each scan is Fourier-transformed
from OPD to fringe frequency space. Considering the wavelengths
present in the band and the rate at which the OPD is changing, the
power is calculated in the correct frequency interval. The total power
of all measured scans with fringes is then averaged and an estimate of
the noise is subtracted. This noise estimate is based on the frames
without fringes. This provides a value of the instrumental visibility
squared of each channel. Contrary to the coherent method, the major
difficulty of this method is that an accurate estimate of the
off-fringe noise power is needed. Since our science targets have small fluxes in
the 10 $\mu$m window, a reliable estimate of the noise power is
difficult to obtain and we focused during data reduction on the
coherent method (see below). Our incoherent analysis was only used to
check the results obtained by a coherent analysis and both methods
give consistent results.
\subsection{coherent analysis}
We first investigated the photometric datasets. The averages of the
target and sky frames are calculated and subtracted, providing a raw
two-dimensional spectrum of the object. The position and width of this
spectrum is determined and a spatial mask is constructed from the
location and average width of the spectrum at each wavelength
position. After multiplication of the detector images with this mask,
the rows are added providing a one dimensional raw spectrum of
the object (i.e. not corrected for the atmospheric transmission and
instrumental efficiency).
The spatial mask is then used to extract the information of the
interferometric observations as well, in the assumption that all
instrumental parameters stay the same between the interferometric and
photometric observation. The two detector spectra with opposite phase,
are subtracted, resulting in one interferometrically modulated
spectrum. In this way, the background is reduced by approximately 90
percent.
Contrary to the incoherent method which allows the summing of scans
where the relative OPD is not known, the coherent method needs an
accurate determination of the atmospheric delay. The large wavelength
coverage of the N-band ensures that this can be accurately done by
measuring the fringes in frequency space \citep[rather than in OPD
space which is done in an incoherent analysis, e.g.][]{Tubbs_2004}. As
a first step, the known instrumental delay is removed from each frame
after which the (previously unknown) atmospheric delay is retrieved
using a group delay estimation. At this point, the data is not yet
fully coherent because of the instrumental phase imposed on the data
(e.g. the varying index of refraction of water vapor imposes
variations in phase that are not removed by a group delay
fitting). These phase shifts are almost constant as a function of
frequency and can be approximated as a constant phase shift over the
N-band \citep{Jaffe_2004}. Finally, the data can be added coherently
to obtain the final visibility amplitude and differential phase.
The instrumental visibility is then calculated dividing the fringe
amplitude by the non-interferometric, photometric exposures.
Repeating this procedure for a calibrator enables to estimate the
instrumental visibility loss and thus determining the calibrated
visibility of the science object.
\subsection{the data}
\subsubsection{photometry}\label{sect_photometry}
A raw spectrum is obtained each night by subtracting the masked target
frames from the masked sky frames. This spectrum is flux calibrated
and corrected for atmospheric transmission using the calibrator
spectra observed during the same night. For the calibrators, the
intrinsic spectra were synthetised from {\sc marcs} atmosphere models
\citep[][ and further updates]{Gustafsson_1975}, using the
temperature, surface gravity and angular diameter determined in
\cite{Vanboekel_2004}. This approach is preferred over a
Rayleigh-Jeans approximation of the calibrator spectrum, since the SiO
first overtone band head is not negligible in a K giant N-band
spectrum. The 12\,$\mu$m flux of this synthetic spectrum is well
within 1\,$\sigma$ of the color corrected IRAS 12\,$\mu$m flux. For
both objects, the absolute flux calibration has been performed with
the data of 10 February only, using the calibrators listed in Table
\ref{tab:photcal}. Both reduced spectra (R\,$\sim 30$) are shown in
Fig.~\ref{fig:totalspectrum}, where they are compared to independent
spectra taken by the ISO/SWS (R\,$\sim 248$) and the SPITZER/IRS
(R\,$\sim 127$) instrument.
\begin{table}
\caption{A summary of the calibrators used to flux calibrate the
spectrum of \object{SX\,Cen} and \object{HD\,52961}. All calibrators listed were
observed on February, 10. The reported flux is the IRAS 12.5 $\mu$m
flux, and the diameters are taken from \cite{Verhoelst_2005}. }
\label{tab:photcal}
\begin{center} \begin{tabular}{l c c c c c} \hline \hline
\multicolumn{1}{c}{calibrator} &
\multicolumn{1}{c}{UT} &
\multicolumn{1}{c}{airmass} &
\multicolumn{1}{c}{spectral} &
\multicolumn{1}{c}{diameter} &
\multicolumn{1}{c}{flux} \\
\multicolumn{1}{c}{target} &
\multicolumn{1}{c}{hh mm} &
\multicolumn{1}{c}{} &
\multicolumn{1}{c}{type} &
\multicolumn{1}{c}{(mas)} &
\multicolumn{1}{c}{(Jy)} \\
\hline
HD\,67582 & 02 37 & 1.09 & K3 III & 2.30 $\pm$ 0.01 & 9.33 \\
HD\,67582 & 03 46 & 1.07 & K3 III & 2.30 $\pm$ 0.01 & 9.33 \\
HD\,49161 & 04 43 & 1.56 & K4 III & 2.44 $\pm$ 0.01 & 10.35 \\
HD\,107446 & 08 16 & 1.24 & K3.5 III & 4.43 $\pm$ 0.02 & 32.42 \\
\hline
\end{tabular}
\end{center}
\end{table}
\subsubsection{interferometry}\label{sec:interferometric_measurement}
We start this discussion with an error estimate on the observed
visibilities. The main source of error is the varying overlap between
the interferometric beams due to imperfect source acquisition and
residual image motion \citep[see e.g.][]{Leinert_2004}. This reduces
the visibility for calibrator and/or science object with an unknown
amount. The shape over the N-band, however, remains the same. The
visibility variation within the spectral band,
is therefore much more reliable than its absolute value.
To get a quantitative estimate of the absolute uncertainty on the
visibility, we look at all calibrators ($\sim$\,point sources)
observed during one night. If the interferometric efficiency is
constant throughout the night, all calibrator measurements should
yield the same instrumental visibility. In
Fig. \ref{fig:instrumental_visibility}, the mean instrumental
visibility of all six calibrators observed in the prism mode during
the night of February 9 is plotted. The variance on the mean is
overplotted. It is clear from this figure that the instrumental loss
of visibility is much higher at 8 $\mu$m than at 13 $\mu$m and that
the uncertainty on the absolute value of the visibility is about 15
\%. However, when calibrating the visibility of the science source
using a calibrator source observed in direct concatenation, this
quantitative error is an upper limit. In the following, we use an
error of 15 \% on the absolute visibility, which is therefore a
conservative estimate.
Calibrated visibilities are obtained dividing the raw visibility by
the instrumental visibility. To calibrate the measurement of \object{SX\,Cen}
observed at 9 February, we used the mean instrumental visibility as
obtained from the three last calibrator measurements. Unfortunately,
such a mean could not be used for the other measurements. Instead, we
employed the calibrator closest in time to calibrate the visibility of
the science source (see Table \ref{tab:log}). The resulting calibrated
visibilities are shown in Figs.~\ref{fig:meanVisSXCEN} and
\ref{fig:visibilities_reduced}.
\section{Discussion}\label{sect:discussion}
\subsection{visibilities}
Because the angle between the projected baselines is the same to
within 5 degrees for both objects, no large effects due to a possible
asymmetry in the source morphology are expected. Therefore, as a
first-order approximation, we modelled the circumstellar environment
of objects using a uniform disc. The visibility in this assumption is
given by $V = 2 J_1(x)/x$, where $x=2\pi \theta B/ \lambda$ with
$\theta$ the diameter of the disc and $B$ the projected baseline
length. This function is smoothly increasing with wavelength, as
long as $x < 1.22 \pi$.
The increase is however different for various amounts in resolving
power, a steeper increase is observed if the source is more
resolved. Assuming a temperature distribution in the disc, with the
colder dust located further away from the star than the hotter dust,
the gradient decreases. For an unresolved source, the value of the
visibility remains constant at unity.
The disc around \object{SX\,Cen} is unresolved in all measurements even using a
45\,m baseline. The visibility is close to unity and shows a flat
distribution over the passband. The mean calibrated visibility for all
measurements is shown in Fig. \ref{fig:meanVisSXCEN}. Because all
measurements are observed at approximately the same projected angle
(ranging from 41 to 46 degrees) this means that in that particular
orientation, the structure is smaller than 11 mas at 8 $\mu$m and 17
mas at 13 $\mu$m in a uniform disc approximation. Using a gaussian
distribution modelling, the FWHM gives respectively 7 mas and 10 mas as
upper limits.
\object{HD\,52961} shows quite a different picture. For this source, the
visibilities are low (see Fig.~\ref{fig:visibilities_reduced}), thus
the source is clearly resolved. Immediately noted is the fact that we
do not get an increase in visibility amplitude which is quite linear
(expected for a uniform disc model in the observed visibility
range). Instead we see a ``bump'' in the visibility pattern ranging
from 9 to 12\,$\mu$m. The geometry of the disc around this object can
clearly not be modelled with the same uniform disc at all wavelenghts
(see also Fig. \ref{fig:angularSize}). Using a uniform disc
approximation for each wavelength independently is however
instructive. For each wavelength bin, we made a $\chi^2$
minimalisation between the observed visibility at both baselines and a
uniform disc model. This fit is shown for three representative
wavelengths in the upper panel of Fig. \ref{fig:angularSize}. The
diameter of the source at all wavelength bins in a uniform disc
approximation is shown in the lower panel of
Fig. \ref{fig:angularSize}. The measurements at both baselines are
very consistent in a uniform disc approximation for each wavelength
independently (the mean reduced chi-square over the wavelength band is
as low as 0.09) and provide a diameter increasing from $\sim$35 mas at
8 $\mu$m to about $\sim$55 mas at 13 $\mu$m. In a gaussian
distribution modelling, the FWHM gives respectively $\sim$23 mas and
$\sim$34 mas (and a mean reduced chi-square of 0.22). The increase
towards longer wavelengths is consistent with a dust-distribution for
which the temparture decreases further away from the star. We however
note that the observed increase in size is not smooth over the
wavelength band. There is an increase in size from 8 $\mu$m to 8.5
$\mu$m and onwards 11.5 $\mu$m, while in between a sort of plateau
exists. We interpret this plateau as resulting from a non-homogenous
distribution of the radiating silicates which contribute most in the
inner regions close to the central star, thus lowering the overall
size (see also the following sections).
For both objects we interpret the small angular scales of the dust
around the objects as another clear indication that the circumstellar
dust is stored in a compact Keplerian disc around the system.
\subsection{spectra}
The SED of both objects shows a significant near-IR excess, indicative
of a hot dust component, while there is no evidence for an ongoing
dusty mass loss of those rather hot stars. In \cite{Deruyter_2006},
this is interpreted as originating from a hot inner rim near dust
sublimation temperature. Because no dust can survive at higher
temperatures, this dust receives head on radiation from the star and
is therefore supposed to be puffed up \citep[e.g. the wall model for
HR4049 elaborated by][]{Dominik_2003}. This is corraborated by the
lower limit on the opening angle of the disc as seen from the star of
13$^o$ for \object{HD\,52961} and 32$^o$ for \object{SX\,Cen} \citep{Deruyter_2006}. In
this model, we expect a highly centerally peaked intensity
distribution which provides that the correlated spectra measured by
the interferometer are dominated by the inner regions of the
disc. However, because of the varying spatial resolution from 8 to 13
$\mu$m, a slope is introduced in the correlated spectrum. To determine
the magnitude of this effect, a detailed modelling has to be
performed, which is out of the scope of this article. For now, we
assume that this effect is a smooth function of wavelength, thus
having only a marginal effect on any mineralogy determination.
Because \object{SX\,Cen} is unresolved at all employed baseline settings, the
correlated spectrum is identical to the single telescope spectrum and
thus no additional information is available for this object. However,
for \object{HD\,52961}, which is clearly resolved in both measurements, the
shape of the two correlated spectra is predominantly determined by the
inner parts of the disc. This means that we can construct independent
spectra of geometrically different areas of dust around \object{HD\,52961}. The
single telescope spectrum provides the full N-band spectrum of all the
dust around \object{HD\,52961}. The correlated spectra sample smaller parts of
the disc. These spectra, each sampling a different geometrical part of
the disc, are shown in Fig. \ref{fig:allspectraatonce}. From this
figure, it is clear that the shape of the correlated spectra is quite
different from the single telescope spectrum. This points to a
different chemical composition of the inner part of the disc and the
outer part of the disc. To quantify this, we have fitted the different
spectra independantly.
\subsection{silicate mineralogy}\label{sec:chemical_composition}
In order to determine the mineralogy and sizes of the emitting dust
grains, we made a fit to the N-band spectra using calculated
emissivities of irregularly shaped, chemically homogeneous dust
grains. The most important dust species causing spectral signature in
the 10\,$\mu$m window are amorphous and crystalline olivine
(Mg$_{2x}$Fe$_{2-2x}$SiO$_4$), amorphous and crystalline pyroxene
(Mg$_{x}$Fe$_{1-x}$SiO$_3$), and amorphous silica (SiO$_2$), where $x$
determines the Mg/Fe ratio ($x=1$ for the crystalline silicates,
$x=0.5$ for the amorphous silicates). The complex refractive indices
for the different grain species were taken from various authors listed
in Table \ref{tab:refractive_indices}. To simulate the effects of
particle irregularity we employ a particular implementation of the
so-called \emph{statistical approach} using a distribution of hollow
spheres. This distribution is very successful in reproducing the
measured absorption spectra of irregularly shaped particles
\citep{Min_2003, Min_2005}. In addition to the dust species causing
the feature, we also add a continuum contribution which accounts for
emission by large grains and/or for the possible presence of
featureless components such as metallic iron and iron sulfide. This
continuum contribution is modeled using a constant mass absorption
coefficient. In the 10 $\mu$m region we are mainly sensitive to the
dust grains smaller than a few $\mu$m. We represent the size
distribution of the particles by two different grain sizes, $0.1$ and
1.5\,$\mu$m. A similar method was successfully employed by, for
example, \citet{Bouwman_2001}, \cite{Honda_2003},
\cite{Honda_2004}, \cite{Vanboekel_2004nature} and
\cite{Vanboekel_2005} to fit 10\,$\mu$m emission spectra of
circumstellar discs. Particles larger than a few $\mu$m contribute
mainly to the continuum. In addition we assume that the thermal
radiation we analyze originates from optically thin parts of the disc,
which allows us to add the contributions from the various components
linearly. For the emission of the outer parts of the disc,
tentatively attributed to layers directly heated by the stellar flux,
this is a reasonable approximation. Because the stellar radiation is
incident under a high inclination, the temperature distribution in the
surface layer of the disc must be very sharp and therefore, the
emission in the N-band comes likely from optically thin parts. For the
inner parts of the disc, the situation is more complex: a large
fraction of the radiation comes from the inner rim, which has regions
of both low and high opacity. The fit, using an optically thin
assumption for the different contributing minerals, is therefore
certainly too simplistic. We use it here as a first order estimate to
show the chemical gradient of the silicates in the disc, but a
detailed 2D radiative transfer model with a gradient in the
physico-chemical condition of the dust grains will be needed to
quantify the results. This is outside the scope of this paper.
We assume all dust grains, including the ones causing the continuum,
to have the same temperature distribution. Due to the limited
wavelength range this temperature distribution can be represented by a
single Planck curve with a characteristic temperature $T_c$. This is
justified because it is very likely that the dust grains of different
species are coagulated, implying thermal contact between the various
components. The characteristic temperatures used in the modeling are
given in Table \ref{tab:composition}.
\begin{table}[!t]
\begin{center}
\linespread{1.3}
\selectfont
\begin{tabular}{lc}
\hline
\hline
grain species & reference \\
\hline
Amorphous Olivine & \cite{Dorschner1995} \\
Amorphous Pyroxene & \cite{Dorschner1995} \\
Forsterite & \cite{Servoin1973} \\
Enstatite & \cite{Jaeger1998} \\
Amorphous Silica & \cite{1960PhRv..121.1324S} \\
\hline
\end{tabular}
\end{center}
\linespread{1}
\caption{A list of the references of the complex refractive indices employed
for the various grain species. }
\label{tab:refractive_indices}
\end{table}
The abundances of the dust components are determined by using a linear
least square fitting procedure with constraints on the weights to
avoid negative values. The temperature of the grains and the
underlying continuum is varied from 0 to 1500\,K until a best fit is
obtained.
The dust parameters derived from the unresolved spectrum of \object{SX\,Cen}
are given in the upper row of Table~\ref{tab:composition}.
The resulting best fit spectrum is shown as a dotted line in
Fig.~\ref{fig:totalspectrum}. The grains in the circumstellar
environment of \object{SX\,Cen} are highly crystalline and also on average
relatively large compared to the interstellar grain population. This
implies a large amount of dust processing in the circumstellar
environment.
For \object{HD\,52961}, we fit the spectrum corresponding to the inner disk
(the correlated spectrum, angular size $\sim$ 20 mas) and that
corresponding to the outer disk (the total disk spectrum from which
the correlated spectrum is subtracted) separately. We focus here on
the correlated spectrum taken with the 40\,m baseline. The resulting
best fit model spectra are shown in Fig.~\ref{fig:chemical_hd52961}
and the composition is given in Table~\ref{tab:composition}. The
varying spatial resolution over the N-band introduces an extra slope
in the spectra which was not corrected. Therefore, the characteristic
temperatures ($T_c$) derived for the inner and outer disc spectra are
not realistic. The influence of this slope on the determined chemical
fractions is however expected marginal (see
e.g. \cite{Vanboekel_2004nature}). The average composition over the
disk can be derived by taking the mass weighted average of the inner
(56\%) and outer (44\%) disk regions. The overall composition of the
dust in \object{HD\,52961} is for $\sim$50\% crystalline, and contains
$\sim$60\% $1.5\,\mu$m grains. This is considerably less than what we
find in \object{SX\,Cen}. It is also clear from Table~\ref{tab:composition}
that the crystalline silicates are not uniformly distributed over the
disk. The inner disk has a much higher crystallinity than the outer
disk.
In order to fit the prominent feature around 9.5\,$\mu$m in the total
and the outer disk spectra of \object{HD\,52961}, we have to add large
(1.5\,$\mu$m) silica grains. We have tried several other dust
components in order to explain this spectral feature, but found no
spectral match using any of them. We have no explanation for the
presence of these amounts of large silica grains, and thus its
detection is debatable. However, its presence is also indicated from
the Spitzer IRS spectrum, which shows a weak feature around 21\,$\mu$m
which is naturally reproduced using large silica grains (not shown).
For a full mineralogy, the broader wavelength range sampled by our
Spitzer data is needed which is outside the scope of this paper.
\begin{table*}[!t]
\begin{center}
\linespread{1.3}
\selectfont
\begin{tabular}{cccccccccccccc}
\hline
\hline
Star & T$_c$ & Cryst. & Large & \multicolumn{2}{c}{Olivine [\%]} & \multicolumn{2}{c}{Pyroxene [\%]} & \multicolumn{2}{c}{Forsterite [\%]} & \multicolumn{2}{c}{Enstatite [\%]} & \multicolumn{2}{c}{Silica [\%]} \\
& 10$^2$\,K & [\%] & grains [\%] & Small & Large & Small & Large & Small & Large & Small & Large & Small & Large\\
\hline
\object{SX\,Cen} & $6.8_{-0.3}^{+0.3}$ & $78_{-23}^{+17}$ & $93_{-4}^{+3} $ & - & $21_{-17}^{+24}$ & - & $1_{-1}^{+6}$ & $7_{-3}^{+3}$ & $25_{-9}^{+9}$ & - & $46_{-15}^{+12}$ & $1_{-1}^{+1}$ & $0_{-0}^{+2}$\\
\object{HD\,52961} (inner)&$12_{-2}^{+2}$ & $79_{-12}^{+10}$ & $63_{-12}^{+7} $ & $6_{-6}^{+13}$ & $4_{-4}^{+16}$ & - & $1_{-1}^{+15}$ & $31_{-3}^{+3}$ & $1_{-1}^{+7}$ & $0_{-0}^{+6}$ & $47_{-11}^{+9}$ & - & $9_{-3}^{+3}$\\
\object{HD\,52961} (outer)&$14_{-1}^{+1}$ & $19_{-3}^{+4} $ & $59_{-17}^{+18}$ & - & $5_{-5}^{+10}$ & $22_{-17}^{+16}$ & $20_{-16}^{+19}$ & $19_{-3}^{+3}$ & - & - & $0_{-0}^{+4}$ & $0_{-0}^{+1}$ & $34_{-3}^{+4}$\\
\hline
\end{tabular}
\end{center}
\linespread{1}
\caption{The composition and grain sizes of the dust in the
circumstellar environments around \object{SX\,Cen} and \object{HD\,52961} as derived
from our fitting procedure. The olivine and pyroxene grains are
amorphous while the forsterite and enstatite are crystalline. For
\object{HD\,52961} 56\% of the dust mass is in the inner disk region. It
should be noted that, as explained in the text, the temperatures
determined for the inner and outer disc spectra of \object{HD\,52961} are not
realistic.}
\label{tab:composition}
\end{table*}
\subsection{formation history of the disc}
The composition of the dust in the circum-binary disc as a function of
distance from the binary can give important clues to its formation
history. In principle, the disc could have been formed by capturing a
"normal" AGB wind, or through non-conservative mass transfer in an
interacting binary. In the wind scenario, it is not unreasonable to
assume that most dust was in the form of amorphous silicates, since
this is the usual dust composition for O-rich AGB stars with a
moderate to high mass loss rate \citep[see e.g.][]{Sloan_1995,Waters_1996L,
Cami_2002}. In the interacting binary scenario, the dust may or may
not have formed before the material entered the circum-binary disc,
but in any case the thermal history of that dust would have been very
different from that of the wind scenario: the grains were likely at
high temperatures for a long period of time, increasing the chances of
a substantial crystallisation. Therefore the wind scenario predicts a
predominantly amorphous silicate composition, while the interacting
binary scenario more likely produces (highly) crystalline discs.
Once in the disc, both grain aggregation and crystallisation may
occur. Grain aggregation is a strong function of density and thus
would be most efficient in the inner disc regions. Large grains settle
quickly to the mid-plane thus creating a cold mid-plane population of
grains, which we believe is responsible for the millimeter continuum
emission \citep[see e.g.][]{Deruyter_2006}. The inner disc reaches
temperatures above the glass temperature, forcing the grains to
anneal. Therefore, in both the wind and in the interacting binary
scenarios the innermost disc regions are expected to be strongly
crystalline. The two scenarios predict strongly differing radial
gradients in crystallinity however.
The present-day orbital parameters of the binary systems with
circum-binary discs strongly suggest that interaction took place when
the current post-AGB star was on the AGB. Therefore it seems difficult
to imagine that a standard stellar wind formed the discs, and one
would expect the discs on average to have much higher crystallinity
than typical AGB winds. The recent spectral survey by
\cite{Deruyter_2006b} indeed suggests that the circumbinary discs are
much more crystalline than typical AGB outflows. One complicating
factor is that at present not much is known about the composition of
the dust in AGB outflows in the dust forming layers: by far most data
are spatially unresolved and present the final outcome of the dust
formation process in O-rich AGB outflows.
Our MIDI observations indeed confirm that the inner disc region of
\object{HD\,52961} is extremely crystalline, and that the outer disc regions
are less so. At first glance this would suggest the wind scenario is
more likely, but the orbital parameters indicate substantial AGB
interaction. These first MIDI observations thus raise interesting
questions: is the outer disc of \object{HD\,52961} really amorphous and what
kind of disc formation scenario could lead to amorphous grains? Does
this hold for all systems, or is there an orbital separation
dependence? Clearly more study is required to answer these questions.
\subsection{comparison with Herbig Ae/Be stars}\label{sec:comparison}
In \cite{Deruyter_2006}, it is argued that the broad-band SED
characteristics of the discs around binary post-AGB objects are very
similar to the those of the Herbig Ae/Be group II sources. Herbig
Ae/Be stars are intermediate mass pre-main sequence stars surrounded
by remnant material of the star formation process. For these objects,
the existence of a passive circumstellar disc is firmly established
\citep[e.g.][ and references therein]{Waters_1998, Eisner_2003}. The
Herbig Ae/Be stars are subdivided in two groups \citep{Meeus_2001}
with the group I sources showing a rising mid-IR flux excess,
while the group II sources only show a modest mid-IR excess. The
difference in SED characteristics between both groups is attributed to
disc geometry. The mid-IR excess of group I sources is indicative of
the flaring of the outer disc, while the inner rim of the group II
sources shadows the whole disc and no flaring occurs
\citep{Chiang_1997, Dullemond_2001}. \cite{Vanboekel_2004} used the
most recent models \citep{Dullemond_2004} of the discs around Herbig
Ae/Be stars to compute the visibilities to be expected in the MIDI
wavelength range. \cite{Leinert_2004} on the other hand made
observations with the MIDI instrument of several of these objects. We
make a comparison of the results obtained in these publications for
the group II sources and our observations under the assumption of the
similarity of both source geometries.
Concerning the continuum radiation, the modelling performed in
\cite{Vanboekel_2004} shows that the size of the disc increases more
rapidly from 8 to 13 $\mu$m than the interferometric resolution
decreases. This provides a visibility curve which is decreasing from 8
to 13 $\mu$m. The observations indeed show this qualitative behaviour,
however some objects, e.g. HD144432, show a rather horizontal
slope. This very similar behaviour is observed for \object{HD\,52961} as
well. The slope of the continuum visibility does not increase as
expected for a uniform disc, it is instead rather constant with
wavelength.
Concerning the visibility in the feature, the modelling performed by
\cite{Vanboekel_2004} suggests a lowered visibility for the silicate
feature than for the continuum. The disc is irradiated by the central
object and therefore the disc surface is hotter than the disc
midplane. Because the opacity in the silicate resonance band is higher
than in the continuum, one looks less deep into the disc in the
resonance. This results in the fact that in the 10 $\mu$m region, a
larger region in the resonance is seen than in the continuum. The
observations of Herbig Ae/Be stars show a similar qualitative
behaviour, however the visibility decrease is less pronounced. In
fact, \cite{Vanboekel_2004nature} finds that for three Herbig Ae stars
of the sample of \cite{Leinert_2004}, there is a large radial gradient
in the processing. The innermost region of the proto-planetary discs
has a substantially higher crystallinity degree with a shape very
similar to that of comets in our solar system, while the outer region
is clearly less processed. Clearly, the homogenous distribution of
dust adopted in the modelling of these discs is a very crude
approximation. For \object{HD\,52961}, no visibility decrease is observed in
the feature, instead an increase is seen. The spatial distribution of
the dust responsible for the resonance is not homogeneously
distributed, instead the hot inner region of the dust is much more
crystalline than the outer parts
(Sect. \ref{sec:chemical_composition}). This qualitative similarity in
the distribution of the chemical species in the discs around some
Herbig Ae stars and \object{HD\,52961} is surprising in the context of the
completely different formation history of both.
\section{Conclusions}\label{sect:conclusions}
The main conclusion of our presented MIDI observations is that they
prove the very compact nature of the circumstellar environments of
\object{HD\,52961} and \object{SX\,Cen}. \object{SX\,Cen} is not resolved using a 45\,m baseline,
which gives an upperlimit of only 18\,AU at the estimated
distance. For the well resolved \object{HD\,52961}, the angular size in the
N-band varies between 35 and 55\,mas in a uniform disc approximation,
which translates to a size of 50 and 80\,AU.
Both stars have an effective temperature in the 6000\,K range and
since there is no evidence for a current dusty mass-loss we
interpret these results as a very stringent proof of the
existence of a stable reservoir near the star. A Keplerian disc seems
the only plausible solution. The dust sublimation temperature is
reached much further out than the binary orbits, hence the discs must
be circumbinary. This is corroborated by the measured size of the
dust-emission region around \object{HD\,52961}.
Given the size of the orbits, the discs were probably formed in a
poorly understood phase of strong binary interaction, when the star
was at giant dimensions. Both discs are O-rich and there is no
evidence for a C-rich component. They were consequently formed prior
to the late AGB evolution where the stars could have changed into
C-stars. The mass of the companion of \object{SX\,Cen} (1.4 -- 1.9 M$_{\odot}$)
is probably within the range of C-star progenitors. We conclude that
the normal single star AGB evolution was shortcut by the presence of a
binary companion. Clearly the formation of a stable Keplerian disc is
a key ingredient in the late evolution of both binaries.
\object{SX\,Cen} is an RV\,Tauri star of photometric class b which shows a long
term variability of the mean magnitude with a period similar to its
orbital period probably due to variable circumstellar extinction in
the line of sight during orbital motion. The inclination cannot be
very small. Additional interferometric data on different projected
angles will be necessary to probe the expected asymmetries.
The characteristics of the dust grains seem to be very different from
normal single star outflows. This is shown in the mineralogy of the
silicate resonance feature which shows for both objects a highly
crystalline component and a size distribution with a much stronger
component of large ($> 1$\,$\mu$m) grains than what is observed in
outflows of AGB stars. It is not clear whether this reflects the
formation history of the disc or this is due to the longer
processing time of the dust in the Keplerian discs. Our analysis of
\object{HD\,52961} shows that the crystallinity is clearly concentrated in the
hotter inner region of the disc. Crystallisation by annealing is very
temperature dependent and a similar picture arises as what is seen in
the discs around some young stellar objects: the grains in the hot
inner region were subject to a much stronger processing while in the
outer region remained less processed. MIDI as spectrally dispersed
N-band interferometer is an ideal instrument to study the
chemo-physical structure of the inner regions of these discs.
\begin{acknowledgements}
The authors would like to thank Jeroen Bouwman for the reduction of
the SPITZER/IRS spectrum of \object{HD\,52961} and Bram Acke for the reduction
of the ISO/SWS spectrum of \object{SX\,Cen}. We also like to thank the referee,
K. Ohnaka, for the many valuable comments. We thank the staff of the Geneva
Observatory and the staff of the Instituut voor Sterrenkunde of the
K.U.Leuven for the generous award of time on the Swiss Euler telescope
at La Silla and the Flemish Mercator telescope at La Roque de los
Muchachos respectively. We also thank our colleagues from the
Instituut voor Sterrenkunde for their contribution to the gathering of
the data. P.D.~and H.V.W.~acknowledge financial support from the Fund
for Scientific Research of Flanders (FWO).
\end{acknowledgements}
\bibliographystyle{aa}
|
Title:
Full polarization study of SiO masers at 86 GHz |
Abstract: We study the polarization of the SiO maser emission in a representative
sample of evolved stars in order to derive an estimate of the strength of the
magnetic field, and thus determine the influence of this magnetic field on
evolved stars. We made simultaneous spectroscopic measurements of the 4 Stokes
parameters, from which we derived the circular and linear polarization levels.
The observations were made with the IF polarimeter installed at the IRAM 30m
telescope. A discussion of the existing SiO maser models is developed in the
light of our observations. Under the Zeeman splitting hypothesis, we derive an
estimate of the strength of the magnetic field. The averaged magnetic field
varies between 0 and 20 Gauss, with a mean value of 3.5 Gauss, and follows a
1/r law throughout the circumstellar envelope. As a consequence, the magnetic
field may play the role of a shaping, or perhaps collimating agent of the
circumstellar envelopes in evolved objects.
| https://export.arxiv.org/pdf/astro-ph/0601098 |
\def\etal{et al.\ }
\def\kms{km\thinspace s$^{-1}$ }
\def\Lsun{L$_\odot$}
\def\water{H$_2$O~}
\def\Msun{M$_\odot$}
\def\ms{m\thinspace s$^{-1}$}
\def\percc{cm$^{-3}$}
\title{Full polarization study of SiO masers at 86 GHz}
\author{F.\,Herpin\inst{1}, A.\,Baudry\inst{1}, C.\,Thum\inst{2}, D.\,Morris\inst{2}$^,$\inst{3} and H.\,Wiesemeyer\inst{2}}
\institute{
Observatoire Aquitain des Sciences de l'Univers, Laboratoire d'Astrodynamique, d'Astrophysique et d'A\'eronomie de Bordeaux, CNRS/INSU UMR n$^{\circ}$ 5804, BP 89, 33270, France
\and
IRAM, 300 rue de la Piscine, Domaine Universitaire, 38406 Saint Martin d'H\`eres, France
\and
Present address: Raman Research Institute, 560080 Bangalore, India
}
\titlerunning{SiO maser polarization}
\abstract{We study the polarization of the SiO maser emission in a representative sample of evolved stars in order to derive an estimate of the strength of the magnetic field, and thus determine the influence of this magnetic field on evolved stars. We made simultaneous spectroscopic measurements of the 4 Stokes parameters, from which we derived the circular and linear polarization levels. The observations were made with the IF polarimeter installed at the IRAM 30m telescope. A discussion of the existing SiO maser models is developed in the light of our observations. Under the Zeeman splitting hypothesis, we derive an estimate of the strength of the magnetic field. The averaged magnetic field varies between 0 and 20 Gauss, with a mean value of 3.5 Gauss, and follows a $1/r$ law throughout the circumstellar envelope. As a consequence, the magnetic field may play the role of a shaping, or perhaps collimating agent of the circumstellar envelopes in evolved objects.}
\textbf{Keywords.} Maser: SiO -- polarization-- survey -- stars: late-type, evolution, magnetic field
\section{Introduction}
The prodigious mass loss observed in numerous and widespread evolved stars make these objects the main recycling agents of the interstellar medium, and thus one of the most important objects in the Universe. Even though our knowledge of evolved stars has considerably improved over recent years, some of their main characteristics remain insufficiently understood (see the review by Herwig 2003): which mechanisms are responsible for their drastic change of geometry when evolving to the Planetary Nebula (hereafter PN) stage ? What is powering so efficiently the mass loss and could the magnetic field play a major role ?
Important information about the physics and chemistry prevailing in the circumstellar envelope (hereafter {\em CSE}) of evolved stars can be retrieved from radiowave line emission of molecules, specially from maser emission (see the review by Bujarrabal 2003). These envelopes can be probed at different depths through the study of three masing molecules, OH, \water and SiO. Our current knowledge indicates that:
\begin{itemize}
\item OH radiation traces the outer part of the envelope, at 1000-10000 AU from the central star;
\item \water molecules are located at intermediate distances, i.e. a few 100 AU;
\item SiO maser emission comes from the inner regions of the envelope, between 5 to 10 AU (a few stellar radii R$_{\star}$).
\end{itemize}
The SiO maser emission is produced in small gas cells, and is known to be polarized. The polarization (circular or linear) and angle of the emission can be measured and thus improve our knowledge of these objects. In addition, studying the maser polarization can shed light on the maser theory itself. As explained further in this paper, several uncertainties in the theory make data interpretation often difficult, and new observational data are helpful. One of the most interesting quantities that can be derived from polarization measurements is the stellar magnetic field. According to theory (e.g. Elitzur 1996 or 2002), measurement of the maser radiation polarization can lead to an estimation of the magnetic field strength B and can reveal its spatial structure (via interferometric observations). In single dish observations, all of the maser components get smeared within the beam and only the mean value of B along the line of sight ($B_{//}$) can be derived. Only SiO masers are capable of tracing the magnetic field as close as $\sim$ 5 AU from the central star. But if SiO masers are to be used as a B-field tracer, we first need to give evidence that SiO masers are reliable B-field tracers. This requires more detailed theories than available today. Nevertheless, we tentatively derive in this work the field strength in the CSE inner layers of several evolved stars.
Research on astronomical masers polarization is very active but is made difficult both by the lack of specific instrumental facilities and by the excitation and propagation of the masers themselves. Until now, numerous polarimetric observations of OH masers have been done, several of \water masers, but few of SiO maser emission. Few SiO polarimetric observations have been done with VLBI giving the very first images of the magnetic field in some objects (e.g. Kemball \& Diamond 1997 in TX Cam). Most of the early SiO studies were done in linear polarization. The first complete SiO polarimetric observations were performed by Johnson \& Clark (1975), then by Troland \etal (1979); emission was found to be typically 15-30 \% linearly polarized and to exhibit no circular polarization. Barvainis, McIntosh \& Predmore (1987), and McIntosh \etal (1989) measured circular polarization of $1-9$ \% in several stars. Circular (0-4 \%) and linear (3.7-9.7 \%) polarizations were measured in VY CMa by McIntosh, Predmore \& Patel (1994). Later, Kemball \& Diamond (1997) made the first image of the magnetic field in the atmosphere of TX Cam, measuring a circular polarization level of 5 \% with some features showing polarization up to 30-40 \%.
It must be stressed that SiO, as H$_2$O, is a non-paramagnetic species. Zeeman splitting exists but the sublevels overlap; the effect is thus undetectable and hence only net polarization can be used to trace the magnetic field. The current status of our knowledge on the magnetic field strength can be summarized as follows:
\begin{itemize}
\item between 1000-10000 AU, $B_{//}\sim 5-20$ mG (OH masers, e.g. Kemball \& Diamond 1997, Szymczak \& Cohen 1997);
\item at a few 100 AU from the star, $B_{//} \sim$ a few 100 mG (\water masers, e.g. Vlemmings, Diamond \& van Langevelde 2001, Vlemmings, van Langevelde \& Diamond 2005);
\item at 5-10 AU, $B_{//} \sim 5-10$ G (SiO masers; Kemball \& Diamond 1997, in TX Cam).
\end{itemize}
The main purpose of this work is to measure and analyze the SiO maser polarization in terms of magnetic field strength in a representative sample of evolved stars. Our observations are presented in Section 2; they include simultaneous spectroscopic measurements of the 4 Stokes parameters. The results for individual stars are discussed in Section 3. In Section 4, we compare our data with predictions from existing SiO maser models and initiate a discussion on the validity of these models. Within the limitations of one of these models we derive the magnetic field strength and try to determine the role of the magnetic field. More broadly, a summary of the magnetic field topic in evolved stars is also given in Section 4. In Section 5 we give some concluding remarks.
\section{Observations}
An electromagnetic plane wave is defined by two components (horizontal and vertical):
\begin{equation}
\label{ }
e_H(z,t)=E_H\ e^{j(\omega t-kz-\delta)}
\end{equation}
\begin{equation}
\label{ }
e_V(z,t)=E_V\ e^{j(\omega t-kz)}
\end{equation}
where $\delta$ is the phase difference between horizontal and vertical components.
Its energy flux is described by the 4 Stokes parameters:
\begin{equation}
\label{ }
I = <{E_H}^2> + <{E_V}^2>
\end{equation}
\begin{equation}
\label{ }
Q = <{E_H}^2> - <{E_V}^2>
\end{equation}
\begin{equation}
\label{ }
U = 2 <E_H E_V cos\ \delta>
\end{equation}
\begin{equation}
\label{ }
V = 2 <E_H E_V sin\ \delta>
\end{equation}
From these parameters, one deduces:
\begin{itemize}
\item the circular polarization rate $p_C = V/I$
\item the linear polarization rate $p_L = \sqrt{Q^2+U^2}/I$
\item the polarization angle $\chi = \frac{\arctan (U/Q)}{2}$
\end{itemize}
The linear$/$circular polarization rate is sometimes called the linear$/$circular fractional polarization.
\scriptsize
\begin{table*} [htb]
\caption{ \label{table} Stars observed in this work. The stellar type is derived from the literature as are the mass loss rates (e.g. Loup \etal 1993) and the period (e.g. AAVSO data). }
{\begin{tabular}{l|c|c|c|c|c|c|c} \hline
{\bf Stars} & RA & DEC & Type & $V_{LSR}$ & dM/dt & Period & Optical \\
& (J2000) & (J2000) & & [\kms] & [M$_{\odot}/$yr]& [days] & phase \\ \hline
IRAS 18055-1433 & 18:08:23.20 & -14:32:43.0 & IR late-type & 180 & unknown & unknown & \\
IRAS 18158-1527 & 18:18:41.50 & -15:26:25.0 & IR late-type & 20 & unknown & unknown & \\
IRAS 18204-1344 & 18:23:17.90 & -13:42:46.0 & IR Supergiant (M8) & 45 & 4.2 $10^{-6}$& unknown & \\
W And & 02:17:32.96 & 44:18:17.8 & Mira (S6,1E-S9,2E$/$M4-M1) & -35 & 8.0 $10^{-7}$ & 395.9 & 0.62 \\
AU Aur & 04:54:15.00 & 49:54:00.3 & Mira (C6-7,3E(N0E)) & 8 & 1.1 $10^{-7}$ & 400 & 0.17 \\
NV Aur & 05:11:19.43 & 52:52:33.6 & Mira (M10) & 2 & 7.6 $10^{-6}$ & 635 & \\
R Aur & 05:17:17.69 & 53:35:10.0 & Mira (M6.5E-M9.5E) & -3 & 9.8 $10^{-7}$ & 457.5 & 0.83 \\
RU Aur & 05:40:07.89 & 37:38:10.7 & SRb (M7E-M9E) & -35 & unknown & 466.4 & 0.25 \\
TX Cam & 05:00:51.15 & 56:10:54.0 & Mira (M8-M10) & 10 & 2.5 $10^{-6}$ & 557.4 & 0.70 \\
V Cam & 06:02:32.30 & 74:30:27.1 & Mira (M7E) & 8 & 1.6 $10^{-6}$ & 522.4 & 0.91 \\
R Cnc & 08:16:33.83 & 11:43:34.6 & Mira (M6E-M9E) & 14 & 6 $10^{-7}$ & 361.6 & 0.33 \\
W Cnc & 09:09:52.63 & 25:14:53.8 & Mira (M6.5E-M9E) & 38 & 3.1 $10^{-8}$ & 393.2 & 0.76 \\
VY CMa & 07:22:58.33 & -25:46:03.2 & Red Supergiant (M3-M4II) & 15 & $10^{-5}$ &
400 & 0.37 \\
S CMi & 07:32:43.08 & 08:19:05.3 & Mira (M6E-M8E) & 52 & 4.1 $10^{-8}$ & 332.9 & 0.35 \\
R Cas & 23:58:24.79 & 51:23:19.5 & Mira (M6E-M10E) & 27 & 1.1 $10^{-6}$ & 430.4 & 0.66 \\
S Cas & 01:19:41.97 & 72:36:39.3 & Mira (S3,4E-S5,8E) & -28 & 3.1 $10^{-6}$ & 612.4 & 0.98 \\
T Cas & 00:23:14.25 & 55:47:33.3 & Mira (M6E-M9.0E) & -7 & 5.1 $10^{-7}$ & 444.8 & 0.22 \\
T Cep & 21:09:31.85 & 68:29:27.6 & Mira (M5.5E-M8.8E) & -1 & 1.4 $10^{-7}$ & 388.1 & 0.94 \\
R Com & 12:04:15.20 & 18:46:56.7 & Mira (M5E-M8EP) & -3 & $10^{-7}$ & 362.8 & 0.43 \\
S CrB & 15:21:23.96 & 31:22:02.7 & Mira (M6E-M8E) & 3 & 5.8 $10^{-7}$ & 360.2 & 0.44 \\
R Crt & 11:00:33.87 & -18:19:29.6 & SRb (M7III) & 10 & 7.5 $10^{-7}$ & 160 & \\
$\chi$ Cyg & 19:50:33.94 & 32:54:50.6 & Mira (S6,2E-S10,4E) & 10 & 5.6 $10^{-7}$ & 408 & 0.37 \\
UX Cyg & 20:55:05.40 & 30:24:53 & irregular variable (M4E-M6.5E) & 1 & 3.2 $10^{-6}$ & 565 & 0.77 \\
R Hya & 13:29:42.82 & -23:16:52.9 & Mira (M6E-M9E(TC)) & -8 & 1.4 $10^{-7}$ & 388.8 & 0.95 \\
W Hya & 13:49:02.03 & -28:22:03.0 & SRa (M7.5E-M9EP) & 42 & 8.1 $10^{-8}$ & 361 & 0.83 \\
X Hya & 09:35:30.26 & -14:41:28.5 & Mira (M7E-M8.5E) & 26 & 4.8 $10^{-8}$ & 301.1 & 0.0 \\
R Leo & 09:47:33.49 & 11:25:44.0 & Mira (M6E-M8IIIE-M9.5E) & 0 & $10^{-7}$ & 309.9 & 0.84 \\
W Leo & 10:53:34.44 & 13:42:54.4 & Mira (M5.5E-M7E) & 49 & unknown & 391.7 & 0.45 \\
R LMi & 09:45:34.28 & 34:30:42.8 & Mira (M6.5E-M9.0E) & 2 & 2.8 $10^{-7}$ & 372.2 & 0.56 \\
T Lep & 05:04:50.84 & -21:54:16.2 & Mira (M6E-M9E) & -29 & 7.3 $10^{-9}$ & 368.1 & 0.59 \\
RS Lib & 15:24:19.78 & -22:54:39.7 & Mira (M7E-M8.5E) & 7 & 1.8 $10^{-8}$ & 217.6 & 0.28 \\
Ap Lyn & 06:34:34.90 & 60:56:33.0 & Mira (M9) & -23 & 4.9 $10^{-6}$ & unknown & \\
U Lyn & 06:40:46.49 & 59:52:01.6 & Mira (M7E-M9.5E) & -10 & unknown & 433.6 & 0.13 \\
GX Mon & 06:52:46.90 & 08:25:20.0 & Mira (M9) & -9 & 5.4 $10^{-6}$ & 527 & \\
SY Mon & 06:37:31.28 & -01:23:43.6 & Mira (M6E-M9) & -57 & unknown & 422.2 & 0.18 \\
V Mon & 06:22:43.58 & -02:11:43.2 & Mira (M5E-M8E) & 5 & unknown & 341 & 0.0 \\
U Ori & 05:55:49.18 & 20:10:30.7 & Mira (M6E-M9.5E) & -38 & 2.8 $10^{-7}$ & 368.3 & 0.24 \\
RR Per & 02:28:28.73 & 51:16:21.1 & Mira (M6E-M7E) & 7 & unknown & 389.6 & 0.17 \\
S Per & 02:22.51.76 & 58:35:11.4 & SRc (M3IAE-M7) & -40 & 1.4 $10^{-6}$ & 822 & 0.58 \\
QX Pup & 07:42:16.83 & -14:42:52.1 & PN (M6) & 34 & 1.1 $10^{-4}$ & unknown & \\
Z Pup & 07:32:38.06 & -20:39:29.2 & Mira (M4E-M9E) & 4 & unknown & 508.6 & 0.74 \\
VX Sgr & 18:08:04.05 & -22:13:26.6 & Red Supergiant (M4EIA-M10EIA) & 6 & 5.5 $10^{-6}$ & 732 & 0.38 \\
AH Sco & 17:11:17.02 & -32:19:30.7 & SRc (M4E-M5IA-IAB) & -7 & $10^{-6}$ & 713.6 & 0.98 \\
RR Sco & 16:56:37.85 & -30:34:48.1 & Mira (M6II-IIIE-M9) & -28 & 1.1 $10^{-8}$ & 281.4 & 0.35 \\
R Ser & 15:50:41.74 & 15:08:01.4 & Mira (M5IIIE-M9E) & 28 & 2.6 $10^{-7}$ & 356.4 & 0.20 \\
S Ser & 15:21:39.53 & 14:18:53.1 & Mira (M5E-M6E) & 20 & $<2.2$ $10^{-7}$ & 371.8 & 0.76 \\
WX Ser & 15:27:47.30 & 19:33:48.0 & Mira (M8E) & 7 & 2.6 $10^{-6}$ & 425.1 & 0.30 \\
\hline
\end{tabular}}
\end{table*}
\normalsize
\addtocounter{table}{-1}
\scriptsize
\begin{table*} [htb]
\caption{ \label{table} (-continued). Stars observed in this work. The stellar type is derived from the literature as are the mass loss rates (e.g. Loup \etal 1993) and the period (e.g. AAVSO data). }
{\begin{tabular}{l|c|c|c|c|c|c|c} \hline
{\bf Stars} & RA & DEC & Type & $V_{LSR}$ & dM/dt & Period & Optical \\
& (J2000) & (J2000) & & [\kms] & [M$_{\odot}/$yr]& [days] & phase \\ \hline
IK Tau & 03:53:28.80 & 11:24:22.7 & Mira (M6E-M10E) & 35 & 4.4 $10^{-6}$ & 470 & 0.80 \\
R Tau & 04:28:18.00 & 10:09:44.8 & Mira (M5E-M9E) & 14 & 6.5 $10^{-8}$ & 320.9 & 0.68 \\
RX Tau & 04:38:14.57 & 08:20:09.4 & Mira (M6E-M7E) & -41 & $<5.8 10^{-8}$ & 331.8 & 0.13 \\
R Tri & 02:37:02.32 & 34:15:51.4 & Mira (M4IIIE-M8E) & 57 & 1.1 $10^{-7}$ & 266.9 & 0.0 \\
R UMi & 16:29:57.87 & 72:16:49.0 & SRb (M7IIIE) & -6 & unknown & 325.7 & 0.35 \\
S UMi & 15:29:34.66 & 78:38:00.2 & Mira (M6E-M9E) & -42 & unknown & 331 & 0.68 \\
R Vir & 12:38:29.95 & 06:59:19.0 & Mira (M3.5IIIE-M8.5E) & -26 & unknown & 145.6 & 0.12 \\
RS Vir & 14:27:16.39 & 04:40:41.1 & Mira (M6IIIE-M8E) & -12 & 3.8 $10^{-7}$ & 353.9 & 0.61 \\
RT Vir & 13:02:37.96 & 05:11:08.5 & SRb (M8III) & 18 & 7.4 $10^{-7}$ & 155 & 0.60 \\
S Vir & 13:33:00.11 & -07:11:41.0 & Mira (M6IIIE-M9.5E) & 12 & 4.1 $10^{-7}$ & 375.1 & 0.27 \\
\hline
\end{tabular}}
\end{table*}
\normalsize
We present here spectroscopic measurements of the 4 Stokes parameters (see Fig. 1). The observations were made with the IF polarimeter installed at the IRAM 30m telescope on Pico Veleta, Spain (Thum \etal 2003). Simultaneous measurements of I, U, Q, V allow us to calculate I, $p_L$, $p_C$ and $\chi$ for each velocity channel. The polarization angle calibration (i.e. the sign of Stokes U) was verified by observations of the Crab Nebula. Moreover, planets (polarization of planets is negligible at our frequency) have been used to check the instrumental polarization on the optical axis.
The instrumental beam polarization is known to be stronger in Stokes Q and U than in Stokes V (known to be $\leq$ 2-3\%, see Thum \etal 2003, comparable to our sensitivity as stated elsewhere). If some detections from sources with weak p$_L$ are from a bad or uncertain pointing, they naturally induce a value of p$_C$ which is weaker than p$_L$.
A strong instrumental polarization in Stokes V would be rather due to a bad phase tracking (the IF polarimeter works in a manner quite similar to that of an adding interferometer, and good phase tracking is essential). From several tests (Thum \etal 2003, Wiesemeyer, Thum \& Walmsley 2004), we know that polarization seen for weak SiO components with (Q,U,V)= (+ - -), (- - +) or (- + -) is instrumental polarization. We see that signature for only 3 objects (R Crt, R UMi and RT Vir). Some instrumental polarization may thus contaminate the observations of these objects.
All instrumental parameters were carefully calibrated through specific procedures described in Thum \etal (2003). The error on $p_{L,C}$ is $\leq 2-3$ \%.
SiO (v=1, J=2-1) line observations at 86.243442 GHz were carried out towards 57 stars in August and November 1999 with the IRAM 30m radiotelescope. The pointing was regularly checked directly on the star itself (for the vast majority of objects). In order to obtain flat baselines, we used the wobbler switching mode. The system temperature of the SIS receiver ranged from 110 to 170 K. The front-ends were the facility receivers A100 and B100, and the back-end was the autocorrelator. The lines were observed with a spectral resolution of 0.3 \kms. The integration times were 4-10 minutes using the wobbler switching. The forward and main beam efficiencies were respectively 0.92 and 0.77 at 3 mm. (Additional SiO (v=1, J=5-4) line observations at 215.596 GHz were also performed in most stars studied here; results will be reported elsewhere.)
Our source sample (see Table 1) consists of 43 Miras, 7 Semi-Regular stars (hereafter {\em SR}), 2 IR late-type stars, 1 irregular variable, 3 supergiants and 1 Planetary Nebula (QX Pup) selected from our SiO maser master catalogue (Herpin \& Baudry, private communication). Coordinates and the main characteristics of the objects are given in Table 1. Nearly 60 \% of stars in this table have been observed with the HIPPARCOS satellite and have thus excellent optical positions; such positions have been adopted in our work.
\scriptsize
\begin{table} [htb]
\caption{ \label{table} Derived parameters of the different components of the SiO
maser emission profile for each star. Only the well identified components are given (distinct peak or strong wing emission separated from the bulk emission). Note that the polarization is fractional. The $\delta P$ is the rms derived from the $p_C$ plot.}
{\begin{tabular}{l|c|c|c|c|c|c} \hline
{\bf Source} & v$_{LSR}$ & F$_{\nu}$ & p$_{c}$ & p$_{L}$ & $\delta$p & $\chi$ \\
& [\kms] & [Jy] & & & & [$^{\circ}$] \\ \hline \hline
{\bf 18055-1433} & 180.6 & 1.02 & -0.10 & 0.11 & $ 0.01$ & 50 \\
& 182.7 & 1.7 & -0.30 & 0.40 & $ 0.01$ & 30 \\ \hline
{\bf 18158-1527} & 15.2 & 2.28 & 0.08 & 0.16 & 0.02 & 90 \\
& 17.5 & 1.86 & -0.10 & 0.20 & 0.02 & 50 \\
& 21.7 & 1.74 & 0.11 & 0.15 & 0.02 & 170 \\
& 24.5 & 0.72 & 0.43 & 0.49 & 0.02 & 46 \\ \hline
{\bf 18204-1344} & 38.3 & 20.94 & 0.0 & 0.06 & $ 0.01$ & 170 \\
& 41.5 & 33.18 & 0.03 & 0.06 & $ 0.01$ &150 \\
& 45.0 & 28.26 & 0.0 & 0.02 & $ 0.01$ &120 \\
& 49.0 & 3.90& 0.04 & 0.08 & $ 0.01$ &110 \\
& 53.7 & 8.34 & $\pm$0.07 & 0.06 & $ 0.01$ &120 \\ \hline
{\bf W And} & -38.0 & 4.98 & 0.08 & 0.20 & 0.01 & 70 \\
& -35.9 & 16.32 & -0.01 & 0.02 & 0.01 & 30 \\
& -34.0 & 9.72 & -0.07 & 0.16 & 0.01 & 175 \\
& -32.3 & 2.52 & 0.12 & 0.28 & 0.02 & 140 \\ \hline
{\bf AU Aur} & 3.4 & 3.72 & -0.09 & 0.17 & $ 0.01$ & 71 \\
& 5.3 & 10.62 & 0.07 & 0.18 & $ 0.01$ & 165 \\
& 7.9 & 27.12 & $\pm$0.02 & 0.09 & $ 0.01$ & 60 \\
& 10.4 & 8.70 & 0.09 & 0.13 & $ 0.01$ & 110 \\ \hline
{\bf NV Aur} & -1.2 & 4.68 & -0.10 & 0.15 & $ 0.01$ & 140 \\
& 1.7 & 17.94 & -0.06 & 0.13 & $ 0.01$ & 150 \\
& 3.0 & 14.04 & -0.04 & 0.08 & $ 0.01$ & 170 \\ \hline
{\bf R Aur} & -7.5 & 12.48 & 0.0 & 0.0 & $ 0.01$ & 60 \\
& -5.6 & 31.56 & -0.09 & 0.27 & $ 0.01$ & 90 \\
& -3.7 & 36.00 & -0.13 & 0.34 & $ 0.01$ & 85 \\
& -1.2 & 38.58 & -0.07 & 0.16 & $ 0.01$ & 90 \\
& -0.1 & 13.14 & -0.12 & 0.27 & $ 0.01$ & 90 \\ \hline
{\bf RU Aur} & -38.2 & 1.85 & 0.08 & 0.15 & 0.02 & 120 \\
& -34.9 & 1.05 & -0.12 & 0.40 & 0.02 & 85 \\
& -33.2 & 0.25 & -0.22 & 0.40 & 0.02 & 40 \\
& -27.1 & 0.2 & 0.0 & 0.35 & 0.02 & 175 \\ \hline
{\bf TX Cam} & 5.0 & 11.52 & -0.06 & 0.16 & $ 0.02$ & 130 \\
& 6.1 & 13.68 & -0.03 & 0.08 & $ 0.02$ & 80 \\
& 8.0 & 23.46 & 0.04 & 0.20 & 0.01 & 175 \\
& 10.1 & 117.06 & -0.01 & 0.17 & 0.01 & 175 \\
& 11.3 & 26.46 & 0.03 & 0.06 & 0.01 & 170 \\
& 13.2 & 22.62 & -0.03 & 0.05 & $ 0.02$ & 50 \\
& 14.9 & 12.78 & 0.02 & 0.18 & $ 0.02$ & 185 \\ \hline
{\bf V Cam} & 3.8 & 3.12 & -0.04 & 0.22 & 0.01 & 0 \\
& 7.6 & 12.72 & 0.0 & 0.04 & 0.01 & 160 \\
& 9.0 & 9.24 & 0.02 & 0.05 & 0.01 & 0 \\ \hline
{\bf R Cnc} & 9.2 & 2.70 & 0.12 & 0.26 & $ 0.01$ & 100 \\
& 10.3 & 5.16 & -0.02 & 0.04 & $ 0.01$ & 160 \\
& 13.7 & 56.46 & 0.02 & 0.17 & $ 0.01$ & 90 \\
& 15.8 & 33.66 & -0.01 & 0.06 & $ 0.01$ & 180 \\ \hline
{\bf W Cnc} & 33.9 & 21.18 & 0.13 & 0.32 & 0.01 & 40 \\
& 41.9 & 2.82 & -0.03 & 0.12 & 0.01 & 150 \\ \hline
\end{tabular}}
\end{table}
\normalsize
\addtocounter{table}{-1}
\scriptsize
\begin{table} [htb]
\caption{ \label{table} (-continued). Derived parameters of the different components of the SiO
maser emission profile for each star. Only the well identified components are given (distinct peak or strong wing emission separated from the bulk emission). Note that the polarization is fractional. The $\delta P$ is the rms derived from the $p_C$ plot.}
{\begin{tabular}{l|c|c|c|c|c|c} \hline
{\bf Source} & v$_{LSR}$ & F$_{\nu}$ & p$_{c}$ & p$_{L}$ & $\delta$p & $\chi$ \\
& [\kms] & [Jy] & & & & [$^{\circ}$] \\ \hline \hline
{\bf VY CMa} & 5.2 & 553.4 & -0.01 & 0.02 & $ 0.05$ & 125 \\
& 8.5 & 973.3 & 0.0 & 0.01 & $ 0.05$ & 160 \\
& 11.3 & 1086.7 & 0.04 & 0.06 & $ 0.05$ & 110 \\
& 14.9 & 559.0 & -0.02 & 0.03 & $ 0.05$ & 80 \\
& 17.6 & 553.0 & -0.03 & 0.08 & $ 0.05$ & 160 \\
& 19.5 & 769.2 & 0.0 & 0.0 & $ 0.05$ & 160 \\
& 22.7 & 1569.0 & -0.04 & 0.13 & $ 0.05$ & 0 \\
& 38.7 & 207.3 & -0.01 & 0.01 & $ 0.05$ & 150 \\ \hline
{\bf S CMi} & 50.1 & 3.61 & 0.07 & 0.36 & 0.01 & 50 \\
& 52.1 & 11.42 & 0.03 & 0.16 & 0.01 & 50 \\
& 54.9 & 3.89 & 0.0 & 0.0 & 0.02 & 75 \\ \hline
{\bf R Cas} & 25.4 & 600.0 & 0.06 & 0.20 & $ 0.01$ & 25 \\
& 27.1 & 780.1 & 0.03 & 0.09 & $ 0.01$ & 70 \\
& 28.4 & 419.6 & -0.02 & 0.08 & $ 0.01$ & 90 \\ \hline
{\bf S Cas} & -32.5 & 1.38 & 0.0 & 0.0 & 0.01 & 85 \\
& -30.0 & 11.52 & -0.32 & 0.52 & 0.01 & 110 \\
& -28.3 & 2.94 & 0.12 & 0.17 & 0.01 & 20 \\
& -27.2 & 4.98 & 0.05 & 0.05 & 0.01 & 20 \\ \hline
{\bf T Cas} & -11.2 & 1.62 & 0.11 & 0.27 & $ 0.01$ & 110 \\
& -8.7 & 11.70 & -0.02 & 0.18 & $ 0.01$ & 100 \\
& -5.1 & 5.82 & 0.08 & 0.12 & $ 0.01$ & 140 \\ \hline
{\bf T Cep} & -2.5 & 63.00 & $\pm$0.03 & 0.05 & $ 0.01$ & 100 \\
&-0.5 & 117.42 & 0.04 & 0.16 & $ 0.01$ & 140 \\
& 0.8 & 36.00& -0.04 & 0.05 & $ 0.01$ & 105\\ \hline
{\bf R Com} & -4.4 & 7.50 & -0.04 & 0.10 & $ 0.01$ & 80 \\
&-3.3 & 8.82 & 0.07 & 0.27 & $ 0.01$ & 20 \\
& -1.4 & 3.42 & 0.02 & 0.35 & $ 0.01$ & 50 \\ \hline
{\bf S Crb} & 0.8 & 41.40 & 0.07 & 0.17 & $ 0.01$ & 75 \\
& 2.0 & 64.44 & 0.0 & 0.22 & $ 0.01$ & 70 \\
& 4.9 & 28.62 & 0.0 & 0.35 & $ 0.01$ & 160 \\ \hline
{\bf R Crt} & 4.5 & 4.02 & -0.07 & 0.13 & 0.01 & 100 \\
& 10.4 & 36.84 & 0.0 & 0.0 & 0.01 & 50 \\
& 16.4 & 4.74 & -0.08 & 0.15 & 0.01 & 120 \\ \hline
{\bf $\chi$ Cyg} & 7.1 & 50.16 & -0.19 & 0.42 & $ 0.01$ & 75 \\
& 10.0 & 206.1 & 0.11 & 0.32 & $ 0.01$ & 140 \\
& 14.6 & 36.72 & 0.0 & 0.0 & $ 0.01$ & 190 \\ \hline
{\bf UX Cyg} & -1.3 & 9.12 & -0.04 & 0.08 & $ 0.01$ & 100 \\
& -0.5 & 21.96 & 0.01 & 0.06 & $ 0.01$ & 110 \\
& 0.8 & 44.22 & 0.02 & 0.06 & $ 0.01$ & 175 \\ \hline
{\bf R Hya} & -11.2 & 31.14 & 0.09 & 0.34 & 0.01 & 135 \\
& -10.0 & 67.08 & 0.10 & 0.34 & 0.01 & 140 \\
& -8.9 & 182.82 & 0.02 & 0.06 & 0.01 & 150 \\
& -6.6 & 146.22 & 0.05 & 0.25 & 0.01 & 130 \\ \hline
{\bf W Hya} & 37.8 & 151.20 & 0.02 & 0.07 & 0.005 & 70 \\
& 41.4 & 934.38 & $\pm$0.02 & 0.03 & 0.005 & 150 \\
& 44.8 & 202.02 & 0.06 & 0.15 & 0.005 & 100 \\ \hline
{\bf X Hya} & 21.8 & 4.26 & 0.01 & 0.70 & 0.01 & 80 \\
& 23.6 & 3.18 & 0.03 & 0.25 & 0.01 & 60 \\
& 28.9 & 23.28 & -0.01 & 0.08 & 0.01 & 170 \\ \hline
{\bf R Leo} & -1.4 & 195.30 & 0.10 & 0.30 & $ 0.01$ & 140 \\
& 0.5 & 1110.6 & -0.09 & 0.36 & $ 0.01$ & 80 \\ \hline
\end{tabular}}
\end{table}
\normalsize
\addtocounter{table}{-1}
\scriptsize
\begin{table} [htb]
\caption{ \label{table} (-continued). Derived parameters of the different components of the SiO
maser emission profile for each star. Only the well identified components are given (distinct peak or strong wing emission separated from the bulk emission). Note that the polarization is fractional. The $\delta P$ is the rms derived from the $p_C$ plot.}
{\begin{tabular}{l|c|c|c|c|c|c} \hline
{\bf Source} & v$_{LSR}$ & F$_{\nu}$ & p$_{c}$ & p$_{L}$ & $\delta$p & $\chi$ \\
& [\kms] & [Jy] & & & & [$^{\circ}$] \\ \hline \hline
{\bf W Leo} & 42.7 & 2.04 & 0.0 & 0.18 & 0.05 & 170 \\
& 44.6 & 2.76 & 0.0 & 0.10 & 0.05 & 100 \\
& 47.9 & 3.30 & 0.0 & 0.20 & 0.05 & 170 \\
& 54.2 & 4.62 & 0.08 & 0.42 & 0.02 & 35 \\ \hline
{\bf R LMi} & 0.3 & 61.98 & 0.0 & 0.02 & 0.01 & 25 \\
& 2.2 & 52.80& 0.12 & 0.36 & 0.01 & 0 \\
& 3.5 & 34. 62 & 0.06 & 0.20 & 0.01 & 15 \\
& 4.5 & 14.16 & 0.04 & 0.08 & 0.01 & 20 \\
& 6.2 & 10.56 & 0.05 & 0.20 & 0.01 & 80 \\ \hline
{\bf T Lep} & -32.0 & 15.48 & -0.11 & 0.34 & $ 0.01$ & 10 \\
& -30.1 & 10.74 & -0.06 & 0.20 & $ 0.01$ & 20 \\
& -27.1 & 42.48 & 0.03 & 0.10 & $ 0.01$ & 95 \\
& -25.4 & 7.80 & 0.10 & 0.14 & $ 0.01$ & 75 \\ \hline
{\bf RS Lib} & 3.2 & 7.62 & 0.01 & 0.08 & 0.01 & 150 \\
& 6.5 & 35.94 & -0.10 & 0.15 & 0.01 & 150 \\
& 9.1 & 12.24 & -0.06 & 0.20 & 0.01 & 150 \\ \hline
{\bf Ap Lyn} & -25.1 & 9.90 & 0.02 & 0.37 & $ 0.01$ & 185 \\
& -23.0 & 36.36 & 0.12 & 0.65 & $ 0.01$ & 165 \\
& -20.9 & 9.24 & 0.03 & 0.20 & $ 0.01$ & 150 \\ \hline
{\bf U Lyn} & -14.0 & 12.00 & -0.02 & 0.04 & 0.01 & 45 \\
& -11.7 & 30.78 & 0.0 & 0.08 & 0.01 & 40 \\
& -4.5 & 7.19 & -0.13 & 0.24 & 0.01 & 80 \\ \hline
{\bf GX Mon} & -10.4 & 29.64 & 0.12 & 0.38 & $ 0.01$ & 100 \\
& -8.5 & 21.60 & -0.03 & 0.08 & $ 0.01$ & 20 \\
& -6.4 & 10.38 & 0.07 & 0.20 & $ 0.01$ & 120 \\
& -4.9 & 7.92 & -0.02 & 0.07 & $ 0.01$ & 160 \\
& -3.0 & 4.98 & 0.01 & 0.02 & $ 0.01$ & 55 \\ \hline
{\bf SY Mon} & -59.6 & 3.01 & 0.12 & 0.58 & 0.02 & 75 \\
& -56.0 & 2.52 & -0.05 & 0.15 & 0.02 & 30 \\ \hline
{\bf V Mon} & 2.0 & 8.58 & 0.05 & 0.10 & 0.02 & 170 \\
& 4.8 & 3.12 & 0.05 & 0.14 & 0.02 & 80 \\ \hline
{\bf U Ori} & -42.4 & 12.42 & 0.05 & 0.15 & 0.01 & 80 \\
& -39.9 & 46.26 & -0.04 & 0.16 & 0.01 & 50 \\
& -37.8 & 13.08 & 0.02 & 0.06 & 0.01 & 145 \\
& -34.2 & 32.40 & 0.13 & 0.40 & 0.01 & 170 \\ \hline
{\bf RR Per} & 5.1 & 10.81 & 0.03 & 0.14 & 0.01 & 165 \\
& 7.5 & 29.46 & 0.11 & 0.43 & 0.01 & 175 \\
& 8.6 & 27.78 & 0.03 & 0.30 & 0.01 & 185 \\
& 11.1 & 13.02 & 0.10 & 0.34 & 0.01 & 180 \\ \hline
{\bf S Per} & -47.4 & 7.02 & -0.03 & 0.03 & 0.02 & 75 \\
& -44.9 & 31.62 & 0.15 & 0.04 & 0.05 & 190 \\
& -43.3 & 35.70 & 0.01 & 0.04 & 0.05 & 190 \\
& -39.2 & 71.16& 0.0 & 0.03 & 0.05 & 150 \\
& -36.0 & 37.79 & $\pm$0.03 & 0.06 & 0.02 & 80 \\ \hline
{\bf QX Pup} & 27.3 & 11.22 & 0.02 & 0.15 & $ 0.01$ & 120 \\
& 29.4 & 6.84 & 0.0 & 0.03 & $ 0.01$ & 70 \\
& 35.3 & 9.12 & 0.17 & 0.41 & $ 0.01$ & 120 \\
& 41.6 & 2.70 & 0.04 & 0.07 & $ 0.01$ & 10 \\ \hline
{\bf Z Pup} & 3.9 & 40.98 & 0.07 & 0.23 & 0.01 & 140 \\ \hline
{\bf VX Sgr} & -6.1 & 25.98 & 0.0 & 0.0 & $ 0.005$ & 140 \\
& 0.9 & 89.16 & 0.01 & 0.02 & $ 0.005$ & 130 \\
& 3.4 & 122.39 & -0.02 & 0.05 & $ 0.005$ & 10 \\
& 5.7 & 178.38 & -0.01 & 0.04 & $ 0.005$ & 10 \\
& 10.1 & 114.61 & -0.03 & 0.10 & $ 0.005$ & 145 \\
& 14.1 & 53.34 & 0.01 & 0.01 & $ 0.005$ & 145 \\
& 16.2 & 80.04 & 0.01 & 0.02 & $ 0.005$ & 160 \\ \hline
\end{tabular}}
\end{table}
\normalsize
\addtocounter{table}{-1}
\scriptsize
\begin{table} [htb]
\caption{ \label{table} (-continued). Derived parameters of the different components of the SiO
maser emission profile for each star. Only the well identified components are given (distinct peak or strong wing emission separated from the bulk emission). Note that the polarization is fractional. The $\delta P$ is the rms derived from the $p_C$ plot.}
{\begin{tabular}{l|c|c|c|c|c|c} \hline
{\bf Source} & v$_{LSR}$ & F$_{\nu}$ & p$_{c}$ & p$_{L}$ & $\delta$p & $\chi$ \\
& [\kms] & [Jy] & & & & [$^{\circ}$] \\ \hline \hline
{\bf AH Sco} & -11.4 & 26.69 & 0.05 & 0.06 & $ 0.005$ &120 \\
& -10.0 & 48.31 & 0.01 & 0.02 & $ 0.005$ &120 \\
& -7.2 & 78.78 & 0.0 & 0.0 & $ 0.005$ &150 \\
& -4.7 & 54.74 & 0.0 & 0.02 & $ 0.005$ &20 \\
& -2.8 & 32.99 & -0.02 & 0.05 & $ 0.005$ &170 \\ \hline
{\bf RR Sco} & -33.9 & 4.02 & 0.10 & 0.14 & 0.02 & 65 \\
& -30.2 & 9.48 & 0.0 & $\pm$0.07 & 0.02 & 85 \\
& -26.5 & 11.16 & 0.04 & 0.10 & 0.02 & 90 \\
& -23.2 & 2.82 & 0.06 & 0.16 & 0.02 & 55 \\ \hline
{\bf R Ser} & 27.8 & 21.61 & -0.07 & 0.30 & $ 0.01$ & 40 \\ \hline
{\bf S Ser} & 18.7 & 11.22 & 0.04 & 0.09 & $ 0.01$ & 180 \\
& 20.9 & 16.92 & 0.0 & 0.06 & $ 0.01$ & 100 \\ \hline
{\bf WX Ser} & 4.1 & 20.34 & -0.05 & 0.15 & $ 0.01$ & 140 \\
& 6.6 & 12.18 & 0.04 & 0.15 & $ 0.01$ & 80 \\
& 9.8 & 8.41 & -0.12 & 0.30 & $ 0.01$ & 140 \\ \hline
{\bf IK Tau} & 31.6 & 46.74 & 0.04 & 0.14 & 0.01 & 180 \\
& 34.7 & 288.24 & 0.05 & 0.16 & 0.01 & 170 \\
& 36.8 & 53.04 & 0.07 & 0.18 & 0.01 & 140 \\
& 39.6 & 27.84 & 0.13 & 0.36 & 0.01 & 125 \\ \hline
{\bf R Tau} & 9.5 & 3.78 & -0.10 & 0.15 & $ 0.01$ & 60 \\
& 11.8 & 19.51 & 0.06 & 0.13 & $ 0.01$ & 150 \\
& 13.1 & 10.26 & 0.07 & 0.19 & $ 0.01$ & 140 \\
& 14.7 & 7.68 & -0.09 & 0.31 & $ 0.01$ & 55 \\
& 16.4 & 6.24& -0.08 & 0.19 & $ 0.01$ & 75 \\
& 19.0 & 3.24 & -0.20 & 0.44 & $ 0.01$ & 55 \\ \hline
{\bf RX Tau} & -44.0 & 3.36 & -0.06 & 0.10 & 0.02 & 120 \\
& -41.0 & 10.98 & -0.01 & 0.01 & 0.02 & 100 \\
& -38.9 & 2.94 & 0.04 & 0.15 & 0.02 & 170 \\ \hline
{\bf R Tri} & 56.7 & 13.62 & -0.05 & 0.24 & $ 0.01$ & 105 \\ \hline
{\bf R UMi} & -6.1 & 1.98 & 0.0 & 0.25 & 0.05 & 110 \\ \hline
{\bf S UMi} & -44.4 & 21.54 & 0.0 & 0.0 & 0.005 & 30 \\
& -43.3 & 30.66 & 0.0 & 0.01 & 0.005 & 20 \\
& -40.8 & 24.24 & -0.02 & 0.10 & 0.005 & 10 \\
& -40.0 & 27.66 & 0.02 & 0.07 & 0.005 & 10 \\ \hline
{\bf R Vir} & -26.8 & 0.84 & 0.15 & 0.30 & 0.05 & 170 \\
& -25.7 & 2.16 & -0.10 & 0.15 & 0.02 & 10 \\ \hline
{\bf RS Vir} & -15.7& 2.64 & -0.11 & 0.18 & 0.01 & 0 \\
& -13.3 & 10.62 & -0.08 & 0.32 & 0.01 & 150 \\
& -12.5 & 8.46 & 0.0 & 0.20 & 0.01 & 0 \\
& -9.1 & 7.08 & -0.06 & 0.52 & 0.01 & 160 \\ \hline
{\bf RT Vir} & 9.1 & 1.20 & 0.0 & 0.02 & $ 0.02$ & 70 \\
& 13.8 & 2.22 & 0.12 & 0.19 & $ 0.02$ & 20 \\
& 15.7 & 3.12 & 0.12 & 0.20 & $ 0.02$ & 140 \\
& 18.0 & 7.98 & 0.0 & 0.06 & $ 0.02$ & 130 \\
& 20.7 & 3.48 & -0.10 & 0.12 & $ 0.02$ & 120 \\
& 22.6 & 3.03 & -0.06 & 0.11 & $ 0.02$ & 60 \\
& 27.7 & 2.34 & -0.04 & 0.10 & $ 0.02$ & 40 \\ \hline
{\bf S Vir} & 10.3 & 13.62 & -0.04 & 0.27 & $ 0.01$ & 180 \\
& 12.2 & 7.79 & 0.03 & 0.06 & $ 0.01$ & 105 \\
& 13.7 & 22.52 & -0.05 & 0.19 & $ 0.01$ & 130 \\ \hline
\end{tabular}}
\end{table}
\normalsize
\section{Results: Polarization Study}
\subsection{Individual results}
Values of the polarization level presented here (see Table 2) are those measured for the different components within the SiO
maser emission profile for each star. Examples are given in Fig.2 for a few stars. The complete Figure 2 with all the observations is available in electronic form at http://www.edpscience.org. Only the well identified components are considered (distinct peaks or strong wing emission well separated from the bulk emission, according to the noise). Some interesting cases are briefly presented below.
Some profiles show isolated emission red$/$blue-shifted from the main emission which are more strongly circularly polarized (e.g. IRAS 18204-1344). These peculiar characteristics imply a different spatial origin for the main and higher/lower velocity components. Sometimes the circular polarization is regularly varying across the profile (e.g. R Leo), but sometimes not.
In T Lep, the SiO emission shows two peaks linked by a plateau; the circular polarization is linearly varying across the profile from -11 to 10 \% (see Fig. 2). Several objects (e.g. S UMi, IK Tau) show the same $p_C$ pattern.
The IR late-type source IRAS18158-1527 exhibits a complex profile with several well defined components, each of them differently polarized indicating a complex maser structure with probably different maser spots contributing to the whole emission. The red wing emission is highly polarized (43 \%). Such a complex multi-component maser line profile and "semi-circle", convex, $p_C$ pattern appear to be characteristic of SR objects (see other similar objects in our sample and R Crt in Fig. 2). Nevertheless, the circular polarization pattern observed in the Mira star U Lyn is a convex profile as encountered in SR objects.
One of the most studied Mira star is R Leo. The profile is made of a strong emission with a blue broad line wing. Main and linewing emissions are strongly polarized (respectively negative and positive $p_C \sim 9-10$ \%). R Leo is a very well studied object exhibiting a bipolar jet throughout its envelope. The clear symmetry observed between the positive and negative circular polarization patterns in the main and wing line emissions suggests that the maser emission comes from the jet lobes. We note that the Mira star RS Lib exhibits an emission and polarization pattern similar to that observed in R Leo. R Leo and RS Lib may have the same spatial structure.
\addtocounter{figure}{-1}
\subsection{Analysis}
The circular polarization level in several of our objects has already been measured by Kemball \& Diamond (1997) or Barvainis, McIntosh \& Predmore (1987). For TX Cam and W Hya, our results are consistent with previous observations:
\begin{itemize}
\item in TX Cam, the bulk of the emission is weakly circularly polarized while its wings show $p_C \sim 3-6 \%$ in good agreement with the VLBI observations of Kemball \& Diamond in 1997 who derived an average value of $p_C \sim 3-5 \%$;
\item in the Semi-Regular object W Hya, the central emission is weakly polarized ($\pm$ 2 \%), while the wings and secondary peaks show $p_C$= 2-6 \%), which is consistent with the 5 \% of Barvainis, McIntosh \& Predmore (1987).
\end{itemize}
On the contrary, the circular polarization level we derive in VY CMa, R Cas, R Leo, and VX Sgr is different from levels measured by Barvainis, McIntosh \& Predmore (1987), respectively 1-4, 2-6, 9-10 and less than 3 \% while they found respectively 6.5, 1.5, 2.4 and 8.7 \%.
This difference is significant and may be due to variability over the fifteen intervening years. Indeed time variability of the polarization remains an open question in the field. Glenn \etal (2003) have shown that the individual maser feature lifetime ranges from a few months or less to more than 2 years, i.e. the characteristic time over which the $Q$ and $U$ spectral features persist. The average linear polarization is 23 \% in Glenn \etal sample with a typical dispersion of 7\%. Cotton et al. (2004) have comparable epoch spacing and do not conclude on the variability. Our observations were repeated at intervals of a few months (August and November 1999) and the polarization tends to remain stable between the two epochs.
We emphasize that we cannot spatially distinguish with a single dish radiotelescope between the different maser spots producing the SiO profile (various masers spots contribute in the various features observed at a given velocity). The whole SiO maser emission region, hence all the maser cells, lie within the 29 arcseconds of the 30m (but not necessarily with a uniform distribution) while the SiO emission covers less than 40 milliarseconds in TX Cam (Kemball \& Diamond 1997) and thus everything is beam averaged. This means that any conclusion on the geometry of the objects observed here would be much uncertain. Only global trends or global geometry can be discussed. One of the consequences of this spatial resolution problem is that if the polarization vectors are distributed isotropically around the object, the average polarization level that we measure is zero, even if the maser emission produced in each SiO cell is well polarized.
A global analysis of our data in Table 2 shows the following. We find that $p_L$ varies between 0 and 70 \%, and $p_C$ between 0 and $\pm$43 \%. Hence, polarization vectors are not distributed isotropically. Emission from Mira-type objects clearly tends to have a relatively high linear ( ${<p_L>}_{Mira} \simeq 30$\%, ${<p_L>}_{SR} \simeq 11$\%) and circular polarization (${<p_C>}_{Mira} \simeq 9$\%, ${<p_C>}_{SR} \simeq 5$\%). Note that the emission from the PN QX Pup is highly polarized, and, on the contrary, maser emission from supergiants shows very weak polarization ($<p_L>=5$\%, $<p_C>=2$\%), with the exception of one maser component in S Per. Moreover, all observations show that the polarization level varies across the maser line profile (see Fig. 2), i.e. the different spectral components of the maser emission producing the profile are coming from different localizations in the SiO shell and have different polarization levels. The highest polarization level for one object can be encountered either in the main peak, or in the other components.
Semi-Regular objects (RU Aur, R Crt, W Hya, S Per, AH Sco, R UMi, RT Vir) have a common circular polarization pattern with the central main emission unpolarized and other peak emission or wings being strongly polarized: a characteristic "semi-circle" (i.e. convex shape) pattern for $p_C$ is observed (see R Crt in Fig. 2). The infrared late-type star IRAS18158-1527 exhibits a similar pattern, thus suggesting that this star is a semi-regular.
A group of objects (W And, NV Aur, T Cas, R Com, T Lep, IK Tau, S Ser, S UMi) shows approximately the same $p_C$ pattern (see T Lep in Fig. 2); the circular polarization varies linearly across the line profile from a positive value to a negative one (or the contrary). The only common spectral characteristic of the SiO emission from these stars is the presence of an plateau-like emission on top of which the narrow emission peaks are located.
\section{Discussion}
In this section, we first discuss our source sample in the frame of the 2-color diagram. Then, we briefly summarize the existing SiO maser polarization theories. Finally, we discuss our data set in this context and estimate the stellar magnetic field strength.
\subsection{2-color diagram}
Stars of our sample can be plotted in a [12]-[25], [25]-[60] color-color diagram (van der Veen \& Habing 1988; [12], [25] and [60] stand respectively for 12, 25 and 60 microns IRAS-fluxes). This diagram is partitioned into several regions (see Fig. 3) defined by van der Veen \& Habing as follows:
Region I, oxygen-rich non variable stars without circumstellar shells;
Region II, variable stars with young O-rich circumstellar shells; Region IIIa,
variable stars with more evolved O-rich circumstellar shells; Region IIIb,
variable stars with thick O-rich circumstellar shells; Region IV, variable stars
with very thick O-rich circumstellar shells; Region V, Planetary Nebulae and non-variable stars with very cool envelopes; Region VIa, non variable stars
with relatively cold dust at large distance; Region VIb, variable stars with
relatively hot dust close to the star and relatively cold dust at large
distance; Region VII, variable stars with more evolved C-rich circumstellar
shells.
On Fig. 3 are represented the linear and circular polarization level for the main SiO emission component from each star. Most of the objects in our sample fall in regions II and IIIa and do not show particular characteristics, except for S Cas (an S-type star) where the circular polarization is high. Mira-type stars are in regions I, II, IIIa and VII. IR late-type objects are in VIb and VII. The SRa semi-regular variable W Hya is in I, while the SRb stars lie in IIIa (RU Aur, RT Vir, R Crt), II (R UMi), and Src in IIIa (S Per) and VII (AH Sco). The Red Supergiants, VY CMa and VX Sgr, and the IR supergiant IRAS 18204-1344 lie in VII. Objects in Region VII do not exhibit strong polarization compared to other objects, perhaps because of their more C-rich circumstellar shells (e.g. AU Aur) or because of the presence of hot dust close to the star implying less SiO abundance and thus weaker emission, making the polarization measurement less significant. The presence of hot dust may also influence the pumping of the SiO molecules and thus the polarization level; the optically thick, hence isotropic, radiation field of hot dust can assist the collisional pumping. This could apply to UX Cyg (an irregular variable), IRAS 18055-1433 and IRAS 18158-1527 in region VIb. QX Pup in region V is a PN and exhibits strong polarized emission. Note that IRAS 18055-1433 and IRAS 18158-1527 show very strong circular polarization in their line wings. We may conjecture here that wing emission comes from more outer layers than those where the main line is excited (Herpin \etal 1998); as a consequence, the SiO cells giving rise to wing emission are less influenced by the presence of hot dust (hot dust preferably lies in the inner layers).
\subsection{The SiO maser polarization theory}
Since SiO is non-paramagnetic, the Zeeman splitting $g\nu_B$ ($g$ is the LandЋ factor) is much less than the Doppler width. Moreover, the degree of saturation is the ratio of the rate $R$ for stimulated emission to the loss rate $\Gamma$ (usually $\Gamma$ is approximated by the inverse radiation lifetime for a vibrational transition, $\Gamma \simeq 5$ s$^{-1}$ for SiO masers, Wiebe \& Watson 1998). Hence, if $R\geq \Gamma$, the maser is saturated. In fact, in the Orion case Plambeck \etal (2003) show that, despite the radiation beam angle is unknown, the 86 GHz SiO maser is saturated. The maser is saturated if the angle averaged intensity $J= \frac {I \Omega_b} {4\pi}$ ($\Omega_b$ is the beaming solid angle) is larger than the saturation intensity $J_S$; $J_S$ is a theoretical quantity. The saturation depends on the angle into which the radiation is beamed, but this angle is unknown (Watson \& Wyld 2001), thus $J$ cannot be directly measured (even if $I$ is measurable when the maser is resolved, the beaming angle $\Omega$ is not an observable).
For more than one decade, two schools have come up against each other to explain SiO maser emission. SiO polarization theory is described in: (i) Watson (e.g. Watson \& Wyld 2001, Wiebe \& Watson 1998, Nedoluha \& Watson 1994); (ii) Elitzur (2002, 1998, 1996, 1994).
The main difference between the two approaches rests in the pumping mechanisms. While anisotropic pumping associated with a weak field produces high $p_L$ and quite significant $p_C$ in Watson's model, a strong magnetic field is necessary with the more classical pumping mechanisms used in Elitzur's model. Details about both models can be found in the Elitzur's review (2002).
We may summarize the main characteristics of Watson's model as follows:
\begin{itemize}
\item non-Zeeman effect;
\item anisotropic pumping;
\item no direct relation between $p_{C}$ and $B$. An estimation of B can only be derived through complete calculation of the radiative transfer. Nevertheless, when maser saturation is not important, the "thermal" spectral line equation $\frac {V}{\delta I/\delta v}=\alpha B \cos \theta$ is applicable (Fiebig \& G\"usten 1989); $I$ is the intensity with respect to Doppler velocity $v$, $\theta$ is the angle between $B$ and the line of sight, $\alpha$ is a constant;
\item the Zeeman splitting parameter $g\Omega$ (in frequency units $g\Omega = 1.5 B[mG] s^{-1}$ for SiO masers) is $\simeq R$;
\item saturated or unsaturated maser (saturated maser increases $p_C$);
\item linear correlation between $p_L$ and $p_C$, and high $p_L$ is needed;
\item intensity dependent circular polarization;
\item $B$ of a few 10 mG varying as $r^{-2,3}$ throughout the envelope.
\end{itemize}
In contrast with Watson's work, Elitzur's model is based on the Zeeman effect and the exponential maser growth in the unsaturated phase; the polarization characteristics are preserved as the radiation is amplified into the saturated regime. This model was improved several times (Elitzur 1994, 1996, 1998) and takes into account the anisotropic pumping. The magnetic field generates circular polarization and the main pumping mechanism for the SiO maser is a "classical" radiative-collisional process. For saturated masers, a direct relation between $p_C$ and $B$ is obtained from simple calculations. The ratio $x_B$ of the Zeeman splitting $\Delta \nu_B$ to the Doppler linewidth $\Delta \nu_D$, can be determined (Elitzur 1996) from $v_{peak}$, the ratio of the Stokes parameter $V/I$ at a given peak feature:
\begin{equation}
\label{ }
x_B=\frac {3\sqrt{2}}{16} ~ v_{peak} \cos \theta
\end{equation}
Following Barvainis, Mc Intosh \& Predmore (1987) and Elitzur (1996) we arbitrarily take $\theta \simeq 45^{\circ}$
\begin{equation}
\label{ }
\Rightarrow x_B= \frac{3}{16} ~ v_{peak}
\end{equation}
\begin{equation}
\label{ }
\Rightarrow x_B= 0.1875 ~10^{-2} ~m_C
\end{equation}
where $m_C$ is the polarization fraction in percentage (i.e. 100 $p_C$).
Moreover:
\begin{equation}
\label{ }
x_B= 14 g \lambda \frac {B}{\Delta v_D}
\end{equation}
where $g$ is the Land\'e factor with respect to the Bohr magneton, $\lambda$ the transition wavelength in cm, $B$ the field in Gauss and $\Delta v_D$ the Doppler width in km$s^{-1}$. Thus,
\begin{equation}
\label{ }
B= 0.1875 ~10^{-2} ~\frac {\Delta v_D}{14 g \lambda} ~m_C
\end{equation}
With $g\simeq 10^{-3}$, $\Delta v_D =1$ \kms and $\lambda=0.2877$ cm, we derive:
\begin{equation}
\label{ }
B\simeq 0.46 ~m_C
\end{equation}
This model predicts that there is no correlation between $p_C$ and $p_L $ (see also Watson \& Wyld 2001); such a mechanism leads to an inferred magnetic field of a few Gauss to 10 Gauss for the SiO maser zone, the field hence varying as $r^{-1,2}$ across the envelope.
\subsection{Theory against present observations}
The relevance of the two different polarization theories can be assessed via a few observational checks:
\begin{itemize}
\item dependence of circular polarization on intensity;
\item linear correlation between $p_C$ and $p_L$;
\item Zeeman effect, i.e. spectral shape of the Stokes parameter $V$;
\item coherence of B strength values inferred from OH and \water observations.
\end{itemize}
As we do not resolve the maser emission into individual spatial components, our current data set is biased by the beam averaging. Therefore, some effects cannot be tested with our data. In the Zeeman effect case, the spectral shape of the Stokes parameter $V$ must follow an antisymmetric {\em S} curve with sharp reversal at line center (Elitzur 1996). Unfortunately the doppler width is less than the resolution of the observations and we cannot conclude.
We may look for any correlation between $p_C$ and $p_L$. As shown in Figs. 2 and 3, it first appears that in all cases $p_L$ is larger than $p_C$ as is predicted by all models. More precisely, $p_C$ is noticeable if and only if $p_L$ is high. If we plot the values of $p_C$ and $p_L$ derived for all maser components in this work (Fig. 4) and make a regression fit to our data we obtain $p_C \simeq 0.015 + 0.25 ~p_L$. The circular polarization level tends to vary approximately linearly with $p_L$ in agreement with the fact that no $p_C$ is detected towards sources with a marginal $p_L$ detection. Note that in Fig. 4, four objects, S Per, IRAS18055-1433, S Cas and $\chi$ Hya, do not follow the same general trend observed for the rest of the sample. (The case of the Supergiant S Per, however, is an exception as it exhibits substantial $p_C$ while $p_L<p_C$.) This observation may in fact favour Watson's model. It must be stressed again that the beam-averaged polarization that we measure makes any conclusion uncertain. In fact, due to this averaging, we should observe no correlation at all, even if such one would exist !
As mentioned earlier (see Section 3.2) we observe relatively high circular polarization rates in several stars (${<p_C>}_{Mira} \simeq 9$\%, ${<p_C>}_{SR} \simeq 5$\%). These values are larger than those predicted from Watson's model (e.g. Nedoluha \& Watson 1994). We also have not been able to find any correlation of $p_C$ with total intensity. Finally, although we adopt the Zeeman case to derive the magnetic field strength (see Section 4.5), we cannot conclude firmly from present observations which maser theory prevails for SiO emission.
\subsection{Magnetic field in AGB stars}
A better knowledge of the stellar magnetic field strength is crucial to understand the last stages in the life of an Asymptotic Giant Branch (hereafter AGB) star. These stages are characterized by a high mass loss process driven by the radiation pressure; they are also influenced by the magnetic field (Palen \& Fix 2000, Blackman, Frank \& Welch 2001, and references therein). A strong magnetic field may rule the mass loss geometry; in particular, it could be the cause of a higher or lower mass-loss rate in the equatorial plane (Soker 2002), and thus determine the global shaping of these objects. But, as the direct dynamical effect of the magnetic activity is much lower than that of the wind (although in local spots the magnetic field can be dominant), the role of the magnetic field might be indirect. Moreover, the observed high mass loss rates are hardly explained by a single process and need a combination of several factors such as rotation, the presence of a companion (binary stars, with our without common envelope, exhibiting mass transfer or tidal effects are common; Soker 1997) and a magnetic field. During its quick transition to the PN stage, the AGB star will completely change its geometry: the quasi-spherical object becomes axisymmetrical, point-like symmetrical or even shows higher order symmetries (Johnson \& Jones 1991, Sahai \& Trauger 1998, Balick \& Frank 2002). The classical or generalized {\em Interacting Stellar Winds} (hereafter ISW or GISW) models (Kwok 2000, Soker \& Livio 1989, Morris 1987) try to explain this shaping, but have serious difficulties in producing complicated structures with peculiar jets or ansae (e.g., CRL 2688, Delamarter 2000) and do not fully address the origin of the wind. Furthermore, recent X-ray studies with the Chandra satellite do not completely agree with GISW predictions for temperatures (Guerrero \etal 2001).
Some recent studies tend to demonstrate the importance of the magnetic field in evolved objects. Bujarrabal \etal (2001) show that for 80\% of the PPNe in their sample the fast molecular flows have too high momenta to be powered by radiation pressure only (1000 times larger in some cases) what may be explained by magnetic field. Moreover, X-ray emission found in evolved stars (e.g. H\"unch, Schmitt \& Schr\"oder 1998) may indicate the presence of a hot corona that possibly results from magnetic activity. Very recently, magnetic field was discovered for the first time in central stars of PN (Jordan \etal 2005) and estimated to kiloGauss, much stronger than what we find here from our SiO data in QX Pup.
New models involving the magnetic field have been developed trying to explain the morphology changes of an object during its transition from the AGB stage to the PN stage; B plays the role of a catalyst and of a collimating agent. The most simple models are based on a moderately weak magnetic field alone ($B \simeq 1$ Gauss at the stellar surface, a few $10^{13}$ cm, i.e. at a radiis of a few AU, Soker 1998). The influence of B is stressed by the work of Smith \etal (2001) and Greaves (2002) in VY CMa. But the role of B can only be decisive when its energy density is greater than the radiative pressure, i.e. when B is greater than around 10 G close to the stellar surface in the SiO region (see Soker \& Zaobi, 2002). Arguing that such a strong field may be very unusual, Balick \& Frank (2002) explain that B alone cannot produce the observed structures, and a combination of several factors has thus to be considered (rotation, magnetic field and presence of a companion).
Soker \& Harpaz (1992) first proposed a model with a weak magnetic field ($\leq 1$ G) and included a slow rotation together with the presence of a companion to transform the envelope (and lead for example to the peculiar geometry observed in NGC 6826 or NGC 6543). Even if the star were not binary, the influence of B is probably important locally (Palen \& Fix 2000). A significant magnetic field can form cold spots on the star's surface and a slow rotation of the star can then increase the field strength to build up a dipolar magnetic field varying as $1/r^{3}$ (Matt \etal 2000); such a field is stronger at the equator and may thus lead to an axisymmetrical mass loss.
The main argument against the dominant influence of the magnetic field on the shaping of the circumstellar envelope is that a strong field seems to be necessary to dominate the dynamics of the gas. However, several authors (Pascoli 1985, 1992, 1997, Chevalier \& Luo 1994, Garc\'{\i}a-Segura 1997, Gurzadyan 1997, Delamarter 2000) have demonstrated the strong influence of a reasonable toro\"{\i}dal magnetic field embedded in the normal radiation-driven stellar wind ({\em Magnetic Wind Bubble} theory, hereafter {\em MWB}). This field has a strength between a few Gauss and a few 10 Gauss at a few stellar radii (the SiO region is believed to be at $\sim 10^{14}$ cm or $\sim 7-10$ AU), varies as $1/r^2$, then as $1/r$ at larger radii; therefore $B\sim1$ mG at $10^{16,17}$ cm or 700-7000 AU. These results are confirmed by the simulations of Garc\'{\i}a-Segura, Lopez \& Franco (2001) for the PN He 2-90. Even if the origin of the wind is not explained by these models, it seems clear that a magnetic field is essential to generate fast collimated outflows (Kastner \etal 2003).
There are many models of magnetic jet production and collimation and some, or all of them, are applicable to various star geometries. One most interesting study was performed by Blackman, Frank \& Welch (2001) in which the magnetic field emerges from the AGB stellar core and the resulting 1G field helps to collimate the radiation-driven wind or a stronger, more anisotropic, magnetically driven wind.
\subsection{Magnetic field in our sample}
The exact interpretation, in terms of magnetic field, of our observations depends on the adopted specific SiO maser model (see Sections 4.2 and 4.3). From the current knowledge of the strength of the magnetic field in the OH and \water layers we expect $B_{//}$ of a few Gauss at least in the SiO maser region (see also Kemball \& Diamond 1997), i.e. at 5-10 AU from the central object. This tends to invalidate Watson's model, and furthermore tends to agree with a field varying in $r^{-1,2}$ as predicted by Elitzur's model. However, Vlemmings \etal (2005) measured the circular polarization of the \water maser emission in a few evolved stars with the VLBA observations and showed that the magnetic field is either a solar-type field (with a $r^{-2}$ field strength dependence) or a dipole magnetic field (with a $r^{-3}$ dependence) in their sample.
In the following, we decide to use Elitzur's theory (Zeeman case) to infer magnetic field strength from the circular polarization levels. From equation (12), we thus calculate the mean value of the magnetic field $B_{//}$ for each star and give results in Table 3.
For our sample B$_{//}$ is between 0 and 20 Gauss, with a mean value of 3.5 G. This value combined with the strength of the field in more outer layers of the envelope (OH and \water masers) agrees with a B field variation law in $1/r$, closer to Elitzur's model. As explained in the Introduction and in Section 4.4, B alone can be the main agent to shape the circumstellar envelope if its value is larger than around 10 Gauss. This means that only S Cas, RU Aur, IRAS 18055-1433 and IRAS 18158-1527 may have a {\em magnetic field ruled geometry} ($B> 10$ G in these objects). The rest of our sample shows that B is sufficiently strong to be dominant at this stage of the AGB star evolution, but it should be associated with rotation and the presence of a companion as suggested in models mentioned in Section 4.4.
Despite our B measurements are beam averaged, they suggest in many cases that they are not too much (not orders of magnitude) below the critical value; local B values may exceed in many cases the critical value, and therefore participate in the shaping of the AGB
envelopes. Our estimated values of B are consistent with the {\em MWB} theory (toro\"{\i}dal magnetic field) or the model of Blackman, Frank \& Welch (2001). %
\begin{table} [h]
\caption{ \label{table} Average magnetic field strengths derived from $p_C$.}
{\begin{tabular}{lc} \hline
{\bf Source} & $B_{//}$ (G) \\ \hline
IRAS18055-1433 & 4.6-13.9 \\
IRAS18158-1527 & 3.7-20.0 \\
IRAS18204-1344 & 0-3.2 \\
W And & 0.4-5.9 \\
AU Aur & 0.9-4.2 \\
NV Aur & 1.9-4.6 \\
R Aur & 0-6.0 \\
RU Aur & 0-10.2 \\
TX Cam & 0.4-2.8 \\
V Cam & 0-1.9 \\
R Cnc & 0.4-5.6 \\
W Cnc & 1.4-6.0 \\
VY CMa & 0-1.9 \\
S CMi & 0-3.2 \\
R Cas & 0.9-2.8 \\
S Cas & 0-14.9 \\
T Cas & 0.9-5.1 \\
T Cep & 1.4-1.9 \\
R Com & 0.9-3.2 \\
S Crb & 0-3.2 \\
R Crt & 0-3.7 \\
$\chi$ Cyg & 0-8.8 \\
UX Cyg & 0.4-1.9 \\
R Hya & 0.9-4.6 \\
W Hya & 0.9-2.8 \\
X Hya & 0.4-1.4 \\
R Leo & 4.2-4.6 \\
W Leo & 0-3.7 \\
R LMi & 0-5.6 \\
T Lep & 1.4-5.1 \\
RS Lib & 0.4-4.6 \\
Ap Lyn & 0.9-5.6 \\
U Lyn & 0-6.0 \\
GX Mon & 0.4-5.6 \\
SY Mon & 2.3-5.6 \\
V Mon & 2.3 \\
U Ori & 0.9-6.0 \\
RR Per & 1.4-5.1 \\
S Per & 0-7.0 \\
QX Pup & 0-7.9 \\
Z Pup & 3.2 \\
VX Sgr & 0-1.4 \\
AH Sco & 0-2.3 \\
RR Sco & 0-4.6 \\
R Ser & 3.2 \\
S Ser & 0-1.9 \\
WX Ser & 1.9-5.6 \\
IK Tau & 1.9-6.0 \\
R Tau & 2.8-9.3 \\
RX Tau & 0.4-2.8 \\
R Tri & 2.3 \\
R UMi & 0 \\
S UMi & 0-0.9 \\
R Vir & 4.6-7.0 \\
RS Vir & 0-5.1 \\
RT Vir & 0-5.6 \\
S Vir & 1.4-2.3 \\ \hline
\end{tabular}}
\end{table}
From Elitzur (1996) and our measurements ${<p_C>}_{Mira} \simeq 9$\%, ${<p_C>}_{SR} \simeq 5$\%, we can estimate (see Eq. (9) in Sect. 4.2) that $<x_B>$ is around 0.017 and $9.4$ $10^{-3}$ respectively for Mira-type objects and semi-regular variables. Moreover, according to Elitzur (1996, see Fig.2), there is no stationary physical solutions for propagation at ${\sin}^2 \theta < \frac {1}{3}$ (i.e. at ${\sin}^2 \theta < \frac {1}{3}$ the radiation is not polarized). As $x_B$ is small ($<0.02$), from Fig.2 of Elitzur (1996) we can estimate the volume of phase space in which propagation of linear polarization in a maser is possible or not ($\theta > 35.3^{\circ}$ or $\theta < 35.3^{\circ}$); we then calculate that the probability for a random magnetic axis to be aligned with a given direction (our line of sight) is better than $35.3^{\circ}$. Our present estimate is that around 18 \%. Therefore, Elitzur's model predicts that 18.4 \% of the SiO 86 GHz masers should not be linearly polarized, because such polarized masers cannot propagate if the magnetic field, although weak, is closer than $35.3^{\circ}$ to the line of sight (propagation direction). Hence non polarized maser emissions do not imply no or weak magnetic field. In our sample, roughly 13 \% of the SiO maser components have no detectable or very weak ($<3\%$) polarization.
We looked without success for a possible correlation between the polarization rates and physical parameters such as the known envelope asymmetry, the presence of SiO maser high velocity linewings (see Herpin \etal 1998), or the mass loss rate. If the magnetic field plays an important role in the shaping of the object, one may expect to find a relationship between the strength of $B_{//}$ (thus $p_C$) and the geometry of the object. Unfortunately, no trend is clearly found in our data. Nevertheless it is known that radiative pressure is driving the wind in AGB objects and it is thus not surprising that we find no correlation between the B strength and a known asymmetry in our sample. Of course, our stellar sample would require new observations with sufficient spatial resolution (VLBI) to confirm the present results; the same type of study should also be conducted toward several Proto-PN and PN objects.
\section{Conclusion}
We have made a study of the SiO maser polarization in a representative sample of evolved stars, simultaneously measuring, for the first time, the 4 Stokes parameters. From our measurements we derive the circular and linear polarization levels and shows that, due to the beam averaging of our polarization measurements, we cannot firmly discriminate between the two dominant theories of SiO maser emission. In particular, VLBI observations of our source sample are absolutely necessary to distinguish between Zeeman or non-Zeeman theories. Nevertheless, the magnetic field strength was derived assuming Elitzur's model. $B_{\\}$ varies between 0 and 20 Gauss, with a mean value of 3.5 G. As a consequence, we suggest that the magnetic field plays a significant role in the evolution of these objects. Within the frame of the Zeeman theory the magnetic field could shape or even collimate the gas layers surrounding the AGB objects. Emission from Mira-type objects clearly tends to have a higher linear ( ${<p_L>}_{Mira} \simeq 30$\%, ${<p_L>}_{SR} \simeq 11$\%) and circular polarization (${<p_C>}_{Mira} \simeq 9$\%, ${<p_C>}_{SR} \simeq 5$\%). Basically, if there is a real correlation between $p_C$ and the strength of the magnetic field, this trend may indicate that the magnetic field may be stronger in Mira objects than in Semi-Regular variables (at least in the inner layers of the circumstellar envelope).
To better understand the mechanisms at work with the magnetic field, complementary studies have to be conducted and in particular the presence of a companion has to be investigated in a large sample of objects. Of course VLBI maps of the magnetic field in these stars are essential.
Another important objective is to investigate the evolution of the magnetic field and its influence during the transition from the AGB star phase to the PN stage.
\acknowledgements{The authors are grateful to M. Elitzur for reading and commenting on this paper. We also thank W.D. Watson for his useful comments and suggestions. The authors are indebted to the staff of the IRAM 30m telescope who most efficiently helped during the observations and to R. Mauersberger who closely followed part of these observations. Finally, we also thank the referee for several useful comments.}
\Online
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Title:
An HLLC Solver for Relativistic Flows -- II. Magnetohydrodynamics |
Abstract: An approximate Riemann solver for the equations of relativistic
magnetohydrodynamics (RMHD) is derived. The HLLC solver, originally developed
by Toro, Spruce and Spears, generalizes the algorithm described in a previous
paper (Mignone & Bodo 2004) to the case where magnetic fields are present. The
solution to the Riemann problem is approximated by two constant states bounded
by two fast shocks and separated by a tangential wave. The scheme is
Jacobian-free, in the sense that it avoids the expensive characteristic
decomposition of the RMHD equations and it improves over the HLL scheme by
restoring the missing contact wave.
Multidimensional integration proceeds via the single step, corner transport
upwind (CTU) method of Colella, combined with the contrained tranport (CT)
algorithm to preserve divergence-free magnetic fields. The resulting numerical
scheme is simple to implement, efficient and suitable for a general equation of
state. The robustness of the new algorithm is validated against one and two
dimensional numerical test problems.
| https://export.arxiv.org/pdf/astro-ph/0601640 |
\date{Accepted ??. Received ??; in original form ??}
\pagerange{\pageref{firstpage}--\pageref{lastpage}} \pubyear{2005}
\label{firstpage}
\begin{keywords}
hydrodynamics - methods: numerical - relativity - shock waves
\end{keywords}
\section{Introduction}
Strong evidence nowadays supports the general idea that relativistic plasmas
may be closely related with most of the violent phenomena observed in astrophysics.
Most of these scenarios are commonly believed to involve strongly
magnetized plasmas around compact objects.
Accretion onto super-massive black holes, for example, is invoked
as the primary mechanism to power highly energetic phenomena observed
in active galactic nuclei, \citep{Macchetto99,ERZ02,McK05,Shapiro05}.
In this respect, the formation and propagation of relativistic jets and
the accretion flow dynamics pose some of the most challenging and interesting
quests in modern theoretical astrophysics.
Likewise, a great deal of attention has been addressed, in the last years,
to the darkling problem of gamma ray bursts
\citep[see for example][]{MR94,McFW99,KG02,RRD03},
whose models often appeal to strongly relativistic collimated outflows
\citep{Aloy_etal00, Aloy_etal02}.
Other attractive examples include pulsar wind nebulae \citep{BZAV05},
microquasars \citep{Meier03,McKG04}, X-ray binaries \citep{VRT02} and
stellar core collapse in the context of general relativity
\citep{Bruenn85, DFM02}.
Theoretical investigations based on direct numerical simulations
have paved a way towards a better understanding of the rich
phenomenology of relativistic magnetized plasmas.
Part of this accomplishment owes to the successful
generalization of existing shock-capturing Godunov-type codes to
relativistic magnetohydrodynamics (RMHD)
\citep[see][and reference therein]{K99, Balsara01,dZBL03}.
Implementation of such codes is based on a conservative formulation
which requires an exact or approximate solution to the
Riemann problem, i.e., the decay of a discontinuity separating
two constant states \citep{Toro97}.
In terms of computational cost, employment of exact relativistic Riemann
solvers may become prohibitive due to the high
degree of intrinsic nonlinearity present in the equations.
This has focused most computational efforts towards the development
of approximate solvers which, nevertheless, require
knowledge of the exact solution, at least on some level
\citep{MM03}.
The presence of magnetic fields further entangles the solution,
since the number of decaying waves increases from
three to seven \citep{AP87,Anile89}. An exact analytical approach
to the solution (which does not allow compound waves) has been
recently presented in \cite{GR05}, while
\cite{RMPIM05} derived a special case where the velocity and magnetic
field are orthogonal.
The trade-off between efficiency, accuracy and robustness of such
approximate methods is still a matter of research.
Solvers based on local linearization have been presented in
\cite{K99} (KO henceforth), \cite{Balsara01} (BA henceforth)
and \cite{KKU02}.
Despite the higher accuracy in reproducing the full wave structure,
these solvers rely on rather expensive
characteristic decompositions of the Jacobian matrix.
Conversely, the characteristic-free formulation of Harten-Lax-van Leer
(HLL) of \cite{HLL83} has gained increasing popularity due to
its ease of implementation and robustness.
The HLL approach has been successfully applied to the RMHD
equations by \citealt{dZBL03} (dZBL henceforth) as well as
to the general relativistic case \citep[see for example][]{GmKT03,DLSS05}
and to the investigation of extragalactic jets, see \cite{LAAM05}.
Besides the computational efficiency, however, the HLL formulation
averages the full solution to the Riemann problem into a single state,
and thus lacks the ability to resolve single intermediate waves such as Alfv{\'e}n,
contact and slow discontinuities.
In \cite{MB05} (paper I henceforth) we proposed an approach that
cured this deficiency by restoring the missing contact wave.
The resulting scheme generalized the HLLC approximate Riemann solver
by \citet{TSS94} to the equations of relativistic hydrodynamics
without magnetic fields.
Here, along the same lines, we propose an extension
of the HLLC solver to the relativistic magnetized case.
Similar work has been presented in the context of classical MHD
by \cite{Gurski04} and \cite{Li05}.
The new HLLC Riemann solver is implemented in the framework of
the corner transport upwind (CTU) method of \cite{Colella90},
coupled with the constrained transport (CT) evolution \citep{EH88}
of magnetic field.
The algorithm naturally preserves the divergence-free condition
to machine accuracy and is stable up to Courant number of $1$.
The paper is organized as follows.
The relevant equations are given in \S\ref{sec:equations}.
In \S\ref{sec:hllc} we derive the new HLLC Riemann solver.
Numerical tests, together with the implementation of the
CTU-CT method are shown in \S\ref{sec:test}.
\section{The RMHD Equations}\label{sec:equations}
The motion of an ideal relativistic magnetized fluid is
described by conservation of mass,
\begin{equation}\label{eq:mass}
\partial_\alpha (\rho u^\alpha) = 0 \;,
\end{equation}
energy-momentum,
\begin{equation}\label{eq:mom}
\partial_\alpha\Big[(\rho h + |b|^2)u^\alpha u^\beta - b^\alpha b^\beta + p\eta^{\alpha\beta}\Big] = 0 \;,
\end{equation}
and by Maxwell's equations:
\begin{equation}\label{eq:maxwell}
\partial_\alpha(u^\alpha b^\beta - u^\beta b^\alpha) = 0 \;.
\end{equation}
see, for example, \cite{AP87} or \cite{Anile89}.
In equations (\ref{eq:mass}), (\ref{eq:mom}) and (\ref{eq:maxwell}) we have
introduced the rest mass density of the fluid $\rho$, the four velocity
$u^\alpha$, the covariant magnetic field $b^\alpha$ and the relativistic specific
enthalpy $h$.
The total pressure $p$ results from the sum of thermal (gas)
pressure $p_g$ and magnetic pressure $|b|^2/2$, i.e., $p = p_g + |b|^2/2$.
In what follows we assume a flat metric, so that
$\eta^{\alpha\beta}= \textrm{diag}(-1,1,1,1)$ is the Minkowski metric tensor.
Greek indexes run from $0$ to $3$ and are customary for covariant
expressions involving four-vectors. Latin
indexes (from $1$ to $3$) describe three-dimensional vectors
and are used indifferently as subscripts or superscripts.
The four-vectors $u^\alpha$ and $b^\alpha$ are related to the spatial
components of the velocity $\A{v} \equiv (v_x, v_y, v_z)$ and
laboratory magnetic field $\A{B} \equiv (B_x, B_y, B_z)$ through
\begin{equation}\label{eq:four-vectors} \begin{array}{ccc}
u^\alpha & = & \DS \gamma\big(1,\; \A{v}\big) \;,\\ \noalign{\medskip}
b^\alpha & = & \DS \gamma\left(\A{v}\cdot\A{B} \,, \quad
\frac{\A{B}}{\gamma^2} + \A{v}(\A{v}\cdot\A{B})\right)\;,
\end{array}
\end{equation}
with the normalizations
\begin{equation}
u^\alpha u_\alpha = -1 \;,\quad
u^\alpha b_\alpha = 0 \;,
\end{equation}
\begin{equation}
|b|^2 \equiv b^\alpha b_\alpha = \frac{|\A{B}|^2}{\gamma^2} + (\A{v}\cdot\A{B})^2 \;,
\end{equation}
where $\gamma = (1 - \A{v}\cdot\A{v})^{-1/2}$ is the Lorentz factor.
We follow the same conventions used in paper I, where velocities are
given in units of the speed of light.
Writing the spatial and temporal components of equation (\ref{eq:maxwell})
in terms of the laboratory magnetic field yields
\begin{equation}\label{eq:induction}
\pd{\A{B}}{t} = \nabla\times(\A{v}\times\A{B}) \;,
\end{equation}
\begin{equation}\label{eq:divB}
\nabla\cdot\A{B} = 0 \; ,
\end{equation}
i.e., they reduce to the familiar induction equation and the
solenoidal condition.
For computational purposes, equations (\ref{eq:mass})--(\ref{eq:maxwell})
are more conveniently put in the standard conservation form
\begin{equation}\label{eq:rmhd_eq}
\pd{\A{U}}{t} + \sum_k \pd{\A{F}^k(\A{U})}{x^k} = 0 \; ,
\end{equation}
together with the divergence-free constraint (\ref{eq:divB}), where
$\A{U} = (D, m_x, m_y, m_z, B_x, B_y, B_z, E)$
is the vector of conservative variables and $\A{F}^k$ are
the fluxes along the $x^k\equiv (x,y,z)$ directions.
The components of $\A{U}$ are, respectively, the laboratory
density $D$, the three components of momentum $m_k$ and
magnetic field $B_k$ and the total energy density $E$.
From equations (\ref{eq:mass}), (\ref{eq:mom}) and the
definitions (\ref{eq:four-vectors}) one has
\begin{eqnarray}
D & = & \rho\gamma \;, \label{eq:cons_var_D}\\ \noalign{\medskip}
m_k & = & (\rho h\gamma^2 + \A{B}^2)v_k - (\A{v}\cdot\A{B})B_k \; ,
\label{eq:cons_var_m}\\ \noalign{\medskip}
E & = & \DS \rho h\gamma^2 - p_g
+ \frac{\A{B}^2}{2} + \frac{\A{v}^2\A{B}^2 - (\A{v}\cdot\A{B})^2}{2}
\label{eq:cons_var_E}\;,
\end{eqnarray}
and
\begin{equation}\label{eq:fluxes}
\A{F}^x(\A{U}) = \left(\begin{array}{c}
Dv_x \\ \noalign{\medskip}
\DS m_xv_x - B_x\frac{b_x}{\gamma} + p \\ \noalign{\medskip}
\DS m_yv_x - B_x\frac{b_y}{\gamma} \\ \noalign{\medskip}
\DS m_zv_x - B_x\frac{b_z}{\gamma} \\ \noalign{\medskip}
0 \\ \noalign{\medskip}
B_yv_x - B_xv_y \\ \noalign{\medskip}
B_zv_x - B_xv_z \\ \noalign{\medskip}
m_x \end{array}\right) \;.
\end{equation}
Similar expressions hold for $\A{F}^y(\A{U})$ and $\A{F}^z(\A{U})$ by
cyclic permutations of the indexes.
Notice that the fluxes entering in the induction equation
are the components of the electric field which, in the infinite
conductivity approximation, becomes
\begin{equation}
\A{\Omega} = -\A{v}\times\A{B} \;.
\end{equation}
The non-magnetic case is recovered by letting $\A{B}\to 0$ in
the previous expressions.
Finally, proper closure is provided by specifying an additional
equation of state. Throughout the following we will
assume a constant $\Gamma$-law, with specific enthalpy given
by
\begin{equation}\label{eq:eos}
h = 1 + \frac{\Gamma}{\Gamma - 1}\frac{p_g}{\rho} \;,
\end{equation}
where $\Gamma$ is the constant specific heat ratio.
\subsection{Recovering primitive variables}
\label{sec:contoprim}
Godunov-type codes are based on a conservative formulation
where laboratory density, momentum, energy and magnetic fields
are evolved in time.
On the other hand, primitive variables, $\A{V} = (\rho, \A{v}, p_g, \A{B})$,
are required when computing the fluxes
(\ref{eq:fluxes}) and more convenient for interpolation
purposes.
Recovering $\A{V}$ from $\A{U}$ is not a straightforward task
in RMHD and different approaches have been suggested by previous authors:
BA used an iterative scheme based on a $5\times 5$ Jacobian sub-block
of the system (\ref{eq:rmhd_eq}); KO solves a $3\times 3$ nonlinear system
of equations; dZBL (the same approach is also used in
\cite{LAAM05}) further reduced the problem to a $2\times 2$
system of nonlinear equations.
Here we reduce this task to the solution of a single nonlinear equation,
by properly choosing the independent variable.
If one sets, in fact, $W = \rho h \gamma^2$, $S = \A{m}\cdot\A{B}$, the
following two relations hold:
\begin{equation}\label{eq:c2p_E}
E = W - p_g + \left(1 - \frac{1}{2{\gamma}^2}\right)|\A{B}|^2
-\frac{S^2}{2 W^2} \;,
\end{equation}
\begin{equation}\label{eq:c2p_m}
|\A{m}|^2 = \left(W + |\A{B}|^2\right)^2
\left(1 - \frac{1}{\gamma^2}\right) -
\frac{S^2}{W^2} \left(2W + |\A{B}|^2\right) \;.
\end{equation}
Since at the beginning of each time step $\A{m}$, $\A{B}$ and $S$ are
known quantities, equation (\ref{eq:c2p_m}) allows one to express the
Lorentz factor $\gamma$ as a function of $W$ alone:
\begin{equation}\label{eq:lorentz}
\gamma = \left(1 - \frac{S^2(2W + |\A{B}|^2) + |\A{m}|^2W^2}{(W + |\A{B}|^2)^2W^2}
\right)^{-\HALF}\;.
\end{equation}
Using the equation of state (\ref{eq:eos}), the thermal pressure
$p_g$ is also a function of $W$:
\begin{equation}\label{eq:c2p_eos}
p_g(W) = \frac{W - D\gamma}{\Gamma_r\gamma^2} \;,
\end{equation}
where $\Gamma_r = \Gamma/(\Gamma - 1)$ and $\gamma$ is now given
by (\ref{eq:lorentz}).
Thus the only unknown appearing in equation (\ref{eq:c2p_E}) is
$W$ and
\begin{equation}\label{eq:press_fun}
f(W) \equiv W - p_g + \left(1 - \frac{1}{2{\gamma}^2}\right)|\A{B}|^2
-\frac{S^2}{2 W^2} - E = 0 \;
\end{equation}
can be solved by any standard root finding algorithm.
Although both the secant and Newton-Raphson methods have
been implemented in our numerical code, we found the latter to be more robust and
computationally efficient and it will be our method of choice.
The expression for the derivative needed in the Newton scheme
is computed as follows:
\begin{equation}
\frac{df(W)}{dW} = 1 - \frac{dp_g}{dW} +
\frac{|\A{B}|^2}{\gamma^3} \frac{d\gamma}{dW} +
\frac{S^2}{W^3} \;,
\end{equation}
where $dp_g/dW$ is computed from (\ref{eq:c2p_eos}), whereas
$d\gamma/dW$ is computed from eq. (\ref{eq:lorentz}):
\begin{equation} \begin{array}{c}
\DS \frac{dp_g}{dW} = \frac{\gamma(1 + Dd\gamma/dW) - 2Wd\gamma/dW}
{\Gamma_r\gamma^3} \;, \\ \noalign{\bigskip}
\DS \frac{d\gamma}{dW} = -\gamma^3\,\frac{2S^2(3W^2 + 3W|\A{B}|^2 + |\A{B}|^4) +
|\A{m}|^2W^3}{2W^3(W + |\A{B}|^2)^3} \;.
\end{array}\end{equation}
Once $W$ has been computed to some accuracy, the Lorentz
factor can be easily found from (\ref{eq:lorentz}),
thermal pressure from (\ref{eq:c2p_eos}) and velocities
are found by inverting equation (\ref{eq:cons_var_m}):
\begin{equation}
v_k = \frac{1}{W + |\A{B}|^2}\left(m_k + \frac{S}{W}B_k\right)
\end{equation}
Finally, equation (\ref{eq:cons_var_D}) is used to determine
the proper density $\rho$.
\subsection{The Riemann Problem in RMHD}\label{sec:riemann}
In the standard Godunov-type formalism, numerical integration of
(\ref{eq:rmhd_eq}) depends on the computation of numerical fluxes
at zone interfaces. This task is accomplished by the
(exact or approximate) solution of the initial value problem:
\begin{equation}\label{eq:riemann}
\A{U}(x,0) = \left\{\begin{array}{ccc}
\A{U}_{L,i+\HALF} & \quad \textrm{if} \; & x < x_{i+\HALF} \;, \\ \noalign{\medskip}
\A{U}_{R,i+\HALF} & \quad \textrm{if} \; & x > x_{i+\HALF} \;, \\ \noalign{\medskip}
\end{array}\right.
\end{equation}
where $\A{U}_{L,i+\HALF}$ and $\A{U}_{R,i+\HALF}$ are assumed to be
piece-wise constant left and right states at zone interface $i+\HALF$.
The evolution of the discontinuity (\ref{eq:riemann}) constitutes
the Riemann problem.
As in classical MHD, evolution in a given direction is governed
by seven equations in seven independent conserved variables.
Integration along the $x$-direction, for example, leaves
$B_x$ unchanged since the corresponding flux is identically zero,
eq. (\ref{eq:fluxes}).
The solution to the initial value problem (\ref{eq:riemann})
results, therefore, in the formation of seven waves: two pairs
of magneto-acoustic waves, two Alfv{\'e}n waves and a contact discontinuity.
The complete analytical solution to the relativistic MHD Riemann
problem has been recently derived in closed form
by \cite{GR05}. A number of properties regarding simple waves are also
well established, see \cite{AP87} and \cite{Anile89}.
\cite{RMPIM05} discuss the case in which the magnetic field of the initial
states is tangential to the discontinuity and orthogonal to the flow velocity.
General guidelines, relevant to the present work, follow below.
Across a magneto-acoustic (fast or slow) shock,
all components of $\A{V}$ can change discontinuously.
Thermodynamic quantities (e.g., $\rho$ and $p_g$)
are continuous through a relativistic Alfv{\'e}n wave
(as in the classical case), but contrary to the classical
counterpart, the magnetic field is elliptically polarized and the normal
component of the velocity is discontinuous \citep{K97}.
Through the contact mode, only density exhibits a jump while
thermal pressure, velocity and magnetic field are continuous.
For the special case in which the component of the magnetic field
normal to a zone interface vanishes, a degeneracy occurs where tangential,
Alfv{\'e}n and slow waves all propagate at the speed of the
fluid and the solution simplifies to a three-wave pattern.
Under this condition, the approximate solution outlined
in paper I can still be applied with minor modifications,
see \S\ref{sec:bx0} in this paper and \cite{MMB05}.
\section{The HLLC Solver}\label{sec:hllc}
The derivation of the HLL and HLLC approximate Riemann solvers has
already been discussed in paper I and will not be repeated hereafter.
Following the same notations, we approximate the
solution to the initial value problem (\ref{eq:riemann}) with
two constant states, $\A{U}^*_L$ and $\A{U}^*_R$, bounded by two fast
shocks and a contact discontinuity in the middle.
We write the solution on the $x/t=0$ axis as
\begin{equation}\label{eq:hllc_states}
\A{U}(0,t) = \left\{\begin{array}{ccc}
\A{U}_L & \quad \textrm{if} & \; \lambda_L \ge 0 \;, \\ \noalign{\medskip}
\A{U}^*_L & \quad \textrm{if} & \; \lambda_L \le 0 \le \lambda^* \;,\\ \noalign{\medskip}
\A{U}^*_R & \quad \textrm{if} & \; \lambda^* \le 0 \le \lambda_R \;,\\ \noalign{\medskip}
\A{U}_R & \quad \textrm{if} & \; \lambda_R \le 0 \;, \\ \noalign{\medskip}
\end{array}\right.
\end{equation}
where $\lambda_L$ and $\lambda_R$ are, respectively, the minimum and maximum characteristic
signal velocities and $\lambda^*$ is the velocity of the middle contact wave.
The corresponding inter-cell numerical fluxes are:
\begin{equation}\label{eq:hllc_flux}
\A{f} = \left\{\begin{array}{ccc}
\A{F}_L & \quad \textrm{if} & \; \lambda_L \ge 0 \;, \\ \noalign{\medskip}
\A{F}^*_L & \quad \textrm{if} & \; \lambda_L \le 0 \le \lambda^*\;, \\ \noalign{\medskip}
\A{F}^*_R & \quad \textrm{if} & \; \lambda^* \le 0 \le \lambda_R\;, \\ \noalign{\medskip}
\A{F}_R & \quad \textrm{if} & \; \lambda_R \le 0 \;. \\ \noalign{\medskip}
\end{array}\right.
\end{equation}
The intermediate fluxes $\A{F}^*_L$ and $\A{F}^*_R$ are expressed
in terms of $\A{U}^*_L$ and $\A{U}^*_R$ through the
Rankine-Hugoniot jump conditions:
\begin{equation}\label{eq:jump_1}\begin{array}{ccc}
\lambda_L \left(\A{U}^*_L - \A{U}_L\right) & = & \A{F}^*_L - \A{F}_L \;,\\ \noalign{\medskip}
\lambda^* \left(\A{U}^*_R - \A{U}^*_L\right) & = & \A{F}^*_R - \A{F}^*_L \;,\\ \noalign{\medskip}
\lambda_R \left(\A{U}_R - \A{U}^*_R\right) & = & \A{F}_R - \A{F}^*_R \;,\\ \noalign{\medskip}
\end{array}\end{equation}
where, in general, $\A{F}^*_{L,R} \neq \A{F}(\A{U}^*_{L,R})$.
The consistency condition is obtained by adding the previous equations
together:
\begin{equation}\label{eq:consistency1}
\frac{(\lambda^* - \lambda_L) \A{U}^*_L +
(\lambda_R - \lambda^*) \A{U}^*_R}{\lambda_R - \lambda_L} = \A{U}^{hll} \;,
\end{equation}
where
\begin{equation}\label{eq:hll_state}
\A{U}^{hll} = \frac{\lambda_R \A{U}_R - \lambda_L\A{U}_L +
\A{F}_L - \A{F}_R}{\lambda_R - \lambda_L} \,,
\end{equation}
is the \emph{state} integral average of the solution to the Riemann
problem.
Similarly, if one divides each expression in eq. (\ref{eq:jump_1}) by the
corresponding $\lambda$'s on the left hand sides and adds the resulting
expressions,
\begin{equation}\label{eq:consistency2}
\frac{\A{F}^*_L\lambda_R(\lambda^* - \lambda_L) +
\A{F}^*_R\lambda_L(\lambda_R - \lambda^*)}{\lambda_R - \lambda_L} = \lambda^*\A{F}^{hll}\;,
\end{equation}
with
\begin{equation}\label{eq:hll_flux}
\A{F}^{hll} = \frac{\lambda_R\A{F}_L - \lambda_L\A{F}_R + \lambda_R\lambda_L
(\A{U}_R - \A{U}_L)}{\lambda_R - \lambda_L} \,.
\end{equation}
being the \emph{flux} integral average of the solution to the Riemann problem.
Since the sets of jump conditions across the contact discontinuity differ
depending on whether $B_x$ vanishes or not, we proceed by separately
discussing the two cases.
In either case, the speed of the contact wave is assumed to be equal to the
(average) normal velocity over the Riemann fan, i.e. $\lambda^* \equiv v^*_x$.
The normal component of magnetic field, $B_x$, is assumed to be continuous at the
interface, so that $B_x^* \equiv B_{x,L} = B_{x,R}$ can be regarded
as a parameter in the solution.
\subsection{Case $B^*_x \neq 0$}\label{sec:bx}
We start by noticing that equations (\ref{eq:consistency1}) and
(\ref{eq:consistency2}) provide a total of 14 relations.
Six additional conditions come by imposing continuity of total pressure,
velocity and magnetic field components across the contact discontinuity.
This gives us a freedom of $20$ independent unknowns, $10$ per state;
we choose to introduce the following set of unknowns for each state
\begin{equation}\label{eq:hllc_vars}
\left\{D^*, \, v_x^*,\, v_y^*,\, v_z^*,\, B_y^*,\, B_z^*,\, m_y^*,\, m_z^*,\, E^*,\, p^*\right\}\;.
\end{equation}
The normal component of momentum ($m_x^*$) is not an independent
variable since we assume, for consistency, that
\begin{equation}\label{eq:mE_rel}
m_x^* = (E^* + p^*)v_x^* - \left(\A{v}^*\cdot\A{B}^*\right)B_x^* \;.
\end{equation}
The previous relation obviously holds between conservative and primitive
physical quantities.
We point out that the choice (\ref{eq:hllc_vars}) is not unique and
alternative sets of independent variables may be adopted.
According to the previous definitions, the state vector solution to the
Riemann problem is written as
\begin{equation}\label{eq:hllc_u*}
\A{U}^* = \Big(D^*, m_x^*, m_y^*, m_z^*, B_y^*, B_z^*, E^*\Big)^t \;.
\end{equation}
while the flux vector, eq. (\ref{eq:fluxes}), becomes
\begin{equation}\label{eq:hllc_f*}
\A{F}^* = \left(\begin{array}{c}
D^*v^*_x \\ \noalign{\medskip}
\DS m_x^*v^*_x - \frac{B_x^*B_x^*}{(\gamma^*)^2}
- B^*_xv^*_x \left(\A{v}^*\cdot\A{B}^*\right) + p^* \\ \noalign{\medskip}
\DS m_y^*v^*_x - \frac{B_x^*B_y^*}{(\gamma^*)^2}
- B^*_xv^*_y \left(\A{v}^*\cdot\A{B}^*\right) \\ \noalign{\medskip}
\DS m_z^*v^*_x - \frac{B_x^*B_z^*}{(\gamma^*)^2}
- B^*_xv^*_z \left(\A{v}^*\cdot\A{B}^*\right) \\ \noalign{\medskip}
B^*_yv^*_x - B^*_xv^*_y \\ \noalign{\medskip}
B^*_zv^*_x - B^*_xv^*_z \\ \noalign{\medskip}
m^*_x
\end{array}\right)
\end{equation}
As in paper I, we adopt the convention that quantities without the
$L$ or $R$ suffix refer indifferently to the left ($L$) or
right ($R$) state.
The six conditions across the contact discontinuity are
\begin{equation}\begin{array}{ccc}
v^*_{x,L} = v^*_{x,R} \;, & v^*_{y,L} = v^*_{y,R}\;, & v^*_{z,L} = v^*_{z,R} \;, \\ \noalign{\medskip}
B^*_{y,L} = B^*_{y,R} \;, & B^*_{z,L} = B^*_{z,R}\;, & p^*_L = p^*_R \;.
\end{array}\end{equation}
For these quantities the suffix $L$ or $R$ is thus unnecessary.
From the transverse components of the magnetic field in the state
consistency condition (\ref{eq:consistency1}), one immediately finds that
\begin{equation}\label{eq:transv_B}
B^*_{y} = B_{y}^{hll} \; ,\quad
B^*_{z} = B_{z}^{hll} \; .
\end{equation}
Thus the transverse components the magnetic field are given by the HLL single
state.
Similarly, from the fifth and sixth components of the flux consistency condition
(\ref{eq:consistency2}) one can express the transverse velocity through
\begin{equation}\label{eq:transv_v}
B^*_xv^*_y = B^*_yv^*_x - F_{B_y}^{hll} \;,\quad
B^*_xv^*_z = B^*_zv^*_x - F_{B_z}^{hll} \;,
\end{equation}
where $F_{B_y}^{hll}$ and $F_{B_z}^{hll}$ are the $B_y$- and $B_z$- components of
the HLL flux, eq. (\ref{eq:hll_flux}).
Simple manipulations of the normal momentum and energy components in equation
(\ref{eq:consistency1}) together with (\ref{eq:mE_rel}) yield the following
simple expression:
\begin{equation}\label{eq:state_eq}
E^{hll} v^*_x + p^*v^*_x - B^*_x\, \big(\A{v}^*\cdot\A{B}^*\big)
= m_x^{hll} \;.
\end{equation}
Similar algebra on the momentum and energy components of the
flux consistency condition (\ref{eq:consistency2}) leads to
\begin{equation}\label{eq:flux_eq}
\Big[F^{hll}_E - B^*_x\,(\A{v}^*\cdot\A{B}^*)\Big]v^*_x -
\left(\frac{B^*_x}{\gamma^*}\right)^2 + p^* - F^{hll}_{m^x} =0 \;.
\end{equation}
where $1/(\gamma^*)^2 = 1 - (v_x^*)^2 - (v_y^*)^2 - (v_z^*)^2$.
Now, if one multiplies equation (\ref{eq:flux_eq}) by $v_x^*$ and subtracts
equation (\ref{eq:state_eq}), the following quadratic equation
may be obtained:
\begin{equation}\label{eq:quadratic}
a(v_x^*)^2 + bv_x^* + c = 0 \;,
\end{equation}
with coefficients
\begin{equation}\begin{array}{ccl}
a & = & F^{hll}_E - \A{B}^{hll}_\perp\cdot\A{F}^{hll}_{\A{B}_\perp} \;,\\ \noalign{\medskip}
b & = & - F^{hll}_{m^x} - E^{hll} + \left|\A{B}_\perp^{hll}\right|^2
+ \left|\A{F}^{hll}_{\A{B}_\perp}\right|^2 \;, \\ \noalign{\medskip}
c & = & m_x^{hll} - \A{B}^{hll}_\perp\cdot\A{F}^{hll}_{\A{B}_\perp} \;.
\end{array}\end{equation}
In the previous expressions $\A{B}^{hll}_\perp \equiv (0,B^{hll}_y,B^{hll}_z)$,
$\A{F}^{hll}_{\A{B}_\perp} \equiv (0,F^{hll}_{B_y}, F^{hll}_{B_z})$.
Similar arguments to those presented in paper I lead to the conclusion
that only the root with the minus sign is physically admissible.
Once $v_x^*$ is known, $v_y^*$ and $v_z^*$ are readily obtained from
(\ref{eq:transv_v}), $p^*$ is computed from (\ref{eq:flux_eq}),
while density, transverse momenta and energy are obtained using the
Rankine-Hugoniot jump conditions across each fast wave:
\begin{eqnarray}
D^* & = & \DS \frac{\lambda - v^x}{\lambda - v_x^*} D
\;, \label{eq:D_jump} \\ \noalign{\medskip}
\label{eq:transv_my}
m^*_y & = &\DS \frac{-B^*_x\left[\frac{B^*_y}{(\gamma^*)^2} + (\A{v}^*\cdot\A{B}^*)v_y^*\right]
+ \lambda m_y - F_{m_y}}{\lambda - v^*_x}
\;, \label{eq:my_jump} \\ \noalign{\medskip}
\label{eq:transv_mz}
m^*_z & = &\DS \frac{-B^*_x\left[\frac{B^*_z}{(\gamma^*)^2} + (\A{v}^*\cdot\A{B}^*)v_z^*\right]
+ \lambda m_z - F_{m_z}}{\lambda - v^*_x}
\;, \label{eq:mz_jump} \\ \noalign{\medskip}
E^* & = & \DS \frac{\lambda E - m_x + p^*v_x^* - (\A{v}^*\cdot\A{B}^*)B^*_x}{\lambda - v_x^*} \;.
\label{eq:E_jump}
\end{eqnarray}
In equations (\ref{eq:my_jump}) and (\ref{eq:mz_jump}), $F_{m_y}$ and $F_{m_z}$ are,
respectively, the $m_y$- and $m_z$- components of the flux, eq. (\ref{eq:fluxes}), evaluated at
the left or right state.
As in paper I, we have omitted the suffix $L$ or $R$ for clarity of exposition.
\subsection{Case $B^*_x = 0$}\label{sec:bx0}
For vanishing normal component of the magnetic field
a degeneracy occurs where the Alfv{\'e}n waves and the two slow
magnetosonic waves propagate at the speed of the contact discontinuity.
For this case the approximate character of the HLLC solver offers a better
representation of the exact solution, since the Riemann fan is comprised of
three waves only.
At the contact discontinuity, however, only the normal component
of the velocity $v_x$ and the total pressure $p$ are continuous
(KO). The remaining variables experience jumps.
This only adds $2$ constraints
to the $14$ jump conditions, leaving a freedom of $8$ unknowns per state.
However, the transverse velocities $v_y$ and $v_z$ do not enter explicitly
in the fluxes (\ref{eq:hllc_f*}) and the jump conditions can be written entirely
in terms of $\{D^*, v^*_x, m^*_y, m^*_z, B^*_y, B^*_z, E^*, p^*\}$,
i.e. $8$ unknowns per state.
Straightforward algebra shows that the coefficients of the quadratic
equation (\ref{eq:quadratic}) are now given by
\begin{equation}
a = F^{hll}_E \;,\quad
b = - F^{hll}_{m^x} - E^{hll} \;, \quad
c = m_x^{hll} \;,
\end{equation}
i.e., they coincide with the expressions derived in paper I.
The root with the minus sign still represents the correct physical
solution.
Once $v_x^*$ is found, the total pressure $p^*$
is derived from
\begin{equation}
p = - F^{hll}_{E}v_x^* + F^{hll}_{m_x}\;,
\end{equation}
and the normal momentum (\ref{eq:mE_rel}) becomes
\begin{equation}
m_x^* = (E^* + p^*)v_x^* \;.
\end{equation}
The remaining quantities are easily obtained
from the jump conditions:
\begin{eqnarray}
D^* & = & \DS \frac{\lambda - v_x}{\lambda - v_x^*} D \;,
\\ \noalign{\medskip}
m^*_{y,z} & = & \DS \frac{\lambda - v_x}{\lambda - v^*_x}\, m_{y,z} \;,
\\ \noalign{\medskip}
E^* & = & \DS \frac{\lambda E - m_x + p^*v_x^*}{\lambda - v_x^*} \;,
\label{eq:E_jump2} \\ \noalign{\medskip}
B^*_{y,z} & = & \DS \frac{\lambda - v_x}{\lambda - v_x^*}\, B_{y,z} \;.
\end{eqnarray}
\subsection{Remarks}\label{sec:remarks}
The expressions derived separately in \S\ref{sec:bx} and
\S\ref{sec:bx0} are suitable in the $B_x\neq0$ and $B_x\to0$ cases,
respectively.
Although other degeneracies may be present (see KO for
a thorough discussion) no other modifications are necessary
to the algorithm.
Before testing the new solver, however, a few remarks are worth of notice:
\begin{enumerate}
\renewcommand{\theenumi}{(\arabic{enumi})}
\item The solutions derived separately for $B_x \neq 0$ and the special
case $B_x = 0$ automatically satisfy the consistency
conditions (\ref{eq:consistency1}) and (\ref{eq:consistency2})
by construction;
\item In the limit of zero magnetic field, the expressions derived
in \S\ref{sec:bx0} reduce to those found in paper I;
\item In the classical limit, our derivation does not coincide with the
approximate Riemann solvers constructed by \cite{Gurski04} or \cite{Li05}.
The reason for this discrepancy stems from the fact that both
\cite{Gurski04} and \cite{Li05} assume that transverse momenta
and velocities are tied by the relation $m^*_{y,z} \equiv \rho^*v_{y,z}^*$.
Although certainly true in the exact solution, this assumption
reduces, in the HLLC approximate formalism, the number of unknowns from
$10$ to $8$ (when $B_x\neq0$) thus leaving the systems of jump conditions
(\ref{eq:jump_1}) overdetermined.
Should this be the case, the number of equations exceeds the number of unknowns
and the integral relations across the Riemann fan inevitably break down.
This explains the inconsistencies found in Li's and Gurski's derivations and
further discussed in \cite{MK05}.
Therefore, in the classical limit, our expressions automatically imply
$m^*_{y,z} \neq \rho^*v_{y,z}^*$
and the correct expressions for the transverse velocities are still given by
(\ref{eq:transv_v}), whereas transverse momenta should be derived from the
jump conditions accordingly.
Furthermore, contrary to Li's misconception, consistency with the jump
conditions requires that the magnetic field components be uniquely determined by
(\ref{eq:transv_B}) and no other choices are thus possible.
\item The reader might have noticed that in the limit
of vanishing $B_x$, some of the expressions given in
\S\ref{sec:bx} do not reduce to the those found in \S\ref{sec:bx0}.
This property also persists in the classical limit, see \cite{Gurski04},
and \cite{Li05}.
The reason for this discrepancy relies on the assumption
of continuity of the transverse components of magnetic field
across the tangential wave $\lambda^*$:
when $B_x \to 0$, a degeneracy occurs where the tangential,
Alfv{\'e}n and slow waves all propagate at the speed of the
fluid and the solution simplifies to a three-wave pattern.
In the exact solution, the continuity of $B_y$ and $B_z$ across
the tangential wave is lost since the middle state
bounded by the two slow waves becomes singular.
\item Lastly, we note that in both the classical and relativistic
case the transverse velocities given by eq. (\ref{eq:transv_v})
become ill-defined as $B_x\to 0$. However,
in the classical case, the terms involving $v^*_y$
or $v^*_z$ in the flux definitions remain finite as
$B_x\to 0$. Conversely, this is not the case
in RMHD for arbitrary orientation of the magnetic field
as one can see, for example, using eq. (\ref{eq:transv_my}):
\begin{equation}
m^*_y \sim \frac{(B_z^{hll}v_x^* - F^{hll}_{B_z} )
(F^{hll}_{B_y} B^{hll}_z - F_{B_z}^{hll}B^{hll}_y)}
{B_x(\lambda - v^*_x)} + O(1)
\end{equation}
as $B_x\to 0$.
Fortunately, for strictly two dimensional flows (e.g. when $B_z = v_z = 0$)
the leading order term vanishes and the singularity
is avoided.
In the general case, however, we conclude that more sophisticated
solvers should allow the presence of rotational discontinuities
in the solution to the Riemann problem.
This has been done, for example, by \cite{MK05} in the context
of classical MHD.
\end{enumerate}
\subsection{Wave Speed Estimate}\label{sec:speeds}
The full characteristic decomposition of the RMHD equation (i.e.
the eigenvalues and eigenvectors of the Jacobian matrix
$\partial\A{F}^x/\partial\A{U}$) was extensively analyzed by
\cite{AP87} and \cite{Anile89}.
In the one-dimensional case the Jacobian matrix can be decomposed
into seven eigenvectors associated with four magnetosonic
waves (fast and slow disturbances), two Alfv{\'e}n waves and one entropy
wave propagating at the fluid velocity.
The eigenstructure is therefore similar to the classical case and
it can be shown that the ordering of the various speeds and corresponding
degeneracies are preserved \citep{Anile89}.
Since the HLLC approximate Riemann solver requires
an estimate of the outermost waves, the right and left-going
fast shock speeds identify the necessary characteristic velocities.
Thus we set \citep{Davis88}:
\begin{equation}\begin{array}{c}\label{eq:wavespeeds}
\lambda_L = \min\big(\lambda_-(\A{V}_L), \lambda_-(\A{V}_R)\big) \;,
\\ \noalign{\medskip}
\lambda_R = \max\big(\lambda_+(\A{V}_L), \lambda_+(\A{V}_R)\big) \;,
\end{array}
\end{equation}
where $\lambda_{-}$ and $\lambda_+$ are the minimum and maximum
roots of the quartic equation
\begin{equation}\label{eq:eigenspeed_eq}
\rho h(1-c_s^2)a^4 = (1-\lambda^2) \left[(|b|^2 + \rho h c_s^2)a^2
- c_s^2{\cal B}^2\right] \;,
\end{equation}
with $a = \gamma(\lambda - v_x)$, ${\cal B} = b^x - \lambda b^0$.
In absence of magnetic field, both the (left and right-going)
slow and fast shocks propagate at the same speed and
equation (\ref{eq:eigenspeed_eq}) reduces to the quadratic equation
(22) shown in paper I.
When $\A{B} \neq \A{0}$, no simple analytical expression is available
and solving (\ref{eq:eigenspeed_eq}) requires numerical or rather
cumbersome analytical approaches.
Recently, \cite{LAAM05} proposed approximate simple lower
and upper bounds to the required eigenvalues.
Here we choose to solve eq. (\ref{eq:eigenspeed_eq}) by means of
analytical methods, where the quartic is reduced to a cubic
equation which is in turn solved by standard methods.
There are special cases where it is possible to handle some
of the degeneracies more efficiently using simple analytical formulae:
\begin{itemize}
\item for vanishing total velocity, equation (\ref{eq:eigenspeed_eq})
reduces to a bi-quadratic,
\begin{equation}
(\rho h + |b|^2)\lambda^4 -
(|b|^2 + \rho hc_s^2 + B_x^2c_s^2)\lambda^2 +
c_s^2B_x^2 = 0
\end{equation}
\item for vanishing normal component of the magnetic field,
equation (\ref{eq:eigenspeed_eq}) yields a quadratic
equation:
\begin{equation}
a_2 \lambda^2 + a_1 \lambda + a_0 = 0
\end{equation}
with $a_2 = \rho h\big[c_s^2 + \gamma^2(1-c_s^2)\big] + {\cal Q}$,
$a_1 = -2\rho h\gamma^2v_x(1-c_s^2)$,
$a_0 = \rho h\big[ - c_s^2 + \gamma^2v_x^2(1-c_s^2)\big] - {\cal Q}$
and ${\cal Q} = |b|^2 - c_s^2(\A{v}_\perp\cdot\A{B}_\perp)^2$.
\end{itemize}
For all other cases we solve the quartic equation (\ref{eq:eigenspeed_eq}).
\subsection{Positivity of the HLLC scheme}\label{sec:positivity}
The set of physically admissible conservative states, $G$, identify
all the $\A{U}$'s yielding positive thermal pressure $p_g$ and total velocity
$|\A{v}| < 1$, according to the procedure outlined in \S\ref{sec:contoprim}.
Thus the positivity of the HLLC approximate Riemann solver requires
that
\begin{itemize}
\item both left and right intermediate states $\A{U}^*_L$ and $\A{U}^*_R$
belong to $G$;
\item the first-order scheme yields updated conservative states that
are in $G$.
\end{itemize}
Unfortunately, the mathematical proof of the positivity of the HLLC scheme
presents remarkable algebraic difficulties.
In absence of the singular behavior described in \S\ref{sec:remarks},
investigations have been carried at the numerical level
by verifying that each intermediate state $\A{U}^*$ correspond to a primitive,
physically admissible state. In all the tests presented in this paper and several
others not discussed here, the scheme did not manifest any loss of positivity.
However, in the general three-dimensional case when $B_x,B_y,B_z\neq0$, the terms
involving $B_x$ in the expressions for the transverse momenta may become
arbitrarily large as $B_x\to 0$ and a loss of positivity can be experienced.
\section{Algorithm Validation}\label{sec:test}
\subsection{Corner Transport Upwind for relativistic MHD}\label{sec:ctu}
The RMHD equations (\ref{eq:rmhd_eq}) are evolved in
a conservative, dimensionally unsplit fashion:
\begin{equation}\label{eq:update}
\A{U}^{n+1}_{i,j} = \A{U}^n_{i,j} + \A{\cal L}^{x,n+\HALF}_{i,j}
+ \A{\cal L}^{y,n+\HALF}_{i,j} \,,
\end{equation}
where the $\A{\cal L}$'s are Godunov operators
\begin{equation}\label{eq:god_op_x}
\A{\cal L}^{x,n+\HALF}_{i,j} =
- \frac{\Delta t}{\Delta x_i}
\left(\A{f}^{x,n+\HALF}_{i+\HALF,j} - \A{f}^{x,n+\HALF}_{i-\HALF,j}\right)\,,
\end{equation}
\begin{equation}\label{eq:god_op_y}
\A{\cal L}^{y,n+\HALF}_{i,j} =
- \frac{\Delta t}{\Delta y_j}
\left(\A{f}^{y,n+\HALF}_{i,j+\HALF} - \A{f}^{y,n+\HALF}_{i,j-\HALF}\right)\,,
\end{equation}
and $\A{U}^n$ is the set of volume-averaged conservative variables
$\A{U}^n = \Big(D, \A{m}, \bar{\A{B}}, E\Big)^n$ at time $t=t^n$.
Here $\bar{\A{B}}$ denotes the zone-averaged magnetic field.
For clarity of exposition we will omit, throughout the following,
integer-valued subscripts $(i,j)$ and retain only the half-integer notation
to denote zone edge values.
The fluxes appearing in equations (\ref{eq:god_op_x}) and (\ref{eq:god_op_y})
are computed by solving, at each zone interface, a Riemann problem
with suitable time-centered left and right input states.
For example, we obtain $\A{f}^{y,n+\HALF}_{j+\HALF}$
as the HLLC flux with input states given
by $\A{V}^{n+\HALF}_{j+\HALF,L}$ and $\A{V}^{n+\HALF}_{j+\HALF,R}$,
respectively.
Computation of time-centered left and right zone edge values
proceeds using the corner transport upwind (CTU) of \cite{Colella90},
recently extended to relativistic hydrodynamics
by \cite{MPB05} and to classical MHD by \cite{GS05}.
Here we generalize the CTU approach to relativistic MHD by following a
slightly different approach, although equivalent to the
guidelines given in \cite{Colella90}.
For the sake of conciseness, only the essential steps will be
described hereafter. The unfamiliar reader is referred to
the work of \cite{Colella90}, \cite{Saltzman94} and \cite{GS05}
for more comprehensive derivations.
In our formulation, second-order accurate left and right states
are sought in the form
\begin{equation}\label{eq:pred_states}
\A{V}^{n+\HALF}_{i\pm\HALF,S} = \A{V}^{x,n+\HALF}
\pm \frac{\delta_x\A{V}^n}{2} \, , \quad
\A{V}^{n+\HALF}_{j\pm\HALF,S} = \A{V}^{y,n+\HALF}
\pm \frac{\delta_y\A{V}^n}{2} \, , \quad
\end{equation}
where we take $S=L$ ($S=R$) with the plus (minus) sign.
The slopes $\delta_x\A{V}^n$ and $\delta_y\A{V}^n$ are
computed at the beginning of the time step
using, for example, the monotonized central-difference (MC)
limiter:
\begin{equation}\label{eq:mc_lim}
\delta_x q^n =
s_i\min\left(2|\Delta q^n_+|, 2|\Delta q^n_-|,
\frac{|q^n_{i+1} - q^n_{i-1}|}{2}\right) \,,
\end{equation}
where $q\in\A{V}$ and
\begin{equation}
\Delta q^n_\pm = \pm\left(q^n_{i\pm1} - q^n_i\right) \,,\;
s_i = \frac{\sign(\Delta q^n_+) + \sign(\Delta q^n_-)}{2} \;.
\end{equation}
An alternative smoother prescription is given by the harmonic mean
\citep{vLeer77}:
\begin{equation}\label{eq:vl_lim}
\delta_x q^n = \frac{2\max\left(0, \Delta q_+\Delta q_-\right)}
{\Delta q_+ + \Delta q_-} \,.
\end{equation}
Equation (\ref{eq:mc_lim}) provides smaller dissipation at
discontinuities, whereas equation (\ref{eq:vl_lim}) was found
to give less oscillatory results.
Interpolation in the $y$-direction is done in a similar manner.
Additional forms of limiting may be adopted if necessary,
see \S\ref{sec:shockflattening} and \S\ref{sec:mdlimit}.
The cell- and time- centered values on the right hand sides
of equations (\ref{eq:pred_states}) are computed from a Taylor
expansion of the conservative variables, i.e.
\begin{equation}\label{eq:cent_pred_x}
\A{U}^{x,n+\HALF} \approx
\A{U}^n + \frac{\Delta t}{2}\pd{\A{U}}{t} =
\A{U}^n - \frac{\Delta t}{2}\left( \pd{\hat{\A{F}}^x}{x}
+ \pd{\A{F}^y}{y}\right) \,,
\end{equation}
\begin{equation}\label{eq:cent_pred_y}
\A{U}^{y,n+\HALF} \approx
\A{U}^n + \frac{\Delta t}{2}\pd{\A{U}}{t} =
\A{U}^n - \frac{\Delta t}{2}\left( \pd{\A{F}^x}{x}
+ \pd{\hat{\A{F}}^y}{y}\right) \,.
\end{equation}
Following \cite{Colella90}, we approximate the spatial derivative in
the direction normal to a zone interface (denoted with a hat)
with the Hancock step already introduced in paper I,
\begin{equation}\label{eq:hancock_diff}
\pd{\hat{\A{F}}^x}{x} \approx
\frac{ \A{F}^x\left(\A{V}^n_{i+\HALF,L}\right)
- \A{F}^x\left(\A{V}^n_{i-\HALF,R}\right)}{\Delta x_i}\,,
\end{equation}
whereas the derivative in the tangential direction is
computed in an upwind fashion using a Godunov operator:
\begin{equation}\label{eq:upwind_diff}
\Delta t\pd{\A{F}^y}{y} \approx -\A{\cal{L}}^{y,n} =
\frac{\Delta t}{\Delta y_j}
\left(\A{f}^{y,n}_{j+\HALF} - \A{f}^{y,n}_{j-\HALF}\right)\,.
\end{equation}
The state $\A{U}^{y,n+\HALF}$ is obtained by similar arguments by
interchanging the role of normal and tangential derivatives.
We would like to point out that the Godunov operators used in the predictor
step involve left and right states computed at $t=t^n$
(and not at $t=t^{n+\HALF}$ as in \cite{GS05}):
\begin{equation}\label{eq:init_states}
\A{V}^n_{i\pm\HALF,S} = \A{V}^n \pm \frac{\delta_x\A{V}^n}{2} \;, \quad
\A{V}^n_{j\pm\HALF,S} = \A{V}^n \pm \frac{\delta_y\A{V}^n}{2} \,.
\end{equation}
This choice still makes the scheme second-order
accurate in space and time and was found, in our experience,
to yield a more robust algorithm.
Besides, our CTU implementation does not require a primitive variable
formulation, thus offering ease of implementation
in the context of relativistic hydro and MHD, where the Jacobian
$\partial\A{F}/\partial\A{U}$ is particularly expensive to evaluate.
Note that a total of four Riemann problems are involved in
the single time step update (\ref{eq:update}).
It can be easily verified that for one-dimensional
flows, the corner transport upwind method outlined
above reduces to the scheme presented in paper I.
Finally, the choice of the time step $\Delta t$ is based on the
Courant-Friederichs-Lewy (CFL) condition \citep{CFL28}:
\begin{equation}
\Delta t = \textrm{CFL} \times \min_{i,j}\left(
\frac{ \Delta x}{\max(|\lambda^x_L|,|\lambda^x_R|)},
\frac{ \Delta y}{\max(|\lambda^y_L|,|\lambda^y_R|)}\right) \,,
\end{equation}
where $0<\textrm{CFL}<1$ is the Courant number and
$|\lambda^x_{L,R}|$, $|\lambda^y_{L,R}|$
are the zone interface wave speeds computed in the $x$ and $y$
directions according to (\ref{eq:wavespeeds}).
\subsubsection{Contrained Transport Evolution of the Magnetic Field}
\label{sec:divB}
It is well known that multidimensional numerical schemes do not
generally preserve the solenoidal condition, eq. (\ref{eq:divB}),
unless special discretization techniques are employed.
In this respect, several approaches have been suggested
in the context of the classical MHD equations \citep{Toth00, LdZ00}
and some of them have been recently extended to the relativistic
case, see dZBL.
Here we adopt the constrained transport (CT)
\citep{EH88} and follow the approach of
\cite{BS99} for its integration in Godunov-type schemes.
In the CT approach a new staggered magnetic field variable is introduced.
In this representation, the components of the magnetic
field are treated as area-weighted averages on the zone
faces to which they are orthogonal.
Thus, $B_x$ is collocated at $(i+\HALF,j)$, whereas
$B_y$ at $(i,j+\HALF)$. No jump is allowed in the normal component
of $\A{B}$ at a zone boundary, consistently with the
well posedness of the Riemann problem presented in \S\ref{sec:riemann}
and \S\ref{sec:hllc}.
Transverse components may be discontinuous.
In this formulation, a discrete version of Stoke's theorem is
used integrate the induction equation (\ref{eq:induction}).
For example, after the predictor steps (\ref{eq:cent_pred_x})
and (\ref{eq:cent_pred_y}), we update the face-centered magnetic
field according to
\begin{equation}\label{eq:stokes}\begin{array}{ccc}
\DS B^{n+\HALF}_{x,i+\HALF} & = & \DS B^{n}_{x,i+\HALF}
- \frac{\Delta t^n}{2\Delta y_j}\Big(\Omega^z_{i+\HALF, j+\HALF} - \Omega^z_{i+\HALF, j-\HALF}\Big)\;,
\\ \noalign{\medskip}
\DS B^{n+\HALF}_{y,j+\HALF} & = & \DS B^{n}_{y,j+\HALF}
+ \frac{\Delta t^n}{2\Delta x_i}\Big(\Omega^z_{i+\HALF, j+\HALF} - \Omega^z_{i-\HALF, j+\HALF}\Big)\;,
\end{array}
\end{equation}
and similarly after the corrector step.
The electromotive force $\Omega$ is collocated at cell corners and
is computed by straightforward arithmetic averaging:
\begin{equation}\label{eq:omega}
\Omega^z_{i+\HALF, j+\HALF} = \frac{ \Omega^z_{i+\HALF,j} + \Omega^z_{i, j+\HALF}
+ \Omega^z_{i+\HALF,j+1} + \Omega^z_{i+1,j+\HALF}}{4} \;,
\end{equation}
where, $\Omega^z_{i+\HALF,j} \equiv -f^{x,n}_{B_y,i+\HALF,j}$ and
$\Omega^z_{i,j+\HALF} \equiv f^{y,n}_{B_x,i,j+\HALF}$ are the $z$ components
of the electric fields available at grid interfaces during the upwind step.
Despite its simplicity, eq. (\ref{eq:omega}) lacks of directional bias
and more sophisticated algorithms may be used to incorporate
upwind information in a consistent way, see \cite{LdZ04}, \cite{GS05}.
For ease of implementation we will not discuss them here.
It is a straightforward exercise to verify that the
$\nabla\cdot\A{B} = 0$ condition is preserved from one time step
to the next one, due to perfect cancellation of terms.
Notice also that, since $B_x$ is continuous at the $(i+\HALF,j)$ interface, only
$\bar{B}_y$ and $\bar{B}_z$ need to be interpolated during the reconstruction
procedure in the $x$-direction. A similar argument applies to $\bar{B}_x$
and $\bar{B}_z$ when interpolating along the $y$ coordinate.
Since equation (\ref{eq:update}) evolves volume-averaged quantities,
the zone-averaged magnetic field, $\bar{\A{B}}$, is computed at the
beginning of the time step from the face-averaged magnetic fields
using linear interpolation:
\begin{equation}\label{eq:bx_average}
\bar{B}_{x} = \frac{B_{x,i+\HALF} + B_{x,i-\HALF}}{2} \;,
\end{equation}
\begin{equation}\label{eq:by_average}
\bar{B}_{y} = \frac{B_{y,j+\HALF} + B_{y,j-\HALF}}{2} \;.
\end{equation}
Equations (\ref{eq:omega}), (\ref{eq:bx_average}) and
(\ref{eq:by_average}) are second-order accurate in space.
\subsubsection{Summary}
We summarize our CTU constrained transport algorithm by the following
steps:
\begin{enumerate}
\renewcommand{\theenumi}{(\arabic{enumi})}
\item At the beginning of the time step, form the volume averages
(\ref{eq:bx_average}) and (\ref{eq:by_average}) from
the face centered magnetic field.
\item Compute $x$ and $y$ limited slopes by interpolating
cell centered primitive variables according to eq. (\ref{eq:mc_lim})
or (\ref{eq:vl_lim}).
\item\label{xpred}
Make a sweep along the $x$ direction. Form left and right states using
the first of eq. (\ref{eq:init_states}) with $B^{n}_{x,i+\HALF,L} =
B^{n}_{x,i+\HALF,R}$ equal to the $x$ component of the face centered
magnetic field;
\begin{itemize}
\item[-] use the Hancock step (\ref{eq:hancock_diff}) to compute
the $x$ derivative in eq. (\ref{eq:cent_pred_x}) and add the resulting
contribution to $U^{x,n+\HALF}$;
\item[-] compute the $\A{\cal L}^{x,n}$ Godunov operator by solving
Riemann problems at the $(i+\HALF,j)$ interfaces and add
the resulting contribution to $U^{y,n+\HALF}_{i,j}$.
\end{itemize}
\item\label{ypred}
Make a sweep along the $y$ direction. Form left and right states using
the second in eq. (\ref{eq:init_states}) with $B^{n}_{y,j+\HALF,L}
= B^{n}_{y,j+\HALF,R}$ equal to the $y$ component of the face centered
magnetic field;
\begin{itemize}
\item[-] obtain the $\A{\cal L}^{y,n}$ Godunov operator (\ref{eq:upwind_diff})
by solving Riemann problems at the $(i,j+\HALF)$ interfaces; add
the resulting contribution to $U^{x,n+\HALF}_{i,j}$.
\item[-] use the Hancock step relative to the $y$ direction to compute
the $y$ derivative and add it to $U^{y,n+\HALF}_{i,j}$;
\end{itemize}
\item Compute the time-centered area weighted magnetic field using
Stoke's theorem (\ref{eq:stokes}).
This concludes the predictor step.
\item Make a sweep along the $x$ direction with left and right
time-centered states given by the first equation in
(\ref{eq:pred_states}) with $B^{n+\HALF}_{x,i+\HALF,L} =
B^{n+\HALF}_{x,i+\HALF,R}$
equal to the time centered face-averaged magnetic field computed via
Stoke's theorem.
Obtain the $\A{\cal L}^{x,n+\HALF}$ Godunov operator.
\item Repeat the previous step by sweeping along the $y$ direction.
Compute the $\A{\cal L}^{y,n+\HALF}$ Godunov operator.
\item Update the cell-centered conservative variables using eq.
(\ref{eq:update}) and the face-averaged magnetic field using
Stoke's theorem.
\end{enumerate}
\subsection{One-dimensional test problems}\label{sec:1d}
One-dimensional problems are specifically designed to verify
the ability of the algorithm in reproducing the exact wave
pattern.
In what follows we present four shock-tube tests, already introduced
by BA and dZBL, with left and right states given in Table \ref{tab:ic}.
Computations are performed on the interval $[0,1]$
and the initial discontinuity is placed at $x = 0.5$.
The final integration time is $t=0.4$.
Note that the constrained transport algorithm
is unnecessary, since eq. (\ref{eq:divB}) is trivially
satisfied in one-dimensional flows.
\begin{table}\begin{center}
\begin{tabular}{ccccccccc}
Test & $\rho$ & $p_g$ & $v_x$ & $v_y$ & $v_z$ & $B_x$ & $B_y$ & $B_z$ \\ \hline\hline
1L & 1 & 1 & 0 & 0 & 0 & 0.5 & 1 & 0 \\
1R & 0.125 & 0.1 & 0 & 0 & 0 & 0.5 & -1 & 1 \\ \hline
2L & 1 & 30 & 0 & 0 & 0 & 5 & 6 & 6 \\
2R & 1 & 1 & 0 & 0 & 0 & 5 & 0.7 & 0.7 \\ \hline
3L & 1 & $10^3$ & 0 & 0 & 0 & 10 & 7 & 7 \\
3R & 1 & 0.1 & 0 & 0 & 0 & 10 & 0.7 & 0.7 \\ \hline
4L & 1 & 0.1 & 0.999 & 0 & 0 & 10 & 7 & 7 \\
4R & 1 & 0.1 & -0.999 & 0 & 0 & 10 & -7 & -7 \\ \hline
\end{tabular}
\caption{Initial conditions for the one-dimensional shock tube problems
presented in the text. In all test problems we adopt
a resolution of $1600$ uniform computational zone, covering
the interval $[0,1]$. Integration is carried until $t=0.4$.}
\label{tab:ic}
\end{center}\end{table}
\subsubsection{Problem 1}\label{sec:p1}
The first test problem, initially proposed by \cite{vP93},
is a relativistic extension of the \cite{BW88} magnetic shock tube.
In analogy with the classical case we use the ideal equation
of state (\ref{eq:eos}) with specific heat ratio $\Gamma = 2$.
The breakup of the initial discontinuity sets up a left-going fast
rarefaction wave, a left-going compound wave, a contact discontinuity,
a right-going slow shock and a right-going fast rarefaction wave.
We compare, in Fig. \ref{fig:flat1}, the results
obtained with the first-order HLL and HLLC solvers on $100$ uniform
computational zones. The exact solution (given by the
solid line) was obtained using the
numerical code available from \cite{GR05}.
The left going compound wave located at $x \approx 0.5$
is only visible in the numerical integration since
the code used to generate the analytical solution
(shown as the solid line in Fig. \ref{fig:flat1}) does
not allow compound structures by construction.
As expected, the HLLC Riemann solver attains sharper representation of
the contact discontinuity when compared to the HLL scheme.
Because of the reduced smearing in proximity of the contact wave,
neighboring structures such as the compound wave on the left and the
slow shock on the right can be better resolved when using the HLLC
solver.
Computations at different resolutions show, in fact, that the
L-1 norm errors in density are reduced by roughly $20\div 30\%$
(see left panel in Fig. \ref{fig:res_1st}), with
$L_1(\%)$ being, respectively, $0.53$ and $0.74$
for the HLLC and HLL solver at the highest resolution
employed ($6400$ zones).
Fig. \ref{fig:sod1} shows the results obtained with the
second-order scheme with the MC limiter, eq. (\ref{eq:mc_lim}),
and the same Courant number, $CFL = 0.8$ on $1600$ grid points.
A direct comparison with the exact solution shows
that all discontinuities are correctly captured and resolved on
few computational zones, owing also to the presence of a
compressive limiter.
In this respect, our second-order HLLC scheme provides similar
results to those obtained with the third-order central ENO-HLL scheme
by dZBL.
The L-1 norm errors computed at different resolutions with
the two different solvers differ by $\approx 10\div 20\%$,
see left panel in Fig. \ref{fig:res_2nd}.
When compared to the more sophisticated, characteristic-based
algorithm presented in BA, our results show slightly sharper
representation of the right-going slow shock and the contact
discontinuity.
Small overshoots appear in the Lorentz factor profile at the left
going compound wave and the right going slow shock.
More diffusive slope limiters do not exhibit this feature.
\subsubsection{Problem 2}\label{sec:p2}
The resulting wave pattern for this configuration is comprised of
two left-going rarefaction fans (fast and slow) and two
right-going slow and fast shocks.
The specific heat ratio used for this calculation is
$\Gamma = 5/3$.
The weak slow rarefaction located at $x\approx 0.53$ and the
slow shock at $x\approx 0.86$ are separated
by a contact discontinuity where the proper density
changes by a factor of $\sim 7$.
The velocity on either side of the contact wave is mildly
relativistic, with a maximum Lorentz factor of $\approx 1.36$.
The improvement offered by the HLLC Riemann solver over the HLL approach
in the resolution of the contact wave is evident from Fig. \ref{fig:flat2},
where we compare the density profiles obtained with the first order
schemes against the analytical solution.
Computations obtained with the second-order limiter (\ref{eq:mc_lim})
show excellent agreements with the analytical profiles, see Fig.
\ref{fig:sod2}.
Our single-step HLLC scheme attain considerably sharper
resolution than the results obtained by previous calculations.
The two right-going shocks, for instance, are smeared over
$\sim 3$ grid points, approximately half of the resolution shown
in BA and dZBL.
Moreover, the smearing of the contact wave is considerably reduced
when compared to the HLL scheme in dZBL ($\sim 10$ zones vs.
$\sim 14$). Similar overshoots, though, appear at the right of
contact mode.
The discrete L-1 errors for different grid sizes are shown
in the right panel of Fig. \ref{fig:res_2nd}, where, at the
maximum resolution employed ($6400$ zones) the HLLC and HLL errors
reduce to $0.17 \%$ and $0.25\%$, respectively.
\subsubsection{Problem 3}\label{sec:p3}
The configuration for this test is similar to the previous problem,
but a higher pressure jump separates the initial left and right states,
see Table \ref{tab:ic}. Only the second-order scheme with the
Van Leer limiter (\ref{eq:vl_lim}) and a Courant number of $0.8$
has been employed. The ideal equation of state (\ref{eq:eos})
with $\Gamma = 5/3$ is used.
The ensuing wave pattern shows a stronger relativistic configuration,
with a maximum Lorentz factor of $\sim 3.37$, see Fig. \ref{fig:sod3}.
The presence of magnetic fields makes the problem even more
challenging than its hydrodynamical counterpart (see test 3 in
paper I), since the contact wave, slow and fast shocks now propagate
extremely close to each other.
As a result, a thin density shell sets up between the contact mode
and the slow shock. The higher compression factor (more than $100$)
follows from a more pronounced relativistic length
contraction effect.
At the resolution of $1600$ grid zones, the relative
error in the density peak ($\rho_{\max} \approx 9.98$) is $1.2 \%$.
A second thin shell-like structure forms between the
slow and fast shocks, as can be seen in the profiles in
Fig. \ref{fig:sod3}.
The peaks achieved in the transverse components of velocity
($\approx -0.37$) and magnetic field ($\approx 8.95$)
achieve, respectively, $87\%$ and $95\%$ of their exact values.
The small shell thickness, however, still prevents
a clear resolution of the two right going shocks,
visible in the exact solution.
This demonstrates that relativistic magnetized flows can develop
rich and complex features difficult to resolve on a grid of fixed
size.
Similar conclusions have been drawn by previous investigators.
Results obtained with the HLL solver (not shown here) indicates that
the resolution attained at the contact discontinuity is
equivalent.
Therefore, as it was also pointed out in paper I, we conclude that,
for strong blast waves where relativistic contraction effects produce
closely moving discontinuities, the HLL and HLLC schemes produce nearly
identical results.
\subsubsection{Problem 4}\label{sec:p4}
The collision of two relativistic streams is considered in
the fourth test problem.
The initial impact produces two strong relativistic fast shocks
propagating symmetrically in opposite direction about the impact
point, $x=0.5$, see Fig. \ref{fig:sod4}.
Two slow shocks delimiting a high pressure region in the center
follow behind.
Computations are carried out with $CFL = 0.8$ and the Van Leer
limiter, eq. (\ref{eq:vl_lim}).
Spurious oscillations in vicinity of strong shocks are reduced
by switching to the more diffusive minmod limiter, see
\S\ref{sec:shockflattening}.
No contact waves are present in the problem and, not
surprisingly, the quality of our solution is essentially
the same obtained by previous authors: the fast shocks
are resolved in $2\div 3$ cells, whereas the slow
shocks are smeared out over $5\div 6$ zones. Very similar
patterns are observed in the work of BA and dZBL.
It is well known that Godunov-type schemes suffer from
a common pathology, often found in these type of problems.
In the classical case, this has been recognized
for the first time by \cite{Noh87}.
The wall heating problem, in fact, consists in an undesired
entropy buildup in a few zones around the point of symmetry.
Our scheme is obviously no exception as it can be
inferred by inspecting the density profile
in Fig. \ref{fig:sod4}.
We repeated the test with the HLL scheme and
found that this pathology is worse when the HLLC scheme
is used. The relative numerical undershoot in density,
in fact, were found to be $\sim 5\%$ for the HLL and
$\sim 12\%$ for the HLLC scheme. Since similar
errors were also reported by BA, and the same conclusions
have been drawn in paper I, we raise the question as
to whether the degree of this pathology grows with the
complexity of the Riemann solver.
Future, more specific works should address this problem.
\subsection{Two-dimensional test problems}\label{sec:multid}
Multi-dimensional numerical computations of magnetized flows
are notoriously more challenging, due to the necessity to preserve the
divergence-free constraint (\ref{eq:divB}).
In what follows, we consider three test problems: a cylindrical blast
wave test, the interaction of a strong magnetosonic shock with a cloud
and the propagation of an axisymmetric jet in cylindrical coordinates.
\subsubsection{Cylindrical Blast Wave}\label{sec:p5}
Cylindrical explosions in cartesian coordinates are
particular useful in checking the robustness of the code and the
algorithm response to different kinds of degeneracies.
Here we follow the same setup adopted by KO, where the
square $[-6, 6]\times[-6, 6]$ is filled with a uniform
($\rho = 10^{-4}$, $p_g=3\cdot10^{-5}$), initially static ($\A{v} = \A{0}$)
medium, threaded by a constant magnetic field $\A{B} = (B_x,0)$.
The circular region $\sqrt{x^2 + y^2} < 0.08$ is initialized
with constant higher density and pressure values,
$\rho = 0.01$ and $p_g = 1$ decreasing linearly
for $0.08\le r \le 1$.
We adopt the ideal equation of state (\ref{eq:eos}) with specific heat
ratio $\Gamma = 4/3$.
We consider two setups, corresponding to a relatively weak
magnetic field $B_x = 0.1$ and a strong field $B_x = 1$.
Figures \ref{fig:blast_lo} and \ref{fig:blast_hi}
show the magnetic field distribution, thermal
pressure and Lorentz factor for the two configurations at $t = 4$.
Computations are carried using the van Leer limiter, eq. (\ref{eq:vl_lim}),
together with the multidimensional limiting procedure described in
\S\ref{sec:mdlimit} on $200\times200$ uniform grid zones.
The Courant number is $0.4$.
The expanding region is delimited by a fast forward shock propagating
(nearly) radially at almost the speed of light.
In the weak field case, a reverse shock delimits the inner region
where expansion takes place radially.
Magnetic field lines are squeezed in the $y$ direction
building up a shell of higher magnetic pressure.
In the $x$ direction the motion of the gas is not
hindered by the presence of the field and it achieves a higher
Lorentz factor ($\gamma_{\max} = 4.39$).
In the strong field case, the expansion is magnetically confined
along the $x$ direction and the outer fast shock has reduced
amplitude. The maximum Lorentz factor is $\gamma_{\max} = 4.02$.
We point out that numerical integrations for this test were
possible only by locally redefining the total energy
at the end of the time step:
\begin{equation}\label{eq:new_E}
E \rightarrow E + \frac{\bar{\A{B}}_{fa}^2 - \bar{\A{B}}_c^2}{2}\;,
\end{equation}
where $\bar{\A{B}}_c$ is the cell-centered magnetic field obtained after
the Godunov step, whereas $\bar{\A{B}}_{fa}$
is the new magnetic field obtained by averaging the face centered
values given by (\ref{eq:stokes}).
Notice that equation (\ref{eq:new_E}) only redefines the energy contribution
of the magnetic field that is not directly coupled to the velocity,
see eq. (\ref{eq:cons_var_E}) and thus may be regarded as a first-order
correction.
In this respect, the energy correction
we propose is the same usually adopted in CT schemes, see \cite{BS99},
\cite{Toth00}.
Although this optional step results in a slight loss of energy conservation
at the discretization level, it was nevertheless found to become
particularly useful in problems where the magnetic pressure dominates over
the thermal pressure by more than two order of magnitudes.
\subsubsection{Relativistic Shock-Cloud Interaction}\label{sec:p6}
The interaction of a strong relativistic fast shock with a
cloud is considered on the unit square $[0,1]\times[0,1]$
in 2-D cartesian coordinates $(x,y)$.
This problem has been extensively used for testing classical MHD codes
see \cite[][and references therein]{DW94,Toth00}.
Here we consider a relativistic extension adopting a somewhat
different initial condition, with magnetic field
orthogonal to the slab plane.
The shock wave travels in the positive $x$-direction and is initially
located at $x=0.6$. Upstream, for $x > 0.6$, the flow is highly
supersonic with pre-shock values given by $(\rho, \gamma_x, p_g, B_z)_\textrm{pre} =
(1, 10, 10^{-3}, 0.5)$, where $\gamma_x = (1 - v_x^2)^{-\HALF}$.
In this reference frame, shocked material is at rest with values given by
\begin{equation}
\left(\begin{array}{c}
\rho \\ \noalign{\medskip}
p_g \\ \noalign{\medskip}
B_z
\end{array}\right)_{\textrm{post}} =
\left(\begin{array}{c}
42.5942 \\ \noalign{\medskip}
127.9483 \\ \noalign{\medskip}
-2.12971 \\ \noalign{\medskip}
\end{array}\right) \;.
\end{equation}
Notice that the magnetic field carries a rotational discontinuity
and the compression factor of density across the shock in not
limited to $7$ (we use $\Gamma = 4/3$) as in the classical case,
but achieves a much higher value ($\approx 43$).
This feature is unique to relativistic flows.
A circular density clump with $\rho = 10$ and radius $r = 0.15$
is placed ahead of the shock front, centered at $(x,y) = (0.8,0.5)$.
Transverse velocities $v_y$ and $v_z$ and the $x$ and $y$ components
of magnetic field are set to zero everywhere.
We use $400\times 200$ computational zones, by assuming reflecting
boundary at $y = 0.5$ and free flow across the remaining boundaries.
The MC limiter, eq. (\ref{eq:mc_lim}), is employed everywhere
except in proximity of strong shocks where we
revert to the minmod limiter, see \S\ref{sec:shockflattening}.
The Courant number is $0.4$.
Shortly after the impact, the cloud undergoes strong
compression with the density rising by a factor of more than $20$.
The collision generates a bow fast shock propagating in the shocked
material and a reverse shock is transmitted into the cloud.
After the transmitted shock reaches the back of the cloud, the
two bent parts of the original incident shock join back together
and complicated wave pattern emerges.
By $t=1$ the cloud is completely wrapped by the incident shock,
and the cloud expands in the form of a mushroom-shaped shell,
see upper half of Fig. \ref{fig:sc}.
The solution computed with the HLL solver (lower half in
Fig. \ref{fig:sc}) show similar structures, although
the amount of numerical viscosity is considerably higher.
Notice that, because of the assumed slab symmetry, the condition
$\A{v}\cdot\A{B} = 0$ is preserved in time
and the solution to the Riemann problem at each interface
consists of a three wave pattern:
two fast waves separated by a tangential discontinuity.
In this regard, our HLLC solver provides a better approximation
of the full wave structure.
\subsubsection{Relativistic Jet}\label{sec:p8}
As a final example, we consider the propagation of an axisymmetric
jet in cylindrical coordinates $(r,z)$.
The configuration adopted here corresponds to model C2-pol-1 in \cite{LAAM05}.
The domain $[0,12]\times[0,50]$ (in units of jet beam) is initially
filled with a static uniform distributions of density, gas pressure and
magnetic field, given respectively by
\begin{equation}
\rho_a = 1 \,,\quad
p_a = \frac{\eta v_b^2}{\Gamma(\Gamma-1)M^2 - \Gamma v_b^2}
\, ,\quad B_z = \sqrt{2p_a}.
\end{equation}
The numerical value of $p_a$ follows from the definitions of
the beam Mach number $M = v_b/c_s = 6$, jet to ambient density
ratio $\eta = 10^{-2}$ and beam axial velocity $v_b=0.99$.
The ideal equation of state (\ref{eq:eos}) is used with $\Gamma = 5/3$.
The jet nozzle is located at the lower boundary $r \le 1$, $z=0$,
where boundary conditions are held constant in time,
$(\rho, v_r, v_z, B_r, B_z, p_g) = (\eta, 0, v_b, 0, B_z, p_a)$.
For $r>1$ we prescribe boundary values with antisymmetric
profiles for axial velocity and radial magnetic field.
Symmetric profiles are imposed on the remaining quantities.
This configuration corresponds to a twin counter jet propagating
in the opposite direction.
Outflow boundaries are imposed on all other sides, except
at $r=0$ where reflecting boundary conditions are used.
We employ a uniform resolution of $20$ zones per beam radius
and carry integration until $t = 126$ with $CFL = 0.4$.
The results are shown in Fig. \ref{fig:jet}, where we display
density logarithm (upper panel), magnetic pressure (middle panel) and
Lorentz factor distributions (lower panel).
In each panel, the upper and lower halves show the solutions obtained
with the HLLC and HLL solvers, respectively.
As we already pointed out in the non magnetic case (Paper I), the
HLLC integration features considerably less amount numerical
diffusion as evident from the richness in small scale structures,
notably in the density distribution.
In fact, density is the physical quantity more sensitive to the
introduction of the tangential wave in the Riemann
solver. Comparing our results with those of \cite[][see their Fig. 5]{LAAM05}
we can observe that our solution has a similar
(or even larger) richness in fine structure details at half the resolution
(20 ppb in our case, 40 ppb in their case).
\section{Conclusions}
An HLLC approximate Riemann solver has been developed
for the relativistic magnetohydrodynamic equations.
The new approach improves over the single state HLL solver
in the ability to capture exactly isolated tangential and
contact discontinuities.
Several test problems in one and two dimensions demonstrate
better resolution properties and a reduced amount of the
numerical diffusion inherent to the averaging process of the
single state HLL scheme.
The solver is well-behaved for strictly two-dimensional flows,
although applications to genuinely three-dimensional problems
may suffer from a pathological singularity when the
component of magnetic field normal to a zone interface
approaches zero. This feature does not persist in the classical
limit.
Multidimensional integration has been formulated in a
versatile and efficient way within the framework
of the corner transport upwind (CTU) method.
The algorithm is stable up to Courant numbers of $1$ and
preserves the divergence-free condition
via constrained transport evolution of the magnetic field.
The additional computational cost and the numerical
implementation in an existing relativistic MHD code are
minimal.
\appendix
\section{}
\subsection{Shock Flattening}\label{sec:shockflattening}
For strong shocks, we found that the
one-dimensional prescriptions (\ref{eq:mc_lim}) or (\ref{eq:vl_lim})
can still produce spurious numerical oscillations eventually
leading to the occurrence of negative pressures.
A weak form of flattening is introduced by replacing eq.
(\ref{eq:mc_lim}) or (\ref{eq:vl_lim}) with the minmod limiter
whenever a strong shock is detected.
In order for the latter condition to hold, we require that
both $\nabla\cdot\A{v} < 0$ and $\chi_{\min} = 0$,
where $\nabla\cdot\A{v}$ is computed by central differences whereas
\begin{equation}
\chi_{\min} = \min\left(\chi^x_{i+1,j},\chi^x_{i,j},\chi^x_{i-1,j},
\chi^y_{i,j+1},\chi^y_{i,j},\chi^y_{i,j-1}\right) \,.
\end{equation}
The switches $\chi^x$ and $\chi^y$ are designed as follows
\begin{equation}
\chi^x_{i,j} = \left\{\begin{array}{cc}
1 & \DS \; \textrm{if}\quad \frac{p_{i+1,j} - p_{i-1,j}}
{\min\left(p_{i+1,j}, p_{i-1,j}\right)} \le \epsilon \;,\\ \noalign{\medskip}
0 & \DS \; \textrm{otherwise} \;,
\end{array}\right.
\end{equation}
\begin{equation}
\chi^y_{i,j} = \left\{\begin{array}{cc}
1 & \DS \; \textrm{if}\quad \frac{p_{i,j+1} - p_{i,j-1}}
{\min\left(p_{i,j+1}, p_{i,j-1}\right)} \le \epsilon \;,\\ \noalign{\medskip}
0 & \DS \; \textrm{otherwise} \;,
\end{array}\right.
\end{equation}
where we set $\epsilon = 5$ in all computations presented
in this paper.
\subsection{Multidimensional Limiting}
\label{sec:mdlimit}
Occasionally, we found that strong shocks propagating
obliquely to the grid in highly magnetized media may
benefit from an additional form of limiting, based
on genuinely multidimensional constraints.
When needed, we enforce the maximum and minimum interpolated values
in each cell $(i,j)$ to lie within the bounds provided by the four
neighboring zones $(i+1,j), (i-1,j), (i,j+1), (i,j-1)$.
Specifically, denote with $\hat{q}^{\max}$ and
$\hat{q}^{\min}$ the maximum and minimum values of
$q\in\A{V}$ in these cells.
Once the limited slopes $\delta_x q$ and $\delta_y q$
have been computed according to (\ref{eq:mc_lim}) or
(\ref{eq:vl_lim}), we apply the correction
\begin{equation}
\delta_xq \rightarrow \tau\delta_xq\;, \quad
\delta_yq \rightarrow \tau\delta_yq\;, \quad
\end{equation}
where the multi-dimensional limiter $\tau$ is constructed
as in \cite{Balsara04}:
\begin{equation}
\tau = \min\left(1,\psi\min\left(
\frac{\hat{q}^{\max} - q}{\delta^{\max}},
\frac{q - \hat{q}^{\min}}{\delta^{\min}}
\right)\right) \;,
\end{equation}
with $\delta^{\max} = \max(|\delta_x q|,|\delta_y q|)$,
$\delta^{\min} = \min(|\delta_x q|,|\delta_y q|)$.
We set $\psi = 2$ for density and magnetic field, $\psi = 3/4$ for
velocity and $\psi = 1$ for thermal pressure.
\label{lastpage} |
Title:
Seeing Star Formation Regions with Gravitational Microlensing |
Abstract: We qualitatively study the effects of gravitational microlensing on our view
of unresolved extragalactic star formation regions. Using a general
gravitational microlensing configuration, we perform a number of simulations
that reveal that specific imprints of the star forming region are imprinted,
both photometrically and spectroscopically, upon observations. Such
observations have the potential to reveal the nature and size of these star
forming regions, through the degree of variability observed in a monitoring
campaign, and hence resolve the star formation regions in distant galaxies
which are too small to be probed via more standard techniques.
| https://export.arxiv.org/pdf/astro-ph/0601667 | command.
\shorttitle{Star Formation \& Gravitational Microlensing}
\shortauthors{Gil-Merino \& Lewis}
\begin{document}
\title{Seeing Star Formation Regions with Gravitational Microlensing}
\author{Rodrigo Gil-Merino\altaffilmark{1} \& Geraint F. Lewis\altaffilmark{2}}
\affil{Institute of Astronomy, School of Physics,
University of Sydney, NSW 2006, Australia}
\altaffiltext{1}{rodrigo@physics.usyd.edu.au}
\altaffiltext{2}{gfl@physics.usyd.edu.au}
\keywords{gravitational lensing -- microlensing -- star forming
regions -- dark halo populations}
\section{Introduction}
Gravitational microlensing is now a well established technique for the
investigation of the distribution of compact (dark) matter in the
universe. Furthermore, it also provides a powerful tool to study
unresolved sources, such as in the case of the structure of QSOs,
through temporal differential magnification (e.g. Yonehara et
al. 1998).
From an observer's point of view, gravitational microlensing can be
naturally divided in two different regimes. In the case of Galactic
microlensing, the optical depth is low and a single star microlenses
another star within the Galactic halo or in one of the galaxies in the
Local Group (Paczy\'nski 1986a). With Extragalactic microlensing,
where the light from a distant quasar shines through a closer galaxy,
the optical depth is roughly unity and many stars contribute to the
overall microlensing effect (Paczy\'nski 1986b). This paper
considers this latter regime, were the source region is populated by a
number of hot, young stars in a star forming region. Such a situation
will occur in strongly lensed, multiply imaged systems, such as the
multiple images seen in galaxy clusters (Mellier 1999), or the case
where a isolated galaxy gravitationally lenses a more distant galaxy
(i.e. Warren et al 1996).
In a similar vein, Lewis \& Ibata (2001) investigated the effect of a
cosmological distribution of compact objects on the surface brightness
distributions of galaxies at $z$$<$0.5, considering a small
microlensing optical depth ($\leq$0.04) and they determined that
low-level fluctuations in surface brightness of $\sim$2\% should
result. Lewis et al. (2000) extended that analysis to distant
galaxies observed through galaxy clusters, assuming dark matter to be
composed of compact objects. Focusing upon Abell~370 as a case study,
concluding that for low-luminosity ($\sim$10$^4$L$_\odot$) stellar
populations would show rapid fluctuations exceeding 10\% of the mean
in the highest cases.
In this contribution we address the question of what microlensing
signatures should be apparent in the case of part of a galaxy which is
lensed by another galaxy. In particular, if the lensed parts of the
source galaxy are regions of star formation, highly dominated by
young, massive stars. Such a situation was recent presented by Smith
et al. (2005) who reported the discovery of a new strong
gravitationally lensed system, with an elliptical galaxy acting as the
lens. The lens galaxy in this system is at redshift $z=0.0345$ and
the source, proposed to be a star formation region, is at $z\sim0.45$,
with the arcs formed by the gravitational mirage showing `knots' of an
extreme blue color of $B-I_c=1.1$ (extinction corrected). This
discovery poses the idea that microlensing in the multiple images of
these systems might be able to distinguish the type of source stars
involved in the mirage and help in the interpretation of its nature.
Within the context of gravitational lensing, a star formation region
would appear as a non-uniform source, composed of a number of bright
points in a more extended background. Hence, the microlensing imprint
of such a source should show quite a different variability imprint
from the uniform sources typically considered in gravitational
microlensing experiments. The nature of this imprint is the basis of
this current contribution.
\section{Microlensing simulations}\label{sim}
For the purpose of this study we performed microlensing simulations by
means of ray-shooting techniques (Paczy\'nski 1986b, Schneider \&
Weiss 1987, Kayser et al. 1986, Wambsganss 1990, Witt 1993, Lewis et
al. 1993). To compute magnification patterns, one has to select
certain values for the convergence ($\kappa$), which represents the
gravitational potential due to matter in the beam, and the shear
($\gamma$), which is the perturbation to the beam due to the large
scale distribution of matter. Typically, these parameters are drawn
from a lens model for a particular system. For this study, however,
representative values of $\kappa=0.55$ and $\gamma=0.55$ are employed,
following Schechter et al. (2004), although other combinations would
illustrate the situation equally; $kappa$ here includes also any form
of compact dark matter, the effects of an smooth dark matter component
are described in Schechter \& Wambsganss (2002). Since high resolution maps are
required, we used a receiving field of 2 Einstein radii\footnote{
The Einstein radius is defined in the source plane as $ER=sqrt{(4GM/c^2)
(D_{s}D_{ls}/D_{l})}$, where $M$ is the mass of the microlens, $D$ is the
angular distance to the source (s), the lens (l) and between the lens and
the source (ls), c is the velocity of light and G the gravitational
constant},
covered by a $2048^2$ pixels area. The
microlenses were randomly distributed and selected to have the same
mass, $M_{\mu lens}=1 M_{\odot}$. Again, the selection of the mass
range is arbitrary for our purposes. However, it is important to note
that rather than covering a large area in the simulations, the key
point remains in the resolution of the magnification patterns, because
we are interested in small flux changes from pixel to pixel, so we
also selected a high number of rays that resulted in over 700 per
pixel on average.
The next step in the simulations is introducing the effect of the
source. To do this, we assumed a source plane at $z=0.5$ and two
different sizes of $0.1$ and $0.5$~Einstein radii (ER), which
corresponds to a physical size of $0.02$~pc and $0.1$~pc respectively
at that distance for the standard $\Lambda$CDM cosmoslogy. Although
star formation regions might be larger than the bigger size
considered, these two examples will illustrate the different effects
due to their sizes and could be seen as clumps of star formations
within larger regions (compact and ultra-compact H~{\small II} regions
as indicators of star formation might be $<$0.1~pc, see e.g. Giveon et
al. 2005 and references therein). We also assumed that our lens plane
is at $z=0.04$ (following the case of Smith et al. 2005). Depending
on the stellar density of the source region, the number of stars in
that region can vary from just a few up to hundreds. Considering
first the $0.1$~Einstein radii region, we `built' three different
sources: one containing 8 stars, another containing 80, and the last
one as a uniform source, i.e., containing one star per pixel in the
region (the number of stars are not representative of any particular
region, and have been chosen arbitrarily).
The results for the first region size are displayed in
Fig.~\ref{fig1}. The upper left-hand pannel corresponds to a
$\sim$1~ER$^2$ region of the original magnifcation pattern. The upper
right-hand pannel is the magnification pattern convolved with a region
of $0.1$~ER containing 8 stars. The lower right-pannel is the same as
the previous one but containing 80 stars. The lower left-hand pannel
is the magnification pannel convolved with an `uniform' source of the
same physical size.
In all the panels the same track has been drawn, in order to compared
the synthetical light curves to each other; these are depicted in
Fig.~\ref{fig3}. The light curves are $\sim$1~ER long, showing the
different expected fluctuations corresponding to the different
scenarios. The magnification distributions for the magnification
patterns corresponding to the different panels in Fig.~\ref{fig1} are
shown in Fig.~\ref{fig2}. Clearly the number of stars in the region
has a significant influence on the resulting light curve; in effect,
the presence of each star produces a ``shift-and-add'' to the
mangification map, greatly increasing the number and overall density
of caustics. This is reflected as additional peaks in the light
curve. As the number of stars in increased to 80, some of the caustic
structure has begun to wash out, leaving small scale fluctuations
superimposed on a more gentle background, whereas the smooth source
(which can be thought of as a very high density of stars) has washed
out all small scale detail.
In Fig.~\ref{fig3b}, for comparison, we consider a
region size of $0.5$~ER containing also 8 stars, 80 stars and a
`uniform' source in the same manner as in Fig.~\ref{fig1}.
The corresponding track shows a completely different light
curves compare to Fig.~\ref{fig3} although their positions are the
same, due to the new caustic structure of the magnification maps
according to the different size of the region considered.
The interpretation of these figures: if the
magnification pattern is convolved with a `uniform' source profile
(Fig.~\ref{fig1}, lower left panel) the result is always a smoother
pattern, with smooth transitions in the value of the magnification
from pixel to pixel; if the source area is made of a number of
point-like objects, the convolution will show many caustics slightly
shifted one another, with no smooth transition between them. This
translates into a rapid variability in the lightcurves of the
corresponding source. Also, the size of the regions considered plays
an active role in the final imprint of microlensing in the
observational light curves.
\section{Applications and Discussion}
The application of the simulations described in the previous Section
can be done in the following manner. If multiple lensed `knots' are
detected in an image (see, e.g., Fig. 3c in Smith et al. 2005) and are
thought to be star-forming regions, the flux will be highly dominated
by young O-stars. In principle, since young massive stars are rare due
to their evolutionary process, only a few are expected in these star
forming knots (an ultraviolet and optical spectral atlas of the Small
Magellanic Cloud includes $<$20 O-stars, see Walborn et al. 2000).
Observing these areas, e.g. in the UV band, which characterizes
regions of star formation, with periodic photometry, the variability
of the observed light curves will be related to the number and
separation of these stars present in the star forming regions (the
contamination by late-type stars in the UV will be almost null). In
practice, one could treat the problem statistically by simulating the
observed variability and thus put limits on the amount and luminosity
of young dominating stars. Knowing the luminosity of these stars
accurately is important, because their masses derived by stellar
evolutionary models and by stellar atmosphere models can be compared.
Stellar evolution theory and initial mass function might take
advantage of these results as well. Gravitational microlensing might
be the only tool to `resolve' these stars in clusters of star
formation, otherwise impossible to investigate in galaxies at
moderate/high redshift.
Spectroscopy of microlensed star-forming regions might help to put
limits on their nature as well. Gravitational microlensing of broad
spectral lines in QSOs has been studied theoretically by a number of
authors (e.g., Abajas et al. 2002, Lewis et al. 2004, Richards et al.
2004) and used to put limits on the size of the broad line emitting
regions of the QSOs. In those cases, the natural shape of a line is
distorted by the complex net of caustics produced by the microlenses
on the source plane. Since microlensing of the broad line region is
expected when its physical size is of the order of the Einstein radius
of the lens projected onto the source plane, microlensed spectral
lines give an idea of such physical sizes. In the same way, observing
typical spectral lines of O-stars (e.g. in the UV, O {\small V}
$\lambda$1371\AA, C {\small III} $\lambda$1176\AA~in the optical, He
{\small II} $\lambda$4686\AA, N {\small III}$\lambda$4634\AA~and
$\lambda$4640\AA) one would expect the lines to be deformed by the
presence of the caustics (due to magnifications/demagnifications), and
these variations in the spectral lines might reveal the size and
populations of the star forming regions.
To illustrate this, we plot in Fig.~\ref{fig4} the spectra of an
O-star (upper pannel) and a solar-like G-star (lower pannel), obtained
from the Kurucz models
database\footnote{http://garnet.stsci.edu/STIS/stis\_models.html}. For
any unresolved star formation region, the (far-)UV range of the
spectrum will be dominated by these O-stars. Late-type stars fluxes in
UV are several orders of magnitude lower and thus they do not
contribute significantly to the total luminosity and the flux
distribution in the upper pannel of Fig.~\ref{fig4} might be a
representative one for that part of the spectrum (different lines
might be present, obviously). Microlensing affecting this part of the
spectrum will only show an enhancement of the flux. However, the
effect is slightly different when using optical range of the
spectrum. In this case, the flux contribution due to late-type stars
starts to be dominant, although O-stars flux is still significantly
present. We construct a toy star formation region model,
merging the spectra of the O-star and the G-star shown in
Fig.~\ref{fig4}, assuming that 90\% of the total flux comes from
solar-like stars and the rest is produced by early-type stars. The toy
model is depicted in Fig.~\ref{fig5} (lower line, spectrum marked as
'O-star$+$G-star') showing only the 1500\AA-5500\AA ~wavelength
interval. When the star formation region travels across the
magnification pattern in Fig.~\ref{fig1}, late-type stars will act as
a constant flux background as a whole and microlensing will affect
mainly O-stars. In this way, the microlensing signature in the spectra
will be a flux ratio variation between O-stars and late-type stars
spectral lines. This is shown also in Fig.~\ref{fig5} (upper line,
spectrum marked as 'O-star$+$G-star$+$microlensing'). There is not
only an enhancement of the flux in the bluest part of the spectrum,
but also a deformation of certain lines due to the different
microlensing effect on the different type of stars. In Fig.~\ref{fig6}
and Fig.~\ref{fig7} we repeat the procedure, but assuming a different
relative flux between the two types of stars. In Fig.~\ref{fig6}, 1\%
of the flux is coming from O-type stars and 99\% is from late-type
stars; in Fig.~\ref{fig7} the percentage is 0.1\% and 99.9\% for early
and late-type stars respectively. As shown in Fig.~\ref{fig3}, high
variability is expected. Both Figures~\ref{fig2} and \ref{fig3} show
that the amount of variability depends on the nature of the star
formation regions (number of stars, size of the regions...). This
means that comparing several consecutive spectra one would be able to
statisticaly determine the relative flux variability of the spectral
lines and continuum due to the presence of the caustics, and thus
compare them with the expected one from the simulations, puting limits
to the number and distribution of the early-type stars.
A key point to note is that to detect these microlensing effects on
star formation regions is the time scale of the events. Considering
the lens configuration describe in Sec.~\ref{sim}, we can estimate a
typical separation between microcaustics in the magnification maps in
Fig.~\ref{fig3}. This separation is $\sim$0.005~ER for the upper
right-hand panel, which corresponds to approximately 10$^{-3}$~pc.
Assuming a transverse velocity for the source galaxy of
$\sim$6000~km/s (see Kayser et al. 1986), the resulting time-scale for
the events is $\sim$50 days. The time-scales of the events get shorter
when the number of O-stars gets higher, although the flux variability
is smaller. This means that six data points in a time period of around
three months should be able to described the type of variability
involved in the gravitational microlensing scenario.
\section{Conclusions}
We described in this contribution how to apply gravitational
microlensing to the observations of unresolved extragalactic star
forming regions. The discussion shows that due to the caustics
configuration in the magnification maps of the region, rapid
monitoring campaigns, both photometric or spectroscopic, would reveal
high variability fluctuations due to the number of early-type
stars. The specific amount of variability will depend on the number of
stars and their distribution in the region, as well as on the exact
configuration of the microlenses in the lensing galaxy. Thus, the
study of a particular system requires the knowledge of a lens model to
perform the right simulations and the analysis of the results should
be based on a statistical approach. The advantage of the method, if
these circumstances take effect, is that we might be able to
investigate star formation regions which are difficult to analyse with
more traditional techniques.
\acknowledgments
|
Title:
The Small-Scale Environment of Quasars |
Abstract: Where do quasars reside? Are quasars located in environments similar to those
of typical L* galaxies, and, if not, how do they differ? An answer to this
question will help shed light on the triggering process of quasar activity. We
use the Sloan Digital Sky Survey to study the environment of quasars and
compare it directly with the environment of galaxies. We find that quasars (M_i
< -22, z < 0.4) are located in higher local overdensity regions than are
typical L* galaxies. The enhanced environment around quasars is a local
phenomenon; the overdensity relative to that around L* galaxies is strongest
within 100 kpc of the quasars. In this region, the overdensity is a factor of
1.4 larger than around L* galaxies. The overdensity declines monotonically with
scale to nearly unity at ~1 Mpc, where quasars inhabit environments comparable
to those of L* galaxies. The small-scale density enhancement depends on quasar
luminosity, but only at the brightest end: the most luminous quasars reside in
higher local overdensity regions than do fainter quasars. The mean overdensity
around the brightest quasars (M_i < -23.3) is nearly three times larger than
around L* galaxies while the density around dimmer quasars (M_i = -22.0 to
-23.3) is ~1.4 times that of L* galaxies. By ~0.5 Mpc, the dependence on quasar
luminosity is no longer significant. The overdensity on all scales is
independent of redshift to z = 0.4. The results suggest a picture in which
quasars typically reside in L* galaxies, but have a local excess of neighbors
within ~0.1 - 0.5 Mpc; this local density excess likely contributes to the
triggering of quasar activity through mergers and other interactions.
| https://export.arxiv.org/pdf/astro-ph/0601522 |
\title{The Small-Scale Environment of Quasars}
\author{Will~Serber\altaffilmark{1}, Neta~Bahcall\altaffilmark{1},
Brice~M\'{e}nard\altaffilmark{2},
Gordon~Richards\altaffilmark{1}\altaffilmark{3}}
\altaffiltext{1}{Princeton University Observatory, Princeton, NJ
08544, USA} \altaffiltext{2}{Institute for Advanced Study, Einstein
Drive, Princeton, NJ 08540, USA} \altaffiltext{3}{Department of
Physics and Astronomy, The Johns Hopkins University, 3400 North
Charles Street, Baltimore, MD 21218-2686}
\slugcomment{Dec15 '05}
\keywords{Quasars: General, Galaxies: Statistics}
\section{Introduction} \label{sec.intro}
For more than two decades, significant effort has been spent
attempting to understand the triggering mechanism of quasar activity,
as well as the relation between quasars and their host galaxies.
Since \citet{bell_1969}, it has become widely accepted that quasars
are fueled by accretion of gas onto super-massive black holes.
Observations have shown that a number of nearby galaxies have a
central black hole whose mass correlates with the luminosity of the
spheroid of the host galaxy. This connection suggests that the
formation of the black hole is linked to the formation of the galaxy
which, in turn, is known to strongly depend on its environment. To
develop a better understanding of the quasar phenomenon, it is
therefore important to investigate and quantify the relation between
quasars and their environments. Despite the importance of this issue,
our knowledge of the quasar environment is still limited.
Quasar environments have been studied on different scales ranging from
those of the host galaxy to those of large scales. Such studies have
provided important but controversial results regarding the environment
of quasars. It has been known for more than three decades that quasars
are associated with enhancements in the spatial distribution of
galaxies \citep{bahcall_1969}. Studies have shown that, in the nearby
universe, quasars reside in environments ranging from small to
moderate groups of galaxies rather than in rich clusters
(\citealt{bahcall_1991b,fisher_1996,mclure_2001}).
Early observations of quasar environments \citep{stockton_1978,
yee_1984, yee_1987, boyle_1988, smith_1990, ellingson_1991}, revealed
a positive association of bright quasars with neighboring galaxies at
a level somewhat higher than that of normal galaxies and comparable to
the environment of small- to intermediate-richness groups of galaxies
\citep[e.g.,][]{bahcall_1991}. Observations of quasar environment from
the Hubble Space Telescope snapshot survey \citep{bahcall_1997}
further support this density enhancement around bright quasars. All
these observations focused on small scales, typically within $\sim
0.5\,$Mpc of the quasars, and used relatively small samples of objects.
Early observations of the clustering properties of quasars themselves,
as measured by the quasar auto-correlation function, suggest that
quasars are significantly more strongly clustered than galaxies on
scales up to $10\,$Mpc and greater \citep[e.g.,][]{shaver_1988,
shanks_1988, chu_1988, chu_1989, crampton_1989}, but less clustered
than rich clusters of galaxies \citep[e.g.,][]{bahcall_1991}. This
finding suggests that quasars are located in high overdensity regions,
more so than $L^*$ galaxies, since higher overdensity regions are
clustered more strongly than lower overdensity regions
\citep[e.g.,][]{bahcall_1983, kaiser_1984, bardeen_1986}. An overdense
environment would indeed be expected if the quasar activity was
triggered by galaxy interactions.
On the other hand, new generation surveys, such as the Two Degree
Field (2dF) and the Sloan Digital Sky Survey (SDSS), have given rise
to different results. Using significantly larger complete samples of
quasars and galaxies, these surveys have shown that on large scales,
i.e. from 1 to $10\,$Mpc, the quasar-galaxy cross-correlation and the
quasar auto-correlation are comparable to the correlation function of
$L^*$ galaxies. This suggests, in conflict with previous results, that
quasars and Active Galactic Nuclei (AGNs) inhabit environments similar
to those of $L^*$ galaxies \citep[e.g.,][]{smith_1995, croom_1999,
croom_2003, miller_2003, kauffmann_2004, wake_2004}. The results
also suggest that the quasar correlation function does not depend
significantly on either quasar luminosity or redshift within the
ranges studied.
Recent work on sub-Mpc scales using the SDSS to find quasar pairs
suggests that the quasar-quasar auto-correlation function may be
enhanced relative to the galaxy-galaxy distribution
\citep{hennawi_2005}, consistent with the earlier results on small
scales, as discussed above.
In this paper we use the SDSS survey to determine the galaxy
environment around quasars as a function of scale. The SDSS is
uniquely suited for this investigation: it is the largest complete
survey available of both galaxies and quasars, carried out in a
well-calibrated, self-consistent manner. The data used in this study
covers 4000 deg$^2$, with $\sim2\times10^3$ quasars of redshift
$z\le0.4$ and ten million photometric galaxies to a magnitude limit of
$i = 21$. We use these data to determine the mean galactic environment
around quasars as a function of quasar luminosity and redshift. For
comparison, the same analysis is then repeated to find the local
environment around $10^5$ spectroscopic galaxies in the SDSS area, as
well as around random positions in the survey. All the analyses are
carried out using the same ten million photometric galaxies. This
technique allows a direct comparison between the environment around
quasars with that around random points as well as with the environment
around $L^*$ galaxies, thereby minimizing potential selection effects and
systematics.
The outline of the paper is as follows: we discuss the data in
Section~\ref{sec.data}, the analysis in Section~\ref{sec.analysis},
and the results in Section~\ref{sec.results}. The conclusions are
summarized in Section~\ref{sec.conclusions}. Throughout this paper, we
use a cosmological model with $H_0\,=\,70\,{\rm km^{-1}\,Mpc^{-1}}$,
$\Omega_M\,=\,0.3$, and $\Omega_{\Lambda}\,=\,0.7$ for both absolute
magnitudes and distance measures. All distances are measured using
comoving coordinates.
\section{Data} \label{sec.data}
We use Sloan Digital Sky Survey (SDSS) data to determine the galactic
environment of quasars and galaxies. The SDSS
\citep{york_2000,stoughton_2002,pier_2003,Abazajian_2003,gunn_2005} is
conducting an imaging survey of $10^4$ square degrees of the sky in
five bands ($u, g, r, i, z$) \citep{fukugita_1996,gunn_1998}, followed
by a spectroscopic multi-fiber survey of the brightest $10^6$ galaxies
and $10^5$ quasars. The spectroscopic targets are selected from the
high quality imaging data using well-defined selection criteria
\citep{lupton_2001,hogg_2001,strauss_2002,richards_2002}. The
drift-scan imaging survey reaches a limiting magnitude of $r < 23$
\citep{fukugita_1996,gunn_1998,lupton_2001}. The main spectroscopic
survey targets galaxies to $r < 17.7$, with a median redshift of $z
\sim0.15$ and a tail reaching $z \sim0.4$ (Strauss et al 2002). The
spectroscopic survey of quasars , with $i < $19, reaches quasar
redshifts out to $z \sim5.4$. For more details on the SDSS see the
above references.
The high quality imaging and spectroscopic survey of quasars and
galaxies provides a unique data set for studying the environment of
quasars and comparing it directly with the environment of galaxies. To
do so, we use the third data release (DR3) of the SDSS spectroscopic
sample of quasars \citep{schneider_2005}, selecting all
spectroscopic quasars with redshift $z \le 0.4$ and $i$-band
Galactic extinction corrected and k-corrected magnitude $-24.2 \le M_i
\le -22.0$ \citep{richards_2002}. We set an upper limit of $-24.2$ on
the luminosity to avoid bright objects that may interfere with
counting nearby galaxies. After applying masks for missing fields (see
the end of this section), a sample of 2028 $z \le 0.4$ quasars is
used, covering an area of approximately 4000 deg$^2$. In addition to
the quasars, we use a sample of spectroscopic galaxies as targets in
our analysis so that we may compare the environment of quasars with
that of galaxies. The spectroscopic galaxy sample used for comparison
is the NYU-LSS sample 12 \citep{blanton_2003a,blanton_2003b}, which is
comprised of a complete spectroscopic sample of galaxies to
$i\,=\,18.5$ corrected for both Galactic extinction and k-correction,
with redshifts in the range $z~\sim0.001$ to $z~\sim0.4$. After
applying masks and limiting the galaxy redshift to $0.08 \le z \le
0.4$ (since there are no quasars with $z < 0.08$), a sample of
$\sim10^5$ spectroscopic galaxies is available over a $2230\,$ deg$^2$
area (mostly overlapping the quasar area). Our galaxy sample has a
median redshift of 0.13 and a median magnitude of $M_i = -21.3$, and
our quasar sample has a median redshift of 0.32 and a median magnitude
of $M_i = -22.5$.
The environment of the above targets - spectroscopic quasars and
spectroscopic galaxies - is then determined using the photometric
galaxies from DR3 of the SDSS imaging survey. We use a sample of over
10 million galaxies with magnitude in the range $14 \le i \le 21$. For
further comparison, we also repeat the environment analysis at
approximately $10^3$ random positions per target using the same method
and background sample of photometric galaxies.
All samples were corrected using the same masks, which remove missing
fields, missing stripes, and regions where the stripe boundaries
extend beyond the extent of the photometric galaxy survey. These masks
were created with SDSSpixel, a pixelization scheme routinely applied
to the SDSS data\footnote{See
http://lahmu.phyast.pitt.edu/\~{}scranton/SDSSPix/ for information on
SDSSpixel}. We also remove all targets (i.e., quasars, spectroscopic
galaxies, and random points) that are closer than $1\,$Mpc to the
boundary or to a mask in the photometric sample in order to ensure
that all targets have a complete field of photometric galaxies within
the scales of interest.
\section{Analysis} \label{sec.analysis}
We study the environment of the spectroscopic targets (quasars and
galaxies) by determining the number of photometric galaxies within
different projected radii from the quasars, from $25\,$kpc to $1\,$Mpc
($h=0.7$). In order to normalize the density of photometric galaxies,
we also estimate the density of photometric galaxies around a large
number of random positions in the survey area. This method allows a
direct comparison of the observed density around quasars and around
galaxies with that found around random positions using the same
observed distribution of photometric galaxies. This latter comparison
yields normalized overdensities, i.e., observed density over random
density, for both the quasars and the spectroscopic galaxies. This
technique then allows a direct comparison of the density of galaxies
around spectroscopic targets.
Throughout the analysis, we treat all of these targets - quasars,
spectroscopic galaxies, and random points - in exactly the same manner
in order to provide a direct and straightforward comparison between
the environment of quasars, galaxies, and random positions, and help
minimize potential biases. For each of the targets, we determine the
number of photometric galaxies within projected comoving radius bins
from $25\,$kpc to $1\,$Mpc ($h=0.7$). The innermost $25\,$kpc
(15.3\arcsec to 3.3\arcsec over our redshift range) is removed in all
density estimations in order to avoid deblending issues at these small
separations. The number of photometric galaxies observed around
quasars and around spectroscopic galaxies, $\ngq$ and $\ngg$
respectively, is divided by the same number found around random
points, $\ngr$; these overdensities, $\ngq/\ngr$ and $\ngg/\ngr$, are
determined for each of the radii specified above. In order to account
for the different redshift distributions of the $z \le 0.4$ targets
(quasars and galaxies), the overdensities are determined for each
individual quasar or galaxy using the mean $\ngr$ appropriate for that
target's redshift. All overdensities are then averaged in the relevant
redshift bins, and are investigated as a function of radius,
luminosity, and redshift. Our environment estimator is less
sensitive to faint galaxies at high redshift, but it is still
informative as we are interested in an excess in quasar environment
density relative to that of galaxies. At low redshift, the average
number of photometric galaxies around quasars ranges from a few
galaxies within $100\,$kpc (typically twice the number found around
random points) to $\sim100\,$ galaxies within $1\,$Mpc.
In order to estimate the density errors and the correlations between
different scales, we have generated $10^5$ bootstrap samples of the
spectroscopic quasar, galaxy and random samples, and measured the
standard deviation among different realizations. As the counts of
photometric galaxies are dominated by projection effects, i.e. objects
uncorrelated to the spectroscopic targets, the error bars are close to
Poisson errors, especially on large-scales.
\section{Results} \label{sec.results}
Our analysis produces normalized overdensities around quasars and
around spectroscopic galaxies. We note that the normalized
densities we use are defined as ratios of the galaxy counts around
each target relative to that around random points (e.g. $\ngq/\ngr$);
a ratio of unity implies no overdensity, i.e. the same density around
quasars as around random positions. Another common definition of
overdensity relative to random can be calculated using
$\ngq/\ngr\,-\,1$. This can be directly obtained from the $\ngq/\ngr$
ratios provided below.
The quasar overdensity
is presented as a function of redshift in Figure~\ref{f.z} within each
of our four standard radii. The mean overdensity around quasars is
shown by the dashed lines. The overdensity is 2.12 within $0.1\,$Mpc
of the quasars; i.e., the density is 2.12 times larger than the
density around random points. The mean overdensity decreases
monotonically with radius; it is 1.57 within $0.25\,$Mpc, 1.27 within
$0.5\,$Mpc, and 1.13 within $1.0\,$Mpc of the quasars. This
overdensity refers to the mean of all quasars with $-24.2 \le M_i \le
-22$. The excess photometric galaxies refers to galaxies within the
magnitude range $14 \le i \le 21$ (Section~\ref{sec.data}). The mean
galaxy overdensity around quasars is independent of redshift for $z
\le 0.4$ (Figure~\ref{f.z}).
In Figure~\ref{f.over} we present the overdensity as a function of
luminosity for quasars and for spectroscopic galaxies within radii of
0.1, 0.25, 0.5, and $1\,$Mpc ($h=0.7$). We find that, at all radii,
the overdensity around galaxies (solid line) increases with galaxy
luminosity. This is expected as brighter galaxies are located, on
average, in higher density regions \citep[e.g.,][]{davis_1988,
hamilton_1988, white_1988, zehavi_2004, eisenstein_2005}. The galaxy
overdensity is greatest on small scales ($0.1\,$Mpc and closer), and
decreases on larger scales, as expected. The luminosity-overdensity
trend is steeper on small scales than on large scales. The overdensity
around $L^*$ galaxies is $1.51\pm0.01$ times larger than random within
a radius of $0.1\,$Mpc; the overdensity is nearly doubled, to
$2.95\pm0.05$, for galaxies that are brighter by 1 magnitude.
The quasar overdensity in Figure~\ref{f.over} shows an increase with
quasar luminosity on the smallest scales, but only for the most
luminous quasars. The trend is considerably weaker than for galaxies
and becomes negligible, with nearly no dependence on quasar luminosity
by $\sim0.5-1\,$Mpc scales as well as for quasars with lower
luminosities (Figure~\ref{f.over}). This indicates that there is
little or no correlation between quasar activity and environment on
these larger scales, and that quasar activity is typically triggered
by an interaction with a neighbor present within $\sim100\,$kpc from
the quasar host galaxy. The mean overdensity around quasars,
$<\ngq/\ngr>$, for all $z \le 0.4$ quasars brighter than $M_i \le -22$
is indicated by the horizontal dashed line in Figure~\ref{f.over}. The
mean overdensity decreases with radius. The results are summarized in
Table~1.
\setcounter{table}{0}
\begin{table}
\caption{Mean Galaxy Overdensity around Quasars ($-24.2 \le M_i \le
-22.0$, $z \le 0.4$) and around $L^*$ Galaxies.}
\begin{tabular}{cccc}
\hline \hline $R_{max}$ (Mpc)&
$\frac{\ngq}{\ngr}$&$\frac{\ngl}{\ngr}$&$\frac{\ngq/\ngr}{\ngl/\ngr}$\\
\hline 0.10 & $2.12\pm0.08$ & $1.507\pm0.010$ & $1.41\pm0.06$\\ 0.25 &
$1.57\pm0.03$ & $1.235\pm0.004$ & $1.27\pm0.03$\\ 0.50 & $1.27\pm0.02$
& $1.144\pm0.003$ & $1.11\pm0.02$\\ 1.00 & $1.13\pm0.01$ &
$1.082\pm0.002$ & $1.05\pm0.01$\\ \hline
\end{tabular}
\end{table}
Figure~\ref{f.over} allows us to compare the environment of quasars
with the environment of galaxies of different magnitudes. We use the
average overdensity around $L^*$ galaxies as a standard by which to
measure the quasar environment. Since quasar luminosity is clearly
physically unrelated to the galactic luminosity, it should be noted
that $L^*$ galaxies are used only as an frame of reference for
comparing the relative environments. We find the mean overdensity
around quasars to be larger than around $L^*$ galaxies (shown by the
vertical dashed line in Figure~\ref{f.over}) on all scales
$<1\,$Mpc. The mean quasar overdensity is larger than the overdensity
around $L^*$ galaxies by factors that range from $1.41\pm0.06$ within
$0.1\,$Mpc, to $1.27\pm0.03$ within $0.25\,$Mpc, $1.11\pm0.02$ within
$0.5\,$Mpc, and $1.05\pm0.01$ within $1\,$Mpc (Table~1). (Using the
alternate overdensity definition,
$[\ngq/\ngr\,-\,1]\,/\,[\ngl/\ngr\,-\,1]$, we find an excess ratio of
$2.2\pm0.16$ within $0.1\,$Mpc decreasing to $1.6\pm0.13$ within
$1\,$Mpc. The interpretation is, of course, the same.) The local
density enhancement around quasars is similar to the local enhancement
around $\sim2L^*$ galaxies. The overdensity around quasars relative
to $L^*$ increases to a factor of $2.8\pm0.56$ closest to the quasars,
at $\sim40\,$kpc (see below). On scales larger than $1\,$Mpc, the
quasar overdensity becomes similar to the environment of $L^*$
galaxies. These results indicate that quasars are located in higher
density regions than are $L^*$ galaxies, but that the overdensity
exists mostly in regions very close to the quasars
($\lesssim0.1\,$Mpc), and is thus a very local excess. This local
excess of galaxies near quasars likely plays an important role in
triggering the quasar activity through mergers and other interactions.
These results are found to be only weakly dependent on redshift in the
$z < 0.4$ range studied here. This is shown in Figure~\ref{f.overz},
where the results, similar to Figure~\ref{f.over}, are presented for
different redshift ranges. As can be seen, the trend of overdensity
with quasar and galaxy luminosity remains consistent within the
statistical uncertainties and no significant trend can be detected as
a function of redshift. The redshift independence is further
illustrated in Figure~\ref{f.overz2}, where the overdensity around
quasars and the overdensity around galaxies are plotted as a function
of luminosity for several redshift bins. The results show a redshift
independent overdensity signal for both quasars and for galaxies. (The
highest luminosity galaxies, which do show some evolution, do not
affect our conclusions as we compare our quasars only to $L^*$ galaxies.)
Our results for the overdensities and the comparison with the
environment of $L^*$ galaxies are unaffected by redshift.
The scale dependence of the galaxy overdensity around spectroscopic
quasars and galaxies is presented as a function of comoving distance
from the target object in Figure~\ref{f.corr}. The quasar
overdensities are shown for both the brightest quasars ($M_i = -23.3$
to $-24.2$) and for fainter quasars ($M_i = -22$ to $-23.3$). We find
that the quasar overdensities are larger than those of $L^*$ galaxies
at all radii less than $\sim0.5\,$Mpc, but the overdensity increases
substantially on smaller scales. The overdensities around the most
luminous quasars are larger than around fainter quasars, mostly on
small scales ($\le\,0.1\,$Mpc). The lower panel of Figure~\ref{f.corr}
presents the ratio of quasar overdensity to that of $L^*$ galaxy
overdensity, as a function of scale. These data illustrate the
transition between the large scales, where quasars and $L^*$ galaxies
inhabit similar environments, and the small scales, where quasars are
located in higher overdensity regions than are $L^*$ galaxies. On
these small scales, the quasar overdensity is comparable to that
around $\sim2L^*$ galaxies (or brighter, for the most luminous
quasars). Our results on scales greater than $0.5\,$Mpc are
consistent with the recent SDSS and 2dF research on the large scale
quasar-galaxy clustering \citep[e.g.,][]{smith_1995, croom_1999,
croom_2003, wake_2004}, while our results on smaller scales are
consistent with the early work discussed in Section~\ref{sec.intro}
\citep[e.g.,][]{shaver_1988, shanks_1988, chu_1988, chu_1989,
crampton_1989} as well as with recent quasar-quasar pair studies
\citep{hennawi_2005}. This difference in the relative quasar
environment between small ($\le\,0.5\,$Mpc) and large ($\ge\,1\,$Mpc)
scales explains some of the previously contradictory results discussed
in Section~\ref{sec.intro}. These results suggest a picture in which
quasars reside in, on average, galaxies with a luminosity comparable
to $\sim L^*$, but with a local excess of neighbors within
$\sim0.1\,-\,0.5\,$Mpc. This local excess is likely associated with
triggering quasar activity.
The different galaxy densities observed for the bright and faint
quasars cannot be due to gravitational lensing. The dark matter
distribution in and around galaxies induce gravitational lensing
effects which locally enlarge the sky solid angle and magnify the
images of background objects. This effect, the magnification bias, can
increase or decrease the density of distant sources around foreground
galaxies depending on the variation of the number of sources as a
function of magnitude \citep{narayan_1989}. It can thus create
correlations between source luminosity and foreground galaxy
density. However, the amplitude of this effect is expected to be only
at the percent level \citep{menard_2002, jain_2003} and recent
observations with the SDSS have confirmed these predictions
\citep{scranton_2005}. This suggests that gravitational lensing does
not significantly affect our density estimations.
\section{Conclusions} \label{sec.conclusions}
We use the SDSS data to investigate the local environment of quasars
and compare it with the environment around galaxies and around random
positions. This study provides a direct comparison between the
environment of quasars and galaxies.
We use bright quasars with $-24.2 \le M_i \le -22$ ($h=0.7$) and
redshift $z \le 0.4$ (2028 quasars over $4000\,$deg$^2$) to study the
density of photometric galaxies with $i = 14$ to 21 (sample of
$\sim10^7$ galaxies) located within $25\,$kpc to $1\,$Mpc ($h=0.7$) of
the parent quasars. We compare the results with the same analysis
carried out at random positions in the SDSS survey, $\ngr$; this
yields the galaxy overdensity, over random, around quasars, i.e.,
$\ngq/\ngr$. We investigate this overdensity as a function of quasar
redshift and luminosity. The same analysis is then repeated for
determining the overdensity around $10^5$ spectroscopic galaxies in
the SDSS data ($i \le 18.5$) in the same redshift range ($z = 0.08$ to
$0.4$). This allows a direct comparison of the quasar overdensity to
the overdensity around galaxies. The overdensities are studied as a
function of scale, luminosity, and redshift. This comparison of quasar
environment with that of galaxies provides a self-consistent
comparison of the host environments.
Our results are summarized below.
1. At all radii, from $25\,$kpc to $\sim1\,$Mpc, quasars are located
in higher density environments than are $L^*$ galaxies. The
overdensity around quasars relative to that of $L^*$ galaxies
increases with decreasing scale: the overdensity is greatest
closest to the quasars. At a distance of $40\,$kpc of the quasars,
the mean overdensity for the brightest quasars ($-24.2 \le Mi \le
-23.3$, $z \le 0.4$) is nearly a factor of three times larger than
the overdensity around $L^*$ galaxies. The mean overdensity around
the brightest quasars relative to that of $L^*$ galaxies decreases
with increasing scale to a value of 2.2 within $0.1\,$Mpc, 1.2 at
$0.3\,$Mpc, and approaches unity at $\sim0.5 - 1\,$Mpc. On these
larger scales, quasars reside in environments similar to those of
$L^*$ galaxies.
2. The brightest quasars are found to be located in higher overdensity
regions than are fainter quasars, especially at small separations
($< 0.1\,$Mpc). On larger scales, bright and faint quasars live in
similarly dense environments.
3. The mean overdensity around quasars 0
($<~\ngq/\ngr~> = 2.12\pm0.08$
within $0.1\,$Mpc, decreasing to $1.13\pm0.01$ within $1\,$Mpc) is
independent of redshift for $z \le 0.4$. The mean overdensity is
also independent of luminosity except for the brightest quasars,
which are located in higher density environments. This dependence
of quasar environment on luminosity, showing enhancement only for
the most luminous quasars, is consistent with recent galaxy merging
models of quasars \citep{hopkins_2005}.
4. The enhanced mean overdensity around quasars is observed to be a
local phenomenon, affecting mostly the $\sim$ $0.1\,$Mpc region
closest to the quasars. On these scales, very close to the quasars,
the high overdensity of galaxies likely affects the formation and
triggering of the quasar activity through mergers and other
interactions. On scales of $\sim 1\,$Mpc, the quasars inhabit
similar environments to those of normal $L^*$ galaxies.
\section{Acknowledgements} \label{sec.acknowledgements}
GTR acknowledges support from a Gordon and Betty Moore Fellowship in
data intensive sciences.
Funding for the creation and distribution of the SDSS Archive has been
provided by the Alfred P. Sloan Foundation, the Participating
Institutions, the National Aeronautics and Space Administration, the
National Science Foundation, the U.S. Department of Energy, the
Japanese Monbukagakusho, and the Max Planck Society. The SDSS Web site
is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium (ARC) for
the Participating Institutions. The Participating Institutions are The
University of Chicago, Fermilab, the Institute for Advanced Study, the
Japan Participation Group, The Johns Hopkins University, the Korean
Scientist Group, Los Alamos National Laboratory, the
Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute
for Astrophysics (MPA), New Mexico State University, University of
Pittsburgh, University of Portsmouth, Princeton University, the United
States Naval Observatory, and the University of Washington.
|
Title:
ELT requirements for future observations of the Intergalactic Medium |
Abstract: We summarise the science cases for an ELT that were presented in the parallel
session on the intergalactic medium, and the open discussion that followed the
formal presentations. Observations of the IGM with an ELT provides tremendous
potential for dramatic improvements in current programmes in a very wide
variety of subjects. These range from fundamental physics (expansion of the
Universe, nature of the dark matter, variation of physical constants),
cosmology (geometry of the Universe, large-scale structure), reionisation
(ionisation state of the IGM at high redshift>6, to more traditional astronomy,
such as the interactions between galaxies and the IGM (metal enrichment,
galactic winds and other forms of feedback), and the study of the interstellar
medium in high redshift galaxies through molecules. The requirements on ELTs
and their instruments for fulfilling this potential are discussed.
| https://export.arxiv.org/pdf/astro-ph/0601637 |
\firstsection %
\section{Introduction}
The advent of echelle spectrographs on 8m class telescopes since the
early 1990's has revolutionised our understanding of the intergalactic
medium (IGM) as observed in quasar spectra. These bright sources have
smooth intrinsic spectra with broad emission lines, yet the {\em
observed} spectra contain hundreds of narrow absorption lines due to
intervening absorbers. The latter can be studied in great detail from
the exquisite, $\rm S/N> 40$, spectra possible with UVES on VLT and HiRes
on Keck.
Most of the absorption in the UV is due to neutral hydrogen left over
from the Big Bang, forming a forest of lines (Bahcall \& Salpter 1965;
Gunn \& Peterson 1965; Lynds 1971). The weaker lines with column
density $N$({H~{\sc i})$\le 10^{15}{\rm cm}^{-2}$ are traditionally
called the \lq
Lyman-$\alpha$ forest\rq, with the strongest lines with column density
$\ge 10^{20.3}{\rm cm}^{-2}$ that show a measurable damping-wing called
\lq Damped Lyman-$\alpha$ systems\rq\, (DLAs). The number of lines as a
function of redshift and column-density, $d^2N/dz/dN$(H~{\sc i}), is close
to a power-law $\propto~N$(H~{\sc i})$^{\beta}$ with $\beta<0$, as function
of column-density, and evolves strongly with redshift as fewer lines are
produced as the mean density decreases due to the expansion of the Universe,
see Rauch (1998) for a review.
The weaker lines originate in the filaments of the cosmic web which
itself is a natural outcome of how structure forms in a dark matter
dominated cosmology (Bi \etal\ 1992; Cen \etal\ 1994; Schaye
2001). The neutral hydrogen fraction is small at redshifts $\le 6$
(Gunn \& Peterson 1965) as the gas is photo-ionised and photo-heated by
the UV-background, with photo-ionisation rate $\Gamma$, produced by
galaxies and quasars (Haardt \& Madau 1996). At lower $z\le 2$, the
forest of lines thins-out into a Lyman-$\alpha$ \lq savanna\rq\,, but
the decline is slowed because $\Gamma$ also decreases as the emissivity
from galaxies and QSOs drops (Theuns \etal\ 1998a; Dav\'e \etal\
1999). Conversely at increasing $z\ge 6$, the mean density increases,
but $\Gamma$ also decreases as many source have yet to form, turning
the forest into a Lyman-$\alpha$ \lq jungle\rq\, which absorbs (nearly)
all light, perhaps signaling the end of reionisation (Becker \etal\
2001; Djorgovski \etal\ 2001). The forest provides a tremendous probe
of how the IGM evolves in the intermediate redshift range $2\le z\le
5$, because the absorbers are only mildly non-linear and hence can be
simulated reliably (Cen \etal\ 1994; Hernquist \etal\ 1996; Theuns \etal\
1998b; Zhang \etal\ 1998; Bryan \etal\ 1999). The combination of
superb data with reliable models makes it possible to constrain models
and determine parameters.
Stronger lines form near galaxies, with the DLAs potential
proto-galaxies or proto-galactic lumps (Wolfe 1995; Haehnelt \etal\
1998; Ledoux \etal\ 1998). Since these systems are discovered in absorption, it is a worry
that even denser systems might be missed because they make the
background QSO too faint to appear in a magnitude-limited survey. For a
recent appraisal of this issue see Ellison \etal\ (2005 and reference
therein). DLAs shield the UV-background and some fraction of the gas
becomes molecular (see, e.g., Srianand this volume). The prospect of
studying star formation in small systems at high redshift which are too
faint to study in emission, is very exciting.
Quasar spectra also contain \lq metal\rq\, lines from highly ionised
species such as \cfour\,, \sifour\, and \osix\, (e.g., Cowie \etal\
1995). These metals were synthesised in stars and managed to diffuse
into the lower density surroundings, either as a result of galactic
winds, or due to an early generation of population~III~ stars.
The next section gives a short overview of recent results, with
emphasis on opportunities for progress with the advent of new
observatories.
\section{IGM observations with ELT: science}
We begin with a short overview of numerical simulations of the IGM, as
these can be used to investigate the main limitations of current
observational strategies, thereby guiding the design for new
instruments. We then discuss current and future science that can be
done with IGM observations. The next section summarises the
corresponding requirements for an ELT.
\subsection{Hydrodynamical simulations}
Most of the weaker ``Lyman$-\alpha$ forest'' lines form in mildly over
dense or under dense structures that can be simulated accurately (Bi \etal\
1992; Cen \etal\ 1994). Fig.~\ref{fig:ts_fig1} displays the gas
distribution in a cosmological hydrodynamical simulation at $z=3$, and
shows that the gas traces the filamentary pattern that results from
structure formation in a dark matter Universe. A sight line through
such a density distribution will most often go through low density
voids, occasionally intersecting a filament which will produce an
absorption line, and even more rarely pass close to or even straight
through a galaxy halo, producing a very strong absorption line.
Mock spectra generated from such simulations look very similar to the
real data, see, e.g., Fig.~\ref{fig:ts_fig2}. Since most of the lines are
due to structures that are only mildly over dense, it is possible to
simulate them quite reliably. Comparison of such simulated spectra with
observed ones makes it possible to constrain the model parameters and
investigate which cosmological parameters determine the line
statistics.
Most of the volume in the simulations is photo-heated by the
UV-background after reionisation, the volume affected by shocks from
structure formation is small. The temperature of the IGM affects the
properties of the lines (Theuns, Schaye \& Haehnelt 2000), because the
widths of the narrowest lines is restricted by thermal
broadening. Detailed comparison with data allows one to constrain the
thermal history of the IGM (Schaye \etal\ 2000; Ricotti \etal\ 2000;
Bryan \& Machacek 2000; McDonald \etal\ 2001), because the thermal
time-scales are long in the low-density IGM. This also puts constraints
on reionisation if photo-heating is the dominant heating mechanism
(Theuns \etal\ 2002b).
The detailed line properties are also sensitive to the nature of the
dark matter, and can for example constrain the mass of a putative warm
dark matter particle (Croft \etal\ 1999; Viel \etal\ 2005). If the
warm dark matter smoothing length is comparable to the width of
filaments, then this will affect the line shape. Note that these scales
become non-linear at lower $z$, making it much harder to put tight
constraints on the dark matter properties.
Sight lines passing close to a galaxy may be affected by
non-gravitational effects such as feedback from star formation or
AGN. Fig.~\ref{fig:ts_fig3} illustrates how supernova feedback causes
bigger galaxies to be embedded in a hot bubble of metal enriched gas,
which expands into the lower density surroundings of halos. Such sight
lines will also show metal line absorption, and it is possible to
compare in detail the metal-hydrogen correlation in simulations and
data to infer the physical properties in the surroundings of high-$z$
galaxies (Adelberger \etal\ 2003; Pieri \etal\ 2005). ELTs will
provide dramatic improvements in this subject, because the bigger
collecting area will allow one to observe fainter sources, and hence
allow a far finer grid of sight lines probing galaxy environs.
\subsection{Fundamental Physics}
High resolution high signal-to-noise spectra of high-$z$ QSOs are
frequently used to test the current theories of cosmology and
fundamental physics.\\
\noindent {\em CODEX:} The observed wavelength of a given absorption line changes
with time due to the expansion of the Universe. The COsmic Dynamics
EXperiment (CODEX) (see Molaro \etal\ , this volume) aims to measure
this change directly by observing many lines in a QSO spectrum at
extreme signal-to-noise and resolution, and repeat the measurement a
few decades later. Such an experiment therefore also requires very
high and long term stability of the spectrograph.
A more indirect measure of the expansion of the Universe is by
determining the CMB temperature at intermediate redshift using the
fine-structure excitation lines of carbon in DLAs, which is excited by
CMB photons (see \cite{srianand00}). Detecting C~{\sc i} absorption
lines from the low density regions, where the collisional excitation
will be sub-dominant using high S/N spectra will allow one to directly
map the redshift evolution of temperature of the CMBR (R. Srianand,
this volume).
Some of the current theories of fundamental physics, such as SUSY, GUT
and Super-string theory, allow possible space and time variations of
the fundamental constants. QSO absorption lines can be used to probe
the time evolution of fundamental constants. The heavy element
absorption lines and H$_2$ Lyman Werner band absorptions lines are used
to investigate the time-variation of the electromagnetic coupling
constant $\alpha$ (see \cite{murphy03}; \cite{chand04} and Mollaro \etal\
this volume) and the proton to electron mass ratio ($\mu$)(see
\cite{ivanchik05}), respectively. The available constraints based on 8m
class telescopes are still much higher than those achieved by
terrestrial techniques. Higher resolution (R$\ge100\,000$) and good
signal-to-noise ratios ($>100$) are needed to improve the precession.
The Square Kilometer Array (SKA) will measure the 21-cm line in most of
the DLAs. The wavelength of the transition depends on $\alpha$, $\mu$
and the proton g-factor and can provide a combined constraint on the
variation of all these fundamental constants (see
\cite{curran04}). Detecting H$_2$ and weak transitions of Mn~{\sc ii},
Ni~{\sc ii} in DLAs with high signal-to-noise and resolution will allow
us to lift the degeneracy between the variation of different fundamental
constants that decide the shift of the 21\,cm absorption line.
To avoid the systematics caused by the small-scale properties of the
lines we require high resolution and high signal-to-noise to improve
current constraints. To study the redshift evolution and to be able to
use different sets of lines from the same system, it is of paramount
importance to have a wide wavelength coverage.
\subsection{Cosmology}
The large-scale flux distribution can be used to infer the dark matter
power spectrum (Croft \etal\ 1998; McDonald \etal\ 2000; Viel \etal\
2004) and constrain the neutrino mass (Croft \etal\ 1999; Viel \etal\
2005). These measurements are currently limited by uncertainties in the
shape of the QSO's underlying continuum, calibration of the echelle
spectrograph for high resolution spectra, and by the statistics of
available spectra, and would not obviously benefit from an
ELT. However, the influence of large-scale structure is also very
prominent in simulations at {\em low} optical depths, below the median
$\tau$. Such observations require much higher S/N than currently
available, since even S/N=50 data cannot recover the median optical
depth at $z\sim 2$.
Observations of the forest along parallel sight lines can constrain the
topology of the Universe via the Alcock-Paczynski (1979) test
(e.g., Rollinde \etal\ 2003). Such a project would benefit greatly from
observing fainter QSOs and bright LBGs to improve the sampling in the
transverse direction to the line of sight.
\subsection{Reionisation}
The Lyman-$\alpha$ forest becomes increasingly opaque above $z\ge 6$,
perhaps signaling the end of reionisation. If the IGM is polluted
through winds from early generations of dwarf galaxies then the
ionisation state of the IGM can be probed through the absorption
produced by C~{\sc iv}, C~{\sc ii}, O~{\sc i} and Si~{\sc ii} (with NIR
wavelength $\lambda<2.1\mu$m for $z<12.5$, 14.7, 15.1 and 15.7,
respectively, e.g., Oh 2002). Given the rapidly declining space density
of QSOs, Gamma Ray Bursts (GRBs) or super novae could be used as
background sources.
GRBs have mean afterglow fluxes of 1.5 to 0.05$\mu$J at $z=10$, 1 to 10
days after the explosion ($K_{AB}=23.6$ to 27). High resolution
($R=4\times 10^4$) and high signal-to-noise ratios ($>50$) are required to
detect individual lines in the NIR. Observations of a very bright
$4\mu$J GRB at $R=10^4$ and S/N=50 gives detection limits for $N$({\rm
C~{\sc ii}})=$4\times 10^{12}{\rm cm}^{-2}$ and $N$(O~{\sc
i})=1$\times 10^{13}{\rm cm}^{-2}$ . Pair-instability SNe of
$M=140-260M_\odot$ pop.~III precursors have $K_{\rm AB}=25$ for
$z=10-15$, and are also potential targets, with a possible time-lag of
weeks between discovery and ELT follow-up spectroscopy (see the
presentation by J. Bergeron in this volume for details).
\subsection{Galaxy-Intergalactic medium interactions and metal enrichment}
The metal density of carbon as inferred from C~{\sc iv} pixel optical
depth analysis (Songaila 2001; Schaye \etal\ 2003) shows little
evidence for evolution over the redshift range $z=2-5$, with possibly a
decline by factor of two above $z=6$. It is possible that not all
metals are seen. Just as most metals are in the hot intra-cluster gas
at $z<1$, metals could be in hot gas resulting from galactic winds at
higher $z$, thereby not producing significant C~{\sc iv} absorption
(Theuns \etal\ 2002a and this volume). The shape of the UV-background,
and its evolution with $z$, is the main uncertainty in converting
optical depth to metallicity. Improved constraints require the
detection of many more transitions to eliminate this uncertainty.
What is the origin of the metals seen in the IGM? Are the metals due to
galactic winds, or is some fraction the result of pop.~III stars? This
important question can be addressed by correlating metals seen in
absorption with the presence of galaxies (e.g., Adelberger \etal\ 2003,
2005; Pieri, Schaye \& Aguirre 2005). This can be done by probing the
IGM with many sight lines, and requires obtaining spectra of fainter
sources, including the brighter Lyman-break galaxies (LBGs)
themselves. Current state of the art (Adelberger \etal\ 2005) is
limited to measuring the mean metallicity in C~{\sc iv} as function of
the galaxy's impact parameter; higher S/N should make it possible to
look for metals in each individual galaxy spectrum and obtain good
redshifts.
A better understanding of galaxy-IGM interactions is needed to
constrain how feedback from stars and AGN affects galaxy formation as
a function of redshift and galaxy mass. The redshift range $z\sim 3$
is well suited for such a study, as there are many lines in the
observed optical-NIR part of the spectrum suited to ground-based
observations, but an ELT is required to be able to observe fainter
QSOs or brighter LBGs and dramatically improve the sampling with many
more sight lines.
\subsection{Molecules at high $z$}
Detecting H$_2$ and other molecules at high-z through their electronic
transitions is important for understanding the physical conditions and
astrochemistry in the interstellar medium of galaxies and
protogalaxies at a very early epoch.
Up to now, H$_2$ has been detected in $\sim15\%$ of DLAs
(\cite{ledoux03}) and only one system shows detectable HD
(Varshalovich \etal\ 2001). H$_2$ can
potentially be detected from LBGs and GRB host galaxies. This will
allow us to understand the interstellar medium in these early
galaxies. As the Lyman Werner band absorptions of H$_2$ are
expected in the \lya forest, it is important to have high resolution R
= 20\,000 and signal-to-noise ($>20$) in the blue spectrum. ELTs with
blue sensitive spectrographs can allow us to search for H$_2$ in
fainter QSOs, GRBs and brighter LBGs. However, in the case of GRBs,
H$_2$ may be in non-equilibrium and it is important to target the
source as quickly as possible to be able to detect the H$_2$ lines and
follow the time variation of H$_2$ column density. This will give
important clues about the GRB hosts.
Detecting other molecules in systems with H$_2$ is also important for
the understanding of astrochemistry in low metallicity gas in the
early universe. CO is not detected in DLAs and the achieved limits
are close to the lowest column measured in Milky Way. As the
metallicities in DLAs are low, to test N(H$_2$) vs. N(CO) relation we
need to push this limit by roughly a factor 50 (see the presentation
by R. Srianand in this volume).
\section{IGM observations with ELT: requirements}
In this section we summarise the ELT requirements that emerge out of
the discussion in the parallel session on IGM.
\noindent{\em {High $z>7$}}: {\bf Reionisation, metals, Lyman-$\alpha$
emitters:} Constraining IGM enrichment and its ionisation state from
metal lines at $z>7$ requires observations at intermediate resolution
of $R=2000$ in the NIR with S/N up to 100. An OH-line suppressor with
multiple IFUs with field-of-view of several arcmin$^2$ is ideal.
Targets are moderately faint QSOs and Lyman-break galaxies of
$m_{AB}\sim 27$, but require an ELT larger than 30m. NIR observations
at $R=10^4$ with S/N up to 100 is possible from average-luminosity
GRBs and pop.~III SNe. Targets need to be found using dedicated
ground and space-based telescopes.
\begin{enumerate}
\item[$\bullet$] NIR, R=2000, S/N=100 (bright QSOs)
\item[$\bullet$] NIR, R=10000, S/N=100 (single target GRBs)
\end{enumerate}
\noindent{\em {Intermediate $z<7$}: {\bf Metals:}} High resolution
$R=40000$ and S/N of 10000 optical spectra of bright QSOs
($z=2-5$, $m_{AB}=16-17$) and bright GRBs ($z$ up to 7, lag is 1 day,
$m_{AB}=20$, S/N=100) in single target mode are required to study the
distribution of metals in the IGM, and its evolution with $z$.
The spectrograph should be {\em blue sensitive} and have a large
wavelength range ($\lambda=3030-9300\AA$) to be able to cover a
large range of transitions and constrain the ionisation
corrections. The latter is the major uncertainty in inferring
metallicity, so a large $\lambda$ range is essential.
\begin{enumerate}
\item[$\bullet$] optical, R=40000, S/N=$10^3-10^4$ (single target
bright QSOs, GRBs). Blue sensitive, large $\lambda$ ($3030-9300\AA$)
coverage.
\end{enumerate}
\noindent{\em {Lower $z<5$}: {\bf galaxy-IGM connection, UV-escape
from galaxies:}} The main gain of an ELT is the possibility of
observing fainter QSOs, which allows one to sample the metal
distribution in the IGM, and its correlation with galaxies
dramatically better by providing a much finer grid of lines along
which the IGM can be probed. The QSOs and bright LBGs can be observed
with optical, high $R=50000$ spectroscopy (S/N=100) to probe the
distribution of metals. A detailed correlation of these metals with
galaxies requires the redshift determination of the fainter LBGs (up
to 0.01$L_\star$) using optical/NIR MOS of $R=2000-5000$, with
multiple IFUs with a total FoV of several arcmin$^2$, centered on LBGs
and QSOs. NIR is required to obtain good redshifts for the galaxies
from stellar and ISM lines, since many of the UV-lines can be
significantly off-set from the redshift of the stars.
\begin{enumerate}
\item[$\bullet$] Optical/NIR, R=2000-5000, S/N=100 (0.01$L_\star$~LBGs)
with multiple IFUs, FoV several arcmin$^2$
\item[$\bullet$] Optical, R=50000, S/N=100 (bright LBGs, QSOs)
\end{enumerate}
Many small programmes could be started at early stages of construction
if the instruments are available. 8m-class telescopes will be used to
find (candidate) LBGs and Lyman-$\alpha$ emitters.
\begin{acknowledgments}
We wish to thank IAU for a travel grant. TT thanks PPARC for the award
of an Advanced Fellowship, and J Schaye, R Bower and I Smail for
comments on the draft.
\end{acknowledgments}
|
Title:
The First Scientific Results from the Pierre Auger Observatory |
Abstract: The southern site of the Pierre Auger Observatory is under the construction
near Malargue in Argentina and now more than 60% of the detectors are
completed. The observatory has been collecting data for over 1 year and the
cumulative exposure is already similar to that of the largest forerunner
experiments. The hybrid technique provides model-independent energy
measurements from the Fluorescence Detector to calibrate the Surface Detector.
Based on this technique, the first estimation of the energy spectrum above 3
EeV has been presented and is discussed in this paper.
| https://export.arxiv.org/pdf/astro-ph/0601035 |
\title{ The First Scientific Results from the Pierre Auger Observatory }
\classification{95.85.Ry, 98.70.Sa
}
\keywords {Cosmic Rays, Ultra-High Energy Particles}
\author{T. Yamamoto}
{
address={KICP, Enrico Fermi Institute, University of Chicago, 5640
S. Ellis Ave, Chicago IL 60637, USA }
}
\author{The Pierre Auger Observatory Collaboration}{
address={}
}
The Pierre Auger Observatory is the largest cosmic ray detector
ever built to study the Ultra-High Energy Cosmic Rays
(UHECR) with unprecedented statistics and high precision
\cite{AUGER}. In particular,
it is important to address whether the cosmic-ray
spectrum continues beyond $10^{20}$ eV. Due to the interaction with microwave
background photons, a steepening is expected around $10^{20}$ eV in the
energy spectrum if the sources are distributed uniformly throughout the
Universe. This conclusion is independent of the composition of the UHECR's.
Recent measurements of the energy spectrum by the AGASA which used
surface detector (SD) array \cite{AGASA} and the HiRes which is using
fluorescence detector (FD) \cite{HyRes} have yielded conflicting
results. There are serious limitations in the use of only the SD or the
FD alone to measure the primary
spectrum. The SD provides high event statistics with high efficiency
and robust exposure estimation. The SD energy estimation, however,
traditionally relies
on Monte-Carlo simulations which require assumptions about the
hadronic-interaction model and the primary-chemical composition. On the
other hand, the FD provides a calorimetric energy measurement but the
estimation of the exposure has a comparatively large uncertainty
relative to the SD.
Based on one year operation of a portion of the Pierre Auger
Observatory, the first scientific results were released this
summer concerning the upper limit of the UHE gamma ray flux
\cite{Photon}, anisotropy
of the arrival directions \cite{Aniso}, and the energy spectrum
\cite{Spect}.
The cumulative exposure, 1750 $km^2$-$sr$-$yr$, is similar to those
achieved by the largest forerunner experiments. Statistical
uncertainties are still too large to draw any firm conclusions ether
rejecting or confirming results obtained by previous
experiments. However, there is an important step achieved in these results.
The Pierre Auger Observatory was designed as a hybrid detector to observe
the shower particles at ground level by the SD and the associated fluorescence
light generated in the atmosphere by the FD. Combining the strengths of
the SD and the FD, we have developed a reliable estimate of the primary
energy spectrum using the full SD exposure without making assumptions
about the primary masses or hadronic model. \\
The southern site of the Pierre Auger Observatory
is now under construction on an Argentinian pampa ($35^{\circ}$ S,
$69^{\circ}$ W, 1400 m.asl, 875.5 g/cm$^2$). The SD consists of 1600
water Cherenkov tanks planed on a triangular 1.5 km grid covering 3000 $km^2$
area with $2\pi$ sky coverage. The construction of
the Southern site is now 60\% complete.
The whole
area of the SD will be overlooked by an FD from 4 sites. Each FD site
has 6 telescopes and each telescope has a $30^{\circ}\times28.6^{\circ}$
field of view with $1.5^{\circ}$ pixel size. Three FD sites are completed
and operating now and one is under construction.
The events recorded in the SD are reconstructed using the arrival time
and the signal size from the shower particles reaching the detectors.
The magnitude of the signal at 1 km from the shower axis, S(1000) in
Vertical Equivalent Muon (VEM), is
estimated from the Lateral Distribution Function fit as a size parameter
of the shower \cite{LDF}.
Two cosmic rays of the same energy, but incident at different
zenith angles, will yield different values of S(1000) due to an
attenuation of the shower in the atmosphere. This attenuation is
measured by the well-established technique of the constant intensity cut
(CIC) method.
The principle of this method is that the nearly isotropic intensity of
cosmic rays means that the integrated intensity above any given energy
must be the same at all zenith angles ($\theta$ degree). One finds the
S(1000) at every zenith angle that corresponds to a single primary
energy by varying S(1000) at each zenith angle to obtain a fixed
integral intensity.
Based on this method, the zenith angle dependence of S(1000), the CIC
curve is obtained as
\begin{equation}
S(1000)_{38^{\circ}} = \frac{S(1000)_{\theta}}{1.049+0.0097\theta -
0.00029\theta^2}
\end{equation}
where $S(1000)_{38^{\circ}}$ VEM is S(1000) adjusted to
$\theta=38^{\circ}$. (The median zenith angle of the showers is
$38^{\circ}$.)
The link between $S(1000)_{38^{\circ}}$ and the primary energy can be
established using data from the FD. On dark dry nights, the
fluorescence signals are observed simultaneously with the SD
events.
The fit to the FD-energy as a function of $S(1000)_{38^{\circ}}$ is
\begin{equation}
log(E) = -0.79+1.06 log(S(1000)_{38^{\circ}})
\label{eq:energy}
\end{equation}
where $E$ is the FD-energy in EeV.
The events detected by the SD are selected as follows:
The estimated energy must be greater than 3 EeV because detection
efficiency is saturated (nearly 100\%) above this energy. The zenith
angle of the
arrival direction must be smaller than 60$^{\circ}$. And the event must fall
within a well-defined fiducial area. The estimate of the SD exposure
is simple. The fiducial area is monitored in the trigger system so that
exposure is calculated as the time integration of the aperture given by
the fiducial area and the 60$^{\circ}$ zenith-angle limit.
The spectrum is then obtained by dividing the number of events in given
energy intervals by the exposure as shown in Figure.\ref{fig:spectrum}.
The systematic uncertainty of the energy spectrum comes mainly from the
energy assignment. In the estimation of the FD-energy, there are several
uncertainties which include the fluorescence yield (15\%), missing
energy carried by high-energy muons and neutrinos (4\%),
the absolute calibration of the FD telescopes (12\%), and
atmospheric condition (10\%). Overall the uncertainty of the FD-energy
is about 25\%. These systematic errors will
be reduced significantly in a year with completion of the FD
calibration and the measurement of the fluorescence yield in
laboratories . The statistical uncertainty in Equation.\ref{eq:energy}
causes additional energy-dependent systematic uncertainty in the energy
estimation. This uncertainty is dominant to the
systematic error in the highest energy and will automatically shrink
with the rapidly-increasing hybrid statistics.
The total systematic error is indicated in the
Figure.\ref{fig:spectrum}.
It should be noted that this energy spectrum
was measured in the southern sky which could differ from that of
northern sky measured in the previous experiments.
The energy scale based on the FD measurements is systematically lower than
that from an SD analysis that uses QGSJetII simulations with proton
primaries. The difference is similar to the conflicting energy scales of
the HiRes and the AGASA collaborations.
The exposure of the southern observatory is expected to increase by a
factor of
5$\sim$7 over the next two years. With completion of the FD calibration,
the statistical and systematic errors will shrink accordingly, permitting
a study of spectral features and the energy scale.
This work was supported in part by the Kavli Institute for Cosmological
Physics through the grant NSF PHY-0114422, by NSF AST-0071235, and
DE-FG0291-ER40606 at the University of Chicago.
\begin{center}
\end{center}
|
Title:
Time delay of SBS 0909+532 |
Abstract: The time delays between the components of a lensed quasar are basic tools to
analyze the expansion of the Universe and the structure of the main lens galaxy
halo. In this paper, we focus on the variability and time delay of the double
system SBS 0909+532A,B as well as the time behaviour of the field stars. We use
VR optical observations of SBS 0909+532A,B and the field stars in 2003. The
frames were taken at Calar Alto, Maidanak and Wise observatories, and the VR
light curves of the field stars and quasar components are derived from aperture
and point-spread function fitting methods. We measure the R-band time delay of
the system from the chi-square and dispersion techniques and 1000 synthetic
light curves based on the observed records. One nearby field star (SBS
0909+532c) is found to be variable, and the other two nearby field stars are
non-variable sources. With respect to the quasar components, the R-band records
seem more reliable and are more densely populated than the V-band ones. The
observed R-band fluctuations permit a pre-conditioned measurement of the time
delay. From the chi-square minimization, if we assume that the quasar emission
is observed first in B and afterwards in A (in agreement with basic
observations of the system and the corresponding predictions), we obtain a
delay of - 45 (+ 1)/(- 11) days (95% confidence interval). The dispersion
technique leads to a similar delay range. A by-product of the analysis is the
determination of a totally corrected flux ratio in the R band (corrected by the
time delay and the contamination due to the galaxy light). Our 95% measurement
of this ratio (0.575 +/- 0.014 mag) is in excellent agreement with previous
results from contaminated fluxes at the same time of observation.
| https://export.arxiv.org/pdf/astro-ph/0601473 |
\title{Time delay of SBS 0909+532}
\author{A. Ull\'an\inst{1} \and L. J. Goicoechea\inst{1} \and A. P. Zheleznyak\inst{2}
\and E. Koptelova\inst{3} \and V. V. Bruevich\inst{3} \and T. Akhunov\inst{4}
\and O. Burkhonov\inst{4}}
\offprints{A. Ull\'an}
\institute{Departamento de F\'{\i}sica Moderna, Universidad de Cantabria,
Avda. de Los Castros s/n, 39005 Santander, Spain\\
\email{aurora.ullan@postgrado.unican.es, goicol@unican.es}
\and
Institute of Astronomy of Kharkov National University,
Sumskaya 35, 61022 Kharkov, Ukraine\\
\email{zheleznyak@astron.kharkov.ua}
\and
Sternberg Astronomical Institute, Universitetski pr. 13,
119992 Moscow, Russia\\
\email{koptelova@xray.sai.msu.ru, bruevich@sai.msu.ru}
\and
Ulug Beg Astronomical Institute of Uzbek Academy of
Science, Astronomicheskaya. Str. 33, 700052 Tashkent, Republic
of Uzbekistan\\
\email{talat77@rambler.ru, boa@astrin.uzsci.net}}
\date{Accepted January 12, 2006}
\titlerunning{Time delay of SBS 0909+532}
\authorrunning{A. Ull\'an et al.}
\abstract{
The time delays between the components of a lensed quasar are basic tools to analyze the
expansion of the Universe and the structure of the main lens galaxy halo. In this paper, we
focus on the variability and time delay of the double system SBS 0909+532A,B as well as the
time behaviour of the field stars. We use $VR$ optical observations of SBS 0909+532A,B and the
field stars in 2003. The frames were taken at Calar Alto, Maidanak and Wise observatories, and
the $VR$ light curves of the field stars and quasar components are derived from aperture and
point--spread function fitting methods. We measure the $R$--band time delay of the system from
the $\chi^2$ and dispersion techniques and 1000 synthetic light curves based on the observed
records. One nearby field star (SBS 0909+532c) is found to be variable, and the other two nearby
field stars are non--variable sources. With respect to the quasar components, the $R$--band
records seem more reliable and are more densely populated than the $V$--band ones. The observed
$R$--band fluctuations permit a pre--conditioned measurement of the time delay. From the $\chi^2$
minimization, if we assume that the quasar emission is observed first in B and afterwards in A
(in agreement with basic observations of the system and the corresponding predictions), we obtain
$\Delta \tau_{BA}$ = $-$ 45 $^{+ 1}_{-11}$ days (95\% confidence interval). The dispersion technique
leads to a similar delay range. A by--product of the analysis is the determination of a totally
corrected flux ratio in the $R$ band (corrected by the time delay and the contamination due to the
galaxy light). Our 95\% measurement $\Delta m_{BA}$ = $m_B(t + \Delta \tau_{BA}) - m_A(t)$ =
0.575 $\pm$ 0.014 mag is in excellent agreement with previous results from contaminated fluxes at
the same time of observation.
\keywords{
Gravitational lensing
-- Quasars: general
-- Quasars: SBS 0909$+$532
-- Stars: variables: general
}
}
\section{Introduction}
The system SBS 0909+532 was discovered by Stepanyan et al. (1991). Some years later,
a collaboration between the Hamburger Sternwarte and the Harvard--Smithsonian Center
for Astrophysics resolved the system into a pair of quasars (A and B) with a direct
$R$--band flux ratio (at the same time of observation) $\Delta m$ = $m_B - m_A$ = 0.58
mag and a separation of about 1\farcs1 (Kochanek et al. 1997). The direct $R$--band
flux ratio was not consistent with the direct flux ratios at other wavelengths: $\Delta m$
= 0.31 mag in the $I$ band and $\Delta m$ = 1.29 mag in the $B$ band. From observations
with the 4.2 m William Herschel Telescope, a Spanish collaboration got spectra for each
component of the system. The data showed that the system consists of two quasars with
the same redshift ($z_s$ = 1.377) and identical spectral distribution, supporting the
gravitational lens interpretation of SBS 0909+532 (Oscoz et al. 1997). Oscoz et al. (1997)
detected a \ion{Mg}{ii} doublet in absorption at the same redshift ($z_{abs}$ = 0.83) in
both components, and they suggested that the absorption features were associated with the
photometrically unidentified lensing galaxy. Through a singular isothermal sphere (SIS)
lens model, the authors also inferred the first constraint on the time delay between the
components: $|\Delta \tau_{BA}| \leq$ 140 days, where $\Delta \tau_{BA}$ is the delay of B
with respect to A and the Hubble constant is assumed to be $H_0$ = 70 km s$^{-1}$ Mpc$^{-1}$.
In recent years, Lubin et al. (2000) indicated the possible nature of the main deflector
(early--type galaxy) and confirmed its redshift ($z_d$ = 0.830). Leh\'ar et al. (2000) reported
on a program including Hubble Space Telescope (HST) observations of SBS 0909+532. They
discovered the main lens galaxy between the components, which has a large effective radius,
with a correspondingly low surface brightness. This lens galaxy is closer to the brightest
component (A), which is not in contradiction with SIS--like lens models when the farther and
fainter component (B) is stronger affected by dust extinction (see below). The colors of
the lens are consistent with those of an early--type galaxy at redshift 0.83. Assuming a
singular isothermal ellipsoid (SIE) model, Leh\'ar et al. predicted a time delay $\Delta
\tau_{BA}$ in the range [$-$ 10, $-$ 87] days ($H_0$ = 70 km s$^{-1}$ Mpc$^{-1}$). At a given
emission time, the sign "$-$" means that the corresponding signal is observed first in B and
later in A. The COSMOGRAIL collaboration provided the distribution of predicted time delays of
the system (Saha et al. 2005). In their histogram (Fig. 10 of Saha et al.), there are two
features: the main feature is an asymmetric peak around $-$ 80 days and the secondary one is
another asymmetric peak around $-$ 45 days. Therefore, if the COSMOGRAIL predictions are right,
the time delay is very probably of 2--3 months (component B leading component A), but we
cannot rule out a delay of about one and a half months. On the other hand, the flux ratio
anomaly pointed by Kochanek et al. (1997) was confirmed and accurately studied by Motta et al.
(2002) and Mediavilla et al. (2005), who reported the existence of differential extinction in
the main lens galaxy. Chartas (2000) and Page et al. (2004) also studied the system in the
X--ray domain.
Time delays are basic tools to discuss the present expansion rate of the Universe and the
structure of the main lens galaxy haloes (e.g., Refsdal 1964; Kochanek, Schneider \& Wambsganss
2004), so that variability studies are crucial. While some time delays have been measured from
radio light curves (PKS 1830$-$211: Lovell et al. 1998; Q0957+561: Haarsma et al. 1999;
B0218+357: Biggs et al. 1999; B1600+434: Koopmans et al. 2000; B1422+231: Patnaik \& Narasimha
2001; B1608+656: Fassnacht et al. 2002) or X--ray variability (e.g., Q2237+0305: Dai et al.
2003), an important set of delays are based on optical monitoring of gravitationally lensed
quasars. Optical frames taken at Apache Point Observatory, Fred Lawrence Whipple Observatory
and Teide Observatory were used to estimate a 14--month delay for the double system Q0957+561
(e.g., Pelt et al. 1996; Kundi\'c et al. 1997; Serra--Ricart et al. 1999; Ovaldsen et al. 2003).
Although the time delay of this first multiple quasar has been confirmed through independent
observations, the measurement is only 5\% accurate, or equivalently, there is an uncertainty of
about 20 days (Goicoechea 2002). The Tel--Aviv University (TAU) group have recently determined
the time delay between the two components of HE 1104$-$1805 (Ofek \& Maoz 2003). The TAU delay
of HE 1104$-$1805 disagrees with the earlier estimation by Gil--Merino, Wisotzki \& Wambsganss
(2002), but it is in excellent agreement with the determination by Wyrzykowski et al. (2003).
Schechter et al. (1997) measured two delays for the quadruply imaged quasar PG 1115+080. The
Belgian--Nordic collaboration carried out a very intense activity during the past five years.
They participated in several monitoring projects and measured several time delays at optical
wavelengths: B1600+434 (Burud et al. 2000), HE 2149$-$2745 (Burud et al. 2002a), RXJ 0911.4+0551
(Hjorth et al. 2002), SBS 1520+530 (Burud et al. 2002b) and FBQ 0951+2635 (Jakobsson et al.
2005). The formal accuracies of these 5 estimations range from 5 to 25\% (the 1$\sigma$ error
bars vary from 4 to 24 days). Kochanek et al. (2005) also measured the time delays between the
components of the quadruple quasar HE 0435$-$1223.
The aim of this paper is to present $VR$ observations of SBS 0909+532 in 2003 conducted by
the University of Cantabria (UC, Spain), the Institute of Astronomy of Kharkov National
University (IAKhNU, Ukraine), the Sternberg Astronomical Institute (SAI, Russia) and the
Ulug Beg Astronomical Institute of Uzbek Academy of Science (UBAI, Uzbekistan). We also
present TAU observations of the field stars in 2003, which have been kindly made available to
us. This new optical monitoring campaign was carried out at the Calar Alto Observatory (Spain),
the Maidanak Observatory (Uzbekistan) and the Wise Observatory (Israel), and the frames were
taken with the 1.5 m Spanish telescope, the 1.5 m AZT$-$22 telescope at Mt. Maidanak and the
Wise Observatory 1 m telescope (Section 2). In Section 3, we describe the methodology to
obtain the fluxes of the quasar components and the field stars. The $VR$ light curves are also
shown in Section 3. Section 4 is devoted to the time delay estimation from the light curves
of A and B (quasar components) in the $R$ band. Finally, in Section 5 we summarize our
conclusions and discuss the feasibility of an accurate determination of the cosmic expansion
rate and the surface density in the main lensing galaxy.
\section{Observations}
We have three different sets of frames for SBS 0909+532. The first set of optical frames cover
the period between 2003 March 4 and June 2, and they are part of a UC project to test the
feasibility of quasar monitoring programs through 1$-$2 m telescopes in Spain (Ull\'an 2005).
These observations were made with the 1.52 m Spanish telescope at Calar Alto Observatory (EOCA),
Almeria, Spain (see Ziad et al. 2005 for a site--testing on Calar Alto). The EOCA is equipped
with a Tektronics 1024$\times$1024 CCD detector, which has pixels with a physical size of
24 $\mu$m, giving a 0.4 arcsec pixel$^{-1}$ angular scale. The gain is 6.55 e$^{-}$/ADU and the
readout noise is 6.384 e$^{-}$. During this first monitoring, exposures in the $V$ and $R$
Johnson--Cousins filters were taken every night when clear, what makes a total of 20 observing
nights. Bad weather in 2003 March and April prevented us from achieving a very dense sampling.
For each monitoring night we have three consecutive frames on each filter, i.e., three 300 s
exposures in the $V$ passband and three 180 s exposures in the $R$ passband. Those were the
maximum exposure times to avoid saturation of selected stars in the field. In Figure 1 we show
a typical frame. In this typical exposure, half a dozen bright and non--saturated stars were
fitted within the field of view (FOV). Following the notation of Kochanek et al. (1997), the
FOV included the gravitationally lensed quasar ("GL") and nearby field stars "a" (South), "b"
(North) and "c" (West). The FOV also included two stars that were introduced by Nakos et al.
(2003) and were labelled as "s1" and "s2". These two stars are placed relatively far from the
gravitational lens system, and they appear close to the North--West edge of the frame (see Fig.
1). A sixth star ("x") appears close to the South--West edge of the typical frame.
The second set of observations include frames in February 2003 as well as during April--May
and October--November 2003. The total number of nights is 18. In this second program the
images were taken with the 1.5 m AZT$-$22 telescope at Maidanak Observatory (Uzbekistan),
with near diffraction--limited optics and careful thermo--stabilization, which allow for
high--angular--resolution imaging. The AZT$-$22 telescope has a LN--cooled (liquid nitrogen
cooled) CCD--camera, SITe- 005 CCD, manufactured in Copenhagen (Denmark). For this camera, the
imaging area is split into 2000$\times$800 pixels, where the pixel size is 15 $\mu$m and the
intrinsic angular scale is 0.26 arcsec pixel$^{-1}$. The frames were taken in the $R$
Bessel filter, which corresponds approximately to the $R$ Johnson--Cousins passband. The poor
tracking system of this telescope allows only exposures up to 3 minutes. To obtain sufficiently
high photometric accuracy, we took several frames each observation night. With respect to the
rectangular FOV of the telescope, the North/South coverage was 2.5 times smaller than the
East/West one, so the "s1", "s2" and "x" stars were not included within the FOV. Figure 2 shows
a zoomed--in image made from one of the best frames in terms of seeing. There are two close
quasar components, but the very faint galaxy is not apparent. The observations at Mt. Maidanak
are part of IAKhNU, SAI and UBAI projects to follow up the variability of gravitationally
lensed quasars.
\begin{table}
\centering
\begin{tabular}{ccc}
\hline\noalign{\smallskip}
Observatory (Telescope) & Frames/night (Filter) & Observation Periods \\
\noalign{\smallskip}\hline\noalign{\smallskip}
Calar Alto (1.5 m) & 3 $\times$ 300 s ($V$) + 3 $\times$ 180 s ($R$) & March--June \\
Maidanak (1.5 m) & (3--11) $\times$ 180 s ($R$) & February, April--May, October--November \\
Wise (1.0 m) & 1 $\times$ 420 s ($R$) & 22 unevenly distributed nights \\
\noalign{\smallskip}\hline
\end{tabular}
\caption{Observations of SBS 0909+532 in 2003.\label{tbl-1}}
\end{table}
For the past six years the TAU group have been monitoring several gravitationally lensed quasars
with the Wise Observatory 1 m telescope. The targets are mainly monitored in the Johnson--Cousins
$R$--band, and the frames are obtained with a cryogenically cooled Tektronix
1024$\times$1024--pixel back--illuminated CCD. The angular scale is 0.7 arcsec per pixel. This
pixel scale and the median seeing ($FWHM$) of about 2\arcsec\ do not allow resolving most of the
lensed objects, e.g., SBS 0909+532. However, the frames of SBS 0909+532 in 2003 are characterized
by wide FOVs, which incorporate the "a--c", "s1--s2" and "x" stars. This fact permits to do
differential photometry between several pairs of field stars, and thus, to test the reliability
of the Calar Alto and Maidanak records.
The pre--processing of the images included the usual bias subtraction, flat fielding using sky
flats, sky subtraction and cosmic ray removal by using the Image Reduction and Analysis
Facility (IRAF) and Munich Image Data Analysis System (MIDAS) environments. Some details about the
whole observational campaign are included in Table 1 (observatories, telescopes, frames/night,
filters and observation periods).
\section{Photometry and $VR$ light curves}
Due to the small angular separation between the two lensed components, about 1\farcs1 (Kochanek
et al. 1997), the photometry of SBS 0909+532 is a difficult task. This task is also complicated
by the presence of the main lensing galaxy between the components, which could make the
computation of individual fluxes even harder. In general, aperture photometry does not work, so
we must look for better approaches. An initial issue is to decide about the inclusion or not
inclusion of a photometric model for the lensing galaxy. In principle, when computing the fluxes
of SBS 0909+532 we may use a galaxy model derived from the HST images of the system. The galaxy
model could also be inferred from the best images in terms of seeing. Once the relevant
information on the galaxy is known, we would apply a PSF fitting method to all optical images,
setting the galaxy properties to those derived from the HST or the best--quality images, and
allowing the remaining parameters to vary (e.g., McLeod et al. 1998; Ull\'an et al. 2003).
Magain, Courbin \& Sohy (1998) also presented an alternative task (deconvolution) that combines
all the frames obtained at different epochs to determine the numerical light distribution of the
lensing galaxy as well as the positions of the point--like sources (quasar components), since
these parameters do not vary with time. The flux of the point--like sources are allowed to vary
from image to image, which produces the light curves. However, these and other procedures have a
reasonable limitation: they only work well when the galaxy light has a significant contribution
to the crowded regions in the individual frames. For a very faint galaxy in a standard (i.e., not
superb) frame, there is confusion between galaxy signal and noise, so the use of a given galaxy
model could lead to biased fluxes of the components. The biases will depend on the quality of
the image (seeing, signal--to--noise ratio, etc), which must produce artificial variability
superposed to the real one. On the other hand, the use of a direct PSF fitting method
(neglecting the galaxy brightness) leads to contaminated fluxes of the components. But if the
galaxy is very faint, the contaminations will be small. Moreover, the variation of the quasar
fluxes, seeing conditions, etc, will cause fluctuations in the contaminations, which are
expected to be below the typical contamination levels. For standard frames of a quasar lensed
by a very faint extended object, it is really difficult to choose between both approaches (with
and without galaxy).
Most of the Calar Alto and Maidanak individual frames of SBS 0909+532 do not show evidences
for a galaxy brightness profile. This fact is due to the faintness of the galaxy, as we corroborate here
below. If we consider an hypothetical astronomer that neglects the galaxy brightness and
does direct PSF fitting (without taking into account the galaxy when doing the computation of the
fluxes), it is possible to attain a rough estimation of the maximum contamination from the galaxy
to the closest component A (at 0\farcs4 from the centre of the deflector). We take into account the
paper about 10 lens systems by Leh\'ar et al. (2000), where, in Table 3, we can find the best
available photometric and astrometric (HST) data of SBS 0909+532. The authors were able to trace the
galaxy light in the $H$ passband, by measuring its position and brightness. If we use the colors in
the same table, we conclude that $m_{gal} \sim$ 19 mag and $m_A \sim$ 16 mag in the $I$ band
(near--IR), and $m_{gal} >$ 20.4 mag and $m_A \sim$ 16.7 mag in the $V$ optical band. Therefore, as
the $R$ filter is placed just between the $I$ filter and the $V$ one, we may assume that $m_{gal} -
m_A \sim$ 3.5 mag in the $R$ band. The difference of 3.5 mag is consistent with a ratio of fluxes
$F_{gal}/F_A$ of about 1/25. Thus, in the case of QSO 0957+561 we found a $R$--band ratio of fluxes
$F_{gal}/F_A$ of about 1/2.5 (Ull\'an et al. 2003), and now we have $F_{gal}/F_A \sim$ 1/25, what
explains our unsuccessful efforts when measuring the flux of the lens galaxy in standard frames. As
a result of that, in an extreme case (when direct PSF fitting leads to a magnitude $m_{A + gal}$
instead of $m_A$, i.e., all the galaxy light is included in the profile of the A component) we find
a relationship: $m_A = m_{A + gal} + F_{gal}/F_A$, where the true flux (in magnitudes) $m_A$ differs
from the contaminated flux through direct PSF fitting ($m_{A + gal}$) in a quantity $F_{gal}/F_A$.
This maximum contamination of A would be only of 40 mmag, and the real contamination of both
components will be less than our upper limit. The artificial fluctuations (caused by variable
contamination) will be even smaller than the typical contamination levels, so we expect they will
not play an important role in analyses of quasar variability (e.g., time delay estimates).
In order to derive the light curves of the components A and B, we decide to use a direct PSF fitting
method and do not consider the galaxy brightness in the fits. The key idea of this procedure is to
obtain the different fluxes we are interested in by using a PSF that comes from a bright star in the
field common to all frames. The point--like objects (quasar components and stars) are modelled by
means of the empirical PSF. Hence, we do not use a theoretical PSF (i.e., Gaussian distribution,
Lorentzian distribution, etc), but the two--dimensional profile of a star in this field (a PSF star).
Apart from a PSF star, we also need a reference star to do differential photometry and to obtain
relative fluxes $m_A - m_{ref}$ and $m_B - m_{ref}$. The good behaviour of the reference star is
usually checked by using a control star, so the fluxes $m_{con} - m_{ref}$ are expected to agree with
a constant level. Nevertheless, since the $R$--band flux ratio is discussed in Section 5, we also want
to obtain a rough estimation of the contaminations from this direct technique. With this aim, a
deconvolution technique (Koptelova et al. 2005) is also applied to a set of frames with good seeing
and signal. The selected frames are fitted to a model including the galaxy, and thus, we are able to
obtain a few clean fluxes of components A and B and compare them with the corresponding contaminated
fluxes (through a direct PSF fitting). The averaged contaminations are used in Section 5.
\subsection{Calar Alto frames and light curves}
We adopt a model of the system including two point--like sources and a constant background. The
model is fitted to each image by adjusting its 7 free parameters (two--dimensional positions of A and
B, instrumental fluxes of both components and background) to minimize the sum of the square residuals,
as described in McLeod et al. (1998) and Leh\'ar et al. (2000). We use windows of 64$\times$64
pixels. Each empirical PSF is a subframe of 64$\times$64 pixels around the PSF star (the "a" star in
Fig. 1), while the lens system is analyzed from a subframe of the same size, but centered on the
double quasar. The instrumental fluxes of the "b", "c", "s1" and "s2" stars are also inferred from
64$\times$64 pixels windows centered on them. We initially focus on the nearby field stars, and take the
"b--c" stars as the control--reference objects. The "a" object is the brightest star in the "a--c"
triangle, and "b" and "c" were spectroscopically identified by Kochanek et al. (1997) and Zickgraf et
al. (2003): "b" is a FG star, whose spectrum includes the G--band and \ion{Ca}{ii} H--K lines, and "c"
is a M3 star. The $R - I$ and $B - R$ colors of the brightest component (A) and the "a" star are similar,
the colors of the faintest component (B) are close to the colors of the "b" star, and the "c" star has
colors different to those of the components and the "a--b" stars (see Table 1 of Kochanek et al. 1997).
On the other hand, after checking the PSFs of the three nearby field stars ("a--c"), we do not find
significant differences between them. This suggests that the global shape of the PSFs around the lens
system does not depend on the position and color of the point--like objects, so the PSF of the "a" star
seems to be a reliable tracer of the PSF associated with any point--like object in the region of
interest.
As a first attempt for obtaining light curves we use the "b" and "c" stars as the control and reference
objects, respectively. Unfortunately, we find clear evidences in favour of variability of the "c" star,
since the three curves $m_A - m_c$, $m_B - m_c$ and $m_b - m_c$ have a similar global behaviour. This
fact forces us to rule out the "c" star as a reference--control object and, thus, to take the "a" and "b"
nearby field stars as the control and reference point--like sources, respectively. In the next subsection,
we analyze the Maidanak--Wise fluxes $m_a - m_b$ and show that both stars ("a" and "b") are non--variable
objects. This result permits to assure the good behaviour of "b". The Calar Alto fluxes $m_a - m_b$
are not included in the analysis, since most Calar Alto data disagree with the Maidanak--Wise
common level of flux. We found an anomaly in the behaviour of the Calar Alto relative fluxes for widely
separate stars (see below), so only the relative fluxes for neighbouring point--like objects are reliable
photometric measurements. Fortunately, the comparison between the quasar components and the "b" nearby
reference star seems to be a feasible approach.
After applying the photometric method to the three individual frames for each filter and night (see
Table 1), we obtain three different measurements of $y_A = m_A - m_b$ and $y_B = m_B - m_b$ in the $V$ and
$R$ passbands for each night. To test the reliability of the instrumental fluxes of A and B, we analyse
the residues in each residual frame. A residual frame is an image after subtracting the fitted background
and point--like objects (PSF fitting method). More properly, we focus on the residual subframe occupied by
the system, and then we estimate the residue--to--signal ratio ($R/S$) in each pixel of interest. A $R/S$
value less than 10\% is acceptable, so a subframe with at least 90\% of pixels having acceptable residues
is considered to be related to reliable photometric solutions. Thus, we classify the individual fits in
two categories: fits leading to $<$ 90\% of pixels having acceptable residues (bad fits, unreliable
results) and good fits that are associated with reliable results ($\geq$ 90\% of pixels having acceptable
residues). As a complementary test, we study the relation between the quality of the fits (in terms of
post--fit residues) and two relevant parameters (image quality). The signal--to--noise at the brightest
pixel of the lens system, $(S/N)_{max}$, and the seeing, $FWHM$ (in \arcsec), are the two parameters to
compare with the fit quality. Some kind of correlation between good fits and good images is expected. In
Figure 3 we draw the $(S/N)_{max}$--$FWHM$ plots for frames in the $R$ filter (top panel) and the $V$
filter (bottom panel). Circles and triangles represent good and bad fits, respectively. The plots in
Fig. 3 indicate that the good fits correspond to images with high or moderate $(S/N)_{max}$ ($\geq$ 30).
Moreover, at moderate $(S/N)_{max}$ ($\sim$ 30--50), most of the good fits seem to be associated with a
relatively good seeing ($<$ 2 \arcsec). To obtain a robust photometry, we finally discard the frames
corresponding to the triangles in Fig. 3. For each filter and night, if there are two or three good
frames (good fits), then we get mean values of $y_A$ and $y_B$, and compute standard deviation of means
as errors. We only consider relative fluxes with uncertainties $\leq$ 40 mmag.
Now we plot $y_A$ (circles) and $y_B - 0.45$ mag (squares) in Figure 4 ($R$--band fluxes). If we
concentrate our attention in the period with the best sampling (after day 2755), the A light curve shows
a moderate decline and the B record shows a moderate rise. Indeed it seems that the "b" star is a good
reference object (constant flux), since there is no zero--lag global correlation between $y_A$ and $y_B$.
In Figure 5 we show the light curves $y_A$ and $y_B -$ 0.65 mag in the $V$ passband. In this case we have
a total of 11 points for the A component (circles) and 10 points for the B component (squares). The
$V$--band and $R$--band light curves of the A component are consistent with each other. A final moderate
decline appears in both curves. The situation is more confused for the B component. The $R$--band final
rise is not clearly reproduced in the $V$ band, and the $V$--band final measurements could have
underestimated formal errors. We note the relative faintness of B in the $V$ band ($\Delta m \sim$ 0.8
mag), and thus, the possibility of systematic uncertainties when the PSF fitting method is applied at
some epochs. The data in both optical filters are available at http://grupos.unican.es/glendama/.
After presenting the records of the double quasar, we concentrate on the Calar Alto light curves of
the field stars that were previously introduced by Kochanek et al. (1997) and Nakos et al. (2003), i.e.,
$y_a = m_a - m_b$, $y_c = m_c - m_b$, $y_{s1} = m_{s1} - m_b$ and $y_{s2} = m_{s2} - m_b$. There are no
previous studies on the variability of the nearby field stars "a--c". On the other hand, the farther
field stars ("s1--s2") were verified to be non--variable by using 76 Wise frames taken from 1999
December 24 to 2002 March 3 (Nakos et al. 2003). As Nakos et al. (2003) found that "s1" and "s2" seem
to be useful reference stars, we check the behaviour of "s1--s2" in 2003. The PSF of the stars in the
surroundings of the double quasar could slightly differ from the PSF of the "s1--s2" stars in a
relatively far region. Therefore, we must be careful when obtaining the instrumental fluxes of the
farther stars. To detect possible anomalies caused by a mismatch between the brightness profile of the
"a" star and the PSF of "s1--s2", the light curves $y_{s1}$ and $y_{s2}$ are derived from both PSF
fitting and aperture methods. The records $y_a$, $y_c$, $y_{s1}$ and $y_{s2}$ in the $R$ filter are
depicted in Figure 6. To guide the eyes, we use some offsets and dashed horizontal lines and put all
the relative records of each pair within a box. Filled and open symbols are associated with PSF fitting
and aperture, respectively. The top box includes the $y_a$ + 2.15 mag fluxes (open squares). The second,
third and fourth boxes (under the top one) correspond to the $y_c$ (filled squares), $y_{s1}$ + 2.38
mag (filled and open triangles) and $y_{s2}$ + 0.29 mag (filled and open circles) records, respectively.
As most of the stars are brighter than the quasar components (A and B) and they are far from other
objects, the typical formal errors in the stellar fluxes are clearly less than the typical uncertainties
in the fluxes of the components (these are usually fainter and are placed in a crowded region). The
stellar error bars in Fig. 6 are often smaller than the sizes of the associated symbols.
When doing aperture photometry on six $R$--band Wise frames covering the first semester of 2003, we
obtain a $y_a$ + 2.15 mag light curve (open triangles in the top box of Fig. 6) that disagrees with the
Calar Alto trend in the overlap period (between days 2710 and 2760). In the next subsection, we show
that the Wise and Maidanak brightnesses are constant and consistent with each other, so the Calar Alto
values of $y_a$ are not true fluxes, but anomalous results. On the contrary, the Wise light curve $y_c$
(open circles in the second box of Fig. 6) agrees with the Calar Alto curve in the overlap period. From
the Wise frames we confirm the flux level during the high--state of "c". Unfortunately, the
small--amplitude variability of "c" (rms fluctuation of $\sim$ 8 mmag) cannot be confirmed from the Wise
data. The rms fluctuation of the Wise fluxes ($\sim$ 9 mmag) is very similar to the Calar Alto variation,
but the formal errors are relatively large ($\sim$ 10 mmag). Moreover, there are no Wise frames in 2003
May (around the day 2780) and, thus, we cannot check (via Wise data) the reliability of the Calar Alto
dip in $y_c$ (80--100 mmag). However, the flux of the "c" star at day 2793 in the $V$ band confirms the
existence of a transition from the low--state to the high--state, which is finished at days 2800--2810
(see the last open circle in the second box of Fig. 6). For the "s1--s2" stars, which are as far from
star "b" as star "a" is, we again find a disagreement between the Calar Alto trends and the Wise records
(open astroids and rhombuses in the third and fourth boxes of Fig. 6). Although aperture curves are
closer to the Wise behaviours, we cannot fairly reproduce the Wise data. Some probes with the "x"
star (using $y_x = m_x - m_b$) also indicate that the Calar Alto and Wise behaviours disagree. It seems
that the differential photometry between widely separate stars may lead to meaningless results, and only
the relative fluxes for neighbouring objects are reliable. To test this conclusion, apart from the
successful results through the neighbouring stars "b" and "c", we also analyze the differential
photometry between the pair "s1--s2" (see Fig. 1). The curves $m_{s2} - m_{s1} -$ 0.20 mag are depicted
in the bottom box of Fig. 6: Calar Alto (filled and open star symbols) and Wise (open crosses). In the
overlap period (from day 2710 to day 2760), there is a reasonable agreement between the results from both
observatories, and the Calar Alto measurements seem to be quite reliable. From the Calar Alto frames,
both photometric techniques are consistent with each other, but a constant flux cannot explain the
observations. When we fit the data sets to a constant, our best solutions are characterized by $\chi^2
\sim$ 162 (PSF fitting) and $\chi^2 \sim$ 6 (aperture). It is a curious fact that aperture photometry on
only one frame per night leads to relative fluxes in rough agreement with a constant level. However, more
refined measurements (aperture or PSF fitting on several frames per night) reveal the variability of one
("s1" or "s2") or both stars.
\subsection{Maidanak frames and global $R$--band light curves of SBS 0909+532}
In the case of the $R$--band Maidanak observations, in order to derive the relative fluxes of the
components of SBS 0909+532, we also use a direct PSF fitting. For a given frame, after to obtain a first
estimate of the free parameters (initial solution), the fit is refined through an iterative procedure,
which works as the CLEAN algorithm ({\O}stensen 1994). The iterative task is done with each individual
image, and the solutions converge after a few cycles. For each night, we take all the available images
and obtain the mean values of $y_A$ and $y_B$. From the standard deviation of the means, we also derive
the errors in $y_A$ and $y_B$. In agreement with the criteria in subsection 3.1, only fluxes with errors
less than or equal to 40 mmag are considered. Apart from the analysis of the lens system, using aperture
photometry, we also measure $y_a$. The relative fluxes $y_a$ are depicted in Figure 7 (open star symbols).
The Maidanak measurements in the first semester of 2003 and the six Wise data of $y_a$ (open triangles;
see here above) are tightly distributed around $-$ 0.842 mag (solid line in Fig. 7). The rms fluctuation
of the data is only of $\sim$ 6 mmag (see the dashed lines in Fig. 7), which is consistent with the
typical error of the measurements. The $y_a$ results in Fig. 7 suggest that both "a" and "b" are
non--variable objects. Through 2003 (first and second semesters) we do not find any evidence in favour
of variability of the "a--b" stars.
We show our global $R$--band light curves of SBS 0909+532 in Figure 8. The open circles (Maidanak) and
filled circles (Calar Alto) are the measurements of $y_A$, whereas the open squares (Maidanak) and filled
squares (Calar Alto) are the values of $y_b -$ 0.45 mag. We have 31 points for the A component (circles)
and 26 points for the B one (squares). The top panel of Fig. 8 contains the results in the winter--spring
of 2003 and the bottom panel of Fig. 8 includes the results in the autumn of 2003. For each component we
test the existence of a bias between the Calar Alto and Maidanak fluxes, e.g., $\beta_A = y_A$ (Calar
Alto) $- y_A$ (Maidanak). Very small biases of $\beta_A$ = + 15 mmag and $\beta_B$ = $-$ 30 mmag are
found, and these corrections are taken into account to make the global records in Fig. 8. The biases are
derived from the comparison between the Maidanak fluxes in a thirty day period (from day 2750 to day
2780) and the Calar Alto fluxes at equal or close dates (see the top panel of Fig. 8).
To roughly estimate the contaminations from the direct PSF fitting technique, we take some of our best
Maidanak images (in terms of seeing conditions, $FWHM \sim$ 1 $\arcsec$) in the $R$ band. A zoom--in of
one of these best frames is shown in Fig. 2. Firstly, we combine the selected frames and derive a
numerical model of the galaxy from a regularizing algorithm. To produce a more stable reconstruction,
the real galaxy profile is assumed to be close to the Sersic profile (Koptelova et al. 2005). Our
deconvolution method differs only slightly from the former deconvolution techniques by Magain, Courbin
\& Sohy (1998) and Burud et al. (1998). Figure 9 presents the galaxy reconstruction obtained from the
stack of the $R$--band selected frames. The box in Fig. 9 is 16\farcs6 on a side. The positions of the
components are labeled with two crosses: A is on the left and B is on the right. The innermost contours
are circular--elliptical rings, whereas the outermost contours show a less definite shape. Secondly,
the selected frames are fitted to a photometric model that includes the galaxy brightness. Therefore,
we are able to infer clean relative fluxes of A and B (without contamination by galaxy light) and to
compare them with the contaminated ones (from direct PSF fitting). As result of the comparison, we
report typical (averaged) contaminations of 18.8 mmag and 4 mmag for the A and B components,
respectively. These very weak contaminations are in reasonable agreement with our preliminary
considerations in the beginning of this section, and are taken into account in the measurement of the
$R$--band flux ratio in Section 5.
\section{Time delay}
To calculate the time delay between both components of SBS 0909+532, we use the $R$--band
brightness records corresponding to the winter--spring of the year 2003. The $R$--band
records are more densely populated than the $V$-band ones. Moreover, the $R$--band time
coverage in the winter--spring of 2003 (about 120 days) is longer than the time coverage in
the autumn of 2003 (about 50 days). Thus we focus on the $R$--band data from day 2670 to day
2790, i.e., 22 points in the A component and 19 points in the B component (see the top panel
of Figure 8). There are different number of points for component A and component B because we only
consider fluxes with uncertainties below 40 mmag (see Section 3). As the B component is fainter,
its photometric uncertainties are larger and the number of final data is smaller. The new light
curves are characterized by a mean sampling rate of one point each six days.
Once we have the data set, a suitable cross--correlation technique is required. Here we mainly
use the $\chi^2$ minimization (e.g., Kundi\'c et al. 1997) and the minimum dispersion ($D^2$) method
(Pelt et al. 1994, 1996). However, although other techniques are probably less robust than the
$\chi^2$ and $D^2$ ones (doing a first delay measurement, without a previous empirical determination),
we also tentatively explore the modified cross--correlation function (MCCF) technique (Beskin \&
Oknyanskij 1995; Oknyanskij 1997). The MCCF combines properties of both standard cross--correlation
functions: the CCF by Gaskell \& Spark (1986) and the DCF by Edelson \& Krolik (1988). We begin our
analysis using the $\chi^2$ method, which is based on a comparison between the light curve $y_A$ (or
$y_B$) and the time shifted light curve $y_B$ (or $y_A$). For a given lag, one can find the magnitude
offset that minimizes the $\chi^2$ difference. From a set of lags, it can be derived a set of minima
(of $\chi^2$), which permits to make a $\chi^2$ spectrum: $\chi^2$ vs lag. The best solution of the
delay is the lag corresponding to the minimum of the $\chi^2$ spectrum. In general, the shifted epochs
$t'_B$ (or $t'_A$) do not coincide with the unchanged epochs $t_A$ (or $t_B$), so we estimate the
values of $y_A(t'_B)$ (or $y_B(t'_A)$) by averaging the A (or B) fluxes within bins centered on times
$t'_B$ (or $t'_A$) with a semiwidth $\alpha$. To average in each bin, it is appropriate the use of
weights depending on the separation between the central time $t'_B$ (or $t'_A$) and the dates
$t_A$ (or $t_B$) in the bin. In principle, we concentrate in the interval [$-$ 90, + 90] days,
which includes the predicted negative delays (see Introduction) as well as a wide range of
unlikely positive delays (positive delays are inconsistent with basic observations
of the system).
Firstly, the curve $y_A$ and the time shifted curve $y_B$ are compared with each other
(using bins in the A component). In order to work with a reasonable time--resolution, we use
$\alpha$ values less than or equal to two times the mean sampling time, i.e., $\alpha \leq$
12 days. The $\chi^2$ value roughly grows with the size of the bin, and $\chi^2 \sim$ 1 for
$\alpha$ = 7$-$9 days. For $\alpha$ = 7$-$9 days, there are best solutions $\Delta \tau_{BA}$
= $+$ 46$-$48 days ($\chi^2$ = 0.97$-$0.98), and we show the corresponding spectra in Figure
10. We have drawn together the spectra for $\alpha$ = 7 days (dashed line), $\alpha$ = 8 days
(solid line) and $\alpha$ = 9 days (dotted line). Apart from the main minima close to + 50 days,
there are other secondary minima at negative and positive lags. In Fig. 10, two secondary
minima seem to stay significant for all the bin sizes: the minima close to $-$ 50 days and the
probable edge effects at + 80$-$90 days. We also compare the curve $y_B$ and the time shifted
curve $y_A$, using bins in the B component. For $\alpha$ = 10 days, we obtain a best solution
$\Delta \tau_{BA}$ = $-$ 44 days ($\chi^2$ = 1.15). Smaller and larger bins lead to solutions
characterized by $\chi^2 <$ 0.7 and $\chi^2 \geq$ 1.2, respectively. In Figure 11, the solid
line represents the spectrum for $\alpha$ = 10 days, while the dashed line represents the
spectrum for $\alpha$ = 9 days and the dotted line traces the spectrum for $\alpha$ = 11 days.
Main minima in the interval $-$ 40$-$50 days appear in all these cases. Unfortunately,
important signals at positive lags and probable border effects at + 80$-$90 days are again
included in the complex spectra. The important structures at positive lags in Figs. 10$-$11 are
probably caused by artifacts in the cross--correlation, so they have no physical origin, but
are due to the 10/20--day gaps and the moderate variability of the components. Therefore,
taking $\alpha$ = 10 days (bins in the B component) and a negative range [$-$ 90, 0] days, we
try to determine a pre--conditioned time delay.
In order to derive uncertainties, we follow a simple approach. We make one repetition
of the experiment by adding a random quantity to each original flux in the light curves. The
random quantities are realizations of normal distributions around zero, with standard
deviations equal to the errors of the fluxes. We can make a large number of repetitions,
and thus, obtain a large number of $\Delta \tau_{BA}$ values. The true value will be included
in the whole distribution of measured delays. From the $\chi^2$ minimization (bins in B and
$\alpha$ = 10 days) and 1000 repetitions, we obtain the histograms in Figure 12. Regarding the
distributions in the top panel (delays) and bottom panel (flux ratios) of Fig. 12, the main
features lead to measurements $\Delta \tau_{BA}$ = $-$ 45 $^{+ 1}_{-11}$ days and $\Delta
m_{BA}$ = 0.590 $\pm$ 0.014 mag (95\% confidence intervals). We note that the main delay peak
is asymmetric, so 55\% of the repetitions correspond to $-$ 44$-$45 days, whereas 40\% of the
repetitions correspond to values $<$ $-$ 45 days. The secondary delay peak (around $-$ 20 days)
represents about 5\% of the repetitions and is associated with the secondary minima in the
negative region of Fig. 11. Therefore, the distribution in the top panel of Fig. 12 permits a
95\% estimation of the time delay of SBS 0909+532.
In Figure 13 (top panel), the A light curve (circles) shifted by the optimal values of the
time delay and the magnitude offset (time--delay--corrected flux ratio), and the unchanged
B light curve (squares) are plotted. The cross--correlation using bins in the B component
($\alpha$ = 10 days) indicates that the initial variations in the brightness of B reasonably
agree with the final fluctuations in the brightness of A. The overlap for a delay of $-$ 80
days (e.g., Saha et al. 2005) also appears in the bottom panel of Fig. 13. However, this last
time delay is clearly rejected by the observations, since the $\chi^2$ value is larger than
10 ($\chi^2 \sim$ 18).
To confirm the results from the $\chi^2$ minimization, we also use the dispersion spectra
introduced by Pelt et al. (1994, 1996). The basic idea is a combination of $y_A$ and $y_B$
into one global record for every lag $\tau$ and magnitude offset $m_0$ by taking all the
values of $y_A$ as they are and shifting the values of $y_B - m_0$ by $\tau$. For each $\tau$
one can find the $m_0$ value that minimizes a dispersion estimate $D^2(\tau,m_0)$, so a
dispersion spectrum $D^2(\tau)$ can be made in a direct way. We focus on the $D^2_{4,2}$
spectra that are called $D^2$ for simplicity (see Pelt et al. 1996 for details). This technique
incorporates a decorrelation length ($\delta$), where $\delta$ plays a role similar to that
of $\alpha$ in the $\chi^2$ method. Considering reasonable values of $\delta$ (from 7 to 11
days, see here above), we are able to make some interesting spectra. In Figure 14 we have
plotted together the spectra for $\delta$ = 7 days (dashed line), $\delta$ = 9 days
(solid line) and $\delta$ = 11 days (dotted line). Although there are main minima in the interval
$-$ 40$-$50 days, there are also significant signals at positive lags and probable border
effects at + 90 days. In the negative region of Fig. 14, a secondary minimum around $-$ 70 days
appears. Using $\delta$ = 9 days and a negative range [$-$ 90, 0] days, we carry out a second
pre--conditioned measurement of the time delay. The uncertainties are deduced from 1000
repetitions of the experiment (see here above), and the relevant histograms are shown in
Figure 15. While the top panel contains the distribution of delays, the bottom panel traces the
distribution of flux ratios. Through the distributions in Fig. 15, we obtain that $\Delta
\tau_{BA}$ = $-$ 48 $^{+ 7}_{-6}$ days and $\Delta m_{BA}$ = 0.585 $\pm$ 0.020 mag (90\%
confidence interval). These $D^2$ results strengthen the conclusions from the $\chi^2$
technique. A marginal measurement (10\% confidence interval) of $\Delta \tau_{BA}$ = $-$
67 $^{+ 1}_{-2}$ days and $\Delta m_{BA}$ = 0.558 $^{+ 0.007}_{-0.008}$ mag is also possible.
However, both this possibility and the $\chi^2$ result of around $-$ 20 days are probably
related to the presence of gaps and the absence of strong variability in the light curves.
A MCCF technique (Beskin \& Oknyanskij 1995; Oknyanskij 1997) is also explored. The MCCF is
a modification of the standard cross--correlation functions (CCF and DCF). When this MCFF is
applied to our data in the lag interval [$-$ 60, + 60] days, the maximum correlation
coefficient (0.907) corresponds to a lag of $-$ 45 days. This last result basically agrees
with the $\chi^2$ and dispersion spectra in Figs. 11 and 14.
\section{Conclusions}
Nowadays several groups are trying to coordinate the rich but scattered research potential
in the field of gravitationally lensed quasar monitoring. The goals are to rationalize the
astronomical work and to catalyze big scientific collaborations so that the astrophysics
community can get a significant progress in the understanding of the central engine in
lensed quasars, the structure of the lensing galaxies and the physical properties of the
Universe as a whole. Some examples about that are the Astrophysics Network for Galaxy
LEnsing Studies (ANGLES, http://www.angles.eu.org/), the Cosmic Lens All-Sky Survey (CLASS,
http://www.aoc.nrao.edu/$\sim$smyers/class.html) and the COSmological MOnitoring of
GRAvItational Lenses (COSMOGRAIL, http://www.cosmograil.org/).
The University of Cantabria group (Spain), three groups of the former Soviet Union
(Institute of Astronomy of Kharkov National University, Ukraine, Sternberg Astronomical
Institute, Russia, and Ulug Beg Astronomical Institute of Uzbek Academy of Science,
Uzbekistan) and the Tel--Aviv University group (Israel) are also carrying out a series of
initiatives to better exploit the recent individual monitoring campaigns as well as to
solidify some future common project. In this paper we present the first collaborative
programme on the variability of the double quasar SBS 0909+532A,B. The $VR$ observations of
the system and the field stars were made with three modern ground--based telescopes in the
year 2003.
The SBS 0909+532c star (N23210036195 in the GSC2.2 Catalogue) at ($\alpha$, $\delta$) =
(09:12:53.59, +52:59:39.82) in J2000 coordinates is found to be variable, with two
different levels of flux. The $VR$ gap between the low--state and the high--state is of
80--100 mmag, and the low--state lasts about one month. In the high--state the star also
seems to vary, but these small--amplitude variations are not so significant as the gap
between states. We want to remark the variability of this nearby star ("c" star), and
to encourage colleagues to follow-up its fluctuations and identify the kind of variable
source. The "c" star cannot be used as the reference object (differential photometry),
because it introduces a zero--lag global correlation between the light curves of the
quasar components A and B. However, the "a--b" nearby stars are non--variable sources,
and we choose the "b" star as the reference candle. On the other hand, the "s1" and "s2"
stars are relatively far objects, which were proposed as good references in a previous
analysis (Nakos et al. 2003). However, the new $R$--band light curve $m_{s2} - m_{s1}$
reveals the variability of one ("s1" or "s2") or both stars. This variability could be
either a very rare phenomenon or a consequence of doing more refined measurements
(aperture or PSF fitting on several frames per night). We warn about the possible
problems with this pair of stars and think it merits more attention. The point--spread
function (PSF) fitting methods permit to resolve the two components of the quasar and
to derive the $VR$ light curves of each component. These new $VR$ light curves represent
the first resolved brightness records of SBS 0909+532. Although the $V$--band curves are
interesting, the $R$--band records seem more reliable and are more densely populated.
The $R$--band curves show a moderate variability through 2003, and the observed fluctuations
are promising for different kinds of future studies.
To estimate the time delay between the components of SBS 0909+532, we use an 120--day piece
of the $R$–-band brightness records, and $\chi^2$ and dispersion ($D^2$) techniques. The
cross--correlation of the two light curves (A and B) leads to complex $\chi^2$ spectra.
However, assuming that the quasar emission is observed first in B and afterwards in A, or in
other words, $\Delta \tau_{BA} <$ 0 (in agreement with basic observations of the
system), 95\% measurements $\Delta \tau_{BA}$ = $-$ 45 $^{+ 1}_{-11}$ days and
$\Delta m_{BA}$ = 0.590 $\pm$ 0.014 mag are inferred from 1000 repetitions of the
experiment (synthetic light curves based on the observed records). From the $D^2$
minimization (Pelt et al. 1996) and 1000 repetitions, we also obtain 90\% measurements $\Delta
\tau_{BA}$ = $-$ 48 $^{+ 7}_{-6}$ days and $\Delta m_{BA}$ = 0.585 $\pm$ 0.020 mag. The $D^2$
uncertainties are derived under the already mentioned assumption that $\Delta
\tau_{BA}$ is negative. There is a clear agreement between the results from both techniques, so
a delay value of about one and a half months is strongly favoured. Our light curves rule
out a delay close to three months, which has been claimed in a recent analysis (Saha et
al. 2005). When we measure the time delay of the system, we simultaneously derive
the time--delay--corrected flux ratio (at the same emission time) in the $R$ band. This
quantity, $\Delta m_{BA}$ = $m_B(t + \Delta \tau_{BA}) - m_A(t)$, is contaminated by light of
the lens galaxy, and taking into account the weak contaminations of A and B (see the end of
subsection 3.2), the totally corrected $R$--band flux ratio is 0.575 $\pm$ 0.014 mag. We
remark that our final $R$ flux ratio is in total agreement with the rough (uncorrected by the
time delay and the contamination by galaxy light) measurement by Kochanek et al. (1997): 0.58
$\pm$ 0.01 mag. To properly determine a flux ratio, one must use clean fluxes at the same
emission time, i.e., fluxes at different observation times and without contamination
(Goicoechea, Gil--Merino \& Ull\'an 2005). Only for particular cases (e.g., faint lens galaxy,
short delay and moderate variability), it may be reasonable to use direct fluxes.
In order to get a reasonably good value of $\chi^2$, we do not need to introduce a time
dependent magnitude offset or a complex iterative procedure (e.g., Burud et al. 2000; Hjorth
et al. 2002), i.e., only a delay and a constant offset are fitted. This is a strong point of
the analysis. The agreement between the results from different techniques is another strong
point. However, the new measurements have some weak points that we want to comment here. The
weakest point is the relatively poor overlap between the A and B records, when the A light
curve is shifted by the best solutions of the time delay and the magnitude offset (e.g., see
the top panel of Fig. 13). Moreover, we carry out pre--conditioned measurements, since a
negative interval [$-$ 90, 0] days is considered in the estimation of uncertainties (component
B leading component A). This second weak point is related to the presence of 10/20--day gaps
and the moderate variability of the components, which does not permit to fairly rule out
positive delays. We nevertheless remark that the negative interval is in good agreement with
the predictions by Leh\'ar et al. (2000) and Saha et al. (2005), and we find $\chi^2$ and $D^2$
minima around $-$ 45 days when the observed data and both negative and positive lags are taken
into account (see Figs. 11 and 14). Of course, as any another first determination of a time delay,
the 1.5--month value should be confirmed from future studies.
Forty years ago, Refsdal (1964) suggested the possibility of determining the current
expansion rate of the Universe (Hubble constant) and the masses of the galaxies from the
time delays associated with extragalactic gravitational mirages. More recently, for a
singular isothermal ellipsoid (SIE), Koopmans, de Bruyn \& Jackson (1998) found that
the time delay can be cast in a very simple form, depending on basic cosmological
parameters, redshifts and image positions. The relevant image positions are the
positions with respect to the centre of the main lens galaxy, and the SIE delay is
similar to the delay for a singular isothermal sphere (SIS). In principle, a singular
density distribution is justified because a small core radius changes the time delay
negligibly, and only a small core radius seems to be consistent with the absence of
a faint central image (e.g., Kochanek 1996). Moreover, individual lenses and lens
statistics are usually consistent with isothermal models (e.g., Witt, Mao \& Keeton
2000 and references therein), so it is common to adopt an isothermal profile. Witt,
Mao \& Keeton (2000) showed that an external shear changes the simple SIS time delay
in proportion to the shear strength. For two--image lenses that have a small shear and
images at different distances from the centre of the lens, the shear should have a
small effect on the time delay. Thus, when one has accurate measurements of image
positions, redshifts and time delay, it is viable an accurate estimation of $H_0$
(using complementary information on the matter/energy content of the Universe).
Very recently, Kochanek (2002) also presented a new elegant approach to the subject.
He modelled the surface density locally as a circular power law, with a mean surface
density $<\kappa>$ in the annulus between the images. Expanding the time delay as a
series in the ratio of the thickness of the annulus to its average radius, it is
derived a delay that is proportional to the SIS time delay. The zero--order expansion
term consists of the SIS delay and a multiplicative factor $2(1 - <\kappa>)$. Kochanek
also incorporated the quadrupoles of an internal shear (ellipsoid) and an external
shear. However, for two-image lenses where the images lie on opposite sides of the
lens, the delay depends little on the quadrupoles. This novel perspective is useful
to infer $<\kappa>$ from observations of the lens system (time delay, image positions
and redshifts) and complementary cosmological data (expansion and matter/energy
content of the Universe).
For SBS 0909+532, although the redshifts are very accurately known and the time delay
is now tightly constrained (or at least there is a first accurate estimation to be
independently confirmed), the inaccurate position of the main lens galaxy does not
permit an accurate measurement the cosmic expansion rate and the surface density of the
main deflector. We have $H_0 \propto \theta_B^2 - \theta_A^2$ and $1 - <\kappa>
\propto (\theta_B^2 - \theta_A^2)^{-1}$, where $\theta_A$ and $\theta_B$ are the image
angular positions with respect to the centre of the main lens galaxy. On the other
hand, using the astrometry in Table 3 of Leh\'ar et al. (2000), it is easy to obtain
$\theta_B^2 - \theta_A^2$ = 0.4 $\pm$ 0.2. Thus we conclude that the accuracy in
$\theta_B^2 - \theta_A^2$ is only 50\%, indicating the necessity of new accurate
astrometry of SBS 0909+532.
\begin{acknowledgements}
The UC members are indebted to J. Alcolea (Observatorio Astron\'omico Nacional, Spain) for
generously granting permission to operate the 1.52 m Spanish telescope at Calar Alto
Observatory (EOCA) in March--June 2003. This 4--month season was supported by Universidad de
Cantabria funds and the Spanish Department for Science and Technology grant AYA2001-1647-C02.
AU thanks the Departamento de F\'isica Te\'orica y del Cosmos de la Universidad de Granada (E.
Battaner) for hospitality during the observational season. The post--observational work and the
spreading of results are supported by the Department of Education and Science grants
AYA2002-11324-E, AYA2004-20437-E and AYA2004-08243-C03-02. We acknowledge the use of data
obtained by the SAI group headed by B. Artamonov. We are also indebted to D. Maoz and E. Ofek
for providing us with the Wise frames of SBS 0909+532. APZ is grateful for the support of
the Science and Technology Center of Ukraine (STCU), grant U127k. The observational work by the
UBAI group (TA and OB) at Mt. Maidanak was supported by the German Research Foundation (DFG),
grant 436 UZB 113/5/0-1. We are also grateful to the referee for several helpful comments.
We acknowledge support by the European Community's Sixth Framework Marie Curie Research Training
Network Programme, Contract No.MRTN-CT-2004-505183 "ANGLES". The GSC-II is a joint project of the
Space Telescope Science Institute (STScI) and the Osservatorio Astronomico di Torino (OAT). STScI
is operated by the Association of Universities for Research in Astronomy, for the NASA under
contract NAS5-26555. The participation of the OAT is supported by the Italian Council for
Research in Astronomy. Additional support is provided by ESO, Space Telescope European
Coordinating Facility, the International GEMINI project and the ESA.
\end{acknowledgements}
{}
|
Title:
The nearest young moving groups |
Abstract: The latest results in the research of forming planetary systems have led
several authors to compile a sample of candidates for searching for planets in
the vicinity of the sun. Young stellar associations are indeed excellent
laboratories for this study, but some of them are not close enough to allow the
detection of planets through adaptive optics techniques. However, the existence
of very close young moving groups can solve this problem. Here we have compiled
the members of the nearest young moving groups, as well as a list of new
candidates from our catalogue of late-type stars possible members of young
stellar kinematic groups, studying their membership through spectroscopic and
photometric criteria.
| https://export.arxiv.org/pdf/astro-ph/0601573 | command.
\shorttitle{Near moving groups}
\shortauthors{L\'opez-Santiago et al.}
\begin{document}
\title{The nearest young moving groups}
\author{J. L\'opez-Santiago\altaffilmark{1,2}, D. Montes\altaffilmark{2},
I. Crespo-Chac\'on\altaffilmark{2} and M.J. Fern\'andez-Figueroa\altaffilmark{2}}
\affil{\altaffilmark{1}INAF - Osservatorio Astronomico di Palermo,
Piazza Parlamento 1, I-90134 Palermo, Italy}
\affil{\altaffilmark{2}Departamento de Astrof\'{\i}sica y Ciencias de la
Atm\'osfera, Facultad de Ciencias F\'{\i}sicas,
Universidad Complutense de Madrid, E-28040 Madrid, Spain}
\keywords{associations and clusters: moving groups ---
stars: kinematics --- stars: stellar activity ---
stars: lithium abundance --- stars: planets}
\section{Introduction}
\label{sec:intr}
In recent years, a series of young stellar kinematic groups
(clusters, associations, and moving groups) of late-type stars
with similar space motion and ages ranging from 8 to 50 Myr
\citep[see][and references therein]{zuc04a}
has been discovered in our neighbourhood:
TW~Hya, $\beta$~Pic, AB~Dor, $\eta$~Cha, $\epsilon$~Cha, Tucana and
Horologium associations. In addition, several more distant young
associations such as MBM~12 \citep{hea00}, Corona Australis \citep{qua01},
and possibly the group of stars with a motion similar to that of HD~141569
\citep{wei00} have also been identified.
In the Galactic velocity space, they situate inside the boundaries of the
\object{Local Association} (see Fig.~\ref{fig:uv}), a mixture of young stellar
complexes ---~OB and T-associations~--- and clusters with different
ages \citep{egg75, egg83a, egg83b, m01}.
These associations of very young stars are excellent laboratories for
investigations of forming planetary systems \citep{zuc04}.
Nevertheless, they are generally situated at distances above 50 pc,
which makes them less accessible to adaptive optics systems even on large
telescopes.
It is well-known that tightly bound, long-lived open clusters can account
for only a few per cent of the total galactic star formation rate
\citep{wie71}. Therefore, either most clusters and associations
disperse very quickly after star formation has started or most are
born in isolation \citep{wic03}.
The existence of very young moving groups (MGs) with a few dozens of stars
showing the same spectroscopic properties ---~i.e. age, metallicity, level
of magnetic activity~--- is in agreement with the first explanation.
Small associations of stars may be dispersed by galactic differential rotation
since they are not gravitationally bounded enough, taking into
account that their nucleus consist of only a few stars as in the case
of the Ursa Major MG (see King et al. 2003, for a recent review) or
the recently discovered AB~Doradus MG \citep{zuc04}.
The location of these young MGs inside the Local Association and its
proximity in the $UV$-plane can be explained as the result of
the juxtaposition of several star forming bursts in adjacent cells of the
velocity field (see Montes et al. 2001, and references therein) or
dynamical perturbations caused by spiral waves \citep{sim04,fam05,qui05}.
Thus, one expects to find groups of coeval stars with similar space motion in
our neighbourhood.
In 2001, the $\beta$~Pic~MG \citep{zuc01} --- a group of stars with an age of
$\sim$~12~Myr \citep{zuc01, ort04} at a mean distance of $\sim$~35~pc
co-moving with the well-known young star $\beta$~Pic --- was confirmed
to be the closest kinematic group up to date.
More recently, \citet{zuc04} have
identified a new group of stars co-moving with the also well-known young star
AB~Dor, at a mean distance of $\sim$~30~pc, and with an age of $\sim$~50~Myr.
Nevertheless, the existence of a nearer association of a few stars was
proposed by \citet{gai98} and studied in detail by \citet{fur04}, though
its existence is quite controversial.
Here we discuss about the fact of the nearest MGs using both spectroscopic
and photometric criteria of membership
for a sample of stars that includes the proposed members
from the literature and our list of young cool stars possible members of
young stellar kinematic groups
\citep{m01, lop05}.
\section{The Hercules-Lyra Association}
\label{sec:herc}
Based on the kinematics of young solar analogues in the solar neighbourhood,
\citet{gai98} confirmed the existence of a group of four stars
(marked with $\dag$ in Table~\ref{tab:MGs}) co-moving in the space towards the
constellation of Hercules. Recently, \citet{fur04} has extended the sample of
late-type stars of this MG up to 15 nearby (d~$<$~25~pc) candidates,
proposing the name Hercules-Lyra since several members show a radiant
``{\it evenly matched}'' with this constellation.
Comparing the level of chromospheric activity of the stars of his sample
with that of the members of the Ursa Major Association and looking for
the existence of lithium in their spectrum, he notices that
several candidates of Hercules-Lyra appear to be coeval of the Ursa Major
stars, for which he gives an age of $\sim$~200~Myr. On the contrary,
other candidates seem to be older (e.g. HD~111395) or younger (HD~17925,
HD~82443, and HD~113449), questioning the existence of Hercules-Lyra as an
entity independent of the Local Association. However, he considers unlikely
that the majority of his sample can originate from the Pleiades alone, or
other clusters of the Local Association since ``{\it they are poorer and more
distant}'' as pointed out by \citet{jef95}. Thus, he confirms
``{\it the bulk}'' of the sample --- formed by the stars HD~166, HD~96064,
HD~97334, HD~116956, HD~139777, HD~139813 and HD~141272 (see his Table~1) ---
to be an entity on its own.
Here we discuss the possible existence of the Hercules-Lyra MG
as an independent association using kinematic (space motion), spectroscopic
(lithium abundance) and photometric (isochrone fitting) criteria. A total of
12 possible members (stars marked with {\it a} in Table~\ref{tab:MGs})
have been added to the initial sample of \citet{fur04} from our
catalogue of {\it Late-type Stars Possible Members of Young Stellar
Kinematic Groups} \citep{m01}.
The candidates have been chosen by their kinematics assuming a total
dispersion of \mbox{$\pm$ 6 km~s$^{-1}$} in $U$ and $V$, respectively;
that is, an average position of \mbox{($U$, $V$) = (-15.4, -23.4)}
km~s$^{-1}$
has been determined using the stars given by \citet{fur04}, and every star
in our catalogue in a radius of \mbox{$\pm$ 6 km~s$^{-1}$} has been
selected. The value of the dispersion has been chosen equal to that of the
\mbox{$\sim 200$ Myrs} old Castor~MG \citep{m01}, coeval of the
Hercules-Lyra Association. No restriction in the $W$ component has been
imposed in this first selection.
In Table~\ref{tab:MGs} we summarize the results obtained by us.
From the whole sample of 27 candidates, eight stars have been discarded
as members by their space motion: HD~25457, located inside the B4
subgroup (see Fig.~\ref{fig:uv}); HD~96064, HD~112733, HIP~67092, the
binary system made up of the F-type star HD~139777 and HD~139813, and
HD~207129 all them with a value in $W$ higher than that of the rest of the
candidates (see Fig.~\ref{fig:uvw_MGs}); and HD~113449, classified as
member of the AB~Dor~MG by \citet{zuc04} (see Table~\ref{tab:MGs} and
$\S$~\ref{sec:abdor} for a more detailed discussion) and
questioned by \citet{fur04} because of its relatively high lithium abundance.
We have also studied the lithium abundance --- measured as the equivalent width
of the lithium line $\lambda$6707.8~\AA, $EW$({Li~{\sc i}}) --- in each one
of the candidates. The values of $EW$(Li~{\sc i}) have been taken from
\citet{lop05} and compared with those of the members of well-known stellar
clusters (see Fig.~\ref{fig:li_MGs}). The results appear to be consistent
with an age of \mbox{150 -- 300~Myr} for seven candidates. However, several
stars (HD~1466, HD~17295, \mbox{1E~0318.5-19.4}, and HD~82443) show an
$EW$(Li~{\sc i}) comparable to that of the members of the Pleiades while
other five (HD~37394, HD~97334B, HD~111395, HD~116956 and HD~141272) are fully
depleted or have a value lower than the expected for a member of the Hercules-Lyra
Association.
For isochrone fitting, we have adopted pre-main sequence models
from \citet{sie00}. For $T_{\rm eff} < 4000$~K,
the models systematically underestimate the age when comparing
with clusters of known age such as the Pleiades and IC~2391
in a $M_{\rm v}$ vs. $V-I$ diagram \citep{lop05} due to the transformation
from flux to colour. Bearing this in mind, the corrected transformation
adopted by \citet{lop03} and \citet{lop05b}
for stars cooler than 4000~K has been used in this work.
The values of $V-I$ have been taken from the Hypparcos Catalogue \citep{esa97}.
The result of comparing the position of the stars with the isochrones
in the colour-magnitude diagram (CMD) (Fig.~\ref{fig:VI}) is again in agreement
with and age of \mbox{$\sim$ 150 - 300 Myrs}. Nevertheless, no conclusions
can be inferred from the CMD alone since isochrones of more than 80~Myr
converge for $V-I \le 1.8$ mag., and ages larger than 300 Myr could be
adopted.
From the combination of the three criteria, the total sample of candidates is
reduced to 10 stars with $EW$(Li~{\sc i}) and position in the CMD compatibles
with an age of \mbox{$\sim$ 200 Myrs}, which could form the bulk of the
Hercules-Lyra Association, and other 15 definitively non members or with a
doubtful classification (Table~\ref{tab:MGs}). The members show a deviation
\mbox{($\sigma_{\rm U}$, $\sigma_{\rm V}$) = (2.46, 1.61) km s$^{-1}$} from
the centre (\mbox{($U$, $V$) = (13.19, 20.64) km s$^{-1}$}) lower than
that of other coeval MGs such as Castor and Ursa Major
\citep[see][and references therein]{m01}.
A similar dispersion (\mbox{$\sigma_{\rm W} \approx 3.4$ km s$^{-1}$}) is
found in W, confirming the results in $U$ and $V$. In the same way, the shape of
the MG in the velocity field is in agreement with the theory of the MGs
\citep{egg65, age79, sku99, asi99b, lop05}. According to this theory, not-gravitational
bounded stars formed in the same forming region and with low sigmas
in $U$, $V$ and $W$ are dispersed during their rotation around the Galactic centre,
inducing a particular shape in both the space and the velocity field since some of
the stars fall behind while others go ahead. The Galactic potential maintains
the group bounded during several hundreds of years, in spite of the initial velocity
dispersion in the molecular cloud, in both the $UV$-plane and the $W$ component.
\section{The AB Dor MG and subgroup B4}
\label{sec:abdor}
Very recently, \citet{zuc04} have identified a large group of stars with the
same space motion than the well-known young K-dwarf AB~Dor (d = 15~pc),
a quadruple system \citep{clo05, gui05} made up of three late-type stars
--- AB~Dor~A (HD~36705),
AB~Dor~Ba and AB~Dor~Bb --- and a very low mass companion which has recently
been object of discussion because of the discrepancy between its
dynamical mass and that predicted by evolutionary models
\citep{clo05}.
All the stars listed in Table~1 of \citet{zuc04} are situated inside
the Local Association (see Fig.~\ref{fig:uv}) near the boundaries of the young
disk stellar population \citep{egg84},
and have at least one indicator of
youth. Taking the intensity of the H$\alpha$ emission line of
these stars and the position in a $V-K_{\rm s}$
diagram of three M-type members of the MG into account,
they estimate an age of $50\pm10$ Myr for the AB~Dor MG.
Very recently, \citet{luh05} and \citet{luh05b}
have showed that the components of
AB~Dor should have an age of 75 -- 150 Myr based on the comparison of both
their position in the $M_{\rm K}$ vs. $V-K_{\rm s}$ diagram
with respect to the Pleiades and IC~2391 clusters,
and the $EW$(Li~{\sc i}) of AB~Dor~A with that of rapidly
rotating K dwarfs in the Pleiades. Moreover, with an age of
$\sim$~100 Myr the discrepancy between observations and models
for the very low-mass companion (AB~Dor~C)
would disappear \citep[eg.][]{clo05}. Taking this into account,
they propose an age range of 75 -- 150 Myr for all the MG.
To the initial sample of \citet{zuc04}, we have added 13 stars (marked with
{\it b} in Table~\ref{tab:MGs}) from our catalogue of {\it Late-type Stars
Possible Members of Young Stellar Kinematic Groups} \citep{m01}.
These stars have been included to both searches: a) for other members
of the group and b) to show the existence of two subgroups of different
ages more clearly.
They have been chosen because of their kinematics, assuming
a total dispersion of \mbox{$\pm 4$ km s$^{-1}$} in $U$ and $V$ respectively,
around the centre of the AB~Dor~MG defined by \citet{zuc04}.
We have imposed no restriction to the $W$ component for this first
selection. The whole sample contains a total of 50 stars.
We have compared the $EW$({Li~{\sc i}}) of every star in the sample with
that of known members of young open clusters (Fig.~\ref{fig:li_MGs}), as well
as their position in the $V-I$ diagram with the isochrones of
\citet{sie00} (Fig.~\ref{fig:VI}). The results reveal the existence
of two subgroups with stars showing different spectroscopic and photometric
features,
mixed in the velocity field (see Fig.~\ref{fig:uvw_MGs}).
The members of the first subgroup, that includes AB~Dor and
PW~And --- a very active young K2-dwarf \citep{lop03} --- show
$EW$({Li~{\sc i}}) similar to that of the high-rotators in the Pleiades
(upper continuous line in Fig.~\ref{fig:li_MGs}) which are
above the values found in the low-rotators of
IC~2602 (lower dashed line in Fig.~\ref{fig:li_MGs}).
Their position between the 30 and the 80 Myr isochrones
in the $V-I$ diagram, together with the first result,
is compatible with an age of 30 - 50~Myr. Moreover, the stars from the
sample of \citet{zuc04} belonging to this subgroup are
situated above the sequence of the Pleiades in the $M_{\rm K}$ vs.
$V-K_{\rm s}$ diagram in \citet{luh05}.
Here we have obtained a dispersion $\sigma \approx 2$ km~s$^{-1}$ in the
W component, quite similar to the one observed in other young stellar
associations such as Tucana or $\epsilon$~Cha
\citep[see][and references therein]{zuc04a}.
For determining the dispersion we have rejected the stars BD+07~1919A and
B (marked with * in Table~\ref{tab:MGs}) since their radial velocities ---
used for calculating the Galactic velocity components --- have not been
corrected for binarity since no orbital solution has been found in the
literature. Nevertheless, although the membership of this system is not
completely reliable taking the value of their $W$ component into account,
it has been included in the sample as possible member because of the
position of the A component in the CMD, which suggests an age of
$\sim$~30~Myr.
The stars in the second subgroup show features, in terms of
$EW$({Li~{\sc i}}) values and position in the CMD,
comparable with that of the members of the Pleiades cluster:
in Fig.~\ref{fig:li_MGs} they are situated slightly above the
lower envelope of the Pleiades, while in Fig.~\ref{fig:VI}
they situate on the (ZAMS) 80~Myrs isochrone.
Its members could be considered as part of
subgroup B4, one of the four subgroups found by \citet{asi99}
inside the Local Association in their study of the space motion
of OB~associations using Hypparcos astrometric data. The
authors find a mean age of $\sim$~150~Myr for this subgroup
using information from the photometry.
The higher dispersion found for the stars of this second subgroup in the
velocity space (Fig.~\ref{fig:uvw_MGs}) is in agreement with the age
estimated by us.
On the other hand,
the results about AB~Dor MG indicate that this quadruple system
has indeed $\sim$~50~Myr.
The value of $EW$(Li~{{\sc i}}) for AB~Dor~A is somewhat above the upper
envelope of the Pleiades but not so high as the one of IC~2602
(Fig.~\ref{fig:li_MGs}). On the other hand, its ($V-I$) colour situates
it between the 30 and 80 Myr isochrones (Fig.~\ref{fig:VI}). The same
result is clearly visible in Fig.~1 of \citet{luh05}, where AB~Dor
is situated above the lower sequence of the Pleiades in the
$M_{\rm K}$ vs. $V-K_{\rm s}$ diagram. With an age of 50~Myr,
the discrepancy between observations and models
for the AB Dor very low-mass companion (AB Dor C)
shown in \citet{clo05} continuous, although it can be solved
if the very low-mass companion were indeed an
unresolved binary system \citep{mar05}.
\section{Discussion and conclusions}
\label{sec:conc}
In Table~\ref{tab:MGs} we list the stars belonging to the nearest moving
groups: Hercules-Lyra Association and AB~Dor MG, and those being part of
the Local Association B4 subgroup.
For the Hercules-Lyra Association, a division between certain members and
candidates with doubtful classification or non members has been made.
In the three groups, new candidates from our catalogue of {\it Late-type Stars
Possible Members of Young Stellar Kinematic Groups} \citep{m01} have been
selected because of their kinematics (see $\S$~\ref{sec:herc}
and $\S$~\ref{sec:abdor}). A total of
75 stars including the known members and the new candidates selected by us
have been analysed. Kinematic, spectroscopic and photometric criteria have
been utilized to discriminate non members from the rest of candidates of
the Hercules-Lyra Association, and to distinguish between the members of
the AB~Dor MG and those of the B4 subgroup.
In the velocity space, Hercules-Lyra is clearly distinguishable from
the rest of the sample (see Figs.~\ref{fig:uv} and~\ref{fig:uvw_MGs}).
The dispersion in $U$, $V$, and $W$ is comparable with that of other
coeval MGs such as Castor and Ursa Major \citep[e.g.][]{m01}, and
compatible with its age (see $\S$~\ref{sec:herc}).
On the other hand, AB~Dor~MG and B4 subgroup are mixed up and age-dating
criteria are necessary to distinguish between the members of both groups.
Nevertheless, the dispersion in $W$ for AB~Dor MG is quite smaller than
the one of B4 subgroup.
Age-dating criteria are also necessary to discriminate non members of
Hercules-Lyra from the certain ones. The results of applying them are
summarized in Table~\ref{tab:MGs}: the Hercules-Lyra Association is formed
by 10 certain members
situated at a mean distance of $\sim$~20~pc and show values of
$EW$(Li~{{\sc i}}) (Fig.~\ref{fig:li_MGs}) and a position in the $V-I$
CMD (Fig.~\ref{fig:VI}) compatible with an age of 150~--~300~Myr;
the members of AB~Dor MG are situated at a mean distance of $\sim$~30~pc and
show lithium abundances typical of stars with 30~--~50~Myr
(Fig.~\ref{fig:li_MGs}), which is in
agreement with their position in the $M_{\rm V}$ vs. $V-I$ diagram
(Fig.~\ref{fig:VI});
finally, a set of stars with $EW$(Li~{{\sc i}}) and positions in the
CMD compatible with an age of 80~--~120~Myr are
mixed with Hercules-Lyra and AB~Dor MG, and have been classified as
other members of the Local Association B4 subgroup (see $\S$~3).
Note that the age estimated using the position of the members of
Hercules-Lyra in the CMD
is a lower limit since the 80~Myrs isochrone is overlapped with the
ZAMS for spectral types earlier than about K5. On the other hand,
the age estimated using the equivalent width of the Li~{\sc i} line
$\lambda$6707.8 \AA \ is more robust
since the 50\% of the stars classified as members have measurements of
the $EW$(Li~{{\sc i}}): the Li indicator is useful only for
spectral type later than G0, but only three of the 25 candidates of the
initial sample are F stars.
Stars in these three subgroups form an excellent list of young cool
stars for studying how planets are formed, since they cover a range of ages
between 30 and 200 Myr, characteristic of the period during which the
Solar System was formed, and they are close enough to be accessible to
adaptive optics.
In addition, they can be taken as targets for direct imaging detection
of sub-stellar companions ---~brown dwarfs and extra-solar giant planets~---
\citep{neu00,mar03,mas05,low05} and for cold dust, debris disks
\citep{gai04,met04,liu04,che05}.
\acknowledgments
This work was supported by the Universidad Complutense de Madrid and
the Spanish Ministerio de Educaci\'on y Ciencia (MEC), Programa
Nacional de Astronom\'{\i}a y Astrof\'{\i}sica under grants AYA2004-03749
and AYA2005-02750.
ICC acknowledges support from MEC under AP2001-0475.
We would like to thanks the referee for useful comments
which have contributed to improve the manuscript.
\clearpage
\clearpage
\clearpage
|
Title:
Did Swift measure GRB prompt emission radii? |
Abstract: The Swift X-Ray Telescope often observes a rapidly decaying X-ray emission
stretching to as long as $ t \sim 10^3$ seconds after a conventional prompt
phase. This component is most likely due to a prompt emission viewed at large
observer angles $\theta > 1/\Gamma$, where $\theta\sim 0.1$ is a typical
viewing angle of the jet and$\Gamma\geq 100$ is the Lorentz factor of the flow
during the prompt phase. This can be used to estimate the prompt emission
radii, $r_{em} \geq 2 t c/\theta^2 \sim 6 \times 10^{15}$ cm. These radii are
much larger than is assumed within a framework of a fireball model. Such large
emission radii can be reconciled with a fast variability, on time scales as
short as milliseconds, if the emission is beamed in the bulk outflow frame,
e.g. due to a random relativistic motion of ''fundamental emitters''. This may
also offer a possible explanation for X-ray flares observed during early
afterglows.
| https://export.arxiv.org/pdf/astro-ph/0601557 |
\date{\today}
\title{Did {\it Swift} measure GRB prompt emission radii?}
\author{M. LYUTIKOV
}
\affil{University of British Columbia, 6224 Agricultural Road,
Vancouver, BC, V6T 1Z1, Canada and
Department of Physics and Astronomy, University of Rochester,
Bausch and Lomb Hall,
P.O. Box 270171,
600 Wilson Boulevard,
Rochester, NY 14627-0171, USA }
\keywords{gamma-rays: burster}
\section{Introduction}
Recently launched
\Swift satellite \citep{Gehrels} together with a network of ground based observations
have been providing scientific community with crucial
information on Gamma Ray Bursts (GRBs). Besides the landmark detection of
afterglows from short GRBs \citep[\eg][]{Gehr05}, \Swift has
gathered crucial data on
developments of GRBs at early times.
This is especially important since early observations
provide clues to the properties
of the ejecta, like its composition, lateral distribution of energy etc.
At late times the energy is mostly transfered to the
forward shock, properties of which can hardly be used to probe the ejecta.
A number of surprising results related to early afterglows have emerged
\citep[\eg][]{Tagliaferri,Nousek,Chincarini,obrien}:
(i) early, $t \leq 10^3$ s, rapidly-decaying
X-ray component, (ii) X-ray flares occurring at
$t \sim 10^2-10^4$ s, (iii) shallower than expected initial decay (or hump) of the afterglow.
These features are common, but the light curves show a large variety.
In this letter we discuss the first two mentioned effects, \ie rapidly-decaying component
and X-ray flares, since both can be related to the prompt emission (as opposed to afterglow)
and can thus be used to probe the ejecta and the central engine.
\section{Prompt emission radii}
The initial fast-decaying part of afterglows can be a ''high altitude'' prompt emission,
coming from angles $\theta > 1/\Gamma$ \citep{Kumar00,Barthelmy05},
where $\theta$ is the angle between the line of sight and the direction from the
center of the explosion towards an emitting point and $\Gamma$ is the Lorentz factor of the outflow.
For a $\delta$-function in time prompt emission pulse, after an initial spike the observed flux
should decay as $t^{-(2+\alpha)}$, where $\alpha \approx 0.5 $ is prompt emission's spectral index,
\citep{Fenimore}, roughly consistent with observations.
One also expects that prompt and early afterglow emission
join smoothly, which seems to be generally observed \citep{obrien}.
[Exceptions, like GRB050219a
\citep{Tagliaferri},
may be due to interfering X-ray flares.]
If we accept the interpretation of the fast decaying part as ''high altitude'' prompt emission,
one can then
determine radii of the prompt emission and compare them with model predictions.
The currently most popular fireball model \cite[\eg][]{Piran04}
relates radii of emission $r_{em}$ to the variability time scale $\delta t$
of the central source
$r_{em} \sim
2 \Gamma_0^2 c \delta t$, where $ \Gamma_0 \sim 100-300$ is the initial Lorentz factor.
Within the framework of the fireball model this is also the variability time scale of the
prompt emission. Observationally,
prompt emission shows variability on time scales as short as milliseconds,
while most power is at a fraction of a second \citep{BeloStern}. Adopting
$\delta t\sim 0.1
$ s, the prompt emission radius is
$r_{em} \sim 6 \times 10^{13}$ cm $(\Gamma_0/100)^2$.
If the emission is generated at $r_{em}$ and is coming to observer
from large angles, $\theta > 1/\Gamma$, its delay with respect to the start
of the prompt pulse
is $t \sim (r_{em}/c) \theta^2/2$. If one can estimate $\theta$, then
this can be used to measure $r_{em}$. This can be done
from late ''jet breaks'',
giving typically $\theta \sim 0.1$ \cite[\eg][]{Frail}.
Then, for the X-ray tail of the prompt emission extending to $t \sim 1000$ seconds,
the implied emission radius is
$r_{em} > 6 \times 10^{15}$ cm.
This is much larger
than is assumed in the fireball model.
To make it consistent with the fireball model and variability on short times scales, the
Lorentz factor of the flow should be huge, $\Gamma_0 \geq 1000$, but this would imply
that emission is strongly de-boosted, $\Gamma_0 \theta \sim 100 $.
Increasing $\theta$ cannot save the day either since
the required viewing angle would be $\theta \sim 1$, implying a jet moving always from an
observer.
Along the similar lines of reasoning,
\cite{Lazzati05} estimated prompt emission radii for a particular case of GRB 050315
for which a possible jet break is identified
\citep{Vaughan} . Steep decay in that case is relatively short and
lasts for 100 s, giving
$r_{em} > 2.5 \times 10^{14}$ cm.
Note,
that any observed duration of the steep decay phase provides only {\it a lower limit}
on the prompt emission radius since
the end of the steep decay may be related to
emergent afterglow emission and not
to the fact that the edge of the jet becomes visible, see Fig. \ref{GRBafter}.
On the other hand,
late jet break time provides an estimate of the total opening angle of the jet.
In any case, GRBs with longer lasting steep decay phase, up to $10^3$ s, provide the
most severe constraints on the models.
Thus, the
interpretation of the fast-decaying initial X-ray light curve
as prompt emission seen at large angles can hardly be
inconsistent with
the fireball model.
We should then either look for alternative possibilities to produce the fast-decaying
part of the X-ray light curve \citep[\eg][]{mr01}, or consider models that advocate production
of prompt emission at larger radii, see \S \ref{Conc}.
\section{Fast variability from large radii}
If prompt emission is produced at distances $\sim 10^{15}-10^{16}$ cm,
how can a fast variability,
on times scales as short as milliseconds, be
achieved? One possibility, is that emission is beamed in the outflow frame, for example
due to a
relativistic motion of (using pulsar physics parlance) ''fundamental emitters'' \citep{lb03}.
To prove this point, we
consider an spherical outflow expanding with a bulk Lorentz factor $\Gamma$
with $N$ randomly distributed emitters moving with respect to the shell rest frame with
a typical Lorentz factor $\gamma_{T}$.
Highly boosted emitters, moving towards an observer,
have a Lorentz factor $
\gamma \sim 2 \gamma_T \Gamma
$ in the observer frame.
If emission is generated at distances $r_{em}$, the observed variability time scale
can be as short as $ \sim (r_{em}/c) /2 \gamma^2 \approx (r_{em}/c) / 8 ( \gamma_T \Gamma)^2$, so
that
modest values of $\gamma_T \sim 5-10 \ll \Gamma \sim 100-300 $
would suffice to produce a short time scale variability from
large distances $r_{em}\sim 10^{15} - 10^{16} $ cm.
The model should satisfy a number of constraints.
First, the number of sub-jets directed towards an observer from viewing angles $\theta< 1/\Gamma$
should be larger than unity (in order to produce at least one true prompt emission spike),
but should not be too large, otherwise prompt emission will be a smooth
envelope of overlapping spikes. If a typical jet opening angle
is $\theta_j$, then
the number of sub-jets seen ''head-on'' from angles $< 1/\Gamma$ is
\be
n_{prompt} \sim { \pi N \over (\Gamma \gamma_{T} \theta_j)^2}.
\ee
This should be larger than $1$.
The second constraint that the model should satisfy relates to the
efficiency of energy conversion.
Suppose that the thickness of an outflowing shell in its rest frame is
$ L_{shell} \sim t_s c \Gamma$, where
$t_s $ is a source activity time ($t_s \sim 30-100$ s for long bursts and $t_s \sim 1 $ s
for short bursts). Suppose then that fundamental emitters operate for a
time $ t_{pulse} = \eta_t L_{shell}$ in the flow frame, where $\eta_t $ is a dimensionless
parameter. During this time the source can tap into energy
contained within volume $ (c t_{pulse})^3$. The ratio
of this volume times the number of emitters to the total volume
of the shell is a measure of efficiency of energy conversion
into radiation:
\be
\eta = { N (c t_{pulse})^3 \over r_{em}^2 \theta_j ^2 t_s c \Gamma}
\ee
Since tapping of energy in the volume $(c t_{pulse})^3$ is a definite upper limit on
conversion efficiency, in the calculations we allow $\eta$ defined above to be slightly
larger than unity.
To produce light curves we calculate the intensity of emission
from sub-jets that are randomly located within the shell and moving in random direction
with random Lorentz factors $1<\gamma_T < \gamma_{T,max} =5 $.
Each emitter is isotropic in its rest frame and is active
for a random time $0<t'_{em} < \eta_T t_s c \Gamma
= t_{pulse,max}$ with $\eta_T =0.5$.
The observed intensity of emission from each sub-jet $\propto \delta^{3+\alpha}$
\citep{BlandfordLind},
where $\delta = 1/\gamma (1-\beta \cos \theta_{sj}) $ is a
total Doppler factor including bulk and
random motion, $\theta_{sj}$ is an angle between the line of sight and direction
of the sub-jet motion.
As the burst progresses, larger angles and more of
sub-jets producing prompt emission become visible. Most of them will be seen from
angles $> 1/\gamma_T$ in the bulk frame, producing a combined smooth curve overlaid
with spikes.
The average Doppler factor decreases with time $\delta \approx
t_s \Gamma / t $ and the average flux decays as $t^{-(2+\alpha)}\approx t^{-2.5}$ for
$\alpha=0.5$.
In Fig. \ref{GRBafter} we plot an example of a prompt light curve
in this model.
\subsection{Lateral dependence of prompt emission}
Variations of the decay rate from the $t^{-(2+\alpha)}$ law may be used to
probe
angular dependence $L(\theta_{axis})$ of the intensity of the
prompt emission, where $\theta_{axis}$ is an angle between the axis of the explosion
and an emitting point. More shallow decays can be due to, \eg,
a structured jet, with $L\sim \theta_{axis}^{-2}$ observed outside of some core:
late time emission then is coming from
the more energetic core part. The effective emission intensity
increases approximately as
$\theta ^2 \propto t$, and will result in an observed decay $t^{-(1+\alpha)}$.
Similarly, if the prompt emission is seen within a core,
late emission comes from less energetic wings, giving in
case of a structured jet a flux $ \propto t^{-(3+\alpha)}$.
Qualitatively, the relativistic internal motion of emitters makes it ''easier'' to see the
high altitude emission.
To show this numerically we parameterize the {\it number density of emitters} as
$n(\theta) \propto 1/(\theta^2 + \theta_0^2)$,
where $\theta_0$ is an angular core radius.
[There are, naturally other possible parameterizations,
e.g. of intensity of each emitter]. The results are presented in Fig. \ref{GRBafterStruct}.
We can also expect deviations from a simple power-law decay due to not exactly spherical
form of the emitting surface. Such distortions are expected due to a development of the
Kelvin-Helmholtz instability during an accelerating phase of the outflow.
They won't be erased during the coasting stage due to causal disconnection of the flow
separated by angles $> 1/\Gamma$.
Additional complications
may come from the way the data analysis is performed, \eg through a choice of
initial time trigger (\cite{Zhang05}, see also \cite{Lazzati05}).
\section{Origin of X-ray flares}
Early X-ray light curves show complex behavior with flares and
frequent changes in a temporal slope \citep[\eg][]{obrien}. Flares
show very short rise and fall times, much shorter than
observation time after the on-set of a GRB, while the
underlying afterglow has the same behavior before and
after the flare \citep{Burrows}
(though there are exceptions).
Both of these observations
argue against a
physical process in the forward shock.
In addition, there is a hardening of the spectrum during X-ray flares \citep{Burrows}.
In the present model we interpret X-ray flares as been due to sub-jets located at
large viewing angles, $\theta > 1/\Gamma$, but
directed towards an observer.
Randomly located, narrow spikes are clearly seen in the model light curves,
Figs. \ref{GRBafter}-\ref{GRBafterStruct}.
In addition, as the flares are less de-boosted than the average high altitude outflow,
they will have a harder spectrum, as observed.
\section{Discussion}
\label{Conc}
In this letter we first point out
that the interpretation of the initial fast-decaying part of the X-ray GRB
light curves as a prompt emission seen at large angles, and a generic estimate of
jets' opening angle allows a measurement of the
radius of prompt emission, which turns out to be relatively large,
$> 10^{15}$ cm.
On basic grounds, $\gamma$-ray emission should be generated before the deceleration radius
$r_{dec} \sim \left( { E_{iso} \over 4 \pi \rho c^2 \Gamma_{dec}^2} \right)^{1/3}\sim 10^{16}-
10^{17}$ cm,
when most energy of the outflow is given to the surrounding medium
(here $ E_{iso}$ is isotropic equivalent energy, $ \rho$ is density of external medium,
$\Gamma_{dec}$ is Lorentz factor at $r_{dec}$).
\footnote{Note, that $r_{dec}$ defined above is {\it independent} of ejecta content, contrary to
the claim in
\cite{zk05}, see \cite{LZK}.}
The inferred emission radius is within this limit.
The estimate of the emission radius is very simple, and, in some sense, generic.
It can hardly be consistent with the fireball model, unless extreme assumptions are made about
the parameters (\eg very large Lorentz factor).
On the other hand,
there are alternative models (\eg the
electromagnetic model \citep{l05}, see also \cite{thompson})
that place prompt emission radii at large distances, just before the
deceleration radius
$r_{dec} $.
Secondly, we show how models placing emission at large radii
may be able to reproduce a short time scale variability of the prompt emission
and explain later X-ray flares. This can be achieved if the prompt emission is beamed
in the rest frame of the outflow, which may be due to an internal relativistic motion
of ''fundamental emitters''.
What can produce a relativistic motion in the bulk frame?
It can be due, for example, to a relativistic Burgers-type turbulence
(a collection of randomly directed shock waves). It is not clear how such turbulence may be generated.
Alternatively,
relativistic internal sub-jets
can result from reconnection occurring in highly magnetized plasma
with $\sigma \gg 1$,
where
$\sigma$ is a plasma magnetization parameter \citep{KC84}.
In this case
the matter outflowing from a reconnection layer reaches
relativistic speeds with $\gamma_{out} \sim \sigma$ \cite[]{lu03}.
Internal synchrotron emission by such jets,
or
Compton scattering of ambient photons, will be strongly beamed in the frame of the outflow.
Note, that {\it this model does not require
late engine activity } to produce flares.
One of the main observational complications is that at observer times larger than the
conventional prompt phase, the X-ray light curve is a sum of the tail of the prompt emission,
coming presumably from internal dissipation in the ejecta, and the forward shock emission.
It is not obvious how to separate the two components.
For example, GRBs which do not show a fast initial decay may be
dominated by the forward shock emission from early on
\citep{obrien}.
This uncertainty also affects estimates of the emission radius
since the end of the steep decay may be related to
the emergent afterglow emission and not to the jet opening angle (or observer's
angle, in case of
a structured jet), see Fig. \ref{GRBafter}.
Another complication is that at these intermediate times,
$10^3 \leq t \leq 10^4$ s, even the
forward shock emission itself often
does not conform to the standard afterglow
models, showing flatter than expected
profiles \citep[\eg][]{Nousek}.
A consequence of the model is that {\it some} short GRBs may be just a single
spike directed towards an observer of a long GRBs. In our model
the shorter spikes are highly beamed, less frequent and produce harder emission.
This can apply only to {\it some} short GRBs since as a class
they are well established to have different origin than long GRBs
(from non-observation of a supernova signature and coming from a distinctly different
host galaxy population).
|
Title:
Development of Gaseous Tracking Devices for the Search of WIMPs |
Abstract: The Time Projection Chamber (TPC) has been recognized as a potentially
powerful detector for the search of WIMPs by measuring the directions of
nuclear recoils, in which the most convincing signature of WIMPs, caused by the
Earth's motion around the Galaxy, appears.
We report on the first results of a performance study of the neutron exposure
of our prototype micro-TPC with Ar-C$_2$H$_6$ (90:10) and CF$_4$ gas of 150
Torr.
| https://export.arxiv.org/pdf/astro-ph/0601568 |
\begin{frontmatter}
\title{Development of Gaseous Tracking Devices for the Search of WIMPs}
\author[KYOTO]{H. Sekiya\corauthref{cor}},
\corauth[cor]{Corresponding author. tel:+81 75 753 3868; fax:+81 75 753 3799.}
\ead{sekiya@cr.scphys.kyoto-u.ac.jp}
\author[KYOTO]{K. Hattori},
\author[KYOTO]{S. Kabuki},
\author[KYOTO]{H. Kubo},
\author[KYOTO]{K. Miuchi},
\author[WASEDA]{T. Nagayoshi},
\author[KYOTO]{H. Nishimura},
\author[KYOTO]{Y. Okada},
\author[KOBE]{R. Orito},
\author[KYOTO]{A. Takada},
\author[ICRR]{A. Takeda},
\author[KYOTO]{T. Tanimori},
\author[KYOTO]{K. Ueno}
\address[KYOTO]{Department of Physics, Graduate School of Science, Kyoto University, Kitashirakawa, Sakyo, Kyoto, 606-8502, Japan}
\address[WASEDA]{Advanced Research Institute for Science and
Engineering, Waseda University, \\
17 Kikui-cho, Shinjuku, Tokyo, 162-0044, Japan}
\address[KOBE]{Department of Physics, Graduate School of Science and Technology, Kobe University, 1-1 Rokkoudai, Nada, Kobe, 657-8501, Japan}
\address[ICRR]{Kamioka Observatory, ICRR, University of Tokyo,\\
456 Higasi-mozumi, Hida-shi, Gifu, 506-1205, Japan}
\begin{keyword}
dark matter\sep WIMP \sep TPC\sep direction sensitive detector
\PACS 95.35.+d\sep 29.40.Gx \sep 29.40.Cs
\end{keyword}
\end{frontmatter}
\section{Introduction}
\label{intro}
It is considered by many that the galactic halo is composed of weakly
interacting massive particles (WIMPs) as dark matter\cite{jung}.
These particles could be detected directly by measuring the nuclear
recoils produced by their elastic scattering off nuclei in detectors.
The most convincing signature of WIMPs appears in the directions of
nuclear recoils. It is provided by the Earth's large velocity through
the isothermal galactic halo ($\sim$230 km/s). Hence, detectors sensitive to
the direction of the recoil nucleus would have a great potential to
identify WIMPs\cite{dir}.
Time Projection Chambers (TPCs) with fine spacial resolutions are among
such devices, and we are developing a micro TPC, which can detect
three-dimensional fine tracks of charged particles\cite{miuchi}.
Since the energy deposits of WIMPs to nuclei are only a few tens of keV
and the range of nuclei is limited, the micro-TPC should be operated
at low pressures.
We also focused on the detection of WIMPs via spin-dependent(SD) interactions
and are interested in operating the micro-TPC with CF$_4$\cite{NA},
because $^{19}$F has a special sensitivity to
SD interactions for its unique spin structure\cite{collar}.
In the present work, in order to examine the response of the
micro-TPC to nuclear recoils at low pressures as a first step,
we irradiated a 150 Torr Ar-C$_2$H$_6$ (90:10 mixture)
gas (one of the standard gases for TPCs) and CF$_4$ with neutrons
from $^{252}$Cf.
The track lengths and deposited energies
of Ar, C, and F recoils were investigated.
\section{The micro-TPC}
The prototype micro-TPC used in this measurements
is shown in Fig.\ref{fig:DC}.
The field cage consists of a drift cathode plane
and nine 0.2 $\mu$m copper wires of 1cm pitch with connections
of 10 M$\Omega$ resistor, which forms a uniform electric field
in the detection volume of $10\times10\times10$ cm$^3$.
The $\mu$-PIC\cite{upic} for 2-dimensional readout is $10\times10$cm$^2$
with 256 anode strips and 256 cathode strips each with a 400 $\mu$m pitch.
We also used a GEM having a 10 cm$\times$10 cm$^2$ sensitive area as a
sub-amplification device between the field cage and $\mu$-PIC, as
illustrated in Fig. \ref{fig:DC}, which enables stable operation
and avoids discharges with low HV operation of
both the $\mu$-PIC and GEM.
The details of this GEM are described in Refs.\cite{gem,hattori}.
The output charges of $256+256$ channels
are pre-amplified (0.7 V/pC) and shaped (with a gain of 7) and
discriminated via ASD chips (4 channels/chip, SONY CXA3653Q)\cite{asd}.
The pre-amplified signals are summed
and digitized by 100 MHz 8bit flash ADCs
in order to determine the deposited energy and the
track direction as the waveforms hold the Bragg curve shapes.
The reference threshold voltage ($0-100$mV) is commonly supplied to
all the ASD chips and all discriminated digital signals are sent to
the position encoding module based on FPGAs with an internal
clock of 100 MHz, so that the anode and cathode coincident position (x,y)
and the timing (z) are recorded in the memory module and the
tracks of charged particles are reconstructed in software.
The tracking performances for electrons, protons, and MIPs
are reported elsewhere\cite{miuchi,hattori}.
\section{Measurements and Results}
As illustrated in Fig. \ref{fig:Setup},
the micro-TPC was set in a 6 mm-thick aluminum vessel of
60 cm diameter $\times$ 20cm height.
In a typical run,
the vessel was evacuated to $\sim8\times10^{-3}$ Torr,
the SAES GETTER$^{\mbox{\scriptsize{\textcircled{\tiny R}}}}$ pump
in the vessel was activated,
and then the vessel was filled with Ar-C$_2$H$_6$ (90:10) or CF$_4$ gas
to a pressure of 150 Torr and sealed for the duration of the measurement.
For measuring the gas gain and the energy calibration,
the gas was irradiated with
$^{109}$Cd 22 keV and $^{133}$Ba 31.0 keV X-rays
through a 1mm thick aluminum window close to the sensitive volume.
We irradiated the micro-TPC with neutrons from a 1 MBq $^{252}$Cf source
on the top of the vessel.
Since one fission decay of $^{252}$Cf emits 3.8 neutrons and 9.7
$\gamma$-rays on average\cite{neut}, the $\gamma$-rays or neutrons
detected by a $10\times10\times2$ cm$^3$ plastic scintillator
were used as the event trigger.
In the $\gamma$/n-triggered events, gamma events would dominate
under normal gas gain ($\sim$10000) operation. Since
the $dE/dx$ values of the neutron events are much larger than
those of gamma events, we operated the $\mu$-PIC and GEM
with a rather low gas gain (below 1000) in order to observe the nuclear recoils.
In such different gas gain measurements,
we fixed the anode voltage of the $\mu$-PIC and changed the voltage
between the GEM electrodes.
Below a gas gain of about 2000, our system was not able to measure the
$^{109}$Cd 22 keV x-ray correctly due to a mismatch of the dynamic
range of the ASD chips and the flash ADC;
therefore, the deposited energy in low-gain operations was extrapolated
from the calibrations with the high gas gain operations.
We evaluated the track length
as a function of the measured electron equivalent energy in the
following way.
\subsection{Ar-C$_2$H$_6$ 150Torr run}
The drift cathode plane was supplied $-1$ kV, which gave a drift field of
60 V/cm and an electron drift speed of 4.0 cm/$\mu$s.
The anode voltage of the $\mu$-PIC was fixed at 350 V.
For nuclear recoil measurements,
the threshold of the discriminator of the ASD chip was set to 80 mV and
the measured track length of events
when the GEM voltage was set to 200 V (gas gain of 3000) and 135 V
(gas gain of 900) is shown in Fig. \ref{fig:ArTE}.
The MC (Geant4\cite{geant}) simulated track length without
consideration of the diffusion, the energy resolution, and
the $dE/dx$ threshold is also indicated for a comparison.
The geometry used for the simulation was in accordance with
Fig. \ref{fig:DC} and Fig. \ref{fig:Setup} and, the neutron energy
spectrum of the spontaneous fissions of $^{252}$Cf was assumed to be
\begin{equation}
\frac{dN}{dE}=\sqrt{E}\exp\left(-\frac{E}{T}\right),
\end{equation}
where $T=1.3$ MeV\cite{fission}.
Under operation with a gas gain of 3000,
electron recoils and proton (of C$_2$H$_6$) recoils were
clearly observed according to their $dE/dx$.
On the other hand, in the operation of the gas gain of 900,
the C and Ar recoils and some proton recoils were observed due to
the high $dE/dx$ threshold.
\subsection{CF$_4$ 150Torr run}
The drift cathode plane was supplied $-2$ kV, which gives a drift field of
120 V/cm and the electron drift speed of 12.0 cm/$\mu$s.
The anode voltage of the $\mu$-PIC was fixed at 600 V.
The measured track length of events
when the GEM voltage was set to 215 V (gas gain of 4500) and
95 V (gas gain of 800) are shown in Fig.\ref{fig:CF4TE}.
The threshold voltage of the discriminator of the ASD chip was
as high as 100mV; therefore, only C and F recoils were clearly
observed under operation with a gas gain of 800.
\section{Discussion and Prospects}
We successfully showed the nuclear recoils
in 150 Torr of Ar-C$_2$H$_6$ (90:10) and CF$_4$ gases
according to their $dE/dx$ by changing the detector threshold.
The energy loss of protons
became lower as the energy increased as opposed to the other nuclei\cite{srim}.
Consequently,
the proton band in Fig. \ref{fig:ArTE}(b) is truncated
at the threshold set in the measurements,
which corresponds to about 5 keV/400$\mu$m.
In terms of $dE/dx$, the tracks in the micro TPC were much easier
to detect for C, F or Ar recoils.
Ultimately, our concern is
the recoil direction of such nuclei below 100 keV
to allow us to observe the signals of WIMPs.
In order to obtain longer tracks and clear Bragg curves,
higher gas gain operations at lower pressures
with low-energy neutron beams are needed.
The measurement of the incident neutron energy
with Time-Of-Flight may also be useful to examine the quenching factor
of nuclear ionization in the micro-TPC.
|
Title:
Multi-Dimensional Simulations of the Accretion-Induced Collapse of White Dwarfs to Neutron Stars |
Abstract: We present 2.5D radiation-hydrodynamics simulations of the accretion-induced
collapse (AIC) of white dwarfs, starting from 2D rotational equilibrium
configurations of a 1.46-Msun and a 1.92-Msun model. Electron capture leads to
the collapse to nuclear densities of these cores within a few tens of
milliseconds. The shock generated at bounce moves slowly, but steadily,
outwards. Within 50-100ms, the stalled shock breaks out of the white dwarf
along the poles. The blast is followed by a neutrino-driven wind that develops
within the white dwarf, in a cone of ~40deg opening angle about the poles, with
a mass loss rate of 5-8 x 10^{-3} Msun/yr. The ejecta have an entropy on the
order of 20-50 k_B/baryon, and an electron fraction distribution that is
bimodal. By the end of the simulations, at >600ms after bounce, the explosion
energy has reached 3-4 x 10^{49}erg and the total ejecta mass has reached a few
times 0.001Msun. We estimate the asymptotic explosion energies to be lower than
10^{50}erg, significantly lower than those inferred for standard core collapse.
The AIC of white dwarfs thus represents one instance where a neutrino mechanism
leads undoubtedly to a successful, albeit weak, explosion.
We document in detail the numerous effects of the fast rotation of the
progenitors: The neutron stars are aspherical; the ``nu_mu'' and anti-nu_e
neutrino luminosities are reduced compared to the nu_e neutrino luminosity; the
deleptonized region has a butterfly shape; the neutrino flux and electron
fraction depend strongly upon latitude (a la von Zeipel); and a quasi-Keplerian
0.1-0.5-Msun accretion disk is formed.
| https://export.arxiv.org/pdf/astro-ph/0601603 |
\title{Multi-Dimensional Simulations of the Accretion-Induced
Collapse of White Dwarfs to Neutron Stars}
\author{L. Dessart\altaffilmark{1},
A. Burrows\altaffilmark{1},
C.D. Ott\altaffilmark{2},
E. Livne\altaffilmark{3},
S.-Y. Yoon\altaffilmark{4},
N. Langer\altaffilmark{5}
}
\altaffiltext{1}{Department of Astronomy and Steward Observatory,
The University of Arizona, Tucson, AZ \ 85721;
luc@as.arizona.edu,burrows@as.arizona.edu}
\altaffiltext{2}{Max-Planck-Institut f\"{u}r Gravitationsphysik,
Albert-Einstein-Institut, Golm/Potsdam, Germany; cott@aei.mpg.de}
\altaffiltext{3}{Racah Institute of Physics, The Hebrew University,
Jerusalem, Israel; eli@frodo.fiz.huji.ac.il}
\altaffiltext{4}{Astronomical Institute ``Anton Pannekoek'', University of Amsterdam,
Kruislaan 403, 1098 SJ, Amsterdam, The Netherlands; scyoon@science.uva.nl}
\altaffiltext{5}{Astronomical Institute, Utrecht University,
Princetonplein 5,3584 CC, Utrecht, The Netherlands; n.langer@astro.uu.nl}
\keywords{hydrodynamics -- neutrinos -- rotation -- stars: neutron --
stars: supernovae: general -- stars: white dwarfs}
\section{Introduction}
Stars can follow a few special evolutionary routes to form an unstable
Chandrasekhar mass core. A main-sequence star of more than
$\sim$8\,\mo evolves to form either a degenerate O/Ne/Mg core (Barkat et al. 1974;
Nomoto 1984,1987; Miyaji \& Nomoto 1987) or a degenerate Fe core (Woosley \& Weaver 1995),
which, due to photodisintegration of heavy nuclei and/or electron capture,
collapses to form a protoneutron star (PNS).
If an explosion ensues, the event is associated with a Type II supernovae (SN).
Less massive stars end their lives as white dwarfs.
White dwarfs located in a binary system may accrete from a companion and
achieve the Chandrasekhar mass, triggering the thermonuclear runaway of the object and leading
to Type Ia SN, leaving no remnant behind.
However, a third class of objects is expected. Theoretically, massive white dwarfs with O/Ne/Mg cores,
due to their high central density ($\sgreat$10$^{10}$\,g\,cm$^{-3}$), experience rapid electron
capture that leads to the collapse of the core. This is accretion-induced collapse (AIC),
an alternative path to stellar disruption through explosive burning, currently associated with
Type Ia SN (Nomoto \& Kondo 1991).
It is presently unclear what fraction of all white dwarfs will lead to AICs,
but of those white dwarfs that evolve to form a Chandrasekhar-mass O/Ne/Mg core,
all will necessarily undergo core collapse.
One formation channel is the coalescence of two white dwarfs (Mochkovitch \& Livio 1989),
with either C/O or O/Ne/Mg cores, although few such binary systems have yet been
observed with a cumulative mass above the Chandrasekhar mass.
There is still uncertainty as to whether such binary systems would not undergo
thermonuclear runaway rather than collapse.
Since the coalescence of two white dwarfs requires a shrinking of the
orbit through gravitational radiation, these systems will take many gigayears to coalesce.
An alternative formation mechanism is via single-degenerate systems, through a combination
of high original white dwarf mass and mass and angular-momentum accretion by mass transfer from a
(non-degenerate) H/He star (Nomoto \& Kondo 1991).
Binary star population synthesis codes predict the occurence of the AIC of white dwarfs
with a galactic rate of 8$\times$10$^{-7}$ yr$^{-1}$ to 8$\times$10$^{-5}$ yr$^{-1}$, depending,
amongst other things, on the treatment of the common-envelope phase and mass transfer
(Yungelson \& Livio 1998).
The set of parameters leading to a Type Ia rate of 10$^{-3}$\,yr$^{-1}$ corresponds to
an AIC rate of 5$\times$10$^{-5}$ yr$^{-1}$. The observed Type Ia rate of
$\sim$3$\times$10$^{-3}$\,yr$^{-1}$ (Madau et al. 1998; Blanc et al. 2004; Manucci et al. 2005)
would imply a galactic AIC rate of 1.5$\times$10$^{-4}$\,yr$^{-1}$.
These rates are likely functions of galaxy and metallicity (Yungelson \& Livio 2000;
Belczynski et al. 2005; Greggio 2005; Scannapieco \& Bildsten 2005).
Based on r-process nucleosynthetic yields obtained from previous simulations of the AIC of white dwarfs,
Fryer et al. (1999) inferred rates ranging from $\sim$10$^{-5}$ to $\sim$10$^{-8}$\,yr$^{-1}$.
Overall, AICs are not expected to occur more than once per 20--50 standard Type Ia events; because
they are intrinsically rarer, they remain to be identified and observed in Nature.
Whatever their origin, their fundamental nature is to accrete both mass
and angular momentum from a companion object. Rotation is, therefore, a key physical component.
Prior to core collapse, differential rotation acts as a stabilizing agent
for shell burning by widening its spatial extent through enhanced mixing and reducing the envelope
density through centrifugal support (Yoon \& Langer 2004).
Mass accretion also leads to an increase of the central density, which may rise up
to a few $10^{10}$ g\,cm$^{-3}$, establishing suitable conditions for efficient electron capture
on Mg/Ne nuclei.
By the time of core collapse and depending on the evolutionary path followed, such white dwarfs may cover a
range of masses from $\sgreat$1.35\,\mo up to $\sim$2\,\mo (2.7\,\mo) in the case of
a non-degenerate (degenerate) companion, potentially well in excess of the standard
Chandrasekhar limit, and possessing an initial rotational energy up to $\sim$10\% of their
gravitational binding energies.
At such values, the centrifugal potential leads to a deformation of the white dwarf from
spherical symmetry, equipotentials and isopressure surfaces adopting a peanut-like shape
in cross section for the largest rotation rates. Such structures are obtained in the
2D differentially rotating equilibrium white dwarf models constructed by Yoon \& Langer
(2005, YL05; see also Liu \& Lindblom 2001).
In the past, the collapse of O/Ne/Mg cores originating from stars in the 8-10\,\mo
range has been studied in 1D by Baron et al. (1987ab), Mayle \& Wilson (1988), and
Woosley \& Baron (1992), who showed that the shock generated at core bounce stalls
rather than leading to a prompt explosion.
However, Hillebrandt et al. (1984) and Mayle \& Wilson (1988) obtained delayed explosions,
the former after 20-30\,ms, and the latter after $\sim$200\,ms, supposedly driven by
neutrino energy deposition behind the stalled shock.
Woosley \& Baron (1992) found the emergence of a sustained neutrino-driven wind, with a mass loss
rate of 0.005\,\mo\,s$^{-1}$ and with an ejecta electron fraction of the order of 0.45, showing promise,
modulo uncertainties, for a contribution to the enrichment of the ISM in r-process elements.
Using 1D/2D SPH simulations, Fryer et al. (1999) reproduced the simulations of Woosley \& Baron (1992),
confirmed that the shock stalls due to the copious neutrino losses associated with core bounce,
did not find prompt explosion, and, depending on the equation of state (EOS) employed, observed a
delayed explosion.
They focused mostly on the early phase, prior to the neutrino-driven wind, and found an ejected mass
of low-$Y_{\rm e}$ material of $\sim$0.05\,\mo, depending on adopted model assumptions.
Their 2D simulations with solid-body rotation showed similar properties to their 1D equivalents,
the authors attributing the small differences to the different grid resolution.
This may partly stem from the essentially spherical explosion triggered just $\la$100\,ms
after core bounce, ejecting the fast-rotating material in the outer mantle.
Consequently, at the end of their 2D simulations, they obtain very slow rotation rates for the
PNS, i.e., of $\sim$1\,s.
Recently, Kitaura et al. (2005) re-inspected the collapse of the progenitor model used by
Hillebrandt et al. (1984). They confirmed again the by-now well-accepted idea that no prompt
explosion occurs, but instead obtain a successful, though
sub-energetic, delayed explosion in spherical symmetry, powered by neutrino heating and a
neutrino-driven wind that sets in $\sim$200\,ms after bounce.
A primary motivation for this work is to improve upon these former investigations that
assumed one-dimensionality, sphericity, and/or zero-rotation, and start instead from the more
physically-consistent 2D models of YL05, thereby fully accounting for the effects of rotation
on the collapse, bounce, and post-bounce evolution of the white dwarf core and envelope, as
well as for the strong asphericity of the progenitor.
Our study uses VULCAN/2D (Livne et al. 2004; Walder et al. 2005) to perform 2D
Multi-Group Flux Limited Diffusion (MGFLD) radiation hydrodynamics simulations.
As we will demonstrate, rotation plays a major role
in the post-bounce evolution, making 1D investigations of such objects of limited utility.
By carrying out the simulations from $\sim$30\,ms before bounce to $\sgreat$600\,ms after
bounce, we capture a wide range of physical processes, including the establishment
of a strong, fast, and aspherical neutrino-driven wind.
We model the centrifugally-supported equatorial regions and the large angular momentum budget
leading to the formation of a sizable accretion disk.
Moreover, an Eulerian investigation is better-suited than a Lagrangean approach to explore
the neutrino-driven wind that develops after $\sgreat$200\,ms.
Finally, despite the relative scarcity of AIC in Nature, these simulations represent interesting examples
for the formation of disks around neutron stars.
The main findings of this work are the following:
We find that the AIC of white dwarfs forms $\sim$1.4-\mo neutron stars, expelling a modest
mass of a few 10$^{-3}$\,\mo mostly through a neutrino-driven wind that develops $\sgreat$200\,ms
after bounce, and that they lead to very modest explosion energies of 5-10$\times$10$^{49}$erg.
Accounting for the rotation and the asphericity of the progenitor white dwarfs reveals a wealth of
phenomena. The shock wave generated at core bounce
emerges through the poles rather than the equator, and it is in this excavated polar region
that the neutrino-driven wind develops. The strong asphericity of the newly-formed protoneutron stars
leads to a latitudinally-dependent neutrino flux, while the effects of rotation modify the relative
flux magnitude of different neutrino flavors.
Besides mass accretion by both the neutron star and mass ejection by the initial blast and the subsequent
wind, we find a sizable component that survives and resides in a quasi-Keplerian disk, which obstructs the wind flow
at low latitudes. This disk will be accreted by the neutron star only on longer, viscous timescales.
In the next two sections, we present the two selected progenitor models in more detail; we also discuss the
radiation-hydrodynamics code VULCAN/2D and the various assumptions made. In \S\ref{sect_results},
we present
the simulation results, focusing on the general temporal evolution from the start until $\sgreat$600\,ms,
a time by which the neutrino-driven wind has reached a steady state.
We then analyse in more detail the various components of these simulations.
In \S\ref{sect_pns}, we discuss the properties of the nascent neutron stars,
with special attention paid to the geometry of the neutrinospheres.
In \S\ref{sect_nu}, we discuss the neutrino signatures, both in terms of luminosity and
energy distribution.
In \S\ref{sect_disk}, we focus on the residual material lying at low latitudes, forming a
quasi-Keplerian disk.
In \S\ref{sect_ener}, we turn to the energetics of the explosion and describe in detail
the main component of the simulations at late times, i.e., the neutrino-driven wind.
In \S\ref{sect_ye}, we analyse the electron fraction of the ejected material and address
the relevance of the AIC of white dwarfs for neutron-rich element pollution
of the interstellar medium.
In \S\ref{sect_gw}, we present the gravitational-wave signal predicted for the AIC of our
white dwarf models.
In \S\ref{sect_conc}, we wrap up with a discussion of the main results of this investigation
and present our conclusions.
\section{Initial models}
\label{sect_progenitor}
In this section, we present the properties of the AIC progenitors selected in our study and
summarize the presentation in \S2 of YL05.
The general assumption for the construction of 2D progenitor models for the AIC of white
dwarf (YL05) is that the resulting structure of the object is essentially independent
of its evolutionary history, the
only factors that matter being the given (final) mass, angular momentum, and central density,
$\rho_{\rm c}$.
Additionally, the angular velocity distribution $\Omega(r,z)$, where $r$ is the cylindrical radius and
$z$ is the distance to the equator, is determined self-consistently, given a number of properties
identified in 1D models: 1) the role played by the dynamical shear instability, 2) the compression
(and spin up) of the surface layers due to mass accretion which puts its peak angular velocity
interior to the surface radius, 3) the adopted surface rotational velocity value (its fraction of
the local Keplerian value), and 4) the geometry of the angular velocity profile, which we assume
to be constant on cylinders, i.e., $\Omega = \Omega(r)$.
Together with the pressure/density dependence $P = P(\rho)$, such rotating stars are called barotropic.
Note that the criterion used in YL05 for the shear rate for the onset of the dynamical shear
instability is determined using the EOS of Blinnikov et al. (1996).
The 2D rotating models then correspond to equilibrium configurations iteratively found from trial
density and angular velocity distributions, using the Self-Consistent-Field method (Ostriker \& Mark 1968;
Hachisu 1986), under the constraint that the density $\rho$ is solely a function of
the effective potential $\Psi(r,z)$, given as the sum of the gravitational potential, i.e.,
$$\Phi(r,z) = -G \int \frac{\rho(r',z')}{|{\bf R}-{\bf R'}|} d^3 R' \,, $$
and the centrifugal potential, i.e., $$\Theta(r) = -\int \Omega^2(r') r' dr' \,,$$
where $R' = \sqrt{r'^2 + z'^2}$ (Tassoul 2000, YL05).
In this paper, we select two progenitors with masses of 1.46 \mo and 1.92 \mo and present their global
characteristics in Table~\ref{tab_aic}.
Both models have an initial central density $\rho_{\rm c}$ equal to 5$\times$10$^{10}$\,g\,cm$^{-3}$.
The 1.46-\mo model serves as a reference for an object with a moderate initial rotational energy $T$
relative to gravitational energy $|W|$, i.e., $T/|W| = 0.0076$, and, indeed, shows only a modest
initial departure from spherical symmetry, with a polar
($R_{\rm p}$) to equatorial ($R_{\rm eq}$) radius ratio $R_{\rm p}/R_{\rm eq} = 0.7$.
At the other end of the white dwarf mass spectrum, the 1.92-\mo model
has considerable rotational energy, with initial $T/|W| = 0.0833$, and the morphology of the
star departs strongly from spherical symmetry, with $R_{\rm p}/R_{\rm eq} = 0.28$.
For later reference, we also provide in Table~\ref{tab_aic} the ratio $T/|W|$ at the end of the simulations.
In practice, the infall of the ambient material causes numerical difficulties soon after the
start of the simulation. These difficulties were resolved by trimming the outer and low-density layers
of both white dwarf progenitors. While the polar radius is hardly affected, the equatorial radius
is reduced to 980\,km (down from 1130\,km) for the 1.46-\mo model and
1860\,km (down from 2350\,km) for the 1.92-\mo model.
The progenitor mass is, however, reduced by less than one part in 10$^6$. Hence, we do not expect
any perceptible effect on the results.
\begin{deluxetable}{lccccccc}
\tablewidth{8cm}
\tabletypesize{\scriptsize}
\tablecaption{Properties of selected AIC progenitors
\label{tab_aic}}
\tablehead{
\colhead{M}&
\colhead{$R_{\rm p}$}&
\colhead{$R_{\rm eq}$}&
\colhead{$J$}&
\colhead{$T$}&
\colhead{$|W|$}&
\colhead{$T/|W|$}&
\colhead{$T/|W|$} \\
\colhead{\mo}&
\colhead{km}&
\colhead{km}&
\colhead{erg$\cdot$s}&
\colhead{erg}&
\colhead{erg}&
\colhead{initial}&
\colhead{final} \\
\colhead{}&
\colhead{}&
\colhead{}&
\colhead{(10$^{50}$)}&
\colhead{(10$^{50}$)}&
\colhead{(10$^{50}$)}&
\colhead{}&
\colhead{}
}
\startdata
1.46 & 800 & 1130 &0.160 & 0.7 & 91.97 & 0.0076 & 0.059 \\
1.92 & 660 & 2350 &1.092 & 10.57 & 126.9 & 0.0833 & 0.262 \\
\enddata
\end{deluxetable}
\section{VULCAN/2D Simulation Code}
\label{sect_VULCAN}
The simulations discussed in this paper were performed with the Newtonian hydrodynamic code
VULCAN/2D (Livne 1993), supplemented with an algorithm for neutrino transport as described
in Livne et al. (2004) and Walder et al. (2005).
The version of the code used here is the same as that discussed in Dessart et al. (2005)
and Burrows et al. (2005), and uses the 2D Multi-Group Flux-Limited Diffusion (MGFLD) method to
handle neutrino transport (see Appendix~A of Dessart et al. 2005). The MGFLD variant of VULCAN/2D
is much faster than
the more accurate, but considerably more costly, multi-angle $S_n$ variant.
Doppler velocity-dependent terms are not included in the transport, although advection terms are.
Frequency redistribution due to the subdominant process of neutrino-electron scattering
is neglected. Our calculations include 16 energy groups logarithmically distributed in energy
from 2.5 to 220\,MeV, take into account the electron and anti-electron neutrinos, and
bundle the four additional neutrino and anti-neutrino flavors into a ``$\nu_{\mu}$'' component.
VULCAN/2D uses a hybrid grid, switching from Cartesian in the inner 20\,km to spherical-polar
further out.
In the simulation of the 1.46-\mo model, mapped onto a 180$^{\circ}$ wedge,
nearly perfect top-bottom symmetry about the equatorial plane was maintained
during the pre- and post-bounce evolution.
Thus, for the 1.92-\mo, we limited the computational domain to just one hemisphere.
The thus reduced number of zones was used to increase the resolution.
To summarize, the 1.46-\mo model uses a grid with a maximum resolution in the Cartesian inner
region of 0.56\,km, and a minimum resolution of 150\,km at a maximum radius of 5000\,km, with 121
regularly spaced angular zones to cover 180$^{\circ}$.
The 1.92-\mo model uses a grid with a maximum resolution in the Cartesian inner region of
0.48\,km, and a minimum resolution of 100\,km at a maximum radius of 4000\,km, with 71
regularly-spaced angular zones covering 90$^{\circ}$.
A tricky part of the set up was to choose the properties of the ``ambient'' medium surrounding the
AIC model. This need arises because our inputs are Lagrangean in spirit, while VULCAN/2D
employs a Eulerian grid.
Ideally, one would like to have material that merely occupies the space that will soon, after bounce,
be replaced by the ejected material following the explosion.
A key requirement is, thus, that this material have
a very low pressure, to influence as little as possible the properties of the blast, but
to allow for a smooth transition from circumstellar to ejected material in a given
region of the Eulerian grid.
To achieve this, we extended our SHEN EOS (Shen et al. 1998) down to very low
densities (10\,g\,cm$^{-3}$) and
low temperatures (10$^8$ K), conditions in which the medium is actually
radiation-dominated and, thus, has a pressure that depends mostly on temperature.
A successful choice was to adopt a density of 1000\,g\,cm$^{-3}$ and a temperature of
4$\times$10$^{8}$\,K for the ambient medium surrounding both white dwarf models.
A major deficiency of the white dwarf progenitor models used here is their unknown initial
thermal structure, which YL05 did not provide.
Given the additional difficulty in handling low temperatures and high densities,
we resorted to using a parameterized function of the local density, i.e.,
$$ T(r,z) = T_{\rm c} (\rho_{\rm c}/\rho(r,z))^{0.35}\,,$$ with
$T_{\rm c} = 10^{10}$\,K for
the 1.46-\mo model, and $T_{\rm c} = 1.3 \times 10^{10}$\,K for the 1.92-\mo model.
Note that, similarly, Woosley \& Baron (1992) were forced to set a central temperature of
1.2$\times$10$^{10}$\,K at the start of their simulation.
Figure~\ref{fig_init} recapitulates the basic properties of the white dwarf
progenitors (left column: 1.46-\mo model; right column: 1.92-\mo model) as
mapped onto our Eulerian grid. In the top panel, we show a color map of the
density on which we superpose
line contours of the effective potential $\Psi$ (defined above), as computed by VULCAN/2D.
Our computation of the gravitational potential is based on a multipole expansion in spherical
harmonics up to $l=33$.
We reproduce the fundamental property of these fast-rotating white dwarf progenitors, in that
the isopressure surfaces and the equipotentials coincide. In the bottom panel, we plot the
initial angular velocity field. To avoid the distortion of streamlines of the infalling ambient
material, the angular velocity is set to zero outside the WD progenitor.
Note how, from zero, the angular velocity rises to a maximum value near the outer
equatorial radius in the 1.46-\mo model (left column), while it is much higher in the center
of the white dwarf in the 1.92-\mo model (right column). As we will show below,
rotation has a much bigger
impact on the collapse phase for the latter configuration. We also overplot line
contours of the temperature for each model, following the prescription outlined in
the above paragraph.
Finally, we adopt an initial electron fraction of 0.5; the high initial
central density permits fast electron capture which soon decreases the $Y_{\rm e}$
in the core, leading to bounce on a timescale ten times shorter than typically
experienced in the collapse of the core of massive star progenitors
(Woosley \& Weaver 1995; Heger et al. 2000; Woosley et al. 2002).
\section{Simulation results}
\label{sect_results}
In this section, we discuss the general properties of the pre- and post-collapse
phases for both models at the same time. We follow
the post-bounce evolution of the 1.46-\mo (1.92-\mo) model for 550\,ms (780\,ms), for a total of
520\,000 (820\,000) timesteps.
In Figs.~\ref{fig_seq46}--\ref{fig_seq92}, we provide entropy (saturated at
20\,k$_{\rm B}$/baryon, where k$_{\rm B}$ is Boltzmann's constant) color maps showing
the key events in the evolution of both white dwarf models, with starting conditions discussed in
the previous section and displayed in Fig.~\ref{fig_init}. We complement these figures
with Fig.~\ref{fig_seqye} for the electron fraction evolution at three reference
times (left column: 1.46-\mo model; right column: 1.92-\mo model).
Note also that we overplot most figures with black arrows representing velocity vectors,
whose maximum length is set to 10\% of the width (or the height) of the image.
The corresponding maximum velocity is then given for each image in the figure captions -
we also mention if we saturate the vector lengths. Having a large central density
of 5$\times$10$^{10}$\,g\,cm$^{-3}$, both models achieve nuclear densities after the same time of $\sim$37\,ms.
Differences in the bounce properties are attributable to the initial inner angular velocity
distributions (Fig.~\ref{fig_init}). Compared to the 1.46-\mo model, the faster rotator
has a lower maximum density at bounce (slightly shifted from the grid center to $r\sim$1\,km),
i.e., 2.2 instead of 3.1$\times$10$^{14}$\,g\,cm$^{-3}$. It manifests an oblate, rather than
a spherical, inner region (the inner tens of kilometers) of low entropy ($\sim$1$k_{\rm B}$/baryon;
visible in red).
The deleptonized material in this inner region is, however, aspherically distributed
in both models, although more so in the 1.92-\mo model, with the lower $Y_{\rm e}$ material lying
in a disk structure along the equatorial direction. The flatter density gradient
in this direction and the longer dwell time near the neutrinosphere
conspire to produce this enhanced deleptonization. At core bounce, the low-density outer
parts of the white dwarf progenitor have not yet started to infall and the progenitor still retains
its original shape.
The collapse of the inner regions, however, creates a rarefaction wave that triggers the infall
of the outer material, with a magnitude that is more pronounced along the poles due to the compact
structure of the white dwarf in these directions.
Although we do not observe a prompt explosion (i.e. occuring on a dynamical timescale of
just a few milliseconds), the shock progresses slowly outwards without conspicuously
stalling. This is slightly different from the core-collapse simulations recently published,
whereby the shock {\it systematically} stalls (Burrows et al. 2005; Buras et al. 2005ab).
In the equatorial direction, facilitated by the centrifugal support of the infalling
material, the shock progresses outwards steadily, and is faster in the faster rotating model,
reaching a few hundred kilometers after $\sim$100\,ms.
Given the initial constant rotation rate on cylinders (see \S\ref{sect_progenitor}),
the {\it radial} inflow of mass
brings angular momentum to the equatorial ($z \sim 0$) region, which becomes more strongly supported,
facilitating further the progress of the shock at low latitudes.
This exacerbates the non-spherical development of the shock structure in both models.
In its wake, we see a few large-scale whirls, resulting from the generation, at mid-latitudes
and due to shock passage, of large vortical motions.
\epsscale{1}
In the polar direction, centrifugal support is absent, but the stalling of the shock
is prevented by the quickly decreasing accretion rate established by the steeper density gradient,
reduced polar radius, and smaller mass budget.
Hence, although the shock slowly migrates along the equatorial regions, it soon traverses the
surface layer of the white dwarf along the poles and escapes outwards into the ambient medium.
Due to the strong asphericity of the progenitor, this occurs only 70\,ms after core bounce in the
1.92-\mo model, $\sim$30\,ms earlier than for the 1.46-\mo model.
The ambient medium is then swept up by this blast, whose opening angle is constrained
by that of the uncollapsed disk of the progenitor.
The outflow expansion rate is larger 30-40 degrees away from the poles than right along
the pole, coinciding in Fig.~\ref{fig_seqye} with the lower $Y_e$ material. The effect
is large for the 1.92-\mo model, giving a butterfly shape in cross section to the faster expanding portions of the
outflow. Off-axis material has more rotational kinetic energy available to convert to 2D planar ($r,z$)
kinetic energy as it streams outward, reducing its rotational velocity while preserving
angular momentum, thereby resulting in an enhanced acceleration compared to that of the material situated
along the poles and lacking rotational energy.
As the shock expands, it wraps around
the disk, typically with a speed near that of the local sound speed. Nearer the pole, the outflow sweeps along
the pole-facing side of the pole-excavated white dwarf, entraining surface material and
effectively loading the outflow with more mass, causing unsteady fallback onto the neutron star.
For the 1.46-\mo progenitor, the shock completely wraps around the low-latitude
regions of the white dwarf, and finally emerges from the outer equatorial regions, as witnessed
by an outward-moving entropy jump. By 250\,ms after bounce, the
shock reaches a few thousand kilometers, assumes a near spherical shape, and the entire
white dwarf material outside the newly-formed neutron star flows nearly radially outward.
In the high-rotation (1.92-\mo) progenitor, the white dwarf possesses a lot more mass at
near zero-latitude and this confines more drastically the emerging shock along the poles.
As the shock migrates outwards, it opens up; it does wrap around the progenitor, but much
later than when the shock escapes in the polar directions.
Despite the reduced ram pressure associated with centrifugal support, the shock stalls
a few hundred milliseconds after bounce along all near-equatorial directions.
Within 100-200\,ms, the newly-formed neutron star has a mass of $\sim$1.4\,\mo, similar in
both models despite the 0.5\,\mo difference in progenitor mass.
Note that the large neutron star asphericity and the sizably lower density at the neutrinosphere for
near-zero latitudes makes this mass definition ambiguous at such early times, especially
in the 1.92-\mo model.
Indeed, the neutron star is not clearly distinguishable from the
surrounding equatorial material, so imposing either a density cut of 10$^{10-11}$\,g\,cm$^{-3}$
or a radius cut in defining the newly-born neutron star appears arbitrary when determining the residual mass.
In the 1.46-\mo model, about 0.06\,\mo remains outside of the neutron star, mostly in the
equatorial disk region; in the 1.92-\mo model, 0.6\,\mo is now lying in this disk-like structure.
The rest of the initial mass is outflowing material, which, if selected according
to an outward radial velocity discriminant of 10000\,\kms (comparable to the escape velocity at
3000\,km), reaches 4$\times$10$^{-3}$\,\mo for the 1.46-\mo model and 3$\times$10$^{-3}$\,\mo
for the 1.92-\mo model. These various components are documented in more detail in
\S\S\ref{sect_disk}--\ref{sect_ye}.
The late-time evolution of both models is characterized by a strong neutrino-driven wind that
sets in about 300\,ms after bounce, replenishing the grid with denser material (on average
10$^4$\,g\,cm$^{-3}$) and large velocities (with a maximum of 30000\,\kms along the poles).
The properties of the neutrino-driven wind are very angle-dependent, the density changing
by 30\% between the pole and the 40$^{\circ}$ latitude, while the radial outflow velocity
varies by a factor of 3 in the 1.92-\mo model.
The latitudinal dependence of the mass flux per unit solid angle is therefore dominated by
a variation in asymptotic velocity. We will discuss this result in more detail in \S\ref{sect_ener}.
By the time we stop the simulations, at 550\,ms and 780\,ms for the 1.46-\mo and 1.92-\mo models,
all the ambient medium originally placed around the white dwarf progenitor has been swept away by the
neutrino-driven wind, which occupies all the space outside the neutron star and the disk.
The electron fraction of the material ejected in the original blast is close to 0.45--0.5,
while subsequently, in the neutrino-driven wind, the values are lower, with a pronounced
decrease towards lower latitudes (note, however, the high $Y_{\rm e}$ right along the pole; see
\S\ref{sect_ye}).
In Fig.~\ref{fig_rho_radslice}, we recapitulate for both models the evolution described
above by showing equatorial and polar slices of the density as a function of time.
Striking features are the distinct polar and equatorial surface radii, the fast infall of
the inner regions to nuclear densities, the slow plowing of the shock along the
equatorial directions, superseded in radial extent and velocity by the shock in the polar direction
as the surface mass shells collapse in, and finally the emergence of a sequence
of density kinks associated with the birth of the fast neutrino-driven wind that sweeps
away the previously shocked material that did not leave the grid.
Similarly, in Fig.~\ref{fig_vr_radslice}, we show slices of the radial velocity, $V_R$,
along the polar (top row)
and equatorial (bottom row) directions for the 1.46-\mo (left column) and 1.92-\mo (right
column) models. Notice the much larger infall velocities, similar along the poles
and the equator for the 1.46-\mo model, but with a strong latitudinal dependence
in the 1.92-\mo model. In that model, the speed contrast between the polar and equatorial
directions is $\sim$30000\,\kms.
Overall, the evolution is more rapid along the poles than on the equator, with larger
asymptotic velocities (30000\,\kms compared with 10000\,\kms), and with the establishment
of a quasi-stationary outflow at late times along the pole.
These radial slices offer a means of better interpreting the fluid velocities, depicted
with vectors, in most color maps shown in this paper. We also show isodensity contours
in most color maps to provide some feeling for the density distribution.
Having described the general properties of the two simulations of the AIC of a
1.46-\mo and 1.92-\mo white dwarf, we now address more specific issues, covering the
properties of the nascent neutron star (\S\ref{sect_pns}), the neutrino signatures (\S\ref{sect_nu}),
the properties of the residual disk and the angular momentum history (\S\ref{sect_disk}),
the neutrino-driven wind and the global energetics (\S\ref{sect_ener}), the electron fraction
of the ejected material (\S\ref{sect_ye}), and, finally, the gravitational wave signatures (\S\ref{sect_gw}).
\section{Neutron Star properties}
\label{sect_pns}
The white dwarf progenitors discussed in this paper, due to their evolution to high central
densities, high rotational kinetic energies, and high mass, are distinctive in that their
cores always collapse to form neutron stars rather than being disrupted by the explosive
burning of carbon and oxygen.
We find that neutron stars formed from the AIC of the progenitor white dwarfs used in this work
are very aspherical (see, for example, Walder et al. 2005; Liu \& Lindblom 2001; Janka \& M\"onchmeyer 1989ab),
although there are significant differences in evolution after bounce between
the two models. In Fig.~\ref{fig_nusphere_46}, we show color maps of the density field 59\,ms
after bounce (top row), as well as for the last time computed ($t=570$\,ms after bounce; bottom row)
in the 1.46-\mo model.
To render more striking the level of asphericity of the neutron star ``surface,''
we overplot the neutrinospheres $R_{\nu}(\varepsilon_{\nu},r,z)$, adopting the definition
$$ \tau(R_{\nu}(\varepsilon_{\nu},r,z)) =
\int_{R_{\nu}(\varepsilon_{\nu},r,z)}^\infty \kappa_{\nu}(\rho,T,Y_{\rm e}) \rho(r',z') dR' = 2/3\,,$$
where $\kappa_{\nu}(\rho,T,Y_{\rm e})$ is the combined material absorption and scattering opacity
to neutrinos, and the integration is carried out along radial rays, with $R' = \sqrt{r'^2 + z'^2}$,
using 30 equally spaced latitudinal directions per quadrant.
Strictly, this definition is most appropriate for the photosphere/neutrinosphere in a plane-parallel
atmosphere, but it gives a sense of the asphericity of the collapsed core.
Note that the above expression contains a dependence both on the neutrino flavor and the
neutrino energy $\varepsilon_{\nu}$.
In the left column of Fig.~\ref{fig_nusphere_46}, line contours correspond to such neutrinosphere radii as
a function of energy group, bracketing the peak of the neutrino energy distribution at the
neutrinosphere, i.e., between 2.5 and 46\,MeV. Material opacity to neutrinos increases with
the square of the energy, so that higher-energy neutrinos have larger neutrinospheres.
Here, for the 1.46-\mo model, these radii vary from $\sim$30 to $\sim$120\,km along the equatorial
direction, with little departure from sphericity (30\% lower values are obtained in the polar direction).
In the right panel, line contours correspond to the neutrinosphere for the three different flavors at
a neutrino energy ($\varepsilon_{\nu}$) of 12.5\,MeV, associated with the peak of the energy distribution
at infinity.
Note that this is approximate, since the neutrino energy distribution hardens
with time and manifests a latitudinal dependence (see \S\ref{sect_nu}).
As in standard 1D and 2D core-collapse computations, we find that the
electron neutrinos decouple from matter
at larger radii than the $\bar{\nu}_e$ and ``$\nu_{\mu}$'' neutrinos. Here, the former decouple
at 90\,km (60\,km) along the equator (pole), the latter two further in but at a similar radius of
70\,km (50\,km).
The neutrinospheres show a similar shape for all three
flavors (and all energy groups), reflecting the corresponding asphericity in the density field.
In the bottom-row panels, we reproduce the above for the last time in the 1.46-\mo
simulation. The departure from sphericity is now considerable, with both an oblateness
and a strong pinching of the neutrinospheres along the polar directions.
Along the equatorial direction, the radial spacing between neutrinospheres of consecutive
and higher energy groups has increased, and the lower (higher) energy groups decouple further
in (out) than in the previous snapshot, with neutrinospheres between 20 and 150\,km
(from 2.5 to 50\,MeV), 80 and 50\,km (for the $\nu_e$ neutrino, and
$\bar{\nu}_e$/``$\nu_{\mu}$'' neutrinos, respectively).
Along the polar direction, neutrinospheres of all neutrino energy groups (and all neutrino
flavors at 12.5\,MeV) shown reside in a narrow range of radii between 20 and 30\,km (22 to 25\,km).
Here again, the neutrinospheres depicted follow very closely the contours of density, which is the primary
factor controlling the neutrino optical depth.
In Fig.~\ref{fig_nusphere_92}, we duplicate Fig.~\ref{fig_nusphere_46} (note the different spatial
scale) for the 1.92-\mo model, showing the same
quantities both early after bounce (59\,ms) and
significantly later at 775.5\,ms after bounce. There are numerous differences with the 1.46-\mo model.
First, the neutrinospheres are aspherical even right after bounce (top row), with equatorial (polar) radii
larger (smaller) by a factor of 2-3 compared with those in the 1.46-\mo model.
All flavors reveal similar neutrinosphere locations.
In the 1.46-\mo model, the later ratio of the equatorial and polar radii is 2.5:1, irrespective of energy
group and flavor. This becomes 15:1 in the 1.92-\mo model, and is thus a considerable departure from
sphericity; the faster rotating model has a neutrinosphere radius of just 14\,km along the pole, but
215\,km along the equator.
This is the most conspicuous difference with the essentially spherical neutron stars seen in non-rotating
simulations of the more standard core collapse of massive stars (Keil et al. 1996;
Swesty \& Myra 2005; Buras et al. 2005a; Dessart et al. 2005), with neutrinosphere radii of the
order of 20-30\,km at comparable times after core bounce.
The very aspherical neutron stars formed through the AIC of a white dwarf make the determination
of the neutron star mass somewhat ambiguous. Rather than taking the enclosed mass within a given spherical
radius, we compute the total mass from all regions above a given mass density.
For a density cut of 10$^{10}$\,g\,cm$^{-3}$, we obtain neutron star masses of 1.42\,\mo
for the 1.46-\mo model, and 1.5\,\mo for the 1.92-\mo model.
However, if we adopt a density cut of 10$^{11}$\,g\,cm$^{-3}$, the neutron
star masses are, respectively, 1.39\,\mo and 1.30\,\mo.
The higher-mass progenitor model has now a smaller neutron star mass, reflecting
the strong asphericity of the density field.
While we might associate the neutrinosphere with the neutron star surface and with a standard
mass density of 10$^{11}$\,g\,cm$^{-3}$, such an association
in the present fast rotating neutron star is inappropriate, since the neutrinospheres extend
well into regions where the density is $\sles$10$^{10}$\,g\,cm$^{-3}$.
These neutron star mass values are reached at 100\,ms after core bounce and remain essentially constant.
The neutrino-driven wind that appears after a few 100\,ms decreases the neutron star mass at
a rate of just a few 10$^{-3}$\,\mo\,s$^{-1}$ (see \S\ref{sect_ener}).
Note also that the enhanced centrifugal support in the faster-rotating, higher-mass model
leads to bounce at a 30\% lower maximum density compared with the 1.46-\mo model,
and both a reduced and a delayed mass accretion rate along the equatorial direction compared with
what would prevail in the absence of rotation. At the end of each simulation, the total
neutron star angular momentum is
1.35$\times$10$^{49}$ erg$\cdot$s (1.13$\times$10$^{49}$ erg$\cdot$s) for the 1.46-\mo model,
using the density cut at 10$^{10}$\,g\,cm$^{-3}$ (10$^{11}$\,g\,cm$^{-3}$) , and
4.57$\times$10$^{49}$\,erg$\cdot$s (2.79$\times$10$^{49}$\,erg$\cdot$s) for the 1.92-\mo model.
However, the accretion rate at a given Eulerian radius is higher and longer-lived
at smaller latitudes, because of the larger amount of mass available and the flatter
density profile in those regions.
Overall, the presence of a massive accretion disk in the fast rotating model complicates
the definition of the neutron star mass at such early times. Evolution over minutes/hours/days
will likely lead to significant accretion onto the neutron star, resulting in a much higher
final mass.
The final rotational to gravitational energy ratio $T/|W|$ is 0.059 for the 1.46-\mo
model and 0.262 for the 1.92-\mo model. These values are large and for the latter model, large enough
to cause the growth of secular and perhaps even dynamical instabilities (Tassoul 2000).
Using realistic post-bounce configurations for a rotating massive star progenitor,
Ott et al. (2005a) find a dynamically unstable spiral mode for $T/|W|$ as low as $\sim$0.08.
Thus, it is likely that the PNS structures found here, especially for the 1.92-\mo model, would
develop some non-axisymmetric instability that would cause, among other things, outward angular momentum transport.
These results are significantly different from those of Fryer et al. (1999), who obtained
a successful explosion $\sles$100\,ms after bounce, a PNS mass of $\sim$1.2\,\mo, and a $\sim$1\,s
period at $\sim$200\,ms after core bounce. Such different conclusions stem
from their adoption of slow, solid-body progenitor rotation, with most of
the angular momentum stored in the outer mantle and blown away by the explosion,
rather than being accreted by the PNS.
\section{Neutrino signatures}
\label{sect_nu}
The first observational signature of an AIC explosion would be the copious emission of
neutrinos immediately after core bounce.
As discussed above, the properties of the bounce of the core and the nascent neutron star
are close enough to those obtained in simulations of the core collapse of massive progenitors
that one expects a neutrino signal with a somewhat similar evolution and character (see, for example,
the predictions for the 11-\mo model of Woosley \& Weaver 1995
in Dessart et al. 2005).
In Fig.~\ref{fig_nuflux}, we show the neutrino luminosities (on a log scale) for the
1.46-\mo (left panel) and 1.92-\mo (right panel) models, with distinct curves for the
different neutrino flavors (solid line: $\nu_e$; dashed line: ${\bar{\nu}_e}$;
dash-dotted line: ``$\nu_{\mu}$''), as well as different colors for the
equatorial (black) and polar (red) directions.
These are luminosities in the sense that the flux in each direction is scaled by 4$\pi R^2$
where $R$ is a spherical radius, of 250\,km for the 1.46-\mo model and 400\,km for the
1.92-\mo model (chosen to be well above the neutrinosphere for the energy at the peak
of the neutrino distribution).
They correspond to the total luminosity that would have obtained had
the selected directional flux been the same in all directions.
The temporal evolution of the various fluxes for both models is comparable.
The total neutrino luminosity reaches a maximum of 5.2$\times$10$^{53}$\,erg\,s$^{-1}$,
mostly due to the $\nu_e$ neutrino contribution, and decreases to $\sim$4$\times$10$^{52}$\,erg\,s$^{-1}$
at 500\,ms after bounce in the 1.46-\mo model, with a further 30\% decrease for the 1.92-\mo model.
At later times, the main reason for this difference is the much lower ``$\nu_{\mu}$''
neutrino luminosity in the 1.92-\mo model.
This reduction has been seen and discussed by Fryer \& Heger (2000) in the context
of the collapse of rotating cores of massive progenitors. The smaller core densities
(weaker bounce) achieved in models with fast rotation lead to smaller temperatures
and, consequently, smaller neutrino emission, with a larger effect for the
$\nu_{\mu}$ and $\nu_{\tau}$ neutrinos (grouped under the name ``$\nu_{\mu}$'' here).
So, while the globally lower neutron star densities in the fast-rotating model induce a reduction
in neutrino luminosity compared to the 1.46-\mo model, the same effect introduces a
latitudinal variation of neutrino fluxes in the faster rotating model, with fluxes,
irrespective of neutrino flavor, larger by a factor of about two along the pole than
along the equator (note that at a radius of 250\,km, the difference is higher and on
the order of three).
This variation, not discussed by Fryer \& Heger (2000), results from the further
variation of the neutrinosphere temperatures with latitude within a given rotating model.
In both models, but more so in the 1.92-\mo model, the temperature gradient and the
temperatures are reduced at the neutrinosphere (for a given energy group)
along the equator compared to the poles,
irrespective of energy group and flavor, as is clearly visible in Fig.~\ref{fig_temp}.
For example, we see that the temperature on the 10$^{10}$\,g\,cm$^{-3}$ contour
is 4\,MeV in the polar direction and 0.5\,MeV in the equatorial direction.
Within the neutrinosphere region, we find fluid velocities that are oriented
preferentially in the $z$-direction, along cylinders, illustrating the weak or absent
convection that results from the stabilizing specific angular
momentum profile (Fryer \& Heger 2000; Heger et al. 2000; Ott et al. 2005b;
see also \S\ref{sect_disk}).
Along a given angular slice, the temperature has a maximum at mid-latitudes, caused
by enhanced ($\nu_e$) neutrino energy deposition in this direction.
These regions offer a tradeoff, since the flux is still higher at such latitudes than
along the equator, while the density is relatively higher than along the poles
(see \S\ref{sect_ener}).
The latitudinal variations seen in the collapsed models of AIC
progenitors are extreme, and, indeed, for the slower rotation rates typically obtained for
massive-star core-collapse progenitors (Heger et al. 2000), a modest anisotropy is found
instead (Walder et al. 2005).
We document further the neutrino signatures of the AIC of white dwarfs by showing in
the top row of Fig.~\ref{fig_nufluxspec}, for the 1.46-\mo (left) and 1.92-\mo (right)
models, the time-integrated neutrino emission at infinity
for each flavor, as a function of neutrino energy. In the bottom three panels of each column,
we show the individual neutrino distributions at three representative times of the simulations
(at bounce, halfway through the simulation, and at the last simulated time).
The overall flux level and hardness of the energy distribution are higher along
the polar direction, the variation at a given time towards higher latitude mimicking the time
evolution seen for non-rotating core collapse simulations of massive star progenitors.
We compute the average neutrino energies, here defined as
$$ \sqrt{\langle\varepsilon_{\nu}^2\rangle} \equiv
\left[
\frac{\int d\varepsilon_{\nu} \varepsilon_{\nu}^2 F_{\nu}(\varepsilon_{\nu},R)}
{\int d\varepsilon_{\nu} F_{\nu}(\varepsilon_{\nu},R)}
\right]^{\frac{1}{2}}\,.$$
For the 1.46-\mo model ($R=250$\,km), we obtain similar values to within
2-3\% along the pole and along the equator with $\langle\varepsilon_{\nu_e}\rangle=10$\,MeV,
$\langle\varepsilon_{{\bar\nu}_e}\rangle=15$\,MeV,
and $\langle\varepsilon_{\nu_{\mu}}\rangle=24$\,MeV.
For the 1.92-\mo model ($R=600$\,km), we obtain systematically lower values than for the 1.46-\mo
model, and for lower latitudes. Along the equator (pole), we obtain
$\langle\varepsilon_{\nu_e}\rangle=9$\,MeV (10\,MeV),
$\langle\varepsilon_{{\bar\nu}_e}\rangle=14$\,MeV (16\,MeV), and
$\langle\varepsilon_{\nu_{\mu}}\rangle=16$\,MeV (21\,MeV).
Note that this definition of the neutrino ``average'' energy is similar to that found in
Thompson et al. (2003), who used the mean intensity $J_{\nu}$ in place of the flux
$F_{\nu}$. Since we are
close to free-streaming regimes at the adopted radii for the peak of the neutrino energy
distribution, the two are equivalent.
In the 1.46-\mo model, and at late times, we see that the neutrino
display is more dramatic (total flux is twice as high) and
is characterised by a harder spectrum than in the 1.92-\mo
model. This is due to the more compact and, thus, hotter neutrinospheres of the neutron star
formed by the lower-mass white dwarf progenitor.
\section{Rotation and the remnant disk}
\label{sect_disk}
Let us now turn our discussion to the angular momentum and angular velocity budget and
profiles in our simulations. As discussed in \S\S\ref{sect_results}--\ref{sect_pns},
we find that the early neutron stars have comparable masses in the two simulations,
the rest residing not so much in the outflow than in a substantial amount of
``circum-neutron star'' disk material, rotating fast, but having little outflow or
inflow velocity (see Fig.~\ref{fig_vr_radslice}).
In Fig.~\ref{fig_w_j}, we plot a temporal sequence of the equatorial radial profile of the
angular velocity (top row) and specific angular momentum (bottom row), for the 1.46-\mo
model (left column) and 1.92-\mo model (right column).
In the 1.46-\mo model, the central angular velocity is $\sim$0.1\,rad\,s$^{-1}$
(or a period $P=63$\,s) at the start of the simulation (initial conditions in the YL05 progenitor),
and $\sim$1000\,rad\,s$^{-1}$ ($P=6.3$\,ms) at the end the simulation, a spin-up factor of 10000.
In the 1.92-\mo model, we start with a much higher angular rotation rate of $\sim$20\,rad\,s$^{-1}$
($P=0.3$\,s), but the final values are comparable with those of the 1.46-\mo model,
being $\sim$2800\,rad\,s$^{-1}$ ($P=2.2$\,ms).
Thus, both simulations lead to the formation of a neutron star with a period of a few milliseconds, although
we expect the neutron star formed in the 1.92-\mo model to further accrete mass and angular momentum,
which may spin-up the residue to even shorter periods.
The general angular velocity and specific angular momentum profiles for both models are quite similar.
Despite wiggles observed in the 1.46-\mo model in the inner 10 kilometers (which we associate with
slight numerical artifacts along the axis - this problem is not present in the 1.92-\mo model, whose
grid covers only 90$^{\circ}$), the neutron star is close to solid-body rotation out to 30\,km,
showing a steady and smooth decline with radius beyond.
In all four panels, we overplot as a broken and black line the corresponding local Keplerian angular velocity,
$\Omega_{\rm Keplerian}(r) = \sqrt{GM/r^3}$, where $G$ is the gravitational constant
and $M$ is the mass interior to the radius (cylindrical, or
spherical). (For these plots, we employ the corresponding neutron star mass.)
The angular velocity or specific angular momentum profiles beyond 30\,km graze the corresponding
line for the Keplerian value, being always lower by a few tens of percent.
In the 1.46-\mo model, the profiles evolve significantly toward this Keplerian limit, angular momentum being
gained along the equatorial direction through the radial infall of non-zero latitude material.
The constancy of the angular velocity with $z$ allows a significant gain from such infall.
In the 1.92-\mo model, the rotational properties along the equator are originally
closer to the Keplerian values, but, accordingly, evolve little.
In both models, angular momentum is transported outwards, first in material blown away
by the shock wave initiated at core bounce, and then in the neutrino-driven wind.
This occurs as the outflowing material wraps around the
progenitor white dwarf and eventually meets along the equator.
There is no ``physical'' viscosity in the code that would permit a proper
modeling of the accretion disk.
Mass accretion should occur in partnership with outward transport of angular momentum
over a longer timescale, yet to be determined.
In the 1.92-\mo model, at the end of the simulation, the near-Keplerian disk extends from 30\,km
out to $\sim$1800\,km, covering a range of densities (temperatures) from 10$^{13}$\,g\,cm$^{-3}$
down to 10$^{8}$\,g\,cm$^{-3}$ (3\,MeV down to 0.1\,MeV; Fig.~\ref{fig_rho_radslice}).
\section{Energetics and the neutrino-driven wind}
\label{sect_ener}
Given that it leads to the formation of a $\sim$1.4-\mo neutron star, the AIC of a
1.4--2.0\,\mo white dwarf is expected to result, in the case of a successful explosion, to an
outflow of modest mass. Furthermore, due to the similarity with the core collapse
of massive star progenitors, their explosion kinetic energy should be lower than
the $\sim$10$^{51}$ erg inferred, e.g., for SN1987A (Arnett 1987).
As discussed in \S\ref{sect_results} and \S\ref{sect_disk}, the total sum of
the neutron star and disk masses is very close to the original progenitor mass, leaving
typically a few 0.001\,\mo for the ejected material. Integrating all the mass that has
left the grid over the course of the simulation, as well as all the material outflowing
with a positive radial velocity greater than 10000\,\kms (which is of the order of the
escape velocity at a radius of 3000\,km) we find a value of 4$\times$10$^{-3}$\,\mo for the 1.46-\mo
model and 3$\times$10$^{-3}$\,\mo for the 1.92-\mo model.
In Fig.~\ref{fig_energy}, we show the evolution of the
corresponding gravitational (blue), thermal (cyan), and kinetic (2D planar: red;
rotational: green) energies for this outflowing material as solid lines,
including the total energy as a black dotted line.
At the last computed time, the total energy is indeed lower than that inferred for standard
core collapse. Adopting a radial velocity cut of 10000\,\kms, we find an energy at the
last simulated time of 2.7$\times$10$^{49}$erg for the 1.46-\mo model and
2$\times$10$^{49}$erg for the 1.92-\mo model.
Note that energy is still being pumped into the wind by the slowly decaying neutrino
luminosity emanating from the neutron star; the trend of the total energy curve suggests that the
total energy of the explosion will be 2-3 times higher, thus $\sim$10$^{50}$erg for the 1.46-\mo model
and $\sim$5$\times$10$^{49}$erg for the 1.92-\mo model.
This is over one order of magnitude smaller than the
explosion energy inferred for normal core-collapse supernovae.
The AIC of white dwarfs is likely to lead generically to underenergetic explosions because
there is too little mass to absorb neutrinos, most of it being quickly accreted while
the rest is centrifugally-supported at large radii, far beyond the region
where there is a positive net gain of electron-neutrino energy.
Interestingly, similar underenergetic explosions are obtained by Kitaura et al. (2005)
and Buras et al. (2005b) for initial main sequence stars of 8.8\,\mo (Nomoto 1984, 1987)
and 11.2\,\mo (Woosley, Heger, \& Weaver 2002). Echoing the properties of AIC progenitors,
the low envelope mass and fast declining density (and, therefore, accretion rate) are
key beneficial components for the success of neutrino-driven explosions, but the same properties are
also why the explosion is necessarily underenergetic.
The various curves also show a few dips and bumps. The first bump, most pronounced
in the 1.92-\mo model, is associated with an early outflow that eventually fell back
to smaller radii, while subsequent bumps are caused by episodic mass loading of the
neutrino-driven wind, which sets in $\sim$200\,ms after core bounce and drives a
8$\times$10$^{-3}$\,\mo\,s$^{-1}$ (5$\times$10$^{-3}$\,\mo\,s$^{-1}$) mass loss rate
in the 1.46-\mo (1.92-\mo; see also Fig.~\ref{fig_mdot}, top panel) model.
The higher-mass flux in the 1.46-\mo model
results from the higher neutrino luminosity, higher mean neutrino energies, and
bigger opening angle of escape for the neutrino-driven wind.
Interestingly, this mass flux is strongly angle-dependent, varying by a factor of a
few between the pole and the angle that grazes the pole-facing side of the disk.
As described in \S\ref{sect_results}, the dynamical effects of the neutrino-driven
wind are to entrain the material lying along this interface, tearing the disk
via Kelvin-Helmholtz shear instabilities and
mass-loading the wind along the corresponding latitudes.
Further in, neither this wind nor the neutrinos have an appreciable dynamical impact in driving
the disk material outwards, a feature only excacerbated by the reduced neutrino flux at low latitudes.
In Fig.~\ref{fig_mdot}, we show in the bottom panel the latitudinal variation of the asymptotic
velocity (solid line) and the density (dotted line).
Both show an overall decrease towards lower latitudes by a factor of three. The dip in density and higher
values of the velocity along the pole to around 70$^{\circ}$ latitude are possibly
due to wind mass loading, in combination with centrifugal support at the neutrinosphere
for off-polar latitudes.
In Fig.~\ref{fig_nuflux_2d}, we show at early times (top row) and at the last simulated
time (bottom row) for the 1.46-\mo model (left column) and the 1.92-\mo model (right column)
color maps of the total neutrino flux in the radial direction, with isodensity contours
overplotted as white curves, and velocity vectors as black arrows.
First, due to the history of the collapse, the neutron star is relatively devoid of overlying material
in the polar direction, while for the higher-mass progenitor a massive ($\sim$0.6\,\mo),
dense (10$^{6-10}$\,g\,cm$^{-3}$ ), near-Keplerian disk obstructs the neutron star at latitudes
$\sles\pm$40$^{\circ}$.
Given this configuration, conditioned essentially by the mass distribution of the progenitor
white dwarf, the dynamical effect of a spherically-symmetric neutrino flux would be enhanced along
the ``excavated'' polar direction. Indeed, we see a strong neutrino-driven wind in the polar
direction that does not exist in directions within $\sim\pm$40$^{\circ}$ of the equator.
However, even in the absence of this anisotropic matter distribution, Fig.~\ref{fig_nuflux_2d}
reveals the strong latitudinal variation of the neutrino flux at a given Eulerian radius, a variation
that is established independently of the configuration of the circum-neutron star disk material.
What controls the flux geometry is the combination of two effects. First, the exceptional elongation of
the neutrinospheres along the equatorial direction leads to a decoupling radius (surface) about 10 (100)
times bigger for a polar observer than for an equatorial observer. The angle-dependent decoupling
radius of neutrinos mitigates this result (Walder et al. 2005), but, as shown in
Fig.~\ref{fig_nuflux_2d}, the latitudinal
variation along different directions persists in the total neutrino flux.
Similarly, Fig.~\ref{fig_nu_flux_vec} shows the anisotropy of the $\nu_e$ neutrino flux,
rendered by the corresponding flux vectors.
Notice how the base of the flux vectors in the high-density central regions is perpendicular
to the local isodensity (or, equivalently, equipotential) contour.
Second, as shown in Fig.~\ref{fig_temp}, the temperature and its radial-gradient along
a given isodensity contour are both significantly lower along the equatorial direction, leading
to reduced diffusive fluxes.
These properties are reminiscent of the effect of gravity darkening (von Zeipel 1924) in fast rotating
(non-compact) stars and the associated scaling of the radiative flux with the local effective gravity
(see Owocki et al. 1996), although this may be the first time it is
reported in the context of a protoneutron star (but see Walder et al. 2005).
With such a polar-enhanced wind, the angular momentum loss rate is
reduced, with consequences for the spin evolution of the PNS.
\section{Ejecta composition}
\label{sect_ye}
In Fig.~\ref{fig_ye_dist}, we show for the two baseline models the electron fraction ($Y_{\rm e}$) distribution
of the material in the ejecta (material outside the neutron star moving outwards with a radial
velocity greater than 10000\,\kms), accounting as well for the mass loss through the
outer grid radius. Such cumulative outflow amounts to
4$\times$10$^{-3}$\,\mo for the 1.46-\mo model and 3$\times$10$^{-3}$\,\mo for the 1.92-\mo model.
We obtain double-peak profiles, the first blast propelling symmetric material
($Y_{\rm e}=0.5$), subsequently followed after 200\,ms by progressively neutron-rich material, i.e.,
$Y_{\rm e} =0.25-0.35$, in the neutrino-driven wind. Note that for these runs we enforced an upper limit of 0.5
to the computed $Y_{\rm e}$ values.
Fryer et al. (1999) obtained an ejecta mass in the vicinity of 0.2\,\mo, two orders of
magnitude larger than our values. Because our ejecta masses are much smaller, we find that
the mass loss rates and the kinetic energies associated with the neutrino-driven
wind are relatively more important for the global energetics of the AIC of white dwarfs.
At late times, the asymptotic electron fraction $Y_{\rm e}^{\rm a}$ of the neutrino-driven wind
varies with latitude
(despite the smooth variation of other quantities at correspondingly larger distances).
Material ejected within 20$^{\circ}$ of the pole has an electron fraction of $\sim$0.5, while towards
the equator, this electron fraction decreases to 0.3, rising again to near 0.5 values
in regions belonging to the disk (see bottom row panels in Fig.~\ref{fig_seqye}).
The $Y_{\rm e}^{\rm a}$ values seen in our simulations are in fact already set when wind material
leaves the vicinity of the neutrinosphere, whose properties depend on the particle
trajectory under scrutiny.
We know from previous studies (Qian \& Woosley 1996; Wheeler et al. 1998; Thompson et al. 2001;
Pruet et al. 2005; Fr\"{o}lich et al. 2005) that the
asymptotic electron fraction of the ejecta is controled by competing factors.
The electron and anti-electron neutrino luminosities, modulated by the hardness of their respective
energy distributions, influence the electron flavor production rates via the reactions
$\nu_e$n$\rightarrow$pe$^{-}$ and $\bar{\nu}_e$p$\rightarrow$e$^+$n and, thereby, the
neutron-richness of the ejecta.
The expansion timescale sets the duration over which interactions between neutrinos and
nucleons can take place.
The starting value of the electron fraction, i.e., at the base of the outflow, is altered by
the above factors and differs from the asymptotic value seen.
In Fig.~\ref{fig_ye_gain}, we show a color map of the electron fraction in the inner 200\,km,
highlighting the butterfly shape of the deleptonized region in cross section, in stark contrast with the
corresponding near-spherical shape seen in core-collapse simulations of both rotating and non-rotating
progenitors (Keil et al. 1996; Walder et al. 2005; Dessart et al. 2005).
Deleptonization obtains preferentially in the vicinity of the dumbbell-shaped neutrinosphere,
and stretches outwards for off-polar latitudes. Along the equator, deleptonization
ceases at smaller radii due to the lower effective temperatures (tied to the neutrino fluxes).
Temperature and neutrino flux are in fact intertwined, since energy deposition by neutrinos
may raise the temperature locally in the so-called gain region.
This is also vividly represented in Fig.~\ref{fig_ye_gain} (right) by the net gain associated with
electron-neutrino energy deposition in this inner region, which also shows the same butterfly shape.
Electron neutrinos emerge from the neutron star, and, due to the dumbbell neutrinosphere morphology,
at much smaller radii along the poles than for off-polar latitudes.
The decreasing neutrino flux (dilution) reduces this energy deposition
beyond $\sim$50\,km, and even at smaller radii along the equator due to the additonal
flux reduction there (Fig.~\ref{fig_nuflux_2d}).
The asymptotic value of the material electron fraction is determined in the vicinity
of the neutrinosphere and, therefore, is directly influenced by this configuration of the
inner $Y_{\rm e}$ distribution. Along the pole, the wind carries initially low $Y_{\rm e}$ material that
absorbs electron neutrinos, whose associated luminosity is one magnitude higher than that of the
anti-electron neutrinos (see Fig.~\ref{fig_nuflux}), raising the electron fraction to the ceiling
value of 0.5 artificially adopted in these calculations.
Away from the pole, the neutrinosphere is located further
out, and despite similar neutrinosphere $Y_{\rm e}$ values, the larger distance from the neutron star implies
a reduced electron-neutrino luminosity and a reduced absorption of neutrinos, leading to asymptotic
values of the electron fraction of only $\sim$0.25, not far from the values at the
corresponding neutrinosphere.
To summarize, the progressive decrease of the electron fraction (and of the entropy) away from
the pole is a result of the reduced electron-neutrino luminosities in the vicinity of the
latitudinal-dependent neutrinosphere radius (and the associated reduced heating and electron-capture
rates).
\section{Gravitational wave signature}
\label{sect_gw}
We estimate the gravitational wave emission from aspherical mass
motions in our models via the Newtonian quadrupole formalism as
described in M\"onchmeier et al. (1991). In addition, we compute
the gravitational wave strain from anisotropic neutrino emission
employing the formalism introduced by Epstein (1978) and developed
by Burrows \& Hayes (1996) and M\"uller \& Janka (1997).
Due to rapid rotation and the resulting oblateness of the core,
one would expect that rotating AIC models would have significant
gravitational wave signatures.
However, though the contribution
to the metric strain in the equatorial plane, $h_+$,
of the aspherical and dynamical matter distributions is not small, we find
that that of the aspherical neutrino field is larger in magnitude, though at
much smaller frequencies. While we calculate that $h_+$(max) for the matter in the
1.46-\mo\ model is $\sim$5.9$\times$10$^{-22}$, with a spectrum that peaks
at $\sim$430 Hz, the corresponding $h_+$(max) due to neutrinos
is $\sim$4.6$\times$10$^{-21}$ (derived from the fluxes at 200 km), but at
frequencies between ($\sim$0.1-10 Hz). The total energy radiated in
gravitational waves is $\sim$5.7$\times$10$^{-10}$ \mo{$c^2$}.
The corresponding numbers for the faster rotating and more massive 1.92-\mo\ model
are $\sim$3.6$\times$10$^{-21}$ (matter),
$\sim$165 Hz, $\sim$2.0$\times$10$^{-20}$ (neutrinos at 300 km), and
$\sim$7.0$\times$10$^{-8}$ \mo{$c^2$}. Note that almost all of the
energy is being emitted by mass motions (99.8\% in the 1.46-\mo\ model and
98.4\% in the 1.92-\mo\ model), since the power scales with the time derivative of
the wave strain $h_+$, which is small for the waves emitted from the
aspherical neutrino field.
We compare the above numbers with those obtained by Fryer, Holz \& Hughes (2002) for an
AIC model of Fryer et al.\ (1999) which was setup with a simple, solid-body
rotation law and had a final T/$|$W$|$ of $\sim$0.06
(for our 1.46-\mo\ model: 0.059; see Table 1). They did not consider anisotropic
neutrino emission. Our more realistic initial models
yield maximum (matter) gravitational wave strains that are 1.5 to 2 orders
of magnitude smaller than those predicted by Fryer, Holz \& Hughes (2002). The total
energy emissions match within a factor of two since our models emit at higher
frequencies.
Based on our results, we surmise that gravitational waves from axisymmetric
AIC events may be detected by current LIGO-class detectors if occurring anywhere
in the Milky Way, but not out to megaparsec distances as suggested by
Fryer, Holz \& Hughes (2002). It is, however, likely (\S 5) that at least the
1.92-\mo\ model will undergo a dynamical rotational instability leading to non-axisymmetric
deformation (which can not be captured by our 2D approach) and, hence, to copious gravitational
wave emission over many rotation periods, greatly enhancing detectability.
\section{Discussion and conclusions}
\label{sect_conc}
We have presented a radiation/hydrodynamic study with the code VULCAN/2D of the
collapse and post-bounce evolution of massive rotating high-central-density white dwarfs,
starting from physically consistent 2D rotational equilibrium configurations
(Yoon \& Langer 2005). The main results of this study are:
\begin{itemize}
\item The AIC of white dwarfs leads to a successful explosion with modest energy
$\sles$10$^{50}$\,erg, thus comparable to the energies obtained through the collapse of
O/Ne/Mg core of stars with $\sim$8-11\,\mo main sequence mass (Kitaura et al. 2005;
Buras et al. 2005b).
This is, however, underenergetic, by a factor of about ten, compared with the inferred value for
the core collapse of more massive progenitors leading to Type II Plateau supernovae.
Although less and less likely to be the engine behind most core-collapse supernova explosions,
the neutrino mechanism can successfully power explosions of low-mass progenitors and AICs due to the
limited mantle mass and steeply declining accretion rate.
\item Due to high-mass and angular-momentum accretion, white dwarf progenitors
leading to AIC can have masses of up to $\sim$2\,\mo and rotate fast, with rotational
to gravitational energy ratios of up to a few percent prior to collapse.
The asphericity of such white dwarfs allows the shock generated at core bounce to
escape along the poles in just a few tens of milliseconds, opening a hole in the white dwarf
along the poles. The blast expands and wraps around the progenitor and escapes the grid
($\sim$5000 km) within a few hundred milliseconds, at which time a neutrino-driven wind has grown in
the pole-excavated region of the white dwarf. Both the original blast and the wind
show strong latitudinal variations, partly constrained by the obstructing uncollapsed equatorial
disk regions of the progenitor, whose centrifugal support prevents it from collapsing on a dynamical
timescale. Rotation in such progenitors, thus, affects both directly and indirectly the morphology
of the explosion.
\item The neutron stars formed have masses on the order of 1.4\,\mo, with rotation periods close
to a millisecond in the rigidly rotating inner $\sim$30\,km. The final rotational to gravitational
energy ratios, for our two test cases, cover 0.06 to 0.26, the latter being large enough to
grow non-axisymmetric instabilities. At the end of our simulations,
the neutron stars are oblate and pinched along the poles, with polar and equatorial radii in the
ratio 1:15 for the faster rotating (1.92-\mo) model.
\item The morphology of the neutron star leads to a latitudinal variation of the
neutrino flux, the net energy gain, and the temperature. In the faster rotating model,
the ``$\nu_\mu$'' neutrino flux is reduced, while the anti-electron neutrino
flux is a factor of ten lower than that of the electron-neutrino.
This raises the electron fraction of the ejected material to values close to 0.5 along the poles,
but to only $\sim$0.25 at lower latitudes, since the corresponding neutrinosphere is more remote and,
thus, the electron-neutrino flux is smaller. This introduces
a latitudinal dependence of the electron fraction of the ejected material, but more importantly
allows neutron-rich material, with entropy on the order of 20-40\,$k_{\rm B}$/baryon, to escape.
Thus, a low-entropy r-process might take place under these conditions.
\item The high original angular momentum of the progenitor follows the mass and is, thus, found
mostly in the neutron star at the end of the simulation. However, rotational energy is
also given to the ejecta, which uses it to gain (planar) radial kinetic energy to escape
the potential well, while the rest is found in a quasi-Keplerian disk of up to 0.5\,\mo in
the 1.92-\mo model. This disk is an essential component of these AICs; it collimates the explosion
and the neutrino-driven wind and also suggests a second stage of long-term accretion onto the compact
remnant.
\item The total ejected mass is only of the order of a few times 0.001\,\mo, with only a quarter in
the form of $^{56}$Ni. The original blast and the short-lived neutrino-driven wind will lead
to a considerable brightening of the object, but the small ejecta mass will quickly become
optically-thin, gamma rays leaking out rather than depositing their energy to power a durable
light curve. Therefore, these explosions should be underluminous and very short lived.
Their appearance may also vary considerably with viewing angle, depending on the mass of
the progenitor and the presence of a sizable disk in the equatorial regions.
\end{itemize}
This study has shown that more consistent, rotating 2D models alter considerably our understanding of
accretion-induced collapse, previously obtained under the simplifying assumptions of spherical symmetry
and/or zero rotation. Further improvements will come by including
a consistent temperature structure for the progenitor white dwarf and by accounting for
the effects of magnetic fields. Due to the rapid differential rotation, magnetic fields
could be amplified considerably and result in MHD jets that might alter yet again
our overall picture of accretion-induced collapse and the energetics of the phenomenon.
Three-dimensional effects may also alter the protoneutron star properties presented here, since
the faster rotating (1.92-\mo) model is expected to experience non-axisymmetric instabilities.
\acknowledgments
We acknowledge discussions with and help from
Jeremiah Murphy and Casey Meakin.
Importantly, we acknowledge support for this work
from the Scientific Discovery through Advanced Computing
(SciDAC) program of the DOE, grant number DE-FC02-01ER41184
and from the NSF under grant number AST-0504947.
E.L. thanks the Israel Science Foundation for support under grant \# 805/04,
and C.D.O. thanks the Albert-Einstein-Institut for providing CPU time on their
Peyote Linux cluster. We thank Jeff Fookson and Neal Lauver of the Steward Computer Support Group
for their invaluable help with the local Beowulf cluster.
This research used resources of the National
Energy Research Scientific Computing Center, which is supported by the
Office of Science of the U.S. Department of Energy under Contract No.
DE-AC03-76SF00098.
|
Title:
F_D-Term Hybrid Inflation with Electroweak-Scale Lepton Number Violation |
Abstract: We study F-term hybrid inflation in a novel supersymmetric extension of the
SM with a subdominant Fayet-Iliopoulos D-term. We call this particular form of
inflation, in short, F_D-term hybrid inflation. The proposed model ties the
mu-parameter of the MSSM to an SO(3)-symmetric Majorana mass m_N, through the
vacuum expectation value of the inflaton field. The late decays of the
ultraheavy particles associated with the extra U(1) gauge group, which are
abundantly produced during the preheating epoch, could lower the reheat
temperature even up to 1 TeV, thereby avoiding the gravitino overproduction
problem. The baryon asymmetry in the Universe can be explained by thermal
electroweak-scale resonant leptogenesis, in a way independent of any
pre-existing lepton- or baryon-number abundance. Further cosmological and
particle-physics implications of the F_D-term hybrid model are briefly
discussed.
| https://export.arxiv.org/pdf/hep-ph/0601080 |
\begin{flushright}
CERN-PH-TH/2006-003\\[-2pt]
{\tt hep-ph/0601080}\\
January 2006
\end{flushright}
\bigskip
\begin{center}
{\LARGE {\bf {\boldmath $F_D$}-Term Hybrid Inflation with}}\\[0.3cm]
{\LARGE {\bf Electroweak-Scale Lepton Number Violation}}\\[1.5cm]
{\large Bj\"orn Garbrecht$^{\, a}$ and Apostolos Pilaftsis$^{\, a,b}$}\\[0.5cm]
{\em $^a$School of Physics and Astronomy, University of Manchester,}\\
{\em Manchester M13 9PL, United Kingdom}\\[0.3cm]
{\em $^b$CERN, Physics Department, Theory Division, CH-1211 Geneva 23,
Switzerland}
\end{center}
\vspace{1.5cm} \centerline{\bf ABSTRACT}
\noindent
We study $F$-term hybrid inflation in a novel supersymmetric extension
of the SM with a subdominant Fayet--Iliopoulos $D$-term. We call this
particular form of inflation, in short, $F_D$-term hybrid inflation.
The proposed model ties the $\mu$-parameter of the MSSM to an
SO(3)-symmetric Majorana mass $m_N$, through the vacuum expectation
value of the inflaton field. The late decays of the ultraheavy
particles associated with the extra U(1) gauge group, which are
abundantly produced during the preheating epoch, could lower the
reheat temperature even up to 1~TeV, thereby avoiding the gravitino
overproduction problem. The baryon asymmetry in the Universe can be
explained by thermal electroweak-scale resonant leptogenesis, in a way
independent of any pre-existing lepton- or baryon-number abundance.
Further cosmological and particle-physics implications of the
$F_D$-term hybrid model are briefly discussed.
\noindent
\medskip
\noindent
{\small PACS numbers: 98.80.Cq, 12.60.Jv, 11.30Pb}
\newpage
\setcounter{equation}{0}
\section{Introduction}
The inflationary paradigm constitutes an ingenious theoretical
framework, in which many of the outstanding problems in standard
cosmology can be successfully addressed~\cite{review}. The recent
WMAP data~\cite{WMAP}, compiled with other astronomical
observations~\cite{MT,Lyman}, improved upon the precision of about a
dozen of cosmological parameters. These include the power spectrum
$P^{1/2}_{{\cal R}}$ of curvature perturbations, the spectral index
$n_s$, the baryon-to-photon ratio of number densities $\eta_B$ and
others. The values of these cosmological observables put severe
constraints on the model-building of successful models of inflation
and their theoretical parameters. For instance, one of the basic
requirements for slow-roll inflation is that the so-called inflaton
potential be flat. In this respect, supersymmetry (SUSY) emerges as a
compelling ingredient in model-building for protecting the flatness of
the inflaton potential against quantum corrections.
In addition to the aforementioned element of naturalness, inflationary
models would have a greater value if they were predictive and testable
as well. One such predictive and perhaps most appealing scenario is
the well-celebrated model of hybrid inflation~\cite{Linde}. In this
model, the inflaton field $\phi$ can start its slow-roll from values
well below the Planck scale $m_{\rm Pl} = 2.4\times 10^{18}$~GeV.
This renders the model very predictive, in the sense that an infinite
set of possible higher-dimensional non-renormalizable operators, being
suppressed by inverse powers of $1/m_{\rm Pl}$, will not generically
contribute significantly to cosmological observables, such as
$P^{1/2}_{{\cal R}}$ and $n_s$. In the hybrid model, inflation ends
through the so-called waterfall mechanism, once the field $\phi$
passes below a critical value $\phi_c$. When this happens, another
field $X$ different from $\phi$, with vanishing initial value,
develops a tachyonic instability and rolls fast down to its true
vacuum expectation value~(VEV). Super-\linebreak symmetric
realizations of hybrid inflation from $F$-terms were first analyzed
in~\cite{CLLSW,DSS}, whereas hybrid inflation triggered by a dominant
Fayet--Iliopoulos~(FI) $D$-term~\cite{FI} was subsequently considered
in~\cite{Halyo}.
In this paper we study $F$-term hybrid inflation in a novel
supersymmetric extension of the Standard Model~(SM) that includes a
subdominant FI $D$-term. We call this scenario for brevity, the
$F_D$-term hybrid model. To account for the low-energy neutrino data,
we introduce 3 singlet neutrino superfields $\widehat{N}_{1,2,3}$ that
contain 3 right-handed neutrinos $\nu_{1,2,3\,R}$ and their
supersymmetric scalar counterparts $\widetilde{N}_{1,2,3}$. Most
importantly, the model ties the $\mu$-parameter of the Minimal
Supersymmetric Standard Model~(MSSM) to an SO(3) symmetric Majorana
mass $m_N$, through the VEV of the inflaton field
$\phi$~\cite{PU2,Francesca}. Hence, the $F_D$-term hybrid model
naturally predicts lepton-number violation at the TeV or even at the
electroweak scale. In this scenario, the non-zero baryon asymmetry in
the Universe (BAU), $\eta_B \approx 6.1 \times 10^{-10}$, can be
explained by leptogenesis~\cite{FY,BAUpapers} and specifically by
thermal electroweak-scale resonant leptogenesis~\cite{APRD,PU2}.
In this paper we also study the constraints on the parameters of the
$F_D$-term hybrid model that result from a reheat temperature $T_{\rm
reh} \stackrel{<}{{}_\sim} 10^9$~GeV, which is necessary to avoid the
well-known gravitino overproduction problem. This consideration puts
severe limits on the size of the superpotential couplings of the
theory, forcing them {\em all} to acquire rather suppressed values,
namely to be smaller than about $10^{-5}$~\cite{SS}. To overcome this
problem of unnaturalness, the presence of a subdominant FI $D$-term in
the $F_D$-term hybrid model is very crucial and provides a new
mechanism of relaxing dramatically the above upper limit. More
explicitly, the size of the $D$-term controls the decay rates of the
ultraheavy fermions and bosons associated with the extra gauge group
U(1)$_X$. In the absence of the $D$-term and any other
non-renormalizable interaction, these ultraheavy gauge-sector
particles are absolutely stable. On the other hand, these particles
are abundantly produced during the preheating epoch, thus dominating
the energy density of the Universe shortly after the period of the
first reheating caused by the perturbative inflaton decays.
Therefore, their late decays induced by a non-vanishing $D$-term could
give rise to a second reheating phase in the evolution of the early
Universe. Depending on the actual size of the FI $D$-term, this second
reheat temperature may be as low as 1~TeV, giving rise to an enormous
entropy release that can dilute the gravitinos produced during the
first reheating to an unobservable level.
The paper is organized as follows: Section~\ref{FDmodel} presents the
model-building aspects of the $F_D$-term hybrid model with
electroweak-scale lepton-number violation. Technical details related
to the radiatively-induced FI $D$-term are relegated to Appendix~A.
In Section~\ref{reheat}, we estimate the reheat temperature from the
perturbative inflaton decays and derive the resulting gravitino
constraint on the theoretical parameters. We then discuss the
non-perturbative production of the supermassive fields associated with
the U(1)$_X$ gauge group during the preheating epoch and how their
late decays can help to lower the reheat temperature even up to~1~TeV.
Section~\ref{inflation} is devoted to inflation. Here we investigate
two regimes: (i) cold hybrid inflation, where dissipative effects can
be ignored, and (ii) warm hybrid inflation, where dissipative effects
dominate over the expansion rate of the Universe. In
Section~\ref{BAU} we illustrate how the $F_D$-term hybrid model can
realize thermal electroweak-scale resonant leptogenesis. Finally,
Section~\ref{conclusions} summarizes our conclusions, including a
brief discussion of further possible implications of the $F_D$-term
hybrid model for particle physics and cosmology.
\setcounter{equation}{0}
\section{The {\boldmath $F_D$}--Term Hybrid Model}\label{FDmodel}
The $F_D$-term hybrid model may be defined by the superpotential
\begin{eqnarray}
\label{Wmodel}
W & =& \kappa\, \widehat{S}\, \Big( \widehat{X}_1
\widehat{X}_2\: -\: M^2\Big)\ +\ \lambda\, \widehat{S} \widehat{H}_u
\widehat{H}_d\ +\ \frac{\rho}{2}\, \widehat{S}\, \widehat{N}_i
\widehat{N}_i\ +\ h^{\nu}_{ij} \widehat{L}_i \widehat{H}_u
\widehat{N}_j\nonumber\\ &&+\ W_{\rm MSSM}^{(\mu = 0)}\; ,
\end{eqnarray} where $W_{\rm MSSM}^{(\mu = 0)}$ is the MSSM superpotential
without the $\mu$ term:
\begin{equation} W_{\rm MSSM}^{(\mu = 0)}\ =\
h^u_{ij}\,\widehat{Q}_i\widehat{H}_u\widehat{U}_j\: +\:
h^d_{ij}\,\widehat{H}_d\widehat{Q}_i\widehat{D}_j\: +\:
h_l\, \widehat{H}_d\widehat{L}_l\widehat{E}_l \; .
\end{equation}
In~(\ref{Wmodel}), $\widehat{S}$ is the SM-singlet inflaton
superfield, and $\widehat{X}_1$ and $\widehat{X}_2$ is a chiral
multiplet pair of the so-called waterfall fields which have opposite
charges under the additional U(1)$_X$ gauge group. The
superpotential~(\ref{Wmodel}) of the model is uniquely determined by
the $R$ transformation: $\widehat{S} \to e^{i\alpha}\, \widehat{S}$,
$\widehat{X}_{1,2} \to e^{\pm i\beta}\,\widehat{X}_{1,2}$,
$\widehat{L} \to e^{i\alpha}\, \widehat{L}$, $\widehat{Q} \to
e^{i\alpha}\, \widehat{Q}$, with $W \to e^{i\alpha} W$, whereas all
other fields remain invariant under an $R$ transformation. As a
consequence of the $R$ symmetry, higher-dimensional operators of the
form $\widehat{X}_1 \widehat{X}_2 \widehat{N}_i \widehat{N}_i/m_{\rm
Pl}$, for example, are not allowed.
In addition, the model contains a subdominant FI $D$-term,
$-\frac{1}{2} g\, m^2_{\rm FI}\, D$, giving rise to the $D$-term
potential
\begin{equation}
\label{Dterm}
V_D\ =\ \frac{g^2}{8}\ \Big( |X_1|^2\, -\, |X_2|^2\, -\, m^2_{\rm
FI}\,\Big)^2\; .
\end{equation}
The FI $D$-term plays no role in the inflationary dynamics, as long as
$g m_{\rm FI} \ll \kappa\, M$. In Appendix~\ref{Dappendix}, we show
how a subdominant $D$-term can be generated radiatively after
Planck-scale heavy degrees of freedom have been integrated out. The
presence of the $D$-term is important to break an accidental discrete
charge symmetry that survives after the spontaneous symmetry breaking
(SSB) of the U(1)$_X$. Such a breaking is crucial to render all
U(1)$_X$ gauge-sector particles unstable. As we will see in
Section~\ref{reheat}, an upper limit on the size of the FI term is
obtained by requiring a sufficiently low reheat temperature,
e.g.~$T_{\rm reh} \stackrel{<}{{}_\sim} 10^9$~GeV, in order to
suppress the gravitino abundance to an unobservable level.
From~(\ref{Wmodel}) it is straightforward to read the Lagrangian of
the inflationary soft SUSY-breaking sector,
\begin{equation}
\label{Lsoft}
-\, {\cal L}_{\rm soft}\ =\ M^2_S S^*S\: +\: \Big(
\kappa A_\kappa\, S X_1X_2\: +\: \lambda A_\lambda S H_u H_d\: \: +\:
\frac{\rho}{2}\, A_\rho\, S \widetilde{N}_i\widetilde{N}_i\:
-\: \kappa a_S M^2 S \: \ +\ {\rm H.c.}\,\Big)\,,
\end{equation}
where $M_S$, $A_{\kappa,\lambda,\rho}$ and $a_S$ are soft
SUSY-breaking mass parameters of order $M_{\rm SUSY} \sim 1$~TeV.
In the regime $|S| \gg M$ relevant to inflation, the dominant
tree-level and one-loop contributions to the renormalized effective
potential may be described by
\begin{eqnarray}
\label{VpotFD}
V_{\rm inflation} & \approx & \kappa^2 M^4\ \Bigg[\, 1\: +\:
\frac{1}{64\pi^2}\, \Big(\, 4\kappa^2 \: +\: 8\lambda^2\: +\:
6\rho^2\,\Big)\, \ln\Bigg(\frac{|S|^2}{M^2}\Bigg)\,\Bigg]\nonumber\\
&&
-\: \Big( \kappa a_S M^2 S\: +\: {\rm H.c.}\Big)\ +\ V_{\rm SUGRA}\ ,
\end{eqnarray}
where $V_{\rm SUGRA}$ denotes the supergravity (SUGRA) correction that
results from the K\"ahler potential. Assuming a minimal K\"ahler
potential, the SUGRA correction of interest to us takes on the simple
form~\cite{CLLSW,CP,LR}
\begin{equation}
\label{Vsugra}
V_{\rm SUGRA}\ =\ \kappa^2 M^4\, \frac{|S|^4}{2\,m^4_{\rm Pl}}\ ,
\end{equation}
where $m_{\rm Pl} = 2.4\times 10^{18}$~GeV is the reduced Planck mass.
Possible one-loop contributions to $V_{\rm inflation}$ from
$A_{\kappa,\lambda,\rho}$-terms become significant only for relatively
low values of $M$, e.g. $M\stackrel{<}{{}_\sim} 10^8$~GeV, for
$\kappa,\lambda,\rho \sim 1$, and may therefore be neglected. At the
tree level, however, only the tadpole term $\kappa a_S M^2\, S$ may
become relevant for values of $\kappa \stackrel{<}{{}_\sim} 10^{-4}$,
whereas the other soft SUSY-breaking terms are negligible during
inflation~\cite{SS}.
We now investigate the stability of the inflationary trajectory in the
presence of the Higgs doublets $H_{u,d}$ and the right-handed scalar
neutrinos $\widetilde{N}_{1,2,3}$. Specifically, the initial
condition for inflation is
\begin{equation}
\label{initial}
{\rm Re}\, S^{\rm in}\ =\ |S^{\rm in}|\ \gg\ M\,,\qquad
X^{\rm in}_{1,2}\ =\ 0\,,\qquad
H^{\rm in}_{u,d}\ =\ 0\,,\qquad
\widetilde{N}^{\rm in}_{1,2,3}\ =\ 0\; .
\end{equation}
At the end of inflation, one should ensure that the waterfall fields
acquire a high VEV, i.e. $X^{\rm end}_{1,2}\ =\ M$, while all other
fields have small VEVs, possibly of the electroweak order. To achieve
this, we have to require that the Higgs-doublet and the sneutrino mass
matrices stay positive definite throughout the inflationary trajectory
up to the critical value $|S_c| \approx M$, whereas the corresponding
mass matrix of $X_{1,2}$ will be the first to develop a negative
eigenvalue and tachyonic instability close to $|S_c|$. In this way, it
will be the fields $X_{1,2}$ which will first start moving away from 0
and set in to the `good' vacuum $X^{\rm end}_1\ =\ X^{\rm end}_2\ =\
M$, instead of having the other fields, e.g.~$H_{1,2}$ and
$\widetilde{N}^{\rm in}_{1,2,3}$, go to a `bad' vacuum where $X^{\rm
end}_{1,2}\ =\ 0$, $H^{\rm end}_{1,2}\ =\ \frac{\kappa}{\lambda}\, M$
and $\widetilde{N}^{\rm in}_{1,2,3} = \frac{\kappa}{\rho}\, M$. To
see this, let us write down the mass matrix in the background
Higgs-doublet field space $(H^\dagger_d\,,\ H_u )$ as
\begin{equation}
\label{Mdoublet}
M^2_{\rm Higgs}\ =\ \left(\! \begin{array}{cc}
|\lambda|^2 |S|^2 & -\,\kappa \lambda (M^2 - X_1 X_2 )\: +\: \lambda
A_\lambda S \\
-\,\kappa^* \lambda^* (M^2 - X^*_1 X^*_2 ) +\: \lambda^*
A^*_\lambda S^* & |\lambda|^2 |S|^2 \end{array}\!\right)\ .
\end{equation}
The requirement of positive definiteness may be translated into the
simple condition:
\begin{equation}
\label{Scondition}
|\lambda|\, |S|^2\ \ge\ |\kappa (M^2 - X_1 X_2 )\: -\:
A_\lambda S|\ .
\end{equation}
From this last inequality, we may see that the condition $\lambda
\stackrel{>}{{}_\sim} \kappa$ is sufficient for ending hybrid
inflation to the `good' vacuum. Likewise, one obtains a condition
analogous to~(\ref{Scondition}) from the sneutrino mass matrix, which
amounts to having $\rho \stackrel{>}{{}_\sim} \kappa$. The above two
restrictions on the superpotential couplings $\lambda$ and $\rho$ will
be imposed throughout our analysis.
As mentioned above, after the end of inflation, one has $X^{\rm
end}_{1,2} = M$, giving rise to a high mass for the inflaton field,
i.e.~$2|\kappa |^2 M^2 |S|^2$. Combining this fact with the soft
SUSY-breaking tadpole $-\kappa a_S M^2 S$ and the trilinear coupling
$\kappa A_\kappa S X^{\rm end}_1 X^{\rm end}_2$, one gets a VEV for
the inflaton~\cite{DLS}:
\begin{equation}
\label{Send}
v_S\ \equiv\ \langle S^{\rm end} \rangle\ =\ \frac{1}{2|\kappa|}\,
\Big|\,A_{\kappa} - a_S\,\Big|\ ,
\end{equation}
where we have neglected the VEVs of the Higgs doublets $H_{u,d}$. The
VEV of $S$ induces an effective $\mu$-term and an SO(3) symmetric
lepton-number-violating Majorana mass $m_N$ of the electroweak
order~\cite{PU2}:
\begin{equation}
\label{mumN}
\mu\ =\ \lambda\, v_S\;, \qquad m_N\ =\ \rho\, v_S\; .
\end{equation}
If $\rho$ and $\lambda$ are comparable in magnitude, then these two
mass parameters are tied together and can naturally be of the
TeV or even of the electroweak scale.
In Sections~\ref{reheat} and~\ref{inflation}, we will derive the
constraints on the key theoretical parameters $\kappa$, $\lambda$,
$\rho$ and $M$ from the requirement of a low reheat temperature,
$T_{\rm reh} \stackrel{<}{{}_\sim} 10^9$~GeV, and successful
inflation.
\setcounter{equation}{0}
\section{Preheating and Second Reheating}\label{reheat}
In the SUGRA framework, the reheat temperature is constrained by the
fact that an overabundant amount of gravitinos may destroy, through
their possible late decays, the successful predictions of Big Bang
nucleosynthesis~\cite{Sarkar}. This possibility is avoided, if the
gravitino abundance $Y_{3/2}$ is small enough, i.e.~$Y_{3/2} <
10^{-12}$. The latter may be translated to an upper limit on the
reheat temperature, i.e.~$T_{\rm reh} \stackrel{<}{{}_\sim} 10^9$~GeV.
If the gravitinos are stable, the above limit may be relaxed by one
order of magnitude to $\sim 10^{10}$~GeV. This depends on whether the
so-called next-to-lightest supersymmetric particle (NLSP) has a small
branching fraction to hadronic decay modes~\cite{FIY}. In addition to
the above upper limit, the reheat temperature $T_{\rm reh}$ is also
constrained from below, depending on the mechanism of baryogenesis.
Thus, for successful electroweak-scale resonant leptogenesis, a lower
bound of order TeV on $T_{\rm reh}$ should be considered.
In the following we will study the post-inflationary dynamics. To this
end, let us define the fields:
\begin{eqnarray}
\label{Xpm}
X_\pm \!&=&\! \frac{1}{\sqrt{2}}\, (X_1\: \pm\: X_2)\ =\
\langle X_\pm \rangle\: +\: \delta X_\pm\; ,\nonumber\\
\delta X_\pm \!&=&\! \frac{1}{\sqrt{2}}\,
(R_\pm\, +\, {\rm i}I_\pm)\; .
\end{eqnarray}
As mentioned in the introduction, inflation ends, once the inflaton
field, $\phi = \sqrt{2}\, {\rm Re}\, S$, rolls below a critical value
$\phi_c \approx \sqrt{2}\, M$. Then, the waterfall regime begins,
where the waterfall fields $S$ and $R_+$ evolve rapidly (we use the
gauge freedom to ensure that all VEVs point to real directions).
Ignoring small corrections due to a non-vanishing FI $D$-term, $m_{\rm
FI}$, the VEVs of $S$ and $R_+$ oscillate around \emph{zero}, whereas
$X_+$ attains its U(1)$_X$-breaking VEV, $\langle X_+ \rangle =
\sqrt{2} M$.
The masses of the waterfall- or $\kappa$-sector fields $\phi$ and
$R_+$ are equal to $m_\kappa = \sqrt 2 \kappa M$. The inflaton $\phi$
decays predominantly into pairs of charged and neutral higgsinos,
$\tilde{h}^\pm_{u,d}$, $\tilde{h}^0_{u,d}$, $\tilde{\bar{h}}^0_{u,d}$,
and into pairs of right-handed Majorana neutrinos
$\nu_{1,2,3\,R}$. The decay width of the inflaton is given by
\begin{equation}
\label{infldecay}
\Gamma_\phi\ =\ \frac{1}{32\pi}\: \Big(\, 4\lambda^2\: +\: 3 \rho^2\,
\Big)\: m_\kappa\; .
\end{equation}
It turns out that the field $R_+$ decays into the scalar SUSY partners
of the aforementioned fields at the same rate. Hence, we find
\begin{equation}
\Gamma_\phi\ =\ \Gamma_{R_+}\ \equiv\ \Gamma_\kappa\; .
\end{equation}
The reheat temperature resulting from the perturbative decays of the
$\kappa$-sector fields may usually be estimated by
\begin{equation}
\label{Treh}
T^\kappa_{\rm reh}\ =\ \left( \frac{90}{\pi^2\, g_*}\right)^{1/4}\,
\sqrt{\Gamma_\kappa\: m_{\rm Pl} }\ ,
\end{equation}
where $g_* = 228.75$ is the number of the relativistic degrees of
freedom in the supersymmetric model under consideration. The gravitino
bound then implies that
\begin{equation}
\label{Tkappa}
\kappa\, \left(\, \lambda^2\: +\: \frac{3}{4}\, \rho^2\, \right)\
\stackrel{<}{{}_\sim}\ 3 \cdot 10^{-15}\,\times\,
\left(\frac{T^\kappa_{\rm reh}}{10^9~{\rm GeV}}\right)^2\,
\left( \frac{10^{16}~{\rm GeV}}{M}\right)\; .
\end{equation}
If $\kappa \approx \lambda \approx \rho$, this amounts to being each
individual coupling smaller than about $10^{-5}$, for $M =10^{16}$~GeV
and $T^\kappa_{\rm reh} \stackrel{<}{{}_\sim} 10^9$~GeV.
So far, we have only considered the post-inflationary dynamics of the
$\kappa$-sector fields, $S$, $R_+$ and $I_+$, to which all the energy
of the inflationary potential is stored at the onset of the waterfall
regime. We now turn our attention to the $g$-sector, namely to the
particles associated with the extra U(1)$_X$ gauge group. This
distinction of the different fields involved after inflation is made
clear in Table~\ref{spectrum}. Thus, the $g$- or U(1)$_X$ gauge-
sector contains the U(1)$_X$ gauge boson $V_\mu$, the Dirac fermion
$\psi_g$, which consists of the gaugino $\lambda$ and the fermionic
superpartner of $X_-$, and the scalars $R_-$ and $I_-$; the field
$I_-$ is a massless would-be Goldstone boson which becomes the
longitudinal component of $V_\mu$. Each of the $g$-sector particles
has a mass $m_g=2^{-1/2} g \langle X_+ \rangle$. In fact, during the
waterfall transition, their masses evolve rapidly from 0 to $g M$. As
we will see below, this rapid non-adiabatic mass variation triggers
the so-called preheating mechanism, through which the $g$-sector
particles can be produced in sizeable amounts. Their decays can only
be induced by the presence of a non-vanishing $D$-term, which breaks
explicitly a discrete charge symmetry in the $F$- and the $D$-term
sectors which would remain otherwise intact even after the SSB of the
U(1)$_X$.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
& & & \\
Sector & Boson & Fermion & Mass\\
& & & \\
\hline\hline
& & & \\
Waterfall &
$S$, $R_+$, $I_+$ &
$\psi_\kappa=
\left(
\begin{array}{c}
\frac{1}{\sqrt2}
\left[
\left(1-\frac{v}{2M}\right)\psi_{X_1}
+\left(1+\frac{v}{2M}\right)\psi_{X_2}
\right]
\\
\psi_S^\dagger
\end{array}
\right)
$
&
$\sqrt 2 \kappa M$
\\
($\kappa$-sector) & & & \\
& & & \\
\hline
& & & \\
U(1)$_X$ Gauge &
$V_\mu$, $R_-$ &
$\psi_g=
\left(
\begin{array}{c}
\frac{1}{\sqrt2}
\left[
\left(1+\frac{v}{2M}\right)\psi_{X_1}
-\left(1-\frac{v}{2M}\right)\psi_{X_2}
\right]
\\
-{\rm i}\lambda^\dagger
\end{array}
\right)
$
&
$g M$
\\
($g$-sector) & & & \\
& & & \\
\hline
\end{tabular}
\end{center}
\caption{\em Particle spectrum of the waterfall and U(1)$_X$ gauge
sectors after inflation, where $V_\mu$ denotes the U(1)$_X$ gauge
boson and $\lambda$~its associate gaugino.}\label{spectrum}
\end{table}
To make this last point explicit, let us express the relevant $F$- and
$D$-term potential in terms of the fields $X_\pm$ defined
in~(\ref{Xpm}):
\begin{equation}
\label{VFD}
V_{FD}\ =\ \frac{\kappa^2}{4}\, \Big|\,X^2_+\: -\: X^2_-\:
-\: 2\,M^2\,\Big|^2\ +\ \frac{g^2}{8}\, \Big(\,
X^*_+ X_-\: +\: X^*_- X_+\: -\: m^2_{\rm FI}\, \Big)^2\; .
\end{equation}
It is obvious that the potential $V_{FD}$ possesses an additional
discrete charge symmetry under the transformation, $X_\pm \to \pm
X_\pm$, if the FI mass term vanishes, $m^2_{\rm FI} = 0$. In the
absence of a FI term, this symmetry will still survive even after the
SSB of the U(1)$_X$ along the flat direction $\langle X_1\rangle =
\langle X_2 \rangle = M$,\footnote{Observe that an analogous discrete
charge symmetry also survives after SSB in the so-called $D$-term
inflationary model~\cite{Halyo}, where $M = 0$ and $m_{\rm FI} \neq
0$. In this case, the waterfall fields $X_{1,2}$ transform as
$X_{1,2} \to \pm X_{1,2}$, while their VEVs after inflation are
$\langle X_1 \rangle = m_{\rm FI}$ and $\langle X_2 \rangle = 0$.} or
equivalently when $\langle X_+ \rangle = \sqrt{2} M$ and $\langle
X_-\rangle = 0$. As a consequence, the U(1)$_X$ gauge boson $V_\mu$,
the scalar field $R_- = \sqrt{2}\,{\rm Re} (X_-)$ and their fermionic
superpartner $\psi_g$ are all stable with a mass $g M$. This feature
is highly unsatisfactory for the hybrid model without a FI term, since
these particles can be produced in large numbers during the preheating
process, and since they are very massive, they could dominate and so
overclose the Universe at later times.
The presence of the FI term $m_{\rm FI}$ breaks explicitly the above
discrete charge symmetry and so provides a new decay mechanism for
making these particles unstable. To leading order in the expansion
parameter $m_{\rm FI}/M$, the potential $V_{FD}$ given in~(\ref{VFD})
can be minimized using the linear field decompositions
\begin{equation}
\label{Xdec}
X_+\ =\ \sqrt{2}\,M\: +\: \delta X_+\,,\qquad
X_-\ =\ \frac{v}{\sqrt{2}}\: +\: \delta X_-\ ,
\end{equation}
where $v = m^2_{\rm FI}/(2M)$. Table~\ref{spectrum} exhibits the
particle spectrum of the waterfall and U(1)$_X$ gauge sectors to
leading order in $m_{\rm FI}/M$. Unlike the case of a vanishing FI
$D$-term, the scalar field $R_-$ of mass $gM$ will now decay into
pairs of two lighter scalars, $R_+$ and $I_+$, of mass
$\sqrt{2}\,\kappa M$, assuming that $g \gg \kappa$. The strength of
this coupling is given by the Lagrangian
\begin{equation}
\label{Lint}
{\cal L}_{\rm int}\ =\ \frac{g^2 m^2_{\rm FI}}{8 M}\; R_-\, (R_+^2 +
I_+^2)\; .
\end{equation}
The $D$-term induced decay width of the $R_-$ particle can readily
be found to be
\begin{equation}
\label{GammaR}
\Gamma_{R_-}\ =\ \frac{g^3}{128 \pi}\, \frac{m^4_{\rm FI}}{M^3}\ ,
\end{equation}
and the same rate also holds true for the decay of $I_-$, or
equivalently for the longitudinal polarization of $V_\mu$.
Correspondingly, the decays of the $g$-sector fermions $\psi_g$ are
induced by the Lagrangian
\begin{equation}
{\cal L}_{\rm int}\ =\ -\, \frac{g}{8}\ \left(\frac{m_{\rm FI}}{M}\right)^2\,
\left(R_+ -{\rm i}I_+\right)\,
\bar\psi_g\, \frac{1-\gamma_5}{2}\, \psi_\kappa\ +\ {\rm H.c.}
\end{equation}
Neglecting soft SUSY-breaking, we find that $\Gamma_{\psi_g} =
\Gamma_{R_-} \equiv \Gamma_g$.
If the decay rate $\Gamma_g$ of the $g$-sector particles is
sufficiently low, they may dominate the energy density of the Universe
at later times, eventually leading to a second reheating phase due to
their out of equilibrium decays. To offer an initial estimate,
consider that, after the first reheating, the energy density
$\varrho_\kappa$ of the waterfall-sector fields gets distributed among
their decay products and so diluted as relativistic radiation $\propto
a^{-4}$, where $a$ is the usual cosmological scale factor describing
the expansion of the Universe. Meanwhile, the energy density
$\varrho_g$ of the ultraheavy $g$-sector particles produced via
preheating scales as $\propto a^{-3}$, such that
$\varrho_g/\varrho_\kappa\propto a$. Moreover, during a
radiation-dominated epoch, the dependence of the Hubble expansion rate
$H$ on $a$ is $H\propto a^{-2}$. Let us therefore denote with $H_{\rm
reh}$ the Hubble rate at the first reheating of the Universe and
$H_{\rm eq}$ the Hubble rate at the time, when
$\varrho_g=\varrho_\kappa$. Then, the U(1)$_X$ gauge-sector particles
will dominate the energy density of the Universe, when
\begin{equation}
\label{condition:domination}
H_{\rm eq}\ =\ H_{\rm reh}\,
\left(\frac{\varrho^0_g}{\varrho^0_\kappa}\right)^2\ \gg\ \Gamma_g\;,
\end{equation}
where the superscript $0$ stands for the energy density right after
preheating. Note that $\varrho_g/\varrho_\kappa$ is conserved until
the time of the first reheating, since both $\varrho_g$ and
$\varrho_\kappa$ scale as $a^{-3}$ during this period.
The $g$-sector particle production via preheating can be computed
numerically~\cite{PREHEATING}, by first solving for the mode functions
and then using these to calculate the Hamiltonian energy density. For
the evolution of the VEVs $\langle X_1 \rangle \approx \langle X_2
\rangle$, we assume that they initially undergo strong damping due to
tachyonic preheating~\cite{TACHYPREH}. This phenomenon can be
mimicked by setting
\begin{equation}
\label{profile}
\langle X_1 \rangle\ =\ \langle X_2 \rangle\
=\ \left\{
\begin{array}{lr}
0\,, & \textnormal{for}\quad t\leq -\pi/(4 \sqrt 2 \kappa M)\,,\\
\frac{1}2\,M\, [1+\sin(\sqrt 2\kappa M t)]\,, & \textnormal{for}\quad
-\pi/(4 \sqrt 2 \kappa M)< t < \pi/(4 \sqrt2 \kappa M)\,,\\
M\, & \textnormal{for}\quad t\geq \pi/(4 \sqrt2 \kappa M)\,,
\end{array}
\right.
\end{equation}
More precise forms of field evolutions may be obtained using numerical
simulations~\cite{TACHYPREH}. For an initial estimate, however, only
the velocity of the transition is important. In
Fig.~\ref{figure:preheating} we display the energy densities
$\varrho_F$ and $\varrho_B$ of the $g$-sector fermions $\psi_g$ and
bosons $R_-$ and $V_\mu$ (produced via preheating), normalized to the
energy density $\varrho_{\rm WF} \equiv \varrho_\kappa$ carried by the
waterfall-sector particles, as functions of the U(1)$_X$ gauge
coupling $g$, for $\kappa = 10^{-3}$. For the given
profile~(\ref{profile}) of field evolutions, these normalized energy
densities depend only very weakly on $\kappa$.
The above results strongly suggest that the U(1)$_X$ gauge-sector
particles, $\psi_g$, $R_-$ and $V_\mu$, if sufficiently long-lived,
will dominate the energy density of the early Universe. We may
estimate the second reheat temperature $T^g_{\rm reh}$ caused by their
late decays, by employing a formula very analogous to (\ref{Treh}).
Solving this last relation for the ratio $m_{\rm FI}/M$ yields
\begin{equation}
\label{ratio:mFI:M}
\frac{m_{\rm FI}}{M} \approx \ 8.4 \cdot 10^{-4}\times
\left( \frac{0.5}{g}\right)^{3/4}
\left(\frac{T^g_{\rm reh}}{10^9~{\rm GeV}}\right)^{1/2}\,
\left( \frac{10^{16}~{\rm GeV}}{M}\right)^{1/4}\; .
\end{equation}
For second reheat temperatures of cosmological interest, i.e.~$1~{\rm
TeV}\leq T^g_{\rm reh}\leq 10^9~{\rm GeV}$, we obtain the combined
constraint for $M = 10^{16}$~GeV:
\begin{equation}
\label{FIcombined}
10^{-6}\ \stackrel{<}{{}_\sim}\ \frac{m_{\rm FI}}{M}\
\stackrel{<}{{}_\sim}\ 10^{-3}\; .
\end{equation}
From our discussion in this section, it is evident that the late
decays of the ultraheavy U(1)$_X$ gauge-sector fields, which are
copiously produced during the preheating epoch, will give rise to a
second reheating phase in the evolution of the early Universe at a
temperature $T^g_{\rm reh} \ll T^\kappa_{\rm reh}$. This makes the
$F_D$-term hybrid model an interesting cosmological scenario that
could even lead to a complete relaxation of the strict
bound~(\ref{Tkappa}) on the couplings $\kappa,\ \lambda,\ \rho$. The
reason is that gravitinos, which are produced very efficiently at high
reheat temperatures $T^\kappa_{\rm reh}>10^9 {\rm GeV}$, will now be
diluted by the large entropy release from the late decays of the
$g$-sector particles. In this way, the so-called gravitino
overproduction problem can be completely avoided. A~detailed study of
this topic will be given elsewhere~\cite{GPP}.
\setcounter{equation}{0}
\section{Inflation}\label{inflation}
In this section we will discuss the additional constraints on the
theoretical parameters of the $F_D$-term hybrid model from the power
spectrum $P_{\cal R}^{1/2}$ and the spectral index $n_s$. We
distinguish two possible regimes of inflation: (i) the cold hybrid
inflation (CHI), where dissipative effects on inflation are
negligible, e.g.~for $\kappa ,\ \lambda ,\ \rho\ \stackrel{<}{{}_\sim}
10^{-2}$ and (ii) the warm hybrid inflation (WHI), where dissipation
might dominate over the expansion rate of the Universe~\cite{AB}.
\subsection{Cold Hybrid Inflation}
In models of hybrid inflation, the spectral index $n_s$ may well be
approximated as follows~\cite{review}:
\begin{equation}
\label{nS}
n_s\: -\: 1\ =\ \frac{d\ln P^{1/2}_{\cal R}}{d\ln k}\ \approx\ 2\eta\; ,
\end{equation}
where $k$ is the comoving wavenumber at the horizon exit and
\begin{equation}
\label{eta}
\eta\ =\ m^2_{\rm Pl}\ \frac{V_{\phi\phi}}{V}\
\end{equation}
is the so-called $\eta$-parameter. In~(\ref{eta}), $V$ denotes the
inflationary potential, and $V_\phi = dV/d\phi$, $V_{\phi\phi} =
d^2V/d\phi^2$ etc. The current WMAP data~\cite{WMAP} show a strong
preference for a red-tilted spectrum, with $n_s - 1 \le 0$, implying
that $V_{\phi\phi} \le 0$. The actual value is $n_s = 0.98 \pm
0.02$~\cite{Lyman}.
The $T_{\rm reh}$ constraint~(\ref{Tkappa}) on the theoretical
parameters imply that $\kappa,\ \lambda,\ \rho \stackrel{<}{{}_\sim}
10^{-5}$. In this case, the radiative correction to the potential
becomes subdominant and may be ignored to a good approximation. The
potential driving inflation simplifies considerably to
\begin{equation}
\label{VCHI}
V_{\rm inflation}\ =\ \kappa^2 M^4\: -\: \sqrt{2}\,\kappa\,a_S M^2 \phi\:
+\: \frac{1}{2}\, M^2_S\, \phi^2\: +\: \frac{\kappa^2\,M^4}{8\,m^4_{\rm
Pl}}\, \phi^4\ ,
\end{equation}
where $\phi = \sqrt{2}\, {\rm Re}\, S$ is the inflaton field
canonically normalized. For $M_S < 1$~TeV, $\kappa \ge 10^{-6}$ and $M
\ge 10^{15}$~GeV, the soft SUSY-breaking term $M_S$ can be omitted.
The inflationary potential $V_{\rm inflation}$ of (\ref{VCHI})
generically leads to a blue-tilted spectrum, i.e.~$n_s - 1 = 2 \eta >
0$, which is slightly disfavoured by the recent WMAP data.
In the following, we will concentrate on the regime where the loop
correction dominates the slope of the potential, such that a negative
value for $n_s - 1$ becomes possible. This possibility arises for
$10^{-4} \stackrel{<}{{}_\sim}\ \kappa ,\ \lambda ,\ \rho\
\stackrel{<}{{}_\sim} 10^{-2}$. Naively, such large values of the
parameters lead to a too high reheat temperature $T_{\rm reh}$,
i.e.~$T_{\rm reh} \stackrel{>}{{}_\sim} 10^{10}$~GeV. However, as we
have discussed in Section~\ref{reheat}, the presence of a subdominant
$D$-term renders the stable U(1)$_X$ gauge-sector fields unstable, and
so a large amount of entropy can be released from their late decays,
leading to a $T_{\rm reh}$ which may even be as low as 1~TeV.
Our results simplify considerably if one assumes that the slope of the
inflationary potential given in~(\ref{VpotFD}) is dominated by the
$\lambda$-dependent term. To be specific, the number of $e$-folds
${\cal N}_e$ is given by
\begin{equation}
\label{Nefold}
{\cal N}_e\ =\ \frac{1}{m^2_{\rm Pl}}\; \int_{\phi_{\rm
end}}^{\phi_{\cal N}}\, d\phi\: \frac{V}{V_\phi}\ \approx\
\frac{2\pi^2}{\lambda^2}\; \frac{\phi^2_{\cal N}}{m^2_{\rm Pl}}\ .
\end{equation}
Notice that at the horizon exit, it is $\phi_{\cal N} = \sqrt{{\cal
N}_e/2}\, (\lambda/\pi)\, m_{\rm Pl}$ and $\phi_{\cal N}
\stackrel{<}{{}_\sim} 10^{-1}\, m_{\rm Pl}$, for $\lambda
\stackrel{<}{{}_\sim} 0.1$ and ${\cal N}_e = 60$. Hence inflation
starts at values of $\phi_{\cal N}$ well below $m_{\rm Pl}$. In terms
of the number of $e$-folds ${\cal N}_e$, the power spectrum
$P^{1/2}_{\cal R}$ of the curvature perturbations may now be given by
\begin{equation}
\label{PRCHI}
P^{1/2}_{\cal R}\ =\ \frac{1}{2\sqrt{3}\, \pi m^3_{\rm Pl}}\;
\frac{V^{3/2}}{|V_\phi|}
\ \approx\ \sqrt{\frac{2{\cal N}_e}{3}}\
\frac{\kappa}{\lambda}\ \Bigg( \frac{M}{m_{\rm Pl}}\Bigg)^2\ =\
5\times 10^{-5}\ .
\end{equation}
Evidently, for ${\cal N}_e = 60$ and $M=10^{16}$~GeV, the parameter
$\lambda$ cannot be by more than one order of magnitude larger than
$\kappa$, i.e.~$\lambda \stackrel{<}{{}_\sim} 10\, \kappa$. Finally,
the spectral index $n_s$ in terms of ${\cal N}_e$ may be expressed as
follows:
\begin{equation}
\label{etaCHI}
n_s\, -\, 1\ =\ -\ \frac{1}{{\cal N}_e}\ \approx\ -\,0.02\ ,
\end{equation}
for ${\cal N}_e = 50$--60. In this CHI regime, the model predicts a
red-tilted spectrum, as currently favoured by the WMAP data.
\subsection{Warm Hybrid Inflation}
It has been extensively argued~\cite{AB} that dissipative effects due
to radiation production of massless particles during inflation may
dominate over the expansion rate $H$ of the Universe. This form of
inflation is known as warm inflation. Although a firm first principles
derivation for the existence of a strong dissipative regime of
inflation is still missing,\footnote{A detailed calculation based on a
two-particle irreducible effective action in an expanding deSitter
background metric would be highly preferable.} it might be worth
presenting tentative results for such a possible situation, using the
semi-empiric formalism on warm inflation developed in~\cite{AB}.
In the framework of WHI, dissipation occurs from the radiation
produced by the decays of the excited $H_u$ doublet of mass $\lambda
S$. Specifically, the interactions relevant to WHI are
\begin{equation}
\label{Lwarm}
-\, {\cal L}_{\rm int}^{\rm WHI}\ =\ |S|^2\, \bigg[\, |\lambda |^2\,
|H_u|^2\: +\: |\rho |^2\, \bigg(\,\sum_{i=1}^3\,
|\widetilde{N}_i|^2\bigg)\,\bigg]\ +\ \Big( h_t\, H_u\, \bar{Q}_t\,
t_R\: +\: h^{\nu}_{ij}\, \bar{L}_i \tilde{h}_u
\widetilde{N}_j\: +\: {\rm H.c.}\Big)\; .
\end{equation}
The dominant decay mode will be $H_u \to Q_t t_R$~\cite{BB}; the other
possible decay channel $\widetilde{N}_j \to L_i \tilde{h}_u$ is
Yukawa-coupling suppressed and kinematically allowed only when $\rho >
\lambda$. Adapting the results of~\cite{AB,BB} to our model, the
dominant friction term for $|S| \gg M$ is given by
\begin{equation}
\label{Ys}
Y_S\ \approx\ \frac{\sqrt{\pi}\, \alpha_\lambda^{3/2}\,
\alpha_t}{20\,\sqrt{2}}\ \phi\; ,
\end{equation}
where $\alpha_\lambda = \lambda^2/(4\pi)$ and $\alpha_t =
h^2_t/(4\pi)$. The dynamics of warm inflation is governed by the
following two equations:
\begin{eqnarray}
\label{phiS}
\ddot{\phi}\ +\ 3H\, ( 1 + r )\,
\dot{\phi}\ +\ V_\phi & = & 0\; ,\\
\label{rhorad}
\dot{\rho}_{\rm rad}\ +\ 4\, H\rho_{\rm rad} & = & Y_S\, \dot{\phi}^2\ ,
\end{eqnarray}
where $r = Y_S/(3H)$, with $H^2 \approx \kappa^2 M^4/(3 m^2_{\rm
Pl})$. In the strong dissipative regime where $r \gg 1$, inflation
usually ends when $\rho_{\rm rad} > \rho_{\rm vac} \approx \kappa^2
M^4$.
Assuming conditions of slow roll during WHI, i.e.~$\eta /r^2 \ll 1$,
we may determine the number of $e$-folds by
\begin{equation}
\label{NeWHI}
{\cal N}_e \ =\ \frac{1}{m^2_{\rm Pl}}\; \int^{\phi_{\cal
N}}_{\phi_{\rm end}}\, d\phi\ \frac{(1+r)\,V }{V_\phi}\ =\
\frac{\pi \alpha_\lambda^{1/2}\, \alpha_t}{60\, \kappa}\ \frac{
\phi^3_{\cal N}}{ m_{\rm Pl}\, M^2}\ .
\end{equation}
In the limit $r\gg 1$, the power spectrum $P_{\cal R}^{1/2}$ due to
WHI is approximately given by
\begin{equation}
\label{PRWHI}
(P_{\cal R}^{1/2})_{\rm WHI}\ \approx\ \left( \frac{3\pi}{4} \right)^{1/4}\,
\sqrt{\frac{T_{\rm rad}}{H}}\;
r^{5/4}\, (P_{\cal R}^{1/2})_{\rm CHI}\ .
\end{equation}
The temperature $T_{\rm rad}$ associated with radiation production can be
calculated from (\ref{rhorad}), by solving the approximate equation
\begin{equation}
\label{Trad}
\rho_{\rm rad}\ =\ \frac{\pi^2}{30}\ g_*\ T^4_{\rm rad}\ \approx\
\frac{3r}{4}\ \dot{\phi}^2\; ,
\end{equation}
where $\dot{\phi} \approx - V_\phi/(3r H)$ is evaluated at the horizon
exit. Putting everything together, we find
\begin{equation}
\label{PRWHI2}
(P_{\cal R}^{1/2})_{\rm WHI}\ \approx\ g_*^{-1/8}\, {\cal N}_e^{5/8}\,
(2\kappa)^{1/4}\, \alpha_\lambda^{5/8}\, \alpha_t^{1/2}\,
\left(\frac{M}{m_{\rm Pl}}\right)^{1/2}\ =\ 5 \times 10^{-5}\ .
\end{equation}
It is interesting to observe that WHI leads to a viable inflationary
scenario even for strong couplings, e.g.~for $\kappa ,\
\alpha_\lambda,\ \alpha_t\ \sim~1$. In this case, the
U(1)$_X$-breaking scale $M$ will be as low as $10^{10}$~GeV, in
agreement with the earlier discussion in \cite{BB}. Obviously, it
would be difficult to associate such a low scale for $M$ with gauge
coupling unification. Finally, the spectral index $n_s$ in WHI is
calculated in terms of ${\cal N}_e$ to be: $n_s - 1 \approx - 5/(4
{\cal N}_e) \approx -0.025$.
\setcounter{equation}{0}
\section{Baryon Asymmetry in the Universe}\label{BAU}
As discussed in Section~\ref{reheat}, the late decays of the U(1)$_X$
gauge-sector particles may lead to a second reheating phase in the
evolution of the early Universe, giving rise to a very low final
reheat temperature~$T_{\rm reh}$. Depending on the size of the
$D$-term, $T_{\rm reh}$ may even be as low as 1~TeV. In such a case,
the BAU may be explained by thermal electroweak-scale resonant
leptogenesis~\cite{APRD,PU2}. The $F_D$-term hybrid model under study
can realize such a scenario even within a minimal SUGRA framework,
where all soft SUSY-breaking parameters are constrained at the
gauge-coupling unification point $M_X$, which can be chosen to be $M =
M_X \approx 10^{16}$~GeV. Instead, electroweak baryogenesis is no
longer viable in minimal SUGRA, since it requires an unconventionally
large hierarchy between the left-handed and right-handed top
squarks~\cite{EWBAU}.
An advantageous feature of resonant leptogenesis is that the
predictions for the BAU are almost independent of any pre-existing
lepton- or baryon-number abundance. This kind of fixed-line attractor
behaviour is a consequence of the quasi-in-thermal equilibrium
dynamics governing the heavy Majorana neutrino sector. It results from
the fact that the heavy neutrino decay widths can be several orders of
magnitude larger than the expansion rate $H$ of the Universe. A
detailed analysis of this dynamics was presented in \cite{PU2}, where
single lepton-flavour and freeze-out sphaleron effects were
systematically considered for the {\em first time}. In particular, it
was shown that single lepton-flavour effects resulting from the
Yukawa-neutrino couplings $h^{\nu}_{ij}$ can have a dramatic impact on
the predictions for the BAU, enhancing its value by many orders of
magnitude. From the model-building point of view, phenomenologically
rich scenarios are now possible with testable implications for
high-energy colliders~\cite{prodN} and low-energy observables, such as
$\mu \to e\gamma$, $\mu \to eee$ and $\mu \to e$ conversion in
nuclei~\cite{LFVN}.
We will not reiterate all these results here, but only underline some
of the key model-building aspects related to the neutrino sector of
the $F_D$-term hybrid model. The $F_D$-term hybrid model contains a
$3\times 3$ Majorana mass matrix $M_S$, which is SO(3) symmetric at
the gauge-coupling unification point $M_X = M \approx 10^{16}$~GeV,
i.e.\ $M_S = m_N {\bf 1}_3$. The parameter $m_N = \rho v_S$ is a
universal Majorana mass whose natural value is of the order of the
soft SUSY-breaking or the electroweak scale, i.e.~$m_N \sim M_{\rm
SUSY}$ or $m_t$. The SO(3) symmetry of the heavy neutrino sector is
broken explicitly by the Yukawa neutrino couplings $h^{\nu}_{ij}$. In
order to explain the low-energy light neutrino data, the breaking of
the SO(3) symmetry should proceed via an intermediate step, namely
SO(3) should first break into its subgroup SO(2) $\simeq$ U(1)$_l$.
This can be achieved by coupling all lepton doublets $L_{e,\mu,\tau}$
to the linear combination: $\frac{1}{\sqrt{2}}\, (\nu_{2R} + i
\nu_{3R})$. These Yukawa couplings could be as large as the
$\tau$-Yukawa coupling $h_\tau$, i.e.~$h^{\nu}_{i2} = i h^{\nu}_{i3}
\sim 10^{-2}$. As a consequence of the U(1)$_l$ symmetry, the
resulting light neutrino mass matrix ${\bf m}^\nu$ vanishes
identically to all orders in perturbation theory. The remaining
U(1)$_l$ symmetry can be broken by smaller Yukawa couplings of the
order of the electron Yukawa coupling $h_e$, i.e.~$h^{\nu}_{i1} =
\varepsilon_i \sim 10^{-6}$--$10^{-7}$, which arise when one couples
$L_{e,\mu ,\tau}$ to $\nu_{1R}$~\cite{PUcomment}.
Further breaking of the U(1)$_l$ symmetry is induced in the
heavy-neutrino sector by renormalization-group and threshold effects
while running $M_S$ from $M$ to $m_t$~\cite{Branco}. Thus, $M_S$ will
generically modify to: $M_S = m_N {\bf 1}_3 + \Delta M_S$, where one
typically has $(\Delta M_S)_{ij}/m_N \sim 10^{-5}$--$10^{-7}$. Taking
the effect of U(1)$_l$-breaking parameters $(\Delta M_S)_{ij}$ and
$\varepsilon_i$ into account, one obtains a light neutrino mass matrix
which can comfortably accommodate the low-energy light neutrino data,
e.g.~with an inverted hierarchical light neutrino spectrum~\cite{PU2}.
On the other hand, the heavy neutrino sector of the $F_D$-term hybrid
model consists of 3 nearly degenerate heavy Majorana neutrinos
$N_{1,2,3}$ of mass $m_{N_{1,2,3}} \approx m_N$, which can give rise
to successful baryogenesis through thermal electroweak-scale resonant
leptogenesis~\cite{PUcomment}.
\setcounter{equation}{0}
\section{Conclusions}\label{conclusions}
We have studied $F$-term hybrid inflation in a novel supersymmetric
extension of the SM, to which a subdominant FI $D$-term was added. We
called this particular form of inflation $F_D$-term hybrid inflation.
The $F_D$-term hybrid model we have been analyzing in this paper ties
the $\mu$-parameter of the MSSM to an SO(3) symmetric Majorana mass
$m_N$, through the VEV of the inflaton field. As a consequence, the
model predicts {\em naturally} lepton-number violation at the
electroweak scale.
In order to obtain predictions for the observables $P_{\cal R}^{1/2}$,
$n_s$ and $\eta_B$ compatible with global cosmological
analyses~\cite{Lyman}, as well as interesting particle-physics
phenomenology that could be tested in laboratory experiments, one
needs to make certain assumptions for the model of $F_D$-term hybrid
inflation:
\begin{itemize}
\item[ (i)] Successful hybrid inflation relies on the assumption that
the inflaton field is displaced from its minimum in the beginning of
inflation, whereas all other non-inflaton fields have zero VEVs,
according to (\ref{initial}).
\item[ (ii)] The present $F_D$-term hybrid scenario utilizes a minimal
K\"ahler potential, where terms of order $H^2 |S|^2$ in the
potential are set to zero or assumed to be negligible. This
consideration introduces some tuning in general SUGRA models with
non-minimal K\"ahler potentials.
\item[(iii)] In order to get a red-tilted spectrum with negative $n_s
- 1$, one has to assume that the radiative corrections dominate the
slope of the inflationary potential. This possibility arises for
superpotential couplings: $10^{-4} \stackrel{<}{{}_\sim} \kappa,\
\lambda,\ \rho \stackrel{<}{{}_\sim} 10^{-2}$.
\item[ (iv)] Even though a bare $D$-tadpole may be present as a bare
parameter in the tree-level Lagrangian, we have considered here,
however, the possibility that such a term is generated radiatively
after heavy degrees of freedom have been integrated out. These
heavy degrees of freedom are assumed to be Planck-mass chiral
superfields which are oppositely charged under the U(1)$_X$ and
which break explicitly the discrete charge symmetry discussed after
(\ref{VFD}) and in Appendix~A.
\item[ (v)] We have assumed that the coupling $\rho$ of the inflaton
to neutrino superfields is SO(3) symmetric or very close to it.
After the inflaton receives a VEV, one ends up with 3 nearly
degenerate heavy Majorana neutrinos with masses at the electroweak
scale. This enables one to successfully address the BAU within the
thermal electroweak-scale resonant leptogenesis framework (see our
discussion in Section~\ref{BAU}). As has also been discussed in
Section~\ref{BAU}, if one assumes that the neutrino-Yukawa couplings
$h^\nu_{ij}$ have a certain hierarchical structure controlled by the
approximate breaking of global flavour symmetries, the model can
have further testable implications for $e^+e^-$ colliders and
low-energy experiments of lepton flavour and/or number violation.
\end{itemize}
The requirement for a sufficiently low reheat temperature $T_{\rm reh}
\stackrel{<}{{}_\sim} 10^9$~GeV, which does not lead to overproduction
of gravitinos, provides an important constraint on the basic
theoretical parameters $\kappa$, $\lambda$ and $\rho$. The naive
limits on these couplings derived from reheating due to perturbative
inflaton decay are very strict, i.e.~$\kappa,\ \lambda,\ \rho
\stackrel{<}{{}_\sim} 10^{-5}$. These limits may be completely avoided
by considering the late decays of the U(1)$_X$ gauge-sector particles
which are induced by a non-vanishing FI $D$-term $m^2_{\rm FI}$.
Their decay rates depend crucially on~$m^2_{\rm FI}$. As menioned
above in point~(iv) and in Appendix~A, the generation of a FI
$D$-tadpole and its size may be engineered by adding Planck-scale
heavy degrees of freedom to the theory and by subjecting these into
extended $R$ symmetries. In this way, a phase of second reheating
takes place in the evolution of the early Universe, which can lead to
a significant lowering of the reheat temperature even up to 1~TeV.
The $F_D$-term hybrid model with electroweak-scale lepton number
violation can easily be embedded within a minimal SUGRA theory, where
all soft SUSY-breaking parameters are constrained at the gauge
coupling unification point $M_X$ which can be chosen to be $M \approx
10^{16}$~GeV. Instead, electroweak baryogenesis is not viable in a
minimal SUGRA scenario of the MSSM. Moreover, the CP-odd soft
SUSY-breaking phases required for successful electroweak baryogenesis
face severe constraints from the non-observation of the electron and
neutron electric dipole moments, even though the latter arise
diagrammatically at the 2-loop level~\cite{CKP}.
The $F_D$-term hybrid model under discussion conserves $R$-parity. The
reason is that all superpotential couplings either conserve the $B-L$
number or break it by even number of units. Specifically, the
operator $\widehat{S}\widehat{N}_i\widehat{N}_i$ breaks explicitly
$L$, as well as $B-L$, by 2~units. Consequently, the lightest
supersymmetric particle (LSP) of the spectrum is expected to be
stable, thus providing a viable candidate to address the so-called
Cold Dark Matter (CDM) problem. The new aspect of our model is that
right-handed sneutrinos could be the LSPs, opening up new
possibilities in the phenomenology of CDM and its detection.
From the particle-physics point of view and in the low-energy limit
where the waterfall sector has decoupled and the $\rho$-coupling
neglected for simplicity, the $F_D$-term hybrid model becomes
identical to the so-called Minimal Nonminimal Supersymmetric Standard
Model (MNSSM) in the decoupling limit of a large tadpole~\cite{PP}.
In particular, in the framework of WHI discussed in
Section~\ref{inflation}, the coupling $\lambda$ can be sizeable,
i.e.~$\lambda \sim 0.6$. In this case, the Higgs phenomenology of the
MSSM will modify drastically, despite the decoupling of the singlet
Higgs states. One striking possibility in the MNSSM is that the
charged Higgs boson $H^+$ could be lighter than the SM-like Higgs
boson~\cite{PP2}, thus pointing to particular collider
phenomenologies~\cite{DP}. However, even within the traditional
scenario of CHI, where $\kappa, \lambda \stackrel{<}{{}_\sim}
10^{-2}$, the $F_D$-term hybrid model will favour particular benchmark
scenarios of the MSSM. For example, if $\lambda \gg \kappa$, the
$F_D$-term hybrid model may account for a possible large value of the
$\mu$-parameter. Specifically, if $\lambda = 4 \kappa$, one gets
from~(\ref{Send}) the hierarchy $\mu \approx 4 M_{\rm SUSY}$, which is
the so-called CPX benchmark scenario~\cite{CPX} describing maximal CP
violation in the MSSM Higgs sector at low and moderate values of $\tan
\beta$.
A possible natural solution to the famous cosmological constant
problem is expected to provide further constraints on the model
building of cosmologically viable models in future. Nevertheless, the
$F_D$-term hybrid model presented in this paper constitutes a first
attempt towards the formulation of a minimal Particle-Physics and
Cosmology Standard Model, whose validity could, in principle, be
tested in laboratory experiments and further vindicated by
astronomical observations.
\bigskip\bigskip
\subsection*{Acknowledgements}
We thank Arjun Berera, Rudnei Ramos and Antonio Riotto for useful
discussions. We also thank Constantine Pallis for collaboration in the
early stages of this project. AP~dedicates this work to the memory of
Darwin Chang, an invaluable friend and collaborator. This work is
supported in part by the PPARC research grants: PPA/G/O/2002/00471 and
PP/C504286/1.
\newpage
\def\theequation{\Alph{section}.\arabic{equation}}
\begin{appendix}
\setcounter{equation}{0}
\section{{\boldmath $D$}--Term Engineering}\label{Dappendix}
The generation and the size of a $D$-term may be engineered by adding
Planck-scale heavy degrees of freedom to the theory and by subjecting
these into extended $R$ symmetries.
To elucidate our point, let us first consider a model augmented by a
pair of oppositely charged superfields $\widehat{\overline{X}}_{1,2}$,
with U(1)$_X$ charges: $Q(\widehat{\overline{X}}_2) = -
Q(\widehat{\overline{X}}_1 ) = Q(\widehat{X}_1) = - Q(\widehat{X}_2 )
= 1$. The extended superpotential $W$ of our interest is
\begin{equation}
\label{Wdterm}
W \ =\ \kappa\, \widehat{S}\, \Big( \widehat{X}_1
\widehat{X}_2\: -\: M^2\Big)\ +\ \xi\, m_{\rm Pl}\,
\widehat{\overline{X}}_1\,\widehat{\overline{X}}_2\ +\
\xi_1\, \frac{ ( \widehat{\overline{X}}_1\widehat{X}_1 )^2}{2\, m_{\rm
Pl}}\ +\ \xi'_1\,
\frac{ ( \widehat{\overline{X}}_2\widehat{X}_2 )^2}{2\, m_{\rm Pl}}\ .
\end{equation}
This form of the superpotential may be enforced by the $R$ symmetry:
$\widehat{S} \to e^{i\alpha}\, \widehat{S}$, $\widehat{X}_{1,2} \to
e^{\pm i\beta}\,\widehat{X}_{1,2}$, $\widehat{\overline{X}}_{1,2} \to
e^{i(\frac{a}{2} \mp \beta)}\, \widehat{\overline{X}}_{1,2}$,
$\widehat{L} \to e^{i\alpha}\, \widehat{L}$, $\widehat{Q} \to
e^{i\alpha}\, \widehat{Q}$, with $W \to e^{i\alpha} W$. As before,
all remaining fields are considered to be neutral under the $R$
symmetry. Notice that the same $R$-symmetry allows for the operator
$\kappa' S (\widehat{X}_1 \widehat{X}_2 )^2/m^2_{\rm Pl}$. The
presence of this superpotential term can trigger shifted hybrid
inflation, where the gauge symmetry U(1)$_X$ is broken along the
inflationary trajectory, thereby inflating away unwanted topological
defects~\cite{JKLS}.
A $D$-term will now be generated after integrating out the
Planck-scale superfields $\widehat{\overline{X}}_{1,2}$. The
loop-induced $D$-tadpole $m^2_{\rm FI}$ is found to be
\begin{equation}
\label{FIdterm}
m^2_{\rm FI}\ \approx\ \frac{\xi^2_1 - \xi'^2_1}{8\pi^2}\
\frac{M^4}{m^2_{\rm Pl}}\ \ln\left(\frac{m_{\rm Pl}}{M}\right)\ .
\end{equation}
For $M = 10^{16}$~GeV, we find that $m_{\rm FI}/M
\stackrel{<}{{}_\sim} 10^{-3}$, for $\xi_1,\ \xi'_1
\stackrel{<}{{}_\sim} 0.3$. Observe that if $\xi_1 = \xi'_1$, the
discrete charge symmetry discussed after (\ref{VFD}) gets restored
again and $m_{\rm FI}$ vanishes identically.
The size of the $D$-term may be suppressed further, if the Planck-mass
chiral superfields $\widehat{\overline{X}}_{1,2}$ possess higher
U(1)$_X$ charges. In general, one may assume that the U(1)$_X$
charges of $\widehat{\overline{X}}_{1,2}$ are:
$Q(\widehat{\overline{X}}_2) = - Q(\widehat{\overline{X}}_1 ) = n$,
where $n\ge 1$. In addition, we require for
$\widehat{\overline{X}}_{1,2}$ to transform under U(1)$_R$ as follows:
\begin{equation}
\label{Rsymn}
\widehat{\overline{X}}_{1,2}\ \to\ e^{\frac{i}2\, [a \, \mp\, (n+1)
\beta ]}\; \widehat{\overline{X}}_{1,2}\; ,
\end{equation}
while $\widehat{S}$, $\widehat{X}_{1,2}$ and all other fields
transform as before. With this symmetry restriction, the
superpotential reads:
\begin{equation}
\label{Wdtermn}
W \ =\ \kappa\, \widehat{S}\, \Big( \widehat{X}_1 \widehat{X}_2\: -\:
M^2\Big)\ +\ \xi\, m_{\rm Pl}\,
\widehat{\overline{X}}_1\,\widehat{\overline{X}}_2\ +\ \xi_n\, \frac{
(\widehat{\overline{X}}_1)^2\, (\widehat{X}_1)^{n+1}}{2\,m^n_{\rm Pl}}\ +\
\xi'_n\, \frac{ (
\widehat{\overline{X}}_2)^2\,(\widehat{X}_2)^{n+1}}{2\,m^n_{\rm Pl}}\ .
\end{equation}
In this case, the loop-induced $D$-term is given by
\begin{equation}
\label{FIdtermn}
m^2_{\rm FI}\ \approx\ \frac{\xi^2_n - \xi'^2_n }{8\pi^2}\
\frac{M^{2(n+1)}}{m^{2n}_{\rm Pl}}\ \ln\left(\frac{m_{\rm Pl}}{M}\right)\ .
\end{equation}
To obtain a small ratio $m_{\rm FI}/M \sim 10^{-6}$, with $\xi_n,\
\xi'_n \sim 1$, one would need $n = 5,\ 6$. Finally, it is important
to remark that the loop-induced $D$-term does not lead to spontaneous
breakdown of global supersymmetry.
\end{appendix}
\newpage
|
Title:
SDSSJ212531.92-010745.9 - the first definite PG1159 close binary system |
Abstract: The archival spectrum of SDSSJ212531.92-010745.9 shows not only the typical
signature of a PG1159 star, but also indicates the presence of a companion. Our
aim was the proof of the binary nature ofthis object and the determination of
its orbital period.We performed time-series photometry of
SDSSJ212531.92-010745.9. We observed the object during 10 nights, spread over
one month, with the Tuebingen 80cm and the Goettingen 50cm telescopes. We
fitted the observed light curve with a sine and simulated the light curve of
this system with the nightfall program. Furthermore, we compared the spectrum
of SDSSJ212531.92-010745.9 with NLTE models, the results of which also
constrain the light curve solution. An orbital period of 6.95616(33)h with an
amplitude of 0.354(3)mag is derived from our observations. A pulsation period
could not be detected. For the PG1159 star we found, as preliminary results
from comparison with our NLTE models, Teff about 90000K, log g about 7.60, and
the abundance ratio C/He = 0.05 by number fraction. For the companion we
obtained with a mean radius of 0.4 +/- 0.1 Rsol, a mass of 0.4 +/- 0.1 Msol,
and a temperature of 8200K on the irradiated side, good agreement between the
observed light curve and the nightfall simulation, but we do not regard those
values as final.
| https://export.arxiv.org/pdf/astro-ph/0601512 |
\title{SDSS\,J212531.92$-$010745.9 - the first definite PG1159 close binary system}
\author{T. Nagel\inst{1} \and S. Schuh\inst{2} \and D.-J. Kusterer\inst{1} \and
T. Stahn\inst{2} \and S.D. H\"ugelmeyer\inst{2} \and
S. Dreizler\inst{2} \and B.T. G\"ansicke\inst{3} \and M.R. Schreiber\inst{4} }
\offprints{T. Nagel}
\mail{nagel@astro.uni-tuebingen.de}
\institute{Institut f\"ur Astronomie und Astrophysik, Eberhard-Karls-Universit\"at T\"ubingen,
Sand 1, 72076 T\"ubingen, Germany
\and
Institut f\"ur Astrophysik, Georg-August-Universit\"at
G\"ottingen, Friedrich-Hund-Platz 1, 37077 G\"ottingen, Germany
\and
Department of Physics, University of Warwick, Coventry, CV4
7AL, Great Britain
\and
Departamento de Fisica y Meteorologia, Facultad de
Ciencias, Universidad de Valparaiso, Valparaiso, Chile
}
\date{Received xx.xx.xx / Accepted xx.xx.xx}
\abstract
{}
{The archival spectrum of SDSS\,J212531.92$-$010745.9 shows not
only the typical signature of a PG\,1159 star, but also indicates the
presence of a companion. Our aim was the proof of the binary nature of
this object and the determination of its orbital period.}
{We performed time-series photometry of SDSS\,J212531.92$-$010745.9. We
observed the object during 10 nights, spread over one month, with the
T\"ubingen 80\,cm and the G\"ottingen 50\,cm telescopes. We fitted the
observed light curve with a sine and simulated the light curve of this
system with the \texttt{nightfall} program. Furthermore, we compared the
spectrum of SDSS\,J212531.92$-$010745.9 with NLTE models, the results of which
also constrain the light curve solution. }
{An orbital period of 6.95616(33)\,h with an amplitude of 0.354(3)\,mag is derived
from our observations. A pulsation period could not be detected. For the
PG\,1159 star we found, as preliminary results from comparison with our NLTE
models, $T_{\rm eff}$\,$\sim$\,90\,000\,K, $\log g$\,$\sim$\,7.60, and the abundance ratio
C/He\,$\sim$\,0.05 by number fraction. For the companion we obtained with a mean radius of
$0.4\pm 0.1\,\rm R_\odot$, a mass of $0.4\pm 0.1\,\rm M_\odot$, and a
temperature of 8\,200\,K on the irradiated side, good agreement between
the observed light curve and the \texttt{nightfall} simulation, but we
do not regard those values as final. }
{}
\keywords{stars: AGB and post-AGB -- white dwarfs -- binaries: close }
\titlerunning{The first definite PG1159 close binary system}
\authorrunning{T. Nagel at al.}
\section{Introduction}
PG\,1159 stars are hot hydrogen-deficient (pre-)white dwarfs with effective
temperatures between 75\,000 and 200\,000\,K, and $\log g$\,=\,5.5--8.0 (Werner 2001).
They are in the transition between the asymptotic giant branch (AGB) and cooling white dwarfs.
Spectra of PG\,1159 stars are dominated by absorption lines of He\,{\sc ii}, C\,{\sc iv}
and O\,{\sc vi}.
Current theory suggests (e.g. Werner 2001) that they are the outcome of a late helium-shell
flash, a phenomenon that drives the currently observed fast evolutionary
rates of three well-known objects (FG~Sge, Sakurai's object, V605
Aql). Flash-induced envelope mixing produces a H-deficient stellar
surface. The photospheric composition then essentially reflects that of the
region between the H- and He-burning shells in the precursor AGB star. The
He-shell flash forces the star back onto the AGB. The subsequent, second
post-AGB evolution explains the existence of Wolf-Rayet central stars of
planetary nebulae and their successors, the PG\,1159 stars.
Currently, 37 PG\,1159 stars are known. Figure\,\ref{pg1159stars} shows their position in a
log\,$T_{\rm eff}$-$\log g$-diagram. Two of them have been found to be
binary stars. These are NGC\,246 (e.g. Bond \& Ciardullo 1999), which is a
resolved visual binary, and PG\,2131+066 (Wesemael \etal 1985). Concerning
the latter, it is still unclear whether it is a close binary (Paunzen \etal
1998) or a resolved visual binary with an M2V star as companion (Reed \etal 2000).
\section{The spectrum of SDSS\,J212531.92$-$010745.9}
The spectrum of SDSS\,J212531.92$-$010745.9 (u=17.15, g=17.54, r=17.75,
i=17.79, z=17.83), taken on Sept. 6th 2002, is from the Sloan Digital
Sky Survey (SDSS) archive Data Release (DR) 4. The spectrum shows significant features
that are typical for PG\,1159 stars, for example the strong C\,{\sc iv}
absorption lines at $4650-4700\AA$ and He\,{\sc ii} at $4686\AA$ (Fig.\,\ref{spectrum}).
Furthermore, the spectrum shows features which indicate the presence of a
companion. The Balmer series of hydrogen is seen in emission, H$_\alpha$ -
H$_\delta$ can clearly be identified. This is probably due to a cool
companion which is heated up by irradiation from the hydrogen-deficient PG\,1159 star.
Figure\,\ref{spectrum} shows the observed spectrum ($t_{\rm exp}=3703\,\rm s$) of
SDSS\,J212531.92$-$010745.9. Overlayed are a PG\,1159 NLTE model spectrum
with $T_{\rm eff}$\,=\,90\,000\,K, $\log g$\,=\,7.60, C/He\,=\,0.05, and N/He\,=\,0.01, a
blackbody model spectrum with $T$\,=\,8\,200\,K for the irradiated
companion, and the sum of the two model spectra. The parameters of both
stellar components are estimates obtained from a qualitative comparison of
our NLTE models to the single SDSS spectrum. Detailed parameters for both stars
need to be derived from a full two-component analysis of orbital
phase resolved spectroscopy. The effective temperature in particular may be
lower or higher by 20\,000\,K. The surface temperature of the companion's irradiated
side was also constrained with \texttt{nightfall} simulations, see below.
The overall shape of the observed spectrum is well fitted with the
combination of a PG\,1159 star and a cool, irradiated companion, but
especially the C\,{\sc iv} spectral lines of the PG\,1159 model atmosphere
are not strong enough. There is another PG\,1159 star showing this
phenomenon (H\"ugelmeyer \etal, in prep.), and also none of the deep
absorption lines which some DO white dwarfs show can be fitted (e.g. Werner \etal 1995).
The spectral signatures of an A star, as one would expect for the companion with
8\,200\,K surface temperature at the irradiated side, cannot be seen in
the observation. This may be because the irradiation from the PG\,1159
leads to a temperature inversion in the upper layers of the companion's
atmosphere up to $\tau_{\rm Ross}=1$, which causes the observed emission line spectrum
(Barman \etal 2004).
\section{Photometry of SDSS\,J212531.92$-$010745.9}
\begin{table}
\centering
\caption{Observation log. All observations are performed with clear filter.}
\begin{tabular}{rrrcc}
\hline
\hline
\noalign{\smallskip}
Date & t$_{\rm exp}[s]$& t$_{\rm cycle}[s]$ & Duration$[s]$ & Telescope\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
2005/09/21 & 90 & 98 & 18900 & 80\,cm\\
2005/09/22 & 90 & 98 & 18899 & 80\,cm\\
2005/09/23 & 90 & 98 & 21758 & 80\,cm\\
2005/09/23 & 180 & 194& 14873 & 50\,cm\\
2005/10/06 & 240 & 248& 10202 & 50\,cm\\
2005/10/07 & 240 & 246& 14897 & 50\,cm\\
2005/10/08 & 240 & 248& 9298 & 50\,cm\\
2005/10/10 & 90 & 98 & 19852 & 80\,cm\\
2005/10/11 & 240 & 248& 17872 & 50\,cm\\
2005/10/18 & 90 & 98 & 16532 & 80\,cm\\
2005/10/26 & 90 & 98 & 20095 & 80\,cm\\
\noalign{\smallskip}
\hline
\end{tabular}\label{tab_obs}
\end{table}
Photometric observations of SDSS\,J212531.92$-$010745.9 were
performed during 10 nights (Tab.\,\ref{tab_obs}) using the T\"ubingen
80\,cm f/8 telescope with an \mbox{SBIG ST-7E} CCD camera and the G\"ottingen
50\,cm f/10 telescope with an \mbox{SBIG STL-6303E} CCD camera. To achieve
good time resolution we chose clear filter exposures with a binning of 2x2
pixels to reduce readout time. The exposure time was t$_{\rm exp}$=90\,s
for the observations with the 80\,cm telescope. In the case of the 50\,cm
telescope, the exposure time was t$_{\rm exp}$=180\,s and t$_{\rm
exp}$=240\,s. The observing conditions were good during the nights,
considering that the telescopes are located in the cities of T\"ubingen and
G\"ottingen.
All images were bias and dark current corrected, then aperture
photometry was performed using our IDL software TRIPP (Time Resolved
Imaging Photometry Package, Schuh \etal 2003). The relative flux of the
object was calculated with respect to the same two comparison stars (SDSS
J212530.60-010921.0 and SDSS J212528.83-010828.5) for all nights, which
were tested for stability. The resulting light curve is displayed in
Fig.\,\ref{lightcurve}.
To analyse the combined light curve of all nights, we used CAFE (Common
Astronomical Fit Environment, G\"ohler, priv. comm.), a collection of
routines written in IDL. The brightness variation is probably caused by a
reflection effect. The companion is, due to the small separation, heated up
on one side by irradiation from the PG\,1159 star, and the orbital motion
then leads to a variable light curve.
We fitted the combined light curve of all nights with a sine, achieving
best results for a period of 6.95616(33)\,h (Fig.\,\ref{lightcurve}). The
observed variability has a mean amplitude of 0.354(3)\,mag.
To check if the observed light curve can be explained by a PG\,1159 star
and an irradiated companion and for an impression of what the system
geometry might look like we simulated the light curve of the binary system
for an orbital period of 6.95616\,h with the program \texttt{nightfall}.
Figure\,\ref{nightfall} shows the simulated and observed light curves of
all nights, folded onto the orbital period. For the PG\,1159 star we assumed
$T_{\rm eff}$\,=\,90\,000\,K, a mass of 0.6\,$\rm M_\odot$ and a radius of
0.1\,$\rm R_\odot$.
For the companion we varied the mass from 0.1\,$\rm M_\odot$ to 0.7\,$\rm
M_\odot$. We found that the observed light curve can be reproduced best
with an M dwarf with an effective temperature of $3\,500\pm 150\,\rm K$,
a mean radius of $0.4\pm 0.1\,\rm R_\odot$ and a mass of about $0.4\pm
0.1\,\rm M_\odot$. For the inclination of this system we obtained
$70\pm 5\,^\circ$.
Due to the irradiation by the PG\,1159 star the surface of the companion
would be heated up to a surface temperature of 8\,200\,K, which, in
combination with the PG\,1159 star, reproduces the overall shape of the
observed spectrum quite well, as can be seen in Fig.\,\ref{spectrum}. The
broad dip at the minimum of the light curve is well reproduced by this
system configuration, too.
In Table \ref{paras} we list all stellar and system parameters assumed and
derived.
We found that ellipsoidal variation due to geometrical deformation of
the stars cannot generate the observed light curve. In the above
configuration, calculated by nightfall according to the geometry in
Djurasevic 1992, the equatorial radius of the M dwarf is only 4.5\,\%
larger than its polar radius, and the PG\,1159 star is not affected by
deformation above the numerical limit of \texttt{nightfall}.
Because the object is positioned in the GW\,Vir instability strip
(Fig.\,\ref{pg1159stars}), we also looked for pulsation periods below two
hours in the light curve of SDSS\,J212531.92$-$010745.9. Therefore, we
calculated a Lomb-Scargle periodogram (Scargle 1982) for the night with the
best S/N (2005/10/26). But observational noise precludes detection of any
periodicity with amplitudes below about 50\,mmag. The amplitudes of
pulsating PG\,1159 stars normally are of the order of a few percent of a
magnitude, and SDSS\,J212531.92$-$010745.9 is fainter than HE1429-1209, for
which we recently discovered pulsation with the T\"ubingen 80\,cm telescope
(Nagel \& Werner 2004). Our 80\,cm telescope might therefore just be too
small to detect pulsation below 50\,mmag in this case.
\begin{table}
\centering
\caption{Stellar and system parameters of SDSS\,J212531.92$-$010745.9,
assumed (normal font) or derived from comparison with NLTE model spectra (boldface),
photometric analysis (*) and nightfall simulation (italic).}
\begin{tabular}{rrrr}
\hline
\hline
\noalign{\smallskip}
Parameter & PG\,1159 & Companion & System \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
$T_{\rm eff}\,\,[\rm K]$ & {\bf $\sim$90\,000} & 3\,500$\pm$150 & \\
$T_{\rm eff,irr} \,\, [\rm K]$ & & {\it 8\,200} & \\
$\log g \,\,[\rm cm/s^2]$ & {\bf $\sim$7.6} & & \\
$m \,\,[\rm M_\odot]$ & 0.6 & {\it 0.4$\pm$0.1} & 1.0$\pm$0.1 \\
$r \,\,[\rm R_\odot]$ & 0.1 & {\it 0.4$\pm$0.1} & \\
$P_{\rm orb} \,\,[\rm h]$ & & & 6.95616(33) *\\
$\Delta m \,\,[\rm mag]$ & & & 0.354(3) * \\
$a \,\,[\rm R_\odot]$ & & & 1.85\\
$i \,\,[^\circ]$ & & & {\it 70$\pm$5}\\
\noalign{\smallskip}
\hline
\end{tabular}\label{paras}
\end{table}
\section{Conclusions}
\begin{enumerate}
\item The spectrum of SDSS\,J212531.92$-$010745.9 from DR4 of the Sloan Digital
Sky Survey shows the signature of a PG\,1159 star plus emission from a
cool irradiated companion.
\item We performed time-series photometry during 10 nights with the
T\"ubingen 80\,cm and the G\"ottingen 50\,cm telescopes and detected a
period of 6.95616(33)\,h with an amplitude of 0.354(3)\,mag. This
represents the orbital period of the binary system. Thus,
SDSS\,J212531.92$-$010745.9 is the first close PG\,1159 binary without any
doubts.
\item From a first comparison with NLTE model spectra we derived, as preliminary
results, an effective temperature of 90\,000\,K, $\log g\,\sim$\,7.60 and
the abundance ratio C/He\,$\sim$\,0.05 for the PG\,1159 component.
A detailed, quantitative NLTE spectral analysis of the PG\,1159 star
and the irradiated companion has to be done next. We will report on the
results in a subsequent paper.
\item We simulated the light curve of the binary system with an orbital
period of 6.95616\,h using \texttt{nightfall}. A good agreement with the
observed light curve was obtained for a mean radius of $0.4\pm 0.1\,\rm
R_\odot$, a mass of \,$0.4\pm 0.1\,\rm M_\odot$ and a temperature of the
irradiated surface of about 8\,200\,K for the companion.
\end{enumerate}
To determine the system parameters more precisely, high-resolution
phase-resolved spectroscopy of SDSS\,J212531.92$-$010745.9 is necessary.
It should then be possible to derive both the companion's variable
light contribution to the overall spectrum as well as dynamical
masses for both components from radial velocity measurements of their
distinct line systems.
\begin{acknowledgements}
We thank T.-O. Husser, R. Lutz and E. Nagel for supporting the
observations. We acknowledge the use of {\texttt CAFE 5.1}, an
astronomical fit enviroment, written by Eckart G\"ohler. We
acknowledge the use of the \texttt{nightfall} program for the
light curve synthesis of eclipsing binaries written by Rainer Wichmann
(http://www.lsw.uni-heidelberg.de/$\sim$rwichman/Nightfall.html). BTG
was supported by a PPARC Advanced Fellowship.
\end{acknowledgements}
|
Title:
Second-order perturbations of a zero-pressure cosmological medium: comoving vs. synchronous gauge |
Abstract: Except for the presence of gravitational wave source term, the relativistic
perturbation equations of a zero-pressure irrotational fluid in a flat
Friedmann world model coincide exactly with the Newtonian ones to the second
order in perturbations. Such a relativistic-Newtonian correspondence is
available in a special gauge condition (the comoving gauge) in which all the
variables are equivalently gauge invariant. In this work we compare our results
with the ones in the synchronous gauge which has been used often in the
literature. Although the final equations look simpler in the synchronous gauge,
the variables have remnant gauge modes. Except for the presence of the gauge
mode for the perturbed order variables, however, the equations in the
synchronous gauge are gauge invariant and can be exactly identified as the
Newtonian hydrodynamic equations in the Lagrangian frame. In this regard, the
relativistic equations to the second order in the comoving gauge are the same
as the Newtonian hydrodynamic equations in the Eulerian frame. We resolve
several issues related to the two gauge conditions often to fully nonlinear
orders in perturbations.
| https://export.arxiv.org/pdf/astro-ph/0601041 |
\draft
\twocolumn[\hsize\textwidth\columnwidth\hsize\csname
@twocolumnfalse\endcsname
\title{Second-order perturbations of a zero-pressure cosmological
medium: \\
comoving vs. synchronous gauge}
\author{Jai-chan Hwang${}^{(a)}$ and Hyerim Noh${}^{(b)}$}
\address{${}^{(a)}$ Department of Astronomy and Atmospheric Sciences,
Kyungpook National University, Taegu, Korea \\
${}^{(b)}$ Korea Astronomy and Space Science Institute,
Daejon, Korea \\
E-mails: ${}^{(a)}$jchan@knu.ac.kr, ${}^{(b)}$hr@kasi.re.kr
}
\vskip2pc]
\section{Introduction}
\label{sec:Introduction}
The general relativistic cosmological linear perturbation theory was
first developed by Lifshitz in 1946 \cite{Lifshitz-1946}. Lifshitz
took the synchronous gauge condition in which the perturbations of
the time-time part and the space-time part of the metric tensor are
equal to zeros; this gauge condition can be taken to fully nonlinear
order without losing any physical degree of freedom \cite{LL}. The
synchronous gauge condition has been popular in the cosmological
perturbation literature despite the complicating fact that, except
for the zero-pressure case, there are remnant gauge modes for both
the spatial and temporal gauge conditions. There exist other spatial
and temporal gauge conditions which fix the gauge transformation
property completely in general situation, thus without any remaining
gauge mode \cite{Harrison-1967,Field-Shepley-1968,Nariai-1969}. This
point was clarified by Bardeen \cite{Bardeen-1980,Bardeen-1988}. In
a zero-pressure medium the density perturbation equation in the
synchronous gauge coincides with the one in the comoving gauge
\cite{Lifshitz-1946,Nariai-1969}. The density perturbation equation
in the comoving gauge condition is known to resemble the Newtonian
equation most closely \cite{Nariai-1969,Bardeen-1980}, and the
equations coincide in the zero-pressure case
\cite{Lifshitz-1946,Bonnor-1957}. Thus, in the zero-pressure case
the density perturbation equation in the synchronous gauge coincides
with the Newtonian one to the linear order
\cite{Lifshitz-1946,Bonnor-1957}.
The synchronous gauge was also used in the nonlinear perturbation
studies \cite{Tomita}, and Kasai \cite{Kasai-1992} has derived
second-order differential equations for density perturbation which
is valid to fully nonlinear order. Although, such an equation in the
synchronous gauge naturally has proper linear limit which
corresponds to the Newtonian equation, it has been unclear whether
such a correspondence continues to the nonlinear situation.
Recently, we have successfully shown an exact relativistic-Newtonian
correspondence of scalar-type perturbations to the second order
based on the comoving gauge
\cite{NL,second-order-CQG,second-order-PRD}. In the zero-pressure
case our comoving gauge condition differs from the conventional
synchronous gauge in the spatial gauge condition. In this work we
will investigate the case in the original synchronous gauge. We will
show that although the equations in the synchronous gauge look
simpler than the ones in the comoving gauge, the variables still
have remaining (spurious) spatial gauge mode to the second order.
The equations in the synchronous gauge, however, are gauge invariant
and can be identified as the Newtonian hydrodynamic equations in the
Lagrangian frame. Whereas, the equations in the comoving gauge can
be identified as the Newtonian hydrodynamic equations in the
Eulerian frame.
Results in Secs. \ref{sec:NL} and the Appendices are valid to fully
nonlinear order in perturbations, and unless mentioned otherwise
results in the remaining sections are valid to the second order in
perturbations. We closely follow notations used in
\cite{NL,second-order-CQG,second-order-PRD}. We set $c \equiv 1$.
\section{Fully nonlinear perturbations}
\label{sec:NL}
The energy conservation equation and the Raychaudhury equation give
\cite{covariant,second-order-CQG,second-order-PRD} \bea
& & \tilde {\dot {\tilde \mu}} + \tilde \mu \tilde \theta = 0,
\label{covariant-eq1} \\
& & \tilde {\dot {\tilde \theta}} + \frac{1}{3} \tilde \theta^2
+ \tilde \sigma^{ab} \tilde \sigma_{ab}
- \tilde \omega^{ab} \tilde \omega_{ab}
+ 4 \pi G \tilde \mu - \Lambda = 0,
\label{covariant-eq2}
\eea where $\tilde \theta \equiv \tilde u^a_{\;\; ;a}$ is an
expansion scalar based on a fluid four-vector $\tilde u_a$; $\tilde
\sigma_{ab}$ and $\tilde \omega_{ab}$ are the shear and the rotation
tensors based on $\tilde u_a$, respectively; tildes indicate the
covariant quantities and the Latin indices indicate spacetime
components. An overdot with tilde is a covariant derivative along
the $\tilde u_a$ vector, e.g., $\tilde {\dot {\tilde \mu}} \equiv
\tilde \mu_{,a} \tilde u^a$. By combining these equations we have
\bea
& & \left( \frac{\tilde {\dot {\tilde \mu}}}{\tilde \mu}
\right)^{\tilde \cdot}
- \frac{1}{3}
\left( \frac{\tilde {\dot {\tilde \mu}}}{\tilde \mu} \right)^2
- \tilde \sigma^{ab} \tilde \sigma_{ab}
+ \tilde \omega^{ab} \tilde \omega_{ab}
- 4 \pi G \tilde \mu
+ \Lambda
= 0.
\nonumber \\
\label{covariant-eq3}
\eea Equations (\ref{covariant-eq1})-(\ref{covariant-eq3}) are fully
nonlinear and covariant, thus valid to all orders in perturbations.
\subsection{Temporal Comoving Gauge}
In this work we will {\it assume} an irrotational fluid, thus
$\tilde \omega_{ab} \equiv 0$. We will consider two different gauge
conditions. In both gauge conditions we will have \bea
& & \tilde u_\alpha = 0,
\eea due to a common temporal gauge condition together with the
irrotational condition; the Greek indices indicate space components.
If we introduce the spatial part of the four-vector as \bea
& & \tilde u_\alpha \equiv a \left( - \hat v_{,\alpha}
+ \hat v^{(v)}_\alpha \right),
\label{u-def}
\eea where $\hat v^{(v)}_\alpha$ is a vector-type perturbation (thus
transverse), the irrotational condition sets $\hat v^{(v)}_\alpha
\equiv 0$ and our temporal comoving gauge sets $\hat v \equiv 0$.
Since $\tilde u_\alpha = 0$ the fluid four-vector in this gauge
coincides with the {\it normal} frame four-vector $\tilde n_a$ with
$\tilde n_\alpha \equiv 0$. Notice that our temporal comoving gauge
condition $\hat v \equiv 0$ (together with the irrotational
condition) implies $\tilde u_\alpha = 0$. This {\it differs} from
the ordinarily known {\it comoving} frame condition which sets
$\tilde u^\alpha \equiv 0$ \cite{Taub-1978}. In our case the
normalized ($\tilde u^a \tilde u_a \equiv -1$) fluid four-vector
$\tilde u_a$ becomes \bea
& & \tilde u_0 = - {1 \over \sqrt{- \tilde g^{00}}}, \quad
\tilde u_\alpha \equiv 0;
\nonumber \\
& &
\tilde u^0 = \sqrt{- \tilde g^{00}}, \quad
\tilde u^\alpha = - {\tilde g^{0\alpha}
\over \sqrt{- \tilde g^{00}}}.
\label{u}
\eea
In the zero-pressure case the momentum conservation equation implies
$\tilde g^{00} = - 1/a^2$ where $a$ is the cosmic scale factor of
the Friedmann background world model. In the ADM approach
\cite{ADM}, our temporal comoving gauge $\hat v = 0$ together with
the irrotational condition implies vanishing momentum vector
$J_\alpha \equiv - \tilde n_b \tilde T^b_\alpha = 0$. The ADM
momentum conservation equation in Eq.\ (13) of \cite{NL} gives
$N_{,\alpha} = 0$ where $\tilde g^{00} \equiv -1/N^2$, thus $N =
N(t)$. In another way, since the acceleration vector $\tilde
a_\alpha \equiv \tilde u_{\alpha ;b} \tilde u^b =
(\ln{N})_{,\alpha}$ vanishes (i.e., geodesic flow) for the
zero-pressure irrotational flow, we have $N = N(t)$; see Eqs.\ (27)
and (42) of \cite{NL}. Without losing generality we can set $N =
a(t)$. Thus we have \bea
& & \tilde g^{00} = - {1 \over a^2}.
\label{g^00}
\eea Thus, Eq.\ (\ref{u}) becomes \bea
& & \tilde u_0 = - a, \quad
\tilde u_\alpha \equiv 0; \quad
\tilde u^0 = {1 \over a}, \quad
\tilde u^\alpha = - a \tilde g^{0\alpha},
\label{u-pressureless}
\eea which is valid to fully nonlinear order. We can show that to
all orders in perturbations the fluid quantities are independent of
the spatial gauge condition which could affect $\tilde g^{0\alpha}$;
see the Appendix A.
\subsection{Nonlinear perturbed equations}
We introduce perturbations \bea
& & \tilde \mu \equiv \mu + \delta \mu, \quad
\tilde \theta \equiv 3 H - \kappa,
\eea where $H \equiv \dot a/a$ and $\delta \equiv \delta \mu/\mu$;
an overdot denotes a time derivative based on background proper-time
$t$. The $\tilde \theta$ is an expansion scalar of the fluid
four-vector which is the same as the normal four-vector because
$\tilde u_\alpha = 0$ in our case. Using Eq.\ (\ref{u-pressureless})
we have \bea
& & \tilde {\dot {\tilde \mu}}
= \dot \mu \left( 1 + \delta \right)
+ \mu \left( \dot \delta - {1 \over a} N^\alpha \delta_{,\alpha}
\right),
\nonumber \\
& &
\tilde {\dot {\tilde \theta}} = 3 \dot H - \left( \dot \kappa
- {1 \over a} N^\alpha \kappa_{,\alpha} \right),
\label{dot-mu}
\eea where $N^\alpha$ is the shift vector in the ADM notation with
$N^\alpha \equiv a^2 \tilde g^{0\alpha}$; the spatial indices of the
ADM variables are based on $h_{\alpha\beta} \equiv \tilde
g_{\alpha\beta}$.
Equations (\ref{covariant-eq1}), (\ref{covariant-eq2}) give \bea
& & \left( \dot \mu + 3 H \mu \right) \left( 1 + \delta \right)
\nonumber \\
& & \quad
+ \mu \left[ \dot \delta
- {1 \over a} \delta_{,\alpha} N^\alpha
- \left( 1 + \delta \right) \kappa \right]
= 0,
\\
& & 3 \left( \dot H + H^2 \right) + 4 \pi G \mu - \Lambda
\nonumber \\
& & \quad
- \left[ \dot \kappa
- {1 \over a} \kappa_{,\alpha} N^\alpha
+ 2 H \kappa
- 4 \pi G \mu \delta
- {1 \over 3} \kappa^2
- \tilde \sigma^{ab} \tilde \sigma_{ab} \right]
\nonumber \\
& & \quad
= 0.
\eea The background parts give \bea
& & \dot \mu + 3 H \mu = 0, \quad
3 \left( \dot H + H^2 \right) + 4 \pi G \mu - \Lambda = 0.
\label{BG-eqs}
\eea The perturbed parts give \bea
& & \hat {\dot \delta} = \left( 1 + \delta \right)\kappa,
\label{NL-eq1} \\
& & \hat {\dot \kappa} + 2 H \kappa = {1 \over 3} \kappa^2
+ \tilde \sigma^{ab} \tilde \sigma_{ab}
+ 4 \pi G \mu \delta,
\label{NL-eq2}
\eea where $\hat {\dot \delta} \equiv \dot \delta - a^{-1}
\delta_{,\alpha} N^\alpha$. By combining these equations we have
\bea
& & \hat {\ddot \delta}
+ 2 H \hat {\dot \delta} - 4 \pi G \mu \delta
= 4 \pi G \mu \delta^2
+ {4 \over 3} {(\hat {\dot \delta})^2 \over 1 + \delta}
+ \left( 1 + \delta \right) \tilde \sigma^{ab} \tilde
\sigma_{ab}.
\nonumber \\
\label{NL-eq3}
\eea These equations are valid to the fully nonlinear orders in
perturbations, subject only to the temporal comoving gauge
condition, the zero-pressure condition, and the irrotational
condition.
\subsection{The synchronous gauge}
Under the synchronous gauge we set $\tilde g_{0\alpha} \equiv 0$
(thus $N^\alpha \equiv 0$) using the spatial gauge condition
(together with the irrotational condition), thus \bea
& & \tilde {\dot {\tilde \mu}}
= \hat {\dot {\tilde \mu}}
= {\dot {\tilde \mu}}.
\eea Thus, Eqs.\ (\ref{NL-eq1})-(\ref{NL-eq3}) simply give \bea
& & \dot \delta = \left( 1 + \delta \right)\kappa,
\label{NL-SG-eq1} \\
& & \dot \kappa + 2 H \kappa = {1 \over 3} \kappa^2
+ \tilde \sigma^{ab} \tilde \sigma_{ab}
+ 4 \pi G \mu \delta,
\label{NL-SG-eq2} \\
& & \ddot \delta + 2 H \dot \delta - 4 \pi G \mu \delta
= 4 \pi G \mu \delta^2
+ {4 \over 3} {\dot \delta^2 \over 1 + \delta}
+ \left( 1 + \delta \right) \tilde \sigma^{ab} \tilde
\sigma_{ab},
\nonumber \\
\label{NL-SG-eq3}
\eea which are valid to the fully nonlinear order. Using $\Delta
\equiv \delta/(1 + \delta)$ Kasai \cite{Kasai-1992} has derived \bea
& & \ddot \Delta + 2 H \dot \Delta - 4 \pi G \mu \Delta
= - {2 \over 3} {\dot \Delta^2 \over 1 - \Delta}
+ \left( 1 - \Delta \right) \tilde \sigma^{ab} \tilde
\sigma_{ab}.
\nonumber \\
\eea To nonlinear order in perturbations the above equations are
incomplete yet because of $\tilde \sigma^{ab} \tilde \sigma_{ab}$
term. Later we will show that these equations in the synchronous
gauge differs from the equations in the comoving gauge to the second
order. Furthermore, although these equations look simple, we will
show that $\delta$ (thus $\Delta$ as well) and $\kappa$ still have
remnant gauge modes to the second order. In Sec.\ \ref{sec:SG} we
will show that to the second order the equations are gauge invariant
and can be identified with the Newtonian hydrodynamic equations in
the Lagrangian frame. In this regard, the equations in the comoving
gauge correspond to the Newtonian hydrodynamic equations in the
Eulerian frame.
\section{Second-order perturbations}
As the metric we take \bea
& & ds^2
= - a^2 \left( 1 + 2 \alpha \right) d \eta^2
- 2 a^2 \beta_{,\alpha} d \eta d x^\alpha
\nonumber \\
& & \quad
+ a^2 \left[ g^{(3)}_{\alpha\beta} \left( 1 + 2 \varphi \right)
+ 2 \gamma_{,\alpha|\beta}
+ 2 C^{(t)}_{\alpha\beta} \right] d x^\alpha d x^\beta,
\label{metric}
\eea where $\alpha$, $\beta$, $\gamma$ and $\varphi$ are spacetime
dependent perturbed-order variables, and $C^{(t)}_{\alpha\beta}$ is
a transverse and tracefree perturbed-order variable. Spatial indices
of perturbed order variables are based on $g^{(3)}_{\alpha\beta}$,
and a vertical bar indicates the covariant derivative based on
$g^{(3)}_{\alpha\beta}$; $g^{(3)}_{\alpha\beta}$ could become
$\delta_{\alpha\beta}$ in a flat Friedmann background. We {\it
ignored} the transverse vector-type perturbation variables. We
introduce $\chi \equiv a ( \beta + a \dot \gamma)$. The perturbed
variables can be regarded as nonlinearly perturbed ones to any order
in perturbations.
To the second order, from Eqs.\ (55), (57) of \cite{NL} we have \bea
& & N^\alpha = - \beta^{,\alpha},
\nonumber \\
& & \tilde \sigma^{ab} \tilde \sigma_{ab}
= \bar K^\alpha_\beta \bar K^\beta_\alpha
= {1 \over a^4} \left[ \chi^{,\alpha|\beta}
\chi_{,\alpha|\beta}
- {1 \over 3} \left( \Delta \chi \right)^2 \right]
\nonumber \\
& & \quad
+ \dot C^{(t)\alpha\beta} \left( {2 \over a^2}
\chi_{,\alpha|\beta} + \dot C^{(t)}_{\alpha\beta} \right),
\label{sigma}
\eea where $N^\alpha$ is evaluated to the linear order; $\bar
K^\alpha_\beta$ is a tracefree part of the extrinsic curvature. We
note that $\tilde \sigma^{ab} \tilde \sigma_{ab}$ is spatially gauge
invariant to the second order, see Sec.\ \ref{sec:gauge-issue}.
Before comparing equations in the two different spatial gauges, we
{\it compare} our $\hat v$ in Eq.\ (\ref{u-def}) with the notation
used in \cite{NL} to the second order in perturbations. In \cite{NL}
we introduced the fluid quantities based on the normal-frame vector
$\tilde n_a$ and provided the relation of fluid quantities between
the energy-frame ($E$) and the normal-frame ($N$). Our fluid
four-vector $\tilde u_a$ is based on the energy frame which sets
$\tilde q_a \equiv 0$. The energy-frame fluid four-vector is
introduced in Eq.\ (53) of \cite{NL}, and using the relations given
in Eqs.\ (87), (88) of \cite{NL} we have \bea
& & \tilde u_\alpha
= a \left( V_\alpha^E - B_\alpha + A B_\alpha
+ 2 V^\beta_E C_{\alpha\beta} \right)
\nonumber \\
& & \quad
= a \left\{ {Q_\alpha^N \over \mu + p}
- {1 \over (\mu + p)^2} \left[
\left( \delta \mu + \delta p \right) Q_\alpha^N
+ Q^\beta_N \Pi_{\alpha\beta} \right] \right\}.
\nonumber \\
\eea Using the decomposition of the normal-frame flux vector
$Q_\alpha^N \equiv (\mu + p) ( - v_{,\alpha} + v_\alpha^{(v)})$ in
Eq.\ (175) of \cite{NL} and setting $v_\alpha^{(v)} \equiv 0$ we
have \bea
& & \hat v_{,\alpha}
= v_{,\alpha}
- {1 \over \mu + p} \left[
\left( \delta \mu + \delta p \right) v_{,\alpha}
+ v^{,\beta} \Pi_{\alpha\beta} \right].
\eea Thus, the temporal comoving gauge $v \equiv 0$ in \cite{NL}
implies $\hat v = 0$ and vice versa.
\subsection{The comoving gauge}
\label{sec:CG}
In \cite{NL,second-order-CQG,second-order-PRD} we took the temporal
comoving gauge and the spatial $\gamma = 0$ gauge \bea
& & v \equiv 0, \quad
\gamma \equiv 0.
\label{CG}
\eea In this work, we call this the {\it comoving} gauge. Thus, we
have $\beta = \chi/a$.
The momentum conservation equation in Eq.\ (105) of \cite{NL} gives
\bea
& & \alpha = - {1 \over 2 a^2} \chi^{,\beta} \chi_{,\beta}.
\label{alpha-CG}
\eea Thus, apparently, $\alpha$ does not vanish to the second order.
Later we will show that if we take $\beta = 0$ as the spatial gauge
condition instead of $\gamma = 0$, we have vanishing $\alpha$.
However, we prefer $\gamma \equiv 0$ as the spatial gauge condition
because it fixes the spatial gauge degree of freedom completely (as
long as we simultaneously take the temporal gauge which removes the
temporal gauge degree of freedom completely, like our $v = 0$), see
Sec.\ VI of \cite{NL}. Whereas, $\beta \equiv 0$ fails to fix the
spatial gauge degree of freedom completely, thus having remaining
gauge degree of freedom even after imposing the gauge condition, see
Sec.\ \ref{sec:two-gauges}.
In our gauge the fluid four-vector in Eq.\ (\ref{u-pressureless})
becomes \bea
& & \tilde u_0 = - a , \quad
\tilde u_\alpha = 0;
\nonumber \\
& &
\tilde u^0 = {1 \over a}, \quad
\tilde u^\alpha = \frac{1}{a^2} \chi^{,\beta}
\left[ \left( 1 - 2 \varphi \right) \delta^\alpha_\beta
- 2 C^{(t)\alpha}_{\;\;\;\;\;\beta} \right].
\label{u-CG}
\eea Thus, although we prefer to call this the temporal comoving
gauge (see \cite{Bardeen-1980,Bardeen-1988}), because $\tilde
u_\alpha = 0$ and $\tilde u^\alpha \neq 0$, our fluid four-vector
corresponds to the normal four-vector rather than the comoving one.
Using Eq.\ (\ref{sigma}), Eqs.\ (\ref{NL-eq1}), (\ref{NL-eq2}) give
\bea
& & \dot \delta
+ {1 \over a^2} \delta_{,\alpha} \chi^{,\alpha}
- \kappa
= \delta \kappa,
\label{delta-eq-CG} \\
& & \dot \kappa
+ {1 \over a^2} \kappa_{,\alpha} \chi^{,\alpha}
+ 2 H \kappa
- 4 \pi G \mu \delta
\nonumber \\
& & \quad
= \left( {1 \over a^2} \chi^{,\alpha|\beta}
+ \dot C^{(t)\alpha\beta} \right)
\left( {1 \over a^2} \chi_{,\alpha|\beta}
+ \dot C^{(t)}_{\alpha\beta} \right).
\label{kappa-eq-CG}
\eea These also follow from the energy-conservation equation and the
trace part of ADM propagation equation in Eqs.\ (104), (102) of
\cite{NL}.
In \cite{second-order-CQG,second-order-PRD} we {\it identified} to
the second order \bea
& & \delta \mu \equiv \delta \varrho, \quad
\kappa \equiv - {1 \over a} \nabla \cdot {\bf u},
\label{identify-second-order}
\eea where $\delta \varrho$ and ${\bf u}$ are Newtonian density and
velocity perturbations, respectively. As we ignore the rotational
mode, the velocity is of potential type with ${\bf u} = \nabla u$.
Apparently, we need $\chi$ to the linear order only, and to that
order we have \cite{second-order-CQG,second-order-PRD} \bea
& & \nabla \chi = a {\bf u},
\label{identify-chi}
\eea where we {\it assume} a flat Friedmann background world model.
With these identifications of the relativistic metric and
energy-momentum perturbation variables (these are equivalently
gauge-invariant combinations, see Sec.\ \ref{sec:two-gauges}) with
the Newtonian hydrodynamic variables, Eqs.\ (\ref{delta-eq-CG}),
(\ref{kappa-eq-CG}) give \bea
& & \dot \delta + {1 \over a} \nabla \cdot {\bf u}
= - {1 \over a} \nabla \cdot \left( \delta {\bf u} \right),
\label{delta-eq-3rd} \\
& & {1 \over a} \nabla \cdot \left( \dot {\bf u}
+ H {\bf u} \right)
+ 4 \pi G \mu \delta
= - {1 \over a^2} \nabla \cdot \left( {\bf u}
\cdot \nabla {\bf u} \right)
\nonumber \\
& & \quad
- \dot C^{(t)\alpha\beta} \left( {2 \over a} \nabla_\beta
u_\alpha
+ \dot C^{(t)}_{\alpha\beta} \right).
\label{u-eq-3rd}
\eea By combining these we have \bea
& & \ddot \delta + 2 {\dot a \over a} \dot \delta
- 4 \pi G \mu \delta
= - {1 \over a^2} {\partial \over \partial t}
\left[ a \nabla \cdot \left( \delta {\bf u} \right) \right]
+ {1 \over a^2} \nabla \cdot \left( {\bf u}
\cdot \nabla {\bf u} \right)
\nonumber \\
& & \quad
+ \dot C^{(t)\alpha\beta} \left( {2 \over a} \nabla_\beta u_\alpha
+ \dot C^{(t)}_{\alpha\beta} \right),
\label{density-eq-3rd}
\eea which also follows from Eq.\ (\ref{NL-eq3}). Except for the
presence of the gravitational waves as source terms Eqs.\
(\ref{delta-eq-3rd})-(\ref{density-eq-3rd}) are valid {\it exactly}
in the Newtonian system \cite{Peebles-1980}.
Although our relativistic equations are valid to the second order,
the Newtonian equations are valid to fully nonlinear order. Thus,
all nonvanishing higher-order perturbation terms in the relativistic
case are pure general relativistic corrections. Recently, we have
presented such pure general relativistic correction terms appearing
in the third order perturbations in \cite{third-order}.
\subsection{The synchronous gauge}
\label{sec:SG}
The synchronous and comoving gauge conditions correspond to taking
\cite{LL} \bea
& & v \equiv 0 , \quad
\beta \equiv 0 ; \quad
\alpha = 0.
\label{SG}
\eea In this work, we call this simply the {\it synchronous} gauge.
Thus, we have $\dot \gamma = \chi/a^2$. If we take $v \equiv 0$ and
$\beta \equiv 0$ as the temporal and the spatial gauge conditions,
respectively, the momentum conservation equation gives $\alpha = 0$
to {\it all} orders in perturbations; although this is well known in
\cite{LL}, we give proofs in the Appendix B. Thus, in the
zero-pressure medium without rotation we can simultaneously impose
the comoving ($v = 0$) and the synchronous ($\alpha = 0$) temporal
gauge conditions as long as we also take $\beta \equiv 0$ as the
spatial gauge condition \cite{LL}; Kasai took these conditions in
his work in \cite{Kasai-1992}.
The original synchronous gauge used by Lifshitz \cite{Lifshitz-1946}
took $\alpha = 0$ and $\beta = 0$ as the temporal and the spatial
gauge conditions, respectively. These gauge conditions are known to
be {\it incomplete} in fixing both the temporal and the spatial
gauge modes even to the linear order. Thus, even after imposing
these gauge conditions we have remaining gauge modes present in the
solutions, in general. Meanwhile, $v \equiv 0$ and $\gamma \equiv 0$
fix the temporal and spatial gauge degree of freedoms completely,
thus no gauge mode is present in the solution, see Sec.\
\ref{sec:gauge-issue}. Since the original synchronous gauge implies
$v = 0$ (the nonvanishing solution of $v$ is the remnant temporal
gauge mode) in the zero-pressure case, we only have to pay attention
to the possible presence of the spatial gauge mode. In this gauge we
have $\tilde g_{00} = - a^2 = 1/\tilde g^{00}$ and $\tilde
g_{0\alpha} = 0 = \tilde g^{0\alpha}$. Thus, the fluid four-vector
in Eq.\ (\ref{u-pressureless}) becomes \bea
& & \tilde u_0 = - a, \quad
\tilde u_\alpha = 0; \quad
\tilde u^0 = {1 \over a}, \quad
\tilde u^\alpha = 0,
\label{u-SG}
\eea which can be compared with Eq.\ (\ref{u-CG}) in the comoving
gauge. Thus, since $\tilde u^\alpha = 0$, our fluid four-vector
corresponds to the conventionally known comoving four-vector
\cite{Taub-1978}, and simultaneously normal because $\tilde u_\alpha
= 0$ as well. All the statements in the above two paragraphs are
valid for all perturbational orders.
Using Eq.\ (\ref{sigma}), Eqs.\ (\ref{NL-SG-eq1}), (\ref{NL-SG-eq2})
give \bea
& & \dot \delta
- \kappa
= \delta \kappa,
\label{delta-eq-SG} \\
& & \dot \kappa
+ 2 H \kappa
- 4 \pi G \mu \delta
\nonumber \\
& & \quad
= \left( {1 \over a^2} \chi^{,\alpha\beta}
+ \dot C^{(t)\alpha\beta} \right)
\left( {1 \over a^2} \chi_{,\alpha\beta}
+ \dot C^{(t)}_{\alpha\beta} \right).
\label{kappa-eq-SG}
\eea These also follow from the energy-conservation equation and the
trace part of ADM propagation equation in Eqs.\ (104), (102) of
\cite{NL}. By combining these equations we have \bea
& & \ddot \delta + 2 H \dot \delta - 4 \pi G \mu \delta
= \dot \delta^2 + 4 \pi G \mu \delta^2
\nonumber \\
& & \quad
+ \left( {1 \over a^2} \chi^{,\alpha\beta}
+ \dot C^{(t)\alpha\beta} \right)
\left( {1 \over a^2} \chi_{,\alpha\beta}
+ \dot C^{(t)}_{\alpha\beta} \right).
\label{ddot-delta-eq-SG}
\eea Apparently, these equations in the synchronous gauge look
simpler than Eqs.\ (\ref{delta-eq-CG}), (\ref{kappa-eq-CG}) in the
comoving gauge. Compared with Eqs.\ (\ref{delta-eq-CG}),
(\ref{kappa-eq-CG}) in the comoving gauge, in Eqs.\
(\ref{delta-eq-SG}), (\ref{kappa-eq-SG}) we lack the
convective-derivative-like terms in the left-hand-sides. By changing
the time derivatives as \bea
& & {\partial \over \partial t} \rightarrow
{\partial \over \partial t} + {1 \over a^2} ( \nabla \chi )
\cdot \nabla
= {\partial \over \partial t} + {1 \over a} {\bf u} \cdot
\nabla,
\label{time-derivative}
\eea we can show that Eqs.\
(\ref{delta-eq-SG})-(\ref{ddot-delta-eq-SG}) become the same ones in
the comoving gauge in Eqs.\ (\ref{delta-eq-CG}),
(\ref{kappa-eq-CG}), and (\ref{density-eq-3rd}); in the last step of
Eq.\ (\ref{time-derivative}) we used Eq.\ (\ref{identify-chi}) which
is valid for a flat background.
{\it If} we make the same identification of the density and velocity
perturbations as in Eqs.\ (\ref{identify-second-order}),
(\ref{identify-chi}), thus assuming a flat background, Eqs.\
(\ref{delta-eq-SG}), (\ref{kappa-eq-SG}) become: \bea
& & \dot \delta
+ {1 \over a} \nabla \cdot {\bf u}
= - {1 \over a} \delta \nabla \cdot {\bf u},
\label{delta-eq-SG2} \\
& & {1 \over a} \nabla \cdot
\left( \dot {\bf u} + H {\bf u} \right)
+ 4 \pi G \mu \delta
= - {1 \over a^2} \left( \nabla^\beta u^\alpha \right)
\left( \nabla_\beta u_\alpha \right)
\nonumber \\
& & \quad
- \dot C^{(t)\alpha\beta}
\left( {2 \over a} \nabla_\beta u_\alpha
+ \dot C^{(t)}_{\alpha\beta} \right).
\label{kappa-eq-SG2}
\eea Ignoring the gravitational waves, these equations can be
identified as the Newtonian hydrodynamic equations in the Lagrangian
frame.
Although equations in the synchronous gauge look simpler than the
ones in the comoving gauge, the presence of additional
convective-like terms in the comoving gauge allows us to make exact
(except for the gravitational waves) correspondence with the
Newtonian hydrodynamic equations in the Eulerian frame
\cite{second-order-CQG,second-order-PRD}. Whereas, the equations in
the synchronous gauge can be identified as the Newtonian equations
in the Lagrangian frame. However, the variables in the synchronous
gauge still have the remnant spatial gauge mode due to incomplete
fixing nature of the spatial gauge condition $\beta \equiv 0$ in
that gauge. That is, to the second order, $\delta$ and $\kappa$ in
the synchronous gauge have the remaining gauge modes, see Sec.\
\ref{sec:two-gauges}.
Now, we can relate the variables in the synchronous ($S$) gauge to
the ones in the comoving ($C$) gauge. {}From Eqs.\ (\ref{CG-SG}),
(\ref{chi-GT}), and (\ref{identify-chi}) we have \bea
& & \delta_S = \delta_C + \left( \int^t {1 \over a^2} \nabla \chi dt
+ \nabla \gamma_{S,{\rm Gauge}} \right) \cdot \nabla
\delta_C,
\nonumber \\
& &
\kappa_S = \kappa_C + \left( \int^t {1 \over a^2} \nabla \chi dt
+ \nabla \gamma_{S,{\rm Gauge}} \right) \cdot \nabla
\kappa_C,
\label{SG-CG-sol}
\eea where $\gamma_{S,{\rm Gauge}}$ is the gauge mode present to the
linear order in $\gamma$; see the next section. In a flat
background, from Eq.\ (\ref{identify-chi}) we have $\nabla \chi = a
{\bf u}$. Notice that, even after ignoring the gauge modes
$\delta_S$ and $\kappa_S$ naturally differ from $\delta_C$ and
$\kappa_C$, respectively, because the final equations are different.
Using Eq.\ (\ref{SG-CG-sol}), Eqs.\
(\ref{delta-eq-SG})-(\ref{ddot-delta-eq-SG}) give Eqs.\
(\ref{delta-eq-CG}), (\ref{kappa-eq-CG}), and
(\ref{density-eq-3rd}).
Although the variables in the synchronous gauge have remnant spatial
gauge mode, somehow the equations in the synchronous gauge coincide
with the Newtonian ones in the Lagrangian frame. Meanwhile, the
Newtonian hydrodynamic equations have nothing to do with the gauge
mode which appears only in the relativistic treatment. We can show
that the situation is consistent in the synchronous gauge. {}From
Eqs.\ (\ref{xi_alpha-SG}), (\ref{GT-SG}) the gauge mode of
$\delta_{S,{\rm Gauge}} = \xi^\alpha \nabla_\alpha \delta_C$ is
proportional to the linear-order solution of $\delta_C$; similarly,
the gauge mode of $\kappa_{S,{\rm Gauge}} = \xi^\alpha \nabla_\alpha
\kappa_C$ is proportional to the linear-order solution of
$\kappa_C$. Thus, the behaviours of the gauge mode cannot be
distinguished from the solutions to the linear order, and can be
absorbed to the linear order solutions. We can also check that the
gauge modes in Eq.\ (\ref{SG-CG-sol}) cancel out in Eqs.\
(\ref{delta-eq-SG}) and (\ref{kappa-eq-SG}). In this sense, Eqs.\
(\ref{delta-eq-SG}) and (\ref{kappa-eq-SG}), thus Eqs.\
(\ref{ddot-delta-eq-SG}), (\ref{delta-eq-SG2}) and
(\ref{kappa-eq-SG2}) as well, are gauge-invariant. Therefore, to the
second order in the synchronous gauge, although the variables have
remnant gauge mode, the equations are gauge invariant; this happens
because the gauge mode temporally behaves exactly like one of the
physical solutions.
A similar situation occurs to the linear order in the original
synchronous gauge which took only $\alpha = 0 = \beta$
\cite{Lifshitz-1946}. Under these gauge conditions Lifshitz derived
\bea
& & \ddot \delta + 2 H \dot \delta - 4 \pi G \mu \delta = 0,
\label{ddot-delta-eq-SG-lin}
\eea which is the LHS of Eq.\ (\ref{ddot-delta-eq-SG}) and concides
with the later derived Newtonian equation \cite{Bonnor-1957}.
However, under these gauge conditions (i.e., without taking $v =
0$), $\delta$ still have the remnant gauge mode due to the
incomplete fixing nature of the temporal gauge condition $\alpha
\equiv 0$. It happens that the temporal behaviour of gauge mode of
$\delta$ is proportional to $H$ which {\it coincides} with one of
the two physical solutions, see \cite{GRG-1991}. Thus, although
$\delta$ has the gauge mode Eq.\ (\ref{ddot-delta-eq-SG-lin}) is
gauge invariant. In our synchronous gauge which also takes $v = 0$
the temporal gauge condition is fixed completely, but a similar
situation repeats due to an incomplete fixing nature of the spatial
synchronous gauge condition ($\beta \equiv 0$) now to the second
order in perturbation.
\section{Gauge issue}
\label{sec:gauge-issue}
\subsection{Gauge transformation}
Under a transformation between two coordinates $\hat x^a = x^a +
\tilde \xi^a$, the gauge transformation properties of all metric and
energy-momentum variables to the second order are presented in Sec.\
VI of \cite{NL}.
Since both the synchronous gauge and the comoving gauge take $v = 0$
we have \bea
& & \tilde \xi^0 = 0.
\eea This follows from Eqs.\ (234) or (238) of \cite{NL}: in the
normal frame by setting $Q_\alpha = 0$ (i.e., $v = 0$) in both
gauges we have $\tilde \xi^0_{\;\;,\alpha} = 0$; or in the energy
frame, by setting $V_\alpha - B_\alpha + A B_\alpha + 2 V^\beta
C_{\alpha\beta} = 0$ (i.e., $\hat v = 0$) in both gauges, we again
have $\tilde \xi^0_{\;\;,\alpha} = 0$. Thus, without losing
generality we can take $\tilde \xi^0 = 0$.
To the second order, with $\tilde \xi^0 \equiv 0$, from Eqs.\ (229),
(232) of \cite{NL} we have \bea
& & \hat \alpha
= \alpha
- \alpha_{,\alpha} \xi^\alpha
- \beta_{,\alpha} \xi^{\alpha\prime}
- {1 \over 2} \xi^{\alpha\prime} \xi_\alpha^\prime,
\nonumber \\
& &
\hat \delta = \delta - \delta_{,\alpha} \xi^\alpha, \quad
\hat \kappa = \kappa - \kappa_{,\alpha} \xi^\alpha,
\label{GT1}
\eea where $\xi^\alpha \equiv \tilde \xi^\alpha$ with the index of
$\xi^\alpha$ based on $g^{(3)}_{\alpha\beta}$; a prime indicates the
time derivative based on the conformal time $\eta$ with $d \eta
\equiv d x^0 \equiv dt/a$. The gauge transformation property of
$\kappa$ follows from the scalar nature of the expansion scalar
$\tilde \theta$ with $\tilde \theta \equiv 3 H - \kappa$ where
$\tilde \theta$ is based on the normal frame; for the gauge
transformation of a scalar quantity, see Eq.\ (239) of \cite{NL}. To
the linear order, from Eq.\ (252) of \cite{NL} we have \bea
& & \hat \beta_{,\alpha} = \beta_{,\alpha} + \xi_\alpha^\prime,
\quad
\hat \gamma_{,\alpha} = \gamma_{,\alpha} - \xi_\alpha.
\label{GT2}
\eea Thus, $\chi \equiv a ( \beta + \gamma^\prime )$ is gauge
invariant to the linear order, and \bea
& & \alpha - \alpha_{,\alpha} \gamma^{,\alpha}
+ {1 \over 2} \beta_{,\alpha} \beta^{,\alpha}, \quad
\delta - \delta_{,\alpha} \gamma^{,\alpha}, \quad
\kappa - \kappa_{,\alpha} \gamma^{,\alpha},
\label{GI} \\
& & \alpha - \alpha_{,\alpha} \gamma^{,\alpha}
- \left( \beta + {1 \over 2} \gamma^\prime \right)^{,\alpha}
\gamma^\prime_{,\alpha},
\label{GI-alpha}
\eea are gauge invariant to the second order.
\subsection{Two gauges}
\label{sec:two-gauges}
In the comoving gauge, by imposing $\gamma \equiv 0$ in all
coordinates (i.e., $\hat \gamma \equiv 0 \equiv \gamma$), from Eq.\
(\ref{GT2}) we have \bea
& & \xi_\alpha = 0.
\eea Thus, the spatial gauge transformation property is fixed
completely. {}From Eq.\ (\ref{GT1}) we have \bea
& & \hat \alpha = \alpha, \quad
\hat \delta = \delta, \quad
\hat \kappa = \kappa,
\eea and each variable in this gauge has unique gauge-invariant
counterpart as $\delta$ and $\kappa$ in Eq.\ (\ref{GI}) and $\alpha$
in Eq.\ (\ref{GI-alpha}). Thus, we can equivalently regard all
variables in this gauge as (spatially and temporally) gauge
invariant ones. {}For example, $\delta_{v,\gamma} \equiv \delta -
\delta_{,\alpha} \gamma^{,\alpha}$ is a unique gauge invariant
combination which is the same as $\delta$ in the $v = 0 = \gamma$
gauge conditions; for an explicit form of $\delta_{v,\gamma}$
including $v$, see Eq.\ (282) in \cite{NL}. We note that these
results (i.e., values remain the same in the comoving gauge
conditions, complete fixing of the gauge degrees of freedom, and
presence of unique corresponding gauge-invariant variables) continue
to be valid even in higher-order perturbations, \cite{NL}.
Whereas, in the synchronous gauge, by imposing $\beta \equiv 0$ in
all coordinates (i.e., $\hat \beta \equiv 0 \equiv \beta$), from
Eq.\ (\ref{GT2}) we have \bea
& & \xi_\alpha^\prime = 0.
\eea Thus, even after imposing the gauge condition we have \bea
& & \xi_\alpha = \xi_\alpha ({\bf x}),
\label{xi_alpha-SG}
\eea which is the remaining gauge mode. Thus, under the synchronous
gauge, from Eqs.\ (\ref{GT1}), (\ref{GT2}) we still have \bea
& & \hat \gamma_{,\alpha} = \gamma_{,\alpha} - \xi_\alpha,
\nonumber \\
& &
\hat \alpha = \alpha - \alpha_{,\alpha} \xi^\alpha, \quad
\hat \delta = \delta - \delta_{,\alpha} \xi^\alpha, \quad
\hat \kappa = \kappa - \kappa_{,\alpha} \xi^\alpha,
\label{GT-SG}
\eea where the transformation of $\gamma$ is valid to the linear
order. In this sense variables in the synchronous gauge have
remaining gauge modes even after imposing the gauge condition.
$\gamma$ has the remaining spatial gauge mode even in the linear
order, and the other variables have remaining gauge modes to the
second order.
\subsection{Transformation between the two gauges}
Using the gauge transformation properties of the variables we can
translate the equations and solutions in one gauge into the ones in
another gauge condition. We indicate the comoving gauge and the
synchronous gauge by subindices $C$ and $S$, respectively. To the
linear order we have \bea
& & \beta_C = \gamma_S^\prime = {1 \over a} \chi.
\label{chi-GT}
\eea We present three different ways to reach the transformation
properties.
{}First, we consider a transformation from the synchronous gauge
(unhat) to the comoving gauge (hat). {}From Eq.\ (\ref{GT1}) we have
\bea
& & \alpha_C
= - {1 \over 2} \xi^{\alpha\prime} \xi_\alpha^\prime,
\nonumber \\
& &
\delta_C
= \delta_S
- \delta_{S,\alpha} \xi^\alpha, \quad
\kappa_C
= \kappa_S
- \kappa_{S,\alpha} \xi^\alpha.
\label{GT-SG-1}
\eea We need to determine the gauge transformation function
$\xi^\alpha$, apparently, only to the linear order. {}From Eq.\
(\ref{GT2}) we have \bea
& & \xi_\alpha = \gamma_{S,\alpha}.
\eea Thus, Eq.\ (\ref{GT-SG-1}) becomes \bea
& & \alpha_C
= - {1 \over 2 a^2} \chi^{,\alpha} \chi_{,\alpha},
\nonumber \\
& &
\delta_C
= \delta_S
- \delta_{S,\alpha} \gamma_S^{,\alpha}, \quad
\kappa_C
= \kappa_S
- \kappa_{S,\alpha} \gamma_S^{,\alpha},
\label{CG-SG}
\eea where we used Eq.\ (\ref{chi-GT}).
Second, we consider a transformation from the comoving gauge (unhat)
to the synchronous gauge (hat). {}From Eq.\ (\ref{GT1}) we have \bea
& & \alpha_C
= {1 \over 2} \xi^{\alpha\prime} \xi_\alpha^\prime
+ \beta_{C,\alpha} \xi^{\alpha\prime},
\nonumber \\
& &
\delta_S
= \delta_C
- \delta_{C,\alpha} \xi^\alpha, \quad
\kappa_S
= \kappa_C
- \kappa_{C,\alpha} \xi^\alpha.
\label{GT-CG-1}
\eea {}From Eq.\ (\ref{GT2}) we have \bea
& & \xi_\alpha = - \gamma_{S,\alpha}.
\eea Thus, Eq.\ (\ref{GT-CG-1}) leads to the same results in Eq.\
(\ref{CG-SG}).
{}Finally, the gauge invariant combination in Eq.\ (\ref{GI})
provides a simpler derivation. {}From the gauge invariance of
combinations in Eq.\ (\ref{GI}) we directly have Eq.\ (\ref{CG-SG}).
Using these gauge transformation properties in Eq.\ (\ref{CG-SG}) we
can derive Eqs.\ (\ref{delta-eq-SG}), (\ref{kappa-eq-SG}) from Eqs.\
(\ref{delta-eq-CG}), (\ref{kappa-eq-CG}), and vice versa.
\section{Discussion}
In this work we have compared the general relativistic weakly
nonlinear cosmological perturbation equations in two different gauge
conditions. In our previous works we have successfully shown that,
except for the coupling with gravitational waves, the relativistic
perturbation equations of a zero-pressure irrotational fluid
coincide exactly with the Newtonian ones to the second order in
perturbations. Such a relativistic-Newtonian correspondence was
available in our special comoving gauge condition in which all the
variables can be equivalently regarded as gauge invariant ones. In
this work we have compared these results with the ones in the
synchronous gauge. The case in the synchronous gauge was previously
studied without noticing the similarity or difference of the
equations with the Newtonian ones to the nonlinear orders.
In this work we compared equations in the synchronous gauge with the
ones in the comoving gauge and in the Newtonian case. Although the
variables in this gauge have remnant spatial gauge modes due to the
incomplete gauge fixing of the spatial gauge condition the equations
are gauge invariant. Ignoring the gravitational waves, the equations
in the synchronous gauge can be identified with the Newtonian
hydrodynamic equations in the Lagrangian frame to the second order,
whereas the equations in the comoving gauge can be identified as the
Newtonian ones in the Eulerian frame. These Eulerian and Lagrangian
correspondences can be understood because the fluid four-vector in
our comoving gauge is in fact normal as in Eq.\ (\ref{u-CG}) whereas
the four-vector in the synchronous gauge is both normal and comoving
(thus Lagrangian) as in Eq.\ (\ref{u-SG}). In our way to clarify the
case in the synchronous gauge we have addressed and resolved several
issues related to the two gauge conditions often to fully nonlinear
orders in perturbations.
\subsection*{Acknowledgments}
We thank Professors J. Richard Bond, Lev Kofman and Misao Sasaki for
insightful and clarifying discussions. H.N. was supported by the
Korea Research Foundation Grant No. R04-2003-10004-0. J.H. was
supported by the Korea Research Foundation Grant No.
2003-015-C00253.
\section*{Appendices}
\subsection*{A. Invariance of fluid quantities}
Here, we {\it show} that the fluid quantities based on the fluid
four-vector in Eq.\ (\ref{u-pressureless}) do not depend on the
choice of $\tilde g^{0\alpha}$ (the spatial gauge condition) to all
orders in perturbations. Using a fluid four-vector $\tilde u_a$ the
energy-momentum tensor is decomposed into fluid quantities as
\cite{covariant,NL} \bea
& & \tilde T_{ab} \equiv \tilde \mu \tilde u_a \tilde u_b
+ \tilde p \left( \tilde g_{ab} + \tilde u_a \tilde u_b \right)
+ \tilde q_a \tilde u_b + \tilde q_b \tilde u_a
+ \tilde \pi_{ab};
\label{Tab} \\
& & \tilde \mu \equiv \tilde T_{ab} \tilde u^a \tilde u^b, \quad
\tilde p \equiv {1 \over 3} \tilde T_{ab} \tilde h^{ab}, \quad
\tilde q_a \equiv - \tilde T_{cd} \tilde u^c \tilde h_a^d,
\nonumber \\
& &
\tilde \pi_{ab} \equiv \tilde T_{cd} \tilde h_a^c \tilde h_b^d
- \tilde p \tilde h_{ab},
\label{fluid-Tab}
\eea where $\tilde h_{ab} \equiv \tilde g_{ab} + \tilde u_a \tilde
u_b$; we have $\tilde u^a \tilde q_a \equiv 0 \equiv \tilde u^a
\tilde \pi_{ab}$, $\tilde \pi_{ab} \equiv \tilde \pi_{ba}$, and
$\tilde \pi^a_a \equiv 0$. The variables $\tilde \mu$, $\tilde p$,
$\tilde q_a$ and $\tilde \pi_{ab}$ are the energy density, the
isotropic pressure (including the entropic one), the energy flux and
the anisotropic pressure (stress) based on the fluid four-vector,
respectively. Let us introduce another four-vector $\tilde U_a$ with
\bea
& & \tilde U_0 = - a, \quad
\tilde U_\alpha \equiv 0; \quad
\tilde U^0 = {1 \over a}, \quad
\tilde U^\alpha = - a \tilde g^{0\alpha}_U.
\label{u-second}
\eea Thus, $\tilde U_a$ is subject to the same conditions as $\tilde
u_a$ in Eq.\ (\ref{u-pressureless}), but with possibly different
spatial gauge condition which could lead to $\tilde g^{0\alpha}_U
\neq \tilde g^{0\alpha}$. The fluid quantities based on $\tilde U_a$
are similarly defined as in Eqs.\ (\ref{Tab}), (\ref{fluid-Tab})
with $\tilde U_a$ replacing $\tilde u_a$: for example, we have
$\tilde \mu^U \equiv \tilde T_{ab} \tilde U^a \tilde U^b$, etc. We
can easily show that if $\tilde p = \tilde q_a = \tilde \pi_{ab} =
0$ we have \bea
& & \tilde \mu^U
\equiv \tilde T_{ab} \tilde U^a \tilde U^b
= \tilde \mu \tilde u_a \tilde u_b \tilde U^a \tilde U^b
= \tilde \mu \tilde u_0 \tilde u_0 \tilde U^0 \tilde U^0
= \tilde \mu,
\eea and $\tilde p^U = \tilde q_a^U = \tilde \pi_{ab}^U = 0$, and
vice versa. This result is also valid to fully nonlinear order.
\subsection*{B. Justification of Eq.\ (\ref{SG})}
Here, we {\it show} that in a zero-pressure irrotational medium we
can take the original synchronous gauge ($\alpha \equiv 0 \equiv
\beta$) together with the temporal comoving gauge ($v \equiv 0$)
simultaneously to all orders in perturbations. This was known in
\cite{LL}, see Sec.\ 97 in \cite{LL}. Here, it is important to take
$\beta \equiv 0$ as the spatial synchronous gauge although we prefer
to take $\gamma \equiv 0$ as the spatial gauge condition because of
the remnant gauge mode in the $\beta \equiv 0$ case. We provide two
different proofs based on the ADM and the covariant formulations.
We begin by taking $v \equiv 0$ and $\beta \equiv 0$ as the temporal
and spatial gauge conditions, respectively. In Eq.\ (\ref{g^00}) we
showed that $\tilde g^{00} = - 1/N^2 = - 1/a^2$. The spatial gauge
condition $\beta = 0$ together with the irrotational condition
implies $\tilde g_{0\alpha} \equiv N_\alpha = 0$. Thus, from Eq.\
(2) of \cite{NL} we have $\tilde g_{00} \equiv - a^2 ( 1 + 2 \alpha
) = - N^2 = - a^2$. This implies that we have $\alpha = 0$.
Now, in the covariant approach, $v = 0$ and irrotational conditions
imply $\tilde u_\alpha = 0$. Since $\tilde u_a$ is the fluid
four-vector we take the energy frame, $\tilde q_a \equiv 0$. The
momentum conservation equation in Eq.\ (27) of \cite{NL} gives
$\tilde a_a = 0$. The spatial gauge condition $\beta = 0$ together
with the irrotational condition implies $\tilde g_{0\alpha} = 0$,
thus $\tilde g^{0\alpha} = 0$ as well. Since $\tilde a_\alpha \equiv
\tilde u_{\alpha;b} \tilde u^b = \tilde \Gamma^0_{0\alpha} = {1
\over 2} \tilde g^{00} \tilde g_{00,\alpha}$, $\tilde a_\alpha = 0$
implies that $\tilde g_{00}$ is a function of time only. Thus, we
have $\alpha = 0$.
If we impose $\alpha \equiv 0$ and $\beta \equiv 0$ as the gauge
condition, instead, we have non-vanishing $J_\alpha$ or $\tilde
u_\alpha$, thus nonvanishing $v$. In a zero-pressure medium this
nonvanishing $v$ can be identified as the remnant temporal gauge
mode, which can be set equal to zero without losing physical degree
of freedom.
|
Title:
Bias-free Measurement of Giant Molecular Cloud Properties |
Abstract: (abridged) We review methods for measuring the sizes, line widths, and
luminosities of giant molecular clouds (GMCs) in molecular-line data cubes with
low resolution and sensitivity. We find that moment methods are robust and
sensitive -- making full use of both position and intensity information -- and
we recommend a standard method to measure the position angle, major and minor
axis sizes, line width, and luminosity using moment methods. Without
corrections for the effects of beam convolution and sensitivity to GMC
properties, the resulting properties may be severely biased. This is
particularly true for extragalactic observations, where resolution and
sensitivity effects often bias measured values by 40% or more. We correct for
finite spatial and spectral resolutions with a simple deconvolution and we
correct for sensitivity biases by extrapolating properties of a GMC to those we
would expect to measure with perfect sensitivity. The resulting method recovers
the properties of a GMC to within 10% over a large range of resolutions and
sensitivities, provided the clouds are marginally resolved with a peak
signal-to-noise ratio greater than 10. We note that interferometers
systematically underestimate cloud properties, particularly the flux from a
cloud. The degree of bias depends on the sensitivity of the observations and
the (u,v) coverage of the observations. In the Appendix to the paper we present
a conservative, new decomposition algorithm for identifying GMCs in
molecular-line observations. This algorithm treats the data in physical rather
than observational units, does not produce spurious clouds in the presence of
noise, and is sensitive to a range of morphologies. As a result, the output of
this decomposition should be directly comparable among disparate data sets.
| https://export.arxiv.org/pdf/astro-ph/0601706 |
\title{Bias-free Measurement of Giant Molecular Cloud Properties}
\author{Erik Rosolowsky\altaffilmark{1}}
\affil{Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, MS-66, Cambridge, MA 02138}
\email{erosolow@cfa.harvard.edu}
\and
\author{Adam Leroy}
\affil{Department of Astronomy, 601 Campbell Hall, University of
California at Berkeley, CA 94720}
\email{aleroy@astro.berkeley.edu}
\altaffiltext{1}{National Science Foundation Astronomy \& Astrophysics Postdoctoral Fellow}
\keywords{ISM:clouds --- methods:data analysis --- radio lines:ISM}
\section{Introduction}
Over the last 15 years, it has become possible to observe molecular
emission in nearby galaxies with sufficient resolution and sensitivity
to distinguish individual giant molecular clouds (GMCs). The immediate
goal of such studies is to determine whether (and how) the GMCs in
other galaxies differ from those seen in the Solar neighborhood. The
most common method used to address this question has been to use
molecular-line tracers of H$_2$, in particular $^{12}$CO($1\to 0$), to
compare the macroscopic properties (size, line width, and luminosity)
of GMCs in other galaxies to those of Milky Way GMCs. Unfortunately, a
wide variety of methods have been used to reduce data from spectral
line data cubes into macroscopic GMC properties. As a result, many of
the differences between GMC populations found in the literature can be
attributed, at least partially, to observational artifacts or
methodological differences. It is therefore difficult to assess what
the real differences between GMC populations are based on the reported
data in the literature.
For GMCs that are either marginally resolved or marginally detected,
observational biases can be severe. Figure \ref{RESBIAS} shows the
variation of the measured spatial size and line width with the
resolution for a model cloud. Typical Galactic GMCs have sizes of a
few 10s of parsecs, comparable to the spatial resolution of many data
sets used to study extragalactic GMCs
\citep[e.g.~][]{vog87,ws90,wr91,is93,wil93,fu99,she00,ros03}. Figure
\ref{RESBIAS} shows that when the size of the beam is comparable to
the size of the object, the measured size is much higher than the true
size of the object. Millimeter spectrometers and correlators often
have excellent frequency resolution, so the spectral resolution bias
is usually less important for GMC studies, but it can become
substantial when data are binned to increase signal-to-noise. Figure
\ref{EXTRAPMOMS} shows that the measured spatial size, line width, and
flux of a real GMC in M~33 \citep[EPRB1 from][]{ros03} are all strong
functions of the sensitivity of the data. We discuss another major
source of bias, the method by which emission is decomposed into GMCs,
in the Appendix.
To place these biases in the the context of real molecular cloud
studies, Figure \ref{DISTCOMP} shows the peak flux and angular size of
typical GMCs \citep[those of][]{srby87} as a function of distance. The
sensitivities and resolutions of a representative sample of molecular
cloud studies have been indicated as horizontal lines in these plots.
Distances to commonly observed objects have also been labeled. Figure
\ref{DISTCOMP} demonstrates that most studies of extragalactic GMCs
are conducted where the clouds of interest are only marginally
resolved and are found at low sensitivity. Even future observations of
GMCs in the Virgo cluster using the Atacama Large Millimeter Array
(ALMA) will be affected by resolution and sensitivity biases.
In this paper, we examine the effects of biases stemming from finite
spatial resolution, spectral resolution, and sensitivity in
molecular-line observations of GMCs. We recommend data analysis
methods to produce a standardized set of observed cloud properties
that account for these biases. Most of the methods used in this paper
have been adopted piecemeal and {\it ad hoc} in previous studies.
Here, we endeavor to justify our choice of methodologies and to
synthesize various author's techniques for approaching the problems of
molecular cloud data analysis. In Section \ref{MOMENTS}, we describe
a standardized method to measure three basic properties of an emission
distribution --- size, line width, and flux --- while accounting for
the sensitivity and resolution of a data set. In Section
\ref{PHYSICAL}, we discuss how these measurements can be transformed
into physical quantities --- radius, line width, luminosity, and
implied mass. Finally, we consider the effects of using
interferometers to derive cloud properties in Section \ref{INTERF}.
The results in these three sections are applicable to all observations
of molecular clouds. In contrast, the methods used for decomposing
emission into molecular clouds vary widely, and there is little basis
for favoring one method over another in all cases. Hence, we defer a
presentation of our decomposition algorithm to the Appendix of the
paper, leaving only a brief discussion of the decomposition problem in
Section \ref{DECOMPSECT}. We conclude the paper by exhibiting several
examples of the application of our standardized methods to previously
observed data and making recommendations for future observations
(Section \ref{APPLICATIONS}).
The methods described in this paper require a computer program to
apply. A documented software version of the decomposition and
measurement algorithms is available from the authors as an IDL
package.
\section{Measuring Molecular Cloud Properties \label{MOMENTS}}
This section describes how to derive the spatial size, line width, and
flux from a region of emission within a spectral line map (a ``data
cube'') while accounting for the finite sensitivity and resolution of
the data set. We use moment methods \citep[e.g.~][]{srby87}, which
make full use of position and intensity information without assuming a
functional form for the cloud. They are therefore robust to
pathologies in the data {\it within} the cloud.
Moments are, however, sensitive to the inclusion of false emission
(noise) at the edge of a cloud. Including noise has the effect of
artificially increasing the values of the moment. Therefore the
methods outlined here should be employed in conjunction with careful
signal identification so that the calculations include as little noise
as possible. We assume throughout this section that the algorithm is
being applied to a distribution of real emission that we label a
``cloud'' (we discuss signal identification and decomposition in the
Appendix).
\subsection{Moment Measurements of Size, Line Width, and Flux}
This subsection describes how to apply moment methods to derive the
size, line width, and flux from a distribution of emission (a
``cloud'') within a position-position-velocity data cube. The data
cube consists of a number of pixels that have sizes of $\delta x$,
$\delta y$, and $\delta v$ in the two spatial dimensions and the
velocity dimension, respectively. The $i$th pixel in the data cube has
positions $x_i$ and $y_i$, velocity $v_i$, and brightness temperature
$T_i$. We assume that the cloud is contiguous and bordered by an
isosurface in brightness temperature of value $T_{edge}$, so that all
of the pixels in the cloud have $T > T_{edge}$ and the pixels outside
the cloud have $T < T_{edge}$ or are separated from the cloud by
emission with $T < T_{edge}$.
We begin by rotating the spatial axes so that the $x$ and $y$ axes
align with the major and minor axis of the cloud, respectively. We
determine the orientation of the major axis using principal component
analysis. We find the eigenvectors of the intensity-weighted
covariance matrix for the cloud,
\begin{equation}
\nonumber
\frac{1}{\sum_i T_i}
\left[\begin{array}{ll}
\sum_i T_i \left( x_i - \bar{x} \right)^2 &
\sum_i T_i \left( x_i - \bar{x} \right) \left( y_i - \bar{y}
\right) \\
\sum_i T_i \left( x_i - \bar{x} \right)
\left( y_i - \bar{y} \right) &
\sum_i T_i \left( y_i - \bar{y} \right)^2 \\
\end{array}\right] \mbox{ .}
\end{equation}
\noindent In the equations above the sum $\sum_{i}$ runs over
all pixels within the cloud and $\bar{x}$ and $\bar{y}$ are the
intensity weighted mean positions within the cloud (defined below). We
define the new $x$ axis to lie along the eigenvector with the largest
eigenvalue --- the major axis of the cloud. The $y$ axis lies
perpendicular to the $x$ axis along the minor axis of the cloud. In
the discussion below, $x$ refers to position along the major axis and
$y$ refers to position along the minor axis. Rotating the axes in this
manner yields information about the axis ratio of the cloud and allows
a more careful deconvolution. This method for determining the
position angle of molecular clouds has also been adopted by
\citet{koda}.
To measure of the size of the cloud, we compute the geometric mean of
the second spatial moments along the major and minor axis. This is
$\sigma_{r}$, the root-mean-squared (RMS) spatial size:
\begin{equation}
\sigma_{r} (T_{edge}) = \sqrt{\sigma_{maj} (T_{edge})~\sigma_{min} (T_{edge})}
\end{equation}
\noindent where $\sigma_{maj} (T_{edge})$ and $\sigma_{min}
(T_{edge})$ are the RMS sizes (second moments) of the intensity
distribution along the two spatial dimensions. We adopt this
particular functional form since it has been used in previous
observational studies \citep{srby87} and explored in depth by
\citet{bert92} with respect to inclination, aspect ratio, and
virialization. We calculate $\sigma_{maj} (T_{edge})$ and
$\sigma_{min} (T_{edge})$ by:
\begin{eqnarray}
\sigma_{maj} (T_{edge}) &= &\sqrt{\frac{\sum_{i}^{cloud} T_i \left[ x_i - \bar{x}
(T_{edge}) \right]^2}{\sum_{i}^{cloud} T_i}}, \mbox{ where} \\
\bar{x}
(T_{edge}) &= &\frac{\sum_{i}^{cloud} T_i x_i}{\sum_{i}^{cloud} T_i}
\mbox{~and} \\
\sigma_{min} (T_{edge}) & = & \sqrt{\frac{\sum_{i}^{cloud} T_i
\left[ y_i - \bar{y} (T_{edge}) \right]^2}{\sum_{i}^{cloud} T_i}} \mbox{, where} \\
\bar{y} (T_{edge}) & = & \frac{\sum_{i}^{cloud} T_i
y_i}{\sum_{i}^{cloud} T_i} \mbox{.}
\end{eqnarray}
\noindent In the equations above the sum $\sum_{i}^{cloud}$ runs over
all pixels within the cloud. We have written each of the moments as a
function of $T_{edge}$ because changing the isosurface that defines
the boundary of the cloud ($T_{edge}$) will change the set of pixels
included in the sum and therefore the values of the moments. Note that
$\sigma_{r}$ is not the RMS distance ($d = \sqrt{x^2 + y^2}$) from the
center of the cloud. Rather it is the analogous to the RMS size of the
cloud along an arbitrarily chosen axis. Thus, $\sigma_{r} = \sigma_{x}
= \sigma_{y}$ for a perfectly round cloud, while the RMS distance from
the center for such a distribution is larger, $\sigma_{d} = \sqrt{2}
\sigma_{x} > \sigma_{x}$. Also note that $\sigma_{maj} / \sigma_{min}$
is the axis ratio of the cloud and will be $\sim 1$ for round clouds
and $\gg1$ for elongated or filamentary clouds.
We calculate the velocity dispersion, $\sigma_v(T_{edge})$ in the same
manner as the size:
\begin{eqnarray}
\sigma_v (T_{edge})& = &\sqrt{\frac{\sum_{i}^{cloud} T_i \left[ v_i -
\bar{v} (T_{edge}) \right]^2}{\sum_{i}^{cloud} T_i}} \mbox{, where} \\
\bar{v} (T_{edge})& = & \frac{\sum_{i}^{cloud} T_i v_i}{\sum_{i}^{cloud}
T_i} \mbox{ .}
\end{eqnarray}
\noindent The sums again run over all emission in the cloud. For a
Gaussian line profile, such as that found for most clouds, the
full-width half-maximum (FWHM) line width, $\Delta V (T_{edge})$ will
be related to $\sigma_v (T_{edge})$ by
\begin{equation}
\Delta V (T_{edge}) =\sqrt{8\ln(2)}~\sigma_v (T_{edge}) \mbox{.}
\end{equation}
\noindent Finally, we calculate the flux of the cloud,
$F_{\mathrm{CO}} (T_{edge})$ using the zeroth moment:
\begin{equation}
F_{\mathrm{CO}} (T_{edge}) = \sum_i T_i~\delta v~\delta x~\delta y
\mbox{.}
\end{equation}
\noindent If $\delta x$ and $\delta y$ are in units of arcseconds,
$\delta v$ in km s$^{-1}$, and $T_i$ in K, then the resulting flux
will have units of K km s$^{-1}$ arcsecond$^{2}$.
\subsection{Correcting for the Sensitivity Bias}
\label{EXTRAPSECT}
The sensitivity of a dataset influences the cloud properties derived
from that data, a fact that we have emphasized in the previous section
by explicitly writing the moments as functions of $T_{edge}$, the
cloud boundary (usually set by the signal-to-noise ratio of the
data). Figure \ref{EXTRAPMOMS} shows the variation of spatial size,
line width, and flux as a function of sensitivity for a bright cloud
in M33. The data for this cloud shows a substantial {\em sensitivity
bias}; all of the derived properties are strong functions of the
boundary isosurface ($T_{edge}$). In order to compare data sets with
different sensitivities, one must account for this bias. In this
section we describe a method to do this by extrapolating the measured
properties of a cloud---$\sigma_{maj} (T_{edge})$, $\sigma_{min}
(T_{edge})$, $\sigma_v (T_{edge})$, and $F_{\mathrm{CO}}
(T_{edge})$---to those we would expect to measure for a cloud within a
boundary isosurface of $T_{edge} = 0$ Kelvin (i.e., perfect
sensitivity).
We estimate the values of the moments at $T_{edge}=0$~K by
extrapolating from higher values of $T_{edge}$. This technique was
originally suggested for inferring total cloud areas by \citet{bt80}
and applied to molecular cloud properties by \citet{syscw}. We
calculate each of the moments for a sample of boundary temperatures,
$T_{edge}$, ranging from near the peak temperature of the cloud to the
lowest boundaries allowed by the data. Thus, we measure the variations
of the moments as a function of the boundary temperature, $T_{edge}$,
within the cloud (this is how we constructed the plot shown in Figure
\ref{EXTRAPMOMS}). Below, we assume that we have measured each of the
four moments for values of $T_{edge}$ ranging from $T_{min}$ (the
minimum allowed by the data, that is the sensitivity limit) to
$T_{max}$ (near the peak temperature of the cloud).
We estimate the value of the moments at $T_{edge} = 0$~K by performing
a weighted, linear least-squares fit to the measured moments. As an
example, we consider $\sigma_{maj} (T_{edge})$. The data are modeled as
\begin{equation}
\sigma_{maj} (T_{edge}) = m~T_{edge} + \sigma_{maj} (0~\mbox{K})
\end{equation}
and the fit determines the extrapolated moment, $\sigma_{maj}
(0~\mbox{K})$. For the fit, each pair of data $\{T_{edge},
\sigma_{maj}(T_{edge})\}$ is assigned a weight proportional to the
number of data in the cloud with $T>T_{edge}$, so that measurements of
the moment using more data are weighted more heavily. Practically,
this means that points to the left in Figure \ref{EXTRAPMOMS} have
higher weights than those to the right. We use this linear
extrapolation for $\sigma_{maj}$, $\sigma_{min}$, and $\sigma_v$, but
we find that a quadratic extrapolation (including a $T_{edge}^2$ term)
gives better results for the zeroth moment, $F_{\mathrm{CO}}$ (though
we revert to a linear extrapolation when the extrapolated flux is
lower than the measured flux). We plot the flux of a Gaussian cloud
as a function of $T_{edge}$ in Figure \ref{FLUXFIG} for an uncorrected
zeroth moment and the linear and quadratic extrapolations to
illustrate the appropriateness of the quadratic fit to the zeroth
moment. At very low sensitivities (signal-to-noise ratios near unity),
the quadratic extrapolation is very noisy, but for signal-to-noise
ratios of $2$ or better it does a dramatically better job of recovering
the true flux of the cloud ($F/F_0 = 1$) than either the linear
extrapolation or no extrapolation.
Figure \ref{EXTRAPMOMS} also shows these extrapolations for a bright
cloud in M~33. The result of this extrapolation is a set of four
moments --- $\sigma_{maj} (0~\mathrm{K})$, $\sigma_{min} (0
~\mathrm{K})$, $\sigma_v (0~\mathrm{K})$, and $F_{\mathrm{CO}} (0
~\mathrm{K})$ --- that correspond to those we would expect to measure
given infinite sensitivity. The values of these moments should be
directly comparable even among datasets with different sensitivities
(values of $T_{min}$).
Note that diffuse emission surrounding a GMC may confuse this
method. If one data set is measured with sensitivity sufficient to
detect diffuse emission surrounding a GMC, while another lacks the
sensitivity to do so then this approach may not be sufficient to
correct for the sensitivity bias. This problem may be particularly
acute when comparing Galactic GMCs observed with very good sensitivity
to extragalactic clouds with worse signal-to-noise
ratios. Interferometric data ``resolves out'' emission significantly
more extended than the synthesized beam (see \S\ref{INTERF}),
representing another bias against detecting diffuse
emission. \citet[][]{polk},
\citet[][]{blitz85}, \citet[][]{ros03}, and \citet[][]{ler05} find
evidence for diffuse emission surrounding GMCs in the Milky Way and
the Local Group Galaxies M~31, M~33, and IC~10, respectively.
\subsection{Correcting for the Resolution Bias}
\label{BEAMDCSECT}
Any astronomical data set represents the convolution of the intensity
of the source with the profile of the instrument used to observe
it. Care must therefore be taken in measuring sizes and line widths
when the extent of the intensity distribution is comparable to the
instrumental profile. In a typical spectral line data cube two
profiles are important: the spatial beam and the width of a velocity
channel. In this section, we describe simple corrections to account
for the effects of finite spatial and spectral resolution.
We ``deconvolve'' the spatial beam from the measured cloud size by
subtracting the RMS beam size, $\sigma_{beam}$, from the extrapolated
spatial moments, $\sigma_{maj} (T_{edge} = 0~\mathrm{K})$ and
$\sigma_{min} (T_{edge} = 0~\mathrm{K})$, in quadrature --- an approach
that is exact for Gaussians. The deconvolved second moment is given
by:
\begin{equation}
\sigma_{r,dc} = \sqrt{ [ \sigma_{maj}^2 \left(0~\mathrm{ K} \right)
- \sigma_{beam}^2 ]^{1/2} ~ [ \sigma_{min}^2 \left(0~\mathrm{
K} \right) - \sigma_{beam}^2 ]^{1/2}}~\mbox{ ,}
\end{equation}
\noindent where $\sigma_{maj} \left(0~\mathrm{ K} \right)$ and
$\sigma_{min} \left(0~\mathrm{K} \right)$ are extrapolated to the 0
Kelvin isosurface as described in \S\ref{EXTRAPSECT}. This
extrapolation is necessary to make this deconvolution valid:
subtracting the full $\sigma_{beam}$ from the spatial moment measured
for only part of the cloud will lead to an overcorrection and thus to
an underestimate of the cloud size. Measuring the spatial size along
the minor axis is also necessary to ensure that the cloud is indeed
resolved in all dimensions. This is an advantage of the choice of axes
(major/minor) described above. With sufficient signal-to-noise, this
method of deconvolution provides a robust measurement of cloud size
even for marginally resolved clouds.
Instrumental resolution also affects the measured line
width. Spectrometers measure the average intensity across a channel,
rather than sampling the intensity at the center (nominal frequency)
of that channel. When the width of the spectral line under
consideration is comparable to the bandwidth of a single channel, the
line strength varies significantly across an individual channel. In
this case, the average value may differ substantially from the value at
line center. The output of the spectrometer is thus a convolution of
the true spectral profile with the profile of an individual
channel. We account for this potential bias towards higher line widths
by a simple deconvolution of the channel width from the extrapolated second
moment:
\begin{equation}
\sigma_{v,dc}=\sqrt{{\sigma_{v}^2 \left(0~\mathrm{
K}\right)-\frac{\Delta V^2_{chan}}{2~\pi}}}
\end{equation}
where $\sigma_{v} \left(0~\mathrm{ K} \right)$ is the second moment of
the cloud in the $v$ dimension extrapolated to 0 Kelvin as described
in \S \ref{EXTRAPSECT} and $\Delta V_{chan}$ is the width of a
velocity resolution element. Although the channel profiles are usually
square in shape and not Gaussian, we simplify the deconvolution by
approximating the channel shape as a Gaussian with integrated area
equal to that of a square channel with width $\Delta V_{chan}$. For
such a Gaussian, $\sigma_{chan}=\Delta V_{chan}/\sqrt{2~\pi}$.
\subsection{Comparison with Other Methods}
\label{compare}
We use extrapolated moments to measure GMC properties rather than
employing an established method from the literature. In this section,
we justify our choice by comparing several methods of measuring GMC
properties. We focus on the performance of these methods at marginal
resolution and low signal-to-noise, conditions typical of
extragalactic GMC observations.
Determining the radius of a cloud is particularly difficult because
GMCs are often asymmetrical with poorly defined boundaries. Several
authors have devised methods to return a single characteristic size
for complicated emission distributions. The intensity-weighted second
moments in the spatial directions have been used in many studies
\citep[e.g.~][]{srby87}, but are sensitive to both noise and
convolution effects. The other commonly used method
\citep[e.g.~][]{clumpfind,hc01} is to infer the radius based on the
area of the cloud:
\[R_e = \sqrt{\frac{A-A_{pt}}{\pi}}.\]
Here $A_{pt}$ is the area of a point source that has been convolved
with a beam and measured with the same signal-to-noise as the emission
in the map. Finally, \citet{ws90} and \citet{tay99} adopted the size
of the cloud as the mean of the deconvolved FWHMs of the emission
distribution along two perpendicular directions.
We compare these three methods to the extrapolated moment method
presented above across a range of resolutions and sensitivities. We
measure the size of a Galactic GMC using each method after convolving
it to a desired resolution and adding noise to produce a particular
signal-to-noise ratio. For the data, we use the $^{12}$CO data from
the Rosette molecular cloud \citep{bs86}, which we clip at
2$\sigma_{RMS}$ (the RMS noise in the original data set) and integrate
in the velocity dimension to produce a map of integrated
intensity. For a range of sensitivities and resolutions, we convolve
this map with a Gaussian beam and add noise. We measure the size of
the cloud in 100 realizations of the noise for each such \{resolution,
sensitivity\} pair using (a) the extrapolated moment method, (b) the
moment of the data without extrapolation, (c) the area method and (d)
the FWHM method. Wherever possible we corrected for the effects of
beam convolution and signal-to-noise for each of the methods. The
results of the analysis are shown in Figure \ref{radplot}.
Figure \ref{radplot} shows the recovered radius as a function of
resolution and sensitivity, with the ``true'' radius defined as that
measured at very high sensitivity and very good resolution (i.e. in
the original data, the top right corner of each plot). The hashed
region of parameter space shows the range of parameters over which
each algorithm recovers a radius within 10\% of the true value. The
extrapolated moment method has the largest hashed region and so is
remarkably robust, recovering the true radius over a large range of
sensitivities and resolutions. Only at low sensitivity
($T_{max}/\sigma_{RMS}<5$) and marginal resolution
($\sigma_{beam}\gtrsim \sigma_0$), does the derived radii depart
systematically from the true radius. Notably, the extrapolated second
moment performs quite well at signal-to-noise ratios from 5 to 10 (in
the integrated intensity map), values typical of extragalactic CO data
sets. By contrast, the uncorrected moment method (panel b in Figure
\ref{radplot}) underestimates the size at low signal-to-noise since
(by construction) the uncorrected moment does not account for emission
below the noise level. Similarly, the area method (panel c) shows
systematic variation at both low signal-to-noise (where emission drops
below the noise level) and low resolution (where the convolved area of
the emission distribution grows disproportionately because of the
filamentary nature of the cloud). Finally, the FWHM method (panel d)
shows large systematic variations since it depends only on the
location of the FWHM contour and not on the remainder of the emission
distribution.
The region of systematic underestimation at low signal-to-noise but
reasonable resolution shows the effects of missing the diffuse
emission mentioned in \S\ref{EXTRAPSECT}. The Rosette includes more CO
emission at low intensities than the extrapolated moment predicts from
the high intensity data. As a result, when that diffuse emission is
not included in the measurement, the algorithm underestimates the true
radius of the GMC. This effect is seen panels (a) and (b) --- the
extrapolated and uncorrected second moments --- and is more pronounced
in the uncorrected second moment, panel (b).
We perform a similar experiment on recovering the line width of an
emission line in noisy data. We measure the recovered line width of a
Gaussian line of known width using three methods (a) the extrapolated
moment method (b) an uncorrected second moment and (c) a Gaussian fit
to the line. For a range of signal-to-noise levels
($T_{max}/\sigma_{RMS}$) and channel widths $\Delta V_{chan}$ we
measure the recovered line width relative to the known line width.
The mean values of the recovered line for 1000 realizations of the
noise are plotted in Figure \ref{dvplot}. The extrapolated moment
does not show the systematic variation with signal-to-noise seen in
the uncorrected moment. The extrapolated moment is nearly as robust a
measure as the Gaussian fit for a perfectly Gaussian line and will
prove superior if the line is not Gaussian. Robust recovery of the
line width using any method requires $\Delta V_{line} / \Delta
V_{chan} > 2$.
\subsection{Assessing Errors in GMC Properties}
The formal uncertainty associated with each moment measurement is
quite small. Cloud identification and extrapolation represent larger
sources of uncertainty, but their effects are difficult to assess
formally. We use bootstrapping methods to estimate the uncertainties
in our measurements of cloud properties. The bootstrapping method
determines errors by generating several trial clouds from the original
cloud data. A trial cloud is generated by considering the cloud to be
a collection of data $\{x_i,y_i,v_i,T_i\}$ for $i=1\dots N$, the
number of points in the cloud. The data are sampled for $N$ random
values of $i$, allowing for $i$ to be repeated. The properties of the
cloud are measured for each trial cloud. We estimate the uncertainty
from the variance of the cloud properties derived from these resampled
and remeasured data sets. The final uncertainty in each property is
the standard deviation of the bootstrapped values scaled up by the
square root of the oversampling rate. The oversampling rate, which is
usually equal to the number of pixels per beam, accounts for the fact
that not all of the data in each cloud are independent. For many
interferometric data sets this is an important effect, since these
data can have $10$ or more pixels per beam.
We compare the uncertainties produced by the bootstrapping to those
derived from repeatedly adding noise to and then reanalyzing a data
set. We use the bright cloud in M~33 shown in Figure
\ref{EXTRAPMOMS}. We conduct 100 realizations of the data plus random
noise. The resulting uncertainties in $\sigma_{maj}$, $\sigma_{min}$,
$\sigma_v$, and the flux are $3\%$, $2\%$, $3\%$, and
$3\%$. Repeatedly bootstrapping the same data set (adjusted to have
the same final noise level) yields average uncertainties of $9\%$,
$9\%$, $11\%$, and $5\%$. The bootstrapping estimates are higher for
this bright cloud because they reflect both the formal uncertainty and
the robustness of the result to the removal of a given piece of
data. In the low signal-to-noise regime, the values for the two
methods converge as noise dominates the uncertainty derived from
bootstrapping --- for example, performing the same experiment in a
dimmer M~33 cloud with $1/4$ the luminosity of the bright cloud and
comparable noise, bootstrapping yields errors of $31\%$, $33\%$,
$32\%$, and $35\%$ in the four moments while repeated realizations
produces scatters of $15\%$, $27\%$, $30\%$, and $40\%$.
The bootstrapping method produces a robust, believable estimate of the
uncertainty in the measurement of the properties of a particular,
defined cloud. It does not account for uncertainties in the assignment
of emission to a cloud either as a result of noise or choice of
algorithm. These uncertainties are more systematic than random in
nature and may be best assessed by analyzing the emission distribution
using several methods. The bootstrapping estimate may be treated as an
accurate estimate of the uncertainties in the results {\em given} that
one adopts the methods presented in this paper.
\section{Deriving Physical Quantities from Moment Measurements
\label{PHYSICAL}}
In this section, we outline how to use the measured size, line width,
and flux to calculate several physical quantities of interest: the
effective spherical radius, the virial mass, and the luminous
mass. Throughout this section we assume that clouds can be described
as self-gravitating spheres with density profiles $\rho \propto
r^{-1}$ and negligible support from magnetic fields or confinement by
external pressure.
We assume below that the data consists of observations of the \co\
\jone\ transition, in units of brightness temperature, but the method
is readily adaptable to analogous data sets.
\subsection{The Spherical Radius}
We define a factor $\eta$ that relates the one-dimensional RMS size,
$\sigma_r$, to the radius of a spherical cloud $R$: $R=\eta \sigma_r$.
It is possible to derive an estimate for $\eta$ based on spherical
cloud of radius $R$ with a density profile $\rho \propto r^{-\beta}$.
In this model,
\begin{eqnarray}
\sigma_r^2 &=& \frac{\int_0^R dr \int_0^{2\pi} d\theta \int_0^{\pi} d\phi~x^2
\rho_0 r^{-\beta} r^2 \sin \phi}{\int_0^R dr
\int_0^{2\pi} d\theta \int_0^{\pi} d\phi~ \rho_0 r^{-\beta} r^2 \sin \phi} \\
\sigma_r^2 &=& \frac{1}{3}~\frac{3 - \beta}{5 - \beta} R^2 \\
\mathrm{so~that~} \eta &=& \sqrt{3~\frac{5 - \beta}{3 -
\beta}}
\end{eqnarray}
\noindent For a cloud with $\beta = 1$, $\eta = \sqrt{6}$, somewhat
higher than the empirical correction of $3.4/\sqrt{\pi}$ derived by
Solomon et al. (1987). The difference arises, in part, from using
$^{12}$CO as a density tracer. Since $^{12}$CO emission saturates in
dense regions and vanishes from low density regions, the apparent
density profile in $^{12}$CO is shallower than the true density
profile. Hence, an appropriate value of $\eta$ likely falls in the
range between $3.4/\sqrt{\pi}$ and $\sqrt{6}$. For tracers like
$^{13}$CO with higher critical densities, a different value of $\eta$
may be appropriate. Since comparison to this ``anchoring'' data set
may be more important than adopting a self-consistent --- but grossly
oversimplified --- model for a cloud, we recommend the Solomon et
al. (1987) definition of the cloud radius, $R \approx 1.91
\sigma_{r}$. Note, that adjusting the definition of the radius
renders the virial mass formula we present below inexact.
\subsection{The Spatial Size, Line Width, Luminosity, and Mass}
We convert the cloud properties $\sigma_r ( 0~\mathrm{K})$, $\sigma_v
( 0~\mathrm{K})$, and $F_{\mathrm{CO}} ( 0~\mathrm{K})$ to the physical
quantities $R$, $\Delta V$, and $L_{\mathrm{CO}}$. For a cloud at a distance
of $d$ (in parsecs), the physical radius will be
\begin{equation}
R[\mathrm{pc}] = \frac{R ( 0~\mathrm{K}) [\mathrm{arcsec}]}{3600}
\times \frac{\pi}{180} \times d [\mathrm{pc}]~\mathrm{,}
\end{equation}
\noindent the FWHM line width will be
\begin{equation}
\Delta V = \sqrt{8\ln (2)}~\sigma_v ( 0~\mathrm{K})~\mathrm{,}
\end{equation}
\noindent and the luminosity of the cloud, $L_{\mathrm{CO}}$, will be
\begin{eqnarray}
\nonumber
L_{\mathrm{CO}} [\mathrm{K~km~s}^{-1}\mathrm{~pc}^2] &=&
F_{\mathrm{CO}}(0~\mathrm{K}) [\mathrm{K~km~
s}^{-1}~\mathrm{arcsec}^2] \\
\nonumber
&\times& (d[\mathrm{pc}])^2\\
&\times& \left(\frac{\pi}{180\cdot 3600}\right)^2.
\end{eqnarray}
\noindent A particular CO luminosity, $L_{\mathrm{CO}}$, implies a mass of
molecular gas, $M_{\mathrm{Lum}}$, of
\begin{eqnarray}
\nonumber
{M_{\mathrm{Lum}}}~[{M}_{\odot}]& = &\frac{X_{\mathrm{CO}}}{2
\times 10^{20} [\mathrm{cm}^{-2}/(\mathrm{K~km~s}^{-1})]} \times
4.4~L_{\mathrm{CO}}\\
&\equiv& 4.4~X_2~L_{\mathrm{CO}}
\end{eqnarray}
\noindent where $X_{\mathrm{CO}}$ is the assumed CO-to-H$_2$
conversion factor. This calculation includes a factor of 1.36 (by
mass) to account for the presence of helium. Including helium is
important to facilitate comparison with the virial mass, which should
reflect all of the gravitating mass in the cloud. We have adopted a
fiducial value of the CO-to-H$_2$ conversion factor of
$X_{\mathrm{CO}}=2\times 10^{20}\mbox{ cm}^{-2} (\mbox{K km
s}^{-1})^{-1}$ and express changes relative to this value in terms of
the parameter $X_2$.
\subsection{The Virial Mass}
We compute the virial masses under the assumption that each cloud is
spherical and virialized with a density profile described by a
truncated power law of the form $\rho \propto r^{-\beta}$ with no
magnetic support or pressure confinement. As with the spherical radius
correction, the exact density profile of the cloud will affect the
correct form of the virial theorem mass. For $\beta = 1$, the virial
mass is given by the formula (Solomon et al. 1987):
\begin{equation}
{M_{\mathrm{VT}}} = 189~M_{\odot}~\Delta V^2 \, R
\end{equation}
\noindent and more generally by
\begin{equation}
{M}_{\mathrm{VT}} = 125~M_{\odot}~\frac{5-2\beta}{3-\beta}~\Delta V^2 \,
R~\mathrm{,}
\end{equation}
\noindent where $\Delta V$ is the FWHM velocity line width in \kmpers,
$R$ is the radius in pc, and the cloud has a density profile of $\rho
\propto r^{-\beta}$.
Clouds exhibit a range of non-spherical geometries and may be
supported by magnetic fields or confined by pressure. Therefore, the
studying the virial parameter may be more useful than the virial mass
itself. The virial parameter is a constant of order unity that
characterizes deviations from the virial theorem applied to a
non-magnetic cloud with no external pressure and constant density.
Following \citet{mckee-vt}, we define the virial parameter as
\begin{equation}
\alpha = \frac{5 \sigma_v^2 R}{M_{\mathrm{Lum}} G} =
\frac{5 \eta \sigma_v^2 \sigma_r}{(4.4 X_2 L_{\mathrm{CO}})
G}~\mbox{.}
\end{equation}
Larger-than-unity virial parameters can result from pressure
confinement, while $\alpha<1$ may result from significant magnetic
support. Incorrect values of the CO-to-H$_2$ conversion factor may
skew the result in either direction. Finding $\alpha<2$ means that the
clouds are gravitationally bound in the absence of significant
magnetic support.
\section{Measuring Cloud Properties from Interferometer Observations}
\label{INTERF}
Millimeter-wave interferometers are required to resolve even the most
massive molecular clouds in galaxies beyond the Magellanic clouds (see
Figure \ref{DISTCOMP}). Unfortunately, interferometers are not
sensitive to spatial frequencies outside the limited region of the
$(u,v)$ plane that they sample. Practically, this means
interferometers do not measure the total flux from the emission
distribution; and structures are resolved out, usually on large angular
scales that correspond to small separations in the $(u,v)$ plane.
Ideally, interferometer observations are combined with single-dish
observations that supply the missing information. In practice such
observations are conducted infrequently and the unknown total flux and
short-spacing information is estimated using deconvolution algorithms
such as CLEAN or maximum entropy
\citep[see][and references therein]{tms}.
\citet{she00} simulate the results of using only an interferometer to
observe Galactic GMCs as if these well-studied clouds were located in
M31. They find that interferometers experience significant (50\%)
flux loss for their simulated observation, primarily from extended
emission around clouds. However, they find that the flux loss does not
change the size and line width of the cloud. \citet{hel02} examine the
recovery of large-scale flux distributions from interferometer
measurements in more depth and explore the effectiveness of
deconvolution algorithms at low signal-to-noise. They find that
deconvolution algorithms recover flux nonlinearly at low
sensitivities, finding much less flux at low sensitivities than one
would expect. Since much of the data on extragalactic GMCs have low
signal-to-noise, this may represent an important bias.
We assess the effects of interferometric biases on the methods
presented here by extending the method of \citet{she00}. We use
$^{12}\mathrm{CO}$ observations of three galactic GMCs: the Orion
molecular complex \citep{wil05}, the Rosette Molecular Cloud
\citep{bs86}, and an excerpt from the Outer Galaxy Survey of
\citet{hc01} which contains the molecular clouds associated with the
W3/W4/W5 \ion{H}{2} regions\footnote{A map of the original Orion data
and typical maps at low resolution and sensitivity appear in the
Appendix.}. We simulate observing these three molecular complexes in
M31 ($D=770$ kpc) with the BIMA interferometer.
We Fourier transform each plane of each data set into the $(u,v)$
domain and resample the data along the $(u,v)$ tracks that would be
sampled by BIMA observations of the data provided the GMCs were in
M31. The $(u,v)$ coverage reflects typical observing strategies for
extragalactic clouds, such as interleaving observations of the source
with calibrators and other sources. We add thermal noise and phase
noise to the $(u,v)$ data, including a phase noise component with a
magnitude that depends on the length of the baseline. We adjust the
scale of the thermal and phase noise to produce the desired peak
signal-to-noise (we find our results depend only weakly on whether the
noise is thermal or phase). We invert the resulting $(u,v)$ data using
the MIRIAD software package \citep{miriad} producing maps separated by
2 km s$^{-1}$, and then we deconvolve the dirty maps using a CLEAN
algorithm that terminates at the 2$\sigma_{RMS}$ level. For each trial
cloud, we then calculate the cloud properties using the methods of
\S\S \ref{MOMENTS} and \ref{PHYSICAL}.
For comparison, we compute cloud properties using the same procedure
to simulate single-dish observations with signal-to-noise and
effective resolution identical to the mock interferometer data. We
generate the mock single-dish observations by sampling the transformed
image data for an equal number of $(u,v)$ points as the interferometer
data, but the points are normally distributed in the $(u,v)$ plane and
one point is forced to lie at $(0,0)$ thus sampling the total
power. The width of the $(u,v)$ point distribution is chosen to give a
beam size similar to that of the mock interferometer data. Random
thermal and phase noise is added to these data in the same fashion as
for the interferometer data. Then the data are inverted using natural
weighting and deconvolved in the exact same fashion as the
interferometer data (though the deconvolution step has little effect).
Again, we extract cloud properties using the methods of
\S\S\ref{MOMENTS} and \ref{PHYSICAL}.
With these simulations, we compare the cloud properties derived from
interferometer data to single-dish data that are equivalent in every
other fashion, thereby isolating the effects of interferometers on the
derived properties. The additional biases imposed by limited
resolution and sensitivity are discussed separately in
\S\ref{compare}. Here we focus on mock BIMA observations of the three
molecular complexes using three antenna configurations: the C array
(extended), the D (compact) array, and a combination of C and D array
\citep[see][for details on the configurations]{wright-configs}.
The synthesized beam sizes for these configurations are $14.4''$,
$6.1''$ and $8.8''$ for the D, C and C+D hybrid array observations
respectively, corresponding in turn to 54, 22 and 33 parsecs at
770~kpc. We conduct 10 sample observations for each cloud in each
array at a range of sensitivities ranging from
$T_{peak}/\sigma_{RMS}=3$ to $>100$.
Figure \ref{interf_props} shows the properties recovered by mock
observations of the Orion molecular complex for each array and a range
of sensitivities. The values of each properties are normalized by the
value recovered by mock single-dish observations at the same
sensitivity and resolution. Thus, the only difference between the
four sets of properties (C, D, C+D, and single dish) is the $(u,v)$
coverage of the simulated observations. We find that the derived
properties from mock single-dish observations follow the same behavior
as the simulations in \S\ref{compare}. Thus, it is possible to
decouple the two sets of biases -- those arising from marginal
resolution and sensitivity and those arising from using
interferometers -- and examine only the latter. We plot the results
for Orion because these observations show the most dramatic variation
of the three complexes studied, but the results are qualitatively the
same for all three data sets. Orion is the most sensitive of the three
to spatial filtering because it consists of three GMCs and therefore
shows more structure than the other two targets.
Based on the results of the mock observations, we make the following
comments regarding the use of interferometer data alone in measuring
cloud properties. Most of these points can be seen visually in Figure
\ref{interf_props}.
\begin{enumerate}
\item Cloud properties measured from interferometric data are
biased. The degree of bias is affected by the sensitivity of the array
as well as the $(u,v)$ coverage of the observations.
\item A minimum signal-to-noise of 10 is required for stable recovery
of cloud properties. Below this level, errors in cloud properties,
can approach 100\% for interferometer data.
\item Even for intermediate signal-to-noise values
($T_{peak}/\sigma_{RMS}=10$) there are significant systematic effects
on the cloud properties. The most extreme effects are on the
luminosity measurement, which can be 40\% lower than a single-dish
observation. This effect is much less pronounced for measurements of
the line width and the radius, which show $\lesssim 10\%$
variations. This result, that the radius and line width are relatively
robust to the spatial filtering of the interferometer, confirms the
qualitative results of \citet{she00}. The values of derived properties
are always {\em underestimated} relative to single-dish observations.
\item Even at high sensitivities, the spatial filtering of
interferometers affects property recovery at the 10\% level. For
example, C-array observations of Orion systematically underestimate
the radius of the cloud by 10\% even for very high signal to noise
ratios and D-array observations underestimate the flux of the Orion by
5\% even at high sensitivity.
\item For interferometer observations, the dynamical mass measurements
of GMCs are more robust than the luminosity measurements. This
behavior will bias interpretations of the virial parameter in
extragalactic observations. Estimates of the CO-to-H$_2$ conversion
factor based on the assumption that GMCs are bound or virialized are
likely to {\em overestimate} $X_{\mathrm{CO}}$.
\item For a given signal-to-noise value, observations with the widest
range of $(u,v)$ coverage provide the most robust measurement of cloud
properties. Thus, in achieving a given sensitivity, observers should
favor arrays with more antennae or observations made in multiple
configurations.
\end{enumerate}
\section{A Note on Decomposition}
\label{DECOMPSECT}
The choice of how to decompose an emission distribution into
individual clouds may be the most important source of bias in GMC
property measurements. Many different methods have been applied to
identify GMCs in blended emission, the most prevalent being
decomposition by eye and the application of the GAUSSCLUMPS
\citep{gaussclumps} or the CLUMPFIND algorithm \citep{clumpfind}.
When comparing GMCs between two data sets, care must be taken to
decompose emission in a consistent way across both data sets,
preferably using the same algorithm on both data sets. Furthermore,
the physical values of any tuning parameters in the algorithms should
be matched where possible so that both algorithms search for peaks in
the emission over the same spatial scale (rather than angular or
resolution-units) or intensity range. This will avoid, for example,
comparing ``clumps'' in a Galactic molecular cloud to GMCs in another
galaxy. A more extreme method to ensure accurate comparison is to
convolve the higher (spatial) resolution data set to the spatial
resolution of the other data \citep[e.g.~][]{she00}. However, this
approach clearly sacrifices accuracy of the derived parameters to
allow a more careful comparison between two data sets.
In the Appendix to this paper we present a robust, conservative, new
decomposition algorithm. This algorithm is designed to avoid creating
spurious clouds from noise and to remain sensitive to non-Gaussian
structures in the data. Additionally, the parameters of the algorithm
are fixed to physical values rather than being determined by the
data. This algorithm is designed explicitly with the goal of
decomposing emission into GMCs (rather than clumps or other
structures) with extragalactic data in mind. The methods for
measurement of cloud properties described above --- including the
sensitivity and resolution corrections --- are independent of the
decomposition algorithm and are important no matter what decomposition
algorithm is chosen. In order to avoid confusion between these two
separate problems, we choose to describe the decomposition algorithm
in the Appendix.
\section{Discussion and Conclusions}
\label{APPLICATIONS}
We conclude the paper by applying these methods to molecular line data
sets that have been previously published. In future studies, the
algorithm will be used to evaluate the differences between GMC
populations between galaxies. Here, we simply present an analysis
designed to demonstrate the method's utility. We present the median
corrections found for a large set of extragalactic (Local Group)
observations and a test application of our methods to Galactic
data. We use all the methods discussed in the previous sections and
the decomposition algorithm discussed in the Appendix.
\subsection{The Effects of Extrapolation and Deconvolution}
We apply the methods outlined here to an array of data from across the
Local Group and measure the properties of 110 spatially resolved
clouds to estimate the typical magnitude of the sensitivity and
resolution corrections for extragalactic data. We use BIMA data on
M~33 and M~31 \citep{ros03,ros05}; NANTEN observations of the LMC
\citep[][]{fu99}; OVRO observations of IC~10 \citep[][]{wa05}; and a
SEST map of N83 in the SMC \citep[][]{bol03}. Table \ref{corrections}
shows the number of clouds measured in each galaxy along with the
median sensitivity and resolution corrections applied to the radius,
line width, and flux. For comparison, we also measure the properties
of a number of clouds in the outer Galaxy (Quadrant 2, see below) from
the \citet[][]{mwco} CO survey of the Milky Way. Table
\ref{corrections} includes all spatially resolved clouds with
masses (derived from the CO luminosity) of $5 \times 10^4$ M$_{\odot}$
or more ($92$ of the $110$ extragalactic clouds are above this mass).
The numbers quoted in Table \ref{corrections} are ``correction
factors,''
\begin{equation}
\frac{R_{corrected}}{R_{uncorrected}},
~\frac{\sigma_{v,corrected}}{\sigma_{v,uncorrected}},\mbox{ and }
\frac{L_{\mathrm{CO},corrected}}{L_{\mathrm{CO},uncorrected}} \mbox{ .}
\end{equation}
Table \ref{corrections} shows that throughout the Local Group data the
corrections suggested in this paper have magnitudes of a few tens of
percent. We draw several conclusions based on these data:
\begin{enumerate}
\item Resolution effects on the size of clouds tend to be significant
--- we would overestimate cloud sizes by $\sim 40$\%\ if we did not
apply a deconvolution. In the Milky Way data, this effect is much
less severe. The sizes of Milky Way clouds are measured to within
$5\%$ before the resolution correction. Unresolved clouds do not
contribute to Table \ref{corrections}, so if the effects of the
resolution bias were completely neglected this would be much larger
(a naive approach would measure these clouds to have the size of the
spatial beam).
\item Resolution effects on the line width are negligible throughout
the Local Group data.
\item Sensitivity effects are also significant. Without a correction
for the sensitivity bias, the size, line width, and luminosity of
clouds would all be significantly underestimated. This sensitivity
bias is least severe --- only about 20 -- 30\%\ --- for the line
width, and most significant (and variable) for the
luminosity. Sensitivity corrections to the luminosity vary from 20\%
to more than 100\%.
\item The resolution and sensitivity biases for the size measurement
tend to cancel out, so that the completely uncorrected radius
measurement is often within 10 -- 20\%\ of the corrected value. This
is a happy coincidence of resolution and sensitivity within the Local
Group, not evidence that sensitivity and resolution corrections are
unimportant.
\item The magnitude of corrections across the Local Group data are
fairly uniform. This is because GMCs near the resolution limit tend to
outnumber higher mass GMCs. Unresolved GMCs are not included in the
analysis, so the median cloud through all the data sets appears
marginally resolved.
\item In order to compare extragalactic data to Galactic data (with
very good sensitivity and resolution and therefore small corrections)
it is crucial to correct for the sensitivity and resolution biases.
\end{enumerate}
\begin{center}
\begin{deluxetable*}{l c c c c c c}
\tablecaption{\label{corrections} Typical Corrections for Local Group Data}
\tablehead{ \multicolumn{1}{l}{Galaxy} & \multicolumn{1}{c}{$N_{Clouds}$} &
\multicolumn{3}{c}{Sensitivity Correction} &
\multicolumn{2}{c}{Resolution Correction} \\
\multicolumn{1}{l}{} & \multicolumn{1}{c}{} &
\multicolumn{3}{c}{$\overbrace{\phm{SpanningSpann}}$} &
\multicolumn{2}{c}{$\overbrace{\phm{SpanningSpann}}$} \\
\multicolumn{1}{l}{} & \multicolumn{1}{c}{} &
\multicolumn{1}{c}{$\frac{R_{corrected}}{R_{uncorrected}}$} &
\multicolumn{1}{c}{$\frac{\sigma_{v,corrected}}{\sigma_{v,uncorrected}}$} &
\multicolumn{1}{c}{$\frac{L_{CO,corrected}}{L_{CO,uncorrected}}$} &
\multicolumn{1}{c}{$\frac{R_{corrected}}{R_{uncorrected}}$} &
\multicolumn{1}{c}{$\frac{\sigma_{v,corrected}}{\sigma_{v,uncorrected}}$} \\}
\startdata
LMC & 46 & $1.4$ & $1.2$ & $1.7$ & $0.8$ & $1.0$ \\
M~31 & 28 & $1.5$ & $1.3$ & $1.6$ & $0.7$ & $1.0$ \\
IC~10 & 17 & $1.7$ & $1.3$ & $2.3$ & $0.7$ & $1.0$ \\
M~33 & 15 & $1.4$ & $1.2$ & $1.5$ & $0.7$ & $1.0$ \\
SMC & 4 & $1.1$ & $1.2$ & $1.2$ & $0.7$ & $1.0$ \\
MW\tablenotemark{a} & 107 & $1.1$ & $1.1$ & $1.4$ & $1.0$ & $1.0$ \\
\enddata
\tablenotetext{a}{Quadrant 2 clouds with $M > 5 \times 10^4$ M$_{\odot}$.}
\end{deluxetable*}
\end{center}
\subsection{Analysis of Second Quadrant CO Data}
The method described in this paper has been designed with
extragalactic data in mind. However, a crucial step in interpreting
extragalactic measurements is to make a fair comparison with Galactic
data. In this section we report some results of applying our
decomposition and measurement algorithms to the survey of the second
Galactic quadrant by \citet{mwco}. We compare the results of this
analysis to the results by \citet[][]{hc01} and show that our analysis
recovers results that are consistent with theirs.
We decompose and analyze $^{12}$CO($1\to 0$) from the second quadrant
\citep[Survey 17 in Table 1 of][]{mwco}. The data set covers the
Galactic plane from $\ell = 70^{\circ}$ to $\ell = 210^{\circ}$ with a
noise level of $0.3$ K. We measure the distance to the molecular
emission using the kinematic distances by adopting a flat rotation
curve with $\Theta_{\mathrm{LSR}} = 220$~km~s$^{-1}$ and
$R_\odot=8.5$~kpc. We omit local emission by discarding all elements
of the data cube with a kinematic distance less than 2 kpc as well as
all elements in the data cube that are connected by significant
emission in position or velocity space to such pixels. We apply the
decomposition algorithm described in the Appendix and measured sizes,
line widths, and luminosities of GMCs using the methods of \S
\ref{EXTRAPSECT} and \S \ref{BEAMDCSECT}. The analysis
recovers 431 clouds with resolved angular sizes and line widths
located within 10 kpc of the Sun. We include the
median sensitivity and resolution corrections for massive ($>5 \times
10^4$ M$_{\odot}$) clouds in Table \ref{corrections} above.
Do the results from our algorithm agree with previous studies of
Galactic GMCs? The data set covers the region studied by \citet{hc01}
using the $45''$ resolution of the FCRAO 14m. That data set has a
lower sensitivity than the \citet[][]{mwco} data, so we apply a
rudimentary sensitivity correction (assuming that the GMCs are
Gaussian and using their peak temperature and boundary isosurface) to
their results and scale to $X_{\mathrm{CO}}=2\times 10^{20}\mbox{
cm}^{-2} (\mbox{K km s}^{-1})^{-1}$ before comparing GMC
properties. We focus on a comparison of the virial parameter between
the two studies --- a full treatment of the Galactic ``Larson's Laws''
is beyond the scope of this paper. We draw several conclusions from
the comparison:
\begin{enumerate}
\item Figure \ref{virparams} shows that the virial parameters derived
in our analysis largely agree with those found by
\citet[][]{hc01}. Below masses of $\approx 3 \times 10^4$ M$_{\odot}$,
both surveys find the same virial parameter in a given mass bin.
\item Above $\approx 3 \times 10^4$ M$_{\odot}$, our analysis of the
\citet[][]{mwco} data set may yield a slightly higher virial
parameter, on average. This may be evidence for the diffuse emission
mentioned in \S\ref{EXTRAPSECT} --- the higher sensitivity
\citet[][]{mwco} data may include diffuse emission surrounding the
GMCs while the \citet[][]{hc01} data may miss this effect. It may also
reflect inadequacies in the simple sensitivity correction we apply to
the \citet[][]{hc01} data. Resolution effects may also play a role ---
the \citet[][]{hc01} data set has $\sim 5$ times better resolution
than the \citet[][]{mwco} data and so the lower resolution data may
tend to lump unbound clouds together. The number of clouds in the high
mass bins is relatively low, so the discrepancy may not be
particularly significant.
\item We apply our algorithm to a small portion of the OGS data and
find our corrections for resolution and sensitivity bias increase the
mean virial parameter to be consistent with the results from the
\citet{mwco} data. This suggests that the differences in the virial
parameters for the high mass clouds in Figure \ref{virparams} may
simply be methodological.
\item Both catalogs of outer Galaxy clouds find an inverse
relationship between luminous mass and the virial parameter,
approximately $\alpha \propto M_{\mathrm{Lum}}^{-0.2}$ --- a
relationship that was also observed by \citet{srby87} for inner Galaxy
molecular clouds.
\end{enumerate}
Thus we find agreement with the results of previous studies of
Galactic molecular clouds. Our methods applied to Galactic data find
the same behavior observed in earlier work and we find agreement among
the method applied to several data sets.
\subsection{Conclusions}
We have presented a method for measuring macroscopic GMC properties
--- spatial size, line width, and luminosity --- from a region of
emission in a spectral line data cube. This method corrects for biases
from limited sensitivity and resolution and produces reliable results
that are directly comparable among a wide variety of data sets. We
correct for limited sensitivity via an extrapolation to a theoretical
0 Kelvin isosurface. We apply a simple quadratic deconvolution to
the extrapolated values to account for resolution biases. We find that
bootstrapping methods yield believable estimates of the uncertainties
in the derived parameters. We present a set of suggestions for
transforming the derived properties into physical quantities of
interest. In the Appendix to this paper we present a new method for
decomposing emission into individual GMCs. This method is conservative
and robust, designed to produce robust results from low
resolution/sensitivity data. In this section we have applied all of
these methods to an array of extragalactic (Local Group) and Galactic
data. We find that the algorithm reproduces established results for
Galactic GMCs and that the sensitivity and resolution biases are
potentially significant --- often $\sim 40\%$ --- for even the most
recent Local Group GMC measurements.
Based on this investigation of the observational biases in measuring
molecular cloud properties, we note several important points that
should be considered in planning observations of GMCs and interpreting
the results. First, resolution and sensitivity biases can be
corrected to $<10\%$ error provided the cloud has a modest peak
signal-to-noise ($T_{peak}/\sigma_{RMS}\gtrsim 10$) and is marginally
resolved $R_{cld} > 0.8\theta_{\mathrm{FWHM}}$, where
$\theta_{\mathrm{FWHM}}$ is the full width at half maximum extent of
the beam. Given current and future telescope capabilities, even the
properties of extragalactic GMCs can be accurately measured. From
Figure \ref{DISTCOMP}, we can see that single dish surveys can
accurately study clouds more massive than $10^4~M_{\odot}$ in the
Magellanic clouds and careful interferometer observations can recover
cloud properties for clouds with mass $M\gtrsim 10^5~M_{\odot}$ in M31
and M33. However, interferometer observations systematically
underestimate molecular cloud properties. All else being equal,
interferometers can underestimate fluxes by 40\% and cloud radii by
10\% relative to single-dish observations. Line widths are largely
unaffected by interferometer observations. The magnitude of the
systematic bias depends on the array configuration and the sensitivity
of the observations. In general, wider coverage of the $(u,v)$ plane
produces better property recovery. To minimize bias, observers should
favor observations from several array configurations or from arrays
with many elements. If possible, the interferometer data should be
supplemented with single-dish observations. Finally, the
decomposition algorithm used to separate emission into physically
relevant structures will systematically affect molecular cloud
properties. To date, there is no algorithm that should be favored in
all circumstances, but any comparative study of GMC properties should
be consistent in the choice of algorithm. For example, the results of
a CLUMPFIND algorithm applied to an extragalactic data set are not
directly comparable to a catalog produced by a simple contouring
method applied to Milky Way data. Provided the same algorithm is used
across multiple data sets, referencing algorithm parameters to a
common physical scale will minimize systematic differences.
We will make a software version of the decomposition and measurement
algorithm available as an IDL package. Further, since methodology can
affect the results of GMC studies so strongly, we encourage authors
working in the field to make their data available to the community
after publication in order to facilitate future rigorous comparisons.
\acknowledgements
We are extremely grateful to Tom Dame and the Millimeter-Wave Group at
the Center for Astrophysics for providing both the Orion data and the
Quadrant 2 data used in the paper. We also thank the NANTEN Group at
Nagoya, especially Yasuo Fukui and Akiko Kawamura, providing us with
the LMC CO data. We are grateful to Fabian Walter for providing us
with the OVRO IC~10 data. Alberto Bolatto, Jason Wright, Jon Swift,
and Leo Blitz all offered helpful comments on drafts of the paper. We
thank Ronak Shah for helping us compare our IDL version of GAUSSCLUMPS
to the original implementation of the algorithm. The informed
comments of an anonymous referee greatly improved the paper,
particularly in encouraging us to explore the effects of
interferometers. ER is grateful for support through the National
Science Foundation Astronomy \& Astrophysics Postdoctoral Fellows
Program (AST-0502605). This work is partially supported by NSF grant
0228963 to the Radio Astronomy Laboratory at UC Berkeley.
\begin{appendix}
\label{ouralg}
\section{Appendix: A New Decomposition Algorithm \label{DECOMPOSITION}}
The choice of what emission to identify as a GMC may be the single
largest source of uncertainty in measuring and comparing GMC
properties. A number of methods have been employed over the years,
from simple contouring methods
\citep[e.g.~][]{sss85,dect86,srby87,hc01} to fitting three-dimensional
Gaussians \citep[GAUSSCLUMPS,][]{gaussclumps,gaussclumps2}, to
modified watershed algorithms \citep[CLUMPFIND and its
kin,][]{clumpfind,clumpfind-mod}.
In this section we present a new decomposition algorithm that is
designed to identify clouds at low sensitivities while avoiding the
introduction of a false clouds due to noise. This algorithm consists
of two parts: identifying regions of significant emission in the data
set and then assigning this emission to individual ``clouds.''
\subsection{Signal Identification}
We first identify regions of contiguous, significant emission in our
position-position-velocity data cube. We estimate the noise in the
data set by measuring the RMS intensity, $\sigma_{RMS}$, from a
signal-free region of the data cube. We then construct a
high-significance mask. This mask includes only adjacent channels that
both have intensities above $4\sigma_{RMS}$. We expand that mask to
include all emission above a lower threshold --- typically two
channels above $2\sigma_{RMS}$ significance --- that is connected to
the original high significance mask through pixels with $\ge
2\sigma_{RMS}$ significance. The resulting mask contains most of the
significant emission in the data cube. Lowering the threshold below
two channels at $\approx 1.5\sigma_{RMS}$ runs the risk of biasing the
moment measurements towards high values by including false emission
(noise with positive values) in the cloud.
\subsection{Cloud Identification}
\label{algorithm}
In this section, we describe the algorithm used to decompose a region
of emission into individual subsections representing the physically
distinct entities in the data (``clouds''). Through the description
of the algorithm, there are several parameters that can be varied to
produce changes in the resulting decomposition. In general, we set
these parameters using physical prior knowledge of the GMCs we are
seeking to catalog. We discuss the choice of these parameters in the
following section.
\begin{enumerate}
\item {\em Discard Small or Low Contrast Regions:} If a region is too
small for us to measure meaningful properties from it, we discard the
region. We require that each region has an area larger than two beam
sizes, so that we can measure its size; and a velocity width of more
than a single channel, so that we can measure its line width. If the
intensity contrast between the peak and the edge of the region is less
than a factor of two, we lack the dynamic range needed to correct the
sensitivity bias and we therefore discard the region. If a region is
not discarded we proceed to the next step.
\item \label{xform}{\em Rescale the Data to Reduce the Effects of
Substructure:} Molecular clouds contain significant substructure that
confuses the decomposition of these sources. The substructure is
often significantly brighter than bulk of the gas in the cloud. We
rescale the data to reduce the contrast between this substructure and
the cloud as a whole. The data are rescaled using the following
transform:
\[ T' = \left\{\begin{array}{cl}
T & ;~T<T_{clip}, \\
T_{clip} [1+{\arctan (T/T_{clip}-1)}] & ;~ T\ge T_{clip}. \\
\end{array} \right.
\]
This transform reduces the contrast pixels with $T\gtrsim 2 T_{clip}$
while preserving the relative brightness distribution. The value of
$T_{clip}$ is left as a free parameter. The transformed data are used
in the decomposition algorithm. Such brightness transforms are
frequently used in the decomposition algorithms used in other fields
such as medical imaging \citep[e.g.~][]{cytom}.
\item {\em Identify Independent Local Maxima:} We identify potential
local maxima, by identifying the elements in the data cube that are
larger than all their neighbors. We consider neighbors to be all data
that lie within a box with side length $D_{max}$ in position and
$\Delta V_{max}$ in velocity centered on the local maximum. These are
our ``candidate maxima.'' The parameters $D_{max}$ and $\Delta
V_{max}$ are free parameters. If we find more than one candidate
maximum in a region of emission, we proceed to the next steps to
further verify each maximum's independence. If we find a single
candidate maximum then we label the region as a cloud and measure its
properties as described in the main paper.\label{locmax}
\item {\em Find Shared Isosurfaces and Reject Small Clouds or those
with Smooth Mergers:} For each {\em pair} of candidate maxima we
calculate the value of the highest intensity isosurface to contain
both maxima. We refer to this highest shared isosurface as the {\em
merge level}. Using this set of highest shared isosurfaces, we
calculate three properties of interest:
\begin{enumerate}
\item The area uniquely associated with each maximum (i.e. the area
above the merge level for that maximum).
\item The antenna temperature interval between the merge level and
each maximum, referred to as the {\em contrast
interval}. \label{decimate}
\item The fractional amount by which each of the (unextrapolated)
moment values changes across each shared isosurface
(i.e. $\frac{\Delta \sigma_{maj}}{\sigma_{maj}}$ for each maximum
across each shared isosurface). We consider the two clouds to merge
smoothly across the isosurface only if:
\begin{enumerate}
\item None of the second moments increase by more than 100\% for both maxima.
\item No two of the second moments increase by more than 50\% for both
maxima.
\item The flux increases by less 200\% for both maxima.
\end{enumerate}
\end{enumerate}
We use these three properties to pare maxima from the region. We
reject maxima associated with small areas (less than two beam sizes,
as for the region above) and contrast intervals less than $\Delta
T_{max}$, a free parameter. Choosing $\Delta T_{max}\ge 2\sigma$
significantly reduces the effects of noise on decomposition
\citep[e.g.~][]{clumpfind-mod} since noise is associated with low
contrast intervals. Finally, when a pair of maxima merge smoothly
across a shared isosurface, we keep the higher intensity maximum and
discard the lower one. This is a conservative choice in the
decomposition algorithm: unless merging the two kernels significantly
alters the properties of one of the clouds associated with the
separate kernels, we assume the kernels are not physically distinct.
The effect of removing kernels that merge smoothly from our data set
is to reduce the algorithm's sensitivity to substructure within
clouds. We iterate this step until we have a set of maxima associated
with the required areas and separated from each other by significant
jumps in their properties.
\item {\em Define Clouds Using Shared Isosurfaces:} The surviving
maxima each correspond to a ``cloud.'' That cloud consists of the
emission within the lowest intensity isosurface uniquely associated
with the cloud. Emission that lies below this isosurface is part of a
``watershed'' shared among clouds and we do not assign it to any
cloud. By not considering contested emission, i.e.~emission that
could be associated with distinct local maxima, we avoid the problem of
how to properly assign this emission to local maxima.
\item {\em Measure Cloud Properties:} Finally, we apply the methods
described in \S2 and 3 to derive spatial sizes, line widths, and
luminosities for each cloud. The transformed data (Step
\ref{xform}) are inverse transformed into the original brightness
units for this analysis.
\end{enumerate}
\subsection{Using Physical Priors to Establish Algorithm Parameters}
Several of the algorithm's parameters are left to the choice of the
user. Without any prior knowledge of the physical objects in the
data, we establish default values for these parameters that will
produce a reasonable decomposition based solely on the characteristics
of the data. These defaults are chosen to provide sensitivity to real
substructure within the data without contamination by noise and can be
regarded as the {\it minimum} appropriate values for these parameters
in most cases. The free parameters in the algorithm are the
brightness transform threshold ($T_{clip}$, Step \ref{xform}) position
and velocity window size used in searching for local maxima ($D_{max}$
and $\Delta V_{max}$, Step \ref{locmax}), and the minimum contrast
temperature ($\Delta T_{max}$, Step \ref{decimate}). Without prior
knowledge of substructure in the data, no brightness transform should
be applied ($T_{clip}=\infty$). The minimum values for the parameters
in the search for initial local maxima are set by the resolution of
the data: $D_{max}=\theta_{beam}$ and $\Delta V_{max}=\Delta V_{chan}$
or else separations between local maxima cannot be resolved. Finally,
we use $\Delta T_{max}=2\sigma_{RMS}$ to prevent the noise in the data
set from being recognized by the algorithm as legitimate structure and
becoming the basis for the decomposition.
In the main section of the paper, we developed methods that measured
physical properties of molecular emission without observational bias.
Ideally, the decomposition algorithm used in the analysis would also
be free of observational bias. A good algorithm would, for example,
decompose the same emission into similar structures regardless of the
resolution and sensitivity of the observations. To progress towards
this goal, we set the parameters of the decomposition algorithm to
have similar values in physical units (pc, km s$^{-1}$, K) rather than
data units (beam widths, channels, $\sigma_{RMS}$).
In the molecular ISM, which has structure on a range of scales, the
choice of the physical values for the algorithm parameters must be
motivated by prior knowledge of the objects that we wish to identify.
The choice of parameters for identifying GMCs would be different from
the choice of parameters for identifying clumps and cores. As a
population, GMCs have size scales of 10s of parsecs, line widths of
several km s$^{-1}$, and brightness temperatures of $T
\lesssim 10$~K \citep[e.g.~][]{srby87}. In contrast, the clumpy
substructure within clouds has a size scale of 1 pc, line widths of
order 1 km s$^{-1}$ and can have brightness temperatures of 30 K or
higher \citep[e.g.~][]{clumpfind}. We select the parameters of the
algorithm to find molecular clouds rather than the clumps within them.
In particular, we set the minimum separation between local maxima as
$D_{max}=15$~pc and the velocity separation to be $\Delta V_{max}
=2$~km s$^{-1}$. We also fix a contrast interval in antenna
temperature of $\Delta T_{max} = 1$~K rather than the data-driven
value of $2 \sigma_{RMS}$. Finally, we must account for the presence
of bright molecular gas substructure within molecular clouds. For
example, Orion A has a few distinct regions separated by $>15$ pc with
antenna temperatures in excess of 15 K \citep{wil05}. These are
associated with hot molecular gas around young stars like the
Trapezium cluster. In general, the typical kinetic temperature of gas
in GMCs is $\sim 10$~K so brightness temperatures in excess of 10 K
are usually associated with structure {\it within} molecular clouds
not changes from cloud to cloud. Hence, to catalog clouds and not
substructure, we must reduce the influence of the high $T_b$ data with
the parameter $T_{clip}$. We use $T_{clip} = 2.5$~K to maintain the
full sensitivity for the gas separating GMCs (which typically has $T_A
\lesssim 5$~K in the \citet{srby87} data) while reducing the influence
of brighter gas associated with a single GMC. Using $T_{clip}=2.5$~K
has little influence on extragalactic data where beam deconvolution
typically averages out the presence of bright substructure within the
clouds.
We summarize our choices of parameters in Table \ref{params} when the
parameters are motivated by the data (Data-Based) or by physical
assumptions about the structures being extracted (GMC Physical Priors). The
resolution or the sensitivity of a data set may be sufficiently poor
that the physical parameters are unattainable in the data set
(e.g.~$\theta_{beam}>15~$pc, $2\sigma_{rms}> 1$~K). In this case, the
decomposition must be regarded with caution as it may not be directly
comparable to other data sets. The adopted values are appropriate for
decomposing data sets of $^{12}$CO emission. The values of $T_{clip}$
and $\Delta T_{max}$ would be different for other tracers.
\begin{deluxetable}{lcc}
\tablecaption{Parameter Choices for Decomposition Algorithm\label{params}}
\tablehead{
\colhead{Parameter} & \colhead{Data-Based} & \colhead{GMC Physical
Priors}}\
\startdata
$T_{clip}$ & $\infty$ & 2.5 K \\
$D_{max}$ & 1 beam width & 15 pc \\
$\Delta V_{max}$ & 1 channel & 2 km s$^{-1}$ \\
$\Delta T_{max}$ & $2\sigma_{RMS}$ & 1 K \\
\enddata
\end{deluxetable}
\subsection{Comparison with Existing Algorithms}
To demonstrate that the proposed algorithm is actually an improvement
on existing methods, we analyze trial data sets with known properties
using this algorithm and compare the results to those from the
GAUSSCLUMPS and CLUMPFIND algorithms to the trial data. We use the
CLFIND algorithm implemented in MIRIAD \citep{miriad} and our own IDL
implementation of the \citet{gaussclumps2} GAUSSCLUMPS algorithm,
which we have compared to the standard algorithm operating on a real
data set with satisfactory results.
We first examine how adept the three algorithms are at decomposing a
pair of blended clouds. We construct a series of data cubes with two
unresolved model clouds separated in position by a variable distance.
The model clouds each have a peak signal-to-noise ratio of 10 with a
Gaussian line profile. For each value of the separation, we decompose
100 data cubes with different realizations of the noise using the
three algorithms, using data based choices for the algorithm
parameters. Figure \ref{nclplot} (left) shows the mean number of
clouds recovered by each algorithm as a function of the separation (we
count clouds with peak signal-to-noise larger than 5 as
``recovered''). As the trial clouds are moved farther apart, the
typical number of clouds detected by each algorithm increases by
1. However, only the algorithm presented here consistently recovers a
single cloud at low separations. GAUSSCLUMPS and CLUMPFIND produce
false clouds from the noise. The jump in the number of clouds detected
by the GAUSSCLUMPS algorithm occurs where the separation equals the
resolution, whereas both the current algorithm and CLUMPFIND are able
to distinguish the clouds only if their separation is over 1.5 times
the resolution. GAUSSCLUMPS appears to be able to distinguish tight
blends of clouds but is the most susceptible to noise.
As a second test of the algorithms, we compared the number of clouds
that the algorithms recover from a single cloud with a circular
top-hat brightness profile ($T_A=\mbox{const. for } r<R_0;~0$
otherwise) and peak signal-to-noise of 10. The size of the cloud is
varied with respect to the resolution of the data set and 100 data
sets for each value of $R_0$ are decomposed by each algorithm. The
non-Gaussian brightness profile confounds all of the algorithms but to
varying degrees. CLUMPFIND detects an increasing number of spurious
clouds as the cloud grows, suggesting that the number of false clouds
grows with the volume studied. Despite the non-Gaussian profile,
GAUSSCLUMPS does surprisingly well with large sources. The current
algorithm, however, does the best job of detecting a single source in
the presence of noise. For analyzing data with a relatively low
signal-to-noise, we find the decomposition algorithm presented in this
Appendix should be favored for identifying clouds.
\subsection{Applying the Algorithm to the Orion-Monoceros Region}
The Orion Molecular Cloud is among the best studied of all GMCs, so it
is an good place to compare the results of our methods to those of
previous work. For this comparison, we use the data and results of
the recent, uniform survey of the entire Orion-Monoceros region by
\citet{wil05}. We analyze their final data set using the methods in
this paper (with physical priors for the decomposition algorithm
parameters) and summarize the results in Figure \ref{orion_fig}, an
integrated intensity map of the region. The decomposition results in
81 molecular clouds, most of which are associated with the Galactic
plane at the top of the Figure.
The results of the algorithms decomposition of the region are good:
major molecular clouds are identified as single entities in the
decomposition. The algorithm succeeds where other cloud identification
schemes would face difficulty. Nearly all of the emission in the data
is connected above a single isosurface so that simple contouring
methods would identify the entire Orion-Monoceros region as a single
molecular cloud, though more complicated algorithms \citep{syscw} may
succeed. CLUMPFIND and GAUSSCLUMPS would isolate individual peaks and
decompose clouds into their substructure --- as was intended by their
design (CLUMPFIND identifies 617 clumps in the same data).
The properties of the clouds agree well with the values published for
a human decomposition of the emission. The results of the analysis
are given in Table \ref{orion_props}. For comparison, we adopt the
distances of \citet{wil05}. We compare the results of the algorithm
to their results, after scaling their mass up by 10\% to account for a
difference in the adopted CO-to-H$_2$ conversion factors. In several
cases the masses agree quite well (to within 5\%, Orion A, Mon R2,
Scissors), while other features show $\sim 30\%$ differences. These
systematic discrepancies arise from differences in how emission is
assigned into structures. For example, the algorithm only identifies
the central region of Orion B as a molecular cloud; it does not
include emission near the location of Orion East that is nominally
part of the Orion B cloud because the assignment of this emission is
contested with neighboring clouds. Similarly, the algorithm
characterizes the Northern Filament region as five distinct clouds
rather than the single large cloud that a human decomposition
produced. Despite these differences, the results of the algorithm are
reassuring -- in most cases, the well-known molecular clouds are
identified as single clouds and there is good agreement between their
derived physical properties and the results of previous studies.
\begin{deluxetable}{lcccccc}
\tablecaption{\label{orion_props} Properties of Major Molecular Clouds in
Orion-Monoceros}
\tablewidth{0pt}
\tablehead{
\colhead{Name} & \colhead{Distance}\tablenotemark{a}
& \colhead{$M_{\mathrm{human}}$}
\tablenotemark{a} &
\colhead{$M_{\mathrm{algorithm}}$} & \colhead{$R_e$} & \colhead{$\sigma_v$} &
\colhead{$\alpha$} \\
& \colhead{(pc)} & \colhead{($10^4 M_\odot$)} &
\colhead{($10^4 M_\odot$)} & \colhead{(pc)} & \colhead{(km s$^{-1}$)} &
}
\startdata
Orion A & 480 & 12. & $11. \pm 0.06$ & $18.6 \pm 0.2$ & $2.9 \pm 0.1$ & $1.6 \pm
0.1$ \\
Orion B & 500 & 9.1 & $5.6 \pm 0.08$ & $12.1 \pm 0.3$ & $1.5 \pm 0.1$ & $0.6 \pm
0.1$ \\
Orion East & 120 & 0.013 & $0.013 \pm 0.001$ & $1.2 \pm 0.1$ & \nodata
& \nodata \\
Mon R2 & 800 & 12. & $11. \pm 0.5$ & $25.9 \pm 1.2$ & $1.6 \pm 0.1$ & $0.7 \pm 0
.1$ \\
Crossbones & 470 & 1.9 & $0.57 \pm 0.02$ & $11.0 \pm 0.6$ & $0.8 \pm 0.1$ & $1.5
\pm 0.3$ \\
Northern Filament & 390 & 1.9 & $1.2 \pm 0.07$ & $8.6 \pm 0.7$ & $1.7 \pm 0.1$ &
$2.6 \pm 0.5$ \\
\enddata
\tablenotetext{a}{Adapted from Table 2 of \citet{wil05}.}
\end{deluxetable}
\subsection{Decomposing Orion-like Clouds in Other Galaxies}
As a final test of the decomposition and analysis algorithm, we apply
the algorithm to simulated observations of the Orion-Monoceros region
with beam sizes and signal-to-noise values typical of extragalactic
GMC observations. We simulate observations by convolving the data set
to the desired resolution and adding a convolved data cube of noise
which is scaled up to give the desired peak signal-to-noise ratio in
the data. We perform this for a peak signal-to-noise ($S/N$) values
of 10 and 30 combined with beam sizes of 10, 20 and 50 pc. The
results of the decompositions are displayed in Figure
\ref{decompdemo}.
At coarse resolution and low sensitivity, the fainter clouds are
undetectable. However, the algorithm identifies each of Orion A,
Orion B, Mon R2 and the Northern Filament in at least one of the trial
data sets. At high resolution (10 pc) and peak signal-to-noise
($S/N=30$), the algorithm successfully identifies the clouds with
properties that are consistent, within the uncertainties determined by
the algorithm, with the masses of the clouds in Table
\ref{orion_props}. The systematic effects of poor resolution manifest
themselves for beam sizes $\gtrsim 20$ pc where the decomposition of
blended emission results in $\sim 20\%$ variations in the properties
of the clouds relative to those found in the original data set for
both high and low $S/N$. For a beam size of 20 pc, the clouds are not
resolved since this is over twice the size of the clouds along their
minor axes. Finally, at 50 pc resolution, Mon R2 is not found by the
algorithm and high $S/N$ is required to distinguish Orion A and Orion
B. There is a variation of 100\% (0.3 dex) in the derived physical
parameters for the clouds; this variation is reflected in the
estimates of the uncertainties. As expected based on the 20 pc
resolution data, the clouds are not spatially resolved for a beam size
of 50 pc. An accurate decomposition of emission into typical Galactic
GMCs requires a beam size $\sim 20$ pc though only a modest
sensitivity is required: $S/N\gtrsim 10$.
\end{appendix}
|
Title:
Tail emission from a ring-like jet: its application to shallow decays of early afterglows and to GRB 050709 |
Abstract: Similar to the pulsar, the magnetic axis and the spin axis of the gamma-ray
burst source may not lie on the same line. This may cause a ring-like jet due
to collimation of the precessing magnetic axis. We analyze the tail emission
from such a jet, and find that it has a shallow decay phase with temporal index
equal to -1/2 if the Lorentz factor of the ejecta is not very high. This phase
is consistent with the shallow decay phase of some early X-ray afterglow
detected by {\it{swift}}. The ring-like jet has a tail cusp with sharp rising
and very sharp decay. This effect can provide an explanation for the
re-brightening and sharp decay of the X-ray afterglow of GRB 050709.
| https://export.arxiv.org/pdf/astro-ph/0601670 |
\title{Tail emission from a ring-like jet: its application
to shallow decays of early afterglows and GRB 050709}
\volnopage{Vol.0 (200x)
No.0, 000--000} \setcounter{page}{1}
\author{Yuan-Chuan Zou and Zi-Gao Dai}
\institute{Department of Astronomy, Nanjing University, Nanjing 210093, China.\\
\email{zouyc@nju.edu.cn, dzg@nju.edu.cn}}
\date{Received~~2006 January 30; accepted~~}
\authorrunning{Y. C. Zou \& Z. G. Dai}
\titlerunning{Tail emission of a ring-like jet}
\section{Introduction}\label{intro}
It is well known that pulsars originate from the core collapse of massive
stars. The average angle between the spinning axis and the magnetic axis of
pulsars is about 27$^\circ$ \citep{leahy91}. Similarly, the spin axis and
magnetic axis of the central engine of the gamma-ray burst may not lie on one
line. As the ejecta may be collimated by the magnetic axis, while the magnetic
axis is processing, so the ejecta may be in a spiral shape at first\citep[][and
reference therein]{fargion05}. As the diversity of the velocities of the ejecta
[as assumed in the standard fireball model of gamma-ray bursts\citep{piran05}],
the spiral ejecta ejected at different times will collide and merge into one
whole shell at last. These collisions just produce internal shocks of gamma-ray
burst. At last, these collisions make the ejecta merge into a ring-shaped jet.
Even if the the ejecta is conical, the baryon-loaded region still be ring-like
\citep{eichler03}. \citet{granot05} and \citet{eichler05} have analyzed
afterglows from ring-like jets. It has also been used to interpret the
$h\nu_{\rm peak}-E_{\rm iso}$ relation\citep{eichler04}.
Tail emission plays an important role at the times when shocks disappear. The
temporal index is $-(2+\beta)$ for a cone-shaped jet, where $\beta$ is the
spectral index of the emission\citep{kumar00, yamazaki05}. Considering the zero
point effect of time, the light curves can be steeper during a short
period\citep{nousek05, zhang05, wu06}.
\citet{nousek05} and \citet{zhang05} have also shown that a shallow decay with
index of about $-1/2$ follows the steep decay for most X-ray afterglows. For
the X-ray afterglow of short burst GRB 050709, there is an unexpected high-flux
point followed by a very steep decay\citep{fox05}. These two observations can
both be explained naturally by considering the tail emission of ring-like jets.
In \S \ref{sec:model} we give the expressions of tail emission from a ring-like
jet. In \S\S \ref{sec:shallow} and \ref{sec:050709}, the shallow decay and
X-ray afterglow of GRB 050709 are analyzed respectively. At last, we summarize
our results in \S \ref{sec:conclusion}.
\section{Model}\label{sec:model}
Considering several ring-like sub-jets emitted from the central engine, they
merge into one whole ring-like jet accompanied with internal shocks. This final
ring with uniform energy density and sharp edges expands with Lorentz factor
$\gamma$, as sketched in Fig. \ref{fig:ring-sketch}. Assuming the radiation
from the ring-like jet begins and ceases at radius $R_c$ (and correspondingly
at time $t_c$) suddenly, we calculate the tail emission from high latitudes of
the ring. The relation is $R_c \simeq 2\eta^2 c t$, where $\eta$ is the mean
Lorentz factor of the internal shocks.
The relation between the latitude angle $\theta$ and the observed time $t$ is
\begin{equation}
R_c(1-\cos\theta)=c(t-t_c)/(1+z),
\end{equation}
where $z$ is the cosmological redshift. Neglecting the depth of the ejecta
and the emission from time equal arrival
surface, and defining the emissivity $I'_{\nu'}$ per unit area in the comoving
frame, which is uniform in the whole ring, the flux density in the observer's
frame is
\begin{equation}
f_{\nu}(t>t_c) =\frac{ I'_{\nu'}}{4\pi D_L^2} \mathscr{D}^2 \frac{{\rm d}S}{{\rm d}t/(1+z)},
\label{eq:f_nu_general}
\end{equation}
where $D_L$ is the luminosity distance, $\mathscr{D}=1/[\gamma(1-\sqrt{1-1/\gamma^2}
\cos\theta)]$ is the Doppler factor, and ${\rm d}S$ is the emitted area during
a period ${\rm d}t$.
At early times when $\theta < \theta_w$, the tail emission is the same as the
case of an on-axis conical jet, which has been investigated by many authors
\citep{kumar00, fan05}. There are two limiting cases: for $\theta \ll
1/\gamma$,
\begin{equation}
f_{\nu}(t>t_c) \propto \delta t^0,
\label{eq:11}
\end{equation}
and for $1 \gg \theta \gg 1/\gamma$,
\begin{equation}
f_{\nu}(t>t_c) \propto \delta t^{-(2+\beta)},
\label{eq:12}
\end{equation}
where $\delta t \equiv t-t_c$. Here we consider the emission as a single
power-law profile $I'_{\nu'} \propto \nu'^{-\beta}$, which is valid for the
high frequency emission $\nu' > \max(\nu'_c, \nu'_m)$, where $\nu'_c$ is the
cooling frequency and $\nu'_m$ is the typical frequency of synchrotron
emission.
In the case $\theta > \theta_w$, the width of the ring can be neglected, and
the flux density
\begin{equation}
f_{\nu}(t>t_c) \propto \mathscr{D}^{-(2+\beta)} \frac{\sin{\theta_p}
\cos{(\theta/2)}}{\sin{\theta} \sqrt{1-\left(\frac{\sin {(\theta/2)}}
{\sin {\theta_p}}\right)^2}}.
\label{eq:f_line}
\end{equation}
There are two limiting cases in which equation (\ref{eq:f_line}) can be
simplified. For $\theta \ll 1/\gamma$,
\begin{equation}
f_{\nu}(t>t_c) \propto \delta t^{-1/2},
\label{eq:21}
\end{equation}
and for $1 \gg \theta \gg 1/\gamma$,
\begin{equation}
f_{\nu}(t>t_c) \propto \delta t^{-(5/2+\beta)}.
\label{eq:22}
\end{equation}
\section{Shallow Decay of Early X-ray Afterglow}\label{sec:shallow}
Statistics of the early X-ray afterglows has shown that there is a shallow
decay phase with temporal index about $-1/2$\citep{nousek05, zhang05}. This
corresponds to the case: $1/\gamma > \theta > \theta_w$, and can be described
by equation (\ref{eq:21}), where the temporal index is just $-1/2$. As the
general shallow decay lasts from $10^2-10^3$s to $10^3-10^4$s \citep[Fig. 1
in][]{zhang05}, this gives limits: the Lorentz factor of the emitting shell
$\gamma < 7.3 (1+z)^{1/2} R_{c,16}^{1/2} \delta t_{3.5}^{-1/2}$, and the width
of the ring-like jet $\theta_w < 1.4\times 10^{-2} (1+z)^{-1/2} R_{c,16}^{-1/2}
\delta t_{2.5}^{1/2}$. (The conventional donation $Q=Q_k\times 10^k$ is used
throughout this paper.) This implies that the shallow decay component
originates from the shocked shells with low Lorentz factors, while these shocks
may be formed due to ejected sub-shells with different Lorentz factors.
This model can answer the following questions:
Firstly, why is there a steep decay before the shallow decay appears in general
case? In \citet{zhang05}, it is general that the temporal index of this steep
decay is less that $-3$. The answer is that the two power law decays originate
from two different emitting shells with different Lorentz factors. The steep
decay corresponds to the greater Lorentz factor shell, which satisfies
$1/\gamma < \theta$, and the temporal indices are $-(2+\beta)$ or
$-(5/2+\beta)$ corresponding to equations (\ref{eq:12}) and (\ref{eq:22})
respectively. The steep decay may become steeper because of the zero time
selection effect\citep{wu06}.
Secondly, why is there no spectral evolution before and after the break time
from the shallow decay phase to the steep decay phase. This is also mentioned
with spectral index value $\sim -1$ in \citet{zhang05}. It is believed that,
after the break, the afterglow becomes a ``normal'' afterglow. It is possible
that the tail emission phase and the ``normal'' afterglow emission phase are
both in the case $\nu_X > \{\nu_m,\nu_c\}$ (corresponding to the spectral index
$-p/2$) and thus the spectra are the same.
Thirdly, since the shallow phase and the steep phase originate from different
sources, how to understand the conjunction at the break time \citep[also can be
seen in Fig. 1 in][]{zhang05}? As time goes on, the case converts from
$1/\gamma > \theta > \theta_w$ to $\theta > 1/\gamma > \theta_w$, and then the
light curve of the tail emission decay has a temporal index $-(5/2+\beta)$.
This is steeper than the ``normal'' afterglow with temporal index $\sim -1.2$.
Some time later, the ``normal'' afterglow will exceed the tail emission
definitely, as in the case GRB 050525a \citep[Fig. 1 in][]{nousek05} (at about
3000s, there is a steep decay). However, GRB 050315\citep{vaughan05} can be
classified into the case that ``normal'' afterglow exceed the shallow tail
emission before the tail emission breaks to steep phase.
\section{X-ray Afterglow of GRB 050709}\label{sec:050709}
GRB 050709 is a short burst with duration 0.3 s, and five points of X-ray
emission after the burst were obtained by Swift and Chandra \citep{fox05}.
Figure \ref{fig:050709} shows the fit by assuming that the latter four points
are the tail emissions from the first point, with parameters $R_c=7.7\times
10^{16}$cm, $\gamma=15.5$, $\beta=1.1$, $\theta_p=0.5$ and $\theta_w=0.005$. As
the first X-ray point occurs at time about 100s, the radius $r \simeq 2\eta^2 c
t/(1+z) \simeq 5.2 \times 10^{16} \eta_{2}^2 t_2$cm, is consistent with the
value of the parameter $R_c$. As a short burst has less total energy than a
long burst does, the ejected shell can be decelerated quickly. The Lorentz
factor at $R_c$ is $\gamma \simeq {E_{\rm{iso}}}/({\pi R_c^3 n m_p c^2}) \simeq
26 E_{\rm{iso},50}^{1/2} n_1^{-1/2} R_{c,16.5}^{-3/2}$, where the external
medium density $n$ is chosen equal to 1$\rm{cm}^{-3}$ because the host is a
star-forming galaxy. Therefore, the parameters chosen to fit the X-ray data are
reasonable for this short burst.
We can see four stages for this tail light curve: first, a horizontal phase
corresponds to the case $\theta < \theta_w < 1/\gamma$; second, a shallow decay
with temporal index $-1/2$ corresponds to the case $\theta_w < \theta <
1/\gamma$; third, a sharper decay with temporal index $-(2.5+\beta)$
corresponds to the case $\theta_w < 1/\gamma < \theta$; and finally, a tail
cusp with sharp rising and very sharp decay, which comes from the end of the
ring.
We should note that the solid line doesn't fit the data very well, especially
that the tail cusp of the model can't reach to the observed data. This shortage
may be overcome by considering a non-uniform ring-like jet or some other
mechanism. However, its unique feature, which the emission after the tail cusp
decays very sharply, is consistent with the last two observed points. On the
other hand, it is possible that the second, third and fifth points in this
figure belong to the ``normal'' afterglow from an external shock.
\section{Conclusions}\label{sec:conclusion}
Enlightened from pulsars, we suggest that the magnetic axis and the spin axis
of a gamma-ray burst source point to different orientations. The ejecta along
the magnetic axis will form a ring finally. Gamma-ray emission will be observed
if the observer locates in the solid angle of the ring. We have investigated
the tail emission from a ring-like jet. We find that the early shallow decay
phase and the late re-brightening of the X-ray emission of GRB 050709 can be
explained.
Note that the shallow decay phase is only possible in the low Lorentz factor
cases. For the case $1/\gamma < \theta_w$, only the steep one appears. As the
tail emission from the shells with high Lorentz factors decays very quickly,
the main emissions will be dominated by the slower shells at later times.
YCZ thanks helpful discussions with Jia Wang and Xuefeng Wu. This work was
supported by the National Natural Science Foundation of China (grants 10233010
and 10221001).
|
Title:
On the accuracy of the ALI method for solving the radiative transfer equation |
Abstract: We solve the integral equation describing the propagation of light in an
isothermal plane-parallel atmosphere of optical thickness $\tau^*$, adopting a
uniform thermalization parameter $\epsilon$. The solution given by the ALI
method, widely used in the field of stellar atmospheres modelling, is compared
to the exact solution. Graphs are given that illustrate the accuracy of the ALI
solution as a function of the parameters $\epsilon$, $\tau^*$ and optical depth
variable $\tau$.
| https://export.arxiv.org/pdf/astro-ph/0601341 |
\title{On the accuracy of the ALI method for solving the radiative
transfer equation} \titlerunning{Accuracy of the ALI method}
\author{L. Chevallier\inst{1} \and F. Paletou \inst{2}\thanks{Present address: Observatoire Midi-Pyr\'{e}n\'{e}es, Laboratoire d'Astrophysique (UMR 5572), 14 avenue E. Belin, F-31400 Toulouse Cedex.} \and B. Rutily
\inst{1}} \institute{ Centre de Recherche Astronomique de Lyon (UMR
5574 du CNRS), Observatoire de Lyon, 9, avenue Charles Andr\'{e},
69561 Saint-Genis-Laval cedex, France\\
\email{loic.chevallier@obs.univ-lyon1.fr} \\
\email{rutily@obs.univ-lyon1.fr} \and Observatoire de la C\^{o}te
d'Azur, D\'{e}partement G. D. Cassini (UMR 6529 du CNRS), BP 4229,
06304 Nice cedex 4, France \\ \email{fpaletou@mail.ast.obs-mip.fr} }
\date{Received 25 April 2003 / Accepted 30 July 2003}
\abstract{We solve the integral equation describing the propagation of light in an isothermal
plane-parallel atmosphere of optical thickness $\tau^*$, adopting a uniform
thermalization parameter $\epsilon$.
The solution given by the ALI method, widely used in the field of stellar atmospheres modelling,
is compared to the exact solution.
Graphs are given that illustrate the accuracy of the ALI solution as a function of the parameters
$\epsilon$, $\tau^*$ and optical depth variable $\tau$.
\keywords{Radiative transfer -- Methods: numerical -- Stars: atmospheres} }
\section{Introduction}
The solution of the radiative transfer equation (RTE) is at the heart
of the stellar atmospheres modelling, since this equation has to be
solved typically thousands of times in order to construct a realistic
model. It is thus crucial to get a clear idea of the accuracy with
which the RTE is solved, and the effect it has on the determination of
the main physical quantities of the model: populations, electron
density, temperature, etc. In this article, we focus on the first
point checking the ALI method for solving the integral form of the RTE, since this method
is nowadays at the basis of most numerical schemes used to determine
the radiation field in stellar atmospheres. We recall that ``ALI''
means Accelerated (or Approximate) Lambda Iteration, the Lambda
operator being defined by Eqs.~(\ref{eq_lambda})-(\ref{eq_e1}) below
for the scattering law we adopt here.
The ALI code used in this paper is a combination of an accelerated iterative method (with a diagonal $\Lambda$-operator) and a formal solver based on parabolic short characteristics. Recent reviews on this approach are Paletou (2001), Hubeny (2003) and section 3 of Trujillo Bueno (2003).
The accuracy of our ALI code is tested while applied to a well-known problem
consisting of a homogeneous, isothermal slab with isotropic and
monochromatic light scattering (Sec.~\ref{sec_2}). Indeed, this
idealized problem can be solved exactly, which allows for a
direct comparison with the solution given by the ALI method.
This problem is very simple on physical grounds but implies analytical and numerical calculations that are far from trivial.
It contains the seeds of most of the difficulties met when solving the RTE
in a thick, highly scattering medium. It thus provides an excellent
test for numerical codes since very accurate analytical solutions
are available (Sec.~\ref{sec_3}). After a brief description of our
ALI code, we move to the numerical tests in Sec.~\ref{sec_4}, which is
the main part of this paper. The link with previous studies on the
subject (Trujillo Bueno \& Fabiani Bendicho 1995, Trujillo Bueno \& Manso Sainz 1999) is finally commented in Sec.~\ref{sec_5}.
\section{The standard radiative transfer problem}\label{sec_2}
This problem consists in solving the RTE in a homogeneous
plane-parallel atmosphere of optical thickness $\tau^*>0$ (possibly
infinite); light scattering is assumed to be isotropic and
monochromatic. It is furthermore supposed that the matter is
in local thermodynamical equilibrium with uniform temperature $T$ through the atmosphere.
The thermal source function at any frequency is then $\epsilon B(T)$,
where $\epsilon$ is the (spatially invariant) photon destruction
probability per scattering and $B(T)$ the Planck function at
temperature $T$ (frequency dependence is not mentioned).
In the absence of any external source of radiation, this problem reduces
to solving the following integral equation for the source function $S$ (Mihalas 1978):
\begin{equation}
\label{eq_s1}
S(\tau)=\epsilon B(T)+(1-\epsilon)(\Lambda S)(\tau) \, ,
\end{equation}
where the $\Lambda$-operator for isotropic and monochromatic scattering is
\begin{equation}
\label{eq_lambda}
(\Lambda S)(\tau)= \frac{1}{2}\int_0^{\tau^*}E_1(\vert \tau-\tau^{\prime}\vert)S(\tau^{\prime}) \,\mathrm{d}\tau^{\prime} \, .
\end{equation}
Here, $E_1$ is the first exponential integral function as defined by
\begin{equation}
\label{eq_e1}
E_1(\tau)=\int_0^1\exp(-\tau/\mu)\frac{\,\mathrm{d}\mu}{\mu}\quad(\tau>0) \, .
\end{equation}
We remind the reader that Eq. (\ref{eq_s1}) models the multiple scattering of photons of frequency $\nu$
assuming that 1) the scattering is monochromatic (or coherent) if $\nu$ belongs to a continuum,
2) the line profile is rectangular (Milne profile) if $\nu$ belongs to a spectral line (see, e.g., Ivanov 1973, p. 57).
The solution to problem (\ref{eq_s1}) is
$S(\tau)=S(\epsilon,\tau^*,\tau)B(T)$, where $S(\epsilon,\tau^*,\tau)$
satisfies the integral equation
\begin{equation}
\label{eq_s2}
S(\epsilon,\tau^*,\tau)=\epsilon+(1-\epsilon)(\Lambda S)(\epsilon,\tau^*, \tau)
\end{equation}
depending on parameters $\epsilon$ and $\tau^*$.
Note that this function is symmetrical about the $\tau$-mid-plane: $S(\epsilon,\tau^*,\tau)= S(\epsilon,\tau^*,\tau^*-\tau)$.
This equation is the integral formulation of the RTE in our model; it
specifies the {\em standard radiative transfer problem} we intend to solve
analytically (Sec.~\ref{sec_3}) and numerically (Sec.~\ref{sec_4}).
\section{Analytical solution of the standard problem}\label{sec_3}
There are many analytical methods for solving the integral equation
(\ref{eq_s2}). The classical approach, recently reviewed by Chevallier
\& Rutily (2003, hereafter Paper I), involves the basic auxiliary
functions of radiative transfer theory in plane-parallel geometry,
namely the $H$-function for a semi-infinite space, and the $X$- and
$Y$-functions for a finite slab (Chandrasekhar 1960). The
$H$-function depends on the parameter $\epsilon$ and on an angular
variable $\mu$, taken as positive hereafter. In addition the $X$- and
$Y$-functions depend on $\tau^*$, and we have
$X(\epsilon,\tau^*,\mu)\to H(\epsilon,\mu)$ and
$Y(\epsilon,\tau^*,\mu)\to 0$ as $\tau^* \to +\infty$.
The zero-order moments of the functions $H$, $X$, and $Y$ yield the
surface values of the solution $S$ to (\ref{eq_s2}).
The moment of the $H$-function is defined and given by
\begin{equation}
\alpha_0(\epsilon)=\int_0^1H(\epsilon,\mu)\,\mathrm{d}\mu=\frac{2}{1+\sqrt{\epsilon}}
\, ,
\end{equation}
and those of the $X$- and $Y$-functions defined as
\begin{equation}
\alpha_0(\epsilon,\tau^*)=\!\int_0^1 \!X(\epsilon,\tau^*\!,\mu)\,\mathrm{d}\mu\;,\quad \beta_0(\epsilon,\tau^*)=\!\int_0^1 \!Y(\epsilon,\tau^*\!,\mu)\,\mathrm{d}\mu
\end{equation}
are related by
\begin{equation}
\left[ 1-\frac{1-\epsilon}{2}\alpha_0(\epsilon,\tau^*) \right]^2 - \left[ \frac{1-\epsilon}{2}\beta_0(\epsilon,\tau^*) \right]^2=\epsilon \, .
\end{equation}
There is no exact expression of these moments.
In a semi-infinite atmosphere, the surface value of the solution $S$ to (\ref{eq_s2}) is
\begin{equation}
S(\epsilon,0)=1-\frac{1-\epsilon}{2}\alpha_0(\epsilon)=\sqrt{\epsilon}
\end{equation}
and it is
\begin{equation}
\label{eq_s4}
S(\epsilon,\tau^*,0)=1-\frac{1-\epsilon}{2}[\alpha_0(\epsilon,\tau^*)+\beta_0(\epsilon,\tau^*)]
\end{equation}
in a finite slab.
As $S$ is symmetrical about the $\tau$-mid-plane, $S(\epsilon,\tau^*,\tau^*)= S(\epsilon,\tau^*,0)$.
These relations were first derived by Sobolev (1957,
1958).
The former result is the famous ``$\!\sqrt{\epsilon}$-law'' for
semi-infinite media. The latter one is less known; it requires a table of
moments $( \alpha_0, \,\beta_0 )$ for numerical applications.
Such tables are available in the literature: see references in Van de Hulst
(1980, p. 225-227). Very accurate surface values of the $S$-function
can also be found in Paper I.
The calculation of the function $S$ within the slab is discussed in
detail in Paper I, which contains ten-figure tables of $S(\epsilon,
\tau^*, \tau)$ for $(\epsilon, \tau^*)$ = $(0.5,2)$, $(10^{-2},20)$,
$(10^{-4},2000)$ and $(10^{-8}, 2\times 10^8)$. In a
half-space, the internal solution is known since the end of the 50's
and it can be expressed in closed-form in terms of the
$H$-function. In a finite slab, the solution involves two
non-classical auxiliary functions $\zeta_+$ and $\zeta_-$, that are
implicitly defined by Fredholm integral equations over $[0, 1]$. These
equations can be solved very accurately, so that the solution in a
finite slab is nearly as accurate as in a half-space. The accuracy is
estimated at better than $10^{-10}$ for any value of $\epsilon,
\tau^*$ and $\tau$, which means that the solution given in Paper I can
safely be used as an accuracy test of the ALI code.
The general behavior of the $S$-function is shown in Fig.~\ref{fig1},
which illustrates the Table 3 of Paper I
($\epsilon = 10^{-4}$ and $\tau^* = 2000$). It can be seen that the
solution $S$ tends to 1 for large values of $\tau$ and that it drops
when $\tau$ is close to the thermalization depth $1/k(\epsilon)\approx
58$ for $\epsilon=10^{-4}$, where $k(\epsilon)$ is defined in Paper
I. It tends steeply to the surface value $S(0)$ as $\tau$ tends to 0,
{\em with an infinite derivative at 0}. It is regrettable that the
generally adopted logarithmic scale in $\tau$ obscures this essential
last point, as seen when comparing the solid and dashed curves of
Fig.~\ref{fig1}. The explanation lies in the fact that $\partial
S/\partial(\log \tau)=\tau \,\partial S/\partial \tau\to 0$ even if
$\partial S/\partial \tau\sim E_1(\tau) \to +\infty$.
\section{Comparison with ALI numerical solutions}\label{sec_4}
In this section we compare in detail the analytical
solution described in the previous section to the one given by our ALI code.
This code uses a diagonal approximate $\Lambda$-operator (Olson et al. 1986).
At each iteration, a formal solver has to be used in order to calculate the transform of the
source function by the $\Lambda$-operator. Inserting the definition
(\ref{eq_e1}) of the $E_1$-function into Eq. (\ref{eq_lambda}) and
inverting the order of integrations, the so-called formal solution to
the RTE is first calculated
\begin{eqnarray}
\label{eq_i}
\lefteqn{I(\tau ,\mu ) = \left\lbrace \!
\begin{array}{ll}\displaystyle
- \frac{1}{\mu } \! \int _{0}^{\tau }S(\tau^{\prime})\exp \left[ (\tau -\tau ^{\prime})/\mu \right] \,\mathrm{d}\tau ^{\prime} & \rm if\;-1\le\mu <0, \\
\displaystyle S(\tau ) & \rm if \;\mu =0, \\
\displaystyle +\frac{1}{\mu } \! \int_{\tau }^{\tau^*}S(\tau ^{\prime})\exp \left[ -(\tau ^{\prime}-\tau )/\mu \right] \,\mathrm{d}\tau ^{\prime} & \rm if\; 0<\mu \le +1.
\end{array} \right. }
\nonumber \\
\end{eqnarray}
Then the $\Lambda$-transform of the source function is derived, since it is here the associated mean intensity
\begin{equation}
\label{eq_ls}
(\Lambda S)(\tau )=\frac{1}{2}\int _{-1}^{+1}I(\tau ,\mu ) \,\mathrm{d}\mu \, .
\end{equation}
The formal solution (\ref{eq_i}) is calculated following the method of short characteristics
whose basic elements can be found in Olson \& Kunasz (1987) and Kunasz \& Auer (1988).
It was further improved by the implementation of
monotonic interpolation for multi-dimensional applications (Auer \& Paletou 1994) and by Fabiani Bendicho \& Trujillo Bueno (1999) for three-dimensional
applications with horizontal periodic boundary conditions.
In the present paper, we used parabolic short characteristics.
The $\mu$-integration in (\ref{eq_ls}) is performed with the help of a Gaussian quadrature.
A numerical acceleration scheme is used so as to improve the rate
of convergence of ALI: this is the so-called Ng-acceleration
introduced in the field of radiative transfer by Auer (1987, 1991; see
also Rybicki \& Hummer 1991).
We have calculated the relative error
\begin{equation}
\label{eq_d}
d(\epsilon, \tau^*,\tau)=\left| \frac{S_\mathrm{ALI}(\epsilon,\tau^*,\tau)-S(\epsilon,\tau^*, \tau)}{S(\epsilon,\tau^*,\tau )}\right|
\end{equation}
at various optical depths, where $S(\epsilon,\tau^*,\tau)$ is the
analytical solution of Sec.~\ref{sec_3} and
$S_\mathrm{ALI}(\epsilon,\tau^*,\tau)$ is the solution given by the
ALI code.
This error corresponds to the ``true error'' defined by Auer, Fabiani Bendicho \& Trujillo Bueno (1994),
who used a finer grid to calculate $S(\epsilon, \tau^*, \tau)$.
We introduce also the maximum value of $d(\epsilon,\tau^*,\tau)$ when the
$\tau$-variable covers the domain $[0,\tau^*]$, viz.
\begin{equation}
\label{eq_dm}
d_\mathrm{M}(\epsilon,\tau^*)=\max_{0\leq\tau\leq\tau^*} d(\epsilon,\tau^*,\tau) \, .
\end{equation}
Of course $d$ and $d_\mathrm{M}$ depend on the number of iterations
$N$ performed by the ALI code during each run. Finally we define $N_\mathrm{c}$ as the
number of iterations used to reach convergence, which is the
smallest value of $N$ satisfying the condition
$|1-d_\mathrm{M}(N)/d_\mathrm{M}(+\infty)| < \varepsilon_\mathrm{c}$,
where $d_\mathrm{M}(+\infty) = d_\mathrm{M}(N=10\,000)$ and
$\varepsilon_\mathrm{c}$ is arbitrarily set to $0.01$ in the present paper.
The slab optical depth is discretized using a logarithmic grid, symmetric
with respect to the mid-plane, with
$n_{\tau}$ points per decade, including the $\tau=0$ point, the next point denoted by $\tau_\mathrm{m}$, and the last point $\tau=\tau^*/2$.
The angular integration in Eq. (\ref{eq_ls}) is performed with a symmetric
grid containing $n_\mu$ Gauss-Legendre points in $[0,1]$. There is no
frequency integration since light scattering has been supposed
monochromatic. In most of our calculations, we chose the values
$\tau_\mathrm{m} = 10^{-4}$, $n_\tau=9$, and $n_\mu=5$ (i.e., values
quite often adopted for stellar atmospheres modelling). Some values
of $(\epsilon,\tau^*)$ may be (0.01, 20) for a continuum, ($10^{-4},
2000$) for an ``average'' spectral line and ($10^{-8}, 2\times10^8$)
for a strong spectral line.
The quantities of interest are the maximum relative error
$d_\mathrm{M}$ and the number of iterations to reach
convergence $N_\mathrm{c}$, which depend on $\epsilon$, $\tau^*$ and numerical parameters
$\tau_\mathrm{m}$, $n_\tau$, $n_\mu$ and $N$. We first study the
variation of $d(\epsilon,\tau^*,\tau)$ with $\tau$. Then, we study the
influence of $\epsilon$, $\tau^*$, $n_\tau$ on $d_\mathrm{M}$ and
$N_\mathrm{c}$, for given $\tau_\mathrm{m}$, $n_\mu$ and for $N=N_\mathrm{c}$.
\subsection{The influence of $\tau$ and $N$}
Figure~\ref{fig2} shows the variation of the relative error $\tau \to
d(\epsilon,\tau^*,\tau)$ for the three selected values of $\epsilon$
and $\tau^*$.
It can be seen that the accuracy (i.e. maximum relative error) of our ALI code is about $5\times 10^{-3}$ for the three cases studied here.
Relative error is close to this accuracy when $\tau$ is smaller than the thermalization depth $1/k(\epsilon)$ of the atmosphere (black dots on the curves), and significantly improves beyond (up to $10^{-8}$).
In photon mean free path units, the thermalization depth is 6, 58 and 5774 for $\epsilon = 10^{-2}, 10^{-4}$ and $10^{-8}$ respectively.
Note that the surface relative error is a good estimator of the accuracy in spectral lines, but not in the continuum.
The iterative algorithm was stopped after $N=1000$ iterations.
Figure \ref{fig3} shows that this number ensures convergence
of the ALI code, the convergence being slower when $\epsilon\to 0$ and
$\tau^* \to +\infty$.
Irregular steps in these curves are due to the Ng acceleration process, here operated every four iterations.
\subsection{The influence of $\tau_\mathrm{m}$}
We point out that $d_\mathrm{M}$ is improved when $\tau_\mathrm{m}$ goes to 0,
up to a given value where the accuracy is constant.
Including the $\tau=0$ point in the grid and choosing $\tau_\mathrm{m} < 10^{-2}$, the best accuracy is warranted.
Excluding the $\tau=0$ point from the grid has no influence on accuracy if we choose $\tau_\mathrm{m} < 10^{-4}$.
The standard choice $\tau_\mathrm{m}=10^{-4}$ is thus correct, and this value will be adopted hereafter.
\subsection{The influence of $n_\tau$ and $n_\mu$}
In Fig.~\ref{fig4} are shown the variations of $d_\mathrm{M}$ in an average line as a function of $n_\tau$ and $n_\mu$.
The maximum relative error $d_\mathrm{M}$ decreases with increasing number $n_\tau$ of $\tau$-grid points, and it is sensitive to the choice of the number $n_\mu$ of angular
grid points up to an optimal value $n_\mu^\mathrm{(opt)}$; the
latter is defined as the smallest value of $n_\mu$ for which the
condition $|1-d_\mathrm{M}(n_\mu)/d_\mathrm{M}(64)| < 0.01$ holds.
The accuracy does not increase with a finer $\mu$-grid.
We note that $n_\mu^\mathrm{(opt)} < n_\tau$ and that we have a linear dependence of this optimal value on $n_\tau$: $n_\mu^\mathrm{(opt)} = 0.8 n_\tau + 2.7$ (dashed curve).
This fit is still valid for strong lines.
As seen in Fig.~\ref{fig5} (same as Fig.~\ref{fig4} for the continuum), the fit for lines cannot be applied to the continuum, for which $n_\mu^\mathrm{(opt)} > n_\tau$.
It is still possible to define and calculate an optimal value for $n_\tau < 18$, using the relation $n_\mu^\mathrm{(opt)} = 1.7 n_\tau + 1.3$ (dashed curve).
It appears that our ALI code is more demanding in angular resolution when solving the problem (\ref{eq_s2}) in a continuum than in a line.
The results of Figs.~\ref{fig4} and \ref{fig5} are detailed in Fig.~\ref{fig6} for the three
chosen values of $(\epsilon,\tau^*)$ and $n_\mu = 64$.
We remark that the accuracy improves with $n_\tau$ for each couple
$(\epsilon,\tau^*)$, more significantly in the continuum than in lines.
Figure~\ref{fig7} gives the number of iterations $N_\mathrm{c}$ used
to reach convergence for $\varepsilon_\mathrm{c} = 10^{-2}$ and $n_\mu
= 64$.
The number $N_\mathrm{c}$ appreciably increases with $n_\tau$ in the lines:
it is indeed well known that the rate of convergence of the one-point ALI
iterative scheme drops for an increasing refinement of the
spatial grid (Olson et al. 1986); however improvements were already
proposed (e.g., Trujillo Bueno \& Fabiani Bendicho 1995) in order
to increase significantly the rate of convergence of ALI-based methods.
\subsection{The influence of $\epsilon$ and $\tau^*$}
The maximum relative error $d_\mathrm{M}$ and number of iterations $N_\mathrm{c}$ are shown
in Figs.~\ref{fig8} and \ref{fig9}
for an extended range of $(\epsilon,\tau^*)$ after the ALI code has
converged ($N=10\,000$, $n_\tau=9$ and $n_\mu=5$ are fixed here).
As seen in Fig.~\ref{fig8}, the accuracy hardly changes as
$\epsilon\to 0$ and $\tau^*\to +\infty$, but the number of iterations needed to achieve
convergence increases substantially (see Fig.~\ref{fig9}).
When $\epsilon>0.1$ the accuracy no longer depends on values of $\tau^*$.
The comparison of Figs.~\ref{fig6} and \ref{fig8} leads to a disagreement
since the parameter $n_\mu$ is set to different values, 64 and 5 respectively.
In Fig.~\ref{fig9}, we have plotted the parameter $N_\mathrm{c}$ as a function of $\epsilon$ and $\tau^*$.
When $\epsilon\to 0$ and $\tau^* \to +\infty$, we note a slowing down of the convergence (already seen in Fig.~\ref{fig3}).
\section{Comments on previous studies}\label{sec_5}
Now we compare our results for the monochromatic scattering problem with those published by Trujillo Bueno \& Fabiani Bendicho (1995) and Trujillo Bueno \& Manso Sainz (1999). Although these two papers concern mainly the development of new iterative methods for radiative transfer applications (for the unpolarized and polarized cases respectively) they give some information on the accuracy of the numerical solutions obtained for spatial grids of increasing resolution.
In Table~\ref{tab1}, good agreement is found between our values of
$N_\mathrm{c}$, $d(\epsilon, \tau^*, 0)$ and those given by these authors; our surface
relative error $d(\epsilon, \tau^*, 0)$ corresponds to their surface
true error $T_\mathrm{e}$.
The observed small discrepancies are possibly due to the different scattering
laws adopted, leading to different $\Lambda$-operators.
\begin{table*}
\caption{Comparison of results obtained with our (ALI+Ng) code and
previous ones.
Our $N_\mathrm{c}$ is defined by $\varepsilon_\mathrm{c} = 0.01$, while values from other authors are
based on a graphical guess $\varepsilon_\mathrm{c} \approx 0.05$.
The optical thickness is $\tau^* = 2\times 10^8$.
Note that in Trujillo Bueno \& Manso Sainz (1999), $N_\mathrm{c}$ values (in parenthesis) are given for a
non-accelerated Jacobi scheme.
These numbers have been divided by 2 in order to estimate the number of iterations when Ng acceleration is used.}
\label{tab1}
\centering
\begin{tabular}{llllllll}
\hline
\noalign{\smallskip}
$\epsilon,n_\tau,n_\mu$ & \multicolumn{2}{l}{JTB \& PFB (1995)} & \multicolumn{2}{l}{JTB \& RMS (1999)} & \multicolumn{3}{l}{This article} \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
& $N_\mathrm{c}$ & $T_\mathrm{e}$ & $N_\mathrm{c}$ & surface $T_\mathrm{e}$ & $N_\mathrm{c}$ & $d_\mathrm{M}(\epsilon,\tau^*)$ & $d(\epsilon,\tau^*,0)$ \\
\noalign{\smallskip}
\hline\noalign{\smallskip}
$10^{-6}$, 9, 1 & 180 & $3.5\times 10^{-3}$ && & 179 & $8.6\times 10^{-2}$ & $4.1\times 10^{-3}$ \\
$10^{-12}$, 9, 1 & 1300 & $3.5\times 10^{-3}$ &&& 985 & $8.6\times 10^{-2}$ & $4.1\times 10^{-3}$ \\
$10^{-4}$, 5, 64 && & 33 (65) & $2\times 10^{-2}$ & 42 & $1.5\times 10^{-2}$ & $1.3\times 10^{-2}$\\
$10^{-4}$, 9, 64 && & 75 (150) & $3\times 10^{-3}$ & 88 & $4.5\times 10^{-3}$ & $3.9\times 10^{-3}$\\
$10^{-4}$, 18, 64 && & 175 (350) & $4\times 10^{-4}$ & 184 & $1.1\times 10^{-3}$ & $9.0\times 10^{-4}$ \\
$10^{-4}$, 36, 64 && & 400 (800) & $5\times 10^{-5}$ & 356 & $2.4\times 10^{-4}$ & $1.9\times 10^{-4}$ \\
\hline
\end{tabular}
\end{table*}
In Fig.~\ref{fig10} it is shown how far the semi-infinite exact result
$S(\epsilon,0)=\sqrt{\epsilon}$ agrees with the finite one.
This comparison is useful since many authors use the $\sqrt{\epsilon}$-law
as a check for their calculations in thick slabs.
We plot the relative difference
$\delta_1=|\,1-S(\epsilon,\tau^*,0)/\sqrt{\epsilon}\,|$ as a function of
$k(\epsilon) \tau^*$, and the quantity
$\exp\left(-k(\epsilon)\tau^*\right)$ which characterizes the
validity of the $\sqrt\epsilon$-law (solid curve). For $k(\epsilon)\tau^* > 100$,
the accuracy limit of our code is reached, which explains that the
solid curve no longer fits the dots. This law is very well
satisfied in lines ($k(\epsilon)\tau^* \approx 34$ in an average line)
but not enough in the continuum ($k(\epsilon)\tau^* \approx 3.3$).
We conclude that the $\sqrt\epsilon$-law can be used as a test for the
ALI code when $k(\epsilon)\tau^* > 10$, since then $\sqrt\epsilon$ is an
approximation to the surface value with an accuracy better than $10^{-4}$, as seen in Fig.~\ref{fig10}.
In Trujillo Bueno \& Fabiani Bendicho (1995), the Eddington approximation is used as the reference solution for a one-point angular quadrature $n_\mu=1$ with $\mu=\pm 1/\sqrt 3$.
The analytical expression of the Eddington approximation in a finite slab is:
\begin{eqnarray}
\label{eq_sedd}
\lefteqn{S_\mathrm{E}(\epsilon,\tau^*,\tau) = 1 - (1-\epsilon) \frac{\exp\left(-\sqrt{3\epsilon}\,\tau\right) + \exp\left(-\sqrt{3\epsilon}\,(\tau^* - \tau)\right)}{1+\sqrt\epsilon+(1-\sqrt\epsilon)\exp\left(-\sqrt{3\epsilon}\,\tau^*\right)}. } \nonumber \\
\end{eqnarray}
This is the exact solution of the monochromatic scattering problem when the mean intensity is calculated with the above-mentioned one-point angular quadrature.
However, as is well-known, it gives only an approximation to the exact (i.e., multi-angle) solution of the full problem (\ref{eq_s1})-(\ref{eq_e1}).
In other words, the true error given by Trujillo Bueno \& Fabiani Bendicho (1995) is relative to the $n_\mu=1$ monochromatic scattering problem only, it does not give information on the error that would have been got by comparing the numerical solution to the $n_\mu=1,3,5, \ldots$ problem to the exact multi-angle solution ($n_\mu=\infty$).
In fact, as given in Table~\ref{tab1} for $n_\mu=1$, when the solution of the $n_\mu=1$ problem is compared to the exact multi-angle solution, we find that the maximum error for $n_\tau=9$ is $8.6\times 10^{-2}$.
The latter represents the maximum relative difference between the Eddington approximation and the exact solution (Fig.~\ref{fig11}).
A similar investigation, but for the two-level atom resonance-line scattering polarization problem, was carried out by Trujillo Bueno \& Manso Sainz (1999), whose Table 3 gives the surface true-error values of the fractional atomic polarization for $n_\mu=3,5,7,11, \ldots, 61$.
\section{Conclusion}
Our ALI code has been subjected to a wide range of tests, revealing at
the same time its capabilities and its limits. Before developing
these two points, we note that our conclusions are relative to the
particular code we have used (based on Jacobi's method), specifically solving the
standard problem (\ref{eq_s1})-(\ref{eq_e1}).
The accuracy of the code is ultimately determined by the accuracy of the
formal solver we have used (parabolic short characteristics).
We have checked the great robustness of our code, which is certainly
its most remarkable feature. It is able to solve the
standard problem (\ref{eq_s1})-(\ref{eq_e1}) for a wide range of input
parameters $\epsilon$ and $\tau^*$, with no important lack of
performance when $\epsilon \to 0$ and/or $\tau^* \to
+\infty$.
However, the lowest accuracy of the ALI numerical solutions happens in
the outermost layers of a star, corresponding to $\tau$ lower than the
thermalization depth $1/k(\epsilon)$, these layers forming, by
definition, the atmosphere of the star.
The accuracy of our code is not better than, say $10^{-2}$, when we choose
$n_\tau = 9$, $n_\mu = 5$ and limit the number of iterations to $N<100$, as it is currently done in stellar atmospheres modelling.
To improve the accuracy of the calculations up to $\sim 10^{-3}$, the parameters $n_\tau$, $n_\mu$ and $N$ should be
set to larger (but today impractical) values when solving the radiative transfer equation on a large frequency spectrum, i.e. at thousands of frequencies.
Indeed we pointed out a truly noticeable improvement of
the accuracy when using finer grids in $\tau$ or $\mu$.
Such an observation was made easier by the use of a very accurate
reference solution.
Of course, increasing the level of refinement of
both spatial and angular quadratures has a strong impact upon
the number of iterations needed for convergence.
However, to overcome this difficulty while keeping the same accuracy on the numerical
solutions, methods based on Gauss-Seidel and successive
over-relaxation iterations were already proposed by Trujillo
Bueno \& Fabiani Bendicho (1995).
Another important question is relative to the propagation of errors in
a stellar atmosphere model: to what extent are the main
quantities provided by the model (populations of heavy particles,
electron density, pressure, etc.) sensitive to the accuracy on the
RTE solution? We intend to tackle this subject in a future work by
constructing an accurate -- but still very idealized -- stellar
atmosphere model, in which the main quantities are first derived from
an exact solution to the RTE, and then from the solution given by a ALI-based numerical method.
\begin{acknowledgements}
The authors wish to thank M. Ahues, A. Largillier, G. Panasenko (Numerical Analysis team of the
University Jean Monnet of Saint-Etienne, France), A. Amosov (Moscow Power Engineering Institute,
Russia) and J. Bergeat (Centre de Recherche Astronomique de Lyon) for some helpful discussions concerning this work.
We also thank Ivan Hubeny and Javier Trujillo Bueno for their valuable comments on a previous version of our manuscript.
\end{acknowledgements}
|
Title:
The Great Observatories Origins Deep Survey VLT/FORS2 Spectroscopy in the GOODS-South Field: Part II |
Abstract: We present the second campaign of the ESO/GOODS program of spectroscopy of
faint galaxies in the GOODS-South field. Objects were selected as candidates
for VLT/FORS2 observations primarily based on the expectation that the
detection and measurement of their spectral features would benefit from the
high throughput and spectral resolution of FORS2. The reliability of the
redshift estimates is assessed using the redshift-magnitude and color-redshift
diagrams and comparing the results with public data. 807 spectra of 652
individua targets have been obtained in service mode with the FORS2
spectrograph at the ESO/VLT, providing 501 redshift determinations. The typical
redshift uncertainty is estimated to be sigma_z ~ 0.001. Galaxies have been
selected adopting three different color criteria and using the photometric
redshifts.The resulting redshift distribution typically spans two redshift
domains: from z=0.5 to 2 and z=3.5 to 6.2. In particular, 94 B435-,V606-,i775
"dropout" Lyman break galaxies have been observed, yielding redshifts for 65
objects in the interval 3.4<z<6.2. Three sources have been serendipitously
discovered in the redshift interval 4.8<z<5.5. Together with the previous
release, 930 sources have now been observed and 724 redshift determinations
have been carried out. The reduced spectra and the derived redshifts are
released to the community through the ESO web page
this http URL Large scale structures are clearly detected
at z=0.666, 0.734, 1.096, 1.221, 1.300, and 1.614. A sample of 34 sources with
tilted [OII]3727 emission has been identified, 32 of them in the redshift range
0.9<z<1.5.
| https://export.arxiv.org/pdf/astro-ph/0601367 |
\newcommand{\magcir}{\ \raise -2.truept\hbox{\rlap{\hbox{$\sim$}}\raise5.truept
\hbox{$>$}\ }}
\newcommand{\mincir}{\ \raise -2.truept\hbox{\rlap{\hbox{$\sim$}}\raise5.truept
\hbox{$<$}\ }}
\title{The Great Observatories Origins Deep Survey}
\subtitle{VLT/FORS2 Spectroscopy in the GOODS-South Field: Part II}
\author{E. Vanzella\inst{1}
\and
S. Cristiani\inst{1}
\and
M. Dickinson\inst{2}
\and
H. Kuntschner\inst{3}
\and
M. Nonino\inst{1}
\and
A. Rettura\inst{5,6}
\and
P. Rosati\inst{5}
\and
J. Vernet\inst{5}
\and
\\
C. Cesarsky\inst{5}
\and
H. C. Ferguson\inst{4}
\and
R.A.E. Fosbury\inst{3}
\and
M. Giavalisco\inst{4}
\and
A. Grazian\inst{9}
\and
J. Haase\inst{3}
\and
L. A. Moustakas\inst{7}
\and
\\
P. Popesso\inst{5}
\and
A. Renzini\inst{8}
\and
D. Stern\inst{7}
\and
the GOODS Team
}
\institute{
INAF - Osservatorio Astronomico di Trieste, Via G.B. Tiepolo 11,
40131 Trieste, Italy.
\and
National Optical Astronomy Obs., P.O. Box 26732, Tucson, AZ 85726.
\and
ST-ECF, Karl-Schwarzschild Str. 2, 85748 Garching, Germany.
\and
Space Telescope Science Institute, 3700 San Martin Drive,
Baltimore, MD 21218.
\and
European Southern Observatory, Karl-Schwarzschild-Strasse 2,
Garching, D-85748, Germany.
\and
Universite' Paris-Sud 11, Rue Georges Clemenceau 15, Orsay, F-91405, France
\and
Jet Propulsion Laboratory, California Institute of Technology,
MS 169-506, 4800 Oak Grove Drive, Pasadena, CA 91109
\and
INAF - Astronomical Observatory of Padova, Vicolo dell'Osservatorio 5,
I - 35122 Padova -ITALY
\and
INAF - Osservatorio Astronomico di Roma, Via Frascati 33,
I-00040 Monteporzio Roma, Italy
\thanks{Based on observations made at the European Southern
Observatory, Paranal, Chile (ESO programme 170.A-0788 {\it The Great
Observatories Origins Deep Survey: ESO Public Observations of the
SIRTF Legacy/HST Treasury/Chandra Deep Field South.}) }
}
\offprints{E. Vanzella, \email{vanzella@oats.inaf.it}}
\date{Received ; accepted }
\abstract
{}
{We present the second campaign of the ESO/GOODS program of spectroscopy
of faint galaxies in the GOODS-South field.}
{Objects were selected as candidates for VLT/FORS2 observations
primarily based on the expectation that the detection and measurement
of their spectral features would benefit from the high throughput and
spectral resolution of FORS2. The reliability of the redshift estimates
is assessed using
the redshift-magnitude and color-redshift diagrams and comparing the results with public data.}
{807 spectra
of 652 individual targets have been obtained in service mode with
the FORS2
spectrograph at the ESO/VLT, providing 501 redshift
determinations.
The typical redshift uncertainty is estimated to be
$\sigma_z \simeq 0.001$.
Galaxies have been
selected adopting three different color criteria and using the photometric redshifts.
The resulting redshift distribution typically spans two redshift domains:
from z=0.5 to 2 and z=3.5 to 6.2. In particular, 94
$B_{435}$-,$V_{606}$-,$i_{775}$-"dropout"
Lyman break galaxies have been observed, yielding redshifts for 65 objects in the
interval 3.4$<$z$<$6.2. Three sources have been serendipitously discovered in the
redshift interval 4.8$<$z$<$5.5.
Together with the previous release, 930 sources have now been observed
and 724 redshift determinations have been carried out.
The reduced spectra and the derived redshifts are released to the community
through the ESO web page $\it{http://www.eso.org/science/goods/}$
\thanks{The catalog (Table~\ref{tab:tblspec}) is available in electronic form
at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5)
or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/}.
Large scale structures are clearly detected at $z \simeq 0.666, 0.734, 1.096, 1.221, 1.300$,
and $1.614$. A sample of 34 sources with tilted [O\,{\sc ii}]3727 emission
has been identified, 32 of them in the redshift range 0.9$<$z$<$1.5.
}
{}
\keywords{Cosmology: observations -- Cosmology: deep redshift surveys
-- Cosmology: large scale structure of the universe -- Galaxies: evolution.
}
\section{Introduction}
The Great Observatories Origins Deep Survey (GOODS) is a public,
multi-facility project that aims to answer some of the most profound
questions in cosmology: how did galaxies form and assemble their
stellar mass? When was the morphological differentiation of galaxies
established and how did the Hubble Sequence form? How did Active
Galactic Nuclei (AGN) form and
evolve, and what role do they play in galaxy evolution? How much do
galaxies and AGN contribute to the extragalactic background light? Is
the expansion of the universe dominated by a cosmological constant? A
project of this scope requires large and coordinated efforts from many
facilities, pushed to their limits, to collect a database of
sufficient quality and size for the task at hand. It also requires
that the data be readily available to the worldwide community for
independent analysis, verification, and follow-up.
The program targets
two carefully selected fields, the Hubble Deep Field North (HDF-N) and
the Chandra Deep Field South (CDF-S), with three NASA Great
Observatories (HST, Spitzer and Chandra), ESA's XMM-Newton, and a wide
variety of ground-based facilities. The area common to
all the observing programs is 320 arcmin$^2$, equally divided between
the North and South fields. For an overview of GOODS, see \cite{dick03},
\cite{renz03} and \cite{giava04a}.
In the last five years
the CDF-S has been the target of several spectroscopic campaigns
(\cite{crist00}, \cite{croom01}, \cite{bunk03}, \cite{stan04},
\cite{stro04}, \cite{vanderwel04}, \cite{dick04}, \cite{szo04},
\cite{fevre05}, \cite{vanz05}).
This is the second paper in a series presenting the results of the GOODS
spectroscopic program carried out with the VLT/FORS2 spectrograph.
For a full description of its aims we refer to the first
paper (\cite{vanz05}, RUN1 hereafter).
Here we recall that the ESO/GOODS spectroscopic program is designed to observe
all galaxies for which VLT optical spectroscopy is likely to allow the redshift
determination.
The program makes full use of the VLT instrument capabilities
(FORS2 and VIMOS),
matching targets to instrument and disperser combinations in order to
maximize the effectiveness of the observations. The magnitude limits
and selection bandpasses to some extent depend on the instrumental
setup being used. The aim is to reach mag~$\sim24-25$ with adequate S/N, with
this limiting magnitude being in the B band for objects observed with
the VIMOS LR-Blue grism, in the V band for those observed in the VIMOS
LR-Red grism, and in the z band for the objects observed
with FORS2.
The second FORS2 spectroscopic campaign (17 masks, RUN2 hereafter) in the
Chandra Deep Field South, was carried out
in the period fall 2003 - early 2004 in service mode.
New FORS2 observations were performed in December 2004 (6 masks, RUN3) mainly
focused on color-selected Lyman break ``dropout'' targets and 5 more masks will be
completed before February 2006 (RUN4) mainly dedicated to sources detected
at 24$\mu$m with the {\it Spitzer Space Telescope} MIPS instrument.
These data will be described in a forthcoming paper.
The VIMOS spectroscopic survey in the GOODS-S field is started and will produce
hundreds of redshift determinations, mainly in the redshift range 0$<$z$\leq$3.5.
The paper is organized as follows. Sect. 2 describes the target selection,
while Sect. 3 describes the observations and data reductions.
The redshift determination is presented in Sect. 4. In Sect. 5 we discuss
the data and in Sect. 6 the conclusions are presented.
Throughout this paper the magnitudes are given in the AB system (\cite{oke77})
(AB~$\equiv 31.4 - 2.5\log\langle f_\nu / \mathrm{nJy} \rangle$),
and the ACS F435W, F606W, F775W, and F850LP filters are designated
hereafter as $B_{435}$, $V_{606}$, $i_{775}$ and $z_{850}$, respectively.
We assume a cosmology with $\Omega_{\rm tot}, \Omega_M, \Omega_\Lambda = 1.0, 0.3, 0.7$
and $H_0 = 70$~km~s$^{-1}$~Mpc$^{-1}$.
\section{Target Selection}
Galaxies were selected as candidates for FORS2 observations
primarily based on the expectation that the detection and measurement
of their spectral features would benefit from the high throughput,
moderately-high spectral resolution, and reduced long-wavelength fringing
of FORS2 relative to other instrument options such as VIMOS.
In particular, the main spectral emission and absorption features for
galaxies at $0.8 < z < 2.0$ appear at very red optical wavelengths
($7000\AA < \lambda < 1\mu$m). Similarly, very faint Lyman break
galaxies at $z \gtrsim 4$, selected as $B_{435}$, $V_{606}$ and
$i_{775}$--dropouts from the GOODS ACS photometry, also benefit greatly
from the red throughput and higher spectral resolution of FORS2.
In practice, several categories of object selection criteria were used to
ensure a sufficiently high density of targets to
efficiently populate masks. These criteria were:
\begin{enumerate}
\item{Primary catalog: $(i_{775}-z_{850}) > 0.6$ and $z_{850} < 25$.
This should ensure redshifts $z \gtrsim 0.7$ for ordinary early-type galaxies
(whose strongest features are expected to be absorption lines),
and higher redshifts for intrinsically bluer galaxies likely to have emission
lines.}
\item{Secondary catalog: $0.45 < (i_{775}-z_{850}) < 0.6$ and $z_{850} < 25$.}
\item{Photometric-redshift sample: $1<$$z_{\rm phot}$$<2$ and $z_{850} < 25$,
from \cite{moba04}.}
\item{$B_{435}$, $V_{606}$ and $i_{775}$--dropouts color selected Lyman break
galaxy candidates (see \cite{giava04b} and \cite{dick04}).}
\item{A few miscellaneous objects, including host galaxies of
supernovae detected in the GOODS ACS observing campaign.}
\end{enumerate}
The targets were selected from a preliminary catalog based on the
v1.0 public release of the GOODS ACS images. This version includes
all five epochs of the GOODS ACS data
\footnote{{\it ftp://archive.stsci.edu/pub/hlsp/goods/catalog$\_$r1/}},
and is a significant improvement
on the previous, 3-epoch v0.5 release that was used to select targets
for the first FORS2 observations (RUN1, \cite{vanz05}).
For this paper and data release, the objects observed with FORS2
have been matched to the public release ACS catalog version r1.1z,
also based on the 5-epoch v1.0 ACS images. The r1.1z catalog
is based on the r1.0z SExtractor run, and merely corrects errors and
omissions in the r1.0z catalog files.
When designing the masks, we generally
tried to avoid observing targets that had already been observed in other
redshift surveys of this field, namely the K20 survey of \cite{cimatti02}
and the survey of X-ray sources by \cite{szo04}.
807 spectra of 652 individual targets have been extracted
from the RUN2 (multiple observations have been performed,
especially for the high redshift candidates).
Out of these 652 targets, 178 are from the primary catalog, 117 are from
the secondary catalog, 141 are from the photometric redshift selection,
94 are from the Lyman break sample, and 3 are from miscellaneous list.
The remaining 119 sources have been serendipitously identified, due to:
a) sources randomly in the slit other than the target or
b) sources put in the slit in the situation where no targets were available
or c) relatively bright objects put in the slit for the alignment of the mask.
The total number of individual sources observed in
the RUN1 + RUN2 is 930 (1203 spectra reduced) with 724 redshift determinations.
The spectroscopic database presented here is incomplete:
none of the above listed categories has been exhaustively observed,
nor any GOODS subarea has been fully covered.
\section{Observations and Data Reduction}
\begin{table}
\centering \caption{Journal of the FORS2 observations (RUN2).}
\begin{tabular}{lccc}
\hline \hline
Mask ID & Date & exp.time (s)\\
\hline
914250 & Aug. 2003 & 17$\times$1200 \cr
905513 & Sept. 2003 & 18$\times$1200 \cr
943018 & Sept. 2003 & 12$\times$1200 \cr
924345 & Sept. 2003 & 12$\times$1200 \cr
945143 & Sept. - Oct. 2003 & 12$\times$1200 + 3$\times$1000 \cr
992438 & Oct. - Dec. 2003 & 12$\times$1200 \cr
985931 & Nov. 2003 & 12$\times$1200 + 2$\times$120 \cr
990204 & Dec. 2003 & 12$\times$1200 \cr
904509 & Dec. 2003 & 12$\times$1200 \cr
991435 & Dec. 2003 & 12$\times$1200 \cr
935030 & Dec. 2003 & 12$\times$1200 \cr
951937 & Dec. 2003 & 12$\times$1200 + 1100 + 500 \cr
960930 & Dec. 2003 & 12$\times$1200 \cr
961839 & Jan. 2004 & 12$\times$1200 \cr
932802 & Jan. 2004 & 12$\times$1200 \cr
993304 & Jan. 2004 & 12$\times$1200 \cr
951526 & Feb. 2004 & 3$\times$1200 \cr
\hline
\label{tab:tblobs}
\end{tabular}
\end{table}
The VLT/FORS2 spectroscopic observations were carried out
in service mode during several nights at the end of 2003 and the beginning
of 2004.
A summary is presented in Table~\ref{tab:tblobs}.
In all cases the $300I$ grism was used as dispersing element without
order-separating filter.
This grism provides a scale of roughly 3.2\AA~pix$^{-1}$. The nominal
resolution of the configuration was
R=$\lambda/\Delta\lambda$=660, which corresponds to
13{\AA} at 8600{\AA}. The spatial scale of FORS2 is $0.126\arcsec$/pixel.
The slit width was always $1\arcsec$.
Dithering of the targets along the slits was applied typically with
steps of 0,$\pm$8 pixels,
in order to effectively improve the sky and fringe subtraction, and remove
CCD blemishes.
\subsection{\it Data Reduction}
Data were reduced with a semi-automatic pipeline
that we have developed on the basis of the MIDAS package (\cite{eso_midas}),
using commands of the LONG and MOS contexts.
The main procedures have been described in the previous paper (\cite{vanz05}).
In the cases of multiple observations of the same source
in different masks, the one dimensional spectra have been co-added
weighing according to the exposure time, the seeing condition and the
resulting quality of each extraction process (defects present in the CCD, object too
close to the border of the slit, etc.). A visual check of the two
dimensional frames has been performed (in some cases the two
dimensional spectra have also been co-added, in order to improve and guide the
visual inspection).
We emphasize here that we opted to observe the science targets {\em without} an
order-sorting filter, implying deleterious effects to the flux calibration.
The second order overlap becomes important at wavelengths above
$\sim$8000\AA~depending on the color of the target.
In Figure~\ref{fig:test_flux} the comparison between two 1-D calibrated spectra
(one blue ($i-z$)$\sim$0 and one red ($i-z$)$\sim$1)
and the correspondent ACS photometry is shown.
The photometric values in the $i$ and $z$ bands are marked with two filled circles
and are consistent with the derived spectral behavior.
For the red objects that dominate the FORS2 target selection, we felt that the
improved wavelength coverage more than compensates for the partial
unreliability of the flux calibration. Due to both this second order
light and uncertain slit losses, we caution against using the
calibrated fluxes for scientific purposes.
Fluxes in the released 1-D spectra are given in units of $10^{-16}$ erg s$^{-1}$
cm$^{-2}$ \AA$^{-1}$.
\section{Redshift Determination}
Spectra of 652 individual objects have been extracted from RUN2.
From them we have determined 501 redshifts.
In the large majority of the cases the redshift has been determined through the
identification of prominent features of galaxy spectra:
depending on the redshift and the nature of the source the 4000\AA\ break, Ca H and K,
g-band, MgII 2798-2802, AlII 3584,
Ly$\alpha$, Si\,{\sc ii} 1260.4\AA, O\,{\sc i} 1302.2\AA, C\,{\sc ii} 1335.1\AA,
Si\,{\sc iv} 1393.8,1402.8\AA, Si\,{\sc ii} 1526.7\AA, C\,{\sc iv} 1548.2, 1550.8\AA~in
absorption and Ly$\alpha$, [O\,{\sc ii}]3727, [O\,{\sc iii}]5007, H$\beta$, H$\alpha$
in emission.
The redshift estimation has been performed cross-correlating the observed
spectrum with templates of different spectral types (S0, Sa, Sb, Sc, Elliptical,
Lyman Break, etc.), using the $rvsao$ package in the IRAF environment.
The redshift identifications are summarized in Table~\ref{tab:tblspec} and
are available at the URL $\it{http://www.eso.org/science/goods/}$.
In Table~\ref{tab:tblspec},
the column {\em ID} contains the target identifier, that is constructed out of the
target position (e.g., $GDS~J$033206.44-274728.8) where GDS stands
for {\bf G}OO{\bf D}S {\bf S}outh.
The coordinates are based on the GOODS v1.1 astrometry.
The v1.1 release is based upon the v1.0 SExtractor run, and merely corrects
errors and omissions in the v1.0 catalog files.
The cataloged sources are identical, in both number and ordering, to the v1.0
release.
The columns $z_{850}$ and ($i_{775}$-$z_{850}$) list the magnitude
(SExtractor ``MAG$\_$AUTO'') and the color (SExtractor ``MAG$\_$ISO'') of the sources
derived from the catalog v1.1. The color has been measured through isophotal
apertures defined in the $z_{850}$ band image (as done in \cite{dick04} and
\cite{giava04b}).
The {\em quality} flag (QF hereafter), indicates the reliability of the
redshift determination. As described in the previous work (\cite{vanz05}, RUN1),
the QF has been divided in three categories: ``A'', ``B'' and ``C''.
An estimation of the confidence level associated to each class ``A'', ``B'' and ``C''
can be derived analyzing the FORS2 measurements in common with independent spectroscopic
estimations available in literature.
This has been done in the previous paper (RUN1) where 39 sources have been analyzed
and in Sect. 5.1 of the present work (98 more sources, see below).
In this way the sample of FORS2 measurements in common with independent spectroscopic
surveys counts 137 galaxies, in which we find 0, 1 and 4 FORS2 wrong redshifts for classes ``A'', ``B''
and ``C'', respectively.
In this way at 1$\sigma$ (\cite{gehrels86}) the confidence level of the ``A'', ``B'' and ``C''
categories turns out to be $\simeq$ 98$\%$, $\simeq$ 97$\%$ and 93$\%$.
There are 291 objects classified with quality
``A'', 119 with quality ``B'', 91 with ``C'', and 151 with ``X'', an inconclusive spectrum.
The flag "{\em class}" groups the objects for which emission line(s) (``em.''),
absorption-line(s) (``abs.'') or both (``comp.'') are detected in the spectrum.
In the present catalog, three sources have been classified as stars.
In 30$\%$ of the cases the redshift is based only on one emission line, usually
identified as [O\,{\sc ii}]3727 or Ly$\alpha$.
In these cases the continuum shape, the presence of breaks, the
absence of other spectral features in the observed spectral range and the broad band
photometry are particularly important in the evaluation. The quality for these
sources ranges from ``A'' to ``C'' depending on the additional information
described above (35$\%$ of the sample with a single emission line have
QF=''A'', with a mean redshift $<z>$=1.21$\pm$0.2).
The {\em comments} column contains additional information relevant
to the particular observation. The most common ones summarize
the identification of the principal lines, the inclination of an emission
line due to internal kinematics, the weakness of the signal
(``faint''), the low S/N of the extracted spectrum (``noisy''), the apparent absence
of spectroscopic lines (``featureless continuum''), etc.
In few cases the spectrum extracted is the combination of more than
one source in the slit and where possible the redshifts of the
``components'' have been estimated. In the RUN1 + RUN2 spectroscopic
data, 11 sources in the GOODS-S field are not present in the ACS
photometric catalog v1.1. Six of them have a redshift estimation (an
example is shown in Figure~\ref{fig:MIXED}). Three out of six appear
to be emission line objects whose continuum is too faint and has not
been detected in the ACS catalogs. The other seven sources are outside
the ACS area.
\begin{table*}
\centering \caption{Spectroscopic redshift catalog. $\dag$}
\begin{tabular}{lcccccl}
\hline \hline
ID(v1.0) & $z_{850}$ & $(i_{775}-z_{850})$ & zspec &class. & Quality & comments \\
\hline
GDS~J033245.99-275108.3 & 23.48 &0.47 &1.238 &em. & B &[O\,{\sc ii}]3727 \cr
GDS~J033246.04-274929.7 & 26.06 &1.77 &5.787 &em. & A &LyA (faint continuum) \cr
GDS~J033246.05-275444.8 & 21.49 &0.53 &0.733 &abs. & A &CaH,g-band,H$\beta$,Mg,CaFe \cr
GDS~J033246.16-274752.3 & 24.46 &0.43 &1.221 &em. & B &[O\,{\sc ii}]3727 \cr
\hline
\multicolumn{6}{l}
{$\dag$ This table is available in its entirety via $\it{http://www.eso.org/science/goods/}$.}\\
\multicolumn{6}{l}
{A portion is shown here for guidance regarding its form and content.}\\
\label{tab:tblspec}
\end{tabular}
\end{table*}
\section{Discussion}
In the following, if not specified, we consider the entire FORS2 sample, including
both RUN1 and RUN2. This sample is summarized in Table~\ref{tab:matrix} where the sources
are divided into different selection categories (see Sec. 2) and by redshift
(or whether a redshift could be determined).
The distribution of the quality flags is also tabulated.
\subsection{Reliability of the redshift - comparison with public data}
A practical way to assess the reliability of the redshifts reported in
Table~\ref{tab:tblspec} is to compare the present results with
independent measurements from other surveys. In the last five years
the CDF-S has been the target of several spectroscopic campaigns
(the surveys with the number of redshifts in parenthesis used in the
comparison are here reported:
\cite{crist00} (5), \cite{croom01} (29), \cite{bunk03} (1),
\cite{stan04} (3), \cite{stro04} (14), \cite{vanderwel04} (6),
\cite{dick04} (1), \cite{szo04} (124),
\cite{fevre05} (748), \cite{vanz05} (234)).
Making use of a publicly available
master compilation of all spectroscopic redshifts in the GOODS/CDF-S
region (Rettura et al. in preparation, available at the URL {\it
http://www.eso.org/science/goods/spectroscopy/CDFS$\_$Mastercat/} we
have been able to compare our redshift determinations with the
existing data in the literature.
There are 98 objects in common with the present second release of the
FORS2 GOODS survey (RUN2).
For 87 cases out of 98 (89$\%$) the agreement is very good, with a mean difference
$<z_{FORS2_{RUN2}} - z_{CDF-S}>$= 0.0042 $\pm$ 0.0095.
15 objects have a redshift determination both in RUN1 and RUN2. The
distribution of the redshift differences has a median
$|z_{FORS2_{RUN2}} - z_{FORS2-{RUN1}} | = 0.0002$ and a difference
between the 82 and 18 percentile of $2.6 \cdot 10^{-3}$.
Assuming equipartition of the redshift uncertainties between RUN1 and
RUN2 we derive a typical error on the redshift determinations
in the FORS2 GOODS spectroscopy of $\sigma_z \simeq 0.001$.
Ten cases show ``catastrophic'' discrepancies between the RUN2 and
the K20, \cite{szo04} and the VVDS surveys,
i.e. $|z_{FORS2_{RUN2}} - z_{CDF-S}|$ greater
than 0.08. In order to compare the redshift estimations we recall
here which is the quality level adopted by other authors. In the K20
survey the QF adopted is 1, 0 or -1 if the redshift determination is
solid, tentative or unconclusive, respectively.
In \cite{szo04} the QF=3 indicates reliable redshift determination
with unambiguous X-ray counterpart, QF=2 corresponds to a reliable
redshift determination and a value of 0.0 indicates no success. QF=1
indicates the detection of $some$ feature in the spectrum (typically a
single narrow emission line). QF=0.5 is used when there is a hint of
some spectral feature. In the VVDS, the flags 2,3,4 are the most
secure with a confidence of 75$\%$, 95$\%$ and 100$\%$
respectively. Flag 1 is an indicative measurement (50$\%$), flag 9
indicates that there is only one secure emission line, and flag 0
indicates a measurement failure with no features identified.
In the following we discuss in detail each discrepant spectrum:
\begin{enumerate}
\item{GDS~J033232.08-274155.2. This is a discrepancy with our previous
identification (RUN1) and the present one (RUN2). In the first run the
redshift determination was tentative (quality ``C'', z= 0.960) and in
the second run we derived z=1.393 (QF=''B''). However the co-addition
of the two produces a featureless continuum, we have changed the
quality to ``X''.}
\item{GDS~J033217.77-274714.9. K20 and VVDS assign redshift 0.729 and
0.731, respectively (and quality 1 and 3). In the FORS2 spectrum
there are three objects in the slit, the GDS~J033217.77-274714.9 is a
serendipitous source at the border of the slit, its exposure time is
reduced of 50$\%$ due to the dithering process. The continuum is
faint and a possible emission line is detected at
7522.6\AA~interpreted to be [O\,{\sc ii}]3727 at z=1.018 (QF=''C'').}
\item{GDS~J033232.18-274534.9. K20 assigns a redshift 0.332 with quality 1. The FORS2 spectrum
shows [O\,{\sc ii}]3727, MgI, CaHK, g-band and the Balmer Break at z=0.523 (QF=''A'').}
\item{GDS~J033239.67-274850.6. Szokoly et al. measure redshift 3.064 with quality 3.
Our spectrum shows a smoothed
break at $\sim$ 6000\AA~and an absorption line at 6789.0\AA, our redshift determination is
tentatively z=3.885, QF=''C''. The spectrum starts at 5600\AA, if it is at redshift 3.064,
the most relevant spectral features are outside the spectral
coverage.
We note that if the redshift is 3.064
the absorption line we measure at 6789.0\AA would be consistent with the Al\,{\sc ii} 1670.8\AA.}
\item{GDS~J033240.84-275546.7. Szokoly et al. measure redshift 0.625 with quality 0.5.
Our spectrum shows a featureless continuum and starts at 5790\AA, a possible emission line is
detected at 8277.6\AA, we assign tentatively z=1.221 QF=''C''.}
\item{GDS~J033222.44-275606.1. VVDS measure redshift 0.490 with quality 2.
The FORS2 spectrum shows a tilted emission line at 7790.3\AA~and a faint-diffuse continuum. We
assign tentatively z=1.090 (QF=''C''). We note that in the FORS2 spectrum the [O\,{\sc ii}]3727,
[O\,{\sc iii}]5007 or H$\beta$ lines at z=0.490 have not been detected.}
\item{GDS~J033225.28-275524.2. VVDS measure redshift 0.923 with quality 1.
The FORS2 spectrum shows [O\,{\sc ii}]3727 (slightly tilted), CaHK, MgI and the Balmer Break
at z=1.017 (QF=''A'').}
\item{GDS~J033230.37-274008.5. VVDS measure redshift 1.083 with quality 2.
The source shows a bright continuum and [O\,{\sc ii}]3727, MgII and the NeIII lines
at z=1.357 (QF=''A'').}
\item{GDS~J033230.50-275312.3. VVDS measure redshift 1.427 with quality 2.
The FORS2 spectrum shows [O\,{\sc ii}]3727 (tilted), CaHK, MgII, g-band at z=1.017 QF=''A''.}
\item{GDS~J033234.82-274721.9. VVDS measure redshift 0.315 with quality 3.
In the FORS2 spectrum an emission line has been detected at 8632.3\AA, interpreted as [O\,{\sc ii}]3727
at z=1.316 with QF=''B''.
The continuum starts at 6260\AA, and if we assume the line to be H$\alpha$ at z=0.315
the H$\beta$ and/or [O\,{\sc iii}]5007 are not present.}
\item{GDS~J033242.97-274649.9. VVDS measure redshift 0.831 with quality 1.
The FORS2 spectrum shows [O\,{\sc ii}]3727, CaHK, NeIII and $H\delta$ (in absorption) at z=1.036
with QF=''A''.}
\end{enumerate}
In summary, 7 out of 10 discrepant redshift determinations turn out to be probably correct in the FORS2
spectroscopy, all with QF better or equal to QF=``B''.
Of the remaining 3 sources (all with QF=''C''), one is uncertain and two are probably wrong
in the FORS2 spectroscopic identification due to the reasons described above.
\subsection{Reliability of the redshifts - diagnostic diagrams}
Figures~\ref{fig:z_vs_mag} and \ref{fig:i_zVSzspec}
show the redshift-magnitude and the color-redshift distributions,
respectively. Figure~\ref{fig:i_zVSzspec} shows the behavior
for galaxies at redshift less than 2 and quality flag ``A'' and ``B''.
The two populations of ``emission-line'' (star-forming) and ``absorption-line''
(typically elliptical) galaxies are clearly separated.
The mean color of the absorption-line objects outline the upper envelop
of the distribution,
consistent but increasingly bluer than the colors of a non-evolving $L^{\star}$
elliptical galaxy (estimated integrating the spectral templates of \cite{cole80}
through the ACS bandpasses).
The emission-line objects show in general a bluer $i_{775}-z_{850}$ color and
a broader distribution than the absorption-line sources.
The broader distribution, with some of the emission-line objects
entering the color region of the ellipticals,
is possibly explained by dust obscuration,
high metallicity or strong line emission in the $z_{850}$ band
(for example emission lines [O\,{\sc iii}]5007, H$\beta$ at redshift 0.8, as measured
for the source GDS~J033219.53-274111.6).
\begin{table*}
\centering \caption{Summary of the spectroscopic catalog as a function of the redshift bin
(first column), categories (from column two to six) and serendipitously identified sources (column seven).
The contribution of the different quality flags (``A'', ``B'' or ``C'') are also reported.
A total of 930 spectra have been analyzed (RUN1 and RUN2).}
\begin{tabular}{lllllll|c}
\hline \hline
z-bin & cat. 1)$_{(A,B,C)}$ & cat. 2)$_{(A,B,C)}$ & cat. 3)$_{(A,B,C)}$ & cat. 4)$_{(A,B,C)}$ & cat. 5)$_{(A,B,C)}$ & seren. & Sum \\
\hline
no redshift & 57 &20 &43 & 37 &0 &49 &206 \cr
stars & 4$_{(0,3,1)}$ &0$_{(0,0,0)}$ &0$_{(0,0,0)}$ & 7$_{(1,2,4)}$ &0$_{(0,0,0)}$ &3$_{(2,0,1)}$&14 \cr
(0..1) & 19$_{(12,1,6)}$ &42$_{(35,6,1)}$ &23$_{(18,3,2)}$ & 2$_{(2,0,1)}$ &1$_{(1,0,0)}$ &121$_{(83,19,18)}$&208 \cr
[1..2) & 193$_{(113,51,29)}$&83$_{(49,24,10)}$&115$_{(76,25,13)}$& 4$_{(2,2,0)}$ &2$_{(2,0,0)}$ &34$_{(10,15,9)}$ &431 \cr
[2..3) & 0$_{(0,0,0)}$ & 1$_{(1,0,0)}$ & 0$_{(0,0,0)}$ & 0$_{(0,0,0)}$ &0$_{(0,0,0)}$ & 0$_{(0,0,0)}$ &1 \cr
[3..4) & 0$_{(0,0,0)}$ & 0$_{(0,0,0)}$ & 0$_{(0,0,0)}$ &26$_{(14,7,4)}$&0$_{(0,0,0)}$ & 0$_{(0,0,0)}$ &26 \cr
[4..5) & 0$_{(0,0,0)}$ & 0$_{(0,0,0)}$ & 0$_{(0,0,0)}$ &23$_{(6,8,9)}$ &0$_{(0,0,0)}$ & 2$_{(0,1,1)}$ &25 \cr
[5..6) & 0$_{(0,0,0)}$ & 0$_{(0,0,0)}$ & 0$_{(0,0,0)}$ &14$_{(7,3,5)}$ &0$_{(0,0,0)}$ & 3$_{(0,1,2)}$ &17 \cr
[6..7) & 0$_{(0,0,0)}$ & 0$_{(0,0,0)}$ & 0$_{(0,0,0)}$ & 2$_{(0,1,1)}$ &0$_{(0,0,0)}$ & 0$_{(0,0,0)}$ &2 \cr
\hline
Sum & 273 &146 & 181 &115 &3 &212 &$\bf{930}$\cr
\hline
\hline
\label{tab:matrix}
\end{tabular}
\end{table*}
\subsection{Redshift distribution and Large Scale Structure}
The top and bottom panels of Figure~\ref{fig:zdistr} show the redshift
distribution of the galaxies at redshift less than 2 and greater than 2, respectively
(solid line QF ``A'' and ''B'', dotted line QF ``C''). In the following sections we
discuss the redshift distribution separating the low ($z<2$) and
high ($z>2$) redshift intervals.
\subsubsection{The sample at $z < 2$}
The redshift distribution is consistent with the criteria
for the target selection (color and photometric redshift selected),
with the majority of the sources having redshifts in the interval 1$<$z$<$2
(see also Table~\ref{tab:matrix}, last column).
In the RUN1+RUN2, out of 181 galaxies selected via photometric redshift, 138
have a spectroscopic redshift identification and 136 with zspec$>$0.8 (115 at zspec$>$1).
Table~\ref{tab:z_properties} shows the fraction of determined redshifts as
a function of the spectral features identified, i.e. emission lines, absorption lines,
emission \& absorption lines. The median of the redshift distribution of each class is close to 1,
with a more populated tail in the redshift interval 1$<z<$2 (see top panel of Figure~\ref{fig:zdistr}).
Obviously, in the presence of emission lines, it is easier to determine a redshift.
As reported in Table~\ref{tab:z_properties}, 537 galaxies (including ``em.''
and ``comp'') show the [O\,{\sc ii}]3727 emission line and assuming as an extreme case that
all the inconclusive redshift determinations (category ``X'') belong to the class ``abs.'',
the number of ``em.'' sources is still dominant, comprising 63$\%$ of the entire target list.
This is a likely reason why the majority of galaxies identified in the present
work belong to the ``em.'' class.
Alternatively, [O\,{\sc ii}]3727 is a classic star forming indicator and the redshift
interval $1<z<2$ corresponds to the peak of the mean star formation intensity of the universe.
There are 102 galaxies identified with absorption lines only (``abs.'' class, mainly Ca H and K,
MgII 2798-2802) in the range of redshift between 0.3-2.0.
28 sources out of 102 with only absorption features detected have been
serendipitously-observed, the redshift distribution of this sample peaks at z=0.68$\pm$0.2.
Six galaxies have been identified at redshift $\sim$2. These sources show
the Mg\,{\sc ii} 2798,2802\AA~in absorption (in three cases the [Fe\,{\sc ii}] 2344,2383\AA~absorption
lines are also present), five of them
(GDS~J033241.84-274657.1 QF=''B'', GDS~J033240.06-274755.4 QF=''A'', GDS~J033228.17-274648.4 QF=''C'',
GDS~J033240.27-274949.7 QF=''C'' and GDS~J033233.84-274520.5 QF=''C'')
have been discovered in the RUN2 and have blue colors ($i_{775}-z_{850}<0.6$).
Two examples of 2D spectra are shown in Figure~\ref{fig:GAL_Z2} and the composite
one-dimensional spectrum is shown in the right panel of Figure~\ref{fig:stack1p61}.
For these sources the [O\,{\sc ii}]3727 emission, if present, is out of the spectral range,
at 11180\AA. The source GDS~J033233.85-274600.2 is an elliptical galaxy at $z$=1.91 already
discussed by \cite{cimatti04}, and has been observed in the RUN1.
\begin{table*}
\centering \caption{Fractions of sources in the redshift interval 0$<$z$<$2 with different spectral features
(RUN1 + RUN2 without stars). The fractions of the different categories observed in Sect.~2 are also shown.}
\begin{tabular}{lccc|cccccc|c}
\hline \hline
Spectral class &$(z_{median})_{-1\sigma}^{+1\sigma}$&$z_{min}$&$z_{max}$& cat.1) & cat.2) & cat.3) & cat.4) &cat.5) & cat.-1) (seren.)&Sum\\
\hline
\cr
emission & (1.13)$_{-0.74}^{+1.33}$ & 0.067 & 1.621 & 117 & 92 & 123 & 3 & 2 & 104 & 441\cr
\cr
absorption & (1.00)$_{-0.67}^{+1.22}$ & 0.337 & 1.998 & 51 & 15 & 5 & 2 & 1 & 28 & 102\cr
\cr
em. \& abs. & (1.02)$_{-0.67}^{+1.29}$ & 0.382 & 1.380 & 44 & 18 & 10 & 2 & 0 & 22 & 96\cr
\\
\hline
Sum & & & & 212 & 125 & 138 & 7 & 3 & 154 & $\bf{639}$ \cr
\hline
\label{tab:z_properties}
\end{tabular}
\end{table*}
441 sources belong to the ``em.'' class (they are dominated by emission lines, mainly [O\,{\sc ii}]3727),
many of them entering the so-called ``spectroscopic desert'' up to z=1.621.
It is interesting to note that 133 galaxies out of 138 with redshift and photo-z selected show the
[O\,{\sc ii}]3727 emission line.
96 sources have been classified as intermediate between ``em.'' and ``abs.'' classes,
where both emission and absorption lines with an evident 4000\AA~break are present.
\subsubsection{Large Scale Structure}
The presence in the CDF-S of large scale structure (LSS) at $z<2$ is indicated by the
peaks in the redshift distribution (see Figure~\ref{fig:LSS}).
To assess the significance of these structures we follow a procedure similar to that
adopted by \cite{gilli03}, who observed features in their X-ray source redshift
distribution.
We have distributed the sources (the ``signal distribution'') in the velocity domain
($V~=~c~ln(1+z)$, so that $dV=\frac{c~dz}{1+z}$) and smoothed with a Gaussian filter with $\sigma_{S}=300~km/s$
(the typical error in the redshift determination).
We have then smoothed the observed distribution with a Gaussian filter with $\sigma_{S}=15000~km/s$
and considered this as the background distribution.
We have searched for possible redshift peaks in the signal distribution, computing a
signal-to-noise ratio defined as $SNR$ = ($\frac{S-B}{B^{0.5}}$), where $S$ is the number
of sources in a velocity interval of fixed width $\Delta V=2000~km/s$ and $B$ is the number
of background sources in the same interval.
Adopting the threshold $SNR>$5 we have found 6 peaks in the signal distribution
(indicated with an arrows in the Figure~\ref{fig:LSS}).
In order to estimate the expected fraction of possibly ``spurious'' peaks arising
from the background fluctuations, we have simulated 10$^{5}$ samples of the same size
of the observed distribution and randomly extracted from the smoothed background distribution
and applied our peak detection method to each simulated sample.
The result is that, with the adopted threshold, the average number of spurious peaks
due to background fluctuations is 0.06. Of the simulated samples, 5.7$\%$ show one spurious
peak, 0.1$\%$ show two spurious peaks, and only one simulation (out of $10^{5}$) has three
spurious peaks. None of the simulated samples have four or more spurious peaks.
\begin{table}
\centering \caption{Peaks detected in the FORS2 source redshift distribution, sorted by increasing
redshift. The signal and background distributions are smoothed with $\sigma_{S}=300km/s$
and $\sigma_{B}=15000km/s$, respectively. Together with the central redshift of each peak,
the number of sources N in each peak and the probability (determined on 10$^{5}$ simulations) to
detect spurious peaks arising from the background distribution with a $SNR$ equal or greater than
the $SNR$ value measured in the signal distribution.}
\begin{tabular}{ccccc}
\hline \hline
z & N & SNR & Prob. \\
\hline
0.666 & 22 & 8.6 & 4.5$\times$10$^{-4}$ \cr
0.734 & 40 & 16.6 & $<$1$\times$10$^{-5}$ \cr
1.096 & 42 & 8.0 & 1.9$\times$10$^{-3}$ \cr
1.221 & 47 & 9.7 & 2.2$\times$10$^{-4}$ \cr
1.300 & 35 & 7.4 & 4.2$\times$10$^{-3}$ \cr
1.614 & 20 & 11.1 & 7.0$\times$10$^{-5}$ \cr
\hline
\label{tab:LSS}
\end{tabular}
\end{table}
The six source peaks detected by our procedure are listed in Table~\ref{tab:LSS}, where for each
peak we give the average redshift, the number of objects (N) in the peak and the
probability (derived from the 10$^{5}$ simulations) of observing a spurious peak with the SNR equal or
greater than the measured SNR of the peak detected in the signal distribution.
The peaks at $z \sim 0.734$ and $z \sim 0.666$ are
already known (\cite{cimatti02}, \cite{gilli03}, \cite{fevre04}).
The other four indications of large scale structures in the CDF-S have been identified at
redshift 1.096, 1.221, 1.300 (also described by \cite{adami05}) and 1.614.
We note that other two peaks have been detected with a $SNR\sim$4.5 at redshift 1.040 and 1.382.
In the current spectroscopic catalog (RUN1 + RUN2) 20 galaxies at $z \sim 1.61$ have
been discovered (Figure~\ref{fig:groupz1.61} shows an example of the
z$\sim$1.61 galaxies discovered in the RUN2).
The number of sources increase if we consider other surveys:
\begin{enumerate}
\item{the observations of \cite{gilli03} who found a peak in the redshift distribution of X-ray sources
at z=1.618 (5 galaxies);}
\item{the three galaxies at z$\sim$1.61 (\cite{cimatti02}, \cite{cimatti04}) which are passively
evolving early type galaxies}
\item{at least 5 more galaxies in the third FORS2 run (from our preliminary reduction);}
\end{enumerate}
Fig.~\ref{fig:space_distr} shows the spatial distribution of the galaxies at $z ~\approx~ 1.61$ using
both the present work and data from the literature.
The current sample contains 28 galaxies apparently distributed in a non-uniform way, the majority of them
have been detected in the upper part of the field and 3 pairs have an angular separation below
4 arcseconds ($\sim$ 30 kpc at $z \sim 1.61$).
At redshift 1.61, the ACS $B_{435}$ band is sampling the 1667\AA~rest-frame UV radiation. As reviewed by
\cite{kenni98}, one can estimate the SFR from the rest-frame UV luminosity density $L_{\nu}$ in the
range 1500-2500~\AA~using the following relation: SFR(M$_{\odot}$yr$^{-1}$) = 1.4 $\times$ 10$^{-28}$
$L_{\nu}$ (ergs s$^{-1}$ Hz$^{-1}$) for a Salpeter IMF, covering the range 0.1 to 100M$_{\odot}$.
This relation applies only to galaxies with continuous star formation over time scales of 10$^{8}$ years
or longer.
We have estimated the rest-frame luminosity density $L_{\nu}$ (in ergs s$^{-1}$ Hz$^{-1}$) of the 20
galaxies at z=1.61 identified in the current FORS2 spectroscopic campaign, using the
apparent $B_{435}$ AB magnitude (the SExtractor ``mag$\_$auto'', \cite{ber96}) and the luminosity distance.
The final luminosity is $L_{\nu,o}$ = $L_{\nu}$ $\times$ 10$^{0.4~Av}$, where $Av$ represents the
amount of dust extinction. Adopting no extinction ($Av$=0), we obtain a lower limit for the mean
star formation rate of $<$SFR$>$4$\pm$2 M$_{\odot}$yr$^{-1}$. Assuming $Av$=1 or $Av$=2 the $<$SFR$>$
increase from 10$\pm$5 M$_{\odot}$yr$^{-1}$ to 24$\pm$14 M$_{\odot}$yr$^{-1}$, respectively.
The rest frame composite spectrum of twenty galaxies at z=1.61 is shown in the left panel of
Figure~\ref{fig:stack1p61}.
The [O\,{\sc ii}]3727 line and the Mg\,{\sc ii} 2798,2802\AA~and [Fe\,{\sc ii}] 2344,2383\AA~are
clearly evident.
\subsection{The Lyman break galaxies}
116 sources in the FORS2 RUN1 and RUN2 belong to the class 4), i.e., objects selected to be at high
redshift by Lyman break color criteria. It is important to divide the first and second run in
order to characterize the success rate.
As already discussed in the previous paper (\cite{vanz05}), in the first FORS2 run
14 candidate dropouts were observed, and only one was confirmed at z=5.83. Another five were found to be
stars and the remaining sources had inconclusive spectra. The photometric
selection of the dropouts galaxies in the first FORS2 run was based on an incomplete photometric
dataset (first three epochs photometry).
In the following, we consider only the results from RUN2, for which dropout
candidates were selected from the full (five epochs) ACS photometry.
94 Lyman break galaxy candidates selected by the $B_{435}$,$V_{606}$ and $i_{775}$-dropout criteria were
observed in RUN2. The redshift distribution measured for 65 of these galaxies is shown in the
lower panel of Figure~\ref{fig:zdistr}.
The 75$\%$, 70$\%$ and 70$\%$ of the observed $B_{435}$, $V_{606}$ and $i_{775}$-dropout color
selected candidates have a redshift estimation. The sources with inconclusive redshift determination
are in general too faint or without evident spectral features.
100$\%$ of the $B_{435}$-dropouts with a measured redshift have been confirmed to be at redshift between
3.4 and 4.6, 90$\%$ of the $V_{606}$-dropouts with a measured redshift are in the range 4.4 and 5.6,
and 93$\%$ of the $i_{775}$-dropouts with a measured redshift are at redshift greater than 5.2
(one source is a probable star).
\begin{table*}
\centering \caption{Fraction of confirmed dropout candidates in the second FORS2 run (RUN2), ``Nobs.'' indicates
the number of sources observed. Four serendipitously-observed high redshift sources are also reported.}
\begin{tabular}{lccc|ccc}
\hline \hline
classes (Nobs.) & confirmed high-z (*)& confirmed low-z & no redshift & (*) ``em.''$_{(A,B,C)}$ &(*) ``abs.''$_{(A,B,C)}$ &(*) ``em.''+''abs.''$_{(A,B,C)}$ \\
\hline
$B_{435}$-drop (44) & 33 (3.418$<$z$<$4.597)& 0 & 11 &12$_{(8,3,1)}$ &20$_{(9,5,6)}$ &1$_{(1,0,0)}$\cr
$V_{606}$-drop (30) & 19 (4.400$<$z$<$5.554)& 2 (z$<$1.4) & 9 &13$_{(5,5,3)}$ & 6$_{(0,2,4)}$ &0$_{(0,0,0)}$\cr
$i_{775}$-drop (20) & 13 (5.250$<$z$<$6.200)& 1 (star) & 6 & 8$_{(3,4,1)}$ & 5$_{(0,0,5)}$ &0$_{(0,0,0)}$\cr
Serend. & 3 (4.838$<$z$<$5.541) & - &-& 3$_{(0,0,3)}$ & 0$_{(0,0,0)}$ &0$_{(0,0,0)}$\cr
\hline
\label{tab:high-z}
\end{tabular}
\end{table*}
Table~\ref{tab:high-z} (and Table~\ref{tab:matrix}) summarize the success rate as a function of
redshift, quality flag, class and selection criteria.
Columns 5, 6 and 7 of Table~\ref{tab:high-z} show the fraction of the confirmed high redshift galaxies
and the ``class'' flag that is related to the features detected in the redshift determination.
Beyond redshift 5, if no spectral lines are present,
the main features indicating the high redshift nature of the source are: the break in the continuum
due to galactic and intergalactic absorption blueward 1215.8\AA, and the flatness of the continuum
redward the 1215.8\AA.
Figure~\ref{fig:panoramic_high_z} shows the two-Dimensional collection of the 18 galaxies at
redshift greater than 5 discovered in the RUN2 and Figure~\ref{fig:z6p097} shows the one-dimensional
spectrum of the galaxy GDS~J033223.84-275511.6 at z=6.097.
In the top of the figure the spectrum of the sky (not flux calibrated)
is shown together with the response curves of the ACS filters $i_{755}$ and the $z_{850}$.
In some cases the Ly$\alpha$ is in emission (marked with a circle)
and the break of the continuum is evident.
Five sources show only the continuum break
(a solid segment marks the possible position of the break).
The mean value of the observed $i_{755} - z_{850}$ for this sample increases with
increasing redshift.
The presence of the Lyman emission line, however,
can affect significantly the resulting color of the galaxy, introducing a scatter in
the blue or in the red directions. For example in the case of the source
GDS~J033218.92-275302.7, the strong Ly$\alpha$ line at z=5.554 produces an
$i_{755} - z_{850}$ = 0.625.
Similarly, in the case of the source GDS~J033223.84-275511.6, the intense Ly$\alpha$ line
falls in the $z_{850}$ band, producing an $i_{755} - z_{850}$ $>$ 4.
The source GDS~J033217.96-274817.0 is an $i_{775}$-dropout candidate.
The FORS2 spectrum is the superposition of two sources.
In Figure~\ref{fig:idrop_blue} the one and two dimensional spectra and the
color ACS image of the sources are shown. One source (GDS J033217.95-274817.5)
is clearly blue ($i_{775}-z_{850}$=-0.25) with respect the $i_{775}$-dropout candidate ($i-z$=1.18).
The one dimensional spectrum shows a break at $\sim$ 7800\AA~and the flatness shape redward the break.
Collapsing $\sim$ 100 columns below and beyond the 7800\AA~break, the two resulting profiles are shifted
of $\sim$ 0.4 arcsecond, consistently with the separation of the two sources measured in the ACS image.
Interpreting this break due to the high redshift nature of the $i_{775}$-dropout source, the redshift
is $\sim$ 5.4 with QF=''C''. We note that the uncertainty of the position of the break
is increased by the presence of the sky absorption A-band at $\sim$7600\AA.
\subsection{Galaxies showing a tilted [O\,{\sc ii}]3727 line}
The current FORS2 spectroscopic catalog contains a sample of sources showing a spatially
resolved [O\,{\sc ii}]3727 line with a characteristic ``tilt'' indicative of a high
rotation velocity. Table~\ref{tab:OII_TILT} lists the 34 sources sorted by increasing redshift,
the majority of them belong to the interval 1$<$z$<$1.5.
Figure~\ref{fig:OIItilted1}
show an example of the two dimensional spectra of the galaxies and the sky lines. The [O\,{\sc ii}]3727 line
is marked with a circle.
As discussed in the previous paper (\cite{vanz05}) the resolution of the FORS2 spectra
favor the detection of high velocity rotational systems. Moreover
a not optimized orientation of the slit suggest that in general the ``true'' maximum velocities
may be significantly higher than the measure value.
As an example, we have analyzed the velocity field of object
J033227.73-275451.8 and estimate the stellar mass from the multi-wavelength dataset.
We first traced the centroid of the
[O\,{\sc ii}]$\lambda\lambda$3726,3729 emission line doublet along the spatial
position. Since the resolution of our spectrum is too low to resolve the
doublet, we fixed the ratio between the two components to 1 and we
checked that the results were fairly insensitive to this assumption.
We then compared this measured rotation curve with a set of synthetic
rotation curves for which the velocity rises linearly up to one disk
scale length and is flat at larger radii. This step takes into account
the inclination of the disk with respect to the line of sight ($i=53\pm3
\deg$), the disk scale length ($r_d=0.405 \pm 0.045 \arcsec$)
(derived from the morphological analysis, Rettura et al., in preparation),
the slit misalignment with respect to the galaxy major axis ($22\pm2 \deg$),
the width of the slit (1") and the seeing ($0.7\arcsec$) (this
method is similar to that of \cite{boehm04}). The best fit rotation
curve gives a rotation velocity of $355\pm50$ {\it km$s^{-1}$}. Using
prescription from \cite{bosch02}, this implies a total halo mass of
$1\pm0.4 \times10^{12}M\sun$.
Dedicated spectroscopic observations specifically designed
for the dynamical mass estimation (higher spectral resolution, optimized slit orientation,
etc.), should be performed/preferred in order to decrease the uncertainties.
We have used the full optical (HST/ACS B,V,i,z), near infrared (VLT/ISAAC J,Ks) to mid-infrared
(Spitzer/IRAC $3.6 \mu, 4.5 \mu, 5.8 \mu, 8.0 \mu$) data to study the Spectral Energy
Distribution (SED) of the same galaxy.
We have adopted $1.5~arcsec$ radius aperture-corrected to $3.5~arcsec$ radius photometry
to account for different instrumental PSFs. We have compared the observed SED with a set of
template computed with P\'EGASE.2 models (\cite{RV97}) via $\chi^{2}$ minimization technique
(the Salpeter IMF has been assumed). A more detailed description of the multi-wavelength cataloging and the
fitting SED technique used here will be presented in Rettura et al. (in preparation) on a
larger sample. We have calculated the errors for the mass estimate by sampling the full
probability distribution in the parameters-space. Results of the SED fit are shown in
Figure~\ref{fig:Vrot} right panel. We find a best-fit stellar mass of
$2.6\times10^{10}~M_{\odot}$ with a 1$\sigma$ confidence interval between
1.7-3.7$\times10^{10}~M_{\odot}$.
The comparison between this estimation and the dynamical halo mass produces a
stellar mass over halo mass ratio of $f*=0.026$. This result is consistent with the estimations
performed by \cite{conselice05} on a large sample of disk galaxies at $z\leq$1.1,
where they find a wide range of $f*$ values (0.004 $\leq f* \leq $ 2).
\begin{table}
\centering \caption{Sample of 34 galaxies (RUN1 + RUN2) with tilted [O\,{\sc ii}]3727.}
\begin{tabular}{lcccc}
\hline \hline
GDS ID & zspec& class & quality \\
\hline
GDS J033254.87-275456.0 & 0.125 &em. & C \cr
GDS J033237.54-274838.9 & 0.665 &em. & A \cr
GDS J033215.88-274723.1 & 0.896 &em. & A \cr
GDS J033227.66-275437.4 & 0.963 &em. & A \cr
GDS J033227.73-275451.8 & 0.966 &comp. & A \cr
GDS J033249.73-275517.4 & 0.981 &em. & A \cr
GDS J033234.56-275543.6$\dag$& 0.983 &em. & A \cr
GDS J033222.44-275606.1$\dag$& 1.090 &em. & C \cr
GDS J033226.03-274856.0 & 1.016 &em. & A \cr
GDS J033230.50-275312.3 & 1.017 &comp. & A \cr
GDS J033225.28-275524.2 & 1.017 &comp. & A \cr
GDS J033235.72-275615.4 & 1.033 &em. & A \cr
GDS J033233.71-274210.2 & 1.043 &em. & B \cr
GDS J033234.42-275405.7 & 1.088 &comp. & A \cr
GDS J033225.86-275019.7 & 1.095 &em. & A \cr
GDS J033246.71-274556.0 & 1.095 &em. & B \cr
GDS J033247.42-274711.1 & 1.098 &em. & A \cr
GDS J033215.23-274437.8 & 1.109 &em. & B \cr
GDS J033223.18-274921.5 & 1.110 &em. & B \cr
GDS J033216.28-274447.6 & 1.183 &em. & C \cr
GDS J033216.26-274703.3 & 1.219 &em. & A \cr
GDS J033238.01-275408.2$\dag$& 1.243 &em. & B \cr
GDS J033224.94-275020.2 & 1.294 &em. & B \cr
GDS J033205.67-274253.5 & 1.296 &em. & A \cr
GDS J033232.42-274150.1$\dag$& 1.296 &em. & B \cr
GDS J033232.47-274151.5$\dag$& 1.296 &em. & B \cr
GDS J033213.21-274158.0 & 1.297 &em. & B \cr
GDS J033240.94-274427.5 & 1.298 &comp. & A \cr
GDS J033244.35-275506.4$\dag$& 1.305 &em. & A \cr
GDS J033230.71-274617.2 & 1.307 &em. & A \cr
GDS J033234.82-274721.9$\dag$& 1.316&em. & B \cr
GDS J033239.66-275406.3 & 1.343 &em. & A \cr
GDS J033240.08-275532.6 & 1.461 &em. & A \cr
GDS J033229.06-275542.8 & 1.469 &em. & A \cr
\hline
\multicolumn{4}{l}
{$\dag$ possible tilted line.}\\
\label{tab:OII_TILT}
\end{tabular}
\end{table}
\section{Conclusions}
As a part of the Great Observatories Origins Deep Survey, a
large sample of galaxies in the Chandra Deep Field South has been
spectroscopically targeted.
After the RUN1 (\cite{vanz05}) and RUN2 (present work) a total of 930 objects
with $z_{850} \mincir 26.8$ have been observed with the FORS2 spectrograph at the
ESO VLT providing 724 redshift determinations.
From a variety of diagnostics the measurement of the redshifts appears
to be precise (with a typical $\sigma_z \simeq 0.001$) and reliable.
The reduced spectra and the derived redshifts are released to the community
($\it{http://www.eso.org/science/goods/}$).
They constitute an essential contribution to reach the scientific goals
of GOODS, providing the time coordinate needed
to delineate the evolution of galaxy masses, morphologies, and star
formation, calibrating the photometric redshifts that can be derived from the
imaging data at 0.36-8$\mu$m and enabling detailed studies
of the physical diagnostics for galaxies in the GOODS field.
\begin{acknowledgements}
We are grateful to the ESO staff in Paranal and Garching who greatly helped
in the development of this programme.
The work of DS was carried out at the Jet Propulsion Laboratory,
California Institute of Technology, under a contract with NASA.
We thank the ASI grant I/R/088/02 (SC, MN, EV).
\end{acknowledgements}
|
Title:
The Paths of Quintessence |
Abstract: The structure of the dark energy equation of state phase plane holds
important information on the nature of the physics. We explain the bounds of
the freezing and thawing models of scalar field dark energy in terms of the
tension between the steepness of the potential vs. the Hubble drag.
Additionally, we extend the phase plane structure to modified gravity theories,
examine trajectories of models with certain properties, and categorize regions
in terms of scalar field hierarchical parameters, showing that dark energy is
generically not a slow roll phenomenon.
| https://export.arxiv.org/pdf/astro-ph/0601052 |
\title{The Paths of Quintessence}
\author{Eric V.\ Linder}
\affiliation{Berkeley Lab, University of California, Berkeley, CA 94720}
\date{\today}
\section{Introduction} \label{sec.intro}
The discovery of the acceleration of the cosmic expansion has thrown
physics and astronomy research into a ferment of activity, from a
search for fundamental theories to investigation of predictions
relating models and the cosmological dynamics, to development of
astrophysical surveys yielding improved measurements.
The acceleration, or more generally the expansion history
of the scale factor evolution with time, $a(t)$, can be equivalently
treated by a dark energy pressure to density, or equation of state,
ratio $w(a)$ \cite{linjen}. One model, Einstein's cosmological constant,
predicts $w=-1$ at all times, but generically the dark energy phenomenon
has dynamics, a time varying $w(a)$.
It is important to note that the
current epoch of accelerated expansion is very different from the early
epoch of inflation. We know a priori that dark energy does not completely
dominate the universe now and we do not know a priori that dark energy
obeys a slow roll approximation (in fact we will see it is unlikely to).
In these senses, dark energy is a more challenging phenomenon than
inflation. We are faced with three ``Goldilocks'' problems: 1) dark
energy is dynamically important, but not fully dominant, with $\sim75\%$
of the total energy density today, 2) the universe is accelerating, as
from a component with $w\lesssim-0.8$, but the field responsible may not
be slowly rolling (unless fine tuned) as it would if nearly completely
potential dominated, and 3) if it's not the cosmological
constant, what happened to the cosmological constant?
To discover whether the physics
is the cosmological constant, and to distinguish between alternate
theories, requires measurement of the possible dynamics.
Caldwell \& Linder \cite{caldlin} (Paper 1) investigated the dynamics in
the phase plane of $w$-$w'$, where $w'=dw/d\ln a$,
for canonical scalar field dark energy, or quintessence, finding that this
reveals important clues to the nature of the new physics. In such a plane,
the time or scale factor variable is a parameter along the paths of
dynamics. They found distinct structure, categorizing fields into those
that at early times are locked by Hubble friction into a cosmological
constant like state, and then move away from this (thaw) as the dark
energy dominates, and those that initially roll due to the steepness
of the potential but later approach the cosmological constant (freeze).
In this article some of these results are put on a firmer
footing, examined in greater detail, and extended to models beyond
canonical scalar fields, including modified gravity theories.
\section{General Dynamics} \label{sec:scadyn}
We begin with a brief, explicit derivation of the key dynamics
equation, Eq.~(\ref{eq:wp}).
A scalar field $\phi$ possesses both potential energy
$V(\phi)$ and kinetic energy $(1/2)\dot\phi^2$. The Lagrangian
density is of the form
\beq
{\mathcal L}=(1/2)g_{\mu\nu}\partial^\mu\phi\partial^\nu\phi-V(\phi),
\eeq
for a canonical, minimally coupled scalar field in a metric $g_{\mu\nu}$.
The equation of motion for the field, the Klein-Gordon equation,
\beq
\ddot\phi+3H\dot\phi+V_{,\phi}=0, \label{eq:kg}
\eeq
where $H=\dot a/a$ is the Hubble parameter,
follows from functional variation of the Lagrangian. (Spatial
inhomogeneities should be negligible on subhorizon scales, but also
see \S\ref{sec:inhom}.) The energy-momentum
tensor is generated through Noether's theorem and one can identify the
energy density and pressure:
\beq
\rho=(1/2)\dot\phi^2+V(\phi)\quad;\quad p=(1/2)\dot\phi^2-V(\phi).
\eeq
It will be convenient to invert these and write the potential and
kinetic energies in terms of $\rho$ and $w$:
\beq
V=(1/2)(1-w)\rho\quad;\quad (1/2)\dot\phi^2=(1/2)(1+w)\rho. \label{eq:vk}
\eeq
Note that $w$ and $\rho$ are both functions of time, or scale factor.
To obtain an equation for the variation $w'$,
take the derivative of $V$ with respect to $\phi$,
\beq
\vp=\dot V/\dot\phi=(1/2)[(1-w)\dot\rho-\rho\dot w]/[(1+w)\rho]^{1/2}.
\label{eq:vdot}
\eeq
Employing the continuity equation $\dot\rho=-3H\rho(1+w)$ and
$H=d\ln a/dt$, one obtains
\beq
w'=-3(1-w^2)-(1-w)(1+w)^{1/2}\sqrt{(3/8\pi) \Omega_\phi(a)}\frac{M_P \vp}{V},
\label{eq:wp}
\eeq
where $\Omega_\phi(a)=8\pi\rho/(3H^2M_P^2)$ is the dimensionless dark energy
density, and $M_P$ is the Planck mass.
\subsection{Distinguishing $\Lambda$ \label{sec:barrier}}
We see that $w'$ is related to the nearness of the equation of state
ratio to the cosmological constant value, i.e.\ $1+w$, and the inverse
of the characteristic field scale of the potential, $\vp/V$
(sometimes phrased as a slow roll parameter, $M_P|\vp/V|\ll1$).
If $w$ is readily distinguished from $-1$, then we know the dark energy
is not a cosmological constant, regardless of the value of the time
variation $w'$. More difficult is the case where $\eps=1+w$ is a small
quantity. Then it will be quite important to determine whether $w'$ is
zero or not. Eq.\ (\ref{eq:wp}) guides us in following the dynamics
in the $w$-$w'$ phase space.
The first point to notice is that the reciprocal of the characteristic
field scale is not generically a small parameter useful for a ``slow roll''
approximation.
Figure \ref{fig:wwpscale} shows curves of constant field scale
\beq
\Phi=V/(-\vp),
\eeq
in the $w$-$w'$ plane. Only a tiny sliver of the phase space, plus a
small hump, satisfies the slow roll approximation;
unless one is exceedingly close to the cosmological constant behavior
there is substantial dynamics in the field.
When $1+w\ll1$, the second term in Eq.~(\ref{eq:wp})
dominates and $w'>0$ (creating the ``humps''), while for less
negative $w$ the first term can
dominate and drive $w'<0$. This driving occurs closer to $w=-1$ for larger
$\Phi$. Within the large-$\Phi$ hump, the dark energy looks similar to a
cosmological constant. Suppose one conjectures some physics limiting
the size of the field scale, equivalently leading to avoidance of
flatness in the potential. An upper bound on $\Phi$ in the scalar
field behavior
would impose a barrier around the cosmological constant $\Lambda$, saying
that the scalar field dynamics must be distinguishable from $\Lambda$,
if it is not $\Lambda$.
The second panel of Fig.~\ref{fig:wwpscale} zooms in to illustrate
this barrier for
$\Phi<M_P$, which rules out any freezing field, and any thawing field
with $1+w<0.004$. So any scalar field theory with field scales barred
from being transPlanckian are distinguishable from $\Lambda$ at this
precision. If the restriction uses, say, $\Phi<M_P/\sqrt{8\pi}$ (i.e.\
the Planck mass is defined in terms of Newton's constant as
$G_N=(8\pi M_P^2)^{-1}$ rather than $G_N=M_P^{-2}$ as
above) then the limit becomes $1+w>0.1$.
The physical origin for the conjectured limit on the characteristic
field scale is not clear, but the implications are important enough
to consider the possibility.
Since the field scale is related to the inverse of the flatness of
the potential, then physics that perturbs a flat direction in the potential
would give this effect. In some supersymmetric models, loop
corrections generate a logarithmic tilt $V\sim\phi^n\ln(\phi/\mu)$
\cite{witten,damourmukhanov}. This would give $\Phi\approx\phi/n$ (or
$\phi\,\ln(\phi/\mu)$ for $n=0$),
and restricting $\phi<M_P$ (for the effective field theory to be valid),
provides the limit $\Phi\lesssim{\cal O}(M_P)$. However, the generic
breakdown of slow roll is independent of any $\Phi$ upper limit conjecture
and we do not consider the latter further.
Hierarchical parameters
to replace slow roll are discussed in \S\ref{sec:slow}.
\subsection{Driving and dragging \label{sec:kglines}}
Returning to the Klein-Gordon equation, we can understand behavior
in the $w-w'$ phase space by first a general and then a specific
analysis of the terms. Writing Eq.~(\ref{eq:kg}) as
$\ddot\phi+3H\dot\phi=-\vp$, we can require $-\vp\ge0$, reflecting
that the field rolls down the potential to large $\phi$. Using
Eq.~(\ref{eq:vk}) for $\dot\phi$, we write this condition in
terms of $w$, $w'$ as $w'\ge-3(1-w^2)$. The boundary defines the
null line $\vp=0$ discussed further below. Similarly, writing the
Klein-Gordon equation as $\ddot\phi+\vp=-3H\dot\phi$ and again using
that the field rolls to larger values with time implies $-3H\dot\phi\le0$.
This reduces to the condition $w\ge-1$.
If we flip the direction of
the field motion and potential slope, i.e.\ the field rolls down the
potential to smaller $\phi$, then these conditions remain.
(I.e.\ $\dot V=\dot\phi\vp$ is still negative, and so the transition
from Eq.~(\ref{eq:vdot}) to Eq.~(\ref{eq:wp}) flips the sign of the
second term on the right hand side of Eq.~(\ref{eq:wp}), canceling
the reversed sign of $\vp$.)
However, if we make the field move up the potential then we have
the relations $w'\le-3(1-w^2)$ and $w<-1$, in the phantom region,
i.e.\ the boundary lines just continue smoothly through $w=-1$.
Finally, we can move $\ddot\phi$ to the right hand side to obtain
$3H\dot\phi+\vp=-\ddot\phi$. This divides the $w-w'$ plane into
regions where the field is accelerating or decelerating, with the
boundary being the coasting behavior $\ddot\phi=0$. This line
corresponds to the condition $w'=3(1+w)^2$. Larger values of $w'$
arise from a field accelerating down the potential, while smaller values
come from a field decelerating (this motion of the field should not
be confused with the accelerating expansion of the universe, which
can hold for either region). These three boundaries -- the null
line $w'=-3(1-w^2)$, coasting line $w'=3(1+w)^2$, and phantom line
$w=-1$ -- define the broad characteristics of the phase plane.
To investigate the dynamics further, we must examine the dominance
of the different terms in the equation of motion (\ref{eq:kg}).
The first term is the acceleration of the field, the second
term a friction term, or Hubble drag, due to the expansion of the
universe, and the third is a driving term due to the steepness
of the potential. We define
\beqa
X&=&\frac{\ddot\phi}{H\dot\phi} \\
Y&=&\frac{\ddot\phi}{\vp}\ .
\eeqa
Figure~\ref{fig:kgterms} shows curves of constant $X$, $Y$ in the
$w-w'$ plane. Note that $X=Y=0$ corresponds to an epoch of coasting
in the scalar field dynamics, $\ddot\phi=0$, as discussed above.
This is nongeneric,
as the field would need to be finely tuned to neither accelerate
due to the slope of the potential nor decelerate due to the Hubble drag,
but be perfectly balanced.
Indeed, the dynamics of scalar fields in Paper 1 avoid this
region, causing the split into the distinct thawing and freezing regions,
respectively above and below this line.
In the accelerating field region, the friction term is the major
determinant of behavior initially, as the field evolves away from a
frozen (cosmological constant-like) state in the matter dominated epoch.
The upper boundary of the thawing region is given by $X=3/2$, where
this value follows directly from the exponent of the expansion history,
$t\sim a^{3/2}$ for matter domination. Thus fields that thaw during
matter domination begin to move along the $X=3/2$ line (see discussion,
and Figure 2, in Paper 1).
We can translate any acceleration to friction ratio $X$ to a $w-w'$
behavior through
\beqa
w'&=&2X(1+w)+3(1+w)^2=(1+w)(3+2X+3w) \nonumber \\
&\approx& 2X(1+w),
\eeqa
where the last approximation is for $1+w\ll1$. Note that the linear
boundaries used in Paper 1 were (good) approximations to the general
parabolic behavior. The value $X=0$ gives the coasting line and $X=-3$
gives the null line.
Thus, the upper thawing boundary $X=3/2$ corresponds to $w'\approx 3(1+w)$
for $1+w\ll1$. If the rolling field then enters a region where the
potential slope is shallower (as usually happens), then the field will
accelerate less and curve toward the $\ddot\phi=0$ line. Since today the
field cannot have rolled so far that $\Omega_{\rm de}>0.8$, the
dynamical track remains within the thawing region $1+w<w'<3(1+w)$,
i.e.\ $1/2<X<3/2$.
For potentials that steepen as the field rolls down, e.g.\ PNGB models
with the field starting near the top of the potential, the tracks instead
lie above the $X=3/2$ or $w'=3(1+w)$ line. The PNGB potential also steepens
more rapidly for small symmetry scales $f$, and indeed, as mentioned in
Linder \cite{osc}, PNGB models roughly follow $w'=F(1+w)$, where $F$
is proportional to the inverse of $f$. (Also see \cite{kalopersorbo}
for discussion of PNGB models, fine tuning, and slow roll.)
In the decelerating field region, the steepness of the potential impacts
the freezing. As the potential becomes shallower, the friction is more
effective. In the limit of a flat potential, one obtains the dynamics
track given by the null curve, $\vp=0$, in Fig.~\ref{fig:kgterms}. As
given in Paper 1, this corresponds to $w'=-3(1-w^2)$, and
the skating model of \cite{lincurv,liddleskate}.
In terms of the friction and driving terms, $X=-3$ and $Y=\infty$.
The lower boundary of the freezing region lies along the $Y=1$ ($X=-3/2$)
line, equivalent to $w'=3w(1+w)$. (See \S\ref{sec:track} for a rationale.)
We will later see that this line also has physical significance.
The general relation between the acceleration to steepness ratio $Y$
and the $w-w'$ track is
\beq
w'=3(1+w)\left[w+\frac{1-Y}{1+Y}\right]=3w(1+w)+3(1+w)\frac{1-Y}{1+Y}.
\eeq
Again we have a parabolic behavior. The coasting line has $Y=0$, and
the null line corresponds to $Y\to\infty$. One could use either variable
$X$ or $Y$ throughout the phase plane, since
\beq
X=-3\frac{Y}{1+Y}\quad ; \quad Y=-\frac {X}{X+3},
\eeq
but this somewhat obscures the physics of friction and steepness. (That
said, we note that the thawing/freezing boundaries are fairly symmetric
in $X$, with the outer boundaries at $X=\pm3/2$ and the inner ones at
$X\approx\pm1/2$.)\footnote{Recall from Paper 1 that the upper bound on
the freezing region is not sharply defined, and extends somewhat above
the $w'=w(1+w)$ line shown for convenience in this paper.}
Note that it is not legitimate to assume tracking behavior (where the
equation of state is constant and related to the dominant component's
equation of state) to impose limits on regions in the phase plane,
as for example \cite{chiba,scherrer} did to try to tighten the
constraints of Paper 1. For one thing, not all freezing models
need start as trackers. Secondly, just because the most negative value
of $w'$ lies above some boundary curve does not ensure that the entire
trajectory remains above the curve.
Most importantly, the tracking
approximation breaks down as the dark energy first becomes significant,
so it is inapplicable for much of the observable dynamical history.
\section{More Specific Dynamics \label{sec:more}}
In addition to analyzing general behavior through the Klein-Gordon equation
terms, we can investigate the properties of the phase plane or specific
track families in terms of other variables. These could include working
from the equation of state $w-w'$ relation directly or the cosmic expansion
history $a(t)$. While not quite as insightful as the physics
motivated driving and drag terms, they can highlight interesting properties.
\subsection{Mocker models \label{sec:mocker}}
Consider a model with dynamics given by $w'=Cw(1+w)$. Note that while
such an equation forms the boundaries of the freezing region, freezing
models do not follow such a trajectory but rather are almost orthogonal
to such tracks (at least initially). So we are talking about
fundamentally different models. The behavior of the dark energy
equation of state and density follow
\beqa
w(a)&=&-1+\left[1-\frac{w_0}{1+w_0}a^C\right]^{-1} \\
\rho_{\rm de}(a)&=&\rho_{\rm de}(1)\,\left[(1+w_0)a^{-C}-w_0\right]^{3/C},
\eeqa
where $w_0$ is the equation of state today.
In the past, $a\ll1$, the component will act like additional nonrelativistic
matter, with $w=0$, while in the future it will approach
a cosmological constant. Since such dark energy sometimes looks like
dark matter and sometimes like a cosmological constant, we call this a
``mocker'' model. These are basically what are known as ``quartessence''
models (see \cite{makleroliveirawaga} for an overview), of which the
Chaplygin gas is one
example. However, we develop them directly from the phase space dynamics
rather than an ansatz for the pressure, so the dynamical behavior is
more general. For example, a
constant pressure model could be a cosmological constant, or could be
a mocker with $C=3$. Note $C=3$ gives precisely the expression for the
lower boundary of the freezing
region (or is $Y=1$ or $X=-3/2$ in the notation of \S\ref{sec:kglines}).
We name the $w'=3w(1+w)$ curve the constant pressure line (also see
\cite{scherrer} in the context of barotropic fluids).
Such combined behavior models are heir to all
the usual problems of trying to unify dark matter and dark energy,
e.g.\ growth instabilities of density perturbations
\cite{sandvik,amendola0509099}. Analysis of perturbations requires
knowledge of the full theory, however.
Merely from the phase plane dynamics, though, we can see trouble arising
for such unified models.
As $C$ gets smaller, the model moves along its trajectory more quickly,
acting less like dark matter except at very early times,
$1+z\gg[-w_0/(1+w_0)]^{1/C}$. Conversely,
as $C$ gets larger, acceleration of the cosmic expansion occurs later
and the model becomes a poorer fit to a host of (purely geometric)
observations such as supernova distances and the distance to the
CMB last scattering surface.
\subsection{Relation to parameterized $w(a)$ \label{sec:parw}}
The approach taken in this article is to examine dark energy dynamics
directly in the phase plane $w-w'$, where a time variable runs along
each trajectory. It is useful to see the relation
of standard parametrizations in terms of the temporal behavior,
i.e.\ $w(a)$, to this approach.
The standard two parameter function $w(a)=w_0+w_a(1-a)$ was shown by
Linder \cite{linprl,linpr} to provide an excellent approximation to
exact solutions of the Klein-Gordon equation in a wide variety of
models. In this ansatz we have $w'=-aw_a=w-w_\infty$, where the
high redshift equation of state $w_\infty=w_0+w_a$.
This describes a straight
line of slope 1 in the $w-w'$ plane, and can be rewritten as
as $w'=(1+w)-(1+w_\infty)$. In particular, if $w_\infty=-1$
we have exactly the behavior of thawing models (lying along lower bound of
that region). From Fig.~2 of Paper 1, we see as well that many tracking
models that fit present data (i.e.\ $\Omega_{\rm de}\sim0.7$ and $w<-0.8$)
are reasonably well described, on average, by a line of slope unity. Of
course this approximation will break down in the future, as the
field freezes more fully, turning toward the cosmological constant;
at the same time this $w(a)$ ansatz loses validity as it moves toward
ever more negative $w$. However, since data only exist toward the past,
we see why the $w_a$ parametrization is an excellent approximation.
To retain boundedness for both the past and present, as well as to
allow more dramatic dynamics (essentially slopes other than unity in
$w-w'$), one could use the ``e-fold'' model of \cite{eospar} or ``kink''
model of \cite{corakink}. Both utilize four parameters for their
description. The e-fold model has a more transparent translation to
$w'=dw/d\ln a$ since it also uses dynamics in terms of $\ln a$:
\beq
w(a)=w_f+\frac{\Delta w}{1+(a/a_t)^\tau},
\eeq
where $\tau$ is the transition rapidity in units of units of
e-folds $\ln a$, $a_t$ is the transition scale factor, $w_f$ is the
asymptotic future value, and $\Delta w=w_p-w_f$ is the difference
between asymptotic past and future values.
In the $w-w'$ phase plane we have
\beq
w'=-\tau(w-w_f)\left(1-\frac{w-w_f}{\Delta w}\right).
\eeq
Note that as we found in the Klein-Gordon equation analysis of
\S\ref{sec:kglines} the equation for $w'$ is quadratic in $w$.
We can identify several special cases. If the asymptotic future
state is deSitter ($w_f=-1$), then $w'=\tau(\Delta w^{-1}-1)(1+w)
+(\tau/\Delta w)\,w(1+w)$. This looks like the sum of a thawing
model and a model in the freezing region, i.e.\ the dark energy can be
viewed as the sum of two components. If we further take $\Delta w=1$,
then we remove the thawing component and end up with $w'=\tau\,w(1+w)$
-- a mocker model with $w_p=0$ and $w_f=-1$.
Starting instead with
an asymptotic past state of $w_p=-1$ gives $w'=\tau\,(1+w)(w-w_f)/(-1-w_f)$.
In the limit $w_f\to\infty$ (i.e.\ not worrying about the region where
there is no data) this gives a thawing model $w'=\tau(1+w)$. Thus the
four parameter e-fold ansatz is also quite versatile. The rapidity
parameter is directly related to both the slope of and the velocity
along the phase space trajectory, and ties in with the steepness of
the scalar field potential, as we saw in \S\ref{sec:kglines} with the
PNGB models where the slope was proportional to the inverse of the
symmetry scale $f$.
Finally, one could invert the situation and go from a
parametrization in the phase plane to
derive the function $w(a)$. For example, a track $w'=A(1+w)+B(1+w)^2$,
which we have seen is a common form, implies
\beq
1+w=(1+w_0)/\left[(1+x)a^{-A}-x\right],
\eeq
where $x=(B/A)(1+w_0)$ defines the present along the trajectory
(equivalently the dimensionless matter density $\om$ today). Note
that while the trajectory has two
parameters, the equation of state has three parameters since we
must have a parameter running along the track. At high redshift,
if $A>0$ then $w\to-1$ and we have a thawing model, asymptotically
independent of $B$. If $A<0$ then $w(z\gg1)=-1+(-A/B)$ and $w'\to0$,
i.e.\ it begins like a tracking model. It reaches a minimum
$w'_{\rm min}=-A^2/(4B)$ at $w_\star=-1-A/(2B)=[-1+w(z\gg1)]/2$;
that is, the trajectory is a parabola from its tracking value of the
equation of state to its future, cosmological constant value. A mocker
model is the special case $C=-A=B$. For
completeness, we give the dark energy density:
\beq
\rho_{\rm de}(a)=\rho_{\rm de}(1)\,(1+x-xa^A)^{3/B}.
\eeq
Of particular interest is the ``leveling'' model where, in loose
physical analogy to the inflationary power spectrum tilt $n-1$ being
driven to zero by large numbers of e-folds of expansion, the
equation of state tilt $1+w$ is driven to zero by the deSitter
expansion as the energy density approaches a certain constant value,
$\rho_f$. That is, take $1+w=D[\rho_{\rm de}(a)-\rho_f]$. This is
equivalent to the above parabolic model with $A=-3D\rho_f$ and $B=-3$.
Another interesting parabolic track is the coasting line $w'=3(1+w)^2$.
This corresponds to not a leveling but a tilting, with
$1+w=(1+w_0)\,[\rho_{\rm de}(1)/\rho_{\rm de}(a)]$, so $w$ is tilted
away from $-1$ as the energy density decreases.
\subsection{Acceleration and jerk \label{sec:jerk}}
One could leave behind the physics of the accelerating phenomenon
and instead use variables in terms of the acceleration itself, though
this seems less appealing. The deceleration parameter
\beq
q=-a\ddot a/\dot a^2=\frac{1}{2}+\frac{3}{2}w\Omega_{de}(a),
\eeq
and the jerk
\beq
j=a^2\, \dddot a/\dot a^3=1-\frac{3}{2}\Omega_{de}(a)\,[w'-3w(1+w)].
\eeq
We also have $j=q+2q^2-q'$. Note that interpreting $q$ and $j$ or $q'$ as
Taylor expansions of the expansion is of strictly limited use (since
observations span $\Delta\ln a\sim{\cal O}(1)$) and can be dangerous
\cite{noqexp}.
Furthermore, there is the same ambiguity there was with using pressure
as a variable. We note that any model where it touches the constant
pressure line
$w'=3w(1+w)$ has $j=1$; this is equivalent to $X=-3/2$ in the notation
of \S\ref{sec:kglines}. The two standard special cases of $j=1$ lie
at the ends of this line: an Einstein-de Sitter universe with $w=0$
``dark energy'' and a $\Lambda$CDM universe with cosmological constant dark
energy.
The constant pressure line is also related to the adiabatic sound speed of
the dark energy. (Note this is not the true sound speed of perturbations
arising from the microphysics of whatever the dark energy is.) The
adiabatic sound speed
\beq
c_a^2=\frac{\dot p}{\dot\rho}=w-\frac{1}{3}\frac{w'}{1+w}, \label{eq:ca}
\eeq
and we see it vanishes for $w'=3w(1+w)$. On the null line, the adiabatic
sound speed equals the speed of light (the same as the true sound speed
for a canonical scalar field). Models below the null line would need
to have $c_a^2>1$.
\section{Beyond Scalar Fields \label{sec:beyond}}
For the cosmic expansion dynamics we can always define an
effective equation of state even if the accelerating mechanism is
not a scalar field \cite{linjen}, through
\beq
w_{\rm eff}=-1-\frac{1}{3}\frac{d\ln\delta H^2}{d\ln a}, \label{eq:weff}
\eeq
where $\delta H^2=(H/H_0)^2-\om a^{-3}$ is the unknown part of the
Hubble parameter, that not due to matter. So it is of interest to
see to what extent the dynamical behaviors we have discussed carry
over to the $w_{\rm eff}-w'_{\rm eff}$ plane. That is, are freezing
and thawing behaviors more general than for scalar fields, and do
the null and coasting lines still play a role?
Due to the diversity of possible accelerating physics we do not
present a general analysis of these important questions
but rather calculate some specific cases.
\subsection{Scalar-tensor gravity \label{sec:scatens}}
Scalar-tensor theories modify the Einstein-Hilbert action with both an
additional scalar field and a coupling to Ricci scalar curvature $R$.
These are of great interest as a comparison in tests of general
relativity and also because gravitational theories involving
a nontrivial function of the scalar curvature can be transformed
to scalar-tensor theories. See \cite{scatensreview} for a general
introduction.
We consider coupling of a general form in the scalar field, but linear
in the curvature. So the general relativistic $R/(8\pi G)\to F(\phi)R$.
One then
obtains the usual Friedmann expansion equations, with extra terms
giving an effective dark energy density \cite{bmp}
\beq
\rho_{\rm ST}=V(\phi)+(1/2)H^2(q-1)(q+5)F_\phi^2+3H^2[(8\pi G)^{-1}-F],
\label{eq:rhost}
\eeq
where $q$ is the deceleration parameter, $F_\phi=dF/d\phi$, and the last
term involves the change of the gravitational strength from Newton's
constant $G$.
From this density one can then define $\delta H^2=8\pi G\rho_{\rm ST}/(3H^2)$
(note that we use the usual $G$ here since the deviation is absorbed into
$\rho_{\rm ST}$ as mentioned above). From this the equation of state
$w_{\rm eff}$ and its variation $w'_{\rm eff}$ can be calculated. A key
quantity will be $F/F_\phi^2\equiv\omega_{JBD}$. This is the
Jordan-Brans-Dicke parameter and its inverse must be very small
according to solar system tests. Expanding $F$ about the present,
\beqa
F(a)&\approx&(8\pi G)^{-1}-(1-a)F_\phi \dot\phi/(aH) \\
&\approx&(8\pi G)^{-1}-z(1-q)F_\phi^2.
\eeqa
So the ratio of the second to the first term ($\sim\omega_{JBD}^{-1}$)
is small, and gravity is nearly Einsteinian.
But this means that the first term in $\rho_{\rm ST}$ dominates (unless
$8\pi GV/H^2\ll1$, but then it doesn't affect the expansion and there is
no acceleration). Thus, the restriction of the scalar-tensor theory by
solar system constraints means that its effective equation of state must
be very close to a cosmological constant -- within $\sim\omega_{JBD}^{-1}$.
Since $\omega_{JBD}^{-1}<2.5\times 10^{-5}$, this would be rather
challenging to distinguish from a cosmological constant with cosmological
observations! One possible loophole is if the solar system limits on
the scalar coupling should not be applied to a cosmological situation
because of the different spacetime backgrounds with very different
scalar curvatures. This arises for example in chameleon scenarios
\cite{chameleon}. The most stringent cosmological bounds on varying $G$
arise from primordial nucleosynthesis and give
$\omega_{JBD}^{-1}\lesssim 3\times10^{-3}$ \cite{gvarycos}
(but see \cite{nsvary}).
Calculation of the effective phase plane parameters finds
\cite{bmplinder}
\beqa
w_{\rm eff}(z=0)&=&-1+0.46/\omega_{JBD} \\
w'_{\rm eff}(z=0)&=&-0.36/\omega_{JBD}.
\eeqa
While even with only the cosmological bounds on $\omega_{JBD}$ these
are quite close to the cosmological constant in the phase plane,
it is interesting to note that the current values lie along
$w'=0.78w(1+w)$, in the freezing region. Indeed its
trajectory is a freezing one, with scalar-tensor theories asymptotically
attracted to general relativity \cite{damourst,rboost}, and to $w=-1$.
One last thing to note, however, is that because scalar-tensor theories
possess anisotropic stress, the growth of density perturbations will
be modified from the quintessence case (see, e.g., \cite{stgrowth} and
references therein).
\subsection{Braneworld cosmology and $H^\alpha$ \label{sec:brane}}
In a braneworld cosmology \cite{dgp,deffayet}, effective acceleration
appears due to a weakening of gravity on large scales as it ``leaks''
from our brane into a higher dimensional bulk. The Friedmann expansion
equation becomes
\beq
H^2-H/r_c=8\pi G\rho_m/3,
\eeq
where $r_c$ is the crossover distance and $\rho_m$ the matter density.
The effective equation of state due to the modification is $w_{\rm eff}=
-[1+\om(a)]^{-1}$ \cite{lueeos}. Its trajectory in the phase plane is
plotted in Fig.~\ref{fig:bw}. Note that it looks like a freezing model,
and will indeed approach a cosmological constant in the asymptotic future.
The position along the
trajectory is a time variable, so taking the present to be, say, $\om=0.2$
would extend the solid curve in Fig.~\ref{fig:bw} slightly further
(since the figure uses $\om=0.3$).
We also see why it is so well approximated by a $w_0-w_a$ model, as
discussed in \S\ref{sec:parw}. (Recall, however, that $w_a$ is actually
defined at $z=1$, not $z=0$, to give the best physics fit \cite{linprl}.)
We can further generalize the modification to $\delta H^2\sim H^\alpha$
\cite{dvaliturner}. Then we find
\beqa
w_{\rm eff}&=&-\left[1-\frac{\alpha}{\alpha-2}\om(a)\right]^{-1} \\
w'_{\rm eff}&=&3w(1+w)[1-(2/\alpha)(1+w)]. \label{eq:wpalpha}
\eeqa
Recall that the braneworld case above corresponds to $\alpha=1$.
For acceleration today (with $\om=0.3$), we require $\alpha<1.57$;
for $w<-0.8$ today we require $\alpha<0.91$. Note that all $H^\alpha$
modified gravity models will look similar (one does require $\alpha<2$
for a negative equation of state at early times). See \S\ref{sec:track}
for discussion
of their tracking behavior. In particular, they all approach
the cosmological constant along $w'=3w(1+w)$, what was called the
constant pressure line for scalar fields. Their tracks must always
lie between $w'=3w(1+w)$ and the $w'=0$ axis. When $\alpha<0$, the
trajectories switch to the phantom regime with $w<-1$, but the bounds
still hold.
\section{Polytropic Dark Energy \label{sec:poly}}
An interesting, if phenomenological, way of obtaining acceleration
is to modify the Friedmann expansion equation but keeping a pure
matter universe. While this leaves open important questions about
its relation to fundamental theory and the growth of density perturbations,
we can investigate some general aspects of the effective equation of state
dynamics.
Consider general functions of the matter density (sometimes referred
to as barotropic models \cite{scherrer})
\beq
H^2=(8\pi G/3)\,g(\rho)=(8\pi G/3)\,[\rho+f(\rho)].
\eeq
The quantity $f(\rho)$ will act like an effective dark energy. Using
Eq.~(\ref{eq:weff}) we can define
\beqa
w_{\rm eff}&=&-1+\frac{d\ln f}{d\ln\rho} \\
w'_{\rm eff}&=&-3\,\frac{d^2\ln f}{d\ln\rho^2},
\eeqa
and identical relations hold for the total equation of state $w_{\rm tot}$
and its variation $w'_{\rm tot}$ upon substituting $g$ for $f$.
The simplest example of such a modification is $f\sim\rho^n$, the
Cardassian model of \cite{freeselewis}. From the above equations we
see that it corresponds to a constant equation of state $w=-1+n$. If
we require $w<-0.9$ (as observations favor for a constant equation of
state), then $n<0.1$; unfortunately $\rho^{1/10}$ does not obviously
appear to be a natural modification of the Friedmann equation.
We can investigate the phase space dynamics by relating $w'$ to $w$:
\beq
w'=3w(1+w)-3\frac{\rho^2}{f}\frac{d^2f}{d\rho^2}.
\eeq
Immediately we see that whether the effective dark energy lies below
the freezing region or not depends on the sign of $d^2f/d\rho^2$.
An equivalent question is whether the effective
pressure is decreasing or increasing with time (since the $w'=3w(1+w)$
line is that of constant pressure).
The model will follow the mocker model track $w'=3w(1+w)$ if $f=A+B\rho$.
This form is equivalent to a redefinition of $\om$, e.g. $\om\to\om(1+B)$,
plus a cosmological constant $A$. As such it has a nonzero minimum in
its effective potential, allowing it to reach the freezing boundary.
It is important to remember that the analysis in this paper and Paper 1
applies to the dynamics of the dark energy itself. Trajectories in the
$w_{\rm tot}-w'_{\rm tot}$ plane convolve the matter and dark energy
components, mixing the dynamics and so not giving rise to the clear
differentiations and regions found. This is why Ref.~\cite{scherrer} appears
to find a violation of the freezing bound and even null bound for some
barotropic models; those are actually phantom models in the dark energy
phase space, but are dragged by the matter to $w_{\rm tot}>-1$. We
make this more explicit later in this section.
For a richer dynamical behavior we propose a class of modifications of
the Friedmann expansion equation we call polytropic models \cite{freese02}.
Here
\beq
H^2/(8\pi G/3)=g(\rho)=\rho\,[1+(\rho/\rhos)^{-\alpha}]^\beta.
\label{eq:polyg}
\eeq
At densities much greater than some crossover value $\rhos$, e.g.\ at
high redshift, the Friedmann equation is standard. At low densities,
the expansion is modified, with $w_{\rm tot}$ asymptotically approaching
$-\alpha\beta$. If we want a future deSitter state, we could choose
$\beta=1/\alpha$. For just the effective dark energy equation of state,
the value in the past is $w_{\rm eff}=-\alpha$, and in the future of
course it dominates so $w_{\rm eff}=-\alpha\beta$.
Figure~\ref{fig:polyw} illustrates the phase plane dynamics. The first
panel takes $\beta=1/\alpha$, so that the future state is deSitter.
Models with $\alpha<1$ have $w\ge-1$ and act like freezing models,
starting from a constant $w=-\alpha$ and today (marked by crosses)
lying in the freezing region, before heading toward the cosmological
constant. Phantom models have $\alpha>1$ and act like mirror images
of freezing models, even to lying within the phantom freezing region today.
In the second panel, we fix $\beta=1/2$. While the models start at
the same phase space point as the previous models with the same $\alpha$,
now their endpoints are different. Indeed for $\alpha<2/3$ the
acceleration of the expansion is a temporary phase. Furthermore, the
trajectories with $\alpha\lesssim1.5$ do not lie in
the freezing regime and both regular and phantom models have
$w'>0$. Such polytropic models without a deSitter future will be clearly
distinguishable from both freezing and thawing quintessence.
Can we give some physics motivation for the polytropic form, aside from
its simplicity and proper asymptotic behavior? When $\beta=0$, there is
no modification; when $\beta=1$ we have the power law modification of
the Cardassian case, with $n=1-\alpha$, hence a constant $w=-\alpha$.
There are some motivations for power law modification from higher dimension
theories (for $n<1$ see \cite{freeselewis}, the nonaccelerating $n=2$
arises in Randall-Sundrum brane scenarios \cite{randallsundrum}).
When $\beta=1/2$, the modification is similar to that from a Chaplygin
gas \cite{chaplygingas}, as we see below; this has claimed motivation
from Born-Infeld actions and brane solutions \cite{bento}. So at
least the polytropic form unifies different prescriptions for dark energy.
Note that as $\beta$ increases from zero, for fixed $\alpha$, the size of
the ``hump'' in the trajectory decreases and the future value of $w$
moves back toward the initial value. At $\beta=1$ the trajectory
collapses to a point at $w=-\alpha$, $w'=0$. For even larger $\beta$,
the hump is flipped (i.e.\ the sign of $w'$ changes) and again
increases in size, with the future value of $w$ drawing away to more
negative values.
When we plot the same models as in the first panel of Fig.~\ref{fig:polyw},
but in the $w_{\rm tot}-w'_{\rm tot}$ plane, in Fig.~\ref{fig:polywtot},
we see that the models
that were phantom in the effective dark energy lie in the region
$w'_{\rm tot}<3w_{\rm tot}(1+w_{\rm tot})$. Furthermore, when $\alpha>2$
they can even lie below what was the null line $w'=-3(1-w^2)$. In a
nice analysis of barotropic models, Scherrer \cite{scherrer} noted
something similar (his barotropic models are a function of an arbitrary
component density not necessarily matter density).
For a barotropic perfect fluid the adiabatic sound speed is the physically
relevant sound speed for perturbations (but not in the quintessence case,
or in a general multicomponent case), and Scherrer's bound of $w'<3w(1+w)$
holds for $c_a^2>0$ (cf.\ Eq.~\ref{eq:ca} here). In general, however,
this is not a violation of the bounds of this article and Paper 1
because it occurs only when the adiabatic assumption holds, e.g.\ when
viewing the total equation of state, not the
properties of a (non-adiabatic) effective dark energy.
Suppose, however, we chose to fit the dark energy component itself,
rather than full energy density entering the Friedmann equation,
by the polytropic form Eq.~(\ref{eq:polyg}). This is somewhat strange
to do, since then the effective dark energy contains a matter-like part,
in addition to the pure matter density, and the polytrope was designed
to be the modified Friedmann equation as a whole. If we do so, though,
then the dark energy equation of state phase plane (and not the total
equation of state) is represented by the curves in Fig.~\ref{fig:polywtot}.
Moreover, the curve with $\alpha=2$, $\beta=1/2$ is the trajectory of
the Chaplygin gas. The generalized Chaplygin gas with pressure
$p\sim-\rho^{-\alpha_{gcg}}$ corresponds to polytropic dark energy
with $\alpha=\alpha_{gcg}+1$, $\beta=1/\alpha$. Whenever $\alpha\beta=1$
(cf.\ \cite{gondolofreese})
we have mocker behavior with $w'=3\alpha w(1+w)$. As stated above,
however, taking the dark energy itself to be polytropic means hiding both
a cosmological constant (if $\beta=1/\alpha$) and a spurious matter
density within the dark sector.
\section{Hierarchy parameters} \label{sec:slow}
Returning to more general dynamics, we saw in \S\ref{sec:scadyn} and
in particular from Fig.~\ref{fig:wwpscale}
that we do not have the standard inflation slow roll perturbative expansion
parameter. Explicitly, we do not have small
$\vp/V$ in the freezing or thawing regions unless $w\lesssim-0.995$;
more generally, $-M_P\vp/V\gtrsim 5\sqrt{1+w}$ today.
We examine here whether we can substitute a physics based hierarchy
of dynamics parameters.
\subsection{Slope parameters \label{sec:slope}}
The Klein-Gordon equation can be rewritten dimensionlessly as
\beq
\phi''+(2-q)\phi'=-\vp/H^2\equiv \eta_1,
\eeq
where as before a prime denotes a derivative with respect to $\ln a$,
and $q=-a\ddot a/\dot a^2$ is the deceleration parameter. In terms of
the $w-w'$ dynamics equation,
\beqa
w'&=&-3(1-w^2)-\sqrt{2(1-w^2)} \,\vp/(HV^{1/2})\\
&\equiv& -3(1-w^2)+\sqrt{2(1-w^2)} \,\eta_2.
\eeqa
So the parameters $\eta_1$, $\eta_2$ are called out by the physics.
If they are small, one could solve the equations perturbatively.
Note that $\eta_2=\eta_1(V/H^2)^{-1/2}$ so we always have
$\eta_2>\eta_1$. We violate the lower bound of the freezing region,
$w'<3w(1+w)$, when
\beq
\eta_2^2<\frac{9}{2}\frac{\epsilon}{2-\epsilon},
\eeq
where $\eps=1+w$ is the tilt parameter.
The analogous condition such that $w'<0$ is $\eta_2^2<(9/2)\eps(2-\eps)$,
and such that field is decelerating ($\ddot\phi<0$) is
$\eta_2^2<18\eps/(2-\eps)$.
Neither $\eta_1$ nor $\eta_2$ are particularly small unless $\eps=1+w$ is.
For example $\eta_2\gtrsim 1.5\sqrt{\eps}$, giving $\eta_2>0.1$ for
$w>-0.995$. Even $\eta_1>0.1$ for $w>-0.94$. The hierarchy among
the parameters is fixed for the region of interest: $\eps<\eta_1<\eta_2$,
so there is no phase space classification in this respect, as there is
for large field, small field, and hybrid models in inflation \cite{kinney}.
However that hierarchy involved the second derivative of the potential,
so it is worth a brief look at that quantity.
\subsection{Tracking parameter \label{sec:track}}
The tracking parameter is defined to be
\beq
\Gamma\equiv\frac{V\vpp}{\vp^2}.
\eeq
This is not generally a small parameter either. Indeed,
models whose energy density tracks \cite{tracker} the evolution of
the dominant energy component fulfill the conditions
\beq
\Gamma>1\qquad;\qquad \frac{d\ln(\Gamma-1)}{d\ln a}\ll 1.
\eeq
Within the class of tracking models (so now a particular subset of scalar
field cosmologies), at high redshifts within the matter dominated epoch
the field obeys
\beq
\Gamma=1-\frac{w}{2(1+w)}. \label{eq:gammaw}
\eeq
(Note the equation of state deviation $1+w$ at high redshift may not be
small.)
In \S\ref{sec:brane} we found that $H^\alpha$ models (including
braneworlds) follow freezing trajectories. This is not surprising
because they are basically trackers. A component starting with
constant $w$ at early times is equivalent there to a modification
$H^\alpha$ with $\alpha=2(1+w)$. The tracking parameter
$\Gamma=(2+\alpha)/(2\alpha)$ and it initially acts like an inverse
power law potential $V\sim\phi^{-n}$ with $n=2\alpha/(2-\alpha)$.
To relate the dynamics of the time variation to the tracking condition,
we invert Eq.~(\ref{eq:gammaw}) to write
\beqa
w&=&-2(\Gamma-1)/[1+2(\Gamma-1)] \\
w'&=&\frac{dw}{d\ln a}=w(1+w)\frac{d\ln(\Gamma-1)}{d\ln a}.
\eeqa
However, this is of limited use since we are unlikely to be able to
probe $w'$ in the high redshift, $z\gg1$, regime where tracking might hold.
Strong acceleration today, with $w\lesssim-0.7$, requires the breakdown of
tracking.
However, the analogy to $H^\alpha$ models presents an important insight into
why the general freezing region is bounded below by $w'=3w(1+w)$. At early
times the contribution to the Friedmann expansion equation by the dark
energy is small and $\delta H^2\sim a^{-3(1+w)}\sim H^{2(1+w)}$. That is,
the effective $\alpha\approx 2(1+w)$. One can generalize this to
$\alpha=2\langle 1+w\rangle$ when time variation of the equation of
state becomes relevant, where angle brackets denote an averaging over
$\ln a$. At late times, the dark energy density dominates the expansion,
$\delta H^2\sim H^2$, as it approaches $w=-1$ (freezes).
For any epoch we can define an instantaneous
value of $\alpha$. Equation~(\ref{eq:wpalpha}) then gives the relation
for $w'$. As freezing models approach $w=-1$, Eq.~(\ref{eq:wpalpha})
indicates they should do so along $w'=3w(1+w)$. Furthermore, since the
bracketed term is less than one, then this trajectory represents a
general lower
bound to the freezing region.\footnote{One caveat involves dark energy
models that possess an internal cosmological constant, i.e.\ nonzero
minimum to the potential, or otherwise act as the sum of two or more
components. These cannot be represented as $H^\alpha$ models and the
freezing bound does not apply.}
\subsection{Dynamics, Mass, and Spatial Inhomogeneities} \label{sec:inhom}
Dynamical models must also possess
spatial inhomogeneities in the field at some level. The equation for
these is given by perturbation of the Klein-Gordon equation (\ref{eq:kg}),
\beq
\delta\ddot\phi+3H\delta\dot\phi+(k^2+\vpp)\delta\phi=-\dot h\dot\phi/2,
\label{eq:pert}
\eeq
where $k$ is the wavenumber and $h$ is the trace of the metric
perturbation \cite{ma}. Just as matter density perturbations are damped
on scales below the Jeans length related to the sound speed in the
background medium, so the spatial inhomogeneities in the scalar field
will be absent on length scales less than that corresponding to the
effective mass $\sqrt{\vpp}$.
Using eq.\ (\ref{eq:wp}) and $\vpp=\dot \vp/\dot\phi$ we can calculate
the critical mass scale (also see \cite{caldwellvpp}), with
\beqa
\vpp/H^2 &=& (2+3w+2q)\frac{w'}{1+w}+
\frac{1}{4}\left(\frac{w'}{1+w}\right)^2 \nonumber \\
&\,& -\frac{1}{2}\frac{w''}{1+w}+\frac{3}{4}(1-w)(5+3w+2q).
\eeqa
Note $w'/(1+w)$ and $w''/(1+w)$ are well behaved and generally nonzero
as $w\to-1$.
This shows that the goal of exploring the temporal and spatial dynamics
of dark energy runs into double jeopardy. If the time variation is
weak, $|w'/(1+w)|\ll1$, then the effective mass $m=\sqrt{\vpp}\lesssim H$.
(The scale $H$ today corresponds to $10^{-33}$ eV; dark energy would be a
very light scalar field).
This means the Compton wavelength of the scalar field perturbations
is larger than the horizon, and so spatial inhomogeneities are also
difficult to detect. Thus, for $|w'|<1+w$, i.e.\ between the thawing and
freezing regions there is a ``dead zone'' of phase space,
where we can detect neither time variation nor spatial inhomogeneity.
For appreciable time variation, $|w'|>1+w$, one can have $m>H$ and
so the possibility of subhorizon clustering. However for models within
the freezing or thawing regions, one is restricted to $m\lesssim 2H$
so this could only occur on the largest scales (largest angles or
lowest multipoles).
Note that as the field approaches $w=-1$, the mass stays nonzero
(except it vanishes along the upper boundary of thawing and along the
null line). However, the amplitude of spatial perturbations vanishes
as can be seen from Eq.~(\ref{eq:pert}) with $\dot\phi=0$.
These properties make scalar field inhomogeneity an extremely begrudging
probe of the nature of dark energy, much less friendly than the dynamics.
\section{Conclusion} \label{sec.concl}
We have deepened and elaborated the understanding of the role that
the dark energy dynamics, through the $w-w'$ phase plane, can play
in leading our understanding of the nature of dark energy. This
includes the foundations of the null line, coasting line,
constant pressure line, and phantom line dividing the phase plane
into distinct, physical regions. We also elucidate the uppermost
and lowermost boundaries of the thawing and freezing regions.
The physical structure has been extended beyond canonical scalar
fields, including specific instances of modified gravity scenarios
such as scalar-tensor, braneworld, and $H^\alpha$ models, and
barotropic and polytropic generalizations of the Friedmann equation.
We outlined similarities and differences with the scalar field case,
showing that many act as freezing fields, and that we should be able
to clearly distinguish certain models that do not possess a deSitter
future. Mocker models, implementing a unification of dark matter
and dark energy, were shown to have difficulties purely from dynamical
considerations, in addition to their problems in structure formation.
Dark energy is demonstrated to be generically not amenable to a slow
roll description -- a major difference from early universe inflation --
as one of its ``Goldilocks'' conundra. This makes the dark energy
problem in some sense even more challenging than the early universe.
However, it also opens the possibility that if some physical bound
can be placed on the flatness of the potential, e.g.\ due to quantum
corrections, then this implies a barrier around the cosmological
constant $\Lambda$ model. This would offer hope, possibly accessible to next
generation experiments, that dark energy could definitely be distinguished
from $\Lambda$, if it is not $\Lambda$. That would be exciting!
The dynamics of the dark energy, in the form of the equation of state
ratio $w$ and its time variation $w'$, provides powerful insight into
the new physics behind cosmic acceleration. Spatial inhomogeneities
in the dark energy are seen to be much weaker and less forthcoming,
unless one entertains direct couplings. While we are not guaranteed
to zero in on the physics -- there is a dead zone of minimal
dynamics and possibly a ``confusion'' zone near the cosmological constant
-- any highly precise and accurate result would be an enormous success
in enlightening us on the dark universe.
$\,$ \\
\section*{Acknowledgments}
I benefited greatly from numerous discussions with my collaborator
Robert Caldwell. I also thank Carlo Baccigalupi, Robert Scherrer, and
participants in the workshop Cosmological Frontiers in Fundamental
Physics at Perimeter Institute (which I thank for hospitality), the
workshop Dark Energy from Fundamentals (DarkFun 3, hosted by the SNAP
Collaboration), and the LBNL particle theory group.
This work has been supported in part by the Director, Office of Science,
Department of Energy under grant DE-AC02-05CH11231.
|
Title:
Weak-Lensing Detection at z~1.3: Measurement of the Two Lynx Clusters with Advanced Camera for Surveys |
Abstract: (Abridged) We present a HST/ACS weak-lensing study of RX J0849+4452 and RX
J0848+4453, the two most distant (at z=1.26 and z=1.27, respectively) clusters
yet measured with weak-lensing. The two clusters are separated by ~4' from each
other and appear to form a supercluster in the Lynx field. Using our deep ACS
F775W and F850LP imaging, we detected weak-lensing signals around both clusters
at ~4 sigma levels. The mass distribution indicated by the reconstruction map
is in good spatial agreement with the cluster galaxies. From the SIS fitting,
we determined that RX J0849+4452 and RX J0848+4453 have similar projected
masses of ~2.0x10^14 solar mass and ~2.1x10^14 solar mass, respectively, within
a 0.5 Mpc (~60") aperture radius.
| https://export.arxiv.org/pdf/astro-ph/0601334 |
\title{WEAK-LENSING DETECTION AT $z\sim1.3$:
MEASUREMENT OF THE TWO LYNX CLUSTERS WITH ADVANCED CAMERA FOR SURVEYS}
\author{M.J. JEE\altaffilmark{1}, R.L. WHITE\altaffilmark{2}, H.C. FORD\altaffilmark{1},
G.D. ILLINGWORTH\altaffilmark{3}
J.P. BLAKESLEE\altaffilmark{4}, B. HOLDEN\altaffilmark{3}, AND S. MEI\altaffilmark{1}}
\altaffiltext{1}{Department of Physics and Astronomy, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218.}
\altaffiltext{2}{Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218.}
\altaffiltext{3}{University of California Observatories/Lick Observatory, University of California, Santa Cruz, CA 95064.}
\altaffiltext{4}{Department of Physics and Astronomy, Washington State University, WA 99164.}
\keywords{gravitational lensing ---
dark matter ---
cosmology: observations ---
X-rays: galaxies: clusters ---
galaxies: clusters: individual (\objectname{RX J0849+4452},~\objectname{RX J0848+4453}) ---
galaxies: high-redshift}
\section{INTRODUCTION}
It has become clear that massive clusters are not extremely rare at high redshifts ($z>0.8$) and
the presence of these large collapsed structures when the age of the
Universe is less than half its present value
is no longer in conflict with our current understanding
of the structure formation, especially in a
$\Lambda$-dominated flat cosmology.
Pursuit of galaxy clusters to higher and higher redshift is important in
the extension of the evolutionary sequences to earlier epochs, when
the effect of the different cosmological frameworks becomes more discriminating.
A great deal of observational efforts have been made in the last decade
in enlarging the sample of high-redshift clusters. X-ray surveys have
provided an efficient method of cluster identification and probe of
cluster properties because a hot intracluster medium (ICM) within the cluster
generates strong diffuse X-ray emission and is believed to be in quasi-equilibrium
with gravity. However, it is questionable how well the clusters
selected by their X-ray excess can provide the unbiased representation of the typical
large scale structure at the cluster redshift. If
the degree of the virialization decreases significantly with redshift and
is strongly correlated with X-ray temperature, the
cosmological dimming $\sim (1+z)^{-4}$ can bias our selection
progressively towards higher and higher mass, relaxed structures.
Among other important approaches to detect high-redshift clusters is a red-cluster-sequence (RCS)
survey using the distinctive spectral feature in cluster ellipticals. This
so-called 4000\AA~break feature is well-captured by
a careful combination of two passbands, and Gladders \& Yee (2005) recently reported
67 candidate clusters at a photometric redshift of $0.9 < z < 1.4$ from the $\sim10$\% subregion
of the total $\sim100~\mbox{deg}^2$ RCS survey field. A related method
but giving a higher contrast of cluster members with respect to the background sources is
to use deep near-infrared (NIR) imaging (e.g., Stanford et al. 1997) for the selection of cluster
candidates. High-redshift clusters identified in these color selection methods are
expected to serve as less biased samples encompassing the lower mass regime
at high redshifts.
In the current paper, we study two $z\sim1.3$ clusters, namely RX J0849+4452
and RX J0848+4453 (hereafter Lynx-E and Lynx-W for brevity),
using the deep F775W and F850LP (hereafter
$i_{775}$ and $z_{850}$, respectively) images obtained with the Advanced Camera for Surveys (ACS)
on the $Hubble$ $Space$ $Telescope$ ($HST$). Interestingly, although these two clusters
are separated by only $\sim4\arcmin$ from each other, they were discovered by
different methods.
Standford et al. (1997) discovered Lynx-W in a NIR survey as an overdense
region of the $J-K > 1.9$ galaxies and spectroscopically confirmed 8 cluster members.
They also analyzed the archival ROSAT-PSPC observation of the region and found
diffuse X-ray emission near the confirmed cluster galaxies. However, they
could not rule out the possibility that the X-ray flux might be coming
from the foreground point sources because of the PSPC PSF is too broad
to identify such objects. The subsequent study of the field using the $Chandra$ observations
showed that, although the previous ROSAT-PSPC observation is severely contaminated
by the X-ray point sources adjacent to the cluster, the cluster
is still responsible for some diffuse X-ray emission. Both
the X-ray temperature and luminosity
of the cluster appear to be low ($T_X\sim1.6$~keV and $L_{bol}\sim0.69\times10^{44}
~ \mbox{ergs}~\mbox{s}^{-1}$; Stanford et al. 2001).
Lynx-E was, on the other hand, first discovered in the ROSAT Deep Cluster Survey (RDCS) as
a cluster candidate and follow-up near-infrared imaging showed an excess of
red ($1.8<J-K<2.1$) galaxies around the peak of the X-ray emission (Rosati et al. 1999).
They also showed that five galaxies around the X-ray centroid have redshifts that are consistent
with the cluster redshift at $z=1.26$ using the Keck spectroscopic observations.
From the $Chandra$ data analysis, Stanford et al. (2001) determined the cluster temperature
and luminosity to be $T_X=5.8_{-1.7}^{+2.8}$ keV and $L_{bol}=3.3_{-0.5}^{+0.9}\times10^{44} \mbox{ergs}~\mbox{s}^{-1}$,
respectively.
The rather large difference in the X-ray properties of these two clusters may be viewed as
representing the characteristics of the sample obtained from
different survey methods. Lynx-E, the X-ray selected cluster, has
much higher X-ray temperature and luminosity than Lynx-W, the
NIR-selected cluster. If the stronger X-ray emission
means higher dynamical maturity, the more compact distribution
of the Lynx-E galaxies provides an alternate support of this
hypothesis. For dynamically relaxed systems the observed X-ray
properties can be easily translated into the mass properties
under the assumption of hydrostatic equilibrium. However,
as we probe into the higher and higher-redshift regime, it is natural
to expect that there will be more frequent occasions when
the equilibrium assumption loses its validity in deriving the
mass properties of the system. In addition, at $z>1$ we expect to
have a growing list of low-mass clusters that are also X-ray dark because
of the evasively low-temperature, as well as the substantial cosmological dimming.
Therefore, it is plausible to suspect
that these two $z\sim1.3$ clusters (especially Lynx-W, the poorer X-ray system)
might lie on a border where the X-ray observations alone start
to become insufficient to infer the mass properties.
Weak-lensing provides an alternative approach to
deriving the mass of a gravitationally bound
system without relying on assumptions about the dynamical state. This
can help us to probe the properties of the high-redshift
clusters in lower mass regimes, where the X-ray measurements
alone may not provide useful physical quantities.
In our particular case, weak-lensing is an important tool to test how
the masses of the two Lynx clusters at $z\sim1.3$
compare with their X-ray measurements. Especially
for Lynx-W, weak-lensing seems to be the unique route for probing
the cluster mass, considering the poor and amorphous X-ray emission.
Another interesting question is whether the low X-ray temperature of Lynx-W arises simply from a low
mass or from a yet poor thermalization of the ICM.
However, the detection of weak-lensing signal at $z\sim1.3$
is difficult and much more so if the lens is not very massive.
In our previous investigation of the two $z\sim0.83$ high-redshift
clusters (Jee et al. 2005a, hereafter Paper I; Jee et al. 2005b, hereafter Paper II),
we were able to detect clear lensing signals. They revealed the
complicated dark matter substructure of the clusters in great detail.
The effective source plane (defined by the effective mean redshift of the background galaxies)
in Paper I and II is at $z_{eff}\sim1.3$, corresponding to
the redshift of the lenses targeted in the current paper! Therefore, the
number density of background galaxies decreases substantially
compared to our $z\sim0.8$ studies and, in addition, the higher fraction of
non-background population in our source sample inevitably dilutes the
resulting lensing signal quite severely. Furthermore, the accurate
removal of instrumental artifacts becomes more critical as stronger
signals come from more distant, and thus fainter and smaller galaxies.
They are more severely affected by the point-spread-function (PSF).
Nevertheless, our
analyses of RDCS 1252-2927 at $z=1.24$ (Lombardi et al. 2005; Jee et al. in
preparation) demonstrate that weak-lensing can still be
applied to clusters even at these redshifts and
reveals the cluster mass distribution with high significance.
Returning to the X-ray properties, the low-energy quantum efficiency (QE) degradation of the $Chandra$ instrument can
cause noticeable biases in cluster temperature
measurements. Although there have been many suggestions regarding this issue,
it was not until recently that a convergent prescription to remedy
the situation has become available from the $Chandra$
X-ray Center\footnote{see http://cxc.harvard.edu/ciao3.0/threads/apply\_acisabs/
or http://cxc.harvard.edu/ciao3.2/releasenotes/}.
Because we suspect that the previous X-ray analyses of the Lynx clusters
suffered from the relatively insufficient understanding of this problem, we have also re-analyzed
the archival $Chandra$ data to enable a fairer
comparison between the weak-lensing and X-ray measurements.
Throughout the paper, we assume a $\Lambda$CDM cosmology favored by the
Wilkinson Microwave Anisotropy Probe (WMAP), where $\Omega_M$, $\Omega_{\Lambda}$, and
$H_0$ are 0.27, 0.73, and 71 $\mbox{km}~\mbox{s}^{-1}~\mbox{Mpc}^{-1}$, respectively.
All the quoted uncertainties are at the 1 $\sigma$ ($\sim68$\%) level.
\section{OBSERVATIONS}
\subsection{ACS Observation \label{subsection_acsobservation}}
Deep ACS/WFC imaging of the Lynx clusters were carried out as part of ACS Guaranteed Time Observation (GTO)
during 2004 March in three contiguous pointings, which cover a strip of $\sim9\arcmin\times3\arcmin$ region.
A slight overlap ($\sim30\arcsec$) was made between the pointings and the strip is oriented in
such a way that the two cluster centers are approximately located near the overlap region.
Each pointing was observed in $i_{775}$ and $z_{850}$
passbands with 3 and 5 orbits of integration, respectively.
We used the ACS GTO pipeline (``APSIS"; Blakeslee et al. 2003) to remove cosmic rays, correct
geometric distortion via drizzle algorithm (Fruchter and Hook 2002), and register different exposures. Apsis
meets all the requirements of weak lensing analysis (Paper I and II), offering
a precise ($\sim0.015$ pixels) image registration via the ``match'' program (Richmond 2002)
after correcting for geometric distortion (Meurer et al. 2003).
In Figure~\ref{fig_lynx_illustration} we present the pseudo-color
image of the entire ACS field with the blow-ups of the two Lynx clusters.
Lynx-E is well-portrayed by the somewhat compact distribution
of the cluster red sequence around the brightest cluster galaxies (BCGs).
It appears that the cluster has a strongly lensed blue giant arc $\sim4.5\arcsec$
south of the BCGs. The spectroscopic
redshift of this arc candidate has not yet been determined.
The red sequence of Lynx-W looks somewhat scattered and there
seem to be no distinct BCGs characterizing the cluster center though
the excess of the early-type galaxies in the region clearly defines
the cluster locus.
The detection image was created by combining the two passband images using inverse variance
weighting. Objects are detected through the SExtractor program (Bertin \& Arnouts 1996) by searching for
at least five connected pixels above 1.5 times the sky rms. The field contains several
bright stars whose diffraction spikes not only induce a false detection, but
also contaminate the neighboring objects. We manually selected and removed these objects.
The catalog contains a total of 8737 galaxies.
\subsection{Chandra Observation}
We retrieved the $Chandra$ observation of the Lynx field from the
Chandra X-ray Center. The field was observed with the Advanced CCD Imaging
Spectrometer I-array (ACIS-I) in the faint mode at a focal temperature of -120 K.
The observation consists of two exposures: $\sim65$ ks and $\sim125$ ks integrations
on 2000 May 3 and 4, respectively.
The raw X-ray events were processed with the $Chandra$ Interactive Analysis of
Observations (CIAO) software version 3.2 and the Calibration Database (CALDB) version 3.1,
which provide the correction for time-dependent gain variation and the low-energy
quantum efficiency degradation without requiring any external guidance.
We identified and flagged hot pixels and afterglow events using the $acis\_build\_badpix$,
$acis\_classify\_hotpix$ and $acis\_find\_hotpix$ scripts while selecting only the standard
$ASCA$ events (0,2,3,4, and 6).
Figure~\ref{fig_xrayoverimage} shows the adaptively smoothed $Chandra$ X-ray contours
of the Lynx field overlaid on the ACS image. This adaptive smoothing
is performed using the CIAO CSMOOTH program with a minimum significance of 3 $\sigma$ and
the contours are spaced in square-root scale. Because of the low counts from the two
high-redshift clusters, the 3 $\sigma$ significance condition can only be met
with rather large smoothing kernels. Therefore, the round appearance
of the contours should not be misinterpreted as indicating the relaxed status of the systems.
When the contours are reproduced with a smaller, constant kernel smoothing, Lynx-W
looks much more irregular than Lynx-E.
The X-ray centroids of Lynx-E and W are in good spatial
agreement with those of cluster optical lights.
The foreground cluster RXJ 0849+4456 (Holden et al. 2001) at z=0.57
appears to be also strong in X-ray emission, but is located
outside the ACS pointings ($\sim5\arcmin$ and $\sim3\arcmin$ apart
from Lynx-E and W, respectively). The multi-wavelength analysis of this
cluster is presented by Holden et al. (2001) and they found that
the cluster can be further resolved into two groups at z=0.57 and 0.54.
Our subsequent X-ray analyses are confined to the two
high-redshift clusters at $\bar{z}=1.265$ present within the current ACS pointings.
\section{ACS DATA ANALYSIS}
As in our previous investigations (Paper I and II), we measure galaxy shapes
and model the point-spread-function of the observation using shapelets (Bernstein \& Jarvis 2002; Refregier 2003).
Readers are referred to Paper I and II for detailed description of the ellipticity measurements.
\subsection{Cluster Luminosity}
Our current spectroscopic catalog of the ACS Lynx field (B. Holden et al. in preparation) contains
150 objects and 32 of them belong to either of the two high-redshift clusters ($1.24<z<1.28$);12 galaxies
are at $z>1.31$ and the rest of them (106 objects) are foreground objects.
We supplemented the cluster member galaxy catalog with
the cluster red sequence (Mei et al. 2005) using
$i_{775}-z_{850}$ colors. In order to minimize the systematics
from internal gradients and the different
PSF sizes (the PSF of $z_{850}$ is $\sim10$\% broader than that of $i_{775}$), the galaxies are deconvolved
with the CLEAN (H\"{o}gbom et al. 1974) algorithm. After an effective radius $R_e$ is determined for each galaxy, we
measured the object colors within a circular aperture defined by $R_e$. When the estimated $R_e$ was less than
three pixels, we used a three pixel aperture instead (the median $R_e$ is $\sim5$~pixels).
At $z\sim1.265$, the 4000\AA~break
is shifted slightly blue-ward of the effective wavelength of the $z_{850}$ filter. Therefore,
this filter combination is less than ideal, but
the red sequence is still visible
down to $z_{850}\sim24$ in the $i_{775}-z_{850}$ versus $z_{850}$ plot (Figure~\ref{fig_cm}).
We visually examined each candidate and discarded the objects
that do not seem to have early-type morphology, or whose
redshifts (if known) are inconsistent with the cluster redshifts.
The final cluster member catalog contains 68 objects.
The rest-frame $B$ band at the cluster redshift is approximately
redshifted to the ACS $z_{850}$ band and we derive the
following photometric transformation from the synthetic photometry
with the Spectral Energy Distribution (SED) templates of Kinney et al. (1996).
\begin{equation}
B_{rest} = z_{850} - (0.70 \pm 0.02) (i_{775}-z_{850}) + (1.08 \pm 0.01) - DM \label{photran},
\end{equation}
\noindent
where DM is the distance modulus of 44.75 at $\bar{z}=1.265$.
From the above selection of the cluster galaxies,
we estimate that Lynx-E and W encloses $L_B\sim1.5\times10^{12}$ and
$\sim0.8\times10^{12} L_{B\sun}$, respectively within 0.5 Mpc ($\sim60\arcsec$) radius.
Of course, these values
correspond to the lower limits because we neglected the
contribution from the blue galaxies (except for the several spectroscopically confirmed
ones), as well
as the less luminous population ($z_{850}>24$).
However, we do not attempt to determine the correction factors in the current
paper because the number of galaxies in both of our spectroscopic
and red sequence samples is insufficient to
support our statistical derivation.
\subsection{PSF Correction}
ACS/WFC has a time- and position-dependent PSF (Paper I) and the
ability to properly model the PSF pattern in the observed cluster field is critical
in subsequent galaxy ellipticity analysis. In paper I and II, we demonstrated that
the PSF of WFC sampled from the 47 Tucanae field can be used to describe the
PSF pattern of the cluster images where only a limited number of stars are available, but can
be used as a diagnosis of the model accuracy.
We selected the stars in the Lynx field via a typical magnitude versus half-light radius plot
(Figure~\ref{fig_starselect}). Figure~\ref{fig_starfield}a show the WFC PSF pattern in the $i_{775}$ image
of the Lynx field, which is
similar to the ones in our previous cluster weak lensing studies. The PSFs are elongated
in the lower-left to upper-right direction. An analogous pattern is also observed
in the $z_{850}$ band. However, the wings of the $z_{850}$
are stretched approximately parallel to the row of the CCD (telescope V2 axis) and
the feature becomes observable when the wings of the PSFs are more heavily weighed (Heymans et al. 2005).
In our calibration of the ACS (Sirianni et al. 2005),
we also observed an opposite pattern (i.e., with an ellipticity nearly perpendicular to
Figure~\ref{fig_starfield}a), it seems that this PSF pattern is more frequently observed, at least
in our GTO surveys of $\sim15$ clusters. Because the focus offsets of different HST visits are
likely to vary, one may desire to find the closest PSF template
for every individual exposure and perform PSF corrections one by one.
However, we find that in our GTO cluster observations the PSF patterns in different exposures do not
vary considerably. Therefore, we chose a single PSF template for each filter and
created a PSF map for the entire $3\times 1$ mosaic image by placing the template PSFs on each pointing.
In order to minimize the model-data discrepancy due to the slight focus variation, we fine-tuned our model
for each exposure by shearing
the PSF by an amount $\delta \eta$, which can be expressed in shapelet notation as
\begin{equation}
b_{pq}^{\prime} = \mbox{\bf{S}}_{\delta \eta} b_{pq},
\end{equation}
\noindent
where $b_{pq}$ is the shapelet componet of the PSF and the evaluation of matrix elements of the shear operator
$\mbox{\bf{S}} _ {\delta\eta}$ can be found in Bernstein \& Jarvis (2002).
Figure~\ref{fig_starfield}b displays
the residual ellipticities of the same stars in the $i_{775}$ when the PSF is circularized
with rounding kernels (Fischer \&
Tyson 1997; Kaiser 2000; Bernstein \& Jarvis 2002). The dramatic reduction
of the PSF anisotropy is also distinct when the ellipticity components ($e_+$ and $e_{\times}$) before
and after the corrections are compared (Figure~\ref{fig_star_anisotropy}).
This rounding kernel test verifies that our PSF models describe the PSF pattern of the
cluster observation very precisely. Although one can continue with this rounding kernel
method and make a subsequent measurement of the galaxy shape in this ``rounded'' images (e.g., Fischer \& Tyson 1997),
we prefer to remove the PSF effect through straightforward deconvolution in $shapelets$ because the latter
gives more satisfactory results for very faint galaxies (Paper I; Hirata \& Seljak 2003). Besides,
the $z_{850}$ PSF is rather complicated because of the ellipticity variation between core and wing metioned above,
and this PSF effect can be more efficiently corrected by the deconvolution.
\subsection{Mass Reconstruction \label{section_mass_reconstruction}}
In order to maximize the weak-lensing signal, it is important to select the source population
in such a way that the source sample contains the minimal contamination from cluster and foreground
galaxies. Because only two passband images of the Lynx field are available,
direct determination of reliable photometric redshift for an individual galaxy is impossible.
Therefore,
we chose to select the background galaxies based on their ($i_{755}-z_{850}$) colors and $z_{850}$
magnitudes. The redshift distribution of this sample can be indirectly inferred when we
apply the same selection criteria to other deep multi-band HST observations such as the
Ultra Deep Field (UDF; Beckwidth et al. 2003) project, for which reliable photometric redshift information is obtainable down
to the limiting magnitude of our cluster observation (D. Coe et al., in preparation).
We selected the $24<z_{850}<28.5$ galaxies whose $i_{775}-z_{850}$ colors are
bluer than those of the cluster redsequence ($i_{775}-z_{850}\lesssim0.7)$ as ``optimal'' background population by examining
the resulting tangential shears around the two $\bar{z}=1.265$ clusters.
This selection yields a total of 6742 galaxies ($\sim204 \mbox{arcmin}^{-2}$).
Assuming that the cosmic variance between the Lynx and UDF is not large,
we estimate that approximately 60 per cent of the selection is behind the Lynx clusters.
Our final ellipticity catalog was created by combining the $i_{775}$ and $z_{850}$ bandpass ellipticities.
Of course, there is a subtlety in this procedure because an object can have intrinsically
different shapes and thus ellipticities in different passbands. We adopted the methodology presented by
Bernstein and Jarvis (2002) to optimally combine the galaxy ellipticities.
In our previous weak-lensing analyses (Paper I and II), we found that this scheme indeed reduced the
mass reconstruction scatters compared to the case when only single passband images were used; the improvement
increases as fainter galaxies are included.
As a consistency check, we compared the shapes and lensing signals from the two passband images, and
confirmed that the results are statistically consistent.
We show the distortion and mass reconstruction of the Lynx field from this combined shape catalog in Figure~\ref{fig_whisker}.
Although the systematic alignments of source galaxies around the
cluster centers are subtle in the whisker plot (left panel), the resulting mass reconstruction (right panel)
clearly shows the dark matter concentration associated with the cluster galaxies.
The mass map is generated using the maximum likelihood algorithm and is
smoothed with a FWHM$\sim40\arcsec$ Gaussian kernel.
We verify that
other methods (e.g., Seitz \& Schneider 1995; Lombardi \& Bertin 1999) also produce
virtually identical results.
The two mass clumps are in good spatial agreement with both the cluster light and X-ray emission.
Within a radius of $1\arcmin$,
both clumps are found to be significant, above the $4 \sigma$ level (determined from bootstrap resampling).
Figure~\ref{fig_high_resolution} shows the high-resolution (smoothed with a FWHM$\sim20\arcsec$ kernel)
version of the mass maps overlaid on the ACS images. The clump associated with Lynx-E is offset $\sim10\arcsec$ from the
BCGs and the Lynx-W clump seems to lie on the western edge of the cluster galaxy distribution.
In Paper I and II, we have reported significant mass-galaxy offsets for two clusters at $z\sim0.83$
and discussed the possibility that those offsets may signal the merging substructures.
Although it is tempting to interpret the mass-galaxy offsets in the current study as also
implying the similar merging of the two Lynx clusters, our investigation of
the mass centroid distribution using the bootstrap resampling shows that the significance is only marginal
(i.e., the $r\sim10\arcsec$ circle roughly encloses $\sim70$\% of the centroid distribution).
It is encouraging to observe that the foreground cluster at $z\sim0.54$ affects
the distortion of source galaxies and reveals itself
in the weak lensing mass reconstruction (Figure~\ref{fig_whisker}) though
most of its galaxies are outside our ACS field (see Figure~\ref{fig_xrayoverimage}
for the location of the X-ray emission from the foreground cluster).
As shown by this foreground cluster and its manifestation in the mass map, light coming from
background galaxies is perturbed by all the objects lying in their paths to the observer.
Considering the high-redshifts ($\bar{z}=1.265$) of the Lynx clusters, the
likelihood of such interlopers is high. In addition, if the masses
of the two high-redshift clusters are not very large, even a moderately massive
foreground object can generate a similar lensing signal
because it has higher lensing efficiency for a fixed source plane (unless
the source plane is located at substantially higher than $z\sim1.3$).
In an attempt to separate this lower-redshift contribution from our weak lensing mass map
presented in Figure~\ref{fig_whisker}b, we created an alternate source sample
by selecting the brighter ($22<z_{850}<25$) galaxies. This time we did not exclude
the galaxies whose $i_{775}-z_{850}$ colors correspond to that of the cluster red-sequence because
they also serve as well-defined source plane at $z\sim1.3$ and their shapes should be
perturbed by any lower-redshift mass clumps. We present this second
version of the mass reconstruction in Figure~\ref{fig_mass_fore}.
It is remarkable to observe that in this version
the two high-redshift clusters disappear whereas many of the assumed foreground
features (including the cluster at $z=0.54$) still remain.
The comparison of this second mass reconstruction with the previous result
also indicates that some of the foreground mass clumps might affect the
shape of the contours of the high-redshift clusters at large radii;
the mass clump of Lynx-E seems to have a neighboring foreground clump
at its southwestern edge, and the southern edge of the Lynx-W clump
also slightly touches the foreground structure (Figure~\ref{fig_high_resolution})
(However, far fewer galaxies were used for this second version of mass
reconstruction and thus the position of these structures have much less
significance). This apparent substructure in projection may bias our measurements of
the total mass. We discuss this issue in \textsection\ref{section_mass_estimate}.
\subsection{Redshift Distribution of Source Galaxies}
As detailed in Paper I, the redshift distribution of the source galaxies of the Lynx field was inferred
from the photometric redshift catalog of the UDF.
We also used the two photometric catalogs created from the Great Observatories Origins Deep
Survey (GOODS; Giavalisco et al. 2004) and the degraded UDF in order to estimate
the contamination of the cluster members in the source sample for $z_{850}<26$ and $z_{850}>26$, respectively.
Figure~\ref{fig_zdist} shows the magnitude distribution of the source galaxies (top panel) with
the estimated mean redshift (bottom panel) for each magnitude bin. It appears that the number density excess
due to the cluster galaxy contamination is not significant throughout the entire magnitude range. However,
we must remember that the sample contains substantial contamination of foreground galaxies, which dilute the lensing signal. We
measure the mean redshift in terms of the following:
\begin{equation}
\beta_{l} = \left < \mbox{max} ( 0, \frac{D_{ls}} {D_s}) \right > \label{eqn_beta},
\end{equation}
\noindent
where $D_s$, $D_l$, and $D_{ls}$ are the angular diameter distance from the observer to the source,
from the observer to the lens and from the lens to the source, respectively.
We obtain $<\beta>=0.155$ for the entire source galaxies. The value corresponds to a single
source plane at $z_{eff}\simeq1.635$ and the critical surface mass density
($\Sigma_c = c^2 (4 \pi G D_l \beta)^{-1}$) has the physical unit of $\sim6180 M_{\sun}/\mbox{pc}^2$
at the redshift of the lens $\bar{z}=1.265$.
\subsection{Weak-lensing Mass Estimation \label{section_mass_estimate}}
A first guess of the mass can be obtained by fitting the SIS model
to the observed tangential shears around the clusters. We chose
the origin of the tangential shears as the centroids
of the mass clumps in Figure~\ref{fig_whisker}. The neighboring
foreground structures at $z\simeq0.54$ as well as the
proximity of the field boundary restrict us to the use of
the tangential shears at radii no greater than $\sim80\arcsec$.
In addition, we discarded the measurements at $r<30\arcsec$
in order to minimize the possible substructure artifact and
the contamination of the lensing signal from the cluster members.
Although this precaution leaves us with only a small fraction of
the total measurements, the lensing signal is clearly detected for both clusters
at the $\sim3\sigma$ level in the tangential shear plots (Figure~\ref{fig_tan_shear}).
It is plausible that the severly decreased shears at $r<30\arcsec$ for Lynx-E
might be in part caused by the aforementioned contamination from the cluster members.
We verified that the lensing signal disappeared when the background galaxies were
rotated by 45$\degr$ (null test).
Note that the uncertainties in Figure~\ref{fig_tan_shear} reflect only the statistical
errors set by the finite number of background galaxies. In Paper II, we demonstrated that the large scale structures
lying in front of and behind the high-redshift cluster MS 1054-0321 ($z\simeq0.83$) were dominant
source of errors in the mass determination, responsible
for approximately 15\% of the total cluster mass. This fractional uncertainty increases
substantially with cluster redshifts because the lensing by the foreground cosmic structures
become more efficient than the lensing by clusters whose redshifts approach those of source galaxies.
However, for the current clusters, we expect that the large statistical errors
still overwhelm the cosmic shear effects.
When we repeat the analysis of Paper II for the current clusters, we estimate that
the uncertainties of the Einstein Radius for the SIS fit marginally increases from
$\sigma_{er}=0\arcsec.75$ and $0\arcsec.77$ to
$\sigma_{er}=0\arcsec.81$ and $0\arcsec.83$ for Lynx-E and W, respectively.
The Einstein radius of $\theta_E=2\arcsec.45\pm0\arcsec.81$ (with respect to
the effective source plane at $z_{eff}\simeq1.635$)
for Lynx-E
corresponds to a mass of $M(r)=(4.0\pm1.3)\times10^{14} (r/\mbox{Mpc})~M_{\sun}$
and a velocity dispersion of $740_{-134}^{+113}\mbox{km}\mbox{s}^{-1}$.
Similar values of $M(r)=(4.2\pm1.4)\times10^{14} (r/\mbox{Mpc})~M_{\sun}$ and
$\sigma_{SIS}=762_{-133}^{+113}\mbox{km}\mbox{s}^{-1}$
are obtained for Lynx-W as implied by its comparable
Einstein Radius $\theta_E=2\arcsec.60\pm0\arcsec.83$.
As mentioned in \textsection\ref{subsection_acsobservation}, we note that there is
a strongly lensed arc candidate at $r\simeq4.5 \arcsec$ for Lynx-E, which
can provide a useful consistency check.
In general,
Einstein radii depend on source redshifts, and the relation steepens if a lens
is at a high redshift. If the Einstein radius of the arc is assumed to be $\theta_E=4.5\arcsec$,
this implies that the redshift of the object should lie at $1.8<z<3.2$ in our adopted cosmology
(the uncertainty reflects only the errors of the Einstein radius from the SIS fit result).
Because we have only $i_{775}$ and $z_{850}$ band images, the photometric redshift estimation
of this arc candidate is unstable. Nevertheless, if we use the HDFN prior and truncate
it below $z=1.2$, the color ($i_{775}-z_{850}=0.098$) of the object is consistent
with the SED of the starburst galaxy at $1.7<z<3.7$.
Alternatively, we can also estimate the cluster mass
based on the two parameter-free methods, namely the aperture mass densitometry and the rescaled mass
reconstruction. Although these two parameter-free approaches need some feedbacks from the above SIS fitting result
to lift the mass-sheet degeneracy, in general they provide
more robust methodology. They are less affected by the cluster substructure or the deviation
from the assumed radial profile. However, one drawback of this approach is that
the measurement is more severely influenced by the cosmic shear effect than
in the case of the SIS fitting because the aperture mass densitometry uses less amount
of information (i.e., decreased tangential shears in outer range).
With the $r=80-90\arcsec$ region as a control annulus for both clusters,
we computed the cluster mass profiles from these two parameter-free methods (Figure~\ref{fig_mass_summary});
from the SIS fit results, we determine the mean mass density in the annulus
to be $\bar{\kappa}=0.014\pm0.004$ and $0.015\pm0.005$ for Lynx-E and W, respectively.
As observed in Paper I and II, the
mass estimation obtained
from the rescaled mass reconstruction (dotted) is in good agreement with the aperture mass densitometry (open
circle). We also note that both methods gives masses consistent with the SIS fit results.
Because we used the SIS fit results above to lift the mass-sheet degeneracy, it is useful to
examine how the result change when an NFW profile is assumed, instead. Unfortunately, the low lensing signal
in the limited range does not allow us to constrain the two free parameters of the NFW profile simultaneously;
the two parameters trade off with each other without significantly altering the quality of the fit.
Freezing the concentration parameter to $c=4$, nevertheless, yields $r_s=180\pm37$ ($187\pm34$) kpc for
Lynx-E (W), predicting the mean mass density of $\bar{\kappa}=0.015\pm0.020$ ($0.016\pm0.021$) in the control
annulus. Different choices for the concentration parameter $c$ do not change these results substantially
(for instance, the choice of $c=6$ gives $\bar{\kappa}\simeq0.012$ for Lynx-E).
In \textsection\ref{section_mass_reconstruction} we demonstrated that
both clusters might have neighboring foreground mass clumps in projection. Therefore, it is
worthwhile to assess how much these foreground structures affect our mass estimation.
Because the redshift information of the foreground masses are not available, we
cannot subtract their contribution directly from our mass map.
Instead, we attempted to minimize their effects by replacing the mass density
of the region that is occupied by the foreground mass clumps
with the azimuthal average from the rest. Of course, we do not expect that this scheme
yields cluster masses that are completely free from foreground contamination, since
the azimuthal averages taken at other regions might be biased.
However, this method is still an important test because
a significant difference in resulting mass estimation must be detected if the foreground
contamination is indeed severe.
The mass-sheet lifted mass map is convenient for this type of analysis. We replaced
the southwestern region ($\sim220\degr<\theta<\sim260\degr$; the angle is measured from
the north axis counterclockwise) of the Lynx-E clump and the southern
region ($\sim130\degr<\theta<\sim195\degr$) of the Lynx-W clump with
the azimuthal averages taken at different angles.
The solid lines in Figure~\ref{fig_mass_summary} represent the mass profiles
obtained from this measurement. For both clusters, this new measurements
give slightly lower values, but the change is only marginal.
We estimate that both Lynx-E and W have
a similar mass of $(2.0\pm0.5) \times 10^{14} M_{\sun}$ within
0.5 Mpc ($\sim60\arcsec$) aperture radius from this approach.
The uncertainties here are estimated from 5000 bootstrap
resampling of the source galaxies and we do not include the cosmic
shear effects because it is non-trivial to estimate the effect
for this rescaled mass map approach.
We adopt the conventional definition of the virial radius, where
the enclosed mean density within the sphere becomes
200 times the critical density $\rho_c(z)=3H(z)^2/8\pi G$ at the redshift of the cluster.
Although the factor 200 above is most meaningful in the mass-dominated flat universe,
we retain this definition so as to enable
a consistent comparison with the values of other clusters found in the literature.
The assumption of the spherical symmetry (SIS)
allows us to estimate $r_{200}\simeq0.75$~Mpc
and $M_{200}\simeq2.0\times 10^{14} M_{\sun}$ for both Lynx clusters.
These virial properties are much smaller than the clusters at $z\sim0.83$
studied in Paper I and II. We reported that CL 0152-1357 has a virial radius of $r_{200}\sim1.1$~Mpc
and a virial mass of $M_{200}\sim4.5\times10^{14}~M_{\sun}$ in Paper I. For
MS 1054-0321, Paper II quoted $r_{200}\simeq1.5$~Mpc and $M_{200}\simeq1.1\times10^{15}~M_{\sun}$.
If we assume that the two Lynx clusters are approaching each other perpendicular to the line of sight at a free-fall speed,
our order-of-magnitude estimation predicts that
the two Lynx cluster will merge into a single cluster whose virial
mass exceeds
$\sim 4.0\times 10^{14} M_{\sun}$ in a time scale of $t\sim2$~Gyrs (or at $z\sim0.8$).
\section{$CHANDRA$ X-RAY ANALYSIS}
\subsection{Cluster Temperature and Luminosity \label{section_temperature}}
The X-ray spectra of Lynx-E and Lynx-W were extracted from the circular regions ($\bar{r}\sim36\arcsec$) positioned
at their approximate X-ray centroids after the point sources (Stern et al. 2002) are
removed. The redistribution matrix file (RMF) and the area response file (ARF) were created using the CIAO tool version 3.2
with the calibration database (CALDB) version 3.1, which
properly accounts for the time-dependent low-energy QE degradation, as well as charge transfer inefficiency (CTI).
The photon statistics is somewhat poor mainly because the clusters are at a high-redshift ($\bar{z}=1.265$) and
thus the differential surface brightness dimming is severe $\sim (1+z)^4$. Especially, as implied by its low
temperature ($T<2$ keV), the Poissonian scatter of the Lynx-W is worse. Therefore, we constructed the spectra
for both clusters with a minimum count of 40 per spectral bin. We think that this choice makes the
spectral fitting stable without diluting the overall shape of photon distribution too much. Because it
is impossible to constrain the iron abundance given the statistics, we fixed the metallicity at 0.36 $Z_{\sun}$.
This assumes that both Lynx clusters possess similar metallicity to RDCS 1252.9-2927 at $z=1.24$ (Rosati et al. 2004).
However, as noted by Stanford et al. (2001), we observed only minor changes even when different values were tried.
The Galactic hydrogen column density was also fixed at $\mbox{n}_H=2.0\times10^{20}\mbox{cm}^{-2}$ (Dickey \& Lockman 1990).
We used the $\chi^2$ minimization modified by Gehrels (1986;CHI-GEHRELS), who extended the conventional
$\chi^2$ statistics so that it can handle the deviation of the Poissonian from the Gaussian at
the low-count limit.
Figure~\ref{fig_spec} shows the best-fit MEKAL plasma spectra (Kaastra \& Mewe 1993; Liedahl, Osterheld, \&
Goldstein 1995) for both clusters. We obtain $T=3.8_{-0.7}^{+1.3}$~keV for Lynx-E with a
reduced $\chi^2$ of 0.79 (19 degrees of freedom). Lynx-W is determined to have $T=1.7_{-0.4}^{+0.7}$~keV
with a reduced $\chi^2$ of 1.16 (7 degrees of freedom).
The observed fluxes are estimated to be $F (0.4-7\mbox{keV})=1.5_{-0.2}^{+0.3}\times10^{-14}\mbox{ergs~cm}^{-2}~\mbox{s}^{-1}$ and
$F (0.4-4\mbox{keV})=7.2_{-0.5}^{+1.4}\times10^{-15}\mbox{ergs}~\mbox{cm}^{-2}~\mbox{s}^{-1}$, which
can be transformed into the rest-frame (also, apertured-corrected to $\sim0.5$~Mpc) bolometric (0.01 - 40 keV)
luminosity of $L_X=(2.1\pm0.5)\times10^{44}$
and $(1.5\pm0.8)\times10^{44}~\mbox{ergs}~\mbox{s}^{-1}$ for Lynx-E and Lynx-W, respectively
(note that the shallow surface brightness profile of Lynx-W requires a rather large aperture
correction factor).
\subsection{X-ray Surface Brightness Profile and Mass Determination \label{section_sb}}
The azimuthally averaged radial profiles were created from the exposure-corrected $Chandra$ image.
In Figure~\ref{fig_betafit} we display these radial profiles with the best-fit isothermal beta models
for both clusters. As is indicated by their X-ray image and cluster galaxy distribution,
Lynx-E has a higher concentration ($\beta=0.71\pm0.12$ and $r_c=13.2\pm3.2$) of the ICM
than Lynx-W ($\beta=0.42\pm0.07$ and $r_c=4.9\arcsec\pm2.8\arcsec$).
Together with the cluster temperatures determined in \textsection\ref{section_temperature}, these
structural parameters can be converted to the cluster mass under the assumption of hydrostatic
equilibrium. In general, many authors report cluster masses within a spherical volume rather
than a cylindrical volume spanning from the observer to the source plane, which however
is the preferred and natural choice in weak-lensing measurements. This different
geometry is often a source of confusion and subtlety in mass comparison between
both approaches. Therefore, in this paper we present our X-ray mass estimates in a cylindrical volume in order
to ensure more straightforward comparison with the weak-lensing result using the following equation (Paper II):
\begin{equation}
M_{ap}(r)= 1.78 \times 10^{14} \beta \left ( \frac{T}{\mbox{keV}} \right )
\left ( \frac{r}{\mbox{Mpc}} \right ) \frac{r/r_c}{\sqrt{1+(r/r_c)^2}} M_{\sun} \label{eqn_xray_mass_2d}
\end{equation}
\noindent
For Lynx-E we obtain $M(r\leq0.5~\mbox{Mpc})=2.3_{-0.4}^{+0.8}\times10^{14} M_{\sun}$, which is in
good agreement with our weak-lensing measurement.
On the other hand, the X-ray mass of Lynx-W ($M(r\leq0.5~\mbox{Mpc})=6.3_{-1.5}^{+2.6}\times10^{13} M_{\sun}$)
is much lower than the weak lensing estimation. We will discuss a few possible scenarios for this discrepancy
in \textsection\ref{summary}.
\section{COMPARISON WITH OTHER STUDIES}
The first attempt to estimate the mass of Lynx-W was made by Stanford et al. (1997)
using the X-ray luminosity from the ROSAT-PSPC observation and
the velocity dispersion obtained from the Keck spectroscopy of 8 galaxies.
They converted the luminosity $L_X\sim1.5\times10^{44} \mbox{ergs}~\mbox{s}^{-1}$ to
$M(r<2.3~ \mbox{Mpc})\sim7.8\times10^{14} M_{\sun}$ assuming $\beta=0.8$. A similar
value of $M(r<2.3 \mbox{Mpc})=5.4_{-2.3}^{+3.1}\times10^{14} M_{\sun}$ was estimated
from the velocity dispersion of $\sigma=700\pm180 \mbox{km}~\mbox{s}^{-1}$ (note
that they adopted $h_{100}=0.65$ and $q_0=0.1$).
Although both masses are consistent with each other,
their X-ray luminosity measurement seems to have suffered a severe contamination
from the neighboring point sources, which are now identified in the $Chandra$ observation.
In their presentation of the $Chandra$ analysis,
Stanford et al. (2001) did not attempt to estimate the mass of Lynx-W
because of the large uncertainty of the temperature measurement, as well as the apparent
asymmetry of the X-ray emission.
Our predicted velocity dispersion of $\sigma_{SIS}=762_{-133}^{+113}\mbox{km}\mbox{s}^{-1}$
from the SIS fit result is consistent with their most recent determination
of the velocity dispersion $\sigma=650\pm170~\mbox{km}~\mbox{s}^{-1}$ from the spectroscopic
redshifts of the 9 member galaxies. Lynx-W was also selected as one of the 28 X-ray clusters for
the study of the X-ray scaling relation at high redshifts by Ettori et al. (2004). From the
re-analysis of the $Chandra$ data, they obtained $\beta=0.97\pm0.43$, $r_c=163\pm70$~kpc, and $T_X=2.9\pm0.8$~keV, which
predicts a projected mass of $M(r\leq 0.5~\mbox{Mpc})= 3.0\pm1.5\times10^{14} M_{\sun}$ (eqn.~\ref{eqn_xray_mass_2d}).
This mass is consistent with our weak-lensing estimation ($2.0\pm0.5) \times 10^{14} M_{\sun}$, but
much higher than the value from our re-analysis of the same $Chandra$ data ($6.3_{-1.5}^{+2.6}\times10^{13} M_{\sun}$).
In general, many detailed steps in the $Chandra$ X-ray analysis such as the QE correction, background
modeling, flare removal, spectral aperture, etc. affect the final result, and much more
if the source is faint. Therefore, it is difficult, if not impossible, to trace the exact causes
of the differences. Nevertheless, we note that there is an important difference in the calibration
of the low-energy quantum efficiency correction between the results. Ettori et al. (2004) used
the ACISABS correction method (Chartas and Getman 2002) to account for the low-energy QE degradation, which is however
now officially disapproved by the $Chandra$ $Data$ $Center$. We also demonstrate in Paper II that
the use of this ACISABS model causes a difference of $\sim1$~keV in the temperature determination of MS1054-0321.
We suspect that the effect should be more important in Lynx-W because of
its low temperature and luminosity.
Stanford et al. (2001) obtained an X-ray temperature of $5.8_{-1.7}^{+2.8}$ keV
for Lynx-E. Combined with their determination of $\beta=0.61\pm0.12$ and
$r_c=11\arcsec.14\pm3\arcsec.41$, this gives
a projected mass of $M(r\leq0.5\mbox{Mpc})=3.1_{-0.9}^{+2.4} \times 10^{14} M_{\sun}$
(eqn.~\ref{eqn_xray_mass_2d}), which is slightly higher
than our X-ray re-analysis of the same $Chandra$ data by $\sim35$\% though the error bars from
both results marginally overlap. Vihklinin et al. (2002) included Lynx-E in their
sample of the 22 distant clusters to study the evolution the X-ray scaling relation.
With the early understanding of the low-energy QE problem of the $Chandra$, they obtained
$T_X=4.7\pm1.0$~keV, $r_c=167$~kpc, and $\beta=0.85\pm0.33$. Using the ACISABS correction. Ettori et al. (2004)
reported $T_X=5.2_{-1.1}^{+1.6}$~keV, $r_c=128\pm40$~kpc, and $\beta=0.77\pm0.19$.
The results from these two papers are statistically consistent with, but
slightly higher than our values ($T=3.8_{-0.7}^{+1.3}$keV, $r_c=111\pm27$, and $\beta=0.71\pm0.12$),
which predicts the lowest projected mass of $M(r\leq0.5\mbox{Mpc})=2.3_{-0.4}^{+0.8}\times10^{14} M_{\sun}$.
As already mentioned in the discussion of the Lynx-W temperature above, we suspect that
the difference in temperatures mainly stems from the different correction methods of the low-energy
QE degradation.
Although our understanding of the $Chandra$ instrument still evolves and this may
neccesitate some updates to our results,
it is encouraging to
note that this X-ray mass is closest to
our independent lensing determination of the cluster mass of
$M(r\leq0.5\mbox{Mpc})=(2.0\pm0.6)\times10^{14} M_{\sun}$ from the SIS fit result.
Our spectroscopic catalog currently provides the redshifts of 11 member galaxies
within a $r=80\arcsec$ radius (B. Holden in prep). Based on Tukey's biweight estimator,
we obtain a velocity dispersion of $720\pm140 \mbox{km}~\mbox{s}^{-1}$ (without
assuming a Gaussian distribution). This direct measurement agrees with the predicted
velocity dispersion of $740_{-134}^{+113}\mbox{km}\mbox{s}^{-1}$ from the lensing
analysis (\textsection\ref{section_mass_estimate}). In addition,
the cluster temperature $T_X=3.8_{-0.7}^{+1.3}$keV with $\beta=0.71$ is translated into
$\sigma_v=662_{-64}^{+106} \mbox{km}\mbox{s}^{-1}$ (from $\beta=\mu m_p \sigma_v^2/kT_X$),
in good agreement with both results.
\section{DISCUSSION AND CONCLUSIONS \label{summary}}
We have presented a weak-lensing analysis of the two Lynx clusters at $\bar{z}=1.265$
using the deep ACS $i_{775}$ and $z_{850}$ images. Our mass reconstruction
clearly detects the dark matter clumps associated with the two high-redshift
clusters and other intervening objects within the ACS field, including
the known foreground cluster at $z=0.57$.
In order to verify the significance of the cluster detection and
to separate the high-redshift signal from the low-redshift contributions,
we performed a weak-lensing tomography by selecting an alternate
lower-redshift source plane. This second mass reconstruction does not
show the mass clumps around the high-redshift clusters, while maintaining
most of the other structures seen in the first mass map. This experiment
strongly confirms that the weak-lensing signals observed in the
first mass reconstruction are real and come from the high-redshift Lynx clusters.
Interestingly, both clusters are found to have similar weak-lensing masses
of $\sim 2.0\times 10^{14} M_{\sun}$ within 0.5 Mpc ($\sim60\arcsec$) aperture radius
despite their discrepant X-ray properties. Our re-analysis of the Chandra archival data
with the use of the latest calibration of the low-energy QE degradation
shows that Lynx-E and W have temperatures of $T=3.8_{-0.7}^{+1.3}$ and
$1.7_{-0.4}^{+0.7}$~keV, respectively.
Combined with the X-ray surface brightness profile measurements, the X-ray temperature of Lynx-E
gives a mass estimate in good agreement with the weak-lensing result. On the other hand,
the X-ray mass of Lynx-W is much smaller than the weak-lensing estimation nearly by a factor of three.
According to our experiment in \textsection\ref{section_mass_estimate},
it is unlikely that any foreground contamination or cosmic shear effect in
weak-lensing measurement causes this large discrepancy.
Apart from a simplistic, but valid possibility that Lynx-W might have a filamentary structure extended along the line of sight,
yielding a substantial, projected mass but with yet only low-temperature thermal emission, we can also
consider the self-similarity breaking (e.g., Ponman et al. 1999; Tozzi \& Norman 2001; Rosati, Stefano, \& Norman et al. 2002)
typically observed for low-temperature X-ray systems. There have been quite a few suggestions that a non-gravitational heating (thus extra
entropy) might prevent the ICM from further collapsing at the cluster core.
The effect is supposed to be more pronounced
in colder systems whose virial temperature is comparable to the temperature created by this non-gravitational heating,
leading to shallower gas profiles than
those of high-temperature systems (e.g., Balogh et al. 1999; Tozzi \& Norman 2001).
Interestingly, our determination of the surface brightness profile of Lynx-W is much shallower ($\beta=0.42\pm0.07$) than that
of Lynx-E ($\beta=0.71\pm0.12$) (however, Ettori et al. (2004) obtained $\beta=0.97\pm0.43$ for Lynx-W).
The relatively loose distribution of the cluster galaxies in Lynx-W without any apparent BCG defining the cluster center
leads us to consider another possibility that the system might be dynamically young and the ICM
has not fully thermalized within the potential well. If we imagine that
the ICM is not primordial, but has been ejected from the cluster galaxies at some recent epoch,
it is plausible to expect that the X-ray temperature of the ICM might yet under-represent the depth of the cluster
potential well. Tozzi et al. (2003) investigated the iron abundance in the ICM
at $0.3<z<1.3$ and argued that the result was consistent with no evolution of the mean iron abundance
out to $z\simeq1.2$. If we assume that, as they suggested, Type Ia SNe are the dominant sources of
this iron enrichment and have already injected their metals into the ICM by $z\sim1.2$,
a significant fraction of clusters at $z\gtrsim 1.2$ may possess dynamically young ICM.
Recently, Nakata et al. (2005) reported with a photometric redshift technique
the discovery of seven other cluster candidates around
these two Lynx clusters possibly forming a $z\sim1.3$ supercluster. Although
further evidence is needed that the individual clumps are dynamically
bound, the clear enhancement of the red galaxies consistent with the color
at the redshift of the two known Lynx clusters is worthy of our attention.
If they are indeed found to be forming groups/clusters at $z\sim1.3$, but
missed by X-ray observations because of their low X-ray contrast,
the detailed studies
of these young high-redshift structures will provide a critical benchmark
in testing our understanding of the structure formation as well as the individual galaxy
evolution in the context of different environments.
Deep two band ($i_{775}$ and $z_{850}$ $HST$/ACS imaging of the five out of the seven group/cluster candidates of
Nakata el al. (2005) are scheduled in $HST$ Cycle 14 (Prop. 10574, PI. Mei). Studies
similar to the current investigation will not only test
whether there exist dark matter clumps around the candidate galaxies, but also
quantify the environments for the investigation of the cluster galaxy
color/morphology evolution.
ACS was developed under NASA contract NAS5-32865, and this research was supported
by NASA grant NAG5-7697. We are grateful for an equipment
grant from Sun Microsystems, Inc.
Some of the data presented herein were obtained at the W.M. Keck
Observatory, which is operated as a scientific partnership among the
California Institute of Technology, the University of California and
the National Aeronautics and Space Administration. The Observatory was
made possible by the generous financial support of the W.M. Keck
Foundation. The authors wish to recognize and acknowledge the very
significant cultural role and reverence that the summit of Mauna Kea
has always had within the indigenous Hawaiian community. We are most
fortunate to have the opportunity to conduct observations from this
mountain.
\clearpage
|
Title:
Radial velocity measurements of B stars in the Scorpius-Centaurus association |
Abstract: We derive single-epoch radial velocities for a sample of 56 B-type stars
members of the subgroups Upper Scorpius, Upper Centaurus Lupus and Lower
Centaurus Crux of the nearby Sco-Cen OB association. The radial velocity
measurements were obtained by means of high-resolution echelle spectra via
analysis of individual lines. The internal accuracy obtained in the
measurements is estimated to be typically 2-3 km/s, but depends on the
projected rotational velocity of the target. Radial velocity measurements taken
for 2-3 epochs for the targets HD120307, HD142990 and HD139365 are variable and
confirm that they are spectroscopic binaries, as previously identified in the
literature. Spectral lines from two stellar components are resolved in the
observed spectra of target stars HD133242, HD133955 and HD143018, identifying
them as spectroscopic binaries.
| https://export.arxiv.org/pdf/astro-ph/0601643 |
\title{Radial velocity measurements of B stars in the Scorpius-Centaurus association}
\author{E.\,Jilinski\inst{1,2}, S. Daflon\inst{1}, K.\,Cunha \inst{1} and
R.\,de la Reza \inst{1} }
\offprints{E.\,Jilinski, Observat\'orio Nacional/MCT, Rua Gal. Jose
Cristino 77, S\~ao Cristov\~ao, Rio de Janeiro, Brazil.
e-mail: jilinski@on.br}
\institute{Observat\'orio Nacional/MCT, Rio de Janeiro, Brazil
\and
Main Astronomical Observatory, Pulkovo, St. Petersburg, Russia}
\date{}
\authorrunning{E.\,Jilinski et al.}
\titlerunning{Radial velocities in the Sco-Cen association}
\abstract{We derive single-epoch radial velocities for a sample of
56 B-type stars members of the subgroups Upper Scorpius, Upper
Centaurus Lupus and Lower Centaurus Crux of the nearby Sco-Cen OB
association. The radial velocity measurements were obtained by
means of high-resolution echelle spectra via analysis of
individual lines. The internal accuracy obtained in the
measurements is estimated to be typically 2-3 km\,s$^{-1}$, but
depends on the projected rotational velocity of the target. Radial
velocity measurements taken for 2-3 epochs for the targets
HD120307, HD142990 and HD139365 are variable and confirm that they
are spectroscopic binaries, as previously identified in the
literature. Spectral lines from two stellar components are
resolved in the observed spectra of target stars HD133242,
HD133955 and HD143018, identifying them as spectroscopic binaries.
\keywords{stars: early-type - stars: binaries: spectroscopic -
stars:
kinematics - stars: radial velocities in open clusters and associations:
individual: Scorpius-Centaurus association. }
}
\section{Introduction}
The Scorpius-Centaurus association is the nearest association of
young OB stars to the Sun. Blaauw (1960, 1964) divided this
association into three stellar subgroups: Upper Scorpius (US),
Upper Centaurus Lupus (UCL) and Lower Centaurus Crux (LCC). LCC
and UCL have roughly similar ages of about $16-20$ Myr, while US
is younger with an estimated age of $\sim$ 5 Myr (Mamajek et
al.2002; Sartori et al. 2003). This complex OB association of
unbound stars is of great interest because, as recently shown, it
is related to the origins of nearby moving groups of low mass
post-T Tauri stars with ages around 10 Myr: the $\beta$ Pictoris
Moving Group, the TW Hydra association, and the $\eta$ and
$\epsilon$ Chamaleonis groups (Mamajek et al. 2000; Ortega et al.
2000, 2004; Jilinski et al. 2005). In addition, the
Scorpius-Centaurus association also appears to be the source of a
large bubble of hot gas in which the Sun is plunged. All these
structures are believed to have been possibly triggered by
supernova explosions taking place in UCL and LCC during the last
~13 Myr (Ma\'iz-Apell\'aniz 2001).
The technique adopted for investigating the origins of the $\beta$
Pictoris Moving Group, for example, consists of tracing back the
3-D stellar orbits of the members of these moving groups until
their main first orbits confinement was found, as well as the past
mean positions of LCC and UCL. This enabled, investigator not only
to determine the dynamical age of this moving group, but also to
investigate properties of their birth clouds (Ortega et al. 2002,
2004). It is also possible to find the past positions of the
possible supernovae that triggered the formation of these groups
by tracing back the orbit of a runaway OB star, which could have
been the result of a supernova explosion in LCC or UCL (see, for
example, Hoogerwerf et al. 2001 and Vlemmings et al. 2004).
While the past evolution of these moving groups of low mass stars
appears to be a relatively simple problem (as the dynamical ages
are not so old), the dynamical evolution of the older and more
numerous subgroups LCC and UCL appears to be more difficult. There
is the possibility of the presence of several generations of hot
stars during the mainstream of the OB association evolution
(Garmany 1994).
Substructure in LCC and UCL was found by de Bruijne (1999), based
on Hipparcos data. The formation of the younger US subgroup could
have been triggered by UCL some 6-8 Myr ago (Preibisch et al.
2001). All these studies require reliable radial velocities in
order to calculate space velocities. In this paper we present
single-epoch radial velocity (RV) measurements for 56 B-type stars
members of LCC, UCL and US subgroups, to contribute to studies of
their dynamics so as to unravel their origins.
\section{Observations and reduction}
A sample of 56 B-type stars from the Scorpius-Centaurus
association was observed during observing runs in May 16-20 and
July 7, 2002, with the 1.52m telescope equipped with the FEROS
echelle spectrograph (Kaufer et al. 2000; resolving power
R=48,000, wavelength coverage between 3900 and 9200\AA.) with a
CCD detector at the European Southern Observatory (ESO) \footnote
{Observations obtained under the ON/ESO agreement}. The target
stars were selected from the list in Humphreys \& McElroy (1984)
and from the comprehensive study of OB associations based on
Hipparcos observations by de Zeeuw et al. (1999). The observed
targets are listed in Table~1. From this sample, according to de
Zeeuw et al. (1999), 15 targets are confirmed members of the LCC,
while 15 stars are members of the UCL and 11 stars are from the
US subgroup. For the remaining 15 stars in our sample, membership
to any of these subgroups was not certain.
The spectra were reduced with the MIDAS reduction package and
consisted of the following standard steps: CCD bias correction,
flat-fielding, extraction, wavelength calibration, correction of
barycentric velocity, as well as spectrum rectification and
normalization. The one-dimensional spectra were then treated by
tasks in the NOAO/IRAF data package. The signal-to noise ratio
obtained in the observed spectra was typically larger than 100 and
typical exposure times varied between 300 seconds for the
brightest stars (V $\sim$ 3) and 1200 seconds for stars with V
$\sim$ 5. Typical spectra are shown in Figure~1 . The top panel
corresponds to the target star HD~122980 and the bottom panel to
HD~112092. Both stars have sharp lines with projected rotational
velocities ($v \sin i$) less than ~40 km\,s$^{-1}$. The spectral
region displayed shows identifications of several lines that were
used in the RV determinations. The FEROS bench spectrograph and
set up have proven to have high spectral stability for RV
measurements as concluded from a study of radial-velocity standard
stars: a r.m.s. of 21 m\,s$^{-1}$ has been obtained for a data set
of 130 individual measurements (Kaufer et al. 2000).
\section{Analysis and discussion}
The cross-correlation technique, which is used for precise RV
determinations in later type stars, when applied to the hotter OB
stars can be problematic as early type star spectra show few
absorption lines. These lines are in many cases, intrinsically
broad (up to a few hundreds km\,s$^{-1}$) due to stellar rotation.
In addition, there is also the possibility of line variability
affecting their line profiles (Steenbrugge et al. 2003).
Therefore, the cross-correlation peak that defines the value of
radial velocity can be very broad and contain important
sub-structures caused by blending of spectral lines that appear to
have different widths. In addition to having high $v \sin i$
values many OB stars are binary and it is not straightforward to
apply the cross-correlation method and to identify them as
double-lined binaries; in order to obtain the orbital solution a
long set of observations is needed. Detailed cross-correlation
technique analyses applied to determinations of radial velocities
of early-type stars has been presented in a number of recent
publications (see, for example, Verschueren et al. 1997;
Verschueren et al. 1999a; Griffin et al. 2000). Griffin et al.
(2000), in particular, discuss in detail the difficulties in
obtaining accurate RV measurements from cross-correlation in
early-type stars spectra.
In this study, having high-resolution observations covering a
large spectral range, radial velocity values for the target stars
were obtained from measurements of the positions of individual
spectral lines of He I, C II, N II, O II, Mg II, Si II and Si III,
relative to their rest wavelengths. Radial-velocity standard stars
were not observed. (The adopted linelists can be found in Daflon
et al. 2001, 2003.) We inspected and identified all unblended
lines visible in the spectral range between 3798 \AA\ (H\,{\sc i})
and 7065 \AA\ (He\,{\sc i}) in each target star: the number of
measurable lines varied between 10 and 74, depending on the star
spectral type, rotation velocity, possible multiplicity, but also
on the signal-to-noise of the obtained spectra. (We note, however,
that for the double lined binary HD 133242, it was possible to
measure positions only for 4 lines in component A and 6 lines in
component B.) Mean radial velocities using all measurable lines
(${\rm RV}$) and respective dispersions were calculated for the
individual target stars.
In Table~1 we assemble our RV results as well as results from the
literature. In the two first columns of this table we list the HD
numbers of the observed stars with the respective spectral types;
in the columns 3 and 4 we list the heliocentric Julian Date (HJD)
and the measured radial velocities, plus the number of measured
lines in brackets. In the other columns we list results from the
literature: columns 5 and 6 list the RV$_{\rm GCRV}$ and
associated error or quality, from the General Catalogue of Radial
Velocities (GCRV; quality flags A to E, or I for insufficient
data); column 7 presents the projected rotational velocity from
Brown \& Verschueren (1997) and, when not available in this
source, the $v\,sin i$ was taken from the compilation of Glebocki
\& Stawikowski (2000); in column 9 we list, when available,
literature references where information about duplicity can be
found for the stars. The stars are separated according to the
different subgroups in the Scorpius-Centaurus association,
following the membership probabilities P(m) listed by de Zeeuw et
al. (1999).
The internal precision of our RV determinations can be represented
by the scatter obtained from the RV measurements line-by-line,
which is listed in column 4 of Table 1. These are typically
smaller than $\sim$2.0 ${\rm km\,s}^{-1}$ for stars with estimated
$v\,sin i$ smaller than 100 ${\rm km\,s}^{-1}$. We note, however,
that when the target $v\,sin i$ are large, the uncertainties in
the derived RVs can be significantly larger due to uncertainties
in defining the line center. This can be seen in Figure~2, where
we show the obtained line-to-line scatter versus the target
projected rotational velocity (as taken from Brown \& Verschueren
1997 and Glebocki \& Stawikowski 2000).
In order to evaluate possible systematic effects that line
selection could have on the RV results, we selected a homogeneous
set of 28 spectral lines of H\,{\sc i}, He\,{\sc i}, Si\,{\sc
iii} and Mg\,{\sc ii}, that could be measured in most of the
studied spectra, and recalculated the mean radial velocity values
for all possible stars. A comparison of the mean radial velocities
${\rm RV}$ with ${\rm RV}_{\rm 28lines}$ (obtained using only the
selected 28 lines) indicates that there are non-significant
systematic differences between the two determinations ${\rm RV} -
\, {\rm RV}_{\rm 28lines}= -1.0 {\rm km\,s}^{-1}$, with $\sigma =
3.1 {\rm km\,s}^{-1}$.
\subsection{Membership}
Table~1 lists the target stars according to their membership the 3
subgroups as assigned by de Zeeuw et al. (1999). Most of the stars
in the Lower Centaurus Crux subgroup are flagged as binaries in
the literature, except for HD103079, HD106490 and HD108483. For
these 3 stars we measured radial velocities of $19.3 {\rm
km\,s}^{-1}$, $15.3 {\rm km\,s}^{-1}$ and $12.8 {\rm km\,s}^{-1}$,
respectively, with RV$_{mean}$=15.8 $\pm$3.2 ${\rm km\,s}^{-1}$.
This mean value is in general agreement with the mean radial
velocity calculated by de Zeeuw et al. (1999) for LCC, which is of
$12 {\rm km\,s}^{-1}$. For the subgroup UCL, our sample has 2
non-binary stars (HD121790 and HD128345) and RV$_{mean}$=$9.6
{\rm km\,s}^{-1}$ which is $\sim 5 {\rm km\,s}^{-1}$ higher than
the de Zeeuw et al. (1999) mean value of $4.9 {\rm km\,s}^{-1}$.
For the Upper Scorpius subgroup all the sample stars have been
flagged as binaries in the literature.
For the five target stars that had not been identified as members
of any of the three subgroups in the Sco-Cen association (listed
as "others" in Table~1) and for which we have no information on
duplicity, we can attempt to discuss their membership status based
on the comparison of the radial velocities measured here and in
the literature. We find that the measured radial velocities for
HD109026 (RV=4.0 ${\rm km\,s}^{-1}$) and HD110335 (RV= 4.8 ${\rm
km\,s}^{-1}$) are consistent with the mean radial velocity for UCL
of 4.9 ${\rm km\,s}^{-1}$s. For the target star HD109026 we have
RV$_{GCRV}$= 2.5 ${\rm km\,s}^{-1}$, therefore it could be
considered initially as having constant RV (within the
uncertainties) and possibly a member of the Upper Centaurus Lupus
subgroup. For HD110335, we find a larger discrepancy between our
measurement (RV=4.8 ${\rm km\,s}^{-1}$) and the RV value in the
GCRV (RV$_{GCRV}$=12.5 ${\rm km\,s}^{-1}$). The RVs, however, are
marginally consistent given the expected uncertainty brackets that
affect the 2 determinations. If this is really the case, HD110335
could be also considered as a possible member of the UCL subgroup.
In addition, the target star HD120640 (RV$_{mean}$=-1.8 ${\rm
km\,s}^{-1}$ from this study and RV$_{GCRV}$=-4.7 ${\rm
km\,s}^{-1}$) can be assumed here to have constant radial
velocity. These measurements are consistent with the mean RV value
of -4.6 ${\rm km\,s}^{-1}$ listed by de Zeeuw et al. (1999) for
this subgroup.
The two other stars in our sample of 'others' (HD115846 and
HD105937) for which we derived RV= -21.8 ${\rm km\,s}^{-1}$ and
RV=22.7 ${\rm km\,s}^{-1}$, respectively, have values in the GCRV
of RV$_{GCRV}$=3.0 ${\rm km\,s}^{-1}$ and RV$_{GCRV}$=15.0 ${\rm
km\,s}^{-1}$. We found no information in the literature about
these stars being confirmed binary stars, but the variation in RV
for HD115846 exceeds the expected uncertainties: this target
probably has a non-constant RV, which prevents further
considerations about it belonging to any of the Sco-Cen subgroups.
HD105937 has an RV only marginally constant within the
uncertainties, but its mean RV is not compatible with any of the
subgroups.
\subsection{Duplicity}
Results from a search for duplicity information for the targets
stars (column 9; Table~1) indicate that a large number of stars in
our sample are flagged as binaries in the literature. For most of
these targets we have only one single-epoch RV measurement and our
results alone cannot be used to infer duplicity. However, the RVs
derived in this study can be added to RV databases and contribute
to long term studies of their orbits. Only a small number of stars
had not been previously flagged as binaries in the different
studies in the literature. For this subsample of 10 stars,
considered a priori as RV constants, it is possible to compare our
RV values with the averaged radial velocities assembled in the
GCRV. This comparison is shown in Figure~3. Our RV determinations
compare favorably with the RVs from the GCRV with a scatter of the
order of the estimated uncertainties. ${\rm RV}_{\rm GCRV} - {\rm
RV} = 0.7$ and $\sigma = 4.9 {\rm km\,s}^{-1}$. (This was
calculated excluding one discrepant star, HD115846, which could be
a binary system.) Taking into account the mean precision of RV
determinations from the GCRV as $\pm 5 {\rm km\,s}^{-1}$, the
external precision of our RV determinations may be evaluated as
approximately $\pm 5 {\rm km\,s}^{-1}$.
For those targets with more than one epoch RV measurement in our
study, a subsample showed radial velocity variations larger than
the expected uncertainties: HD120307, HD142990 and HD139365. Since
these had been previously identified as SBs in the literature
(Levato et al. 1987 and Batten et al. 1989), our results confirm
their duplicity. Four other stars with multiple epoch observations
in this study showed a constant RV within the uncertainties:
HD116087, HD130807, HD132200 and HD120640.
For 3 targets in our sample (HD133242, HD133955 and HD143018) we
were able to separate and identify lines of two stellar
components, classifying them as double lined spectroscopic
binaries. Their combined spectra showing spectral lines from two
stars are shown in Fig~4. Two of these stars (HD143018 and
HD133955) were previously identified in Batten et al. (1999) as a
spectroscopic binaries.
\begin{acknowledgements}
We thank the anonymous referee for suggestions that significantly
improved the paper. E.G.J. thanks FAPERJ and MCT Brazil for
financial support.
\end{acknowledgements}
{}
\newpage
\begin{table}
\begin{center}
\begin{tabular}{llcrcccc}
\multicolumn{8}{c}{Table 1. Radial velocities of observed Sco-Cen OB stars}\\
\hline
\hline
\,\,\, HD &\,\,\,\,Sp& HJD &RV\,[N]\,\,\,\,\,\,\,\,& RV$_{\rm GCRV}$\,\,\, & eRV* & V$\sin$ i & Duplicity \\
& & & (km\,s$^{-1}$)\,\,\,\,\,\, & (km\,s$^{-1}$) & (km\,s$^{-1}$) & (km\,s$^{-1}$) & \\
\hline
\multicolumn{8}{c}{Lower Centaurus Crux} \\
\,\,\,98718&B5Vn & 52411.0285 & 26.8$\pm$2.7 [19] & 9.4 & C & 340$^a$ & ST \\
103079 & B4V & 52411.0425 & 19.3$\pm$1.0 [58] & 20.6 & B & 47$^b$ & \\
105382 & B6IIIe & 52413.0429 & 15.8$\pm$1.1 [50] & 16.4 & B & 75$^b$ & ST \\
106490 & B2IV & 52413.0617 & 15.3$\pm$1.5 [35] & 22.0 & B & 135$^b$ & \\
106983 & B2.5V & 52411.0787 & 12.1$\pm$0.8 [50] & 15.8 & A & 65$^b$ & VDH \\
108257 & B3Vn & 52411.0875 & 19.9$\pm$2.7 [19] & 5.0 & C & 298$^b$ & VDH \\
108483 & B2V & 52411.1008 & 12.8$\pm$1.4 [29] & 8.0 & C & 169$^b$ & \\
109668 & B2IV-V & 52411.1127 & $-$0.1$\pm$1.4 [38] & 13.0 & C & 114$^b$ & ST \\
110879 & B2.5V & 52411.1177 & 56.9$\pm$2.5 [33] & 42.0 & D & 139$^b$ & VDH \\
110956 & B3V & 52411.1229 & 15.6$\pm$0.7 [69] & 16.4 & B & 26$^b$ & VDH \\
112091 & B5Vne & 52411.1449 & 15.9$\pm$1.9 [19] & 13.0 & C & 242$^b$ &VDH\\
112092 & B2IV-V & 52411.1389 & 14.4$\pm$0.6 [72] & 13.9 & A & 34$^b$ & VDH \\
113703 & B5V & 52415.1623 & 1.2$\pm$1.4 [29] & 6.0 & C & 140$^b$ & VDH,ST \\
113791 & B1.5V & 52411.1615 & 58.5$\pm$0.8 [72] & 14.3 & C & 15$^b$ & SB8 \\
116087 & B3V & 52411.2216 & 12.3$\pm$1.6 [21] & 6.0 & C & 233$^b$ & VDH,ST \\
& & 52414.1034 & 9.4$\pm$3.9 [10] & & & & \\
\multicolumn{8}{c}{Upper Centaurus Lupus} \\
120307 & B2IV & 52413.1160 & 25.4$\pm$0.9 [54] & 9.1 & & 65$^b$ & SB8,L87\\
& & 52413.1562 & 7.4$\pm$0.8 [39] & & & & \\
& & 52414.1143 & 11.0$\pm$0.7 [31] & & & & \\
121743 & B2IV & 52414.1453 & 9.6$\pm$0.8 [57] & 5.3 & 1.4 & 79$^b$ & L87\\
121790 & B2IV-V & 52414.1523 & 9.2$\pm$1.4 [35] & 4.8 & B & 124$^b$ & \\
122980 & B2V & 52414.1594 & 10.5$\pm$0.6 [74] & 9.6 & 2.8 & 15$^b$ & L87 \\
128345 & B5V & 52414.1828 & 9.9$\pm$1.9 [25] & 8.0 & D & 186$^b$ & \\
129056 & B1.5III& 52414.1909 & 18.3$\pm$0.8 [72] & 5.4 & 0.6 & 16$^b$ & VDH\\
130807 & B5IV & 52414.1957 & 7.1$\pm$0.6 [67] & 7.3 & A & 27$^b$ & VDH,ST\\
& & 52414.2150 & 7.2$\pm$0.5 [56] & & & & \\
132200 & B2IV & 52414.2097 & 4.6$\pm$0.6 [70] & 8.0 & 0.9 & 32$^b$ &VDH,L87,ST\\
& & 52414.2266 & 4.9$\pm$0.6 [55] & & & & \\
133242A\,& B5IV & 52414.2599 &$-$52.5$\pm$0.7 [4] & 4.5 & C & 140$^a$ & VDH\\
133242B\,& & & 81.5$\pm$2.1 [6] & & & & \\
133955A\,& B3V & 52414.2696 & 61.2$\pm$1.4 [10] & 9.8 & B & 135$^b$ & SB8\\
133955B\,& & &$-$31.6$\pm$1.0 [10] & & & & \\
134687 & B3IV & 52414.2777 & 23.7$\pm$0.8 [74] & 13.5 & D & 13$^b$ & SB8 \\
136504 & B2IV-V & 52414.2904 & $-$5.7$\pm$1.1 [60] & 7.9 & C & 41$^b$ & SB8,ST \\
137432 & B4Vp & 52414.2976 & 6.3$\pm$1.0 [41] & $-$0.8 & E & 77$^b$ & VDH,SB8 \\
139365 & B2.5V & 52414.1535 & 33.3$\pm$2.3 [23] &$-$14.0 & E & 134$^b$ & SB8 \\
& & 52414.3140 & $-$5.8$\pm$1.1 [25] & & & & \\
140008 & B5V & 52414.3210 & 2.6$\pm$0.8 [56] & 3.9 & B & 11$^b$ & SB8,ST \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}
\begin{center}
\begin{tabular}{llcrcccc}
\multicolumn{8}{c}{Table 1. Continued} \\
\hline
\hline
\,\,\, HD &\,\,\,\,Sp& HJD &RV\,[N]\,\,\,\,\,\,\,\,& RV$_{\rm GCRV}$\,\,\, & eRV* & V $\sin$ i & Duplicity \\
& & & (km\,s$^{-1}$)\,\,\,\,\,\, & (km\,s$^{-1}$) & (km\,s$^{-1}$) & (km\,s$^{-1}$) & \\
\hline
\multicolumn{8}{c}{Upper Ssorpius} \\
142669 & B2IV-V & 52415.2944 & 2.5$\pm$1.0 [41] & 3.3 & E & 98$^b$ & SB8 \\
142883 & B3V & 52415.3002 &$-$54.3$\pm$0.5 [70] &$-$27.5 & E & 14$^b$ & SB8 \\
142990 & B5V & 52481.0065 & $-$5.6$\pm$3.5 [33] &$-$12.1 & 3.4 & 178$^b$ & L87 \\
& & 52481.1270 &$-$10.9$\pm$2.9 [34] & & & & \\
143018A\,& B1V+ & 52414.3559 & 113.8$\pm$2.8 [25] &$-$11.7 & D & 100$^b$ & SB8 \\
143018B\,& & &$-$173.6$\pm$3.2 [21]& & & & \\
144217 & B0.5V & 52414.3646 & 9.1$\pm$1.3 [43] & $-$1.0 & & 91$^b$ &SB8,VDH,L87,ST\\
144470 & B1V & 52415.2873 & $-$0.6$\pm$1.1 [39] & $-$4.4 & 3.0 & 100$^b$ & L87 \\
147165 & B1III & 52414.3498 &$-$25.6$\pm$1.2 [50] & 2.5 & & 56$^b$ &SB8,L87,ST\\
147888 & B3/B4V & 52481.1481 & $-$3.8$\pm$2.3 [27] & $-$6.8 & 2.9 & 175$^a$ & L87 \\
147932 & B5V & 52481.2006 & $-$2.8$\pm$2.7 [23] &$-$11.0 & 2.4 & 186$^a$ & L87 \\
148184 & B2Vne & 52481.1797 & $-$4.7$\pm$2.1 [34] &$-$19.0 & & 148$^b$ &SB8,L87\\
149438 & B0V & 52414.3446 & 1.6$\pm$0.8 [43] & 1.7 & 0.8 & 10$^b$ & L87 \\
\multicolumn{8}{c}{Other} \\
104841 & B2IV & 52411.0594 & $-$8.9$\pm$0.5 [73] & 16.1 & I & 25$^b$ & SB8 \\
105435 & B2IVne & 52411.0725 & 3.8$\pm$2.8 [15] & 11.0 & C & 298$^b$ & VDH\\
105937 & B3V & 52413.0558 & 22.7$\pm$1.5 [33] & 15.0 & C & 129$^b$ & \\
109026 & B5V & 52411.1069 & 4.0$\pm$1.6 [31] & 2.5 & D & 188$^b$ & \\
110335 & B6IVe & 52481.0996 & 4.8$\pm$1.8 [23] & 12.5 & B & 250$^a$ & \\
111123 & B0.5IV & 52411.1357 & 9.8$\pm$0.7 [53] & 10.3 & A & 40$^b$ & VDH\\
115846 & B3IV & 52481.97 &$-$21.8$\pm$1.6 [27] & 3.0 & 4.0 & 168$^b$ & \\
116072 & B2.5Vn & 52411.1714 & 18.8$\pm$2.5 [21] & 3.0 & C & 233$^b$ & VDH\\
118716 & B1III & 52413.1107 & 14.0$\pm$1.2 [32] & 3.0 & B & 114$^b$ & VDH\\
120640 & B2Vp & 52413.1209 & $-$2.1$\pm$0.8 [63] & $-$4.7 & 0.8 & 21$^b$ & \\
& & 52414.1219 & $-$1.5$\pm$0.6 [49] & & & & \\
126341 & B2IV & 52414.1712 &$-$21.1$\pm$1.0 [73] &$-$21.5 & B & 15$^b$ & VDH \\
132058 & B2III & 52414.2050 & 0.1$\pm$1.0 [48] & 0.2 & 0.9 & 92$^b$ & L87 \\
132955 & B3V & 52414.2424 & 5.1$\pm$0.5 [70] & 3.7 & 2.1 & 8$^b$ & VDH \\
144218 & B2V & 52414.3716 & 0.6$\pm$0.8 [57] & $-$5.6 & 0.8 & 56$^b$ &VDH,L87,ST,SB8\\
151985 & B2IV & 52414.3352 & 1.9$\pm$0.7 [56] & 1.3 & 0.7 & 52$^b$ & L87,SB8 \\
\hline
\end{tabular}
\end{center}
\begin{list}{}{}
\item * A: errors $\le$ 2.5 ${\rm km\,s}^{-1}$; B: 2.5 $<$ errors $\le$ 5.0 ${\rm km\,s}^{-1}$;
C: 5.0 $<$ errors $\le$ 10.0 ${\rm km\,s}^{-1}$;
D: errors $\ge$ 10 ${\rm km\,s}^{-1}$;
E: too uncertain (from Table 3 of Barbier-Brossat \& Figon \cite{bbf00})
\item a: from Glebocki \& Stawikowski (\cite{gle00}) and
b: from Brown \& Verschueren (\cite{ver97})
\item SB8 - Eighth Orbital Elements of Spectroscopic Binaries (Batten et al. \cite{bat89})
\item L87 - Levato et al. (1987)
\item VDH - Visual Double Stars in Hipparcos (Dommanget \& Nys \cite{dom00})
\item ST - Shatsky \& Tokovinin (2002)
\end{list}
\end{table}
|
Title:
Quantifying the Luminosity Evolution in Gamma-ray Bursts |
Abstract: We estimate the luminosity evolution and formation rate for over 900 GRBs by
using redshift and luminosity data calculated by Band, Norris, $&$ Bonnell
(2004) via the lag-luminosity correlation. By applying maximum likelihood
techniques, we are able to infer the true distribution of the parent GRB
population's luminosity function and density distributions in a way that
accounts for detector selection effects. We find that after accounting for data
truncation, there still exists a significant correlation between the average
luminosity and redshift, indicating that distant GRBs are on average more
luminous than nearby counterparts. This is consistent with previous studies
showing strong source evolution and also recent observations of under luminous
nearby GRBs. We find no evidence for beaming angle evolution in the current
sample of GRBs with known redshift, suggesting that this increase in luminosity
can not be due to an evolution of the collimation of gamma-ray emission. The
resulting luminosity function is well fit with a single power law of index
$L'^{-1.5}$, which is intermediate between the values predicted by the
power-law and Gaussian structured jet models. We also find that the GRB
comoving rate density rises steeply with a broad peak between $1<z<2$ followed
by a steady decline above $z> 3$. This rate density qualitatively matches the
current estimates of the cosmic star formation rate, favoring a short lived
massive star progenitor model, or a binary model with a short delay between the
formation of the compact object and the eventual merger.
| https://export.arxiv.org/pdf/astro-ph/0601146 |
\title{Quantifying the Luminosity Evolution in Gamma-ray Bursts}
\author{Dan Kocevski \altaffilmark{1} and Edison Liang\altaffilmark{}}
\altaffiltext{1}{Physics Department, University of California, Berkeley, Berkeley, Ca 94709 }
\altaffiltext{2}{Department of Physics and Astronomy, Rice University, Houston, Tx 77005 }
\email{kocevski@rice.edu}
\email{liang@spacibm.rice.edu}
\keywords{gamma rays: bursts---gamma rays: theory}
\section{Introduction} \label{sec:Introduction_ch3}
There are currently roughly three dozen gamma-ray burst events
(GRBs) for which we have independently measured redshifts. Most of
these redshift determinations come from either identification of
absorption lines in the afterglow spectra, attributed to the gas in
the host galaxy, or from observations of emission lines from the
host galaxy. The combination of these techniques has resulted in a
small but growing GRB sample with redshifts ranging from 0.0085 to
4.5 and a distribution peaking around $z \sim 1$. From this small
sample, it is already abundantly clear that the isotropic equivalent
energy $E_{iso}$ released in the prompt GRB phase is not a standard
candle. The total radiated energy taken at face value (i.e. when not
correcting for a beaming factor $d\Omega$) clearly spans several
orders of magnitude, ranging from $10^{47}$ for the closest event,
GRB 980425 at $z=0.0085$ \citep{Kulkarni98}, to $10^{54}$ for GRB
990123 at $z=1.6004$ (Kulkarni et al. 1999). Recently
\citet{Sazonov04} and \citet{Soderberg04} have reported on gamma-ray
observations of a nearby underluminous GRB occurring at redshift of
$z=0.106$. These new findings have added to the speculation that
there is either a substantial under luminous population of GRBs
which cannot be seen at large distances and/or that nearby events
($z <$ 0.15) are underluminous compared to distant counterparts,
pointing to the evolution of the average energy emitted by a GRB
with time.
A measure to the extend to which luminosity evolution exists in the
GRB population, along with their true luminosity function and
density distribution, may yield important clues regarding the nature
of gamma-ray bursts and how they're progenitors have evolved with
time. Although the physics of the underlying GRB engine is hidden
from direct observation and is yet uncertain, the total GRB energy
budget is most likely linked to the mass and/or rotational energy of
the GRB progenitor. Understanding of how this energy budget has
changed with time may offer constraints on progenitor properties and
may ultimately point to the physics leading to their explosions.
Since GRB progenitors are most likely linked to compact objects
(supermassive rotating star, black hole or neutron star mergers)
understanding how the GRB luminosity function evolves with time may
give insight to the host environment in the early universe, namely
the star formation rate or initial stellar mass functions at high
redshifts.
Any attempt at quantifying the evolution of intrinsic source
properties of parent populations must account for Malquist type
biases. Detection thresholds prevent events below a certain flux
from being observed, resulting in the detection of only bright
objects at large distances. Combined with the fact that bright
events are typically rare, it is very easy in astronomy to
incorrectly conclude that the distant universe is filled with
extremely bright rare objects. Any attempt at measuring the
correlation between luminosity and redshift without properly
accounting for selection effects will grossly overestimate the
correlation strength between the two variables. Flux limited samples
are a classic problem in astronomy, which manifested prominently in
early quasar studies. Fortunately, straight forward methods have
been devised to account for such effects based on maximum likelihood
techniques. These methods allow for the correct estimation of the
correlation strength between a truncated data set as well as an
estimate on the underlying parent population. The "catch" of such
techniques is that the overall normalization of the resulting parent
distributions cannot be determined, although their functional forms
are constructed in such a way to account for the data truncations.
These techniques also have to limitation of requiring a large sample
sizes and more importantly, an extremely good understanding of the
survey's detection thresholds (i.e. the flux cutoff for magnitude
limited samples). The use of the current sample of GRBs with known
redshift is limited by both of these restrictions. The current size
of a little over two dozen bursts does not lend itself well to
producing statistically robust results, especially in the high and
low redshift regimes for which only a handful of events have been
detected. Furthermore, the sample is an accumulation of
observations from several different spacecraft, all of varying
detector thresholds. It would seem that these limitations could
only be overcome by the accumulation of a larger data set with
consistent detector thresholds which is expected to come from the
Swift spacecraft and the upcoming GLAST mission.
Fortunately, several authors have announced empirical Cepheid like
correlations linking intrinsic burst properties, such as luminosity
\citep{norris00} and the total radiated energy
\citep{amati02,ghirlanda04a} to other GRB observable. These
correlations may allow for the determination of burst redshifts directly
from the gamma-ray data, which has the advantage of being relatively
insensitive to extinction and observable at far greater distances
than afterglow line measurements. The first of these correlations
was reported by \citet{norris00}. Using 6 BATSE detected bursts with
known redshift, they found an anti-correlation between the {\it
source} frame lag between the 25-50 keV and 100-300 keV emission and
the absolute luminosity of the GRB. More recently, \citep{ghirlanda04a} reported an empirical correlation between the collimation correction total energy $E_{\gamma}$ radiated by the burst and the rest frame energy at which most of the prompt radiation is emitted $E_{pk}$. Using these relationships, it is now possible to estimate "pseudo" redshifts for a much larger number
of GRBs detected by the BATSE instrument which perviously lacked any
information as to their distance. More importantly, the BATSE
detector threshold is relatively well understood for the entire
sample, making the resulting pseudo redshift data excellent for
statistical analysis.
In this paper, we examine the issue of luminosity and density
evolution by using a sample of over 900 BATSE GRBs for which the
luminosity and redshift where recently estimated by \citet{band04}
through the use of the lag-luminosity correlation. We limit our analysis to the lag-luminosity correlation primarily due to the lack of jet opening angle $\theta_{j}$ information that is required for the use of the $E_{\gamma}$-$E_{pk}$ relation. This relationship requires knowledge of $\theta_{j}$ in order to determine the collimation factor, which is only known for bursts with measured jet break times and hence cannot be used with the BATSE sample in consideration for this paper. We found that the more general, and much broader, correlation between intrinsic $E_{pk}$ and $E_{iso}$ reported by \citet{amati02} did not provide redshift constraints for a majority of the bursts in our sample. This is consistent with recent observations by \citet{nakar} and \citet{bandpreece} who also found large fractions of their BATSE samples to be inconsistent with the Amati correlation. Therefore we limit the current analysis to distances estimated through the use of the lag-luminosity relationship.
To our sample, we apply statistical techniques developed by \citet{Lynden-Bell71} and
\citet{efron92} and first applied to GRB analysis by \citet{lloyd02}
to measure the underlying luminosity and density distribution in a
way that properly accounts for the detection thresholds of the BATSE
instrument. We find a strong (11.63 $\sigma$) correlation between
luminosity and redshift that can be parameterized as $L(z) =
(1+z)^{1.7 \pm 0.3}$. The resulting cumulative luminosity function
$N(L')$ is well fit by double power law separated by a break energy
of about $10^{52}$ ergs s$^{-1}$, with the differential luminosity
function $dN/dL'$ exhibiting a power law shape of $L^{-1.5}$ below
this luminosity. We show that the GRB comoving rate density
increases roughly as $\rho_{\gamma}(z) \propto (1+z)^{2.5}$ to a
redshift of $z \approx 1$ followed by a flattening and eventual
decline above $z>3$. This rate density is in qualitatively
agreement with recent photometric estimates of the cosmic star
formation rate (SFR), as would be expected from massive short lived
progenitors.
In $\S 2$, we describe the data set that we use in our study. In
$\S 3$ we discuss the statistical methods applied to this data to
estimate the GRB luminosity function and comoving rate density as
well as to test for any correlation between luminosity and redshift.
In $\S 4$ we present the resulting demographic distribution
functions of this analysis followed in $\S 5$ by a discussion of the
implications of the shape and evolution of the luminosity function
and comoving rate density on various jet profile. We show that
there is no evidence for beaming angle evolution in the current
sample of GRBs with known redshift, suggesting that the variation of
the observed luminosity with redshift can not be due to an evolution
of the collimation of gamma-ray emission. We conclude by examining
how the similarity between the SFR and the GRB comoving rate density
tentatively favors short lived progenitor models.
\section{Data} \label{sec:data_ch3}
For this analysis we utilize data for 1438 BATSE detected GRBs
presented in \citet{band04}, hereafter BNB04. This sample includes
peak photon flux $f_{pk}$ in the 50-300 keV band on a 256 ms
timescale, the burst duration $T_{90}$, and measured lags and their
uncertainties for each burst. From these lag measurements, the
authors infer each burst's luminosity and redshift by use of the
lag-luminosity correlation, allowing also for an estimation of the
intrinsic $E_{pk}$ and $E_{iso}$ for each burst. Of these 1438
bursts, 1218 have positive lags making them suitable for this
analysis. This data is shown in Figure \ref{fig:data} with an
imposed flux cut set at 0.5 photon cm$^{-2}$ s$^{-1}$, leaving a
total of 985 bursts.
The lags measurements used in this sample where made using a
cross-correlation analysis similar to that previously employed by
\citet{band93b} and \citet{norris00}. The cross correlation method
has been widely used in x-ray and gamma-ray astronomy, and is well
suited for timing analysis between two signals. In this application,
the normalized discrete cross correlation function is given by
\begin{equation} \label{eq:CCF}
CCF(\tau)=\sum_{i}^{N-1}\frac{f_{i}(t)*g_{i}(t-t')}{\sigma_{f}\sigma_{g}}
\end{equation}
where $t'$ is commonly referred to as the lag between $f(t)$ and
$g(t)$ and $\sigma_{f}=\langle f(t)^{2}\rangle^{1/2}$. By maximizing
the CCF function (i.e. by maximizing the area of the product of the
two functions) as a function of $t'$, an estimate of the temporal
offset of the two signals can be made. If $g(t)$ leads $f(t)$ by
$t_{0}$ (i.e. $f(t)=g(t+t_{0})$) than the CCF curve peaks at
$t'=t_{0}$.
In BNB04, the authors utilize 64ms count data gathered by BATSE's
Large Area Detectors (LADs) which provide discriminator rates with
64 ms resolution from 2.048 s before the burst to several minutes
after the trigger \citep{fish94}. The discriminator rates are
gathered in four broad energy channels covering approximately 25-50,
50-100, 100-300, and 300 to about 1800 keV allowing for excellent
count statistics since the photons are collected over a wide energy
band. BNB04 measure the temporal offset or lag between channel 3
(100-300 keV) and channel 1 (25-50 keV) light curves to produce the
CCF31 lags listen in their sample.
The shifting of the GRB spectra out of (or into) the observers frame, otherwise known as the k-correction, was accounted for in the analysis performed in BNB04. They perform spectral fits for most of the bursts in their sample and for those which cannot yeild a fit, a "Band" spectral model with average parameters is assumed for the spectra. The effects of time dilation and k-correction are then used to obtain the source frame lag and also applied to the energy flux to obtain a bolometric luminosity.
As was the case in the original \citet{norris00} paper, the CCF method used in BNB04 can result in lag measurements which are less than the 64ms time resolution of the BATSE instrument. In these cases, the associated errors of these values tend to be quite large, reducing the significance of their associated luminosity and redshift values. These errors are taken into consideration in the maximum likelihood techniques performed in our analysis. Therefore, bursts with extremely short lags (and hence high luminosity's) are weighted accordingly. A plot of the lag-luminosity plane for the events under consideration along with the errors in the lag measurements are shown in Figure \ref{fig:laglumplane}.
\subsection{Estimating Redshifts} \label{sec:redshifts}
Using these lag measurements, BNB04 utilize the lag-luminosity
correlation to estimate the luminosity of each event. This
empirical correlations was reported by \citet{norris00} who used the
CCF method to measure the lag between BATSE's channel 3 and channel
1 energy light curves for 6 GRBs with independently measured
redshift. They concluded that there was an anti-correlation between
the {\it source} delay in the low and high energy emission and the
absolute luminosity of the GRB showing that high luminosity events
exhibited very small intrinsic (source frame) lag, whereas fainter
bursts exhibited the largest time delay. This empirical correlation
can be expressed as
\begin{equation} \label{eq:laglum}
L = 2.51\times10^{51} (\Delta t'/0.1)^{-1.15}
\end{equation}
where $\Delta t'$ is the source frame lag related to the observed
lag $\Delta t'_{obs}$ by a time dilation factor of $(1+z)^{-1}$. The
fact that the lag-luminosity correlation relates two source frame
quantities (i.e. luminosity and intrinsic lag) would make it seem
that knowledge of the redshift is needed \emph{a priori}. As it
turns out this is not the case. A simple numerical iteration
routine can be used to solve for the redshift of a GRB which lacks
any information as to its distance. This is done by first making an
initial guess for $z$ (say $z \sim 1$) to obtain the lag in the
comoving frame $\Delta t' = \Delta t'_{obs}/(1+z)$. This in turn
gives us an initial value for the luminosity through the use of the
lag-luminosity relation. This luminosity is then used in combination
with the burst's energy flux to obtain a value for the luminosity
distance $D_{L}$ through the standard relation
\begin{equation} \label{eq:DL1}
D_{L} = \sqrt{\frac{L/d\Omega}{f_{256}}}
\end{equation}
where $f_{256}$ is the peak flux in the 256 ms timescale and
$d\Omega$ is the beaming factor. This distance is then compared to
the $D_{L}$ that can be calculated directly from the guessed
redshift $z$ by assuming standard cosmological parameters ($H_{o} =
65$ km s$^{-1}$, $\Omega_{m} = 0.3$, $\Omega_{\Lambda} = 0.7$) and
using the expression
\begin{equation} \label{eq:DL2}
D_{L} =
(1+z)\frac{c}{H_{0}}\int_{0}^{z}\frac{dz}{\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}}}
\end{equation}
The value for $z$ is then varied until the luminosity distances
obtained from the two separate methods converge to within some
predetermined precision.
We note that it has been suggested by \citet{salmonson01} and
\citet{norris02} that the lag-luminosity relationship should be a
broken power law in order to accommodate GRB 980425. This burst was
associated with SN 1998bw and when using the distance to the
supernova, the GRB appears under luminous compared to the other
bursts that fall on the lag-luminosity correlation. In their
analysis, BNB04 note that this break has been suggested to fit a
single point, which may or may not be associated with the SNe event
and hence decide to use a single power law of -1.15.
The physical origin of the lag-luminosity correlation is not
immediately clear. Fundamentally, this observed lag is due to the
evolution of the GRB spectra to lower energies, so a relationship
between the rate of spectral decay and luminosity is expected
\citet{kocevski03a}. This implies that the mechanisms resulting in
the "cooling" of the GRB spectra is intimately related to the total
energy budget of a GRB or its collimation factor. Other purposed
theories attempt to explain the lag-luminosity correlation as being due
to the effect of the viewing angle of the GRB jet \citep{krm02,
Ioka01}, and or kinematic effects \citep{salmonson00}. In any case,
the use of this correlation is similar to methods used to calibrate
Type Ia supernova luminosities based on the empirical correlation
between their peak magnitude and rate of light curve decay (Phillips
1999). The lack of a clear physical interpretation of these
correlations does not immediately preclude their use in determining,
or refining, luminosity estimates.
\section{Analysis} \label{sec:analysis_ch3}
The luminosity and redshift data calculated by BNB04 gives us an
enormous sample from which to investigate the evolution of the GRB
luminosity function. As with any cosmological source, it is
important and revealing to understand of how the average luminosity
and density has evolved with cosmic time. Attempting to do so by
simply measuring the correlation coefficient between the flux
truncated luminosity and redshift data in the BNB04 sample without
properly accounting for the detector selection effects would grossly
overestimate the correlation strength. This is true whenever an
estimate of correlation is made between two variables that suffer
from data truncations, with the resulting correlation coefficient
representing the truncation itself and not the underlying relation.
There have been several methods developed in astronomy to account
for such selection effects, based largely on maximum likelyhood
techniques (see Petrosian 1992 for a review). In our analysis, we
use a nonparametric statistical technique originally proposed by
\citet{Lynden-Bell71} for applications in flux limited quasar
studies. This so called C-Method has been used successfully to
reconstruct underlying parent distributions for quasars and GRBs
samples by \citet{Maloney99}, and \citet{lloyd02} respectively. The
parent luminosity and redshift distributions which the method
estimates allows for the construction of a GRB luminosity function,
a measure of the number of bursts per unit luminosity, and an
estimate on the comoving rate density, a measure of the number of
bursts per unit comoving volume and time.
The C-Method has two important limitations, or stipulations, to its
use. First, the truncation limit below which no observations can be
made must be well known. This is not a problem in our case, since
the detector threshold of the BATSE instrument is well understood
and BNB04 quantify the truncation limit of their sample. Secondly,
the parent luminosity and redshift distributions can only be
estimated in a bivariate manner if the two variables are
uncorrelated. This is a limitation of all nonparametric techniques
which rely on the assumption of stochastic independence. Therefore,
it is necessary to first determine the degree of correlation between
the two variables, in our case luminosity and $Z = 1+z$, and then
produce an uncorrelated data set through the transformation $L
\rightarrow L' = L/g(z)$, where $g(z)$ parameterizes the luminosity
evolution. Using this uncorrelated data set, it is then possible to
apply the C-Method to estimate the underlying parent luminosity and
redshift distributions. To estimate the degree of correlation we
use a simple test of independence for truncated data put forth by
\citet{efron92} which is based in part on Lynden-Bell's C-Method.
Below we describe the details of both Lynden-Bell's C-Method and the
Efron $\&$ Petrosian independence test and how they are applied in
our analysis.
\subsection{Test of Independence} \label{sec:efron}
If the variables $x$ and $y$ in a data set are independent, then the
rank $R_{i}$ of $x_{i}$ within that set should be distributed
uniformly between 1 and $N$ with an expected mean $E=(1/2)(N+1)$ and
variance $V=(1/12)(N^{2}-1)$. It is common practice to normalize
the rank $R_{i}$ such that for independent variables $R_{i}$ has a
mean of 0 and a variance of 1 by defining the statistic
$T_{i}=(R_{i}-E)/V$. A specialized version of the Kendell $\tau$
statistic can be constructed to produce a single parameter whose
value directly rejects or accepts the hypothesis of independence.
This quantity is commonly defined as
\begin{equation} \label{eq:tau0}
\tau = \frac{\Sigma_{i} (R_{i}-E)}{\sqrt{\sum_{i}V}}
\end{equation}
Based on this definition, a $\tau$ of 1 indicates a 1 $\sigma$
correlation whereas a $\tau$ of 0 signifies a completely random data
set. See \citet{efron92} for a more detailed (and elucidating)
proof of the applicability of normalized rank statistics.
The modified version of this rank statistic proposed by
\citet{efron92} to test the independence of truncated data is based
on a simple concept. Instead of measuring the ranks $R_{i}$ for the
entire set of observables, rather deal with data subsets which can
be constructed to be independent of the truncation limit suffered by
the entire sample. This is done by creating "associated sets" which
include all objects that could have been observed given a certain
limiting luminosity. We can define an associated set as
\begin{equation} \label{eq:Ji}
J_{i} \equiv \{j:L_{j} > L_{i}, L_{lim,j} < L_{i}\}
\end{equation}
In other words, for each burst $i$ there can be constructed a data
subset that includes all events within the range $L_{i} < L <
\infty$ and $0 < z < z_{max}(L_{i})$. The boundaries of an
associated set for a given burst $i$ are shown as dotted lines in
Figure \ref{fig:associatedsets}. In this scenario, we expect the rank
$R_{i}$ of $z_{i}$ within the associated set
\begin{equation} \label{eq:Ri}
R_{i} \equiv \{j\in J_{i} : z_{j} \leq z_{i}\}
\end{equation}
to be uniformly distributed between 1 and $N_{j}$, where $N_{j}$ is
the number of points in the associated set $J_{i}$. Using these new
ranks, we can again construct the mean and variance, except that now
we replace $N$ with $N_{j}$ such that $E=(1/2)(N_{j}+1)$ and
$V=(1/12)(N_{j}^{2}-1)$. The specialized version of Kendell's $\tau$
statistic is now given by
\begin{equation} \label{eq:tau}
\tau = \frac{\Sigma_{i} (R_{i}-E_{i})}{\sqrt{\sum_{i}V_{i}}}
\end{equation}
where the mean and variance are calculated separately for each
associated set and summed accordingly to produce a single value for
$\tau$. This parameter represents the degree of correlation for the
entire sample with proper accounting for the data truncation. With
this statistic in place, it is a simple matter to find the
parametrization that best describes the luminosity evolution. This
is accomplished by first choosing a functional form for the
luminosity evolution, which in our case we choose a simple power law
dependence $g(z) = (1+z)^{\alpha}$. We can then make the
transformation $L \rightarrow L' = L/g(z)$ and vary $\alpha$ until
$\tau \rightarrow 0$.
An example of how well these methods are able to estimate underlying
correlations in truncated data is shown in Figure
\ref{Fig:fakedata}. Here we have plotted a distribution of fake
luminosity and redshift data with some known power law dependence $L
\propto (1+z)^{p}$ which is subjected to a flux cut $L_{lim} \propto
(1+z)^{q}$ represented by the red dashed line. The crosses show the
observable data whereas the dots represent the data that would
otherwise be undetectable. The long dashed line is the best fit to
the truncated data without any knowledge of the flux cut whereas the
dash dot line is the reconstructed correlation when taking into account
the flux threshold. This method fails when the undetected data
points become significantly larger than the observable data set,
with the exact boundary at which this occurs depending on the
difference in the power law indices between the underlying
correlation and the flux threshold. Since these quantities
cannot be known a priori, it is explicitly assumed that a large data
sample contains a sufficient amount of events above the flux
threshold for the method to work. A histogram of the difference
between the known correlation index and the reconstructed index
$(p-q)$ for multiple such simulations is shown in Figure
\ref{Fig:alphadist}. The error, or difference between the known $p$
and the measured $q$ is peaked about zero with a fwhm which roughly
matches that error estimates that correspond to the 1 $\sigma$ range
for this parameter given by the condition $|\tau|<1$.
\subsection{Determination of Distribution Functions} \label{sec:distributions}
Once a parametric form that removes the the correlation between $L$
and $z$ is known, it is possible to use nonparametric maximum
likelyhood techniques to estimate the underlying parent luminosity
and redshift distributions. This luminosity distribution
$\Phi(L_{i})$ represents the cumulative GRB luminosity function with
the redshift distribution $\sigma(z_{i})$ representing the GRB
density evolution. \citet{Petrosian92} has shown that many, if not
most, of the familiar nonparametric methods used in astronomy to
produce $\Phi(L_{i})$ and $\sigma(z_{i})$ reduce fundamentally to
Lynden-Bell's C-Method. Consider the area, or number of events, in
the box produced by the associated set shown in Figure
\ref{fig:associatedsets}. If $N_{1}$ represents the number of points with $L
\geq L_{1}$, then let $dN_{1}$ represent the number of points in the
infinitesimal column between $L_{1}$ and $L_{1}+dL_{1}$. The general
premise behind the C-Method is that if the two variables $(L,z)$ are
stochastically independent, then the ratio between $N_{1}$ and
$dN_{1}$ should equal the ratio between $d\Phi$ and the true
cumulative distribution function $\Phi(L_{1})$
\begin{equation} \label{eq:dnn}
\frac{dN_{1}}{N_{1}} = \frac{d\Phi}{\Phi_{1}}
\end{equation}
which can then be integrated to find $\Phi(L)$. In the case of
discrete data points, this integration becomes a summation, yielding
the solution
\begin{equation} \label{eq:lyndenbell1}
\Phi(L_{i}) = \prod_{k=2}^{j}\left(1+\frac{1}{N_{j}}\right)
\end{equation}
where $N_{j}$, is the number of bursts in the box defined by
$0<z<z_{max}(L_{j})$ and $L_{j}<L<\infty$. The value $N_{j}$ is the
same as Lynden-Bell's $C^{-}_{j}$ in that it does not count the
$L_{i}$ object that is used to form the associated set. Similarly,
we can construct the underlying cumulative redshift distribution
function $\sigma(z_{i})$ by reversing the definition of the
associated set such that $M_{j}$ represents the number of bursts in
the box $0<z<z_{i}$ and $L_{min}(z_{i})<L<\infty$. Then
\begin{equation} \label{eq:lyndenbell2}
\sigma(z_{i}) = \prod_{k=2}^{j}\left(1+\frac{1}{M_{j}}\right)
\end{equation}
As mentioned in $\S$ 1, there are several important limitations to
the C-method. First, the overall normalization of $\Phi(L_{i})$ and
$\sigma(z_{i})$ is arbitrary, so information regarding the absolute
numbers and densities cannot be obtained. Despite this, the shape
of the bivariate distribution is constructed in such a way that it
accounts for the data truncations. Due to this limitation, all
distributions presented in this paper will have arbitrary
normalizations. Secondly, it is clear from Equation
\ref{eq:lyndenbell1} and \ref{eq:lyndenbell2} that the cumulative
distribution function is not defined when either $N_{j}$ or $M_{j}$
are zero. This limitation restricts the use of the C-method to
samples with a data size sufficiently large to ensure that all
associated sets greater than $j=2$ contain a nonzero number of
points.
\section{Results} \label{sec:results_ch3}
\subsection{Luminosity Evolution} \label{sec:lumevolution}
We apply the test of independence outlines in the $\S$ 3.1 to the
entire BNB04 GRB sample to test for luminosity evolution. For this
analysis we use the flux threshold suggested by BNB04 of $f_{min}$ =
0.5 photons cm$^{-2}$ s$^{-1}$, decreasing the sample size to 985
bursts. Applying this method, we find evidence for a strong 11.63
$\sigma$ correlation between luminosity and redshift. This
evolution is well parameterized by a power law of the form $g(z) =
(1+z)^{\alpha}$, with an optimal value for the power law index (i.e
when $\tau(\alpha)$=0 given the transformation $L \rightarrow L' =
L/g(z)$) of $\alpha$=1.7 $\pm$ 0.3. The error estimates on $\alpha$
correspond to the 1 $\sigma$ range for this parameter given by the
condition $|\tau|<1$. A plot of $\tau(\alpha)$ vs. $\alpha$ with the
corresponding 1 $\sigma$ levels are shown in Figure \ref{fig:taualpha}.
These findings indicate that the average luminosity (modulo a
beaming factor $d\Omega$) of GRBs in the universe has evolved with
time. Because of the lack of beaming information, it may also be
possible that the luminosity is remaining constant while the beaming
factor $d\Omega$ is actually evolving. As will be discussed in $\S$
5, there is no observational evidence to suggest that this is the
case.
It should also be noted that $\tau(\alpha)$ appears to be strongly
affected by the choice of the flux threshold assumed for the sample.
Plotted in Figure \ref{fig:alphacut} is the optimal value for $\alpha$
vs. $f_{min}$. Not surprisingly, if we assume no flux threshold
(i.e. $f_{min}=0$), $\tau$ approaches the overestimated value
received from the standard Kendell $\tau$ statistic. $\alpha$
similarly approached the value obtained by simply performing a
power-law fit to the truncated data. $\alpha$ decreases steeply with
increasing $f_{min}$, never reaching a stable plateau as one would
hope would happen as the $f_{min}$ approaches the $\emph{true}$
threshold of the detector. This underscores the importance of
having a good understanding the thresholds of the detector used to
collect the sample. BNB04 make a strong case for a threshold of
$f_{min}$=0.5 photons cm$^{-2}$ s$^{-1}$ based on where they see a
strong drop off of detected events in the $L-Z$ plane (see their
Figures \ref{fig:alphacut} $\&$ \ref{fig:alphacut}) and we adopt this
value for all analysis presented in this paper.
\subsection{Luminosity Function} \label{sec:lumfunc}
The deduced parametric form describing the luminosity evolution
allows us to use the C-method on the uncorrelated parameters $L'$
and $Z$ to obtain the cumulative luminosity function $\Phi(L')$.
Shown in Figure \ref{fig:cumluminosity} is the cumulative $\Phi(L')$
distribution plotted as $\Phi(>L')$ as a function of $L'$ for all
985 bursts. Because the luminosity evolution has been explicitly
removed, this distribution represents the luminosity function in the
present epoch. Fitting a double power law to the curve yields
$\Phi(>L') \propto L'^{-0.623}$ and $\Phi(>L') \propto L'^{-1.966}$
for the low and high luminosity ranges respectively, separated by a
break at a luminosity of roughly $\sim 10^{52}$. These slopes are
very similar to those reported by \citet{lloyd02} who found a GRB
cumulative luminosity function with power law slopes of
$k_{1}=-0.51$ and $k_{2}=-2.33$ below and above a break at about
$L'=5.9\times10^{51}$. These values can also be compared to the
luminosity functions found by \citet{Maloney99} who employ the
C-method to account for selection effects in quasar samples. They
find that the quasar luminosity function exhibits a double power law
form with indices of $k_{1}=-1.16$ and $k_{2}=-3.59$.
Next, we differentiate the cumulative luminosity function with a
3-point Lagrangian interpolation to find the differential luminosity
function $d\Phi/dL'$, or what is commonly referred to as simply the
luminosity function $\psi(L')$. This function represents the total
number of bursts with luminosity between $L'$ and $L'+dL'$. A plot
of the $\psi(L')$ vs. $L'$ is shown in Figure \ref{fig:lumfunction}. The
function falls roughly as $\psi(L') \propto L'^{-1.5}$ below the
break energy of $\sim 10^{52}$ with a sharp decline for higher $L'$.
This power law index is identical to the slope found by
\citet{lloyd02} who found $L'^{-1.5}$ and similar to the index found
by \citet{Schaefer01} who found $L^{-1.7 \pm 0.1}$ from $(L,z)$ data
estimated from a combined use of the lag-luminosity function and
variability-luminosity function, although the latter did not account
for any selection biases in their data set. This value is also
similar to results of several studies that used the measured flux
distribution with an assumed density distribution $\rho(z)$, such as
\citep{Schmidt01} who uses the star formation rate to infer a
$\rho(z)$ and finds $\psi(L') \propto L^{-1.4}$. The shape of the
GRB luminosity function has important implications to jet model
theories which predict specific power law indices for various jet
structures. A comparison between theorized shapes and our deduced
values will be discussed in more detail in $\S$\ref{sec:discussion_ch3}.
\subsection{Density Evolution} \label{sec:densityevolution}
Using the alternative definition of the associated set, we can
construct the cumulative density distribution
$\sigma(z)=\int_{0}^{z}\rho(z)(dV/dz)dz$, or the total number of
GRBs per comoving volume, up to a given redshift. The cumulative
distribution is shown in Figure \ref{fig:cumdensityvolume} plotted as
$\sigma(>z)$ as a function of $z$. The distribution of GRBs appears
to increase smoothly with $z$, without a pronounced break at any
distance, but with a flattening at high redshift indicating a drop
off of events between $5\leq z \leq 10$. To get a better look at
this density evolution, we can plot the cumulative density
distribution $\sigma(z)$ as a function of comoving volume $V(z)$ as
seen in Figure \ref{fig:cumdendist} If the density of GRBs per
comoving volume $V(z)$ is constant, i.e. $\rho(z)=\rho_{0}$, then it
should follow that $\sigma(z) \propto V(z)$. We can test for
evolution by fitting $\sigma(z)$ vs $V(z)$ to a simple power law
$\sigma(z) \propto V(z)^{\beta}$ and looking for deviations from the
constant density case. An index of $\beta \neq 1$ indicates the
presence of density evolution, with $\beta > 1$ and $\beta < 1$
signifying an increasing and decreasing population respectively.
Using the definition of $V(z)$ in a flat universe of
\begin{equation} \label{eq:volume}
V(z) =
\frac{4\pi}{3}\left[\frac{c}{H_{0}}\int_{0}^z\frac{dz}{\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}}}\right]^{3}
\end{equation}
we find that the cumulative density distribution increase with $z$
roughly as $\sigma(z) \propto V^{1.25}$ at low redshifts before
falling off at higher redshifts. From these results we can deduce
that the GRB density has undergone complicated evolution, increasing
as $\rho \sim V^{0.25}$ before peaking between $z \sim 1-2$ and then
decreasing. To obtain a more quantitative look at the shape of the
comoving rate density $\rho(z)$, we again use a 3-point Lagrangian
interpolation routine on $\sigma(z)$ to find the differential
cumulative distribution $d\sigma/dZ$. We can then convert this
differential distribution into a comoving rate density through the
relation:
\begin{equation} \label{eq:rho}
\rho(Z)=\frac{d\sigma}{dZ}(1+z)\left(\frac{dV}{dZ}\right)^{-1}
\end{equation}
In Figure \ref{fig:comovingratedensity} we show the resulting comoving rate
density plotted as a function of $z$. It can be seen that the GRBs
density function increases out to a redshift between $1\leq z \leq$
then flattens before beginning to show signs of a turn over at a
redshift of $z > 3$. This is in contrast to previous estimates of
the comoving rate density by \citet{Schaefer01}, \citet{lloyd02},
and \citep{Yonetoku04} all of who find a flattening of the GRB
population with no apparent turn over out to a redshift of $z \sim
10$. It is also in contrast to results reported by \citet{Murakami} who also used the lag-luminosity correlation to estimate the GRB formation rate. There the authors find the GRB formation rate increases steadily out to a redshift of at least 4, but it should be noted that this work did not take into account the detector selection effects discussed above so a direct comparison may not be appropriate. As opposed to these previous findings, the turn over observed in our data quantitatively matches the global behavior of the star formation rate of the universe which has
been observed to peak between $1 \leq z \leq 2$ followed by a steady
decline \citep{madau96,Steidel99}. A more detailed comparison
between the GRB comoving rate density and the supernova and star
formations rates will be continued in $\S 5$.
\section{Discussion} \label{sec:discussion_ch3}
We find an 11.63 $\sigma$ correlation between the luminosity and
redshift data deduced from the lag-luminosity correlation, strongly
suggesting an evolution of the average luminosities of GRBs. We
show that this correlation can be parameterized as a power law as
$L(z) = (1+z)^{1.7 \pm 0.3}$. This value agrees extremely well with
the results presented in \citet{lloyd02} who found a power law index
of $\alpha = 1.4$ after performing a similar analysis on $(L,z)$
data estimated using the variability-luminosity correlation. These
results imply that the average energy emitted per unit time per unit
solid angle by GRBs was much higher in the distant past compared to
relatively recent events. This is consistent with previous studies
showing strong source evolution and also recent observations of
under luminous nearby GRBs. Due to our lack of knowledge regarding
the beaming angle of the bursts in our sample, it is also possible
that the increase in the apparent luminosity is due to an increasing
collimation at higher redshifts. As we will discuss in more detail
below, we find no evidence for beaming angle evolution in the
current sample of GRBs with known redshift and jet opening angle,
suggesting that this increase in luminosity can not be due simply to
an evolution of the collimation of the gamma-ray emission.
\subsection{Comparison to Other Objects} \label{sec:comparisons}
Such a steep luminosity evolution is not uncommon in other
astrophysical objects that show evolution with redshift.
\citet{Maloney99} perform a similar analysis using the statistical
techniques described in this paper on a combination of several
quasar samples and find that the quasar luminosity function evolves
as $L(z) = (1+z)^{2.58}$ up to a redshift of at least 2. There is
evidence that this evolution may then become constant up to a
redshift of at least 3 \citep{Boyle93}. We find no such break in
the luminosity evolution of GRBs, which in our case can be
adequately fit by a single power law between at least $0 < z < 10$.
The authors also find a density evolution of $\sigma(z) \propto
V^{1.19}$ similar to the power law of $\sigma(z) \propto V^{1.25}$
that we find in GRBs at low redshifts. A more detailed look at
their comoving rate density estimate shows that the quasar density
rises as $\rho \sim (1+z)^{2.5}$ before peaking at $z \approx 2$ and
then declining rapidly as roughly $\rho(z) ~ (1+z)^{-5}$ for $z
> 2.0$. This is qualitatively similar to the trend we deduce from
the GRB sample, which rises as $\rho \sim (1+z)^{2.4}$ to a $z
\approx 1$ although the proceeding decline is much more shallow as
$\rho \sim (1+z)^{-0.6}$ and extends to at least a redshift of $z
\approx 6$ before dropping off sharply.
There is also evidence for significant evolution in the luminosities
of star forming galaxies, which is perhaps a more relevant
comparison to GRBs because of their suggested association with
active star forming regions \citep{Djorg98}. Hopkins (2004) used a
compilation of recent star formation rate density measurements as a
function of redshift to constrain the evolving luminosity function
of star-forming galaxies. He finds that the preferred evolution in a
standard cosmology is given by $L(z) = (1+z)^{2.70 \pm 0.60}$ out to
a redshift as high a $z \approx 6$. At the same time he finds
evidence for a very shallow density evolution given by $\rho(z) \sim
(1+z)^{0.15 \pm 0.60}$, markedly different from steep density
evolution $\rho(z) \sim (1+z)^{2.5}$ that we estimate for GRBs at
low redshift. This would indicate that GRB luminosities have evolved
at a slower rate, but that their density in the past rises much more
steeply compared to the number of star forming galaxies. It could
also mean that the number of GRBs per star forming galaxy has
evolved rapidly with cosmic time.
Perhaps more interesting is the comparison between the GRB comoving
rate and luminosity densities with the global star formation rate
history of the universe. Because GRBs suffer little extinction and
are potentially detectable out to redshifts of $z \approx 10$, they
could offer a unique tracer to the SFR history. They would allow for
a more complete sampling of dust enshrouded star forming regions
that may be missed in traditional SFR estimates based on the UV
"drop-out" technique that is currently employed to identify Lyman
break galaxies. The shape of the SFR at low redshifts $z<1$ is
relatively well understood, showing an order of magnitude increase
from $0\leq z \leq 1$ \citep{madau96,fall96}. These early estimates
suggest that the star formation activity peaks around $z \sim 1$
followed by a rapid decline at higher redshifts. However, further
observations of hundreds of Lyman break galaxies at redshifts of $z
\sim 3$ and 4 have shown that the SFR may remain constant after
reaching a maximum around $1 \leq z \leq 2$ \citep{Steidel99}.
Recent deep surveys with the Subaru \citep{Iwata03} and Hubble Space
Telescopes \citep{Bouwens03} out to $z \sim 5$ and 6 show evidence
for a mild evolution of the SFR at redshifts $z>3$, with
measurements based on photometric redshifts showing a constant SFR
out to $z \approx 6$ \citep{Fontana03}. These recent SFR estimates
qualitatively match the deduced GRB comoving rate density shown in
Figure \ref{fig:comovingratedensity}. At low redshifts $z < 1$, the GRB rate
density increases as $\rho(z) \sim (1+z)^{2.5}$ roughly matching the
rise in the SFR over the same range, with a peak somewhere between
$1\leq z\leq 2$. The following flattening and decline between
$2\leq z\leq 6$ in the GRB $\rho(z)$ matches the global properties
of the SFR estimated from the recent deep surveys.
Of course the comparisons between the GRB comoving rate density and
the SFR are simply phenomenological, since we have as of yet no way
of connecting the amount of star formation for a given amount of
GRBs. Ultimately this conversion factor depends on knowledge of the
GRB progenitor and the initial stellar mass function (IMF) and how
it changes with redshift. In the case of the collapsar model
\citep{woosley00, macfadyen99}, the rate of GRBs produced for a
given SFR would increase sharply with redshift, as is the case for
all core collapse events, due primarily to the redshift dependence
on the IMF. However, the connection between the GRB $\rho(z)$ and
the SFR would be straightforward since the progenitors would consist
of massive stars with short lifetimes making them direct indicators
of the SFR at that redshift. If the mass range of the progenitors
and the redshift dependence of the IMF is know (or assumed) then it
would be possible to calculate a constant that directly relates
$\rho(z)$ to the SFR. The case is more complicated for binary
merger models since there would be a substantial delay between the
formation of the progenitor star and the final merger event that
produces the GRB. The distribution in the delay times is not well
known for SNe events, much less GRBs, but it is expected to be large
enough to dissociate the GRB $\rho(z)$ and the active SFR at a given
redshift. The peak we observed in our deduced values for $\rho(z)$
matching the peak of the current SFR estimates hardly seems
coincidental, tentatively favoring the core collapse models.
It is interesting then to compare our demographic results to that of
various types of supernovae. There is overwhelming observational
evidence and theoretical discussion suggesting a GRBs-SNe
connection, including observations of supernovae bumps in afterglow
lightcurves \citep{stanek03,hjorth} and a deduced collimation
corrected energy that is narrowly clustered around the typical SNe
energy of $10^{51}$ ergs \citep{Frail01,bloom03b}. Although the
intrinsic luminosity of type Ia SNe are \emph{a priori}
\emph{assumed} to be constant with redshift (hence no luminosity
evolution), we can still compare the formation rates of SNe Ia/b/c
to GRBs, although the b/c events are obviously of more relevance to
GRB models. Unfortunately, very little SNe data is available for the
high $z$ universe with only 7 SNe at $z>1.25$ of the 42 SNe detected
in the redshift range of $0.2 \leq z \leq 1.6$ by the Advanced
Camera for Surveys (ACS) on the Hubble Space Telescope
\citep{riess04}. Data on core collapse supernova accounts for only
17 events of this sample, going out to a maximum range of
$z\approx0.7$. \citet{dahlen04} use this data to estimate the core
collapse SNe (CC SNe) rate between $0.3 \leq z \leq 0.7$ and find a
steep (about a factor of $\sim 7$) increase in the SNe
$\rho(z\approx0.7)$ compared to the local rate presented by
\citet{capp99}. Shown in Figure \ref{fig:comovingratedensity} are their data
points for CC SNe plotted over the GRB comoving density, both rates
normalized to 1 at $z=0.7$. The two data points, although limited,
do agree with the rise of the GRB $\rho(z)$. A direct SNe-GRB
comparison at higher redshifts will have to wait until the launch of
the SNAP spacecraft which is predicted to find thousands of
supernovae, including a significant number at high redshift.
\subsection{The Nature of the Luminosity Evolution} \label{sec:lumevolution2}
The observed luminosity evolution that we observed in the $(L,z)$
data leads to the conclusion that the GRB progenitor population has
most likely evolved in such a way as to create more energetic or
more narrowly beamed bursts in the distant past. Speculations on the
nature of this evolution are dependant on the progenitor model and
how the properties of their population are affected by the
conditions of the early universe. In the case of highly rotating
massive stars (i.e. the collapsar model), the overall progenitor
mass and/or rotation rate could be the determining factor. There is
ample evidence suggesting that the so called population III stars
were much more massive than their present day counterparts. This is
suggested by recent work showing that the stellar initial mass
function (IMF) has evolved with time, having a much higher value in
the distant past. This higher IMF is due to various factors,
although it primarily is due to the lower metallicity in the early
universe. The amount of material lost to stellar ejecta has also
been shown to be dependant on the stellar metallicity, causing these
early stars to retain more of their mass until their eventual
collapse.
Although the relationship between progenitor mass and emitted energy
and/or beaming angle is not straight forward, there are reasons to
think that this increase in average mass could result in an increase
in the total energy budget available to a burst. \citet{macfadyen99}
show that under the right conditions, the collapsar model could
produce more energetic bursts with increasing stellar mass, up to
some limit dictated by the energy needed by the GRB jet to punch
through the stellar envelope. Proponents of black hole accretion
disk models have also shown that the rate of accretion onto the
central engine of the GRB increases dramatically as a function of
the progenitor mass, increasing the overall energy available to the
burst.
Unfortunately, a simple increase in the overall energy budget cannot
by itself explain the deduced luminosity evolution. \citet{Frail01}
and \citet{bloom03} have recently shown observational evidence
suggesting that the collimation corrected GRB energies $E_{\gamma}$
are actually narrowly cluster around the $10^{51}$ ergs typical of
SNe explosions. They come to this conclusion by correcting the
observed prompt isotropic equivalent energy release $E_{iso}$ of
several GRBs with known redshift by a factor of $1 - \cos
\theta_{j}$, where $\theta_{j}$ is the canonical jet opening angle.
These angles are derived from broadband breaks observed in the
afterglow light curves attributed to the slowing of the GRB jet to
the point where the relativistic beaming angle of the radiation
$1/\Gamma$ becomes greater then $\theta_{j}$. \citet{bloom03}, using
a larger sample of bursts, show that the corrected energies cluster
around $1.33\times10^{51}$ ergs with a variance of 0.35 dex, or a
factor of 2.2. \citet{guetta03} has reported a similar result when
correcting for the isotropic luminosity $L_{iso}$, although not as
narrow as the $E_{\gamma}$ results. If the collimation corrected
energy and luminosity are indeed invariant with redshift, it
directly implies that the brightening of the apparent isotropic
equivalent luminosity is actually due largely to an increase in the
beaming factor as a function of redshift and not an increase in the
overall energy of the burst. There are physical arguments that can
be made in the case of the collapsar model that would suggest that
more massive progenitor stars could indeed produce more collimated
jet outflows.
Plotted in Figure \ref{fig:fluencevsredshift} are the $E_{\gamma}$ estimates
from \citet{Friedman05} for a little over two dozen GRBs. Furthermore, plotted in Figure \ref{fig:beamingangle} is the canonical jet opening angle for the same two dozen GRBs. By apply a standard Kendall rank order $\tau$ statistic we can measure the degree of correlation in these two samples in a nonparametric way (i.e. without assuming an underlying correlation type). We find a correlation strength of $\tau = 0.093$ between $E_{\gamma}$ and redshift and $\tau = 0.163606$ between $\theta_{j}$ and redshift, where a $tau$ of 1 signifies a significant correlation. Therefore, there is no deduced redshift dependency that would suggest any evolution of the jet opening angle or $E_{\gamma}$ with redshift in the pre-Swift data set. This lack of redshift dependency stands to be tested in the Swift era as more GRBs with measured jet break times are observed over a broader redshift range, but if it is confirmed then it would imply that the evolution of some jet property other than the collimation factor must be responsible for the brightening of GRBs with redshift. Speculations on the nature of this evolution are
dependant on the jet model used to explain the emission. The
simplest model assumes a uniform energy distribution per solid angle $\epsilon(\theta)$ across the jet with a sharp drop beyond $\theta_{j}$. In this scenario, the observed distribution in GRB
energies is directly due to the diversity of jet opening angles, as
is the observed values of the jet break time $t_{j}$. The lack of
any concrete evidence for an evolution of $\theta_{j}$ with redshift
combined with the observation that the collimation corrected
$E_{\gamma}$ and $L_{\gamma}$ are very narrowly clustered, create
difficulty for the uniform jet model to explain any kind of
evolution in luminosity. One of the observed conditions above would
have to be broken in order to accommodate any such evolution with
this model.
In a structured jet model, the GRB jets are identical having a
quasi-universal shape with a fixed opening angle and a nonuniform
energy distribution per solid angle. The diversity in the observed
jet break time and isotropic energies would then be a result of
varying viewing angles away from the jet axis $\theta_{v}$.
Furthermore, an observer viewing the GRB at a small $\theta_{v}$
would see an extremely powerful burst, with the observed luminosity
declining as some function of increasing $\theta_{v}$. A jet
structure with a functional form of $\epsilon(\theta)^{-2}$ is
required to reproduce the observed $t_{j} \propto E_{iso}$, i.e the
\citet{Frail01} and \citet{bloom03} results. If the requirement of
a narrow $E_{\gamma}$ and $L_{\gamma}$ distribution is broken, then
any power law structure $\theta^{k}$ could still produce the
observed steepening in the afterglow light curve. In this case, the
luminosity evolution would manifest itself not as an narrowing of
$\theta_{j}$ but rather as an overall increase in the normalization
of $\epsilon(\theta_{j})$. Another possibility would be an evolution
of the morphology of $\epsilon(theta)$ as a function of redshift. If
the power law index $\epsilon(\theta_{j}) \sim \theta_{j}$ has
evolved with time, or if $\epsilon(\theta_{j})$ has evolved from a
non-power-law shape (e.g. a Gaussian profile), such that
$\epsilon(\theta_{j})$ varies more slowly with viewing angle, then
there would be a markedly different luminosity distribution at high
redshift. A third, rather implausible, explanation is a
preferentially small viewing angle $\theta_{v}$ at higher redshift,
although there is no physical reason to think that this is at all
possible. Therefore, it would seem that evidence of luminosity
evolution in the presence of the observation that $E_{\gamma}$ and
$L_{\gamma}$ are narrowly distributed and the lack of any evidence
of an evolution of $\theta_{j}$ with redshift, favors a
quasi-universal jet model over a uniform jet model. This is
primarily due to the inability of the uniform jet model to explain
any kind of luminosity evolution with redshift without a parallel
evolution in the jet opening angle, something that is not currently
observed.
\subsection{Luminosity Functions and Jet Model Discrimination} \label{sec:jetmodels}
Because the energy distribution $\eta(\theta_{j})$ of the structured
jet model is well defined, it can make specific predictions
regarding the GRB luminosity function $\phi(L)$. In the case of
power-law structured jets $\epsilon(\theta_{j}) \propto
\theta_{j}^{-k}$, resulting in a predicted luminosity function with
a slope of $\gamma = 1 + 2/k$ \citep{zhang02}. The "canonical" $k=2$
model would yield a luminosity function $\propto L^{-2}$
\citep{rossi02}, whereas the quasi-universal gaussian structured jet
model predicts $\propto L^{-1}$ \citep{lloyd04}. Although the
uniform jet model also exhibits a well defined
$\epsilon(\theta_{j})$, it cannot make any firm predictions about
the shape of the GRB luminosity function due to the random variation
$\theta_{j}$.
The luminosity function deduced from our analysis is well fit by a
single power low with an index of $\propto L'^{-1.5}$ for
luminosities below about 10$^{52}$, with a sharp decline for higher
luminosities. This value is intermediate between the expected value
for the $k=2$ power law and gaussian structure models. An index of
$L'^{-1.5}$ actually predicts a power law model with $k=4$ which is
much steeper than the $k=2$ shape needed to keep $E_{iso}\theta^{2}
\sim$ constant. It is also possible that a simple gaussian or power
law profile for $\eta(\theta)$ is simply an oversimplification. It
has been pointed out that by \citet{lamb03} that $\eta(\theta)$
would have to fall off steeper than $\theta^{-2}$ at large angles if
the quasi-uniform jet model is to explain the relative numbers of
x-ray flashes to GRBs. This may explain why so many studies have
found a sharp decline above some break energy possibly indicating a
different value for $k$ at low and high luminosities, i.e large and
small opening angles respectively. In any case, it would not be
unfeasible to think that the jet morphology is more complicated than
a simple power or gaussian profile, as is suggested by our results.
\section{Conclusions} \label{conclusion:ch3}
In this work we perform demographic studies on a large sample of
luminosity and redshift data found through the use of the
lag-luminosity correlation. By applying maximum likelihood
techniques, we are able to obtain an estimate of the luminosity
evolution, luminosity function, and density distributions in a way
that accounts for detector selection effects. We find that there
exists a strong (11.63 $\sigma$) correlation between luminosity and
redshift that can be parameterized as $L(z) = (1+z)^{2.58}$. The
resulting cumulative $\Phi(L')$ and differential $d\Phi/dL'$
luminosity functions are well fit by double power laws separated by
a break energy of about $10^{52}$ ergs s$^{-1}$, with $d\Phi/dL'$
exhibiting a power law shape of $L^{-1.5}$ below this luminosity.
This value does not immediately discriminate against any proposed
structured jet models, but it may indicate that a more complicated
jet profile is need to explain the observed luminosity function of
GRBs. The GRB comoving rate density is found to increase as
$\rho_{\gamma}(z) \propto (1+z)^{2.5}$ to a redshift of $z \approx
1$ followed by flattening and eventual decline above $z>6$.
Although, the conversion between $\rho_{\gamma}(z)$ and an estimate
of the SFR cannot be quantitatively made due to the uncertainty
regarding the GRB progenitors and their initial stellar mass
functions, it can be said that $\rho_{\gamma}(z)$ does qualitatively
follow recent photometric estimates of the SFR, as would be expected
from massive short lived progenitors. We stress that these
conclusions are based on the validity of the lag-luminosity
correlation, which still stands to be confirmed as new redshift data becomes available. A full
confirmation, and most probably further calibration, of this
distance indicator will have to wait for addition detections with
the Swift spacecraft, which should have the collecting area
necessary to obtain high signal-to-noise energy dependant light
curves from which to measure statistically significant lags. Even
with the slew of directly measured luminosity and redshift data
expected to come from the Swift mission, empirical distance
indicators may still play an important roll in expanding the
available GRB data set through the use of archival BATSE and
BepooSAX data for statistical studies such as the analysis performed
in this work.
\bigskip
\section*{Figure Captions}
\bigskip
{\bf Fig. 1.} - The luminosity and redshift data used in our analysis as
deduced from the lag-luminosity correlation. The dashed line
represents an imposed 0.5 photons cm$^{-2}$ s$^{-1}\ldots$ cut to
the original 1438 bursts analyzed by BNB04, producing the sharp
cutoff in the data. This leaves a total of 985 bursts with a median
redshift of 1.64.
\bigskip
{\bf Fig. 2.} - A plot of the lag-luminosity plane for the events under consideration. The error in the lag and flux measurements are used to determine the uncertainty in the luminosity values which in effect is used as a weight in the maximum likelihood technique that estimates the correlation strength between $L$ and $z$.
\bigskip
{\bf Fig. 3.} - A representation of the associated sets used in the Lynden-Bell
technique. For each data point with $(L_{i},z_{i})$, the solid line
represents the minimum luminosity or maximum redshift that the burst
could have had and still have been observed. Employing the
Lynden-Bell technique with associated sets defined as $N_{j}$
($L_{j} < L < \infty$, $0 < z < z_{max}(L_{i})$) produces the
cumulative distribution for the vertical axis, whereas the $M_{j}$
associated set produces the distribution for the data represented on
the horizontal axis.
\bigskip
{\bf Fig. 4.} - Generated luminosity and redshift data used to test the
ability of the Efron $\&$ Petrosian method to estimate the
correlation strength and functional dependence for data of a given
flux cut.
\bigskip
{\bf Fig. 5.} - A histogram of the difference between the known correlation
index and the reconstructed index $(p-q)$ for a large set of generated $(L,z)$
data with arbitrarily imposed flux thresholds. The error in the
method is tightly centered around $p-q=0$.
\bigskip
{\bf Fig. 6.} - The correlation statistic $\tau$ is plotted as a function
of power law index $g_{\alpha}(z)=(1+z)^{\alpha}$ parameterizing the
luminosity evolution. The solid line represents the $\alpha$ index
that minimizes the correlation between $L'$ and $z$. The dotted
lines show the 1 $\sigma$ \emph{statistical} error in the $\alpha$
parameter.
\bigskip
{\bf Fig. 7.} - The correlation parameter $\alpha$ is plotted as a function of the flux
cut applied to the BNB04 sample. The optimal $\alpha$ is highly
dependent on the choice of the cut value, showing the importance of
a good understanding of the detector flux threshold.
\bigskip
{\bf Fig. 8.} - The luminosity and redshift data used in our analysis as
deduced from the lag-luminosity correlation. The dashed line
represents an imposed 0.5 photons cm$^{-2}$ s$^{-1}\ldots$ cut to
the original 1438 bursts analyzed by BNB04, producing the sharp
cutoff in the data. This leaves a total of 985 bursts with a median
redshift of 1.64.
\bigskip
{\bf Fig. 9.} - The present epoch GRB luminosity function $\psi(L') =
d\Phi/dL'$, representing the number of events between the luminosity
$L'$ and $L'+dL'$. The function falls as roughly $\propto
L'^{-1.5}$ for luminosities below the break energy of $10^{52}$ ergs
s$^{-1}$.
\bigskip
{\bf Fig. 10.} - The cumulative density distribution
$\sigma(z)=\int_{0}^{z}\rho(z)(dV/dz)dz$, representing the total
number of events up to a given redshift. The flattening at high
redshift indicates a drop off of events around $z \sim 5-10$.
\bigskip
{\bf Fig. 11.} - The
cumulative density function $\sigma(z)$ plotted as a function of
comoving volume. The dashed line represents the increase in the
number of sources if the GRB density were constant throughout the
history of the universe. The GRB density has increased as $\rho
\sim V^{0.25}$ before peaking between $z \sim 1-2$ and then
decreasing.
\bigskip
{\bf Fig. 12.} - The comoving rate density $\rho(z)$ as a function of
redshift. The rate density of sources can be seen to follow the
evolution deduced from Figure \ref{fig:cumdensityvolume}, increasing to a
redshift of 1-2 then flattening before decreasing at higher
redshifts. The circles represent high redshift cc rates
from \citet{dahlen04} whereas the square point is the local rate
found \citet{capp99}. The increase in the cc event rate with
redshift qualitatively matches the overall increase in the GRB
comoving rate density.
\bigskip
{\bf Fig. 13.} - Plot of the collimation corrected total emitted energy of
23 GRBs with known redshift and beaming angles. No significant
correlation can be seen with redshift.
\bigskip
{\bf Fig. 14.} - The beaming angles $\theta_{j}$ of 23 GRBs with known
redshift. The lack of a significant correlation with redshift is
quite evident.
\bigskip
\section*{Figures}
|
Title:
Low radiative efficiency accretion at work in active galactic nuclei: the nuclear spectral energy distribution of NGC4565 |
Abstract: We derive the spectral energy distribution (SED) of the nucleus of the
Seyfert galaxy NGC4565. Despite its classification as a Seyfert2, the nuclear
source is substantially unabsorbed. The absorption we find from Chandra data
(N_H=2.5 X 10^21 cm^-2) is consistent with that produced by material in the
galactic disk of the host galaxy. HST images show a nuclear unresolved source
in all of the available observations, from the near-IR H band to the optical U
band. The SED is completely different from that of Seyfert galaxies and QSO, as
it appears basically ``flat'' in the IR-optical region, with a small drop-off
in the U-band. The location of the object in diagnostic planes for low
luminosity AGNs excludes a jet origin for the optical nucleus, and its
extremely low Eddington ratio L_o/L_Edd indicates that the radiation we observe
is most likely produced in a radiatively inefficient accretion flow (RIAF).
This would make NGC4565 the first AGN in which an ADAF-like process is
identified in the optical. We find that the relatively high [OIII] flux
observed from the ground cannot be all produced in the nucleus. Therefore, an
extended NLR must exist in this object. This may be interpreted in the
framework of two different scenarios: i) the radiation from ADAFs is sufficient
to give rise to high ionization emission-line regions through photoionization,
or ii) the nuclear source has recently ``turned-off'', switching from a
high-efficiency accretion regime to the present low-efficiency state.
| https://export.arxiv.org/pdf/astro-ph/0601629 | command.
\newcommand{\myemail}{chiab@stsci.edu}
\slugcomment{Submitted to ApJ}
\shorttitle{Low efficiency accretion in NGC~4565}
\shortauthors{Chiaberge et al.}
\begin{document}
\title{Low radiative efficiency accretion at work in active galactic
nuclei: the nuclear spectral energy distribution of NGC~4565}
\author{M. Chiaberge\altaffilmark{1}}
\affil{Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218}
\email{chiab@stsci.edu}
\author{R. Gilli\altaffilmark{2}}
\affil{INAF - Osservatorio astronomico di Bologna, Via Ranzani 1, 40127 Bologna, Italy}
\author{F. D. Macchetto\altaffilmark{3}}
\affil{Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218}
\and
\author{William B. Sparks}
\affil{Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218}
\altaffiltext{1}{On leave from INAF, Istituto di Radioastronomia, via P. Gobetti 101, I-40129 Bologna}
\altaffiltext{2}{Visiting Programmer, Space Telescope Science Institute}
\altaffiltext{3}{On assignment from ESA}
\keywords{galaxies: active --- galaxies: nuclei --- accretion, accretion disks --- galaxies: individual (NGC~4565)}
\section{Introduction}
Low luminosity active galactic nuclei (LLAGN) are believed to be
powered by accretion of matter onto the central supermassive black
hole, similarly to powerful AGN. In a large fraction of LLAGN, the
central black hole is as massive as in powerful distant quasars
($M_{BH} \sim 10^{8}- 10^{9} M_\sun$), thus their very low nuclear
luminosity implies that accretion occurs with very low radiative
efficiency \citep[or at very low rates; see
e.g.][]{ho04coz,papllagn}. If so, the physics of the accretion process
may be different from the ``standard'' optically thick, geometrically
thin accretion disks. Starting from the ``ion-supported tori'' of
\citet{rees82}, a number of theoretical models have been developed to
describe such radiatively inefficient accretion flows \citep[RIAF,
e.g. advection-dominated accretion flows, ADAF, advection-dominated
inflow-outflow solutions, ADIOS, convection-dominated accretion flows,
CDAF][]{narayanyi94,quataert99,abramowicz02}. But because of the very
low radiation they emit at all wavelengths, these objects (if they at
all exist) are very difficult to be observed. Recently, the AGN
nature of optical nuclear components seen in HST images of a sample of
LLAGN have been unambiguously established. \citet{Maoz05} have shown
that among a sample of 17 LLAGN, 15 of them show variability over a
timescale of a few months, which demonstrate their non--stellar
origin. However, it is still unclear whether the radiation is from a
jet or from the accretion flow. LLAGN have also been found to lie
on the so-called ``fundamental plane of black hole activity''
\citep{merloni03,falcke04}, which attempts to unify the emission from
all sources around black holes, over a large range of masses and
luminosities, from Galactic sources to powerful quasars. But the
origin of such a ``fundamental plane'' and its relationship with the
origin of the radiation is still a matter of debate
\cite[e.g][]{bregman05,koerding06}.
RIAF models have been applied to several sources belonging to
different classes, such as low-luminosity radio galaxies, ``normal''
ellipticals, the Galactic center Sagittarius~A
\citep[e.g.][]{quataert99,dimatteo00,dimatteo03}. However, for most
of these objects, the models cannot be properly constrained, mostly
because the nuclear radiation is swamped by other processes. This is
particularly problematic in the optical band, which appears to be
crucial to fix the models, where the stellar emission from the host
galaxy is substantial. It is indeed in the IR-to-UV region that
different accretion disk models are expected to show the largest
difference in spectral shape. RIAFs should lack both the "big
blue-bump" and the IR (reprocessed) bump, which instead characterize
optically thick, geometrically thin accretion disk emission and the
surrounding heated dust. For example, in low luminosity radio
galaxies non-thermal emission from the jet dominates the optical
nuclear radiation \citep{pap1}, while the Galactic center is not
visible in the optical because it is hidden by a large amount of dust.
Therefore, neither of the above mentioned classes of objects appear to
be suitable laboratories to test models of low-efficiency accretion
through the analysis of their overall SED. Recently, among LLAGN
which show very low Eddington ratios $L_{bol}/L_{Edd} << 10^{-3}$,
\citet{papllagn} have found that a class of LLAGN, mainly composed by
LINERs and low-luminosity Seyfert~1 galaxies, show faint optical
unresolved nuclei in HST images that may be interpreted as direct
radiation from a very low efficiency accretion flow. In fact, when
the radio-optical properties of LLAGN are considered, Seyfert, LINERs
and low luminosity radio galaxies separate into different regions of
diagnostic planes, according to the properties of their nuclei. If
this interpretation is correct, we now have a powerful tool to
identify the nature of the nuclear radiation (i.e. jet-dominated or
accretion-dominated). The best possibility of detecting radiation
from an ADAF-like process would then be to study unobscured Seyferts
of lowest luminosity, as well as a sub-class of LINERs. In all other
objects other radiation processes dominate.
In this paper we further test the picture outlined in
\citet{papllagn} by studying the nuclear spectral energy distribution
of a galaxy, \object{NGC~4565}, that seems to be a perfect candidate
for hosting a RIAF around the central supermassive black hole. The
object is part of the ``Palomar sample'' of LLAGN \citep{ho97}, and it
is included in both the \citet{merloni03} and \citet{falcke04} samples
that were used to define the ``fundamental plane of black hole
activity''. It is worth mentioning that NGC~4565 does not show any
significant peculiarity in that plane. NGC~4565 is a nearby (d=9.7
Mpc) LLAGN classified as a Seyfert~1.9 because of the possible
presence of a faint, relatively broad (FWHM = 1750 km s$^{-1}$)
H$\alpha$ line. However, as \citet{ho97broad} have pointed out, the
detection of a broad component is highly uncertain. As we show in the
following, although it is a Type 2 Seyfert, this object is only
moderately absorbed, and the nuclear radiation is visible in the
optical spectral region. NGC~4565 may thus represent the first clear
example of low-luminosity accretion onto a supermassive black hole in
the optical band.
In Section~\ref{obs} we describe the {\it Chandra} and {\it HST}
observations, the data analysis procedures and flux measurements, in
Section~\ref{discussion} we present the results, we derive the
spectral energy distribution and we discuss its interpretation. In
Section~\ref{conclusions} we give a summary of our findings and we
draw conclusions.
\section{Observations and data analysis}
\label{obs}
We use X-ray data taken with \facility{Chandra} satellite, and IR
through optical \facility{HST} images. In the following we describe
the data and the analysis procedures.
\subsection{{\it Chandra} data}
A 60 ksec ACIS-S observation of NGC~4565 (performed in 2003, PI
D. Wang) is publicly available in the {\it Chandra} archive. We
retrieve the {\it Chandra} data and analyze them using standard CIAO
3.2.2 procedures, applying the latest calibration files in the CALDB
3.1.0 database. The X-ray image reveals a wealth of pointlike sources,
many of which located along the NGC~4565 disk. The two brightest
sources correspond to an off-nuclear source at $\sim 50$ arcsec from
the nucleus, and to the nucleus itself \citep[see also the XMM image
in][]{foschini02}.
To avoid contamination from faint nearby X-ray sources, the 0.4-7 keV
nuclear spectrum is extracted in a circular aperture of 6 pixel radius
($\sim 3$ arcsec, corresponding to an encircled energy fraction of
$>97\%$ at 1.5 keV). The background is evaluated in a large annulus
around the nucleus. Faint X-ray sources are not masked out from the
background region, since their presence has a negligible impact on our
results (the total background flux including faint sources is less
than $1\%$ of the nuclear flux). Given the moderate nuclear count rate
(0.036 counts/sec), X-ray photon pileup is under control ($\lesssim
4\%$).
Spectral analysis is carried out with XSPEC v11.3.1, with the column
density of our Galaxy fixed to $1.3\:10^{20}$ cm$^{-2}$
\citep{dickey90}. The spectrum is re-binned to have at least 20 photons
per bin to allow use of the $\chi^2$ statistics, errors are quoted at
the $90\%$ c.l. for one interesting parameter. We find that an
absorbed power law model provides a very good description of the data
($\chi^2/dof=51/81$), the best fit photon index and column density
being $\Gamma=1.91^{+0.22}_{-0.19}$ and $N_H=2.5\pm0.6\;10^{21}$
cm$^{-2}$, respectively. The observed source fluxes in the 0.5-2 keV (soft)
and 2-10 keV (hard) bands are $9.0\;10^{-14}$ erg cm$^{-2}$ s$^{-1}$ and
$2.1\;10^{-13}$ erg cm$^{-2}$ s$^{-1}$, respectively. When corrected
for absorption, these correspond to intrinsic nuclear luminosities of
$1.9\;10^{39}$ erg s$^{-1}$ and $2.5\;10^{39}$ erg s$^{-1}$ in the
soft and hard band, respectively. We note that the derived X-ray
spectral parameters, fluxes and luminosities are in good agreement
with those measured in a 14 ksec XMM-$Newton$ observation performed in
2001 \citep{cappi06}.
\subsection{{\it HST} data}
\begin{deluxetable}{l r c c c}
\tabletypesize{\scriptsize}
\tablewidth{0pt}
\tablecaption{Nuclear fluxes from HST observations \label{fluxes}}
\tablehead{\colhead{Instrument/Filter} & \colhead{T$_{exp}$} & \colhead{Program ID} & \colhead{Wavelength} & \colhead{$F_\lambda$} \\
\colhead{} & \colhead{[s]} & \colhead{} & \colhead{[\AA]} & \colhead{ }\\}
\startdata
ACS-HRC/F330W & 1200 & 9379 & 3367 & 8.1 \\
WFPC2/F450W & 600 & 6092 & 4575 & 18 \\
WFPC2/F555W & 320 & 6685 & 5468 & 21 \\
WFPC2/F814W & 480 & 6092 & 8023 & 11 \\
NICMOS/F160W & 384 & 7331 & 16074 & 5.9 \\
\enddata
\tablecomments{Fluxes (in units of $10^{-17}$ erg cm$^{-2}$s$^{-1}$ \AA$^{-1}$) have been corrected for local extinction using $N_H=2.5\times
10^{21} cm^{-2}$ and standard $A_V/N_H = 5 \times 10^{-22}$ ratio.}
\end{deluxetable}
HST data are available in the MAST archive at STScI from the near IR
to the optical U-band. Images were taken as part of different
programs, with the following instruments and filters: NICMOS (F160W),
WFPC2 (F814W, F555W, F450W), ACS/HRC (F330W). These filters
approximate the H,I,V,B and U bands in the HST system. The
images are processed with the standard on-the-fly reprocessing
calibration pipeline \citep[see][]{acshandbook}.
The optical images show the bulge of the galaxy partially covered by a
prominent dust lane or disk seen almost edge-on (Fig. \ref{galaxy})
The inclination of the ``disk'' is such that the
central region of the bulge is not covered by a large amount of dust,
and a faint nuclear compact source (to which we refer as the {\it
nucleus}) is visible in all images.
\subsubsection{Nuclear photometry}
In the U-band, the emission from the bulge stars is low, and the
nucleus is the by far the brightest source in the field of view of the
ACS/HRC (Fig. \ref{profiles}). Photometry of the nucleus is thus
straightforward in the U-band, also thanks to the higher resolution,
smaller projected pixel-size of the HRC. On the other hand, in the IR
(NICMOS) and optical I,V and B band WFPC2 images (the target is always
located in one of the WF cameras) the contrast with the underlying
stellar background is low, thus the measurement of the nuclear flux is
more problematic. For the photometry of nuclear unresolved sources
superimposed to the stellar emission of the host galaxy, we undertake
two different approaches, as described in the following.
1) Aperture photometry with the {\it IRAF } task {\it radprof},
measuring the background close to the unresolved nucleus, at a
distance of $\sim 0.4^{\prime\prime}$ from the center of the point
source, and setting the aperture radius at the same distance. Note
that the ``background'' here is the stellar emission of the galaxy in
the vicinity of the nucleus. Therefore, this approach works well for
nuclei in elliptical galaxies with flat radial brightness profiles,
i.e. Nuker-law ``core'' galaxies \citep{faber97} which have a flat
($\gamma < 0.3$, $\Sigma \propto r^{-\gamma}$, where $\Sigma$ is the
surface brightness) slope in the inner region \citep[see also the
discussion in][]{barbara}. Clearly, this is because in this case
the ``background'' measured at a distance of $\sim 0.4^{\prime\prime}$
is a good estimate of the stellar emission at the center of the
nucleus. On the other hand, for both ``power-law'' ellipticals and
spirals bulges, the profile in the innermost regions (i.e. in the
central 1-2 arcsec) is significantly steeper ($\gamma \sim 0.8$). In
this latter case, the measurement and even the identification of faint
nuclei is more difficult, because a ``peaked'' brightness profile of
the bulge may hamper the detection of the central emission from the
AGN. Furthermore, for the IR images, which have a lower angular
resolution, the background cannot be measured close enough to the
center of the nucleus, and thus may be significantly underestimated.
We find that this would lead to overestimate the nuclear flux by a
factor as large as $\sim 5$. Thus, while we used this method to
measure the nuclear flux in the F330W image \citep[aperture correction
was taken into account, following the prescriptions given
in][]{sirianniacs}, we had to adopt a different strategy for the
WFPC2 and NICMOS images.
2) An alternative approach consists in deriving the radial brightness
profile of the galaxy, and measure any nuclear excess. Multiple
component models are often used to reproduce the central regions of
galaxies and measure the flux of nuclear sources \citep[see e.g. the
discussion in][]{quillen01}. But since here we are not interested
in modeling the galaxy bulge on large scales, we only derive the
radial brightness profile in the central $\sim 2$ arcsec. Then we
produce a model galaxy with the same slope as observed in the region
$R > 0.2$ arcsec and we assume that the profile can be extrapolated to
the center of the bulge, all the way to $R=0$. As shown in Fig.
\ref{profiles} (solid line), the effect of the finite resolution of
HST ($\sim 0.1$ arcsec) produces a flattening of the observed profile,
at a distance of $\sim 0.15$ arcsec. The observed profile, obtained
by fitting ellipses to the galaxy image using the {\it IRAF} task {\it
ellipse}, shows a significant excess (filled circles). We produce
synthetic PSF's using the {\it TinyTim} software \citep{krist}, which,
for WFCPC2, produces an accurate representation of the central region
of the PSF, thus appropriate for our purpose. We align the synthetic
PSF's with the position of the nucleus, we multiply the PSF image by
an appropriate constant $K$ and we subtract the two images. We change
the value of $K$ until the profile of the nuclear regions do not
produce a ``hole'' at the center of the galaxy. The obtained profile
is shown in Fig. \ref{profiles} as the empty circles. To convert the
flux of the nucleus from counts to physical units, we multiply the
count rate ($CR = K/t_{\rm exp}$) by the keyword {\it PHOTFLAM} in the
image header (for NICMOS $CR=K$). In Fig.~\ref{profiles} we also show
the radial profiles obtained by subtracting PSFs that are 20\%
brighter and 20\% fainter than our reference value.
It is not straightforward to estimate the error on the flux
measurements obtained using method 2, because the main uncertainty is
the assumption that the radial profile of the galaxy can be
extrapolated all the way to the center, at R=0. After subtracting
synthetic PSFs with different total counts and comparing the resulting
profiles with our model profile, we prefer to adopt a rather
conservative value of $\sim 20\%$ for the error of the IR and optical
fluxes. With future observations, which should be taken using a
dithering strategy aimed at improving the PSF sampling, the error
could be significantly reduced. The statistical error on the F330W
flux (obtained with method 1) is 7\%. A summary of the photometry is
given in Table~\ref{fluxes}.
The F450W and F555W filter pass bands include relatively strong
emission lines (mainly [OIII]5007 and H$\beta$). However, since the
pass bands are $\sim 1000$\AA~~ wide, the observed flux is likely to be
dominated by continuum emission (see also Sect. \ref{discussion}).
Since the near-IR and the U-band images were not taken simultaneously
to the optical data, the SED may be affected by variability. We
checked for variability of the nuclear source in the optical (F814W
and F450W), for which two sets of observations with the same filters,
taken at a distance of $\sim 1$ year, are available. The nuclear
fluxes are consistent within the errors, thus no variability is found
between the 2 observations. However, our estimated $20\%$ error on
the optical fluxes does not allow us to exclude variations of smaller
amplitude, as observed in other LLAGN \citep{Maoz05}.
\section{Results and discussion}
\label{discussion}
\subsection{The nuclear SED of NGC~4565}
The absorption corrected nuclear spectral energy distribution is shown
in Fig.~\ref{sed}. The HST data are de-reddened using $N_H = 2.5
\times 10^{21}$ cm$^{-2}$, which converts to $A_V = 1.25$, assuming
Galactic gas-to-dust ratio. Although in AGN the gas-to-dust ratio may
differ from the local value, we believe that this choice is justified
in the case of NGC~4565. As it is clear from the large field of view
image of the galaxy (Fig.~\ref{galaxy}), this is a spiral seen almost
edge on. Therefore, it is reasonable to assume that a significant
amount of dust and gas in the disk of the galaxy project on our
line-of-sight to the nucleus.
Assuming a circular geometry, from the observed ellipticity of the
disk in the image we find that the orientation of the disk is likely
not to exceed $\sim 10^\circ$. For comparison, we can check the
absorption we find in our Galaxy for $10^{\circ}$ Galactic
Latitude. We obtain $A_V \sim 1.0 - 1.5$, where the lower value is
found for Galactic longitude $\sim 180^\circ$, the higher value is for
$\sim 0^\circ$ (from NED). These values may actually increase
substantially if we observe the same Galactic Latitude from the Galaxy
center. This simple check shows that the absorption we measure in the
X-rays is compatible with that provided by galactic dust in the disk.
This supports our hypothesis that the moderate absorption observed to
the nucleus of NGC~4565 is not produced locally, in the vicinity of
the nucleus. In this case, Galactic dust-to-gas ratio may be used to
convert $N_H$ derived from the X-rays to optical $A_V$.
The nuclear SED appears basically flat ($\alpha \sim 1$, $F_\nu
\propto \nu^{-\alpha}$) from the 1.6$\mu$m to 4500 \AA, with possibly
a small peak between 5000 and 4000 \AA~~ and a small drop-off in the
U-band. This peak may be real, or due to a possible contamination
from emission lines (mainly [OIII] and H$\beta$) that fall in the
F555W and F450W filters pass bands. Unfortunately, since neither
images with narrow-band filter nor nuclear spectra are available to
date, a certain ambiguity persists. However, all other filters are
free from strong lines, thus the intrinsic SED cannot be dramatically
different from what we show here. Whatever the nature of such a small
peak, it is clear that neither a significant UV bump nor IR thermal
emission from hot dust, which are characteristic of AGNs, are visible
in NGC~4565. Furthermore, note that the luminosity in the X-ray is not
higher than in the optical, even after absorption has been taken into
account (see Fig.~\ref{xspec}).
For comparison, in Fig.~\ref{sed} we show the nuclear SED of two
specific objects, together with the average SED of radio-quiet QSO as
taken from \citet{elvis94}. The two objects are a Seyfert~1 galaxy
(\object{NGC~3516}) and of a Seyfert~2 (\object{Fairall~49}), for
which the absorption, estimated from X-ray observations, is $N_H = 1.4
\times 10^{22}$ \citep{iwasawa04}. These two Seyferts have similar
bolometric luminosity, but they are both clearly more powerful than
NGC~4565, by $\sim 3$ orders of magnitude. The Type~1 object clearly
shows a concave spectrum, which is interpreted as the signature of the
presence of the blue bump in the UV and of dust heated by the central
AGN in the near-IR. The Type~2 galaxy, instead, is very bright in the
IR, while the flux is dramatically reduced for higher (optical)
frequencies, as a result of absorption.\footnote{It might seem
surprising to detect the nucleus in Fairall~49, which is a Seyfert~2
absorbed by a large amount of $N_H$ in the X-rays. In fact, if
converted to $A_V$ using standard Galactic dust-to-gas ratio, this
would correspond to 7 mag extinction in V and 11.5 mag in the F330W
filter. Two possible explanations have been proposed: i) part of the
optical nuclear flux in Fairall~49, if not all, might not be radiation
from the accretion disk seen directly (through a moderate amount of
dust). Instead, the nucleus might be in part (or completely) obscured
in the optical-to-UV band, and the bulk of the observed emission may
be scattered light; ii) the properties of the circumnuclear absorber
are different from the Galactic dust and this would result in a
non-standard $A_V/N_H$ ratio. This has been discussed by various
authors \citep[e.g.][]{granato97,maiolino01}.} Not all Seyferts show
such a clear behavior, but these objects serve as good examples to be
compared with the peculiar SED of NGC~4565.
\subsection{A low radiative efficiency accretion disk}
How do we interpret the nuclear emission in NGC~4565? As pointed
out in the introduction, the source is included in the sample of both
\citet{merloni03} and \citet{falcke04}. The object does not appear to
show any peculiarity, and its location along the ``fundamental plane
of black hole activity'' does not provide us with information on the
emission process. Therefore, in order to answer the above question,
we use diagnostics introduced by \citet{papllagn}. As already
mentioned in Section~1, we found that the radio to optical nuclear
luminosity ratio (i.e. the ``nuclear radio-loudness'') $L_r/L_o$
for LLAGNs gives us important information on the nature of the source.
In particular, we can infer whether we are observing synchrotron
emission in both bands (if $\log (L_r/L_o)\sim 3$), or in the
optical we have some kind of excess radiation which, in the case of
unabsorbed Seyferts is most likely interpreted as radiation from the
accretion process. Furthermore, when the optical luminosity to
Eddington luminosity ratio $L_o/L_{Edd}$ is plotted against the
nuclear radio loudness (Fig. \ref{r_edd}) different classes of LLAGNs
nicely separate into three different regions of the diagram. Seyfert
nuclei with relatively high accretion efficiency objects occupy the
top-left part of the diagram ($L_o/L_{Edd} \sim 10^{-2}-10^{-3}$ and
$\log(L_r/L_o) \sim 1$); LINERs separate into two subclasses,
which we named according to their nuclear radio-optical ratio as
``radio--quiet'' LINERs (bottom-left side, $L_o/L_{Edd} < 10^{-4}$ and
$\log(L_r/L_o) \sim 1$), and ``radio--loud'' LINERS
(bottom-right); radio galaxies (bottom-right part of the diagram) have
the same Eddington ratio as for radio--quiet LINERs, but a much higher
$\log(L_r/L_o)$. Note that in the plane of Fig. \ref{r_edd}, the
objects in which we observe an extra-component in the optical, in
excess of synchrotron emission, are those that lie on the left side.
Therefore, those are the objects in which emission from the accretion
process can be detected. In fact, in perfect agreement with the
shape of its SED, NGC~3516 lies in top-left panel of the plane, where
relatively high efficiency accretion disks are. On the other hand, we
did not mark the location of the Seyfert~2 galaxy Fairall~49 on the
plot of Fig.~\ref{r_edd}, since the nuclear emission is most likely to
be affected by significant obscuration. However, we point out
that assuming that the IR flux can be used as instead of the optical,
the radio to IR ratio would be similar to the Seyfert 1s in the plot
(at $\log L_r/L_o = 0.88$).
Let us explore how this applies to NGC~4565. First of all, we
calculate the ratio between the nuclear radio flux and the optical
flux. \citet{neil05} measured a radio core flux of 3.2 mJy, which
implies that $\log(L_r/L_o) = 1.6$. Therefore, NGC~4565 has an
excess in the optical of at least 2 dex with respect to the expected
synchrotron emission (i.e. the optical counterpart of the radio core
should be $> 2$ dex fainter than the measured optical flux, unless the
radio-to optical spectral index has unreasonable values for
synchrotron radiation). Thus it is reasonable to interpret the
optical nucleus as radiation from the accretion process. In this
case, assuming a central black hole mass of $2.8 \times 10^{7} M_\sun$,
as derived from the M-$\sigma$ relation of \citet{tremaine02}, and
using $\sigma$ value from the LEDA
database\footnote{http://leda.univ-lyon1.fr/}, the resulting Eddington
ratio $L_o/L_{Edd}$ is extremely low, $2 \times 10^{-6}$. It is
important to note that in the case of ``typical'' Seyferts, which show
a blue bump, a significant bolometric correction is needed (however,
this should not exceed a factor of $\sim 15$). For NGC~4565 a big
blue bump is clearly not present, thus our value of $L_o$ is a
good estimate of $L_{bol}$.
In the diagnostic diagram of Fig.~\ref{r_edd}, NGC~4565 lies in the
lower-right quadrant, among ``radio-quiet'' LINERs \citep[the other
two Seyferts in the same region of the plot are M~81 and NGC~4639, as
discussed in][]{papllagn}. In order to reconcile NGC~4565 with other
Seyferts, which are confined in the top-left quadrant, the central
black hole mass would have to be at least a factor of $\sim 100$
lower. This would substantially violate the $M_{BH} - \sigma$
relation. We conclude that the nucleus of NGC~4565 is a very
low-efficiency accretion object and that we are observing the
accretion process directly in the optical. This is extremely
important since models of advection-dominated accretion flows are
particularly sensitive to the optical-UV spectral region. For
example, as shown by \citet{quataert99} the presence of winds in the
disk can dramatically change the shape of the observed SED in the
range of frequencies between $\nu \sim 10^{13}$ Hz and $\nu \sim
10^{15}$ Hz. However, a detailed comparison with a set of models
is beyond the scope of this paper. RIAF models are very sensitive to
many different physical parameters, therefore detailed modeling is
useful only if the SED is well determined, from the radio-mm to the
X-ray band and, if possible, when simultaneous data are available.
A similar study of the nuclear emission has been performed by
\citet{moran99} for NGC~4395, ``the least luminous Seyfert~1''. In
that case, the nuclear luminosity is even lower than in NGC~4565, but
the central black hole mass in NGC~4395 is dramatically lower. A
recent estimate based on reverberation mapping gives a value of
$M_{BH} = 3.6 \times 10^{5} M_\sun$ \citep{peterson05}. Such a low
black hole mass implies $L_{bol}/L_{Edd} \sim 10^{-3}$ or higher if,
as \citet{moran99} point out, intrinsic nuclear absorption is present.
This seems in fact to be the case since \citep{moran05} obtain $N_H
\sim 10^{22}$ cm$^{-2}$ analyzing {\it Chandra} data, although most of
the absorption might be produced by ionized gas. Therefore, although
it is clear that even if NGC~4395 displays peculiar characteristics,
its Eddington ratio is not different from the average of low
luminosity Seyferts in the Palomar and CfA samples
\citep{papllagn,ho04coz}. Instead, NGC~4565 has completely different
physical properties, as appears from its extremely low value of the
Eddington ratio, and it is a perfect candidate for hosting a radiative
inefficient accretion process.
\subsection{Where is the narrow line region in NGC~4565?}
One further implication of the observations we present here is worth
mentioning. The flux of the [OIII]5007 emission line ($F_{[OIII]} =
1.5 \times 10^{-14}$ erg s$^{-1}$ cm$^{-2}$), as measured from the
ground with a 2'' beam size \citep{ho97}, and the diagnostic line
ratios are typical of Seyfert galaxies. Even considering the
interesting (although only slightly different) classification scheme
for LLAGN proposed by \citet{kewley06} based on SDSS data, NGC~4565
still falls in the region occupied by Seyferts. However, if we
assume that all of the [OIII] flux is produced in the unresolved
nucleus, this would result in a count rate higher by factor of 5 and
20 than we measure in the nucleus in the F555W and F450W,
respectively. This implies that the narrow emission line region (NLR)
must be extended. It is particularly interesting to investigate the
properties of the NLR relatively to the nuclear properties, since it
is sometimes assumed that radiative inefficient accretion cannot
provide a sufficient photon field to ionize the surrounding medium and
create the NLR. If this is true, we can speculate that NGC~4565 may
have recently transitioned from a relatively high-efficiency accretion
state (as in ``normal'' Seyferts) to a very low-efficiency accretion
process. This might also reconcile its classification as Seyfert with
the fact that its nucleus is located among LINERs in the plane of
Fig.~\ref{r_edd}. The spectral classification is in fact based on the
large-scale properties of the emission-line gas, that may still be
powered by a higher radiation field (possibly having also a different
spectral shape), because of light travel time effects. However, the
equivalent width of the [OIII] line, as measured with respect to the
nuclear continuum emission, is $EW_{[OIII]} \sim 100$\AA. This value
is in the range normally spanned by Type~1 AGN \citep[see
e.g.][]{kinney91,pap4,marziani03}. Therefore, this may indicate that
the continuum emission from RIAFs is sufficient to account for the
observed [OIII] flux, and Seyferts' NLR can be powered by radiatively
inefficient accretion flows. However, in the scenario in which the
accretion disk has changed its state, $EW_{[OIII]}$ may assume a low
value if only a fraction of the NLR gas is still highly ionized under
the effect of the ``past'' high--efficiency state of the accretion.
Clearly, only high spatial-resolution images with narrow band filters,
and nuclear spectra, can provide further information to understand the
recent history and ionizing mechanism of both the nucleus and NLR of
NGC~4565.
\section{Summary and conclusions}
\label{conclusions}
We have derived the spectral energy distribution of a peculiar
low-luminosity Seyfert~2 galaxy which, despite its spectral
classification, basically shows no evidence for local nuclear
absorption. The SED is peculiar, as it is almost flat in a $\log \nu -
log (\nu F\nu)$ representation, with no sign of both a UV bump and
thermally reprocessed IR emission. The very low luminosity of the
source associated with a relatively high central black hole mass imply
an extremely small value of the Eddington ratio ($L_o/L_{Edd} \sim
10^{-6}$). This, together with the position occupied by this object on
diagnostic planes for low luminosity AGN, represents clear evidence
for a low radiative efficiency accretion process at work in the
innermost regions of NGC~4565. The direct detection of optical
emission from such radiative inefficient processes is particularly
important for providing constraints to ADAF models or similar.
NGC~4565 is therefore a perfect candidate for studying RIAFs, and more
observations aimed at achieving a complete coverage of the SED, from
the radio-mm to the X-ray bands, should be taken in order to test the
models. As part of a ``search for RIAFs in LLAGN'', it would also
be extremely useful to derive the SED of objects that are located in
the same region of the diagnostic planes as NGC~4565.
The fact that the [OIII] emission line flux is substantial in this
object implies that an extended narrow line region, similar to other
Seyfert galaxies, is still present in NGC~4565. A possible intriguing
scenario is that the active nucleus has recently ``turned-off'',
switching from a high efficiency, standard, accretion disk, to a
radiative inefficient accretion process. However, since the EW of the
[OIII]5007 emission line is rather small, with the present data we
cannot rule out that the amount of ionizing photons from the RIAF is
sufficient to produce the observed [OIII] flux.
\acknowledgments
We thank the anonymous referee for her/his comments that greatly
improved the paper. We acknowledge Dave Axon, Alessandro Capetti and
Alice Quillen for useful comments. RG
acknowledges support from the STScI Visitor Program.
This research has made use of the NASA/IPAC Extragalactic Database
(NED) which is operated by the Jet Propulsion Laboratory, California
Institute of Technology, under contract with the National Aeronautics
and Space Administration.
{\it Facilities:} \facility{HST}, \facility{Chandra}.
|
Title:
Plane waves in metric-affine gravity |
Abstract: We describe plane-fronted waves in the Yang-Mills type quadratic
metric-affine theory of gravity. The torsion and the nonmetricity are both
nontrivial, and they do not belong to the triplet ansatz.
| https://export.arxiv.org/pdf/gr-qc/0601074 |
\title{Plane waves in metric-affine gravity}
\author{Yuri N.~Obukhov\footnote{On leave from:
Dept. of Theoret. Physics, Moscow State University, 117234 Moscow, Russia}}
\address{Institute for Theoretical Physics, University of Cologne,
50923 K\"oln, Germany}
\bigskip
\noindent PACS: 04.50.+h, 04.30.-w, 04.20.Jb.
\section{Introduction}
Although Einstein's general relativity theory is satisfactorily supported
by experimental tests on a macroscopic level, the gravitational interaction
on a microscopic scale is not well understood. The gravitational gauge
models provide an alternative description of gravitational physics in
the microworld \cite{PR}. A variety of models arise within the framework of
the gauge approach to gravity (Poincar\'e, teleparallel, metric-affine,
supergravity, to mention but a few), and their corresponding {\it kinematic}
schemes are well established at present. However, the {\it dynamic} aspects
of the gauge gravity models have been rather poorly studied up to now. This
includes the choice of the basic Lagrangian of the theory, as well as the
detailed analysis of possible physical effects. The derivation of a new
exact solutions for these models may bring new insight to the understanding
of gravitational physics on small scales.
The plane-fronted gravitational waves represent an important class of exact
solutions which generalize the basic properties of electromagnetic waves
in flat spacetime to the case of curved spacetime geometry. The relevant
investigation of the gravitational waves in general relativity has a
long and rich history, see, e.g., \cite{peres,pen1,pen2,griff,vdz,exact}.
The discussion of the possible generalizations of such solutions revealed
the exact wave solutions in Poincar\'e gauge gravity
\cite{adam,chen,sippel,vadim,singh,babu}, in teleparallel gravity
\cite{tele}, in generalized Einstein theories \cite{gurses,lovelock},
in supergravity \cite{sg1,sg2,sg3,sg4,sg5}, as well as, more recently,
in superstring theories \cite{gimon,ark1,ark2,str1,str2,str3,str4}.
Some attention has also been paid to the higher-dimensional generalizations
of the gravitational wave solutions \cite{coley1,coley2,hervik,ndim}.
It was demonstrated \cite{dirk1,dirk2,king,vas1,vas2,vas3} that gravitational
wave solutions are also admitted in the metric-affine theory of gravity (MAG)
with the propagating torsion and nonmetricity fields. The latter results are,
however, restricted either to the case of torsion waves only, or to the
triplet class of solutions with a specific ansatz for torsion and
nonmetricity \cite{eff,tri} and for a special form of the Lagrangian.
The aim of this paper is to describe the plane gravitational waves for the
general Yang-Mills type quadratic MAG Lagrangian with nontrivial torsion and
nonmetricity configurations that do not belong to the triplet ansatz.
The motivation is twofold. On the one hand, the systematic study of the
space of solutions represents a significant aspect of the development of
any field-theoretic model. On the other hand, the wave phenomena as such
are of fundamental importance, and the construction and comparison of the
wave solutions in different models may clarify the physical contents of and
the relations between the microscopic and macroscopic gravitational theories
(in particular, general relativity, Poincar\'e gauge gravity and MAG).
The metric-affine spacetime is described by the metric
$g_{\alpha\beta}$, the coframe 1-forms $\vartheta^{\alpha}$, and the linear
connection 1-forms $\Gamma_{\beta}{}^{\alpha}$. These are interpreted as
generalized gauge potentials, while the corresponding field strengths
are the nonmetricity 1-form $Q_{\alpha\beta}=-Dg_{\alpha\beta}$ and the
2-forms of torsion $T^{\alpha}=D\vartheta^{\alpha}$ and curvature
$R_{\beta}{}^{\alpha}=d\Gamma_{\beta}{}^{\alpha} + \Gamma_{\gamma}{}^{\alpha}
\wedge\Gamma_{\beta}{}^{\gamma}$. The metric-affine geometry reduces to a
purely Riemannian one as soon as torsion and nonmetricity both vanish. The
teleparallel geometry arises when the curvature is trivial, $R_{\beta}
{}^{\alpha} = 0$, whereas a vanishing nonmetricity $Q_{\alpha\beta}=0$
yields the Riemann-Cartan geometry of spacetime. It is well known that for
every metric $g_{\alpha\beta}$ there exists a unique torsion-free and
metric-compatible connection represented by the Christoffel symbols. We
will denote this Riemannian connection by $\widetilde{\Gamma}_{\beta}
{}^{\alpha}$, and hereafter the tilde will denote purely Riemannian
geometrical objects and covariant differentials constructed from them.
Our general notations and conventions for the basic geometric objects,
the holonomic and anholonomic indices, the choice of the metric signature
are that of \cite{PR}.
The plan of the paper is as follows. In the next Sec.~\ref{emwave}, we
recall the definition of the ordinary electromagnetic wave. This is used then
in Sec.~\ref{ansatz} for the description of the corresponding ansatz for
a gravitational plane wave in MAG. The properties of the resulting
curvature, torsion and nonmetricity are discussed in Sec.~\ref{NTC}.
Finally, in Sec.~\ref{eqs} we demonstrate that the proposed ansatz provides
the exact solution for the general quadratic MAG model. The conclusions are
outlined in Sec~\ref{conclusion}.
\section{Electromagnetic plane waves}\label{emwave}
An electromagnetic plane wave is described by a 1-form $u$ which
satisfies
\begin{eqnarray}
d\,{}^\ast du &=& 0,\label{ddu}\\
k\wedge{}^\ast du = 0,\qquad k\wedge du &=& 0.\label{kdu}
\end{eqnarray}
The propagation 1-form $k$ is null (i.e., $k\wedge{}^\ast k =0$) and
geodetic, $k\wedge{}^\ast dk =0$, and it corresponds to a congruence with
zero shear, expansion and rotation. This is typical for the plane wave, and
the equations (\ref{kdu}) represent the so-called radiation conditions
imposed on the electromagnetic field. With the electromagnetic potential
$A = u$, the electromagnetic field strength $F = du$ satisfies the vacuum
Maxwell equation (\ref{ddu}).
We do not specify the spacetime metric (and hence the Hodge operator ${}^\ast$)
as the flat Minkowski one. It will be convenient not to fix the spacetime
geometry at this stage.
Such an electromagnetic wave construction underlies the derivation of the
corresponding gravitational wave solutions as described in
\cite{pleb,orr,pod1,pod2,pod3}, for example. In our study, we will use
the similar constructions by extending the Riemannian results to their
non-Riemannian counterparts.
\section{Wave ansatz for metric, coframe and affine connection}\label{ansatz}
Let us denote the local spacetime coordinates as $x^i = \{\sigma, \rho, z^2,
z^3\}$. The upper case Latin indices, $A,B,\dots = 0, 1$, will label the first
2 spacetime dimensions which are relevant to a $pp$-wave. In particular,
$x^A = \{\sigma, \rho\}$ are the wave coordinates with the wave fronts
described by the surfaces of constant $\sigma$, and $\rho$ is an affine
parameter along the wave vector of the null geodesic. The lower case Latin
indices, $a,b,\dots = 2,3$, refer to the remaining spatial coordinates:
$x^a = \{z^2, z^3\}$. The Greek indices, $\alpha, \beta, \dots = 0,\dots, 3$,
label the local anholonomic (co)frame components. We denote separate frame
components by a circumflex over the corresponding index in order to
distinguish them from coordinate components.
The can now formulate the wave ansatz for the MAG gravitational potentials
$g_{\alpha\beta}$, $\vartheta^{\alpha}$, and $\Gamma_{\beta}{}^{\alpha}$ as
follows. We choose the half-null metric
\begin{equation}
g_{\alpha\beta} = \left(\begin{array}{cc}g_{AB}&0\\ 0&g_{ab}\end{array}
\right),\qquad g_{AB} = \left(\begin{array}{cc}0&1\\ 1&0\end{array}\right),
\quad g_{ab} = \delta_{ab}.\label{met}
\end{equation}
The components of the coframe 1-form are given by
\begin{equation}\label{cof}
\vartheta^{\widehat{0}} = -\,d\sigma,\qquad \vartheta^{\widehat{1}} = {\frac
12}\,H(\sigma, z^a)\,d\sigma + d\rho,\qquad \vartheta^a = dz^a,\quad a=2,3.
\end{equation}
Finally, the ansatz for the affine connection reads:
\begin{equation}\label{conn}
\Gamma^{\alpha\beta} = k^{[\alpha}\varphi^{\beta]}\,k + k^\alpha k^\beta\,u.
\end{equation}
The corresponding dual frame basis (such that $e_\alpha\rfloor\vartheta^\beta
= \delta_\alpha^\beta$) reads:
\begin{equation}
e_{\widehat{0}} = -\,\partial_\sigma + H\,\partial_\rho,\qquad
e_{\widehat{1}} = \partial_\rho,\qquad e_a = \partial_a.\label{frame}
\end{equation}
We now make a crucial assumption that the 1-forms $u$ and $k$ above fulfill
the radiation conditions (\ref{ddu}), (\ref{kdu}). Moreover, using the
local coordinate system adapted for a plane wave and the half-null nature of
the coframe (\ref{cof}), we can put $k = \vartheta^{\widehat{0}}$ without
loss of generality. As usual, we introduce the components as $k_\alpha
= e_\alpha\rfloor k$. Finally, the components of $\varphi_\alpha$ are
determined by the function $H$ as follows:
\begin{equation}
\varphi_{\widehat{0}} = 0,\qquad \varphi_{\widehat{1}} = 0,\qquad
\varphi_a = \partial_a H.
\end{equation}
Although this choice looks to be rather ad hoc, it is actually well
motivated by the corresponding Riemannian solution, cf. \cite{ndim}.
Indeed, the ansatz (\ref{met}) and (\ref{cof}) for the metric and the
coframe is exactly the same as in the purely Riemannian case, whereas
the connection (\ref{conn}) minimally extends the Christoffel connection
(given in Appendix of \cite{ndim}) via the term proportional to the
non-Riemannian parameter $u$. In the next section we demonstrate that
both the torsion and the nonmetricity are determined by this 1-form.
It is worthwhile to note that the wave 1-form is closed, $dk = 0$, whereas
the wave covector is covariantly constant,
\begin{equation}
D k_\alpha = -\,\Gamma_\alpha{}^\beta\,k_\beta = 0,\qquad
D k^\alpha = \Gamma_\beta{}^\alpha\,k^\beta = 0.\label{Dk}
\end{equation}
Here we used the fact that this covector is null, $k^\alpha k_\alpha =0$
and orthogonal to $\varphi_\alpha$, i.e. $k^\alpha\varphi_\alpha =0$.
\section{Nonmetricity, torsion and curvature}\label{NTC}
Given the above ansatz for the MAG potentials -- metric (\ref{met}), coframe
(\ref{cof}) and linear connection (\ref{conn}) -- it is straightforward to
find the corresponding gauge field strengths. The nonmetricity, torsion and
curvature read, explicitly:
\begin{eqnarray}
Q_{\alpha\beta} &=& 2k_\alpha k_\beta\,u,\label{nonm}\\
T^\alpha &=& k^\alpha\,u\wedge k,\label{tor}\\
R^{\alpha\beta} &=& 2\gamma^{[\alpha}k^{\beta]}\wedge k
+ k^\alpha k^\beta\,du.\label{curv}
\end{eqnarray}
Here the covector-valued 1-form is defined by $\gamma_\alpha = - {\frac 12}
\,\underline{d}\varphi_\alpha$, where the differential $\underline{d}$ is
taken with respect to the $z^a$ coordinates only. This 1-form has the
obvious properties: $k^\alpha\gamma_\alpha = 0$, $\vartheta^\alpha\wedge
\gamma_\alpha = 0$ and $e_A\rfloor\gamma_\alpha = 0$, $A = 0,1$.
When $u = 0$, we find zero torsion and nonmetricity. In this sense, we may
consider the nontrivial $u$ to represent the true post-Riemannian geometric
structures which we are primarily interested in.
Let us compute the irreducible parts of the curvature 2-form. It is well
known \cite{PR} that the curvature for the general linear connection can
be decomposed into 11 irreducible parts. Following \cite{PR}, we first
decompose the curvature 2-form into the skew-symmetric and symmetric forms,
\begin{equation}
W^{\alpha\beta} = R^{[\alpha\beta]} = 2\gamma^{[\alpha}k^{\beta]}\wedge k,
\qquad Z^{\alpha\beta} = R^{(\alpha\beta)} = k^\alpha k^\beta\,du.\label{WZ}
\end{equation}
Then we have to calculate the interior products with the frame and exterior
products with the coframe. In tensor language, this corresponds
to computing the various contractions of the curvature tensor.
All the contractions of the symmetric curvature are trivial. Namely, $Z =
Z_\alpha{}^\alpha = 0$ in view of the nullity of the wave vector ($k_\alpha
k^\alpha = 0$), whereas
\begin{equation}
e_\alpha\rfloor Z^{\alpha\beta} = -\,k^\beta\,{}^\ast(k\wedge{}^\ast du) =0,
\qquad \vartheta_\alpha\wedge Z^{\alpha\beta} = k^\beta\,k\wedge du =0,
\end{equation}
due to the fact that $k_\alpha\vartheta^\alpha = k$ and using the radiation
conditions (\ref{kdu}). As a result, all the irreducible parts of the
symmetric curvature form are zero except for the first piece:
\begin{equation}
Z_{\alpha\beta} = {}^{(1)}\!Z_{\alpha\beta}.\label{Z1}
\end{equation}
In tensor language this means that all the contractions of the symmetric
part of the curvature tensor are trivial, i.e., this tensor is totally
trace-free and dual trace-free. The symmetric part of the curvature has
no Riemannian counterpart, this is a totally post-Riemannian object.
For the skew-symmetric curvature we find straightforwardly:
\begin{equation}
e_\alpha\rfloor W^{\alpha\beta} = (e_\alpha\rfloor\gamma^\alpha)\,k^\beta\,k,
\qquad \vartheta_\alpha\wedge W^{\alpha\beta} = 0.
\end{equation}
Accordingly, if we demand that the zero-form $e_\alpha\rfloor\gamma^\alpha$
vanishes, then all the contractions of the skew-symmetric curvature are also
trivial. This condition imposes the partial differential equation on the
unknown function $H$:
\begin{equation}\label{ddH}
e_\alpha\rfloor\gamma^\alpha = -{\frac 12}\,\partial_a\partial^a H = 0.
\end{equation}
Provided that $H(\sigma, z^a)$ is a solution of the Laplace equation
(\ref{ddH}), we ultimately find that all the irreducible parts of the
skew-symmetric curvature form are zero except for the first piece
\begin{equation}
W_{\alpha\beta} = {}^{(1)}\!W_{\alpha\beta}.\label{W1}
\end{equation}
This is again the pure tensor part which is totally trace-free and
dual trace-free, in complete analogy with the symmetric curvature. The
2-form ${}^{(1)}\!W_{\alpha\beta}$ is a direct non-Riemannian generalization
of the Weyl tensor.
\section{Field equations: quadratic MAG model}\label{eqs}
Let us consider the general Yang-Mills type (curvature quadratic) Lagrangian
for the MAG model which was studied recently in the literature
(see, for example, \cite{dirk1,dirk2,king,vas1,vas2,vas3}):
\begin{eqnarray}
V_{\rm MAG}&=& -\,{\frac{1}{2}}\,R^{\alpha\beta}
\wedge{}^*\!\left(\sum_{I=1}^{6}w_{I}\,^{(I)}W_{\alpha\beta}
+ w_7\,\vartheta_\alpha\wedge(e_\gamma\rfloor
{}^{(5)}W^\gamma{}_{\beta} ) \nonumber\right.\\
&& +\,\left.\sum_{I=1}^{5}{z}_{I}\,^{(I)}Z_{\alpha\beta} + z_6
\,\vartheta_\gamma\wedge (e_\alpha\rfloor ^{(2)}Z^\gamma{}_{\beta})
+\sum_{I=7}^{9}z_I\,\vartheta_\alpha\wedge(e_\gamma\rfloor
^{(I-4)}Z^\gamma{}_{\beta} )\right)\label{QMA}\,.
\end{eqnarray}
The 16 dimensionless coupling constants $w_1, \ldots w_7$, $z_1, \ldots z_9$
describe the contributions of all possible quadratic invariants which can be
constructed from the components of the curvature in a general MAG theory
\cite{remark}.
The vacuum gravitational field equations of the MAG theory read \cite{PR}:
\begin{eqnarray}
DH_{\alpha}- E_{\alpha}&=& 0,\label{first}\\
DH^{\alpha}{}_{\beta}-E^{\alpha}{}_{\beta}&=& 0.\label{second}
\end{eqnarray}
The gravitational gauge field momenta are introduced by partial
differentiation,
\begin{equation}
H_\alpha = - \frac{\partial V}{\partial T^\alpha}\,,\quad
H^\alpha{}_\beta= -\frac{ \partial V}{\partial R_\alpha{}^\beta}\,,\quad
M^{\alpha\beta} = - 2\frac{\partial V}{\partial Q_{\alpha\beta}}\,,\label{3}
\end{equation}
whereas the canonical gauge field currents of the gravitational
energy--momentum and of the hypermomentum, respectively, are defined as
the following expressions, linear in the Lagrangian and in the gauge field
momenta:
\begin{eqnarray}
E_{\alpha} & := & \frac{\partial V}{\partial\vartheta^\alpha}
=e_{\alpha}\rfloor V + (e_{\alpha}\rfloor T^{\beta}) \wedge
H_{\beta} + (e_{\alpha}\rfloor R_{\beta}{}^{\gamma})\wedge
H^{\beta}{}_{\gamma} + {1\over 2}(e_{\alpha}\rfloor Q_{\beta\gamma})
M^{\beta\gamma}\,,\\ E^{\alpha}{}_{\beta} & := &\frac{\partial
V}{\partial\Gamma_\alpha{}^\beta}= - \vartheta^{\alpha}\wedge
H_{\beta} - g_{\beta\gamma}M^{\alpha\gamma}\,.
\end{eqnarray}
For the purely curvature quadratic Lagrangian (\ref{QMA}), we obviously have
$H_\alpha =0$ and $M^{\alpha\beta} = 0$. Hence $E^{\alpha}{}_{\beta} = 0$,
and as a result the field equations (\ref{first}) and (\ref{second}) reduce to
\begin{eqnarray}
E_{\alpha} = e_{\alpha}\rfloor V + (e_{\alpha}\rfloor
R_{\beta}{}^{\gamma})\wedge H^{\beta}{}_{\gamma} &=& 0,\label{first2}\\
D H^\alpha{}_\beta &=& 0.\label{second2}
\end{eqnarray}
For the above gravitational wave ansatz, we have verified that all the
irreducible parts of the curvature are trivial except for the pure tensor
pieces (\ref{Z1}) and (\ref{W1}). Accordingly, the direct computation of
the gravitational hypermomentum 3-form then yields
\begin{equation}
H^\alpha{}_\beta = {}^\ast\!\left(w_1\,{}^{(1)}W^\alpha{}_\beta
+ z_1\,{}^{(1)}Z^\alpha{}_\beta\right).\label{Hab}
\end{equation}
Let us now demonstrate that the gravitational wave ansatz above provides
an exact solution for the the gravitational field equations (\ref{first2})
and (\ref{second2}). The following two facts are crucial for this. The first
one is the property of the curvature 2-form (\ref{curv})
\begin{equation}\label{kR}
k_\alpha R^\alpha{}_\beta = 0,\qquad k^\beta R^\alpha{}_\beta = 0,
\end{equation}
which is obviously satisfied due to the null nature of the wave vector
$k$ and its orthogonality to the 1-form $\gamma_\alpha$.
The second fact concerns the well-known double duality property for the
first irreducible part of the curvature:
\begin{equation}
{}^{\ast (1)}W_{\alpha\beta}= {\frac 1 2}\eta_{\alpha\beta\mu\nu}
\,{}^{(1)}W^{\mu\nu},\label{ddW1}
\end{equation}
We begin with the first equation (\ref{first2}). The property (\ref{kR})
obviously yields for the gravitational wave configuration
$V_{\rm MAG}= -\,{\frac{1}{2}}\,R^{\alpha\beta}\wedge H_{\alpha\beta} = 0$
as well as the contraction $(e_{\alpha}\rfloor R_{\beta}{}^{\gamma})\wedge
H^{\beta}{}_{\gamma} = 0$. Hence, our ansatz solves the first equation.
Finally, we turn to the second equation (\ref{second2}). Substituting
(\ref{Hab}), we notice that the second term vanishes,
\begin{equation}
z_1\,D\,{}^\ast\!\left({}^{(1)}Z^\alpha{}_\beta\right) =
z_1\,k^\alpha k_\beta\,d\,{}^\ast du = 0,
\end{equation}
due to the property (\ref{Dk}) and the radiation conditions (\ref{ddu}).
As a result, the second equation (\ref{second2}) reduces to
$w_1\,D\,{}^\ast\!\left({}^{(1)}W^\alpha{}_\beta\right) = 0$. Since
${}^{(1)}W^\alpha{}_\beta = g^{\alpha\gamma}\,{}^{(1)}W_{\gamma\beta}$,
we have $D\,{}^\ast\!\left({}^{(1)}W^\alpha{}_\beta\right) = g^{\alpha\gamma}
\,D\,{}^\ast\!\left({}^{(1)}W_{\gamma\beta}\right) + Q^{\alpha\gamma}\wedge
{}^\ast\!\left({}^{(1)}W_{\gamma\beta}\right)$. Using the explicit form the
nonmetricity (\ref{nonm}), we prove that the last term vanishes. Thus we
find, in the end,
\begin{equation}
w_1\,D\,{}^\ast\!\left({}^{(1)}W_{\alpha\beta}\right) = 0.
\end{equation}
Now we can use the double duality identity (\ref{ddW1}) and obtain
\begin{equation}\label{dW}
{\frac {w_1} 2}\,\eta_{\alpha\beta\mu\nu}\,D\,{}^{(1)}W^{\mu\nu} = 0,
\end{equation}
where we used the fact that the covariant derivative of the Levi-Civita
tensor $D\eta_{\alpha\beta\mu\nu} = -\,2Q\,\eta_{\alpha\beta\mu\nu} = 0$
vanishes due to the absence of the Weyl covector, $Q = {\frac 14}
Q^\alpha{}_\alpha = 0$.
In order to demonstrate that our ansatz solves (\ref{dW}), we can use the
MAG Bianchi identity which reads $DR_\alpha{}^\beta = 0$. We find $D\,
(g_{\alpha\gamma}\,R^{\gamma\beta}) = g_{\alpha\gamma}D R^{\gamma\beta}
- Q_{\alpha\gamma}\wedge R^{\gamma\beta}$. Because of (\ref{nonm}) and
(\ref{curv}), the last term vanishes for the gravitational wave
configuration. Thus, in view of the Bianchi identity, the gravitational
wave curvature satisfies $D R^{\alpha\beta} = 0$. It now remains to use
the explicit formulas (\ref{curv}), (\ref{WZ}), (\ref{Z1}) and (\ref{W1})
to verify that
\begin{equation}
D{}^{(1)}W^{\alpha\beta} = 0,
\end{equation}
since the covariant derivative of the symmetric curvature vanishes
identically $D{}^{(1)}Z^{\alpha\beta} = k^\alpha k^\beta\,ddu \equiv 0$.
Consequently, (\ref{dW}) is satisfied by the gravitational wave ansatz, and
this completes the proof that such a configuration is, indeed, an exact
solution of the MAG field equations.
\section{Discussion and conclusion}\label{conclusion}
The extension of the Riemannian geometry of Einstein's general relativity
to the post-Riemannian structures of the metric-affine gravity can be
motivated by a number of reasons. Among them, we mention the problem of
quantization (see the discussion of the renormalizable MAG models in
\cite{lee1,lee2}), the theory of defects in the continuous media with
microstructure (for a overview, see \cite{frank} and \cite{PR}), the
physics of hadrons in terms of extended structures (see \cite{nee1,nee2,nee3}
and more details and references in \cite{PR}), the study of the early
universe (in particular, relating the post-Riemannian structures to the
dark matter problem, see \cite{dirk3,dirk4,dirk5}). Finally, one can show
that the MAG models may arise as the effective theories in the context of
the dilaton-axion-metric low-energy limit of the string theory (see, e.g.,
\cite{dil1,dil2,dil3,dil4}). The study of the exact solutions of the MAG
field equations is important for understanding and development of the
physical aspects mentioned above.
In this paper, we have derived a new plane wave solution of the general
Yang-Mills type (curvature quadratic) metric-affine theory of gravity. This
extends the previous study of the waves in the Yang-Mills type models of
the Poincar\'e gauge gravity \cite{adam,chen,sippel,vadim,singh,babu}. As
compared to the other exact wave solutions available in the literature
\cite{dirk1,dirk2,king,vas1,vas2,vas3}, the new configuration has
the following characteristic properties: (i) the spacetime metric
is not a flat Minkowski one but the metric of the Riemannian gravitational
plane wave determined by the single harmonic function $H(\sigma, z^a)$,
(ii) there are not only torsion waves present but the nonmetricity has
a nontrivial wave behavior as well, (iii) the post-Riemannian sector
of the torsion and nonmetricity does not belong to the triplet ansatz.
It is worthwhile to note that the triplet ansatz might be considered as
a useful tool which helps to avoid a possible problem of the well-posedeness
of the field equations by reducing them to the effective Einstein-Maxwell
system of equations. However, this ansatz is applicable only to a quite
narrow class of the MAG models, namely to those Lagrangians (\ref{QMA})
where the only nontrivial coupling constant $z_4\neq 0$ is allowed. Our
results apply to the general case with all the 16 nontrivial coupling
constants $w_1, \ldots w_7$, $z_1, \ldots z_9$. The well-posedness of the
general MAG model was never studied in the literature, and this question
clearly represents an open potentially interesting and important problem
within the metric-affine approach to gravity.
The results obtained have a number of interesting mathematical and physical
applications. To begin with, the curvature quadratic Lagrangian is
potentially important for a quantized theory of gravity. Furthermore, the
long-distance character of the wave solutions makes them a convenient tool
for the tests of the additional properties of matter besides the mass
(energy-momentum), namely, the hypermomentum which includes the spin and
the dilaton/shear charges. This is of particular interest for the study
of the elastic media with defects, and for the physics of hadrons (see the
references quoted above).
It is worthwhile to stress that the new solutions are obtained as the direct
generalization of the general-relativistic wave solutions. When the 1-form
$u$ vanishes, the post-Riemannian geometric quantities disappear. On
the other hand, the Riemannian (metric) sector of the solution has the same
form as in general relativity, and thus all the earlier mathematical and
physical analyses \cite{peres,pen1,pen2,griff,vdz,exact} are directly
applicable to our case. In physical terms this means that the usual general
relativistic detectors (with mass as the only gravitational charge) will not
distinguish between the gravitational waves of the Einstein theory and
the new MAG waves. This is in complete agreement with the correspondence
principle which underlies the dynamical structure of the metric-affine gravity:
MAG is not supposed to replace the general relativity theory in the well
established macroscopic domain, but rather to extend the latter in the
microscopic domain by taking into account the additional physical properties
of matter (such as the spin, dilation and hypermomentum currents). The above
conclusion is based on the fact that the equations of motion in MAG for the
test particles without the spin, dilaton and proper hypermomentum exactly
coincide with the equations of motion of the test massive matter in general
relativity \cite{nee}. Furthermore, it is worthwhile to recall that within
the Poincar\'e gauge gravity the equations of motion of matter are also known
to coincide with the general-relativistic equations of motion for the bodies
with the trivial average spin value \cite{yass}. The corresponding
generalization is expected to be valid in MAG for the macroscopic bodies
with vanishing average spin, dilaton and proper hypermomentum, although
the detailed relevant analysis is still missing in the literature.
The new solution gives a natural generalization of the definitions of
a gravitational plane-fronted wave. In accordance with \cite{orr},
a gravitational wave is defined by the existence of a null, geodetic,
shear-, twist, and expansion-free vector field $k$ and the Weyl 2-form
subject to the algebraic conditions
\begin{equation}
{}^{(1)}W^{\alpha\beta}\,k_\beta = 0,\qquad
{}^{(1)}W^{[\alpha\beta}\,k^{\gamma]} =0.\label{wavedef1}
\end{equation}
According to \cite{sippel}, a gravitational wave is defined by a
covariantly constant null field and a quadratic algebraic condition on
the components of the curvature tensor
\begin{equation}
R_{\mu\nu\alpha}{}^\beta\,R_{\rho\sigma\beta}{}^\alpha = 0.\label{wavedef2}
\end{equation}
As we can immediately verify, the non-Riemannian curvature (as well as its
Riemannian constituent) satisfies both (\ref{wavedef1}) and (\ref{wavedef2}).
In \cite{king,vas1,vas2,vas3} the notion of the pseudo-instanton solutions
of the MAG field equations was introduced. The latter are described by
a metric-compatible linear connection for which only one of the
eleven irreducible parts of the curvature is nontrivial. Our gravitational
wave solution provides a minimal generalization of the pseudo-instanton,
in the sense that the nonmetricity does not vanish and that the curvature
has {\it two} purely tensor irreducible parts (\ref{Z1}) and (\ref{W1}).
The applications mentioned above refer to the fundamental MAG theory. However,
we recall that MAG also arises as an effective theory within the framework of
the dilaton-axion-metric low energy limit of the string models. Accordingly,
one can use our solution as a technical tool to construct the exact wave
configurations in the string motivated models where the plane waves play
essential role \cite{gimon,ark1,ark2,str1,str2,str3,str4}. This construction
will be described in detail elsewhere.
\bigskip
{\bf Acknowledgment} This work was supported by the Deutsche
Forschungsgemeinschaft (Bonn) with the grant HE~528/20-1. I thank
Friedrich Hehl for the reading of the manuscript and for the discussion
of the results obtained.
|
Title:
Large Scale CO Observations of a Far-Infrared Loop in Pegasus; Detection of a Large Number of Very Small Molecular Clouds Possibly Formed via Shocks |
Abstract: We have carried out large scale 12CO and 13CO observations with a mm/sub-mm
telescope NANTEN toward a loop-like structure in far infrared whose angular
extent is about 20x20 degrees around (l, b) ~ (109, -45) in Pegasus. The 12CO
distribution is found to consist of 78 small clumpy clouds whose masses range
from 0.04 Mo to 11 Mo. About 83% of the 12CO clouds have very small masses less
than 1.0 Mo. 13CO emission shown in the 19 of the 78 12CO clouds was detected
in the region where the column density of H2 derived from 12CO is greater than
5x10(20) cm(-2), corresponding to Av of ~ 1 mag, which takes into account that
of HI. We find no indication of star formation in these clouds in IRAS and
2MASS Point Source Catalogs. The very low mass clouds, M < 1 Mo, identified are
unusual in the sense that they have very weak 12CO peak temperature of 0.5 K to
2.7 K and that they aggregate in a region of a few pc with no main massive
clouds of ~ 100 Mo. A comparison with a theoretical work on molecular cloud
formation (Koyama & Inutsuka 2002) suggests that the very low-mass clouds may
have been formed in the shocked layer through the thermal instability. The star
HD886 (B2IV) may be the source of the mechanical luminosity via stellar winds
to create shocks, forming the loop-like structure where the very low-mass
clouds are embedded.
| https://export.arxiv.org/pdf/astro-ph/0601315 |
\title{Large Scale CO Observations of a Far-Infrared Loop in Pegasus; Detection of a Large Number of Very Small Molecular Clouds Possibly Formed via Shocks}
\author{H.\ Yamamoto\altaffilmark{1}, A.\ Kawamura\altaffilmark{1}, K.\ Tachihara\altaffilmark{2}, N.\ Mizuno\altaffilmark{1}, T.\ Onishi\altaffilmark{1} and Y.\ Fukui\altaffilmark{1}}
\altaffiltext{1}{Department of Astrophysics, Nagoya University, Chikusa-ku, Nagoya, Japan 464-8602; hiro@a.phys.nagoya-u.ac.jp}
\altaffiltext{2}{Graduate School of Science and Technology, Kobe University, 1-1 Rokko-dai, Nada-ku, Kobe, Japan 657-8501}
\keywords{ISM: clouds --- ISM: individual(High Latitude Clouds) --- radio lines: ISM --- stars: formation --- stars: winds}
\section{INTRODUCTION}
High Galactic latitude molecular clouds (hereafter HLCs) are typically located at $\mid$b$\mid$ $\gtrsim$ 20$\degr$--30$\degr$. Since the Gaussiun scale height of CO is estimated to be $\sim$ 100 pc in the inner Galactic disk (e.g., Magnani et al. 2000), HLCs are likely located very close to the Sun, within a few hundred pc or less. Their proximity to the Sun and the low possibility of overlapping with other objects along the line of sight enable us to study them with a high spatial resolution and to compare CO data unambiguously with the data at other wavelengths. HLCs have lower molecular densities compared with dark clouds where the optical obscuration is significant. Therefore, HLCs are often called as translucent clouds (e.g., van Dishoeck \& Black 1988) and most of the known HLCs are not the sites of active star formation, although a few of them are known to be associated with T Tauri stars (e.g., Magnani et al. 1995; Pound 1996; Hearty et al. 1999).
Given the very small distances of HLCs, it is a challenging task for observers to make a complete survey for HLCs over a significant portion of the whole sky. $^{12}$CO ($J$ = 1--0) emission has been used to search for HLCs because the line emission in the mm band is strongest among the thermally or sub-thermally excited spectral lines of interstellar molecular species. It is however difficult to cover an area as large as tens of square degrees subtended by some of the HLCs because of the general weakness of the $^{12}$CO emission, typically $\sim$ a few K (e.g., Magnani et al. 1996), with existing mm-wave telescopes in a reasonable time scale. HLCs have been therefore searched for by employing various large-scale datasets at other wavelengths including the optical obscuration (Magnani et al. 1985; Keto \& Myers 1986), the infrared radiation (Reach et al. 1994), and the far-infrared excess over H{\small \,I} (=FIR excess)(Blitz et al. 1990; Onishi et al. 2001). On the other hand, unbiased surveys in CO at high Galactic latitudes have been performed at very coarse grid separations of 1$\degr$ resulting in a small sampling factor of a few \% (Hartmann et al. 1998; Magnani et al. 2000). Most recently, Onishi et al. (2001) discovered 32 HLCs or HLC complexes. This search was made based on the FIR excess, demonstrating the correlation among FIR excess clouds with CO clouds is a useful indicator of CO HLCs.
Previous CO observations of individual HLCs at higher angular resolutions show that HLCs exhibit often loop-like or shell-like distributions having filamentary features with widths of several arc min or less (Hartmann et al. 1998; Magnani et al. 2000; Bhatt 2000), and in addition that HLCs often compose a group, whose angular extent is $\sim$ 10 degrees or larger. In order to better understand the structure of HLCs and to pursue the evolution of HLC complexes, CO observations covering tens of square degrees at a high angular resolution are therefore crucial. The past observations of such complexes of HLCs are limited to a few regions including Polaris flare (Heithausen \& Thaddeus 1990), Ursa Major (Pound \& Goodman 1997) and the HLC complex toward MBM 53, 54, and 55 (Yamamoto et al. 2003). Pound \& Goodman (1997) showed an arc-like structure of the molecular cloud system and suggested that the origin of such structures could be some explosive events. Most recently, Yamamoto et al. (2003) carried out extensive observations of the molecular cloud complex including MBM 53, 54, and 55 and suggest that the HLCs may be significantly affected by past explosive events based on the arc-like morphologies of molecular hydrogen (see also Gir et al. 1994).
The region of MBM 53, 54, and 55 is of particular interest among the three, because it is associated with a large H{\small \,I} cloud of $\sim$ 590 $M_{\sun}$ at a latitude of $-$35 degrees and because there is a newly discovered HLC of 330$M_{\sun}$, HLCG92$-$35, which is significantly H{\small \,I} rich with a mass ratio $M$(H$_{2}$)/$M$(H{\small \,I}) of $\sim$ 1, among the known HLCs (Yamamoto et al. 2003). This cloud was in fact missed in the previous surveys based on optical extinction (Magnani et al. 1985). Subsequent to these observations we became aware of that the region is also very rich in interstellar matter as shown by the 100$\mu$m dust features (Kiss et al. 2004). There is a loop-like structure shown at 100 $\mu$m around ($l$, $b$) $\sim$ (109$\degr$, $-$45$\degr$). Toward the center of the loop, an early type star HD886(B2IV) is located and may play a role in creating the loop. Its proper motion is large at a velocity of a few km s$^{-1}$, suggesting that the stellar winds of the star might have continued to interact with the surrounding neutral matter over a few tens of pc in $\sim$ a few Myr. Magnani et al. (1985) and Onishi et al. (2001) yet observed only a small part of this region. In order to reveal the large scale CO distribution of the region, we have carried out observations toward ($l$, $b$) $\sim$ (109$\degr$, $-$45$\degr$) by $^{12}$CO ($J$ = 1--0) and $^{13}$CO ($J$ = 1--0) with NANTEN 4-meter millimeter/sub-mm telescope of Nagoya University at Las Campanas, Chile. We shall adopt the distance of 100 pc from the sun to the loop-like structure which is equal to the distance of the B2 star in the center of the loop, and is also a typical value for the HLCs.
\section{OBSERVATIONS}
$^{12}$CO ($J$ = 1--0) and $^{13}$CO ($J$ = 1--0) observations were made with the 4-meter telescope, NANTEN, of Nagoya University at Las Campanas Observatory of Carnegie Institutions of Washington, Chile. The front-end was an SIS receiver cooled down to 4 K with a closed-cycle helium gas refrigerator (Ogawa et al. 1990). The backend was an acousto-optical spectrometer with 2048 channels, and the total bandwidth was 40 MHz. The frequency resolution was 35 kHz, corresponding to a velocity resolution of $\sim$ 0.1 km s$^{-1}$. A typical system noise temperature was $\sim$ 200 K (SSB) at 115.271 GHz and $\sim$ 150 K (SSB) at 110.201 GHz. The half-power beam width was about 2$\farcm$6, corresponding to 0.076 pc at a distance of 100 pc. The pointing accuracy was better than 20$\arcsec$, as established by radio observations of Jupiter, Venus, and the edge of the Sun in addition to optical observations of stars with a CCD camera attached to the telescope.
The observed region in $^{12}$CO was $\sim$ 240 square degrees toward the whole area of the loop-like structure centered at around ($l$, $b$) $\sim$ (109$\degr$, $-$45$\degr$) shown in a 100 $\mu$m map by Schlegel et al. (1998). First, the $^{12}$CO observations were made at a grid spacing of 8$\arcmin$$\times$cos($b$) and 8$\arcmin$ in Galactic longitude and latitude, respectively. Then, the regions where the $^{12}$CO emission is significantly detected were observed at a grid spacing of 4$\arcmin$$\times$cos($b$) and 4$\arcmin$ in Galactic longitude and latitude, respectively. The $^{13}$CO observations were made in and around the whole area where the peak temperature of $^{12}$CO emission is higher than 2.0 K at a grid spacing of 2$\arcmin$$\times$cos($b$) and 2$\arcmin$ in Galactic longitude and latitude, respectively. The periods of $^{12}$CO observations were several sessions between 2002 May and November and those of $^{13}$CO were those between 2003 April and August. All the observations were made by frequency switching whose interval is 20 MHz, corresponding to $\sim$ 50 km s$^{-1}$. The integration times per point of $^{12}$CO and $^{13}$CO observations were typically $\sim$ 30 s and $\sim$ 75 s, respectively, resulting in typical rms noise temperatures per channel of $\sim$ 0.35 K and $\sim$ 0.15 K in the radiation temperature, $T_{\rm R}^*$, respectively. In reducing the spectral data, we subtracted forth-order polynomials for the emission-free parts in order to ensure a flat spectral baseline. Total numbers of observed points of $^{12}$CO and $^{13}$CO are 16890 and 3100, respectively.
We employed a room-temperature blackbody radiator and the sky emission for the intensity calibration. An absolute intensity calibration and the overall check of the whole system were made by observing Orion KL [$\alpha$(1950) = 5$^{\rm h}$32$^{\rm m}$47.$^{\rm s}$0, $\delta$(1950) = $-$5$\degr$24$\arcmin$21$\arcsec$] every
2 hours. We assumed the $T_{\rm R}^*$ of Orion KL to be 65 K for $^{12}$CO and 10 K for $^{13}$CO.
\section{RESULTS}
\subsection{$^{12}$CO Observation}
\subsubsection{Distribution and Past Detection of $^{12}$CO Clouds}
Figure 1 shows the distribution of the velocity-integrated intensity map of $^{12}$CO emission. We defined a $^{12}$CO cloud as a collection of more than two contiguous observed positions whose integrated intensity exceeds 0.77 K km s$^{-1}$ (5$\sigma$). Based on the definition, we identified 78 molecular clouds in this region. Molecular clouds are concentrated from ($l$, $b$) $\sim$ (107$\degr$, $-$37$\degr$) to (116$\degr$, $-$45$\degr$) and around ($l$, $b$) $\sim$ (114$\degr$, $-$52$\degr$). Most of the molecular clouds are very small, having size of $\lesssim$ 1$\degr$. Figure 2 shows the distribution of the CO superposed on the SFD 100 $\mu$m (Schlegel et al. 1998), which was derived from a composite of the COBE/DIRBE and IRAS/ISSA maps, with the foreground zodiacal light and confirmed point sources removed. CO clouds are distributed along the infrared loop whose diameter is $\sim$ 25 pc. We detected little CO emission within the loop-like structure, while toward some of the local peaks of SFD 100 $\mu$m there is no CO emission.
Figure 3 shows the peak radial velocity distribution derived from the present $^{12}$CO data set. The velocity in Figure 3 is derived by a single gaussiun fitting from all CO spectra.
The velocity range of the molecular clouds is from $-$18.3 km s$^{-1}$ to 0.3 km s$^{-1}$ and there is no systematic large scale velocity gradients.
Some of the molecular clouds have already been known by previous observations. Molecular clouds toward ($l$, $b$) $\sim$ (110$\fdg$18, $-$41$\fdg$23) and (117$\fdg$36, $-$52$\fdg$28) are identified by Magnani et al. (1985) and named as MBM 1 and MBM 2, respectively. DIR117$-$44 and DIR105$-$38 identified by Reach et al. (1998) are also identified in CO toward ($l$, $b$) $\sim$ (116$\fdg$5, $-$44$\fdg$0) and (105$\fdg$0, $-$38$\fdg$0) by Onishi et al. (2001). Magnani et al. (1986) detected CO emission at ($l$, $b$) $\sim$ (112$\degr$, $-$40$\degr$). Magnani et al. (2000) also covered this region even though they made observations on a locally Cartesian grid with 1$\degr$(true angle) spacing in longitude and latitude for a beam size of 8$\farcm$8, they detected CO emission at eight positions of ($l$, $b$) $\sim$ (103$\fdg$2, $-$38$\fdg$0), (103$\fdg$2, $-$39$\fdg$0), (104$\fdg$4, $-$39$\fdg$0), (106$\fdg$8, $-$37$\fdg$0), (108$\fdg$0, $-$52$\fdg$0), (109$\fdg$5, $-$51$\fdg$0), (110$\fdg$4, $-$41$\fdg$0), and (111$\fdg$0, $-$50$\fdg$0) in the present region while they missed the present small molecular clouds whose sizes are less than several arc min in Figure 1 due to the coarse grid spacing.
\subsubsection{Physical Properties of $^{12}$CO Molecular Clouds}
Seventy-eight $^{12}$CO molecular clouds are identified in the present region. For each molecular cloud, $\Delta V$ derived from single Gaussian fitting was from 0.5 to 3.7 km s$^{-1}$, and the radial velocity, $V_{\rm LSR}$, ranges from $-$15.7 to $-$0.1 km s$^{-1}$. The maximum brightness temperature, $T_{\rm R}^*$($^{12}$CO) ranges from 0.5 to 5.7 K. The radius of a cloud, $R$, which is defined as the radius of an equivalent circle having the same area, i.e., and $R$(pc)$=\sqrt{A/\pi}$ where A is the total cloud surface area within the 5$\sigma$--contour level, ranges from 0.07 to 0.79 pc. The peak column density of molecular hydrogen, $N$(H$_{2}$), in each cloud derived by assuming a conversion factor of 1.0$\times$10$^{20}$ cm$^{-2}$/(K km s$^{-1}$) (Magnani et al. 2000) ranges from 8.0$\times$10$^{19}$ to 1.7$\times$10$^{21}$ cm$^{-2}$ with the present detection limit, 7.7$\times$10$^{19}$ cm$^{-2}$, corresponding to mass detection limit of 0.014 $M_{\sun}$. We estimate the molecular mass, $M$($^{12}$CO), by using the following formula
\begin{equation}
M(^{12}{\rm CO}) = \mu m_{{\rm H}} \Sigma[D^2 \Omega N({\rm H}_{2})],
\end{equation}
where $\mu$ is the mean molecular weight, assumed to be 2.8 by taking into account a relative helium abundance of 25\% in mass, $m_{{\rm H}}$ is the mass of the atomic hydrogen, $D$ is the distance from the Sun to the molecular clouds, and $\Omega$ is the solid angle subtended by a unit grid spacing of (4$\arcmin$)$\times$(4$\arcmin$$\times$cos($b$)). $M$($^{12}$CO) ranges from $\sim$ 0.04 to $\sim$ 11 $M_{\sun}$ and the total mass of molecular clouds is $\sim$ 64 $M_{\sun}$. These physical properties are listed in Table 1 and the histograms of $T_{\rm R}^*$($^{12}$CO), $\Delta V$, log($R$), and log($N$(H$_{2}$)) of these clouds are shown in Figure 4. Histograms in Figure 4 are divided into three different categories, \textit{Usual Cloud} (hereafter UC) whose mass is greater than 1 $M_{\sun}$, \textit{Small Cloud} (hereafter SC) whose mass is between 0.1 and 1 $M_{\sun}$, and \textit{Very Small Cloud} (hereafter VSC) whose mass is less than 0.1 $M_{\sun}$. It is remarkable that there are a number of molecular clouds having mass less than 1 $M_{\sun}$ and that the fractions of SC and VSC are 43/78 $\sim$ 55\% and 22/78 $\sim$ 28\% in the present region, respectively. In addition, the sizes of SC and VSC are equal to or less than 0.1 pc. We also note that the peak temperatures of SC and VSC are typically in a range from 0.5 K to 2.7 K, well below that of UC in the same region.
\subsection{The Detection and Physical Properties of the $^{13}$CO Molecular Clouds}
Figure 5 shows the distribution of the velocity-integrated intensity map of the $^{13}$CO emission superposed on the $^{12}$CO distribution. The total area of the $^{13}$CO observations is $\sim$ 29 square degrees toward 38 of the 78 $^{12}$CO clouds. We observed all of 13 UCs, 24 of 43 SCs, and 3 of 22 VSCs. We detected $^{13}$CO emission at 11 of the 13 UCs, 8 of the 24 SCs, and none of the 3 VSCs, indicating a trend that the $^{13}$CO intensity increases with $^{12}$CO cloud mass.
A $^{13}$CO cloud is defined in the same way as for a $^{12}$CO cloud except for the lowest integrated intensity level, 0.3 K km s$^{-1}$ (3$\sigma$). Based on the definition, we identified 33 $^{13}$CO clouds. For the 33 $^{13}$CO molecular clouds, $\Delta$$V$ derived from single Gaussian fitting is $\sim$ 1.5 km s$^{-1}$ and $V_{\rm LSR}$ of them ranges from $-$13.1 to $-$1.9 km s$^{-1}$. Other physical properties, the maximum brightness temperature, $T_{\rm R}^*$($^{13}$CO), and $R$ range from 0.3 to 2.3 K and from 0.04 to 0.21 pc, respectively. The physical parameters including the molecular column density and mass (hereafter $M_{\rm LTE}$) are derived on the assumption of local thermodynamic equilibrium (LTE). To derive the column density of molecular hydrogen, the optical depth of $^{13}$CO is estimated by using the following equations,
\begin{equation}
\tau(^{13}{\rm CO})={\rm ln}\left[1-\frac{T_{\rm R}^{*}(^{13}{\rm CO})}{5.29}\left\{\frac{1}{{\rm exp}(5.29/T_{\rm ex})-1}-0.164\right\}^{-1}\right],
\end{equation}
where $T_{\rm ex}$ is the excitation temperature of the $J$ = 1--0 transition of CO in K and was derived from
\begin{equation}
T_{\rm ex}=\frac{5.53}{{\rm ln}\left\{1+5.53/\left[T_{\rm R}^{*}(^{12}{\rm CO})+0.819\right]\right\}}.
\end{equation}
$T_{\rm ex}$ was estimated to be 9.4 K from our $^{12}$CO data. The $^{13}$CO column density, $N$($^{13}$CO), is estimated by
\begin{equation}
N(^{13}{\rm CO})=2.42\times10^{14}\nonumber\times\frac{\tau(^{13}{\rm CO})T_{\rm ex}({\rm K})\Delta V({\rm km \hspace{0.1cm} s^{-1}})}{1-{\rm exp}[-5.29/T_{\rm ex}({\rm K})]} ({\rm cm^{-2}}).
\end{equation}
The ratio of $N$(H$_{2}$)/$N$($^{13}$CO) was assumed to be 7$\times$10$^{5}$ (Dickman 1978). The $M_{\rm LTE}$ of a cloud from $N$(H$_{2}$) is derived by the same way as $^{12}$CO (see equation (1)). The column density and $M_{\rm LTE}$ range from 2.3$\times$10$^{20}$ to 1.7$\times$10$^{21}$ cm$^{-2}$ and 0.03 to 1.41 $M_{\sun}$, respectively, where the detection limit in the column density is 2.0$\times$10$^{20}$ cm$^{-2}$, coressponding to mass limit of 0.009 $M_{\sun}$, smaller than that of $^{12}$CO because the observations of $^{13}$CO were made by higher grid sampling and lower rms noise fluctuations than those of $^{12}$CO, respectively. Figure 6 shows the histograms of each physical property. The virial mass, $M_{\rm vir}$, of a cloud was derived by using the following equation, assuming isothermal, spherical, and uniform density distribution with no external magnetic pressure:
\begin{equation}
M_{\rm vir} = 209 \times R \times \Delta V_{\rm comp}^{2},
\end{equation}
where $R$ and $\Delta V_{\rm comp}$ are the radius (pc) and line width (km s$^{-1}$) of the composite profile obtained by averaging all the spectra within a cloud, respectively (for details of the line width of composite profiles, see Yonekura et al. 1997; Kawamura et al. 1998). From this equation, $M_{\rm vir}$ is estimated to be in a range from 4.7 to 197 $M_{\sun}$. These physical properties are also listed in Table 2.
\section{CORRELATIONS AMONG THE CLOUD PHYSICAL PARAMETERS}
\subsection{Mass Spectrum and Size Linewdith Relation}
Figure 7a and 7b show the mass spectrum of the present $^{12}$CO and $^{13}$CO clouds. The spectra have been fitted by the maximum-likelihood method (Crawford et al. 1970), and it is found that they are well fitted by a single power law as follows; $dN/dM$ $\propto$ $M^{-1.53\pm0.13}$ for the $^{12}$CO clouds and $dN/dM$ $\propto$ $M^{-1.36\pm0.10}$ for the $^{13}$CO clouds. These values of the spectral indices seem to be similar to those for the higher mass range (e.g., Yonekura et al. 1997).
Figure 8 shows a plot of size, $R$, versus line width, $\Delta V$, of the $^{13}$CO clouds in this region and for a comparison with other HLCs, MBM 53, 54, and 55 complex (Yamamoto et al. 2003). We can make fitting as follows by using a least-squares fitting, log($\Delta V$) = (0.22$\pm$0.43) $\times$ log($R$) + (0.37$\pm$0.52) (c.c.=0.23) for the present region and log($\Delta V$) = (0.43$\pm$0.32) $\times$ log($R$) + (0.53$\pm$0.28) (c.c.=0.37) for MBM 53, 54, and 55 complex. The low correlation coefficient (c.c.) indicates that there is no significant correlation between $R$ and $\Delta V$ because of a small range of $R$. Here we do not show the same relationship for $^{12}$CO, because the non-circular shape of the $^{12}$CO clouds may not be appropriate to derive reliable $R$.
\subsection{$M_{\rm LTE}$ vs. $M_{\rm vir}$}
Figure 9 shows a plot of $M_{\rm LTE}$ versus $M_{\rm vir}$. The present $^{13}$CO clouds are located far above the equilibrium line where $M_{\rm LTE}$ is equal to $M_{\rm vir}$, indicating that the $^{13}$CO clouds are not in the virial equilibrium. This indicates that none of the molecular clouds are gravitationally bound. These parameters can be fitted by using a least-squares fitting as follows, log($M_{\rm vir}$) = (0.91$\pm$0.30) $\times$ log($M_{\rm LTE}$) + (2.23$\pm$0.29) (c.c.=0.66) for present molecular clouds and log($M_{\rm vir}$) = (0.77$\pm$0.13) $\times$ log($M_{\rm LTE}$) + (0.16$\pm$0.08) (c.c.=0.74) for MBM53, 54, and 55 complex. As mentioned in Yamamoto et al. (2003), the present molecular clouds also tend to be more virialized as the mass increases.
For the Gemini and Auriga, and Cepheus-Cassiopeia regions, the indices of $M_{\rm LTE}$ for $M_{\rm vir}$ of $^{13}$CO clouds are estimated to be 0.72$\pm$0.03 and 0.62$\pm$0.03 for the cloud mass range of $M_{\rm LTE}$ $<$ 10$^{4}$ $M_{\sun}$ and 10$^{2}$ $M_{\sun}$ $<$ $M_{\rm LTE}$ $<$ 10$^{5}$ $M_{\sun}$, respectively (Kawamura et al. 1998; Yonekura et al. 1997). Although the mass ranges of MBM 53, 54, and 55 complex and of this region are 10$^{-1}$ $M_{\sun}$ $<$ $M_{\rm LTE}$ $<$ 10$^{2}$ $M_{\sun}$ and 10$^{-2}$ $M_{\sun}$ $<$ $M_{\rm LTE}$ $<$ 1 $M_{\sun}$, respectively, difference in the power-law indices among these regions is small and a tendency that the SCs have large ratios of $M_{\rm vir}$/$M_{\rm LTE}$ is commonly seen.
\section{COMPARISON WITH OTHER WAVELENGTH DATA}
\subsection{No Sign of Star Formation}
In order to look for sings of star formation associated with the present molecular clouds, we searched the IRAS point source catalog for candidates of protostellar objects satisfying the following criteria: (1) point sources having a data quality flag better than 2 in 4 bands, (2) flux ratios at 12, 25, 60 $\mu$m satisfying both log($F_{12}$/$F_{25}$) $<$ $-$0.3 and log($F_{25}$/$F_{60}$) $<$ 0, and not identified as galaxies or planetary nebulae and stars. We find that there are no cold IRAS point sources satisfying these criteria in the present molecular clouds. We also find that there are no IRAS point sources having a spectrum like a T-Tauri type star or no YSOs identified from Point Source Catalog of Two-Micron All-Sky Survey in this region. Here we select the 2MASS sources whose signal to noise ratio of valid measurements in all bands are greater than 10 and extract the sources which have the spectra like T Tauri stars in ($J$$-$$H$)--($H$$-$$K$) color--color diagram (e.g., Meyer et al. 1997) . These results suggest that the present molecular clouds are not the site of recent star-formation, or that the region is not remnants of past star formation. Such a low level of star formation is similar to the other HLCs including MBM 53, 54, and 55 complexes.
\subsection{Comparison with HI}
Figure 10 shows the integrated intensity map of H{\small \,I} taken from a Leiden-Dwingeloo H{\small \,I} survey (Hartmann \& Burton 1997) superposed on the integrated intensity of CO. The integrated velocity range is from $-$16 to 0 km s$^{-1}$, corresponding to the velocity range of the $^{12}$CO emission. Because an angular resolution of $\sim$ 30$\arcmin$ is coarser than that of the present CO observations by a factor of $\sim$ 10, we discuss here only the overall comparison between CO and H{\small \,I} distributions. The H{\small \,I} distribution is loop-like and the molecular clouds are distributed nearly along the H{\small \,I} loop.
Figure 11 shows the position-velocity diagram of H{\small \,I} integrated from $-$40$\fdg$5 to $-$39$\fdg$5 and $-$52$\fdg$5 to $-$51$\fdg$5 in Galactic latitude, respectively. The hole like structures can be seen in H{\small \,I}, suggesting that these H{\small \,I} clouds are expanding. These expanding structures in Figures 11(a) and 11(b) correspond to the Galactic northern and southern HI shells shown as the thick dashed (semi) ellipses in Figure 10, respectively, while the expanding motion is not seen in CO (See Figure 3). These two expanding shells are also identified by an infrared radiation (Kiss et al. 2004). From Figure 10, an H{\small \,I} cloud around ($l$, $b$) $\sim$ (109$\degr$, $-$52$\degr$) seems to be located on the left side of the southern expanding shell and the molecular clouds are distributed nearby. It is difficult to distinguish with which H{\small \,I} shell the molecular clouds located around ($l$, $b$) $\sim$ (109$\degr$, $-$52$\degr$) are associated because the H{\small \,I} velocities of the two shells are similar with each other. The shape of $^{12}$CO and SFD100 $\mu$m radiation around ($l$, $b$) $\sim$ (109$\degr$, $-$52$\degr$) in Figure 2 is also similar to the left side of the expanding shell. If this is true, there may be two expanding structures in the present region. HD886 (109$\fdg$43, $-$46$\fdg$68) is located near the center of the northern expanding shell, indicating that HD886 may be affecting the northern expanding shell. The parallax of HD886 has been measured to be 9.79$\pm$0.81 mas (Perryman et al. 1997), corresponding to a distance from the Sun of 102$^{+9}_{-8}$ pc. The proper motion has also been measured to be $\mu\alpha*$=0$\farcs$0047 yr$^{-1}$, $\mu\delta$=$-$0$\farcs$0082 yr$^{-1}$ by Perryman et al. (1997). From this proper motion, the velocity of HD886 in the L-B map is estimated to be $\sim$ 4.3$\times$10$^{-6}$ pc yr$^{-1}$ at $\sim$ 100 pc (see Figure 10). We use typical values of the stellar wind for B2(IV) star on $dM$/$dt$=10$^{-9}$ $M_{\sun}$ yr$^{-1}$ and $V_{\infty}$ = 1000 km s$^{-1}$ (e.g., Snow 1982). From these parameters, the energy injected to the northern shell is estimated to be $\sim$ 10$^{47}$ ergs in a few $\times$10$^{6}$ yr. The expanding energy of the northern shell is estimated to be $\sim$ 10$^{48}$ ergs from the atomic and molecular hydrogen, using that the masses of amomic and molecular hydrogen associated with the northern shell are $\sim$ 400 and 42 $M_{\sun}$, respectively, the expanding velocity is $\sim$ 7 km s$^{-1}$ which is estimated from Figure 11 and the equation of $E_{\rm exp}$=1/2$M$$V_{\rm exp}^{2}$. Since the expanding energy of the atomic and molecular hydrogen is comparable to the energy from HD886, additional source of energy other than HD886 such as photo evaporation is needed to explain the expanding energy because the energy conversion efficiency of the stellar wind is $\lesssim$ 10\%. The energy of the southern shell is estimated to be $\sim$ 10$^{47}$ ergs, by using that the masses of atomic and molecular hydrogen are $\sim$ 1000 and 16 $M_{\sun}$, respectively, and that the expanding velocity is $\sim$ 9 km s$^{-1}$. We could not find possible candidates of the energy source for the expanding feature in the literature (SIMBAD) and there are no counterparts in optical or X-ray wavelength. Although we could not identify the possible candidates, we cannot exclude a possibility that these objects may have escaped from the region in a few $\times$ 10$^{6-7}$ yr after forming these structures.
\section{DISCUSSION}
\subsection{Physical States of the Small Clouds}
The present observations have revealed numerous molecular clouds having very small mass of less than 1 $M_{\sun}$. It is of considerable interest to pursue the physical states of these SCs from view points of cloud physics and chemistry as well as of the origin of molecular clouds.
We shall hereafter focus on the low mass $^{12}$CO clouds whose mass is less than 1 $M_{\sun}$. The total number of such clouds is 65 among 78. The $^{13}$CO emission has been searched for toward 24 of the 43 $^{12}$CO low-mass clouds whose mass is in a range of 0.1--1.0 $M_{\sun}$ and has been detected from 8 of them.
Figure 12 shows correlations of the molecular column density, estimated from $^{12}$CO and $^{13}$CO, of the clouds where both $^{12}$CO and $^{13}$CO emission are detected. It is seen that almost all of the $^{12}$CO clouds having molecular column density greater than 5$\times$10$^{20}$ cm$^{-2}$, corresponding to the visual extinction of 0.55 mag if we use the relationship of $N(\rm H_2)$=9.4$\times$10$^{20}$$\times$$A$v cm$^{-2}$ (Bohlin et al. 1978; Hayakawa et al. 1999), show significant $^{13}$CO emission. We note that there are 22 $^{12}$CO clouds whose mass is less than 0.1 $M_{\sun}$; for those it is doubtful that the $^{13}$CO emission is so significant as those whose mass is a range of 0.1--1.0 $M_{\sun}$ although only 3 of them were searched for the $^{13}$CO emission in the present study. We note that we ignore the possible contribution of atomic hydrogen in the above relationship. If we take into account the distribution of atomic hydrogen as $N$(H{\small \,I}), the visual extinction would increase by 0.2 $\sim$ 0.3 mag. This may be explained as that the $^{13}$CO emitting regions become significant when $A$v becomes larger than $\sim$ 1 mag, marginally enough to shield the ultraviolet radiation to protect $^{13}$CO molecules (e.g., Warin et al. 1996), although $^{13}$CO molecules may be also affected from the ultraviolet radiation because the $N(\rm H_2)$ derived from $^{13}$CO is lower than that derived from $^{12}$CO in most of molecular clouds.
The peak intensity ratio of $^{12}$CO and $^{13}$CO is around 5, much smaller than the terrestrial abundance ratio of 89, indicating that the $^{12}$CO emission is optically thick in the clouds where the both emissions are detected. The maximum $^{12}$CO peak temperature of the present brightest $^{12}$CO emission is 6 K, and this suggests that the excitation temperature is consistent with the kinetic temperature of $\sim$ 10 K, typical to the local dark clouds. The low mass clouds show lower $^{12}$CO peak temperatures down to 1 K, significantly less than the brightest peak intensities. It is not clear if this is due to the lower excitation temperatures or due to smaller filling factors significantly less than 1. In order to clarify this point we need observations of the present low mass clouds at much higher angular resolutions.
\subsection{Origin of the Very Small Clouds}
Figure 13 shows the histograms of mass and sizes of $^{12}$CO clouds. In the present region, we have detected a large number of molecular clouds of mass less than 0.1 $M_{\sun}$ and sizes less than 0.1 pc not detected so far in the other regions. We may ask why molecular clouds of mass less than 0.1 $M_{\sun}$ and size less than 0.1 pc such as the present clouds have not been detected so far. The main reason for this is perhaps the paucity of high-resolution observational studies of nearby molecular clouds. Most of the observations of the local high latitude clouds were made at lower resolutions of 10 arc-min or at a coarse grid spacing of 1 degree, both of which are unable to detect and resolve the present low mass clouds. This suggests that the low mass clouds similar to the present ones may not be uncommon in the interstellar space and warrant more extensive searches for them in the other parts of the sky.
It is interesting to compare the physical parameters of the present VSCs with theoretical studies. The typical size of these HLCs, $\sim$ 0.1 pc, is significantly less than the Jeans length of 1.3 pc -- 7 pc for molecular gas with $T$=10 K and $n$(H$_{2}$)=10--100 cm$^{-3}$. If we take temperature higher than 10 K, the length becomes even larger. Figure 14 shows the radial distribution of mass surface density which is derived by dividing the mass in circular annulus of a radius by the area of the circular annulus. Mass surface density in the present region is fairly flat in radius because there is no massive molecular cloud in the center of the VSCs. On the other hand, mass surface density in the other regions has a gradient for radius, indicating that there are small molecular clouds around a massive molecular cloud whose mass is several dozen $M_{\sun}$ or greater (e.g., Sakamoto 2002, Sakamoto \& Sunada 2003).
In these regions, the gravity of the massive molecular cloud may contribure to the formation of these small clouds by increasing the pressure in the surroundings.
These suggest that mechanisms other than gravitational instability might contribute to the formation of present VSCs. A theory of molcular cloud formation is discussed by Koyama \& Inutsuka (2002). According to them, molecular clouds smaller than the Jeans-length can be formed in the shocked layer through the thermal instability. We shall present some considerations by comparing the observational results with their theoretical results.
Present VSCs are likely to be affected by HD886 in the last few $\times$10$^{6}$ yr (for details, see section 5.2). Although the mechanical luminosity from HD886 injected to the loop-like structure during a few $\times$10$^{6}$ yr is low to explain the expanding of the interstellar matter, it is a possibility that the stellar wind of HD886 is the source of shock. Koyama \& Inutsuka (2002) assumed shock velocity of 26 km s$^{-1}$ and density of 0.6 cm$^{-3}$ as an initial pre-shock condition and find that the region of density greater than 100 cm$^{-3}$ grows in size to $\sim$ 0.2 pc $\times$ 0.1 pc in 1.06$\times$10$^{6}$ yr and the internal structure consists of some filaments. The velocity dispersion of CO derived by Koyama \& Inutsuka (2002) is a few km s$^{-1}$.
The size of the smallest molecular clouds and the velocity dispersion of CO are comparable to those derived from Koyama \& Inutsuka (2002). We cannot resolve the internal structure of the VSCs because the present VSCs are detected with only a few points for each. The typical column density derived by Koyama \& Inutsuka (2002) is 2$\times$10$^{20}$ cm$^{-2}$, while the column density of the VSCs is estimated to be $\sim$ 1.6$\times$10$^{20}$ cm$^{-2}$ (see Figure 4).
The surface filling factor of the region of density greater than 100 cm$^{-3}$ in Koyama \& Inutsuka (2002) is roughly estimated to be 30--40\%. In this surface filling factor the column density estimated from the observations is consistent with that derived from Koyama \& Inutsuka (2002) and the low temperature of the VSCs is consistent with the result that their peak temperature is lower than that of the typical local dark clouds. These results indicate a potential that the present VSCs are formed in the shock compressed layer through thermal instability.
In order to compare the observational results with the theoretical simulation on internal temperature and density structure in more details, the observations of higher resolutions are needed.
\section{CONCLUSIONS}
We have made a large-scale survey of high Galactic latitude molecular clouds in the $J$ = 1--0 lines of $^{12}$CO and $^{13}$CO toward a large scale structure located around ($l$, $b$) $\sim$ (109$\degr$, $-$45$\degr$) with NANTEN. This survey spatially resolved the distribution of molecular gas associated with the large scale structure. The main conclusions of the present study are summarized as follows:
\begin{enumerate}
\item The $^{12}$CO observation covered the entire large loop-like structure. The loop-like structure consits of very small clumpy clouds. The $^{12}$CO clouds are concentrated on the north to north-west of the loop-like structure and toward the south of that. We identified 78 $^{12}$CO clouds in the observed region. The total mass is estimated to be $\sim$ 64 $M_{\sun}$ if we assume the conversion factor from CO intensity to $N$(H$_2$) as 1.0$\times$10$^{20}$ cm$^{-2}$/(K km s$^{-1}$).
\item We performed $^{13}$CO observations in and around the whole area where the peak temperature of $^{12}$CO is more than 2.0 K. We identified 33 $^{13}$CO clouds and derived physical properties under the assumption of LTE.
\item The mass spectra are well fitted by a power law, $dN/dM$ $\propto$ $M^{-1.53\pm0.13}$ for the $^{12}$CO clouds and $dN/dM$ $\propto$ $M^{-1.36\pm0.10}$ for the $^{13}$CO clouds. These spectral indices are similar to those derived in the other regions.
\item The size and the line width relation of $^{13}$CO clouds is fitted by a least-squares method, log($\Delta V$) = (0.22$\pm$0.43) $\times$ log($R$) + (0.37$\pm$0.52) (c.c.=0.23), but the correlation is not good.
\item Present $^{13}$CO clouds are far from the virial equilibrium, indicating that $^{13}$CO clouds are not gravitationally bound. $M_{\rm vir}$ and $M_{\rm LTE}$ relation can be fitted by a least-squares method as log($M_{\rm vir}$) = (0.91$\pm$0.30) $\times$ log($M_{\rm LTE}$) + (2.23$\pm$0.29) (c.c.=0.66). This index is slightly different from the indices in the other regions although the tendency that molecular clouds are more vilialized as the mass increases is consistent with the other regions.
\item There is no sign of star formation from the comparison of IRAS point sources and Point Source Catalog of Two-Micron All-Sky survey in the present region. This suggests that molecular clouds in this region are not the site of present star formation or the remnants of past star formation.
\item There may be two expanding shells in the present region as inferred from H{\small \,I} although we cannot identify them from CO. The total mechanical luminosity of HD886 during the last few $\times$ 10$^{6}$ yr is comparable to the expanding energy of the northern expanding H{\small \,I} shell. This indicates that some additional source of energy other than HD886 is needed to explain the expanding energy.
\item $^{13}$CO emission is significantly detected in the $^{12}$CO clouds having molecular column density greater than 5$\times$10$^{20}$ cm$^{-2}$. This may be explained as that the $^{13}$CO emitting regions become significant when $A$v becomes larger than $\sim$ 1 mag, marginally enough to shield the ultraviolet radiation to protect $^{13}$CO molecules.
\item There is a possibility that very small clouds have been formed in the shoked layer through the thermal instability. The stellar wind of HD886 may be the source to creat shocks, forming the loop-like structure where the very small clouds are embedded.
\end{enumerate}
\acknowledgments
We greatly appreciate the hospitality of all staff members of the Las
Campanas Observatory of the Carnegie Institution of Washington. The
NANTEN telescope is operated based on a mutual agreement between
Nagoya University and the Carnegie Institution of Washington. We also
acknowledge that the operation of NANTEN can be realized by
contributions from many Japanese public donators and companies. Three
of the authors (N.M., T.O., and Y.F.) acknowledge financial support
from the scientist exchange program under bilateral agreement between
JSPS (Japan Society for the Promotion of Science) and CONICYT (the
Chilean National Commission for Scientific and Technological
Research). This research has made use of the SIMBAD astronomical database
operated by CDS, Strasbourg, France. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. This research has made use of the IRAS point sources from the NASA/IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
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\catcode`@=\active \def@{\phantom{0}}
\begin{deluxetable}{ccccccccc}
\tablewidth{0pt}
\tablecaption{Physical Properties of $^{12}$CO Clouds \label{tbl-1}}
\tablehead{
\colhead{No.} &
\colhead{$l$} &
\colhead{$b$} &
\colhead{$T_{\rm R}^{\ast}$} &
\colhead{$\Delta V$} &
\colhead{$V_{\rm LSR}$} &
\colhead{$R$} &
\colhead{$N$(H$_2$)} &
\colhead{$M_{\rm CO}$} \\
\colhead{(1)} &
\colhead{(2)} &
\colhead{(3)} &
\colhead{(4)} &
\colhead{(5)} &
\colhead{(6)} &
\colhead{(7)} &
\colhead{(8)} &
\colhead{(9)}
}
\startdata
@1 & @95.52 & $-$50.00 & 3.2 & 1.3 & $-$10.7 & 0.31 & @4.7 & @1.07 \\
@2 & @96.11 & $-$53.13 & 1.1 & 1.3 & @$-$9.4 & 0.09 & @0.8 & @0.07 \\
@3 & 101.43 & $-$41.53 & 1.5 & 1.5 & @$-$9.5 & 0.17 & @1.9 & @0.27 \\
@4 & 101.91 & $-$41.20 & 0.7 & 0.5 & @$-$9.5 & 0.08 & @0.9 & @0.04 \\
@5 & 102.09 & $-$41.20 & 1.9 & 1.4 & @$-$8.9 & 0.10 & @3.4 & @0.12 \\
@6 & 102.19 & $-$41.07 & 0.9 & 1.1 & @$-$8.9 & 0.08 & @1.0 & @0.04 \\
@7 & 102.80 & $-$40.27 & 2.5 & 1.6 & @$-$8.4 & 0.16 & @4.5 & @0.30 \\
@8 & 103.32 & $-$40.33 & 1.1 & 1.3 & @$-$8.3 & 0.11 & @1.6 & @0.11 \\
@9 & 103.47 & $-$40.60 & 1.0 & 1.2 & @$-$7.3 & 0.08 & @1.0 & @0.04 \\
10 & 103.74 & $-$39.33 & 5.2 & 2.9 & $-$10.1 & 0.59 & 12.3 & @7.02 \\
11 & 104.20 & $-$38.93 & 0.8 & 1.0 & $-$15.7 & 0.10 & @2.0 & @0.09 \\
12 & 104.48 & $-$38.53 & 1.4 & 3.5 & @$-$8.6 & 0.16 & @4.5 & @0.44 \\
13 & 104.81 & $-$38.80 & 1.4 & 1.2 & @$-$5.0 & 0.10 & @1.5 & @0.08 \\
14 & 105.07 & $-$38.80 & 1.1 & 1.4 & @$-$5.7 & 0.08 & @1.8 & @0.06 \\
15 & 105.08 & $-$52.27 & 1.0 & 1.2 & @$-$0.1 & 0.07 & @1.2 & @0.04 \\
16 & 105.10 & $-$38.07 & 2.5 & 2.4 & $-$12.7 & 0.26 & @5.7 & @1.19 \\
17 & 105.22 & $-$53.20 & 1.2 & 1.3 & @$-$3.7 & 0.09 & @1.1 & @0.05 \\
18 & 105.40 & $-$48.13 & 1.6 & 1.8 & @$-$9.0 & 0.12 & @3.0 & @0.20 \\
19 & 105.77 & $-$38.40 & 5.6 & 1.9 & @$-$5.0 & 0.47 & 10.5 & @5.60 \\
20 & 105.80 & $-$39.60 & 1.7 & 1.8 & @$-$9.8 & 0.24 & @3.1 & @0.70 \\
21 & 105.82 & $-$54.00 & 1.1 & 1.4 & @$-$4.8 & 0.07 & @1.3 & @0.04 \\
22 & 106.76 & $-$36.53 & 2.7 & 3.0 & @$-$9.8 & 0.53 & @8.3 & @4.29 \\
23 & 106.97 & $-$37.87 & 1.7 & 1.7 & @$-$1.2 & 0.12 & @3.4 & @0.17 \\
24 & 107.32 & $-$37.60 & 1.2 & 1.5 & @$-$2.3 & 0.12 & @1.6 & @0.11 \\
25 & 107.38 & $-$52.00 & 1.0 & 2.6 & @$-$7.5 & 0.12 & @2.1 & @0.13 \\
26 & 107.87 & $-$53.93 & 0.8 & 1.6 & @$-$4.9 & 0.11 & @1.8 & @0.11 \\
27 & 108.21 & $-$53.73 & 2.2 & 1.4 & @$-$5.1 & 0.16 & @2.9 & @0.25 \\
28 & 108.33 & $-$53.13 & 2.2 & 1.2 & @$-$0.2 & 0.07 & @3.5 & @0.08 \\
29 & 108.78 & $-$52.60 & 5.7 & 2.4 & @$-$6.6 & 0.79 & 16.6 & 11.13 \\
30 & 109.00 & $-$52.13 & 3.3 & 1.9 & @$-$1.5 & 0.31 & @5.1 & @1.43 \\
31 & 109.00 & $-$50.07 & 2.5 & 1.6 & @$-$4.7 & 0.13 & @5.1 & @0.26 \\
32 & 109.11 & $-$50.60 & 0.9 & 2.2 & @$-$5.9 & 0.09 & @1.6 & @0.06 \\
33 & 109.17 & $-$37.60 & 4.0 & 1.5 & @$-$4.4 & 0.36 & @5.6 & @1.83 \\
34 & 109.55 & $-$52.87 & 1.3 & 2.0 & @$-$7.1 & 0.14 & @2.5 & @0.16 \\
35 & 109.68 & $-$38.27 & 4.6 & 1.4 & @$-$8.1 & 0.17 & @5.4 & @0.52 \\
36 & 109.76 & $-$38.00 & 2.7 & 2.0 & @$-$4.9 & 0.21 & @4.7 & @0.63 \\
37 & 109.78 & $-$53.33 & 2.8 & 1.5 & @$-$6.4 & 0.18 & @2.6 & @0.30 \\
38 & 109.84 & $-$50.73 & 4.0 & 2.6 & @$-$7.8 & 0.18 & @5.0 & @0.40 \\
39 & 109.85 & $-$38.60 & 0.8 & 1.8 & @$-$8.1 & 0.12 & @2.1 & @0.13 \\
40 & 110.05 & $-$50.40 & 2.1 & 3.1 & @$-$7.4 & 0.10 & @3.6 & @0.14 \\
41 & 110.06 & $-$41.27 & 4.4 & 2.9 & @$-$6.1 & 0.47 & 10.2 & @4.21 \\
42 & 110.32 & $-$48.93 & 2.9 & 0.9 & @$-$4.2 & 0.12 & @2.6 & @0.17 \\
43 & 110.94 & $-$41.00 & 1.3 & 3.4 & @$-$6.3 & 0.17 & @5.7 & @0.67 \\
44 & 110.98 & $-$39.13 & 1.2 & 2.2 & @$-$9.6 & 0.12 & @2.9 & @0.20 \\
45 & 111.08 & $-$50.13 & 1.6 & 1.0 & @$-$7.2 & 0.13 & @1.7 & @0.11 \\
46 & 112.04 & $-$39.87 & 3.4 & 2.3 & @$-$5.4 & 0.26 & @6.4 & @1.11 \\
47 & 112.40 & $-$40.07 & 1.3 & 2.6 & @$-$7.2 & 0.10 & @3.6 & @0.19 \\
48 & 112.55 & $-$39.60 & 1.2 & 1.9 & @$-$3.6 & 0.16 & @3.0 & @0.29 \\
49 & 112.64 & $-$50.07 & 0.7 & 1.5 & @$-$6.5 & 0.14 & @1.7 & @0.14 \\
50 & 112.72 & $-$41.13 & 1.2 & 1.9 & @$-$6.2 & 0.11 & @2.4 & @0.13 \\
51 & 112.72 & $-$41.13 & 1.2 & 1.9 & @$-$6.2 & 0.11 & @2.4 & @0.13 \\
52 & 112.72 & $-$39.67 & 1.2 & 2.6 & @$-$5.3 & 0.10 & @1.9 & @0.09 \\
53 & 112.84 & $-$40.27 & 1.3 & 1.7 & @$-$5.1 & 0.08 & @2.9 & @0.11 \\
54 & 113.15 & $-$39.47 & 1.5 & 1.2 & @$-$4.7 & 0.16 & @2.5 & @0.23 \\
55 & 113.23 & $-$52.07 & 4.4 & 1.3 & @$-$7.3 & 0.30 & @5.7 & @1.32 \\
56 & 113.42 & $-$42.33 & 3.4 & 2.4 & $-$10.5 & 0.23 & @5.9 & @0.89 \\
57 & 113.47 & $-$39.20 & 0.5 & 1.9 & @$-$5.7 & 0.10 & @1.4 & @0.08 \\
58 & 113.56 & $-$49.93 & 1.3 & 0.9 & @$-$7.5 & 0.12 & @1.1 & @0.09 \\
59 & 113.94 & $-$51.67 & 1.4 & 1.6 & @$-$9.7 & 0.07 & @3.2 & @0.07 \\
60 & 114.38 & $-$51.73 & 3.9 & 1.6 & @$-$9.7 & 0.17 & @6.7 & @0.55 \\
61 & 114.49 & $-$50.80 & 1.5 & 1.8 & @$-$7.8 & 0.09 & @3.2 & @0.12 \\
62 & 114.52 & $-$41.53 & 2.2 & 1.1 & @$-$7.8 & 0.15 & @2.1 & @0.20 \\
63 & 114.83 & $-$51.07 & 0.8 & 1.8 & @$-$8.1 & 0.10 & @1.4 & @0.07 \\
64 & 114.90 & $-$41.73 & 1.5 & 1.2 & @$-$8.3 & 0.08 & @1.1 & @0.04 \\
65 & 115.10 & $-$43.80 & 1.6 & 1.0 & @$-$2.1 & 0.11 & @1.5 & @0.09 \\
66 & 115.16 & $-$45.33 & 1.2 & 1.6 & @$-$8.4 & 0.17 & @1.6 & @0.25 \\
67 & 115.24 & $-$43.40 & 1.8 & 2.0 & @$-$3.5 & 0.23 & @2.7 & @0.57 \\
68 & 115.63 & $-$43.60 & 0.8 & 1.0 & @$-$3.3 & 0.08 & @1.6 & @0.06 \\
69 & 115.74 & $-$46.20 & 1.5 & 1.0 & @$-$6.8 & 0.13 & @2.3 & @0.18 \\
70 & 116.20 & $-$43.73 & 3.2 & 1.5 & @$-$2.6 & 0.16 & @3.6 & @0.31 \\
71 & 116.33 & $-$44.80 & 4.2 & 2.3 & @$-$3.9 & 0.59 & 10.0 & @7.13 \\
72 & 116.45 & $-$50.53 & 2.3 & 1.1 & @$-$7.3 & 0.17 & @3.5 & @0.35 \\
73 & 116.64 & $-$52.33 & 0.9 & 1.5 & @$-$7.6 & 0.07 & @2.1 & @0.05 \\
74 & 116.86 & $-$43.87 & 0.8 & 3.7 & @$-$4.7 & 0.18 & @2.9 & @0.41 \\
75 & 117.01 & $-$50.73 & 2.7 & 0.8 & @$-$7.5 & 0.10 & @2.0 & @0.10 \\
76 & 117.11 & $-$44.33 & 2.3 & 1.1 & @$-$2.6 & 0.14 & @5.1 & @0.25 \\
77 & 118.12 & $-$52.13 & 1.1 & 1.1 & @$-$6.7 & 0.12 & @1.8 & @0.12 \\
78 & 118.23 & $-$52.67 & 4.2 & 2.0 & @$-$7.8 & 0.43 & @8.8 & @3.12 \\
\enddata
\tablecomments{Col. (1) : Cloud number, Col. (2)--(3) : Cloud
peak ($l,b$) position in degree, Col. (4) : Peak temperature in K, Col. (5)
: Line
width of the composite spectrum in km s$^{-1}$, Col. (6) : Peak velocity of the
composite spectrum in km s$^{-1}$, Col. (7) : Radius of the molecular cloud
in pc, Col. (8) : Column density of peak position in 10$^{20}$ cm$^{-2}$,
Col. (9) : Mass of the molecular cloud in $M_{\sun}$. Col. (4) to (6) are derived
by using a single Gaussiun fitting.
}
\end{deluxetable}
\clearpage
\begin{deluxetable}{ccccccccccc}
\tablewidth{0pc}
\tablecaption{Physical Properties of $^{13}$CO Clouds}
\tablehead
{
\\
\colhead{No.} &
\colhead{$l$} &
\colhead{$b$} &
\colhead{$T_{\rm R}^{\ast}$} &
\colhead{$\Delta V$} &
\colhead{$V_{\rm LSR}$} &
\colhead{$R$} &
\colhead{$\tau(^{13}\rm CO$)} &
\colhead{$N(\rm H_2)$} &
\colhead{$M_{\rm LTE}$} &
\colhead{$M_{\rm vir}$} \\
\colhead{(1)} &
\colhead{(2)} &
\colhead{(3)} &
\colhead{(4)} &
\colhead{(5)} &
\colhead{(6)} &
\colhead{(7)} &
\colhead{(8)} &
\colhead{(9)} &
\colhead{(10)} &
\colhead{(11)}
}
\startdata
16a & 105.11 & $-$38.03 & 0.7 & 1.7 & $-$11.4 & 0.06 & 0.11 & @4.0 & 0.08 & @34.6 \\
16b & 105.12 & $-$37.80 & 0.7 & 1.0 & $-$13.1 & 0.07 & 0.12 & @3.6 & 0.09 & @13.5 \\
19a & 105.54 & $-$38.63 & 0.8 & 1.1 & @$-$5.5 & 0.09 & 0.14 & @2.5 & 0.17 & @23.2 \\
19b & 105.77 & $-$38.37 & 2.3 & 1.7 & @$-$4.0 & 0.21 & 0.46 & 16.6 & 1.41 & 129.8 \\
20@ & 103.70 & $-$39.33 & 1.2 & 1.5 & @$-$9.7 & 0.13 & 0.22 & 10.0 & 0.56 & @57.9 \\
22@ & 106.93 & $-$36.40 & 0.7 & 2.1 & @$-$9.4 & 0.05 & 0.13 & @3.7 & 0.06 & @45.2 \\
26@ & 107.87 & $-$53.93 & 0.3 & 1.8 & @$-$7.7 & 0.07 & 0.13 & @2.3 & 0.09 & @46.4 \\
27@ & 108.10 & $-$53.77 & 0.8 & 3.1 & @$-$6.2 & 0.09 & 0.13 & @3.7 & 0.21 & 176.1 \\
29a & 107.82 & $-$51.70 & 1.2 & 1.6 & @$-$5.0 & 0.08 & 0.21 & @7.0 & 0.21 & @40.7 \\
29b & 108.78 & $-$52.63 & 1.4 & 2.0 & @$-$4.6 & 0.14 & 0.26 & 11.1 & 0.87 & 120.6 \\
29c & 108.84 & $-$52.03 & 1.2 & 0.7 & @$-$6.2 & 0.11 & 0.21 & @5.9 & 0.29 & @10.6 \\
29d & 108.89 & $-$52.17 & 1.3 & 1.1 & @$-$6.9 & 0.08 & 0.23 & @3.8 & 0.13 & @18.4 \\
29e & 109.00 & $-$52.40 & 1.1 & 2.1 & @$-$7.8 & 0.12 & 0.20 & @6.0 & 0.41 & 110.6 \\
29f & 109.00 & $-$52.13 & 0.9 & 1.6 & @$-$1.9 & 0.04 & 0.15 & @5.8 & 0.05 & @22.5 \\
29g & 109.16 & $-$51.87 & 1.0 & 1.7 & @$-$5.8 & 0.10 & 0.17 & @2.3 & 0.25 & @57.6 \\
29h & 109.53 & $-$51.17 & 0.8 & 0.9 & @$-$6.5 & 0.05 & 0.15 & @2.9 & 0.04 & @@8.7 \\
29i & 109.64 & $-$51.57 & 0.3 & 2.4 & @$-$7.0 & 0.05 & 0.12 & @3.2 & 0.04 & @59.7 \\
29j & 109.80 & $-$51.37 & 0.4 & 1.5 & @$-$6.7 & 0.05 & 0.14 & @2.5 & 0.04 & @22.3 \\
33a & 109.17 & $-$37.60 & 0.8 & 1.0 & @$-$4.4 & 0.05 & 0.13 & @2.4 & 0.05 & @10.7 \\
33b & 109.21 & $-$37.80 & 0.9 & 0.9 & @$-$4.1 & 0.06 & 0.16 & @4.7 & 0.08 & @10.6 \\
33c & 109.34 & $-$37.80 & 0.7 & 0.7 & @$-$4.5 & 0.05 & 0.13 & @3.2 & 0.05 & @@4.7 \\
35@ & 109.72 & $-$38.27 & 0.9 & 0.7 & @$-$8.1 & 0.05 & 0.16 & @3.2 & 0.05 & @@5.1 \\
38@ & 109.90 & $-$50.73 & 1.1 & 1.8 & @$-$7.7 & 0.11 & 0.19 & 10.5 & 0.48 & @75.3 \\
41a & 110.11 & $-$41.27 & 1.0 & 2.9 & @$-$5.6 & 0.11 & 0.23 & 10.1 & 0.41 & 197.4 \\
41b & 110.19 & $-$41.07 & 0.6 & 2.0 & @$-$7.6 & 0.04 & 0.10 & @2.3 & 0.03 & @33.1 \\
43@ & 111.12 & $-$41.00 & 0.7 & 1.1 & @$-$5.6 & 0.04 & 0.12 & @4.3 & 0.03 & @@9.4 \\
46@ & 112.08 & $-$39.87 & 0.8 & 0.9 & @$-$5.1 & 0.06 & 0.14 & @3.7 & 0.07 & @@9.3 \\
55@ & 113.17 & $-$52.03 & 1.2 & 0.7 & @$-$7.2 & 0.11 & 0.22 & @4.3 & 0.24 & @10.6 \\
56@ & 113.17 & $-$42.60 & 0.8 & 0.9 & $-$10.9 & 0.04 & 0.14 & @3.7 & 0.04 & @@6.5 \\
60@ & 114.32 & $-$51.70 & 0.8 & 1.2 & @$-$9.7 & 0.08 & 0.15 & @4.8 & 0.19 & @24.9 \\
71a & 115.49 & $-$44.40 & 0.9 & 1.0 & @$-$3.5 & 0.10 & 0.16 & @3.7 & 0.20 & @22.2 \\
71b & 116.21 & $-$44.97 & 1.2 & 1.6 & @$-$3.9 & 0.18 & 0.22 & @9.1 & 1.04 & @93.9 \\
78@ & 118.19 & $-$52.70 & 1.5 & 1.2 & @$-$8.1 & 0.12 & 0.28 & @9.6 & 0.53 & @36.7 \\
\enddata
\begin{flushleft}
{\footnotesize Note---Col. (1) : Cloud number of $^{13}$CO taken from that of $^{12}$CO cloud with
which the $^{13}$CO cloud is associated. If plural $^{13}$CO clouds are associated with one $^{12}$CO cloud,
a sequential alphabet is added, Col. (2)--(3) : Cloud peak ($l$, $b$) position in degree, Col. (4) : Peak temperature in K, Col. (5) : Line
width of the composite spectrum in km s$^{-1}$, Col. (6) : Peak velocity of the
composite spectrum in km s$^{-1}$, Col. (7) : Radius of the molecular cloud in pc,
Col. (8) :
Optical depth of $^{13}$CO, Col. (9) : Column density of peak position in 10$^{20}$
cm$^{-2}$,
Col. (10) : Mass of the molecular cloud assuming the LTE in $M_{\sun}$, Col. (11) :
Virial mass of the molecular cloud in $M_{\sun}$. Col. (4) to (6) are derived by using
a single Gaussiun fitting.}
\end{flushleft}
\end{deluxetable}
|
Title:
On the variation of the fine-structure constant: Very high resolution spectrum of QSO HE 0515-4414 |
Abstract: We present a detailed analysis of a very high resolution (R\approx 112,000)
spectrum of the quasar HE 0515-4414 obtained using the High Accuracy Radial
velocity Planet Searcher (HARPS) mounted on the ESO 3.6 m telescope at the La
Silla observatory. The HARPS spectrum, of very high wavelength calibration
accuracy (better than 1 m\AA), is used to search for possible systematic
inaccuracies in the wavelength calibration of the UV Echelle Spectrograph
(UVES) mounted on the ESO Very Large Telescope (VLT). We have carried out
cross-correlation analysis between the Th-Ar lamp spectra obtained with HARPS
and UVES. The shift between the two spectra has a dispersion around zero of
\sigma\simeq 1 m\AA. This is well within the wavelength calibration accuracy of
UVES (i.e \sigma\simeq 4 m\AA). We show that the uncertainties in the
wavelength calibration induce an error of about, \Delta\alpha/\alpha\le
10^{-6}, in the determination of the variation of the fine-structure constant.
Thus, the results of non-evolving \Delta\alpha/\alpha reported in the
literature based on UVES/VLT data should not be heavily influenced by problems
related to wavelength calibration uncertainties. Our higher resolution spectrum
of the z_{abs}=1.1508 damped Lyman-\alpha system toward HE 0515-4414 reveals
more components compared to the UVES spectrum. Using the Voigt profile
decomposition that simultaneously fits the high resolution HARPS data and the
higher signal-to-noise ratio UVES data, we obtain,
\Delta\alpha/\alpha=(0.05\pm0.24)x10^{-5} at z_{abs}=1.1508. This result is
consistent with the earlier measurement for this system using the UVES spectrum
alone.
| https://export.arxiv.org/pdf/astro-ph/0601194 |
\title{On the variation of the fine-structure constant:
Very high resolution spectrum of QSO HE {0515$-$4414}
\author{Hum Chand\inst{1},
Raghunathan Srianand\inst{1},
Patrick Petitjean\inst{2,3},\\
Bastien Aracil\inst{2,4},
Ralf Quast\inst{5},
Dieter Reimers\inst{5}
}
\thanks{
Based on observations collected
at the European Southern Observatory (ESO), under Programe
ID No. 072.A-0244 with HARPS on the 3.6~m
telescope operated at the La Silla Observatory and Programe
ID 066.A-0212 with
UVES/VLT at the Paranal observatory.}
}
\titlerunning{Variation of the fine-structure constant}
\authorrunning{H. Chand et al.}
\author{Hum Chand\inst{1},
Raghunathan Srianand\inst{1},
Patrick Petitjean\inst{2,3},\\
Bastien Aracil\inst{2,4},
Ralf Quast\inst{5},
Dieter Reimers\inst{5}
}
\offprints{H. Chand \\~\email{hcverma@iucaa.ernet.in}}
\institute{$^1$IUCAA, Post Bag 4,
Ganeshkhind, Pune 411 007, India\\
$^2$Institut d'Astrophysique de Paris,
UMR7095 CNRS, Universite Pierre \& Marie Curie,
98 bis boulevard Arago,
75014 Paris.\\
$^3$LERMA, Observatoire de Paris,
61 Rue de l'Observatoire,
F-75014 Paris, France\\
$^4$Department of Astronomy,
University of Massachusetts,
710 North Pleasant Street,
Amherst, MA 01003-9305, USA\\
$^5$Hamburger Sternwarte, Universitat Hamburg, Gojenbergsweg 112,
D-21209 Hamburg, Germany\\
}
\date{Received date/ Accepted date}
\abstract
{}
{We present a detailed analysis of a very high resolution
(R~$\approx 112,000$) spectrum of the quasar HE {0515$-$4414}
obtained using the High Accuracy Radial velocity Planet Searcher
(HARPS) mounted on the ESO 3.6~m telescope at the La Silla observatory.
The main aim is to use HARPS spectrum of very high
wavelength calibration accuracy (better than 1~m\AA), to constrain
the variation of $\alpha=e^2/\hbar c$ and investigate
any possible systematic inaccuracies in the
wavelength calibration of the UV Echelle Spectrograph (UVES) mounted on the ESO Very
Large Telescope (VLT).}
{A cross-correlation analysis between the Th-Ar lamp
spectra obtained with HARPS and UVES is carried out
to detect any possible shift between the two spectra.
Absolute wavelength calibration accuracies, and how that translate
to the uncertainties in \dela are computed using Gaussian fits
for both lamp spectra. The value of \dela at \zabs~=~1.1508 is obtained
using Many Multiplet method, and simultaneous
Voigt profile fits of HARPS and UVES spectra.}
{We find the shift between the HARPS and UVES spectra has mean around zero with a dispersion
of $\sigma\simeq 1$ m\AA.
This is shown to be well within the wavelength calibration accuracy of UVES
(i.e $\sigma\simeq 4$ m\AA). We show that the uncertainties in the
wavelength calibration induce an error of about,
\dela $~\le 10^{-6}$, in the determination of the
variation of the fine-structure constant.
Thus, the results of non-evolving \dela reported in the
literature based on UVES/VLT data should not be heavily influenced by
problems related to wavelength calibration uncertainties.
Our higher resolution spectrum
of the \zabs~=~1.1508 Damped Lyman-$\alpha$ system toward HE {0515$-$4414}
reveals more components compared to the UVES spectrum.
Using only \feii lines of \zabs~=~1.1508 system, we obtain
\dela~=~${ (0.05\pm0.24)\times10^{-5}}$. This result is consistent with
the earlier measurement for this system using the UVES spectrum alone.}
{}
\keywords{
{\em Quasars:} absorption lines --
{\em cosmology:} observations
}
\section{Introduction}
\label{sect:Int}
Some of the modern theories of fundamental physics, such as SUSY,
GUT and Super-string theory, allow possible space and time variations of
the fundamental constants, thus motivating an
experimental search for such a variation (Uzan 2003 and 2004
for a detail review on the subject). Murphy et al. (2003),
applying the Many Multiplet method (MM method) to 143
complex metal line systems, claimed
a non-zero variation of the fine-structure constant, $\alpha=e^2/\hbar c$:
$\langle\Delta\alpha/\alpha\rangle = (-0.57\pm0.11)\times10^{-5}$
for $0.2\le z \le 3.5$,
where $\Delta\alpha/\alpha=(\alpha_{z}-\alpha_{0})/\alpha_{0}$, with
$\alpha_{0}$ being the present value and $\alpha_{z}$ its value at redshift $z$.
This result, if true, would have very
important implications to our understanding of fundamental
physics and has therefore motivated new activities in the field.
Search for the possible time-variation of $\alpha$ using alkali doublets
has started long ago (Bahcall et al. 1967).
The alkali-doublet method is a clean method for constraining
the variation in $\alpha$ using spectral lines because it uses transitions
from the same species
(Wolfe et al. 1976; Levshakov 1994; Potekhin et al. 1994; Cowie \&
Songaila, 1995; Varshalovich et al. 1996; Varshalovich et al. 2000;
Murphy et al. 2001a; Martinez et al. 2003; Chand et al. 2005).
The tightest constraint obtained using this method till date
is \dela = (0.15$\pm$0.44) $\times10^{-5}$ at
$z \sim 2$ (Chand et al. 2005).
\par
Studies based on heavy element molecular absorption lines seen in the
radio/mm wavelength range are more sensitive than that
based on optical/UV absorption lines.
They usually provide constraints on the variation of a combination of
the fine-structure constant, the proton g-factor ($G_{p}$) and
the electron-to-proton mass ratio ($\mu$). Murphy et al. (2001b)
have obtained \dela = $(-0.10\pm0.22)\times10^{-5}$ at
z = 0.2467 and \dela = $(-0.08\pm0.27)\times
10^{-5}$ at z = 0.6847, assuming a constant proton g-factor ($G_{p}$).
It has been pointed out that OH lines are very useful in
simultaneously constraining various fundamental constants
(Chengalur \& Kanekar 2003; Kanekar et al. 2004;
Darling 2003, 2004). These studies have provided
\dela = $(0.6\pm1.0)\times 10^{-5}$ for an absorption system
at \zabs = 0.247 toward PKS~1413+135. Such
studies have not been performed yet at higher redshift (i.e $z$~$\ge1$)
due to the lack of molecular absorption systems. \par
Constraints on the variations of $\alpha$ are also obtained from
terrestrial measurements.
The most stringent constrain has been obtained from the analysis
of the Oklo phenomenon. Fujii et al. (2000) find that $\Delta\alpha/\alpha =
(-0.8\pm1.0)\times10^{-8}$
over a period of about 2 billion years (or $z\simeq0.45$).
Laboratory experiments also give very stringent constraints on the
local variation of $\alpha$. Marion et al. (2003) have obtained, $\Delta\alpha/\alpha\Delta t =
(-0.4\pm16)\times10^{-16}\,{\rm yr}^{-1}$, by
comparing the hyperfine transition in $^{87}$Rb and $^{133}$Cs over a
period of 4 years assuming no variation in the magnetic moments.
Fischer et al. (2004) have obtained, $\Delta\alpha/\alpha\Delta t =
(-0.9\pm2.9)\times10^{-16}\,{\rm yr}^{-1}$, by comparing the absolute
$1S-2S$ transition of atomic hydrogen to the ground state of
Cesium. A linear extrapolation gives a constraint of
$-1.3\times10^{-6}\le$ \dela$\le1.9\times10^{-6}$ at $z = 1$ for
the most favored cosmology ($\Omega_m = 0.27$, $\Omega_{\Lambda} =0.73$ and $ h=0.71$).
\par
Clearly all the experimental results summarized above are
consistent with no variation of $\alpha$. However, these results
do not directly conflict with the positive detection by Murphy et al. (2003)
either because of the insufficient sensitivity of the method (as in the case of
alkali doublets) or because of the different redshift coverage (as in
the case of radio and terrestrial measurements).
However, recent attempts using the MM method (or its modified version)
applied to very high quality UVES spectra have resulted in null detections.
The analysis of Fe~{\sc ii} multiplets and Mg~{\sc ii} doublets
in a homogeneous sample of 23 systems has yielded a stringent constraint,
\dela = $(-0.06\pm0.06)\times10^{-5}$ (Chand et al. 2004;
Srianand et al. 2004). Modified MM method analysis of \zabs = 1.1508
toward HE 0515$-$4414 that avoids possible complications due to isotopic
abundances has resulted
in \dela = $(0.01 \pm 0.17)\times 10^{-5}$ (Quast et al. 2004).
Levshakov et al. (2005b) have
re-analysis this system using the single ion differential
alpha measurement method as described in Levshakov et al. (2005a),
and obtained \dela = $(-0.007\pm0.084)\times10^{-5}$.
Clearly all studies based on VLT-UVES data are in contradiction with the conclusions
of Murphy et al. (2003).
\par
A first possible concern about these studies is the real
accuracy and robustness of the various calibration procedures.
A second possible source of uncertainty comes from the multi-component Voigt-profile
decomposition.
It is very important to check how sensitive the derived constraints are to
the profile decomposition. This can be done by performing the analysis on data
of higher resolution than typical UVES (or HIRES) spectra.
The best way to investigate all this is to compare data taken by
UVES (or HIRES) with data on the same object taken with another
completely independent, well controlled, and higher spectral resolution instrument.
The advent of HARPS mounted on the ESO 3.6~m telescope makes this possible.
Unfortunately this is only possible on the brightest quasar in the
southern sky, HE~0515$-$4414.
\par
This forms the basic motivations of this work.
We report the analysis of the \zabs = 1.15 DLA system
toward QSO HE 0515$-$4414 (De la Varga et al. 2000, Quast et al. 2004, 2005)
using very high resolution (R$\sim112,000$) spectra obtained with
HARPS mounted on the ESO 3.6~m telescope.
The organization of the paper is as follows. The HARPS
observations of HE 0515$-$4414 are described in Section 2.
Calibration accuracy and comparison with the UVES observations
are discussed in Section 3. In Section 4 we present the
joint analysis of the HARPS and UVES spectra. Results
are summarized and discussed in Section 5.
\section{Observations}
\label{sect:Obs}
The spectrum of HE 0515$-$4414 used in this work was obtained with the
High Accuracy Radial velocity Planet Searcher (HARPS)
mounted on the ESO 3.6~m telescope at the La Silla observatory.
HARPS is a fiber-fed spectrograph and is therefore less affected by
any fluctuation in the seeing conditions (Mosser et al. 2004).
It is installed in the Coud\'e room of the 3.6~m telescope building and is
enclosed in a box in which vacuum and constant temperature are maintained.
The instrument has been specifically designed to guarantee stability
and high-accuracy wavelength calibration.
\par
The observations were carried over four nights in classical
fiber spectroscopy mode, with one fiber on the target and
the other on the sky.
The CCD was read in normal low readout mode without binning.
The echelle order extraction from the raw data frame is done
using the HARPS reduction pipeline.
The error spectrum is computed by modeling the photon
noise with a Poisson distribution and
CCD readout noise with a Gaussian distribution.
The calibrated spectrum is converted to vacuum wavelengths according
to Edl\'en (1966) and the heliocentric velocity correction is done
manually using the dedicated MIDAS (ESO-Munich Image Data Analysis Software) procedure.
Special attention was given while merging the orders.
While combining overlapping regions, higher weights were assigned to the
wavelength ranges toward the
center of the order compared to the one at the edges.
The resulting 1-D spectrum covers the wavelength range from 3800 to 6900
\AA, with a gap between 5300 to 5330 \AA~ caused by the transition between
the two CCDs used in HARPS.
In total, we obtained 14 individual exposures, each
of duration between 1 and 1.5 hour. Combination of individual exposures
is performed using a sliding window and weighting the signal by the
errors in each pixel.
The final error spectrum was obtained
by adding quadratically in each pixel the extracted errors
and the rms of the 14 individual measurements. The final
combined spectrum has a S/N ratio of about 30 to 40 per pixel
of size $\sim$0.015 \AA~ and a spectral resolution of R$\approx$ 112,000.\par
To make quantitative comparisons, as will be discussed in
the next section, we have also used the UVES spectrum of this QSO.
The details of the UVES observation and data reduction can be found
in Quast et al. (2004). However we have used our procedures for air-to-vacuum
wavelength conversion, heliocentric velocity correction
and for the addition of individual exposures
as in the case of the HARPS spectrum.
\section{Accuracy of wavelength calibration}
\label{sect:comp_cal}
In this Section we investigate (i) the cross-correlation between
the Th-Ar lamp spectra obtained with HARPS and UVES, (ii) the absolute wavelength
calibration accuracies of HARPS and UVES and (iii) how the uncertainties
in the wavelength calibration translate into uncertainties in \dela measurements
in the case of HARPS and UVES.
\subsection{Cross-correlation of UVES and HARPS Th-Ar spectra}
\label{crossCorr:subsec}
To estimate how well the UVES and HARPS wavelength scales agree,
one can in principle use the narrow heavy element absorption lines
seen in the spectra of the QSO.
However not only the number of such lines is small but also,
due to differences in the resolutions and S/N ratios, spurious
shifts can be introduced in the analysis. In order to
avoid this, we perform a cross-correlation analysis between the
Th-Ar lamp spectra obtained with UVES and HARPS.
We have 4 and 14 Th-Ar lamp exposures respectively for UVES
and HARPS observations in the setting that covers the
wavelength range where Fe~{\sc ii} and Mg~{\sc ii} absorption lines from
the \zabs~=~1.1508 absorption system are seen.
We have combined
all the extracted Th-Ar exposures after subtracting a smooth
continuum corresponding to the background light.
\par
The cross-correlation analysis was performed
on groups of five consecutive unblended emission lines that
are clearly seen in both the UVES and HARPS spectra.
For this, both spectra were re-sampled to an uniform wavelength scale using
cubic spline and the pixel-by-pixel cross correlation was performed
by shifting the UVES spectrum with respect to the HARPS spectrum. The results of the
cross-correlation at places where absorption lines at \zabs~=~1.1508 are
redshifted are shown in Fig.~\ref{crossc.fig}.
All the curves shown in this figure have their peak at zero pixel shift
with a typical pixel size of 15 m\AA.
In order to derive sub-pixel
accuracy in the cross-correlation, we have fitted a Gaussian to
the cross correlation curves as is shown by dotted lines
(Fig.~\ref{crossc.fig}) and derive its centroid accurately.
The corresponding values are given in each panel. The
relative shifts between the two spectra are less than 1 m\AA~
except in one case where it is 1.7~m\AA. We note that the quadratic refinement
technique (instead of a Gaussian fitting) also gives similar results.
To derive the global trend of the relative shift, we have extended our
cross-correlation analysis, to the entire wavelength range.
The result of the analysis is shown in Fig.~\ref{crossHARPSUV.fig}.
The shifts are obtained in the same way as in Fig.~\ref{crossc.fig}.
The average of the mean relative shifts
over the entire wavelength range is 0.01 m\AA~ with an rms deviation of 1.09 m\AA.
In what follows we investigate the absolute wavelength calibration accuracies of
the two instruments.
\subsection{Testing absolute wavelength calibration error of UVES and HARPS}
\label{abscal:subsec}
To test the absolute wavelength calibration accuracy we compare
the central wavelength of strong un-blended emission lines in
the extracted Th-Ar lamp spectrum with the wavelengths tabulated in Cuyper
et al. (1998). We model the emission lines by a single Gaussian function.
The best-fit line-centroid along
with other parameters of the models and errors are determined by a
\chisq minimization procedure. In many cases we
find it difficult to fit the lines with reduced $\chi^{2}\approx 1$.
In such cases we have scaled the flux errors by square root
of the reduced \chisq and re-run the fitting procedure. In this way, we
have avoided any underestimation of the errors on the best fit parameters,
assuming that the actual errors on the flux of the Th-Ar lamp spectrum
was somehow underestimated.\par
The difference between the best-fit line centroid,
in the extracted lamp spectra and the wavelength quoted by Cuyper et al. (1998)
is plotted in Fig.~\ref{dlam.fig}. The wavelength range shown in this
figure is the one covered by the main \feii and \mgii lines of the
\zabs~=~1.1508 system. We find the rms of the deviation
($\Delta\lambda$ in Fig.~\ref{dlam.fig}) around zero to be,
respectively, 0.87~m\AA~ and 4.08~m\AA~ for the HARPS and UVES lamp spectra.
This clearly demonstrates that the shifts between the HARPS and the
UVES lamp spectra measured from the cross-correlation analysis (i.e
$\le1$ m\AA) are well within the wavelength calibration accuracy
of UVES.\par
In addition, we have used the best-fit FWHM of the Gaussian fit of the lamp lines
to derive the spectral resolution ($R=\lambda/FWHM$) of the spectrum. The
resolution measurements are shown in Fig.~\ref{reso.fig}. The mean resolution
and standard deviation for HARPS and UVES
are found to be $R= 112,200$ and $\sigma=8,400$; $R= 55,100$ and $\sigma=7,600$ respectively.\par
\subsection{Effect of calibration error on $\Delta\alpha/\alpha$
measurement}
\label{calDal:subsec}
Next we investigate how the scatter in wavelength calibration
($\Delta\lambda$) translates into a scatter in \dela.
We follow the method used by Murphy et al. (2003) for this purpose.
We randomly choose 3 emission lines in the
lamp spectrum, with a rest wavelength close to each of the observed wavelengths
of the \feii and \mgii lines used in the analysis of the variation of $\alpha$.
There are two \mgii lines, $\lambda$2796 and $\lambda$2803, and
five \feii lines, $\lambda$2344, $\lambda$2374, $\lambda$2382, $\lambda$2586,
and $\lambda$2600. Thus we have 21 ($7\times3$) lines per realization.
By choosing 3 lines, we mimic 3 distinct components in the actual absorption system.
We assume that the measured shift in the emission line centroid away from
the actual value is caused by the variation in $\alpha$.
To estimate this variation, we use the analytic fitting function given by
Dzuba et al. (2002),
\begin{equation}
w = w_o + q x.
\end{equation}
Here, $w_o$ and $w$ are, respectively, the vacuum wave number
(in units of cm$^{-1}$) measured in the laboratory and
the modified wave number due to a change in $\alpha$;
$x=(\Delta\alpha/\alpha+1)^2-1$ and $q$ is the
sensitivity coefficient.
At each chosen lamp emission line we assign the $q$ value of the neighboring
metal absorption transition.
All the lamp emission lines in each realization are fitted
simultaneously with Gaussians,
for one fixed value of \dela. Here, the \dela value is used to
modify the rest wavelength of the emission lines using the $q$
coefficients given by Dzuba et al. (2002) for the corresponding metal lines.
This procedure is repeated for a range of \dela, from
$-2.0\times10^{-5}$ to $2.0\times 10^{-5}$ in steps of $0.02\times10^{-5}$
to achieve \chisq as a function of \dela. The \chisq versus \dela curve is used
to extract the best fitted \dela (with error-bars) in a similar way
as is used in the absorption system (discussed in the next Section). The
measured spurious \dela for 100
random realizations are plotted in Fig.~\ref{dalpha.fig} both for
HARPS (left-hand side middle panel) and UVES (left-hand side lower panel) lamp spectra.
In the top panel we give the results for similar analysis of UVES spectrum
considering 6 Fe~{\sc ii} lines (i.e including Fe~{\sc ii}$\lambda$1608 instead
of Mg~{\sc ii} doublet) alone.
\par
We notice that the measured values of \dela obtained in this experiment
have a Gaussian-shape distribution with $\sigma$~of~$0.02\times 10^{-5}$ for HARPS
and $\sigma \simeq 0.1\times10^{-5}$ for UVES.
As the system under consideration
is known to have much more than 3 components, the above quoted values
are conservative errors
due to uncertainties in the wavelength calibration.
Murphy et al. (2003) have also carried out such analysis
for HIRES Th-Ar lamp spectra. Their weighted mean from the
sample of 128 sets of Th-Ar lines resulted in
$\langle\Delta\alpha/\alpha\rangle_{ThAr}=(0.4\pm0.8)\times10^{-7}$. If one
assumes a Gaussian distribution for the individual values, then
the central limits theorem implies that the typical $\sigma$ from
one set of Th-Ar lines in the case of HIRES should be around
$0.09\times10^{-5}$ ($\equiv0.8\times10^{-7}\times\sqrt{128}$), which is
similar to our value for UVES Th-Ar lamp spectra
(i.e $\sigma = 0.1\times10^{-5}$).\par
\subsection{Effect of using different Th-Ar line tables on wavelength calibration}
Th-Ar reference wavelengths are taken from the compilations of
Palmer et al. (1983) for Thorium lines and
Norl\'en et al. (1973) for Argon lines. The line lists
built from these compilations and commonly used for echelle
spectroscopy calibration are available on the web-pages of the European Southern
Observatory (ESO\footnote{http://www.eso.org/instruments/uves/tools/tharatlas.html})
and the National Optical Astronomy Observatory
(NOAO\footnote{http://www.noao.edu/kpno/specatlas/thar/thar.html}).
The two tables differ slightly, because the
ESO Th-Ar line table is not accurate up to 4 decimal
places as is the case with NOAO Th-Ar line table.
For the extraction of UVES lamp spectra
we have used the Th-Ar line table provided by NOAO.
To investigate whether the use of ESO table
could induce systematic shifts in \dela, we have also
extracted the same UVES Th-Ar lamp
spectrum using the Th-Ar line table provided by ESO.
We fit a Gaussian function to the un-blended Th-Ar line as
described in sub-section~\ref{abscal:subsec} and get the deviation,
$\delta\lambda_{fit}$, of the best-fit centroid with respect to the
corresponding value in the NOAO Th-Ar table. The deviation
($\delta\lambda_{fit}$) is plotted in
Fig.~\ref{tab_diff.fig} as a function of the difference in the
wavelengths tabulated by ESO and NOAO, $\Delta\lambda_{tab}$.
If the wavelength uncertainties caused by the inaccurate
wavelengths listed in ESO Th-Ar table for some of the
Th-Ar lines are larger than the
errors allowed by the dispersion solution, then we expect a
correlation between $\delta\lambda_{fit}$ and $\Delta\lambda_{tab}$.
The lack of such a correlation and the larger scatter of
$\delta\lambda_{fit}$ compared to $\Delta\lambda_{tab}$ in the
figure, show that the effect of inaccurate rest-wavelengths of a few
lines in the ESO line list is negligible.\par
To complement this,
we perform the cross-correlation between the lamp
spectra calibrated using the two wavelength tables.
The cross-correlation is performed in a similar way as described
in sub-section~\ref{crossCorr:subsec}.
Here we have shifted the UVES lamp spectrum
calibrated using the ESO Th-Ar table over the same lamp spectrum
calibrated using the NOAO Th-Ar line table. The result of the cross-correlation
is shown in the upper panel of Fig.~\ref{crossNoaoEso.fig}. From
the figure it can be seen that the relative shift is not completely random.
However the relative shift is most of the time less than 2m\AA~and even 1m\AA, which is
well within the UVES calibration accuracy.\par
We also repeat the exercise to derive how these wavelength calibration
uncertainties translate into \dela as described in detail in
sub-section~\ref{calDal:subsec} for the case when one uses
for calibration the ESO Th-Ar line table (Fig.~\ref{dalpha.fig} for UVES lamp uses NOAO table).
The result is shown in the lower
left-hand side panel of the Fig.~\ref{crossNoaoEso.fig} for 100 realizations.
The histogram shown
in the lower right-hand side panel shows that the fiducial \dela is distributed
like a Gaussian. As a result, we can conclude that the \dela measurements in the
literature (Chand et al. 2004 \& 2005, Quast et al. 2004) using the ESO
Th-Ar line table, should not be significantly affected by this possible
systematic effect.\par
\section{Analysis}
\label{sect:Ana}
In this section we present the results on the measurement of
\dela using the HARPS and UVES spectra. The details of the analysis used
here, validation of the procedure using simulated spectra
and the error budget from $\chi^2$ analysis can be
found in Chand et al. (2004, 2005). Here, we mainly concentrate on
(i) comparing the methods used by Chand et al. (2004, 2005) to derive \dela
with that used by Quast et al. (2004) and (ii) understanding the effect of
the decomposition of the absorption profiles into multiple narrow Voigt-profile.
\par
\subsection{Re-analysis of the red sub-system in the UVES data}
In the analysis of Chand et al (2004, 2005)
\dela is not explicitly used as fitting parameter. Instead \chisq versus \dela
curve is used to get the best fitted value of \dela. However, Quast et al.
(2004) use the Voigt profile analysis keeping \dela also as a fitting
parameter in addition to $N$, $b$ and $z$. Chand et al. (2005), using
analytic calculations, have shown that both the approaches should
give the same result. Here we check this by re-analysing the
absorption lines of the \zabs~=~1.1508 system toward HE {0515$-$4414}
using \chisq versus \dela curve.\par
The absorption lines of this system is spread over
about 730 \kms (Quast et al. 2004).
We have divided the whole system in two well
detached blue and red sub-systems. The blue sub-system covers the velocity
range $-$570 to $-$100 \kms and
the red sub-system covers the velocity range $-$20 to $+$110 \kms
with respect to \zabs=1.1508.
Our best fit Voigt-profiles to the blue and red
sub-system using the UVES spectrum, is shown respectively in
the left and right-hand side panels of Fig.~\ref{uves.fig}.
The vertical dotted lines are best fitted velocity components
obtained in this study and the long dashed vertical lines mark
the velocity components of the Quast et al. (2004). Apart form
the component around $\sim90$~\kms, we find almost perfect
matching between the components obtained
with two different fitting codes.
The variation of $\chi^2$ as a function
of \dela using this initial fit (Fig.~\ref{uves.fig}) is shown
in the left-hand side panel of Fig.~\ref{res.fig}.
The scatter seen in the \chisq curve is mainly due
to low column density of many components in blue sub-system
(see the discussion in Chand et al. 2004).
The position of the minimum
in the \chisq curve remains uncertain till either we smooth the curve
or fit some smoothing polynomial to it.
Therefore we have fitted a polynomial function of 4th order minimizing the
rms deviation.
The best fit of the \chisq curve is shown by the solid line
(left-hand side panel of Fig.~\ref{res.fig}).
Its minimum gives
\dela~=~${(0.10\pm0.22)\times10^{-5}}$, using $\chi^2_{min}+1$ statistics.
The derived position of the minimum does not depart significantly
when we use a 2nd or 3rd order polynomial fit
to the $\chi^2$ data points.
Our best fitted value, \dela = ${(0.10\pm0.22)\times10^{-5}}$,
is very much consistent
with that obtained by Quast et al. (2004)
(\dela~=~${[0.01\pm0.17]\times10^{-5}}$).
The best fitted column densities
and Doppler parameters in individual components also agree well
(see Fig.~\ref{compfit}).
\begin{table*}
{\large
\caption{Results of the Voigt profile fit of Fe~{\sc ii} lines at \zabs = 1.1508 toward HE {0515$-$4414}.}
\begin{tabular}{rrrcr}
\hline\hline
\\
{C.N} & \multicolumn{1}{c}{\zabs} & \multicolumn{1}{c}{b}
& \multicolumn{1}{c}{log[N(Fe~{\sc ii})]} & \multicolumn{1}{c}{V$^{a}$} \\
& & \multicolumn{1}{c}{(\kms)}
& \multicolumn{1}{c}{(cm$^{-2}$)}& \multicolumn{1}{c}{(\kms)} \\
\\
\hline\hline
\\
1 &$1.146938\pm0.00000^{\dagger}$ &$ 1.70\pm0.22$ &$ 11.38\pm 0.14$ &$ -538.79\pm 00.00$ \\
2 &$1.146969\pm0.000098$ &$ 2.34\pm0.25$ &$ 12.30\pm 0.03$ &$ -534.46\pm 13.71$ \\
3 &$1.147008\pm0.00000^{\dagger}$ &$ 4.47\pm0.75$ &$ 11.90\pm 0.06$ &$ -529.02\pm 00.00$ \\
4 &$1.147117\pm0.001030$ &$ 7.45\pm1.01$ &$ 12.01\pm 0.04$ &$ -513.80\pm 143.7$ \\
5 &$1.147169\pm0.000410$ &$ 4.25\pm0.88$ &$ 11.58\pm 0.09$ &$ -506.54\pm 57.27$ \\
6 &$1.147249\pm0.000106$ &$ 4.63\pm0.22$ &$ 11.92\pm 0.04$ &$ -495.37\pm 14.83$ \\
7 &$1.147312\pm0.00000^{\dagger}$ &$ 4.90\pm0.45$ &$ 11.23\pm 0.17$ &$ -486.57\pm 00.00$ \\
8 &$1.147416\pm0.000096$ &$ 4.70\pm0.19$ &$ 11.93\pm 0.04$ &$ -472.05\pm 13.33$ \\
9 &$1.147587\pm0.000255$ &$ 4.49\pm0.67$ &$ 11.24\pm 0.15$ &$ -448.18\pm 35.65$ \\
10 &$1.147809\pm0.000113$ &$ 3.47\pm0.22$ &$ 11.91\pm 0.04$ &$ -417.19\pm 15.84$ \\
11 &$1.147911\pm0.000133$ &$ 3.39\pm0.25$ &$ 11.81\pm 0.04$ &$ -402.96\pm 18.58$ \\
12 &$1.147980\pm0.000215$ &$ 3.75\pm0.57$ &$ 12.12\pm 0.10$ &$ -393.33\pm 30.04$ \\
13 &$1.148101\pm0.000543$ &$ 4.99\pm1.12$ &$ 11.75\pm 0.07$ &$ -376.44\pm 75.84$ \\
14 &$1.148501\pm0.000218$ &$ 7.52\pm0.44$ &$ 11.56\pm 0.10$ &$ -320.62\pm 30.47$ \\
15 &$1.148783\pm0.000287$ &$ 2.97\pm0.57$ &$ 11.09\pm 0.18$ &$ -281.27\pm 40.03$ \\
16 &$1.149088\pm0.000096$ &$ 2.11\pm0.21$ &$ 12.44\pm 0.03$ &$ -238.72\pm 13.32$ \\
17 &$1.149112\pm0.000057$ &$ 6.46\pm0.11$ &$ 12.52\pm 0.03$ &$ -235.38\pm 07.97$ \\
18 &$1.149489\pm0.000398$ &$ 4.30\pm0.43$ &$ 12.03\pm 0.03$ &$ -182.79\pm 55.50$ \\
19 &$1.149547\pm0.000470$ &$ 5.50\pm0.53$ &$ 12.20\pm 0.02$ &$ -174.70\pm 65.55$ \\
20 &$1.149817\pm0.000061$ &$ 4.14\pm0.12$ &$ 12.08\pm 0.03$ &$ -137.05\pm 08.56$ \\
21 &$1.149915\pm0.000108$ &$ 5.12\pm0.21$ &$ 12.01\pm 0.03$ &$ -123.38\pm 15.02$ \\
22 &$1.150548\pm0.00000^{\dagger}$ &$ 0.26\pm0.0^{\ddagger}$&$ 11.21\pm 0.16$ &$ -35.13\pm 00.00$ \\
23 &$1.150659\pm0.00000^{\dagger}$ &$ 17.85\pm0.0^{\ddagger}$&$ 12.21\pm 0.06$ &$ -19.65\pm 00.00$ \\
24 &$1.150688\pm0.000107$ &$ 2.98\pm0.18$ &$ 12.58\pm 0.02$ &$ -15.61\pm 14.94$ \\
25 &$1.150747\pm0.00000^{\dagger}$ &$ 4.62\pm0.78$ &$ 12.47\pm 0.32$ &$ -7.39\pm 00.00$ \\
26 &$1.150792\pm0.000102$ &$ 1.95\pm0.18$ &$ 13.26\pm 0.03$ &$ -1.11\pm 14.20$ \\
27 &$1.150819\pm0.00000^{\dagger}$ &$ 8.16\pm1.67$ &$ 13.46\pm 0.07$ &$ 2.65\pm 00.00$ \\
28 &$1.150864\pm0.000126$ &$ 1.07\pm0.25$ &$ 12.88\pm 0.05$ &$ 8.92\pm 17.53$ \\
29 &$1.150903\pm0.00000^{\dagger}$ &$ 3.65\pm2.03$ &$ 12.58\pm 0.37$ &$ 14.36\pm 00.00$ \\
30 &$1.150962\pm0.000190$ &$ 4.21\pm0.21$ &$ 13.47\pm 0.03$ &$ 22.58\pm 26.50$ \\
31 &$1.151063\pm0.000207$ &$ 6.68\pm0.39$ &$ 13.09\pm 0.02$ &$ 36.66\pm 28.89$ \\
32 &$1.151113\pm0.00000^{\dagger}$ &$ 6.00\pm1.61$ &$ 12.34\pm 0.11$ &$ 43.63\pm 00.00$ \\
33 &$1.151152\pm0.00000^{\dagger}$ &$ 3.37\pm0.83$ &$ 12.25\pm 0.08$ &$ 49.06\pm 00.00$ \\
34 &$1.151218\pm0.000235$ &$ 7.13\pm0.38$ &$ 13.29\pm 0.02$ &$ 58.26\pm 32.68$ \\
35 &$1.151314\pm0.000158$ &$ 6.21\pm0.17$ &$ 13.56\pm 0.02$ &$ 71.64\pm 22.06$ \\
36 &$1.151406\pm0.00000^{\dagger}$ &$ 15.40\pm0.0^{\ddagger}$&$ 12.72\pm 0.02$ &$ 84.46\pm 00.00$ \\
\\
\hline
\multicolumn{5}{l}{`$^{a}$' relative velocity with respect to $z_{\rm abs}=1.1508$.}\\
\multicolumn{5}{l}{`$^{\dagger}$' The redshift ($z$) of these components are kept fixed.}\\
\multicolumn{5}{l}{`$^{\ddagger}$' The Doppler parameter, $b$, of these components are kept fixed.}\\
\label{model.tab}
\end{tabular}
}
\end{table*}
The larger errors in the measured
quantities in the present study is mainly due to higher values of
the error assigned to the flux in individual pixels.
Thus the analysis presented here clearly shows that
the analysis used by us in Chand et al (2004, 2005) produces
consistent results.\par
In addition we have also performed the analysis
of UVES spectra by excluding the weaker \feii lines
from the blue sub-system and heavily saturated
strong \feiistr lines from the red-subsystem,
(see discussion in Chand et al. 2004).
In this case the \chisq curve is found relatively less
fluctuating as compare to the left-hand side panel of Fig.~\ref{res.fig},
and has resulted in \dela = $(0.00\pm0.26)\times 10^{-5}$.
\subsection{\dela from the HARPS data}
The decomposition of the absorption profiles in sub-components is
expected to be better defined from the HARPS spectrum because
of its superior spectral resolution.
In Fig.~\ref{comp_stru.fig} we compare the profiles of the
\feii lines in the red sub-system
as observed with HARPS and UVES. The best multi-component
Voigt-profiles fit using the UVES spectrum alone is over plotted.
To fit the HARPS data we need additional components,
as is apparent in the region around $-$20 to $+$30 \kms where consistent
differences are seen for all profiles between the HARPS spectrum and the fit to the
UVES data alone.
However, the UVES spectrum has the advantage of having higher S/N. Thus,
in our analysis we fitted simultaneously
both HARPS and UVES data using the same component structure
and the appropriate instrumental functions.
We initially fitted the HARPS data and used the derived parameters to fit the UVES data.
The process was repeated until the residuals along the profiles are symmetrically
distributed around zero and the best-fit parameters from these two data
sets are consistent with one another within measurement uncertainties.
In this exercise we have not included the line \feiia (covered only in the UVES spectrum)
so that our derived component structure is not artificially bias towards \dela$=0$.
\par
Our best-fit Voigt-profile components that simultaneously fit
the HARPS and UVES spectra are shown in Fig.~\ref{fit_blueboth.fig},\ref{fit_redboth.fig}
respectively for the blue and red sub-systems.
The best-fit parameters are listed in Table.~\ref{model.tab}.
The component identification number (C.N), redshift ($z$), velocity
dispersion ($b$), and Fe~{\sc ii} column density ($N$), for each component
are listed respectively in columns 1, 2, 3 and 4. The last column of the table lists the
relative velocity of the components with respect to \zabs=1.1508.
We find that the blue and red sub-system (Fig.~\ref{fit_blueboth.fig},\ref{fit_redboth.fig})
require respectively 3 and 6 extra components
compared to the minimum number required to
fit the UVES spectrum alone with $\chi^2=1$.\par
We evaluate the best-fit \dela value using the high resolution HARPS
spectrum for the five main Fe~{\sc ii} lines and the UVES spectrum for
Fe~{\sc ii}$\lambda1608$ considering both the blue and red
sub-systems simultaneously.
Here it should be noted that the
\feiia is crucial for \dela measurement due to its
opposite sensitivity for \dela (negative $q$ coefficient)
compared to the other main \feii lines. However as its
observed wavelength range ($\approx 3460$\AA) is not
covered by the HARPS spectral coverage (3800 - 6900\AA),
we have to use it from the UVES spectrum for constraining the
\dela value.
The $\chi^2$ versus \dela curve is shown in the
right-hand side panel of Fig.~\ref{res.fig}.
The scatter seen in the \chisq curve is mainly due
to the low S/N ratio and low column density of many components as can be seen from
Table~\ref{model.tab} (see the discussion in Chand et al. 2004).
The continuous curve gives the 4th order
polynomial fit to the $\chi^2$ data points
using rms minimisation.
Its minimum gives
\dela~=~${(0.05\pm0.24)\times10^{-5}}$, using $\chi^2_{min}+1$ statistics.
This result is consistent with the Quast et al. (2004) measurement
(\dela~=~${[0.01\pm0.17]\times10^{-5}}$) based on the UVES spectrum and
lesser number of components. Thus in this particular case lack of
information on the additional components in the UVES spectrum does not
seem to affect the final result.
\section{Result and discussion}
\label{sect:result}
In this paper, we present a very high resolution (R~=~112,000) spectrum of
QSO HE 0515$-$4414 obtained using HARPS.
We have used the high wavelength calibration accuracy and high
spectral resolution capabilities of HARPS to address the following
issues.\par
We compare the lamp spectra obtained with UVES and HARPS.
Using cross-correlation analysis we show that any possible relative
shift between the two spectra are within 2~m\AA.
Using Gaussian fits to unblended lamp emission lines, we find that the
absolute wavelength calibration of HARPS is very robust with rms
deviation of 0.87~m\AA~with respect to the wavelengths tabulated in Cuyper et al. (1998).
This is about a factor of 4 better than that of
UVES ($\sigma=4.08$ m\AA,~see Fig.~\ref{dlam.fig}).
Thus the small shifts noted between the HARPS and UVES lamp spectra
are well within the typical wavelength calibration accuracy of UVES.
We have derived the error on \dela measurements
due to the calibration accuracy alone. For UVES and HARPS spectra this
is found to be respectively
$\sigma=0.96\times10^{-6}$ and $\sigma=0.19\times10^{-6}$ for a typical
system with three well detached components.
The value obtained for the UVES spectrum is
also consistent with that of HIRES (Murphy et al. 2003).
\par
This shows that HARPS is the ideal instrument for this kind of measurement.
Unfortunately it is mounted on the 3.6~m telescope at La~Silla and
only HE~0515$-$4414 is bright enough to be observed in a reasonable
amount of time. This shows as well that the UVES spectra reduced (or calibrated)
with the UVES pipeline and used in the literature to constrain
\dela (Srianand et al. 2004 and Chand et
al. 2004, Quast et al. 2004, Chand et al. 2005)
do not suffer from major systematic error in the
wavelength calibration. \par
We have obtained the accurate
multi-component structure using the higher resolution data
(R~$\approx 112,000$ for HARPS compared to $\approx 55,000$
for UVES). The best fit to the profiles obtained by fitting
simultaneously the HARPS data (of higher resolution) and
the UVES data (of better S/N ratio) require additional
components as compared to the fit using the UVES data alone
(Quast et al. 2004). Using this new sub-component decomposition
and both HARPS and UVES data, we find
\dela~$=(0.05\pm0.24)\times10^{-5}$. This is
consistent with the results derived by Quast et al. (2004)
from the UVES data alone. Indeed, we have in addition
re-analyzed the UVES data which was used in
Quast et al. (2004) (without using the component structure from HARPS data),
to estimate the effect of different
independent algorithms used to obtain error spectra, to combine the data,
to fit the continuum and to fit the absorption lines. We find that the best-fit
parameters as well as the
\dela measurement (\dela~=~${[0.10\pm0.22]\times10^{-5}}$),
obtained by our independent analysis, are consistent with that of
Quast et al. (2004) (\dela~=~${[0.01\pm0.17]\times10^{-5}}$).\par
We note that the precision on the \dela measurement obtained
using the HARPS spectrum, which is of high resolution and low S/N ratio,
is similar to that obtained from the UVES spectrum, which is of lower resolution
and higher S/N ratio.
Therefore, the improvement in the wavelength calibration
accuracy by an order of magnitude using HARPS will be effective to
improve the constrain on \dela only if high S/N ratio can also be obtained.
This could be
possible if an instrument such as HARPS can be mounted on bigger
telescopes.
\section*{Acknowledgments}
HC thanks CSIR, INDIA for the grant award
No. 9/545(18)/2KI/EMR-I. RS and PPJ gratefully acknowledge support from the Indo-French
Centre for the Promotion of Advanced Research (Centre Franco-Indien pour
la Promotion de la Recherche Avanc\'ee) under contract No. 3004-3.
PPJ also thanks IUCAA (Pune, India) for hospitality during the time part of this
work was completed. RQ has been supported by the DFG under Re353/48.
|
Title:
Low and intermediate mass star yields.II: The evolution of nitrogen abundances |
Abstract: We analyze the impact on the Galactic nitrogen abundances of using a new set
of low and intermediate mass star yields. These yields have a significant yield
of primary nitrogen from intermediate mass stars. We use these yields as an
input to a Galactic Chemical Evolution model and study the nitrogen abundances
in the halo and in the disc, and compare them with models obtained using other
yield sets and with a large amount of observational data. We find that, using
these new yields, our model adequately reproduce the observed trends. In
particular, these yields solve the historical problem of the evolution of
nitrogen, giving the right level of relative abundance N/O by the production of
a primary component in intermediate mass stars. Moreover, using different
evolutionary rates in each radial region of the Galaxy, we may explain the
observed N dispersion.
| https://export.arxiv.org/pdf/astro-ph/0601326 |
\title{Low and intermediate mass star yields.\\
II: The evolution of nitrogen abundances}
\author{Marta Gavil\'{a}n\inst{1},
Mercedes Moll\'{a} \inst{2} and James F. Buell\inst{3}}
\offprints{Marta Gavil\'{a}n}
\institute{Departamento de F\'{\i}sica Te\'{o}rica, Universidad Aut\'onoma
de Madrid, 28049 Cantoblanco, Spain\\
\email{marta.gavilan@uam.es}\\
\and Departamento de Investigaci\'{o}n B\'{a}sica,
C.I.E.M.A.T., Avda. Complutense 22, 28040 Madrid, Spain \\
\email{mercedes.molla@ciemat.es}\\
\and Department of Mathematics and Physics, Alfred State College,
Alfred, NY 14802, USA \\
\email{BuellJF@alfredstate.edu}\\
}
\date{Received ; accepted }
\titlerunning{Low and intermediate mass star yields}
\authorrunning{Gavil\'{a}n, Moll\'{a} \& Buell }
\abstract
{}
{We analyze the impact on the Galactic nitrogen abundances of using a
new set of low and intermediate mass star yields. These yields have a
significant yield of primary nitrogen from intermediate mass stars.}
{We use these yields as an input to a Galactic Chemical Evolution
model and study the nitrogen abundances in the halo and in the disc,
and compare them with models obtained using other yield sets and with
a large amount of observational data.}
{ We find that, using these new yields, our model adequately reproduce
the observed trends. In particular, these yields solve the historical
problem of the evolution of nitrogen, giving the right level of
relative abundance N/O by the production of a primary component in
intermediate mass stars. Moreover, using different evolutionary rates
in each radial region of the Galaxy, we may explain the observed N
dispersion.}
{}
\keywords{ stars: -- galaxies: abundances -- galaxies: evolution--
galaxies: spirals}
\section{Introduction}
Most elements are created in the interiors of stars by nucleosynthesis
processes \citep[see][for a review]{wall97}, starting with hydrogen
and progressing toward heavy elements. These processes are called {\sl
primary production}. Some elements, however, can be formed from
nuclei heavier than hydrogen originally present in the star. They are
called {\sl secondary}. This is the case of nitrogen, that can be
created during the CNO cycle using seeds of original carbon and/or
oxygen. From a theoretical point of view, it has been considered that
massive stars produce secondary nitrogen \citep{pei87}, while low and
intermediate mass (LIM) stars have mechanisms, like the third
dredge-up and the Hot Bottom Burning processes, to produce both,
primary and secondary nitrogen \citep{edm78,all79}. The third
dredge-up event is a consequence of the thermal pulses in the star,
and transport C and He to the outer layers. The Hot Bottom Burning
occurs when the CNO cycle takes place at the base of the convective
envelope.
Observationally, there are several open questions about the primary or
secondary character of nitrogen that up to now remain unsolved. When N
and O data are represented as log(N/O) {\sl vs} log(O/H), including
the galactic stars, H{\sc ii} regions for the Milky Way Galaxy (MWG),
external galaxies \citep{gar95,gar99,vze98,izo99} and the high
redshift data \citep[][and references therein]{pet02,pro02,cen03}, a
clear positive slope appears for abundances larger than $\rm
12+log(O/H)=7.8-8$ dex which indicates a secondary behavior, but the
plot shows a flat slope for low metallicities that can only be
explained with a primary component of nitrogen. Taking into account
that this flat slope occurs for low abundances, the first idea
proposed, shared by some authors \citep{pag79,dia86,dah95}, is that
observations would be reproduced if the nitrogen ejected by massive
stars would be primary, while intermediate mass stars might have both
primary and secondary components.
Thus, some authors have tried to look for mechanisms that explain how
massive stars could produce primary nitrogen. This is the
case of \citet{mey02} that have recently proposed rotation as a
possible source of primary nitrogen, since low metallicity stars show
a bigger rotation than high metallicity ones. \cite{chi03-1} have used
these yields in their chemical evolution models, concluding that they
are only a lower limit for the primary nitrogen production since the
Hot Bottom Burning is not considered in their calculation. In fact,
\cite{chi05} find that an extra-- production of N in low metallicity
massive stars by a large factor, between 40 and 200 along the mass
range, is necessary to explain the data of very metal-poor halo stars
since these yields do not produce a sufficient amount of primary
N. Moreover, if the production of primary nitrogen would proceed from
massive stars, the left side of the (N/O) {\sl vs} (O/H) plot should
not show any scatter. Although some authors claim to observe
\citep{izo99,pil03} this no-scatter, recent observations from low
metallicity objects \citep{pet02,pro02,cen03,isr04,spi05} do show a
clear dispersion.
\citet{ser83} already claimed that a secondary production by
intermediate mass stars must exist and suggested that the zero slope
may be explained by two factors: 1) a delay in the ejection of N to
the ISM due to the different mean-lifetimes of stars and 2) the gas
infall effects. The advantage of taking a delay into account is that
the great data scatter can be explained by considering different
evolutionary states for each galaxy and, therefore, this possibility
has been supported by a large number of authors
\citep{vila93,pil92,pil93,vze98b,hen00}. These last ones also include
gas flows --infall or/and outflow--, and low efficiency for the star
formation rate (the equivalent mechanism to produce a delay) in the
low evolved regions, in order to reproduce the flat slope in the (N/O)
vs (O/H) plot. They conclude that the secondary production of
nitrogen should dominate in high metallicity environments while the
primary one should act at low metallicities.
Some new yields for LIM stars have been given in \citet[][hereinafter
Paper I]{gav05}, where they were adequately evaluated and calibrated
by using them in a Galaxy chemical evolution model. It was shown that
the results about C and O abundances adequately reproduce the Galactic
and Solar Neighborhood data. The purpose of this work is to analyze
the impact of these stellar yields on the nitrogen abundances. In
particular, we check, using the same prescriptions of Paper I, if the
contribution to N given by these yields for LIM stars is sufficient to
justify the amount of primary nitrogen the observations point out.
We describe the yields in Section 2, analyzing in particular the
primary and secondary components of the nitrogen production. In
section 3 we describe briefly the chemical evolution model. Section 4
is devoted to the results, and the conclusions are presented in
section 5.
\section{Low and intermediate mass yields:
The secondary and primary components of nitrogen}
\label{prim-t-sec}
The aim of this work is the study of the nitrogen behavior, using the
same set of yields as in Paper I, that we call BU yields. For
comparison purposes we also take the LIM stars yields from
\cite{hoe97} and \cite{mar01} that we call VG and MA, respectively.
The complete table of yields BU was already given in Paper I for five
metallicities: -0.2, -0.1, 0.0, +0.1 and +0.2, expressed as $\rm
log(Z/Z_{\odot})$, where solar abundances are taken from \cite{gre98}
\footnote{The use of these solar abundances implies that
$Z_{\odot}=0.02$. Recently, \cite{asp05} have obtained lower
abundances which lead to a value $Z_{\odot}=0.012$. However, these new
determinations are still questioned by some authors
\citep{bah05,dra05,ant05} because they do not fit the
helioseismological constraints.}.
We summarize the behavior of the carbon and nitrogen yields for LIM
stars, as shown in Fig.~\ref{yields}. In panel a) we see that
$^{12}C$ yield is extremely small for stars with mass lower than 2
$\rm M_{\odot}$, since they do not experience third dredge up events.
However, stars begin to suffer these kinds of events for smaller
masses at lower metallicity. In other words, in the low mass range,
the metallicity and the $^{12}C$ yield are anti-correlated. When the
stars have enough mass to undergo Hot Bottom Burning (HBB), the
$^{12}C$ yield drops abruptly because of the conversion of carbon into
nitrogen. The $^{14}N$ yield presents a local maximum in the mass
range from 3.5 to 5 M$_{\odot}$, depending on the metallicity, then
decreases before beginning to increase again as a function of stellar
mass. The largest amount of nitrogen is produced by stars of
intermediate mass because HBB and the 2$^{\rm nd}$ dredge-up occur
only in stars with $M>3.5-5 M_{\odot}$. As the HBB increases the
luminosity and the mass-loss rate, stars that suffer this process have
shorter TP-AGB lifetimes. The local maximum occurs in the transition
between stars with HBB and those without. The increase at higher
masses is due to the shortened time between third dredge-up
events. The yields at the lowest masses are due to the 1$^{st}$
dredge-up.
The most important difference among the used yields resides in the
contribution of primary and secondary components of nitrogen by LIM
stars. In Fig.~\ref{prim_total} we represent the fraction of primary
$^{14}N$ for the three used sets, as labeled, as a function of mass
(M $\leq$ 8 $M_{\odot}$, except for MA for which M $\leq 5$
$M_{\odot}$). In panel a) we show the results for solar abundances.
All of them show a similar behavior with a maximum for masses around
3.5-4 $M_{\odot}$.
We must clear some points about the components of N. The only
difference between primary and secondary nitrogen is the origin of the
carbon atom producing it. Although the idea is conceptually clear, it
is not so simple to separately compute both components. Thus, although
BU and MA give the two components separately for each model, VG do
not. These authors, however, show their yields in each phase of
stellar evolution. If we consider that all the nitrogen created in
the AGB phase is primary, about $ \sim 90$ \% of the N ejected by
LIM stars will be primary. This is sometimes assumed when
these yields are used. This hypothesis, that we call {\it AGB
technique} leads to a primary N component excessively large and is not
totally adequate.
Let us return to the definition: secondary N proceeds from the burning
of original $^{12}C$. If a fraction of the original carbon is burned
in the pre-AGB phase, it produces secondary N. Sometimes, this gives
a negative $^{12}C$ yield. But, not all the initial carbon is consumed
before the AGB phase. If we take, as an example, a star of
4$M_{\odot}$ of solar abundance, that is with $X(^{12}C)=0.28\times
10^{-2}$, it has an initial $^{12}C$ abundance $4 \msun X(^{12}C) =
1.12 \times 10^{-2} \msun$ The pre-AGB phase carbon yield is
$yC12_{pre} = 0.300\times 10^{-4}$, so the mass of this element
present in the star before the AGB begins is:
\begin {equation}
M(^{12}C)=yC12_{pre}M_{ini}+M_{end} X(^{12}C)
\end{equation}
where $M_{end}$ is the mass of the star at the end of this first
phase: 3.95 $\msun$. Therefore, there is a mass $\sim 1.118\times
10^{-2} M_{\odot}$ of $^{12}C$, from which $M_{end}X(^{12}C) \sim
1.102\times 10^{-2}M_{\odot}$ corresponds to original carbon. This
implies that a quantity of the original carbon is still available to
form nitrogen in the following phases. Thus, a fraction of the total
nitrogen produced in the AGB phase (given by the addition of the two
values given by VG in their tables denoted AGB yields and final AGB
yields) may be secondary. In order to calculate this component from
the total AGB yields we use the fraction, called $r$, between the
secondary to the total nitrogen yield, $r=^{14}N_{S}/^{14}N$. Taking
into account that the secondary N proceeds from the existing carbon
used as a seed, we assume that $r$ is equal to the ratio between the
old carbon and the new plus old carbon:
\begin{equation}
r= \frac{^{14}N_{S}}{^{14}N} =
\frac{^{12}C_{old}}{^{12}C_{old}+^{12}C_{new}}
\end{equation}
This method (hereinafter called $r$ method), may only be applied to
stars which suffer the HBB and produce primary $^{14}N$, that is,
those for which the core mass before the HBB is larger than
$M_{HBB}=0.8 M_{\odot}$, usually stars with $M> 3.5-4
M_{\odot}$. Otherwise, the nitrogen yield is all secondary. The
results of VG shown in Fig.~\ref{prim_total} proceed from this
calculation.
We have then computed the integrated yields of $^{14}N$ produced by
LIM stars that we present in Fig.~\ref{yields_integrados} as a
function of metallicity Z. In panel a), we represent the BU results
as solid circles to which we have performed a least-squares fit shown
by the solid (red) line. This integrated yield for $^{14}N$,
equivalent to the yield produced by a single stellar population, is
located between the two other sets in this panel, with a similar
dependence on Z that VG \footnote{The integrated yields for VG are
slightly different than those obtained by \cite{hen00} due to the
Initial Mass Function (IMF) used by us, from \cite{fer92}} but with
lower absolute values.
More significant however, is the dependence on metallicity of the
ratio of primary to total integrated yields, $^{14}N_{P}/^{14}N$,
shown in panel b). This ratio increases for decreasing metallicity for
all sets, as expected, although for VG the integrated yield is quite
different if we consider AGB technique than $r$ method.
This metallicity effect can be easily explained: low metallicity stars
have smaller radii and take longer to reach super-winds, so they have
more time to experience more third dredge-up events than solar
metallicity stars. As a consequence, they have more fresh $^{12}{\rm
C}$ in their envelopes and they can make more primary nitrogen by the
HBB process. On the other hand, due to the lower amount of original
carbon, they produce less secondary nitrogen. For VG yields the ratio
is almost constant at a value of 15\% when the $r$ technique,
represented by the short-dashed (blue) line, is used, for
metallicities greater then 0.01, although it also increases for
metallicities lower than this value. While it is $ \sim 90$ \% when
the $AGB$ technique, represented by a dot-short-dashed (blue) line, is
used, showing a smaller variation with Z. It is interesting that the
integrated yield for solar abundance in the BU case \footnote{For $Z<
0.0126$ we use the same yields set, that is, this one from
$Z=0.0126$. It is always possible to extrapolate the trend obtained
for the other Z sets, that we show with the straight line joining the
points corresponding to $Z=0.0126 $ and $Z=0.0159$} is around 20\%,
very similar to the value computed by \citet{all79} two decades ago on
the basis of the observations available at that time.
All these considerations indicate that the primary nitrogen appears at
a different time scale in the ISM depending on the value of Z. The first
primary N will be ejected when stars of 8 (5 for MA) $M_{\odot}$ die,
while Z is still low.
\section{The chemical evolution model and its calibration}
\subsection{Description}
The model used in this work is the Multiphase Chemical Evolution Model
described in \citet{fer92,fer94}, in the version presented in
\citet{mol05} and in Paper I. For LIM stars, we use the same yields BU
than in these two last papers, and for comparison purposes those from
MA and VG. For massive stars we have chosen \cite{por98} and
\citet[][hereinafter PCB and WW, respectively]{woo95}. We have run
different models computed with different combination of yields: BU +
WW, VG + WW and MA + PCB, which we distinguish as BU, VG and MA,
respectively.
The nitrogen study is usually done by comparing its behavior relative
to iron and oxygen, so it is very important to have a careful
calibration for these two elements. Oxygen calibration was done in
Paper I. The SNIa are the main manufacturer of iron. The yields for
type Ia supernova (SNIa) explosions are taken from \cite{iwa99} and
\cite{bra86}. The evolution of this element in the model is
quasi-independent of the normal stars yields. However, since iron is
mainly produced by SNIa, even if its yield is very well known, its
abundance is very dependent on the method to compute the rate of these
explosions. For this purpose we analyzed the results obtained with
different possibilities in order to eliminate, if possible,
uncertainties in the iron abundance evolution. This point is
relatively important because the Age-metallicity relation and the
G-dwarf metallicity distribution are usually used as calibration
methods for chemical evolutions models. In our case, furthermore, we
compared our results with observed stellar nitrogen abundances, most
of which are given as [N/Fe], and so, we checked that the Iron
evolution is adequately reproduced by our models before this
comparison can be made
We used three methods to compute the SNIa rates as given by the
following authors: the classical one \citep{mat86,fer93}, the one
given by \cite{tor89}, and other, more recent, described in
\cite{ruiz00}, hereinafter named MAT, TOR and RL, respectively. The
first authors estimate the SN rates by using only the Initial Mass
Function. The method, well described in depth in both cited works, is
summarized as follows: a proportion of the stellar masses in a given
range [$M_{min}$-- M$_{max}$] will be in binary systems and a fraction
of them will develop type Ia supernova. Based on this idea, a mass
function for the secondary stars is computed from the original
one. Finally the SNIa rate depends on the number of secondary stars
that died in each time step, which implies that the time scale for the
iron appears in the ISM is controlled by the mean lifetimes of these
secondary stars.
Actually, this time scale does not depend only on the secondary mean
lifetimes, since there are other processes that also participate in
the conversion of a binary system into a SNIa explosion. It is
necessary to take into account the effects of the distances between
both stellar components, the orbital velocities and other parameters
to finally obtain the time taken for the system to explode since the
moment of its formation. \cite{tor89} performed these calculations
for several combinations of possible candidates of binary system or
SNIa scenarios (Double Degenerate, Single Degenerate, etc...),
providing the supernova rate as a function of time normalized for a
binary system of 1 $\rm M_{\sun}$. All the physical processes and
assumptions are included in their calculations, so we only need to
include the selected functions in our code and multiply them by the
number of binary systems, avoiding the need of computing the secondary
and primary initial mass function as defined in the previously
described method. A similar technique has also been performed more
recently by \cite{ruiz00}. A numerical table has been provided to us
by Ruiz-Lapuente ( private communication) with the time evolution of
the supernova rates for a single stellar population, computed under
updated assumptions about different scenarios and probabilities of
occurrence. We have computed the supernova rates using the three
methods, thus producing three models MAT, TOR and RL. These different
techniques affect mostly the iron abundances, the other elemental
abundances being equal for all of them. Therefore, we will compare the
three type of SN rate calculations by using only BU yields in the
analysis of the Iron abundance evolution, as well as in the
calibration of the model (next section). We will compare the three set
of LIM stars yields when N be studied, using only the RL technique and
only for the comparison of the relative abundance [N/Fe] we will show
the nine possible combinations of models.
The main disparity among the three techniques described above resides
in the different evolution of the SN rate in time. As we see in
Fig.~\ref{tasas}, MAT is the technique that presents highest values of
SNIa/SNII at any time, reaching the maximum at 2.5 Gyr. RL has a
maximum between 2 and 5 Gyr, with values approximately 1/2 or 1/3 of
those given by MAT. Nevertheless it still is within the error bar
given by observations \citep{capp99,capp04,mann05}. Note that this
value has been reduced for the most recent determinations compared
with the oldest ones. TOR model is the only one with low values.
Even if it presents a maximum before the first Gyr, this will not be
seen in the results because its value is very small. From the first
Gyr, SNIa/SNII has positive slope and it almost reaches the observed
value at the present time. \footnotesize
\begin{flushleft}
\begin{table*}
\begin{tabular}{lccccccc}
\hline
\noalign{\smallskip}
Reference & Fe & C & N & O & R & Age \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\cite{ake04}(AKE) & X & X & --- & X & --- & --- \\
\cite{bar88} & X & --- & --- & X & --- & --- \\
\cite{bar89} & X & --- & --- & X & --- & --- \\
\cite{barry88}(BA) & X & --- & --- & -- & --- & X \\
\cite{boe99} & X & --- & --- & X & --- & --- \\
\cite{car87}(CARB) & X & X & X & --- & --- & --- \\
\cite{car98}(CARR) & X & --- & --- & --- & --- & X \\
\cite{car00-2} & X & X & X & X & --- & --- \\
\cite{cav97} & X & --- & --- & X & --- & --- \\
\cite{che00} & X & --- & --- & X & --- & X \\
\cite{cle81} & X & X & X & X & --- & --- \\
\cite{daf04}(DAF) & --- & X & X & X & X & --- \\
\cite{dep02} & X & X & X & X & --- & --- \\
\cite{ecu04} & X & --- & X & --- & --- & --- \\
\cite{edv93}(EDV) & X & --- & X & X & X & X \\
\cite{gus99} & --- & X & --- & --- & X & X \\
\cite{fri90} & X & X & --- & --- & --- & --- \\
\cite{gra00} & X & X & X & X & --- & --- \\
\cite{gum98} & --- & X & X & X & X & --- \\
\cite{isr98,isr01} & X & --- & --- & X & --- & --- \\
\cite{isr04}(ISR) & X & --- & X & X & --- & --- \\
\cite{lai85} & X & X & X & --- & --- & --- \\
\cite{mel01}, & X & X & --- & --- & --- & --- \\
\cite{mel02} & & & & & & \\
\cite{mis00} & X & --- & --- & X & --- & --- \\
\cite{nis02,nis02b} & X & --- & --- & X & --- & --- \\
\cite{red03}(RED) & X & X & X & --- & --- & X \\
\cite{roc00,roc00b}(RO) & X & --- & --- & --- & --- & X \\
\cite{rol00}, & X & X & X & X & X & --- \\
\cite{sma97}, & & & & & & \\
\cite{sma01} & & & & & & \\
\cite{shi02} & X & X & X & --- & --- & --- \\
\cite{smi01} & X & --- & --- & X & --- & --- \\
\cite{spi05}(SPI) & X & X & X & X & --- & --- \\
\cite{tom84}, & X & X & X & X & --- & --- \\
\cite{tom86,tom95}, & & & & & & \\
\cite{twa80}(TW) & X & --- & --- & --- & --- & X \\
\cite{wes00} & X & X & X & X & --- & --- \\
\hline
\noalign{\smallskip}
\end{tabular}
\caption{References for CNO stellar abundances used for the comparison
with model results.}
\label{authors}
\end{table*}
\end{flushleft}
\normalsize
\subsection{Calibration of the model: Iron evolution in the Solar Vicinity}
The results for iron abundance obtained with these three methods are
shown in Fig.~\ref{AMR}, the Age-Metallicity Relation (AMR) and in
Fig.~\ref{G_dwarf}, the G-Dwarf distribution, for the Solar Vicinity.
For this comparison we have shown only BU yields, keeping in mind that
the set of yields will have only small effects on this relation.
Nevertheless, this model uses WW yields for massive stars and these
authors claimed that this could produce too much iron, and advised in
\cite{tim95} to divide the iron ejections at least by two. In order to
calculate how much that WW iron excess will be, we have calculated
four different models for BU yields and RL technique, where massive
stars iron production is divided by 1, 1.5, 2 and 3. Results are
represented in panel a) where it is clearly shown that a factor of 2
is a good compromise that we will use in panel b). In this last panel
we present the Age-Metallicity relation for the three SNIa cases,
where all of them are in reasonable agreement with data, given their
large dispersion. Although there are very little differences between
models, it can be seen that the iron appears later and takes a little
more time to reach high values when MAT and RL techniques are used,
than for the TOR SNIa method, but all of them reach the solar
abundance.
Regarding the G-dwarf distribution, represented in Fig.~\ref{G_dwarf},
differences appear mainly between TOR and the others because it
provides a narrower distribution than the others. The three models
are able to reproduce the low metallicity tail without showing any
G-dwarf problem.
In Fig.~\ref{feo} we show the relation between iron and oxygen. As
before, in panel a) the BU + RL model is presented varying the massive
stars iron ejection. In this case the differences are clearer than in
the AMR case. We chose the model Fe/2 that we will use for the rest
of the paper. In panel b) we plot the model results using BU yields
with the three SNIa techniques. As in the previous case, the LIM stars
yields do not change the results because oxygen is ejected by massive
stars and iron is mainly produced by SNIa events. The three models
have very similar behavior.
\section{Results analysis: The nitrogen abundances}
We devote this section to analyzing the results obtained with our
models for nitrogen abundances and comparing them with the available
observational data. We divide these results in four parts: a) the
evolution of Nitrogen over time for the Solar Vicinity (assumed
located at a galactocentric distance of 8 kpc), b) the radial
distributions of elements in the disc, c) the relation of log(N/O)
with the oxygen evolution and d) the relation of [N/Fe] with iron.
\subsection{Time evolution of nitrogen}
Fig.~\ref{abunt} shows the evolution of nitrogen with the solar
abundance values --large filled symbols-- taken from \cite{gre98} --
circle--, \cite{hol01} --square-- and \cite{asp05}, --cross--, by
assuming an age of 4.5 Gyr for the sun. For the interstellar medium
abundances at 13.2 Gyr, large empty symbols, we use the abundances
given by \cite{mey97,mey98}, --circle--, and \cite{pei99}
--square--. The small open circles are abundances for objects with
given stellar ages, at a radial galactocentric distance between 7.5
and 9.5 kpc.
Models BU, VG and MA are represented by the (red) solid, the (blue)
short-dashed and the (green) long-dashed lines, respectively. Both
models VG, following the two possible techniques to calculate the
proportion of primary nitrogen, techniques $r$ and $AGB$ described in
Section~ \ref{prim-t-sec}, yield results indistinguishable for times
larger than 1 Gyr, so we represent only the results for the first one.
In panel a) MA and VG models give a greater value than BU since at the
lowest metallicity their nitrogen yield are higher than the
corresponding one from BU (see Fig.\ref{prim_total}). Then, once an
abundance higher than $\sim$0.004 is reached, it continues increasing
smoothly until the present time, reproducing both, solar and ISM,
abundances. The shape shown by the three models are similar and all of
them reproduce both the solar value and the ISM value.
The same kind of information can also be extracted from the relative
abundances represented in panel b). In panel b), we show the
time evolution of log(N/O). Since there is a good agreement in
fitting the abundance of oxygen for all models, \citep[see][]{gav05},
the differences in this plot must be only due to the nitrogen
production. The disagreement between the different models is important
for times shorter than 1.5 Gyr, when intermediate mass and massive
stars are the main contributors and the distinct primary/secondary ratios
effects are evident there. Model MA has a strong increase in the first
Gyr due the primary component and then it flattens up. Model BU has a
ratio of primary nitrogen larger than MA for all Z except for the
lowest one, what produces a smoother evolution, and, finally, N
remains below MA. The resulting final N/O ratios are similar in both
models and in agreement with observations. The behavior mostly primary
of model VG, when all AGB nitrogen is considered as such, implies a
very strong increase of the abundance at the earliest times. After
that, both methods give a smooth slope, reaching an absolute value
around -0.5 dex higher than the observed one.
The good behavior of BU yields is also evident in panel c) where
log(N/C) is shown. MA model presents a maximum at the first Gyr, the
decrease is due to the higher amount of carbon ejected in that model
(see Paper I); so the absolute value at the present time is only
marginally reached. The shape for VG model is similar than the BU one
but with a nitrogen excess. All models seem to fit the solar and ISM
data but the model BU is the best one in reproducing the stellar data,
and, more importantly, it is the best one at fitting all of the data
at the same time.
\subsection {Nitrogen abundance in the Galactic disc}
Now, we will explore the radial distributions of nitrogen over the
galactic disc, shown in Fig.~\ref{gran}. Data correspond to H{\sc ii}
regions from references as labeled in the figure and to stars from
references in Table~\ref{authors}.
The radial distribution is more or less reproduced within the errors
by all models. Actually, the shape of the radial distribution is well
fitted in all cases, independently of the absolute values, since this
results as an effect of the ratio infall/SFR along the galactocentric
radius, produced by the scenario of our MWG model, and therefore is
rather independent of the yields used. However, the slope of the
radial distribution at the two ends of the disc, in the center and in
the outer regions, is a matter of discussion. Thus, \cite{vil96} claim
that the gradients are not as steep in these regions as in the rest of
the galactic disc. The same occurs in the inner disc where the most
recent data from \cite{sma01} show that the distribution flattens. Our
models have been tuned to fit these two sets of data, and due to that,
the resulting overall gradient is smaller than that obtained by other
authors. As can be seen in Fig.~\ref{gran} MA and VG models produce a
gradient flatter than observed.
In Fig.~\ref{gran}b) and c), the radial distributions log(N/O) and
log(N/C) are plotted, as they are considered important for the study
of different yields. In panel b) the radial distribution of log(N/O)
showed by data presents a clear slope, although there is some data
that shows a flatter distribution in the outer regions. A steep radial
distribution for N/O is expected because oxygen is produced by massive
stars. If nitrogen was ejected by massive stars, its secondary
character would cause it to enter the ISM after the oxygen. Instead,
if it would be ejected by intermediate stars, the time needed for
their evolution would be larger. Thus, in both cases, the nitrogen
appears in the ISM after the oxygen does.
Once again, the large dispersion of the data prevents a clear
selection of {\sl the best model}, however the BU model seems most
adequate to reproduce the H{\sc ii} regions data from \cite{vil96} and
\citet{fic91}. The MA model shows a radial gradient flatter than
indicated by observations, with higher absolute values compared to the
mean values of data, mostly in the outer disc, and VG models, as
before, show higher values. The same arguments are also valid when
panel c) is analyzed. In this case a slight negative gradient is shown
for log(N/C). The small amount of data and the great dispersion
prevent the selection of any model as better than the others, although
it seems clear that MA remains below most of them as corresponds to
the large production of C, and that VG lies in the upper side of the
data. It is apparent that BU model shows a better behavior compared
to the data. Once again we stress the importance of using adequate
yields to reproduce the whole set of data at the same time. Yield BU
seems to be in the adequate range of production of N, C and O, since
the model appears in the zone occupied by of data in the three panels.
It is necessary to remember that open dots represent stellar
abundances. We have tried to select only those corresponding to young
stars, but we do not know the age of the complete set of stars with
available data. In this case we have preferred to use the available
abundances; thus, it is possible that some data does not correspond to
young enough stars.
\subsection {Nitrogen {\sl vs} Iron}
In this case, as the iron evolution may have also an influence over
the model results, we have presented the relation between nitrogen
and iron, Fig.~\ref{nfe}, with a different panel, a), b) and c), for
each set of yield, MA, BU and VG, respectively. The three possible
methods to compute the SNIa, RL, TOR, and MAT are shown with solid
(red), short-dashed (blue) and long-dashed (green) lines,
respectively, in each panel. The first thing we observe is that the
effects of the different SNIa techniques are almost indistinguishable.
Therefore, the main features of each model at those metallicities are
due to yields. In other words, we may analyze the behavior of the
nitrogen corresponding to each yield set disregarding the accuracy in
the SNIa calculations.
\label{nfe}
All results agree in the sense that the first nitrogen to be ejected
is secondary, as due to the massive stars, so the initial slope is
positive and large, --although this behavior is not shown in the
figure because it occurs when oxygen abundances is lower that 5 dex--
but they differ in when the slope begins to change. When the N ejected
by LIM stars appear, there is a strong increase due to the change from
a secondary to a primary behavior. In MA yields LIM stars eject less
primary nitrogen, and later, since it is ejected as secondary for
stars up to 5 $M_{\odot}$. Then the main contributors to primary N are
the stars with masses between $2 M_{\odot}$ and $3 M_{\odot}$. For
this reason the slope does not change until it reaches $\rm [Fe/H] =
-1.5$, the moment in which these stars begin to die. When BU yields
are used, the trend changes earlier in the evolution due to the
contribution of the primary nitrogen ejected by stars in the range
4--8 $M_{\odot}$. As their life is so brief, the ejection occurs at
$\rm [Fe/H] = -4$. From then, the slope is close to zero: the
signature of primary nitrogen. The case of VG shows a behavior
similar to BU.
We would like to remark that the data dispersion is so great that all
the models lie in the data area, regardless of their big
discrepancies, although the region of the metal-rich objects ($[Fe/H]
> -1.5$) is particularly well fitted in panel b) by Model BU. It is
necessary to use the very low metallicity data to clarify which model
works better. In fact, the most recent observations from
\cite{isr04,spi05} show a slope flatter than before, (even with a
negative slope) which is a behavior more consistent with our model BU
than with the model obtained with MA yields. This last model might be
considered acceptable when the available low metallicity data were
only those from \cite{car87}, but when using the new determinations of
N abundances for this kind of objects, the conclusion is that BU
better reproduces the generic trend of data. It is also necessary to
take into account that most of the metal-poor objects do not belong to
the disc but to the halo. In this way, we represent the halo model
results for the zone that infall over the disc at galactocentric
distance equal to 8 Kpc, at the right panels b), d) and f) of
Fig.\ref{nfe}. We see that the trend shown by the recent observations
from \cite{isr04,spi05} is more compatible with BU and VG than MA.
\subsection {Nitrogen {\sl vs} oxygen}
Finally, we show in Fig.~\ref{no} the classical and well known graphic
of the relative abundance of N {\sl vs} O as $\log{(N/O)}$ {\sl vs}
$12 +\log{(O/H)}$. We show the final results for the Solar
Neighborhood of the computed models. Model BU reproduces well the
expected behavior of N when the whole data is taken into account. Not
only the level of N is so adequate but also the shape is smoother than
the one shown by the other two models. The observed trend at low
metallicity can be well reproduced with BU yields because they have
the appropriate primary to secondary ratio, and the adequate
integrated nitrogen yields. MA yields have also a primary nitrogen
component, but the integrated nitrogen yield has a metallicity
dependence in the opposite way as BU for the lowest Z, so the trend
shown by data can not be well reproduced. Both VG models have the
right shape and are almost the same for $12 +\log{(O/H)} \ge 8$. The
problem is that the integrated yields are high. It would be necessary
to change the input parameters as the infall rate or the efficiencies
to form stars in order to fit the solar abundances. In that case,
probably, other data will not be reproduced. Only the BU model shows
simultaneously the good shape and adequate absolute abundances. For
comparison purposes, we have also shown the resulting model using VG
yields but assuming that the Nitrogen is completely secondary.
If the flat behavior were caused by massive stars, the data
dispersion would be very small. A problem arises when the metal-poor
objects \citep{isr04,spi05} are included in the figure, as can be seen
in Fig.~\ref{no}. Some values follow the described trend over the flat
line, but there exist some lower abundances which are around
$log(N/O)\sim -2$. This behavior is not compatible with a primary
component proceeding only from massive stars.
We show in Fig.\ref{no_bis} the evolution given by BU model for four
different radial regions of the Galaxy: two inner ($\sim$ 2 and 4 kpc)
more evolved regions --(magenta) dotted and (blue) short-dashed
lines--, the solar vicinity ($\sim$ 8 kpc) as in the previous figure,
the (red) solid line, and an outer one ($\sim$ 18k pc), --the (green)
long-dashed line--, where the evolution takes place slowly. In panel
a) we show the results for the halo zones and in panel b) for the disk
regions. In both panels we have included the stellar data for the MWG.
The numbers on the graph indicate the evolutionary time, in million
years, that corresponds to that point of the line. This is necessary
because the $12+log(O/H)$ value is not the same for each radius at the
same value of time.
The halo regions have similar evolutions independent of their distance
from the center of the Galaxy. All of them reproduce well the recent
data from \cite{isr04,spi05} obtained for $5.5 < 12+log(O/H) < 8$, and
the disk regions fit the stellar data obtained for $12+log(O/H) > 8$
of the disk. Their evolutionary tracks, however, are very different,
as corresponds to their distinct input parameters (infall rates,
initial gas masses, efficiencies to form stars...) which are
translated into very different star formation histories. Thus, the
dispersion of the MWG data can be well explained on the basis of a
primary production of nitrogen from LIM stars, higher for the lowest
metallicities, and with different star formation efficiencies in the
different regions.
We represent the same results for the disk regions in Fig.\ref{no_hii}
compared with data referring to Galactic HII regions, taken from the
same authors than those of Fig.~\ref{no}, but without limiting the
possible galactocentric distance. Other galaxy data
\citep{gar95,gar99,vze98} and \cite{izo99} are also shown. The large
open triangle is the recent estimate obtained from \cite{izo05} for
the lowest metallicity known galaxy. We have also added the DLA
objects data from \cite{pet02,pro02,cen03} as solid points. We want
to remark that the disk regions evolve in good agreement with all of
them, showing the inner regions a steeper evolution while the outer
one shows a very flat evolution with a high and constant value
$log(N/O)\sim -1.2$ dex, similar to the behavior of dwarf galaxies.
These results suggest that the observed dispersion in this kind of
plot, when other galaxies data (such as dwarf or DLA galaxies) are
included, might be reproduced if different star formation histories
have occurred in different galaxies. This argument has already been
invoked by other authors, in particular by \cite{hen00} and
\cite{pra03}. It was even demonstrated by \cite{pil03}, who analyzed
data for different radial regions in spiral galaxies and showed the
changes of the evolutionary track in the plane N/O {\sl vs} O/H for
each one of them. It is evident that this kind of behavior may be
represented by our models and that the new yields may reproduce better
the whole set of data. In fact, these yields have already been used in
a grid of chemical evolution models for a large number of theoretical
galaxies \cite{mol05}. A discussion about the resulting N/O abundances
and its possible dispersion for different objects is done in
\cite{mol05b}.
This figure and the behavior of (C/O) vs O/H, shown in Paper I, are
the main clues to consider the present yields the most adequate to
represent the evolution of galaxies. The production of carbon by LIM
stars is sufficient to obtain an increase in C/O without the need to
invoke mass loss by massive star winds, and the N/O behavior may be
well reproduced with different star formation efficiencies due to the
adequate level of the primary component produced by LIM stars and to
the right dependence of this component with Z.
\section{Conclusions}
Our conclusion can be summarized as follows:
\begin{enumerate}
\item The primary component of nitrogen, necessary to explain the
trend of N/O with O/H, may be mostly produced by LIM stars, and
adequately fits all the data, including the observed dispersion. In
this way the integrated yield produced by LIM stars must be directly
proportional to Z, while the ratio $N_{P}/N_{tot}$ must increase for Z
decreasing. A primary component, larger for lowest metallicities, has
an important effect on explaining low abundance range data.
\item The dependence of the N yield on stellar mass would have a
maximum around 5-6 $M_{\odot}$, while the primary component shows
other around 3.5-5 $M_{\odot}$. This constraints the time where these
contributions have important effects on the evolution.
\item The high dispersion on N/O data for low and high metallicity
galactic regions may be explained with these yields as we have
demonstrated with the radial regions of the disc models which have
different star formation efficiencies. Our findings go in the same
address than \cite{hen00} and \cite{pra03} using VG yields, but we
claim that it is easier to reproduce the whole data set when BU yields
are used. Our model for MWG with small efficiencies to form stars is
consistent with data.
\item These results seem suggest that models with differences in the
star formation histories for different types of galaxies, such as
those calculated in \cite{mol05}, might produce final abundances
with high dispersion, in agreement with the observed one when dwarfs
galaxies or DLA galaxies are included in a plot N/O-O/H, such as we
will show in \cite{mol05b}.
\item As \cite{chi03-2}, we also support that the halo and the disc
have different evolutions. The set of stellar data [C/Fe], [N/Fe] and
[N/C] may be divided into two trends. The first one is well reproduced
by our disc models, while the second one is well fitted by our halo
results.
\item Due to the primary N component of BU yields, and since the
intermediate stars have short lifetimes, it is possible to produce
high [N/Fe] abundances even at low metallicities which are in perfect
agreement with the recent halo stars data obtained by
\cite{isr04,cay04,spi05}.
\item We claim that the fit of the whole set of data with only one
model is not an easy task. We may reproduce the observed trend with
BU yields combined with yields from WW. In summary, our model BU
reproduce reasonably well the whole CNO data set.
\end{enumerate}
\begin{acknowledgements}
This work has been partially supported by the Spanish PNAYA project
AYA2004--8260-C03-03. We acknowledge Pilar Ruiz-Lapuente for her
personal contribution in the SNIa rates data. We also thanks Jose
Manuel V\'{\i}lchez for his valuable suggestions and the referees,
Leonid S. Pilyugin and Angeles I. D\'{\i}az their comments that have
improved this paper.
\end{acknowledgements}
\bibliographystyle{aa}
\bibliography{bibliografia} |
Title:
Late Light Curves of Normally-Luminous Type Ia Supernovae |
Abstract: The use of Type Ia supernovae as cosmological tools has reinforced the need
to better understand these objects and their light curves. The light curves of
Type Ia supernovae are powered by the nuclear decay of $^{56}Ni \to ^{56}Co \to
^{56}Fe$. The late time light curves can provide insight into the behavior of
the decay products and their effect of the shape of the curves. We present the
optical light curves of six "normal" Type Ia supernovae, obtained at late times
with template image subtraction, and the fits of these light curves to
supernova energy deposition models.
| https://export.arxiv.org/pdf/astro-ph/0601088 |
\runauthor{Lair,et al.}
\begin{frontmatter}
\title{Late Light Curves of Normally-Luminous Type Ia Supernovae}
\author[Clemson University]{Jessica C. Lair}
\author[Clemson University]{Mark D. Leising}
\author[Steward Observatory]{Peter A. Milne}
\author[Steward Observatory]{G. Grant Williams}
\address[Clemson University]{Department of Physics and Astronomy, Clemson University, Clemson, SC 29634}
\address[Steward Observatory]{Steward Observatory, University of Arizona, Tucson, AZ 85721}
\begin{keyword}
supernovae
\end{keyword}
\end{frontmatter}
\section{Introduction}
Type Ia Supernovae (SNe Ia) are thought to be the thermonuclear explosion of a white dwarf \citep[see][and references therein]{2000A&ARv..10..179L}. The light curves of SNe Ia are powered by deposition in the SN ejecta of the $\gamma$-ray and positron products of the $^{56}Ni\rightarrow ^{56}Co\rightarrow ^{56}Fe$ decay \citep*{1969ApJ...157..623C}. The extreme brightness and seemingly uniform light curves of SNe Ia make them good candidates for use as standard candle distance indicators. In more recent years, it has been shown that Type Ia supernovae do not have uniform light curve magnitudes, shape or spectra. The light curves can, however, be normalized to account for this inhomogeneity, thus allowing these objects to be used at standard candle distance indicators \citep[e.g.][]{1993ApJ...413L.105P,1996ApJ...473...88R}.
Between 100-200 days after the explosion the ejecta become transparent to the $\gamma$-rays and the light curve is powered by the deposition of the positron kinetic energy into the ejecta. The escape of a fraction of these positrons from the ejecta has been suggested as a possible source of the Galactic 511 keV annihilation radiation \citep*{1999ApJS..124..503M}.
There are currently two methods of modeling the late emission of SNe Ia. One is radiation transport with complete and instantaneous trapping of the positrons. \cite{1980PhDT.........1A} showed, by comparing a model to the late time spectra of SN 1972E, that the ejecta will cool leading to an increased fraction of the emission coming out in the infrared, the so named ``infrared catastrophe" (IRC). Other studies of radiation transport \citep[e.g.][]{1996ssr..conf..211F,2004A&A...428..555S}, have reproduced the IRC, but they also predict the abrupt fall off of the optical light curves as the emission shifts into the NIR and ultimately into the IR, which is not seen in observed light curves.
The other method consists of positron energy deposition modeling without radiation transport. In this type of modeling \citep[e.g.][]{1980ApJ...237L..81C,1997A&A...328..203C,1998ApJ...500..360R,1999ApJS..124..503M} , optical band light curves are used as tracers of the bolometric luminosity and fit to model energy deposition curves. The results show model curves with varying degrees of positron escape fitting the light curves. One weakness in this model fitting technique is in using the optical bands as tracers of bolometric. \citet{2001ApJ...559.1019M} constructed bolometric curves using BVRI bands and showed those curves roughly fitting the positron escape energy deposition curves.
\section{BVRI Photometry using Template Subtraction}
We preformed aperture photometry on six ``normal" SNe Ia at late epochs, SN 2000E, SN 2000ce, SN 2000cx, SN 2001C, SN 2001bg, SN 2001dp. Some of these SNe were located in very complicated regions in their host galaxies. For this reason, we chose to do template image subtraction, on all but SN 2000cx, before preforming the aperture photometry. All data reduction, image subtraction and aperture photometry was performed using the Image Reduction and Analysis Facility (IRAF) software \footnote{IRAF is distributed by the National Optical Astronomy Observatory. http://iraf.noao.edu}. The combined light curves can be seen in Figures \ref{radmodel} \& \ref{posmodel}, where they are normalized to be zero magnitude at 200days past explosion assuming an 18d rise time to peak light. The data set for SN 2000E includes photometry from \cite{2003ApJ...595..779V}, and the data set for SN 2000cx includes data from \cite{2001PASP..113.1178L}, \cite{2002PhDT........10J}, \cite{2003PASP..115..277C}, and \cite{2004A&A...428..555S}, where the data from our observations are plotted as the filled symbols.
\section{Light Curve Decline Rates}
The decline rates, the slope of the light curve, between 200-500 days were calculated for these light curves and the averages are shown in Figure \ref{slopes}, where the solid line is average for the six SNe. The calculated averages for B,V,R,\& I bands were 1.43 (0.07), 1.46 (0.04), 1.36(0.04), 0.95 (0.06), respectively, in magnitudes per day. The shaded bar represents the average decline rate for 16 normal/super-luminous SNe Ia from \cite{2001ApJ...559.1019M} with a $1\sigma$ error bar. In the R-band, there is a second average, represented by the dot-dashed line, which is the average decline rate leaving out SN 2000ce and SN 2001C. This was done only to show the agreement with the Milne et al. averages.
As shown in Figure \ref{slopes}, the B,V,\& R bands have decline rates of $\sim 1.4$ mag/day but the I-band has a much shallower slope of 0.95 mag/day. This is in agreement with the decline rates of SN 2000cx as shown by \cite{2004A&A...428..555S}. These results suggest that a slower I-band decline rate is a general feature of the late light curves of normal/super-luminous SNe Ia, and is possibly suggesting a shift in the late emission to longer wavelengths. A major result of \cite{2004A&A...428..555S} was the constant late time emission seen in the NIR curves of SN 2000cx, which supports the idea that the emission is moving into the NIR and eventually into the IR resulting in an IRC. Our results from the analysis of these SNe reinforce the need for more observations of SNe Ia in the NIR in an attempt to reproduce what was seen in SN 2000cx and also in SN 1998bu \citep{2004A&A...426..547S}
\section{Discussion}
Figure \ref{radmodel} shows the combined light curves of the six SNe plotted on the radiation transport models of \cite{2004A&A...428..555S}. The V-band model light curve has been normalized to be zero magnitude at 200d along with the data. The B, R, \& I band model light curves have been adjusted so that the colors of the model are preserved. The dotted curve is the model including photoionization and the dot-dashed curve is the model without photoionization.
Figure \ref{posmodel} shows the combined light curves of the six SNe plotted on the positron energy deposition curves of \cite{2001ApJ...559.1019M}. The model curves have been normalized to be zero magnitude at 200d. The solid curve is the energy deposition with the positron kinetic energy trapped and deposited into the ejecta and the dashed curve is the energy deposition curve with a radially combed magnetic field allowing a fraction of the positrons to escape the ejecta without depositing their kinetic energy.
As can be seen in these figures, the shape of the B, V, \& R band light curves could be explained by either the color evolution in the radiation transport model or the escape of positrons from the ejecta, while the I-band has a slower decline rate than both models. This suggests that a model combining radiation transport with positron transport would be preferred to attempt to explain the late light curves of SNe Ia. One thing is clear from the light curves; these SNe show very little deviation from each other in a given band, implying that within this class of normally-luminous SNe Ia there is only one answer for the question of positron escape from SNe Ia ejecta.
|
Title:
SINFONI's take on Star Formation, Molecular Gas, and Black Hole Masses in AGN |
Abstract: We present some preliminary (half-way) results on our adaptive optics
spectroscopic survey of AGN at spatial scales down to 0.085arcsec. Most of the
data were obtained with SINFONI which provides integral field capability at a
spectral resolution of R~4000. The themes on which we focus in this
contribution are: star formation around the AGN, the properties of the
molecular gas and its relation to the torus, and the mass of the black hole.
| https://export.arxiv.org/pdf/astro-ph/0601417 |
\title*{SINFONI's take on Star Formation, Molecular Gas, and Black
Hole Masses in AGN}
\titlerunning{Star Formation, Molecular Gas, \& Black Holes Masses in AGN}
\author{R.~Davies, R.~Genzel, L.~Tacconi, F.~M\"uller~Sanchez,
J.~Thomas, \and S.~Friedrich}
\authorrunning{R. Davies et al.}
\institute{Max Planck Institut f\"ur extraterrestrische Physik,
Postfach 1312, 85741, Garching, Germany}
\section{The AGN Sample}
\label{dav:sec:sample}
The primary criteria for selecting AGN were that
(1) the nucleus should be bright enough for adaptive optics
correction,
(2) the galaxy should be close enough that small spatial scales can be
resolved, and
(3) the galaxies should be ``well known'' so that complementary
data can be found in the literature.
These criteria were not applied strictly, since some targets were also
of particular interest for other reasons.
The resulting sample of 9 AGN is listed in Table~\ref{dav:tab:sample}.
The observations of these are now completed, and while the data for
some objects has been fully analysed, others are still in a
preliminary stage.
Additional AGN will likely be added once the Laser Guide Star Facility
is commissioned.
\begin{table}
\begin{centering}
\caption{AGN sample}
\label{dav:tab:sample} %
\begin{tabular}{llrlrll}
\hline\noalign{\smallskip}
Target & Classification & Dist. & \ \ \ & \multicolumn{3}{c}{Observations} \\
& & (Mpc) && Date & \ \ \ & Instrument\\
\noalign{\smallskip}\hline\noalign{\smallskip}
Mkn 231$^1$ & ULIRG / Sy 1 / QSO & 170 && May '02 && Keck / NIRC2 \\
NGC 7469$^2$ & Sy 1 & 66 && Nov '02 && Keck / NIRSPAO \\
IRAS 05189-2524 \ \ & ULIRG / Sy 1 & 170 && Dec '02 && VLT / NACO \\
Circinus$^3$ & Sy 2 & 4 && Jul '04 && VLT / SINFONI \\
NGC 3227$^4$ & Sy 1 & 17 && Dec '04 && VLT / SINFONI \\
NGC 3783 & Sy 1 & 42 && Mar '05 && VLT / SINFONI \\
NGC 2992 & Sy 1 & 33 && Mar '05 && VLT / SINFONI \\
NGC 1068 & Sy 2 & 14 && Oct '05 && VLT / SINFONI \\
NGC 1097 & LINER / Sy 1 & 18 && Oct '05 && VLT / SINFONI \\
\noalign{\smallskip}\hline
\end{tabular}
\end{centering}
$^1$ Davies et al. 2004a \cite{dav:dav04a};
$^2$ Davies et al. 2004b \cite{dav:dav04b};
$^3$ M\"uller Sanchez et al. 2006 \cite{dav:mul06};
$^4$ Davies et al. 2006 \cite{dav:dav06};
\end{table}
One immediate result, which has a bearing on the classifications in
the table, is the frequent detection of broad Br$\gamma$ --
i.e. with FWHM at least 1000\,km\,s$^{-1}$.
An example of this is given in Fig.~\ref{dav:fig:n2992}.
In only 3 galaxies was no broad Br$\gamma$ detected: Circinus,
NGC\,1068, and NGC\,1097 (in which even the narrow Br$\gamma$ is so
weak that it is almost lost in the stellar absorption features).
\section{Star Formation}
\label{dav:sec:starform}
The topics we address here are the spatial scales on which
stars exist around the AGN, the age and star formation history of
these stars, and their contribution to the bolometric luminosity with
respect to that of the AGN itself.
The stellar K-band (or equivalently H-band) continuum can be
distinguished from the non-stellar continuum
via the depth of stellar absorption features such as the CO bandheads,
because for any ensemble of stars the intrinsic depth will not vary
much once late-type stars appear (see Davies et al. 2006 \cite{dav:dav06}
for a more detailed discussion of this).
Doing so immediately allows one to assess the physical size
scale of the stellar population close to the AGN (see Fig.~\ref{dav:fig:plot3}).
In addition it permits a lower limit to be put on
the bolometric luminosity originating in stars.
This is because, while a stellar population which is still
forming stars will have $L_{\rm bol}/L_{\rm K} \sim 50$ (or even higher
if it is very young), even an old passively evolving population has
$L_{\rm bol}/L_{\rm K} \sim 20$.
In most cases we are able to apply tighter constraints than this by
considering other diagnostics.
For example, from the morphology and kinematics one can estimate the
fractions of the narrow Br$\gamma$ flux that are associated with stars
and with the AGN's narrow line region.
Similarly, it is often possible to estimate the fractions of the
radio continuum associated with the AGN and stars:
the former will be unresolved and have very high brightness
temperatures (see Condon et al. 1991 \cite{dav:con91}).
The ratio of either of these to the stellar K-band continuum can provide
strong constraints on the star formation time scales and hence the
bolometric luminosity from stars close around the AGN.
Our preliminary results are:
\begin{itemize}
\item
In all 9 cases we have resolved a stellar population around the AGN on
the scales we have achieved (0.08--0.3$''$); and the stellar
luminosity increases as one approaches the AGN.
\item
In the 5 cases we have analysed in detail so far (Mkn\,231, NGC\,7469,
IRAS\,05189-2524, Circinus, NGC\,3227), the stellar
population is young: the range of ages we find is 40--120\,Myr
\item
The (young) stellar luminosity is comparable to that of the AGN on
scales of 1\,kpc (Mkn\,231, IRAS\,05189-2524);
is 10--50\% of the AGN on scales of 50--100\,pc (NGC\,7469, NGC\,3227);
and is a few percent of the AGN on scales of 10--20\,pc (Circinus).
\end{itemize}
\section{Molecular Gas}
\label{dav:sec:torus}
The H$_2$ morphologies traced by the 1-0\,S(1)
line show a much greater diversity than the stellar distributions, as
typified in Fig.~\ref{dav:fig:plot3}.
This might be expected since it is known that distribution of gas is
strongly influenced by dynamical resonances and outflows.
However, when analysing the morphologies on $\sim$10\,pc scales, one
needs to remember that the 1-0\,S(1) line traces only hot
(typically 1000--2000\,K) gas, and hence the very local environment
will have an important impact on the observed luminosity
distribution: for example, is there
a particularly massive star cluster nearby or has there been a recent
supernova?
With this caveat in mind, our preliminary results are:
\begin{itemize}
\item
the 1-0\,S(1) emission is stronger closer to the AGN (with
the exception of NGC\,1068) indicating the gas distribution is also
concentrated towards the nucleus on scales of 10--50\,pc.
\item
the kinematics show ordered rotation (again excepting NGC\,1068) but
also remarkably high velocity dispersion -- in the
range $\sigma = 70$--140\,km\,s$^{-1}$, giving
$V_{\rm rot}/\sigma \sim 1$.
This means that the gas must be rather turbulent, most likely due to
heating from the AGN and/or star formation, and as a result is
probably geometrically thick.
\item
Given the size scales on which models predict the molecular torus
around AGN should exist (10--100\,pc, e.g. most recently Schartmann et
al. 2005 \cite{dav:sch05}), and the fact that the torus
must have a large enough scale height to collimate ionisation cones,
it is reasonable to propose that the gas we have seen in these data is
associated with the torus.
\end{itemize}
\section{Black Hole Masses}
\label{dav:sec:bhmass}
Since it was first discovered, the
relation between the mass of the supermassive
black hole $M_{\rm BH}$ and the velocity
dispersion $\sigma_*$ of the surrounding spheroid has become a
cornerstone of galaxy evolution and black hole growth in the
cosmological context.
However, almost without exception the `reliable' black hole masses
(typically based on stellar
kinematics and resolving the black hole's radius of influence) have
been derived only for nearby bulge
dominated E/S0 quiescent galaxies (see the review by Ferrarese
\& Ford 2005 \cite{dav:fer05}).
While extremely challenging, it is therefore crucial to determine
stellar dynamical black hole masses in AGN -- not only to verify
that the $M_{\rm BH} - \sigma_*$ relation holds for galaxies which
are by definition active, but to assess its scatter for these
galaxies, and to provide a comparison to reverberation masses which
might then allow one to constrain the geometry of the broad line region.
The high spatial resolution and integral field capability of SINFONI
provide an ideal combination to do this, and we have successfully derived
$M_{\rm BH}$ in NGC\,3227 from stellar kinematics -- the first time
for a Seyfert~1 -- using Schwarzschild orbit superposition techniques.
Details of the specific code, which is based on that used by the
Nuker team, are given in Thomas et al. (2004) \cite{dav:tho04}.
While the inclination and mass-to-light ratios are often uncertain
parameters, for NGC\,3227 they are relatively well constrained.
Nevertheless, we have explored the range of values which the modelling
would permit and find it to be consistent with those expected,
giving us confidence that the results are physically meaningful and
reasonably robust.
The resulting range of permissible black hole masses is
$M_{\rm BH} = 5\times10^6$--$2\times10^7$\,$M_\odot$.
The range is a result of the degeneracy between the black hole mass
and the `effective' mass-to-light ratio of the stellar population,
which includes the contribution of the gas mass.
If the gas is significantly less concentrated than the stars, then the
higher $M_{\rm BH}$ is possible;
on the other hand if the gas is strongly centrally concentrated in a
similar way to the stars, then $M_{\rm BH}$ must be correspondingly
lower.
That the mass we find is within a factor of 2--3 of the masses found
by other methods suggests that all are satisfactory to this level of
accuracy.
However, the fact that the mass is also likely to be a factor of a few
below that implied by the $M_{\rm BH} - \sigma_*$ relation, while in
contrast the stellar dynamical mass of Cen\,A (Silge et al. 2005,
\cite{dav:sil05}) is a factor of several greater, may indicate that for AGN
the scatter around this relation could be very considerable.
\printindex |
Title:
Star formation in the nearby universe: the ultraviolet and infrared points of view |
Abstract: This work presents the main ultraviolet (UV) and far-infrared (FIR)
properties of two samples of nearby galaxies selected from the GALEX ($\lambda
= 2315$\AA, hereafter NUV) and IRAS ($\lambda = 60\mu$m) surveys respectively.
They are built in order to get detection at both wavelengths for most of the
galaxies. Star formation rate (SFR) estimators based on the UV and FIR
emissions are compared. Systematic differences are found between the SFR
estimators for individual galaxies based on the NUV fluxes corrected for dust
attenuation and on the total IR luminosity. A combined estimator based on NUV
and IR luminosities seems to be the best proxy over the whole range of values
of SFR. Although both samples present similar average values of the birthrate
parameter b, their star-formation-related properties are substantially
different: NUV-selected galaxies tend to show larger values of $b$ for lower
masses, SFRs and dust attenuations, supporting previous scenarios for the star
formation history (SFH). Conversely, about 20% of the FIR-selected galaxies
show high values of $b$, SFR and NUV attenuation. These galaxies, most of them
being LIRGs and ULIRGs, break down the downsizing picture for the SFH, however
their relative contribution per unit volume is small in the local Universe.
Finally, the cosmic SFR density of the local Universe is estimated in a
consistent way from the NUV and IR luminosities.
| https://export.arxiv.org/pdf/astro-ph/0601235 |
\title{Star formation in the nearby universe: the ultraviolet and infrared points of view}
\author{J. Iglesias-P\'{a}ramo\altaffilmark{1,10}, V. Buat\altaffilmark{1}, T. T. Takeuchi\altaffilmark{1}, K. Xu\altaffilmark{2},
S. Boissier\altaffilmark{3}, A. Boselli\altaffilmark{1}, D. Burgarella\altaffilmark{1}, B. F. Madore\altaffilmark{3}, A. Gil de Paz\altaffilmark{3}, L. Bianchi\altaffilmark{4}, T. A. Barlow\altaffilmark{2},
Y.-I. Byun\altaffilmark{5}, J. Donas\altaffilmark{1}, K. Forster\altaffilmark{2}, P.G. Friedman\altaffilmark{2}, T. M. Heckman\altaffilmark{6}, P. N. Jelinski\altaffilmark{7}, Y.-W. Lee\altaffilmark{5},
R. F. Malina\altaffilmark{1}, D. C. Martin\altaffilmark{2}, B. Milliard\altaffilmark{1}, P. F. Morrissey\altaffilmark{2}, S. G. Neff\altaffilmark{8}, R. M. Rich\altaffilmark{9}, D. Schiminovich\altaffilmark{2},
M. Seibert\altaffilmark{2}, O. H. W. Siegmund\altaffilmark{7}, T. Small\altaffilmark{2}, A. S. Szalay\altaffilmark{6}, B. Y. Welsh\altaffilmark{7}
}
\and
\author{T. K. Wyder\altaffilmark{2}}
\altaffiltext{1}{
Laboratoire d'Astrophysique de Marseille, 13376 Marseille, FRANCE}
\altaffiltext{2}{
Space Astrophysics Laboratory, Mail Stop 405-47, California Institute of Technology, 1200 East California Boulevard, Pasadena, CA 91125}
\altaffiltext{3}{
Observatories of the Carnegie Institution of Washington, 813 Santa Barbara Street, Pasadena, CA 91101}
\altaffiltext{4}{
Center for Astrophysical Sciences, Johns Hopkins University, 3400 North Charles Street, Baltimore, MD 21218}
\altaffiltext{5}{
Center for Space Astrophysics, Yonsei University, Seoul 120-749, Korea}
\altaffiltext{6}{
Department of Physics and Astronomy, Johns Hopkins University, Homewood Campus, Baltimore, MD 21218}
\altaffiltext{7}{
Space Sciences Laboratory, University of California at Berkeley, 601 Campbell Hall, Berkeley, CA 94720}
\altaffiltext{8}{
Laboratory for Astronomy and Solar Physics, NASA Goddard Space Flight Center, Greenbelt, MD 20771}
\altaffiltext{9}{
Department of Physics and Astronomy, University of California, Los Angeles, CA 90095}
\altaffiltext{10}{
Instituto de Astrof\'{\i}sica de Andaluc\'{\i}a, 18008 Granada, SPAIN}
\keywords{surveys: GALEX --- ultraviolet: galaxies --- infrared: galaxies}
\section{Introduction}
What is the best way to measure the SFR of galaxies on large scales
and at different redshifts? The possibility of estimating the SFR of
a galaxy directly from the luminosity at a single wavelength would be
a major advantage for anyone wanting to compute the SFR per unit
volume at a given redshift. This quantity could be derived directly
from the luminosity function (LF) at this wavelength and at this
redshift. Under these conditions, large area surveys at single
wavelengths might suffice.
The recent SFR of a galaxy is often measured from the light emitted by the young stars: given their short lifetimes their luminosity is directly proportional to the rate at which they are currently forming.
The UV and FIR luminosities of star forming galaxies are both closely related to the recent star formation: most of the UV photons are originally emitted by stars younger than $\sim 10^{8}$~yr, but many of these photons are re-processed by the dust present in galaxies and re-emitted at FIR wavelengths.
Strictly speaking, neither of these fluxes can be used alone to estimate the SFR independently of the other one (e.g. Buat \& Xu 1996; Hirashita et al. 2003; Iglesias-P\'{a}ramo et al. 2004). Because of the previous lack of data at both wavelengths, attempts have been made using only the rest frame UV
(Lilly et al. 1996; Madau et al. 1996; Steidel et al. 1999; and more recently Schiminovich et al. 2005 with GALEX data), or just FIR data (Rowan-Robinson et al. 1997; Chary \& Elbaz 2001), but only a few authors have compared both (Flores et al. 1999, Cardiel et al. 2003).
The SFR estimator based on the UV luminosity suffers from attenuation by dust and it has to be corrected in order to properly trace the SFR: for UV-selected samples of galaxies the attenuation can reach more than 1~mag (e.g. Iglesias-P\'{a}ramo et al. 2004; Buat et al. 2005). On the other hand the FIR emission is not free of problems because the dust can also be heated by old stars, and can be a non-negligible correction for many star forming galaxies (Lonsdale \& Helou 1987; Sauvage \& Thuan 1992).
Neither of these two indicators taken alone is an accurate estimator of the SFR except perhaps for starburst galaxies where (a) the dust attenuation was found to follow a tight relation with the slope of the spectrum at UV wavelengths (Meurer et al. 1999), thus allowing one to estimate the dust attenuation with only information on UV fluxes, and (b) the contribution to the dust emission coming from old stars can be neglected (Sauvage \& Thuan 1992). In the most general case, the best estimator of the SFR should contain combined information of the luminosities at both wavelength ranges (Hirashita et al. 2003).
The UV and FIR fluxes are thus complementary for tracing star formation and it is well known that the FIR/UV ratio is a proper indicator of the dust attenuation (Buat et al. 1999; Witt \& Gordon 2000; Panuzzo et al. 2003).
Although other indicators of the recent SFR of galaxies have been extensively used in the literature, a detailed discussion of their quality as SFR tracers will not be discussed in this paper.
The GALEX mission (Martin et al. 2005a) is imaging the high-Galactic
latitude sky at two UV wavelengths ($\lambda=1530$\AA, FUV;
$\lambda=2315$\AA, NUV) and is providing the astronomical community
with unprecedented data (both in quantity and quality). The UV data
combined with existing FIR datasets (from the IRAS, ISO or
Spitzer missions) now allow us to carry out detailed studies of the UV and
FIR properties of galaxies, with special emphasis on the derivation of
the dust attenuation and star formation activity in star forming
galaxies.
With this purpose in mind we selected two samples of galaxies: one from the GALEX-All Imaging Sky (AIS) and the other from the IRAS PSCz (redshifts, infrared and optical photometry, and additional information for 18,351 IRAS sources, mostly selected from the Point Source Catalog) and FSC (Faint Source Catalog)
for which UV and FIR fluxes are available. With these datasets in hand we undertaken a study of the properties related to their emission at these wavelengths.
Both samples were extracted from the same region of the sky ($\sim 600$~degrees$^{2}$, constrained by the status of the GALEX survey when this work was initiated). From the GALEX catalog we built a complete sample of galaxies down to $AB_{NUV}=16$~mag\footnote{AB magnitudes are defined as: $AB_{\nu} = -2.5 \log f_{\nu} -48.6$, where $f_{\nu}$ is the monochromatic flux density expressed in erg~s$^{-1}$~cm$^{-2}$~Hz$^{-1}$.} and cross-correlated it with the IRAS database (FSC), allowing non-detections at 60$\mu$m (fluxes lower than 0.2 Jy).
The FIR-selected sample was built from the IRAS catalog (PSCz) with a limiting flux at 60$\mu$m of 0.6~Jy as the only constraint. The resulting list of PSCz sources was cross-correlated with the GALEX database,
allowing again non-detections in the NUV.
Both the NUV and the 60$\mu$m limits used to build the samples were chosen to allow for a very small number of non-detections and to sample the galaxy population over a large range of values of the dust attenuation. Besides an analysis of the SFR, for these
NUV and FIR-selected samples of galaxies (chosen with well-defined selection criteria) can also be used to place important constraints on models designed to predict the statistical properties of galaxy populations.
The attenuation of star light by interstellar dust, and its
emission in the far infrared are usually computed very crudely in
models of galactic evolution. Dust attenuation is usually deduced in such
approaches from other quantities such as the mass gas and the
metallicity (e.g. Guiderdonni \& Rocca-Volmerange 1987; Devriendt \& Guiderdoni 2000; Balland et al. 2003).
The properties of large samples of galaxies observed both
in the UV and FIR with clear selection criteria, such as the ones presented
in this paper, provide an important statistical constraint for the
calibration of the treatment of dust in such models.
The first paper in this series (Buat et al. 2005) based on these samples was mainly devoted to the dust attenuation properties. In the present work we discuss various aspects relating to the NUV and 60$\mu$m emission of our sample galaxies, including systematic differences in the 60$\mu$m and NUV luminosities and we will focus on their star formation related properties.
This paper is organized as follows:
the samples are presented in section~2. The relation between NUV and 60$\mu$m luminosities is discussed in section~3. Section~4 is devoted to the derivation of the SFR and to a comparison of various estimators as well as to a discussion on the star formation activity related properties of the samples.
The derivation of the local cosmic SFR density by different methods is discussed in section~5.
The main conclusions are presented in section~6.
Throughout this paper we adopt the following cosmological parameter set: $(h, \Omega_0, \lambda_0)=(0.7,0.3,0.7)$, where
$h\equiv H_0/100\; (\mbox{km\,s}^{-1}\mbox{Mpc}^{-1})$.
\section{Observational dataset}
\subsection{NUV-selected sample \label{uvsel_sam}}
In order to build the NUV-selected sample, we used 833 fields of the GALEX All sky Imaging Survey
(AIS) each having an exposure time equal to or larger than 50s. We used only the central 0.5~deg radius
circles in each field to ensure a uniform image quality: the resulting sky coverage is
615 deg $^2$. Within this area we selected all the galaxies of the GALEX AIS survey with
$AB_{NUV} \leq 16$~mag.
This bright limit was chosen in order to ensure IRAS detections of all the galaxies with
attenuation larger than $\sim 0.3$~mag (for a limit of 0.2 Jy at 60 $\mu$m from the IRAS FSC using
the calibration of Buat et al. 2005) and highly significant upper limits for the less attenuated galaxies.
On the one hand, the moderate angular resolution of GALEX (FWHM $\sim 6$~arcsecs) does not allow
for a secure discrimination of stars from small galaxies.
On the other hand the GALEX pipeline can induce some shredding of larger galaxies. This results in multiple detections which (cumulatively) correspond to a single galaxy because of the misidentification of star forming regions as if they were individual galaxies. The main consequence of this is the underestimation of the fluxes of large galaxies.
Corollary catalogs were thus required in order to perform a reliable selection of galaxies.
Our starting point was the catalog of NUV detections produced by the GALEX pipeline\footnote{version 0.2.0, September 2003, with the correction to the NUV and FUV magnitudes reported in Buat et al. 2005}, which made use of the Sextractor code (Bertin \& Arnouts 1996). We made the assumption that all the galaxies brighter than $AB_{NUV}=$16~mag, even if they are shredded,
should contain at least one Sextractor detection brighter than $AB_{NUV}=18$~mag.
Then we associated an object from databases of well known stars and galaxies (SIMBAD, 2MASS, LEDA) with each of the Sextractor detections brighter than $AB_{NUV}=18$~mag.
The problem of shredding was mostly resolved by using LEDA. As this database contains the optical diameters of the galaxies, NUV detections corresponding to shredded galaxies can be associated with
their parent galaxies provided they are located within the aperture defined by the optical diameters and the position angle listed in LEDA.
For the detections of galaxies not shredded we used SIMBAD and 2MASS
in order to classify them as stars or galaxies. Finally, objects not associated with any known source were classified as ``dubious''.
In order to test the quality of our selection method, we cross-correlated our final catalog with the SDSS DR1, which spatially overlap one fifth of our sample. The spectral and photometric information of the SDSS together with its higher angular
resolution made possible an optimal classification of all the objects detected in the field:
all the objects present in both GALEX and SDSS catalogs were found to be properly
classified.
Dubious objects were found to be noise detections or ghosts generated mainly near the edges of the GALEX frames.
Thus we ended up with a catalog of {\em bona fide} galaxies or fragments of galaxies (i.e. belonging to the large galaxies shredded by Sextractor) brighter than $AB_{NUV}=18$~mag. The next step was to obtain aperture photometry of these objects in order to select out only the galaxies brighter than $AB_{NUV}=16$~mag.
Photometry of our sample galaxies was performed in the GALEX NUV and FUV bands. Since the selection criterion for our sample galaxies was imposed in the NUV band, we also took the NUV images as our reference for the
total photometry.
We performed aperture photometry using a set of elliptical apertures, the total photometry corresponding to the aperture where convergence of the growth curve is achieved.
Once we determined the total NUV flux, the photometry in the FUV band was obtained by performing background subtracted aperture photometry within the same elliptical aperture where convergence of
the NUV growth curve was achieved. This way, the NUV$-$FUV colors are consistent.
Some galaxies were contaminated by the flux of nearby bright stars or galaxies. The contaminating sources were then masked in the NUV and FUV frames in order to obtain proper NUV and FUV fluxes for our galaxies.
Table~\ref{error} shows the typical uncertainties of the NUV and FUV magnitudes.
NUV and FUV magnitudes were corrected for Galactic extinction using the Schlegel et
al. (1998) dust map and the Cardelli et al. (1989) extinction curve.
In the end, a total of 95 non-stellar objects brighter than $AB_{NUV}=16$~mag were found. One of them, classified as a QSO, was excluded of the sample.
FIR fluxes at 60$\mu$m were obtained from the IRAS Faint Source Catalog (FSC, Moshir et al. 2000) for 68 of our 95
galaxies.
We discarded all these sources for which the cirrus parameter (as listed by the FSC) is larger than 2 because
it can result in uncertain fluxes.
The Scan Processing and Integration Facility (SCANPI) was used to obtain the FIR
fluxes for the remaining 27 objects. Three of these galaxies (UGC~11866, UGCA~438 and UGC~12613) were not detected at 60$\mu$m.
We adopted a conservative upper limit of 0.2Jy at 60$\mu$m (as given in the FSC) for these galaxies.
Four galaxies of the sample were not covered by the IRAS survey.
\subsection{FIR-selected sample \label{firsel_sam}}
The FIR-selected sample was extracted from the IRAS PSCz (Saunders et al. 2000)
over the 509~$\deg^{2}$ in common with the GALEX AIS fields having exposure
times larger than 90~sec.
In order to keep only good quality FIR data we discarded those galaxies
for which the probability of a correct optical identification
of the FIR-selected galaxies was lower than 50\%, as listed in the PSCz.
As for the NUV-selected sample, galaxies for which the cirrus parameter (as listed in the PSCz)
was larger than 2 were discarded.
A total of 163 galaxies were selected; all but two of them (Q00443+1038 and Q23367-0448)
have published radial velocities.
As galaxies were selected from the PSCz, the imposed limiting flux at 60$\mu$m was 0.6~Jy.
This limit, combined with the one estimated for the GALEX AIS at NUV ($AB_{NUV} \sim 20.5$~mag, Morrissey et al. 2005), results in detections at NUV for all the galaxies with dust attenuation as large as $\sim 4.4$~mag (Buat et al. 2005). Indeed, only two galaxies (Q00443+1038 and Q00544+0347)
were not detected in the NUV frames and a total of 23 were not detected in the FUV frames.
The NUV and FUV photometry of the FIR-selected galaxies was performed using
the same technique as for the NUV-selected galaxies.
\subsection{Completeness and bias of the samples}
Before drawing conclusions about the properties of the samples we have to check on just how representative they are. Because of the reduced statistics of the samples, if the sampled volumes are not large enough it could be that some luminosity ranges are oversampled or undersampled with respect to reference samples defined over larger volumes of the local Universe.
We check the representative nature of our samples by building LFs and comparing them to the standard ones at $z=0$, constructed from larger samples of galaxies (NUV LF of Wyder et al. (2005) and 60$\mu$m LF of Takeuchi et al. (2003)).
We calculate both NUV and 60$\mu$m LFs of our sample by the Lynden-Bell method
(Lynden-Bell 1971), implemented to obtain the normalization using the
formulation of Takeuchi et al. (2000).
The calculation of the uncertainty is based on a bootstrap resampling method (Takeuchi et al.~(2000)).
We note that the Lynden-Bell method is robust against density
inhomogeneities, and hence we can trust the LF so determined (see Takeuchi et al.~2000 for details).
The results are shown in Figure~\ref{nlf}.
The error bars correspond to 1$\sigma$ uncertainties.
The agreement between the LFs of our samples and the corresponding LFs from
larger samples is very good, so we are confident that in spite of their
small size our samples are representative of flux-limited NUV and FIR
samples in the local Universe.
We also compare the volumes from which the samples were extracted.
Figure~\ref{histo_velo_all}a shows the redshift distributions
for both samples.
The median values are 0.013 and 0.027 for the NUV and FIR-selected
samples respectively.
At a first glance, this means that the FIR selection samples a volume 8
times larger than the corresponding NUV selection.
However, we must recall that the redshift distribution of a flux limited
sample is strongly dependent on the shape of the LF,
and as shown in Figure~\ref{nlf}, the NUV and 60$\mu$m LFs are very different.
In Figure~\ref{histo_velo_all}b we show the theoretical redshift distributions
for NUV and FIR-selected samples with the same limiting fluxes as our two
samples\footnote{Details on the calculation are given in
Appendix~\ref{depth}.}.
As can be seen, the theoretical median values of the redshift for both samples are consistent with the ones obtained from the observational data.
A limiting magnitude of $AB_{NUV} = 18$~mag is required to obtain similar median values of the redshift distributions of both samples, as we show in
Figure~\ref{histo_velo_all}c.
And in this case most of the galaxies detected in NUV will not have any
counterpart in FIR.
This behavior of the redshift distribution can be understood intuitively.
Indeed, the flux density selection procedure omits intrinsically
low-luminosity objects from the sample, whereas bright objects are hardly
affected by the flux selection.
To the degree that the LF is well reconstructed from the flux/magnitude-limited
samples, these samples can be said to be representative, with respect to
the luminosity and/or flux density, and it is indeed the case for
the present work.
\section{Relation between $L_{60}$ and $L_{NUV}$ \label{tsu_vero}}
The physical link between the FIR and UV luminosities of galaxies is rather
complex. On the one hand, both are related to the light of young stars, so one
expects a correspondence between the two. On the other hand, the FIR emission is due to
the absorption of the UV light thereby leading to an anti-correlation. Since our samples were built to avoid upper limits -- i.e. most of the galaxies selected in NUV (or at 60 $\mu$m) are also detected at 60 $\mu$m (or in NUV) -- we are able to discuss statistically the intrinsic relation between both the two wavelengths and outline the specifics of NUV versus FIR selection effects.
In Figure~\ref{l60_lnuv} we plot $L_{NUV}$ versus $L_{60}$ for both samples
\footnote{Throughout the paper the NUV and 60 $\mu$m luminosities $L_{NUV}$ and $L_{60}$ will be calculated as $\nu L_{\nu}$ expressed in
solar units. The adopted value for the bolometric solar luminosity is $L_{\odot} = 3.83 \times 10^{33}$~erg~s$^{-1}$.}.
The two samples exhibit very different behaviors:
the NUV-selected galaxies show a good correlation between both luminosities,
with a correlation coefficient (in log units) of $r
\simeq 0.8$. On the contrary
the dispersion is very large for FIR-selected galaxies and the
correlation coefficient is low: $r \simeq 0.3$.
NUV-selected galaxies appear intrinsically less luminous
at 60$\mu$m than FIR-selected ones.
This is also true for the sum of both luminosities, $L_{tot} = L_{NUV} +
L_{60}$, which is supposed to be a crude estimate of the bolometric luminosity of galaxies
related to recent star formation (e.g. Martin et al. 2005b). Although the luminosity
distribution within each sample is the combined effect of the LFs
and selection criteria,
this result is confirmed by other studies and from the comparison of the 60$\mu$m and NUV LFs themselves (e.g. Martin et al. 2005b; Buat et al. 2005).
Both distributions flatten at higher luminosities, reflecting a general increase of
the dust attenuation already pointed out in the literature by several authors (Wang \& Heckman 1996; Buat et al. 1999; Sullivan et al. 2001; Vijh et al. 2003).
One could argue that the difference in luminosity between the two samples is a consequence of bias in the sampling.
We show in Figure~\ref{l60_lnuv} the lines corresponding to our lower (upper) limit of the NUV attenuation above (below) which the NUV (FIR) selected sample is complete. Thus, the fact that only very few low-luminosity and low-attenuation FIR-selected galaxies are detected must be taken as real. Low-luminosity, high-attenuation galaxies should have been detected by our FIR survey if they were present. For the same reason, very luminous galaxies should have been detected in the NUV survey if they existed.
The good correlation found between $L_{NUV}$ and $L_{60}$ for the NUV-selected galaxies has to be related to their rather low dust attenuation: in these galaxies both $L_{NUV}$ and $L_{60}$ represent a significant part of the total luminosity of the galaxies. This result holds for intrinsically faint galaxies ($L_{tot} \leq 2 \times 10^{9} L_{\odot}$). The very loose correlation found for FIR-selected galaxies may also be explained by the effect of the dust attenuation. With a mean dust attenuation larger than 2~mag, the NUV luminosity becomes a poor tracer of their total
luminosity whereas the 60$\mu$m luminosity is not very different from the bolometric emission of the young stars.
Some fluctuations in the percentage of NUV photons escaping the galaxies can induce large variations in the NUV
observed luminosity on an absolute scale without any strong physical
difference on the scale of the total luminosity of the galaxies.
We make a final comment on the so-called ``UV luminous galaxies'' (UVLGs, defined as those with $L_{FUV} \geq 2 \times 10^{10}$~L$_{\odot}$ in Heckman et al. 2005). We found 3 ULVGs in our NUV-selected sample and a total of 8 (including the previous 3) in the FIR-selected sample. All but one of these galaxies are more luminous at 60$\mu$m than in the NUV (in fact most of them are LIRGs), and their attenuation is typically larger than 1~mag. This means that these galaxies are not only UV luminous but also very luminous from a bolometric point of view.
\section{Selection effects on observational quantities and physical properties of galaxies}
The main aim of this section is to show the effect of the selection criteria of samples of galaxies on observational and physical quantities. We will now show that the selection criteria of a sample of galaxies play an important role in defining the nature of the galaxies selected and thus, in their averaged properties. Accordingly we warn against the unqualified comparison of results obtained from samples of galaxies selected on the basis of different criteria.
In order to reduce
the uncertainties associated with the FIR and NUV fluxes we
impose further constraints on our galaxy sample:
\begin{itemize}
\item Ellipticals, S0s as well as AGNs (Seyferts and QSOs) were excluded since the origin
of their 60$\mu$m and NUV fluxes is clearly not associated to recent star formation.
The necessdary classification information is available for most of the NUV-selected galaxies; but this turned
out not to be the case for the FIR-selected galaxies, so contamination of the sample by ellipticals
and/or AGN among these galaxies cannot be totally excluded.
Galaxies with extraneous radio sources (from NVSS and/or FIRST) within the IRAS beam
were also excluded since part of the FIR flux of these galaxies could be due to
contaminating background objects.
\item Multiple galaxies, not resolved by the IRAS
beam but clearly resolved into various components in the GALEX frames were excluded, since a one-to-one
60$\mu$m-NUV association is not possible for them.
\end{itemize}
After applying these criteria we ended up with 59 and 116 galaxies from the original NUV and FIR-selected samples, respectively.
Hereafter we will use these restricted subsamples for our subsequent analysis of the star formation related properties, although we will keep the terminology FIR and NUV-selected samples to refer to the restricted subsamples.
Given that all the galaxies were extracted from the same region of the sky,
some of them belong to both subsamples. Their basic properties are listed in Table~2: (1) Identifier of the galaxy; (2) ``Y'' (``N'') indicates whether the galaxies is included (or not) in the NUV-selected sample; (3) ``Y'' (``N'') indicates whether the galaxies is included (or not) in the FIR-selected sample; (4) R.A.(J2000 equinox) of the source ; (5) Declination (J2000 equinox); (6) Radial velocity in km~s$^{-1}$, obtained from NED or LEDA; (7) Distance to the source in Mpc, corrected for the Local Group Infall to Virgo and $H_{0} = 70$~km~s$^{-1}$~Mpc$^{-1}$; (8) Morphological type, from NED or LEDA; (9) IRAS identifier: ``F'' for FSC; ``Q'', ``O'' or ``R'' for PSCz; ``SCANPI'' for absence in both catalogs.
Table~\ref{photo} gives some useful photometric data for the galaxies in the restricted subsamples: (1) Optical identifier; (2) NUV magnitude corrected for Galactic extinction; (3) FUV magnitude corrected for Galactic extinction; (4) Flux density at 60$\mu$m in Jy; (5) Flux density at 100$\mu$m in Jy; (6) $H$ magnitude from 2MASS Extended Source Catalog. For galaxies with no detection by 2MASS we adopt the limiting value of $H = 13.9$~mag (3~mJy) as given by Jarrett et al. (2000); (7) NUV attenuation in mag, derived as in Buat et al. (2005); (8) FUV attenuation derived as indicated in Buat et al. (2005).
For some galaxies Eqs.~\ref{attenuv_eq} and \ref{attefuv_eq} gave negative values of the NUV and FUV attenuations which is, of course, unphysical. In fact, this is an artifact of the polynomial fitting used to derive a functional form for the attenuation in Buat et al. (2005). Throughout this paper they will be considered as zero.
Given that the FIR fluxes were extracted from different catalogs (PSCz for the FIR-selected sample and FSC/SCANPI for the NUV-selected sample), for those galaxies belonging
to both samples we list the FIR entries corresponding to the PSCz catalog.
For those galaxies not present in the FSC and not detected by SCANPI at 60$\mu$m, we list
$f_{60} \leq 0.2$Jy, which is the nominal limiting flux of the FSC. No estimate of an upper
limit at 100 $\mu$m is given for these galaxies.
For galaxies with no detection in 2MASS we adopt the limiting value of $H = 13.9$~mag (3~mJy) as published by Jarrett et al. (2000).
In Table~\ref{sfr_tab} we list some star-formation properties which will be used in the forthcoming discussion: (1) Identifier of the galaxy; (2) $SFR_{NUV}$ from Eq.~\ref{sfrnuv_eq}; (3) $SFR_{FUV}$ from Eq.~\ref{sfrfuv_eq}; (4) $SFR_{dust}$ from Eq.~\ref{sfrfir_eq}; (5) $SFR_{tot}(NUV)$ from Eq.~\ref{sfrtot_eq}; (6) $SFR_{tot}(FUV)$ from Eq.~\ref{sfrtot_eq} but modified by using $SFR^{0}_{FUV}$ instead of $SFR^{0}_{NUV}$; (7) $\left< SFR \right>$ averaged over the galaxy's lifetime, estimated as indicated in Appendix~\ref{apendice_b}.
\subsection{SFR derivations}
This section is devoted to a detailed comparison of the recent SFR as seen in the FIR and NUV-selected samples.
Although other estimators of the recent SFR can be found in the literature (see Kennicutt 1998 for an interesting review on several methods to derive the SFR), we focus
on only two of them, those using the NUV and FIR fluxes.
Our aim is to compare commonly used recipes to derive SFR from the UV and FIR luminosity of the galaxies. Therefore we will make very classical calculations, as described below. For consistency we re-derive the calibrations in a homogeneous way, adapted to the GALEX bands: the formulae are found to be very similar to those of Kennicutt (1998).
The underlying physical justification for deriving the SFR of a galaxy from the UV luminosity is the following: most of the UV photons emerging from a galaxy originate in the atmospheres of stars younger than $\sim 10^{8}$~yr. Thus, the SFR is proportional to the UV luminosity emitted by the young stars under the assumption that the SRF is approximately constant over this timescale. This is reasonable given that Salim et al. (2005) and Burgarella et al. (2005) found that the intensity of the youngest burst in large samples of nearby galaxies contributes typically less than 5\% to the total.
However, the presence of dust absorbs a part of the UV light escaping from galaxies and breaks down the proportionality between the SFR and the observed UV luminosity.
As star-forming galaxies may present a large variety of relative geometries between stars and dust, the scattering of the stellar photons through the interstellar medium may introduce a fraction of them in the line of sight before they escape the galaxy. Thus, the effect of the dust differs from a pure extinction but is a complex combination of absorption and scattering. Following Gordon et al. (1997) we will use the term `dust attenuation' for this global process at work in galaxies.
The most commonly accepted method to estimate the dust attenuation at UV wavelengths is to use the ratio of FIR-to-UV fluxes (Buat \& Xu 1996; Meurer et al. 1999; Gordon et al. 2000). Several analytical expressions are already available in the literature for different UV wavelengths (Panuzzo et al. 2003; Kong et al. 2004; Buat et al. 2005).
All these expressions are fairly consistent except at high values of the dust attenuation, where some dispersion appears (e.g. Meurer et al. 1999, Kong et al. 2004, Buat et al. 2005 at $\lambda \sim 1500$\AA.).
In this work we use the prescription of Buat et al. (2005) to obtain the corrected NUV and FUV luminosities:
\begin{equation}
A_{NUV} = -0.0495 x^{3} + 0.4718 x^{2} + 0.8998 x + 0.2269
\label{attenuv_eq}
\end{equation}
where $x = \log L_{IR}/L_{NUV}$ and
\begin{equation}
A_{FUV} = -0.0333 y^{3} + 0.3522 y^{2} + 1.1960 y + 0.4967
\label{attefuv_eq}
\end{equation}
where $y = \log L_{IR}/L_{FUV}$.
Once the observed NUV and FUV luminosities have been corrected for dust attenuation, the SFRs can be derived using the following expressions\footnote{This formula has been derived from Starburst99 (Leitherer et al. 1999) and assuming solar metallicity, and a Salpeter IMF from 0.1 to 100~$M_{\odot}$.}:
\begin{equation}
\log SFR_{NUV} (M_{\odot}~yr^{-1}) = \log L_{NUV,corr} (L_{\odot}) - 9.33
\label{sfrnuv_eq}
\end{equation}
\begin{equation}
\log SFR_{FUV} (M_{\odot}~yr^{-1}) = \log L_{FUV,corr} (L_{\odot}) - 9.51
\label{sfrfuv_eq}
\end{equation}
In Figure~\ref{sfrnuvsfrnuvfuv} we show the ratio of $SFR_{NUV}/SFR_{FUV}$ as a function of $L_{tot}~(= L_{NUV} + L_{60})$,
which traces the bolometric luminosity related to recent star formation and has the advantage of being a purely observational quantity. As this figure shows, both quantities are equivalent with a dispersion of about 20\%. Since our sample is NUV-selected, hereafter we will use NUV as our reference wavelength for star formation related properties.
The luminosity at IR wavelengths provides a different avenue to the derivation of the SFR.
Dust absorbs photons at UV wavelengths and re-emits most of them at IR wavelengths ($8 - 1000\mu$m).
Under the hypothesis that all the UV photons are absorbed by dust, the IR luminosity would be a direct tracer of the SFR of a galaxy.
One source of uncertainty is the difficulty in estimating the total IR luminosity from the FIR flux at only one or two wavelengths. In this paper we use the prescription of Dale et al. (2001) and derive $L_{IR}$ by using $f_{60}$ and $f_{100}$. For the galaxies for which only $f_{60}$ is available we use the mean value of $f_{60}/f_{100}$ estimated using the galaxies detected at both wavelengths.
If we assume the same scenario as for Eq.~\ref{sfrnuv_eq}, the SFR can be expressed as:
\begin{equation}
\log SFR_{dust} (M_{\odot}~yr^{-1}) = \log L_{IR} (L_{\odot}) - 9.75
\label{sfrfir_eq}
\end{equation}
However, Eq.~\ref{sfrfir_eq} is a good approximation only for the most extreme starbursts, since many of the FIR-selected galaxies are, in fact, detected at UV wavelengths. A further limitation of this method concerns the fraction of the total IR luminosity heated by old stars (the cirrus component, hereafter represented by $\eta$), which should be removed before applying Eq.~\ref{sfrfir_eq}. This quantity is known to depend on the morphological type of galaxies (Sauvage \& Thuan 1992), but a precise estimate for individual galaxies is subject to large uncertainties (Bell~2003).
The SFRs estimated from these methods are often compared in the
literature for individual objects or for large samples of galaxies. In
order to see whether they are consistent with each other we show here
a comparison of the two using the galaxies of our two
samples. Figure~\ref{sfrnuvdustb} shows the ratio
$SFR_{NUV}/SFR_{dust}$ as a function of $L_{tot}$; each sample shows a
different behavior. For the NUV-selected sample (blue, filled
circles), $SFR_{NUV}$ is always larger than $SFR_{dust}$ (and the ratio
can be as high as 3) but the discrepancy is lowered as $L_{tot}$ (and
$A_{NUV}$) increase. This result is expected since we have seen in
Section~3 that low luminous galaxies are brighter in the NUV than at
60$\mu$m. This affirms that $SFR_{dust}$ cannot give a proper
estimation of the SFR for these galaxies.
The FIR-selected galaxies extend the trend found for the NUV-selected sample to higher luminosities.
For values of $L_{tot} \geq 3 \times 10^{10}$ (and for higher values of the dust attenuation), where no NUV-selected galaxies are present, $SFR_{dust}$ systematically exceeds $SFR_{NUV}$ by a factor of $\sim 2$.
One reason that could play a role in this inconsistency between the two estimators is that the dust attenuation is not properly estimated for very dusty galaxies. In any case it does not make sense to use the corrected UV luminosity to measure the SFR for these IR bright galaxies. In fact, Charmandaris et al. (2004) have reported decoupled IR and UV emissions for some dusty galaxies, which could be at the basis of the discrepancy found between $SFR_{NUV}$ and $SFR_{dust}$ found in this work for galaxies with large attenuation.
The conclusion of this analysis seems to be that $SFR_{NUV}$ is a good tracer of the SFR for low values of the attenuation, and in the opposite extreme $SFR_{dust}$ must be used for very heavily attenuated galaxies. There is no obvious way to delimit these two different regimes, or to chose which and which of the two indicators should be used in the intermediate cases. And so we warn users about any undiscriminated comparison of $SFR_{NUV}$ and $SFR_{dust}$ for samples of galaxies selected with different criteria.
An alternative tracer of the SFR containing information from NUV and IR luminosities has already been discussed in the literature (Hirashita et al. 2003; Iglesias-P\'{a}ramo et al. 2004, Bell 2003):
\begin{equation}
SFR_{tot} = SFR^{0}_{NUV} + (1 - \eta) \times SFR_{dust}
\label{sfrtot_eq}
\end{equation}
where $\eta$ accounts for the IR cirrus emission and $SFR^{0}_{NUV}$ is obtained following Eq.~\ref{sfrnuv_eq} but using $L_{NUV,obs}$ (that is the observed NUV luminosity) instead of $L_{NUV,corr}$.
This estimator has the advantages of being free of the model dependence of the attenuation correction, and it contains information of the observed NUV and the IR luminosities.
One limitation of this estimator, $\eta$, is the adopted value of the
IR cirrus contribution. Hirashita et al. (2003) and
Iglesias-P\'{a}ramo et al. (2004) reported a value of $\eta \sim 0.4$
for normal disk galaxies. Accurate values of $\eta$ for individual
galaxies are not easily obtained and instead, averaged values are often
used. However, this parameter is strongly dependent on the sample of
galaxies under study and cannot be easily generalized. Whereas an
average value of $\eta \sim 0.4$ seems to apply for normal disk
galaxies, a value of $\eta \sim 0$ seems to better represent the
properties of starbursts (Hirashita et al. 2003). Bell (2003) also
proposed a cirrus correction for a compilation of galaxies from the
literature with FUV, optical, IR and radio luminosities. He found
$\eta \sim 0.32 \pm 0.16$ for galaxies with $L_{IR} \leq
10^{11}$~L$_{\odot}$ and $\eta \sim 0.09 \pm 0.05$ for galaxies with
$L_{IR} > 10^{11}$~L$_{\odot}$. For our NUV-selected sample (similar
to the normal star forming galaxies of Hirashita et al.) a value of
$\eta \sim 0.2$ gives similar values for $SFR_{NUV}$ and
$SFR_{tot}(NUV)$. Although our NUV-selected sample must contain
galaxies more active than that of Hirashita et al. (since their
selection is based on optical fluxes rather than on UV fluxes), this
result gives an idea of the uncertainties related to the determination
of $\eta$. For practical issues, throughout this paper we will use
the value of $\eta$ of Bell (2003) -- not far from that of Hirashita
et al. (2003) -- when computing $SFR_{tot}$, but keeping in mind that
the uncertainties reported by this author are of the order of 50\%.
Another limitation of $SFR_{tot}$ is that it depends on the wavelength at which we measure the UV flux. In order to illustrate this point we show in Figure~\ref{ldustnuvldustfuv} the ratio of $SFR_{tot}(NUV)/SFR_{tot}(FUV)$ as a function of $SFR_{tot}(NUV)$ for both samples.
As can be seen, for the NUV-selected galaxies $SFR_{tot}(NUV)$ is systematically larger than $SFR_{tot}(FUV)$ by about $20\%$. This
discordance for the NUV-selected galaxies is due to the fact that the UV attenuation is not grey: $A_{NUV} \leq A_{FUV}$ for most galaxies (see Buat et al. (2005) and Table~\ref{photo}), and since we showed in Figure~\ref{sfrnuvsfrnuvfuv} that $SFR_{NUV} \approx SFR_{FUV}$, it is obvious that $SFR^{0}_{NUV} \geq SFR^{0}_{FUV}$.
On the contrary, for the brightest FIR-selected galaxies the agreement between $SFR_{tot}(NUV)$ and $SFR_{tot}(FUV)$ is good since for these galaxies $SFR_{tot}$ is dominated by $SFR_{dust}$.
We conclude that $SFR_{tot}$ is stable to within 20\% for whatever UV wavelength at which we measure the UV flux.
We compare now $SFR_{tot}(NUV)$ to the classical estimators $SFR_{NUV}$ and $SFR_{dust}$, in order to set their domain of applicability.
Figure~\ref{sfrtotsfrnuv}a shows the comparison between $SFR_{NUV}$ and $SFR_{tot}$. At low values of the SFR both quantities are almost identical for the NUV-selected galaxies. This is expected since for these galaxies both $A_{NUV}$ and $L_{IR}$ are almost negligible and $SFR_{NUV} \approx SFR_{NUV}^{0}$. As the SFR grows, we note an increase of $SFR_{NUV}$ with respect to $SFR_{tot}$, but always within $\sim 15\%$. This increase could be due to the choice for the cirrus correction and/or to the fact that $A_{NUV}$ does not exactly corresponds to the dust emission (since factors other than absorption do play a role in the attenuation, like for example the relative geometry between stars and dust). Finally, the NUV-selected galaxies with the largest values of SFR show a decrease of $SFR_{NUV}$ with respect to $SFR_{tot}$. We stress that these galaxies have $L_{IR} > 10^{11}$~L$_{\odot}$ and so their cirrus correction is different from for the rest. All in all we find that for the NUV-selected galaxies, basically those with $SFR_{NUV} \leq 15$~$M_{\odot}$~yr$^{-1}$, $SFR_{NUV}$ and $SFR_{tot}$ are equivalent to within $\sim 15\%$.
The FIR-selected galaxies show a different behavior. Whereas those with $L_{IR} < 10^{11}$~L$_{\odot}$ show an $\sim 15\%$ excess of $SFR_{NUV}$ with respect to $SFR_{tot}$, similar to the NUV-selected galaxies, for those with $L_{IR} > 10^{11}$~L$_{\odot}$, $SFR_{NUV}$ is well below $SFR_{tot}$. This is easily understood as a consequence of the already mentioned discrepancy between $SFR_{dust}$ and $SFR_{NUV}$ for galaxies dominated by their IR emission.
In Figure~\ref{sfrtotsfrnuv}b we compare $SFR_{dust}$ and $SFR_{tot}$. The NUV-selected galaxies follow a very dispersed trend with $SFR_{dust}/SFR_{tot}$ increasing with $SFR_{tot}$. This behavior is due to the fact that $SFR_{dust}$ lacks the UV contribution which is dominant in these galaxies. The FIR-selected galaxies obey two different trends: for galaxies with $SFR_{tot} \leq 15$~$M_{\odot}$~yr$^{-1}$ the ratio $\log SFR_{dust}/SFR_{tot} \sim 0$, although with a dispersion of $\sim 0.2$~dex. This large dispersion is due to the contribution of the NUV luminosity to $SFR_{tot}$, which is important for the less attenuated galaxies. On the contrary, at large values of $SFR_{tot}$ the average value of $\log SFR_{dust}/SFR_{tot} \sim 0.04$~dex with a very small dispersion. This is a consequence of the fact that most of these galaxies have $L_{IR} > 10^{11}$ and are dominated by their IR emission, so the difference between $SFR_{dust}$ and $SFR_{tot}$ corresponds basically to the cirrus correction applied to $SFR_{tot}$, which is minimal.
Bell (2003) proposed a calibration of the SFR similar to the one described in Eq.~\ref{sfrtot_eq} but using FUV as the reference UV wavelength. His method is based on the relation he found between $L_{IR}/L_{FUV}$ and $L_{IR}$ ($L_{IR}/L_{FUV} \sim \sqrt{L_{IR}/10^9}$)
for a compilation of galaxies from the literature with FUV, optical, IR and radio luminosities.
One can see in Figure~\ref{lfuvtirltir} that our NUV-selected galaxies follow well the Bell's relation whereas it is not the case for the FIR-selected sample. The galaxy sample used by Bell is therefore closer to a UV selection than to an IR one. Again we emphasize the importance of the selection biases in deriving SFRs.
The overall conclusion emerging from this study is that $SFR_{tot}$ seems to be a proper estimator of the SFR of galaxies whatever their dust content is, since it avoids the main problems of the clasical estimators $SFR_{NUV}$ and $SFR_{dust}$ and is consistent with them within their respective domains of applicability to within $\sim 15\%$.
We again warn against indiscriminate comparisons of the SFR of galaxies estimated from these classical estimators since the results could be strongly affected by selection biases as we have illustrated in this section.
The combined uncertainty of $SFR_{tot}$ due to the choice of the UV wavelength at which we measure the UV flux and to the cirrus contribution to the IR luminosity is $\lesssim 55\%$. Throughout this paper, we adopt $SFR_{tot}(NUV)$ as our proxy to trace the recent SFR.
\subsection{Star formation history}
The determination of the SFR of a galaxy gives information about the total number of young stars that are being formed. But this does not necessarily mean that the light coming from this galaxy is dominated by these young stars given that most galaxies are composed of a mixture of various stellar populations of different ages.
This parameter is of major importance in understanding the SFH of the Universe. Recent results based on large amounts of SDSS data suggest that the higher the mass of a galaxy, the earlier its stars were formed (Heavens et al. 2004), thus supporting the so-called ``downsizing'' explanation for the SFH of galaxies already proposed by several authors (e.g. Cowie et al. 1996; Brinchmann \& Ellis 2000; Boselli et al. 2001). We devote this section to the study of the SFH of the galaxies in our samples.
A quantitative estimation of the SFH of a galaxy requires information relating the relative contribution of young and old stars.
The birthrate parameter (hereafter $b$) has been proposed as a quantitative estimator of the recent SFH of a galaxy (Scalo 1986). It is defined as the ratio between the current and the past-averaged SFR:
\begin{equation}
b = \frac{SFR}{\left< SFR \right>}
\label{b_eq}
\end{equation}
Since $b$ depends on the overall SFH of the galaxy, an accurate estimation
from observational quantities is complex and involves several approximations.
A detailed derivation of $b$ following the prescriptions of Boselli et al. (2001) can be found in Appendix~\ref{apendice_b}. As explained in Section~4.1, the NUV luminosity is sensitive to the SFR over a timescale of $\sim 10^{8}$~yr, and thus $b$ is not sensitive to shorter-timescale variations in the SFH. However, this is not a serious problem since Burgarella et al. (2005) have shown that less than 20\% of the galaxies in either sample have bursts younger than $10^{8}$~yr.
Figure~\ref{histob} shows the distributions of $b$ for both samples of galaxies. The median values of both distributions are similar: 0.50 and 0.58 for the NUV and FIR-selected galaxies respectively.
In Figure~\ref{bvssfrtot}a we show the relation between $SFR_{tot}$ and $b$ for both samples.
In the range of overlap between the two samples (approximately $0.5 \leq \log SFR_{tot} \leq 1.5$) the values of $b$ are consistent, but beyond this region two different trends are seen: the NUV-selected galaxies show no trend of b with the SFR, whereas the FIR-selected galaxies show an increase of $b$ for high SFR.
This bimodal behavior of $b$ is also seen in Figure~\ref{bvssfrtot}b, where $b$ is plotted as a function of the attenuation. Again, for the NUV-selected galaxies $b$ at lower values of the attenuation, although this trend is very dispersed.
The opposite holds for the FIR-selected galaxies, with galaxies with high $b$ being the most attenuated.
Thus, the picture emerging from this study is that galaxies dominated by young stellar populations fall into two categories: those showing low SFRs and low attenuation, which naturally appear in UV surveys, and those with high SFRs and large attenuation, mainly detected in FIR surveys.
\subsection{The link between the $H$ luminosity and star formation properties of galaxies}
The baryonic mass of galaxies is a key parameter in understanding their evolution. It has been proposed as the parameter which governs the SFH, rather than the morphological type, for example (Boselli et al. 2001). In addition, it is often used to derive some properties like the dust attenuation in semi-empirical models of formation and evolution of galaxies. For this reason we devote this section to a discussion of the effects of the sample selection on the relation between the mass and the star formation related properties of galaxies. As explained in the previous section, we will use the $H$-band luminosity as a tracer of the galaxy mass.
First we show in Figure~\ref{lhsfrtot} $SFR_{tot}$ as a function of the $H$-band luminosity. Both samples show a positive relation between these two quantities, which means that more massive galaxies are also currently forming more young stars.
This result is expected since we are comparing two extensive quantities.
However, whereas the relation followed by the NUV-selected galaxies shows a small dispersion, the FIR-selected galaxies exhibit a more dispersed relation, especially at the most massive end. At high galaxian masses the range in SFR spans almost two orders of magnitude, which is not seen in the NUV-selected sample.
In Figure~\ref{hvsanuv} we show the dust attenuation as a function of the $H$-band luminosity. Two different trends are seen in this plot.
The NUV-selected galaxies show a fairly good correlation between the two quantities, with the dispersion increasing towards high $H$-band luminosities. On the contrary, the FIR-selected galaxies span an interval of almost 5~mag in dust attenuation and no correlation at all is shown with the galaxian mass. While the trend followed by the NUV-selected galaxies could be interpreted as a result of the mass -- metallicity relation reported for samples of spiral and irregular galaxies (Garnet \& Shields 1987; Zaritsky 1993) in the sense that more metallic galaxies contain more dust, there is no simple explanation for the lack of any trend shown by the FIR-selected galaxies.
Finally, we show in Figure~\ref{lh_b} the $b$ parameter as a function of the $H$-band luminosity. The NUV-selected galaxies follow the classical trend that low-luminosity galaxies have larger values of $b$ (e.g. Boselli et al. 2001). Some of the FIR-selected galaxies also follow this trend, although
about 20\% of them that show large masses and large values of $b$. As shown in Figure~\ref{hvsanuv}, these galaxies are among the most attenuated of the FIR-selected sample.
Overall, our galaxies are shifted towards higher values of $b$ with respect to the sample of galaxies of Boselli et al. (2001). These authors adopt a slightly different IMF than we do ($M_{up} = 80$~M$_{\odot}$ against 100~M$_{\odot}$) and different evolutionary synthesis codes. Nevertheless, the large shift in $b$ found between the samples can probably not be explained by these differences alone.
The correction for dust attenuation could also partially explain the shift in $b$, since Boselli et al. assume average values of 0.20~mag for Sds and later types, and 0.60~mag for types earlier than Sd. For our NUV and FIR-selected samples, the values of the dust attenuation estimated from the FIR/UV flux ratio of each of the individual galaxies show higher averaged values for the two categories of morphologies than those of Boselli et al., which would imply higher SFRs. However, this effect is diluted by the fact that for many objects in their sample, Boselli et al. estimate the SFR as the mean value of $SFR_{\rm H \alpha}$ and $SFR_{UV}$.
A further factor that could be responsible for the shift in $b$ is the different selection effect of the sample: the sample of Boselli et al. (2001) is drawn from the nearby clusters Virgo, Cancer, Coma and A1367 and from the Coma-A1367 supercluster. Although not a unique selection criterion was applied, this sample can be defined as an optically selected sample of galaxies with a normal H{\sc i} content. Thus, in their sample there is a non-negligible fraction of Sa-Sab, bulge-dominated galaxies, which tend to lower the average value of $b$ (see their Figure~2). Since our selections are based on NUV and FIR fluxes, we argue that we are surely avoiding these kind of objects. Anyway, one important point is that the NUV-selected galaxies follow the same relation between mass and $b$ as the optically selected ones (disregarding the absolute calibration of both quantities) and that a fraction of the FIR-selected galaxies do not follow this trend.
We have seen that the relation of the star-formation-related properties with the mass of galaxies strongly depends on the selection procedure of the sample: whereas for NUV-selected galaxies low-luminosity galaxies are also low mass, show low attenuation and have high values of $b$. A selection based on the FIR fluxes yields a different result: a population having high attenuation, high mass and strong star-formation activity appears. This population is absent in the NUV-selected sample. Since these galaxies present very high values of the attenuation (most of them are LIRGs and/or ULIRGs), their UV (and optical) fluxes are strongly dimmed and for this reason they are often excluded from flux limited surveys. However, even if these galaxies show such extreme properties, they do not put into question the downsizing picture for the SFH of galaxies since their contribution to the local cosmic SFR density is very low (see Takeuchi et al. 2005).
\section{The local cosmic SFR density from different estimators}
We saw in Section~4.1 that a proper estimation of the SFR is not possible with information restricted only to either NUV or FIR fluxes. However, big surveys usually provide large amounts of data only at single wavelengths, thus an estimation of the density of SFR over cosmological volumes has to be carried out under these constraints. The accuracy of the cosmic density of SFR has already been discussed by Hirashita et al. (2003) using FOCA UV data. In this section we make a similar analysis using the new GALEX data.
The usual way to estimate the average SFR density is to construct the monochromatic LF and then to weight the corresponding contribution to the SFR at a given luminosity with the probability of finding a galaxy with this luminosity.
\begin{equation}
\rho_{\lambda} = \kappa_{\lambda} \int L_{\lambda} \phi(L_{\lambda}) dL_{\lambda}
\label{sfrdensity}
\end{equation}
where $\rho_{\lambda}$ is the SFR density estimated from the flux at a given $\lambda$, $\kappa_{\lambda}$ is the conversion factor between $SFR_{\lambda}$ and $L_{\lambda}$, and $\phi(L_{\lambda})$ is the LF.
A simple calculation using Eqs.~\ref{sfrnuv_eq}, \ref{sfrdensity} and the NUV LF of Wyder et al. (2005) yields a value of $\rho_{NUV} = 0.009^{+0.007}_{-0.004}$~M$_{\odot}$~yr$^{-1}$~Mpc$^{-3}$ for the local cosmic SFR density. Unfortunately, there is not a simple relation linking the observed NUV luminosity and the attenuation (see Figure~3 in Buat et al. 2005), so we adopt the median attenuation of our NUV-selected sample, which is $A_{NUV} = 0.78$~mag.
After correcting for this median attenuation we obtain $\rho_{NUV,corr} = 0.018^{+0.013}_{-0.008}$~M$_{\odot}$~yr$^{-1}$~Mpc$^{-3}$.
Takeuchi et al. (2005) report a value of the local cosmic star formation density derived from $L_{IR}$ of $\rho_{IR} = 0.013^{+0.008}_{-0.005}$~M$_{\odot}$~yr$^{-1}$~Mpc$^{-3}$. However, we recall that this quantity does not account for the fraction of UV escaping photons or for the cirrus IR heating. For our FIR-selected sample, the median contribution of the NUV escaping luminosity to $SFR_{tot}$ is 17\%
which leads to $\rho'_{IR} = 0.016^{+0.010}_{-0.006}$~yr$^{-1}$~Mpc$^{-3}$.
After correcting for the cirrus contribution assuming $\eta = 0.32$ we obtain $\rho_{IR,corr} = 0.011^{+0.007}_{-0.004}$~M$_{\odot}$~yr$^{-1}$~Mpc$^{-3}$ which is well below $\rho_{NUV,corr}$, although still within the 1-$\sigma$ uncertainty. Although we have previously accepted that a cirrus correction of $\eta = 0.32$ is valid for galaxies with $L_{IR} \leq 10^{11}$~L$_{\odot}$ and $\eta = 0.09$ for brighter galaxies, we argue that adopting just $\eta = 0.32$ for all the galaxies is not a bad approximation for this calculation since it can be seen in Takeuchi et al. (2005) that the contribution of $L_{IR} \phi(L_{IR})$ to the total $\int L_{IR} \phi(L_{IR}) dL_{IR}$ of galaxies with $L_{IR} > 10^{11}$~L$_{\odot}$ is very low and can hardly affect our calculations.
Proceeding in an analogous way as in Section~4.1, we can add both contributions to get the total cosmic SFR, and we get $\rho_{tot} = \rho_{NUV} + (1 - \eta) \times \rho_{IR}~(= 0.009 + 0.009) = 0.018$~M$_{\odot}$~yr$^{-1}$~Mpc$^{-3}$, which is in good agreement with $\rho_{NUV,corr}$, which means that correcting $\rho_{NUV}$ with a median attenuation is a good approximation.
We also stress that the $\rho_{tot}$ is almost equally shared between the NUV and the IR contribution.
The discrepancy found with $\rho_{IR}$ alone may be due to the average corrections applied. In order to obtain a better agreement between $\rho_{NUV,corr}$, $\rho_{tot}$ and $\rho_{IR,corr}$ two points should be studied in more detail:
\begin{itemize}
\item A detailed knowledge of the cirrus component is required since this contribution is probably multivalued for a given value of $L_{IR}$. Although the morphological type seems to drive this parameter (Sauvage \& Thuan 1992), it could also be dependent on $b$ since this parameter also measures the relative weight of the young and old stellar populations.
A more detailed study of a large samples of galaxies, covering a wide range of galaxian properties, could shed light on its fractional contribution to the total cosmic $L_{IR}$ density.
\item The bivariate LF $\phi(L_{NUV},L_{IR})$ appears to be the best way to estimate the fraction of UV photons escaping from the galaxy, required to correct $\rho_{IR}$. It is also required since for large values of $L_{NUV}$, the attenuation can take on multiple values, and thus an average value, as the one used in this work, might be not the most appropriate.
\end{itemize}
Under these conditions, the cosmic SFR densities would be expressed as:
\begin{equation}
\rho_{NUV,corr} = \int\int \kappa_{NUV} \times L_{NUV} \times 10^{A_{NUV}(L_{NUV},L_{IR})/2.5} \times \phi(L_{NUV},L_{IR}) dL_{NUV} dL_{IR}
\end{equation}
and
\begin{equation}
\rho_{IR,corr} = \int\int \kappa_{IR} \times \left[1 - \eta(L_{IR},L_{NUV})\right] \times L_{IR} \times (1 + (\kappa_{NUV}/\kappa_{IR}) \times (L_{NUV}/L_{IR})) \times \phi(L_{NUV},L_{IR}) dL_{NUV} dL_{IR}
\end{equation}
or if we use the approximation of Eq.~\ref{sfrtot_eq}
\begin{equation}
\rho_{tot} = \int\int \left( \kappa_{IR} \times \left[1 - \eta(L_{IR},L_{NUV})\right] \times L_{IR} + \kappa_{NUV} \times L_{NUV} \right) \times \phi(L_{NUV},L_{IR}) dL_{NUV} dL_{IR}
\end{equation}
\section{Conclusions}
We have performed a detailed study of the star formation properties of two samples of galaxies selected on the basis of their NUV and FIR fluxes, which were found to be representative of the nearby Universe when compared to samples drawn from larger volumes.
The main conclusions of this work are:
\begin{enumerate}
\item $L_{NUV}$ and $L_{60}$ are tightly correlated for the NUV-selected galaxies. The opposite holds for FIR-selected galaxies, which span a large range of $L_{60}$ for a given value of $L_{NUV}$ and show larger values of attenuation. Intrinsically bright galaxies are more luminous at FIR than at NUV wavelengths, including the UV luminous galaxies (UVLGs), and they show moderate to high attenuation.
\item The SFR deduced from the NUV fluxes, corrected for the dust attenuation ($SFR_{NUV}$), are not found to be consistent with those calculated using the total dust emission ($SFR_{dust}$). Whereas $SFR_{NUV}$ is larger than $SFR_{dust}$ for galaxies with low attenuation ($A_{NUV} \lesssim 1$~mag) the inverse is found for bright, but highly extinguished galaxies, mostly selected in IR: $SFR_{NUV}$ is likely to underestimate the actual SFR in these galaxies by a factor $\sim 2$.
A combined estimator based on UV and IR luminosities with a cirrus correction depending on the IR luminosity seems to be the best proxy over the whole range of values of SFR. As a practical recipe we found that $SFR_{tot}$ and $SFR_{NUV}$ yield similar results for $SFR_{tot} \lesssim 15$~M$_{\odot}$~yr$^{-1}$, whereas $SFR_{tot}$ and $SFR_{dust}$ are almost equivalent for $SFR_{tot} \gtrsim 15$~M$_{\odot}$~yr$^{-1}$.
\item NUV-selected galaxies follow a trend whereby low-mass galaxies show lower SFRs, low attenuation and higher values of $b$, indicating the existence of a dominant young stellar population. On the contrary,
about 20\% of
the FIR-selected sample shows high attenuation, high SFRs and also large values of $b$, most of them being LIRGs and/or ULIRGs. In spite of their discordant properties, these galaxies are not sufficiently abundant in the local Universe to question the downsizing picture for the SFH seen at $z = 0$ from optical surveys.
\item The cosmic SFR densities of the local Universe, estimated from the NUV and IR luminosities, are consistent to within 1-$\sigma$, although the difference between the two values is large, when average corrections for the attenuation, UV escaping photons and IR cirrus component are applied. The sum of the individual contributions is quite consistent with the value obtained from the NUV luminosities corrected for average attenuation.
A better knowledge of the cirrus contribution to $L_{IR}$ and of the bivariate LF is required in order to better understand the large differences found between the monochromatic estimators of the local SFR density.
\end{enumerate}
\acknowledgments
GALEX is a NASA Small Explorer, launched in 2003 April. We gratefully acknowledge NASA's support for construction, operation, and science analysis for the GALEX mission, developed in cooperation with the Centre National d'Etudes Spatiales of France and the Korean Ministry of Science and Technology.
This publication makes use of data products from the Two Micron All Sky Survey,
which is a joint project of the University of Massachusetts and the Infrared
Processing and Analysis Center/California Institute of Technology,
funded by the National Aeronautics and Space Administration and the National
Science Foundation. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
The Lyon Extragalactic Database (LEDA) is available at http://leda.univ-lyon1.fr/.
TTT has been financially supported by the Japan Society for the Promotion of
Science.
\appendix
\section{Relation of the Mean Redshift of Galaxies and the LF:
Representativity of a Survey Depth\label{depth}}
We show the strong dependence of the mean redshift on the shape of
the LF.
This means that the mean distance of galaxies does not represent the
depth of a survey, but rather reflects an intrinsic property of the sample.
Since we treat a local sample of galaxies, we first approximate their
distance by the classical Hubble's law:
\begin{eqnarray}
cz \simeq \frac{r}{H_0} \;,
\end{eqnarray}
where $c$ is the velocity of light, and $r$ represents the distance.
We define ${\cal N}$ to be surface density of galaxies on the sky, and
denote the LF as an explicit function of the characteristic luminosity, $L_*$
as $\phi(L/L_*)$.
Then, ${\cal N}$ is written as (Peebles 1993):
\begin{eqnarray}
{\cal N} = \int \int \phi \left[ \frac{L(r)}{L_*} \right]
\frac{dL(r)}{L_*} r^2 dr \;.
\end{eqnarray}
Using the detected flux density $S$, this can be expressed as
\begin{eqnarray}
{\cal N} &=& \int \int \phi \left[ \frac{4\pi}{L_*}
\left( \frac{c}{H_0}\right)^2 z^2 S \right]
\frac{4\pi}{L_*} \left( \frac{c}{H_0}\right)^2 z^2 dS
\left( \frac{c}{H_0}\right)^3 z^2 dz \nonumber \\
&=& \int \int \phi \left[ \frac{4\pi}{L_*}
\left( \frac{c}{H_0}\right)^2 z^2 S \right]
\frac{4\pi}{L_*} \left( \frac{c}{H_0}\right)^5 z^4 dS dz \;.
\end{eqnarray}
We observe the distribution function of the sources
with a {\sl fixed} flux density $S$ as
\begin{eqnarray}\label{eq:dif_nc}
\frac{d^2{\cal N}}{dzdS} = \phi \left( \alpha z^2 S \right)
\alpha \left( \frac{c}{H_0}\right)^3 z^4 \;,
\end{eqnarray}
where
\begin{eqnarray}
\alpha \equiv \frac{4\pi}{L_*}\left(\frac{c}{H_0}\right)^2\;,
\end{eqnarray}
The mean redshift of a flux-limited survey (limiting flux density $S$),
$\langle z \rangle_{>S}$, is then
defined as\footnote{Note that this is a different quantity defined by
Equation~(5.134) of Peebles (1993).}
\begin{eqnarray}\label{eq:meanz}
\langle z \rangle_{>S} \equiv \frac{\displaystyle \int_S^\infty \int_0^\infty
\frac{d^2{\cal N}}{dzdS'} zdzdS'}{\displaystyle \int_S^\infty \int_0^\infty
\frac{d^2{\cal N}}{dzdS'} dzdS'} \;.
\end{eqnarray}
The numerator is obtained as
\begin{eqnarray}\label{eq:numer}
\int_S^\infty \int \frac{d^2{\cal N}}{dzdS'} zdzdS' &=&
\frac{1}{2\alpha^2} \left(\frac{c}{H_0}\right)^3
\left[ \int_0^{\infty} \phi (x) x^2 dx\right]
\int_S^\infty {S'}^{-3} dS'\nonumber \\
&=& \frac{1}{2\alpha^2} \left(\frac{c}{H_0}\right)^3
\left[ \int_0^{\infty} \phi (x) x^2 dx\right] \frac{S^{-2}}{2}\;.
\end{eqnarray}
Here, $x \equiv \alpha z^2 S'$, a luminosity normalized by the characteristic
luminosity of the LF, $L_*$ and expressed in terms of flux density $S'$.
Since the luminosity must be positive, the lower bound on the
integration with respect to $x$ is 0, and upper bound is
large, effectively taken to be $+\infty$.
Similarly the denominator becomes
\begin{eqnarray}\label{eq:denom}
\int_S^\infty \int_0^\infty \frac{d^2{\cal N}}{dzdS'} dzdS' &=&
\frac{1}{2\alpha^{3/2}} \left(\frac{c}{H_0}\right)^3
\left[ \int_0^\infty \phi (x) x^{3/2} dx\right]
\int_S^\infty {S'}^{-5/2} dS' \nonumber \\
&=& \frac{1}{2\alpha^2} \left(\frac{c}{H_0}\right)^3
\left[ \int_0^\infty \phi (x) x^2 dx\right] \frac{2S^{-3/2}}{3}\;.
\end{eqnarray}
Both the numerator and the denominator of this part is the moment of the LF.
This means that this is a function of its shape.
Combining Equations~(\ref{eq:meanz}), (\ref{eq:numer}), and (\ref{eq:denom}),
we have
\begin{eqnarray}\label{eq:meanz2}
\langle z \rangle_{>S} &=& 3\alpha^{-1/2}
\frac{\displaystyle \int_0^\infty \phi (x) x^2 dx}{
\displaystyle \int_0^\infty \phi (x) x^{3/2} dx} S^{-1/2} \nonumber \\
&=& 3 \left(\frac{L_*}{4\pi}\right)^{1/2}\left(\frac{H_0}{c}\right)
\frac{\displaystyle \int_0^\infty \phi (x) x^2 dx}{
\displaystyle \int_0^\infty \phi (x) x^{3/2} dx} S^{-1/2} \;.
\end{eqnarray}
Let us carefully examine Equation~(\ref{eq:meanz2}).
First, the dependence of the mean redshift on the limiting flux density $S$
is a power of $-1/2$.
Also, it has the same order of dependence on $L_*$.
In contrast, the integral part of Equation~(\ref{eq:meanz2}) has an important
meaning.
Since this part depends on the second-order moment,
the tail of the LF affects the value very strongly.
As we mentioned, we integrate over the (normalized) luminosity up to a
certain very large value, and this part is a ratio between the moments of
order $3/2$ and 2.
Hence the contribution from a large value of $x$ controls the value.
Consequently, the mean redshift $\langle z \rangle_{>S}$ is very
sensitive to the LF shape.
This aspect is clearly seen in the comparison of the expected redshift
distributions in NUV and $60\mu$m calculated from the LFs and the limiting
flux density or magnitude, because the shapes of the LFs at these wavelengths
are very different (Buat \& Burgarella 1998; Takeuchi et al. 2005).
We show the comparison in Figures~\ref{histo_velo_all}b and ~\ref{histo_velo_all}c.
In these figures we fix the limiting flux density at $60\mu$m to
be 0.6~Jy, while we change the limiting magnitude from $AB_{NUV}=16.0$~mag
(the actual value we adopt in this work) to 18.0~mag.
The medians for the two wavelengths are very different when we adopt a limiting value of $AB_{NUV}=16.0$~mag. Only when we adopt a limiting value of $AB_{NUV}=18.0$~mag, which corresponds to a very sensitive survey, the median values approach for the NUV and FIR-selected samples.
\section{Estimating the birthrate parameter \label{apendice_b}}
Boselli et al. (2001) give a detailed recipe to estimate $\left< SFR \right>$, based only on observable quantities and/or adopted values for the parameters (see Gavazzi et al. 1996). Here we follow their prescriptions, using the $H$-band luminosity to estimate the past averaged SFR and the same parameters as Boselli et al. (2001):
\begin{equation}
b = \frac{SFR \times t_{0} \times (1 - R)}{L_{H} \times (M_{tot}/L_{H}) \times DM_{cont}}
\end{equation}
where $SFR$ in this work is averaged over $10^{8}$~yr, $t_{0}$ is the age of the disk (assumed to be equal to 13~Gyr), $R$ is the fraction of gas re-injected by stars through stellar winds into the interstellar medium during their lifetime (taken to be equal to 0.3 for a Salpeter IMF), $L_{H}$ is the $H$-band luminosity estimated as $\log L_{H} = 11.36 - 0.5 \times H + 2 \times \log D$ (in solar units) where $D$ is the distance to the source (in Mpc), $M_{tot}$ is the dynamical mass at the B-band 25~mag~arcsec$^{-2}$ isophotal radius, $M_{tot}/L_{H}$ is taken to be constant and equal to 4.6 (in solar units) and $DM_{cont}$ is the dark matter contribution to the $M_{tot}/L_{H}$ ratio at the optical radius, assumed to be equal to 0.5.
\clearpage
\newpage
\begin{table}
\footnotesize
\begin{center}
\caption{Typical uncertainties of the NUV and FUV magnitudes as a function of the magnitude.\label{error}}
\begin{tabular}{ccc}
\tableline\tableline
AB Magnitude & $\sigma$(NUV) & $\sigma$(FUV) \\
interval & (mag) & (mag) \\
\tableline
$\leq 16$ & 0.01 & 0.02 \\
$16 ~ ... ~ 18$ & 0.03 & 0.05 \\
$18 ~ ... ~ 20$ & 0.09 & 0.15 \\
$20 ~ ... ~ 22$ & 0.26 & 0.40 \\
\tableline
\end{tabular}
\end{center}
\end{table}
\begin{table}
\footnotesize
\begin{center}
\caption{Basic properties of the sample galaxies: (1) Name; (2) Flag indicating membership to the NUV-selected subsample; (3) Flag indicating membership to the FIR-selected subsample; (4) R.A. (J2000); (5) Dec. (J2000); (6) Heliocentric velocity; (7) Distance derived from the velocity corrected for the Local Group infall onto Virgo and $H_{0} = 70~km~s^{-1}~Mpc^{-1}$; (8) Morphological type from NED; (9) IRAS identificator: F for FSC origin; Q,O,R for PSCz origin; SCANPI for absence in both catalogs.\label{pro}}
\begin{tabular}{lcccccrcl}
\tableline\tableline
Name & UVsel & FIRsel & R.A. & Dec. & vel. & Dist & Type & IRAS Id.\\
& & & \multicolumn{2}{c}{(J2000.0)} & (km~sec$^{-1}$) & (Mpc) & & \\
\tableline
MRK~544 & Y & N & 0 4 48.70 & $-$ 1 29 54.60 & 7110 & 101.39 & S? & F00022-0146 \\
NGC~10 & Y & Y & 0 8 34.56 & $-$33 51 27.25 & 6811 & 94.62 & Sbc & Q00060-3408 \\
NGC~35 & Y & Y & 0 11 10.46 & $-$12 1 14.74 & 5964 & 83.95 & Sb & Q00086-1217 \\
NGC~47 & Y & Y & 0 14 30.42 & $-$ 7 10 6.28 & 5700 & 80.54 & Sbc & Q00119-0726 \\
NGC~101 & Y & N & 0 23 54.72 & $-$32 32 9.06 & 3383 & 45.92 & Sc & F00214-3248 \\
\tableline
\end{tabular}
\end{center}
Note: The distances to UGCA~438 and UGC~12613 were taken from Karachentsev et al. (2002) and Hoessel et al. (1990) respectively.
\end{table}
\begin{table}
\footnotesize
\begin{center}
\caption{Photometric properties of the sample galaxies: (1) Optical identifier; (2) NUV magnitude corrected for Galactic extinction; (3) FUV magnitude corrected for Galactic extinction; (4) Flux density at 60$\mu$m in Jy; (5) Flux density at 100$\mu$m in Jy; (6) $H$ magnitude from 2MASS Extended Source Caztalog. For galaxies with no detection at 2MASS we adopt the limiting value of $H = 13.9$~mag (3~mJy) as given by Jarrett et al. (2000); (7) NUV attenuation in mag, derived as indicated in Buat et al. (2005); (8) FUV attenuation derived as indicated in Buat et al. (2005).
For some galaxies Eqs.~\ref{attenuv_eq} and \ref{attefuv_eq} gave negative values of the NUV and FUV attenuations which is senseless. In fact, this is an artifact of the polynomial fitting used to derive a functional form for the attenuation in Buat et al. (2005). Throughout this paper they will be considered as zero.\label{photo}}
\begin{tabular}{lccccccc}
\tableline\tableline
Name & $AB_{NUV}$ & $AB_{FUV}$ & $f_{60}$ & $f_{100}$ & $H$ & $A_{NUV}$ & $A_{FUV}$ \\
& \multicolumn{2}{c}{(mag)} & \multicolumn{2}{c}{(Jy)} & (mag) & \multicolumn{2}{c}{(mag)} \\
\tableline
MRK~544 & 15.75 & 15.97 & 0.49 & 1.11 & 12.50 & 0.71 & 0.96 \\
NGC~10 & 15.48 & 15.98 & 0.66 & 2.97 & 9.50 & 1.59 & 2.10 \\
NGC~35 & 15.55 & 15.89 & 1.31 & 2.34 & 11.91 & 1.29 & 1.67 \\
NGC~47 & 15.46 & 15.97 & 0.85 & 2.57 & 10.25 & 1.02 & 1.49 \\
NGC~101 & 14.78 & 14.98 & 0.55 & 1.75 & 10.74 & 0.50 & 0.72 \\
\tableline
\end{tabular}
\end{center}
\end{table}
\begin{table}
\footnotesize
\begin{center}
\caption{Star formation related properties of the sample galaxies: (1) Identifier of the galaxy; (2) $SFR_{NUV}$ from Eq.~\ref{sfrnuv_eq}; (3) $SFR_{FUV}$ from Eq.~\ref{sfrfuv_eq}; (4) $SFR_{dust}$ from Eq.~\ref{sfrfir_eq}; (5) $SFR_{tot}(NUV)$ from Eq.~\ref{sfrtot_eq}; (6) $SFR_{tot}(FUV)$ from Eq.~\ref{sfrtot_eq} but using $SFR^{0}_{FUV}$ instead of $SFR^{0}_{NUV}$; (7) $\left< SFR \right>$ averaged over the galaxy's lifetime, estimated as indicated in Appendix~\ref{apendice_b}.\label{sfr_tab}}
\begin{tabular}{lcccccc}
\tableline\tableline
Name & $\log SFR_{NUV}$ & $\log SFR_{FUV}$ & $\log SFR_{dust}$ & $\log SFR_{tot}(NUV)$ & $\log SFR_{tot}(FUV)$ & $\log \left< SFR \right>$ \\
& \multicolumn{6}{c}{($M_{\odot}$~yr$^{-1}$)} \\
\tableline
MRK~544 & 0.83 & 0.85 & 0.60 & 0.80 & 0.75 & 0.81 \\
NGC~10 & 1.23 & 1.24 & 1.07 & 1.09 & 1.02 & 1.95 \\
NGC~35 & 0.98 & 1.00 & 0.78 & 0.84 & 0.79 & 0.88 \\
NGC~47 & 0.87 & 0.86 & 0.78 & 0.85 & 0.77 & 1.51 \\
NGC~101 & 0.45 & 0.46 & 0.13 & 0.43 & 0.38 & 0.83 \\
\tableline
\end{tabular}
\end{center}
\end{table}
\clearpage
\clearpage
\clearpage
\clearpage
\clearpage
\clearpage
\clearpage
\clearpage
\clearpage
\clearpage
\clearpage
\clearpage
\clearpage
|
Title:
Bubbles in Planetary Nebulae and Clusters of Galaxies: Jet Bending |
Abstract: We study the bending of jets in binary stellar systems. A compact companion
accretes mass from the slow wind of the mass-losing primary star, forms an
accretion disk, and blows two opposite jets. These fast jets are bent by the
slow wind. Disregarding the orbital motion, we find the dependence of the
bending angle on the properties of the slow wind and the jets. Bending of jets
is observed in planetary nebulae which are thought to be the descendants of
interacting binary stars. For example, in some of these planetary nebulae the
two bubbles (lobes) which are inflated by the two opposite jets, are displaced
to the same side of the symmetry axis of the nebula. Similar displacements are
observed in bubble pairs in the center of some clusters and groups of galaxies.
We compare the bending of jets in binary stellar systems with that in clusters
of galaxies.
| https://export.arxiv.org/pdf/astro-ph/0601032 |
\title{BUBBLES IN PLANETARY NEBULAE AND CLUSTERS OF GALAXIES:
JET BENDING}
\author{Noam Soker and Gili Bisker}
\affil{Department of Physics, Technion$-$Israel
Institute of Technology, Haifa 32000 Israel;
soker@physics.technion.ac.il.}
\keywords{stars: mass loss --- binaries: close --- planetary nebulae: general --—
intergalactic medium --— ISM: jets and outflows --- galaxies: clusters: general}
\section{INTRODUCTION}
The nebular gas in planetary nebulae (PNs) originates in the envelope of asymptotic
giant branch (AGB) stars that are the descendants of intermediate mass
stars (initial masses $\sim 1-8 M_\odot$).
Such stars rotate very slowly, and their mass loss is expected to be spherical.
Indeed, AGB stellar winds usually consist of a more or less spherically
symmetric outflow at rates of $\sim 10^{-7}-10^{-5} M_\odot \yr ^{-1}$.
Most PNs, though, possess a global axisymmetrical structure rather than a spherical
structure in their inner region, indicating a non-spherical shaping process.
Among the several PN shaping models (Balick \& Frank 2002), one
of the most successful is the jet-shaping model.
If the jets are not well collimated they are termed
collimated fast wind (CFW).
The presence of jets in PNs was deduced from observations
more than 20 years ago (e.g., Feibelman 1985).
Gieseking et al. (1985) found collimated outflow in the PN NGC 2392,
and noted the similarity of these jets with that of young stellar objects,
and speculated that such outflows exist in many similar PNs.
On the theoretical side, Morris (1987) suggested that two jets
blown by an accreting companion (the secondary star) can form bipolar nebulae.
This model is strongly supported by the similarity of bipolar PNs to
many bipolar symbiotic nebulae which are known to be shaped by jets
(e.g., Schwarz et al. 1989; Corradi \& Schwarz 1995).
Soker (1990) proposed that the two fast low-ionization emission blobs
(FLIERs or {\it ansae}) along the symmetry axis of many elliptical PNs are formed
by jets blown during the last phase of the AGB or the post-AGB phase of
the PN progenitor.
The high quality HST images led Sahai \& Trauger (1998) to suggest that
in many PNs the non-spherical structures are formed solely by jets.
Projecting from similar astronomical objects, the formation of massive jets,
to distinguish from magnetized low density pulsar jets, require
the presence of accretion disks.
The only source of angular momentum sufficient to form accretion disks in evolved
stars is the orbital angular momentum of a stellar (or in some cases substellar)
companion.
The disk can be formed around the progenitor during the late post-AGB phase,
when it is already small (Soker \& Livio 1994), or, more likely, around a
stellar companion accreting mass, forming an accretion disk, and blowing two jets.
The past seven years have seen further consolidation of the bipolar jet-shaping
model in binary systems, addressed both in observations
(e.g., Parthasarathy et al. 2000; Sahai \& Nyman 2000; Miranda et al. 2001a,b;
Corradi et al. 2001; Guerrero et al. 2001; Vinkovic et al. 2004; Huggins et al. 2004;
Pena et al.\ 2004; Balick \& Hajian 2004; Arrieta et al. 2005;
Oppenheimer et al. 2005; Sahai et al. 2005)
and in theory (e.g., Soker 2002, 2005; Lee \& Sahai 2003, 2004; Livio \& Soker 2001;
Garcia-Arredondo\& Frank 2004; Velazquez et al. 2004; Riera et al. 2005).
Many of the PNs in the observations listed above posses point symmetric morphology,
i.e., several symmetry axes rotate with respect to each other through a common origin,
indicating precessing jets.
The most likely explanation for precession is an accretion disk in the presence of
a companion.
Soker \& Rappaport (2000) further discussed the jet shaping process and have shown
that the statistical distribution of bipolar PNs can be accounted for in
the binary model.
Further support for the formation of jets in binary systems comes from X-ray
observations hinting at jets in a PN (Kastner et al. 2003) similar to X-ray jets in symbiotic systems
(Kellogg et al. 2001; Galloway \& Sokoloski 2004).
Garcia-Arredondo \& Frank (2004) were the first to conduct 3D numerical simulations
of the interaction of jets launched by a secondary star with the slow primary wind.
Their high quality results strengthen the general stellar-binary jets model, and in
particular the conjecture (Soker \& Rappaport 2000) that a narrow waist can be formed
by jets.
It should be stressed that not all PNs are shaped by jets, but bubble pairs are
formed by jets.
X-ray images of active galactic nuclei in clusters of galaxies indeed show
that double-jets, observed in the radio band, can form a bubble pair with
a narrow waist between them, similar to narrow waists in PNs with no need
for enhanced equatorial mass loss rate,
although enhanced equatorial mass loss rate might occur in many PNs.
The subject of the similarity between some morphological structures in
clusters of galaxies, as revealed via X-ray observations, and in PNs,
as revealed in the visible band, was studied in a series of four papers.
\newline
{\it Paper 1} ( Soker 2003b).
In that paper ( see also Soker 2003a, and section 5 in Soker 2004c)
the similarity in morphological structures was discussed
\footnote{The similar morphologies are compared in the appendix
of the astro-ph version of the present paper.}.
This similarity is not trivial. Two opposite jets are observed in many young stellar
objects (YSOs), however, bubbles pairs similar to those in PNs and in
clusters of galaxies are not usually observed around YSOs.
\newline
{\it Paper-2 } (Soker 2004a).
It was found that to inflate fat, more or less spherical, bubbles the
opening angle of the jets should be large; the half opening
angle measured from the symmetry axis of each jet should typically be
$\alpha \gtrsim 40 ^\circ$, or the jets might precess.
\newline
{\it Paper-3} (Soker 2004b).
Paper 3 studies the stability of off-center low-density fat bubbles
in clusters of galaxies and in PNs to the Rayleigh-Taylor instability.
\newline
{\it Paper 4} (Pizzolato \& Soker 2005).
Pizzolato \& Soker examined the point symmetric structure
of the bubble pair in the cluster MS 0735.6+7421 (McNamara et al. 2005) and
compared it to the point symmetric structure of PNs.
Point symmetric PNe are thought to be shaped by stellar binary interactions;
namely, the presence of a companion to the PN’s progenitor star is required.
Pizzolato \& Soker (2005) suggested that similar point-symmetric structures
in the X-ray deficient cavities of galaxy clusters might be associated with
the presence of massive binary black holes.
In this paper, the fifth in the series, we examine the bending of the
two jets, and the subsequent bending of the two bubbles inflated by the jets,
to the same side of their original symmetry axis (jets' axis).
Such displacement relative to the symmetry axis of bubbles touching the center
is seen, for example, in the Perseus cluster of galaxies (Fabian et al. 2000;
Dunn et al. 2006), and in the PN NGC 3587 (Guerrero et al. 2003).
Dunn et al. (2006) discuss the departure of the two bubbles from their alignment
along a cluster center and explain this departure by the two opposite bubbles
detaching from the precessing jets at different times.
We consider this displacement to result from the ram pressure of
the intra cluster medium (ICM).
Displacement of bubbles at a distance from the center are seen
in the PN NGC 6886 (Terzian \& Hajian 2000), and the group of galaxies
HCG 62 (Vrtilek et al. 2002).
We focus on PNs and related binary stellar objects, e.g.,
the massive binary stellar system $\eta$ Carinae ($\S 2.1$).
The departure of PNs and related binary systems from axisymmetry has
been previously studied (Soker \& Hadar 2002 and references therein).
Our goal here is to derive a simple expression for the bending of jets
in binary stellar systems ($\S 2.2$).
This expression is not a substitute for future numerical simulations.
The results for typical binary systems ($\S 2.2$) can account for some
morphological structures in PNs and related systems.
Readers interested in only using the relations and the results, can skip
$\S 2.1$ and go directly to $\S 2.2$.
In $\S 3$ we compare the situation with jet bending in cooling
flow clusters, and $\S 4$ is a summary.
\section{BENDING IN A BINARY STELLAR SYSTEM}
\subsection{Assumptions and Equations}
When a compact secondary star accretes from the AGB (or post-AGB) stellar wind
only part of the AGB wind is accreted, and the rest expands outward and forms the
medium that the jets expand into.
In addition, when the jet is still close to the binary system, the AGB wind hits
the jet on its side, causing the jet to deflect (Soker \& Rappaport 2000).
Like precession, this can have large effects on the descendant PN morphology.
However, while precession leads to point-symmetric nebula, the deflection of the two
oppositely ejected jets is to the same side, leading the two opposite lobes
to be bent to the same side; this is the {\it bent} departure from axisymmetry
according to the classification of Soker \& Hadar (2002).
The bending interaction can clear the way to radiation, possibly ionizing radiation,
from the central binary system to more strongly illuminate
the same side in both lobes (bubbles).
Due to the orbital motion, this structure forms a revolving light source.
Livio \& Soker (2001) suggested such a revolving ionizing source model to explain the
positional shift of the bright knots in the inner nebular lobes of the M2-9 nebula
(Doyle et al. 2000).
Soker \& Rappaport (2000) derived a simple expression for the bending angle
of a narrow jet.
In this section we relax some of the assumptions made by Soker \& Rappaport and
derive a more accurate expression for the bending angle, while still keeping the
expression simple.
The goal is to derive a simple approximate relation that will give the jet's bending
angle upon specifying the jet's parameters and slow wind parameters.
The bending interaction is drawn schematically by Soker \& Rappaport (2000) and
Livio \& Soker (2001), and it is shown in Figure \ref{draw1};
3D images of numerical simulations are presented by Garcia-Arredondo \& Frank (2004).
The slow wind has a spherical mass loss rate of $\dot M_s$ and a relative
speed to the primary star of $v_s$.
A small fraction of this wind is accreted by the secondary star, forms an accretion
disk that blows two jets, with a mass loss rate of $\dot M_j$ into the
two jets together, and with a speed of $v_j$ perpendicular to the equatorial plane
relative to the secondary star.
Although the jets can have a large opening angle and, in many cases, are likely to
have a large opening angle, in the present study we assume a narrow jet with
a half opening angle $\alpha \ll 1$, and also assume that the jet is bent
as one entity (sound crossing time across the jet is very short).
The density per unit length along the jet axis is
\begin{equation}
m_j = \frac {\dot M_j}{2 v_j}
\label{mj}
\end{equation}
(recall that $\dot M_j$ is the mass loss rate into the two jets together).
We move to a frame of reference attached to the secondary star in its orbital
motion, with a velocity relative to the primary of
${\bf {v}}_{\rm orb}=v_r \hat r + v_\theta \hat \theta$, where
$v_\theta \simeq r \dot \theta$, $\theta$ is the relative angle of the two stars
in the equatorial plane, and $r$ is the projected distance from the primary to the jet
on the equatorial plane; $v_r<0$ when the two stars approach each other.
We consider a narrow jet's segment at a height $z$ above (or below) the equatorial plane.
The slow wind segment that hit this segment left the primary at an angle $\beta$
to the equatorial plane (see fig. \ref{draw1})
\begin{equation}
\sin \beta = \frac{z} {(z^2+r^2)^{1/2}}.
\label{stheta}
\end{equation}
The slow wind that hits the jet at a high $z$ above the equatorial plane has
a relative velocity to the jet of
\begin{equation}
v_{rel}= [v_\theta^2+(v_s \cos \beta-v_r)^2+(v_s \sin \beta)^2]^{1/2}.
\label{vrel}
\end{equation}
We consider a fast jet $v_j \gg v_s$ that initially expands perpendicularly
to the orbital plane, but is then bent by the ram pressure of the slow wind
and acquires a velocity parallel to the equatorial plane $v_p$.
The ram pressure exerted by the slow wind on the jet in a direction
parallel to the equatorial plane is
\begin{equation}
P_{ram}=
\rho \left\{ [v_\theta^2+(v_s \cos \beta-v_r)^2]^{1/2}-v_p \right\}^2,
\label{pram1}
\end{equation}
where the density of the slow wind
\begin{equation}
\rho = \frac {\dot M_s}{4 \pi v_s (r^2+z^2)}.
\label{rho}
\end{equation}
The equation for accelerating the jet in a direction parallel to the equatorial
plane (perpendicular to the initial direction of the jet), under the assumption
of a fast jet, $v_p \ll v_j$, reads
\begin{equation}
\frac{dv_p}{dt}= \frac{P_{ram} 2 z \tan \alpha}{m_j}
\label{dvpdt}
\end{equation}
Under the assumption of a fast jet, $z=v_j t$ and $dt=dz/v_j$.
We also scale velocities by the slow wind speed
\begin{equation}
u_r \equiv \frac{v_r}{v_s}; \quad u_\theta \equiv \frac{v_\theta}{v_s}; \quad
u_p \equiv \frac{v_p}{v_s}; \quad u_j \equiv \frac{v_j}{v_s}.
\label{defvs}
\end{equation}
The equation of motion reads
\begin{equation}
\frac{du_p}{dz}= A \left\{
\left[ u_\theta^2+\left( \frac{r}{\sqrt{r^2+z^2}}-u_r \right)^2 \right]^{1/2}
-u_p \right\}^2 \frac{z}{r^2+z^2},
\label{dvpdz}
\end{equation}
where
\begin{equation}
A= \frac{\tan \alpha}{\pi} \frac{\dot M_s}{\dot M_j}
\label{adef}
\end{equation}
The meaning of the different terms in equation (\ref{dvpdz}) are as follows.
(1) The factor $A$ is proportional to the ratio of colliding masses.
Bending efficiency increases with $A$.
(2) The terms $u_\theta$ and $u_r$ result from the orbital motion of the
secondary star, which blows the jets, relative to the slow wind.
(3) The term $r/(r^2+z^2)^{1/2}$ results from the ram pressure of the
slow wind on the jet. The slow wind moves at a velocity $v_s$; but since velocity was
scaled by $v_s$, a factor of unity multiplies this term.
(4) The numerator in the last term is due to the increase in the jet cross section,
and it increases the bending efficiency as the jet expands.
(5) The denominator in the last term is the decrease in the slow wind density,
and it makes bending less efficient as distance from the primary star grows.
\subsection{Results for Impulsive Jets}
We consider a case in which the jets are blown by a secondary star
that is less massive than the primary star.
companion. The slow wind is blown by the primary star, residing close to the
center of mass of the binary system.
The formulation derived above is applicable to continuously blown jets, or
jets blown impulsively.
However, for the bubbles in PNs, or similar object, to be significantly
displaced by the mechanism discussed in $\S 2.1$, the jet should be
blown during a short time compare to the orbital period.
(Significant displacement from axisymmetry for continuously blown jets can
be acquired if the binary system has a large eccentricity;
see references in Soker \& Hadar 2002.).
In many PNs, the jets' ejection (PN jets refer to the jets blown
by the PN progenitor) can take place over a short time period
(e.g., Meaburn 2006), which we take to be shorter than the orbital period.
For example, the orbital period can be 5-50 years (orbital separation of
$\sim 3-20 \AU$), and the ejection event a few years, as in
symbiotic-nova outbursts on an accreting WD companion.
The mass accretion rate from the primary stellar wind, $\dot M_2$, by a companion
of mass $M_2$ at an orbital separation $r_0$ is
\begin{equation}
\frac{\dot M_2}{\dot M_s} \simeq
0.05
\left( \frac {M_2}{0.6 M_\odot} \right)^{2}
\left( \frac {v_{rel}}{15 \km \s^{-1}} \right)^{-4}
\left( \frac {r_0}{10 \AU} \right)^{-2}.
\end{equation}
If in impulsive jets' ejection $\dot M_j \sim 0.2 \dot M_2$, then
for the above mass accretion rate $A \simeq 5.6 \alpha/10^\circ$.
In short eruption events, like disk instability or nova-like outbursts on an
accreting WD, it might be that $\dot M_j > 0.2 \dot M_2$, and $r_0$ span a
range of $\sim 1-30 \AU$.
Therefore, we consider $A$ to be be in the range $A \sim 0.1-100$.
The jet is bent, according to equation (\ref{dvpdz}), and $u_p$, the velocity component
parallel to the equatorial plane and perpendicular to initial velocity of the jet
reaches an asymptotic velocity of
\begin{equation}
u_{pa} = \left[ u_\theta^2+\left( \frac{r}{\sqrt{r^2+z^2}}-u_r \right)^2 \right]^{1/2}.
\label{upa}
\end{equation}
where $u_{pa}$ is in unit of the slow wind speed $v_s$.
The asymptotic (final) velocity $u_p$ due to the orbital tangential velocity $u_\theta$
does not depend on the factor $A$ or the jet speed $v_j$ (or $u_j=v_j/v_s$).
This is approximately true for the radial orbital component $u_r$ as well,
meaning that the initial jet velocity component along the secondary stellar orbital
motion is quite efficiently reduced to zero.
The departure from axisymmetry due to the orbital motion of the star blowing
the jet will be small.
Therefore, in imposing a noticeable large-scale departure from axisymmetry,
where the two jets are bent to the same side, the bending due to the
slow wind outflow from the primary star must be considered.
This bending is less efficient because the slow wind velocity is
not perpendicular to the jet velocity after the jet leaves the equatorial
plane, as seen by the decreasing of the term $r/(r^2+z^2)^{1/2}$.
Ignoring the orbital motion, equation (\ref{dvpdz}) reads
\begin{equation}
\frac{du_p}{dz}= A
\left[ \frac{r}{(r^2+z^2)^{1/2}} -u_p \right]^2 \frac{z}{r^2+z^2},
\label{dup}
\end{equation}
This equation is supplemented by another equation for the jet propagation
along the direction perpendicular to the equatorial plane.
For a fast jet, $v_j \gg v_s$, this reads,
\begin{equation}
\frac{dr}{dz}= \frac{u_p}{u_j}.
\label{dr}
\end{equation}
Figure \ref{upf1} presents the numerical solutions of the last two coupled equations
for initial jet's speed $u_j\equiv v_j/v_s=6$ and for three values of $A$
as function of the distance from the equatorial
plane $z$ in units of the orbital separation $r_0$.
The velocity $u_p$ is plotted in the upper panel, in the middle panel
the projection of the jet distance on the equatorial plane $r$ is drawn
(in units of $r_0$), while the lower panel presents the acceleration $du_p/dz$.
In Figure \ref{upaf}, we show the asymptotic velocity $u_{pa}$ as a function of $A$
for $u_j=6$ (the thick line).
Changing the initial jet's speed $u_j$ does not change the solution
for $u_p$, while the quantity $r-r_0$ is proportional to $u_j^{-1}$, because the
bending angle is given by $\tan \phi =u_p/u_j$, so that for faster jets
the bending angle decreases; the dependence is $\phi \propto v_j^{-1}$.
This can be understood as follows. As the jet speed $v_j$ increases,
the time of accelerating the jet by the slow wind's ram pressure along
a distance $dz$ decreases as $v_j^{-1}$;
however, the density in the jet decreases as $v_j^{-1}$ as well.
Hence the total change in $v_p$ (or $u_p$) along a distance $dz$ does
not depend on $v_j$ under our assumptions, in particular the assumption
$v_j$ (or $u_j$) is constant and is not influenced by the
interaction with the slow wind.
As seen from Figure \ref{upf1}, for a fast jet (and for typical values used here)
$u_p$ almost reaches its terminal speed while $r \simeq r_0$, where $r_0$
is the initial orbital separation.
Note that most of the bending occurs for $r$ not much larger than $r_0$.
For $r=r_0$ equation (\ref{dup}) reads then
\begin{equation}
\frac{du_p}{dz}= A
\left[ \frac{r_0}{(r_0^2+z^2)^{1/2}} -u_p \right]^2 \frac{z}{r_0^2+z^2},
\label{dup0}
\end{equation}
The solution near the origin, when $u_p \ll 1$ is
\begin{equation}
u_{p0} (r \sim r_0) \simeq
\frac{A}{2} \frac{z^2}{r_0^2+z^2}.
\label{up0}
\end{equation}
The asymptotic velocity is reached when the numerical value inside the
square brackets in equation (\ref{dup0}) is very small, or
\begin{equation}
u_{pac} (r \sim r_0) \simeq \frac{r_0}{(r_0^2+z^2)^{1/2}}.
\label{upaa}
\end{equation}
The change of behavior between the solution near the jets' origin
and the asymptotic solution takes place when $u_{p0} \sim u_{pa}$, which
by equations (\ref{up0}) and (\ref{upaa}) is
\begin{equation}
u_{pac} \sim \left( 1+ \frac{1}{A^2} \right)^{1/2} -\frac{1}{A}.
\label{upas}
\end{equation}
This very crude expression for the asymptotic transverse velocity
is drawn by a thin line on Figure \ref{upaf}.
As the jet leaves the launching accretion disk, it is very dense and no
bending occurs, namely, $du_p/dz$ is very small.
At large distances from the jet's origin, the angle $\delta$ is small and bending
is no longer efficient.
The bending is most efficient at some intermediate value of $z$, after the density of
the jets decreases as they expand, but before the angle $\delta$ decreases much.
Practically, this intermediate value of $z$ is quite close to the jet's origin,
$z \la r_0$, as is seen in the lower panel of Figure \ref{upf1}.
If the jet pair in a binary system is known to be blown in a time period much shorter
than the orbital period (an impulsive jet pair),
and a bending is observed, the bending angle can be used with the thick line
in Figure \ref{upaf} to find the constant $A$ given in equation (\ref{adef}).
Thus the relation between the three quantities:
the primary stellar mass loss rate $\dot M_s$, the secondary stellar
mass loss rate to the two jet $\dot M_j$, and the half opening angle of
the jets $\alpha$, can be found.
\section{BENDING IN CLUSTERS}
\label{clben}
Many jet pairs blown by radio galaxies are observed to be bent as a result of
the relative motion of the galaxy and the ICM (e.g., Bliton et al. 1998).
Radio jets which are strongly bent are called narrow-angle tailed (NAT)
radio galaxies, while those with slightly bent jets are called
wide-angle tailed (WAT) radio galaxies.
Many of the WAT radio galaxies are dominant galaxies in clusters, like
cD galaxies (Owen \& Rudnick 1976; Burns et al. 1979).
A bulk motion of the IC, e.g., as a result of cluster merger, can efficiently
bend radio jets (Bliton et al. 1998).
A bulk ICM motion relative to the central cD galaxy can exist as a result of merging
with a sub-cluster (group of galaxies), as found in several cases
(Dupke \& Bregman 2005, 2006; Fujita et al. 2006).
The bending process of jets by the ICM was extensively studied
(e.g. Balsara \& Norman 1992); the calculations are not repeated here.
Basically, the bending of jets in clusters is characterized by the
curvature radius $R_c$ of the bent jet.
Because the ambient density changes slowly with distance from the cluster center,
unlike the case in PNs, a constant ambient density is assumed in the region
where most of the jet's bending occurs.
An approximate expression for the radius of curvature is (Sarazin et al. 1995)
\begin{equation}
R_{\rm {curv}} \sim 2
\left( \frac {L_{2j}}{3 \times 10^{43} \erg \s^{-1}} \right)
\left( \frac {R_j}{1 \kpc} \right)^{-1}
\left( \frac {v_j}{0.1 c} \right)^{-1}
\left( \frac {v_a}{300 \km \s^{-1}} \right)^{-2}
\left( \frac {n_e}{0.1 \cm^{-3}} \right)^{-1} \kpc
\label{rcur}
\end{equation}
where $n_e$ is the ambient electron density,
$R_j$ is the jet's radius where most bending takes place, the total
mechanical (kinetic) power of the two jets, $L_{2j}$, was scaled according to
Birzan et al. (2004), and the relative ambient to jet speed $v_a$ according to
Malumuth (1992), with a $90^\circ$ angle between the
initial jet velocity and the ambient flow.
To achieve a noticeable bending in cD clusters we require $R_{\rm curv} \la 10 \kpc$.
This implies that even if the relative velocity of the cD galaxy to the
ambient medium component perpendicular to the jet axis is $\sim 100 \km \s^{-1}$
we get the required bending, as observed in some jets, or bubbles, blown by cD
galaxies.
This required bending will occur also for a narrower jet or a faster jet with
$v_j$ close to the speed of light $c$,
Our primary interest is to compare the bending process in binary stars to that of
jets blown by the dominant cD galaxies at the centers of cooling flow clusters
(or galaxies).
Common to bending of jets in binary progenitors of PNs (referred to as bending
in PNs) and bending of jets at the centers of clusters of galaxies
is that the jets are bent by the ram pressure due to the relative motion of
the medium the jets expand to and interact with, both in PNs and clusters.
This is unlike point-symmetric structures which result from precession; in
both PNs and clusters precession is due to the accretion disk that launches
the jets and not due to the ambient medium.
There are some differences between bending in clusters and bending in binary systems,
such as PNs' progenitors.
\begin{enumerate}
\item {\it Relative velocity.} The bending considered in PNs is due to the
outflow velocity of the slow wind blown by the primary star. This implies that the
angle between the jets velocity and the ambient medium velocity, $\delta$ in
Figure \ref{draw1}, decreases very fast. It decreases even if
bending does not occur.
In clusters the velocity is due to the motion of the cD galaxy relative to the ICM.
The angle decreases only because of the bending, and it decreases slowly.
\item {\it Densities.} The jet's density decreases as the jet moves outward,
both in clusters and in stellar binary systems.
However, in PNs the density of the bending slow wind
(see Fig. \ref{draw1}) decreases as well, $\rho_s \propto (r^2+z^2)$
(denominator of last term in eq. \ref{dvpdz}), while near the center of clusters the
ICM density profile is much shallower and decreases slowly with increasing distance.
\item {\it Asymmetry in clusters.} The bending process in PNs is the same for the
two opposite jets, but this is not necessarily the case in clusters of galaxies.
If the jets are not blown perpendicular to the relative velocity between the ICM
and the galaxy, then the jet expanding to the same direction the galaxy moves
to will feel a larger ram pressure opposing its expansion velocity,
and it will be slowed down more efficiently.
More important, as this jet is bent, the angle of the relative velocity between
the ICM and the galaxy to the jet's axis will increase to $90^\circ$
before decreasing.
In the opposite jet
this angle decreases continuously. Therefore, the jet expanding against the ICM motion
will be bent more than the other jet.
Asymmetry between the two jets in clusters can be also caused by the presence of
asymmetric strong magnetic fields in the ICM (Soker 1997),
and/or density inhomogeneities such as clouds (Sarazin et al. 1995).
\item {\it Late stages of the bending process. }
After reaching their asymptotic bending angle $\phi$, jets in binary stellar systems
will not bend any more.
If a jet inflates a bubble, it will move outward radially along the streaming
slow wind material.
In clusters the situation is different because of the flow structure mentioned in
points (1) and (2) above and because the low density bubbles buoyant outward.
The result is that although the radio jets of cD galaxies are not bent much,
after they become subsonic the bending is very efficient
(Eilek et al. 1984; Odea \& Owen 1986), and the asymmetry between the two sides
can substantially increase (Burns et al. 1986).
\item{\it The effective bending location.} From the differences in points 1,2, and 4
it turns out that in stellar binary systems most of the bending occurs close to the
jets' origin (lower panel of fig. \ref{upf1}).
In clusters the bending becomes more efficient as the jet expands and slows down.
In particular, if the jet inflate bubbles, they move slowly, have very low density,
and large cross section. Thus, in clusters the departure from axisymmetry will be most noticeable
in bubbles.
\end{enumerate}
Despite the differences listed above, there are some striking morphological
similarities of bubbles displaced from the symmetry axis in PNs and clusters;
two cases are mentioned in $\S 1$ (see appendix of the astro-ph version of the
paper).
\section{SUMMARY}
In recent years the jet shaping model for many PNs and similar objects,
like the massive binary star $\eta$ Carinae, acquired considerable acceptance.
It should be stressed that not all PNs were shaped by jets, and not
all morphological structure in PNs were formed by jets.
Bubble pairs, though, are most likely inflated by double jets, and
the jets are probably blown by a stellar secondary star.
The secondary star accretes mass from the primary's slow wind, forms an accretion
disk and blow two jets, either continuously or impulsively,
that is, during a time shorter than the orbital period.
In some PNs, the line joining the centers of the two bubbles in a pair
does not pass through the center of the nebula, meaning that the bubbles are
displaced such that the nebular structure departs from axisymmetry.
The explanation is that the two jets that inflate the bubbles were bent to
the same side by the ram pressure of the slow wind (Fig. \ref{draw1}).
We therefore set the goal of deriving a simple and approximate relation between
the bending angle of the jets and the properties of secondary stellar jets and the
primary slow wind.
For fast jets, $v_j \gg v_s$, the important factor is the quantity $A$
defined in equation (\ref{adef}).
The relation between the jet's asymptotic transverse speed $v_{pa}$
(see fig. \ref{draw1}) and $A$ is presented by the thick line in
Figure \ref{upaf}, and a very crude approximation is given in equation (\ref{upas})
($v_{pa}$ is in units of the slow wind speed $v_s$).
If the jets are impulsive, then the bending will be easier to observe;
otherwise it is averaged over different directions as the binary system rotate.
If $A$ is not too small, and the orientation of the nebula is such that the
bending is not along the line of sight, then observations may reveal
the two jets or the bubbles (lobes) inflated by the jets to be displaced to
the same side of the symmetry axis.
Examples of such PNs are listed and classified by Soker \& Hadar (2002).
In some clusters, X-ray-deficient bubble (cavity) pairs that were inflated by
jets blown by the central cD galaxy, show displacement from axisymmetry similar
to visible-deficient bubble (lobe) pairs observed in PNs (see appendix
of the astro-ph version of this paper).
We therefore set a second goal of comparing the bending process of jets
in these two groups of objects.
Two factors of the bending process are common to these two classes of objects:
1) the bending results from the ram pressure perpendicular to the jet axis,
and 2) the ram pressure is exerted by the same external medium the jets
expand to and interact with.
However, there are some significant differences listed in \S\ref{clben}.
(1) Because the bending in binary systems results from the slow wind blown by the
primary star, the ambient density decreases faster with distance than the ambient
density of the ICM in the centers of clusters.
(2) Also, the angle $\delta$ (see Fig. \ref{draw1}) between the jet velocity
and ambient slow wind velocity in binary systems decreases with distance
along the jet axis, even when bending does not occur.
In clusters the relative velocity is due to the bulk ICM motion and
changes only because of bending.
(3) In binary systems the two opposite jets are likely to be blown perpendicular
to the orbital plane, thus they will be bent in the same way.
In clusters, the jets' axis need not be perpendicular relative to the bulk
motion of the ICM relative to the central black hole that blows the jets, so
the jet facing the ICM flow will be bent more efficiently.
In binary systems such an asymmetry between the two jets can occur if the
jets (more specifically the accretion disk that launches the jets) precess.
(4) In clusters, after the bubbles (cavities; lobes) are inflated, they buoy outward.
They are more susceptible than the jets to the ram pressure, and departure from
axisymmetry may substantially increase. This process does not exist in binary systems
because the circumbinary ambient matter is not in hydrostatic equilibrium, but rather
the ambient matter expands at a high Mach number.
(5) In binary stars most of the bending occurs when the jets are at a distance
$z \la r_0$, where $r_0$ is the orbital separation.
In clusters the bending becomes more efficient at larger and larger distances.
We hope that the study presented in this paper will motivate researchers to
pay more attention to the departure from axisymmetry of bubble (cavity; lobe)
pairs in both clusters of galaxies and PNs.
\acknowledgments
\acknowledgments This research was supported in part by the Asher
Fund for Space Research at the Technion.
|
Title:
Thermal Structure and Radius Evolution of Irradiated Gas Giant Planets |
Abstract: We consider the thermal structure and radii of strongly irradiated gas giant
planets over a range in mass and irradiating flux. The cooling rate of the
planet is sensitive to the surface boundary condition, which depends on the
detailed manner in which starlight is absorbed and energy redistributed by
fluid motion. We parametrize these effects by imposing an isothermal boundary
condition $T \equiv T_{\rm deep}$ below the photosphere, and then constrain
$T_{\rm deep}$ from the observed masses and radii. We compute the dependence of
luminosity and core temperature on mass, $T_{\rm deep}$ and core entropy,
finding that simple scalings apply over most of the relevant parameter space.
These scalings yield analytic cooling models which exhibit power-law behavior
in the observable age range $0.1-10 {\rm Gyr}$, and are confirmed by
time-dependent cooling calculations. We compare our model to the radii of
observed transiting planets, and derive constraints on $T_{\rm deep}$. Only HD
209458 has a sufficiently accurate radius measurement that $T_{\rm deep}$ is
tightly constrained; the lower error bar on the radii for other planets is
consistent with no irradiation. More accurate radius and age measurements will
allow for a determination of the correlation of $T_{\rm deep}$ with the
equilibrium temperature, informing us about both the greenhouse effect and
day-night asymmetries.
| https://export.arxiv.org/pdf/astro-ph/0601317 |
\title{Thermal Structure and Radius Evolution of Irradiated Gas Giant
Planets}
\author{Phil Arras and Lars Bildsten}
\affil{ Kavli Institute for Theoretical Physics \\
Kohn Hall, University of California,
\\ Santa Barbara, CA 93106; arras@kitp.ucsb.edu, bildsten@kitp.ucsb.edu}
\keywords{planetary systems--planets and satellites:general}
\section{Introduction}
\begin{deluxetable*}{lccccccr}
\tabletypesize{\small}
\tablewidth{0pt}
\tablecaption{Transiting Extrasolar Planets \label{tab1} }
\tablehead{
\colhead{object} &
\colhead{$a$(au)} &
\colhead{$M_{\rm p}$($M_{\rm J}$)} &
\colhead{$R_{\rm p}$($R_{\rm J}$)} &
\colhead{ $T_{\rm eq}[K]$ \tablenotemark{a} } &
\colhead{ $T_{\rm deep}[K]$\tablenotemark{b}} &
\colhead{Age (Gyr)} &
\colhead{Reference} }
\startdata
OGLE-TR-132 & 0.031 & $1.19 \pm 0.13$ & $1.13 \pm 0.08$ &
2100& $\leq 2200$ & 0--1.4 & 1 \\
OGLE-TR-56 & 0.023 & $1.24 \pm 0.13$ & $1.25 \pm 0.08$ & 2100
& $ 1000-3100$ & $3 \pm 1$ & 2,3,12 \\
HD 209458 & 0.046 & $0.69 \pm 0.05$ & $\rm
1.31^{+0.05}_{-0.05}$ & 1500 & 2200-2800 & 4--7 & 4,5\\
OGLE-TR-10 & 0.042 & $0.63 \pm 0.14$ & $1.14 \pm 0.09$ &
1500 & $\leq 2600$ & -- & 6,12 \\
OGLE-TR-113 & 0.023 & $1.35 \pm 0.22$ & $\rm
1.08^{+0.07}_{-0.05}$ & 1300 & $\leq 2100$ & -- & 7 \\
TrES-1 & 0.039 & $0.73 \pm 0.04$ & $\rm
1.08^{+0.05}_{-0.05}$ & 1200 & $\leq 1000$ & $2.5 \pm 1.5$ & 5,8 \\
OGLE-TR-111 & 0.047 & $0.52 \pm 0.13$ & $\rm
0.97 \pm 0.06$ &1000 & $\leq 1200$ & -- & 9,12 \\
HD 149026 \tablenotemark{c} & 0.042 & $0.36\pm 0.04$ & $0.725 \pm
0.05$ & 1700
& & $2.0 \pm 0.8$ & 10 \\
HD 189733 & 0.031 & $1.15\pm0.04$ & $1.26\pm 0.03$ & 1200 & $\leq 3200$ &
& 11 \\
\enddata
\tablenotetext{a}{Here $T_{\rm eq} \equiv T_{\rm
*}(R_*/2a)^{1/2}$. See the discussion following eq.\
(\ref{eq:Teq}).}
\tablenotetext{b}{Allowed range of $T_{\rm deep}$ given range of mass,
radius and age. If no age range given in the literature, we
(arbitrarily) give the maximum value of $T_{\rm deep}$ for an age less
than 10Gyr. However, given an accurate age range, the figures in
\S \ \ref{sec:applications} can be used to obtain stronger
constraints than given here.}
\tablenotetext{c}{HD 149026's small radius clearly indicates a large
core size or heavy element abundance. The present paper does not
include heavy element cores, so we do not discuss HD 149026 further.}
\tablerefs{(1) \cite{2004A&A...424L..31M}, (2)
\cite{2004ApJ...609.1071T}, (3) \cite{2003ApJ...596.1327S},
(4) \cite{2002ApJ...569..451C}, (5) \cite{2005ApJ...621.1072L},
(6) \cite{2005ApJ...624..372K}, (7) \cite{2004A&A...421L..13B},
(8) \cite{2004ApJ...616L.167S}, (9) \cite{2004A&A...426L..15P},
(10) \cite{2005ApJ...633..465S},(11)\cite{2005astro.ph.10119B},
(12) \cite{2006astro.ph..1024S} }
\end{deluxetable*}
Following the discovery of the planet orbiting 51 Peg
\citep{1995Natur.378..355M,1995AAS...187.7004M}, more than 160 planets
have been found around nearby stars
using precision Doppler spectroscopy.
\footnote{For up to date catalogs, see http://exoplanets.org/ and
http://obswww.unige.ch/~udry/planet/planet.html.} Theories of planet
formation now have the demanding task of explaining the existence of
gas giants with semi-major
axes one hundred times smaller than Jupiter, others with order unity
orbital eccentricities, a detailed spectrum of (minimum) planet
masses, and metallicity correlations with the parent star.
The discovery of transiting planets in the last five
years (see Table \ref{tab1}) challenges not only theories for the
origin of short-period gas giants, but also their structure and
thermal evolution, spectrum, and interior fluid dynamics. Measurements
of planetary mass, radius, and (stellar) age test cooling models
which predict radius as a function of mass and age. The atmospheres of
two planets have been directly observed. For HD 209458b, absorption
lines (due to stellar photons passing through planet's atmosphere)
have been found \citep{2002ApJ...568..377C, 2003Natur.422..143V,
2004ApJ...604L..69V}, and the first detections of photons
emitted by planets outside our solar system have been made for the
thermal emission from HD 209458b \citep{2005Natur.434..740D} and
TrES-1 \citep{2005ApJ...626..523C}. These observations directly
constrain the atmospheric structure, temperature profile and chemical
composition near the photosphere.
Evolution of the short orbital period transiting exoplanets is
significantly different than for Jupiter and Saturn due to proximity
of the parent star \citep{1996ApJ...459L..35G}. Irradiation increases
the photospheric temperature by nearly an order of magnitude relative to
an isolated planet. Irradiation also decreases the cooling rate, and
hence the rate of shrinkage, by altering the surface boundary
condition \citep{2000ApJ...534L..97B}. This is immediately apparent in
Table \ref{tab1} as many transiting extra-solar giant planets (EGP's)
have radii significantly larger than Jupiter. As short period planets
are expected to be tidally synchronized
\citep{1996ApJ...459L..35G,1997ApJ...481..926M},
the strong day-night
temperature contrast will drive winds to transport heat from the day
to the night side \citep{2002A&A...385..166S}. Hence the atmospheric
temperature profile depends on a combination of detailed
radiative transfer calculations for absorption of starlight, and
hydrodynamics to model day-night winds and dissipation of wind kinetic
energy. Lastly, tides raised on the planet by the parent star may
significantly affect its thermal evolution
\citep{2001ApJ...548..466B,2002A&A...385..166S}. The free energy
available by synchronizing the planet's spin or circularizing the
orbit are comparable or larger than the thermal energy. Hence {\it if}
the heat can be deposited sufficiently deep in the planet in less than
a cooling timescale, the cooling can be slowed, or even reversed.
However, it is uncertain if tides can deposit heat deep in
the planet \citep{1997ApJ...484..866L, 2004ApJ...610..477O,
2004astro.ph..7628W} .
Evolutionary models show that the cooling rate is quite sensitive to
the uncertain surface boundary condition
\citep{2002A&A...385..156G}. This boundary condition has been
implemented using various approximations. Full radiative transfer
calculations \citep{2001ApJ...556..885B, 2003ApJ...594.1011H} of {\it
static} atmospheres include the stellar irradiation self-consistently,
and determine the temperature structure for a given cooling flux from
the deep interior. These calculations compute (rather than assume) the
albedo, and determine the temperature rise due to absorption of
starlight (the greenhouse effect). Such detailed radiative transfer
solutions have been incorporated as boundary conditions for some
evolutionary calculations \citep{2003A&A...402..701B,
2003ApJ...594..545B}. However, as day-night and equator-pole winds are
not included, assumptions must be made about how the stellar flux is
deposited over the surface of the planet (only day-side versus evenly
over the entire surface, etc.) which directly affect the
temperature profile. Other evolutionary calculations
(e.g. Bodenheimer et al. 2003) solve the radiation diffusion
equation and set the temperature at (infrared) optical depth $2/3$ to
be the equilibrium temperature, ignoring additional temperature
increase due to absorption of starlight. Lastly, a number of groups
\citep{2002A&A...385..166S, 2003ApJ...587L.117C, 2005ApJ...618..512B,
2005ApJ...629L..45C, 2005A&A...436..719I} are beginning to model the
day-night winds on tidally locked, short orbital period
planets, and the role of clouds
\cite{2003ApJ...589..615F} . As we
stress here, the crucial parameter for the cooling rate is the
temperature at the radiative-convective boundary, which is orders of
magnitude deeper in pressure than the photosphere.
The plan of the paper is as follows. The uncertain surface boundary
condition is discussed in \S \ \ref{sec:surfacebc},
motivating the surface isotherm used in our models. Details of cooling
models and microphysical input are described in
\S \ \ref{sec:numerics}. In \S \ \ref{sec:luminosity} we compute the
dependence of
the luminosity on planet mass, core entropy and irradiation. An
analytic solution for the temperature profile in the radiative zone,
and the position of the radiative-convective boundary are
derived in \S \ \ref{sec:transition}.
These results are collected together in
\S \ \ref{sec:cooling} to derive an analytic cooling model which
exhibits simple power-law dependence on time. The radii of irradiated
gas giant planets are discussed in \S \ \ref{sec:radiusev}, and the
analytic formula for the radius given in eq.\ (\ref{eq:Rtfit}). We
apply our models to the observed transiting planets and give
constraints on the temperature of the deep surface isotherm in \S \
\ref{sec:applications}. Our main conclusions are summarized in \S \
\ref{sec:conclusions}.
\section{ Surface Boundary Condition }
\label{sec:surfacebc}
The surface boundary condition we adopt is to set the temperature $T \equiv
T_{\rm deep}$ at a sufficiently large optical depth that the stellar
light is fully absorbed, and the radiation diffusion approximation is
valid. This choice of surface boundary condition has also recently
been advocated by
\citet{2005A&A...436..719I}, based on the results of time-dependent
radiative models for the atmosphere of HD 209458b. We motivate
our choice with a simple
toy problem, and then discuss its relation to detailed radiative
transfer solutions for the atmosphere.
The atmosphere is heated by absorption of starlight, and possibly
dissipation of day-night winds and tidal flows. Let there be an energy
deposition rate $\varepsilon$ per unit volume in a radiative region of
thickness $\Delta z$. Choose boundary conditions $T=0$ at the top (for
simplicity) and outward flux $F=0$ at the base of the heated layer. The
latter choice is required in steady state so that the temperature deeper
in the atmosphere not increase in time. The flux generated in the layer,
which exits the planet, is $F = \varepsilon \Delta z$, and the temperature
of the deep atmosphere is $T_{\rm deep} \sim (\tau F/\sigma)^{1/4}$,
where $\tau=\kappa \rho \Delta z$ is the optical depth, $\rho$ is the
density and $\kappa$ is the opacity. Hence an atmosphere subject to
intense heating is expected to develop a deep isothermal region below the
heated layer, the temperature determined primarily by the energy flux and
depth of the layer, through $\tau$. This estimate of the deep isotherm
temperature is similar to that found for absorption of starlight for
the proper choice of $\tau$ \citep{2003ApJ...594.1011H}.
We now discuss the temperature profile
for static atmospheres in more detail. In the absence of external
irradiation, the photosphere of a planet will
cool to a temperature $T_{\rm cool} \sim (F_{\rm cool}/\sigma)^{1/4}
\sim 100\ {\rm K}$ in a few Gyr's, where $F_{\rm cool}$ is the flux
from the deep interior. A characteristic temperature at small
optical depth for an
irradiated planet can be defined by balancing absorbed and emitted
energy flux. For a star with mass $M_*$, radius $R_*$ and effective
temperature $T_*$ a distance $a=(GM_*P_{\rm orb}^2/4\pi^2)^{1/3}$
away, this ``equilibrium'' temperature is
\be
&& T_{\rm eq} \equiv T_*
(R_*/2a)^{1/2} \nonumber \\ & \simeq & 1400\ {\rm K}
\left(\frac{{\rm 3\ day}}{P_{\rm orb}}\right)^{1/3}
\left(\frac{T_*}{6000\ {\rm K}}\right)
\left(\frac{R_*}{R_\odot}\right)^{1/2}
\left(\frac{M_\odot}{M_*}\right)^{1/6},
\label{eq:Teq}
\ee
an order of magnitude larger than for an isolated planet. Hence
the surface boundary condition is drastically altered from the
isolated case. In general, the irradiated boundary condition will
cause the planet to cool slower \citep{2000ApJ...534L..97B}, as we
discuss in detail. As significant horizontal temperature variation is expected
above the photosphere, $T_{\rm eq}$ is an average temperature which
gives
the correct outgoing flux.
Eq.\ (\ref{eq:Teq}) assumes zero
reflection of the stellar photons, and should be multiplied by
$(1-A)^{1/4}$ for nonzero Bond albedo $A$.
The (optical) incoming stellar photons not scattered back out of the
planet are absorbed at the starlight's photosphere, typically at a
pressure $\la 10^6\ {\rm dyne\ cm^{-2}}$. Radiative balance implies
an outgoing (infrared) flux $F \sim (T_{\rm eq}/T_{\rm cool})^4 F_{\rm
cool} \sim 10^4 F_{\rm cool}$ generated by thermal emission. This large
flux may lead to a significant increase in temperature above $T_{\rm
eq}$ (the greenhouse effect, e.g. Hubeny et al. 2003). This situation
continues to a depth at which the starlight is fully absorbed, at which
point the temperature profile becomes isothermal. Hence, a semi-infinite
atmosphere subject to external irradiation, and with no internal flux
deep in the atmosphere, becomes isothermal at large optical depth. We
label the temperature of this deep isotherm $\Tdeep$. Now including the
internal cooling flux $F_{\rm cool}$, the temperature will again rise
toward the interior, the gradient eventually becoming large enough for
convection to occur.
Since the cooling luminosity is generated in deep layers with
sufficiently large optical depth that the stellar light is fully
absorbed, the radiation diffusion approximation is valid
there. Furthermore, we will show in \S \ \ref{sec:transition} that the
temperature profile becomes isothermal within a pressure scale height
of the radiative-convective boundary. Hence the problem of determining
the cooling luminosity is insensitive to many of the details of the
absorption of starlight. The only input needed from the full radiation
transfer problem near the photosphere is the temperature of the deep
isotherm, $T_{\rm deep}$. \footnote{ We expect that the degree to which this
layer is isothermal depends on the number of pressure scale heights
separating the optical photosphere from the radiative-convective
boundary. Larger irradiation and lower core entropy should make this
layer more nearly isothermal.}
For tidally locked planets, the day side will be significantly hotter
than the night side in static atmospheres with negligible day-night
winds. A more uniform temperature distribution results if winds can
carry heat from the day to the night side without suffering radiative
losses (e.g. Iro et al. 2005). We will show that the
cooling luminosity is determined in deep layers with thermal time
$t_{\rm th} \ga 10^3\ {\rm yr}$. While significant day-night
temperature asymmetries may exist near the optical photosphere, winds
moving at even a tiny fraction, $\sim 10^{-5}$, of the sound speed could
deposit heat on the night side in less than a thermal time at the
depths where the cooling luminosity is determined. Hence we have a
strong expectation of a near-spherically symmetric, isothermal
temperature profile deep in the radiative layer.
\section{ Numerical Models for the Interior }
\label{sec:numerics}
In the deep interior where the diffusion approximation is valid, we
solve the mechanical and thermal structure equations
\citep{1959flme.book.....L}
\be
\frac{dm}{dr} & = & 4\pi r^2 \rho, \\
\frac{dP}{dr} & = & - \frac{Gm\rho}{r^2}, \\
\frac{dT}{dr} & = & \frac{dP}{dr} \frac{T}{P} \nabla, \\
\frac{dl}{dr} & = & \frac{dm}{dr} \left( \varepsilon - T \frac{\partial
S}{\partial t} \right)
\label{eq:entropy},
\ee
for the interior mass $m$, pressure $P$, temperature $T$, and
outward luminosity $l$, as a function of radius $r$. Here $S$ is the entropy
per gram,
and $\nabla=d\ln T/d\ln P$ is the logarithmic temperature
gradient. The energy generation $\varepsilon$ is set to zero
throughout this paper, as we study passively cooling planets.
The subscript ``cool'' on the luminosity will be assumed for
the rest of the paper. As
the eddy turnover time is much shorter than the cooling time and
convection is quite efficient, entropy is very nearly constant in
space in the convection zone, but decreases in time due to
cooling. Hence we treat $\partial S/\partial t$ as a constant in the
convection zone. For numerical convenience, we use this same value of
$\partial S/\partial t$ in the surface radiative zone. A negligible
luminosity is generated there however, so this error does not affect
our results. While the entropy equation (\ref{eq:entropy}) is valid on
timescales longer than an eddy turnover time ($\sim {\rm yrs}$) in the
convective core, the assumption of nearly spatially constant
luminosity is only
valid in the radiative envelope on timescales longer than the thermal
time ($\sim 10^3\ {\rm yr}$) there. As this is much shorter than the
global cooling time, we expect our numerical cooling models to be
as accurate as a relaxation (Henyey-type) code.
The equation of state (EOS) from \cite{1995ApJS...99..713S} (SCVH) is used
with a mixture of $70\%$ hydrogen and $30\%$ helium, ignoring metals, and
using the tables which smooth over the plasma phase transition. There
are been several improvements to SCVH \citep{1999astro.ph..9168S,
2004ApJ...608.1039F} using recent laser shock- compression data and
including the effects of helium phase separation, which change the radii
at the few percent level. We use the solar composition ``condensed''
phase opacities from \citet{2001ApJ...556..357A}, which includes the
effects of grains in the equation of state, but ignores their opacity,
as is appropriate if the grains have condensed out. Mixing length theory
is used to calculated $\nabla$ in convective regions, and radiative
diffusion in radiative regions. The mixing length is set equal to the
pressure scale height.
Two boundary conditions are needed at the surface. First, the surface
temperature is set to $\Tdeep$, the temperature of the deep isotherm
discussed in \S \ \ref{sec:surfacebc}. The second boundary
condition is that we specify the surface to be at the (arbitrarily
chosen) pressure $P=10^4\ {\rm dyne\ cm^{-2}}$. As the surface layer
is isothermal, the contribution to the radius from near-surface layers
is larger than for the radiative zero temperature profile, hence care
is needed when comparing the radii computed here with previous
work. As the radius is somewhat dependent on the problem at hand
(optical photosphere versus infrared photosphere, corrections due to
geometry in a transit, etc.) we have made this arbitrary choice of the
surface for simplicity. The change in radius between pressures $P_1$
and $P_2$ is $\Delta R=\int_{P_1}^{P_2} d\ln P (k_bT/\mu m_p g)
\simeq (k_bT/\mu m_p g) \ln(P_2/P_1)$. For example, the radius must
be decreased by $\Delta R=-0.022R_J$ for an outer boundary condition
$P=10^6\ {\rm dyne\ cm^{-2}}$ for $T=1000\ {\rm K}$ and mean molecular
weight $\mu=2.43$ ($70\%$ molecular hydrogen and $30\%$ neutral helium).
We do not include a solid core in the present calculations.
We make a single model of a planet as follows. Planet mass $M$, core
entropy $S$, and surface temperature $\Tdeep$ are treated as fixed
parameters. Assuming values for the planet's radius $R$, cooling
luminosity $L=l(R)$, $\partial S/\partial t$, and central pressure
$P_c$, we integrate outward from the center and inward from the
surface. The four parameters are adjusted to make the
integration variables ($m$, $P$, $T$, and $l$) continuous at a fitting
radius. Given the subroutine to solve for a single model, evolving
the planet in time is trivial. As we specify the core entropy $S$, and
have solved for $\partial S/\partial t$, we compute the time it
takes to cool from one entropy to the next.
\section{ Impact of irradiation on heat loss }
\label{sec:luminosity}
We now show the dependence of the cooling luminosity on the depth of
the radiative-convective boundary, emphasizing the role of the opacity
deep in the planet. Many of the luminosity dependences can be
understood with purely local arguments, without the need to build a
global planet model. Hence, qualitative statements can be made about
cooling of EGP's under irradiation just given EOS and opacity
tables. We make comparisons between the local arguments and the global
numerical calculations as well.
The convective core is capable of transporting enormous
luminosities through fluid motion. Hence it is the large thermal
resistance of the outer radiative envelope that determines the
cooling flux. For an opacity which increases inward from the surface, this
resistance is largest at the base of the radiative layer, hence it is
the radiative-convective boundary that determines the cooling
flux. This boundary is moved to higher pressures by irradiation
\citep{1996ApJ...459L..35G}.
The outward flux carried by radiative diffusion is
\be
F & = & - \frac{16\sigma T^3}{3\kappa \rho} \frac{dT}{dr},
\label{eq:flux}
\ee
where $\kappa$ is the Rosseland mean opacity.
The maximum flux which can be carried by radiative diffusion is found
using the adiabatic temperature gradient $dT/dr|_{\rm
ad}=(\nabla_{\rm ad}T/P)(-Gm\rho/r^2)$, where $\nabla_{\rm ad}=\partial
\ln T/\partial \ln P |_S$ ($=2/7$ for
an ideal gas with five degrees of freedom) is the adiabatic
temperature gradient.
Multiplying by $4\pi r^2$,
the maximum luminosity per unit mass which can be carried by
radiative diffusion at a local temperature $T$, pressure
$P$, opacity $\kappa(T,P)$, and enclosed mass $m \simeq M$ is
\be
\frac{L}{M} & = & \frac{64\pi G}{3}
\frac{\sigma T^4}{\kappa P} \nabla_{\rm ad}.
\label{eq:Lmax}
\ee
Choosing an entropy $S$, the right hand side of eq.\ (\ref{eq:Lmax}) can
be evaluated along an adiabat out from the center, yielding the
cooling flux for a specified temperature $T_{\rm rcb}$ at
the radiative-convective boundary. Eq.\ (\ref{eq:Lmax}) shows that the
luminosity per unit mass depends solely on the entropy and
irradiating flux, and that the luminosity is proportional to the
planet's mass. \footnote{It is commonly stated that the luminosity {\it
decreases} with increasing mass. This is true at fixed core
temperature, rather than fixed entropy. The derivatives can be related
by $\partial \ln L/\partial \ln M|_{T_c}=\partial \ln L/\partial \ln
M|_S + \partial \ln L/\partial S|_M \partial S/\partial \ln
M|_{T_c}$. At fixed core temperature, entropy increases for
decreasing mass (see Figure \ref{fig:Tcore_vs_M}).}
Figure \ref{fig:T_vs_P} shows the run of temperature versus pressure from
numerical models for a range of $M$, $S$ and $\Tdeep$. Choosing $S$
and $\Tdeep$, the temperature profile must follow the adiabat deep in
the planet and the isotherm near the surface. An even stronger
statement can be made, however. The temperature profile over the
entire planet from the center to the top of the deep isotherm depends
only on $S$ and $\Tdeep$, and is independent of $M$ (since
eq.\ (\ref{eq:Tprofile}) and (\ref{eq:Pdeep}) depend only on $F/g
\propto L/M$).
Next, for a given irradiation flux (fixed $\Tdeep$), the
radiative-convective transition burrows deeper into the planet with
time (decreasing $S$). Increasing the irradiation flux at fixed $S$
also moves the radiative-convective region deeper into the
planet.
Temperature changes in response to small changes in flux in optically thick
regions occur on the thermal time, estimated from eq.\ (\ref{eq:entropy}) to be
\be
t_{\rm th} & = & \frac{PC_pT}{gF} \simeq 10^4\ {\rm yr}
\left(\frac{P}{10^8\ {\rm dyne\ cm^{-2}}} \right)
\left( \frac{T}{10^3\ {\rm K}} \right)
\nonumber \\ &&
\left( \frac{10^3\ {\rm cm\ s^{-2}}}{g} \right)
\left( \frac{10^4\ {\rm erg\ cm^{-2}\ s^{-1}}}{F} \right).
\ee
Here we have used typical numbers from Figure \ref{fig:T_vs_P} for the
radiative-convective boundary. Note that this
estimate is much longer than the adjustment time near the optical
photosphere ($\sim {\rm days}$, e.g. Iro et al. 2005),
as the cooling flux is
$\sim 10^4$ times smaller than the stellar flux, and the heat content
increases $\propto TP$. As the thermal time at the
radiative-convective boundary is so much longer than the horizontal
sound travel time ($\sim {\rm days}$), we expect the day-night
temperature asymmetry to be small there.
Figures \ref{fig:LoverM_vs_T} and \ref{fig:LoverM_vs_T_int}
show the local calculation of $L/M$ evaluated along adiabats, and the
global calculation of $L/M$ versus $\Tdeep$, respectively. The x-axis
in Figure \ref{fig:LoverM_vs_T} is the local temperature, which should be
interpreted as $T_{\rm rcb}$, the temperature of the
radiative-convective boundary. Care must be taken in
Figure \ref{fig:LoverM_vs_T} in regions where $L(T)$ increases inward. As
we show in \S \ \ref{sec:cooling}, $L(T)$ must {\it decrease} inward in
order for convection to begin. Hence, if the chosen isotherm
intersects a region of positive slope, such as the bump near
$T=2000-2500\ {\rm K}$, the convection zone actually begins at a
deeper point at which the slope $L(T)$ is again negative. Such regions
correspond to the flat parts of the curves in
Figure \ref{fig:LoverM_vs_T_int}. The result is that the luminosity
generally decreases with irradiation temperature, or is roughly
constant, but should not increase. This is the origin of the result
found by previous investigators \citep{2000ApJ...534L..97B} that
irradiated planets cool
slower. Comparison of Figures \ref{fig:LoverM_vs_T} and
\ref{fig:LoverM_vs_T_int} show rough agreement in regions where $L(T)$
is decreasing, the main discrepancies due to the ratio $T_{\rm
rcb}/\Tdeep$ not being precisely a constant (see
Figure \ref{fig:T_vs_P}).
Figure \ref{fig:LoverM_vs_S_num} shows luminosity versus core entropy
for the numerical models. If $\Tdeep$ is constant during the
evolution, Figure \ref{fig:LoverM_vs_S_num} shows the change in
luminosity as the planet cools. Comparison of lines with different
$\Tdeep$ clearly shows the monotonic decrease in luminosity as the
irradiation temperature is increased. Aside from models with large mass
($M=3.2M_J$) and irradiation temperature ($\Tdeep=3500\ {\rm K}$) at
entropies so low ($S<8k_b/m_p$) as
to be unreachable in a Hubble time, the luminosity is proportional to
the mass and the curves overlie each other.
\section{ Radiative-convective boundary }
\label{sec:transition}
We now derive a simplified analytic model for the temperature
profile at the transition from the surface radiative zone to the core
convection zone. We relate $T_{\rm deep}$
to $T_{\rm rcb}$, the radiative-convective boundary temperature where
the cooling luminosity is determined. The
scalings derived here are used in \S \ \ref{sec:cooling} to
derive the scalings of the cooling luminosity.
We assume constant gravity $g$, ideal gas
pressure $P=\rho k_b T/\mu m_p$ and power-law opacity \footnote{
Significant features in the opacity may be treated as broken power-laws.}
$\kappa\equiv\kappa_0 \rho^a T^b\equiv\kappa_1 P^a T^{b-a}$. In the radiative
zone,
\be
F & = & \frac{16\sigma T^3 g}{3\kappa}\frac{dT}{dP}
= \frac{a+1}{4+a-b} \frac{16\sigma g}{3\kappa_1}\frac{dT^{4+a-b}}{dP^{a+1}}.
\ee
When integrating this equation, it's essential to retain the constant
of integration. Defining the temperature gradient for a radiative zero
solution $\nabla_\infty=(a+1)/(4+a-b)$, we find
\be
T^{4+a-b} & =& {\rm constant} + \nabla_\infty^{-1}
\left(\frac{3\kappa_1 F}{16\sigma g} \right) P^{a+1}.
\ee
At small pressure, $T \simeq \Tdeep$, so we write the
temperature profile as
\be
T & =& \Tdeep \left[ 1 + \left(P/P_{\rm deep}
\right)^{a+1} \right]^{1/(4+a-b)},
\label{eq:Tprofile}
\ee
which becomes isothermal below the pressure
\be
P_{\rm deep} & = & \left( \nabla_\infty \frac{16 \sigma g
\Tdeep^{4+a-b}}{3\kappa_1 F} \right)^{1/(a+1)}.
\label{eq:Pdeep}
\ee
The logarithmic temperature gradient is then
\be
\nabla & = & \nabla_\infty \frac{ \left(P/P_{\rm deep} \right)^{a+1} }{1
+ \left(P/P_{\rm deep} \right)^{a+1} },
\label{eq:nabla}
\ee
which decreases sharply over a pressure scale height. A plot of
$\nabla$ versus $P$ is shown in Figure \ref{fig:del_vs_P_allard_cond}
for several values of $\Tdeep$. The upper envelope of the curves is
set by the adiabatic gradient in the convection zone. Increasing
$T_{\rm deep}$ moves the boundary inward along the adiabat,
$P_{\rm deep} \propto \Tdeep^{1/\nabla_{\rm ad}}$, aside from regions
where the opacity changes irregularly. Eq.\ (\ref{eq:nabla}) agrees
well with the numerical integrations in regions where the opacity is
smooth.
To solve for the
transition from radiative to convective zone, we set
$\nabla=\nabla_{\rm ad}$. As $a+1>0$, one must have the
inequality $\nabla_\infty \geq \nabla_{\rm ad}$ for a convection zone
to exist. We find the temperature and pressure at the boundary are
\be
T_{\rm rcb} & = & \Tdeep \left(
\frac{\nabla_\infty}{\nabla_\infty-\nabla_{\rm ad}}
\right)^{1/(4+a-b)}
\label{eq:Ttr}
\nonumber \\
P_{\rm rcb} & = & P_{\rm deep} \left(
\frac{\nabla_{\rm ad}}{\nabla_\infty-\nabla_{\rm ad}} \right)^{1/(a+1)},
\ee
so they differ by a factor of order unity from $\Tdeep$ and
$P_{\rm deep}$ unless $|\nabla_{\rm
ad}-\nabla_\infty|<<\nabla_\infty$. The decrease of $\nabla_{\rm ad}$
at large pressures seen in Figure \ref{fig:del_vs_P_allard_cond} makes
the ratio $T_{\rm rcb}/\Tdeep$ closer to unity for large $T_{\rm deep}$
and $P$, as seen in Figure \ref{fig:T_vs_P}. Also note that at fixed $S$
and $\Tdeep$, $T_{\rm rcb}$ is largely independent of mass.
\section{ Analytic cooling model }
\label{sec:cooling}
In \S \ \ref{sec:luminosity} we found that the luminosity scales
with core entropy and irradiating flux over much of the
relevant parameter space. Here we show that the core temperature also
scales simply with mass and entropy when sufficiently degenerate. As a
consequence, we derive an analytic model in which entropy has a
simple power-law time dependence at late times. We compare the
power-law model against numerical time integrations.
The scaling of luminosity with $\Tdeep$ and $S$
can be found by substituting $P(T,S)$ into eq.\ (\ref{eq:Lmax}).
Using the thermodynamic relation
\be
\frac{dS}{C_p} & = & \frac{dT}{T} - \nabla_{\rm ad} \frac{dP}{P},
\label{eq:thermo}
\ee
and expanding about a reference point
$T_{\rm ref}$, $P_{\rm ref}$ and $S_{\rm ref}$, the adiabat is
\be
P & \simeq & P_{\rm ref} \left( \frac{T}{T_{\rm ref}}
\right)^{1/\nabla_{\rm ad}} \exp\left(-\frac{\Delta S}{C_p\nabla_{\rm
ad}} \right),
\label{eq:adiabat}
\ee
where $\Delta S=S-S_{\rm ref}$, and we approximate $\nabla_{\rm
ad}$ and $C_p$ as constants. For an ideal gas, $C_p \nabla_{\rm
ad}=k_b/\mu m_p$, but particle interactions and molecular
dissociation reduce $C_p \nabla_{\rm ad}$ below the ideal value.
Inserting eq.\ (\ref{eq:adiabat}), (\ref{eq:Ttr}), and the power-law
form of the opacity into eq.\ (\ref{eq:Lmax}), we find
\be
L & \simeq & L_{\rm ref} \left( \frac{T_{\rm deep}}{T_{\rm ref}}
\right)^{-\alpha}
\exp\left[ \beta \frac{(S-S_{\rm ref})}{k_b/m_p} \right]
\label{eq:Lpl}
\ee
where the exponents are (Figures \ref{fig:LoverM_vs_T_int} and
\ref{fig:LoverM_vs_S_num})
\be
\alpha & \simeq & (4+a-b)\left(\frac{\nabla_\infty}{\nabla_{\rm ad}} -
1\right) \simeq 0.0-10.0,
\nonumber \\
\beta & \simeq & (a+1) \frac{k_b/m_p}{C_p \nabla_{\rm ad}} \simeq 2.5-3.5.
\ee
Examination of the exponent $\alpha$ shows that irradiation slows the
cooling, i.e. luminosity decreases as $T_{\rm deep}$ increases. The
condition $\nabla_\infty > \nabla_{\rm ad}$ is required for a core
convection zone to exist, hence $\alpha \geq 0$. Evaluation of
$\alpha$ depends on the detailed density and temperature dependence of
the opacity, which can be found in
Figure \ref{fig:LoverM_vs_T}. Features of note are the positive slope
near $2000-3000\ {\rm K}$ at which point $\alpha$ become small, and
also the steep decrease for $T_{\rm rcb} \geq 3000\ {\rm K}$.
The exponent $\beta$ can be estimated for an ideal gas (solar mixture,
molecular hydrogen and neutral helium) and density independent opacity
to be $\beta \simeq \mu \sim
2.4$. This ideal limit is expected for small $T_{\rm deep}$ and
hence low density. As $T_{\rm deep}$ is increased,
molecular interactions make the gas less ideal, reducing the value
of $C_p \nabla_{\rm ad}$ and increasing $\beta$. This qualitative
trend may be seen in Figure \ref{fig:LoverM_vs_S_num}.
The core temperature increases during the initial contraction phase
when the core is non-degenerate. A maximum is reached when $k_bT_c
\simeq E_F$, the Fermi energy, and subsequently $T_c$ decreases as
entropy decreases. In this degenerate phase, the core temperature
depends mainly on mass and entropy, with only a weak dependence on
irradiation. Figure \ref{fig:Tcore_vs_M} shows core temperature versus mass
for four adiabats and a range of irradiation temperatures. The
dependence on $\Tdeep$ gives only a slight broadening of each
adiabat. The dependence on mass is quite simple when sufficiently
degenerate. Figure \ref{fig:Tc_vs_S} shows the dependence of core
temperature on entropy for a range of masses and irradiation
temperatures, showing a simple exponential dependence at low
entropy. For the degenerate phase we write $T_c$ in the form (see
Figure \ref{fig:Tcore_vs_M})
\be
T_c(M,S) & = & T_{c,\rm ref}\left( \frac{M}{M_{\rm ref}}\right)^\gamma
\exp\left[ \delta \frac{(S-S_{\rm ref})}{k_b/m_p}\right].
\label{eq:Tcpl}
\ee
Using hydrostatic balance $P
\propto M^2/R^4$, and parameterizing $R \propto M^\lambda$, we estimate
the exponents to be
\be
\gamma & \simeq & \nabla_{\rm ad}(2-4\lambda)\simeq 0.6-0.7
\nonumber \\
\delta & \simeq & k_b/C_pm_p\simeq 0.5.
\ee
Next we solve for the change in core entropy with time for the
analytic model. We treat $T_{\rm deep}$ and $M$ as constants
during the evolution. The entropy equation integrated over the convective
core gives
\be
\frac{\partial S}{\partial t} & = & - \frac{L/M}{fT_c},
\label{eq:intentropy}
\ee
where $f=\int (dm/M)(T/T_c)\simeq 0.6-0.7$ and we have treated $\partial
S/\partial t$ as constant in space. Plugging eq.\ (\ref{eq:Tcpl}) and
(\ref{eq:Lpl}) into eq.\ (\ref{eq:intentropy}), we find the following
solution for the entropy with time
\be
\exp\left[\frac{S-S_{\rm ref}}{k_b/m_p} \right] & = & \left( 1 +
\frac{t}{t_{\rm S}} \right)^{-1/(\beta-\delta)},
\label{eq:Svst}
\ee
where the characteristic cooling time is
\be
&& t_{\rm S}(M,T_{\rm deep}) \nonumber \\
& = & \left( \frac{f}{\beta - \delta } \right)
\left( \frac{ k_b T_c/m_p}{ L/M } \right)_{\rm ref}
\left( \frac{T_{\rm deep}}{T_{\rm deep,ref}}
\right)^\alpha \left( \frac{M}{M_{\rm ref}} \right)^\gamma.
\label{eq:tS}
\ee
This solution has a number of notable features:
\begin{list}{}
\item (1) The fiducial evolution time, $Mk_b T_c/m_p L$, is just the
{\it initial} time to radiate away the core's thermal energy.
\item (2) At late times, $t \geq t_S$, the solution is a
power-law in time. As the initial cooling
time is very short, the power-law occurs for most of the
planet's lifetime.
\item (3) The exponent of the power-law involves the change of luminosity
and core temperature with respect to entropy.
\item (4) Evolution timescale is slowed for large irradiation or large planet
mass. The dependence on planet mass comes purely from the dependence
of $T_c$ on planet mass (at fixed entropy). The dependence on
irradiation is primarily through the luminosity.
\end{list}
Figure \ref{fig:S_vs_time} shows entropy versus time for a range of mass
and irradiation. Note the large spread in $S$ at a given age. For low
irradiation, $S \simeq 6-8k_b/m_p$ in the age range $1-10\ {\rm
Gyr}$, while $S$ can be as high as $\simeq 10k_b/m_p$ for
$\Tdeep=3500\ {\rm K}$. While the range of $\Tdeep$ shown here gives
fairly good power-laws, we note that at $\Tdeep<500\ {\rm K}$ there is
a break occurring at $\simeq 1\ {\rm Gyr}$, due to the increase in
luminosity seen in Figure \ref{fig:LoverM_vs_S_num} below $S=8k_b/m_p$.
While we have used the SCVH EOS and Allard et.al.(2001) opacities for
numerical estimates, the analytic solution makes it particularly clear
which quantities need by evaluated for a given opacity table and
EOS. The luminosity is sensitive only to the local conditions at the
radiative-convective boundary, while the core temperature involves
building static (i.e. not time-dependent) models.
\section{ Radius Evolution }
\label{sec:radiusev}
Planets are initially nondegenerate in their core, and undergo rapid
contraction until the core is
degenerate. If $T_{\rm eff}$ is
constant during
the contraction, the energy equation $d/dt(-3GM^2/7R)=-4\pi R^2 \sigma
T_{\rm eff}^4$ is solved to find the change in radius with
time (see, e.g. Bildsten et al. 1997)
\be
R(t) & \simeq & 8\ R_J\ \left( \frac{M}{M_J} \right)^{2/3} \left(
\frac{300\ {\rm K}}{T_{\rm eff}} \right)^{4/3} \left( \frac{1\ {\rm
Myr}}{t} \right)^{1/3}.
\ee
The core temperature $T_c \simeq GM\mu m_p/Rk_b \propto t^{1/3}$ is
increasing during the non-degenerate phase, and reaches a maximum when
$k_bT \simeq E_F$. For an ideal gas,
$k_bT/E_F$ is a function only of entropy, so that the maximum
temperature would occur at the same entropy for all planet
masses. Coulomb interactions suppress the
value of $\nabla_{\rm ad}$ below $2/5$, so that if $T_c \propto
M^{4/3}$ and $P_c \propto M^{10/3}$, the entropy at maximum
temperature will increase a bit with mass. This can be seen in
Figure \ref{fig:Tc_vs_S}, as the two higher masses have maxima at higher
entropy (off the plot) than the lowest mass.
Once degeneracy sets in, the radius is primarily determined in the
degenerate core of the planet, although as irradiation is increased
the contribution from the outer
envelope becomes larger due to the increased scale height. This is
clarified by writing the radius as an integral over pressure
\be
r(P) & = & \int^{P_c}_P d\ln P \left( \frac{P}{\rho g} \right) .
\ee
In the degenerate core, $P/\rho \propto E_F$ while in the
nondegenerate envelope $P/\rho \propto k_bT$. The
contribution from the core is larger when $E_F \gg k_b T$ unless the
number of pressure scale heights in the envelope is much larger than
the core.
The equation of state in the core for $M<M_J$ is
complicated by strong Coulomb interactions. For illustrative purposes,
an approximate equation of
state including the leading order contributions from Coulomb
interactions as well as ideal ion pressure is
\be
P & = & n_e \left( \frac{2}{5} E_F - \frac{3}{10} \frac{Z^2e^2}{a_i} \right) +
\frac{\rho k_b T}{\mu_i m_p},
\label{eq:simpleeos}
\ee
where $a_i=(4\pi \rho/\mu_i m_p)^{1/3}$ is the mean ion spacing and
$\mu_i m_p$ is the mean ion mass. The energy scales relevant for the
core are $E_F=(\hbar^2/2m_e)(3\pi^2
\rho/\mu_e m_p)^{2/3} \simeq 26\ {\rm eV}(\rho\mu_e^{-1}{\rm g\
cm^{-3}})^{2/3}$, $k_b T
\simeq 0.9\ {\rm eV} (T/10^4\ {\rm K})$, and $E_{\rm coul}=Z^2e^2/a_i
\simeq 20\ {\rm eV} Z^2 (\rho\mu_i{\rm g\ cm^{-3}})^{1/3}$ is the Coulomb
interaction energy between a nucleus and uniform electron cloud.
Ignoring the ion pressure term, the density
at zero pressure is $\rho_{\rm zp}\simeq 0.2 \mu_e Z^2
{\rm g\ cm^{-3}}$. As the central density in Jupiter mass objects is
near $\rho_{\rm zp}$, Coulomb interactions (and further the tendency
to form bound states) stiffen the EOS, and are important in
determining the radius.
Figure \ref{fig:R_vs_M_irr} shows mass versus radius for a range of core
entropy and irradiation. The effects of irradiation are seen to be
most severe at low mass and low entropy, since $\Tdeep$ is becoming a
significant fraction of the core temperature. At $M \simeq M_J/2$ and
low entropy, the range of irradiation temperatures shown here can
change the radius by as much as $50\%$. Radii for fully adiabatic
planets (not shown here) agree well with the $\Tdeep=500\ {\rm K}$
lines.
Figure \ref{fig:R_vs_S} shows radius versus entropy for a range of
masses and irradiation temperature. At late times in the evolution
when the entropy is small, the radius is converging to some constant
value which depends on {\it both} $M$ and $\Tdeep$. If the planet were
allowed to cool under a constant irradiation field indefinitely, it
would approach an isothermal state \citep{1977Icar...30..305H} at
$T=\Tdeep$ with a radius \footnote{In principle, $R_0$ can be
calculated by integrating the structure equations for a given EOS. In
practice, such low temperatures and high densities are not covered by
the SCVH EOS. In this paper, we compute the isothermal radius by
fitting evolutionary curves of radius versus entropy, defining $R_0$
by extrapolating to the small entropy limit.} $R=R_0$. Although in
practice planets will never reach this isothermal state, it is the
{\it minimum} radius to which the planet is evolving. Furthermore, it
is the {\it deviation} around the isothermal radius, $\delta R=R-R_0$
which is changing with age. As we now show, $\delta R$ has a
particularly simple behavior with time over the entire observable
range $\delta R \leq R$.
To motivate the following numerical calculations, we first discuss the
change in radius for a fluid element in mass shell $m$ as the
entropy is changed. The radius of a mass shell in the convection
zone can be written
\be
r^3(m,S) & = & \frac{3}{4\pi} \int_0^m \frac{dm'}{\rho(m',S)},
\ee
hence for fixed interior mass the change in radius with respect to
entropy is
\be
\frac{\partial r}{\partial S} & = & - \frac{1}{4\pi r^2}
\int_0^m \frac{dm'}{\rho(m',S)} \frac{\partial \rho(m',S)}{\partial
S} \rfloor_{m'}.
\ee
Given an equation of state $\rho(P,S)$, and switching to radius as the
integration variable, we find
\be
\frac{\partial r}{\partial S} & = & - \frac{1}{r^2}
\int_0^r {r'}^2 dr' \left( \frac{1}{C_p}
\frac{\partial \ln \rho}{\partial \ln T}\rfloor_{P}
+ \Gamma_1^{-1} \frac{\partial \ln P}{\partial S}\rfloor_{m'} \right)
\label{eq:drds}
\ee
where eq.\ (\ref{eq:thermo}) has been used. The second term in
eq.\ (\ref{eq:drds}) mainly corresponds to a uniform shift in pressure in
the core, due to the radius changing. Near the surface this term
must go to zero since pressure is proportional to external mass, which
is fixed. Consequently, the first term is most
important. From eq.\ (\ref{eq:simpleeos}), the volume expansion term is
$\partial \ln \rho /\partial \ln T|_P \propto k_bT/E_F$, with a
significant correction due to Coulomb interactions which acts to
increase the expansion since the electron pressure is effectively
lowered. Hence the change in radius in the core is proportional
\footnote{ Eq.\ (\ref{eq:simpleeos}) has ignored contributions to the
Coulomb correction which depend on temperature, and do not scale
linearly with temperature. Using the EOS in
\citet{2000PhRvE..62.8554P}, we find the contribution of these terms
to the
volume expansion seems to be somewhat smaller than the ideal ion
pressure. } to
$T_c$. As $T_c$ depends exponentially on the entropy
(eq.\ [\ref{eq:Tcpl}]), the contribution to the radius from the degenerate
core depends exponentially on entropy. In the nondegenerate envelope,
$\partial \ln \rho /\partial \ln T|_P \simeq -1$. Plugging this result
into eq.\ (\ref{eq:drds}) implies that the change in radius due to the
nondegenerate envelope scales linearly with entropy. As a consequence,
it is less important than the exponential dependence from the core.
A suite of evolutionary calculations has been done for
$M/M_J=0.32,1.0,3.2$ and $\Tdeep[K]=500, 1000, ..., 3500$ starting
from high entropy and evolved to ages greater than $15\ {\rm
Gyr}$. Given the run of $R(S)$, we fit a function
\be
R(S) & = & R_0+\delta R_0\exp(\eta m_p S/k_b)
\label{eq:Rfit}
\ee
to determine the isothermal radius $R_0$, coefficient $\delta
R_0$,
and exponent $\eta$. The coefficients $R_0$, $\delta R_0$ and $\eta$
depend
on $M$ and $\Tdeep$. The small
entropy points were more heavily weighted to force the fit to agree
there. The weighting was adjusted until the fit agreed for
as large a region in $S$ as possible (for the plots here we used
weighting $\propto R^{10}$.) A comparison of the fit against the data
for one example is given in Figure \ref{fig:goodfitexample}. The
agreement is good at small entropies, and gets worse for large entropy
as degeneracy is lifted. We find good agreement between $\eta \sim
0.5-0.7$ and $\delta$ from eq.\ (\ref{eq:Tcpl}), as expected if
$\delta R \propto T_c$.
The deviation of the radius about the isothermal value is plotted for
all runs over the age range $0.1-10\ {\rm Gyr}$ in
Figure \ref{fig:dR_vs_S}. Recall that $R_0$ is different for each
line. Note that each line is approximately a power-law, even to
$\delta R/R_0 \simeq 1$, where the degenerate approximation breaks
down. Hence the fitting formula often works better than naively
expected.
\begin{deluxetable}{rrrrr}
\tablecolumns{5}
\tablewidth{0pc}
\tablecaption{Parameters for the Fitting Function $R(t)$ in
eq.\ (\ref{eq:Rtfit}). }
\tablehead{
\colhead{$M/M_J$} & \colhead{$\Tdeep [K]$} &
\colhead{$\eta/(\beta-\delta)$} & \colhead{$R_0/R_J$} &
\colhead{$\delta R_1/R_J$} }
\startdata
0.316 & 500 & 0.31 & 0.836 & 0.308 \\
0.316 & 1000 & 0.25 & 0.881 & 0.330 \\
0.316 & 1500 & 0.23 & 0.905 & 0.359 \\
0.316 & 2000 & 0.23 & 0.952 & 0.361 \\
0.316 & 2500 & 0.32 & 1.05 & 0.401 \\
0.316 & 3000 & 0.43 & 1.15 & 0.867 \\
0.316 & 3500 & 0.50 & 1.30 & 2.49 \\
1.00 & 500 & 0.16 & 0.825 & 0.320 \\
1.00 & 1000 & 0.16 & 0.894 & 0.273 \\
1.00 & 1500 & 0.15 & 0.902 & 0.280 \\
1.00 & 2000 & 0.16 & 0.930 & 0.266 \\
1.00 & 2500 & 0.24 & 0.992 & 0.264 \\
1.00 & 3000 & 0.25 & 0.993 & 0.439 \\
1.00 & 3500 & 0.35 & 1.09 & 0.727 \\
3.16 & 500 & 0.20 & 0.915 & 0.243 \\
3.16 & 1000 & 0.16 & 0.934 & 0.238 \\
3.16 & 1500 & 0.17 & 0.947 & 0.227 \\
3.16 & 2000 & 0.18 & 0.962 & 0.217 \\
3.16 & 2500 & 0.26 & 0.994 & 0.239 \\
3.16 & 3000 & 0.30 & 1.02 & 0.379 \\
3.16 & 3500 & 0.41 & 1.10 & 0.673 \\
\enddata
\label{tab:fits}
\end{deluxetable}
We now combine the power-law cooling result in eq.\ (\ref{eq:Svst}) and
(\ref{eq:tS}) with the fit for the radius in eq.\ (\ref{eq:Rfit})
to find
\be
R(t) & = & R_0 + \delta R_0 \exp(\eta m_pS_{\rm ref}/k_b) \left( 1 +
\frac{t}{t_{\rm S}} \right)^{-\eta/(\beta-\delta)}.
\ee
At late times $t \gg t_S$, the deviation in radius from the isothermal
value is a power-law in time. In order to provide useful fits to our
evolutionary tracks, we parametrize this late time power-law as
\be
R(t) & = &
R_0 + \delta R_1 \left(
\frac{{\rm 1\ Gyr}}{t} \right)^{\eta/(\beta-\delta)},
\label{eq:Rtfit}
\ee
where $R_0$ is again the isothermal radius and $\delta R_1=\delta R_0
\exp(\eta m_pS_{\rm ref}/k_b) (t_{\rm S}/{\rm 1\
Gyr})^{\eta/(\beta-\delta)}$ is the
deviation at an age of $1\ {\rm Gyr}$. We fit tracks of $R(t)$ to find
the coefficients $R_0$, $\delta R_1$ and $\eta/(\beta-\delta)$ in the
same way as the fits for $R(S)$ in eq.\ (\ref{eq:Rfit}). The coefficients
are given in Table \ref{tab:fits}. Comparison between the numerical
evolutionary tracks for $R(t)$ and the analytic fit in
eq.\ (\ref{eq:Rtfit}) are given in Figure \ref{fig:R_vs_age}.
The agreement is generally very good.
Approximate values and scalings of the coefficients in Table
\ref{tab:fits} can be understood as follows. The expected power-law
index $\eta/(\beta-\delta) \simeq 0.6/3.0 = 0.20$ agrees well with the
temperature range $\Tdeep=1000-2000\ {\rm K}$ where $L$ is
independent of $\Tdeep$. At large irradiation, Figure \ref{fig:dR_vs_S}
shows $\eta$ increases and Figure \ref{fig:LoverM_vs_S_num} shows that
$\beta$ decreases, explaining the increase in
$\eta/(\beta-\delta)$. Since $\delta R_1 \propto
t_S^{\eta/(\beta-\delta)}
\propto \Tdeep^{\alpha \eta/(\beta-\delta)}$, regions of constant
(decreasing) slope in Figure \ref{fig:LoverM_vs_T_int} correspond to
$\delta R_1$ being constant (increasing). The magnitude of $\delta
R_1$ can be estimated from Figure \ref{fig:dR_vs_S} and
eq.\ (\ref{eq:tS}). Interestingly, $R_0$ can be somewhat bigger for
$M=0.32M_J$ than for the higher masses. While a larger radius is
expected for strong irradiation, we caution the reader about
interpretation of the exact values for $R_0$. It would be interesting
to compare the values obtained by fitting tracks with actual
calculations of isothermal planets given a sufficiently accurate low
temperature EOS.
Given measurements of planetary mass, radius and age, $T_{\rm deep}$
can be constrained. Figure \ref{fig:Tdeep_vs_age} shows the value of
$T_{\rm deep}$ required to explain a planet of a given mass and radius,
as represented by different lines, as a function of age. The lines slope
up to the right since the cooling must be slower (higher $T_{\rm deep}$)
to reach the same radius at larger age. As the lines are not horizontal
or vertical, there is significant degeneracy between $T_{\rm deep}$
and age. Large radii in the age range $1-10\ {\rm Gyr}$ can only be
explained by large irradiation temperatures for the mass range $0.32-3.2
M_J$. For each mass and radius, there is a minimum age which is set
by the unirradiated planet, resulting in a steep slope down to the left.
We shall use Figure \ref{fig:Tdeep_vs_age} in \S \
\ref{sec:applications} to constrain $T_{\rm deep}$ for the observed
transiting planets.
\section{Applications to Transiting Planets}
\label{sec:applications}
We now compare our theory to the observed masses and radii of
the transiting planets (Table \ref{tab1}). Figure
\ref{fig:R_vs_M_data} shows radius versus mass for
the observed transiting planets. The points with errorbars are the
data. The three different hatched regions show $T_{\rm deep}=500, 2500,
3000\ {\rm K}$ from bottom to top. The change is gradual from $T_{\rm
deep}=500$ to $2500$, and then accelerates for higher temperatures (see
Figure \ref{fig:LoverM_vs_T_int}). Within each hatched region, a spread of
ages from $1$ (top) to $10\ {\rm Gyr}$ (bottom) is shown. The radius of
HD 149026 is so small as to be well outside the plot. It clearly has a
large abundance of heavy elements. The radii of the other eight planets
can be broadly explained with solar composition, ages in the range $1-10\
{\rm Gyr}$, and temperatures deep in the atmosphere $T_{\rm deep}
\leq 3000\ {\rm K}$. The largest radii requiring the most irradiation
are HD 209458, HD 189733 and OGLE-TR-56.
There are significant uncertainties in fitting
stars on the main sequence to find stellar ages. Hence there is
motivation to understand how a range of ages affects the range of
observed radii. Figure \ref{fig:dR_vs_S} shows deviation from the
isothermal radius by factors $1.1-2$ in the age range $0.1-10\ {\rm
Gyr}$. The length of each track gives an idea of the uncertainty in
radius due to an uncertainty in age. Using the fitting formula in eq.\
(\ref{eq:Rtfit}), the fractional difference in radius between ages
$t_1$ and $t_2$ is $\simeq [\eta/(\beta-\delta)](\delta
R_1/R_0)\ln(t_2/t_1)$. For the strongly irradiated case, if we choose
characteristic values $\eta/(\beta-\delta)=0.5$, $\delta R_1/R_0=0.5$,
and a factor of two error in age $t_2=2t_1$, the fractional error in
radius is $17\%$. Hence, the age dependence for strongly irradiated
planets is important because (i) the decrease in time is steep, and
(ii) strong irradiation increases the size of $\delta R_1$ relative to
$\delta R_0$.
Next, the parameter $T_{\rm deep}$ is crucial for the cooling rate,
but is not directly measurable. Here we constrain $T_{\rm deep}$
using measured mass, radius and age. We then compare $T_{\rm
deep}$ to the equilibrium temperature.
We interpolate over the ${\rm age}-T_{\rm deep}$ tracks in Figure
\ref{fig:Tdeep_vs_age} for the mass and radii appropriate for each
planet (except HD 149026, which we do not discuss). Since the
uncertainty in $T_{\rm deep}$ due to the error bar in planet mass is
smaller than that due to the error bar in radius, we fix the
planet mass at the central value and only vary the radius. For those
planets with an age range quoted in the literature, we show the age
range in the plot, and derive the range of $T_{\rm deep}$ consistent
with the age range. These values are listed in Table \ref{tab1}. For
those planets with no age determination, we find the maximum value of
$T_{\rm deep}$ consistent with an age less than $10\ {\rm Gyr}$.
The constraints on $T_{\rm deep}$ for all planets except HD 149026 are
shown in Figures \ref{fig:HD209458TRES1}, \ref{fig:OGLETR56OGLETR10},
\ref{fig:OGLETR111OGLETR113} and \ref{fig:OGLETR132HD189733}, and
summarized in Table \ref{tab1}. The $T_{\rm deep}$ of HD 209458 is
best constrained due to the small error bar on mass and radius, as
well as detailed fitting of the parent star to find the age. From
Figure \ref{fig:HD209458TRES1} we find $T_{\rm deep}=2200-2800\ {\rm
K}$; HD 209458b is not consistent with an un-irradiated
planet. OGLE-TR-56 has a weak lower limit on $T_{\rm deep}$ which is far
less than the equilibrium temperature. All
other planets with age constraints have only upper limits, set by the
upper limit on the radius, since the lower limit on the radius is
consistent with no irradiation. Plots are provided for those planets
with no age constraints at the present time.
The equilibrium temperature $T_{\rm eq} \equiv
T_{\ast}(R_\ast/2a)^{1/2}$ is measurable, but plays no part in our
model. On the other hand, the temperature of the deep isotherm is not
measurable, but is crucial for the cooling rate. From radiative
transfer models, we expect these two temperatures to be roughly
proportional, the exact ratio determined by the size of the greenhouse
effect (\S \ref{sec:surfacebc}). Hence, they should be strongly
correlated. Figure \ref{fig:TeqvsTdeep} shows a plot of $T_{\rm eq}$
versus $T_{\rm deep}$. The large error bars on $T_{\rm deep}$, due to
large error bars on the radius, prevent one from drawing robust
conclusions. It is in principle possible for a correlation (sloping up
to the right) to exist given the current error bars, however, it is
not required.
Tighter constraints on $T_{\rm deep}$ require the following:
\begin{itemize}
\item Significantly smaller error bars on the radii of OGLE-TR-56 and
OGLE-TR-132.
\item An age estimate is needed for HD 189733. If it is found to have an age
$\geq 1\ {\rm Gyr}$, $T_{\rm deep}$ will be well constrained with a
value much larger than $T_{\rm eq}$, similar to HD 209458b.
\item Age estimates are needed for OGLE-TR-10, OGLE-TR-111, and
OGLE-TR-113. However, given the present error bar on radii of
OGLE-TR-10 and OGLE-TR-113, $T_{\rm deep}$ will be constrained only
at the factor of two level. OGLE-TR-111 is an interesting case, as
it must be older than $\sim 3-4\ {\rm Gyr}$ to be consistent with our model.
\end{itemize}
We encourage efforts in these directions.
\section{conclusions}
\label{sec:conclusions}
We have presented calculations of cooling and radius
evolution for strongly irradiated planets. Novel aspects of this model
are the following:
\begin{itemize}
\item We argue that the generic outcome of strong surface heating, whether it
be due to absorption of stellar flux or dissipation of winds and
tidal flow, is that a deep isothermal region exists above the
radiative-convective boundary. The
thermal time in this layer is sufficiently long that the temperature
profile is approximately spherically symmetric, irrespective of the
size of the asymmetry near the photosphere. We assign this region
the temperature $T_{\rm deep}$ and treat it as a boundary
condition for the cooling models.
\item We show that the cooling flux is determined at the
radiative-convective boundary, which is much deeper than the
photosphere. Scalings of the flux with
core entropy, $T_{\rm deep}$, and mass are computed.
\item These scalings allow us to derive an analytic model for the cooling,
which shows power-law decrease over a large range of parameter
space. The part of the radius which changes in time (the deviation from
the isothermal planet) is also a
power-law. An analytic formula for radius evolution is given in
eq.\ (\ref{eq:Rtfit}), with coefficients in Table \ref{tab:fits}.
\item While we have used the SCVH EOS and Allard et.al.(2001)
opacities for numerical estimates, the analytic solution makes it
particularly clear which quantities need to be evaluated for a given
opacity table and EOS. The luminosity is sensitive only to the local
conditions at the radiative-convective boundary, while the core
temperature involves building static (i.e. not time-dependent)
models.
\end{itemize}
We have compared our theory to observed masses and radii for the
transiting planets in Table \ref{tab1} (except for HD 149026, which
clearly has a large abundance of heavy elements). Our findings are as follows:
\begin{itemize}
\item Figure \ref{fig:R_vs_M_data} shows mass versus radius for eight
transiting planets, compared to our model. The radii can be broadly
explained with solar composition, ages in the range $1-10\ {\rm
Gyr}$, and temperatures deep in the atmosphere $T_{\rm deep} \leq
3000\ {\rm K}$. The largest radii requiring the most irradiation to
explain are HD 209458, HD 189733 and OGLE-TR-56.
\item Figures \ref{fig:HD209458TRES1}, \ref{fig:OGLETR56OGLETR10},
\ref{fig:OGLETR111OGLETR113} and \ref{fig:OGLETR132HD189733} show
constraints on $T_{\rm deep}$ using measured masses, radii, and ages
(when available), and their uncertainties. We find that only HD
209458b is well constrained, with $T_{\rm deep}=2200-2800\ {\rm
K}$. OGLE-TR-56 has a weak lower limit, and the other six planets
have only upper limits, due to the
large measurement uncertainty in the radius, or lack of an age
determination. These constraints are summarized in Table \ref{tab1}.
\item The equilibrium temperature $T_{\rm eq}$ is measurable, but
plays no part in our model. The deep isothermal temperature $T_{\rm
deep}$ is not measurable, but is crucial for the cooling
rate. Radiative transfer calculations find these two temperatures
should be strongly correlated. Figure \ref{fig:TeqvsTdeep} shows
$T_{\rm eq}$ versus $T_{\rm deep}$. As only upper limits on $T_{\rm
deep}$ are available for all but HD 209458b and OGLE-TR-56,
it is difficult to draw
conclusions at the present time. It is in principle possible for a
correlation to exist given the current error bars, however, it is
not required.
\end{itemize}
We hope that our models have illuminated the need for more accurate
ages and radii. Once those are in hand, our calculations
will provide a measurement of $T_{\rm deep}$ of adequate accuracy to
compare to $T_{\rm eq}$, thus constraining greenhouse physics and
day-night transport.
\vspace{4cm}
\acknowledgements
This project arose out of a lunchtime conversation with Adam Burrows
discussing cooling models for gas giant planets. We thank Tristan
Guillot and France Allard for helpful advice on opacities. We also
thank Omer Blaes, Shane Davis, Eric Pfahl and Evan Scannapieco for
useful discussions. We would also like to thank the referee for constructive
comments which improved the presentation of this paper. Phil Arras was
supported by the NSF Astronomy and Astrophysics Postdoctoral Fellowship,
and the Kavli Institute for Theoretical Physics during this project.
This work was supported by the National Science Foundation under
grants PHY99-07949 and AST02-05956.
|
Title:
The flat synchrotron spectra of partially self-absorbed jets revisited |
Abstract: Flat radio spectra with large brightness temperatures at the core of AGN and
X-ray binaries are usually interpreted as the partially self-absorbed bases of
jet flows emitting synchrotron radiation. Here we extend previous models of
jets propagating at large angles to our line of sight to self-consistently
include the effects of energy losses of the relativistic electrons due to the
synchrotron process itself and the adiabatic expansion of the jet flow. We also
take into account energy gains through self-absorption. Two model classes are
presented. The ballistic jet flows, with the jet material travelling along
straight trajectories, and adiabatic jets. Despite the energy losses, both
scenarios can result in flat emission spectra, however, the adiabatic jets
require a specific geometry. No re-acceleration process along the jet is needed
for the electrons. We apply the models to observational data of the X-ray
binary Cygnus X-1. Both models can be made consistent with the observations.
The resulting ballistic jet is extremely narrow with a jet opening angle of
only 5". Its energy transport rate is small compared to the time-averaged jet
power and therefore suggests the presence of non-radiating protons in the jet
flow. The adiabatic jets require a strong departure from energy equipartition
between the magnetic field and the relativistic electrons. These models also
imply a jet power two orders of magnitude higher than the Eddington limiting
luminosity of a 10 solar mass black hole. The models put strong constraints on
the physical conditions in the jet flows on scales well below achievable
resolution limits.
| https://export.arxiv.org/pdf/astro-ph/0601103 |
\title[Flat spectra of self-absorbed jets]{The flat synchrotron spectra of partially self-absorbed jets revisited}
\author[C.R. Kaiser]{C. R. Kaiser\thanks{crk@soton.ac.uk}\\
School of Physics \& Astronomy, University of Southampton, Southampton SO17 1BJ
}
\begin{keywords}
radiation mechanisms: non-thermal -- radio continuum: general -- methods: analytical -- galaxies: active -- stars: individual: Cygnus X-1 -- stars: outflows
\end{keywords}
\section{Introduction}
The centres or cores of many AGN show a flat radio spectrum in the sense that for the flux density as a function of frequency $\nu$ we observe $F_{\nu} \propto \nu^{\alpha}$ with $\alpha \sim 0$. The high surface brightness temperature associated with these spectra suggests a synchrotron origin of the emission. Observations with high spatial resolution reveals that the flat spectrum arises in the base of jet flows which continue to much larger scales \citep[for a review see][]{tc91}. Similar flat or inverted ($\alpha > 0$) radio spectra are also observed in X-ray binaries in the low-hard state \citep[e.g.][]{rf01}. If optically thin, the flat synchrotron spectrum would imply a power-law energy distribution of the radiating relativistic electrons with a slope of unity. Such a distribution is very unlikely to arise for the usually assumed mechanism for the acceleration of the electrons at shock fronts \citep[e.g.][]{ab78}.
A magnetised plasma containing very energetic electrons with a power-law energy distribution will produce a power-law spectrum at high frequencies. The slope of the spectrum, typically $\alpha < 0$, is determined by the slope of the energy distribution. However, below a critical frequency the radiating electrons will re-absorb some of the photons. In this self-absorbed, optically thick regime the spectrum has a power-law slope of $5/2$, independent of the slope of the electron energy distribution \citep[e.g.][]{rl79}. The spectrum of a uniform, self-absorbed synchrotron source therefore shows a pronounced peak. \citet{bk79} pointed out that in a jet the plasma conditions are changing along the flow and therefore the peaks of the self-absorbed spectra of different parts of the jet can occur at different frequencies. If the plasma conditions change such that the spectra peak at the same level, then the overall spectrum, observed with a spatial resolution insufficient to resolve the individual parts of the jet, will be flat. Their model has become the standard tool for interpreting observations of flat radio spectra from jetted sources.
In the \citet{bk79} model the jet is assumed to have a conical geometry, i.e. the velocity with which the jet is expanding sideways, is constant. The bulk velocity of the jet material along the jet axis is also assumed to be constant. The magnetic field is assumed to be directed perpendicular to the jet axis and `frozen' into the jet plasma. Adiabatic losses of the electrons are mentioned by the authors, but are assumed to be replenished by an unknown, continuous re-acceleration process along the entire jet. The same assumption is made for radiative energy losses associated with the emission of synchrotron radiation. The subsequent model of \citet{am80} includes a simplified treatment of energy losses of the electrons due to adiabatic expansion and radiative processes. It also allows for more confined jets, i.e. the jets are not necessarily conical. With these assumptions, the model is unable to produce a flat emission spectrum. A similar model was developed by \citet{hj88}. They consider adiabatic, but not radiative, energy losses of the electrons. The jet geometry is again conical, but they also investigate a more confined jet. The model can predict flat spectra, but \citet{hj88} point out that these may only arise under special circumstances, particularly in the case of confined jets. The model of \citet{gm98} includes a detailed treatment of the energy losses of the relativistic electrons, but it concentrates only on the optically thin part of the spectrum of jets propagating close to the line of sight for which numerical solutions are presented. The perhaps most comprehensive study of jet emission models is that of \citet{sr82} which includes the effects of energy losses on the electron population, but neglects the effects of self-absorption on the electron energy spectrum.
In this paper we extend the previous models by including adiabatic and radiative energy losses and gains (due to absorption) for the electrons as well as investigating various possibilities for the evolution of the magnetic field. We consider two distinct cases: The ballistic and the adiabatic jet models. In the ballistic case the jet material follows straight trajectories and does not behave like a fluid, because individual fluid elements do not interact with each other. In many ways this model is similar to the \citet{bk79} model, but we show that because of self-absorption effects we do not need to invoke a re-acceleration process to achieve flat emission spectra. in the adiabatic jet model the relativistic electrons suffer from adiabatic energy losses as well as radiative losses. Again we show that the models can produce flat spectra without re-acceleration of the electrons, but only for a very specific jet geometry. The emphasis of our treatment is on the construction of analytical models and so we concentrate on jets propagating at large angles to the line of sight, i.e. the viewing angle is larger than the inverse of the Lorentz factor of the jet flow.
In Section \ref{conical} we briefly discuss the basic properties of our jets in terms of their geometry, the evolution of the magnetic field and that of the relativistic electrons. We present the first fully analytical solution of the equations governing partially self-absorbed synchrotron emission from a jet in Section \ref{radiation}. Section \ref{simple} summarises the model results for the case without radiative energy losses as studied in many previous models. In Section \ref{cutoff} we develop the formalism for including radiative energy losses in the model and the resulting spectra are discussed in Section \ref{losses}. We apply the model to the data obtained for Cygnus X-1 in Section \ref{obs} and derive the properties of this jet. Finally, we summarise our conclusions in Section \ref{conc}.
\section{The model}
In this Section we derive the emission properties of partially self-absorbed jets neglecting radiative energy losses of the relativistic electrons. Note that we are concentrating on jets at comparatively large angles to our line of sight, $\vartheta$. As we will point out further down this greatly simplifies the determination of the optical depth of the jet material.
\subsection{The basic jet properties}
\label{conical}
\subsubsection{Jet geometry and velocity}
We take the $x$-axis as the centre of a jet that is rotationally symmetric about this axis. The geometrical shape of the jet is then given by a one-dimensional function $r(x)$ defining the jet radius with respect to the $x$-axis. Analogous to previous work we parameterize this function as $r(x)= r_0 \left(x / x_0 \right)^{a_1} = r_0 l^{a_1}$, where $x_0$ is an arbitrary position along the $x$-axis defining the dimensionless coordinate $l$ and $r_0$ is a constant scaling factor. The value of the exponent $a_1$ depends on the details of the confinement of the jet. Confinement by external pressure is the simplest mechanism \citep{br74}, but can lead to problems with the collimation of the jet \citep{bbr84}. Confinement by magnetic fields has also been suggested by various authors, but it is unlikely that magnetic fields alone, without additional gas pressure, can collimate the jet on large scales \citep{mb95}. For our purposes here we do not need to specify the details of the jet confinement and we will assume that $0\le a_1 \le1$. In principle one could also envisage highly overpressured jets with an accelerating expansion rate, i.e. $a_1 > 1$. However, the pressure in such jets would fall very rapidly and they would quickly evolve to a situation where $a_1 \le 1$.
The extreme case of a highly overpressured jet is that of a jet flow expanding into a (near) vacuum. In such a ballistic jet, as opposed to the adiabatic, confined jet discussed above, the jet freely expands in the direction perpendicular to the jet axis. In this process random, `thermal' energy of the jet material is converted to ordered, kinetic energy associated with the sideways expansion. The random energy of the electrons giving rise to the synchrotron emission is reduced by this adiabatic expansion.
However, in Section \ref{cyg} we apply the ballistic jet model to the observations of the jet in Cygnus X-1. There we will find a very small opening angle for the ballistic jet implying very small adiabatic expansion losses. Therefore we can assume that in the limiting case of a ballistic jet studied here the relativistic electrons do not suffer energy losses other than those associated with radiation processes.
We assume in this paper that the velocity of the jet material along the jet axis, $v_{\rm j}$, is constant. While this is justified in the case of the ballistic jet, the confined, adiabatic jets can be accelerated, for example, by a pressure gradient in the confining medium. In the model of \cite{br74} the Lorentz factor of the bulk velocity is proportional to $p_{\rm x}^{-1/4}$, where $p_{\rm x}$ is the pressure of the external medium. As long as the external pressure gradient is shallow, the Lorentz factor of the jet flow will be only a very weak function of the position along the jet axis. Similar arguments hold for a magnetically confined jet. The constant bulk velocity of the jet also implies that a given volume element $\Delta V$ travelling with the jet flow will only expand sideways according to $\Delta V \propto r^2$. A constant jet velocity also simplifies the model greatly as we can ignore the effects of varying length contraction along the jet axis \citep{gk04}.
\subsubsection{Magnetic field}
The strength of the magnetic field changes during the sideways expansion of the jet material. In general, we parameterize the evolution of the magnetic field as $B = B_0 l^{-a_2}$. For flux freezing of the magnetic field and using flux conservation we have that the field component parallel to the jet axis, $B_{\parallel}$, is proportional to $r^{-2}$. Also, the magnetic field components perpendicular to the jet axis, $B_{\perp}$, are proportional to $r^{-1}$. For an initially mixed field, $B_{\perp}$ will always become the dominant component and so $a_2 = a_1$. The perpendicular magnetic field may also contribute to the confinement of the jet. For completeness we also consider a purely parallel configuration of the magnetic field with $a_2 = 2a_1$. Finally, if the magnetic field is constantly tangled by turbulent motions in the jet material on scales smaller than the jet radius, then it can remain isotropic and it behaves like a relativistic fluid with $B = B_0 l^{-4 a_1 / 3}$ and $a_2 = 4 a_1 /3$ \citep[e.g.][]{hb00}. This is analogous to the behaviour of the magnetic field in an isotropic expansion \citep[e.g.][]{ml94}, but is clearly incompatible with flux freezing. The last case of a permanently isotropic field cannot be realised in the ballistic jet as it would require that the jet material behaves like a fluid.
\subsubsection{Relativistic electrons}
In order to produce synchrotron emission the jets must contain a population of relativistic electrons. We assume that the latter has a power-law energy distribution of the form
\begin{equation}
N(E)\,{\rm d}E = \kappa E^{-p} \, {\rm d}E,
\end{equation}
where $E$ is the electron energy, $E=\gamma m_{\rm e} c^2$, and $\kappa$ is a scaling independent of $E$. $\gamma$ is the Lorentz factor associated with the relativistic motion of the electrons. In this section we do not impose a high-energy cut-off on the energy distribution and we neglect radiative energy losses. Even so the energy distribution of the electrons changes as the jet expands. We represent the evolution of the electron distribution by setting $\kappa = \kappa _0 l^{-a_3}$.
For a given volume of jet material $\Delta V$ particle conservation demands that
\begin{equation}
\Delta V \kappa \gamma ^{-p} \, {\rm d}\gamma = \Delta V_0 \kappa _0 \gamma _0^{-p} \, {\rm d}\gamma_0,
\label{dist}
\end{equation}
where all quantities with subscript `0' refer to their values at $x=x_0$. Therefore for the ballistic jet we have $a_3 = 2 a_1 = 2$.
For the adiabatic jet we need to include energy losses due to the jet expansion. Since most of the electrons are highly relativistic we have \citep[e.g.][]{ml94}
\begin{equation}
\frac{\partial \gamma}{\partial t} = - \frac{1}{3} \gamma \frac{\partial \ln \left( \Delta V \right)}{\partial t},
\label{adiabat}
\end{equation}
which has the solution
\begin{equation}
\gamma = \gamma _0 \left( \frac{\Delta V}{\Delta V_0} \right)^{-1/3},
\label{adsol}
\end{equation}
and it follows that
\begin{equation}
\frac{\partial \gamma _0}{\partial \gamma} = \left( \frac{\Delta V}{\Delta V_0} \right)^{1/3}.
\end{equation}
Re-arranging equation (\ref{dist}) and substituting yields
\begin{equation}
\kappa \gamma ^{-p} \, {\rm d} \gamma = \frac{\Delta V_0}{\Delta V} \kappa _0 \left( \frac{\Delta V}{\Delta V_0} \right)^{-p/3} \gamma^{-p} \left( \frac{\Delta V}{\Delta V_0} \right)^{1/3} \, {\rm d} \gamma.
\end{equation}
Collecting terms and remembering that $\Delta V \propto r(x)^2$ we find $a_3 = (4+2p) a_1/ 3$.
\subsubsection{Individual models}
On the basis of the discussion above we formulate five individual models distinguished by the magnetic field behaviour. The ballistic jet models, B1 and B2, as well as the adiabatic models, A1 and A2, correspond to a perpendicular and parallel field structure, respectively. The adiabatic model A3 represents the case of an isotropic magnetic field in the jet. The relevant coefficients describing the jet geometry and the behaviour of the magnetic field and relativistic particles are summarised in Table \ref{expo}.
\begin{table}
\begin{tabular}{llccccc}
& & $a_1$ & $a_2$ & $a_3$ & $a_4$ & $ a_5$\\
\hline
ballistic & B1 & 1 & 1 & 2 & $-4-p$ & $\frac{-5}{4+p}$\\[1.5ex]
& B2 & 1 & 2 & 2 & $-6-2p$ & $\frac{-3}{2p+3}$\\[1.5ex]
adiabatic & A1 & $a$ & $a$ & $\frac{\left(4 + 2p \right) a}{3}$ & $\frac{-a \left(8 + 7p \right)}{3}$ & $\frac{-3 \left( 3a +2 \right)}{a \left(8 +7p \right)}$\\[1.5ex]
& A2 & $a$ & $2a$ & $\frac{(4+2p)a}{3}$ & $\frac{-2 a\left( 7 +5p \right)}{3}$ & $\frac{-3 \left(2a +1 \right)}{a \left(7+5p \right)}$\\[1.5ex]
& A3 & $a$ & $\frac{4 a}{3}$ & $\frac{(4+2p)a}{3}$ & $\frac{-2a \left(5+4p \right)}{3}$ & $ \frac{-5a - 3}{a \left(5 +4p \right)}$\\
\hline
\end{tabular}
\caption{Exponents of the model parameters used in this paper. See text for details.\label{expo}
}
\end{table}
\subsection{Partially self-absorbed synchrotron emission from a jet}
\label{radiation}
From the expressions for $r$, $B$ and $\kappa$ defined in the previous Section, we can now build a model for the emission from the jet. For this purpose we split the jet into small segments of length ${\rm d} x$ along the $x$-axis. We assume that the segments move along the jet axis at a constant velocity $v_{\rm j} = \beta _{\rm j} c$ corresponding to a Lorentz factor $\gamma _{\rm j}$. The jet axis is at an angle $\vartheta$ to the line of sight of the observer and so the Doppler factor for an approaching (`$-$') or receding (`$+$') jet is $\delta _{\mp} = \left[ \gamma _{\rm j} \left( 1 \mp \beta _{\rm j} \cos \vartheta \right)\right]^{-1}$. The observable monochromatic intensity of one such segment taking into account absorption is
\begin{equation}
I_{\nu} = \delta _{\mp}^3 \frac{J_{\nu}}{4 \pi \chi_{\nu}} \left( 1 - e^{-\chi_{\nu} r(x)} \right),
\end{equation}
where $J_{\nu}$ is the emissivity per unit volume and $\chi_{\nu}$ is the absorption coefficient. For convenience in the development of the model the frequency $\nu$ is measured in the restframe of the jet material. It is related to the observing frequency by $\nu_{\rm ob} = \delta_{\mp} \nu$. Here and in the following we assume that the average path of a photon through the jet has the length $r(x)$. An exact calculation of the radiative transfer of photons through various jet elements would have to take into account relativistic aberration effects. It is therefore complex and impossible in an analytical model. The assumption of an average path length $r$ will not introduce a large error as long as the angle to the observer's line of sight is large.
The jet segment has a surface area of $2 \pi r(x)\,{\rm d} x$ and so the observable flux density of the segment is given by
\begin{equation}
{\rm d} F_{\nu} = \delta _{\mp}^3 \frac{r(x) J_{\nu}}{2 D^2 \chi_{\nu}} \left( 1 - e^{-\chi_{\nu} r(x)} \right) \, {\rm d} x,
\label{pflux}
\end{equation}
where $D$ is the distance of the jet from the observer.
Substituting the dimensionless variable $l=x/x_0$, we can express $J_{\nu}$ and $\chi_{\nu}$ in SI units as \citep{ml94}
\begin{eqnarray}
J_{\nu} & = & J_0 \nu^{\left(1-p \right)/2} l^{-a_3-a_2 \left( p+1 \right)/2}\nonumber\\
\chi_{\nu} & = & \chi _0 \nu^{\left(-p-4 \right)/2} l^{-a_3-a_2 \left(p+2 \right)/2},
\label{jk}
\end{eqnarray}
with
\begin{eqnarray}
J_0 & = & 2.3 \times 10^{-25} \left( 1.3 \times 10^{37} \right)^{\left(p-1 \right)/2} c_1(p) \nonumber \\
& & B_0^{\left( p + 1\right)/2} \kappa _0 \,{\rm W\,m^{-3}\,Hz^{-1}}\nonumber\\
\chi_0 & = & 3.4 \times 10^{-9} \left( 3.5 \times 10^{18} \right)^p c_2(p) B_0^{\left( p +2 \right)/2} \kappa _0 \, {\rm m^{-1}},
\label{jk0}
\end{eqnarray}
and the constants $c_1(p)$ and $c_2(p)$ given by equations 18.49 and 18.74 in \citet{ml94}. Substituting into equation (\ref{pflux}) and integration gives the total flux density of the jet as
\begin{equation}
F_{\nu} = \delta _{\mp}^2 \frac{x_0 r_0 J_0}{2 D^2 \chi_0} \nu^{5/2} \int _{l_{\rm min}}^{l_{\rm max}} l^{a_1+a_2/2} \left[ 1 - e^{-\tau} \right] \, {\rm d}l,
\label{lflux}
\end{equation}
where $l_{\rm min}$ and $l_{\rm max}$ are the physical limits of the jet flow along the $x$-axis and the optical depth of the jet material is given by
\begin{equation}
\tau (l)= \chi_{\nu} r(x) = \chi _0 r_0 \nu^{\left(-p-4\right)/2} l^{a_1-a_3-a_2 \left(p+2 \right)/2}.
\label{depth}
\end{equation}
The reduction in the number of Doppler factors arises from our assumption of a steady state of the jet flow. In principle further relativistic corrections must be applied in the case of mixed optically thin and thick emission, but these corrections are small \citep{tc91} and we neglect them here for simplicity.
It is convenient to recast equation (\ref{lflux}) with the help of equation (\ref{depth}) as an integration over optical depth,
\begin{equation}
F_{\nu} = \delta _{\mp}^2 \frac{x_0 r_0 J_0}{a_4 D^2 \chi _0} \nu^{5/2} \tau _0^{-a_5} \int _{\tau _{\rm max}}^{\tau _{\rm min}} \tau^{a_5-1} \left( 1- e^{-\tau} \right) \, {\rm d} \tau,
\label{flux}
\end{equation}
where
\begin{equation}
a_4 = 2a_1 -2a_3-\left(p+2\right) a_2
\end{equation}
and
\begin{equation}
a_5 = \frac{2a_1+a_2+2}{a_4}.
\end{equation}
The coefficients $a_4$ and $a_5$ are listed for the ballistic and adiabatic jets in Table \ref{expo}. $\tau_0$ is given by setting $l=1$ in equation (\ref{depth}) while $\tau_{\rm max} = \tau (l_{\rm min})$ and $\tau_{\rm min} = \tau (l_{\rm max})$, which reflects the fact that the optical depth is always greatest in the innermost regions of the jet. The solution of equation (\ref{flux}) is given by
\begin{equation}
F_{\nu} = \delta _{\mp}^2 \frac{x_0 r_0 J_0}{a_4 D^2 \chi_0} \nu^{5/2} \tau_0^{-a_5} \left[ \Gamma \left(a_5, \tau \right)+\frac{1}{a_5} \tau^{a_5} \right]_{\tau_{\rm max}}^{\tau_{\rm min}}.
\label{sol}
\end{equation}
Here, the incomplete $\Gamma$-function is defined as
\begin{equation}
\Gamma \left( a , z \right) = \int_z^{\infty} t^{a-1} e^{-t} \, {\rm d} t.
\end{equation}
\subsection{Spectra in the absence of radiative energy losses and without a high-energy cut-off}
\label{simple}
We can immediately recover the well-known solutions for an entirely optically thin ($\tau _{\rm min} \ll 1$ and $\tau _{\rm max} \ll 1$) and an entirely optically thick ($\tau_{\rm min} \gg 1$ and $\tau_{\rm max} \gg 1$) jet. We note that for all choices of $a_1$ discussed above and for physical reasonable values for the power-law exponent $2\le p \le 3$ we find $a_5 < 0$ (see Table \ref{expo}). The incomplete $\Gamma$-function has the series representation \citep{gr00}
\begin{equation}
\Gamma \left(a, z \right) = \Gamma \left( a \right) - \sum_{n=0}^{\infty} \frac{\left( -1 \right)^n z^{a+n}}{n! \left( a+n \right)}.
\label{series}
\end{equation}
For $\tau \ll 1$ we ignore all terms beyond $n=1$ and thus obtain
\begin{equation}
\left[ \Gamma \left( a_5, \tau \right)+\frac{1}{a_5} \tau^{a_5} \right]_{\tau_{\rm max}}^{\tau_{\rm min}} \sim \frac{1}{1+a_5} \left( \tau _{\rm min}^{1+a_5} - \tau _{\rm max}^{1+a_5} \right).
\end{equation}
Because of equation (\ref{depth}) we have $\tau_0 \propto \tau_{\rm min} \propto \tau _{\rm max} \propto \nu^{\left(-4-p\right)/2}$, and from equation (\ref{sol}) it then follows that $F_{\nu} \propto \nu^{\left(1-p \right)/2}$ as expected.
For large optical depths we can use the limit for the incomplete $\Gamma$-function, $\lim_{z\rightarrow \infty} \Gamma \left(a, z \right) =0$ \citep{gr00}. Thus
\begin{equation}
\left[ \Gamma \left( a_5, \tau \right)+\frac{1}{a_5} \tau^{a_5} \right]_{\tau_{\rm max}}^{\tau_{\rm min}} \sim \frac{1}{a_5} \left( \tau_{\rm min}^{a_5} - \tau_{\rm max}^{a_5} \right),
\label{largetau}
\end{equation}
and from the proportionality of the optical depths terms it then follows that $F_{\nu} \propto \nu^{5/2}$, again as expected.
The final special case is that of a spatially very extended jet or `long' jet. If the physical dimensions of the long jet, $l_{\rm min}$ and $l_{\rm max}$, are such that $\tau_{\rm max} \rightarrow \infty$ and $\tau_{\rm min} \rightarrow 0$, then for $-1 < a_5 < 0$ we have
\begin{equation}
\left[ \Gamma \left( a_5, \tau \right)+\frac{1}{a_5} \tau^{a_5} \right]_{\tau_{\rm max}}^{\tau_{\rm min}} \sim \Gamma \left( a_5 \right),
\end{equation}
which implies
\begin{equation}
F_{\nu} \propto \nu^{\left[5 +\left( p+4 \right) a_5\right]/2}.
\label{longslope}
\end{equation}
Using the results summarised in Table \ref{expo}, we recover the result of \citet{bk79} that the ballistic jet with the magnetic field perpendicular to the jet axis (model B1) has a flat spectrum, i.e. $F_{\nu}$ is independent of $\nu$, if it is extended and $-1 < a_5 < 0$. The ballistic jet with a parallel magnetic field, model B2, can never produce a flat spectrum for positive $p$ because equation (\ref{longslope}) predicts $F_{\nu} \propto \nu^ {\left(7p +3 \right) / \left[2 \left( 2p + 3\right) \right]}$. For the adiabatic jet models we can substitute the expressions for $a_5$ and find that a flat spectrum is predicted if the exponent $a$ for the geometrical shape of the jet, $r(x) \propto x^a$, is given by
\begin{eqnarray}
{\rm Model\ A1:} && a = \frac{3p +12}{13p +2}\nonumber\\
{\rm Model\ A2:} && a = \frac{3p+12}{19p+11}\\
{\rm Model\ A3:} && a = \frac{3p+12}{15p+5} \nonumber.
\label{flatrel}
\end{eqnarray}
Figure \ref{flat} plots the relation for model A3. It is interesting that for all adiabatic jets geometrical shapes described by exponents $a$ in the range $1/3 \ltappeq a \ltappeq 2/3$ are required for flat spectra.
Note that for $a_5 < -1$ the $n=1$ term in the series in equation (\ref{series}) dominates for $\tau_{\rm min} \rightarrow 0$. In that case we have for the long jet
\begin{equation}
\left[ \Gamma \left( a_5, \tau \right)+\frac{1}{a_5} \tau^{a_5} \right]_{\tau_{\rm max}}^{\tau_{\rm min}} \sim \frac{1}{1+a_5} \tau _{\rm min}^{1+a_5},
\end{equation}
similar to the entirely optically thin jet. The spectrum of the long jet is then also optically thin, i.e. $F_{\nu} \propto \nu^{\left(1-p \right)/2}$, and a flat spectrum is not possible for physically reasonable values of the exponent $p$.
\begin{table}
\begin{tabular}{lc}
Model parameter & Value\\
\hline
$x_0$ & $47$\,AU\\
$r_0$ & $8.9 \times 10^7$\,m\\[1ex]
$p$ & $2.5$\\
$B_0$ & $2.4$\,mT\\[1ex]
$\kappa _0$ & $3.3\times10^{-7}$\,J$^{1.5}$\,m$^{-3}$\\
$D$ & $2$\,kpc\\[1ex]
$l_{\rm min}$ & $4.3\times 10^{-5}$\\
$l_{\rm max}$ & $200$\\[1ex]
$\gamma_{\rm max} \left( t_{\rm min} \right)$ & $10^6$\\
$v_{\rm j}$ & $0.97\,c$\\
$\vartheta$ & $40^{\circ}$\\
\hline
\end{tabular}
\caption{Parameters used to illustrate the spectral properties of the jet models. These model parameters are also used to explain the observational data of Cygnus X-1 in Section \ref{cyg} when using a ballistic jet model with a magnetic field perpendicular to the jet axis, model B1. \label{modpara}}
\end{table}
Unless the jet is exceedingly short, there will always be a range of frequencies for which $\tau_{\rm min} \rightarrow 0$ and $\tau_{\rm max} \rightarrow \infty$ and the long jet scenario applies. An example is shown in Figure \ref{tauspec} where we plot the integrand in equation (\ref{lflux}) and the optical depth of the jet material as a function of $l$ for a single frequency for the ballistic jet. The contribution to the overall flux of the jet peaks close to $\tau=1$. For the long jet scenario to apply the jet must be long enough so that substantial emission from either side of the peak contributes to the overall flux.
Below the frequency range of the long jet the spectrum will follow the optically thick case, $F_{\nu} \propto \nu^{5/2}$, and above this range the optically thin case applies with $F_{\nu} \propto \nu^{(1-p)/2}$. The solid line in Figure \ref{illustration2} illustrates this generic overall shape of the jet spectrum for the ballistic jet model B1. For this Figure and the following we have used the model parameters summarised in Table \ref{modpara}. The observations of the partially self-absorbed jet of Cyg X-1 are well explained by the model for these parameters (see Sections \ref{losses} and \ref{cyg}). For comparison, the solid line in Figure \ref{adillu2} shows the spectrum of the adiabatic jet model A3 with $a=0.46$ for the same set of parameters. The value of $a$ was chosen according to equation (\ref{longslope}) to allow for a flat spectrum at intermediate frequencies.
It is interesting to note that the frequencies for which the optical depth of the jet material is unity at $l_{\rm min}$ and $l_{\rm max}$ are located well within the optically thick and thin regimes respectively. The transition from the, in this case, flat spectrum of the long jet occurs closer to $\tau \left( l_{\rm max} \right) \sim 10^{-4}$ and $\tau \left( l_{\rm min} \right) \sim 100$. Obviously, for sufficiently short jets the frequency range over which the long jet case applies may vanish altogether.
\section{Including energy losses of the electrons}
In the previous Section we did not consider the effect of radiative energy losses of the relativistic electrons on the predicted spectra. The adiabatic jet models include the effect of adiabatic energy losses on the overall energy distribution of the relativistic electrons. However, because we did not impose a high-energy cut-off to this distribution, we did not have to consider the effect of adiabatic losses on such a cut-off. In this Section we introduce a high-energy cut-off at $\gamma _{\rm max}$ and include the effect of adiabatic energy losses on this cut-off.
\subsection{Evolution of the high-energy cut-off}
\label{cutoff}
\subsubsection{Adiabatic and synchrotron losses}
Other than adiabatic energy losses, the radiative losses due to synchrotron radiation modify the energy distribution of the relativistic electrons away from a simple power-law, unless $p=2$ \citep[e.g.][]{nk62}. In the following we will make the simplifying assumption that the energy losses only shift the sharp high-energy cut-off to lower energies while not altering the power-law shape or exponent of the power-law distribution. This approximation does not introduce a large error as the deviation from the original power-law is significant only near the cut-off. Also, the expressions for the synchrotron emissivity and absorption coefficient given in equations (\ref{jk}) and (\ref{jk0}) are strictly valid only for power-law energy distribution extending from $\gamma =1$ to $\gamma_{\rm max} \rightarrow \infty$. However, the expressions involved in the derivation of $J_{\nu}$ and $\chi_{\nu}$ decay sufficiently quickly for $\gamma \gtappeq 10$ that the results for finite $\gamma _{\rm max}$ do not deviate greatly from those presented in the previous Section for $\gamma_{\rm max} \rightarrow \infty$ \citep{rl79,ml94}.
In the optically thin regime the evolution of the Lorentz factor of a given relativistic electron in the rest frame of the jet material is described by \citep[e.g.][]{ml94}
\begin{equation}
\dot{\gamma} = -\frac{4}{3} \frac{\sigma _{\rm T} u_0}{m_{\rm e} c} \left( \frac{t}{t_0} \right)^{-2 a_2} \gamma^2 - \frac{2a_1}{3t} \gamma,
\label{evol}
\end{equation}
where the first term on the right describes the energy losses due to synchrotron radiation and the second term reproduces equation (\ref{adiabat}) for the adiabatic losses where we substituted for $\Delta V$. Because of our assumption of a constant bulk velocity for the jet material along the $x$-axis, $v_{\rm j}$, we can express the dimensionless coordinate $l=x/x_0$ also as a time variable, i.e. $l=\gamma _{\rm j}v_{\rm j}t/x_0$. Because of time dilation, we have to include $\gamma_{\rm j}$. Thus $t=l x_0 / \left( v_{\rm j} \gamma _{\rm j} \right)$ and $t_0=x_0/ \left( v_{\rm j} \gamma _{\rm j} \right)$. $\sigma _{\rm T}$ is the Thomson cross section and $u_0=B_0^2/\left(2 \mu_0 \right)$ is the energy density of the magnetic field at $x_0$. The solution of equation (\ref{evol}) is found as
\begin{equation}
\gamma \left( t \right) = \frac{\gamma \left( t_{\rm min}\right) t^{-2 a_1 /3}}{t_{\rm min}^{-2 a_1/3} + \frac{4 \sigma _{\rm T} u_0}{3 a_6 m_{\rm e} c} t_0^{2a_2} \gamma \left( t_{\rm min}\right) \left( t^{a_6}-t_{\rm min}^{a_6} \right)},
\label{gamadiabat}
\end{equation}
with $a_6 = 1-2a_2-2a_1/3$ and $t_{\rm min} = l_{\rm min} x_0 / \left( v_{\rm j} \gamma _{\rm j} \right)$. Electrons which were injected into the jet at $t_{\rm min}$ or, equivalently, $l_{\rm min}$ with a Lorentz factor $\gamma \left( t_{\rm min}\right)$, have a Lorentz factor $\gamma \left( t \right)$ at $t$ or, equivalently, $l$. For the ballistic jet the adiabatic, second term on the right of equation (\ref{evol}) vanishes and we have instead
\begin{equation}
\gamma \left( t \right) = \frac{\gamma \left( t_{\rm min}\right)}{1+\frac{4 \sigma _{\rm T} u_0}{3 \left(1-2a_2 \right) m_{\rm e} c} t_0^{2a_2} \gamma \left( t_{\rm min} \right)\left( t^{1-2a_2}-t_{\rm min}^{1-2a_2} \right)}.
\label{gamrad}
\end{equation}
Note that the exponents $a_6$ and $1-2a_2$ are usually negative. This implies in the case of the ballistic jet models (no adiabatic losses) that the Lorentz factors of electrons do not necessarily decrease forever, but converge to a finite value for $t\gg t_{\rm min}$. The somewhat surprising result simply reflects the fact that the synchrotron losses rapidly decline in the decreasing magnetic field of the expanding jet. For the adiabatic case the adiabatic losses continue at all times and so $\gamma \propto t^{-2a_1 /3} \propto l^{-2 a_1 / 3}$ for $t\gg t_{\rm min}$.
For optically thin conditions the evolution of the high-energy cut-off $\gamma _{\rm max}$ also obeys equations (\ref{gamadiabat}) and (\ref{gamrad}). Below we will refer to the high-energy cut-off in the optically thin regime as $\gamma _{\rm thin}$. However, for large parts of the spectrum the jet is optically thick. Electrons with a given Lorentz factor $\gamma$ emit most of their radiation at the critical frequency $\nu \sim \nu _{\rm g} \gamma^2$, where the gyro-frequency is defined as $\nu_{\rm g} = e B / \left(2 \pi m_{\rm e}\right)$. An electron emitting at a critical frequency for which the jet is optically thick gains energy through synchrotron self-absorption. Ideally we would include an energy gain term for the self-absorption effect into equation (\ref{evol}) and then derive the electron evolution as before. While this approach leads to analytic solutions when only considering the systematic energy gain of electrons of a single energy \citep{mr67}, it is not applicable in most cases because the stochastic energy gain for power-law energy distributions is comparable to the systematic term. In this case, only numerical solutions are possible, because the stochastic term depends on the entire energy distribution \citep{rm69}.
The full numerical treatment of synchrotron losses and gains in the optically thick regime is beyond the scope of this paper. However, electrons radiating mainly at frequencies for which the jet is optically thick, do on average not lose or gain energy due to radiative effects, even if they are relatively close to the surface of the jet \citep{rm69}. Thus the high-energy cut-off in the optically thick regime, $\gamma _{\rm thick}$, is given by the requirement that $\tau \left( \gamma _{\rm thick} \right) \sim 1$. The electrons at this cut-off emit mainly at a frequency $\nu_{\rm thick} = \nu_{\rm g} \gamma _{\rm thick}^2$ and so we find from equation (\ref{depth})
\begin{equation}
\gamma _{\rm thick} = \left[ \frac{4 \pi m_{\rm e}}{3 e B_0} \left( \chi_0 r_0 \right)^{2 / \left(p+4 \right)} l^{a_7} \right]^{1/2},
\label{gamthick}
\end{equation}
with
\begin{equation}
a_7 = \left(a_1 -a_3 - a_2 \frac{p+2}{2}\right) \frac{2}{p+4}+a_2.
\end{equation}
For the ballistic jet with perpendicular magnetic field, model B1, $a_7 =0$ and so $\gamma _{\rm thick}$ is constant along the entire length of the jet. In other words, electrons with Lorentz factors equal or below $\gamma _{\rm thick}$ never loose their energy to radiation unless they are very close to the jet surface. The existence of a constant high-energy cut-off is required for a flat spectrum from the jet. While \citet{bk79} invoked an unknown re-acceleration mechanism to ensure $\gamma _{\rm max} ={\rm constant}$, we have shown here that such a process is unnecessary because of the energy gains associated with synchrotron self-absorption. For a parallel magnetic field in the ballistic jet, model B2, we have $a_7= 2/ \left(p+4 \right)$. The Lorentz factor of relativistic electrons for which the jet is optically thick is {\em increasing\/} for increasing $l$ in this model. Therefore, if $l_{\rm thick}$ is the position along the jet where $\gamma _{\rm thin} = \gamma _{\rm thick}$, we have $\gamma_{\rm max} =\gamma_{\rm thick}$ at this position and $\gamma _{\rm max} = {\rm constant}$ for all $l>l_{\rm thick}$. In the adiabatic cases we find
\begin{eqnarray}
{\rm Model\ A1:} && a_7 = a \frac{4 \left( 1 - p \right)}{3 \left( p + 4 \right)}\nonumber\\
{\rm Model\ A2:} && a_7 = a \frac{2 \left( 5 - 2p \right)}{3 \left( p + 4 \right)}\\
{\rm Model\ A3:} && a_7 = a \frac{2 \left( 3 -2 p \right)}{3 \left(p+4 \right)}.\nonumber
\end{eqnarray}
Even for optically thick conditions the electron energy distribution does not deviate greatly from the original power-law with a high-energy cut-off $\gamma _{\rm max}$ for exponents $2\le p \le 3$ \citep{rm69}. Thus, for each position $l$ along the jet we can now determine $\gamma_{\rm max}$ and thereby the entire electron energy distribution. For small $l$ the cut-off is given by $\gamma _{\rm max} =\gamma _{\rm thin}$. Further down the jet $\gamma _{\rm thin}$ will first become equal to and then fall below $\gamma _{\rm thick}$ and for the ballistic jet models $\gamma _{\rm max} = \gamma _{\rm thick}$ afterwards. In the adiabatic jet models two competing effects can diminish $\gamma _{\rm max}$ further after passing through the point at which $\gamma _{\rm thick} = \gamma _{\rm thin}$. In most cases equation (\ref{gamthick}) implies a further reduction of $\gamma _{\rm thick}$ for increasing $l$. This means that $\gamma _{\rm max}$ would also decrease. At the same time, adiabatic losses lead to $\gamma _{\rm max} = \gamma _{\rm thick} \left( l / l_{\rm thick} \right) ^{-2 a_1 / 3}$. It is straightforward to show that for all adiabatic models the adiabatic losses of $\gamma _{\rm max}$ are the dominant effect.
By again making the assumption that all electrons only emit at their critical frequency, we can now define for a given frequency $\nu$ a maximum distance $l'_{\rm max}$ along the jet axis where the jet material is still contributing to the overall emission,
\begin{equation}
l'_{\rm max} = \left(\frac{2 \pi m_{\rm e} \nu}{e B_0 \gamma _{\rm max}^2} \right)^{-1/a_2}.
\label{ldmax}
\end{equation}
Clearly, as long as $l'_{\rm max}$ is larger than the physical extent of the jet, $l_{\rm max}$, the jet spectrum is not affected by energy losses of the relativistic electrons at frequency $\nu$ and we can use the results of the previous Section. For $l'_{\rm max} < l_{\rm max}$ we have to take into account energy losses of the electrons by using $l'_{\rm max}$ instead of $l_{\rm max}$ in the calculation of $\tau_{\rm min}$.
\subsubsection{Losses due to Compton scattering}
In the optical thick parts of the jet Compton scattering of the synchrotron photons off the relativistic electrons may become important. We do not include energy losses of the relativistic electrons due to Compton scattering in the jet in our calculations as these would require a full treatment of radiative transfer. However, it is obviously necessary to test whether these losses are important when applying the model to observational data and so we give the necessary expressions below.
The relevant limit for the energy density of the synchrotron photon field is most conveniently expressed in terms of the brightness temperature \citep[e.g.][]{rl79},
\begin{equation}
T_{\rm b} = \frac{c^2 I_{\nu}}{2 \delta_{\mp}^3 k_{\rm B} \nu^2},
\end{equation}
where $k_{\rm B}$ is the Boltzman constant and $\nu$ is the emitted frequency rather than the observing frequency. For $T_{\rm b} \ltappeq 10^{12}$\,K Compton losses are not important compared to the energy losses due to synchrotron radiation. The maximum brightness temperature for a given frequency is reached at the position along the jet where the optical depth of the jet material roughly equals unity for photons of this frequency. Hence in our model we have
\begin{equation}
T_{\rm b, max} = \frac{c^2 J_0}{8 \pi k_{\rm B} \chi _0} \left( \chi _0 r_0 \right)^{1/ \left( p+4 \right)} \left( 1 - e^{-1} \right) l^{ \left( a_1 + a_2 - a_3 \right) / \left( p + 4 \right)}.
\label{bright}
\end{equation}
The maximum brightness temperature is either constant or only a weak function of $l$ in all our models. Also, using the dependencies of $J_0$ and $\chi _0$ given in equation (\ref{jk0}) we find that $T_{\rm b, max}$ depends only weakly on the other model parameters $B_0$, $\kappa _0$ and $r_0$.
In some jets relativistic induced Compton scattering may be more important than direct Compton scattering discussed above. Induced Compton scattering causes significant energy losses for the relativistic electrons if \citep{sk94}
\begin{equation}
\frac{k_{\rm B} T_{\rm b}}{m_{\rm e} c^2} \tau _{\rm T} \ge 1,
\label{crit}
\end{equation}
where $\tau_{\rm T}$ is the Thomson depth of the jet material,
\begin{equation}
\tau_{\rm T} = n_{\rm e} \sigma_{\rm T} r.
\end{equation}
Here, $n_{\rm e}$ is the number density of electrons, $\sigma _{\rm T}$ is the Thomson cross-section and we have again assumed that the average path length a photon travels through the jet material is equal to the jet radius, $r$. For our power-law energy distribution of the electrons an upper limit for the electron density is given by $n_{\rm e} \le \kappa \left( m_{\rm e} c^2 \right)^{1-p}$. The Thomson depth of the jet materials in our models is then limited by
\begin{equation}
\tau_{\rm T} \le \left(m _{\rm e} c^2 \right)^{1-p} \sigma _{\rm T} \kappa _0 r_0 l^{a_1-a_3}.
\label{thom}
\end{equation}
Equations (\ref{bright}) and (\ref{thom}) can be used to ensure that the models are applicable to a given observational data set, i.e. that they do not suffer from Compton losses which are not included in the models.
\subsection{Spectra with energy losses and a high-energy cut-off}
\label{losses}
\subsubsection{Iterative construction of model spectra}
For a given set of model parameters we can construct a model spectrum. In practice this will involve the determination of $\tau_{\rm min}$ and $\tau _{\rm max}$ from equation (\ref{depth}) to be substituted into equation (\ref{sol}). For a given frequency $\nu$, we set $l= l_{\rm min}$ and calculate $\tau_{\rm max}$. Determining $\tau _{\rm min}$ is more involved as it requires the calculation of $l'_{\rm max}$. This calculation involves an implicit equation and so cannot be done analytically. Here we describe one possible iterative procedure for the determination of $l'_{\rm max}$.
The first step is to choose a trial distance $l$ such that $l_{\rm min} \le l \le l_{\rm max}$. The strength of the magnetic field in the jet material at $l$ is $B \left( l \right) = B_0 l^{-a_2}$. The maximum frequency at which jet material located at $l$ is still contributing to the emission is given by $\nu _{\rm max} \left( l \right) = \nu_{\rm g} \left( l \right) \gamma _{\rm max} \left( l \right)$. For the next iteration we need to compare $\nu_{\rm max} \left( l \right)$ with $\nu$. Therefore we must next derive $\gamma _{\rm max} \left( l \right)$, the high-energy cut-off of the electron energy distribution at $l$.
From equations (\ref{gamadiabat}, adiabatic jet models) or (\ref{gamrad}, ballistic jet models) we can determine $\gamma _{\rm thin} \left( l \right)$. We calculate $\gamma _{\rm thick}$ from equation (\ref{gamthick}). If $\gamma_{\rm thin} \left( l \right) > \gamma _{\rm thick}$, then $\gamma _{\rm max} \left( l \right)= \gamma _{\rm thin} \left( l \right)$. Otherwise, for the ballistic jet models $\gamma _{\rm max} \left( l \right) = \gamma _{\rm thick}$. For the adiabatic jet models $\gamma _{\rm max} \left( l \right) = \gamma _{\rm thick} \left( l / l_{\rm thick} \right)^{-2a_1 /3}$. The required distance $l_{\rm thick}$ must be found from the implicit equation resulting from setting $\gamma \left( t \right) = \gamma _{\rm thick}$ in equation (\ref{gamadiabat}). Finally, if $\nu_{\rm max} \left( l \right) > \nu$, then the trial distance in the next iteration should be larger than the current one. In the case of $\nu_{\rm max} \left( l \right) < \nu$, the trial distance should be decreased. The iterations can be stopped when $\nu_{\rm max} \left( l \right) \sim \nu$ within the required accuracy and at that point we can set $l'_{\rm max} =l$ and then proceed to calculate $\tau_{\rm min}$.
\subsubsection{Example spectra}
The model parameters in Table \ref{modpara} were chosen to explain the observations of the jet in Cygnus X-1 with the ballistic jet model B1 including radiative energy losses (see Section \ref{cyg}). This does not imply that energy losses will always be important in all jets and at all frequencies.
The spectrum of the ballistic and adiabatic jets with energy losses of the electrons are shown in Figures \ref{illustration2} for model B1 and \ref{adillu2} for model A3. When energy losses of the electrons are taken into account, then we cannot in general expect that $\tau_{\rm min} \ll 1$ for a given frequency. Therefore the spectrum of the jet will not necessarily be that of the long jet described by equation (\ref{longslope}). In fact, in most cases $\tau_{\rm min}$ will considerably exceed unity. If $\tau_{\rm max} \rightarrow \infty$, then equation (\ref{largetau}) applies and we find
\begin{equation}
F_{\nu} \sim \delta_{\mp}^2 \frac{x_0 r_0 J_0}{a_4 a_5 D^2 \chi_0} \nu^{5/2} \left( \frac{\tau_{\rm min}}{\tau_0} \right)^{a_5}.
\label{reduced}
\end{equation}
For the ballistic jet models we have argued in the previous Section that $\gamma _{\rm max} ={\rm constant}$ for $l \ge l_{\rm thick}$. For rapid radiative energy losses of the electrons at $l$ in the range $l_{\rm min} < l < l_{\rm thick}$, the distance $l_{\rm thick}$ itself will not depend strongly on the observing frequency. From equation (\ref{ldmax}) we then find $l'_{\rm max} \propto \nu ^{-1/a_2}$ and substituting into equation (\ref{depth}) we get
\begin{equation}
\tau_{\rm min} \propto \nu^{\left( a_3 -a_1-a_2 \right) / a_2}.
\label{taumin}
\end{equation}
Note that the exponent does not depend on the slope of the electron energy distribution, $p$, nor on the geometrical shape of the jet described by $a$ (see Table \ref{expo}). Finally, from equation (\ref{reduced}) we obtain the slope of the spectrum as
\begin{equation}
F_{\nu} \propto \nu^{\left( 4a_2-2a_1-2 \right) / \left( 2 a_2 \right)}.
\end{equation}
For the ballistic jet with perpendicular magnetic field, model B1, we have $F_{\nu} ={\rm constant}$ as in the case without energy losses of the electrons, which is confirmed by Figure \ref{illustration2}. For a parallel field structure, model B2, the spectrum would follow $F_{\nu} \propto \nu$. Model B2 is incompatible with a flat spectrum. The slope in the optically thin part of the spectrum in Figure \ref{illustration2} is steeper compared to the case of no energy losses because of the effect of the high-energy cut-off in the electron energy spectrum.
In the case of the adiabatic jet models $\gamma _{\rm max} \propto l^{-2 a_1 /3}$ and so $l'_{\rm max} \propto \nu^{-3 / \left( 4 a_1 + 3 a_2 \right)}$. Again substituting into equation (\ref{depth}) yields
\begin{equation}
\tau_{\rm min} \propto \nu^{- \left(7 a_1 + 3 a_2 \right) / \left( 4 a_1 + 3 a_2 \right)},
\end{equation}
where we also used $a_3 = \left( 4 + 2p \right) a_1 /3$ as appropriate for the adiabatic jet models. Again the exponent of this expression does not depend on $p$ or $a_1$. The shape of the spectrum is now
\begin{equation}
F_{\nu} \propto \nu^{\left( 7 a_1 + 6 a_2 -3 \right) / \left(4 a_1 +3 a_2 \right)}.
\end{equation}
The exponent of the power-law spectrum predicted by model A3 for $a_1 = 0.46$ is then 1.06 which is confirmed by the slope of the spectrum in Figure \ref{adillu2} below about $10^{12}$\,Hz. The emission at high frequencies comes from the innermost parts of the jet close to $l_{\rm min}$. Radiative losses had not enough time there to completely change the electron energy distribution. This explains the peak in the spectrum. At the highest frequencies only optically thin parts of the jet contribute to the overall emission and lead to a negative power-law similar to the case without energy losses. Note however that the slope of this power-law is somewhat steeper due to the decreasing high-energy cut-off.
The adiabatic jet models are all consistent with a flat spectrum provided the shape of the jet described by $a_1$ takes a suitable value (Model A1: $a_1 = 3/13$, A2: $a_1 = 3/19$ and A3: $a_1 = 1/5$). In all three adiabatic models the jet needs to be strongly confined, i.e. $a_1 < 1/4$, to achieve a flat emission spectrum. The slope of the spectrum is quite sensitive to the value of $a_1$. For example, a change of $a_1$ in model A3 from $1/5$ to $1/3$ results in a change of the power law exponent of the spectrum from zero to 0.75.
For the following discussion we note that all the spectral slopes calculated above, with the only exception of the purely optically thin case, are all independent of the exponent $p$ of the power-law describing the energy distribution of the electrons. Therefore we can readily apply the model to observational data even if the optically thin part of the spectrum is not observed and thus we do not know the value of $p$.
\section{Application to observations}
\label{obs}
The jet emission models depend on a number of parameters. Some of these parameters can be constrained by applying general considerations and others may be inferred from applying the model predictions to observational data with a view to determining the physical conditions within the jet. Here we discuss all of the relevant parameters in turn and demonstrate below how observations of the jet in Cygnus X-1 may be used to infer the properties of this object.
The scale height $x_0$ can always be chosen arbitrarily to provide a convenient location along the jet axis at which to define the exact values of other quantities. In many cases we will be mainly interested in that part of the jet spectrum which is strongly affected by absorption. As we have seen in the previous Section, we then do not need to know the exponent of the power-law energy distribution of the electrons, $p$, as it does not influence the slope of the predicted spectrum. However, estimates for other quantities derived from the model, for example the strength of the magnetic field, depend weakly on $p$. The distance of the jet, $D$, the bulk velocity of the jet material, $v_{\rm j}$, and the viewing angle of the jet axis to our line of sight, $\vartheta$, cannot normally be determined by the model itself and need to be measured by other means.
The parameters describing the geometrical shape and size of the jet, $r_0$, $l_{\rm min}$ and $l_{\rm max}$, could in principle be determined from observations. However, $l_{\rm min}$ is probably too small to be resolvable even with a large improvement on current resolution limits. Currently only upper limits exist for the radius of jets in Galactic X-ray binaries (Miller-Jones et al., in preparation) while for AGN jets $r_0$ is sometimes resolved \citep[e.g.][]{jbl99}. The maximum extent of a jet at a given observing frequency is sometimes measured and an example is provided by the observations of the jet in Cygnus X-1 of \citet{ssf01} which we use in the following Section. It should be borne in mind that at one observing frequency we can always only measure $l'_{\rm max}$ given by equation (\ref{ldmax}) rather than the physical extent of the jet flow $l_{\rm max}$. However, $l_{\rm max}$ only determines the low frequency cut-off of the spectrum, but is not important for the model otherwise. In the case of the adiabatic jet models resolved observations can, in principle, also determine the shape of the jet as described by the parameter $a$. However, in practice it is easier to infer the value of $a$ from the slope of the observed self-absorbed spectrum as this is a strong function of $a$.
Finally, the normalization of the electron energy distribution, $\kappa _0$, and the strength of the magnetic field, $B_0$, cannot be determined directly from observations, but must be inferred from the model. We can reduce the number of free parameters by assuming that the energy densities of the magnetic field and of the relativistic electrons are initially in equipartition. In this case,
\begin{equation}
u_0 = \frac{B_0^2}{2 \mu _0} = \int_{E_{\rm min}}^{E_{\rm max}} \kappa _0 E^{1-p} \, {\rm d}E.
\end{equation}
For an energy distribution extending over all physically meaningful Lorentz factors ($1 \le \gamma \le \infty$) we then have
\begin{equation}
\kappa _0 \sim \left( p-2 \right) \frac{B_0^2}{2 \mu _0} \left( m_{\rm e} c^2 \right)^{p-2}.
\end{equation}
With these considerations we can now apply the model to observations and determine relevant parameters for the observed jet.
\subsection{Application to Cygnus X-1}
\label{cyg}
In the following we apply the model to the jet observed in the X-ray binary Cygnus X-1. \citet{ssf01} report a resolved jet extending to about 15\,mas from the position of the X-ray binary system at an observing frequency of 8.4\,GHz. We set the bulk velocity of the jet material to 0.97\,$c$ and the viewing angle to our line of sight to $\vartheta = 40^{\circ}$ \citep{ssf01}. For a distance of 2\,kpc \citep{gzp99} the observed, projected jet length then corresponds to a real jet length of roughly 47\,AU. For convenience we set $x_0$ equal to this value and so $l'_{\rm max} = 1$ at 8.4\,GHz. Only one jet is observed and it is therefore reasonable to assume that this is the approaching jet. Note that the bulk velocity of the jet and the viewing angle imply $\delta _- \sim 1$ and so $\nu \sim \nu_{\rm ob}$. The total flux density at the same frequency is 13\,mJy.
In a map at 15\,GHz from an earlier observing epoch, the jet may also be marginally extended along the same axis \citep{ssg98,ssf01}. The extension is 2\,mas or less which corresponds to $l'_{\rm max} \left( 15\,{\rm GHz} \right) \le 0.13$. In the following we will mainly concentrate on the observational data at 8.4\,GHz.
There are no simultaneous observations at any frequency other than 8.4\,GHz and so we cannot be certain what the spectral slope of the jet emission was. However, during the low/hard X-ray state the source usually shows a flat spectrum extending up to at least 220\,GHz \citep{fpdtb99}. We assume here that the spectral slope at the time the radio jet at 8.4\,GHz was observed was zero. The extent of the flat spectrum to high frequencies is also not known. However, for GX 339-4 the flat jet spectrum is observed to near-IR wavelengths \citep{cf02} while for XTE J1118+480 it may extend to the near-UV \citep{hmh00}. For the purpose of illustrating the model, we assume here that the flat spectrum extends to the near-IR of wavelengths of about 1\,$\mu$m. The flat spectrum always arises from regions in the jet which are at least partially self-absorbed. Thus the exact value of $p$ is not very important and we set $p=2.5$.
\subsubsection{Ballistic jet models}
We have seen above that the ballistic jet with a parallel magnetic field configuration, model B2, is inconsistent with a flat spectrum. In this Section we therefore concentrate on model B1, a ballistic jet with a magnetic field perpendicular to the jet axis. With the assumptions made above equation (\ref{reduced}) reduces to
\begin{equation}
F_{\nu} \sim 5.9 \times 10^{-7} \delta _{\mp}^2 x_0 r_0 D^{-2}B_0^{-1/2}\nu^{5/2} {l'_{\rm max}}^{5/2} \, {\rm mJy}
\end{equation}
for the ballistic jet model B1. Here and in the following, all quantities are measured in SI units unless indicated otherwise. Substituting the measurements discussed above we get an expression for $r_0$ as a function of $B_0$. A second equation relating the same quantities can be found from substituting $\gamma _{\rm thick}$ from equation (\ref{gamthick}) for $\gamma_{\rm max}$ in equation (\ref{ldmax}) resulting in
\begin{equation}
l'_{\rm max} = 2.0 \times 10^{11} \nu^{-1} B_0^{9/13} \kappa _0^{4/13} r_0^{4/13} = 1,
\end{equation}
where we again made use of the observed quantities. For initial equipartition we can eliminate $\kappa _0$ and solve for $B_0$. We can then calculate all other model parameters and they are summarised in Table \ref{modpara}. The model spectrum is plotted as the dashed line in Figure \ref{illustration2}.
The maximum brightness temperature of the jet emission does not depend on $l$ for model B1 and from equation (\ref{bright}) we find $T_{\rm b, max} =1.1\times10^{10}$\,K. We can also calculate at what distance $l$ along the jet the jet material is dense enough so that relativistic induced Compton scattering becomes important. From equation (\ref{crit}) we find that this loss process would only play a role for $l < 1.5\times10^{-7}$ which is well inside $l_{\rm min}$. We therefore conclude that our model can be applied to the jet of Cygnus X-1.
The jet radius $r_0$ is very small. The ballistic jet model is conical and therefore we can define a jet opening angle as $\theta = 2 r_0 / x_0 = 5"$ which is much smaller than the observational upper limit of $2^{\circ}$ \citep{ssf01}. It is of course possible to assume a larger radius for the jet by dropping the assumption of equipartition. However, setting the jet radius equal to the observational upper limit would imply that the jet material is out of equipartition by several orders of magnitude.
The physical limits of the jet flow, $l_{\rm min}$ and $l_{\rm max}$, can be derived from equation (\ref{depth}) by setting $\tau \left( l_{\rm min} \right) = \tau \left( l_{\rm max} \right) = 1$ for those frequencies at which the flat spectrum is required to break to the optically thick or thin regime, respectively. In our example of Cygnus X-1 we used a lower break of just under 100\,MHz and an upper break of $3\times 10^{14}$\,Hz corresponding to a wavelength of 1\,$\mu$m. The resulting limit $l_{\rm min}=4.3\times10^{-5}$, associated with the break to optical thin conditions, corresponds to a distance of about 6000 Schwarzschild radii from the central black hole with a mass of 10\,M$_{\odot}$ \citep{gzp99}. Since for the ballistic jet model B1 $l_{\rm min}$ decreases linearly with the break frequency, the lower physical limit decreases to about 500 Schwarzschild radii if the flat spectrum extends to the near-UV. Clearly the determination of the high frequency break of the flat part of the spectrum can put interesting constraints on the distance from the central black holes at which jets become ballistic. Similarly the low frequency break constrains the overall extent of the ballistic jet flow.
The observable extent of the jet flow depends linearly on the observing frequency. Since $l'_{\rm max} \left(8.4\,{\rm GHz} \right) = 1$, we would expect that $l'_{\rm max} \left( 15\,{\rm GHz} \right) \sim 0.6$. This prediction significantly exceeds the tentative extension of the Cygnus X-1 jet of $l'_{\rm max} \left( 15\,{\rm GHz} \right) \le 0.13$ reported in \citet{ssg98}. However, the observations at 15\,GHz were not simultaneous with those at 8.4\,GHz.
The strength of the magnetic field, $B_0$, at $x_0$ implies a field strength of around 50\,T at the distance $l_{\rm min}$. If $l_{\rm min}$ is reduced because the flat spectrum extends to the near-UV, then the magnetic field in the jet has a strength of 500\,T about 500 Schwarzschild radii away from the black hole. Again this demonstrates that our model can provide useful constraints on the conditions at the very base of the observed jets despite them not being spatially resolved.
The strength of the magnetic field, $B_0$, and the constant in the expression for the density of relativistic electrons, $\kappa _0$, can be used to estimate the power of the jet. We find that the energy transport rate associated with the magnetic field and the relativistic particles alone is $5.2\times10^{25}$\,W. A further $1.1 \times 10^{25}$\,W is added by the kinetic energy of the electrons. Not surprisingly these numbers are comparable to the estimates using the \citet{bk79} model \citep{fpdtb99}. If there is a cold proton for every relativistic electron in the jet, then its energy transport rate in terms of kinetic energy is $1.1\times 10^{28}$\,W. This power is about one order of magnitude below the time-averaged energy transport rate of the Cygnus X-1 jet, recently estimated from the observed interaction of the jet with the surrounding ISM \citep{gfk05}. The flux density of the flat jet spectrum does not vary significantly between available observations. Unless the jet power varies considerably over timescales longer than the timespan since the first available radio observations of Cygnus X-1, then our estimate strongly suggests the presence of a proton-electron plasma in the jet.
\subsubsection{Adiabatic jet models}
All three adiabatic jet models are consistent with a partially flat emission spectrum. In fact, if we choose the geometrical parameter $a_1$ appropriate for a flat spectrum for each model, then the differences between the adiabatic models become small as the differences in the behaviour of the magnetic field as a function of $l$ between them is compensated for by the different degrees of confinement of the jet. Therefore and to simplify the discussion below, we focus on the adiabatic model A3 with an isotropic magnetic field. None of the conclusions change greatly for models A1 and A2.
For the adiabatic jet model A3 our assumptions with equation (\ref{reduced}) lead to
\begin{equation}
F_{\nu} \sim 8.8 \times 10^{-7} \delta _{\mp}^2 x_0 r_0 D^{-2}B_0^{-1/2}\nu^{5/2} {l'_{\rm max}}^{5/3} \, {\rm mJy},
\end{equation}
where we have set $a=1/5$ to allow for a flat section in the spectrum. The calculation of $\gamma _{\rm max}$ now requires $l_{\rm thick}$, which must be determined from the implicit equation (\ref{gamadiabat}) for a given magnetic field strength, $B_0$. Figure \ref{lmaxest} demonstrates that the allowed minimum for $l'_{\rm max}$ exceeds unity for the assumption of initial equipartition. Since observations require $l'_{\rm max} =1$ at $8.4$\,GHz, the adiabatic models are incompatible with observations unless we drop the requirement of equipartition.
We now introduce a reduction factor $f$ such that the initial energy density of the relativistic electrons is a fraction $f$ of the initial energy density of the magnetic field. An example of the results for $l'_{\rm max}$ for $f=10^{-6}$ is shown in Figure \ref{lmaxest}. There are now two possible solutions for the strength of the magnetic field. However, we also require that the lower size limit of the jet, $l_{\rm min}$, accommodates a break of the flat spectrum to the optically thin regime in the near-IR. This lower limit cannot lie inside the last stable orbit of the central black hole. Thus we obtain another constraint on the solution because $l_{\rm min} \ge 3 R_{\rm S}$, where $R_{\rm S} = 3 \times 10^4$\,m is the Schwarzschild radius of a 10\,M$_{\odot}$ black hole. The additional constraint is also plotted in Figure \ref{lmaxest}. Only one of the two possible solutions for $f=10^{-6}$ is consistent with this constraint. It is also interesting to note that the solution requiring initial equipartition is also inconsistent with a physical meaningful value of $l_{\rm min}$.
Using the remaining solution for $f=10^{-6}$ as an example, we can compute all remaining model parameters which are summarised in Table \ref{failedtab}. Figure \ref{failedspec} shows the resulting spectrum. It is flat over a wide range of frequencies with the required flux level. The dip at around $10^{14}$\,Hz is a result of the diminishing optical depths of the jet to radiation emitted by electrons with the limiting Lorentz factor $\gamma _{\rm max} = \gamma_{\rm thick} \left( l / l_{\rm thick} \right)^{-2 a_1/3}$. For higher frequencies those parts of the jet close to $l_{\rm min}$ which still contain electrons with Lorentz factors in excess of $\gamma _{\rm thick}$ contribute to the emission and cause the peak. The position of the peak is located at the frequency with optical depth $\tau_{\rm max} \sim 1$ at $l_{\rm min}$. Clearly the spectrum is not flat from radio to near-IR frequencies because of the emission peak. A flat spectrum extending over the entire radio to near-IR range could be achieved by moving the peak to higher frequencies. However, an appropriate adjustment of the model parameters would also tighten the constraints on the reduction factor $f$ and thereby on $B_0$.
\begin{table}
\begin{tabular}{lc}
Model parameter & Value\\
\hline
$x_0$ & $47$\,AU\\
$r_0$ & $1.3 \times 10^9$\,m\\[1ex]
$p$ & $2.5$\\
$B_0$ & $1.2$\,T\\[1ex]
$\kappa _0$ & $7.6\times10^{-8}$\,J$^{1.5}$\,m$^{-3}$\\
$D$ & $2$\,kpc\\[1ex]
$l_{\rm min}$ & $2.0\times 10^{-8}$\\
$l_{\rm max}$ & $200$\\[1ex]
$\gamma_{\rm max} \left( t_{\rm min} \right)$ & $10^6$\\
$v_{\rm j}$ & $0.97\,c$\\
$\vartheta$ & $40^{\circ}$\\
\hline
\end{tabular}
\caption{Parameters for the adiabatic jet model A3 without initial equipartition. The model spectrum is shown in Figure \ref{failedspec}.\label{failedtab}}
\end{table}
The maximum brightness temperature of the adiabatic model used here arises at $l_{\rm min}$ and is with $7.3\times 10^9$\,K well below the limit for efficient Compton scattering. The distance at which relativistic induced Compton scattering would become important in the adiabatic jet is $l=1.7\times 10^{-16}$. Again we are justified to neglect energy losses due to Compton scattering.
The jet radius, $r_0$, for the adiabatic jet is larger than for the ballistic jet. Formally, we cannot define a jet opening angle for adiabatic jets discussed here, because their shape is not conical. However, if a conical shape was assumed, then the opening angle inferred from the radius at $x_0$ would be 1.3'.
In our example, the lower limit of the physical extent of the jet, $l_{\rm min}$, corresponds to 4.7\,$R_{\rm S}$. While this is close to the theoretical limit and gravitational redshift would affect the spectrum, a further decreased value of the reduction factor would increase this limit at the expense of requiring an even stronger magnetic field. For adiabatic jets the constraints on the nature of the jet flow extend to even closer distances from the central black hole than in the ballistic case.
For the adiabatic jet model A3 the observable extent of the jet flow is proportional to $\nu^{-3 / \left( 8 a \right)}$. In our example this relation implies that $l'_{\rm max} \left( 15\,{\rm GHz} \right) \sim 0.3$, which is still larger than the observed value of \citet{ssg98}, but closer than the prediction of the ballistic model. However, as mentioned above, the observations are not simultaneous and that may explain the discrepancy in both cases.
The strength of the magnetic field of 1.2\,T is high at $x_0$. However, because of the much more collimated geometry of the jet, the field strength only increases to 140\,T at $l_{\rm min}$. Nevertheless, the energy transport rate of the jet due to the magnetic field is very large with $10^{34}$\,W determined at $l_{\rm min}$. Due to the small value for the reduction factor $f$, the contribution of the relativistic electrons to any energetic considerations is negligible, even if the kinetic energy of possibly associated protons is taken into account. The derived jet power exceeds by far all previous estimates and is inconsistent with the time-averaged jet power \citep{gfk05}, unless the jet flow is suppressed for long periods on very long timescales. Note also, that this jet power corresponds to one hundred times the Eddington limiting luminosity of a 10\,M$_{\odot}$ black hole. The large jet power is caused by the significant reduction in the radiative efficiency of the synchrotron process well away from equipartition conditions.
\section{Conclusions}
\label{conc}
We construct a model for the synchrotron emission of partially self-absorbed jets. The model does not invoke a re-acceleration process for the relativistic electrons. All electrons are accelerated only once at the lower physical limit of the jet, $l_{\rm min}$. It is not necessary to postulate an unknown re-acceleration mechanism as was suggested in previous work \citep{bk79}, because synchrotron self-absorption counteracts excessive energy losses.
Two classes of models are considered. The ballistic jets have a conical geometry and their contents do not suffer adiabatic energy losses. This situation may arise when jets are initially highly overpressured with respect to their environments. They expand unimpeded and random thermal energy is converted into bulk kinetic energy, but not dissipated to any external medium. At the end of this very rapid expansion the mean free path of the jet material exceeds the physical dimensions of the jet itself and follows ballistic trajectories.
The other class of models considered here are adiabatic jets confined by either the pressure of the gas surrounding them, surface magnetic fields or both. Their geometrical shape is dictated by the details of the confinement mechanism which is not the subject of this paper. The jet material dissipates energy to the external gas during its adiabatic expansion.
Both classes of models can predict flat emission spectra if energy losses of individual electrons are neglected. The ballistic jet with a magnetic field perpendicular to the jet axis produces a flat spectrum without further assumptions. The adiabatic jet models require a specific jet geometry to allow for flat emission spectra.
Both model classes can predict flat emission spectra, even when taking energy losses of the electrons and the magnetic field into account. Synchrotron self-absorption prevents radiative energy losses below a critical Lorentz factor $\gamma _{\rm thick}$. In the ballistic case $\gamma _{\rm thick}$ is constant along the jet flow and because adiabatic energy losses are absent, the energy distribution of the relativistic electrons remains stationary. The emission properties of the jet in this case are essentially identical to the model of \citet{bk79}. For the adiabatic jet the with a perpendicular magnetic field electrons with Lorentz factors at and below $\gamma _{\rm thick}$ continue to lose energy because of the sideways expansion of the jet. However, for a geometrical shape given by $r \propto x^{1/5}$ the resulting spectrum is again flat. Other configurations of the magnetic field can also lead to flat emission spectra. The spectral slope is very sensitive to the jet shape. For example, slightly changing the jet shape to $r \propto x^{1/4}$ results in a spectrum with $F_{\nu} \propto \nu^{0.38}$.
We show an application of the model to observations of a resolved jet in the X-ray binary Cygnus X-1 \citep{ssf01}. As input we use the flux density of the jet in the flat part of the spectrum, the physical extent of the resolved jet and an assumed break of the flat spectrum to optically thin conditions in the near-IR. Both model classes can be made consistent with the observational constraints. However, in doing so the adiabatic jet models require a significant departure from energy equipartition between the magnetic field and the relativistic particles. The associated reduced radiative efficiency of the jet plasma implies extremely high energy transport rates for the jet of around $10^{34}$\,W. This jet power exceeds the Eddington limiting luminosity of a 10\,M$_{\odot}$ black hole by two orders of magnitude.
The ballistic jet is consistent with current observations and requires energy transport rates well below the time-averaged jet power \citep{gfk05}. This result holds even if the kinetic energy of one non-radiating proton per relativistic electron is taken into account. Unless Cygnus X-1 ceases to produce a jet for long periods of time, then the ballistic model requires a proton-electron jet plasma to explain the large accumulated energy in the region where the jet interacts with the surrounding gas.
The jet opening angle of 5" required by the ballistic model is very small. The velocity perpendicular to the jet axis established in the jet acceleration region must be very small for this model to work. If the ballistic jet results from a rapid initial expansion of a highly overpressured jet, then the requirements are even more severe. This result places stringent limits on the acceleration process.
Both model classes can probe the conditions in jets on scales unresolvable with current telescopes. The most important measurement in this respect is the high frequency cut-off of the flat spectrum which is formed closest to the jet acceleration region. In order to resolve the remaining uncertainties with the model, the most helpful measurements would be to resolve the jet along its axis in quasi-simultaneous observations at two different frequencies. As mentioned above, even a factor 2 between the observing frequencies employed would help to constrain the geometrical shape of the jet and thereby help to decide which of the model classes is compatible with observations.
Obviously from the work presented here we cannot rule out the possibility of continued re-acceleration of the radiating electrons in the jet. However, such an energetisation process is not necessary to produce flat spectra from self-absorbed, synchrotron emitting jets.
\section*{Acknowledgments}
It is a pleasure to thank E. Gallo, J. Miller-Jones and R.P. Fender for helpful discussions. I also thank the referee A. Marscher for many helpful comments that improved the paper.
\def\newblock{\hskip .11em plus .33em minus .07em}
\bibliography{crk}
\bibliographystyle{mn2e}
|
Title:
Planets and Asteroids in the gamma Cephei System |
Abstract: The binary star system gamma Cephei is unusual in that it harbours a stable
giant planet around the larger star at a distance only about a tenth of that of
the stellar separation. Numerical simulations are carried out into the
stability of test particles in the system. This provides possible locations for
additional planets and asteroids. To this end, the region interior to the
planet is investigated in detail and found to permit structured belts of
particles. The region between the planet and the secondary star however shows
almost no stability. The existence of an Edgeworth-Kuiper belt analogue is
found to be a possibility beyond 65 au from the barycentre of the system,
although it shows almost no structural features. Finally, the region around the
secondary star is studied for the first time. Here, a zone of stability is seen
out to 1.5 au for a range of inclinations. In addition, a ten Jupiter-mass
planet is shown to remain stable about this smaller star, with the habitability
and observational properties of such an object being discussed.
| https://export.arxiv.org/pdf/astro-ph/0601609 |
\label{start}
\begin{keywords}
stars: individual: $\gamma$ Cephei (HR 8974) -- planetary systems --
binaries: general -- methods: \textit{N}-body simulations
\end{keywords}
\section{Introduction}
$\gamma$ Cephei is one of the closest separation binary systems that
contains a planet. The primary and secondary stars are separated by
18.5 au. The planet (designated $\gamma$ Cep Ab) has a minimum mass of
1.7 $\MJ$ and follows an eccentric orbit about the larger star
alone. The observational parameters of $\gamma$ Cephei, as determined
by \citet{Ha03}, are summarised in~Table~\ref{tab:orbels}.
Although the data are reasonably reliable, there is some uncertainty
regarding the masses of the components. The inclination to the line of
sight of a spectroscopic binary is usually indeterminate. This means
that the mass of the primary is generally reckoned from spectral
type. Then, the mass function (e.g., Smart 1977) permits a minimum
mass to be assigned to the secondary, albeit crudely (Griffin,
Carquillat \& Ginestet 2002).
For $\gamma$ Cephei, the mass of the primary is $\approx 1.57
M_{\odot}$ from photospheric modelling \citep{Fu04}. The most recent
determination from stellar evolution models is $\approx 1.7
M_{\odot}$~\citep{Af05}. The mass of the secondary is given by
\citet{Dv03} as $\approx 0.4 M_{\odot}$. However, assuming the
original value of the primary's mass of 1.57 $M_{\odot}$, a minimum
mass can be derived as $\approx 0.34$ $M_{\odot}$ from the velocity
amplitude fitted by \citet{Ha03}.
There have been a number of studies of the dynamics of the $\gamma$
Cephei system to date. Foremost is the numerical investigation of
\citet{Dv03}, which does use an earlier, and slightly different, set
of orbital parameters (see Table \ref{tab:altorbels}). They used
Burlisch-Stoer and Fast Liapunov Indicator methods to show that the
$\gamma$ Cephei system is stable over Myr time-scales. They carried
out test particle integrations as a guide to the possible existence of
further planets. The results show a stable inner region between 0.5
and 0.8 au and then an additional stable zone at low inclination
around 1 au. \citet{Dv03} noted that this coincides with the 3:1 mean
motion resonance. This resonance is stable in the $\gamma$ Cephei
system, but is unstable in the Solar system. \citet{Dv03} conclude
that Earth-mass planets (up to 90 $\ME$) can exist in this region
which is fortunately just on the edge of the habitable zone
\footnote{The habitable zone has boundaries which depend on
assumptions regarding stellar luminosity and effective temperature, as
well as the climate model adopted for the hypothetical habitable
planet.}. An extension to this work by \citet{Dv04} showed that the
planet's eccentricity is less important than that of the two stars for
the stability of a second planet. \citet{Ha05} also carried out
numerical studies of the system's stability using a Burlisch-Stoer
integrator, for a range of possible configurations of the planet's and
binary's semimajor axes, eccentricities and inclinations, confirming
and extending the results of \citet{Dv03}.
Here, we use both numerical simulations and analytic calculations to
study zones of stability as possible locations of additional planetary
companions to either star and asteroid or Edgeworth-Kuiper belt
analogues. Edgeworth-Kuiper belts are of particular interest in
binary systems, as they may be being observed (indirectly) in
exosystems such as $\tau$ Ceti and $\eta$ Corvi (Greaves et al. 2004;
Wyatt et al. 2005). The results fall into three categories: possible
planetary companions and asteroid belts centred on the primary star
(\S 2), planetary companions around the secondary star (\S 3) and
finally possible Edgeworth-Kuiper belt about both stars (\S 4).
\begin{table*}
\centerline{
\scriptsize
\begin{tabular}{|l|r@{ $\pm$ }l|r@{ $\pm$ }l|r@{ $\pm$ }l|}
\hline
Name &\multicolumn{2}{|c|}{$\gamma$
Cep A} & \multicolumn{2}{|c|}{$\gamma$ Cep B} &
\multicolumn{2}{|c|}{$\gamma$ Cep Ab } \\
\hline
Class &\multicolumn{2}{|c|}{K1IV sub-giant star} & \multicolumn{2}{|c|}{M dwarf star } & \multicolumn{2}{c}{Planet}\\
Mass & 1.59 & 0.12 $M_{\odot}$
& \multicolumn{2}{|c|}{0.4 $M_{\odot}$} & 1.7 & 0.4 $M_{J}$\\
Period (days) &\multicolumn{2}{|c|}{--}
& 20750.6579 & 1568.6 & 905.574 & 3.08\\
Semimajor axis (au) &\multicolumn{2}{|c|}{--}
& 18.5 & 1.1 & 2.13 & 0.05\\
Eccentricity &\multicolumn{2}{|c|}{--} & 0.361 & 0.023 & 0.12 & 0.05\\
Longitude of periastron ($^{\circ}$) &\multicolumn{2}{|c|}{--}
& 158.76 & 1.2 & 49.6 & 25.6\\
Time of periastron passage (JD) &\multicolumn{2}{|c|}{--}
& 2448429.03 & 27.0 & 2453121.925 & 66.9\\
\hline
\end{tabular}}
\large
\caption{Best fit orbital parameters for the $\gamma$ Cephei system
from \citet{Ha03}. Mass of star B is from \citet{Dv03}.}
\label{tab:orbels}
\end{table*}
\begin{table}
\centerline{
\scriptsize
\begin{tabular}{|l|c|c|c|}
\hline
Name & $\gamma$ Cep A & $\gamma$ Cep B & $\gamma$ Cep Ab\\
\hline
Class & K1IV sub-giant star & M dwarf star & Planet\\
Mass & 1.6$M_{\odot}$ & 0.4$M_{\odot}$ & 1.76$M_J$\\
Period (days) & -- & 25567.5 & 902.2 \\
Semimajor axis (au) & -- & 21.36 & 2.15\\
Eccentricity & -- & 0.44 & 0.209\\
\hline
\end{tabular}}
\large
\caption{Orbital parameters for the $\gamma$ Cephei system used by
\citet{Dv03}.}
\label{tab:altorbels}
\end{table}
\section{Planets and Asteroid Belts around the Primary}
\subsection{Algorithm}
\label{sec:algorithm}
For all the simulations, we use the parameters of the $\gamma$ Cephei
system given in Table~\ref{tab:orbels}. Here and elsewhere in the
paper, the equations of motion are integrated using a conservative
Burlisch-Stoer method provided in the {\tt MERCURY} software package
\citep{Ch99}. Although not as fast as sympletic methods, this was
chosen because of its ability to provide close encounter data and
handle highly eccentric objects. The two stars and planet are
simulated as point masses for gravitational interactions. Any
additional objects are taken as massless test particles to decrease
integration times. Test particles are removed from the simulations
when they collide with the primary star, or pass an ejection distance
of the order of several hundred au. A collision with the primary
means that the test particle has a position that lies within the body
of the primary, as judged by its stellar radius of $0.02$ au (Hatzes
et al. 2003). Close encounters are allowed to occur, and are defined
as taking place whenever a test particle enters within one Hill radius
$\RH$ of the secondary star or the planet, defined as
\begin{equation}
\RH = a \left( {\frac{m}{3M}} \right)^{1/3}
\end{equation}
where $m$ is the mass of the secondary or planet and $M$ is the mass
of the primary. This works out as $\approx 8.1$ au for the secondary and
$\approx 0.15$ au for the planet.
To maintain accuracy, the variation in the system's total energy and
angular momentum is monitored throughout each simulation. Using this
to constrain the initial timestep to 1 day and the tolerance in the
Burlisch-Stoer algorithm \citep{Pr99} to $10^{-12}$ leads to an
overall fractional change in the system's energy $\Delta E/E$ of about
$10^{-8}$ over a 100 Myr period. This can therefore be considered the
maximum time the system can be accurately followed. All the
simulations presented here are typically 1 Myr in time-scale, for which
$\Delta E/E \approx 10^{-11}$ or better.
It is straightforward to show that the $\gamma$ Cephei system has long
term stability. A 100 Myr integration shows no major variation in the
orbits. Regular short period variations do occur, for example, a
slight oscillation of the planet's semimajor axis over time-scales
equal to both its orbital period and the binary's orbital period. An
additional secular variation is seen over a period of about 5500
years, evident in the eccentricity and longitude of the planet
only. This secular period is in good agreement with the results of
quadrupole theory (see eq. (36) of \cite{Fo00}).
\subsection{Test Particles Interior to the Orbit of the Planet}
\label{sec:tpip}
Figure~\ref{fig:inc_illus} shows two possible configurations of test
particles in the system considered here. In the first, the test
particles are inclined to the common plane of the binary and
planet. In the second, both the test particles and planet are
coplanar, yet inclined relative to the plane of the binary. The
binary is always assumed to be viewed edge-on (that is, it lies in a
plane perpendicular to the plane of the sky). If the planet is at an
inclination $i_{\rm Ab}$ relative to this, its mass must be increased
accordingly by dividing by $\sin i$ where $i = 90^\circ - i_{\rm Ab}$.
With the planet and binary coplanar, we begin by investigating the
stability of test particles in the region interior to the known
planet. A grid of particles with semimajor axis from 0.5 to 1.85 au
and inclination from 0$^{\circ}$ to 50$^{\circ}$ with resolution 0.05
au and 5$^{\circ}$ respectively is integrated for 1 Myr. Thirty-six
particles are started at each grid point with varying initial
longitudes of pericentre ($\omega = 0^\circ, 60^\circ, \dots
300^\circ)$ and longitudes of ascending node ($\Omega = 0^\circ,
60^\circ, \dots 300^\circ$). The orbits are initially circular. The
stability is then determined by computing the mean survival time $\ts$
in Myrs at each grid point averaged over the 36 test particles. The
results are shown in Figure~\ref{fig:stabmapinner} and can be compared
to figure 2 of \citet{Dv03}. Note that our stability index is slightly
different to the criteria used by \citet{Dv03}, who removed test
particles after they become orbit crossing. This may miss the
occasional test particle that is stable, for example, if it lies in a
Trojan-like orbit. We only remove test particles if they collide with
the central star or are ejected from the system.
The map shows test particles with semimajor axes less than $\approx
1.4$ au are stable. However, there is a strip of instability between
roughly 0.8 and 1.0 au, creating an island of stability at low
inclinations between 1.0 and 1.3 au. Some of the structure in the map
can be clearly related to the positions of the mean motion resonances
(MMRs) with the planet (indicated in Fig.~\ref{fig:stabmapinner}). The
4:1, 3:1, 5:2 and 2:1 resonances seem to mark transitions from
stability to instability. For example, the 5:2 MMR divides the island
at $\approx 1.15$ au. The lack of effect of some of the higher order
MMRs, such as the 5:1 case, is probably due to their comparative
weakness and narrowness. The lack of stability beyond $\approx 1.4$ au
is readily explained. The gravitational reach of the planet as a
multiple of the Hill radius (Jones, Underwood \& Sleep 2005) places
the limit on stability at 1.31 au, a good match with the results
here. Note that in calculating this limit the maximum
eccentricity of the planet obtained during the simulation has been
used.
Many of the test particles in the high inclination region of
Figure~\ref{fig:stabmapinner} show evidence of Kozai cycles, as
illustrated by the orbits in Figure~\ref{fig:orbitsb}. The
\citet{Ko62} instability is well-known from studies of high
inclination comets and asteroids in the Solar system. It sets in at
inclinations greater than a critical value of $i_{\rm crit} = {\rm
asin} \sqrt{0.4} \approx 39.23^\circ$. During Kozai cycles, the
eccentricity and inclination vary so as to maintain approximate
constancy of the integral of motion $I_{\rm K} = \sqrt{1-e_{\rm tp}^2}
\cos i_{\rm tp}$, where $e_{\rm tp}$ is the test particle's
eccentricity. As the semimajor axis increases, the amplitude of
eccentricity and inclination librations becomes larger, thus
accounting for the increased instability evident in this region of
Figure~\ref{fig:stabmapinner}.
As already seen, the MMRs with the planet are important in shaping the
regions of stability. However, one major feature unexplained by this
is the instability strip between roughly 0.8 and 1.0 au. Although at
zero inclination the edges are marked by the 4:1 and 3:1 MMRs, the
instability strip shows a pronounced evolution with inclination which
is suggestive of a secular resonance instead. The classical
Laplace-Lagrange linear theory \citep{Mu00}, although derived for low
eccentricity and inclination regimes around a dominant central mass,
can be applied to give a first approximation of the locations of these
resonances. A secular resonance for a test particle occurs when its
precession rate has exactly the same magnitude as an eigenfrequency of
the system. The eigenfrequencies are easily calculable for the three
body system made up of the two stars and planet and are the
eigenvalues of the 2x2 matrices $\mathbf{A}$ and $\mathbf{B}$
respectively, which have components
\begin{eqnarray}
A_{jj} & = & + n_j \frac{1}{4} \frac{m_k}{M+m_j} \alpha \bar{\alpha} b^{(1)}_{3/2}(\alpha),\nonumber\\
A_{jk} & = & - n_j \frac{1}{4} \frac{m_k}{M+m_j} \alpha \bar{\alpha} b^{(2)}_{3/2}(\alpha),\nonumber\\
B_{jj} & = & - A_{jj},\\
B_{jk} & = & - A_{jk} \frac{ b^{(1)}_{3/2}(\alpha)}{ b^{(2)}_{3/2}(\alpha)}.\nonumber
\end{eqnarray}
Here, $n_j$ is the mean motion of object $j$ (1 represents the planet
and 2 represents the secondary), $m_j$ and $a_j$ are the mass and
semimajor axis of object $j$, $M$ is the mass of the primary (the
central object), $\alpha = a_1/a_2$ and $b^{(1)}_{3/2}(\alpha)$ and
$b^{(2)}_{3/2}(\alpha)$ are Laplace coefficients. Using the values for
the masses and semimajor axes given in Table \ref{tab:orbels} along
with $n_1 = 145.2^\circ \yr^{-1}$, $n_2 =6.337 ^\circ \yr^{-1}$ and
$\alpha=0.1151$ gives the Laplace coefficients as
$b^{(1)}_{3/2}(\alpha)=0.3542^\circ \yr^{-1}$ and
$b^{(2)}_{3/2}(\alpha)=0.05089^\circ \yr^{-1}$, employing the {\tt
MATHEMATICA} routines of \citet{Mu00}. Calculating the matrices and
solving for the eigenfrequencies gives
\begin{eqnarray}
g_1 &=&0.04338^\circ \yr^{-1},\nonumber\\
g_2 &=& 0.00005211^\circ\yr^{-1},\\
f &=& -0.04343^\circ\yr^{-1},\nonumber
\end{eqnarray}
where $g_1$ and $g_2$ are the eigenvalues of $\mathbf{A}$ and $f$ is
the degenerate eigenvalue of $\mathbf{B}$. Note that the $g_1$ and $f$
eigenfrequencies have about the same magnitude, whilst $g_2$ is almost
zero due to the large mass ratio between the planet and secondary
star. The precession rate of the test particle is given by
\begin{equation}
A_{\rm tp} = n \frac{1}{4} \left(\frac{m_1}{M}\alpha_1\bar{\alpha}_1b^{(1)}_{3/2}(\alpha_1) + \frac{m_2}{M}\alpha_2\bar{\alpha}_2b^{(1)}_{3/2}(\alpha_2) \right)
\end{equation}
where $n$ is the particle's mean motion. For the region interior to
the planet $\alpha_j = \bar{\alpha}_j= a/a_j$, where $a$ is the test
particles semimajor axis. Plotting $A_{\rm tp}$ as a function of $a$
from 0.5 to 1.85 au shows a resonant location where the $g_1$ and $f$
eigenfrequencies intersect the curve at $\approx 0.8$ au. This
supports the idea that the location of the inner edge of the
instability strip on the map coincides with a secular resonance.
Comparing our results with those of \citet{Dv03}, it is easy to see
that the broad trends are similar, despite slightly differing orbital
elements. The main stable regions are slightly closer to the star in
\citet{Dv03}. This may be due to a higher eccentricity of both the
planet and the secondary, which means that they approach the central
star more closely, reducing separations with the test particles.
\citet{Dv03} find that the 3:1 MMR is stable, in contrast to asteroids
in the Solar System in the same resonance with Jupiter. Here, we find
that the resonance is unstable, with a particle following a fairly
steady evolution until switching to a mode where its eccentricity is
rapidly driven to unity on a time-scale of 10 kyrs, as shown in
Figure~\ref{fig:orbitsa}.
The case where the planet is also inclined ($i_{Ab} \neq 0$) has not
been previously studied. This is a more likely configuration for a
system that has formed in a common protoplanetary disc. Here, we
investigate this case by using the same grid of test particles as
before, but with the planet sharing the same inclination as the test
particles and with $\Omega_{Ab} = 0$. This means that, to reproduce
the same radial velocity dataset, the mass of the planet must be
increased by dividing by $\sin i$, where $i = 90^\circ - i_{Ab}$.
The results are displayed in Figure~\ref{fig:stabmapincl}. As compared
to the earlier case of Figure~\ref{fig:stabmapinner}, the unstable
region has expanded, especially at high inclinations. This is partly
caused by the change in the extent of the gravitational reach of the
planet, as shown by the dotted curve in
Figure~\ref{fig:stabmapincl}. Although this does scale with the
increasing planetary mass, the sharp change at $40^\circ$ inclination
is due to the planet becoming subject to Kozai cycles. The large
increase in eccentricity here means that the planet's periastron is
much closer to the star, and hence its gravitational influence is
larger. At $50^\circ$, the planet is unstable, colliding with the
central star after $\approx 0.5$ Myr.
At first sight, it may seem that the rest of the increased instability
evident in Figure~\ref{fig:stabmapincl} is caused by the increased
mass of the planet. However, experiments show that this is not so.
For example, an integration of the $30^\circ$ case with the planet
inclined but at minimum mass ($1.7 \MJ$) shows very little difference
to Figure~\ref{fig:stabmapincl}. This suggests that the cause lies in
the relative inclination of test particle and planet to the plane of
the binary. In the case of Figure~\ref{fig:stabmapinner}, the
amplitude of libration of $i_{\rm tp}$ and $e_{\rm tp}$ of test
particles is generally modest for all cases below $i_{\rm crit}$, the
critical value for the Kozai instability. For test particles in
Figure~\ref{fig:stabmapincl}, the amplitude is no longer small and
becomes larger with increasing inclination, rapidly driving particles
into the regime where the Kozai instability is effective. This is
understandable as in the former case, the forces due to the masses in
the system are always directed towards the plane of the binary,
whereas in the latter case this is not true. As the inclination
increases, the misalignment between the forces due to the stars and
the force due to the planet increases, and so the amplitude of
libration increases.
\subsection{Test Particles Interior to the Orbit of the Binary}
\label{sec:tpib}
Holman \& Wiegert (1999) have already studied the stability of test
particles in binary systems. These may orbit either a single star or
both stars. Here, we study test particles around the primary star.
For this case, Holman \& Wiegert (1999) introduced the notion of a
critical semimajor axis $a_{\rm crit}$, which is the largest circular
orbit around the primary for which a ring of test particles survives
for at least $10^4$ binary periods ($\approx 600$ kyrs in the case of
$\gamma$ Cephei). Using their eq~(1), we find $a_{\rm crit} = 4.0 \pm
0.6$ au. In other words, our expectation is that all test particles
starting out at semimajor axes greater than 4.0 au will be rapidly
swept out. The existence of the comparatively large and eccentric
planet $\gamma$ Cephei Ab will cause further de-stabilization.
To investigate this, simulations of test particles in the region from
0.5 to 20.0 au are carried out. They have initially zero eccentricity
and are set up on a grid with resolution 0.5 au in starting semimajor
axis. The range of inclinations is restricted from $0^\circ$ to
$30^\circ$ for the prograde case, and from $150^\circ$ to $180^\circ$
for the retrograde case, both in steps of $10^\circ$. Seventy-two
particles are started at each grid point with varying initial
longitudes of pericentre ($\omega = 0^\circ, 30^\circ, \dots
330^\circ)$ and longitudes of ascending node ($\Omega = 0^\circ,
60^\circ, \dots 300^\circ$). The orbits of the test particles are
followed for 1 Myr, and any close encounters are recorded. Although
this is a limited range of inclinations, it is expected that those not
investigated are largely unstable. This receives confirmation from
exploration integrations in the case of $70^\circ$ inclination, for which
very few test particles survive throughout the entire region.
Figures~\ref{fig:inner} and \ref{fig:innerAb} show the ultimate fates
of the test particles within 5 au. They differ in that the planet is
in the plane of the binary and test particles are inclined in
Figure~\ref{fig:inner}, whilst both planet and test particles are
similarly inclined in Figure~\ref{fig:innerAb}. Four panels showing
the results of selected simulations are displayed in each case,
corresponding to prograde with $i_{\rm tp} =0^\circ$ and $30^\circ$,
retrograde with $i_{\rm tp} = 180^\circ$ and $150^\circ$.
In the inner regions, test particles close to the planet are swept out
on a precession time-scale ($\approx 5.5$ kyrs). Within $\approx 3$
au, prograde test particles are either ejected or collide with the
central star after a close encounter with the planet. This is evident
from the particles coloured orange and red in the upper panels of both
Figures~\ref{fig:inner} and \ref{fig:innerAb}. The retrograde case is
different. The lower panels of Figure~\ref{fig:inner} show swathes of
stable particles coloured either brown or green, according to whether
they reside within the Hill sphere of the secondary or not. Note that
the secondary's periastron is at $11.8$ au, so particles out to $3.7$
au are within its Hill sphere at some point. In the case $i_{\rm tp} =
180^\circ$, the retrograde stability zone extends out to $\approx 7$
au. As the inclination decreases ($i_{\rm tp} \rightarrow
150^\circ$), the stability zone shrinks to the annuli between $0.5$ to
$1.0$ au and $3.0$ to $5.0$ au. The lower panels of
Figure~\ref{fig:innerAb} show the case when both test particles and
planet are retrograde. In the case $i_{\rm tp} = 180^\circ$, the
large stability region seen previously has almost completely
disappeared. There are only a few remaining test particles that
survive the 1 Myr integration. The similarity of the two right-hand
panels of this figure shows that the inner region's evolution is
almost entirely controlled by the planet.
\begin{table*}
\caption{\label{table:pevtable} The number of test particles interior
to the orbit of the binary at each starting semimajor axis and
inclination that survive for 1 Myrs. There are 72 test particles
initially for all cases except $i=0^\circ$ and $i=180^\circ$, which
have 12.
}
\centerline{ \scriptsize
\begin{tabular}[width=0.9\textwidth]{c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|c||c|c|c|c|c} \hline
\null &\multicolumn{20}{c}{Starting Semimajor Axis [in au]}\\
Inclination & 0.5 & 1.0 & 1.5 & 2.0 & 2.5&3.0 & 3.5 & 4.0 & 4.5 & 5.0 & 5.5 & 6.0 & 6.5 & 7.0 & 7.5 & 12.0 & 12.5 & 13.0 & 13.5 & 14.0 \\
\hline
$0^\circ$ & 12 & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - \\
$10^\circ$ & 72 & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & 2 \\
$20^\circ$ & 72 & - & - & - & - & - & - & - & - & - & - & - & - & - & - & 1 & - & 2 & - & - \\
$30^\circ$ & 72 &72 & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - \\
$150^\circ$ & 72 &67 & - & - & - &23 &72 &70 &67 &72 &60 &11 &14 & - & - & - & - & - & - & - \\
$160^\circ$ & 72 &72 &72 & - & - &54 &72 &72 &72 &72 &62 &39 &25 & 3 & 1 & - & - & - & - & - \\
$170^\circ$ & 72 &72 &72 & - &24 &72 &72 &72 &72 &72 &68 &49 &15 & 4 & 1 & - & - & - & - & - \\
$180^\circ$ & 12 &12 &12 & - &11 &12 &12 &12 &12 &12 &12 &12 & 8 & 4 & 1 & - & - & - & - & - \\
\hline
$0^\circ$ & 12 & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - \\
$10^\circ$ & 72 & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - \\
$20^\circ$ & 60 &17 & - & - & - & - & 1 & 2 & - & - & - & - & - & - & - & 1 & - & 1 & 1 & - \\
$30^\circ$ & 35 & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - \\
$150^\circ$ & 37 & - & - & - & - & - & - & - & 5 & - & 1 & - & - & - & - & - & - & - & - & - \\
$160^\circ$ & 69 & - & - & - & - & - & 2 & 1 &10 &13 &15 &10 & 4 & 1 & - & - & - & - & - & - \\
$170^\circ$ & 72 & - & - & - & - & - &25 &16 & 4 & 9 &11 &11 & - & - & - & - & - & - & - & - \\
$180^\circ$ & 12 &12 & - & - & - & - & 6 & 5 & 4 & 2 & 2 & - & - & - & - & - & - & - & - & - \\\hline
\end{tabular}
}
\end{table*}
The numbers of test particles surviving after 1 Myr at each semimajor
axis are given in Table~\ref{table:pevtable}. Shown in
Figure~\ref{fig:survivtimes} are the mean survival times plotted
against semimajor axis for a variety of inclinations. The mean
survival time $\ts$ can be computed by averaging over the results for
the differing longitudes of ascending node and pericentre at fixed
semimajor axis and inclination. The averaging is over 12 test
particles for $i_{\rm tp} = 0^\circ$ and $180^\circ$ and 72 for all
other cases. Test particles that survive to the end of the
integrations are included with a survival time of 1 Myr, so that the
computed mean survival time is a lower limit in these cases.
Figure~\ref{fig:survivtimes} shows that there is a region of enhanced
stability with $\ts \approx 100$ kyrs -- for all the studied
inclinations -- centred at $\approx 3.5$ au, just beyond the
gravitational reach of the planet but within the critical semimajor
axis and just within the region affected by the secondary. Most test
particles here (and also beyond this region) suffer a close encounter
with the secondary before ejection or collision with the primary. The
effect of the secondary star becomes increasingly important with
increasing semimajor axis and $\ts$ falls to $\approx$ 1 kyr. There
is, for the prograde cases, a region of enhanced stability around $12$
au for some inclinations, clearly visible in
Figure~\ref{fig:survivtimes}. This is due to a few particles surviving
here for the full length of the integration.
As stated in the introduction, the mass of star B is uncertain. By
altering this parameter and rerunning some of the simulations the
importance of the uncertainty can be seen. For the coplanar
configuration of test particles described in this section, changing
the mass of star B by $\sim 20 \%$ results in almost no difference in
global statistical properties, such as average survival times. This
would indicate that the system's dynamics are not significantly
affected by the uncertainty in the secondaries mass.
\section{Planets and Asteroid Belts around the Secondary}
\label{sec:b}
Here, we investigate whether planets could exist around star B. This
possibility has not been investigated before for this system. The
critical semimajor axis for stability, as defined by Holman \& Wiegert
(1999), is $1.9 \pm 1.0$ au. To investigate further, test particles
are set up from $0.5$ to $2.5$ au in steps of $0.05$ au for
inclinations $0^\circ$ to $30^\circ$ and $150^\circ$ to $180^\circ$ in
steps of $10^\circ$. The particles are on initially circular orbits
again, and spaced in longitude of perihelion and ascending node by
$60^\circ$ as before. The results of two of the prograde cases are
shown in Figure~\ref{fig:starB}. Prograde test particles are not
stable beyond $1.5$ au. In the case of $10^\circ$, $20^\circ$ and
$30^\circ$ inclinations, the unstable test particles generally survive
at least ten times longer than those in the $0^\circ$ case. The
unstable test particles within about $2$ au all collide with the
primary, whilst those beyond this point have a range of fates. Holman
\& Wiegert's critical semimajor axis does not match up as well as in
Section~\ref{sec:tpib}, but still agrees within the rather large
uncertainty. In the retrograde cases, almost all the test particles
are stable, with the exception of 7 particles at $2.45$ au in the
$150^\circ$ case. Integrating more distant particles shows that the
retrograde stability reaches out to about $3.5$ au.
The stability of the test particles can be further investigated by
plotting the evolution of their eccentricity and semimajor axis, as
shown in Figure~\ref{fig:starBae}. For all the simulations, there is
not much change in the inclination of the test particles. The
variation in eccentricity and semimajor axis of stable particles
increases as the initial distance from star B increases, and is
similar for all the prograde cases. However, the variation is much
larger for the (more stable) retrograde cases. When $i_{\rm tp} =
150^\circ$, the variations are no longer smoothly increasing and show
abrupt jumps, indicating that this case may not be long term stable.
As test particles can remain stable around the secondary, this raises
the question of whether a massive planet could also persist here. So,
it is interesting to consider whether this could be detectable in the
radial velocity curve of the primary star. To investigate this, the
simple case of a $10$ Jupiter-mass planet in a coplanar, circular
orbit with initial semimajor axis $0.5$ au and initial longitude
$0^\circ$ is integrated. The orbital elements of the secondary are
adjusted so that the centre of mass of it and its putative planet
orbits the primary with the elements shown in Table 1 for star B
alone. As expected from the test particle results, the planet remains
stable for the full Myr length of the integration and shows very
little variation in its orbit. However, the 1 Myr integrations here
may overestimate the extent of the stability zone for planets, as
instabilities can appear even after 100 Myr of apparent stability
(Jones, Sleep \& Chambers 2001). To calculate the radial velocity
curve, the system is assumed to be at zero inclination relative to the
line of sight ($i = 90^\circ$). Figure~\ref{fig:rv} shows the radial
velocity curve of the primary, together with the residuals with
respect to the case with no extra planet. There is no detectable
signal with the period of the extra planet. The small variations
shown, which have the period of the binary, amount to 25 ms$^{-1}$
over 1000 yr timespans, would be undetectable.
\section{Edgeworth-Kuiper Belt Analogues}
\label{sec:kb}
The final region remaining to be studied is that exterior to the
binary. This may host particles analogous to the Edgeworth-Kuiper belt
in our own Solar system. There is observational evidence for the
existence of extrasolar Edgeworth-Kuiper belts from infrared imaging
of dusty discs around other stars (e.g., Wyatt 2003, Greaves et al.
2004), so the longevity of such a structure around $\gamma$ Cephei is
worth investigation.
Numerical integrations of test particles around binary systems suggest
that they will remain stable in this region (e.g., Harrington
1977). In addition to the criterion in Section~\ref{sec:tpib}, Holman
\& Wiegert (1999) also provide an empirical rule-of-thumb to determine
prograde test particle stability exterior to the binary. Here, the
critical semimajor axis $a_{\rm crit}$ is that beyond which almost all
test particles survive for at least $10^4$ binary periods. Using
eq. (3) of their paper, we find that $a_{\rm crit} = 64 \pm 2$ au.
However, Holman \& Wiegert (1999) caution that there can sometimes be
islands of instability beyond the critical radius associated with
$n$:1 MMRs. There is an older criterion due to Harrington (1977), who
also considers the case of retrograde test particles. Harrington's
equation suggests that prograde test particles are stable for $a \gta
56$ au and retrograde stable for $a \gta 45$ au. By stability,
Harrington means that the particles show bounded motion with no
secular or large periodic variations in their elements over his -- by
nowadays standards -- very small integration timespans.
Using the same method as in Sections~\ref{sec:tpip} and
\ref{sec:tpib}, test particles are set up with inclinations $0$ to
$30^\circ$ and $150^\circ$ to $180^\circ$ in the (barycentric) region
from $20$ to $150$ au in steps of 5 au. The mean survival times are
shown in Figure~\ref{fig:outer}. The prograde test particles exhibit a
sharp cut-off, with those beyond $65$ au being stable, independent of
the starting inclination. This figure also shows that the retrograde
test particles survive to the end of the integration for starting
semimajor axes beyond 40 au for $180^\circ$ inclination, 45 au for
$170^\circ$ and $160^\circ$ inclinations and 60 au for $150^\circ$
inclination. The range of initial conditions for which retrograde
particles survive is larger than for prograde (e.g., Harrington 1977).
For all inclinations, unstable particles from about $20$ to $40$ au
are removed quickly and generally are ejected after a close encounter
with the secondary. Unstable particles from about $40$ to $65$ au are
less rapidly removed and tend to collide with star A or be ejected
from the system, without experiencing a prior close encounter. We see
that both Harrington's (1977) and Holman \& Wiegert's (1999) stability
criteria seem to give a reasonable description of the results of our
simulations.
The average maximum eccentricity and change in semimajor axis is shown
in Figure~\ref{fig:baryboth} as a function of initial semimajor axis
for all eight inclinations. All the test particles show a small
secular variation in their orbital elements.
The Edgeworth-Kuiper belt in our Solar system shows some structure due
to locations of MMRs with the giant planets (e.g., Luu \& Jewitt
2002). However, the results in Figure~\ref{fig:outer} do not indicate
any such resonant features. This is most likely due to the coarse
semimajor axis grid employed. To investigate this, the coplanar case
was re-run, but now with a spacing in semimajor axis of $1$ au. The
mean survival times and fates of individual particles for this
simulation are shown in Figure~\ref{fig:14}. The first location where
all test particles survive is now $64$ au, agreeing exactly with
Holman \& Wiegert's value of $a_{\rm crit}$. There is then an unstable
band from $67$ to $70$ au that seems to match up to the 7:1 MMR. The
plot gives no evidence for any other resonant effects beyond this
location. This is understandable since the locations of the other
major resonances are outside the region of Hill stability (for
example, the 3:2 resonance that is important in our own Solar system),
leaving only those of very high order to affect stable test particles.
The distinct difference in fates of test particles is obvious in this
plot, with the blue coloured points, indicating close encounters with
star B, not occurring beyond about $45$ au. At this point, the test
particles are yellow or grey, indicating either ejection or collision
with star A without undergoing any close encounter.
\section{Conclusions}
In this paper, we have carried out a suite of test particle
integrations for the $\gamma$ Cephei system, as a guide to possible
locations for additional planets. For test particles in the plane of
the binary and planet, there are three zones of stability for 1 Myr
timespans at least. These are [1] the region interior to the planet,
which is stable within the bands $1.2$ and $1.3$ au, $1.05$ to $1.15$
au, and $0.75$ to at least $0.5$ au, [2] the region around star B from
$1.5$ au into at least $0.5$ au and [3] the region around both stars
extending out from about $65$ au. These can be used to constrain
possible locations of additional planetary companions.
The region interior to the planet has a complicated structure. Low
order resonances, such as the 4:1, 3:1 and 5:2, mark transitions
between stable and unstable regimes. There is a secular resonance
located at $0.8$ au from the primary that also plays a role in the
dynamics of the test particles. The results for this region match up
well with the previous work of Dvorak et al. (2003) despite the
differences in the parameters of the system and the methods used. This
implies that the slight improvements in the observations of the system
have not significantly altered its dynamical characteristics. One
difference though is that, unlike Dvorak et al. (2003), we find that
test particles close to the 3:1 resonance are unstable, just as for
the asteroid belt in the Solar System. In agreement with Dvorak et
al. (2003), we find that small planets interior to $\gamma$ Cephei Ab
are long-lived. Another as yet unconsidered possibility seen from the
results here is the existence of an asteroid belt interior to the
giant planet. However, the stability seen for test particles in this
region disappears when $\gamma$ Cephei Ab is inclined in the same
plane as the test particles. Now, the test particles are rapidly
driven to inclinations which are subject to the Kozai instability.
The region interior to the binary and exterior to the planet is
unpromising. It seems unlikely that any prograde planet or asteroid
can remain stable between $\gamma$ Cephei Ab and the binary, although
retrograde test particles in this region are more long-lived. The
region around the secondary is stable out to $1.5$ au for test
particles. We have shown that planets up to 10 Jupiter-masses in
circular orbits at $0.5$ au can survive for at least 1 Myr. This
seems to be a promising place for additional planets to reside, in a
similar manner to satellites about a planet around a single star.
In the final region, that exterior to the binary, test particles are
also stable. Although planets would be able to reside out here, the
existence of an Edgeworth-Kuiper belt structure is also a possibility.
Once again the retrograde particles are more stable than those that
are prograde. Whilst in every region studied this is true, such
objects are perhaps unlikely, if the origin of the system was a common
disk. If particles are captured from elsewhere, then this may become
a possibility. Retrograde asteroids and comets certainly exist in the
Solar system.
Since the survival of additional planets in the system has been shown
to be a possibility, it is interesting to consider their habitability.
There are a number of estimates of the habitable zone around star A in
the literature. For example, Dvorak et al. (2003) place it at $1.0$ to
$2.2$ au, in which case habitable planets could exist at the very edge
of the zone. However, Haghighipour (2005) places the habitable zone
at $3.1$ to $3.8$ au, while Jones et al. (2005) place it at $2.07$ to
$4.17$ au. These different locations reflect differences in the
criteria for the location of the habitable zone or the assumed stellar
luminosity and effective temperature. The zone $2.07$ to $4.17$ au is
unstable for all except retrograde test particles. So, this would
permit the possibility of a habitable planet only if retrograde. The
detection of any such small planet is challenging, given the small (of
the order of a few ms$^{-1}$) radial velocity signatures that they
cause. In addition, a retrograde planet appears in radial velocity
curves merely as a prograde one with a $180^\circ$ phase difference.
Although retrograde objects -- especially planets -- are thought to be
unlikely, it has been shown that giant planets in binary systems can
end up with retrograde orbits after close encounters within a
planetary system (Marzari et al. 2005).
A more interesting possibility is that of a habitable planet around
star B, since it has been shown that planets can survive
here. Theories of planetary formation do not preclude this possibility
(e.g., Armitage, Clarke \& Palla 2003). The habitable zone for an M
dwarf extends out to about 0.3 au (Kasting, Whitmire \& Reynolds
1993), which is within the stable zone found here. The large primary
star nearby might also act as a `shield' from comets and asteroids
similar to Jupiter for the Earth. As seen in the results in
Sections~\ref{sec:b} and \ref{sec:kb}, very few test particles collide
with the secondary star once the region interior to the binary has
been rapidly cleared. Edgeworth-Kuiper belt objects that are perturbed
into the inner regions of the system are also likely to suffer
encounters with the primary, thus leaving the secondary and its
environs largely unscathed.
The detection of such a habitable planet around star B has,
unfortunately, been shown to be virtually impossible from the radial
velocity signature of star A alone. Should the secondary be resolved
this would change, and any companion giant planet would be easily
detectable. The region around star B, none the less, remains as a
promising place for the existence of a habitable planet in this
system.
\section*{Acknowledgments}
PEV acknowledges financial support from the Particle Physics and
Astronomy Research Council. We thank the referee for his helpful
report.
\label{lastpage}
|
Title:
Stable Models of superacceleration |
Abstract: We discuss an instability in a large class of models where dark energy is
coupled to matter. In these models the mass of the scalar field is much larger
than the expansion rate of the Universe. We find models in which this
instability is absent, and show that these models generically predict an
apparent equation of state for dark energy smaller than -1, i.e.,
superacceleration. These models have no acausal behavior or ghosts.
| https://export.arxiv.org/pdf/astro-ph/0601517 |
\bibliographystyle{prsty}
\preprint{UCI-TR-2006-1}
\title{Stable Models of Super-acceleration}
\author{Manoj\ Kaplinghat and Arvind\ Rajaraman}
\affiliation{Department of Physics and Astronomy\\
University of California, Irvine, California 92697, USA}
\date{\today}
\pacs{98.70.Vc}
\section{Motivation\label{sec:motivation}}
Observations of distant Type Ia supernovae \cite{riess98,
perlmutter99} and the cosmic microwave background \cite{spergel03}
together strongly prefer an accelerated
expansion of the universe in the recent past. In the standard
cosmological model this is accommodated by introducing ``dark
energy'', a component which has a significantly negative pressure
causing the expansion of the universe to accelerate.
In the standard cosmological model, dark energy is completely
decoupled from the rest of the matter in the universe except for
its gravitational effects. It is interesting to consider more
general models in which the dark matter and dark energy have a
coupling. Such models could have new nontrivial signatures in
cosmology and structure formation.
One simple class of such models is a model in which the vacuum
energy density depends on the matter density. We shall consider a
class of these models in which the dark energy responds to changes
in the matter density on a time scale shorter than the expansion
time scale. For example, one can consider models with scalar field
dark energy coupled to matter (\eg
\cite{casas91,anderson97,amendola99,bean01,comelli03,farrar03,chimento03}),
in which the mass of the scalar field is much larger than the
expansion rate (for example, the MaVaN scenario \cite{fardon03}).
As we show below, these models generically suffer from an
instability which we label AZK-instability. The AZK-instability
was pointed out in the context of mass-varying neutrinos
(MaVaN) \cite{afshordi05}. A similar effect was identified in the
context of unified dark energy models \cite{beca05}. This
instability can also occur in models of dark energy coupled to
matter, such as the MaVaN scenario \cite{fardon03}, the Chameleon
dark energy scenario \cite{brax04} and the Cardassian expansion
scenario \cite{freese02}. Not all models in the above scenarios
are necessarily unstable (for example,
\cite{fardon06,koivisto05,takahashi06}). This will become clear
when we discuss the instability.
In this paper, we will construct a large class of models in which
this instability is avoided. We find that these models generically
predict an apparent equation of state (pressure over energy
density) $\wde$ which is less than -1 (such a phase is labeled
super-acceleration~\cite{kaplinghat03b}). That is, a model of
interacting dark energy can be incorrectly interpreted as a theory
with super-acceleration if the interactions are not taken into
account.
For example, the coupling of dark energy to matter could be such that
the total matter density decreases more slowly than $1/a^3$ (where
$a$ is the scale factor of the universe). When we interpret
observations in such a universe with a canonical matter density term
(that decreases with expansion as $1/a^3$) and dark energy, we would
infer an equation of state for dark energy more negative than it
truly is \cite{huey04,das05}. There is no physical reason why this
inferred equation of state cannot be below -1.
This is particularly interesting because current data seem to
favor a dark energy density which is almost constant or even
increasing with time
\cite{caldwell99,schuecker02,tonry03,knop03,choudhury03,alam03,melchiorri03,majerotto04,astier06,schaefer05}.
and exciting results can be expected in the future
\cite{weller01,frieman02,linder03,kratochvil04}. SNIa observations
currently favor a phase of super-acceleration. Future SNIa and CMB
observations have the potential to detect super-acceleration
\cite{kaplinghat03b}. No other combination has been shown to
robustly detect the signature of super-acceleration, although
combining SNIa and baryon oscillation \cite{astier06} or weak
lensing data set seem promising. Note that a measurement of just
the average equation of state \cite{saini03} is not sufficient for
this purpose \cite{maor01}. This was made explicit recently
\cite{csaki05} using a simple single scalar field model.
Scalar field models with canonical kinetic terms always produce
$\wde > -1$. Effective models with the opposite sign kinetic term
\cite{caldwell99,schulz01} imply $\wde < -1$ but are unstable
\cite{carroll03} unless more than one scalar field
\cite{feng04,guo04,hu04,wei05,urena05} or quantum effects
\cite{onemli04} are considered. Models with higher derivative terms or
scalar-tensor theories can give rise to an apparent $\wde < -1$
\cite{boisseau00}, but are constrained
\cite{carroll04,vikman04,abramo05}. Interpreting an alternative
gravity theory in the context of 4-d GR can also lead to
super-acceleration
\cite{mcinnes01,sahni02,pietroni02,elizalde04,nojiri05,martin06}. Some
Cardassian models may have $\wde <-1$
\cite{wang03,koivisto04,freese05} while still satisfying the dominant
energy condition. Another possible way to get super-acceleration with
no instabilities is to appeal to photon-axion mixing (conversion of
photons to axions) in a universe dominated by a cosmological constant
(or quintessence) \cite{csaki04}.
In our models, the superacceleration arises due to interactions of
dark energy and matter. Our models therefore provide
super-acceleration with none of the attendant problems that plague
most of the above models. Furthermore, the interactions are generic;
we do not need to fine-tune couplings in order to avoid theoretical
pitfalls or observational constraints. We therefore believe that
considering interactions of dark energy is the best way to generate
models of superacceleration.
\section{AZK-instability\label{sec:instability}}
In this section we will consider a general class of models in
which the dark energy density is coupled to the non-relativistic
matter density. For an example of how this could occur, suppose
that non-relativistic matter particles are coupled to a scalar
field. Thus the local density of the matter particles can
influence the vacuum expectation value (vev) of the scalar field.
The change in the potential of the scalar then affects the dark
energy, thus coupling matter and dark energy.
In this class of models, the matter fields will be taken to have
a matter density $\rhom$. They are coupled to a scalar field
$\chi$ (dark energy) through Yukawa like couplings. We take the
potential to be
\bea E&=&\int d^3x\ V(\chi,\rhom)\,,\\
&=&\int d^3x\ \left[ V_0(\chi)+m\rhom+\lambda g(\chi)\rhom
\right]\,.\label{eq:defineV}\eea We will assume that \(
m_\chi^2=V''_0(\chi_0)+\lambda g''(\chi_0)\rhom\), the
mass-squared of the scalar field about its vev $\chi_0$, is very
large so that the $\chi$ field always sits at the minimum of its
effective potential. This is the central assumption of our paper.
The mass will certainly have to be larger than the expansion rate
of the universe to be consistent with this assumption. We will
also assume that the mass is large enough to satisfy the
constraints imposed by experiments that probe the strength of a
fifth force.
In the absence of the last term, this is the potential energy of
two decoupled fluids. The first term corresponds to a cosmological
constant term (since we have assumed that the field $\chi$ is
always at the minimum). The second term is the energy density of a
dark matter fluid with density $\rhom$ and particle mass $m$.
The last term couples these two fluids, and leads to interesting
effects. In particular $\chi_0$, the value of the scalar field at
its minimum is now found by solving the equation \bea
V'_0(\chi_0)+\lambda g'(\chi_0)\rhom =0 \,,\label{eq:potmin} \eea
where $V_0'$ and $g'$ are derivatives of $V_0$ and $g$ with
respect to $\chi$. Thus $\chi_0$ is now a function of $\rhom$.
We can make the dependence of $\chi_0$ on $\rhom$ explicit in the
following way. Consider small deviations in $\rhom$. The vev of
the scalar field shifts to account for this change in $\rhom$.
Taking a further derivative, we find \bea (V''_0(\chi_0)+\lambda
g''(\chi_0)\rhom){\partial\chi_0\over\partial\rhom}+\lambda
g'(\chi_0) =0 \,.\label{eq:dchi0dn}\eea This explicitly shows how
$\chi_0$ varies as $\rhom$ varies.
In writing Eq.~\ref{eq:defineV}, we neglected the kinetic term in
comparison to the potential. This is necessary if the scalar field
is to behave as dark energy and, as we now show, consistent with
our assumption of a large mass for the scalar field. Note that
$\dot{\chi}=\dot{\rhom}\partial \chi_0/\partial \rhom$. Working
out this expression, we find that $\dot{\chi}^2/V$ for
$\chi=\chi_0$ is given by $(V_0^{\prime 2}/V
m_\chi^2)(\dot{\rhom}/m_\chi\rhom)^2$. Lets look at changes to the
scalar field potential around $\chi=\chi_0$. Unless there are
strong fine-tunings and cancellations, we will have
$V_0'\pert{\chi} < V$ and $m_\chi^2(\pert{\chi})^2/2 < V$, which
together imply that $2V_0^{\prime 2}/V m_\chi^2 < 1$. Hence the
natural expectation is that $\dot{\chi}^2/V \sim H^2/m_\chi^2$.
For large enough $m_\chi$, the kinetic term is negligible.
We now show that there is an instability in this system. We start
with a configuration where the dark matter is evenly distributed,
and the $\chi$ field is at its minimum $\chi_0$ everywhere. Now
consider small fluctuations in the matter density $\d \rhom$ which
preserve $\int_\tau d^3x\ \d \rhom=0$, i.e., the total number in
volume $\tau$. The integral is over some region $\tau$, much
smaller than the Hubble volume, over which the fluctuations are
coherent. Such a fluctuation leads to a change in the total
energy. The energy change proportional to $\d \rhom$ vanishes
because of Eq.~\ref{eq:potmin} and the condition that $\int_\tau
d^3x\ \d \rhom=0$. The energy change to next order is \bea
\pert{E}= {1\over 2}\int d^3x
(\pert{\rhom})^2\left({\partial\chi_0\over\partial\rhom}\right)\left(
m_\chi^2
{\partial\chi_0\over\partial\rhom}
+2\lambda
g'(\chi_0)\right)
\\
= -{1\over 2}\int d^3x
(\pert{\rhom})^2\lambda^2{\left[g'(\chi_0)\right]^2 \over m_\chi^2
}~~~\eea
Therefore the leading correction to the energy is always negative,
implying that the configuration is unstable to the growth of these
fluctuations. We dub this the AZK-instability. This instability
was first noted in the context of the MaVaN scenario
\cite{afshordi05}.
We have neglected gravity and the expansion of the universe in the
above analysis. We neglected gravity because the relevant length
scales are much smaller than the Jeans length; the instability
occurs on all scales and hence the effect is most severe on
microscopic scales. The analysis above was thus for a region
$\tau$ much smaller than that where gravity would be important. We
neglected the expansion of the universe because the relevant time
scales are much smaller than the age of the universe. In addition
our setup started with a smooth distribution of matter. For this
one must go to scales smaller than the free-streaming scale of
dark matter particles. For example, the comoving free-streaming
scale of a typical neutralino dark matter particle is of the order
of parsec. We do not study this system on larger cosmologically
relevant scales. It is, however, unlikely that the system will
still able to drive the accelerated expansion of the universe
since the generic AZK instability is intimately related to the
adiabatic sound speed of the fluid \cite{afshordi05}.
The result above assumes that the scalar field is much heavier
than the expansion rate of the universe. This constraint is easy
to satisfy and the large mass makes the model more robust to
radiative corrections (for example, see \cite{fardon06}).
Secondly, the calculation is only valid for modes which have a
wavelength much larger than $1/m_\chi$; for shorter wavelengths,
we cannot assume that the scalar field relaxes to the minimum
quickly enough.
\section{Avoiding the AZK-instability\label{sec:avoiding-instability}}
To avoid this instability, we look at more general couplings.
Consider now a model where the total energy is \bea E=\int d^3x\
\left[ V_0(\chi)+m \rhom+\lambda g(\chi)\rhom^n \right] \,,\eea
and we choose $\lambda > 0$ without loss of generality.
Again we assume that the scalar field tracks the minimum of the
potential and hence we have, \bea V'_0(\chi_0)+\lambda
g'(\chi_0)\rhom^n=0 \,,
\\
(V''_0(\chi_0)+\lambda
g''(\chi_0)\rhom^n)\left({\partial\chi_0\over\partial\rhom}\right)+\lambda
g'(\chi_0)n\rhom^{n-1}=0\,.\eea
Following our earlier calculation, we find \bea \pert{E}=
{1\over 2}\int d^3x
\left({\pert{\rhom} \over \rhom}\right)^2 \left(-{\left[n \lambda
g'(\chi_0) \rhom^n \right]^2 \over
m_\chi^2}\right. \nonumber \\
\left. +\lambda n(n-1)g(\chi_0)\rhom^n { \over }\right)\,,\eea
Therefore, the instability is avoided if \bea -n^2\lambda^2\rhom^n
{\left[g'(\chi_0)\right]^2 \over m_\chi^2} +n(n-1)\lambda
g(\chi_0) > 0 \,.\label{eq:condition-on-n} \eea
We note that the first term is always negative and gets large with
$\rhom$ unless $g'(\chi_0)$ decreases fast enough. Looking at the
second term we note that any value of $0 < n\leq 1$ is unstable
independent of the form of $g(\chi)$ except for the requirement
that $g(\chi_0) > 0$ which is required anyway for the potential to
be bounded from below.
A robust way to avoid the instability is to choose $n<0$, which
makes the second term positive. This is, of course, not sufficient
to guarantee the inequality in Eq.~\ref{eq:condition-on-n}. We
need the magnitude of the second term to be larger than that of
the first. This is easy to arrange. We again look at changes to
the potential as we vary $\chi$ about $\chi_0$. If the potential
is not fine-tuned to give rise to cancellations between terms in
the Taylor expansion, then $n\lambda g' \rhom^n \pert{\chi} < V$
and also $m_\chi^2(\pert{\chi})^2/2 < V$. Putting these two
expressions together yields $2n^2\lambda^2 (g')^2
\rhom^{2n}/m_\chi^2 < V \sim \lambda g \rhom^n$. Hence we see that
it is natural, if $n<0$, for the inequality in
Eq.~\ref{eq:condition-on-n} to be satisfied.
It is also possible to avoid the instability by choosing $n>1$.
However, this region of model space will be heavily constrained by
observations. In situations where the matter density gets large,
i.e., in collapsed structures, the last term in the potential
dominates. It would make the dark energy density in galaxies
large, change structure formation and clustering properties of
dark matter halos. Therefore, these kinds of models would be
tightly constrained. In order for these models to be viable,
$\lambda$ would have to be small and the model would essentially
be the same as that with two decoupled fluids.
Thus the requirement of AZK-stability and observational
constraints naturally lead us to consider models where $n<0$. We
now look at observational consequences of such a coupling.
\section{AZK-stability and Super-acceleration\label{sec:super-acceleration}}
The coupling term above with $n<0$ introduces a very interesting
effect: this model has super-acceleration. That is, observations
will seem to show a phase with dark energy equation of state less
than -1.
To see this, we first note that the observational quantity that is
important is the pressure. We will fit to the observations a model
with matter that scales with the expansion as $1/a^{3}$, and dark
energy with some equation of state $w_{\rm DE}$. Note that adding
or removing a component of energy density that scales as $1/a^3$
does not change the pressure of the fluid. Hence very generally
$P_{\rm tot}=P_{\rm DE}$. $P_{\rm tot}$ is defined by the equation
$\dot{V}=-3H(V+P_{tot})$ from which we find
$P_{tot}=-V_0(\chi_0)+\lambda g(\chi_0)\rhom^n(n-1)$. We set the
equation of state $\wde \equiv P_{\rm tot}/(V-m\rhom)$ and find,
\bea
\wde =-1+{n\lambda g(\chi_0)\rhom^n\over V_0(\chi_0)
+\lambda g(\chi)\rhom^n}\,.\eea
Now since $n<0$, the second term is actually negative, and we have
$w_{DE}<-1$ i.e. super-acceleration.
We emphasize that this super-acceleration is {\it not} accompanied by
any of the problems normally associated with theories with
equation of state less than -1. There is no acausal behavior, and
there are no ghosts. This is because the
super-acceleration in our model results from an interaction which
is ignored in the fitting of theory to observations. If we fit our
observations using a canonical matter density term and dark
energy, then the interaction has the effect of making the the
effective equation of state for dark energy more negative.
\section{Sound speed\label{sec:sound-speed}}
Here we present an alternative derivation of the instability in
terms of the sound speed of the combined fluid. A negative sound
speed squared would signal instability.
On length scales much larger than $m_\chi^{-1}$, the evolution of
the system is adiabatic and hence the sound speed is
\begin{equation}\label{eq:define-c}
c_a^2 = {\dot{P}_{\rm tot} \over \dot{V}}\,.
\end{equation}
The adiabatic sound speed in this theory can then be expressed as
\begin{eqnarray}
c_a^2 & = & { \rhom \partial \wde / \partial \rhom + \wde(1+\wde)
\over
1+\wde+m\rhom/(V-m\rhom)} \,.\label{eq:c-wde}\\
&=&{\rhom \over M} \left[ {\partial^2 V(\chi_0,\rhom) \over
\partial \rhom^2} - m_\chi^2\left({\partial \chi_0 \over
\partial \rhom}\right)^2 \right]\,,\label{eq:c-V}\\
& = & { \rhom \partial \wdem / \partial \rhom + \wdem(1+\wdem)
\over
1+\wdem}
\,,\label{eq:c-wdem}
\end{eqnarray}
where $\wdem \equiv P_{\rm tot}/V$ is the equation of state of the
total fluid.
For a universe with an accelerating expansion $\wdem < -1/3$. For
a wide class of models with $\wdem<0$ and either the $\rhom
\wdem'$ term sub-dominant or negative, we have $c_a^2 < 0$ and the
system is unstable. This is just the AZK-instability.
Lets now look in more detail at Eq. \ref{eq:c-wde}. First,
consider the case where $\wde>-1$: the denominator is positive and
if the $\wde'$ term is sub-dominant or negative, then
AZK-instability sets in. It is clear that this instability may not
be present in models with $\wde < -1$. We also note that this
instability will likely set in well before the current epoch
because at early times $\rhom/(V-\rhom) \gg 1$. For this case
where $\wde(1+\wde) > 0$, the sign and magnitude of the
$\rhom\wde'$ term is important. In particular, the requirement
that the $\rhom\wde'$ term is sub-dominant may not be trivial to
obtain \cite{linder06}.
While the above derivation shows us how the instability arises, it
does not provide us with an intuitive understanding of what
happens to the matter. In order to better understand that we look
at the Boltzmann equation for the matter coupled to a scalar
field. The scalar field gives the matter a mass term that can vary
spatially and temporally. Following AZK \cite{afshordi05}, we
write down the Boltzmann equation for matter neglecting gravity
and hence only valid on small scales. These are the scales of
interest since we have assumed $m_\chi \gg H$. We write down the
first order perturbations to this equation and expand the
perturbations in plane wave modes. Denoting the effective mass of
the matter particle by $M(\chi)$ we find, \bea \omega
\pert{f}({\vec p},{\vec k}) &-&
(\gamma M)^{-1}{\vec p}\cdot{\vec k}\pert{f}({\vec p},{\vec k}) \\
&+& \gamma^{-1}\pert{M}({\vec k}){\vec k}\cdot\nabla_{\vec
p}f({\vec p})=0 \,.\label{eq:boltzmann} \eea We then find the
perturbation to the matter density $\pert{\rhom}({\vec k})$ using
the above equation. In the limit that matter is non-relativistic,
the resulting equation has a simple form. We find that the
variation in effective mass of the particle is given by
$\pert{M}({\vec k}) = (M / \rhom) c_s^2 \pert{\rhom}({\vec k})$
where we have defined $c_s=\omega/k$, the sound speed of matter.
The above equation is valid for perturbations $\pert{M}$ on all
scales at which our assumptions hold. As pointed out in
\cite{afshordi05}, there is no scale in the equation for $c_s^2$
because we are studying scales where it is correct to assume that
the scalar field adjusts to changes in the matter density, and
gravity is unimportant.
We now turn to the fluid description and write
$M=V(\chi_0,\rhom)/\partial \rhom$. Using Eq.~\ref{eq:dchi0dn} for
$d\chi_0/d\rhom$, one may then obtain perturbations in $M$ as
$\pert{M} = (M / \rhom) c_a^2 \pert{\rhom}$ where $c_a^2$ is given
by Eq.~\ref{eq:c-V}. In the framework of a scalar degree of
freedom coupled to matter, both descriptions must be valid and
hence we find that $c_s^2=c_a^2$. The instability may therefore be
analyzed in terms of $c_a^2$. All of our analyses in earlier sections
go through if we work with $c_a^2$ and we conclude that models with
super-acceleration provide a generic way to avoid the AZK
instability.
\section{Conclusions\label{sec:conclusions}}
In this paper, we have explored the possibility that dark energy
may interact with matter. Such a hypothesis is natural if the
explanation for dark energy requires extra scalar degrees of
freedom. Unfortunately, as we have shown here, these models suffer
from a generic instability when the mass of the scalar field is
very large. We have verified that this instability is also present
in scalar-tensor theories where the scalar plays the role of dark
energy, and also in models with multiple scalar fields.
We then looked for models where this instability could be avoided,
and found a large class of such models. Most interestingly, we
found that in these models, the {\em apparent} equation of state
of the dark energy density is generically smaller than -1. This
super-acceleration is a result of the fact that we fit
observations with models that have non-interacting matter and dark
energy fluids.
There is a theoretical prejudice against models of $\wde<-1$ due
to their apparent theoretical problems. The observational data
certainly do not disfavor $\wde< -1$. Indeed a large region of the
parameter space allowed by SNIa observations corresponds to a
constant $\wde < -1$. Here we have shown that stable models with
$\wde< -1$ may be constructed without encountering ghosts or
acausal behavior. These models are no more fine-tuned than
quintessence models. Thus theoretical bias against $\wde <-1$
should be treated with circumspection, and not be given any weight
when interpreting observational data.
|
Title:
The ultra-cool white dwarf companion of PSR J0751+1807 |
Abstract: We present optical and near-infrared observations with Keck of the binary
millisecond pulsar PSR J0751+1807. We detect a faint, red object - with
R=25.08+-0.07, B-R=2.5+-0.3, and R-I=0.90+-0.10 - at the celestial position of
the pulsar and argue that it is the white dwarf companion of the pulsar. The
colours are the reddest among all known white dwarfs, and indicate a very low
temperature, Teff~4000 K. This implies that the white dwarf cannot have the
relatively thick hydrogen envelope that is expected on evolutionary grounds.
Our observations pose two puzzles. First, while the atmosphere was expected to
be pure hydrogen, the colours are inconsistent with this composition. Second,
given the low temperature, irradiation by the pulsar should be important, but
we see no evidence for it. We discuss possible solutions to these puzzles.
| https://export.arxiv.org/pdf/astro-ph/0601205 |
\title{The ultra-cool white dwarf companion of PSR~J0751+1807}
\author{C. G. Bassa\inst{1}
\and M. H. van Kerkwijk\inst{2}
\and S. R. Kulkarni\inst{3}}
\institute{Astronomical Institute, Utrecht University, PO Box 80\,000,
3508 TA Utrecht, The Netherlands\\
\email{c.g.bassa@astro.uu.nl}
\and Department of Astronomy and Astrophysics, University of
Toronto, 60 Saint George Street, Toronto, ON M5S 3H8,
Canada
\and Palomar Observatory, California Institute of Technology
105-24, Pasadena, CA 91125, USA%
}
\offprints{C.G. Bassa}
\date{Received / Accepted}
\abstract{We present optical and near-infrared observations with Keck
of the binary millisecond pulsar PSR~J0751+1807. We detect a faint,
red object -- with $R=25.08\pm0.07$, $B-R=2.5\pm0.3$, and
$R-I=0.90\pm0.10$ -- at the celestial position of the pulsar and
argue that it is the white dwarf companion of the pulsar. The
colours are the reddest among all known white dwarfs, and indicate a
very low temperature, $T_\mathrm{eff}\approx4000$\,K. This implies that
the white dwarf cannot have the relatively thick hydrogen envelope
that is expected on evolutionary grounds. Our observations pose two
puzzles. First, while the atmosphere was expected to be pure
hydrogen, the colours are inconsistent with this
composition. Second, given the low temperature, irradiation by the
pulsar should be important, but we see no evidence for it. We
discuss possible solutions to these puzzles.
\keywords{Pulsars: individual (\object{PSR~J0751+1807})
-- binaries: close
-- stars: neutron
-- white dwarfs}
}
\section{Introduction}\label{sec:intro}
Among the pulsars in binaries, the largest group, the low-mass binary
pulsars, has low-mass white-dwarf companions. Before the companions
became white dwarfs, their progenitors filled their Roche lobe and
mass was transferred to the neutron stars, thereby spinning them up and
decreasing their magnetic fields. Considerations of the end of this
stage, where the white dwarf progenitor's envelope becomes too tenuous
to be supported further, allow one to make predictions for relations
between the orbital period and white dwarf mass, and orbital period
and eccentricity (for a review, e.g., \citealt{pk94,sta04}). Furthermore,
after the cessation of mass transfer, two clocks will start ticking at
the same time: the neutron star, now visible as a millisecond pulsar,
will spin down, while the secondary will contract to a white dwarf and
start to cool. Consequently, the spin-down age of the pulsar should
equal the cooling age of the white dwarf.
From optical observations of white-dwarf companions to millisecond
pulsars one can estimate the white-dwarf cooling age and compare it
with the pulsar spin-down age. Initial attempts to do this
\citep{hp98a,hp98b,sdb00} revealed a dichotomy in the cooling
properties of white dwarfs in the sense that some white dwarf
companions to older pulsars have cooled less than those of younger
pulsars. In particular, the companions of PSR~J0437$-$4715
\citep{dbv93,sbb+01} and PSR~B1855+09 \citep{kbkk00,rt91} have
temperatures of about 4000--5000\,K, with characteristic pulsar ages
of 5\,Gyr. This is in contrast to the companion of PSR~J1012+5307
\citep{llfn95,kbk96,cgk98}, which has a higher temperature (8600\,K),
while it orbits an older pulsar (8.9\,Gyr).
A likely cause for this dichotomy is the difference in the thickness
of the envelope of hydrogen surrounding the helium core of the white
dwarf \citep{ashp96}. After the cessation of mass transfer, the white
dwarfs have relatively thick ($\sim\!10^{-2}$\,M$_\odot$) hydrogen
envelopes which are able to sustain residual hydrogen shell-burning,
keeping the white dwarf hot and thereby slowing the cooling
\citep{dsbh98}. The shell burning, however, can become unstable and
lead to thermal flashes which can reduce the mass of the
envelope. White dwarfs with such reduced, relatively thin
($\la10^{-3}$\,M$_\odot$) hydrogen envelopes cannot burn hydrogen and,
as a result, cool faster. The transition between thick and thin
hydrogen envelopes was predicted to lie near 0.18--0.20\,M$_\odot$
(where heavier white dwarfs have thin envelopes;
\citealt{ashp96,seg00,asb01}).
Until recently, PSR~J1012+5307, with an orbital period
$P_\mathrm{b}=0.60$\,d, was the only system for which a thick hydrogen
envelope was required to match the two timescales. Given the relation
between the white dwarf mass and the orbital period
\citep{jrl87,rpj+95,ts99}, companions in similar or closer orbits
should have similar or lower mass, and thus have thick hydrogen
envelopes as well. This was confirmed by the recent discovery of two
new, nearby, binary millisecond pulsars with orbital periods similar
to that of PSR~J1012+5307; PSR~J1909$-$3744 (1.53\,d,
\citealt{jhb+05}) and PSR~J1738+0333 (0.354\,d, Jacoby et al., in
prep.; see \citealt{kbjj05} for preliminary results). For both, the
temperatures and characteristic ages are similar to those of PSR
J1012+5307, and thus one is led to the same need for a thick hydrogen
envelope. These discoveries, combined with the thin envelopes inferred
for PSR~J0034$-$0534 (1.59\,d) and binaries with longer periods,
suggest that the transition occurs at a mass that corresponds to an
orbital period just over 1.5\,d (\citealt{kbjj05}). All systems with
shorter orbital periods should have thick hydrogen envelopes.
The two known millisecond pulsars with white dwarf companions that
have shorter orbital periods than PSR~J1012+5307 but do not have
optical counterparts, are PSR~J0613$-$0200, with a 1.20\,d period, and
PSR~J0751+1807, which has the shortest orbital period of all binary
millisecond pulsars with $M_\mathrm{c}>0.1$\,M$_\odot$ companions,
0.26\,d \citep{lzc95}. The latter system is of particular interest
because the companion mass has been determined from pulse timing
($M_\mathrm{WD}=0.19\pm0.03$\,M$_\odot$ at 95\% confidence;
\citealt{nss+05}), so that one does not have to rely on the
theoretical period-mass relationship. Intriguingly, for PSR
J0751+1807, optical observations from \citet{lcf+96} set a limit to
the temperature of 9000\,K, which is only marginally consistent with
it having a thick hydrogen envelope. Based on this, \citet{esa01},
suggested the hydrogen envelope may have been partially lost due to
irradiation by the pulsar.
The faintness of the companion to PSR~J0751+1807 aroused our curiosity
and motivated us to obtain deep observations to test the theoretical
ideas discussed above. We describe our observations in
Sect.~\ref{sec:observations}, and use these to determine the
temperature, radius and cooling history in Sect.~\ref{sec:tandr}. In
Sect.~\ref{sec:irradiation}, we investigate irradiation by the pulsar,
finding a surprising lack of evidence for any heating. We discuss our
results in Sect.~\ref{sec:discussion}.
\section{Observations and data reduction}\label{sec:observations}
The PSR~J0751+1807 field was observed with the 10~meter Keck I and II
telescopes on Hawaii on five occasions. On December 11, 1996 the Low
Resolution Imaging Spectrometer (LRIS, \citealt{occ+95}) was used to
obtain $B$ and $R$-band images, while the Echellette Spectrograph and
Imager (ESI, \citealt{sbe+02}) was used on December 21, 2003 to obtain
deeper $B$ and $R$-band, as well as $I$-band images. The $R$-band
filter used that night was the non-standard ``Ellis $R$'' filter. The
observing conditions during the 1996 night were mediocre, with
0\farcs8--1\farcs1 seeing and some cirrus appearing at the end of the
night. The conditions were photometric during the 2003 night, and the
seeing was good, 0\farcs6--0\farcs8. The third and fourth visit were
with LRIS again, now at Keck I, on January 7 and 8, 2005. The red arm
of the detector was used to obtain $R$-band images. The seeing on the
first night in 2005 was rather bad, about 1\farcs5 and improved to
about 1\farcs0 on the second night. The conditions on these nights
were not photometric. Finally, a series of 36 dithered exposures, each
consisting of 5 co-added 10\,s integrations, were taken through the
$K_\mathrm{s}$ filter with the Near Infrared Camera (NIRC;
\citealt{ms94}) on January 26, 2005. The conditions were photometric
with 0\farcs6 seeing. Standard stars (\citealt{lan92,ste00}) were
observed in 1996 and 2003, while a 2MASS star \citep{csd+03} in the
vicinity of PSR~J0751+1807 was observed to calibrate the NIRC data. A
log of the observations is given in Table~\ref{tab:obs}.
The images were reduced using the Munich Image Data Analysis System
(MIDAS). The $BRI$ images were bias-subtracted and flat-fielded using
dome flats. The longer exposures in each filter were aligned using
integer pixel offsets, and co-added to create average images. The
near-infrared images were corrected for dark current using dark frames
with identical exposure times and number of co-adds as those used for
the science frames. Next, a flatfield frame was created by median
combining the science frames. After division by this flatfield, the
science frames were registered using integer pixel offsets and
averaged.
\begin{table}
\begin{minipage}[t]{\columnwidth}
\centering
\caption[]{Observation log.}\label{tab:obs}
\renewcommand{\footnoterule}{}
\begin{tabular}{l@{\hspace{0.5cm}}
c@{\hspace{0.5cm}}
c@{\hspace{0.5cm}}
c@{\hspace{0.5cm}}
c@{\hspace{0.5cm}}
}
\hline
Field & Time (UT) & Filter & $t_\mathrm{int}$ (s) & $\sec z$ \\
\hline
\multicolumn{5}{l}{December 11, 1996, LRIS} \\[0.2ex]
SA\,95 & 08:23--08:25 & $R$ & $2+10$ & 1.07 \\
& 08:27--08:29 & $B$ & $2+10$ & 1.07 \\[0.1ex]
SA\,95 & 09:28--09:31 & $B$ & $2+10$ & 1.07 \\
& 09:33--09:35 & $R$ & $2+10$ & 1.08 \\[0.1ex]
PSR~J0751+1807 & 09:45 & $R$ & $10$ & 1.39 \\
& 09:47--09:59 & $R$ & $2\times300$ & 1.36 \\
& 10:01 & $R$ & $600$ & 1.31 \\
& 10:13 & $B$ & $600$ & 1.26 \\[0.8ex]
\multicolumn{5}{l}{December 21, 2003, ESI} \\[0.2ex]
PSR~J0751+1807 & 10:06--10:27 & $R$ & $3\times360$ & 1.14 \\
& 10:29--10:57 & $I$ & $6\times240$ & 1.08 \\
& 11:00--11:33 & $B$ & $3\times600$ & 1.04 \\[0.1ex]
NGC\,2419 & 11:40 & $B$ & $10+30$ & 1.06 \\
& 11:44 & $R$ & $10+30$ & 1.06 \\
& 11:47 & $I$ & $10+30$ & 1.06 \\[0.8ex]
\multicolumn{5}{l}{January 7, 2005, LRIS} \\[0.2ex]
PSR~J0751+1807 & 11:54--12:53 & $R$ & $5\times600$ & 1.05 \\[0.8ex]
\multicolumn{5}{l}{January 8, 2005, LRIS} \\[0.2ex]
PSR~J0751+1807 & 11:42--12:51 & $R$ & $6\times600$ & 1.05 \\[0.8ex]
\multicolumn{5}{l}{January 26, 2005, NIRC} \\[0.2ex]
PSR~J0751+1807 & 08:06--08:56 & $K_\mathrm{s}$ & $36\times50$ & 1.07 \\
2MASS star\footnote{2MASS\,J07510621+1807253} & 08:59 & $K_\mathrm{s}$ & $0.4$ & 1.02\\
\hline
\end{tabular}
\end{minipage}
\end{table}
\subsection{Astrometry}
For the astrometric calibration, we selected 14 stars from the second
version of the USNO CCD Astrograph catalogue (UCAC2; \citealt{zuz+04})
that overlapped with the 10\,s $R$-band LRIS image of December
1996. Of these, 11 were not saturated and appeared stellar and
unblended. The centroids of these objects were measured and corrected
for geometric distortion using the bi-cubic function determined by
J.~Cohen (1997, priv.\
comm.)\footnote{http://alamoana.keck.hawaii.edu/inst/lris/coordinates.html}. We
fitted for zero-point position, plate scale and position angle. The
inferred uncertainty in the single-star measurement of these 11 stars
is $0\farcs057$ and $0\farcs083$ in right ascension and declination,
respectively, and is consistent with expectations for the UCAC
measurements of approximately $0\farcs020$ for stars of 14th magnitude
and $0\farcs070$ for stars 2 magnitudes fainter.
This solution was transferred to the 10\,min $R$-band LRIS image using
91 stars that were present on both images and were stellar,
unsaturated and not blended. Again the zero-point position, plate
scale and position angle were left free in the fit and the final
residuals were $0\farcs016$ and $0\farcs019$ in right ascension and
declination. The UCAC is on the International Celestial Reference
System (ICRS) to $\la\!0\farcs01$, and hence the final systematic
uncertainty with which our coordinates are on the ICRS is dominated by
our first step, and is $\sim\!0\farcs03$ in each coordinate.
Our images, with the position of PSR~J0751+1807 \citep{nss+05}
indicated, are shown in Figure~\ref{fig:fc}. On the 10\,min LRIS
$R$-band images from 1996 and 2005, we find a faint object, hereafter
star X, at the position of the pulsar. It is also, though marginally,
present in the two 5\,min $R$-band images from 1996, but not detected
in the 10\,min $B$-band LRIS image of that observing run. Star X is
clearly present in the 2003 ESI $R$ and $I$-band images, and
marginally in the $B$-band image. It is not detected in the
near-infrared observations (Fig.~\ref{fig:fc}).
Positions for star X and other objects inferred using the astrometry
of the 10\,min LRIS $R$-band image are listed in Table~\ref{tab:phot}.
The pulsar position at the time of the 1996 LRIS observation, using
the \citet{nss+05} position and proper motion, is
$\alpha_\mathrm{J2000}=07^\mathrm{h}51^\mathrm{m}09\fs1574(1)$,
$\delta_\mathrm{J2000}=+18\degr07\arcmin38\farcs624(10)$. We find
that star X is offset from the pulsar position by
$-0\farcs01\pm0\farcs06$ in right ascension and
$0\farcs04\pm0\farcs06$ in declination, well within the $1\sigma$
uncertainties (including those on the pulsar position). Given the low
density of about 47 stars per square arcminute and the excellent
astrometry, the probability of a chance coincidence in the 95\%
confidence error circle, which has a radius of $0\farcs24$, is only
0.1--0.2\%. Since, as we will see, it is hard to envisage how the
companion could be fainter than the object detected, we are confident
that star X is the companion of PSR~J0751+1807.
\subsection{Photometry}\label{sec:photometry}
The DAOPHOT II package (\citealt{ste87}), running inside MIDAS, was
used for the photometry on the averaged
images. We followed the recommendations of
\citet{ste87}: instrumental magnitudes were obtained through point
spread function (PSF) fitting and aperture photometry on brighter
stars was used to determine aperture corrections.
For the calibration of the optical images, instrumental magnitudes of
the standard stars, determined using aperture photometry, were
compared against the values of \citet{ste00}. We used the standard
Keck extinction coefficients of 0.17, 0.11 and 0.07\,mag per airmass
for $B$, $R$ and $I$, respectively. Colour terms were not required for
the LRIS $B$ and $R$ bands, but were significant for the ESI bands:
$0.107 (B-R)$ for $B$, $0.083 (B-R)$ for $R$, and $-0.004 (R-I)$ for
$I$, i.e., the ESI $B$, $R$ are redder than the standard bands, while
ESI $I$ is slightly bluer. The root-mean-square residuals of the ESI
calibrations are about 0.05\,mag in $B$, and 0.03\,mag in $R$ and $I$,
while those of the LRIS calibration are 0.08\,mag in $B$ and 0.05\,mag
in $R$; we adopt these as the uncertainty in the zero-points. The
near-infrared observations were calibrated through aperture photometry
with 1\farcs5 (10\,pix) apertures using the 2MASS star, fitting for a
zero-point only, as the difference in airmass between the science and
calibration images is small. We adopt an uncertainty in the
$K\mathrm{s}$ zero-point of 0.1\,mag.
Calibrated ESI magnitudes for star X and selected other stars in the
field are listed in Table~\ref{tab:phot}. Star X is barely above the
detection limit of the ESI $B$-band observations, hence the large
error. It is not detected in the LRIS $B$-band and the NIRC
$K_\mathrm{s}$-band observations, and, scaling from the magnitude of a
star with a signal-to-noise ratio of about 10 and 6, we estimate the
3$\sigma$ detection limits at $B=26.8$ and $K_\mathrm{s}=21.3$,
respectively. The former is consistent with the ESI detection. None
of the stars in Table~\ref{tab:phot} are covered by the small
$38\arcsec\times38\arcsec$ field-of-view of NIRC, hence we do not have
near-infrared magnitudes for these.
The 1996 LRIS $R$-band magnitude is $25.13\pm0.11$, which is
consistent with the ESI measurement. Since the conditions during the
1996 LRIS observations may not have been photometric, however, this
may be a coincidence. To check for variability, we tied the
instrumental LRIS $R$ band magnitudes directly to the ESI $R$ and $I$
ones, using 38 stars that both images had in common and that had
magnitude uncertainties below 0.1 mag. As expected given the
non-standard ``Ellis $R$'' filter on ESI, we required a large colour
term, $-0.302(R_\mathrm{inst}-I_\mathrm{inst})$, but with this the fit
was adequate, with root-mean-square residuals of 0.14 mag. Compared to
the fit, the ESI minus LRIS difference in $R$-band magnitude is
insignificant, $-0.03\pm0.13$\,mag. Similarly, comparing instrumental
$R$-band magnitudes from 2005 January 7 with those taken 2005 January
8 and 1996 December 11, fitting for an offset only, results in
magnitude differences of $0.03\pm0.07$ and $-0.16\pm0.12$\,mag,
respectively. Thus, no large variations in brightness are seen; we will
see in Sect.~\ref{sec:irradiation} that this is somewhat surprising.
\begin{table}
\caption[]{LRIS Astrometry and ESI photometry of the companion of
PSR~J0751+1807 and stars in the field. The nomenclature of the
stars is according to Fig.~\ref{fig:fc}. The uncertainties
listed in parentheses are instrumental, i.e., they do not include
the zero-point uncertainty in the astrometric tie (about
$0\farcs03$ in each coordinate) or of photometric calibration
(0.05 mag in $B$ and 0.03 mag in both $R$ and $I$).}
\label{tab:phot}
\begin{tabular}
{l@{\hspace{0.15cm}}
l@{\hspace{0.15cm}}
l@{\hspace{0.15cm}}
l@{\hspace{0.15cm}}
l@{\hspace{0.15cm}}
l@{\hspace{0.15cm}}
}
\hline
\hline
ID & \multicolumn{1}{c}{$\alpha_\mathrm{2000}$} &
\multicolumn{1}{c}{$\delta_\mathrm{2000}$} &
\multicolumn{1}{c}{$B$\phantom{0}} &
\multicolumn{1}{c}{$R$\phantom{0}} &
\multicolumn{1}{c}{$I$\phantom{0}} \\
& $\phantom{00}^\mathrm{h}\phantom{00}^\mathrm{m}\phantom{00}^\mathrm{s}$
&$\phantom{00}\degr\phantom{00}\arcmin\phantom{00}\arcsec$ & & & \\
\hline
X & 07 51 09.158(4) & 18 07 38.66(6) & 27.56(25) & 25.08(7) & 24.18(7) \\[0.2em]
A & 07 51 09.933(1) & 18 07 05.97(1) & 21.73(1) & 19.30(1) & 18.31(1) \\
B & 07 51 10.844(1) & 18 07 52.91(1) & 22.80(1) & 21.03(1) & 20.32(1) \\
C & 07 51 10.891(1) & 18 07 35.69(1) & 24.30(2) & 21.81(1) & 20.63(1) \\
D & 07 51 10.739(1) & 18 07 32.79(1) & 24.28(6) & 22.50(5) & 21.99(6) \\
E & 07 51 08.519(1) & 18 07 59.89(2) & 24.56(7) & 22.87(5) & 22.38(8) \\
F & 07 51 08.859(2) & 18 07 08.83(3) & 24.94(4) & 24.00(5) & 23.29(4) \\
G & 07 51 08.908(4) & 18 07 35.71(5) & 25.65(8) & 24.51(5) & 23.85(6) \\
H & 07 51 10.691(3) & 18 07 24.69(6) & 25.69(7) & 24.94(9) & 24.34(8) \\
\hline
\end{tabular}
\end{table}
\section{Temperature, radius, and cooling history}\label{sec:tandr}
We use our observations of star X, the companion of PSR~J0751+1807, to
constrain its temperature, radius, and atmospheric constituents, and
discuss our result that the white dwarf does not have the expected
thick hydrogen envelope.
\subsection{Colours, temperature, and atmospheric composition}
We first use the colours of star X to constrain its temperature. The
red colours are largely intrinsic, as the maximum reddening towards
PSR~J0751+1807 ($l=202.73$, $b=21.09$) is small, $E_{B-V}=0.05\pm0.01$
\citep{sfd98}. This value is consistent with the low value found for
the interstellar absorption
$N_\mathrm{H}\sim4\times10^{20}$\,cm$^{-2}$, as estimated from {\it
ROSAT} X-ray observations of PSR~J0751+1807 by \citet{btl+96}. For
comparison, the relation by \citet{ps95} predicts an
$N_\mathrm{H}\approx3\times10^{20}$\,cm$^{-2}$ for the above
reddening. Given the distance of $\sim\!0.6\,$kpc \citep{nss+05}, we
expect most of the reddening to be in the foreground to the
pulsar. Hence, the dereddened colours are $(B-R)_0=2.40\pm0.27$ and
$(R-I)_0=0.86\pm0.10$.
In Fig.~\ref{fig:tcd}a, we compare the intrinsic colours of star X with
those of other white-dwarf companions of millisecond pulsars, other
white dwarfs, and models. We find that the colours of star X are the
reddest for any known millisecond pulsar companion or white dwarf.
The pulsar companion that comes closest is that of PSR~J0437$-$4715
($B-R=2.12\pm0.06$, $R-I=0.56\pm0.02$ [\citealt{dbv93}] and negligible
extinction\footnote{As inferred from the dust maps of \cite{sfd98};
\cite{dbv93} estimate $E_{B-V}=0.07$ from the work of \cite{knu79}.});
the most similar white dwarf is WD~0346+246 ($B-R=2.2\pm0.1$,
$R-I=0.76\pm0.08$, \citealt{osh+01}). Thus, star X is likely as cool
or even cooler than the $T_{\rm eff}\simeq3700\,$K inferred for those
two sources (PSR~J0437$-$4715: \citealt{dbv93}; Hansen 2002, priv.\
comm.; WD~0346+246: \citealt{osh+01,ber01}).
Also shown in Fig.~\ref{fig:tcd}a are colours expected from model
atmospheres of \citet{sarb01} and of Hansen (2004, priv.\ comm.),
which are specifically tailored to the low-mass, helium-core
companions of millisecond pulsars, as well as those for updated
low-gravity ($\log g=7$), pure hydrogen atmosphere models\footnote{For
updated versions of the \citet{bwb95} models, see
http://www.astro.umontreal.ca/\~{}bergeron/CoolingModels/} of
\citet{bwb95}. One sees that the colours of the companion of
PSR~J0437$-$4715, as well as those of the hotter companions of
PSR~J1012+5307 and J0218+4232, are consistent with these models. For
star X, however, the colours are not consistent, as the models never
venture redwards of $R-I\approx0.7$ and $B-R\approx2.0$.
The change in direction of the tracks is seen in all models for
hydrogen-rich, metal-free atmospheres; it reflects a change in the
dominant source of opacity, from bound-free absorption of H$^-$ at
higher temperatures to collision-induced absorption of H$_2$ at lower
ones (\citealt{lcs91,sbl+94,han98}). The latter process is highly
non-grey, and leads to absorption predominantly longward of the
$R$-band. As a result, the $R-I$ colour becomes bluer with decreasing
temperatures, while $B-R$ remains roughly constant.
Could star X have a different composition? Due to the high gravity of
white dwarfs, metals settle out of the atmosphere. However, some white
dwarfs have atmospheres dominated not by hydrogen, but by helium. For
the latter, the opacity sources are all fairly grey, and hence the
colours continue to redden with decreasing temperatures. Indeed, the
colours of star X are consistent with the predictions of the updated
$\log g=7$ pure helium models after \citet{bwb95} at
$T_\mathrm{eff}\simeq4200$\,K (Fig.~\ref{fig:tcd}a).
From an evolutionary perspective, however, a pure helium atmosphere is
not expected. Low-mass white dwarfs such as the companions to
millisecond pulsars are all formed from low-mass stars whose evolution
was truncated by mass transfer well before helium ignition (for recent
models, see \citealt{ts99,ndm04}). As a result, they should have
helium cores surrounded by relatively thick, 0.01 to 1\% of the mass,
hydrogen envelopes (\citealt{dsbh98,asb01}). Indeed, among the
low-mass white-dwarf companions to pulsars \citep{kbjj05} as well as
among low-mass white dwarfs in general \citep{blr01}, only
hydrogen-dominated atmospheres have been observed.
In principle, at low temperatures, the hydrogen envelope might become
mixed in with the helium core. Even if fully mixed, however, the
remaining amounts of hydrogen would strongly influence the spectrum.
Indeed, the effects of collision-induced absorption {\em increase}
with increasing helium abundance up to $N({\rm He})/N({\rm
H})\simeq10^5$ (\citealt{bl02}).
From Fig.~\ref{fig:tcd}a, it is clear that the predictions for
hydrogen-dominated atmospheres are also a somewhat poor match to the
colours of the cooler normal white dwarfs with hydrogen in their
atmospheres (as inferred from absorption at H$\alpha$,
\citealt{blr01}; filled circles in the figure). For most, this appears
to be due to missing blue opacity in the models (see \citealt{blr01}
for a detailed study); the visual through infrared fluxes are
reproduced well by the models, and show unambiguously that
collision-induced absorption by H$_2$ is important. Indeed, the
absorption is evident in the optical colours of some objects, in
particular LHS~3250 (shown in Fig.~\ref{fig:tcd}) and
SDSS~J133739.40+000142.8 (\citealt{bl02} and references therein).
For our purposes, however, the case of the ultra-cool white dwarf
WD~0346+246 is most relevant. For this source, the colours cannot be
reproduced with either pure hydrogen or helium, but require a mixed
atmosphere, dominated by helium (with fractional hydrogen abundances
ranging from $10^{-9}$ to $10^{-1}$, depending on assumptions about
the contribution of other opacity sources; \citealt{osh+01,ber01},
though recent work puts these abundances in to doubt, P.\ Bergeron
2005, priv.\ comm.). For all cases, the temperature is around
3700\,K. The similarity in the colours of WD~0346+246 and star~X would
suggest that star~X has a similar, maybe slightly lower, temperature.
From the above, we find that we cannot determine the temperature of
the companion of PSR~J0751+1807 with certainty, since we do not know
its composition. Most likely, however, it is somewhere between the
temperature inferred for WD~0346+246 and that indicated by the (pure
helium) models, i.e.\ in the range of, say 3500--4300\,K.
A more stringent test could be provided by the near-infrared
observations, as the $R-K$ colour (which is similar to
$R-K_\mathrm{s}$) differs for different predictions. At a temperature
of 4000\,K the $\log g=7$ \citet{bwb95} models predict $R-K$ colours
of 2.7 and 1.6 for pure helium and pure hydrogen atmospheres,
respectively. For the same temperature, $R-K=1.6$ is predicted by the
0.196\,M$_\odot$ model by \citet{sarb01}. Finally, for WD~0346+246,
with presumably a mixed hydrogen/helium atmosphere, \citet{osh+01}
observed $R-K=-0.7$. Unfortunately, our near-infrared observations
only limit the colour to $R-K<3.8$, which does not constrain any of
these predictions.
\subsection{Brightness, distance and radius}
So far, we have only discussed the colours and temperature. We now
turn to the absolute magnitude and radius. In Fig.~\ref{fig:cmd}b, we
show $M_R$ as a function of $R-I$. For star~X, we computed the
absolute $R$-band magnitude $M_R$ using the parallax of
$\pi=1.6\pm0.8$\,mas as measured through radio timing \citep{nss+05}.
The resulting distance of $0.6^{+0.6}_{-0.2}$\,kpc is consistent with
that estimated from the dispersion measure which predicts
$1.1\pm0.2$\,kpc, using a dispersion measure of
$30.2489\pm0.003$\,pc\,cm$^{-3}$ \citep{nss+05} and the recent model
of the Galactic electron distribution of \citet{cl02}. Correcting for
the reddening, this implies $M_R=15.97^{+0.88}_{-1.51}$.
Given the similarities in the above absolute magnitude of star~X and
that of WD~0346+246 ($M_R=16.1\pm0.3$; \citealt{hsh+99,osh+01}), and
assuming similar temperature, one finds that the radius of star~X
should be comparable to the $R=0.010$\,R$_\odot$ for WD~0346+246
\citep{ber01}. However, the large uncertainty in the parallax of
PSR~J0751+1807 allows radii between 0.007--0.021\,R$_\odot$. For the
white-dwarf mass of $\sim\!0.19$\,M$_\odot$ inferred from pulse timing
\citep{nss+05}, this is consistent the $\sim\!0.022$\,R$_\odot$
expected from the 0.196\,M$_\odot$ model by \citep{sarb01}.
As can be seen in Fig.~\ref{fig:cmd}, the absolute magnitude is also
consistent with the predicted values from the $\log g=7$ pure helium
model by \citet{bwb95}. At a temperature of $T_\mathrm{eff}=4250$\,K,
this model has a radius of 0.020\,R$_\odot$ and a mass of
0.15\,M$_\odot$, somewhat smaller than the observed
0.19\,M$_\odot$. To correct for the small difference in mass, we
computed white dwarf radii for the observed temperature and mass of
the companion and used these to scale the absolute magnitudes of the
pure helium track in Fig.~\ref{fig:cmd}. At 0.19\,M$_\odot$ and
$T_\mathrm{eff}=4000$\,K, the \citet{pab00} helium core white dwarf
mass-radius relation predicts 0.021\,R$_\odot$. This is very similar
to the radius predicted by the \citet{bwb95} $\log g=7$ pure helium
models, and as such, the absolute magnitudes are comparable. We
conclude that, with in the large uncertainties on the parallax
distance, the absolute magnitude and radius that we derive for the
companion of PSR~J0751+1807 are consistent with the predictions for a
pure helium atmosphere.
We note that of the models presented in Fig.~\ref{fig:cmd}, those of
\citet{bwb95} have been extensively tested to explain the population
of nearby white dwarfs \citep{blr01,ber01,bl02} and use a very
detailed description of the white dwarf atmosphere combined with the
latest opacities (P.\,Bergeron 2005, priv.\ comm.). This is not the
case for the models of Serenelli et al.\ and Hansen, and thus we
should be careful in using their models quantitatively. Indeed, as can
be seen from Fig.~\ref{fig:cmd}, their models do not reproduce the
observations of cool white dwarfs well. For instance, for the
companion of PSR~J0437$-$4715, which has a well-determined mass of
$0.236\pm0.017$\,M$_\odot$ and distance of $139\pm3\,$pc
\citep{sbb+01}, the models of \cite{sarb01}, while consistent with the
observed $B-R$ and $R-I$ well, do not reproduce $R-I$ and $M_R$
simultaneously. In contrast, the $0.2$\,M$_\odot$ model of Hansen
(2004, priv.\ comm.) does pass through the $R-I$, $M_R$ point, but
cannot reproduce both colours. It may be that both problems reflect
uncertainties in the model atmospheres used by Hansen and
\citet{sarb01}. It would be worthwhile to couple the evolutionary
models of these authors with the updated, very detailed atmospheric
model of \citet{bwb95}.
\subsection{Cooling history and nature of the envelope}\label{sec:cooling}
Despite the uncertainty in the models and in the composition of the
atmosphere, our observations show that the companion of PSR~J0751+1807
has cooled much more than expected if the amount of hydrogen was thick
enough for significant residual nuclear burning
(Sect.~\ref{sec:intro}). Indeed, the temperature is as expected if no
residual hydrogen burning occurred. For instance, at the
characteristic age of the pulsar, $\tau=7.1$\,Gyr \citep{nss+05}, the
0.196\,M$_\odot$ of \citet{sarb01}, which has a thin envelope,
predicts a temperature of about 3200\,K, which is roughly consistent
with what is observed. With a pure helium atmosphere, a slightly
colder temperature, of $\sim\!2500$\,K, is expected, though this is a
less secure estimate due to uncertainties in the opacities
\citep{hp98a}
The presence of a thin (or no) hydrogen envelope is not expected,
however, since thick envelopes are inferred for other optically
identified companions in short-period systems (see
Sect.~\ref{sec:intro}). What could be wrong with this expectation? It
was based on two theoretical ideas: (i) that below a certain critical
mass, no shell flashes occur and hydrogen layers will be thick; and
(ii) that the companion mass monotonously increases with increasing
orbital period. These assumptions appeared to be confirmed by the
available data: for PSR~J0751+1807, with a period of 0.26\,d, the
companion mass of 0.16--0.21\,M$_\odot$ (95\% conf.; \citealt{nss+05})
is similar to what is found for two other short-period systems with
companions for which thick hydrogen envelopes are inferred, and less
than the masses for longer period systems with thin-envelope
companions. Specifically, PSR~J1012+5307 (0.60\,d,
0.12--0.20\,M$_\odot$) and PSR~J1909$-$3744 (1.53\,d,
0.19--0.22\,M$_\odot$) have thick envelopes while PSR~J0437$-$4715
(5.74\,d, 0.20--0.27\,M$_\odot$) and PSR~B1855+09 (12.33\,d,
0.24--0.29\,M$_\odot$) have thin envelopes (see Fig.~\ref{fig:pbm2}
and \citealt{kbjj05} and reference therein). Thus, while the
uncertainties do not exclude that the companion of PSR~J0751+1807 is
so massive that it its envelope was diminished by shell flashes, the
existing data make it unlikely.
Two explanations for a thin envelope remain. First, there may be
differences in metallicity among the progenitors of pulsar companions.
\citet{sarb02} studied the evolution of low-mass pulsar companions
with sub-solar metallicity and found that, since the thermonuclear
flashes are induced by the reactions of the CNO-cycle, the threshold
mass between thin and thick hydrogen envelopes increases with
decreasing metallicity of the white dwarf progenitor. Thus, it may be
that the companion of PSR~J0751+1807 had a sufficiently higher
metallicity that it was above the threshold for shell flashes, while
companions in other short-period systems had lower metallicity and
hence were below the threshold, despite having higher masses.
The next possibility is that the white dwarf was indeed formed with a
thick envelope, which was subsequently removed by an action other than
shell flashes. Based on the upper limit on the temperature of
\citet{lcf+96}, \citet{esa01} already argued that the pulsar companion
could not have the thick hydrogen envelope, and they proposed a
scenario where part of the envelope was removed by pulsar
irradiation. \citeauthor{esa01} found that irradiation driven
mass-loss could remove as much as 0.01\,M$_\odot$ from the thick
hydrogen envelope (mostly while the companion is contracting following
the cessation of mass transfer).
A possible problem with the above suggestions, is that none predict
the removal of the entire hydrogen envelope, while the observed
colours seem most consistent with a pure helium or at least
helium-dominated atmosphere.
\section{Irradiation by the pulsar?}\label{sec:irradiation}
Above, we have treated the companion as if it were an isolated
object rather than member of a binary system. Might the presence of a
relatively energetic pulsar influence our observations?
The observed pulsar period and period derivative imply a spin-down
luminosity $L_\mathrm{SD}=(2 \pi)^2 I
\dot{P}/P^3=7.5\times10^{33}\,I_{45}$\,ergs\,s$^{-1}$ \citep{lzc95,nss+05},
where $I=10^{45}\,I_{45}{\rm\,g\,cm^2}$ is the pulsar moment of
inertia. For a $2.1$\,M$_\odot$ pulsar and a $0.19$\,M$_\odot$
companion, the orbital separation is $a=2.3$\,R$_\odot$, and,
consequently, the irradiative flux of the pulsar wind incident on the
companion is $f_{\rm
irr}=2.1\times10^{10}\,I_{45}$\,erg\,s$^{-1}$\,cm$^{-2}$. This is about
twice the flux of the companion itself, $f_{\rm th}=\sigma
T_\mathrm{eff}^4=1.06\times10^{10}$\,erg\,s$^{-1}$\,cm$^{-2}$ for
$T_\mathrm{eff}=3700$\,K. Therefore, the presence of the pulsar and
its irradiation may be important.
Given the irradiation, one would expect the side of the companion
facing the pulsar to be brighter than the side facing away from it.
Thus, from Earth, the companion should appear faintest at phase 0.25
and brightest at phase 0.75 (using the convention that at phase 0, the
pulsar is at the ascending node). This is indeed seen in other pulsar
binaries, with the black widow pulsar PSR~B1957+20 perhaps the most
spectacular example \citep{vpasc+88,fgld88}.
For star X, assuming a fraction $\eta$ of the incident flux is
absorbed and reradiated as optical flux, the flux from the bright side
of the companion should be a factor $1+\frac{2}{3}\eta f_{\rm
irr}/f_{\rm th}$ brighter (here, the factor $\frac{2}{3}$ reflects
projection effects). Observationally, the inferred values of $\eta$
range from 0.1 to 0.6 (\citealt{ok03}, and references therein), and
thus one expects a maximum change in bolometric flux by a factor 1.13
to 1.8. For the $R$-band flux, the range is 1.2 to 2.2 (assuming it
scales like a black-body spectrum, $\propto\!T^6$ around 3700\,K). We
confirmed this using a detailed light-curve synthesis model (described
briefly in \citealt{sklk99}).
For star~X, no effect is seen. Using the PSR~J0751+1807 ephemeris from
\citet{nss+05}, we find that during the ESI $R$-band observations the
orbital phase ranged from 0.22 to 0.25, while the 1996 LRIS $R$-band
images were taken at phases 0.86--0.90, and the 2005 LRIS images at
phases 0.01--0.14 on January 7, and 0.77--0.93 on January 8. Thus,
these observations span the orbital phases necessary to test for any
modulation in brightness. Indeed, using the inclination inferred from
timing, $i=66^{+4}_{-7}\,$deg \citep{nss+05}, we find that during the
ESI observations only 4 to 5\% of the irradiated part of the companion
surface was in view, while during the 1996 LRIS observations is was
78\% to 85\%. As a consequence, we expect to see nearly the maximum
change in brightness. Nevertheless, in Sect.~\ref{sec:photometry}, we
found no significant variation, $R_{\rm LRIS}-R_{\rm
ESI}=0.03\pm0.13$; thus, to $\sim\!99$\% confidence, the variation is
smaller than 0.3\,mag, which implies $\eta<0.15$.
The lack of observed modulation could be taken to indicate that the
irradiation is not very effective, e.g., because the albedo is large
(i.e., $\eta$ is small), the pulsar emission is non-isotropic, or the
spin-down luminosity is overestimated. We believe these options are
not very likely (for a discussion in a slightly different context, see
\citealt{ok03}), which leads us to consider the only alternative, that
one of the assumptions underlying the above estimates is wrong.
In particular, we assumed implicitly that irradiated flux is
reprocessed and re-emitted instantaneously, i.e., transfer of flux
inside and around the companion are assumed to have negligible effect.
For the companions of black-widow pulsars, this is reasonable, since
for these relatively large objects, tides will have ensured
synchronous rotation. Any flux transfer would thus have to be due to
winds and/or convection, which plausibly happens on a timescale long
compared to the thermal time of the layer in which the pulsar flux is
reprocessed.
The companion of PSR~J0751+1807, however, is well within its
Roche-lobe, and tidal dissipation should be negligible. We can
estimate its current rotation period from its prior evolution,
following the reasoning used by \cite{kk95} for the companion of PSR
B0655+64. Briefly, during mass transfer, the companion filled its
Roche-lobe and tides ensured the system was synchronised and
circularised. Once mass transfer ceased and the companion started to
contract to a white dwarf, however, the tides became inefficient, and
the rotational evolution of the companion was determined by
conservation of angular momentum.
For our estimates, we split the total moment of inertia of the
progenitor into two parts, one from the core, $I_\mathrm{core}=k_{\rm
core}^2M_\mathrm{core} R_\mathrm{core}^2$ and one from the envelope,
$I_\mathrm{env}=k_{\rm env}^2M_\mathrm{env} R_\mathrm{L}^2$; here $k$
is the radius of gyration and $R_{\rm L}$ is the radius of the Roche
lobe. After contraction of the envelope, one is left with a white
dwarf with $I_\mathrm{WD}=k_{\rm WD}^2 M_\mathrm{WD}
R_\mathrm{WD}^2$. If we now assume that $I_\mathrm{core} \simeq
I_\mathrm{WD}$ and ignore differences in radius of gyration,
conservation of angular momentum yields $\Omega_\mathrm{rot} /
\Omega_\mathrm{orb} \simeq 1+ M_\mathrm{env} R_\mathrm{L}^2 /
M_\mathrm{WD} R_\mathrm{WD}^2$. In reality, likely the envelope will
be more centrally concentrated than the white dwarf, i.e.,
$k_\mathrm{env}<k_{\rm WD}$, and tidal dissipation will be important
in the initial stages of the contraction. This will reduce the
spin-up. On the other hand, the hot core of the progenitor will be
larger than the white dwarf, i.e., $I_{\rm core}>I_{\rm WD}$. In any
case, it follows that unless the envelope mass is very small, the
white dwarf should be significantly spun up.
Model predictions for the envelope mass of helium-core white dwarfs
differ. The $0.196$\,M$_\odot$ model by \citet{sarb01}, has an
envelope mass of $6.7\times10^{-3}$\,M$_\odot$ (as given in
\citealt{asb01}), whereas a model of similar mass
($M_\mathrm{WD}=0.195$\,M$_\odot$) by \citet{dsbh98} has one of
$3.1\times10^{-2}$\,M$_\odot$. Using these values, taking
$M_\mathrm{WD}=0.19$\,M$_\odot$, $R_\mathrm{WD}=0.021$\,R$_\odot$ and
$R_\mathrm{L}=0.48$\,R$_\odot$, and ignoring differences in $k$, we
find current rotation periods a factor 18--85 faster than the
orbital period, or 20 to 5 minutes. Given that thick envelopes seem
inconsistent with the low observed temperature
(Sect.~\ref{sec:tandr}), the slower end of the range seems more
likely.
To estimate the timescale on which the pulsar flux is reprocessed, we
assume that the incident particles are predominantly highly energetic,
and that they penetrate to, roughly, one Thompson optical depth. This
corresponds to a column depth of $N=1.5\times10^{24}{\rm\,cm^{-2}}$,
for which the thermal timescale $t\simeq NkT/\sigma
T_\mathrm{eff}^4\simeq1\,$min, where the numerical estimate is for
$T=T_\mathrm{eff}=3700\,$K. This is shorter than the rotation periods
estimated above, suggesting that rotation may not be too important.
On the other hand, our estimate is very rough. For instance, at one
Thompson depth, the opacity at optical wavelengths is much smaller
than unity for the cool temperatures under consideration
\citep{sbl+94}. Thus, the material likely radiates less efficiently
than a black body, which would make the thermal timescale longer.
Furthermore, the irradiation will change the temperature and
ionisation structure of the atmosphere, further complicating matters.
(Indeed, could this be the underlying cause for the fact that the
colours deviate so strongly from those expected for a pure hydrogen
atmosphere?) Finally, it might induce strong winds which equalise the
temperature on both hemispheres (as is the case for Jupiter).
\section{Conclusions}\label{sec:discussion}
We have optically identified the white dwarf companion of the binary
millisecond pulsar PSR~J0751+1807. We find that the companion has the
reddest colours of all known millisecond pulsar companions and white
dwarfs. These colours indicate that the companion has a very low
(ultra-cool) temperature of
$T_\mathrm{eff}\sim\!3500-4300$\,K. Furthermore, the colours suggest
that the white dwarf has a pure helium atmosphere, or a helium
atmosphere with some hydrogen mixed in, as invoked for the field white
dwarf WD~0346+246 which has similar colours \citep{osh+01,ber01}.
Our observations are inconsistent with evolutionary models, from which
one would expect a pure hydrogen atmosphere. Indeed, as for other
short-period systems, the hydrogen envelope is expected to be thick
enough to sustain significant residual hydrogen burning, leading to
temperatures far in excess of those observed. It may be that the mass
of the envelope was reduced due to shell flashes or irradiation by the
pulsar, as was proposed by \citet{esa01}.
However, we see no evidence for irradiation, despite the fact that the
pulsar spin-down flux inpinging on the white dwarf is roughly double
the observed thermal flux. Clues to what happens might be found from
more detailed studies of the spectral energy distribution, or more
accurate phase-resolved photometry.
Finally, a deeper observation at infrared wavelengths would allow one
to distinguish between the different atmosphere compositions for the
companion: for a pure helium atmosphere, black-body like colours are
expected, while if any hydrogen is present, the infrared flux would be
strongly depressed (as is seen for WD 0346+246). With adaptive optics
instruments, such observations should be feasible.
\begin{acknowledgements}
We thank Norbert Zacharias for providing preliminary UCAC2 data. We
also would like to thank the referee, Pierre Bergeron, for his useful
suggestions and for pointing out the existence of his updated
models. The observations for this paper were taken at the W. M. Keck
Observatory, which is operated by the California Association for
Research in Astronomy, a scientific partnership among the California
Institute of Technology, the University of California, and the
National Aeronautics and Space Administration. It was made possible by
the generous financial support of the W. M. Keck Foundation. MIDAS is
developed and maintained by the European Southern Observatory. This
research made use of the SIMBAD and ADS data bases and of data
products from the Two Micron All Sky Survey, which is a joint project
of the University of Massachusetts and the Infrared Processing and
Analysis Center/California Institute of Technology, funded by the
National Aeronautics and Space Administration and the National Science
Foundation. We acknowledge support from NWO (C. G. B.), NSERC
(M. H. v. K.), and from NASA and NSF (S. R. K.).
\end{acknowledgements}
\bibliographystyle{aa}
|
Title:
The Bolocam 1.1 mm Lockman Hole Galaxy Survey: SHARC II 350 micron Photometry and Implications for Spectral Models, Dust Temperatures, and Redshift Estimation |
Abstract: We present 350 micron photometry of all 17 galaxy candidates in the Lockman
Hole detected in a 1.1 mm Bolocam survey. Several of the galaxies were
previously detected at 850 microns, at 1.2 mm, in the infrared by Spitzer, and
in the radio. Nine of the Bolocam galaxy candidates were detected at 350
microns and two new candidates were serendipitously detected at 350 microns
(bringing the total in the literature detected in this way to three). Five of
the galaxies have published spectroscopic redshifts, enabling investigation of
the implied temperature ranges and a comparison of photometric redshift
techniques.
Lambda = 350 microns lies near the spectral energy distribution peak for z =
2.5 thermally emitting galaxies. Thus, luminosities can be measured without
extrapolating to the peak from detection wavelengths of lambda > 850 microns.
Characteristically, the galaxy luminosities lie in the range 1.0 - 1.2 x 10^13
L_solar, with dust temperatures in the range of 40 K to 70 K, depending on the
choice of spectral index and wavelength of unit optical depth. The implied dust
masses are 3 - 5 x 10^8 M_solar. We find that the far-infrared to radio
relation for star-forming ULIRGs systematically overpredicts the radio
luminosities and overestimates redshifts on the order of Delta z ~ 1, whereas
redshifts based on either on submillimeter data alone or the 1.6 micron stellar
bump and PAH features are more accurate.
| https://export.arxiv.org/pdf/astro-ph/0601582 | command.
\shorttitle{Bolocam 1.1 mm Lockman Hole Galaxy Survey}
\shortauthors{Laurent et~al.}
\newcommand{\deriv}[2]{\frac{d{#1}}{d{#2}}}
\newcommand{\D}{\,d}
\newcommand{\Avg}[1]{\left\langle #1 \right\rangle}
\newcommand{\vect}[1]{\ensuremath{\mathbf{#1}}}
\begin{document}
\title{The Bolocam 1.1 mm Lockman Hole Galaxy Survey: SHARC II 350 $\mu$m Photometry and Implications for Spectral
Models, Dust Temperatures, and Redshift Estimation}
\author{
G.~T. Laurent\altaffilmark{1,2},
J. Glenn\altaffilmark{1},
E. Egami\altaffilmark{3},
G.~H. Rieke\altaffilmark{3},
R.~J. Ivison\altaffilmark{4},
M.~S. Yun\altaffilmark{5},
J.~E. Aguirre\altaffilmark{6,1},
P.~R. Maloney\altaffilmark{1}, \&
D.~Haig\altaffilmark{7}}
\altaffiltext{1}{Center for Astrophysics and Space Astronomy \& Department of Astrophysical and Planetary Sciences, University of Colorado,
593 UCB, Boulder, CO 80309-0593}
\altaffiltext{2}{glaurent@colorado.edu}
\altaffiltext{3}{Steward Observatory, University of Arizona, 933 North Cherry Avenue, Tucson, AZ 85721}
\altaffiltext{4}{UK Astronomy Technology Centre, Royal Observatory, Blackford Hill, Edinburgh EH9}
\altaffiltext{5}{Department of Astronomy, University of Massachusetts, Amherst, MA 01002}
\altaffiltext{6}{Jansky Fellow, National Radio Astronomy Observatory}
\altaffiltext{7}{Physics and Astronomy, Cardiff University, 5 The Parade, P.O. Box 913, Cardiff CF24 3YB, Wales, UK}
\keywords{galaxies: high-redshift ---
galaxies: starburst --- submillimeter}
\section{Introduction}
Surveys at submillimeter and millimeter wavelengths have detected hundreds of galaxy candidates by their thermal dust emission. The
galaxies (hereafter referred to as submillimeter galaxies) characteristically have redshifts $z > 1$ and inferred luminosities of $L \sim 10^{13}$ L$_\odot$ and star
formation rates of $10^3$ M$_\odot$ per year (assuming dust heating by young stars). Such enormous luminosities and star formation rates, or accretion rates in the case
of super-massive black hole growth, make submillimeter galaxies strong candidates for the progenitors of massive galaxies at the current epoch. Clearly, it is crucial
to characterize the spectral energy distributions (SEDs) where their emission peaks ($\lambda$ = 350 $\mu$m for 40 K dust at a redshift of $z = 2.5$), measure their
redshifts and luminosity functions, determine their power sources, and integrate them into theories of galaxy formation.
Although submillimeter galaxy SEDs peak at a few hundred microns for all but the highest redshifts, $\lambda \ge$ 850 $\mu$m surveys have been most successful at detecting
galaxies because of the lower atmospheric noise and greater transmission, and less stringent telescope surface requirements. Most of the detections have been low
signal-to-noise ratio (just over thresholds of 3-4 $\sigma$), necessitating multiwavelength confirmation. Furthermore, the SEDs have been extrapolated shortward from
850-1200 $\mu$m over the peak, or between 850-1200 $\mu$m and the far-infrared, to estimate dust temperatures, luminosities, and star formation rates.
Clearly, 350 $\mu$m photometry can confirm galaxy candidates and sample the SEDs near their peaks for more precise inferences of physical parameters.
Similarly, because of the difficulty in obtaining spectroscopic redshifts of large numbers of highly obscured galaxies, various photometric redshift estimation techniques have emerged, notably
based on the far-infrared to radio luminosity relation in ULIRGs \citep{carilli99,yun02} and the stellar continuum bump in the infrared for {\it Spitzer}-detected galaxies
\citep{egami04,sawicki02}. However, despite the difficulty, candidate spectroscopic redshifts have been obtained for $\sim$ 73 galaxies \citep{chapman05}. Thus, with
well-determined dust-emission SEDs, including 350 $\mu$m, photometric techniques can be compared to spectroscopic redshifts.
In this paper, we present 350 $\mu$m photometry of all 17 submillimeter galaxy candidates from the Bolocam Lockman Hole survey \citep{laurent05},
some with previous 850 $\mu$m and 1200 $\mu$m detections. Two new galaxy candidates were serendipitously detected at 350 $\mu$m, bringing the total number of 350
$\mu$m-discovered galaxies to three \citep{khan05}. We combine these data with infrared and radio data to derive improved luminosities, explore the range of
implied dust temperatures and spectral indices, and compare photometric redshift techniques. This comparison is timely with the imminent launch of the {\it Herschel
Space Observatory} (scheduled for August 2007), which will detect thousands of galaxies at far-infrared and submillimeter wavelengths, but for which spectroscopic redshifts will be attainable for
only a small fraction. Throughout the paper, a cosmology of H$_0 = 70$ km s$^{-1}$ Mpc$^{-1}$, $\Omega_M = 0.3$, and $\Omega_\Lambda$ = 0.7 is assumed.
\section{The 350 $\mu$m SHARC II Galaxy Survey}
\label{section:sharc}
Observations at multiple submillimeter wavelengths are vital both to confirm the Bolocam sources (as 6 false detections are
expected from Monte-Carlo simulations) and to make photometric
redshift and temperature estimates. The 350 $\mu$m SHARC-II observations combined with the Bolocam 1.1 mm galaxy survey provides a
flux density ratio that is strongly dependent on redshift for a given temperature. This is because the rest wavelength corresponding to the observed
wavelength of 350 $\mu$m with SHARC-II is near the peak of the grey-body spectrum (for a $z \sim 2$ galaxy at 40 K), and Bolocam's 1.1 mm observations
climb the steep $\nu^{2+\beta}$ ($\beta$ $\approx$ 1.5) modified Rayleigh-Jeans side of the spectral energy distribution.
Follow-up observations of each of the Bolocam Lockman Hole
galaxy candidates \citep{laurent05} were taken with the Submillimeter High Angular Resolution Camera (SHARC
II) at the Caltech Submillimeter Observatory. The observations were taken on three observing runs: 2004 March-April, 2005 January, and 2005 February. The brightest Bolocam
sources (1 and 2) were observed over 8 hours of total integration time during the 2004 March-April run, although most of the run was lost due to poor
weather. Bolocam sources 1 and 16 were observed over 18 hours of integration time during the 2005 January run, again with much of the run lost due
to poor weather. The 2005 February run was characterized by much better weather, and all the Bolocam sources except for 1, 5, 8 and 16 were
observed over 35 hours of integration time\footnote{The primary weather measurement correlated with the SHARC II mapping speed is the CSO 225 GHz heterodyne, narrowband,
``tipper tau'' monitor, which measures the zenith atmospheric attenuation. The 2004 March-April, 2005 January, and 2005 February Lockman Hole
observations yielded $\tau_{225\mathrm{GHz}}$ ranges and 75th percentiles of $\tau_{225\mathrm{GHz}}=0.046-0.087$, $\tau_{75\%}=0.076$,
$\tau_{225\mathrm{GHz}}=0.044-0.120$, $\tau_{75\%}=0.093$ and $\tau_{225\mathrm{GHz}}=0.030-0.074$, $\tau_{75\%}=0.047$, respectively.}. When combined with
the observations of Bolocam sources 5 and 8 by \cite{kovacs05}, the entire Bolocam sample was observed over these observing runs.
Observations with SHARC II were taken in the point source observing mode, with a Lissajous (parametric sinusoidal curve) scan pattern using the SWEEP command of telescope. The
Lissajous pattern was scanned in altitude and azimuth, with amplitudes of 30$\arcsec$ and 20$\arcsec$, respectively. When combined with the 2.6$\arcmin$ $\times$ 1.0$\arcmin$ SHARC
II field-of-view and 9$\arcsec$ FWHM instrument beam size, this resulted in a uniform coverage region of 95$\arcsec$ $\times$ 18$\arcsec$, with a border of additional coverage (60$\arcsec$
$\times$ 40$\arcsec$) outside this region.
Each observation had a fixed length of 10 minutes to ensure uniform coverage even on individual scans. Integration times and the resulting depths of each of the SHARC II
fields are listed in Table \ref{table:sharcdetections}.
\setlength{\tabcolsep}{1mm}
\begin{deluxetable}{ccccccccccc}
\tabletypesize{\scriptsize}
\tablecaption{SHARC II Photometry and New Galaxy Candidates}
\tablewidth{0pt}
\tablehead{
\colhead{} & \colhead{} & \colhead{} & \colhead{SHARC II/Bolocam} & \colhead{Bolocam 2 $\sigma$} & \colhead{SHARC II} & \colhead{SHARC II} &
\colhead{} & \colhead{} & \colhead{} \\
\colhead{Bolocam} & \colhead{SHARC II} & \colhead{SHARC II} & \colhead{Offset} & \colhead{Error Circle} & \colhead{R.A} & \colhead{Dec} &
\colhead{S/N} & \colhead{S$_\nu$} & \colhead{$\sigma$} \\
\colhead{Source} & \colhead{Source} & \colhead{\# 10 min. Scans} & \colhead{(")} & \colhead{(")} & \colhead{(J2000.0)} & \colhead{(J2000.0)} &
\colhead{} & \colhead{(mJy)} & \colhead{(mJy)} \\
}
\startdata
1&7 &30,16&13.2 &21&10:52:57.1&57:21:01&3.7&38.0&14.0\\
2&8 &17,23&15.8 &21&10:51:18.6&57:16:36&3.5&20.9&7.9\\
3&10 &20&9.1 &21&10:52:13.0&57:15:46&3.2&14.0&5.6\\
"&11&20&17.3 &21&10:52:14.0&57:16:02&3.1&15.1&6.2\\
4&21&20&(18.3) &22&10:52:04.8&57:18:39&&$\le$ 15.4&\\
5&6 (Kov\'{a}cs 4)&&15.0&22&10:52:30.9&57:22:06&5.9&40.4&8.6\\
6&1 & 5&12.0 &22&10:51:14.1&57:14:21&6.8&63.6&18.4\\
"&9 & 5&18.0 &22&10:51:17.8&57:14:20&3.3&27.6&10.8\\
7&20&15&(16.7) &22&10:51:28.6&57:30:50&&$\le$ 10.7&\\
8&5 (Kov\'{a}cs 5)&&16.3&23&10:52:38.8&57:24:38&6.2&40.5&8.1\\
9&18&14&(15.4) &23&10:53:05.0&57:15:08&&$\le$ 19.6&\\
10&16&36&(22.4) &23&10:51:31.0&57:23:35&&$\le$ 25.1&\\
11&19&13&(19.9) &23&10:52:49.0&57:13:01&&$\le$ 18.9&\\
12&14&18&(1.5) &23&10:51:15.5&57:15:22&&$\le$ 20.5&\\
13&17&13&(20.5) &23&10:52:34.9&57:18:13&&$\le$ 23.7&\\
14&2 &16&20.5 &24&10:52:01.7&57:24:43&5.9&24.1&7.3\\
15&15&15&(24.0) &24&10:51:47.9&57:28:57&&$\le$ 20.0&\\
16&13 &93&1.6 &25&10:52:27.3&57:25:13&3.0&44.0&18.3\\
17&12&16&6.0 &26&10:52:00.6&57:24:21&3.1&15.5&6.3\\
\hline
(9)&3 &14&- &23&10:53:08.3&57:15:01&4.8&28.4&9.2\\
(16)&4 &93&- &25&10:52:32.3&57:24:48&3.8&37.0&13.4\\
\enddata
\tablecomments{SHARC II detections and 3 $\sigma$ upper limits at each of the Bolocam sources, in order of descending brightness at 1.1 mm. The Bolocam sources in parentheses
correspond to SHARC II detections well outside of the Bolocam 2 $\sigma$ positional error circle, and are therefore believed not to be associated with the Bolocam source. Two
SHARC II detections from \cite{kovacs05} are also included. The SHARC II source numbers are listed in order of 350 $\mu$m S/N.
}
\label{table:sharcdetections}
\end{deluxetable}
\setlength{\tabcolsep}{2mm}
The reduction of the raw SHARC II data was accomplished with the use of the "deep" cleaning utility of the Comprehensive Reduction Utility for SHARC
II (CRUSH\footnote{http://www.submm.caltech.edu/$\sim$sharc/crush/index.htm}). Observations of pointlike galaxies, quasars, protostellar sources, H$_\mathrm{II}$ regions,
and evolved stars were used to construct pointing models
for each of the observing runs. Observations of the pointing sources were taken with a scan strategy identical to that of the science fields.
A subset of the pointing sources were used for flux density calibration, with reference 350 $\mu$m flux densities obtained from the SHARC II
website\footnote{http://www.submm.caltech.edu/$\sim$sharc/}.
Source extraction was performed on the CRUSH-cleaned maps, with each map (corresponding to a single Bolocam candidate) consisting of all of the
individual scans co-added together. The algorithm was begun by doing a cut on the uniform coverage region, defined as the set of pixels for which the
coverage is $\ge$ 60\% of the maximum per-pixel coverage. The uniform coverage region is a contiguous region in the center of each map. Next, an RMS in
sensitivity units (the flux density of each pixel times the square root of the integration time for that pixel in units of mJy s$^{1/2}$) was computed in
the uniform coverage region. This RMS is valid for the entire uniform coverage region since variations in coverage have been
accounted for by the $t_i^{1/2}$ coverage normalization, where $t_i$ is the total integration time for pixel $i$. All
pixels with coverage-normalized flux densities exceeding 3 $\sigma$ (``hot pixels'')
were flagged as potential sources. Then hot pixels were grouped into multi-pixel sources by making the maximal group of adjacent hot pixels, including
those within $\sqrt{2}$ pixels (i.e., diagonally adjacent). The peak flux density, right ascension and declination of the source candidates were computed
by centroiding two-dimensional Gaussians on the groups. The uncertainty in the flux density of each source is given by the
pixel-to-pixel RMS at the centroid location of the source.
\section{Positional Uncertainties}
\label{section:positionalerrors}
The large beam sizes of submillimeter and millimeter wave instruments (31$\arcsec$, 14$\arcsec$, 11$\arcsec$, and 9$\arcsec$ FWHM for Bolocam, SCUBA, MAMBO, and SHARC II, respectively)
makes it difficult to identify likely optical and radio counterparts to the galaxy candidates. Despite the large beam sizes, however, individual sources
can be centroided to much higher precision than the quoted beam size. To help constrain this issue of source matching between the various surveys, a
positional error circle was estimated for each of the submillimeter and millimeter band detections. For the Bolocam detections, Monte Carlo simulations
were performed by injecting sources into the timestream and running the reduction pipeline and source extraction algorithm. This simulation was repeated
for a range of source flux densities. The resulting centroiding error as a function of flux density (5.4 - 9.1$\arcsec$) was added in quadrature with the RMS telescope
pointing error (9.1$\arcsec$) to yield a range of 2 $\sigma$ positional error circles of 21 - 26$\arcsec$. A similar approach was used to estimate the centroiding errors for
SCUBA \citep{scott02} and MAMBO \citep{greve04}, yielding 2.2 - 10.4$\arcsec$ and 1.2 - 4.3$\arcsec$ respectively. When added in quadrature to the quoted pointing errors (4$\arcsec$ and
3$\arcsec$, respectively), this yields 2 $\sigma$ positional errors of 9.2 - 22$\arcsec$ and 6.5 - 13$\arcsec$, respectively. As the centroiding error as a function of flux density for
the SHARC II
observations was not
available, values of 3.0 - 4.0$\arcsec$ were empirically determined from the SHARC II centroiding fits of the Bolocam sources, which, when added in quadrature with the pointing error of
3.8$\arcsec$,
yields 2 $\sigma$ positional errors of 9.8 - 11$\arcsec$. These error circles were used to correlate the sources between the
different surveys to find coincident detections.
\section{Results}
\subsection{SHARC II 350 $\mu$m Detections}
Postage stamp images of each of the SHARC II fields are shown in Figure \ref{figure:postage}. Each image has been cropped to 60$\arcsec$ $\times$ 60$\arcsec$,
centered on the Bolocam source positions (dotted circle). The SHARC II source candidate list is presented in Table \ref{table:sharcdetections}, where the sources are
listed in order of Bolocam source number. Seven Bolocam galaxy candidates were detected by SHARC II at $> 3 \sigma$ (Bolocam 1, 2, 3, 6,
14, 16, 17). Two of these sources (Bolocam 3, 6) were found to have two SHARC II counterparts. Two Bolocam candidates (5 and 8) were observed by \cite{kovacs05}
and are also included in the list.
3 $\sigma$ upper limits are given for each of the Bolocam fields with no positive detections. Note that the flux density uncertainties in the last column of
Table \ref{table:sharcdetections} include uncertainties of $\sim$ 20\% due to calibration error (as determined by the dispersion of the calibration
source flux densities). The correlation between the two source lists based upon the positional error circles and detection offsets is
lower than expected, as only 2 of the 7 Bolocam sources with a single SHARC II counterpart have errors within 1 $\sigma$ (5 are expected from a normal
distribution). This may be due to underestimating the Bolocam pointing error (cf.\ \S\ \ref{section:positionalerrors} and Laurent et al.\ 2005).
An additional two sources were detected in the survey (in the fields of Bolocam 9 and 16); they are not associated with the Bolocam sources because their locations are well outside of the
Bolocam positional error circles (see \S\ \ref{section:positionalerrors}). While 0.8 false detections are expected from Gaussian statistics in the SHARC II survey (given the 60\% uniform
coverage cut and the 9\arcsec\ SHARC II beam), these serendipitous sources nevertheless may be real. SHARC II sources 3 and 4 were detected at high significance at 350 $\mu$m and lie in
regions of positive flux density in the Bolocam map, with S/N ratios of 2.9 and 2.1, respectively.
\subsection{SHARC II / Bolocam Correspondence}
\label{section:summary}
In addition to the Bolocam 1.1 mm and SHARC II 350 $\mu$m detections, existing multiwavelength coverage (submillimeter,
radio, infrared, optical, and X-ray) of the Lockman Hole was used to identify likely counterparts and characterize coincident sources.
A detailed description of each of these surveys is found in Appendix \ref{section:coverage}. A comprehensive summary of the counterparts to the Bolocam detections (including the coverage by each
survey) is listed in Table \ref{table:detections}. Given the large size of the Bolocam beam, identifying likely counterparts requires a certain amount of judgement. Detailed maps of the sources
can be found in Figure \ref{fig:circles}. Additional notes on individual objects are discussed below.
Bolocam.LE.1100.1 --
We conclude that Bolocam source 1 is likely to be a submillimeter galaxy given the coincident Bolocam, MAMBO, and SHARC II detections. In addition, highly
plausible radio, {\it Spitzer}, and faint optical counterparts exist.
Bolocam.LE.1100.2 -- The SHARC II detection falls 16$\arcsec$ to the ENE, but contains both radio sources in its error circle. The two 20 cm radio sources
\citep{yun05,biggs06} are unrelated due to their separation. We treat each of the radio sources separately when fitting photometric redshifts. The coincident northern radio source with
bright optical counterparts has a low photometric redshift (see next section) and is likely to be a low-redshift galaxy as the SDSS survey concludes. Bolocam and
SHARC II may also be detecting the \cite{biggs06} southern radio source, which has a very faint optical counterpart.
Bolocam.LE.1100.3 -- The position of this 6.0 mJy Bolocam source is at the edge of the good coverage region of the SCUBA survey. All of the optical
counterparts to the radio sources are relatively bright (22 - 24 magnitude), although curiously, the SDSS catalog classifies the optical counterpart to the northeast radio source as a star. (Given the
radio and 350 $\mu$m SHARC II counterparts, we conclude that the SDSS classification may be incorrect.) Given that both of the SHARC II sources have radio, {\it Spitzer}, and optical
counterparts, each (or both) are likely candidates as submillimeter galaxies and each could contribute to the flux density of the Bolocam source. Note that when estimating photometric redshifts
where source confusion may be present (for this Bolocam source and elsewhere), no attempt was made to partition the Bolocam flux density among multiple submillimeter sources (due to the large
uncertainties in position).
Bolocam.LE.1100.5 -- Four radio detections fall within the Bolocam positional error circle: one within the SCUBA and SHARC II error circles and just outside the
edge of the MAMBO error circle, one on the edge of the MAMBO error circle, and the other two near the edge of the Bolocam positional error circle.
The southwest radio source has 5-color SDSS photometry and is classified as a galaxy (extended). The fact that three of the radio sources lie outside both the SCUBA and SHARC II error circles
makes them unlikely to be the correct counterpart of the submillimeter detections. We therefore choose the northeast radio source to be the more likely counterpart
(which is confirmed by the fact that Chapman et al.\ 2005 were able to obtain a spectroscopic reshift for this submillimeter galaxy at
this radio position, as discussed in Appendix \ref{section:previousredshift}).
\noindent A MAMBO detection located just outside the Bolocam error circle was not detected by SHARC II or SCUBA (3 $\sigma$ upper limit of 35.4
mJy), but has a \cite{chapman05} spectroscopic redshift of 1.956. Due to the large size of the Bolocam beam, the Bolocam flux densities and positions may be influenced by source confusion.
Bolocam.LE.1100.6 -- Two SHARC II counterparts fall within the Bolocam positional error circle, each with radio counterparts (with the eastern source containing two radio counterparts).
The SDSS survey classifies both the radio source associated with the western SHARC II source and the eastern radio source associated with the eastern SHARC II source as galaxies.
Each of the three radio sources may be contributing to the Bolocam flux density due to source confusion. We treat each of the radio sources separately when fitting photometric redshifts.
Bolocam.LE.1100.8 -- Given the fact that the southern radio source (of the pair of two radio sources oriented N-S) has Bolocam, SCUBA, MAMBO, SHARC II, and {\it
Spitzer} detections, along with a spectroscopic redshift, we conclude that this Bolocam source is real. Nevertheless, the northern radio
source cannot be ruled out as a galaxy also contributing to the submillimeter fluxes.
Bolocam.LE.1100.14 -- This 4.4 mJy Bolocam detection is likely influenced by source confusion, given three closely spaced submillimeter sources (SCUBA sources
1, 4, and 8, with the latter two lying
near, but outside of the Bolocam positional error circle). The location of the southern radio source relative to the Bolocam position is greater than the 16$\arcsec$ radius used for the
{\it Spitzer}
counterparts catalog, and thus no {\it Spitzer} data is available. This set of coincident sources is the most likely counterpart to the Bolocam
source. Just outside of the Bolocam error circle lies another 850 $\mu$m SCUBA detection (LE850.4) to the northeast, with a coincident \cite{ivison02} 20 cm
radio source as well as published 3.6, 4.5, 5.8, and 8.0 $\mu$m {\it Spitzer} counterparts.
The SCUBA source that coincides with Bolocam source 14 (LH850.1) is also the brightest SCUBA source and has been the target of many published multi-wavelength studies. In addition to the
extensive radio, infrared, optical and X-ray surveys discussed in Appendix \ref{section:coverage}, a faint (K $\simeq$ 23.5) galaxy counterpart was positively identified \citep{lutz01} at
the radio position. The source was found to be extended (20-30 kpc), clumpy (on subarcsecond scales) and very red ({\it I - K} $>$ 6.2).
Bolocam.LE.1100.16 -- We conclude that this Bolocam source is in fact a submillimeter galaxy, given the large number of multiwavelength detections and a radio source with a confirmed
spectroscopic redshift.
Bolocam.LE.1100.17 -- This 4.0 mJy Bolocam detection is likely influenced by source confusion, given two nearby submillimeter sources. The 850 $\mu$m SCUBA,
1.1 mm MAMBO, and 850 $\mu$m SHARC II coincident detections to the northeast are the likely counterparts to Bolocam source 14 and are
discussed in detail in Bolocam.LE.1100.14. The SDSS catalog curiously classifies the northwest radio source as a star. We conclude that both the southeast and northwest radio sources associated
with the Bolocam source may be submillimeter galaxies, given the large number of multiwavelength detections. While a confirmed spectroscopic redshift exists near the southeast radio source,
self-consistent photometric redshifts and multiple optical counterparts at the radio position suggest that the spectroscopic redshift may not correspond to the radio / submillimeter sources (see
\S\ \ref{subsection:bolocam17}).
\subsection{SHARC II Non-Detections}
From extensive Monte-Carlo simulations of the Bolocam data set \citep{laurent05}, 6 false detections (Poisson distributed) are expected. This represents a large fraction (6/17) of
the overall source catalog and is a consequence of the relatively low 3 $\sigma$ detection threshold used in the source detection algorithm. Eight of the Bolocam sources (4, 7, 9, 10,
11, 12, 13, and 15) were found to show no secure counterparts at 350 $\mu$m, although two of the sources (Bolocam 9, 12) exhibit flux densities just below the 3 $\sigma$ detection
threshold (with a coincident radio detection for Bolocam source 9). Here we describe each of the SHARC II non-detections of the Bolocam sources in detail.
Bolocam.LE.1100.4 -- A single radio counterpart \citep{biggs06} lies near the edge of the MAMBO positional error circle, with an SDSS classification of the optical counterpart as a
galaxy. While well within the Bolocam positional error circle, the location of the radio source
relative to the Bolocam position is greater than the 16$\arcsec$ radius used for the {\it Spitzer} counterparts catalog, and thus no {\it Spitzer} data is available. Given the coincident
Bolocam and Mambo
detections, along with a lack of SHARC II and SCUBA detections, this source could possibly be a very high redshift galaxy ($z > 4$), such that the SED falls below the 3 $\sigma$
detection threshold of the SCUBA 850 $\mu$m survey.
Bolocam.LE.1100.7 -- The lack of multiwavelength observations makes it difficult to determine whether this Bolocam detection is real (and associated with the coincident radio detection).
The lack of {\it Spitzer} and SHARC II counterparts to the radio source, however, leads us to believe that the Bolocam source may be a spurious detection.
Bolocam.LE.1100.9 -- It is interesting to point out that the SHARC II upper limit in the Bolocam error circle is just below the
3 $\sigma$ detection flux density threshold and coincides with the radio position. The lack of multiwavelength observations makes it difficult to determine whether this Bolocam detection is real.
The presence of {\it Spitzer} counterparts and a possible dim SHARC II detection, however, leads us to believe that the Bolocam source may be real.
Bolocam.LE.1100.10 -- Given the lack of counterparts, we conclude that there is little evidence to suggest that this detection represents a submillimeter galaxy and is likely a
spurious detection.
Bolocam.LE.1100.11 -- The lack of more multiwavelength data makes it difficult to determine whether this Bolocam detection is real.
While lacking a SHARC II detection, the radio source with an SDSS classification as a galaxy (extended object) leads us to believe that the Bolocam source
may be real.
Bolocam.LE.1100.12 -- A portion of the Bolocam error circle lies outside the deep \cite{ivison05} optical Subaru R-band field. Similar to Bolocam source 9, we point out that the
SHARC II upper limit in the Bolocam error circle is just below the 3 $\sigma$ detection flux density threshold. The lack of more multiwavelength data makes it
difficult to confirm whether this Bolocam detection is real.
Bolocam.LE.1100.13 -- This Bolocam source lacks 850 $\mu$m SCUBA detections (although a SCUBA source is located just
outside of the Bolocam positional error circle). We suggest that there is little evidence that the Bolocam detection represents a submillimeter galaxy and is likely a spurious
detection.
Bolocam.LE.1100.15 --
The lack of MAMBO and SHARC II counterparts makes it difficult to confirm the Bolocam detection. Nevertheless, the SDSS classification of the position coincident with the radio source
as a galaxy (extended) leads us to believe that the Bolocam source may be real.
\subsection{Submillimeter Spectral Energy Distributions}
\label{subsection:correlations_between_spectra}
The submillimeter spectral energy distributions (SEDs) of the coincident SHARC / Bolocam detections is shown in Figure \ref{figure:sed_all}. Five of the 17 Bolocam galaxy candidates
(5, 8, 14, 16, 17) have spectroscopic redshifts from Chapman et al. (2005, see Appendix \ref{section:previousredshift}). In order to properly compare the SEDs, it is necessary to shift each of the SEDs to a
common redshift. Thus, each observed SED was brought to a redshift of 2.0 (the mean redshift of the five Bolocam galaxies) using the spectroscopic redshifts. The composite SED of these five Bolocam
galaxies can be seen in Figure \ref{figure:revertzoom}.
In addition to redshifting the SEDs to align their rest wavelengths, a cosmological dimming term was applied by assuming a flat
($\Omega_k=0$), $\Omega_\Lambda$ = 0.7 cosmology.
Finally, to account for variations in their intrinsic brightnesses, we normalize the flux densities of these five Bolocam galaxies by tying
together their SEDs at the observed Bolocam wavelength of 1.1 mm. To account for the spread of the redshifted wavelengths of the 1.1 mm
Bolocam observations, the flux densities were normalized to the \cite{laurent05} model based on the observations cited in the \cite{blain02}
paper. The model assumes a single dust temperature of 40 K ($\beta=1.6$) and is overplotted as a solid line in
Figure \ref{figure:revertzoom}. Note that only the Bolocam observations are constrained to pass through this model.
Upon inspection, we find that at least four of the five Bolocam galaxies with spectroscopic redshifts (5, 8, 14, 16) exhibit very similar
SEDs in the submillimeter and millimeter wavelengths. They are modeled adequately by the 40 K composite SED based on nearby dusty IRAS
galaxies, high redshift submillimeter galaxies, gravitationally lensed high-redshift galaxies, and high redshift AGN. The Bolocam
galaxy 17, however, appears to peak at a much higher wavelength and lower flux density than the others. We believe that there is enough
source confusion to question whether the \cite{chapman05} redshift for this galaxy ($z$ = 0.689) corresponds to the SED shown (see \S\ \ref{subsection:bolocam17}). If the
spectroscopic redshift is valid, the SED is modeled much better by a T = 20 K ($\beta=1.0$) grey-body dust spectrum.
\section{Redshifts}
\label{section:photometric_redshifts}
\subsection{Introduction}
With the multiwavelength photometry of the Bolocam sources, we fit photometric redshifts using various models based on
different portions of the SED. Photometric redshifts based on the far-IR-to-radio correlation were derived using the models
of \cite{carilli99} and \cite{yun02}. The shape of the submillimeter and millimeter part of the spectrum was also fit
without the radio points, assuming a blackbody emission spectrum modified by a dust emissivity term \citep{wiklind03,laurent05}. A brief description of each of the models, fitting
methods, and the redshift results are discussed in the next section.
\subsection{Redshift Techniques}
\label{section:redshift_techniques}
This section attempts to briefly describe each of the five photometric redshift techniques used in this paper and the results of the fits when applied to the Bolocam galaxy candidates. The
following section will compare the relative merits of each of the photometric redshift fitting techniques and discuss the results of redshift distributions.
1) FIR-Radio Spectral Index -- \cite{carilli99} used the semianalytic, linear relationships
derived by Condon (1992) between the massive star formation rate and the radio synchrotron luminosity and far-IR dust emission from active star-forming galaxies to
show that the spectral index between these two frequencies, $\alpha^{350}_{1.4}$, is a well behaved function of redshift:
\begin{eqnarray}
\alpha^{350}_{1.4} = -0.24 - [0.42 \times (\alpha_{\mathrm{radio}} - \alpha_{\mathrm{submm}}) \times \mathrm{log} (1+z)],
\end{eqnarray}
where we adopt the standard value in Condon (1992) of -0.8 for $\alpha_{\mathrm{radio}}$, and a value of +3.2 for $\alpha_{\mathrm{submm}}$ (an average of the spectral indices between 270
and 850 GHz for
M82 and Arp 220. The relation is believed to be a result of relativistic electrons accelerated in supernova remnants (producing synchrotron radiation) and dust heated by the interstellar
radiation field (with a thermal peak of $\sim 380 \mu$m for a galaxy with $z = 2$ and T = 40 K). Photometric redshifts determined using only the Bolocam and radio flux densities are listed in Table
\ref{table:redshifts_yunandcarilli}. Redshift results from Bolocam sources with multiple radio counterparts are listed using the higher S/N detection
in the case of coincident detections by independent surveys or are listed together in the case of multiple counterparts detected by a single group.
The error bars listed in Table \ref{table:redshifts_yunandcarilli} (and elsewhere throughout this paper) were obtained from Monte-Carlo simulations of the
fits and represent statistical errors due to measurement uncertainty in the flux densities. The flux densities at each observed wavelength were varied
about their mean value assuming a Gaussian distribution of flux errors. Each Monte-Carlo SED was
then fit to the photometric redshift models with a standard, least-squares minimization fitting routine. Each simulation was repeated 1000 times, with the
error bars quoted being the minimum-length 1 $\sigma$ confidence intervals from the resulting histogram of redshifts. It should be noted that these
confidence intervals represent only the statistical goodness of fit and that uncertainties in the templates themselves are expected to dominate
the photometric redshift errors.
2) Entire FIR-Radio SED -- \cite{yun02} utilized the entire Far-IR to radio spectral energy distribution to estimate photometric redshifts and SFRs. The
redshift template is based upon the theoretical models of thermal dust emission, thermal bremsstrahlung (free-free) emission, and
nonthermal synchrotron emission for dusty starburst galaxies. Photometric redshift fits of the five Bolocam galaxy candidates (5, 8, 14,
16, 17) with spectroscopic redshifts \citep{chapman05} are shown in Figure \ref{figure:photoz},
with best-fit redshifts (and errors) also listed in Table \ref{table:redshifts_yunandcarilli}. The solid lines in the figure represent
the best fit spectrum to the submillimeter, millimeter, and radio point shown. The dotted line
represents a second fit using the \cite{yun02} model, this time fixing the spectroscopic redshift and normalizing (varying only the SFR) to the submillimeter
points.
\begin{deluxetable}{ccccccccccc}
\tabletypesize{\tiny}
\tablecaption{Photometric Redshifts}
\tablewidth{0pt}
\tablehead{
\colhead{Bolocam} & \colhead{N$_{\mathrm{Radio}}$} & \colhead{N$_{\mathrm{Submm}}$} & \colhead{N$_{\mathrm{{\it Spitzer}}}$} &
\colhead{Carilli \& Yun} & \colhead{Yun \& Carilli} & \colhead{Wiklind} & \colhead{Laurent} & \colhead{MRK231} & \colhead{ARP220} & \colhead{Chapman}\\
\colhead{1.1 mm} & \colhead{} & \colhead{} & \colhead{} &
\colhead{1999} & \colhead{2002} & \colhead{2003} & \colhead{et al.\ 2005} & \colhead{} & \colhead{} & \colhead{et al.\ 2005}\\
\colhead{Number} & \colhead{} & \colhead{} & \colhead{} &
\colhead{z$_{\mathrm{phot}}$} & \colhead{z$_{\mathrm{phot}}$} & \colhead{z$_{\mathrm{phot}}$} & \colhead{z$_{\mathrm{phot}}$} & \colhead{z$_{\mathrm{phot}}$} & \colhead{z$_{\mathrm{phot}}$} & \colhead{z$_{\mathrm{spec}}$}
}
\startdata
1 &2(S) &3 &3 &4.6$^{+0.3}_{-0.3}$ &4.1$^{+0.3}_{-0.3}$ &3.2$^{+0.4}_{-0.5}$ &3.2$^{+0.3}_{-0.4}$ &15$^{+1.4}_{-1.5}$ &3.0$^{+0.2}_{-0.3}$ &\\
" &1(N) &" &3 &4.6$^{+0.3}_{-0.3}$ &4.1$^{+0.3}_{-0.3}$ &" &" &1.8$^{+1.0}_{-0.8}$ &2.0$^{+0.7}_{-0.7}$ &\\
2 &2,1(N,S)&2 &0 &2.2$^{+0.2}_{-0.2}$, 3.9$^{+0.3}_{-0.3}$ &0.6$^{+0.1}_{-0.1}$, 2.9$^{+0.7}_{-0.8}$ &4.2$^{+0.6}_{-0.8}$ &3.9$^{+0.4}_{-0.4}$ & & &\\
3 &2(NE) &2 &5 &2.6$^{+0.2}_{-0.2}$ &0.7$^{+0.1}_{-0.1}$ &4.7$^{+0.8}_{-0.7}$ &4.2$^{+0.4}_{-0.5}$ &0.2$^{+0.2}_{-0.2}$ &3.1$^{+0.2}_{-0.2}$ &\\
" &2(SW) &2 &3 &3.7$^{+0.3}_{-0.3}$ &1.6$^{+0.3}_{-0.4}$ &4.9$^{+0.8}_{-0.7}$ &4.3$^{+0.5}_{-0.4}$ &10$^{+0.9}_{-0.8}$ &2.3$^{+0.7}_{-0.2}$ &\\
4 &1 &2 &0 &5.7$^{+0.7}_{-0.8}$ &5.1$^{+0.7}_{-0.9}$ & & & & &\\
5 &3(E) &4 &5 &4.3$^{+0.4}_{-0.4}$ &3.4$^{+0.2}_{-0.3}$ &2.5$^{+0.5}_{-0.4}$ &2.6$^{+0.4}_{-0.5}$ &1.2$^{+0.2}_{-0.2}$ &2.6$^{+0.2}_{-0.2}$ &2.611\\
6 &2,2(E,W)&2(E) &0 &4.2$^{+0.5}_{-0.4}$, 3.1$^{+0.3}_{-0.3}$ &3.5$^{+0.4}_{-0.5}$, 2.0$^{+0.3}_{-0.3}$ &3.2$^{+0.7}_{-0.7}$ &3.2$^{+0.5}_{-0.6}$ & & &\\
" &2 &2(W) &0 &1.6$^{+0.2}_{-0.2}$ &0.8$^{+0.1}_{-0.1}$ &1.9$^{+0.4}_{-0.5}$ &2.0$^{+0.5}_{-0.6}$ & & &\\
7 &1 &1 &0 &4.2$^{+0.6}_{-0.8}$ &3.6$^{+0.6}_{-1.0}$ & & & & &\\
8 &2(N) &4 &4 &5.1$^{+0.5}_{-0.6}$ &4.7$^{+0.3}_{-0.3}$ &3.1$^{+0.4}_{-0.6}$ &3.1$^{+0.3}_{-0.5}$ &0.6$^{+0.3}_{-0.3}$ &2.4$^{+0.6}_{-0.2}$ &\\
" &1(S) &" &4 &5.1$^{+0.6}_{-0.6}$ &4.7$^{+0.4}_{-0.4}$ &" &" &0.7$^{+0.4}_{-0.1}$ &3.1$^{+0.2}_{-0.6}$ &3.036\\
9 &2 &1 &5 &3.9$^{+0.4}_{-0.4}$ &3.2$^{+0.5}_{-0.5}$ & & &1.3$^{+0.2}_{-0.2}$ &1.8$^{+0.2}_{-0.5}$ &\\
10 &0 &1 &0 & & & & & & &\\
11 &1 &1 &2 &3.2$^{+0.4}_{-0.5}$ &2.5$^{+0.4}_{-0.6}$ & & &0.4$^{+0.2}_{-0.2}$ &0.8$^{+0.3}_{-0.2}$ &\\
12 &0 &1 &0 & & & & & & &\\
13 &2,1(NE,SE)&1 &0 &3.9$^{+0.4}_{-0.4}$, 4.8$^{+0.6}_{-0.6}$ &3.2$^{+0.5}_{-0.4}$, 4.4$^{+0.8}_{-0.8}$ & & & & &\\
" &1(E) &" &4 &5.0$^{+0.6}_{-0.7}$ &4.6$^{+0.8}_{-1.0}$ & & &0.7$^{+0.3}_{-0.2}$ &2.4$^{+0.3}_{-0.5}$ &\\
14 &4 &5 &5 &3.7$^{+0.4}_{-0.3}$ &3.1$^{+0.2}_{-0.2}$ &3.2$^{+0.3}_{-0.4}$ &3.2$^{+0.3}_{-0.3}$ &7.3$^{+0.6}_{-0.2}$ &3.0$^{+0.2}_{-0.3}$ &2.148\\
15 &1 &1 &5 &4.1$^{+0.5}_{-0.4}$ &3.5$^{+0.6}_{-0.5}$ & & &0.6$^{+0.2}_{-0.2}$ &1.0$^{+0.2}_{-0.2}$ &\\
16 &2 &4 &5 &4.1$^{+0.5}_{-0.5}$ &3.2$^{+0.2}_{-0.2}$ &1.9$^{+0.3}_{-0.5}$ &2.0$^{+0.4}_{-0.4}$ &0.4$^{+0.2}_{-0.2}$ &2.3$^{+0.3}_{-0.2}$ &2.142\\
17 &3(SE) &4 &10 &3.8$^{+0.5}_{-0.4}$ &2.8$^{+0.3}_{-0.3}$ &3.4$^{+0.5}_{-0.7}$ &3.3$^{+0.4}_{-0.4}$ &3.4$^{+0.3}_{-0.3}$ &3.1$^{+0.2}_{-0.2}$ &0.689\\
" &1(NW) &" &5 &5.3$^{+0.7}_{-0.8}$ &4.1$^{+0.4}_{-0.5}$ &" &" &0.3$^{+0.2}_{-0.2}$ &3.0$^{+0.2}_{-0.6}$ &\\
\enddata
\tablecomments{Best fit photometric redshifts of the Bolocam galaxy candidates using the models of \cite{carilli99}, \cite{yun02}, \cite{wiklind03}, and \cite{laurent05}, and
cool and warm ULIRGs Arp 220 and MRK 231.
N$_{\mathrm{Radio}}$, N$_{\mathrm{Submm}}$, and N$_{\mathrm{{\it Spitzer}}}$ are the number of coincident radio, submillimeter and {\it Spitzer} infrared points, respectively.}
\label{table:redshifts_yunandcarilli}
\end{deluxetable}
3) Modified Blackbody -- \cite{wiklind03} found that observations of local ULIRGs exhibit a remarkably low dispersion in the far-IR to millimeter wavelengths
($\lambda > 50 \mu$m), independent of whether the power source of the thermal emission is due to AGN or intense star formation.
\cite{wiklind03} fit a simple blackbody emission spectrum (modified by a dust emissivity term) to sample of 37 local ULIRGs from \cite{klaas01}:
\begin{eqnarray}
\label{equation:greybody}
f_\nu \propto \epsilon_\nu B_\nu(T) \propto [1-\exp{(-\tau_\nu)}] B_\nu(T),
\end{eqnarray}
where $B_\nu(T)$ is the Planck function evaluated at dust temperature, $T$, and frequency, $\nu$, and $\tau_\nu$ is the optical depth
of the dust:
\begin{eqnarray}
\nonumber
\tau_\nu = \left(\frac{\nu}{\nu_0}\right)^\beta.
\end{eqnarray}
\cite{wiklind03} made no assumption about the Wien side of the spectrum, as only the submillimeter ($\ge$ 450 $\mu$m) and millimeter points were fit. Using the best-fit parameters from
\cite{wiklind03}: $\beta = 1.8$, $\nu_0 = 1.2 \times 10^{12}$ Hz (250 $\mu$m), and $T_d =
68$ K, we fit photometric redshifts to the SHARC II 350 $\mu$m, SCUBA 450 and 850 $\mu$m, Bolocam 1.1 mm, and MAMBO 1.2 mm flux densities
of the galaxies detected in our Bolocam survey. The two parameter fit (redshift and overall flux density normalization) yields redshifts for 9 of the 17 bolocam
galaxies with $\ge$ 2 submillimeter/millimeter points. Seven of the Bolocam galaxies (Bolocam 7, 9, 10, 11, 12, 13, 15) have no
counterpart in the submillimeter/millimeter and one (Bolocam 4) has detections only at 1.1 and 1.2 mm, which is an insufficient
wavelength spread in order to properly constrain the galaxy redshift using this two parameter model.
Redshift results for each of the 9 Bolocam galaxies are listed in Table \ref{table:redshifts_yunandcarilli}. The best fit models to
the five galaxies with \cite{chapman05} redshifts (Bolocam 5, 8, 14, 16, 17) are shown in Figure \ref{figure:wiklind}.
Similar to the method of \cite{wiklind03}, \cite{laurent05} created a composite SED of nearby dusty {\it IRAS} galaxies,
high-redshift submillimeter galaxies, gravitationally lensed high-redshift
galaxies, and high-redshift AGNs \citep[][and references therein]{blain02}, and found fit parameters of $T$ = 40 K, $\nu_0$ = 3700
GHz, and $\beta$ = 1.6 for Equation \ref{equation:greybody}. Redshift results for each of the 9 Bolocam galaxies are listed in Table
\ref{table:redshifts_yunandcarilli}. The best fit models to the five galaxies with \cite{chapman05} redshifts (Bolocam 5, 8, 14,
16, 17) are shown in Figure \ref{figure:wiklind}.
4) Near IR Stellar Bump -- \cite{egami04} used the multiband imaging capabilities of the IRAC and MIPS IR cameras of the {\it Spitzer Space Telescope} to
observe 38 VLA radio sources in the Lockman Hole. They classified the resulting IR SEDs into two types: those showing a clear near-IR
stellar continuum hump at a rest wavelength of 1.6 $\mu$m (due to the minimum opacity of the H$^-$ ion at 1.6 $\mu$m from photo-detachment and free-free transitions, which results in
a local maximum in the the SEDs of cool stars, Sawicki 2002), and those with a featureless power-law continuum (from AGN). We fit the {\it
Spitzer} IR counterparts of the Bolocam galaxies with each of these spectra, using the \cite{egami04} models of a cool ULIRG
Arp 220 \citep[from][]{silva98} and a warm (dominated by an AGN) ULIRG Mrk 231. Only two fit parameters were used: the redshift, and
an overall normalization. Photometric redshifts were fit for each of the 12 Bolocam galaxies (Bolocam 1, 3, 5, 8, 9, 10,
11, 13, 14, 15, 16, 17) with $\ge$ 2 IR {\it Spitzer} points. Three Bolocam galaxies (Bolocam 2, 6, 12) were outside of the field
surveyed by \cite{egami04}. Two galaxies (Bolocam 4, 7) have {\it Spitzer} counterparts, but due to the high density of {\it
Spitzer} sources in the field, they could not be uniquely associated with the Bolocam sources (because of a lack of another coincident
detection in the submillimeter and/or radio). The best-fit photometric redshifts for both Arp 220 and Mrk 231 (fitting only the {\it Spitzer} near- and mid-infrared points) are shown in
Figure \ref{figure:spitzer}, with the resulting redshifts also listed in Table \ref{table:redshifts_yunandcarilli}.
\subsection{Comparison of Photometric Redshift Techniques}
\label{subsection:discussion}
Comparing the results of each of the photometric redshift techniques with the spectroscopic redshifts of \cite{chapman05}
yields widely varying degrees of agreement. Histograms of redshift errors for each of the photometric redshift models \citep[other than][in which coincident radio detections were
treated separately]{carilli99} are shown in Figure \ref{figure:histogram}. The histogram from fitting models of Arp 220 and Mrk 231 to the {\it Spitzer} IRAC and MIPS
observations are the fits that result in the lowest residual $\chi^2$ (Arp 220 for Bolocam 5, 8, 14, and 16, and Mrk 231 for Bolocam 17).
The \cite{yun02} model \citep[as well as][]{carilli99} yields systematically high photometric redshifts compared to the spectroscopic redshifts by \cite{chapman05}.
The comparison of model SEDs for photometric (solid line) and spectroscopic (dotted line) redshifts in Figure \ref{figure:photoz} suggests that SED
data points at the extreme ranges of wavelength coverage strongly influence the model fit and that the systematic tendency to derive a high redshift is primarily
driven by the lower than expected radio continuum flux density. This is supported by the fact that fits to only the submillimeter and millimeter-wave points yield much more accurate photometric
redshifts (see below). This is perhaps not surprising, given recent evidence \citep{chapman05} suggesting a large degree of dispersion in the radio-to-far-IR correlation at higher redshift.
Nevertheless, it is unlikely that the \cite{yun02} model template, which is derived from the ensemble average of 23 infrared luminous galaxies in the local universe, is systematically biased by radio
bright objects because $\ge$98\% of all FIR-selected galaxies follow the well known and tight radio-FIR correlation, independent of FIR luminosity \citep{yun01}. Aside from Bolocam source 17 whose
spectroscopic redshift by \cite{chapman05} appears suspect (see \S\ \ref{subsection:bolocam17}), these comparisons suggest that the observed radio continuum in Bolocam galaxies is 2-5 times
fainter than predicted by the synchrotron flux densities (which dominate thermal brehmsstrahlung by a factor of $\sim$ 13 at 20 cm) from the low-redshift ULIRGs
for which the local FIR-radio correlation was derived (see further discussions in \ref{subsection:radio}).
In contrast to the far-IR-to-radio correlation photometric redshift techniques, both the \cite{wiklind03} and \cite{laurent05}
modified blackbody curves correctly estimate the redshifts for three of the five Bolocam galaxies with spectroscopic redshifts (within the uncertainties of the photometric redshift techniques). The
strength of the submillimeter / millimeter-only photometric redshift technique is twofold. First, while abandoning the radio
points limits the number of points (as few as two, in some cases) to which we may fit a model, we ensure that the physics
that dominates the region of the spectrum to which we are fitting is directly relevant to star formation-heated dust emission.
Synchrotron radiation, by contrast, is dependent upon high energy electrons streaming through interstellar magnetic fields, whose
properties may vary as a function of environment (e.g.\ inverse-Compton losses for cosmic rays with higher CMB energy densities at high redshift -- see \S\ \ref{subsection:radio}). Indeed, these
galaxies are typically at least an order of magnitude more luminous than the low-redshift, infrared-luminous
galaxies from which the FIR-radio correlation was derived.
Second, having the Bolocam
1.1 mm flux densities on the Rayleigh-Jeans side of the spectrum and the SHARC II 350 $\mu$m flux densities near the peak of the SED makes
the 350 $\mu$m/1.1 mm flux density ratio a strong function of redshift. This can be seen in Figure \ref{figure:fluxratio}, which
shows the flux density ratios between various wavebands based on the \cite{laurent05} model SED. The importance of the
SHARC II 350 $\mu$m waveband is apparent. For intermediate to high ($z < 5$) redshifts, the SHARC II flux density drops rapidly as
a power law ($\sim \nu^{-1.7}$ due to the hotter components of dust) with redshift on the Wien side of the spectrum, while the millimeter-wave climbs up the steep
Rayleigh-Jeans portion of the SED. Photometric redshifts using the 450 or 850 $\mu$m wavebands are less sensitive than the
350 $\mu$m / 1.1 mm wavebands. Extending this analysis shows the discriminatory power of the BLAST and {\it
Herschel} space telescope 250 $\mu$m bands in conjunction with a millimeter waveband, although the far-IR waveband begins to probe a range of hotter dust temperatures. (The flux density ratio of
these wavebands may not be well correlated, as discussed in \S\ \ref{subsection:irspectrum}.) BLAST \citep[Balloon-borne Large-Aperture Submillimeter Telescope,][]{devlin01} is a
balloon-based instrument which incorporates a 2-meter primary mirror and is equipped with large-format bolometer cameras operating at 250, 350, and 500 $\mu$m which, when complete, will
provide the first sensitive large-area (0.5-40 deg$^2$) submillimeter surveys at these wavelengths. The bolometer arrays are prototypes of the Spectral and Photometric Imaging Receiver
(SPIRE) focal plane cameras for the {\it Herschel} satellite \citep{griffin01}, which will further investigate the formation and evolution of AGNs and star formation in
high redshift submillimeter galaxies.
It is important to note that while both the \cite{wiklind03} and \cite{laurent05} modified blackbody curves both produce reasonably accurate photometric redshifts (and produce nearly identical SEDs on
the Rayleigh-Jeans portion of the spectrum), they make substantially different assumptions about the dust properties: $T_{\mathrm{dust}}$ and $\beta$ are 68 K and 1.8 for the \cite{wiklind03}
model SED and 40 K and 1.6 for the \cite{laurent05} model SED. This points to the degeneracy of the dust temperature and the grain emissivity index. While the shapes of the submillimeter SEDs are
reasonably modeled by either dust model and thus predict photometric redshifts with some accuracy, essentially no information about the dust temperatures can be inferred. In fact,
representing the dust SED with two (or more) components produces similar $\chi^2$ values (and thus similar redshifts), with both temperatures {\it lower} than that of the single dust
temperature model \citep{wiklind03}.
The photometric redshifts determined by fitting the {\it Spitzer} infrared observations yield redshifts that are equivalent to both the \cite{laurent05}
and \cite{wiklind03} model SEDs. This confirms the conclusions of \cite{egami04}, in which starburst-dominated galaxies ("cold") show remarkably similar SEDs
in the infrared. The resulting photometric redshifts are highly sensitive to the 1.6 $\mu$m continuum hump (and PAH features), with a sharp minimum in $\chi^2$.
(The ARP 220 fits may be biased towards particularly good fits as 4 of the 5 galaxies with spectroscopic redshifts lie between 2 $\lesssim z \lesssim$ 3, which is optimal
for the 8 $\mu$m PAH feature to be shifted into the observed 20 $\mu$m IRAC waveband.) AGN-dominated ("warm") galaxies also show very similar SEDs, but lacking a strong
continuum feature in the infrared, are subject to larger redshift fitting uncertainties; a brighter, higher redshift galaxy is
characterized by a similar shape in the infrared portion of the SED as a cooler, low-redshift galaxy, with the Wien side of the spectrum well-modeled with a power-law \citep{blain99}.
\subsection{Bolocam Source 17: Spectroscopic Misidentification?}
\label{subsection:bolocam17}
We bring special attention to photometric redshift analysis of Bolocam source 17 (corresponding to SCUBA source 8), as the \cite{laurent05}, \cite{wiklind03}, and {\it Spitzer} IR
models are consistent in overpredicting the redshift of this galaxy ($z_{\mathrm{spec}}$ = 0.689) by $\ge 4 \sigma$ ($z_{\mathrm{phot}}$ = 3.3$^{0.4}_{0.5}$, 3.4$^{0.6}_{0.6}$, and
3.4$^{0.3}_{0.3}$, respectively). We point out that the large offset between the spectroscopic and photometric redshifts is possibly the result of source confusion, as
two radio sources (both with {\it Spitzer} and optical counterparts) fall near the center of the Bolocam, SCUBA, MAMBO, and SHARC II error circles, within 4$\arcsec$ of each other. (Egami
et al.\ 2004
refer to the northwest and southeast radio sources as LE850.8a and LE850.8b, respectively.) The northwest
radio source is believed by \cite{lehmann01} to be the counterpart to the ROSAT X-ray emission, who find a redshift of 0.974 using optical Keck spectroscopy. Using XMM-Newton
observations, however, \cite{ivison02} conclude that the X-ray source corresponds to the southeast radio source. Indeed, the linear fit of a combination of ARP 220 and MRK 231 ULIRG models to the
{\it Spitzer} points coincident with the southeast radio source yields a 100\% warm (AGN dominated) component. It is near this radio position that \cite{chapman05} find a
spectroscopic redshift of 0.689. In fact, the X-ray emission observed with both instruments appears to fall between these two radio sources. Optical R-band images from both \cite{yun05}, and
\cite{ivison05} show multiple optical counterparts at the southeast radio source position. Furthermore, the spectroscopic position quoted by \cite{chapman05}
appears to coincide with an optical source $\sim$ 2$\arcsec$ to the south of the southeast radio source, a source detected with four-color SDSS photometry (in addition to Yun et
al.\ 2005 and Ivison et al.\ 2005 R-band photometry) and cataloged as a low-redshift galaxy. We conclude that it is possible that the submillimeter detections may either be suffering from
source confusion from two or more galaxies, or that the \cite{chapman05} redshift corresponds to a source other than that of the southeast radio detection. If the latter is true,
then the consistent redshifts predicted by the Laurent et al./Wiklind (2005, 2003) and {\it Spitzer} IR models may further point to the accuracy of these photometric
redshift techniques.
\section{Discussion}
\subsection{IR Spectrum}
\label{subsection:irspectrum}
SEDs over the entire IR-radio spectral range of the five Bolocam galaxies with spectroscopic redshifts are shown in Figure
\ref{figure:revert}. These spectra have the same redshift, cosmological dimming, and normalization corrections as in Figure
\ref{figure:revertzoom}. While four of the five galaxies have closely correlated spectra in the submillimeter region of the spectrum,
the infrared spectra ({\it Spitzer} 3.6, 4.5, 5.8, and 8.0 $\mu$m IRAC and 24 $\mu$m MIPS observations) exhibit a large degree of
dispersion. This dispersion may be the
result of several things: 1) Because the spectra have been normalized to a T = 40 K ($\beta=1.6$) spectrum at their 1.1 mm Bolocam flux densities to
account for intrinsic brightness variation between the galaxies, the normalization will result in an artificial reduction in the
submillimeter flux density dispersions. This effect is not likely to dominate, as the flux density normalization has a $\sim$ 10\% effect on the flux densities
of the galaxies. 2) The {\it Spitzer} detection associated with Bolocam source 14 lies systematically low compared to the other three
galaxies well-modeled by a T=40 K dust spectrum. While these {\it Spitzer} observations \citep{egami04} are from a different data
set than the remaining {\it Spitzer} observations (this work), it is unlikely that their flux densities are
systematically uncertain by nearly an order of magnitude. 3) The 40 K dust temperature model of the submillimeter portion of the
spectrum for Bolocam sources 5, 8, 14, and 16 assume a single dust temperature for each of the four sources. The temperature fit
parameter is somewhat degenerate with other fit parameters, including the critical frequency, $\nu_0$, where the optical depth of the
dust is unity. Thus, if Bolocam source 14 has a lower characteristic dust temperature, then the infrared portion of the 40 K model will
significantly overestimate the infrared flux density. \cite{chapman05} estimates the temperature of Bolocam source 14 to be 33 K from two
photometric points (850 $\mu$m SCUBA and 1.4 GHz VLA radio observations) and the dust SED templates of \cite{dale02}. This
temperature uncertainty likely dominates our uncertainty in matching the infrared flux densities. 4) In addition to heating by the
ultraviolet and optical flux density from young stars associated with ongoing star formation, the thermal dust emission responsible
for the bright submillimeter flux densities may be contributed to by an energetic AGN. While not dominating the total bolometric output from
the galaxy, they may have a non-negligible (20\%) contribution \citep{alexander05}. If this is the case, then the shape of the
infrared continuum may be vary according to the relative contribution of star formation rates for these galaxies. 5) Another possible
explanation for the larger dispersion in {\it Spitzer} infrared flux density as compared to our single dust temperature model may be from the fact
that the infrared flux densities trace separate epochs of star formation within the galaxy. It is plausible that the dust heated
by the ultraviolet and optical flux density from from current star formation is not well correlated to the current infrared flux density of older stars
(from previous star formation). Furthermore, models of UV to millimeter emission of star clusters embedded in optically thick giant molecular clouds (GMCs) suggest that the near-infrared to
far-infrared portion of starburst galaxy SEDs vary considerably with age of the starburst \citep{efstathiou00}.
\subsection{Radio Spectrum / FIR-Radio Correlation}
\label{subsection:radio}
The composite SED (including the radio points) of the 5 Bolocam sources with spectroscopic redshifts
is shown in Figure~\ref{figure:revert}. Two interesting observations
can be made about the radio continuum emission associated with these
galaxies: (1) the radio continuum is lower (or the submillimeter continuum is higher) on average than the well
established radio-FIR correlation for the local universe by Yun, Reddy, \& Condon (2001, cf.\ \S\ \ref{subsection:discussion});
and (2) like the infrared flux densities, the 6 and 20 cm VLA radio flux densities
for the four galaxies with closely correlated spectra in the submillimeter region of the
spectrum show a large degree of dispersion (factor of 5 in 20 cm flux density). This scatter is much larger than the
quoted uncertainties of the VLA radio observations (which constrain the 20 cm radio flux density of each galaxy to better than 20\%) This degree of dispersion
is also larger than the factor of 3 scatter in the radio-FIR correlation seen among the FIR selected galaxies in
the local universe. In fact, while the spectroscopic redshifts are similar for the four galaxies (2.1 $< z <$ 3.0),
varying the 20 cm flux density over the observed range causes the best fit photometric redshifts of \cite{yun02}
to vary from $z = 2.7$ to 4.8. This dispersion undoubtably contributes to the large errors of the photometric redshifts
discussed in \S\ \ref{section:photometric_redshifts}.
Deep radio continuum imaging using the VLA is a
technically challenging task, and the disparate 20 cm flux densities
by $\sim$ 50\% reported for Bolocam sources 2 and 6 (cf.\ \S\ \ref{section:summary})
exemplifies the difficulty of the photometry at radio wavelengths.
Most systematic noises in interferometry tend to suppress the brightness
of astronomical sources, and part of the lower radio continuum flux
density might be related to the imaging and photometry problems.
It is important to note that due to the low S/N ratios at which the sources have been detected
in the submillimeter wavebands ($\le 4 \sigma$ for the vast majority of SHARC, SCUBA, Bolocam, and MAMBO detections),
flux bias \citep{laurent05} plays a major role in overestimating the flux densities at these wavelengths.
While this factor indeed results in a systematic shift of the entire submillimeter portion of the spectrum to higher flux densities,
it is unlikely that the magnitude of this effect ($\sim$ 20\% for the Bolocam flux densities) could fully account for the
lower than expected (from the FIR-radio correlation) radio continuum flux densities. Calibration errors may also contribute to a systematic overestimate of the submillimeter flux densities.
Finally, as two or more independent radio sources are found within the Bolocam error circle in 8 out of 17 cases in Figure 2, source confusion
or source blending may also contribute to the apparently lower radio continuum flux if only one radio source is identified as the counterpart.
The lower radio continuum flux density and the larger scatter may also
reflect an actual breakdown in the radio-FIR correlation.
Inverse-Compton losses for the high energy cosmic rays responsible
for the synchrotron radiation is thought to be significant at $z > 2$,
and a possible breakdown in the radio-FIR correlation has been
considered previously (see Condon 1992, Carilli \& Yun 2000).
This effect is demonstrated in Figure \ref{figure:spitzer}, in which models
of local ULIRGs Arp 220 and Mrk 231 systematically overestimate the radio flux
densities of these submillimeter galaxies. Higher quality data on a larger sample of high redshift systems are needed to
examine the importance of inverse-Compton loss and the possible breakdown in the radio-FIR correlation.
X-ray heating of the circum-nuclear gas and dust is an important source of
luminosity in the far-IR if a luminous AGN is present (Maloney, Hollenbach, \& Tielens 1996).
Radio-quiet AGN FIR emission could reduce the 1.4 GHz flux density with
respect to the FIR heating. Alexander et al.\ (2005, and references
therein) make the case using X-ray detections and spectral indices that many, perhaps most,
submillimeter galaxies have AGN but that they are not bolometrically important. However,
the statistics of X-ray detected submillimeter galaxies for which hard/soft ratios can be measured is not large and it cannot be ruled out that many submillimeter
galaxies are Compton thick (N$_\mathrm{H}$ $>$ 1.5 $\times$ 10$^{24}$ cm$^{-2}$).
We conclude that the generally low radio flux densities in our sample could be due to small number statistics, source confusion, or
generally depressed radio emission, perhaps due to quenching of high energy cosmic rays,
although radio-quiet, Compton-thick AGN contributions to the dust heating cannot be ruled out.
There are new AGN versus star formation spectral diagnostics emerging (Ivison et al., Egami
et al.), and it is possible that to definitively settle the issue may ultimately require
ALMA, Constellation-X, and interferometric FIR spectral line diagnostic capability.
\subsection{Inferred Luminosities}
To obtain the intrinsic bolometric luminosities of the five Bolocam galaxies with spectroscopic redshifts, the spectra of
\S\ \ref{subsection:correlations_between_spectra} (based on the Laurent et al.\ 2005 model) were
integrated for each galaxy in their respective rest frames. The resulting bolometric
luminosities are listed in Figure
\ref{figure:revertzoom}. The four galaxies well-modeled by a 40 K dust spectrum have luminosities ranging from $L = (1.0-1.2)
\times 10^{13}$ $\mathrm{L}_\odot$. The lower redshift galaxy (Bolocam source 17) has an inferred luminosity two orders of magnitude
lower ($L = 1.3 \times 10^{11}$ $\mathrm{L}_\odot$). If the spectroscopic redshift of 0.689 does not apply to this galaxy and it instead lies at $z$ = 3.4 (the photometric redshift predicted by the
Laurent et al. 2005 / Wiklind 2003 and {\it Spitzer} IR models), then its luminosity of $L = 8.2 \times 10^{12}$ $\mathrm{L}_\odot$ agrees well with the others.
\subsection{Stellar and Dust Masses Implied from the Integrated Submillimeter Luminosities}
Three hundred and fifty micron observations combined with far-infrared and millimeter-wavelength photometry enables accurate measurements of luminosity for
galaxies near $z$ = 2 because no interpolation across the peak of the SED is required. Characteristically, integration of the SEDs of
the galaxies in our sample from far-infrared to millimeter-wavelengths yields luminosities of $\sim$ 1 $\times$ 10$^{13}$ L$_\odot$. Assuming: 1) that the
luminosity derives from star formation (young stars, which may overestimate stellar masses due to contribution from intermediate mass giants),
2) a characteristic (Salpeter) form of the initial
mass function (IMF) from \cite{chabrier03}, and 3) all of the optical and ultraviolet radiation is reprocessed to long wavelengths by dust, enables the stellar mass content of the galaxies to be
approximately estimated.
We adopt the $M^{3.5}$ luminosity function of \cite{demircan91}.
Because the luminosity function is so steep, the derived mass depends strongly on the assumed upper mass limit of integration. The lower limit of
integration is not well constrained by the data, although masses less than 0.3 M$_\odot$ are not likely to dominate the mass because the IMF flattens considerably at low
mass. Lower mass limits of 0.7 and 1.0 M$_\odot$ could be relevant because: 1) \cite{dwek98} argued that a Salpeter IMF for $z$ $>$ 1 galaxies cannot
extend much lower than this without producing too many low mass stars that would be present today and 2) \cite{chabrier03} suggests (with caution) that the
high-z IMF could cut off $>$ 1 M$_\odot$ based on multiple circumstantial lines of evidence. Varying $m_l$ from 0.3 to 1.0 M$_\odot$ and limiting $m_u$ to $\le$ 50 M$_\odot$ yields a mininum
stellar mass of 10$^{10}$ M$_\odot$ and a maximum of a few $\times$ 10$^{11}$ M$_\odot$, consistent with the stellar mass content of large elliptical galaxies, as previously pointed out by many
authors \citep[e.g.,][]{smail02, lilly99}. This range must be considered an upper limit because AGN could be responsible for
some of the dust heating.
For all of the galaxies with secure 350 $\mu$m detections, especially those with \cite{chapman05} spectroscopic redshifts, it is clear that the Bolocam 1.1 mm observations
lie on the optically thin Rayleigh-Jeans side of the SED, and therefore enable dust mass estimates. The flux density, $S_\nu$, of a galaxy at an observed frequency, $\nu$, is related to the dust
mass, M, by
\begin{eqnarray}
\nonumber
S_\nu=B_{\nu'}(T) \frac{\kappa_\nu M (1+z)}{D_L^2},
\end{eqnarray}
where $D_L$ is the luminosity distance to redshift z, $B_{\nu'}(T)$ is the Plank function evaluated at the emitted frequency, $\nu'$, and $\kappa_\nu$ is the dust opacity. Using the range
of
observed Bolocam flux densities (4.0 - 6.8 mJy), assuming a redshift of 2.1, and applying the dust
cross section of $\kappa_\nu$ = 12.4 cm$^2$/g \citep{ossenkopf94} most relevant for high mass star formation (high gas density and thin ice mantle model), leads to dust masses of
3 - 5 $\times$ 10$^8$ M$_\odot$ (2 - 3 $\times$ 10$^8$ M$_\odot$) for a dust temperature of 40 K (50 K). Blindly applying a dust-to-gas mass ratio of 1/100 implies gas masses of
3 - 5 $\times$ 10$^{10}$ M$_\odot$. These gas masses are comparable to those from a recent sample of 8 submillimeter galaxies of \cite{genzel04} and
\cite{neri03} which yield median molecular gas masses (from CO emission) of 2.2 $\times$ 10$^{10}$ M$_\odot$ and 2.8 $\times$ 10$^{10}$ M$_\odot$, with median dynamical masses of 1.1 $\times$
10$^{11}$ M$_\odot$ and 6.2 $\times$ 10$^{10}$ M$_\odot$ (assuming the most probable inclination angle of sin $i = 2/\pi$), respectively. These gas mass estimates are uncertain to at least a factor
of a few due to 1) the Bolocam flux density bias, which causes the measured flux densities to be overestimated by 10 - 30\%, 2) our assumed values of $\kappa$
and T, which may vary by a factor of a few and $\pm$ 20 K, respectively, and 3) increasing our assumed redshift of 2.1 (the mean of the 5 Chapman spectroscopic
redshifts of the Bolocam galaxies) to $z$ = 2.4 {\it increases} our calculated gas mass by 30\%.
Nevertheless, taking these factor into account still imply that a considerable fraction of the mass could already by in stars and substantial gas remains for star formation. This
major epoch of galaxy formation at approximately $z$ = 2 is consistent with the conclusions of \cite{fontana04} from spectral fitting of a sample of 500 elliptical galaxies at 0.2 $\le$ $z$ $\le$
2.5 that approximately 35\% of elliptical galaxy stellar mass was assembled by $z$ = 2 and approximately 80\% by $z$ = 1.
\section{Conclusions}
We have obtained 350 $\mu$m SHARC II observations toward galaxy candidates from the Bolocam Lockman Hole survey. The
Lockman Hole has rich, deep, multiwavelength observations enabling detailed studies of galaxies. The 350 $\mu$m photometry is
near the peaks of the SEDs of galaxies with characteristic temperatures of $\sim$ 50 K and redshifts of $z$ $\sim$ 2 to 3. They therefore
enable measurements of luminosities and estimates of temperatures and photometric redshifts without interpolating over the
peak of the FIR thermal SEDs. Seven galaxies detected at 1.1 mm with Bolocam were detected at 350 $\mu$m, two of which have two
350 $\mu$m counterparts; these were combined with two 350 $\mu$m detections from the survey of \cite{kovacs05}, bringing the
total number of Bolocam galaxies detected with SHARC II to nine. Two additional galaxies not associated with the Bolocam sources
were also detected. The SHARC II detections range in significance from 3.0 $\sigma$ to 6.8 $\sigma$, with flux densities
ranging from 14 mJy to 64 mJy.
We combined our observations with 850 $\mu$m and 1.2 mm photometry from the literature to fit the
submillimeter/millimeter-wave spectra to thermal dust models. We found that two models with significantly different dust
temperatures (40 K and 68 K) and spectral indices $\beta$ (1.6 and 1.8, respectively) yielded similar quality fits owing to the
degeneracy in T and $\beta$, rendering them indistinguishable without better SED sampling. However, there is little
consequence of the degeneracy to the derived luminosities, photometric redshifts, and dust masses within the statistical
uncertainties. Five of the galaxies have spectroscopic redshifts in the literature, with four ranging from $z$ = 2.1 to 3.0 and
one at $z$ = 0.689. The four high-z galaxies have luminosities of (1.0 - 1.2) $\times$ 10$^{13}$ L$_\odot$, while the $z$ = 0.689 galaxy is
best fit by a 20 K, $\beta$ = 1.0, spectrum with a much lower luminosity: 1.3 $\times$ 10$^{11}$ L$_\odot$. (Given the source confusion in the optical and radio, along with consistent photometric
redshifts, we suggest that the $z$ = 0.689 spectroscopic redshift of Bolocam source 17 may be a misidentification.) The characteristic dust masses
for the four high-$z$ spectroscopic galaxies are 4 $\times$ 10$^8$ M$_\odot$, implying gas masses of 4 $\times$ 10$^{10}$
M$_\odot$. The dominant uncertainties in this estimation are the dust opacity and the gas-to-dust conversion factor, which
make the estimation uncertain to a factor of a few. Assuming a Salpeter IMF and that the submillimeter emission derives
completely from star formation yields stellar masses of 10$^{10}$ to a few times 10$^{11}$ M$_\odot$, broadly consistent with the
stellar content of modern-day elliptical galaxies.
The photometric redshifts of the full sample of seven galaxies span the range of $z$ = 2.0 to $z$ = 4.3, with statistical
uncertainties of $\Delta z$ = 0.3 to 0.6 (1 $\sigma$). Photometric redshifts utilizing composite radio/FIR spectra representative
of local star-forming ULIRGs yields systematically higher redshifts, on the order of $\Delta z$ = 1. For the four galaxies
with optical spectroscopic redshifts the anomolously high redshifts arise from systematically low 1.4 GHz observed flux
densities. The discrepancy could arise from small number statistics, inverse-Compton losses of high energy cosmic rays off the CMB,
heating by radio-quiet AGN, or suppressed synchrotron emission from supernova remnants in the unusually luminous galaxies. For comparison,
photometric redshifts derived using only Spitzer MIPS and IRAC data points yielded slightly more precise and accurate
redshifts than the submillimeter/millimeter-wave data alone, with discriminatory power between heating by AGN and star
formation (albeit with limited bolometric luminosity constraints).
\acknowledgments
We thank Attila Kov\'{a}cs for providing us with SHARC II detections of Bolocam sources 5 and 8.
We also acknowledge the support of the CSO director and staff, the support of Kathy Deniston, and
helpful conversations with Steven Eales.
This work was supported in part by NSF grants AST-0098737, AST-9980846, and AST-0206158 and
PPARC grants PPA/Y/S/2000/00101 and PPA/G/O/2002/00015. G.\ T.\ L.\ acknowledges NASA for GSRP Fellowship NGT5-50384, D.\ J.\ Haig and D.\ Dowell for their
assistance during the SHARC II observing runs, and the entire Bolocam instrument team.
|
Title:
Enhanced Mass-to-Light Ratios in UCDs through Tidal Interaction with the Centre of the Host Galaxy |
Abstract: A recent study of ultra-compact dwarf galaxies (UCDs) in the Virgo cluster
revealed that some of them show faint envelopes and have measured mass-to-light
ratios of 5 and larger, which can not be explained by simple population
synthesis models. It is believed that this proves that some of the UCDs must
possess a dark matter halo and may therefore be stripped nuclei of dwarf
ellipticals rather than merged star cluster complexes. Using an efficient
N-body method we investigate if a close passage of a UCD through the central
region of the host galaxy is able to enhance the measured mass-to-light ratio
by tidal forces leaving the satellite slightly out of virial equilibrium and
thereby leading to an overestimation of its virial mass. We find this to be
possible and discuss the general problem of measuring dynamical masses for
objects that are probably interacting with their hosts.
| https://export.arxiv.org/pdf/astro-ph/0601330 |
\label{firstpage}
\title[Enhanced $M/L$-ratios in UCDs]{Enhanced Mass-to-Light Ratios in
UCDs through Tidal Interaction with the Centre of the Host Galaxy}
\author[M. Fellhauer and P. Kroupa] {M. Fellhauer$^{1,2,3}$
\thanks{madf@ast.cam.ac.uk} and P. Kroupa$^{1,2}$ \thanks
{pavel@astro.uni-bonn.de} \\
$^{1}$ Argelander Institute for Astronomy, University Bonn, Auf dem
H\"{u}gel 71, 53121 Bonn\\
$^{2}$ The Rhine Stellar-Dynamical Network \\
$^{3}$ Institute of Astronomy, University of Cambridge, Madingley Road,
Cambridge CB3 0HA}
\pagerange{\pageref{firstpage}--\pageref{lastpage}} \pubyear{2005}
\begin{keywords}
galaxies: dwarfs -- galaxies: interactions -- galaxies: kinematics
and dynamics -- methods: N-body simulations
\end{keywords}
\section{Introduction}
\label{sec:intro}
\citet{has05} investigated ultra-compact dwarf galaxies (UCDs) around
M87, the central galaxy of the Virgo cluster. By measuring the
surface brightness profiles and assessing the projected velocity
dispersion of the objects they concluded that some UCDs of their
sample have mass-to-light ratios of the order $5$--$9$. Furthermore,
they find that some of the UCDs show faint envelopes. This supports
the notion that these UCDs may be stripped nuclei of dwarf
ellipticals.
UCDs were first discovered by \citet*{hil98,hil99} during a study of
globular clusters and dwarf galaxies around the central galaxy in the
Fornax cluster. These objects are compact with effective radii of
about $15$--$25$~pc and a hundred to several hundred pc in extension,
and they are massive with masses of a few $10^{6}$ up to a few
$10^{7}$~M$_{\odot}$.
There are several theories about the origin of UCDs: (I) They could be
the most luminous end of the distribution function of very massive
globular clusters \citep{hil99,mie02,dir03}. (II) They could be the
remnants of stripped dwarf ellipticals \citep*{bek01,bek03,dep05}. In
this 'threshing' scenario a nucleated dwarf elliptical looses its
envelope and most of its dark matter content due to tidal interaction
with the host galaxy, such that only the 'naked' nucleus remains.
(III) They could be amalgamated young massive star clusters formed in
a star cluster complex during the star-burst caused by the tidal
perturbation and possible disruption of a gas-rich galaxy
\citep{fel02,fel05}. This scenario is well-established theoretically
and was first proposed by \citet{kro98}. It must have been
profusively active during the early hierarchical structure formation
epoch when gas-rich substructures merged to the present-day major
galaxies.
\begin{table*}
\centering
\begin{minipage}{10.5cm}
\caption{Table of our initial model parameters. The columns
denote the initial mass of our model ($M_{\rm pl} = M_{\rm
ini}$), the scale length (Plummer radius, $R_{\rm pl}$;
analytical \& measured as described in Sect.~\ref{sec:res}), the
characteristic crossing time ($T_{\rm cr}$), the central
projected (line-of-sight) velocity dispersion ($\sigma_{0,p}$;
analytical \& measured as described in Sect.~\ref{sec:res}), and
the scaling factor ($A$) to compute the virial mass (see
Eq.~\ref{eq:virplum}).}
\label{tab:para}
\begin{tabular}[t!]{rrrrrrr} \hline Mass [M$_{\odot}$] & $R_{\rm
pl}$ [pc] & measured & $T_{\rm cr}$ [Myr] & $\sigma_{0,p}$
[km\,s$^{-1}$] & measured & $A$ \\ \hline \hline $10^{7}$ & $25$
& $23.7$ & $3.69$ & $15.92$ & $16.20$ &
$1608$ \\
$10^{7}$ & $50$ & $46.6$ & $10.43$ & $11.25$ & $11.71$ &
$1565$ \\
$10^{7}$ & $100$ & $94.6$ & $29.50$ & $7.96$ & $8.43$ &
$1487$ \\
$10^{7}$ & $250$ & $221.0$ &$116.61$ & $5.03$ & $6.30$ & $1140$
\\ \hline $10^{8}$ & $25$ & $25.2$ & $1.17$ & $50.33$ & $50.2$ &
$1573$ \\
$10^{8}$ & $50$ & $47.4$ & $3.30$ & $35.59$ & $36.8$ &
$1558$ \\
$10^{8}$ & $100$ & $92.8$ & $9.33$ & $25.16$ & $26.9$ &
$1489$ \\
$10^{8}$ & $250$ & $213.0$ & $36.88$ & $15.92$ & $19.91$ &
$1184$ \\ \hline
\end{tabular}
\end{minipage}
\end{table*}
It is still under debate whether the UCDs, which fill the gap between
globular clusters and dwarf galaxies, follow the fundamental plane
(effective radius / velocity dispersion -- total luminosity) relation
for globular clusters or for dwarf galaxies \citep*{has05,evs05}.
While, in the $M_{V}$--$\sigma_{0}$-plane \citep[see e.g.][their
Fig.~7]{has05}, they lie closer to the globular clusters, their
relation seems to rather follow the one of dwarf galaxies.
Furthermore they occupy exactly the space between dwarf ellipticals
and their nuclei. This seems to point to formation theory (II). But
the surface brightness profiles of the bright UCDs in Fornax are much
more extended (i.e.\ larger effective radii) compared with nuclei of
dEs \citep{dep05}.
On the other hand, \citet{mar04} found an intermediate age object (W3,
age $300$--$500$~Myr) in the merger remnant galaxy NGC~7252. The mass
($M = 8 \cdot 10^{7}$~M$_{\odot}$), size ($r_{\rm eff} = 17.5$~pc, and
velocity dispersion of $45$~km\,s$^{-1}$) strongly suggest this object
to be a UCD rather than a globular cluster. The age of this object,
which corresponds to the time elapsed since the major interaction,
unambiguously shows that this object can not be a stripped nucleus of
a dwarf elliptical. A dwarf elliptical can not be stripped in a time
interval of only $500$~Myr. \citet{fel05} showed that W3 could be the
merger object of a massive star cluster complex which was formed
during the interaction. The subsequent merging of star clusters forms
an object with properties similar to those of W3. The evolution of
the simulated super cluster shows that it transforms into a UCD such
as those found in Fornax \citep{hil99,phi01}, Abell~1689
\citep{mie04}, and Virgo \citep{has05,evs05}. Moreover, \citet{fel05}
showed that, due to its high mass, the object was able to retain an
envelope of bound stars which initially were expelled from the
individual clusters during the merger process. Thus, an envelope
around a UCD is not a proof of its cosmological origin as a dwarf
elliptical. Still the puzzle of high mass-to-light ratios of some of
the UCDs in Virgo remains. These mass-to-light ratios (5--9) are
found for the UCDs lying closest to the host galaxy. This suggests
that deviations from virial equilibrium may play a role.
The reasoning is that dwarf galaxies are on radial rather than
circular orbits, if the UCDs formed as star-cluster complexes during
the merging of major gas-rich substructures. This allows the
satellites to pass close to the galactic centre. Tidal forces in the
central region of the host galaxy are rather strong. Hence, the dwarf
object looses stars and subsequently departs from virial equilibrium.
Some of the lost stars (stars which are no longer bound to the object)
do not immediately leave the object or its vicinity but disperse
slowly along the orbit. Thus, a line-of-sight velocity dispersion
measurement may be contaminated by these unbound stars which inflate
the velocity dispersion. Furthermore, the gravitational shock of the
central passage leads to an expansion of the dwarf galaxy which is
later reversed again. But still this expansion through tidal heating
may result in a measurable increase of the core radius once the dwarf
object is again outside the host galaxy. Both of these effects may
lead to an overestimation of the measured dynamical mass resulting in
a higher mass-to-light ratio. These effects are not new to the
astronomical community and are studied intensively by various authors
(e.g. \citep{kro97} for dwarf spheroidals, \citep{may01,may02} for
dwarf discs and dwarf spheroidals or \citep{bek03} for dwarf
ellipticals). With this paper we want to extend these studies onto
massive (compared to globular clusters) and compact (compared to other
dwarf galaxies) objects like UCDs.
It is definitely clear that these effects must be very strong if an
object comes close to the galactic centre. But on the other hand
these very close passages may be highly unlikely. By means of
numerical simulations we intend to find out which sets of orbits allow
an enhanced mass-to-light ratio, i.e.\ how close the UCD has to
approach to the centre of its host galaxy, for which we choose the
parameters of M87 to account for the UCDs with high mass-to-light
ratios found around the central galaxy of Virgo.
\section{Setup}
\label{sec:setup}
We model the parent galaxy as an analytical potential because on one
single passage dynamical friction for a dwarf galaxy of mass $M \leq
10^{8}$~M$_{\odot}$ affects the orbit to at most $2$--$3$~per cent,
estimated using Chandrasekhar's formula \citep{cha43} as described in
\citet{por02}. We also look for only one central passage because
unbound stars disperse along the orbit and are lost at subsequent
passages. Some of them might still be around the object after the
second or third central passage but definitely not for dozens of
orbits.
As a model for the analytical potential we choose the parameters for
M87, the central galaxy in the Virgo cluster, consisting of a
NFW-profile for the dark halo, a Hernquist profile (H) for the visible
matter (stars) and a central super-massive black hole (BH)
\citep*{mcl99,ves03,dim03}. The density profile of the host galaxy is
\begin{eqnarray}
\label{eq:pot}
\rho_{\rm tot}(r) & = & \rho_{\rm NFW}(r) + \rho_{\rm H}(r)
\nonumber \\
& = & \frac{\rho_{0,{\rm NFW}} \ r_{\rm s,NFW}} {r \left( 1 +
\frac{r}{r_{\rm s,NFW}} \right)^{2}} + \frac{M_{\rm H} \ r_{\rm
s,H}} {2 \pi \ r \left( r_{\rm s,H} + r \right)^{3}},
\end{eqnarray}
with the following parameters,
\begin{eqnarray}
\label{eq:param}
\rho_{0,{\rm NFW}} & = & 3.17 \cdot 10^{-4} \ {\rm M}_{\odot}/{\rm
pc}^{3}, \\
r_{\rm s,NFW} & = & 560 \ {\rm kpc}, \\
M_{\rm H} & = & 8.1 \cdot 10^{11} \ {\rm M}_{\odot}, \\
r_{\rm s,H} & = & 5.1 \ {\rm kpc}, \\
M_{\rm BH} & = & 3 \cdot 10^{9} \ {\rm M}_{\odot}.
\end{eqnarray}
The above parameters denote from top to bottom the characteristic
density and the scale-length of the NFW-profile, the total mass and
the scale-length of the Hernquist-profile and finally the mass of the
central super-massive black hole.
We model the UCD as a Plummer-sphere \citep{plu11} in the numerical
realisation described by \citet*{aar74},
\begin{eqnarray}
\label{eq:plummer}
\rho_{\rm pl}(r) & = & \frac{3M_{\rm pl}}{4\pi R_{\rm pl}^{3}}
\left( 1 + \frac{r^{2}}{R_{\rm pl}^{2}}\right)^{-5/2},
\end{eqnarray}
with varying scale-lengths, $R_{\rm pl}$, and initial masses, $M_{\rm
pl}$, because \citet{dep05} found a Plummer-profile to fit four out
of five UCDs in the Fornax cluster quite well. Furthermore, the
Plummer model is analytically simple and the Plummer radius is not
only the scale-length of the model but also its half-light radius (the
projected radius from within which half of the light of the object is
emitted).
The masses of UCDs range between several million M$_{\odot}$ up to
several tens of millions. So we choose for our models $10^{7}$ and
$10^{8}$~M$_{\odot}$ as initial masses ($M_{\rm pl} = M_{\rm ini}$).
We vary the initial scale-length of our models to be $R_{\rm pl} =
25$, $50$, $100$ and $250$~pc to determine the influence of the
concentration of the objects. The cut-off radius ($R_{\rm lim}$; the
radius where we truncate the Plummer distribution of our object) of
all our models was kept constant at $500$~pc which is larger than the
initial tidal radius. A detailed list of our model parameters can be
found in Table~\ref{tab:para}. The Plummer spheres are modelled using
$10^{6}$ particles. The object is set-up and integrated in isolation
until equilibrium, as measured by the constancy of the $90$~\%
Lagrangian radius, is reached \citep{kro97}.
The UCD model is then placed at a distance of $10$~kpc to the centre of
the host galaxy with no radial velocity. This means that the effect
of the central passage is the most harmful possible because it is the
slowest possible. The faster the passage would be (if the satellite
was to start further out) the less tidal influence the central passage
would have. No UCD is found closer than $10$~kpc from the centre of
its host and therefore it acts as a minimum apogalacticon, which has
the strongest tidal effect possible. To vary the minimum distance
(perigalacticon) we give our models different tangential velocities
which are listed in Tab.~\ref{tab:vel}. We assume the problem to be
spherically symmetric, so we are able to place the trajectory of our
model in the $x$-$y$-plane of our simulation area without restricting
the problem.
\begin{table}
\centering
\caption{List of minimum distances and the corresponding tangential
velocities at the start of the simulation.}
\label{tab:vel}
\begin{tabular}[t!]{rr} \hline
$D_{\rm min}$ [pc] & $v_{\rm tan}$ [km\,s$^{-1}$] \\ \hline \hline
0 & 0.0 \\
50 & 6.0 \\
100 & 10.9 \\
150 & 15.9 \\
250 & 25.4 \\
500 & 48.1 \\
1000 & 89.3 \\
1500 & 126.0 \\
2000 & 158.9 \\ \hline
\end{tabular}
\end{table}
We use the particle-mesh code {\sc Superbox} to carry out the
simulations. {\sc Superbox} has high-resolution sub-grids which stay
focused on the core of the dwarf object while it is moving through
the host galaxy. The resolution of the innermost grid containing the
core is $3$~pc. For a detailed description of the code see
\citet{fel00}.
\section{Results}
\label{sec:res}
We carried out a parameter survey of 76 simulations covering different
dwarf galaxy objects and different orbits. The parameter range is
shown in Tables~\ref{tab:para} and~\ref{tab:vel}. For each parameter
set one simulation over two central passages is carried out. For the
determination of our results we look at the satellite when it reaches
apogalacticon again after the first central passage.
\subsection{The analytical method}
\label{sec:theo}
The surface density profile of the satellite is fitted by a Plummer
profile of the form
\begin{eqnarray}
\label{eq:plum-surf}
\Sigma(r) & = & \Sigma_{0} \left( 1 + \frac{r^{2}} {R_{\rm
pl}^{2}}\right)^{-2},
\end{eqnarray}
by applying a non-linear least-squares Marquardt-Levenberg algorithm.
From this procedure we take the fitted Plummer radius for our
determination of the virial mass. It can be shown that the Plummer
radius exactly coincides with the half-light (projected half-mass)
radius of a Plummer sphere.
Furthermore we determine the line-of-sight velocity dispersion
profile. In cases where it is possible, i.e.\ the profile is not too
contaminated by unbound stars, we again fit the Plummer profile for
the line-of-sight velocity dispersion,
\begin{eqnarray}
\label{eq:plum-sig}
\sigma_{\rm p}(r) & = & \sigma_{\rm 0,p} \left( 1 + \frac{r^{2}}
{R_{\rm pl}^{2}}\right)^{-1/4}.
\end{eqnarray}
The projected velocity dispersion, as well as the surface density, is
measured along all three Cartesian coordinates in logarithmically
spaced, concentric rings centred on the object. For the measurement
all stars within a certain distance $R_{\rm max}$ in front and behind
the centre of the object are taken into account. The procedure is
done along all three Cartesian axes and we take the arithmetic mean
value (i.e.\ a mean profile), because we do not know which orientation
our objects have with respect to the observer. Therefore effects may be
very strong measuring along the trajectory of the object but almost
not visible in the perpendicular direction. Any random direction is
likely to yield an intermediate result.
For the surface density we always fit a Plummer profile even if the
object is completely destroyed and does not follow a Plummer
distribution at all. But in most of our models the remaining object
is still fairly well represented by a Plummer profile even if it is on
the way to complete destruction. In cases where we are not able to
fit the velocity dispersion profile with a Plummer profile (see e.g.\
first panel of Fig.~\ref{fig:rmax}) we take an average of all measured
line-of-sight velocity dispersion values within the innermost $10$~pc
in projected radius to determine the central line-of-sight velocity
dispersion.
We now determine the virial mass taking the formula from \citet{has05}
which is based on the theoretical work of \citet{kin66},
\begin{eqnarray}
\label{eq:vir}
M_{\rm vir} & = & \frac{9}{2 \pi G} \ \frac{\nu}{\alpha p} \ R_{\rm
c} \ \sigma_{\rm 0,p}^{2},
\end{eqnarray}
where $R_{\rm c}$ is the core radius and $\sigma_{\rm 0,p}$ is the
central value of the projected velocity dispersion. The parameters
$\nu$, $\alpha$, and $p$ are dependent on the kind of King model one
wants to fit. Because we are using Plummer spheres we accumulate
these parameters together with the constants into one single parameter
$A$, which we determine for the isolated Plummer sphere before we
start the simulation,
\begin{eqnarray}
\label{eq:virplum}
M_{\rm vir} & = & A \ R_{\rm pl} \ \sigma_{\rm 0,p}^{2},
\end{eqnarray}
where the values of the scaling factor $A$ can be found in
Table~\ref{tab:para}. As one can see these values are lower for
higher masses and increase for larger scale-lengths. Nevertheless we
use the same $A$ determined for the isolated model for all final
models stemming from this initial model, even if the mass-loss is
significant. This may lead to a slight underestimation of the final
virial mass.
On the other hand {\sc Superbox} calculates the number of bound
particles (energy below zero) at each time-step. Therefore we know
the real bound mass $M_{\rm b}$.
With these two masses we can determine a virial-mass-to-bound-mass
ratio ($M_{\rm vir} / M_{\rm b}$ or short M/M) which can then be
multiplied by a 'normal' mass-to-light ratio for a population of stars
with the determined age and metallicity and without dark matter to
obtain the dynamically measured $M/L$-ratio.
This M/M-ratio is plotted in Fig.~\ref{fig:all}, when the satellites
have reached their apogalactica again. At this point the satellites
are almost back to virial equilibrium again if they are not on the way
to complete dissolution.
The results show clearly that only objects which are not compact
and/or have a low mass can be influenced enough by one central passage
to enhance the M/M to account for the high mass-to-light ratios found
in UCDs. But these satellites will also not survive the next central
passage or have not survived the first one at all.
The best representation of real UCDs is the model with an initial mass
of $10^{7}$~M$_{\odot}$ and an initial Plummer radius of $25$~pc. The
results for this model are shown as tri-pointed stars in the first
panel of Fig.~\ref{fig:all} and the lowest line in the first panel of
Fig.~\ref{fig:result}. The results show clearly that either the
satellite gets completely dissolved if $D_{\rm min}$ is closer than
$50$--$100$~pc or shows no deviation in M/M at all
(Fig.~\ref{fig:result}).
In Fig.~\ref{fig:result} we distinguish the models according to their
survival of the passage and if they show enhanced M/M-ratios. The
panels show clearly that only dissolved or almost dissolved objects
show enhanced M/M-ratios. All surviving objects show no deviation
from virial equilibrium.
\subsection{The 'observational' method}
\label{sec:obs}
The results in Sect.~\ref{sec:theo} are based on the exact knowledge
of the theoretical central line-of-sight velocity dispersion.
Observers on the other hand do not have this information. The usual
way to determine the velocity dispersion of a marginally resolved
object is to place a slit on the object and obtain a spectrum. From
this spectral information one chooses one or two spectral lines and
fits a template spectrum convolved with the instrumental line-width
function and a Gaussian velocity distribution. This Gaussian velocity
distribution measures the line-of-sight velocity dispersion of the
whole object. Based on assumed theoretical models this value is then
corrected to the central line-of-sight velocity dispersion. In the
case of the simple Plummer model one has to multiply the result for
the whole object by a factor of 1.25. Thus, the central velocity
dispersion is
\begin{eqnarray}
\label{eq:gauss1}
\sigma_{0,p}^{\rm pl} & = & \sqrt{\frac{3 \pi G M_{\rm pl}} {64
R_{\rm pl}}},
\end{eqnarray}
while the projected velocity dispersion integrated over the whole
Plummer sphere is
\begin{eqnarray}
\label{eq:gauss2}
\sigma_{\rm obs,p}^{\rm pl} & = & \frac{1}{M_{\rm pl}} \
\int_{0}^{\infty} 2 \pi r' \Sigma(r') \sigma_{p}(r') {\rm d}r',
\nonumber \\
& = & \sqrt{\frac{3 \pi G M_{\rm pl}} {100 R_{\rm pl}}},
\end{eqnarray}
so that
\begin{eqnarray}
\label{eq:gauss3}
\frac{\sigma_{0,p}^{\rm pl}} {\sigma_{\rm obs,p}^{\rm pl}} & = & 1.25.
\end{eqnarray}
$\sigma_{0,p}^{\rm pl}$ denotes the central line-of-sight velocity
dispersion and $\sigma_{\rm obs,p}^{\rm pl}$ the weighted
line-of-sight velocity dispersion integrated over the whole object.
As one can see in Fig.~\ref{fig:rmax} it can be a crucial point how
strongly a measurement of the line-of-sight velocity dispersion is
contaminated by unbound stars. The observational values can highly
overestimate the real central value. Unbound stars have a different
velocity distribution than the bound ones. They are either travelling
in front or behind the object (seen along the trajectory; this is
shown in Fig.~\ref{fig:contour}) and are faster or slower. But the
velocity distribution of these stars which are still located in the
vicinity of the object and stem from one central passage, peak on
either side of the Gaussian distribution of the bound stars, leading
to an effective broadening of the measured velocity distribution.
Fitting just a single Gaussian will therefore lead to an enhanced
measured velocity dispersion and an overestimation of the central
velocity dispersion value of the satellite. This is demonstrated in
Fig.~\ref{fig:gauss} where the velocity distribution of all stars is
shown together with the best fitting single Gaussian and the fitting
curve of a triple Gaussian which takes the unbound stars into account.
The measured values for the central line-of-sight velocity dispersion
are shown in Table~\ref{tab:sig}.
\begin{table}
\centering
\caption{Central velocity dispersion along the x-axis derived by
different methods. Shown are the values of the same three
simulations plotted in Figs.~\ref{fig:rmax} to~\ref{fig:gauss}
now labelled 'dissolved', 'enhanced' and 'massive', respectively.
The rows show
the different values of the central velocity dispersion. First
row is the direct measurement of the central dispersion, 'single'
denotes the central dispersion derived from the fit of a single
Gaussian to the data and 'triple (c)' denotes the value of a fit
using three Gaussians where the central one (c) is used to
compute the velocity dispersion of the bound object.}
\label{tab:sig}
\begin{tabular}[t!]{r|rrr} \hline
[km/s] & dissolved & enhanced & massive \\ \hline \hline
true actual & $10.6 \pm 3.1$ & $9.49 \pm 0.05$ & $35.7 \pm 0.2$
\\
single & $44.6 \pm 0.3$ & $15.90 \pm 0.30$ & $38.4 \pm 0.1$ \\
triple (c) & $13.3 \pm 0.1$ & $12.35 \pm 0.04$ & $35.3 \pm 0.3$
\\ \hline
\end{tabular}
\end{table}
Clearly this effect is strongest in the plane of the orbit and is not
visible perpendicular to it. This can be seen in Fig.~\ref{fig:rmax}
where the symbol for the $z-$axis-value in all three panels shows the
same value as the direct measurement. Actually for strongly disturbed
systems (i.e.\ high mass-loss) the values in the $z$-direction are
below the line because we do not account for the change of the
constant $A$ which should increase for lower masses. But in any
random orientation of the object with respect to the observer the
effect of the enhanced M/M-ratio should at least be partly visible.
Summing up our results we state the following: Satellites which are
out of virial equilibrium, i.e.\ in the state of dissolution show
enhanced virial masses and therefore the real mass content is
overestimated. Figure~\ref{fig:obs} shows in the first panel the
M/M-ratio plotted against the ratio of final to initial bound mass of
the object. The dividing dashed line separates objects which have
lost more than $50$~per cent of their mass during the central passage
and are already dissolved or are not likely to survive the next
passage, from objects which are stable for several more passages
through the centre. While on the left side the derived M/M-ratios
(using the observational method) can climb up to very high values, the
stable objects show only slight enhancements of the ratio if at all.
But in the second panel one already sees that even if the object
itself is in virial equilibrium ($M_{\rm vir}/M_{\rm b} \approx 1.0$)
there are satellites which show a broadening of the velocity
distribution leading to an enhanced mass-ratio if measured the
observational way. The third panel finally shows an enlargement of
this area and one finds M/M-ratios overestimated by up to a factor
of five. Surviving objects which show an enhancement in their
M/M-ratio, if measured with this observational method, are already
marked in Fig.~\ref{fig:result} with a small 'e'. As one can see
there is a range of critical distances ($D>100$~pc and $D<1$~kpc) to
the centre of the host galaxy where an UCD like the ones found in
Virgo could show an overestimated mass-to-light ratio.
\subsection{Observability}
\label{sec:observ}
In the previous section we claimed that with the 'observational'
method the mass-to-light ratios of UCDs could be overestimated,
because the velocity distribution is not Gaussian any more but
'contaminated' by unbound stars around the object. In this section we
show that there is almost no chance for an observer to detect
this 'non-Gaussianity' of the velocity distribution.
When measuring a spectrum of a distant object, observers have to deal
with two major shortcomings. First there is the intrinsic line-width
produced mainly by the instrument. State-of-the-art instruments like
UVES or Flames can reduce this line-width down to about
$2$~km\,s$^{-1}$. But the observations of the UCDs in Virgo were made
with an instrumental line-width of about $25$~km\,s$^{-1}$.
The second effect an observer has to take into account is noise in the
spectrum.
Taking the velocity distribution from the enhanced M/M-ratio
simulation we fold it with a Gaussian of the width $\sigma_{\rm i}$ to
mimic the instrumental line-width and determine at which line-width
the 'features' of our distribution are washed out. This happens at an
instrumental line-width of $\sigma_{\rm i}=7.5$~km\,s$^{-1}$ (as shown
in Fig.~\ref{fig:dist}). Then we take the best state-of-the-art
line-width of $\sigma_{\rm i}=2$~km\,s$^{-1}$ and add random white
noise to the distribution until again the 'features' are almost
invisible again. This happens already at a signal-to-noise ratio of
$20$ (see Fig.~\ref{fig:dist} lower left panel). In the final panel
of Fig.~\ref{fig:dist} we fold our distribution with the line-width of
the observations of the Virgo-UCDs and add noise to mimic the same
S/N-ratio as in the observations. There is no deviation from
Gaussiantity visible any more.
\section{Discussion \& Conclusion}
\label{sec:disc}
We have shown with our models that dwarf satellites around a giant
elliptical galaxy like M87 can have an enhanced mass-to-light ratio
due to close passages to the centre of the host galaxy. While very
close passages lead to the destruction of the satellite there are
orbits which allow for enough 'damage' to the satellite to enhance the
mass-ratio (measured the same way an observer would do) without
completely destroying the object.
The amount of destruction is larger if the satellite is less massive
and less concentrated. But the loss of about $20$~per cent of the
initial mass is enough to have the object surviving several more close
passages and to mimic an enhanced mass-to-light ratio. For models
comparable to the UCDs in Virgo and Fornax the range of possible
minimum distances during central passages is about $100$ to $1000$~pc.
All passages closer to the centre lead to complete destruction and all
passages further away show no measurable effect at all, except for a
mass-loss of the order of a few per cent. We therefore conclude that
the enhanced M/L-ratios measured for the Virgo UCDs by \citet{has05}
may not be due to dark matter.
A more general result of our study is the discrepancy between the
derived virial masses if one has access to the correct properties of
the satellites compared to the virial masses derived the way an
observer would measure. While very massive objects which are almost
unaffected by tidal forces show the same results within the
uncertainties one has to be careful if objects are less massive and
are surrounded by a cloud of tidally stripped stars. These stars are
either faster or slower in the mean but their mean values are not too
different to the bulk velocity of the bound stars to clearly
disentangle the 'populations' (populations in the sense of within, in
front or behind the object). Especially if the broadening of a
spectral line is estimated by fitting the template line folded with
a Gaussian for the instrumental line-width and a single Gaussian for
the velocity distribution can the deduced central velocity dispersion
be too high thus leading to a mass-to-light ratio that is too large by
up to a factor of ten.
Effects like this have to be given serious consideration when
measuring velocity dispersions of faint and distant objects. \\
\noindent {\bf Acknowledgements:}
\noindent MF thankfully announces financial support through DFG-grant
KR1635/5-1 and PPARC. We also want to thank M. Hilker and T. Richtler
for useful comments regarding how to mimic observations.
\label{lastpage}
|
Title:
Upper limit on the ultra-high-energy photon flux from AGASA and Yakutsk data |
Abstract: We present the interpretation of the muon and scintillation signals of
ultra-high-energy air showers observed by AGASA and Yakutsk extensive air
shower array experiments. We consider case-by-case ten highest energy events
with known muon content and conclude that at the 95% confidence level (C.L.)
none of them was induced by a primary photon. Taking into account statistical
fluctuations and differences in the energy estimation of proton and photon
primaries, we derive an upper limit of 36% at 95% C.L. on the fraction of
primary photons in the cosmic-ray flux above 10^20 eV. This result disfavors
the Z-burst and superheavy dark-matter solutions to the GZK-cutoff problem.
| https://export.arxiv.org/pdf/astro-ph/0601449 |
\title{Upper limit on the ultra-high-energy photon flux from AGASA and
Yakutsk data}
\author{G.I.~Rubtsov$^1$, L.G.~Dedenko$^{2,3}$, G.F.~Fedorova$^3$,
E.Yu.~Fedunin$^3$, A.V.~Glushkov$^4$, D.S.~Gorbunov$^1$,
I.T.~Makarov$^4$, M.I.~Pravdin$^4$,
T.M.~Roganova$^3$, I.E. Sleptsov$^4$
and S.V.~Troitsky$^1$}
\affiliation{$^1$Institute for Nuclear Research of the Russian
Academy of Sciences,
Moscow 117312, Russia}
\affiliation{$^2$Faculty of Physics, M.V.~Lomonosov Moscow State
University,
Moscow 119992, Russia}
\affiliation{$^3$D.V.~Skobeltsin Institute of Nuclear Physics,
M.V.~Lomonosov Moscow State University,
Moscow 119992, Russia}
\affiliation{$^4$Yu.G.~Shafer Institute of Cosmophysical Research and
Aeronomy,
Yakutsk 677980, Russia}
\date{January 13, 2006}
\pacs{98.70.Sa, 96.40.De, 96.40.Pq}
\section{Introduction}
\label{sec:intro}
One of the most intriguing puzzles in astroparticle physics is
the observation of air showers initiated by particles with
energies beyond the cutoff predicted by Greisen and by Zatsepin and
Kuzmin~\cite{gzk}. Compared to lower energies, the energy losses of
protons
increase sharply at $\approx 5\times 10^{19}$~eV since pion production on
cosmic microwave background photons reduces the proton mean free path by
more than two orders of magnitude. This effect is even stronger for heavier
nuclei, while photons are absorbed
due to pair production on the radio background
with the mean free path of a few Mpc. Thus, the cosmic-ray (CR) energy
spectrum should dramatically steepen at $\approx 7\times 10^{19}$~eV for
any homogeneous distribution of CR sources.
Despite the contradictions in the shape of the spectrum,
the existence of air showers with energies
in excess of $10^{20}$~eV is firmly established by several independent
experiments using different
techniques (Volcano Ranch~\cite{exp}, Fly's
Eye~\cite{Bird}, Yakutsk~\cite{YakutskExperiment},
AGASA~\cite{agasares}, HiRes~\cite{HiRes} and Pierre Auger~\cite{Auger}
experiments). Some explanations for these showers, like the $Z$-burst or
top-down models, predict a significant fraction of photons above typically
$8\times 10^{19}$~eV (for reviews see, e.g., Refs.~\cite{reviews}).
Indications for the presence of neutral particles at lower energies were
found in Refs.~\cite{neutral}. Thus, the determination of the photon
fraction in the CR flux is of crucial importance, and the aim of this work
is to derive a stringent limit on this fraction in the integral CR flux
above $10^{20}$~eV. To this end, we compare the reported information on
signals measured by scintillation and by muon detectors for observed
showers with those expected by air shower simulations. We
focus on the surface detector signal density at 600 meters $S(600)$ (known
as charged particle density) and the muon density at 1000 m,
$\rho_{\mu}(1000)$, which are used in experiments as primary energy and
primary mass estimators, respectively.
We study individual events of
AGASA~\cite{AGASA_Eest} and of the Yakutsk extensive air shower array
(Yakutsk in what follows)~\cite{YakutskExperiment} with \textit{reconstructed}
energies above $8\times 10^{19}$~eV and measured muon content. We reject
the hypothesis that any of showers considered was initiated by a photon
primary at the 95\% confidence level (C.L.). We then derive as our main
result an upper limit of 36\% (at 95\%~C.L.) on the fraction
$\epsilon_\gamma$ of primary photons with \textit{original} energies above
$10^{20}$~eV (the difference between original and reconstructed energies
is discussed in Sec.~\ref{sec:data}).
The rest of the paper is organized as follows. In Sec.~\ref{sec:data} we
discuss the experimental data set which we use for our study. In
Sec.~\ref{sec:simulations}, the details of the simulation of the artificial
shower libraries and comparison of the simulated and real data are given.
This section contains the description of our method and the main
results. We discuss how robust these results are with respect to
changes in assumptions, to analysis procedure, and to variations in the
experimental data, in Sec.~\ref{sec:robustness}. In
Sec.~\ref{sec:comparison}, we discuss the differences between our
approach and previous studies, which allowed us to put a
significantly more stringent limit on the gamma-ray fraction. Our
conclusions are briefly summarized in Sec.~\ref{sec:conclusions}.
\section{Experimental data}
\label{sec:data}
AGASA was operating from 1990 to 2003 and consisted of 111
surface scintillation detectors (covering an area of about $100$~km$^2$)
and 27 muon detectors. The areas of the AGASA muon detectors
varied between 2.8 and 20~m$^2$. The detectors consisted of 14--20
proportional counters aligned under a shield of either 30~cm of iron or
1~m of concrete and were placed below or close to scintillation detectors.
The threshold energy was 0.5~GeV$/\cos\theta_\mu$ for muons with zenith
angle $\theta_\mu$~\cite{AGASAmu}. During 14 years of operation, AGASA had
observed 11 events with reported energies above $10^{20}$~eV and zenith
angles $\theta<45^\circ$~\cite{agasares,AGASAarrDir}. Among them, six
events had $\rho_{\mu}(1000)$ determined~\cite{AGASAmu}.
Yakutsk is observing CRs of highest energies since 1973,
with detectors in various configurations. With
{$\theta< 60^\circ$}, it has observed three events
above $10^{20}$~eV, all with measured muon content. Before 1978, only
one muon detector with the area of 8~m$^2$ and threshold energy
$0.7$~GeV$/\cos\theta_\mu$ was in operation. Later, it has been
replaced by six detectors with areas up to 36~m$^2$ and the threshold
energy of $1.0$~GeV$/\cos\theta_\mu$~\cite{Yakutsk_mu}.
In our study, we combine the AGASA and Yakutsk datasets, motivated
by the following. First, both datasets are obtained from
surface array experiments operated with similar plastic scintillation
detectors. Second, the energy estimation procedures of the two
experiments are compatible, within the reported systematic errors at
$\sim 10^{20}$~eV, if differences in the observational conditions are
taken into account~\cite{SakakiThesis}. Finally, the values of the CR
flux at $10^{20}$~eV reported by the two experiments are consistent
within their $1\sigma$ errors.
The shower energy estimated by an experiment (hereafter denoted as
$E_{\rm est}$) is in general different
from the true primary energy (denoted as $E_0$) because of natural
shower fluctuations, etc.
Moreover, the energy estimation algorithms used by surface-array
experiments normally assume that the primary is a proton. While the
estimated energy for nuclei depends only weakly on their mass number,
the difference between photons and hadrons is significant.
For photons, the effects of geomagnetic field~\cite{GMF} result in
directional dependence of the energy reconstruction. Thus,
the event energy reported by the experiment should be treated with
care when we allow the primary to be a photon. In this study we include
events with $E_{\rm est} \ge 8\times 10^{19}$~eV because of possible
energy underestimation for photon-induced showers; these events contribute
to the final limit, derived for $E_0>10^{20}$~eV, with different weights.
For AGASA, we use the events given in Ref.~\cite{agasares} that
pass the ``cut B'' defined in Ref.~\cite{AGASAmu}, that is having at
least one \footnote{We thank K.~Shinozaki for bringing a misprint in
Ref.~\cite{AGASAmu} to our attention: ``more than one'' was written
there.}
muon detector hit between 800~m and 1600~m from the shower axis. The
$\rho_\mu(1000)$ of the individual events can be read off from Fig.~2 of
Ref.~\cite{AGASAmu}. Yakutsk muon detectors have larger area and are more
sensitive both to weak signals far from the core and to strong signals for
which AGASA detectors might become saturated. This allowed the Yakutsk
collaboration to relax the cuts, as compared to AGASA, and to obtain
reliable values of $\rho_\mu(1000)$ using detectors between 400~m and
2000~m from the shower
axis~\cite{Knurenko,Yakutsk-muon-new}. Providing these cuts, six
AGASA and four Yakutsk events entered the dataset in our study (see
Table~\ref{events} for the event details).
\begin{table*}
\caption{\label{events}
Description of the individual events used in this work. Columns: (1),
event number; (2), experiment; (3), date of the event detection (in the
format dd.mm.yyyy); (4), the reported energy assuming a hadronic primary
(in units of $10^{20}$~eV); (5), the zenith angle (in degrees); (6) the
azimuth angle (in degrees, $\phi =0$ corresponds to a particle coming from
the South, $\phi =90^\circ$ -- from the West); (7) number of muon
detectors used to reconstruct muon density; (8) muon density at 1000~m from
the shower axis (in units of m$^{-2}$); (9), probability that this event
was initiated by a photon with $E>10^{20}$~eV; (10), probability that this
event was initiated by a non-photon with $E>10^{20}$~eV, assuming correct
energy determination. The sum $p_1^{(i)} +p_2^{(i)}$ gives the weight of
this event in the final limit on $\epsilon _\gamma $. The probability
that the primary had the energy $E<10^{20}$~eV is $1-p_1^{(i)}-p_2^{(i)}$.}
\begin{center}
\begin{ruledtabular}
\begin{tabular}{cccdddcddd}
$i$ & Experiment&Date & \multicolumn{1}{c}{$E_{\rm obs}$}
& \multicolumn{1}{c}{$\theta$} &
\multicolumn{1}{c}{$\phi$}&
\multicolumn{1}{c}{$n_{\rm det}$}&
\multicolumn{1}{c}{$\rho_{\mu}^{(i)}(1000)$}&
\multicolumn{1}{c}{$p_1^{(i)} $}& \multicolumn{1}{c}{$p_2^{(i)}$}\\
(1)&
(2)&(3)&
\multicolumn{1}{c}{(4)}&
\multicolumn{1}{c}{(5)}&
\multicolumn{1}{c}{(6)}&
\multicolumn{1}{c}{(7)}&
\multicolumn{1}{c}{(8)}&
\multicolumn{1}{c}{(9)}&
\multicolumn{1}{c}{(10)}\\
\hline
1 &AGASA & 10.05.2001 &2.46 & 36.5& 79.2 & 3& 8.9 &0.000 & 1.000\\
2 &AGASA & 03.12.1993 &2.13 & 22.9& 55.5 & 1& 10.7 &0.001 & 0.998\\
3 &AGASA & 11.01.1996 &1.44 & 14.2& 27.5 &$>1$&8.7 &0.013 & 0.921\\
4 &AGASA & 06.07.1994 &1.34 & 35.1&234.9 & 1& 5.9 &0.003 & 0.887\\
5 &AGASA & 22.10.1996 &1.05 & 33.7&291.6&$>1$&12.6 &0.000 & 0.581\\
6 &AGASA & 22.09.1999 &1.04 & 35.6&100.0 &$>1$&9.3 &0.000 & 0.565\\
7 &Yakutsk& 18.02.2004 &1.60 & 47.7&180.8 & 5& 19.6 &0.000 & 0.876\\
8 &Yakutsk& 07.05.1989 &1.50 & 58.7&230.6 & 5& 11.8 &0.000 & 0.868\\
9 &Yakutsk& 21.12.1977 &1.10 & 46.1&346.8 & 1& 8.0 &0.000 & 0.645\\
10&Yakutsk& 02.05.1992 &0.85 & 55.7&163.0 & 5& 4.7 &0.000 & 0.303\\
\end{tabular}
\end{ruledtabular}
\end{center}
\end{table*}
\section{Simulations and results}
\label{sec:simulations}
In order to interpret the data, for each of the ten events, we generated a
shower library containing 1000 showers induced by primary photons
\footnote{For the illustration in Fig.~\ref{fig:event3}, 500
proton-induced showers were simulated and processed in a similar way.}.
Thrown energies
$E_0$ of the simulated showers were randomly selected (see below the
discussion of the initial spectra) between $5\times 10^{19}$~eV and $5\times
10^{20}$~eV to take into account possible deviations of $E_{\rm est}$ from
$E_0$. The arrival directions of the simulated showers were the same as
those of the corresponding real events. The simulations were performed
with CORSIKA~v6.204~\cite{Heck:1998vt}, choosing
QGSJET~01c~\cite{Kalmykov:1997te} as high-energy and
FLUKA~2003.1b~\cite{fluka} as low-energy hadronic interaction model.
Electromagnetic showering was implemented with EGS4~\cite{Nelson:1985ec}
incorporated into CORSIKA. Possible interactions of the primary photons
with the geomagnetic field were simulated with the PRESHOWER option of
CORSIKA~\cite{Homola:2003ru}. As discussed in
Sec.~\ref{sec:robustness:models}, this choice of the interaction models
results in a conservative limit on gamma-ray primaries. As suggested
in Ref.~\cite{Thin}, all simulations were performed with thinning level
$10^{-5}$, maximal weight $10^6$ for electrons and photons, and $10^4$ for
hadrons.
For each simulated shower, we determined $S(600)$ and $\rho_{\mu}(1000)$.
For the calculation of $S(600)$, we used the detector response functions
from Refs.~\cite{Sakaki,YakutskGEANT}.
For a given arrival direction, there is one-to-one correspondence
between $S(600)$ and the quantity called estimated energy,
$E_{\rm est}$. The relation is determined by
the standard analysis procedure of the two
experiments~\cite{AGASA_Eest,Yakutsk_Eest}.
This allows us to select simulated showers compatible with the
observed ones by the signal density. The quantity $S(600)$ is
reconstructed not precisely. In terms of estimated energy,
for AGASA events, the reconstructed energies are
are distributed with a Gaussian in $\log
\left(E_{\rm est}/\bar E_{\rm rec}\right)$;
the standard deviation of $E_{\rm est}$ is
$\sigma\approx 25\%$ \cite{SakakiThesis}. For Yakutsk events, the
corresponding $\sigma$ has been determined event-by-event and is typically
30--45\% \cite{PravdinICRC2005}.
To each simulated shower, we assigned a weight $w_1$ proportional to
this Gaussian probability distribution in $\log E_{\rm est}$ centered at
the observed energy $\bar E_{\rm rec}=E_{\rm obs}$.
Additionally, each simulated shower was weighted with $w_2$ to reproduce
the thrown energy spectrum $\propto E_0^{-2}$ which is typically predicted
by non-acceleration scenarios (see Sec.~\ref{sec:robustness:spectrum} for
a discussion of the variations of the spectral index). For each of the ten
observed events, we obtained a distribution of muon densities $\rho _\mu
(1000)$ representing photon-induced showers compatible with the observed
ones by $S(600)$ and arrival directions. To this end, we calculated $\rho
_\mu (1000)$ for each simulated shower by making use of the same muon
lateral distribution function as used in the analysis of real
data~\cite{AGASAmu,Yakutsk_mu}. To take into account possible experimental
errors in the determination of the muon density, we replaced each
simulated $\rho _\mu (1000)$ by a distribution representing possible
statistical errors (50\% and 25\% Gaussian for AGASA cut
B~\cite{AGASAmu50} and Yakutsk~\cite{Yakutsk-muon-new}, respectively). The
distribution of the simulated muon densities is the sum of these Gaussians
weighted by $w_1w_2$.
A typical distribution of simulated $\rho _\mu (1000)$ is given in
Fig.~\ref{fig:event3},
for gamma-
and proton-induced simulated showers compatible with the event 3 by
$S(600)$ and the arrival direction. We will see below that this particular
event has the largest probability of gamma interpretation among all ten
events in the data set; still the proton interpretation looks perfect for
it. This is the case for all events except event 7, which has too high
$\rho _\mu (1000)$ for a proton; possible nature of its primary particle will
be discussed elsewhere.
To estimate the allowed fraction $\epsilon_\gamma $ of primary
photons among CRs with $E_0>10^{20}$~eV,
we compare, for each observed event, two possibilities: (i)~that it was
initiated by a photon primary with $E_0>10^{20}$~eV and (ii)~that it was
initiated by any other primary with $E_0>10^{20}$~eV for which the
experimental energy estimation works properly.
Let us consider the $i$th observed event. Denote by $M$
the weighted number of showers contributed to the $\rho _\mu
(1000)$ distribution for the simulated photon-induced showers compatible
with the $i$th event by arrival direction and $S(600)$ (throughout
this paragraph, the weighted number is the sum of corresponding weights,
that is $M$ is the sum of weights of all 1000 showers simulated for the
$i$th event). Some of the simulated showers contributed to the part of the
distribution for which $\rho _\mu (1000)>\rho _\mu ^{(i)}(1000)$, where
$\rho _\mu ^{(i)}(1000)$ is the observed value for this event. The weighted
number of these showers is $M'$. Some part $l$ of this $M'$
corresponds to showers with
$E_0>10^{20}$~eV, the rest
($M'-l$) to $E_0<10^{20}$~eV. The probability
$p_1^{(i)}$ of case (i) is $p_1^{(i)}=l/M$, while the probability that the
event is consistent with a photon of $E_0<10^{20}$~eV is
$p_1^{\prime(i)}=(M'-l)/M$. Moreover, the probability that the event is
described by any other primary is
$1-p_1^{(i)}-p_1^{\prime(i)}=1-M^\prime/M$. We assume that the
experimental energy estimation works well for non-photon primaries and
determine the fraction $\xi$ of events with $E>10^{20}$~eV simply from the
Gaussian $\log(E_{\rm est})$ distribution, so the probability of the case
(ii) is $p_2^{(i)}=\xi(1-M^\prime/M)$. The values of $p_{1,2}^{(i)}$
are presented in Table~\ref{events}. Note that $p_1^{(i)}+p_2^{(i)}<1$
because of a non-zero probability that a simulated shower is initiated by
a primary with $E_0<10^{20}$~eV. This happens especially for events with
reported energies close to $10^{20}$~eV and reduces considerably the
effective number of events contributing to the limit on $\epsilon_\gamma$:
since we are interested in the limit for $E_0>10^{20}$~eV only, each event
contributes to the result with the weight $(p_1^{(i)}+p_2^{(i)})$.
Inspection of Table~\ref{events} demonstrates that the total effective
number of events with $E_0>10^{20}$~eV (the sum of $p_1^{(i)}$ and
$p_2^{(i)}$ over all ten events) is 7.67.
If the $i$th primary particle was a photon with $E_0>10^{20}$~eV with the
probability $p_1^{(i)}$ and a non-photon with $E_0>10^{20}$~eV with the
probability $p_2^{(i)}$, one can easily calculate the probability
${\cal P}(n_1,n_2)$ to have $n_1$ photons and $n_2$ non-photons in the set
of $N=10$ observed events ($0\le n_1+n_2 \le N$, the rest $N-n_1-n_2$
events have $E_0<10^{20}$~eV). From the set of $N$ events, one should take
all possible non-overlapping subsets of $n_1$ and $n_2$ events and sum up
probabilities of these realisations (since $p_{1,2}^{(i)}\ne
p_{1,2}^{(j)}$, these probabilities are different for different
realisations with the same $n_1$ and $n_2$).
Now, suppose that the fraction of the primary photons at $E_0>10^{20}$~eV
is $\epsilon _\gamma $. Then, the probability to have $n_1$ photons and
$n_2$ non-photons at $E_0>10^{20}$~eV is $
\epsilon _\gamma ^{n_1} \left(1-\epsilon _\gamma
\right)^{n_2}
$,
and the probability that the observed muon densities were obtained with a
given $\epsilon _\gamma $ is
$$ {\cal P}(\epsilon _\gamma )=\sum\limits_{n_1,n_2=0}^N
{\epsilon_\gamma} ^{n_1} \left(1-\epsilon_\gamma \right)^{n_2}
{\cal P}(n_1,n_2) \,
$$
(cf.\ Ref.~\cite{Homola} for a particular case $n_1+n_2=N$; note that the
combinatorial factor is included in the definition of ${\cal P}(n_1,n_2)$).
The cases $n_1+n_2<N$ reflect the possibility that some of
the $N$ events correspond to primaries with $E_0<10^{20}$~eV. In our
case, the probability ${\cal P}(\epsilon_\gamma)$ is a monotonically
decreasing function of $\epsilon_\gamma$. Thus the upper limit on
$\epsilon_\gamma$ at the confidence level $\alpha'$ is obtained by solving
the equation ${\cal P}(\epsilon_\gamma )=1-\alpha'$. For our dataset, the
95\%~C.L.\ upper limit on the photon fraction is $\epsilon_\gamma<0.33$.
The limit on $\epsilon_\gamma$ is rather weak compared to the
individual values of $p_1^{(i)}$ because of the small number of
observed events.
However, some of the photon-induced showers may escape from our
study because they may not pass the muon measurement quality cuts or
their estimated energy is below $8\times 10^{19}$~eV.
Possible reasons for an underestimation of the energy may be either
the LPM effect~\cite{LPM} or substantial attenuation of gamma-induced
showers at large zenith angles.
To estimate the fraction of these
``lost'' events, we have simulated 1000 gamma-induced showers for each
experiment with arrival directions distributed according to the
experimental acceptance. We find that the fraction of the ``lost'' events
is $\sim 3.5\%$ for AGASA and $\sim 15\%$ for Yakutsk. The account
of these fractions, weighted with the relative exposures of both
experiments, results in the final upper limit,
$$
\epsilon_\gamma <36\% ~~(95\%~{\rm C.L.}).
$$
In Fig.~\ref{fig:limits}, we present our limit on $\epsilon_\gamma $
(AY) together with previously published limits on the same quantity.
Also, typical theoretical predictions are shown for the superheavy
dark-matter, topological-defect
and $Z$-burst models.
Our limit on $\epsilon_\gamma $ is currently the
strongest one at $E_0>10^{20}$~eV. It disfavours some of the
theoretical models such as the $Z$-burst and superheavy dark-matter
scenarios.
\section{Robustness of the results}
\label{sec:robustness}
In this section, we discuss systematic uncertainties of our
limit that are related to the air shower simulations, to the data
interpretation and to selection cuts.
\subsection{Systematic uncertainty in the $S(600)$ and energy
determination}
\label{sec:robustness:energy}
The systematic uncertainty in the absolute energy
determination is 18\% and 30\% for AGASA~\cite{AGASA_Eest} and
Yakutsk~\cite{YakutskExperiment}, respectively. These systematic errors
originate from two quite different sources: (a)~the measurement of
$S(600)$ and (b)~the relation between $S(600)$ and primary energy.
The probabilities $p_1^{(i)}$ that a particular event may allow for a
gamma-ray interpretation are not at all sensitive to the
$S(600)$-to-energy conversion because we select simulated events by
$S(600)$ and not by energy. These probabilities may be affected by
relative systematics in determination of $\rho _\mu (1000)$ and $S(600)$.
On the other hand, in the calculation of $p_2^{(i)}$ we assumed that the
experimental energy determination is correct for non-photon primaries; the
values of $p_2^{(i)}$ and the effective number of events contributing to
the limit on $\epsilon _\gamma $ at $E_0>10^{20}$~eV would change if the
energies are systematically shifted. In our case (all $p_1^{(i)}\approx 0$),
the reported value of $\epsilon _\gamma $ would be applicable to the
shifted energy range in that case.
Thus, the 95\%~C.L. conclusion that none of the ten events considered here
was initiated by a photon is robust with respect to any changes in the
$S(600)$-to-energy conversion. As for the limit on $\epsilon _\gamma $ we
report, instead of $E_0>10^{20}$~eV, it would be applicable to a different
energy range if all experimental energies are systematically shifted. One
should note that theoretical predictions, e.g. the curves shown in
Fig.~\ref{fig:limits}, would also change because they are normalised to
the observed AGASA spectrum.
\subsection{Interaction models and simulation codes}
\label{sec:robustness:models}
Our simulations were performed entirely in the CORSIKA framework, and any
change in the interaction models or simulation codes, which affects either
$S(600)$ or $\rho _\mu (1000)$, may affect our limit. We have studied the
model dependence of our results by comparing different low- and high-energy
hadronic interaction models (GHEISHA~\cite{GHEISHA} versus FLUKA, SIBYLL
2.1~\cite{SIBYLL} versus QGSJET). Our
result is quite stable with respect to these changes. In all cases,
individual values of $p_1^{(i)}$
are always close to zero, thus the limit on $\epsilon_\gamma$ is
not affected. The change of the low energy model does not at all affect the
reported values. In use of SIBYLL compared with QGSJET, $\rho_\mu(1000)$
is $\sim 20\%$ smaller for photon-
induced showers. While $S(600)$ is almost unchanged,
events in our dataset are better explained by showers initiated by heavier
nuclei and the probability of photon-induced showers is even smaller. A
similar effect is expected for the coming interaction model
QGSJET~II~\cite{QGSJET-II}.
We also performed simulations with the help of
the hybrid code~\cite{hybrid} which reproduced the CORSIKA results with
high accuracy.
Another popular simulation code, AIRES~\cite{Aires}, differs from
CORSIKA mainly in the low-energy hadronic interaction model (which
is fixed in AIRES to be the Hillas splitting algorithm), hence we hope
that simulations with AIRES would not significantly affect our results.
Comparison with AIRES will be presented elsewhere.
The values presented here were
obtained for the standard parameterization of the photo-nuclear cross
section given by the Particle Data Group~\cite{PDG} (implemented as default
in CORSIKA). The muon content of gamma-induced showers is in principle
sensitive to the extrapolation of the photonuclear cross section to high
energies. The hybrid code~\cite{hybrid} allows for easy variations of the
cross section; we checked that the results are stable for various
reasonable extrapolations, in agreement with Ref.~\cite{RissePgamma}.
\subsection{Primary energy spectrum}
\label{sec:robustness:spectrum}
For our limit, we used the primary photon spectrum $E_0^{-\alpha }$ for
$\alpha =2$. While the individual probabilities $p_{1,2}^{(i)}$ are not
affected by the change of the spectral index $\alpha$ because the
simulated events are selected by $S(600)$ anyway, the value of $\alpha $
changes the fraction of ``lost'' photons and, correspondingly, the final
limit on $\epsilon _\gamma $. Variations of $1\le \alpha \le 3$ result in
the photon fraction limits between 36\% and 37\% (95\%~C.L.).
\subsection{Width of the $\rho _\mu $ distribution}
\label{sec:robustness:width}
Clearly, the rare probabilities of high values of $\rho _\mu (1000)$ in
the tail of the distribution for primary photons depend on the width of
this distribution. The following sources contribute to this width:
\begin{itemize}
\item
variations of the primary energy compatible with the observed $S(600)$
(larger energy correspond to larger muon number and $\rho _\mu (1000)$);
\item
physical shower-to-shower fluctuations in muon density for a given energy
(dominated by fluctuations in the first few interactions, including
preshowering in the geomagnetic field);
\item
artificial fluctuations in $S(600)$ and $\rho _\mu (1000)$ due to thinning;
\item
experimental errors in $\rho _\mu (1000)$ determination.
\end{itemize}
While the first two sources are physical and are fully controlled by the
simulation code, the variations of the last two may affect the results.
\subsubsection{Artificial fluctuations due to thinning}
\label{sec:robustness:thinning}
It has been noted in Ref.~\cite{Badagnani} that
the fluctuations in $\rho _\mu (1000)$ due to thinning may affect strongly
the precision of the composition studies.
For the thinning parameters we use, the relative size of these fluctuations
is~\cite{our-thinning} $\sim 10\%$ for $\rho _\mu (1000)$ and $\sim
5\%$ for $S(600)$. Thus with more precise simulations, the distributions of
muon densities should become more narrow, which would reduce the
probability of the gamma-ray interpretation of each of the studied events even
further.
\subsubsection{Experimental errors in $\rho _\mu (1000)$
determination}
\label{sec:robustness:muon-errors}
The distributions of $\rho _\mu (1000)$ we use accounted for
the error in experimental determination of this quantity. The size of the
errors was taken from the original experimental
publications~\cite{Yakutsk-muon-new,AGASAmu50}. In principle, this error
depends on the event quality and on the muon number itself, which is lower
for simulated gamma-induced showers than for the observed ones. However,
e.g.\ Ref.~\cite{AGASAmu} states that for the AGASA cut A (two or more
muon detectors), the error is 40\%, lower than 50\% we
use~\cite{AGASAmu50}. Note that Ref.~\cite{AGASAmu} discusses muon
densities as low as 0.04~m$^{-2}$ and even 0~m$^{-2}$, much lower than
$\sim 1$~m$^{-2}$ typical for our simulated gamma-induced events. Still,
we tested the stability of our limit by taking artificially high values of
experimental errors in muon density: 100\% for AGASA and 50\% for
Yakutsk. The limit on $\epsilon _\gamma $ changes to 37\% (95\%~C.L.) in
that case.
\subsection{Data selection cuts}
\label{sec:robustness:cuts}
Since all events in the data set are unlikely to be initiated by
primary photons (all $p_1^{(i)}\approx 0$), the limit on $\epsilon _\gamma
$ is determined by statistics only and is affected if the number of events
is changed. Here, we discuss possible variations of the data set
corresponding to more stringent quality cuts which reduce the event number
and weaken the limit.
\subsubsection{Zenith angle}
\label{sec:robustness:ZA}
All Yakutsk events in the data set have zenith angles $45^\circ<\theta
<60^\circ$, so the cut $\theta<45^\circ$ imposed by AGASA reduces the
sample to six AGASA events which results in the limit $\epsilon _\gamma
<50\%$ (95\%~C.L.). One should note however that AGASA muon detectors are
not sensitive to inclined showers, which is not the case for Yakutsk.
\subsubsection{Core inside array}
\label{sec:robustness:event7}
Another cut imposed on the AGASA published dataset is the location of the
core inside array. The event number 7 does not satisfy this criterion; its
exclusion from the data set results in $\epsilon _\gamma <40\%$
(95\%~C.L.).
\subsubsection{More than one muon detector}
Reconstruction of the muon density at 1000~m from a single muon detector
reading requires extrapolation of the lateral distribution function with
an averaged slope. Though it is well-studied, the data points
corresponding to events with a single muon detector hit might be
considered less reliable than those with two or more hits. With the
account of the events with two or more hits only, we are left with seven events (four AGASA and
three Yakutsk) which weakens the 95\%~C.L.\ limit to $\epsilon _\gamma
<48\%$.
\section{Comparison with other studies}
\label{sec:comparison}
Some of the previous studies used the AGASA~\cite{AGASAmu,Homola} and
Yakutsk~\cite{Knurenko} muon data to limit the gamma-ray primaries at high
energies. Our results differ from the previous ones not only because we
join the data sets of the two experiments. Two major distinctive
features of our approach allowed us to put the stringent limit:
\begin{itemize}
\item
both $\rho _\mu (1000)$ and $S(600)$ were tracked for simulated showers
within framework of a {\em single } simulation code (CORSIKA in our case);
\item
each event was studied individually, without averaging over arrival
directions.
\end{itemize}
In Refs.~\cite{AGASAmu,Knurenko}, no conclusion was derived about
$\epsilon _\gamma $ at $E>10^{20}$~eV, and the data points corresponding
to highest-energy events were found to be quite close to the gamma-ray
domain. To our opinion, the main source of this effect is averaging over
arrival directions which introduced additional fluctuations for gamma-ray
primaries due to direction-dependent preshowering (see
Fig.~\ref{fig:2events} for an illustration).
In Ref.~\cite{Homola} which discussed the same six AGASA events,
all simulated showers for an event with the observed energy $E_{\rm obs}$
had energies
$1.2 E_{\rm obs}$ (up to the energy reconstruction uncertainty of 25\%).
This conversion had been obtained as the average over $\theta<36^\circ$ in
Ref.~\cite{AGASAmu} using AIRES simulation code~\cite{Aires}. That is, not
only the average results were applied to individual showers, but
effectively muon densities were simulated with CORSIKA while energies --
with AIRES, though the two codes result in a systematically different
relations between energy and $S(600)$. Artificially high energies resulted
in higher, closer to observed, muon densities for simulated photonic
showers. In our event-by event simulations with CORSIKA, the energies of
gamma-ray primaries whose $S(600)$ were compatible to observed values,
were not higher by a factor 1.2, but in fact even lower than $E_{\rm obs}$
for some of the events: besides the difference in simulation codes, this
is partially due to non-uniform distribution of the highest-energy AGASA
events on the celestial sphere~\cite{AGASAarrDir,NorthSouth} which makes
the usage of averaged energies poorly motivated.
The impact of two other sources of
difference between our approach and that of Ref.~\cite{Homola} is less
important for the final result: (i)~Ref.~\cite{Homola} does not account
for the ``lost'' photons and (ii)~the detector error is applied in our
study to the simulated events while in Ref.~\cite{Homola} -- to the
observed ones.
The difference with Ref.~\cite{Homola} is illustrated in
Fig.~\ref{fig:DifferentEnergies}, where $\rho _\mu (1000)$ is plotted
versus $E_0$ for simulated gamma-induced showers with the arrival
direction of the event \#1. For simulated events compatible with the real
event by $S(600)$, the average point is shown together with one sigma
error bars. Horizontal error bars correspond to variations in $E_0$
compatible with $S(600)$. Vertical error bars include variations in
simulated $\rho _\mu (1000)$ and 50\% detector error. The point
corresponding to simulated showers with $E_0=1.2E_{\rm obs}$ from
Ref.~\cite{Homola} has a larger $\rho _\mu (1000)$.
Horizontal error bars correspond to the energy reconstruction accuracy.
Vertical
error bars include variations in simulated $\rho _\mu (1000)$ reported in
Ref.~\cite{Homola} and 40\% detector error applied to the observed value,
added in quadrature.
We see that the main source of the
disagreement is in the values of $E_0$ which push, for the case of
Ref.~\cite{Homola}, the simulated muon densities closer to the observed
one.
\section{Conclusions}
\label{sec:conclusions}
To summarize, we have studied the possibility that the highest-energy
events observed by the AGASA and Yakutsk experiments were initiated by
primary photons. Comparing the observed and simulated muon content of
these showers, we reject this possibility for each of the ten events
at $E>8\cdot 10^{19}$~eV at least at the 95\% C.L. An important
ingredient in our study is the careful tracking of differences between
the original and reconstructed energies. This allows us to put an
upper bound of 36\% at 95\% C.L.\ on the fraction $\epsilon_\gamma $
of primary photons with original energies $E_0>10^{20}$~eV, assuming
an isotropic photon flux and $E_0^{-2}$ spectrum. This limit is the
strongest one up to date. It strongly disfavors the $Z$-burst and
constrains severely superheavy dark-matter models. The method that we
have used is quite general and may be applied at other energies and to
other observables.
We are indebted to
M.~Kachelrie\ss,
K.~Shinozaki and M.~Teshima
for numerous helpful discussions and collaboration at initial stages of
this work.
We thank L.~Bezroukov, V.~Bugaev, R.~Engel, D.~Heck, A.~Ringwald, M.~Risse,
V.~Rubakov, D.~Semikoz and P.~Tinyakov for helpful discussions and
comments on the manuscript. This study was
performed within the INTAS project 03-51-5112. We acknowledge also support
by fellowships of the Russian Science Support Foundation and of the
Dynasty foundation (D.G.\ and S.T.), by the grants NS-2184.2003.2 (D.G.,
G.R.\ and S.T.), NS-1782.2003.2, RFFI 03-02-16290 (L.D., G.F., E.F.\ and
T.R.), NS 748.2003.2, RFFI 03-02-17160, RFFI 05-02-17857, FASI
02.452.12.7045 (A.G., I.M., M.P.\ and I.S.). G.R.\ and S.T.\ thank the
Max-Plank-Institut f\"ur Physik (Munchen), where a significant part of
this work was done, for warm hospitality. Computing facilities of the
Department of Theoretical Physics, Institute for Nuclear Research
(Moscow), were used to perform the simulations of air showers.
|
Title:
Sersic Properties of Disc Galaxy Mergers |
Abstract: Sersic parameters characterising the density profiles of remnants formed in
collision-less disc galaxy mergers are obtained; no bulge is included in our
simulations. For the luminous component we find that the Sersic index is n ~
(1.5,5.3) with <n> ~ 3 +/- 1 and an effective radius of R_e ~ (1.6,12.9) kpc
with <R_e> ~ 5 +/- 3 kpc. A strong correlation of n with the central projected
density I_0 is found [n ~ I_0^(-0.14)] which is consistent with observations.
No positive linear correlation between the size (R_e) and structure (n) of our
remnants is found; we do not advocate the existence of this. The photometric
plane (PHP) of the luminous component [n ~ R_e^(0.05) I_0^(0.15)] agrees well,
within the uncertainties and the assumption of a constant mass-to-light ratio,
with those observationally determined for ellipticals. We found that the
surface defined by Sersic parameters (n, R_e, mu_0) in log-space is not a true
plane, but a pseudo-plane with a small curvature at low values of n owed to
intrinsic properties of the Sersic model. The dark haloes of the remnants have
a 3-dimensional Sersic index of <n> ~ 4 +/- 0.5 that are smaller than the ones
obtained for dark haloes in LCDM cosmologies <n> ~ 6 +/- 1. A tight dark Sersic
``plane'' (DSP) is also defined by the parameters of the remnants haloes with n
\~ r_e^(0.07) rho_0^(0.10). We conclude that collision-less merger remnants of
pure disc galaxies have Sersic properties and correlations consistent with
those of observed in early-type galaxies and local remnants. It seems that a
``primordial'' bulge in spirals is not a necessary condition to form bona fide
ellipticals on grounds of the Sersic properties of remnants.
| https://export.arxiv.org/pdf/astro-ph/0601412 |
\date{Accepted ------. Received ------; in
original form ------}
\pagerange{\pageref{firstpage}--\pageref{lastpage}} \pubyear{2004}
\label{firstpage}
\begin{keywords}
galaxies: kinematics and dynamics -- galaxies: formation -- galaxies:
fundamental parameters -- galaxies: interactions -- galaxies:
elliptical -- methods: $N$-body simulations.
\end{keywords}
\section{Introduction}
Hierarchical galaxy formation theory (e.g. Cole~et~al.~2000,
De~Lucia~et~al.~2005, Bower~et~al.~2005) considers that early-type galaxies
have an accretion/merger origin, as was originally suggested by Toomre (1977).
Observational (e.g. Schweizer~1998, Struck~2005, Rothberg \& Joseph~2006, Kaviraj~et~al.~2006) and theoretical (e.g. Naab \& Burkert 2003,
Meza~et~al.~2005, Naab~et~al.~2005) evidence supports this picture
although several topics remain
unsolved (e.g. Peebles~2002, Tantalo \&
Chiosi~2004).
Early-type galaxies show several correlations among
their colours, luminosities,
velocity dispersions, effective radii and surface brightness (e.g.
Baum~1990, Faber~\&~Jackson~1976, Kormendy~1977, Djorgovski~\& Davis~1987,
Dressler~et~al.~1987, Bernardi~et~al.~2003). These
correlations provide
constraints to any theory of formation and evolution of
these galaxies.
Furthermore, their properties
are linked with the
distribution of luminous and dark matter, that would be
important when comparing with models of formation of
elliptical galaxies.
Observational studies [e.g. Caon, Capaccioli \& D'Onofrio 1993 (CCD93),
Graham \& Colles~1997, Binggeli \& Jerjen 1998, D'Onofrio~2001 (D01),
Trujillo~et~al.~2004] have found that the surface
brightness density profiles
of early-type galaxies are better described by a S\'ersic (1968)
$R^{1/n}$--profile than the classical de~Vaucouleurs (1948)
$R^{1/4}$--profile. The index $n$ is directly related with
the cur\-va\-ture and
``concentration'' of the light profile (Trujillo, Graham \& Caon~2001).
Several observational relationships have been found between
the index $n$ and, for example, the
total luminosity ($L$),
effective radius ($R_{\rm e}$)
and central velocity dispersion (e.g. CCD93, Prugniel \& Simien~1997,
Graham \& Guzm\'an~2003).
Also, it has been found a linear relation among
$\log n$, $\log R_{\rm e}$ and
$\mu_0$ (central brightness) termed the Photometric Plane (PHP)
for early-type galaxies [e.g. Khosroshahi~et~al.~2000 (K00), Graham~2002],
analogous to the Fundamental Plane (Djorgovski \& Davis 1987,
Dressler~et~al.~1987). Recently,
Rothberg~\&~Joseph (2004, RJ04) have found that nearby merger remnants
have a peak in the $n$-distribution
at $n\approx 2$ with most values in the range of $1<n<6$, and in
some cases it is found that $n > 8$.
On other hand, theoretical studies of S\'ersic properties of merger
remnants have appeared recently. For example,
G\'onzalez-Garc\'{\i}a~\&~Balcells (2005, GGB)
and Naab~\&~Trujillo~(2005, NT) find in
collision-less simulations that
bulge-less progenitors lead to ranges of
$n\! \in\! (2.4,3.2)$ and $(1.2,3.1)$, respectively; when
a single S\'ersic function is used to fit the entire remnant.
For progenitors with a bulge component they obtain
about the same range of S\'ersic index,
$n\! \in \!(3,8)$.
Since ``bona fide'' ellipticals have values $n\gta 4$, they
reach the conclusion
that collision-less merger remnants of pure disc galaxies do not
lead to concentrations, indicated by $n$, similar to those
found in intermediate or giant elliptical galaxies (e.g. Graham~et~al.~1996).
The above findings suggest that a
primordial bulge in spirals is a necessary condition to
form bona fide ellipticals in the hierarchical merging scenario.
However, we show below, collision-less mergers of pure discs
can cover the range of observed values of the shape parameter $n$,
and can reproduce adequately other observational correlations.
S\'ersic model in a de-projected form has been recently used
to represent the dark matter distribution in $\Lambda$CDM haloes
(Navarro~et~al.~2004,
Merritt~et~al.~2005, Graham~et~al.~2005, Prada~et~al.~2005), in order
to have a better estimation of the inner asymptotic logarithmic derivative.
A mean value of a 3D S\'ersic
index $\approx 6$, with a scatter of $\approx 1$, has been found
in these works.
So it is of interest to determine the three-dimensional S\'ersic
parameters that characterise our remnants.
In this work, we study the structural properties of remnants
as provided by fitting a S\'ersic profile to their luminous and dark
mass distribution.
The paper has been organised as follows: in $\S$\ref{sec:model} we present a
summary of the properties of our progenitors, some details
of the simulations performed, as well as some basic characteristics
of S\'ersic profile;
both projected and deprojected. In $\S$\ref{sec:results}
we present distributions and
correlations, in two and three-dimensions, found among the different
S\'ersic parameters for our remnants, and compare them with observations.
S\'ersic properties of the dark haloes of the remnants are
determined, some correlations presented, and
compared with those obtained in cosmological simulations.
Some final comments are given in $\S$\ref{sec:discussion}
and a summary of our conclusions.
\section{Simulations and S\'ersic Functions}\label{sec:model}
\subsection{Galaxy models}
The galaxy models used in this work have been already described in Aceves \&
Vel\'azquez (2005) and follow the method outlined by
Shen, Mo \& Shu (2002) to
obtain the global properties of the discs, once the haloes properties are
known. Our numerical galaxies do \emph{not} include a bulge-like component.
The dark haloes
follow a modified NFW (Navarro, Frenk \& White~1997) model with an exponential
cutoff. The discs have a typical exponential density profile, and satisfy
the Tully-Fisher relation at redshift $z\!=\! 1$; roughly a look-back time of
$8\,$Gyr in a $\Lambda$CDM cosmology with Hubble parameter $h=0.7$.
Only discs satisfying the Efstathiou, Lake \& Negroponte~(1982) stability
criterion were used.
In this work, an additional simulation to those reported in Table~1 of
Aceves \&~Vel\'azquez
(2005) has been done. This is a merger from the resulting remnants of
$M01$ and $M05$, label $MM$.
All simulations were carried out using a parallel version of {\sc
gadget}-1.1 code, a tree base code (Springel, Yoshida \& White 2001), and
evolved for $\approx \! 8\,$Gyr with conservation of energy better than
$0.25$ percent.
\subsection{Density Profiles}
We fit only S\'ersic profiles to our merger remnants; no bulge-disc
decomposition is attempted since progenitors lack any bulge component.
The S\'ersic surface luminous-mass density profile is given by
\begin{equation}
\Sigma(R) = \Sigma_0 \, {\rm e}^{ -b (R/R_{\rm e})^{1/n}}\;,
\label{eq:sersic}
\end{equation}
where $R$ is the projected spherical radius, $R_{\rm e}$ is the effective
radius, $n$ the index of the profile, $b=b(n)\approx 2n - 0.324$ (Ciotti
\& Bertin 1999) and $\Sigma_0$ the central surface density.
Index $n$ is associated with the curvature and the concentration of the
profile (Trujillo, Graham \& Caon~2001);
$n\!=\!1$ corresponds to an exponential
profile while the classical de Vaucouleurs (1948)
profile is obtained for $n=4$.
The accumulated projected luminous mass, $M_{\rm
L}(R)$, is given by
\begin{equation}
M_{\rm L}(R)= \int_0^R \Sigma(R) {\rm d}(\pi R^2) = \frac{2\pi n \gamma(\alpha,x)}{b^{2n}}\, \Sigma_0 R^2_{\rm e} \,,
\end{equation}
where $\alpha \equiv 2n$, $x \equiv b(R/R_{\rm e})^{1/n}$, and
$\gamma (\alpha,x)$ is the incomplete gamma function.
The total projected luminosity mass is given by
\begin{equation}
M_{\rm L} = \frac{2\pi n}{b^{2n}} \Gamma(2n) \, \Sigma_{\rm 0} R^2_{\rm e}\, ,
\label{eq:Lmass}
\end{equation}
being $\Gamma(\alpha)$ the complete gamma function.
A summary of S\'ersic projected profile properties
is given by Graham \& Driver (2005). When comparing our simulations with
observations we assume a constant mass-to-light ratio, so that $\Sigma
\!\propto \! I$; where $I$ refers to the surface brightness.
The three-dimensional (3D) S\'ersic profile
is
\begin{equation}
\rho(r) = \rho_{\rm 0} \, {\rm e}^{ -d (r/r_{\rm e})^{1/n}}\;,
\label{eq:3Dsersic}
\end{equation}
where $r$ is the spatial radius, $d\approx 3n-1/3+0.005/n^2$
(Graham~et~al.~2005) such that
$r_{\rm e}$ is the half-mass spatial radius. The total mass is determined
from
\begin{equation}
M_{\rm t} = \frac{4\pi n}{d^{3n}} \Gamma(3n) \, \rho_0 r^3_{\rm e} \,.
\label{eq:3Dmass}
\end{equation}
S\'ersic parameters for
the luminous component were computed along 400
different random line-of-sights. To each projection a circularly averaged
density profile $\Sigma(R)$ was determined, and a S\'ersic profile
(\ref{eq:sersic}) fitted by $\chi^2$--minimisation using the
Levenberg-Marquardt method (Press~et~al.~1992) to obtain $\{n,R_{\rm
e},\Sigma_0\}$. S\'ersic parameters for dark haloes are obtained by a similar
procedure, but using equation (\ref{eq:3Dsersic}).
\subsubsection{Fitting Range}
The fitting set of parameters depend on the
methodology used to obtain them. In particular, there have been indications
that these parameters depend on both
the covered range of surface brightness range
(e.g. Capaccioli, Caon \& D'Onofrio 1992) and the spatial
radial interval for fitting (e.g. Kelson~et~al.~2000).
Also, the determination of fitted parameters degrades when
the inner parts of a galaxy are not well considered. For example, the index $n$
tends more to be a representation of the outer slope of the profile than of
the curvature of the luminosity distribution (Graham~et~al.~1996).
The treatment and quality of data has also an effect on the fitted parameters.
For example, CCD93 obtain higher values of $n$ for NGC~4406, NGC~4552 and
NGC~1399 (14.9, 13.9, 16.8) in comparison with D01 ($6.5$, $7.2$, $6.1$).
We have considered two radial intervals for our fits in
order to asses their effect on the S\'ersic parameters.
The first radial interval, $I_1$, is taken from our numerical resolution value
$\xi_{\rm i}=100\,$pc to the outer radius $\eta_{95}$, which encloses 95 percent of the projected luminous mass and is
determined directly from the simulations; thus,
$I_1=[\xi_{\rm i},\eta_{95}]$.
The second one, $I_2$, uses another inner point at
$\xi_{\rm f}=10\xi_{\rm i}$,\footnote{ \footnotesize For reference, in a $\Lambda$CDM cosmology with $h=0.7$ we have
that $1''=464\,{\rm pc}$ at the distance of the Coma cluster
($z\!=\!0.023$), $977\,$pc at $z\!=\!0.05$, and $4.5\,$kpc at $z\!=\!0.3\,$.}
and outer point at $\eta_{70}$; this enclosing 70 percent of the luminous mass. For each line-of-sight used, two uniform random numbers $\xi \in [\xi_{\rm i},\xi_{\rm f}]$ and
$\eta \in [\eta_{70},\eta_{95}]$ are generated that
in turn define $I_2=[\xi,\eta]$. In the Appendix we discuss some effects the radial range of a fit has on the parameters estimated using synthetic models.
\section{Results}\label{sec:results}
In this section we present the results of the fittings done,
both ``luminous'' and dark, to the merger remnants, as well
as several relationships among them based in observational studies.
Table~\ref{tab:global} lists different
global physical properties of our
remnants obtained directly from the $N$-body simulations.
Column (2) is the total half-mass radius $R_{\rm h}$,
(3) the virial radius $R_{\rm v}$, (4) the virial velocity $V_{\rm v}$,
(5) the total luminous mass ${\rm M}_{\rm lum}$ and (6)
the total bounded mass
${\rm M}_{\rm tot}$, and column (7) is the virial ratio at the end of
the simulation. The last column (8) provides the ratio of the total mass
of the secondary to the primary galaxy in the simulations.
The merger labelled as $MM$ corresponds to the simulation where the
resulting remnants of $M01$ and $M05$ were merged together in a parabolic encounter.
\begin{table}
\begin{minipage}{80mm}%
\caption{Physical properties of remnants}\label{tab:global}
\begin{tabular}{lcrrrrll}
\hline
{\sc id}& $R_{\rm h}$ & $R_{\rm v}$ & $V_{\rm v}$ &
$ \frac{{\rm M}_{\rm lum}}{10^{10}}$
& $ \frac{ {\rm M}_{\rm tot}}{10^{11} }$ & $\frac{2T}{|W|}$ & $\frac{2}{1}$\\
& [kpc] & [kpc] & [km/s] & [M$_\odot$] & [M$_\odot$] & & \\
\hline
$M01$ & 66.9 & 156.1 & 213.0 & 10.00 & 16.60 & 0.99 & 0.32\\
$M02$ & 29.8 & 71.4 & 108.5 & 0.60 & 1.95 & 0.99 & 0.46 \\
$M03$ & 24.6 & 56.4 & 99.0 & 0.54 & 1.29 & 0.99 & 0.53\\
$M04$ & 41.6 & 96.2 & 132.8 & 1.55 & 3.98 & 0.99 & 0.74\\
$M05$ & 22.2 & 48.6 & 100.3 & 0.83 & 1.15 & 1.00 & 0.93\\
$M06$ & 27.3 & 63.4 & 96.9 & 0.81 & 1.41 & 0.99 & 0.87\\
$M07$ & 24.0 & 55.3 & 105.8 & 1.02 & 1.45 & 0.99 & 0.51\\
$M08$ & 33.8 & 80.5 & 92.5 & 0.37 & 1.62 & 0.98 & 0.97\\
$M09$ & 28.7 & 66.1 & 103.5 & 1.41 & 1.66 & 0.99 & 0.98\\
$M10$ & 33.3 & 74.9 & 110.8 & 1.66 & 2.19 & 0.99 & 0.70\\
$M11$ & 32.4 & 76.6 & 178.2 & 4.47 & 5.62 & 1.00 & 0.14\\
$M12$ & 32.1 & 74.9 & 147.0 & 2.39 & 3.72 & 1.01 &0.18\\
$MM$ & 68.2 & 163.1 & 216.7 & 10.72 & 17.78 & 1.02 & 0.07 \\
\hline
\end{tabular}
\end{minipage}
\end{table}
\begin{table}
\centering
\begin{minipage}{140mm}
\caption{Mean parameters using radial range $I_1$}\label{tab:fits1}
\begin{tabular}{lcrccc}
\hline
{\sc id}& $n$ & $R_{\rm e}$ & $-\mu_0$ & $-M_{T}$ & {\sc rms} \\
& & [kpc] & [${\rm M}_\odot/{\rm kpc}^2$] & [M$_\odot$] & \\
\hline
$M01$ & $ 4.3 \pm 0.4$ & $ 9.2 \pm 1.9$ & $28.5 \pm 0.5$ & $27.6$ & $0.10$ \\
$M02$ & $ 2.1 \pm 0.1$ & $ 2.7 \pm 0.2$ & $23.3 \pm 0.3$ & $24.4$ & $0.10$ \\
$M03$ & $ 1.9 \pm 0.1$ & $ 2.5 \pm 0.0$ & $23.0 \pm 0.1$ & $24.3$ & $0.12$ \\
$M04$ & $ 2.8 \pm 0.1$ & $ 3.9 \pm 0.2$ & $25.1 \pm 0.2$ & $25.5$ & $0.14$ \\
$M05$ & $ 3.9 \pm 0.2$ & $ 1.7 \pm 0.1$ & $28.4 \pm 0.4$ & $24.8$ & $0.10$ \\
$M06$ & $ 2.5 \pm 0.1$ & $ 4.2 \pm 0.3$ & $23.6 \pm 0.4$ & $24.7$ & $0.12$ \\
$M07$ & $ 3.1 \pm 0.2$ & $ 2.6 \pm 0.3$ & $26.1 \pm 0.5$ & $25.0$ & $0.10$ \\
$M08$ & $ 2.6 \pm 0.1$ & $ 2.1 \pm 0.2$ & $24.5 \pm 0.3$ & $23.9$ & $0.12$ \\
$M09$ & $ 2.7 \pm 0.2$ & $ 8.1 \pm 0.8$ & $23.2 \pm 0.6$ & $25.3$ & $0.20$ \\
$M10$ & $ 3.2 \pm 0.2$ & $ 6.4 \pm 0.7$ & $24.8 \pm 0.6$ & $25.5$ & $0.14$ \\
$M11$ & $ 1.6 \pm 0.1$ & $ 9.3 \pm 1.4$ & $22.0 \pm 0.5$ & $26.6$ & $0.08$ \\
$M12$ & $ 3.2 \pm 0.2$ & $ 4.3 \pm 0.6$ & $26.0 \pm 0.6$ & $25.9$ & $0.19$ \\
$MM$ & $ 2.4 \pm 0.1$ & $ 9.1 \pm 0.7$ & $24.5 \pm 0.1$ & $27.6$ & $0.19$ \\
\hline
\end{tabular}
\end{minipage}
\end{table}
\begin{table}
\centering
\begin{minipage}{140mm}
\caption{Mean parameters using random radial range $I_2$}\label{tab:fits2}
\begin{tabular}{lcrccc}
\hline
{\sc id}& $n$ & $R_{\rm e}$ & $-\mu_0$ & $-M_{T}$ & {\sc rms} \\
& & [kpc] & [${\rm M}_\odot/{\rm kpc}^2$] & [M$_\odot$] & \\
\hline
$M01$ & $ 5.8 \pm 1.3$ & $13.2 \pm 5.5$ & $30.9 \pm 2.0$ & $27.8$ & $0.05$ \\
$M02$ & $ 2.1 \pm 0.3$ & $ 2.4 \pm 0.3$ & $23.7 \pm 0.8$ & $24.4$ & $0.05$ \\
$M03$ & $ 2.2 \pm 0.4$ & $ 2.3 \pm 0.2$ & $23.9 \pm 0.8$ & $24.3$ & $0.05$ \\
$M04$ & $ 3.3 \pm 1.0$ & $ 3.1 \pm 0.5$ & $26.5 \pm 2.1$ & $25.4$ & $0.07$ \\
$M05$ & $ 3.2 \pm 1.1$ & $ 2.1 \pm 0.6$ & $26.6 \pm 2.1$ & $24.8$ & $0.04$ \\
$M06$ & $ 2.7 \pm 0.4$ & $ 3.7 \pm 0.3$ & $24.3 \pm 0.9$ & $24.7$ & $0.06$ \\
$M07$ & $ 2.6 \pm 0.7$ & $ 3.0 \pm 1.0$ & $24.8 \pm 1.4$ & $25.0$ & $0.06$ \\
$M08$ & $ 2.1 \pm 0.3$ & $ 2.3 \pm 0.5$ & $23.3 \pm 0.8$ & $23.9$ & $0.04$ \\
$M09$ & $ 2.9 \pm 0.5$ & $ 6.9 \pm 0.9$ & $23.8 \pm 1.2$ & $25.3$ & $0.10$ \\
$M10$ & $ 3.5 \pm 0.6$ & $ 5.8 \pm 0.7$ & $25.7 \pm 1.4$ & $25.5$ & $0.06$ \\
$M11$ & $ 1.6 \pm 0.3$ & $ 9.6 \pm 1.9$ & $22.0 \pm 0.8$ & $26.6$ & $0.06$ \\
$M12$ & $ 3.2 \pm 1.4$ & $ 6.3 \pm 4.0$ & $25.5 \pm 2.4$ & $26.0$ & $0.16$ \\
$MM$ & $ 3.2 \pm 0.6$ & $ 9.2 \pm 1.5$ & $26.3 \pm 1.0$ & $27.6$ & $0.07$ \\
\hline
\end{tabular}
\end{minipage}
\end{table}
Tables~\ref{tab:fits1} and~\ref{tab:fits2} summarise the mean values of
the fitted S\'ersic parameters $\{n,R_{\rm e},\mu_0\}$
($\mu_0\!=\!-2.5\log \Sigma_0$),
the total ``magnitude''
($M_T\!\equiv\! -2.5\log M_{\rm L}$) and the {\sc rms} of
the fit, for the different projections for both radial intervals $I_1$
and $I_2$; respectively.
Here $M_{\rm L}$ is
determined from the fitted values using equation (\ref{eq:Lmass}).
Standard deviations are listed for the S\'ersic parameters.
The values of $M_{\rm L}$ determined from the
fits agree very well with $M_{\rm lum}$.
\subsection{Luminous Distributions}
\subsubsection{Shape parameter}\label{ssec:n}
Figure~\ref{fig:Nglxs}~({\it top})
shows the frequency distribution of $n$
for a set of observational data in optical wave bands (D01,
La~Barbera~et~al.~2005) and in the near-infrared ($K$) band
[La~Barbera~et~al~2005, Ravikumar~et~al.~2005 (R05)].
A total of 169 galaxies in the optical and 156 in the $K$
band were used here.
The frequency distribution of 41 merger remnants observed in the
$K$-band by
Rothberg \& Joseph (2004) are also indicated as a shaded histogram.
The mean and standard deviations of these data sets are indicated, as
well as their median.
In Figure~\ref{fig:Nglxs}~({\it bottom})
we show the distribution of $n$
for our merger remnants using the radial fitting intervals $I_1$ and $I_2$.
The frequency distribution for our $N$-body remnants peak at a value
$n\approx 3$ in both cases; although
using $I_2$ it shows a somewhat broader distribution.
For $I_1$ it is found that $n\in (1.5,5.3)$ and for $I_2$ that
$n\! \in \! (1.4,9.5)$. These values are in good agreement with those found
in intermediate mass ellipticals (e.g. Graham~\&~Guzm\'an~2003, de
Jong~et~al.~2004, Trujillo, Burkert \& Bell~2004, Ellis~et~al.~2005),
some brightest cluster galaxies (e.g. Graham~et~al.~1996), dwarf ellipticals
(e.g. Binggeli \& Jerjen~1998, Young \& Currie 2001), and the local merger remnants of RJ04.
Our results using $I_1$, and \emph{no} bulge, are consistent with the values found by
NT and GGB for their models with a bulge in the progenitors. Furthermore,
using interval $I_2$ lead to some values $n \! \approx \! 9$.
This does not seem to be due to the methodology in the
computation of the surface density profiles.
NT construct artificial
images analogous to the observational procedure while GGB fit ellipses
to isodensity contours,
both considering a wide range in the radial fitting range,
and obtaining similar ranges for $n$.
It is likely that
differences in the way models of the progenitors are set up be
probably one of the
reasons behind the differences with our results; see
$\S$\ref{sec:discussion}.
\subsubsection{Effective Radius}
Figure~\ref{fig:Nre} ({\it top}) shows the
observed frequency distribution of effective radii $R_{\rm e}$ for the
data considered in $\S$\ref{ssec:n} and
that corresponding to our remnants ({\it bottom}).
For the fitting radial range $I_1$ we obtain
$R_{\rm e}\! \in \!(1.6,12.9)\,$kpc, and for $I_2$ we have
$R_{\rm e}\! \in \! (1.6,34.5)\,$kpc.
The average value of the observational data is about
$4\,$kpc and for our remnants is about
$5\,$kpc. It can be noticed that $R_{\rm e}$ shows a larger dispersion of
values than the index $n$ depending on the fitting interval. This was also noticed by Binggeli \& Jerjen (1998).
Our remnants have a lower bound of
$R_{\rm e}\!\approx \!1.5\,$kpc, while
the observational data considered here can reach smaller values
$R_{\rm e} \! \gta \! 0.5\,$kpc. We
are not able to reproduce the small values of $R_{\rm e}$ mainly because
in our sample of initial conditions no pairs of small progenitors were
included.
On other hand, values of $R_{\rm e}\gta 10\,$kpc can be reproduced by
our more massive remnants ($M01$ and $MM$); see
Tables~\ref{tab:fits1} and~\ref{tab:fits2}.
A unique comparison with the distribution of$R_{\rm e}$ values found by
NT and GGB is not possible, since their models can be scaled to arbitrary
physical units; a thing that is not possible here due to the way our disc
galaxy progenitors were built up. Nonetheless, if we use the range of
dimensionless values found by NT ($1\! < \! R_{\rm e} \! < \! 1.7$)
for systems classified as pure ``bulges'',
and use a length unit of $3.5\,$kpc
(i.e., the radial scale-length of the Milky Way)
to transform their results to physical units,
we find that both results are consistent.
Also, we obtain qualitatively the same behaviour
as the one shown in their Figure~18 where a sharp cut
at the lower-end of the distribution, as well as an
extended tail at larger values.
Considering the observational values of $R_{\rm e}$
and those in our $N$-body remnants, we can establish with confidence that the
simulations can reproduce quite well the observed range of values. Even some
large values of $R_{\rm e}$ found in giant ellipticals (e.g.,
Graham~et~al.~1996) are reproduced.
\subsection{Luminous Correlations}
Several works (e.g. CCD93, D01, R05)
have found a series of correlations among S\'ersic
parameters in early-type galaxies. We now turn to study some of these
and compare them
with the properties of our numerical remnants. Firstly, we consider
two-dimensional correlations, and then turn to consider the so called
Photometric Plane (PHP) [e.g. K00].
\subsubsection{Two Dimensional Correlations}\label{sec:2Dcorrel}
In the work of Caon~et~al.~(1993) it was stated that a linear positive
correlation between $n$ and $R_{\rm e}$ exists for early-type galaxies;
they find that
$n\propto R_{\rm e}^{0.52}$ for early-type galaxies in Virgo.
A similar conclusion was reached by
D'Onofrio, Capaccioli \& Caon (1994) analysing galaxies in Fornax.
Combining the data of both works
one finds $n\propto R_{\rm e}^{0.50}$ with
a Pearson's linear correlation
coefficient $r=0.72$.
The statement of CCD93 that
structure (as indicated by $n$) of an elliptical
depends on its size
$R_{\rm e}$ has been supported
by the analysis of Trujillo, Graham \& Caon (2001).
Figure~\ref{fig:NRcorrel} shows
index $n$ against $R_{\rm e}$ for some of
the data considered here, as
well as the values obtained for our disc galaxy merger remnants. A
linear least-square fit to the data of D01 leads to
$n\propto R_{\rm e}^{0.37}$, and for RJ04 mergers
$n\propto R_{\rm e}^{0.26}$; with linear correlation
coefficients $r=0.73$ and $0.39$, respectively.
A similar fit
to our remnants yields $n\propto R_{\rm e}^{0.22}$ with $r=0.39$.
However, the observational
data plotted in Figure~\ref{fig:NRcorrel} shows a large scatter
around the assumed linear correlation; a fact already noticed by
other authors (e.g. Trujillo~et~al.~2001). These fluctuations
are quantified by considering the {\sl coefficient of
determination} ($r^2$) that measures
the proportion of the variance of one variable that is predictable
from the other (e.g. Ryan~1997).
As indicated in Figure~\ref{fig:NRcorrel},
the coefficients of determination are rather small, and the {\sc rms} of the
fits are large, so its is not clear
that a {\it true} linear correlation exists
between $\log n$ and $\log R_{\rm e}$.
We notice also
that the $N$-body remnant $M05$, being the smallest one, has the second
largest $n$ value in our simulations. These results lead us to state that
there is {\it no} linear positive correlation between
the ``structure'' and size of an elliptical. It seems that the values of $n$
and $R_{\rm e}$ are restricted by some physical mechanism to a finite region
of the S\'ersic parameter space; an option also indicated by
Trujillo~et~al.~(2001).
On other hand, a stronger observational correlation
has been found between $n$ and the central
brightness $\mu_0$ in ellipticals (e.g. K00, Graham \& Guzm\'an~2003) and
it appears to extend to dwarf ellipticals (e.g. Binggeli \& Jerjen 1998,
R05).
In Figure~\ref{fig:NMu0correl} we plot these quantities for the observational
data of D01, R05, RJ04, and for comparison our $N$-body remnants.
A linear fit $\log n$--$\mu_0$ to
these data leads to $n\propto I_0^{0.17}$ for D01, $n\propto I_0^{0.14}$
for both RJ04 and the ellipticals in R05, and $n\propto I_0^{0.14}$
for our merger remnants. All fits have $r^2\gta 0.9$ and an
{\sc rms}~$\lta 0.07$, that lead us to conclude that $\log n$--$\mu_0$ is
a true linear correlation, at least for the range of $n$ values considered.
As shown, the numerical remnants presented here
are able to reproduce very well the $\log n$--$\mu_0$
correlation and its tightness. We recall that we have assumed
a constant mass-to-light ratio to convert $\Sigma_0$ to $I_0$ in
order to compare with observations.
Hence, it appears that at least in the range
of masses of our remnants (Table~\ref{tab:global}), there is no need
to assume a dependence of the mass-to-light ratio dependence with
mass (or luminosity) to reproduce the observations.
\subsubsection{The Photometric ``Plane''}
Several authors [e.g. K00, Graham~2002,
Khos\-ro\-sha\-hi~et~al.~2004 (K04), La~Barbera~et~al.~2005, R05]
have found that S\'ersic parameters $\{n, R_{\rm e},
\mu_0\}$ of early-type galaxies define a plane in log-space
of the form
\begin{equation}
\log n = a \log R_{\rm e} + b \, \mu_0 + c\,,
\label{eq:php}
\end{equation}
which is termed the ``photometric plane'' (PHP).
Some authors instead of $\mu_0$ use the mean effective brightness
$\langle \mu \rangle_{\rm e}$ (e.g. Graham~2002, La~Barbera~et~al.~2005).
The different 2D correlations of $\S$\ref{sec:2Dcorrel} can be considered
then projections of the PHP.
We have computed, by a linear-square fit procedure, the coefficients of
this PHP for our remnants under the assumption of
a constant mass-to-light ratio. Since we find that using
$\langle \mu \rangle_{\rm e}$ leads to about twice the {\sc rms} in $\log n$
than using $\mu_0$, we restrict ourselves to an expression of the
form (\ref{eq:php}).
\begin{table}
\centering
\begin{minipage}{140mm}
\caption{Photometric Plane coefficients}\label{tab:php}
\begin{tabular}{lccrc}
\hline
{\sc id}& $a$ & $-b$ & $c$ & {\sc rms}$_n$ \\
\hline
K00 & $0.17 \pm 0.03$ & $0.069\pm 0.007$ & $1.2\pm 0.1$ & 0.04 \\
D01 & $0.09 \pm 0.02$ & $0.058\pm 0.004$ & $1.3\pm 0.1$ & 0.06 \\
K04 & $0.21 \pm 0.09$ & $0.074\pm 0.013$ & $1.7\pm 0.3$ & 0.13 \\
R05E & $0.15 \pm 0.02$ & $0.066\pm 0.003$ & $1.1\pm 0.0$ & 0.04 \\
R05dE & $0.16 \pm 0.04$ & $0.082\pm 0.004$ & $1.6\pm 0.1$ & 0.07 \\
RJ04 & $0.11 \pm 0.04$ & $0.054\pm 0.004$ & $1.0\pm 0.0$ & 0.06 \\
M($I_1$) & $0.05 \pm 0.00$ & $0.057\pm 0.001$ & $-1.0\pm 0.0$ & 0.05 \\
M($I_2$) & $0.05 \pm 0.01$ & $0.057\pm 0.001$ & $-1.0\pm 0.0$ & 0.05 \\
\hline
\end{tabular}
\end{minipage}
\end{table}
In Table~\ref{tab:php} we list the values of the coefficients of (\ref{eq:php})
found in the works of K00, K04, RJ04, the elliptical and dwarf ellipticals
of R05, and
those we obtain from the data of D01. Also shown are the coefficients
found for
our merger remnants for both fitting
radial intervals $I_1$ and $I_2$;
M($I_1$) and M($I_2$), respectively.
In Figure~\ref{fig:PHPrem} we plot the PHP from these data, using for
illustrative purposes the values from the data of D01 to
define the abscissa axis.
The remnants' coefficient $b$ in (\ref{eq:php}), associated with $\mu_0$,
is rather consistent with the observed ones, aside of those found
in dEs (R05).
The coefficient $a$, associated with $R_{\rm e}$, is less well reproduced.
This is not surprising taking into account
the large dispersion
in the $n$--$R_{\rm e}$ relation (see Figure~\ref{fig:NRcorrel}).
As several authors have pointed out (e.g. K00, K04, R05)
a slight curvature towards small values of $n$ is observed, a feature that
tends to be reproduced here by the effect of merger $M11$ that has
$n\! \approx \! 1.5$.
The {\sc rms}
is similar for both the observational data and our simulations. The
best overall agreement is obtained with the
data of D01.
We consider that our $a$ and $b$ values are rather
consistent with the whole set of values
listed in Table~\ref{tab:php}, and that the numerical remnants are able to
reproduce the PHP. In $\S$\ref{sec:discussion} we argue that the PHP is not
really a plane, but a ``pseudo-plane'' with a small curvature at low
values of $n$ due to the intrinsic properties of S\'ersic model.
\subsection{Dark Haloes}\label{sec:dark}
Dark haloes in cosmological simulations are started to being described
by a 3D S\'ersic function
(Merritt~et~al.~2005, Prada~et~al.~2005,
Graham~et~al.~2005), of the form
\begin{equation}
\rho(r) = \rho_0 \exp[ -d_n (r/r_{\rm e})^{1/n} ]
\label{eq:3dsersic}
\end{equation}
with $r$ being the spatial radius, $r_{\rm e}$ a 3D ``effective radius'', and
$d_n \approx 3n-1/3+0.005/n^2$ (Graham~et~al.~2005).
It has been
found that (\ref{eq:3dsersic}) provides a better fit to dark haloes
than the typical NFW or M99 (Moore~et~al.~1999)
density profiles.
S\'ersic indices of about $6$, with a scatter of $\approx 1$, are found for the cosmological dark haloes.
\begin{table}
\centering
\begin{minipage}{140mm}
\caption{Dark haloes 3D S\'ersic fits}\label{tab:darkies}
\begin{tabular}{lccrcc}
\hline
{\sc id}& $n$ & $r_{\rm e}$ & $\log \rho_0$ & & {\sc rms} \\
& & [kpc] & [M$_\odot/{\rm kpc}^3$] & & \\
\hline
$M01$ & 3.3 & 86.7 & 9.08 & & 0.05 \\
$M02$ & 3.5 & 28.4 & 9.81 & & 0.03\\
$M03$ & 4.5 & 26.5 & 10.90 & & 0.04\\
$M04$ & 4.0 & 42.1 & 10.17 & & 0.08\\
$M05$ & 4.3 & 20.4 & 10.93 & & 0.06\\
$M06$ & 4.4 & 25.2 & 10.88 & & 0.05\\
$M07$ & 3.9 & 23.0 & 10.32 & & 0.05\\
$M08$ & 3.8 & 28.9 & 9.99 & & 0.05\\
$M09$ & 4.5 & 29.0 & 10.92 & & 0.07\\
$M10$ & 4.0 & 31.9 & 10.24 & & 0.12\\
$M11$ & 3.7 & 41.3 & 10.03 & & 0.06\\
$M12$ & 3.3 & 39.5 & 9.38 & & 0.05\\
$MM$ & 3.0 & 82.6 & 8.78 & & 0.04\\
\hline
\end{tabular}
\end{minipage}
\end{table}
We have fitted 3D S\'ersic profiles (\ref{eq:3dsersic})
to the dark haloes of our remnants. The
radial range of the fit was from the convergence radius $r_{\rm c}$ (e.g.
Power~et~al.~2003) to
the dynamical virial radius of the remnant (Table~\ref{tab:global}).
However, instead
of using the orbital period at the $r_{200}$ radius
to determine $r_{\rm c}$ as done in cosmological
simulations, we used
the orbital period at the virial radius.
In Figure~\ref{fig:darkprofiles}
we display the fitted 3D S\'ersic profiles, along with their residuals.
In Table~\ref{tab:darkies} the values of the 3D S\'ersic
parameters are listed for each remnant, as well as the corresponding
{\sc rms}.
It can be seen that
the 3D S\'ersic profile (\ref{eq:3dsersic}) is a very good representation
of the density distribution up to the virial radius of each remnant. This
is in concordance with the behaviour of the 3D S\'ersic profile for
characterising cosmological haloes.
The haloes of the $N$-body remnants
have a mean 3D S\'ersic index $\langle n \rangle\! = \!
3.9 \pm 0.5$; the uncertainty being the standard deviation.
A value that
is lower than that found for cosmological haloes.
However, it is consistent with the mean value of
the dark haloes of the progenitors $\langle n \rangle\! = \! 3.7\pm
0.3$; as expected from the pre\-ser\-vation of the cuspyness of dark haloes
in mergers (e.g. Boylan-Kolchin \& Ma~2004, Aceves~\&~Vel\'azquez~2006,
Kazantzidis, Zentner \& Kravtsov~2006).
Differences in results are expected since the outer radius of the fits are not the same; we use a dynamical virial radius while for cosmological haloes the fits are done up to $r_{200}$ or further out (e.g. Prada~et~al.~2005).
It should be noticed that
S\'ersic fits (\ref{eq:3dsersic}) by
Graham~et~al.~(2005) have values for
$r_{\rm e}$ (see their Table~1) in some cases
larger than their cosmological virial radius;
these last ones listed in Table~1 of Diemand,
Moore \& Stadel~(2004).
For example,
their haloes G02 and G03 have
$r_{\rm e}\!=\! 391.4$~kpc and 405.6~kpc, respectively,
while their virial radii are $337\,$kpc and $299\,$kpc;
halo B09 shows even a more larger discrepancy.
Unfortunately, they do not provide their numerical
half-mass radii to make a direct comparison with
the values $r_{\rm e}$ they obtained.
Also, Merritt~et~al.~(2005) and Prada~et~al.~(2005) do not provide the fitted
values $r_{\rm e}$ and the numerical half-mass radii.
This makes uncertain any comparison of our results with these works.
Figure~\ref{fig:darkies} shows different
relations among the 3D S\'ersic parameters
for the haloes of our remnants:
$n$--$\rho_0$ and $\rho_0$--$r_{\rm e}$, and in analogy to the PHP we have
constructed a 3D dark S\'ersic plane (DSP). We find, assuming a log-linear
correlation,
that $n\propto \rho_0^{0.08}$ and $\rho_0\propto r_{\rm e}^{-3.10}$ with
coefficients of determination $0.96$ and $0.70$, respectively.
This indicates that
$n$--$\rho_0$ can be considered with confidence a true log-linear positive
correlation, as its was in the corresponding 2D case, but we do not deem
on the same level $\rho_0$--$r_{\rm e}$.
The DSP found for the remnants is
\begin{equation}
\log n \! = \! (0.07 \pm 0.02) \log r_{\rm e} - (0.04 \pm 0.00) \bar{\mu}_0 -
(0.48 \pm 0.07)\;
\label{eq:dsp}
\end{equation}
where $\bar{\mu}_0\!=\!-2.5 \log \rho_0$. This turns out to be
a very tight correlation, with a
coefficient of determination of $0.98$ and {\sc rms} of $0.007$, for the range of haloes masses considered in our simulations (see Table~\ref{tab:global}).
\section{Final Remarks and Conclusions}\label{sec:discussion}
The results found here, as well as those of NT and GGB, show that
the merger scenario is capable of reproducing
the S\'ersic properties of observed
elliptical galaxies.
This work shows, however, that
the presence of a ``primordial'' bulge in the progenitors is not
necessary to satisfy, for example, the observed values of
the shape parameter $n$; as was suggested by NT and GGB.
It is likely that the different results with NT and GGB have their
origin on the initial
properties of the progenitors; in particular, their
dark matter distribution. We have used cuspy (NFW-type) dark haloes in
contrast to those used by NT (pseudo-isothermal) and GGB (Lowered Evans) that
have a constant density core. It is probable that the higher concentration of dark matter used here, affected the distribution of luminous matter in a way to increase the index $n$ which is correlated with the luminous concentration (Trujillo~et~al.~2001). Other initial conditions of the progenitors and of the encounters such as energy and angular momentum, both intrinsic and orbital, may have played a role in the final concentration of luminous matter of the remnants, as indicated by the index $n$. A systematic study of the way different dynamical elements determine the S\'ersic index is, however, beyond the scope of the present work.
We have shown that haloes of remnants define a tight dark S\'ersic plane
(DSP) analogous to that of the luminous matter and with less dispersion.
No indication of curvature is present, at difference to what is noted
in the PHP at low values of $n$.
We argue here that this curvature is real and is related to an intrinsic
property of a S\'ersic profile.
Consider the expression for the total luminous
matter associated with a S\'ersic profile (\ref{eq:Lmass}). This
can be written in log-space as
\begin{equation}
\log L_{\rm T} = \log n - 0.4\, \mu_0 + 2 \log R_{\rm e} + \log f_2(n)
\label{eq:plane2}
\end{equation}
with the ``form factor'' $f_2(n)= 2 \pi \Gamma(2n)/b^{2n}$.
A given set of galaxies with equal $L_{\rm T}$ and different
S\'ersic parameters would define an exact log-plane, except for the presence
of the $\log f_2(n)$ term.
In the 3D
S\'ersic function (\ref{eq:3dsersic}) the analogous form factor is
$f_3(n) = 4 \pi \Gamma(3n)/d^{3n}$.
These non-constant terms
introduce
a systematic change in a PHP-like expression.
The importance of the form factor is
larger for values
$n\lta 1$ and smaller for $n \gta 1$;
both
$f_2(n)$ and $f_3(n)$ are shown in
Figure~\ref{fig:Nfactor}. Thus, the form factor of the S\'ersic
model determines the curvature observed in the PHP.
This explains why no curvature is found by La~Barbera~et~al.~(2005) whose
galaxies show $n \gta 2$, but this can be seen in dwarf ellipticals with
several values of $n\lta 1$ (K04). Also, the DSP does not show such
curvature since $n \gta 3$ (see Figure~\ref{fig:darkies}).
Furthermore, the dispersion
about these ``planes'' is determined by the luminosity or dark mass range of
the galaxy sample.
It remains to study the central phase-space densities of the remnants, to see
if they are consistent with the estimates for ellipticals
(e.g. Carlberg~1986), and to analyse their kinematical properties with those
observed in elliptical galaxies. We plan to study these topics in a future work. In summary, our main conclusions are as follows:
\begin{enumerate}
\item Collision-less mergers of \emph{pure} disc galaxies yield values and
distributions of S\'ersic parameters consistent with those observed
for bona fide ellipticals. The existence of a bulge in merging spirals
does not appear to be a necessary condition on grounds
of S\'ersic properties of the remnants.
\item
The suggested
positive log-linear correlation between
the size ($R_{\rm e}$) and structure ($n$)
in ellipticals is not supported.
However,
the strong $\log~n$--$\mu_0$ linear correlation found in observational
studies is supported by our merger simulations. On other hand,
the PHP is fairly well reproduced. For these results
a constant mass-to-light ratio is assumed.
\item The final dark haloes of remnants show values of $n\approx 4$ lower than
those found in cosmological simulations $n\approx 6$. The
difference may be attributable to the non-equivalence outer radius,
where the dynamical virial radius was used in our case to carry out the
fitting by a S\'ersic profile.
Haloes define a tight Dark S\'ersic Plane (DSP) in three
dimensions, with no indication of curvature at the level of the smaller $n$
obtained.
\item The curvature observed in the PHP
at low values of $n$ is an intrinsic manifestation
of the properties of S\'ersic model, due to the presence
of a non-constant term dependent of $n$.
\end{enumerate}
\section*{Acknowledgments}
This research was funded by CONACyT-M\'exico project 37506-E.
We appreciate the kindness of Chazhiyat Ravikumar for providing us
observational data used in this work.
\section*{Appendix}
We briefly discuss here the effect of the radial interval on the fitting
process of a S\'ersic profile to a mass distribution.
To do this we generate exact
S\'ersic profiles with $R_{\rm e}\!=\! 1$, $L_{\rm T}\!=\!1$, and
$n=2,4,8$. Also, random fractional errors $\le \{1,10,20\}$\% are
introduced.
The radial fitting interval is chosen as follows.
A random inner point $\xi$ is selected from the interval
$[0.03,0.5]R_{\rm e}$ while the outer radius, $\eta$, is randomly generated
from the interval $[R_{70},R_{95}]$; where $R_{70}$ and $R_{95}$ correspond to
the radii containing 70\% and 95\% percent of the projected mass. These two
points define our radial fitting interval $[\xi,\eta]$.
To corroborate the importance the importance of the underlying mass
distribution we use
also a Hernquist (1990) mass model with an without errors.
\subsubsection*{S\'ersic Distribution}
Table~\ref{tab:app1} lists the mean
values of $\{n,R_{\rm e},\mu_0\}$ obtained from fitting
1000 Monte Carlo experiments for three S\'ersic models with errors
as indicated above.
Each line lists, in order of the ascending error introduced to the
theoretical S\'ersic profile, the parameters recovered from the fit inside the random interval
$[\xi,\eta]$. The standard deviation for each
quantity is provided.
These results show that the determination of S\'ersic
parameters is very robust, for errors $\lta 10$\%, against the size of
the fitting region. As the error in the ideal S\'ersic distribution
increases the dispersion grows. This is more clearly appreciated for
index $n$.
In the limit of zero error, even for a random radial fitting interval,
the model parameters are recovered exactly. For this case,
we conclude that the radial fitting range does not
has an important effect on S\'ersic fitted parameters.
\begin{table}
\centering
\begin{minipage}{8cm} %
\caption{S\'ersic fits.}\label{tab:app1}
\begin{tabular}{cccc}
\hline
$n_{\rm{true}}$ & $n$ & $R_{\rm e}$ & $-\mu_0$ \\
\hline
2 & $2.000\pm 0.015$ & $1.000 \pm 0.003$ & $\hphantom{0}0.956\pm \hphantom{0}0.030$ \\
& $2.014\pm 0.162$ & $1.002 \pm 0.031$ & $\hphantom{0}0.981 \pm \hphantom{0}0.319$ \\
& $2.061\pm 0.418$ & $1.009 \pm 0.082$ & $\hphantom{0}1.067 \pm \hphantom{0}0.819$ \\
\hline
4 & $4.001\pm 0.037$ & $1.000 \pm 0.004$ & $\hphantom{0}4.940 \pm \hphantom{0}0.080$ \\
& $4.041\pm 0.408$ & $1.001 \pm 0.043$ & $\hphantom{0}5.026 \pm \hphantom{0}0.876$ \\
& $4.222\pm 1.749$ & $1.010 \pm 0.139$ & $\hphantom{0}5.407 \pm \hphantom{0}3.719$ \\
\hline
8 & $8.002\pm 0.087$ & $0.999 \pm 0.007$ & $13.261 \pm \hphantom{0}0.196$ \\
& $8.119\pm 1.042$ & $0.998 \pm 0.071$ & $13.521 \pm \hphantom{0}2.316$ \\
& $8.663\pm 4.776$ & $1.006 \pm 0.251$ & $14.719 \pm 10.407$ \\
\hline
\end{tabular}
\end{minipage}
\end{table}
\subsubsection*{Hernquist Distribution}
We now consider that the case where the underlying mass distribution follows
a Hernquist model. Here, $R_{\rm hl}$ denotes its theoretical projected
half-light (mass) radius.
It is found that in the fitting interval
$[0.03,2.79]R_{\rm hl}$ , the underlying Hernquist's profile is fitted by
a S\'ersic profile with index $n=2.6$ and $R_{\rm e}=0.82$ in agreement with
NT. For a radial fitting interval of
$[0.03,14.5]R_{\rm hl}$ we find that
$n\!=\!3.67$ and $R_{\rm e}\!=\!1.10$. This indicates that the process of
fitting a S\'ersic profile is far more sensitive when the underlying mass
distribution does not follows a S\'ersic one.
The above was already noted by Boylan-Kolchin, Ma \& Quataert~(2005),
where a systematic change in S\'ersic parameters was found when trying to
fit a Hernquist profile.
In Figure~\ref{fig:app1} ({\it left}) we reproduce this systematic effect, for
the S\'ersic index $n$, $R_{\rm e}$, and the mean effective
surface brightness $\langle I_{\rm e}\rangle$.
If a random error $\le 10$\%
is introduced the trend is preserved but a large
dispersion results; especially as the inner radius of radial interval of the
fit is increased.
Figure~\ref{fig:app2} shows the distribution of fitted values
with no error ({\it solid line}) and with a random error
$\le 10$\% ({\it dashed line}) for an underlying Hernquist profile where
the radial interval was obtained from
$\xi \in [0.06,0.50]R_{\rm hl}$ and $\eta\in [\eta_{70},\eta_{95}]$.
The mean and standard deviations of the distribution are indicated.
For comparison, the histogram of values corresponding to a S\'ersic
model with $n=4$ with a random error
$\le 10$\% is also shown ({\it dotted line}); see Table~\ref{tab:app1}, second line in the entry for $n=4$.
From the above results, it follows that when the underlying mass distribution
is {\it not} of a S\'ersic type, the fitted values have a rather large
dispersion even in the presence of no error.
In particular, higher values of the index $n$ are obtained for different
radial ranges of the fit.
In order to have a confident estimate of $n$, and other parameters,
one has to sample rather deep inside and outside the luminous (mass)
distribution; from about $0.1$ to $6\,R_{\rm hl}$.
In practise, for example, sampling very near the centre of a galaxy may pose
problems due to resolution effects. This is a particular problem for
observations of galaxies at different redshifts, using the same angular
resolution but representing different physical scales, and can lead
to uncertain S\'ersic parameters.
\bsp
\label{lastpage}
|
Title:
The effect of environment on the UV colour-magnitude relation of early-type galaxies |
Abstract: We use \textit{GALEX} (Galaxy Evolution Explorer) near-UV (NUV) photometry of
a sample of early-type galaxies selected in \textit{SDSS} (Sloan Digital Sky
Survey) to study the UV color-magnitude relation (CMR). $NUV-r$ color is an
excellent tracer of even small amounts ($\sim 1$% mass fraction) of recent
($\la 1$ Gyr) star formation and so the $NUV-r$ CMR allows us to study the
effect of environment on the recent star formation history. We analyze a
volume-limited sample of 839 visually-inspected early-type galaxies in the
redshift range $0.05 < z < 0.10$ brighter than $M_{r}$ of -21.5 with any
possible emission-line or radio-selected AGN removed to avoid contamination. We
find that contamination by AGN candidates and late-type interlopers highly bias
any study of recent star formation in early-type galaxies and that, after
removing those, our lower limit to the fraction of massive early-type galaxies
showing signs of recent star formation is roughly $30 \pm 3%$ This suggests
that residual star formation is common even amongst the present day early-type
galaxy population.
We find that the fraction of UV-bright early-type galaxies is 25% higher in
low-density environments. However, the density effect is clear only in the
lowest density bin. The blue galaxy fraction for the subsample of the brightest
early-type galaxies however shows a very strong density dependence, in the
sense that the blue galaxy fraction is lower in a higher density region.
| https://export.arxiv.org/pdf/astro-ph/0601036 |
\title{The Effect of Environment on the UV Color-Magnitude Relation of Early-type Galaxies}
\author{
K. Schawinski,\altaffilmark{1}
S. Kaviraj,\altaffilmark{1}
S. Khochfar,\altaffilmark{1}
S.-J. Yoon,\altaffilmark{2,1}
S. K. Yi,\altaffilmark{2,1,10}
J.-M. Deharveng,\altaffilmark{3}
A. Boselli,\altaffilmark{3}
T. Barlow,\altaffilmark{4}
T. Conrow,\altaffilmark{4}
K. Forster,\altaffilmark{4}
P. G. Friedman,\altaffilmark{4}
D. C. Martin,\altaffilmark{4}
P. Morrissey,\altaffilmark{4}
S. Neff,\altaffilmark{5}
D. Schiminovich,\altaffilmark{6}
M. Seibert,\altaffilmark{4}
T. Small,\altaffilmark{4}
T.Wyder,\altaffilmark{4}
L. Bianchi,\altaffilmark{7}
J. Donas,\altaffilmark{3}
T. Heckman,\altaffilmark{7}
Y.-W. Lee,\altaffilmark{2}
B. Madore,\altaffilmark{8}
B. Milliard,\altaffilmark{3}
R. M. Rich\altaffilmark{9}\&
A. Szalay\altaffilmark{7}
}
\altaffiltext{1}{Department of Physics, University of Oxford,
Oxford OX1 3RH, UK}
\altaffiltext{2}{Center for Space
Astrophysics, Yonsei University, Seoul 120-749, Korea}
\altaffiltext{3}{Laboratoire d'Astrophysique de Marseille, 13376
Marseille Cedex 12, France}
\altaffiltext{4}{California
Institute of Technology, MC 405-47, Pasadena, CA 91125}
\altaffiltext{5}{Laboratory for Astronomy and Solar Physics, NASA
Goddard Space Flight Center, Greenbelt, MD 20771}
\altaffiltext{6}{Department of Astronomy, Columbia University, MC
5246, New York, NY 10027}
\altaffiltext{7}{Department of Physics and Astronomy, Johns
Hopkins University, Baltimore, MD 21218}
\altaffiltext{8}{IPAC,770 S. Wilson Ave., Pasadena, CA 91125}
\altaffiltext{9}{Department of Physics and Astronomy,
University of California, Los Angeles, CA 90095}
\altaffiltext{10}{Send offprint request to yi@yonsei.ac.kr}
\keywords{galaxies: elliptical and lenticular, cD -- galaxies:
evolution -- galaxies: formation -- galaxies: fundamental
parameters}
\section{Introduction}
There is observational evidence pointing to a very simple
evolutionary model for early-type galaxies. This model of
\textit{Monolithic Collapse} was first proposed by
\citet*{1962ApJ...136..748E} to explain the origin of the Milky
Way halo. According to this model, the Milky Way halo formed through
the rapid collapse of a cloud of gas very early on in the history
of the universe, forming all of its stars in an initial burst
followed by a passive evolution of the stellar population. A
similar model is often invoked as the simplest explanation for the
old and seemingly homogeneous stellar populations seen in early-type
galaxies \citep{1975MNRAS.173..671L}.
The apparently universal relationship between galaxy color and
luminosity in early-type galaxies was first studied in detail
by \citet*{1977ApJ...216..214V}, even though the relation
had been observed before
\citep{1959PASP...71..106B, 1961ApJS....5..233D, 1968AJ.....73.1008M}.
This \textit{Color-Magnitude Relation} (CMR) is often used as a tool
for understanding the formation and evolution of early-type
galaxies.
A seminal investigation on the optical CMR was undertaken by
\citet*{1992MNRAS.254..589B} on the elliptical galaxies in the
Virgo and Coma clusters. Their study revealed a remarkably small
intrinsic scatter around the mean relation. In the context of the
monolithic paradigm, they interpreted the small scatter as the
result of a small age dispersion amongst galaxies of the same age
and the slope as a result of a mass-metallicity relation
\citep{1997A&A...320...41K}.
Further, they concluded that massive early-type galaxies did not
have any major episodes of star formation at redshifts z $< 2$.
More massive galaxies are likely to be in deeper potential wells and are therefore
more able to retain metals ejected from supernovae from the
initial generations of young stars at high redshift, leading to
the observed mass-metallicity relation
\citep{1974MNRAS.169..229L}. In addition to this, the observed
levels of $\alpha$-enhancement
\citep{1992ApJ...398...69W,1993MNRAS.265..553C,
1994MNRAS.270..743C, 1994MNRAS.270..523C,
1997A&A...320...41K, 2000AJ....120..165T} in many
giant ellipticals imply that the initial formation starburst was
of a very short duration of less than 1 Gyr
\citep{1993ApJ...405..538B, 1999MNRAS.302..537T}.
Later studies have found that there is no significant evolution in
the optical CMR out to z=1 and further
\citep{1997ApJ...483..582E, 1998ApJ...501..571G,1998ApJ...492..461S,
2000ApJ...541...95V, 2003ApJ...596L.143B, 2005ApJ...635..243F}.
All this adds up to a picture of
massive early-type galaxies forming in an initial, intense
starburst at high redshift followed by a relatively-passive
evolution.
However, we know since the simulations of
\citet*{1972ApJ...178..623T} that the product of a spiral-spiral
merger can be an elliptical galaxy \citep{1983MNRAS.205.1009N,
1988ApJ...331..699B, 1992ApJ...400..460H, 2003ApJ...597..893N}.
An alternative approach to
understanding early-type galaxies takes into account
dynamical interactions and mergers. In the \textit{Hierarchical
Merger} paradigm, small galaxies form first and later assemble
into larger objects \citep{1978MNRAS.183..341W}.
The advent of semi-analytical models (SAMs) in the 1990s has
greatly enhanced our understanding of galaxy evolution in such a
hierarchical universe.
\citet{1996MNRAS.283L.117K} find that in the Canada-France
Redshift Survey, only 1/3 of elliptical and lenticular galaxies at
redshift z=1 were fully assembled and showed colors expected of
old passively evolving systems.
There is of course older evidence for a strong dependence
of the population of early-type galaxies on density and redshift.
\citet{1980ApJ...236..351D} found that approximately 80\% of
galaxies in a sample of 55 clusters were of early-type morphology,
a much higher fraction than in the field suggesting that the
denser cluster environment does affect galaxy evolution. When
\citet{1984ApJ...285..426B} looked at higher redshift clusters,
they found that the fraction of blue, spiral galaxies in cluster
environments increased with redshift. Later studies confirmed that this
trend was not a selection effect
\citep{1997ApJ...490..577D, 2000ApJ...541...95V}. This
evolution is accompanied with an increase in merger rates
\citep{1998ApJ...497..188C, 1999ApJ...520L..95V, 2001ApJ...561..517K}.
In a purely monolithic collapse model, the star formation history of
early-type galaxies is almost trivial, as they are composed of
uniformly old stars. As soon as we allow for any sort of
hierarchical merging, the story becomes much more complex. Rather
than being uniform, the star formation histories become highly
degenerate, as disparate stellar populations from progenitor
galaxies are mixed together. Beyond this simple addition, the
merging history of the galaxy and its progenitors adds further
complication as entirely new populations are created during
interactions and mergers. Thus, assigning a single age to the
stellar population of an early-type galaxy is misleading - there
is no single age. The typically-derived luminosity-weighted ages
are in this sense nontrivial to interpret.
We know now that the combined effects of age, dust, metallicity and
- potentially - a multitude of progenitors are highly degenerate.
\citet*{1992MNRAS.254..589B} took the apparent uniformity and low
intrinsic scatter as a very strong constraint on the evolution of
the Virgo \& Coma Early-type population.
While monolithic evolution is the simplest
possible explanation of these observations, however, it does not necessarily
exclude other interpretations.
\citet{2005MNRAS.360...60K} have argued using merger models that
the optical early-type CMR is useful for constraining evolution
models \textit{only} if we believe \textit{a priori} in a
monolithic model. The effect of progenitor bias - the fact that a
progressively larger fraction of the progenitor set of present-day
ellipticals is contained in late-type star-forming galaxies at
higher redshift - means that we are \textit{not} probing the
entire star formation history of early-type galaxies, but rather a
progressively more biased subset. Besides, the level of
star formation predicted by SAMs incorporating
AGN and supernova feedback is very low; on the order of a few percent by
stellar mass. Optical filters, including $U$-band, are not sufficiently
sensitive to detect such a low-level star-forming activity. This is why we
must turn to the UV.
The \textit{Galaxy Evolution Explorer (GALEX)}
\citep{2005ApJ...619L...1M} near-UV filter is
capable of detecting even a small ($\sim 1\%$ mass fraction)
young stellar population and so ideal for tracing the recent
star formation history of early-type galaxies. The UV
color-magnitude relation allows us to identify the last important
episode of star formation in galaxies.
\citet{2005ApJ...619L.111Y}
have already shown using \textit{GALEX} information that a significant
fraction of massive early-type galaxies at low redshift exhibit
levels of star formation undetectable in the optical but visible in the UV.
Our paper presents the results of our search for the effect of environment
on the recent star formation.
We assume a standard $\Lambda$CDM cosmology with $(\Omega_{M},
\Omega_{\Lambda}) =(0.3, 0.7)$ and a Hubble constant of $H_{0}=70\,
{\rm km\,s^{-1}\,Mpc^{-1}}$ \citep{2003ApJS..148..175S}.
\section{Sample Selection \label{sample_selection}}
The \textit{GALEX} \textit{Medium Imaging Survey} (MIS) is a
wide-area survery with limiting magnitudes of 22.6 AB in the
far-UV ($FUV; 1344-1786 A $) and 23.0 AB in the near-UV filter
($NUV; 1771-2831 A $) (\cite{2005ApJ...619L...7M})
with substantial overlap with the
\textit{Sloan Digital Sky Survey} DR3 (\cite{2002AJ....123..485S},
\cite{2000AJ....120.1579Y}, \cite{2005AJ....129.1755A}) .
We define a sample of early-type
galaxies within \textit{SDSS} and then cross-match it to
\textit{GALEX} detections. We use the \textit{GALEX}
Internal Release 1.1 MIS data.
\subsection{Early-type Galaxy Selection in \textit{SDSS} \label{criteria}}
A fundamental problem in the study of early-type galaxies is that
there are no fixed criteria for their classification.
In terms of the Hubble Sequence, everything equal to or earlier than
a lenticular is an early-type galaxy, but even
this innocent definition is highly subjective, varies between different
observers, and strongly depends
on the image quality used to evaluate it. There is danger in
classifying galaxies as early-types using the properties that
are based on the presumption that early-type galaxies are old, red,
dead, uniform and dustless, e.g, colors or spectral features. By making
such supposition, any sub-population of early-type galaxies departing
from this set of prejudices is liable to be rejected. Such a sample is then
biased against \textit{precisely those} early-type galaxies that
can tell us the most about galaxy evolution.
In order to create an unbiased, volume-limited sample, we
match \textit{GALEX} NUV detections to a catalog of early-type
galaxies identified in the \textit{SDSS}.
The paramount effort of \cite{2003AJ....125.1817B}
(hereafter B03) has already generated such a catalog of $\sim 9000$
galaxies. They were selected on a number of \textit{SDSS} pipeline
parameters.
Such a catalog is of no doubt extremely useful to study the overall properties
of galaxies in a statistical sense but less than perfect
to our investigation which is searching for ``abnormality'' of early-type
galaxies. For instance, B03 uses the Principal Component Analysis Technique
which is biased strongly against star-forming ellipticals
(e.g. \citet{2004ApJ...601L.127F}).
Second, the sample generated this way is bound to be
contaminated by late-type interlopers despite the effort of
cleaning the sample in various ways (see B03 for details).
In a visual inspection of a
sample of bright ($M_{r} < -22$) early-types from the B03 catalog, we
found up to 30\% contamination. These were not only Sa galaxies
with small or faint spiral arms, but also edge-on disk and a
number of face-on spirals. Such late-types are
generally actively star-forming and should be
removed from our sample. Besides, it is difficult to estimate
the rate of false rejections (that is, early-types falsely rejected)
if we use a catalog generated by a different group a priori.
Some of these false rejections may be due to the
\textit{spectral} part of the B03 criteria. \citet{2005ApJ...619L.107R}
find the same contamination problem when they employed a method
similar to B03.
To avoid these problems, we define a simple set of
\textit{morphology-driven} criteria with no assumptions at all on
color or spectral energy distribution (SED).
We define early-type galaxies to be those
bulge-dominated galaxies that lack clearly visible spiral arms. We
use these criteria to create an \textit{inclusive} rather than
\textit{exclusive} sample to avoid rejecting too many genuine
early-types. In order to select early-type galaxies over
late-types, we consider the surface-brightness profiles in three
bands and select those which have a very high likelihood of being
a de Vaucouleurs profile rather than an exponential profile. To do
this, we use the \textit{fracDev} parameter, which is the weight of
the deVaucouleurs profile of the linear combination which best fits
the image in each band. We
select galaxies in DR3 with:
\begin{enumerate}
\item \textit{SED Quality}:
The spectrum is of good quality ($S/N> 10$).
\item \textit{fracDev\textunderscore g} $> 0.95$
We use the $g$ profile as it is sensitive to blue disk and
arm stellar populations to ensure that Spiral galaxies are
rejected.
\item \textit{fracDev\textunderscore r} $> 0.95$
The $r$ band traces bulge populations and so will select
bulges that follow an $r^{1/4}$ profile.
\item \textit{fracDev\textunderscore i} $> 0.95$
The $i$ band strengthens the constraint derived from the
$r$ band profile.
\end{enumerate}
For relatively bright galaxies ($r < 16.8$), this method is
very reliable. The number of galaxies accepted that do not
appear to be early-types upon visual inspection is on the order of
$\sim 15\%$. Similarly, the number of galaxies that appear to be
early-types amongst those which are rejected due to low values
of $fracDev$ is $\sim 10\%$, which
gives us confidence that we are not excluding a significant part
of the early-type population. This level of contamination
nevertheless requires a careful visual inspection, which is
performed after the matching process.
\subsection{Matching to \textit{GALEX}-MIS}
The initial selection of early-type galaxies in \textit{SDSS} DR3
yields a total of 89248 galaxies without any constraints on
luminosity or redshift. The detections in the \textit{GALEX} MIS
survey are then cross-matched to this catalog. All early-type
galaxies within each \textit{GALEX} field of view (FOV) are
flagged and retained. We then perform a simple proximity search
algorithm to find all those \textit{GALEX} detections that are
within the $4 \arcsec$ angular resolution limit of \textit{GALEX}
of each \textit{SDSS} early-type. All unique matches are flagged
and kept together with all galaxies within \textit{GALEX} fields
that are not detected.
\subsection{Visual Inspection of Galaxy Morphology}
The most dangerous contaminant when constructing a sample of
supposed early-type galaxies are Sa galaxies. We set the
difference between S0 (which we keep) and Sa (which we reject) to
be \textit{the presence of distinct spiral arms}. This can be
challenging when the galaxies in question are at higher redshift
or faint.
In order to quantify how well we can distinguish Sa galaxies based on SDSS images
alone, we compare these to the \textit{COMBO-17} S11 field, which
overlaps with \textit{SDSS} DR3 and has a number of galaxies at $
0.10 \la z \la 0.13$. This image is significantly deeper (24 000
sec) and has better seeing ($\sim 0.7\arcsec$) than \textit{SDSS} images,
so they allow us to identify morphology with much higher accuracy.
We selected the brightest galaxies in the S11 field, ranging in
$R$-band magnitude from 16.56 to 17.31. From this experiment, we conclude that
the reliability of visual inspection is dependent first on
redshift and seeing and second on apparent magnitude. In order
to set a reliable redshift and magnitude limit, we limit our sample
to $z < 0.1$ and $r < 16.8$.
\subsection{The Volume-limited Sample}
In order to create an unbiased sample, we need to take into
account a number of factors. At $z < 0.05$, \textit{SDSS}
spectroscopy begins to be incomplete for bright galaxies, so $z =
0.05$ is our lower limit
\citep{2002AJ....123..485S, 2002AJ....124.1810S}.
The \textit{GALEX} MIS limiting magnitude in NUV is 23.0 AB
\citep{2005ApJ...619L...7M}, but many fields have longer exposure
and some have been visited multiple times and co-added, giving us
no uniform NUV magnitude limit. This is a problem since if we
wished to probe the reddest early-types out to $NUV-r \sim 7.5$,
we could only probe the most massive galaxies within a small
redshift slice. In order to maximise the range in absolute
magnitude to a reasonable part of the high end of the luminosity
function, we must leave the reddest galaxies incomplete in some
fields. We nevertheless retain them as non-detections.
If we choose $r < 16.8$ as an apparent magnitude limit out to
which visual inspection can be done reliably, the $NUV = 23.0$
hard limit guarantees us completeness to $NUV-r = 6.2$ which
corresponds roughly to the top of the red sequence which was
introduced by Yi et al. (2005, Figure 3). However, the
fact that many images go up to a magnitude deeper than $NUV = 23.0$
means we can still probe the red end of the UV color-magnitude
relation. Colors redder than $NUV-r \sim 6.5$ cannot be produced by
an old stellar population of any age on its own; these galaxies
must contain dust to achieve such red colors. Since we are
primarily interested in studying those early-type galaxies that
show signs of recent star formation, this is a safe limit.
In addition to this, there is a significant fraction of
\textit{SDSS} galaxies in the \textit{GALEX} field of view that
are not matched to \textit{any} \textit{GALEX} counterpart (Figure
\ref{fig1}). Even when matching to a sample of
spectroscopic galaxies of \textit{all} morphologies in
\textit{SDSS} roughly $10\%$ do not have \textit{GALEX} detections
(Figure \ref{fig1}). Thus we must assume that they
are either too faint in the UV, too dusty, or a combination of
both and so can be assumed to be red on the UV color-magnitude
diagram for the purpose of deriving the fractions
of UV-bright galaxies. Some of these non-detections might also be due to
mechanical problems in astrometry near the edge of the detector.
Nevertheless, by making a number of assumptions on these
non-detections, we can still derive some information from them.
At $z = 0.1$, $r = 16.8$ is equivalent to an absolute
magnitude limit of $M_{r} = -21.5$, so the limits on our
sample are $z =[0.05, 0.10]$ and the color-magnitude relation can
be probed out to $M_{r} = -21.5$. For comparison, $M_* = -20.83$
for all morphologies in an \textit{SDSS} sample \citep{2001AJ....121.2358B}.
We then perform a visual inspection of all matched galaxies in our
sample and place them into one of three categories:
\begin{enumerate}
\item Elliptical galaxies (847)
\item Lenticular galaxy (112)
\item Other (126)
\end{enumerate}
The ``Other'' category includes all galaxies rejected for either
non early-type morphology or for the presence of nearby, bright blue
stars which might contaminate the UV flux. The apparently low number
of lenticular galaxies is due to the fact that we were very stringent
about giving out the label ``lenticular''. If there was any doubt
between elliptical and S0, we placed it in the elliptical category.
In a study of 146 early-type galaxies of the Coma cluster,
\citet{1994ApJ...433..553J} find that the separation of early-type
galaxies into elliptical and lenticular is very difficult and that
many face-on lenticulars have been misclassified as elliptical galaxies.
In Section \ref{ba}, we discuss the relationship between recent star formation
and axis ratio, where this effect becomes important.
\subsection{Discussion of Random $\&$ Systematic Errors}
The random errors in the $NUV-r$ color are dominated by the errors
in the NUV. The mean 1-$\sigma$ error is 0.17 magnitudes, which is
much smaller than the overall scatter of the observed colors (Figures
\ref{fig3}, \ref{fig6}). The \textit{GALEX} photometry
are taken from Internal Release 1.1, which is known to underestimate
the errors. The errors are recalculated for our analysis following
the instruction given iin the GALEX WEB site.
Virtually all of our galaxies are
unresolved in \textit{GALEX} NUV due to the large size of the NUV
point spread function (4$\arcsec$ FHWM). Due to this large difference between
the optical and UV resolutions, we do not attempt to use matched
apertures. Since we use total fluxes, we do not expect color gradients
to affect $NUV-r$ colors.
\subsection{AGN Contamination $\&$ Removal}
The other major problem is the presence of AGN. In
the local universe, AGN hosts are preferentially massive
elliptical galaxies. A strong AGN can easily produce a
UV flux similar to that of a small mass fraction of young stars.
In order to minimize the contamination from the galaxies whose UV
fluxes are possibly dominated by an AGN non-thermal spectrum rather
than a thermal stellar spectrum we apply two methods.
First, we perform a BPT analysis \citep{1981PASP...93....5B}
wherein galaxies are classified using a number of emission line
ratios into either quiescent, star-forming or AGN. We employ a
method similar to the one devised by Kauffmann et al. (2003). The
line ratios used are [NII]/H$\alpha$ and [OIII]/H$\beta$. A full
description of our method can be found in Kaviraj et al. (2005, in
preparation).
Classification using such a BPT diagram is only reliable when all
four emission lines have sufficient $S/N$. The $S/N$ cut we employ in
this study is $S/N > 3$ for all four lines. We reject all galaxies
consequently classified as Seyfert, LINER or transition objects
and only retain those which are quiescent or star-forming. It is
interesting to note that most objects classified as
star-forming were in fact galaxies rejected by the visual
inspection as late-types (see Figure \ref{fig2}).
Most of our early-type galaxies do not appear in \ref{fig2} because
they do not show emission lines with $S/N>3$.
For a discussion
of how the AGN were identified, see \citet{2003MNRAS.346.1055K}.
This process removed 11\% of our volume-limited sample after
visual inspection. In
order to ensure that we have as few AGN as possible left in
our sample, we checked if any strong radio sources were left. The
VLA FIRST survey \citep{1995ApJ...450..559B} covers about 80 \% of
our galaxies at 1.4 GHz with $5\arcsec$ resolution. We removed all
strong radio sources with a luminosity $L_{\rm 1.4 Ghz} > 10^{23}
{\rm W\,Hz^{-1}}$. This cutoff was chosen as it is often assumed that
below this luminosity, AGN
activity and star formation are degenerate, whereas above it, most
sources are AGN. We cross-checked this value to be consistent with
the radio luminosities of our BPT-selected AGN. We only identify 8
further sources, which gives us confidence in the reliability of our BPT
diagnostics. In total, this leaves us with a sample of 839
early-type galaxies to analyze. A catalog of these 839 galaxies can be
made available upon request.
We now construct the UV color-magnitude relation using
this sample. In Figure \ref{fig3}, we show
the entire sample of \textit{GALEX}-\textit{SDSS} matches with
their classification into early-types, rejected late-types
and AGN candidates.
\section{Method \label{method}}
In this section we describe our methods for classifying environment and
for how we separate recent star formation (RSF) galaxies from
``UV-upturn'' galaxies.
\subsection{Defining a Parameter for Local Environment \label{env_section}}
We wish to define \textit{a quantitative way for measuring
environment} that makes as much use of the information we are given
as possible. Two-dimensional projected number densities would offer
some information, but without redshift information, they can easily be
rendered meaningless for anything but the most nearby clusters
(e.g. Coma) due to fore- and background contaminants. It is
possible to apply statistical methods to correct for this,
but since \textit{SDSS} spectroscopy is available to us for all
our galaxies and their surroundings, we can make use of
spectroscopic redshifts to determine proximity.
The high redshift accuracy of \textit{SDSS} spectroscopy
($\sigma_{z} = 1.7 \times 10^{-4} \pm 2 \times 10^{-5}$ for our
sample , corresponding to $\sim 0.5 Mpc$ in our redshift range)
allows us to compute the number density of neighboring galaxies
\citep{2002AJ....124.1810S}. The SDSS spectroscopic completeness
limit of $r = 17.77$ imposes a cut-off in absolute magnitude of
$M_{r} = -20.55$ at our upper redshift limit of $z=0.1$. This
allows us to sample the luminosity function to about $M_*$,
which for an \textit{SDSS} sample is $M_* = -20.83$
\citep{2001AJ....121.2358B}.
Any method that relies on number density has to deal with the
fact that dense clusters give rise to peculiar velocities that
can translate to shifts of up to
several hundred ${\rm km\,s^{-1}}$, which can correspond to
shifts of up to $\sim 10$ Mpc. \cite{2004ApJ...601L..29H} for
example use a cylindrical volume elongated along the z-axis to
$16h^{-1}$ Mpc to deal with this. Thus, their method corrects as
well as is possible for dense environments. However, such a fixed-
volume method
does not take into account the density-dependence of peculiar
velocity.
We therefore attempt to correct for this by employing an adaptive
volume: for each galaxy, we initially count all neighbors within
a certain radius $\sigma$, ignoring the fact that the distance along
redshift may be distorted. This number is capped at 10.
We use this number $n$ as a guide to adaptively
change the extent of our redshift search radius. We define the scale
factor $c_{z}$ as follows:
\begin{equation}
c_{z} = 1+0.2n
\label{cz}
\end{equation}
The scale factor is used to scale the value of $\sigma$ along the
redshift axis by up to a factor 3 for the highest density environments
to compensate for the ``finger of god'' effect. This is only a zeroth order
approximation however and modelling will be needed to devise a more
reliable method for scaling to a realistic volume.
We then employ a Gaussian distribution
to give more weight to closer neighbors and use $c_{z}$ to increase
the extent of this Gaussian along the z-axis. We define the
\textit{Adaptive Gaussian Environment Parameter} $\rho_{g}$ as the sum
over all neighbors within the ellipse defined by
\begin{equation}
{(\frac{r_{a}}{3 \sigma})^2 + (\frac{r_{z}}{3 c_{z} \sigma})^2 \leq 1},
\end{equation}
that is, we search out to $3 \sigma$:
\begin{equation}
\rho_{g}(\sigma) = \frac{1}{\sqrt{2 \pi}\sigma } \exp \left[ -\frac{1}{2} \left(\frac{r_{a}^2}{\sigma^2} + \frac{r_{z}^2}{c_{z}^2 \sigma^2} \right) \right]
\label{rho_g1}
\end{equation}
where $r_{a}$ is the angular distance in Mpc to each surrounding galaxy,
$r_{z}$ is the distance along the line-of-sight in Mpc to each surrounding
galaxy, and $\sigma$ is an arbitrary dospersion parameter.
This weighting scheme is biased towards nearby galaxies
and so is a more realistic measure than a raw number density or
overdensity. When measuring this parameter for a particular
galaxy, the galaxy itself is \textit{not} counted towards
the total. For this project, we adopt a fiducial value
of $\sigma$ of 2.0 Mpc.
The choice of $\sigma = 2$\,Mpc is somewhat arbitrary. We chose it so that the
scale-length of our measure was focused approximately on the scale of
large groups and small clusters. Perturbing $\sigma$ does not change
our results within 1-$\sigma$.
But what does this parameter actually measure? It is blind as to
whether the structure around it is gravitationally bound or in
equilibrium, so it is not a way of separating clusters from the
field. Rather, it is a measure of the number and proximity of
galaxies around a point in space, a more sophisticated number
density. Despite that, it is useful to have physical sense on the values
of $\rho_g$.
First of all, the spatial distribution of our galaxies is mapped in Figure
\ref{map1}\footnote{Created using POV-ray (http://www.povray.org)}.
The birghter regions are denser.
For comparison, we compute $\rho_g$ for the bright cluster
galaxies in the C4 Catalog \citep{2005AJ....130..968M}
within our redshift range.
Indeed, most of them have high values of $\rho_g$ (Figure \ref{C4_density}).
Typically, the central galaxy of a typical cluster with 10
$L_*$ galaxies randomly distributed within a $r \sim 3$\,Mpc sphere
would have $\rho_g \sim 1$.
A galaxy at the edge of the same cluster however would have
$\rho_g \sim 0.5$.
All galaxies in a group with three $L_*$ galaxies within a
$r \sim 1$\,Mpc sphere would have similar values of $\rho_g$ to the
cluster outskirts. Typical field galaxies would have $\rho_g \la 0.2$.
We divide our final sample into three numerically equal
environment bins, which we arbitrarily label ``low'', ``medium''
and ``high'' density (see Table \ref{environment_bins}).
The (1) low, (2) medium, and (3) high density roughly correspond to (1) fields,
(2) groups and cluster outskirts and (3) cluster centers, respectively.
We also tested whether a mass weight could improve our measure. We
tested a weight of the form:
\begin{equation}
\rho_{g}(\sigma) = \frac{f(mass)}{\sqrt{2 \pi}\sigma }\, \exp \left[ -\frac{1}{2} \left(\frac{r_{a}^2}{\sigma^2} + \frac{r_{z}^2}{c_{z}^2 \sigma^2} \right) \right]
\label{rho_g2}
\end{equation}
where we chose $f$ to be a linear function of absoulte $r$-band
magnitude such that a galaxy at the lower limit of $M_{r} = -20.55$
counted as 1 and the most massive neighbors of $M_{r} \sim -23$ counted
three times as much. This made no difference within error to our result, so we
do not use such a mass weight to avoid introducing unnecessary
complication, so we adopt $f(mass) = 1$.
\begin{table}[h]
\caption{Environment Bins \label{environment_bins}}
\begin{center}
\begin{tabular}{c|ccc}
\hline
Bin & $\rho_{g}(\sigma = 2.0 Mpc)$ & Label\\
\hline
\hline
$0 - \frac{1}{3}$ & 0.00 $<$ $\rho_{g}$ $\le$ 0.21 & Low density\\
$\frac{1}{3} - \frac{2}{3}$ & 0.21 $<$ $\rho_{g}$ $\le$ 0.58 & Medium density\\
$\frac{2}{3} - 1$ & 0.58 $<$ $\rho_{g}$ $\le$ 4.68 & High density\\
\end{tabular}
\end{center}
\tablecomments{These bins are derived by splitting our sample of
839 galaxies into three euqal-number bins. The values of
$\rho_{g}$ represend the boundaries between them.}
\end{table}
\subsection{Recent Star Formation and the UV-Upturn Phenomenon \label{upturn}}
Many early-type galaxies exhibit the UV-upturn
phenomenon (\cite{1979ApJ...228...95C},
\cite{1988ApJ...328..440B}) characterized by unusually strong
UV flux rising with decreasing wavelength in the range
($1000-2500 A$). The UV-upturn phenomenon is thought to be due to
the presence of low-mass, core helium-burning horizontal branch
(HB) and evolved HB stars (\cite{1997ApJ...486..201Y}). We therefore
face the problem that the moderate UV flux that we see in many of
our early-type galaxies may in fact be due to such an old stellar
population, or, even more difficult to resolve, due to both old
and young stars.
There is however a limit to how much NUV flux an early-type galaxy
can produce via UV-upturn. This limit can be explored using both
theoretical and observational methods. Ideally, we wish to combine
both to derive a conservative limit beyond which we can be certain
to probe recent star formation only. However, the UV upturn theory
is still debated and thus observational
evidence should take precedence.
The IUE satellite conducted a survey of UV spectra of nearby
elliptical galaxies \citep{1988ApJ...328..440B}. Among the strongest
known nearby UV-upturn galaxies is NGC 4552, which has an
$NUV-r$ color of 5.4 mag. We therefore choose $NUV-r
= 5.4$ as a conservative lower boundary in color. At $NUV-r < 5.4$,
all galaxies are considered to have experienced a recent episode of
star formation, although part of their UV flux may come from a
UV-upturn. Above this limit, a galaxy might either (or both)
be forming stars or (and) exhibiting UV-upturn - we
cannot distinguish the two using \textit{GALEX} NUV alone.
Considering that the IUE SEDs were obtained from the UV-bright
central regions of galaxies, our $NUV-r=5.4$ cut is conservative
and puts some fraction of star-forming galaxies into the quiescent
galaxy bins.
\subsection{Comparison Between the Optical and UV-CMR \label{comparison}}
In Figure \ref{fig6}, we plot the optical $u-r$ and
$g-r$ color-magnitude relations on the same scale as the $NUV-r$.
We label galaxies not classified as AGN by the
BPT diagram above $NUV-r= 5.4$ as Quiescent (QST) and those bluer
as Recent Star Formation (RSF).
We do not include a slope in this cut-off, although one might suggest
a slope based on the red-sequence slope for example as found by Yi et al.
(2005, Figure 3), because any slope over our magnitude
range would likely be very small and complex (albeit not impossible)
to explain theoretically.
We can see that the $g-r$ is completely insensitive.
It cannot be used to detect recent star formation in early-type
galaxies. Even $u-r$ color does not break this degeneracy. While
the scatter of the UV-bright RSF galaxies is slightly greater, the bulk of
them are indistinguishable from quiescent ones.
In order to properly study recent star formation in early-type galaxies
the UV information is essential.
In total, $30\% \pm 3$ of our 839 early-type galaxies with
$M_{r} < -21.5$ are classified as RSF using this scheme.
This RSF galaxy fraction is probably a lower limit, first because of our
conservative UV-upturn criterion and because we do not correct
our UV data for internal extinction.
\section{The Effect of Environment on Early-type Galaxies
\label{env_dep}}
In this Section, we investigate two related questions:
does the UV color-magnitude relation depend on environment? And
does the $fraction$ of early-type galaxies showing signs of recent
star formation depend on environment?
\subsection{The Color-Magnitude Relation \& Environment}
It is well known that more massive early-type galaxies reside in
denser environments \citep{1980ApJ...236..351D, 1984ApJ...281...95P}
even though the slope and zero-point of their color-magnitude
relations do not appear to depend on environment.
\citet{2004ApJ...601L..29H} find that (1) the color-magnitude relation
for their sample of 55,158 early-type galaxies in \textit{SDSS}
does not depend on environment and that (2) the most luminous galaxies
reside preferrentially in the most high-density environments. (2)
is not surprising as the most massive ellipticals are known to
reside at the centers of clusters \citep{1983ApJ...274..491B}.
In their analysis, \citet{2003AJ....125.1882B} also find little
dependence of the color-sigma relation on environment. Further,
\citet{2005AJ....129...61B} suggest that the color-magnitude
relation is entirely a consequence of the fact that both the
luminosities and colors are correlated with sigma, a proxy to mass; that the
color-sigma relation is in fact the more fundamental relation.
In Figure \ref{color_cmr}, we show the UV color-magnitude relation
for the three equal-number environment bins defined in Table
\ref{environment_bins}. From this, we can see that there are two
obvious differences between the low, medium and high density
color-magnitude relations. As expected, the higher-density CMR
extends to more massive galaxies. However,
the low-density CMR extends to bluer colors than the
high-density one. This is observational evidence for a change in
the range of color of the UV-CMR with environment.
We test the statistical significance of both of these environmental
differences.
\subsection{The Dependence of $NUV-r$ Color on Environment \label{color_dep}}
The first quantity we consider is $NUV-r$ color. In Figure
\ref{fig8}, we show how $NUV-r$ color varies with
$\rho_{g}$. The range in $NUV-r$ remains more
or less constant over the entire range of $\rho_{g}$, but the
distribution itself varies with $\rho_{g}$. We can make this
variation more apparent by plotting the cumulative color
distribution of the three environment bins in Figure
\ref{fig8}. In this plot, we not only show the cumulative
distribution itself, but also a Monte Carlo re-simulation of the
color distribution. In order to assess to what extent the difference
between the environment bins is, we regenerate the distribution
by randomly changing the color by the error and recomputing
the distribution.
The ``medium'' and ``high'' density curves are statistically
indistinguishable. On the other hand, the low density bin (the
yellow line in Figure \ref{fig8}) diverges from
the other two at blue colors. We test the significance of this
difference using both Kolmogorov-Smirnov (KS) and
Kuiper test. The test significances are the probability that
one of the distribution is drawn from a different parent distribution.
The results are shown
in Tables \ref{color_KS} and \ref{color_kuiper}.
\begin{table}[!ht]
\caption{KS-Test of NUV-r color dependence on environment \label{color_KS}}
\begin{center}
\begin{tabular}{c|ccc}
\hline
bin & low & medium & high \\
\hline
\hline
low & - & 89.220\% & 99.239\% \\
medium & 89.220\% & - & 25.130\% \\
high & 99.239\% & 25.130\% & - \\
\end{tabular}
\end{center}
\tablecomments{Table of Kolmogorov-Smirnov test significance comparing the
distribution of $NUV-r$ color in the three environment
bins.}
\end{table}
\begin{table}[!ht]
\caption{Kuiper Test of NUV-r color dependence on environment
\label{color_kuiper}}
\begin{center}
\begin{tabular}{c|ccc}
\hline
bin & low & medium & high \\
\hline
\hline
low & - & 83.428\% & 99.657\% \\
medium & 83.428\% & - & 55.753\% \\
high & 99.657\% & 55.753\% & - \\
\end{tabular}
\end{center}
\tablecomments{Table of Kuiper test significance comparing the
distribution of $NUV-r$ color in the three environment
bins.}
\end{table}
\subsection{The Dependence of Mass on Environment \label{mass_env}}
\begin{table}[!ht]
\caption{KS-Test of r-band absolute magnitude dependence on environment
\label{mr_KS}}
\begin{center}
\begin{tabular}{c|ccc}
\hline
bin & low & medium & high \\
\hline
\hline
low & - & 99.139\% & 90.816\% \\
medium & 99.139\% & - & 50.728\% \\
high & 90.816\% & 50.728\% & - \\
\end{tabular}
\end{center}
\tablecomments{Table of Kolmogorov-Smirnov test significance comparing the
distribution of $M_{r}$ in the three environment
bins.}
\end{table}
Figure \ref{fig9} shows how
$M_{r}$ varies with $\rho_{g}$. If we then plot the
cumulative $M_{r}$ distribution for the three environment
bins (Figure \ref{fig9}), we see a clear dependence of
absolute magnitude on environment. Even in our volume-limited sample
with a narrow baseline in luminosity
there is a clear trend for brighter galaxies to
be in higher-density environments. In Table
\ref{mr_KS}, we give the KS-test significance for the differences
between the $M_{r}$ distributions in each bin.
\section{The Dependence of Recent Star Formation Activity on Environment}
The fact that the distribution of $NUV-r$ color of massive
early-type galaxies changes between low- and high-density
environments may suggest that the recent star formation history of
those galaxies is different. In order to quantify this, we use the
criterion for recent star formation outlined in Section
\ref{upturn}. It is not possible to directly convert NUV flux into
an actual star formation rate, chiefly due to our inability to
quantify dust extinction, to which the near-UV is extremely
sensitive. We therefore merely classify our galaxies as RSF
and QST and calculate the RSF galaxy fractions
for subsamples in different environments in an attempt to find
general trends.
We calculate the recent star-forming fraction of
early-types by dividing the number of galaxies
bluer than $NUV-r = 5.4$ (RSF) by the total number of galaxies in
this bin - that is, both those bluer and redder than $NUV-r = 5.4$
as well as those not detected by \textit{GALEX} but classified as
early-type galaxies during the visual inspection. We include these
non-detections as QSTs on the assumption that they are red galaxies
further reddened by dust beyond the MIS detection limit. It is an intriguing
possibility that at least some of these galaxies are dusty because
they are actually forming stars, but we cannot make this
distinction using \textit{GALEX}.
In total, $30\% \pm 3$ of our 839 early-type galaxies with
$M_{r} < -21.5$ are classified as RSF. The
ellipticals, the bulk of our sample, have an RSF fraction of
$29\% \pm 3$, while the lenticulars have an RSF fraction of
$39\% \pm 5$. The division into ellipticals and lenticulars is based
on visual inspection.
We mentioned in \S 2.4 that our visual classification was generous to
ellipticals. Hence, if a half of our ellipticals were in truth
lenticulars, and if 39\% were the true RSF galaxy fraction for lenticulars,
the RSF fraction for true ellipticals would be as low as 20\%.
\subsection{RSF and Axis Ratio \label{ba}}
The RSF galaxy
fraction of those galaxies identified as lenticulars in the visual
inspection is higher than that of the ellipticals. While there is no natural
way to distinguish ellipticals and lenticulars \citep{1994ApJ...433..553J}, we
can look at the change in UV properties with axis ratio. This still
suffers from the fact that orientation can obscure true axis ratio.
In Figure \ref{fig10}, we show the distribution of $NUV-r$ color with
$r$-band axis ratio together with the RSF percentage as a function of $b/a$.
Even amongst the roundest elliptical galaxies
such as E0/1, there still is a significant fraction of star-forming galaxies.
The RSF percentage appears to have a weak dependence on $b/a$
rising upto $\sim 50$\% for the most flattened galaxies (which corresponds
to the 39\% we find for the visually identified lenticulars) but the trend
is statistically insignificant. All this should be viewed in light of the
bias against visually identifying face-on lenticular galaxies
\citep{1994ApJ...433..553J}; it is likely
that a fraction of the round early-types are such mis-classified
objects and that the RSF fraction for genuine, round ellipticals is lower.
\subsection{The Dependence of the RSF Galaxy Fraction on Environment}
We now divide our sample into the three equal-number environment
bins (see Table \ref{environment_bins}) to see whether the RSF
fraction depends on environment. As expected from the results in
Section \ref{color_dep}, the low-density environment bin shows a
pronounced enhancement of the fraction of galaxies showing signs
of recent star formation (see Figure \ref{fig11}). The medium-
and high-density bins are consistent with having the same
fraction.
In order to constrain this enhancement further, we then divide our
sample into 5 equal-number bins to see at what values of
$\rho_{g}$ this increase lies and in particular whether there is
any change at very low or high values. The red 5-bin curve in
Figure \ref{fig11} shows that there is no change at high density
and that the enhancement of the fraction of RSF galaxies begins at
values of $\rho_{g} \sim 0.4$. This corresponds roughly to one
$M_*$ galaxy per cubic Megaparsec, a loose definition of the
``field''. Thus, the enhancement of star formation in our sample
is primarily due to the galaxies in the field.
Our environment parameter on the other hand only probes
neighbors down to $M_{r} \sim -20.5$, so these galaxies may well merely
lack large neighbors - that they may simply be the dominant galaxy in a
small group.
Large surveys of galaxy
star formation rates show a strong dependence on environment.
Studies in both \textit{2dF} (\cite{2002MNRAS.334..673L}) and
\textit{SDSS}. (\cite{2003ApJ...584..210G}) find that above a
certain ``break'' local density, star formation rapidly declines.
This ``break'' or ``characteristic'' local galactic density is given
as $\sim 1 h^{2} {\rm Mpc}^{2}$, so the enhancement we see is similar.
However, they also find a continuing decrease in star formation
rate with increasing galactic density, which we do \textit{not}
see. A direct comparison to our result is not possible however,
since we cannot trace actual star formation rates, but rather
only the fraction of galaxies showing signs of $recent$ star formation.
\subsection{Breaking the Mass-Environment Degeneracy \label{mass_deg}}
In Section \ref{mass_env}, we have shown the well-known fact that
more massive galaxies prefer higher-density environments. It is
also well-known that smaller galaxies tend to have higher
star formation rates and are bluer, i.e. that the color-magnitude
relation has a slope. This
raises the possibility that the dependence of the RSF fraction for
our entire volume-limited sample is nothing but an effect of
mass. In order to test whether this is the case, we have to break
the mass-environment degeneracy.
Similarly to the environment bins, we divide our sample
into three equal-number absolute magnitude bins.
Together with the three environment bins, this gives us three
curves of RSF percentage as a function of environment like Figure
\ref{fig12}. These nine sub-samples are indicated by the dashed
lines on Figure \ref{fig9}.
The resulting curves are shown in Figure
\ref{fig12}. From this, it is apparent that the effect of
environment that we are seeing is not due to a stellar mass effect, as all
three curves follow almost the same trend of a high RSF fraction
at low density and a low RSF fraction at high density.
Intriguingly, the high mass bin ($-23.82 < M_{r} \leq -22.13$)
departs from the others at high density, though this remains just
above a 1-$\sigma$ result.
It should also be noted that the strongest density dependence is
found among the brightest galaxies.
\section{Summary}
We have used the UV color-magnitude relation of low redshift,
massive early-type galaxies to study their recent star formation
history. Our sample is volume-limited, ranging in redshift from z=0.05
to 0.1 and is limited in absolute magnitude to $M_{r} < -21.5$. Our
sample is highly unlikely to be contaminated by any significant
number of late-type galaxies, as all our
galaxies have been visually inspected.
In order to classify galaxies by their environment, we have devised a
method for measuring environment that takes the proximity, and not just
the number density of neighboring galaxies, into account (see Section
\ref{env_section}). This method can easily be modified to different
samples within SDSS and can take into account a larger part of the
luminosity function if restricted to lower redshift limits than
$z=0.1$. Our measure works very well for the brightest cluster galaxies
in the C4 cluster catalog and in addition also performs for field
galaxies.
In our sample of 839 early-type galaxies with $M_{r} < -21.5$,
the recent star formation (RSF) galaxy fraction is $30 \pm 2$\%. Our
ellipticals, the bulk of our sample, have an RSF fraction of
$29\% \pm 3$, while the lenticulars show
$39\% \pm 5$. This implies that \textit{residual star formation is common
amongst the present day early-type galaxy population}.
Our estimates are very likely lower limits on
the true fractions, as our criteria for RSF are
conservative in the consideration of internal extinction
and the UV contribution from the old populations.
The UV color-magnitude relation differs from the optical
color-magnitude relation \citep{1992MNRAS.254..589B, 2004ApJ...601L..29H}
in that it does vary more clearly with environment.
The recent star formation history of early-type galaxies also
varies with environment. It is well-known that more massive galaxies
reside in higher-density environments (Figure
\ref{fig9}), but we show for the first time that UV-bright early-type
galaxies preferentially reside in low density environments. The RSF
fraction is a function of environment and drops by 25\%
from field to group but then puzzlingly remains
relatively constant at higher densities,
even when split into luminosity bins (Figure
\ref{fig12}). Interestingly, the most massive galaxies
($-23.82 < M_{r} \leq -22.13$) show the strongest dependence
on environment and alone exhibit a further drop in RSF fraction from
medium to high density.
One possible way to understand the drop in the RSF fraction between
low and medium density is in the context of ram pressure stripping.
Galaxies moving fast in the deep gravitational potential
of a galaxy cluster are bound to lose most of their gas during their
orbital motion \citep{1972ApJ...176....1G}.
The density dependence of gas content in galaxies has long been
established empirically as well \citep{1981ApJ...247..383G}.
Our RSF fraction-density relation is in the right direction. The gas
goes into the ICM and so could potentially explain the star formation
we do see in high density environments.
Another noteworthy observation is the fact that those early-type
galaxies which have been identified as AGN - by emission lines
and/or radio - are significantly bluer than those who are not (see
Figure \ref{fig3}). We have removed these AGN from our sample
since we cannot disentangle the UV flux from the AGN from that of
a possible young stellar population. However, the blueness of the
AGN colors is intriguing - are we really just seeing the AGN
itself, or is this from the star formation triggered by the jets and
outflows from the AGN \citep{1998A&A...331L...1S}?
In the latter case, the RSF fraction would increase further from
our estimates, and the AGN regulations might present a possible
physical mechanism responsible for the
star formation that we observe. In fact, \citet{1995ApJ...452..549H}, using IUE
observations of nearby Type 1 and 2 Seyferts, suggest
that at most 20\% of the UV continuum emission seen in them can
originate from the nucleus itself. This means that the vast majority
of our AGN candidate (removed) would qualify as RSF galaxies.
It is important to note that we only deal with \textit{fractions} of
star-forming galaxies and not actual star-formation rates. Thus, our
RSF fractions simply give us an indication of how likely an early-type
galaxy with certain properties and in a certain environment is to have
experienced recent star formation. Neither environment, luminosity nor
axis ratio seems to be the primary physical quantity that
regulates recent star formation in early-type galaxies.
The relative insensitivity to environment in any environment
denser than the field is also surprising and warrants further study.
The observational trends presented here give us new
constraints for theoretical models of galaxy evolution.
\acknowledgements
Special thanks are given to M. Bernardi who kindly supplied her
early-type galaxy catalog which provided us with a great insight
on our catalog generation.
We warmly thank C. Wolf for making the \textit{COMBO-17} S11 field
image available to us. We would also like to thank E. Gawiser,
L. Miller, S. Rawlings, J. Silk, R. Davies,
I. Jorgensen, M. Sarzi, J. Magorrian, S. Salim, M. Urry
and K. Kotera for helpful comments and discussions.
GALEX (Galaxy Evolution Explorer) is a NASA Small Explorer, launched
in April 2003. We gratefully acknowledge NASA's support for construction,
operation, and science analysis for the GALEX mission, developed in
cooperation with the Centre National d'Etudes Spatiales
of France and the Korean Ministry of Science and Technology.
This was supported by Yonsei University Research Fund of 2005 (S.K.Yi).
\bibliographystyle{astroads}
\bibliography{bibliography}
|
Title:
Constraints on SN Ia progenitor time delays from high-z SNe and the star formation history |
Abstract: We re-assess the question of a systematic time delay between the formation of
the progenitor and its explosion in a type Ia supernova (SN Ia) using the
Hubble Higher-z Supernova Search sample (Strolger et al. 2004). While the
previous analysis indicated a significant time delay, with a most likely value
of 3.4 Gyr, effectively ruling out all previously proposed progenitor models,
our analysis shows that the time-delay estimate is dominated by systematic
errors, in particular due to uncertainties in the star-formation history. We
find that none of the popular progenitor models under consideration can be
ruled out with any significant degree of confidence. The inferred time delay is
mainly determined by the peak in the assumed star-formation history. We show
that, even with a much larger Supernova sample, the time delay distribution
cannot be reliably reconstructed without better constraints on the
star-formation history.
| https://export.arxiv.org/pdf/astro-ph/0601454 |
\date{Accepted 2006 March 01. Received 2006 February 28; in original form 2006 January 19}
\label{firstpage}
\begin{keywords}
supernovae: general --- cosmology: observational.
\end{keywords}
\section{Introduction}
Type Ia supernovae have been used extensively as standard distance indicators
and have provided the best evidence to date for an acceleration of the
Universe \citep{rie98, per99, rie04}. Future missions,
e.g. \emph{GAIA} and \emph{SNAP}, will greatly increase the number of
detected SNe Ia and significantly reduce the statistical errors in the
determination of cosmological parameters. However, the nature of the
progenitors of type Ia supernovae is still unknown and the empirically
calibrated \emph{Phillips relation} \citep{phi93} is not fully
understood physically.
Several progenitor scenarios are under discussion, but there is no
consensus due to uncertainties in the evolutionary processes
\citep{HKN96, HKN99, LV97, lan00, HP04} and the explosion mechanism
\citep{HN00, roe05, gam05}. One of the signatures of the various
scenarios is the distribution of time delays between the formation of
the progenitor systems and their explosion, which could give rise to a
significant difference between the redshift dependence of the
supernova rate (SNR) and the star-formation history (SFH).
\citet{str04}, hereafter S04, aimed to detect this difference by studying
the distribution of 25 high-z SNe Ia in the \emph{Hubble} Higher-z
Supernova Search sample \citep{rie04}. Their approach was to infer the
mean time delay of the distribution using a Bayesian analysis, which
assumed different parametrized time-delay distributions and adopted
the star-formation history (SFH) from \citet{gia04}, hereafter
G04. They concluded that mean time delays shorter than $\sim 2$~Gyr
ought to be excluded with a 95 per cent confidence level, ruling out
essentially all progenitor scenarios currently under
discussion. In a recent re-assessment of the constraints,
\citet{str04erratum} obtained a 95 per cent lower limit ranging
from 0.2 to 1.6~Gyr for different time-delay distributions. In the
corrected best-fitting model, the 95 per cent confidence interval
ranged from 1 to 4.4~Gyr with the most likely value at 3.4~Gyr.
Unlike core collapse SNe (SNe II, Ib/c) that originate from massive
progenitors with relatively short main-sequence (MS) lifetimes ($\sim
3-20$~Myr), SNe Ia are believed to be thermonuclear explosions of
white dwarf stars (WDs) whose progenitors have MS lifetimes ranging
from $\sim 30$~Myr to several billion years. This implies a minimum
time delay for SNe Ia of the order of $\sim 30$~Myr.
Most of the SN Ia progenitor scenarios that have been proposed involve
mass transfer on to a CO WD in a binary system, either through the
expansion and Roche lobe overflow of an evolved companion
(\emph{single degenerate [SD] scenarios}) or through the slow release
of gravitational waves, orbital shrinking, Roche lobe overflow and
merging of a compact double WD system (\emph{double-degenerate [DD]
scenarios}). Both scenarios have associated time-delay distributions
that have been estimated with binary population synthesis codes (BPS),
where the properties of binary systems are followed from their birth
up to the explosion stage through the many different evolutionary
paths. Independently of the particular treatment of the binary
interactions, the resulting time-delay distributions differ
considerably since their characteristic time-scales have different
origins.
The SD scenario is controlled by the process of mass accretion, which
has to occur at just the correct critical rate in order to allow the
growth of the mass of the companion WD up to the Chandrasekhar limit
\citep{nom91}. The dominant evolutionary path seems to occur via the
accretion of matter on to a CO WD from a slightly evolved MS star, the
CO WD + MS -- SD scenario \citep{vH92, rap94, LV97, lan00, HP04}. In
this channel, the accretion rate is determined mainly by the mass of
the donor star, which must lie in a narrow range in order to satisfy
the required accretion-rate constraints. As a consequence, the
distribution of MS lifetimes and the time-delay distribution of the
channel are relatively narrow, peaking at $\sim 670$~Myr and rapidly
becoming negligible after $\sim 1.5$~Gyr.
Although recent simulations have suggested that other evolutionary
paths within the SD framework are of minor importance
\citep{HP04}, it is quite possible that their contribution has been
underestimated. This is particularly important for the CO WD + RG --
SD scenario, where a red-giant (RG) star accretes matter on to a CO WD
star \citep{HKN96}; in this channel the time-delay distribution
extends up to several Gyr.
The DD scenario \citep{IT84, web84}, in contrast, is controlled by the
time that it takes for the binary system to coalesce, which depends
roughly on the fourth power of the separation of the double-degenerate
system \citep{sha83}. As a result, the time-delay distribution can be
described by a low time-delay cutoff ($\sim 30-100$~Myr) and an
approximately power-law decline up to the age of the Universe. The
lower time-delay cutoff can be explained by the time required to form
the most massive degenerate systems with the shortest MS lifetimes,
whereas the power-law tail can be explained by the power-law relation
between coalescence time and separation of the double-degenerate
systems.
However, the expected accretion rates in the DD scenario are a
problem: they are so high that present calculations suggests that this
leads to accretion-induced collapse (AIC) and the formation of a
compact object rather than a thermonuclear explosion \citep{NI85,
SN85, SN98, TWT94, nom91}.
Therefore, the currently generally most favoured progenitor scenario
is the SD scenario. Because the seemingly dominant evolutionary path
of this channel would need to be discarded if the mean time-delay were
found to be higher than 2~Gyr, it is important to confirm the
significance of the S04 results.
In this work we have studied the SN Ia time-delay distribution using
the sample of S04 and the same basic analysis, but introducing
alternative SFHs found in the literature, avoiding binning effects as
much as possible and using a Goodness of Fit (GoF) test that is
generally recommended for small samples. We discuss the data and
analysis in Section \ref{sec:analysis}, show the results and
Monte Carlo simulations in Sections \ref{sec:results} and
\ref{sec:Monte Carlo} and discuss their significance in Sections
\ref{sec:discussion} and
\ref{sec:conclusions}. Throughout his paper, we adopted a value for the
Hubble constant of $H_{\rm 0} = 70$ km s$^{-1}$ Mpc$^{-1}$, present
ratios of matter, curvature and dark energy density over the critical
density of $\Omega_M = 0.3$, $\Omega_K = 0$ and $\Omega_{\Lambda} = 0.7$,
respectively, and a `dark energy' pressure over density ratio ('equation
of state') of $w =-1$.
\section{Data and Analysis} \label{sec:analysis}
The analysis is based on the \emph{Hubble} Higher-z Supernova Search
sample \citep{rie04, dah04, str04}, which contains 25 SNe Ia found in
the GOODS field (13 in the Hubble Deep Field North, HDFN, and 12 in
the Chandra Deep Field South, CDFS) in the redshift range 0.21 to
1.55. The SNe were discovered in four difference images that were
produced by observing both fields five times in intervals of
approximately one month, comparable to the typical duration of
the main SN Ia light curve peak.
To infer the underlying time-delay distribution of SNe Ia, S04
compared the observed redshift distribution in the sample with a
parametrized predicted distribution, derived from the G04 SFH
convolved with three alternative time-delay distributions.
Each distribution was parametrized by its mean time delay, which was
recovered using a Bayesian analysis. Among these, the distribution
that best fit the data was a `narrow Gaussian', which after being
corrected was centred on 3.4~Gyr with a FWHM of $\sim 1.5$~Gyr. The 95
per cent confidence interval for the mean time delay ranged from 1.0 to
4.4~Gyr. The alternatives `wide Gaussian' and e--folding distributions
had a mean time delay above 0.2 and 1.6~Gyr, respectively, with more
than 95 per cent confidence.
Only the shapes of the distributions are compared, i.e. the analysis
is scale--free, and the associated efficiencies of SNe per unit formed
mass are calculated later by normalising the models to the SN numbers
and are not used to constrain the models. This means that the sample
must ideally span a redshift range that includes both the rising and
declining parts of the SNR, i.e. where the SNR is not approximately
linear. A recent study \citep[see][Fig.~6--8]{BT05} could not fully
exploit information on the SN redshift distribution because their
sample did not reach to a sufficiently high redshift ($z>1$), as the
authors indicate in the text. A similar situation is found in the work
of \citet{GYM04}.
Thus, because the \emph{Hubble} Higher-z Supernova Search sample is
the deepest SN sample available, it is the most suitable one for
constraining the time-delay distribution of SNe Ia.
However, the formal errors quoted in S04 reflect only the limited size
of the sample and not other systematic uncertainties, such as
those associated with the SFH.
In the following Sections \ref{sec:SNR} to \ref{sec:tc} we introduce
the formalism that gives the SNR, the number of detected SNe per unit
redshift and the control times used in the derivations. In Section
\ref{sec:timedist} we discuss alternative time-delay distributions
and in Section \ref{sec:SFH} alternative SFHs. The Bayesian
analysis is described in Section \ref{sec:Bayes} and further
modifications concerning binning effects and the GoF test are
discussed in Section \ref{sec:newanalysis}.
\subsection{The SN Ia rate} \label{sec:SNR}
The rate of SNe Ia per unit time per unit co-moving volume ($SNR_{\rm
Ia}$) is given by the star-formation rate per unit time per unit
co-moving volume ($SFR$) convolved with the normalised distribution of
explosions per unit time of the progenitor channel (the time-delay
distribution, $\phi$), and multiplied by the number of SNe per unit
formed mass (the efficiency, $\nu$). We assume that neither $\nu$ nor
$\phi$ evolve with redshift:
\begin{align}
SNR_{\rm Ia}(z) = \nu \int_{\rm t(z_R)}^{t} SFR(t')\ \phi(t - t',
\tau) \ dt' \label{eq:SNR},
\end{align}
where $t = t(z)$, $\tau$ is some characteristic time-scale defined in
Section ~\ref{sec:timedist} and $z_R$ is the redshift associated with
the time when the first stars formed, approximately the epoch of
reionisation. We assumed $z_R =10$, as in S04.
\subsection{Distribution of detected SNe} \label{sec:nIa}
The number of detected SNe Ia per unit redshift interval ($n_{\rm
Ia}$) is given by the multiplication of the rate of SNe Ia per unit
time per unit co-moving volume ($SNR_{\rm Ia}$), a time dilation
factor, $(1+z)^{-1}$, the control time of the survey ($t_{\rm c}$) and
the volume per unit redshift being surveyed, $\displaystyle{
\frac{dV}{dz d\omega} \Delta \omega}$:
\begin{align}
n_{\rm Ia}(z) = \frac{SNR_{\rm Ia}(z)}{1+z} \ t_{\rm c}(z) \
\frac{\mathrm{d}V(z)}{\mathrm{d}z \mathrm{d}\omega} \ \Delta\omega \label{eq:nIa},
\end{align}
where in our cosmology the volume derivative formula simplifies to:
\begin{align}
\frac{\mathrm{d}V}{\mathrm{d}z \mathrm{d}\omega} = d_C^2 \frac{\mathrm{d}(d_C)}{\mathrm{d}z}\text{, where}
\end{align}
\begin{align}
d_C = c H_0^{-1} \int_0^z \mathrm{d}u \left[ (1+u)^3 \Omega_M +
\Omega_\Lambda \right] ^{-1/2}
\label{eq:dV1},
\end{align}
and hence,
\begin{align}
\frac{\mathrm{d}V(z)}{\mathrm{d}z \mathrm{d}\omega} = c H_0^{-1} \left[ (1+z)^3
\Omega_M + \Omega_\Lambda \right] ^{-1/2} d_C^2(z)
\label{eq:dV2}.
\end{align}
\subsection{The control time} \label{sec:tc}
The control time can be understood as the total observing time
multiplied by the probability of detecting a SN at a given
redshift. We used the same values as S04, that were calculated taking
into account the expected extinction, spectra, light curve shapes and
peak magnitude dispersion of SNe Ia, the way each field was revisited,
and the efficiency of the detection algorithm (but see Section
\ref{sec:binning}). The control times are defined by
\begin{align}
t_{\rm c}(z) = \iiint P(t |
M_\lambda, A_\lambda, z) \ P(M_\lambda) \ P(A_\lambda) \ dM_\lambda \
\mathrm{d}A_\lambda \ \mathrm{d}t,
\end{align}
where $P(t | M_\lambda, A_\lambda, z)$ is the probability of detecting
a new SN at time t, given its rest-frame luminosity and its host
galaxy extinction and redshift. It depends on the assumed spectra
through K--corrections, the sensitivity of the survey and the
efficiency of the detection algorithm.
$P(M_\lambda)$ is the probability of having a given SN rest-frame
luminosity. It was estimated based on the characteristic relation
between peak luminosity and light curve shape of SNe, and the observed
dispersion of SN Ia peak luminosities.
$P(A_\lambda)$ is the probability of having a given host galaxy
extinction at the given rest-frame wavelength. It was assumed to be
proportional to $e^{- A_\lambda}$.
For more details see the original discussion in S04.
\subsection{Time delay distributions} \label{sec:timedist}
All the time delay distributions were parametrized by their mean time
delays, $\tau$. S04 used an exponential distribution and two families
of Gaussian distributions whose width scale with the mean time
delay. The e-folding distributions are given by:
\begin{align}
\phi(t,\tau) = \frac{e^{-t/\tau}}{\tau}.
\end{align}
The two alternative Gaussian distributions are grouped into the
families of `narrow' ($\sigma_{\tau} = 0.2 \tau$) and `wide'
($\sigma_{\tau} = 0.5 \tau$) distributions, of the form:
\begin{align}
\phi(t,\tau) = \frac{1}{\sqrt{2 \pi \sigma_{\tau}^2}} e^{ -\tfrac{(t - \tau)^2}{2
\sigma_{\tau}^2} }.
\end{align}
The previous distributions are defined only for positive
values. However, negative time delays must be allowed in order to
avoid statistical bias in a small sample and to get confidence
intervals that do not artificially discard short time
delays. Moreover, a preference for negative time delays would signal
SFHs that peak too late in time. Thus, we considered a fourth time
delay distribution, a Gaussian distribution with fixed width ($\sigma
= 0.5$~Gyr) that allows either for positive or negative time delays:
\begin{align}
\phi(t,\tau) = \frac{1}{\sqrt{2 \pi \sigma^2} }
e^{ -\tfrac{ (t-\tau)^2 }{ 2 \sigma^2 }}.
\end{align}
We also added a log-normal distribution, which is associated with
processes where the source of uncertainty has multiplicative effects
rather than additive ones, as is the case for Gaussian
distributions. The best-fitting models to the theoretical time delays
were in most cases log-normal distributions, whose width $\sigma$, in
units of $\rm{log}(t)$, was kept fixed and determined by the
best-fitting model of the theoretical time delays:
\begin{align}
\phi(t,\tau) = \frac{1} {\sqrt{2 \pi \rm{ln}(\sigma)^2}} e^{-
\tfrac{\rm{ln}(t/\tau)^2} {2\ \rm{ln}(\sigma)^2} } \frac{1}{t}.
\end{align}
The theoretical time delay distributions from \citet{HP04} were also
examined with a GoF test. They were produced assuming either the CO WD
+ MS -- SD scenario or the DD scenario with different binary evolution
parameters. In Fig.~\ref{fig:loglog-TheoryDelay} the theoretical time
delay distributions of the SD and DD scenario together with two time
delay distributions with different mean time delays are plotted,
including the best-fitting model from S04.
\subsection{The Star-Formation History (SFH)} \label{sec:SFH}
Because we consider alternative prescriptions for the SFH that have
incomplete redshift information, further complications arise. The
ideal redshift coverage of the SFH should range from zero to the
redshift of the first star formation, farther than the highest-redshift
object presently known in the Universe. This is a consequence
of the redshift range of the detected SNe and the long time delays
that have to be considered. If high-redshift SNe only exploded after
long time delays, their progenitors would need to form at redshifts up
to $\sim 30$, as Fig.~\ref{fig:SNeHisto} shows.
If the determination of the SFH did not cover the required redshift
range, we used as an approximation either a power-law extrapolation in
time or a scaled version of the optical--UV derivation. We found that
the method is not very sensitive to this approximation when the
position of the peak of the SFH is well constrained, since it is the
difference between the peaks of the SFH and the SNR that mainly
constrains the best-fitting models. The alternative prescriptions of
the SFH we have used are the following:
\begin{itemize}
\item The SFH by \citet{gia04}, G04. We have used continuous
approximations for the extinction corrected and not corrected models,
inferred from deep optical--UV observations of galaxies in the GOODS
field. Both versions differ by a factor of $\sim 3$ in the redshift
range of interest. The continuous approximations are the ones used
in S04, which peak at $z \sim 2.7$ in the extinction corrected model
(M1) and at $z \sim 1.8$ in the model that is not corrected for
extinction (M2).
\item The best-fitting model of \citet{CE01}, hereafter CE01. It was
derived from the integrated cosmic infrared background (CIRB) and
covers the redshift range from 0 to 4.5. At $\rm{z > 4.5}$ we tried
a power-law extrapolation in time or a scaled version of G04 at
$\rm{z > 3}$. Mainly because the peak of the SFH occurs very late in
time, at $z \sim 0.8$, we found that the inferred time delays are
not very sensitive to this approximation. However, a constant SFH
model is within the error bars at high--z.
\item The SFH from \citet{hea04}, hereafter H04, inferred from the
`fossil record' of stellar populations in the \emph{Sloan Digital
Sky Survey} (SDSS). We interpolated a Spline function to the binned
SFH, which peaks at $z \sim 0.4$, in order to obtain a smooth
approximation. It is not usually recommended to approximate data in
this way, but we think that our approximation preserves the general
differences between this SFH and the alternative prescriptions. We
also tried a scaled version of G04 at $\rm{z > 3}$, or a constant
star-formation history for all times, since this SFH is very flat at
$z \gtrsim 1$ and peaks very late in time with respect to the
detected SNe. Both alternatives gave very similar results to the fit
to the original binned data.
\end{itemize}
Additionally, we considered one of the most recent determinations of
the SFH using \emph{Spitzer} data, presented by \citep{PG05}. However,
both its limited redshift coverage and its dependence on the assumed
galaxy luminosity function makes the high-redshift extrapolation
ambiguous. For this reason, we did not try a continuous approximation
of this SFH in the analysis, although it must be considered a reliable
result. The variance between its different versions only demonstrates
the persisting uncertainties in our knowledge of the SFH.
In Fig.~\ref{fig:loglog-SFH} we show the four continuous
approximations of the SFH and the binned SFH from PG05. The continuous
approximations are based on the extinction corrected and not corrected
SFH from G04, the best-fitting model from CE01 with a power-law
extrapolation in time at high redshift and a continuous approximation
of the best-fitting model from H04, which is constant at high
redshift. It is apparent that there is a range of SFHs in the
literature that do not agree and, importantly, peak at very different
times. Thus, it is important to understand the systematic errors
associated with this uncertainty. For a recent estimation of the
uncertainties on the SFH see also Fig.~2 and 4 from \citet{HB06}.
\subsection{The Bayesian probability} \label{sec:Bayes}
Using Bayes theorem with a uniform prior, the probability of a mean
time delay $\tau$ with a time-delay distribution $\phi(t, \tau)$ and a
SFH in the form of $SFR(t')$, given the set of SN redshifts, $\{ z_i
\}$, is proportional to:
\begin{align}
P( SFR(t'), \phi(t, \tau), \tau \vert \{z_i\}) \varpropto P( \{z_i\} \vert
SFR(t'), \phi(t, \tau), \tau).
\end{align}
Thus, it is proportional to the probability of the particular SN
redshift distribution:
\begin{align}
P( \{z_i\} \vert SFR(t'), \phi(t, \tau), \tau) \varpropto \prod_{i=1}^{25}
n_{\rm Ia}(z, \tau),
\end{align}
where $n_{\rm Ia}(z,\tau)$ has been normalised for every $\tau$
because the analysis is scale free. It depends on $\tau$ through the
time-delay distributions of Section \ref{sec:timedist} and equations
\ref{eq:SNR} and \ref{eq:nIa}.
Hence, for a given combination of SFH, time-delay distribution and
mean time delay, the predicted number of SNe per unit redshift can be
expressed as a probability distribution in redshift. Subsequently, the
probability of the set of SNe can be calculated for every $\tau$.
\subsection{New analysis} \label{sec:newanalysis}
The main differences between the S04 calculations and this work are a
result of considering a range of alternative SFHs. Further
differences are as follows:
\subsubsection{Redshift binning effects} \label{sec:binning}
The main advantage of using the observed redshift distribution of SNe
Ia instead of the corresponding SNR \citep{GYM04} is that the analysis
can be done in a way that avoids binning and the subsequent loss of
information.
Moreover, we have found that the analysis is very sensitive to the way
the volume derivative is calculated in equation \ref{eq:nIa}. Because
the SN sample is small, binning the data is not recommended, and all
the calculations should be done continuously. Binning can introduce a
relative overestimation of the volume derivative at low redshift,
effectively pushing the most probable time delays to higher values. As
a result, the lower limit of the Bayesian analysis can be
overestimated by more than $\sim 1$~Gyr and the peak of the Bayesian
probability distribution by $\sim 500$~Myr, according to our
calculations. This is consistent with the corrected results in
\citet{str04erratum}.
However, in order to get a continuous version of the control times we
have used an interpolation of the values calculated in S04 at redshift
intervals of $0.2$. Recalculating the control times with more redshift
resolution would be a better approach, but we have not tried it in
this analysis.
\subsubsection{Goodness of fit test (GoF)}
A maximum likelihood analysis must be accompanied by a GoF test to
check that the parametrized model has an appropriate form to start
with, and only then can the confidence intervals be trusted at
all. Accordingly, a $\chi^2$ test was used to check consistency
between the predicted and observed redshift histograms of SNe Ia in
S04. However, the $\chi^2$ test is not reliable when the number of
elements per bin is not greater than five in 80 per cent of the bins
\citep{WJ03}. Instead, we have used a Kolmogorov--Smirnov test (KS
test) as our goodness of fit test, which is the recommended test to
use when the sample size is small and because the analysis is done
continuously. Furthermore, selecting the best-fitting models with the
KS test can be used as an alternative parameter estimator.
\subsubsection{Confidence intervals} \label{sec:ConfInt}
In S04 the confidence intervals were obtained starting from the mode
(the maximum) of the Bayesian probabilities, partly because in the
original results the probabilities at low time delays were negligible.
However, because in some cases the probability distributions are
relatively flat, or not negligible at zero time delay, the definition
of the confidence intervals becomes important. In our calculations,
taking 95 per cent confidence intervals around the maximum vs. from the
median can make a difference of typically $\sim 500$~Myr.
One way to avoid this problem it to use the parameter region that is
not rejected by the GoF test with a certain confidence level. In this
approach, we obtain 95 and 68 per cent confidence intervals that are
unambiguously defined.
\subsubsection{Photometric redshifts}
Because the spectra of SNe are characterised by many blended lines
broadened by high velocity dispersion, SN redshifts are determined from
their host galaxies. Of the 25 SNe Ia in the sample, six have only
photometric redshifts, three of them in each field. We found that the
photometric redshift of SN 2003al, $0.91 \pm 0.2$, has a better
estimate in the public COMBO--17 catalogue \citep{wol04} of $0.82 \pm
0.04$. Additionally, in \citet{str05}, the photometric redshift of SN
2003lu, $0.11 \pm ^{0.13}_{0.11}$, has a better estimate of $0.14 \pm
0.01$.
\section{Results} \label{sec:results}
\subsection{Varying the SFH} \label{sec:resultsSFH}
If alternative SFHs are allowed, the Bayesian probabilities
associated with a given time-delay distribution have a wide range of
preferred time delays. The Bayesian probabilities and KS test
associated rejection probabilities show a preference for values
ranging from very long ($\sim 4$~Gyr) to very short, and even negative
($\sim -3$~Gyr) if the SFH peaks very late in time (see
Fig.~\ref{fig:probs}). The relation between the peak of the SFH and
the peak of the Bayesian probability distribution is, to zeroth order,
such that later peaked SFHs give shorter time delays.
Interestingly, inspection of Fig.~\ref{fig:probs} (upper-left panel)
shows two types of maxima in the Bayesian probabilities: one whose
position decreases with later peaked SFHs and another that is fixed at
approximately $\sim 3.5$~Gyr, even for different SFHs. The first peak
approximately reflects the time difference between the peaks of the
SFH and the SNR. The second peak reflects the relative absence of SNe
at high--z. Because no SNe were detected between the epoch of
reionisation and $z \sim 1.5$, or between $t \sim 0.5$~Gyr and $t \sim
4$~Gyr, the Bayesian analysis marginally favours models that do not
produce SNe in the first $\sim 3.5$~Gyr after the assumed epoch of
reionisation. The upper plot of Fig.~\ref{fig:SNeHisto} is
illustrative of this effect.
With the current data, it is the first peak which is statistically
dominant for all the SFHs, but this may change with deeper and wider
SN surveys in the future.
\subsection{Kolmogorov--Smirnov test}
With the KS test we find best-fitting mean time
delays and confidence intervals that are free from the problems
explained in Section \ref{sec:ConfInt}. The rejection probabilities
for the `fixed width Gaussian' and e-folding time-delay
distributions are shown in Fig.~\ref{fig:probs}.
We found that all the combinations of SFH and time-delay distributions
had an associated parameter region that is accepted by the KS test,
which validates the use of the Bayesian analysis. Additionally, the
parameter estimation seems robust in the sense that it gives results
that are consistent with what the Bayesian analysis shows. Moreover,
the addition of this test shows that the negative time-delay peak for
the H04 SFH is favoured over the long time-delay peak (see
Fig.~\ref{fig:probs}), which may be an indication that this SFH is not
compatible with the SN data.
\subsection{Confidence intervals}
The confidence intervals were defined as the parameter region that
cannot be rejected with a certain confidence level based on the KS
test. We found that only the extinction corrected SFH from
\citet{gia04} has a 95 per cent confidence lower limit greater than zero,
i.e. around $1$~Gyr. All the alternative SFHs did not result in a
lower limit for the time delays greater than zero. A summary of the
confidence intervals obtained with the Gaussian and e-folding time
delay distributions is shown in Fig.~\ref{fig:summaries}.
\subsection{Varying the time-delay distribution}
The non-rejection regions of the five time-delay distributions tested
in this work can be grouped into three families of results: one family
associated with the `fixed width Gaussian' (that allows for negative
time delays) and `narrow Gaussian' distributions, another with the
`wide Gaussian' and log-normal distributions and one associated with
the e-folding distribution.
As a general rule, the narrower the test time-delay distribution, the
narrower the associated Bayesian probabilities. However, it is in the
long time-delay region where the changes are more noticeable, as can
be seen in Fig.~\ref{fig:summaries}. This is because the abrupt
transition that occurs in the SFH at the epoch of reionisation is
reflected in a less smooth SNR when narrower time-delay distributions
are assumed. Hence, the wider the time-delay distribution, the less
pronounced the second peak in the Bayesian probabilities (see Section
~\ref{sec:resultsSFH}) and the longer the time delays allowed.
\subsection{Theoretical time-delay distributions}
We performed KS tests of the theoretical time-delay distribution
varying the BPS parameters and the assumed SFHs. As a result, if the
extinction corrected SFH from G04 is assumed, the CO WD + MS -- SD
scenario alone has a 3 per cent probability of not being rejected and
the best-fitting models are obtained for a DD scenario which has a
very high mass transfer efficiency. Conversely, if any of the
theoretical models are assumed to be true, the best-fitting models
are, in almost all the combinations, obtained with the SFH from
\citet{CE01}. In Table \ref{tab:summary_models} we show the
non-rejection probabilities for the different combinations of SFH,
theoretical scenario and BPS parameters tested in this work. The BPS
parameters are: $\alpha_{\rm CE}$, the common--envelope ejection
efficiency efficiency as in \citet{HP04}; $\alpha_{\rm RLOF}$, the
Roche lobe overflow mass transfer efficiency, and $Z$, the
metallicity.
\begin{table}\centering
\caption{Summary of the KS non-rejection probabilities (per cent) for
different combinations of SFH, theoretical time-delay distribution
and BPS parameters. Unless stated otherwise, the standard parameters are
$\alpha_{\rm CE} = 1.0$~, $\alpha_{\rm RLOF} = 1.0$~ and $Z =
0.02$~.}
\begin{tabular}{@{}l ccc c ccc@{}}
\cmidrule{1-8}
&
\multicolumn{3}{c}{SD scenario -- $\alpha_{\rm{CE}}:$} & &
\multicolumn{3}{c}{DD scenario -- $\alpha_{\rm{CE}}:$} \\
\cmidrule{2-8}
SFH & 0.5 & 0.75 & 1.0 & & 0.5 & 0.75 & 1.0\\
\cmidrule{1-8}
G04 (M1) & 3.1 & 3.0 & 3.1 & & 5.7 & 11.4 & 25.9 \\
G04 (M2) & 12.2 & 11.7 & 12.4 & & 15.8 & 29.1 & 51.6 \\
CE01 & 49.8 & 47.8 & 50.8 & & 48.1 & 69.2 & 85.4 \\
H04 & 28.1 & 28.1 & 28.0 & & 25.7 & 24.2 & 21 \\
\cmidrule{1-8}
&
\multicolumn{3}{c}{DD scenario -- $Z:$} & &
\multicolumn{3}{c}{DD scenario -- $\alpha_{\rm{RLOF}}:$} \\
\cmidrule{2-8}
SFH & 0.001 & 0.004 & 0.02 & & 0.5 & 0.75 & 1.0 \\
\cmidrule{1-8}
G04 (M1) & 11.0 & 13.2 & 25.9 & & 25.9 & 47.4 & 56.5 \\
G04 (M2) & 30.1 & 31.2 & 51.6 & & 51.6 & 82.6 & 78.5 \\
CE01 & 70.2 & 71.2 & 85.4 & & 85.4 & 80.7 & 62.5 \\
H04 & 25.2 & 23.1 & 21.0 & & 21.0 & 18.1 & 17.3 \\
\cmidrule{1-8}
\end{tabular}
\label{tab:summary_models}
\end{table}
\subsection{Constraints on the SFH assuming the theoretical models}
In addition to the previous constraints on the time delay distribution
we followed the approach of \citet{GYM04} to constrain the SFH. The
method assumes a particular time delay distribution and a SFH that
consists of a broken power--law smoothly joined at the transition
redshift $z_0$, proportional to $\sim (1+z)^\alpha$ at high $z$ and to
$\sim (1+z)^\beta$ at low $z$, i.e.:
\begin{align}
SFR(z) \propto \left\{ \left( \frac{1+z_0}{1+z} \right)^{5 \alpha} +
\left ( \frac{1+z_0}{1+z} \right) ^{5 \beta} \right\} ^{-1/5}.
\label{eq:SFRGY}
\end{align}
For more details see \citet{GYM04}. We assumed the theoretical time
delay distributions and performed a KS test for different combinations
of power--law indices and transition redshift, $z_0$. We marginalised
the probabilities on $\alpha$ because it was found that the dependency
on this parameter is weak, which is expected because the SN sample
contains very few objects at high redshift. The resulting probability
distributions for different values of $z_0$ and $\beta$, assuming
either the SD or DD scenario is shown in
Fig.~\ref{fig:KS_z0beta}. Virtually all measurements of the SFH
suggest $SFR(z) \propto (1+z)^\beta$, with $\beta$ in the range
$\sim$~2 to 4 \citep{wil02, PG05}. For $\beta \gtrsim 2$ we find that
the peak of the SFH is most likely between $z \sim 0.7$ and
$1.2$. However, any rejection at 90 per cent confidence is only
possible for a peak location at $z \gtrsim 2$. The flatter the recent
decline in the SFH (lower $\beta$), the less clearly constrained is
the peak of the SFH.
\section{Monte Carlo simulations} \label{sec:Monte Carlo}
In this section we examine whether our approach is subject to systematic
biases due to the method itself and the choice of the SFH.
To assess the robustness of the method, we performed Monte Carlo
simulations for each progenitor scenario separately, where we took
large sets of simulated SNe (typically 10,000) to minimize statistical
errors. Because the DD scenario can extend to times comparable to the
age of the universe, its time delay distribution cannot be sampled
completely; we therefore expect that the recovered values will always
be biased towards shorter mean time delays. We found the following:
(1) such an expected bias is indeed present for the DD scenario, but
is always below $0.5$~Gyr; (2) for large sample sizes, the smallest
statistical errors are obtained using the e-folding time delay
distribution, independent of the theoretical scenario assumed.
To quantitatively estimate the systematic errors associated with the
choice of SFH, we proceeded in the following way: (1) in order to
minimize the statistical errors and study the systematics, we
performed Monte Carlo simulations with 10,000 mock SNe drawn from the
theoretical scenarios, using the e-folding time-delay distribution in
the analysis; (2) we produce a mock sample of SNe with a SFH that
differs from the one used in the Bayesian analysis; (3) a comparison
between the different recovered mean time delays then gives an
estimate of the systematic error due to the choice of SFH (see
Fig.~\ref{fig:Monte Carlo} and Fig.~\ref{fig:Monte Carlo2}). The result
is consistent with what can be concluded from
Section~\ref{sec:results}, i.e. that the bias on the mean time delay
can be of the order of $\sim 2$~Gyr, even ignoring the H04 SFH.
We then tested the robustness of the algorithm with small samples. In
order to do this, we performed Monte Carlo simulations with 10,000
different sets of 25 mock SNe drawn from the theoretical models. We
repeated the Bayesian and GoF analysis and found the following: (1)
when the theoretical time delays are relatively short and the SFH
peaks early in time, or when the time delays are long and the SFH
peaks late in time, the Bayesian probabilities tend to have very
positive or very negative skewness and, consequently, the mode of the
Bayesian probabilities either overestimates or underestimates the
theoretical mean time delays; (2) the distribution of non-rejection
probabilities using the KS test is flat, meaning that there is no
significant bias towards short or long time delays. Hence, we conclude
that the mode of the Bayesian probability is not a good estimator for
the mean time delays, while the KS test confidence intervals are
robust even with small samples, general results consistent with the
discussion in \citet{pre92}.
\section{Discussion} \label{sec:discussion}
\subsection{Large scale structure}
Large-scale structure (LSS) effects are important in small pencil beam
surveys when studying the time delay of SNe Ia, even when the SFH and
the SNR have been measured in the same field. Usually, the SFR is
measured as an `instantaneous' observed rate at a particular location
in space and time, where for this study a prediction of the SNR will
be needed for comparison. However, this prediction ought to be based
on the SFR in the same position in space, not only at the time when
the SNe are seen to explode, but at earlier times as well.
Since the latter figure can not be measured directly, we have to
obtain the SFR for earlier times by looking at a higher redshift,
i.e. at a different location in space. Cosmic variance and the
dependence of star formation on local environment will lead to a SFR
measurement that is somewhat different from the past SFR of the
target location in the same field. This SFR variance will appear
whether the SFR is measured from the same fields or not, since the
locations in space where the SFR and SNR are measured will be unrelated. A
1~Gyr time delay already translates into proper distance differences
of $\sim 500$~Mpc at $z=1$.
So, if we do not know the individual star-formation histories of
galaxies in the supernova field, we are best advised to use our best
knowledge of the cosmic SFH, while LSS still leaves an imprint on the
observed SNR history. We investigate the large scale structure in the
CDFS where the spectroscopic redshift survey \emph{VIMOS VLT Deep
Survey} \citep[][VVDS]{LF04} overlaps almost precisely with the
GOODS field and is complete to $M_{\rm{V}} < -19.5$ at $z<1$. In
Fig.~\ref{fig:vvds_histo} we show a histogram of galaxy redshifts with
$M_{\rm{V}}<-19.5$ in bin sizes chosen to contain a constant co-moving
volume of $(25\ \rm{Mpc})^3$. A non-evolving and homogeneous galaxy
distribution should appear flat in this representation. Well-known
over-densities or wall-like structures are clearly apparent,
especially in the redshift range from 0.5 to 0.75. Two wall-like
structures near $z \approx 0.67$ and $z \approx 0.74$ are conspicuous
in the distribution and have been observed in both x-ray and optical
surveys \citep[see e.g.][for independent confirmation]{G03, wol04}.
Also shown is a histogram of SNe Ia with secure (spectroscopic)
redshift determinations. SNe with photo-z's only have been omitted due
to the large error in redshift which makes any association with
structures in the galaxy distribution difficult.
The average bin in the redshift range from 0 to 1 contains 7.8
galaxies, whereas the average SNe-weighted bin contains 16.7 galaxies
(2.14 times the normal average), a possible indication that the
inferred SNR is affected by LSS in this field and redshift range. In
order to understand how these numbers depend on the choice of our
bins, we have repeated the calculations for bin widths of $(\Delta
V)^{1/3} = (20,21.. 25)$~Mpc and found ratios of 2.52, 2.38, 2.15,
2.22, 2.00 and 2.14, respectively. Interestingly, the ratios that
result using only SNe Ia-weighted bins are even higher: 2.87, 2.68,
2.88, 3.29, 2.55 and 2.97.
The imprint of LSS in the SNR distribution could be corrected in the
analysis of time delays. If the underlying mass density on a given
direction $\hat n$ were of the form $\rho(\hat n,z)=\rho_0(z) (1 +
\delta(\hat n,z))$, the over-densities could be corrected by
multiplying the \emph{control times} by the same factor, $1 +
\delta(\hat n, z)$. However, we do not correct for density
variations, because we lack a good determination of the LSS with
consistent quality in both fields.
\subsection{SN rates and efficiencies}
In the process of matching the observed number of SNe with a model, it
is possible to explicitly calculate the SN production efficiencies and
the supernova rate (SNR). Not only does the shape of the SNR history
constrain the progenitor models via the delay times, but also the SN
efficiency must be compatible with the model.
As a first consistency test, we compare the directly measured SNRs
with the parametrized estimates. In Fig.~\ref{fig:SNR} we show two
versions using the results of the parameter estimation with the G04
(M1) and CE01 SFHs. Clearly, the extinction corrected SFH from G04
with the `narrow Gaussian' distribution is consistent with long time
delays ($\sim 3.5$~Gyr), whereas the SFH from CE01 matches best with
an e-folding distribution of shorter time delays ($\sim 1.5$~Gyr).
The second alternative is only marginally favoured by the KS test, so
both possibilities seem equally plausible.
It is important to mention that in \citet{str05} a deeper SN search in
the smaller Ultra Deep Field and its parallel fields (UDF/P) resulted
in the detection of four additional SNe with $z<1.4$. The lack of
high-redshift SNe is one of the predictions of the S04 best-fitting
model and hence it supports the result. However, the authors also
considered the SFH from CE01, but without any time delay, and
concluded that it is not possible to rule out this SFH with more than
50 per cent confidence.
The SN efficiencies required to explain the observed SNR pose a problem
for the theoretical SD scenario, which produces too few SNe. This is true
for all SFHs and combination of parameters, and amounts to a shortfall of
$4\times$ to $10\times$ depending on the assumed SFH. The DD scenario
can reproduce the required efficiencies for some combinations of BPS
parameters. However, changes in the little constrained binary fraction
and mass ratio distributions, or possibly in the initial mass function
(IMF), may solve this problem in the future.
One additional challenge is that the time-delay distribution or the
supernova efficiencies may evolve with time: if the delay distribution
is made up of several components from different SN production
channels, their relative contribution could change with time; the
supernova efficiencies themselves may evolve with time, especially
considering that the accretion processes could depend on environmental
factors such as the metallicity \citep{kob98}. The relative absence of
SNe at high redshift could be a result of a metallicity effect, rather
than a reflection of the time delays. If this is truly the case, our
error bars on the time delays would be largely underestimated. The use
of compatible star formation and chemical enrichment histories will be
required to tackle this problem.
\subsection{\emph{Spitzer} SFH}
We have shown that it is crucial to determine the SFH that is to be
used in the analysis. The recent determination of the SFH from
\citet{PG05}, using three different extrapolations of the galaxy
luminosity function, was shown in Fig.~\ref{fig:loglog-SFH}. This SFH
was obtained by combining infrared \emph{Spitzer} observations with
optical--UV data of galaxies in the GOODS field. The resemblance
between this SFH and the CE01 best-fitting model justifies the use of
the latter.
Interestingly, the recent near--infrared and sub--millimetric
determination of the SFH from \citet{WCB05} peaks at $z \sim 1$,
similarly to what is obtained in CE01. These new results emphasize the
persistent uncertainties in our knowledge of the SFH and the problem
of extinction corrections in the optical.
\section{Conclusions} \label{sec:conclusions}
We have found that systematic errors associated with the use of
alternative star-formation histories are comparable if not larger than
the statistical errors reported in \citet{str04}. The position of the
peak of the SFH was found to be the crucial parameter for the
recovered time delays: Later peaked SFHs result in lower time delays
and vice versa. Furthermore, the confidence intervals for the time
delays depend on the functional form of the delay distributions
assumed in the analysis. The use of wider time-delay distributions, in
particular the e-folding model, gives considerably longer upper limits
for the time delays.
For the data set under investigation, we found that the KS test is
better suited to obtain confidence intervals than a Bayesian analysis.
The KS test confidence intervals are unambiguously defined, and we have
confirmed their validity using Monte Carlo simulations. In contrast, the
skewness of the Bayesian probabilities and the small sample size result
in somewhat arbitrary confidence intervals.
A KS test using the shape of the theoretical time-delay distributions
shows that the extinction corrected model from G04 is incompatible with
the CO WD + MS -- SD scenario, although other SFHs are compatible. The
DD scenario cannot be rejected at 95 per cent confidence with any combination
of SFH and time-delay distribution. Starting from theoretical time-delay
distributions, they consistently favour the SFH from CE01.
If we wish to constrain the time-delay distribution and possibly discard
progenitor scenarios from the redshift distribution of supernovae, it is
of foremost importance to determine the cosmic star-formation history
more accurately. Otherwise, the uncertainty in the SFH will continue to
limit the interpretation of SN data sets of any conceivable size.
Secondly, we would need a better understanding of our progenitor
models: if we ignore factors affecting efficiencies, their cosmic
evolution will make the delay time distributions evolve and produce a
situation where it is hard to disentangle these different effects
without qualitatively different observations.
Only after having solved these issues would deeper and wider surveys help to
constrain the delay times and progenitor models by decreasing statistical
noise and reducing the influence of environmental cosmic variance on the
supernova samples.
Different observations have already produced some evidence for shorter
time delays: Recent work by \citet{man05} showed that the most
efficient host galaxies for SNe Ia production among all galaxies are
irregulars, and \citet{dv05} showed that among elliptical galaxies it
is in particular the radio-loud galaxies, which are also believed to
be associated with recent star formation. Also, \citep{gar05} have
shown that SNe Ia of normal luminosity occur particularly in those
elliptical galaxies with quite substantial star-formation rates. Also,
SNe Ia in galaxy clusters seem to indicate time delays that are
shorter than 2~Gyr \citep{MGY04}.
Finally, it is clear that, if the SNe Ia phenomenon is composed of several
production channels, all conclusions we have drawn apply to the dominant
channel. A generally less common channel with different characteristics
could again be the dominant channel in a subset of galaxies with older-age
or higher-metallicity stellar populations. It is already established that
under-luminous 91bg-type SNe Ia preferably occur in non-star-forming hosts,
such as ellipticals. It remains to be explored theoretically, whether the
CO WD + RG -- SD scenario could be related specifically to old populations.
We should also consider the possibility that we may not even have found
the dominant progenitor channel for normal SNe Ia \citep{ham03, tou05}.
Even if the SFH peaks at redshift $\sim 1$ and the recovered time
delays are consequently low, the associated SNR (see
Fig.~\ref{fig:SNR}, lower panel) tends to be over-estimated in the
highest-z bin. This effect could be interpreted as the signature of a
long time delay component that does not contribute to the total SNR
when the universe is too young. However, because the highest-z bin
only contains two SNe, this does not lead to low non-rejection
probabilities. A study of bimodal time delay distributions could only
be done with this method if the uncertainties in the SFH were
significantly reduced and the metallicity cutoff on the efficiencies
was properly quantified.
If different channels produce SNe Ia from progenitors of distinctly
different age or metallicity, then an increase of the supernova sample
could greatly help the identification of the various plausible
progenitors. However, such a data set would most be beneficial if it
is complemented with host galaxy characterisation
\citep[see][]{vdb05} and spectra with better signal
\citep[see][]{ben05}.
\section*{Acknowledgments}
We are indebted to Louis-G.Strolger for fruitful discussions and
providing us with the control times required in the analysis. We also
thank Guillaume Blanc, Ranga-Ram Chary, Ben Panther, Alan Heavens and
Pablo P\'erez-Gonz\'alez for discussions related to the SFH and Klaus
Meisenheimer and an anonymous referee for comments that significantly
improved the manuscript. F.F. was supported by a Fundaci\'on Andes --
PPARC Gemini studentship. C.W. was supported by a PPARC Advanced
Fellowship. This work was in part supported by a Royal Society
UK-China Joint Project Grant (Ph.P and Z.H.), the Chinese National
Science Foundation under Grant Nos. 10521001 and 10433030 (Z.H.) and a
European Research \& Training Network on Type Ia Supernovae
(HPRN-CT-20002-00303).
|
Title:
General stability criterion of inviscid parallel flow |
Abstract: A more restrictively general stability criterion of two-dimensional inviscid
parallel flow is obtained analytically. First, a sufficient criterion for
stability is found as either $-\mu_1<\frac{U''}{U-U_s}<0$ or
$0<\frac{U''}{U-U_s}$ in the flow, where $U_s$ is the velocity at inflection
point, $\mu_1$ is the eigenvalue of Poincar\'{e}'s problem. Second, this
criterion is generalized to barotropic geophysical flows in $\beta$ plane.
Based on the criteria, the flows are are divided into different categories of
stable flows, which may simplify the further investigations. And the
connections between present criteria and Arnol'd's nonlinear criteria are
discussed. These results extend the former criteria obtained by Rayleigh,
Tollmien and Fj{\o}rtoft and would intrigue future research on the mechanism of
hydrodynamic instability.
| https://export.arxiv.org/pdf/physics/0601043 |
\preprint{APS/123-QED}
\title{General stability criterion of two-dimensional inviscid parallel flow }
\author{Liang Sun}
\email{sunl@ustc.edu.cn; sunl@ustc.edu} \affiliation{Dept. of
Modern Mechanics, and School of Earth and Space Sciences,
\\
University of Science and
Technology of China, Hefei, 230026, China.}
\date{\today}
\pacs{47.20.-k, 47.20.Cq, 47.20.Ft, 47.15.Ki }
The stability due to shear in the flow is one of the
fundamental and the most attracting problems in many fields, such
as fluid dynamics, astrophysical fluid dynamics, oceanography,
meteorology et al. The shear instability has been intensively
investigated, which is to the greatly helpful understanding of
other instability mechanisms in complex shear flows. For the
inviscid parallel flow with horizontal velocity profile of $U(y)$,
the general way is to investigate the growth of linear
disturbances by means of normal mode expansion, which leads to the
famous Rayleigh's equation \cite{Rayleigh1880}. Using this
equation, Rayleigh \cite{Rayleigh1880} first proved a necessary
criterion for instability, i.e., Inflection Point Theorem. Then,
Fj{\o}rtoft \cite{Fjortoft1950} found a stronger necessary
criterion for instability. These criteria are well known and have
been applied to understanding the mechanism of hydrodynamic
instability \cite{Drazin1981,Huerre1998,CriminaleBook2003}.
Unfortunately, both criteria are only necessary criteria for
instability, except for some special cases of the symmetrical or
monotone velocity profiles. Tollmien \cite{Tollmien1935} gave a
heuristic result that the criteria are also sufficiency for
instability in these special cases.
The stable criteria also provide a way to categorize the velocity
profiles of the flows. According to Rayleigh's criterion, the
flows are stable if $U''(y)\neq 0$, where $U''(y)$ denotes
$d^2U/dy^2$. And according to Fj{\o}rtoft's criterion, there is
another kind of stable flows if $U''(U-U_s)>0$, where $U_s$ is the
velocity at the inflection point $U''_s=0$. Then if
$U''(U-U_s)<0$, can the flow still be stable? Is there another
kind of stable flows besides the above flows? To answer these
questions, a more restrictive criterion is needed. And the
criterion itself is important for both theoretic researches and
real applications. The aim of this letter is to obtain such a
stability criterion. and other instabilities may be understood via
the investigation here.
For this purpose, Rayleigh's equation for an inviscid parallel
flow is employed
\cite{Rayleigh1880,Drazin1981,Huerre1998,SchmidBook2000,CriminaleBook2003}.
For a parallel flow with mean velocity $U(y)$, the streamfunction
of the disturbance expands as a series of waves (normal modes)
with real wavenumber $k$ and complex frequency
$\omega=\omega_r+i\omega_i$, where $\omega_i$ denotes the grow
rate of the waves. The flow is unstable if and only if
$\omega_i>0$. We study the stability of the disturbances by
investigating the growth rate of the waves, this method is known
as normal mode method. The amplitude of waves, namely $\phi$,
satisfies
\begin{equation}
(\phi''-k^2 \phi)-\frac{U''}{(U-c)}\phi=0,
\label{Eq:stable_parallelflow_RayleighEq}
\end{equation}
where $c=\omega/k=c_r+ic_i$ is the complex phase speed. The real
part of complex phase speed $c_r=\omega_r/k$ is the wave phase
speed. In fact, Rayleigh's equation is the vorticity equation of
the disturbance \cite{Drazin1981,Huerre1998}. This equation is to
be solved subject to homogeneous boundary conditions
\begin{equation}
\phi=0 \,\, at\,\, y=a,b.
\label{Eq:stable_parallelflow_RayleighBc}
\end{equation}
There are three main categories of boundaries: (i) enclosed
channels with both $a$ and $b$ being finite, (ii) boundary layer
with either $a$ or $b$ being infinite, and (iii) free shear flows
with both $a$ and $b$ being infinite.
It is obvious that the criterion for stability is $\omega_i=0$
($c_i=0$), for that the complex conjugate quantities $\phi^*$ and
$c^*$ are also a physical solution of
Eq.(\ref{Eq:stable_parallelflow_RayleighEq}) and
Eq.(\ref{Eq:stable_parallelflow_RayleighBc}). Multiplying
Eq.(\ref{Eq:stable_parallelflow_RayleighEq}) by the complex
conjugate $\phi^{*}$ and integrating over the domain $a\leq y
\leq b$, we get the following equations
\begin{equation}
\displaystyle\int_{a}^{b}
[(\|\phi'\|^2+k^2\|\phi\|^2)+\frac{U''(U-c_r)}{\|U-c\|^2}\|\phi\|^2]\,
dy=0%
\label{Eq:stable_parallelflow_Rayleigh_Int_Rea}
\end{equation}
and
\begin{equation}
\displaystyle c_i\int_{a}^{b}
\frac{U''}{\|U-c\|^2}\|\phi\|^2\,dy=0.
\label{Eq:stable_parallelflow_Rayleigh_Int_Img}
\end{equation}
Rayleigh used only
Eq.(\ref{Eq:stable_parallelflow_Rayleigh_Int_Img}) to prove his
theorem. Fj\o rtoft noted that
Eq.(\ref{Eq:stable_parallelflow_Rayleigh_Int_Rea}) should also be
satisfied, then he obtained his necessary criterion. To find a
more sufficient criterion, we shall investigate the conditions for
$c_i=0$. Unlike the former investigations, we consider this
problem in a totally different way: if the velocity profile is
stable ($c_i=0$), then the hypothesis $c_i\neq0$ should result in
contradictions in some cases. Following this, some more
restrictive criteria can be obtained.
To find a stronger criterion, we need to estimate the ratio of
$\int_{a}^{b} \|\phi'\|^2 dy$ to $\int_{a}^{b} \|\phi\|^2 dy$.
This is known as Poincar\'{e}'s problem:
\begin{equation}
\int_{a}^{b}\|\phi'\|^2 dy=\mu\int_{a}^{b}\|\phi\|^2 dy,
\label{Eq:stable_parallelflow_Poincare}
\end{equation}
where the eigenvalue $\mu$ is positive definition for any $\phi
\neq 0$. The smallest eigenvalue value, namely $\mu_1$, can be
estimated as $\mu_1>(\frac{\pi}{b-a})^2$, like Tollmien
\cite{Tollmien1935} did.
Then using Poincar\'{e}'s relation
Eq.(\ref{Eq:stable_parallelflow_Poincare}), a new stability
criterion may be found: the flow is stable if
$-\mu_1<\frac{U''}{U-U_s}<0$ everywhere.
To get this criterion, we introduce an auxiliary function
$f(y)=\frac{U''}{U-U_s}$, where $f(y)$ is finite at the inflection
point. We will prove the criterion by two steps. At first, we
prove proposition 1: if the velocity profile is subject to
$-\mu_1<f(y)<0$, then $c_r\neq U_s$.
\iffalse Proof: Otherwise, $f(y)$
\begin{equation}
-\mu_1<\frac{U''}{U-U_s}=\frac{U''(U-U_s)}{(U-U_s)^2}\leq\frac{U''(U-U_s)}{(U-U_s)^2+c_i^2},
\end{equation}
\fi
Proof: Since $-\mu_1<f(y)<0$, then
\begin{equation}
-\mu_1<\frac{U''}{U-U_s}=\frac{U''(U-U_s)}{(U-U_s)^2}\leq\frac{U''(U-U_s)}{(U-U_s)^2+c_i^2}.
\label{Eq:stable_parallelflow_Rayleigh_inequ}
\end{equation}
Substitution of $c_r=U_s$ and
Eq.(\ref{Eq:stable_parallelflow_Rayleigh_inequ}) into
Eq.(\ref{Eq:stable_parallelflow_Rayleigh_Int_Rea}) results in
\begin{equation}
\displaystyle\int_a^b
[\|\phi'\|^2+k^2\|\phi\|^2+\frac{U''(U-U_s)}{\|U-c\|^2}\|\phi\|^2]\,
dy > 0.
\end{equation}
This contradicts
Eq.(\ref{Eq:stable_parallelflow_Rayleigh_Int_Rea}). So proposition
1 is proved.
Then, we prove proposition 2: if $-\mu_1<f(y)<0$ and $c_r\neq
U_s$, there must be $c_i^2=0$.
Proof: If $c_i^2\neq0$, then multiplying
Eq.(\ref{Eq:stable_parallelflow_Rayleigh_Int_Img}) by
$(c_r-U_t)/c_i$, where the arbitrary real constant $U_t$ does not
depend on $y$, and adding the result to
Eq.(\ref{Eq:stable_parallelflow_Rayleigh_Int_Rea}), it satisfies
\begin{equation}
\displaystyle\int_a^b
[(\|\phi'\|^2+k^2\|\phi\|^2)+\frac{U''(U-U_t)}{\|U-c\|^2}\|\phi\|^2]\,
dy=0.
\label{Eq:stable_parallelflow_Sun_Int} \end{equation}
But the above Eq.(\ref{Eq:stable_parallelflow_Sun_Int}) can not
hold for some special $U_t$. For example, let $U_t=2c_r-U_s$, then
there is $(U-U_s)(U-U_t)<\|U-c\|^2$, and
\begin{equation}
\frac{U''(U-U_t)}{\|U-c\|^2}=
f(y)\frac{(U-U_s)(U-U_t)}{\|U-c\|^2}>-\mu_1.
\label{Eq:stable_parallelflow_Sun_Ust}
\end{equation}
This yields
\begin{equation} \int_a^b
\{\|\phi'\|^2+[k^2+\frac{U''(U-U_t)}{\|U-c\|^2}]\|\phi\|^2\} dy>0,
\end{equation}
which also contradicts Eq.(\ref{Eq:stable_parallelflow_Sun_Int}).
So proposition 2 is also proved.
Using 'proposition 1: if $-\mu_1<f(y)<0$ then $c_r\neq U_s$' and
'proposition 2: if $-\mu_1<f(y)<0$ and $c_r\neq U_s$ then $c_i =
0$', we find a stability criterion. If the velocity profile
satisfies $-\mu_1<\frac{U''}{U-U_s}<0$ everywhere in the flow, it
is stable. Moreover, the above proof is still valid for $0<f(y)$,
which is equivalent to Fj\o rtoft's criterion. Thus we have the
following theorem.
Theorem 1: If the velocity profile satisfies either
$-\mu_1<\frac{U''}{U-U_s}<0$ or $0<\frac{U''}{U-U_s}$, the flow is
stable.
This criterion is more restrictive than Fj\o rtoft's criterion. As
known from Fj\o rtoft's criterion, the necessary condition for
instability is that the base vorticity $\xi=-U'$ has a local
maximal in the profile. Noting that $U''/(U-U_s)\approx
\xi_s''/\xi_s$ near the inflection point, where $\xi_s$ is the
vorticity at inflection point, it means that the base vorticity
$\xi$ must be convex enough near the local maximum for
instability, i.e., the vorticity should be concentrated somewhere
in the flow for instability. A simple example can be given by
following Tollmien's way \cite{Tollmien1935}. As shown in
Fig.\ref{Fig:vorticity_profile}, there are three vorticity
profiles within the interval $-1\leq y\leq 1$, which have local
maximal at $y=0$. Profile 2 ($U=-2\sin(\pi y/2)/\pi$) is neutrally
stable, while profile 1 ($U=-\sin(y)$) and profile 3
($U=-\sin(2y)/2$) are stable and unstable, respectively.
Moreover, the stabile criterion for the parallel inviscid flows
can be applied to the barotropic geophysical flows in $\beta$
plane, like Kuo did \cite{KuoHL1949}. This is a generalized stable
criterion, we state it as a new theorem.
Theorem 2: The flow is stable, if the velocity profile satisfies
either $-\mu_1<\frac{U''-\beta}{U-U_s}<0$ or
$0<\frac{U''-\beta}{U-U_s}$ in the flow, where $U_s$ is the
velocity at the point $U''=\beta$.
The criteria proved above may shed light on the investigation of
vortex dynamics. Both Theorem 1 and
Fig.\ref{Fig:vorticity_profile} show that it is the vorticity
profile rather than the velocity profile that dominates the
stability of the flow. This means that the distribution of
vorticity dominates the shear instability in parallel inviscid
flow, which is essential to understanding the role of vorticity in
fluid. So an unstable flow might be controlled just by adjusting
the vorticity distribution according to present results. This is
an very fascinating problem, but can not be discussed in detail
here.
To show the power of the criteria obtained above, we consider the
stability of velocity profile $U=\tanh(\alpha y)$ within the
interval $-1\leq y\leq 1$, where $\alpha$ is a constant. This
velocity profile is an classical model of mixing layer, and has
been investigated by many researchers (see
\cite{Huerre1998,SchmidBook2000,CriminaleBook2003} and references
therein). Since $U''(U-U_s)=-2\alpha^2\tanh^2(\alpha
y)/\cosh^2(\alpha y) <0$ for $-1\leq y\leq 1$, it might be
unstable for any $\alpha$ according to both Rayleigh's and Fj\o
rtoft's criteria. But it can be derived from Theorem 1 that the
flow is stable for $\alpha^2<\pi^2/8 $. For example, we choose
$\alpha_1=1.1$ and $\alpha_2=1.3$ for velocity profiles $U_1(y)$
and $U_2(y)$. The growth rate of the profiles can be obtained by
Chebyshev spectral collocation method \cite{SchmidBook2000} with
100 collocation points, as shown in Fig.\ref{Fig:Growth}. It is
obvious that $c_i=0$ for $U_1$ and $c_i>0$ for $U_2$, which agrees
well with the criteria obtained above. This is also a
counterexample that Fj\o rtoft's criterion is not sufficient for
instability. So this new criterion for stability is more useful in
real applications.
The present stable criteria give a affirmative answer to the
questions at the beginning, i.e., there are some stable flows if
$U''(U-U_s)<0$. Based on the former criteria, the velocity
profiles can be categorized as follows: (\romannumeral1) without
inflection point (Reyleigh's criterion), (\romannumeral2) $f(y)>0$
(Fj{\o}rtoft's criterion), and (\romannumeral3) $\mu_1<f(y)<0$
(present criterion). Then the flow might be unstable only for
$f(y)<\mu_1$ and $f(y)$ changing sign within the interval.
However, if $f(y)$ changes sign somewhere within the interval
$[a,b]$, then the flow is stable. For that $f(y)$ changing sign
implies $U'''_s=0$ but $U''''_s\neq0$, so $U''$ does not change
sign near the inflection point. Thus $c_i$ must vanish in
Eq.(\ref{Eq:stable_parallelflow_Rayleigh_Int_Img}), i.e., the flow
is stable for $f(y)$ changing sign within the interval. In this
way, the flow might be unstable only for $f(y)<\mu_1$ somewhere,
which will intrigue further studies on this problem. In fact,
there are still stable flows if $\mu_1<f(y)$ is violated.
Recall the proof of theorem 1, it is found that the following
Rayleigh's quotient $I(f)$ plays a key role in determination the
stability of the flows.
\begin{equation}
I(f)=\min_{\phi} \frac{\int_{a}^{b}
[\,\|\phi'\|^2+f(y)\|\phi\|^2\,]\, dy}{\int_{a}^{b} \|\phi\|^2}
\label{Eq:stable_paralleflow_sun_Energy}
\end{equation}
Noting that the proof of theorem 1 is still valid in the case of
$I(f)>0$. We have such result: the flows are stable if $I(f)>0$.
Though this criterion is more restrictive than that in theorem 1,
it is inconvenient for the real applications due to unknown value
of Rayleigh's quotient $I(f)$. Theorem 1 is more convenient for
the real applications in different research fields.
The idea of categorization the velocity profiles of the flows may
simplify the investigation of stability problem. It can be seen
from Rayleigh's equation
Eq.(\ref{Eq:stable_parallelflow_RayleighEq}) that the stability of
profile $U(y)$ is not only Galilean invariant, but also
independent from the the magnitude of $U(y)$ due to linearity. So
the stability of $U(y)$ is the same as that of $AU(y)+B$, where
$A$ and $B$ are any arbitrary nonzero real numbers. As the value
of $U''(U-U_s)$ in Fj\o rtoft's criterion is only Galilean
invariant but not magnitude free, it satisfies only part of the
Rayleigh's equation's properties. On the other hand the value of
$U''/(U-U_s)$ satisfies both conditions, this is the reason why
the criteria in both Arnol'd's theorems and present theorems are
the functions of $U''/(U-U_s)$. Since the stability of inviscid
parallel flow depends only on the velocity profile's geometry
shape, namely $f(y)$, and the magnitude of the velocity profile
can be free, then the instability of inviscid parallel flow could
be called "geometry shape instability" of the velocity profile. As
the above investigation shows that the inviscid shear instability
is only associated with the geometry of velocity profile. The
concept of "geometry shape instability" would be help in further
investigations. This distinguishes from the viscous instability,
which is also associated with the magnitude of the velocity
profile.
As mentioned above, we have investigated the stability of the
flows via Rayleigh's equation, while Arnol'd \cite{Arnold1969}
considered the hydrodynamic stability in a totally different way.
He studied the conservation law of the inviscid flow via Euler's
equations and found two nonlinear stability conditions by means of
variational principle. So what is the relationship between the
linear criteria and the nonlinear ones?
It is very interesting that the linear stability criteria match
Arnol'd's nonlinear stability theorems very well. Applying
Arnol'd's First Stability Theorem to parallel flow, the stable
criterion is $0<C_1<(U-U_s)/U''<C_2<\infty$ everywhere in the
flow, where $C_1$ and $C_2$ are constants. This corresponds to
Fj\o rtoft's criterion for linear stability, and is well known
\cite{Drazin1981,Dowling1995}. Here we find that Theorem 1 proved
above corresponds to Arnol'd's Second Stability Theorem, i.e., the
stable criterion is $0<C_1<-(U-U_s)/U''<C_2<\infty$ everywhere in
the flow. Given $C_1=1/\mu_1$, Arnol'd's Second Stability Theorem
is equivalent to Theorem 1. Moreover, the proofs here are similar
to Arnol'd's variational principle method. For the arbitrary real
number $U_t$, which is like a Lagrange multiplier in variational
principle method, plays a key role in the proofs. So that the
above Theorem 1 is similar to Arnol'd's theorems.
Unfortunately, Arnol'd's nonlinear stability theorems, though
quite useful in the geophysical flows \cite{Dowling1995}, are
seldom known by the scientists in other fields. The main reason is
that the proofs of Arnol'd's theorems are very advanced and
complex in mathematics for most general scientists in different
fields to understand. Although Dowling \cite{Dowling1995}
suggested that Arnol'd's idea need to be added to the general
fluid-dynamics curriculum, his suggestion has not been followed
even 10 years later. Compare with Arnol'd's theorems, the
theorems proved here are equivalent in some sense but much simpler
and easier to understand, therefore it is more convenient to use
our new results in applications.
In summary, the general stability criteria are obtained for
inviscid parallel flow. These results, which are equivalent to
Arnol'd's nonlinear theorems, extend the former theorems proved by
Rayleigh, Tollmien and Fj\o rtoft. Based on the criteria, the
velocity profiles are divided into different categories, which may
simplify the further investigations. In general, these criteria
would intrigue future research on the mechanism of hydrodynamic
instability and to understand the mechanism of turbulence. And it
also sheds light on the flow control and investigation of the
vortex dynamics.
The author thanks Prof. Sun D-J at USTC, Dr. Yue P-T at UBC
(Canada) and two anonymous referees for their useful comments.
This work was original from author's dream of understanding the
mechanism of instability in the year 2000, when the author was a
graduated student and learned the course of hydrodynamic stability
by Prof. Yin X-Y at USTC.
\iffalse
\bibliography{MSH1}
\fi
|
Title:
Stellar Multiplicity and the IMF: Most Stars Are Single |
Abstract: In this short communication I compare recent findings suggesting a low binary
star fraction for late type stars with knowledge concerning the forms of the
stellar initial and present day mass functions for masses down to the hydrogen
burning limit. This comparison indicates that most stellar systems formed in
the galaxy are likely single and not binary as has been often asserted. Indeed,
in the current epoch two-thirds of all main sequence stellar systems in the
Galactic disk are composed of single stars. Some implications of this
realization for understanding the star and planet formation process are briefly
mentioned.
| https://export.arxiv.org/pdf/astro-ph/0601375 | .
\begin{document}
\title{Stellar Multiplicity and the IMF: Most Stars Are Single}
\author{Charles J. Lada\altaffilmark{1}}
\altaffiltext{1}{Harvard-Smithsonian Center for Astrophysics, 60 Garden Street,
Cambridge, MA 02138, USA; clada@cfa.harvard.edu}
\keywords{stars: binary, formation}
\section{Introduction} \label{sec:introduction}
Ever since Mitchell (1767) pointed out that the observed frequency of visual
double stars was too high to be due to random chance, the study of binary stars
has occupied an important place in astrophysics. William Herschel (1802)
discovered and cataloged hundreds of visual pairs and produced the first
observations of a rudimentary binary orbit. In doing so he established that the
double stars were indeed physical pairs and that Newtonian physics operated
nicely in the distant sidereal universe. By the beginning of the twentieth
century tens of thousands of binary stars were known and cataloged (e.g., Burnham
1906). By the middle to late twentieth century the first systematic attempts to
establish the binary frequency of main sequence F and G stars suggested that a
very high fraction (70 - 80\%) of all such stellar systems consist of binary or
multiple stars (Heintz 1969; Abt \& Levy 1976; Abt 1983). The most comprehensive
and complete study of the multiplicity of G stars was performed by Duquennoy \&
Mayor (1991) who argued that two-thirds of all such stellar systems are
multiple.
It has often been assumed but never clearly demonstrated that similar statistics
applied to stars of all spectral types. This assumption has led to the commonly held
opinion that most all stars form in binary or multiple systems with the Sun (and its
system of planets) being atypical as a single star.
But how robust is the assumption
that the binary statistics for G stars is representative of all stars?
Over the last decade two important developments have occurred in stellar research which
directly bear on this question. First, the functional form of the stellar initial mass
function (IMF) has been better constrained by observations of both field stars (e.g.,
Kroupa, 2002) and young embedded clusters (e.g., Muench et al. 2002). The IMF has been
found to peak broadly between 0.1 - 0.5 \msun, indicating that most stars formed in the
Galactic disk are M stars. Second, surveys for binary stars have suggested that the binary
star frequency may be a function of spectral type (e.g., Fischer \& Marcy 1992). In
particular, there have been a number of attempts to ascertain the binary frequency of M type
stars and even for L and T dwarfs, objectss near and below the hydrogen burning limit. These
studies suggest that the binary frequency declines from the G star value, being only around
30\% for M stars (e.g., Leinert et al. 1997; Reid \& Gizis 1997; Delfosse et al. 2004;
Siegler et al. 2005) and as much as a factor of 2 lower for L and T dwarfs (e.g., Gizis et
al. 2003). I argue in this communication that these two facts together suggest that most
stellar systems in the Galaxy consist of single rather than binary or multiple stars.
\section{The Single Star Fraction and Spectral Type}
\label{sec:observations}
In this section I use data compiled from the literature to examine the single
star fraction as a function of stellar spectral type, in particular for the range
spanning G to M stars. I consider the single star fraction (SSF) to be the
fraction of stellar systems without a {\it stellar} companion, that is, primary
stars without a companion whose mass exceeds 0.08 \msun. Figure 1 displays the
single star fraction as a function of spectral type for G and later type stars.
This plot suggests that the SSF is significantly greater for M stars than for G
stars. Indeed the SSF for M stars appears to be at least 70\%. It is difficult
to evaluate the significance of this difference at face value given that the
differing binary surveys suffer from differing biases and varying degrees of
incompleteness. The systematic differences that can arise between the surveys
mostly derive from varying sensitivities to primary/secondary separations and
mass ratios. Below I attempt to evaluate the results from the surveys used to
construct Figure 1.
In their seminal study, Duquennoy \& Mayor (1991) obtained a spectroscopic survey
of a distance-limited complete sample of F7-G9 stars in the Northern Hemisphere
and within 22 pc of the Sun. They examined radial velocities obtained for these
stars over a 13 year period. They combined their detections of spectroscopic
binaries with known visual binaries and common proper motion pairs to examine 164
primaries for evidence of multiplicity. They derive multiplicity ratios of
57:38:4:1 for single:double:triple:quadruple systems, respectively. They
considered all the various detection biases to estimate the incompleteness of
their study and concluded that there was a slight bias against detecting low mass
companions, this resulted in a 14\% upward correction to the multiplicity
fraction such that 57\% of systems were estimated to be multiple for a
primary/companion mass ratio, q $>$ 0.1. They further extrapolated this
incompleteness correction to include substellar secondaries and estimated a
multiplicity fraction of 2/3 and a single star fraction of 1/3 for their sample.
However, in recent years sensitive and precise radial velocity surveys of 1330
single FGKM stars have indicated a paucity of substellar companions within 5 AU
of the primary stars (Marcy \& Butler 2000; Marcy et al. 2005). In addition
coronographic imaging surveys have found a similar dearth of substellar
companions around GK and M stars over separations between 75 and 300 AU (McCarthy
\& Zuckerman 2004). The existence of this so-called ``brown dwarf desert''
indicates that Duquennoy \& Mayor may have overestimated the multiplicity
fraction of G stars and the true value is likely 57\% or even somewhat smaller.
For the purposes of this paper I adopt 57\% as the multiplicity fraction of G
type stars and thus 43\% for the SSF.
The first extensive examination of the multiplicity of M stars was performed by Fischer \&
Marcy (1992) who studied radial velocity, speckle and visual binary data for a sample of
stars within 20 pc. The full range of separations, $a$ $<$ 10$^4$ AU, was examined, similar
to the G star study. These authors pointed out that M star surveys suffer less from the
effects of incompleteness than G star surveys because the M star sample is on the whole a
factor of 2 closer in distance and M star primaries are sufficiently faint to enable
detection of very faint companions more readily. They derived a SSF of 58\% which is higher
than the G star value.
Reid \& Gizis (1997) determined the SSF for a volume complete sample of 79
M2-M4.5 primary stars within 8 pc of the Sun and derived a SSF of 70 $\pm$ 12\%
for this sample. The range of binary separations they were able to probe was 0.1
- 10$^4$ AU. A similar volume complete search for M dwarf binaries within 5 pc
of the Sun was performed by Leinert et al. (1997) who reported a SSF of 74 $\pm$
19\%. However, their sample of 29 stars is smaller than the Reid \& Gizis (1997)
and Fischer \& Marcy (1997) samples accounting for the larger uncertainty. More
recently Delfosse et al. (2004) presented statistics for a much larger sample of
100 M dwarfs which they estimated was 100\% complete for stellar mass companions
over the entire separation range and out to 9 pc from the Sun. Delfosse et al.
(2004) derive a multiple star fraction of 26 $\pm$ 3 \% which corresponds to a
SSF of 74 $\pm$ 6\%. This may represent the most accurate determination for the
M star SSF yet made. I note here that even if one considers substellar
companions this estimate for the SSF will not likely alter significantly since as
mentioned earlier, surveys have revealed a dearth of substellar companions to G,
K {\it and} M stars (Marcy \& Butler 2000; McCarthy and Zuckerman 2004).
Surveys for multiplicity among very late M stars and even L and T dwarfs have
also been recently reported. These studies typically explore more limited
separation ranges and somewhat smaller samples of stars. The multiplicity
fractions they find are however all lower than that reported for the earlier type
M stars. For example, Siegler et al. (2005) examined a magnitude-limited survey
of 36 M6 - M 7.5 stars and derived a binary fraction of 9 $\pm$ 4\% corresponding
to a SSF of 91 $\pm$ 5\%. However this sample is not volume limited and may be
incomplete. Thus the inferred SSF is likely an upper limit. Despite this
limitation Siegler et al. were able to conclude that wide (a$>$20 AU) binaries
are very rare among these stars. Although not considered for inclusion in Figure
1 because of the large fraction of brown dwarfs in their samples, surveys by
Gizis et al. (2003) and Bouy et al. (2003) find similarly small binary
fractions for ultra low mass objects. For example, Gizis et al. examined 82
nearby late M and L dwarfs and derived a (incompleteness corrected) binary
fraction of 15 $\pm$ 5\% (corresponding to a SSF of 85 $\pm$ 14\%) for
separations, $a > $ 1.6 AU. Estimating the possible contribution of
companions at smaller separations they suggest a binary star fraction (BSF)
of 15 $\leq$ BSF $\leq$ 25 \% corresponding to 75 $\leq$ SSF $\leq$ 85 \%
for these
objects near and just below the hydrogen burning limit. Bouy et al. (2003)
examined the binary statistics for a sample of 134 late M and L field dwarfs and
estimated a binary fraction for a separation range of about 2 - 140 AU of only
10\% corresponding to a SSF of 90\% for these objects. They also noted a dearth
of companions with wide (i.e., $a >$ 15 AU) separations. Although these surveys
of very low mass and substellar objects suffer from some degree of incompleteness
it is quite unlikely that sensible corrections for such effects would decrease
the estimated single star fraction to a value similar to that of G stars or even
typical M stars.
The observations discussed above lead to the conclusion that the single
star fraction is a function of spectral type and increases from about
43\% for G stars to $\sim$ 85\% for brown dwarfs. The most secure estimate
for M stars appears to be about 74\% based on the complete volume-limited
sample of Delfosse et al. (2004) for M stars with stellar companions.
\section{M Stars and the IMF} \label{sec:results}
The stellar IMF is one of the most fundamental
distribution functions in astrophysics. A great deal of effort has been expended
in determining its form since the first attempt to measure its shape by Salpeter
(1954). He found that the IMF is a power-law which decreases with stellar mass
for field stars with masses in the range between 1-10 \msun. More recent
determinations of the IMF for field stars and young embedded clusters have
expanded the mass range covered by Salpeter. These studies have found the IMF
to break from a single power-law shape near 0.5 \msun\ and to have a broad peak
between $\sim$ 0.1 - 0.5 \msun. On either side of this peak the IMF falls off
rapidly (e.g., Miller \& Scalo 1979; Kroupa 2002; Muench et al. 2002; Chabrier
2003; Luhman et al. 2006).
The broad peak of the IMF encompasses the M stars and indicates that these stars
are the most numerous objects created in the star formation process. This is
illustrated in Figure 2 which shows the cumulative fraction of all stars above
the hydrogen burning limit given by the IMF. Two different IMFs are plotted
which span the range of modern day determinations of this function. One is the
log-normal field star IMF derived by Miller \& Scalo (1979) and the other
represents a determination of the IMF for the embedded Trapezium cluster in Orion
in which the IMF is characterized by a series of broken power-laws (Muench et al.
2002). This latter IMF is very similar to that determined for the field by
Kroupa (2002) but is more sensitive to substellar masses (not plotted). The
vertical dashed line shows the boundary for the M star population. The fraction
of all stars {\it above the hydrogen burning limit (HBL)} that are M stars is
73\% for the Muench et al. IMF and 78\% for the Miller-Scalo IMF. (It is
important to note here that these two IMFs are essentially primary star IMFs,
that is, IMFs that do not include companion star masses.) This analysis
indicates that roughly 3/4 of all stars formed are M stars.
The IMF represents the frequency distribution of stars at birth and differs from
the present day mass function (PDMF) which represents the frequency distribution
of all stars currently living within the Galactic disk. Stellar evolution has
significantly depleted the high mass end of the PDMF relative to the IMF.
Therefore, the fraction M stars in the PDMF is somewhat higher than the fraction
in the IMF. Indeed, for the PDMF derived by Miller \& Scalo (1979) we find from
Figure 2 that 84\% of all stars in the Galactic disk are M stars.
\vskip 0.2in
\section{The Total Single Star Fraction} \label{sec:SSF}
To estimate the total fraction of single stars, I assume that all stars earlier
than M are characterized by the single star fraction for G stars determined by
Duquennoy \& Mayor (1991), that is, $SSF_{<M} =$ 43\%. The single star
fraction for M-type stars (i.e., $SSF_{M}$) is assumed to be that (74\%)
determined by Delfosse et al. (2004) for a complete, volume limited sample. The
total SSF is then simply given by:
\small
$$ {\rm SSF(total)} = SSF_{<M} \times ETF + SSF_{M} \times MTF $$
\normalsize
\noindent
Here $MTF$ is the M-type fraction, that is, the fraction of all stars that are M-type stars
and $ETF = 1 - MTF$ is the early-type fraction, that is the fraction of all stars that have
spectral types earlier than M. To determine the SSF for all stars produced at any one time
by the star formation process I adopt the Muench et al. and Miller-Scalo IMFs,
specifically, MTF = 0.73 and 0.78, respectively. The total SSF is found to be 66\% and 67\%
for these two IMFs, respectively. Therefore, single stars must ultimately account for as
many as two-thirds of all stellar systems that formed at any one time in the Galaxy.
Similarly, if we consider the MTF (0.84) for the Miller-Scalo PDMF we find the total SSF to
be 69\%. Thus, {\it two thirds of all (main sequence) primary stars currently residing in
the Galactic disk are single stars}.
\section{Discussion and Conclusions}
\label{sec:discussion}
The primary result of this paper is the recognition that most stellar systems in
the Galaxy consist of single rather than binary stars. This fact has important
consequences for star and planet formation theory. For example, contrary to the
current accepted paradigm that most, if not all, stars form in binary or
multiple systems (e.g., Larson 1972, 2001; Mathieu 1994), this result could
indicate that the theoretical frameworks developed to explain the formation of
single, sunlike stars (e.g., Shu, Adams \& Lizano 1987) have wide applicability.
Indeed, when appropriately modified for a cluster-forming environment (e.g.,
Myers 1998; Shu, Li \& Allen 2004), they may even describe most star forming
events in the Galaxy. On the other hand, most stars could still initially form
in binary or multiple systems provided that most such systems promptly
disintegrate via dynamical interactions or decay in an early, perhaps even
protostellar, stage of evolution (e.g., Kroupa 1995; Sterzik \& Durisen 1998,
Reipurth 2000).
The current paradigm that most, if not all stars, form in binaries was
strengthened by early multiplicity surveys of pre-main sequence (PMS) stars. In
particular, surveys of the PMS population of the Taurus cloud indicated a binary
fraction that was twice that of field G stars (Ghez et al. 1993; Leinert et al.
1993; Reipurth \& Zinnecker 1993). However, most field stars are now known to
have formed in embedded clusters, environments quite different than represented
by the Taurus PMS population (e.g., Lada \& Lada 2003). Binary surveys of both
young embedded and Galactic clusters have revealed binary fractions
indistinguishable from that of the field (e.g., Petr et al. 1998; Duch\^ene,
Bouvier \& Simon 1999; Patience \& Duch\^ene 2001). The most simple and
straightforward hypothesis to explain these two facts and the finding of
a high SSF in this paper is that the most common outcome of the star formation
process is a single rather than multiple star.
Observations of dust emission and extinction of molecular cloud cores have found
that the shape of the primordial or dense core mass function is very similar to
that of the stellar IMF except that the core mass function is offset to higher
mass by a factor of 2-3 (e.g., Stanke et al. 2005, Alves, Lombardi \& Lada
2005). These observations indicate that a 1-to-1 mapping of core mass to
stellar mass, modified by a more or less constant star formation efficiency of
30-50\%, is possible, if not likely. This idea is consistent with single
star systems being most often produced once the cores undergo collapse.
The fact that stellar multiplicity is a function of stellar mass, however, may provide
important clues to the nature of the physical process of star formation. For example,
Durisen, Sterzik \& Pickett (2001) have shown that if individual protostellar cores can
further fragment and produce small N clusters, the dynamical decay of these clusters into
binary and single stars can in certain circumstances produce a binary star fraction that
declines with decreasing primary mass, similar to what is observed. However, to be
consistent with the SSF derived here and to simultaneously produce reasonable binary
component separations, such models would require N $\geq$ 5, within a region $\sim$ 300
AU in size (Sterzik \& Durisen 1998). This would correspond to a stellar surface density
($\sim$ 7.5 $\times$ $10^5$ stars pc$^{-2}$) about two orders of magnitude higher than the
peak density (7.2 $\times$ 10$^3$ stars pc$^{-2}$) measured for the rich Trapezium cluster
(Lada et al. 2004). Such ultra-dense protostellar groups have not yet been identified, but
could be revealed with high resolution infrared imaging surveys of deeply embedded
candidates. A related possibility, proposed by Kroupa (1995) and collaborators, posits that
all stars are formed in binaries in modestly dense embedded clusters. Dynamical
interactions between these systems can disrupt some binaries and modify the separations of
others. These models can produce the observed dependance of binary frequency with mass, but
at the expense of a SSF (50\%) that is too low to be consistent with that derived here.
These models could be made consistent with the high Galactic SSF by assuming more compact
configurations for the birth clusters, however it is unclear whether
the required higher cluster densities would remain consistent with observed values.
Another possibility is that binary star formation is related to the initial
angular momentum content of the primordial cores. In this case the
initial angular momentum of a protostellar core would be expected to be a
function of core mass, with low mass cores being endowed with considerably less
angular momentum than high mass cores. A systematic molecular-line survey of
cores of varying mass within a molecular cloud could test this idea. A related
possibility is that turbulence may play a role in the propensity for a core to
fragment. For example, Shu, Li \& Allen (2004) posit that the break in the
stellar IMF at 0.5 \msun\ is a result of the transition from turbulent to
thermal support of the envelopes of dense pre-collapse cloud cores. The more
massive the core, the more turbulence is required to insure its support.
Ammonia observations of dense cores in fact do suggest that massive cores are
more turbulent than low mass cores (Jijina, Myers \& Adams 1999). Perhaps
increased cloud turbulence in the more massive dense cores can also promote, in
some fashion, more efficient core fragmentation and a higher incidence of binary
star formation. In this context it would be interesting to know if the trend of
increasing stellar multiplicity with stellar mass continues to the more massive
A, B and O stars, as has been suggested in some studies (e.g., Preibisch,
Weigelt, \& Zinnecker 2001, Shatsky \& Tokovinin 2002).
Finally I note that the large fraction of single star systems in the field is
consistent with the idea that most stars could harbor planetary systems
unperturbed by binary companions and thus extra-solar planetary systems
that are characterized by architectures and stabilities similar
to that of the solar system could be quite common around M stars, provided
planetary systems can form around M stars in the first place.
\vskip -0.1in
\acknowledgments
I am indebted to August Muench for constructing the cumulative IMFs presented in
Figure 2 and many useful discussions. I thank David Latham and Bo Reipurth for
their careful reading of the paper and detailed suggestions and Kevin Luhman,
Geoff Marcy, Frank Shu and Pavel Kroupa for useful comments which
improved the paper.
|
Title:
A Chandra View of Dark Matter in Early-Type Galaxies |
Abstract: We present a Chandra study of mass profiles in 7 elliptical galaxies, of
which 3 have galaxy-scale and 4 group-scale halos demarcated at 1E13Msun. These
represent the best available data for nearby objects with comparable X-ray
luminosities. We measure ~flat mass-to-light (M/L) profiles within an optical
half-light radius (Reff), rising by an order of magnitude at ~10Reff, which
confirms the presence of dark matter (DM). The data indicate hydrostatic
equilibrium, which is also supported by agreement with studies of stellar
kinematics in elliptical galaxies. The data are well-fitted by a model
comprising an NFW DM profile and a baryonic component following the optical
light. The distribution of DM halo concentration parameters (c) versus Mvir
agrees with LCDM predictions and our observations of bright groups.
Concentrations are slightly higher than expected, which is most likely a
selection effect. Omitting the stellar mass drastically increases c, possibly
explaining large concentrations found by some past observers. The stellar M/LK
agree with population synthesis models, assuming a Kroupa IMF. Allowing
adiabatic compression (AC) of the DM halo by baryons made M/L more discrepant,
casting some doubt on AC. Our best-fitting models imply total baryon fractions
\~0.04--0.09, consistent with models of galaxy formation incorporating strong
feedback. The groups exhibit positive temperature gradients, consistent with
the "Universal" profiles found in other groups and clusters, whereas the
galaxies have negative gradients, suggesting a change in the evolutionary
history of the systems around Mvir=1E13 Msun.
| https://export.arxiv.org/pdf/astro-ph/0601301 |
\title{A Chandra View of Dark Matter in Early-Type Galaxies}
\author {\href{mailto:phumphre@uci.edu}{Philip J. Humphrey}\altaffilmark{1}, David A. Buote\altaffilmark{1}, Fabio Gastaldello\altaffilmark{1}, Luca Zappacosta\altaffilmark{1}, James S. Bullock\altaffilmark{1}, Fabrizio Brighenti\altaffilmark{2,3} and William G.~Mathews\altaffilmark{3}}
\altaffiltext{1}{Department of Physics and Astronomy, University of California at Irvine, 4129 Frederick Reines Hall, Irvine, CA 92697-4575}
\altaffiltext{2}{Dipartimento di Astronomia, Universit\`{a} di Bologna, Via Ranzani 1, Bologna 40127, Italy}
\altaffiltext{3}{University of California Observatories, Lick Observatory, University of California at Santa Cruz, Santa Cruz, CA 95064}
\keywords{Xrays: galaxies--- galaxies: elliptical and lenticular, cD---
galaxies: halos--- galaxies: ISM--- dark matter}
\section{Introduction}
The nature and distribution of dark matter (DM) in the Universe is one
of the fundamental problems facing modern physics. Cold DM lies at the
heart of our current (\lcdm) cosmological paradigm, which predicts
substantial DM halos for objects at all mass-scales from galaxies to
clusters. Although \lcdm\ has been remarkably successful at
explaining large-scale features
\citep[\eg][]{spergel03a,perlmutter99a},
observations of galaxies have been more problematical for the theory.
Dissipationless dark matter simulations find that dark matter
halos are well characterized by a
``Universal'' mass density profile \citep[][hereafter NFW]{navarro97}
over a wide range of Virial masses (\mvir) \citep[e.g.][]{bullock01a}.
Low mass halos tend to form first in hierarchical cosmologies
and are consequently more tightly concentrated than their
later forming, high mass counterparts.
This tendency produces a predicted correlation
between the DM halo concentration parameter (c, which is ratio between
Virial radius, \rvir, and the characteristic scale of the density profile)
and \mvir \citep{navarro97}. However, since mass and formation
epoch are not perfectly correlated, we expect a significant
scatter at fixed Virial mass \citep{jing00a,bullock01a,wechsler02a}.
The tight link between halo formation epoch and concentration
implies that the precise relation between c and \mvir\
is sensitive to the underlying Cosmological parameters,
including \sigmaeight\ and the dark energy equation of state
\citep{kuhlen05a}, making an observational test of this relation
potentially a very powerful tool for cosmology.
The mass profiles of galaxies also may provide valuable clues
as to the way in which galaxies form in DM halos.
In particular, as baryons cool and collapse into stars, the
associated increase in the central mass density should in turn
modify the shape of the DM halo. This process is typically
modelled assuming adiabatic contraction (AC) of the DM particle orbits
\citep[\eg][]{blumenthal86a,gnedin04a}.
If the galaxy halo subsequently evolves by major mergers,
simulations are unclear as to whether these features would persist
\citep[\eg][]{gnedin04a} or whether the merging process may
destroy this imprint of star formation, or even
mix the DM and baryons sufficiently to produce a {\em total}
gravitating mass profile more akin to NFW \citep{loeb03a,elzant04a}.
Observational tests of the predictions of \lcdm\ have proven
controversial. In clusters of galaxies there is overwhelming evidence
for DM, and an increasing body of work verifying the
predictions of \lcdm. In particular recent, high-quality
\chandra\ and \xmm\ observations have revealed mass profiles
in remarkable agreement with the Universal profile from
deep in the core to a large fraction of
\rvir\ \citep[\eg][]{lewis03a,zappacosta06a,vikhlinin05b},
and a distribution of c {\em versus} \mvir\ in good agreement with
\lcdm\ \citep{pointecouteau05a}.
In galaxies, however, the picture is much less clear. Rotation curve
analysis of low surface brightness (LSB) disk galaxies has suggested
significantly less cuspy density profiles than expected
\citep[\eg][]{swaters00a}.
Although this discrepancy led to a serious discussion of
modifications to the standard paradigm
\citep[\eg][]{hogan00a,spergel00a,zentner02a,kaplinghat05a,cembranos05a},
recent results, taking account of observational bias and the 3-dimensional
geometry of the DM halos, have done much to resolve the discrepancy
\citep[\eg][]{swaters03a,simon05a}. However, some
significant discrepancies remain, not least of which
is that the DM halos of these galaxies appear less concentrated
than expected \citep[\eg][]{gonzalez00a,kassin06a}.
A possible explanation is that LSB galaxies are preferentially
found in low-concentration halos \citep{bullock01a,bailin05a,wechsler05a}, making
additional constraints at the galaxy scale extremely important. %
In many respects, kinematical mass measurements
are far more challenging for early-type than spiral galaxies.
As essentially pressure-supported systems little is known
{\em a priori} about the velocity anisotropy tensor of
the stars in elliptical galaxies, which is problematical for
the determination of the mass from stellar motions.
Nonetheless, stellar kinematical measurements
have widely been used as a means to measure the gravitating matter
within $\sim$the optical half-light radius (\reff) of
elliptical galaxies \citep[\eg][]{binney90a,vandermarel91a,gerhard01a}.
These studies tend to find
relatively flat mass-to-light (M/L) ratios within \reff,
implying that most of the matter within this radius is baryonic.
Consideration of the tilt in the fundamental plane can also
lead to the same conclusion \citep{borriello03a}.
In contrast, \citet{padmanabhan04a} pointed out that
dynamical M/L ratios within \reff\ are much larger than
predicted by realistic stellar population synthesis models for
stars alone, allowing
\gtsim 50\% of the mass within \reff\ to be dark. %
Attempts to extend kinematical studies of elliptical galaxies
to larger radii, where DM should be dominant,
have proven controversial.
In particular \citet{romanowsky03a} argued against the existence
of DM in a small sample of elliptical galaxies, based on
planetary nebulae dynamics within $\sim$5\reff.
We note that this sample was heavily biased towards very X-ray
faint objects, which might hint at low-mass halos
since they have not held onto their hot gas.
In any case \citet{dekel05a} pointed out that their
conclusions were very sensitive
to the uncertainty in the velocity anisotropy tensor,
for plausible values of which the data were consistent
with substantial DM halos.
In fact globular cluster dynamics in one of these systems,
NGC\thin 3379, does imply a significant amount of DM
\citep{pierce06a,bergond06a}.
As more kinematical studies of early-type galaxies at
large radii are appearing, it is becoming clear
that at least some elliptical galaxies host considerable
DM halos \citep[\eg][]{statler99a,romanowsky05a}. There
persist some questions, however, as to the extent
to which all galaxies have DM halos consistent with \lcdm.
In particular \citet{napolitano05a} argued
that a substantial number of early-type galaxy halos
appear less concentrated than expected.
Gravitational lensing provides further evidence
that, at least some, early-type galaxies possess substantial
DM halos \citep[\eg][]{kochanek95a,fischer00a,rusin02a}.
Since weak lensing of galaxies only provides useful mass constraints
in a statistical sense, the relatively rare instances of
strong lensing are required to study DM in individual systems.
Nonetheless it has been possible in a few cases to decompose
the mass into stellar and DM components, albeit with strong
assumptions or additional observational constraints
\citep[\eg][]{rusin03a,treu04a}.
X-ray observations of the hot gas in early-type galaxies provide a
complementary means to infer the mass-profiles {\em via}
techniques similar to those used in studying clusters.
Since the X-ray emission from early-type galaxies is typically
not very bright, prior to the
advent of \chandra\ and \xmm\ this was limited
by the relatively sparse information on the
radial temperature and density profiles of the hot gas
which could be determined by prior generations of
satellites.
Notwithstanding this limitation, large M/L ratios (consistent
with substantial DM) were
inferred for a number of X-ray bright galaxies,
albeit with strong assumptions concerning the temperature and density
profiles \citep[\eg][]{forman85a,loewenstein99b}.
Using a novel technique which relied, instead, on the
ellipticity of the X-ray halo, \citet{buote94} were able robustly to
detect DM in the isolated elliptical
NGC\thin 720 \citep[see also][]{buote96a,buote98d,buote02b}.
Detailed measurements of the radial mass distribution
were, however, largely restricted
to a few massive systems, which may be entwined with a group
halo \citep[\eg][]{irwin96,brighenti97a}.
Nevertheless \citet{brighenti97a}
were able to decompose the mass profiles of two systems, NGC\thin 4472
and NGC\thin 4649, into stellar and DM components.
\citet{sato00a} investigated the \mvir-c relation using \asca\ for a
sample of objects ranging from massive clusters to $\sim$3 elliptical
galaxies. The limited spatial resolution of \asca\ necessitated
some assumptions about the density profiles
and, crucially, the authors neglected any stellar mass component
in their fits. This omission may explain the very steep
\mvir-c relation (with c$_{200}$\gtsim 30 for the galaxies) found by
these authors, in conflict with \lcdm\ \citep{mamon05a}.
Although mass profiles of early-type galaxies are beginning to appear
which exploit the improved sensitivity and resolution of \chandra\ and \xmm,
many of the most interesting constraints on DM are still restricted to
massive systems, which may be at the centres of groups.
For example, \citet{fukazawa06a} reported \chandra\ and \xmm\ M/L profiles for
$\sim$50 galaxies and groups,
confirming $\sim$flat profiles within \reff\ which rise at larger
radii. However, the constraints at large radii were dominated by
the massive (group-scale) objects so the implications for the DM
content of normal galaxies are unclear. Furthermore,
the authors included a substantial number of
highly disturbed systems, in which hydrostatic equilibrium
may be questioned, and failed to account for the unresolved
sources which dominate the emission in the lowest-\lx\ objects
in their sample\footnote{Although the authors account for unresolved
sources when measuring the gas temperature, they do not account for
it when computing the gas density, where its effect is more pronounced}.
Recently, however, detailed \chandra\ and \xmm\ mass profiles have begun to
appear for isolated early-type galaxies, also confirming the presence
of massive DM halos \citep[\eg][]{osullivan04b,khosroshahi04a}.
This paper is part of a series
\citep[see also][]{gastaldello06a,zappacosta06a,buote06b,buote06a}
using high-quality \chandra\ and \xmm\ data to investigate the
mass profiles of galaxies, groups and clusters. This provides
an unprecedented opportunity to place definitive constraints upon
the \mvir-c relation over $\sim$2 orders
of magnitude in \mvir. In this paper, we focus on the temperature,
density and mass profiles of seven galaxies and poor groups
chosen from the \chandra\ archive.
In order to compare to theory we perform spherically-averaged
analysis, leaving a discussion of the ellipticities of the
X-ray halos to a future paper.
In \S~\ref{sect_targets} we discuss the target selection. The
data-reduction is described in \S~\ref{sect_reduction} and
the X-ray morphology is addressed in \S~\ref{sect_imaging}.
We discuss the spectral analysis in \S~\ref{sect_spectra}, the
mass analysis in \S~\ref{sect_mass}, the systematic uncertainties
in our analysis in \S~\ref{sect_systematics}
and reach our conclusions in \S~\ref{sect_discussion}.
The three systems for which we find \mvir$<10^{13}$\msun\ are
optically isolated and so we refer to them as ``galaxies'', and
the other systems in our sample as groups. We discuss this in
more detail in \S~\ref{discussion_groups}.
In this paper, all error-bars quoted represent 90\% confidence limits,
unless otherwise stated, and we computed Virial quantities assuming
a ``critical overdensity''
factor for the DM halos of $\rho_{\rm halo}/\rho_{\rm crit} = 103$
(where $\rho_{\rm halo}$ is the mean density of a sphere of
mass \mvir\ and radius \rvir).
\section{Target Selection} \label{sect_targets}
We chose, for this initial study, to focus on objects observed with
\chandra. \chandra\ data are particularly valuable
for the study of galaxies since the unprecedented spatial resolution
makes it possible to resolve the temperature and density profiles
deep into the galaxy core, allowing us to disentangle the stellar
and dark matter, and resolve them into discrete components.
We initially chose a set of potential target systems from detections
listed in the X-ray
catalogue of \citet{osullivan01a} which have non-grating ACIS
data in the \chandra\ archive. To eliminate bright
groups and cluster cDs in the sample, we excluded galaxies with
\lx\gtsim $10^{43}$\ergps. In order to perform the required spatially-resolved
spectroscopy, we required at least $\sim$5000 hot gas photons.
The potential targets were processed
and the 0.1--10.0~keV image examined for evidence of large-scale
disturbances (\S~\ref{sect_imaging}). We included some systems
with low-amplitude asymmetries which should not strongly
disturb hydrostatic equilibrium (we discuss this in more detail in
\S~\ref{sect_asymmetry}).
Preliminary analysis was conducted to estimate the Virial mass of
the object (\S~\ref{sect_mass}). Since we aimed to focus on
lower-mass objects, systems for which a fit
using a simple NFW profile yielded \mvir\gtsim $10^{13}$\msun were
discounted. Massive objects of this type are the focus of another study
\citep{gastaldello06a}. The most promising
candidates for study found {\em via} this method were chosen for detailed
analysis. The properties of the 7 objects in our sample
and the \chandra\ exposures are shown in Table~\ref{table_obs}.
Our selection criteria naturally bias the sample towards
X-ray bright galaxies. One might expect that galaxies sitting in
deep potential wells are more likely to retain hot gas than those
with little dark matter, and so our results may be biased somewhat
towards those galaxies with substantial dark halos \citep[in contrast
to the opposite bias in the analysis of][]{romanowsky03a}.
As we are selecting objects which are not heavily disturbed, we are
also biased towards galaxies which have not recently undergone a
major merger. For the purposes of this paper, however, we do not
require statistical completeness, and we will discuss how to
take account of these selection effects in \citet{buote06b}.
\begin{deluxetable*}{lllllllrrr}
\tablecaption{The galaxy sample\label{table_sample}}
\tabletypesize{\scriptsize}
\tablehead{
\colhead{Galaxy} & \colhead{Type} & \colhead{\lb} & \colhead{\lk} & \colhead{Dist} &
\colhead{Scale} & \colhead{\reff}
& \colhead{ObsID} & \colhead{Date} & \colhead{Exposure} \\
\colhead{} & \colhead{} & \colhead{($10^{10}$\lsun)} &\colhead{($10^{11}$\lsun)} & \colhead{(Mpc)} & \colhead{(\arcsec\ kpc$^{-1}$)} & \colhead{(kpc)} &
\colhead{} & \colhead{(dd/mm/yy)} & \colhead{(ks)}
}
\startdata
NGC\thin 720 & E5 & 3.1 & 1.7 & 25.7 & 8.1 & 3.1 & 492 & 12/10/00 & 17 \\
NGC\thin 1407 & E0 & 6.4 & 3.1 & 26.8 & 7.8 & 4.4 & 791 & 16/08/00 & 38 \\
NGC\thin 4125 & E6 pec Liner & 4.7 & 1.8 & 22.2 & 9.4 & 3.3 & 2071& 09/09/01 & 63 \\
NGC\thin 4261 & E2-3 Liner Sy3& 4.4 & 2.2 & 29.3 & 7.1 & 3.4 & 834 & 06/05/00 & 34 \\
NGC\thin 4472 & E2/S0(2) Sy2 & 7.5 & 3.2 & 15.1 & 14 & 4.0 & 321 & 12/06/00 & 34 \\
NGC\thin 4649 & E2 & 5.1 & 2.5 & 15.6 & 13 & 3.2 & 785 & 20/04/00 & 21 \\
NGC\thin 6482 & E Liner & 10.9 & 3.2 & 58.8 & 3.6 & 3.4 & 3218& 20/05/02 & 18
\enddata
\tablecomments{The galaxies in the sample. Distances were obtained from
\citet{tonry01}, corrected for the the new Cepheid zero-point
\citep{jensen03},
except for NGC\thin 6482, for which we adopted the kinematical distance
modulus from \leda. \lb\ was obtained from \leda, corrected to our distance.
\ks-band luminosities (\lk) and
effective radii (\reff) were obtained from \twomass. We assumed
${\rm M_{B\odot}=5.48}$ and ${\rm M_{K\odot}=3.41}$
\citep[\eg][]{maraston98a}. We also list the image scale
(Scale), which is the number of arc seconds corresponding to 1 kpc.
We list the observation ID (ObsID) and total exposure times, after
having eliminated flaring intervals.}\label{table_obs}
\end{deluxetable*}
\section{Data reduction} \label{sect_reduction}
For data reduction, we used the \ciao\ 3.2.2 and \heasoft\ 5.3 software
suites, in conjunction with \chandra\ calibration database (\caldb)
version 3.1.0. Spectral-fitting was conducted with \xspec\ 11.3.1w.
In order to ensure the most up-to-date calibration, all data were
reprocessed from the ``level 1'' events files, following the standard
\chandra\ data-reduction threads\footnote{\href{http://cxc.harvard.edu/ciao/threads/index.html}{http://cxc.harvard.edu/ciao/threads/index.html}}.
We applied the standard correction to take account of the time-dependent gain-drift
as implemented in the standard \ciao\ tools. To identify periods of enhanced
background (``flaring''), which seriously degrades the signal-to-noise (S/N)
and complicates background subtraction \citep{markevitch02}
we accumulated background lightcurves for each exposure from
low surface-brightness regions of the active chips. We
excluded obvious diffuse emission and data in the vicinity of any detected
point-sources (see below). Periods of flaring were identified by eye and
excised. Small amounts of residual flaring not removed by this procedure
can be important in low surface-brightness regions at large radii,
but this was taken into account in our treatment of the
background (\S~\ref{sect_bkd}).
The final exposure times are listed in Table~\ref{table_obs}.
Point source detection was performed using the \ciao\ tool
{\tt wavdetect} \citep{freeman02}. Point sources were identified
in full-resolution images of the \acis\ focal-plane, containing all active
chips (except the S4 chip, which suffers from serious ``streaking'', which can
lead to false detections). To maximise the likelihood of identifying
sources with peculiarly hard or soft spectra, images were created in three
energy bands, 0.1--10.0~keV, 0.1--3.0~keV and 3.0--10.0~keV. Sources were
detected separately in each image. In order to minimize spurious detections at
node or chip boundaries we supplied the detection algorithm with
exposure-maps generated at
energies 1.7~keV, 1.0~keV and 7~keV respectively (although the precise
energies chosen made little difference to the results). The
detection algorithm searched for structure over pixel-scales of 1, 2, 4, 8 and
16 pixels, and the detection threshold was set to ensure $\sim$0.1
spurious detections per image.
The source-lists obtained within each energy-band were combined and
duplicated sources removed, and the final list was checked
by visual inspection of the images.
The data in the vicinity of any detected point source
were removed so as not to contaminate the diffuse emission.
As discussed in \citet[][see also \citealt{kim03a}]{humphrey04a}
a significant fraction of faint X-ray binary sources
will not have been detected by this procedure, and so we include
an additional component to account for it in our spectral fitting
(\S~\ref{sect_spectra}).
For each galaxy, we extracted spectra in a number
of concentric annuli, centred on the nominal X-ray centroid. We determined
the centroid iteratively by placing a 0.5\arcmin\ radius aperture at the nominal
galaxy position (obtained from \ned) and computing the X-ray centroid
within it. The aperture was moved to the newly-computed centroid, and the
procedure repeated until the computed position converged. Typically the
X-ray centroid agreed with that from \ned. The widths of
the annuli were chosen so as to contain approximately the same number of
background-subtracted photons and ensure there were sufficient photons in each
to perform useful spectral-fitting. The data in the vicinity of any
detected point-sources were excluded, as were the data from the vicinity
of chip gaps, where the instrumental response may be uncertain. We extracted
products from all active chips, excluding the S4, since it suffers from
considerable ``streaking'' noise.
Appropriate count-weighted spectral response
matrices were generated for each annulus
using the standard \ciao\ tasks {\bf mkwarf} and {\bf mkacisrmf}.
\subsection{Background estimation} \label{sect_bkd}
One of the chief difficulties in performing spectral-fitting of
diffuse emission is the proper treatment of the background.
A set of standard blank-field ``template'' files are available for
\chandra\ as part of the \caldb. We found, however, that
the background template files are not sufficiently accurate to use
in the very low surface brightness regions at large radii, which are
crucial to determine interesting global mass constraints.
The background comprises cosmic, instrumental and non X-ray
(particle) components.
The cosmic component is known to vary from field to field, while
the non X-ray background exhibits long-term secular variability.
To mitigate the latter effect, several authors have
adopted the practice of renormalizing the background
template to ensure good agreement with their data at high
energies (\gtsim 10~keV). Such a procedure, however, also
renormalizes the (uncorrelated) cosmic X-ray background and instrumental
line features, which can lead to serious over- or under-subtraction.
Given these reservations we chose to use an alternative
background estimation procedure.
Our method involved modelling the background, somewhat akin to
the approach of \citet{buote04c}. All of the targets were centred on the
\acis-S3 chip, which is back-illuminated (BI). To obtain constraints on the
background, we extracted spectra from a
$\sim$2\arcmin\ region centred on the S1 chip, which is also BI, and
from an annulus centred at the galaxy centroid and with an inner and outer
radii typically $\sim$2.5\arcmin\ and 3.3\arcmin.
We excluded data from the vicinity
of any point-sources found by the source detection algorithm.
Although the diffuse emission from each galaxy typically had a very low
surface-brightness on the S1 CCD, we found that using two regions
in this way with different contributions of source emission enabled the
background components to be most cleanly disentangled from the source.
The \acis\ focal plane also consists of front-illuminated (FI) chips,
which have significantly different (and lower) background.
To obtain an estimate of the background for these chips,
we extracted spectra from the entirety of each chip, excluding
detected sources and data towards the edge of the chips where the
exposure-map may be uncertain.
In order to constrain the model, we fitted all
spectra simultaneously, without background subtraction, using \xspec.
Our model consisted of
a single \apec\ plasma (to take account of the diffuse emission from
the galaxy; the ``source''),
plus background components. These comprised
a power law with $\Gamma=1.41$ (to account for the hard X-ray background),
two \apec\ models with solar abundances and kT$=$ 0.2 and
0.07~keV (to account for the soft X-ray background) and, to model
the instrumental and particle contributions, a broken power law
model and two Gaussian lines with energies 1.7 and 2.1~keV and negligible
intrinsic widths. We have found that this model
can be used to parameterize adequately the template background spectra.
In general, the instrumental contributions of the FI chips were very
similar in shape. Therefore, the background components of all the FI
chips were tied, assuming the normalization scaled with the spectral
extraction area. For the BI chips, there was some evidence that the
S1 chip background can be somewhat larger at energies \gtsim 5~keV
(although this is variable).
In order to disentangle the source and background components,
given the general lack of
photons in these spectra, we tied the abundances and
temperatures of the ``source'' \apec\ components between the
extraction regions, but allowed the normalizations to be free.
Where there was a significant improvement in the fit-statistic
if this assumption was relaxed, we allowed the abundances or
temperatures to fit freely. Notwithstanding, this
assumption should not significantly affect our results.
This model was able to fit all of the data well.
In our subsequent spectral analysis, we did
not background-subtract the data using the standard templates, but took
into account the background by using appropriately scaled versions
of the models fitted to each CCD, which were added according to the
overlap between the source region and the CCD.
We found that the standard background templates fared
much worse than these modelled background estimates when the data
were from regions of low surface-brightness. We discuss the impact of
the background treatment on our results in \S~\ref{sect_systematics_bkd}.
\section{X-ray images} \label{sect_imaging}
The X-ray image of each galaxy was examined to identify any
obvious surface-brightness disturbances or asymmetries
which would be indicative of clear deviations from hydrostatic equilibrium.
We note that low-level X-ray asymmetries,
such as the ``fingers of emission'' identified by
\citet{randall03} in the adaptively-smoothed images of NGC\thin 4649,
probably do not merit concern\footnote{Although the authors suggested these
may arise from bulk convective flow, the spectra do not agree with
simulations of such.}, as, provided care is taken to avoid
seriously disturbed emission regions, reliable mass profiles can
be inferred even in mildly disturbed systems \citep{buote95a}.
In Fig~\ref{fig_images} we show the 0.1--10.0~keV
\acis-S3 images of each of the systems.
These images were first processed to remove point-sources,
using the \ciao\ tool {\em dmfilth}, which replaces photons in the vicinity
of each point-source with a locally-estimated background.
NGC\thin 4261 contains an AGN which appears as a bright central X-ray
source and there is evidence of a small, low surface-brightness
jet \citep{zezas05a}. We have also removed these
sources from the image. The images were flat-fielded with the
1.7~keV monochromatic exposure-map (although this analysis is insensitive
to the choice of energy), and then smoothed by convolution with a 5\arcsec\
gaussian, to make large-scale structure more apparent.
Due to the
low surface-brightness nature of the emission at large radii, it
is difficult to appreciate X-ray emission outside $\sim$a few arc minutes
in many of the images. However, detailed
spectral analysis and azimuthally-averaged surface brightness analysis
reveals substantial hot gas extending beyond the edge of the S3 chip
in each system.
None of the objects show very obvious disturbances in their X-ray emission
on the \acis-S3 chip \citep[such as those found in NGC\thin 4636:][]{jones02a}.
Some low-amplitude features are evident such as the faint
jet in NGC\thin 4261 (which is not visible in the above images),
a possible north-south asymmetry in NGC\thin 1407 and some asymmetry, in particular
an off-axis X-ray enhancement, in NGC\thin 4125. Based on adaptively-smoothed
\xmm\ images, \citet{croston05a} argued that the X-ray emission in
NGC\thin 4261 is anti-correlated with the galaxy radio lobes.
By inspection of the \xmm\ images, this actually appears to be a very
low-amplitude effect. It is not obvious in the
\chandra\ images, although the X-ray isophotes do align somewhat perpendicularly
to the jet.
In any case, this does not appear to have significantly disturbed
hydrostatic equilibrium, since there is excellent agreement between our
inferred mass profile and a model comprising stellar plus DM components
(\S~\ref{sect_mass}), which would be an extraordinary coincidence
if hydrostatic equilibrium had been strongly disturbed.
The limited field-of-view makes it difficult to assess asymmetries and
disturbances on the other chips. NGC\thin 4472 is known, however,
to exhibit a disturbance outside $\sim$6\arcmin\ \citep{irwin96}, but
mass analysis inside this radius should be reliable. We assess the impact
of all these features in \S~\ref{sect_asymmetry}.
\section{Spectral Analysis} \label{sect_spectra}
Spectral-fitting was carried out in the energy-band 0.5--7.0~keV,
to avoid calibration uncertainties at lower energies
and to minimize the instrumental background, which dominates at high
energies. The spectra were rebinned to ensure a S/N ratio of at least
3 and a minimum of 20 photons per bin (to validate $\chi^2$ fitting).
We fitted data from all annuli simultaneously using
\xspec. To model the hot gas we adopted a {\bf vapec} component,
plus a bremsstrahlung component for all annuli within the
twenty-fifth magnitude isophote (\dtwentyfive) of each galaxy, taken
from the Third Reference Catalog of Bright Galaxies
\citep[RC3:][]{devaucouleurs91}, to
account for undetected point-sources \citep[this model gives a good fit to
the composite spectrum of the detected sources in nearby
galaxies:][]{irwin03a}.
We used a slightly modified form of the existing \xspec\ {\bf vapec}
implementation so that \zfe\ is determined directly, but for the remaining
elements the abundance ratios (in solar units) were directly
determined with respect to Fe. This was useful since, in general, the data
did not enable us to determine any abundance {\em ratio} gradients and
so we tied the abundance ratios between all annuli.
Where abundances or abundance ratios could not be constrained,
they were fixed at the Solar value.
We adopted the solar photospheric
abundances standard of \citet{asplund04a}. We refer the interested
reader to \citet{humphrey05a} for a detailed discussion of this
choice and how to convert our results to older abundance
standards. In the interests of
physically reasonable results, we constrained all abundances and
abundance ratios to the range 0.0--5.0~times solar.
The absorbing column density (\nh) was fixed at the Galactic value
\citep{dickey90}; the effect of varying \nh\ is discussed in
\S~\ref{sect_systematics_spectra}.
For NGC\thin 4261, our innermost annulus contained substantial
contamination from the central AGN. However, this was sufficiently
absorbed that the thermal emission from the gas can be clearly
disentangled from it. To account for the AGN emission, we fitted
a highly absorbed (\nh${\rm =10^{+6}_{-4}\times 10^{22} cm^{-2}}$)
power law component ($\Gamma=1.4\pm0.8$). We discuss the impact
of including this annulus on our fits in \S~\ref{sect_asymmetry}.
To account for projection effects, we used the {\bf projct} model implemented
in \xspec. This model, unfortunately, does not take into account the
emission from gas outside the outermost shell, which is also projected
into the line-of-sight. To take account of this effect, we assumed that
the emission outside this shell has the same spectral shape as the
emission in that shell and a density profile well-described by a $\beta$-model
\citep[\eg][]{buote00c}.
We included an extra spectral component to our fits of each annulus to account
for projected emission from this gas.
To estimate the parameters of the $\beta$-profile, we
we fitted the galaxy surface brightness, using dedicated software,
in the 0.1--3.0~keV band. Although a single $\beta$-model did not
always match the fine detail of the surface brightness profiles, it
adequately parameterized the data for our purposes (our results are
not expected to be strongly dependent upon the parameters of this fit).
We obtained good fits to the spectra of each galaxy with this model.
The best-fitting abundances
were in excellent agreement with those of other early-type galaxies
\citep{humphrey05a}, and are shown in Table~\ref{table_abundances}.
We note that \citet{randall05a} found \zsi/\zfe$\simeq$1.7 for
NGC\thin 4649 (adjusting to our abundances standard) when
fitting the data from single, large aperture, which they
argued points to substantial enrichment from SN~II,
in stark contrast to the predominantly SN~Ia enrichment
we found in such galaxies \citep{humphrey05a}. From our analysis,
however, \zsi/\zfe$\simeq$1, which is more
consistent with our results for other systems. The discrepancy
appears to be related to the ``Fe bias'' \citep[where \zfe\ is
systematically underestimated if one assumes multi-temperature gas
is isothermal:][]{buote00c} which has suppressed their large aperture
\zfe\ in comparison to their spatially-resolved results (which
agree better with our measurement).
Error-bars were computed {\em via} the Monte-Carlo technique which
we have extensively used in past analyses \citep[\eg][]{buote03a}.
We simulated spectra from the best-fit models, which were then
fitted exactly analogously to the real data. We performed 25
simulations, which were sufficient to assess the distribution
of the fit parameters about the best-fit values; the standard
deviation of this distribution corresponds to the 1-$\sigma$
confidence region.Assuming that we have found the global minimum,
and the fit statistic follows a $\chi^2$ distribution this is
statistically equivalent to searching the parameter space for
changes in the fit statistic.
Temperature and density profiles are discussed below
(\S~\ref{sect_temp_profiles} and \S~\ref{sect_mass_results})
\begin{deluxetable*}{lrrrrrrrr}
\tabletypesize{\scriptsize}
\tablecaption{Emission-weighted average abundances\label{table_abundances}}
\tablehead{
\colhead{Galaxy} & \colhead{$\chi^2$/dof} & \colhead{\zfe} & \colhead{\zo/\zfe} & \colhead{\zne/\zfe} &
\colhead{\zmg/\zfe} & \colhead{\zsi/\zfe} & \colhead{\zs/\zfe} & \colhead{\zni/\zfe}
}
\startdata
NGC\thin 720$^1$ & 383.4/357& $0.80^{+0.45}_{-0.24}$ & 0.30$\pm 0.28$ & 0.68$\pm0.67$ & 1.26$\pm0.35$ & \ldots& \ldots & \ldots \\
NGC\thin 1407$^1$ & 222/221& 2.1$^{+1.1}_{-0.9}$$^\dagger$& 0.37$^{+0.21}_{-0.25}$ & \ldots & 1.10$\pm 0.23$ & 1.21$^{+0.31}_{-0.27}$ & 2.2$\pm1.1$ & 3.3$^{+1.7}_{-1.3}$\\
NGC\thin 4125 & 327/307 & 0.55$^{+0.22}_{-0.13}$ & 0.29$^{+0.13}_{-0.09}$ & 0.62$\pm$0.14 & 0.33$\pm0.20$ & \ldots & \ldots & \ldots \\
NGC\thin 4261 & 307/319 & 1.72$\pm0.50^\dagger$ & $<$0.23 & 0.36$^{+0.79}_{-0.36}$ & 0.83$\pm$0.23& 1.2$\pm$0.4& \ldots & 1.8$^{+2.3}_{-1.8}$ \\
NGC\thin 4649 & 563/491 & 2.32$^{+0.87}_{-0.37}$ & $<0.15$ & \ldots & 0.97$\pm$0.13 & 1.02$\pm$0.13 & \ldots & 1.42$^{+0.85}_{-0.73}$ \\
NGC\thin 4472$^1$& 785/740 & $1.4^{+1.7}_{-0.4}$$^\dagger$& 0.51$\pm$0.12 & 0.95 $\pm0.44$ & 1.02$\pm$0.11 & 1.25$\pm$0.11 & 2.36$\pm$0.33 & 3.28$\pm0.61$\\
NGC\thin 6482 & 256/262 & $>$2.5 & 0.34$\pm$0.20 & \ldots & 1.15$\pm$0.18 & 1.3$\pm$0.3 & \ldots & 3.2$^{+1.5}_{-1.2}$
\enddata
\tablecomments{The best-fitting globally-averaged emission-weighted abundances
and abundance ratios for each galaxy, shown along with the quality of fit.
Statistical errors represent the 90\% confidence region.
Where we were able to constrain an abundance gradient, we estimated an
emission-weighted \zfe, extrapolated
over a large aperture \citep[see][]{humphrey05a};
those affected galaxies are marked ($^\dagger$).
$^1$---results taken from \citet{humphrey05a}.
Where parameters could not be constrained, they were fixed
at the Solar value, and listed as ``\ldots''.}
\end{deluxetable*}
\section{Mass modelling} \label{sect_mass}
\subsection{Assumed potential method} \label{sect_potential}
We adopted two complementary approaches in order to determine the
mass profiles of the galaxies
in the sample. The first method, discussed here, was found to be
less sensitive to the assumptions of the modelling and therefore was
adopted as our default. We discuss our alternative approach in
\S~\ref{sect_xmass}.
Starting with a parameterised
model for the temperature (T) and gravitating mass ($M_{grav}$)
profiles, the equation of
hydrostatic equilibrium can be solved for \rhog\ thus:
\begin{equation}
\ln \left( \frac{\rho_g}{\rho_{g0}} \right) = - \ln \left( \frac{T}{T_0} \right)
- G \mu m_p \int^R_{R_0} \frac{M_{grav}(<R)}{kT R^2} dR
\label{eqn_hydrostatic_rho}
\end{equation}
where R is the radius from the centre of the gravitational potential,
\rhog\ is the gas density, $\rho_{g0}$ and $T_0$ are density and temperature
at some ``reference'' radius $R_0$,
k is Boltzmann's constant, G is the universal gravitational constant,
$m_p$ is the atomic mass unit and $\mu$ is the mean atomic weight of the gas.
In our fitting we explicitly ignored the contribution of the gas to
the gravitating mass, but we subsequently verified this contributed
\ltsim 1\% of the total gravitating matter within 100~kpc, justifying
this assumption.
We developed software to fit \rhog\ and temperature
profiles simultaneously using this procedure.
For speed we assumed that the density and temperature data-points were
each evaluated at a single point, the radius of which was given by:
\begin{equation}
\bar{R_i} = \left( 0.5*(Rin_i^{1.5}+Rout_i^{1.5}) \right)^{2/3} \label{eqn_radius}
\end{equation}
where $Rin_i$ and $Rout_i$ were the inner and outer radius of the bin
\citep[see][]{lewis03a}.
\subsection{Temperature profiles} \label{sect_temp_profiles}
There were considerable differences
in the temperature profiles from object to object
(Fig~\ref{fig_temp}), so that we were not
able to adopt a ``universal'' profile for all of the systems.
{\em A priori} we do not expect any particular form for the temperature
profile and so we determined appropriate functional forms for our
temperature models empirically. Based on experience, the following
``toolbox'' of models provided adequate flexibility to ensure
at least one model can describe
the temperature profiles reasonably well \citep[see][]{buote06a}:
\begin{eqnarray}
T & = & T_0 + T_1 \left[1+x^{-\epsilon}\right]^{-1} \label{eqn_trise2}\\
T & = & \left[T_0 + T_1 x^{p_1}\right]e^{-x^{p_e}} + T_2 x^{p_2} \left[1 - e^{-x^{p_e}}\right]
\label{eqn_pow2expcut2} \\
T & = & \frac{A}{A+B}\left[ T_0+T_1\left( \frac{x_1}{1+x_1}\right)^{p_1} \right]
+ \nonumber\\ & & \frac{B}{A+B}\left[ T_2+T_3(1+x_2)^{-p_2} \right] \label{eqn_twophase2}
\end{eqnarray}
where $x=(r/r_c)$, $x_1=(r/r_{c1})$, $x_2=(r/r_{c2})$,
$A=(1+r/r_{t1})^{-3\beta_1}$ and $B=\epsilon (1+r/r_{t2})^{-3\beta_2}$.
$T_0$, $T_1$, $T_2$, ${T_3}$, $r_c$, $r_{c1}$, $r_{c2}$,
$r_{t1}$, $r_{t2}$, $p_1$, ${p_2}$, $p_e$ and $\epsilon$ are parameters
of the fit.
For NGC\thin 4261, we ignored temperature data-points from 15--25~kpc,
which were poorly-determined and seemed erroneous. We experimented with
fitting the projected (rather than deprojected) spectra, and found no
evidence of any features (in either temperature or density) around
this range of radii, strongly implying that they
arise solely due to deprojection noise. The temperature profiles and best-fit
models are shown in Fig~\ref{fig_temp}.
Our deprojected temperature profiles generally agree with those
appearing in the literature for these objects. (Although most
of these are projected profiles, typically deprojection does not
strongly alter the overall character of the temperature profile.)
\citet{osullivan03a}
reported \rosat\ profiles for all of the galaxies which, although
substantially less well-constrained, agree well with our results.
Likewise our NGC\thin 4472 temperature profile agrees well with
the (less well-constrained) \rosat\ profile of \citet{irwin96}.
Our profile for NGC\thin 4649 is in reasonable agreement with the
projected \xmm\ measurements of \citet{randall05a}, and likewise our
measured profile of NGC\thin 6482 agrees with the deprojected
results of \citet{khosroshahi04a}. Our temperature profiles for
NGC\thin 1407, NGC\thin 720 and NGC\thin 4472 were also in agreement
with those we reported in \citet{humphrey05a}.
\subsection{Mass-fitting results} \label{sect_mass_results}
\begin{deluxetable*}{llll}
\tablecaption{Quality of the mass fits\label{table_chisq}}
\tabletypesize{\scriptsize}
\tablehead{ \colhead{Galaxy} & \colhead{NFW} & \colhead{NFW+stars} & \colhead{AC NFW+stars}
}
\startdata
NGC\thin 720 & 1.9/9& 1.0/8& 0.9/8\\
NGC\thin 1407 & 26.7/9& 20.7/8& 20.7/8\\
NGC\thin 4125 & 23.4/11 & 9.5/10 & 10.8/10 \\
NGC\thin 4261 & 22.6/12 & 14.0/11 & 14.0/11 \\
NGC\thin 4472 & 35.2/20 & 34.9/20 & 35.2/20 \\
NGC\thin 4649 & 30.2/7 & 11.0/6 & 11.5/6 \\
NGC\thin 6482 & 0.5/5 & 2.4/4 & 1.7/4
\enddata
\tablecomments{The $\chi^2$/dof of the fits to the density and
temperature profiles used to infer the mass, for the three
basic mass-models adopted. For the NFW+stars and AC NFW+stars models,
we constrain \fbaryons\ to Eq~\ref{eqn_baryons}.}
\end{deluxetable*}
\begin{deluxetable*}{lllll|rrrr}
\tablecaption{Best-fitting NFW+stars results\label{table_results}}
\tabletypesize{\scriptsize}
\tablehead{ \colhead{Galaxy} & \multicolumn{4}{l}{\fbaryons=Eq~\ref{eqn_baryons}} & \multicolumn{4}{l}{0.032$\leq$\fbaryons$\leq$0.16}\\
\colhead{} & \colhead{\mvir ($10^{12}$\msun)} & \colhead{\rvir (kpc)} & \colhead{c} & \colhead{\fbaryons} &\colhead{\mvir ($10^{12}$\msun)} & \colhead{\rvir (kpc)} & \colhead{c} & \colhead{\fbaryons} }
\startdata
NGC720 &$6.6^{+2.4}_{-3.0}$ &$480^{+50}_{-90}$ &$18.^{+30.}_{-8.}$ &$0.044^{+0.037}_{-0.003}$ &$6.6^{+6.0}_{-4.3}$ &$480\pm 120$ &$18.^{+49.}_{-10.}$ &$0.044^{+0.095}_{-0.012}$ \\
NGC1407 &$16.\pm 6.$ &$650^{+80}_{-100}$ &$18.^{+11.}_{-7.}$ &$0.065^{+0.041}_{-0.001}$ &$21.\pm 15.$ &$720^{+140}_{-200}$ &$15.^{+16.}_{-6.}$ &$0.032^{+0.130}_{-0.001}$ \\
NGC4125 &$6.2^{+0.8}_{-2.3}$ &$470^{+20}_{-70}$ &$10.^{+5.}_{-2.}$ &$0.039^{+0.035}_{-0.001}$ &$7.2^{+1.4}_{-4.9}$ &$500^{+30}_{-160}$ &$9.3^{+11.}_{-2.1}$ &$0.032^{+0.13}_{-0.001}$ \\
NGC4261 &$67.^{+41.}_{-15.}$ &$1040^{+200}_{-90}$ &$3.7\pm 1.7$ &$0.14^{+0.01}_{-0.03}$ &$57.^{+260}_{-15.}$ &$990^{+760}_{-100}$ &$4.0\pm 2.0$ &$0.16^{+0.00}_{-0.13}$ \\
NGC4472 &$33.^{+6.}_{-10.}$ &$820^{+50}_{-100}$ &$13.^{+4.}_{-2.}$ &$0.084^{+0.037}_{-0.001}$ &$63.^{+17.}_{-44.}$ &$1020^{+90}_{-300}$ &$10.0^{+7.}_{-2.}$ &$0.032^{+0.13}_{-0.00}$ \\
NGC4649 &$35.^{+7.}_{-13.}$ &$840^{+60}_{-120}$ &$21.^{+6.}_{-3.}$ &$0.086^{+0.037}_{-0.001}$ &$93.^{+26.}_{-73.}$ &$1200^{+100}_{-500}$ &$15.^{+11.}_{-3.}$ &$0.032^{+0.12}_{-0.00}$ \\
NGC6482 &$7.1^{+4.4}_{-1.7}$ &$500^{+90}_{-40}$ &$18.^{+13.}_{-8.}$ &$0.075^{+0.013}_{-0.032}$ &$3.6^{+5.5}_{-1.5}$ &$390^{+140}_{-70}$ &$38.^{+76.}_{-24.}$ &$0.16^{+0.00}_{-0.10}$
\enddata
\tablecomments{The best-fitting results for the NFW+stars model. All
error-bars shown correspond to 90\% confidence regions. The
fit results for the AC NFW+stars model are very similar, and are
shown in Fig~\ref{fig_confidence}. Results are shown for the fits
using the two different constraints on \fbaryons\ we adopted (see text).}
\end{deluxetable*}
We tested three different mass-models against the data.
In order to investigate the suggestion that historically large
c values found based on X-ray analysis were an artefact of the
omission of the stellar mass, as well as to investigate the
scenario of \citet{loeb03a}, we first tested a model comprising a
single NFW profile. Although stellar kinematical results would
seem to rule out the \citeauthor{loeb03a} picture, our analysis
of more massive systems \citep{gastaldello06a} does suggest that the
stellar mass may not be uniformly required in all systems.
In order to take into account the stellar mass, we fitted a
model comprising an NFW DM component, plus
a \citet[][hereafter H90]{hernquist90} mass component, the \reff\
of which being fixed to that measured in the \ks-band
(Table~\ref{table_obs}). The H90 model is, in projection, a good
approximation to the familiar de Vaucouleurs profile of elliptical
galaxies. To test whether the DM halos retained any
evidence of their response to baryonic
condensation, we further adopted an H90 component, plus an NFW component
modified by the adiabatic contraction model of \citet{gnedin04a}\footnote{Available publicly from \href{http://www.astronomy.ohio-state.edu/$\sim$ognedin/contra/}{http://www.astronomy.ohio}-state.edu/$\sim$ognedin/contra/}.
Hereafter, we refer to these three models as, respectively, NFW,
NFW+stars and AC NFW+stars. Our computed \mvir\ for each system included both
dark and stellar mass. For NGC\thin 4472 and NGC\thin 4649, which lie
in Virgo, there is the possibility that the DM halo may have experienced
some tidal truncation at a radius $<$\rvir. Our measured Virial quantities
relate to the original halo prior to truncation.
We also experimented with replacing the NFW component with the less
cuspy \citet[][hereafter N04]{navarro04a} model, which gives
an improved fit to DM halos in high-resolution N-body simulations.
However, since the \mvir-c relation was calibrated using
the NFW model we treat this choice as a systematic effect and it is
discussed in \S~\ref{sect_n04}.
For NGC\thin 4261 we ignored a deviant \rhog\ data-point at $\sim$11~kpc,
in addition to the excluded temperature data-points discussed above. %
In Fig~\ref{fig_density} we show the density profiles (along with the
best-fitting AC NFW+stars model, which is described below).
In Fig~\ref{fig_confidence}(a) we show the best-fitting
1-$\sigma$ contours of c {\em versus}
\mvir\ for the NFW model fitted to each galaxy. The fits were
typically, but not uniformly, poor (Table~\ref{table_chisq}).
We found very large ($\gg$20) values for c, completely
inconsistent with the expectation of N-body simulations.
Good constraints on the global halo properties typically require interesting
density and temperature constraints over as large a radial range as possible.
In our case, the absence of data outside $\sim$50--100~kpc ($\sim$0.1--0.2\rvir)
therefore
makes the inner data-points critical in determining the profile of the
halo. Unfortunately, since the scale radius of a galaxy-size DM halo is
$\sim$10--30~kpc, there is some degeneracy between the DM and
stellar mass components at small radii.
As we discuss in \S~\ref{sect_mass_to_light} there are considerable
uncertainties in estimating a reliable mass-to-light (M/L) ratio from
the characteristics of the stellar population. We found that the
results are extremely sensitive to the stellar M/L adopted; we found
that varying this ratio by as little as 20\% could cause \mvir\
variations of $\sim$50--100\% \citep[see][]{humphrey05b}.
It was therefore necessary to allow the stellar mass to be determined as a
parameter of the fit. This, unfortunately, made it very difficult
to constrain \mvir\ or c, unless additional constraints were applied
to the fit.
One way to achieve this is to constrain the fit to lie on the mean
\mvir-c relation predicted from N-body simulations \citep[\eg][]{bullock01a}.
Although this would prevent our measuring \mvir\ and c independently,
it would enable us to determine whether the
data are consistent with the mean relation.
However, this relation was determined for an unbiased sample of
DM halos, whereas our selection criteria (\S~\ref{sect_targets}) should
bias us towards systems which have not recently had a merger (implying
earlier-forming, hence more concentrated, objects).
Furthermore, individual halos are not expected
to lie exactly on the mean \mvir-c relation, but be scattered
about it. Nevertheless, we experimented with applying this constraint.
The data for each galaxy were consistent with this model, but
we found \mvir\ was generally poorly constrained, and extremely sensitive
to any scatter we introduced about the mean \mvir-c relation.
A far more useful way to constrain the fit was to restrict the
total baryon fraction (\fbaryons) in the system.
Such a constraint is useful since we found that, for a given system,
\fbaryons\ determined from our fits was strongly anti-correlated with
the measured \mvir. To estimate \fbaryons, we
computed the gas mass by extrapolating our \rhog\ model from the
centre of the innermost radial bin to the Virial radius.
The contribution of stars to the total baryon fraction was derived
from the stellar mass found by our fits.
We crudely took into account the fact that
not all of the stellar mass within \rvir\ is necessarily
contained in the central galaxy by scaling this mass by the ratio
of the total B-band light of all putative ``group'' members listed
in the catalogue of \citet[][hereafter G93]{garcia93} to that of the central galaxy.
This is likely to overestimate slightly the stellar mass content, since
it assumes the same stellar M/L ratio for all low-mass companions/ group
members,
whereas some fraction of these are likely to have substantial young stellar
populations, with lower M/L ratios.
We discuss the impact of this assumption in \S~\ref{sect_systematics_baryon_fraction}.
G93 lists NGC\thin 4649 as belonging to the NGC\thin 4472
``group'', whereas they both have distinct X-ray halos, indicating
they are, in fact, distinct systems. As a zeroth order approximation, we
therefore assumed that the total B-band luminosity was divided between
the two ``subgroups'' in proportion to the central galaxy's B-band
luminosity. In practice, between $\sim$25\% (for NGC\thin 4472) and
84\% (for NGC\thin 4125) of the B-band light of the system resides in the
central galaxy. NGC\thin 6482 was not listed in G93,
but as it is known to be relatively isolated \citep{khosroshahi04a},
we assumed that $\sim$80\% of its mass is in the central galaxy, consistent
with the other relatively isolated systems.
Based on hydrodynamical simulations incorporating gas cooling and
supernovae feedback, \citet{kay03a} predicted \fbaryons\ as a function
of Virial temperature for systems with \mvir\gtsim a few $\times 10^{12}$\msun.
Fitting their data by eye, converting from Virial temperature to mass,
and assuming a Universal baryon fraction of 0.16, we estimate
\begin{eqnarray}
f_{b} = & 0.062 \log_{10}(M_{14})+0.13 & (M_{14}<1.02) \label{eqn_baryons} \\
f_{b} = & 0.14 & (M_{14}>1.02)\nonumber
\end{eqnarray}
where ${M_{14}=M_{vir}/10^{14} M_\odot}$, with an approximate scatter of
$\pm$0.02.
Adopting this constraint (including the allowed range of scatter)
and fitting the NFW+stars model resulted
in significant improvements in the fit quality over the simple NFW
model (Table~\ref{table_chisq}).
We show the resulting c-{\em versus}-\mvir\ contours in
Fig~\ref{fig_confidence}(b), and
summarise our results in Table~\ref{table_results}.
Clearly adding the stellar mass component
allows the DM halos to be substantially less concentrated, since
less DM is required in the centre of the halo.
These results were in much better agreement with the results
of N-body simulations than those obtained with the NFW model.
There is, however, a slight trend towards
more concentrated halos than \lcdm.
Since the results of simulations can be sensitive to the rather uncertain
process of feedback, we additionally adopted, as a
somewhat less restrictive constraint on \fbaryons, 0.03$<$\fbaryons$<0.16$,
the lower limit being $\sim$the lowest values found in
\citeauthor{kay03a}'s simulations.
The shallower potential of very low-mass halos
makes it more difficult for them to hold onto their hot gas,
and so our lower limit on \fbaryons\ may be an overestimate if
the Virial mass is small. However, imposing a lower limit on
\fbaryons\ in the fitting algorithm actually works to exclude
the most massive solutions, which we would expect to be closer
to baryonic closure \citep{mathews05a}. It is conceivable that some
more massive (\mvir\gtsim $10^{13}$\msun) systems are rare examples
of ``dark'' groups which have unusually low baryon fractions,
a possibility we return to in \S~\ref{sect_ngc1407}.
Notwithstanding, the results are shown in Figs~\ref{fig_confidence}(c),
which are qualitatively similar to those obtained applying the more
restrictive constraint on \fbaryons, although with larger
uncertainty.
In Fig~\ref{fig_mass_to_light}
we show the gravitating mass to K-band light (\mgrav/\lk) ratio profile
implied by our best-fit models for each system.
In each case we found that the X-ray emission was considerably
more extended than the optical light.
We also show data-points estimated from ``parameterized profile''
mass modelling (\S~\ref{sect_xmass}),
which tend to agree reasonably well; the slight systematic
differences between the profiles are an artefact of the assumptions used
to derive the data-points and we discuss this in detail in \S~\ref{sect_xmass}.
Clearly \mgrav/\lk\ increases
very slowly with radius within \reff, rising very steeply outside this
range. This arises naturally from the very different shapes of the stellar
and DM halos, and is similar to M/L profiles seen from stellar
kinematics and the results of \citet{brighenti97a} for NGC\thin 4472
and NGC\thin 4649.
By \rvir, \mgrav/\lk\ reaches as high as $\sim$20--40
\msun/\lsun\ for the galaxy-scale systems or
$\sim$100-200 \msun/\lsun\ for the group-like objects. We stress
that this only includes the light of the central galaxy which,
for the group-like systems may be a little as $\sim$25\% of the total
luminosity.
\subsection{Parameterized profile mass modelling} \label{sect_xmass}
We briefly discuss here an alternative technique to determine the
mass profiles of X-ray bright objects which we have extensively
employed in our previous studies, as well as the companion
papers to this present work
\citep[\eg][]{lewis03a,buote06a,gastaldello06a,zappacosta06a}.
This technique, which we here dub the ``parameterized profile'' method
involves parameterizing independently the temperature and density
profiles of the system with simple, empirical models. These functions
were then inserted into the equation of hydrostatic equilibrium,
which we solved for the mass enclosed within any given radius.
The temperature profiles were parameterized with the models
discussed in \S~\ref{sect_temp_profiles}, and to fit \rhog\ we
adopted, where appropriate a, $\beta$-model, a ``double-$\beta$'' model
or a ``cusped-$\beta$'' model, defined, respectively, as:
\begin{eqnarray}
\rho_g & = & \rho_{g0} \left[ 1+ (r/r_c)^2 \right]^{-3\beta /2} \label{eqn_beta}\\
\rho_g & = & \sqrt{\rho_{g0} \left[ 1+ (r/r_c)^2 \right]^{-3\beta} + \rho_{g1} \left[ 1+ (r/r_{c2})^2 \right]^{-3\beta_2}} \label{eqn_twobeta} \\
\rho_g & = & \rho_{g0} 2^{3\beta/2-\epsilon/2}(r/r_c)^{-\epsilon}\left[ 1+ (r/r_c)^2 \right]^{-3\beta /2+\epsilon/2} \label{eqn_cusp}
\end{eqnarray}
Where the parameters $\rho_{g0}$, $\rho_{g1}$, $r_c$, $r_{c2}$,
$\beta$, $\beta_2$ and $\epsilon$ are determined by the fit.
Fitting these models to the simulated temperature and density profiles
discussed in \S~\ref{sect_spectra} (which were used therein to estimate the
error-bars on kT and \rhog\ in each data-bin) allowed us to estimate the
scatter in the mass data-points arising from statistical noise, and
hence the error-bars. %
For a full discussion of this technique, we refer the interested
reader to \citet{buote06a}, who demonstrate the good agreement
typically found between this method and the assumed potential
modelling of \S~\ref{sect_potential}, when fitting high-quality data.
However, the mass data-points, especially at the innermost and outermost
radii, are rather sensitive to the parameterized models adopted to fit
the temperature and, especially, \rhog. The systematic uncertainty
introduced by the choice of \rhog\ model can be considerably
larger than the statistical error. For our purposes the absence of data
at very large radii, which are vital to constrain the curvature of the mass
model, exacerbated by the uncertainty introduced at small radii due to
the uncertain stellar mass-to-light ratio, magnified the impact of these
systematic effects. Notwithstanding these reservations, it is still interesting
to compare the results obtained {\em via} both mass-fitting methods.
We show in Fig~\ref{fig_mass} the mass data-points computed using parameterized
potential modelling, along with the best-fitting mass models
found in \S~\ref{sect_potential}. Clearly there is good overall agreement
between the two methods although there are some systematic differences,
which reflect the systematics inherent in our choice of
parameterized model for \rhog.
\subsection{Stellar mass-to-Light ratios} \label{sect_mass_to_light}
\begin{deluxetable*}{llll|rr}
\tablecolumns{5}
\tablecaption{Stellar mass-to-light ratios\label{table_mass_to_light}}
\tabletypesize{\scriptsize}
\tablehead{ \colhead{Galaxy} & \colhead{\lk/\lb} &
\multicolumn{2}{c}{Fitted \mstars/\lk\ (\msun/\lsun)} &
\multicolumn{2}{c}{Pop.\ synthesis \mstars/\lk\ (\msun/\lsun)}\\
\colhead{} & \colhead{} & \colhead{NFW+stars} & \colhead{AC NFW+stars} &
\colhead{Salpeter IMF} & \colhead{Kroupa IMF}
}
\startdata
NGC\thin720 & 5.5 & 0.77$^{+0.52}_{-0.71}$& 0.54$^{+0.42}_{-0.48}$& 0.54$\pm$0.11& 0.35$\pm$0.07\\
NGC\thin 1407 & 4.8 & 0.52$^{+0.25}_{-0.32}$& 0.35$\pm$0.25& 1.6$\pm$0.2& 1.1$\pm$0.1\\
NGC\thin 4125 & 3.8 & 0.72$\pm$0.11& 0.53$\pm$0.11& 1.7$\pm$0.5& 1.1$\pm$0.4\\
NGC\thin 4261 & 5.0 & 1.2$\pm$0.1& 1.0$\pm$0.1& 1.9$\pm$0.1& 1.3$\pm$0.1\\
NGC\thin 4472 & 4.3 & 0.51$^{+0.24}_{-0.31}$& 0.36$\pm$0.1& 1.3$\pm$0.3& 0.83$\pm$0.15\\
NGC\thin 4649 & 4.9 & 0.90$\pm$0.13& 0.65$\pm$0.12& 1.7$\pm$0.2& 1.1$\pm$0.1\\
NGC\thin 6482 & 2.9 & $0.73^{+0.18}_{-0.27}$& 0.52$^{+0.19}_{-0.23}$& 1.58$\pm$0.02& 1.05$\pm$0.02\\
\enddata
\tablecomments{K-band stellar mass-to-light ratios measured from our
fits to the data using both the NFW+stars and the AC NFW+stars models. Since AC tends
to increase the cuspiness of the DM profiles, \mstars/\lk\ is substantially
lower for the AC NFW+stars models. We also show the predicted \mstars/\lk\ values
derived from simple stellar population synthesis, assuming either the
Salpeter or \citet{kroupa01a} IMF.}
\end{deluxetable*}
It is interesting to compare the stellar M/L ratios (\mstars/L)
determined by
our fitting to the expectations of stellar population synthesis
models. In order to ensure that the optical light traces the
stellar mass as closely as possible, we opted to perform this
comparison in the K-band.
Table~\ref{table_mass_to_light}
shows \mstars/\lk\ determined from our models using eq~\ref{eqn_baryons}
to constrain \fbaryons.
Since AC tends to increase the cuspiness of the DM profile
we found a significantly lower mass-to-light ratio for the AC NFW+stars model
than for the NFW+stars model.
To compare our measured \mstars/\lk\ to single burst stellar population synthesis
predictions, we first estimated a mean emission-weighted
stellar age and metallicity for each galaxy, as
outlined in Appendix~\ref{sect_stars}.
We linearly interpolated synthetic \mstars/\lk\ values based on
the stellar population models of \citet{maraston98a} from
updated model-grids made available by the author\footnote{\href{http://www-astro.physics.ox.ac.uk/~maraston/Claudia's_Stellar_Population_Models.html}{http://www-astro.physics.ox.ac.uk/$\sim$maraston/Claudia's}-\\\_Stellar\_Population\_Models.html}.
For typical early-type galaxies, K-band and
\twomass\ \ks-band magnitudes should
differ by $<$0.1~magnitudes \citep{carpenter01a}, so we were
able to compare directly the synthetic K-band M/L ratios with
our measured \mstars/\lk\ ratios.
The predicted \mstars/\lk\
ratios are shown in Table~\ref{table_mass_to_light} for different
assumptions about the stellar IMF, which is poorly-known in
early-type galaxies. In this case we show predicted \mstars/\lk\ assuming
a standard Salpeter IMF, and for the IMF of \citet{kroupa01a}.
It is immediately clear that these ratios
are very sensitive to this choice; \mstars/\lk\ is typically
$\sim$50--60\% higher if the Salpeter IMF is used.
Our measured \mstars/\lk\ for the NFW+stars models are typically
$\sim$20\% lower than the synthetic M/L ratios, assuming the
Kroupa IMF. Using the AC NFW+stars models, the discrepancy is $\sim$40\%.
Assuming a Salpeter IMF, the discrepancies for both models
are considerably larger. This would seem to rule out the
Salpeter IMF, in agreement with the conclusions of
\citet{padmanabhan04a}.
The best-fitting \mvir\ and c are sensitive to
\mstars/\lk. If we fix \mstars/\lk\ to the synthetic value, this
essentially pushes all the galaxies, except NGC\thin 4261 (for which
the measured and synthetic values are in excellent agreement) and
NGC\thin 720 in the direction of the high-\mvir\ range of their
confidence contours shown in Fig~\ref{fig_confidence}.
For NGC\thin 720, \mvir\ is lowered and c increased. The
fits are then typically much worse ($\Delta \chi^2 \sim$7--35),
and the loci in the \mvir-c plane slightly more discrepant with
simulations.
There are a considerable number of systematic uncertainties
in the computation of the synthetic M/L ratios, not the least
of which is the very uncertain IMF, which could
probably account for the modest discrepancy with our NFW+stars
results (see \S~\ref{discussion_mass_to_light}).
In the case of NGC\thin 720, the rather young age inferred for
the stellar population ($\sim$3~Gyr) leads to a significantly lower
synthetic \mstars/\lk\ than measured. Fitting template models
to spatially-resolved spectra of this system, \citet{rembold05a}
found evidence of a significant age gradient, which falls from
$\sim$12~Gyr in the centre to $\sim$3~Gyr by 1~kpc. This may, therefore,
represent a system in which a relatively small fraction of the
stellar component, produced in a modest, recent star-formation
event (``frosting'') dominates the optical line emission. In this case,
the synthetic \mstars/\lk\ may be underestimated. We return to this issue
in \S~\ref{discussion_mass_to_light}.
\section{Systematic errors} \label{sect_systematics}
\begin{deluxetable*}{lllllllllll}
\tablecaption{Systematic error budget}
\tabletypesize{\scriptsize}
\tablehead{ \colhead{Galaxy} & \colhead{Best-fit} & \colhead{$\Delta$stat} &
\colhead{$\Delta$N04} & \colhead{$\Delta$stars} & \colhead{$\Delta$bkd} &
\colhead{$\Delta$asym} & \colhead{$\Delta$temp} & \colhead{$\Delta$spectra}
& \colhead{$\Delta$dist} & \colhead{$\Delta$baryons}
}
\startdata
\multicolumn{11}{c}{\mvir/$10^{12}$\msun}\\ \hline
NGC720 &$6.6$ &$^{+2.4}_{-3.0}$ &$+0.06$ &$^{+0.5}_{-0.3}$ &$-1.7$ &$-0.2$ &$-0.9$ &$^{+0.7}_{-0.2}$ &$^{+0.9}_{-0.6}$ &$-0.01$ \\
NGC1407 &$16$ &$\pm6$ &$-0.1$ &$^{+1}_{-0.5}$ &$-3$ &$+3$ &$^{+1}_{-3}$ &$^{+1}_{-3}$ &$^{+3}_{-4}$ &$-0.3$ \\
NGC4125 &$6.2$ &$^{+0.8}_{-2.3}$ &$-0.2$ &$^{+1.7}_{-2.1}$ &$^{+0.09}_{-0.2}$ &$^{+0.2}_{-0.03}$ &$-1.0$ &$^{+0.5}_{-2.0}$ &$^{+0.7}_{-0.6}$ &$-0.3$ \\
NGC4261 &$67$ &$^{+41}_{-15}$ &$+2$ &$^{+15}_{-24}$ &$+9$ &$^{+5}_{-0.5}$ &$^{+0.08}_{-0.6}$ &$^{+13}_{-0.04}$ &$^{+4}_{-7}$ &$-3$ \\
NGC4472 &$33$ &$^{+6}_{-10}$ &$+0.5$ &$^{+6}_{-0.05}$ &$+2$ &$+12$ &$-4$ &$^{+8}_{-1}$ &$\pm 3$ &$-5$ \\
NGC4649 &$35$ &$^{+7}_{-13}$ &$-8$ &$^{+16}_{-3}$ &$+2$ &$-1$ &$^{+63}_{-2}$ &$^{+6}_{-2}$ &$\pm 3$ &$-19$ \\
NGC6482 &$7.1$ &$^{+4.4}_{-1.7}$ &$-0.8$ &$^{+1.9}_{-0.3}$ &$-0.1$ &$+0.8$ &$^{+0.1}_{-3.6}$ &$^{+0.3}_{-0.8}$ &$^{+0.9}_{-0.8}$ &$-0.2$ \\
\hline\multicolumn{11}{c}{c}\\ \hline
NGC720 &$18$ &$^{+30}_{-8}$ &$+0.1$ &$\pm2$ &$+6$ &$+0.2$ &$+2$ &$^{+2}_{-1}$ &$\pm 2$ &$+0.03$ \\
NGC1407 &$18$ &$^{+11}_{-7}$ &$-0.8$ &$^{+2}_{-3}$ &$^{+4}_{-3}$ &$-4$ &$^{+4}_{-1}$ &$^{+6}_{-3}$ &$^{+5}_{-3}$ &$+0.02$ \\
NGC4125 &$10$ &$^{+5}_{-2}$ &$-1$ &$^{+5}_{-4}$ &$^{+0.9}_{-0.1}$ &$^{+0.05}_{-1.0}$ &$+2$ &$^{+2.}_{-0.7}$ &$^{+2}_{-1}$ &$+0.3$ \\
NGC4261 &$3.7$ &$\pm 1.7$ &$-1.4$ &$^{+3.6}_{-1.2}$ &$^{+0.10}_{-0.7}$ &$-0.3$ &$^{+0.08}_{-0.03}$ &$-1.5$ &$^{+0.8}_{-0.1}$ &$+0.08$ \\
NGC4472 &$13$ &$^{+4}_{-2}$ &$-2$ &$^{+0.4}_{-2}$ &$-0.7$ &$-5$ &$+1$ &$^{+0.2}_{-3.}$ &$\pm 2.$ &$+1.0$ \\
NGC4649 &$21$ &$^{+6}_{-3}$ &$-2$ &$^{+3}_{-6}$ &$-0.9$ &$+2$ &$^{+3}_{-4}$ &$^{+0.4}_{-3}$ &$\pm 3$ &$+7$ \\
NGC6482 &$18$ &$^{+13}_{-8}$ &$+1$ &$^{+2}_{-6}$ &$+3$ &$+6$ &$^{+51.}_{-0.4}$ &$+2$ &$\pm 3$ &$-0.7$ \\
\hline \multicolumn{11}{c}{\mstars/\lk (NFW+stars)}\\ \hline
NGC720 &$0.77$ &$^{+0.52}_{-0.71}$ &$+0.17$ &$^{+0.28}_{-0.18}$ &$-0.15$ &$+0.07$ &$-0.04$ &$\pm 0.10$ &$\pm 0.16$ &$-0.001$ \\
NGC1407 &$0.52$ &$^{+0.25}_{-0.32}$ &$+0.06$ &$^{+0.71}_{-0.16}$ &$^{+0.12}_{-0.04}$ &$+0.14$ &$-0.19$ &$^{+0.09}_{-0.18}$ &$^{+0.10}_{-0.08}$ &$+0.001$ \\
NGC4125 &$0.72$ &$\pm 0.11$ &$+0.04$ &$^{+0.62}_{-0.20}$ &$^{+0.01}_{-0.03}$ &$+0.06$ &$-0.04$ &$+0.05$ &$\pm 0.15$ &$-0.003$ \\
NGC4261 &$1.2$ &$\pm 0.1$ &$+0.05$ &$^{+0.7}_{-0.8}$ &$^{+0.008}_{-0.001}$ &$^{+0.04}_{-0.05}$ &$^{+0.0009}_{-0.006}$ &$^{+0.09}_{-0.010}$ &$\pm 0.2$ &$-0.006$ \\
NGC4472 &$0.51$ &$^{+0.24}_{-0.31}$ &$+0.06$ &$^{+0.45}_{-0.06}$ &$+0.05$ &$+0.20$ &$-0.01$ &$^{+0.13}_{-0.02}$ &$^{+0.11}_{-0.09}$ &$-0.01$ \\
NGC4649 &$0.91$ &$\pm 0.13$ &$+47$ &$^{+0.80}_{-0.19}$ &$+0.02$ &$-0.06$ &$-0.16$ &$^{+0.05}_{-0.009}$ &$\pm 0.18$ &$-0.03$ \\
NGC6482 &$0.73$ &$^{+0.18}_{-0.27}$ &$+0.13$ &$^{+0.50}_{-0.04}$ &$-0.06$ &$-0.15$ &$-0.51$ &$-0.05$ &$\pm 0.15$ &$-0.01$ \\
\hline \multicolumn{11}{c}{\mstars/\lk (AC NFW+stars)}\\ \hline
NGC720 &$0.54$ &$^{+0.42}_{-0.48}$ &$-0.12$ &$\pm 0.16$ &$-0.14$ &$+0.06$ &$-0.05$ &$^{+0.03}_{-0.10}$ &$^{+0.12}_{-0.09}$ &$-0.003$ \\
NGC1407 &$0.35$ &$\pm 0.25$ &$-0.06$ &$^{+0.41}_{-0.08}$ &$^{+0.12}_{-0.05}$ &$+0.12$ &$-0.14$ &$^{+0.07}_{-0.13}$ &$^{+0.07}_{-0.05}$ &$-0.001$ \\
NGC4125 &$0.53$ &$\pm 0.11$ &$-0.04$ &$^{+0.46}_{-0.14}$ &$^{+0.01}_{-0.03}$ &$+0.06$ &$-0.04$ &$+0.04$ &$^{+0.11}_{-0.09}$ &$-0.002$ \\
NGC4261 &$1.0$ &$\pm 0.1$ &$+0.02$ &$\pm0.6$ &$^{+0.02}_{-0.004}$ &$^{+0.03}_{-0.04}$ &$-0.006$ &$^{+0.1}_{-0.003}$ &$\pm 0.2$ &$-0.009$ \\
NGC4472 &$0.36$ &$\pm0.1$ &$-0.04$ &$^{+0.27}_{-0.03}$ &$+0.04$ &$+0.17$ &$-0.02$ &$^{+0.09}_{-0.01}$ &$^{+0.08}_{-0.06}$ &$-0.010$ \\
NGC4649 &$0.65$ &$\pm 0.12$ &$-0.05$ &$^{+0.57}_{-0.12}$ &$+0.02$ &$-0.06$ &$-0.13$ &$^{+0.06}_{-0.008}$ &$\pm 0.13$ &$-0.05$ \\
NGC6482 &$0.52$ &$^{+0.19}_{-0.23}$ &$-0.07$ &$^{+0.37}_{-0.03}$ &$-0.05$ &$-0.13$ &$^{+0.005}_{-0.03}$ &$-0.04$ &$^{+0.11}_{-0.09}$ &$-0.01$ \\
\enddata
\tablecomments{The estimated error-budget for each of the galaxies.
Excepting the statistical error, these values estimate a likely
upper bound on the sensitivity of the (best fit) value of each
parameter to various data-analysis choices, and should {\em not}
be added in quadrature with the statistical errors.
The systematic uncertainties on \mvir\ and c are estimated for
the NFW+stars model.
In addition to the best-fit values, we show the 90\% confidence
interval for each parameter ($\Delta$stat). We also show
estimated upper-limits on the systematics likely to arise by making
various changes to our default analysis choices. This includes
adopting the N04 DM model ($\Delta$N04), varying the shape of the
stellar mass component ($\Delta$stars), varying the background
($\Delta$bkd), excluding data in the vicinity of asymmetries
($\Delta$asym), adopting alternative temperature models
($\Delta$temp), changing spectral analysis choices ($\Delta$spectra),
varying the distance ($\Delta$dist) or assuming that all of the
stellar baryons are in the central galaxy ($\Delta$baryons).} \label{table_syserr}
\end{deluxetable*}
In this section we address the sensitivity of our results to various
data-analysis choices which were made. An estimated upper limit
on the sensitivity of our results to these choices is shown in
Table~\ref{table_syserr}. These numbers reflect the
sensitivity in the best-fit parameter to
each potential source of systematic error, and we stress they
should {\em not} be added in quadrature with the statistical
errors. We outline in detail below how each of these
systematics were estimated. Those readers
uninterested in the technical details of our analysis may like
to proceed directly to \S~\ref{sect_discussion}.
\subsection{DM profile shape} \label{sect_n04}
As discussed above, we experimented with replacing the NFW model by
the revised N04 model, which is less cuspy.
We caution that the \mvir-c relation was
derived assuming NFW.
We fixed the $\alpha$ parameter for this model to 0.17, the
mean value determined from simulations since the inner slope
of the DM halo is degenerate to some degree with the stellar mass.
The quality of the fits (N04+stars, AC N04+stars) were typically similar to
those using the simple NFW profile. There were some slight systematic
differences
in the inferred \mvir\ and c as compared to NFW. It is interesting
to note that this model, which is less cuspy than NFW, gave
slightly larger \mstars/\lk, although not sufficiently to bring our
measurements completely into agreement with the synthetic estimates.
For the adiabatically-compressed N04 model, \mstars/\lk\ did not
increase, but this is unsurprising since the stellar component
significantly modifies the shape of the inner DM halo in this
model. We note that the predicted typical inner slope for
\lcdm\ halos is still under debate. If, instead of the N04 profile
we had adopted the cuspier profile of \citet{diemand05a},
then the resultant \mstars/\lk would have been even smaller, and in
worse agreement with stellar population models.
\subsection{Shape of the stellar potential} \label{sect_syserr_potential}
To account for the stellar component, we adopted an H90 model,
the effective radius of the model being fixed to that
determined by \twomass. However, it is not entirely clear
that the H90 model is an adequate description of the stellar
mass. There are some deviations between H90 and the
de Vaucouleurs model fitted as the {\em de facto} standard
to the optical light profiles of elliptical galaxies,
particularly in the critical central regions.
Furthermore, the K-band light profiles of elliptical galaxies may,
in fact, be better described by the \sersic\ profile
\citep[\eg][]{brown03a}.
To investigate the sensitivity of our results to the H90 assumption,
therefore, we experimented with adopting a \sersic\ stellar mass
potential \citep[\eg][]{prugniel97a}.
To determine the two parameters of this model
(the \sersic\ index and half-light
radius) we obtained the \ks-band \twomass\ images of each galaxy
from \ned, and fitted the surface brightness profiles
using dedicated software. A \sersic\ model fitted the
\ks-band light profile of each galaxy
in the radial range 5\arcsec--3\arcmin\ reasonably well.
The fitted profiles tended to be slightly more centrally peaked than
H90, which resulted primarily in
slightly {\em lower} inferred \mstars/\lk\ ratios when adopted
as mass models. We also experimented with replacing the H90 model
with a de Vaucouleurs model \citep{mellier87a},
and adopting \reff\ values from \citet{pahre99a}.
Elliptical galaxies exhibit radial colour gradients, which may reflect
gradients in the metallicity or age of the stellar population
(see discussion in \S~\ref{discussion_mass_to_light}). These
may therefore imply a radial gradient in the stellar M/L ratio.
It is beyond the scope of this present work to take such a gradient
into account. However, we investigated the sensitivity of
our results to the precise shape of the optical light profile
we adopted by experimenting with replacing the K-band \reff\
for each galaxy with the (typically larger) B-band value listed in RC3.
For NGC\thin 6482, for which \reff\ is not listed in RC3, we simply
increased \reff\ by 50\%.
\subsection{Background subtraction} \label{sect_systematics_bkd}
One of the major potential sources of systematic uncertainty
in measuring the mass profiles of galaxies is the background
subtraction technique. This is especially important in the
low surface-brightness regime at large radii in our galaxies.
In order to
estimate the likely magnitude of uncertainty arising from
our modelling, when initially fitting the background
components (\S~\ref{sect_bkd}) we artificially adjusted the
slope of the instrumental background components, which dominate
at high energy, to the limits
of their 90\% confidence regions, refitting the other components
and then refitted all the spectra with these revised background
models.
\subsection{X-ray asymmetries} \label{sect_asymmetry}
We note that there are some low-level asymmetries in the X-ray
surface brightness profiles (\S~\ref{sect_imaging}). In order to
assess the potential impact of these features, we experimented
with excluding or including the features. In particular, we
tried
excluding data from the vicinity of the jet and AGN in
NGC\thin 4261. We also excluded data from an off-axis X-ray
asymmetry in NGC\thin 4125 and excluded data for NGC\thin 4472
outside 6\arcmin, where \citet{irwin96} pointed out that the
X-ray data become asymmetric.
These choices did not dramatically affect our results, indicating
that these features did not indicate a significant
violation of hydrostatic equilibrium, at least in an
azimuthally-averaged sense.
To gain an insight into possible asymmetries in other sources, we
tried re-extracting all our spectra, and re-deriving the mass
profiles, from suitably-oriented semi-annuli
(thereby excluding one half of the emission from each system).
\subsection{Temperature profile}
In principle multiple temperature profiles may be able to fit the
same data adequately well but give rise to slightly different
global halo parameters. In particular our constraints upon
\fbaryons, the computation of which requires the extrapolation of the density
(and hence temperature) profiles to large radius, may make
\mvir\ and c sensitive to this effect. To test this, we
experimented with cycling through each of our adopted temperature profiles
(eq~\ref{eqn_trise2}--\ref{eqn_twophase2}). Provided the fits
were of comparable quality to our preferred choice, the impact on
the best-fit parameters reflect the systematic uncertainty in this
choice. Furthermore, we also experimented with excluding the central
bin from the temperature profiles of NGC\thin 1407 and NGC\thin 4649,
which may indicate a central disturbance (although there is no
obvious X-ray morphological disturbance in this region).
These choices did not strongly affect our results.
\subsection{Spectral-fitting choices} \label{sect_systematics_spectra}
A variety of choices are made in the spectral-fitting, each of
which can affect, to some degree, the inferred \rhog\ and temperature
in each radial bin. A thorough discussion of these
effects is given in \citet{humphrey05a}.
{\em Column density.} In order to take account of possible
local deviations in the line-of-sight \nh\ from the value
of \citet{dickey90}, we experimented with allowing \nh\ to vary
by $\pm$25\%.
{\em Bandwidth.} To estimate the impact of the bandwidth on
our fits, we experimented with fitting the data in the energy
ranges 0.7--7.0~keV, 0.5--2.0~keV and 0.4--7.0~keV, in addition
to our preferred choice 0.5--7.0~keV.
{\em Plasma code.} There are some uncertainties in the correct
modelling of the individual emission lines, in particular those of Fe. This
can systematically lead to differences in the inferred temperature
and density depending on choice of plasma code. We therefore
experimented with replacing the APEC model with the MEKAL plasma model.
{\em Unresolved source component.} We included a 7.3~keV bremsstrahlung
component to account for unresolved point sources
within \dtwentyfive. This model is generally a good fit to the
resolved point sources in early-type galaxies, but is an
empirical result which may not be appropriate to model
all unresolved sources in all early-type galaxies. We therefore
tested the sensitivity of our results
to this model, by replacing the bremsstrahlung component
with a simple power law (with $\Gamma=$1.5) or varying the temperature
of the component by $\pm$25\% %
\subsection{Distance uncertainty}
The estimated distance to the object enters into our mass determination
(Eq~\ref{eqn_hydrostatic_rho}) primarily through the impact on the
radial scale.
To assess its impact on our fitting,
we experimented by varying the distance by $\pm$20\%.
\subsection{Stellar baryon fraction} \label{sect_systematics_baryon_fraction}
In our analysis, we restricted \fbaryons\ to enable interesting
constraints on \mvir\ and c. For the stellar contribution to the
baryon fraction, we assumed that mass is divided among group members
following the B-band light, which is not formally correct since
\mstars/L ratios are very sensitive to the age of the stellar
population. To estimate how much impact this makes to our fits,
we experimented with assuming that all the stellar mass is in the central
galaxy, which should place an upper limit on the uncertainty
arising from this choice.
\section{Discussion} \label{sect_discussion}
\subsection{Hydrostatic equilibrium}
Our fit results provide strong evidence that the gas is in
hydrostatic equilibrium in these systems. Despite highly
nontrivial temperature and density profiles, we were able to recover
smooth mass profiles in remarkably good agreement with
expectation for these systems, using two complementary
techniques. If the gas is significantly out of
hydrostatic equilibrium, this would represent a remarkable
``conspiracy'' between the density and temperature profiles.
It is unsurprising that the gas is close to hydrostatic equilibrium
in these systems, since we took care to choose objects with relaxed
X-ray morphology. Based on N-body/ hydrodynamical analysis,
X-ray measurements are expected to give reliable constraints
on the DM in systems without obvious substructure
\citep{buote95a}.
Further support for hydrostatic equilibrium is provided by
the general agreement between our measured
\mstars/\lk\ ratios and those predicted by SSP models,
coupled with the agreement between the measured \mvir-c relation
and that expected.
Similarly a comparison between our results and masses determined
from stellar dynamics provides even more evidence that our
measured mass profiles are reliable.
Dynamically-determined \mgrav/\lb\ within the B-band \reff\ are
typically found to be $\sim$4--10 \citep{gerhard01a,trujillo04a}.
We found \mgrav/\lb\ within the B-band \reff\
(taken from RC3 or \citealt{faber89}) for our systems ranged
from $\sim$3 to $\sim$8, in good agreement
with this result.
For NGC\thin 4649, outside \reff\ there is excellent agreement
between our measured \mgrav/L profile and that obtained from globular
cluster kinematics, although at small radii the X-ray data
lie $\sim$30\% lower (K.\ Gebhardt et al, in preparation).
\citet{vandermarel91a} constructed stellar kinematical models
for 5 galaxies in our sample (NGC\thin 720, NGC\thin 1407 and
NGC\thin 4261, NGC\thin 4472 and NGC\thin 4649), under the assumption
of a constant M/L profile. Strictly speaking a direct comparison cannot
be made between their \mgrav/\lb\ measurements and our results since our
data indicate this assumption is incorrect. However, if we simply assume
that these M/L ratios represent those integrated out to \reff, the
X-ray inferred
masses vary from $\sim$40\% lower to $\sim$10\% higher than those
from kinematics. \citet{kronawitter00a}
report \mgrav/\lb$\sim$8$\pm1.5$ for NGC\thin 4472 within $\sim$50\arcsec,
at which radius our X-ray determined value is $\sim$50\% lower. The
discrepancies between the X-ray and dynamical masses are only modest
(the X-ray mass being on average $\sim$20\% lower), indicating that
the data must be close to hydrostatic equilibrium.
Turbulence is expected to contribute only $\sim$10\% pressure support
in clusters, which are believed to be more turbulent than galaxies,
\citep{rasia06a}.
Therefore, on a case-by-case
basis, the observed differences are most likely a manifestation of the
mass-anisotropy degeneracy \citep[\eg][]{dekel05a}.
\subsection{Mass profiles}
We obtained detailed mass profiles for 3 galaxies and 4 group-scale
systems, out to $\sim$10\reff. The data clearly show M/L profiles
which are $\sim$flat within \reff\ and rise considerably outside
this range. This confirms the presence of substantial DM in
at least some early-type galaxies and indicates that a stellar mass
component dominates within $\sim$\reff.
This is consistent with studies of
stellar kinematics and similar to the mass decomposition analysis
of \citet{brighenti97a}.
The data are well-fitted by a model comprising a stellar
mass (H90) component and an NFW DM profile.
Omitting the stellar
mass component led to systematically poorer fits, smaller
\mvir\ and vastly larger c ($\gg$20), which are inconsistent with
the predictions of \lcdm. This effect is easy to understand--- if
we add a compact stellar mass component to an (extended) NFW profile,
we increase the mass in the core which, by definition, makes the
halo more concentrated.
However, it is not entirely clear whether this
effect, pointed out by \citet{mamon05a},
can completely account for the significantly steeper
\mvir-c relation found by \citet{sato00a}.
Based on our analysis of group-scale halos \citep{gastaldello06a}
we found that the inclusion of the stellar mass component
does not have a strong effect on c in most systems with
\mvir \gtsim 2$\times 10^{13}$\msun,
provided the data are fitted to
a sufficiently large fraction of \rvir.
The data did not allow us to distinguish statistically between
the simple NFW+stars model and
scenarios in which the DM halo experiences adiabatic compression
due to star formation (however, see \S~\ref{discussion_mass_to_light}),
or the NFW profile was replaced with the alternative N04 profile.
Comparing our inferred \mvir\ and c to the predictions of \lcdm\
we find general agreement. There is some evidence, however, that
the concentrations are systematically higher than one would
expect, although the error-bars are typically large.
Such a trend is also seen in our analysis of groups
\citep{gastaldello06a}. Whilst the slope of the \mvir-c relation
therefore implied by our data is difficult to explain by varying the
cosmological parameters within reasonable limits \citep{buote06b},
we suspect that the discrepancy can be resolved by taking into
account the selection function of our galaxies.
Our
data were not selected in a statistically complete manner and, by choosing
objects with relatively relaxed X-ray morphologies we are probably
selecting objects which have not had a recent major merger.
This systematically biases us towards early-forming, hence higher
concentration halos.
In fact, it is striking that all three {\em de facto} galaxies
in our sample are relatively isolated systems (\S~\ref{discussion_groups}).
Such systems preferentially might be expected to occupy high-c
halos \citep{zentner05a},
which does appear to be the case for 2 out of 3 of the galaxies.
We will return to these issues in detail in \citet{buote06b}.
\subsection{Galaxies, Groups and Fossil Groups} \label{discussion_groups}
All three of the lowest-mass systems in our sample are very isolated
optically. NGC\thin 6482 matches the isolation criteria adopted
to identify so-called ``fossil groups'' \citep{khosroshahi04a}.
NGC\thin 4125 and NGC\thin 720 are both listed as ``groups'' in
G93, but closer inspection actually reveals they are also
very isolated. Excepting the central galaxy, only one of the putative
members of the NGC\thin 720 ``group'' listed in the G93
catalogue \citep[which omits the dwarf galaxy population
studied by][]{dressler86a},
actually lies within the projected \rvir\ (but outside 0.75$\times$\rvir)
and it is 2.4 magnitudes fainter in B than the central galaxy.
\citeauthor{dressler86a} remarked upon the optical isolation of this galaxy.
Of the two putative companion galaxies to NGC\thin 4125 given
in G93 which lie within the projected \rvir\ (but outside
0.67$\times$\rvir), both are much fainter (by 2.3 and 3.9 magnitudes,
respectively) in B than the central galaxy.
In contrast, the four remaining systems in our sample are much less
optically isolated. \citet{schindler99a} show the clear over-density of
early-type galaxies around NGC\thin 4649 and NGC\thin 4472, and almost
60 group members are associated with these systems by G93.
\citet{gould93a} identified at least 10 members of the NGC\thin 1407
group, from the dynamics of which he inferred a mass broadly consistent
with our measured \mvir\ (\S~\ref{sect_ngc1407}).
\citet{helsdon03a} report 57 galaxies associated with the NGC\thin 4261
group within $\sim$1~Mpc projected radius, which is consistent with
our measured \rvir.
Rather than an isolated galaxy \citet{khosroshahi04a} identify NGC\thin 6482
as a ``fossil group''. Fossil groups are group-sized X-ray halos centred on
essentially a
single elliptical galaxy \citep{ponman94,vikhlinin99a,jones03}. The typical
interpretation of these objects is groups in which all of the \lstar\ members
have merged. Confusingly, using almost the same selection criteria,
\citet{osullivan04b} classify the galaxy NGC\thin 4555 as an
``isolated elliptical
galaxy'' and posit a very different formation scenario.
This object appears to be more massive than NGC\thin 6482;
the authors found \mgrav $\sim 3\times 10^{12}$\msun\ within 60~kpc
which, assuming an NFW profile with c$=$15 would imply
\mvir $\sim 2\times 10^{13}$\msun.
Nonetheless, both of these systems have more in
common (both optically and in the X-ray)
with each other, and the other isolated ellipticals in our sample,
than, for example, the massive (\mvir\gtsim$10^{14}$\msun), hotter (kT$\sim$2~keV)
fossil groups considered by \citet{vikhlinin99a}.
We suspect that the distinction made between ``isolated elliptical'' and
``fossil group'' for these two systems is largely semantic,
and consider NGC\thin 6482 more properly an isolated galaxy, too.
The clear division in the galaxy content of our sample clearly lends itself
to the nomenclature ``galaxies'' for the three lowest-mass systems,
and ``groups'' for the others. Strikingly, this separation
between galaxies and groups also appears consistent with a difference in
temperature profiles (\S~\ref{discussion_temp}).
That this distinction appears commensurate with
\mvir$\sim 10^{13}$\msun\ is suggestive that this mass-scale may be a useful
yard-stick with which to compare to other systems.
The error-bars on our mass estimates are sufficiently large that
the 90\% confidence regions of several of the objects (notably
NGC\thin 720, NGC\thin 6482 and NGC\thin 1407) actually straddle
$10^{13}$\msun. However, it is clear that {\em on average}, the systems
with \mvir \ltsim $10^{13}$\msun\ are galaxies. We note that the \mvir\ adopted here
is that {\em before}
any tidal truncation which is almost certainly occurring as NGC\thin 4472
and NGC\thin 4649 merge with Virgo (their untruncated \rvir\ would stretch
much of the distance to M\thin 87).
\mvir\ does not exactly correlate with formation epoch,
so that lower-mass halos may still be in the process of forming
(hence contain multiple galaxies of similar magnitude), and more
massive halos may contain single, dominant ellipticals (fossil groups).
Nonetheless, classifying halos primarily on the basis of
\mvir\ provides a straightforward way to locate them in the formation hierarchy.
Traditionally, galaxy-like and group-like systems are distinguished
on the basis of local over-densities of galaxies. However,
placing optically-identified groups into a cosmological context
requires a firm understanding not only of the formation of DM halos
but also how galaxies populate them, which is much less
well-understood \citep[\eg][]{kravtsov04a}.
This problem is compounded by the difficulties faced by optical group-finding
algorithms in identifying very poor groups \citep[\eg][]{gerke05a}. Not only
can a significant fraction of putative groups be chance superpositions of
galaxies, particularly along filaments, but adjacent groups can be merged,
such as happened for NGC\thin 4649 and NGC\thin 4472 in G93. If there are
only a few identified members, small-number statistics and the treatment of
interlopers can affect their interpretation \citep[\eg][]{gould93a}.
To complicate matters further, some authors refer to
{\em any} over-density of galaxies as a group, even a Milky Way-sized
galaxy and its dwarf satellites \citep[\eg][]{tully05a}.
\subsection{Stellar Mass-to-Light Ratios} \label{discussion_mass_to_light}
Comparing our measured stellar M/L ratios
to the predictions of simple stellar population (SSP)
models, we found reasonable agreement provided one assumes
a \citet{kroupa01a} IMF. There is modest disagreement,
even when the less-cuspy N04 DM model was adopted.
Considering the uncertainties in the SSP modelling
(discussed below), however, we believe this discrepancy
is not significant.
If we allowed the DM
profile to be modified by adiabatic compression, we obtained
substantially smaller \mstars/\lk\ values from our data,
(since it increases the cuspiness of the halo)
which are more discrepant with the SSP models. This result
casts doubt on AC being as significant an effect as currently modelled.
However, the data
alone did not allow us statistically to distinguish between
the NFW+stars and AC NFW+stars models.
Nonetheless, this result is joining a growing body of literature
which similarly calls into question whether AC operates as
predicted \citep{zappacosta06a,kassin06a,sand04a}.
There are a number of major uncertainties in the
computation of the stellar mass-to-light ratios from the SSP models.
Specifically, the results are very dependent upon the
assumed IMF, which is not confidently known in early-type
galaxies.
Furthermore there is some evidence that early-type galaxies
frequently contain multiple stellar populations of different
ages, including a significant young population
\citep[\eg][]{rembold05a,nolan06a}. Depending on the mass
fraction of the young component, this may substantially
reduce \mstars/\lk\ in the galaxy, hence possibly reconciling
the data and the AC NFW+stars model.
A small amount of star formation may also give rise to a
population of stars which can dominate the light in the galaxy
core, giving rise to significantly lower synthetic \mstars/\lk\
than measured. This may be the case in NGC\thin 720 (see
\S~\ref{sect_mass_to_light}).
More problematically, there
are known to be significant abundance, or possible age,
gradients in the stellar populations of early-type galaxies
\citep[\eg][]{trager00a,kobayashi99a,rembold05a}, which would translate
into stellar \mstars/\lk\ gradients. Our simple modelling
did not allow us to account for such an effect {\em per se}.
Although we suspect that such gradients
will primarily lead to a \mstars/\lk\ value which reflects
an average for the galaxy, \mstars/\lk\ does depend to
some extent upon the shape of the assumed stellar potential.
Properly taking account of this effect is beyond the scope
of this present work, but may bring the synthetic M/L ratios
and our results into better agreement. Clearly this is only
one of a number of other systematic effects which may also
reconcile the slight discrepancy
(Table~\ref{table_syserr}).
\subsection{Baryon fractions}
An interesting result from our analysis is that these systems,
despite having masses \gtsim 5$\times 10^{12}$\msun,
do not appear in general to be baryonically closed.
To some extent this trend was enforced by applying
Eq~\ref{eqn_baryons} to constrain the data. However,
the excellent fits we obtained by this method, in
conjunction with
the good agreement between the measured \mvir-c relation
and the predictions of \lcdm\ and, crucially,
our measurements at the group scale (which do not
employ this restriction: \citealt{gastaldello06a}),
indicate that the inferred \fbaryons ($\sim$0.04--0.09; Table~\ref{table_results})
are accurate.
Furthermore, if we relaxed this constraint
and instead restricted \fbaryons\ to a finite range, we
also found that the data tended to favour modest values of \fbaryons. In particular,
for any given system, the measured \mvir\ and \fbaryons\ were
strongly anti-correlated, so that our upper \mvir\ constraint
is in part imposed by the {\em lower} limit we place on \fbaryons.
Given the shapes of the \mvir-c contours (Fig~\ref{fig_confidence}),
it is clear that good agreement with the \mvir-c relation predicted
from simulations tends, therefore, to require rather modest values of \fbaryons.
This would suggest that strong feedback plays an important
role in the evolution of these objects.
\subsection{Temperature profiles} \label{discussion_temp}
By inspection of the temperature profiles (Fig~\ref{fig_temp}) it
is immediately clear that, for all of the galaxy-scale systems
in our sample the temperature profiles have negative gradients.
In contrast the group-scale objects have positive temperature gradients,
similar to observations of other
X-ray bright groups and clusters
\citep{gastaldello06a,vikhlinin05b,piffaretti05a}. This radical difference
in the temperature
profiles seems consistent with our division of galaxies and groups
at \mvir $\sim 10^{13}$\msun.
The origin of this distinct demarcation between objects around
$10^{13}$\msun\ is unclear, however.
Negative temperature gradients are expected for
isolated galaxies containing relatively cool gas, such as that
arising from stellar mass-loss.
In the deep stellar potential well,
compressive heating of the gradually inflowing gas can dominate
over radiative cooling to produce a negative temperature slope.
In contrast, if hotter ($\sim$1--2~keV) baryons are allowed
to flow in, radiative cooling dominates to produce a positive
temperature gradient \citep{mathews03a}.
It is by no means clear, however, why the hot baryons appear
to be present only in the systems with \mvir\gtsim $10^{13}$\msun.
One possibility is the local environment;
all of the galaxy-scale objects are rather isolated, whereas
the groups NGC\thin 4472 and NGC\thin 4649, in particular,
are found in a relatively dense cluster environment, which could
provide a reservoir of hot baryons. However, such an explanation
cannot easily account for the positive temperature gradient in
NGC\thin 1407, which is comparatively isolated, or the
isolated system NGC\thin 4555, which appears
only slightly more massive than our galaxies.
It is possible that selection effects may have played some role in
the bimodal temperature profile behaviour, since both
NGC\thin 4125 and NGC\thin 6482 are
classified in \ned\ as LINERS, and NGC\thin 720 has a dominant
young stellar population (Appendix~\ref{sect_stars}).
However, none of these systems
show strong X-ray morphology disturbances in the core,
which might indicate a substantial energy input from
star-formation or AGN activity.
In any case, the cooling time in the core of NGC\thin 720 is only
$\sim$200~Myr, substantially less than the
implied time since the last major burst of star formation,
and so the negative temperature gradient cannot simply be
related to energy injection during a starburst.
Furthermore, at least two of the group-scale
systems also harbour AGN and do not show obvious negative temperature
gradients in the core.
Another example of an object we believe to be a galaxy
(rather than a group) which exhibits a negative temperature
gradient is the S0 NGC\thin 1332 \citep{humphrey04b}.
A possible counter-example to this trend might by the
``isolated elliptical galaxy'' NGC\thin 4555, which exhibits
a temperature profile akin to the groups in our sample
\citep{osullivan04b}. However, as we discuss in
\S~\ref{discussion_groups}, this probably has comparable \mvir\
to the groups.
Another intriguing feature of two of the group scale
objects is a central temperature peak, similar to a feature
we found in the cluster A\thin 644 \citep{buote05a}.
In that system, we found a significant offset between the
X-ray centroid and the emission peak in an otherwise fairly
relaxed object. We suggested that both of these features
may be related to the cD ``sloshing'' in the potential well
of the cluster, which is relaxing following disturbance by,
for example, a merger.
We do not find obvious evidence of a similar offset in either
NGC\thin 1407 or NGC\thin 4649. However, these groups may
be in a comparably more relaxed (evolved) state than A\thin 644.
Alternatively, the central peaks may be related to past AGN activity
heating the gas in the core of the galaxies, from which the system
has had time to relax dynamically but not cool completely.
\subsection{Is NGC\thin 1407 a ``dark group''?} \label{sect_ngc1407}
Based on the group member dispersion velocity \citet{gould93a}
suggested that NGC\thin 1407 may lie in a massive (\gtsim a few
$\times 10^{13}$\msun) DM halo. Although such a conclusion was
strongly dependent on the association of the galaxy NGC\thin 1400,
which exhibits a large peculiar velocity, with the group, we can
now confirm the presence of a substantial DM halo around this system.
Both the temperature profile and our best-fit mass are similar to the
bright X-ray group NGC\thin 5044 \citep{buote06a}, and yet
it is almost 2 orders of magnitude fainter in \lx.
NGC\thin 5044 appears to be close to baryonic closure
\citep{mathews05a}, and so has likely retained most of
its large gaseous halo. On the other
hand NGC\thin 1407 is not baryonically closed (we estimate
\fbaryons$\simeq$0.06) and so the loss of much of its hot gas
envelope easily explains its lower \lx/\lb. Since the masses
of the two systems are not considerably different, this
points to substantial variation in the evolutionary
history of these two groups. In particular, feedback may
have operated more efficiently in evacuating the gas from
NGC\thin 1407.
\citeauthor{gould93a}'s preferred mass estimate
($\sim10^{14}$\msun) would imply a remarkably high M/L ratio
for the system (\mvir/\lb$\sim$900\msun/\lsun), making NGC\thin 1407
a bona fide ``dark group''. The existence of such an object
would provide a valuable insight into the process of star
formation in DM halos, as it would imply star formation was
somehow inhibited in that system. This mass estimate is, however,
considerably larger than our preferred value
$\sim 1.5 \times 10^{13}$\msun, which implies a more modest
M/L ratio (\mvir/\lb$\sim$140\msun/\lsun).
To some extent, though, our constraint
on \fbaryons, which was necessary to obtain interesting \mvir\
constraints, has probably enforced this behaviour.
Such a restriction
may not be valid in a system with an unusual star-formation history
and so we experimented with freeing \fbaryons.
To enable \mvir\ to be constrained, we
restricted c to lie on the
best-fit \mvir-c relation found by \citet{bullock01a}.
The best-fitting mass,
\mvir$=(9.7^{+17.8}_{-6.2})\times 10^{13}$\msun, was in good agreement
with \citet{gould93a}'s values, but implies a baryon fraction
only of $\sim$0.003. Since this fit was statistically
indistinguishable from the preferred model, we cannot
determine which mass estimate is more likely.
\section{Summary}
Using \chandra\ we have obtained detailed mass profiles
centred on 7 elliptical galaxies,
of which 3 were found to have {\em de facto} galaxy-scale halos,
with \mvir $< 10^{13}$\msun, and 4 had group-scale
($10^{13}$\msun$<$\mvir$< 10^{14}$\msun) halos.
These represent the best available data for nearby objects
with comparable \lx. In summary:
\begin{enumerate}
\item The M/L ratio profiles were $\sim$flat within \reff\ and rose sharply
outside this region, indicating substantial DM in all 7 systems.
\item The data were well-described by a two component model, comprising
an NFW potential for the DM and a H90 stellar mass model.
We were not able statistically to distinguish
between this scenario and one in which the DM profile was modified by ``adiabatic
compression'' due to baryonic infall. Similarly, we could not
distinguish between the NFW and the revised N04 DM halo profiles.
\item The distribution of the galaxies in the \mvir-c plane was
in broad agreement with the predictions of \lcdm, although with a
slight trend toward more concentrated halos, in good agreement
with our modelling of X-ray bright groups and poor clusters
\citep{gastaldello06a}.
This probably represents a galaxy
selection bias to earlier-forming systems,
and we will discuss how we might account for it
in \citet{buote06b}. Allowing AC to modify the shape of the DM halo
did not appreciably affect the \mvir-c relation.
\item Omitting the stellar mass component resulted in
systematically poorer fits, smaller \mvir\ and unphysically large c,
confirming the conclusions of \citet{mamon05a}. This may explain
very large values of c found by some previous X-ray observers
\citep[\eg][]{sato00a,khosroshahi04a}.
\item For the NFW+stars model, \mstars/\lk\ was found to be
in approximate agreement with the predictions of simple stellar
population synthesis models, assuming a \citet{kroupa01a} IMF. The AC NFW+stars
models have significantly lower \mstars/\lk\ which seems to cast
doubt on the AC scenario, although this conclusion is sensitive to
the considerable uncertainties in the theoretical modelling.
\item Despite having \mvir \gtsim 5$\times 10^{12}$\msun, typically
\fbaryons $\sim$0.04--0.09 for each galaxy,
implying that feedback has played an important role in the evolution
of these systems.
\item The temperature profiles of the galaxy-scale systems all
exhibited negative radial gradients, whereas the group-scale objects
exhibited positive gradients, similar to the ``Universal'' temperature
profiles being found in other X-ray bright groups and clusters.
This implies a strict line of demarcation between systems at \mvir $\sim 10^{13}$\msun.
\item In two of the groups, we found central temperature peaks, similar
to that found in the cluster A\thin 644 \citep{buote05a}, but
no obvious central disturbances in X-ray morphology. This may
relate to past AGN activity, following which the heated gas in the
core of the galaxy has relaxed but not
cooled.
\item We confirm the suggestion of \citet{gould93a} that the
elliptical galaxy NGC\thin 1407 lies at the centre of a massive
DM halo, possibly making it a ``dark
group'' with an unusually large M/L.
Our best-fitting \mvir\ is considerably lower than that
of \citeauthor{gould93a}, implying M/L more consistent
with normal groups.
Nonetheless, if we relax
the assumptions of our modelling very large masses
(\mvir$\sim 10^{14}$\msun) are allowed.
\end{enumerate}
\begin{acknowledgements}
We would like to thank Oleg Gnedin for making available his adiabatic compression code. We would also like to thank Karl Gebhardt for communicating with us
results from his paper in preparation.
We thank Louisa Nolan for interesting
discussions on the stellar populations of galaxies.
This research has made use of data obtained from the High Energy Astrophysics
Science Archive Research Center (HEASARC), provided by NASA's Goddard Space
Flight Center.
This research has also made use of the NASA/IPAC Extragalactic Database (\ned)
which is operated by the Jet Propulsion Laboratory, California Institute of
Technology, under contract with NASA.
In addition, this work also made use of the HyperLEDA database
(\href{http://leda.univ-lyon1.fr}{http://leda.univ-lyon1.fr}).
Support for this work was provided by NASA under grant
NNG04GE76G issued through the Office of Space Sciences Long-Term
Space Astrophysics Program.
\end{acknowledgements}
\appendix
\section{Stellar population parameters} \label{sect_stars}
\begin{deluxetable*}{lllllll}
\tablecaption{Stellar population parameters\label{table_ssp}}
\tabletypesize{\footnotesize}
\tablehead{
\colhead{Galaxy} & \colhead{indices} & \colhead{ref.} & \colhead{${\rm [\alpha/Fe]}$} & \colhead{age} & \colhead{${\rm [Z/H]_0}$}
& \colhead{$\rm <[Z/H]>$} \\
\colhead{} & \colhead{} & \colhead{} & \colhead{} & \colhead{(Gyr)} & \colhead{} & \colhead{}}
\startdata
NGC\thin720$^\dagger$ & H$\beta$, Mgb, Fe5270, Fe5335 & 2 & 0.37$\pm0.05$ & 2.9$^{+1.3}_{-0.3}$ & 0.65$\pm$0.13& 0.48$\pm$0.18 \\
NGC\thin 1407$^\dagger$ & H$\beta$, Mgb, Fe5270, Fe5335 & 1 & 0.33$\pm 0.02$ & 12$\pm2$ & 0.35$\pm0.06$ & 0.08$\pm0.06$\ddag \\
NGC\thin 4125 & H$\beta$, Mgb, Fe5270, Fe5335 & 3 & 0.33$\pm 0.16$ & 13$\pm8$ & 0.16$\pm$0.25 & -0.11$\pm$0.25\ddag \\
NGC\thin 4261 & H$\beta$, Mgb, Fe5270, Fe5335 & 2 & 0.25$\pm 0.02$ & 15$\pm1$ & 0.30$\pm0.03$ & -0.03$\pm$0.10 \\
NGC\thin 4472$^\dagger$ & H$\beta$, Mgb, Fe5270, Fe5335 & 2 & 0.25$\pm0.03$ & 9$\pm2$ & 0.36$\pm0.05$ & 0.17$\pm$0.12 \\
NGC\thin 4649 & H$\beta$, Mgb, Fe5335 & 2 & 0.25$\pm0.02$ & 13$\pm2$ & 0.41$\pm0.04$ & 0.23$\pm$0.15 \\
NGC\thin 6482 & Mgb, Fe5270, Fe5335 & 3 & 0.30$\pm0.15$ & 12 & 0.28$\pm0.15$ & 0.06$\pm0.15$\ddag \\
\enddata
\tablecomments{Stellar population parameters determined from Lick
index fitting. The indices used in fitting are listed (indices),
as is the reference (ref) whence they were taken.
Those mean stellar abundances (${\rm <[Z/H]>}$) marked
\ddag\ were estimated from the central abundance (${\rm [Z/H]_0}$)
adopting the mean abundance gradient ${\rm [Z/H]_0}-{\rm <[Z/H]>}=0.27$\citep[see][]{humphrey05a}.
Where no error-bar
is given, the parameter was frozen.
Table references: 1--- \citet{beuing02a}, 2--- \citet{trager00a}, 3--- \citet{trager98a}
Results for galaxies marked $^\dagger$ were taken from \citet{humphrey05a}
}
\end{deluxetable*}
The mass-to-light ratio of a stellar population is dependent upon both the
age and the metal abundance ([Z/H]) of the stars. To estimate these
quantities we searched the literature
to obtain Lick/IDS indices for each galaxy, which we fitted with
the simple stellar population (SSP) models of \citet{thomas03a}, using the
technique outlined in \citet{humphrey05a}. Briefly, we constructed
a model by linearly interpolating the SSP models as a function of
stellar age, metallicity and $\alpha$-element to Fe ratio, which was
then fitted {\em via} a $\chi^2$ minimization technique to
those indices shown in Table~\ref{table_ssp}. \citet{trager00a} provided
indices measured in two apertures, which enabled us to take account
of any abundance gradients, as outlined in \citet{humphrey05a}.
Where only a central Lick index was available, we estimated the total
emission-weighted abundance by correcting the central metallicity by
-0.27 dex. We did not attempt to take account of possible age gradients.
The results, including the
Lick indices adopted and the reference whence the indices were obtained,
are shown in Table~\ref{table_ssp}.
This method implicitly assumes that all the
stars were created in a single burst of star formation, which may
be over-simplistic if there are, in fact, multiple bursts of star
formation in early-type galaxies \citep[\eg][]{rembold05a,nolan06a}.
We note that we were not able to obtain acceptable solutions for
NGC\thin 6482 if we used the H$\beta$ index, which is the most sensitive
age indicator. This galaxy is classified in \ned\ as a LINER
and is rather blue for an old stellar population
(${\rm M_B-M_K=3.4\pm0.2}$, whereas a 12~Gyr, solar abundance population is
expected to have ${\rm M_B-M_K\simeq3.9}$: \citealt{maraston98a}).
Both of these factors might suggest the presence of a significant young
population of stars \citep[although see][]{cidfernandes04a}.
It is beyond the scope of this present work, however,
to attempt to take account of this effect.
\bibliographystyle{apj_hyper}
\bibliography{paper_bibliography.bib}
|
Title:
PBH and DM from cosmic necklaces |
Abstract: Cosmic strings in the brane Universe have recently gained a great interest. I
think the most interesting story is that future cosmological observations
distinguish them from the conventional cosmic strings. If the strings are the
higher-dimensional objects that can (at least initially) move along the
compactified space, and finally settle down to (quasi-)degenerated vacua in the
compactified space, then kinks should appear on the strings, which interpolate
between the degenerated vacua. These kinks look like ``beads'' on the strings,
which means that the strings turn into necklaces. Moreover, in the case that
the compact manifold is not simply connected, the string loop that winds around
a non-trivial circle is stable due to the topological reason. Since the
existence of degenerated vacua and a non-trivial circle is the common feature
of the brane models, it is important to study cosmological constraints on the
cosmic necklaces and their stable winding states in the brane Universe.
| https://export.arxiv.org/pdf/hep-ph/0601014 |
\title{%
PBH and DM from cosmic necklaces
}
\author{%
Tomohiro Matsuda\footnote{E-mail:matsuda@sit.ac.jp}
}
\address{%
$^1$Theoretical Physics Group,
Saitama Institute of Technology,
Saitama 369-0293, Japan
}
\abstract{Cosmic strings in the brane Universe have
recently gained a great interest.
I think the most interesting story is that future cosmological
observations distinguish them from the conventional cosmic strings.
If the strings are the higher-dimensional objects
that can (at least initially) move along the compactified space, and
finally settle down to (quasi-)degenerated vacua in the compactified
space, then kinks should appear on the strings, which interpolate
between the degenerated vacua.
These kinks look like ``beads'' on the strings, which means that the
strings turn into necklaces.
Moreover, in the case that the compact manifold is not simply connected,
the string
loop that winds around a non-trivial circle is stable due to the
topological reason.
Since the existence of degenerated vacua and a non-trivial
circle is the common feature of the brane models, it is important to
study
cosmological constraints on the cosmic necklaces and their stable winding
states in the brane Universe.}
\section{Introduction}
In this talk we will explain the cosmological consequences of the
production of Dark Matter(DM) and Primordial Black Hole(PBH) from the
loops of the cosmic necklaces.
To begin with, I think it is fair to explain why
necklaces\cite{vilenkin_book} are produced
in brane models, since in many papers it is discussed that ``only strings
are produced in the brane Universe''\cite{tutorial}.
Of course I think this claim is not wrong, however somewhat misleading
for non-specialists.
To explain what is misleading in the ``standard scenario'', we have a
figure in Fig.\ref{fig:fig0}.
In general, the distance between branes may appear in the
four-dimensional effective action as a Higgs field of the effective gauge
dynamics.
At least in this case, it is natural to consider the cosmological
defects coming from the spatial variation of the Higgs field,
which corresponds to the ``deformation'' of the
branes\cite{matsuda_deformation}.
Is the spatial variation of the Higgs field unnatural in the brane
Universe?
The answer is, of course, no.
One should therefore consider at least two different kinds of defects in
brane models:
One is induced by the brane creation that is due to the spatial
variation of the tachyon condensation, while the other is induced by the
brane
deformation that is due to the spatial variation of the brane distance.
Along the line of the above arguments, it is possible to construct
Q-ball's counterpart in brane models\cite{matsuda_Q-balls}, which can be
distinguished from the conventional Q-balls by their decay process.
We therefore have an expectation that strings can be distinguished from
the conventional ones, if one properly considers their characteristic
features.
Now let us discuss about the validity of the conventional Kibble
mechanism.
Of course the Kibble mechanism is an excellent idea that explains the
nature of the cosmological defect formation.
However, if there is oscillation of the brane distance that may be
induced by the brane inflation or by a later phase transition
that changes the brane distance, the four-dimensional counterpart of the
brane distance (i.e. the Higgs field) oscillates in the effective
action.
In the four-dimensional counterpart, defect production induced by such
oscillation is already discussed by many
authors, including the production of the sphaleron domain walls which
otherwise cannot be produced in the Universe\cite{matsuda_deformation}.
The defect production induced by such oscillation may or may not be
explained by the Kibble mechanism, however it should be fair to
distinguish it from the ``conventional'' Kibble mechanism.
Let us summarize the above discussion about the defect production in
the brane Universe.
Actually, it is possible to produce all kinds of defects in the brane
Universe, however it is impossible to produce defects other than the
strings {\bf simply as the result of the brane creation that is induced by
the conventional Kibble mechanism}.
One should therefore be careful about the assumption that is made in the
manuscript, which may or may not be explicit.
The necklaces are produced as the hybrid of the brane creation and the
brane deformation.
It should be noted that the stable loops of the necklaces that we will
discuss in this talk
may appear in the four-dimensional gauge dynamics, irrespective of the
existence of the branes\cite{050906x}.
The stabilization of the necklace loops is first discussed in
Ref.\cite{matsuda_necklace} for brane models and in Ref.\cite{050906x}
for necklaces embedded in four-dimensional gauge dynamics.
In order to produce necklaces in the brane Universe, the motion in the
compactified direction is important.
I know that in the ``standard scenario'' it is sometimes discussed that
the position of the strings
are fixed by the potential that is induced by the supersymmetry
breaking, and the position is a homogeneous parameter of the Universe
because all the decay products (Typically, they are F, D, and $(p,q)$
strings) lie (at least initially) along the same plane of the original
hypersurface on which the tachyon condensation took place.
However, in this case one may hit upon the idea that the potential
for a string cannot be identical to all the other kinds of the strings.
One may therefore obtain many kinds of strings that may move
independently along different
hypersurfaces, with exponentially small intersection ratios.
Moreover, I think it is not reasonable(but may be possible) to
assume that the string motion is utterly restricted by the potential
even in the most energetic epoch just after inflation.
Please remember that in general the moving (inflating) brane carries
kinetic energy, and the
brane annihilation should be an energetic process, although one may
admit that there could be exceptional scenarios.
I therefore think that the decay products should have kinetic energy,
which is enough to climb up the potential hill at least just after
brane inflation.
\section{PBH and DM from necklaces}
The scenario of the PBH formation from strings is initiated by
Hawking\cite{Hawking}, who utilizes the huge kinetic energy of the
shrinking loops.
However, the probability of finding loops that can
shrink into their small Schwarzschild radius is very rare due to their
random shape and motion, which weakens
the obtained bound for the string tension up to $G\mu < 10^{-6}$.
In our previous paper hep-ph/0509061\cite{0509061}, we have just extended
Hawking's idea to the networks that include monopoles attached
to the strings.
It should be emphasized that both in the above scenarios, the kinetic
energy of the shrinking object plays crucial role.
Now let us discuss about our new idea for producing PBHs.
The most obvious discrepancy is that in our new scenario we discard the
benefit of the kinetic energy.
We consider stable relics that are produced from necklace loops, whose
mass is large enough to turn into black holes, even after they have
dissipated their kinetic energy during the loop oscillation.
The stability of the loops is due to their windings around the
compactified space.
Of course the production of PBHs is delayed compared
to the Hawking's scenario, however
the obtained bound is much stronger than the original scenario due to
the high ($\sim 1$) production ratio.
This new mechanism of PBH production is first advocated in
hep-ph/0509062\cite{050906x}.
Let us explain how one can count the winding number of a necklace
loop.
Please see Fig.\ref{fig:count}.
We introduce $\chi(t)$, which is the step length between each random
walk that corresponds to the right or the left movers in the
compactified direction.
Since the left and the right movers can annihilate on the necklaces,
the actual distance between ``beads'' becomes much larger than
$\chi(t)$.
We therefore introduce another parameter $d(t)$, which is the typical
length between the remaining ``beads''.
Although the annihilation could be efficient, the simple statistical
argument shows that the typical number of the beads that
remain after annihilation is about $n^{1/2}$, if the initial number of
the random walk is given by $n$.
If the strings are in the scaling epoch, the typical length of the loops
is $l(t)\sim \alpha t$ when they are disconnected from the string
networks.
Then one can obtain the typical mass of the stable relics, $M_{coils}\sim
n(t)^{1/2}m \sim [l(t)/\chi(t)]^{1/2}m$, where $m$ is the typical mass
of the beads.
Here we should note that disregarding the annihilation, $\chi(t)\propto
t^{-1}$ is already obtained in Ref.\cite{vilenkin-necklace}, which means
that $d(t)$ should evolve as $d(t)\sim t^{-1}$ at least during the short
periods between each annihilation.
Of course $d(t)$ is discontinuous at each annihilation, however the
underlying parameter $\chi(t)$ is continuous and depends on time as
$\chi(t) \sim t^{-1}$.
Using the above ideas, we can calculate the typical mass of the stable
relics that are produced from necklace loops.
The calculation of the PBH density is straightforward.
We have obtained the result
\begin{equation}
G\mu < 10^{-21} \times
\left[\frac{p}{10^{-2}}\right]^{4/5}
\left[\frac{\gamma}{10^{-2}}\right]^{1/5}
\left[\frac{t_n}{M_p/\mu}\right]^{3/5}
\left[\frac{d(t_n)}{M_p/\mu}\right]^{3/5}
\left[\frac{m}{10^{16} GeV}\right]^{-6/5},
\end{equation}
where we have assumed $\alpha \sim \gamma G\mu$, and $t_n$ is the time
when necklaces are produced.
$p$ is the reconnection ratio that should be $\sim 1$ for the
conventional cosmic strings, but $p \ll 1$ is possible in our case.
The string loops are produced at any time, and
the typical mass of the stable relics depends on the time when they are
produced, because the typical length scale of
the string network increases with time both in the friction-dominated
and in the scaling epoch.
Therefore, the relics that are produced in an earlier epoch may be too
light to turn into black holes.
The ``light'' relics are the ``monopoles'' if the cosmic strings are
D-branes.
However, the ``magnetic charge'' of the ``monopoles'' may or may not be
identical to the conventional magnetic charge of the electromagnetism.
Therefore, they are the candidate of DM, and possibly the origin of
the troublesome monopole problem only if they carry the conventional
magnetic charge.
In our paper hep-ph/0509064\cite{050906x}, we have examined if the DM
relics can put significant bound on the tension of the cosmic strings.
We have obtained the result for $m \sim M_{GUT} \sim 10^{16}GeV$,
\begin{equation}
\label{finalresult}
G\mu< 10^{-23}\times \left[\frac{p}{10^{-2}}\right]^{9/10}
\left[\frac{1}{\beta_s}\right]^{9/10}
\left[\frac{10^{-3}}{r}\right]^{9/10}
\end{equation}
where $r$ is the mass ratio between the string part and the beads on the
necklaces, which becomes a constant in the scaling epoch\cite{050906x}.
The difficulty in lowering the typical energy scale is discussed in
Ref.\cite{matsuda_baryo} for baryogenesis and
Ref.\cite{matsuda_CMB} for the mechanism to generate density
perturbations.
|
Title:
Stochastic Gravitational Wave Production After Inflation |
Abstract: In many models of inflation, the period of accelerated expansion ends with
preheating, a highly non-thermal phase of evolution during which the inflaton
pumps energy into a specific set of momentum modes of field(s) to which it is
coupled. This necessarily induces large, transient density inhomogeneities
which can source a significant spectrum of gravitational waves. In this paper,
we consider the generic properties of gravitational waves produced during
preheating, perform detailed calculations of the spectrum for several specific
inflationary models, and identify problems that require further study. In
particular, we argue that if these gravitational waves exist they will
necessarily fall within the frequency range that is feasible for direct
detection experiments -- from laboratory through to solar system scales. We
extract the gravitational wave spectrum from numerical simulations of
preheating after $\lambda \phi^4$ and $m_{\phi}^2 \phi^2$ inflation, and find
that they lead to a gravitational wave amplitude of around $\Omega_{gw}h^2\sim
10^{-10}$. This is considerably higher than the amplitude of the primordial
gravitational waves produced during inflation. However, the typical wavelength
of these gravitational waves is considerably shorter than LIGO scales, although
in extreme cases they may be visible at scales accessible to the proposed BBO
mission. We survey possible experimental approaches to detecting any
gravitational wave background generated during preheating.
| https://export.arxiv.org/pdf/astro-ph/0601617 |
\title{Stochastic Gravitational Wave Production After Inflation}
\author{Richard Easther} \author{Eugene A. Lim}
\affiliation{Department of Physics, Yale
University, New Haven CT 06520, USA \\ Email: {\tt
richard.easther@yale.edu} {\tt eugene.lim@yale.edu}}
\section{Introduction}
In recent years, the Cosmic Microwave Background (CMB) has been our primary window into the primordial universe. Scale invariant density perturbations, sourced by quantum fluctuations during inflation and processed by the photon-baryon plasma, are by far the most satisfying explanation for the observed temperature anisotropies in the CMB. Inflation also predicts the existence of a scale invariant spectrum of primordial gravitational waves, sourced by the same quantum fluctuations that underlie the scalar perturbations. Gravitational waves are only weakly coupled to matter fields, and move freely through the universe from the moment they are produced. They are thus the deepest probe of the early universe of which we currently know \cite{Starobinsky:1979ty,Rubakov:1982df,Fabbri:1983us,Abbott:1984fp,Allen:1987bk,Turner:1993vb,Boyle:2005se}.
In the short term, the best hope for observing primordial gravitational waves is via their contribution to the B-mode of the CMB polarization. However, foreground weak lensing puts a fundamental limit on a B-mode signal sourced by gravitational waves \cite{Knox:2002pe,Hirata:2003ka,Seljak:2003pn}. Consequently, attention has recently focussed upon direct detection experiments \cite{Boyle:2005se,Ungarelli:2005qb,Crowder:2005nr,Smith:2005mm} which, while enormously challenging, may ultimately be more sensitive to a stochastic gravitational wave background than the CMB B-mode. These experiments are sensitive to frequencies far higher than those that contribute to the CMB, since the physical sizes of detectors necessarily range from laboratory scales through to solar system scales. However, as the inflationary spectrum is almost scale-free, the amplitude at short scales is not dramatically different from that seen at CMB scales, at least for simple models of inflation.
The primordial spectrum is not alone: gravitational waves are produced whenever there are large, time-dependent inhomogeneities in the matter distribution. These more pedestrian gravitational waves are generated ``classically'', much like radiation produced by the oscillation of an electron, whereas the inflationary tensor perturbations are sourced by quantum fluctuations in the spacetime background. During the evolution of the universe, there are several hypothetical mechanisms which would generate large, local inhomogeneities. These include first order phase transitions \cite{Kamionkowski:1993fg}, brane-world models \cite{Sahni:2001qp,Sami:2004xk}, networks of cosmic strings \cite{Vilenkin:1981bx}, or inhomogenous neutrino diffusion \cite{Dolgov:2001nv}. Here, we investigate a further possibility: gravitational waves produced during preheating
\cite{Felder:2000hq,Garcia-Bellido:1997wm,Greene:1997ge,Greene:1997fu,Traschen:1990sw,Kofman:1994rk,Greene:1998nh,Giudice:1999fb,Khlebnikov:1997di,Garcia-Bellido:1998gm,Kofman:1997yn,Bassett:1998wg,Easther:1999ws,Parry:1998pn,Finelli:1998bu,Liddle:1999hq,Bassett:2005xm,Suyama:2004mz,Tsujikawa:2002nf,Finelli:2001db,Tilley:2000jh,Henriques:2003ga}, a period of non-thermal evolution following the end of inflation.
While the detailed dynamics of resonance and preheating is a complicated, nonlinear problem, the basic picture is very simple. For a given combination of fields, parametric resonance occurs when some subset of Fourier modes have exponentially growing solutions, driven by the oscillating inflaton field. The resonant modes are quickly pumped up to a large amplitude $\Phi^2 \approx \langle \chi^2 \rangle$. Resonance ends when nonlinearities render further growth kinematically expensive, leaving the universe far from thermal equilibrium. Thermalization occurs as the excited modes dissipate their energy over the entire spectrum via self-interaction. The rapid rise in the mode amplitude can be associated with an exponentially growing occupation number (at least for bosonic species). If one transforms into position-space, the highly pumped modes correspond to large, time dependent inhomogeneities, ensuring the matter distribution has a non-trivial quadrupole moment, sourcing the production of gravitational radiation.
This topic was first addressed by Khlebnikov and Tkachev \cite{Khlebnikov:1997di}, and has not been widely discussed within the experimental community. We believe that the time is ripe for revisiting this question. Since \cite{Khlebnikov:1997di} was written, technological approaches to gravitational wave detection have advanced considerably. Moreover, there have been significant advances in the theoretical understanding of preheating in more complicated inflationary models. Finally, numerical simulations benefit from the gains in computational power over this interval. It will turn out that for ``typical'' cosmological parameters, gravitational radiation sourced by preheating has a peak frequency in the MHz band. Coincidentally, a new generation of detectors has been proposed which is tuned to gravitational waves in this range \cite{Ballantini:2005am}, although their strain sensitivity would need improve over current values by around $10^6$ in order to detect the spectra we compute here. This is clearly ambitious, but probably not unreasonable when compared to the 25 year interval anticipated before BBO [Big Bang Observer] will be in a position to detect the inflationary gravitational wave spectrum \cite{Phinney}.
Since preheating occurs in many (although not all) inflationary models, any gravitational wave signal associated with preheating would provide a new and currently unexplored window into inflationary physics. It is worth emphasizing that the physical principles that underlie the calculations in this paper are well established and do not rely on exotic physics -- in particular the mechanism that governs the generation of the gravitational waves is simply the usual quadrupole related emission.\footnote{The possibility that preheating dynamics are directly affected by ``back-reaction'' from metric perturbations has also been discussed \cite{Finelli:1998bu,Parry:1998pn,Easther:1999ws,Bassett:1998wg,Finelli:2001db,Suyama:2004mz} but we do not address this effect in this work.} Finally, this signal carries information about the epoch at the end of inflation, opening a new window into the early universe. With the stakes this high any possible observational opportunity must be carefully explored.
Unlike the primordial spectrum, gravitational waves induced by preheating (or any subsequent phase transition) will not be scale-free; indeed their spectrum is indicative of the complicated processes that generate them. This is both a boon and a possible pitfall. A scale dependent spectrum necessarily contains more information than a scale-free spectrum, so the detection and subsequent mapping of a gravitational wave background induced by preheating would yield a rich trove of information. However, gravitational wave detectors are subject to physical limitations, since they all ultimately measure deformations in (an array of) physical objects induced by passing gravitational waves. Thus, even in principle, it is hard to imagine the direct detection of gravitational waves below atomic scales, or beyond solar system scales. To be sure, this is a large range but it is very much shorter than any relevant cosmological scale. Fortunately, by a happy numerical coincidence, any gravitational wave spectrum generated during preheating will peak at scale between a few meters and millions of kilometers -- which overlaps with the ``golden window'' open to direct detection experiments.
In Section (\ref{sect:preheating}), we briefly review preheating and discuss previous work. Following that, in Section (\ref{sect:lengthscales}) we demonstrate that preheating gravitational waves from inflation scales ranging from TeV to the GUT scale will peak around $1$ Hz to $10^8$ Hz. In Section (\ref{sect:production}) we discuss our methodology for numerically computing the gravitational wave spectrum. In Section (\ref{sect:results}) we present our results for $\lambda \phi^4$ and $m^2 \phi^2$ inflation, confirming and extending the results of \cite{Khlebnikov:1997di}. We note that while the inflationary dynamics of the two models are very similar, their resonant behaviors diverge considerably. Finally, we discuss the theoretical and observational implications of our results and lay out future plans to further improve the computational methodology in Section (\ref{sect:conclusions}).
\section{Preheating} \label{sect:preheating}
A key problem facing any inflationary model is to ensure that inflation ends. This issue is highlighted in Guth's foundational paper, describing what is now known as old inflation, which is driven by a meta-stable false vacuum does not successfully terminate \cite{Guth:1980zm}. In new or chaotic (slow-roll) inflation \cite{Linde:1981mu,Albrecht:1982wi,Linde:1983gd}, it was thought that inflation was followed by a period of {\em reheating\/}, where energy slowly bleeds from the inflaton field as it oscillates about the minimum of its potential \cite{Albrecht:1982mp}. Generating the correct perturbation spectrum typically requires that inflaton self-coupling is extremely weak, and this small coupling must be protected from loop corrections. Consequently, the coupling between the inflaton and other particles is necessarily tiny ($\lesssim 10^{-6}$ for $\lambda \phi^4$), ensuring that reheating proceeds slowly.
Preheating provides a vastly more efficient mechanism for extracting energy from the inflaton field, and it proceeds non-perturbatively and non-thermally via a process known as parametric resonance \cite{Traschen:1990sw,Kofman:1994rk}. This is akin to stimulated emission in a laser: during preheating, individual momentum modes of fields coupled to the inflaton (or the inflaton itself, in some cases) have exponentially growing amplitudes. Consider the following action
\begin{eqnarray}
S&=&\int dx^4 \sqrt{-g}\left[\frac{\mpl ^2 R}{16\pi }-\frac{1}{2}(\partial \phi)^2 - V(\phi)- \right. \nonumber \\
&&\left.\frac{1}{2}(\partial \chi)^2-\frac{1}{2}g^2\phi^2\chi^2\right]. \label{eqn:action}
\end{eqnarray}
As usual the Hubble parameter, $H$ and scale factor, $a$, are related by $H = \dot{a}/a$ and during inflation the dynamics are described by%
\begin{eqnarray}
H^2 &=& \frac{8 \pi}{3 \mpl^2} \left[ \frac{\dot{\phi}^2}{2} + \frac{\dot{\chi}^2}{2} + {\cal V}(\phi,\chi) \right] \, , \\
\dot{H} &=& -\frac{8 \pi}{ \mpl^2} \left[ \frac{\dot{\phi}^2}{2} + \frac{\dot{\chi}^2}{2} \right] \, , \\
\ddot{\phi} &+& 3 H \dot{\phi} + \frac{\partial {\cal V}}{\partial \phi} =0 \, ,\\
\ddot{\chi} &+& 3 H \dot{\chi} + \frac{\partial {\cal V}}{\partial \chi} =0 \, ,
\end{eqnarray}
where ${\cal V}=V(\phi)+1/2g^2\phi^2\chi^2$ is a shorthand for the potential terms in (\ref{eqn:action}). For simplicity we consider only couplings to scalar fields although fermionic preheating has also been investigated \cite{Greene:1998nh,Giudice:1999fb}.
In this action, $\phi$ is the inflaton and the inflationary dynamics are fixed by the potential $V(\phi)$, which is assumed to possess a minimum. At the end of inflation, the potential energy of the field is quickly converted into kinetic energy, and the field oscillates with frequency $m_{\phi} = \sqrt{d^2V(\phi)/d\phi^2}$ evaluated at the minimum of $V(\phi)$. The solution for $\phi$ is then approximately
\begin{equation}
\phi(t)=\Phi(t)\sin m_{\phi} t.
\end{equation}
Meanwhile, the equation of motion for the $\chi$ field after expanding it in Fourier modes is \cite{Kofman:1994rk}
\begin{equation}
\chi_k''+3H\chi_k'+(A(k)-2q\cos 2z)\chi_k=0 \label{eqn:eom}.
\end{equation}
In the limit where we can ignore the expansion of the universe this is the Mathieu equation which is simply a harmonic oscillator with a periodic forcing function. Here we have made the identification $A(k)=k^2/(m_{\phi}^2 a^2)+2q$, rescaled the time to $z=m_{\phi}t$ and used a prime to denote differentiation with respect to $z$. The crucial \emph{resonance parameter} is
\begin{equation}
q=g^2\Phi^2/(4m_\phi^2) \, . \label{eqn:resonanceparameter}
\end{equation}
The Mathieu equation possesses both oscillatory and exponential solutions; for each individual mode $k$ one can compute $A(k)$ and $q$ to determine whether or not it goes into resonance \cite{Abramowitz}. Roughly speaking, for broad resonance where a large number of modes are excited we need $q>1$ \cite{Kofman:1997yn}.
As first described by \cite{Traschen:1990sw,Kofman:1994rk}, some $\chi_k$ will have exponentially growing solutions for realistic parameter values. A full treatment of \emph{parameteric resonance} in an expanding universe is complicated, but we can make several generic statements. Firstly, preheating is very efficient and proceeds much more rapidly than reheating, which relies on tree level couplings between the inflaton and other matter fields (which, at minimum are provided by gravitational interactions). Parametric resonance typically lasts less than a Hubble time, and in some models will complete in a few oscillations of the inflaton. This is because the resonant modes are rapidly pumped up to an amplitude $\langle \chi \rangle^2 \approx \Phi^2$, cutting off resonance as it becomes kinematically expensive. Once preheating ends, the pumped-up modes dissipate their energy via self-interaction with other modes, thermalizing the universe.
Preheating leads to an initially non-thermal distribution of energy in the $\chi_k$ states. At high frequencies, corresponding to Fourier modes that are much shorter than the size of the post-inflationary Hubble horizon, the effective mass of the $\chi_k$ state is much larger than the function amplitude and resonance does not occur. Meanwhile, at low frequencies, causality ensures that modes longer than the Hubble horizon are unlikely to be in resonance.\footnote{It is actually not impossible to have resonance at very small $k$, but it does not occur in generic models of preheating. For a summary see \cite{Tsujikawa:2002nf}.} Consequently, we expect the spectrum to be narrow and centered around a wavelength which is dependent on the energy scale at the end of inflation. For the models we study in detail here, the gravitational wave spectrum induced by preheating peaks at scales $1\sim 2$ orders of magnitude shorter than the energy scale at the end of inflation. This comoving scale can be converted into a physical scale in the present universe once the post-inflationary behavior of the scale factor $a(t)$ is specified. In Section (\ref{sect:lengthscales}) we show that the peak wavelength has a physical wavelength that scales as $H_e$
\begin{equation}
l_0 \propto \frac{1}{\sqrt{H_e}} \propto \frac{1}{ V(\phi_e)^{1/4}} . \label{eqn:positionscaling}
\end{equation}
where the $e$ subscript denotes the value of $\phi$ and $H$ at the end of inflation. Lowering the inflation scale reduces the reheating scale, and reddens the gravitational wave spectrum. If the longest possible modes are excited in GUT scale inflation, the signal peaks around $10^{7} \sim 10^{8}$ Hz, although the excited modes are generally slightly shorter. As we will see below, this is a very challenging frequency range for any direct detection experiment. Reducing the inflation scale pushes the signature towards more easily observable frequencies. Given that parametric resonance naturally cuts off at both small and large scales, the spectrum of any gravitational waves will cover a fixed range of wavelengths. As the power is thus restricted to a relatively narrow band, the total gravitational radiation remains safely below the bound from big bang nucleosynthesis \cite{Maggiore:1999vm}.
The bottom line is that for a short moment the universe is highly inhomogenous, providing a fertile ground for the generation of gravitational waves. Needless to say, preheating is a highly non-linear process and analytical estimates can only take us so far. Fortunately, given an action such as (\ref{eqn:action}), this is a problem that can be solved numerically; we simply derive the equations of motion and evolve them numerically on an expanding lattice. This mimics the growth of the universe, but ignores the back-reaction of metric perturbations on the field evolution -- an assumption that is self-consistent, as while $\delta \rho/\rho$ can be large, the metric perturbations are typically small. We use a modified version of the publicly available package {\sc LatticeEasy} \cite{Felder:2000hq} for the numerical computations.
To our knowledge, the generation of gravitational waves by preheating has been thoroughly examined only once before, by Khlebnikov and Tkachev \cite{Khlebnikov:1997di}. Their work is the starting point for this paper: we elaborate and expand upon their treatment, considering a broader range of models, and taking recent developments in detector technology into account. In particular, preheating can occur at a very broad range of scales, for example via hybrid inflation \cite{Linde:1991km,Garcia-Bellido:1997wm,Garcia-Bellido:1998gm}. If the scale gets low enough, the peak wavelength can be close to the scales probed by next generation observatories such as BBO \cite{Phinney}. In Section (\ref{sect:results}), we reproduce numerical results of \cite{Khlebnikov:1997di} for the $\lambda \phi^4$ model, and present new results for the $m^2 \phi^2$ model. While the inflationary behavior of these two models is similar, their resonance structure is very different, providing a useful crosscheck on the generality of our analytic estimates. In future work we will extend these numerical calculations to hybrid inflation and fermionic preheating.
\section{Peak wavelength and amplitude} \label{sect:lengthscales}
We begin our detailed analysis with a general discussion of the different parameters that determine the amplitude and wavelength of any gravitational waves generated during preheating. Consider the ``usual'' gravitational wave power spectrum generated by quantum fluctuations of the background,
\begin{equation}
\Omega_{gw,inf}(k)h^2 = \Omega_{r}h^2\frac{32}{9}\left(\frac{V_e}{\mpl^4}\right) \left(\frac{g_0}{g_*}\right)^{1/3}, \label{eqn:quantumgrav}
\end{equation}
where $V^{1/4}_{inf}$ is the energy scale of inflation and $\Omega_r h^2\approx 4 \times 10^{-5}$ is the total density of radiation today. The effective number of degrees of freedom in the radiation at matter-radiation equality and today are given by $g_*$ and $g_0$ respectively. This form of the power spectrum is slightly non-standard as tensor modes are usually expressed via $P_h=8\pi H^2/\mpl^2$. There are two salient features to this spectrum. The first is that it is (almost) scale-invariant, since each mode is frozen out at an approximately constant energy scale, namely the inflation scale. The second is that the power is minute; for the most optimistic scenario where inflation occurs around the GUT scale, $\Omega_{gw,inf}(k) h^2 < 10^{-14}$. The current upper limit on the scale of inflation from WMAP observations for single-field inflation models is $V^{1/4}<3.3 \times 10^{16}$ GeV, corresponding to $\Omega_{gw,inf}h^2<2\times 10^{-15}$ \cite{Peiris:2003ff,Smith:2005mm}
The gravitational waves which we are considering in this paper are not directly sourced by quantum fluctuations; instead they are generated by the classical motion of particles during preheating. As is well-known, accelerated motion generates a quadrupole moment, leading to the generation of gravitational radiation. During preheating at the end of inflation, large inhomogeneities in the matter fields are generated by the selective pumping of modes in parametric resonance. These large inhomogeneities, as first shown in \cite{Khlebnikov:1997di} for the $\lambda \phi^4$ model, are sufficiently large to produce gravitational waves with amplitudes many orders of magnitude larger than those produced by the quantum fluctuations. An analytical estimate \cite{Khlebnikov:1997di}, for an inflaton with an effective oscillation frequency $\bar{m}$, coupled to a massless scalar field $\chi$ with a $g^2\phi^2\chi^2$ term, yields the peak amplitude at the resonance mode $k \sim H_r$
\begin{equation}
\Omega_{gw}(k\sim H_e)h^2\approx \Omega_{r}h^2\frac{\bar{m}^2}{g^2 \mpl^2}\left(\frac{g_0}{g_*}\right)^{1/3}. \label{eqn:simpleamplitude}
\end{equation}
In other words, the amplitude probes the oscillation scale, in contrast to the primordial spectrum which probes the inflation scale. If we plug in the usual field values for chaotic inflation $\bar{m}=m_{\phi}$ at the end of inflation $\phi\approx \mpl$ such that $V_e=m_{\phi}^2\phi^2/2 = m_{\phi}^2\mpl^2 /2 $, we see from equation (\ref{eqn:quantumgrav}) that the amplitude of the gravitational waves generated by preheating is $1/g^2$ larger than the inflationary spectrum. This is a significant boost, as we expect $g^2 \lesssim 10^{-6}$. Note that in models such as hybrid inflation \footnote{We note that in hybrid inflation, preheating amplification of the perturbations is achieved through a combination of parametric resonance and tachyionic instabilities. We thank Gary Felder for pointing this out to us.} one can decouple the oscillation frequency $\bar{m}$ from the inflaton mass $m$, and this simple relationship has to be revisited \cite{Garcia-Bellido:1997wm,Garcia-Bellido:1998gm}.
However, only a finite range of modes excited during preheating. If the power was generated at scales corresponding to, say, atomic distances today, then our hope of detecting any gravitational waves induced by preheating would be dashed. On causal grounds, we expect that resonant modes have a wavelength roughly equal to or less than the Hubble length at the end of inflation, $1/H_e$:
\begin{equation}
H_e \sim \frac{ \sqrt{V_e}}{\mpl} \, ,
\end{equation}
where $V_e$ is the inflationary potential. After inflation, the universe reheats to a temperature $\Trh$. During the subsequent radiation dominated phase, the Hubble parameter scales as $H = H_*(a_*/a_e)^2$ until matter-radiation equality at $T_*$. Meanwhile, the scale factor evolves as $a_* = a_0 (g_0/g_*)^{1/2} (T_0/T_*)$ from matter-radiation equality until today when $a_0\equiv 1$. Thus for a physical length $l$ \cite{Khlebnikov:1997di} and \emph{physical} wavevector $k$ we have
\begin{eqnarray}
l&=&\frac{1}{k} \frac{g_*^{1/2}}{g_0^{1/3}}\left(\frac{8\pi^3}{90}\right)^{-1/4}\frac{\sqrt{H_e M_p}}{T_0} \nonumber \\
& \approx& 0.5\frac{\sqrt{M_p H_e}}{k} ~\mathrm{cm}
\end{eqnarray} %
or
\begin{equation}
f=6 \times 10^{10} \frac{k}{\sqrt{M_p H_e}} ~\mathrm{Hz} \label{eqn:frequency}
\end{equation}
where we have used $g_0/g_* = 1/100$ in the second line. Plugging in the lowest excitable frequency $k = H_e$, where we more or less expect peak gravitational waves production to occur, we obtain the scaling relation (\ref{eqn:positionscaling}), as claimed earlier.
The inverse scaling is particularly important: it means that the pertinent wavelengths are longer for smaller inflationary energy scales. If we assume instantaneous reheating after inflation for GUT scale inflation $H_e \approx 10^{13}$ GeV, $l\approx 1-10$ meters, and $f\approx 10^{7}$ to $10^{8}$ Hz. Lowering the inflationary scale reduces power in the primordial gravitational wave spectrum making it harder to detect, as quantified by equation (\ref{eqn:quantumgrav}). However, this also reddens the peak power of any preheating generated gravitational waves, making them easier for us to observe. This follows because the strain sensitivity $\tilde{h}_f$ of a detector scales as $\Omega_{gw}/f^3$ \cite{Maggiore:1999vm}, i.e. for the same value of $\Omega_{gw}$ we have to build a more sensitive detector if the frequencies are higher. In addition, if inflation occurs at a lower scale, then the gravitational wave energy density will be diluted less by expansion following preheating, again increasing our chance of observing them. On the other hand, if the gravitational waves are generated at a lower scale the off-diagonal terms of $T_{\mu\nu}$ are smaller for fixed $\delta \rho / \rho$, and will be less efficient sources of gravitational radiation. On the basis of the limited calculations performed in this paper, we see that the last two effects roughly cancel and $\Omega_{gw}$ does not depend strongly on $H_e$. However, further work will be needed before we can safely say that this is true of all models which undergo preheating.
In this naive analysis, a few subtle points have been glossed over. The peak resonance modes are usually not exactly at the Hubble scale; instead they are frequently $1\sim 2$ orders of magnitude smaller \cite{Greene:1997fu,Kofman:1997yn,Greene:1997ge}. This has the effect of pushing the observable modes to a bluer band. On the other hand, preheating does not always start immediately after inflation ends; peak particle production occurs when the amplitude of the field perturbations $\delta \phi/\phi$ grows to order unity, which need not happen quickly. In the models looked at here, the Hubble parameter during peak gravitational wave production is about $1\sim 2$ orders of magnitude smaller than $H_e$, shifting the observable modes to a redder band.
It is also worth noting that the gravitational waves are generated causally within the Hubble volume, and thus the phases of the individual modes are uncorrelated -- unlike the primordial spectrum. This is a generic feature of all causally generated perturbation spectra, and is a powerful discriminant \cite{Dodelson:2003ip}. Unfortunately, direct detection experiments cannot dinstinguish the coherence (or lack of) of the gravitational waves as their signal is an integral over some time interval greater than the frequency scale. To do this, one must find a processed \emph{imprint} on a fixed time-slice.
While there is an upper bound on the inflationary energy scale from the contribution of tensor modes to the CMB, the lower bound is very weak. At minimum, the post-inflationary universe must be hot enough to permit baryogenesis and nucleosynthesis. We conservatively assume that the former occurs via electroweak scale processes, so we can easily have $V_e^{1/4}$ as low as the TeV scale. Nuclear reactions necessarily take place at MeV scales and ensuring successful nucleosynthesis provides an absolute lower limit on the reheating temperature. This corresponds to gravitational wave peak wavelength scales ranging from laboratory scales through to solar system scales.
\section{Gravitational Wave Production} \label{sect:production}
Equation (\ref{eqn:simpleamplitude}) suggests that the preheating induced gravitational wave spectrum is larger than its primordial counterpart. To obtain an actual power spectrum, the highly nonlinear physics of preheating forces us to turn to numerical methods. We use {\sc LatticeEasy\/} \cite{Felder:2000hq} to simulate the evolution of the early universe, solving the equations of motion for a set of interacting scalar fields in a flat Friedmann-Robertson-Walker (FRW) Universe. The fields become highly inhomogeneous, which is important for the generation of gravitational waves. This does not immediately make the {\em metric\/} perturbations large \cite{Ishibashi:2005sj}. Consequently, we can solve the nonlinear field evolution numerically while assuming a rigid spacetime background, and then extract the spectrum of gravitational radiation produced during preheating.
We extended {\sc LatticeEasy\/} to compute the gravitational wave spectrum generated during preheating. We follow the approach of \cite{Khlebnikov:1997di} (see also \cite{Kamionkowski:1993fg}), reproducing their results for the quartic $\lambda \phi^4 /4 + g^2\phi^2\chi^2/2$ model. In addition, we compute the gravitational wave spectrum for the quadratic inflation model $m^2\phi^2/2 +g^2\phi^2\chi^2/2$. We leave the simulation of other models such as the negative coupling $-g^2\phi^2\chi^2/2$ \cite{Greene:1997ge} or hybrid inflation models to future work.
We now sketch the approach we use to compute the spectrum. We begin by considering the energy radiated in gravity waves in a frequency interval $d\omega$ and a solid angle $d\Omega$, given by
\begin{equation}
\frac{dE}{d\Omega}=2G\Lambda_{ij,lm}\omega^2 T^{ij*}(\vec{\mathbf{k}},\omega)T^{lm}(\vec{\mathbf{k}},\omega) d\omega \label{eqgrav}
\end{equation}
where $T^{ij}$ is the stress tensor describing the source matter fields. Here $i$,$j$ run over the spatial indices and the projection tensor is given by \cite{WeinbergBook}
\begin{eqnarray}
\Lambda_{ij,lm}(\hat{k})=\delta_{ij}\delta_{lm}-2\hat{k}_j\hat{k}_m\delta_{il}+\frac{1}{2}\hat{k}_i\hat{k}_j\hat{k}_l\hat{k}_m \nonumber \\
-\frac{1}{2}\delta_{ij}\delta_{lm}+\frac{1}{2}\delta_{ij}\hat{k}_l\hat{k}_m+\frac{1}{2}\delta_{jl}\hat{k}_i\hat{k}_m
\end{eqnarray}
with unit vector $\hat{k}\equiv \vec{\mathbf{k}}/\omega$. Strictly speaking, this formula is only valid for linearized gravity in Minkowski space. A more accurate calculation will involve solving the equations of motion for linearized gravity on a curved background.
To see why the use of this formula is justified, consider the gravitational wave energy emitted by a three dimensional box of \emph{conformally flat} spacetime with physical size $l\times l\times l$. The energy density in this box is
\begin{equation}
d\rho(\omega)=8\pi G l^{-3} \Lambda_{ij,lm}\omega^2 T^{ij*}(|\mathbf{\vec{k}}|,\omega)T^{lm}(|\mathbf{\vec{k}}|,\omega) d\omega \label{eqn:gravdensity}
\end{equation}
where we have assumed that the spectrum is isotropic.\footnote{Isotropy allows us to choose any direction for the projection vector $\Lambda_{ij,lm}$: we chose for simplicity $\hat{k}=(1,0,0)$. We checked that this assumption is robust by showing that the final simulation results are not sensitive to different choices of direction.} From causal arguments alone, only modes of wavelengths equal to or shorter than $1/H$ will be generated, imposing a natural cut-off at long scales. Thus, provided we choose $l\geq 1/H$ we will effectively capture the essential physics. Depending on how efficient preheating is in a particular model, the entire phase can last for several Hubble times. However, the gravitational waves are produced near the end of preheating, as the inhomogeneities in the fields become large. We therefore expect the gravitational wave power to be generated in a short burst, and numerical simulations confirm this suspicion. Thus it is a good approximation to assume that the gravitational wave source is localized in the box \cite{Kosowsky:1991ua}.
In our simulations, we begin our computations at the end of inflation, near the beginning of the parameteric resonance phase. We end our simulations when the fields are stabilized and parameteric resonance ends. We subdivide the the spacetime into discrete 4-D boxes of spatial sizes $L^3$ and time interval $\tau=L$, where $L$ and $\tau$ are the conformal length and time respectively. Our physical box size thus scales roughly as $a^3$. The choice of $\tau=L$ is purely operational, allowing us to fix our Fourier variables to be the conformal frequency and conformal wavevector such that $|\mathbf{\vec{k}}_{conf}|=\omega_{conf}$. One can in principal decompose the conformal time differently, but that would unnecessarily complicate matters. We fix $L$ so that during the period of gravitational wave production $aL \geq 1/H$ and the box is larger than the effective Hubble horizon.
We assume that each ``box'', labeled $\alpha$, is a localized source, and compute the total gravitational wave density produced for each box $\rho_{gw}^{(\alpha)}$ using equation (\ref{eqn:gravdensity}). We then sum them up, diluting them appropriately as follows
\begin{equation}
\frac{d\rho_{gw}(a_{e})}{d\ln \omega}=\sum_{\alpha} \frac{d\rho_{gw}^{(\alpha)}(a_{\alpha})}{d\ln \omega}\left(\frac{a_{e}}{a_{\alpha}}\right)^{4} \label{eqn:finalpower}
\end{equation}
where $a_{\alpha}$ is the scale factor taken at the middle of the box in conformal time and $a_{e}$ is the scale factor at the end of inflation. Meanwhile, for each box $\alpha$
\begin{equation}
\frac{d\rho_{gw}(a_{\alpha})}{d\ln \omega}=8\pi G \omega^3 l_{a_{\alpha}}^{-3}\Lambda_{ij,lm}T^{ij*}T^{lm}
\end{equation}
where $l_{a_{\alpha}}^3$ is the physical size of the box at time $a_{\alpha}$ and $\omega$ is the physical frequency.
Finally, putting everything together, the total density of gravitational waves today is given by
\begin{equation}
\Omega_{gw}h^2=\Omega_r h^2 \frac{d\rho_{gw}(a_{e})}{d\ln \omega}\left(\frac{g_0}{g_*}\right)^{1/3} \label{eqn:finalpowertoday}
\end{equation}
We should mention that in equation (\ref{eqn:finalpower}) and hence equation (\ref{eqn:finalpowertoday}), we have implicitly assumed that the universe is radiation dominated at the end of preheating, which is not true for certain chaotic models.
\section{Numerical Results} \label{sect:results}
In this section, we give numerical results for the gravitational wave spectrum produced during resonance in two different models: $\lambda \phi^4$ and $m^2\phi^2$. While the inflationary dynamics of these two systems are very similar, there is considerable divergence in the resonance structure between the models, making this a useful generalization of \cite{Khlebnikov:1997di}.
\subsection{Quartic Inflation ($\lambda \phi^4$)}
To test our code, we reproduce Khlebnikov and Tkachev's results \cite{Khlebnikov:1997di} for $\lambda \phi^4$ with a $\phi^2\chi^2$ term
\begin{equation}
{\cal V}(\phi,\chi)=\frac{\lambda}{4}\phi^4 + \frac{1}{2}g^2\phi^2 \chi^2.
\end{equation}
From the perspective of preheating, this model is atypical \cite{Greene:1997fu} as it possess only a weak resonance band. Even so, we still see significant production of gravity waves.
Following \cite{Khlebnikov:1997di}, we set $\lambda=10^{-14}$ and $g^2/\lambda=120$, corresponding to a resonance parameter $q\approx 120$ from equation (\ref{eqn:resonanceparameter}). In this model, inflation ends around the GUT scale, where $\phi_0\approx \mpl$, or $H_{end}\approx 10^{12}$ GeV. We begin our simulation on a $256^3$ size lattice from that time and run it until preheating ends around $H\approx 10^{7}$ GeV. Parameteric resonance peaks around $H_{peak}\approx 10^{8}$ GeV, and the size of the box is chosen to ensure that its physical size at this time $l \approx 1/H_{peak}$. With a $\lambda \phi^4$ potential, the background spacetime scales like a radiation dominated universe during parametric resonance.
Using (\ref{eqn:frequency}), the present frequency associated with the Hubble parameter during preheating is $10^{6}$Hz. From figure (\ref{fig:quarticstdresult}), we see that the peak frequency is actually $10^{7}\sim 10^{8}$ Hz, suggesting that the peak resonance modes are about two orders of magnitude smaller than the Hubble wavelength. The amplitude of the gravitational waves peaks at around $\Omega_{gw}\approx 10^{-9}$, consistent with equation (\ref{eqn:simpleamplitude}).
From the plot, we see that even at the lower end of the relevant frequency range which is easier to detect, $\Omega_{gw} \sim10^{-11}$. This is 3 orders of magnitude larger than the primordial spectrum. Beyond that at lower frequencies, we expect the spectrum to undergo a steep $k^3$ decline. This $k^3$ superhorizon tail is a common property for a spectrum which is causally generated inside the Hubble horizon \cite{Liddle:1999hq}. In cases where the inflationary scale and thus the intrinsic stochastic background is very low (so it does not mask the signal) this $k^3$ tail might be easier to detect than the peak wavelengths, given the physical limitations on realistic detectors.
\subsection{Quadratic Inflation ($m_{\phi}^2\phi^2$)}
We now turn to the $m_{\phi}^2\phi^2$ model \cite{Linde:1983gd} with the same interaction term as before
\begin{equation}
{\cal V}(\phi,\chi)=\frac{1}{2}m_{\phi}^2\phi^2 + \frac{1}{2}g^2\phi^2 \chi^2.
\end{equation}
The amplitude of the CMB temperature anisotropy requires $m_{\phi}\approx 10^{13}$ GeV. At the end of inflation $\phi \approx \mpl$. Choosing $g^2=2.5 \times 10^{-7}$, gives a resonance parameter of $q\approx 2.5\times 10^{5}$, via (\ref{eqn:resonanceparameter}). Again we begin our simulation after inflation ends at $H_{end}\approx 10^{13}$ GeV through the peak preheating phase at $H_{peak}\approx 10^{11}$ GeV, until the end of preheating. Since $m_{\phi}$ is non-zero, the universe evolves as if it was matter dominated\footnote{The equation of state $w=\langle p \rangle / \langle \rho \rangle$ fluctuates rapidly between $1$ and $-1$ around a center value of $0$.} with the scale factor growing 30-fold. At the end of preheating, we assume that the universe reheats normally and enters a radiation dominated phase. This is in principle not a valid assumption, as some numerical results have shown that it is difficult for quadratic inflation to reheat to radiation domination without a trilinear coupling \cite{Podolsky:2005bw}. In this paper, we are using it as a toy model to illustrate preheating for a inflation model with a different mass scale.
We simulated this model on a $256^3$ lattice, with the results as shown in figure (\ref{fig:chaoticstdresults}).
Using (\ref{eqn:simpleamplitude}), we expect $\Omega_{gw}h^2\approx 10^{-10}$ at peak which matches the result of our detailed calculation. Although inflation ends at $H_{end}\approx 10^{13}$ GeV, peak resonance occurs at $H_{peak}\approx 10^{11}$ GeV, which sets the comoving size of our lattice. Figure (\ref{fig:chaoticstdresults}) shows that the peak frequency is actually $10^{8}\sim 10^{9}$ Hz, a couple of orders of magnitude smaller than the Hubble parameter during reheating.
Finally, we present the results for a model with a lower mass, $m_{\phi}=10^{12}$ GeV in figure (\ref{fig:chaoticstdresults2}). This model is ruled out by CMB data, but it demonstrates the way in which the gravitational wave spectrum generated by preheating depends on the inflationary scale. In this model $H\approx 10^{12}$ GeV, and as expected from equation (\ref{eqn:positionscaling}), the peak location is reddened by a factor of $\sqrt{10}$. The observable power remains comparable to the previous model. The emitted power is reduced, but since the overall expansion of the universe is reduced by the lower reheating temperature, the values of $\Omega_{gw}h^2$ today is roughly fixed. It is tempting to conjecture that the cancellation between these two effects will be seen in other preheating induced gravitational wave spectra, and that the $\Omega_{gw} h^2 \sim 10^{-10}$ seen here will prove to be a generic value \cite{Garcia-Bellido:1997wm,inpreparation}.
\section{Summary and Future Prospects} \label{sect:conclusions}
We have carefully investigated the production of gravitational waves during preheating, reproducing the work of Khlebnikov and Tkachev \cite{Khlebnikov:1997di} for the $\lambda \phi^4$ inflation model and extending it to the $m_{\phi}^2\phi^2$ case. For both models we show numerically that preheating is a sizable source of gravitational waves with frequencies of around $10^{6}\sim 10^{8}$ Hz, and peak power of $\Omega_{gw}h^2\approx 10^{-9} \sim 10^{-11}$. We present simple scaling arguments to predict the overall properties of the spectrum for a broader class of inflationary models. We see that the spectrum of gravitational waves induced by preheating peaks at a scale proportional to $1/\sqrt{H}$, where $H$ is the Hubble parameter during preheating, and generally somewhat smaller than the scale of inflation. Thus, lowering the inflationary scale reddens the spectrum and makes it easier to observe. This is in contrast to the primordial inflationary spectrum, which is roughly scale invariant and becomes harder to observe as inflationary scale is lowered.
We now ask what we can learn about inflation if we detect a spectrum of gravitational waves generated during preheating. Its two most basic features, the peak frequency and the amplitude, represent the reheat and the oscillation scales respectively. As the reheat scale is lower than the inflation scale, its detection would impose a lower bound on the inflationary scale. Its usefulness as a probe of inflation is amplified if we have a separate probe of the scale of inflation, say from the CMB B-mode observations.
The oscillation scale is harder to interpret, as it is often highly model dependent. For single scalar field inflation, such as the models considered here, knowledge of the reheat scale would constrain the coupling constant $g^2$. More optimistically, the \emph{structure} of the spectrum encodes information about resonance and preheating, so if we can predict the structure accurately we potentially probe the detailed mechanics of preheating. Such an endeavour will require more careful computations and simulations than we present in this paper.
Further progress on this problem can be made in two ways \cite{inpreparation}. The first is to further refine the code to accommadate a larger class of models, particularly hybrid inflation models which have an essentially arbitrary inflationary scale. Alternatively, as alluded to in Section (\ref{sect:production}), a more sophisticated theoretical calculation would be to directly solve the evolution equations for the off-diagonal parts of the perturbed Einstein tensor, which are sourced by $T_{ij}$. This would avoid any ambiguity concerning the use of a formula that is only strictly applicable in flat space, and it would avoid the need to run the code for a finite number of ``boxes'', since we would only need to take a Fourier transform at the end of the computation.
Investigating these gravitational waves is timely, since there is currently considerable interest in the direct detection of gravitational waves. At the moment, several proposals based on different technologies are being actively pursued. At the solar-system scale, the space-based interferometer LISA \cite{LISA} will probe frequencies from $10^{-2}$ Hz, which is probably too small for the gravitational waves we are considering. The proposed BBO \cite{Phinney} and also the Deci-hertz Interferometer Gravitational Wave observatory (DECIGO) \cite{Seto:2001qf} missions are sensitive to frequencies on the order of $1$ Hz, and would probe gravitational waves arising from preheating after TeV scale inflation.
The array of terrestrial interferometers also probes frequencies corresponding to preheating following low-scale inflation. These experiments include GEO600 \cite{GEO600}, LIGO \cite{LIGO}, TAMA \cite{TAMA} and VIRGO \cite{VIRGO}. These are sensitive to scales between $100\sim 1000$ Hz and may be able to probe a stochastic background in the interesting range $\Omega_{gw}h^2 \sim 10^{-10}$, if they are correlated. At even higher frequencies in the KHz range, we have a slew of resonant bars detectors \cite{ALLEGRO,AURIGA,EXPLORER,Blair:1997as}. Once correlated, these resonant bars have a potential to reach $\Omega_{gw}h^2\approx 10^{-5}$ \cite{Maggiore:1999vm}, which puts the signals we see here out of their reach. However, theoretical studies suggest that correlating hollow spherical detectors may eventually allow us to reach $\Omega_{gw}h^2\approx 10^{-9}$ \cite{Coccia:1997gy}.
At even higher frequencies from $10^{3} \sim 10^{5}$ Hz, there has been a proposal to build a superconducting resonant cavity detector called the Microwave Apparatus for Gravitational Wave Observation (MAGO) \cite{Pegoraro:1977uv,Reece:1984gv,Ballantini:2005am}. Although the strain sensitivity $\tilde{h}_f \approx 10^{-21} \textrm{Hz}^{-1/2}$ for the prototype is expected to be comparable to resonant bar detectors at $4\times 10^{3}$ Hz, the large $f^3$ suppression from the relation $\Omega_{gw} h^2 \propto \tilde {h}_f^2 f^3$ means that at these frequencies we can only reach $\Omega_{gw} h^2\approx {\cal{O}}(1)$. An improvement of $5\sim 6$ orders of magnitude in the strain sensitivity is needed to reach $\Omega_{gw}h^2\approx 10^{-10}$, a possibility which may be achieved by further refinements to the prototype and/or construction of an array of such detectors \cite{Ballantini:2005am}. By way of comparison, we note that the best hope for observing a primordial gravitational wave background is currently provided by BBO, which has a lead time of at least 20-25 years. In this context hoping for a large extrapolation of detector technologies at high frequencies is perhaps not excessively optimistic. In Figure~\ref{fig:sensitivityplot} we sketch the sensitivities of the leading interferometric detectors along with the expected stochastic background produced during preheating for the models we discuss in detail here.
Finally, if one was to ever make a concerted attempt to detect a gravitational wave spectrum associated with preheating, one would need to be understand other potential sources that could supply a stochastic background of gravitational waves. This includes any first order phase transition in the early universe (such as the electroweak scale), or decays from cosmic strings. In addition, the presence of a rising component in the spectrum illustrates the dangers of using a (locally) positive spectral index of any detected stochastic gravitational wave background to rule out inflation in favour of alternative cosmogenesis ideas such as ekpyrosis \cite{Khoury:2001wf,Boyle:2003km} or pre-big bang scenarios \cite{Gasperini:1992em}.
The potential for gravitational waves to provide a clean probe of inflation has rightfully drawn considerable attention, and strongly motivates attempts to detect the primordial gravitational spectrum. However, cosmic evolution is seldom tidy and gravitational waves are produced as long as large inhomogeneities are present. Preheating is a mechanism which will generate large inhomogeneities, and will necessarily be accompanied by the generation of a stochastic background of gravitational waves. The challenge now is to better determine their properties, and to assess possible strategies for their detection.
\section*{ Acknowledgments}
We thank Latham Boyle, Gary Felder, Gianluca Gemme, Tom Giblin, Will Kinney, Hiranya Peiris, Geraldine Servant, Igor Tkachev, and David Wands for a number of useful discussions. We are particularly indebted to Gary Felder and Igor Tkachev for their work on {\sc LatticeEasy\/}. This work is supported in part by the United States Department of Energy, grant DE-FG02-92ER-40704.
|
Title:
Helicity-Rotation-Gravity Coupling for Gravitational Waves |
Abstract: The consequences of spin-rotation-gravity coupling are worked out for linear
gravitational waves. The coupling of helicity of the wave with the rotation of
a gravitational-wave antenna is investigated and the resulting modifications in
the Doppler effect and aberration are pointed out for incident high-frequency
gravitational radiation. Extending these results to the case of a
gravitomagnetic field via the gravitational Larmor theorem, the rotation of
linear polarization of gravitational radiation propagating in the field of a
rotating mass is studied. It is shown that in this case the linear polarization
state rotates by twice the Skrotskii angle as a consequence of the spin-2
character of linear gravitational waves.
| https://export.arxiv.org/pdf/gr-qc/0601054 |
\title{Helicity-Rotation-Gravity Coupling for Gravitational Waves}
\author{Jairzinho Ramos}
\affiliation{Physics Department, Drexel University,
Philadelphia, Pennsylvania 19104, USA}
\author{Bahram Mashhoon}
\affiliation{Department of Physics and Astronomy, University of Missouri-Columbia, Columbia, Missouri 65211, USA}
\pacs{04.20.Cv}
\section{Introduction}
In a recent paper on the purely gravitational spin-rotation coupling, Shen \cite{Shen} has treated the coupling of graviton spin to the gravitomagnetic
field. In this way, spin-gravity coupling has been extended to include gravitational waves. The subject of spin-rotation-gravity coupling for a
particle of spin $s$ has been reviewed in Refs. \cite{Mash} and \cite{Ryder} and discussions of more recent advances are contained in Refs. [4-12];
however, these treatments have ignored the $s=2$ case. Shen's field-theoretical approach is based on a weak-field approximation scheme that emphasizes the
self-interaction of the nonlinear gravitational field \cite{Shen}.
The purpose of the present paper is to investigate the consequences of the helicity-gravitomagnetic field coupling for weak gravitational waves.
To provide a comprehensive treatment, we begin with the analysis of the propagation of free gravitational waves in a Minkowski spacetime background
from the standpoint of a uniformly rotating observer. We then generalize our helicity-rotation coupling results to the propagation of gravitational
waves in the field of a rotating astronomical mass via the gravitational Larmor theorem.
The plan of this paper is as follows. In Section II, we study the reception of a free gravitational wave by a rotating observer in Minkowski
spacetime. Section III deals with the modification of Doppler effect and aberration for gravitational waves caused by the helicity-rotation coupling.
The results are extended to the gravitational case in Section IV using the gravitational Larmor theorem. It is then possible to study the influence of
the gravitomagnetic field of a rotating source on the propagation of high-frequency gravitational radiation. The rotation of the linear polarization
state of gravitational radiation propagating in the field of a rotating source is directly calculated in Section V using the eikonal approximation.
Section VI contains a discussion of our results. In this paper, we choose units such that $c=1$, moreover, the signature of the metric is $+2$ in our convention.
\renewcommand{\theequation}{2.\arabic{equation}}
\setcounter{equation}{0}
\section*{II. HELICITY-ROTATION COUPLING}
Imagine a class of uniformly rotating observers ${\cal O}'$ in a global inertial frame of reference. For the sake of concreteness, we choose Cartesian coordinates such that
the observers rotate with a frequency $\Omega$ about the $z$ axis, each on a circle of radius $\rho$, $0 \leq \rho < 1/\Omega$, parallel to the $(x,y)$ plane of the
inertial system. The local orthonormal tetrad frame of each observer is given in the $(t,x,y,z)$ system by
\begin{eqnarray}
\Lambda^{\mu}_{(0)}&=&\gamma(1,-v \sin\phi,v \cos\phi,0), \label{b1} \\
\Lambda^{\mu}_{(1)}&=&(0,\cos\phi,\sin\phi,0), \label{b2} \\
\Lambda^{\mu}_{(2)}&=&\gamma(v,-\sin\phi,\cos\phi,0), \label{b3} \\
\Lambda^{\mu}_{(3)}&=&(0,0,0,1),
\label{b4}
\end{eqnarray}
where $\phi=\Omega t=\gamma\Omega\tau$, $v=\Omega\rho$, $\gamma$ is the Lorentz factor corresponding to $v$ and $\tau$ is the proper time such that $\tau=0$ at
$t=0$. Regarding the motion of each observer, we note that $\Lambda^{\mu}_{(1)}$ and $\Lambda^{\mu}_{(2)}$ indicate the radial and tangential directions, respectively,
in cylindrical coordinates.
Let us first consider an incident plane monochromatic gravitational wave of frequency $\omega$ and wave vector ${\bf k}=\omega(0,0,1)$, so that each observer rotates about
the direction of wave propagation. The gravitational potential of the incident radiation is given by the symmetric tensor $h_{\mu\nu}$, which represents a small perturbation
of the background Minkowski metric $\eta_{\mu\nu}$. Therefore, only terms that are linear in $h_{\mu\nu}$ will be considered throughout. In the transverse-traceless gauge,
$h_{0\mu}=0$ and the potential for circularly polarized gravitational radiation is given by the matrix $(h_{ij})={\rm Re}(P_{\pm})$, where
\begin{equation}
P_{\pm}=(\epsilon_{\oplus} \pm i\epsilon_{\otimes})\hat{h}({\bf k})e^{i\omega(z-t)}.
\label{b5}
\end{equation}
The upper (lower) sign corresponds to positive (negative) helicity and the two independent linear polarization states are denoted by
\begin{equation}
\epsilon_{\oplus}=\left(
\begin{array}{ccc}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & 0
\end{array}\right), \,\,\,\,\,\,\,\,\,\,\,\
\epsilon_{\otimes}=\left(
\begin{array}{ccc}
0 & 1 & 0 \\
1 & 0 & 0 \\
0 & 0 & 0
\end{array}\right).
\label{b6}
\end{equation}
All of the operations involving wave functions are linear; therefore, only the real part of the relevant quantities will be of any physical significance.
Consider the measurement of spacetime curvature by the static observers ${\cal O}$; the Riemann tensor can be expressed as a $6 \times 6$ matrix $({\cal R}_{AB})$, where
the indices $A$ and $B$ range over the set $\{01,02,03,23,31,12\}$. The gravitational field is given by the measured components of the Riemann tensor and for a Ricci-flat
field,
\begin{equation}
{\cal R}=\left(
\begin{array}{cc}
E & H \\
H & -E
\end{array}\right),
\label{b7}
\end{equation}
where $E$ and $H$ are $3 \times 3$ symmetric and traceless matrices corresponding to the electric and magnetic components of the curvature tensor. It turns out that for all
of the circularly polarized gravitational waves considered in this section, the gauge-invariant Riemann tensor is of the form of Eq. (\ref{b7}) with $H=\pm i E$,
which, taking due account of the proper definitions of curvature-based gravitoelectric and gravitomagnetic fields \cite{Mash3}, corresponds exactly to the analogous
electromagnetic case. It is therefore sufficient to focus attention only on the electric part of the Riemann tensor. For the static observers ${\cal O}$, the field of
circularly polarized radiation is thus given by $E=\frac{1}{2}\omega^{2}P_{\pm}$.
The gravitational field as determined by the rotating observers ${\cal O}'$ is given by
$R_{\mu\nu\rho\sigma}\Lambda^{\mu}_{(\alpha)}\Lambda^{\nu}_{(\beta)}\Lambda^{\rho}_{(\gamma)}\Lambda^{\sigma}_{(\delta)}$. These may be expressed via the real part of
${\cal R}'={\cal L}{\cal R}{\cal L}^{\dagger}$, where ${\cal L}$ is a real $6 \times 6$ matrix that can be determined from Eqs. (\ref{b1})-(\ref{b4}). We find that
\begin{equation}
{\cal L}=\left(
\begin{array}{cc}
{\cal A} & {\cal B} \\
-{\cal B} & {\cal A},
\end{array}\right),
\label{b8}
\end{equation}
where ${\cal A}$ and ${\cal B}$ are given by
\begin{equation}
{\cal A}=\left(
\begin{array}{ccc}
\gamma\cos\phi & \gamma\sin\phi & 0 \\
-\sin\phi & \cos\phi & 0 \\
0 & 0 & \gamma
\end{array}\right),
{\cal B}=v\gamma\left(
\begin{array}{ccc}
0 & 0 & -1 \\
0 & 0 & 0 \\
\cos\phi & \sin\phi & 0
\end{array}\right).
\label{b9}
\end{equation}
It is then possible to express ${\cal R}'$ as
\begin{equation}
{\cal R}'=\left(
\begin{array}{cc}
C_{\pm} & \pm iC_{\pm} \\
\pm iC_{\pm} & -C_{\pm}
\end{array}\right),
\label{b10}
\end{equation}
where $C_{\pm}$ is given by
\begin{equation}
C_{\pm}=\frac{1}{2}\omega^{2}\left(
\begin{array}{ccc}
\gamma^{2} & \pm i\gamma & \pm iv\gamma^{2} \\
\pm i\gamma & -1 & -v\gamma \\
\pm iv\gamma^{2} & -v\gamma & -v^{2}\gamma^{2}
\end{array}\right)\hat{h}({\bf k})e^{i\omega z-i(\omega \mp 2\Omega) t}.
\label{b11}
\end{equation}
Thus the frequency measured by the rotating observers via the temporal dependence of Eq. (\ref{b11}) is
\begin{equation}
\omega'=\gamma(\omega \mp 2\Omega),
\label{b12}
\end{equation}
which clearly exhibits the contribution of helicity-rotation coupling \cite{Mansouri}. This expression is the exact spin-2 analog of the electromagnetic result
\cite{Mash1,Neutze} that has been observationally verified for $\omega\gg\Omega$. Let us note that a simple application of the Doppler formula would lead to the transverse
Doppler frequency $\omega'_{D}=\gamma\omega$; however, the Doppler formula must be modified by taking into account the helicity-rotation coupling as in Eq. (\ref{b12}).
For a packet of free gravitational waves propagating along the axis of rotation of the observer, the frequency of each Fourier component would be affected as in
Eq. (\ref{b12}). A complete discussion of the physical implications of Eq. (\ref{b12}) will not be given here, since such treatment would be entirely analogous to the
electromagnetic case that has been discussed in detail in Ref. \cite{Mash}.
Let us now consider the extension of helicity-rotation coupling to the case of {\it oblique} incidence. Expressing the incident plane wave in terms of spherical waves whose
dependence upon time $t$ and the azimuthal coordinate $\varphi$ is of the form ${\rm exp}(-i\omega t+im\varphi)$, and taking into account the fact that a transformation to
the frame of the rotating observer involves $(r,\vartheta, \varphi) \rightarrow (r,\vartheta, \varphi')$ in terms of spherical polar coordinates such that $\varphi=\varphi'+
\Omega t$, we find that
\begin{equation}
\omega'=\gamma(\omega-m\Omega), \,\,\,\,\,\,\,\ m=0,\pm 1, \pm 2,\ldots .
\label{b13}
\end{equation}
Here $m$ is the multipole parameter such that $m\hbar$ is the total (orbital plus spin) angular momentum of the radiation field along the direction of rotation
of the observer. When this direction coincides with the direction of wave propagation, only the spin contributes to $m$ in Eq. (\ref{b13}) and hence we recover Eq. (\ref{b12})
for radiation of definite helicity $(m=\pm 2)$. It proves interesting to consider the other special case where the contribution of the orbital angular momentum of the
obliquely-incident radiation field vanishes, namely, the reception of the radiation by a rotating observer ${\cal O}'_{0}$ at ${\bf x}=0$. In this case, the noninertial
observer is at rest at the origin of spatial coordinates, but refers its measurements to axes rotating with frequency $\Omega$. The tetrad frame for ${\cal O}'_{0}$ is given
by Eqs. (\ref{b1})-(\ref{b4}) with $v=0$ and $\gamma=1$. Imagine therefore an incident plane circularly-polarized gravitational wave with wave vector
${\bf k}=\omega{\bf \hat{k}}$, where ${\bf \hat{k}}=(0,-\sin\theta, \cos\theta)$. According to the static inertial observers, the natural orthonormal triad for the wave is
$({\bf \hat{x}},{\bf \hat{n}},{\bf \hat{k}})$, where ${\bf \hat{n}}={\bf \hat{k}} \times {\bf \hat{x}}=(0,\cos\theta,\sin\theta)$. It is straightforward to express the
potential of the wave in the transverse-traceless gauge as $(\tilde{h}_{ij})={\rm Re}(\tilde{P}_{\pm})$, where
\begin{equation}
\tilde{P}_{\pm}=(\tilde{\epsilon}_{\oplus} \pm i\tilde{\epsilon}_{\otimes})\hat{h}({\bf k})e^{i({\bf k}\, .\, {\bf x}-\omega t)}
\label{b14}
\end{equation}
and
\begin{eqnarray}
\tilde{\epsilon}_{\oplus}&=&\left(
\begin{array}{ccc}
1 & 0 & 0 \\
0 & -\cos^{2}\theta & -\sin\theta\cos\theta \\
0 & -\sin\theta\cos\theta & -\sin^{2}\theta
\end{array}\right), \nonumber \\
\tilde{\epsilon}_{\otimes}&=&\left(
\begin{array}{ccc}
0 & \cos\theta & \sin\theta \\
\cos\theta & 0 & 0 \\
\sin\theta & 0 & 0
\end{array}\right).
\label{b15}
\end{eqnarray}
The gravitational field measured by the static inertial observers, $\tilde{{\cal R}}$, has the standard form and its electric part is given by $\frac{1}{2}\omega^{2}
\tilde{P}_{\pm}$. Similarly, the field according to the special rotating observer ${\cal O}'_{0}$ is $\tilde{{\cal R}}'$ with electric components given by
$\frac{1}{2}\omega^{2}\tilde{P}'_{\pm}$. To determine $\tilde{P}'_{\pm}$, let
\begin{equation}
R_{\Omega}=\left(
\begin{array}{ccc}
\cos\phi & \sin\phi & 0 \\
-\sin\phi & \cos\phi & 0 \\
0 & 0 & 1
\end{array}\right)
\label{b16}
\end{equation}
be the rotation matrix that relates the spatial frame $({\bf \hat{x}}',{\bf \hat{y}}',{\bf \hat{z}}')$ of the rotating observer to that of the static inertial observers.
We find that
\begin{equation}
\tilde{P}'_{\pm}=R_{\Omega}\tilde{P}_{\pm}R^{\dagger}_{\Omega}.
\label{b17}
\end{equation}
It then follows after some algebra that $\tilde{P}'_{\pm}$ is given by
\begin{equation}
\tilde{P}'_{\pm}=\hat{h}({\bf k})\sum^{2}_{m=-2}\mu_{m}{\cal M}^{(m)}e^{-i(\omega-m\Omega)t},
\label{b18}
\end{equation}
so that in this case the measured frequencies are $\omega'_{0}=\omega-m\Omega$, where $m=0,\pm 1,\pm 2$. Here ${\cal M}^{(m)}$ are $3 \times 3$ symmetric and traceless
matrices given by ${\cal M}^{(0)}={\rm diag}(\frac{1}{2},\frac{1}{2},-1)$,
\begin{equation}
{\cal M}^{(\pm 1)}=\left(
\begin{array}{ccc}
0 & 0 & \pm i \\
0 & 0 & -1 \\
\pm i & -1 & 0
\end{array}\right),\,\
{\cal M}^{(\pm 2)}=\left(
\begin{array}{ccc}
1 & \pm i & 0 \\
\pm i & -1 & 0 \\
0 & 0 & 0
\end{array}\right);
\label{b19}
\end{equation}
moreover, the coefficients $\mu_{m}$ can be expressed as
\begin{eqnarray}
\mu_{0}&=&\sin^{2}\theta, \,\,\ \mu_{1}=\pm a_{\pm}\sin\theta, \,\,\ \mu_{2}=(a_{\pm})^{2}, ~~~~~~~~ \label{b20} \\
\mu_{-1}&=&\mp a_{\mp}\sin\theta, \,\,\ \mu_{-2}=(a_{\mp})^{2},
\label{b21}
\end{eqnarray}
where
\begin{equation}
a_{\pm}=\frac{1}{2}(1\pm \cos\theta).
\label{b22}
\end{equation}
It follows from these results that ${\cal O}'_{0}$ can express its measurements as a Fourier sum of frequencies $\omega'_{0}=\omega-m\Omega$ with amplitudes given in
Eqs. (\ref{b18})-(\ref{b22}). It is reasonable to assume that the weight $W_{m}$, or intensity, assigned to each Fourier component is given by the sum of the squares
of the absolute magnitudes of the elements of the corresponding matrix. Thus, $W_{m}=|\hat{h}({\bf k})|^{2}w_{m}$, where
\begin{eqnarray}
w_{0}&=&\frac{3}{2}\sin^{4}\theta, \,\, w_{1}=4(a_{\pm})^{2}\sin^{2}\theta, \,\, w_{2}=4(a_{\pm})^{4}, ~~~~~~~~~~ \label{b23} \\
w_{-1}&=&4(a_{\mp})^{2}\sin^{2}\theta, \,\,\,\,\,\,\,\,\,\,\,\ w_{-2}=4(a_{\mp})^{4},
\label{b24}
\end{eqnarray}
and one can show that
\begin{equation}
\sum_{m}w_{m}=4.
\label{b25}
\end{equation}
Let us now define the {\it relative} weight of each frequency $(\omega-m\Omega)$ in the Fourier sum to be
\begin{equation}
\wp_{m}=\frac{W_{m}}{\sum_{m} W_{m}}=\frac{1}{4}w_{m};
\label{b26}
\end{equation}
therefore, the average frequency measured by the observer ${\cal O}'_{0}$ can be computed and the result is
\begin{equation}
\langle \omega'_{0} \rangle = \sum_{m}(\omega-m\Omega)\wp_{m}=\omega \mp 2\Omega \cos\theta.
\label{b27}
\end{equation}
These considerations can be given a proper physical interpretation based upon the eigenstates of a particle with spin $2\hbar$. That is, according to the representation
theory of the rotation group, the eigenstates of the particle with respect to the coordinate system $({\bf \hat{x}},{\bf \hat{n}},{\bf \hat{k}})$ can be transformed to the
$({\bf \hat{x}},{\bf \hat{y}},{\bf \hat{z}})$ system using the matrix $({\cal D}^{(j)}_{mm'})$ for $j=2$ \cite{Edmonds}. Each element of this matrix is given, up to a phase
factor, by $d^{(j)}_{mm'}(\theta)$ with $j=2$. This latter matrix can be expressed \cite{Edmonds} as
\begin{equation}
\left( \begin{array}{ccccc}
a^{2}_{+} & -b_{+} & b_{0} & -b_{-} & a^{2}_{-} \\
b_{+} & c_{+} & c_{0} & c_{-} & -b_{-} \\
b_{0} & -c_{0} & a_{0} & c_{0} & b_{0} \\
b_{-} & c_{-} & -c_{0} & c_{+} & -b_{+} \\
a^{2}_{-} & b_{-} & b_{0} & b_{+} & a^{2}_{+}
\end{array}\right),
\label{b28}
\end{equation}
where $a_{+}$ and $a_{-}$ are given by Eq. (\ref{b22}) and
\begin{equation}
a_{0}=\frac{1}{4}(1+3\cos2\theta), \,\,\ b_{0}=\frac{\sqrt{6}}{4}\sin^{2}\theta, \,\,\ b_{\pm}=-a_{\pm}\sin\theta, ~~~~~~~~~~~
\label{b29}
\end{equation}
\begin{equation}
c_{0}=\frac{\sqrt{6}}{4}\sin2\theta, \,\,\,\,\,\,\,\,\ c_{\pm}=\frac{1}{2}(\cos\theta \pm \cos2\theta).
\label{b30}
\end{equation}
Consider an incident graviton with definite helicity $\pm 2\hbar$ as in Eq. (\ref{b14}). The probability amplitude that the graviton has an angular momentum $m\hbar$ along the
$z$ axis, corresponding to a frequency $\omega'_{0}=\omega-m\Omega$ as measured by ${\cal O}'_{0}$, is given up to a phase factor by $\zeta_{m}$, where
\begin{equation}
(\zeta_{m})=\left(
\begin{array}{c}
a^{2}_{\pm} \\
\pm b_{\pm} \\
b_{0} \\
\pm b_{\mp} \\
a^{2}_{\mp}
\end{array}\right).
\label{b31}
\end{equation}
Here the upper (lower) sign refers to an initial incident state of positive (negative) helicity. Equation (\ref{b31}) is obtained from the first and last columns of the
matrix (\ref{b28}), since the state of the particle in the $({\bf \hat{x}},{\bf \hat{y}},{\bf \hat{z}})$ system is obtained from the application of the matrix (\ref{b28})
on the helicity states of the incident graviton. It follows that the probability that an incident graviton of helicity $\pm 2\hbar$ has spin $m\hbar$, $m=0,\pm 1,\pm 2$,
along the direction of rotation of the observer is given by $|\zeta_{m}|^{2}$, where
\begin{equation}
\wp_{m}=|\zeta_{m}|^{2}
\label{b32}
\end{equation}
based upon the comparison of Eqs. (\ref{b26}) and (\ref{b31}). Thus $\wp_{m}$ is indeed the probability that observer ${\cal O}'_{0}$ would measure
frequency $\omega - m\Omega$ and hence the average measured frequency is in fact $\langle \omega'_{0} \rangle$ given by Eq. (\ref{b27}).
Introducing the helicity vector ${\bf \hat{H}}=\pm {\bf \hat{k}}$, the average frequency measured by the observer can be written as
$\langle \omega'_{0} \rangle=\omega-2{\bf \hat{H}}\, .\, {\bf \Omega}$, which is another expression of helicity-rotation coupling. In view of Eq. (\ref{b12}), we
may interpret the expression for $\langle \omega'_{0} \rangle$ as follows: the rotation frequency of the observer ${\bf \Omega}=\Omega {\bf \hat{z}}$ may be decomposed
into a component of magnitude $\Omega \cos\theta$ parallel to the wave vector ${\bf k}$ and a component of magnitude $\Omega \sin\theta$ perpendicular to
${\bf k}$. On the average, the latter component does not contribute to the measured frequency; hence $\langle \omega'_{0} \rangle=\omega \mp 2(\Omega\cos\theta)$
in agreement with Eq. (\ref{b12}).
The average frequency measured by the observer is expected to be the same as the result that would be obtained in the JWKB regime in accordance with the quasi-classical
approximation. In the case under consideration, this corresponds to the high-frequency regime for gravitational waves; that is, waves with $\omega\gg\Omega$. To see how this
comes about explicitly, we follow the approach that has been developed for the electromagnetic case \cite{Mash1} and adapt it to the gravitational case under consideration
here. It is first necessary to define a triad $({\bf \hat{\alpha}},{\bf \hat{\beta}},{\bf \hat{k}})$ that can provide a more natural polarization basis for the rotating
observer ${\cal O}'_{0}$ such that ${\bf \hat{\alpha}}$ and ${\bf \hat{\beta}}$ remain ``fixed'' in the rotating frame as much as possible. These unit vectors are defined by
\begin{eqnarray}
{\bf \hat{\alpha}}&=&{\bf \hat{x}}\cos\Phi + {\bf \hat{n}}\sin\Phi, \label{b33} \\
{\bf \hat{\beta}}&=&-{\bf \hat{x}}\sin\Phi + {\bf \hat{n}}\cos\Phi,
\label{b34}
\end{eqnarray}
where $\Phi$ is given by
\begin{equation}
\sin\Phi=\frac{1}{D}\cos\theta\sin\Omega t, \,\,\,\,\ \cos\Phi=\frac{1}{D}\cos\Omega t
\label{b35}
\end{equation}
and $D>0$ can be obtained from
\begin{equation}
D^{2}=\cos^{2}\theta + \sin^{2}\theta \cos^{2}\Omega t.
\label{b36}
\end{equation}
We note that $\Phi$ reduces to $\Omega t$ for $\theta=0$ and to $-\Omega t$ for $\theta=\pi$, while $\Phi=0$ for $\theta=\pi/2$.
The orthonormal triad $({\bf \hat{\alpha}},{\bf \hat{\beta}},{\bf \hat{k}})$ is related to $({\bf \hat{x}},{\bf \hat{y}},{\bf \hat{z}})$ by a rotation; in fact, the
transformation that takes $({\bf \hat{x}},{\bf \hat{y}},{\bf \hat{z}})$ to $({\bf \hat{\alpha}},{\bf \hat{\beta}},{\bf \hat{k}})$ is given by
\begin{equation}
T=\left(
\begin{array}{ccc}
\cos\Phi & \cos\theta\sin\Phi & \sin\theta\sin\Phi \\
-\sin\Phi & \cos\theta\cos\Phi & \sin\theta\cos\Phi \\
0 & -\sin\theta & \cos\theta
\end{array}\right).
\label{b37}
\end{equation}
We wish to find the new polarization basis for linearly polarized gravitational waves based on the new triad; this can be achieved by a similarity transformation of the basis
given in Eq. (\ref{b6}), and the result is
\begin{equation}
\xi=T^{\dagger}\epsilon_{\oplus}T, \,\,\,\,\,\,\ \nu=T^{\dagger}\epsilon_{\otimes}T.
\label{b38}
\end{equation}
It is possible to connect this new basis with Eq. (\ref{b15}) as follows
\begin{eqnarray}
\xi&=&\tilde{\epsilon}_{\oplus} \cos2\Phi + \tilde{\epsilon}_{\otimes} \sin2\Phi, \label{b39} \\
\nu&=&-\tilde{\epsilon}_{\oplus} \sin2\Phi + \tilde{\epsilon}_{\otimes} \cos2\Phi.
\label{b40}
\end{eqnarray}
The rotation by $2\Phi$ takes a simple form for the circular polarization basis, namely,
\begin{equation}
\xi \pm i \nu = (\tilde{\epsilon}_{\oplus} \pm i\tilde{\epsilon}_{\otimes})e^{\mp 2i\Phi}.
\label{b41}
\end{equation}
Using this relation in Eq. (\ref{b14}), we have
\begin{equation}
\tilde{P}_{\pm} = (\xi \pm i\nu)\hat{h}({\bf k})e^{-i\omega t \pm 2i\Phi},
\label{b42}
\end{equation}
so that from Eq. (\ref{b17}),
\begin{equation}
\tilde{P}'_{\pm}=(\xi' \pm i\nu')\hat{h}({\bf k})e^{-i\omega t \pm 2i\Phi},
\label{b43}
\end{equation}
where the new polarization basis involving
\begin{equation}
\xi'=R_{\Omega}\xi R^{\dagger}_{\Omega}, \,\,\,\,\ \nu'=R_{\Omega}\nu R^{\dagger}_{\Omega},
\label{b44}
\end{equation}
is naturally adapted to the rotating observer in the sense that the temporal dependence in Eq. (\ref{b43}) has been transferred to the phase of the wave as much as possible.
Let us note that $\xi'$ and $\nu'$ are obtained from $\epsilon_{\oplus}$ and $\epsilon_{\otimes}$, respectively, by a unitary transformation involving the orthogonal matrix
$R_{\Omega}T^{\dagger}$.
According to Eq. (\ref{b43}), observer ${\cal O}'_{0}$ receives a circularly polarized wave with a phase $-\omega t \pm 2\Phi$. The frequency of the wave is defined to be
the negative gradient of the phase with respect to time; hence, $\omega'_{0}=\omega \mp 2\partial \Phi / \partial t$. It follows from Eq. (\ref{b36}) that
\begin{equation}
\frac{\partial \Phi}{\partial t}=\frac{\Omega \cos\theta}{D^{2}}.
\label{b45}
\end{equation}
In practice, the frequency determination would necessitate the reception of at least a few oscillations of the wave. During such a period of time $t$, assuming that
the observations begin at $t=0$, $\epsilon'=\Omega t \ll 1$ due to the fact that $\Omega \ll \omega$; moreover,
\begin{equation}
D^{-2}=1+\epsilon'^{2}\sin^{2}\theta + O(\epsilon'^{4})
\label{b46}
\end{equation}
by Eq. (\ref{b36}). Thus $\omega'_{0}=\omega \mp 2\Omega\cos\theta$ in the high-frequency regime. This result is equivalent to the intuitive expectation that from the
standpoint of the rotating observer, the spin of the radiation field should precess in the opposite sense. This circumstance can be restated in terms of the rotation
of the state of linear polarization of the gravitational wave as explained in Section IV.
Henceforward, we limit our considerations to the high-frequency regime $(\omega \gg \Omega)$. It is then possible to generalize the main result that we have obtained
for the fixed rotating observers ${\cal O}'_{0}$ to the case of arbitrary rotating observers and thereby determine the modifications in the Doppler effect and aberration
that are brought about by the helicity-rotation coupling. This is done in the next section.
\renewcommand{\theequation}{3.\arabic{equation}}
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\section*{III. MODIFIED DOPPLER AND ABERRATION FORMULAS FOR GRAVITATIONAL WAVES}
In a background global inertial frame, we first consider the class of noninertial observers that are at rest in the inertial frame but refer their measurements to rotating
axes as discussed in Section II. Each of these observers carries a tetrad frame of the form
\begin{eqnarray}
{\lambda}'^{\mu}_{(0)}&=&(1,0,0,0), \,\,\ {\lambda}'^{\mu}_{(1)}=(0,\cos\phi,\sin\phi,0), \label{c1} \\
{\lambda}'^{\mu}_{(2)}&=&(0,-\sin\phi,\cos\phi,0), \,\,\ {\lambda}'^{\mu}_{(3)}=(0,0,0,1), ~~~~~~~
\label{c2}
\end{eqnarray}
where $\phi=\Omega t$ as before. We are interested in the determination of frequency and wave vector of incident high-frequency $(\omega\gg\Omega)$ gravitational waves.
To this end, we focus attention on the noninertial observer ${\cal O}'_{0}$ at the origin of spatial coordinates and imagine the class of observers that are at the rest in
the rotating system of ${\cal O}'_{0}$, each with a tetrad of the form of Eqs. (\ref{b1})-(\ref{b4}). The gravitational field measured by this class of rotating observers
is given by $\tilde{{\cal R}}'={\cal L}\tilde{{\cal R}}{\cal L}^{\dagger}$. A complete Fourier analysis of $\tilde{{\cal R}}'$ in time and space is needed to determine the
frequency and wave-vector content of the incident radiation according to ${\cal O}'_{0}$. In the high-frequency regime $(\omega \gg \Omega)$, the measurements of
${\cal O}'_{0}$ can be restricted in space to the cylindrical domain of radius $\ll \Omega^{-1}$. The analysis of frequency determination for $\omega\gg\Omega$ has been
given in Section II and a corresponding analysis of the wave-vector determination will not be carried out here, since it is entirely analogous to the electromagnetic case
presented in detail in Ref. \cite{Hauck}. It follows from an analysis similar to the one given in Section 2 of Ref. \cite{Hauck} that ${\bf k}'_{0}={\bf k}_{0}$.
Thus we conclude that to lowest order
\begin{equation}
\omega'_{0}=\omega - s{\bf \hat{H}}\, .\, {\bf \Omega} , \,\,\,\ {\bf k}'_{0}={\bf k}
\label{c3}
\end{equation}
for ${\cal O}'_{0}$ in the high-frequency regime. The dispersion relation for ${\cal O}'_{0}$ is then $\omega'_{0}=k'_{0}\mp s{\bf \hat{k}}'_{0}\, . \, {\bf \Omega}$, where
$k'_{0}=|{\bf k}'_{0}|$. Here $s=1$ for electromagnetic waves and $s=2$ for gravitational waves. Indeed, these results hold for each member of the rotating class of
observers that are at rest and carry the tetrad frame $\lambda'^{\mu}_{(\alpha)}$ given by Eqs. (\ref{c1})-(\ref{c2}).
The generalization of Eq. (\ref{c3}) to the case of rotating observers that are not at rest in the background global inertial frame can be simply obtained from the
observation that at each event along the circular path of an observer ${\cal O}'$, its tetrad $\Lambda^{\mu}_{(\alpha)}$ is related to the tetrad $\lambda'^{\mu}_{(\alpha)}$
of the static rotating observer at that event by a Lorentz boost
\begin{eqnarray}
\Lambda^{\mu}_{(0)}=\gamma[\lambda'^{\mu}_{(0)}+v\lambda'^{\mu}_{(2)}], \,\,\ \Lambda^{\mu}_{(1)}=\lambda'^{\mu}_{(1)}, \label{c4} \\
\Lambda^{\mu}_{(2)}=\gamma[\lambda'^{\mu}_{(2)}+v\lambda'^{\mu}_{(0)}], \,\,\ \Lambda^{\mu}_{(3)}=\lambda'^{\mu}_{(3)}.
\label{c5}
\end{eqnarray}
It follows from Lorentz invariance that $(\omega',{\bf k}')$ for ${\cal O}'$ are related to $(\omega'_{0},{\bf k}'_{0})$ by the standard Doppler and aberration formulas; that
is
\begin{eqnarray}
\omega'&=&\gamma(\omega'_{0}-{\bf v}\, .\, {\bf k}'_{0}), \label{c6} \\
{\bf k}'&=&{\bf k}'_{0}+\frac{(\gamma-1)}{v^{2}}({\bf v}\, .\, {\bf k}'_{0}){\bf v}-\gamma\omega'_{0}{\bf v}.
\label{c7}
\end{eqnarray}
Substituting Eq. (\ref{c3}) in Eqs. (\ref{c6})-(\ref{c7}), we obtain the modified Doppler and aberration formulas for gravitational waves in the high-frequency regime
\begin{eqnarray}
\omega'&=&\gamma[(\omega-s{\bf \hat{H}}\, .\, {\bf \Omega})-{\bf v}\, .\, {\bf k}], \label{c8} \\
{\bf k}'&=&{\bf k}+\frac{(\gamma-1)}{v^{2}}({\bf v}\, .\, {\bf k}){\bf v}-\gamma(\omega-s{\bf \hat{H}}\, .\, {\bf \Omega}){\bf v}. ~~~~~~~
\label{c9}
\end{eqnarray}
Eq. (\ref{c8}) may be interpreted in terms of Eq. (\ref{b13}) in the eikonal approximation, namely,
\begin{equation}
\omega'=\gamma(\omega-{\bf j}\, .\, {\bf \Omega}), \,\,\,\ {\bf j}={\bf r}\times {\bf k}+s{\bf \hat{H}},
\label{c10}
\end{equation}
where $\hbar{\bf j}$ is the total angular momentum of the graviton $(s=2)$ or the photon $(s=1)$. The results of this section for $s=2$ are expected to be of importance
in experiments involving the reception of gravitational waves by antennas rotating with frequency $\Omega\ll\omega$. Clearly, current large-scale Earth-fixed
gravitational-wave antennas rotate with the frequency of rotation of the Earth.
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\section*{IV. HELICITY-GRAVITY COUPLING}
It is possible to employ Einstein's principle of equivalence in order to extend the helicity-rotation coupling discussed in previous sections to the propagation of
gravitational waves in a curved spacetime background. For this purpose, we consider a solution of the linearized gravitational field equations. We assume that this
background field is due to isolated gravitating sources that move slowly compared to the speed of light in vacuum. It is possible to describe such a background field
in terms of gravitoelectromagnetism (``GEM'') in close analogy with Maxwell's electrodynamics. GEM has been thoroughly reviewed in Ref. \cite{Mash3} and references cited
therein. In this approach to GEM, the metric of the curved spacetime background is of the form
\begin{equation}
-(1-2f)dt^{2}-4({\bf S}\, .\, d{\bf x})dt+(1+2f)\delta_{ij}dx^{i}dx^{j},
\label{d1}
\end{equation}
where in our convention $f$ and ${\bf S}$ are respectively the gravitoelectric
and gravitomagnetic potentials subject to a (``Lorentz'') gauge condition
\begin{equation}
\frac{\partial f}{\partial t}+{\bf \nabla}\, .\, (\frac{1}{2}{\bf S})=0.
\label{d2}
\end{equation}
These potentials are connected to sources via the linearized gravitational field equations. For the treatment in this section and in conformity with our general
linear approach, we assume that in the presence of an incident gravitational wave the deviation from the Minkowski spacetime is a linear superposition of the perturbations
due to the isolated sources and the incident wave.
The GEM fields are defined by
\begin{equation}
{\bf F}=-{\bf \nabla}f-\frac{1}{2}\frac{\partial {\bf S}}{\partial t}, \,\,\,\ {\bf B}={\bf \nabla}\times {\bf S},
\label{d3}
\end{equation}
in our special convention \cite{Mash3} that has been designed to provide the closest possible connection with the standard formulas of classical electrodynamics.
Eqs. (\ref{d2}) and (\ref{d3}) together with the linearized gravitational field equations lead to the Maxwell equations for the GEM fields \cite{Mash3}.
It follows from a detailed discussion of Einstein's principle of equivalence within the context of GEM that the gravitoelectric field is in effect locally equivalent to
a translationally accelerated system, while the gravitomagnetic field is in effect locally equivalent to a rotating system \cite{Mash3}. Traditionally, Einstein's heuristic
principle of equivalence refers to the accelerated ``elevator'' in relation with the gravitoelectric field of the source; however, the rotation of the elevator is in general
necessary as well to take due account of the corresponding gravitomagnetic field.
In keeping with the electromagnetic analogy, this application of Einstein's principle of equivalence is the content of the gravitational Larmor theorem \cite{Larmor, Mas}. It
turns out that a spinning particle at rest in the exterior field of a rotating mass precesses with frequency ${\bf B}$, which is equivalent to what would be observed from a
local frame of reference rotating with frequency ${\bf \Omega}=-{\bf B}$. Following this general line of thought, we may conclude from the results of Section III that for
observers at rest in the exterior GEM background, the local dispersion relation
\begin{equation}
\omega=k \pm s{\bf \hat{k}}\, .\, {\bf B}
\label{d4}
\end{equation}
is approximately valid for high-frequency incident gravitational ($s=2$) or electromagnetic ($s=1$) waves. Eq. (\ref{d4}) follows from Eq. (\ref{c3}) with
${\bf \Omega} \rightarrow-{\bf B}$ in accordance with the gravitational Larmor theorem. Recognizing that the {\it local} dispersion relation (\ref{d4}) ignores the usual
global GEM effects, such as the bending of the incident beam of radiation, one may nevertheless employ Eq. (\ref{d4}) globally in order to uncover the specific consequences
of helicity-gravity coupling. The results may then be superposed on the standard GEM effects in line with our general linear perturbative approach.
An interesting consequence of the coupling of helicity with the gravitomagnetic field is the rotation of the state of linear polarization of a gravitational wave that
propagates in the field of a rotating mass. To illustrate this effect, we assume that the background field is {\it stationary} and is due to a rotating astronomical source.
The GEM fields have been determined in this case \cite{Tey}.The curl of the gravitomagnetic field ${\bf B}$ vanishes in the exterior of the source; hence,
\begin{equation}
{\bf B}=-{\bf \nabla}Q,
\label{d5}
\end{equation}
where $Q$ is the gravitomagnetic scalar potential. Far from the source
\begin{eqnarray}
f &\sim& \frac{GM}{r}, \,\,\,\ {\bf S} \sim \frac{G {\bf J}\times {\bf r}}{r^{3}},\label{d6}\\
Q &\sim& \frac{G {\bf J}\, .\, {\bf r}}{r^{3}}, \,\,\,\,\ {\bf B} \sim \frac{GJ}{r^{3}}[3({\bf \hat{J}}\, .\, {\bf \hat{r})}{\bf \hat{r}}-{\bf \hat{J}}],
\label{d7}
\end{eqnarray}
where $M$ is the mass and ${\bf J}=J{\bf \hat{z}}$ is the angular momentum of the source.
Consider a linearly polarized gravitational wave starting at $z=z_{0}$ far from the source and propagating outward along its rotation axis. Let $\Pi^{\mu\nu}_{1}$ and
$\Pi^{\mu\nu}_{2}$ be the linear polarization tensors for the wave. Assuming that at $z=z_{0}$ the state of the wave is given by the real part of
$\psi^{\mu\nu}=\hat{\psi}\Pi^{\mu\nu}_{1}{\rm exp}(-i\omega t)$, where $\hat{\psi}, |\hat{\psi}| \ll 1$, is a constant amplitude, then for any $z$
\begin{equation}
\psi^{\mu\nu}=\frac{1}{2}\hat{\psi}[(\Pi^{\mu\nu}_{1}+i\Pi^{\mu\nu}_{2})e^{i{\cal S}_{+}-i\omega t}+(\Pi^{\mu\nu}_{1}-i\Pi^{\mu\nu}_{2})e^{i{\cal S}_{-}-i\omega t}].
\label{d8}
\end{equation}
Here ${\cal S}$ is given by
\begin{equation}
{\cal S}_{\pm}=\int^{z}_{z_{0}}k_{\pm}dz,
\label{d9}
\end{equation}
where $k_{+}$ and $k_{-}$ are the wave numbers of the positive and negative-helicity components of the gravitational wave. Expressing ${\cal S}_{\pm}$ as
\begin{equation}
{\cal S}_{+}={\cal S}_{0}-\Delta, \,\,\,\ {\cal S}_{-}={\cal S}_{0}+\Delta,
\label{d10}
\end{equation}
we find that $\psi^{\mu\nu}$ can be written as
\begin{equation}
\psi^{\mu\nu}=\hat{\psi}(\Pi^{\mu\nu}_{1}\cos\Delta + \Pi^{\mu\nu}_{2}\sin\Delta)e^{i{\cal S}_{0}-i\omega t}.
\label{d11}
\end{equation}
Inspection of this equation reveals that as the wave propagates, the linear polarization state rotates by an angle $\Delta$ given by
\begin{equation}
\Delta=\frac{1}{2}\int^{z}_{z_{0}}(k_{-}-k_{+})dz.
\label{d12}
\end{equation}
To compute this angle, we write Eq. (\ref{d4}) for the positive and negative helicity components of the wave,
\begin{equation}
\omega=k_{+}+sB_{z}, \,\,\,\,\ \omega=k_{-}-sB_{z},
\label{d13}
\end{equation}
so that we find
\begin{equation}
\Delta=s\int^{z}_{z_{0}}B_{z}dz=s[Q(z_{0})-Q(z)].
\label{d14}
\end{equation}
Using Eq. (\ref{d7}), we finally have
\begin{equation}
\Delta=sGJ(\frac{1}{z^{2}_{0}}-\frac{1}{z^{2}}).
\label{d15}
\end{equation}
For electromagnetic radiation $(s=1)$, this gravitomagnetic rotation of the plane of polarization was first studied by Skrotskii \cite{Skr}; detailed discussions and
references are contained in Ref. \cite{Kop} and references cited therein. The angle of rotation for gravitational radiation $(s=2)$ is twice the Skrotskii angle.
Let us note that $\Delta$ vanishes for waves propagating from $-\infty$ to $+\infty$ along the $z$ axis; this is an instance of a general result discussed in Appendix
A, where the consequences of Eq. (\ref{d4}) are worked out in a more general context.
The angle of rotation of the state of linear polarization $\Delta$ is independent of the frequency (or wavelength) of the radiation; therefore, Eq. (\ref{d15}) is
valid in the limit of vanishing wavelength, namely, the JWKB (or eikonal) limit. In this limit, gravitational waves propagate along a null geodesic; in this case, a
full treatment is contained in the next section.
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\section*{V. ROTATION OF LINEAR POLARIZATION}
The purpose of this section is to compute (within the eikonal approximation scheme) the rate of rotation of the state of linear polarization of high-frequency gravitational
waves propagating in the exterior background field of a rotating astronomical source.
Consider the propagation of gravitational radiation on a background spacetime such that the waves cause a small perturbation. Let $\bar{g}_{\mu\nu}(x)$ be the metric tensor
of the background field in a given coordinate system and $g_{\mu\nu}=\bar{g}_{\mu\nu}+h_{\mu\nu}$ be the spacetime metric tensor. Under an infinitesimal coordinate
transformation $x'^{\mu}=x^{\mu}-\epsilon^{\mu}(x)$,
\begin{equation}
h'_{\mu\nu}(x)=h_{\mu\nu}(x)+\epsilon_{\mu | \nu}+\epsilon_{\nu | \mu},
\label{e1}
\end{equation}
where the vertical bar denotes covariant differentation with respect to $\bar{g}_{\mu\nu}$. We also raise and lower indices, etc., with $\bar{g}_{\mu\nu}$. Thus the
perturbation is determined up to a gauge transformation given by Eq. (\ref{e1}). Introducing the trace-reversed potential
\begin{equation}
\psi_{\mu\nu}=h_{\mu\nu}-\frac{1}{2}\bar{g}_{\mu\nu}\bar{g}^{\rho\sigma}h_{\rho\sigma},
\label{e2}
\end{equation}
we find that under a gauge transformation
\begin{equation}
\psi'_{\mu\nu}=\psi_{\mu\nu}+\epsilon_{\mu | \nu}+\epsilon_{\nu | \mu}-\bar{g}_{\mu\nu}\, \epsilon^{\sigma}_{\,\ | \sigma}.
\label{e3}
\end{equation}
It is convenient to impose the transverse gauge condition
\begin{equation}
\psi^{\mu\nu}_{\,\,\,\,\ | \nu}=0,
\label{e4}
\end{equation}
which does not fix the gauge completely, however. It turns out that any solution of
\begin{equation}
\epsilon^{\mu \,\,\ \nu}_{\,\ | \nu}+\bar{R}^{\mu}_{\,\ \sigma}\epsilon^{\sigma}=0
\label{e5}
\end{equation}
in Eq. (\ref{e3}) leads to $\psi'^{\mu\nu}_{\,\,\,\,\,\,\ | \nu}=0$ if Eq. (\ref{e4}) is assumed. In the transverse gauge, the gravitational field equations imply that
$\psi_{\mu\nu}$ satisfies the wave equation \cite{Eisenhart}
\begin{equation}
\psi^{\,\,\,\,\,\,\,\,\,\,\ \sigma}_{\mu\nu | \sigma}+2\bar{R}_{\mu\rho\nu\sigma}\psi^{\rho\sigma}=0,
\label{e6}
\end{equation}
where $\bar{g}_{\mu\nu}$ is assumed to be Ricci-flat in the spacetime region under consideration here.
To describe the propagation of the wave function $\psi_{\mu\nu}$ in the eikonal approximation, we seek a solution of Eq. (\ref{e6}) in the form
\begin{equation}
\psi_{\mu\nu}={\rm Re}\{\tilde{\chi}_{\mu\nu}(x;\epsilon)e^{i \epsilon^{-1}\sigma(x)}\},
\label{e7}
\end{equation}
where $\epsilon$, $0<\epsilon \ll 1$, is directly proportional to the wavelength of the radiation. In the eikonal
approximation, $\tilde{\chi}_{\mu\nu}(x;\epsilon)$ is expressed as an asymptotic series in powers of $\epsilon$
\begin{equation}
\tilde{\chi}_{\mu\nu}(x;\epsilon)=\chi_{\mu\nu}(x)+\epsilon \rho_{\mu\nu}(x)+\epsilon^{2}\kappa_{\mu\nu}(x)+....
\label{e8}
\end{equation}
Let $k_{\mu}=\partial \sigma(x) / \partial x^{\mu}$ be the propagation vector of the wave; then, the substitution of equations (\ref{e7}) and (\ref{e8})
in the gauge condition (\ref{e4}) and propagation equation (\ref{e6}) results in series that contain powers of $1/ \epsilon$ in addition to powers of $\epsilon$.
It follows from Eq. (\ref{e4}) that there is only one such term involving $1/ \epsilon$ and in the eikonal limit $(\epsilon \rightarrow 0)$, the coefficient
of this term must vanish; therefore,
\begin{equation}
\chi_{\mu\nu}k^{\nu}=0.
\label{e9}
\end{equation}
Moreover, the propagation equation (\ref{e6}) involves $1/ \epsilon^{2}$ and $1/ \epsilon$ terms
and the coefficients of these terms must also vanish in the eikonal limit, hence we have respectively
\begin{equation}
k_{\mu}k^{\mu}=0,
\label{e10}
\end{equation}
and
\begin{equation}
i(2\chi_{\mu\nu | \sigma}k^{\sigma}+\chi_{\mu\nu}k^{\sigma}_{\,\ | \sigma})-k^{\sigma}k_{\sigma}\rho_{\mu\nu}=0.
\label{e11}
\end{equation}
Let us first note that equation (\ref{e10}) implies that the radiation follows a null geodesic in the eikonal limit. It follows from
$k_{\mu}=\partial \sigma(x) / \partial x^{\mu}$ that $k_{\mu | \nu}=k_{\nu | \mu}$. Thus, taking covariant derivative of Eq. (\ref{e10}),
we get that $k_{\mu | \nu}k^{\mu}=0$; hence,
\begin{equation}
k_{\nu | \mu}k^{\mu}=0.
\label{e12}
\end{equation}
The geodesic equation follows from Eq. (\ref{e12}) and $k^{\mu}=dx^{\mu}/ d \lambda$, where $\lambda$ is an affine parameter along the path.
Equations (\ref{e9}) and (\ref{e11}) describe the propagation of the wave amplitude $\chi_{\mu\nu}$ along the null geodesic, since it follows from
Eq. (\ref{e11}) that
\begin{equation}
2\chi_{\mu\nu | \sigma}k^{\sigma}+\chi_{\mu\nu}k^{\sigma}_{\,\ | \sigma}=0.
\label{e13}
\end{equation}
An immediate consequence of this relation is that $\Sigma^{0}=\chi^{*}_{\mu\nu}\chi^{\mu\nu}$ satisfies the conservation law
\begin{equation}
(\Sigma^{0}k^{\sigma})_{ | \sigma}=0,
\label{e14}
\end{equation}
which can be interpreted as the conservation of the ``graviton'' number
along the null geodesic congruence. These results have been based on terms involving $1/ \epsilon$ and $1/ \epsilon^{2}$; taking account of the other
terms in the eikonal series, i.e., those involving $\epsilon^{n}, n=0,1,2,...$, would simply specify the manner in which the general wave amplitude
$\tilde{\chi}_{\mu\nu}(x;\epsilon)$ varies along the null geodesic. This eikonal (or JWKB) treatment of gravitational radiation has been previously considered in
Ref. \cite{Isaacson}. A critical assessment of the eikonal approximation scheme is contained in Ref. \cite{Mash4}.
In the eikonal approximation scheme, the curves $x^{\mu}=x^{\mu}(\lambda)$ that have $k^{\mu}=d x^{\mu}/ d \lambda$ as tangent vectors are null geodesics orthogonal to
the surfaces of constant phase $\sigma$. Imagine a bundle of such null rays in a congruence characterized by the propagation vector $k^{\mu}$. Let us define
a null tetrad system $(k^{\mu},l^{\mu}, n^{\mu}, n^{* \mu})$ such that $k^{\mu}l_{\mu}=-1$ and $n^{\mu}n^{*}_{\mu}=1$ are the only nonvanishing scalar products among the
four null vectors. Starting from an observer's orthonormal tetrad frame $\lambda^{\mu}_{(\alpha)}$, the null frame is constructed as follows:
\begin{eqnarray}
k^{\mu}=\frac{a}{\sqrt{2}}[\lambda^{\mu}_{(0)}+\lambda^{\mu}_{(3)}] , \,\ l^{\mu}=\frac{1}{a\sqrt{2}}[\lambda^{\mu}_{(0)}-\lambda^{\mu}_{(3)}] ~~~~~~~~~
\label{e15}
\end{eqnarray}
are real, while $n^{\mu}$ and its complex conjugate $n^{* \mu}$ are complex, since $n^{\mu}$ is defined by
\begin{equation}
n^{\mu}=\frac{1}{\sqrt{2}}[\lambda^{\mu}_{(1)}+i\lambda^{\mu}_{(2)}].
\label{e16}
\end{equation}
Here $a=-\sqrt{2}\, k_{\mu}\lambda^{\mu}_{(0)}$ is a nonzero constant. The tetrad system is assumed to be parallel propagated along the congruence.
The symmetric and transverse tensor $\chi_{\mu\nu}$ can be locally expressed in terms of the parallel-propagated null tetrad as
\begin{equation}
\chi_{\mu\nu}=\Phi_{+}n_{\mu}n_{\nu}+\Phi_{-}n^{*}_{\mu}n^{*}_{\nu}+k_{\mu}\Gamma_{\nu}+k_{\nu}\Gamma_{\mu}.
\label{e17}
\end{equation}
It turns out that in this expansion $n_{\mu}n_{\nu}$ corresponds to a positive helicity wave and $n^{*}_{\mu}n^{*}_{\nu}$ corresponds to a negative
helicity wave; thus, $\chi_{\mu\nu}$ consists in general of a positive helicity part with amplitude $\Phi_{+}$, a negative helicity part with amplitude $\Phi_{-}$
and a gauge part involving a transverse vector $\Gamma_{\mu}$ such $k^{\mu}\Gamma_{\mu}=0$ (see the appendix of Ref. \cite{Mash5}).
We note that $\langle \psi^{\mu\nu}\psi_{\mu\nu}\rangle=\Sigma^{0}/2$ and
\begin{equation}
\Sigma^{0}=\chi^{\mu\nu}\chi^{*}_{\mu\nu}=|\Phi_{+}|^{2}+|\Phi_{-}|^{2},
\label{e18}
\end{equation}
so that the intensity of the wave depends only on its {\it irreducible} part
\begin{equation}
\hat{\psi}_{\mu\nu}=\Phi_{+}n_{\mu}n_{\nu}+\Phi_{-}n^{*}_{\mu}n^{*}_{\nu}.
\label{e19}
\end{equation}
Using this irreducible part of $\chi_{\mu\nu}$, we find from Eq. (\ref{e13}) that
\begin{equation}
\frac{d \Phi_{+}}{d\lambda}+\hat{\theta}\, \Phi_{+}=0, \,\,\ \frac{d \Phi_{-}}{d\lambda}+\hat{\theta}\, \Phi_{-}=0,
\label{e20}
\end{equation}
where $\hat{\theta}=(1/2)k^{\sigma}_{\,\ | \sigma}$ is the {\it expansion} of the null congruence $k^{\mu}$. Let $A$ be the area of the cross section of a bundle of
null rays in the congruence; then,
\begin{equation}
\frac{d A}{d\lambda}=2\hat{\theta} A.
\label{e21}
\end{equation}
It follows that $|\Phi_{+}|^{2} A$ and $|\Phi_{-}|^{2} A$ are separately conserved along the trajectory; that is,
the number of positive or negative helicity null rays (``gravitons'') is independently conserved along the congruence. This approach to the polarization of
gravitational waves in the eikonal limit demonstrates that the parallel transport of polarization tensors $n_{\mu}n_{\nu}$ and $n^{*}_{\mu}n^{*}_{\nu}$ for
positive and negative helicities, respectively, is a natural interpretation of Eq. (\ref{e13}). This circumstance is a spin-2 analog of the spin-1 electromagnetic case, where
the irreducible part of the wave amplitude corresponding to the vector potential is of the form $\phi_{+}n_{\mu}+\phi_{-}n^{*}_{\mu}$. Furthermore,
it is possible to define Stokes parameters $\Sigma^{\alpha}$ for gravitational radiation such that $\eta_{\alpha \beta}\Sigma^{\alpha}\Sigma^{\beta}=0$ and
\begin{equation}
\Sigma^{1}=\Phi_{+}\Phi^{*}_{-}+\Phi_{-}\Phi^{*}_{+}, \,\,\ \Sigma^{2}=-i(\Phi_{+}\Phi^{*}_{-}-\Phi_{-}\Phi^{*}_{+}),
\label{a22}
\end{equation}
\begin{equation}
\Sigma^{3}=|\Phi_{+}|^{2}-|\Phi_{-}|^{2},
\label{e23}
\end{equation}
along the lines developed in Ref. \cite{Mash5}.
It is important to discuss the uniqueness of the representation (\ref{e17}). At any given event, the observer tetrad frame $\lambda^{\mu}_{(\alpha)}$
is unique up to a Lorentz transformation. We are interested in a subgroup of the Lorentz group that preserves $k^{\mu}$; in fact, this is the little group of
$k^{\mu}$ that is isomorphic to the Euclidean group in the plane. Therefore, under the action of the little group, $k'^{\mu}=k^{\mu}$,
\begin{eqnarray}
l^{' \mu}&=&l^{\mu}+b\, n^{\mu}+b^{*}n^{* \mu}+|b|^{2}k^{\mu}, \label{e24} \\
n^{' \mu}&=&e^{-i\Theta}(n^{\mu}+b^{*}k^{\mu}).
\label{e25}
\end{eqnarray}
Here $\Theta$ is real and corresponds to the constant angle of rotation in the $(\lambda^{\mu}_{(1)},\lambda^{\mu}_{(2)})$ plane. For $\Theta=0$, Eqs. (\ref{e24}) and
(\ref{e25}) with a complex constant $b$ correspond to the Abelian subgroup of the little group under which
\begin{equation}
\hat{\psi}'_{\mu\nu}=\hat{\psi}_{\mu\nu}+k_{\mu}G_{\nu}+k_{\nu}G_{\mu},
\label{e26}
\end{equation}
where $k^{\mu}G_{\mu}=0$ and $G_{\mu}$ is given by
\begin{equation}
G_{\mu}=\Phi_{+}b^{*}n_{\mu}+\Phi_{-}b\, n^{*}_{\mu}+\frac{1}{2}(\Phi_{+}b^{* 2}+\Phi_{-}b^{2})k_{\mu}.
\label{e27}
\end{equation}
Equation (\ref{e26}) amounts to a gauge transformation of $\hat{\psi}_{\mu\nu}$; therefore, under this gauge subgroup the Stokes parameters for the
radiation field remain invariant. On the other hand, with $b=0$
and under rotation of angle $\Theta$, $\hat{\psi}_{\mu\nu}$ remains invariant if
\begin{equation}
\Phi'_{+}=e^{2i\Theta}\Phi_{+}, \,\,\ \Phi'_{-}=e^{-2i\Theta}\Phi_{-}
\label{e28}
\end{equation}
which demonstrates that $\Phi_{+}$ ($\Phi_{-}$) is the amplitude of the graviton in the positive (negative) helicity state. Moreover, the Stokes parameters
undergo a rotation as well, since a rotation of the angle $\Theta$ in the ($\lambda^{\mu}_{(1)},\lambda^{\mu}_{(2)}$) plane induces a rotation of angle $-4\Theta$ in
the ($\Sigma^{1},\Sigma^{2}$) plane \cite{Mash5}.
It proves useful to define real linear polarization tensors $\Pi^{\mu\nu}_{1}$ and $\Pi^{\mu\nu}_{2}$ such that
\begin{equation}
n^{\mu}n^{\nu}=\frac{1}{\sqrt{2}}(\Pi^{\mu\nu}_{1}+i\Pi^{\mu\nu}_{2}).
\label{e29}
\end{equation}
Here $\Pi_{1}$ and $\Pi_{2}$ are independent polarization states such that $\hat{\psi}^{\mu\nu}$ can be written as
\begin{equation}
\hat{\psi}^{\mu\nu}=L_{1}\Pi^{\mu\nu}_{1}+L_{2}\Pi^{\mu\nu}_{2},
\label{e30}
\end{equation}
where $L_{1}$ and $L_{2}$ are the linear polarization amplitudes. Under a constant rotation of angle $\Theta$ in a plane perpendicular to the spatial direction of
propagation of the wave,
\begin{eqnarray}
\Pi'^{\mu\nu}_{1}&=&\Pi^{\mu\nu}_{1}\cos2\Theta + \Pi^{\mu\nu}_{2}\sin2\Theta, \label{e31} \\
\Pi'^{\mu\nu}_{2}&=&-\Pi^{\mu\nu}_{1}\sin2\Theta + \Pi^{\mu\nu}_{2}\cos2\Theta,
\label{e32}
\end{eqnarray}
so that one linear polarization state turns into another under a rotation of $\Theta=\pi/4$. In general, $\Theta=\pi/(2s)$ is the angle ``between'' the linear polarization
states $\Pi_{1}$ and $\Pi_{2}$.
The constant parameter $a$ in the definition of $k^{\mu}$ and $l^{\mu}$ in Eq. (\ref{e15}) is related to the choice of the affine parameter $\lambda$, which is defined
up to a linear transformation $\lambda \rightarrow \lambda'={\rm constant}+\lambda/A_{0}$ where $A_{0}\neq 0$ is a constant. Under this affine transformation
$\lambda^{\mu}_{(\alpha)}$ is unchanged, but $a \rightarrow aA_{0}$ and hence $k^{\mu}
\rightarrow A_{0}k^{\mu}$, $l^{\mu}\rightarrow A^{-1}_{0}l^{\mu}$ and $n^{\mu}$ is unchanged. A null rotation of the null tetrad is defined to be a 4-parameter
group that consists of the combined action of the little group together with an affine transformation. It is the most general transformation that leaves the
{\it spatial direction} of wave propagation vector $k^{\mu}$ invariant.
It is important to note that under a rotation with angle $\Theta$, the linear polarization states of a massless spin-1 field would rotate by $\Theta$, while that of a
spin-2 field would rotate by $2\Theta$. It has been shown in Ref. \cite{Kop} that in a general gravitomagnetic field ${\bf B}$, the plane of linear polarization of
electromagnetic radiation rotates by an angle
\begin{equation}
\alpha_{Skrotskii}=\int {\bf B}\, .\, d{\bf x},
\label{e33}
\end{equation}
where the integral is evaluated along the spatial path of the null ray; see Section VII of Ref. \cite{Kop} for a detailed derivation. The Skrotskii effect \cite{Skr}
is the gravitomagnetic analog of the Faraday effect. Therefore, the state of linear polarization of gravitational radiation would rotate by an angle $2\alpha_{Skrotskii}$
in a general gravitomagnetic field. Thus, we recover, in the eikonal limit, the result of Section IV.
\section*{VI. DISCUSSION}
It is useful to provide estimates of the spin-rotation-gravity coupling effects presented in this work. For Earth-based gravitational-wave antennas that rotate with the
Earth, the effective rotation frequency is therefore about $10^{-5} {\rm Hz}$ in Eqs. (\ref{c8})-(\ref{c9}), so that the incident gravitational waves that would be
relevant in this case satisfy the high-frequency condition, namely, $\omega\gg\Omega$. Ignoring the helicity-rotation coupling would introduce a small systematic Doppler
bias of magnitude $2\Omega/ \omega$.
Let us next consider, as in Section IV, a linearly polarized gravitational wave that propagates outward to infinity starting from the north pole of a rotating astronomical
system of radius $R_{0}$. We assume that $\omega \gg B_{0}$, where $B_{0}=2GJ/R^{3}_{0}$ is the gravitomagnetic (Larmor) frequency of the system.
The rotation of the linear polarization state of the wave is in the same sense as the rotation of the source and the net angle of rotation is given by Eq. (\ref{d15}), namely,
$\Delta=2GJ/R^{2}_{0}$. For a homogeneous sphere of mass $M$ rotating with frequency $\Omega$, $\Delta=0.8GM\Omega$. This amounts to
$\Delta \approx 0.025 \, {\rm rad}$ (or about $1.5^{\circ})$ for a millisecond pulsar of mass $M \approx M_{\odot}$ and rotational period $\approx 10^{-3} \, {\rm s}$.
For the Earth, however, the corresponding result would be negligibly small, that is, $\Delta \approx 10^{-15} \, {\rm rad}$.
\renewcommand{\theequation}{A\arabic{equation}}
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\section*{APPENDIX A}
The purpose of this appendix is to work out {\it to first order in the helicity-gravity coupling} the solution of Hamilton's equations for the dispersion relation
(\ref{d4}). The Hamiltonian for the ray motion is given in this case by
\begin{equation}
{\cal H}({\bf r},{\bf k})=k \pm s{\bf \hat{k}}\, . \, {\bf B}({\bf r}).
\label{a1}
\end{equation}
Hamilton's equations are
\begin{eqnarray}
\frac{d{\bf r}}{dt}&=&{\bf v}_{g}, \label{a2} \\
\frac{d{\bf k}}{dt}&=&-{\bf \nabla}[\pm s{\bf \hat{k}}\, . \, {\bf B}({\bf r})],
\label{a3}
\end{eqnarray}
where
\begin{equation}
{\bf v}_{g}=\frac{\partial \omega}{\partial {\bf k}}=\frac{1}{\omega}({\bf k} \pm s{\bf B})
\label{a4}
\end{equation}
is the group velocity of the rays to first order in the coupling to the gravitomagnetic field. The background is stationary; therefore, for any ray that is a solution
of equations (\ref{a2}) and (\ref{a3}), $\omega={\cal H}({\bf r},{\bf k})$ is a constant of the motion.
Let $({\bf r}_{+},{\bf k}_{+})$ denote the solution of the equations of motion for a positive-helicity ray and $({\bf r}_{-},{\bf k}_{-})$ denote the corresponding solution
for a negative-helicity ray. Working to first order in the helicity-gravity coupling, we let
\begin{eqnarray}
{\bf k}_{+}&=&{\bf k}_{0}+{\bf \kappa}(t), \,\,\,\,\ {\bf k}_{-}={\bf k}_{0}-{\bf \kappa}(t), \label{a5} \\
{\bf r}_{+}&=&{\bf r}_{0}+{\bf q}(t), \,\,\,\,\ {\bf r}_{-}={\bf r}_{0}-{\bf q}(t),
\label{a6}
\end{eqnarray}
where ${\bf k}_{0}$ is constant, $\omega=|{\bf k}_{0}|$ and
\begin{equation}
\frac{d{\bf r}_{0}}{dt}={\bf \hat{k}}_{0}
\label{a7}
\end{equation}
is the equation of motion of the unperturbed ray in the absence of spin-gravity coupling. Substituting Eqs. (\ref{a5})-(\ref{a7}) in the equations of motion (\ref{a2})
- (\ref{a3}), we find
\begin{eqnarray}
\frac{d{\bf q}}{dt}&=&\frac{1}{\omega}({\bf \kappa}+s{\bf B}), \label{a8} \\
\frac{d{\bf \kappa}}{dt}&=&-s{\bf \nabla}({\bf \hat{k}}_{0}\, . \, {\bf B}).
\label{a9}
\end{eqnarray}
Using the fact that ${\bf B}=-{\bf \nabla}Q$, we can write ${\bf \nabla}({\bf \hat{k}}_{0}\, . \, {\bf B})=({\bf \hat{k}}_{0}\, . \, {\bf \nabla}){\bf B}$, which can be
expressed as $d{\bf B}/dt$ via Eq. (\ref{a7}) in our approximation scheme. Hence it follows from Eq. (\ref{a9}) that ${\bf \kappa}+s{\bf B}$ is a constant of the motion.
This implies that the right-hand side of Eq. (\ref{a8}) is a constant as well; therefore,
\begin{equation}
{\bf \kappa} + s{\bf B}=\omega{\bf V}_{0},
\label{a10}
\end{equation}
where ${\bf V}_{0}=d{\bf q}/dt$ is a constant and ${\bf B}$ is evaluated along the average (unpolarized) ray. The requirement that $\omega$ be the same for both rays can be
implemented using Eqs. (\ref{a1}) and (\ref{a5}) and the result is
\begin{equation}
{\bf \hat{k}}_{0}\,\ . \, ({\bf \kappa}+s{\bf B})=0,
\label{a11}
\end{equation}
so that ${\bf v}_{g}\, . \, {\bf \hat{k}}_{0}=1$. This completes the solution of the equations of motion.
It follows from ${\bf q}(t)={\bf q}(0)+{\bf V}_{0}t$ and Eq. (\ref{a6}) that the positive and negative helicity rays diverge away from the path of the average (unpolarized)
radiation, since ${\bf \hat{k}}_{0}\, . \, {\bf V}_{0}=0$. The integration constant ${\bf V}_{0}$ must be determined from the boundary conditions. Consider, for instance, the
emission of rays of radiation that originate at $z=z_{0}$ on the axis of rotation of an astronomical mass. Let the initial direction of propagation of the radiation be normal
to the $z$ axis. Then, it follows from these initial data that ${\bf V}_{0}=2sG{\bf J}/(\omega z^{3}_{0})$. Thus there will be a differential deflection of the radiation such
that the positive and negative helicity rays separate at a constant rate about the average direction of propagation ${\bf \hat{k}}_{0}$. The total gravitomagnetic splitting angle between the rays would be $2V_{0}=4sGJ/(\omega z^{3}_{0})$, which amounts to a few degrees for gravitational waves of frequency $10^{3}\, {\rm Hz}$ grazing the north pole of a millisecond pulsar.
We emphasize that the polarization-dependent differential deflection of rays will occur, in our treatment, only for radiation that originates near a rotating source. The gravitomagnetic field ${\bf B}$ rapidly falls off to zero far from the source; therefore, there will be no differential deflection of rays in scattering situations. This conclusion agrees in the electromagnetic case $(s=1)$ with the results of recent investigations \cite{Gua}.
One can use the results of this appendix to estimate the polarization-dependent time delay in the arrival of rays originating near a rotating astronomical source \cite{Mas}. Moreover, one can generalize the treatment of the rotation of linear polarization given in Section IV. In fact, it follows from Eqs. (\ref{a5}), (\ref{a10}) and (\ref{a11}) that
\begin{equation}
\frac{1}{2}({\bf k}_{-}-{\bf k}_{+})\, . \, {\bf \hat{k}}_{0}=s{\bf \hat{k}}_{0}\, . \, {\bf B},
\label{a12}
\end{equation}
where ${\bf \hat{k}}_{0}\, . \, {\bf B}=-dQ/dt$ along the unperturbed ray. Therefore, we find, following a method analogous to that used in Section IV, that
the angle of rotation of linear polarization is
\begin{equation}
\Delta = s[Q({\bf r}_{i})-Q({\bf r}_{f})]
\label{a13}
\end{equation}
for a linearly polarized ray of radiation traveling from ${\bf r}_{i}$ to ${\bf r}_{f}$. Thus for rays that are incident from infinity and travel to infinity, the net angle of rotation of linear polarization vanishes.
Finally, we should mention that one can think of the effects discussed here in terms of the helicity dependence of the index of refraction of an effective medium for the
propagation of rays. From the definition
\begin{equation}
|{\bf k}_{\pm}|=\omega n_{\pm},
\label{a14}
\end{equation}
we find that
\begin{equation}
n_{\pm}=n_{0}\mp \frac{s}{\omega}{\bf \hat{k}}\, . \, {\bf B},
\label{a15}
\end{equation}
where $n_{0}=1$, since we have neglected here the polarization-independent bending of rays given by $n_{0} \approx 1+2f$.
|
Title:
Avoiding Dark Energy with 1/R Modifications of Gravity |
Abstract: Scalar quintessence seems epicyclic because one can choose the potential to
reproduce any cosmology (I review the construction) and because the properties
of this scalar seem to raise more questions than they answer. This is why there
has been so much recent interest in modified gravity. I review the powerful
theorem of Ostrogradski which demonstrates that the only potentially stable,
local modification of general relativity is to make the Lagrangian an arbitrary
function of the Ricci scalar. Such a theory can certainly reproduce the current
phase of cosmic acceleration without Dark Energy. However, this explanation
again seems epicyclic in that one can construct a function of the Ricci scalar
to support any cosmology (I give the technique). Models of this form are also
liable to problems in the way they couple to matter, both in terms of matter's
impact upon them and in terms of the long range gravitational force they
predict. Because of these problems my own preference for avoiding Dark Energy
is to bypass Ostrogradski's theorem by considering the fully nonlocal effective
action built up by quantum gravitational processes during the epoch of
primordial inflation.
| https://export.arxiv.org/pdf/astro-ph/0601672 |
\title*{Avoiding Dark Energy with 1/R Modifications of Gravity}
\titlerunning{1/R Modifications of Gravity}
\author{R. P. Woodard}
\institute{Department of Physics, University of Florida, Gainesville, FL
32611-8440, USA
\texttt{woodard@phys.ufl.edu}}
\section{Introduction}
\label{sec:1}
The case for alternate gravity is easily made. The best that can be done from
observing cosmic motions is to infer the metric $g_{\mu\nu}$ in some
coordinate system. From this one can reconstruct the Einstein tensor and then
ask whether or not general relativity predicts it in terms of the observed
sources of stress-energy,
\begin{equation}
\Bigl(R_{\mu\nu} - \frac12 g_{\mu\nu} R\Bigr)_{\rm rec} = 8 \pi G \Bigl(
T_{\mu\nu}\Bigr)_{\rm obs} \; ?
\end{equation}
One way of explaining any disagreement is by positing the existence of an
unobserved, ``dark'' component of the stress-energy tensor,
\begin{equation}
\Bigl(T_{\mu\nu}\Bigr)_{\rm dark} \equiv \frac1{8\pi G} \Bigl(R_{\mu\nu}
- \frac12 g_{\mu\nu} R\Bigr)_{\rm rec} - \Bigl(T_{\mu\nu}\Bigr)_{\rm obs} \; .
\end{equation}
This always works, but recent observations make it seem epicyclic.
The theory of nucleosynthesis implies that no more than about 4\% of the
energy density currently required to make general relativity agree with all
observations can consist of any material with which we are presently familiar
\cite{BBN} --- and only a fraction of this 4\% is observed. Just to
make general relativity agree with the observed motions of galaxies and
galactic clusters we must posit that {\it six times} the mass of ordinary
matter comes in the form of nonbaryonic, cold dark matter \cite{CDM}.
Although there are some plausible candidates for what this might be, no
Earth-bound laboratory has yet succeeded in detecting it.
I belong to the minority of physicists who feel that this factor of six
already strains
credulity. Easing that strain is what led Milgrom to propose MOND \cite{MM},
which can be viewed as a phenomenological modification of gravity in the
regime of very small accelerations. There is an impressive amount of
observational data in favor of this modification \cite{SM} --- although see
\cite{GSKVK}. Bekenstein has recently constructed a fully relativistic field
theory \cite{JDB} which reproduces MOND, and a preliminary analysis of the
resulting cosmology works better than many experts thought possible
\cite{SMFB}.
However, the worst problem for conventional gravity comes on the largest
scales. To make general relativity agree with the Hubble plots of distant
Type Ia supernovae \cite{SNCP,SNST,SNLS}, with the power spectrum of
anisotropies in the cosmic
microwave background \cite{CMB} and with large scale structure surveys
\cite{LSS}, one must accept an additional component of ``dark energy'' that
is about {\it eighteen times} larger than that of ordinary matter. This
would mean that 96\% of the current universe's energy exists in forms which
have so far only been detected gravitationally! Even people who believe
passionately in dark matter (and hence accept the factor of six) find this
factor of $6 \!+\! 18 \!=\! 24$ difficult to swallow. That is why there has
been so much recent interest in modifying gravity to make it predict
observed cosmic phenomena without the need for dark energy, and sometimes
even without the need for dark matter.
I want to stress that the issue is one of plausibility. There is no
problem inventing field theories which give the required amount of dark
energy. The simplest way of doing it is with a minimally coupled scalar
\cite{CW,PR},
\begin{equation}
\mathcal{L} = -\frac12 \partial_{\mu} \varphi \partial_{\nu} g^{\mu\nu}
\sqrt{-g} - V(\varphi) \sqrt{-g} \; . \label{quint}
\end{equation}
The usual procedure is to begin with a scalar potential $V(\varphi)$ and
work out the cosmology, but it is easy to start with whatever cosmological
evolution is desired and {\it construct} the potential which would support
it. I will go through the construction here, both to make the point and so
that it can be used later.
On the largest scales the geometry of the universe can be described in terms
of a single function of time known as the scale factor $a(t)$,
\begin{equation}
ds^2 = -dt^2 + a^2(t) d\vec{x} \cdot d\vec{x} \; .
\end{equation}
The logarithmic time derivative of this quantity gives the Hubble parameter,
\begin{equation}
H(t) \equiv \frac{\dot{a}}{a} \; .
\end{equation}
If we specialize to a solution $\varphi_0(t)$ of the scalar field equations
which depends only upon time, the two nontrivial Einstein equations are,
\begin{eqnarray}
3 H^2 & = & 8 \pi G \Bigl(\frac12 \dot{\varphi}_0^2 + V(\varphi_0)\Bigr) \; ,
\label{E1} \\
-2 \dot{H} - 3 H^2 & = & 8 \pi G \Bigl(\frac12 \dot{\varphi}_0^2 -
V(\varphi_0)\Bigr) \; . \label{E2}
\end{eqnarray}
Let us assume $a(t)$ is known as an explicit function of time, and construct
$\varphi_0(t)$ and $V(\varphi)$. By adding (\ref{E1}) and (\ref{E2}) we obtain,
\begin{equation}
-2 \dot{H} = 8 \pi G \dot{\varphi}_0^2 \; . \label{twoeqns}
\end{equation}
The weak energy condition implies $\dot{H}(t) \leq 0$ so we can take the
square root and integrate to solve for $\varphi_0(t)$,
\begin{equation}
\varphi_0(t) = \varphi_I \pm \int_{t_I}^t dt' \sqrt{\frac{-2 \dot{H}(t')}{
8 \pi G}} \; . \label{phi}
\end{equation}
One can choose $\varphi_I$ and the sign freely.
Because the integrand in (\ref{phi}) is always positive, the function
$\varphi_0(t)$ is monotonic. This means we can invert to solve for
time as a function of $\varphi_0$. Let us call the inverse function
$T(\varphi)$,
\begin{equation}
\psi = \varphi_0\Bigl(T(\psi)\Bigr) \; . \label{inv}
\end{equation}
By subtracting (\ref{E2}) from (\ref{E1}) we obtain a relation for the
scalar potential as a function of time,
\begin{equation}
V = \frac1{8\pi G} \Bigl( \dot{H}(t) + 3 H^2(t)\Bigr) \; .
\end{equation}
The potential is determined as a function of the scalar by substituting the
inverse function (\ref{inv}),
\begin{equation}
V(\varphi) = \frac1{8\pi G} \Biggl\{ \dot{H}\Bigl(T(\varphi)\Bigr)
+ 3 H^2\Bigl(T(\varphi)\Bigr) \Biggr\} \; .
\end{equation}
This construction gives a scalar which supports any evolution $a(t)$ (with
$\dot{H}(t) < 0$) all by itself. Should you wish to include some other,
known component of the stress-energy, simply add the energy density
and pressure of this component to the Einstein equations,
\begin{eqnarray}
3 H^2 & = & 8 \pi G \Bigl(\frac12 \dot{\varphi}_0^2 + V(\varphi_0) + \rho_{\rm
known}\Bigr) \; , \\
-2 \dot{H} - 3 H^2 & = & 8 \pi G \Bigl(\frac12 \dot{\varphi}_0^2 -
V(\varphi_0) + p_{\rm known}\Bigr) \; .
\end{eqnarray}
Provided $\rho_{\rm known}$ and $p_{\rm known}$ are known functions of either
time or the scale factor, the construction goes through as before.\footnote{
This construction seems to be due to Ratra and Peebles \cite{PR}. Recent
examples of its use include \cite{TW2,SRSS,NO0}.}
Using this method one can devise a new field $\varphi(x)$ which will
support {\it any} cosmology with $\dot{H}(t) < 0$. However, the introduction
of such a ``quintessence'' field raises a number of questions:
\begin{enumerate}
\item{Where does $\varphi$ reside in fundamental theory?}
\item{Why can't $\varphi$ couple to fields other than the metric? And if
it does couple to other fields, why haven't we detected its influence in
Earth-bound laboratories?}
\item{Why did $\varphi$ come to dominate the stress-energy of the universe
so recently in cosmological time?}
\item{Why is the $\varphi$ field so homogeneous?}
\end{enumerate}
When a phenomenological fix raises more questions than it answers people
are naturally drawn to investigate other fixes. One possibility is that
general relativity is not the correct theory of gravity on cosmological
scales.
In this talk I shall review gravitational Lagrangians of the form,
\begin{equation}
\mathcal{L} = \frac1{16 \pi G} \Bigl(R + \Delta R[g]\Bigr) \sqrt{-g} \; ,
\label{ansatz}
\end{equation}
where $\Delta R[g]$ is some local scalar constructed from the curvature
tensor and possibly its covariant derivatives. Examples of such scalars
are,
\begin{equation}
\frac1{\mu^2} R^{\alpha\beta} R_{\alpha\beta} \qquad , \qquad \frac1{\mu^4}
g^{\mu\nu} R_{,\mu} R_{,\nu} \qquad , \qquad \mu^2 \sin\Bigl(\frac1{\mu^4}
R^{\alpha\beta\rho\sigma} R_{\alpha\beta\rho\sigma}\Bigr) \; .
\end{equation}
I begin by reviewing a powerful no-go theorem which pervades and constrains
fundamental theory so completely that most people assume its consequence
without thinking. This is the theorem of Ostrogradski \cite{MO}, who
essentially showed why Newton was right to suppose that the laws of physics
involve no more than two time derivatives of the fundamental dynamical
variables. The key consequence for our purposes is that the only viable
form for the functional $\Delta R[g]$ in (\ref{ansatz}) is an algebraic
function of the undifferentiated Ricci scalar,
\begin{equation}
\Delta R[g] = f(R) \; .
\end{equation}
I review the Ostrogradski result in section 2, and hopefully immunize you
against some common misconceptions about it in section 3. In section 4 I
explain why $f(R)$ theories do not contradict Ostrogradski's result. I also
demonstrate that, in the absence of matter, $f(R)$ theories are equivalent
to ordinary gravity, with $f(R) = 0$, plus a minimally coupled scalar of
the form (\ref{quint}). Then I use the construction given above to show how
one can choose $f(R)$ to enforce an arbitrary cosmology. This establishes
that an $f(R)$ can be found to support any desired cosmology. In section 5
I discuss problems associated with the particular choice function $f(R) =
-\frac{\mu^4}{R}$. Section 6 presents conclusions.
\section{The Theorem of Ostrogradski}
\label{sec:2}
Ostrogradski's result is that there is a linear instability in the Hamiltonians
associated with Lagrangians which depend upon more than one time derivative in
such a way that the dependence cannot be eliminated by partial integration
\cite{MO}. The result is so general that I can simplify the discussion by
presenting it in the context of a single, one dimensional point particle whose
position as a function of time is $q(t)$. First I will review the way the
Hamiltonian is constructed for the usual case in which the Lagrangian involves
no higher than first time derivatives. Then I present Ostrogradski's
construction for the case in which the Lagrangian involves second time
derivatives. And the section closes with the generalization to $N$ time
derivatives.
In the usual case of $L = L(q,\dot{q})$, the Euler-Lagrange equation is,
\begin{equation}
\frac{\partial L}{\partial q} - \frac{d}{dt} \frac{\partial L}{\partial
\dot{q}} = 0 \; . \label{ELE1}
\end{equation}
The assumption that $\frac{\partial L}{\partial \dot{q}}$ depends upon
$\dot{q}$ is known as {\it nondegeneracy}. If the Lagrangian is nondegenerate
we can write (\ref{ELE1}) in the form Newton assumed so long ago for the laws
of physics,
\begin{equation}
\ddot{q} = \mathcal{F}(q,\dot{q}) \qquad \Longrightarrow \qquad q(t) =
\mathcal{Q}(t,q_0,\dot{q}_0) \; . \label{newt}
\end{equation}
From this form it is apparent that solutions depend upon two pieces of initial
value data: $q_0 = q(0)$ and $\dot{q}_0 = \dot{q}(0)$.
The fact that solutions require two pieces of initial value data means that
there must be two canonical coordinates, $Q$ and $P$. They are traditionally
taken to be,
\begin{equation}
Q \equiv q \qquad {\rm and} \qquad P \equiv \frac{\partial L}{\partial \dot{q}}
\; . \label{ctrans}
\end{equation}
The assumption of nondegeneracy is that we can invert the phase space
transformation (\ref{ctrans}) to solve for $\dot{q}$ in terms of $Q$ and $P$.
That is, there exists a function $v(Q,P)$ such that,
\begin{equation}
\frac{\partial L}{\partial \dot{q}} \Biggl\vert_{q = Q \atop \dot{q} = v}
= P \; . \label{invct}
\end{equation}
The canonical Hamiltonian is obtained by Legendre transforming on $\dot{q}$,
\begin{eqnarray}
H(Q,P) & \equiv & P \dot{q} - L \; , \\
& = & P v(Q,P) - L\Bigl(Q,v(Q,P)\Bigr) \; .
\end{eqnarray}
It is easy to check that the canonical evolution equations reproduce the
inverse phase space transformation (\ref{invct}) and the Euler-Lagrange
equation (\ref{ELE1}),
\begin{eqnarray}
\dot{Q} & \equiv & \frac{\partial H}{\partial P} = v + P \frac{\partial v}{
\partial P} - \frac{\partial L}{\partial \dot{q}} \frac{\partial v}{\partial P}
= v \; , \\
\dot{P} & \equiv & -\frac{\partial H}{\partial Q} = -P \frac{\partial v}{
\partial Q} + \frac{\partial L}{\partial q} + \frac{\partial L}{\partial
\dot{q}} \frac{\partial v}{\partial P} = \frac{\partial L}{\partial q} \; .
\end{eqnarray}
This is what we mean by the statement, ``the Hamiltonian generates time
evolution.'' When the Lagrangian has no explicit time dependence, $H$ is also
the associated conserved quantity. Hence it is ``the'' energy by anyone's
definition, of course up to canonical transformation.
Now consider a system whose Lagrangian $L(q,\dot{q},\ddot{q})$ depends
nonde\-gen\-er\-ate\-ly upon $\ddot{q}$. The Euler-Lagrange equation is,
\begin{equation}
\frac{\partial L}{\partial q} - \frac{d}{dt} \frac{\partial L}{\partial
\dot{q}} + \frac{d^2}{dt^2} \frac{\partial L}{\partial \ddot{q}} = 0 \; .
\label{ELE2}
\end{equation}
Non-degeneracy implies that $\frac{\partial L}{\partial \ddot{q}}$ depends
upon $\ddot{q}$, in which case we can cast (\ref{ELE2}) in a form radically
different from Newton's,
\begin{equation}
q^{(4)} = \mathcal{F}(q,\dot{q},\ddot{q},q^{(3)}) \qquad \Longrightarrow \qquad
q(t) = \mathcal{Q}(t,q_0,\dot{q}_0,\ddot{q}_0,q^{(3)}_0) \; .
\end{equation}
Because solutions now depend upon four pieces of initial value data there must
be four canonical coordinates. Ostrogradski's choices for these are,
\begin{eqnarray}
Q_1 \equiv q \qquad & , & \qquad P_1 \equiv \frac{\partial L}{\partial \dot{q}}
- \frac{d}{dt} \frac{\partial L}{\partial \ddot{q}} \; , \label{ct1} \\
Q_2 \equiv \dot{q} \qquad & , & \qquad P_2 \equiv \frac{\partial L}{\partial
\ddot{q}} \; . \label{ct2}
\end{eqnarray}
The assumption of nondegeneracy is that we can invert the phase space
transformation (\ref{ct1}-\ref{ct2}) to solve for $\ddot{q}$ in terms of
$Q_1$, $Q_2$ and $P_2$. That is, there exists a function $a(Q_1,Q_2,P_2)$
such that,
\begin{equation}
\frac{\partial L}{\partial \ddot{q}} \Biggl\vert_{{q = Q_1 \atop \dot{q} =
Q_2} \atop \ddot{q} = a} = P_2 \; . \label{invct2}
\end{equation}
Note that one only needs the function $a(Q_1,Q_2,P_2)$ to depend upon {\it
three} canonical coordinates --- and not all four --- because
$L(q,\dot{q},\ddot{q})$ only depends upon three configuration space
coordinates. This simple fact has great consequence.
Ostrogradski's Hamiltonian is obtained by Legendre transforming, just as in the
first derivative case, but now on $\dot{q} = q^{(1)}$ and $\ddot{q} = q^{(2)}$,
\begin{eqnarray}
\lefteqn{H(Q_1,Q_2,P_1,P_2) \equiv \sum_{i=1}^2 P_i q^{(i)} - L \; , } \\
& & = P_1 Q_2 + P_2 a(Q_1,Q_2,P_2) - L\Bigl(Q_1,Q_2,a(Q_1,Q_2,P_2)\Bigr) \; .
\label{Host}
\end{eqnarray}
The time evolution equations are just those suggested by the notation,
\begin{equation}
\dot{Q_i} \equiv \frac{\partial H}{\partial P_i} \qquad {\rm and} \qquad
\dot{P}_i \equiv - \frac{\partial H}{\partial Q_i} \; .
\end{equation}
Let's check that they generate time evolution. The evolution equation for
$Q_1$,
\begin{equation}
\dot{Q}_1 = \frac{\partial H}{\partial P_1} = Q_2 \; ,
\end{equation}
reproduces the phase space transformation $\dot{q} = Q_2$ in (\ref{ct2}).
The evolution equation for $Q_2$,
\begin{equation}
\dot{Q}_2 = \frac{\partial H}{\partial P_2} = a + P_2 \frac{\partial a}{
\partial P_2} - \frac{\partial L}{\partial \ddot{q}} \frac{\partial a}{\partial
P_2} = a \; ,
\end{equation}
reproduces (\ref{invct2}). The evolution equation for $P_2$,
\begin{equation}
\dot{P}_2 = -\frac{\partial H}{\partial Q_2} = -P_1 - P_2 \frac{\partial a}{
\partial Q_2} + \frac{\partial L}{\partial \dot{q}} + \frac{\partial L}{
\partial \ddot{q}} \frac{\partial a}{\partial Q_2} = -P_1 + \frac{\partial L}{
\partial \dot{q}} \; ,
\end{equation}
reproduces the phase space transformation $P_1 = \frac{\partial L}{\partial
\dot{q}} - \frac{d}{dt} \frac{\partial L}{\partial \ddot{q}}$ (\ref{ct1}). And
the evolution equation for $P_1$,
\begin{equation}
\dot{P}_1 = -\frac{\partial H}{\partial Q_1} = -P_2 \frac{\partial a}{\partial
Q_1} + \frac{\partial L}{\partial q} + \frac{\partial L}{\partial \ddot{q}}
\frac{\partial a}{\partial Q_1} = \frac{\partial L}{\partial q} \; ,
\end{equation}
reproduces the Euler-Lagrange equation (\ref{ELE2}). So Ostrogradski's system
really does generate time evolution. When the Lagrangian contains no explicit
dependence upon time it is also the conserved Noether current. By anyone's
definition, it is therefore ``the'' energy, again up to canonical
transformation.
There is one, overwhelmingly bad thing about Ostrogradski's Hamiltonian
(\ref{Host}): it is {\it linear} in the canonical momentum $P_1$. This means
that no system of this form can be stable. In fact, there is not even any
barrier to decay. Note also the power and generality of the result. It applies
to {\it every} Lagrangian $L(q,\dot{q},\ddot{q})$ which depends
nondegenerately upon $\ddot{q}$, independent of the details. The only
assumption is nondegeneracy, and that simply means one cannot eliminate
$\ddot{q}$ by partial integration. This is why Newton was right to assume
the laws of physics take the form (\ref{newt}) when expressed in terms of
fundamental dynamical variables.
Adding more higher derivatives just makes the situation worse. Consider a
Lagrangian $L\left(q,\dot{q},\dots,q^{(N)}\right)$ which depends upon the
first $N$ derivatives of $q(t)$. If this Lagrangian depends nondegenerately
upon $q^{(N)}$ then the Euler-Lagrange equation,
\begin{equation}
\sum_{i=0}^N \left(-{d \over dt}\right)^i {\partial L \over \partial q^{(i)}}
= 0 \; , \label{ELEN}
\end{equation}
contains $q^{(2N)}$. Hence the canonical phase space must have $2N$
coordinates. Ostrogradski's choices for them are,
\begin{equation}
Q_i \equiv q^{(i-1)} \qquad {\rm and} \qquad P_i \equiv \sum_{j=i}^N \Bigl(-
\frac{d}{dt}\Bigr)^{j-i} \frac{\partial L}{\partial q^{(j)}} \; .
\end{equation}
Non-degeneracy means we can solve for $q^{(N)}$ in terms of $P_N$ and the
$Q_i$'s. That is, there exists a function $\mathcal{A}(Q_1,\ldots,Q_N,P_N)$
such that,
\begin{equation}
\frac{\partial L}{\partial q^{(N)}} \Biggl\vert_{q^{(i-1)} = Q_i \atop
q^{(N)} = \mathcal{A}} = P_N \; . \label{nondeg}
\end{equation}
For general $N$ Ostrogradski's Hamiltonian takes the form,
\begin{eqnarray}
H & \equiv & \sum_{i=1}^N P_i q^{(i)} - L \; , \\
& = & P_1 Q_2 + P_2 Q_3 + \cdots + P_{N-1} Q_N + P_N \mathcal{A} -
L\Bigl(Q_1,\ldots,Q_N,\mathcal{A}\Bigr) \; . \label{HN}
\end{eqnarray}
It is simple to check that the evolution equations,
\begin{equation}
\dot{Q}_i \equiv \frac{\partial H}{\partial P_i} \qquad {\rm and} \qquad
\dot{P}_i \equiv -\frac{\partial H}{\partial Q_i} \; ,
\end{equation}
again reproduce the canonical transformations and the Euler-Lagrange equation.
So (\ref{HN}) generates time evolution. Similarly, it is Noether current for
the case where the Lagrangian contains no explicit time dependence. So there
is little alternative to regarding (\ref{HN}) as ``the'' energy, again up to
canonical transformation.
One can see from (\ref{HN}) that the Hamiltonian is linear in $P_1, P_2, \ldots
P_{N-1}$. Only with respect to $P_N$ might it be bounded from below. Hence
the Hamiltonian is necessarily unstable over half the classical phase space
for large $N$!
\section{Common Misconceptions}
\label{sec:3}
The no-go theorem I have just reviewed ought to come as no surprise. It
explains why Newton was right to expect that physical laws take the form
of second order differential equations when expressed in terms of fundamental
dynamical variables.\footnote{The caveat is there because one can always get
higher order equations by solving for some of the fundamental variables.}
Every fundamental system we have discovered since Newton's day has had this
form. The bizarre, dubious thing would be if Newton had blundered upon a tiny
subset of possible physical laws, and all our probing over the course of the
next three centuries had never revealed the vastly richer possibilities.
However --- {\it deep sigh} --- particle theorists don't like being told
something is impossible, and a definitive no-go theorem such as that of
Ostrogradski provokes them to tortuous flights of evasion. I ought to know, I
get called upon to referee the resulting papers often enough! No one has so
far found a way around Ostrogradski's theorem. I won't attempt to prove that
no one ever will, but let me use this section to run through some of the
misconceptions which have been in back of attempted evasions.
To fix ideas it will be convenient to consider a higher derivative
generalization of the harmonic oscillator,
\begin{equation}
\mathcal{L} = -\frac{g m}{2 \omega^2} \ddot{q}^2 + \frac{m}2 \dot{q}^2 -
\frac{m\omega^2}2 q^2 \; . \label{HDO}
\end{equation}
Here $m$ is the particle mass, $\omega$ is a frequency and $g$ is a small
positive pure number we can think of as a coupling constant. The Euler-Lagrange
equation,
\begin{equation}
-m \Bigl( \frac{g}{\omega^2} q^{(4)} + \ddot{q} + \omega^2 q\Bigr) = 0 \; ,
\label{HDE}
\end{equation}
has the general solution,
\begin{equation}
q(t) = A_+ \cos(k_+ t) + B_+ \sin(k_+ t) + A_- \cos(k_- t) + B_- \sin(k_- t)
\; . \label{gensol}
\end{equation}
Here the two frequencies are,
\begin{equation}
k_{\pm} \equiv \omega \sqrt{ \frac{1 \mp \sqrt{1 \!-\! 4 g}}{2 g} } \; ,
\end{equation}
and the initial value constants are,
\begin{eqnarray}
A_+ = \frac{k_-^2 q_0 \!+\! \ddot{q}_0}{k_-^2 \!-\! k_+^2} \qquad & , & \qquad
B_+ = \frac{k_-^2 \dot{q}_0 \!+\! q^{(3)}_0}{k_+ (k_-^2 \!-\! k_+^2)} \; , \\
A_- = \frac{k_+^2 q_0 \!+\! \ddot{q}_0}{k_+^2 \!-\! k_-^2} \qquad & , & \qquad
B_- = \frac{k_+^2 \dot{q}_0 \!+\! q^{(3)}_0}{k_- (k_+^2 \!-\! k_-^2)} \; .
\end{eqnarray}
The conjugate momenta are,
\begin{eqnarray}
P_1 = m \dot{q} + \frac{g m}{\omega^2} q^{(3)} \qquad & \Leftrightarrow &
\qquad q^{(3)} = \frac{\omega^2 P_1 \!-\! m \omega^2 Q_2}{g m} \; , \\
P_2 = - \frac{g m}{\omega^2} \ddot{q} \qquad & \Leftrightarrow &
\qquad \ddot{q} = -\frac{\omega^2 P_2}{g m} \; .
\end{eqnarray}
The Hamiltonian can be expressed in terms of canonical variables,
configuration space variables or initial value constants,
\begin{eqnarray}
H & = & P_1 Q_2 - \frac{\omega^2}{2 g m} P_2^2 - \frac{m}2 Q_2^2 + \frac{m
\omega^2}2 Q_1^2 \; , \label{H1} \\
& = & \frac{g m}{\omega^2} \dot{q} q^{(3)} - \frac{g m}{2 \omega^2}
\ddot{q}^2 + \frac{m}2 \dot{q}^2 + \frac{m \omega^2}2 q^2 \; , \label{H2} \\
& = & \frac{m}2 \sqrt{1 \!-\! 4 g} \, k_+^2 (A_+^2 \!+\! B_+^2) -
\frac{m}2 \sqrt{1 \!-\! 4 g} \, k_-^2 (A_-^2 \!+\! B_-^2) \; . \label{H3}
\end{eqnarray}
The last form makes it clear that the ``$+$'' modes carry positive energy
whereas the ``$-$'' modes carry negative energy.
\subsection{Nature of the Instability}
\label{sub:3.1}
It's important to understand both how the Ostrogradskian instability manifests
and what is physically wrong with a theory which shows this instability.
Because the Ostrogradskian Hamiltonian (\ref{HN}) is not bounded below with
respect to more than one of its conjugate momenta, one sees that the problem
is not reaching arbitrarily negative energies by setting the dynamical
variable to some {\it constant value}. Rather it is reaching arbitrarily
negative energies by making the dynamical variable have a certain {\it time
dependence}. People sometimes mistakenly believe they have found a higher
derivative system which is stable when all they have checked is that the
Hamiltonian is bounded from below for constant field configurations. For
example, from expression (\ref{H2}) we see that our higher derivative
oscillator energy is bounded below by zero for $q(t) = {\rm const}$! Negative
energies are achieved by making $\ddot{q}$ large and/or making $q^{(3)}$
large while keeping $\dot{q} \!+\! g q^{(3)}/\omega^2$ fixed.
Another crucial point is that the same dynamical variable typically carries
both positive and negative energy degrees of freedom in a higher derivative
theory. For our higher derivative oscillator this is apparent from expression
(\ref{gensol}) which shows that $q(t)$ involves both the positive energy
degrees of freedom, $A_+$ and $B_+$, and the negative energy ones, $A_-$ and
$B_-$. And note from expression (\ref{H3}) that I really mean positive and
negative {\rm energy}, not just positive and negative frequency, which is the
usual case in a lower derivative theory.
People sometimes imagine that the energy of a higher derivative theory decays
with time. That is not true. Provided one is dealing with a complete system,
and provided there is no external time dependence, the energy of a higher
derivative system is conserved, just as it would be under those conditions
for a lower derivative theory. This conservation is apparent for our higher
derivative oscillator from expression (\ref{H3}).
The physical problem with nondegenerate higher derivative theories is not that
their energies decay to lower and lower values. The problem is rather that
certain sectors of the theory become arbitrarily highly excited when one is
dealing with an interacting, continuum field theory which has nondegenerate
higher derivatives. To understand this I must digress to remind you of some
familiar facts about the Hydrogen atom.
If you consider Hydrogen in isolation, there is an infinite tower of stationary
states. However, if you allow the Hydrogen atom to interact with
electromagnetism only the ground state is stationary; all the excited states
decay through the emission of a photon. Why is this so? It certainly is {\it
not} because ``the system wants to lower its energy.'' The energy of the full
system is constant, the binding energy released by the decaying atom being
compensated by the energy of the recoil photon. Yet the decay always takes
place, and rather quickly. The reason is that decay is terrifically favored
by entropy. If we prepare the Hydrogen atom in an excited state, with no
photons present, there is {\it one} way for the atom to remain excited,
whereas there are an {\it infinite} number of ways for it to decay because
the recoil photon could go off in any direction.
Now consider an interacting, continuum field theory which possesses the
Ostrogradskian instability. In particular consider its likely particle
spectrum about some ``empty'' solution in which the field is constant.
Because the Hamiltonian is linear in all but one of the conjugate momenta
we can increase or decrease the energy by moving different directions in
phase space. Hence there must be both positive energy and negative energy
particles --- just as there are in our higher derivative oscillator. Just as
in that point particle model, the same continuum field must carry the creation
and annihilation operators of {\it both} the positive and the negative energy
particles. If the theory is interacting at all --- that is, if its Lagrangian
contains a higher than quadratic power of the field --- then there will be
interactions between positive and negative energy particles. Depending upon
the interaction, the empty state can decay into some collection of positive
and negative energy particles. The details don't really matter, all that
matters is the counting: there is {\it one} way for the system to stay empty
versus a continuous {\it infinity} of ways for it to decay. This infinity is
even worse than for the Hydrogen atom because it includes not only all the
directions that recoil particles of fixed energies could go but also the fact
that the various energies can be arbitrarily large in magnitude provided they
sum to zero. Because of that last freedom the decay is instantaneous. And the
system doesn't just decay once! It is even {\it more} entropicly favored
for there to be two decays, and better yet for three, etc. You can see
that such a system instantly evaporates into a maelstrom of positive and
negative energy particles. Some of my mathematically minded colleagues would
say it isn't even defined. I prefer to simply observe that no theory of this
kind can describe the universe we experience in which all particles have
positive energy and empty space remains empty.
Note that we only reach this conclusion if the higher derivative theory
possesses both interactions and continuum particles. Our point particle
oscillator has no interactions, so its negative energy degree of freedom is
harmless. Of course it is also completely unobservable! However, it is
conceivable we could couple this higher derivative oscillator to a discrete
system without engendering an instability. The feature that drives the
instability when continuum particles are present is the vast entropy of
phase space. Without that it becomes an open question whether or not there is
anything wrong with a higher derivative theory. Of course we live in a
continuum universe, and any degree of freedom we can observe must be
interacting, so these are very safe assumptions. However, people sometimes
delude themselves that there is no problem with continuum, interacting higher
derivative models of the universe on the basis of studying higher derivative
systems which could never describe the universe because they either lack
interactions or else continuum particles.
In this sub-section we have learned:
\begin{enumerate}
\item{The Ostrogradskian instability does not drive the dynamical variable to
a special, constant value but rather to a special kind of time dependence.}
\item{A dynamical variable which experiences the Ostrogradskian instability
will carry both positive and negative energy creation and annihilation
operators.}
\item{If the system interacts then the ``empty'' state can decay into a
collection of positive and negative energy excitations.}
\item{If the system is a continuum field theory the vast entropy at infinite
momentum will make the decay instantaneous.}
\end{enumerate}
\subsection{Perturbation Theory}
\label{sub:3.2}
People sometimes mistakenly believe that the Ostrogradskian instability is
avoided if higher derivatives are segregated to appear only in interaction
terms. This is not correct if one considers the theory on a fundamental level.
One can see from the construction of section {\ref{sec:2} that the fact of
Ostrogradski's Hamiltonian being unbounded below depends only upon
nondegeneracy, irrespective of how one organizes any approximation technique.
However, there is a way of imposing constraints to make the theory agree with
its perturbative development. If this is done then there are no more higher
derivative degrees of freedom, however, one typically loses unitarity,
causality and Lorentz invariance on the nonperturbative level.
I constructed the higher derivative oscillator (\ref{HDO}) so that its
higher derivatives vanish when $g \!=\! 0$. If we solve the Euler-Lagrange
equation (\ref{HDE}) exactly, without employing perturbation theory, there
are four linearly independent solutions (\ref{gensol}) corresponding to a
positive energy oscillator of frequency $k_+$ and a negative energy
oscillator of frequency $k_-$. However, we might instead regard the parameter
$g$ as a coupling constant and solve the equations perturbatively. This means
substituting the ansatz,
\begin{equation}
q_{\rm pert}(t) = \sum_{n=0}^{\infty} g^n x_n(t) \; , \label{pertan}
\end{equation}
into the Euler-Lagrange equation (\ref{HDE}) and segregating terms according
to powers of $g$. The resulting system of equations is,
\begin{eqnarray}
\ddot{x}_0 + \omega^2 x_0 & = & 0 \; , \label{E0} \\
\ddot{x}_1 + \omega^2 x_1 & = & -\frac1{\omega^2} x^{(4)}_0 \; , \\
\ddot{x}_2 + \omega^2 x_2 & = & -\frac1{\omega^2} x^{(4)}_1 \; ,
\end{eqnarray}
and so on. Because the zeroth order equation involves only second derivatives,
its solution depends upon only two pieces of initial value data,
\begin{equation}
x_0(t) = q_0 \cos(\omega t) + \frac{\dot{q}_0}{\omega} \sin(\omega t) \; .
\end{equation}
The first correction is,
\begin{equation}
x_1(t) = -\frac{\omega t}2 q_0 \sin(\omega t) + \frac{t}2 \dot{q}_0
\cos(\omega t) - \frac1{2 \omega} \dot{q_0} \sin(\omega t) \; ,
\end{equation}
and it is easy to see that the sum of all corrections gives,
\begin{equation}
q_{\rm pert}(t) = q_0 \cos(k_+ t) + \frac{\dot{q}_0}{k_+} \sin(k_+ t) \; .
\label{pertsol}
\end{equation}
What is the relation of the perturbative solution (\ref{pertsol}) to the
general one (\ref{gensol})? The perturbative solution is what results if
we change the theory by imposing the constraints,
\begin{eqnarray}
\ddot{q}(t) = - k_+^2 q(t) \qquad & \Longleftrightarrow & \qquad P_2 =
\frac{m}2 \Bigl(1 \!-\! \sqrt{1 \!-\! 4g}\Bigr) Q_1 \; , \label{C1} \\
q^{(3)}(t) = - k_+^2 \dot{q}(t) \qquad & \Longleftrightarrow & \qquad P_1 =
\frac{m}2 \Bigl(1 \!+\! \sqrt{1 \!-\! 4g}\Bigr) Q_2 \; . \label{C2}
\end{eqnarray}
Under these constraints the Hamiltonian becomes,
\begin{equation}
H_{\rm pert} = \sqrt{1 \!-\! 4g} \Bigl( \frac{m}2 Q_2^2 + \frac{m k_+^2}2 Q_1^2
\Bigr) \; ,
\end{equation}
which is indeed that of a single harmonic oscillator. From the full theory,
perturbation theory has retained only the solution whose frequency is well
behaved for $g \rightarrow 0$,
\begin{equation}
k_+ = \omega \Bigl(1 + \frac12 g + \frac78 g^2 + O(g^3) \Bigr) \; .
\label{lowk}
\end{equation}
It has discarded the solution whose frequency blows up as $g \rightarrow 0$,
\begin{equation}
k_- = \frac{\omega}{\sqrt{g}} \Bigl(1 -\frac12 g - \frac58 g^2 + O(g^3)\Bigr)
\; . \label{highk}
\end{equation}
So what's wrong with this? In fact there is nothing wrong with the procedure
for our model. If the constraints (\ref{C1}-\ref{C2}) are imposed at one
instant, they remain valid for all times as a consequence of the full equation
of motion. However, that is only because our model is free of interactions.
Recall that this same feature means the positive and negative energy degrees
of freedom exist in isolation of one another, and there is no decay to
arbitrarily high excitation as there would be for an interacting, continuum
field theory.
When interactions are present it is more involved but still possible to impose
constraints which change the theory so that only the lower derivative,
perturbative solutions remain. The procedure was first worked out by Ja\'en,
Llosa and Molina \cite{JLM}, and later, independently, by Eliezer and me
\cite{EW}. To understand its critical defect suppose we change the
``interaction'' of our higher derivative oscillator from a quadratic term to a
cubic one,
\begin{equation}
-\frac{g m}{2 \omega^4} \, \ddot{q}^2 \longrightarrow -\frac{g m}{6 \ell
\omega^4} \, \ddot{q}^3 \; .
\end{equation}
Here $\ell$ is some constant with the dimensions of a length. As with the
quadratic interaction, the new equation of motion is fourth order,
\begin{equation}
-m \Biggl[ \frac{d^2}{dt^2} \Bigl(\frac{g \ddot{q}^2}{2 \ell \omega^4}\Bigr) +
\ddot{q} + \omega^2 q \Biggr] = 0 \; ,
\end{equation}
Its general solution depends upon four pieces of initial value data. However,
by isolating the highest derivative term of the free theory,
\begin{equation}
\ddot{q} =- \omega^2 q -\frac{d^2}{dt^2} \Bigl(\frac{g \ddot{q}^2}{2 \ell
\omega^4} \Bigr) \; , \label{rewrite}
\end{equation}
and then iteratively substituting (\ref{rewrite}), we can delay the
appearance of higher derivatives on the right hand side to any desired order
in the coupling constant $g$. For example, two iterations frees the right
hand side of higher derivatives up to order $g^2$,
\begin{eqnarray}
\ddot{q} & = & -\omega^2 q -\frac{d^2}{dt^2} \Biggl\{ \frac{g}{2 \ell \omega^4}
\Biggl[ -\omega^2 q - \frac{d^2}{dt^2} \Bigl( \frac{g \ddot{q}^2}{2 \ell
\omega^4}\Bigr)\Biggr]^2\Biggr\} \; , \\
& = & -\omega^2 q + \frac{g}{\ell} \Bigl( \omega^2 q^2 \!-\! \dot{q}^2\Bigr)
+ \frac{g^2}{2 \ell^2 \omega^4} \, q \frac{d^2}{dt^2} \Bigl( \ddot{q}^2\Bigr)
\nonumber \\
& & \hspace{.5cm}
- \frac{g^2}{2 \ell^2 \omega^6} \frac{d^2}{dt^2} \Bigl[ q \frac{d^2}{dt^2}
\Bigl( \ddot{q}^2 \Bigr)\Bigr] - \frac{g^3}{8 \ell^3 \omega^{12}} \frac{d^2}{
dt^2} \Biggl[ \frac{d^2}{dt^2} \Bigl( \ddot{q}^2\Bigr) \Biggr]^2 \; .
\end{eqnarray}
This obviously becomes complicated fast! However, the lower derivative
terms at order $g^2$ are simple enough to give if I don't worry about the
higher derivative remainder,
\begin{equation}
\ddot{q} = -\omega^2 q + \frac{g}{\ell} \Bigl( \omega^2 q^2 \!-\! \dot{q}^2
\Bigr) + \frac{g^2}{\ell^2} \Bigl( -6 \omega^2 q^3 \!+\! 14 q \dot{q}^2\Bigr)
+ O(g^3) \; .
\end{equation}
If we carry this out to infinite order, {\it and drop the infinite derivative
remainder}, the result is an equation of the traditional form,
\begin{equation}
\ddot{q} = f(q,\dot{q}) \; .
\end{equation}
The canonical version of this equation gives the first of the desired
constraints. The second is obtained from the canonical version of its
time derivative.
The constrained system we have just described is consistent on the perturbative
level, but not beyond. It does not follow from the original, exact equation.
That would be no problem if we could define physics using perturbation theory,
but we cannot. Perturbation theory does not converge for any known interacting,
continuum field theory in $3\!+\!1$ dimensions! The fact that the constraints
are not consistent beyond perturbation theory means there is a nonperturbative
amplitude for the system to decay to the arbitrarily high excitation in the
manner described in sub-section \ref{sub:3.1}. The fact that the constraints
treat time derivatives differently than space derivatives also typically leads
to a loss of causality and Lorentz invariance beyond perturbation theory.
A final comment concerns the limit of small coupling constant, i.e., $g
\rightarrow 0$. One can see from the frequencies (\ref{lowk}-\ref{highk}) of
our higher derivative oscillator that the negative energy frequency diverges
for $g \rightarrow 0$. Disingenuous purveyors of higher derivative models
sometimes appeal to people's experience with {\it positive energy} modes by
arguing that, ``the $k_-$ mode approaches infinite frequency for small
coupling so it must drop out.'' That is false! The argument is quite correct
for an infinite frequency {\it positive} energy mode in a stable theory. In
that case exciting the mode costs an infinite amount of energy which would
have to be drawn from de-exciting finite frequency modes. However, a {\it
negative} energy mode doesn't decouple as its frequency diverges. Rather it
couples {\it more strongly} because taking its frequency to infinity opens up
more and more ways to balance its negative energy by exciting finite frequency,
positive energy modes.
\subsection{Quantization}
\label{sub:3.3}
People sometimes imagine that quantization might stabilize a system against
the Ostrogradskian instability the same way that it does for the Hydrogen atom
coupled to electromagnetism. This is a failure to understand correspondence
limits. Conclusions drawn from classical physics survive quantization unless
they depend upon the system either being completely excluded from some region
of the canonical phase space or else inhabiting only a small region of it. For
example, the classical instability of the Hydrogen atom (when coupled to
electromagnetism) derives from the fact that the purely Hydrogenic part of the
energy,
\begin{equation}
E_{\rm Hyd} = \frac{\Vert \vec{p}\Vert^2}{2m} - \frac{e^2}{\Vert \vec{x}\Vert}
\; .
\end{equation}
can be made arbitrarily negative by placing the electron close to the nucleus
at fixed momentum. Because this instability depends upon the system being
in a very small region of the canonical phase space, one might doubt that it
survives quantization, and explicit computation shows that it does not.
In contrast, the Ostrogradskian instability derives from the fact that
$P_1 Q_2$ can be made arbitrarily negative by taking $P_1$ either very negative,
for positive $Q_2$, or else very positive, for negative $Q_2$. {\it This covers
essentially half the classical phase space!} Further, the variables $Q_2$ and
$P_1$ commute with one another in Ostrogradskian quantum mechanics. So there
is no reason to expect that the Ostrogradskian instability is unaffected by
quantization.
\subsection{Unitarity vs. Instability}
\label{sub:3.4}
Particle physicists who quantize higher derivative theories don't typically
recognize a problem with the stability. They maintain that the problem with
higher derivatives is a breakdown of unitarity. In this sub-section I will
again have recourse to the higher derivative oscillator (\ref{HDO}) to explain
the connection between the two apparently unrelated problems.
Let us find the ``empty'' state wavefunction, $\Omega(Q_1,Q_2)$ that has the
minimum excitation in both the positive and negative energy degrees of
freedom. The procedure for doing this is simple: first identify the positive
and negative energy lowering operators $\alpha_{\pm}$ and then solve the
equations,
\begin{equation}
\alpha_+ \vert \Omega \rangle = 0 = \alpha_- \vert \Omega \rangle \; .
\label{wavef}
\end{equation}
We can recognize the raising and lowering operators by simply expressing the
general solution (\ref{gensol}) in terms of exponentials,
\begin{eqnarray}
\lefteqn{q(t) = \frac12 (A_+ \!+\! i B_+) e^{-ik_+ t} + \frac12 (A_+ \!-\!
i B_+) e^{ik_+ t} } \nonumber \\
& & \hspace{2cm} + \frac12 (A_- \!+\! i B_-) e^{-ik_-t } + \frac12 (A_- \!-\!
i B_-) e^{ik_- t} \; .
\end{eqnarray}
Recall that the $k_+$ mode carries positive energy, so its lowering operator
must be proportional to the $e^{-ik_+ t}$ term,
\begin{eqnarray}
\alpha_+ & \sim & A_+ + i B_+ \; , \\
& \sim & \frac{m k_+}2 \Bigl(1 \!+\! \sqrt{1 \!-\! 4g}\Bigr) Q_1 + i P_1
- k_+ P_2 - \frac{i m}2 \Bigl(1 \!-\! \sqrt{1 \!-\! 4g}\Bigr) Q_2 \; .
\end{eqnarray}
The $k_-$ mode carries negative energy, so its lowering operator must be
proportional to the $e^{+i k_- t}$ term,
\begin{eqnarray}
\alpha_- & \sim & A_- - i B_- \; , \\
& \sim & \frac{m k_-}2 \Bigl(1 \!-\! \sqrt{1 \!-\! 4g}\Bigr) Q_1 - i P_1
- k_- P_2 + \frac{i m}2 \Bigl(1 \!+\! \sqrt{1 \!-\! 4g}\Bigr) Q_2 \; .
\end{eqnarray}
Writing $P_i = -i \frac{\partial}{\partial Q_i}$ we see that the unique
solution to (\ref{wavef}) has the form,
\begin{equation}
\Omega(Q_1,Q_2) = N \exp\Biggl[-\frac{m \sqrt{1 \!-\! 4g}}{2 (k_+ \!+\! k_-)}
\Bigl(k_+ k_- Q_1^2 + Q_2^2\Bigr) - i \sqrt{g} m Q_1 Q_2\Biggr] \; .
\label{true}
\end{equation}
The empty wave function (\ref{true}) is obviously normalizable, so it gives a
state of the quantum system. We can build a complete set of normalized
stationary states by acting arbitrary numbers of $+$ and $-$ raising operators
on it,
\begin{equation}
\vert N_+ , N_-\rangle \equiv \frac{(\alpha_+^{\dagger})^N_+}{\sqrt{N_+ !}}
\frac{(\alpha_-^{\dagger})^N_-}{\sqrt{N_- !}} \vert \Omega \rangle \; .
\end{equation}
On this space of states the Hamiltonian operator is unbounded below, just as
in the classical theory,
\begin{equation}
H \vert N_+ , N_- \rangle = \Bigl(N_+ k_+ - N_- k_-\Bigr) \vert N_+ , N_-
\rangle \; .
\end{equation}
This is the correct way to quantize a higher derivative theory. One evidence
of this fact is that classical negative energy states correspond to quantum
negative energy states as well.
Particle physicists don't quantize higher derivative theories as we just have.
What they do instead is to regard the negative energy lowering operator as a
positive energy raising operator. So they define a ``ground state'' $\vert
\overline{\Omega} \rangle$ which obeys the equations,
\begin{equation}
\alpha_+ \vert \overline{\Omega} \rangle = 0 = \alpha_-^{\dagger} \vert
\overline{\Omega} \rangle \; . \label{falsewave}
\end{equation}
The unique wave function which solves these equations is,
\begin{equation}
\overline{\Omega}(Q_1,Q_2) = N \exp\Biggl[-\frac{m \sqrt{1 \!-\! 4g}}{2 (k_-
\!-\! k_+)} \Bigl(k_+ k_- Q_1^2 - Q_2^2\Bigr) + i \sqrt{g} m Q_1 Q_2 \Biggr]
\; . \label{wrong}
\end{equation}
This wave function is {\it not} normalizable, so it doesn't correspond to a
state of the quantum system. At this stage we should properly call a halt to
the analysis because we aren't doing quantum mechanics anymore. The
Schrodinger equation $H \psi(Q) = E \psi(Q)$ is just a second order
differential equation. It has two linearly independent solutions {\it for
every} energy $E$: positive, negative, real, imaginary, quaternionic --- it
doesn't matter. The thing that puts the ``quantum'' in quantum mechanics is
requiring that the solution be normalizable. Many peculiar things can happen
if we abandon allow normalizability \cite{RPW1,TW0}.
However, my particle theory colleagues ignore this little problem and define
a completely formal ``space of states'' based upon $\vert\overline{\Omega}
\rangle$,
\begin{equation}
\vert \overline{N_+ , N_-}\rangle \equiv \frac{(\alpha_+^{\dagger})^{N_+}}{
\sqrt{N_+ !}} \frac{(\alpha_-)^{N_-}}{\sqrt{N_- !}} \vert \overline{\Omega}
\rangle \; .
\end{equation}
None of these wavefunctions is any more normalizable than $\overline{\Omega}(
Q_1,Q_2)$, so not a one of them corresponds to a state of the quantum system.
However, they are all positive energy eigenfunctions,
\begin{equation}
H \vert \overline{N_+ , N_-} \rangle = \Bigl(N_+ k_+ + N_- k_-\Bigr) \vert
\overline{N_+ , N_-} \rangle \; .
\end{equation}
My particle physics colleagues typically say they {\it define} $\vert
\overline{\Omega}\rangle$ to have unit norm. Because they have not changed the
commutation relations,
\begin{equation}
[\alpha_+,\alpha_+^{\dagger}] = 1 = [\alpha_-,\alpha_-^{\dagger}] \; ,
\end{equation}
the norm of any state with odd $N_-$ is negative! The lowest of these is,
\begin{equation}
\langle \overline{0,1} \vert \overline{0,1}\rangle =
\langle \overline{\Omega} \vert \alpha_-^{\dagger} \alpha_- \vert \overline{
\Omega} \rangle = - \langle \overline{\Omega} \vert \overline{\Omega} \rangle
\; .
\end{equation}
As I pointed out above, the reason this has happened is that we aren't
doing quantum mechanics any more. We ought to use the normalizable, but
indefinite energy eigenstates. What particle physicists do instead is to
reason that because the probabilistic interpretation of quantum mechanics
requires norms to be positive, the negative norm states must be excised from
the space of states. At this stage good particle physicists note that that
the resulting model fails to conserve probability \cite{KS}. Just as the
correctly-quantized, indefinite-energy theory allows processes which mix
positive and negative energy particles, so too the indefinite-norm theory
allows processes which mix positive and negative norm particles. It only
conserves probability on the space of ``states'' which includes both kinds of
norms. If we excise the negative norm states then probability is no longer
conserved.
So good particle physicists reach the correct conclusion --- that nondegenerate
higher derivative theories can't describe our universe --- by a somewhat
illegitimate line of reasoning. But who cares? They got the right answer! Of
course {\it bad} particle physicists regard the breakdown of unitarity as a
challenge for inspired tinkering to avoid the problem. Favorite ploys are the
Lee-Wick reformulation of quantum field theory \cite{LW} and nonperturbative
resummations. The analysis also typically involves the false notion that
high frequency ghosts decouple, which I debunked at the end of sub-section
\ref{sub:3.2}. When the final effort is written up and presented to the world,
some long-suffering higher derivative expert gets called away from his research
to puzzle out what was done and explain why it isn't correct. {\it Sigh}. The
problem is so much clearer in its negative energy incarnation! I could list
many examples at this point, but I will confine myself to citing a full-blown
paper debunking one of them \cite{TW1}. It is also appropriate to note that
Hawking and Hertog have previously called attention to the mistake of
quantizing higher derivative theories using nonnormalizable wave functions
\cite{HH}.
\subsection{Constraints}
\label{sub:3.5}
The only way anyone has ever found to avoid the Ostrogradskian instability on a
nonperturbative level is by violating the single assumption needed to make
Ostrogradski's construction: nondegeneracy. Higher derivative theories for
which the definition of the highest conjugate momentum (\ref{nondeg}) cannot be
inverted to solve for the highest derivative can sometimes be stable. An
interesting example of this kind is the rigid, relativistic particle studied by
Plyushchay \cite{MSP,DZ}.
Degeneracy is of great importance because {\it all theories which possess
continuous symmetries are degenerate,} irrespective of whether or not they
possess higher derivatives. A familiar example is the relativistic point
particle, whose dynamical variable is $X^{\mu}(\tau)$ and whose Lagrangian is,
\begin{equation}
L = -m \sqrt{-\eta_{\mu\nu} \dot{X}^{\mu} \dot{X}^{\nu}} \; .
\end{equation}
The conjugate momentum is,
\begin{equation}
P_{\mu} \equiv \frac{m \dot{X}_{\mu}}{\sqrt{-\dot{X}^2}} \; .
\end{equation}
Because the right hand side of this equation is homogeneous of degree zero
one can not solve for $\dot{X}^{\mu}$. The associated continuous symmetry is
invariance under reparameterizations $\tau \rightarrow \tau'(\tau)$,
\begin{equation}
X^{\mu}(\tau) \longrightarrow X^{\prime \mu}(\tau) \equiv X^{\mu}\Bigl({\tau'
}^{-1}(\tau)\Bigr) \; .
\end{equation}
The cure for symmetry-induced degeneracy is simply to fix the symmetry by
imposing gauge conditions. Then the gauge-fixed Lagrangian should no longer
be degenerate in terms of the remaining variables. For example, we might
parameterize so that $\tau = X^0(\tau)$, in which case the gauge-fixed
particle Lagrangian is,
\begin{equation}
L_{\rm GF} = -m \sqrt{1 - \dot{\vec{X}} \cdot \dot{\vec{X}} } \; .
\end{equation}
In this gauge the relation for the momenta is simple to invert,
\begin{equation}
P_i \equiv \frac{m \dot{X}_i}{\sqrt{1 - \dot{\vec{X}} \cdot \dot{\vec{X}} }}
\qquad \Longleftrightarrow \qquad \dot{X}^i = \frac{P^i}{\sqrt{m^2 + \vec{P}
\cdot \vec{P}}} \; .
\end{equation}
When a continuous symmetry is used to eliminate a dynamical variable, the
equation of motion of this variable typically becomes a {\it constraint}.
For symmetries enforced by means of a compensating field --- such as local
Lorentz invariance is with the antisymmetric components of the vierbein
\cite{RPW2} --- the associated constraints are tautologies of the form $0 = 0$.
Sometimes the constraints are nontrivial, but implied by the equations of
motion. An example of this kind is the relativistic particle in our synchronous
gauge. The equation of the gauge-fixed zero-component just tells us the
Hamiltonian is conserved,
\begin{equation}
\frac{d}{d\tau} \Biggl( \frac{m \dot{X}_0}{\sqrt{-\eta_{\mu\nu} \dot{X}^{\mu}
\dot{X}^{\nu}}} \Biggr) = 0 \longrightarrow \frac{d}{dt} \Bigl(
\sqrt{m^2 + \vec{p} \cdot \vec{p} } \Bigr) = 0 \; .
\end{equation}
And sometimes the constraints give nontrivial relations between the canonical
variables that generate residual, time-independent symmetries. In this case
another degree of freedom can be removed (``gauge fixing counts twice,'' as
van Nieuwenhuizen puts it). An example of this kind of constraint is Gauss'
Law in temporal gauge electrodynamics.
Were it not for constraints of this last type, the analysis of a higher
derivative
theory with a gauge symmetry would be straightforward. One would simply fix
the gauge and then check whether or not the gauge-fixed Lagrangian depends
nondegenerately upon higher time derivatives. If it did, the conclusion would
be that the theory suffers the Ostrogradskian instability. However, when
constraints of the third type are present one must check whether or not they
affect the instability. This is highly model dependent but a very simple rule
seems to be generally applicable: {\it if the number of gauge constraints is
less than the number of unstable directions in the canonical phase space then
there is no chance for avoiding the problem}. Because the number of constraints
for any symmetry is fixed, whereas the number of unstable directions increases
with the number of higher derivatives, one consequence is that gauge
constraints can at best avoid instability for some fixed number of higher
derivatives. For example, the constraints of the second derivative model of
Plyushchay are sufficient to stabilize the system \cite{MSP,DZ}, but one would
expect it to become unstable if third derivatives were added.
People sometimes make the mistake of believing that the Ostrogradskian
instability can be avoided with just a single, global constraint on the
Hamiltonian. For example, Boulware, Horowitz and Strominger \cite{BHS} showed
the energy is zero for any asymptotically flat solution of the higher
derivative field equations derived from the Lagrangian,
\begin{equation}
\mathcal{L} = \alpha R^2 \sqrt{-g} + \beta R^{\mu\nu} R_{\mu \nu} \sqrt{-g}
\; .
\end{equation}
As I explained in sub-section \ref{sub:3.1}, the nature of the Ostrogradskian
instability is not that the energy decays but rather that the system evaporates
to a very highly excited state of compensating, positive and negative energy
degrees of freedom. As long as $\beta \neq 0$, there are six independent,
higher derivative momenta at each space point, whereas there are only four
local constants --- or five if $\alpha$ and $\beta$ are such as to give local
conformal invariance. Hence there are two (or one) unconstrained instabilities
per space point. There are an infinite number of space points, so the addition
of a single, global constraint does not change anything. I should point out
that Boulware, Horowitz and Strominger were aware of this, cf. their discussion
of the dipole instability.
The case of $\beta = 0$ is special, and significant for the next section. If
$\alpha$ has the right sign that model has long been known to have positive
energy \cite{AAS0,AS}. This result in no way contradicts the previous analysis.
When $\beta = 0$ the terms which carry second derivatives are contracted in
such way that only a single component of the metric carries higher derivatives.
So now the counting is {\it one} unstable direction per space point versus four
local constraints. Hence the constraints can win, and they do if $\alpha$ has
the right sign.
\subsection{Nonlocality}
\label{sub:3.6}
I would like to close this section by commenting on the implications of
Ostrogradski's theorem for fully nonlocal theories. In addition to nonlocal
quantum field theories \cite{KW0,CHY,JJ} this is relevant to string field
theory \cite{GJ1,GJ2,GJ3}, to noncommutative geometry \cite{NC,HPR}, to
regularization techniques \cite{EMKW,KW1,KW2} and even to theories of cosmology
\cite{TW2,SW1,BMS}. The issue in each case is whether or not we can think of
the fully nonlocal theory as the limit of a sequence of ever higher derivative
theories. When such a representation is possible the nonlocal theory must
inherit the Ostrogradskian instability.
The higher derivative representation is certainly valid for string field
theory because, otherwise, there would be cuts and poles that would interfere
with perturbative unitarity. So string field theory suffers from the
Ostrogradskian instability \cite{EW}. The same is true for theories where the
nonlocality is of limited extent in time \cite{RPW3}, although not everyone
agrees \cite{JL,RPW4}. However, when the nonlocality involves inverse
differential operators there need be no problem \cite{EW,SW1}. Indeed, the
effective action of any quantum field theory is nonlocal in this way
\cite{BGVZ,BM}! Nor is there necessarily any problem when the nonlocality
arises in the form of algebraic functions of local actions \cite{BNW}.
\section{$\Delta R[g] = f(R)$ Theories}
\label{sec:4}
From the lengthy argumentation of the previous two sections one might conclude
that the only potentially stable, local modification of gravity is a
cosmological constant,
$\Delta R[g] = - 2\Lambda$. However, a close analysis of sub-section
\ref{sub:3.5} reveals that it is also possible to consider algebraic functions
of the Ricci scalar. In this section I first explain why such theories can
avoid the Ostrogradskian instability. I then demonstrate that they are
equivalent to general relativity with a minimally coupled scalar, provided
we ignore matter. Finally, I exploit this equivalence, with the construction
described in the Introduction, to show how $f(R)$ can be chosen to enforce
any evolution $a(t)$.
\subsection{Why They Can Be Stable}
\label{sub:4.1}
The alert reader will have noted that the $R + R^2$ model \cite{AAS0,AS}
avoids the Ostrogradskian instability. It does this by violating Ostrogradski's
assumption of nondegeneracy: the tensor indices of the second derivative terms
in the Ricci scalar are contracted together so that only a single component of
the metric carries higher derivatives. This component does acquire a new,
higher derivative degree of freedom, and the energy of this degree of freedom
is indeed opposite to that of the corresponding lower derivative degree of
freedom, just as required by Ostrogradski's analysis. However, that lower
derivative degree of freedom is the {\it Newtonian potential}. It carries
negative energy, but it is also completely fixed in terms of the other metric
and matter fields by the $g_{00}$ constraint. So the only instability
associated with it is gravitational collapse. Its higher derivative counterpart
has positive energy, at least on the kinetic level; it can still have a
bad potential, and the model is indeed only stable for one sign of
the $R^2$ term.
None of these features depended especially upon the higher derivative term
being $R^2$. Any function for the Ricci scalar would work as well. Note that
we cannot allow derivatives of the Ricci scalar, because Ostrogradski's theorem
says the next higher derivative degree of freedom would carry negative energy
and there would be no additional constraints to protect it. We also cannot
permit more general contractions of the Riemann tensor because then other
components of the metric would carry higher derivatives. These components are
positive energy in general relativity, so their higher derivative counterparts
would be negative, and there would again be no additional constraints to
protect the theory against instability.
\subsection{Equivalent Scalar Representation}
\label{sub:4.2}
The general Lagrangian we wish to consider takes the form,
\begin{equation}
\mathcal{L} = \frac1{16 \pi G} \Bigl( R + f(R)\Bigr) \sqrt{-g} \; .
\end{equation}
If we ignore the coupling to matter the modified gravitational field equation
consists of the vanishing of the following tensor,
\begin{equation}
\frac{16 \pi G}{\sqrt{-g}} \frac{\delta S}{\delta g^{\mu\nu}} = [1 \!+\! f'(R)]
R_{\mu\nu} - \frac12 [R \!+\! f(R)] g_{\mu\nu} + g_{\mu\nu} [f'(R)]^{;\rho}_{
~\rho} - [f'(R)]_{;\mu\nu} \; . \label{MGR}
\end{equation}
There is an old procedure for reformulating this as general relativity with a
minimally coupled scalar. I don't know whom to credit, but I will give the
construction.
The first step is to define an ``equivalent'' theory with an auxiliary field
$\phi$ which is defined by the relation.
\begin{equation}
\phi \equiv 1 + f'(R) \qquad \Longleftrightarrow \qquad R = \mathcal{R}(\phi)
\; .
\end{equation}
Inverting the relation determines the Ricci scalar as an algebraic function
of $\phi$. We can then define an auxiliary potential for $\phi$ by Legendre
transformation,
\begin{equation}
U(\phi) \equiv \Bigl(\phi \!-\! 1\Bigr) \mathcal{R}(\phi) -
f\Bigl(\mathcal{R}(\phi)\Bigr) \qquad \Longrightarrow \qquad U'(\phi) =
\mathcal{R}(\phi) \; .
\end{equation}
Now consider the equivalent scalar-tensor theory whose Lagrangian is,
\begin{equation}
\mathcal{L}_{\rm E} \equiv \frac1{16 \pi G} \Bigl(\phi R - U(\phi)\Bigr)
\sqrt{-g} \; .
\end{equation}
Its field equations are,
\begin{eqnarray}
\frac{16 \pi G}{\sqrt{-g}} \frac{\delta S_{\rm E}}{\delta \phi} & = & R -
U'(\phi) = 0 \; , \label{E11} \\
\frac{16 \pi G}{\sqrt{-g}} \frac{\delta S_{\rm E}}{\delta g^{\mu\nu}} & = &
\phi R_{\mu\nu} - \frac12 \Bigl(\phi R \!-\! U(\phi)\Bigr) g_{\mu\nu} +
g_{\mu\nu} \phi^{;\rho}_{~\rho} - \phi_{\mu\nu} = 0 \; . \label{E12}
\end{eqnarray}
The scalar equation (\ref{E11}) implies $\phi \!=\! 1 \!+\! f'(R)$, whereupon
the tensor equations (\ref{E12}) reproduce the original modified gravity
equations (\ref{MGR}).
The final step is to define a new metric $\widetilde{g}_{\mu\nu}$ and a new
scalar $\varphi$ by the change of variables,
\begin{eqnarray}
\widetilde{g}_{\mu\nu} \equiv \phi \, g_{\mu\nu} \qquad & \Longleftrightarrow &
\qquad g_{\mu\nu} = \exp\Bigl[-\sqrt{\frac{4\pi G}3} \, \varphi\Bigr] \,
\widetilde{g}_{\mu\nu} \; , \label{gtrans} \\
\varphi \equiv \sqrt{\frac3{4\pi G}} \, \ln(\phi) \qquad & \Longleftrightarrow
& \qquad \phi = \exp\Bigl[\sqrt{\frac{4\pi G}3} \, \varphi\Bigr] \; .
\end{eqnarray}
In terms of these variables the equivalent Lagrangian takes the form,
\begin{equation}
\mathcal{L}_E = \frac1{16 \pi G} \widetilde{R} \sqrt{-\widetilde{g}}
-\frac12 \partial_{\mu} \varphi \partial_{\nu} \varphi \,
\widetilde{g}^{\mu\nu} \sqrt{-\widetilde{g}} - V(\varphi)
\sqrt{-\widetilde{g}} \; , \label{finalL}
\end{equation}
where the scalar potential is,
\begin{equation}
V(\varphi) \equiv \frac1{16 \pi G} U\Biggl(\exp\Bigl[
\sqrt{\frac{4 \pi G}3} \, \varphi\Bigr]\Biggr) \exp\Bigl[-\sqrt{\frac{16 \pi
G}3} \, \varphi\Bigr] \; .
\end{equation}
This is general relativity with a minimally coupled scalar, as claimed.
\subsection{Reconstructing $f(R)$ from Cosmology}
\label{sub:4.3}
I want to show how to choose $f(R)$ to support an arbitrary $a(t)$.\footnote{
For a somewhat different construction which achieves the same end, see
\cite{NO0,NO00}.} Recall from the Introduction that one can choose the
potential of a quintessence model such as (\ref{finalL}) to support any
homogeneous and isotropic cosmology for its metric $\widetilde{g}_{\mu\nu}$.
However, we cannot immediately exploit this construction because it is the
metric $g_{\mu\nu}$ which is assumed known, not $\widetilde{g}_{\mu\nu}$. We
must explain how to infer the one from the other without knowing $f(R)$.
Because the relation (\ref{gtrans}) between $g_{\mu\nu}$ and $\widetilde{g}_{
\mu\nu}$ is a conformal transformation, it makes sense to work in a coordinate
system in which each metric is conformal to flat space. This is accomplished
by changing from co-moving time $t$ to conformal time $\eta$ though the
relation, $d\eta = dt/a(t)$,
\begin{equation}
ds^2 = -dt^2 + a^2(t) d\vec{x} \cdot d\vec{x} = a^2 \Bigl(-d\eta^2 + d\vec{x}
\cdot d\vec{x}\Bigr) \; .
\end{equation}
The $\widetilde{g}_{\mu\nu}$ element takes the same form in conformal
coordinates, but note that its different scale factor implies a different
co-moving time,
\begin{equation}
d\widetilde{s}^2 = \widetilde{a}^2 \Bigl(-d\eta^2 + d\vec{x} \cdot
d\vec{x}\Bigr) = -d\widetilde{t}^{~2} + \widetilde{a}^2(\widetilde{t}\,)
d\vec{x} \cdot d\vec{x} \; .
\end{equation}
From relation (\ref{gtrans}) we infer,
\begin{equation}
a(t) = \widetilde{a}(\widetilde{t}\,) \exp\Bigl[-\sqrt{\frac{\pi G}3} \,
\varphi_0(\widetilde{t}\,) \Bigr] \; . \label{keyrel}
\end{equation}
We denote differentiation with respect to $\eta$ by a prime, and one should
note the relation between derivatives with respect to the various times,
\begin{equation}
\frac{\partial}{\partial \eta} = a \frac{\partial}{\partial t} =
\widetilde{a} \frac{\partial}{\partial \widetilde{t}} \; .
\end{equation}
Differentiating the logarithm of (\ref{keyrel}) with respect to $\eta$ and
using the relation (\ref{twoeqns}) between $\widetilde{a}$ and $\varphi_0$
gives,
\begin{equation}
\frac{a'}{a} = \frac{\widetilde{a}'}{\widetilde{a}}
-\sqrt{\frac{\pi G}3} \, \varphi_0'
= \frac{\widetilde{a}'}{\widetilde{a}} -\sqrt{-\frac1{12} \widetilde{a}'} \; .
\end{equation}
This is a nonlinear but first order differential equation for the variable
$\widetilde{a}$ in terms of the known function, $a(t(\eta))$. At the worst
it can be solved numerically.
Once we have $\widetilde{a}$ the potential $V(\varphi)$ can be constructed
using the procedure explained in the Introduction. We then compute the
auxiliary potential,
\begin{equation}
U(\phi) = 16 \pi G \phi^2 V\Bigl( \sqrt{\frac3{4\pi G}} \, \ln(\phi)\Bigr) \; .
\end{equation}
The auxiliary field can be expressed in terms of the Ricci scalar from the
algebraic relation,
\begin{equation}
U'(\phi) = R \qquad \Longleftrightarrow \qquad \phi = \Phi(R) \; .
\end{equation}
And we finally recover the function $f(R)$ by Legrendre transformation,
\begin{equation}
f(R) = \Bigl(\Phi(R) \!-\! 1\Bigr) R - U\Bigl(\Phi(R)\Bigr) \; .
\end{equation}
\section{Problems with $f(R) = -\frac{\mu^4}{R}$}
\label{sec:5}
In view of the construction of sub-section \ref{sub:4.3} it is not surprising
but rather {\it inevitable} that an $f(R)$ can be found to support late time
acceleration, or indeed, any other evolution. However, the method is not
guaranteed to produce a simple model, so the discovery that $f(R) = -\mu^4/R$
works is quite noteworthy \cite{CDTT,CCT}.\footnote{Although extensions
involving $R^{\mu\nu} R_{\mu\nu}$ and $R^{\rho\sigma\mu\nu}
R_{\rho\sigma\mu\nu}$ have also been studied \cite{CDDETT}, they must be ruled
out on account of the Ostrogradskian instability.} It may also be significant
that models of this type seem to follow from fundamental theory \cite{NO1}.
To derive acceleration in this model consider its field equations,
\begin{equation}
\Bigl(1 \!+\! \frac{\mu^4}{R^2}\Bigr) R_{\mu\nu} - \frac12 \Bigl(1 \!-\!
\frac{\mu^4}{R^2}\Bigr) R g_{\mu\nu} + \Bigl(g_{\mu\nu} \square - D_{\mu}
D_{\nu}\Bigr) \frac{\mu^4}{R^2} = 8 \pi G T_{\mu\nu} \; . \label{theeqn}
\end{equation}
Setting $T_{\mu\nu} \!=\! 0$ and searching for constant Ricci scalar solutions
gives,
\begin{equation}
\Bigl(1 \!+\! \frac{\mu^4}{R^2}\Bigr) R_{\mu\nu} - \frac12 \Bigl(1 \!-\!
\frac{\mu^4}{R^2}\Bigr) R g_{\mu\nu} = 0 \qquad \Longleftrightarrow \qquad
R_{\mu\nu} = \pm \frac{\sqrt{3}}4 \mu^2 g_{\mu\nu} \; .
\end{equation}
The plus sign corresponds to acceleration.
In addition to proposing the model, Carroll, Duvvuri, Trodden and Turner
\cite{CDTT} also showed that it suffers from a very weak tachyonic instability
in the absence of matter. Because the only new higher derivative degree of
freedom resides in the Ricci scalar, we may as well derive an equation for it
alone from the trace of (\ref{theeqn}),
\begin{equation}
-R + \frac{3\mu^4}{R} + \square \Bigl(\frac{3\mu^4}{R^2}\Bigr) = 0 \; .
\end{equation}
Now perturb about the accelerated solution,
\begin{equation}
R = +\sqrt{3} \mu^2 + \delta R \quad \Longrightarrow \quad -2 \delta R
-\frac{2}{\sqrt{3} \mu^2} \square \delta R + O(\delta R^2) = 0 \; .
\end{equation}
By comparing the linearized equation for $\delta R$ with that of a positive
mass-squared scalar,
\begin{equation}
(\square - m^2) \varphi = 0 \; , \label{comp}
\end{equation}
we see that $\delta R$ behaves like a tachyon with $m^2 = -\sqrt{3} \mu^2$.
However, because explaining the current phase of acceleration requires
$\mu \sim 10^{-33}~{\rm eV}$, the resulting instability is not very serious.
I should note that the existence of a tachyonic instability in no way
contradicts the Ostrogradskian analysis that this model's higher derivative
degree of freedom carries positive kinetic energy.
\subsection{Inside Matter}
\label{sub:5.1}
Dolgov and Kawasaki \cite{DK} showed that a radically different result emerges
when this model is considered inside a static distribution of matter,
\begin{equation}
T_{\mu\nu} = \rho \delta_{\mu}^0 \delta_{\nu}^0 \qquad {\rm with} \qquad
8 \pi G \rho \equiv M^2 \gg \mu^2 \; . \label{Dolgov}
\end{equation}
In that case the trace of (\ref{theeqn}) gives,
\begin{equation}
-R + \frac{3 \mu^4}{R} + \square \Bigl(\frac{3 \mu^4}{R^2} \Bigr) = - M^2 \; .
\end{equation}
As might be expected, the static Ricci scalar solution in this case is
dominated by $M$ rather than $\mu$,
\begin{equation}
R_0 = \frac12 \Bigl(M^2 \!+\! \sqrt{M^4 \!+\! 12 \mu^4}\Bigr) \simeq M^2 \; .
\end{equation}
Perturbing about this solution gives,
\begin{equation}
R = R_0 + \delta R \quad \Longrightarrow \quad - \delta R -\frac{3\mu^4}{
R_0^2} \delta R -\frac{6 \mu^4}{R_0^3} \square \delta R + O(\delta R^2) = 0\; .
\end{equation}
Comparing with the reference scalar (\ref{comp}) now reveals an enormous
tachyonic mass,
\begin{equation}
m^2 = -\frac{R_0}2 -\frac{R_0^3}{6 \mu^4} \simeq -\frac{M^6}{6 \mu^4} \; !
\end{equation}
Plugging in the numbers for the density of water ($\rho \sim 10^3~{\rm kg/m}^3$)
gives $M \sim 10^{-18}~{\rm eV}$, implying a tachyonic mass of magnitude
$\vert m\vert \sim 10^{12}~{\rm eV} = 10^3~{\rm GeV}$!
As disastrous as this problem might seem, Dick \cite{RD} and Nojiri and
Odintsov \cite{NO2} have shown that it can be avoided by changing the model
slightly,
\begin{equation}
f(R) = -\frac{\mu^4}{R} + \frac{\alpha}{2 \mu^2} R^2 \quad \Longrightarrow
\quad -R + \frac{3 \mu^4}{R} + 3 \square \Bigl( \frac{\mu^4}{R^2} +
\frac{\alpha}{\mu^2} R \Bigr) = 0 \; . \label{Ext}
\end{equation}
Because an $R^2$ term has global conformal invariance, it makes no contribution
to the trace for constant $R$. Hence the cosmological solution of $R = +
\sqrt{3} \mu^2$ is not affected, nor is the static solution inside the matter
distribution (\ref{Dolgov}). However, the equation for linearized perturbations
inside matter changes to,
\begin{equation}
-\delta R - \frac{3 \mu^4}{R_0^2} \delta R + 3 \Bigl(-\frac{2 \mu^4}{R_0^3}
\!+\! \frac{\alpha}{\mu^2} \Bigr) \square \delta R = 0 \; .
\end{equation}
The instability of Dolgov and Kawasaki was driven by the smallness of $2\mu^4/
R_0^3$. By simply taking $\alpha$ positive and of order one the tachyon becomes
a positive mass-squared particle of $m^2 \sim \mu^2/\alpha$.
\subsection{Outside Matter}
\label{sub:5.2}
Marc Soussa and I analyzed force of gravity outside a matter distribution
\cite{SW2}. Although our analysis was for the original $f(R)= -\mu^4/R$
model, there would be only slight differences for the extended model
(\ref{Ext}). So our result seems to foreclose this possibility, but see
\cite{NO3}.
The tachyonic instability could be studied using the perturbed Ricci scalar,
but the gravitational force requires use of the metric. We perturbed about the
de Sitter solution with Hubble constant $H = \mu/(48)^{\frac14}$ in co-moving
coordinates,
\begin{equation}
ds^2 = -(1 \!-\! h_{00}) dt^2 + 2 a(t) h_{0i} dt dx^i + a^2(t) (\delta_{ij}
\!+\! h_{ij}) dx^i dx^j \quad {\rm with} \quad a(t) = e^{H t} \; .
\end{equation}
In the gauge,
\begin{equation}
h_{\mu\nu}^{~~,\nu} - \frac12 h_{\mu} + 3 h_{\mu}^{~\nu} [\ln(a)]_{,\nu} = 0
\; ,
\end{equation}
with $h \equiv -h_{00} \!+\! h_{ii}$, the perturbed Ricci scalar takes the
form,
\begin{equation}
\delta R = -\frac12 \partial^2 h + 2 H \partial_0 h \; . \label{deltaR}
\end{equation}
Our strategy was first to solve the de Sitter invariant equation for the
perturbed Ricci scalar, then reconstruct the gauge-fixed metric.
We assumed a matter density of the form,
\begin{equation}
\rho(t,\vec{x}) = \frac{3 M}{4 \pi R_g^3} \theta\Bigl(R_g - a(t) \vert \vec{x}
\vert \Bigr) \; .
\end{equation}
The exterior field equation has a simple expression in terms of the coordinate
$y \equiv a(t) H \vert \vec{x}\vert$,
\begin{equation}
\Biggl[\Bigl(1 \!-\! y^2\Bigr) \frac{d^2}{dy^2} + \frac2{y} \Bigl(1 \!-\! 2 y^2
\Bigr) \frac{d}{dy} + 12 \Biggr] \delta R = 0 \; .
\end{equation}
The solution takes the form,
\begin{equation}
\delta R = \beta_1 f_0(y) + \beta_2 f_{-1}(y) \; , \label{genform}
\end{equation}
where $f_0$ and $f_{-1}$ are hypergeometric functions whose series expansions
are,
\begin{eqnarray}
f_0(y) & = & 1 - 2 y^2 + \frac15 y^4 + \ldots \; , \\
f_{-1}(y) & = & \frac1{y} \Bigl( 1 - 7 y^2 + \frac{14}3 y^4 + \ldots \Bigr)
\; .
\end{eqnarray}
We only need the behavior for small $y$ because $y \!=\! 1$ is the Hubble
radius! Matching to the source at $y = H R_g$ determines the combination
coefficients to be,
\begin{equation}
\beta_1 \simeq \frac{3 G M}{R_g^3} \qquad , \qquad \beta_2 \simeq -12 G M H^3
\; .
\end{equation}
This last step might seem bogus because we needed to regard the mass density as
a small perturbation on the cosmological energy density $\mu^4$, whereas the
opposite would be the case for galaxies or clusters of galaxies. However, this
will only make changes of order one in the $\beta_i$'s. In particular, the
asymptotic solution must still take the form (\ref{genform}).
The next step is solving for the trace of the perturbed metric. It turns out
that relation (\ref{deltaR}) can also be expressed very simply using the
variable $y$,
\begin{equation}
\Biggl[ \Bigl(y^2 \!-\! 1\Bigr) \frac{d}{dy} + \frac1{y} \Bigl(5 y^2 \!-\!2
\Bigr) \Biggr] h'(y) = \frac{2}{H^2} \delta R \; .
\end{equation}
We only need to solve for the derivative of $h$ because that is what gives
the gravitational force in the geodesic equation. The solution is,
\begin{equation}
h'(y) = -\frac{2 G M}{H^2 R_g^3} y + O(y^3) \; .
\end{equation}
This should be compared to the general relativistic prediction,
\begin{equation}
h'_{\rm GR}(y) = -\frac{4 G M H}{y^2} + O(1) \qquad \Longrightarrow \qquad
\frac{h'}{h'_{\rm GR}} = \frac12 \Bigl(\frac{\Vert \vec{x}\Vert}{R_g}\Bigr)^3
\; .
\end{equation}
One consequence is that the force between the Milky Way and Andromeda galaxies
would be about a million times larger than predicted by general relativity!
\section{Conclusions}
\label{sec:6}
The potential of a quintessence scalar can be chosen to support any cosmology,
but the epicyclic nature of this construction suggests we consider modifications
of gravity. Ostrogradski's theorem \cite{MO} limits local modifications of
gravity to just algebraic functions of the Ricci scalar. Models of this form
can give a late phase of cosmic acceleration such as we are currently
experiencing. However, they can be tuned to give anything else as well. They
seem every bit as epicyclic as scalar quintessence. Further, the $f(r) =
-\mu^4/R$ model is problematic, both inside and outside matter
sources.\footnote{Observations also rule out the somewhat different version
of this model that results from regarding the connection and the metric
as independent, fundamental variables in the Palatini formalism \cite{AEMM}.}
An interesting and largely overlooked possibility for modifying gravity is
the fully nonlocal effective action that results from quantum gravitational
corrections. In weak field perturbation theory it has long been known that
the most cosmologically significant one loop corrections are not of the $R^2$
form usually studied but rather of the form $R \ln(\square) R$ \cite{EMV}.
More potentially interesting is the possibility of very strong infrared
effects from the epoch of primordial inflation \cite{TW3,RBM}.
It can be shown that quantum gravitational corrections to the inflationary
expansion rate grow with time like powers of $\ln(a)$. Although suppressed by
very small coupling constants, the exponential growth in $a(t)$ during
inflation must eventually cause the effect to become nonperturbatively strong
\cite{TW4,TW5}. Similar secular growth occurs as well for minimally coupled
scalar field theories \cite{OW1,OW2}, in which context Starobinski\u{\i} has
developed a technique for summing the leading powers of $\ln(a)$ at each
loop order \cite{AAS,SY}. If Starobinski\u{\i}'s technique can be generalized
to quantum gravity \cite{RPW5,TW6} it might result in a nonlocal effective
gravity theory for late time cosmology in which a large, bare cosmological
constant is almost completely screened by a nonperturbative quantum
gravitational effect. In such a formalism the current phase of acceleration
might result from a very slight mismatch between the bare cosmological
constant and the quantum effect which screens it. It is even conceivable that
one could reproduce the phenomenological successes of MOND \cite{MM,SM} with
such a nonlocal metric theory, although it would have to unstable against
decay into galaxy-scale gravitational waves \cite{SW3}.
\vskip 1cm
\centerline{Acknowledgements}
It is a pleasure to acknowledge conversations and correspondence on this
subject with S. Deser, A.D. Dolgov, D.A. Eliezer, S. Odintsov, M.E. Soussa,
A. Strominger and M. Trodden. This work was partially supported by NSF grant
PHY-244714 and by the Institute for Fundamental Theory at the University of
Florida.
\printindex |
Title:
Separation of dwarf and giant stars with ROTSE-IIId |
Abstract: 136 stars which were known to be the members of open cluster NGC 752 were
observed at R band with ROTSE-IIId telescope located at the Turkish National
Observatory (TUG) site. The data had been evaluated together with BV and 2MASS
photometric data. A new practical method for separating dwarf and giant was
described and applied. Evaluating the colour magnitude--diagrams with Padova
isochrones revealed metallicity similar to the Sun and an age of 1.41 Gyr for
the open cluster NGC 752.
| https://export.arxiv.org/pdf/astro-ph/0601681 |
\lhead[\thepage]{A.N. Bilir et al.: Separation of dwarf and giant stars with ROTSE-IIId}
\rhead[Astron. Nachr./AN~{\bf XXX} (200X) X]{\thepage}
\headnote{Astron. Nachr./AN {\bf 32X} (200X) X, XXX--XXX}
\title{Separation of dwarf and giant stars with ROTSE--IIId}
\author{S. Bilir
\and T. G\"uver
\and M. Aslan
}
\institute{Istanbul University Science Faculty,
Department of Astronomy and Space Sciences,
34119, University-Istanbul, Turkey}
\date{}
\abstract{136 stars which were known to be the members of open cluster NGC 752
were observed at $R$ band with ROTSE--IIId telescope located at the Turkish
National Observatory (TUG) site. The data had been evaluated together with BV
and 2MASS photometric data. A new practical method for separating dwarf and
giant was described and applied. Evaluating the colour magnitude--diagrams with
Padova isochrones revealed metallicity similar to the Sun and an age of 1.41 Gyr
for the open cluster NGC 752.
\keywords{Galaxy: open cluster and associations, stars: colour-magnitude
diagrams, stars: giants}
}
\correspondence{sbilir@istanbul.edu.tr}
\section{Introduction}
One of the problems of the Galactic astronomy is the estimation of Galactic model
parameters of giant stars in our Galaxy. In many studies, the Galactic model
parameters are estimated without any discrimination between dwarfs and giants,
whereas some researchers estimated model parameters only for certain star categories
(e.g. Pritchet 1983, Bahcall \& Soneira 1984, Buser \& Kaeser 1985 and Mendez \&
Altena 1996). A very recent work is an example for this where the Galactic model
parameters were estimated using only giants (Cabrera-Lavers, Garzon \& Hammersley
2005). The separation of field dwarfs and field giants plays an important role for such
kinds of works. The most efficient classical methods of identifying dwarf and giant
stars utilize spectroscopy. By inspecting spectral line profiles, one has to
estimate surface gravity to discriminate between higher and lower pressure stellar
atmospheres to be sure for the identification. This is, however time consuming and
tiring. A rather easier procedure is to separate dwarfs and evolved stars (subgiants
or giants) such as to obtain a luminosity function consistent with the local
luminosity function of nearby stars due to Gliese \& Jahreiss (1991) and Jahreiss
\& Wielen (1997). The procedure of this separation is based on the fact that the
local luminosity functions obtained for many fields indicates a systematic excess
of star counts relative to the luminosity function of nearby stars for the fainter
segment, i.e. $M(V)\geq5^{m}.5$, and a deficit for brighter segment, $M(V)<5^{m}.5$.
(in $RGU$ system $M(G)\geq6^{m}$ and $M(G)<6^{m}$, respectively). The works of
Karaali (1992); Ak, Karaali \& Buser (1998); Karata\c{s}, Bilir \& Karaali
(2000); Karaali et al. (2000); Karata\c{s}, Karaali \& Buser (2001); Karaali,
Bilir \& Buser (2004); Bilir, Karaali \& Buser (2004) and Karata\c{s} et al.
(2004) can be given as examples for application of this procedure.
Recently, a new method were suggested by Bilir et al. (2006) for separating the
field dwarfs and field giants. This new method is based on the comparison of the Two
Micron All Sky Survey (2MASS, hereafter) $J$, $H$, $K_{s}$ with the $V$ magnitudes
down to the limiting magnitude of $V=16$. In this work we extend application of
this method to the open cluster NGC 752 observed by a robotic telescope ROTSE-IIId
(Akerlof et al. 2003).
This paper is organized as follows. In Section 2 the BV, 2MASS and ROTSE data
are presented. In Section 3 the method is applied to ROTSE and 2MASS data, and
the separation of dwarf and giant stars is tested. In Section 4 colour-magnitude
diagrams (CMDs) of NGC 752 is compared to the Padova isochrones. Finally, the
conclusion is given in Section 5.
\begin{figure}
\center
\resizebox{5cm}{7cm}{\includegraphics*{fig01.eps}}
\caption {$V \times (B-V)$ CMD of 136 probable member stars in NGC 752.}
\end {figure}
\begin{figure*}
\center
\resizebox{17cm}{5.51cm}{\includegraphics*{fig02.eps}}
\caption {V and 2MASS magnitudes of 136 stars in NGC 752. (a) $J_{0} \times V_{0}$,
(b) $H_{0} \times V_{0}$, and (c) $Ks_{0} \times V_{0}$. The solid lines in diagrams
were drawn according to eqs. (1), (2), and (3).}
\end {figure*}
\section{Observations}
\subsection{The BV and 2MASS Data}
NGC 752=C0154+374 ($\alpha=01^{h}57^{m}41^{s}$, $\delta=+37^{o}47^{'}06^{''}$;
$l=137^{o}.13$, $b=-23^{o}.25$; J2000) has been subject of many studies, because
it is the nearest intermediate-age cluster, with 427 pc (Dzervitis \& Paupers
1993) distance from the Sun. It is usually considered as metal-deficient with respect
to the Sun, $[Fe/H]=-0.15 \pm$ 0.05 dex, slightly reddened $E(B-V)=0.035 \pm 0.005$,
with distance module $(m-M)=8.25\pm$0.10 (Daniel et al. 1994). Accurate proper motion
and radial velocity measurements show that there are 136 probable member stars of the
open cluster (Daniel et al. 1994). $V$ magnitudes and $(B-V)$ colour indices used in
this study were taken from Daniel et al. (1994) and they were given in Table 1. The
$V\times(B-V)$ CMD in Fig. 1 shows that stars with $V<10$ and $(B-V)>0.80$ are giants.
Recently the 2MASS, including the Point-Source Catalogue and Atlas, has produced huge
amounts of data to be explored in the coming years (Skrutskie et al. 1997). The
photometric system comprises Johnson's $J$ (1.25 $\mu$m) and $H$ (1.65 $\mu$m) bands with
the addition of $K_{s}$ (2.17 $\mu$m), slightly bluer than Johnson's K. The 2MASS sky
coverage, homogeneity and depth will certainly make this set of filters a photometric
standard reference for the future.
2MASS data of the 136 probable member stars in NGC 752 are obtained by
Vizier\footnote{http://vizier.u-strasbg.fr/viz-bin/Vizier?-source=2MASS} in CDS
and they are given in Table 1. We used the equations of Fiorucci \& Munari (2003)
for the determination of the total absorptions for the bands $V$, $J$, $H$ and
$K_{s}$, i.e. $A(V)=3.1E(B-V)$, $A(J)=0.887E(B-V)$, $A(H)=0.565E(B-V)$ and
$A(K_{s})=0.382E(B-V)$. Thus the de-reddened magnitudes were obtained as follows:
$V_{0}=V-A(V)$, $J_{0}=J-A(J)$, $H_{0}=H-A(H)$ and $(K_{s})_{0}=K_{s}-A(K_{s})$.
The subscript ``0'' indicates de-reddened magnitude.
$J_{0} \times V_{0}$, $H_{0} \times V_{0}$ and $(K_{s})_{0} \times V_{0}$ diagrams
for the cluster stars are given in Fig. 2. The solid lines represent the equations
in Bilir et al. (2006), i.e.
\begin{equation}
J_{0} = 0.957V_{0} - 1.079,
\end{equation}
\begin{equation}
H_{0} = 0.931V_{0} - 1.240,
\end{equation}
\begin{equation}
(K_{s})_{0} = 0.927V_{0} - 1.292.
\end{equation}
14 stars below the lines are the giants in Fig. 1, whereas 122 stars above the
lines are the dwarfs of the same cluster. The distribution of different star
categories at different sides of the lines in three figures were presented here
to confirm separation of dwarfs and giants.
\subsection{ROTSE data}
The Robotic Optical Transient Experiment (ROTSE-III) consists of four 0.45m
worldwide robotic, automated telescopes situated at different locations on Earth.
They are designed for fast ($\sim$ 6 sec) responses to Gamma-Ray Burst (GRB)
triggers from satellites such as Swift. Each ROTSE telescope has a $1.85 \times 1.85$
deg$^{2}$ field of view, and uses a Marconi 2048 $\times$ 2048 back illuminated
thinned CCD. These telescopes operate without filters, and have wide passband which
peaks around 550 nm (Akerlof et al. 2003). In this work, we present optical
observations of NGC 752 performed by ROTSE-IIId, telescope located at Turkish National
Observatory (TUG) site, Bak$\i$rl$\i$tepe, Turkey. The observations took place between
MJD 53637 (September 2005) and MJD 53649 (October 2005). A total of about 217 CCD
frames were analyzed. After determining the instrumental magnitudes (Bertin \& Arnouts,
1996), they were reduced to ROTSE magnitudes via comparing all the field stars with
the USNO A2.0 $R$-band catalog. All the processes were done in an automated mode.
\begin{figure}
\center
\resizebox{6.51cm}{6.22cm}{\includegraphics*{fig03.eps}}
\caption {$R_{0} \times V_{0}$ magnitudes of 136 stars in NGC 752.}
\end {figure}
\begin{figure*}
\center
\resizebox{17cm}{5.51cm}{\includegraphics*{fig04.eps}}
\caption {$R_{0}$ and 2MASS magnitudes of 136 stars in NGC 752. (a) $J_{0} \times R_{0}$,
(b) $H_{0} \times R_{0}$, and (c) $Ks_{0} \times R_{0}$. The solid lines in diagrams were
drawn according to eqs. (5), (6), and (7).}
\end {figure*}
$R$-band magnitudes of the 136 stars are given in Table 1. ROTSE magnitudes are also
de-reddened in order to homogenize the data. The total absorption for the $R$ band
could be determined by the equation of Fiorucci \& Munari (2003), i.e.
$A(R)=2.613E(B-V)$. Thus the de-reddened magnitude in $R$ becomes $R_{0}=R-A(R)$.
\section{Application of the method to ROTSE-IIId data}
We used the following relation between the $V_{0}$ magnitude and the ROTSE-IIId
magnitude $R_{0}$ for 136 stars of the open cluster NGC 752 (Fig. 3) in order to
apply the method to ROTSE-IIId data:
\begin{equation}
V_{0}=(1.019\pm0.006)R_{0}+0.092\pm0.070~~(\sigma=0.10).
\end{equation}
Thus, substituting the value of $V_{0}$ in (4) into (1), (2), and (3) we obtain the
relations between 2MASS and ROTSE-IIId magnitudes, i.e.
\begin{equation}
J_{0} = 0.975R_{0} - 0.991,
\end{equation}
\begin{equation}
H_{0} = 0.949R_{0} - 1.154,
\end{equation}
\begin{equation}
(K_{s})_{0}= 0.945R_{0} - 1.207.
\end{equation}
The diagrams $J_{0} \times R_{0}$, $H_{0} \times R_{0}$ and $(K_{s})_{0} \times R_{0}$
for the NGC 752 cluster stars and the line corresponding to the eqs. (5), (6), and (7)
are given in Fig. 4. One can see that dwarfs and giants lie at opposite sides of the
line in these figures, especially the relation between $(K_{s})_{0} \times R_{0}$
(Fig. 4c) is the most successful in separating of the two different star categories.
\section{Age estimation for the open cluster NGC 752 via two photometries}
We estimated the age of the open cluster NGC 752 by the means of BV and 2MASS
data to show the advantage. The absolute magnitudes of the stars were determined
by the corresponding apparent magnitudes and the distance module of the cluster.
The Padova isochrones were taken from Girardi et al. (2002) and Bonatto, Bica \&
Girardi (2004) for BV\footnote
{http://pleiadi.pd.astro.it/isoc\_photsys.02/isoc\_photsys.02.html} and 2MASS\footnote{
http://pleiadi.pd.astro.it/isoc\_photsys.01/isoc\_2mass/index.html}
photometry, respectively. The isochrone sets were computed with updated opacities,
and equations of state, and a moderate amount of convective overshoot. The basic
isochrone set presented in Girardi et al. (2002) covers a very wide range of initial
masses (from 0.15 to $\sim 100 M_{\odot}$), metallicities, and photometric systems,
being well suited for studies of clusters of all ages.
The isochrones mentioned above were fitted to the CMDs in Figs. 5-7 for two sets
of chemical compositions, i.e. Z=0.019, Y=0.273 (panel a) and Z=0.008, Y=0.250
(panel b). The isochrones in Fig. 5a fits to the main-sequence and turn-off segments
of the $M_{V} \times (B-V)_{0}$ diagram and reveal an age of $t=1.26$ Gyr, whereas
in Fig. 5b, the isochrones could be fitted only to the giant branch of the same CMD,
resulting a larger age, i.e. $t=1.78$ Gyr. The isochrones could not be fitted to
all segments of the $M_{J} \times (J-H)_{0}$ diagram in Fig. 6 either. In Fig. 6a
the fit is better to the main-sequence and turn-off segments, however it is only to
the giant branch in Fig. 6b. The best fit is accomplished with the isochrone of age
$t=1.41$ Gyr to the $M_{J} \times (J-K_{s})_{0}$ in Fig. 7a. In fact, the isochrone
fits to all segments, main-sequence, turn-off and giant branch, for a metallicity
close to the solar one which is expected (Daniel et al. 1994). Thus, the comparison
of the six diagrams in Figs. 5-7 reveals that the CMD $M_{J} \times (J-K_{s})_{0}$
is the best one which fits for the age estimation.
\begin{figure}
\center
\resizebox{8cm}{6.92cm}{\includegraphics*{fig05.eps}}
\caption {$M_{V} \times (B-V)_{0}$ CMDs of NGC 752. (a) Z=0.019 and 1.12, 1.26 and 1.41
Gyr and (b) Z=0.008 and 1.58, 1.78 and 2.00 Gyr of Padova isochrones.}
\end {figure}
\begin{figure}
\center
\resizebox{8cm}{6.92cm}{\includegraphics*{fig06.eps}}
\caption {$M_{J} \times (J-H)_{0}$ CMDs of NGC 752. (a) Z=0.019 and 1.00, 1.12 and 1.26
Gyr and (b) Z=0.008 and 1.41, 1.59 and 1.78 Gyr of Padova isochrones.}
\end {figure}
\begin{figure}
\center
\resizebox{8cm}{6.92cm}{\includegraphics*{fig07.eps}}
\caption {$M_{J} \times (J-Ks)_{0}$ CMDs of NGC 752. (a) Z=0.019 and 1.26, 1.41 and 1.59
Gyr and (b) Z=0.008 and 1.59, 1.78 and 1.99 Gyr of Padova isochrones.}
\end {figure}
\section{Conclusion}
In this study, we have reduced the relations between the 2MASS and $V$ magnitudes by
which field dwarfs and giants can be separated, to the USNO A2.0 $R$-band magnitudes.
The $R_{0}$ magnitudes of 136 stars in open cluster NGC 752 were transferred to
the $V_{0}$ magnitudes, and the relations between $J_{0}$, $H_{0}$, $(K_{s})_{0}$
and $R_{0}$ were derived by means of the relations between $J_{0}$, $H_{0}$,
$(K_{s})_{0}$ and $V_{0}$ given by Bilir et al. (2006). Dwarf and giant stars
identified by the CMD of open cluster NGC 752 lie at different sides of the line
representing the relation between the 2MASS and $R_{0}$ magnitudes, in the
$J_{0} \times R_{0}$, $H_{0} \times R_{0}$ and $(K_{s})_{0} \times R_{0}$
diagrams. The best one is the last diagram, i.e. $(K_{s})_{0} \times R_{0}$.
Thus, dwarf-giant separation could be carried out also in the ROTSE-IIId data.
Proven to be successful, this practical method can provide good contributions to
the studies of Galactic model parameters in which separation of dwarfs and giants
were needed.
A set of Padova isochrones were fitted to the CMDs of the open cluster NGC
752 using BV and 2MASS photometric data. It turned out that the isochrone with
chemical composition Z=0.019 and Y=0.273 which reveals an age of 1.41 Gyr
for the open cluster NGC 752 could be fitted to all segments, i.e.
main-sequence, turn-off and giant branch, of the $M_{J} \times (J-K_{s})_{0}$
two-colour diagram. This result is very close to the age 1.24$\pm$0.20
Gyr which Salaris, Weiss \& Percival (2004) calculated from the morphology
of 71 open clusters in our Galaxy. The metal-abundance of the cluster given
by Daniel et al. (1994), i.e. $[Fe/H]=-0.15\pm0.05$ dex, is a strong
confirmation for our result.
\begin{acknowledgements}
We thank international ROTSE collaboration and TUG for the optical facilities
(Project number: TUG-ROTSE.05.14). This research has made use of the SIMBAD
database, operated at CDS, Strasbourg, France. This publication makes use of
data products from the Two Micron All Sky Survey, which is a joint project of
the University of Massachusetts and the Infrared Processing and Analysis
Center/California Institute of Technology, funded by the National Aeronautics
and Space Administration and the National Science Foundation. We would also
like to thank Dr. Salih Karaali for helpful comments and suggestions, Dr. Zeki
Eker for reading the whole manuscript and correction and Dr. Tansel Ak for
various helps. This work was supported by the Research Fund of the University
of Istanbul. Project number: BYP 914.
\end{acknowledgements}
\begin{table*}
{\scriptsize
\center
\caption{BV, ROTSE and 2MASS magnitudes and its errors of 136 probable member stars of open cluster NGC 752. ID name is the same as Daniel et al. (1994) and the coordinates are for the epoch 2000.}
\begin{tabular}{rcccccccccccccc}
\hline
ID & $\alpha$ & $\delta$ & \multicolumn{2} {c} {V} & \multicolumn{2} {c} {B-V}& \multicolumn{2} {c} {R} & \multicolumn{2} {c} {J}& \multicolumn{2} {c} {H}& \multicolumn{2} {c} {$K_{s}$}\\
\hline
143 & 01 54 31.04 & +37 29 31.6 & 11.140 & & 0.470 & & 10.970 & 0.010 & 10.343 & 0.019 & 10.184 & 0.024 & 10.151 & 0.020 \\
215 & 01 54 56.71 & +37 58 28.9 & 11.400 & & 0.430 & & 11.200 & 0.012 & 10.615 & 0.021 & 10.416 & 0.017 & 10.378 & 0.018 \\
222 & 01 54 59.65 & +37 28 59.8 & 11.651 & 0.009 & 0.489 & 0.007 & 11.365 & 0.011 & 10.607 & 0.021 & 10.377 & 0.018 & 10.293 & 0.021 \\
237 & 01 55 02.85 & +38 16 38.1 & 12.418 & 0.000 & 0.519 & 0.000 & 12.110 & 0.017 & 11.364 & 0.021 & 11.105 & 0.017 & 11.094 & 0.020 \\
245 & 01 55 07.13 & +37 32 36.9 & 14.190 & & 0.660 & & 13.830 & 0.026 & 12.838 & 0.022 & 12.464 & 0.021 & 12.397 & 0.021 \\
259 & 01 55 12.62 & +37 50 14.4 & 9.496 & 0.013 & 0.954 & 0.006 & 9.092 & 0.016 & 7.818 & 0.023 & 7.370 & 0.023 & 7.228 & 0.016 \\
264 & 01 55 15.29 & +37 50 31.2 & 9.569 & 0.005 & 0.996 & 0.003 & 9.090 & 0.017 & 7.795 & 0.023 & 7.317 & 0.016 & 7.197 & 0.018 \\
300 & 01 55 26.18 & +38 08 22.0 & 13.610 & & 0.700 & & 13.200 & 0.019 & 12.182 & 0.018 & 11.798 & 0.021 & 11.743 & 0.021 \\
305 & 01 55 27.68 & +37 34 04.5 & 10.152 & 0.003 & 0.411 & 0.001 & 9.940 & 0.011 & 9.293 & 0.018 & 9.103 & 0.015 & 9.062 & 0.016 \\
308 & 01 55 27.67 & +37 59 55.2 & 9.285 & 0.011 & 0.966 & 0.009 & 8.902 & 0.020 & 7.618 & 0.021 & 7.163 & 0.021 & 7.039 & 0.020 \\
313 & 01 55 29.29 & +37 50 26.2 & 9.968 & 0.013 & 0.470 & 0.001 & 9.718 & 0.014 & 9.001 & 0.023 & 8.792 & 0.018 & 8.696 & 0.017 \\
350 & 01 55 39.37 & +37 52 52.4 & 8.922 & 0.002 & 1.011 & 0.006 & 8.439 & 0.024 & 7.151 & 0.020 & 6.674 & 0.016 & 6.547 & 0.016 \\
356 & 01 55 42.39 & +37 37 54.3 & 9.161 & 0.003 & 1.010 & 0.001 & 8.687 & 0.020 & 7.395 & 0.019 & 6.900 & 0.020 & 6.800 & 0.018 \\
361 & 01 55 44.75 & +37 54 42.5 & 13.899 & 0.000 & 0.724 & 0.000 & 13.497 & 0.025 & 12.438 & 0.023 & 12.054 & 0.023 & 11.988 & 0.019 \\
363 & 01 55 44.93 & +38 08 21.4 & 10.420 & & 0.360 & & 10.189 & 0.013 & 9.619 & 0.021 & 9.464 & 0.021 & 9.410 & 0.020 \\
372 & 01 55 47.40 & +37 42 26.4 & 9.890 & 0.005 & 0.464 & 0.000 & 9.656 & 0.011 & 8.971 & 0.029 & 8.741 & 0.027 & 8.717 & 0.018 \\
391 & 01 55 53.54 & +37 49 26.6 & 13.890 & & 0.730 & & 13.501 & 0.024 & 12.457 & 0.023 & 12.072 & 0.021 & 11.985 & 0.023 \\
397 & 01 55 55.53 & +37 28 33.2 & 9.831 & 0.000 & 0.477 & 0.005 & 9.585 & 0.011 & 8.839 & 0.021 & 8.636 & 0.020 & 8.576 & 0.016 \\
413 & 01 55 59.44 & +37 40 48.5 & 12.303 & 0.016 & 0.517 & 0.022 & 12.035 & 0.013 & 11.181 & 0.021 & 10.924 & 0.017 & 10.842 & 0.016 \\
429 & 01 56 02.95 & +37 36 32.7 & 14.270 & & 0.850 & & 13.791 & 0.033 & 12.741 & 0.023 & 12.338 & 0.022 & 12.225 & 0.021 \\
435 & 01 56 03.69 & +37 59 22.4 & 11.467 & 0.017 & 0.447 & 0.010 & 11.223 & 0.013 & 10.572 & 0.020 & 10.384 & 0.018 & 10.324 & 0.018 \\
455 & 01 56 08.96 & +37 39 52.6 & 10.512 & 0.013 & 0.395 & 0.003 & 10.298 & 0.020 & 9.657 & 0.021 & 9.496 & 0.017 & 9.442 & 0.014 \\
461 & 01 56 10.30 & +37 44 59.9 & 10.054 & 0.002 & 0.384 & 0.006 & 9.752 & 0.041 & 9.239 & 0.021 & 9.115 & 0.018 & 9.015 & 0.019 \\
465 & 01 56 11.11 & +37 45 11.1 & 11.229 & 0.012 & 0.412 & 0.007 & 10.896 & 0.045 & 10.419 & 0.021 & 10.235 & 0.019 & 10.194 & 0.018 \\
472 & 01 56 12.88 & +38 01 43.2 & 11.060 & & 0.460 & & 10.813 & 0.140 & 10.127 & 0.020 & 9.897 & 0.018 & 9.866 & 0.020 \\
475 & 01 56 13.70 & +37 15 56.9 & 12.847 & 0.000 & 0.624 & 0.000 & 12.496 & 0.016 & 11.778 & 0.022 & 11.514 & 0.026 & 11.435 & 0.022 \\
477 & 01 56 13.96 & +37 47 04.7 & 10.572 & 0.006 & 0.364 & 0.006 & 10.375 & 0.015 & 9.778 & 0.020 & 9.627 & 0.019 & 9.589 & 0.021 \\
479 & 01 56 14.28 & +37 58 14.2 & 10.938 & 0.011 & 0.442 & 0.005 & 10.696 & 0.014 & 10.013 & 0.021 & 9.828 & 0.018 & 9.777 & 0.017 \\
486 & 01 56 15.51 & +37 38 41.5 & 10.074 & 0.000 & 0.493 & 0.006 & 9.824 & 0.012 & 9.115 & 0.026 & 8.918 & 0.017 & 8.853 & 0.016 \\
505 & 01 56 18.63 & +37 37 39.3 & 10.778 & 0.005 & 0.399 & 0.008 & 10.543 & 0.014 & 9.903 & 0.021 & 9.719 & 0.017 & 9.669 & 0.016 \\
506 & 01 56 18.90 & +37 58 00.4 & 8.971 & 0.018 & 1.003 & 0.011 & 8.456 & 0.044 & 7.174 & 0.018 & 6.723 & 0.024 & 6.606 & 0.023 \\
512 & 01 56 21.65 & +37 36 08.2 & 9.375 & 0.022 & 1.029 & 0.004 & 8.896 & 0.018 & 7.558 & 0.020 & 7.045 & 0.020 & 6.915 & 0.018 \\
517 & 01 56 22.57 & +37 39 17.8 & 14.230 & & 0.830 & & 13.833 & 0.030 & 12.681 & 0.021 & 12.265 & 0.024 & 12.163 & 0.022 \\
520 & 01 56 23.10 & +37 38 03.0 & 12.850 & & 0.570 & & 12.530 & 0.016 & 11.684 & 0.021 & 11.390 & 0.023 & 11.342 & 0.022 \\
542 & 01 56 29.45 & +37 55 14.7 & 14.350 & & 0.490 & & 14.045 & 0.032 & 13.214 & 0.024 & 12.878 & 0.025 & 12.828 & 0.026 \\
552 & 01 56 32.05 & +37 34 22.2 & 12.921 & 0.016 & 0.581 & 0.036 & 12.425 & 0.025 & 11.696 & 0.021 & 11.417 & 0.023 & 11.351 & 0.020 \\
555 & 01 56 32.96 & +37 56 46.4 & 11.786 & 0.001 & 0.482 & 0.006 & 11.514 & 0.014 & 10.779 & 0.020 & 10.519 & 0.015 & 10.489 & 0.018 \\
563 & 01 56 34.46 & +38 08 49.4 & 13.680 & & 0.910 & & 13.216 & 0.020 & 12.020 & 0.019 & 11.565 & 0.017 & 11.483 & 0.022 \\
575 & 01 56 36.88 & +37 45 12.7 & 13.840 & & 0.760 & & 13.477 & 0.029 & 12.384 & 0.020 & 12.030 & 0.019 & 11.944 & 0.018 \\
580 & 01 56 39.22 & +37 51 41.1 & 10.398 & 0.007 & 0.373 & 0.005 & 10.194 & 0.015 & 9.630 & 0.020 & 9.458 & 0.017 & 9.416 & 0.018 \\
619 & 01 56 47.60 & +37 24 30.4 & 10.280 & 0.022 & 0.415 & 0.005 & 10.054 & 0.013 & 9.412 & 0.020 & 9.226 & 0.018 & 9.173 & 0.019 \\
622 & 01 56 48.61 & +37 29 11.2 & 10.503 & 0.001 & 0.391 & 0.007 & 10.321 & 0.013 & 9.714 & 0.020 & 9.584 & 0.021 & 9.533 & 0.020 \\
626 & 01 56 49.77 & +38 01 21.7 & 9.158 & 0.016 & 0.439 & 0.006 & 8.902 & 0.032 & 8.202 & 0.026 & 8.105 & 0.057 & 8.005 & 0.024 \\
630 & 01 56 50.44 & +38 01 58.1 & 8.961 & 0.011 & 0.811 & 0.020 & 8.661 & 0.059 & 7.379 & 0.021 & 6.936 & 0.020 & 6.824 & 0.023 \\
641 & 01 56 53.06 & +37 52 09.3 & 10.270 & 0.004 & 0.434 & 0.004 & 10.036 & 0.014 & 9.371 & 0.020 & 9.164 & 0.015 & 9.122 & 0.018 \\
648 & 01 56 54.33 & +37 23 51.9 & 12.108 & 0.000 & 0.585 & 0.000 & 11.782 & 0.014 & 10.952 & 0.021 & 10.693 & 0.019 & 10.652 & 0.018 \\
653 & 01 56 55.38 & +38 04 45.8 & 12.410 & & 0.540 & & 12.088 & 0.013 & 11.262 & 0.019 & 10.986 & 0.017 & 10.929 & 0.020 \\
654 & 01 56 55.77 & +37 47 59.3 & 11.196 & 0.000 & 0.396 & 0.014 & 11.022 & 0.015 & 10.385 & 0.020 & 10.216 & 0.017 & 10.155 & 0.018 \\
655 & 01 56 56.15 & +38 08 16.2 & 13.040 & & 0.720 & & 12.665 & 0.017 & 11.706 & 0.021 & 11.370 & 0.019 & 11.294 & 0.018 \\
659 & 01 56 56.35 & +37 39 51.3 & 10.116 & 0.003 & 0.424 & 0.001 & 9.898 & 0.014 & 9.220 & 0.018 & 9.039 & 0.017 & 8.959 & 0.018 \\
667 & 01 56 57.59 & +37 23 20.5 & 10.925 & 0.021 & 0.377 & 0.004 & 10.720 & 0.170 & 10.131 & 0.021 & 9.990 & 0.021 & 9.925 & 0.018 \\
682 & 01 57 02.51 & +37 53 07.7 & 11.255 & 0.012 & 0.447 & 0.002 & 11.007 & 0.014 & 10.332 & 0.019 & 10.205 & 0.023 & 10.178 & 0.024 \\
684 & 01 57 02.81 & +38 14 03.5 & 12.480 & & 0.500 & & 12.153 & 0.015 & 11.415 & 0.021 & 11.180 & 0.019 & 11.123 & 0.020 \\
687 & 01 57 03.12 & +38 08 02.6 & 8.927 & 0.002 & 1.026 & 0.004 & 8.424 & 0.024 & 7.136 & 0.024 & 6.663 & 0.015 & 6.537 & 0.017 \\
689 & 01 57 03.19 & +37 55 44.5 & 11.788 & 0.005 & 0.455 & 0.009 & 11.529 & 0.015 & 10.838 & 0.019 & 10.641 & 0.019 & 10.582 & 0.020 \\
694 & 01 57 03.64 & +38 05 11.7 & 11.779 & 0.011 & 0.448 & 0.007 & 11.520 & 0.013 & 10.720 & 0.021 & 10.478 & 0.019 & 10.439 & 0.019 \\
699 & 01 57 04.89 & +38 07 33.1 & 13.001 & 0.024 & 0.627 & 0.048 & 12.647 & 0.016 & 11.652 & 0.021 & 11.322 & 0.018 & 11.222 & 0.018 \\
701 & 01 57 05.47 & +37 50 42.8 & 13.060 & & 0.690 & & 12.700 & 0.017 & 11.684 & 0.019 & 11.339 & 0.021 & 11.241 & 0.020 \\
720 & 01 57 10.50 & +38 02 06.6 & 12.367 & 0.020 & 0.494 & 0.034 & 12.064 & 0.013 & 11.322 & 0.020 & 11.066 & 0.019 & 11.019 & 0.018 \\
722 & 01 57 10.35 & +37 25 55.3 & 13.500 & & 0.910 & & 13.221 & 0.021 & 12.178 & 0.019 & 11.813 & 0.019 & 11.724 & 0.018 \\
723 & 01 57 10.53 & +37 27 26.6 & 13.770 & & 0.870 & & 13.339 & 0.033 & 12.030 & 0.019 & 11.557 & 0.021 & 11.489 & 0.020 \\
728 & 01 57 12.13 & +37 59 24.8 & 9.420 & 0.005 & 0.471 & 0.004 & 9.169 & 0.011 & 8.485 & 0.020 & 8.296 & 0.016 & 8.246 & 0.024 \\
731 & 01 57 12.17 & +37 56 04.7 & 11.956 & 0.000 & 0.595 & 0.000 & 11.608 & 0.013 & 10.722 & 0.019 & 10.426 & 0.019 & 10.335 & 0.020 \\
745 & 01 57 14.27 & +37 46 51.0 & 9.874 & 0.011 & 0.400 & 0.007 & 9.659 & 0.016 & 9.032 & 0.021 & 8.878 & 9.995 & 8.790 & 0.018 \\
756 & 01 57 17.11 & +37 26 08.8 & 10.207 & 0.012 & 0.428 & 0.004 & 9.971 & 0.013 & 9.302 & 0.018 & 9.122 & 0.017 & 9.047 & 0.020 \\
768 & 01 57 19.42 & +37 59 23.5 & 12.072 & 0.011 & 0.490 & 0.000 & 11.811 & 0.014 & 11.080 & 0.020 & 10.874 & 0.018 & 10.830 & 0.019 \\
772 & 01 57 20.73 & +37 51 43.1 & 10.188 & 0.009 & 0.476 & 0.003 & 9.923 & 0.012 & 9.213 & 0.019 & 9.022 & 0.019 & 8.937 & 0.018 \\
783 & 01 57 22.29 & +37 36 23.2 & 12.240 & & 0.610 & & 11.946 & 0.015 & 11.054 & 0.018 & 10.771 & 0.019 & 10.717 & 0.024 \\
786 & 01 57 22.97 & +37 38 21.8 & 13.170 & & 0.730 & & 12.746 & 0.016 & 11.703 & 0.018 & 11.327 & 0.017 & 11.261 & 0.020 \\
790 & 01 57 23.81 & +37 52 11.9 & 12.267 & 0.012 & 0.529 & 0.018 & 11.938 & 0.015 & 11.117 & 0.018 & 10.860 & 0.019 & 10.756 & 0.018 \\
\end{tabular}
}
\end{table*}
\begin{table*}
{\scriptsize
\begin{tabular}{rcccccccccccccc}
\hline
ID & $\alpha$ & $\delta$ & \multicolumn{2} {c} {V} & \multicolumn{2} {c} {B-V}& \multicolumn{2} {c} {R} & \multicolumn{2} {c} {J}& \multicolumn{2} {c} {H}& \multicolumn{2} {c} {$K_{s}$}\\
\hline
791 & 01 57 24.01 & +38 06 10.4 & 12.684 & 0.017 & 0.543 & 0.017 & 12.370 & 0.015 & 11.566 & 0.020 & 11.306 & 0.019 & 11.261 & 0.021 \\
798 & 01 57 26.01 & +37 43 19.7 & 10.454 & 0.005 & 0.419 & 0.004 & 10.226 & 0.013 & 9.565 & 0.018 & 9.401 & 0.019 & 9.322 & 0.020 \\
799 & 01 57 26.17 & +37 39 20.3 & 11.304 & 0.004 & 0.423 & 0.004 & 11.073 & 0.013 & 10.419 & 0.018 & 10.225 & 0.015 & 10.184 & 0.020 \\
806 & 01 57 27.47 & +37 35 10.4 & 10.756 & 0.000 & 0.385 & 0.000 & 10.555 & 0.015 & 9.948 & 0.018 & 9.817 & 0.017 & 9.748 & 0.018 \\
814 & 01 57 28.26 & +37 24 02.6 & 10.219 & 0.011 & 0.367 & 0.007 & 10.015 & 0.014 & 9.427 & 0.021 & 9.287 & 0.019 & 9.209 & 0.020 \\
823 & 01 57 30.93 & +37 54 57.9 & 10.273 & 0.003 & 0.417 & 0.002 & 10.037 & 0.014 & 9.379 & 0.018 & 9.225 & 0.019 & 9.165 & 0.020 \\
824 & 01 57 31.86 & +37 53 40.6 & 11.629 & 0.011 & 0.440 & 0.005 & 11.375 & 0.015 & 10.720 & 0.019 & 10.533 & 0.019 & 10.452 & 0.018 \\
828 & 01 57 32.58 & +37 42 05.8 & 13.943 & 0.000 & 0.681 & 0.000 & 13.537 & 0.024 & 12.452 & 0.031 & 12.095 & 0.035 & 12.051 & 0.020 \\
847 & 01 57 35.91 & +37 58 23.1 & 14.200 & & 1.030 & & 13.704 & 0.025 & 12.385 & 0.021 & 11.891 & 0.023 & 11.769 & 0.018 \\
849 & 01 57 36.23 & +37 45 10.0 & 9.917 & 0.010 & 0.424 & 0.005 & 9.682 & 0.013 & 9.049 & 0.024 & 8.860 & 0.023 & 8.773 & 0.019 \\
857 & 01 57 37.69 & +37 49 00.7 & 10.028 & 0.011 & 0.470 & 0.002 & 9.780 & 0.013 & 9.072 & 0.021 & 8.841 & 0.020 & 8.774 & 0.019 \\
858 & 01 57 37.62 & +37 39 37.8 & 8.958 & 0.003 & 1.085 & 0.004 & 8.416 & 0.030 & 7.070 & 0.021 & 6.555 & 0.018 & 6.412 & 0.018 \\
859 & 01 57 37.77 & +37 49 50.4 & 13.200 & & 0.670 & & 12.842 & 0.016 & 11.735 & 0.021 & 11.364 & 0.023 & 11.244 & 0.018 \\
864 & 01 57 38.78 & +38 08 30.3 & 12.885 & 0.013 & 0.578 & 0.017 & 12.553 & 0.017 & 11.711 & 0.019 & 11.424 & 0.019 & 11.374 & 0.016 \\
867 & 01 57 38.97 & +37 46 12.2 & 9.041 & 0.007 & 1.004 & 0.006 & 8.556 & 0.019 & 7.263 & 0.020 & 6.813 & 0.024 & 6.669 & 0.031 \\
868 & 01 57 39.46 & +37 52 25.8 & 10.465 & 0.014 & 0.381 & 0.005 & 10.262 & 0.014 & 9.676 & 0.021 & 9.485 & 0.021 & 9.418 & 0.017 \\
888 & 01 57 43.97 & +37 51 42.1 & 10.447 & 0.007 & 0.427 & 0.003 & 10.208 & 0.012 & 9.557 & 0.021 & 9.371 & 0.021 & 9.279 & 0.017 \\
889 & 01 57 44.46 & +38 11 06.8 & 12.802 & 0.015 & 0.551 & 0.018 & 12.475 & 0.014 & 11.654 & 0.018 & 11.395 & 0.019 & 11.323 & 0.016 \\
890 & 01 57 44.74 & +37 59 18.4 & 10.087 & 0.011 & 0.453 & 0.006 & 9.837 & 0.014 & 9.172 & 0.021 & 8.985 & 0.021 & 8.901 & 0.018 \\
897 & 01 57 46.07 & +38 04 28.4 & 10.506 & 0.037 & 0.407 & 0.003 & 10.382 & 0.170 & 9.653 & 0.018 & 9.481 & 0.019 & 9.407 & 0.014 \\
901 & 01 57 47.15 & +37 47 30.3 & 10.981 & 0.006 & 0.388 & 0.007 & 10.763 & 0.014 & 10.126 & 0.021 & 9.985 & 0.019 & 9.897 & 0.018 \\
917 & 01 57 51.42 & +37 39 52.4 & 14.310 & & 0.520 & & 13.590 & 0.026 & 12.596 & 0.019 & 12.203 & 0.019 & 12.089 & 0.020 \\
921 & 01 57 52.00 & +37 27 46.0 & 12.644 & 0.011 & 0.553 & 0.007 & 12.349 & 0.013 & 11.507 & 0.018 & 11.275 & 0.017 & 11.226 & 0.021 \\
935 & 01 57 55.20 & +37 52 46.0 & 11.620 & & 0.450 & & 11.417 & 0.014 & 10.750 & 0.021 & 10.553 & 0.021 & 10.520 & 0.018 \\
937 & 01 57 54.97 & +37 20 26.6 & 10.980 & & 0.380 & & 10.762 & 0.014 & 10.161 & 0.018 & 10.015 & 0.017 & 9.971 & 0.019 \\
941 & 01 57 56.45 & +37 50 01.0 & 10.706 & 0.006 & 0.424 & 0.001 & 10.462 & 0.014 & 9.807 & 0.021 & 9.640 & 0.023 & 9.556 & 0.019 \\
950 & 01 57 57.79 & +37 48 22.3 & 11.467 & 0.020 & 0.480 & 0.007 & 11.160 & 0.077 & 10.393 & 0.020 & 10.225 & 0.021 & 10.163 & 0.019 \\
952 & 01 57 58.24 & +37 26 06.4 & 12.650 & & 0.540 & & 12.356 & 0.014 & 11.556 & 0.019 & 11.282 & 0.019 & 11.268 & 0.019 \\
953 & 01 57 58.85 & +37 41 26.8 & 12.352 & 0.038 & 0.600 & 0.008 & 12.012 & 0.014 & 11.144 & 0.018 & 10.870 & 0.019 & 10.797 & 0.020 \\
955 & 01 57 59.37 & +37 54 53.8 & 9.968 & 0.010 & 0.445 & 0.001 & 9.730 & 0.013 & 9.084 & 0.021 & 8.885 & 0.023 & 8.804 & 0.018 \\
964 & 01 58 02.79 & +38 02 30.4 & 12.912 & 0.016 & 0.582 & 0.014 & 12.568 & 0.015 & 11.668 & 0.019 & 11.385 & 0.019 & 11.273 & 0.018 \\
983 & 01 58 06.31 & +37 38 06.6 & 13.110 & & 0.610 & & 12.709 & 0.017 & 11.756 & 0.019 & 11.401 & 0.019 & 11.314 & 0.020 \\
988 & 01 58 07.71 & +37 39 57.0 & 10.934 & 0.008 & 0.373 & 0.004 & 10.714 & 0.016 & 10.144 & 0.019 & 9.987 & 0.019 & 9.953 & 0.017 \\
993 & 01 58 09.26 & +37 28 35.5 & 13.590 & 0.000 & 0.708 & 0.000 & 13.226 & 0.018 & 12.271 & 0.021 & 11.923 & 0.021 & 11.855 & 0.021 \\
999 & 01 58 10.66 & +37 24 05.9 & 13.550 & & 0.670 & & 13.189 & 0.019 & 12.236 & 0.021 & 11.899 & 0.023 & 11.819 & 0.019 \\
1000 & 01 58 11.43 & +37 39 33.4 & 11.410 & 0.012 & 0.419 & 0.006 & 11.175 & 0.014 & 10.568 & 0.021 & 10.379 & 0.019 & 10.332 & 0.018 \\
1003 & 01 58 12.27 & +37 32 38.2 & 11.193 & 0.003 & 0.472 & 0.003 & 10.933 & 0.014 & 10.251 & 0.021 & 10.013 & 0.018 & 9.942 & 0.017 \\
1007 & 01 58 13.40 & +38 11 41.5 & 13.018 & 0.025 & 0.556 & 0.042 & 12.641 & 0.016 & 11.742 & 0.020 & 11.467 & 0.017 & 11.375 & 0.016 \\
1008 & 01 58 12.71 & +37 34 40.4 & 10.959 & 0.005 & 0.371 & 0.004 & 10.743 & 0.015 & 10.188 & 0.021 & 10.009 & 0.018 & 9.964 & 0.019 \\
1012 & 01 58 12.94 & +37 15 20.2 & 12.417 & 0.015 & 0.539 & 0.000 & 12.113 & 0.016 & 11.336 & 0.020 & 11.083 & 0.023 & 11.047 & 0.019 \\
1017 & 01 58 15.33 & +37 33 19.6 & 13.260 & & 0.660 & & 12.920 & 0.017 & 12.014 & 0.021 & 11.673 & 0.019 & 11.602 & 0.019 \\
1023 & 01 58 16.88 & +37 38 15.9 & 11.250 & 0.015 & 0.424 & 0.010 & 11.005 & 0.013 & 10.407 & 0.021 & 10.203 & 0.019 & 10.159 & 0.019 \\
1026 & 01 58 19.00 & +38 32 14.0 & 11.089 & 0.016 & 0.377 & 0.017 & 10.851 & 0.010 & 10.303 & 0.020 & 10.134 & 0.023 & 10.079 & 0.017 \\
1027 & 01 58 18.42 & +38 06 54.0 & 12.597 & 0.020 & 0.541 & 0.036 & 12.282 & 0.022 & 11.497 & 0.020 & 11.237 & 0.017 & 11.184 & 0.018 \\
1083 & 01 58 27.61 & +37 35 22.2 & 11.920 & 0.011 & 0.484 & 0.007 & 11.676 & 0.014 & 10.977 & 0.021 & 10.721 & 0.019 & 10.656 & 0.018 \\
1089 & 01 58 29.84 & +37 51 37.4 & 9.303 & 0.007 & 0.963 & 0.008 & 8.836 & 0.021 & 7.621 & 0.026 & 7.166 & 0.026 & 7.040 & 0.021 \\
1107 & 01 58 34.42 & +37 40 15.1 & 13.660 & & 0.660 & & 13.273 & 0.020 & 12.299 & 0.022 & 11.923 & 0.019 & 11.884 & 0.021 \\
1117 & 01 58 36.91 & +37 45 10.6 & 9.598 & 0.002 & 0.406 & 0.002 & 9.370 & 0.012 & 8.747 & 0.027 & 8.559 & 0.026 & 8.502 & 0.016 \\
1123 & 01 58 38.12 & +37 32 15.7 & 11.507 & 0.000 & 0.417 & 0.007 & 11.278 & 0.014 & 10.647 & 0.020 & 10.467 & 0.019 & 10.395 & 0.019 \\
1129 & 01 58 40.07 & +37 38 05.1 & 11.910 & 0.011 & 0.481 & 0.007 & 11.629 & 0.014 & 10.911 & 0.021 & 10.691 & 0.019 & 10.599 & 0.018 \\
1151 & 01 58 47.96 & +38 26 08.2 & 10.063 & 0.013 & 0.444 & 0.014 & 9.788 & 0.014 & 9.158 & 0.021 & 8.947 & 0.018 & 8.912 & 0.019 \\
1161 & 01 58 50.00 & +37 59 46.6 & 14.600 & & 0.680 & & 14.060 & 0.040 & 12.715 & 0.019 & 12.215 & 0.019 & 12.108 & 0.020 \\
1165 & 01 58 50.44 & +37 20 52.0 & 10.462 & 0.000 & 0.390 & 0.000 & 10.239 & 0.013 & 9.643 & 0.020 & 9.485 & 0.023 & 9.436 & 0.020 \\
1172 & 01 58 52.93 & +37 48 57.0 & 9.060 & 0.003 & 1.036 & 0.006 & 8.529 & 0.036 & 7.270 & 0.019 & 6.795 & 0.023 & 6.643 & 0.018 \\
1178 & 01 58 53.94 & +37 34 42.6 & 13.390 & & 0.830 & & 13.028 & 0.015 & 11.711 & 0.021 & 11.268 & 0.019 & 11.197 & 0.022 \\
1196 & 01 58 57.32 & +37 39 40.9 & 13.810 & & 0.910 & & 13.339 & 0.019 & 12.150 & 0.021 & 11.703 & 0.017 & 11.611 & 0.022 \\
1204 & 01 58 59.87 & +38 01 18.9 & 11.680 & & 0.460 & & 11.446 & 0.014 & 10.796 & 0.021 & 10.632 & 0.023 & 10.577 & 0.019 \\
1263 & 01 59 14.82 & +38 00 55.2 & 9.006 & 0.011 & 1.019 & 0.007 & 8.483 & 0.025 & 7.199 & 0.019 & 6.767 & 0.018 & 6.648 & 0.016 \\
1270 & 01 59 18.04 & +37 49 49.4 & 14.090 & & 0.700 & & 13.457 & 0.060 & 12.552 & 0.019 & 12.159 & 0.021 & 12.069 & 0.022 \\
1284 & 01 59 19.91 & +37 23 23.1 & 12.893 & 0.000 & 0.677 & 0.000 & 12.476 & 0.013 & 11.472 & 0.021 & 11.140 & 0.021 & 11.052 & 0.020 \\
1296 & 01 59 26.08 & +37 40 39.9 & 14.750 & & 0.530 & & 14.111 & 0.035 & 13.123 & 0.023 & 12.767 & 0.022 & 12.676 & 0.027 \\
1304 & 01 59 29.63 & +38 16 04.3 & 11.341 & 0.008 & 0.412 & 0.011 & 11.076 & 0.014 & 10.485 & 0.022 & 10.293 & 0.022 & 10.251 & 0.020 \\
1365 & 01 59 47.27 & +37 49 53.8 & 13.290 & & 0.710 & & 12.935 & 0.018 & 12.005 & 0.022 & 11.696 & 0.021 & 11.669 & 0.019 \\
1407 & 01 59 56.83 & +37 58 10.5 & 12.949 & 0.000 & 0.552 & 0.000 & 12.630 & 0.016 & 11.782 & 0.027 & 11.575 & 0.034 & 11.454 & 0.023 \\
1474 & 02 00 21.98 & +38 02 41.0 & 10.698 & 0.009 & 0.355 & 0.016 & 10.441 & 0.013 & 9.894 & 0.027 & 9.730 & 0.030 & 9.692 & 0.022 \\
1602 & 02 01 05.97 & +37 42 23.6 & 9.961 & 0.011 & 0.462 & 0.014 & 9.671 & 0.013 & 8.978 & 0.020 & 8.786 & 0.032 & 8.733 & 0.023 \\
\hline
\end{tabular}
}
\end{table*}
|
Title:
Constraints on UED KK-neutrino dark matter from magnetic dipole moments |
Abstract: Generically, universal extra dimension (UED) extensions of the standard model
predict the stability of the lightest Kaluza-Klein (KK) particle and hence
provide a dark matter candidate. For UED scenarios with one extra dimension, we
model-independently determine the size of the induced dimension-five magnetic
dipole moment of the KK-neutrino, $\nu^{(1)}$. We show that current
observational bounds on the interactions of dipole dark matter place
constraints on UED models with KK-neutrino dark matter.
| https://export.arxiv.org/pdf/hep-ph/0601161 |
\preprint{
\hfill
\begin{minipage}[t]{3in}
\begin{flushright}
\vspace{0.0in}
OUTP-0605P
\end{flushright}
\end{minipage}
}
\title{Constraints on UED KK-neutrino dark matter from magnetic dipole moments}
\author{Thomas Flacke}
\email{t.flacke1@physics.ox.ac.uk}
\affiliation{Rudolf Peierls Centre for Theoretical Physics, University of Oxford,
1 Keble Road, Oxford OX1 3NP, United Kingdom}
\author{David W.~Maybury}
\email{d.maybury1@physics.ox.ac.uk}
\affiliation{Rudolf Peierls Centre for Theoretical Physics, University of Oxford,
1 Keble Road, Oxford OX1 3NP, United Kingdom}
\date{January 24, 2006}
\pacs{12.60.-i, 14.60.St, 95.30.Cq, 95.35.+d}
\keywords{extra dimensions, dark matter, dipole moments}
\section{Introduction}
While the standard model proves an excellent framework for fundamental interactions at energy scales up to at least the sub TeV range, it nevertheless leaves a number of fundamental problems. Theoretically, one of the most outstanding puzzles centers on the origin and mechanism of electroweak symmetry breaking, and the quantum mechanical stability of the hierarchy generated between the electroweak scale and the Planck scale. In addition, recent astrophysical observations \cite{DarkMatter} concord with $0.094 < \Omega_{CDM}h^2 < 0.129$, indicating the presence of cold non-baryonic dark matter as the principle form of matter in the Universe, of which the standard model provides no explanation. The most popular candidate for dark matter assumes a non-standard model, stable, electrically neutral, and weakly interacting particle -- the WIMP hypothesis. Clearly, from both a theoretical and phenomenological perspective, the standard model requires extension. In this letter we wish to explore the consequences of the universal extra dimension (UED) \cite{Appelquist:2000nn} extension of the standard model.
In the UED scenario, all standard model particles can freely propagate in the bulk of one or more extra dimensions and thus each standard model particle is associated with a Kaluza-Klein (KK) tower of states. Each state in the KK tower has the same spin as its standard model counterpart. An important consequence of UED models concerns the existence of a conserved discrete symmetry, KK-parity, which guarantees the stability of the lightest KK particle (LKP) and thus provides a dark matter candidate. Suitable thermal relic dark matter candidates that have been studied extensively \cite{Servant:2002aq} include the first KK-excitations of the hypercharge boson, the photon, and the neutrino \ie $B^{(1)}$, $\gamma^{(1)}$, or $\nu^{(1)}$.
The tree-level mass spectrum of the KK-excitations of UED models reveals a nearly degenerate spectrum. As an example, a UED model with one extra-dimension compactified on an $S_1/Z_2$ orbifold of radius $R$, leads to the tree level mass relation
\be\label{mass} m^{(n)} = \sqrt{(n/R)^2 + (m^{(0)})^2}
\ee
for the n-th KK mode, where $m^{(0)}$ constitutes the zero-mode mass (\ie the standard model particle value). Quantum corrections typically dominate over zero mode level contributions and therefore the resulting mass spectrum depends crucially on radiative effects. In general, a moderately split UED mass spectrum \cite{Cheng:2002ej} develops. In the minimal UED model (MUED) \cite{Cheng:2002ej}, one-loop calculations suggest that the LKP is well approximated by the KK hypercharge boson, $B^{(1)}$, and numerous studies have examined the thermal production and prospects of direct and indirect detection of $B^{(1)}$ and $\gamma^{(1)}$ LKP dark matter \cite{Servant:2002aq,KKDMcollection}.
However, the non-renormalizability of UED models imply the existence of an ultraviolet cut-off, typically of the order of a few tens of TeV, at which point the model requires UV completion. As such, UED models must be regarded as an effective theory. The presence of incalculable boundary terms arising from the UV complete theory can potentially change the mass spectrum, resulting in different LKP candidates. Recent studies of the relic density of $B^{(1)}$ LKP dark matter with the full MUED spectrum \cite{coannmat,coannkribs} reveal substantial observational tension with constraints from electroweak precision data \cite{Flacke:2005hb}. Thus, non-minimal models with brane-localized terms appear as a likely alternative if UED models are to provide a successful phenomenology. Furthermore, model independent studies \cite{Servant:2002aq} show that $B^{(1)}$, $\gamma^{(1)}$, and $\nu^{(1)}$ can all be thermally produced with abundances sufficient to provide the dark matter. Constraints on minimal UED models from limits on weak neutral current nucleon-$\nu^{(1)}$ elastic scattering in direct searches, together with thermal dark matter production mechanisms, disfavour $\nu^{(1)}$ dark matter \cite{Servant:2002hb}. However, given the need for possible new non-minimal interactions, we consider further consequences of KK-neutrino dark matter model-independently.
While there exists compelling evidence for dark matter in the form of WIMPS, there also exists strong constraints on possible electromagnetic interactions of dark matter, even in the limit of complete charge neutrality. A neutral Dirac fermion can posses both a permanent magnetic dipole moment, $\mu$, and a permanent electric dipole moment, $d$, arising from the dimension-five operator,
\be
\mathcal{L}_{D} = \frac{i}{2} \bar f
\sigma_{\mu\nu}\left(\mu +\gamma_5 d \right)f F^{\mu\nu}.
\ee
While the presence of the magnetic dipole moment does not violate any discrete symmetries, the electric dipole moment requires the violation of parity and CP. Severe constraints exist on the dipole moments of $\sim 1$ TeV dark matter WIMPS \cite{Sigurdson:2004zp}. (We are aware that the authors of \cite{Sigurdson:2004zp} are currently revising their estimates and, as a result, the strength of the constraints in \cite{Sigurdson:2004zp} may change substantially \cite{private}.) Thus, if a model predicts Dirac fermionic dark matter, it is important to determine the strength of the induced dipole moment.
Since the KK-neutrino of UED models is a Dirac fermion from the 4-dimensional perspective, UED models that assume KK-neutrino dark matter are constrained by the strength of the induced dipole operator.
In this letter we derive model independent bounds on KK-neutrino dark matter by examining the induced dipole moment and comparing the predictions with the current observational bound. Section II provides a brief review on the properties of KK-fermions in UED models along with a discussion on dipole moments relevant to the calculation presented in section III. Finally, in section IV we present our conclusions.
\section{KK-neutrino LKP and induced dipole moments}
In UED models, standard model fields become identified with the zero modes of KK towers of states once the extra dimensions are integrated out of the theory. For concreteness, we will restrict our discussion to five dimensional UED models compactified on an $S_1/Z_2$ orbifold. Five dimensional theories do not posses a chirality condition since $\gamma_5$ becomes part of the five dimensional Clifford Algebra. Thus, in order to arrive at a chiral theory at the zero mode level, we require one five dimensional \emph{Dirac} spinor for every \emph{Weyl} spinor of the standard model. By use of the orbifold boundary conditions half the number of states project out of the spectrum leaving a zero mode level chiral theory. In the case of the lepton doublet, the decomposition of the corresponding five dimensional Dirac spinor reads (\cf \eg
\cite{Appelquist:2000nn}),
\be \label{fermiondecomp}
\begin{split}
\hat{\mathcal{L}}(x^{\mu},y) = \frac{1}{\sqrt{\pi R}}
P_L\mathcal{L}(x^\mu) + \sqrt{\frac{2}{\pi
R}}\sum_{n=1}\left[P_L\mathcal{L}(x^\mu)\cos(\frac{ny}{R})\right.&\\
\left.+P_R\mathcal{L}(x^\mu)\sin(\frac{ny}{R})\right]&\\
\end{split}
\ee
where $\hat{\mathcal{L}}$ denotes the five dimensional Dirac spinor, $\mathcal{L}$ denotes a four dimensional \emph{4
component} spinor with $P_{L,R} = (1\pm\gamma_5)/2$, and $\hat{\mathcal{L}}$ is associated with the (4D left-handed) lepton doublet via the identification $P_L\mathcal{L}(x^\mu)\leftrightarrow (\nu_L,e_L)$. We see from eq(\ref{fermiondecomp}) that $\hat{\mathcal{L}}$ contains a purely left-handed zero mode while all higher KK modes contain both chiralities. Therefore, the non-zero mode KK-level fermions appear as Dirac spinors from the 4-dimensional perspective. Specifically, the KK-neutrino charged under the weak $SU(2)$ appears with both chiralities.
Translational invariance in the fifth direction implies (once the extra dimension becomes integrated out) that a UED model compactified solely on $S_1$ conserves KK-excitation-number in every vertex. The orbifold reduces the conserved KK-number to a discrete $Z_2$ symmetry, called KK-parity. The conservation of KK-parity in every vertex implies the stability of the lightest KK-particle.
Since the fermions receive mass through Yukawa couplings after electroweak symmetry breaking and through the KK-level expansion itself, the theory, in general, requires a unitary transformation to connect mass and gauge eigenstates. For example, in the lepton sector at the $j$th KK-level we have,
\be
\left(\begin{array}{c} \mathcal{E}^j \\ \mathcal{L}^j \end{array} \right) =
\left(\begin{array}{cc}-\gamma_5\cos\alpha_j & \sin\alpha_j\\
\gamma_5\sin\alpha_j & \cos\alpha_j
\end{array}\right)\left(\begin{array}{c} \mathcal{E}^{\prime j} \\
\mathcal{L}^{\prime j} \end{array} \right)
\ee
where $\mathcal{E}^j$ and $\mathcal{L}^j$ denote the $j^{\mbox{th}}$ KK mode of the lepton $SU(2)$ singlet and doublet respectively, and where \be \tan (2\alpha_j) = \frac{m_l^{(0)}}{j/R}. \ee As the lepton masses are small compared to $j/R$, we ignore the effects of the mixing matrix for the remainder of this letter. Furthermore, we ignore the effects of lepton flavour violation and neutrino mixing.
A Dirac fermion can posses a dipole moment, derived from the transition amplitude (\cf \eg \cite{neutrinobook}),
\be
T =
-i\epsilon^\mu q^\nu \bar f (p^\prime) \sigma_{\mu\nu} \left(F_2 + G_2
\gamma_5\right) f(p)
\ee
where $q=p^\prime -p$. The magnetic moment is defined by $\mu = F_2(0)$ while the electric dipole is defined by $ d = G_2(0)$. At low energies compared to the mass of the particle, the photon does not distinguish between $\mu$ or $d$ provided that one ignores other time-reversal violating observables. We will focus on the limits established on $\mu$ throughout. On dimensional grounds, we naively expect the induced magnetic dipole moment to scale as,
\be
\mu \lesssim e \frac{M_{\nu^{(1)}}}{R^{-2}} \simeq \frac{e}{M_{\nu^{(1)}}} = 1.022\times
10^{-6}\mu_B\left(\frac{\mathrm{TeV}}{M_{\nu^{(1)}}}\right)
\ee
where $M_{\nu^{(1)}}$ denotes the mass of KK-neutrino, $\nu^{(1)}$, and $R$ indicates the radius of compactification.
\section{Calculation}
The KK-neutrino develops a magnetic dipole moment through the diagrams tabulated in figure \ref{Figgraphs}.
In general, the entire Kaluza-Klein tower of states participate, however we estimate the leading order effect by considering only the first level KK excitations. Furthermore, we restrict our calculation to KK-number conserving graphs since KK-number violation proceeds with volume suppression. For simplicity, we ignore flavour violation in the lepton sector and we assume that the lightest KK-neutrino is $\nu_e^{(1)}$. Relaxing these assumptions will not significantly alter our conclusions.
As we make no assumptions on the exact UED spectrum, we consider the mass of the KK-$W$ ($W^{(1)}$) and the KK-electron ($e^{(1)}$) as free parameters. In our numerical calculations we do not consider a KK-electron/KK-neutrino mass difference in excess of 5\% as any substantial splitting will lead to unacceptably large contributions to the $T$ parameter.
The relevant UED Feynman rules are listed in \cite{Buras:2002ej} and we calculate in the Feynman-'t Hooft gauge. In the limit of exact $M_{\nu^{(1)}}$-$M_{e^{(1)}}$ degeneracy and where the effects of Yukawa couplings are ignored, we arrive at the semi-analytic result,
\be
\label{dipole_result}
\begin{split}
\mu = & \frac{eg^2}{(4\pi)^2}\frac{1}{M_{\nu^{(1)}}} \times\\
&\left\{\frac{3}{2}\ln (\epsilon)+r+\frac{7}{2}+\frac{1}{2r}-\frac{5}{2}(r-1)\ln \left(\frac{r}{r-1}\right)\right.\\
&\left. -(r-1)^2\ln \left(\frac{r}{r-1}\right) + \mathcal{O}(\sqrt{\epsilon})\right\}
\end{split}
\ee
with the approximation $\epsilon\equiv M_{W^{(0)}}^2/M_{\nu^{(1)}}^2\ll 1$ and $r\equiv M_{W^{(1)}}^2/M_{\nu^{(1)}}^2\simeq 1$.
Numerically, we find agreement with our semi-analytical result as seen in figure \ref{plot_1}.
In figure \ref{plot_1} we display the result of the dipole moment as a function of the KK-neutrino mass. The upper curve illustrates the magnetic dipole moment with exact degeneracy $M_{e^{(1)}}$-$M_{\nu^{(1)}}$ while holding the KK-$W$ mass at its tree level value. The middle curve plots the effect of a KK-electron 5\% heavier than the KK-neutrino. This has an $\mathcal{O}(1)$ effect on the dipole moment. The predicted value of the dipole moment exceeds the current upper bound for a KK-neutrino mass $M_{\nu^{(1)}}\sim 1 \hspace{1mm}\mathrm{TeV}$ by more than five orders of magnitude \cite{Sigurdson:2004zp}. The lower curve displays the effect of maintaining $M_{\nu^{(1)}}$-$M_{e^{(1)}}$ degeneracy while varying the KK-$W$ mass. Allowing the mass difference, $M_{W^{(1)}}$-$M_{\nu^{(1)}}$, to vary by up to 5\% has at most an $\mathcal{O}(10)$ effect.
We should note that the calculation presented above determines only the radiatively induced part of the KK-neutrino magnetic dipole moment. The presence of boundary terms or effects arising from the UV complete theory may also contribute a non-renormalizable dimension-five dipole operator which, a priori, may be of the same order as the radiative part itself.
\section{Conclusion}
UED models have attracted attention as a possible extension to the standard model. A particular appealing feature of the model class centers on the existence of plausible dark matter candidates as the result of KK-parity conservation. While the minimal UED model suggests $B^{(1)}$ dark matter \cite{Cheng:2002ej}, detailed studies of the relic abundance in the minimal UED model \cite{coannmat,coannkribs, Matsumoto:2005uh} in combination with electroweak precision constraints \cite{Flacke:2005hb} show strong observational tension, and thus provide motivation for new possible UED model building avenues. The need for non-minimality has been reported \cite{Hewett:2004py}. Extensions of the MUED scenario by incalculable boundary terms arising from the UV completion of the model can lead to a different LKP and therefore different possible dark matter candidates. We have taken a model independent approach, following \cite{Servant:2002aq}, and examined the consequences of UED KK-neutrino dark matter.
As the KK-neutrino is a Dirac fermion, UED models predict an induced KK-neutrino dipole moment. We find that the induced dipole moment, typically $\mu \lesssim 10^{-7} \mu_B$, strongly conflicts -- by over five orders of magnitude -- with the current observational bounds stated in \cite{Sigurdson:2004zp} for TeV scale dipole dark matter. We reiterate that the constraints provided by \cite{Sigurdson:2004zp} are currently under revision, and the strength of the stated bounds are expected to change \cite{private}. The constraints on magnetic dipole moments given in \cite{Sigurdson:2004zp} would provide the strongest limits on KK-neutrino dark matter. Even in the absence of the strong limits provided by \cite{Sigurdson:2004zp}, the bounds on dipole moments remain an important constraint on future model building. Not only will new, non-minimal models that predict KK-neutrino LKP need to circumvent the constaints provided by \cite{Servant:2002hb}, but will also have to evade the constraints provided by radiatively induced magnetic dipole moments, which are generically at least as large as the current experimental bounds.
While we have restricted our discussion to five-dimensional models compactified on $S_1/Z_2$, we expect that the qualitative features carry over to UED models with multiple extra dimensions. We have also assumed the absence of any fine-tuning between the radiatively induced magnetic dipole moment and possible non-renormalizable dimension-five dipole operators arising from the UV complete theory or boundary terms.
Our results indicate that observational limits on dipole dark matter can place significant constraints on UED scenarios where the KK-neutrino is the LKP dark matter candidate.
\section{Acknowledgments}
We would like to thank J.~March-Russell, G.~Starkman, B.~A.~Campbell, and K.~Sigurdson for useful discussions. DM wishes to acknowledge the support of the Natural Science and Engineering Research Council of Canada and the Canada-United Kingdom Millennium Research Fellowship. The work of TF is supported by ``Evangelisches Studienwerk Villigst e.V." and PPARC Grant No. PPA/S/S/2002/03540A. This work was also supported by the ``Quest for Unification" network, MRTN 2004-503369.
|
Title:
Detection and Fundamental Applications of Individual First Galaxies |
Abstract: First galaxies formed within halos of mass M=E7.5-E9 Msun at z=30-40 in the
standard cold dark matter (CDM) universe may each display an extended hydrogen
21-cm absorption halo against the cosmic microwave background with a brightness
temperature decrement of del T=-(100-150)mK at a radius 0.3 < r < 3.0 comoving
Mpc, corresponding to an angular size of 10-100 arcseconds. A 21-cm tomographic
survey in the redshift shell z=30-40 (at 35-45MHz), which could be carried out
by the next generation of radio telescopes, is expected to be able to detect
millions of first galaxies and may prove exceedingly profitable in enabling (at
least) four fundamental applications for cosmology and galaxy formation. First,
it may yield direct information on star formation physics in first galaxies.
Second, it could provide a unique and sensitive probe of small-scale power in
the standard cosmological model hence physics of dark matter and inflation.
Third, it would allow for an independent, perhaps "cleaner" characterization of
interesting features on large scales in the power spectrum such as the baryonic
oscillations. Finally, possibly the most secure, each 21-cm absorption halo is
expected to be highly spherical and faithfully follow the Hubble flow. By
applying the Alcock-Paczynski test to a significant sample of first galaxies,
one may be able to determine the dark energy equation of state with an accuracy
likely only limited by the accuracy with which the matter density can be
determined independently.
| https://export.arxiv.org/pdf/astro-ph/0601010 |
\title{Detection and Fundamental Applications of Individual First Galaxies}
\author{Renyue Cen\altaffilmark{1}}
\altaffiltext{1} {Princeton University Observatory,
Princeton University, Princeton, NJ 08544;
cen@astro.princeton.edu}
\received{\date}
\accepted{ }
\keywords{galaxies - radio - intergalactic medium - cosmology: theory}
\section{Introduction}
It is of wide interest to detect and understand the first generation of galaxies,
expected to form in the redshift range $z=30-50$
in the standard CDM universe (Spergel \etal 2003).
Extensive literatures on 21-cm properties of neutral hydrogen in the dark ages
and during cosmological reionization
have long focused on large-scale fluctuations of the intergalactic neutral hydrogen
and global spectral features (e.g., Hogan \& Rees 1979; Scott \& Rees 1990).
In this {\it Letter} we point out a unique
feature possessed by the first {\it individual} galaxies of mass
$10^{7.5}-10^9\msun$ formed at $z=30-40$ --- a large hydrogen 21-cm absorption halo
against the cosmic microwave background (CMB).
Each 21-cm absorption halo has a size
$10^{''}-100^{''}$ with a brightness temperature decrement of $\delta T=-(100-150)$~mK
at $35-45$MHz,
which could serve as a visible proxy for each galaxy that otherwise may
be undetectable.
The next generation of radio telescopes, such as LOFAR,
may be able to detect such a signal.
A range of fundamental applications is potentially possible with
a redshift (i.e., 21-cm tomographic) survey of the first galaxies
in the redshift shell $z=30-40$,
which may hold the promise to revolutionize the field of cosmology
and shed illuminating light on dark matter, dark energy and inflation physics.
Throughout, a standard (Wilkinson Microwave Anisotropy Probe)
WMAP-normalized CDM model
is used (unless indicated otherwise): $\Omega_M=0.31$, $\Lambda=0.69$,
$\Omega_b=0.048$, $H_0=69$km/s/Mpc, $n_s=0.99$ and $\sigma_8=0.90$.
\section{Large 21-cm Absorption Halos of First Galaxies}
A first-generation galaxy is expected to emit UV and X-ray radiation,
each carving out an H II region of size (ignoring recombination):
$r_{\hbox{HII}} \sim 43 ({M_h\over 10^7\msun})^{1/3} ({c_*\over 0.1})^{1/3} ({f_{esc}\over 0.1})^{1/3} ({N_p\over 8\times 10^4})^{1/3}\kpc$ comoving,
where $M_h$ is the halo mass,
$c_*$ the star formation efficiency,
$f_{esc}$ the ionizing photon escape fraction into the intergalactic medium (IGM),
and $N_p$ the number of hydrogen ionizing photons
produced by each baryon formed into stars,
($\sim 10^{4.5-5}$ for a massive metal-free Population III IMF;
Bromm, Kudritzki, \& Loeb 2001).
Hard X-ray photons ($\ge 1$keV) produced escape deep
into the IGM with a distance of $\sim 500-1000$ comoving megaparsecs,
building an X-ray background.
Sandwiched between small H II regions and the X-ray sea
sits a quite large \lya\ scattering region (Loeb \& Rybicki 1999),
resulting in a four-layer structure, as depicted in Figure 1.
The IGM in the vicinity %
of a galaxy interacts with ionizing UV and soft X-ray
as well as near \lya\ photons emanating from the galaxy, which can be computed.
The most important and relevant physical processes
are (1) the interaction between
neutral gas and near \lya\ photons emitted by the central host galaxy,
which couples the spin temperature of the IGM to its kinetic temperature
(Wouthuysen 1952; Field 1958),
and (2) the interaction between neutral gas and ionizing
UV and soft X-ray photons emanating from the host galaxy,
which provides a heating source for the otherwise cold IGM
up to some small radius.
In the absence of heating the kinetic temperature $T_k$ of the IGM
would be equal to $T_{\hbox{IGM}} \approx 18 \left({1+z\over 31}\right)^2~\hbox{K}$
(for the redshift range of interest here) since
beginning of decoupling with CMB at $z\sim 200$ (Peebles 1993).
We perform spherically the transfer of UV and soft X-ray radiation from
the host galaxy outward to compute
(1) the development of the HII region around the galaxy as a function of time,
(2) the evolution of the temperature of the surrounding IGM,
subject to UV and soft X-ray heating by the host galaxy,
as a function of radius and time,
and (3) the \lya\ coupling coefficient $y_\alpha$ as a function of radius and time.
We assume a uniform IGM density equal to
the mean gas density of the universe (i.e., $\Delta=0$ in equation 1 below).
Two scenarios of metal-free star formation in first galaxies are considered:
(1) all stars have a single mass of $200\msun$ (VMS),
advocated by Oh \etal (2001) and Qian \& Wasserburg (2002),
and
(2) an IMF has the Salpeter slope of $2.35$ with a lower cutoff of
$25\msun$ and an upper cutoff of $120\msun$ (SAL),
close to what favored by Umeda \& Nomoto (2003),
Tumlinson, Venkatesan, \& Shull (2004) and Tan \& McKee (2004).
We adopt the library of stellar spectra and ages from Schaerer (2002);
we use a black-body radiation spectrum for each star of chosen mass
with an effective surface temperature from Table 3 of Schaerer (2002)
but slightly adjusted so as to
produce the correct ratio of the number of photons above helium II Lyman limit
to the number of photons above hydrogen Lyman limit,
averaged over the lifetime of each star (see Table 4 of Schaerer 2002).
We use an escape fraction for Lyman limit photons $f_{esc}$ from the host galaxy
and self-consistently a frequency dependent
escape fraction for other UV and soft X-ray photons assuming that
they are subject to the same absorbing column in the galaxy.
All escaped photons (emerging from the virial radius)
are then subject to the (time-dependent) combined absorption
of H I, He I and He II in the IGM,
self-consistently computed.
Since the \lya\ scattering region is mostly neutral
with a residual ionized fraction of $2\times 10^{-4}$ left from
recombination (Peebles 1993),
we assume that $\eta=14\%$ of X-ray energy is used to heat the gas
(Shull \& Van Steenberg 1985) with the heating
rate per hydrogen atom at radius $r$ computed with the following formula:
${dE\over dt} = \int_0^\infty {\eta L_\nu\over 4\pi h\nu r^2}(\sigma_\nu(HI)(h\nu-h\nu_{H}) + \xi\sigma_\nu(HeI)(h\nu-h\nu_{HeI})
+\xi\sigma_\nu(HeII)(h\nu-h\nu_{HeII})) e^{-\tau_\nu} d\nu$,
where
$L_\nu$ is the luminosity per unit frequency of the galaxy;
$\nu_{HI}$, $\nu_{HeI}$ and $\nu_{HeII}$ are
ionization potentials
of H, He I and He II, respectively;
$\sigma_\nu(HI)$
$\sigma_\nu(HeI)$ and
$\sigma_\nu(HeII)$ are
photo-ionization cross-sections
of H, He I and He II, respectively;
$\xi$ is ratio of helium number density to hydrogen number density;
$\tau_\nu$ is the optical depth from the galaxy
to radius $r$ at frequency $\nu$.
The spin temperature of neutral hydrogen (Field 1958, 1959) is then given by
$T_{\hbox{s}}={T_{\hbox{cmb}}+y_\alpha T_k + y_c T_k\over 1+y_\alpha+y_c}$,
where $y_c\equiv {C_{10}\over A_{10}} {T_*\over T_k}$
is the collisional coupling coefficient with
the collisional de-excitation rate
$C_{10}={4\over 4} \kappa (1-0) n_H$,
$\kappa (1-0)$ taken from Zygelman (2005),
$n_{\rm H}$ is the mean hydrogen density
and
$T_*=0.0682$~K is the hydrogen hyperfine energy splitting.
In the expression for the \lya\ coupling coefficient
$y_\alpha = {P_{10}T_*\over A_{10}T_k}$,
$A_{10}=2.87\times 10^{-15}$~s$^{-1}$
is spontaneous emission coefficient of the 21-cm line,
the indirect de-excitation rate $P_{10}$ of the hyperfine structure
levels is related to the total \lya\ scattering rate
$P_\alpha$ by $P_{10}=4P_\alpha/27$ (Field 1958).
Here $P_\alpha=\int F_\nu \sigma(\nu) d\nu$
with $F_\nu$ being the \lya\ photon flux (in units of cm$^{-2}$~s$^{-1}$)
and $\sigma(\nu)=\sigma_\alpha \phi(\nu) = {3\over 8\pi}\lambda_\alpha^2 A_\alpha \phi(\nu)$
being the cross section for \lya\ scattering (MMR),
where $\lambda_\alpha=1.216\times 10^{-5}$~cm
is the wavelength of the \lya\ line,
$A_\alpha=6.25\times 10^8$~s$^{-1}$ is the spontaneous Einstein coefficient
for \lya\ line and $\phi(\nu)$ is the normalized \lya\ line (Voigt) profile with $\int \phi(\nu) d\nu=1$.
Most of the \lya\ scattering is accomplished
by UV photons slightly on the blue side of the \lya\
that redshift into \lya\ resonant line due to the Hubble expansion
(note that $\Delta\nu/\nu\sim 10^{-3}$ due to Hubble expansion
at $r\sim 1$Mpc comoving),
not the intrinsic \lya\ line photons that escape from the host
galaxy and redshift to the damping wing
(Madau, Meiksin, \& Rees 1997; MMR).
Additional physical processes that
were not treated previously, including higher-order
Lyman lines that result in cascade in two-photon emission,
fine structure of \lya\ resonance and spin-flip scattering,
introduce corrections of order unity
to $P_\alpha$ (CM; Hirata 2005; Chuzhoy \& Shapiro 2005)
but all these corrections terms
are insignificant for our case,
and we only apply the relatively large
correction term $S_c$ ($\sim 1.5$) as shown in Figure 4 of CM
due to a spectral shape change near \lya\ .
The observed brightness temperature increment/decrement against the CMB is
\begin{eqnarray}
\delta T = 41 (1+\Delta)x_H ({T_{\hbox{s}}-T_{\hbox{cmb}}\over T_{\hbox{s}}})({\Omega_b h^2\over 0.02})({0.15\over \Omega_Mh^2})^{1/2} ({1+z\over 31})^{1/2}~\hbox{mK},
\end{eqnarray}
\noindent
where $\Delta$ is gas overdensity relative to the mean,
$x_H$ neutral hydrogen fraction,
$T_{\hbox{cmb}}=2.73(1+z)$~K CMB temperature
and other symbols have their usual meanings.
Figure 2 shows the profile of
$\delta T$ for four cases of halo masses with each choice of IMF.
Let us examine each of the four regions (sketched in Figure 1)
with respect to 21-cm observations.
Inside the virial radius (the red circle in Figure 1)
the gas is overdense with $\delta\ge 100$ and
a positive large-amplitude emission signal may result,
if a significant amount of neutral hydrogen gas exists within.
However, the size of this regions falls below $0.1^"$
and its signal is unlikely to be detectable in the foreseeable future.
The H II region (inside the blue circle in Figure 1)
is ionized hence $\delta T=0$.
In the region outside the \lya\ scattering region (exterior to
the green region in Figure 1)
the spin temperature of the IGM has been progressively
attracted to the temperature of the CMB with gradually weakening
coupling to the gas kinetic temperature by atomic collisions,
producing a small but non-negligible 21-cm absorption signal
at the redshift of interest ($z\sim 30-40$) (Loeb \& Zaldarriaga 2004).
It is the \lya\ scattering region
that is of most interest here.
In the inner part of the \lya\ scattering region
(shown in magenta in Figure 1)
the IGM is significantly heated by UV and soft X-ray photons
to exceed the CMB temperature, while
its spin temperature is very strongly coupled to its kinetic temperature
by \lya\ scattering.
As a result, the inner radial region at
$0.01-0.04$~Mpc/h comoving for minihalos and $0.04-0.4$~Mpc/h
comoving for large halos
displays an emission signal against CMB with an amplitude of $\delta T\sim 30$~mK
(a shark dorsal fin-like feature in Figure 2).
Going outward (shown in green in Figure 1),
the soft X-ray heating abates (because the
cumulative optical depth to these photons increases quickly) but
the \lya\ scattering remains strong,
up to a distance of about $10$~Mpc/h comoving.
Consequently, a strong 21-cm absorption signal against the CMB
with an amplitude of $\delta T= -(100-150)$~mK at $\sim 35-45$MHz (for $z=30-40$)
on a scale of $0.3-3$~Mpc/h comoving, corresponding to an angular scale
of $10^{''}-100^{''}$, is produced for large halos.
This is the 21-cm absorption halo --- a unique and strong feature for the
large first galaxies.
We note that the absorption signal cast by minihalos
(the two top sets of thin curves in each panel
in Figure 2) is relatively weak
due to a combination of low mass and low star formation efficiency.
We will therefore focus on large halos for practical observability purposes.
Besides stars,
no other soft X-ray source in the galaxy is assumed to concurrently exist.
We shall examine the validity of this assumption in detail.
The density of the interstellar
medium plays an important role for
some of the potentially relevant processes considered here
and it is assumed to be $n(z) = n_0 (1+z)^3$,
where local interstellar density $n_0=1$~cm$^{-3}$.
This assumption should hold in
hierarchical structure formation model for the following reasons.
The mean gas density scales as $(1+z)^3$ and
halos at low and high redshift in cosmological
simulations show similarities when
density and length are measured in their respective comoving units
(e.g., Navarro, Frenk, \& White 1997; Del Popolo 2001).
The spin parameters
(i.e., angular momentum distribution) of both high and low
redshift halos have very similar
distributions
peaking at a nearly identical value $\lambda\sim 0.05$
(Peebles 1969; White 1984;
Barnes \& Efstathiou 1987;
Ueda \etal 1994;
Steinmetz \& Bartelmann 1995;
Cole \& Lacey 1996;
Bullock \etal 2001).
Thus, cooling gas in galaxies at low and high redshift
should collapse by a similar factor
before the structure becomes dynamically stable
(e.g., rotation support sets in),
resulting in interstellar densities scaling as $(1+z)^3$.
Direct simulations (Abel \etal 2002; Bromm \etal 2002)
suggest a gas density of $10^3-10^4$cm$^{-3}$ by
the end of the initial free fall for minihalos at $z\sim 20$,
verifying this simple analysis.
We will estimate each of several possible
types of soft X-ray emission sources in turn.
First, let us estimate soft X-ray emission from supernova remnants.
Assuming the standard cooling curve (Sutherland \& Dopita 1993)
we find that at $z=30$
a supernova blastwave with an initial explosion energy of $5\times 10^{52}$~erg
(for a star of mass $200\msun$; e.g., Heger \& Woosley 2002)
would enter its rapid cooling phase at a temperature of $2.7\times 10^7$~K.
This implies that the energy emitted at $\sim 100-300$~eV
from the cooling shell is about $7\%$.
A $200\msun$ mass would release $2.5\times 10^{54}$~erg total energy
due to nuclear burning, out of which $0.3\%$ is released
in photons at $100-300$eV for our adopted radiation spectrum.
Therefore, the ratio of total photon energy from the supernova remnant
to that from the star is $0.4-0.5$.
Thus, for the VMS IMF, stellar soft X-ray appears to dominate
over that from its supernova remnant.
The soft X-ray contribution from supernova remnant cooling
increases relatively compared to that
from the star itself with decreasing stellar mass and we estimate that
the overall contribution from the two components may become comparable
for the SAL IMF case, averaged over time.
Second, we will examine X-rays produced
from cooling of supernova-accelerated relativistic electrons by CMB photons
via inverse Compton (IC) process (e.g., Oh 2001).
For adiabatic shocks, as is appropriate in our case,
the IC spectral energy distribution
has a two-power-law form:
$L_\nu\propto$~constant at $E<E_{break}$
and
$L_\nu\propto \nu^{-1}$ at $E>E_{break}$.
The break energy is
$E_{break}=70$~keV, independent of redshift
with the assumed scaling of the interstellar medium density with redshift.
Then the ratio of energy from IC to that from stars
is found to be $10^{-4}-10^{-3}$ in the $100-300$eV band,
depending on the exact upper energy cutoff (assuming
$10\%$ of supernova explosion energy is utilized to accelerate
relativistic electrons in shocks).
Clearly, contribution to soft X-rays from IC process is unimportant.
Third, X-ray binaries during the relatively short lifetime of massive
stars may be rare, for top-heavy IMFs of concern here.
We can make an estimate based on
the calculation by Rappaport, Podsiadlowski \& Pfahl (2005),
who give an ultra luminous X-ray binary formation
rate of $3\times 10^{-5}$ per supernova.
It is clear that even if each X-ray binary is able to release
as much energy as in a supernova explosion itself and all in the
soft X-ray band, the resulting contribution will be less than
a fraction of a percent of that from stars.
Fourth, stellar mass black holes (BH) of $\sim 10-100\msun$
may be produced in significant numbers with a top-heavy IMF
as well as a central galactic BH.
It seems that stellar BH accretion is likely
significantly suppressed and small due to
feedback effect from stars on surrounding gas
(e.g., Mori, Umemura, \& Ferrara 2004;
Alvarez, Bromm, \& Shapiro 2005).
A concomitant contribution of soft X-rays from central BH accretion
in the lifetime of a $200\msun$ star is approximately
$(M_{BH}/M_*)\times (1/0.007)) \times (t_*/t_E) \times (f_{BH,SX}/f_{*,SX})
=0.6 f_{BH,SX} (M_{BH}/M_*/0.003)$,
after inserting soft X-ray emission fraction for a $200\msun$ stellar spectrum
of $f_{*,SX}=0.003$, stellar lifetime $t_*=2.2\times 10^6$~yrs
and Eddington time $t_E=4.4\times 10^8$~yrs,
where $f_{BH,SX}$ is the energy fraction
released by the BH accretion in the soft X-ray band ($100-300$eV).
Thus, if the ratio of black hole mass to (bulge) stellar mass
follows the local Magorrian (Magorrian \etal 1998) relation,
then, unless most of the accretion energy
is released in the soft X-ray band, contribution from central BH accretion
to the soft X-ray band is relatively small.
Fifth, thermal bremsstrahlung emission from gravitational
shock heated gas is likely negligible due to a low
gas temperature ($T\sim 10^4$K).
Finally, soft X-rays from massive first stars themselves are thought to be
produced by stellar winds and quite uncertain.
Recent work suggests that the winds hence soft X-ray emission
from metal-free stars are expected to be insignificant (e.g., Krticka \& Kubat 2005).
In summary, soft X-rays from neglected, possible
sources other than that from the stellar photospheres
would, at most, make a modest correction to what is adopted in our calculation.
To ascertain our conclusion,
we compute a case with the amplitude of soft X-ray intensity
at $h\nu \ge 100$~eV artificially raised by a factor of $10$
and do not find any significant effect that would qualitatively
change our results (Figure 2).
The reason is that the IGM quickly becomes optically thick to
a few $100$eV soft X-ray photons at $\sim 1$~Mpc comoving.
Therefore, our results should be quite robust.
Since we are concerned with gas of relatively low temperature $\sim 20$K,
heating by a cumulative (hard) X-ray background may become relevant at some redshift.
We estimate when this may happen in the CDM model.
While an X-ray background may be generated by a variety of processes,
black hole accretion at the centers of galaxies
are thought to be the most dominant (e.g., Ricotti \& Ostriker 2003;
Kuhlen \& Madau 2005),
estimated as follows.
Let us suppose the energy extraction efficiency from black
accretion is $\alpha$ and a fraction $f_x$ of the released energy
is in the form of hard X-rays.
Then, the X-rays may collectively
heat up the IGM temperature at most (ignoring Compton cooling and
assuming all X-ray photons in the background are consumed by the IGM) by an increment
\begin{equation}
\Delta T_{\hbox{xray}} = 1.1({f_{\hbox{coll}}\over 10^{-6}}) ({c_*\over 0.1})({\alpha\over 0.1}) ({M_{BH}/M_*\over 0.003}) ({f_x\over 0.029})~\hbox{K}
\end{equation}
\noindent
(assuming 14\% of X-ray energy is used to heat the IGM; Shull \& Van Steenberg 1985),
where $f_{\hbox{coll}}$ is the fraction of matter that
has collapsed to halos where stars have formed.
Under the reasonable assumption that the parameters
have their adopted fiducial values
(for $\alpha$ see Yu \& Tremaine 2002,
$f_x$ see Elvis \etal 1994,
for $c_*$ see Gnedin 2000),
it becomes evident that,
for the very first galaxies formed in the universe
that comprise a collapsed
mass fraction less than $10^{-6}$,
heating of the IGM by an X-ray background radiation field may be small.
Figure 3 shows cumulative halo mass functions at redshift $z=(20,30,40)$,
based on Press-Schechter (1974) formalism
(using $\delta_c=1.67$ and the standard model of Spergel \etal 2003),
which should be accurate for the exponentially falling regime of interest here
(Sheth \& Tormen 1999; Jenkins \etal 2001).
Figure 3 suggests that at redshift $z\ge 30$
heating of the IGM by an X-ray background is small.
Neglected contributions to the X-ray background
from other sources (X-ray binaries,
supernova remnants, etc) (e.g., Oh 2001, Cen 2003)
will likely just add a modest numerical
correction factor for equation (2) and the net effect, if any,
would push the epoch for significant heating by an X-ray background
to a slightly higher redshift
(note that structure formation is exponentially increasing
with decreasing redshift at $z=30-40$).
But even a factor of a few upward correction to equation (2)
would still leave the temperature of
IGM $z=30$ relatively unaffected by an X-ray background.
Another critical issue is whether heating of IGM by \lya\ photons is important.
In a recent accurate calculation based on Fokker-Planck approximation,
Chen \& Miralda-Escud\'e (2004; CM) show that the heating rate by \lya\ photons
is much lower than previous estimates (MMR).
We recast their important result (equation 17 of CM and using Figure 3 of CM),
the heating rate per hydrogen atom and per Hubble time, $\beta$,
in the following way:
\begin{equation}
\beta\equiv {\Gamma_c\over H n_{H} k_{B}} = 0.08 ({P_\alpha\over P_{th}})~\hbox{K}
\end{equation}
\noindent
at $z=30$, where
$k_{\rm B}$ is the Boltzmann constant,
$H$ is the Hubble constant
and $P_{th}=2.4\times 10^{-11}({1+z\over 31})$~s$^{-1}$
is the thermalization rate for \lya\ scattering at $z=30$,
above which \lya\ scattering brings down
the spin temperature to the gas kinetic temperature (MMR).
The Hubble time is $1.4\times 10^8$~yrs at $z=30$.
So, over the duration of a stellar lifetime $6\times 10^6$~yrs
(of the least massive stars in our model, $25\msun$),
the gas will be heated up by $0.003$~K at
${P_\alpha\over P_{\rm th}}=1$.
For the regime of interest where we see the strong
21-cm absorption signal (Figure 2)
we find ${P_\alpha\over P_{\rm th}}=1-10$.
Thus, heating of surrounding IGM by \lya\ photons
emanating from the host galaxy can be safely neglected.
In addition,
since we are concerned with early times when
the universe is far from being ionized and the number of \lya\ photons
per hydrogen atom is significantly less than unity,
indicating that heating by the background \lya\ photons can also be safely neglected
(CM). Furthermore, heating rate by high order Lyman series photons
is still lower and thus negligible (Pritchard \& Furlanetto 2005).
\section{Fundamental Applications}
We have demonstrated a unique feature of first galaxies.
A large 21-cm survey of the first galaxies will be invaluable.
Demanding that each of the physical quantities
be resolved by a factor of $10$
would translate to the following requirements:
an angular resolution of $\sim 1^{''}$,
a spectral resolution of $\sim 4$kHz
($\Delta\nu\sim 40$kHz across a radius of $1$Mpc/h at $z=30$ due to the Hubble expansion)
and a sensitivity of $\sim 10$~mK at $35-45$~MHz.
Among the next generation of radio telescopes currently under construction/consideration,
LOFAR (http://www.lofar.org) appears to be best positioned
to be able to carry out such a survey,
at least for some of the brighter and larger galaxies.
LOFAR is currently designed to
reach a frequency as low as $10$MHz with an angular resolution
of $3^{''}-4^{''}$ at $35-45$MHz, a sensitivity of $10$~mK
and a spectral resolution (i.e., processing capability)
of $1$~kHz (e.g., Rottgering 2003),
while MWA (http://web.haystack.mit.edu/arrays/MWA/index.html),
PaST (http://web.phys.cmu.edu/$\sim$past/index.html)
and SKA (http://www.skatelescope.org) appear to be placed
out of the $35-45$MHz range, as they stand now.
Figure 4 shows the density of galaxies versus
the maximum absorption cross section
(i.e., in the plane
of the large circle centered on the galaxy perpendicular to the line of sight)
with $\delta T<-100$~mK.
Here we point out four fundamental
and potentially ground-breaking applications
regarding cosmology and galaxy formation,
if a 21-cm tomographic survey of galaxies in the redshift shell
$z=30-40$ is carried out,
which may be able to detect millions of galaxies.
First, a characteristic sharp fall-off at $5-10$ square arcseconds and
a characteristic peak of the number of 21-cm absorption halos is expected,
as seen in Figure 4 due to a lower star formation efficiency in (and low mass of)
minihalos, as suggested by available simulations (Abel \etal 2002).
This should yield direct information
on physics of cooling and star formation in first galaxies,
which may be unobtainable otherwise by any other means in the foreseeable
future.
Note that the left cutoff and the peak density
are functions of $c_*$ and $g({\rm IMF})$ (noting
the differences between SAL and VMS cases in Figure 4),
where $g({\rm IMF})$ denotes dependence on the properties
of IMF such as stellar lifetime and spectrum.
A full parameter space exploration will be given in a separate paper with
more detailed treatments.
We expect that $c_*$ and $g({\rm IMF})$
may be determined separately,
when jointly analyzed with
the density of absorption halos in the context of the standard CDM model.
Second, we see that the density of strong 21-cm absorption halos
depends strongly on $n_s$,
as testified by the large difference
(a factor of $\sim 50$) between solid and dotted curves in Figure 4.
One may then obtain a constraint on $n_s$,
which is made possible because the effect due to
difference in IMF may be isolated out, as discussed above,
thanks to the features in the density of absorption halos
(e.g., peak location and sharp fall-off at the low end).
Let us estimate a possible accuracy of such measurements.
At $n\sim 10^{-6}$h$^3$Mpc$^{-3}$ one would find
0.1 million galaxies in the redshift shell between $z=28$ and $z=32$,
giving a relative fraction (Poisson) error
of $0.3\%$.
By comparing the solid and dotted curves in Figure 4,
we find that a constraint on $n_s$
with $\Delta n_s=0.01$ ($\sim 3\sigma$) may be achieved.
This may have the potential to discriminate between
inflationary theories (e.g., Liddle \& Lyth 1992; Peiris \etal 2003).
In addition,
the constraint placed on the temperature (or mass) of dark matter particles
or running of the spectral index
may be still tighter, because a significant, finite dark matter temperature
or a running index tends to
suppress small-scale power exponentially thus amplify the effects.
The high-sensitivity constraint on small-scale power
is afforded by the physical fact that we are dealing with
rare $\ge 5-6\sigma$ peaks in the matter distribution.
Third, clustering of galaxies may be computed
using such a survey containing potentially hundred of thousands
to millions of galaxies
in a comoving volume of size $\sim 100$Gpc$^3$ (for the redshift
shell $z=28-32$).
Both the survey volume and the number of observable galaxies within
are large enough to allow for accurate determinations
of the correlations of first galaxies,
particularly on large scales.
It may then provide an independent,
perhaps ``cleaner" characterization of interesting features
in the power spectrum such as the baryonic oscillations,
with the advantage that they are not subject to
subsequent complex physical processes, including
cosmological reionization, gravitational shock heating
of the IGM and complex interplay
between galaxies and IGM, which in turn might
introduce poorly understood biases in galaxy formation.
A comparison between clustering of first galaxies
and local galaxies (e.g., Eisenstein \etal 2005)
will provide another, high-leverage means to gauge
gravitational growth and other involved processes between $z=30$ to $z=0$.
Finally, the scale ($\sim 1$Mpc comoving)
of the 21-cm absorption halo signals is
much greater than the nonlinear scale and virial radius (both $\sim 1$kpc comoving).
Thus, the IGM region in the 21-cm absorption halo
is expected to closely follow the Hubble flow.
Since near \lya\ photons (between \lya\ and Ly$\beta$)
are not subject to absorption by hydrogen (and helium) atoms whose distribution might be complex,
they escape into the IGM in a spherical fashion.
Additionally, since the dependence on $\Delta$ is linear (see equation 2),
density inhomogeneities are likely to average out (to zero-th order)
and results do not depend sensitively on uncertain linear density fluctuations in the IGM.
Although there might exist ``pores" in the domain of 21-cm absorption halo due to
fluctuations in local IGM temperatures which may be
caused by local shock heating due to formation of minihalos,
the overall effect is likely negligible,
because the mass fraction contained in all halos down to
a mass as small as $M_h=10^{5}\msun$ is about $10^{-4}$ at $z=30$.
Furthermore, at $n=10^{-6}$h$^3$Mpc$^{-3}$
the mean separation between the galaxies
is $100$~Mpc/h, much larger than
the size of \lya\ scattering regions of size $\sim 1$Mpc/h,
so overlapping of the latter should be very rare (taking
into account the known fact that they are strongly clustered
in the standard cosmological model with {\it gaussian} random fluctuations;
Mo \& White 1996).
For these reasons, each 21-cm absorption halo
is expected to be highly spherical in real space.
Therefore, 21-cm absorption halos
are ideal targets to apply the Alcock-Paczy\'nski (1979) test.
Accurate measurements of angular size $\Delta\theta$ and radial depth
$\Delta v$ for a sample of galaxies
would yield a sample of $d_A(z) H(z)$,
where $d_A(z)$ and $H(z)$
are the angular diameter distance and Hubble constant, respectively,
both of which are, in general, functions of $\Omega_M$, $w$($\equiv p/\rho$) and $k$,
with $w$ describing the equation of state for dark energy and
$k$ being the curvature of the universe.
As an example, let us assume
that $\Omega_M$($\approx 0.3$) has been fixed
exactly by independent observations and $k=0$
and that $w\approx -1$.
Then one obtains $|d ln[d_A(z) H(z)]/dw|=0.45$ at $z=30$
(Huterer \& Turner 2001).
Let us suppose a relative measurement error
on each individual $d_A(z) H(z)$ is $20\%$,
then with ten thousand galaxies,
one could obtain a highly accurate
constraint on $w$ with $\Delta w = 20\%/0.45/\sqrt{10000} \sim 0.004$.
Likely, the accuracy of $w$ determined by this method may eventually be limited by
the accuracy with which $\Omega_M$ (and $k$) can be determined
by independent observations, due to the degenerate nature.
We stress that this method
is valid for each individual first galaxy and unaffected
by uncertainties, for example,
in the precise abundance of such galaxies.
In post-survey analyzes
one faces the practical issue of extracting the wanted
signals from the raw data, whose amplitude is expected to be dominated by
foreground radio sources, including galactic synchrotron radiation,
galactic and extragalactic free-free emission, and extragalactic point sources
(e.g., Di Matteo, Ciardi, \& Miniati 2004).
While seemingly daunting,
it has already been shown that signals of the amplitude
proposed here may be recovered with relatively high fidelity,
when one takes into account the expected, potent differences
in the spectral and angular properties between
the 21-cm signal and foreground contaminants
(e.g., Zaldarriaga, Furlanetto, Hernquist 2004;
Santos, Cooray, \& Knox 2005;
Wang \etal 2005).
Since the 21-cm absorption halos are expected to be rather regular and simple,
one might be able to significantly enhance the signal by
using additional techniques, such as matched filter algorithm,
in combination with foreground ``cleaning" methods.
Finally,
the amount of data in such a high spatial and frequency resolution
3-dimensional survey will be many orders of magnitude larger than that of WMAP.
Computational challenges for analyzing it
will be of paramount concern and most likely demand new
and innovative approaches.
\section{Conclusions}
It is shown that a first galaxy
hosted by a halo of mass $M=10^{7.5}-10^9\msun$ at $z=30-40$
possesses a large 21-cm absorption halo against the CMB
with a brightness temperature decrement $\delta T=-(100-150)$~mK
and an angular size of $10^{''}-100^{''}$.
A 21-cm tomographic survey of galaxies in the redshift shell
at $z=30-40$ may detect millions of galaxies
and may yield critical information on cosmology and galaxy formation.
A successful observation may need
an angular resolution of $\le 1^{''}$,
a spectral resolution of $\le 4$kHz,
and a sensitivity of $\le 10$~mK at $35-45$~MHz.
LOFAR appears poised to be able to
execute this unprecedented task, at least for the high end
of the distribution.
At least four fundamental applications may be launched with such a survey,
which could potentially revolutionize cosmological study
and perhaps the field of astro-particle physics.
First, it may provide unprecedented
constraint on star formation physics in first galaxies,
for there is a proprietary sharp feature
related to the threshold halo mass for efficient atomic cooling.
Second, it may provide a unique and sensitive probe
of the small-scale power in the cosmological model
hence physics of dark matter and inflation,
by being able to, for example,
constrain $n_s$ to an accuracy of $\Delta n_s=0.01$
at a high confidence level.
Constraints on the nature of dark matter particles,
i.e., mass or temperature, or running of index could be still tighter.
Third, clustering of galaxies that may be computed
with such a survey will provide an independent set
of characterizations of potentially interesting features
on large scales in the power spectrum including the baryonic oscillations,
which may be compared to local measurements
(Eisenstein \etal 2005) to
shed light on gravitational growth
and other involved processes from $z=30$ to $z=0$.
Finally, the 21-cm absorption halos
are expected to be highly spherical
and trace the Hubble flow faithfully,
and thus are ideal systems for an application of the Alcock-Paczy\'nski test.
Exceedingly accurate determinations of
key cosmological parameters, in particular,
the equation of state of the dark energy,
may be finally realized.
As an example, it does not seem excessively difficult
to determine $w$ to an accuracy of
$\Delta w\sim 0.01$, if $\Omega_M$ has been determined
to a high accuracy by different means.
If achieved,
it may have profound ramifications pertaining dark energy and fundamental
particle physics (e.g., Upadhye, Ishak, \& Steinhardt 2005).
If a null detection of the proposed signal is found,
as it might turn out,
implications may be as profound.
It might be indicative of some heating and/or reionizing sources
in the early universe ($z=30-200$)
that precede or are largely unrelated to structure formation,
possibly due to yet unknown properties of dark matter particles or dark energy.
Alternatively, star formation and/or BH accretion in first galaxies
may be markedly different from our current expectations.
\acknowledgments
I thank Dr. Daniel Schaerer for helpful information on Pop III stars.
This research is supported in part by grants
AST-0206299, AST-0407176 and NAG5-13381.
|
Title:
Radial velocities of population II binary stars. II |
Abstract: Here we publish the second list of radial velocities for 91 Hipparcos stars,
mostly high transverse velocity binaries without previous radial velocity
measurements. The measurements of radial velocities are done with a
CORAVEL-type radial velocity spectrometer with an accuracy better than 1 km/s.
We also present the information on eight new radial velocity variables - HD
29696, HD 117466AB, BD +28 4035AB, BD +30 2129A, BD +39 1828AB, BD +69 230A, BD
+82 565A and TYC 2267-1300-1 - found from our measurements. Two stars (HD
27961AB and HD 75632AB) are suspected as possible radial velocity variables.
| https://export.arxiv.org/pdf/astro-ph/0601212 |
\begin{center}
\vbox{\scriptsize
\tabcolsep 2pt
\begin{tabular}{rlcrc|rlcrc}
\multicolumn{10}{c}{\parbox{120mm}{\baselineskip=8.5pt
{\smallbf Table 3.}{ \small Individual radial velocities.}}}\\
\noalign{\vskip1mm}
\tablerule
No. & ~~Star name & HJD & ${V_r}$~~ & ${\varepsilon}1 $ &
No. & ~~Star name & HJD & ${V_r}$~~ & ${\varepsilon}1 $ \\
& & +2400000 & km/s & km/s &
& & +2400000 & km/s & km/s \\
\tablerule
\noalign{\vskip1mm}
1. & HD\,225220\,AB & 51794.481 & --24.3 & 0.6 & ~~34. & & 52994.830 & --6.9 & 0.6 \\
& & 52950.633 & --24.4 & 0.6 & ~~35. & HIP\,17876 & 52992.880 & --10.3 & 0.8 \\
2. & HIP\,375 & 51783.569 & --9.8 & 0.7 & ~~36. & HD\,26735\,A & 51893.445 & 64.3 & 0.7 \\
& & 51794.486 & --8.6 & 0.7 & & & 51893.453 & 64.8 & 0.8 \\
& & 52950.672 & --9.4 & 0.6 & & & 52991.865 & 63.0 & 0.7 \\
3. & HDE\,236325 & 51784.493 & 40.7 & 0.6 & ~~37. & HD\,26735\,B & 51893.441 & 62.8 & 0.8 \\
& & 51795.494 & 40.5 & 0.6 & & & 51893.449 & 63.6 & 0.8 \\
4. & BD\,+38 35\,AB & 51786.503 & --9.1 & 0.8 & & & 52991.879 & 64.4 & 0.6 \\
& & 51794.507 & --9.7 & 0.8 & & & 52994.842 & 63.6 & 0.6 \\
5. & HDE\,232169 & 51784.502 & --3.6 & 0.7 & ~~38. & HD\,27961\,AB & 52949.913 & 36.3 & 0.7 \\
& & 51795.501 & --4.3 & 0.7 & ~~39. & BD\,+03 609 & 52990.846 & 53.4 & 0.7 \\
6. & BD\,+23 80\,A & 51786.522 & --17.1 & 0.7 & & & 52995.791 & 52.5 & 0.7 \\
& & 51794.525 & --17.1 & 0.7 & ~~40. & HIP\,21089\,AB & 51893.465 & 32.0 & 1.0 \\
7. & BD\,+23 80\,B & 51786.526 & --17.7 & 1.2 & & & 52990.859 & 32.4 & 0.7 \\
& & 51794.531 & --17.1 & 1.5 & & & 52995.799 & 32.5 & 0.7 \\
8. & HDE\,236523\,AB & 51784.537 & --9.7 & 0.8 & ~~41. & HIP\,21842 & 52990.901 & 4.4 & 0.8 \\
& & 51795.508 & --7.7 & 0.8 & & & 52995.805 & 4.7 & 0.7 \\
& & 52950.689 & --10.7 & 0.6 & & & 52995.813 & 5.4 & 0.7 \\
9. & HD\,6094\,AB & 52950.700 & 8.7 & 0.7 & ~~42. & BD\,+44 1142\,AB & 51892.484 & 0.2 & 0.8 \\
10. & HD\,6448\,A & 51784.552 & --25.3 & 0.6 & & & 52949.965 & --1.3 & 0.6 \\
11. & HD\,6448\,B & 51784.559 & --25.6 & 0.8 & ~~43. & LP\,360-6\,AB & 51892.500 & --1.5 & 1.0 \\
12. & BD\,+83 28\,A & 51794.546 & --18.2 & 0.7 & & & 52940.027 & --1.4 & 0.7 \\
13. & BD\,+14 232\,AB & 51786.563 & 25.1 & 0.7 & & & 52940.031 & --2.5 & 0.8 \\
& & 51794.593 & 24.5 & 0.7 & & & 52950.018 & --4.1 & 0.7 \\
14. & BD\,+75 65\,A & 51786.569 & --22.8 & 0.7 & ~~44. & HIP\,25413 & 52990.944 & 24.1 & 0.9 \\
& & 51794.572 & --22.4 & 0.7 & & & 53000.810 & 23.9 & 0.8 \\
15. & HIP\,8607 & 51795.541 & 25.0 & 0.8 & ~~45. & HDE\,244359\,AB & 52991.897 & 98.8 & 0.7 \\
& & 52953.828 & 23.9 & 0.8 & & & 53001.834 & 99.4 & 0.7 \\
& & 52959.766 & 23.3 & 0.7 & ~~46. & HD\,36195\,AC & 52991.918 & --21.7 & 0.6 \\
16. & HD\,14106 & 51795.554 & 51.8 & 0.7 & ~~47. & WDS\,05329+5208\,B & 52991.926 & --22.0 & 0.8 \\
17. & HD\,14202 & 51784.580 & --10.9 & 0.7 & ~~48. & HD\,248330 & 52991.946 & 24.9 & 0.6 \\
& & 51795.563 & --11.7 & 0.7 & ~~49. & HD\,39448 & 52991.968 & 45.6 & 0.6 \\
18. & HIP\,10774\,B & 51784.575 & --10.4 & 0.7 & ~~50. & HD\,40412 & 52991.981 & --46.5 & 0.7 \\
& & 51795.567 & --10.4 & 0.7 & ~~51. & HIP\,120002 & 52991.992 & --46.5 & 0.7 \\
19. & HD\,14511\,AB & 51784.587 & 13.1 & 0.7 & ~~52. & BD\,+77 250 & 51795.611 & --52.6 & 0.8 \\
& & 51786.582 & 12.2 & 0.7 & & & 51894.547 & --53.5 & 0.8 \\
& & 51795.574 & 12.5 & 0.7 & ~~53. & HD\,49622\,AB & 51892.477 & 10.5 & 0.9 \\
20. & BD\,+00 494\,AB & 52990.792 & 31.8 & 0.7 & & & 52940.039 & 10.8 & 0.7 \\
21. & BD\,+39 692\,AB & 52991.736 & --18.2 & 0.6 & & & 52950.026 & 8.6 & 0.6 \\
& & 52994.801 & --18.3 & 0.6 & ~~54. & BD\,+71 380\,A & 51894.535 & 25.8 & 0.8 \\
22. & BD\,+01 549\,AB & 52990.807 & 53.1 & 0.7 & ~~55. & BD\,+65 559\,AB & 51894.563 & 52.3 & 0.9 \\
& & 52994.769 & 52.5 & 0.6 & ~~56. & HD\,56244\,AB & 52990.994 & --36.8 & 0.6 \\
23. & BD\,+00 523\,AB & 52990.816 & --4.0 & 0.7 & & & 52994.004 & --37.1 & 0.6 \\
& & 52994.777 & --3.7 & 0.7 & ~~57. & BD\,+28 1365 & 52991.009 & 38.9 & 0.8 \\
24. & BD\,+00 549\,A & 51893.340 & 88.1 & 0.9 & & & 53001.981 & 37.6 & 0.7 \\
25. & BD\,+15 452\,AB & 52953.840 & 25.3 & 0.7 & ~~58. & BD\,+39 1967\,AB & 52991.027 & 33.7 & 1.0 \\
& & 52958.846 & 25.0 & 0.6 & & & 53001.993 & 37.5 & 0.8 \\
26. & HD\,20289\,AB & 52991.744 & 16.8 & 0.6 & ~~59. & HD\,60820\,AB & 51892.590 & 63.0 & 0.6 \\
& & 52994.794 & 17.4 & 0.6 & ~~60. & BD\,+25 1788\,AB & 52991.047 & 58.4 & 0.7 \\
27. & HD\,20369\,AB & 52990.830 & --65.1 & 0.6 & & & 53002.002 & 59.6 & 0.7 \\
& & 52994.785 & --65.7 & 0.7 & ~~61. & HD\,71185 & 51894.578 & --65.2 & 0.7 \\
28. & BD\,+12 472\,AB & 52991.755 & --14.9 & 0.8 & ~~62. & BD\,+40 2062\,AB & 52992.028 & 44.2 & 0.9 \\
& & 52994.808 & --14.4 & 0.7 & & & 53002.009 & 44.7 & 0.9 \\
29. & HIP\,16332\,AB & 51795.589 & --55.2 & 0.8 & ~~63. & HD\,73889 & 52993.021 & 56.0 & 0.6 \\
30. & HDE\,232816 & 51786.599 & 4.3 & 0.7 & ~~64. & HD\,74861\,AB & 52951.025 & --25.5 & 0.6 \\
& & 51795.515 & 4.0 & 0.7 & & & 52952.030 & --25.9 & 0.6 \\
31. & BD\,+16 492\,AB & 52992.824 & 23.5 & 0.6 & & & 52953.027 & --23.6 & 0.7 \\
& & 52994.822 & 23.0 & 0.6 & & & 52953.030 & --24.4 & 0.7 \\
32. & HD\,23439\,A & 52949.936 & 50.5 & 0.6 & ~~65. & BD\,+46 1436 & 52991.056 & 58.6 & 0.7 \\
& & 52958.865 & 50.4 & 0.7 & & & 52996.065 & 58.8 & 0.7 \\
33. & HD\,23439\,C & 52949.952 & --0.5 & 0.7 & ~~66. & HD\,75632\,AB & 52985.016 & 48.0 & 0.7 \\
& & 52958.899 & --1.6 & 0.7 & & & 52987.017 & 45.8 & 0.6 \\
34. & HD\,23865\,AB & 52992.836 & --8.0 & 1.0 & ~~67. & BD\,+49 2588 & 51784.328 & 24.9 & 0.7 \\
& & 52992.841 & --9.7 & 0.9 & & & 51785.297 & 26.3 & 0.7 \\
\end{tabular}
}
\end{center}
\newpage
\begin{center}
\vbox{\scriptsize
\tabcolsep 2pt
\begin{tabular}{rlcrc|rlcrc}
\multicolumn{10}{c}{\parbox{125mm}{\baselineskip=8.5pt
{\smallbf Table 3.}{ \small \hskip 1.5mm Continued}}}\\
\noalign{\vskip1mm}
\tablerule
No. & ~~Star name & HJD & ${V_r}$~~ & ${\varepsilon}1 $ &
No. & ~~Star name & HJD & ${V_r}$~~ & ${\varepsilon}1 $ \\
& & +2400000 & km/s & km/s &
& & +2400000 & km/s & km/s \\
\tablerule
\noalign{\vskip1mm}
68. & BD\,+68 986\,A & 51784.338 & 1.5 & 0.8 & ~~81. & & 51022.427 & 16.6 & 0.7 \\
& & 51794.337 & 2.0 & 0.7 & & & 51784.436 & 17.3 & 0.7 \\
69. & BD\,+75 641 & 51016.382 & --118.8 & 0.7 & & & 51795.384 & 18.0 & 0.7 \\
& & 51018.403 & --119.4 & 0.7 & ~~82. & BD\,+14 4697\,AB & 51779.467 & --57.0 & 0.7 \\
& & 51018.423 & --119.3 & 0.7 & & & 51786.446 & --55.6 & 0.7 \\
& & 51022.408 & --119.0 & 0.7 & ~~83. & HDE\,239892\,AB & 51021.457 & --19.0 & 1.1 \\
& & 51785.285 & --117.5 & 0.7 & & & 51021.467 & --22.2 & 1.0 \\
& & 51795.370 & --117.3 & 0.7 & & & 51022.448 & --18.6 & 1.0 \\
70. & HD\,177349 & 51780.375 & --1.9 & 0.8 & & & 51784.447 & --21.3 & 1.3 \\
& & 51785.339 & --0.9 & 0.7 & & & 51795.447 & --20.5 & 0.9 \\
71. & HIP\,93600 & 51780.383 & --30.6 & 0.9 & ~~84. & HDE\,235919\,AB & 51784.459 & --30.4 & 0.7 \\
& & 51785.343 & --31.4 & 0.7 & & & 51795.458 & --30.3 & 0.7 \\
72. & BD\,+49 3113 & 51784.352 & --8.9 & 0.7 & ~~85. & HDE\,240021 & 51021.481 & --1.9 & 1.1 \\
& & 51785.310 & --8.8 & 0.7 & & & 51021.496 & --3.7 & 1.4 \\
73. & HDE\,239292\,AB & 51784.364 & --83.3 & 0.7 & & & 51022.464 & --2.2 & 1.1 \\
& & 51794.363 & --83.3 & 0.7 & & & 51784.471 & --5.2 & 1.0 \\
74. & HDE\,356668\,AB & 51780.396 & 10.8 & 0.9 & & & 51795.468 & --3.0 & 1.2 \\
& & 51785.376 & 10.8 & 0.7 & ~~86. & HDE\,236100 & 51779.512 & --38.8 & 0.7 \\
75. & HDE\,340730 & 51784.387 & --10.5 & 0.8 & & & 51783.420 & --39.7 & 0.7 \\
& & 51785.391 & --10.8 & 0.7 & ~~87. & BD\,+21 4948 & 51783.461 & 3.3 & 0.7 \\
76. & GSC\,2161-755 & 51785.403 & --35.5 & 0.9 & & & 51786.453 & 4.4 & 0.8 \\
77. & HDE\,235300 & 51784.377 & --110.9 & 0.7 & ~~88. & PPM\,11774 & 51794.464 & --33.3 & 0.7 \\
& & 51794.374 & --111.5 & 0.7 & ~~89. & HDE\,236223 & 51021.515 & 1.0 & 0.7 \\
78. & BD\,+13 4614\,AB & 51780.406 & --24.1 & 0.8 & & & 51022.479 & 1.7 & 0.7 \\
& & 51786.381 & --25.1 & 0.8 & & & 51784.482 & 2.3 & 0.7 \\
79. & BD\,+25 4462\,A & 51780.449 & --21.4 & 1.0 & & & 51795.482 & 1.6 & 0.7 \\
& & 51786.389 & --20.7 & 1.1 & ~~90. & HDE\,223523\,AB & 52950.644 & --90.5 & 0.6 \\
80. & G\,212-24\,AB & 51780.481 & --101.7 & 0.9 & & & 52953.684 & --91.7 & 0.7 \\
& & 51786.415 & --102.0 & 0.8 & ~91. & BD\,+43 4596\,AB & 51783.555 & 9.2 & 0.8 \\
81. & HDE\,235634 & 51010.522 & 17.5 & 0.7 & & & 51786.461 & 9.1 & 0.7 \\
& & 51016.403 & 17.0 & 0.8 & & & 51794.447 & 9.9 & 0.7 \\
\tablerule
\end{tabular}
}
\end{center}
|
Title:
Large dust grains in the inner region of circumstellar disks |
Abstract: CONTEXT: Simple geometrical ring models account well for near-infrared
interferometric observations of dusty disks surrounding pre-main sequence stars
of intermediate mass. Such models demonstrate that the dust distribution in
these disks has an inner hole and puffed-up inner edge consistent with
theoretical expectations. AIMS: In this paper, we reanalyze the available
interferometric observations of six intermediate mass pre-main sequence stars
(CQ Tau, VV Ser, MWC 480, MWC 758, V1295 Aql and AB Aur) in the framework of a
more detailed physical model of the inner region of the dusty disk. Our aim is
to verify whether the model will allow us to constrain the disk and dust
properties. METHODS: Observed visibilities from the literature are compared
with theoretical visibilities from our model. With the assumption that
silicates are the most refractory dust species, our model computes
self-consistently the shape and emission of the inner edge of the dusty disk
(and hence its visibilities for given interferometer con gurations). The only
free parameters in our model are the inner disk orientation and the size of the
dust grains. RESULTS: In all objects with the exception of AB Aur, our
self-consistent models reproduce both the interferometric results and the
near-infrared spectral energy distribution. In four cases, grains larger than
1.2 micron, and possibly much larger are either required by or consistent with
the observations. The inclination of the inner disk is found to be always
larger than 30 deg, and in at least two objects much larger.
| https://export.arxiv.org/pdf/astro-ph/0601438 |
\title{Large dust grains in the inner region of circumstellar disks}
\author{
Andrea Isella \inst{1,2},
Leonardo Testi \inst{1}
and
Antonella Natta \inst{1}
}
\institute{
Osservatorio Astrofisico di Arcetri, INAF, Largo E.Fermi 5,
I-50125 Firenze, Italy
\and
Dipartimento di Fisica, Universit\'a di Milano, Via Celoria 16,
20133 Milano, Italy
}
\offprints{isella@arcetri.astro.it}
\date{Received ...; accepted ...}
\authorrunning{ISELLA, TESTI \& NATTA}
\titlerunning{Large grains in circumstellar disks}
\abstract
{ Simple geometrical ring models account well for
near-infrared interferometric observations of dusty disks surrounding
pre-main sequence stars of intermediate mass. Such models
demonstrate that the dust distribution in these
disks has an inner hole and puffed-up inner edge consistent with
theoretical expectations. }
{ In this paper, we reanalyze the available interferometric
observations of six intermediate mass pre-main sequence stars (CQ
Tau, VV Ser, MWC 480, MWC 758, V1295 Aql and AB Aur) in the
framework of a more detailed physical model of the inner region of
the dusty disk. Our aim is to verify whether the model will allow us
to constrain the disk and dust properties.}
{ Observed visibilities from the literature are compared with
theoretical visibilities from our model. With the
assumption that silicates are the most refractory dust species,
our model computes self-consistently the shape and emission of the
inner edge of the dusty disk (and hence its visibilities for given
interferometer configurations). The only free parameters in our
model are the inner disk orientation and the size of the dust
grains.}
{ In all objects with the exception of AB Aur, our self-consistent
models reproduce both the interferometric results and the
near-infrared spectral energy distribution. In four cases, grains
larger than $\sim$1.2\um, and possibly much larger are either
required by or consistent with the observations.
The inclination of the inner disk is found to be always
larger than $\sim 30^{\circ}$, and in at least two objects much
larger.}
\keywords{}
\section {Introduction}
Understanding the properties and evolution of the dust grains
contained in proto-planetary disks around pre-main sequence stars is
important because they are the seeds from which planets may form. We
have now strong evidence that grains in disks are very different from
the grains in the diffuse interstellar medium and in the molecular
clouds from which disks form, as reviewed, e.g., by Natta et
al. (2006). In many objects, observations with millimeter
interferometers have provided strong evidence that the grains in the
outer and cooler regions of the disk (further than 50AU from the star)
have been hugely processed, and have grown from sub-micron
sizes to millimeter and centimeter ones. Closer to the star, however,
in the regions were planets are more likely to form, observational
evidence has been confined to grains close to the disk surface. For
these, which however account for a tiny fraction of the total dust
mass, emission in the silicate features has shown a correlation
between the shape of the feature and its strength that is interpreted
as due to growth of the grains from size $a\sim0.1\mu$m to
$a\sim1\mu$m (van Boekel et al. 2003, 2004; Meeus et al. 2003). In
this inner disk, the properties of the grains in the disk midplane are
still unknown.
In the last few years, due to the new long baseline near-infrared
interferometers, many important steps forward in the study of the
internal regions of circumstellar disks have occurred. The available
near-infrared interferometric observations of T Tauri (TTS)
and Herbig Ae (HAe) stars (Eisner et al. 2003, 2004; Millan-Gabet et
al. 2001; Tuthill et al. 2001; Monnier et al. 2005) confirm the idea
that the inner disk properties are controlled by the dust evaporation
process which produce a ``puffed-up'' inner rim at the dust
destruction radius (Natta et al. 2001; Dullemond, Dominick \& Natta
2001, hereafter DDN01). In these models, the location and shape of the
rim depends on the properties of grains located not on the disk
surface but on its midplane.
Isella \& Natta (2005, hereafter IN05) have recently proposed models
of the ``puffed-up''inner rim which include a self-consistent
description of the grain evaporation and its dependence on the gas
density. IN05 have explored a large range of grain properties, and
discussed how the location of the rim depends on grain properties. In
this paper, we will use the IN05 models to analyze the existing
interferometric data of the best observed HAe stars to explore, in
practice, the constraints on grain properties provided by this
technique and their uncertainties.
As a byproduct of the modeling process, one obtains also the
orientation of the inner disk (i.e. its inclination with
respect to the line of sight and its position angle); this can be
compared with the orientation of the outer disk, obtained from
millimeter observations of the molecular gas and dust emission and/or
scattered light in the optical.
The paper is organized as follows. In \S2 we describe the available
interferometric observations of the target stars. The IN05
model for the inner rim is briefly summarized in \S3 and used
to fit the observations of the individual objects in \S4.
A comparison of the results with previous analysis of the same data
is presented in \S5. Our results are discussed in \S6. Conclusions
follow in \S7.
\section{Target stars and observations}
Our sample is composed of six HAe stars (AB Aur, CQ Tau, VV Ser, MWC
480, MWC 758 and V1295 Aql), for which near-infrared interferometric
observations exist in the literature. Table \ref{tab.sources}
summarizes the physical properties of the target stars. All the stars
are classified as young stellar objects with masses ranging from 1.5
to 4.3 solar mass and a spectral type between A0/B9 and A8/F2. CQ Tau
and VV Ser belong to the family of UXORs and are characterized by
large and irregular variability.
We use visibility measurements of the target stars from the literature,
obtained with interferometric observations carried out with PTI (Palomar
Testbed Interferometer) in K band ($\lambda_0=2.2\mu$m, $\Delta\lambda
= 0.4\mu$m) described in Eisner et al. (2004). For AB Aur and
V1295Aql, IOTA observations are also available (Millan-Gabet et
al. 2001) for the K'($\lambda_0=2.16\mu$m, $\Delta\lambda = 0.32\mu$m)
and H ($\lambda_0=1.65\mu$m, $\Delta\lambda = 0.30\mu$m) bands.
\begin{table*}
\centering
\begin{minipage}[t]{12.5cm}
\caption{Stellar parameters.}
\label{tab.sources}
\begin{tabular}{c c c c l l l l} %
\hline\hline
Source & Alternate Name & Spectral Type & d & T & L & M & Av \\
~ & ~ & ~ & (pc) & (K) & (\Lsun) & (\Msun) & ~ \\
\hline
AB Aur & HD 31293 & A0pe & 144 & 9772 & 47 & 2.4 & 0.5 \\
MWC 480 & HD 31648 & A2/3ep+sh & 140 & 8700 & 25 & 2.2 & 0.25 \\
MWC 758 & HD 36112 & A5IVe & 230 & 8128 & 22 & 2.0 & 0.22 \\
CQ Tau & HD 36910 & A8 V/F2 IVea & 100 & 8000 & 5 & 1.5 & 1.00 \\
VV Ser & HBC 282 & B9/A0 Vevp & 260 & 10200 & 49 & 3.0 & 3.6 \\
V1295 Aql & HD 190073 & B9/A0 Vp+sh & 290 & 8912 & 83 & 4.3 & 0.19 \\
\hline
\end{tabular}
\\\\
Stellar parameters are from Hillenbrand et al. (1992), van den Ancker
et al. (1998), Chiang et al. (2001), Strizys et al. (1996), Mannings
et al. (2000) and references therein.
\end{minipage}
\end{table*}
\section{Model description}
\label{sec:model}
We use a model based on the assumption that the near-infrared
emission of HAe stars originates in the ``puffed up'' inner rim which
forms in the circumstellar disk at the dust evaporation radius (Natta
et al 2001; DDN01).
In IN05 we revised the concept of the ``puffed up''inner rim,
introducing the dependence of the dust evaporation temperature on the
local gas density, following the dust model of Pollack et
al. (1994). The main result is that the surface of the rim presents a
curved shape (see Fig.1), whose features are summarized in the
following.
\subsection{The dust evaporation and the ``puffed-up'' inner rim}
\label{sec:mod_1}
Following the suggestion of Natta et al. (2001), we assume that
the dust component of a circumstellar accreting disk is internally
truncated by the dust evaporation process, forming an inner hole of
radius $R_{evp}$ inside of which only gas can survive. If the
radiation absorption due to this inner gas is negligible (as is
often the case; see, e.g., Muzerolle et al. 2004), dust evaporation
occurs where the equilibrium temperature $T_d$ of grains embedded
in the unattenuated stellar radiation field, equals their evaporation
temperature $T_{evp}$. In IN05 we used an analytical solution of the
radiation transfer problem (Calvet et al. 1991, 1992) to calculate the
grain temperature inside the disk and we showed that evaporation
occurs at a distance from the star that can be expressed as:
\begin{equation}
\label{eq:Revp}
R_{evp}[AU] = 0.034 \cdot \left( \frac{1500}{T_{evp}} \right)^2
\sqrt{ \frac{L_{\star}}{L_\odot} \left( 2+\frac{1}{\epsilon} \right) },
\end{equation}
where $L_{\star}$ is the stellar luminosity and $\epsilon$ is the
ratio of the Planck mean opacity at $T_{evp}$ to that at the
stellar effective temperature $T_\star$,
$\epsilon=\kappa_P(T_{evp})/\kappa_P(T_{\star})$. The quantity
$\epsilon$ measures the cooling efficiency of the grains; it depends
on the wavelength dependence of the absorption efficiency of the
grains and varies with grain composition and size.
If the dust in the proto-planetary disk is composed of different
types of grains, the location and structure of the inner rim
depends on the properties of the grains with the highest evaporation
temperature. In the dust model proposed by Pollack et
al. (\cite{PH94}), the most refractory grains are silicates for which
$T_{evp}$ is given by the relation
\begin{equation}
\label{eq:Tevp}
T_{evp}(r,z) = 2000 \cdot [\rho_g(r,z)]^{0.0195},
\end{equation}
valid for the gas density $\rho_g$ in the range between $10^{-18}g\,
cm^{-3}$ and $10^{-5}g\, cm^{-3}$.
In the following analysis, we will therefore assume that the inner
disk dust is made of silicates, with optical properties given by
Weingartner \& Draine (2001); thus, $\epsilon$ is uniquely defined by
the grain radius $a$, and we will use $a$, rather than $\epsilon$, as
a model parameter.
Assuming that the proto-planetary disk is in hydrostatic equilibrium
in the gravitational field of the central star and that it is
isothermal in the vertical direction $z$, the gas density
$\rho_g(r,z)$ has its maximum value on the midplane and decreases
with $z$ as
\begin{equation}
\label{eq:rhog}
\rho_g(r,z)=\rho_{g,0}(r) \exp(-z^2/2h(r)^2) ,
\end{equation}
where $h$ is the pressure scale height of the disk. The midplane
density can be expressed as a power-law of $r$ $\rho_{g,0}(r) =
\rho_{g,0}(r_0) (r_0/r)^{\gamma}$, with $\gamma$ of the order of
2--3 (see, e.g., Chiang \& Goldreich 1997).
The decrease of $\rho_g$ with $z$, combined with Eq.\ref{eq:Tevp},
implies that the silicate evaporation temperature varies from, i.e.,
$\sim1500K$ on the midplane (assuming a typical gas density of
$\sim10^{-7}$g/cm$^3$) to $\sim 1000K$ at $z/h=6.4$ and $\sim 800K$ at
$z/h=8$. Since $T_{evp}$ decreases with $z$ , it is immediately clear
from Eq.\ref{eq:Revp} that the distance from the star at which dust
evaporates increases with $z$, describing a curved surface as shown in
Fig.\ref{fig:disk}.
The dependence of $T_{evp}$ on the gas density is an important
factor when computing the shape of the rim in the vertical direction,
where the gas density varies by many orders of magnitude while the
distance from the star is practically unchanged. In the radial
direction, we expect relatively small variations of $\rho_{g,0}$, for
any reasonable value of the disk mass, so that the distance of the rim
from the star, measured in the midplane, is practically independent of
the gas density.
The emission of the rim is computed assuming that it originates from
the surface characterized by an effective temperature
$T_{eff}=T(\tau_d=2/3)$, where $\tau_d$ is the optical depth for the
emitted radiation. The $T_{eff}$ surface, therefore, defines the
observed location and shape of the rim. In IN05, we discussed how the
$T_{evp}$ and the $T_{eff}$ surfaces behave for small and large
silicate grains, and showed that the $T_{eff}$ surface moves closer
to the star for increasing grain size until a critical value, which
for silicates is about 1.2 \um. Larger grains produce rims with
$T_{eff}$ surfaces practically independent of $a$. Therefore, for a
fixed stellar luminosity, silicates with $a\sim 1.2$\um\ give the
minimum value of the distance of the rim from the star. Conversely, if
the measured rim distance is equal to this minimum value, one can
derive from it only a lower limit ($\sim 1.2$\um) to the grain size.
The rim emission peaks at near-infrared wavelengths. At $\lambda
\simless 5-7$\um, one can assume that the observed flux is the sum of
the stellar + rim emission, with only negligible contribution from the
outer disk (see, e.g., DDN01). We model the stellar photospheric
flux using standard Kurucz model atmospheres.
\subsection{Visibility model}
Due to the limited coverage of the $u-v$ plane of the existing
near-infrared interferometers, it is not possible at present to
recover full images from the available data, and one has to resort to
the analysis of the visibilities on given interferometric baselines.
Starting from the synthetic images of the inner rim (see IN05), we
compute the predicted visibility values using a Fast Fourier Transform
recipe. For face-on inclination, due to the circular symmetry of the
rim image, the visibility depends only on the length of the baseline
$B$. For inclination greater than zero, the image of the rim has an
``elliptical'' shape: the minor axis decreases with increasing inclination
and the upper half of the rim becomes brighter than the lower part.
For baselines oriented along the direction of the minor axis of the
rim image, the visibility decreases more slowly than for those oriented
along the major axis. For all other orientations of the baseline, the
visibility will have values intermediate between these two (see
Fig.\ref{fig:vis}). Moreover, due to the Earth rotation during the
observation, the baseline corresponding to a fixed telescope pair
moves in the u-v plane describing an ellipse. Along this ellipse,
each point is related to the hour angle $HA$ of the target object in
the sky. In the next section we use the $V^2$-$B$ plot and $V^2$-$HA$
plot, to show how the models fit the observations.
The visibility model takes into account the emission of the central
star, modeled as a uniform disk of radius $R_{\star}$. If $F_{\star}$
and $F_{d}$ are respectively the stellar and the inner rim flux at the
wavelength of the observation, the total visibility is given by the
relation:
\begin{equation}
V^2 = \left( \frac{F_\star V_\star + F_d V_d}{F_\star + F_d}
\right) ^2,
\end{equation}
where $V_\star$ and $V_d$ are the visibility values of the star and of
the disk. Note that for the PTI configuration (baselines between 84m
and 100m) $V_\star$ is in all cases very closed to 1.
\section {Comparison with the observations}
\label{sec:fit}
\subsection{Model parameters}
Once the stellar and dust properties are fixed, the model-predicted
visibilities depend on the dust grain radius $a$, which completely
defines the rim structure, and two parameters (observational parameters
in the following) that describe the orientation of the disk, namely
the inclination $\iota$ and position angle PA.
For each star, we firstly compute the predicted rim structure varying
the size of the grains from very small to very large values. As
discussed in \S3, we assume that silicates are the most refractory
component; we take the optical properties of astronomical silicates
defined by Weingartner \& Draine (\cite{WD01}). Other disk parameters
(i.e., mass and density radial profile) can be neglected in this
analysis. We fix $\rho_{g,0}(R_{rim})\sim 10^{-7}$ g/cm$^3$ which
gives a total disk mass of about 0.1\Msun, for a fiducial value of
$\gamma=2.5$ and an outer disk radius of the disk of 200AU.
Once the structure of the ``puffed up'' inner rim is calculated, the
predicted visibility depends on the orientation of the disk in
the sky, defined by the inclination $\iota$ of the midplane of the
disk with respect to the line of sight and its position angle $PA$,
measured from north to east and relative to the major axis of the
projected image of the disk on the sky. The inclination is defined
so that $\iota=0^{\circ}$ identifies a face-on disk while
$\iota=90^{\circ}$ corresponds to an edge-on disk. For inclinations
higher than 80$^\circ$ the rim emission is likely absorbed by the
outer regions of the disk and the IN05 model can not be applied.
In practice, we compute visibility models for each object
varying $a$, $\iota$ and $PA$ independently. We then select the best
models calculating the reduced $\chi^2$ between the visibility data
and the theoretical values calculated at the same points in the u-v
plane. The observed near-infrared fluxes, and the IOTA data when
available, are then ``a posteriori'' used to check the quality
of the fit and, when possible, to reduce the degeneracy due to the
small number of visibility points. For some stars, the existing data
do not constrain the parameters, but still define a range, outside of
which the fit to the data is very poor.
Tab.\ref{tab:fit} shows in column 2 the best values of the
astronomical silicate radius $a$ and, in column 3, the corresponding
values of the radius of the inner rim $\Rrim$. The two observational
parameters $i$ and $PA$ are given in columns 4 and 5,
respectively. Note that the free parameters are in boldface; $\Rrim$
is a derived quantity.
\begin{table*}
\centering
\begin{minipage}[t]{17.5cm}
\caption{Best fitting model parameters. }
\label{tab:fit}
\begin{tabular}{l|l l l l | l l l | l l }
\hline\hline
~ & \multicolumn{4}{c}{IN05 model} &
\multicolumn{3}{c}{Eisner et al. (2004)} & \multicolumn{2}{c}{outer
disk} \\
Source &{\boldmath $a$ } & {$\Rrim$} & {\boldmath
$\iota$} & {\bf PA}& $\Rrim$ & $\iota$ & PA & $\iota$ & PA \\
~ &($\mu$m) & (AU) & (deg) &(deg) & (AU) & (deg)
&(deg) & (deg) & (deg) \\
\hline
MWC 758 &{\boldmath $\geq 1.2$} & 0.32 & {\bf 40} & {\bf 145} & 0.21 &
$36^{+3}_{-2}$ & $127^{+4}_{-3}$ & 46 & $ 116^{+6 \, (a)}_{-5}$\\
VV Ser & {\boldmath $\geq 1.2$} & 0.54 & {\boldmath$50-70$} &
{\boldmath$60-120$}& 0.47 & $42^{+6}_{-2}$ & $166^{+17}_{-6}$&
$72\pm5$ & $13\pm5 ^{(b)}$ \\
CQ Tau & {\boldmath $0.3 - \geq 1.2$} &
$0.16 - 0.25$ & {\boldmath $40 - 55$} & {\boldmath$145-
190$} & 0.23& $48^{+3}_{-4}$ & $ 106^{+4}_{-5}$
&$63^{+10}_{-15}$ & $2\pm13^{(c)}$ \\
V1295 Aql & {\boldmath $0.3 - \geq1.2$} & $0.7 -
1.2$ & {\boldmath$40 - 65$} & &0.55 &
$23^{+15}_{-23}$ & & ~ &\\
MWC 480 & {\boldmath $0.2 - 0.3$} & $0.53
- 0.63$ & {\boldmath$30 - 65$} & ~& 0.23 & $28^{+2}_{-1}$
& $145^{+9}_{-6}$ & $ 20 - 40$ &$147 - 180^{(a,d)}$ \\
AB Aur & \multicolumn{3}{c}{impossible to fit} && 0.25 &
$8^{+7}_{-8}$ & &$15 - 35$ &$50 - 110 ^{(e,f,g,h)}$ \\
\hline
\end{tabular}
\\\\
From column 2 to 5 are reported the best fit parameters for the
``puffed-up'' inner rim, obtained with the IN05 model: the grain
radius $a$, the radius of the inner rim $\Rrim$, the inclination
$\iota$ and the position angle $PA$. The free parameters of the model
are presented in bold face. Columns 6,7 and 8 show the values of the
radius of the inner rim, the inclination and the position angle,
obtained by Eisner et al. (2004). Finally, the last two columns show
the available estimates of inclination and position angle for the
external region of the disk: $(a)$ Mannings et al. (1997); $(b)$
Pontoppidan et al. (2006); $(c)$ Testi et al. (2001, 2003); $(d)$
Simon et al. (2000); $(e)$ Fukagawa et al.(2004); $(f)$ Grady et
al. (1999); $(g)$ Corder et al. 2005; $(h)$ Pi\'etu et al. (2005).
\end{minipage}
\end{table*}
\subsection{MWC 758}
\label{sec:MWC758}
PTI visibilities are fitted by a family of models, with parameters
varying between the two extreme cases shown in
Fig.\ref{fig:MWC758_vis2}. In one case, the disk has small grains of
radius $a=0.17$ $\mu$m, $\iota= 48^{\circ}$ and $PA=134^{\circ}$; in
the other, big grains with $a \geq 1.2 \mu$m, $\iota =40^{\circ}$ and
$PA= 145^{\circ}$. Models with $a$ values within this range
will fit the observed visibilities equally well, provided that we vary
$\iota$ and $PA$ in an appropriately way.
However, if we consider also the constraints set by the SED at
near-infrared wavelengths, we find that only models with big grains
fit it reasonably well (see the right panel of Fig.\ref{fig:MWC758_vis2}). The
best-fitting model $(\chi^2_r=2.0)$ has then $a \geq 1.2$\um, inner rim radius
is $\Rrim=0.32$AU, rim effective temperature (at $z$=0) is 1460K. The
near-infrared flux, $L_{NIR}$, integrated between 2$\mu$m and 7$\mu$m
is $25\%$ of the total stellar luminosity, similar to the observed
value. Once we fix $a$, the formal uncertainties on $\iota$,
estimated from the surface where the reduced $\chi^2$ equals
$\chi^2_{min}+1$, are quite small, $\pm 3^{\circ}$. More realistic
uncertainties are of the order of $10^{\circ}$ for both $\iota$ and
$PA$.
Note that in MWC 758 the PTI visibilities define quite well the
orientation of the disk, even when the SED is not
used to constrain the grain size. In particular, the inclination
cannot be lower than about $30^\circ$.
\subsection{VV Ser}
The results for the star VV Ser are shown in
Fig.\ref{fig:VVSer_HA}. As for MWC 758, the interferometric
observations allow different sets of model parameters.
Namely, we obtain similar values of the reduced $\chi^2$
$(\sim1.2)$ for all grain sizes $a\simgreat 0.4$\um. Over this
range of $a$, inclination and position angle vary in the
intervals $45^\circ - 80^\circ$ and $60^\circ - 120^\circ$,
respectively, with lower inclinations for larger grains.
The correlation between $\iota$ and $PA$ is very strong, and
the uncertainties in these two parameters remain very large even for
fixed $a$.
Although the fit is never very good, the VV Ser SED is better
accounted for by large grains (see the right panel of
Fig.\ref{fig:VVSer_HA}) and in Table 2 we show the best values of the
parameters for $a\geq 1.2$\um. The rim effective temperature is 1400
K, the near-infrared excess is 21\% of \Lstar.
\subsection{CQ Tau}
The limited number of visibility points does not allow us to constrain
all the parameters of the disk. Fig.\ref{fig:CQTau_HA} shows two
models, with the same level of confidence $(\chi^2_r \sim 1)$;
the two models have similar orientations (inclinations of 52$^{\circ}$
and 46$^{\circ}$ with position angles of 168$^{\circ}$ and
164$^{\circ}$ respectively) but very different grain sizes
($a=0.3\mu$m and $a \geq 1.2 \mu$m) and radii of the inner rim (0.25AU and
0.16AU, respectively). For $a \geq 1.2 \mu$m, the effective
temperature of the rim (at z=0) is $T_{eff}=1480K$, while
$T_{eff}=1050$ for $a=0.3\mu$m. The SEDs of the two models are
compatible with the observed fluxes, with $L_{NIR}=13\%-18\%$
\Lstar. All the models with $a$ within this range, and similar
$\iota$ and $PA$, have similar $\chi^2$ values. Outside this range,
models give a much poorer fit to the data.
As for MWC 758, the visibility data constrain well the orientation of the
disk on the sky. The inclination, in particular, has to be quite
large, $40^{\circ} \simless \iota \simless 55^{\circ}$.
\subsection{V1295 Aql}
\label{sec:V1295Aql}
The PTI observations of V1295 Aql are characterized by a very small number
of visibility points and the disk parameters are hardly constrained. Even
adding the IOTA data does not help due to the big
errors that affect these observations. Note also that PTI and IOTA
observations are performed at different wavelengths, K and H respectively.
As for CQ Tau, we show the models for the two extreme sets of
parameters that give an equally good fit (Fig.\ref{fig:V1295_HA})
with $\chi^2_r \sim 1$. All the intermediate combinations of $a$,
$\iota$ and PA can explain the observations as well. In order to fit
the values of visibility, an inclination ranging between $40^\circ$
and $65^\circ$ is required. More face-on systems can not in general
reproduce the visibility spread in the IOTA data and the $V^2-HA$
behaviour of the PTI points. The grain radius varies from
$a\geq1.2\mu$m ($\Rrim=0.7$AU, $T_{eff}=1400K$) to $a=0.3\mu$m
($\Rrim=1.2$AU, $T_{eff}=970K$) while the position angle can not be
defined at all.
The right panel of Fig.\ref{fig:V1295_HA} shows the comparison between
the observed and the predicted SED. More inclined disks can in general
reproduce better the photometric measurements around 1.5$\mu$m. The
near-infrared emission of disk with an inclination of less than
50$^\circ$ is peaked at about $4\mu$m and can not reproduce the
infrared excess between 1$\mu$m and 2$\mu$m; in both models, the
integrated near-infrared flux is $L_{NIR}\sim 20\%$ \Lstar.
\subsection{MWC 480}
Also in this case, due to the narrow range of available baselines, the
PTI visibilities of MWC 480 are consistent with different sets of
parameters at the same level of confidence $(\chi^2_r\sim2)$.
Fig.\ref{fig:MWC480_vis2} shows the two extreme disk configurations
characterized by quite similar parameters for the dust grain size
($a=0.2\mu$m$-0.3\mu$m; $\Rrim=0.63$AU-0.53AU and $T_{eff}\simeq
1250K$), but very different values of the inclination
($\iota=35^{\circ}$ and $\iota=60^{\circ}$) and the position angle
($\psi=60^{\circ}$ and $\psi=168^{\circ}$). Several intermediate
configurations reproduce the observed data as well. The degeneracy can
not be removed even using the SED (Fig.\ref{fig:MWC480_vis2}) which is
similar in the two models ($L_{NIR}/L_{\star} = 0.18\%-0.14\%$) and it
is only roughly consistent with the photometric values.
\subsection{AB Aur}
\label{sec:fitABAur}
AB Aur is the only HAe star that cannot be fitted with the IN05
models. The PTI visibilities require a face-on rim, consistent with
the inclination derived from large-scale images in scattered light
(Grady et al.~1999; Fukagawa et al.~2004) and at millimeter
wavelengths (Corder et al.~2005; Pi\'etu et al.~2005). For these
inclinations, the $V^2$ data imply a very small inner radius, about
two times smaller that the smallest $R_{rim}$ obtained using the IN05
model (see Fig.\ref{fig:ABAur_vis2}).
If, to put the discrepancy in a more physical context, we take
$T_{evp}$ as a free parameter, we find good agreement with the PTI
data for $T_{evp}\sim 2800K$ (dashed line), a value by far too high
not only for silicates but also for any other type of grains (e.g.,
Pollack et al.~1994).
The situation becames even less clear if we consider also the IOTA
observations (squared points), since they seem to indicate the
presence of a more inclined disk with an inner radius between 0.26AU
and 0.51AU. No additional information can be obtained from the
analysis of the spectral energy distribution, since all the inner rim
models with an effective temperature between 1500K and 2500K are
compatible with the photometric data. We will come back to AB Aur in
\S 6.
\section {Comparison with previous analysis}
Fits to the same interferometric data analyzed in \S\ref{sec:fit} have
been obtained by Eisner et al. (2004, hereafter E04) assuming a
toroidal shape for the inner ``puffed-up'' rim, based on the
simplified DDN01 model. In these fits, the free parameters are the
location of the rim $\Rrim$ and the two observational parameters,
$\iota$ and $PA$. The E04 results are shown in Table 2.
We note that for three objects (MWC 758, VV Ser and CQ Tau) the E04
inclinations are in agreement within the errors with the values
obtained with the IN05 model, while for the other two (V1295 Aql and
MWC 480) the E04 $\iota$ estimates are consistent with the lowest
value of the range derived in this paper.
The largest differences are in the derived values of $R_{rim}$: the
IN05 inner radii are always larger than E04 results, with a maximum
difference of a factor $\sim 3$ if we consider our maximum $\Rrim$ in
the MWC 480 system. While for CQ Tau the two values are almost the
same, for all the other stars the difference is a factor 1.5 and 2.
This discrepancy is mainly due to the difference between IN05 and E04
models. In particular, in IN05, the curved shape of the emitting
surface is self-consistently calculated, allowing a more correct
determination of the dependence of the rim emission on the inclination
of the disk.
Moreover, the IN05 model takes into account the effect of the
radiation transport within the disk, even if in an approximate way (see
Appendix A in IN05). This supplementary heating is neglected
by E04, who calculate the dust temperature taking into account only
the direct stellar radiation. The ratio between the two values of
$\Rrim$ is given by the relation
\begin{equation}
\label{eq:R/R}
\frac{ \Rrim }{ \hat{R}_{rim} } = \sqrt{\hat{\epsilon} \left(2 +
1/\epsilon \right) },
\end{equation}
where the $\hat\epsilon$, $\hat{R}_{rim}$ and $\hat{T}_{evp}^2$ are the
values used by E04 and the inner radius $\hat{R}_{rim}$ is given by the
relation
\begin{equation}
\label{eq:R04}
\hat{R}_{rim} = \frac{1}{\hat{T}_{evp}^2} \sqrt{ \frac{L_{\star}}{4\pi\sigma}
{\frac{1}{\hat{\epsilon} } } }.
\end{equation}
Assuming the same value of the dust emissivity
($\epsilon=\hat{\epsilon}$), the ratio $\Rrim/\hat{R}_{in}$ is $\sim1$
for $\epsilon<<1$. The difference increases for larger $\epsilon$ and
is maximum when $\epsilon$ and $\hat\epsilon$ are very different.
Finally, there also differences due to the fact that E04 assumes
$T_{evp}$ (or, equivalently, $\Rrim$) as a free parameter, while in
the IN05 model $T_{evp}$ is self-consistently determined starting from
the choice of the type of grains and the gas density in the disk (see
Eq.\ref{eq:Tevp}). For all our target stars, the resulting values of
$T_{evp}$ vary between 1370K and 1460K, and are in some cases
significantly different from those given by E04.
\section{Discussion}
The results presented in \S\ref{sec:fit} show that, with the exception
of AB Aur, the IN05 self-consistent models of the ``puffed-up'' inner
rim can explain the available observations, both visibilities and
SEDs, of HAe stars. They can be used to derive information
about the properties of the dust present in the innermost region of
the circumstellar disk, the location of the inner rim and the
orientation of the disk on the sky, using a minimum number of
assumptions.
\subsection{Presence of large grains}
\label{sec:large}
As shown in \S\ref{sec:fit}, the IN05 models reproduce the
interferometric data under the assumption that the most refractory
dust in the inner disk is made of silicates, with properties typical
of astronomical silicates (Weingartner \& Draine \cite{WD01}).
In four cases, grain sizes larger than $\sim 1.2$\um\ are either
required by or consistent with the observations. Only in one
case are the data better fitted with $a \sim
0.2-0.3\mu$m. Grains in the rim are thus larger, and often much
larger, than grains in the interstellar medium ($a=0.01-0.1\mu$m,
Weingartner \& Draine \cite{WD01}), confirming that grain growth has
taken place in the innermost disk regions (van Boekel et
al. 2004).
Even if the predicted near-infrared excess
agrees well with the photometric observations, some interesting
differences exist between the theoretical and the observed spectral
energy distributions. With the exception of CQTau, the predicted SEDs
always peak at a wavelength slightly longer than found in the
observations: the flux at short wavelengths (between 1.5\um\
and 2.2\um) is thus generally underestimated while the flux
between 2.2\um\ and 7\um\ is overestimated. This may be due to the
fact that in our models the SED is computed assuming that each point
on the surface of the rim emits as a black body at the local effective
temperature. This approximation is energetically correct but may not
reproduce the exact wavelength dependence of the emitted radiation
(see Appendix in IN05).
The rim models fail only in the case of AB Aur, where silicates,
of whatever size, produce rims that are too distant from the star to
be consistent with the observations. As shown in \S 4.7, PTI and IOTA
data give somewhat contradictory results, and more interferometric
data are clearly required. However, unless further observations
drastically change the present picture, the discrepancy between
the rim model predictions and the data is highly significant, and
some of the basic underlying assumptions need to be changed. It is
possible that in AB Aur grains more refractory than silicates dominate
the dust population in the inner disk; however, the $T_{evp}$ required
($\sim 2800$K) is too high for any dust species likely present in
disks (e.g., Pollack et al. 1994).
It is more likely that gas in the dust-depleted inner region absorbs a
significant fraction of the stellar radiation, shielding the dust
grains which are therefore cooler than in our models. This
requires a high gas density in the inner disk, as expected if the
accretion rate is high; in general, the accretion rates of HAe
stars (including AB Aur) are low enough to ensure that the gaseous
disk remains optically thin (Muzerolle et al.~2004). However, our
knowledge of the accretion properties and gas disks of HAe stars is
very poor, and should be investigated further.
AB Aur may be more than just an oddity. The presence of optically
thick gas inside the inner rim has been proposed to explain the
near-infrared interferometric observations of some very bright
Herbig Be stars, for which the visibility data suggest inner disk
radii many times too small to be consistent with the ``puffed-up'' rim
models (Malbet et al. 2005, Monnier et al. 2005). If this is the case,
AB Aur could be the low-luminosity tail of the same phenomenon, which,
given its small distance and large brightness, could be used to
understand a whole class of objects.
\subsection{Inclination and position angle}
Inclination and $PA$ are well constrained by the existing data only in
one case (MWC~758). We want to stress, however, that values of the
parameters outside the ranges given in Table 2 do not fit the data at
all. In particular, there are no objects consistent with face-on
disks, or in general with a centro-symmetric brightness
distribution. This rules out models, such as those of Vinkovic et
al. (2005), where most of the near-infrared flux is contributed by a
spherically symmetric shell around the star, rather than by a
circumstellar disk.
Two stars (CQ Tau and VV Ser) belong to the group of UXOR
variables, which are interpreted as objects with disks seen close to
edge-on (Grinin et al. 2001, Natta \& Whitney 2001, Dullemond et
al. 2003). We derive for them large inclinations, in agreement with
this interpretation.
For some of our targets, there are in the literature estimates of the
orientation of the outer disk on the plane of the sky obtained
with millimeter interferometers (Testi el al. 2001, 2003; Manning \& Sargent
1997; Pi\'etu et al.~2005; Corder et al.~2005). These determinations
refer to the outer disk, i.e., to spatial scales of 50--100 AU at
least. The comparison with the values derived in the
near-infrared for the inner disk (on scales of less than 1 AU) can
provide information on possible distortions in the disk,
e.g. variations of the inclination with radius.
For the three disks with millimeter data (MWC 758, CQ Tau and MWC 480),
there is agreement (within the uncertainties) between the inclination
obtained by infrared and millimeter observations
However, it is certainly premature to exclude the existence of
disk distortions, given the large uncertainties that affect both the
millimeter and the near-infrared estimates. More accurate
interferometric observations in the two wavelength ranges and
self-consistent models of the disk at all physical scales are
required.
An interesting case is that of VV Ser, whose disk has recently
been imaged as a shadow seen against the background emission in the
11.3 PAH feature (Pontoppidan et al. 2006); These authors derive an
inclination (of the outer disk) of about 70$^{\circ}$ and a position
angle of $13^\circ \pm 5^\circ$. While the inclination is consistent
with the upper limit of the range we obtain for the inner disk, the
position angle is off by almost $90^\circ$. This discrepancy is
intriguing, and deserves further investigation.
\subsection{Improving the model constrains}
\label{sec:Const}
Near-infrared interferometric observations of disks around pre-main
sequence stars are still few and sparse. It is clear from our analysis
that even in the most favorable cases more visibility data at
different baselines are necessary to narrow the range of possible disk
inclinations and grain properties.
Given the huge demands of telescope time that interferometric
observations require, it is useful to make use of model
predictions in preparing the observations and in choosing the baseline
configurations that can constrain the disk structure.
CQ Tau represents an example of how the IN05 model can be used in this
context. To better constrain the inner rim structure, one will
need observations with baselines longer than 130m (available
with the VLT interferometer), for which the predicted values of the
squared visibility parameters are very different for different disk
models (see Fig.\ref{fig:CQTau_HA}). On
the other hand, observations with baseline shorter than 60m could
better constrain the inner radius of MWC 480, since a degeneracy in
the models is present at longer baselines.
V1295 Aql represents a still different case, in which the degeneracy
in the values of the predicted squared visibility can be removed
observing at distant hour angles for the same baseline configurations,
in order to determine the visibility variations at the same baseline,
due to the inclination of the disk.
\section{Summary and conclusions}
In this paper we have analyzed the near-infrared interferometric
observations of the six best observed HAe stars using the rim models
developed by Isella \& Natta (2005). Our aim was to explore the
potential of near-infrared interferometry to constrain the properties
of the grains in the inner disks of these stars.
The basic assumptions of the IN05 rim models are that the inner
disk structure is controlled by the evaporation of dust in the
unattenuated stellar radiation field, as expected if the gaseous disks
have low optical depth, and that the most refractory grains are
silicates. The IN05 self-consistent models for the ``puffed-up''
inner rim reproduce both the interferometric observations and the
near-infrared spectral energy distribution of all the objects we have
studied, with the exception of AB Aur, which we have briefly
discussed.
For the five stars where we are able to obtain a good fit to the data,
we can estimate the grain sizes in the rim, i.e., in the midplane of
the inner disk. We find that in four cases grains larger than $\sim
1.2$\um\ are either required by or consistent with the
data. Only in one case do we find that the existing data require
$a\sim 0.2-0.3$\um. Note that this value of $a=1.2$\um\ is a lower
limit to the grain size: grains can be much larger, since the
rim location and shape do not change significantly if the grains grow
further.
As a result of the model-fitting, one derives also the inclination and
position angle of the disk on the plane of the sky. We find that, in
general, these parameters are not well constrained by the
existing data. However, in all cases we can fit, inclinations lower than
$30^{\circ}$ are not consistent with the observations and the surface
brightness distribution can not be circularly symmetric. This rules
out a spherical envelope as the dominant source of the
near-infrared emission.
For some objects, estimates of the inclination of the outer disk have
been obtained from millimeter interferometric observations; within the
uncertainties, they agree with the values obtained for the inner disk.
Our analysis shows that near-infrared interferometry is a very
powerful tool for understanding the properties of the inner disks, in
particular when combined with physical models of these
regions. However, at present the existing data are for many objects
still too sparse in their coverage of the u-v plane to allow an
accurate determination of the disk parameters. We expect that this
will be improved in the future. In this context, since near-infrared
interferometry is and will remain a very time demanding technique, we
stress the importance of using physical models of the inner
region of the disk in planning future observations.
\begin{acknowledgements}
We are indebted to Josh Eisner and Rafael Millan-Gabet for
providing us with the PTI and IOTA data. The authors acknowledge partial
support for this project by MIUR PRIN grant 2003/027003-001.
\end{acknowledgements}
|
Title:
Evolution of the Color-Magnitude Relation in High-Redshift Clusters: Blue Early-Type Galaxies and Red Pairs in RDCS J0910+5422 |
Abstract: The color-magnitude relation has been determined for the RDCS J0910+5422
cluster of galaxies at redshift z = 1.106. Cluster members were selected from
HST ACS images, combined with ground--based near--IR imaging and optical
spectroscopy. The observed early--type color--magnitude relation (CMR) in
(i_775 -z_850) versus z_850 shows intrinsic scatters in color of 0.042 +/-
0.010 mag and 0.044 +/- 0.020 mag for ellipticals and S0s, respectively. From
the scatter about the CMR, a mean luminosity--weighted age t > 3.3 Gyr (z > 3)
is derived for the elliptical galaxies.
Strikingly, the S0 galaxies in RDCS J0910+5422 are systematically bluer in
(i_775 - z_850) by 0.07 +/- 0.02 mag, with respect to the ellipticals. The
ellipticity distribution as a function of color indicates that the face-on S0s
in this particular cluster have likely been classified as elliptical. Thus, if
anything, the offset in color between the elliptical and S0 populations may be
even more significant.
The color offset between S0 and E corresponds to an age difference of ~1 Gyr,
for a single-burst solar metallicity model. A solar metallicity model with an
exponential decay in star formation will reproduce the offset for an age of 3.5
Gyr, i.e. the S0s have evolved gradually from star forming progenitors.
The early--type population in this cluster appears to be still forming. The
blue early-type disk galaxies in RDCS J0910+5422 likely represent the direct
progenitors of the more evolved S0s that follow the same red sequence as
ellipticals in other clusters.
Thirteen red galaxy pairs are observed and the galaxies associated in pairs
constitute ~40% of the CMR galaxies in this cluster.
| https://export.arxiv.org/pdf/astro-ph/0601327 | command.
\newcommand{\vdag}{(v)^\dagger}
\newcommand{\myemail}{smei@pha.jhu.edu}
\def\etal{et~al.}
\def\hst{{\it HST}}
\def\vi{\ifmmode(V{-}I)\else$(V{-}I)$\fi}
\def\viz{\ifmmode(V{-}I)_0\else$(V{-}I)_0$\fi}
\def\gz{\ifmmode(g_{475}{-}z_{850})\else$(g_{475}{-}z_{850})$\fi}
\def\gzz{\ifmmode(g_{475}{-}z_{850})_0\else$(g_{475}{-}z_{850})_0$\fi}
\newcommand\lta{\mathrel{\rlap{\lower 3pt\hbox{$\mathchar"218$}}
\raise 2.0pt\hbox{$\mathchar"13C$}}}
\newcommand\gta{\mathrel{\rlap{\lower 3pt\hbox{$\mathchar"218$}}
\raise 2.0pt\hbox{$\mathchar"13E$}}}
\def\mbari{\ifmmode\overline{m}_I\else$\overline{m}_I$\fi}
\def\mbarz{\ifmmode\overline{m}_z\else$\overline{m}_z$\fi}
\def\mbar{\ifmmode\overline{m}\else$\overline{m}$\fi}
\def\Mbar{\ifmmode\overline{M}\else$\overline{M}$\fi}
\def\Mbarz{\ifmmode\overline{M_z}\else$\overline{M}_z$\fi}
\shorttitle{CL0910}
\shortauthors{Mei et al.}
\tolerance=100000000
\begin{document}
\title{Evolution of the Color-Magnitude Relation in High-Redshift Clusters:
Blue Early-Type Galaxies and Red Pairs in RDCS~J0910+5422}
\author{S. Mei\altaffilmark{1},
J. P. Blakeslee\altaffilmark{1},
S. A. Stanford\altaffilmark{2,3},
B. P.~Holden \altaffilmark{4},
P. Rosati\altaffilmark{6},
V. Strazzullo\altaffilmark{20,6},
N.~Homeier\altaffilmark{1},
M. Postman\altaffilmark{1,5},
M. Franx\altaffilmark{12},
A. Rettura\altaffilmark{6},
H. Ford\altaffilmark{1},
G. D. Illingworth \altaffilmark{4},
S. Ettori\altaffilmark{19},
R.J.~Bouwens\altaffilmark{4},
R. Demarco\altaffilmark{1},
A.R. Martel\altaffilmark{1},
M. Clampin\altaffilmark{5},
G.F. Hartig\altaffilmark{5},
P. Eisenhardt\altaffilmark{7},
D.R.~Ardila\altaffilmark{1},
F. Bartko\altaffilmark{8},
N. Ben\'{\i}tez\altaffilmark{18},
L.D. Bradley\altaffilmark{1},
T.J. Broadhurst\altaffilmark{9},
R.A. Brown\altaffilmark{5},
C.J. Burrows\altaffilmark{5},
E.S. Cheng\altaffilmark{10},
N.J.G. Cross\altaffilmark{17},
P.D. Feldman\altaffilmark{1},
D.A. Golimowski\altaffilmark{1},
T. Goto\altaffilmark{1},
C. Gronwall\altaffilmark{13},
L. Infante\altaffilmark{14}
R.A. Kimble\altaffilmark{11},
J.E. Krist\altaffilmark{5},
M.P. Lesser\altaffilmark{15},
F. Menanteau\altaffilmark{1},
G.R. Meurer\altaffilmark{1},
G.K. Miley\altaffilmark{12},
V. Motta\altaffilmark{14},
M. Sirianni\altaffilmark{5},
W.B. Sparks\altaffilmark{5},
H.D. Tran\altaffilmark{16},
Z.I.~Tsvetanov\altaffilmark{1},
R.L. White\altaffilmark{5},
\& W. Zheng\altaffilmark{1}}
\altaffiltext{1}{Dept.\ of Physics \&Astronomy, Johns Hopkins University, Baltimore, MD 21218; smei@pha.jhu.edu}
\altaffiltext{2}{Department of Physics, University of California, Davis, CA 94516}
\altaffiltext{3}{Institute of Geophysics and Planetary Physics, Lawrence Livermore National Lab, Livermore, CA 94551}
\altaffiltext{4}{Lick Observatory, University of California, Santa Cruz, CA 95064}
\altaffiltext{5}{Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218}
\altaffiltext{6}{European Southern Observatory, Karl-Schwarzschild-Str. 2, D-85748 Garching, Germany}
\altaffiltext{7}{Jet Propulsion Laboratory, CalTech, 4800 Oak Grove Drive, Pasadena, CA 91125}
\altaffiltext{8}{Bartko Science \& Technology, 14520 Akron Street,
Brighton, CO 80602.}
\altaffiltext{9}{School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel}
\altaffiltext{10}{Conceptual Analytics, LLC, 8209 Woburn Abbey Road, Glenn Dale, MD 20769.}
\altaffiltext{11}{NASA Goddard Space Flight Center, Code 681, Greenbelt, MD 20771.}
\altaffiltext{12}{Leiden Observatory, Postbus 9513, 2300 RA Leiden,
Netherlands.}
\altaffiltext{13}{Dept.\ of Astronomy \& Astrophysics,
Penn State University, University Park, PA 16802.}
\altaffiltext{14}{Dept.\ de Astronom\'{\i}a y Astrof\'{\i}sica,
Pontificia Universidad Cat\'{\o}lica, Casilla 306, Santiago
22, Chile.}
\altaffiltext{15}{Steward Observatory, University of Arizona, Tucson,
AZ 85721.}
\altaffiltext{16}{W. M. Keck Observatory, 65-1120 Mamalahoa Hwy.,
Kamuela, HI 96743}
\altaffiltext{17}{Royal Observatory Edinburgh, Blackford Hill, Edinburgh, EH9 3HJ, UK}
\altaffiltext{18}{Instituto de Astrof\'\i sica de Andaluc\'\i a (CSIC), Camino Bajo de Hu\'etor 50, Granada 18008, Spain }
\altaffiltext{19}{INAF - Osservatorio Astronomico, via Ranzani 1, 40127 Bologna, Italy }
\altaffiltext{20}{Dipartimento di Scienze Fisiche, Universit\`a Federico II, I-80126 Napoli,Italy}
\keywords{galaxies: clusters: individual (RDCS~J0910+5422) --
galaxies: elliptical and lenticular ---
galaxies: evolution}
\section{Introduction}
The Advanced Camera for Surveys (ACS; Ford et al. 2002), by virtue
of its high spatial resolution and sensitivity,
allows us to study galaxy clusters in great detail up to
redshifts of unity and beyond. At these redshifts, galaxy
clusters are still assembling and galaxies are evolving towards the
populations that we observe today. Recent results
from our ACS Intermediate Redshift Cluster Survey (Blakeslee et al. 2003a; Lidman et
al. 2004; Demarco et al. 2005; Goto et al. 2005; Holden et al. 2005a;
Holden et al. 2005b; Homeier et al. 2005; Postman et al. 2005) have
shown that galaxy clusters at redshift around unity show many
similarities with local clusters, in terms of galaxy populations and
their distribution, but also significant differences in galaxy
morphology, ellipticity, and mass--luminosity ratios.
The strongest evolution observed in the early--type population is
a deficit of a S0 population in this sample when compared to lower
redshift samples (Postman et al. 2005). This would give evidence
that the formation of the S0 population is still under way in clusters
at redshift unity.
One of the most
striking similarities is that the tight relation between early-type
galaxy colors and luminosities that applies locally
(the color--magnitude relation; CMR) is already in place at redshifts as
high as $z \sim 1.3$ (e.g. Stanford et al.\ 1997; Mullis et al.\ 2005). The
CMR in local samples of galaxy clusters presents universal properties,
in terms of scatter and zero point (Bower et al. 1992; van Dokkum et
al. 1998, Hogg et al. 2004; L\'opez--Cruz et al. 2004; Bell et
al. 2004; Bernardi et al. 2005; McIntosh et al. 2005) that evolve back
in time in agreement with passively evolving models
(Ellis et al.\ 1997; Stanford, Eisenhardt, \& Dickinson 1998; van
Dokkum et al. 2000, 2001; Blakeslee et al. 2003a; Holden et al. 2004;
De Lucia et al. 2004; Blakeslee et al. 2005). ACS enables accurate measurement of the scatter
around the CMR, with enough precision to seriously constrain galaxy formation age,
which is impossible to obtain from ground--based data (see for example Holden et al. 2004). The measurement of the CMR scatter of the first cluster in our ACS
cluster survey, RXJ1252.9-292, permitted us to constrain the mean luminosity--weighted age for the ellipticals to be $> 2.6$~Gyr ($z > 2.7$) (Blakeslee et
al. 2003a), based on simple modeling. In this paper,
we extend the results obtained in Blakeslee et
al. (2003a) to RXJ~0910+5422.
RXJ~0910+5422 is part of the ACS cluster survey (guaranteed time
observation, GTO, program \#9919), that includes eight
clusters in the redshift range at $0.8 < z < 1.3$, selected in
the X--ray, optical and near--IR (Ford et al.\ 2004).
RXJ~0910+5422 was selected from the ROSAT
Deep Cluster Survey (Rosati et al. 1998) and confirmed with near-IR and
spectroscopic observations by Stanford et al. (2002). Extensive
followup spectroscopy at the Keck Observatory has been carried out in
a magnitude limited sample reaching $K_s = 20.0$~mag in the central
$3$~arcmin (Stanford et al.\ 2005; in preparation). The mean redshift of the cluster
was measured to be $z=1.106$ (Stanford et al. 2002). In this paper,
we combine ACS imaging
with ground--based spectroscopy and near--IR imaging to constrain galaxy
ages and formation histories from the study of their color--magnitude relation.
We discuss the properties of the elliptical (E) and lenticular (S0)
populations separately in the light of simple galaxy formation scenarios.
\section{Observations}
RXJ~0910+5422 was observed in March 2004 with the ACS WFC (Wide
Field Camera) in the F775W ($i_{775}$) and F850LP
($z_{850}$) bandpasses, with total exposure times of 6840~s and 11440~s,
respectively. The ACS WFC scale is 0.05\arcsec/pixel, and its field of
view is $210\arcsec \times 204\arcsec$. The APSIS pipeline (Blakeslee
et al. 2003b), with a {\it Lanczos3} interpolation kernel, was used
for processing the images. The ACS photometric zero--points (AB system)
are 25.654~mag and 24.862~mag in $i_{775}$ and $z_{850}$, respectively
(Sirianni et al. 2005). A Galactic reddening of $E(B-V)=0.019$ towards
RXJ~0910+5422 was adopted (Schlegel et al. 1998), with $A_{i775}=0.039$
and $A_{z850}=0.029$ (Sirianni et al. 2005). The ACS WFC field covers
an area that at the redshift of this cluster, $z=1.106$, corresponds
to $\approx 1 Mpc^2$ in the WMAP cosmology (Spergel et al. (2003):
$\Omega_m =0.27$, $\Omega_{\Lambda} =0.73$, $h=0.71$, adopted as our
standard cosmology hereafter). Fig.~\ref{cluster} shows the ACS color
image with X--ray contours from Chandra ACIS (Advanced CCD Imaging
Spectrometer) data that have been adaptively smoothed (Stanford et
al. 2002).
Near-IR $JK_s$ and optical $i$-band images were obtained at Palomar
Observatory as described in detail by Stanford et al. (2002).
Optical spectroscopy of galaxies in RXJ~0910+5422 was obtained using the Low
Resolution Imaging Spectrometer (LRIS; Oke et al. 1995) on the Keck 1
and 2 telescopes (Stanford et al. 2005; in preparation).
Our typical errors in redshift correspond to errors in velocity between
100 and 300~km/s.
Objects for spectroscopy were chosen
initially from the catalog of objects with $K_s < 20.0$~mag (Vega
magnitudes) within the IR imaging area; outside of this area objects
were chosen with $i > 21$~mag from the $i$-band
image to fill out masks. Our final sample included 66\% of the
objects with $K_s < 20.0$~mag.
Spectra were obtained using the 400 lines mm$^{-1}$ grating for all
runs except for the initial two discovery masks as reported in
Stanford et al.\ (2002). Nine more masks were observed using LRIS
during four runs between January 2001 and February 2003. Usually each
mask was observed in a series of four 1800~s exposures, with small
spatial offsets along the long axis of the slits. On average, the
seeing was 0.9~$\arcsec$. The blue side data were generally not used since the rest
frame wavelengths probed at $z = 1.1$ fall far to the blue of the
spectral features of interest for galaxies in the cluster.
In total, 149 redshifts were obtained.
The slit mask data were separated into slitlet spectra and then
reduced using standard long-slit techniques. A fringe frame was
constructed for each exposure from neighboring exposures, each offset from
the previous by 3$\arcsec$, in an
observing sequence for each mask,
and then subtracted from each
exposure to greatly reduce fringing in the red. The exposures for
each slitlet were reduced separately and then co-added.
One-dimensional spectra were extracted for each targeted object, as
well as the occasional serendipitous source. Wavelength calibration
of the 1-D spectra was obtained from arc lamp exposures taken
immediately after the object exposures. A relative flux calibration
was obtained from long-slit observations of the standard stars HZ44,
G191B2B, and Feige 67 (Massey \& Gronwall 1990).
\section{Object selection and photometry}
SExtractor (Bertin \& Arnouts 1996)
was used to find objects in the $i_{775}$ and $z_{850}$ images and measure their magnitudes.
Threshold and deblending settings were used as in Ben\'{\i}tez et al. (2004).
Although we have extensive spectroscopy, the ACS imaging
reaches considerably deeper along the cluster luminosity function. Thus, we have
chosen to use colors ($i_{775}$ - $z_{850}$) and $(J-K_s)$ to isolate
a set of probable cluster members.
In Fig.~\ref{colage}, the ($i_{775}$ - $z_{850}$) and $(J-K_s)$
colors are shown as a function of galaxy age, using BC03 stellar population models,
redshifted to $z{=}1.106$. Early--type cluster members would have ages
of at least 0.5~Gyr, corresponding to $(i_{775} - z_{850}) >$0.8~mag and
$(J-K_s) > 1.45$~mag.
At first, we give a larger color margin and select as potential cluster members all morphologically-classified early-type galaxies with $0.5 <(i_{775} -
z_{850}) < 1.2$~mag and $(J-K_s) > 1.45$~mag, down to $z_{850} = 24$~mag (the limiting magnitude
of Postman et al. (2005) morphological classification, that included all clusters
in our sample at redshift unity). Our results in this paper are based on this morphological classification and a detailed discussion of the uncertainties in this classification can be found in that work.
This selected sample
includes 38 galaxies within the ACS field.
Our final colors
were measured within galaxy effective radii ($R_e$), to avoid biases
due to galaxy internal gradients, following the approach in Blakeslee
et al. (2003a) and van Dokkum et al. (1998, 2000). $R_e$ values were
derived with the program GALFIT (Peng et al. 2002), constraining the
{\it Sersic} index $n \le 4$ (as in Blakeslee et al. 2003a).
To remove differential blurring effects
(the PSF is $\sim10\%$ broader in the $z_{850}$ band) each galaxy image in
both $i_{775}$ and $z_{850}$ was deconvolved using the CLEAN algorithm
(H{\"o}gbom et al. 1974). The $(i_{775} - z_{850})$ colors were
measured on the deconvolved images within a circular aperture of radius equal to
$R_e$, or 3 pixels, whichever is larger. Our median $R_e$ is $\approx$~5.5~pixels
($\approx$~13kpc at z=1.106).
Our final results do not change (within the uncertainties) if the
effective radii are calculated via a two component
(Sersic bulge + exponential disk) surface brightness decomposition
technique using GIM2D (Marleau \&
Simard 1998; Rettura et al., in preparation), that permits us to better fit
the galaxy light profile.
The photometric uncertainties
due to flat fielding, PSF variations, and the pixel-to-pixel
correlation for ACS (Sirianni et al. 2005) were estimated by measuring
the standard deviation of photometry in the background for circular
apertures in the range of the measured effective radii. These
photometric errors were added in quadrature to the Poisson
uncertainties in the measured fluxes for each object. The derived
errors in the colors are between 0.01 and 0.03~mag down to $z_{850} =
24$~mag. SExtractor MAG\_AUTO were used for the $z_{850}$ magnitude in
the color--magnitude relation; these are fairly robust, though may
systematically miss a small fraction of the light (Ben\'{\i}tez et al.\ 2004).
We finally color--selected 34 early--type (E, S0 and S0/a) galaxies with $0.8
<(i_{775} - z_{850}) < 1.1$~mag within 2$\arcmin$ from the cluster center,
taken as the center of the X--ray emission (Stanford et al.\ 2002).
Images of the color--selected galaxies are shown in
Fig~\ref{epostage}, Fig~\ref{sopostage} and Fig~\ref{sapostage}.
Moreover, there are late--type galaxies with
luminosities that are similar to the red--sequence bright
early--type galaxies (Fig.~\ref{spipostage}).
Of the 34 color--selected galaxies, 15 are spectroscopically confirmed cluster members,
one (S0/a, with magnitude $z_{850} = 24.2$~mag) is a confirmed non-member,
and the others were not targeted for spectroscopy. The
selection in $(i_{775} - z_{850})$ at $z=1.1$ therefore appears to be
robust: only one of the 16 selected galaxies with measured redshifts
is a non-member.
We expect few of the other 18 to be interloper field galaxies.
\section{Color-Magnitude Relation}
The color--magnitude relation for the final color--selected objects
is shown in Fig~\ref{cmd}. Red dots
are ellipticals, orange squares and stars are S0 and S0/a galaxies,
respectively. Smaller black symbols represent early--type galaxies that do
not lie on the red sequence. Small triangles are late-type
galaxies. Boxes are plotted around confirmed cluster members.
Confirmed interlopers are circled in the figure.
Surprisingly, the two brightest
cluster members are not ellipticals, but S0. The brightest of these
two galaxies lies $\approx$~700~kpc ( $\approx$1.2$\arcmin$) from
the cluster center, and the other bright S0 at $\approx$~300~kpc
($\approx$~0.6$\arcmin$). Moreover, there are late--type galaxies with
luminosities that are similar to the red--sequence bright
early--type galaxies. Two of them lie on the red--sequence and are
confirmed cluster members, at $\approx$~80~kpc from the cluster center
(see also below in the discussion of the color and morphology
distribution as a function of distance from the cluster center).
We fitted the following linear color--magnitude relation to various
subsamples of the galaxies:
\begin{equation}
i_{775} - z_{850} = c_0 + Slope (z_{850} - 23)
\end{equation}
The solid line in Fig~\ref{cmd} is the fit to the
color--magnitude relation for the ellipticals, the black dotted line is
the fit to the CMR for the S0s, and the dashed--dotted line
the fit to the full sample, within 2$\arcmin$ from the cluster center (see discussion below). The
dashed line is the fit to the full sample of early--type galaxies in
RXJ1252.9-292 from Blakeslee et al. (2003a), scaled to this
redshift with BC03 evolved stellar population models, with solar
metallicity and a formation age of 2.6~Gyr (since Blakeslee et al. 2003a
obtains elliptical mean ages ~$>$~2.6~Gyr). The long--dashed vertical line is the magnitude limit
of the morphological classification $z_{850} = 24$~mag.
The results for different morphological samples are given in Table~\ref{results}.
A robust linear fit based on Bisquare weights (Tukey's biweight; Press et al. (1992)) has been used to fit the color--magnitude relation. Uncertainties on the parameters were estimated by bootstrapping on 10,000 simulations. The scatter around the fit was estimated from a biweight scale estimator (Beers, Flynn \& Gebhardt 1990), that is insensitive to outliers, in the same set of bootstrap simulations.
The internal color scatter ($\sigma_{int}$) was measured in two ways:
1) to the scatter around the fit, we have subtracted in quadrature the
average uncertainty due to the galaxy color error; and 2) we have
calculated the internal scatter for which the $\chi^2$ of the fit
would be unity. Both methods give us internal scatters consistent to
within a few 0.001~mag. All galaxies in this sample lie within
three sigma from the fit.
The X--ray distribution appears to be very symmetric, and largely confined
within 1\farcm5 from the cluster center. We
calculated the CMR zero point and scatter within 1$\arcmin$ (which
corresponds to a scale of $\approx$~0.5~Mpc at this redshift), and within
1\farcm5 ($\approx$0.7~Mpc), and within 2$\arcmin$
($\approx$~1~Mpc, the scale
used for the analysis of RXJ1252.9-292). According to the results in
Table~\ref{results}, the internal color scatter increases when adding
populations between 1$\arcmin$ and 2$\arcmin$, especially for the S0
and S0/a populations, as also observed in local samples (e.g. van Dokkum
et al. 1998), with only a small increase in sample size.
We will therefore focus on the results obtained for color--selected galaxies
within 1$\arcmin$ from the cluster
center (where 90\% of the color-selected galaxies lie). The slope of the
elliptical CMR ($-0.033 \pm 0.015$) is slightly
steeper than the
observed slope in RXJ1252.9-292 ($-0.020 \pm 0.009$), and in Coma when
the latter are shifted to the observed colors at $z \sim 1.1$, using
non-evolving BC03 stellar population models, but still consistent within the uncertainties. We do
not find a flatter (with respect to Coma) slope as in Stanford et
al. (2002). However, the S0 sample shows a much shallower slope
($0.005 \pm 0.023$) than the ellipticals, resulting in a much flatter slope
for ellipticals and S0s together ($-0.024 \pm 0.020$). This can explain why a
shallower slope was found in that work, in which elliptical and
S0s were not separated. The spectroscopically-selected elliptical plus
S0 slope ($-0.010 \pm 0.034$) is also flattened by the S0 population,
while the spectroscopically-selected ellipticals have a slope ($-0.021 \pm 0.046$) similar to that of RXJ1252.9-292 and Coma. All the difference in slope are
however within the uncertainties and are statistically insignificant.
Using Bruzual \& Charlot (2003; BC03) stellar population models, as in
Blakeslee et al. (2003a), we derive a constraint on the age of the stellar
populations in the galaxies from galaxy colors and the
scatter of the CMR (van Dokkum et al. 2001; Blakeslee et
al. 2003a). Two simple models have been considered and our conclusions will
depend on the chosen models. The first model is a {\it single burst} model,
in which galaxies form in single bursts at random times $t_f$,
between the age of the cluster and the recombination epoch. The second is a
model with {\it constant star formation} in a range of time between $t_1$
and $t_2$, randomly chosen to be between the age of the cluster and
the recombination epoch. Colors for 10,000 galaxies were simulated
with their scatter around the CMR to be
dependent on the burst age.
In Fig~\ref{scatterage}, we show the simulated scatters as a function
of burst age, with solar, half solar and twice solar
metallicity model. We will assume solar metallicity in what follows.
From the scatter ($\sigma_{int}=0.042\pm
0.010$) in the colors of the galaxies classified as ellipticals, we obtain ages $>2.1$~Gyr
($z >2$), with a mean luminosity--weighted age $\overline t=3.31$~Gyr
($z_f \approx$~3.1), assuming the random single burst model. From the constant star
formation model, we obtain ages $>1.6$~Gyr ($z >1.7$), with a mean
luminosity--weighted age $\overline t=3.26$~Gyr
($z_f \approx$~3). This agrees with the conclusion (e.g. Blakeslee et
al. 2003a; Holden et al. 2004; Lidman et al. 2004; De Lucia et
al. 2005) that the elliptical population in clusters of galaxies
formed at $z_f >$~3, and has evolved mainly passively until
$z = 1.1$.
In the (U-B)--rest frame (using BC03 stellar population models with solar metallicity and age equal to 4~Gyr), a scatter in $(i_{775}-z_{850})$ of $0.042\pm
0.010$ corresponds to $0.050 +/- 0.011$. As pointed out in van Dokkum (2000) and Blakeslee et al. (2003a), CMR scatters vary little with redshift. The Blakeslee et al. (2003a) scatter for the elliptical CMR in RDCS~1252--2927 ($0.024 \pm 0.008$), correspond to a scatter of $0.042 \pm 0.014$ in the (U-B)--rest frame, indistinguishable within the uncertainties from our result.
The scatter in the CMR for galaxies classified as S0
($\sigma_{int}=0.044\pm 0.020$) is
comparable to the one in the E CMR, but the galaxy colors are bluer
and are not compatible with a population that is as old as the
ellipticals. All the S0s lie below the elliptical color--magnitude
relation. In fact, between the elliptical and the S0 CMR fits there
is a zero point difference of $0.07 \pm 0.02$~mag, with the S0s being
bluer than the ellipticals. One of the three S0/a galaxies has a color
that is 0.07~mag redder than the elliptical CMR, and another has a
color that is $\approx$~0.15 bluer than the CMR relation for Es.
The inclusion of the S0/a galaxies does not significantly change the
fitted CMR for the S0s.
When we consider all galaxies within 2$\arcmin$ from the cluster center,
the S0 and total early--type slopes are similar, while the color offset
in the CMR is still present ($0.05 \pm 0.02$~mag).
The S0 population of this cluster has a
very peculiar CMR with respect to the average cluster of galaxies.
In fact, for several other studies the CMR of the S0 population has a similar
zero point and on average a larger scatter with respect to the
elliptical population (van Dokkum et al. 1998; Blakeslee et al. 2003a;
Holden et al. 2004; De Lucia et al. 2004), quite different from our results.
We will discuss this peculiar behavior in detail in the rest of the paper,
including an examination of orientation effects on the classification
In Fig.~\ref{ircolors}, the near-IR (Vega magnitudes) and $(i_{775} -
z_{850})$ (AB magnitudes) colors are shown compared with single burst stellar
population model predictions from BC03. The S0 colors are consistent
with young ($<$~2~Gyr) solar metallicity, or older ($<$~3.5~Gyr), half
solar metallicity populations. If the difference in E and S0 mean
colors is mainly due to metallicity, then even if the
two populations were formed at the same epoch, ellipticals must have been able
to retain more metals than the S0s, i.e., they were more massive at a
given luminosity (given the observed mass--metallicity relation for
early-type galaxies, e.g., Tremonti et al. 2004, Bernardi et al. 2005,
and references therein). This would imply higher mass--to--light ratios for
the ellipticals with respect to the S0s. However, the lack of
strong evolution in the slope and scatter of the CMR from the present out
to $z{\sim}1$ suggests that the CMR is mainly the result of a metallicity--mass
(i.e. metallicity--magnitude) relation
(e.g., Kodama \& Arimoto 1997; Kauffman \& Charlot 1998
Vazdekis et al. 2001; Bernardi et al. 2005).
So, at a given magnitude we do
not expect large metallicity variations.
If the offset is due to a
different star formation history, a model with solar metallicity
and with an exponential decay of the star formation will reproduce the offset
at a galaxy mean age of $\approx$~3.5~Gyr. This age is consistent with
the small scatter observed in the S0 CMR.
We would then be observing galaxies
that followed different star formation: single burst and passive
evolution for the ellipticals and exponentially decaying star formation
for the S0s. An exponential decay in the star formation is observed
in field spiral samples (Rowan--Robinson 2001). If this is the case,
our S0 population might
be the evolved product of an old spiral population that was already in place
in this cluster when the ellipticals formed and then gradually lost
available gas for star formation.
If the E vs S0 color difference is mainly due to a difference in age,
for a solar metallicity and a single burst BC03 template with age 4~Gyr,
the color difference
corresponds to an age difference of $\sim$1~Gyr. For clusters of
galaxies at z~$>$~1, the cluster members on the red sequence are only a
part of all the progenitors of present--day early--type galaxies.
Some of today's galaxy progenitors would have been bluer than the red
sequence at these redshifts (van Dokkum \& Franx 2001). In the S0
population of this cluster, we may be seeing the transitional
progenitor population that in $\sim$1~Gyr will evolve onto the same
red sequence as now occupied by the ellipticals.
Either of the latter two scenarios would be consistent with
the Postman et al. (2005) observed deficit
of the S0 population of our ACS cluster sample,
when compared to lower
redshift samples, implying that part of the S0 population is still forming in clusters at redshifts around unity.
\section{Galaxy shape properties}
Since the galaxies classified as S0 in RXJ~0910+5422 are found to be
systematically bluer (with respect to the red sequence) than the
S0 populations observed in previous studies, we wish to examine further the
properties of these galaxies in terms of their shapes and light distributions,
and how they compare to the elliptical and spiral samples in this and other clusters.
The shape parameters that we will consider are Concentration and Asymmetry
(Abraham et al. 1996; Conselice et al. 2004), Sersic index $n$,
and galaxy axial ratios.
\subsection{Asymmetry and Concentration}
In Fig~\ref{cas} (left), we compare the Asymmetry A and the Concentration C
for ellipticals, S0s, and spirals with $(i_{775} - z_{850})$ colors between 0.5
and 1.2~mag. The Asymmetry parameter is obtained by subtracting a 180--degree
rotated image from each original galaxy image, summing the residuals and
including a correction for the background. The Concentration parameter is
defined as in Abraham et al. (1996) as the sum of the galaxy flux within an
aperture $r_{0.3}$ divided the total flux. $r_{0.3}$ is calculated using the
SExtractor fit to the galaxies at 1.5~$\sigma$ above the background. The
obtained semi--major and semi--minor axes from this fit were multiplied by
0.3 to derive the $r_{0.3}$ aperture (see also Homeier et al. 2005b).
Early--type and late--type galaxies lie on different regions in this A vs C
plane. All our red sequence S0s have $A < 0.3$ and $C > 0.3$. All but one
have $A < 0.2$ and $C > 0.3$. This is the same locus in the A--C plane that
is occupied by most early--type galaxies in Abraham et al. (1996) and in our
Postman et al. (2005) low--redshift sample (Fig~\ref{cas}, right). This last
sample includes 5 strong lensing clusters observed as part of our ACS GTO
program [Zw1455+2232 ($z{=}0.258$), MS1008-1224 ($z{=}0.301$), MS1358+6245,
CL0016+1654 ($z{=}0.54$), and MS J0454-0300 ($z{=}0.55$)]. Visual and automated
classification for this cluster in the $i_{775}$--band for all galaxies with
$i_{775}< 22.5$ was performed by Postman et al. (2005). We conclude that the
S0 population presents statistical parameters typical of an early--type
population (low asymmetry and high compactness).
\subsection{Sersic Indices}
Fig~\ref{justsersic} plots Sersic index $n$ as a function of galaxy effective
radius $R_e$, both from our GALFIT modeling, for
ellipticals, S0s and spirals with $(i_{775} - z_{850})$ colors between 0.5 and
1.3~mag. Red sequence ($(i_{775} - z_{850})$ color between 0.8 and 1.1~mag)
galaxies are shown by large diamonds. Most spirals have $n<2$ and
most early--types have $n>2$, as expected. However, the $n$ values do not
permit us to discriminate between S0s and ellipticals in a unique way, unless they are combined with goodness--of--fit information for the Sersic model (e.g. the {\it Bumpiness} parameter introduced by Blakeslee et al. 2005)
\section{Axial Ratios}
\subsection{Axial Ratio Distribution}
In Fig~\ref{ab} we compare the apparent axial ratio from SExtractor
versus effective radii for
elliptical and S0s with $(i_{775} - z_{850})$ colors between 0.5 and 1.3~mag.
The axial ratios have been verified by using ELLIPROF (the isophotal
fitting software that is used for Surface Brightness Fluctuations analysis in
Tonry et al. 1997 and Mei et al. 2005) on each galaxy image, after the cleaning
procedure.
As above, red sequence galaxies are shown by large diamonds. The red sequence
ellipticals and the bluer S0s have different axial ratio distributions,
with all
red sequence S0s showing axial ratios $\frac{b}{a} \lesssim 0.7$, and nearly
all red sequence ellipticals with $\frac{b}{a} > 0.7$.
Assuming axisymmetric
disks (oblate ellipsoids) viewed with random orientation, and with a Gaussian distributed intrinsic axial ratio (with mean equal to $0.3 \pm 0.1$ (extreme thin disk), $0.5 \pm 0.1$ (early--type galaxy) and $0.75 \pm 0.1$ (elliptical) (Jorgensen \& Franx 1994)), one would expect $\gtrsim\,$40\%, $\gtrsim\,$60\%, and $\gtrsim\,$90\%, respectively, of the S0s to have axial ratios above 0.7.
Just 9\% (1 out of 11)
of the red sequence S0s are observed to have an axial ratio this large
(or 22\% for the full S0 sample in this cluster field) (Fig~\ref{ab2}; top).
The random probability that the S0 axial ratios
would show such a low fraction with $\frac{b}{a} > 0.7$ is less than 1\%.
This is a very simple model, but it points out a lack
of round S0s,
indicating either that there is some orientation bias in the classification,
or that this class of objects is intrinsically prolate in shape.
Jorgensen \& Franx (1994) found a similar deficit of round S0s in the center of the Coma cluster. They concluded that part of the face--on S0s were classified as elliptical galaxies. Fabricant et al. (2000) also found a deficit of round S0s in the cluster CL~1358+62, at z=0.33. Their analysis shows that ellipticities and bulge--to--total--light--ratio do not allow us to distinguish elliptical from S0 galaxies.
The other two ACS GTO clusters at $z{\,>\,}1$
(RXJ1252.9-292 and RX~J0848+4452) do not show a similar lack of round S0s,
as 90\% of the red sequence ellipticals (out of $\approx\,$70) and 47\% of
the S0s (out of $\approx\,$35) have $\frac{b}{a} > 0.7$ (Fig~\ref{ab2}; bottom). %
This bias is also not observed in other clusters of our ACS Intermediate Redshift Cluster Survey (see Fig~5 from Postman et al. 2005), for which more than 40\% of the S0 galaxies have axial ratios above 0.7.
The observed peculiarity of the RXJ~0910+5422
S0 axial ratio distribution might
call into question our result above that the S0s have a significant color
offset with respect to the ellipticals. For instance, if there is a
bias in our color measurement procedure which causes elongated objects
to have colors that are too blue, then the color offset found above may
be artificial. Such a color bias might occur if the high inclination angles
bias our $R_e$ measurements to higher values, and if the S0s become
progressively bluer at larger radii. We first examine this possibility,
then proceed to discuss resolutions to the peculiarity of the S0
axial ratio distribution, with the aim to establish if a misclassification of
face--on S0s as ellipticals would bias the measurement of the offset between the elliptical and S0 CMR zero points.
\subsubsection{Internal color gradients}
It is conceivable that our
$(i_{775} - z_{850})$ colors could be biased by aperture effects in
the nearly edge--on S0s, for which the (possibly) bluer outer disks might
contribute more to the galaxy colors than in the rounder ellipticals population.
If this effect were severe enough, it might mimic the offset in color of the
S0s and ellipticals found above. We test this possibility here.
S0 and elliptical internal color profiles are
shown in Fig~\ref{gradso} and Fig~\ref{grade}. The gradients have
been calculated with aperture photometry on the same images used to calculate
our $(i_{775} - z_{850})$ colors. Circles are $(i_{775} - z_{850})$ colors at
different radii, the cross the $(i_{775} - z_{850})$ color at the effective
radius, used for the CMR.
The S0 colors profiles do not show strong gradients. In particular,
the $(i_{775} - z_{850})$ colors calculated at the effective radii are
not systematically bluer than colors determined at smaller radii.
Most of S0 galaxies have flat profiles; one has a blue inward gradient
(ACS ID 1621; $z_{850}=23.72$~mag, $(i_{775} - z_{850})=1$~mag).
In two galaxies (ACS ID 1393 and 3177)
the colors in the central 0.15~$\arcsec^2$ are redder than the color
at the effective radius. When compared with elliptical gradients, on average S0
colors do not appear biased towards higher effective radii and bluer colors than
the ellipticals.
We note that three elliptical galaxies show blue inward gradients (ACS ID
1753, 1519, 3323).
\subsubsection{Orientation or intrinsic shape: Axial ratios vs $(i_{775} - z_{850})$ colors}
Orientation biases are known to occur in the classification
of ellipticals and S0s in local galaxy samples. For instance,
Rix \& White (1990, 1992) showed, based on both isophotal and
dynamical modeling, that a large fraction of ellipticals contain
a disk component with at least $\sim\,$20\% of the light,
but which is hidden due to projection effects.
Jorgensen \& Franx (1994) found a strong deficit of round S0s
in a sample of 171 galaxies in the central square degree of the
nearby Coma cluster, and concluded that inclination angle played
a large part in the classification of Es and S0s. Michard (1994)
proposed that, except for the bright boxy ellipticals without
rotational support, early-type galaxies comprise a single class
of oblate rotators with orientation being the main criterion for
classification as either E or S0. On the other hand, van den Bergh (1994)
explained the predominance of flattened S0s by invoking two distinct
subpopulations: bright disky objects intermediate between ellipticals
and spirals, and a fainter population of prolate objects.
We now address the question of whether the galaxies classified as
S0 in RXJ~0910+5422 are preferentially flattened in shape because
of an orientation bias in the classifications or intrinsically prolate
shapes. If it is an orientation bias,
then this could mean either that (1) face-on S0s have been misclassified as
ellipticals because their disks are not apparent, or (2) that edge-on spirals
tend to be called S0s because the spiral structure is obscured.
Either would result in a predominance of flattened S0s.
However, in the former case, the misclassification
of face-on S0s as ellipticals would tend to blur any color separation
between the two classes, while in the latter case, a color offset
might be introduced between the two classes because of contamination
by bluer spirals. Because we do observe a color offset between galaxies
classified as E and S0, with the S0s being bluer,
it is possible to look for the ``missing'' population of face-on blue
galaxies by examining ellipticity versus color. If a population
of round blue objects is found, we can then determine the nature of
the classification bias, and whether it biases our color offset measurement.
Histograms of the axial ratio distributions for ellipticals and S0s
are shown in the top panel of Fig.~\ref{ab2}.
We find that 60\% of the early-type red sequence galaxies have
$\frac{b}{a} > 0.7$, but 95\% of these low-ellipticity galaxies
are classified as Es.
However, if we split galaxies instead by color,
using $(i_{775} - z_{850}) < 0.99$~mag as the separation point,
then we find that
54\% of all galaxies bluer than this separation have $\frac{b}{a} > 0.7$,
while 43\% of the red sequence galaxies bluer than this
[$0.8 < (i_{775} - z_{850}) < 0.99$~mag] have $\frac{b}{a} > 0.7$
(Fig.~\ref{ab3}).
Thus, the deficit of round S0 galaxies (which are also significantly bluer
than the mean of the E class) is not found when the
early-type galaxies are split based purely on color. This suggests that
some of the bluer round galaxies are the face-on counterparts of those
classified as S0.
Fig~\ref{blue} shows axial ratios vs $(i_{775}- z_{850})$ color residuals
with respect to the total early-type galaxy CMR relation, for all galaxies
in the RXJ~0910+5422 red sequence. Galaxy types are coded with different
symbols; using the color residuals in this way takes out the effect of the
magnitude dependence of the colors. There are five round blue ellipticals,
i.e., with $\frac{b}{a} > 0.7$ and on the blue side of the early-type CMR.
If these are the face-on counterparts of the blue S0s, then the S0
axial ratio distribution becomes much more in line with expectations
for a randomly oriented disk population (35\% are rounder than
$\frac{b}{a}{\,=\,}0.7$). Further, we note that
four other ellipticals were classified E/S0 in Postman et al.\ (2005);
if these are also taken as face-on S0s, then the axial ratio distribution
comes in very close agreement with expectations.
We conclude that the face-on S0s in RXJ~0910+5422 are classified as
ellipticals, just as in local early-type galaxy samples.
From our axial ratio simulations and statistics from our z$\approx$1
ACS GT0 sample, the number of blue S0
galaxies might be between 25 and 40\% higher than estimated in
section~5.3.1. Moreover, the inclusion of a few blue S0s in the elliptical
red--sequence will have the effect of increasing very slightly the observed
CMR scatter for the ellipticals.
The uncertainty in the Postman et al. (2005) S0 fraction in RXJ~0910+5422 is significantly larger than 40\% and hence a systematic change by this amount for one cluster does not alter any of the results or conclusions in that paper.
Throughout the paper,
we continue to use elliptical and S0 classifications from Postman et al. (2005), but keeping in mind the
presence of this possible projection effect.
\section{Color trends, velocity dispersion and merging activity}
Stanford et al. (2002) have analyzed the X--ray and near-IR
properties of this cluster. This system appears to be fairly
relaxed based on its regular X--ray profile; however, they find
indications that the cluster is in an early phase of formation. In fact, the
Chandra ACIS data show evidence for temperature structure, possibly
due to an infalling group or mass streaming along a filament.
The soft component of the X--ray emission (0.5--2~keV) dominates the
X--ray center of the cluster, while to the south there is a harder
component (2--6~keV) (see Fig. 6 from Stanford et al. 2002).
The cluster does not have a central BCG or cD galaxy, and the X--ray
emission center does not correspond to an optical grouping of galaxies;
rather a number of luminous confirmed cluster members are linearly
distributed, at least as projected on the sky (as shown in
Fig~\ref{cluster}). We do not observe any strong trend of the galaxy
color $(i_{775} - z_{850})$ with distance from the cluster X--ray center
(Fig.~\ref{colors}). Surprisingly, though not statistically
significant, the bluer galaxies are concentrated toward the cluster
center, instead of the outskirts, as in the other ACS intermediate
redshift cluster sample (Demarco et al. 2005; Goto et al. 2005;
Homeier et al. 2005;
Postman et al. 2005). This tendency might
support the hypothesis that we are observing sub-groups of galaxies in
an edge--on sheet, e.g a group of bluer disk galaxies on a redder, older population of ellipticals.
There are 10 confirmed elliptical and 5 confirmed S0 members in the
center of the cluster (R~$<$~500~kpc). The average S0 redshift is
1.102~$\pm$~0.002 and the average elliptical redshift is
1.105~$\pm$~0.007 (the given uncertainties are standard deviations
around the mean). This indicates small relative velocities (the two
redshifts are indistinguishable given the errors) between the
classes and is true regardless of the possible classification
bias discussed above. Unfortunately, our
spectroscopic sample does not permit us to track the cluster central
structure in detail. The average relative velocity between the
confirmed E and S0s of $\approx$~500~km/s (that also corresponds to
the cluster velocity dispersion; see below) is fairly small. If
merging of two distinct groups of galaxies is happening along the
line of sight, we expect much higher
velocity dispersions and/or relative velocities between the
infalling S0s and the ellipticals.
Stanford et al. (2002) also suggest that active galaxy-galaxy merging
should be observed, based on the X--ray temperature structure.
To investigate any on--going
dynamical activity, we calculated the cluster velocity dispersion, and
the merger rate. From the 25 spectroscopically confirmed members (all galaxy types included) the
line--of--sight rest--frame velocity dispersion is $\sigma =
675 \pm 190$~km/s, using the software ROSTAT from
Beers, Flynn, \& Gebhardt (1990).
The available Chandra data give X-ray temperatures ranging from $kT =
7.2^{+ 2.2}_{-1.4}$~keV (Stanford et al.\ 2002) to $kT = 6.6^{+
1.7}_{-1.3}$~keV (Ettori et al.\ 2004). Wu et al. (1999; see also
Rosati et al. 2002) gives the relationship between $kT$ and $\sigma$
for relaxed clusters, which predicts that the velocity dispersion
corresponding to the measured X-ray temperature should be
$\approx$~1000~km/s, considerably higher than we have found.
Again, in the case of merging groups (along the line of sight)
we would also have expected a higher velocity dispersion.
\section{Red galaxy pairs}
The quality of the ACS data allows us to discern
merging activity among the cluster galaxies. If we assume
that galaxy pairs with projected separations less than $20
h^{-1}_{70}$~kpc are physically associated, we observe 13 associated
early--type galaxies, nine galaxies of which lie on a filamentary
structure about $\approx$~100~kpc from the cluster center
(Fig.~\ref{cluster}). As noted above, RXJ~0910+5422 lacks any
cD galaxy near the center of the X--ray emission
(see also Fig.~\ref{cluster}), but rather has a filamentary
group of galaxies around the X--ray center.
The nine early-type interacting galaxies
within this filamentary structure (at radius of
$\sim100 h^{-1}_{70}$~kpc from the X--ray center) include 3 unique pairs
(yellow arrows in Fig.~\ref{cluster}), plus a galaxy triplet
(the three components marked with red arrows in Fig.~\ref{cluster}).
Each of the 3 pairs consists of a bright elliptical with a smaller
companion (all closer than $10 h^{-1}_{70}$~kpc), while the triplet is a large E with two smaller S0s
(also closer than $10 h^{-1}_{70}$~kpc; one of these two S0s is ACS ID 1621, the S0 with an inward blue gradient).
One of the pairs (the middle pair in the figure) and the two nearest
galaxies in the triplet, have essentially zero relative velocity and
thus are likely merger candidates.
Also two of the ellipticals with blue inward gradients lie on the central filaments, and they are both small satellites of a larger galaxy.
The other two pairs, which do not lie on the filamentary structure,
are at ~$250h^{-1}_{70}$~kpc (two E with similar size, both with weak O~II
emission (Stanford et al. 2002) and ~$300 h^{-1}_{70}$~kpc (one S0/a
and one E of similar size) from the X--ray center, and have relative
velocities of ~10,000~km/s and ~2000~km/s, respectively.
The presence of the low-velocity pairs is consistent with the
low velocity dispersion, and provides evidence for the
on--going hierarchical growth of the cluster (e.g., van Dokkum
et al. 1999).
The pairing of the bright ellipticals with smaller S0s also argues against
the view that we are observing a blue S0-dominated group infalling
into a red cluster elliptical population, but rather complex stellar
population evolution within a filamentary structure.
The red--sequence S0/a confirmed members also lie within this
filamentary structure.
The observations of a significant number of red galaxy pairs in a
cluster at z~$\sim$~1 is interesting in the context of the
recent findings by van Dokkum (2005) of red galaxy interactions
in $\approx$~70\% of 86 early--type galaxies in a selected sample of nearby red galaxies from the MUSYC (Multiwavelength Survey by Yale--Chile; Gawiser et al. 2005) and the NOAO Deep Wide--Field Survey.
This work concluded that most of the ellipticals in local samples were
assembled by red galaxy--galaxy mergers, denominated {\it dry} mergers because they would involve
gas--poor early--type galaxies.
At higher redshift, Tran et al. (2005b) confirmed red galaxy mergers first observed by van Dokkum (1999) in MS~1054-03 at z=0.83. Tran et al. selected mergers as associated pairs with projected separation less than $10 h^{-1}_{70}$~kpc and relative line--of--sight velocities less than 165~km/s. As in RXJ~0910+5422, the red early--type galaxies involved in these mergers are among the brightest cluster members. Their results suggest that most early--type galaxies grew from passive red galaxy--galaxy mergers.
In our sample we observe a triplet and three red galaxy pairs with projected distances less than $10 h^{-1}_{70}$~kpc. Of those, the triplet and one galaxy pair show zero relative velocity. Of the two other pairs, composed of a bright and a fainter companion, redshifts are not available for the faint companions.
\section{Cluster luminosity function}
To obtain a deeper understanding of the RXJ~0910+5422 galaxy population,
we constructed galaxy luminosity functions in the following way. We start with the original
Sextractor catalog (described in Section~3). All objects with
magnitudes brighter than $i_{F775W} = 21.1$~mag were considered as
foreground objects. Nine of these bright objects are confirmed
non-members. The remaining seven are objects that do not belong to the
red sequence and whose sizes and luminosities are much larger
than those of the confirmed members, in particular those of the
bright red sequence galaxies; therefore they are very
unlikely to be at the cluster redshift.
The contribution to the luminosity function from both foreground and
background field galaxies (hereinafter, the field) has been estimated from the galaxy counts in a
reference field. The control region is taken from the GOODS-S
(Great Observatories
Origins Deep Survey--South; Giavalisco et al. 2004) ACS field,
observed in the same filter as the cluster field. Point--like
objects were eliminated in a consistent way in the cluster and in the
control field, by identification of the stellar locus in the
diagnostic plot of the SExtractor parameters MAG\_AUTO vs FLUX\_RADIUS
(the selected objects have FWHM equal to the PSF in the image).
Cluster and control field luminosity functions were normalized to the
cluster area. Both cluster and field counts were binned with a bin
size of 0.5~mag. For each bin, the field counts are subtracted from
the cluster counts, taking into account the extensive spectroscopic
sample (more than 60\% of the objects used for the LF determination
brighter than M$^{*}$ have measured redshifts). Known interlopers
were excluded from the analysis. The uncertainties in the cluster
counts after subtraction of the field contribution are calculated by adding in quadrature
Poissonian uncertainties.
The luminosity functions are shown in Fig~\ref{lumfun}.
The filled circles with errors are the total background--corrected cluster
luminosity function. We do not include errors from cosmic variance due to
the choice of the background control region. The red sequence elliptical and
spheroidal (S0 and S0/a) luminosity functions are shown respectively in
blue and green.
The red histogram is the luminosity function of all early--type
galaxies with color $0.8 <(i_{775} - z_{850}) < 1.1$~mag, excluding
confirmed non--members. The histogram of red sequence galaxies in
RXJ1252.9-292 is shown as the dashed red line. The background contribution is very small
for the early-type sample. The red arrows show the histogram values
after background subtraction.
The rest-frame $B$~magnitudes are shown on the top of the plot,
calculated from colors obtained from the BC03
stellar population model and templates (Sbc, Scd) from Coleman, Wu \&
Weedman (1980). The solid black line is a Schechter function fit
to the total cluster luminosity function. It is obtained by
calculating the C (Cash 1979) statistic (a
maximum likelihood statistic to fit data with Poissonian errors)
on a grid in the M$^{*}$ --
$\alpha$ plane for each combination of M$^{*}$ and $\alpha$: first the
normalization ($\phi^{*}$) is calculated in order to reproduce the
observed number of galaxies in the observed magnitude range, then the
C statistic is computed as $C=-2\Sigma_{i_{bin}} n_{i} \ln m_{i} -
m_{i} - \ln n_{i}!$, where $n_{i}$ is the observed number of galaxies
in the $i\,$th bin and $m_{i}$ is the number of galaxies predicted in
that bin by the Schechter function with parameters M$^{*}$, $\alpha$,
and $\phi^{*}$.
The combination M$^{*}$,$\alpha$ which minimizes the C statistic is
taken as the best-fit. If the C statistic is defined as
above, the 1- 2- and 3-$\sigma$ confidence levels for M$^{*}$ and $\alpha$
can be estimated from $\Delta C = 2.3, 6.17,11.8$. We obtain
M$^{*}=22.6^{+0.6}_{-0.7}$~mag and $\alpha=-0.75\pm0.4$.
Most of the faint--end population is composed of S0 and S0/a galaxies.
The two brightest galaxies in the red sequence are S0.
With respect to RXJ1252.9-292, a bright population of
red sequence ellipticals is missing in RXJ~0910+5422.
However, the large Poissonian errors on the bright end of the
cluster population prevent us from definitively excluding the hypothesis
that
the two clusters could be drawn from the same parent population.
Similarly, small number statistics do not permit us to study the
luminosity functions of
the different red and blue faint populations in this cluster.
\section{Discussion and Conclusions}
In this paper we have studied the color--magnitude relations of
galaxies in RXJ\,0910+5422 to constrain their ages and formation histories.
Our results show that the color--magnitude relation for the elliptical
galaxies is consistent both in slope and scatter with that of RXJ\,1252.9-292
(Blakeslee et al. 2003a, Lidman et al. 2004) and recent results from Holden et
al. (2004) and De Lucia et al. (2004), confirming that elliptical
galaxies in galaxy clusters show a universal color--magnitude relation
consistent with an old passively evolving population even at $z \sim 1$.
From the color--magnitude relation of the ellipticals, we derive a mean
luminosity--weighted age $\overline t > 3.3$~Gyr
($z_f > $~3).
We find that the S0s in RXJ~0910+5422 define a color--magnitude sequence
with a scatter similar to that found for the ellipticals, but shifted bluer
by $0.07 \pm 0.02$~mag. This is peculiar with respect to previous cluster
studies, which more typically found that the S0s followed the same CMR
as the ellipticals, but with somewhat larger scatter (Bower et al. 1992; Ellis et al.\ 1997; Stanford et al.\ 1997; L\'opez--Cruz et al. 2004; Stanford, Eisenhardt, \& Dickinson 1998; van Dokkum et al. 2000, 2001; Blakeslee et al. 2003a; Holden et al. 2004). Only one earlier study, van Dokkum et al.
(1998) found a significantly bluer S0 population.
We examine this
population of blue S0s in some detail, noting that there is a strong
predominance of flattened systems with axial ratios $\frac{b}{a} > 0.7$,
and conclude that the face-on members of the population have likely been
classified as ellipticals. If so, the color offset between the two
classes would become even more significant, and the true CMR scatter
for the ellipticals would be slightly lower than we have estimated.
This peculiarity is not observed in other clusters of our ACS
Intermediate
Redshift
Cluster Survey, and its amplitude is smaller
than the uncertainties adopted in Postman et al. (2005).
If the observed color difference between the ellipticals and S0s is mainly
due to metallicity at the same age, this would imply that the redder
ellipticals
were able to retain more metals than the S0s, i.e., they are more massive.
However, current data suggest that the CMR is mainly the result of a metallicity--mass
(i.e. metallicity--magnitude) relation
(e.g., Kodama \& Arimoto 1997; Kauffman \& Charlot 1998
Vazdekis et al. 2001; Bernardi et al. 2005).
This implies that we do
not expect large metallicity variations at a given magnitude.
If, instead, the offset is mainly due to age, then the implied age difference would be
$\sim$1~Gyr for single-burst solar metallicity BC03 models. It could
also result from different star formation histories, with the S0s experiencing
a more extended period of star formation. A model with solar metallicity
and with an exponential decay of the star formation reproduces the offset
at a galaxy mean age of $\approx$~3.5~Gyr.
The blue S0s may comprise a group infalling from the field onto a
more evolved red cluster population, or they may be
a transitional cluster population not yet evolved all the way onto the
elliptical red sequence (van Dokkum \& Franx 2001). Assuming
passive evolution, they will reach this red sequence after about 1~Gyr.
High fractions of faint blue late type galaxies were observed in
substructures infalling in a main cluster (e.g. Abraham et al. 1996; Tran et al. 2005a), and proposed as the progenitors of faint S0s in clusters.
The view of this
cluster as a structure still in formation is supported by X--ray
observations of the cluster temperature structure (Stanford et al. 2002),
the lack of a cD galaxy, and its
filamentary structure that suggests merging of substructures. However,
we derive a small cluster velocity dispersion, unusual for merging
substructures.
Moreover, the blue S0s in this sample span the same
luminosity range of the bright ellipticals, are distributed towards the center of the
cluster, and some of the faintest ones are physically associated with
brighter ellipticals belonging to the central filamentary structure.
These elements would argue against the bluer S0s being a young group merging with an existing red cluster population, and support the hypothesis that we are observing a transitional blue S0 population in a cluster core that
is still evolving onto the elliptical red sequence.
This result is also consistent with the deficit of S0s observed in our ACS cluster sample, when compared to lower
redshift samples, that implies that the S0 population is not
yet in place but still forming in clusters at redshifts around unity (Postman et al. 2005).
Interestingly, we also observe in this cluster potential progenitors for bright S0 galaxies: four bright spirals (spectroscopically confirmed cluster members)
with $z_{850}$ brighter than 22.5~mag and $(i_{775}-z_{850})$
between 0.5 and 1.3~mag.
Red galaxy pairs are also observed. A triplet and three red galaxy pairs have projected distances less than $10 h^{-1}_{70}$~kpc, and of those, the triplet and one pair show zero relative velocity. This would be the evidence of red galaxy mergers at z$\sim$~1. van Dokkum (2005) and Tran et al. (2005b) have observed mergers of red galaxies in a nearby elliptical sample and in MS 1054-03 at z=0.83, respectively. They suggested a scenario in which most of the early--type galaxies were formed from passive red galaxy--galaxy mergers, called {\it dry} mergers, because they involve gas--poor early--type galaxies.
Future papers will analyze the ages and masses of the cluster members
using our optical spectroscopy along with newly obtained Spitzer IRAC imaging.
A larger sample would be needed to draw firmer conclusions about the formation
of S0s.
\begin{acknowledgements}
ACS was developed under NASA contract NAS 5-32865, and this research
has been supported by NASA grant NAG5-7697 and
by an equipment grant from Sun Microsystems, Inc.
The {Space Telescope Science
Institute} is operated by AURA Inc., under NASA contract NAS5-26555.
We are grateful to K.~Anderson, J.~McCann, S.~Busching, A.~Framarini, S.~Barkhouser,
and T.~Allen for their invaluable contributions to the ACS project at JHU.
We thank W. J. McCann for the use of the FITSCUT routine for our color images.
SM thanks Tadayuki Kodama for useful discussions.
\end{acknowledgements}
\newpage
\clearpage
\begin{small}
\begin{table*}
\begin{center}
\caption{Color--Magnitude Relations \label{results}}
\vspace{0.25cm}
\begin{tabular}{cccccccccccc}
\tableline \tableline\\
Sample &$N$&$c_0$&$Slope$ & $\sigma_{int}$\\
&&(mag)&&(mag)&\\
\tableline \tableline\\
E+S0+S0/a$^1$& 31& 0.99 $\pm$ 0.01& -0.030 $\pm$
0.020& 0.060 $\pm$ 0.008\\
E+S0$^{1a}$& 14& 1.00 $\pm$ 0.02& -0.010 $\pm$
0.033& 0.054 $\pm$ 0.009\\
E$^1$& 19& 1.02 $\pm$ 0.01& -0.033 $\pm$
0.015& 0.042 $\pm$ 0.011\\
E$^{1a}$& 10& 1.02 $\pm$ 0.04& -0.020 $\pm$
0.044& 0.047 $\pm$ 0.022\\
S0$^1$& 9& 0.95 $\pm$ 0.02& 0.005 $\pm$
0.023& 0.044 $\pm$ 0.02\\
S0+S0/a$^{1}$& 12& 0.95 $\pm$ 0.02& -0.007 $\pm$
0.027& 0.057 $\pm$ 0.015\\ \tableline \\
E+S0+S0/a$^2$& 32& 0.99 $\pm$ 0.01& -0.032 $\pm$
0.019& 0.060 $\pm$ 0.008\\
E+S0$^{2a}$& 15& 0.99 $\pm$ 0.02& -0.021 $\pm$
0.034& 0.054 $\pm$ 0.009\\
E$^2$& 19& 1.02 $\pm$ 0.01& -0.033 $\pm$
0.015& 0.042 $\pm$ 0.011\\
E$^{2a}$& 10& 1.02 $\pm$ 0.04& -0.020 $\pm$
0.044& 0.047 $\pm$ 0.022\\
S0$^2$& 10& 0.95 $\pm$ 0.02& -0.012 $\pm$
0.036& 0.051 $\pm$ 0.018\\
S0+S0/a$^{2}$& 13& 0.96 $\pm$ 0.02& -0.015 $\pm$
0.033& 0.065 $\pm$ 0.015\\ \tableline \\
E+S0+S0/a$^3$& 34& 0.99 $\pm$ 0.01& -0.036 $\pm$
0.018& 0.059 $\pm$ 0.008\\
E+S0$^{3a}$& 15& 0.99 $\pm$ 0.02& -0.022 $\pm$
0.035& 0.054 $\pm$ 0.009\\
E$^3$& 20& 1.01 $\pm$ 0.01& -0.032 $\pm$
0.015& 0.044 $\pm$ 0.010\\
E$^{3a}$& 10& 1.02 $\pm$ 0.04& -0.021 $\pm$
0.046& 0.047 $\pm$ 0.022\\
S0$^3$& 11& 0.96 $\pm$ 0.02& -0.022 $\pm$
0.038& 0.053 $\pm$ 0.015\\
S0+S0/a$^{3}$& 14& 0.96 $\pm$ 0.02& -0.024 $\pm$
0.034& 0.065 $\pm$ 0.013\\
\tableline \tableline
\end{tabular}
\end{center}
$1$: within 1~$\arcmin$ \\
$2$: within 1.5~$\arcmin$ \\
$3$: within 2~$\arcmin$ \\
$a$: only confirmed members
\end{table*}
\end{small}
|
Title:
First scattered light images of debris disks around HD 53143 and HD 139664 |
Abstract: We present the first scattered light images of debris disks around a K star
(HD 53143) and an F star (HD 139664) using the coronagraphic mode of the
Advanced Camera for Surveys (ACS) aboard the Hubble Space Telescope (HST). With
ages 0.3 - 1 Gyr, these are among the oldest optically detected debris disks.
HD 53143, viewed ~45 degrees from edge-on, does not show radial variation in
disk structure and has width >55 AU. HD 139664 is seen close to edge-on and has
belt-like morphology with a dust peak 83 AU from the star and a distinct outer
boundary at 109 AU. We discuss evidence for significant diversity in the radial
architecture of debris disks that appears unconnected to stellar spectral type
or age. HD 139664 and possibly the solar system belong in a category of narrow
belts 20-30 AU wide. HD 53143 represents a class of wide disk architecture with
characteristic width >50 AU.
| https://export.arxiv.org/pdf/astro-ph/0601488 |
\title{First scattered light images of debris disks around HD 53143 and HD 139664}
\author{Paul Kalas\altaffilmark{1}, James R. Graham\altaffilmark{1}, Mark C. Clampin\altaffilmark{2}, \& Michael P.
Fitzgerald\altaffilmark{1}}
\affil{}
\altaffiltext{1}{Astronomy Department, University of California,
Berkeley, CA 94720}
\altaffiltext{2}{Goddard Space Flight Center, Greenbelt, MD 20771}
\keywords{stars: individual(\objectname{HD 53143, HD 139664}) - circumstellar matter}
\section{Introduction}
The configuration of our solar system is perhaps the most significant starting
point for our understanding of planet formation. Therefore a
fundamental question is whether or not the architecture of our solar system is
common relative to other planetary systems.
One point of comparison is the structure of our Kuiper Belt relative to other
systems, which are typically seen as debris disks in scattered light or thermal emission.
In scattered light, some debris disks, such as $\beta$ Pic and
AU Mic, have central holes, but are radially extended to hundreds of
AU radii \citep{smith84, kalas04}. Other debris disks, such as HR 4796A and Fomalhaut
consist of relatively narrow rings with sharp inner and outer
boundaries \citep{schneider99, kalas05a}.
However, a narrow-belt architecture has not previously been
detected in scattered light among stars similar in spectral type and age to the Sun.
HD 53143 (K1V) and HD 139664 (F5V) are two stars $\sim$18 parsec from the Sun
known to have circumstellar dust due to excess thermal emission at far-infrared
wavelengths (Aumann 1985; Stencel \& Backman 1991; Table 1).
Various indicators place the age of
HD 53143 at $1.0 \pm 0.2$ Gyr \citep{decin00, song00, nord04}, whereas
HD 139664 may be a younger system with age $0.3^{+0.7}_{-0.2}$ Gyr \citep{lach99, montes01, mallik03, nord04}.
For these two stars the infrared excess corresponds
to a dust mass 3 - 10 times smaller than that of $\sim$10 Myr-old systems such
as AU Mic and $\beta$ Pic \citep{kalas04}.
Direct imaging of debris disks with masses this small is observationally
challenging, but it is now feasible using the optical coronagraph in ACS.
\section{Observations \& Data Analysis}
We utilized the HST ACS High Resolution Camera (HRC)
with a 1.8$\arcsec$ diameter
occulting spot to artificially eclipse each star (Table 1; Fig. 1).
Five F stars were observed in consecutive orbits, as were five K stars,
in order to minimize differences in the point spread function (PSF) due to telescope thermal variations.
Each PSF was subtracted using the four other stars in
each set of observations.
The relative intensity scaling and registration between images was iteratively
adjusted until the residual image showed a mean radial profile equal to zero
intensity.
After the excess nebulosity was detected around HD 53143 and HD 139664,
we determined that no surface brightness asymmetries were detected between
each side of each disk. To improve the signal-to-noise, we mirror averaged the data (Figs. 2 \& 3).
Mirror averaging splits the image into two halves along the axis that is
perpendicular to the disk midplane and bisects the disk. One side is transposed
onto the other side and the data are then averaged.
Asymmetries due to scattering phase function effects will be coadded \citep{kalas96}.
In effect mirror averaging doubles the integration time
spent on the circumstellar disk given that the broad features between each
side are symmetric. As a test, we also subtracted the two disk halves from
each other and confirmed that the assumption of symmetry is valid.
\section{Results}
The two disks have different morphologies due to different inclinations
and intrinsically different architectures. To quantify the
viewing geometries and structural properties of the disks, we produce a series
of simulated scattered-light disks that explore the parameters of inclination to the line of sight,
inner and outer disk radius, and the radial and vertical
variation of dust number density \citep[Fig. 2;][]{kalas96}. We reinsert the simulated
disks across each star in a direction orthogonal
to the observed midplanes and select those models that
most closely resemble the properties of the observed disks.
Table 1 summarizes our findings.
The shape of the midplane surface brightness distribution differs significantly for
each system (Fig. 3). The HD 53143 midplane surface brightness decreases monotonically with projected radius, approximately
as $r^{-3}$, where $r$ is the projected radius. In the simulated disk, the
radial number density distribution decreases as $q^{-1}$, where $q$ is radius in the disk cylindrical
coordinate system.
Our solar system's Zodiacal dust complex has
a comparable dependence of grain number density as a function of
radius ($q^{-1.34}$; Kelsall et al. 1998), controlled mainly by the force of
Poynting-Robertson (PR) drag that causes small grains to spiral into the Sun \citep{burns79}.
Therefore the HD 53143 disk may simply represent
a population of unseen parent bodies that collisionally replenish dust
that is redistributed radially
by PR drag.
Our simulations show that the outer radius of the observed disk is a sensitivity-limited
value at approximately 6$\arcsec$ radius (110 AU).
The inner radius is also sensitivity-limited to 3$\arcsec$ radius (55 AU). Therefore, the debris disk
around HD 53143 is at least 55 AU wide.
Material surrounding HD 139664, on the other hand,
is confined to a narrow belt, as indicated by a turnover
in the midplane surface brightness profile
between 4.5$\arcsec$ and 5.5$\arcsec$ (79 - 96 AU; Fig. 3).
Our disk simulations show that the peak in the dust distribution occurs
at 83 AU, decreasing as $q^{-2.5}$ from 83 - 109 AU, and with a sharp
outer truncation at 109 AU (Figs. 2 \& 3).
We tested model disks that have outer radii $>$109 AU and found that these disks would have been
detectable in our data as far as 10$\arcsec$ radius (175 AU; Fig. 3).
Given an absence
of significant gas, the belt-like nature of HD 139664 is most likely
a structure related to planet formation. \citet{kenyon04} find that
a dust belt with peak surface brightness at $\sim$80 AU radius forms within a planetesimal disk
at age 400 Myr. The appearance of this belt signals the recent formation
of a $\geq$1000 km planet that gravitationally stirs planetesimals in
its viscinity. \citet{liou99} and \citet{moro02}, on the other hand, simulate the concentrations of
dust in trans-Neptunian space that arise due to trapping in mean motion resonances.
A natural explanation for the belt-like morphology of HD 139664
is that the $\sim$83 AU peak in the dust distribution
corresponds to either an interior or exterior mean motion resonance
created by a companion to HD 139664. Large grains may
dominate the belt's peak at 83 AU, with smaller grains
passing quickly through the resonance regions due to radiation forces.
\section{Discussion}
Taking a census of eight debris disks resolved in scattered light and the predicted
dust distribution in our Kuiper Belt,
we observe two basic architectures
that are not correlated with stellar mass and luminosity,
but must depend on other environmental factors (Table 2).
Debris systems are either narrow belts or wide disks. Both types have central dust depletions,
but the distinguishing characteristic
is the presence or absence of a distinct outer edge. HR 4796A,
Fomalhaut, HD 139664 and the Sun
are examples of narrow-belt systems. The belt systems appear
to have radial widths ranging between 20 and 30 AU, and the inner edges may begin as close as
25 AU (Sun), or as far as 133 AU (Fomalhaut).
HD 32297, $\beta$ Pic, HD 107146,
HD 53143 and AU Mic are examples of disks with sensitivity-limited outer edges
that imply disk widths $>$50 AU.
The F, G, and K stars are interesting because {\it a priori} we might expect to find planetary
systems similar to our own, yet we discover significant diversity in the outer regions that correspond
to Neptune and our Kuiper Belt.
With age $\sim$1 Gyr, HD 53143 is among the oldest known extrasolar debris disks,
yet its wide-disk architecture resembles that of the $\sim$10 Myr old systems
of $\beta$ Pic and AU Mic. The lingering dust mass (Table 1) throughout the system
could signal the absence of giant planets that otherwise sweep clear the
parent bodies (comets and asteroids) responsible for the
producing the dust disk. Yet the presence of a dust disk out to at least 110 AU
radius shows that the primordial circumstellar disk probably contained the prerequisite mass
of gas and dust to form giant planets.
By contrast, the younger system HD 139664 has already developed
a narrow-belt architecture.
Narrow-belt architectures for the underlying population of planetesimals
may originate from early stochastic dynamical events,
such as a close stellar flyby, that strip disk mass and dynamically heat the
surviving disk \citep{ida00, adams01}. Theoretical simulations show that a reduction in disk mass,
combined with dynamical heating, produces a less stable planetary system that is more likely to
eject giant planets from their formation site to much larger radii, as has been proposed for
the origin of Neptune \citep{thommes99,tsiganis05}. Therefore, planetesimal belts not only evolve
into a narrow structures because of external stochastic events, but they may be found
at large distances from the central star due to the subsequent
outward migration of interior planets.
The collisionally replenished dust population will spread away
from any narrow belt of planetesimals. If the scattered light appearance continues to manifest
a narrow structure, then both the inner and outer edges are probably maintained by other
gravitational perturbers such as stellar or sub-stellar companions. However, only HR 4796
has a known stellar companion that may truncate the outer radius of the dust belt \citep{aug99}.
If there is no confinement
mechanism for the outer radius, then the architecture will manifest as a wide-disk. For example,
$\beta$ Pic and AU Mic may have narrow belts of planetesimals \citep{aug01, strubbe05}, but the
observed dust disk widths extend to hundreds of AU.
Though the predicted structure of our Kuiper Belt places the solar system
among the narrow disk architectures, it is conceivable that the dust component
extends to greater radii \citep{trujillo01}. Thus the Sun's classification as a narrow-disk
system is tentative.
\section{Summary}
We present the first optical scattered light images of debris disks surrounding
relatively old main sequence F and K stars.
Material around HD 139664 is concentrated at 83 AU radius, with a distinct
outer edge at 109 AU, and a depleted, but not empty, region at $<$83 AU radius.
Dust surrounding HD 53143 has a monotonic $q^{-1}$ variation in grain number density, and
the disk edges from 55 AU to 110 AU are sensitivity-limited values.
The different radial widths appear consistent with a more general grouping of
debris disks into either narrow or wide architectures. These two categories
are probably an oversimplification of significant diversity in the formation and
evolution of debris disks.
Future observations should test for common traits
among these stars, such as stellar multiplicity and the existence of planets.
\acknowledgements
{\bf Acknowledgements:} Based on observations with the NASA/ESA Hubble Space Telescope obtained
at the Space Telescope Science Institute (STScI), which is operated by the Association of Universities for
Research in Astronomy. Support for Proposal number GO-9475 was provided by NASA through a grant from STScI
under NASA contract NAS5-26555.
\clearpage
\clearpage
\clearpage
\begin{deluxetable}{lllll}
\tabletypesize{\scriptsize}
\tablecaption{Star (rows 1-7) and disk (rows 8-15) properties
\label{tbl-1}}
\tablewidth{0pt}
\tablehead{
\colhead{} & \colhead{HD 53143} & \colhead{HD139664}
}
\startdata
Age (Gyr) &1.0$\pm$0.2 &$0.3^{+0.7}_{-0.2}$\\
Spectral Type &K1V & F5IV-V \\
Mass (M$_\odot$) &0.8 &1.3\\
T$_{eff}$ (K) &5224 & 6653\\
Luminosity (L$_\odot$) &0.7 &3.3\\
Distance (pc) &18.4 & 17.5 \\
$m_V$ (mag) &6.30 & 4.64 \\
&&\\
Peak disk surf. bright. (mag/arcsec$^{2}$) &22.0$\pm$0.3 &20.5$\pm$0.3\\
Disk position angle (degrees) &147$\pm$2 &77 $\pm$ 0.5\\
Inclination (degrees) &40 - 50 &85 - 90\\
Disk number density gradients ($q^{\alpha}$) &-1 &+3.0 \& -2.5 \\
Inner dust depletion (AU) & $<55$ &83 \\
Maximum outer radius (AU) & $>110$ &109 \\
Optical depth from IRAS data\tablenotemark{a} &$2.5\times10^{-4}$ & $0.9\times10^{-4}$ \\
Optical depth from HST data\tablenotemark{b} &$>1.6\times10^{-5}$ &$1.0\times10^{-5}$ \\
Total Dust Mass (g)\tablenotemark{c} &$>7.1\times10^{23}$& $5.2\times10^{23}$ \\
\enddata
\tablenotetext{a}{The fractional dust luminosity
\citep{zuck04}.
}
\tablenotetext{b}{Derived from the model disks.
Since the disks are optically thin, we sum the cumulative
light from the model disk, $m_d$, and quote optical depth as $10^{(m_d - m_V) / -2.5)}$. We assume
albedo=1 and the optical depth will scale inversely with the assumed albedo.
}
\tablenotetext{c}{Dust mass follows from the HST optical depth and assumes a uniform particle
radius of 30 $\mu$m, density 2.5 g cm$^{-3}$ and albedo=1.0.
}
\end{deluxetable}
\clearpage
\begin{deluxetable}{lllllll}
\tabletypesize{\scriptsize}
\tablecaption{Debris disk architectures from scattered light properties\tablenotemark{a}
\label{tbl-1}}
\tablewidth{0pt}
\tablehead{
\colhead{Name} & \colhead{$r_{in}$ (AU)} & \colhead{$r_{out}$ (AU)} & \colhead{Width (AU)} & \colhead{d (pc)} & \colhead{SpT} & \colhead{References}
}
\startdata
$\beta$ Pic &$\sim$90 &$>$1835 & \bf{$>$1745} & 19.3 & A5V & 1, 2\\
HD 32297 &$<$40 &$>$1680 & \bf{$>$1640} & 112 & A0 & 3, 4\\
AU Mic &$\sim$12 &$>$210 & \bf{$>$198} & 9.9 & M1Ve & 5, 6\\
HD 53143 &$<$55 &$>$110 & \bf{$>$55} & 18.4 & K2V & This work.\\
HD 107146 &$\sim$130 &$>$185 & \bf{$>$55} & 28.4 & G2V & 7\\
&&\\
HD 139664 & 83 & 109 & \bf{26} & 18.5 &F5V & This work.\\
Fomalhaut & 133 & 158 & \bf{25} & 7.7 & A3V & 8\\
HR 4796A & 60 & 80 & \bf{20} & 67.1 & A0V & 9\\
Sun\tablenotemark{b} & 25 & 50 & \bf{25} & -- & G2V & 10, 11
\enddata
\tablenotetext{a}{We do not include systems younger than 10 Myr that are likely to possess significant primordial circumstellar gas.
Column 2 gives the inner disk radius corresponding to the approximate
peak of dust number density. Column 3 gives the outer radius.
}
\tablenotetext{b}{From simulations of relatively large grains
trapped in resonances with Neptune.
}
\tablecomments{REFERENCES: (1) \citet{pantin97}; (2) \citet{larwood01};
(3) \citet{ssh05}; (4) \citet{kalas05b}; (5) \citet{krist05}; (6) \citet{kalas04}; (7) \citet{ardila04};
(8) \citet{kalas05a}; (9) \citet{schneider99}; (10) \citet{liou99}; (11) \citet{moro02}
}
\end{deluxetable}
|
Title:
Morpho-kinematic modeling of gaseous nebulae with SHAPE |
Abstract: We present a powerful new tool to analyse and disentangle the 3-D geometry
and kinematic structure of gaseous nebulae. The method consists in combining
commercially available digital animation software to simulate the 3-D structure
and expansion pattern of the nebula with a dedicated, purpose built rendering
software that produces the final images and long slit spectra which are
compared to the real data. We show results for the complex planetary nebulae
NGC 6369 and Abell 30 based on long slit spectra obtained at the San Pedro
Martir observatory.
| https://export.arxiv.org/pdf/astro-ph/0601585 |
\section{Introduction}
\label{sec:introduction}
In recent years, the discovery of a variety of complex structures in
planetary nebulae has opened many questions regarding the origin and evolution
of these objects (e.g. L\'opez 2000). Deviations from simple expanding shells
can include collimated outflows, poly-polar and
point-symmetric structures, rings or disks. These observations have lead to
a wealth of theoretical research into the effects of stellar magnetic fields,
rapid rotation and binarity of the central stars, and their evolutionary path
from spherically symmetric to bipolar mass-loss (e.g. Balick \& Frank, 2002,
and references therein).
In the absence of spherical symmetry, the tilt of the nebula with respect to
the line of sight and the location and position angle of the slit on the
nebula can often result in complicated position-velocity (P-V) diagrams
that can be difficult to interpret. The correct interpretation of
the nebular 3-D geometry and kinematic structure of PNe
is key to the understanding of the dynamics ruling their origin and evolution.
Modeling of line emission intensity maps have been used to obtain density
distributions over the face of the nebula in order to assess 3-D structures,
assuming pure photoionization from the central star (e.g. Monteiro et al. 2004;
Morisset, Stasi\'nska \& Pe\~na, 2005) but without incorporating kinematic
information or assuming simple velocity laws (e.g. Ragazzoni et al. 2001).
In this paper we present a new interactive 3-D modeling tool called \shape
which combines the versatility of commercial 3-D modeling software with a
rendering module specifically developed for application
in astrophysical research. The application of this method yields a 3-D
emissivity and velocity distribution for the object. Furthermore, different
velocity laws can be applied to different sections of the nebula to reproduce
the complex velocity patterns often observed in PNe. We exemplify the power
of this new method with models of the particularly complex planetary nebulae
Abell 30 and NGC 6369.
\section{Problem Definition}
The problem we attack with \shape is to characterize the current 3-D morphology
and velocity field of a nebula based on imagery and spectral kinematic
information. Detailed knowledge of this information leads to a better
understanding of the physical structure and dynamical evolution of a
gaseous nebula.
The projected image on the sky of an extended nebula provides
bidimensional spatial information of its structure. On the other hand,
the velocity field provides information on the radial component
of the velocity vector along the line of sight and conveys limited
but useful information on its depth or third spatial dimension.
However, an unambiguous solution of the complete 3-D structure at least
requires full knowledge of the velocity field. This situation is usually
not given, although topological and symmetry information
apparent in the images and spectra may help resolve ambiguities.
The simplest case occurs if the velocity of a volume element is constant over most of the
expansion time. In complex objects, this type of velocity distribution can
be expected if the nebula has evolved from a relatively short mass-loss
event and is now moving ballistically (e.g. Zijlstra et al. 2001) or from a continuous
interaction of a wind with small scale structures (Steffen \& L\'opez, 2004).
In these cases, after a sufficiently long time, the velocity pattern becomes
proportional to the distance from the center (a hubble-like velocity law).
The expansion of such a nebula is self-similar, i.e. the global shape is
conserved over time.
Under such conditions, the velocity vector is proportional to the position
vector of every material element in the nebula. The shape of
the nebula along the line of sight
is mapped linearly into the corresponding component of the velocity vector.
Hence, the Doppler-shift, which is equivalent to the velocity component
along the line of sight, is a map of the structure that is lost in a
direct image, i.e. the long-slit spectrum allows a
view of the nebula from a direction perpendicular to the line of sight.
This situation can clearly be appreciated in the case of axisymmetric bipolar
nebulae, where the line profiles also show a bipolar structure that represents
the depth or third dimension of the 2-D image on the sky.
In many planetary nebulae an expansion velocity proportional to distance
from central star seems to be a reasonable
approximation at least for the brightest regions (e.g. Wilson 1950, Weedman 1968).
However, more complex velocity structures can
be expected when one or more mass loss events arise over a significant timescale
compared with the age of the nebula.
Sabbadin et al. (2000)
have used the assumption of a radial velocity field proportional to the
distance from the central star to reconstruct the 3-D structure of a
number of nebulae with a "tomographic" method. This tomography works
well as long as there are no significant deviations from the
hubble-like velocity law.
\section{The 3D modeling system}
In this section we
describe our new version of the code \shape as a tool to find the 3-D structure
and kinematics of gaseous nebulae.
Originally \shape was based on a description of structure and kinematics
using parametric geometrical equations on a regular 3-D grid
(Steffen et al. 1996). This code was adapted with a simple graphical interface
by Harman et al. (2004). It has been applied to a variety of objects
from individual knots in planetary nebulae (L\'opez, Steffen \& Meaburn, 1997)
to moderately complex structures in active galactic nuclei (Steffen et al. 1996).
In the present upgraded version we have devised a completely different
approach based on particle systems, rather than a regular grid, in combination
with a commercial 3-D animation package, which we describe next.
As our 3-D modeling software we use {\em Autodesk 3DStudio Max 7} ({\em
3DStudio Max} is a trademark of {\em Autodesk Media and Entertainment}, see
the website {\em www.discreet.com} for detailed software information).
We apply the available tools of this software to create
a particle and velocity distribution in space and time in order
to model an object. In particular we use the {\it ParticleFlow} particle
system to generate particle distributions which are then exported and
rendered in \shape.
\shape renders images and spectral information from the kinematics of the model
particle distribution. Key parameters such as orientation of the object on the
sky, location and width of the slit, seeing values and spectral and spatial
resolution are handled interactively in the graphical interface of \shape.
The general modeling process is as follows.
With the inspection of available observations
one obtains an initial rough idea of the structure and topology of the
object, which is then reproduced in the modeling software. For this
purpose one produces a distribution of particles in space with its corresponding
velocities. The particles may be distributed over a topologically complex surface
or throughout a volume. The resulting emissivity is integrated along the line
of sight. The modeling software allows a very complex object to be
built and since all the features may be variable in time, the time
evolution of the object may also be explored.
As a guide during modeling, a spectal preview feature has been developed.
For a limited number of particles it allows a rough version of
the P-V diagrams to be visualized during modeling
in the viewport of {\em 3DStudio Max}.
The particle data are then exported into an external file.
This file is read by the core code of \shape, which produces the rendered image
and corresponding P-V diagram.
In contrast to the real-time feature described above, the core code of \shape is
not limited by the number of particles. It has been tested with up to one
million particles. The rendered images are then compared with the observations.
To improve a model, changes may be introduced at any point of the procedure
until, after a number of iterations, a satisfactory approximation to
the observations are obtained. Alternative solutions to an ambiguous dataset
will not be automatically found, but may be sought for separately.
The processing of the data, including the viewing direction, the emissivity,
internal structure, size and other parameters of individual particles is
controled via a dedicated graphical interface. This interface has been programmed
as a module of the modeling software {\em 3DStudio Max} and is fully integrated in
its interface.
An important analytic feature of \shape is the possibility to apply different colors to
sections of a complex object. This allows a clear distinction
of them in the P-V diagrams, which helps considerably in the interpretation of
the observed spectra. By combining models of various emission lines, color images
are obtained to be compared with similar images from observations
(see Figure \ref{fig:images}). A red-blue coloring mode allows a clear
distinction between red and blue-shifted regions in the model image.
Sequences of varying parameters like slit position and width, as well as
orientation of the object allow a systematic search for the
best parameters that match the observations or the production of image
sequences for animations which help considerably to visualize the 3-D structure of
the object. The P-V diagrams of objects that have been modeled as evolving in
time can also be visualized as a time sequence in animation form.
The most important current limitation of \shape is that only optically thin
nebulae can be modeled. The code does not perform any physical
radiation transport or line emission calculation.
What it does is to directly assign a relative emissivity
distribution, which is sufficient for its main purpose, the characterization
of the structure and kinematics of a nebula. Moreover, \shape can also be
used to produce complex density distributions and kinematics from which
photoionization models can be calculated using codes like NEBU\_3D (Morisset et
al, 2005). The current apparent drawbacks can hence be largely overcome by
combining \shape with codes which calculate radiation transport. We aim at
this next step in the near future.
\subsection{The rendering code}
The core code of \shape renders images and P-V diagrams from the position and
velocity data provided with {\em 3DStudio Max}. Each particle is read from a file
and its emission mapped to the image and spectrum. The intensity and color of
the emission from a particle may depend on the position in the object or on the
substructure of which it is part. The particles may also have a finite size larger
than single image pixels, as a well as an internal emissivity structure.
At this time this internal emissivity distribution may be either constant,
an exponential or a gaussian fall-off. This is useful mostly
for the initial modeling stages, before the convolution with the seeing parameters
are applied, because convolution eliminates all internal particle structure.
Particle sizes smaller than the seeing together with a sufficiently large number
of particles ensure adequate sampling of the object structure.
The image blurring due to seeing is modeled by convolution with a gaussian point-spread
function and including emission from within one
FWHM of the seeing disk outside the spectrograph slit. The contribution of this
emission decreases with distance from the edge of the slit.
The instrumental resolution is included
in the convolution of the raw image and P-V diagram with gaussian kernals of FWHM
corresponding to the spatial and velocity resolution of the instruments.
A Fast-Fourier-Transform algorithm is applied to calculate the convolutions
with gaussian kernels. After convolution, the P-V diagrams can be used to obtain
one-dimensional line profiles adding all or sections of the emission along the
spacial domain, which are often useful
in work with low spatial resolution on the nebulae.
The core code of \shape has been programmed updating the Fortran code of the
original version.
In the following sections we present two example
models to reproduce observations of objects with different degrees of
complexity. An additional good example for an application of \shape
on the complex structure of NGC~6302 can be found in Meaburn et al. (2005).
\section{Observations and models}
\subsection{NGC~6369}
Our model of NGC~6369 is based on a long-slit spectrum obtained
with MES (Meaburn et al. 2003) on
the 2.1 m telescope at the San Pedro Martir Observatory. The spectral
resolution is 10 km s$^{-1}$ and the seeing 1.5 arcsec. The image has been
obtained from the Hubble Heritage Team, NASA, STScI (Figure \ref{fig:images}).
The spectrograph slit was located in the east-west direction, which
is coincident with the main axis of the
object (as indicated in the model image in Figure \ref{fig:images}).
The slit width corresponds to 1.5 arcsec on the sky.
The observed spectra (Figure \ref{fig:spectra}, left) show that the central
"barrel" and the "lobes" are connected and probably conform a single topological
closed surface. Therefore we have started the modeling process with an initially
spherical shape and deformed it such that it matches our initial estimate
of the 3-D shape according to the image and spectrum. We then applied a brightness map
to the surface which approximates the emissivity distribution from
the H$\alpha$ and NII emission lines.
The structure and emission distribution is then adjusted interactively until
a satisfactory match was found. Figure \ref{fig:modvec} shows the
3-D structure of NGC~6369 depicted as a central ellipsoid with two opposite,
off-axis, protruding lobes.
We have modeled the nebula's expansion with two different velocity laws.
Based on the observation that
the highest and lowest projected velocities in the spectra are similar for
the barrel and the lobes, we considered a constant expansion velocity directed
perpendicularly to the surface. This corresponds roughly to an energy driven
bubble (in this case three different bubbles, the barrel and the two lobes).
In this case we did not find any shape that would reasonably resemble
the image and spectra simultaneously.
The second model assumes that the velocity is proportional to distance and
the direction of motion is radial from the center of the nebula. The resulting
surface model with representative velocity vectors is shown in Figure
\ref{fig:modvec}. For the H$\alpha$ emission we assume that emission comes from
a region somewhat inside this surface and for NII slightly outside, with some
overlapping around the surface that is shown (as seen in the observed spectra and
images). The rendered image and spectrum are shown in Figures (\ref{fig:images}) and
(\ref{fig:spectra}), respectively. The particle number density is proportional to the
brightness on the 3-D surface (see Fig.\ref{fig:modvec}), which is not
accurately represented in the image due to lighting effects of this
visualization, which emphasizes the topology of the 3-D structure.
Our modeling indicates, that the line of sight is close to
the ellipsoid's symmetry axis (tilt $\approx 15^\circ$)
and the axis of the lobes is tilted approximately $40^\circ\pm 10^\circ$
with respect to that axis. The ratio between the height and the
diameter of the barrel is not well constrained by the current modeling and
observations. We estimate that this ratio is of the order $3/2$. The model images
and P-V diagrams have spatial and spectral resolution corresponding to those
in our observations, which are 1.5 arcsec and 10 km s$^{-1}$ using a slitwidth of
1.5 arcsec in the simulation.
We find that the model with the velocity law proportional to distance
produces an acceptable fit to the observed image and spectrum.
This suggests that this flow is in a relaxed momentum driven state; though
the lobes appear to move slightly slower than expected in this case
(Figure \ref{fig:spectra}). This might be an indication for a short-lived
collimated outflow, which is not acting on the lobes anymore,
and/or a stronger interaction of the
lobes with the ambient medium due to their lower densities.
Monteiro et al. (2004) proposed a ``diabolo''-type structure for
NGC~6369 based on the analysis of spectral imaging data of a number of
ions. The diabolo model in this case necessarily implies a narrow waist
in the equatorial plane with its symmetry axis close to the line of sight.
With this orientation such a narrow torus-like waist
is expected to have very low velocity along the line of sight, and
in the H$\alpha$ long\-slit spectrum it should appear as a narrow feature
near the systemic velocity. The observations presented in Figure
\ref{fig:spectra} do not show evidence for a narrow low-velocity waist.
However, bright emission regions near the systemic velocity in the
line profiles are apparent. In our model this is due to both, an
intrinsic equatorial density enhancement as seen in the image (Figure \ref{fig:images})
as well as a long tangential line of sight through the spheroidal main nebula.
Thus, our model does not require a waist to reproduce this feature.
NGC~6886 and NGC~6565 are two remarkably similar nebulae to NGC~6369,
both in terms of morphology and line-profile structure. Turatto et al. (2002)
have modeled the structure of NGC~6565 with their tomographic method obtaining
similar results to ours.
\subsection{Abell 30}
Abell~30 is a hydrogen deficient planetary nebula with a spherical
[OIII] shell and complex knotty and filamentary structures in the
central region. The inner region contains
unusual cometary knots with large velocity spikes in the
P-V diagrams. Meaburn \& L\'opez (1996) describe its kinematics as "dramatic".
In this section we present a model produced with \shape as a
further illustration of its flexibility to handle complex kinematic
structures which may be very different from a hubble-like velocity law.
Observations of Abell~30 have been adopted from figures 1 and 2a in
Meaburn \& L\'opez (1996) which are reproduced here in
Figure \ref{fig:abell30_obs}. The observations may be compared with our model
model image and P-V diagram for one slit position as shown in Figure
(\ref{fig:abell30}).
The outer shell was modeled as a sphere with some variations in brightness and
a constant radial expansion. A random distribution of knots
and filaments was used for the inner regions. For the cometary knots,
particles where emitted from discrete points which then interacted with a
central spherically symmetric wind, accelerating the particles outwards
and producing the high-speed features in the P-V diagrams. Support for the
existence of such a wind in Abell~30 comes from observations of X-rays
(Chu, Chang \& Conway, 1997).
For the model we adapted the parameters
similar to those of the observations with seeing between 1 and 2 arcsec and
a width of the slit of 1.9 arcsec. For the model we used 1.5 and 1.9 arcsec
respectively. Since the object is basically spherically symmetric, except
for the random distribution of brightness variations of the outer shell and
the distribution of knots in the central region, the specific orientation
of the model is not a fundamentally important parameter.
At this time no attempt was
made to match individual small-scale features with those in Abell~30.
Still, the spectrum in Figure \ref{fig:abell30} matches the observed one
very well. This model shows how different methods can be used to combine
substructures which have a variety of complexity and kinematic signatures.
\section{Discussion and Conclusions}
In this paper we have presented the upgraded version of \shape, with a
novel approach to the determination of the
three-dimensional structure and kinematics of gaseous nebula.
\shape combines the
capabilities of commercial 3-D modeling software with a purpose-built rendering
software and graphical control interface for its application to astrophysical
nebulae. \shape produces images and P-V diagrams for highly complex nebulae. The
results can be compared directly with observations and help to understand
structure, kinematics and orientation of the objects.
The kinematics as observed
from long-slit spectra often provides sufficient information about the
3-D structure and topology of an object for the modeling with \shape to
provide a self-consistent 3-D structure. In addition the \shape models
can be applied directly as input density distributions for photo-ionization codes.
In \shape one can define and combine different velocity laws that fit complex
structures. Solutions are expected to be most reliable if the object shows evidence
for a significant degree of symmetry, then the full 3-D structure and
kinematics can be deduced unambiguously.
In other cases the object may be devided into regions or subsystens
which allows them to be solved separately.
We are currently building a catalogue of of synthetic emission line profiles
with \shape that should be a useful reference to interpret long-slit observations
of PNe with diverse morphologies and orientations.
In this paper we showed applications of \shape to the planetary nebulae
Abell~30 and NGC~6369. As a result of our application of \shape to new
observations of the planetary nebula NGC~6369, we propose that its basic
structure is that of a ellipsoidal or barrel-shaped main nebula with bipolar
protrusions at a large angle to the symmetry-axis of the main nebula.
Abell~30 has been modeled combining different velocity patterns to fit the
expanding shell and the inner high-velocity knots.
\acknowledgements
We acknowledge support from DGAPA-UNAM projects IN111803 and IN112103 as well as
CONACYT projects 37214 and 43121.
|
Title:
Two-body problem with the cosmological constant and observational constraints |
Abstract: We discuss the influence of the cosmological constant on the gravitational
equations of motion of bodies with arbitrary masses and eventually solve the
two-body problem. Observational constraints are derived from measurements of
the periastron advance in stellar systems, in particular binary pulsars and the
solar system. Up to now, Earth and Mars data give the best constraint, Lambda <
10^{-36} km^{-2}; bounds from binary pulsars are potentially competitive with
limits from interplanetary measurements. If properly accounting for the
gravito-magnetic effect, this upper limit on $\Lambda$ could greatly improve in
the near future thanks to new data from planned or already operating
space-missions.
| https://export.arxiv.org/pdf/astro-ph/0601612 |
\title{Two-body problem with the cosmological constant and observational constraints}
\author{Philippe Jetzer}
\email{jetzer@physik.unizh.ch}
\author{Mauro Sereno}
\email{sereno@physik.unizh.ch}
\affiliation{Institut f\"{u}r Theoretische Physik, Universit\"{a}t Z\"{u}rich,
Winterthurerstrasse 190, CH-8057 Z\"{u}rich , Switzerland}
\date{November 21, 2005}
\pacs{04.25.Nx,04.80.Cc,95.10.Ce,95.30.Sf,95.36.+x,96.30.-t,97.60.Gb,98.80.Es}
\keywords{cosmological constant, pulsars, solar system}
\section{Introduction}
Aged nearly one century, Einstein's cosmological constant $\Lambda$
still keeps unchanged its cool role to solve problems. $\Lambda$,
despite being just one number, was able to respond to very different
needs of the scientific community, from theoretical prejudices about
universe being static (which provided the original motivation for
introducing $\Lambda$ in 1917) to observational hints that the
universe is dominated by unclustered energy density exerting negative
pressure, as required by data of exquisite quality which became
available in the last couple of decades. Although it is apparently
plagued by some theoretical problems about its size and the
coincidence that just in the current phase of the universe the energy
contribution from $\Lambda$ is of the same order of that from
non-relativistic matter, the cosmological constant still provides the
most economical and simplest explanation for all the cosmological
observations \cite{pad05}. The interpretation of the cosmological
constant is a very fascinating and traditional topic.
$\Lambda$ might be connected to the vacuum density, as suggested by
various authors (see \cite{pe+ra03} for an historical account), and could offer the greatest contribution from cosmology to fundamental physics.
The big interest in the cosmological constant has recursively raised
attempts in putting observational bounds on its absolute value from
completely different phenomena. $\Lambda$, supposed to
be $\sim 10^{-46}\mathrm{km}^{-2}$ from observational cosmology analyses, is obviously of relevance on cosmological scales but it could play some role also in local
problems. Up to now, no convincing methods for constraining $\Lambda$
in a laboratory have been proposed \cite{je+st05}, but interesting
results have been obtained considering planetary motions in the solar
system \cite{isl83,wri98,ker+al03}. The effects of $\Lambda$ become
stronger for diluted mass conglomeration but they get enhanced also
through various mechanisms \cite{now+al02,ba+no05}. As an example,
conditions for the virial equilibrium can be affected by $\Lambda$ for
highly flattened objects \cite{now+al02}. On the scale of the Local Volume, a cosmological constant could have observable consequences by producing lower velocity dispersion around the Hubble flow \cite{tee+al05}.
Up to now, local physical consequences of the existence of a
cosmological constant were investigated studying the motion of test
bodies in the gravitational field of a very large mass. This one-body
problem can be properly considered in the framework of the spherically
symmetric Schwarzschild vacuum solution with a cosmological constant,
also known as Schwarzschild-de Sitter or Kottler space-time. The rotation of the central source can also be
accounted for using the so-called Kerr-de Sitter space-time
\cite{ker+al03}. Here, we carry out an analysis of the gravitational
$N$-body problem with arbitrary masses in the weak field limit with a
cosmological constant. This study is motivated by the more and more
central role of binary pulsars, from the discovery of the pulsar PSR
B1913+16 in 1974 \cite{hu+ta75}, in testing gravitational and relativistic
effects. The gravitational two-body equations of motion for arbitrary
masses were first derived in absence of spin by Einstein, Infeld and
Hoffmann (EIH) \cite{ein+al38}. The problem was later addressed in
more general cases, subsequently accounting for spins and quadrupole
moments \cite[and reference therein]{ba+oc75}. Here, we take the
further step to consider a cosmological constant.
The paper is as follows. In section~\ref{sec:field}, we discuss the
gravitational weak field limit in presence of a
cosmological constant and introduce the relevant approximations.
Section~\ref{sec:EIHeq} presents the generalization of the EIH equations of motion,
whereas section~\ref{sec:twobo} is devoted to the
study of the two-body problem. In section~\ref{sec:obser}, we review
how measurements of precession of pericentre in stellar system can
constrain $\Lambda$. In particular, we consider binary pulsars and the
solar system. Section~\ref{sec:concl} contains some final
considerations.
\section{Field equations with a cosmological constant in post-Newtonian approximation}
\label{sec:field}
Einstein's equations with the cosmological constant are
\begin{equation}
R_{\mu \nu} - \Lambda g_{\mu \nu} = \frac{8 \pi G}{c^4} S_{\mu \nu} ,
\end{equation}
where $G$ is the gravitational constant, $c$ the vacuum speed of light
and
\begin{equation}
S_{\mu \nu} \equiv T_{\mu \nu} - \frac{1}{2} g_{\mu \nu}
T^{\lambda}_{\lambda},
\end{equation}
with $ T_{\mu \nu}$ being the energy-momentum tensor. The weak field expansion can start by introducing a nearly Lorentzian
system for weak, quasi-stationary fields, in which
\begin{equation}
g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu}, \ \ \ |h_{\mu \nu}| \ll 1.
\end{equation}
Actually, the Minkowski metric $\eta_{\mu \nu}$ is not a vacuum
solution of the field equations with a cosmological constant, but for
$ | \Lambda | \ll 1$ an approximate solution in a finite region can
still be found by an expansion around $\eta_{\mu \nu}$. In the
post-Newtonian (pN) approximation, metric components can be expanded
in powers of
\begin{equation}
\varepsilon \sim \left( \frac{G M}{c^2 R} \right)^{1/2} \sim \frac{v}{c} \sim
\frac{p}{\rho},
\end{equation}
where $M$, $R$, $v$, $p$ and $\rho$ represent typical values for the mass,
length, velocity, pressure and energy density of the system, respectively. In what follows, $^{(n)}g_{\mu \nu}$ and $^{(n)}T_{\mu \nu}$ will
denote terms of order $\varepsilon^n$ and $\varepsilon^n (M/R)$,
respectively. To perform a proper treatment in presence of a $\Lambda$-term we have to consider the suitable approximation order for $\Lambda$. We assume that the size of the contributions due to the cosmological constant is at most comparable to the post-Newtonian terms, i.e, ${\cal{O}}(\Lambda g_{00}) \gs {\cal{O}} (G\ ^{(2)} T^{00}/c^2)$. This condition can be rewritten as
\begin{equation}
\label{fie1a}
\Lambda \ls \frac{R_\mathrm{g}^2}{R^4} ,
\end{equation}
where $R_\mathrm{g} \equiv G M/c^2$ is the gravitational radius. Eq.~(\ref{fie1a}) is easily satisfied by gravitational bound systems with $M \sim M_\odot$ and $ R\sim 1-10^2~\mathrm{AU}$ if $\Lambda \ls 10^{-33}~\mathrm{km}$, a value well above the estimated one from cosmological constraints and also greater than the limits we will derive in section \ref{sec:obser} considering stellar systems. Hereafter, we will put $c=1$. With such an approximation order, we can use classical results within the standard pN gauge. Following
\citet{str04}, the approximate field equations read
\begin{eqnarray}
\Delta^{(2)} g_{00} & = & -8 \pi G \ ^{(0)} T^{00},
\\
\Delta^{(4)} g_{00} & = & \ ^{(2)}g_{ij}\ ^{(2)}g_{00,ij}+
^{(2)}g_{ij,j}\ ^{(2)} g_{00,i}
- \frac{1}{2}\ ^{(2)} g_{00,i}\ ^{(2)} g_{00,i}
- \frac{1}{2}\ ^{(2)} g_{00,i}\ ^{(2)} g_{jj,i}
\\
& -& 8 \pi G \left(\ ^{(2)} T^{00}-2^{\ (2)}g_{00}\ ^{(0)}T^{00}
+^{\ (2)}T^{ii} \right) +2 \Lambda , \nonumber
\\
\Delta^{(3)} g_{oi} & = & - \frac{1}{2}\ ^{(2)}g_{jj,0i}+
^{\ (2)}g_{ij,0j} + 16 \pi G ^{\ (1)} T^{i0} ,
\\
\Delta^{(2)} g_{ij} & = & - 8 \pi G \delta_{ij} \ ^{(0)} T^{00} .
\end{eqnarray}
The components of the metric can be expressed in terms of potentials.
Let $\phi_\mathrm{N}$ be the Newtonian potential,
\begin{equation}
\phi_\mathrm{N} = -G \int \frac{^{(0)} T^{00} (t,\bfx^{'})}{|\bfx-\bfx^{'}|}d^3 x^{i} .
\end{equation}
According to our approximation order, the cosmological constant
appears only in the equation for $^{(4)} g_{00}$. This can be
re-arranged to give
\begin{equation}
\Delta^{(4)} ( g_{00} +2 \phi_\mathrm{N}^2 )= -8 \pi G \left(\ ^{(2)}
T^{00} + ^{(2)} T^{ii} \right) +2 \Lambda
\end{equation}
Together with the classical pN potential $\psi$,
\begin{equation}
\psi = -G \int \frac{d^3 x^{'}}{|\bfx-\bfx^{'}|} \left( \ ^{(2)}T^{00}+
^{(2)} T^{ii} \right) ,
\end{equation}
we introduce $\phi_\Lambda$, solution of
\begin{equation}
\Delta \phi_\Lambda = -\Lambda .
\end{equation}
In presence of a cosmological constant, there is an upper limit on the
maximum distance within which the Newtonian limit holds and boundary
conditions must then be chosen at a finite range \citep{now01}. When
these boundary conditions are chosen on a sphere whose origin
coincides with the origin of the coordinate system, $\phi_\Lambda$ can
be expressed as
\begin{equation}
\phi_\Lambda = -\frac{1}{6} \Lambda |\bfx|^2 ,
\end{equation}
where we have neglected correction terms which appear because of
boundary conditions. Due to a positive cosmological constant, the origin of the coordinate system has a distinguished dynamical role with a radial force directed away from it
\cite{adl+al65}. Since the choice of the origin is arbitrary, any point in the
space will experience repulsion
from any other point. Finally, introducing the standard pN potentials,
\begin{eqnarray}
\xi_i & = & - 4 G \int \frac{d^3 x^{'}}{|\bfx-\bfx^{'}|}\ ^{(1)} T^{i0}
(t,\bfx^{'}), \\
\chi & = & - \frac{G}{2} \int |\bfx-\bfx^{'}| ^{(0)} T^{00} d^3 x^{'}
(t,\bfx^{'}),
\end{eqnarray}
the metric components read
\begin{eqnarray}
^{(2)}g_{00} & = & -2 \phi_\mathrm{N} , \label{metr1} \\
^{(4)}g_{00} & = & -2 ( \phi_\mathrm{N}^2 +\psi + \phi_\Lambda ), \label{metr2} \\
^{(2)}g_{ij} & = & -2 \delta_{ij} \phi_\mathrm{N} , \label{metr3} \\
^{(3)}g_{0i} & = & \xi_i +\chi_{,i0} . \label{metr4}
\end{eqnarray}
For a point-like mass at the centre of the coordinate system, the
above expressions reduce to the weak field limit at large radii of the
Kottler space-time.
\subsection{Equations of motion for a test particle}
The motion of a particle in an external gravitational field can be
described by the Lagrangian
\begin{equation}
{\cal L}= 1- \sqrt{ - g_{\mu \nu} \left( \frac{d x^{\mu}}{dt} \right)
\left( \frac{d x^{\nu}}{dt} \right)}.
\end{equation}
Using the metric components in equations~(\ref{metr1}-\ref{metr4}), we get
\begin{equation}
{\cal {L}} \simeq \frac{1}{2} v^2+ \frac{1}{8} v^4 -\phi_\mathrm{N} -\frac{1}{2}
\phi_\mathrm{N}^2 -\psi -\phi_\Lambda -\frac{3}{2}\phi_\mathrm{N} v^2 +v^i \left( \xi_i +
\frac{\partial \chi}{\partial t \partial x^i} \right) .
\end{equation}
The corresponding Euler-Lagrange equations of motion in a
3-dimensional notation read,
\begin{equation}
\frac{d {\bf v}}{dt} \simeq -\nabla \left( \phi_\mathrm{N} +2\phi_\mathrm{N}^2+\psi \right) -
\frac{\partial \bfxi}{\partial t} - \frac{\partial^2 }{\partial t^2} \nabla \chi +
\bfv {\times} (\nabla {\times} \bfxi) + 3 \bfv \frac{\partial \phi_\mathrm{N}}{\partial t}
+4 \bfv (\bfv {\cdot} \nabla)\phi_\mathrm{N}-\bfv^ 2\nabla \phi_\mathrm{N} +\frac{\Lambda}{3}
\bfx .
\end{equation}
The above expression reduces to equation (20) in \cite{ker+al03} when
neglecting pN corrections.
\section{The Einstein-Infeld-Hoffmann equations}
\label{sec:EIHeq}
Since the contribution from the cosmological constant is of higher-order,
it does not couple with other corrections. The Lagrangian of an $N$-body
system of point-like particles can be written as
\begin{equation}
{\cal L} \simeq {\cal L}_{(\Lambda =0)} + \delta{\cal L}_\Lambda,
\end{equation}
where ${\cal L}_{\Lambda =0}$ is the total Lagrangian in absence of
$\Lambda$. The Lagrangian ${\cal L}_a $ of particle $a$ in the field of
other particles is
\begin{equation}
{\cal L}_a \simeq {\cal L}_{a(\Lambda =0)} +\frac{\Lambda}{6} \bfx_a^2,
\end{equation}
where ${\cal L}_{a(\Lambda =0)}$ is given in equation (5.94) in
\cite{str04} The total Lagrangian reads
\begin{equation}
{\cal L} \simeq {\cal L}_{(\Lambda =0)} + \sum_a \frac{\Lambda}{6} m_a
\bfx_a^2,
\end{equation}
with ${\cal L}_{(\Lambda =0)}$ given in \cite[equation~(5.95)]{str04}.
The corresponding Euler-Lagrange equations are the Einstein-Infeld-Hoffmann equations corrected for a $\Lambda$ term,
\begin{equation}
\dot{\bfv}_a = - G \sum_{b \neq a} \left( \frac{\bfx_{ab}}{r_{ab}} \right)+
\delta \mathbf{F}_{\mathrm{pN} (\Lambda=0)} + \frac{\Lambda}{3} \bfx_a
\end{equation}
where $\mathbf{F}_{\mathrm{pN} (\Lambda=0)}$ is the post-Newtonian perturbing
function \cite[equation~(5.96)]{str04}.
\section{The two-body problem}
\label{sec:twobo}
The total Lagrangian for two particles can be written as
\begin{equation}
{\cal L} \simeq \frac{1}{2}m_\mathrm{a} v_\mathrm{a}^2 +G \frac{m_\mathrm{a} m_\mathrm{b}}{x}+ \frac{1}{2}m_\mathrm{b}
v_\mathrm{b}^2 + \delta {\cal L}_{\mathrm{pN} (\Lambda=0)} + \delta {\cal L}_{\Lambda}
\end{equation}
where $\bfx \equiv \bfx_\mathrm{a} - \bfx_\mathrm{b} $ is the separation vector and $\delta {\cal L}_{\mathrm{pN} (\Lambda=0)}$ and $\delta {\cal L}_{\Lambda}$ are the pN and $\Lambda$-contributions, respectively. It is \cite{str04}
\begin{equation}
\delta {\cal
L}_{\mathrm{pN} (\Lambda=0)} = \frac{1}{8} (m_\mathrm{a} v_\mathrm{a}^4 + m_\mathrm{b} v_\mathrm{b}^4) + G
\frac{m_\mathrm{a} m_\mathrm{b}}{2 r} \left[ 3 (v_\mathrm{a}^2+v_\mathrm{b}^2) - 7 \bfv_\mathrm{a} \dot \bfv_\mathrm{b} -
(\bfv_\mathrm{a} \dot {\bf n})(\bfv_\mathrm{b} \dot {\bf n}) \right]
-\frac{G^2}{2}\frac{m_\mathrm{a} m_\mathrm{b} (m_\mathrm{a}+m_\mathrm{b})}{x^2}
\end{equation}
with ${\bf n} \equiv {\bf x}/x$ and
\begin{equation}
\delta {\cal L}_{\Lambda} = \frac{\Lambda}{6} (m_\mathrm{a} x_\mathrm{a}^2 + m_\mathrm{b} x_\mathrm{b}^2) .
\end{equation}
Due to cosmological constant, the energy of the system is modified by
a contribution $-\delta {\cal L}_{\Lambda}$. The pN and $\Lambda$
corrections are additive and can be treated separately. We are
interested in examining the effect of a non vanishing $\Lambda$ term.
Let us consider the centre of mass and relative motions. Introducing
${\bf X} \equiv \left( m_\mathrm{a} \bfx_\mathrm{a} +m_\mathrm{b}
\bfx_\mathrm{b} \right)/M $, with $M \equiv
m_\mathrm{a}+m_\mathrm{b}$, the Lagrangian can be re-written as
\begin{equation}
{\cal L} \simeq \frac{1}{2} M V^2 + \frac{\Lambda}{6} M X^2 +\frac{1}{2} \mu
v^2 + \frac{\Lambda}{6} \mu x^2 + G \frac{M \mu}{x},
\end{equation}
with $\mu \equiv m_\mathrm{a} m_\mathrm{b}/M$. Due to cosmological constant, the centre of mass of the
system is subject to an effective repulsive force given by $\Lambda {\bf X}/3$ per unit mass.
The equations for the relative motion are those of a test particle in
a Schwarzschild-de~Sitter space-time with a source mass equal to the
total mass of the two-body system. Since the perturbation due to
$\Lambda$ is radial, the orbital angular momentum is conserved and the
orbit is planar. The main effect of $\Lambda$ on the orbital motion is
a precession of the pericentre \cite[and references
therein]{kr+wh03,ker+al03}. Following the analysis of the Rung-Lenz
vector in \cite{ker+al03} and restoring the $c$ factors, we get for the contribution to the precession angular velocity due to $\Lambda$,
\begin{equation}
\dot{\omega}_\Lambda = \frac{\Lambda c^2 P_\mathrm{b}}{4 \pi}
\sqrt{1-e^2},
\end{equation}
where $e$ is the eccentricity and $P_\mathrm{b}$ the Keplerian period of
the unperturbed orbit. This contribution should be considered together with the post-Newtonian
periastron advance, $\dot{\omega}_{\mathrm{pN}} = 3 (2 \pi/P_\mathrm{b})^{5/3} (G
M/c^3)^{2/3} (1-e^2)^{-1} $. The ratio between these two contributions
can be written as,
\begin{equation}
\frac{ \dot{\omega}_\Lambda } {\dot{\omega}_{\mathrm{pN}}} =
\frac{\bar{R}}{R_\mathrm{g}} \frac{\rho_\Lambda }{ \rho }
= \frac{1}{6}\frac{\bar{R}^4}{R_\mathrm{g}^2} \Lambda ,
\end{equation}
where $\bar{R} = a (1-e^2)^{3/8}$ is a typical orbital radius with $a$ the semi-major radius of the unperturbed orbit, $\rho \equiv M/(4 \pi \bar{R}^3/3)$ is a typical density of the system and $\rho_\Lambda
\equiv c^2 \Lambda /8 \pi G $ is the energy density associated to
the cosmological constant. The effect of $\Lambda$ can be significant for
very wide systems with a very small mass.
\section{Observational constraints}
\label{sec:obser}
In this section, we derive observational limits on $\Lambda$ from
orbital precession shifts in stellar systems and in the solar system.
\subsection{Interplanetary measures}
\label{sec:obserA}
Precessions of the perihelia of the solar system planets have provided
the most sensitive local tests for a cosmological constant so far
\cite{isl83,wri98,ker+al03}. Estimates of the anomalous perihelion
advance were recently determined for Mercury, Earth and Mars \cite{pit05a,pit05b}. Such ephemerides were constructed integrating
the equation of motion for all planets, the Sun, the Moon and largest
asteroids and including rotations of the Earth and of the Moon,
perturbations from the solar quadrupole mass moment and asteroid ring
in the ecliptic plane. Extra-corrections to the known general relativistic predictions can
be interpreted in terms of a cosmological constant effect. We considered the 1-$\sigma$ upper bounds. Results are listed in Table~\ref{tab:plan}. Best constraints come from Earth and
Mars observations, with $\Lambda \ls 10^{-36}\mathrm{km}^{-2}$. Major sources of systematic errors come from uncertainties about solar oblateness and from the
gravito-magnetic contribution to secular advance of perihelion but
their effect could be in principle accounted for \citep{ior05}. In
particular, the general relativistic Lense-Thirring secular precession
of perihelia is compatible with the determined extra-precessions
\citep{ior05}. The accuracy in determining the planetary orbital
motions will further improve with data from space-missions like
BepiColombo, Messenger and Venus express. By considering a
post-Newtonian dynamics inclusive of gravito-magnetic terms, the
resulting residual extra-precessions should be reduced by several
orders of magnitude, greatly improving the upper bound on $\Lambda$.
The orbital motion of laser-ranged satellites around the Earth has been also considered to confirm general relativistic predictions. Observations of the rates of change of the nodal longitude of the LAGEOS satellites allowed to probe the Lense-Thirring effect with an accuracy of $\sim10\%$, i.e. about half a milliarcsecond per year \cite{ci+pa04}. Other proposed missions, such as the LARES/WEBER-SAT satellite \cite{ciu04}, should further increase this experimental precision. In general, since effects of $\Lambda$ become significant only for very dilute systems, even very accurate measurements of orbital elements of Earth's satellites can not help in constraining the cosmological constant. For a satellite with a typical orbital semi-major axis of about 12,000~km, in order to get a bound on $\Lambda$ as accurate as those inferred from Earth and Mars perihelion shifts (i.e. $\Lambda \ls 10^{-36}~\mathrm{km}^{-2}$), changes in orbital elements should be measured with a today unattainable precision of a few tens of picoseconds of arc per year, about six order of magnitude better than today accuracy.
\begin{table}
\caption{\label{tab:plan} Limits on the cosmological constant
due to extra-precession of the inner planets of the solar system.}
\begin{ruledtabular}
\begin{tabular}{lrrr}
Name & $\delta\dot{\omega}\footnotemark[1]$ (arcsec/year)&
$\dot{\omega}_\Lambda$ ($\deg$/year)& $\Lambda_\mathrm{lim}~(\mathrm{km}^{-2}) $ \\
\hline
Mercury & $-0.36(50)\times 10^{-4}$ & $9.61{\times} 10^{25} \Lambda /(1~\mathrm{km}^{-2})$ &
$4 {\times} 10^{-35}$
\\
Venus & $0.53(30)\times 10^{-2}$ & $2.51{\times} 10^{26}\Lambda /(1~\mathrm{km}^{-2})$ &
$9 {\times} 10^{-33}$
\\
Earth & $-0.2(4)\times 10^{-5}$ & $4.08{\times} 10^{26}\Lambda /(1~\mathrm{km}^{-2})$ & $1{\times}
10^{-36}$
\\
Mars & $0.1(5)\times 10^{-5}$ & $7.64{\times} 10^{26} \Lambda /(1~\mathrm{km}^{-2})$& $ 2{\times}
10^{-36}$
\\
\end{tabular}
\end{ruledtabular}
\footnotetext[1]{From \cite{pit05a}.}
\end{table}
\subsection{Binary pulsars}
\begin{table}
\caption{\label{tab:1} Binary pulsars with known post-Keplerian parameter
$\dot{\omega}$ and corresponding limits on the cosmological constant.
The identification of the companion is often uncertain. We refer to
the original papers for a complete discussion.}
\begin{ruledtabular}
\begin{tabular}{lllllcc}
PSR Name & $P_\mathrm{b}$ (days) & $e$ & $\dot{\omega}$ ($\deg$/year)&
$\dot{\omega}_\Lambda$ ($\deg$/year)& $ \Lambda_\mathrm{lim}~(\mathrm{km}^{-2})$ & ref.\\
\hline
\multicolumn{7}{c}{Double neutron star binaries} \\
\hline
J1518+4904 & 8.634000485 & 0.2494849 & 0.0111(2) &
$9.335 {\times} 10^{24}\Lambda /(1~\mathrm{km}^{-2})$ & $2 {\times} 10^{-29}$ &
\cite{nic+al96}
\\
B1534+12 & 0.2736767 & 0.420737299153 &
1.755805(3) & $2.772 {\times} 10^{23}\Lambda /(1~\mathrm{km}^{-2})$ & $1 {\times}
10^{-29}$ & \cite{wil05} \\
B1913+16 & 0.323 &
0.617 & 4.226595(5) & $2.838{\times} 10^{23} \Lambda /(1~\mathrm{km}^{-2})$ &
$2
{\times} 10^{-29}$ & \cite{wil05}
\\
J1756-2251 & 0.319633898 & 0.180567 & 2.585(2) & $3.510
{\times} 10^{23} \Lambda /(1~\mathrm{km}^{-2})$ & $6 {\times} 10^{-27}$ & \cite{fau+al05}
\\
J1811-1736 & 18.779168 & 0.82802 & 0.009(2) &
$1.176{\times} 10^{25} \Lambda /(1~\mathrm{km}^{-2})$ & $2 {\times} 10^{-28}$ & \cite{lyn+al00}
\\
J1829+2456 & 1.176028 & 0.13914 & 0.28(1) & $1.300 {\times}
10^{24} \Lambda /(1~\mathrm{km}^{-2})$ & $ 8 {\times} 10^{-27}$ & \cite{cha+al04}
\\
B2127+11C & 0.68141 & 0.335282052 & 4.457(12) &
$7.168{\times} 10^{23}\Lambda /(1~\mathrm{km}^{-2})$ & $2{\times} 10^{-26}$ & \cite{wil05}
\\
B2303+46 & 12.34 & 0.65837 & 0.01019(13) & $1.037 {\times}
10^{25} \Lambda /(1~\mathrm{km}^{-2}) $ & $1 {\times} 10^{-29}$ & \cite{th+ch99} \\
\hline
\multicolumn{7}{c}{Neutron star/white dwarf binaries} \\
\hline
J0621+1002 & 8.3186813 & 0.00245744 & 0.0116(8) &
$9.288 {\times} 10^{24} \Lambda /(1~\mathrm{km}^{-2})$ & $9 {\times} 10^{-29}$ & \cite{spl+al02}
\\
J1141-6545 & 0.171876 & 0.1976509587 & 5.3084(9) & $1.881
{\times} 10^{23}\Lambda /(1~\mathrm{km}^{-2}) $ & $5 {\times} 10^{-27}$ & \cite{wil05}
\\
J1713+0747 & 67.82512987 & 0.0000749406 &
0.0006(4)\footnotemark[1]
& $7.573{\times} 10^{25} \Lambda /(1~\mathrm{km}^{-2}) $ & $8 {\times} 10^{-30}$ &
\cite{spl+al05}
\\
B1802-07 & 2.617 & 0.212 & 0.0578(16) &
$2.856{\times} 10^{24}\Lambda /(1~\mathrm{km}^{-2}) $ & $ 6 {\times} 10^{-28}$ &
\cite{th+ch99}
\\
J1906+0746 & 0.085303(2) & 0.165993045(8) & 7.57(3) &
$9.392{\times}10^{22} \Lambda /(1~\mathrm{km}^{-2})$ & $ 3 {\times} 10^{-25}$ & \cite{lor+al05}
\\
\hline
\multicolumn{7}{c}{Double pulsars} \\
\hline
J0737-3039 & 0.087779 & 0.102251563 & 16.90(1) &
$9.750{\times} 10^{22} \Lambda /(1~\mathrm{km}^{-2}) $ & $1 {\times} 10^{-25}$ & \cite{wil05}
\\
\hline
\multicolumn{7}{c}{Unknown companion}
\\
\hline
B1820-11 & 357.7622(3) & 0.79462(1) & 0.01\footnotemark[1] &
$2.425 {\times} 10^{26} \Lambda /(1~\mathrm{km}^{-2}) $ & $4 {\times} 10^{-29}$ & \cite{ly+mc89}
\\
\end{tabular}
\end{ruledtabular}
\footnotetext[1]{Upper limit}
\end{table}
Binary pulsars have been providing unique possibilities of probing
gravitational theories. Relativistic corrections to the binary
equations of motion can be parameterized in terms of post-Keplerian
parameters \cite{wil93}. As seen before, the advance of periastron of
the orbit, $\dot{\omega}$, depends on the total mass of the system and
on the cosmological constant. In principle, because Keplerian orbital parameters
such as the eccentricity $e$ and the orbital period $P_\mathrm{b}$ can
be separately measured, the measurement of $\dot{\omega}$ together
with any two other post-Keplerian parameters would provide three
constraints on the two unknown masses and on the cosmological
constant. As a matter of fact for real systems, the effect of $\Lambda$ is much smaller
than $\dot{\omega}_{\mathrm{pN}}$, so that only upper bounds on the
cosmological constant can be obtained by considering the uncertainty
on the observed periastron shift. We considered binary systems with
measured periastron shift, see Table~\ref{tab:1}. The effect of $\Lambda$
is maximum for B1820-11 and PSR J1713+0747. Despite of the low accuracy in the
measurement of $\dot{\omega}$, PSR J1713+0747 provides the best
constraint on the cosmological constant, $\Lambda \ls 8
{\times}10^{-30}\mathrm{km}^{-2}$. Uncertainties as low as $\delta\dot\omega
\gs 10^{-6}$ have been achieved for very well observed systems, such
as B1913+16 and B1534+12. Such an accuracy for B1820-11 would allow
to push the bound on $\Lambda$ down to $10^{-33}\mathrm{km}^{-2}$.
Better constraints could be obtained by determining post-Keplerian
parameters in very wide binary pulsars. We examined systems with known
period and eccentricity as reported in \cite{lor05}. The binary pulsar
having the most favourable orbital properties for better constraining
$\Lambda$ is the low eccentricity B0820+02, located in the Galactic
disk, with $\dot{\omega}_\Lambda
\sim 1.4{\times}10^{27}\Lambda / (1~\mathrm{km}^{-2})\deg/\mathrm{days}$. For binary pulsars J0407+1607, B1259-63, J1638-4715 and J2016+1948,
the advance of periastron due to the cosmological constant is between
7 and $9{\times}10^{26}\Lambda/ (1~\mathrm{km}^{-2}) \deg/\mathrm{days}$. All of these shifts are
of similar value or better than the Mars one. A determination of
$\dot{\omega}$ for B0820+02 with the accuracy obtained for B1913+16, i.e.
$\delta \dot{\omega}\gs~10^{-6} \deg/\mathrm{days}$ would allow to
push the upper bound down to $ 10^{-34}-10^{-33}\mathrm{km}^{-2}$.
\section{Conclusions}
\label{sec:concl}
We considered the $N$-body equations of motion in presence of a
cosmological constant. The impact of $\Lambda$ on the two-body system
was explicitly derived. Due to the anti-gravity effect of the
cosmological constant, the barycentre of the system drifts away. The
relative motion is like that of a test particle in a Schwarzschild-de
Sitter space-time with a source mass equal to the total mass of the
two-body system. The main effect of $\Lambda$ is the precession of the
pericentre on the orbital motion.
We determined observational limits on the cosmological constant
from measured periastron shifts. With respect to previous
similar analyses performed in the past on solar system planets,
our estimate was based on a recent determination of the planetary
ephemerides properly accounting for the quadrupole moment of the Sun
and for major asteroids. The best constraint comes from Mars and Earth,
$\Lambda \ls 1-2\times 10^{-36}\mathrm{km}^{-2}$.
Due to the experimental accuracy, observational limits on
$\Lambda$ from binary pulsars are still not competitive with
results from interplanetary measurements in the solar system. Accurate
pericentre advance measurements in wide systems with orbital periods
$\gs 600~\mathrm{days}$ could give an upper bound of $\Lambda \ls
10^{-34}-10^{-33}\mathrm{km}^{-2}$, if determined with the accuracy performed for B1913+16, i.e.
$\delta\dot\omega \gs 10^{-6}\deg/\mathrm{years}$. For some binary pulsars, observations with an accuracy comparable to that achieved in the solar system could allow to get an upper limit on $\Lambda$ as precise as one obtains from Mars data.
The bound on $\Lambda$ from Earth or Mars perihelion shift is nearly $\sim
10^{10}$ times weaker than the determination from observational cosmology, $\Lambda
\sim 10^{-46}\mathrm{km}^{-2}$, but it still gets some relevance. The
cosmological constant might be the non perturbative trace of some
quantum gravity aspect in the low energy limit \cite{pad05}. $\Lambda$
is usually related to the vacuum energy density, whose properties
depends on the scale at which it is probed \cite{pad05}. So that, in
our opinion, it is still interesting to probe $\Lambda$ on a scale of
order of astronomical unit. Measurements of periastron shift should be
much better in the next years. New data from space-missions should get
a very high accuracy and might probe spin effects on the orbital
motion \cite{oco04,ior05}. A proper consideration of the
gravito-magnetic effect in these analyses plays a central role to
improve the limit on $\Lambda$ by several orders of magnitude.
{\it Note added}. After submission of this work, L.~Iorio \cite{ior05b} presented an analysis of solar system data similar to our results in section~\ref{sec:obserA}.
\begin{acknowledgments}
The authors thank N. Straumann for stimulating discussions. M.S. is
supported by the Swiss National Science Foundation and by the Tomalla
Foundation.
\end{acknowledgments}
|
Title:
The Vertical Structure of the Outer Milky Way HI Disk |
Abstract: We examine the outer Galactic HI disk for deviations from the b=0 plane by
constructing maps of disk surface density, mean height, and thickness. We find
that the Galactic warp is well described by a vertical offset plus two Fourier
modes of frequency 1 and 2, all of which grow with Galactocentric radius.
Adding the m=2 mode accounts for the large asymmetry between the northern and
southern warps. We use a Morlet wavelet transform to investigate the spatial
and frequency localization of higher frequency modes; these modes are often
referred to as "scalloping." We find that the m=10 and 15 scalloping modes are
well above the noise, but localized; this suggests that the scalloping does not
pervade the whole disk, but only local regions.
| https://export.arxiv.org/pdf/astro-ph/0601697 |
\title{The Vertical Structure of the Outer Milky Way HI Disk}
\author{E.S. Levine\altaffilmark{1}, Leo Blitz\altaffilmark{1}, and Carl Heiles\altaffilmark{1}
}
\affil{Department of Astronomy, University of California at Berkeley, Mail Code
3411, Berkeley, CA 94720 USA}
\email{elevine@astron.berkeley.edu}
\keywords{Galaxy: disk --- Galaxy: structure --- Galaxy: kinematics and dynamics --- ISM: structure --- radio lines: general}
\section{Introduction}
Although the topography of the gas disk of the Milky Way has previously been mapped \citep[and many others]{W1957, HJK1982,BT1986}, its complete Fourier structure has never been quantitatively described, though \citet{BM1998} have approximated its three lowest frequency terms.
Which spatial oscillation frequencies are most important?
Is the scalloping a local or global effect?
This paper constitutes an in depth analysis of the shape of the outer HI disk, as well as the first quantitative analysis of the scalloping.
The large-scale warp in the gas disk of the Milky Way has been known since 1957 \citep{B1957,K1957,W1957,KHC1957}.
The warp has a large amplitude, rising to a height greater than 4 kpc at a Galactocentric radius of 25 kpc in the northern data. It is also asymmetric; in the south the gas falls about 1 kpc below the plane before rising back to it. By observing other galaxies, it is possible to develop a general understanding of how warps behave.
\citet{B1991} found that at minimum half of all galaxies are warped, and that galaxies with smaller dark matter halo core radii are less likely to be warped.
\citet{B1990} claimed that a warp's line of nodes starts out straight, and at a transition begins to advance in the direction of rotation, with some exceptions.
Another survey has shown that warps are common in galaxies with HI disks that are extended compared to their optical components, and are often asymmetric \citep{GSK2002}. In the Milky Way, many components other than HI also participate in the warp. A partial list includes dust \citep{FBD1994}, CO \citep{WBB1990}, solar neighborhood stars \citep{D1998}, and IRAS point sources \citep{DS1989}.
Many efforts have been directed toward understanding the warp on a theoretical basis. Bending modes have long been suspected as the mechanism creating and maintaining the warp. Early work studying the evolution of bending mode oscillations was hampered by a lack of knowledge regarding galactic halos, but showed that the shape of the density falloff near the edge of the disk plays an important role in the stability of bending modes \citep{HT1969,T1983}.
The distribution of matter in the halo controls the ability of the disk to sustain long-lived bending wave warps, so studying the properties of the warp will tell us about the shape of the halo \citep{B1978,S1984}. \citet{SC1988} argued that bending mode oscillations are plausible when the halo and the self-gravity of the disk are taken into account, but \citet{BJD1998} showed that in this situation the warp will wind up within a few dynamical times. Previous work has largely been concerned with the existence and behavior of $m=1$ warps only, though \citet{S1995} demonstrated the stability of $m=0$ modes in an axisymmetric halo.
Several other mechanisms have been suggested as possibilities for creating and maintaining a warp \citep{KG2000}. Gravitational interaction with satellites such as the Magellanic Clouds is a promising candidate \citep{B1957,W1998,WB2005}, but there is a longstanding debate of whether tidal effects are strong enough to produce the observed effect \citep{K1957}. Indeed, even in galaxies with companions similar to the LMC, tidal amplification may not be strong enough to account for the size of the warp \citep{GKD2002}.
Accretion of matter onto the halo is another plausible cause \citep{JB1999}, or matter can accrete directly onto onto the disk and torque the gas orbits \citep{LBB2002,S2004}. Intergalactic magnetic fields can act on a slightly ionized gas disk to produce a warp \citep{BFS1990}, or the intergalactic medium can excite a warp through a wind \citep{KW1959}.
Throughout this paper, we will refer to shorter wavelength (5--25 kpc) oscillations in azimuth as scalloping. \citet{GKW1960} first noticed a ``waviness'' in the gas layer of the inner Galaxy; observational evidence for scalloping turns out to be prominent in the outer Galaxy \citep{HJK1982,KBH1982}. \citet{F1983} investigated the possibility that the Milky Way scalloping results from the Kelvin-Helmholz instability, and \citet{S1995} suggested that the scalloping should only be present in the outer parts of Galactic disks. \citet{SF1986} find azimuthal corrugations in HI and other components along the spiral arms after removing the warp. In an N-Body simulation, \citet{EE1997} found evidence for spiral corrugations due to gravitational interaction with a satellite galaxy.
In this paper we will only investigate oscillation in Galactocentric azimuth, and not in radius.
In \S 2 we describe the method of transforming the Heliocentric data cube into Galactocentric coordinates, and present maps of the surface density, average height off the $b=0\degr$ plane, and vertical thickness of the outer Galaxy. While these maps are constructed using new data, they are not significantly different from previous work. In \S 3 we perform global and local analyses on these maps to better understand the warp and the scalloping.
\section{Method}\label{sec:proc}
\subsection{Data Processing}
We use the 21 cm Leiden/Argentine/Bonn (LAB) data \citep{LAB,HB1997,BALMPK2005,ABLMP2000} to conduct a quantitative study of the warp and scalloping. The LAB survey is a combination of the LDS data set \citep{HB1997} with Southern sky observations from the IAR \citep{ABLMP2000}; however, much attention has been paid toward ensuring a uniform data set. The data are corrected for stray radiation. The combined survey maps the entire sky within $-450\le v_r\le 400$ km s$^{-1}$ with a resolution of 1.3 km s$^{-1}$; this velocity range includes all of the gas in the Galaxy in circular rotation. We used the Hanning smoothed data, which have a velocity resolution of 1.9 km s$^{-1}$, and used only data with $|b|\le30\degr$. The signal of the warp in the north can be weakly traced beyond this elevation limit \citep{B1985}, but the vast majority of the warp signal is included within our $b$ range.
The LAB survey contains a large number of angularly small emission features, especially at high latitudes, which are not contiguous with the gas in the disk, and may not even be in circular rotation. Features like these are particularly troublesome at larger Galactic latitudes since they can contain enough gas to contaminate our calculations, especially at large Galactocentric radii. These objects are removed with a median filter so we can focus on the gas in the disk. Points with brightness temperature $T_b\le 0$ are temporarily filled with the value $0.01$ K for this filtering only. We then cycle through the LAB data cube and
calculate the median of each point and its 12 nearest neighbors in $\ell$ and $b$ at the same line-of-sight velocity $v_r$ (a two dimensional diamond shaped filter); call this median $T_m(\ell,b,v_r)$. Any point with $T_b>10~T_m$ is replaced with 10 $T_m$. Two examples of objects caught by this filter are M31 and NGC 6822.
The survey grid in $\ell,b,$ and $v_r$ is not convenient for analyzing Galactic properties. Ideally, our measurements would be equally spaced in the Galactocentric cylindrical coordinates $R,\phi,$ and $z$; we must interpolate a Galactocentric grid from the LSR-centered data. The Galactic azimuth $\phi$ is defined such that it converges with galactic longitude $\ell$ at large $R$. To convert from observed $\ell,b,$ and $v_r$ to $R,\phi$, and $z$ we use the following mapping functions:
\begin{eqnarray}\label{eqn:transform}
\ell&=&\sin^{-1}\left[\frac{R}{r'}\sin\phi\right]\nonumber\\
b&=&\tan^{-1}\frac{z}{r'}\nonumber\\
v_r&=&\sin\ell\cos b\left[\frac{R_0}{R}\Theta(R)-\Theta_0\right]\nonumber\\
&&+v_\Pi(R)\cos\phi\left(1-\frac{R_0^2}{R^2}\sin^2\ell\right)^{1/2}\cos b.
\end{eqnarray}
Here, $\Theta(R)$ is the Galactic rotation curve, which we assume to be 220 km s$^{-1}$ everywhere \citep{BB1993}. $\Theta_0$ and $R_0$ are 220 km s$^{-1}$ and 8.5 kpc, respectively. $\bf{r}$ is the vector connecting the Sun's location to the point under investigation; $\bf{r'}$ is the projection of this vector onto the plane of the disk (with magnitude $r'$).
Equation \ref{eqn:transform} is the transformation that results from an assumption of elliptical gas orbits with major axis
along the $\phi=90\degr, 270\degr$ line. To derive this, we assume the gas moves on the orbit:
\begin{eqnarray}\label{eqn:ellipse}
v_\phi&=&220~\mathrm{km~s^{-1}}\nonumber\\
v_R&=&v_\Pi(R)\cos\phi.
\end{eqnarray}
$v_R$ is the magnitude of the velocity in the Galactocentric radial direction. $v_\Pi$ is a parameterization of the ellipticity of the orbit, which is free to vary with Galactocentric radius; we discuss our method of calculating $v_\Pi(R)$ in the Appendix. At all points, $v_R/v_\phi<0.1$. These equations are simply the epicyclic approximation for an orbit with epicyclic frequency 1 and the angle of ellipse orientation fixed. Although the gas orbits in the Galaxy are not likely to correspond to the fixed ellipse orientation we describe, this configuration minimizes the correction to $v_r$ for gas far from $\ell=0\degr$ or $180\degr$ (see \citet{V1999} for more detail on elliptical gas orbits).
Without a correction for radial motion of the gas, there is a large asymmetry between the surface densities at Galactic longitudes on either side of $\ell =0\degr$ and $\ell =180\degr$ \citep{HJK1982}. This must be taken into account, or features in these two regions such as the surface density will appear discontinuous and distorted. Assuming an outward velocity for the Local Standard of Rest (LSR) will correct this discontinuity to some degree \citep{K1962,KW1965}. However, it seems that the best fit for the motion relative to the LSR changes with radius, implying that the effect is global, rather than local \citep{BS1991}. To reduce the magnitude of the discontinuity using gas orbits, one can use a Galactocentric radial velocity roughly of the form $\cos\phi$ (or $\cos\ell$) \citep{KT1994}.
We exclude all points that lie within $345\degr\le\ell\le15\degr$ or $165\degr\le\ell\le195\degr$. Points in these two wedges have velocities along the line of sight that are too small with respect to their random velocities to establish reliable distances. All points in this region are set to $T_b = 0$.
Using (\ref{eqn:transform}) we construct a Galactocentric grid $T_b(R,\phi,z)$ by trilinear-interpolating from the grid $T_b(\ell,b,v_r)$. We do this by calculating the coordinates of an $(R,\phi,z)$ point in $(\ell,b,v_r)$ space, and interpolating from $T_b(\ell,b,v_r)$. The resolution of the Galactocentric grid is set by the spacing of the LSR centered grid, but in this paper we are not interested in small-scale disk structure. A grid of 100 points in 10 kpc $\le R \le$ 30 kpc, 350 points in $-\pi\le\phi\le \pi$, and 141 points in -20 kpc$\le z \le$ 20 kpc gives us sufficient resolution to answer the questions we are interested in.
Undersampled grids fail to utilize all of the information in the data; our choice of grid spacing is both is an undersampling and an oversampling of the information in $T_b(\ell,b,v_r)$ depending on the position in the disk. Consider two points near $R=10$ kpc and $\ell = 15\degr$, where we have poor Galactocentric resolution in $\phi$. The LAB survey has $\Delta \ell = 0.5\degr$; at this location this corresponds to $\Delta\phi\approx0.9\degr$. The spacing in our Galactocentric grid is larger than $1\degr$, thus we are undersampled everywhere in the $\phi$ dimension. In \S \ref{sec:lomb} we will perform an azimuthal frequency analysis of each ring; none of the frequencies we examine approach the Nyquist frequency of the data. Near the midplane at $R=30$ kpc and $\ell =15\degr$, lines of constant $b$ are separated by around 300 pc, and near the top of the grid they are separated by around 400 pc; there is some oversampling in the $z$ dimension by no more than a factor of 2. The most severe case of oversampling is in the $R$ dimension, where along $\ell = 15\degr$ we have only 14 velocity resolution elements in our $R$ range for the 100 grid points. On the other hand, at $\ell=90\degr$, there are more than 60 resolution elements in our range. We have chosen a grid spacing in $R$ that oversamples to a varying degree depending on location in the disk.
From $T_b(R,\phi,z)$, we can recover $\rho(R,\phi,z)$ using the method outlined in \citet{K1968}. We assume $T_s$, the spin temperature, is 155 K everywhere. This is slightly higher than the maximum brightness temperature found in the LAB survey in the region we are concerned with. Choosing a larger number to force the optically thin limit ($T_b\ll T_s$) makes a difference only in a small number of areas in the inner radii of our grid. The vast majority of the points are optically thin with any reasonable $T_s$, and are not affected by this choice.
The transformation to a density grid depends on $|dv_r/dr|$. Using elliptical orbits makes calculating this quantity slightly more difficult than with a flat rotation curve. Although the full derivative can be written analytically, we just calculate it numerically.
Points with $T_b<0$ are set to $\rho =0$.
\subsection{Surface Density and Mean Height maps}
The grid $\rho(R,\phi,z)$ contains information about the density of HI in the Galaxy, minus whatever has been removed by the median filter and the excluded regions. Previous studies have proceeded by calculating a mean height $\bar{z}(R,\phi)$ for the gas. However, the Galaxy is a complicated place that contains a variety of HI structures in addition to the disk. In particular, there are many extended clouds located near the disk as well as spurs that split off from the disk. None of these will have been removed by the median filter, which acts only on comparatively small areas of the sky. \citet{V1999} developed an alternative method to remove some of these features; he masked out a map of high velocity cloud complexes.
Since we are only interested in the shape of the disk itself, these additional components must be filtered out before our calculation of the mean height. We perform a dispersion filter that operates as follows.
\begin{enumerate}
\item Calculate the total mass surface density\begin{equation}M(R,\phi)=\sum_{i=1}^{N=141} \rho(R,\phi,z_i)\Delta z\end{equation}
where $\Delta z$ is the $z$ bin size.
\item Calculate \begin{equation}\bar{z}(R,\phi)=\frac{\sum_{i=1}^{N=141} z_i \rho(R,\phi,z_i) \Delta z}{M(R,\phi)}.\end{equation}
\item Calculate the second moment\begin{equation}d^2(R,\phi)=\frac{\sum_{i=1}^{N=141}[z_i-\bar{z}(R,\phi)]^2 \rho(R,\phi,z_i) \Delta z}{M(R,\phi)}.\end{equation}
\item Run through each point in $(R,\phi,z)$ space.
For any point that does not lie within $2d$ of $\bar{z}$, set $\rho = 0$. Call the grid that results from this dispersion filter $\rho_d(R,\phi,z)$.
\end{enumerate}
For step 4, we experimented with several different cutoffs (in the range $1-3d$); our results do not depend strongly on which cutoff we choose.
We construct the Galactic disk surface density and mean height maps from $\rho_d(R,\phi,z)$. For example, we can sum the density over the $z$ dimension to construct the surface density associated with the disk (Figure \ref{fig:sigma}):
\begin{equation}
\Sigma(R,\phi)= \sum_{i=b}^t \rho_d(R,\phi,z_i) \Delta z
\end{equation}
The indices $t$ and $b$ represent the top and bottom $z_i$ that have not been zeroed out by the dispersion filter.
The resulting figure clearly demonstrates the falloff of the disk surface density with radius. The contour lines are nicely continuous across the $\ell=0\degr$ and $\ell=180\degr$ lines because of our use of elliptical orbits; see the Appendix for a version of this figure without correcting for these orbits. The jagged nature of the contours with $R\la 18$ kpc is due in part to spiral arms \citep{LBH2006}. There is a region near $R\approx 27$ kpc and $\phi\approx90\degr$ with a smaller surface density than other regions at the same radius; this region has somewhat unusual features in all of our maps.
Also notice the excluded regions near the Sun-Galactic center line; these gaps will appear in all of our plots.
We will be looking closely at the mass weighted mean height of the gas disk (Figure \ref{fig:height}),
\begin{equation}
h(R,\phi)=\frac{1}{\Sigma(R,\phi)}\sum_{i=b}^{t} z_i \rho_d(R,\phi,z_i) \Delta z.
\end{equation}
This height is calculated with respect to the Galactic midplane defined by $b=0\degr$. The Galactic warp is the most immediately evident feature in this map; the gas in the northern hemisphere peaks at $h\approx5$ kpc, while the southern gas descends only to $h\approx-1.5$ kpc, consistent with previous maps of the Galaxy. At least three vertical oscillations of magnitude $\approx 1$ kpc can be seen in the south from $\phi\approx-120\degr$ to $\phi\approx-20\degr$; these have previously been called the ``scalloping''. The approximate extent of the scalloping is marked with an arc connecting two ``S'' labels. The region of low surface density noted in the discussion of Figure \ref{fig:sigma} ($R=27$ kpc, $\phi=90\degr$) has a height that seems anomalous when compared to surrounding gas (it is marked with an ``X''). Several features are elongated along lines of constant $\ell$ indicating some level of contamination by turbulent velocities and/or local gas. In the past these have been dubbed ``fingers of God'' because they all point back to the Sun.
In \S \ref{sec:warp} we will need a measure of the uncertainty in the mean height of each point in the map. We define this error using the sum of squared residuals, as is usual for least squares fits:
\begin{equation}
e^2(R,\phi)=\frac{\sum_{i=b}^{N=t}(z_i-h(R,\phi))^2 \rho_d(R,\phi,z_i) \Delta z}{(t-b+1)\Sigma(R,\phi)},
\end{equation}
where $t-b+1$ is the number of points in the calculation of the mean and surface density. This is only an approximation because it does not account for any of the uncertainty introduced in $T_b(\ell,b,v_r)$, $v_\Pi(R)$, the interpolation to $\rho(R,\phi,z)$, or the dispersion filter. Typically, $e(R)/R\approx0.01$.
Because of effects like turbulence, anomalous velocities, and spiral arms, the features in the mean height map may not correspond to the actual shape of the Galaxy. The severity of this effect can be seen by looking at a contour map of $\mathrm{d}r/\mathrm{d}v_r$ (see Figure \ref{fig:uncert}). While similar to previously published plots \citep{BT1986}, this figure also includes the correction for elliptical gas orbits. Given a turbulent velocity of magnitude $v_t$, features with coherent scales less than $v_t \mathrm{d}r/\mathrm{d}v_r$ could potentially be false signals. Furthermore, even real features will be blurred out or even incorrectly positioned over the same length scale. A typical turbulent velocity is around 8 km s$^{-1}$. Small regions can differ from the flat rotation curve by 20-30 km s$^{-1}$ \citep{BB1993}. We do not expect that these distortions will significantly effect our Fourier analysis. The steep contours near $\ell=0\degr$ and $\ell=180\degr$ demonstrate the need for the excluded regions, because small irregularities in velocity there result in large changes in distance.
\subsection{Thickness map}
We also construct a measure of the thickness of the disk. In contrast to the previous section, we keep all of the complicated features of the HI in our calculation of the thickness. We do this because these features do contain information regarding pressure and gravitational force, although they are a nuisance when calculating the mean height. In particular, our method for finding the thickness of the disk relies heavily on the tails of the vertical density distribution. Thus, in this section, we will work with $\rho$ instead of $\rho_d$.
Following \citet{HJK1982}, we define the first and third quartile points as $z_{j1}$ and $z_{j3}$ as the smallest and largest indices, respectively, that satisfy:
\begin{eqnarray}
\sum_{i=0}^{j1}\rho(R,\phi,z_i)\Delta z&\ge&M(R,\phi)/4\nonumber\\
\sum_{i=j3}^{141}\rho(R,\phi,z_i)\Delta z&\ge&M(R,\phi)/4.
\end{eqnarray}
We then define the half thickness (a factor of 2 smaller than \citet{HJK1982} to ease comparison with recent work): $T_h(R,\phi)=(z_{j3}-z_{j1})/2.$ Note that this implies that $T_h$ is quantized by $\Delta z/2$ (about 140 pc). This can lead to inaccuracies in $T_h$ in places where the thickness is small, i.e. $R\approx10$ kpc. $T_h(R,\phi)$ is shown in Figure \ref{fig:disp}. Some points near the sun actually have calculated half-thicknesses of zero; this is due to the comparatively small thickness of the disk in that region combined with our limited LAB survey range $|b|\le30\degr$ and poor grid resolution in $z$.
The flaring of the disk with radius is immediately evident, as $T_h$ increases in magnitude by a factor of about 8 between $R\approx10$ and 30 kpc; this flaring was first seen in \citet{LK1963}. Asymmetry between the northern and southern halves of the disk is also prominent; the southern half of the Galaxy has a lower average thickness than the northern. This asymmetry was also evident in \citet{HJK1982} and \citet{BT1986}.
The low surface density region around $R\approx27$ kpc has a very large thickness; the gas in the region has clearly been disturbed and dispersed by some mechanism. There are regions of increased thickness close to $\ell=15\degr$ and $345\degr$ near $R\approx 10$ kpc probably caused by local gas.
Figure \ref{fig:rfuncs} plots the surface density and half-thickness of the HI layer averaged over $\phi$ as a function of $R$. This figure confirms our visual impressions from the surface density and dispersion maps by showing the falloff in the surface density and the flaring of the thickness with radius. The contamination by local gas described in the previous paragraph as well as the poor $z$ resolution problem mentioned at the beginning of the section account for the rise in the average of $T_h$ with decreasing $R$ near $R\approx 10$ kpc; there is no evidence the thickness of the disk actually behaves this way.
A gradient-expansion least-squares routine \citep{M1963} gives the best fit for the surface density beyond 14 kpc (where the exponential falloff begins) as:
\begin{equation}
\Sigma(R)=4.5\times\exp[-(R-14~\mathrm{kpc})/ 4.3~ \mathrm{kpc}] ~\mathrm{M}_\odot~ \mathrm{pc}^{-2}.
\end{equation}
\section{Analysis}
\subsection{Global Structure}
Using the maps we have constructed, we conduct a quantitative investigation of the disk vertical structure. Since we are studying a disk, we use a method that is independent of rotation in $\phi$ and treats $\phi =0$ and $\phi = 2\pi$ as the same point. Furthermore, the data are unevenly sampled due to the excluded regions within $15\degr$ of the Sun-Galactic center line. We complete the analysis without extrapolating $h(R,\phi)$ in these regions, to avoid introducing any artifacts into the signal. Throughout this paper, we refer to different frequency oscillations in the disk. These frequencies will always be labeled by the number of oscillations they will complete in a full $2\pi$; thus the $m=1$ mode has a one maximum and one minimum in the disk.
\subsubsection{Galactic Warp}\label{sec:warp}
The Galactic warp is the most prominent feature in Figure \ref{fig:height}.
A Lomb periodogram analysis of each radius ring (see \S \ref{sec:lomb}) reveals that the power in each of the $0,1,$ and 2 modes is consistently larger than that in any other mode for $R\ga 20$ kpc. At some radii $m=3$ is the next strongest mode, and at others it is $m=4$.
Accordingly, we characterize the warp by an offset in the $z$ direction, plus two Fourier modes with frequency 1 and 2. We fit each ring with the function:
\begin{equation}\label{eqn:fit}
W(\phi)=W_0 + W_1\sin(\phi -\phi_1)+W_2\sin(2\phi-\phi_2).
\end{equation}
Each of the three amplitudes $W_i$ and two phases $\phi_i$ in this fit is a function of radius, because we fit each radius ring independently.
We use the gradient-expansion fitting algorithm to perform this fit, weighting each point by the inverse of the squared estimate of the uncertainty in the mean height, $e^2(R,\phi)$. The results of this fit for the rings at $R=16,22,$ and $28$ kpc are shown in Figures \ref{fig:warp16},\ref{fig:warp22}, and \ref{fig:warp28}. Error bars in these plots represent $e(R,\phi)$. Each of these plots is a good fit; the offset and the two Fourier components are both necessary and sufficient to describe the large-scale structure of the disk. For illustrative purposes, we will follow the ring at $R=28$ kpc through each step of the analysis.
Following the evolution of the fit parameters at different places in the disk will tell us how the warp changes with radius.
The three amplitude parameters each increase monotonically, with the $m=0$ mode possibly reaching an asymptotic value near the far end of our radius range (Figure \ref{fig:warpamp}). At $R\approx11$ kpc, the $m=1$ mode dominates the shape of the warp; the other two modes do not become important until $R\approx 15$ kpc. This plot implies that the $m=1$ mode has power even at the edge of our grid, thus we cannot establish the onset of the warp. We do a linear least-squares fit on the growth of each warp parameter using the function
\begin{equation}\label{eqn:warpfit}
W_n=k_0+k_1\left(R-R_k\right)+k_2\left(R-R_k\right)^2
\end{equation}
where only points at $R_k$ and beyond are weighted in the fit. $R_k$ is arbitrarily chosen to be near where each of the three modes starts growing. The value of $k_0$ for each fit is strongly correlated to the choice of $R_k$.
\begin{deluxetable}{rrrrr}
\tablecaption{\label{tab:warpfit} Parameters resulting from a linear least-squares fit to the warp}
\tablehead{
\colhead{$m$}&\colhead{$R_k$ (kpc)} &\colhead{$k_0$ (pc)}&\colhead{$k_1$ (pc kpc$^{-1}$)}&\colhead{$k_2$ (pc kpc$^{-2}$)}}
\startdata
0&15&-66&150&-0.47\\
1&10&9&197&-3.1\\
2&15&-70&171&-5.3
\enddata
\end{deluxetable}
\citet{BM1998} (hereafter BM) discuss an approximation to the warp that is of similar form to our fit. They also fit the first three modes, but they fix the line of zeros to lie along the Sun-Galactic center line. In our fit, this would be equivalent to setting $\phi_1$ and $\phi_2$ to zero. Also, BM fix $W_0$ and $W_2$ to be the same. Note that our warp data are adjusted for elliptical gas orbits, and are filtered according to the procedure described in \S\ref{sec:proc}, whereas BM's data are not. Fig.~\ref{fig:binney} compares the mode amplitudes calculated from the data, the fit from BM, and our fit. BM overestimate the strength of the warp starting at around $R\approx22$ kpc for the $m=2$ mode, 24 kpc for the $m=0$, and 27 kpc for the $m=1$. However, the BM fit matches our data fairly well for the radii where the warp is growing most rapidly.
The line of maxima for the $m=1$ mode and one of the two lines of maxima for the $m=2$ mode are marked on the contour map of the warp fit (Figure \ref{fig:phase}). Since $\phi_1$ and $\phi_2$ are actually the line of zeros, the lines of maxima are shifted $90\degr$ and $45\degr$ from these values in our fit, respectively. The parameter $\phi_2$ is not well determined at small radii in our fit because the amplitude of the $m=2$ Fourier component is very small in that region. For this reason, we do not plot $\phi_2$ in the region where $W_2$ is less than 150 pc. There is little evidence for precession in the lines of maxima for the two modes, and the line of maxima for the $m=1$ mode is roughly aligned with one of the lines of maxima of the $m=2$ mode; for each radius their difference in $\phi$ is less than $12\degr$. \citet{B1988} has examined how the two lines of zeros for the mean height change in $\phi$ as a function of radius; they appear to stay roughly aligned with the Sun-Galactic center line as $R$ changes from $R_0$ to 26 kpc. Fig.~\ref{fig:phase} shows that the line of zeros for the sum of our three warp modes falls within the excluded region, consistent with the earlier work.
The three component fit shown in Figure \ref{fig:phase} does a good job of reproducing the large-scale features in the mean height map. This is not surprising, given that the $m=0,1,$ and 2 modes are the strongest in the Lomb periodogram; differences between the two plots are due to power in high frequency modes. We will now examine the differences between the mean height map and the warp fit.
\subsubsection{Scalloping}\label{sec:lomb}
From the fit to the warp, we can determine a function for the scalloping: $s(R,\phi)=h(R,\phi)-W(R,\phi)$. We continue to assume no information regarding the shape of the disk in the excluded regions.
We can now look for higher $m$ modes of oscillation that may be present in $s(R,\phi)$. Such a signal would be present if the gas were coherently moving up and down as a function of $\phi$. The Lomb periodogram is a useful numerical method for detecting periodic signals in unevenly spaced data \citep{NumRec}. It provides the same amplitude we would determine by using a linear least squares fit, even if the signal is not sinusoidal, while allowing for a straightforward error analysis. The data are weighted equally per point, which is necessary to deal with unevenly sampled data. To ease comparison with the warp component amplitudes in Fig.~\ref{fig:warpamp}, we use an unnormalized Lomb periodogram and take the square root of the power to get the amplitude.
We use a Monte Carlo algorithm to determine the noise level for the Lomb periodogram. For each ring, the null hypothesis is Gaussian white noise with the same dispersion as the data in $s(R)$. We construct $10^3$ sets of noise for each ring, run the Lomb periodogram on each set, and record the highest peak in the Lomb amplitude. We then determine the distribution of peaks in these amplitudes, and define the 95\% confidence interval as the amplitude just larger than 95\% of the noise amplitude peaks.
We use the same technique to determine the 99\% confidence level. To conclude that a signal is real and not caused by noise, the signal strength must cross these confidence levels.
We found nearly identical noise levels by scrambling the order of the data instead of using Gaussian white noise, indicating that these calculations are robust.
The Lomb periodogram of $s(R,\phi)$ for the $R=28$ kpc ring is shown in Figure \ref{fig:lombring28}. 95\% and 99\% confidence levels are marked as thresholds in amplitude. At this radius, there is significant strength in modes 4-6, 10, and 15. These modes are markedly weaker than the warp components at the same radii, which have amplitudes of 1-2 kpc.
In the case of unevenly spaced or missing data points, simple sine waves are not eigenmodes of the system. One way to see this is to take a Lomb periodogram of a pure sine wave on our $\phi$ grid with missing data; a small amount of the power will leak into other frequencies. Thus subtracting out the warp from $h(R,\phi)$ influences the amplitudes of the higher order modes from the Lomb periodogram because individual $m$ modes are not independent. We argue that the subtraction is nonetheless acceptable because the warp and the scalloping appear to be due to physically distinct phenomena; studying them is much easier once they have been separated. Removing the strongest modes will also result in a more accurate power spectrum of the weaker frequencies, since we eliminate the power leakage from the stronger to the weaker modes.
Figure \ref{fig:lombfreqs} shows how the amplitudes in several different $m$ modes evolve with $R$. Modes 4-6, 10, and 15 appear to increase in strength at outer radii, while mode 3 is strongest in the intermediate radii in our map. Other modes have no detectable strength because they do not cross the 95\% confidence level at any radii. Also notice how the confidence thresholds rise with radius because of the increase in the dispersion of $s(R,\phi)$.
Another consequence of using the Lomb periodogram analysis on unevenly spaced data is that it is possible to be fooled by a false signal due to ringing from interference between different $m$ modes. It is difficult to protect against this, but if two strong modes were interfering with each other to produce a third signal, we would expect at least two of the signals to increase in strength at the same radius. The strongest modes in Figure \ref{fig:lombfreqs} become significant at different radii and have visually different evolution with radius; we conclude that these modes are real.
\subsection{Local Structure}\label{sec:wave}
The Lomb periodogram cannot be used to study the local structure of the disk since it cannot determine where in azimuth each mode is strong. Imagine a situation where, like a falling stone creating ripples on the surface of a pond, something passes through the HI disk and excites a local series of vertical oscillations. The oscillation will add power to some frequencies in the Lomb periodogram, but this effect may be dwarfed by oscillations elsewhere in the disk. We wish to detect these sorts of perturbations and learn where in the disk they are prominent.
One way to draw out this type of structure is to use a wavelet analysis. Wavelets are ideal for our purposes because they are localized in both position and frequency space; in other words, they will show where in the disk a given mode of oscillation is dominant. In wavelet theory, it is beneficial to choose a mother wavelet that is similar in shape to the signals you are searching for. Since we are looking for sinusoidal perturbations, we use the normalized Morlet wavelet with $\omega_0=6$; this is simply a sine wave times a Gaussian envelope. In the same way that a Fourier transform breaks down the frequency structure of a signal using sines and cosines, a wavelet transform breaks down a signal in terms of a set of these Morlet wavelets centered at different spatial positions and with a range of frequencies. However, in a continuous wavelet transform like the one we perform, the different wavelet functions are not orthogonal.
To avoid having to interpolate or zero out the excluded regions, we will examine the northern and southern halves of the Galaxy separately. As in \S \ref{sec:lomb}, we will work with the mean height function once the warp has been subtracted, $s(R,\phi)$.
Points in frequency and position space that would involve the excluded regions are defined as being inside the ``cone of influence''; points in this region are subject to edge effects, and are therefore discarded. The points affected by this are not just those in the excluded regions but also those adjacent to these regions, within some range set by the wavelength (and thus the frequency) of the particular mode \citep{TC1998}. Thus, lower frequency modes will have a larger portion of the disk fall inside the cone of influence, and have to discard a larger range of points.
It is also important to have some analytic measure of which peaks in the filtered power spectrum are significant. Significance levels are discussed in detail in \citet{TC1998}. We construct a combination of parameters that have a $\chi^2$ distribution, and count as significant those that cross the 95\% confidence level threshold.
Again, we use Gaussian white noise to model the randomness in the height function.
The wavelet power spectrum, $\overline{W_n}^2$, for the northern half of the $R=28$ kpc ring is plotted in Figure \ref{fig:specring}; the southern half is shown in Figure \ref{fig:specring2}. The mathematical details of the wavelet transform are summarized in the Appendix.
This ring has several regions with significant power. These regions have a large width in both dimensions because wavelets do not have precise resolution in either position or frequency space. Much like the uncertainty principle, the cost of using a technique that gives both position and frequency information is mediocre resolution in both. The power around $\phi\approx90\degr$ comes from the sharp dip at that azimuth; as the figure shows, this causes ringing for a large range of frequencies. The power at $-45\degr\ga\phi\ga-90\degr$ is due to the ``scalloping'' in that region.
The filtered wavelet power spectra for several different bands are plotted in Figure \ref{fig:waveplot}. The band that is labeled $m=7$ is actually the filtered sum of the power spectra that satisfy $6.5\le m \le 7.5$, and so on. Note that since we calculate the power spectra for the northern and southern halves independently, the significance contours are different for the two hemispheres, in addition to being a function of $R$. In practice, this occurs because the variance of $s(\phi)$ for the northern half of the Galaxy is larger than that for the southern, and the strength required of the power spectrum to cross the 95\% confidence level is directly proportional to the variance.
\section{Discussion}\label{sec:disc}
We grouped the $m=0,1$ and 2 modes together in the warp because of their similar magnitudes in a Lomb periodogram. A close look at their dependence with $R$ (Fig.~\ref{fig:warpamp}) demonstrates that there are additional similarities. Both the 0 and the 2 mode are near zero until about 15 kpc from the Galactic Center. The 0 mode grows linearly from this point outwards while the 2 mode
declines a small amount, and then grows approximately linearly by about 1.2 kpc. In comparison, the $m=1$ mode starts out fairly large ($\approx 300$ pc), declines slightly and then grows by about an order of magnitude. Though the $m=2$ mode does break off from this pattern at larger radii, all three components grow essentially monotonically, approximately linearly, and with similar slopes over the range $15\la R\la 22$ kpc. This may be a clue that their origins involve the same physics, and helps to justify our classification of higher frequency modes as scalloping. Studying the radial dependence of the warp inside the solar circle seems a fruitful way to learn more about these three components.
The filtered wavelet power spectrum maps (Fig.~\ref{fig:waveplot}) are a representation of the scalloping in the outer Galaxy. Due to frequency-position uncertainty relations, it is impossible to establish precisely the oscillation frequency of any local disturbance. This uncertainty manifests itself in the wavelet transform by perturbations that are only somewhat localized in frequency and position space, and therefore have some width in both.
These maps demonstrate that the $m\approx10$ scalloping found near $\ell\approx 310\degr$ and $R\approx 25$ kpc mentioned in previous work \citep{HJK1982,KBH1982} is real. The strong power around $m=10$ in the wavelet transform is accompanied by a large amplitude in the Lomb periodogram for $m=10$ at the corresponding radii.
Wavelet transforms are also subject to the same problems that bedevil traditional Fourier transform approaches. For example, there is a large amount of power around $R\approx30$ kpc and $\phi\approx90\degr$ in $8\le m\le15$. This power is most likely not a result of scalloping in all of these different frequencies over some range of $\phi$; instead, it is probably due to ringing.
As in a Fourier decomposition, sharp changes in height will cause ringing in all frequencies of a wavelet transform. Indeed, the height map (Fig.~\ref{fig:height}) does have an abrupt drop near $\phi\approx90\degr$ which could cause this ringing; we mark this region with an `X'. The same feature can be seen after the warp has been subtracted in the ring at 28 kpc plotted in Figure \ref{fig:specring}.
We use the local and global analyses in conjunction to determine whether there is any scalloping mode that is present over a full $2\pi$ ring in the outer Galaxy. This is important because it will determine whether the mechanism that causes scalloping operates on a global or local scale. For this task, both techniques are necessary because even a pure $m=10$ oscillation will have some width in frequency space when put through a wavelet transform. However, this same oscillation will have a sharp peak in the Lomb periodogram. Therefore, in order to state that some mode exists over a full $2\pi$, we require both significant strength in the Lomb periodogram and significant power over a large range of $\phi$ in the wavelet transform. This combination will conclusively determine whether the scalloping is a global or local phenomenon.
For modes with $m\le6$ this technique is not useful. These modes are clearly important because their Lomb periodograms show significant power for $3\le m\le 6$. Unfortunately, these modes have large enough wavelengths that the excluded regions always interfere with the wavelet transform regardless of where in the disk we look. A method of reducing the size of the excluded regions would alleviate this problem. However, this is likely to be difficult because eliminating the excluded regions would require a detailed knowledge of both $v_R$ and the distribution of gas velocities due to turbulence in the disk. Portions of these region are optically thick, which will make density determination impossible.
With these constraints, the Lomb periodogram leads us to conclude that the modes $m=10$ and 15 are the most fruitful places to look for scalloping that travels a full circle around the disk. Other modes do have interesting features, but their smaller Lomb amplitudes imply that we could be fooled by the imperfect frequency resolution of the wavelet power spectrum. For example, the modes $7\le m\le9$ have significant wavelet power for large portions of the northern half of the galaxy, but none of these modes has significant signal for the corresponding radius range in the Lomb periodogram.
For the $m=10$ mode the only large region with significant power in the south is the one around $\ell\approx310\degr$ that we noted previously. The power in this perturbation has fallen below the 95\% confidence level by $\ell\approx270\degr$, implying that this scalloping is a local effect.
The same appears to be true for the $m=15$ oscillations, although this mode does have a region of high significance near $R\approx30$ kpc over a large range of $\phi$. However, the northern part of this signal is the region where we believe ringing to play an important role. It is therefore possible that the $m=15$ mode does have significant wavelet power over the full $2\pi$ near $R\approx 30$ kpc, but the evidence is not conclusive because the northern part of the wavelet power is most likely not due to scalloping.
With the exception of the $m=15$ mode, no frequency we examine with the wavelet transform carries significant power around an entire ring along with a correspondingly significant Lomb periodogram strength. For this reason, we conclude that scalloping generally appears to be a local phenomenon. The modes $3\le m \le 6$ remain a possible exception, since we were not able to study them with a wavelet transform.
It remains unclear what mechanism acts as the energy source for the scalloping. One possible cause is a massive object passing through the disk that excites local vertical oscillations in the HI gas. Azimuthally traveling wavefronts can be created by the magnetic field threaded through the disk; if the field is primarily azimuthal in nature, vertical oscillations will have larger phase and group velocities in the azimuthal direction than in the radial direction. This could lead to scalloping such as that seen towards $\ell\approx 310\degr$.
\section{Conclusions}
We fit the global shape of the warp on our grid of concentric rings using a vertical offset and two sinusoidal modes. Outside of $R\approx20$ kpc, each of these three modes has more power than any of the higher frequencies we look at. The amplitude increases with radius over our entire radius range for the 1 mode, and starting from around 15 kpc for the 0 and 2 modes. The growth of the 0 and 2 modes results in asymmetry in the warp; this growth begins near where the stellar disk ends.
The line of maxima of the $m=1$ mode is essentially coincident with one of the lines of maxima of the $m=2$ mode. There is little evidence for precession or winding of these two modes.
A global analysis with the Lomb periodogram shows that each $m$ mode evolves differently with radius. The most interesting include $m=3-6,10$, and 15, each of which start at small radii below the 95\% significance level and then cross it further out in the disk. An analysis combining the global Lomb periodogram and the local wavelet transform shows that none of the modes with $7\le m\le15$ have strength over a full ring of the disk. Using a wavelet transform, we show that the scalloping observed by previous authors near $\ell\sim 310\degr$ is real. We therefore conclude that the scalloping is a local effect.
Lower frequency modes proved impossible to study with wavelets due to the presence of the excluded regions.
\acknowledgments
Wavelet software was provided by C.~Torrence and G.~Compo and is available at URL: http://paos.colorado.edu/research/wavelets/ \citep{TC1998}. Many thanks to Peter Kalberla for providing a copy of the LAB data set. Thanks to Eugene Chiang for discussions of dynamics. ESL and LB are supported by NSF grant AST 02-28963. CH is supported by NSF grant AST 04-06987.
\appendix
\section{Measuring radial motion of the gas}
This appendix describes our method for finding a functional form for $v_\Pi(R)$, the magnitude of the elliptical corrections to circular rotation. A naive surface density plot using only circular rotation has a large degree of asymmetry across the lines $\ell=0\degr$ and $180\degr$ \citep{KBH1982}. Figure \ref{fig:novlsr} shows the calculated surface density without any correction for elliptical gas orbits, but with the filters we describe in \S\ref{sec:proc}. Observational experience tells us that the surface density should not have discontinuities in cardinal directions; the elliptical orbits we describe in \S \ref{sec:proc} provide the strongest corrections in the directions where the $\cos \phi$ term in (\ref{eqn:transform}) is large, and smaller corrections elsewhere. The algorithm is based on matching the surface density of the HI disk on either side of the excluded region centered at $\ell = 180\degr$. We draw confidence from the fact that this fit also does a good job matching the contours around $l=0\degr$ even though they are not included in the fit.
In order to calculate a surface density, we must first know $v_\Pi(R)$ because it enters into the derivative $|dv_r/dr|$. Thus, determining $v_\Pi(R)$ from anything connected to $\rho$ is a circular problem. We must assume some form of $v_\Pi(R)$ and check to see if it results in surface densities that are well matched. We choose the functional form:
\begin{equation}
v_\Pi(R)=\alpha\frac{(R-R_0)}{R_0}+\beta\frac{(R-R_0)^2}{R_0^2},
\end{equation}
and do not include a zeroth order term to ensure that $v_\Pi(R)$ passes through zero for the solar circle. \citet{RS1980} showed that the LSR does not have a radial velocity with respect to the Galactic center.
Our algorithm follows these steps:
\begin{enumerate}
\item Construct $v_\Pi(R)$ for some combination of $\alpha$ and $\beta$
\item Interpolate from the LAB survey to find $T_b(R,\ell,z)$ for a grid in $R_i$ and $z_j$ over the survey range $\left|b\right| \le 30\degr$ and $155\degr \le \ell \le 165\degr$ or $195\degr\le \ell \le205\degr$ using $v_\Pi(R)$
\item Sum over $z$ to find the surface density $\Sigma(R,\ell)$
\item Average over $\ell$ for the two subsets of $\ell$ to find $\Sigma_{165}(R)$ and $\Sigma_{195}(R)$
\item Calculate a modified $\chi^2$ statistic to determine how well the surface densities are matched
\end{enumerate}
The modified $\chi^2$ is defined as
\begin{equation}
\chi^2=\sum_{R_i} \left[\frac{\Sigma_{165}(R_i)-\Sigma_{195}(R_i)}{\Sigma_{165}(R_i)<\Sigma_{195}(R_i)}\right]^2
\end{equation}
where the $<$ operator returns the smaller of its two operands.
Using this algorithm we search for the values of $\alpha$ and $\beta$ that minimize the modified $\chi^2$. We find good matches for $\alpha=8.67$ and $\beta=-1.08$.
We also tried a weighting that only fit points beyond the perturbations of the spiral arms, i.e. $R>2R_0$. Although this did change $v_\Pi(R)$ by 25\% or so, it had no qualitative effect on our other results.
\section{Wavelets}
The continuous wavelet transform for a discrete series of points $z_n$ is given by
\begin{equation}\label{eqn:wavetrans}
W_n(s)=\sum^{N-1}_{n'=0} z_{n'} \psi^*\left[
\frac{(n'-n)\Delta\phi}{s}\right]
\end{equation}
where $N$ is the number of points in the series, $n$ is a position index, $\Delta\phi$ is the spacing of the points in $\phi$ space, and $\psi^*$ is the complex conjugate of the normalized Morlet wavelet \citep{TC1998}:
\begin{equation}
\psi\left(\eta\right)
=\left(\frac{\Delta\phi}{s}\right)^{1/2}\pi^{-1/4}e^{i\omega_0\eta}e^{-\eta^2/2}.
\end{equation}
The scale of the transform is $s$; for the Morlet wavelet this is simply related to the wavelength by $\lambda=1.03s$. Evenly spaced points are necessary in order to use this transform, but since we are treating the two halves of the Galaxy separately, $s(R,\phi)$ is split into two halves, each with equally spaced points. We perform this transform for the dense set of frequencies given by
\begin{equation}
s_j=s_02^{j\Delta j}, ~~~j=0,1,\ldots,J.
\end{equation}
Here, $s_0$ is the smallest scale that can be sampled, $2\Delta\phi$. $\Delta j$ is a measure of how densely we sample in scale space; because computation time is not large, we choose relatively dense sampling throughout: $\Delta j = 0.0125$.
From (\ref{eqn:wavetrans}), we construct the wavelet power spectrum, $\left|W_n(s)\right|^2$.
We calculate the wavelet power spectrum for the dense set of frequencies, and then filter over a range of scales to find the power in a frequency band. The filtered power spectrum is given by
\begin{equation}
\overline{W_n}^2=\frac{\Delta j\Delta \phi}{C_\delta}\sum_{j=j_1}^{j_2}\frac{\left|W_n(s_j)\right|^2}{s_j}
\end{equation}
where $C_\delta$ is a reconstruction factor that depends on the choice of wavelet. For the Morlet wavelet with $\omega_0 =6$, $C_\delta=0.776$.
|
Title:
Recovery of the global magnetic field configuration of 78 Virginis from Stokes IQUV line profiles |
Abstract: The surface magnetic field configuration of the Ap star HD 118022 (78 Vir)
has been reconstructed in the framework of the magnetic charge distribution
(MCD) method from the analysis of Stokes $IQUV$ spectra obtained using the
MuSiCoS spectropolarimeter at Pic du Midi Observatory. Magnetically-sensitive
Fe~{\sc ii} lines were primarily employed in the analysis, supposing that iron
is evenly distributed over the stellar surface. We show that the Stokes $IQUV$
profile shapes and variations of 78 Vir can be approximately fit assuming a
global magnetic field configuration described by a slightly decentered,
inclined magnetic dipole of polar surface intensity approximately 3.3~kG. The
derived inclinations of the stellar rotational axis to the line of sight
$i=24\pm 5\degr$ as well as to the magnetic dipole axis $\beta=124\pm5\degr$
are in good agreement with previous estimations by other authors, whereas the
sky-projected position angle\thanks{$\Omega$ increases clockwise from the axis
to the North Celestial Pole and relates to the azimuth angle $\Theta$ specified
by Landolfi at al.~(\cite{Landolfi+93}) as $\Omega=360\degr-\Theta$.} of the
stellar rotation axis $\Omega\sim110\degr$ is reported here for the first time.
In addition, several lines of Cr~{\sc ii} and Ti~{\sc ii} were studied,
yielding evidence for non-uniform surface distributions of these elements, and
magnetic field results similar to those derived from Fe.
| https://export.arxiv.org/pdf/astro-ph/0601677 |
\title{Recovery of the global magnetic field configuration of\\
78 Virginis from Stokes $IQUV$ line profiles}
\author{V.R. Khalack \inst{1, 2} and G.A. Wade \inst{3}}
\offprints{V. Khalack \\ \email{khalakv@umoncton.ca}}
\institute{ D\'{e}partement de physique et d'astronomie, Universit\'{e}
de Moncton, Moncton, N.-B., E1A 3E9, Canada
\and
Main Astronomical Observatory, 27 Zabolotnoho Str., 03680, Kyiv, Ukraine
\and
Department of Physics, Royal Military College of Canada, PO Box 17000 stn
'FORCES', Kingston, Ontario, Canada K7K 4B4
}
\date{Received {\it date will be inserted by the editor}\
Accepted {\it date will be inserted by the editor}
}
\abstract{The surface magnetic field configuration of the Ap star
HD~118022 (78 Vir) has been reconstructed in the framework of the
magnetic charge distribution (MCD) method from the analysis of
Stokes $IQUV$ spectra obtained using the MuSiCoS
spectropolarimeter at Pic du Midi Observatory.
Magnetically-sensitive Fe~{\sc ii} lines were primarily employed
in the analysis, supposing that iron is evenly distributed over
the stellar surface. We show that the Stokes $IQUV$ profile shapes
and variations of 78 Vir can be approximately fit assuming a
global magnetic field configuration described by a slightly
decentered, inclined magnetic dipole of polar surface intensity
approximately 3.3~kG. The derived inclinations of the stellar
rotational axis to the line of sight $i=24\pm 5\degr$ as well as
to the magnetic dipole axis $\beta=124\pm5\degr$ are in good
agreement with previous estimations by other authors, whereas the
sky-projected position angle\thanks{ $\Omega$ increases clockwise
from the axis to the North Celestial Pole and relates to the
azimuth angle $\Theta$ specified by Landolfi at al.~(\cite{Landolfi+93})
as $\Omega=360\degr-\Theta$.} of the stellar rotation axis
$\Omega\sim110\degr$ is reported here for the first time.
In addition, several lines of Cr~{\sc ii} and Ti~{\sc ii} were
studied, yielding evidence for non-uniform surface distributions
of these elements, and magnetic field results similar to those
derived from Fe.
\keywords{stars: chemically peculiar --
stars: magnetic fields -- line: polarisation -- stars:
individual: HD~118022, 78~Virginis}}
\titlerunning{Recovery of the 78 Vir global magnetic field}
\authorrunning{V.R. Khalack and G.A. Wade}
\section{Introduction
\label{intro}}
78 Virginis (HD~118022) is a bright Ap star, and the first star other than the
sun in which magnetic field was discovered (Babcock~\cite{Babcock47}). The longitudinal
magnetic field of 78 Vir, as first observed by Babcock, is variable, and
constantly negative. 78 Vir is also marginally variable in broadband light,
in radial velocity, and shows variations of its line profiles
(Preston~\cite{Preston69}). As suggested by Preston (\cite{Preston69}), the observed
properties of 78 Vir can be explained in the framework of the oblique rotator
model (Stibbs~\cite{Stibbs50}), in which the star rotates with a period of
approximately 3.7 days and has a rotation axis forming a small angle with
respect to the line of sight.
The first comprehensive modelling of the surface magnetic field structure of
78 Vir was performed by Borra (\cite{Borra80}). Borra obtained high-resolution
measurements of circular polarisation across the Fe\,{\sc ii} $\lambda$4520.2\AA\,
line, which he attempted to reproduce using nine different magnetic field
configuration models. As argued by Borra (\cite{Borra80}), the profiles were
well reproduced with an inclined dipole geometry that contains a moderate
quadrupolar component. Nevertheless, he also pointed out that a decentered
($a$=0.2) dipole model provides {essentially} the same fit quality.
Borra also concluded that the 78 Vir
presents to an observer a remarkably uniform magnetic field over most of its
visible disk, at most phases. It is unfortunate that due to the small
inclination of the rotational axis (about $25\degr$) a significant part of the stellar surface
remains hidden - a part which might have a less uniform field geometry.
The analysis of broadband linear polarization (BBLP) measurements of several Ap
stars (including 78 Vir) by Leroy~et~al.~(\cite{Leroy+96}) suggests that the
magnetic fields of many Ap stars
exhibit departures from the standard oblique rotator model assuming a pure
dipole field geometry. According to those
authors, significant discrepancies between the BBLP observations and the
``canonical model'' results {(Landolfi~et~al. \cite{Landolfi+93})} required the
assumption of local departures from a dipolar field. In particular,
Leroy~et~al. (\cite{Leroy+96}) showed that the BBLP variations of most stars
could be reproduced assuming a dipole magnetic field geometry with slightly
expanded field lines over some parts of the magnetic equator. For 78 Vir, the
observed BBLP variations do not show especially strong departures from the
dipolar case, although according to Leroy~et~al.~(\cite{Leroy+96}) a modified
dipolar model provides a somewhat better fit to the BBLP curves for this star.
In order to better constrain the magnetic field configuration of
78 Vir, in this paper we undertake a more detailed modelling of
the surface magnetic field structure based on the analysis of high
resolution line profiles in all 4 Stokes parameters. In
Sec.~\ref{obs} we discuss the properties of obtained Stokes $IQUV$
spectra, and in Sec.~\ref{fund} we summarise the fundamental
characteristics of 78 Vir. In Sec.~\ref{mod} we describe the
features of the magnetic field modelling framework, and derive the
global magnetic field configuration of the star, comparing
observed and computed Stokes profiles for various spectral
features. In Sec.~\ref{discuss} we discuss the derived global
magnetic field characteristics and their agreement with earlier
studies.
\section{Observations}
\label{obs}
Spectropolarimetric observations of 78 Vir were obtained in 1997
February, 1998 February and 1999 January using the 2m T\'elescope Bernard Lyot
at Observatoire du Pic du Midi (Wade~et~al.~\cite{Wade+00a}). The MuSiCoS
cross-dispersed \'echelle spectrograph (Baudrand \& B\"{o}hm~\cite{BB92}) and
dedicated polarimeter module (Donati et al.~\cite{Donati+99}) were employed for
the observations.
The MuSiCoS spectrograph is a table-top instrument, which allows the
acquisition of a stellar spectrum in a given polarization state (Stokes $V$,
$Q$ or $U$) throughout the spectral range from 4500 to 6600~\AA\ with a resolving
power of about 35 000, in a single exposure. The spectrograph is fed by a
double optical fibre directly from the Cassegrain-mounted polarimeter. The
optical characteristics of the polarization analyser, as well as the
spectropolarimeter observing procedures, are described in detail by Donati et
al.~(\cite{Donati+99}). Observing details specific to the acquisition,
reduction and analysis of the 78 Vir spectra are provided by
Wade~et~al.~(\cite{Wade+00a}).
The journal of spectropolarimetric observations is reported by Wade~et~al.~(\cite{Wade+00a}).
The 52 Stokes $V$, $Q$ and $U$ spectra, obtained on 18 different nights, cover the whole
rotational period of 78 Vir approximately uniformly, and provide an average S/N of about 370.
\section{Fundamental parameters of 78 Vir}
\label{fund}
78 Vir was classified by Cowley et al.~(\cite{C2J2}) as A1pCrSrEu.
Adelman~(\cite{Adelman73a, Adelman73b}) performed both a line
identification and abundance analysis of this star.
The distance to 78 Vir $d=48\pm15$ pcs and its radius $R=1.77\pm0.68R_{\rm \sun}$
are derived by Monier~(\cite{Monier92}) using the Infrared Flux Method. On the
other hand, taking into account the mean angular diameter
$\theta_{\rm a}$= 0.343~{milliarcsec} obtained by
Monier~(\cite{Monier92}) for this star and its distance
$d=56.2\pm2.5$~pc derived from the Hipparcos parallax (ESA~\cite{ESA97}),
we find a somewhat larger value for the stellar radius $R=2.06\pm0.17~R_{\rm \sun}$.
Using the Hipparcos visual magnitude and parallax we have found the absolute
visual magnitude of 78 Vir, $M_{\rm v}= 1.18\pm0.10$. Taking into
account the bolometric correction (Flower~\cite{Flower96}), which
corresponds to Monier's (1992) $9200\pm 290$~K effective temperature
and the bolometric zeropoint correction BC$_{\rm V}$=-0.07,
which is obtained assuming $M_{bol}^{\sun}$=4.74 for
the Sun (Bessell~et~al.~\cite{Bessell+98}), we can find
the absolute luminosity for 78 Vir $L_{\rm \star}=(27.3\pm2.5)L_{\rm \sun}$.
For 78 Vir Monier~(\cite{Monier92}) also estimates the integrated flux
$f_{\star}=(2.7\pm 0.27)\times 10^{-7} erg\; sm^{-2} s^{-1}$. Taking into account
the distance to the star, this provides the luminosity
$L_{\rm \star}=(26.5\pm2.8)L_{\rm \sun}$.
Finally, employing the stellar radius and
the effective temperature (see Table~\ref{tab2})
we obtain a third estimate of the absolute luminosity
$L_{\rm \star}=27.4\pm4.5L_{\rm \sun}$. All of these values are in good mutual agreement.
The luminosity $L_{\rm \star}=(27.3\pm2.5)~L_{\rm \sun}$
together with the effective temperature (see Table~\ref{tab2}) allow us to
find the stellar mass $M_{\rm \star}=2.18\pm0.06M_{\rm \sun}$ and age
$3.0^{+0.7}_{-1.3}\times10^{8}$ years (see Fig.~\ref{HRdiaram}) by
spline interpolation (Sandwell~\cite{Sandwell87})
in the model evolutionary tracks of Schaller~et~al.~(\cite{Schaller+92})
for metallicity $Z$=0.02. The derived age suggests that 78 Vir
has completed approximately 37\% of its main sequence life.
The mass and radius provide $\log g = 4.16\pm 0.07$.
This value is marginally inconsistent with that of the Monier~(\cite{Monier92}), but it
is in good agreement with the value $\log g =4.20$ reported by King~et~al.~(\cite{King+03}).
A recent investigation of the Ursa Major stream characteristics performed by
King~et~al.~(\cite{King+03}) derived $Z=0.016-0.02$ and age $(5\pm1)\times10^{8}$ years
for the stream. They found that 78 Vir is a "certain member" of the stream
from photometric data, while the kinematic characteristics of the star argue
for its "probable non-membership".
The derived age for 78 Vir is marginally inconsistent with
the UMa stream age. The published metallicity of 78 Vir
is $\log(Fe/H)_{\star}-\log(Fe/H)_{\sun}=^{-0.13}_{+1.58}$
(Cayrel~de~Strobel~et~al.~\cite{Cayrel+97}) and it is
not clear if this star truly has a solar metallicity. For this reason the
derived uncertainties of the mass and age may be somewhat larger than indicated here.
\begin{table}[t]
\caption[]{Fundamental parameters of 78 Vir.}
\begin{tabular}{lcc}
\hline
Parameter & Value & Reference\\
\hline
Spectral type& A1pCrSrEu&Cowley et al.~(\cite{C2J2})\\
Age & $3.0^{+0.7}_{-1.3}\times10^{8}$y.& this work\\
Distance & 56.2$\pm$2.5 pcs& Hipparcos (ESA~\cite{ESA97})\\
Period & 3$.^{d}$7220$\pm0.^{d}$003 & Preston~(\cite{Preston69}) \\
$T_{\rm eff}$&9200$\pm$290K &Monier~(\cite{Monier92}) \\
$\log{g}$ &4.50$\pm$ 0.25 & ibidem \\
$M_{\rm v}$ & 1.18$\pm$0.10 &this work\\
$L_{\rm \star}$& 27.3$\pm$2.5 $L_{\sun}$&this work\\
$R_{\rm \star}$& 2.06$\pm$0.17$R_{\sun}$& this work\\
$M_{\rm \star}$& 2.18$\pm$0.06$M_{\sun}$& this work \\
$V_{\rm e}\sin{i}$& 12$\pm$1 km s$^{-1}$& this work \\
$i$ & 25$\degr\pm$5$\degr$ &Leroy~et~al.~(\cite{Leroy+96}) \\
$\beta$ & 120$\degr\pm$5$\degr$&ibidem \\
\hline
\end{tabular} \label{tab2}
\end{table}
Babcock~(\cite{Babcock47}) observed that the longitudinal magnetic field of
78 Vir was variable, but always negative. Analysing Babcock's data along with
his own measurements, Preston~(\cite{Preston69}) determined
the periodic character of the magnetic field variability and derived the
ephemeris:
\begin{equation}\label{ephemeris}
{\rm JD}\ {\rm (magnetic\;\; maximum)}= 2 434 816.9 + 3.^{d}7220\cdot {\rm E}.
\end{equation}
\noindent Several authors have confirmed this period on the basis of further
magnetic field measurements (Wolff~\&~Wolff~\cite{W+W71};
Wolff~\&~Bonsack~\cite{W+B72}; Wolff~\cite{Wolff78}; Borra~\cite{Borra80};
Borra~\& Landstreet~\cite{B+L80}; Landstreet~\cite{Landstreet82};
{Wade~et~al.~\cite{Wade+00b}), while
Landstreet~(private communication) has estimated the accuracy of the period
determination to be $\pm0.^{d}003$. Preston~(\cite{Preston69}) also found that
the crossover effect is very pronounced and is present throughout most of the
magnetic cycle. This feature is particularly remarkable at phase 0.85
(Wolff~\&~Bonsack~\cite{W+B72}, see also Wade et al.~\cite{Wade+00a}).
Applying the oblique rotator model with the distorted dipole
approximation, %
Leroy~(private communication) found that they must adopt a period 3.7218 days
in order to fit the observed BBLP variations. As discussed in Sect.~\ref{intro},
their model, using local modifications of the axisymmetric magnetic field imposed
near the magnetic equator, differs only mildly from a dipole configuration. The
derived rotational axis inclination and dipole obliquity $i=25\degr$ and
$\beta=120\degr$ provide a good match to the longitudinal field and linear
polarisation variations. Nevertheless, the period inferred is somewhat shorter
than that obtained by other authors (Eq.~\ref{ephemeris}), and
Wade~et~al.~(\cite{Wade+00b}) reported that this period is only barely
consistent with the available longitudinal field measurements.
78 Vir also shows photometric variability in the visible and the infrared
with the same period (Eq.~\ref{ephemeris}). Combining the visual flux
variability in the $uvby$ bands, obtained by Wolff~\&~Wolff~(\cite{W+W71}), with their own
new observations, Catalano~\&~Leone~(\cite{C+L94}) have tried to improve the
period determination. Their analysis results in the new ephemeris
\begin{equation}\label{ephemeris-new}
{\rm JD}\ {\rm (y\; min.)}=2 434 816 + (3.^{d}722084 \pm 0.^{d}000042) {\rm E}, %
\end{equation}
\noindent which was recently confirmed by Leone~\& Catanzaro~(\cite{L+C01}).
These authors have shown with the help of Hipparcos photometry that the light
variations of 78 Vir are not purely sinusoidal. Taking into account all
archival magnetic field observations as well as their own data,
Leone~\&~Catanzaro~(\cite{L+C01}) have shown that the longitudinal field
measurements are in phase only when the 3.722084 day period is adopted.
Moreover, by adopting the 3.7218 day period suggested by
Leroy~(private communication), they find a 0.12-cycle phase shift between the
Hipparcos light curves and the other light curves.
Monier~(\cite{Monier92}) has derived for 78 Vir from the simulation
of the energy distribution in the spectral range from 1200\AA\, to 22000\AA\, an
effective temperature $T_{\rm eff}=9200\pm290$~K, a surface gravity $\log{g}=4.5$
and a photospheric metallicity $[M/H]=10$. In addition, using a model atmosphere
with these values of $\log{g}$ and $[M/H]$, he has found
{that the effective temperature $T_{\rm eff}=9300$K provides the best
description of the spectral energy distribution in the region from 1200\AA\,
to 8000\AA\, at rotational phase 0.0, while $T_{\rm eff}=9200$K seems to
be the best effective temperature, at phase 0.5. These temperatures are
somewhat lower than most previous estimates, ranging from 9700K to 10700K
(Mihalas~\&~Henshaw~\cite{M+H66}; Wolff~\cite{Wolff67};
Jugaku~\&~Sargent~\cite{J+S68}), probably because the models used by these
authors were unblanketed and calculated for solar abundances.
The fundamental parameters of 78 Vir are summarised in Table~\ref{tab2}.
\section{Modelling of the Stokes $IQUV$ spectra}
\label{mod}
According to the discussion of Sect.~\ref{fund}, an ATLAS9 model
(Kurucz~\cite{Kurucz94}) with parameters $T_{\rm eff}=9250$K,
$\log{g}=4.5$, [M/H]=0 and
microturbulent velocity $v_{\rm t}$=0 km s$^{-1}$ %
successfully approximates the stellar atmosphere of 78 Vir.
We describe the structure of the surface magnetic field of 78 Vir (assumed to
be a rigidly rotating and spherically symmetric star) using the {\em magnetic
charge distribution} (MCD) method (Gerth~et~al.~\cite{gerth+};
Khalack~et~al.~\cite{khalack+}). Originally, the MCD method considered a
system of spatially separated {\it point field sources} with {\it virtual
magnetic charges} in the stellar interior. {In order to provide zero magnetic
flux through the stellar surface the sum of ``magnetic charges'' should be kept to zero.}
These sources produce a magnetic
field whose potential at each point on the stellar surface is specified by
the superposition of potentials of individual sources (Gerth~et~al.~\cite{gerth+};
Gerth~\&~Glagolevskij~\cite{gerth+01, gerth+04}). Since the number of sources is usually
more than one, we actually operate with a system of several magnetic dipoles
located in the stellar interior. In order to minimize the number of free model
parameters, the most convenient way is to consider a system of two sources, which
mathematically formulate a magnetic dipole (Khalack~et~al.~\cite{khalack+}).
When the dipole is centered on the stellar centre and the dipole size is
much smaller than the stellar radius (Khalack~\cite{khalack02}), the MCD model
transforms to the conventional model of a symmetric inclined magnetic rotator
(Stibbs~\cite{Stibbs50}). Otherwise, we deal with a more complex magnetic
field configuration.%
The mathematical verification of the MCD model as well as the procedure of
specification of the angle $\beta$ between the rotational and magnetic dipole
axes, the coordinates and the field strengths of the positive and negative
magnetic poles on the basis of the derived free model parameters is described
in detail by Khalack~et~al.~(\cite{khalack+03}).
\begin{table}[t]
\caption{Here the individual columns specify
the ion, the wavelength, the quantum number $J$ and the Land\'e factor
for lower and upper atomic levels, the observed Stokes $I$ profile depth and the
line scaling factor (LSF) (see Sect.~\ref{integral}), derived from the
comparison of simulated equivalent width of the Stokes $Q$ and $U$ Zeeman profiles
with the BBLP data (Leroy~\cite{Leroy95}). Asterisks marks the
LSF calculated from Stokes $Q$ and $U$ profiles simulated for a field
structure that is derived from only Stokes $I$ and $V$ profile variability
(see Table.~\ref{fe2sm}).}
\begin{tabular}{lccccccl}
\hline
Ion& $\lambda$, \AA &$J_{\rm lo}$&$g_{\rm lo}$&$J_{\rm up}$&$g_{\rm up}$&$I_{\rm obs}$&LSF\\
\hline
Fe\,{\sc ii} & 4620.52& 3.5 & 1.21 & 3.5 & 1.40 & 0.33&0.035\\
Fe\,{\sc ii} & 4635.32& 2.5 & 1.20 & 3.5 & 1.13 &0.32&0.036$^*$\\
Fe\,{\sc i} & 4635.85& 1.0 & 1.49 & 2.0 & 1.89 & & \\
Fe\,{\sc ii} & 4923.93& 2.5 & 2.00 & 1.5 & 2.40 &0.58&0.079\\
Ti\,{\sc i} & 5017.95& 4.0 & 1.50 & 5.0 & 1.18 & & \\
Cr\,{\sc i} & 5018.15& 2.0 & 1.16 & 3.0 & 1.09 & & \\
Fe\,{\sc ii} & 5018.44& 2.5 & 2.00 & 2.5 & 1.87 &0.60&0.102\\
Cr\,{\sc ii} & 5018.84& 3.5 & 1.24 & 3.5 & 1.42 & & \\
Fe\,{\sc ii} & 5100.61& 4.5 & 1.54 & 4.5 & 1.34 & & \\
Fe\,{\sc ii} & 5100.66& 4.5 & 1.31 & 3.5 & 1.40 & & \\
Fe\,{\sc ii} & 5100.73& 4.5 & 1.54 & 5.5 & 1.36 & & \\
Fe\,{\sc ii} & 5100.85& 1.5 & 0.80 & 1.5 & 0.72 &0.47&0.083$^*$\\
Fe\,{\sc i} & 5168.90& 3.0 & 1.50 & 3.0 & 1.75 & & \\
Fe\,{\sc ii} & 5169.03& 2.5 & 2.00 & 3.5 & 1.70 &0.59&0.178\\
Fe\,{\sc i} & 5169.30& 4.0 & 1.26 & 3.0 & 1.32 & & \\
Fe\,{\sc ii} & 5197.48& 2.5 & 1.20 & 1.5 & 0.72 & & \\
Fe\,{\sc ii} & 5197.58& 2.5 & 0.57 & 1.5 & 0.44 &0.39&0.073\\
Fe\,{\sc ii} & 5362.74& 3.5 & 1.15 & 4.5 & 1.24 & & \\
Fe\,{\sc ii} & 5362.87& 4.5 & 1.15 & 3.5 & 1.40 &0.45&0.112$^*$\\
Cr\,{\sc i} & 5362.96& 3.0 & 1.33 & 3.0 & 1.06 & & \\
Fe\,{\sc ii} & 5362.97& 3.5 & 1.44 & 4.5 & 1.34 & &\\
Cr\,{\sc ii} & 5363.88& 3.5 & 1.71 & 3.5 & 1.39 & & \\
Fe\,{\sc ii} & 6247.35& 1.5 & 0.40 & 2.5 & 0.62 & & \\
Fe\,{\sc ii} & 6247.56& 2.5 & 1.33 & 1.5 & 1.72 &0.36&0.036$^*$\\
Fe\,{\sc ii} & 6432.68& 2.5 & 2.00 & 2.5 & 1.65 &0.24&0.026\\
Fe\,{\sc ii} & 6516.08& 2.5 & 2.00 & 3.5 & 1.58 &0.25&0.032\\
\hline
\end{tabular} \label{tab3}
\end{table}
\subsection{Procedure
\label{proc}}
From previous analyses of the Stokes $IQUV$ spectra of 78 Vir, it has been
found by Wade~et~al.~(\cite{Wade+00a})
that the {three} strong Fe\,{\sc ii}
lines $\lambda$4923.93\AA, $\lambda$5018.44\AA\, {and $\lambda$5169.03\AA\,}
show the strongest Zeeman signatures of any lines in the optical spectra of
CP stars. The real magnetic field structure of 78 Vir is expected to differ only
marginally from a centered magnetic dipole (based on previous modeling efforts:
Borra~\cite{Borra80}).%
The most attention is therefore paid
to the variability of the Stokes $I$ and $V$ profiles, which contain the most information
about the global surface magnetic field configuration.
The Stokes $I$ variation provides information about the surface distribution of
the chemical abundance, as well as the stellar radial velocity $V_{\rm r}$ and
projected rotational velocity $V_{\rm e}\sin{i}$.
The Stokes $V$ profiles are sensitive to the global field configuration,
in particular the lower-order multipolar components. The Stokes $Q$ and $U$ profiles
also provide some constraint on the global field morphology, but are most sensitive to
the smaller-scale structure of the field. Unfortunately, the available
Stokes $Q$ and $U$ profiles have low relative S/N ratio (Wade~et~al.~\cite{Wade+00a}),
and in this study they are used primarily to check the results of the Stokes $I$ and $V$
profile simulations
and to determine the sky-projected position angle of the rotational
axis, $\Omega$. According to Khalack et al.~(\cite{khalack+, khalack+03})
this angle ($0\leq\Omega<360\degr$)
is counted clockwise from the rotation axis to the North
Celestial Pole in the plane of the sky, and is related to the azimuth
angle $\Theta$ specified by Landolfi et al.~(\cite{Landolfi+93}) by
$\Omega=360\degr-\Theta$. The opposite counting of the position angle
in the MCD model is compensated for by the negative sign in
definition of the $B_{\rm y}$-component of the local magnetic field.
Hence a right-handed reference frame is applicable.
We have examined the spectra for additional lines with
prominent polarisation signatures, in order to compile a list of lines
that are especially appropriate for this task. To perform the line
identification we use the VALD-2 resources (Kupka~et~al.~\cite{Kupka+99};
Ryabchikova~et~al.~\cite{Ryab+99}). The adopted list contains Fe\,{\sc ii},
Fe\,{\sc i}, Cr\,{\sc ii}, Cr\,{\sc i}, Ti\,{\sc ii}, Ti\,{\sc i} and
Mg\,{\sc ii} lines, but for the present study we will concentrate primarily on the
strongest Fe\,{\sc ii} lines. The main reason for this is that Fe is presumably
almost uniformly distributed over the stellar surface (see Sect.~\ref{res}) and
in this way we exclude from the fitting procedure a model of the surface
abundance distribution. Table~\ref{tab3} presents the final list of Fe\,{\sc ii}
lines used in the simulation procedure, together with lines of some other
elements that are responsible for blends.
Some of the Fe\,{\sc ii} lines listed in Table~\ref{tab3} have
no detectable features in the Stokes $Q$ and $U$ spectra. Nevertheless, they are
comparatively strong lines (in the Stokes $I$ spectra) with clear
variability in the Stokes $V$ spectra, and are analysed without the linear
polarization data in order to check our final results.
All line profiles are simulated using the {\sc Zeeman2} polarised spectrum
synthesis code (Landstreet~\cite{Landstreet88}; Wade~et~al.~\cite{Wade+01}).
The code has been modified to include a magnetic field described within the
framework of the MCD method (Khalack~et~al.~\cite{khalack+03}), and to allow
for an automatic minimization of the model parameters using the {\it downhill
simplex method} (Press~et~al.~\cite{press+}).
The relatively poor efficiency of the downhill simplex method, requiring a large
number of function evaluations, is a well-known problem.
Repeating the minimization routine 3$\div$4 times in the vicinity of a supposed
minimum in the parameter space allows us to check if the method
converges to a global minimum.
In our case, this technique requires a
comparatively long computational time due to the large amount of analysed
observational data and the large number of free parameters.
\begin{table*}[th]
\parbox[t]{\textwidth}{
\caption[]{Results of Fe\,{\sc ii} lines simulation for $T_{\rm eff}=9250$K,
$\log{g}=4.5$ and $v_{\rm t}$=0 km~s$^{-1}$. %
The first column indicates the free parameters, while the other columns specify
the parameter values for the given line profile. The last
column provides the averaged estimation errors for each parameter.
The first 6 rows show the fit
quality for the each analysed Stokes profile (Eq.~\ref{chi2}), weighted and
unweighted (Eqs.~\ref{chi2wa}-\ref{chi2o}) $\chi^2$-function.
The following 12 rows show the best fit values of the free
model parameters, while the final 9 rows provide the characteristics of the
magnetic dipole poles, which are derived from the free parameters. }
\label{fe2sm}
\vspace{0.in}
\begin{tabular}{l|cccccccccccc}
\hline\hline
Line, \AA & 4620 & 4635 & 4923 & 5018 & 5100 & 5169 & 5197 & 5362 & 6247 & 6432 & 6516 &
$\sigma_{\rm er}$\\
\hline
$\chi^2_{I}$ & 2.92& 10.91& 8.96& 6.97& 7.40& 13.10& 16.06& 5.54& 2.27& 8.83& 5.94& \\
4$\chi^2_{V}$& 6.14& 7.86& 9.14& 8.69& 6.97& 9.09& 8.71& 6.64& 5.00& 12.24& 7.39& \\
6$\chi^2_{Q}$& 7.75& - & 7.73& 5.76& - & 8.23& 6.35& - & - & 12.15& 9.30& \\
6$\chi^2_{U}$& 7.42& - & 10.04& 6.82& - & 10.79& 6.80& - & - & 9.99& 6.19& \\
$\chi^2_w$ & 6.06& 9.39& 8.97& 7.06& 7.19& 10.30& 9.45& 6.09& 3.64& 10.81& 7.20& \\
$\chi^2$ & 1.75& 6.44& 3.55& 2.81& 4.57& 4.64& 5.11& 3.60& 1.76& 3.90& 2.59& \\
\hline
Qr, kG & 180 & 254 & 303 & 202 & 230 & 221 & 151 & 192 & 398 & 173 & 289 & 50\\
$a_{1}, 10^{-3}$&8.4& 6.7 & 4.3 & 6.7 & 5.3 & 7.3 & 12.6 & 9.7 & 5.4 & 8.9 & 6.4 & 1.2\\
$\lambda_{1}$& 13\dr& 24\dr& 34\dr& 41\dr& 33\dr& 15\dr& -4\dr& 19\dr& 69\dr& 23\dr& 20\dr& 5\dr\\
$\delta_{1}$ &-45\dr&-46\dr&-52\dr&-51\dr&-60\dr&-30\dr&-40\dr&-44\dr&-39\dr&-45\dr&-39\dr& 8\dr\\
$a_{2}, 10^{-3}$&3.1& 3.7 & 3.5 & 5.2 & 4.3 & 2.5 & 1.6 & 3.7 & 5.1 & 3.8 & 2.7 & 0.3\\
$\lambda_{2}$&118\dr&125\dr&144\dr&135\dr&145\dr& 85\dr&111\dr&101\dr&113\dr& 96\dr&111\dr& 10\dr\\
$\delta_{2}$ &-16\dr& -9\dr& -9\dr&-14\dr&-12\dr&-11\dr&-11\dr&-26\dr&-11\dr&-27\dr&-11\dr& 10\dr\\
$\Omega$ &109\dr& - &109\dr&109\dr& - &109\dr&102\dr& - & - &164\dr&128\dr& 17\dr\\
$i$ & 22\dr& 24\dr& 22\dr& 23\dr& 25\dr& 21\dr& 29\dr& 27\dr& 26\dr& 29\dr& 24\dr& 5\dr\\
$\log(Fe/N_{\rm tot})$& -3.36& -2.82& -3.26& -3.24& -3.20& -3.12& -3.25& -2.86& -3.17& -2.97& -3.50& 0.22\\
$V_r$ & -8.78& -8.42& -7.82& -7.51& -8.69& -7.23& -7.93& -8.25& -7.99& -6.57& -6.74& 1.18\\
$V_{\rm e}\sin{i}$& 10.9 & 11.1 & 11.9 & 12.0 & 11.5 & 12.6& 11.9 & 13.0 & 11.9 & 13.7 & 11.5 & 0.9\\
\hline
$\beta$ &125\dr&124\dr&122\dr&120\dr&123\dr&118\dr&129\dr&126\dr&123\dr&126\dr&122\dr& 5\dr\\
$a_{0}, 10^{-3}$&4.5& 3.9 & 2.9 & 6.0 & 3.4 & 4.3 & 6.5 & 5.8 & 4.8 & 5.6 & 3.6 & 1.6 \\
$a, 10^{-3}$ & 4.5 & 3.8 & 2.6 & 0.7 & 3.4 & 3.4 & 6.2 & 4.4 & 2.1 & 3.8 & 3.4 & 1.6\\
$B_{p}$, kG & 3.25 & 3.95 & 3.47 & 3.18 & 3.13 & 3.02 & 4.00 & 3.45 & 3.42 & 2.69 & 3.92 & 0.7\\
$\lambda_{p}$& -9\dr&-10\dr& -6\dr& -7\dr&-10\dr& -8\dr&-11\dr& -8\dr&-11\dr& -8\dr& -8\dr& 5\dr\\
$\delta_{p}$ &-35\dr&-33\dr&-30\dr&-30\dr&-33\dr&-28\dr&-37\dr&-36\dr&-34\dr&-36\dr&-32\dr& 12\dr\\
$B_{n}$, kG & -3.18& -3.82& -3.46& -3.16& -3.12& -2.95& -3.85& -3.36& -3.41& -2.62& -3.86& 0.7\\
$\lambda_{n}$&170\dr&170\dr&174\dr&173\dr&170\dr&172\dr&168\dr&171\dr&169\dr&171\dr&172\dr& 5\dr\\
$\delta_{n}$ & 35\dr& 33\dr& 30\dr& 30\dr& 33\dr& 28\dr& 37\dr& 35\dr& 33\dr& 36\dr& 32\dr& 12\dr\\
\hline\hline
\end{tabular}
}
\end{table*}
The free model parameters employed in the line profile simulation can be
divided into two groups: those that describe the magnetic field structure, and
those that describe the stellar geometry and atmospheric characteristics. The
first group includes the parameters $Q_{\rm r}$, $a_{\rm 1}$ and $a_{\rm 2}$
(which specify the modulus of ``magnetic charges" and their distance from the
center of the star, expressed in the units of stellar radius), and the
parameters $\lambda_{\rm 1}, \delta_{\rm 1}$ and $\lambda_{\rm 2}, \delta_{\rm
2}$ (which specify the spherical coordinates of the ``charges" in the rotational
reference frame of the star). Meanwhile, the second group includes the
parameters $\Omega$ (the position angle),
$i$ (the rotational axis inclination with respect to the line
of sight), $V_{\rm r}$ (the heliocentric radial velocity), $V_{\rm e}\sin{i}$
(the projected rotational velocity), and $\log(N_{\rm x}/N_{\rm tot})$ (the
abundance(s) of the element(s) forming the line to be modeled). Instead of
$V_{\rm e}\sin{i}$ we can obviously use only $V_{\rm e}$ as a free parameter,
but $V_{\rm e}\sin{i}$ is preferable here, because it allows us to compare our
estimate of this parameter directly with the results of other authors. The
minimum number of free model parameters is 11 for the MCD model if we do not
take into account the linear polarization data, and 12 if we do. Sometimes, to
obtain a good fit to the line profiles we need to include in the simulation
lines of other chemical elements (see Table~\ref{tab3}). In this case the
number of free model parameters grows with the number of chemical elements
under investigation, but usually does not exceed 15 parameters.
Initially we specify arbitrarily the values of the free model parameters in the
range of their possible values and simulate the Stokes $IQUV$ spectra at the
phases of the observations. We then compare the simulated profiles with the
observed spectra, and calculate the reduced $\chi^\mathrm{2}$, which we adopt
as a measure of the fit quality. The expression for the $\chi^\mathrm{2}$
function, which reflects the agreement between (for example) the simulated
Stokes $I$ ($I_{\rm \lambda_j}(\varphi_{\rm i})$) and observed Stokes $I$
($I^{obs}_{\rm i,\lambda_j}$) spectra is given by:
\begin{equation} \label{chi2}
\chi^{2}_{I} = \displaystyle \frac{1}{N_{I}} \sum^{N_{I}}_{i=1}
\displaystyle \frac{1}{N_{i}} \sum^{N_{i}}_{j=1}
\left( \displaystyle \frac{I^{obs}_{i,\lambda_j}-I_{\lambda_j}(\varphi_{i})}
{\sigma[I^{obs}_{i,\lambda_j}]}\right)^{2},
\end{equation}
\noindent where $\sigma[I^{obs}_{\rm i,\lambda_j}]$ corresponds to the measurement errors,
$\varphi_{\rm i}$ is the rotational phase of the observation, $N_{\rm I}$
specifies the number of spectra, while $N_{\rm i}$ is the number of pixels in
each analysed line profile.
In fact, although the simulated spectra are calculated with approximately the
same resolving power as the observed spectra, this does not provide a
direct coincidence of wavelengths in the simulated and observed spectra.
Therefore, during the $\chi^\mathrm{2}$ function evaluation the simulated
spectral intensity at the exact observed wavelength is calculated using a
linear interpolation.
The scale of variability and the measurement errors are different for each of
the four Stokes $IQUV$ spectra (Wade~et~al.~\cite{Wade+00a}). In order to
balance (from a statistical point of view) the information flow from each of the
four Stokes spectra, {the respective $\chi^\mathrm{2}$ contributions are weighted.
The weights are derived from the comparison of the best fit values of
$\chi^2_{\rm V}, \chi^2_{\rm Q}, \chi^2_{\rm U}$ with $\chi^2_{\rm I}$
(when the ordinary $\chi^{2}$ function is minimized) and provide approximately
the same weighted value of these functions for the majority of analysed lines
(see Table~\ref{fe2sm}). The weighted $\chi^\mathrm{2}_{\rm w}$
functions is specified} in the following way:
\begin{equation} \label{chi2wa}
\chi^{2}_{w}=\displaystyle \frac{1}{4}[\chi^{2}_{I}+
4\chi^{2}_{V}+6(\chi^{2}_{Q}+\chi^{2}_{U})],
\end{equation}
\noindent or
\begin{equation} \label{chi2wIV}
\chi^{2}_{w}=\displaystyle \frac{1}{2}[\chi^{2}_{I}+4\chi^{2}_{V}],
\end{equation}
\noindent if we work only with the Stokes $I$ and $V$ spectra. The value of the
$\chi^{2}_{\rm w}$ function is reduced throughout the minimization procedure,
varying the values of all free model parameters. When it reaches its
global minimum, we can also calculate the ordinary $\chi^{2}$ function:
\begin{equation} \label{chi2o}
\chi^{2}=\displaystyle \frac{1}{4}[\chi^{2}_{I}+
\chi^{2}_{V}+\chi^{2}_{Q}+\chi^{2}_{U}].
\end{equation}
\noindent We begin by simulating the strong
Fe\,{\sc ii} $\lambda$4923.927~\AA, $\lambda$5018.44
and $\lambda$5169.03 lines, for which the minimization process
has been repeated for 13$\div$15 times starting from different locations
in the free model parameter space in order to avoid local minima.
In the case of 78 Vir, the downhill simplex method converges to
four global minima, which are caused by the decentered dipole model
symmetry in the MCD method. Two global minima correspond to the
parameter sets $i, \beta, \Omega$ and $i, \beta, 180\degr+\Omega$
which differ only by the angle $\Omega$ and provide exactly the same value
of the $\chi^{2}_{\rm w}$-function. Models with the opposite direction of stellar
rotation ($180\degr-i, 180\degr-\beta, \Omega$ or
$180\degr-i, 180\degr-\beta, 180\degr+\Omega$) do not result in minima.
The other configurations ($i, 180\degr-\beta, \Omega$ and $i,
180\degr-\beta, 180\degr+\Omega$) correspond to models in which
the positive magnetic pole faces the observer. These
configurations are not applicable to the case of 78 Vir, where we
always see the negative magnetic pole, while the positive pole is
partially visible just for phases close to $\varphi=0.0$
(Babcock~\cite{Babcock47}; Borra~\cite{Borra80}). The other two
global minima have parameter sets which differ from the previous
ones by the location of the ``magnetic charges". The two
``magnetic charges" can be located in two different stellar
hemispheres (separated by the equatorial plane), or in a single
hemisphere. If they are not significantly shifted from the stellar
centre, they can form the same dipole axis and will produce almost
the same surface magnetic field configuration. The
differences between the two models can be revealed by the
sensitivity of $\chi^{2}_{\rm w}$ function on the parameter
values, but the minima will still exist. Preference is given to
the deepest global minimum, obtained for the model with the two
``magnetic charges" located in the same stellar hemisphere.
For the other lines the minimization process was performed only
3$\div$4 times (although if we tested the contribution of blends
to the analysed profiles, we ran the minimization routine several
more times). Supposing that all analysed lines contain the
signatures of the same magnetic field structure, we chose the
initial locations in parameter space not far from the parameter
set obtained from the strong line analysis.
In order to evaluate the fit errors (and therefore the uncertainties on the
derived free parameters), we calculate deviations of the simulated profiles
produced as a result of small variations of each of the free parameters, thus
introducing a small shift along one axis in the $\chi^\mathrm{2}$ hyper-space
from the point of the function minimum value. Using this procedure, and taking
into account the uncertainties of the observational data and the obtained
minimal value of $\chi^\mathrm{2}$-function, we can estimate the errors of the
best-fit parameters.
\subsection{Results}
\label{res}
Our initial assumption that iron is uniformly distributed over the surface of
78 Vir appears to be valid, given that the analysed Fe\,{\sc ii} lines reveal
no significant variability of the Stokes $I$ profiles with
rotational phase. Each Fe\,{\sc ii} line (or group of lines) shown in
Table~\ref{tab2} was analyzed independently of the others, using a
stellar atmosphere model with
$T_{\rm eff}=9250$K, $\log{g}=4.5$, $v_{\rm t}$=0 km s$^{-1}$.
For each best-fit simulation the derived free model parameters are given in
Table~\ref{fe2sm}.
During the simulation of some Fe\,{\sc ii} lines the
Stokes $Q$ and $U$ profiles were not taken into account,
and consequently the angle $\Omega$ is not determined for those lines.
The other data in Table~\ref{fe2sm}
are derived from the free
model parameters (Khalack~et~al.~\cite{khalack+03}). Here $\beta$ defines the
angle between the magnetic dipole axis and the stellar rotation axis (the angle
exists if these two axes cross each other). The distance of the magnetic dipole
center from the center of the star and one-half of the magnetic dipole size are
represented by variables $a_{\rm 0}$ and $a$, respectively. The variables
($B_{\rm p}, \lambda_{\rm p}, \delta_{\rm p}$) and ($B_{\rm n}, \lambda_{\rm n},
\delta_{\rm n}$) specify the location of the positive and negative magnetic poles
at the stellar surface and the respective strength of the magnetic field.
\subsubsection{The strong lines
\label{strong}}
The Fe\,{\sc ii} $\lambda$4923.927, $\lambda$5018.44
and $\lambda$5169.03 lines are the most
prominent Fe lines in the spectrum of 78 Vir, and show the clearest
Stokes $Q$ and $U$ signatures.
The first profile is formed essentially by the single line Fe\,{\sc ii}
$\lambda$4923.927 (see Table~\ref{tab3}) and appears to contain no
important blends. The agreement
between the observed and simulated data for Fe\,{\sc ii} $\lambda$4923.927
is similar to that shown in Fig.~\ref{fe5018IVQU} (for Fe~{\sc ii}~$\lambda 5018$).
Independent Stokes~$I$ spectra (each corresponding to a slightly different phase)
were obtained with each of the Stokes $V$, $Q$ and $U$ spectra, and
although only one set of Stokes $I$ profiles are presented in the figure, all
were taken into account during the calculation of the
$\chi_{\rm I}^2$-function (Eq.~\ref{chi2}). The best-fit values of
the Stokes $IVQU$ $\chi^2$-functions for the Fe\,{\sc ii} $\lambda$4923.927 line
are given in the third column of Table~\ref{fe2sm}.
The second line profile is composed mainly of the Fe\,{\sc ii} $\lambda$5018.44\AA\,
line, but is also contaminated by contributions from the Ti\,{\sc i}
$\lambda$5017.95, Cr\,{\sc i} $\lambda$5018.15 and
Cr\,{\sc ii} $\lambda$5018.84 lines. It seems that the
Cr\,{\sc ii} $\lambda$5018.84\AA\, is responsible for the
blend in the red wing of the Stokes~$I$ and $V$ profiles (Fig.~\ref{fe5018IVQU}).
Supposing a uniform Cr distribution,
this blend is
well fit for a Cr abundance of $\log Cr/N_{\rm tot}$=-3.38 dex.
The best fit $\chi^2$ values for the Stokes $IVQU$ profiles of Fe\,{\sc ii}
$\lambda$5018.44 are given in the fourth column of Table~\ref{fe2sm}.%
In the case of Fe\,{\sc ii} $\lambda$5169.033, the profile is blended by
the weaker Fe\,{\sc i} $\lambda$5168.898 and $\lambda$5169.296 lines
(see Table~\ref{tab3}). According to Kochukhov et al.~(\cite{Kochukhov+04})
$\log gf$=-0.786 given in the GRIFON list for the Fe\,{\sc i} $\lambda$5169.296\AA\,
line is too high to match the solar spectrum, and they recommend to use a
decreased $\log gf$=-2.15. This value is employed here for the respective
profile simulation and provides much better agreement with the observed data.
The behaviour of the best fit Stokes~$IVQU$ profiles with stellar rotational
phase for this particular line is very similar to that shown in
Fig.~\ref{fe5018IVQU} for Fe\,{\sc ii} $\lambda$5018.44. The sixth column of
Table~\ref{fe2sm}
presents the best fit parameters and the
model characteristics for Fe\,{\sc ii} $\lambda$5169.033.
The concordance of the observed and simulated Stokes~$I$ and $V$ profile
variations as a function of stellar rotation is extremely good for these lines.
On the other hand, the fit to the
Stokes $Q$ and $U$ profiles is only approximate.
The simulated profiles show generally the same intensity,
qualitative structure and variability as the observations.
However, even with these relatively noisy data, it is clear
that the model does not reproduce the observations within the
errors at some phases (for example, see phases 0.1174, 0.5339 and 0.9851 in Fig. 2).
In order to verify the assumption of a uniform surface distribution of iron in
the atmosphere of 78 Vir, we have calculated the difference in radial velocity
between the observed and simulated Stokes $I$ profiles for the strong
Fe\,{\sc ii} lines $\lambda$4923.927, $\lambda$5018.44 and
$\lambda$5169.033\AA\, (see Fig.~\ref{VelocityDiff}).
The mean (averaged for all the available observational phases) radial
velocities obtained from the simulation are taken into account in the calculation
of the velocities of the respective simulated profiles.
Fig.~\ref{VelocityDiff} shows that the derived differences in $V_{\rm r}$
almost coincide for all three lines. A moderate disagreement may exist only
for the data obtained in the vicinity of rotational phase $\varphi$=0.5, when
the negative magnetic pole is most visible. The variation of radial
velocity, which shows a reasonably coherent variation with phase from
about -1 to +1~km/s, may reflect a mildly non-uniform distribution of Fe,
unmodelled structure in the magnetic field, or other unaccounted-for physical
processes in the stellar atmosphere.
\subsubsection{The moderate strength lines}
From the list of the Fe\,{\sc ii} lines selected for analysis (see Table~\ref{tab3})
the $\lambda$4620.52, $\lambda$5197.58,
$\lambda$6432.68 and $\lambda$6516.08 lines are weaker
than the $\lambda$4923.927, $\lambda$5018.44
and $\lambda$5169.03 lines.
Due to the weaker polarised signal, a few spectra
with comparatively large observational errors have been excluded from the simulation
(phases 0.5757, 0.5847 and 0.5914).
The line Fe\,{\sc ii} $\lambda$4620.521 is primarily responsible for the
formation of the observed profile. No blends were taken into account during the
simulation in this case. %
The Fe\,{\sc ii} $\lambda$5197.577 line is blended by the
weak Fe\,{\sc ii} $\lambda$5197.48 line, but provides the main contribution
to the observed profile. The behaviour of the best fit Stokes~$IVQU$ profiles with
stellar rotational phase for these lines is very similar to that
shown in Fig.~\ref{fe6516IVQU} for Fe\,{\sc ii} $\lambda$4620.521\AA. The first
and seventh columns of Table~\ref{fe2sm} %
present the best fit parameters and the model characteristics for Fe\,{\sc ii}
lines $\lambda$4620.521\AA\, and $\lambda$5197.577\AA, respectively.
The Fe\,{\sc ii} $\lambda$6432.68\AA\, line is also essentially unblended. The
agreement between the observed and simulated Stokes~$IVQU$ data for this
particular line is very similar to that presented by
Fig.~\ref{fe6516IVQU}. The Fe\,{\sc ii} $\lambda$6516.08\AA\, line is the
primary contributor to the formation of the corresponding observed
Stokes~$IVQU$ profiles. The agreement of the best-fit simulation with the
observed data for Fe\,{\sc ii} $\lambda$6516.08\AA\, line is shown at
Fig.~\ref{fe6516IVQU}. The tenth and eleventh columns at the Tables~\ref{fe2sm}
contain the best fit parameters and model characteristics for the
Fe\,{\sc ii} $\lambda$6432.68\AA\, and $\lambda$6516.08\AA\, lines,
respectively. The best fit of the simulated Stokes~$IVQU$ profiles for the
Fe\,{\sc ii} $\lambda$6516.08\AA\, line is statistically better than that
obtained for Fe\,{\sc ii} $\lambda$6432.68\AA.
\subsubsection{Separate analysis of %
the Stokes $I$ and $V$ spectra}
The other Fe\,{\sc ii} lines $\lambda$4635.316\AA, the $\lambda$5100\AA-group,
$\lambda$5362.87\AA\, and $\lambda$6247.56\AA\, show no variability
in the Stokes~$Q$ and $U$ spectra and hence only the Stokes~$I$ and $V$ spectra
are taken into account during the simulation. These lines are selected for the
analysis in order to check the resulting magnetic field structure of 78 Vir on
the basis of a more thorough line list.
The Fe\,{\sc ii} $\lambda$4635.316 line is the main contributor to the
formation of the corresponding observed Stokes~$I$ and $V$ profiles, although
it is blended by the comparatively weak Fe\,{\sc i} $\lambda$4635.846\AA\,
line. The agreement between the observed and simulated data for this particular
line is similar to that shown in Fig.~\ref{fe5100IV}. The superposition of the
Fe\,{\sc ii} $\lambda$5100.607, $\lambda$5100.664, $\lambda$5100.727
and $\lambda$5100.852 lines is responsible for the formation of Stokes~$I$
and $V$ profiles at $\sim 5100.7$~\AA\ (see Table~\ref{tab3}).
Fig.~\ref{fe5100IV} shows the simulated Stokes~$I$ and $V$ profiles, which fit
well the observed profiles (see also the respective $\chi^2$ value in the fifth
column of Table~\ref{fe2sm}).
In the region of the Fe\,{\sc ii} $\lambda$5362.87 line the observed
Stokes~$I$ and $V$ profiles are formed mainly by this line and by a weak
contribution from the Fe\,{\sc ii} $\lambda$5362.74, $\lambda$5362.98,
Cr\,{\sc i} $\lambda$5362.87 and Cr\,{\sc ii} $\lambda$5363.88 lines.
The Fe\,{\sc ii} $\lambda$6247.557 line also
provides the main contribution to the observed Stokes~$I$ and $V$ profiles and is
blended by the Fe\,{\sc ii} $\lambda$6247.35 line.
The best fit simulated data show almost the same fit quality as is
shown in Fig.~\ref{fe5100IV}.
Simulation of the Stokes~$I$
and $V$ profiles in the regions of these lines results in almost the same
configuration of the magnetic field structure as obtained from the analysis of the
stronger Fe\,{\sc ii} lines, for which all four Stokes parameters
were taken into account during the simulation.
\subsection{Integral magnetic field characteristics}
\label{integral}
In order to check the agreement of the derived magnetic field model with other
available magnetic field data for 78 Vir, we calculate the intensity-weighted,
averaged (over the visible stellar disk) longitudinal magnetic field $B_{\rm l}$
and the normalized equivalent widths of the Stokes~$Q$ and $U$ profiles for all
the analysed phases. The normalization procedure is performed in accordance
with the method described by Wade~et~al. (\cite{Wade+00b}) over the passband of
the analysed line profiles.
All the available longitudinal magnetic field measurements for 78 Vir are
plotted in Fig.~\ref{Blon1} of Leone~\&~Catanzaro~(\cite{L+C01}). This figure shows
the good agreement of the $B_{\rm l}$ measurements obtained by
Wade~et~al.~(\cite{Wade+00b}) using the Least-Squares Deconvolution (LSD)
technique (Donati~et~al.~\cite{Donati+97}) with the majority of other
observational data. Therefore, in this paper we just compare our results with
the LSD longitudinal field data.
As demonstrated by Wade~et~al. (\cite{Wade+00b}), the 78 Vir Stokes~$Q$ and $U$
LSD equivalent widths are proportional to the BBLP measured at that phase
(Leroy~\cite{Leroy95}) with a line scaling factor. This factor might be
different for the analysed lines (see the last column in Table~~\ref{tab3}),
but is the same for the both Stokes~$Q$ and $U$ equivalent widths for a
particular spectral line.
Neither the $B_{\rm l}$ measurements nor the BBLP data are included directly
into the model minimization procedure. However, given that the model
successfully reproduces the Stokes profiles with which these quantities are
fundamentally related, we should expect that the model magnetic field
configuration is capable of reproducing them. Using the same stellar atmosphere
model, the longitudinal field calculated from the model fits to
the strongest Fe\,{\sc ii} line $\lambda$5018.44\AA\, and to
the weak Fe\,{\sc ii} line $\lambda$6432.68\AA\, show a
good agreement with the LSD $B_{\rm l}$ data (see Fig.~\ref{Blon1}). The
longitudinal fields calculated from the simulation results for the other
Fe\,{\sc ii} lines appear to be shifted downward in longitudinal
field intensity relative to the LSD $B_l$ variation (see Fig.~\ref{Blon1}).
\begin{table*}[th]
\parbox[t]{3.5in}{
\center{\caption[]{The same as at Table~\ref{fe2sm},
but for Cr\,{\sc ii} and Ti\,{\sc ii} lines}
\label{crti}}
\vspace{0.in}
\begin{tabular}{l|ccccccc|ccc}
\hline\hline
Line & Cr\,{\sc ii}& Cr\,{\sc ii}& Cr\,{\sc ii}& Cr\,{\sc ii}& Cr\,{\sc ii}&Cr\,{\sc ii}&
& Ti\,{\sc ii}& Ti\,{\sc ii}& \\
\AA & 4592 & 4634 & 5237 & 5310 & 5407 & 5421 & $\sigma_{\rm er}$ & 5188 & 5336 &
$\sigma_{\rm er}$ \\
\hline
$\chi^2_{\rm I}$ & 14.82& 13.12& 10.51& 6.47& 6.20& 9.48& & 21.53& 12.45& \\
$4\chi^2_{\rm V}$& 8.83& 6.39& 12.70& 8.64& 11.11& 11.50& & 24.28& 9.30& \\
$6\chi^2_{\rm Q}$& 5.62& 6.18& - & 8.18& 8.55& 11.76& & 8.77& 5.79& \\
$6\chi^2_{\rm U}$& 6.48& 5.88& - & 6.63& 9.17& 12.33& & 7.45& 6.45& \\
$\chi^2_{\rm w}$ & 8.95& 7.89& 11.60& 7.49& 8.76& 11.27& & 15.51& 8.50& \\
$\chi^2$ & 4.76& 4.18& 6.84& 2.77& 2.98& 4.09& & 7.58& 4.20& \\
\hline
Qr, kG & 175 & 186 & 185 & 182 & 161 & 171 & 30 & 287 & 271 & 40 \\
$a_{\rm 1}, 10^{-3}$&10.2& 9.5 & 9.7 & 8.9 & 9.6 & 8.6 & 1.5 & 3.7 & 3.8 & 1.4 \\
$\lambda_{\rm 1}$& 19\dr& 15\dr& 16\dr& 25\dr& 17\dr& 21\dr& 5\dr& 28\dr& 31\dr&17\dr\\
$\delta_{\rm 1}$ &-41\dr&-42\dr&-45\dr&-49\dr&-40\dr&-46\dr& 5\dr&-39\dr&-41\dr&19\dr\\
$a_{\rm 2}, 10^{-3}$&4.4& 4.1 & 3.8 & 3.9 & 3.6 & 3.7 & 0.3 & 2.8 & 3.5 & 1.0 \\
$\lambda_{\rm 2}$&103\dr&120\dr&111\dr&115\dr&109\dr&120\dr& 8\dr&113\dr&113\dr&19\dr\\
$\delta_{\rm 2}$ &-17\dr&-18\dr&-17\dr&-12\dr& -2\dr&-19\dr&10\dr& -8\dr&-14\dr&12\dr\\
$\Omega$ &109\dr&117\dr& - &123\dr&115\dr&109\dr&14\dr& 97\dr& 90\dr&19\dr\\
$i$ & 26\dr& 26\dr& 26\dr& 26\dr& 22\dr& 27\dr& 5\dr& 25\dr& 18\dr& 5\dr\\
$\log(El/N_{\rm tot})$
&-3.90 & -4.02& -4.04& -3.91& -4.18& -4.17& 0.19& -5.48& -5.68& 0.22\\
$V_{\rm r}$ & -8.25& -8.02& -8.80& -8.41& -8.05& -8.25& 1.23& -5.61& -7.91& 1.26\\
$V_{\rm e}\sin{i}$&11.7 & 10.8 & 11.8 & 11.5 & 10.9 & 12.2 & 1.1 & 13.4 & 13.3 & 1.5 \\
\hline
$\beta$ &123\dr&119\dr&126\dr&131\dr&127\dr&124\dr& 5\dr&117\dr&112\dr& 5\dr\\
$a_{\rm 0}, 10^{-3}$&6.1& 5.2 & 5.5 & 5.2 & 5.1 & 4.9 & 1.4 & 2.5 & 2.9 & 1.7 \\
$a, 10^{-3}$ & 5.0 & 5.1 & 5.0 & 4.6 & 5.1 & 4.4 & 1.4 & 2.1 & 2.2 & 1.7 \\
$B_{\rm p}$, kG & 3.52 & 3.86 & 3.71 & 3.36 & 3.32 & 3.05 & 0.4 & 2.46 & 2.41 & 0.5 \\
$\lambda_{\rm p}$&-11\dr&-10\dr&-11\dr& -8\dr& -8\dr& -7\dr& 8\dr&-18\dr&-23\dr& 8\dr\\
$\delta_{\rm p}$ &-33\dr&-29\dr&-36\dr&-41\dr&-37\dr&-34\dr&10\dr&-27\dr&-22\dr&11\dr\\
$B_{\rm n}$, kG & -3.44& -3.78& -3.62& -3.29& -3.24& -2.99& 0.4 & -2.45& -2.41& 0.5 \\
$\lambda_{\rm n}$&169\dr&170\dr&169\dr&172\dr&171\dr&173\dr& 8\dr&161\dr&156\dr& 8\dr\\
$\delta_{\rm n}$ & 33\dr& 29\dr& 35\dr& 40\dr& 37\dr& 34\dr&10\dr& 27\dr& 22\dr&11\dr\\
\hline\hline
\end{tabular}
}
\end{table*}
The calculated Stokes~$Q$ and $U$ equivalent widths for all the analysed line
profiles are multiplied by the respective line scaling factor in order to fit
them to the LSD Stokes~$Q$ and $U$ data. For each analysed line a scaling
factor is determined with the help of the Least Squares method. The respective
results are given in Table~\ref{tab3}, taking into account that the LSD
equivalent widths of Stokes~$Q$ and $U$ profiles have a line scaling factor of 0.1
(Wade~et~al.~\cite{Wade+00b}) from comparison with the BBLP data
(Leroy~\cite{Leroy95}). Fig.~\ref{Wade-4923} presents the Stokes $Q$ and $U$
equivalent widths variability with phase, derived
from the best fit simulations of Fe\,{\sc ii} $\lambda$4923.927
(with $\Omega=109\degr$) and Fe\,{\sc ii} $\lambda$6516.08 (with $\Omega=128\degr$).
They are scaled by a factor of 0.08 (for $\lambda 4923$) and 0.03 (for $\lambda 6516$)
and plotted over the
Stokes $Q$ and $U$ equivalent widths derived from LSD profiles. It is remarkable
that the simulated polarimetric data for Fe\,{\sc ii} $\lambda$4923.927\AA\,
and $\lambda$5018.44\AA\, lines have almost
the same scaling factor as the LSD Stokes $Q$ and $U$ equivalent widths.
Theoretically, the line scaling factor depends on the Land\'e factor and on the
degree of line saturation (Wade~et~al.~\cite{Wade+00a}). Table~\ref{tab3}
shows that in general the lines with a high mean Land\'e factor have also a
comparatively high line scaling factor. Nevertheless, this dependence is not
clearly pronounced by the sample of derived data. This is because, for some
lines, this effect is reduced by blend contamination of the analysed
Stokes $Q$ and $U$ profiles
(for example the Fe\,{\sc ii} $\lambda$5100 line group,
$\lambda$5197.03 and $\lambda$5362.87 lines) or by weak Stokes $Q$
and $U$ profile variability,
which results in a comparatively low precision of the calculated equivalent widths.
The LSD net linear polarisation (in the same way as the BBLP)
reflect the
contributions of many spectral lines with significant polarization. These lines
belong to a number of chemical elements, which apparently have non-uniform and
usually quite different abundance distributions. Such a composition of polarised
features results in LSD (or BBLP) data that may differ substantially from the
Stokes $Q$ and $U$ equivalent widths derived for a
particular Fe\,{\sc ii} line (see Fig.~\ref{Wade-4923}).
This is illustrated by the rather large differences between
the two simulated curves in Fig.~\ref{Wade-4923}. Moreover, lines of elements
with non-uniform abundance distributions can provide substantially different values
of the sky-projected position angle of the stellar rotation axis (see
Sect.~\ref{other}), and can exhibit totally different net linear polarisation
variations (e.g. 180$\degr$ out of phase with those shown here). The spectral
resolution and the observational errors of
the available spectra do not allow us to map confidently the detailed
distribution of chemical abundances nor the local patterns of the surface
magnetic field. The uncertainties in the details of the abundance distributions
and the magnetic field structure limit the precision of $\Omega$ as well as the
quality of simulated Stokes $Q$ and $U$ equivalent widths. Given these limitations,
we would characterise the agreement of the observed and simulated variations shown
in Fig.~\ref{Wade-4923} as very acceptable.
\subsection{Lines of other elements
\label{other}}
78 Vir is a chromium-rich star (Cowley~et~al.~\cite{C2J2}) and has some strong
Cr\,{\sc ii} lines, which are located in the observed spectral range. The most
prominent of them, Cr\,{\sc ii} $\lambda$4592.049, $\lambda$4634.07,
$\lambda$5237.329, $\lambda$5310.687,
$\lambda$5407.604 and $\lambda$5420.922\AA\,, have been selected for simulation.
Atomic parameters of spectral lines were extracted from the VALD database
(Kupka~et~al.~\cite{Kupka+99}) and from Raassen~\&~Uylings~(\cite{R+U98})
({\it ftp://ftp.wins.uva.nl/pub/orth}) in the case of Cr\,{\sc ii} lines.
These lines are not significantly blended by lines of other elements and
show variability of the Stokes $IV$ parameters with
phase. Some show marginally-detected Stokes $QU$ signatures as well.
We analyse also the two strong Ti\,{\sc ii} $\lambda$5188.68
and $\lambda$5336.771 lines.
These Ti lines are not significantly blended (although we take into account
a small contribution of the V\,{\sc i} $\lambda$5188.885 blend to the
profile formed by Ti\,{\sc ii} $\lambda$5188.68).
Table~\ref{crti} presents the results of the best fit simulations for the
Cr and Ti lines. %
The best fit quality is on average similar to that
obtained for the Fe\,{\sc ii} lines. The Stokes $I$ profiles for these particular
lines vary weakly with rotational phase, suggesting the presence of non-uniform
distributions of Cr\,{\sc ii} and Ti\,{\sc ii} in the atmosphere of 78 Vir.
To take into account the non-uniform abundance distributions the stellar
surface is divided into 30 areas, each characterised by an independent local
abundance of the analysed chemical element. The local abundances
are included as free parameters in the simulation procedure and the model
operates in this case with 41 parameters.
The best fit simulations of the aforementioned lines tentatively suggest that
chromium is enhanced in the vicinity of the negative magnetic pole,
while titanium is underabundant in this region (see Fig.~\ref{Abun}). The improvement
in the fit to Cr~{\sc ii} $\lambda 5310$ resulting from a non-uniform abundance distribution
is highly significant (reduction of $\chi^2_I$ by over 40\%), although the improvement for
Ti~{\sc ii} $\lambda 5336$ is much less so (just 10\%).
The averaged chromium and titanium abundances are given in Table~\ref{crti}.
It is remarkable that the best fit simulations of Cr\,{\sc ii} lines
provide almost the same global magnetic field structure and the
sky-projected position angle of the stellar rotation axis as the Fe lines
(see Tables~\ref{fe2sm},~\ref{crti}). Meanwhile, the Ti\,{\sc ii} simulation
results in somewhat lower values of $\Omega$ and $\beta$.
These differences may results from the
non-uniform Ti\,{\sc ii} distribution in the stellar atmosphere.
\section{Discussion}
\label{discuss}
The abundance of Fe\,{\sc ii} derived in this study is $\log{Fe/N_{\rm tot}}=-3.16\pm 0.20$
(see Table~\ref{fe2sm}).
From analysis of spectra of 78 Vir in the regions
3850-3870~\AA\ and 3868-4650~\AA\
Adelman~(\cite{Adelman73b}) obtained an Fe\,{\sc ii} abundance of
$\log{Fe/N_{\rm H}}=-2.89$dex.
The differences in the iron abundances
{may be due to the inclusion/exclusion of polarized transfer,
microturbulence, effects of Balmer line wings, etc.}, as well as from
the different effective temperatures that have been employed for the
abundance analysis (for 78 Vir Adelman~(\cite{Adelman73a}) applied
$T_{\rm eff}=9950$K, significantly higher than the temperatures employed
in our model. %
For the 11 different Fe features studied here, we observe no systematic difference
in strength between weak and strong lines. Therefore, the results of this study are
consistent with the absence of important Fe stratification in the atmosphere of 78~Vir.
According to our results the global magnetic field structure of 78
Vir is well-described by a slightly decentered magnetic dipole.
Usually the starting point of a simulation (in the free parameter
hyperspace) is chosen to correspond to a central magnetic dipole,
but for all the analysed lines the final model results in a
slightly decentered magnetic dipole. This fact is in good
qualitative agreement with the results previously obtained by
Borra (\cite{Borra80}). In that paper Borra considered the
classical decentered dipole model, where the magnetic dipole size
is insignificant in comparison with the stellar radius, and
obtained $a_{\rm 0}$=0.2. Our model provides a smaller
$a_{\rm 0}=0.006\pm 0.002$ due to the non-zero dipole size
($2a=0.008\pm 0.003$),
that allows
us to take into account the non-symmetrical (with respect to the
stellar center) magnetic field configuration of the star. The
magnetic field intensity and location of the positive and negative
magnetic poles in the stellar rotational reference frame are:
\begin{equation}\label{poles}
\begin{array}{lll}
B_{p}\!=\!3.4\pm 0.7~{\rm kG},&\!\lambda_{p}\!=\!-9^\circ\!\pm 5^\circ,&
\!\delta_{p}\!=\!-33^\circ\!\pm 12^\circ; \\
B_{n}\!=\! -3.3\pm 0.7~{\rm kG},&\!\lambda_{n}\!=\!171^\circ\!\pm 5^\circ,&
\!\delta_{n}\!=\!33^\circ\!\pm 12^\circ.
\end{array}
\end{equation}
\noindent Due to the shift of the dipole from the stellar centre, the positive
magnetic pole has a slightly stronger field intensity than the negative pole,
but not all the analysed lines are equally sensitive to this difference.
Nevertheless, the final model is hardly distinguishable
from a symmetric central dipole model, which provides a
slightly poorer agreement between the simulated and
observed data.
For Fe\,{\sc ii} lines %
the centered dipole
model results in a $\chi^{2}$-function which exceeds by 3\%$\div$5\% the best
fit result from the decentered dipole model. This fact can be related to the quality
of the data and to the model sensitivity.
The dipole offset could be a result of
the particular geometry of 78 Vir,
because we never completely see the positive magnetic pole of the star.
The modeled global magnetic field configuration of 78 Vir is illustrated in
Fig.~\ref{structure} according to the best fit results obtained for Fe\,{\sc ii}
$\lambda$5018\AA.
Five of the Fe\,{\sc ii} lines analysed using the entire Stokes
vector result in a plane-of-sky orientation of the rotational axis of
$\Omega=110\degr\pm17\degr$, while the last two lines (Fe\,{\sc ii}
$\lambda$6432\AA\, and $\lambda$6516\AA), with comparatively weaker variability
of the observed Stokes $Q$ and $U$ profiles, provide higher values of this
angle. Fig.~\ref{XOmega} shows the dependence of the $\chi^{2}_{\rm Q}$ and
$\chi^{2}_{\rm U}$ functions on the angle $\Omega$ for the three
strong Fe\,{\sc ii} lines (see Subsec.~\ref{strong}). Both
functions reach their minima at $\Omega=109\degr$ and $\Omega=289\degr$.
The second value describes a model which is indistinguishable from the configuration
specified in Table~\ref{fe2sm} due to the definition of the Stokes Q and U
parameters.
As discussed in Sect. 3, 78 Vir is a probable member of the Ursa Major stream.
It approaches us with a mean radial
velocity $V_{\rm r}=-8.1\pm1.0$ km~s$^{-1}$
(Wade~et~al.~2000a). This velocity varies by about $\pm2$ km~s$^{-1}$ with
the phase of stellar rotation (Preston~\cite{Preston69}, this work).
The possible variability of the radial velocity is not taken into account
in the simulation procedure. Respectively, our estimations of the radial
velocity for different Fe\,{\sc ii} lines are distributed around the
aforementioned value within the range $\pm1.5$ km~s$^{-1}$.
Nevertheless, the $V_{\rm r}$ variability through the rotational cycle has
been determined from the strong Fe\,{\sc ii} lines $\lambda$4923.927\AA\,
$\lambda$5018.44\AA\, and $\lambda$5169.033\AA\, (see Subsec.~\ref{strong}) and
for the other Fe\,{\sc ii} lines as well. We show that the mean radial
velocity derived from the simulation varies from line to line (see
Table~\ref{fe2sm}). The other Fe\,{\sc ii} lines provide a similar
$\Delta V_{\rm r}$ variability. The precision of the mean radial velocity
determination depends on the observational errors and on the quality of
the description of the surface magnetic field structure. For the blended profiles
it also depends on the accuracy of oscillator strengths.
The analysed lines show weak, coherent variations of
$\Delta V_{\rm r}$ with the period of stellar rotation (see Fig.~\ref{VelocityDiff}).
This suggests a possible moderately non-uniform distribution of Fe\,{\sc ii}.
Nevertheless, the non-uniform iron distribution will not lead to a significantly
different surface magnetic field structure. The simulation of Cr\,{\sc ii} lines
with the assumption of a non-uniform chromium distribution resulted in almost the
same field structure that we obtained from the simulation of Fe\,{\sc ii} lines.
The other best fit parameters are the rotational axis inclination
$i=24\degr\pm5\degr$ and the dipole axis obliquity $\beta=124\degr\pm5\degr$,
which are in a good agreement with the estimates of Leroy~et~al.~(\cite{Leroy+96}).
The derived $V_{\rm e}\sin{i}=12\pm1~{\rm km~s^{-1}}$
provides $V_{\rm e}=29\pm 4~{\rm km~s^{-1}}$ and is consistent with that
of Preston~(\cite{Preston71}).
The derived surface magnetic field variability interval
ranges from 2.1~kG to 3.2~kG and covers the value $B_s$=2.9~kG estimated by
Preston~(\cite{Preston71}). Besides, as Fig.~\ref{Blon1} shows, the longitudinal
magnetic field variation obtained from the two
Fe\,{\sc ii} lines simulation is also in good agreement with the LSD $B_{\rm l}$
data (Wade~et~al.~\cite{Wade+00b}). These facts justify the applicability of the
slightly decentered magnetic dipole model with the aforementioned values of free
parameters for the global magnetic field structure description at 78 Vir.
Unfortunately in the case of 78 Vir the intensity of the Stokes $Q$ and $U$
profiles is similar to the observational errors.
The data are unsuitable for performing an analysis of the small-scale structure of
the magnetic field, using a more sophisticated technique such as
Magnetic Doppler Imaging (MDI; Kochukhov~\&~Piskunov~\cite{K+P02}).
However, it appears, given the relatively good agreement between the intensity,
structure and variability of the observed and simulated Stokes $QU$ profiles,
that the real magnetic field of 78 Vir does not depart strongly from the configuration
derived here. At the same time, we have noted that the Stokes $QU$ profiles are not,
at some phases, fit to within their errors. Therefore these data, notwithstanding their
relatively low S/N, already suggest the limitations of the MCD model framework.
A similar analysis of the Stokes $IV$ profile variability was performed by
Kochukhov et al.~(\cite{Kochukhov+02}) for $\alpha^2$~CVn.
Those authors applied MDI
using ``multipolar regularization'',
analysing the Stokes $IV$ line profiles and obtained a good agreement between
observed and computed profiles for Fe\,{\sc ii},
Cr\,{\sc ii} and Si\,{\sc ii} lines.
The quality of the Stokes $I$ and $V$ profile fits obtained in this
paper is only marginally poorer
than that obtained by Kochukhov et al.~(\cite{Kochukhov+02}), although remarkably
the magnetic field model framework employed here is much simpler.
New spectro-polarimetric observations of 78 Vir with higher spectral resolution and signal-to-noise
ratio (especially for the Stokes $Q$ and $U$ spectra) are required
for further refinement of the global magnetic field
structure. With such data, a model with additional parameters (such as MDI) could be applied,
in order to study the local magnetic field topology and the detailed relationship between
the magnetic field and the abundance distributions.
\begin{acknowledgements}
The authors are grateful to Prof. John Landstreet and to the referee, Dr. O. Kochukhov,
for their valuable remarks and advice that have led to the improvement of this paper.
The authors acknowledge grant support from the Natural Sciences and Engineering Research Council of
Canada, and the Department of National Defence of Canada (DND-ARP).
\end{acknowledgements}
|
Title:
Massive and Red Objects predicted by a semianalytical model of galaxy formation |
Abstract: We study whether hierarchical galaxy formation in a concordance $\Lambda$CDM
universe can produce enough massive and red galaxies compared to the
observations. We implement a semi-analytical model in which the central black
holes gain their mass during major mergers of galaxies and the energy feedback
from active galaxy nuclei (AGN) suppresses the gas cooling in their host halos.
The energy feedback from AGN acts effectively only in massive galaxies when
supermassive black holes have been formed in the central bulges. Compared with
previous models without black hole formation, our model predicts more massive
and luminous galaxies at high redshift, agreeing with the observations of K20
up to $z\sim 3$. Also the predicted stellar mass density from massive galaxies
agrees with the observations of GDDS. Because of the energy feedback from AGN,
the formation of new stars is stopped in massive galaxies with the termination
of gas cooling and these galaxies soon become red with color $R-K>$5 (Vega
magnitude), comparable to the Extremely Red Objects (EROs) observed at redshift
$z\sim$1-2. Still the predicted number density of very EROs is lower than
observed at $z\sim 2$, and it may be related to inadequate descriptions of dust
extinction, star formation history and AGN feedback in those luminous galaxies.
| https://export.arxiv.org/pdf/astro-ph/0601685 |
\title{Massive and Red Objects predicted by a semianalytical model of
galaxy formation}
\author{X. Kang$^{1,2}$, Y. P. Jing$^{1}$, J. Silk$^{2}$}
\affil{$^1$ Shanghai Astronomical Observatory, Nandan Road 80, Shanghai, China}
\affil{$^2$ Astrophysics, University of Oxford, Denys Wilkinson
Building, Keble Road, Oxford OX1 3RH, UK}
\affil{e-mail: kangx@astro.ox.ac.uk}
\keywords{galaxies: formation---galaxies: evolution---galaxies:
luminosity function,mass function}
\section{Introduction}
There are many recent observations of high-redshift galaxies that
probe the star formation history of the Universe. The finding of many
massive galaxies, especially massive Extreme Red Objects (EROs), at
high redshift is particularly interesting. These observations show
that some EROs are passive ellipticals, and were already in place at
redshift z$\sim 2$. It is usually argued that in a Cold Dark Matter
(CDM) universe, structures form via a hierarchical formation process
in which small galaxies form first at early times, and massive
galaxies form later through the continuous mergers of the smaller
systems. With representative semi-analytical models (SAMs; Kauffmann
et al. 1999, Somerville \& Primack 1999, Cole et al. 2000), it was
found that in the concordance $\Lambda$CDM universe, it is difficult
to produce enough massive and red galaxies that look like those
observed(e.g. Cimatti et al. 2002a, Glazebrook et al. 2004). On the
other hand, the existence of the observed massive galaxies at high
redshift is not necessarily in conflict with the concordance
$\Lambda$CDM model, because the conversion of just ten percent of
baryons in dark matter halos of mass $M >10^{13}M_{\odot}$ to stars is
sufficient to produce the number of observed massive galaxies
(Somerville 2004a).
Many authors have studied the formation of these massive, red objects
using SAMs or Smoothed Particle Hydrodynamics (SPH) simulations. It
was shown that the SAMs (Kauffmann et al. 1999, Somerville \& Primack
1999, Cole et al. 2000) cannot produce enough massive/red objects at
redshift $z>1$ (e.g. Firth et al 2002, Somerville et al. 2004b, Daddi
et al. 2005). The SPH simulations (e.g. Nagamine et al. 2004, 2005)
have succeeded in producing massive and red galaxies at high redshift,
but at the cost of introducing more uncertainties. First, it is
unknown if these SPH simulations can produce the local galaxy
luminosity function. It seems that these simulations produce too many
bright galaxies at $z=0$ (Nagamine et al. 2004). Secondly, Nagamine
et al. (2005) used a high dust extinction for the entire galaxy population,
but the observations show that some EROs are passive
ellipticals with little dust extinction (Cimatti et al. 2002b).
The main reason that the SAMs fail to produce enough massive and
luminous galaxies at high redshift is that the gas cooling and star
formation in early massive halos is over-suppressed. In previous SAMs,
the gas cooling in massive halos is switched off in order not to
produce more luminous central galaxies than observed at redshift
$z=0$. The suppression of gas cooling is also motivated by the X-ray
observations that massive cooling flows are not observed in groups and
clusters (e.g. Peterson et al. 2003). But as the consequence, the gas
cooling may be over-suppressed at high redshift if a simplified
prescription is used for the cooling cutoff. For example, in the
Munich group model and also in Kang et al. (2005), the gas cooling is
shut off by hand in halos with the virial velocity greater than
$350km/s$. Since the halo mass is much lower at high redshift than at
the present for a given virial velocity, the gas cooling is suppressed
in this model for halos with the virial mass greater than 2.5$\times
10^{12}M_{\odot}$ at z $=$ 3. This artificial cooling switch-off seems
to be the main reason that these models do not produce as many massive
galaxies as observed.
In this paper, we implement a new model in which the energy from AGN
is used to suppress the cooling of hot gas in halos. Following
Kauffmann \& Haehnelt (2000) we use a simple model wherein black holes
gain most of their mass during major mergers. Our implementation of
the feedback from AGN is very similar to that used recently by Croton
et al. (2006) and Bower et al. (2005), and resembles a combination of
their models. In our model, the total energy from the AGN is
proportional to the Eddington luminosity of the central black hole and
the efficiency of reheating the gas is proportional to a power of the
virial velocity of the galaxy. Then the energy compensates for the
radiative energy of the cooling gas, and the actual cooling rate is
determined by the ratio between the two energies. The cooling is
totally suppressed if the energy from AGN is larger than the energy
radiated by the cooling gas. Compared with the previous model used by
Kang et al. (2005) with an artificial cut-off of the gas cooling in
the halos with the virial velocity larger than $350km/s$, the gas
cooling and AGN feedback in the new model are treated in a more
self-consistent way. The $M_{\rm bh}$-$\sigma$ relation of black hole
mass $M_{\rm bh}$ and the bulge velocity dispersion $\sigma$ implies
that massive black holes are present only in massive spheroids. In our
present model, the energy feedback from AGN indeed is efficient in
galaxies with a massive spheroid. We also require that the star
formation rate in quiescent disks is reduced at high redshift as
motivated by the observed evolution of cosmological cold gas content
with redshift (Keres et al. 2005); thus the gas-rich mergers result in
earlier formation of supermassive black holes in massive central
bulges. Once the energy feedback is enough to suppress the gas
cooling, the termination of new star formation will soon make the
galaxies red. We will compare the model prediction of the number density
of luminous galaxies with the K20 survey, and find that good agreement
holds up to z$\sim$3, beyond which there is little observational
data. Compared with previous SAMs, our present model can also produce
some very red ($R-K>5$, magnitudes are given in the Vega system unless
otherwise stated) passive ellipticals which are observed by the Great
Observatories Origins Deep Survey (GOODS) at z$\sim 1-2$.
We arrange our paper as follows. In section 2, we briefly introduce
our new model with AGN feedback and compare our model predictions with
the local galaxy population. In section 3, we give the model
predictions and compare them with the observations at high redshift.
Finally, we discuss our results and conclude our work in section 4.
\section{Model}
The SAM that we use here was described in detail by Kang et
al. (2005) who studied the local galaxy population. The merger tree is
constructed based on a high-resolution N-body simulation (Jing \& Suto
2002) of 512$^{3}$ particles in a box of 100$h^{-1}{\rm Mpc}$. The
cosmological parameters adopted there are $\Omega_{m} = 0.3$,
$\Omega_{\Lambda} = 0.7$, $h=0.7$, $\sigma_{8} = 0.9$. Here we still
use this simulation, but the SAM model is modified in two ways.
1. We adopt a star formation efficiency $\alpha \sim (1+z)^{-1}$ in a
quiescent disk that was shown to give a better match with the
evolution of cosmological cold gas content with redshift (Kauffmann \&
Haehnelt 2000, P$\acute{\rm e}$roux et al. 2003, Keres et al. 2005).
In the recent model of Durham group (Baugh et al. 2005, Bower et
al. 2005), they adopt a constant star formation timescale for the
disk. The star formation timescale used in our model is the dynamical
time of the disk which scales with redshift as $(1+z)^{-1.5}$. So the
star formation rate ($\dot{M_{\ast}}=\alpha M_{cold}/t_{dyn}$) of our
model differs from that of the Durham model only slightly. Note that
the relatively lower star formation rate in quiescent disks leaves
more cold gas which helps to produce massive black holes during galaxy
mergers at high redshift.
2. We include a model for the growth of black holes and for the energy
feedback from AGN to suppress the gas cooling. As the $M_{\rm
bh}$-$\sigma$ relation indicates that the central black holes grow
with the growth of the spheroid components, it is plausible that the
black holes get their mass through major mergers. But it is far from
clear about the exact way that the black holes accrete the surrounding
material. Here following Kauffmann \& Haehnelt (2000), we use a
simple parameterised form to describe the cold gas accreted by the
black hole during a major merger,
\begin{equation}
\Delta M_{bh} = F_{acc} \frac {M_{cold}} {1+(280km/s/V_{vir})^{2}}
\end{equation}
where $M_{cold}$ is the total cold gas in merging galaxies, and
$V_{vir}$ is the virial velocity of the post-merger host halo. We
normalize the parameter $F_{acc}$ by best matching the observed
$M_{bulge}-M_{bh}$ relation at z=0 (H\"aring \& Rix 2004). During the
gas accretion by black holes, part of the gravitational energy will be
converted into radiations which in turn will heat the surrounding cold
gas. But it is again unclear in a quantitative way about how much the
radiation is produced and how efficiently the cold gas is re-heated.
Croton et al. (2006) use a simple phenomenological model to describe
the accretion rate which depends on the hot gas fraction and circular
velocity of the halo, but the efficiency of heating the gas by AGN are
the same in all halos of different mass. Sijacki \& Springel (2006)
have shown that heating efficiency from a AGN bubble is lower in low
mass halos. Here we simply assume that the energy from the central AGN
is proportional to the Eddington luminosity $L_{edn}$ and the heating
efficiency is proportional to a power of the virial velocity of
the host halo. Thus the heating rate ejected into the gas is taken as,
\begin{equation}
L_{reheat}=F_{0}(V_{vir}/V_{\star})^{n}L_{edn}\,.
\end{equation}
If we denote the cooling rate in a halo of gas temperature $T$ by
$\dot{M}_{0,cool}$ in the case of no AGN feedback, then the cooling
rate $\dot{M}_{cool}$ in the presence of AGN feedback is:
\begin{equation}
\frac {\dot{M}_{cool}} {\dot{M}_{0,cool}} = 1 - \frac {L_{reheat}} {\frac {3} {4}\dot{M}_{0,cool}V_{vir}^{2}}.
\end{equation}
If the heating rate from AGN $L_{reheat}/\frac{3}{4}V_{vir}^2$ is
larger than the radiative cooling rate ${\dot{M}_{0,cool}}$, the gas
cooling is totally suppressed. We normalize the parameters $F_{0}$,
$V_{\star}$ to get a good match to the galaxy luminosity function at
z=0. In our model we obtain $F_{0}=2\times 10^{-5}$ and
$V_{\star}=200km/s$ and $n=4$.
In Fig.~\ref{fig:Bh_Bulge} we plot the relation between the bulge mass
and the black hole mass. The data points show for the model galaxies
and the solid line the best fit to the observations by H$\ddot{\rm
a}$ring \& Rix (2004). Here $F_{acc}$ is taken to be $0.01$. It is
seen that a simple model of black hole growth with a free parameter
can reproduce the observed $M_{bulge}-M_{bh}$ relation. After the
black hole mass is normalized, we then tune the parameters in equation
2 to get good fits to the local galaxy luminosity functions. In
Fig.~\ref{fig:LF_z0} we show the luminosity function at B$_{j}$ and K
bands. The upper panel shows a comparison with the 2dFGRS at B$_{j}$
band. The solid circles show the observational data of 2dFGRS, and
the thick solid histogram associated with Poisson errors is our model
prediction.
The lower panel shows the comparison at K band where the circles are
from Cole et al. (2001) and squares are the observations by Huang et
al. (2003). We find that the new model can produce the local galaxy
luminosity functions at blue and near-IR bands which are respectively
sensitive to the current star formation rate and the total stellar
mass in the galaxies. It has been shown (Croton et al. 2006, Bower et al. 2005)
that without an effective energy feedback, the predicted
luminosity functions at the bright end are too flat with many more
luminous galaxies predicted than observed. Note that here our model
predictions at high luminosity ends are still slightly higher than
observed. This might point to the fact that a more detailed model is
needed for AGN heating in massive halos which we will address in
future work.
\section{Results at high redshift}
As discussed in Section 1, the gas cooling in our new model is not
suppressed artificially but by heating due to the energy injected from
AGN in the galaxy center. So compared to previous SAMs without
AGN, the gas cooling and star formation can continues until a massive
spheroid forms at the galaxy center. It is expected that this model
can produce more massive and luminous galaxies at high redshift. In
Fig.~\ref{fig:K20_LF} we show the predicted rest-frame K band
luminosity function at z$\sim 1.5$. The squares with error bars are
the observational results from K20 (Pozzetti et al. 2003). The solid
circles are the predictions by the new model and the triangles show
the results predicted by Kang et al. (2005) where
they adopted a artificial shut off of gas cooling in galaxies with
$V_{vir}>350km/s$. We also re-plot the results of K band luminosity
function at z=0 by the solid line, taken from from lower panel of
Fig.\ref{fig:LF_z0}. It is clearly seen from the plot that the new
model produces more massive galaxies and the agreement with the
observations is very good. Also note that the good agreement holds for
faint galaxies as well, whereas it was reported previously that SAM
models produce more faint galaxies than observed (Pozzetti et
al. 2003).
Another test, firstly proposed by Kauffmann \& Charlot (1998), is the
evolution of the surface number density of galaxies at a fixed
limiting magnitude, which also widely used to constrain the
models. There are plenty of data from GOODS that are already publicly
available (Giavalisco et al. 2004). In Fig.~\ref{fig:GOODS_num} we
show the predicted redshift surface number density of galaxies with K$<20$.
The square points show the results of K20 and triangles are the data
from GOODS. The new model predictions are shown as the solid line,
and the dashed line shows the prediction by the model of Kang et
al. (2005). Here we find that compared with Somerville et al. (2004b)
who predicted much fewer luminous galaxies at $z>1.5$, the agreement
between our model and the observations holds much better up to z$\sim
3$. Here we also show how dust extinction will change the result. The
dotted line is the new model with the simple dust extinction model of
Calzetti et al. (2000) with $E(B-V)=0.1$. Clearly dust extinction has
no significant effect on the predicted number of galaxies in the
observed-frame K band up to z=3.
Though the predicted numbers of luminous galaxies agree with the
observations, it would be interesting to check the predicted color
distributions. The color is dependent on the star formation history
and on the dust extinction. At high redshift the galaxy mergers are
very frequent and the dust extinction is significant in these
starburst galaxies, but no reliable model of dust extinction is
available for such galaxies. Observations show that at z$\sim 1-2$
the EROs have contributions both from passive ellipticals with little
dust and from dust-enshrouded starburst galaxies (Cimatti et al. 2002b, Cimatti et al. 2003,
Yan \& Thompson 2003, Yan et al. 2004, Moustakas et al. 2004). Because there are
significant uncertainties in the dust extinction modelling for the
starburst galaxies, we think that the predicted number density of
passive ellipticals should set a more meaningful constraint on the
galaxy formation model. Here we take a simple model of dust
extinction. We classify the galaxies with starbursts produced during
the major mergers in the past 0.1Gyr as young starburst galaxies
and those otherwise as passive galaxies. We then use the Calzetti et
al. (2000) reddening law to model the dust extinction effect on the
galaxy color. The amount of dust in passive and young starburst
galaxies is difficult to assess, and here we simply assume a small
reddening $E(B-V)=0.05$ for the passive galaxies. The dust extent in
young starburst galaxy is expected to be high. Observations of
EROs show that some extremely red galaxies have heavy dust
extinction with $E(B-V)=0.4$. But the average extinction should be
lower. Here we assume a Gaussian distribution of $E(B-V)$ with a mean of
0.1 and a dispersion of 0.05 for the young starburst galaxies. Our
main motivation is to see if a simple dust reddening model can produce
the main features of the observed color distribution.
In Fig.~\ref{fig:GOODS_color} we show the observed $R-K$ (both in the
AB magnitude system, $(R-K)_{AB} \simeq (R-K)_{Vega}-1.65$) color distribution with a comparison with the
data which are from the GOODS Southern field in an area of 160
arcmin$^{2}$ (Somerville et al. 2004b). The upper panel shows the GOODS
data, which is from Figure 2 of Somerville et al. (2004b). The model
galaxies are selected using the magnitude cut and are normalized to
the same area of 160 arcmin$^{2}$. The total number of galaxies
selected in our model is 1595 which is $6\%$ higher than the GOODS
data points used here. The lower panel shows the model predictions.
In each panel we also show the evolution track of single burst stellar
populations with solar metallicity, the Salpeter IMF, and the ages (at
$z=0$) of 13.35 and 11.7 Gyrs (i.e. $z_{f}=26, 2.6$) based on the
model of BC03 (Bruzual \& Charlot 2003). From the figure, our model
can reproduce the main features of the observed galaxies: 1) many
extremely red galaxies ($R-K>4$) at $z>1$; 2) the bimodal color
distribution, red passive and young starburst galaxies at
$z>1.5$. Still there are some discrepancies. The predicted numbers of blue
galaxies are too prominent at z$<1.5$ and this might be due to the
inadequate treatment of star formation rate, stellar initial mass
function, or the dust extinction model. Also the predicted number of
extremely red galaxies with $(R-K)_{AB}>3.35$ at $z \sim 2$ is still
lower than observed. In our model there are enough luminous galaxies
but insufficient number of very red galaxies, which means that the
star formation (at $\sim 2$) in the current model are still high. There are two
possible reasons for this discrepancy. First the star formation is
not strong enough in the past in our model, as we do not include any
star formation during minor mergers which are also frequent at early
times. Second the energy from central AGN is not high enough to
suppress the hot gas cooling. Observations have shown that there are
already massive black holes ($\sim 10^{9}M_{\odot}$) at $z \sim 6$
(Fan et al. 2001), so the growth of black holes in massive galaxies
might be much quicker at early time than in our model in which the
fraction of cold gas accreted by black hole is constant with time. We
will address this in a forthcoming paper (Kang et al. 2006).
Glazebrook et al. (2004) used the Gemini Deep Deep Survey (GDDS) to
obtain the stellar mass distribution from $z \simeq$ 0.7 to 2. The
evolution of stellar mass density does place important constraints on
the formation model of massive spheroids. But due to the uncertainties
in fitting the multi-broad band colors of high redshift galaxies
including those of the IMF and dust extinction, the constraints are
weak. In Fig.~\ref{fig:stellar_GDDS}, we show the stellar mass
density of galaxies with stellar mass above certain
limits. The lines show the predicted stellar density in galaxies with
stellar mass in the range indicated in the plot. Black lines
are for this model and blue lines are from the model of Kang et
al. (2005) where they used an artificial cut of gas cooling in the halos
with $V_{vir} > 350 km/s$. We can still see a good match between the
model and the data. Although it seems that the stellar mass density
with $M_{\star}> 10^{10.46}M_{\odot}$ is higher than the data points,
it agrees with the integral of the star formation rate (see figure 4
of Glazebrook et al. 2004). Note that galaxies with
$M_{\star}>10^{11}M_{\odot}$ are in the sharply declining tail of the
mass function, therefore a small uncertainty in the estimated stellar
mass can introduce a very large uncertainty in the number density.
The hexagon in the plot shows the stellar mass density of massive galaxies with
$M_{\star}>10^{11}M_{\odot}$ recently obtained by van Dokkum et
al. (2006) making use of the deep multi-wavelength GOODS, FIRES and MUSYC surveys.
It is seen from the black dashed lines that our
model prediction is slight lower than the data by a factor of 2. At
high redshift the cosmic variance is so large in the observed catalogs
(about $60\%$, Somerville et al. 2004c) that the discrepancy might not
be serious.
\section{Discussion}
Here we have implemented a new semi-analytical model in which the
energy from AGN suppresses hot gas cooling in massive
halos. The growth of black holes and bulges, and the gas cooling, are
determined in a self-consistent way. In our description, the AGN
feedback becomes efficient in massive galaxies after a massive black
hole is formed in the galaxy center. The AGN feedback model has drawn
much recent attentions. The main motivation is that in massive groups
and clusters cooling flows are not observed. There should be some
physical process to reheat the cooling region, and the energy from AGN
has been proposed as an effective source (e.g. B$\ddot{\rm o}$hringer
et al. 2002, Begelmen et al. 2002, Sijacki \& Springel 2006). At the
same time, the AGN feedback models have also been incorporated into
the SAMs recently and it has been shown that AGN feedback can produce
a break of the luminosity function at the bright end and produce the
color-magnitude relation observed in SDSS (Croton et al. 2006, Bower
et al. 2005). Our model of AGN feedback is very similar to theirs in
spirit, but the detailed prescription is different. In this paper
we use this model to address some issues about the number distribution
and color distribution of galaxies at high redshift. We compare the
model predictions with the K20 and GOODS surveys. Our conclusions are
as follows.
\begin{itemize}
\item The predicted number distribution of $K<20$ galaxies matches
well with that of the GOODS and K20 galaxies up to a redshift of z
$\sim$ 3;
\item The predicted color distribution is similar to that observed in
the surveys and many extremely red galaxies ($R-K_{AB}>4$) are
produced, which has not been seen in previous models (Somerville et
al. 2004b). At $z > 1.5$ the galaxy population already displays a
bimodal color distribution;
\item The predicted stellar mass density can marginally agree with the
GDDS observation even with the uncertainties in the IMFs;
\end{itemize}
These results demonstrate that it is not difficult to produce massive
and red galaxies at z $\sim$ 1-2 in the concordance CDM universe. The
stellar mass in galaxy centers continues to grow until the energy from
central AGN is high enough to suppress the gas cooling. In our model
the black holes acquire most of their mass during major mergers, so
the AGN energy feedback is expected to be effective after the last
major merger which led to massive bulge formation at galactic centers.
In our model we can produce some of those passive ellipticals at
z$\sim 1-2$ with extremely red colors $(R-K)_{AB}>4$.
Many observations have shown that the star formation rate was higher
in massive galaxies at high redshift and these support the
"downsizing" formation scenario (Cowie et al. 1996). It is often argued that hierarchical galaxy
formation cannot reproduce the downsizing formation process. But
recent works (de Lucia et al. 2005, Bower et al. 2005, Scannapieco et
al. 2005) have shown that models with AGN feedback in the hierarchical
universe can reproduce the downsizing process in which the massive
galaxies forms earlier. In this paper, we also find that the predicted
luminous and massive galaxies are increased to the degree that is in
agreement with the observations, though the predicted number of red
galaxies may still be fewer than observed. Once more observations are
available on the dust extinction in these galaxies, the number density
and evolution of red passive ellipticals will put more stringent
constraints on the galaxy formation models. It is also possible that a
new ingredient is needed, such as the star formation induced by AGN
feedback prior to disruption of the cold gas supply (Silk 2005), in
order to make bulge formation more efficient and to account for the
chemical evolution of massive early-type galaxies.
\acknowledgments
We thank Mashiro Nagashima for kindly providing the GOODS data, and
Manfred Georg Kitzbichler for the binned data of GOODS and K20. Xi
Kang acknowledge support from the Royal Society China Royal Fellowship
Fellowship scheme. This work is supported in part by
NSFC(No. 10373012, 10533030) and by Shanghai Key Projects in Basic
research (04jc14079, 05xd14019).
\newpage
\clearpage
\clearpage
\clearpage
\clearpage
\clearpage
\clearpage
|
Title:
Lensed Quasar Hosts |
Abstract: Gravitational lensing assists in the detection of quasar hosts by amplifying
and distorting the host light away from the unresolved quasar core images. We
present the results of HST observations of 30 quasar hosts at redshifts 1 < z <
4.5. The hosts are small in size (r_e <~ 6 kpc), and span a range of
morphologies consistent with early-types (though smaller in mass) to
disky/late-type. The ratio of the black hole mass (MBH, from the virial
technique) to the bulge mass (M_bulge, from the stellar luminosity) at 1<z<1.7
is broadly consistent with the local value; while MBH/M_bulge at z>1.7 is a
factor of 3--6 higher than the local value. But, depending on the stellar
content the ratio may decline at z>4 (if E/S0-like), flatten off to 6--10 times
the local value (if Sbc-like), or continue to rise (if Im-like). We infer that
galaxy bulge masses must have grown by a factor of 3--6 over the redshift range
3>z>1, and then changed little since z~1. This suggests that the peak epoch of
galaxy formation for massive galaxies is above z~1. We also estimate the duty
cycle of luminous AGNs at z>1 to be ~1%, or 10^7 yrs, with sizable scatter.
| https://export.arxiv.org/pdf/astro-ph/0601391 |
\begin{frontmatter}
\title{Lensed Quasar Hosts\thanksref{label1}}
\thanks[label1]{Observations presented in this paper were obtained using the
Hubble Space Telescope, operated by the Space Telescope Science Institute under
contract to NASA.}
\author[STScI]{Chien Y. Peng},
\author[Steward]{Chris D. Impey},
\author[MPIA]{Hans-Walter Rix},
\author[CfA]{Emilio E. Falco}
\author[Rutgers] {Charles R. Keeton},
\author[OSU]{Chris S. Kochanek},
\author[CfA]{Joseph Leh\'ar}, \&
\author[CfA]{Brian A. McLeod}
\address[STScI]{Space Telescope Science Institute, 3700 San Martin Drive,
Baltimore, MD 21218}
\address[Steward]{Steward Observatory, Univ. of Arizona, 933 N. Cherry
Ave., Tucson, AZ 85721}
\address[MPIA]{Max-Planck-Institut f\"{u}r Astronomie, K\"onigstuhl 17,
Heidelberg, D-69117, Germany}
\address[CfA]{Harvard-Smithsonian Center for Astroph., 60 Garden
St., Cambridge, MA 02138}
\address[Rutgers]{Department of Physics \& Astronomy, Rutgers University, 136
Frelinghuysen Road, Piscataway, NJ 08854}
\address[OSU]{Department of Astronomy, The Ohio State University, 4055
McPherson Lab, 140 West 18th Avenue, Columbus, OH 43210}
\begin{keyword}
quasar \sep host galaxy \sep gravitational lensing \sep evolution
\sep supermassive black hole
\end{keyword}
\end{frontmatter}
\section{Introduction}
Elsewhere in these proceedings the reader can find summaries of previous work
on quasar hosts. We concentrate here on the benefits and challenges of using
gravitational lensing as a technique for measuring the host galaxy, especially
in regimes where traditional direct imaging methods are difficult -- at high
redshift, or when the host is either sub-luminous or compact. The context for
this work is the coevolution of galaxies and supermassive black holes. In the
local universe, a tight relation is observed between bulge mass (\mbulge) and
black hole mass (\mbh) measured with stellar kinematics \citep{Geb00, Fer00}.
We now extend the relation to $z \gtrsim 1$ using the virial technique to
estimate $M_{BH}$ (e.g. Kaspi et al., 2000; Vestergaard \& Peterson, 2006),
and the stellar luminosity to estimate \mbulge\ \citep{Kor95,Mag98}. The
existence, slope, and scatter of a \mbh/\mbulge\ relation at high redshift can
be used to analyze the relative growth rates of galaxies and their (presumably
ubiquitous) central engines.
The CfA-Arizona Space Telescope Lens Survey (CASTLES) is a project to image
all known, multiply-imaged, quasars in a homogeneous set of optical and near
infrared passbands using the {\it Hubble Space Telescope} ({\it HST}). With
the number of lens systems now near 100, data are in hand for 80 targets. The
observations are shallow, 1-2 orbits per filter, but the excellent surface
brightness sensitivity of the {\it HST} leads to a host detection in most
cases. The overall CASTLES project is described by \citet{Fal01}; early
results in the lensed host search are given by \citet{Rix01} in the same
conference proceedings; and detections and models of the hosts of several
individual quasars have been published \citep{Imp98,Koc00,Kee00}.
Gravitational lensing is a large and growing field of astrophysics so only the
rudiments can be given here; a number of excellent book and reviews are
available for the full formalism and diverse applications
\citep{Bla92,Sch92,Cla02}. About one in 500 quasars has a sight-line passing
close enough to the central potential of a massive galaxy for multiple image
formation. It has taken surveys of tens of thousands of radio and optically
selected quasars to yield the sample of $\sim$100 objects (see the CASTLES web
site\footnote{http://www.cfa.harvard.edu/castles} and that of Li\'ege
group\footnote{http://vela.astro.ulg.ac.be/themes/extragal/gravlens}). Even
though adaptive optics techniques from the ground are improving, stable point
spread functions, obtained with {\it HST}, are still essential for reliable
modeling.
In gravitational lensing, the AGN is magnified into multiple images, but
remains unresolved, whereas the extended light from the host galaxy maps into
arcs or Einstein rings (ER). A lens model is needed to extract the full
information content of the lensed host light. In principle, a typical ER
having a radius $\sim$ 1 arcsec means that {\it HST} imaging potentially
yields 50-100 resolution elements in the host galaxy in the deep images from
the survey.
\begin {figure}
\hskip 1.2cm
\vskip -0.1in
\vbox{
\hbox{
\hskip +1.0in
\psfig{file=pg1115.ps,height=3.5truein,angle=0}
}
}
\vskip -0.1in
\caption {An example of the lens modeling technique. (a) Original NIC2 image
of PG~1115+080 ($z_{\mbox QSO} = 1.72$). (b) The host galaxy Einstein ring,
after removing the best fit lensing galaxy and quasar point sources. (c) The
best fitting residuals. (d) The parametric model of the host galaxy in the
source plane.}
\label{fig:pg1115}
\end {figure}
\section{Modeling}
The image modeling (Fig.~\ref{fig:pg1115}) uses a custom-built program called
LENSFIT (Peng et al. 2006, in prep.), which is based on a methodology that
has been well-tested with the GALFIT algorithm \citep{Pen02}. The model for
the light profiles of the host and foreground (lens) galaxy uses a S\'ersic
model with a concentration index $n$ that is often used to quantify the gross
morphology of galaxies (e.g. $n=1$ for late-type, while $n=4$ for
early-type). Both the quasar point source and the host galaxy light profile
are propagated through the lens model to produce the image distortion, and
multiple images. External shear is included to model the tidal influence due
to neighbors. All the parameters are simultaneously varied to reduce the
$\chi^2$ on a pixel-by-pixel basis. The models are often very robust in well
resolved systems ($\theta \ge 1$ arcsec), due to the spatial separation
between the host and the lens, and their different shapes.
\begin {figure}
\hskip 1.2cm
\vskip -0.1in
\vbox{
\hbox{
\hskip +1.0in
\psfig{file=radio.ps,height=3.3truein,angle=0}
}
}
\vskip -0.1in
\caption {The (deprojected) host galaxy luminosity with redshift. Open
points: radio loud quasar hosts. Solid points: radio quiet or unknown
(most likely radio quiet). The stellar evolution tracks correspond to
$z_{form} = 5$, all normalized to $L^*$ by $z=0$. The E/S0 and Sb/c
tracks have an initial burst of stars, followed by star formation
rates that produce colors of E/S0 and Sb/c type galaxies by $z=0$.
The ``No Evolution'' model corresponds to a redshifted spectrum of a
$z=0$ E/S0 galaxy.
}
\label{fig:radio}
\end {figure}
\section{Results}
Here we present a summary of our findings, which are detailed along with a
description of our analysis techniques elsewhere (Peng et al. 2006, in prep.).
We select on lensing geometry size ($\theta \gtrsim 0.7$ arcsec), which does
not a priori bias the intrinsic AGN luminosity selection, or the host
luminosities. Therefore, we expect the sample of the AGNs to be randomly
drawn from the AGN luminosity function, where the lower limits are determined
by various lensing search programs. The heterogeneity of different surveys,
however, will not affect our primary conclusions about the relationships
between \mbh\ and \mbulge, since both quantities are measured in the {\it
same} objects.
\subsection {General Properties}
Figure \ref{fig:radio} shows the $H$-band host galaxy luminosities (lensing
distortion removed) versus redshift. Overall, the host luminosities from
CASTLES appear to agree well with non-lensing studies \citep{Kuk01,Rid01}.
The host luminosities range from 1 to 20 times $L^*_V$ today, while the AGNs
are 0 to 3 magnitudes brighter than the host in restframe $B$ to $V$ band.
Despite their brightnesses, the host galaxies appear to be fairly small in
size (typical $r_e \lesssim 6$ kpc) for their central AGNs. As we shall see
later, when coupled with information about their \mbh, both the host
luminosities and sizes lead to the conclusion that the bulges may be
undermassive compared to present-day normal galaxies. Lastly, the S\'ersic
index values suggest that while a number of quasar hosts at $z\gtrsim 1.5$
have steep central concentrations consistent with the presence of a bulge,
many (30\% -- 50\%) also have low S\'ersic values ($n \le 2$) more analogous
to later-type galaxies today. Even those galaxies with high S\'ersic indices
may not qualify as bona fide -- fully formed and passively evolving --
ellipticals, given their small sizes and black hole masses.
{\it Radio-loud (RLQ) vs. Radio-quiet (RQQ) hosts}\ \ \ \ \ Quantifying
differences between RLQ and RQQ hosts has historically been controversial.
Figure~\ref{fig:radio} shows that there is not a clear difference in the host
luminosities between RLQ (open) and RQQ (closed) AGNs in the lensing sample.
The diverse selection criteria from disjointed surveys are, however, hard to
quantify. It is worth to keep in mind, however, that in a study by
\citet{Kuk01} RLQs were drawn from the rare and extreme radio-loud sources
which may require atypically large BHs to produce. Consequently, the finding
of luminous hosts in those RLQs may reflect a correlation between
\mbh\ and \mbulge\ at high redshifts. The issue of radio correlation with
host properties remains unsettled.
{\it AGN Duty Cycle}\ \ \ \ \ We can estimate the duty cycle of nuclear
activity for each object with a host detection. A rough estimate of the duty
cycle is: $D \sim\Phi_Q(L_{QSO},z)/\Phi_G(L_{Gal},z)$, where $\Phi_Q$ and
$\Phi_Q$ are the luminosity functions of quasars and galaxies, respectively,
appropriate to a given redshift. At $z\gtrsim 1$, the median duty cycle is
1\%, or $10^7$ years, with a sizable scatter.
\subsection {Black Hole vs. Bulge Evolution}
Based on quasar and host luminosities we can study the \mbh\ vs. bulge
properties at $z\gtrsim 1$ (see Peng et al. 2005 for details), where \mbh's
are obtained using the virial technique \citep {Kas00,Ves06}.
{\it $1.7 \lesssim z\lesssim 4.5$}\ \ \ \ \ Fig.~\ref{fig:highz} shows the
\mbh\ vs. the restframe $R$-band bulge luminosity (\lr) for the host galaxies
at $1.7 \lesssim z \lesssim 4.5$. Determining \lr\ requires $K$-corrections,
computed using an Sbc SED; the dashed lines shows the small systematic effect
of using an E/S0 (right) or Im (left) SED. It is clear that a correlation
between \mbh\ and bulge luminosity was already present at a lookback time of
10--12 Gyr. Remarkably, the high-redshift hosts appear to lie on the {\em
same} relation as $z=0$ normal galaxies, implying that the high-$z$ hosts are
{\em undermassive} in comparison. To explain why, Fig.\ 3b shows the host
luminosities after we account for passive evolution of $dM_R/dz = -0.8$ mag;
specifically, we hold $M_{BH}$ fixed and shift the color points in Fig.\ 3a to
the left. Now the $z\gtrsim 1.7$ hosts are displaced from the local host
relation (solid line) by a factor of 3--6 in luminosity, which translates into
a mass deficit of a factor of 3--6 in the quasar hosts compared to local
galaxies with the same \mbh.
{\it $1\lesssim z \lesssim 1.7$}\ \ \ \ \ In contrast, by $z\approx 1$,
Fig.~\ref{fig:lowz} shows that the mass deficit of the hosts is mostly reduced
(to within a factor of unity in mass), after accounting solely for passive
evolution. Thus, massive bulges that correspond to luminous E/S0 galaxies
today may have been nearly assembled by $z\approx 1$.
By requiring that the hosts evolve onto the local \mbh\ vs. \lr\ of
Figs.~\ref{fig:highz} and \ref{fig:lowz}, we can illustrate a growth in the
\mbh/\mbulge\ ratio with redshift relative to today, shown in
Figure~\ref{fig:growth}. We find that the ratio of \mbh/\mbulge\ increases
roughly to a factor of 10 higher than today (assuming an Sbc-type SED) out at
$z=4$. This conclusion depends somewhat on the assumption of the SED and
evolutionary history: the ratio at earlier times would be higher than shown
for a bluer SED than Sb/c, or for a faster fading rate than passive evolution.
\begin {figure}
\hskip 1.2cm
\vskip -0.1in
\vbox{
\hbox{
\hskip +0.4in
\psfig{file=bh-bulge-highz.ps,height=4.5truein,angle=-90}
}
}
\vskip -0.1in
\caption {The relationship of the black hole mass, \mbh, vs. bulge absolute
luminosity (\lr, bottom axis; \mr, top axis), at low $z$ (solid round
points) and $z\gtrsim 1.7$ (open points). Solid lines: fit to
$z\approx0$ solid points. All open points assume a modern-day Sbc-type
SED for $K$-correction and their average is represented by dotted lines.
Dashed lines illustrate assumptions of bluer(Im, left)/redder(E, right)
SEDs. Open circle: gravitationally lensed quasar hosts. Open
triangles: \citet{Rid01}. Open squares: \citet{Kuk01}. Vertical line
in points: a possible lower limit in \mbh\ due to AGNs being broad
absorption line QSOs. Criss-crossed points: potential problem with
lens identification, host detection, radio-loud quasar, or narrow line
AGN. Panel ({\it a}): The observational data. Panel ({\it b}): The
same data in {\it a}, but the open points are shifted horizontally by
assuming that the hosts evolve {\it passively} with $z_f=5$ by
$d$\mr/$dz$ = $-0.8$ mag. See also \citet{Pen05} for details.
}
\label{fig:highz}
\end {figure}
\begin {figure}
\hskip 1.2cm
\vskip -0.1in
\vbox{
\hbox{
\hskip +0.4in
\psfig{file=bh-bulge-lowz.ps,height=4.5truein,angle=-90}
}
}
\vskip -0.1in
\caption {The same diagram as Figure \ref{fig:highz}, except for redshift of
$1 \lesssim z \lesssim 1.7$ quasar hosts. The vertical line in the
square points indicates that the AGN either has a strong narrow
Mg~{\sc ii} line component or is strongly absorbed in the wings,
causing a potentially low \mbh\ estimate. The dotted line is
displaced from the solid line, representing the local \mbh-\lr\
relation, by $-0.5$ (Fig. {\it a}) and $+0.5$ (Fig. {\it b})
magnitude. Note the very slight bias between the non-lensed and lensed
datasets, which might be explained by the difference in the median
redshift of $\left<z\right>_{\mbox{med}} = 1.45$ for the lens sample
and $\left<z\right>_{\mbox{med}} = 0.94$ for the non-lenses. However,
three of the non-lensed data points may also have a lower limit on the
\mbh\ estimate, as noted above.}
\label {fig:lowz}
\end {figure}
\begin {figure}
\hskip 1.2cm
\vskip -0.1in
\vbox{
\hbox{
\hskip +0.12in
\psfig{file=growth1.ps,height=2.4truein,angle=0}
\hskip +0.16in
\psfig{file=growth2.ps,height=2.4truein,angle=0}
}
}
\vskip -0.1in
\caption {The growth of the \mbh/\mstar\ ratio as a function of ({\it a})
redshift and ({\it b}) age of the universe in Gyrs. Circles are
gravitational lens data points, while triangles are from direct
imaging of hosts using {\it HST} NICMOS $H$-band \citep{Rid01,Kuk01}.
Point styles are the same as Fig.~\ref{fig:highz}. The \mbh/\mstar\
ratio appears to rise quickly beyond $z\approx 1$ and may slow, and
perhaps flatten, to a factor of $6-10\times$ local value by $z\approx
3$. A fading rate of $dM_R/dz=-0.8$ is assumed here (passive
evolution since $z_{form} = 5$).
}
\label{fig:growth}
\end {figure}
\section{Conclusions}
Detailed modeling of 30 well-observed systems from a total sample of 80 lensed
quasars has provided new insights into the properties of host galaxies at $1 <
z < 4.5$. About half have S\'ersic model fits indicative of early type
galaxies. However, combined with their small sizes of $r_e < 6$ kpc,
luminosities, and \mbh, it appears that luminous, fully-formed, ellipticals
are in a minority as hosts of luminous quasars at $z\gtrsim 2$. No difference
is seen between the luminosities of radio-loud and radio-quiet quasars in the
sample, with a caveat on sample selection. Even at $z \gtrsim 2$, the host
galaxies follow nearly the same relationship between \mbh\ and luminosity as
at low redshifts, but the bulges must gain in mass by a factor of 3-6 between
$1\lesssim z \lesssim 4.5$. However, by $z\approx1$, the mass deficit is
mostly gone. Thus massive bulges at $z\approx 1$ may be consistent with being
passively evolving, or may still grow by at most a factor of 1. Our estimate
of the AGN duty cycle is $\approx 1\%$, or $10^7$ years. Ongoing work
includes using color information to constrain the host star formation
histories, obtaining sub-millimeter data to measure star formation rate, and
characterizing detailed host morphology with deeper {\it HST} imaging.
|
Title:
Constraints on the coupled quintessence from cosmic microwave background anisotropy and matter power spectrum |
Abstract: We discuss the evolution of linear perturbations in a quintessence model in
which the scalar field is non-minimally coupled to cold dark matter. We
consider the effects of this coupling on both cosmic microwave background
temperature anisotropies and matter perturbations. Due to the modification of
the scale of cold dark matter as $\rho_{c} = \rho_{c}^{(0)} a^{-3 + \xi}$, we
can shift the turnover in the matter power spectrum even without changing the
present energy densities of matter and radiation. This can be used to constrain
the strength of the coupling. We find that the phenomenology of this model is
consistent with current observations up to the coupling power $n_{c} \leq 0.01$
while adopting the current parameters measured by WMAP. Upcoming cosmic
microwave background observations continuing to focus on resolving the higher
peaks may put strong constraints on the strength of the coupling.
| https://export.arxiv.org/pdf/astro-ph/0601333 |
\baselineskip=16pt
\begin{titlepage}
\rightline{astro-ph/0601333} \rightline{January 2006}
\begin{center}
\vspace{0.5cm}
\large {\bf Constraints on the coupled quintessence from cosmic
microwave background anisotropy and matter power spectrum}
\vspace*{5mm} \normalsize
{\bf Seokcheon Lee$^{\,1}$}, {\bf Guo-Chin Liu,$^{\,2}$ and
{\bf Kin-Wang Ng$^{\,1,2}$}}
\smallskip
\medskip
$^1${\it Institute of Physics,\\ Academia Sinica,
Taipei, Taiwan 11529, R.O.C.}
$^2${\it Institute of Astronomy and Astrophysics,\\
Academia Sinica, Taipei, Taiwan 11529, R.O.C.}
\smallskip
\end{center}
\vskip0.6in
\centerline{\large\bf Abstract} We discuss the evolution of linear
perturbations in a quintessence model in which the scalar field is
non-minimally coupled to cold dark matter. We consider the effects
of this coupling on both cosmic microwave background temperature
anisotropies and matter perturbations. Due to the modification of
the scale of cold dark matter as $\rho_{c} = \rho_{c}^{(0)} a^{-3
+ \xi}$, we can shift the turnover in the matter power spectrum
even without changing the present energy densities of matter and
radiation. This can be used to constrain the strength of the
coupling. We find that the phenomenology of this model is
consistent with current observations up to the coupling power
$n_{c} \leq 0.01$ while adopting the current parameters measured
by WMAP. Upcoming cosmic microwave background observations
continuing to focus on resolving the higher peaks may put strong
constraints on the strength of the coupling.
\vspace*{2mm}
\end{titlepage}
\section{Introduction}
\setcounter{equation}{0}
Analysis of the Hubble diagram of high redshift Type Ia supernovae
(SNe Ia) has discovered that the expansion of the Universe is
currently accelerating \cite{SCP}. In addition, combining
measurements of the acoustic peaks in the angular power spectrum
of the cosmic microwave background (CMB) anisotropy which indicate
the flatness of the Universe \cite{CMB} and the matter power
spectrum of large scale structure (LSS) which is inferred from
galaxy redshift surveys like the Sloan Digital Sky Survey (SDSS)
\cite{SDSS} and the $2$-degree Field Galaxy Redshift Survey
($2$dFGRS) \cite{2dFGRS} has confirmed that a component with
negative pressure (dark energy) should be added to the matter
component to make up the critical density today.
The cosmological constant and/or a quintessence field are the most
commonly accepted candidates for dark energy. The latter is a
dynamical scalar field leading to a time dependent equation of
state, $\omega_{\phi}$. Also, this scalar field has a fluctuating,
inhomogeneous component in order to conserve the equivalence
principle corresponding to the response of the new component to
the inhomogeneities in the surrounding cosmological fluid
\cite{CDS}.
Several new observational effects produced by the existence of
quintessence are imprinted in the CMB anisotropies and the matter
power spectrum when compared to models with the cosmological
constant. The locations of the acoustic peaks in the CMB angular
power spectrum are shifted due to their dependence on the amount
of dark energy today and at last scattering as well as
$\omega_{\phi}$ \cite{Doran, SLee}. Usually the Universe is
dominated by the quintessence at late times ($z \sim {\cal
O}(1)$), when the gravitational potential associated with the
density perturbations is changed due to a time dependent
$\omega_{\phi}$. This enhances the CMB anisotropies at large
angular scales by the integrated Sachs-Wolfe effect (ISW)
\cite{ISW}. Thus the amplitudes of both CMB angular and matter
power spectra decrease at large scale for the fixed Cosmic
Background Explorer Satellite (COBE) normalization compared to those
in the cold dark matter model with a cosmological constant
($\Lambda$CDM)~\cite{Dodelson1}.
The possibility that a scalar field at early cosmological times
follows an attractor-type solution and tracks the evolution of the
visible matter-energy density has been explored \cite{Ratra}. This
may help alleviate the severe fine-tuning associated with the
cosmological constant problem. However, this still cannot explain
the reason why the dark energy and the dark matter have comparable
energy densities at present. Recently models considering the
coupling of quintessence to dark matter have been investigated as
a possible solution for this late time coincidence problem
\cite{coupQ}. However, several of these models using a simple
coupling can be ruled out by observational constraints
\cite{coupQ1}.
These non-minimally coupled quintessence models have several different
observational effects compared to the minimally coupled models.
One of the most important effects is a different scaling of the
cold dark matter (CDM) compared to that of CDM of the minimally
coupled case. Since CDM scales as $\rho_{c} = \rho_{c}^{(0)} a^{-3
+ \xi}$ where $\xi < 0$, there will be more CDM energy density at
early epoch compared to the case with $\xi = 0$ ({\it i.e.} $n _c
=0$). As the coupling is increased, the locations of the acoustic
peaks are also shifted to smaller scales. However, the amplitudes
of the odd-number peaks decrease due to the decrease of the baryon
density at early time and the increasing ISW effect as the
coupling scaled with the COBE normalization. The amplitudes of the
even-number peaks increase due to the increase of the CDM energy
density. Even though the ratios of the height of the first peak to
those of higher peaks between different couplings are quite
similar to one another, these are quite different from the
$\Lambda$CDM model \cite{KMT}. The location of the turnover in the
matter power spectrum corresponds to the scale that entered the
Hubble radius when the universe became matter-dominated
($a_{eq}$). Thus the coupling might shift the turnover scale.
This paper is organized as follows. In the next section we show
the basic equations of linear perturbations of the coupled
quintessence model. In Sec. 3, we derive the formal entropy
perturbation due to multiple fluids. We also consider the
possibility of isocurvature perturbations of the quintessence. We
check the coupling effects on CMB and matter power spectrum in
Sec. 4. The effect of coupling on the metric perturbation is
considered in Sec. 5. Our conclusion is in the last section.
\section{Linear perturbations}
\setcounter{equation}{0}
We will consider the perturbation effect of the quintessence model
which is coupled to the CDM. First we start from the
metric in the conformal Newtonian (longitudinal) gauge, which is
restricted only for the scalar mode of the metric perturbations
\cite{Perturb}. The line element is given by \be ds^2 = a^2(\eta)
\Biggl[ -\Bigl(1 + 2 \Psi(\eta, \vec{x}) \Bigr) d\eta^2 + \Bigl(1
- 2 \Phi(\eta,\vec{x}) \Bigr) dx^i dx_i \Biggr], \label{CNG} \ee
where $\eta$ is the conformal time, $\Psi$ is the amplitude of
perturbation in the lapse function, and $\Phi$ is the amplitude of
perturbation of a unit spatial volume. We will consider only a
flat universe case. If the coupling is derived by a Brans-Dicke
Lagrangian, the radiation is decoupled from the dark energy
\cite{Amendola0}. Due to the strong constraint on the coupling to
the baryons from the local gravity experiments such as radar
time-delay measurements, we will assume that the baryons are
decoupled from quintessence \cite{Damour1}. Also from the
violation of the weak equivalence principle, even though it is
in a still
way that is locally unobservable, we can have the
species-dependent couplings \cite{Damour2}. So we will make an
assumption that the scalar field $\phi$ is coupled only to CDM by
means of a general function $\exp[B_{c}(\phi)]$ and there is no
coupling to
the baryons or the radiation \cite{Coupled}.
We can write the
general equation including this interaction as
\be S = - \int d^4x
\sqrt{-g} \Biggl\{ \frac{\bar{M}^2}{2} [\partial^{\mu} \phi
\partial_{\mu} \phi - R] + V(\phi) - {\cal L}_{c} - {\cal L}_{r}
- {\cal L}_{b} \Biggr\}, \label{lagrangian} \ee where $\bar{M} =
1/ \sqrt{8\pi G}$ is the reduced Planck mass and ${\cal L}_{i}$s
denote respectively CDM, radiation, and baryons. If we regard the
matter as a gas of pointlike particles with masses $m_{c}$ and
paths $x_{c}^{\nu}(t)$, then we can write ${\cal L}_{c}$ as \be
{\cal L}_{c} = - \frac{m_{c}}{\sqrt{-g}} \delta(\vec{x} -
\vec{x}_{c}(t)) (- g_{\mu\nu} \dot{x}_{c}^{\mu}
\dot{x}_{c}^{\nu})^{1/2}, \label{Lm} \ee where $m_{c} =
e^{B_{c}(\phi)} m^{*}_{c}$ ($m_{c}^{*}$ being a bare mass of the
CDM) ~\cite{Peebles,LOP}. Each fluid element has an
energy-momentum tensor $T^{\mu}_{(\beta) \nu}$ where $\beta$
includes all of the species. The total energy-momentum tensor is
covariantly conserved, however
the
energy-momentum transfer between CDM and quintessence is written
as
\ba \sum_{\beta} \nabla_{\mu} T^{\mu}_{(\beta) \nu} &=& 0, \label{dTbeta} \\
\nabla_{\mu} T^{\mu}_{(\gamma) \nu} &=& 0, \label{dTgamma} \\
\nabla_{\mu} T^{\mu}_{(d) \nu} &=& Q_{(d) \nu}, \label{dTd} \ea
where $\gamma$ denotes baryons or radiation,
$d$ denotes CDM or quintessence, and $Q_{(d) \nu}$ is the
energy-momentum transfer vector which shows the energy transfer to
the $d$-fluid \cite{Kodama}. This transfer vector is constrained
as \be \sum_{d} Q_{(d) \nu} = 0. \label{sumQ} \ee From the above
action (\ref{lagrangian}) we have the following equation which
will give the constraint equation for the interaction between the
quintessence and the CDM: \be \bar{M}^2 \Box \phi - \frac{\partial
V}{\partial \phi} - \frac{\partial B_{c}}{\partial \phi} {\cal
L}_{c} = 0. \label{boxphi} \ee From this we can find the unperturbed part of
the field equation, \be \phi'' + 2 {\cal H}
\phi' + \frac{a^2}{\bar{M}^2} \frac{\partial V(\phi)}{\partial
\phi} = - \frac{a^2}{\bar{M}^2} \frac{\partial
B_{c}(\phi)}{\partial \phi} \rho_{c}, \label{phieq} \ee where the prime
means $d/d\eta$, ${\cal H} = a'/a$, and $\rho_{c}$ means the energy
density of CDM. As we mentioned before the energy-momentum of each
species may not be conserved due to the scalar field coupling even
though the total energy momentum does conserve. If we use this
fact, then we can rewrite Eqs.~(\ref{dTbeta}),
(\ref{dTgamma}), and (\ref{dTd}) as
\ba \rho_{\tot}' &=& - 3 {\cal H} (\rho_{\tot} + p_{\tot}), \label{rhotot'} \\
\rho_{\gamma}' &=& - 3 {\cal H} (\rho_{\gamma} + p_{\gamma}), \label{rhogamma'} \\
\rho_{c}' &=& - 3 {\cal H} (\rho_{c} + p_{c}) + B_{c,\phi} \phi'
\rho_{c} \equiv -3 {\cal H}(\rho_{c} + p_{c})(1 - {\cal B}_{c}), \label{rhoc'} \\
\rho_{\phi}' &=& -3 {\cal H}(\rho_{\phi} + p_{\phi}) - B_{c,\phi}
\phi' \rho_{c} \equiv -3 {\cal H}(\rho_{\phi} + p_{\phi})(1 -
{\cal B}_{\phi}), \label{rhophi'} \ea where $B_{c,\phi} = \partial
B_{c}/ \partial \phi$. Henceafter we will adopt the potential and the coupling
as
given in Ref.~\cite{LOP}: \ba V(\phi) &=& V_{0} \exp \Bigl(
\frac{\lambda\phi ^2 }{2} \Bigr), \label{V} \\ \exp [B_{c}(\phi)]
&=& \Biggl(\frac{b_c+V(\phi)/V_0}{1+b_c}\Biggr)^{n_c}, ~~~ {\rm
with}~~ b_c+1>0, \label{BF} \ea
where $V_0$, $\lambda$, $b_c$, and $n_c$ are constant parameters.
Here we will consider only $b_{c} =0$ case.
\begin{center}
\end{center}
From these we have the evolution of the background quantities as
shown in
the first panel of Fig.~\ref{fig:Qnxi}.
The parameters we use in
this figure for the
present energy density contrasts of the quintessence and
the CDM are $\Omega_{\phi}^{(0)} = 0.76$ and $\Omega_{c}^{(0)} =
0.20$, respectively. To be compatible with observational data, the
energy density of the quintessence must be subdominant during the
big-bang
nucleosynthesis (BBN) $\Omega_{\phi}^{\rm{BBN}}(a
\sim 10^{-10}) < 0.2$ at $T \sim 1\,{\rm MeV}$~\cite{Ratra},
and the energy
density at present must be comparable to
the preferred range for
dark energy,
$0.60 \leq \Omega_{\phi}^{(0)}(a=0) \leq 0.85$
~\cite{CMB}.
The stronger bound on the energy density of
the quintessence during BBN,
$\Omega_{\phi}^{\rm{BBN}}(a \sim 10^{-10}) < 0.045$~\cite{Bean},
is also well satisfied in this model.
We note that the equation of state (EOS) of the quintessence
$\omega_{\phi}
\equiv p_\phi/\rho_\phi \approx \omega_r = 1/3$
during the radiation dominated era as in other tracker solutions.
Instead of
falling from the tracking value of $1/3$ towards $-1$, as in the
uncoupled case, $\omega_{\phi}$ increases towards higher values
($\leq +1$), before ultimately dropping to $-1$ at later times.
This is due to the fact that when the matter density increases,
the effect on the effective potential $V_{eff}(\phi)= V(\phi) +
\rho_{c}$ given in Eq. (\ref{phieq}) drives the faster evolution
of $\phi$. From Eq.~(\ref{rhoc'}), we find that the scaling of the
CDM energy density differs from the usual $a^{-3}$ due to the
coupling : \ba \rho_{c}(a) &=& \rho_{c}^{(0)} a^{-3} \Bigl(
\exp[B_{c}(\phi(x)) - B_{c}(\phi(0))] \Bigr) \equiv \rho_{c}^{(0)}
a^{-3 + \xi}, \label{rhoca} \\ \xi \cdot x &=& B_{c}(\phi(x)) -
B_{c}(\phi(0)), \label{xi} \ea where we set the present scale
factor as one ($a^{(0)} =1$), $x = \ln a$, and $\xi$ is the
deviation of the CDM redshift as a result of the coupling to a
scalar field. This is shown in the second panel of
Fig.~\ref{fig:Qnxi}. As the coupling is increased, the magnitude
of $\xi$ is increased. The $\xi$ depends also on the scale factor.
Consequently, the location of the turnover in the matter power
spectrum will be shifted due to the coupling as well (see below).
We can find the time-component of the source term in the
unperturbed background ($\bar{Q}_{(d) 0}$) from Eqs.~(\ref{dTd}),
(\ref{sumQ}), (\ref{rhoc'}), and (\ref{rhophi'}) which is given by
\be \bar{Q}_{(c) 0} = - \bar{Q}_{(\phi) 0} = - B_{c,\bar{\phi}}
\bar{\phi}' \bar{\rho}_{c}. \label{Q0} \ee Due to the conservation
of the total energy-momentum (\ref{dTbeta}), we can find the
constraint equation of ${\cal B}_{c}$ and ${\cal B}_{\phi}$, \be
(\rho_{c} + p_{c}) {\cal B}_{c} + (\rho_{\phi} +
p_{\phi}) {\cal B}_{\phi} = 0. \label{bconstrain} \ee %
To consider the perturbation we can decompose the scalar field as
\be \phi(\eta, \vec{x}) = \bar{\phi}(\eta) + \delta \phi (\eta,
\vec{x}) \label{pphi}, \ee where $\bar{\phi}$ is the unperturbed
part and $\delta \phi$ is the perturbed part of the scalar field.
We will express the perturbed parts of each quantities by means of
Fourier expansions. So the above perturbed scalar field will be
expressed as \be \delta \phi (\eta, \vec{x}) = \sum_{k} \delta
\phi_{k}(\eta) e^{i\vec{k} \cdot \vec{x}}, \label{Fourier1} \ee
where \be \delta \phi_{k}(\eta) = \frac{1}{V} \int \delta
\phi(\eta, \vec{x}) e^{-i \vec{k} \cdot \vec{x}} d^3{\vec x}.
\label{Fourier2} \ee Therefore, the
energy-momentum tensor of the scalar field can be decomposed into
the unperturbed part and the perturbed one. The background
energy-momentum tensors are \ba ^{(0)}T^{0}_{(\phi)0} &=& - \Bigl(
\frac{\bar{M}^2}{2 a^2} \bar{\phi'}^{2} + V(\bar{\phi}) \Bigr)
\equiv - \bar{\rho}_{\phi},
\label{0rhophi} \\
^{(0)}T^{i}_{(\phi)j} &=& \Bigl(\frac{\bar{M}^2}{2 a^2}
\bar{\phi'}^{2} - V(\bar{\phi})\Bigr) \delta^{i}_{j} \equiv
\bar{p}_{\phi} \delta^{i}_{j} \equiv \omega_{\phi}
\bar{\rho}_{\phi} \delta^{i}_{j}, \label{0pphi} \ea and
the first-order perturbed parts are \ba \delta T^{0}_{(\phi)0}
&=& \frac{1}{a^2} \Bigl( \bar{M}^2 \bar{\phi'}^{2} \Psi -
\bar{M}^2 \bar{\phi}' \delta \phi' - a^2 \frac{\partial
V(\bar{\phi})}{\partial \bar{\phi}} \delta \phi
\Bigr), \label{deltarhophi} \\
\delta T^{i}_{(\phi)j} &=& \frac{1}{a^2} \Bigl( - \bar{M}^2
\bar{\phi'}^{2} \Psi + \bar{M}^2 \bar{\phi}' \delta \phi' - a^2
\frac{\partial V(\bar{\phi})}{\partial \bar{\phi}} \delta \phi \Bigr)
\delta^{i}_{j}, \label{deltapphi} \\
\delta T^{0}_{(\phi)i} &=& -\frac{\bar{M}^2}{a^2} \bar{\phi}'
\partial_{i} (\delta \phi). \label{delta0iphi} \ea We can repeat
the similar consideration for the other components. We will regard
the CDM as a perfect fluid of energy density $\rho_{c}$ and pressure
$p_{c}$. To the linear order in the perturbations the energy
momentum tensors are given by \ba T^{0}_{(c) 0} &=& -\Biggl(
\bar{\rho}_{c} + \frac{\partial B_{c}(\bar{\phi})} {\partial
\bar{\phi}} \delta \phi \bar{\rho}_{c} + \delta \rho_{c} \Biggr)
\equiv -\Biggl(
\bar{\rho}_{c} + \delta \bar{\rho}_{c} \Biggr), \label{rhoc} \\
T^{0}_{(c) i} &=& ( \bar{\rho}_{c} + \bar{p}_{c})
{\it v}_{(c) i} = -T^{i}_{(c) 0}, \label{T0ic} \\
T^{i}_{(c) j} &=& \Biggl( \bar{p}_{c} + \frac{\partial
B_{c}(\bar{\phi})} {\partial \bar{\phi}} \delta \phi \bar{p}_{c} +
\delta p_{c} \Biggr) \delta^i_j + \Sigma^i_{(c) j} \equiv \Biggl(
\bar{p}_{c} + \delta \bar{p}_{c} \Biggr) \delta^i_j +
\Sigma^i_{(c) j}, \label{pc} \ea where ${\it v}^{i} =
dx^{i}/d\eta$ and $\Sigma^i_{j}$ is an anisotropic shear
perturbation. Here we define the perturbed part of the CDM as
$\delta \bar{\rho}_{c} = B_{c,\bar{\phi}} \delta \phi \bar{\rho}_{c} +
\delta \rho_{c}$, where the first term of the right hand side
is due to the coupling
of the scalar field to t
the CDM~\footnote{We can show this as follows: $\rho_{c} =
e^{B_c(\phi)} \rho_{c}^{*} = ( e^{B_c(\bar{\phi})} +
B_c(\bar{\phi})_{,\bar{\phi}} \delta \phi e^{B_c(\bar{\phi})})
(\bar{\rho}_{c}^{*} + \delta \rho_{c}^{*}) \sim \bar{\rho}_{c} +
B_c(\bar{\phi})_{,\bar{\phi}} \delta \phi \bar{\rho}_{c} + \delta
\rho_{c}$, where we means $B_c(\bar{\phi})_{,\bar{\phi}} =
\partial B_c(\bar{\phi})/ \partial \bar{\phi}$ and $\rho_{c}^{*}$ is the bare
energy density of the CDM.}. For the baryons and the radiation we
have the following equations, \ba T^{0}_{(\gamma) 0} &=& -\Bigl(
\bar{\rho}_{\gamma} + \delta \rho_{\gamma} \Bigr), \label{rhogamma} \\
T^{0}_{(\gamma) i} &=& \Biggl( \bar{\rho}_{\gamma} +
\bar{p}_{\gamma} \Biggr) {\it v}_{(\gamma)i} = - T^{i}_{(\gamma) 0},
\label{T0igamma} \\
T^{i}_{(\gamma) j} &=& \Bigl( \bar{p}_{\gamma} + \delta p_{\gamma}
\Bigr) \delta^i_j + \Sigma^i_{(\gamma) j}. \label{pgamma} \ea For a
flat Friedmann-Robertson-Walker universe we can collect the
unperturbed equations for each species: \ba 3 {\cal H}^2 &=&
\frac{a^2}{\bar{M}^2} \Bigl( \bar{\rho}_{r} + \bar{\rho}_{b} +
\bar{\rho}_{c} + \bar{\rho}_{\phi} \Bigr) \equiv
\frac{a^2}{\bar{M}^2} \sum_{\beta} \bar{\rho}_{\beta} \equiv
\frac{a^2}{\bar{M}^2} \bar{\rho}_{cr}, \label{H} \\
{\cal H}' &=& - \frac{a^2}{6 \bar{M}^2} \sum_{\beta} (1 + 3
\omega_{\beta}) \bar{\rho}_{\beta} = - \sum_{\beta} \frac{(1 + 3
\omega_{\beta})}{2} \bar{\Omega}_{\beta} {\cal H}^2, \label{H'} \\
\bar{\rho}_{\gamma}' &=& - 3 {\cal H} (\bar{\rho}_{\gamma} + \bar{p}_{\gamma})
,
\label{barrhogamma'} \\
\bar{\rho}_{c}' &=& - 3 {\cal H} (\bar{\rho}_{c} + \bar{p}_{c}) +
B_{c,\bar{\phi}} \bar{\phi}' \bar{\rho}_{c}
\equiv -3 {\cal H}(\bar{\rho}_{c} + \bar{p}_{c})(1 - \bar{{\cal B}}_{c}), \label{barrhoc'} \\
\bar{\rho}_{\phi}' &=& -3 {\cal H}(\bar{\rho}_{\phi} +
\bar{p}_{\phi}) - B_{c,\bar{\phi}} \bar{\phi}' \bar{\rho}_{c}
\equiv -3 {\cal H}(\bar{\rho}_{\phi} + \bar{p}_{\phi})(1 -
\bar{{\cal B}_{\phi}}). \label{barrhophi'}\ea If we include terms
up to the first order of the energy transfer vector (\ref{dTd}),
then we can write $Q_{(d) \nu}$ as \ba Q_{(d) 0} &=& -
\bar{Q}_{(d)} (1 + \Psi) - \delta Q_{(d)}, \label{Q0p} \\ Q_{(d)
i} &=& \Bigl( f_{(d)} + \bar{Q}_{(d)} v_{(d) i} \Bigr)_{,i}.
\label{Qip} \ea where $\delta Q_{(d)}$ and $f_{(d)}$ is the energy
and momentum transfer of the CDM or quintessence, respectively.
These terms should be included in the coupled quintessence models
when we use the conformal Newtonian gauge \cite{Kodama}. If we
missed this term, then we would have $2 \Psi B_{c,\bar{\phi}}$
instead of $3 \Psi B_{c,\bar{\phi}}$ in the last term of Eq.
~(\ref{deltaphi}). With using Eqs.~(\ref{Q0p}) and (\ref{Qip}), we
can find the perturbed part of the energy momentum conservation
equations in $k$-space. The perturbed equations for the scalar
field and the fluids which can be obtained from
Eqs.~(\ref{dTgamma}) and (\ref{dTd}) are given by \ba \delta \phi
'' &+& k^2 \delta \phi + 2 {\cal H} \delta \phi' + 2 \Psi
\frac{a^2}{\bar{M}^2} V_{,\bar{\phi}} + \frac{a^2}{\bar{M}^2}
V_{,\bar{\phi}\bar{\phi}} \delta \phi - (\Psi' + 3 \Phi')
\bar{\phi}' \nonumber
\\ &=& - \frac{a^2}{\bar{M}^2} \bar{\rho}_{c} \Biggl(
B_{c,\bar{\phi}\bar{\phi}} \delta \phi + B_{c,\bar{\phi}}^2
\delta \phi + B_{c,\bar{\phi}} \delta_{c} + 3 \Psi
B_{c,\bar{\phi}} \Biggr) \nonumber
\\ &=& - \frac{a^2}{\bar{M}^2} \bar{\rho}_{c} \Biggl(
B_{c,\bar{\phi}\bar{\phi}} \delta \phi + B_{c,\bar{\phi}}
\bar{\delta}_{c} + 3 \Psi B_{c,\bar{\phi}} \Biggr)
,
\label{deltaphi} \\
\frac{\delta \rho_{c}'}{\bar{\rho}_{c}} &=& -(1 + \omega_{c})
(\theta_{c} - 3 \Phi') - 3 {\cal H} \Bigl(\frac{\delta
p_{c}}{\delta \rho_{c}} + 1 \Bigr) \delta_{c} + B_{c,\bar{\phi}}
\bar{\phi}' \Bigl( \delta_{c} + \Psi \Bigr), \label{deltarhoc} \\
\frac{\delta \bar{\rho}_{c}'}{\bar{\rho}_{c}} &=& -(1 +
\omega_{c}) (\theta_{c} - 3 \Phi') - 3 {\cal H} \Bigl(\frac{\delta
\bar{p}_{c}}{\delta \bar{\rho}_{c}} + 1 \Bigr) \bar{\delta}_{c} +
B_{c,\bar{\phi} \bar{\phi}} \bar{\phi}' \delta \phi \nonumber
\\ &+& B_{c,\bar{\phi}} \Bigl(
\bar{\phi}' \bar{\delta}_{c} + \delta \phi' + \bar{\phi}' \Psi
\Bigr), \label{deltabarrhoc} \\
\delta_{c}' &=& -(1 + \omega_{c})(\theta_{c} - 3 \Phi') - 3 {\cal
H} \Bigl(\frac{\delta p_{c}}{\delta \rho_{c}}
- \omega_{c} \Bigr) \delta_{c} + B_{c,\bar{\phi}} \bar{\phi}' \Psi, \label{deltac} \\
\bar{\delta}_{c}' &=& -(1 + \omega_{c})(\theta_{c} - 3 \Phi') - 3
{\cal H} \Bigl(\frac{\delta p_{c}}{\delta \rho_{c}} - \omega_{c}
\Bigr) \bar{\delta}_{c} + B_{c,\bar{\phi} \bar{\phi}}
\bar{\phi}' \delta \phi + B_{c,\bar{\phi}} \delta \phi'
\nonumber \\ &+& B_{c,\bar{\phi}} \bar{\phi}' \Psi, \label{bardeltac} \\
\theta_{c}' &=& - {\cal H} (1 -3 \omega_{c}) \theta_{c} -
\frac{\omega_{c}'}{1 + \omega_{c}} \theta_{c} + \frac{\delta
p_{c}/\delta \rho_{c}}{ 1 + \omega_{c}} k^2 \delta_{c} - k^2
\sigma_{c} + k^2 \Psi \nonumber \\ &+&
\fr{\omega_{c}}{(1+\omega_{c})} k^2 B_{c,\bar{\phi}} \delta \phi
- \fr{\omega_{c}}{(1+\omega_{c})} B_{c,\bar{\phi}} \bar{\phi}'
\theta_{c} \label{thetac} \\
&=& - {\cal H} (1 -3 \omega_{c}) \theta_{c} - \frac{\omega_{c}'}{1
+ \omega_{c}} \theta_{c} + \frac{\delta \bar{p}_{c}/\delta
\bar{\rho}_{c}}{ 1 + \omega_{c}} k^2
\bar{\delta}_{c} - k^2 \sigma_{c} + k^2 \Psi \nonumber \\
&-& \fr{\omega_{c}}{(1+\omega_{c})} B_{c,\bar{\phi}} \bar{\phi}'
\theta_{c}, \label{barthetac} \ea where we define $\delta_{c} =
\delta \rho_{c}/ \bar{\rho}_{c}$, $\bar{\delta}_{c} = \delta
\bar{\rho}_{c}/ \bar{\rho}_{c}$, $\theta_c=i \vec{k} \cdot
\vec{v}$, $\omega_c$ denotes the equation of state parameter (EOS)
of the CDM, and $\sigma_c$ is related to the CDM anisotropic
stress perturbation $\Pi_{c}$, by $\sigma_c = 2 \Pi_{c} \,
\bar{p_{c}}/3(\bar{\rho}_c + \bar{p}_c)$. We express both
$\delta_{c}'$ and $\bar{\delta}_{c}'$ explicitly in order to show
that the equations of the energy density perturbation can be
different with different definitions. However, it is the coupled
energy density which is measured in observations. As such, we will
use $\bar{\delta}_{\beta}$ as the energy density contrast of each
species. %
Every term containing $B_{c}(\phi)$ in the above equations comes
from the coupling. If we drop all these terms, then they are
obviously identical to the expressions Ref.~\cite{MaB}. If we have
more than one ideal gas components, then we have entropy
perturbation ($\delta S$) which can be expressed as \be \delta
p_{\tot} = \sum_{\beta} \Bigl[ (\partial p_{\beta}/\partial
\rho_{\beta})|_{S} \, \delta \rho_{\tot} + (\partial p_{\beta}/
\partial S_{\beta})|_{\rho} \, \delta S_{\tot} \Bigr] \equiv c_{\tot}^2 \delta \rho_{\tot} +
p_{\tot} \Gamma_{\tot} , \label{deltaS} \ee where $c_{\tot}^2$ is
the overall adiabatic sound speed squared and $\Gamma_{\tot}$ is
the total entropy perturbation. We will consider this more
carefully in the following section. We can write the perturbed
equations for the CDM from the above generic perturbation
equations (\ref{bardeltac}) and (\ref{barthetac}). However due to
the coupling between the baryons and the photons we also need to
consider the Thomson scattering term in the baryon-photon fluid.
After including this we have the following equations.
\ba \delta_{b}' &=& -\theta_{b} + 3 \Phi', \label{deltab} \\
\theta_{b}' &=& -{\cal H} \theta_{b} + c_{s}^2 k^2 \delta_{b} +
k^2 \Psi + \frac{4 \bar{\rho}_{\gamma}}{3 \bar{\rho}_{b}} a n_{e}
\sigma_{T} (\theta_{\gamma} - \theta_{b}), \label{thetab} \\
\delta_{c}' &=& -\theta_{c} + 3 \Phi' + B_{c,\bar{\phi}} \bar{\phi}' \Psi, \label{deltac2} \\
\bar{\delta}_{c}' &=& -\theta_{c} + 3 \Phi' + B_{c,\bar{\phi}
\bar{\phi}} \bar{\phi}' \delta \phi +
B_{c,\bar{\phi}} \delta \phi' + B_{c,\bar{\phi}} \bar{\phi}' \Psi, \label{bardeltac2} \\
\theta_{c}' &=& -{\cal H} \theta_{c} + k^2 \Psi, \label{thetac2} \\
\delta_{r}' &=& - \frac{4}{3} \theta_{r} + 4 \Phi'
,
\label{deltar} \\
\theta_{r}' &=& k^2 \Bigl( \frac{1}{4} \delta_{r} - \sigma_{r}
\Bigr) + k^2 \Psi + a n_{e} \sigma_{T} (\theta_{b} -
\theta_{\gamma}), \label{thetar} \ea where $n_{e}$ is the electron
number density, $\sigma_{T}$ is the cross section for the Thomson
scattering.
\section{Entropy perturbation}
\setcounter{equation}{0}
Let us start from the definition of the total energy density
perturbation and that of the total pressure perturbation, \ba
\delta \bar{\rho}_{\rm{tot}} &=& \sum_{\beta} \delta
\bar{\rho}_{\beta}, \label{deltarhotot} \\
\delta \bar{p}_{\rm{tot}} &=& \sum_{\beta} \delta \bar{p}_{\beta}
. \label{deltaptot} \ea For a given $p_{\rm{tot}}(\rho,S)$, the
pressure fluctuation can be expressed as \be \delta p_{\tot}
\equiv c_{\tot}^2 \delta \rho_{\tot} + p_{\tot} \Gamma_{\rm{int}}
+ p_{\tot} \Gamma_{\rel}, \label{deltaptot2} \ee where $S$ is an
entropy, $c_{\tot}^2$ is the overall adiabatic sound speed
squared, $\Gamma_{\rm{int}}$ and $\Gamma_{\rel}$ are the intrinsic
and the relative entropy perturbations respectively. We have \ba
p_{\tot} \Gamma_{\rm{int}} &=& \sum_{\beta} p_{\beta}
\Gamma_{\beta} ,
\label{Gammaint} \\
p_{\tot} \Gamma_{\rel} &=& \sum_{\beta} (c_{\beta}^2 - c_{\tot}^2)
\delta \rho_{\beta}, \label{Gammarel} \\
c_{\tot}^2 &=& \frac{\sum_{\beta}
c_{\beta}^2\rho_{\beta}'}{\rho_{\tot}'}, \label{ctot} \ea where
$\Gamma_{\rm{int}}$ is the sum of the intrinsic entropy
perturbation of each fluid and $\Gamma_{\rel}$ arises from the
relative evolution between fluids with different sound speeds. As
we mentioned before the energy momentum of each species may not be
conserved due to the scalar field coupling even though the total
energy momentum does conserve. We can rewrite the equation for the
total adiabatic sound speed (\ref{ctot}) as \be c_{\tot}^2 =
\sum_{\beta} c_{\beta}^2 (1 -{\cal B}_{\beta}) \frac{\rho_{\beta}
+ p_{\beta}}{\rho_{\tot} + p_{\tot}}. \label{ctot2} \ee Now we can
rewrite the relative entropy perturbation as \be p_{\tot}
\Gamma_{\rel} = \frac{1}{2} \sum_{\beta,\alpha}
\frac{(\rho_{\beta} + p_{\beta})(\rho_{\alpha} + p_{\alpha})}
{\rho_{\tot} + p_{\tot}}(c_{\beta}^2 - c_{\alpha}^2) S_{\beta
\alpha} + \sum_{\beta} {\cal B}_{\beta} c_{\beta}^2 (\rho_{\beta}
+ p_{\beta}) \Delta, \label{Gammarel2} \ee where \ba S_{\beta
\alpha} &=& \Delta_{\beta} - \Delta_{\alpha}, \label{S}
\\ \Delta_{\beta} &=& \frac{\delta \rho_{\beta}}{\rho_{\beta} +
p_{\beta}}, \label{Deltabeta} \\
\Delta &=& \frac{\delta \rho_{\tot}}{\rho_{\tot} + p_{\tot}} ,
\label{Delta} \ea where $S_{\beta \alpha}$ is the entropy
perturbation \cite{Kodama}. Due to the $\Delta$-term the relative
entropy perturbation is non-vanishing even without the
non-adiabatic perturbation. This is improper, so we redefine the
new quantities as \ba \hat{\Delta}_{\beta} &=& \frac{\delta
\rho_{\beta}}{(1 - {\cal
B}_{\beta})(\rho_{\beta} + p_{\beta})}, \label{hatDeltabeta} \\
\hat{S}_{\beta \alpha} &=& \hat{\Delta}_{\beta} -
\hat{\Delta}_{\alpha}. \label{S2} \ea With these quantities we can
rewrite Eq.~(\ref{Gammarel2}) as
\be p_{\tot} \Gamma_{\rel} = \frac{1}{2} \sum_{\beta,\alpha}
\frac{(1 - {\cal B}_{\beta})(1 - {\cal B}_{\alpha})(\rho_{\beta} +
p_{\beta})(\rho_{\alpha} + p_{\alpha})} {\rho_{\tot} +
p_{\tot}}(c_{\beta}^2 - c_{\alpha}^2) \hat{S}_{\beta \alpha}.
\label{Gammarel3} \ee
\begin{center}
\end{center}
\subsection{Isocurvature condition}
Due to the out of thermal equilibrium nature of the quintessence,
we need to check the isocurvature evolution of the scalar field
perturbation \cite{Isocur}. We can analytically show this in the
tracking region. First of all the adiabatic sound speed squared
$c_{\phi}^2$ of the quintessence can be represented as \be
c_{\phi}^2 = \frac{\bar{p}_{\phi}'}{\bar{\rho_{\phi}}'} = 1 +
\frac{2 \bar{\phi}' V_{,\bar{\phi}}}{3 {\cal H} (\bar{\rho}_{\phi}
+ \bar{p}_{\phi})( 1 - {\cal B}_{\phi})} = \omega_{\phi} -
\fr{\omega_{\phi}'}{3 {\cal H} ( 1 + \omega_{\phi})(1 - {\cal
B}_{\phi})}, \label{cphi} \ee where we have used
Eq.~(\ref{rhophi'}). From this equation we can find the second
derivative of the potential, \be \frac{a^2}{\bar{M}^2}
V_{,\bar{\phi} \bar{\phi}} = \frac{3}{2} {\cal H} c_{\phi}^{2'}(1
- {\cal B}_{\phi}) + \frac{3}{2} {\cal H}^2(c_{\phi}^2 - 1)(1 -
{\cal B}_{\phi}) \Bigl[ \frac{{\cal H}'}{{\cal H}^2} - 1 -
\frac{3}{2}(c_{\phi}^2 + 1)(1 - {\cal B}_{\phi}) \Bigr] -
\frac{3}{2} {\cal H}(c_{\phi}^2 - 1) {\cal B}_{\phi}'.
\label{Vdouble} \ee The relation between ${\cal H}^2$ and ${\cal
H}'$ can be found from Eq.~(\ref{H'}) : \ba \fr{{\cal
H}'}{{\cal H}^2} &=& - \fr{1}{2} ( 1 + 3 \omega_{\tot}), \label{HH} \\
\omega_{\tot} &=& \sum_{\beta} \omega_{\beta} \Omega_{\beta},
\label{omegatot} \ea where $\omega_{\tot}$ is the weighted EOS. As
long as we use the background as shown in the pervious model, the
tracking region is well established during the radiation dominated
era~\cite{LOP}. During this era we can find the following
relation, \be c_{\phi}^2 =
\frac{\bar{p}_{\phi}'}{\bar{\rho}_{\phi}'} = \omega_{\phi} =
\omega_{r}. \label{trackingcphi} \ee This is shown in the first
panel of Fig.~\ref{fig:cpBn}. As the coupling is increased we can
have the non-monotonic behavior of $c_{\phi}^2$ as shown in the
$n_{c} = 10^{-2}$ case. In addition, we can rewrite the coupling
term as \be {\cal B}_{\phi} = - \frac{ B_{c,\bar{\phi}}
\bar{\phi}' \bar{\rho}_{c}}{3 {\cal H} (\bar{\rho}_{\phi} +
\bar{p}_{\phi})} = - \frac{{\cal H}}{\bar{\phi}'} B_{c,\bar{\phi}}
\bar{\Omega}_{c} . \label{calBphi} \ee As shown in the second
panel of Fig.~\ref{fig:cpBn}, this term is negligible during the
radiation dominated epoch. The coupling drives a faster evolution
of $\phi$ when matter energy dominates the Universe and the
magnitude of ${\cal B}_{\phi}$ depends on the energy density of
the CDM. With these facts we can have the approximate expression
of Eq.~(\ref{Vdouble}), \be \frac{a^2}{\bar{M}^2} V_{,\bar{\phi}
\bar{\phi}} \simeq \frac{3}{2} {\cal H}^2(c_{\phi}^2 - 1) \Bigl[
\frac{{\cal H}'}{{\cal H}^2} - \frac{3}{2}(c_{\phi}^2 + 1) \Bigr]
= - \frac{3}{4} {\cal H}^2(c_{\phi}^2 - 1) \Bigl[ 3 \omega_{\tot}
+ 3 c_{\phi}^2 + 4 \Bigr], \label{Vdouble2} \ee where we have used
Eq.~(\ref{HH}) in the second equality. The isocurvature mode in
the radiation dominated epoch can be obtained from
Eq.~(\ref{deltaphi}) and using the fact that $a \propto \eta$
during the radiation dominated era. To check this we can put $\Phi
= 0$ and then we have \be \delta \phi'' + \frac{4}{(3 \omega_{r} +
1) \eta} \delta \phi' + \Bigl[ k^2 - (c_{\phi}^2 - 1) \frac{3}{(3
\omega_{r} + 1)^2 \eta^2} (3 \omega_{r} + 3 c_{\phi}^2 + 4 )
\Bigr] \delta \phi = 0. \label{deltaphi2} \ee If we use the
tracking solution (\ref{trackingcphi}), then we can rewrite the
above equation as \be \delta \phi'' + \frac{2}{\eta} \delta \phi'
+ \Bigl[ k^2 + \frac{3}{\eta^2} \Bigr] \delta \phi = 0.
\label{deltaphi3} \ee The solutions of this equation are Bessel
functions, \be \delta \phi(\eta) = {\rm const} ~ \eta^{-1/2} ~
J_{\pm |i \sqrt{11}/2|} (k \eta) . \label{soldeltaphi1} \ee Thus
both solutions decay in time. At the superhorizon scale the
$k$-dependent term can be neglected and we obtain the power-law
solutions, $\delta \phi \propto
\eta^{\nu}$,
where the power index $\nu = (-1 \pm i \sqrt{11})/2$.
Therefore, any initial nonzero isocurvature fluctuation of the
quintessence is damped to zero with time. We will use
only the adiabatic perturbations in the following section.
\begin{center}
\end{center}
\section{Effects of coupling}
\setcounter{equation}{0}
The non-minimally coupled quintessence models have been
investigated as a possible solution for the late time coincidence
problem \cite{coupQ}. The coupling gives rise to the additional
mass and source terms of the evolution equations for CDM and scalar field
perturbations. This also affects the perturbation of radiation
indirectly through the background bulk ${\cal H}$ and the metric
perturbations \cite{Bean2}. The value of the energy density
contrast of the CDM ($\Omega_{c}$) is increased in the past when the
coupling is increased.
\subsection{CMB} \setcounter{equation}{0}
The temperature anisotropy measured in a given direction of the
sky can be expanded in spherical harmonics as \be \Theta \equiv
\frac{\Delta T}{T}(\hat{n})=\sum_{\ell m}a_{\ell m}Y_{\ell m}
(\hat{n}), \label{Theta} \ee where $\Theta$ is the temperature
brightness function that is the fractional perturbation of the
temperature of the photons $T = T_{0}(1 + \Theta)$, $\hat{n}$ is
the direction of the photon momentum, and $a_{\ell m}$ are the
multipoles. We can also expand the brightness function as a
Legendre polynomial, $P_{\ell}$ : \be \Theta = \sum_{\ell}
(-i)^{\ell} \, (2 \ell + 1) \, \Theta_{\ell} \, P_{\ell}
.
\label{Theta2} \ee From the inflationary scenario, the multipoles
are Gaussian random variables which satisfy \be \langle a^*_{\ell
m} \, a_{\ell' m'}\rangle = C_{\ell} \, \delta_{\ell \ell'} \,
\delta_{m m'}. \label{Cl} \ee The angular power spectrum
($C_{\ell}$) contains all the information about the statistical
properties of CMB and is defined as \cite{Hu} \be \frac{2 \ell +
1}{4 \pi} C_{\ell} = \frac{1}{2 \pi^2} \int d \eta \frac{dk}{k}
\frac{k^3 |\Theta_{\ell}(k,\eta)|^2}{2 \ell +1}. \label{spectrum}
\ee In the standard recombination model the acoustic oscillations
will be frozen into the CMB. A generalization of the
free-streaming equation in a flat universe gives the resulting
anisotropies : \be \fr{\Theta_{\ell}(\eta)}{2 \ell + 1} =
[\Theta_0 + \Psi](\eta_{ls}) j_{\ell}(k(\eta - \eta_{ls})) +
\Theta_{1}(\eta_{ls}) \fr{1}{k} \fr{d}{d \eta} j_{\ell}(k(\eta -
\eta_{ls})) + \int_{\eta_{\,ls}}^{\eta} (\Psi' - \Phi') j_{\ell}
(k(\eta - \tilde{\eta})) d \tilde{\eta} \label{Spectrum2} \ee
where $j_{\ell}$ is the spherical Bessel function and
$\eta_{\,ls}$ is the conformal time at last scattering. Photon
density perturbation is related to the temperature perturbation in
the matter rest frame : \be \delta_{r} = 4 \Theta_0 + 4 {\cal H}
\fr{\theta_{\tot}}{k^2}, \label{deltarm} \ee and the gravitational
potentials will be given in Eq. (\ref{deltaG00}) -
(\ref{deltaGij}) in the following section. We use the fact that in
the absence of anisotropic stress, the two scalar potentials
$\Psi$ and $\Phi$ defined in the conformal Newtonian gauge
(\ref{CNG}) are equal and they coincide with the usual
gravitational potential in the Newtonian limit.
Now, we investigate the effects of non-minimal coupling of a
scalar field to the CDM on the CMB power spectrum. Firstly, the
Newtonian potential at late times changes more rapidly as the
coupling increases, as shown in Eq.~(\ref{deltaG00}). This leads
to an enhanced ISW effect as indicated in the last term of
Eq.~(\ref{Spectrum2}). Thus we have a relatively larger $C_{\ell}$
at large scales ({\it i.e.} small $\ell$). Thus, if the CMB power
spectrum normalized by COBE, then we will have smaller quadrupole
\cite{COBE}. This is shown in the first panel of
Fig.~\ref{fig:cl}. One thing that should be emphasized is that we
use different parameters for the $\Lambda$CDM and the coupled
quintessence models to match the amplitude of the first CMB
anisotropy peak. The parameter used for the quintessence model is
indicated in Fig.~\ref{fig:cl} ({\it i.e.} $\Omega_{\phi}^{(0)} =
0.76$, $\Omega_{m}^{(0)} = 0.191$, $\Omega_{b}^{(0)} = 0.049$, and
$h = 0.7$, where $h$ is the present Hubble parameter in the unit
of $100 {\rm km s^{-1} Mpc^{-1}}$). However, these parameters are
well inside the $1 \, \sigma$ region given by the WMAP data. We
use the WMAP parameters for the $\Lambda$CDM model ({\it i.e.}
$\Omega_{\phi}^{(0)} = 0.73$, $\Omega_{m}^{(0)} = 0.23$,
$\Omega_{b}^{(0)} = 0.04$, and $h = 0.72$). In both models we use
the same spectral index $n_{s} =1$. The heights of the acoustic
peaks at small scales ({\it i.e} large $\ell$) can be affected by
the following two factors. One is the fact that the scaling of the
CDM energy density deviates from that of the baryon energy
density. Therefore for the given CDM and baryon energy densities
today, the energy density contrast of baryons at decoupling
($\Omega_{b}^{(ls)}$) is getting lower as the coupling is being
increased. This suppresses the amplitude of compressional (odd
number) peaks while enhancing rarefaction (even number) peaks. The
other is that for models normalized by COBE, which approximately
fixes the spectrum at $\ell \simeq 10$, the angular amplitude at
small scales is suppressed in the coupled quintessence. This is
shown in the second panel of Fig.~\ref{fig:cl}. The third peak in
this model is smaller than that in the $\Lambda$CDM model. The
WMAP data do not show the value of the third peak but quote a
compilation of other experiments \cite{Wang}. The ratio of the
amplitude between the second and the third peaks is $1.03 \pm
0.02$. In the $\Lambda$CDM model this value is $0.986$ and in our
model these values are $1.08$ and $1.11$ for without and with the
coupling equal to $n_c=0.01$ respectively. We also show the $n_{c}
= 0.02$ case in this figure. With the same parameters this case
can be ruled out from the current data. In all cases, we have used
the CMBFAST code~\cite{CMBFAST} with the modified Boltzmann
equations to compute the CMB power spectrum.
Secondly, for $\ell>200$ we can see that the locations of the
acoustic peaks are slightly shifted to smaller scales ({\it i.e.}
larger $\ell$). This can be explained as follows. The locations of
peaks and troughs can be parametrized as \be \ell_{m} = \ell_{A}
(m - \varphi_m) = \ell_{A} (m - \bar{\varphi} - \delta \varphi_{m}
), \label{lm} \ee where $\ell_{A}$ is the acoustic scale dependent
on the geometry of the Universe, $\bar{\varphi}$ is the overall
peak shift, and $\delta \varphi_{m}$ is the relative shift of the
$m$-th peak relative to the first one~\cite{Doran}. The overall
peak shift, $\bar{\varphi}$ is given by \be \bar{\varphi} \simeq
0.267 \Bigl(\fr{r_{ls}}{0.3} \Bigr)^{0.1}, \label{varphi} \ee
where $r_{ls} = \rho_r^{(ls)} / \rho_m^{(ls)}$ is the ratio of the
energy densities of radiation to matter at last scattering. The
shift is due to both driving effects from the decay of the
gravitational potential and contributions from the Doppler shift
of the oscillating fluid. The acoustic scale $\ell_{A}$ depends on
both the sound horizon $s_{ls}$ at decoupling and the angular
diameter distance $D$ to the last scattering surface: \be \ell_{A}
= \pi \fr{D}{s_{ls}} = \pi \fr{\eta_{0} - \eta_{\,ls}}{\bar{c}_{s}
\eta_{\, ls}} , \label{la} \ee where $\bar{c}_{s}$ is the average
sound speed before last scattering : \be \bar{c}_{s} =
{\eta}_{\,ls}^{-1} \int_{0}^{\eta_{\,ls}} c_{s} d \eta
\hspace{0.2in} {\rm with} \hspace{0.2in} c_s^{-2} = 3 + \fr{9}{4}
\fr{\rho_{b}}{\rho_{r}} . \label{cs} \ee Also from the Hubble
parameter, \be \Biggl(\frac{d a}{d \eta}\Biggr)^2 = H_{0}^2
\Biggl\{ \Omega_{r}^{(0)} + \Omega_{b}^{(0)} a + \Omega_{m}^{(0)}
a^{1+ \xi} + \fr{\rho_{\phi}}{\rho_{cr}^{(0)}} a^{4} \Biggr\},
\label{da1} \ee where $H_{0}$ is the present value of the Hubble
parameter, we can find the angular diameter distance, \be
D=\eta_{0} - \eta_{ls} = H_{0}^{-1} \int_{a_{ls}}^{a^{(0)}}
\fr{da}{\sqrt{\Omega_{r}^{(0)} + \Omega_{b}^{(0)} a +
\Omega_{m}^{(0)} a^{1+ \xi} + \rho_{\phi} / \rho_{cr}^{(0)} a^4}}.
\label{da2} \ee The sound horizon is not affected by the coupling
and the effect of coupling on the angular diameter distance is
also quite small in our model. However, the overall shift and the
relative shift are affected by the coupling. As the coupling is
increased, the value of $r_{ls}$ is decreased. Hence, the
locations of the peaks are shifted to the right. However, this
shift is quite small. We show this in the second panel of
Fig.~\ref{fig:cl}. The heights and the locations of the acoustic
peaks in various models are summarized in Table~\ref{tab:1}. There
is a significant difference for the heights of the peaks of the
second and the third peaks between the models. Thus upcoming
observations continuing to focus on resolving the higher peaks may
constrain the strength of the coupling.
\begin{table}[htb]
\begin{center}
\caption{Summary of the heights and the locations of the
considered models. The heights of peaks, $A_{i}$ have been scaled
by a factor of $10^{10}$. $A_{i:j}$ is the ratio of the $i$-th
peak height to the $j$-th. The locations of peaks, $\ell_{i}$.
\label{tab:1}} \vskip .3cm
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline Model & $A_{1}$ & $A_{2}$ & $A_{3}$ & $\ell_{1}$ &
$\ell_{2}$ & $\ell_{3}$ & $A_{1:2}$ & $A_{2:3}$ \\
\hline $\Lambda$CDM & $7.26$ & $3.25$ & $3.30$ & $220$ & $536$ & $812$
& $2.23$& $0.986$ \\
\hline $n_{c} = 0$ & $7.04$ & $3.23$ & $3.00$ & $217$ & $532$ &
$811$ & $2.18$& $1.08$ \\
\hline $n_{c} = 10^{-2}$ & $7.02$ & $3.36$ & $3.03$ & $218$ &
$535$ & $818$ & $2.09$& $1.11$ \\
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Matter power spectrum}
\begin{center}
\end{center}
The effects of quintessence on structure formation in several
other models have been investigated \cite{WS, DSW}. Structure as a
function of physical scale size is usually described in terms of a
power spectrum : \be P(k) = \langle|\delta_{k}|^2\rangle = A
k^{n_{s}} T^{2}(k), \label{Pk} \ee where $A$ is the COBE
normalization, $n_s$ is a power index, and $T(k)$ is the transfer
function. The coupling of quintessence to the CDM can change the
shape of matter power spectrum because the location of the
turnover corresponds to the scale that entered the Hubble radius
when the Universe became matter-dominated. This shift on the scale
of matter and radiation equality is indicated in the second panel
of Fig.~\ref{fig:Qnxi}: \be a_{eq} \simeq
\fr{\rho_{r}^{(0)}}{\rho_{c}^{(0)}} \exp [B_{c}(\phi_{0}) -
B_{c}(\phi_{eq})], \label{aeq} \ee where $\rho_{r}^{(0)}$ and
$\rho_{c}^{(0)}$ are the present values of the energy densities of
radiation and CDM respectively, and the approximation comes from
the fact that the present energy density of CDM is bigger than
that of baryons ($\rho_{c}^{(0)} > \rho_{b}^{(0)}$). Increasing
the coupling shifts the epoch of matter-radiation equality further
from the present, thereby moving the turnover in the power
spectrum to smaller scale. If we define $k_{eq}$ as the wavenumber
of the mode which enters the horizon at radiation-matter equality,
then we will obtain \be k_{eq} = \fr{2 \pi}{\eta_{eq}}.
\label{keq} \ee However, from the previous subsection, we notice
that the value of $\eta_{eq}$ remains unchanged for different
couplings and this degeneracy is indicated in Fig.~\ref{fig:mp}.
We have used different parameters for the $\Lambda$CDM and
quintessence models; the matter power spectra look different
between the models. There is a slight suppression in the
quintessence models. Note that a bias factor could resolve the
discrepancy and perhaps a parameter fitting may also help this.
However, the detailed parameter fitting is beyond the scope of
this paper.
We can write the equation of the matter fluctuation in the
synchronous gauge during the matter dominated epoch : \be
\bar{\delta}_{c}'' + {\cal H} \bar{\delta}_{c}' - \fr{3}{2} {\cal
H}^2 \fr{( \delta \bar{\rho}_{c} + \delta \bar{p}_{c}
)}{\bar{\rho}_{cr}} - \fr{(a F(\phi))'}{a} = 0, \label{deltacsyn}
\ee where the coupling term $F(\phi)$ is given by
\be F(\phi) = B_{c,\bar{\phi}
\bar{\phi}} \bar{\phi}' \delta \phi + B_{c,\bar{\phi}} \delta
\phi' = n_c \lambda (\bar{\phi} \delta \phi)'. \label{coupc} \ee
This equation can be rewritten by the structure growth exponent
$f$, which is defined as \be f(a) = \fr{d \ln
\bar{\delta}_{c}(a)}{d x}. \label{fa} \ee Hence, we have the
following equation for $f(a)$ which is identical to Eq.
(\ref{deltacsyn}): \be \fr{d f}{d x} + f^2 + \Bigl( 2 + \fr{d \ln
({\cal H} / a )}{d x} \Bigr) f - \fr{3}{2} \Omega_{c} = \fr{d
\Bigl[a F(\phi) \Bigr]}{d x} \fr{1}{a {\cal H} \bar{\delta}_{c}},
\label{faeq} \ee If we use Eq.~(\ref{rhoca}), then we can find
that $a \propto \eta^{2/(1-\xi)}$ and ${\cal H} = 2/(1- \xi)
\eta^{-1}$ for a matter dominated universe. As we can see from the
above Eq.~(\ref{faeq}) that the coupling term is negligible
because $\phi$ varies much slower than ${\cal H}$. Thus we can
ignore the last term in Eq.~(\ref{deltacsyn}). From this we can
rewrite the above equation in a matter dominated era : \be
\bar{\delta}_{c}'' + \fr{2}{(1-\xi) \eta} \bar{\delta}_{c}' -
\fr{6}{(1-\xi)^2 \eta^2} \bar{\delta}_{c} = 0. \label{deltacsyn2}
\ee This equation has two solutions, \be \bar{\delta}_{c}^{\pm} =
c_{\pm} \eta^{\nu_{\pm}}, \label{deltacsol} \ee where $c_{\pm}$
are arbitrary constants and \be \nu_{\pm} = \fr{-(1+\xi) \pm
\sqrt{24 + (1 + \xi)^2}}{2(1 - \xi)}. \label{nupm} \ee
$\bar{\delta}_{c}^{+}$ indicates a growing mode, which is only
relevant today because $|\bar{\delta}_{c}^{+}|$ is small at early
time and we can ignore the decaying mode $\bar{\delta}_{c}^{-}$.
If we remove the coupling effect in this solution, then we can
recover the well known solution $\delta_{c}^{+} = \eta^{2/3}$.
This effect is shown in Fig.~\ref{fig:mp}.
As the coupling is increased, we have little more matter at early
times and this increases the height of the matter power spectrum. Again
this effect is tiny and we can hardly see the difference between
various couplings.
\section{Metric perturbation}
\setcounter{equation}{0}
In addition to these, the perturbed equations of the metric can be
obtained from the Einstein equations:
\ba k^2 \Phi + 3 {\cal H} \Bigl( \Phi' + {\cal H} \Psi \Bigr) &=&
\frac{3}{2} \frac{{\cal H}^2}{\bar{\rho}_{tot}} \delta
T^0_{(tot)0}, \label{deltaG00} \\
\Phi'' + {\cal H} ( \Psi' + 2 \Phi') + \Bigl(2{\cal H}' + {\cal
H}^2 \Bigr) \Psi + \frac{k^2}{3} ( \Phi - \Psi) &=& \frac{1}{2}
\frac{{\cal H}^2}{\bar{\rho}_{tot} } \delta T^i_{(tot)i}, \label{deltaGii} \\
k^2 \Bigl( \Phi' + {\cal H} \Psi \Bigr) &=& \frac{3}{2} {\cal H}^2
\sum_{\beta} (1 + \omega_{\beta}) \bar{\Omega}_{\beta} \theta_{\beta}
,
\label{deltaG0i} \\
k^2 ( \Phi - \Psi) &=& \frac{9}{2} {\cal H}^2 \sum_{\beta} (1 +
\omega_{\beta}) \bar{\Omega}_{\beta} \sigma_{\beta}
,
\label{deltaGij} \ea
where we have used the unperturbed equations~(\ref{H}) and (\ref{H'})
.
From Eqs.~(\ref{deltaG00}) and (\ref{deltaGii}), we
can find the metric perturbation equation,
\ba \Phi'' + {\cal H}(\Psi' + 5 \Phi') + 2 ( {\cal H}' + 2{\cal
H}^2) \Psi + \frac{k^2}{3}(4\Phi - \Psi) \nonumber \\
= -\frac{a^2}{2\bar{M}^2} [ (1 - c_{tot}^2) \delta \rho_{tot} -
p_{tot} \Gamma_{int} - p_{tot} \Gamma_{rel} ]. \label{metricper}
\ea
If we use Eq.~(\ref{deltaG00}),
this equation can be rewritten as
\ba \Phi'' + {\cal H} \Bigl[\Psi' + (2 + 3 c_{tot}^2) \Phi' \Bigr]
+ \Bigl[ 2 {\cal H}' + {\cal H}^2(1 + 3c_{tot}^2) \Bigr] \Psi +
\frac{k^2}{3} \Bigl[ (1 + 3 c_{tot}^2) \Phi - \Psi \Bigr]
\nonumber \\ = \frac{a^2}{2\bar{M}^2} \Bigl( p_{tot} \Gamma_{int}
+ p_{tot} \Gamma_{rel} \Bigr) \equiv \frac{a^2}{2\bar{M}^2}
p_{tot} \Gamma_{tot}, \label{metricper2} \ea where we define
$\Gamma_{tot} = \Gamma_{int} + \Gamma_{rel}$ in the last equality.
The last term in this equation comes from the coupling of
the
scalar field. So even if we start from the adiabatic condition (
$p_{tot} \Gamma_{tot} =0$), we can analytically solve the above
equation for the specific case. Let us first put $\Psi = \Phi$ (no
anisotropic stress) and $k^2 c_{tot}^2 \Phi = 0$ (consider the
superhorizon scale), then Eq.~(\ref{metricper2}) is
simplified as
\be \Phi'' + 3 {\cal H} (1 + c_{tot}^2) \Phi' + \Bigl[ 2 {\cal H}'
+ {\cal H}^2(1 + 3c_{tot}^2) \Bigr] \Phi = \frac{a^2}{2\bar{M}^2}
p_{tot} \Gamma_{tot}. \label{metricper3} \ee
If you use the background equations,
this equation can be
rewritten as
\be \Phi'' - \Bigl( \ln[\rho_{tot} + p_{tot}] \Bigr)' \Phi' +
\Bigl( \ln[\rho_{tot}/(\rho_{tot} + p_{tot})] \Bigr)' {\cal H}
\Phi = \frac{3 {\cal H}^2}{2} \frac{p_{tot}}{\rho_{tot}}
\Gamma_{tot}. \label{metricper4} \ee If we consider the adiabatic
condition ($p_{tot} \Gamma_{tot} = 0$), then the above equation is
identical to the equation in Bardeen's article~\cite{Bardeen}.As such,
the
metric perturbation can be rewritten as the curvature
perturbation~\cite{Lyth},
\be \zeta = \frac{({\cal H}^{-1} \Phi' + \Phi)}{(1 +
\omega_{tot})} + \frac{3}{2} \Phi. \label{zeta} \ee With this we
can express Eq.~(\ref{metricper3}) in the adiabatic
case as
\be \zeta' = 0. \label{etaprime} \ee
This result looks the same to the
minimally coupled case and there seems to have no difference from the
non-minimally coupled case. Nevertheless, the coupling
information is absorbed in both $c_{tot}^2$ and the perturbation equation
of each species.
\section{Conclusions}
\setcounter{equation}{0}
We have analyzed the linear perturbations of the cosmological
models for the scalar field with its self-interaction potential,
$V(\phi) = V_{0} \exp(\lambda \phi^2 /2)$, and its coupling to the
CDM, $\exp[B_{c}(\phi)] = \exp(n_{c} \lambda \phi^2 /2)$. The
evolution of the non-perturbed background scalar field
$\bar{\phi}$ occurred in the tracking regime throughout the
radiation dominated epoch. The full analysis of the non-perturbed
and perturbed equation of each species in the conformal Newtonian
gauge has been done including the proper energy-momentum transfer
vector due to the coupling, $Q_{(d) \nu}$. The Boltzmann equations
have been modified as to account for the coupling between a scalar
field and
the CDM.
We have seen that the energy-momentum of each species may not be
conserved as a result of the coupling of the scalar field to the
CDM. Thus we have redefined the concepts of entropy perturbations.
We have shown that the isocurvature perturbation of the
quintessence has been damped to zero with time in the tracking
regime during the radiation epoch. Thus we have constrained our
considerations on the adiabatic perturbation.
We have considered the CMB anisotropy spectrum and the matter
power spectrum for the non-minimally coupled models. Additional
mass and source terms in the Boltzmann equations induced by
the coupling give the rapid changes of the Newtonian potential
$\Phi$ and enhance the ISW effect in the CMB power spectrum. The
modification of the evolution of the CDM, $\rho_{c} =
\rho_{c}^{(0)} a^{-3 + \xi}$, changes the energy density contrast
of the CDM at early epoch. We have adopted the current
cosmological parameters
measured by WMAP within $1 \sigma$ level. With the COBE
normalization and the WMAP data we have found the constraint of
the coupling $n_{c} \leq 0.01$. The locations and the heights of
the CMB anisotropy peaks have been changed due to the coupling.
Especially, there
is a significant difference for the heights of the
second and the third peaks among the models. Thus upcoming
observations continuing to focus on resolving the higher peaks may
constrain the strength of the coupling. The suppression of the
amplitudes of the matter power spectra could be lifted by a bias
factor. However, a detailed fitting is beyond the scope of this
paper. The turnover scale of the matter power spectrum may be used
to constrain the strength of the coupling $n_{c}$.
Finally, we have investigated the metric perturbations including
the coupling between the scalar field and the CDM. There is no
difference to the curvature perturbation $\zeta$ for the different
couplings in the adiabatic case. However, the effects of the
coupling have been absorbed in the Boltzmann equations already.
\section{Acknowledgements}
\setcounter{equation}{0}
This work was supported in part by the National Science Council,
Taiwan, ROC under the Grant NSC94-2112-M-001-024 (K.W.N.).
|
Title:
Stellar and Gas properties of High HI Mass-to-Light Ratio Galaxies in the Local Universe |
Abstract: We present a multi-wavelength study (BVRI band photometry and HI line
interferometry) of nine late-type galaxies selected from the HIPASS Bright
Galaxy Catalog on the basis of apparently high HI mass-to-light ratios (3
M_sun/L_sun < M_HI/L_B < 27 M_sun/L_sun). We found that most of the original
estimates for M_HI/L_B based on available photographic magnitudes in the
literature were too high, and conclude that genuine high HI mass-to-light ratio
(>5 M_sun/L_sun) galaxies are rare in the Local Universe. Extreme high M_HI/L_B
galaxies like ESO215-G?009 appear to have formed only the minimum number of
stars necessary to maintain the stability of their HI disks, and could possibly
be used to constrain galaxy formation models. They may to have been forming
stars at a low, constant rate over their lifetimes. The best examples all have
highly extended HI disks, are spatially isolated, and have normal baryonic
content for their total masses but are deficent in stars. This suggests that
high M_HI/L_B galaxies are not lacking the baryons to create stars, but are
underluminous as they lack either the internal or external stimulation for more
extensive star formation.
| https://export.arxiv.org/pdf/astro-ph/0601321 |
\title{Stellar and Gas properties of High \hi{} Mass-to-Light Ratio Galaxies in the Local Universe}
\author{Bradley E. Warren\altaffilmark{1} and Helmut Jerjen}
\affil{Research School of Astronomy and Astrophysics, Australian National
University, Mount Stromlo Observatory, Cotter Road, Weston ACT 2611,
Australia}
\email{bewarren@mso.anu.edu.au, jerjen@mso.anu.edu.au\\}
\and
\author{B\"arbel S. Koribalski}
\affil{Australia Telescope National Facility, CSIRO, PO Box 76, Epping NSW 1710, Australia}
\email{Baerbel.Koribalski@csiro.au}
\altaffiltext{1}{Affiliated with the Australia Telescope National Facility, CSIRO.}
\keywords{galaxies: irregular --- galaxies: dwarf --- galaxies: evolution --- galaxies: photometry --- galaxies: ISM --- galaxies: kinematics and dynamics --- galaxies: individual (\esoq{}, ESO\,572-G009, ESO\,428-G033, \mcg{}, ESO\,473-G024, IC\,4212, ESO\,348-G009, ESO\,121-G020, ESO\,505-G007, \atg{})}
\section{Introduction}
\label{sec:intro}
The number of low mass dark matter halos predicted by models of a CDM dominated Universe far exceeds the quantity of observed dwarf galaxies, typically by several orders of magnitude \citep*[see][]{kau93,moo99,kly99}. In this context we consider a galaxy to be a dark matter halo that contains baryons. Consequently the slope of dark matter mass functions generally rises much more steeply than observed galaxy luminosity functions (\citealp*{tre02,hil03,bla03}; but see also \citealp{bla04}).
While there are some physical processes that could help narrow this discrepancy \citep*{kly99,sha04}, the theoretical low mass halo frequency is not reduced enough to reconcile them with observations. So we are left with the conclusions that either the current most favored cosmological models significantly over-estimate the number of low mass dark matter halos present in the Local Universe, or the observations have failed to find the vast majority of low mass galaxies to date. If the latter were true, then it is important to look at why dwarf galaxies could be missed and how we could detect them.
Two reasons why galaxies might not have been found yet in optical surveys are that they could exhibit low stellar densities ($<$ 1\Msun{}\,pc$^{-2}$) or most of their baryons are in invisible form. In the extreme case, they do not contain any baryonic matter at all and are in fact ``empty'' dark matter halos. These possibilities could be dark matter halos in which the star formation from accreted gas has been halted, suppressed, never began, or there were simply no baryons to form stars to begin with. Such objects lacking stars might be considered as being ``dark galaxies,'' which could be easily missed in surveys biased towards optical wavelengths \citep{dis76}. However, dark galaxies have yet to be found in the blind \hi{} surveys such as HIPASS \citep{kor04,doy05}.
If they do exist, large numbers of low mass dark galaxies could naturally steepen the mass functions from observations, which are mostly derived from optical or near-infrared galaxy luminosity functions without consideration of other baryonic matter. Recent large scale observations of neutral hydrogen gas (\hi{}) are now allowing mass functions to be derived based on non-stellar properties. \citet{zwa03} produced one of the most extensive \hi{} mass functions based on a catalogue of the 1000 \hi{}-brightest galaxies in the Southern hemisphere and found a similar low mass end slope to other observational studies, again in contradiction with $\Lambda$CDM models.
Galaxies that have failed to convert most of their primordial gas into stars, and thus retained a high proportion of \hi{}, may well provide a partial solution. While not entirely ``dark'' these galaxies are hard to detect optically, but may be detectable through 21cm line observations. The \hi{} mass-to-light ratio (\mlr{}, which compares the \hi{} mass to the {\em B} band luminosity) of these objects could be significantly higher than the typical ratios measured for late-type galaxies, so that they would be in a lower mass bin in a luminosity function than they would be for a baryonic mass function. If high \mlr{} galaxies existed in significant numbers then they could help correct the discrepancy in two ways, by including more previously unknown galaxies, and by shifting known galaxies to higher mass bins than they would be placed in with purely optical results.
An example of an extreme \hi{} mass-to-light ratio object, which could be described as a ``dim'' galaxy, is the nearby dwarf irregular \esoq{} with \mlr{} = $22 \pm 4$\mls{} \citep*[][ hereafter \pI{}]{war04}. This faint low surface brightness dwarf irregular was found to be spatially isolated (1.7~Mpc from the nearest neighbor), with a low current star formation rate ($\la 2.5 \times 10^{-3}$\Msun{}\,yr$^{-1}$). It has an extended regularly rotating \hi{} disk, which can be traced out to over six times the Holmberg radius of the optical galaxy, making it one of the most extended \hi{} envelopes relative to the optical extent.
\pI{} included an analysis of the \hi{} gas surface density of \esoq{}. The azimuthally averaged surface density at all radii was below the critical gas surface density needed for large scale star formation as defined by the \citet{too64} stability criteria \citep{ken89,mar01}. It was proposed in \citet{ver02} that a large fraction of low mass halos may form \citeauthor{too64} stable gas disks and become ``dark'' galaxies, possibly 95\% of objects with halo masses of $\la 10^{10}$\Msun{}. If so \esoq{} would be just the tip of the iceberg. If we can find more galaxies similar to \esoq{}, where the gas density after gravitational collapse is too low for efficient star formation, it may go some way to explaining the discrepancy between the dark matter halo mass function and the observed galaxy luminosity function.
To continue our study of the stellar and gas properties of galaxies with high \mlr{} we have selected a sample of nine galaxies from the BGC in the approximate range 3\mls{} $<$ \mlr{} $<$ 27\mls{}. In this paper, \S~\ref{sec:sample} looks at what was previously known about the sample galaxies. \S~\ref{sec:obs} summarizes our 21cm and optical observations. \S\S~\ref{sec:radio} and \ref{sec:optp} present the results of the \hi{} line observations and optical photometry, respectively. \S~\ref{sec:dis-indi} compares the properties of individual galaxies. \S~\ref{sec:dis} contains the discussion of these results and the possible implications, while \S~\ref{sec:conc} gives our conclusions.
\section{Galaxy Selection}
\label{sec:sample}
\begin{deluxetable}{lccccccccc}
\tabletypesize{\scriptsize}
\tablecaption{Summary of Previously Measured Galaxy Properties.
\label{tab:prop}}
\tablewidth{0pt}
\tablehead{\colhead{Name} & \colhead{Center} & \colhead{Galactic} & \multicolumn{3}{c}{Bright Galaxy Catalog} & \colhead{LEDA} & \colhead{SFD98} & \multicolumn{2}{c}{BGC + LEDA} \\
\colhead{HIPASS Name} & \colhead{$\alpha$(J2000.0)} & \colhead{$l$} & \colhead{\vsys{}} & \colhead{$D$} & \colhead{\FHI{}} & \colhead{\mB{}} & \colhead{\AB{}} & \colhead{\MB{}} & \colhead{\mlr{}} \\
& \colhead{$\delta$(J2000.0)} & \colhead{$b$} & \colhead{(\kkms{})} & \colhead{(Mpc)} & \colhead{(\jjks{})} & \colhead{(mag)} & \colhead{(mag)} & \colhead{(mag)} & \colhead{(\mmls{})} \\
\colhead{(1)} & \colhead{(2)} & \colhead{(3)} & \colhead{(4)} & \colhead{(5)} & \colhead{(6)} & \colhead{(7)} & \colhead{(8)} & \colhead{(9)} & \colhead{(10)}}
\startdata
\mcg{} & $00^{\rm h}\,19^{\rm m}\,11^{\rm s}$ & 62\fdg6 & $669 \pm 3$ & 9.5 & $16.0 \pm 2.5$ & $15.8 \pm 0.2$ & $0.08 \pm 0.01$ & $-14.1 \pm 0.2$ & $4.8 \pm 1.6$ \\
HIPASS\,J0019--22 & --22\degr\,40\arcmin\,14\arcsec{} & --81\fdg4 \\ \\
ESO\,473-G024 & $00^{\rm h}\,31^{\rm m}\,23^{\rm s}$ & 75\fdg7 & $540 \pm 4$ & 7.6 & $7.2 \pm 1.8$ & $16.2 \pm 0.2$ & $0.08 \pm 0.01$ & $-13.3 \pm 0.2$ & $3.0 \pm 1.3$ \\
HIPASS\,J0031--22 & --22\degr\,46\arcmin\,02\arcsec{} & --83\fdg7 \\ \\
ESO\,121-G020 & $06^{\rm h}\,15^{\rm m}\,53^{\rm s}$ & 266\fdg5 & $577 \pm 5$ & 4.1 & $14.1 \pm 2.9$ & $17.0 \pm 0.4$ & $0.17 \pm 0.03$ & $-11.3 \pm 0.4$ & $11 \pm 6$ \\
HIPASS\,J0615--57 & --57\degr\,43\arcmin\,24\arcsec{} & --27\fdg3 \\ \\
ESO\,428-G033 & $07^{\rm h}\,25^{\rm m}\,49^{\rm s}$ & 244\fdg2 & $1727 \pm 4$ & 19.5 & $12.8 \pm 2.7$ & $17.9 \pm 0.5$ & $1.10 \pm 0.18$ & $-14.7 \pm 0.5$ & $10 \pm 7$ \\
HIPASS\,J0725--30B & --30\degr\,55\arcmin\,05\arcsec{} & --6\fdg9 \\ \\
\esoq{} & $10^{\rm h}\,57^{\rm m}\,30^{\rm s}$ & 284\fdg1 & $598 \pm 2$ & 4.2 & $104.4 \pm 11.5$ & $16.4 \pm 0.4$ & $0.95 \pm 0.15$ & $-12.6 \pm 0.4$ & $24 \pm 12$ \\
HIPASS\,J1057--48 & --48\degr\,11\arcmin\,02\arcsec{} & 10\fdg5 \\ \\
ESO\,572-G009 & $11^{\rm h}\,53^{\rm m}\,23^{\rm s}$ & 284\fdg1 & $1745 \pm 3$ & 20.4 & $8.5 \pm 1.9$ & $17.4 \pm 0.2$ & $0.16 \pm 0.03$ & $-14.3 \pm 0.2$ & $10 \pm 4$ \\
HIPASS\,J1153--18 & --18\degr\,09\arcmin\,59\arcsec{} & 42\fdg6 \\ \\
ESO\,505-G007 & $12^{\rm h}\,03^{\rm m}\,30^{\rm s}$ & 289\fdg5 & $1785 \pm 4$ & 20.8 & $20.5 \pm 3.2$ & $17.7 \pm 0.2$ & $0.36 \pm 0.06$ & $-14.2 \pm 0.2$ & $27 \pm 9$ \\
HIPASS\,J1203--25 & --25\degr\,28\arcmin\,22\arcsec{} & 36\fdg1 \\ \\
IC\,4212 & $13^{\rm h}\,12^{\rm m}\,09^{\rm s}$ & 312\fdg0 & $1484 \pm 2$ & 18.1 & $47.5 \pm 4.6$ & $16.4 \pm 1.1$ & $0.19 \pm 0.03$ & $-15.0 \pm 1.1$ & $23 ^{ +45}_{ -16}$ \\
HIPASS\,J1311--06 & --06\degr\,58\arcmin\,31\arcsec{} & 55\fdg5 \\ \\
ESO\,348-G009 & $23^{\rm h}\,49^{\rm m}\,23^{\rm s}$ & 349\fdg8 & $648 \pm 4$ & 8.4 & $13.4 \pm 2.2$ & $16.7 \pm 0.7$ & $0.06 \pm 0.01$ & $-13.0 \pm 0.7$ & $9 \pm 7$ \\
HIPASS\,J2349--37 & --37\degr\,46\arcmin\,23\arcsec{} & --73\fdg2 \\
\enddata
\end{deluxetable}
The HIPASS Bright Galaxy Catalog \citep[][ hereafter BGC]{kor04} lists the 1000 \hi{}-brightest extragalactic sources (by \hi{} peak flux density) in the Southern hemisphere ($v_{\rm sys} < 8\,000$\kms{}). Photometric and structural parameters for the BGC's {\it optical} counterparts were obtained in 2002 from the Lyon-Meudon Extragalactic Database \citep[LEDA,][ and references therein, now moved to HyperLEDA]{pat97} to study the statistical properties and various scaling relations of these galaxies. First estimates of \hi{} mass-to-light ratios were obtained for 789 BGC galaxies that had mean apparent {\em B} band photographic magnitudes from LEDA, using the equation:
\begin{equation}
\frac{{\cal M}_{\rm HI}}{L_{\rm B}} = 1.5 \times 10^{-7} F_{\rm HI}~ 10^{0.4(m_{\rm B}-A_{\rm B})}~~\frac{{\cal M}_{\sun}}{L_{\sun,{\rm B}}} ,
\label{eqn:mlr}
\end{equation}
where \MHI{} is the \hi{} mass in solar units, \LB{} is the {\em B} band luminosity in solar units, \FHI{} is the integrated \hi{} flux density in \jjks{}, \mB{} is the apparent {\em B} magnitude, and \AB{} is the {\em B} band Galactic extinction. Extinction correction from the host galaxy is not included for reasons explained in \S~\ref{sec:dis-extinct}. Fig.~\ref{fig:mlmbt} shows the log(\mlr{}) distribution for these galaxies as a function of their absolute {\em B} magnitude:
\begin{equation}
M_{\rm B,0} = m_{\rm B} - A_{\rm B} - 5\log(D) - 25 ~~{\rm mag} ,
\label{eqn:abmag}
\end{equation}
where the galaxy distances, $D$ (Mpc), were calculated from the Local Group velocities given in the BGC. Throughout this paper we adopt H$_0$ = 75\kms{}\,Mpc$^{-1}$.
The relation between these two quantities seems to suggest that many of the low luminosity galaxies listed in the BGC have high \mlr{}, up to 27\mls{}, well above typical values for late-type galaxies of less than 1\mls{} \citep[median \mlr{} of 0.78\mls{} in][ for type Sm/Im galaxies]{rob94}. The nine filled circles mark the positions of the galaxies subject to our detailed follow up observations, including \esoq{} (\pI{}). The galaxies were chosen for various reasons, mostly because of a high estimated \mlr{}, but also for reasons of unusual morphology, and after initial ATCA follow up observations showed some unexpected results (see \S~\ref{sec:structure}).
Previously measured properties of those nine galaxies are summarised in Table~\ref{tab:prop}. The columns are as follows: (1) commonly used galaxy name and HIPASS source name; (2) J2000.0 Right Ascension and Declination as given in the RC3 \citep{dev91}; (3) Galactic longitude and latitude; (4) \hi{} systemic velocity as given in the BGC; (5) galaxy distance derived from the velocity relative to the barycentre of the Local Group as given in the BGC; (6) total integrated \hi{} flux density as given in the BGC; (7) apparent {\em B} band photographic magnitude as listed in LEDA; (8) \citet*[ hereafter SFD98]{sch98} Galactic dust extinction in the {\em B} band; (9) absolute {\em B} band magnitude calculated as in eqn.~\ref{eqn:abmag} using the LEDA magnitude and SFD98 extinction; (10) preliminary estimate of \hi{} mass-to-light ratio calculated using eqn.~\ref{eqn:mlr} from the LEDA, BGC and SFD98 data.
\section{Observations}
\label{sec:obs}
Each galaxy was observed in two different wavelength regimes. Optical CCD photometry was obtained with the Australian National University (ANU) 2.3-meter Telescope at the Siding Spring Observatory. \hi{} (21cm) line data were obtained with the Australia Telescope Compact Array (ATCA).
\begin{deluxetable}{lcccccccc}
\tabletypesize{\scriptsize}
\tablecaption{Summary of Observations for each Galaxy.
\label{tab:obs}}
\tablewidth{0pt}
\tablehead{
\colhead{Name} & \multicolumn{3}{c}{Optical} & \colhead{\phantom{0000}} & \multicolumn{4}{c}{Radio} \\
\colhead{} & \colhead{Band} & \colhead{Exposure Time} & \colhead{Seeing} & & \colhead{Arrays} & \colhead{Time On Source} & \colhead{Central Freq.} & \colhead{Phase Cal.} \\
\colhead{} & \colhead{} & \colhead{(seconds)} & \colhead{(arcsec)} & & \colhead{} & \colhead{(hours)} & \colhead{(MHz)} & \colhead{} \\
\colhead{(1)} & \colhead{(2)} & \colhead{(3)} & \colhead{(4)} & & \colhead{(5)} & \colhead{(6)} & \colhead{(7)} & \colhead{(8)} }
\startdata
\mcg{} & {\em B} & 3000 ($10\times300$) & 2\farcs2 & & H75B & $\sim1.5$ & 1417 & PKS\,0023--263 \\
& {\em V} & 2400 ($8\times300$) & 2\farcs0 & & H168B & $\sim8.6$ & 1417 & PKS\,0023--263 \\
& {\em R} & 1800 ($6\times300$) & 1\farcs9 \\
& {\em I} & 1800 ($6\times300$) & 2\farcs2 \\ \\
ESO\,473-G024 & {\em B} & 3000 ($10\times300$) & 2\farcs5 & & H75B & $\sim1.5$ & 1417 & PKS\,0023--263 \\
& {\em V} & 2400 ($8\times300$) & 3\farcs3 & & H168B & $\sim1.0$ & 1417 & PKS\,0023--263 \\
& {\em R} & 1800 ($6\times300$) & 3\farcs3 \\
& {\em I} & 1800 ($6\times300$) & 2\farcs0 \\ \\
ESO\,121-G020 & {\em B} & 3000 ($10\times300$) & 2\farcs1 & & 750D & $\sim10.5$ & 1417 & PKS\,0407--658 \\
& & & & & 1.5B & $\sim10.9$ & 1417 & PKS\,0407-658 \\
& {\em R} & 1800 ($6\times300$) & 1\farcs9 & & EW352 & $\sim2.4$ & 1416 & PKS\,0537--441 \\ \\
ESO\,428-G033 & {\em B} & 3000 ($5\times600$) & 2\farcs2 & & 750D & $\sim10.6$ & 1412 & PKS\,0614--349 \\
& {\em V} & 2400 ($4\times600$) & 2\farcs0 & & 1.5B & $\sim10.3$ & 1412 & PKS\,0614--349 \\
& {\em R} & 1800 ($3\times600$) & 1\farcs9 & & EW352 & $\sim1.1$ & 1414 & PKS\,0614--349 \\
& {\em I} & 1800 ($3\times600$) & 3\farcs0 & & EW367B & $\sim0.8$ & 1414 & PKS\,0614--349 \\ \\
\esoq{} & {\em B} & 3000 ($10\times300$) & 1\farcs9 & & EW352 & $\sim11.6$ & 1417 & PKS\,1215--457 \\
& {\em V} & 2400 ($8\times300$) & 1\farcs9 & & 750A & $\sim10.6$ & 1417 & PKS\,1215--457 \\
& {\em R} & 1800 ($6\times300$) & 1\farcs8 & & 6A & $\sim11.5$ & 1417 & PKS\,1215--457 \\
& {\em I} & 1800 ($6\times300$) & 2\farcs0 \\ \\
ESO\,572-G009 & {\em B} & 1800 ($3\times600$) & 1\farcs9 & & H75B & $\sim1.0$ & 1412 & PKS\,1127--145 \\
& {\em V} & 2400 ($8\times300$) & 2\farcs7 \\
& {\em R} & 1800 ($6\times300$) & 2\farcs2 \\ \\
ESO\,505-G007 & {\em B} & 3000 ($10\times300$) & 2\farcs0 & & H75B & $\sim1.0$ & 1412 & PKS\,1127--145 \\
& {\em V} & 1800 ($6\times300$) & 2\farcs2 & & H168B & $\sim1.9$ & 1412 & PKS\,1127--145 \\
& {\em R} & 1800 ($6\times300$) & 2\farcs2 & & H75B & $\sim9.2$ & 1412 & PKS\,1151--348 \\ \\
IC\,4212 & {\em B} & 3000 ($5\times600$) & 2\farcs9 & & H75B & $\sim8.8$ & 1413 & PKS\,1308--220 \\
& {\em V} & 2400 ($4\times600$) & 2\farcs3 \\
& {\em R} & 1800 ($3\times600$) & 1\farcs9 \\ \\
ESO\,348-G009 & {\em B} & 3000 ($10\times300$) & 1\farcs7 & & 750D & $\sim10.6$ & 1417 & PKS\,0008--421 \\
& {\em V} & 2400 ($8\times300$) & 1\farcs4 & & 1.5B & $\sim9.8$ & 1417 & PKS\,0008--421 \\
& {\em R} & 1800 ($6\times300$) & 1\farcs5 & & EW352 & $\sim1.7$ & 1417 & PKS\,0008--421 \\
& {\em I} & 1800 ($6\times300$) & 1\farcs4 \\
\enddata
\end{deluxetable}
\subsection{Radio Observations}
\label{sec:obs-rad}
ATCA \hi{} line observations of the selected galaxies were carried out between June 2002 and June 2003. The galaxies ESO\,121-G020, ESO\,428-G033, and ESO\,348-G009 ($\delta < -30$\degr{}) were observed for $2 \times \sim12$ hours in different East-West arrays, while \esoq{} was observed for $3 \times \sim$12 hours. For the other five galaxies ($\delta > -30$\degr{}) we used the compact hybrid arrays that include antennas on the Northern spur, resulting in a rather large synthesized beams. The galaxies \mcg{}, ESO\,505-G007, and IC\,4212 were observed for $\sim$10 hours, while ESO\,572-G009 and ESO\,473-G024 were only observed in snapshot mode ($\sim1 - 2$ hours taken over a 12 hour period). \hi{} snapshot observations that were initially taken for the other galaxies (except \esoq{} and IC\,4212) were added to the other observations. Details of the \hi{} observations for each galaxy are given in Table~\ref{tab:obs}. The columns are as follows: (5) ATCA configurations used; (6) approximate time on source for each array; (7) central observing frequency; and (8) phase calibrator. We used a bandwidth of 8 MHz, divided into 512 channels, resulting in a channel width of 3.3\kms{}. The velocity resolution of the \hi{} data is $\sim$4\kms{}. The primary calibrator for all observations was PKS\,1934--638.
Data reduction and analysis were performed with the {\sc MIRIAD} package using standard procedures, with further analysis using {\sc AIPS}, {\sc GIPSY}, and {\sc KARMA}. Channels affected by Galactic \hi{} emission were discarded where appropriate. After continuum subtraction, the \hi{} data were Fourier-transformed using ``natural'' weighting and a channel width of 4\kms{}. The data were cleaned and restored with the synthesized beam (the size of which is given in Table~\ref{tab:radio} for each galaxy). Primary beam correction was applied. \hi{} distributions (0th moment) were obtained for all galaxies using cutoffs between 3 and 4$\sigma$ and are shown in Fig.~\ref{fig:himap}, while the corresponding \hi{} spectra are shown in Fig.~\ref{fig:hispectra}. For ESO\,121-G020, ESO\,428-G033, \esoq{}, and ESO\,348-G009 mean velocity fields and dispersion maps were also produced as they had sufficient resolution.
\subsection{Optical Photometry}
\label{sec:obs-opt}
{\em BVRI} band CCD images were obtained at the 2.3m telescope as a series of 300\,s or 600\,s exposures during observing runs between April 2002 and February 2004 using the Nasmyth Imager (SITe $1124 \times 1024$ thinned CCD). The imager has a circular field of view with a diameter of 6\farcm62 and a pixel size of 0\farcs59. Table~\ref{tab:obs} gives a summary of the observations taken for each galaxy in each band. The columns are as follows: (2) broad band (Cousins) filters used; (3) total observing time in each of the optical bands including the number of individual exposures; and (4) atmospheric seeing in the final optical images. Most observations were taken at low airmass. Twilight sky flat fields in all bands and bias images were obtained at the same time. On each photometric night several \citet{lan92} standard stars were taken together with shallow 120\,s {\em BVRI} images of the galaxy fields to perform the photometric calibration of the deeper images.
Data reduction, photometric calibration, and analysis were carried out within IRAF using standard procedures. After overscan subtraction, bias subtraction, and flatfielding, individual sets of images were registered and the sky level was subtracted. The images for each band were then combined into a single image (to increase signal-to-noise, remove cosmic rays, etc.) and the photometric calibration applied. Fig.~\ref{fig:optimage} shows the resulting master images in the {\em B} band for all nine galaxies.
Foreground stars were removed by replacing them with the surrounding sky so that only the galaxy remained. Special care was taken to restore the light distribution under any stars superimposed onto the galaxies, e.g. using the mirror image from across the galaxies center. For more details of this technique see \citet{jer03}. To illustrate the final result Fig.~\ref{fig:starsub} shows the {\em B} band images of the galaxies after cleaning.
\section{Radio Properties}
\label{sec:radio}
ATCA \hi{} follow-up observations are needed to obtain accurate positions of the targeted HIPASS BGC sources and to reliably identify their optical counterparts. For those galaxies where we have high angular resolution \hi{} observations (ESO\,121-G020, ESO\,428-G033, \esoq{}, and ESO\,348-G009), we also analyse their \hi{} structure and kinematics, including the galaxy rotation curve.
\subsection{\hi{} Structure}
\label{sec:structure}
\begin{deluxetable}{lcccccc}
\tabletypesize{\scriptsize}
\tablecaption{ATCA \hi{} Results.
\label{tab:radio}}
\tablewidth{0pt}
\tablehead{ \colhead{Name} & \colhead{Beam (\hi{})} & \colhead{\Speak{}} & \colhead{\FHI{}} & \colhead{\vsys{}} & \colhead{\whalf{}} & \colhead{\wxx{}} \\
\colhead{} & \colhead{(arcsec)} & \colhead{(Jy)} & \colhead{(\jjks{})} & \colhead{(\kkms{})} & \colhead{(\kkms{})} & \colhead{(\kkms{})} \\
\colhead{(1)} & \colhead{(2)} & \colhead{(3)} & \colhead{(4)} & \colhead{(5)} & \colhead{(6)} & \colhead{(7)} }
\startdata
\mcg{} & $193\times129$ & $0.232 \pm 0.009$ & $16.2 \pm 0.6$ & $670 \pm 2$ & $112 \pm 2$ & $126 \pm 2$ \\ \\
ESO\,473-G024 & $359\times205$ & $0.17 \pm 0.02$ & $5.7 \pm 0.9$ & $542 \pm 3$ & $37 \pm 2$ & $50 \pm 3$ \\ \\
ESO\,121-G020 & $32\times30$ & $0.204 \pm 0.006$ & $9.1 \pm 0.3$ & $583 \pm 2$ & $47 \pm 2$ & $61 \pm 4$ \\
~~\atg{} & $32\times30$ & $0.075 \pm 0.006$ & $2.7 \pm 0.2$ & $554 \pm 4$ & $36 \pm 3$ & $56 \pm 8$ \\ \\
ESO\,428-G033 & $45\times30$ & $0.179 \pm 0.005$ & $14.5 \pm 0.3$ & $1728 \pm 2$ & $94 \pm 2$ & $110 \pm 2$ \\ \\
\esoq{} & $38\times35$ & $2.128 \pm 0.005$ & $122 \pm 4$ & $597 \pm 1$ & $64 \pm 2$ & $90 \pm 4$ \\ \\
ESO\,572-G009 & $477\times327$ & $0.25 \pm 0.03$ & $7.2 \pm 1.3$ & $1740 \pm 4$ & $36 \pm 2$ & $49 \pm 2$ \\ \\
ESO\,505-G007 & $413\times304$ & $0.347 \pm 0.009$ & $21 \pm 3$ & $1776 \pm 2$ & $69 \pm 3$ & $88 \pm 5$ \\ \\
IC\,4212 & $444\times357$ & $0.342 \pm 0.009$ & $46.0 \pm 1.0$ & $1476 \pm 1$ & $158 \pm 2$ & $172 \pm 2$ \\ \\
ESO\,348-G009 & $44\times27$ & $0.165 \pm 0.005$ & $13.1 \pm 0.3$ & $648 \pm 1$ & $86 \pm 2$ & $100 \pm 3$ \\
\enddata
\end{deluxetable}
The \hi{} distributions of all observed galaxies are shown in Fig.~\ref{fig:himap} overlaid onto second generation Digitized Sky Survey (DSS\,{\sc II}) {\em R} band images. The synthesized beam sizes for the individual observations differ significantly due to the use of either East-West or northern hybrid arrays. The ATCA \hi{} spectra of the observed galaxies are shown in Fig.~\ref{fig:hispectra}, with the BGC spectra plotted for comparison.
The ATCA \hi{} observations of the galaxy ESO\,121-G020 reveal a previously uncatalogued galaxy at a projected distance of only 3\arcmin{} (see Fig.~\ref{fig:himap}{\em c}). The two galaxies are well resolved spatially, and their systemic velocities differ by only $29 \pm 6$\kms{}, less than the 50\% velocity width of either galaxy. In the following we refer to this companion as \atg{} according to its \hi{} centre position. The \hi{} spectra of the two galaxies as well as the global spectrum of the galaxy pair is shown in Fig.~\ref{fig:121spect} (Fig.~\ref{fig:hispectra}{\em c} shows the global spectrum only). \atg{} has no previous optical measurements and we included it in our optical follow-up observations (see \S~\ref{sec:optp}). Because \atg{} contributes $\sim$20\% to the \hi{} flux density of HIPASS J0615--57, the \hi{} mass to light ratio for ESO\,121-G020 decreases slightly.
The galaxy ESO\,505-G007 likewise has a neighbor, the catalogued galaxy ESO\,505-G008, which lies at a projected distance of $\sim7$\arcmin{}. However, the two galaxies are not entirely spatially or spectrally resolved in our \hi{} observations. ESO\,505-G007 appears to have a low inclination in the optical images and its \hi{} line is relatively narrow. In contrast, the galaxy ESO\,505-G008 is seen close to edge-on and shows a broad \hi{} line, as seen in Fig.~\ref{fig:hispectra}{\em g}. By fitting two point sources to the low-resolution \hi{} distribution we determine an \hi{} flux density of $21 \pm 3$\jks{} for ESO\,505-G007, in excellent agreement with the BGC value for HIPASS\,J1203--25 despite the confusion, and $8 \pm 3$\jks{} for ESO\,505-G008.
The ATCA \hi{} maps of the other seven galaxies match the optical counterparts identified in the BGC. For \mcg{} (Fig.~\ref{fig:himap}{\em a}) the \hi{} distribution is extended North-South and aligned with the stellar distribution (see \S~\ref{sec:optp}), although little other structure is distinguishable with the large beam. For IC\,4212 (Fig.~\ref{fig:himap}{\em h}) the source appears extended compared to the large beam, and deconvolution with imfit in {\sc MIRIAD} indicates that the \hi{} extends $\sim$200\arcsec{}. We find that for all galaxies, except \esoq{} (see \pI{}), the measured ATCA \hi{} flux densities are in agreement with the BGC values (including the combined ESO\,121-G020/\atg{} system).
Our results for the ATCA \hi{} observations of the nine galaxies are listed in Table~\ref{tab:radio}. The columns are as follows: (1) galaxy name (with \atg{} included); (2) size of the synthesized beam; (3) \hi{} peak flux density; (4) integrated \hi{} flux density; (5) \hi{} systemic velocity from the \hi{} line; (6) velocity width of the \hi{} line at 50\% of the peak flux density; and (7) velocity width of the \hi{} line at 20\% of the peak flux density.
\subsection{\hi{} Gas Dynamics}
\label{sec:velo}
\begin{deluxetable}{lccccc}
\tabletypesize{\scriptsize}
\tablecaption{Rotation curve fit for galaxies with high-resolution ATCA \hi{} data.
\label{tab:rot}}
\tablewidth{0pt}
\tablehead{ \colhead{Name} & \colhead{\vsys{}} & \colhead{{\it PA}} & \colhead{{\em i}} & \colhead{\vmax{}} & \colhead{$r_{\rm max}$} \\
\colhead{} & \colhead{(\kkms{})} & \colhead{(degrees)} & \colhead{(degrees)} & \colhead{(\kkms{})} & \colhead{(arcsec)} \\
\colhead{(1)} & \colhead{(2)} & \colhead{(3)} & \colhead{(4)} & \colhead{(5)} & \colhead{(6)}}
\startdata
ESO\,121-G020 & $584.5 \pm 1.0$ & $262 \pm 2$ & $78 \pm 5$ & $21 \pm 2$ & $80 \pm 10$ \\ \\
ESO\,428-G033 & $1717 \pm 3$ & $295 \pm 5$ & $11 \pm 5$ & $200^{+160}_{-60}$ & $130 \pm 5$ \\ \\
\esoq{} & $597 \pm 1$ & $119 \pm 2$ & $36 \pm 10$ & $51 \pm 8$ & $370 \pm 20$ \\ \\
ESO\,348-G009 & $646 \pm 2$ & $245 \pm 3$ & $80 \pm 5$ & $50 \pm 5$ & $160 \pm 15$ \\
\enddata
\end{deluxetable}
The \hi{} velocity fields (1st moment) of the galaxies ESO\,121-G020/\atg{}, ESO\,428-G033, \esoq{} (\pI{}), and ESO\,348-G009 are shown in Fig.~\ref{fig:hivfield}. All galaxies show clear signs of rotation. The velocity field of the galaxy ESO\,428-G033 appears similar to that of \esoq{}, displaying fairly regular, undisturbed gas dynamics. The extreme velocity contours on both the approaching and receding sides close, suggesting that either the rotation curve turns down at large radii or the galaxy is warped. The position angle ($PA$) of ESO\,428-G033, as measured from its \hi{} velocity field, is aligned with the bright optical emission, although the nature of the stellar feature is unclear (see \S~\ref{sec:dis-compd}). Likewise, the position angle of the galaxy ESO\,348-G009 aligns well with the optical disk (see \S\S~\ref{sec:optp} and \ref{sec:dis-comp} for more on the optical details of the galaxies).
The analysis of the rotation curve for \esoq{} using {\sc rocur} in {\sc AIPS} was presented in \pI{}. For the other three galaxies we instead used the equivalent procedure {\sc rotcur} in {\sc GIPSY} \citep[both use the tilted ring algorithm described by][]{beg89}. We used the standard procedure of narrowing the free parameters (centre position, \vsys{}, position angle, and inclination) down one at a time until the best fit to all parameters was obtained and a rotation curve could be produced. All three fits were done with 10\arcsec{} rings (the \esoq{} fit used 12\arcsec{} rings). After an initial fit which included both sides of the galaxy, the fit was done individually for the approaching and receding sides of the galaxy to check for any asymmetry. Of the three new fits, the only galaxy which had significant differences between the two sides was ESO\,428-G033, where the inclination fit was lower on the approaching side, which may have been a result of the very low inclination (this was included in the uncertainties).
The results of fitting rotation curves to the \hi{} velocity fields of these four galaxies is listed in Table~\ref{tab:rot}. The columns are as follows: (1) galaxy name; (2) systemic velocity; (3) position angle of the galaxy's receding side; (4) inclination angle; (5) maximum rotation velocity; and (6) maximum radius. The final rotation curves are shown in Fig.~\ref{fig:hirotcur}. The very low inclination of ESO\,428-G033 produced a high uncertainty in the rotation velocity values as shown by the curves plotted. In the case of ESO\,348-G009 the curve appears to still be rising at the last points of the rotation curve, suggesting that the \hi{} is not tracing the galaxy out to the radius at which the maximum rotation velocity is reached (so \vmax{} should be considered a lower limit in this case). Similarly for ESO\,121-G020, where we may just be reaching the point where the curve flattens out. In the other two galaxies we appear to have reached the flat part of the curve.
\section{Optical Properties}
\label{sec:optp}
\begin{deluxetable}{lccccccc}
\tabletypesize{\scriptsize}
\tablecaption{2.3m Telescope Optical Results.
\label{tab:opt}}
\tablewidth{0pt}
\tablehead{ \colhead{Name} & \colhead{Band} & \colhead{$m_{T}$\tablenotemark{a}} & \colhead{$\mu_{0}$\tablenotemark{a}} & \colhead{$\langle \mu \rangle _{\rm eff}$\tablenotemark{a}} & \colhead{$r_{\rm eff}$} & \colhead{$r_{\rm H,0}$} & \colhead{\AG{}} \\
\colhead{} & \colhead{} & \colhead{(mag)} & \colhead{(mag arcsec$^{-2}$)} & \colhead{(mag arcsec$^{-2}$)} & \colhead{(arcsec)} & \colhead{(arcsec)} & \colhead{(mag)} \\
\colhead{(1)} & \colhead{(2)} & \colhead{(3)} & \colhead{(4)} & \colhead{(5)} & \colhead{(6)} & \colhead{(7)} & \colhead{(8)}}
\startdata
\mcg{} & {\em B} & $15.32 \pm 0.06$ & $22.34 \pm 0.02$ & $23.77 \pm 0.04$ & $19.6 \pm 0.7$ & $51.0 \pm 2.0$ & $0.08 \pm 0.01$ \\
& {\em V} & $14.91 \pm 0.05$ & $21.89 \pm 0.01$ & $22.98 \pm 0.04$ & $16.4 \pm 0.6$ & -- & $0.06 \pm 0.01$ \\
& {\em R} & $14.40 \pm 0.05$ & $21.60 \pm 0.01$ & $22.68 \pm 0.05$ & $18.1 \pm 0.7$ & -- & $0.05 \pm 0.01$ \\
& {\em I} & $13.60 \pm 0.06$ & $21.17 \pm 0.01$ & $22.76 \pm 0.05$ & $27.1 \pm 1.0$ & -- & $0.04 \pm 0.01$ \\ \\
ESO\,473-G024 & {\em B} & $16.38 \pm 0.06$ & $24.63 \pm 0.04$ & $25.30 \pm 0.04$ & $24.3 \pm 0.8$ & $32.4 \pm 0.8$ & $0.08 \pm 0.01$ \\
& {\em V} & $15.44 \pm 0.07$ & $23.73 \pm 0.03$ & $24.46 \pm 0.03$ & $25.4 \pm 0.7$ & -- & $0.06 \pm 0.01$ \\
& {\em R} & $15.19 \pm 0.03$ & $23.51 \pm 0.03$ & $24.18 \pm 0.02$ & $25.1 \pm 0.4$ & -- & $0.05 \pm 0.01$ \\
& {\em I} & $14.76 \pm 0.07$ & $23.42 \pm 0.08$ & $24.35 \pm 0.02$ & $33.0 \pm 0.9$ & -- & $0.04 \pm 0.01$ \\ \\
ESO\,121-G020 & {\em B} & $15.27 \pm 0.05$ & $23.36 \pm 0.03$ & $23.95 \pm 0.02$ & $21.7 \pm 0.4$ & $47.0 \pm 2.0$ & $0.17 \pm 0.03$ \\
& {\em R} & $14.62 \pm 0.04$ & $22.71 \pm 0.03$ & $23.29 \pm 0.02$ & $21.6 \pm 0.5$ & -- & $0.11 \pm 0.02$ \\
~~\atg{} & {\em B} & $17.01 \pm 0.06$ & $22.81 \pm 0.04$ & $23.34 \pm 0.02$ & $7.4 \pm 0.3$ & $18.6 \pm 0.6$ & $0.17 \pm 0.03$ \\
& {\em R} & $16.36 \pm 0.06$ & $22.07 \pm 0.02$ & $22.72 \pm 0.02$ & $7.5 \pm 0.2$ & -- & $0.11 \pm 0.02$ \\ \\
ESO\,428-G033 & {\em B} & $16.90 \pm 0.10$ & $23.33 \pm 0.11$ & $24.69 \pm 0.02$ & $14.4 \pm 0.5$ & $37.2 \pm 1.2$ & $1.10 \pm 0.18$ \\
& {\em V} & $16.13 \pm 0.10$ & $23.12 \pm 0.02$ & $24.23 \pm 0.05$ & $16.7 \pm 0.8$ & -- & $0.85 \pm 0.14$ \\
& {\em R} & $15.61 \pm 0.08$ & $22.56 \pm 0.02$ & $23.48 \pm 0.05$ & $15.0 \pm 0.9$ & -- & $0.68 \pm 0.11$ \\
& {\em I} & $15.04 \pm 0.09$ & $22.01 \pm 0.03$ & $22.95 \pm 0.04$ & $15.2 \pm 0.7$ & -- & $0.50 \pm 0.08$ \\ \\
\esoq{} & {\em B} & $16.13 \pm 0.07$ & $24.97 \pm 0.03$ & $25.48 \pm 0.02$ & $29.7 \pm 0.6$ & $57.6 \pm 0.6$ & $0.95 \pm 0.15$ \\
& {\em V} & $14.89 \pm 0.06$ & $23.65 \pm 0.03$ & $24.14 \pm 0.02$ & $28.3 \pm 0.7$ & -- & $0.73 \pm 0.12$ \\
& {\em R} & $14.38 \pm 0.05$ & $23.16 \pm 0.02$ & $23.64 \pm 0.03$ & $28.4 \pm 0.5$ & -- & $0.59 \pm 0.09$ \\
& {\em I} & $13.76 \pm 0.06$ & $22.91 \pm 0.04$ & $23.40 \pm 0.03$ & $33.9 \pm 0.8$ & -- & $0.43 \pm 0.07$ \\ \\
ESO\,572-G009 & {\em B} & $16.79 \pm 0.05$ & $24.96 \pm 0.05$ & $26.03 \pm 0.02$ & $28.1 \pm 0.7$ & $30.6 \pm 1.2$ & $0.16 \pm 0.03$ \\
& {\em V} & $15.65 \pm 0.07$ & $23.69 \pm 0.02$ & $25.12 \pm 0.02$ & $31.1 \pm 0.8$ & -- & $0.12 \pm 0.02$ \\
& {\em R} & $15.40 \pm 0.04$ & $23.42 \pm 0.03$ & $24.75 \pm 0.01$ & $29.6 \pm 0.5$ & -- & $0.10 \pm 0.02$ \\ \\
ESO\,505-G007 & {\em B} & $14.20 \pm 0.06$ & $23.99 \pm 0.04$ & $24.00 \pm 0.01$ & $36.4 \pm 0.7$ & $82.8 \pm 1.2$ & $0.36 \pm 0.06$ \\
& {\em V} & $14.48 \pm 0.05$ & $23.89 \pm 0.04$ & $23.96 \pm 0.01$ & $31.3 \pm 0.6$ & -- & $0.28 \pm 0.04$ \\
& {\em R} & $13.97 \pm 0.04$ & $23.46 \pm 0.04$ & $23.66 \pm 0.02$ & $34.6 \pm 0.5$ & -- & $0.22 \pm 0.04$ \\ \\
IC\,4212 & {\em B} & $14.11 \pm 0.04$ & $22.38 \pm 0.02$ & $24.42 \pm 0.02$ & $46.0 \pm 1.0$ & $98.4 \pm 0.6$ & $0.19 \pm 0.03$ \\
& {\em V} & $13.69 \pm 0.05$ & $22.00 \pm 0.03$ & $23.98 \pm 0.01$ & $45.7 \pm 0.9$ & -- & $0.14 \pm 0.02$ \\
& {\em R} & $13.29 \pm 0.06$ & $21.53 \pm 0.02$ & $23.53 \pm 0.02$ & $44.5 \pm 0.9$ & -- & $0.12 \pm 0.02$ \\ \\
ESO\,348-G009 & {\em B} & $14.81 \pm 0.07$ & $23.78 \pm 0.04$ & $24.79 \pm 0.03$ & $39.5 \pm 0.9$ & $72.0 \pm 0.6$ & $0.06 \pm 0.01$ \\
& {\em V} & $14.68 \pm 0.06$ & $23.42 \pm 0.03$ & $24.40 \pm 0.03$ & $35.1 \pm 1.0$ & -- & $0.04 \pm 0.01$ \\
& {\em R} & $14.48 \pm 0.05$ & $23.13 \pm 0.03$ & $24.03 \pm 0.03$ & $32.4 \pm 0.8$ & -- & $0.04 \pm 0.01$ \\
& {\em I} & $13.75 \pm 0.10$ & $22.75 \pm 0.03$ & $23.72 \pm 0.02$ & $39.3 \pm 1.1$ & -- & $0.03 \pm 0.01$ \\
\enddata
\tablenotetext{a}{Correction for Galactic extinction not applied.}
\end{deluxetable}
The observed {\em B} images taken on the 2.3m Telescope of all nine galaxies are in Fig.~\ref{fig:optimage}, while Fig.~\ref{fig:starsub} shows the same images after star subtraction. Star contamination is obviously a problem for \esoq{} and especially ESO\,428-G033, both of which are close to the Galactic Plane, so special care was taken with these images. Some of the galaxies (notably \mcg{}, \esoq{}, ESO\,505-G007 and ESO\,348-G009) were affected by moderately bright foreground stars sitting on top of the galaxy that had to be removed with care. As mentioned in \S~\ref{sec:radio}, a companion galaxy to ESO\,121-G020 was found in the \hi{} imaging. The optical image for ESO\,121-G020 also includes \atg{}, and the photometry for that galaxy is included with the other results here. For each galaxy a growth curve was measured on the star-subtracted images from the luminosity weighted center in 2 pixel ($\sim$1\farcs2) circular aperture rings to obtain the total intensity and a surface brightness profile.
Our results for the optical {\em BVRI} photometry taken on the 2.3m Telescope are in Table~\ref{tab:opt}. No Galactic extinction correction was applied to the apparent magnitude and surface brightness. The columns are as follows: (1) galaxy name; (2) broad band (Cousins) filter used; (3) total apparent magnitude; (4) central surface brightness; (5) effective surface brightness (i.e. the average surface brightness out to the half light radius); (6) half light (effective) radius; (7) radius out to $\mu = 26.6$~mag arcsec$^{-2}$ (i.e. the Holmberg radius in the {\em B} band, extinction corrected); and (8) Galactic extinction correction from SFD98.
Surface brightness profiles for the nine initially selected galaxies in each of the observed bands are in Fig.~\ref{fig:sbpro}, while the profile of \atg{} is in Fig.~\ref{fig:sbpatca}. All profiles have been corrected for Galactic extinction, but no attempt to correct for inclination was made as it is difficult to calculate given the morphology of many galaxies, and as the correction for edge on galaxies (such as ESO\,348-G009) would be unrealistic without correction for the thickness of the stellar disk. The profiles for several galaxies show some of the underlying morphology. \mcg{} (Fig.~\ref{fig:sbpro}{\em a}) has two distinct components, an inner bright bulge region which is prominent in the optical image, and a surrounding low surface brightness disk (almost 4~mag fainter in surface brightness). IC\,4212 (Fig.~\ref{fig:sbpro}{\em h}) appears to have a small central bulge, and the effect of the spiral arms are evident in the bumps of the profile. ESO\,572-G009 (Fig.~\ref{fig:sbpro}{\em f}) also seems to have a small central bulge, which is seen in the image (Fig.~\ref{fig:starsub}{\em f}). ESO\,348-G009 and ESO\,428-G033 (Figs.~\ref{fig:sbpro}{\em d} and {\em i}) exhibit pure exponential disks, while ESO\,473-G024 (Fig.~\ref{fig:sbpro}{\em b}) is similar but with a flatter central region.
The various light profiles of ESO\,505-G007 (Fig.~\ref{fig:sbpro}{\em g}) are flat in the central region as the luminosity weighted centre of this galaxy is a faint region between many denser sites that are probably star formation regions. The optical structure of this galaxy (see Fig.~\ref{fig:starsub}{\em g}) possibly reflects a highly disturbed galaxy with increased star formation due to interaction with the neighboring galaxy ESO\,505-G008 (the source which contaminates its \hi{} spectrum). The new galaxy \atg{} notably has a much steeper exponential profile than its companion ESO\,121-G020, as well as a higher central surface brightness. This denser structure may be the result of recent star formation (there is a bright region in the centre of the galaxy which could be a star formation site), and may have lead to it previously being missed in optical surveys if it were mistaken for a star \citep[it is falsely identified as a star in the USNO star catalogue,][]{mon03}.
For only two galaxies, ESO\,473-G024 and \esoq{}, did we find {\em B} band apparent magnitudes that are in agreement with the values quoted in LEDA (even within their sometimes large error bars). In the case of all seven other galaxies our value of \mB{} was brighter than that given in LEDA, sometimes by several magnitudes. The effect this has on the \hi{} mass-to-light ratios and why this occurred so often in our sample will be discussed in the following sections (\S\S~\ref{sec:dis-comp} and \ref{sec:dis-slide}).
\section{Discussion of Individual Galaxies}
\label{sec:dis-indi}
\subsection{Comparing Optical and \hi{} Properties}
\label{sec:dis-comp}
Now that we have more accurate optical and \hi{} measurements we can recalculate many of the physical properties of the galaxies, including \mlr{}. Table~\ref{tab:summary} summarises new parameters from our observations. The columns are as follows: (1) galaxy name; (2) {\em B} band absolute magnitude; (3) {\em B} band luminosity; (4) \hi{} mass; (5) \hi{} mass-to-{\em B} band luminosity ratio (\mlr{}); (6) total dynamical mass; (7) \hi{} mass-to-total mass ratio; and (8) total mass-to-{\em B} band luminosity ratio. The latter three values (\Mtot{}, \mtmr{}, and \tmlr{}) are only given for the four galaxies where we fit rotation curves.
\begin{deluxetable}{lccccccc}
\tabletypesize{\scriptsize}
\tablecaption{Summary of Derived Galaxy Properties from ATCA and 2.3m Data.
\label{tab:summary}}
\tablewidth{0pt}
\tablehead{ \colhead{Name} & \colhead{\MB{}} & \colhead{\LB{}} & \colhead{\MHI{}} & \colhead{\mlr{}} & \colhead{\Mtot{}} & \colhead{\mtmr{}} & \colhead{\tmlr{}} \\
\colhead{} & \colhead{(mag)} & \colhead{($\times 10^{7}$\Lsun{})} & \colhead{($\times 10^{7}$\Msun{})} & \colhead{(\mmls{})} & \colhead{($\times 10^{9}$\Msun{})} & \colhead{} & \colhead{(\mmls{})} \\
\colhead{(1)} & \colhead{(2)} & \colhead{(3)} & \colhead{(4)} & \colhead{(5)} & \colhead{(6)} & \colhead{(7)} & \colhead{(8)}}
\startdata
\mcg{} & $-14.65 \pm 0.06$ & $11.3 \pm 0.6$ & $34.5 \pm 1.3$ & $3.0 \pm 0.3$ \\ \\
ESO\,473-G024 & $-13.10 \pm 0.06$ & $2.7 \pm 0.1$ & $7.8 \pm 1.2$ & $2.8 \pm 0.6$ \\ \\
ESO\,121-G020 & $-12.97 \pm 0.06$ & $2.39 \pm 0.13$ & $3.61 \pm 0.12$ & $1.50 \pm 0.13$ & $0.16 \pm 0.05$ & $0.23 \pm 0.11$ & $7 \pm 4$ \\
~~\atg{} & $-11.23 \pm 0.07$ & $0.48 \pm 0.03$ & $1.07 \pm 0.08$ & $2.2 \pm 0.3$ \\ \\
ESO\,428-G033 & $-16.7 \pm 0.2$ & $28 \pm 5$ & $130 \pm 3$ & $4.5 \pm 0.9$ & $110^{+190}_{-70}$ & $0.012^{+0.020}_{-0.008}$ & $\sim380$ \\ \\
\esoq{} & $-12.9 \pm 0.2$ & $2.3 \pm 0.4$ & $50.8 \pm 1.7$ & $22 \pm 4$ & $4.5 \pm 1.6$ & $0.11 \pm 0.04$ & $200 \pm 110$ \\ \\
ESO\,572-G009 & $-14.92 \pm 0.06$ & $14.5 \pm 0.8$ & $71 \pm 13$ & $4.8 \pm 1.1$ \\ \\
ESO\,505-G007 & $-17.75 \pm 0.08$ & $196 \pm 14$ & $210 \pm 30$ & $1.1 \pm 0.2$ \\ \\
IC\,4212 & $-17.36 \pm 0.05$ & $137 \pm 6$ & $356 \pm 8$ & $2.55 \pm 0.17$ \\ \\
ESO\,348-G009 & $-14.87 \pm 0.07$ & $13.8 \pm 0.9$ & $21.8 \pm 0.5$ & $1.56 \pm 0.16$ & $3.8 \pm 1.1$ & $0.06 \pm 0.02$ & $27 \pm 10$ \\
\enddata
\end{deluxetable}
\subsubsection{\mcg{}}
Our {\em B} band apparent magnitude for this galaxy is $\sim0.5$~mag brighter than that expressed in LEDA, which was based on the APM plate scan results of \citet{mad90}. The galaxy has a distinctive optical morphology (Fig.~\ref{fig:starsub}{\em a}). A shallow CCD image shows only a small, bright, centrally concentrated circular region like a BCD. However, deeper observations reveal a second, low surface brightness disk component extending well beyond this central bulge, which is clearly visible in our surface brightness profile for this galaxy (Fig.~\ref{fig:sbpro}{\em a}). Difficulty in measuring the full extent of this faint disk could account for the disagreement between our measurement and the previous result of \citet{mad90}. The \hi{} flux density result on the other hand is in excellent agreement with the BGC and the observations of \citet{fou90}, suggesting we have recovered almost all the Parkes flux density. The result of our combined \hi{} and optical values is that the \hi{} mass-to-light ratio drops moderately to $3.0 \pm 0.3$\mls{}, still higher than ``typical'' late-type galaxies but not unusually so. Despite the large beam for the \mcg{} observations, we can clearly see the \hi{} envelope is extended in a North-South direction. This is in the same direction as the outer low surface brightness optical disk, and the two correlate quite well in both shape and extent.
\subsubsection{ESO\,473-G024}
Of the nine sample galaxies, ESO\,473-G024 had the best agreement with the {\em B} band apparent magnitude quoted by LEDA, also agreeing with the results in \citet{lau89} and \citet{mad90}. Our \hi{} flux density is in agreement with both the BGC and \citet{fou90}. Consequently, \mlr{} remained at a moderate ratio of $2.8 \pm 0.6$\mls{}, one of the higher ratios of this sample after accurate measurements. It is a well studied dwarf irregular from the Sculptor group, and the \mlr{} has been noted before \citep{ski03a,ski03b}. In the optical the galaxy is extended in the North-South direction, and has a number of potential star formation regions (see Fig.~\ref{fig:starsub}{\em b}), some of which were studied by \citet[][ see also \S~\ref{sec:dis-implicat}]{ski03a,ski03b}.
\subsubsection{ESO\,121-G020 and \atg{}}
The unexpected discovery of the companion galaxy \atg{} to the South East of ESO\,121-G020 must affect the \mlr{} previously calculated for this galaxy, as the BGC \hi{} flux density measurement for HIPASS\,J0615--57 includes the \hi{} emission from both galaxies. But, more importantly, our optical {\em B} magnitude determined here is $\sim1.7$~mag brighter than the value in LEDA, although it is closer to the \citet{lau89} total magnitude ($15.85 \pm 0.09$~mag). The new ratio for the combined system is $1.62 \pm 0.18$\mls{}, and for the individual galaxies it is $1.50 \pm 0.13$\mls{} for ESO\,121-G020 and $2.2 \pm 0.3$\mls{} for \atg{}. Down to our sensitivity limits there is neither an \hi{} nor a stellar bridge between the two galaxies detected.
\subsubsection{ESO\,428-G033}
\label{sec:dis-compd}
At only 6\fdg9 from the Galactic Plane, ESO\,428-G033 suffers from relatively high Galactic extinction \citep[$1.10 \pm 0.18$~mag in {\em B},][]{sch98} and foreground star contamination (see Fig.~\ref{fig:optimage}{\em d}). Our {\em B} band magnitude of $16.90 \pm 0.10$~mag is $\sim1$~mag brighter than the LEDA value (which has a high uncertainty of 0.5~mag), but agrees with the \citet{lau89} measurement ($16.83 \pm 0.09$~mag). Despite this, with our new data the galaxy still has a high \hi{} mass-to-light ratio of $4.5 \pm 0.9$\mls{}, which is in good agreement with the results of \citet{kra92}. The optical morphology of the galaxy is difficult to determine due to the high foreground star obscuration. We can only discern an elongated stellar feature (see Fig.~\ref{fig:starsub}{\em d}), which could be a galaxy disk. Alternatively, this could be a central bar, while the fainter spiral arms remain undetected. The \hi{} distribution of ESO\,428-G033 is nearly circular, indicating a disk that is close to face on, which favors the optical feature being a bar rather than a disk. The position angle of velocity field is aligned with the optical emission.
\subsubsection{\esoq{}}
This low surface brightness dwarf irregular galaxy was discussed in detail in \pI{}, where we confirmed that it did have the unusually high \mlr{} as initially suggested by the combination of the BGC results and the magnitudes listed in LEDA. It is included here for comparison with the other galaxies and we refer the reader to the previous work for more details on it. After obtaining new estimates of \mB{} and \FHI{} for all our sample galaxies, \esoq{} remains as the stand out galaxy with \mlr{} = $22 \pm 4$\mls{}. To our knowledge it has one of the highest \hi{} mass-to-light ratios that has been confirmed by accurate measurement to date for any galaxy system, being approximately double the ratio of the best example in the literature DDO\,154 \citep{car89}, and about four times the ratio of our next highest sample objects (ESO\,572-G009 and ESO\,428-G033). Like DDO\,154 and another known high \mlr{} galaxy NGC\,3741 \citep{beg05}, \esoq{} has a highly extended \hi{} envelope, over six times the optical Holmberg radius. \citet{beg05} also point out that all three galaxies are isolated and have low tidal indexes from nearby galaxies. The nearest neighbor we can identify to \esoq{} is approximately 1.7 Mpc away in the Centaurus A Group.
\subsubsection{ESO\,572-G009}
Although our \hi{} flux density was in good agreement with the BGC and \citet{fou90}, our apparent {\em B} magnitude is about half a magnitude brighter than that given in LEDA and that of \citet{lau89}. Despite this correction ESO\,572-G009 remains one of the few galaxies in our sample to retain a high \mlr{} at $4.8 \pm 1.1$\mls{}. Morphologically it is a faint low surface brightness galaxy and appears to have two stellar components, a brighter cigar shaped central region (likely to be a central bar) surrounded by a fainter disk that extends only a short distance (see Fig.~\ref{fig:starsub}{\em f}).
\subsubsection{ESO\,505-G007}
A previous measurement of the integrated \hi{} flux density taken on the Effelsberg Radio Telescope by \citet[ $19.3 \pm 2.2$\jks]{ric87} agrees with both the BGC's and our ATCA results. However, literature \mB{} measurements vary wildly \citep[both $17.64 \pm 0.09$~mag and $15.95 \pm 0.09$~mag from][ dependent on the isophotal level it is measured to]{lau89}. Our measurement is $\sim$3.5~mag brighter than that listed in LEDA (corresponding to a 25 times more luminous). The severe underestimate of the {\em B} band apparent magnitude quoted by LEDA for this galaxy has a dramatic effect on the \hi{} mass-to-light ratio. From the highest ratio of the 789 BGC galaxies with {\em B} magnitudes in LEDA, \mlr{} for ESO\,505-G007 has dropped down to a more typical ratio of $1.18 \pm 0.12$\mls{}, the lowest of our sample galaxies. We note that optically ESO\,505-G007 has an unusual irregular morphology (see Fig.~\ref{fig:starsub}{\em g}), with several large clumpy structures and some ragged spiral arm-like features, which could suggest recent disruption and star formation triggered by interaction with ESO\,505-G008.
\subsubsection{IC\,4212}
IC\,4212 is an unusual galaxy to have in our sample since its optical morphology is that of a face on spiral rather than the dwarf irregular we might expect for the magnitude given in LEDA. It has two bright, loosely wound arms, several fainter arms and a small bright central bar (see Fig.~\ref{fig:starsub}{\em h}). The uncertainty in the apparent magnitude given in LEDA of $\pm 1.1$~mag is quite large. Our optical measurements find that IC\,4212 is a much brighter galaxy than suggested by the \mB{} value LEDA lists (over two magnitudes). This means that the \mlr{} is not the extreme value initally suggested (the third highest ratio of the 789 BGC galaxies). However, IC\,4212 retains at a ratio of $2.55 \pm 0.17$\mls{}, which relatively high for a galaxy with such a distinct spiral structure, typical values for late type spirals being less than 1\mls{} \citep{rob94}. We measure a deconvolved \hi{} diameter of $\sim$200\arcsec{}, suggesting it may extend significantly beyond the optical disk.
\subsubsection{ESO\,348-G009}
The apparent {\em B} magnitude we measured for ESO\,348-G009 was again brighter than the value listed in LEDA, while the \hi{} flux density is consistent with the BGC result. This means we end up with an \hi{} mass-to-light ratio of $1.56 \pm 0.16$\mls{}, down from the preliminary value of $9 \pm 7$\mls{}. The optical image shows an edge on disk galaxy extending East-West, with some clumpy structures visible along the length of the disk (see Fig.~\ref{fig:starsub}{\em i}). The position angle of the \hi{} velocity field is aligned closely with the stellar disk.
\subsection{Distance Uncertainties}
\label{sec:dis-dist}
As was discussed in \pI{}, the use of distances calculated from Local Group velocities for galaxies in the Local Universe can be problematic as the peculiar velocities in nearby groups are potentially of similar order to the redshifts themselves, and the local Hubble flow can differ from the cosmological expansion. Evidence from studies on the Sculptor group \citep{jer98} suggests that the local velocity-distance relationship is much steeper in the direction of this group than for galaxies further out due to the probable gravitational influence of the Local Group. If we use the Hubble constant of \citet[ H$_0$ = 119\kms\,Mpc$^{-1}$]{jer98} for the three Sculptor Group members in our sample \citep[membership confirmed using][]{cot97} then the Local Group velocity distances to these galaxies would be less than listed in Table~\ref{tab:prop}, with 6.0 Mpc for \mcg{}, 4.8 Mpc for ESO\,473-G024, and 5.3 Mpc for ESO\,348-G009. This would put all three on the far side of the Sculptor group \citep{jer98}, and would mean that the distance dependent quantities in Table~\ref{tab:summary} (everything except \mlr{}) would require adjustment.
\section{Discussion}
\label{sec:dis}
\subsection{The revised \hi{} mass-to-light ratios}
\label{sec:dis-slide}
The plot in Fig.~\ref{fig:slidenine} shows \mlr{} versus \MB{} (as in Fig.~\ref{fig:mlmbt}) with the new positions of the nine target galaxies resulting from our observations. The lines connect our results (large points with error bars) to the initial estimates (open circles). While the new measurements resulted in lower \mlr{} values for all selected galaxies, the decrease is particularly significant for galaxies which had preliminary values of \mlr{} $> 5$\mls{}. While disappointing, our result is not too surprising given that we selected galaxies from the BGC with initially the most extreme \mlr{} values and large uncertainties in their optical magnitudes, so we preferentially selected galaxies with underestimated \mB{}. For many of our target galaxies optical magnitudes exist that agree with our results \citep[e.g.][]{lau89}, but the mean magnitudes available from LEDA were generally highly underestimated.
All but one of the selected galaxies, \esoq{}, now have revised \hi{} mass-to-light ratios in the range $\sim$1-5\mls{}. Such revisions are by no means uncommon among claims of high \mlr{} galaxies as we discussed in \S~6.2 of \pI{}, and as seen in \citet{vzee97} and \citet{chu02}. The faint luminosity of these galaxies and the difficulty in getting high quality data in both the optical and radio regime make examples of high \mlr{} galaxies difficult to find. Only a few other galaxies with confirmed high \mlr{} are in the literature, most notably DDO\,154 \citep[9.4\mls{}][]{car89,hof93}, UGCA\,292 \citep[7.0\mls{}][]{you03}, and NGC\,3741 \citep[5.8\mls{}][]{beg05}.
While we must be careful of small number statistics, the trend of all the findings of current studies strongly suggest that there do not appear to be large numbers of ``dim'' galaxies like \esoq{} in the local Universe. Therefore high \mlr{} galaxies cannot account for much of the discrepancy between observations and theoretical predictions of low mass galaxy numbers. But there are other possible ways that galaxies could be missed observationally, and several suggestions how they might be detected. Some methods have been proposed for finding ``empty'' dark matter halos, such as the suggestion to use the Milky Way halo microlensing statistics to look for dark matter satellite influence \citep{wid98}, or to analyse the gravitational lensing of quasars to determine the dark matter sub-halos of the lensing object \citep{moo99,dal02}. However, the existence of halos without baryons is still highly speculative.
True ``dark'' galaxies in the form of isolated, rotating, extragalactic \hi{} clouds have so far proven elusive \citep{rya02,kor04,doy05}. Some isolated \hi{} sources have been found in HIPASS \citep{kil00,ryd01,ryd04} and were interpreted as high velocity clouds or tidal debris by the respective authors. A recent claim of a ``dark galaxy'' close to the one-armed spiral NGC\,4254, in the outskirts of the Virgo Cluster \citep{min05} also appears to be tidal debris \citep{bek05}; see also \citep{oos05}. \citet{tay05} have discussed theoretically that an isolated \hi{} cloud which formed without a stellar component is likely to be unstable to star formation, and therefore would not remain dark.
\subsection{The importance of dust extinction}
\label{sec:dis-extinct}
As well as the accuracy of the {\em B} band photometry and \hi{} flux density it is also important to discuss the one contributor to \mlr{} that is beyond the scope of our observations, the dust extinction due to both our Galaxy (see Table~\ref{tab:opt}) and the host galaxy (``internal'' extinction). As we noted in \pI{} Galactic extinction is particularly important for \esoq{} due to its sky position only 10\fdg5 from the Galactic Plane. ESO\,428-G033 at $b$ = -6\fdg9 is the only other galaxy that is subject to similar Galactic extinction (\AB{} = $1.10 \pm 0.18$~mag, SFD98).
The uncertainties in the SFD98 Galactic extinction grow proportionally to the value. This means that it only contributes a significant fraction to the total error in \mlr{} for \esoq{} and ESO\,428-G033. Close to the Galactic Plane the dust distribution can be patchy. Fig.~\ref{fig:schlegel} shows the SFD98 dust extinction maps in terms of \AB{} for the regions around ESO\,428-G033 and \esoq{}, with the last \hi{} contour from Fig.~\ref{fig:himap} superimposed for reference. Both maps show relatively low variation in the amount of extinction over the field around the galaxy, less than the uncertainty in \AB{} in both cases. This suggests that the SFD98 value at the position of both galaxies is an accurate representation of the true Galactic extinction. ESO\,428-G033 was studied by \citet{kra92} as part of an investigation of galaxies in a region of the Galactic Plane with reduced dust extinction. Their results are in agreement with our measurements, and in general they found that the properties of their sample objects were typical of other samples of nearby galaxies despite the relatively high extinction. It is worth noting that even if the Galactic extinction was much higher at the position of \esoq{}, say by 0.5~mag, the galaxy would still have a very high \mlr{} of $14 \pm 4$\mls{}.
While we can at least get a relatively accurate estimate of Galactic extinction, dust extinction internal to the host galaxy itself is much harder to quantify. Dust extinction in late-type galaxies is poorly understood but thought to be lower than in early-type spirals due to low metalicity and ineffective dust accretion processes \citep{dwe98,hir99}. We would expect a disk galaxy to have significant extinction when viewed close to edge on, like ESO\,348-G009 ($i = 80$\degr{}$\pm 5$\degr{}, see Table~\ref{tab:rot}), due to the geometry of the dust distribution. We discussed \esoq{}'s possible internal extinction in \pI{}, concluding that it is most likely low for this close to face on galaxy. The galaxy ESO\,121-G020 may have a high inclination angle of $78$\degr{}$\pm 5$\degr{} (Table~\ref{tab:rot}), in contrast to the optical dimensions, and a moderate amount of internal extinction. The optical emission of the galaxy ESO\,428-G033 is highly obscured by foreground stars. Due to the current inability to estimate internal extinction we have not accounted for it in our \mlr{} calculations.
\subsection{Physical Characteristics of High \mlr{} Galaxies Implications for Their Existence}
\label{sec:dis-implicat}
Several common elements between the galaxies with the most extreme \mlr{}'s are becoming more evident. The three best examples, \esoq{} ($22 \pm 4$\mls{}, \pI{}), DDO\,154 \citep[9.4\mls{},][]{car89,hof93}, and NGC\,3741 \citep[5.8\mls{},][]{beg05}, all have \hi{} envelopes that are 5 to 8 times the optical Holmberg radius, as IC\,4212 may also have. This may be because the gas is at a low density and is in a stable state, as was seen for \esoq{} in \pI{}. All these galaxies also have low tidal indexes \citep{kar04}, indicating {\em they are isolated in space} and have little external stimulation to form stars. Despite having very low stellar content for their dynamical masses, the baryonic masses of the galaxies are always of the order of $\sim10\%$ of the total dynamical mass, which is a typical fraction seen in galaxies from $L_{*}$ to the dwarf regime \citep{beg05}. This suggests {\em that high \mlr{} galaxies are not lacking the baryons to create stars, but are underluminous as they lack either the internal or external stimulation for further star formation}.
\citet{ski03b} obtained spectra of \hii{} regions within five Sculptor group galaxies including two in our sample, ESO\,473-G024 and ESO\,348-G009. For ESO\,473-G024, for which we found a moderately high \mlr{}, they were able to produce oxygen and nitrogen abundances. The oxygen abundance indicated a low metalicity which is typical of other late type galaxies. Normally dwarf galaxies with a similar low metalicity have low nitrogen to oxygen ratios in a narrow range around an average of $\log{N/O} \simeq -1.6$ \citep{izo99}. It is thought this is because these galaxies are undergoing their first burst of star formation, and that the nitrogen from this burst has not yet had time to dissipate into the ISM \citep[coming from type {\sc ii} supernovae of intermediate mass stars, while oxygen comes from higher mass stars, see][]{ski03b}. However, \citet{ski03b} found that the N/O ratio for ESO\,473-G024 was relatively high. Most importantly for our study, they compared these results to a study of the high \mlr{} galaxy DDO\,154 by \citet{ken01} which shows the same trend for N/O. In both galaxies nitrogen from any past star formation events has had time to disperse and there is no current burst of star formation to reduce the N/O ratio, neither galaxy having a particularly high current star formation rate \citep[][ respectively]{ski03a,ken01} and like \esoq{} they may be considered quiescent galaxies. In fact, their current SFR and luminosity \citep[which can be used as a rough estimate of the average past star formation rate,][]{tin80} are similar to what was found for \esoq{} in \pI{}. This suggests that {\em high \hi{} mass-to-light ratio galaxies may have been forming stars at a low, constant rate over their lifetimes.} In order to understand further the significance of this result we would need to expand this study to spectra of \hii{} regions in other high \mlr{} galaxies (especially \esoq{}) and look at other ratios which may indicate the timing of star formation events. Metalicity may also be an important element in determining the fraction of baryons which remain in gas form \citep{tay05}.
A close look at the plots of \mlr{} versus \MB{} (Figs.~\ref{fig:mlmbt} and \ref{fig:slidenine}) shows that there is potentially an upper envelope to a galaxy's \hi{} mass-to-light ratio at a given luminosity. Low luminosity galaxies appear to be able to have a higher portion of their detectable baryons in the form of neutral hydrogen than galaxies around $L_{*}$, where the baryonic mass is dominated by stars even for the most gas rich galaxies. What this suggests is that {\em there is a minimum quantity of stars a galaxy will form that goes as a function of initial baryonic mass}. Support for this idea can also be found in the theoretical work of \citet{tay05}. Whether or not a galaxy forms more than this minimum is likely to be a function of such factors as environment and the galaxy's dark matter properties.
\citet{tay05} developed models to determine whether a neutral gas disk without stars (a ``dark galaxy'') could remain dynamically stable or if some gas will collapse and form stars. They found that without an internal radiation field the majority of the gas in the disk will become gravothermally unstable, even for galaxies with very low baryonic masses (down to $5\times10^{6}$\Msun{}). They also found that the fraction of unstable gas decreases as the baryonic mass decreases. This may provide an explanation to why we see the slope in the upper envelope for \mlr{}, the lower mass galaxies only having to convert a much smaller fraction of their baryons to stars in order to become stable. Galaxies such as \esoq{} and DDO\,154 are close to our upper envelope, and may define the extreme cases of galaxies which have formed only the minimum number of star required in order to remain stable and have not experienced any other events which may trigger star formation. In this way they might be used to distinguish between various models for galaxy collapse by defining the minimum star formation required for stability. We will further explore this possibility, and other properties that vary with \mlr{}, with a larger sample of 37 late type dwarf galaxies in an upcoming paper.
\section{Conclusions}
\label{sec:conc}
We obtained accurate optical CCD apparent magnitudes and \hi{} flux densities for nine late type dwarf galaxies and recalculated their \hi{} mass-to-light ratios. The new \mlr{} values are:
\begin{itemize}
\item $22 \pm 4$\mls{} for \esoq{},
\item $4.8 \pm 1.1$\mls{} for ESO\,572-G009,
\item $4.5 \pm 0.9$\mls{} for ESO\,428-G033,
\item $3.0 \pm 0.3$\mls{} for \mcg{},
\item $2.8 \pm 0.6$\mls{} for ESO\,473-G024,
\item $2.6 \pm 0.2$\mls{} for IC\,4212,
\item $1.6 \pm 0.2$\mls{} for ESO\,348-G009,
\item $1.5 \pm 0.1$\mls{} for ESO\,121-G020, and
\item $1.2 \pm 0.1$\mls{} for ESO\,505-G007.
\end{itemize}
Many of these \hi{} mass-to-light ratios are significantly below the initial estimates, due to inaccurate magnitude estimates in the literature. This strongly emphasises the importance of having accurate observations in both the \hi{} line and the optical. Based on the new \hi{} mass-to-light ratio distribution we conclude that genuine ``dim'' galaxies with high ratios (\mlr{}$>$5\mls{}) are rare in the local Universe.
A previously uncatalogued companion galaxy to ESO\,121-G020 was found at a projected distance of 3\arcmin{}. \atg{} has an \hi{} mass of $\sim10^{7}$\Msun{} and \mlr{} of $2.2 \pm 0.3$\mls{}. This was the only such companion detected, and is well within the beam of the Multibeam instrument used by the HIPASS survey. Despite our low resolution \hi{} observations we were able to separate the galaxy ESO\,505-G007 from its close neighbor ESO\,505-G008 and determined \hi{} flux densities of $21 \pm 3$ and $8 \pm 3$\jks{}, respectively.
The best examples of high \mlr{} dwarf galaxies in the literature all have highly extended \hi{} disks, are spatially isolated and have normal baryonic content for their dynamical masses. The galaxies are not lacking the baryons to create stars, but are underluminous as they lack either the internal or external stimulation for further star formation. Future examination of element abundances within star formation sites of high \mlr{} galaxies may reveal important clues about their star formation history. Recent observations \citep{ski03b,ken01} support the idea that high \mlr{} galaxies may have been forming stars at a low, constant rate over their lifetimes as proposed in \pI{}. There may be a minimum quantity of stars a galaxy will form that depends on the initial baryonic mass, which is supported by the theoretical work of \citet{tay05}. If this is true then maybe high \hi{} mass-to-light ratio galaxies have over their lifetimes only formed the minimum number of stars necessary to maintain the stability of their \hi{} gas disk.
\section*{Acknowledgments}
We are grateful for the assistance of Ken Freeman and Lister Staveley-Smith in this project, especially for their assistance with observations. We would like to thank Erwin de Blok for his help with various aspects of the \hi{} data reduction and interpretation. We would also like to thank Marilena Salvo and Gayandhi de Silva for their observing assistance. Our thanks also go to the anonymous referee for their useful comments, especially regarding the \hi{} spectra. The 2.3-meter Telescope is run by the Australian National University as part of Research School of Astronomy and Astrophysics. The Australia Telescope Compact Array and the Parkes Radio Telescope are part of the Australia Telescope that is funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO. This research has made use of the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. The Digitized Sky Survey (DSS) was produced at the Space Telescope Science Institute under U.S. Government grant NAG W-2166, based on photographic data obtained using the UK Schmidt Telescope.
\clearpage
|
Title:
The late time evolution of Gamma-Ray Bursts: ending hyperaccretion and producing flares |
Abstract: We consider the properties of a hyperaccretion model for gamma-ray bursts
(GRBs) at the late time when the mass supply rate is expected to decrease with
time. We point out that the region in the vicinity of the accretor and the
accretor itself can play an important role in determining the rate and time
behavior of the accretion and ultimately the energy output. Motivated by
numerical simulations and theoretical results, we conjecture that the energy
release can be repeatedly stopped and then restarted by the magnetic flux
accumulated around the accretor. We propose that the episode or episodes when
the accretion resumes correspond to X-ray flares discovered recently in a
number of GRBs.
| https://export.arxiv.org/pdf/astro-ph/0601272 |
\title[]
{The late time evolution of Gamma-Ray Bursts:
ending hyperaccretion and producing flares}
\author[]
{\parbox[]{6.in} {Daniel Proga$^1$ and Bing Zhang$^1$ \\
\footnotesize
$^1$ Department of Physics, University of Nevada, Las Vegas,
NV 89154, USA, e-mail: dproga@physics.unlv.edu; bzhang@physics.unlv.edu}
\date{Accepted . Received ; in original form }}
\label{firstpage}
\begin{keywords}
accretion, accretion discs -- gamma rays: bursts --
methods: numerical -- MHD
\end{keywords}
\section{Introduction}
Gamma-Ray Bursts (GRB) are generally believed to be powered by hyperaccretion
onto a compact, stellar mass object. The total amount of the available fuel
is considered to be the key factor determining the burst duration.
Within merger scenarios for short-duration GRBs, a neutron star (NS)
is accreted onto another NS
or onto a stellar mass black hole (BH; e.g, Paczy\'nski 1986, 1991;
Eichler et al. 1989; Narayan et al. 1992; Fryer et al. 1999).
Within the collapsar model for long-duration GRBs,
up to 20 $\MSUN$ of a stellar envelope collapses onto the star's core
which is a NS or a BH (e.g., Woosley 1993; Paczy\'nski 1998;
MacFadyen \& Woosley 1999; Popham, Woosley, \& Fryer 1999; Proga et al. 2003).
For short- and long-duration events, the accretion
rate, $\MDOT_a$ must be of order of 1~$\MSUN~{\rm s^{-1}}$, yielding a duration
of less than a few seconds for the former and a duration as long as tens
to hundreds of seconds for the latter. These duration estimates are made
under the assumption
that all the available fuel is accreted during the GRB activity
at a time-averaged constant rate.
Recent GRB observations obtained with {\it Swift} motivate us to review
the above assumption and some other aspects of GRB models.
In particular, early X-ray afterglow lightcurves of
nearly half of the long-duration GRBs show X-ray flares
(Burrows et al. 2005; Romano et al. 2006; Falcone et al. 2006).
X-ray flares are also
found to follow the short-duration GRB 050724 (Barthelmy et al. 2005)
whose host galaxy is early-type, which is consistent with the merger
origin. The flares generally rise and fall rapidly, with typical
rising and falling time scales much shorter than the epoch when the
flare occurs. This time behavior strongly supports the ``internal'' origin
of the flares (Burrows et al. 2005; Zhang et al. 2006; Fan \& Wei 2005),
in contrast to
the ``external'' origin of the power-law decay afterglows. The
internal model not only offers a natural interpretation of the rapid
rise and decay behavior of the flares, but also demands a very small
energy budget (Zhang et al. 2006). Within this picture, the data
require a {\it restart} of the GRB central engine (i.e., a restart
of accretion).
Fragmentations in the collapsing star (King et al. 2005) or in the
outer parts of the accretion disc (Perna et al. 2006) have been
suggested to be responsible for the observed episodic flaring behavior.
These two flare models appeal to one of the basic ingredients of an accretion
powered engine -- the mass accretion rate -- and conjecture that
the episodic energy output is driven by changes in the mass supply and
subsequently accretion rate. In this picture, the inner
part of the accreting system {\em passively} responds
to changes in the accretion flow at larger radii.
Here, we point out that the region in the vicinity of the accretor
and the accretor itself can play an important role of determining
the rate and time behavior of the accretion and the energy output.
In particular, we conjecture that the energy release can be repeatedly
stopped and then restarted, provided the mass supply rate decreases with time
even if the decrease is smooth. For both merger and collapsar
GRB models, a decrease of the mass supply
rate is expected, especially in the late phase of activity,
because the mass density decreases with increasing radius.
In our model, we appeal to the fact that, as mass is being accreted onto
a BH, the magnetic flux is accumulating in the vicinity of the BH.
Eventually, this magnetic flux must become dynamically important
and affect the inner accretion flow, unless the magnetic field
is very rapidly diffused. In the remaining part of the paper
we list and discuss theoretical arguments and results from
a variety of numerical magnetohydrodynamic (MHD) simulations
of accretion flows that support our model. We also provide
analytic estimates to show that our model can quantitatively account
for the observed features of the flares.
\section[]{Magnetic model for GRBs and their flares}
\subsection{Insights from numerical models}
Generally, our model for the flares is based on the results from the numerical
simulation of an MHD collapsar model for GRBs carried out by Proga et al.
(2003) and the results from a number of simulations of
radiatively inefficient accretion flows (RIAFs) onto a BH
(Proga \& Begelman 2003, PB03 hereafter;
Igumenshchev, Narayan, \&Abramowicz 2003, INA03 hereafter).
Proga et al. (2003) performed time-dependent axisymmetric MHD simulations of
the collapsar model. These MHD simulations included a realistic equation
of state, neutrino cooling, photodisintegration of helium,
and resistive heating. The progenitor was assumed to be
spherically symmetric but with spherical symmetry broken by the
introduction of a small, latitude-dependent angular momentum
and a weak split-monopole magnetic field.
The main conclusion from the simulations is that,
within the collapsar model, MHD effects alone are able to launch,
accelerate and sustain a strong polar outflow.
The MHD outflow provides favorable initial conditions for the subsequent
production of a baryon-poor fireball
(e.g., Fuller, Pruet \& Abazajian 2000;
Beloborodov 2003; Vlahakis \& K$\ddot{\rm o}$nigl 2003;
M\'{e}sz\'{a}ros 2002), or a magnetically dominated ``cold
fireball'' (Lyutikov \& Blandford 2002), though the specific
toroidal magnetic field geometry Proga et al. derived differs from some
of these models (e.g., Vlahakis \& K$\ddot{\rm o}$nigl 2003;
Lyutikov \& Blandford 2002).
The latest Swift UV-Optical Telescope (UVOT) observations indicate that
the early reverse shock emission is generally suppressed (
Roming et al. 2005), which is consistent with the suggestion that at least
some GRBs are Poynting-flux-dominated outflows (Zhang \& Kobayashi 2005).
To study the extended GRB activity, one would like to follow the collapse
of the entire star. However, such studies are beyond current computer
and model limits. Therefore, we explore instead the implications of
the published simulations and consider the physics of the collapsing star
to infer the properties and physical conditions in the vicinity of a BH
during the late phase of evolution, i.e., when a significant
fraction of the total available mass is accreted.
The long time evolution of axisymmetric MHD accretion flows
was studied by PB03 who explored simulations very similar to those
performed by Proga et al. (2003) but with much simpler micro physics
(i.e., an adiabatic equation of state, no neutrino cooling
or photodisintegration of helium). Proga et al. (2003) found that
despite the more sophisticated micro physics of the MHD collapsar
simulations the flow cooling is dominated by advection not
neutrino cooling.
As a result, the early phase of the time evolution,
and the dynamics of the innermost flow, are very similar in both
the RIAF simulations and the collapsar simulations.
In particular, after an
initial transient behavior, the flow settles into a complex convolution
of several distinct, time-dependent flow components
including an accretion torus, its corona and outflow, and
an inflow and outflow in the polar funnel (see the left panel in Fig. 1 for
a schematic picture of such a flow).
The accretion through the torus is facilitated by the magnetorotational
instability (MRI, e.g., Balbus \& Halwey 1991) which also
dominates the overall dynamics of the inner flow.
In the remaining part of the paper, we will assume that
the late evolution of the MHD collapsar simulations
is similar to the late evolution of the RIAFs simulations.
This assumption is justifiable because the flows in the collapsar and RIAFs
simulations are similar during the early phase of the evolution
(i.e., their dymanics and cooling are dominated by MRI and advection,
respectively)
The late evolution of RIAFs shows that the torus accretion can
be interrupted for a short time by a strong poloidal magnetic field in
the vicinity of a BH. This result is the main motivation for this paper,
as it shows that the extended GRB activity may be a result
of an accretion flow modulated by
the ``magnetic-barrier'' and gravity. Because this barrier halts
the accretion flow intermittently (see Figs.~6 \& 8 in PB03),
the accretion rate is episodic (see Fig.3 of PB03).
This potentially gives a natural mechanism
for flaring variability in the magnetic-origin models of GRBs as
we first mentioned in Fan, Zhang \& Proga (2005;
see the middle panel of Fig. 1 here, for a cartoon picture of
the accretion halted by the magnetic-barrier.)
The importance of accumulating of the magnetic flux
has been explored and observed by others in various astrophysical
contexts (e.g., Bisnovatyi-Kogan \& Ruzmaikin 1974, 1976;
Narayan, Igumenshchev \& Abramowicz 2003; INA03).
In particular, INA03 carried out a three-dimensional (3D)
MHD simulation (their model B) to late model times.
They found that the magnetic flux accumulates, initially
near the BH and then farther out, and the field
becomes dynamically dominant. At late times, mass is able
to accrete only via narrow streams, in a highly nonaxisymmetric
manner (see also Narayan et al. 2003).
The main difference between PB03's and INA03's results is the extent
and duration of the magnetic dominance. In PB03, the magnetic dominance
is a {\em transient} whereas in INA03 is a {\em persistent} state.
The reason for this difference is the treatment of the magnetic field:
for the initial conditions, PB03 used the split-monopole magnetic field
and any changes in the magnetic flux near the BH during the evolution
are due to the chaotic, small-scale fields generated in the disc.
The detailed analysis show that the disc
properties in PB03's simulations are determined by MRI.
In particular, MRI is responsible for
the complex field structure and
for the disc toroidal field being one or even two orders of magnitude
higher than the poloidal field (see figs. 9 and 10 in PB03
and fig. 3 in Proga et al. 2003).
On the other hand, in their model B, INA03 set up a poloidal
field configuration in the injected gas in such a way that the portion
of the material that accretes always carries in the same sign
of the vertical component of the magnetic field.
The simulations carried out by PB03 and INA03 differ also
in the assumed geometry (axisymmetric versus fully 3D).
INA03 and PB03
do not explore all cases including the case where the external or initial
field has zero net flux or the field with the poloidal component changing
sign on length scales much smaller than the size of
the mass reservoir \footnote{
In the case where the initial or external flux has zero-net flux,
a large scale coherent field might in some circumstances be generated by MRI
(e.g., Livio, Pringle, \& King 2003). If so the central magnetic flux
could vary with time but still be dynamical signifacant for
some periods of time.}. Additionally, these simulations
also do not give definitive answers to the problems for which they were
designed. Nevertheless, they give interesting insights
into the general problem of MHD accretion flows. In particular,
they suggest that magnetic
fields can provide an important parameter determining the time scale for
the accretion; i.e., it can be significantly longer than
the local dynamical time scale.
This can have important implications for the observed X-flares in GRBs,
as we argue here, and for X-ray spectral states for BH binaries
as discussed by Spruit \& Uzdensky (2005, SU05 hereafter).
In fact, the work by SU05 describes very well the general physics and theory
of magnetic flux accumulated by an accretion flow.
Therefore we now turn our attention to some theoretical aspects of the problem
as presented by SU05.
\subsection{Theory of the magnetic barrier and accretion flow}
SU05 considered a new mechanism of efficient inward transport of
the large-scale magnetic field through a turbulent accretion disc.
The key element of the mechanism is concentration of the external field
into patches of field comparable in strength to the MRI turbulence
in the disc. They focused on how to increase the magnetic flux at the center
in the context of BH binaries. In particular, they argue that the capture of
external magnetic flux by accretion disc and its subsequent compression
in the inner regions of the disc may explain both changes in the radiation
spectrum and jet activity in those objects.
However, their model and physical arguments are generic
and applicable to our problem.
One can expect that as the strength of the magnetic field
increases at the center, the field may eventually suppress MRI turbulence
and reduce the mass accretion rate and the power in the outflow.
This should be the case especially for GRBs because the mass inflow
rate at the late time is most likely much lower than at the early time.
The disc may become a Magnetically-Dominated Accretion Flow (MDAF)
as proposed by Meier (2005) or the fields in the polar funnel can expand
toward the equator and reconnect as in PB03's simulations.
In the latter, the torus is pushed outward by the magnetic field.
At this time, the gas starts to pile up outside the barrier;
eventually it can become unstable to interchange instabilities
at the barrier outer edge as suggested by SU05 or the gas in the torus
can squash the magnetic field (compare
Fig.~5 and 6 in PB05 or the middle and right panel in Fig.1 here).
When interchange instabilities operate, magnetic flux from the bundle mixes
outward into the disc while the disc material enters the barrier.
In the accretion disk context, interchange instabilities
have been studied by a few authors
(e.g., Spruit et al. 1995; Lubow \& Spruit 1995;
Stehle 1996; Stehle \& Spruit 2001; Li \& Narayan 2004).
These studies showed that the onset of small-scale modes typical
of interchanges (as in Rayleigh-Taylor instabilities) takes place only
at rather large field strengths, due to a stabilizing effect of the Keplerian
shear. The interchange instability operates at moderate field strengths,
but only at low shear rates (less than Keplerian). However for most of
the time, we expect high shear rates in a torus because a low shear torus
quickly becomes Keplerian due to MRI (e.g., PB03 and Proga et al. 2003).
We note that SU05 interpreted INA03 accretion through the barrier,
in the form of blobs and streams as a product of interchange instabilities.
SU05 also suggested that the field strength at which these instabilities
become effective is most usefully expressed in terms of the degree of support
against gravity provided by the magnetic stress $B_R B_Z$.
According to SU05, the instabilities become effective when the radial magnetic
force, $F_m\sim 2 B_R B_Z/4\pi$, is of the order of a few percent
of the gravitational force, $F_g=GM\Sigma/R^2$, where $M$ is the central mass,
$R$ is the radius, and $\Sigma$ is the surface density.
For $B_R \approx B_Z$, there is a range in field strengths between
the value at which MRI turbulence is suppressed and the value where dynamical
instability of the barrier itself sets in, where no known instability operates
(Stehle \& Spruit 2001). In this range, the disk material
cannot mix or penetrate the magnetic field accumulated at the center
(e.g., the middle panel of Fig. 1).
Instead, mass builds up outside a region with such field
strengths until the magnetic field at the center is compressed enough
for instability to set in.
Thus, both numerical work and theoretical models of
magnetized accretion flows show that the inner most
part of the flow and accretor can respond {\em actively} to
changes of the accretion flow at larger radii.
In particular, the inner most accretion flow can be halted
for a very long time as shown by INA03 or it can be
repeatedly halted and reactivated as shown in PB03.
\subsection{Analytic estimates}
We finish this section with order-of-magnitude estimates of a few key features
of our X-ray flare model. We start by estimating the strength and flux of
magnetic field required to support the gas.
The gas of the surface density, $\Sigma$ can be supported against
gravity by the magnetic tension if $F_g\sim F_m$.
The surface density can be estimated from
$\Sigma=\MDOT /2\pi R \epsilon v_{ff}~{\rm g~cm^{-2}}$,
where $\epsilon v_{ff}$ is the flow radial velocity assumed to be
a fraction $\epsilon$ of the free fall velocity, $v_{ff}$.
Assuming $B_r \approx B_z=B$, the force balance yields the field strength
$B \sim 2\times 10^{16}~\epsilon_{-3}^{-1/2} r^{-5/4} \MDOT_1^{1/2} M_3^{-1}
$~G, where $\epsilon_{-3} \equiv 10^{3} \epsilon$,
$r \equiv R/R_S=R/(2GM_{BH}/c^2)$,
$\MDOT_1=\MDOT/1~\MSUN~{\rm s^{-1}}$, and
$M_3=M/3 \MSUN$. We estimate the magnetic flux as
$\Phi\sim\pi r^2 R_S^2 B(r)= 5\times10^{28}~\epsilon_{-3}^{-1/2}r^{3/4}\MDOT_
1^{1/2} M_3~{\rm cm^2~G}$ from which we obtain an estimate
to the magnetospheric radius
$r_m \approx 60~\epsilon_{-3}^{2/3} \MDOT_1^{-2/3} M_3^{-4/3} \Phi_{30}^{4/3}$,
where $\Phi_{30}\equiv \Phi/(10^{30}~{\rm cm^2 G})$.
Substituting the expression for $B$ into the expression for the surface
density, one finds that a given magnetic flux can support
the gas with the surface density of
$\Sigma_B=5\times10^{19}~\Phi_{30}^2 M_3^{-3}r^{-2}~{\rm g~cm^{-2}}$.
To stop accretion with the hyper rate of $1\MSUN~{\rm s^{-1}}$ onto
a 3$\MSUN$ black hole at r=3 (i.e., for $r_m$ to be 3),
the magnetic flux of order $\Phi_{30} \sim 0.11$ is required.
We now assume that such a magnetic flux is accumulated during hyperaccretion
and that it does not change with time. Under these assumptions, $r_m=300$ for
the mass supply rate of $10^{-3}~\MSUNYR$ representative of
the late time evolution .
This relatively large radius demonstrates one of
our key points that the innermost part an accreting system
can actively respond, via magnetic fields,
to changes in the inflow at large radii.
To estimate the conditions needed to restart accretion,
the accretion energetics and related time scales, we ask what is the mass of
a disc with $\Sigma$ high enough to reduce $r_m$ from 300 to 3 or so.
To answer this question, we adopt Popham et al.' (1999) model
of neutrino-dominated discs. Popham et al. assumed that
neutrino cooling produces a thin disc (Shakura \& Sunyaev 1973)
for accretion rates require to power GRBs.
Using the disc solution for the
density and height (eqs. 5.4 and 5.5 in Popham et al. 1999),
we can express the disc surface density as
$\Sigma_\alpha=1.8\times10^{19}~\alpha^{-1.2} M_3^{-0.8}\MDOT_1r^{-1.25}$~g,
where $\alpha$ is the dimensionless parameter
scaling the stress tensor and the gas pressure (Shakura \& Sunyaev 1973).
Equating $\Sigma_B$ with $\Sigma_\alpha$,
one can estimate the mass accretion rate of an $\alpha$ disc
and compute $M_D$ by integrating $\Sigma_\alpha$ over radius.
For $\Phi_{30}=0.11$ and $\alpha=10^{-2}$
the accretion rate through the $\alpha$ disc is $0.03~\MSUN~s^{-1}$
and $M_D$ for $r$ between 3 and 300 is 0.32~$\MSUN$. This
mass accretion rate is more than one order of magnitude lower
than the rate of $\sim 1~\MSUN~s^{-1}$ typical for
the early time evolution. Thus, our estimates
are consistent with the fact that the X-ray flare luminosity
is at least one or two orders of magnitude lower the prompt
gamma-ray emission (see section 3).
If this disc mass is a result of slow mass accumulation during
the late evolutionary stage, then it will take about 400 s to
rebuild the disc for the mass supply rate of $10^{-3}~\MSUN~s^{-1}$
and 12 s to accrete all this mass at the disc accretion rate of
$0.03~\MSUN~s^{-1}$. The latter is a lower estimate for
the flare duration because, for simplicity, we assumed a relatively high,
{\em constant} disc accretion rate. It is very likely that the rate
changes with time as the shape of the light curve
during the flares indicates.
In our model, the mass supply rate controls
the epochs when the flares happen:
the disc is rebuilt on the time scale which increases with time
because the mass supply slowdowns. Additionally, the flare duration
is coupled to the epoch through the mass of the rebuilt disc.
Thus our model is capable of accounting for the observed
duration - time scale correlation.
\section{Discussion and conclusions}
The detailed analysis of the X-ray flares revealed that
they generally have lower luminosities (by at least one or two orders
of magnitude) than the prompt gamma-ray emission. Additionally,
the total energy of the flare is also typically smaller than that of
the prompt emission, although in some cases both could be comparable
(e.g. for GRB 050502B, Falcone et al. 2006).
Moreover multiple flares are observed in some GRBs and
the durations of these flares seem to be positively correlated with
the epochs when the flares happen, i.e. the later the epoch,
the longer the duration (O'Brien et al. 2005; Falcone et al. 2006;
Barthelmy et al. 2005).
The flare analysis also showed that the later the epoch
the lower the flare luminosity.
The above qualitative properties of the flares provide important constraints
on models of them.
Perna et al.'s (2006) disc fragmentation model promises to
account for the duration - time scale correlation and the duration - peak
luminosity anticorrelation. However, the physical process or processes
causing fragmentation are uncertain. It is also uncertain that
the conditions for the disc fragmentation are met in GRB progenitors.
This seems to be the case especially for the collapsar
model as a relatively high rotation of the progenitor
is required. We also note that magnetic fields can suppress or even prevent
disc fragmentation (e.g., Banerjee \& Pudritz 2006).
Here, we propose that the X-ray flares in GRBs are consequences
of the fact that during the late time evolution of a hyperaccretion
system the mass supply rate should decrease with time while
the magnetic flux accumulating around a BH should increase.
In particular, we point out that the flux accumulated during
the main GRB event can change the dynamics of the inner accretion flow.
We argue that the accumulated flux is capable of
halting intermittently the accretion flow. In our model,
the episode or episodes when the accretion resumes correspond to
X-ray flares.
A comparison of our analytic estimates from Section 2.3
with the observed X-ray flare characteristics, shows that our model is
not only physically based but also can both qualitatively and quantitatively
account for some aspects of the flares -- such as the peak times.
In general, our model fits under the general label of
the magnetic jet model for GRBs as we appeal to the magnetic effects to play
the key role not only during the main event but also during the late evolution.
The importance of the magnetic effects for the X-ray flares can be argued based
on energy budget of the accretion model (Fan et al. 2005).
The X-ray flares discovered in GRBs are relatively new and unexpected
phenomena. They give a strong incentive to apply the existing models
of hyperaccretion systems to circumstances where the mass supply is reduced.
Studies of this kind should reveal whether one needs
to introduce additional physics in order
to explain the flares. If so one should
explore the effects of this on the early evolution of GRBs
and check whether they are consistent with GRBs observations.
Our X-ray flare model has the advantage that it is essentially the same
as the MHD collapsar model for GRBs, with
only one justifiable change in a
key physical property of the collapsar model:
a decrease of the mass supply rate with time.
\section*{Acknowledgments}
We thank D. Meier, D. Uzdensky, and a referee for useful comments.
This work is supported by NASA under grants
NNG05GB68G (DP) and NNG05GB67G (BZ).
\bsp
|
Title:
Silicate Emission in the Spitzer IRS spectrum of FSC 10214+4724 |
Abstract: We present the first MIR spectrum of the z=2.2856 ultraluminous, infrared
galaxy FSC 10214+4724, obtained with the Infrared Spectrograph onboard the
Spitzer Space Telescope. The spectrum spans a rest wavelength range of 2.3-11.5
microns, covering a number of key diagnostic emission and absorption features.
The most prominent feature in the IRS spectrum is the silicate emission at
rest-frame 10 microns. We also detect an unresolved emission line at a rest
wavelength of 7.65 microns which we identify with [NeVI], and a slightly
resolved feature at 5.6 microns identified as a blend of [Mg VII] and [Mg V].
There are no strong PAH emission features in the FSC 10214+4724 spectrum. We
place a limit of 0.1 micron on the equivalent width of 6.2 micron PAH emission
but see no evidence of a corresponding 7.7 micron feature. Semi-empirical fits
to the spectral energy distribution suggest about 45% of the bolometric
luminosity arises from cold 50 K dust, half arises from warm (190 K) dust, and
the remainder, 5%, originates from hot (640 K) dust. The hot dust is required
to fit the blue end of the steep MIR spectrum. The combination of a red
continuum, strong silicate emission, little or no PAH emission, and no silicate
absorption, makes FSC 10214+4724 unlike most other ULIRGs or AGN observed thus
far with IRS. These apparently contradictory properties may be explained by an
AGN which is highly magnified by the lens, masking a (dominant) overlying
starburst with unusually weak PAH emission.
| https://export.arxiv.org/pdf/astro-ph/0601061 |
\title{Silicate Emission in the {\it Spitzer}$^1$\ IRS$^2$\ spectrum of FSC 10214+4724}
\altaffiltext{1}{based on observations obtained with the {\it Spitzer Space Telescope}, which is
operated by JPL, California Institute of Technology for the National Aeronautics and Space Administration}
\altaffiltext{2}{The IRS is a collaborative venture between Cornell
University and Ball Aerospace Corporation that was funded by NASA
through JPL.}
\author{H. I. Teplitz\altaffilmark{3},
L. Armus\altaffilmark{3},
B.T. Soifer\altaffilmark{3},
V. Charmandaris\altaffilmark{4,5,6},
J. A. Marshall\altaffilmark{4},
H. Spoon\altaffilmark{4},
C. Lawrence\altaffilmark{7},
L. Hao\altaffilmark{4},
S. Higdon\altaffilmark{4},
Y. Wu\altaffilmark{4},
M. Lacy\altaffilmark{3},
P. R. Eisenhardt\altaffilmark{7},
T. Herter\altaffilmark{4},
J.R. Houck\altaffilmark{4}
}
\altaffiltext{3}{Spitzer Science Center, MS 220-6, Caltech, Pasadena, CA 91125. hit@ipac.caltech.edu}
\altaffiltext{4}{Astronomy Department, Cornell University, Ithaca, NY 14853}
\altaffiltext{5}{Chercheur Associ\'e, Observatoire de Paris, F-75014, Paris, France}
\altaffiltext{6}{University of Crete, Dept. of Physics, GR-71003 Heraklion, Greece}
\altaffiltext{7}{Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109}
\keywords{
cosmology: observations ---
galaxies: evolution ---
galaxies: high-redshift ---
galaxies: individual (FSC 10214+4724)
}
\section{Introduction}
FSC 10214+4724, at a redshift of $z=2.2856$\ \citep{Rowan-Robinson
1991}, was initially thought to be the most luminous object in the
Universe, but was later revealed to be gravitationally lensed by a
foreground galaxy \citep{Broadhurst 1995, Graham 1995, Serjeant 1995,
Eisenhardt 1996}. The lensing model of \cite{Eisenhardt 1996}\
suggests that the central (optical/UV) source is magnified by a factor
of $\sim 100$, and that the lensing arc is an image of the central
$\sim 0^{\prime \prime}.005$\ (40 pc) of the source at B-band rest
wavelength. They further conclude that the bolometric luminosity of
the source is produced over a larger region (240 pc), implying a
bolometric magnification factor of 30 and an intrinsic luminosity of
$\sim 2 \times 10^{13}$\ \Lsun\ , placing it in the class of
ultraluminous infrared galaxies (ULIRGs). The large magnification
makes FSC 10214+4724 a unique subject for the study of high redshift
ULIRGs, offering greater effective sensitivity and spatial resolution
than possible in observations of unlensed sources.
A central question in understanding FSC 10214+4724 is the relative
contribution of starburst and AGN components to its luminosity. The
rest-frame UV-optical spectrum appears to be similar to that of
Seyfert 2 galaxies \citep{Elston 1994}. Strong UV polarization shows
that much of the UV continuum results from scattered light, and broad
lines in the polarization spectrum identify the presence of an AGN
\citep{Lawrence 1993, Goodrich 1996}. The bulk of the luminosity,
however, is emitted in the infrared \citep[as much as
99\%,][]{Rowan-Robinson 1991}. CO observations point to a large
reservoir of molecular gas \citep{Solomon 1992, Scoville 1995}.
Sub-mm and millimeter detections \citep{Downes 1992, Rowan-Robinson
1993,Benford 1999} suggest substantial emission from cold dust,
usually associated with extended star formation. Recent {\it
Chandra}\ observations show the object to have weak X-ray emission,
consistent with vigorous star formation or a Compton-thick AGN \citep{Alexander 2005}. Lens
models suggest differential amplification, so that the central
(AGN-dominated) region is more highly magnified than the surrounding
starburst \citep[e.g.,][]{Eisenhardt 1996, Lacy 1998}.
The sensitivity and large wavelength coverage of the {\it Spitzer}\
Infrared Spectrograph \citep[IRS][]{Houck 2004}\ makes it possible to
explore the dust emission and absorption features in the rest-frame
mid-infrared spectra of dusty galaxies at low and high redshift, and
thus identify the power sources which may be hidden in the UV and
optical. In this paper, we present the first MIR spectrum of FSC
10214+4724. We describe the observations and data reduction in
Section 2, present the spectrum and a dust emission model fit to the
spectral energy distribution in Section 3, and discuss the
implications in Section 4. Throughout, we assume a
$\Lambda$-dominated flat universe, with $H_0=70$\ km s$^{-1}$\
Mpc$^{-1}$, $\Omega_{\Lambda}=0.7, \Omega_{m}=0.3$.
\section{Observations and Data Reduction}
Spectra of FSC 10214+4724 were obtained with the IRS on 19 April 2004.
The data were taken in the first order of the low resolution, short
wavelength module (SL-1; 7.5-14.2 \mic) and in both orders of the low
resolution, long wavelength module (LL-1 and LL-2, 14.2-21.8 and
20.6-38 \mic, respectively). The spectral resolution varies from 60
to 120 across each order. Individual ramp durations were 60 seconds
in SL-1 and 120 in LL-1 and LL-2. Spectra were taken in the standard
``staring'' mode, with four exposures at each of two positions,
separated by one third of the slit length. A total of 8 individual
spectra were taken in each sub-slit, for total on-source integration
times of 480, 960, and 960 seconds.
Spectra were reduced using the S11 pipeline at the {\it Spitzer}\
Science Center, which includes ramp fitting, dark sky subtraction,
droop correction, linearity correction, and wavelength calibration.
One-dimensional spectra were extracted from the un-flatfielded
two-dimensional spectra using the SMART data reduction package
\citep{Higdon 2004}. The extractions are then flux calibrated with
the IRS spectrum of a star ($\alpha$\ Lac), extracted in an identical
manner using SMART. The data have been sky-subtracted by
differencing the two nod positions along the slit, before spectral
extraction. As a final step, we have scaled the SL-1 and LL-2 1D
spectra by 3\% and 8\%, respectively, to match the LL-1 spectrum in
the overlap region and produce a single low-resolution spectrum from
$7.5-38\mu$m (observed frame).
Mid-IR photometry of FSC10214+4724 at 3.6, 4.5, 5.8 and 8.0 $\mu$m was
obtained on 20 May 2004 using the Infrared Array Camera
\citep[IRAC;][]{Fazio 2004}. The galaxy was observed in full array
mode with a cyclic 5 point dither pattern resulting in a total on
source integration time of 1 minute per filter. The source was
unresolved and its Full-Width at Half Maximum (FWHM) varied between
1.5\arcsec and 2\arcsec. We performed photometry on the final mosaics
produced by the SSC pipeline (S11.0.2) using an aperture of 3.6 arcseconds
radius following the method described in the IRAC data Handbook.
The resulting flux densities are accurate to $<$5\%. No attempt was made to
correct the IRAC photometry for the contribution of the lensing
galaxy which is unresolved from the source.
\section{Results}
We plot the IRS spectrum of FSC 10214+4724 in Figure \ref{fig: spectrum}.
The most prominent feature is a steep rise at the red end ($\sim
8-10$\ \mic\ rest wavelength). Extending the spectral energy
distribution (SED) with the IRAS 60 and 100 \mic\ photometry
\citep{Moshir 1990} makes clear that there is not simply a steeply
rising continuum, but rather is a broad emission feature on top of a
somewhat less steep continuum. We identify this feature as silicate
emission, centered at $\sim 10$\ \mic. It is difficult to measure an
accurate equivalent width for the feature, because the IRS wavelength
coverage does not extend far enough to the red.
Three weak emission lines are also present. We identify the narrow
emission line at rest wavelength 7.65 \mic\ as the [Ne VI] fine
structure line. This line has a rest-frame equivalent width (EW) of
0.02 \mic, roughly comparable to that observed in low redshift
Seyferts \citep{Sturm 1999, Lutz 2000}. The emission feature observed
at $\sim 5.5$\ \mic\ in the rest-frame appears to be marginally
resolved. We identify it as a blend of [Mg VII] and [Mg V] at 5.503
and 5.610 \mic, respectively. These lines have been seen in nearby
AGN \citep{Sturm 2002}\ and have ionization potentials of about 186
and 105 eV, similar to the 158 eV ionization potential of [Ne VI].
While there is also a rotational transition of H$_{2}$ at $5.51\mu$m
(the S(7) line) which is often quite strong in ULIRGs, this feature is
always accompanied by much stronger emission from the other, lower
transition, H$_{2}$ rotational lines, which are not present. The
rest-frame equivalent width of the [Mg VII] and [Mg V] lines combined
is $\sim 0.03\pm 0.01$\ \mic. While uncertain, this EW appears to be
a factor of 3-5 higher than in local AGN \citep{Sturm 2002}. The EW
of the Ne and Mg lines is a factor of a few higher than the strongest
of the narrow, high-excitation emission lines (C IV, He II, [Ne IV])
seen in the rest-frame UV spectrum of FSC 10214+4724
\citep{Rowan-Robinson 1991, Goodrich 1996}.
There is a marginal detection of broad emission at 6.2 \mic\
rest-frame, corresponding to the wavelength of polycyclic aromatic
hydrocarbon (PAH) emission. Although the feature is clearly visible
in Figure \ref{fig: spectrum}, examination of the individual
extractions shows that it is more prominent in LL-2 than in LL-1. The
observed wavelength (20.3 \mic) places the feature near the noisy blue
end of LL-1, but at a wavelength that is usually regarded as reliable.
We take the measurement, $EW \sim 0.1$\ \mic\ in the rest-frame,
as an upper limit. We also note that the corresponding 7.7 and 8.6
\mic\ PAH emission features are not seen, despite being redshifted
into a clean part of the spectrum and the fact that they are usually
1.5--2 times stronger than the 6.2 \mic\ feature in
starburst-dominated low redshift ULIRGs. The low ratio of 7.7 to 6.2
\mic\ PAH equivalent width, while highly unusual, is not impossible
under certain conditions \citep{Kessler-Silacci 2005}. The 11.3 \mic\
PAH feature might also be expected, but that line falls near the red end
of LL-1 where the noise precludes a meaningful limit. Nonetheless,
the lack of other PAH lines indicates that the 6.2 \mic\ feature
should be treated with caution.
\section{Discussion}
The shape of the SED of FSC 10214+4724 is dominated by
dust at a variety of temperatures. Substantial
cold dust must be present, given the strong emission at long
wavelengths \citep[$> 40$\ \mic, rest-frame; e.g., ][]{Rowan-Robinson 1993}.
At the same time, warmer dust will be required to explain the
5-10 \mic\ continuum. Dust warmer than 100 K can produce
the silicate emission \citep[e.g.,][]{Li 2001}, but a hot dust
component (several hundred K) is required to explain the $\sim 5$\
\mic\ continuum. The rest-frame near-infrared (NIR) will have a contribution
from both star light and hot dust.
We have fit the SED with a
multi-component model which includes three graphite and silicate dust
grain components and a 3500K blackbody stellar component \citep{Marshall 2006}.
The three dust components in the model are not meant to represent
three distinct physical structures, but rather they are indicative of
the average temperature ranges within the source. The model was fit
to the SED from observed frame NIR to millimeter wavelengths (see
Figure \ref{fig: SED fit}. NIR data included the photometry of
\cite{Soifer 1991}\ and the IRAC data described in Section 2. Longer
wavelength data included IRAS photometry at 60 and 100 \mic, the 350
\mic\ detection of \cite{Benford 1999}, and the sub-mm and mm data of
\cite{Rowan-Robinson 1993}\ and \cite{Downes 1992}.
A minimum of three dust components are required to fit the FSC
10214+4724 SED, one at $\sim 50$\ K (cold), one at $\sim 190$\ K
(warm), and one at $\sim 640$\ K (hot). Their relative contributions
to the bolometric luminosity are given in Table \ref{tab: SED fit} .
Note that hot dust contribution is an upper limit, because it
is dominated by the IRAC data and no correction has been made for
contamination by the foreground lensing galaxy.
The stellar emission is a negligible contribution to the bolometric
luminosity, but is needed to fit the shortest wavelength data.
component. The cold dust component ($51 \pm 6$\ K) is in good
agreement with the estimate of \cite{Benford 1999}, 55 K. Each
component contains a distribution of grains with different equilibrium
temperatures depending on their size and composition. Additionally,
each component contains emission from dust at different radial
distances, and therefore temperatures, from the illuminating source.
We define the characteristic temperature of a component to be the
temperature of the most luminous grain size at the distance from the
source contributing the majority of the luminosity. This luminosity
dominating distance corresponds to a $\tau(UV) \sim 0.5$, where
approximately half of the UV-source photons have been absorbed. Each
component therefore contains dust above and below the characteristic
temperature. With this definition, the characteristic temperature
roughly corresponds to the expected peak in the dust modified
(grey-body) Planck function.
The observed SED of FSC 10214+4724 appears consistent with low
redshift, AGN-dominated ULIRGs. Such sources can have cold dust
components which account for up to 40\%\ of the bolometric luminosity,
due to both AGN-heated cold dust far from the nucleus and the presence
of a small underlying starburst \citep{Armus 2004, Armus 2006}.
\cite{Rowan-Robinson 2000} also estimates that the AGN contributes
most of the luminosity of FSC 10214+4724. In addition, FSC 10214+4724
shows silicate emission, similar to other AGN observed with the IRS,
and a hot (graphite) dust continuum. However, the hot dust
contribution is quite small compared to many AGN-dominated ULIRGs or
QSOs. The only obvious fine-structure lines are [Ne VI], [Mg VII] and
[Mg V], high-ionization species not observed in starburst galaxies.
There is little if any PAH emission. The upper limit to the $6.2\mu$m
EW is approximately a factor of $5-10$ lower than most pure starburst
galaxies \citep{Brandl 2005}\ and ULIRGs dominated by star formation
\citep{Armus 2004, Armus 2006}.
However, differential magnification is likely to be enhancing the central AGN,
reducing the PAH EW and making the silicate emission more obvious in
the IRS spectrum. \cite{Eisenhardt 1996}\ estimate a magnification
factor of $\sim 100$\ for the central 40 pc, but only a factor of
$\sim 30$\ out to at least 240 pc. The intrinsic contribution of the
starburst to the bolometric luminosity could be larger than that
inferred by our current model. We assume that the warm and hot dust
heated by the AGN are magnified by an additional factor of 3.3, but
that both the cold dust heated by the AGN and any dust heated by
starburst are not. The ratio of cold to warm dust in local AGN- and
starburst-dominated ULIRGs is approximately 2:3 and 3:1, respectively
(Armus et al. 2006, in preparation). Taking these assumptions
together, we estimate the intrinsic AGN contribution to the bolometric
luminosity of FSC 10214+4724 to be $\sim 35$\% after correction for
the differential magnification. If the differential magnification
factor is correct, this AGN contribution is probably an upper limit
because AGN-dominated ULIRGs often have measurable starbursts.
If the starburst contributes 65\%\ of the intrinsic luminosity, it is
surprising that little or no PAH emission is seen. Taking our limit
of 0.1 \mic\ $>EW_{\mbox{rest}}$\ and a differential magnification
factor of 3.5 would place the limit of the EW for PAH emission within
the range of other star-forming ULIRGs\citep[Armus et al. 2005, in
prep;][]{Brandl 2005}. However, the uncertainty in our PAH measurement leaves
this possibility unconfirmed; a lower signal to noise ratio would have
made our limit higher. Furthermore, the lack of evidence for
comparably strong PAH emission at 7.7 \mic\ makes it likely that the
EW at 6.2 \mic\ is overestimated. Nonetheless, the limit demonstrates
that substantial star formation can be present in FSC 10214+4724 and
not visible in the (differentially magnified) IRS spectrum.
In figure \ref{fig: SED compare}, we compare the rest frame
MIR spectrum of FSC 10214+4724 to the IRS spectra of three
other sources. The intrinsic spectrum of FSC 10214+4724 is redder
than the observed spectrum, given the differential magnification, so
the differences seen in the figure would be greater if corrected for
lensing. The comparison sources are an AGN-dominated ULIRG (FSC
15307+3252), a silicate-emitting QSO (PG 1351+640) and a pure
starburst galaxy (NGC 7714). PG 1351+640 has similar silicate
emission, but has a much stronger hot dust continuum at $\sim 5$\
\mic\ \citep{Hao 2005}. In fact, the cold dust emission in FSC
10214+4724 gives it a much steeper mid-to-far infrared slope than most
AGN with silicate emission. The 30 \mic\ to 6
\mic\ flux density ratio of 46 in FSC 10214+4724, is higher than the
reddest of the quasars in \cite{Hao 2005}\ or \cite{Siebenmorgen
2005}, with a ratio of 11, and the reddest AGN in \cite{Weedman
2005}, NGC 1275, which has a ratio of 30. The ULIRG FSC 15307+325
has a similar MIR slope to FSC 10214+4724 in the 2-10 \mic\ region,
but has silicate absorption rather than emission, and a much weaker
cold dust continuum at longer wavelengths. The continuum of NGC7714
is very red with a 30 \mic\ to 6 \mic\ flux density ratio of 120, and
a strong stellar contribution at the short wavelengths.
In many AGN-dominated ULIRGs, the MIR spectrum is even bluer, with a
strong continuum in the $\sim 5$\ \mic\ region \citep[e.g.,
][]{Laurent 2000}. ULIRGs with steeper mid-infrared continua tend
also to have silicate absorption, not emission. We see no evidence in
the spectrum of FSC 10214+4724 for an underlying silicate absorption feature
at 9.7 \mic.
Furthermore, the profile of the silicate emission is consistent with
that seen in most other AGN observed with the IRS \citep{Hao 2005,
Siebenmorgen 2005, Weedman 2005}; although see \cite{Sturm 2005}\
for a counter example and discussion of the factors influencing the
profile shape. While some absorption may be filled in with emission,
the absorption and emission profiles are not necessarily identical. A
coincidence of opacity, temperature, and grain mixture in the emitting
and absorbing regions would be required.
The AGN contribution to the bolometric luminosity in FSC 10214+4724 is
apparently substantial but does not dominate. Nonetheless, the PAH
emission expected for a starburst is weak or absent. An unusual dust
geometry would be required for an AGN alone to explain the SED, given
the amount of cold dust. The other available evidence points to a
standard AGN configuration. The shape of the spectrum between
$2-12\mu$m is qualitatively similar to dusty torus models \citep{Pier
and Krolik 1992}\ with inner radius to height ratios of $a/h\sim
0.3$, and opening angles of $\sim 50$\deg -- consistent with a
high-luminosity equivalent of Seyfert 2 galaxy, like NGC 1068. The
similarlity to NGC 1068 hass previously been
noted\citep[e.g.,][]{Barvainis 1995}. The presence of silicate
emission in the IRS spectrum rules out models where the torus is seen
edge-on, or where the AGN is completely obscured by foreground (cold)
dust.
\acknowledgements
This work is based in part on observations made with the {\it Spitzer
Space Telescope}, which is operated by the Jet Propulsion
Laboratory, California Institute of Technology under NASA contract
1407. Support for this work was provided by NASA through an award
issued by JPL/Caltech.
\clearpage
\begin{deluxetable}{lll}
\tablecaption{SED fit parameters \label{tab: SED fit}}
\tablehead{
\colhead{Dust Component} &
\colhead{Temperature (K) } &
\colhead{$L/L_{tot}$\ (\%)}
}
\startdata
Hot & $638\pm20$ & 5.6 \\
Warm & $191\pm 2$ & 51.3 \\
Cold & $51\pm 6$ & 43.3
\enddata
\end{deluxetable}
\clearpage
\clearpage
\clearpage
|
Title:
NLTE spectral analysis of GW Vir pulsators |
Abstract: GW Vir variables are the pulsating members in the spectroscopic class of PG
1159 stars. In order to understand the characteristic differences between
pulsating and non-pulsating PG 1159 stars, we analyse FUSE spectra of eleven
objects, of which six are pulsating, by means of state-of-the-art NLTE model
atmospheres. The numerous metal lines in the FUV spectra of these stars allow a
precise determination of the photospheric parameters. We present here
preliminary results of our analysis.
| https://export.arxiv.org/pdf/astro-ph/0601566 |
\title{NLTE Spectral Analysis of GW~Vir Pulsators}
\author{E. Reiff,$^1$ D. Jahn,$^1$ T. Rauch,$^{1}$ K. Werner,$^1$ and
J. W. Kruk$^2$}
\affil{$^1$Institut f\"ur Astronomie und Astrophysik, Universit\"at
T\"ubingen, Germany \\
$^2$Department of Physics and Astronomy, The Johns Hopkins University,
Baltimore, USA}
\section{Introduction}
\label{sec:introduction}
GW~Vir variables \index{GW~Vir variables} belong to the spectroscopic
class of the PG~1159 stars \index{PG~1159 stars} (Wesemael, Green \&
Liebert 1985), which is named after
the prototype \index{PG~1159$-$035} PG~1159$-$035. These objects are
strongly hydrogen-deficient post-AGB stars \index{post-AGB stars}
which pass through the hottest stage of stellar evolution. Their
effective temperatures range between 75~000 and 200~000\,K, surface
gravities vary from $\log g = 5.5-8.0\,\,[\mathrm{cm\,s}^{-2}]$. The
so-called born-again scenario (a late thermal pulse which
transferred these objects back to the AGB followed by a second
post-AGB evolution, Iben et al\@. 1983) is mainly accepted as an
explanation for the H-deficiency and can reproduce well the observed
abundances. PG~1159 stars have spectra which are dominated by lines of
\ion{He}{ii}, \ion{C}{iv}, and \ion{O}{vi} (Werner et al. 2004), their
atmospheres show a typical surface composition of He:C:O = 33:50:17 by
mass. Beside these main constituents there are several lines of trace
elements, such as neon, nitrogen, silicon, sulfur, phosphorus, and
fluorine
(Reiff et al. 2005, Werner et al. 2005). Presently 37 PG~1159 stars
are known, eleven of them proved to be pulsators. The pulsating members
of the PG~1159 class are referred as GW~Vir variables. They are
non-radial g-mode pulsators with periods from 300\,s up to 1000\,s, in
some cases exceeding even 2000\,s (Nagel \& Werner 2004). The favored
excitation mechanism for the pulsations is the
$\kappa$-mechanism associated with cyclic ionization of carbon and
oxygen (Quirion, Fontaine \& Brassard 2004). In the $\log
T_{\mathrm{eff}} - \log g$ diagram the GW~Vir variables are located
among the PG~1159 stars in the so-called GW~Vir instability
strip. Spectral analyses of pulsating and non-pulsating PG~1159 stars
were used by Dreizler \& Heber (1998) to define empirically the edges
of this instability strip. But it is still puzzling that also
non-pulsating PG~1159 stars are located within the instability
strip. In our analysis we try to find more characteristic properties
to distinguish between pulsating and non-pulsating members of this
class.
\section{Observations and First Results}
\label{sec:observations_results}
For our analysis we selected pulsating and non-pulsating PG1159 stars
for which high resolution (R $\approx$ 20~000) FUV spectra obtained
with the Far Ultraviolet Spectroscopic Explorer (FUSE) are
available. The resulting sample comprises eleven objects. The FUSE
spectra are processed within the standard Calfuse pipeline process. A
log of all observations used for this analysis is listed in Table
\ref{tab:log}. Besides the FUV
spectra we also used spectra obtained with STIS, GHRS and IUE as well
as optical spectra. The model atmospheres and synthetic
line profiles are computed with the T\"ubingen Model Atmosphere Package
(Werner et al\@. 2003, Rauch \& Deetjen 2003). The
line-blanketed NLTE model atmospheres are in radiative and in
hydrostatic equilibrium. Besides the main constituents of the
atmospheres of PG~1159 stars, helium, carbon,
and oxygen, our model atmospheres also contain neon and
nitrogen. For the abundances of these elements we use atmospheric
parameters taken from the literature which are summarized in Table
\ref{tab:parameters}. For neon an abundance of 2\% mass fraction was
assumed for all models, according to Werner \& Rauch (1994) and Werner
et al\@. (2004). Although the abundances in the literature were mostly
determined in analyses of optical spectra the synthetic spectra can
also fit the FUV spectra well, which confirms the literature values
for
abundances in most cases. In Fig.\,\ref{fig:pg1159} we display the FUSE
spectrum of PG\,1159$-$035 together with our synthetic
spectrum. As lines of sulfur and silicon were also identified in
several objects we included those elements in the synthetic spectra,
too. Both were treated with line formation calculations without
back-reaction on the atmospheric structure. We assumed
solar abundances for both elements.
\begin{table}[t]
\caption{Log of the FUSE observations used for this analysis.}
\label{tab:log}
\footnotesize
\begin{center}
\begin{tabular}{l l r c}
\noalign{\smallskip}
\tableline
\noalign{\smallskip}
Object & Observation ID &\multicolumn{1}{c}{$t_\mathrm{exp}$} & Aperture \\
\noalign{\smallskip}
\tableline
\noalign{\smallskip}
RX\,J2117.1+3412\index{RX\,J2117.1+3412} & P1320501 & 8232\,s & LWRS\\
PG\,1144+005\index{PG\,1144+005} & P1320201 & 6859\,s & LWRS\\
PG\,1520+525\index{PG\,1520+525} & P1320101 & 3648\,s & LWRS\\
PG\,1159$-$035\index{PG\,1159-035} & Q1090101 & 6321\,s & LWRS\\
K\,1$-$16\index{K\,1-16} & M1031010 &11271\,s & HIRS\\
HS\,2324+3944 \index{HS\,2324+3944} & P1320601 & 4004\,s & LWRS\\
Abell 78\index{Abell 78} & B1100101 & 9972\,s & LWRS\\
& B1100102 & 7894\,s & LWRS\\
NGC 7094\index{NGC 7094} & P1043701 &23183\,s & LWRS\\
Abell 43\index{Abell 43} & B0520202 &12150\,s & LWRS\\
PG\,1424+535\index{PG\,1424+535} & P1320301 &11132\,s & LWRS\\
PG\,1707+427\index{PG\,1707+427} & P1320401 &14599\,s & LWRS\\
\noalign{\smallskip}
\tableline
\end{tabular}
\end{center}
\end{table}
Silicon is detectable in at least three objects, which are
PG~1159$-$035, and the two cooler stars PG~1424+535 and
PG~1707+427. Models with a solar Si abundance can fit the doublets at
1122/1128\,\AA\ and 1393/1402\,\AA\ well. In all spectra sulfur lines are
detected, but our preliminary fits also suggest abundances less than
solar. In Fig. \ref{fig:SiS_lines} we display part of the FUSE spectrum
of PG~1424+535 with a preliminary fit of the sulfur and silicon lines,
both with solar abundances.
In former analyses by Dreizler \& Heber (1998) it was suggested that
the nitrogen abundance is a characteristic difference between
pulsating and non-pulsating PG~1159 stars, as nitrogen was detected in
all GW~Vir pulsators with a rather high abundance of 1\% by mass,
while in stable PG~1159 stars no nitrogen could be detected, except
for PG~1144+005 (which is considered outside the instability
strip). In order to confirm previously determined N abundances
we tried to fit the N resonance doublet at 1238/1242\,\AA. For this
purpose we also analysed the STIS spectrum of PG~1159$-$035, which has
a high resolution (0.1\,\AA) and high S/N. In this spectrum the interstellar
component of the resonance doublet is clearly separated from the
photospheric component. This allows to determine the N abundance much
more precisely than before and it seems to turn out that the N abundance
is also significantly lower, about 0.1\% by mass, than suggested by
Dreizler \& Heber (1998). Fig.
\ref{fig:N_comparison} shows the N resonance doublets of three
objects, the pulsators PG~1159$-$035 and PG~1707+427 and the
non-pulsator PG~1424+535. While the comparison of PG~1707+427 and
PG~1424+535 seems to confirm the characteristic difference in the N
abundance, the new fit to the photospheric components in the STIS
spectrum of PG~1159$-$035 shows that the N abundance is only 0.1\% by
mass, but still higher than in the non-pulsator PG~1424+535. Further
analyses are necessary
to confirm these preliminary results.
\begin{table}
\caption{Summary of the atmospheric parameters of our program stars
taken from the literature. All objects are of spectral type PG1159
except for Abell\,78, which is a [WC]-PG1159 transition object. The
last column indicates whether the star is pulsating or not.}
\label{tab:parameters}
\footnotesize
\begin{center}
\begin{tabular}{l r c r c r r r c}
\noalign{\smallskip}
\tableline
\noalign{\smallskip}
Object & $T_{\mathrm{eff}}$ & $\log g$ &\multicolumn{1}{c}{H} & He
&\multicolumn{1}{c}{C} & \multicolumn{1}{c}{N} & \multicolumn{1}{c}{O} & Puls.\\
\noalign{\smallskip}
\cline{4-8}
\noalign{\smallskip}
& $[$kK$]$ & (cgs) &\multicolumn{5}{c}{(mass fractions)} &\\
\noalign{\smallskip}
\tableline
\noalign{\smallskip}
RX\,J2117.1+3412 & 170 & 6.0 & & 38.0 & 56.0 & & 6.0 & $\times$\\
PG\,1144+005 & 150 & 6.5 & & 39.0 & 58.0 & 1.5 & 1.6 & \\
PG\,1520+525 & 150 & 7.5 & & 44.0 & 39.0 & & 17.0 & \\
PG\,1159$-$035 & 140 & 7.0 & & 33.0 & 49.0 & 1.0 & 17.0 & $\times$\\
K 1$-$16 & 140 & 6.4 & & 33.0 & 50.0 & & 17.0 & $\times$\\
HS\,2324+3944 & 130 & 6.2 & 21.0 & 41.0 & 37.0 & & 1.0 & $\times$\\
Abell 78 & 110 & 5.5 & & 33.0 & 50.0 & 2.0 & 15.0 & \\
NGC 7094 & 110 & 5.7 & 36.0 & 43.0 & 21.0 & & & \\
Abell 43 & 110 & 5.7 & 36.0 & 43.0 & 21.0 & & & $\times$ \\
PG\,1424+535 & 110 & 7.0 & & 50.0 & 44.0 & & 6.0 & \\
PG\,1707+427 & 85 & 7.5 & & 43.0 & 38.5 & 1.5 & 17.0 & $\times$\\
\noalign{\smallskip}
\tableline
\end{tabular}
\end{center}
\end{table}
\acknowledgements{This research is supported by the DFG under grant
WE\,1312/30-1 (E.R.), by the DLR under grant 50\,OR\,0201 (T.R.) and
the FUSE project, funded by NASA contract NAS5-32985 (J.W.K.).}
{}
|
Title:
A high accuracy computed water line list |
Abstract: A computed list of H$_{2}$$^{16}$O infra-red transition frequencies and
intensities is presented. The list, BT2, was produced using a discrete variable
representation two-step approach for solving the rotation-vibration nuclear
motions. It is the most complete water line list in existence, comprising over
500 million transitions (65% more than any other list) and it is also the most
accurate (over 90% of all known experimental energy levels are within 0.3
cm$^{-1}$ of the BT2 values). Its accuracy has been confirmed by extensive
testing against astronomical and laboratory data.
The line list has been used to identify individual water lines in a variety
of objects including: comets, sunspots, a brown dwarf and the nova-like object
V838 Mon. Comparison of the observed intensities with those generated by BT2
enables physical values to be derived for these objects. The line list can also
be used to provide an opacity for models of the atmospheres of M-dwarf stars
and assign previously unknown water lines in laboratory spectra.
| https://export.arxiv.org/pdf/astro-ph/0601236 |
\date{Accepted XXXX. Received XXXX; in original form XXXX}
\pagerange{\pageref{firstpage}--\pageref{lastpage}} \pubyear{2005}
\label{firstpage}
\begin{keywords}
water, line list, BT2, molecular spectra
\end{keywords}
\section{Introduction}
Water is the most abundant molecule in the universe after H$_{2}$ and
CO. It is present in many astrophysical environments including the
atmospheres of: M dwarfs (Allard et al.1994), brown dwarfs (Allard et
al. 1996), K and M giants and supergiants (Jennings and Sada 1998;
Ryde et al. 2002; Tsuji 2001) and oxygen-rich AGB stars (Barlow et al.
1996). It occurs in: sun-spots (Wallace et al. 1992; Polyansky et
al. 1997), nova outflows (Banerjee et al. 2005), Mira variables
(Hinkle and Barnes 1979), T Tauri eruptive variables (Shiba et
al. 1993), dark molecular clouds (Gensheimer et al. 1996), young
stellar objects (Carr et al. 2004), comets (Mumma et al. 1996; Dello
Russo et al. 2000), the ISM (Cernicharo 1994), masers (Cheung et
al. 1969; Gonz\'{a}lez-Alphonso et al. 1995) and planetary
atmospheres. An accurate water line list is thus essential for
interpreting spectra from all of these sources and in modelling
stellar atmospheres at temperatures up to 4,000 K.
The importance of water has given rise to many laboratory
investigations of its spectrum. Ludwig 1971; Camy-Peyret et al. 1977;
Bernath 1996 all investigated hot water line positions. However,
techincal problems and the huge number of transitions (many of which
appear blended) mean that only in the region of 80,000 (out of a total
of more than a billion) transitions are known experimentally and there
are few hot water lines for which intensities have been determined.
The spectrum of water, which extends over a wide wavelength range
from millimetre to near ultra-violet, is due to quantised changes in
the rotation-vibration energy of the atomic nuclei moving in the
electronic potential well. Essentially, the water molecule is only
able to absorb or emit in its ground electronic state as the energy of
the first stable excited electronic state is above the dissociation
energy. In practice, some emissions do occur from short-lived excited
electronic states. H$_{2}$O electronic transitions from diffuse
interstellar clouds provide an example (Smith et al. 1981), but these
transitions, which occur in the ultra-violet, can be disregarded in
almost all other situations.
Water is a triatomic asymmetric top molecule. Its rotation-vibration
spectrum is more complicated than those of most other triatomic
molecules. In common with all non-linear triatomics, H$_{2}$O has six
degrees of internal freedom (three of rotation and three of
vibration). However, the lightness of the hydrogen atoms means that
the rotation constants are large, and this gives rise to an open
spectrum that extends over a wide frequency range. Moreover, the
`floppy' nature of the molecule means that the movement of the
hydrogen atoms is generally anharmonic and consequently transitions
involving changes of more than one vibrational quanta often
occur. Also, since many of the vibrational frequencies are nearly
resonant with other frequencies, it is common for vibrational bands to
overlap and for states to interact in ways that cannot easily be
predicted by perturbation theory but are amenable to a variational
approach.
The importance of the H$_{2}$O molecule in astronomy and the
complexity of its spectrum have created a great deal of interest in
the possibility of generating the spectrum synthetically. Previous
synthetic line lists include: MT (Miller et al. 1994), VT1 (Viti et
al. 1997) VT2 (Viti Ph.D.thesis 1997), PS, otherwise called AMES,
(Partridge \& Schwenke 1997) and SCAN (J{\o}rgensen et al. 2001). All
previous synthetic line lists have suffered from a number of problems
that are discussed below. The most successful attempts have employed
similar variational nuclear motion procedures to that used in
producing the BT2 water line list. However, for reasons detailed in
the next section, none of the earlier lists is considered to be
satisfactory. We have addressed these problems and consequently the
BT2 line list is an accurate tool for astronomers working in a variety
of fields.
\section{Background to the calculations}
\subsection{Variational Techniques}
Variational techniques represent the best approach to solving the the
nuclear motion problem (Tennyson 1992), and they are examined in detail in
Ba\v{c}i\'{c} and Light (1989). Here we use a discrete variable
representation (DVR).
In a DVR, the wavefunctions are defined by a complete set of weighted,
orthogonal grid points, each wavefunction having a different set of
weightings. This method is capable of generating accurate solutions,
the accuracy being determined by the the number and appropriateness of the
points. It is efficient for a large number of situations. The DVR
approach has the advantage that the potential matrix elements are
diagonal, and hence are easily evaluated. Even more important is the
fact that the dipoles can be reduced to a similar form (Tennyson et al.
2004).
The current work is not unique in employing a DVR approach to solve
the nuclear motion problem for water (see for example, Viti et
al. 1997). However, improved physics, in the form of a highly accurate
potential energy surface (PES), the methodology embodied in the DVR3D
program suite and increases in computational power have made it
possible to produce a line list that is more complete and more
accurate than any previous list, even those employing similar
methodolodies.
The DVR3D approach generates four rotation-vibration symmetry blocks
for the H$_{2}$O molecule, which we label ee, eo, oe and oo. The first
e/o term is the vibrational basis symmetry `q' and the second, e/o is
the standard quantum number `p', the rotational parity. These
symmetry blocks are not the same as the C$_{2v}$(M) symmetry
blocks $\Gamma_{rv}$: A$_1$, A$_2$, B$_1$, B$_2$, but are related to
them (see standard texts on molecular spectroscopy, such as Bunker
\& Jensen 2005).
The nuclear permutation operation in which the two identical protons
comprising the hydrogen nuclei are interchanged gives rise two two
dicrete states of the molecule: ortho (O) and para (P).The nuclear
spins may couple symmetrically or antisymmetrically. The
antisymmetric coupling that gives rise to the O form of the H$_{2}$O
molecule is triply degenerate, whilst the symmetric coupling, that
gives rise to the P form of the molecule is undegenerate.
For any J, apart from J=0, there are two ortho and two para symmetry
blocks (in the case of J=0 there is one O and one P block)
There are two possible arrangements, depending on whether J is odd or
even and these are detailed in Table 1.
\begin{table}
\begin{center}
\caption{Symmetry Blocks}
\begin{tabular}{cccccccccc}
\hline
{ }& \multicolumn{4}{c}{\bf J even}&&\multicolumn{4}{c}{\bf J odd}\\
\cline{2-5}\cline{7-10}
\bf q & e & e & o & o && e & e & o & o\\
\bf p & e & o & e & o && e & o & e & o\\
\bf O/P & P & O & O & P && O & P & P & O\\
\bf Code & 1 & 3 & 4 & 2 && 3 & 1 & 2 & 4\\
\hline
\end{tabular}
\end{center}
{\footnotesize q is the vibrational basis symmetry and p is the
rotational parity. These are labelled as either symmetric (e),
or antisymmetric (o) states. O/P = Ortho/Para. The `Code' is the
notation for symmetry used in the Levels File (Table 2).}
\end{table}
The difference in degeneracies of the O and P states impacts on the
partition function of the molecule as well as on line intensity. Also,
the fact that O-P and P-O transitions are forbidden, has spectroscopic
consequences.
\subsection{PES and the energy levels}
The energies of the quantised rotation-vibration states are
the eigenvalue solutions that satisfy the Shr\"{o}dinger equation for
the oxygen and two hydrogen nuclei moving within the electronic
potenial. However, the electronic potential within which the charged
nuclei are moving is itself a function of the actual internuclear
geometry. Therefore, in order to solve for the nuclear motion, it is
necessary to have an accurate model of how the potential varies with
the nuclear geometry. The problem is rendered tractable by adopting
the Born-Oppenheimer approximation which separates the nuclear and
electronic motions and has as its basis the fact that, due to their
lightness, the electrons may be considered to react immediately to any
changes in the nuclear geometry. The most accurate PESs are computed
using an \emph{ab initio} starting point, with the resulting surface
being empirically adjusted to improve the agreement between the
computed energies and experimental data (Partridge and Schwenke
1997).
We used the potential energy surface fit B of Shirin et al. (2003),
which is based on a highly accurate \emph{ab initio} surface with
adjustments for electronic relativistic and adiabatic (also known as
the Born-Oppenheimer diagonal correction) effects and fitted to the
available experimental data. At the time of writing, this surface is
the most accurate available and it is the single most important factor
affecting the accuracy of our results.
Non-adiabatic corrections to the Born-Oppenheimer approximation are
also important in the case of water (Schwenke 2003). However, a full
theoretical non-adiabatic adjustement to the PES has been shown to
produce no significant benefits compared to simplified approaches
(Tennyson et al., 2002). These authors examine two such approaches:
using separate reduced masses for the vibrational and rotational
motions (this systematically over-corrects bending motions and
under-corrects stretches), and a simplified version of the full
correction which includes only terms that scale with the kinetic
energy terms in $\theta$ and r. The latter approach is preferred and
the ajustment is effected through the DVR3D program rather than
changes to the PES.
\subsection{Transition intensities}
The intensities of the allowed transitions between rotation-vibration
states are determined by the dipole transition moments for these pairs
of states. They are:
\begin{equation}
\label{Eq:A}
\quad\qquad \qquad \qquad \qquad \langle
\psi^{\prime}|\overline\mu|\psi^{\prime\prime}\rangle
\end{equation}
where $\psi^{\prime}$ and $\psi^{\prime\prime}$ are the wavefunctions
of the two states,and $\overline{\mu}$ is the electronic dipole moment
vector. Therefore, in order to calculate intensities, a dipole moment
surface (DMS) is required.
The parameter computed by the DVR3D program suite is the Einstein A
coefficient for, A$_{if}$, for each transition. This is the
coefficient of spontaneous emission between the upper and lower
states. It is related to the dipole transition moment for the pair of
states and to J for the upper state. A$_{if}$, is independent of
temperature, relates to a single molecule, has units of s$^{-1}$ and
is given by:
\begin{equation}
\label{Eq:B}
\qquad A_{if} =
\frac{64\pi^{4}}{3c^{3}h}\nu^{3}g_{i}(2J^{\prime\prime}+1)|\langle
\psi^{\prime} | \overline{\mu} | \psi^{\prime\prime} \rangle|^2
\end{equation}
where prime and double prime represent the upper, i, and lower, f,
states respectively.
The quantity usually derived from observation, is the line intensity,
I, which has units of cm/molecule. I is temperature-dependent and in
emission is related to A$_if$ by the expression:
\begin{equation}
\label{Eq:C}
I = \frac{C(2J^{\prime}+1)}{Q_{vr}(T) \nu^{2}
g_i}\exp\left(\frac{-hcE^{\prime\prime}}{kT}\right)\left[1-\exp\left(\frac{-hc\nu}{kT}\right)\right]A_{if}
\end{equation}
where $\nu$ is the frequency in cm$^{-1}$, E$^{\prime\prime}$ is the
energy of the lower ro-vibrational level in cm$^{-1}$. $Q_{vr}$(T) is
the ro-vibrational partition function and is dimensionless, and $g_i$
is the nuclear spin degeneracy and caries only one subscript since
transitions between different nuclear spin states are not allowed
(Miani and Tennyson 2004). Boltzmann's constant, k, has units of
JK$^{-1}$ and the constant C has the value (8$\pi$c)$^{-1}$ =
1.3271x10$^{-12}$ s cm$^{-1}$.
Since the BT2 line list includes A$_{if}$ for each transition, the
above equation enables the line intensities to be computed at any
given temperature.
Unlike the PES, where the most accurate are \emph{ab initio} surfaces
that have been fitted to the available experimental data, in the case
of DMS, limitations on the accuracy of experimental line strengths
means that at the current time the most accurate surfaces are purely
\emph{ab initio} (Lynas-Gray et al. 1995). Two such DMS were tested
in the intensity part of our calculations. Our initial BT1 line list
was computed using the same PES as BT2 and a preliminary Lynas-Gray et
al. (in preparation) DMS. The line positions in BT1 are identical to those in
BT2 (the same PES having being used for both). However, we observed
that the Einstein A coefficients of the weaker lines generated using
the Lynas-Gray et al. DMS were often too large when compared with
experiment. This part of the computation was therefore repeated using
the DMS of Schwenke and Partridge (2000). The results, which are
contained in BT2, show much better agreement between the computed
strengths of weak lines and experiment. The Einstein A coefficients of
the stronger lines in BT1 and BT2 generally agree to within two
percent and both agree reasonably well with experimental values. In
addition to being superior to the Lynas-Gray et al. surface that we
tested, Schwenke and Partridge's DMS represents a major improvement on
the earlier DMS of Partridge and Schwenke (1997) and from our
analysis, as well from the same authors, is the most accurate in
existence.
\section{Calculating the line list}
The BT1 and BT2 line lists were computed using the DVR3D suite of
programs (Tennyson et al. 2004) on three Sun 5 Microsystem V880
mainframe computers: Enigma and Ra which are clustered using high
speed interconnects, each having 8 processors and 32 Gb of RAM and 432
Gb of disk storage and PSE, which has 24 processors and 96 Gb of RAM
and 1,296 Gb of disk storage at UCL's Hiperspace computing centre.
The final, `DIPOLE' stage of the suite was amenable to parallelisation
with little time penalty. Other parts of the program were run on
single processors, which avoided coding problems and was more
efficient in computer time. The total number of processor hours
employed in generating the BT1 line list (including the preliminary
convergence testing which is discussed below) was 55,000 hrs and a
further 10,000 hrs. were used in repeating the `DIPOLE' runs to
generate BT2 and in testing the outputs.
DVR3D calculates the bound rotation-vibration energy levels, the
wavefunctions on a grid in three dimensional space, and the dipole
transition strengths of the allowed transitions. The final part of the
DVR3D suite, `SPECTRA' is able to compute temperature-dependent
spectra over any selected frequency range, convolved with either the
natural line width or some other selected profile, such as that given
by the resolving power of a particular spectrometer. DVR3D uses an
exact (within the Born-Oppenheimer approximation) kinetic energy
operator. The program uses a discrete variable notation (DVR) with two
radial and one angular co-ordinate for the nuclear motion problem
(vibrational and rotational motions being treated spearately).
In establishing the working parmeters for our calculations, we aimed
to provide a line list that would be complete and accurate at the
temperatures of late series K-series stars (up to 4,000 K) and at
wavelengths down to 0.8 $\mu$m. Preliminary calculations showed that
in order to achieve this it would be necessary to include all states
lying at energies up to 30,000 cm$^{-1}$ relative to the ground state
of the system. Previous workers (Miller et al. 1994 and
Partridge and Schwenke 1997) selected lower energy cut-offs.
Our cut-off for the total angular momentum was J=50. We calculate that
the highest value of J which has ro-vibrational energies of less than
30,000 cm$^-1$ is 58. However, we also estimate that by terminating
our calculation at J=50 we omit less than 500 levels out of a total of
more than 505 million and none of the omitted levels has an energy
less than 23,490 cm$^{-1}$, with the majority being at energies above
28,000 cm$^{-1}$. Consequently, even at a temperature of 4,000 K these
missing levels contribute less than 0.02\% of the total partition
function. Moreover by omitting Js above 50 we saved in the region of
8,000 processor hours.
In order to generate accurate eigenvalues, it is essential that all
calculations are fully converged within the limitations imposed by
computing power and time. The lack of convergence in earlier lists has
already been noted (Polyansky et al. 1997).
DVR3D has the option of using Jacobi or Radau co-ordinates. The latter
was selected as being more appropriate for water. Our Radau grid is
defined in terms of two radial co-ordinates, r$_{1}$ and r$_{2}$, each
with 28 points, and one angular co-ordinate, $\theta$ having 44
points. These numbers are determined by convergence testing at
J$=$20. The eigenvalue solutions are particularly sensitive to the
number of radial points, r$_n$. However, the computation time rises very
rapidly as this number is increased, so the selection of r$_n$
represents a compromise and is the principal factor impacting on
convergence.
The angular grid points are arrived at using associated Legendre
polynomials for the underlying basis sets, whilst the radial grid
points are set up using Morse oscillator-like underlying basis sets,
which are defined in terms of parameters, r$_{e}$, $\omega_{e}$ and
D$_{e}$. These must be entered into `DVR3DRJZ', the first module of
the DVR suite. The parameters have physical counterparts which are
respectively: the equilibrium bond length, harmonic frequency and
dissociation energy of the water molecule. However, the parameters in
the Morse oscillator function differ from the physical values and must
be determined by empirical testing.
Although in principle, DVR3D is not strictly a variational method, in
practice it is found that the Variational Principle does apply; this
fact is employed in obtaining values for r$_{e}$, $\omega_{e}$ and
D$_{e}$. The method is to alter the three parameters in a systematic
manner until DVR3DRJZ generates a set of pure vibrational states (the
J=0, para states), the sum of whose energies is a minimum (251
eigenvalues were employed). We conducted this part of the process
manually.
The investigation was complicated by the existence of local minima,
which are not global minima for the three variables. The set of
parameters giving this result was: r$_{e}$ = 2.05, $\omega_{e}$ = 0.008
and D$_{e}$ = 0.20 all in atomic units (the equivalent dissociation
energy is 43,895 cm$^{-1}$). A good choice of these parameters (which
are a function of the system and of the particular energy range that
is of interest) is fundamental to the accuracy of the ensuing
calculations. The eigenvalues are particluarly sensitive to r$_{e}$
and $\omega_{e}$ and it was observed that differences of as little as
0.05 in the first of these two parameters and 0.002 in the second
could affect the energy levels by in the region of 0.01cm$^{-1}$ for
states with energies in the region of 10,000 cm$^{-1}$. Larger
deviations in these basis functions from their determined optimum
values produced correspondingly greater errors and in the case of
states with energies over 20,000 cm$^{-1}$, a bad choice of parameters
could easily result errors in the individual levels in excess of
20 cm$^{-1}$. Consequently considerable time was spent in determining
the values of the Morse oscillator-like basis set.
Two other inputs are required by the vibrational module,
DVR3DRJZ. These are: the maximum size of the intemediate Hamiltonian,
which we chose after testing as 2500 and the number of eigenvectors to
be saved for use in the rotational module, `ROTLEV3B', the optimum
value for which was found by testing to be 700.
ROTLEV3B also requies one variable to be determined; this is IBASS,
the size of the Hamiltonian in the rotation module. IBASS varies with
J. Its value was established by convergence testing at J=20 as being
$530\times(J+1-p)$, where p is the rotational parity and has the value
0 for even parity states and 1 for odd parity states. Hamiltonians of
this size are expensive in terms of computing time at high
Js. Nevertheless, it was easy to demonstrate that lower values of
IBASS produce results that are not converged. This is significant, as
earlier workers using similar DVR techniques have used lower IBASS
values. Viti et al. 1997, for example, used $200\times(J+1-p)$, and PS
used an even lower effective number, particularly at high J.
In selecting the various parameters referred to above, we regularly
tested for convergence. We estimate that our choice of r$_{n}$=28
accounted for approximately half of our total convergence error, which
we estimated as being less than 0.01 cm$^{-1}$ at 10,000 cm$^{-1}$ and
in the region of 0.02 cm$^{-1}$ at 20,000 cm$^{-1}$.
\section{Results}
The BT2 water line list is available electronically in compressed form
at: \emph {ftp://cdsarc.u-strasbg.fr/cats/VI/119} The data are in two
parts. The first, the `` Levels File'' is a list of 221,097 energy
levels, ordered by J and symmetry block. About 25,000 of these energy
levels have been labelled with the appropriate angular momentum
(J,$K_a,K_c$) and vibrational ($\nu_1$,$\nu_2$,$\nu_3$) quantum
numbers. An extract from the Levels File with an explanation of the
contents of each of the 11 columns in the file is given in Table 2.
The second part of BT2 is the ``Transitions File''. This has
505,806,202 entries. Each transition references upper and lower energy
levels in the Levels File and gives the Einstein $A_{if}$ coefficient
for the transition. An extract from the Transitions File is given in
Table 3.
In uncompressed form the BT2 Transitions File is 12.6 Gb of
data. Therefore, in order to facilitate use of the list, the
transitions have been ordered by frequency and separated into 16
smaller files, each representing a specific frequency range.
In addition to the files containing the actual line list, the
Strasbourg site contains a Fortran program, spectra-BT2.f90, that will
enable users to generate emission or absorption spectra from BT2 by
specifying various parameters including: temperature, frequency range,
cut-off intensity and line width. There is also a facility to generate
spectra with full ro-vibrational assignments if required. The method
of using spectra-BT2.f90 is detailed in a `readme-spectra' file and
there are also examples of a job file and an output file.
\begin{table}
\caption{Extract from the BT2 Levels File}
\begin{tabular}{|cccrccccccc|}
\hline
A & B & C & D & E & F & G & H & I & J & K \\[0.5ex]
\hline
2284 & 2 & 2 & 5 & 3885.718672 & 0 & 0 & 1 & 2 & 2 & 1 \\
2285 & 2 & 2 & 6 & 4777.145956 & 0 & 3 & 0 & 2 & 1 & 1 \\
2286 & 2 & 2 & 7 & 5332.258064 & 1 & 1 & 0 & 2 & 1 & 1 \\
2287 & 2 & 2 & 8 & 5472.371851 & 0 & 1 & 1 & 2 & 2 & 1 \\
2288 & 2 & 2 & 9 & 6254.694085 & 0 & 4 & 0 & 2 & 1 & 1 \\
2289 & 2 & 2 &10 & 6876.917089 & 1 & 2 & 0 & 2 & 1 & 1 \\
2290 & 2 & 2 &11 & 7027.396535 & 0 & 2 & 1 & 2 & 2 & 1 \\
2291 & 2 & 2 &12 & 7293.201639 & 2 & 0 & 0 & 2 & 1 & 1 \\
2292 & 2 & 2 &13 & 7376.617020 & 1 & 0 & 1 & 2 & 2 & 1 \\
2293 & 2 & 2 &14 & 7536.864373 & 0 & 0 & 2 & 2 & 1 & 1 \\
\hline
\end{tabular}
{\footnotesize
A: Row in file, B: J, C: Symmetry (1-4: see Table 1), D: Row in block,
E: $\nu$ in cm$^{-1}$ F, G, H: $\nu_1$, $\nu_2$, $\nu_3$. I, J, K: J,
K$_a$,K$_c$. }
\end{table}
\begin{table}
\begin{center}
\caption{Extract from BT2 Transitions File}
\begin{tabular}{|ccc|}
\hline
A & B & C \\[0.5ex]
\hline
1000 & 239 & 9.671E+01 \\
1001 & 239 & 1.874E+00 \\
1002 & 239 & 4.894E-03 \\
1003 & 239 & 1.140E-04 \\
1004 & 239 & 1.707E-02 \\
1005 & 239 & 8.473E-08 \\
1006 & 239 & 6.535E-04 \\
1007 & 239 & 7.157E+00 \\
1008 & 239 & 6.403E-06 \\
1009 & 239 & 9.861E-05 \\
\hline
\end{tabular} \\
{\footnotesize A, B: Row numbers in the Levels File
(upper and lower levels are not identified as the program tests for
these). C: A$_{if}$ (s$^{-1}$).}
\end{center}
\end{table}
\begin{table}
\begin{center}
\caption{Comparison of BT2 and PS (Partridge \& Schwenke 1997) with 14,889 experimentally-determined energy levels}
\begin{tabular}{|ccc|}
\hline
Within & BT2 & PS \\
cm$^{-1}$ & \% & \% \\
\hline
0.1 & 48.7 & 59.2 \\
0.3 & 91.4 & 85.6 \\
1.0 & 99.2 & 92.6 \\
3.0 & 99.9 & 96.5 \\
5.0 & 100 & 97.0 \\
10.0& 100 & 98.1 \\
\hline
\end{tabular} \\
\end{center}
\end{table}
\subsection{Comparing the BT2 and Partridge \& Schwenke line lists}
Several water line lists are in regular use by astronomers and the the
most accurate list previously is that of Partridge and Schwenke (PS).
Table 4 compares BT2 and PS energy levels with known experimental
values (Tennyson et al. 2001). It will be seen from this table that
although the PS list is more accurate than BT2 in the cases where
agreement with experiment is better than 0.1 cm$^{-1}$, this is not the
case generally. Specifically, based on a sample of 14,889 levels,
whilst 99.9\% of the BT lines are within 3.0 cm$^{-1}$ of experiment
3.5\% of the PS lines are outside this range. Other line lists (MT,
VT2, SCAN) perform significantly worse than this.
Examining deviations from experiment by energy is even more revealing,
for it is seen that PS is increasingly unreliable above 10,000
cm$^{-1}$, which is the region with the greatest number of transitions.
\begin{table}
\begin{center}
\caption{Distribution of levels in the BT2 and PS (Partridge \&
Schwenke 1997) disageeing with experiment by more than 2 cm$^{-1}$ -
by frequency}
\begin{tabular}{rccc}
\hline
Level Energy & Number & BT2 & PS \\
cm$^{-1}$ & in range & No. & No. \\
\hline
20,000 - 26,300 & 575 & 9 & 334 \\
15,000 - 20,000 & 2,813 & 10 & 105 \\
10,000 - 15,000 & 6,323 & 8 & 58 \\
7,000 - 10,000 & 3,263 & 3 & 9 \\
$<$ 7,000 & 1,914 & 0 & 0 \\
\hline
Total & 14,889 & 30 & 506 \\
\hline
\end{tabular} \\
\end{center}
\end{table}
\subsection{Labelling the levels}
As with observed water lines, the production of a synthetic line list
prompts the question of how to assign quantum numbers to the
transitions. The BT2 line list contains over 505 million transitions,
but the DVR3D suite only provides data on J and the symmetry block of
each energy level. Since each individual line is a transition between
two energy states, the problem reduces to one of labelling the 221,097
energy levels, but this is still a large task.
Many papers have been devoted to assigning quantum numbers to energy
levels that have been deduced from experimental line frequencies (e.g.
Zobov et al. 2000). So far, less than 15,000
experimentally-determined levels have been labelled with their 3
rotational and 3 vibrational quantum numbers, even though much effort
has been expended on the task. 25,870 levels in the BT2 list have been
labelled using methodologies detailed below. BT2 labels 270 of the 416
levels which have J=0 and energies below 30,000 cm$^{-1}$. However,
the proportion labelled is less at higher Js.
The process of labelling BT2 started by identifying particular
energies with experimental levels that have already been
determined. In addition, we have labelled many of the energy levels
that are unknown experimentally, using several different methods. A
large number of vibrational states for J=0 have been labelled by
visual inspection of the nodal structure of the wavefunction as
described in Mussa and Tennyson (1998). This method has the
disadvantage that observations can only easily be made on
two-dimensional sections of the three-dimensional wavefunction and the
procedure can be misleading. In addition, Mussa and Tennyson oberved
that at energies in the region of our 30,000 cm$^{-1}$, a high
proportion of the wavefunctions are irregular with no identifyable
nodal structure. .
A second method involved examining the $A_{if}$ coefficients for pure
rotational transitions between an unlabelled vibrational state and a
known vibrational state; the strongest transitions being those where
standard selection rules are obeyed. We found this method (see
Tolchenov et al 2005) to be useful for J$<$10
A third method that was found to be useful in labelling higher
J states involved the use of an algorithm to identify sets of levels
within the same parity block having the same K$_{a}$, $\nu_{1}$,
$\nu_{2}$, $\nu_{3}$ quantum numbers, but different values of J and
K$_{c}$. The method was originally developed for labelling the energy
levels of the HCN and HNC isomers (Barber et al. 2000). However,
because of the density of the energy levels in water, an extra term
was introduced when labelling the water levels. The algorithm used in
this case was:
\begin{equation}
\label{Eq;6}
E_{J_{n}}\cong 4E_{J_{n-1}} -6E_{J_{n-2}} +4E_{J_{n-3}} -E_{J_{n-4}}
\end{equation}
However, when resonance between levels caused the behaviour pattern to
be erratic, or no E$_{J_{n-4}}$ value existed, the original, simpler
algorithm was used:
\begin{equation}
\label{Eq;7}
E_{J_{n}}\cong 3E_{J_{n-1}} -3E_{J_{n-2}} +E_{J_{n-3}}
\end{equation}
where E$_{J_{n}}$ is the energy of the state in the same symmetry block
having J = n and the same set of K$_{a}$, $\nu_{1}$, $\nu_{2}$,
$\nu_{3}$ quantum numbers.
The results of the labelling exercise are included in the BT2 Level
File. This means that when synthetic spectra are generated many of the
transitions are fully labelled. This feature is useful when generating
synthetic spectra for astronomical or laboratory applications as is
discussed in the next section.
\subsection{Completeness}
If we compare the partition function, Q(T) for water computed at a
particular temperature using the BT and PS line lists with the most
accurately known value at this temperature, it is possible to estimate
the the completeness of the line lists and the amounts of opacity that
are missing in spectra generated by the two line lists at the selected
temperature.
A calculation of the partition function of water at 3,000K using the
221,097 energy levels in BT2 yields a value that is 99.9915 percent of
the Vidler \& Tennyson (2000) value, which indicates that levels
missing from BT2 only contribute about 85 parts in a million to the
partition function of water at this temperature, the reason being that
there is a diminishingly small probability of states above 30,000
cm$^{-1}$ being occupied at this temperature. For comparison, the PS
line list, which has 28,000 cm$^{-1}$ cut-off gives a partition
function at 3,000 K that is only 99.493$\%$ of the Vidler \& Tennyson
value.
Although the exclusion of levels above 30,000 cm$^{-1}$ does not
materially affect the completeness of the BT2 list, it does affect
absorption at shorter wavelengths. If we consider a photon of of
wavelength 1 $\mu$m (energy 10,000 cm$^{-1}$). This is able to be
absorbed by a water molecule in a particular rotation-vibration
eigenstate provided that there is another eigenstate exactly 10,000
cm$^{-1}$ above this lower state into which the molecule may be
excited. It follows that since the BT2 line list has an upper cut-off
of 30,000 cm$^{-1}$, none of the energy levels in the list above
20,000 cm$^{-1}$ are capable of being excited by a 1$\mu$m photon,
since there is no corresponding upper level.
If we examine the extent to which Q(3,000K) computed from BT2, but
excluding all levels above 20,000 cm$^{-1}$, falls short of the Vidler
\& Tennyson value we will have an indication (this is an upper limit
as it takes no account of blending effects) of the opacity that has
been excluded by adopting a 30,000 cm$^{-1}$ cut-off. Performing the
calculation gives a shortfall of 0.83$\%$.
In the case of PS, only energy levels below 18,000 cm$^{-1}$ are able
to absorb a 1$\mu$m photon, and computing Q(3,000K) using only PS
states up to 18,000 cm$^{-1}$ shows that these comprise only
98.37$\%$ of the Vidler value. Hence it will be seen that the opacity
deficit at 1$\mu$m is in the region of twice as great in the PS list
as in BT and the ratio increases at shorter wavelengths.
\section{Sample Applications}
Although BT2 shows good agreement with experimentally known lines
this is not a sufficient test of its accuracy as the PES used to
generate BT2 was fitted to the known experimental data (this is also
true of the PS line list).
The most effective way of checking the accuracy of a line list is to
test its ability to predict or identify previously unknown lines in
astronomical or laboratory spectra. BT2 has been used successfully as
outlined below. It should be noted here that some of the earlier
spectroscopic applications used BT1 as they predate BT2. However, the
line positions of the two lists are identical, and the intensities of
the strong lines are similar for the two lists.
\subsection{Astronomical Spectra}
\subsubsection{$\epsilon$ Indi Ba}
A synthetic spectrum generated at 1,500K using BT1 was able to
reproduce the previously unknown absorption feature observed at 1.554
$\mu$m in the spectrum of the early T dwarf $\epsilon$ Indi Ba,
discovered by Volk et al. (2003), as being a blend of six water lines
with no individual line contributing more than 25$\%$ of intensity
(Smith et al. 2003).
\subsubsection{Comet 153P/Ikeya-Zhang (2002 C1)}
BT1 was used to compute the frequencies and Einstein A coefficients of
the 64 transitions (up to J=7) that make up each of the 7 hot bands of
water detected in Comet 153P/Ikeya-Zhang. Dello Russo et al. (2004)
applied these data in determining the rotational temperature of the
comet on three dates.
\subsubsection{Temperatures of comet forming regions in early solar
nebula}
The hot-band transitions identified by BT1 in Dello Russo et
al. (2004) were classified into ortho and para using the symmetry
information contained within the BT2 energy file (see column C Table
1). Transitions between different nuclear spin parities can be ignored
(Miani and Tennyson 2004). Dello Russo et al. (2005) were able to
deduce the primordial O/P water composition of three comets: C/1999
H1, C/1999 S4 and C/2001 A2 and hence the temperatures of the
different regions of the early solar nebula in which the comets were
formed. The normal O/P ratio is 3:1, but since the lowest ortho level
lies 23.8 cm$^{-1}$ ($\sim$ 34 K) above the ground state (which is a
para state), the O/P ratio is sub-normal at temperatures below $\sim$
50 K. A comparison of the Einstein A coefficients in BT2 with those
actually used in BT1 shows that Dello Russo et al.'s results would
have been the same had the later line list been used.
\subsubsection{Detection of water lines in nova-like object V838 Mon}
Synthetic spectra generated by BT2 were used to identify absorption
features observed in the 1.73 to 1.75 $\mu$m region of the spectrum of
the nova-like object, V838 Monocerotis on five separate dates as being
due to blended water lines. Quantum numbers were assigned to the 17
strong transitions that comprise the 5 absorption features (Banerjee
et al. 2005). Sixteen of the lines were found to be in the (0 0
0)-(0 1 1) band. This is the first time that individual water lines
have been identified in a nova-like outflow region.
In addition, BT2 was used to compute the theoretical intensities of
the five absorption features as a function of temperature and column
density (assuming LTE). The results indicated that the water features
were arising from a cool $\sim$ 750--900 K region around V838 Mon that
was cooling at a rate of $\sim$ 100 K per year. Column
densities computed for the five dates also showed a reduction with
time.
\subsubsection{Sunspot spectra}
Following their assignment of high temperature laboratory water lines
in which extensive use was made of BT2 (see `Laboratory Spectra'
below), Coheur et al. (2005) revisited the sunspot absorpton spectrum
in the 9.89--12.95 $\mu$m region of Wallace et al. (1995) which had
been partially assigned by Polyansky et al. 1997 and Zobov et
al. 1999. Coheur et al. used their high temperature laboratory
assignments to identify a substantial number of previously unassigned
sunspot lines.
\subsection{Laboratory Spectra}
Hot water spectra have been analysed in the laboratory over the last
thirty years. Methods generally involve high resolution Fourier
transform spectrometry of vapour which may be at an elevated
temperature, such as in an oxy-acetylene flame (Camy-Pyret et
al. 1977, Coheur et al. 2005).
The BT2 line list has already been used on a number of occasions to
analyse spectra generated at both high and low temperatures at various
wavelengths and this has has added considerably to the existing
database of experimentally known energy levels and transitions.
Coheur et al. (2005) working with experimental spectra generated at
3,000 K in the 5--20 $\mu$m region labelled about 600 previously
unidentified levels using the BT2 line list. The identification of
these levels was an important factor in their subsequent assignment of
8,500 of the 10,100 lines that they observed in the 0.385-1.081 $\mu$m
region. Most of the states that were labelled by Coheur et al. were
either high J states having low bending modes or else lower J states
with higher bending modes. These states had defied earlier analysis
because the previous line lists used in earlier work, were unable to
accurately predict the energies of states above 15,000 cm$^{-1}$ or to
treat accurately the high bending mode states, even those below 15,000
cm$^{-1}$.
They note that whilst the agreement between BT2 and
observation is generally within 0.1 cm$^{-1}$ for the energy levels
that they observe for low J states, the disageement can be as great as
0.8 cm$^{-1}$ for some of the very high J states. Nevertheless, they
comment that since for a given vibrational and K$_a$ state the
diffence between the BT2 list and experiment increases smoothly with
J, they are able to use BT2 to predict the positions of unknown higher
J levels with an accuracy of 0.02 cm$^{-1}$. This is considerably less
than the experimentally determined line widths (full widths at half
maximum) which were in the range 0.05--0.10 cm$^{-1}$ due to
broadening at 1 atm. and T $=$ 3000 K.
In a follow-up paper analysing the 2--5$\mu$m region Zobov et al.,
2006 use BT2 to label approximately 700 previously
unidentified ro-vibrational energy levels for water, observed in
laboratory torch spectra.
Tolchenov et al. 2005 analysed long-pathlength room temperature
spectra. They use three separate line lists in their work. However,
only the BT2 list is found to be reliable over all transition
frequencies and in the case of lines having frequncies above 16,000
cm$^{-1}$ it is the only one that they use. In addition, Tolchenov et
al. encounter difficulties with the labelling adopted in previous
studies, finding, for example, that in some cases different states
have been labelled with the same quantum numbers. They therefore
undertake a systemmatic re-labelling exercise using the BT2
energies. These labels are incorporated into the BT2 list for states
with J $\le$ 9.
Dupr\'{e} et al.(2005) found BT2 similarly reliable for predictions of
long-pathlength room temperature spectra in the near
ultra-violet. They observed 62 R-brach transitions in the 8$\nu$
polyad and were able to determine 36 energy levels, previously unknown
experimentally.
\section{Conclusion}
We present a new synthetic water line list which gives the energies of
221,097 states with cut-offs of J=50 and E=30,000 cm$^{-1}$. 25,870 of
the lower energy levels have been labelled with a full set of three
rotational and three vibrational quantum numbers. BT2 lists
505,806,202 trasitions. It has been extensively tested against
experimental observations and also compared with other lists. It has
been shown to be the most complete and accurate water line list in
existence.
We make our results freely available in electronic form via
\emph{ftp://cdsarc.u-strasbg.fr/cats/VI/119}, in the hope that BT2
line list will be a vauable tool for astronomers in both spectroscopy
and atmospheric modelling applications.
One of the problems facing the modellers of the atmospheres of cool
stars and brown dwarfs is the disageement between observation and
model. Nevertheless, considerable progress has been made in recent
years in such areas as: convection, molecular abundances, non-LTE
effects and dust (Hauschildt et al. 1999; Tsuji 2002; Alexander et
al. 2003). At the same time, there have been advances in computing the
opacity effects of the various species that are included in the
models.
At the temperature of late M-dwarfs (2,500K), water is by far the most
important contributor to stellar atmospheric opacity, typically
contributing over 60$\%$ of all opacity in the infra-red. It is also
an extremely important contributor to the opacity of L and T brown
dwarfs, although at these lower temperatures (down to 900K), methane
and ammonia play an increased role.
As part of our on-going work in applying BT2 we plan to compare
observed M-dwarf spectra with synthetic spectra produced with the
Phoenix model (Hauschildt, Allard \& Baron 1999) using several water
line lists, including BT2. This is an extension of work already
conducted on modelling oxygen rich cool stars. Jones et al. (2005)
compare the CO 2-0 bands in the 2.297-2.210 $\mu$m region of several
M dwarfs and an L dwarf with opacity calculations using the PS and BT
line lists and conclude that for this particular wavelength range
`While the Partridge-Schwenke line list is a reasonable spectroscopic
match [for the BT line list] at 2,000K, by 4,000K it is missing around
25 $\%$ of the water vapour opacity.'
\section{Acknowledgments}
R.J.Barber has been sponsored in this work by PPARC.\\
Calculations were performed at the UCL Hiperspace computing centre and
we thank its manager, Callum Wright, for his assistance.
We also thank Svetlana Voronina for performing consistency checks on our energy
level labels.
\label{lastpage}
|
Title:
Self-Gravitating Phase Transitions: Point Particles, Black Holes and Strings |
Abstract: We compute the quantum string entropy S_s(m,j) of the microscopic string
states of mass m and spin j in two physically relevant backgrounds: Kerr
(rotating) black holes and de Sitter (dS) space-time. We find a new formula for
the quantum gravitational entropy S_{sem} (M, J), as a function of the usual
Bekenstein-Hawking entropy S_{sem}^(0)(M, J). We compute the quantum string
emission by a black hole in de Sitter space-time (bhdS). In all these cases:
(i) strings with the highest spin, and (ii) in dS space-time, (iii) quantum
rotating black holes, (iv) quantum dS regime, (v) late bhdS evaporation, we
find a new gravitational phase transition with a common distinctive universal
feature: A square root branch point singularity in any space-time dimensions.
This is the same behavior as for the thermal self-gravitating gas of point
particles (de Vega-Sanchez transition), thus describing a new universality
class.
| https://export.arxiv.org/pdf/astro-ph/0601601 |
\centerline{\bf SELF-GRAVITATING PHASE TRANSITIONS:}
\centerline{\bf POINT PARTICLES, BLACK HOLES AND STRINGS}
\begin{center}{\bf Norma G. Sanchez}
\footnote{Norma.Sanchez@obspm.fr}
\end{center}
\begin{center}{Observatoire de Paris, LERMA, Laboratoire Associ\'e au CNRS UMR 8112, 61, Avenue de l'Observatoire, 75014 Paris, France.}
\end{center}
\centerline{\bf Abstract}
We compute the quantum string entropy $S_s(m,j)$ of the microscopic string states of mass $m$ and spin $j$ in two physically relevant backgrounds: Kerr (rotating) black holes and de Sitter (dS) space-time. We find a {\bf new} formula for the quantum gravitational entropy $S_{sem} (M, J)$, as a function of the usual Bekenstein-Hawking entropy $S_{sem}^{(0)}(M, J)$. We compute the quantum string emission by a black hole in de Sitter space-time (bhdS). In {\bf all} these cases: (i) strings with the highest spin, and (ii) in dS space-time, (iii) quantum rotating black holes, (iv) quantum dS regime, (v) late bhdS evaporation, we find a {\bf new} gravitational phase transition with a common distinctive {\bf universal} feature: A square root {\bf branch point} singularity in any space-time dimensions. This is the same behavior as for the thermal self-gravitating gas of point particles (de Vega-Sanchez transition), thus describing a new {\bf universality class}.
Nous calculons l'entropie $S_s(m, j)$ des \'etats microscopiques de masse $m$ et spin $j$ des cordes quantiques dans deux espaces-temps physiquement relevants: Le trou noir en rotation (de Kerr) et l'espace-temps de de Sitter (dS). Nous trouvons une {\bf nouvelle} formule pour l'entropie gravitationnelle quantique $S_{sem} (M, J)$ comme fonction de l'entropie de Bekenstein-Hawking $S_{sem}^{(0)}(M)$. Nous calculons l' \'emission quantique des cordes par un trou noir avec constante cosmologique (bhdS). Dans {\bf tous} ces cas: (i) cordes avec le spin maximal, et (ii) dans l'espace-time de dS, (iii) trous noirs de grand moment angulaire, (iv) regime quantique de dS, (v) derni\`ere \'etape d'\'evaporation bhdS, nous trouvons une {\bf nouvelle} transition de phase avec la m\^eme caract\'eristique distinctive {\bf universelle} : un {\bf point de ramification} racine carr\'ee \`a la transition, pour toute dimension de l'espace-temps. C'est le m\^eme comportement que pour le gaz auto gravitant de particules ponctuelles (transition de de Vega-Sanchez), d\'efinissant ainsi une nouvelle {\bf classe d'universalit\'e}.
\\
{\bf Key words}: gravitational phase transitions, rotating black holes, quantum strings, de Sitter, self-gravitating gaz
\\
A theory of quantum gravity, such as string theory, should account for an unified and consistent description of both black holes and elementary particles, and the physics of the early universe as well. \\
A central object in string theory is $\rho_s (m,j)$, the microscopic string density of states of mass $m$ and spin $j$. We find $\rho_s (m,j)$ by solving the non-linear and quantum string {\bf dynamics} in the curved space-times considered, refs. \cite {1}-\cite{5}. We compute the quantum string entropy from the microscopic density of states $\rho_s (m,j)$ in two physically relevant space-times: the rotating (Kerr) black hole and the de Sitter space-time, refs \cite{6},\cite{7}. Combination of the two effects at the transition: black hole and cosmological constant are also treated for a black hole in de Sitter space-time (bhdS), ref. \cite{7}.
\\
The Hawking temperature $T_{sem}$ and the string (Hagedorn) temperature $T_s$ are the same concept in different (semiclassical and quantum) gravity regimes, ref. \cite{2}. Similarly, it holds for the Bekenstein-Hawking entropy and the string entropy, ref. \cite{2}. $T_s$ is the precise quantum dual of $T_{sem}=\hbar c /(2\pi k_B L_{c\ell})$, $ L_{c\ell}$ being the classical radius: $2R_S/(D-3)$, ($R_S$ is the black hole radius), or $c/H$ for Sitter space-time, $H$ being the Hubble constant, (in any $D$ space-time dimensions). For $L_{c\ell}\gg \ell_{Planck}$, ie. for low curvature or low $H\ll c/\ell_{Planck}$, (semiclassical gravity regime), the Bekenstein-Hawking entropy $S_{sem}^{(0)}$ is the leading term of the whole entropy $S_{sem}$, {\bf but} for high curvature or very small radius near $\ell_{Planck}$, (quantum gravity regime), a {\bf new} phase transition operates and the whole entropy $S_{sem}$ is drastically {\bf different} from the Bekenstein-Hawking entropy $S_{sem}^{(0)}$, refs \cite{6},\cite{7}.
\\
The phase transitions we found for quantum strings in Minkowski, Kerr and de Sitter backgrounds, as well as for the quantum regimes of rotating black holes and de Sitter space, are {\bf all} of the same nature: a {\bf square root} branch point singularity, (the value of the critical temperature being different in each case). This phase transition does not occur in Anti de Sitter space-time (AdS): in the AdS background, the density of mass states is finite, the negative cosmological constant "pushes" the critical temperature to infinity and there is no singularity at all, refs \cite{1}, \cite{2}.
\\
The origin of this string or gravitational {\bf square root} phase transition is gravity at finite temperature or in the presence of a repulsive force (angular momentum, positive cosmological constant). This behavior is similar to that found for the selfgravitating gas of point particles (de Vega-Sanchez transition), refs.\cite{8}, \cite{9}. In our context here, the temperature is not an external parameter, but {\bf intrinsic} to the theory, determined by the mass of the system (strings, or space-time backgrounds). Furthemore, our treatment here is not statistical mechanical , but {\bf microscopic} or dynamical.
\\
Here, we focalizes on the phase transition we found for strings and the quantum black hole and de Sitter regimes. More results and implications on these new issues can be found in refs. \cite{6},\cite{7}. Black hole evaporation ends as quantum string decay into elementary particles (most massless), ie pure (non mixed) quantum states refs \cite {2},\cite{6}, \cite{7}.\\
{\bf Quantum String Entropy in the Kerr background}: The string entropy $S_s(m,j)$ in a Kerr black hole background is given by
$\rho_s(m,j)= e^{\frac{S_s(m,j)}{k_B}} $, where the microscopic string density of states of mass $m$ and spin $j$, $\rho_s(m, j)$, has been found in ref. \cite{6} and can be expressed as
$$
\rho_s(m, j) \sim \Big(\frac{S_s^{(0)}}{k_B}\Big)^{-a} ~ e^{\Big(\frac{S_s^{(0)}}{k_B}\Big)} ~F(S_s^{(0)}, j)~~,
$$
$$ F(S_s^{(0)}, j) = \Delta_s^{-a-1} e^{\frac{S_s^{(0)}}{k_B} \frac{(1 - \Delta_s) ^2}{2 \Delta_s}}
\cosh^{-2} \Bigg( \frac{S_s^{(0)}}{4 k_B} \frac{(1 - \Delta_s^2)}{\Delta_s} \Bigg)
$$
~~~~~~~~~~$\Delta_s = \sqrt{1 - \frac{4 j }{m^2 \alpha^{'} c}}=\sqrt{1 - \frac{j}{\hbar} \Big( \frac{k_B~b}{S_{s}^{(0)}} \Big)^2}~~,~~~4j\leq m^2 \alpha^{'} c$~.
$S_s^{(0)}$ is the zero order string entropy for $j=0$: $S_s^{(0)} = \frac{1}{2} \frac{mc^2}{t_s}$ ~~~(closed~ strings), ~~~~
$S_s^{(0)} = \frac{mc^2}{t_s}$~~~(open~ strings), and $t_s$ is the string Hagedorn temperature, ~~~~$t_s = \frac{m_s c^2}{k_B b}$, ~~~being ~~$b = 2 \pi \sqrt{ \frac{D- 2 }{6}}$, $a = D$ (closed strings),~~ $a = (D-1)/2$ (open strings). $D$ is the number of space-time dimensions. $m_s$ is the fundamental string mass: $m_s = \sqrt{\frac{\hbar}{\alpha'c}}\equiv~\frac{\ell_s}{\alpha'}$. $\alpha'$ is the fundamental string constant ($\alpha'^{-1}$ is a mass linear density), $\ell_{s}$ is the fundamental string length. $\Delta_s$ can be expressed as :
$$
\Delta_s= \sqrt{1 - \frac{j}{\hbar} ~\Big(\frac{m_s}{m} \Big)^2} = \sqrt{1 - \frac{j}{\hbar}~ \Big(\frac {t_s}{T} \Big)^2} ~~,~~ T = \frac{m c^2}{k_B b}.
$$
$F({m}, j)$ takes into account the effect of the angular modes $j$, being $ F({m},j=0) = 1$. Therefore, the string entropy $S_s(m,j)$ in the Kerr background is given by
$$
S_s(m,j) = S_s^{(0)} - a ~k_B~\ln \Big (~\frac{S_s^{(0)}}{k_B}~ \Big)~+~k_B~\ln F(S_s^{(0)}, j)
$$
The last new term $k_B\ln F(S_s^{(0)},j)$ is enterely due to the spin $j \neq 0$. Interestingly enough, this formula expresses the string entropy $S_s(m,j)$ for mass $m$ and spin $j$ in terms of the string entropy $S_s^{(0)}$ for $j=0$. For $j=0$, we recover the usual expression. The logarithmic terms have a negative sign, the effect of the spin is to reduce the entropy. $S_s(m,j)$ is maximal for $j=0$ (ie, for $\Delta_s = 1$).\\
For $j < (m/m_s)^2$, and $m \gg m_{s}$, that is for low $j$ and very excited string states, $S_s^{(0)}(m,j)$ is the leading term, but for high $j$, that is $j \rightarrow m^2 \alpha^{'} c$, ie $\Delta_s \rightarrow 0$, the situation is {\it very different} as we see it below. Moreover, $S_s(m,j)$ allows us to write the full expression for the gravitational Kerr black hole entropy $S_{sem}$, as a function of the Bekenstein-Hawking entropy $S_{sem}^{(0)}$.\\
{\bf Extremal String States. A New Transition}: In ref. \cite {6} we considered a new kind of string states, the states in which $j$ reachs its {\bf maximal} value, that is $j = m^2 \alpha^{'} c$, we call these states {\bf "extremal string states"}. In this case, $\Delta_s = 0$. For $\Delta_s \rightarrow 0$, the entropy $S_s(m,j)$ behaves as
$$
S_s(m,j)_{extremal} = -(a + 1)~k_B~ \ln \frac {\Delta_s}{2}~ +~ k_B~\ln 2~ +~ \Delta_s ~\Big(~\frac{3} {4}~S_s^{(0)}~-~ak_B \Big)~+ ~ O (\Delta_s^{2})
$$
In the extreme limit $(j/\hbar) \rightarrow (m/m_s)^2 $:~~$\Delta_{s ~extremal}= \sqrt{2}\sqrt{1 - \Big(\frac{j}{\hbar}\Big)^{1/2} ~\frac{m_s}{m}}$\\ and $S_s(m,j)_{extremal}$ is dominated by
$$
S_s(m,j)_{extremal} = -(a + 1)~k_B~ \ln ~ \sqrt{1 - \Big(\frac{j}{\hbar}\Big)^{1/2} ~\frac{t_s}{T}}~~+~O(1)
$$
This shows a {\bf phase transition} takes place at $T \rightarrow \sqrt{(j/\hbar)}~t_s$, we call it {\bf extremal} transition. This is {\bf not} the usual (Hagedorn/Carlitz) string phase transition occuring for $m \rightarrow \infty $, $T \rightarrow t_s$; such transition is also present for $j\neq 0$ since $\rho (m,j)$ has the same $m \rightarrow \infty $ behaviour as $\rho(m)$. The extremal transition we found here is a {\bf gravitational} like phase transition: the square root {\bf branch point} behaviour near the transition is analogous to that found in the thermal self-gravitating gas of (non-relativistic) particles (by both mean field and Monte Carlo methods), refs. \cite {8},\cite{9}. And this is also the same behaviour found for the microscopic density of states and entropy of strings in de Sitter background, refs. \cite{2}, \cite{7}.\\
As pointed out in ref.\cite {2}, this string behaviour is {\bf universal}: the logarithmic singularity in the entropy (or pole singularity in the specific heat) holds in any number of dimensions, and is due to the gravitational interaction in the presence of temperature, similar to the Jeans's like instability at finite temperature but with a more complex structure.
A particular new aspect here is that the transition shows up at high angular momentum, (while in the thermal gravitational gaz or for strings in de Sitter space, angular momentum was not considered, (although it could be taken into account)).
Since $j\neq 0$, the extremal transition occurs at a temperature $ T_{sj}~=~ \sqrt{j/\hbar}~T_s $, {\bf higher} than the string temperature $T_{s}$.
That is, the angular momentum which acts in the sense of the string tension, appears in the transition as an "effective string tension" $(\alpha^{'}_j~)^{-1}$: a smaller $\alpha^{'}_j~ \equiv ~ \sqrt{\hbar/j}~ \alpha^{'}$ ~ (and thus a {\bf higher} tension).\\
{\bf Quantum String Entropy in de Sitter Background}: The entropy of quantum strings in de Sitter background is defined by
$ \rho_{s}(m, H) = e^{\frac{S_{s}(m,H)}{k_B}}$,
where $\rho_{s}(m, H)$ is the microscopic string density states of mass $m$ in de Sitter space time, and $H$ is the Hubble constant. In terms of the zero order string entropy in flat space time ($H=0$) $S_s^{(0)}$,
the string density of mass states $\rho_s(m, H)$ for both open and closed strings can be expressed as :$$\rho_s (m, H) =\Big( \frac{S_s^{(0)}}{k_B} \sqrt{f(x)} \Big)^{-a}~ e^{\Big(\frac{S_s^{(0)}}{k_B}\sqrt{f(x)}\Big)}~F(x)
~~~~,~~~~ F(x) = \frac{1}{\Delta_s \sqrt{f(x)}},$$ ~~~~
$$f(x)= \frac{2}{1+\Delta_s}~~, ~~a=\frac{(D-1)}{2} ~~(open),~~ a=D ~~(closed) , $$
$x$~ being the dimensionless variable~~$x(m, H)\equiv \frac{1}{2}\Big(\frac{m}{M_s}\Big)=
\frac{m_s}{b M_s}\frac{S_s^{(0)}}{k_B}$~~ and ~~$\Delta_s$~~ is given by:
$$\Delta_s~~=~~\sqrt{1-4x^2}~~=~~\sqrt{1 -\Big(\frac{m}{M_s}\Big)^2}$$
$M_s$ is the highest string mass in de Sitter space time ~\cite{1},\cite{2},\cite{3},\cite{5}$$
M_s = \frac{L_{c\ell}}{\alpha'} = \frac{c}{H ~ \alpha'};~~~~\Bigg(\frac{m_s}{M_s} \Bigg) = \frac{\ell_{s}}{L_{c\ell}}= \frac{H}{c}\ell_{s}$$
$L_{c\ell}= c/H$ being the classical de Sitter radius. Furthermore, $M_s$ defines the quantum string de Sitter length $L_s$ and the de Sitter string temperature $T_s$:
$$
L_s = \frac{\hbar }{M_s ~c}= \frac{\ell_s^2}{L_{c\ell}}=\frac{\hbar\alpha'}{c^2}H~~,~~
T_s = \frac{1}{2\pi k_B} ~ M_s~c^2 = \frac{\hbar c}{2 \pi k_B} ~\frac{1}{L_s}= \frac{1}{2 \pi k_B} ~\frac{c^3}{H \alpha'}
$$
$T_s$ appears to be a true critical temperature as we will see below. \\
For small $x$, (small $H m\alpha '/c$), $f(x)$ can be naturally expressed as a power expansion in $x$. In particular, for $H=0$, we have $x=0$ and $f(x)=1$, and we recover the flat space time string solution: $
\rho_s(m, H=0) \simeq \Big( \frac{S_s^{(0)}}{k_B} \Big)^{-a} ~e^{\big( \frac{S_s^{(0)}}{k_B} \big)}$.
From $\rho_s(m,H)$, we can read the full string entropy in de Sitter space :
$$
S_s(m,H) = \hat {S_s}^{(0)}(m,H)
-a~k_B~\ln ~\big(\frac{\hat {S_s}^{(0)}(m, H)}{k_B}\big) + k_B ~\ln F(m,H)
$$
$$\hat {S_s}^{(0)}(m,H)\equiv S_s^{(0)}\sqrt{f(x)} $$
The mass domain is $ 0 \leq m \leq M_{s}$, ie. $ 0 \leq x \leq 1/2 $, (which implies $ 0\leq \Delta_s \leq 1$, ie. $1 \leq f(x)\leq 2 $). All terms in the entropy $S_s(m,H)$ except the first one have negative sign. For $\Delta_s \neq 0$, (ie. $ m \neq M_{s}$), the entropy $S_s(m,H)$ of string states in de Sitter space is smaller than the string entropy for $H=0$. The effect of the Hubble constant is to reduce the entropy.\\
For low masses $m \ll M_{s}$, the entropy is a series expansion in $(H m\alpha' /c \ll 1)$, like a low H expansion around the flat $H=0$ solution, $S_s^{(0)}$ being its leading term. But for high masses $m \rightarrow M_s$, that is $(H m \alpha^{'}/c) \rightarrow 1$, (i.e. $\Delta_s \rightarrow 0$), the situation is {\bf very different} as we see it below. Moreover, $S_s(m,H)$ allows us to write the whole expression for the semiclassical de Sitter entropy $S_{sem}(H)$, as a function of the (Bekenstein-Hawking) de Sitter entropy $S_{sem}^{(0)}(H)$.\\
{\bf The String de Sitter Phase Transition}: For high masses $m \sim M_s$, (or in terms of temperature $T \sim T_s$), the entropy behaves as:
$$
S_s(T,H)_{T \sim T_s} = k_B \ln \sqrt{1 - \frac{T}{T_s}}~-k_B\ln~2 ~+~k_B \frac{ b}{\sqrt{2}}~(\frac{T_s}{t_s})~-~ a k_B \ln~(\frac{T_s}{t_s})
$$
where ~ $T = m c^2/2 \pi k_B $. We see that a phase transition takes place at $m = M_s$, ie, $T = T_s$. This is a {\bf gravitational} like phase transition: the square root {\bf branch point} behaviour near the transition is analogous to the thermal self-gravitating gas phase transition of point particles, refs. \cite {8},\cite{9}. This is also the same behaviour of the microscopic density of states and entropy of strings with the spin modes included ref. \cite{6}. \\
The transition occurs at the string de Sitter temperature $T_{s}$ {\bf higher} than the (flat space) string temperature $t_{s}$: $(T_s/t_s)= (b M_s /2\pi m_s) = (b L_{c\ell}/2\pi \ell_s)$.
This is so since in de Sitter background, the flat space-time string mass $m_s$, (or Hagedorn temperature $t_s$) is the scale mass, (or scale temperature), in the {\bf low} $Hm$ regime. For high masses, the critical string mass, (temperature), in de Sitter background is $T_s$, instead of $m_s$, $(t_s)$. In de Sitter space, $H$ "pushes" the string temperature beyond its flat space Hagedorn value $t_s$ . That is, $H$, which acts in the sense of the string tension, does appear in the transition temperature as an "effective string tension" $(\alpha^{'}_H) ^{-1} $ : a smaller $\alpha^{'}_H = (\hbar/c)(~H \alpha'/c)^2$, (and thus a {\bf higher} tension). The effect of $H$ in the transition is similar to the effect of angular momentum $j$. \\
Furthemore, for high masses ($m \sim M_s$), the {\bf partition function} $\ln Z$ of a {\bf gas of strings} in de Sitter space-time, behaves as
$$
(\ln Z)_{T \sim T_s}\sim V_{D-1}\left(\frac{k_B T_{sem}}{\hbar c}\right)^{D-1}~~
\sqrt{1 - \frac{T_{sem}}{T_s}}
$$
where $T_s$ is the string de Sitter temperature and $ T_{sem}$ is the semiclassical (Gibbons-Hawking) de Sitter temperature. $\ln Z$ shows a singular behavior for $T_{sem} \rightarrow T_{s}$ which is general for {\bf any} space-time dimensions $D$. Again, this is a {\bf square root branch point} at $T_{sem}= T_s$. That is, a {\bf phase transition} takes place for $T_{sem} \rightarrow T_s$, which implies $M_{c\ell}\rightarrow m_{s}$, $L_{c\ell}\rightarrow \ell_{s}$.\\
Furthemore, the high mass behaviour of $\ln Z$ implies that $T_{sem}$ has to be bounded by $T_s$, $(T_{sem} < T_s)$, which means: $L_{c\ell} > \ell_s, ~~i.e., ~~H < {c}/{\ell_s}$. In the de Sitter string phase transition $T_{sem} \rightarrow T_{s}$, $H$ reachs a maximum value sustained by the string tension $\alpha'^{-1}$ (and the fundamental constants $\hbar$, $c$ as well):
$$H_s = c ~\sqrt{\frac{c}{\alpha'\hbar}}, ~~~~ i.e.,~~\Lambda_s = \frac{1}{2 {\ell_s}^2}(D-1)(D-2)$$
The highly excited $m \rightarrow M_s$ string gas in de Sitter space undergoes a phase transition at high temperature $T_{sem} \rightarrow T_{s}$, into a condensate stringy state. This means that the background itself becames a string state. \\
These results also allow to consider the string regimes of a black hole in a de Sitter (or asymptotically) de Sitter background. This allow to study the effects of the cosmological constant $\Lambda$ on the quantum string emission by black holes, and the string bounds on the semiclassical (Gibbons-Hawking) {\bf black hole-de Sitter} (bhdS) temperature~ $T_{sem~bhdS}$, ref. \cite{7}. We computed the quantum string emission cross section $\sigma_{string}$ by a black hole in de Sitter (or asymptotically de Sitter) space-time (bhdS), ref. \cite{7}. For $T_{sem~bhdS}\ll T_{s}$, (early evaporation stage), it shows the QFT Hawking emission with temperature $T_{sem~bhdS}$, (semiclassical regime). For $T_{sem~bhdS}\rightarrow T_{s}$, $\sigma_{string}$~exhibits a phase transition into a string de Sitter state of size $L_s = \ell_s^2/L_{c\ell}$, ($\ell_s= \sqrt{\hbar \alpha'/c}$), and string de Sitter temperature $T_s$. For high masses ($m \sim M_s$), we find for the $\sigma_{string}$ leading behavior:
$$
\sigma_{string} ~~(T \sim T_s) \sim V_{D-1}~\Gamma_A~\left(\frac{k_B T_{sem~bhdS}}{\hbar c}\right)^{D-1}
\sqrt{1~-~\frac{T_{sem~bhdS}}{T_s}}
$$
Instead of featuring a single pole singularity in the temperature (Carlitz transition), it features a square root {\bf branch point} (de Vega-Sanchez transition, refs.\cite{8},\cite{9}). {\bf New} bounds on the black hole radius $r_g$~ emerge in the bhdS string regime: it can become $r_g = L_s/2$, or it can reach a more quantum value, $r_g = 0.365 ~\ell_s$, ref. \cite{7}.\\
{\bf Semi-classical and quantum (string) black hole regimes}:
Our analysis of the string canonical partition function, the black hole quantum string emission and the string bounds on the black hole, shows that a semiclassical black hole (BH) with size $L_{c\ell}$, mass $M$, temperature $ T_{sem}$, density of states $ \rho_{sem}$ and entropy $S_{sem}$, namely $(BH)_{sem}= (L_{c\ell}, M, T_{sem}, \rho_{sem}, S_{sem})$, evolves through evaporation into a quantum string state of size $L_{s}$, mass $m$, temperature $T_{s}$, density of states $\rho_{s}$ and entropy $S_{s}$, namely $(BH)_{s}$ = $(L_{s}, m , T_{s}, \rho_{s}, S_{s})$. The quantities in the set $(BH)_{sem}$ are precisely the semiclassical expressions of the respective ones in the set $(BH)_{s}$. In the quantum string regime, the black hole size $L_{c\ell}$ becomes the string size $L_{s}$, the Hawking temperature $T_{sem}$
becomes the string temperature $T_s$, the black hole entropy $S_{sem}$ becomes the string entropy $S_s$. The sets $(BH)_{s}$ and $(BH)_{sem}$ are the same quantities but in different (quantum and semiclassical/classical) domains. That is, $(BH)_{s}$ is the quantum dual of the semiclassical set $(BH)_{sem}$ in the precise sense of the wave-particle (de Broglie) duality. This is the usual classical/quantum duality but in the gravity domain, which is {\bf universal}, not linked to any symmetry or isommetry nor to the number or the kind of space-time dimensions.
From the semiclassical and quantum (string) black hole regimes $(BH)_{sem}$ and $(BH)_{s}$, we can write the full gravity entropy $S_{sem}(M, J)$ for the {\bf Kerr black hole} such that it becomes the string entropy $S_s(m,j)$ in the quantum string regime, namely:
$$
S_{sem}(M, J) = S_{sem}^{(0)}(M, J) - a ~k_B~\ln \Big (~\frac{S_{sem}^{(0)}(J, M)}{k_B}~ \Big)~+~k_B~\ln~ F(S_{sem}^{(0)}, J)
$$
where $S_{sem}^{(0)}(M, J)$ is the Kerr black hole Bekenstein-Hawking entropy and $F(S_{sem}^{(0)}, J)$ is given by
$$
F = \Delta^{-1} \Big(\frac{1 + \Delta}{2\Delta}\Big)^{a}~e^{\Big(\frac{1 -\Delta}{1 +\Delta}\Big) \frac{S_{sem}^{(0)}(M, J)}{\Delta k_B}}
\cosh^{-2} \Bigg(\frac{(1 - \Delta)}{\Delta}\frac{S_{sem}^{(0)}(M, J)}{2 k_B}\Bigg)
$$
For $J=0:~~~~~~F = 1 ~~~~, ~~~~ S_{sem}^{(0)}(M,J=0)~\equiv~ S_{sem}^{(0)} ~=~ 4 \pi k_{B} \Big(\frac{M}{m_{Pl}}\Big)^2$
$S_{sem}^{(0)}$ being the Schwarzschild black hole Bekenstein-Hawking entropy. $\Delta$ is given by :
$$
\Delta = \sqrt{1 - \Big(\frac{J}{\hbar}\Big)^2 \Big(\frac{4 \pi k_B~}{S_{sem}^{(0)}} \Big)^2}=~ \sqrt{1 - \Big(\frac{J}{\hbar}\Big)^2 \Big(\frac{m_{Pl}}{M} \Big)^4}~=~ \sqrt{1 - \Big(\frac{J}{\hbar}\Big)^2 \Big(\frac{T_{sem}}{T} \Big)^2}
$$
where $T = M c^2 / 8\pi k_B$ and $T_{sem}$ is the Schwarzshild Hawking temperature. $m_{Pl}$ is the Planck mass. Notice the {\bf new} term $k_B\ln F(S_{sem}^{(0)},J)$ in $S_{sem}(M, J)$ enterely due to $J \neq 0$.
For $\Delta \neq0$, the effect of the angular momentum is to reduce the entropy. $S_{sem}(M, J)$ is maximal for $J=0$ (ie, for $\Delta = 1$); for $ J=0$ we have: $S_{sem}(M)~ = ~S_{sem}^{(0)}~-~ a ~k_B~\ln S_{sem}^{(0)}$.\\
$S_{sem}(M,J)$ provides the full Kerr black hole entropy as a function of the Kerr Bekenstein-Hawking entropy $S_{sem}^{(0)}(M,J)$. Interestingly enough, the full Kerr black hole entropy $S_{sem}(M, J)$ can be also written enterely in terms of the Schwarschild Bekenstein-Hawking entropy $S_{sem}^{(0)}$, this is done in ref. \cite{6}. For $M\gg m_{Pl}$ and $J < GM^2/c$, $S_{sem}^{(0)}$ is the leading term of this expression, {\bf but} for high angular momentum, (nearly extremal or extremal case $J= GM^2/c$), a gravitational {\bf phase transition} operates and the whole entropy $S_{sem}$ is drastically {\bf different} from the Bekenstein-Hawking entropy $S_{sem}^{(0)}$, as we precisely see it below.\\
{\bf A New feature. The Extremal Black Hole Phase Transition:}
When $J$ reachs its {\bf maximal} value, that is $J = M^2 G/c^2$, then $\Delta = 0$ and the term $S_{sem}^{(0)}(M,J)$ is minimal :
$S_{sem}^{(0)} (M,J)_{extremal} ~ = ~\frac{1}{2}~S_{sem}^{(0)}$.
But for $\Delta \rightarrow 0$, the last term in the expression for $S_{sem}(M,J)$ substracts the first one, the poles in $\Delta$ cancel out, yielding:
$$
S_{sem}(M,J)_{extremal} = -(a + 1)~k_B~ \ln \frac {\Delta}{2}~ +~ k_B~\ln 2~ +~ \Delta ~\Big(~\frac{3} {4}~S_{sem}^{(0)}~-~ak_B~\Big)~+ ~ O(\Delta^{2})
$$
In the extreme limit $(J/\hbar) \rightarrow (M/m_{Pl})^2 $ :~~ $\Delta_{extremal} = ~ 2 ~\sqrt{1 - \Big(\frac{J}{\hbar}\Big)^{1/2}~ \frac {m_{Pl}}{m}}$\\and $S_{sem}(M,J)_{extremal}$ is dominated by
$$ S_{sem}(M,J)_{extremal} = -(a + 1)~k_B~ \ln ~\sqrt{1 - \Big(\frac{J}{\hbar}\Big)^{1/2}~\frac{t_{Pl}}{T}}~+~O(1)
$$ This shows that a {\bf phase transition} takes place at $T \rightarrow \sqrt{(J/\hbar)}~t_{Pl} $, equivalently , at $T \rightarrow (J/\hbar)~T_{sem} $, we call it {\bf extremal transition}. $t_{Pl}$ is the Planck temperature.
The characteristic features of this gravitational transition can be discussed on the lines of the extremal string transition we analysed for the extremal string states. Our discussion on the string case translates into the respective black hole quantities but we do not extend on these new features and implications here.\\
{\bf Semi-classical and quantum (string) de Sitter regimes}:
From the microscopic string density of mass states $\rho_s(m, H)$, we have shown that for $m\rightarrow M_s$,~ i.e. $T\rightarrow T_s$, the string undergoes a phase transition into a semiclassical phase with mass $M_{cl}$ and temperature $T_{sem}$. Conversely, from the string canonical partition function in de Sitter space and from the quantum string emission by a black hole in de Sitter space, we have shown that for $T_{sem}\rightarrow T_s$, the semiclassical (Q.F.T) regime with Hawking-Gibbons temperature $T_{sem}$ undergoes a phase transition into a string phase at the string de Sitter temperature $T_{s}$. This means that in the quantum string regime, the semiclassical mass density of states $\rho_{sem}$ and entropy $S_{sem}$ become the string density of states $\rho_s$ and string entropy $S_{s}$ respectively. Namely, a semiclassical de Sitter state, $(dS)_{sem}= (L_{c\ell}, M_{c\ell}, T_{sem}, \rho_{sem}, S_{sem})$, undergoes a phase transition into a quantum string state $(dS)_{s}$ = $(L_{s}, m , T_{s}, \rho_{s}, S_{s})$.
The sets $(dS)_{s}$ and $(dS)_{sem}$ are the same quantities but in different (quantum and semiclassical/classical) regimes. This is the usual classical/quantum duality but in the gravity domain, which is {\bf universal}, not linked to any symmetry or isommetry nor to the number or the kind of dimensions.
From the semiclassical and quantum de Sitter regimes $(dS)_{sem}$ and $(dS)_{s}$, we can write the full de Sitter entropy $S_{sem}(H)$, with quantum corrections included, such that it becomes the de Sitter string entropy $S_s(m,H)$ in the string regime: the full de Sitter entropy $S_{sem}(H)$ is given by $$ S_{sem}~(H) = \hat{S}_{sem}^{(0)}~(H)
-a~k_B~\ln ~(\frac{\hat{S}_{sem}^{(0)}~(H)}{k_B})+ k_B \ln~F(H)$$
$$\hat{S}_{sem}^{(0)}~(H)\equiv S_{sem}^{(0)}~(H) \sqrt{f(X)}~~~,~~~F(H) = \Delta \sqrt{f(X)}~~,~~ a=D~~,~~
f(X)= \frac{2}{1+\Delta},$$
$$\Delta ~\equiv~\sqrt{1-4X^2}~=~ \sqrt{1 -\Big(\frac{\pi k_B}{S_{sem}^{(0)}(H)}\Big)^2}, ~~X(H)\equiv \frac{\pi k_B}{2 S_{sem}^{(0)}(H)}= \frac {M_{sem}}{M_{cl}}= \Big(\frac{m_{Pl}}{M_{cl}}\Big)^2.
$$
$S_{sem}^{(0)}(H)$ is the usual Bekenstein-Hawking entropy of de Sitter space. $M_{cl}= c^3/GH $ is the de Sitter mass scale, $M_{sem}= m_{pl}/M_{cl}$ is the semiclassical mass. \\
In this expression, the mass domain is $m_{pl} \leq M_{c\ell} \leq \infty$, that is, $0\leq X \leq 1/2$. (ie. $0\leq \Delta \leq 1$). The same formula but with $\hat{X}(H) = S_{sem}^{(0)}(H)/2\pi k_B$, instead of $X(H)$, describes $S_{sem}(H)$ in the mass domain $0 \leq M_{cl} \leq m_{pl}$. This provides the whole de Sitter entropy $S_{sem}(H)$ as a function of the Bekenstein-Hawking entropy $S_{sem}^{(0)}(H)$.\\
The limit $X \rightarrow 0$ corresponds to $M_{cl}\gg m_{Pl}$, that is $L_{c\ell}\gg \ell_{Pl}$ , (low $H \ll c/\ell_{Pl}$ or low curvature regime). In this limit, $\Delta \rightarrow 1$, $f(X)\rightarrow 1$ and $S_{sem}^{(0)}(H)$ is the leading term of $S_{sem}(H)$, with its logarithmic correction:
$$
S_{sem}(H) = S_{sem}^{(0)}(H) -ak_B~\ln \Big(\frac{S_{sem}^{(0)}(H)}{k_B}\Big)
$$
But for {\bf high} Hubble constant, $H \sim c/\ell_{Pl}$, (ie. $M_{cl}\sim m_{Pl}$), (that is {\bf high} curvature or quantum gravity regime), $S_{sem}^{(0)}(H)$ is sub-dominant, a gravitational {\bf phase transition} operates and the whole entropy $S_{sem}(H)$ is drastically {\bf different} from the Bekenstein-Hawking entropy $S_{sem}^{(0)}(H)$, as we precisely see below.\\
{\bf The de Sitter Gravitational Phase Transition} : For $\Delta \rightarrow 0$, that is for $ M_{c\ell}\rightarrow M_{sem}$, the full de Sitter entropy $S_{sem}(H)$ behaves as:$$
S_{sem}(H)_{\Delta \sim 0}~~ =~~ k_B~ \ln \Delta ~+~O(1)$$
In this limit, the Bekenstein-Hawking entropy $S_{sem}^{(0)}(H)$ is sub-leading, O(1), (~$S_{sem}^{(0)}(H)_{\Delta = 0} = \pi k_B$~). In terms of the mass, or temperature:$$
\Delta~ =~ \sqrt{1 - \Big(\frac{m_{Pl}}{M_{cl}} \Big)^4}~=~ \sqrt{1 - \Big(\frac{T_{sem}}{T} \Big)^2} ~~,~~~~T = \frac{1}{2\pi k_B}M_{cl} c^2$$
$T_{sem}$ is the semiclassical (Gibbons-Hawking) de Sitter temperature . In the limit $ M_{c\ell}\rightarrow M_{sem}$, which implies $ M_{cl} \rightarrow m_{Pl} $, $S_{sem}(H)$ is dominated by
$$
S_{sem}(H)_{\Delta \rightarrow 0} = -~k_B~ \ln ~\sqrt{1 - \frac{T_{sem}}{T}}~~+~O(1)
$$
This shows that a {\bf phase transition} takes place at $T \rightarrow T_{sem}$. This implies that the transition occurs for $M_{cl} \rightarrow m_{Pl}$, ie $T \rightarrow t_{Pl}$, (that is for high $H \rightarrow c/\ell_{Pl}$).
This is a {\bf gravitational} like transition, similar to the de Sitter string transition we analysed above: the signature of this transition is the square root {\bf branch point} behavior at the critical mass (temperature) {\bf analogous} to the thermal self-gravitating gas behavior of point particles, refs \cite {8}, \cite{9}, and to the string gas in de Sitter space, ref.\cite {7}. This is also the same behaviour as that found for the entropy of the Kerr black hole in the high angular momentum $ J\rightarrow M^2G/c$ regime, (extremal transition), ref.\cite{6}. This is {\bf universal}, in any number of space-time dimensions.
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