\section{Introduction}\label{sec:intro}

In a hierarchical universe, a large galaxy will undergo many mergers during its lifetime.  The amount and properties of accreted material observed in present-day galaxies can be used to test hierarchical formation scenarios.  Stellar halos provide an ideal environment for investigating the detailed merger history of an individual galaxy.  The sparse stellar populations of a galaxy's halo combined with long dynamical times make it possible for tidal debris features to remain identifiable in phase-space for billions of years.  In contrast, it is difficult to separate material formed in situ from accreted material in a galaxy's relatively dense disk and bulge.

Recent sophisticated numerical and
semianalytic simulations of stellar halo formation have made great strides in
characterizing the properties of stellar halos that are built up through the tidal
stripping of accreted satellites \citep[e.g.,][]{johnston1996,johnston1998,helmi1999,helmi2000,bullock2001}.
\citet{bullock2005} (hereafter BJ05) used a combination of $N$-body and
semi-analytic techniques to simulate the formation via accretion of 11
MW-analog stellar halos in a cosmological context. These high resolution
simulations allow detailed analysis of the expected physical properties of stellar
halos composed primarily of tidal debris and their associated satellite systems
\citep{font2006apjsats,font2006apjhalos,font2008,johnston2008}.

The stellar halos of the Milky Way (MW) and Andromeda
(M31) galaxies both show ample evidence of recent and on-going accretion events.
Among the most prominent of these substructures are the Sagittarius
stream \citep{ibata1994,majewski2003,newberg2003} and the
Monoceros stream \citep{yanny2003,rocha-pinto2003}
in the MW, and the giant southern stream
\citep[GSS;][]{ibata2001nature} in M31, which likely pollutes much of M31's
spheroid interior to $\sim 20$\,--\,30~kpc \citep{fardal2007,gilbert2007,richardson2008}.

In addition to these prominent
stellar streams, recent large observational surveys are increasing
the number of known tidal debris features by pushing to lower surface brightness
limits.  Analysis of the Sloan
Digital Sky Survey has revealed many faint tidal debris features in the MW's
stellar halo \citep[e.g.,][]{grillmair2006,grillmair2006b,belokurov2006,belokurov2007b},
while large-scale photometric studies of M31 are revealing similarly large numbers
of tidal debris features in M31's stellar halo \citep{ibata2001nature,ibata2007}.
Spectroscopic surveys of red giant branch (RGB) stars in M31
\citep{irwin2005,kalirai2006halo,gilbert2006} are enabling the
discovery of tidal debris features through the kinematics of M31's stellar
populations \citep[e.g.,][]{ibata2005,kalirai2006gss,gilbert2007,chapman2008,gilbert2009gss}.

A classic signature
of substructure is an abnormally high
surface brightness relative to the value expected from the
observed surface brightness profile of the stellar halo.  Due to the long
dynamical times in the sparse outer regions of galaxy halos,
tidal debris from relatively recent mergers
(within the last several Gyr) will not have had time to become
spatially well-mixed.
The kinematical distribution of a well-mixed stellar halo population is
expected to be broad (dynamically hot) out to large distances from
the center of the galaxy.
In the MW and M31, the measured line-of-sight velocity dispersion is
$\sigma_v\sim 120$ to 150~\,km~s$^{-1}$\ near the galaxy's center
\citep{battaglia2005,chapman2006,gilbert2007} and decreases to
$\sigma_v\sim $~100~\,km~s$^{-1}$\ at $R\sim 60$~kpc \citep{battaglia2005,chapman2006}.
In contrast, tidal debris from a single accretion event should be
dynamically cold, since the local velocity dispersion of the debris should
decrease as it mixes along the progenitor satellite's orbit: as the stars
spread apart, decreasing their spatial density, their density in velocity-space
must increase in order to conserve their phase-space density and satisfy
Liouville's theorem \citep{helmi1999}.
Hence, kinematical substructure in a survey is another signature of
hierarchical structure formation.   A
difference in chemical abundance between stars in the kinematically hot
and cold populations or between two closely spaced fields is also evidence
of substructure, as the progenitors of recent accretion events are likely to be chemically distinct from the ensemble of satellites that were accreted at early times.

We have utilized the kinematical and chemical signatures of substructure to identify multiple,
kinematically cold tidal debris features \citep{guhathakurta2006,kalirai2006gss,gilbert2007,gilbert2009gss}
during the course of the Spectroscopic and Photometric Landscape of Andromeda's Stellar Halo (SPLASH) Survey which includes extensive Keck/DEIMOS\footnote{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.
} spectroscopic survey of M31's stellar halo
\citep[e.g.,][]{guhathakurta2005,kalirai2006halo,gilbert2006,gilbert2007}.
Kinematical signatures of substructure enable the detection of faint and diffuse
features which are not easily identified in starcount maps.  For example,
a shelf feature \citep{gilbert2007} along M31's southeast minor axis
($R_{\rm proj}\sim 12$\,--\,18~kpc) went unnoticed on star count maps
\citep[in fact, the region was specifically targeted
for deep HST photometry based on its apparent smoothness;][]{brown2003}.
The observed distribution of stars in line-of-sight velocity vs.\ position
space and its striking similarity to model predictions led to the
discovery of the SE shelf.

We can use the kinematically cold tidal debris features found in our survey
to study general trends of tidal debris in stellar halos.
The recent advances in
simulations of stellar halo formation have made it possible to undertake detailed comparisons of the properties of observed and simulated substructure \citep{bell2008,font2008}.
In this contribution, we present a relationship between the surface brightness and metallicity
of tidal debris features observed in our M31 data set, and provide a physical interpretation of the observed trend through comparisons of the observations with simulated stellar streams in the BJ05 simulations.

The paper is organized as follows.  In \S\,\ref{sec:data} we briefly discuss the
observations and reduction techniques and present the fields in our M31 survey in
which substructure has been identified.  We present the kinematical and chemical abundance
distributions of the stars in these fields as well as the surface brightness and average metallicity of the observed kinematical substructures.  The methods used in the BJ05 simulations are outlined
in \S\,\ref{sec:sims}, and the
trends of surface brightness with chemical abundance of stellar streams
in the simulated BJ05 stellar halos are presented.  We compare the observations with the simulations and discuss the resulting physical implications in \S\,\ref{sec:comparison}.  Finally, we summarize our results in \S\,\ref{sec:sum}.

\section{Data}\label{sec:data}
\subsection{Photometric and Spectroscopic Observations}
The photometric and spectroscopic observations and data reductions employed
in our SPLASH M31 survey have been discussed in detail in previous papers
\citep{guhathakurta2006,kalirai2006gss,gilbert2006,gilbert2007}
and will be only briefly outlined here.  The
observations are organized into ``fields'' in M31's stellar halo based
on the spatial proximity of the spectroscopic slitmasks (Table~\ref{table1}).
A single photometric
pointing is large enough to support multiple, non-overlapping DEIMOS slitmask
observations; the spectra from slitmasks based on a single photometric pointing
are in general analyzed together and considered a field.  The
exception to this are fields f207 and H13s, which are from a single
photometric pointing but are treated as separate fields due to the
fact that they are located at
different positions along M31's GSS; the debris from
the GSS is centered at a different mean velocity in
each of these fields.

\begin{deluxetable*}{lccrrl}
\tabletypesize{\scriptsize}
\tablecolumns{6}
\tablewidth{0pc}
\tablecaption{M31 Spectroscopic Fields with Identified Substructure.}
\tablehead{\multicolumn{1}{c}{Field} & \multicolumn{1}{c}{\#\ Spectroscopic} & \multicolumn{1}{c}{Projected}          &
  \multicolumn{1}{c}{\#\ Stars\tablenotemark{a}} & \multicolumn{1}{c}{\#\ of M31} & Reference \\
&   \multicolumn{1}{c}{Masks}      & \multicolumn{1}{c}{Radius}     &  &
\multicolumn{1}{c}{Stars\tablenotemark{b}} & \\
&     & \multicolumn{1}{c}{(kpc)} &  &  &}
\startdata
SE shelf & 4 & 12\,--\,18 & 510 & 428 & \citet{gilbert2007} \\
f207 & 1 & 16\,--\,19 & 57 & 49 & \citet{gilbert2009gss} \\
H13s & 2 & 20\,--\,23 & 212 & 178 & \citet{kalirai2006gss}, \\
     &   &            &     &     & \citet{gilbert2009gss}\\
a3 & 3 & 31\,--\,34 & 88 & 69 & \citet{guhathakurta2006}, \\
     &   &            &     &     & \citet{gilbert2009gss}\\
a13 & 4 & 55\,--\,63 & 109 & 44 & \citet{gilbert2009gss} \\
m4 & 5 & 53\,--\,60 & 154 & 53 & \citet{gilbert2009gss} \\
\enddata
\tablenotetext{a}{The total number of unique stellar spectra for which we recovered velocities.}
\tablenotetext{b}{The number of M31 RGB stars is defined as the number of stars
that are identified as secure and marginal M31 RGB stars by the \citet{gilbert2006}
diagnostic method.}
\label{table1}
\end{deluxetable*}

Photometry and astrometry for the SE shelf, f207, and H13s fields
were derived from images in the $g'$ and $i'$ bands taken with the MegaCam instrument
on the 3.6-m Canada-France-Hawaii Telescope (CFHT).\footnote{MegaPrime/MegaCam is
a joint project of CFHT and CEA/DAPNIA, at the Canada-France-Hawaii Telescope
which is operated by the National Research Council of Canada, the Institut
National des Science de l'Univers of the Centre National de la Recherche
Scientifique of France, and the University of Hawaii.}  The SExtractor program
was used for object detection, photometry, and morphological classification \citep{bertin1996} and observations of Landolt photometric standard stars were used to transform
from instrumental $g'$ and $i'$ magnitudes to Johnson-Cousins $V$ and $I$ magnitudes \citep{kalirai2006gss}.

Photometry and astrometry for the a3, a13, and m4 fields
were derived by \citet{ostheimer2003} from images in the Washington $M$ and
$T_2$ bands and the intermediate-width
DDO51 band taken with the Mosaic camera on the 4-m Kitt Peak National Observatory (KPNO) telescope.\footnote{Kitt Peak National
Observatory of the National Optical Astronomy Observatory is operated by the
Association of Universities for Research in Astronomy, Inc., under cooperative
agreement with the National Science Foundation}  The DDO51 filter includes the
surface-gravity sensitive Mg\,$b$ and MgH stellar absorption features.
The combination of these three filters allows photometric selection
of stars that are likely to be M31 red giant branch (RGB) stars rather
than MW dwarf stars along the line-of-sight to M31 \citep{majewski2005}.  The
photometric transformation relations in \citet{majewski2000} were used to
convert the $M$ and $T_2$ magnitudes to Johnson-Cousins $V$ and $I$ magnitudes.

Objects were selected for spectroscopy based on their derived $I$ magnitudes and
morphological properties.  In the outer fields (a3, a13, and m4), objects were further prioritized by
their probability of being an M31 RGB star based on the $M$, $T_2$, and DDO51
photometry; this is vital for increasing the spectroscopic efficiency of our survey
in the sparse outer regions of M31's stellar halo.  Spectroscopic slitmasks were
observed with the DEIMOS instrument on the Keck~II telescope for 1 hr each,
using the 1200 line mm$^{-1}$ grating (which has a dispersion of 0.33~\AA\
pixel$^{-1}$) centered at 7800~\AA.  This led to an average spectral coverage
of 6450\,--\,9150~\AA.

Modified versions of the {\tt spec2d} and {\tt spec1d} software\footnote{http://astron.berkeley.edu/$\sim$cooper/deep/spec2d/primer.html,
\newline\indent
http://astron.berkeley.edu/$\sim$cooper/deep/spec1d/primer.html
} developed by the DEEP2 team
at the University of California, Berkeley were used to reduce and analyze the
spectroscopic data \citep[see][]{simon2007,gilbert2007}.  Reduced one-dimensional spectra are cross-correlated with
a library of template stellar spectra to determine the redshift of the object.
Each spectrum is visually inspected and assigned a quality code based on the
number and quality of the absorption lines.  A spectrum must have at least two
identifiable spectral features to be considered to have a secure
redshift measurement \citep{guhathakurta2006,gilbert2007}.  A
heliocentric correction and a correction for possible miscentering of the star
within the slit (based on the position of the telluric A-band relative to
atmospheric emission lines in the observed spectrum) is applied to the
measured velocities \citep{simon2007,sohn2007}.
The median velocity error is $\sim 6$~\,km~s$^{-1}$, based on error estimates from
the cross-correlation routine and repeat measurements of individual stars.

\subsection{Selection of M31 RGB Stars}

The line-of-sight velocity distributions of foreground MW dwarf stars
and M31 RGB halo stars overlap, making it nontrivial to identify
individual stars as M31 red giants or MW dwarfs. We use the diagnostic
method described in \citet{gilbert2006} to isolate M31 RGB stars
from foreground MW dwarf stars in our spectroscopic sample.  Empirical
probability distribution functions in five photometric and spectroscopic
diagnostics are used to determine the likelihood an
individual star is an M31 red giant.  These diagnostics include
(1) line-of-sight velocity,
(2) photometry in the $M$, $T_2$, and DDO51 bands (when available),
(3) the EW of the Na\,{\smcap i}\ doublet at 8190~\AA\ versus the $(V-I)_0$\ color of the star,
(4) position of the star in an ($I,\,V-I$)\ color magnitude diagram, and
(5) photometric versus spectroscopic metallicity estimates.
The M31 RGB sample used in this analysis includes all stars that are
more probable to be M31 red giants than MW dwarfs.

\subsection{Metallicity Estimates}\label{sec:data_met}
Metallicity estimates ($\rm[Fe/H]$) are derived by comparing a star's position in the ($I,\,V-I$)\
color magnitude diagram with theoretical isochrones adjusted
to the distance of M31 \citep[783~kpc;][]{stanek1998,holland1998}.  Interpolation within the
grid of 12~Gyr, [$\alpha$/Fe]=0, \citet{vandenberg2006} isochrones yields an $\rm[Fe/H]$\ measurement
for each M31 RGB star \citep{kalirai2006halo}.

Spectroscopic estimates of the metallicity
of an individual star can also be obtained using the equivalent width of the Ca\,{\smcap ii}\
triplet and the calibration relations of \citet{rutledge1997pasp1,rutledge1997pasp2}.  On average, the
two estimates are in good agreement for the M31 survey data \citep{kalirai2006halo}, although there is
significantly larger scatter in the spectroscopic metallicity estimates
due to large measurement errors in the Ca\,{\smcap ii}\ EW measurements
\citep{kalirai2006halo,gilbert2006}.  This is due primarily to the relatively
low average signal-to-noise ratio of M31 RGB spectra in our spectroscopic
survey (typical values of $S/N\sim 10$~pixel$^{-1}$).  There is
smaller scatter in the spectroscopic metallicity estimates and very good
agreement with photometric metallicity estimates for our highest $S/N$ M31
RGB spectra.  Due to the large measurement
uncertainties in the spectroscopic
metallicity estimates for individual stars, we choose to use photometric
metallicities for this analysis.

The use of photometric metallicities introduces systematic biases due to the isochrone set used and to
the adoption of a single age and alpha-enrichment for the stellar populations.
Estimates of $\rm[Fe/H]$\ based on different sets of isochrones vary by less than 0.15~dex \citep{kalirai2006halo}.
Although the alpha-enrichment of M31's stellar halo has yet to be
observationally constrained, deep HST/ACS observations of fields in M31's stellar halo which
reach the main sequence turnoff \citep[e.g.,][]{brown2006apj,brown2007} have shown
that the stellar population of M31's halo spans a wide range in age.
In the SE shelf and H13s fields, which are contaminated by GSS debris, the stellar population ranges from 6\,--\,13~Gyr in age.
The younger stars are found to be the most metal-rich, while the
old population is relatively metal-poor \citep{brown2006apjl,brown2006apj}.  In an HST/ACS M31 halo field at 21~kpc, a spread in ages of $\sim 8$\,--\,13~Gyr is found.  Once again, the younger stars are the most metal-rich while the old stellar population is relatively metal-poor.
We therefore expect our metallicity estimates to be affected mostly at the metal-rich end by the assumption of a single, old stellar population.  Varying the age from 12 to 6~Gyr results in an $\approx +0.15$~dex
shift in the $\rm[Fe/H]$\ estimates derived from isochrone fitting for metal-rich ($\rm[Fe/H]$$>-1$) stars \citep{gilbert2009gss}.

\subsection{Identification of Substructure: Kinematics and Metallicity}\label{sec:data_ident}

Figure~\ref{fig:data} presents line-of-sight velocity distributions and
$\rm[Fe/H]$\ versus line-of-sight velocity for the stars in each of six fields
in which our survey of M31's stellar halo has discovered kinematical substructure.  The distributions of CMD-based $\rm[Fe/H]$\ values (\S\,\ref{sec:data_met})
in Figure~\ref{fig:data} show the presence of a relatively metal-poor
($\rm[Fe/H]$$\lesssim -1$) population with a broad velocity distribution in
each field, while the kinematically cold substructure tends to be relatively
metal-rich ($\rm[Fe/H]$$> -1$).
Detailed analyses of the kinematics, chemical
properties, spatial distributions, and plausible origins of the various substructures are presented
elsewhere [SE shelf region in \citet{gilbert2007}; a3 in \citet{guhathakurta2006};
H13s in \citet{kalirai2006gss}; fields f207, H13s, a3, a13, and m4 in \citet{gilbert2009gss}].

\begin{figure}
\epsscale{1.0}
\plotone{f1.eps}
\caption{Line-of-sight velocity distributions and $\rm[Fe/H]$\ versus line-of-sight
velocity for the fields discussed in this paper.  Shaded velocity
histograms and open circles represent stars classified as M31 red giants
by the \citet{gilbert2006} diagnostic method, while crosses represent
stars classified as MW dwarfs. Open velocity histograms show the
velocity distribution of all the stars (M31 RGB and MW dwarf) in our
fields.  The solid curves show the best-fit velocity distributions in each field (\S\,\ref{sec:data_ident}).
In general, the observed kinematically cold populations are more metal-rich
than the rest of the stellar population.  }
\label{fig:data}
\end{figure}

The primary signature used to identify substructure in a field is the
presence of a kinematically
cold population; the line-of-sight velocity distributions of these
fields are inconsistent with being drawn from a single, broad
Gaussian centered at M31's systemic velocity \citep[$\langle v\rangle$=$-300$~\,km~s$^{-1}$,
$\sigma_v=129$~\,km~s$^{-1}$;][]{gilbert2007}.
Fields a13 and m4
are at larger projected radial distances from the center of M31 ($R_{\rm proj}\sim60$~kpc) than
the study of \citet{gilbert2007}.  Their
line-of-sight velocity distributions are also inconsistent with being drawn
from a Gaussian with parameters $\langle v\rangle$=$-300$~\,km~s$^{-1}$, $\sigma_v=100$~\,km~s$^{-1}$, the velocity dispersion implied for the distance of these fields by the study of \citet{chapman2006}.

The velocity distribution in each field has been fit by a combination
of Gaussians using a maximum-likelihood technique (red curves in
Figure~\ref{fig:data}).  In general, the number of Gaussians used in the
multi-Gaussian fits is the minimum required to produce a maximum-likelihood
fit that is consistent with the data, although
additional evidence of substructure in a field is also taken into account in
determining the number of kinematical components in a given spectroscopic field \citep{gilbert2007,gilbert2009gss}.
The Gaussian components of the underlying kinematically
hot population were held fixed ($\langle v\rangle^{\rm sph}$\,$=-300$~\,km~s$^{-1}$, $\sigma^{\rm sph}_{\rm v}$\,$=129$~\,km~s$^{-1}$)
while the properties of the
Gaussians representing the kinematically cold populations (mean velocity and
velocity dispersion) and the fraction
of stars in each cold population were allowed to vary.
The SE shelf, f207, H13s, a3, and a13 fields all contain
substructure related to the GSS in M31
\citep{kalirai2006gss,guhathakurta2006,gilbert2007,gilbert2009gss}.
Fields f207 and H13s both have
 a second kinematically cold component which may be unrelated to
the GSS.  The cold components with mean velocities of $-527$~\,km~s$^{-1}$\ and $-490$~\,km~s$^{-1}$\
in f207 and H13s, respectively, are those associated with the GSS \citep{gilbert2009gss}.
Although there are some lines of evidence that
suggest the substructure in m4 and the secondary substructures in fields f207 and H13s
may also be associated with the GSS \citep{fardal2008,gilbert2009gss}, the origins
of these features are currently unknown.

\begin{table}[tb!]
\begin{center}
\caption{Fraction of M31 RGB Stars in Kinematically Cold Components.}
\vskip 0.3cm
\begin{tabular}{lll}
\hline
\hline
\multicolumn{1}{c}{Field} & \multicolumn{1}{c}{$\langle v\rangle^{\rm sub}$} & \multicolumn{1}{c}{$N_{\rm KCC}/N_{\rm tot}$\tablenotemark{a}}\\
& \multicolumn{1}{c}{(\,km~s$^{-1}$)} & \\
\hline
SE shelf & $-285$ & $0.41^{+0.10}_{-0.09}$ \\
f207 & $-524$ & $0.31^{+0.08}_{-0.07}$ \\
f207 & $-426$ & $0.31^{+0.11}_{-0.10}$ \\
H13s & $-490$ & $0.48^{+0.07}_{-0.06}$ \\
H13s & $-388$ & $0.27^{+0.07}_{-0.08}$ \\
a3 &  $-441$ & $0.59^{+0.11}_{-0.12}$ \\
a13 & $-301$ & $0.72^{+0.15}_{-0.21}$ \\
m4 & $-355$ & $0.45^{+0.14}_{-0.13}$ \\
m4 & $-255$ & $0.16^{+0.10}_{-0.08}$ \\
\hline
\hline
\end{tabular}
\tablenotetext{a}{Quoted errors are the 90\% confidence limits from the maximum likelihood analysis (\S\,\ref{sec:data_ident}).}
\label{table2}
\end{center}
\end{table}

\subsection{Characteristics of Observed Substructure: Surface Brightness versus Metallicity}\label{sec:char_obs_sub}
Figure~\ref{fig:mufeh_data1} displays surface brightness versus mean
metallicity for the M31
stellar halo data presented in Figure~\ref{fig:data}.  Each kinematical
component identified in the 6 fields shown in Figure~\ref{fig:data}
is represented: the kinematically cold
components related to the GSS (solid circles),
the kinematically cold components that are not known to be associated with the
GSS (solid triangles), and the kinematically
hot population (open squares).

\begin{figure}
\plotone{f2.eps}
\caption{Surface brightness versus average metallicity for each of the
kinematical components in Figure~\ref{fig:data}: kinematically cold components likely associated
with the GSS (solid circles), kinematically cold components which are not known to be associated
with the GSS (solid triangles), and the kinematically hot components (open squares).
The error bars represent the Poisson error for the surface brightness
estimates and the error in the mean $\rm[Fe/H]$\ value.
The mean and root-mean-square (rms) [Fe/H] and $\mu_{V}$ of the lower surface brightness ($\mu_{V}>29$~mag arcsec$^{-2}$) and higher surface brightness ($\mu_{V}<29$~mag arcsec$^{-2}$) kinematically cold components are also shown (open hexagons and error bars).
Higher surface brightness features are
on average more metal-rich than lower surface brightness features, and kinematically cold substructures are on average more metal-rich than the kinematically hot populations.  Lower surface brightness features display a larger spread in metallicity than higher surface brightness features.  The dotted line denotes the faintest identifiable tidal streams to date, given the number of stars in our DEIMOS survey.  }
\label{fig:mufeh_data1}
\end{figure}

Table~\ref{table2} lists the maximum-likelihood
fraction of M31 RGB stars in each of the kinematically cold components
(from the fits shown in Fig.~\ref{fig:data}).  These
fractions were used to determine the number of M31 RGB stars in each component.
The ratio
of M31 RGB to MW dwarf stars, multiplied by the expected number of MW
dwarf stars along the line of sight to each field \citep[based on the Besancon
Galactic population model;][]{robin2003} is used to estimate the surface brightness of each
component, as described in \citet{guhathakurta2005}.  Correction factors that account for the rate of
successful velocity measurements as a function of magnitude and
varying M31 RGB target efficiency due to photometric pre-selection of stars likely to
be red giants are applied to each field (Gilbert et al., in preparation).
The spectroscopic surface brightness estimates are normalized using photometric surface brightness estimates
from \citet{pritchet1994}.  Error bars
represent Poisson error estimates based on the number of stars in each component.

The $\rm[Fe/H]$\ for the kinematically cold components is the mean $\rm[Fe/H]$\
of stars within $\pm2\sigma_v$ of the mean velocity of the cold component
($|v_{\rm star}-$\,$\langle v\rangle^{\rm sub}$$|<2$\,$\sigma^{\rm sub}_v$), while the $\rm[Fe/H]$\ for the
kinematically hot components is the mean $\rm[Fe/H]$\ of stars with velocities
greater than 2$\sigma^{\rm sub}_v$\ from the mean velocity of the cold components
($|v_{\rm star}-$\,$\langle v\rangle^{\rm sub}$$|>2$\,$\sigma^{\rm sub}_v$; in
fields with two cold components, stars must meet this criterion for
both).  The numbers of stars which are thus used for the $\rm[Fe/H]$\
measurement for each component are consistent with the number of stars
expected to be in each component based on the maximum-likelihood fits
shown in Figure~\ref{fig:data}.  Error bars represent the error in the mean $\rm[Fe/H]$\ value.

Since stars cannot be identified individually as belonging to a particular
kinematical component, these velocity cuts are chosen to maximize the number of
stars involved in the $\rm[Fe/H]$\ measurement of the kinematically cold component,
while minimizing contamination in the
mean $\rm[Fe/H]$\ measurement from other kinematical components.
Even so, the metallicity estimate of the substructure
will be biased by contamination from the kinematically hot component.
This effect will be greatest for the least dominant substructures, and will in
general bias the mean metallicity of the kinematically cold substructures to lower
values.  In the SE shelf fields, where we have a large statistical sample,
accounting for the contamination by the kinematically hot component
in the measurement of the mean
metallicity of the kinematically cold component changes the mean
metallicity by only $+0.03$~dex \citep{gilbert2007}; however, it must be noted that
the $\rm[Fe/H]$\ distribution of the kinematically hot population in the SE shelf fields
is very similar to the $\rm[Fe/H]$\ distribution of the kinematically cold population.

The assumption of a single, old stellar population likely provides a larger source
of systematic error in the photometric (CMD-based) metallicity
estimates.  As discussed in
\S\,\ref{sec:data_met}, this assumption is known to be wrong for at
least some of our fields and is expected to affect our metallicity estimates mostly on the
metal-rich end.  Indeed, a comparison of the mean Ca\,{\smcap ii}-based spectroscopic metallicity
estimates (measured in bins of photometric metallicity) of stars associated with the kinematically
hot and cold populations shows that the kinematically cold populations do, on average,
have a higher spectroscopic abundance than the kinematically hot population at a  given
photometric abundance.  This indicates that the substructure in M31's stellar halo is
on average younger and/or more enriched in [$\alpha$/Fe] than M31's kinematically smooth stellar halo.
Using 6~Gyr instead of 12~Gyr isochrones changes the $\rm[Fe/H]$\ estimates by $\sim +0.15$ for more metal-rich
stars ($\rm[Fe/H]$$\gtrsim-1.0$) and by as much as $\sim +0.3$ for metal-poor stars ($\rm[Fe/H]$$\lesssim-1.5$).
Thus, the true metallicity spread is likely greater than Figure~\ref{fig:mufeh_data1} suggests.

The data in Figure~\ref{fig:mufeh_data1} show a clear correlation
between surface brightness and metallicity: higher surface brightness
features are more metal-rich than lower surface brightness features.  The
kinematically cold substructures are on average more metal-rich
than the underlying kinematically hot components, as was
implied in Figure~\ref{fig:data}.   Figure~\ref{fig:mufeh_data1} also shows
evidence for a larger spread in metallicities among lower surface brightness
features ($\mu_{V}>29$~mag arcsec$^{-2}$) than higher surface brightness features ($\mu_{V}<29$~mag arsec$^{-2}$).  The root-mean-square (rms)
deviation of average [Fe/H] values among higher surface brightness, kinematically cold
components is only 0.08~dex, while the rms deviation of average [Fe/H] values among
lower surface brightness, kinematically cold components is 0.32~dex.

Figure~\ref{fig:mufeh_data2} displays surface brightness
versus mean metallicity for fields from our M31 halo spectroscopic survey with
kinematically-identified substructure (those presented in Figure~\ref{fig:data};
solid circles), fields
without kinematically-identified substructure (open triangles), and fields for which
we cannot determine if substructure is present due to small
numbers of M31 RGB stars ($\lesssim 10$ M31 stars per field; open squares).
The fields with small numbers of M31 RGB stars are all located in the outer
 regions of M31's stellar halo ($R_{\rm proj}\gtrsim 80$~kpc), while the fields without
 kinematically-identified substructure span a similar range in projected distance
 from M31's center as the fields with kinematically-identified substructure.
Fields with identified substructure are on average more metal-rich than
fields without identified substructure, as would be expected if kinematically cold
substructures are more metal-rich on average than the kinematically hot population
(Fig.~\ref{fig:mufeh_data1}).

\begin{figure}
\plotone{f3.eps}
\caption{The same as Figure~\ref{fig:mufeh_data1} but for entire fields from our M31 spectroscopic survey.  Fields are divided into three categories: fields
with (solid circles; Fig.~\ref{fig:data}) and without (open triangles)
kinematically identified substructure, and fields with small numbers
of M31 RGB stars
($\lesssim 10$), for which we cannot determine if substructure is present.
The large symbols show the mean and rms (errorbars) [Fe/H] and $\mu_{V}$ of fields with and without identified substructure.  Fields with identified substructure are on average more metal-rich than fields without identified substructure.
The dashed
line represents the current surface brightness limit achieved in our
M31 spectroscopic survey, while the dotted line denotes the faintest
tidal stream identified in our survey (Fig.~\ref{fig:mufeh_data1}).
}
\label{fig:mufeh_data2}
\end{figure}

\section{Simulated Stellar Halos}\label{sec:sims}

Simulations of hierarchical stellar halo formation provide a physical explanation of the surface brightness versus metallicity trend observed in our M31 halo data.  In this section, we briefly describe the simulations (\S\,\ref{sec:sims_method}), present the abundance properties of substructure in the simulated stellar halos (\S\,\ref{sec:sims_properties}), and discuss the physical mechanisms behind the trends (\S\,\ref{sec:sims_interp}).  We will compare the abundance trends in the simulations with our M31 observations in \S\,\ref{sec:comparison}.

\subsection{Methods}\label{sec:sims_method}
The BJ05 simulations follow the formation of 11 MW-analog stellar halos
via accretion in a cosmological context.
For each host halo, a merger tree was generated using the extended
Press-Schechter formalism \citep{somerville1999,lacey1993}.
Only merger histories without recent high-mass mergers were selected, resulting
in $z=0$ halos suitable for hosting a large disk galaxy like the Milky Way or M31.
For each merger, an $N$-body simulation was used to track the evolution
of the dark matter halo of the satellite galaxy in the
host halo potential.  The potential of the host galaxy is modeled with
a smoothly growing, analytic function, and satellite-satellite interactions
are neglected.  Overall, these assumptions will lead to an unnatural enhancement of the  level of substructure in the simulated halos, especially in the inner halo.
Thus, in the
following analysis we discuss only the properties of the simulated stellar
halos exterior to $R=20$~kpc.

Gas accretion onto each satellite halo tracks its mass
accretion history and the star formation rate within each object is
proportional to its instantaneous gas content over a fixed timescale.
The timescale is chosen
so that dwarfs infalling today would contain similar gas fractions to those observed in
Local Group field objects (see BJ05 for details).
Given the gas infall, star formation rates, and mass of a satellite as a function of time, the method of
\citet{robertson2005} is used to follow the chemical evolution of the stellar
populations \citep[see also][]{font2006apjhalos}.
The \citet{robertson2005} prescriptions
include enrichment from Type Ia and Type II supernovae and the effects of feedback
(from supernovae and stellar winds).
The free parameters in the prescriptions are
constrained by requiring consistency with observations: the Fe and $\alpha$-element wind
efficiencies are chosen separately by requiring matches to the observed stellar
mass-metallicity relation for dwarf galaxies \citep[e.g.,][]{larson1974,mateo1998,dekel2003}
and the $\alpha-$element abundance patterns in satellite dwarf spheroidals \citep[e.g.][]{venn2004}.
Both gas accretion and star formation are
assumed to truncate once the satellite is accreted onto the host halo.
A variable mass-to-light ratio is assigned to every dark
matter particle in order to embed the stellar components within the
dark matter halos of the satellites in a way that reproduces the observed distribution of structural properties of Local Group dwarfs.
Each stellar particle in the simulations is randomly assigned $\rm[Fe/H]$\ and
[$\alpha$/Fe] ratios based on the satellite's star
formation history \citep{font2006apjhalos}.  Thus each model
satellite contains a realistic intrinsic metallicity spread, but no radial
metallicity gradient.

The resulting simulated halos have similar luminosities and stellar density
profiles to the Milky Way's halo, and the surviving satellites are
comparable in number, luminosity distribution, and structural properties to those of the Milky Way (discovered prior to 2005).  The models are also able
to reproduce the observed difference in [$\alpha$/Fe] ratios between the
Milky Way's stellar halo and dwarf satellites. The lower [$\alpha$/Fe]
ratios in the surviving simulated dwarf galaxies are due to their late accretion times
and the prolonged, less bursty star formation histories of these systems
compared with those of the progenitors of the halo \citep{robertson2005,font2006apjsats}. In the models, the kinematically cold
streams tend to have distinct chemical abundances (i.e. higher [Fe/H] and
lower [$\alpha$/Fe]) than that of the smooth, more well-mixed halo \citep{font2006apjhalos}. This is also due to the fact that the progenitors of these streams were accreted following the formation of the bulk of the halo.

Of course, in the framework of the hierarchical formation of stellar halos,
there is no clear demarcation between tidal streams and the ``smooth''
halo. In reality, present day stellar halos are expected to display a
natural progession in the kinematical properties of their unbound
substructure, ranging from thin and kinematically cold to more
disperse, kinematically hot substructure. In light of the above findings,
the prospect of using chemical abundances as a complementary dimension to
phase-space to disentangle various substructures is encouraging.

\subsection{Predicted Abundance Properties of Substructure}\label{sec:sims_properties}

Figure~\ref{fig:sims} shows the projected surface brightness
({\it top}), metallicities ({\it middle}), and alpha-enrichment ({\it bottom})  for two of the BJ05
simulated stellar halos.
The projected surface brightness reflects the total luminosity in each
0.25~kpc~$\times$~0.25~kpc
pixel, summed over all the stellar populations.  The $\rm[Fe/H]$\ and [$\alpha$/Fe] values
are the stellar-mass-weighted average of the stellar populations in each pixel.  These maps reveal that
the highest surface brightness streams tend to be more metal-rich and less alpha-enhanced than the lower surface brightness streams.

\begin{figure}
\begin{center}
\plottwo{f4a.eps}{f4b.eps}\\
\plottwo{f4c.eps}{f4d.eps}\\
\plottwo{f4e.eps}{f4f.eps}
\end{center}
\caption{
Maps of projected surface brightness ({\it top}),
average metallicity ({\it middle}),
and average alpha-enrichment ({\it bottom})
for 2 of the BJ05 simulated stellar halos.  Each map depicts a 300$\times$300~kpc region.
The $\rm[Fe/H]$\ and [$\alpha$/Fe] values are the stellar-mass-weighted average of
all the stars in each pixel.  A comparison of the surface brightness
and metallicity maps reveals that the brightest features tend to be
more metal-rich and less alpha-enhanced than the rest of the halo. Image credit: Sanjib Sharma.
}
\label{fig:sims}
\end{figure}

This is demonstrated quantitatively in Figure~\ref{fig:mfsim},
which displays maximum
surface brightness versus mean metallicity ({\it left panels}) and alpha-enhancement ({\it right panels}) for
individual stellar streams produced by accretion events in all 11 BJ05
stellar halos, taken from the analysis of \citet{johnston2008}.
Only
stellar streams that contribute at least 50\% of the total light in the halo
at the position of their highest surface brightness point and which are at
radial distances greater than 30~kpc from the center of the galaxy are
included.  Although for any given maximum surface brightness stellar streams exhibit a range of mean $\rm[Fe/H]$\ and [$\alpha$/Fe] values, the higher surface brightness streams are in general more metal-rich and less alpha-enhanced than the lower surface brightness streams.

\begin{figure}
\epsscale{1.0}
\plotone{f5.eps}
\caption{Maximum surface brightness versus average metallicity ({\it left panels})
and alpha enhancement ({\it right panels}) for stellar streams in the 11 BJO5
simulated stellar halos. The top panels are color coded according to the luminosity
of the progenitor satellite [$L<10^6$\mbox{\,$L_{\odot}$}\ ({\it black}); $10^6$\mbox{\,$L_{\odot}$}\,$<L<10^7$\mbox{\,$L_{\odot}$}\
({\it red}); $10^7$\mbox{\,$L_{\odot}$}\,$<L<10^8$\mbox{\,$L_{\odot}$}\ ({\it blue}); $10^8$\mbox{\,$L_{\odot}$}\,$<L<10^9$\mbox{\,$L_{\odot}$}\ ({\it yellow})]
while the bottom panels are color coded according to the time of accretion
of the progenitor satellite
onto the host halo [ $3<t_{\rm acc}<6$ Gyr
({\it purple}); $6<t_{\rm acc}<9$ Gyr ({\it cyan}); $9<t_{\rm acc}<12$~Gyr
({\it orange})].   The distribution in the left panels is
a reflection of the mass-metallicity relationship observed for
Local Group dwarfs, while the
distribution in the right panels reflects the
time-sensitivity of alpha elements.  The bottom panels also show a
slight tendency for more recent accretion events to be brighter.
}
\label{fig:mfsim}
\end{figure}

\subsection{Physical Interpretation}\label{sec:sims_interp}

The points in Figure~\ref{fig:mfsim} are
color-coded by mass of the progenitor satellite ({\it top panels}) and time
of accretion of the progenitor satellite onto the host halo ({\it bottom panels}; see figure caption for details).  The top left panel clearly shows the effect of the
assumption of a mass-metallicity relation for the progenitor satellites (based on observations of dwarf galaxies; \S\,\ref{sec:sims_method}): more massive satellites are more metal-rich, thus the tidal streams produced as massive satellites fall into the host halo are also more metal-rich.  Since the more massive progenitors also typically produce brighter stellar streams, it follows that the surface brightness and metallicity of stellar streams are correlated.

The alpha-enhancement of the stellar streams traces the time of accretion of the progenitor onto the host halo, as is
demonstrated in the bottom right panel of Figure~\ref{fig:mfsim}.
Alpha elements are produced in Type II supernovae; their abundances become subsequently
diluted by the elements produced in Type I supernovae.  Thus, progenitors accreted
early are alpha-enriched relative to progenitors that are accreted later and have had longer star formation histories.
The surface brightness of stellar streams observed today is a function
not only of the mass of the progenitor but also of the time since accretion,
therefore the bottom right panel shows a slight trend for stellar streams produced by progenitors that were accreted at early times to have lower
surface brightnesses than progenitors that were accreted at later times.  This leads to the anti-correlation between surface brightness and alpha-enrichment.

\section{Comparison of Observations and Simulations}\label{sec:comparison}

\subsection{Observing the Simulations}\label{sec:comp_obssims}
Figure~\ref{fig:mfsim} provides useful insight into the physical mechanisms underlying the expected general relationships
between the chemical enrichment and maximum surface brightness of stellar streams.  The data in
Figure~\ref{fig:mufeh_data1} appear to agree qualitatively
with the characteristics seen in the left panels of
Figure~\ref{fig:mfsim}; identifiable (and thus fairly dominant and
high surface brightness) substructures
tend to be relatively metal-rich.  However, our Keck/DEIMOS spectroscopic
masks provide information on the characteristics
of the population in isolated lines-of-sight through M31's stellar halo,
rather than global information about individual streams.
The data cannot be compared directly with Figure~\ref{fig:mfsim},
since the maximum surface brightness of an observed stellar stream can only
be determined by tracing the full extent of the stream in space.
The only stellar stream for which this measurement could
currently be made in M31 is the GSS.

A more direct comparison between the data and simulations can be made
by analyzing stellar populations along individual lines-of-sight through
the simulated stellar halos. Figure ~\ref{fig:sim_feh_alpha} displays
mean stellar metallicity and alpha-enhancement as a function of
surface brightness for sight lines through one of the BJ05 simulated
stellar halos; each point represents a line-of-sight that covers a
projected area of 0.4~kpc~$\times$~0.4~kpc.
Only lines of sight beyond 20~kpc in projection from the center of the galaxy are included (\S\,\ref{sec:sims_method}).
Lines-of-sight through the simulated stellar halos whose stellar populations are dominated by debris from one or two
accretion events are denoted by red solid circles in Figure~\ref{fig:sim_feh_alpha}, while lines-of-sight whose stellar populations are largely comprised of debris from multiple ($\ge3$)
accretion events are denoted by blue open squares.

\begin{figure}
\plotone{f6.eps}
\caption{Surface brightness versus stellar-mass-weighted average metallicity ({\it left}) and
alpha-enhancement ({\it right}) for
individual lines-of-sight through one of the 11 BJ05 simulated stellar halos,
divided into two categories based on the
number and prominence of individual stellar streams.  ({\it a}) Lines-of-sight where the majority of the stellar population derives from one or two accretion events (solid red circles).  ({\it b}) Lines-of-sight where the majority of the stellar population derives from multiple ($\ge3$) accretion events  (open blue squares).  ({\it c}) The two types of sight-lines superimposed.
The overall trends of surface brightness with metallicity and alpha-enhancement (panel {\it c}),
are similar to those in Figure~\ref{fig:mfsim}, although with a somewhat steeper slope.
For a given surface brightness, lines-of-sight whose stellar populations are dominated by debris
from one or two
accretion events are on average more metal-rich
and less alpha-enhanced than lines-of-sight whose stellar populations are
composed primarily by debris from multiple
accretion events.  The surface
brightness of the faintest tidal debris that has
been identified as kinematically cold substructure in our Keck/DEIMOS survey
is denoted by the dotted line (Fig.~\ref{fig:mufeh_data1}) while the
current surface brightness limit of our spectroscopic survey is denoted by the
dashed line (Fig.~\ref{fig:mufeh_data2}).
}
\label{fig:sim_feh_alpha}
\end{figure}

The same general trends seen in individual streams (Fig.~\ref{fig:mfsim}) are repeated
here, with a steeper slope: higher surface brightness lines-of-sight are on average more
metal-rich and less alpha-enhanced than lines-of-sight of lower surface
brightness.  The same physical mechanisms discussed in \S\,\ref{sec:sims_interp} are relevant here, however
each point in Figure~\ref{fig:sim_feh_alpha} is averaging over stellar
populations from {\it multiple} satellite progenitors.
Thus, the trends in metallicity
and alpha-enhancement with surface brightness are
not as strong.

\subsection{Common Trends}\label{sec:comp_trends}

We can't know a~priori the number of streams that are along any
particular observed sight-line in M31's stellar halo.
However, separating the kinematically cold and hot components in fields
with identifiable substructure, as is done in Figure~\ref{fig:mufeh_data1},
effectively separates a single field into the equivalents of lines-of-sight dominated by stellar populations arising from a single accretion event
(kinematically cold components) and lines of sight dominated by stellar populations arising from many accretion events (kinematically
hot components).   In a stellar halo built in a purely dissipationless
fashion, a population with a large spread in velocities is most likely to be
formed from the superposition of debris originating in multiple satellites
that merged early with the host galaxy.
The dotted line in Figure~\ref{fig:sim_feh_alpha} denotes the surface
brightness of the faintest tidal debris that is
identifiable as kinematically cold substructure in our Keck/DEIMOS survey
(Fig.~\ref{fig:mufeh_data1}).

The simulations (Fig.~\ref{fig:sim_feh_alpha}) show the same general trends seen in
our M31 data (Fig.~\ref{fig:mufeh_data1}): (1) Higher surface brightness features are on average more metal-rich, and (2) at a given surface brightness
lines-of-sight dominated by debris from one or two accretion events tend to be
more metal-rich than those dominated by debris from multiple accretion events.

We have discussed the first trend in \S\S\,\ref{sec:sims_properties}\,--\,\ref{sec:sims_interp}, and shown that in a stellar halo formed via accretion, it can be explained via the mass-metallicity relationship for dwarf galaxies: larger galaxies are more metal-rich and also produce the brightest tidal streams.

This same physical mechanism can also explain the second trend. In order
for tidal debris from a single satellite to dominate the stellar population
in a line-of-sight through a halo, the tidal debris must be part of a particularly
bright stellar stream relative to the average surface brightness at
its location in the halo.  It thus must originate from a relatively massive
satellite (relatively metal-rich) and/or recent accretion event (low relative alpha-enhancement) compared to the accretion events
in a line-of-sight of equivalent surface brightness but composed primarily
of debris from multiple progenitors.

With multiple physical mechanisms (e.g., the size and orbit of the progenitor and the time since its accretion onto the host halo) affecting the surface brightness of individual streams, a significant spread in the chemical properties of individual sight-lines of a given surface brightness, as seen in Figure~\ref{fig:sim_feh_alpha}, is to be expected.  This is particularly relevant for faint lines-of-sight, since faint stellar streams can be formed by the recent accretion of a low mass progenitor or early accretion events whose debris streams have spatially dissipated, or both.  The brightest lines of sight, however, must be composed of debris from very massive and/or recent events, resulting in a narrower range in chemical properties.

The data also show evidence of a narrower spread in metallicity among higher surface brightness streams/lines-of-sight and a larger spread in metallicity among fainter streams/lines-of-sight (Figs.~\ref{fig:mufeh_data1} \& \ref{fig:mufeh_data2}, \S\,\ref{sec:char_obs_sub}).  The rms deviation of mean [Fe/H] values for the high surface brightness ($\mu_{V}<29$~mag arcsec$^{-2}$) and low surface brightness ($\mu_{V}>29$~mag arcsec$^{-2}$) tidal streams observed in our M31 spectroscopic data are 0.08~dex and 0.32~dex, respectively (although the observed spread is likely underestimated due to systematic errors, \S\,\ref{sec:char_obs_sub}).  To compare the spread in [Fe/H] seen in the observations with that seen in the simulations, we impose surface brightness limits on the simulations based on the surface brightness of tidal streams observed in our survey (Fig.~\ref{fig:mufeh_data1}), and measure the rms devation of the mean [Fe/H] values of lines-of-sight through the BJ05 simulated stellar halo which have a stellar population dominated by one or two accretion events (open circles in Fig.~\ref{fig:sim_feh_alpha}).  The rms deviation of [Fe/H] values for high surface brightness lines-of-sight ($27<\mu_{\rm V}<29$~mag arcsec$^{-2}$) is 0.07~dex, while the rms deviation of [Fe/H] values for low surface brightness lines-of-sight ($29<\mu_{\rm V}<31.7$~mag arcsec$^{-2}$) is 0.13~dex.  The spread in [Fe/H] values in our observations and the simulations is similar for high surface brightness lines-of-sight, while the spread in [Fe/H] values in the observations is greater than in the simulations for lower surface brightness lines-of-sight.  The rms deviation of [$\alpha$/Fe] values in the simulations for the magnitude limits used above are 0.02~dex and 0.05 dex, respectively.

We note that the absolute [Fe/H] scale is uncertain for both the observations (due to the age-metallicity degeneracy of the RGB as well as systematic errors in the generation of theoretical isochrones, \S\,\ref{sec:data_met}) and the simulations (where the output [Fe/H] is dependent on the input chemical enrichment prescriptions, \S\,\ref{sec:sims_method}).  Therefore, we do not attempt to directly compare the metallicities of the data and simulations.

\subsection{Discussion}\label{sec:comp_disc}

The common trend of increasing metallicity with increasing surface brightness in lines-of-sight through M31's stellar halo and simulated stellar halos is encouraging.  It gives us confidence that such simulations can lead to meaningful interpretation of stellar halo observations.

The surface brightness of M31's stellar halo is observed to decrease with increasing distance from the center of the galaxy \citep{guhathakurta2005,irwin2005,ibata2007}, and the inner region of M31's stellar halo is observed to be more metal-rich than the outer region \citep{kalirai2006halo,chapman2006,koch2008}.  Thus, the general increase in mean [Fe/H] with increasing surface brightness of the kinematically broad components in our M31 data (Fig.~\ref{fig:mufeh_data1}, open squares) is expected.  In the hiearchical framework, this global trend in stellar halo metallicity could be explained with the same physical mechanisms invoked above (\S\,\ref{sec:comp_trends}).
Although the radial extent over which tidal debris from an individual accretion event is spread depends on the orbit of the progenitor, the effect of dynamical friction on the progenitor's orbit is more important for more massive progenitors, causing them to sink to the center of the gravitational potential of the host galaxy more quickly than less massive progenitors.  Therefore, more massive progenitors will tend to deposit most of their relatively metal-rich debris in the interior regions of the host halo.

If the number of M31 stars detected in an observational field is small ($\lesssim 20$), it can be difficult to
determine if the field is dominated by contributions from one or two
kinematical components or if the field
contains contributions from multiple kinematical components (i.e., contains a relatively hot population).
The results of the simulations can be used to examine the probable number of stellar streams
in a given field. If the mean $\rm[Fe/H]$\ of a field is
relatively high compared to other fields of comparable surface brightness, the field may be dominated
by one or two stellar streams, while if the mean $\rm[Fe/H]$\ is relatively low, the field is more likely to be dominated by
multiple superposed stellar streams along the line of sight (i.e., the
stellar population along the line of sight is approaching a dynamically hot
population).

There are several fields in our M31 spectroscopic survey in which we have
spectra of $\lesssim 10$ M31 RGB stars (open squares, Fig.~\ref{fig:mufeh_data2});
all are in M31's sparse outer halo ($R_{\rm proj}\gtrsim 80$~kpc).
Based on the above discussion (\S\,\ref{sec:comp_trends}), the outer halo fields with the
highest mean $\rm[Fe/H]$\ values are the most likely to be dominated by debris from
one or two high mass progenitors, while the fields with lower mean $\rm[Fe/H]$\ are more likely to be
dominated by debris from many lower mass progenitors.  Future observations will be able to
test this conjecture.  If fields with lower mean $\rm[Fe/H]$\ are found to have stars with a wide range of
velocities, it will support the hypothesis that the field is dominated by debris from many progenitors.
Although the presence of a significant old,
metal-poor stellar population in a higher mass progenitor may somewhat obscure this effect,
the mean $\rm[Fe/H]$\ of a field dominated by
such a progenitor would still be expected to be higher on average than the mean $\rm[Fe/H]$\ of a field
dominated by low mass progenitors which did not have an appreciable number of metal-rich stars.

There are currently no observational constraints on the alpha-enrichment of
stars in M31's stellar halo.  However, if a mean [$\alpha$/Fe] can be
measured for the M31 RGB stars in a given field (i.e., by measuring the
relative abundances of elements in a composite spectrum formed by coaddition
of spectra from multiple stars), it could be used to constrain the average time
of accretion of the stellar population.  Measurement of the mean [$\alpha$/Fe] would greatly facilitate
the interpretation of the surface brightness of specific fields by lessening the degeneracy
between the effect of the luminosity of accreted satellites versus the time of accretion onto the host halo.

\section{Summary}\label{sec:sum}

We have presented the general trends of surface brightness with chemical enrichment of RGB stars observed in the course of our M31 Keck/DEIMOS spectroscopic survey and of stellar streams in simulations of stellar halos formed in a hierarchical framework.  Kinematically identified substructure in M31's stellar halo is metal-rich relative to the kinematically hot population, and higher surface brightness features are more metal-rich than lower surface brightness features.

Similar trends are found in the simulated BJ05 stellar halos, which are formed entirely via accretion.  Higher surface brightness lines-of-sight through the simulated stellar halos are in general more metal-rich than lower surface brightness lines-of-sight, and lines-of-sight dominated by stars from one or two
accretion events are in general more metal-rich than lines-of-sight whose stars come from many events.  The comparison of the observations and simulations provides a physical explanation for the observed trends in surface brightness and metallicity of stellar halo structure: more massive progenitors, which are also the most metal-rich, produce the brightest stellar halo substructures.  This results in the dominance of metal-rich streams in galactic halos.

Both the data and the simulation results show a significant spread in metallicity at a given surface brightness for lines-of-sight through stellar halos.  This is a result of the varying time since accretion of the tidal streams' progenitors onto the host halo.  Tidal streams produced by more recent accretion events are, on average, brighter than tidal streams produced by less recent accretion events.

The simulations show a relationship between the alpha-enhancement and surface brightness of tidal streams is expected: fainter tidal debris is on average more alpha-enhanced than brighter tidal debris.  This arises from the different times of accretion of progenitor satellites onto the host halo, as well as the assumption of the truncation of star formation upon accretion onto the host halo.  Satellites accreted early had less time for chemical enrichment via Type Ia SNe to reduce their alpha enrichment, while their tidal debris has had more time to spread along the progenitor's orbit within the host halo.  The ability to measure the alpha-enrichment of M31 RGB stars would thus lead to constraints on both the luminosity and the time of accretion onto the host halo of the progenitors of M31's tidal streams.

\acknowledgments
The authors would like to thank James Bullock, Brant Robertson, Steve Majewski, Marla Geha, and Sanjib Sharma
for their contributions to the research programs whose products were used
in this work.
The authors of the paper benefited from the hospitality of the Aspen Center for
Physics in June 2006.  This work was supported by
NSF grants AST-0307966, AST-0507483, and
AST-0607852 (K.M.G., P.G.) and a UCSC Chancellor's Division Dissertation-Year Fellowship (K.M.G.), NSF CAREER award AST-0733966 and NSF grant AST-0734864 (K.V.J.), and an STFC  Fellowship at the Institute for Computational Cosmology in Durham (A.S.F.).


\section{Introduction}
A large fraction of edge-on spiral galaxies have been observed to have
boxy or peanut-shaped (hereafter B/PS) bulges.  As the observations
have improved and the samples have grown, the percentage of galaxies
with these bulge shapes has increased from less than 20\%
\citep{Jarvis86,Shaw87,deSouza87} to up to 45\% \citep{Lutticke00a}.
Because the observed frequency of B/PS bulges is so high, significant
effort has been invested in determining the formation mechanism
responsible for these bulge shapes.

\citet{Binney85} and \citet{Rowley88} demonstrated that the accretion
of satellite galaxies could result in a galaxy bulge looking boxy or
peanut-shaped.  The currently favored formation mechanism, however, is
the buckling and subsequent vertical thickening of a bar
\citep[e.g.][]{Combes81,Combes90,Raha91}.

This is a plausible explanation, as bars are known to be common
features in disk galaxies.  \citet{Whyte02} found that 79\% of disk
galaxies are barred in the near-infrared and 74\% are barred in the
optical \citep[see also][]{Eskridge00,Marinova07}.  Barred galaxies
are also known to occur in a variety of environments. Using a sample
of 930 galaxies, \citet{vandenbergh02} showed that the presence
of a bar does not depend on whether the galaxy is in the field or is a
member of a group or a cluster.  Similarly, \citet{Verley07} studied the
bar frequency in isolated galaxies and found roughly equal numbers of
barred and unbarred systems.

To confirm the bar buckling hypothesis, bars must be detected in
edge-on galaxies with B/PS bulges.  While bars can be easily seen in
face-on galaxies, it is much more difficult, however, to unambiguously
photometrically confirm the presence of a bar in an edge-on galaxy.
This problem was simplified when \citet{Kuijken95} and
\citet{Merrifield99} demonstrated that bars in edge-on galaxies could
be detected kinematically from features like parallelograms and
figures-of-eight in the position-velocity diagrams.  Subsequent
observations and simulations \citep[e.g.][]{BF99,BA99,AB99,CB04,BA05}
have confirmed the relationship between B/PS bulges, complex
kinematics, and bars.

We have obtained the optical long-slit spectrum and SparsePak Integral
Field Unit (IFU) velocity field, as well as optical and near-IR images,
of NGC~2683, an isolated, nearly edge-on Sb galaxy with a B/PS bulge.
NGC~2683 is highly inclined; \citet{Funes02} report an inclination of
78$\degr$, while others suggest a \textit{minimum} inclination of
80$\degr$ \citep[e.g.][]{Barbon75, Broeils94}.  The complex kinematic
and photometric features of the data are suggestive of a bar in the
galaxy.  Although previous authors have observed NGC~2683 and linked
its complex velocity structure and bulge shape with a bar, there has
been little discussion about the bar orientation.

In this paper we present our spectroscopic and photometric
observations as evidence that NGC~2683 contains a bar that is 6$\degr$
on the sky away from the galaxy major axis.  In $\S$ 2 we review the
data for NGC~2683 in the literature.  Our new observations and data
reduction are discussed in $\S$ 3.  We present the spectroscopic data
in $\S$ 4 and the photometric data in $\S$ 5.  We describe in each
section how these data support the hypothesis that a bar is present in
the galaxy, and use the features of the kinematic and photometric
signatures to constrain the orientation of the bar.  The results for
NGC~2683 are compared in $\S$ 6 with the results for M31 by
\citet{AthanassoulaBeaton}.  A comparison to M31 is insightful, as it
has photometric and kinematic characteristics, as well as an
inclination, that are similar to NGC~2683. Finally, a summary is
presented in $\S$ 7.

\section{Previous Observations}
Because of its prominent boxy/peanut-shaped bulge and interesting
kinematics, NGC~2683 has been a well-studied object.  In 1974, de
Vaucouleurs classified the bulge as peanut-shaped. \citet{Ford71}
obtained a long-slit spectrum of the galaxy and noted that outside of
the nucleus, the [\mbox{N\,{\sc ii}}]$\lambda$6584\ emission was fainter than the \Ha\ emission,
whereas inside the nucleus, it was brighter than the \Ha\ emission.
\citet*{Barbon75} presented \textit{B-}band photometry and optical
emission line spectroscopy.  Based on the galaxy image, they found the
bulge to be peanut-shaped and the dust-obscured NW side of the galaxy
to be the near side. The spectroscopic data showed trailing spiral
arms, a [\mbox{N\,{\sc ii}}]/\Ha\ ratio in the outer parts of the galaxy consistent
with what is seen in H{\sc ii} regions ([\mbox{N\,{\sc ii}}]\ emission that is fainter
than \Ha\ emission), and strong evidence for deviations from circular
motions.  \citet{deSouza87} classified the bulge+disk shapes of 72
galaxies and categorized NGC~2683 as a BS-II galaxy: a system in which
the disk and bulge components are not sharply distinct and the bulge
is an elongated ellipsoid that is sometimes rectangular.  An H{\sc i}\
rotation curve of the galaxy was presented by \citet{Casertano91}, and
noncircular motions were noted.  \textit{J-} and \textit{K-}band
photometry, as well as \textit{(J-K)} color, eccentricity, position
angle and $\cos$~4$\theta$ profiles as functions of radius were
presented by \citet{Shaw93}, as were models of the observed luminosity
distribution.  \citet{Rubin93} found the \Ha\ and [\mbox{N\,{\sc ii}}]\ emission to
show multi-valued, figure-of-eight velocity structure in the inner
60$\arcsec$ of the galaxy.  Light from bulge stars or foreground gas
warping out of the galaxy plane and into the line-of-sight were
suggested as possible explanations for the observed complex velocity
structure.  An H{\sc i}\ position-velocity map of NGC~2683 was presented by
\citet*{Broeils94}.

The link between boxy bulges and bars was investigated by
\citet{Merrifield99} using a sample of 10 galaxies, including
NGC~2683.  They found that the emission line kinematics of galaxies
with boxy bulges show complex velocity structure; the
position-velocity diagrams can be described as being X-shaped or
containing features similar to figures-of-eight or parallelograms.
The complex kinematics of NGC~2683 were also observed by
\citet{Pompei99} who noted the presence of a counter-rotating stellar
system and two gas components, one of which was rotating in the same
sense as the stellar component, and the other rotating in the opposite
direction.  A nuclear bar close to the minor axis or a merger event
were two scenarios given to explain the counter-rotation.
Additionally, \citet{Pompei99} observed the bulge isophotes to be boxy
inside 50$\arcsec$.  Optical and NIR photometry also led
\citet{Lutticke00a,Lutticke00b} to classify the bulge of NGC~2683 as
box-shaped.  Stellar and ionized gas velocity curves as well as
velocity dispersions were presented by \citet{VegaBeltran01}.  They,
too, detected kinematically distinct stellar and gas components, but
remarked that they were unable to resolve the fast and slow rotating
components in the gas.  \citet{Funes02} presented a position-velocity
diagram for NGC~2683 derived from \Ha, [\mbox{N\,{\sc ii}}]$\lambda$6584, and [\mbox{O\,{\sc
iii}}]$\lambda$5007 emission.  Due to the low \textit{S/N} of their
data, the figure-of-eight shape in the position-velocity diagram was
faint, but they did observe two spatially distinct gas components.

\textbf{ }

\section{New Observations and Data Reduction}
\subsection{Spectroscopy}
We observed NGC~2683 with the RC Spectrograph on the Kitt Peak
National Observatory (KPNO) 4-meter telescope during the nights of
2007 October 16 and 18. We used the T2KB CCD with the 860 line
mm$^{-1}$ grating in second order, centered near \Ha.  The slit width
was 1.5$\arcsec$, giving $\sim$ 1.0 \AA\ spectral resolution and a
spatial scale of 0.69$\arcsec$ pixel$^{-1}$.  We centered the slit on
a nearby bright star and then offset to the optical center of the
galaxy.  Two 600 s exposures were taken with the slit aligned along
the major axis of the galaxy (position angle $=$ 41.5$\degr$).  A
HeNeAr lamp was observed before and after each science exposure to
provide wavelength calibration.

We also observed NGC~2683 with the SparsePak IFU on the 3.5-meter WIYN
$\footnote{Based on observations obtained at the WIYN Observatory.
The WIYN Observatory is a joint facility of the University of
Wisconsin-Madison, Indiana University, Yale University, and the
National Optical Astronomy Observatory.}$ telescope at KPNO on the
night of 18 February 2009.  SparsePak is a 70$\arcsec$ $\times$
70$\arcsec$ fixed array of 82 5$\arcsec$ diameter fibers.  We used the
STA1 CCD with the 316@63.4 grating in eighth order, centered near
H$\alpha$, giving a 40 km s$^{-1}$ velocity resolution.  The SparsePak
array was aligned with the major axis of the galaxy and five pointings
were used to cover the entire length of the galaxy.  Each exposure was
1200 s, and two exposures were taken at each pointing.  A ThAr lamp
was observed to provide wavelength calibration.

The spectral data were reduced using standard reduction routines in
IRAF$\footnote{IRAF is distributed by the National Optical Astronomy
Observatory, which is operated by the Association of Universities for
Research in Astronomy (AURA), Inc., under agreement with the National
Science Foundation.}$.  The data were bias-subtracted and flattened.
The IRAF task \texttt{dohydra} was used to extract the IFU spectra.
The galaxy spectra were wavelength-calibrated using a wavelength
solution created from the observations of the comparison HeNeAr and
ThAr lamps.  To increase the signal-to-noise and remove cosmic rays,
the two galaxy frames per pointing were combined. In general, the
galactic emission lines in the spectra are much stronger than the
night-sky emission lines, and the night-sky lines bracket, rather than
overlap, the galaxy lines.  Sky subtraction was not performed for
either data set and we subsequently used the night-sky emission lines
as the reference wavelengths \citep{Osterbrock96} by which the
velocities of the galactic emission lines were measured.  The
velocities were measured by fitting Gaussians to both the sky lines
and the five galactic emission lines of interest: \Ha, [\mbox{N\,{\sc ii}}]$\lambda$6548, [\mbox{N\,{\sc ii}}]$\lambda$6584,
[\mbox{S\,{\sc ii}}]$\lambda$6717, [\mbox{S\,{\sc ii}}]$\lambda$6731.  There was less scatter between the measured galactic
emission line velocities when using the night-sky calibration than
either the HeNeAr or ThAr calibrations.

\begin{figure*}[ht]
\epsscale{0.8}
\plotone{f1.eps}
\caption{\textit{Top:} Observed \Ha\ and [\mbox{N\,{\sc ii}}]\ region of the long-slit
position-velocity diagram.  The emission traces two parallelogram
shapes: an outer parallelogram extending to about $\pm$45$\arcsec$,
and a compact, inner parallelogram extending to about
$\pm$5$\arcsec$. \textit{Bottom:} Observed velocities of the \Ha,
[\mbox{N\,{\sc ii}}]$\lambda$6584, [\mbox{S\,{\sc ii}}]$\lambda$6717, and [\mbox{S\,{\sc ii}}]$\lambda$6731\ emission lines.  The error on the velocity at
a given position is equal to the scatter in the measured velocities of
the observed emission lines, $\sim$10 \kms.  At radii $\lesssim$
45$\arcsec$, the velocities are multi-valued.  The measured velocities
occupy the two forbidden quadrants of the PVD; there is red-shifted
emission on the blue-shifted side of the galaxy (upper left quadrant)
and blue-shifted emission on the red-shifted side of the galaxy (lower
right quadrant).}
\end{figure*}

\subsection{Photometry}
The galaxy was also imaged at the KPNO 2.1-meter telescope during the
night of 2007 March 17.  The galaxy was imaged in $B$, $V$, $R$, and
$I$, with total exposure times of 600 s, 600 s, 300 s, and 300 s,
respectively.  Two exposures were taken in each band to correct for
cosmic rays.  The T2KB CCD was used, and the spatial resolution was
0.305$\arcsec$ pixel$^{-1}$.  These data were taken under
non-photometric conditions.  The optical broad-band images were
bias-subtracted, flattened, and combined using standard IRAF
routines. 2MASS \textit{J-} and \textit{K-}band images of NGC~2683 are
also available, and were downloaded from the on-line 2MASS catalog
\citep{2MASS}.

\section{Kinematic Signatures}
\subsection{Long-slit Position Velocity Diagram}
In Figure 1 we show the observed \Ha\ and [\mbox{N\,{\sc
ii}}]$\lambda$$\lambda$6548, 6584 region of the long-slit
position-velocity diagram (PVD) of NGC~2683.  The PVD shows complex
structure out to radii of $\sim$ 45$\arcsec$, almost half of its
entire length.  The \Ha\ and [\mbox{N\,{\sc ii}}]\ emission (as well as the [\mbox{S\,{\sc ii}}]$\lambda$6717\ and
[\mbox{S\,{\sc ii}}]$\lambda$6731\ emission that are not shown) trace a parallelogram shape
extending to $\sim$ 45$\arcsec$ on each side. The edges of this region
are relatively bright compared to the faint interior emission.  At
radii $\lesssim$ 45$\arcsec$, the velocities are double, sometimes
triple, valued.  The maximum velocity reached is comparable to the
velocities reached in the outer parts of the galaxy.  The
parallelogram velocities are seen to populate the ``forbidden''
quadrants of the PVD; there are red-shifted velocities on the
blue-shifted side of the galaxy and vice versa.

There is also a second parallelogram feature seen in the PVD.  In
approximately the inner $\pm$ 5$\arcsec$ there is a steeper and more
compact parallelogram that is brighter in the [\mbox{N\,{\sc ii}}]$\lambda$6584\ emission than \Ha.
The velocities reached in this region do not exceed the velocities in
the more extended parallelogram or those of the outer disk, and there
is a smooth transition between the compact, inner and extended, outer
parallelogram-shaped features.

This complex ``figure-of-eight'' PVD can be explained by the presence
of a bar in the galaxy.  In a series of papers, \citet[][hereafter
BA99]{BA99}, \citet[][hereafter AB99]{AB99}, and \citet[][hereafter
BA05]{BA05} used simulations of edge-on ($i$ = 90$\degr$) galaxies to
demonstrate that features in PVDs like those shown in Figure 1 can be
explained by gas (and stars) moving in the \textit{x$_{1}$} and
\textit{x$_{2}$} families of periodic orbits of barred potentials.
Each of these orbit families produces unique signatures in the PVD
and, by virtue of the orientations of the orbits with respect to the
bar, the appearance of the PVD features can be used to put constraints
on the position angle of the bar.

\defcitealias{BA99}{BA99}
\defcitealias{AB99}{AB99}
\defcitealias{BA05}{BA05}

\begin{figure*}[ht]
\plotone{f2.eps}
\caption{Schematic representation of the shape of the
position-velocity diagram produced by \textit{x$_{1}$} orbits for
different lines of sight.  The major axes of both the bar and
\textit{x$_{1}$} orbits are parallel to the observer's line-of-sight
when the bar is viewed end-on, and are perpendicular to the observer's
line-of-sight when the bar is viewed side-on.}
\end{figure*}

\citetalias{BA99}, \citetalias{AB99}, and \citetalias{BA05} showed
that in edge-on galaxies \textit{x$_{1}$} orbits produce extended
features in the PVD that range from bow tie-shaped to
parallelogram-shaped depending on the orientation of the bar with
respect to the observer's line-of-sight (see Figure 2).  The
\textit{x$_{1}$} orbits are elongated parallel to the major axis of
the bar, so that when the bar is viewed end-on, the line-of-sight is
parallel to the major axis of both the bar and \textit{x$_{1}$}
orbits.  Viewed at this angle, the ``fast'' parts of the elongated
\textit{x$_{1}$} orbits are pointing directly toward or away from the
observer and very high radial velocities at small projected distances
are observed.  This produces a bow tie-shaped feature in the PVD.

When the bar is viewed side-on, the elongated axis of the
\textit{x$_{1}$} orbits is perpendicular to the line-of-sight.  The
``slow'' parts of the \textit{x$_{1}$} orbits are thus pointing
directly toward or away from the observer, and as a result, much lower
radial velocities are observed.  In addition, because the long axis of
the orbits is across the line-of-sight, these observed velocities will
extend to larger projected distances than when the bar (and
\textit{x$_{1}$} orbits) is viewed end-on.  The resulting feature in
the PVD is therefore a relatively thin, diagonal band of velocities
extending to large projected distances.

When the bar in an edge-on galaxy is not viewed directly end-on or
side-on, the PVD will contain a parallelogram-shaped feature with
velocities appearing in the ``forbidden'' quadrants.  This velocity
structure arises because the orbits are elongated and not circular.
The line-of-sight does not fall along either the major axis or minor
axis of the orbits. Thus, if they are viewed at an angle other than
end-on or side-on, the positions along the orbits where the tangential
velocity is greatest (and radial velocity is zero) will be offset from
the center.  This means that the velocities transition from being
blue-shifted to red-shifted, and vice versa, at radii greater than
zero.  This leads to red(blue)-shifted velocities being observed on
the blue(red)-shifted side of the galaxy, thus falling in the
``forbidden'' quadrants.  The closer the bar is to end-on, the sharper
and more peaked the parallelogram is at small projected radii (and the
more it looks like a bow tie).  The interior of these PVDs also tend
to be brighter than the edges.  As the bar angles closer to side-on,
the corners of the parallelogram round out and the parallelogram
becomes thinner as it extends to larger projected distances and rises
to lower radial velocities.

In the observed PVD of NGC~2683, the broad ($\pm$45$\arcsec$)
parallelogram shape corresponds to the \textit{x$_{1}$} feature.  The
simulated PVDs of \citetalias{BA99}, \citetalias{AB99}, and
\citetalias{BA05} can serve only as a \textit{guide} to the bar
orientation in our galaxy because (a) NGC~2683 is not exactly edge-on
($i$ $\gtrsim$ 78$\degr$), and (b) the models are not tailored to
specifically match NGC~2683 (bar strength, pattern speed, mass
distribution, etc).  With that caveat in mind, both the bright
emission along the edges of the \textit{x$_{1}$} parallelogram and a
lack of clear central peaks suggest that the bar is oriented at an
intermediate angle (perhaps between 10$\degr$ and 45$\degr$) on the
sky away from the galaxy major axis and therefore viewed closer to
side-on than end-on.

At the center of the observed PVD of NGC~2683 is the signature of the
\textit{x$_{2}$} orbits: a compact feature that is brighter in [\mbox{N\,{\sc ii}}]\
emission than \Ha\ emission \citep[Figure 1; see also][]{Ford71}.
This relatively large [\mbox{N\,{\sc ii}}]\ emission may be caused by the shocks that
develop at the transition between the \textit{x$_{1}$} and
\textit{x$_{2}$} orbits (Baldwin, Phillips, \& Terlevich 1981; Dopita
\& Sutherland 1996; AB99), as well as the AGN that is known to be at
the center of NGC~2683 \citep{Irwin}.  Without additional diagnostic
emission lines such as [\mbox{O\,{\sc i}}]\ or [\mbox{O\,{\sc iii}}], we cannot say for certain that
it is the AGN rather than shocks that is causing the bright [\mbox{N\,{\sc ii}}]\
emission.  It is not uncommon for barred galaxies to host
AGN. \citet{Philips83} observed a number of relatively face-on SB
galaxies with large [\mbox{N\,{\sc ii}}]\//\Ha\ ratios and confirmed the presence of
active nuclei.  \citet{Knapen00} have also found that galaxies with
active centers are more often barred (79\%) than galaxies without
active nuclei (59\%) \citep[see also][]{Laurikainen04}.

The \textit{x$_{2}$} orbits are elongated perpendicular to the bar
(making them also perpendicular to the \textit{x$_{1}$} orbits) and
are less radially extended than the \textit{x$_{1}$} orbits.  Because
the major axes of the \textit{x$_{2}$} and \textit{x$_{1}$} orbits are
perpendicular, when one orbit family reaches its maximum radial
velocity, the other is at its minimum.  Thus, the radial velocities of
the \textit{x$_{2}$} orbits peak when the bar is viewed side-on and
are lowest when the bar is viewed end-on.

Rather than using the shape of the \textit{x$_{2}$} feature in the
PVD, \citetalias{AB99} advocate the ratio of the maximum observed
radial velocity of the \textit{x$_{2}$} orbits to the maximum velocity
observed in the outer parts of the galaxy ($V_{x2}$/$V_{disk}$) as the
best measure of bar viewing angle.  When the bar is viewed close to
end-on, the maximum velocity of the \textit{x$_{2}$} orbits will be
low compared to the outer disk (the line-of-sight is perpendicular to
the elongated axis of the \textit{x$_{2}$} orbits and the ``slow''
parts of the orbit are observed); $V_{x2}$/$V_{disk}$ $\lesssim$ 1 in
this case.  The ``fast'' parts of the \textit{x$_{2}$} orbits are
observed when the bar is viewed side-on; it therefore follows that
$V_{x2}$/$V_{disk}$ $\gtrsim$ 1 in this orientation.  In NGC~2683, we
find that $V_{x2}$/$V_{disk}$ $\approx$ 0.6. This value of
$V_{x2}$/$V_{disk}$ suggests that the bar is viewed nearer to end-on
than side-on.

\begin{figure*}[ht]
\epsscale{0.8}
\plottwo{f3a_color.eps}{f3b_color.eps}
\caption{Observed \Ha\ SparsePak velocity field of NGC~2683.  The
entire velocity field is shown in the left panel while a zoomed-in
version with the same color/velocity scale is shown on the right.  The
galaxy major and minor kinematic axes are clearly skewed from
perpendicular.  Diamonds in the left panel indicate fibers containing
multi-valued \Ha\ velocities.  In the full velocity field, isovelocity
contours are spaced in 50 \kms\ intervals. To highlight the twist of
the minor axis, the isovelocity contours are spaced in 25 \kms\
intervals in the right panel.  In both panels, a 45$\arcsec$ line
indicates the radius of the parallelogram in the long-slit PVD in
Figure 1.  North is up and East is to the left.  (A color version of
this figure is available in the online journal.)}
\end{figure*}

\begin{figure}
\epsscale{0.8}
\plotone{f4.eps}
\caption{Variation in position angle (PA) across the velocity field.}
\end{figure}

This result contrasts with the other indicators we have, both
photometric and kinematic, that suggest a more side-on orientation.
The $V_{x2}/V_{disk}$ indicator advocated by \citetalias{AB99} appears
to be one of the more promising indicators of bar orientation.
However, it must depend on the details of the disk mass distribution.
Some galaxies have a distinct peak in their rotation curves before
declining to a flat level while others rise continuously.
Consequently, $V_{disk}$ of this particular galaxy may not correspond
well to that of the AB99 model.  More generally, it may be an
indication that we still do not have a complete understanding of the
complex kinematics in NGC 2683.

\subsection{SparsePak IFU Velocity Field}
In Figure 3 we show the observed \Ha\ SparsePak velocity field of
NGC~2683.  The major and minor kinematic axes are not perpendicular
and there is an \textsf{S}-shaped twist to the minor axis.  These are
characteristic of velocity fields with oval distortions or bars
\citep[e.g.][]{Bosma81}.  Similar \textsf{S}-shaped velocity fields
can be seen in observations of more face-on barred galaxies such as
NGC~6300 \citep[\textit{i}=52$\degr$,][]{Buta87} and NGC~5383
\citep[\textit{i}=50$\degr$,][]{Peterson78}.  As in the long-slit
spectrum, double and triple-valued \Ha\ velocities (as well as [\mbox{N\,{\sc ii}}]\
and [\mbox{S\,{\sc ii}}]\ velocities) are detected in some of the IFU fibers.  The
multi-valued velocities are more striking in the long-slit spectrum
than in the velocity field.

Analysis of the velocity field with the tilted-ring fitting program
\texttt{ROTCUR} \citep{Teuben,Begeman} shows that the position angle
(PA) varies by $\sim$ 25$\degr$ across the velocity field, with the
largest deviation from the PA of the outer galaxy occuring at $\sim$
45$\arcsec$, the length of the \textit{x$_{1}$} parallelogram in the
long-slit PVD (see Figure 4).  If we assume that there should be a
steep velocity gradient (i.e. the isovelocity contours bunch up)
across the bar and that the central minor axis velocity contours run
along the length of the bar, then the center of the velocity field
qualitatively indicates (visually) that the bar is angled not far
(perhaps $\sim$ 15$\degr$) from the galaxy major axis on the sky,
running from northeast to southwest.  This is supported by the change
in PA determined by \texttt{ROTCUR}; the bar is a maximum of 25$\degr$
from the major axis.  This angle implies that the bar is viewed more
side-on than end-on, in agreement with the constraints from the
parallelogram in the long-slit PVD produced by \textit{x$_{1}$}
orbits.

\begin{figure}
\plotone{f5_color.eps}
\caption{\Ha\ image of NGC~2683 created from the SparsePak IFU data
with velocity contours overlaid.  The brightest and most concentrated
\Ha\ emission coincides with the ends of the bar. (A color version of
this figure is available in the online journal.)}
\end{figure}

In Figure 5 we show the \Ha\ image of NGC~2683 constructed from the
IFU data with isovelocity contours from the velocity field
overplotted.  We find that while there is diffuse emission throughout
the galaxy, the brightest concentrations of emission coincide with the
bends in the \textsf{S}-shape (the ends of the bar).  The NE (upper)
end of the bar is slightly brighter than the SW (lower) end and there
is no strong emission along the length of the bar.

\begin{figure*}[ht]
\plotone{f6.eps}
\caption{Contour plots of the bulge region of NGC~2683, ordered by
wavelength from longest to shortest.  For comparison, isophotes are at
the same relative levels for each frame.  The bulge of NGC~2683 is
boxy: the isophotes are parallel to the major axis and do not show a
pinch along the minor axis.  The arrows indicate $\pm$45$\arcsec$, the
length of the \textit{x$_{1}$} extended parallelogram feature in the
position-velocity diagram. North is towards the upper left corner,
East to the lower left corner.  In the optical images there is
significant dust on the northwest side of the galaxy.}
\end{figure*}

\begin{figure*}[ht]
\plotone{f7_color.eps}
\caption{$K$-band surface brightness measured along
5$\arcsec$$\times$140$\arcsec$ slits positioned parallel to the major
axis.  There is an enhancement in the light profile in the N$-$S
direction, suggesting the presence of a bar at a mild angle with
respect to the major axis.  (A color version of this figure is
available in the online journal.)}
\end{figure*}

\begin{figure*}[ht]
\plotone{f8_color.eps}
\caption{$K$-band surface brightness measured along
5$\arcsec$$\times$140$\arcsec$ slits oriented at different position
angles.  There is an enhancement in the light above the level of the
major axis along the +84$\degr$ slit. (A color version of this figure
is available in the online journal.)}
\end{figure*}

Strong \Ha\ emission at the bar ends and relatively weak emission
along its length has also been seen in the relatively face-on galaxies
NGC~1530 \citep{Zurita04}, NGC~5383 \citep{Sheth00}, and NGC~6300 and
NGC~1433 \citep{Buta01}.  Similar to what is seen in NGC~2683,
\citet{Zurita04} found that the brightest \Ha\ emission in NGC~1530
coincides with the regions of lowest velocity gradient perpendicular
to the bar (regions were the isovelocity contours are not bunched up).

\section{Photometric Signatures}
Boxy/peanut-shaped bulges in edge-on galaxies have been shown to
correlate with the presence of a bar
\citep[e.g.][]{Combes81,Combes90,BF99}.  It is thought that these
bulge shapes are a result of a bar that has buckled and thickened due
to vertical instabilities \citep[e.g.][]{Combes90}.  Depending on the
bar strength and the orientation of the bar with respect to the
line-of-sight, the bulge may look round (the bar is viewed end-on),
boxy (the bar is viewed at an intermediate angle), or peanut-shaped
(the bar is viewed side-on) \citep[e.g.][]{CB04,BA05}.

In Figure 6, we show contour plots of the bulge region of the
\textit{B, V, R, I, J,} and \textit{K} images of NGC~2683.
\citet{Lutticke00a} define peanut-shaped bulges as having isophotes
that pinch inwards along the minor axis on both sides of the galaxy
major axis, and boxy bulges as having isophotes that remain parallel
to the major axis.  Applying these definitions to Figure 6, it is
clear that the bulge of NGC~2683 is boxy.  This is most easily seen in
the \textit{K-}band image of the galaxy where the obscuration from
dust is at a minimum.  On both sides of the major axis, the isophotes
are parallel and remain flat as they cross the minor axis.  Even as
the galaxy is viewed through progressively bluer filters and the
effects of dust on the northwest side of the galaxy become more
severe, the isophotes on the less-obscured side of the galaxy do not
show signs of pinching along the minor axis that are characteristic of
a peanut-shaped bulge.  The classification of the bulge shape is
independent of wavelength \citep[see also][]{Lutticke00a}.

Two of us (RKD and SSM) made independent ellipse fits with different
fitting routines (\texttt{ellipse} in IRAF and ARCHANGEL
\citep{Schombert07}).  While these ellipse fits are broadly
consistent, they give slightly different results for the
boxiness/pointiness parameter $b_4$.  Thorough examination of the
images and residuals of the fits shows that this can be traced to
isophotes that are both boxy and pointy.  This is most apparent in the
$J$ and $K$-band images in Figure 6.  Along the minor axis of the
bulge, the isophotes are flatter than a pure ellipse, indicating a
boxy bulge.  Along the major axis, the isophotes become pointy, so
that the entire isophote shape is both pointy and boxy.  We interpret
the pointiness to be due to the bar itself extending out into the
disk; see Figures 7 and 8.  The length of the boxy region,
corresponding to the thick inner part of the bar, is about
$\pm$23$\arcsec$ in the $K$-band.  The pointy isophotes extend to
roughly $\pm$54$\arcsec$ and correspond to the thin outer part of the
bar.

It is not uncommon for barred galaxies to have features that connect
to the ends of the bar such as ansae (``handles'') or rings
\citep[e.g.][]{Martinez07,Buta01}.  Because NGC~2683 is so highly
inclined, it is difficult to determine if these structures are also
present.

The boxy bulge shape of NGC~2683 indicates that the bar is not so
strong as to induce the pinched isophotes of a peanut and that the bar
is positioned at an intermediate viewing angle.  To determine the
general orientation of the bar, we plot in Figure 7 the observed
surface brightness measured along 5$\arcsec \times 140\arcsec$ slits
positioned parallel to the galaxy major axis.  We find that the
surface brightness profiles are asymmetric around the minor axis.
Specifically, in the slices above the major axis (the northwest
slices), there is a ``hump'' in the profile indicating excess light to
the left (north) of the galaxy minor axis and in the slices below the
major axis (the southeast slices), the ehancement in the light is to
the right (south) of the galaxy minor axis.  We interpret this
asymmetry in the light profiles as an indication that the bar is
aligned in a N-S direction, consistent with the orientation derived
from the SparsePak velocity field.  Consistent with the approximate
lengths of the boxy and pointy isophote regions, the excess light
begins around $\pm$ 20$\arcsec$$-$25$\arcsec$, peaks around
45$\arcsec$, and drops off between 50$\arcsec$ and 60$\arcsec$.  Based
on the photometry, we estimate the length of the bar to be $\sim$
55$\arcsec$.

In Figure 8, we confirm this N-S orientation by plotting surface
brightness profiles along slits oriented at different position angles.
We define 0$\degr$ as the minor axis and 90$\degr$ as the major axis.
There is an enhancement in the light along the slits rotated
counterclockwise (positive angles) from the galaxy minor axis.  The
humps in the profile reach a maximum above the level of the light
along the major axis at a position angle of +84$\degr$ from the minor
axis (or, equivalently, 6$\degr$ from the major axis).  This angle
coincides with the asymmetric, elongated isophotes seen most clearly
in the \textit{K}-band image.  In agreement with the kinematic data,
the humps in the surface brightness profiles extend to $\sim$
45$\arcsec$$-$50$\arcsec$.  Based on the photometric signatures, we
conclude that the bar is oriented $\sim$ 6$\degr$ from the galaxy
major axis on the sky, running from north to south.

\section{Comparison to M31}

It is instructive to compare the observations of NGC~2683 with
observations of other galaxies having similar inclinations ($i$
$\gtrsim$ 78$\degr$) and bar orientations ($\varphi_{phot}$ $\sim$
6$\degr$, $\varphi_{kin}$ $\sim$ 10$\degr$ $-$ 25$\degr$).  One such
example is M31.  \citet{AthanassoulaBeaton} recently used kinematic
and photometric data to argue for the presence of a bar in the galaxy
\citep[see also][]{Lindblad56,Stark94}.  They find that M31 ($i$ =
77$\degr$) contains a medium-strength bar that is on the order of
$\varphi$ $\sim$ 7$\degr$ on the sky away from the galaxy major axis
(PA$_{disk}$ = 38$\degr$, PA$_{bar}$ $\sim$ 45$\degr$).

The observed PVDs of NGC~2683 and M31 show similar complex kinematics.
The observed H{\sc i}\ PVD of M31 (their Figure 9) is more bow tie-shaped
than the long-slit \Ha\ PVD of NGC~2683.  As discussed in $\S$4.1, in
edge-on galaxies, the PVD appears more bow tie-like as the bar angles
farther away from the major axis.  Taken at face value, the observed
PVDs would therefore suggest that the bar in M31 is seen more end-on
than the bar in NGC~2683.  The simulations described in $\S$4.1,
however, are \textit{ideal} and in reality, the observed kinematics
are highly influenced by physical conditions in the galaxy, such as
the presence of dust or the density of the gas.  Without further
modeling, we cannot say whether the differences in the PVDs originate
from these physical differences or from different bar orientations or some
combination of both.

The isophotes seen in the images of M31 (their Figure 2) are similar
to the isophotes of the $K$-band image of NGC~2683; the isophotes are
boxy rather than pinched in a peanut shape.  Based on isophote shapes
in $N$-body simulations tailored to the orientation of M31, they argue
that the absence of peanut-shaped isophotes means the bar is not a
strong bar \citep[see also][]{Lutticke00b,Athan02}.  By this standard, the
bar in NGC~2683 is not strong.  Similar to NGC~2683, M31 has
elongated isophotes that are angled away from the major axis and
indicate the PA of the bar.  Additionally, they find similar
asymmetric enhancements in the light profile in slits parallel to the
major axis (their Figures 5 and 6).

It is encouraging that there are so many similarities between the
observations of these two galaxies.  Bars in nearly edge-on galaxies
cannot be unambiguously visually confirmed as such, but strong
evidence for their presence comes from the comparison of observed
kinematic and photometric signatures to the results of numerical
simulations.  That simulations are consistent with numerous observed
galaxies increases our confidence in using bars to explain boxy/PS
bulges and figure-of-eight PVDs.

\section{Summary}
We have presented new spectroscopic and photometric observations of
the near edge-on spiral galaxy NGC~2683.  The long-slit PVD displays a
complex, figure-of-eight distribution of velocities, and the SparsePak
IFU velocity field shows characteristics of an oval distortion.
Isophotes in optical and near-IR images of the galaxy are boxy.  We
find an asymmetric enhancement in the light profile in slits placed
parallel to, and offset from, the major axis.

We argue that the kinematic and photometric signatures are evidence
that the galaxy hosts a bar.  We find that the data support a bar that
is viewed closer to side-on than end-on (i.e. closer to the galaxy
major axis than the minor axis). Based on kinematic and photometric
constraints, we determine that the bar is 6$\degr$ away from the major
axis on the sky.  Our results are consistent with previous
observations of NGC~2683 in the literature, as well as recent results
for M31, and add to the growing body of evidence linking B/PS bulges
and complex PVDs with bars in edge-on galaxies.

\acknowledgements
We would like to thank the referee for helpful comments.  R.K.D. is
supported by an NSF Astronomy \& Astrophysics Postdoctoral Fellowship
under award AST 07-02496.  S.S.M. and M.J.Z. are supported by NSF
grant AST 05-05956.  We would like to thank L.~Athanassoula for
helpful conversations.  R.K.D. would also like to thank Brian
Marsteller and Misty Bentz for useful discussions.  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.