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\slugcomment{Submitted to the Astronomical Journal}
\lefthead{Layden \& Sarajedini}
\righthead{M54 and the Sagittarius Dwarf Galaxy}
 
\begin{document}

\title{Photometry of the Globular Cluster M54 and the 
Sagittarius Dwarf Galaxy: the Age-Metallicity Relation}
\author{Andrew C. Layden\altaffilmark{1,2}}
\affil{Physics \& Astronomy Dept., Bowling Green State University, 
Bowling Green, OH 43403}
\authoremail{layden@kracken.bgsu.edu}
%\affil{Dept. of Astronomy, University of Michigan, Ann Arbor, MI 48109-1090}
%\authoremail{layden@astro.lsa.umich.edu}
\author{Ata Sarajedini}
\affil{Astronomy Dept., Wesleyan University, Middletown, CT 06459-0123}
\authoremail{ata@astro.wesleyan.edu}

\slugcomment{To appear in the Astronomical Journal, April 2000}

\altaffiltext{1}{Hubble Fellow.}

\altaffiltext{2}{Observations obtained at the Cerro Tololo
Inter-American Observatory (CTIO), one of the National Optical
Astronomy Observatories, which are operated by the Association of
Universities for Research in Astronomy, Inc., under cooperative
agreement with the National Science Foundation.}

\begin{abstract}

We present deep $VI$-band photometry of the globular cluster M54, a
nearby field in the Sagittarius dwarf galaxy, and a control field.
The color-magnitude diagrams reach well below the oldest main sequence
turn-offs, thus enabling an analysis of the galaxy's age-metallicity
relation with unprecedented clarity.  We also study the variable stars
in the direction of M54.  From 67 RR Lyrae variables, we confirm and
improve on our previous estimates of the cluster horizontal branch
magnitude, foreground reddening, and horizontal branch morphology.
These values are used in determining the ages of M54 and the
Sagittarius field populations.  We confirm our previous result that
M54 is the same age as Galactic globular clusters of similar
metallicity.  We also derive ages on a self-consistent scale for the
other three globular clusters in Sagittarius.  We find strong evidence
for multiple episodes of star formation (or continuous star formation
with a variable rate) in the field of Sagittarius.  We characterize
the principal episodes with the ages 11, 5, and 0.5 through 3 Gyr, and with
$[Fe/H]$ values of --1.3, --0.7, and --0.4, respectively.  On this
scale, M54 has an age of 15 Gyr.  Surprisingly, the age-metallicity
relation we have derived for the galaxy as a whole is described quite
well by a closed box chemical evolution model.  We also find that the
populations associated with the Sgr field are clumped spatially around
M54, and we consider several explanations for this phenomenon.  We again
speculate that Sagittarius is a nucleated dwarf elliptical galaxy with
M54 as its nucleus.

\end{abstract}


\keywords{galaxies: stellar content --- galaxies: star clusters ---
galaxies: individual (Sagittarius) --- globular clusters: individual
(M54) --- stars: variables: general}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}

The study of the star formation history of dwarf galaxies is
interesting for several reasons.  First, dwarf galaxies are among the
most frequently-occurring galaxies in the universe.  Second, they may
account for much of the blue galaxy population seen at intermediate
redshift (e.g., \markcite{dri95}Driver et al. 1995).  Finally,
theories of structure formation in the cold dark matter paradigm
involve building luminous galaxies from numerous smaller fragments;
observational evidence suggests that the dwarf spheroidal galaxies we
observe today may represent such fragments which were not assimilated
into larger galaxies (e.g., \markcite{sz78}Searle \& Zinn 1978).
%these may be the predecessors of the dwarf spheroidal and
%dwarf irregular galaxies we observe today (e.g., Searle \& Zinn 1978).

% Cohen, J.G., Blandford, R. Hogg, D.W., Pahre, M.A., \& Shopbell,
% P.L. 1999, ApJ, 512, 30

The study of dwarf galaxies is hampered by their large distances.  In
the early 1990's, only six dwarf spheroidals were known to lie within
150 kpc of the Galaxy, the nearest being Ursa Minor at 66 kpc
(\markcite{mm98}Mateo 1998).
%, and the nearest dwarf irregular is over 400 kpc away (Phoenix;
%excluding LMC \& SMC).
Color-magnitude diagram studies of their stellar populations were
largely limited to the horizontal branch and brighter before the
repaired {\em Hubble Space Telescope} was able to reach their main
sequence turn-off levels.

It was therefore very exciting when \markcite{igi94}Ibata et
al. (1994) discovered a gas-poor galaxy lying behind the bulge of the
Milky Way at a distance of only $\sim$25 kpc.  In addition to mapping
out the field population of this galaxy, now known as the Sagittarius
Dwarf Galaxy (hereafter ``Sgr''), they suggested that four
previously-known globular clusters (M54, Arp 2, Ter 7, \& Ter 8) are
members of this galaxy.  Sgr thus provides an unprecedented
opportunity to study the star formation history of a gas-poor dwarf.
The main complication is confusion in the color-magnitude diagrams
from the vast number of foreground bulge stars which served to veil
our nearest neighbor for so long.

%DaCosta & Armandroff => new Vrad, improve membership probability?
%Mateo et al. (IGI2 & others?) => maps showlarger extent of Sgr, cover clusters
%
%MUSKKK => msto CCD photom of Sgr, ages of field
%Fahlman => deeper, ages of field
%Buonanno etc +> msto photom of other GCs, ages

\markcite{igi97}Ibata et al. (1997) reviewed much of the early work on
Sgr.  Several relevant points include the radial velocity study of
\markcite{dca95}Da Costa \& Armandroff (1995) which confirmed the
membership of the globular cluster M54 (NGC 6715), and the first CCD
color-magnitude diagram (CMD) of this cluster by
\markcite{sl95}Sarajedini \& Layden (1995, hereafter referred to as
SL95).  They found M54 to be metal poor ($[Fe/H] = -1.79$ dex) and to
have a blue horizontal branch typical of old Galactic globular
clusters.  The other globular clusters have been studied by
\markcite{rb95a}Buonanno et al. (1995a, Arp 2),
\markcite{rb95b}Buonanno et al. (1995b, Ter 7), and
\markcite{mont98}Montegriffo et al. (1998, Ter 8).

\markcite{muskkk}Mateo et al. (1995a, hereafter referred to as MUSKKK)
were the first to obtain main sequence turn-off photometry of the Sgr
field population.  They characterized it with a metallicity of $[Fe/H]
= -1.1$ dex and an age of $\sim$10 Gyr, and also noted the presence of
a small population of younger-aged stars.  \markcite{mm96}Mateo et
al. (1996) and \markcite{fah96}Fahlman et al. (1996) found similar
ages and metallicities for the dominant Sgr field population at a
location far distant from the dense, central regions of Sgr.
\markcite{mar98}Marconi et al. (1998) found their Sgr field photometry
to closely resemble the CMD of the young globular Ter 7.
\markcite{ls97}Layden \& Sarajedini (1997) found Sgr to be at least 3
Gyr younger than typical Galactic globular clusters.

More recently, \markcite{bel99a}Bellazzini et al. (1999a,
\markcite{bel99b}1999b) have obtained deeper photometry of the Sgr
field populations at three locations in the galaxy.  They found the
bulk of the Sgr field stars to have an age $\sim$4 Gyr younger than
those of the Galactic globular clusters.  They also suggested the
young population discovered by \markcite{muskkk}MUSKKK has an age of
1--4 Gyr and $[Fe/H] \approx -0.7$ dex.  They raised the possibility
that the dominant Sgr main sequence turn-off (MSTO) region, previously
considered to be a single population, may in fact be composed of two
or more populations with different ages and metallicities.  However,
the limited photometric depth and large errors near the MSTO in their
survey prevented them from
%making any strong conclusions on this subject, or 
quantifying the ages of these populations.  They discussed their
results in the context of the global star formation history of Sgr.

The globular cluster M54 must play an important role in the star
formation history of Sgr.  It lies in the highest density region of
Sgr (\markcite{igi97}Ibata et al. 1997), it is one of the most
luminous globular clusters known (\markcite{sl95}SL95), and as we
shall show, it marks one of the earliest epochs of star formation in
Sgr.  \markcite{sl95}SL95 suggested that M54 may be akin to the
nuclear star clusters seen in nucleated dwarf elliptical galaxies.

\markcite{mar98}Marconi et al. (1998) presented photometry of M54
which just reached the MSTO region.  From it, they estimated that the
age of M54 is similar to that of the Galactic globular clusters.
Deeper photometry by \markcite{ls97}Layden \& Sarajedini (1997,
hereafter referred to as LS97) provided more convincing evidence that
the age of M54 is typical of Galactic globular clusters with the same
metallicity.



In this paper, we discuss in detail our deep photometric observations
of M54 which were presented in preliminary form by
\markcite{ls97}LS97.  We also present observations of a nearby field
in Sgr, and of a control field selected to represent the contaminating
foreground populations.  In \S 2 we describe our observations, and in
\S 3 we detail the photometric reduction techniques.  We describe the
resulting color-magnitude diagrams in \S 4 and perform statistical
subtractions of the contaminating populations.  We digress in \S 5 to
study the variable stars in our M54 field, and derive from them in \S
6 several important cluster parameters, including the mean apparent
magnitude of the cluster RR Lyrae stars, the foreground reddening, and
the horizontal branch morphology of M54.  In \S 7 we estimate the ages
of M54 and the other globulars in Sgr.  In \S 8 we present clear
evidence for temporally-extended star formation in the Sgr field, and
characterize the ages of these populations.  We consider the
age-metallicity relation of the Sgr galaxy, including its globular
clusters, in \S 9, while in \S 10 we demonstrate that the Sgr field
populations are clumped around M54.
%reconsider whether M54 is the nucleus of Sgr.
We summarize our results and consider their larger meaning in \S 11.

 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Observations}

Observations of the bright sequences ($M_V \lesssim +2.0$) of M54 were
obtained with the 0.9-m telescope at CTIO, as reported in
\markcite{sl95}SL95.  These included photometry obtained with the
Tek\#1 1024$\times$1024 pixel CCD (Field of View (FOV) $\approx 6.8$
arcmin) on a field roughly centered on M54, and on another field 12
arcmin North of M54 (hereafter referred to as the Sgr field).
Separate datasets were presented for these fields from observations
made with the Tek\#3 2048$\times$2048 pixel CCD (FOV $\approx 13.5$
arcmin).  The quality of the latter sets was compromised by small,
spatially-dependent variations in throughput which varied with time.
We hereafter disregard the \markcite{sl95}SL95 Tek 2048 photometry and
when referring to the \markcite{sl95}SL95 work, and consider only the
Tek 1024 data.

In an effort to obtain photometry to the MSTO, we secured deep images
of M54 using the CTIO 4.0-m telescope at prime focus during the nights
of 1995 June 28 and 29.  We employed the Tek\#4 2048$\times$2048 pixel
CCD, which provided a scale of 0.43 arcsec pixel$^{-1}$.  The filters
were matched to the CCD response to closely approximate the standard
Johnson $V$ and Kron-Cousins $I$ passbands.  

Each ``pointing'' consisted of one 600 sec $V$ exposure and two 300
sec $I$ exposures.  Nine such pointings, centered on M54, were
obtained during the course of the run.  The fwhm seeing ranged from
1.0 to 1.5 arcsec (median = 1.2 arcsec).  The equatorial and galactic
coordinates of the field center are shown in Table 1.
%The coordinates of the on-cluster field center are $l^{\rm II} =
%5.61^{\circ}$, $b^{\rm II} = -14.09^{\circ}$, or $\alpha_{2000} = 18^{\rm
%h} 55^{\rm m} 03^{\rm s}$, $\delta_{2000} = -30^{\circ} 28^{\prime}
%45^{\prime\prime}$.  
Eight pointings of the Sgr field
%(centered 15 arcmin North and 3 arcmin East of M54) 
were taken, with seeing ranging from 1.2 to 2.1 arcsec (median = 1.4 arcsec).
%The coordinates of the Sgr field center are $l^{\rm II} =
%5.86^{\circ}$, $b^{\rm II} = -14.03^{\circ}$, or $\alpha_{2000} = 18^{\rm
%h} 55^{\rm m} 17^{\rm s}$, $\delta_{2000} = -30^{\circ} 13^{\prime}
%55^{\prime\prime}$.  
Four pointings near the \markcite{muskkk}MUSKKK bulge control field
were also obtained, with seeing ranging from 1.1 to 1.3 arcsec (median
= 1.1 arcsec).
%The coordinates of our control field center are $l^{\rm II} =
%-6.30^{\circ}$, $b^{\rm II} = -14.35^{\circ}$, or $\alpha_{2000} = 18^{\rm
%h} 33^{\rm m} 31^{\rm s}$, $\delta_{2000} = -41^{\circ} 10^{\prime}
%38^{\prime\prime}$.

%MUSKKK FIELDS
% SGR   RA=19:06:11  DEC=-30:25:22  l= 6.6 b=-16.3  (galactic => +6.63 -16.27)
% CTRL     18:33:26      -40:57:57    -5.8   -14.8  (galactic => -6.11 -14.25)
%
% OUR FIELDS
% M54  l=+5.61  b=-14.09
% SGR    +5.86    -14.03
% CNTRL  -6.30    -14.35

Short exposures of M54 were obtained in order to ensure a good
photometric transformation between the \markcite{sl95}SL95 and 4.0-m
data, and to provide further information on the bright variable stars
in the cluster.  Five paired frames in $V$ and $I$ were obtained
during the run, along with six additional, unpaird $V$ frames.
Exposure times were typically 30 sec in $V$ and 25 sec in $I$.

%Finally, observations of four bright, well-studied clusters were
%obtained in order to provide $VI$ photometry at the MSTO for age
%comparisons with M54 and Sgr.  The clusters, their metallicities,
%ages, and references to previous studies are listed in Table 1.  The
%numbers and exposure times of the frames obtained herein are also
%listed in this table.
%
%=======================================================================
%Table 1: Comparison Clusters
%
%NGC	[Fe/H]	Source	Age	Source	Previous_Photom	T_exp,V T_exp,I
%
%104	-0.76	Harris	.	.	.		.	.
%4833	-1.79	Harris	.	.	.		.	.
%6397	-1.91	Harris	.	.	.		.	.
%6752	-1.61	Harris	.	.	.		.	.
%=======================================================================

The frames were overscan corrected, bias subtracted, and trimmed using
the IRAF\footnote{IRAF is written and distributed by NOAO.} task
QUADPROC.  Twilight sky flats were used to flat field the data.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Photometry}

All the photometry described in this paper has been computed using the
DoPHOT photometry program (\markcite{dophot}Schechter et al. 1993).  A version
employing a 2-dimensionally variable point spread function enabled us
to get good results despite the presence of significant radial
variation in the width, shape, and orientation of the stellar images.
Small variations not accounted for by the variable profile fitting
function appeared as a 2-dimensionally variable zero-point offset in
the photometry, and must be corrected for, either through a
2-dimensionally variable aperture correction or through other means as
described below.

 
%-------------------------------------------------------------------
\subsection{General Procedure}

The high stellar densities in the Sgr field, and particularly in the
M54 field, present a distinct observational challenge.  Preliminary
tests showed that while PSF fitting routines employing a small fitting
radius ($\sim$fwhm) produced high quality instrumental magnitudes, the
crowding of faint neighbor stars prevented us from obtaining accurate
aperture corrections in our deep images.

Fortunately, in the M54 and Sgr fields, there exists $VI$ CCD
photometry of the bright stellar sequences.  Crowding of the bright
aperture correction stars on these frames is less severe, so accurate
aperture corrections were obtained.

We therefore transformed the instrumental DoPHOT photometry ($v$, $i$)
directly onto the existing photometry ($V$, $I$) using the large
number of stars in common.  This transformation took the form
%\[v - V = c_0 + c_1(V-I) + c_2X + c_3X^2 + c_4X^3 +
%c_5XY + c_6X^2Y + c_7XY^2 + c_8Y^3 + c_9Y^2 + c_{10}Y\]
\[v - V = c_0 + c_1(V-I) + c_2X + c_3X^2 + c_4X^3 + c_5XY + \]
\[c_6X^2Y + c_7XY^2 + c_8Y^3 + c_9Y^2 + c_{10}Y\]
and
\[i - I = d_0 + d_1(V-I) + d_2X + d_3X^2 + d_4X^3 + d_5XY + \]
\[d_6X^2Y + d_7XY^2 + d_8Y^3 + d_9Y^2 + d_{10}Y,\]
where $X$ and $Y$ are the two spatial coordinates, and $c_i$ and $d_i$
are the $i$-th fitted coefficients.
%included a zero-point offset, a linear
%term in $V-I$, and a cubic surface fit ($c_1X + c_2X^2 + c_3X^3 +
%c_4XY + c_5X^2Y + c_6XY^2 + c_7Y^3 + c_8Y^2 + c_9Y$, where $X$ and $Y$
%are the two spatial coordinates and $c_i$ is the $i$-th fitted
%coefficient).
The details of the fits are described in the following
subsections.

Note that the M54 photometry of \markcite{mar98}Marconi et al. (1998)
is in good agreement with that of \markcite{sl95}SL95.  This
strengthens our confidence in the results of the photometric
transformation procedure.

%A detraction of this approach is that we cannot utilize the full area
%of the $2048^2$ pixel CCD employed at the 4.0-m telescope; we are
%limited to the area of the existing photometry.  In practice, we found
%the spatial component of the transformations to vary slowly and
%smoothly, so we allowed ourselves to extrapolate the functions
%%$\sim$100 pixels beyond the limits of the existing photometry fields
%beyond the limits of the existing photometry fields
%in every direction.


%-------------------------------------------------------------------
\subsection{The Bright Sequences of M54 Revisited}

We obtained five $VI$ frame pairs with short exposure times centered
on M54.  Here, we use these images to extend slightly the areal
coverage of the bright-sequence photometry relative to that presented
in \markcite{sl95}SL95.  Not only does this provide us with more stars in the bright
sequences, it is an important link for photometrically transforming
the deeper MSTO photometry of M54, since much of the inner region
sampled by \markcite{sl95}SL95 is saturated or severely crowded in the deeper frames.

For each frame, we trimmed off the edges to remove the worst effects
of the image variations, leaving a 11.1 arcmin FOV centered on the
cluster.  We then computed $v$ and $i$ instrumental stellar magnitudes
using the DoPHOT program.  For each frame pair, we matched the $v$ and
$i$ photometry of the spatially coincident stellar images, thus
creating five lists of instrumental photometry.  Next, we performed
($X$, $Y$) coordinate transformations to put the coordinate frame of
each photometry list onto a common system.  We then used the equations
shown in \S 3.1 to transform the instrumental magnitudes onto the
standard $VI$ magnitude system of \markcite{sl95}SL95 (their Table 1).  For each
frame, the RMS scatter in the transformation was typically $<$0.030
mag, and the transformation always employed $>$800 stars covering $0.0
\lesssim V-I \lesssim 2.8$ mag and spatially spanning the image.  Since
the area of the 4.0-m frames is larger than that of the \markcite{sl95}SL95 Tek 1024
data, we had to extrapolate the photometric transformation into the
regions not covered by the \markcite{sl95}SL95 photometry.  However, the ($X$, $Y$)
variations were small and well-behaved, so the extrapolation should
not incur large errors.

We then compiled the photometry of spatially coincident stars from the
five photometry lists, keeping only those stars which were detected in
at least three of the five lists.  For each star, we computed the
weighted mean $V$ and $I$ magnitudes (the individual magnitudes were
weighted by their DoPHOT errors).  We also computed the standard error
of the mean of the 3--5 $V$ or $I$ observations, which we treat as the
error in the $V$ or $I$ magnitude.  Table 2 presents, star by star,
our final bright-sequence photometry set.  The columns contain (1) an
identification number, (2) the $X$ and (3) $Y$ coordinates in the
system defined by our Figure 1, (4) the weighted $V$ magnitude and (5)
its error, (6) the weighted $I$ magnitude and (7) its error, and (8)
the number of frames on which the star was detected.  The table
contains data for 20,346 stars.

Figure 1 presents a $V$-band image of the cluster.  The $XY$
coordinate system is the same as presented in Table 2, and several
stars from this table are identified on the image.  We used the IRAF
task ELLIPSE to find the center of M54 by fitting ellipses to a
heavily boxcar-smoothed, short exposure image of M54.  The mean center
in the coordinate system shown in Figure 1 is ($X$, $Y$) = (726, 806)
pix.  This agrees with a visual estimate of the center to within 2
pixels.

% FIGURE 1:  A 30 second $V$-band exposure of M54.  The field is 
%11.1 arcmin on a side.  North is to the right and East is up.  The
%coordinate system employed in Tables 2 and 3 correspond to this
%figure, with ($X$, $Y$) = (1, 1) at the lower-left and (1551, 1551) at
%the upper-right.  The cluster center is at ($X$, $Y$) = (726, 806)
%pixels.  Three stars from Table 2 (shallow photometry) are marked with
%their ID numbers (37, 46, 49), and three stars from Table 3 (deep
%photometry) are also marked (15, 51, 70).


%=======
%Table 2: M54 Bright-Sequence Photometry
%========

Figure 2 shows differences between our final bright-sequence
photometry set and the \markcite{sl95}SL95 photometry of their Table 1.
Figure 1a indicates the presence of a small $V$ magnitude
difference as a function of magnitude.
%, at the level of $\sim$0.01 mag mag$^{-1}$.  This could be due 
%to a slight non-linearity in one or the other CCD used.  
Since it is not clear whether the fault lies with the \markcite{sl95}SL95 or the
present photometry, and since the effect is small and not well
defined, we have not corrected for it in our 4.0-m photometry.  Note
that the stars with $V < 17$ mag are not used directly in any of the
results derived in this paper -- both the RR Lyrae variables and the
stars used to calibrate the deep M54 photometry are fainter than this
level.  On the whole, the transformation to the photometric system of
\markcite{sl95}SL95 was successful.

%Figure 2:  see CAPTIONS.

%Figure 2 shows the CMD of the photometry derived herein.  It
%demonstrates that we did not compromise our photometry by
%extrapolating the photometric transformation outside the region
%covered by the SL95 photometry.  Both figures show that that the
%transformation to the photometric system of SL95 was successful, and
%that this data set may be used to transform the deeper MSTO photometry
%of M54 to the SL95 system.

%Fig2: CMD of brt sequences?  (different symbols for extrapolated
%stars?  stick to mags above msto to weed out bad photom (no mag cut allowed))


%-------------------------------------------------------------------
\subsection{Deep Photometry of M54}

We followed a similar procedure to obtain photometry from the deep
frames of M54.  We trimmed the deep frames to cover the same region of
the sky as the shallow frames, thus no spatial extrapolation of the
photometric transformation (\S 3.1) is required in this section.  We
then computed instrumental stellar magnitudes using the DoPHOT
program, and matched the instrumental photometry of spatially
coincident stellar images within each pointing (contemporaneous set of
$V$, $I$, $I$ images).

An ($X$, $Y$) coordinate transformation brought the coordinate frame
of each instrumental photometry list onto the same coordinate system
used for the bright M54 sequences (see Figure 1).  We then used the
equations shown in \S 3.1 to transform these instrumental magnitudes
onto the standard $VI$ magnitude system defined by the data in Table
2.  The RMS scatter in the transformations were typically $\sim$0.020
mag, and the transformations always employed $>$900 stars, and spanned
the entire region of color--$X$--$Y$ space.

We then combined the photometry of spatially coincident stars in the
resulting photometry lists.  Experiments showed that the assembled
photometry of the five pointings with the best seeing (median = 1.1
arcsec) produced the CMD with the tightest sequences; including the
poorer seeing data degraded the photometric quality in the crowded
inner regions of the cluster.  Therefore, we used data only from the 
five pointings with the best seeing.  For each star appearing in three or
more of the five photometry lists, we computed the weighted mean $V$ and $I$
magnitudes (weighting the individual magnitudes by their DoPHOT
errors).  We also computed the standard error of the mean of the 3--5
$V$ or $I$ observations, which we treat as the error in the $V$ or $I$
magnitude.  These final photometric data are presented in Table 3, in
the same format as Table 2.  There are 30,081 stars in this data set.

% Fig showing differences wrt brt seq photom (or PP stating that they
%  are flat?)...NAH, THERE IS NO EXTRAPOLATION HERE AND ITS 4m -> 4m,
%  SO PEOPLE WILL PROBALBY TRUST THAT ITS A FINE TRANSFORM.  SAVE A TREE.


%-------------------------------------------------------------------
\subsection{Deep Photometry of Sgr}

We did not obtain short exposure frames of the Sgr field, so we
transformed the MSTO instrumental photometry directly onto the
standard $VI$ photometry presented in Table 2 of \markcite{sl95}SL95.  We began by
trimming the 4.0-m images down to a 8.2 arcmin FOV centered on the
\markcite{sl95}SL95 Sgr field.  This left a relatively small (0.7 arcmin) strip
around the outside of the image over which the photometric
transformation's spatial fit would be extrapolated.  The trimmed field
lay near the physical corner of the CCD, and in some frames the image
quality in the South-West corner was sufficiently poor that we masked
out these regions.  Thus, the final photometry data set is deficient
in stars in the that corner of the field.

We then computed instrumental magnitudes using the DoPHOT program, and
matched the instrumental photometry of spatially coincident stellar
images within each pointing.  We performed the ($X$, $Y$) coordinate
transformation to bring the coordinate frame of each instrumental
photometry list onto the $XY$ coordinates of the image shown in Figure
3 (we also transformed the $XY$ coordinates of the short exposure
data from \markcite{sl95}SL95, their Table 2, onto this coordinate system; we present
those data in Table 4 for convenience).  We then used the equations
shown in \S 3.1 to transform these instrumental magnitudes onto the
standard $VI$ magnitude system defined by the short exposure data in
Table 2 of \markcite{sl95}SL95 (our Table 4).  The RMS scatter in the transformations
were typically $\sim$0.024 mag, and the transformations always
employed $>$220 stars.

% FIGURE 3:  A 600 second $V$-band exposure of our Sgr field.  The image is 
%8.2 arcmin on a side.  North is to the right and East is up.  The
%coordinate system employed in Tables 4 and 5 correspond to this
%figure, with ($X$, $Y$) = (1, 1) at the lower-left and (1150, 1150) at
%the upper-right.  Three stars from Table 4 (shallow photometry) are
%marked with their ID numbers (20, 49, 68), and three stars from Table
%3 (deep photometry) are also marked (5, 12, 86).

We then combined the photometry of spatially coincident stars in the
resulting photometry lists.  In this case, experimentation showed that
including data from the six pointings with the best seeing (median =
1.2 arcsec) produced the best CMD.  We used only these six pointings
in our final analysis.  For each star which appeared in four or more
of the six photometry lists, we computed the weighted mean $V$ and $I$
magnitudes and their errors as in \S 3.2.  These data are presented
in Table 5, in the same format as Table 2.  There are 10,147 stars in
this data set.

Figure 4 shows differences between our final Sgr photometry and the
\markcite{sl95}SL95 photometry listed in Table 4.  The rise in $\Delta V$ at $V
< 18.5$ seen in panel (a) mimics that seen in the bright sequence
data (Figure 2(a)).
%results from a small systematic error incurred by DoPHOT as more 
%pixels at the center of a stellar image become saturated.  
However, only stars with $I > 17.3$ ($V \gtrsim 18.3$) were used in
the photometric transformation, so the affects of this rise are
negligible.
%only affect of this rise in the final data is the slight (0.02 mag) 
%underestimate of a few bright stars.  
Overall, the transformation to the \markcite{sl95}SL95 $VI$ system was successful.


%-------------------------------------------------------------------
\subsection{Photometry in the Bulge Control Field}

No previous photometry exists for our bulge control field, so we can
not transform our instrumental photometry to standard $VI$ photometry
using the method employed in the previous subsections.  Fortunately,
the sky conditions on 1995 June 29 were photometric, and we obtained
five short exposure $VI$ image pairs of M54 which spanned the airmass
and Universal Time of the control field observations.  We used these
observations to derive photometric transformation equations which we
applied to the deep observations of the control field -- essentially,
we used the M54 stars as secondary photometric standards.  In
practice, the airmass-dependence of the transformation was
minimal, and only a color transformation was employed: \[m_f - M_f =
\alpha_f + \beta_f~(V-I),\] where $m$ and $M$ are the instrumental and
standard magnitudes, respectively, and the subscript $f$ refers to the
filter, $V$ or $I$.  The coefficients $\alpha_f$ and $\beta_f$ were
derived from the short-exposure M54 observations, and applied (below)
to the control field observations.  Application of the method to a
single, short-exposure $V$ image of M54 taken during the same time
interval reproduced the standard magnitudes accurately, thus
confirming the validity of this transformation method and the
photometric quality of the night.

We trimmed the control field images down to a 10.7 arcmin square FOV
to avoid poor image quality near the corners of the frames.  We then
computed instrumental magnitudes using DoPHOT, and matched the
instrumental photometry of spatially coincident stellar images within
each pointing.  We performed the ($X$, $Y$) coordinate transformation
to bring the coordinate frame of each instrumental photometry list
onto the coordinate system of the frame with the best seeing.  We then
applied the photometric transformation equations described in the
previous paragraph to convert the instrumental magnitudes to the
standard $VI$ system.  Next, we combined the photometry of spatially
coincident stars in the four photometry lists (one list per pointing),
retaining only the stars which appeared in three or more of the lists.
We computed the weighted mean $V$ and $I$ magnitudes and their errors
as in \S 3.2.  These data are presented in Table 6, in the same
format as Table 2.  There are 7917 stars in this data set.  The median
seeing of the frames employed was 1.2 arcsec.  
%Precise equatorial 
%coordinates for three stars, derived from the Digitized Sky Survey,
%are presented in Table 6 to aid in field identification.

The purpose of the control field is to quantify the positions of
foreground bulge stars in the M54 (on-cluster) and Sgr CMDs.  The
photometry provided herein should be adequate for that purpose,
despite the fact that a different technique was used in the
photometric transformations, relative to the M54 and Sgr fields.  We
emphasize that no such difference exists between the M54 and Sgr
photometry.  Furthermore, we can use the \markcite{muskkk}MUSKKK bulge control field
%($l^{\rm II} = -6.11^{\circ}$, $b^{\rm II} = -14.25^{\circ}$) as an
(see Table 1) as an additional control sample.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Color Magnitude Diagrams}
 
The color-magnitude diagrams (CMDs) of our M54 and Sgr fields can be
used to determine the ages of these systems by using techniques such
as isochrone fitting and comparison with CMDs of clusters of known
age.  However, the foreground bulge population complicates the
interpretation of the Sgr CMD, while the bulge and Sgr populations
together complicate the CMD of M54.  In this section, we will attempt
to simplify the CMDs by statistically subtracting away the
contaminating populations, and where appropriate, by excluding stars
at certain radii.


%------------------------------------------------------------------------
\subsection{The Sagittarius CMD}

We begin with the simpler case of Sgr.  To get the widest possible
range in magnitudes, we have combined the deep photometry from Table
5 with the shallower photometry from \markcite{sl95}SL95 (our Table 4); the splice
occurs at $V = 18.7$.  Because the \markcite{sl95}SL95 data covers a slightly smaller
field of view, the number of bright stars is slightly underrepresented
relative to the number of faint stars.  Figure 5(a) shows the CMD of
the 9,540 stars which have errors in $V$ and $V-I$ less than 0.10 and
0.14 mag, respectively.

% FIG 5: The CMDs of (a) the Sgr field, and (b) the M54 field.
% Deep and shallow photometry sets have been spliced together as
% described in the text.  Only stars with errors in $V$ and $V-I$ less than
%0.10 and 0.14, respectively are shown.  
% (cmdcomb.sgr2.e1 and cmdcomb.photcut.e1r9 -- sm.cmd2.sgr_m54)

Figure 6(a) shows this data again, zoomed in on the MSTO
region.  Panel (b) of this figure presents the CMD of our control
field data (from Table 6) which has errors in $V$ and $V-I$ less than
0.10 and 0.14 mag, respectively.  Since we have no photometry of the
brighter stars in this field, the CMD is cut off above $V = 18$ mag.
The bulge MSTO is seen at $V \approx 19$ mag and $V-I = 0.75$, while
the ridgeline of stars extending to fainter magnitudes and redder
colors marks the bulge main sequence.  This same pattern is evident in
panel (a), albeit with fewer stars, since the Sgr data was derived
from a smaller field of view ($8.2\times8.2$ versus $10.7\times10.7$
arcmin).  The bulge MSTO region appears to have a narrower color width
in panel (a) than in panel (b), suggesting that differential reddening
across the control field may be larger.  Also, the bulge MSTO region
appears to be about 0.1 mag bluer in panel (b), suggesting a lower
mean reddening in that field.  Fortunately, we can use the \markcite{muskkk}MUSKKK
control field photometry as a ``second opinion'' on the foreground
contamination.  The \markcite{muskkk}MUSKKK data extends to brighter magnitudes, but
has a lower signal-to-noise ratio at a given faint magnitude.

%FIG 6:  the CMDs from (a) the Sgr field, and (b) our control field.
%Only stars with errors in $V$ and $V-I$ less than 0.10 and 0.14,
%respectively, are shown.  In (c), a subset of the stars from (b) have been
%subtracted statistically from those in (a), resulting in a clearer
%view of the Sgr sequences.  In (d), a subset of the control sample
%of MUSKKK has been subtracted from the stars in (a), providing a
%complementary view of Sgr.

The excess of Sgr stars in Figure 6(a) over the bulge contribution is
obvious.  The primary Sgr MSTO occurs near $V \approx 21.4$ and $V-I =
0.75$, well separated from most of the bulge stars.  A large fraction
of the Sgr stars seem to form a population which turns red-ward onto
the subgiant branch (SGB) at $V = 21.1$ mag, though a significant
number of stars contribute to one or more MSTOs at brighter
magnitudes.  The SGBs of the Sgr populations can be seen as slight
density enhancements crossing the bulge main sequence, and turning up
onto a sparsely-populated RGB.  The Sgr red clump is at $V = 18.3$ and
$V-I = 1.13$ mag.  Also notable in panel (a) are the stars brighter
and bluer than the Sgr MSTO.  These stars are not present in the
comparison fields, and may represent either a very young population or
blue straggler stars in Sgr (\markcite{muskkk}MUSKKK,
\markcite{bel99a}Bellazzini et al. 1999a).

We can statistically subtract the stars in panel (b) from those in
panel (a) to get a clearer picture of the various stellar populations
in Sgr.  First, we need to characterize two parameters, the reddening
difference between the fields, and the density normalization.
\markcite{sl95}SL95 estimated $E(V-I) = 0.18 \pm 0.02$ in our Sgr
field, using the shape and color of the Sgr RGB.  We have no
corresponding reddening estimate from our control field.  The
reddening maps of \markcite{sch98}Schlegel et al. (1998) give values
of $E(V-I) = 0.19$, 0.10, and 0.13 at the positions of our Sgr field,
our control field, and the \markcite{muskkk}MUSKKK control field,
respectively.  The $E(B-V)$ values from \markcite{sch98}Schlegel et
al. have been converted to $E(V-I)$ using the relations of
\markcite{dwc78}Dean et al. (1978).

We check these reddening differences by comparing the color histograms
of stars near the MSTO in the three fields.  We extract from the CMD
of each field all stars with $16.0 < V_{VI} < 18.0$ mag, where
$V_{VI}$ is the reddening-independent parameter defined as $V -
2.6~(V-I)$ by \markcite{sta94}Stanek et al. (1994).  The red wing of
the color histogram for the Sgr field is contaminated by Sgr RGB
stars.  We therefore compare the colors of the blue edge of the
histograms.  They coincide when our control field data are reddened by
0.07 mag, and when the \markcite{muskkk}MUSKKK control field data are
reddened by 0.05 mag.  These correspond well with the reddening
differences from the \markcite{sch98}Schlegel et al. (1998) dust maps.
We therefore adopt $\Delta~E(V-I) = 0.08$ for our control field, and
0.04 mag for the \markcite{muskkk}MUSKKK control field.

The density of bulge stars in each field can be estimated from the
number of stars in a region along the bulge main sequence, around $V =
21.8$ and $V-I = 1.3$ mag.  The relative numbers are used to normalize
the control field samples to the Sgr field.  Since the signal-to-noise
and detection-completeness functions vary between the different
fields, and since the tightness of the bulge sequence may be affected
by reddening variations within a field, these normalizations are only
approximate, and we treat them as adjustable parameters in our
subtractions.

Figure 6(c) shows the data from panel (a) after the subtraction of a
subset of stars from panel (b) which had been reddened by 0.08 mag.
The subtraction procedure involved identifying for a given star in the
control CMD the star in the Sgr CMD with the smallest ``distance'',
$\rho$, and removing this star from the Sgr CMD.  We define $\rho$ as
%\begin{eqnarray}
%\rho = (~(V_c - V_s) + \alpha~\{~(V-I)_c  - (V-I)_s\}~)^{0.5}.
%\end{eqnarray}
\[ \rho = (~(V_c - V_s) + \alpha~\{~(V-I)_c  - (V-I)_s\}~)^{0.5}. \]
The subscripts $c$ and $s$ refer to the control and Sgr data sets,
respectively; $V$ and $(V-I)$ are the corresponding magnitude and
color; $\alpha$ accounts for the aspect ratio of the CMD.  We found
that $\alpha = 3$ produced good results.

Figure 6(c) shows the Sgr populations much more clearly.  In
particular, the SGB region is more clearly defined.  It shows a large
vertical spread which may indicate multiple epochs of field star
formation in Sgr, or a long duration of continuous star formation (see
\S 8).  The population of bright, blue stars is also clearly shown.
%The stars brighter than $V = 18.8$ and redder than $V-I = 0.8$ are
%the result of limitations in the subtraction procedure.

We also performed subtractions employing the \markcite{muskkk}MUSKKK
control field data.  Subtractions using the preferred reddening shift
of 0.04 produced satisfactory results, but tended to oversubtract
stars on the Sgr RGB, and leave bulge stars $V < 20$ and $V-I < 0.8$
mag.  A reddening shift of 0.03 mag produced a much better
subtraction, which we show in Figure 6(d).  The same features appear
as in panel (c), confirming that they are real features of the Sgr
populations.  A slight excess of unsubtracted bulge stars remains
below and to the right of the Sgr SGB.  The knot of stars at $V =
18.7$ and $V-I = 0.85$ in both panels (c) and (d) is spurious: the
result of the signal-to-noise discontinuity at the joint of the deep
and shallow Sgr photometry sets.


%------------------------------------------------------------------------
\subsection{The M54 CMD}

As in the previous section, we merged the deep and shallow M54
photometry sets from Tables 3 and 2, respectively.  The splice was
made at $V = 19.0$.  The field sizes are the same, so the relative
numbers of bright and faint stars should be consistent across the
boundary.  Figure 5(b) shows the CMD of the 28,950 stars which have
errors in $V$ and $V-I$ less than 0.10 and 0.14 mag, respectively.
The features of the M54 CMD, including a blue HB, a more vertical RGB,
and the MSTO, are superimposed on the Sgr and bulge features seen in
panel (a).  Figure 7(a) shows the M54 data again, zoomed in on the
MSTO region.  Panel (b) of this repeats the Sgr data from 6(a).

% FIG 7: the CMDs from (a) the M54 field at all radii, (b) the Sgr
%field, and (c) the M54 field with radii between 300 and 600 pixels.
%Only stars with errors in $V$ and $V-I$ less than 0.10 and 0.14,
%respectively, are shown.  In (d), the stars from (b) have been
%subtracted statistically from those in (c), resulting in a clearer
%view of the M54 sequences alone.  Note, that some traces of the young
%Sgr populations remain.

It is useful to plot the CMD of stars with positions inside an annulus
around the globular cluster center, where the inner radius is chosen
to minimize photometric errors due to crowding, and the outer radius
is chosen to minimize contamination by field stars.  Figure 7(c) shows
the CMD of M54 stars lying within 300--600 pix (2.2--4.3 arcmin) of
the cluster center, ($X$, $Y$) = (726, 806) pix.  Note that this
region is mostly contained within the area covered by the
\markcite{sl95}SL95 photometry, so the photometric transformations
discussed in Secs. 3.2 and 3.3 are particularly trustworthy.  The
sequences associated with M54 become clearer in this panel, though
there is still a strong contribution from the Sgr and bulge
populations.
% 12,975 stars in e1r3.

In Figure 7(d) we present the result of statistically subtracting the
data in panel (b) from that in panel (c).  We made no reddening
correction, since \markcite{sl95}SL95 found identical reddenings from
the shape and position of the RGBs, and since the centroids of the red
clumps coincide to within 0.01 mag.  The areas of these two fields are
nearly identical, but normalization factor computed from counts of
main sequence bulge stars called for a reduction in the Sgr
contribution by about 10\%.  This is explained by the enhanced
crowding in the M54 field: some fraction of the faint bulge stars will
be overwhelmed by bright stars belonging to the cluster, and thus not
detected.  However, to eliminate successfully the Sgr population in
panel (d) required an enhancement of the Sgr field contribution by
20\%, and even then, traces of the brightest Sgr MSTO, SGB, and RGB
remain.  This supports the idea raised in \markcite{sl95}SL95 that the
young, metal-rich component is concentrated toward the cluster.  We
will pursue this question in \S 10.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Variable Stars}

The large number of CCD frames which we have accumulated in our study
of M54 gives us an excellent opportunity to study the variable stars
in this cluster.  The most recent survey for variable stars in this
cluster was the photographic work of \markcite{rn59}Rosino \& Nobili
(1959, hereafter referred to as ``RN59'').  They identified 82
variable stars in this cluster (see \S 5.4).

In our investigation, we assume no {\it a priori} knowledge of the
variable stars found by \markcite{rn59}RN59 -- we begin from scratch.
Table 7 describes the five observing runs involved in our study.
Columns 1--5 give the designation and the UT dates of the run, the
telescope and CCD employed, and the field of view in arcmin.  Columns
6 and 7 list the number of frames obtained during that run in the $V$
and $I$ bands, respectively.  Different plot symbols will be used to
distinguish data taken during the different runs; these symbols are
defined in the last column on Table 7.

%Table 7:  Observing Runs Used in Variable Star Study
% col1 : Run = 94jun1 ...
% col2 : Dates = 1994/06/nn-mm ...
% col3 : Telescope = CTIO 0.9-m ...
% col4 : CCD = Tek2k (20' FOV)
% col5 : N_V = Number of V-band frames obtained
% col6 : N_I = Number of I-band frames obtained
% col7 : plot symbol in var* figures

Instrumental photometry for each of the individual frames was computed
using DoPHOT, and was then transformed onto the ($X$, $Y$) coordinate
system defined by the stars on the short-exposure CTIO 4-m images (see
\S 3.2, Figure 1).  Then it was transformed photometrically onto
the bright sequence photometry of Table 2 as described in \S 3.2.
Photometry from the 15 $VI$ frame pairs with the best seeing were
assembled, i.e., we spatially matched stars from the different frames
and combined the photometry for each star using an error-weighted
mean.

For each star, we also computed the \markcite{ws93}Welch \& Stetson
(1993) variability index, $I_{\rm WS}$, to characterize the likelihood
that the star is variable.  The value of $I_{\rm WS}$ is computed for
each star from the individual $V$ and $I$ magnitudes and their errors.
For a non-variable star, $I_{\rm WS}$ approaches zero as the number of
observations becomes large, whereas for a variable, $I_{\rm WS}$
approaches a constant value.

The morphology of the points in the ($I_{\rm WS}$, $V$) plane led us
to classify stars as probable and possible variables.  When plotted on
the cluster CMD, it became clear that three classes of variables are
present.  Most of the variable candidates fell near the middle of the
cluster horizontal branch, and are very probably RR Lyrae variables in
M54 or in Sgr.  A number of candidates lay near the tips of
the Sgr and M54 red giant branches.  A few other variables scattered
around the CMD, mostly blue-ward of $V-I = 1$, suggesting that they
are RR Lyrae or eclipsing binaries in the foreground field.

Our main goal in this study is to use the M54 RR Lyrae stars to define
the distance and reddening to the cluster.  We therefore focused
closely on the stars near the M54 HB, in the region $V-I < 1.0$ mag,
$17.8 < V < 18.9$ mag.  We found 82 such stars with $I_{\rm WS} > 30$.
We also considered possible foreground stars in the region $V-I < 1.0$
and $V < 17.8$ mag.  Eleven such stars were found with $I_{\rm WS} >
50$.  Plots of magnitude as a function of time for these 93 stars showed
that, almost universally, they had periods shorter than about a day,
consistent with their being RR Lyrae or contact binary stars.  We also
searched for variability in the bright, red stars with $V-I > 1.4$, $I
< 15$, and $I_{\rm WS} > 10$.  We found 26 candidates, whose
characteristics will be discussed in detail in \S 5.2.


%----------------------------------------------------------------------
\subsection{Short Period Variables: the RR Lyrae}

In an effort to determine the types of variable stars present among
the 93 blue, short-period variable star candidates discussed above, we
attempted to find the period of each star and create the star's mean
light curve.  Classical period finding techniques such as Fourier
transforms or Phase Dispersion Minimization perform poorly, if at all,
when the data are sparse and irregularly spaced.  We therefore used
the method of template fitting described in \markcite{acl98}Layden
(1998) and \markcite{acl99}Layden et al. (1999).

Briefly, a set of template light curves representing the types of
variable stars expected are fit to the data, which are folded by a set
of periods ranging over the expected values.  At each period in the
series, the observed $V$-band data are folded and fitted by the each
of the templates (three free parameters are used: magnitude
zero-point, phase zero-point, and magnitude amplitude), and the
$\chi^2$ of the fit is computed.  Small values of $\chi^2$ indicate
possible periods.  For each template, a plot of $\chi^2$ as a function
of period allows us to select the most probable periods.  

In practice, we used the six templates shown in \markcite{acl98}Layden
(1998) to represent ab-type RR Lyrae (RRab, fundamental mode
pulsators).  We also employed templates for a c-type RR Lyrae (RRc,
first-overtone pulsator)\footnote{The RRc template is an average of
ten high-quality light curves of RRc stars in M68
\markcite{aw94}(Walker 1994).}, a cosine curve, a curve derived from
the Fourier coefficients of \markcite{ruc93}Rucinski (1993, his Figure
1) to represent a W Ursae Majoris (W UMa) contact binary, and a
schematic template of a detached (Algol) eclipsing binary.
%These templates are shown in Figure MMM.  
While the binary star templates do not represent the entire range of
light curves displayed by these variables, we think they are
sufficient to identify eclipsing binaries.  We will not derive useful
information from these stars, so we are content to cull them out of
the RR Lyrae sample.  Indeed, given the limited number of
observations, it is difficult to derive reliable periods for
Algol-type binaries, and it is likely that some exist in our field
which have not been detected.

% FIGURE:  4 non-RRab templates.

We fit the templates over periods ranging from 0.2 to 1.0 days,
increasing the period by 0.03\% from one step to the next.  If no
suitable minima were found, we searched on the period interval 1--3
days, and in some cases, 0.1--0.2 days.  Periods longer than 3 days
are evident in the time-magnitude plots.  For each variable, we
inspected the light curves produced with periods from the best
ten or so minima.  In almost all cases, the period with the smallest
$\chi^2$ value in the array of periods and templates produced the
light curve most pleasing to the eye, and this period and template
were adopted.  In most cases, changing the period by 0.00005 days
resulted in a noticeable degradation of the light curve, but in many
cases a nearby minimum produced a satisfactory light curve as well.

% ``SOME STARS'' = 4 SVs + 28 blends
Periods could not be determined for 32 stars.  Possible reasons
include, (1) photometric errors due to crowding produced light curves
with unacceptable scatter, (2) the star has multi-periodic behavior,
(3) the star undergoes changes in light curve shape or amplitude
(e.g., the Blazhko effect) from cycle-to-cycle, and (4) the templates
we employed did not reflect the star's actual light curve shape.  Upon
inspection of the images, 28 of these stars were seen to have stars
very close by; the stellar images blended on poor-seeing frames, and
separated on good-seeing frames, mimicking variability.  We do not
consider these stars farther.  The three candidates which {\em do}
appear to be true variables, but for which we could not determine a
period, are listed in Table 8 as SV1--3.

%SV4 was here, but later I realized it was the Algol candidate V91.

% TABLE XYZ:  suspected variables.
% Snn  Xpix  Ypix  Vbar  V-Ibar  Vampl  Comments

We were able to derive periods for 60 of the 93 short-period variable
candidates.  We obtained photometry for 11 more known variables after
identifying them with the finder chart from \markcite{rn59}RN59.
Figure 8 shows the $V$-band light curves of these stars, folded by
their adopted periods, and with the preferred template overplotted.
In Table 9, we present the data derived from these stars: the first
column gives the variable star name; $X_{\rm pix}$ and $Y_{\rm pix}$
are the positions on Figure 1; Per is the period in days; Typ is the
variable type; $\langle V \rangle$ and $\langle I \rangle$ are the
intensity-mean magnitudes in $V$ and $I$; $V_{\rm max}$ and $V_{\rm
min}$ are the $V$-band magnitudes at maximum and minimum light;
$E_{\rm max}$ is the epoch of maximum light (Heliocentric Julian Date
- 2,449,000 days); $N_{\rm obs}$ is the number of $V$ frames on which
the variable was measured; $N_{\rm tmp}$ is the template number used
in the adopted fit; $rms$ is the root-mean-squared scatter of the data
points about the fit; $(V-I)_{\rm min}$ is the magnitude-mean $V-I$
color at minimum light, defined as $0.5 < {\rm Phase} < 0.8$
(\markcite{stu66}Sturch 1966); $\epsilon (V-I)_{\rm min}$ and $N_{\rm
min}$ are the standard error of the mean in $(V-I)_{\rm min}$, and the
number of observations used in its computation.  The a mark in the
final column of Table 9 indicates that a note regarding that star is
available in the Appendix.  The quantities $\langle V \rangle$,
$V_{\rm max}$, $V_{\rm min}$, $E_{\rm max}$, and $rms$ were derived
from the fitted light curve, the others are from the observed data
points.

% FIGURE 8: $V$-band light curves for 71 short-period variables,
%listed by variable star designation.  The points represent
%observations and the curve represents the best-fit template.  The
%point symbols indicate the observing run during which that observation
%was obtained, as defined in Table 7.  Solid and open symbols
%indicate seeing better and worse than the median, respectively.  In
%each panel, the magnitude range is 1.7 mag and the minor tick marks
%are separated by 0.1 mag, except in the case of the eclipsing
%binary V91.

% TABLE 9: short period variables (see /Kr4/.../PERIOD/LC/tab.phot2tab.out)

Table 9 contains 71 stars, 60 of which are RRab, and 7 are RRc.  V01
appears to be a Population II Cepheid, and V02 is probably an
Anomalous Cepheid (see Appendix).  There is also one probable W UMa
contact binary (EW, V99), and one probable Algol detached binary (EA,
V91).


%----------------------------------------------------------------------
\subsection{Long Period Variables}

In addition to the RR Lyrae and other short period variables, we
detected 26 bright, red variable star candidates in our data.  Figure
9 shows a plot of magnitude versus time for a typical example, V25.
The magnitudes vary slowly, if at all, during the course of a single
run, but jump noticeably between runs, indicating that the stars have
long periods.  Given their colors and periods, they are very likely to
be Irregular, Semi-Regular, or Mira type pulsating variables.  Indeed,
three of our Long Period Variable (LPV) candidates appear in the
\markcite{hsh73}Sawyer-Hogg (1973) catalog (V25, V66, and V71), and
all three are classified as ``SR'' (Semi-Regular) therein.  The fourth
SR listed in \markcite{hsh73}Sawyer-Hogg (1973), V64, is outside our
field of view.  Careful inspection of all the magnitude-time plots
showed that eight of the candidates were probably non-variable stars
whose photometry had been compromised by crowding in the
densely-packed cluster center.

% NB: ditch this figure if doesn't use argument about anon-RHB=RRL@min
%!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

% FIGURE 9: FIG_LPV/mag_time.lpv_V25.sm
% caption:  magnitude as a function of time for the Long Period Variable
% V25.  The time axis is broken into the five observing runs as marked.
% The top set of plots are for $V$ magnitude, and the bottom set are 
% for $I$ magnitude.  The long period variations are clear.

% FIGURE 9: FIG_LPV/mag_time.lpv_V25.sm
% caption:  magnitude as a function of time for the Long Period Variable
% V25.  The time axis is broken into the five observing runs as marked.
% The top set of plots are for $V$ magnitude, and the bottom set are 
% for $I$ magnitude.  The long period variations are clear.

The 18 remaining LPV candidates are listed in Table 10, along with
their ($X$,$Y$) positions in Figure 1, their arithmetic mean $V$
magnitudes and $V-I$ colors, the maximum difference in $V$ magnitudes
observed ($\Delta V_{\rm min}^{\rm max}$), and in some cases a
comment.  Stars with larger values of $\Delta V_{\rm min}^{\rm max}$
are likely to be true LPVs, and the probability decreases with
decreasing amplitude, as noted in the comments column.  Additional
uncertainty is encountered in some stars in crowded environments,
where increased photometric errors are incurred.  
% Some spurious detections of variability due to DoPHOT photometry of
% some stars near saturation limit of some frames?

% Table 10 = TABLS/tab_vars_lpv.tex

We emphasize that our list of LPV stars is likely to be incomplete for
several reasons.  First, there may be some bulge LPV stars in the
field which are brighter than our saturation limit.  Second, the
distribution of our observations in time is not ideal for detecting
variables with periods longer than a few days; it is possible that
some true variables did not vary enough over the course of our
observations to be detected.  We also note that the mean photometry
listed in Table 10 is only approximately representative on the true
values, since the stars' light cycles are poorly sampled.  Clearly,
long-term observations of the LPVs would be useful.


%----------------------------------------------------------------------
\subsection{Faint Variables from Deep Exposures}

For completeness, we also searched the deep M54 and Sgr frames for
variable stars.  The poor time sampling, 8--9 frames over two nights,
means that we will not be able to determine periods, and that
significant numbers of actual variables may go undetected.
Nevertheless, it is interesting to identify any variables as
motivation for more complete study in the future.

In the M54 frames, we detected two eclipsing variables which we
designate V116 and V117.  Table 11 gives their $(X, Y)$ positions in
Figure 1, their arithmetic mean $V$ magnitude and $V-I$ color, their
maximum difference in $V$ magnitude ($\Delta V_{\rm min}^{\rm max}$),
and a comment.  Time series data for these stars is found in
Table 12.

In the Sgr frames, we detected two more variables, which we designate
Sgr1 and Sgr2 (see Table 11).  Sgr1 has the right light curve shape,
magnitude, and color to be an RR Lyrae star in the Sgr field.  Sgr2 is
an eclipsing binary.  Time series data for these stars is found in
Table 12.


%----------------------------------------------------------------------
\subsection{Previously Known Variable Stars}

The most recent, significant study of the variable stars in M54 is the
photographic work of \markcite{rn59}RN59.  Results from this and other
works are summarized in \markcite{hsh73}Sawyer-Hogg (1973, hereafter
``SH73''), and have been updated by Christine Clement\footnote{The
updated electronic catalog is available upon request from Dr. Clement
(cclement@astro.utoronto.ca).}.  \markcite{hsh73}SH73 lists 82
variable stars in M54, but less than half of these stars have well
determined periods or variability types.  Finder charts of variables
V1--V82 are presented in \markcite{rn59}RN59.

% Out of our FOV: V8, 9, 10, 11, 13, 16, 17, 18, 19, 24, 64, 68, 70 - N=13

Of these 82 stars, 13 are outside our field of view (V8, V9, V10, V11,
V13, V16, V17, V18, V19, V24, V64, V68, and V70).  We could not
identify two stars (V21 and V26; see Appendix).  Five stars appeared
not to be variable, despite being identified positively on the \markcite{rn59}RN59
charts (V20, V22, V27, V53, V72; see Appendix).  

We did detect variability in the remaining 62 stars, three of which
were long period variables.  We could not determine periods for the
six stars (see \S 5.1; V67, V69, V73, V75, V79, V81; see Appendix).
However, we did obtain good periods and light curves for the remaining
53 known cluster variables.

% Vnn stars w/ variation but no period: V67, V69, V73, V75, V79, V81 - N=6

%We also detected 33 new variables stars.  Eighteen of them have short
We also detected 35 new variables stars.  Twenty of them have short
periods, and appear in Table 9, while fifteen have long periods, and
appear in Table 10.  We also detected five new stars which appear to
be variable, but for which we could not obtain a period or light curve
(V116, V117, and SV1--3 in Table 8).  We begin naming the new
variables at V83.  This brings the total number of known or suspected
variables toward M54 to 120.
%variables toward M54 to 118.

We present the magnitude data as a function of time in Table 12.  The
columns contain: (1) the star name, (2) the reduced Heliocentric
Julian Date (HJD - 2,449,000) in days, (3) the adopted phase, (4) the
$V$ magnitude and (5) its error, (6) the $I$ magnitude and (7) its
error, and (8) the symbol with which the data are plotted in Figures 8
and 9.\footnote{This code is a Super-Mongo point type.  The first
digit indicates the symbol shape and run number (see Table 7): 3 =
{\em triangle} = Run1, 4 = {\em square} = Run2, 5 = {\em pentagon} =
Run3, 9 = {\em circle} = Run4.  The second digit indicates whether the
seeing was better or worse than the median seeing (3 or 1,
respectively).  The exception to this rule is code=41, which indicates
Run5.}  Epochs for which $I$ frames were not obtained are assigned the
values 88.888 in columns (6) and (7).  In Figure 10, we provide a
``postage stamp'' finder chart for each variable in our field of view.
Figure 11 shows the mean magnitudes and colors of the variable stars
near the HB superimposed on the cluster CMD.

%% TABLE 11:  star  HJD  phase  V  Verr  I  Ierr  plot-symbol

% FIGURE 10: finder charts for each variable star in our field of
%view, listed by variable star designation.  Each finder is 20 arcsec
%on a side, and is oriented with North to the top and East to the
%left.

% Figure 11: the CMD from stars in Table 2 with errors $\sigma_V <
%0.10$ and $\sigma_{V-I} < 0.14$ and with radii greater than
%150 pixels.  The mean magnitudes and 
%colors of the variable stars marked as follows:
%ab-type RR Lyrae ({\em circles}), c-type RR Lyrae ({\em triangles}),
%Cepheids ({\em stars}), eclipsing binaries ({\em asterisks}), long
%period variables ({\em squares}).  For the RR Lyrae, filled and open
%symbols indicate stars with template fits having rms values less than
%and greater than 0.033 mag, respectively.  For the long period
%variables, the open squares indicate stars listed as ``probable'' or
%``possible'' LPV stars in Table 10.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Cluster Properties Derived from the RR Lyrae Stars}

We can derive two important quantities from the RR Lyrae stars in M54:
the cluster distance modulus, and the foreground reddening.  The
former comes from the observed apparent magnitudes of cluster RRab and
RRc stars, while the latter is derived from all the RRab stars
available, be they members of the cluster, of Sgr, or of the Galactic
foreground.  Before proceeding, we must attempt to determine which
stars are members of which population.  We will discuss a number of
other interesting results regarding the HB of M54 as we progress
toward measuring its distance and reddening.

%%%%%
\subsection{RR Lyrae Membership in Sgr and M54}

Assigning membership in this three-population case is more complicated
and prone to uncertainty than the usual two-population case (cluster
versus field).  The foreground RRL stars will be uniformly scattered
over the field of view, and display a range of apparent magnitudes.
The Sgr member RRL will also be scattered over the field, but will all
have very similar magnitudes.  The M54 RRL members will be radially
concentrated toward the cluster center, but will have the same
magnitudes as the Sgr RRL, since the distances to M54 and Sgr are
indistinguishable (\markcite{sl95}SL95).

We begin by trying to identify RR Lyrae stars in the foreground field.
In a plot of $V$ magnitude as a function of radius, there are only
four clear outliers from the swath of stars defining the M54/Sgr
locus: V12, V55, V65, and V93.  V55 is 0.4 mag fainter than the
cluster RR Lyrae, while the other three are significantly brighter.
The four RR Lyrae about 0.2 mag brighter than the clump of M54/Sgr
RRab in Figure 11 all lie within 1.3 arcmin of the cluster center, so
are almost certainly cluster members.  Their enhanced brightness is
probably due to image blending in this croweded region.

Having culled the field RRL from our sample, we now attempt to
separate M54 members from Sgr members.  One way to accomplish this is
by comparing the radial distributions of the various HB components in
our CMD.  Figure 12 shows the cumulative radial distributions of the
blue HB stars (BHB), RR Lyrae stars (RRL), red clump stars (RC), and
stars belonging to the ``anonymous red HB'' (AHB) discussed by \markcite{sl95}SL95.
Here, we defined AHB stars as those with $18.17 < V < 18.33$ and $0.75
< V-I < 0.90$ mag.  \markcite{sl95}SL95 showed that the BHB stars belong to the
cluster, and that the RC stars belong to Sgr, while the source of the
AHB stars was tentatively attributed to Sgr.  In Figure 12, it is
clear that the AHB star distribution follows that of the RC stars,
again suggesting that the AHB stars belong to Sgr.  Meanwhile, the RRL
are distributed more like the BHB stars, indicating that most of the
RRL belong to M54.

The RRL distribution does flatten a bit at $R > 500$ pix, suggesting
an additional contribution of stars at large radius relative to the
BHB stars.  We experimented with subtracting random samples of RRL
stars from the RRL distribution outside this radius, and found that
the RRL and BHB distributions matched better when 3--9 RRL were
removed.  Given that RRL are known to exist in the Sgr field (e.g.,
\markcite{muskkk}MUSKKK, \markcite{ala96}Alard 1996), it is reasonable
to believe that these excess stars are members of Sgr.  The region
outside $R = 500$ pix comprises about $\slantfrac{2}{3}$ of the area
of our frame, so the number of Sgr RRL in our entire frame is probably
between 5 and 14.  For comparison, \markcite{muskkk}MUSKKK found 6--7
Sgr RRL in a $15\times15$ arcmin field about 2.4 degrees away, in a
lower density region of Sgr according to the \markcite{igi94}Ibata et
al. (1994) isopleths.

% FIGURE 12:  The cumulative radial distributions of blue HB 
% stars (BHB), RR Lyrae stars (RRL), red clump stars (RC), and
% ``anonymous red HB'' (AHB) stars in our field of view.  The AHB
% was defined as all stars with $18.17 < V < 18.33$ and $0.75 < V-I 
% < 0.90$.  This definition was based on the CMD in Figure 6a of
% SL95, upon which the AHB was first identified.  SL95 showed that
% the BHB stars belong to M54, while the RC stars belong to Sgr.

%%%
\subsection{The Period--Amplitude Diagram and Oosterhoff Type}

Is it possible to determine exactly which RRL belong to Sgr and which
belong to M54?  One means may be through the stars' positions in the
Period-Amplitude diagram.  It has long been thought that stars with
lower metal abundances shift to longer periods in this diagram
(\markcite{san93}Sandage 1993 and references therein).  Since M54, at
$[Fe/H] = -1.55$ to $-1.79$, is significantly more metal-poor than the
old Sgr population responsible for the RRL ($[Fe/H] \approx 1.3$), we
expect the M54 RRL to have systematically longer periods than the Sgr
RRL.  \markcite{san82}Sandage's (1982, Eqn. 2)
% Sandage, A. 1982, ApJ, 252, 553; Eqn. 2)
relation predicts the shift to be ${\rm log} P \approx -0.06$ dex.

% FIGURE 13: the $V$-band amplitudes of our RR Lyrae stars as a
% function of log(period).  Symbols are as for Figure 11, except that
%here, squares indicate the four field RRL.  Symbols overstruck
%with a {\em cross} lie more than 500 pix from the cluster center.  The
%solid lines show the period-amplitude relations for M3 and M9 derived
%by Clement \& Shelton (1999), and the dotted line is the M3 relation
%shifted to the position expected for RRab in a population with $[Fe/H]
%\approx -1.1$ (Sandage 1982).

Figure 13 shows the positions of our RRL (the four field RRL are
plotted as squares) in the Period-Amplitude diagram.  The majority of
stars seem to follow the relation derived by \markcite{cs99}Clement \&
Shelton (1999) for M3, though there might be a slight difference in
slope or zero point.  Ten of the RRab seem to follow the relation
expected for Sgr from the Sandage period-shift relation (dotted line),
and two of these belong to the field (V55 and V93).  Eight Sgr RRL
agrees well with the estimate of 5--14 from the flattening of the RR
Lyrae cumulative radial distribution.  However, two points conflict
with this interpretation.  First, only three of the eight stars lie
more than 500 pix from the cluster center, whereas we would expect the
number to be 5--6 if the population is distributed uniformly across
the field.  The radial distribution of the eight stars is like that of
the M54 stars.  Second, if the period shift is due solely to
metallicity alone, presence of six or so stars shifted to {\em longer}
periods would suggest the presence of an additional {\em metal-poor}
field population in Sgr, at $[Fe/H] \approx -2.1$ dex.  While
\markcite{bel99a}Bellazzini et al. (1999a, \markcite{bel99b}1999b)
suggest that a minor fraction of the Sgr field has this abundance,
($i$) it is not clear whether those stars produce RRL, and ($ii$) most
of the longer-period RRL in Figure 13 lie within 500 pix of the
cluster center, and thus probably belong to M54.

\markcite{cs99}Clement \& Shelton (1999) provide a different
interpretation of the period-shift effect.  They claim that the shift
is not a continuous function of metallicity, but is instead a
two-valued distribution directly related to the Oosterhoff dichotomy.
That is, all Oosterhoff type-I clusters follow the relation defined by
M3, and all Oosterhoff type-II clusters follow the relation defined by
M9.  They obtained narrow sequences of stars in the Period-Amplitude
diagram by identifying and rejecting all RRab with ``peculiar'' light
curves (e.g., the Blazhko effect) using the compatibility test of
\markcite{jk96}Jurcsik \& Kovacs (1996).  The peculiar stars generally
lay below the relation defined by the ``normal'' stars.

Clearly, most of the RRab in Figure 13 lie near the M3 relation,
suggesting M54 is a Oosterhoff type-I cluster.  The mean periods of
the RR Lyrae confirm this: $\langle P_{\rm ab} \rangle = 0.579 \pm
0.007$ and $\langle P_{\rm c} \rangle = 0.342 \pm 0.016$ days.  For
comparison, the average values for Oosterhoff type-I clusters (OoI)
are $\langle P_{\rm ab} \rangle = 0.55$ and $\langle P_{\rm c} \rangle
= 0.32$ days, while for Oosterhoff type-II clusters (OoII) they are
$\langle P_{\rm ab} \rangle = 0.64$ and $\langle P_{\rm c} \rangle =
0.37$ days (\markcite{hs95}Smith 1995, Table 3.2).

According to the picture of \markcite{cs99}Clement \& Shelton (1999),
the RR Lyrae stars in Sgr and M54 should both follow the M3 relation
(presumably being more metal-rich than M54, Sgr is also a OoI system),
and we would expect most of the stars lying near the dotted line in
Figure 13 to have peculiar light curves.  Unfortunately for us, the
\markcite{jk96}Jurcsik \& Kovacs (1996) compatibility test involves
high-order Fourier decompositions of the RRab light curves, and we do
not have enough data points per star to provide reliable
determinations of the stars' Fourier coefficients.  Still, one might
expect stars with light curves which repeat faithfully from cycle to
cycle to produce a relatively small rms scatter about a best-fit
template, while stars with variable light curve shapes or amplitudes
would tend to have larger values of rms.  In the
\markcite{cs99}Clement \& Shelton view, we would then expect the
low-rms stars (solid circles) to lie on the M3 relation, while the
high-rms stars (open circles) would lie below it.  This does not
appear to be the case in Figure 13.

However, the six longer period RRab in Figure 13 {\em do} have a
reasonable explanation in the \markcite{cs99}Clement \& Shelton (1999) picture.
Those authors note three stars in M3 with normal light curves which
obey the OoII (M9) relation.  They suggest that these stars evolved
off the blue HB and through the instability strip at higher
luminosity, and thus longer period, than the other M3 variables, which
lie closer to the zero-age HB.  Since the blue HB of M54 is more
heavily-populated than in M3, it is reasonable to expect a higher
fraction of OoII-like variables in M54.

At this stage, the uncertainties in our data do not allow us to
support firmly either the \markcite{san82}Sandage (1982) or
\markcite{cs99}Clement \& Shelton (1999) explanation of the
period-shift effect.  Nor we can use the Period-Amplitude diagram to
identify RRL which are members of the Sgr field population.  An
observational dataset with more observational epochs would allow us to
do reliable Fourier decompositions, and clarify the Period-Amplitude
diagram using the \markcite{jk96}Jurcsik \& Kovacs (1996)
compatibility test.  Below, we show that the inclusion of eight or so
Sgr stars in the M54 sample does not change significantly the derived
value of $\langle V({\rm RR}) \rangle$ for M54.


%%%%
\subsection{$\langle V({\rm RR}) \rangle$ of M54}

After removing the four field RRL from our sample, we determined the
mean $V$ magnitude of the 63 remaining RRL using a $3-\sigma$ clipping
algorithm.  We obtained $\langle V({\rm RR}) \rangle = 18.17 \pm 0.01$
mag, with an rms of 0.06 mag.  We obtain the same value if we remove
the eight candidate Sgr RRL which lie near the dotted line in Figure
13.  The systematic uncertainty in the photometric zero point is about
0.02 mag (\markcite{sl95}SL95).  In \markcite{sl95}SL95, we estimated
$\langle V({\rm RR}) \rangle = 18.17 \pm 0.05$ mag from 16 RRL which
were poorly sampled in light curve phase.

Most researchers studying the change in RR Lyrae absolute magnitude,
$M_V({\rm RR})$, with $[Fe/H]$ find a slope of $\sim$0.23
mag~dex$^{-1}$ (\markcite{bri99}Chaboyer 1999).  If we assume that M54
and Sgr are equidistant, and that they have a metallicity difference
of $\sim$0.5 dex, any Sgr variables should be about 0.12 mag fainter
than their counterparts in M54.  If 15\% of our RRL sample are members
of Sgr, the $\langle V({\rm RR}) \rangle$ estimate for M54 quoted
above may be systematically too faint by $\sim$0.02 mag.


%%%%
\subsection{Reddening toward M54}

In \markcite{sl95}SL95, we determined the reddening toward M54 to be
$E(B-V) = 0.13 \pm 0.02$ mag from the shape and color of the M54 red
giant branch (\markcite{raj94}Sarajedini 1994).  The dust-based
extinction map of \markcite{sch98}Schlegel et al. (1998) gives $E(B-V)
= 0.15 \pm 0.02$ mag, while the map of \markcite{bh82}Burstein \&
Heiles (1982), which is based on {\sc Hi} gas and galaxy counts, gives
$0.14 \pm 0.02$ mag.

We can obtain another, independent reddening estimate from the color
of the RRL.  \markcite{muskkk}MUSKKK showed that the intrinsic colors
of ab-type RRL, measured at minimum light (phases between 0.5 and
0.8), are universal: $\langle (V-I)_{0,min} \rangle = 0.58 \pm 0.03$
mag, independent of metallicity and period.  From Table 9, we find the
apparent color of the 29 RRab in M54 with four or more observations
during minimum light to be $\langle (V-I)_{min} \rangle = 0.76 \pm
0.01$ mag.  The value does not change if we include stars with fewer
data points at minimum light.  The color distribution is very narrow,
the standard deviation being 0.03 mag, and there is no correlation
with obvious parameters such as period or distance from the cluster
center.

The color excess toward M54 is thus $E(V-I) = 0.18 \pm 0.04$ mag,
where we have included a systematic error of 0.02 mag in the mean
apparent color due to uncertainty in the photometric zero point
(\markcite{sl95}SL95).  Using the relations of \markcite{dwc78}Dean et
al. (1978), this reddening corresponds to $E(B-V) = 0.14 \pm 0.03$
mag.
%Dean, Warren & Cousins (1978, MNRAS, 183, 569)  
Thus the RRL-based result is in excellent agreement with the three
previous reddening estimates.  In the remainder of this paper, we
adopt $E(V-I) = 0.18 \pm 0.02$ mag and $A_V / E(V-I) = 2.6$.


%%%%
\subsection{Distance Modulus}

%m - M - Av = 5 log d - 5
In order to compute the distance modulus to M54, we must assume a
value for the RR Lyrae absolute magnitude, $M_V({\rm RR})$.
\markcite{bri99}Chaboyer (1999) reviews the recent, post-{\it
Hipparcos} results.  We adopt his relation \[ M_V({\rm RR}) =
0.23~([Fe/H] + 1.6) + 0.56.\] The slope is fairly reliable, but the
zero-point has an uncertainty of 0.12 mag stemming from lingering
disagreements between the various methods used to obtain $M_V({\rm
RR})$.  We note that a systematic change in the zero-point affects all
our results in concert, e.g., affecting the relative ages less than
the absolute ages.

Adopting $[Fe/H] = -1.79$ for M54 (\markcite{sl95}SL95) and using the
above values for $\langle V({\rm RR}) \rangle$ and $E(V-I)$, we obtain
a true distance modulus of $(m-M)_0 = 17.19 \pm 0.03$ (internal) $\pm
0.12$ (systematic) mag, or a distance of 27.4 kpc.  This result is in
excellent agreement with \markcite{sl95}SL95.

In \S 4.1, we showed that the reddening in our Sgr field is
indistinguishable from that in our M54 field.  In \markcite{sl95}SL95,
we showed that the distance to stars in the Sgr field are
indistinguishable from those in M54.  In the remainder of this paper,
we use $(m-M)_0 = 17.19$ and $E(V-I) = 0.18$ mag for both M54 and for
Sgr.

%%%%
\subsection{Horizontal Branch Morphology}

In \markcite{sl95}SL95, we estimated the HB morphology of M54 to be
$(B-R)/(B+V+R) = 0.87$, where $B$, $V$, and $R$ are the numbers of
blue HB stars, RR Lyrae variables, and red HB stars, respectively
(\markcite{ldz94}Lee et al. 1994).  We estimated $V$ from the
\markcite{hsh73}Sawyer-Hogg (1973) catalog.  In the present study, we
have discovered new RR Lyrae members of M54, so we can update this
number.

In \markcite{sl95}SL95, we also assumed that $R = 0$.  Though there is
evidence for a red HB in Figure 3 of \markcite{sl95}SL95, we
attributed this feature to the Sgr field population, rather than to
M54, based on three pieces of evidence.  First, the luminosity
functions of our Sgr field and the Sgr field of
\markcite{muskkk}MUSKKK had corresponding peaks at the magnitude level
of the ``anonymous'' red HB (AHB) in the M54 field (Figure 10 of
\markcite{sl95}SL95), indicating that the feature was ubiquitous in
Sgr.  Second, the cumulative radial distribution of AHB stars (Figure
11 of \markcite{sl95}SL95, also Figure 12 herein) matched those of the
Sgr red clump and red giant branch, and disagreed with those of the
M54 blue HB and red giant branch.  Thus the AHB stars were not
spatially concentrated around the cluster.  Finally, we noted that the
AHB was $\sim$0.15 mag fainter than the HB fiducial of M68, when the
latter was fitted to the blue HB stars in M54 (Figure 6 of
\markcite{sl95}SL95).

Recently, this interpretation was challenged by \markcite{srn98}Smith
et al.  (1998).  They employed statistical CMD subtraction techniques
on the data from \markcite{sl95}SL95.  In one test, they subtracted
stars with $R > 200$ arcsec from the cluster center away from the
stars inside that radius (see their Figure 2c).  However,
\markcite{tra95}Trager et al. (1995) found the tidal radius of M54 to
be 460 arcsec, so in this test, \markcite{srn98}Smith et al. were
subtracting one sample of cluster stars away from another.  This
biases the appearance of the HB, and explains why their subtracted CMD
has so few HB stars.  Clearly, this result and the conclusions derived
from it are incorrect.

In a second test, \markcite{srn98}Smith et al. (1998) subtracted the
Sgr-field CMD of \markcite{sl95}SL95 off the M54 CMD.  This procedure
is more nearly correct.  However, there are still many Sgr red clump
and red giant branch stars visible in the CMD (see their Figure 2b),
indicating that too few Sgr stars were subtracted away (see \S 10).
In our subtractions (\S 4.2), the anonymous red HB disappears.
However, as we lower the Sgr/M54 normalization, a hint of the red HB
reappears.  Thus, our subtractions can not ascertain without doubt
whether the anonymous red HB belongs to Sgr or to M54.

Recently, additional evidence in favor of the AHB belonging to Sgr has
come from the wide-field photometry of \markcite{bel99a}Bellazzini et
al. (1999a).  In their fields ``SGR12'' and ``SGR34'', a weak AHB can
be seen at $V-I < 1.0$ mag.  Photometry of stars in the core of M54
are scheduled to be obtained soon with the {\em Hubble Space
Telescope} (GO 6701).  The presence or absence of red HB stars in
these data should settle the question of the membership of this
population.  Until then, the exact HB morphology of M54 remains
uncertain.

We compute two provisional values for $(B-R)/(B+V+R)$, one including
and one excluding a contribution from red HB stars.  From a CMD
generated from stars lying between 1.0 and 4.0 arcmin from the cluster
center, we count $B = 123 \pm 19$ blue HB stars, where the error
enhancement over Poisson statistics reflects the uncertainty in
identifying blue HB stars along the faint end of the extreme blue HB.
In this same radial range, we count 44 RR Lyrae with the correct
magnitude to be cluster members.  Roughly $3\pm 2$ of the RRL in our
annulus are Sgr members (see \S 6), so $V = 41 \pm 7$ stars.
% err = sqrt { ( sqrt(44) )**2 + (2)**2 } = 6.9
Counting the red HB stars is complicated by the heavy contamination
by foreground stars.  We created a luminosity function by counting
stars with $0.72 < V-I < 0.95$ in bins of $\Delta V = 0.25$ mag.  The
excess of stars in the bin containing the red HB ($18.05 < V < 18.30$)
was found by fitting a line to the eight surrounding points in the
luminosity function, and subtracting off the amount predicted at
$V = 18.20$ mag (50 stars) from the observed amount (85 stars).  Thus
$R = 35 \pm 10$ if the red HB is associated with M54, and $R = 0$ if
it is not.
% err = sqrt { ( sqrt(85)**2 + (rms=3.12)**2 } = 9.7
From these values, we compute $(B-R)/(B+V+R) = 0.44 \pm 0.09$ in the
former case, and $0.75 \pm 0.04$ in the latter.

The $R = 0$ case agrees reasonably well with the \markcite{sl95}SL95
value of 0.87.  The difference is due mostly to the newly discovered
RR Lyrae presented in \S 5.  The other case agrees quite well with the
``far field subtraction'' of \markcite{srn98}Smith et al. (1998), but
disagrees strongly with their ``near field subtraction'' result, which
we showed above to be incorrect.  \markcite{srn98}Smith et al. (1998)
adopted a final value by averaging the HB morphology estimates from
their two subtractions, obtaining $(B-R)/(B+V+R) = 0.20$.
%This result was instrumental to their claim that the Sgr globulars
%are fundamentally different from their Galactic counterparts, and
%that the Galactic halo can not be constructed of accreted dwarf galaxies.
Our much bluer HB morphology estimates place M54 back among the
Galactic globulars in a plot of HB morphology {\it versus} [Fe/H] (see
Figure 1 of \markcite{srn98}Smith et al.).  If $R = 0$, the cluster
may be associated with the ``old halo'' Galactic globulars
(\markcite{rjz96}Zinn 1996), while if the redder HB morphology value
is correct, an association with the ``young halo'' clusters is
preferred.  We re-emphasize that if the red HB does belong to M54, an
explanation will be required for its faintness relative to the blue HB
and RRL stars.
%The best association is made with the ``young halo'' clusters, which 
%are thought to have originated in dwarf galaxies that were later 
%accreted by the growing Galaxy (Zinn 1993), but an association with 
%the ``old halo'' Galactic globulars can not be ruled out if $R = 0$.  
Nevertheless, it is clear that whether or not the red HB belongs to
M54, the three metal-poor Sgr globulars (M54, Arp 2, and Ter 8)
present no complication to the accretion picture of Galactic halo
formation, contrary to the claims of \markcite{srn98}Smith et
al. (1998).

The unremarkable position of M54 in the HB morphology {\it versus}
[Fe/H] diagram is also reassuring given that the main sequence
turn-off age of this cluster is typical of Galactic globulars at its
metallicity (\S 7, \markcite{ls97}LS97, \markcite{mar98}Marconi et
al. 1998).  Metallicity and age effects are sufficient to describe its
HB morphology, and no additional ``second'' parameters need to be
invoked.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Age of M54 and the Sgr Star Clusters}

\subsection{Comparison with Cluster Fiducial Sequences}

One method by which we can probe the age of M54 is by direct
comparison with the fiducial sequences of other clusters. For this
purpose, we have selected M92 ($[Fe/H] = -2.24$) and M5 ($[Fe/H] =
-1.40$), which bracket M54 in metallicity and possess high quality
$V-I$ CCD photometry available from the literature. Figure 14
%(m54\_fids.eps) 
shows our field-decontaminated M54 photometry from Figure 7(d). The
dashed line is the fiducial sequence of M92 and the solid line is the
fiducial of M5, both taken from \markcite{jb98}Johnson \& Bolte
(1998).  These have been registered to the observational HR Diagram
using the $V({\rm HB})$, $[Fe/H]$, and $E(V-I)$ values given in Table
13, along with our adopted luminosity-metallicity relation
(\markcite{bri99}Chaboyer 1999).

The fiducial sequence comparison reveals that the age of M54 is
comparable to those of M92 and M5, with the M54 data located between
the two fiducials as one would expect based on the relative
metallicities.  This analysis supports the results of similar work by
\markcite{ls97}LS97, who used M68 and M5 as comparison clusters.  M68
appears to be among the youngest metal-poor globulars (13 Gyr), while
M92 is perhaps the oldest (18 Gyr), and M5 is typical of
intermediate-metallicity clusters (15 Gyr; \markcite{cds96}Chaboyer et
al. 1996).  Thus, M54 falls nicely within the age-metallicity relation
defined by the Galactic globular clusters (e.g.,
\markcite{cds96}Chaboyer et al. 1996, Figure 1).  In the next section,
we will apply a more precise relative age technique that is likely to
shed more light on this issue.

\subsection{The Magnitude of the Subgiant Branch}

Since the subgiant branch (SGB) of M54 is so clearly delineated in the
CMDs of Fig. 7, 
%i.e. the decontaminated CMD 
we have investigated the use of this feature to probe the age of
M54. We should measure the $V$ magnitude of the SGB at a specific
$V-I$ color chosen so as to minimize contamination from main sequence
turnoff stars and lower RGB stars scattered onto the SGB. We find that
a $V-I$ color 0.21 mag bluer than $(V-I)_g$ (the color of the RGB at
the level of the HB) allows the construction of the luminosity
function (LF) which optimally isolates the location of the SGB. Using
Gaussian fits to the cluster LF at this color yields the value of
$V({\rm SGB})$ for M54 listed in Table 14.  The quoted uncertainty
represents the standard error of the mean. In a similar manner, we
measured the $V({\rm SGB})$ values of Terzan 8
(\markcite{mont98}Montegriffo et al. 1998), M92, and M5
(\markcite{jb98}Johnson \& Bolte 1998). In the latter two clusters
(and for Ter 7), the SGBs are not populated adequately to facilitate
the construction of the LFs; however, the photometry defines the SGB
exceptionally well allowing us to simply compute the median magnitude
of stars within $\Delta$$(V-I) = \pm 0.02$ around the $(V-I)_g - 0.21$
value for each cluster. Table 14 lists these measured quantities, and
Figure 15 shows the Gaussian fits to the observed LFs.

For the remaining two clusters in Sgr, Arp 2 and Ter 7, published
photometry does not exist in the $VI$ filter passbands, but it does
exist in $BV$ (\markcite{rb95a}Buonanno et al. 1995a,
\markcite{rb95b}1995b).  In order to measure $V({\rm SGB})$ in these
clusters and remain consistent with the values for the clusters
measured thus far, we rely upon the $(V-I)_0$--$(B-V)_0$
transformation established by \markcite{zb96}Zinn \& Barnes (1996). To
test the veracity of this technique, we utilize the $BVI$ CCD
photometry of M5 from \markcite{esa96}Sandquist et al. (1996). The
$V({\rm SGB})$ of M5 was measured from the $VI$ photometry as well as
the $BV$ data after transformation to $VI$ via the \markcite{zb96}Zinn
\& Barnes equation. These two determinations of $V({\rm SGB})$
differed by only 0.01 mag, well within the errors, thereby supporting
the use of the \markcite{zb96}Zinn \& Barnes (1996) transformation to
convert the $BV$ photometry of Arp 2 and Ter 7 to the $VI$ system.
The resultant values of $V({\rm SGB})$ for these clusters are also
given in Table 14, and the age diagnostic $\Delta$$V_{SGB-HB}$ is
presented for each cluster.

To convert these $\Delta$$V_{SGB-HB}$ values to relative ages, we rely
on the Revised Yale Isochrones (RYI, \markcite{gdk87}Green et
al. 1987) to calibrate age as a function of $[Fe/H]$ and $M_V({\rm
SGB})$, which is valid in the range $-2.3 < [Fe/H] < -0.70$.  Since we
are required to measure $M_V({\rm SGB})$ at $(V-I)_g - 0.21$, we have
used relations between $(V-I)_{o,g}$ and $[Fe/H]$
(\markcite{raj94}Sarajedini 1994) to infer the theoretical value of
$(V-I)_{o,g}$ which is then utilized to measure $M_V({\rm SGB})$ at
the same point on all of the isochrones. It is important to point out
that use of the \markcite{raj94}Sarajedini (1994) relations between
$(V-I)_{o,g}$ and $[Fe/H]$ maintains the internal consistency of our
method because the abundances of M54, Arp 2, Ter 8, and Ter 7 have
been determined using these same relations. In any event, we note that
the exact algorithm used to set the location of the fiducial
$(V-I)_{o,g}$ point for the isochrones is largely irrelevant; this is
because the SGBs of isochrones with different ages are essentially
parallel leaving the relative ages unchanged (the SGBs are no longer
parallel for ages $\lesssim 6$ Gyr).  Table 14 shows the ages of M92,
Ter 8, Arp 2, M54, M5, and Ter 7, assuming the $M_V({\rm
RR})$--$[Fe/H]$ relation of \markcite{bri99}Chaboyer (1999). For Ter
7, which possesses a red HB, we offset the synthetic red HB models of
\markcite{raj95}Sarajedini et al. (1995) so that they are consistent
with our adopted $M_V({\rm RR})$ relation.  Column 6 lists the ages
assuming scaled solar abundances. In summary, we find that the oldest
Sgr globular clusters are similar in age to the Galactic globular
clusters, while Ter 7 is some 5 to 6 Gyr younger than these clusters.
In the case of M54, these results support our findings based on the
fiducial sequence comparisons.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Ages of the Sgr Field Populations}

\subsection{Isochrone Fits}

We showed in Figure 6(a) the CMD of the Sgr field located 12
arcminutes north of M54. Several features of this diagram are relevant
to the question of the age(s) of the Sgr field. First, the vertical
extent of the dominant main sequence turnoff (MSTO) appears to span
$\sim$1 mag with at least two SGBs present, one at $V\sim$20.4 and the
other at $V\sim$20.8. These features are also present in the
field-subtracted CMDs shown in Figures 6(c) and 6(d). The presence of
two or more SGBs indicates that star formation in Sgr did not occur in
a single event.  \markcite{bel99b}Bellazzini et al. (1999b) also
suggested this possibility, but the poorer quality of their data near
the MSTO made a definitive statement impossible; our data show the
evidence for extended star formation much more clearly.

The fact that the brighter (younger) SGB extends from a turnoff with
roughly the same $V-I$ color as the fainter (older) SGB indicates that
the stellar population associated with the former is likely to be more
metal rich than that of the latter.

With these points as the foundation, we now discuss the results of
isochrone comparisons to the Sgr CMD. We make use of the theoretical
isochrones published by \markcite{ber94}Bertelli et al. (1994).  We
employ our estimates of the distance modulus and reddening from \S
6.5.  Results in the literature suggest a metallicity range in Sgr
from $[Fe/H]\sim-0.5$ to $[Fe/H]\sim-1.3$ (e.g., \markcite{sl95}SL95,
\markcite{igi97}Ibata et al. 1997 and references therein).
\markcite{bel99a}Bellazzini et al. (1999a, \markcite{bel99b}1999b)
suggest the presence of a trace population of stars at $[Fe/H]
\lesssim -2$, but so few stars would be poorly represented near the
MSTO, and we can not reliably identify them in our CMDs.

Figure 16(a) shows the field-subtracted Sgr CMD from Figure 6(d),
together with selected \markcite{ber94}Bertelli et al. (1994)
isochrones for $[Fe/H] = -1.3$ dex.  Figures 16(b) and 16(c) show the
same observational data plotted with isochrones for $[Fe/H] = -0.7$
and --0.4 dex, respectively.

%CAPTION - FIGURE NN: (a) The squares are the field-subtracted Sgr CMD
%from Figure 3d?.  The curves are Bertelli et al. (1994) isochrones
%for $[Fe/H] = -1.3$ dex and ages (from left to right) 0.004, 1.0,
%2.0, 3.2, 4.0, 6.3, 8.0, 10.0, 12.0, and 13.8 Gyr.  (b) The same as
%in panel (a) but the isochrones are for $[Fe/H] = -0.7$ dex.  (c) The
%same as (a) but the isochrones are for $[Fe/H] = -0.4$ dex.  (d) The
%squares are the field-subtracted M54 CMD from Figure 4d.  The lower
%set of isochrones are for $[Fe/H] = -1.7$ and ages 10.0, 12.0, 13.8, 
%and 15.8 Gyr.  The upper set of isochrones are for $[Fe/H] = -0.4$ and
%ages 2.0, 2.5, and 3.2 Gyr


The fainter MSTO/SGB sequence, with $V({\rm SGB}) \approx 20.8$, appears to
be fit very well by the isochrone with $[Fe/H] = -1.3$ and an age of
10 or 11 Gyr.  The curvature of the more metal-rich isochrones around
the SGB do not match the observations, nor do the RGB colors, though
we cannot rule out $[Fe/H] = -0.7$ and an age of 7 Gyr.

By contrast, none of the $[Fe/H] = -1.3$ isochrones fit the brighter
MSTO/SGB sequence, with $V({\rm SGB}) \approx 20.4$; the isochrone
colors are too blue near the MSTO.  This sequence falls between the 4
and 6 Gyr isochrones for $[Fe/H] = -0.7$, and near the 4 Gyr isochrone
for $[Fe/H] = -0.4$ dex.  The color of the RHB seems to favor $[Fe/H]
= -0.7$ for this population.  A explanatory note about the brighter
sequence is required.  One tends to see the SGB as the line of points
which slopes upwards as it gets redder across $V-I = 0.95$, parallel
to the lower sequence.  However, the isochrones slope sharply downward
here, indicating that the true SGB is the downward sloping line of
points at $V-I = 0.90$ and $V = 20.6$ mag.  The brighter line of
points is probably an artifact of the subtraction --- an extension of
the ridge of unsubtracted stars near ($V-I$, $V$) = (1.3, 21.6) and
the unsubtracted clump at (0.85, 18.7) mag.  The 5 Gyr isochrone for
$[Fe/H] = -0.7$ follows the claimed SGB sequence nicely.

\markcite{muskkk}MUSKKK, \markcite{mar98}Marconi et al. (1998), and
\markcite{bel99a}Bellazzini et al. (1999a) have noted the presence of
bright, blue stars in Sgr.  Though it is possible that they are blue
stragglers, these stars are generally interpreted as being members of
a very young population in Sgr.  In Figure 16, the stars with $V-I <
0.6$ and $V < 21$ do not form a tight sequence indicative of a coeval
population.  Many of these stars lie near the 1 Gyr, $[Fe/H] = -0.4$
isochrone, but the dispersion of stars both blue-ward and red-ward of
this curve suggests that Sgr has experienced ongoing, low-level star
formation over the last 2--3 Gyr.  The spur of MSTO stars brighter
than the upper MSTO/SGB sequence, terminating at $V \approx 20$, is
consistent with a 2.5 Gyr isochrone at $[Fe/H] = -0.4$ (or somewhat
younger if it is more metal-rich).  We can not tell from our CMDs
whether the 1.0 and 2.5 Gyr populations are discrete or represent a
continuous period of star formation.

Figure 16(d) shows the field-subtracted CMD of M54 presented in Figure
7(d).  The lower set of isochrones are for $[Fe/H] = -1.7$, very close
to the metallicity of M54.  Though the RGB color is slightly
mismatched, the SGB and MSTO regions of M54 appear to be represented
nicely by the isochrones with ages of 14--16 Gyr.

A spur of MSTO stars rises above the M54 MSTO in Figure 16(d), up to
$V \approx 20$, very similar to the spur seen in the Sgr CMD (Figure
16c).  The upper set of isochrones, for $[Fe/H] = -0.4$, demonstrate
that 2.5 Gyr is a reasonable estimate for the age for this population.
Hints of the SGB and upper RGB can be seen along this isochrone.  The
region where the SGB and RGB join is not populated, but the density of
bulge stars is high here (see Figure 6b), so the elimination of a
sparse population is not surprising (\markcite{bel99b}Bellazzini et
al. 1999b).  We believe that the 2.5 Gyr population seen in Figures
16(c) and 16(d) are the same population.  However, since the Sgr CMD
observed 12 arcmin from M54 was subtracted from the M54 data, the fact
that this population appears in Figure 16(d) indicates that the
relative density of the 2.5 Gyr population must be higher near M54.
We shall investigate this further in \S 10.

To summarize this section, we find that M54 has an age of $\sim$15 Gyr
on the \markcite{ber94}Bertelli et al. (1994) scale.  The Sgr field is
composed of several different populations, which we characterize as
follows: (a) $[Fe/H] = -1.3$ and an age of 10--11 Gyr, (b) $[Fe/H] =
-0.7$ and an age of 5 Gyr, and (c) a sprinkling of younger stars with
$[Fe/H] \approx -0.4$ and ranging in age from perhaps 0.5 to 2.5 or 3
Gyr.  Given the photometric errors and the complicated mix of stellar
populations which overlap both spatially and in the CMD, it is
difficult to provide definitive ages and metallicities for these
populations --- we give our best estimates.  Furthermore, it is
difficult to say whether the features in the CMD depict discrete star
formation events of short duration, more extended epochs, or even
continuous star formation with a variable rate.  Finally, we
acknowledge the difficulty in obtaining accurate absolute ages (e.g.,
\markcite{cds96}Chaboyer et al. 1996).  The relative ages determined
in this section are more secure than the absolute ages.  In the
following section, we compare the observed Sgr sequences with those of
several well-studied Galactic clusters in an effort to corroborate our
isochrone-based findings.


\subsection{Fiducial Comparisons}

To further explore the ages and metallicities of Sgr, it is
instructive to compare our statistically decontaminated CMD with
fiducial sequences of open and globular clusters with similar
properties.  The upper panel of Figure 17 shows the fiducials of the
Galactic globular clusters M92 and M5 (\markcite{jb98}Johnson \& Bolte
1998) and the lower panel illustrates the sequences for the open
clusters M67 (\markcite{km93}Montgomery et al. 1993) and NGC 2204
(\markcite{kas97}Kassis et al. 1997). The reddenings for the globular
clusters come from \markcite{har96}Harris (1996) while the distances
have been computed given the data in Table 13 and the RR Lyrae
luminosity--metallicity relation of \markcite{bri99}Chaboyer
(1999). In the case of the open clusters, we have adopted the
reddenings and distance moduli from the study of
\markcite{raj99}Sarajedini (1999). \markcite{twa99}Twarog et
al. (1997) indicate that NGC 2204 has a metallicity of $[Fe/H] =
-0.34$ and M67 has $[Fe/H] = 0.00$.

From Figure 17, we can draw the following conclusions. The top panel
indicates that the mean stellar population in Sgr has a metal
abundance that is higher than that of M92 and M5 as evidenced by the
mismatch in the RGB positions. The oldest population in Sgr has a main
sequence turnoff that is brighter than that of M5 suggesting an age
(10-12 Gyr) that is slightly younger than the canonical age (14-16
Gyr) of the Galactic globulars. 

The open cluster fiducial comparisons indicate that a solar metal
abundance (i.e., M67) is likely to be too high for the mean population
in Sgr whereas an abundance of $[Fe/H] \approx$ --0.4 to --0.3 or less
is more appropriate. As for the ages, the youngest stars in Sgr are
younger than NGC 2204 ($\sim$2 Gyr, \markcite{raj99}Sarajedini 1999),
but it is difficult to find $VI$ photometry for a cluster with
metallicity near $[Fe/H]$$\sim$ --0.4 and an age of 1 Gyr.  The
intermediate age population appears to have an age close to that of
M67 ($\sim$4 Gyr, \markcite{raj99}Sarajedini 1999).  All in all, the
ages and metallicities of the Sgr stellar populations as indicated by
the fiducial sequence comparisons are in line with those suggested by
the isochrone comparisons.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Age-Metallicity Relation of Sgr}

So far in our discussion, we have determined ages and metallicities
for the star clusters and field populations in Sgr. The former were
studied using the difference in magnitude between the HB and the SGB
calibrated with the Revised Yale Isochrones, while the latter were
investigated via comparison with \markcite{ber94}Bertelli et
al. (1994) isochrones.  The calibration of the SGB age method could
not be done using the Bertelli isochrones, since the most metal-poor
available are for $[Fe/H] = -1.7$, while the clusters go as low as
--2.2 (M92).

In order to interpret the results for the cluster and field components
of Sgr, it is necessary to place them on the same age scale. To this
end, we devised a calibration between age, $[Fe/H]$, and $M_V({\rm
SGB})$ using the isochrones of \markcite{ber94}Bertelli et al. (1994).
For the two clusters in our sample with metallicities in the relevant
range (i.e., between $-1.7 < [Fe/H] < -0.7$), Ter 7 and M5, we find
ages of $8.3 \pm 1.8$ Gyr and $13.6 \pm 0.7$ Gyr, respectively. These
are $\sim$1.0 Gyr younger than those derived via the RYI.  We
therefore subtract 1.0 Gyr from the RYI ages in Table 14 to give ages
on the Bertelli scale.  Note that the age thus derived for M54, $14.7
\pm 0.5$ Gyr, is in excellent agreement with the age of 15 Gyr from
the isochrone match shown in Figure 16(d).

Our final age estimates for the field populations and individual
clusters in Sgr are listed in Table 15.  The error estimates on the
ages reflect the uncertainty in the age {\it relative} to a population
of similar abundance.  The error in the absolute age of a given object
is about 15\%.  If we were to adopt a different $M_V({\rm
RR})$--$[Fe/H]$ relation, the entire manifold of ages would shrink or
expand proportionally.
% (i.e, like an accordion).

We are now in a position to synthesize the ages and metallicities of
the field and cluster populations of Sgr into a coherent picture of
its star formation history. Figure 18(a) shows the ages and
metallicities of Sgr cluster (open symbols) and field populations
(filled symbols).  It is evident from the top panel of Figure 18 that
there is a strong correlation between age and metallicity with the
most metal-poor components of Sgr being the oldest. This behavior is
qualitatively similar to that shown in Figure 8 of
\markcite{mont98}Montegriffo et al.  (1998) and in
\markcite{bel99b}Bellazzini et al. (1999b).

Utilizing the formalism of \markcite{bm98}Binney \& Merrifield (1998), we can
investigate the predictions of the closed-box model of chemical
evolution in the case of Sgr. Assuming that the yield of each
generation of stars is constant, the variation of metallicity ($Z$)
with time is expressed as
%\begin{eqnarray}
%Z(t) = -p ~~ ln \frac{M_g(t)}{M_g(0)}.
%\end{eqnarray}
\[ Z(t) = -p ~~ ln \frac{M_g(t)}{M_g(0)}. \]
We have assumed a constant star formation rate with time so that the
ratio of gas masses [$M_g(t)/M_g(0)$] is unity at $t = 0$ and is equal
to zero at the present time ($t = 15$ Gyr). The yield is represented
by $p$ and has been adjusted to 0.005 so as to match the zero-point of
the Sgr age-metallicity relation.

Figure 18(b) shows that the observed age-metallicity relation of Sgr is
consistent with the predictions of a closed box continuous star
formation model. This in turn argues that, if we accept the
possibility that Sgr formed stars continuously, then it is likely that
it did so in relative isolation with little or no infall or release of
its interstellar gas.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Stellar Population Gradients in Sgr}

\markcite{sl95}SL95 and \markcite{mar98}Marconi et al. (1998) noted that the more metal-rich
% see sec 5 pp3 of Marconi for their take on it.
component of Sgr is somewhat clustered around M54.
\markcite{bel99a}Bellazzini et al. (1999a) used their wide-field
observations near our Sgr field and near the field of
\markcite{muskkk}MUSKKK $\sim$2 degrees away to indicate that the
density of Sgr stars peaks near M54.  We begin our investigation of
the spatial distribution of the various Sgr field population by
considering the density of RGB stars in our different fields.

%%%%
\subsection{Density Gradients of RGB Stars}

Using the CMDs in Figure 7, we have defined several loci in which to
count RGB stars.  The first contains the metal-poor RGB stars
associated with M54\footnote{The vertices of the z-poor box are
($V-I$, $V$) = (1.10, 17.50), (1.23, 16.20), (1.59, 14.92), (1.77,
15.05), (1.37, 16.26), (1.21, 17.53).}.  For brevity, we will refer to
this locus as the ``z-poor box''.  The ``z-rich box'' contains the
metal-rich ($[Fe/H] = -0.5$) RGB described by
\markcite{sl95}SL95\footnote{The vertices of the z-rich box are
($V-I$, $V$) = (1.29, 17.57), (1.65, 16.26), (2.50, 15.86), (3.20,
17.00), (1.84, 16.45), (1.38, 17.66).}.  The ``mid-z box'' contains
the scattering of RGB stars between these boxes which
\markcite{sl95}SL95 characterized as an intermediate metallicity
($[Fe/H] \approx -1.3$) population in Sgr.  The lack of a distinct RGB
led \markcite{sl95}SL95 and \markcite{bel99b}Bellazzini et al. (1999b)
to suspect that this population exhibits a significant spread in
metallicity.

We shall count stars in four different spatial regions: (1) the M54
field between 0.72 and 4.44 arcmin from the cluster center, (2) the
M54 field outside 4.44 arcmin, (3) the Sgr field about 12 arcmin from
the cluster center, and (4) the \markcite{muskkk}MUSKKK control field
(the RGB stars in our control field are saturated in our photometry).
The areas of these regions and the number of stars in each of the
three CMD boxes are shown in Table 16.

%===============================================================
%TABLE 1.  Red Giant Star Counts.
%
%			Area	Z-Poor	Mid-Z	Z-Rich
%		     (arcmin^2)	
%
%M54 (100-620pix) 	60.4	227	33	43
%
%M54 (>620 pix)		61.4	42	21	26
%
%Sgr			67.9	11	17	12
%
%MUSKKK Control		225.6	30	6	6
%===============================================================

In Figure 19, we express these counts as surface densities in units of
stars per square arcminute.  The error bars reflect simple Poisson
statistics.  We have placed the control sample at $R = 19$ arcmin for
convenience of display.  The top panel shows the distribution of M54
RGB stars.  As expected, the density of stars in the $R = 12$ arcmin
field is small and consistent with the density seen in the control
field.

In panels (b) and (c), the density at $R = 12$ arcmin is significantly
enhanced over the control field --- Sgr exists, as shown initially by
\markcite{igi94}Ibata et al. (1994) and subsequently by many authors
using this technique.  More interestingly, the density of Sgr stars
rises as we approach the center of M54.  The effect is subtle for the
intermediate-metallicity population; the inner point for M54 is
enhanced over the density at 12 arcmin by a factor of 2.2, a
2.5$\sigma$ result.  The effect is much stronger for the z-rich
population; the enhancement factor of 3.9 is significant at the
4.4$\sigma$ level.

To conclude, we find that the density of Sgr field stars increases
with increasing proximity to M54.  The $[Fe/H] \approx -1.3$
population shows a mild concentration toward M54, while the metal-rich
population is more strongly concentrated.  If this pattern persists at
larger radii, it explains why observations in more distant fields
(e.g., \markcite{muskkk}MUSKKK, \markcite{bel99a}Bellazzini et al
(1999a, \markcite{bel99b}1999b) fail to find as many stars with
$[Fe/H] = -0.5$ as did \markcite{sl95}SL95 and \markcite{mar98}Marconi
et al. (1998) in the viscinity of M54.  Furthermore, it suggests that
subsequent generations of stars may have formed with greater proximity
to M54.

%%%%%
\subsection{Density Gradients of MSTO Stars}

In theory, we could investigate this supposition in more detail by
examining the density distributions of stars near the various main
sequence turn-offs discussed in \S 8.  In practice, the interpretation
becomes more difficult for a number of reasons.  The main complication
is in defining boxes which sample single-aged populations.  Only
the most massive MSTO stars can be isolated completely; for lower
masses, the main-sequences of the different populations blend together
(see the isochrones in Figure 16).  Thus, we can isolate stars
belonging to the bright, blue MSTO stars with ages $\sim$1 Gyr, and
for the ``spur'' of stars above the dominant Sgr MSTOs which we
estimated to be $\sim$2.5 Gyr old.  The older populations are
hopelessly intermingled.

%The additional complications include the following.  The crowding of
%the various populations in the CMD demands the use of very small
%boxes.  The boxes are much smaller with respect to the photometric
%errors of the faint MSTO stars than in the case of the RGB stars.  The
%counts within the small boxes are also more sensitive to small
%displacements of the boxes (i.e., where the boxes are defined to be)
%than are the RGB counts.  Photometric incompleteness also affects the
%star counts at the MSTO more than at the RGB.  Finally, the MSTOs of
%M54 and the oldest Sgr field population overlap (Figure 16), making
%any trustworthy separation of the two impossible.  The only advantage
%of studying the MSTO stars is their greater abundance, since they
%sample a region of the CMD where stars are evolving more slowly than
%on the RGB.

Again using the CMDs in Figure 7, we have defined two boxes in which
to count MSTO stars.  The first contains the $\sim$1 Gyr
population\footnote{The vertices of the 1.0 Gyr box are ($V-I$, $V$) =
(0.32, 18.98), (0.63, 20.62), (0.61, 21.34), (0.30, 19.70).}.  The
second box contains stars in the $\sim$2.5 Gyr old ``spur''
population\footnote{The vertices of the 2.5 Gyr box are ($V-I$, $V$) =
(0.65, 19.90), (0.74, 19.80), (0.79, 20.36), (0.63, 20.62).}.
%The ``5 Gyr box'' contains stars near the ``upper Sgr MSTO''
%discussed in \S N\footnote{The vertices of the 5 Gyr box are 
%($V-I$, $V$) = (0.63, 20.62), (0.79, 20.36), (0.82, 20.7), (0.616
%21.124).}.  %The ``old box'' contains both the ``lower Sgr MSTO'' 
%(age $\approx 11$ Gyr, \S N) and the MSTO of M54.\footnote{The 
%vertices of the ``old box'' are ($V-I$, $V$) = (0.616, 21.124), 
%(0.82, 20.7), (0.89, 21.48), (0.596, 21.84).}.  These boxes are 
%designed to capture the upper MSTO regions of the populations in 
%question, but the contents of a given box should
%not be construed as containing only stars in the population for which
%it was intended.  In particular, lower mass, less evolved main
%sequence stars from the younger populations will be present in the box
%for a given MSTO.
We count stars in same four radial regions described in \S 1.1, though the
boundaries of the M54 annuli were shifted outward to 2.15 and 4.66
arcmin to avoid crowding effects on the faint MSTO stars.  In
addition, we count the MSTO stars in our control field.  The areas of
these regions and the number of stars in each of the two CMD boxes
are shown in Table 17.

%===============================================================
%TABLE 2.  MSTO Star Counts.
%
%			Area	1.0Gyr	2.5Gyr	5Gyr	Old
%		     (arcmin^2)	
%
%M54 (300-650pix) 	53.7	64	189	481	2660
%
%M54 (>650 pix)		55.2	35	102	205	1311
%
%Sgr			67.9	44	88	243	1058
%
%Control		131.5	4	91	76	111
%
%MUSKKK Control		225.6	10	106	103	182
%===============================================================

In Figure 20, we express these counts as surface densities.  The error
bars and control sample radii are as described for Figure 19.  As for
the RGB case discussed above, the two youngest populations are
concentrated around M54, with density enhancements of $\sim$2 inside 6
arcmin from the cluster center.  It is worth noting that these radial
patterns are not artifacts of photometric detection incompleteness; if
the Sgr populations were uniformly distributed across the field, our
counts would tend to drop at smaller radii, where the stellar crowding
due to M54 stars increases.  If anything, the true density rise at
small radius is underestimated in Figure 20.

Finally, we note that counts inside a box designed to isolate the 5
Gyr population discussed in \S 8 show a similar radial pattern.
However, it is difficult to estimate the contaminating effects of the
two younger populations, and we do not consider the 5 Gyr population
further.

Figure 20 demonstrates that the two youngest populations are centrally
concentrated around M54, as expected from the distribution of
metal-rich ($[Fe/H] \approx -0.5$) RGB stars shown in Figure 19(c).
It is unfortunate that we can not reliably test the radial
distributions of the older populations, since they might indicate
whether the spatial extent of star formation varied from one episode
to the next.

%%%%
\subsection{M54 as the Nucleus of Sgr}

Given the evidence presented above, it is clear that the Sgr field
populations are concentrated around M54 with a characteristic scale of
a few arcminutes.  \markcite{bel99a}Bellazzini et al. (1999a) found
evidence for clumping around M54 on the scale of several degrees, and
\markcite{igi94}Ibata et al. (1994) showed that M54 resides globally
in the densest region of Sgr.  One can easily envision that this old,
massive globular cluster lies at the bottom of Sgr's gravitational
potential well (the current interaction between Sgr and the Milky Way
has probably disrupted the previously smooth and well-ordered stellar
distributions in Sgr).
%-- at least, the bottom prior to the current encounter between Sgr 
%and the Milky Way.  
Gas released from the oldest generations of evolving stars in Sgr
would tend to dissipate its orbital energy and settle around M54, thus
forming a preferred site for subsequent generations of metal-enriched
stars.  A complication to this picture is that stars from the 1 Gyr
population have been detected several degrees away from M54
(\markcite{muskkk}MUSKKK, \markcite{bel99a}Bellazzini et al. 1999a).
Perhaps some metal-rich stars formed during the collision of
evolution-expelled gas clouds far from the bottom of the potential
well.

Was the density enhancement acquired, rather than natal?
\markcite{srn98}Smith et al. (1998) raised the possibility that M54
might have captured stars from the Sgr field.  They referred to the
simulations of \markcite{bic97}Bica et al. (1997), who found that a
$10^5 M_\odot$ globular cluster orbiting in the Galactic bulge might
capture several thousand bulge stars over the course of 1 Gyr.
\markcite{srn98}Smith et al. (1998) noted that the larger mass of M54
and the lower velocity dispersion of the Sgr field would increase the
rate of stellar capture in M54.  Detailed capture simulations designed
for the case of M54/Sgr would put this possibility on a firmer
footing.

In a similar vein, \markcite{vdb96}van den Bergh (1996) suggested that
M54 may represent the merger of two globular clusters, the larger an
old, metal-poor system (15 Gyr, $[Fe/H]=-1.8$) and the smaller a
younger, metal-rich system like Terzan 7.  \markcite{vdb96}van den
Bergh notes that the ability for two clusters to bleed off angular
momentum by ejecting stars and spinning up enhances their ability to
merge, relative to the capture of a single star by a cluster.  While
intriguing, this scenario does not explain the fact that both the 1
Gyr and 2.5 Gyr populations are centrally concentrated, unless the
merger involved two young, metal-rich clusters.  This seems less
likely, though multiple cluster mergers at the potential well center
is considered a viable mechanism for the formation of nuclei in dwarf
elliptical galaxies (\markcite{mil98}Miller et al. 1998).
%is considered a viable mechanism for the formation of nuclei in
%dwarf elliptical galaxies (Miller et al. 1998).

\markcite{sl95}SL95 discussed the possibility that Sgr is a nucleated
dwarf elliptical galaxy, with M54 as its nucleus.  Their main
objection to this conclusion was the blue color of M54 relative to the
red color of the surrounding, more metal-rich field populations; the
evidence then current indicated that nuclei tend to have the same
color as the surrounding galaxy light (cf. \markcite{kcf93}Freeman
1993).  More recently, cases of blue galactic nuclei have surfaced.
For example, the galactic nucleus of VCC 1254 in the Virgo Cluster is
bluer than its galaxy light (\markcite{pd97}Durrell 1997).  Given this
and the improved evidence that the density of stars and mix of stellar
populations in the Sgr field changes with proximity to M54, we are
again encouraged to suggest that M54 is the nucleus of the Sgr dwarf
elliptical galaxy.


%\section{M54 as the Galactic Nucleus of Sgr}
% \subsection{Radial Light and Color Profiles}
% \subsection{Comparison with Globular Clusters and Galactic Nuclei}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Summary and Conclusions}
 
In this paper, we describe the deep $VI$ photometry of M54 originally
presented in \markcite{ls97}LS97.  We also present and describe
similar quality photometry of a field $\sim$12 arcmin North of M54,
whose bright stars we discussed in \markcite{sl95}SL95, along with
photometry of a control field on the opposite side of the Galactic
bulge.  We produce color-magnitude diagrams of these fields, and
perform statistical subtractions to remove the foreground populations.
We use these diagrams to derive the ages of the various stellar
populations in Sgr and thereby outline the history of star formation
in this nearby dwarf galaxy.

As a prelude to the age analysis, we determined several important
cluster parameters from the variable stars in the cluster.  Ours is
the first CCD-based variable star survey for M54.  In it, we detected
and obtained light curves for 60 ab-type RR Lyrae and seven c-type RR
Lyrae, along with two additional short period pulsating variables,
four eclipsing binary stars, and 18 candidate long period variables
(Mira, SR, or IR).  Together with the previous surveys, the number of
known or suspected variables in the direction of M54 now totals 120.

The mean periods of the RRL in M54 indicate that it is an Oosterhoff
type-I cluster.  To our surprise, we were not able to isolate RRL
belonging to Sgr in the M54 field using the period-amplitude diagram,
thought is is well-established that Sgr contains field RRL.  We
discuss this result in the context of the period-shift interpretation
of \markcite{cs99}Clement \& Shelton (1999).  We reevaluate the
horizontal branch morphology of M54, which was recently questioned by
\markcite{srn98}Smith et al. (1998).  We find $(B-R)/(B+V+R) = 0.44$
if the ``anonymous RHB'' described by \markcite{sl95}SL95 is
associated with M54, or 0.75 if it belongs to Sgr.
\markcite{sl95}SL95 and the data of \markcite{bel99a}Bellazzini et
al. (1999a) provide support for the latter.  In either case, our value
indicates a much bluer morphology than found by \markcite{srn98}Smith
et al. (1998).  Contrary to their claims, M54 lies among the Galactic
globular clusters in the HB-morphology {\em versus} metallicity plot.
This is consistent with the globular cluster-like age determined for
M54 (\markcite{mar98}Marconi et al. 1998, \markcite{ls97}LS97,
herein), and no additional ``second parameters'' need to be invoked.
Also, the three old globulars in Sgr appear indistinguishable from the
Galactic globulars, so the picture of Galactic halo formation via the
accretion of dwarf galaxies is not violated.

Using 63 RRL in M54, we determine $\langle V({\rm RR}) \rangle = 18.17
\pm 0.02$ mag with an rms dispersion of 0.06 mag.  This value is
robust against the presence of moderate numbers of unidentified Sgr
RRL in the field.  From the apparent color of 29 ab-type RRL at
minimum light, we determine the reddening toward M54 to be $E(B-V) =
0.14 \pm 0.03$ mag, in excellent agreement with results from the shape
and color of the M54 RGB (\markcite{sl95}SL95) and from the dust maps
of \markcite{sch98}Schlegel et al. (1998).  By adopting the RR Lyrae
luminosity-metallicity relation of \markcite{bri99}Chaboyer (1999), we
determine the true distance modulus of M54 and Sgr to be $(m-M)_0 =
17.19 \pm 0.03$ (internal) $\pm 0.12$ (systematic) mag, or $d = 27.4$
kpc.

We use this information, together with the CMDs described above, to
determine the ages of the clusters and field populations in Sgr.  For
the clusters, we have developed an age dating technique which measures
the magnitude difference between the HB and the SGB, and used
published isochrones to calibrate the magnitude difference in terms of
age.  We find that M54, Ter 8, and Arp 2 approximately coeval, and are
as old as typical Galactic globular clusters of comparable
metallicity.  Comparison with the fiducial sequences of well-studied
Galactic globulars, and directly with isochrones, supports this
result.

For the Sgr field populations, we interpret the ages directly from
isochrones which were shifted to the distance modulus and reddening
values discussed above (i.e., no fitting was performed).  We find that
the dominant Sgr population, originally characterized by \markcite{muskkk}MUSKKK as 10
Gyr old with $[Fe/H] = -1.1$ dex, is actually composed of several
MSTOs, or perhaps a continuum of MSTOs indicating continuous star
formation with a variable rate.  The two strongest MSTO signatures
match the best with isochrones of 11 Gyr at $[Fe/H] = -1.3$ and 5 Gyr
at $[Fe/H] = -0.7$ dex, respectively.  A sparser population rises
above the dominant Sgr MSTOs; we characterize this ``spur'' in the CMD
with the parameters 2.5 Gyr and $[Fe/H] = -0.4$ dex.  The extremely
bright, blue stars noted by \markcite{muskkk}MUSKKK and others are matched reasonably
with an isochrone of 1 Gyr and $[Fe/H] = -0.4$ dex.  However, neither
of the two younger populations presents a tight sequence in the CMD,
suggesting that both might have formed over a significant period of
time.  Indeed, the entire Sgr CMD hints that star formation in Sgr was
a continuous process, punctuated by peaks and valleys in the star
formation rate, rather than by discrete episodes.

The age-metallicity relation defined by the clusters and field
populations (Figure 18) is surprisingly well defined, compared with
those of \markcite{mont98}Montegriffo et al. (1998) and
\markcite{bel99b}Bellazzini et al. (1999b).  The main difference is
the age of Arp 2; those authors place it 3--4 Gyr younger than M54 and
Ter 8, whereas we find the three clusters to be nearly coeval.

We find that a simple closed-box model of chemical evolution fits our
age-metallicity relation very well.  Taken at face value, this
suggests several things.  First, Sgr must have experienced little
gas-infall after its initial formation.  This is consistent with the
picture of dwarf galaxies as primordial building blocks of larger
galaxies (e.g., \markcite{sz78}Searle \& Zinn 1978).  Second, Sgr must
have been been massive enough to retain its gas both against internal
energy inputs such as supernovae and stellar winds, and against
external influences like tidal interaction with the Milky Way.  Sgr is
a fairly luminous, hence massive, dwarf galaxy (\markcite{igi97}Ibata
et al. 1997), and its gas retention is qualitatively consistent with
the energy injection simulations of \markcite{mmml99}Mac Low (1999).
\markcite{igi97}Ibata et al. (1997) suggested that Sgr has an
especially large dark matter halo to maintain coherence during its
many perigalacticon passages.  Alternately,
\markcite{bel99b}Bellazzini et al. (1999b) and others have suggested
that the short-period orbit favored by Ibata and collaborators is
incorrect, and that Sgr is making its first perigalacticon passage.

A full description of the star formation history of a galaxy requires
an estimate of the star formation rate as a function of time, in
addition to the age-metallicity relation we have derived here (e.g.,
\markcite{eva98}Grebel 1998).  \markcite{bel99b}Bellazzini et
al. (1999b) give several estimates of this function, but their results
depend heavily on their assumed age-metallicity relation.  In a
forthcoming paper, we will use population synthesis techniques to
constrain the star formation rates of the different Sgr field
populations described above directly from the density of these stars
in the CMD (\markcite{liu01}Liu et al. 2001).

Finally, we showed that the populations generally associated with Sgr
are spatially concentrated around M54.  In particular, the metal-rich,
1.0--2.5 Gyr populations are the most strongly clumped.  This explains
why \markcite{sl95}SL95 found a higher metallicity for Sgr ($[Fe/H] =
-0.5$ dex) than other studies which observed stars farther from M54;
the relative densities of the different-metallicity populations vary
radially, and both metallicity estimates are correct in their
respective regions of Sgr.  It is not currently clear what mechanism
is responsible for this clumping.  Does it reflect new generations of
stars formed from chemically evolved gas which had settled to the
bottom of the galactic potential well, or were these stars captured
from the Sgr field or pre-existing clusters by the tidal influence of
M54?  In any case, it seems likely that the luminous globular cluster
M54 is analogous to the central star clusters seen in nucleated dwarf
elliptical galaxies.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgments

We thank Stefanie Wachter for obtaining several images of M54 during
her observing run in June 1996, and for finding the Rosino \& Nobili
(1959) article in the CTIO library and mailing a copy to us.  We
appreciate the efforts of Benjamin Bowes in determining preliminary
light curves for the M54 variable stars.  We thank the following
people for productive discussions: Gary Bernstein, Christine Clement,
and Mario Mateo.

ACL acknowledges support from NASA grant HF-01082.02-96A from the Space
Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract
NAS5-26555.  ACL also appreciates the kindness and generosity of the
Dept. of Astronomy at the University of Michigan, which hosted him
for two years, during which time much of this work was accomplished.



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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% TABLES
%\clearpage

\begin{deluxetable}{lrrrr}
\tablewidth{0pt}
\tablecaption{Coordinates of Field Centers \label{tbl-1}}
\tablehead{
\colhead{Field}                 & 
\colhead{$RA_{\rm J2000}$}      &
\colhead{$Dec_{\rm J2000}$}     & 
\colhead{$l^{\sc II}$}          &
\colhead{$b^{\sc II}$}
}
\startdata
%
M54          & 18:55:03 &--30:28.8 &  5.61 &--14.09 \nl
Sgr          & 18:55:17 &--30:13.9 &  5.86 &--14.03 \nl
Control      & 18:33:31 &--41:10.6 &--6.30 &--14.35 \nl 
MUSKKK Cntrl & 18:33:26 &--40:58.0 &--6.11 &--14.25
%
\enddata
\end{deluxetable}

% TAB02

\begin{deluxetable}{crrrrrrc}
\tablewidth{0pt}
\tablecaption{Photometry of M54 from Shallow Frames \label{tbl-2}}
\tablehead{
\colhead{ID}                    & 
\colhead{$X_{\rm pix}$}         &
\colhead{$Y_{\rm pix}$}         &
\colhead{$V$}                   & 
\colhead{$\sigma_V$}            &
\colhead{$I$}                   &
\colhead{$\sigma_I$}            &
\colhead{$N_{\rm obs}$}
}
\startdata
% & 
    1  & 1240.31  &   18.67  & 14.078  &  0.047  & 14.031  &  0.215  &  3 \nl
    2  & 1162.75  &  684.69  & 14.390  &  0.009  & 13.605  &  0.209  &  4 \nl
    3  & 1167.17  &  634.97  & 14.396  &  0.004  & 13.621  &  0.008  &  3 \nl
    4  &  983.93  &  704.52  & 14.493  &  0.005  & 13.751  &  0.012  &  4 \nl
    5  & 1380.84  &  495.20  & 14.557  &  0.007  & 13.752  &  0.016  &  4 \nl
    6  &  169.81  &  408.57  & 14.594  &  0.017  & 14.117  &  0.014  &  5 \nl
    7  & 1385.77  &  914.20  & 14.601  &  0.006  & 13.767  &  0.018  &  4 \nl
    8  & 1182.45  &  476.76  & 14.609  &  0.006  & 13.746  &  0.014  &  4 \nl
    9  & 1123.60  &  550.27  & 14.616  &  0.010  & 13.507  &  0.006  &  3 \nl
%   10  &  191.22  &  474.32  & 14.631  &  0.004  & 13.450  &  0.008  &  3
%
\enddata
%
\tablecomments{This table is presented in its entirety in the
electronic edition of the Astronomical Journal.  A portion is shown
here for guidance regarding its form and content.}
%
\end{deluxetable}

%TAB03

\begin{deluxetable}{crrrrrrc}
\tablewidth{0pt}
\tablecaption{Photometry of M54 from Deep Frames \label{tbl-3}}
\tablehead{
\colhead{ID}                    & 
\colhead{$X_{\rm pix}$}         &
\colhead{$Y_{\rm pix}$}         &
\colhead{$V$}                   & 
\colhead{$\sigma_V$}            &
\colhead{$I$}                   &
\colhead{$\sigma_I$}            &
\colhead{$N_{\rm obs}$}
}
\startdata
% & 
 1  & 1354.35  & 1031.66  & 17.575  &  0.034  & 16.940  &  0.001  &  3 \nl
 2  & 1462.94  & 1094.82  & 17.652  &  0.064  & 16.597  &  0.126  &  3 \nl
 3  &  564.35  & 1536.87  & 17.713  &  0.004  & 16.889  &  0.014  &  3 \nl
 4  & 1436.04  &  711.26  & 17.747  &  0.023  & 16.807  &  0.009  &  4 \nl
 5  &  153.00  & 1451.39  & 17.749  &  0.017  & 16.735  &  0.042  &  3 \nl
 6  &  514.81  & 1543.79  & 17.751  &  0.014  & 16.771  &  0.016  &  3 \nl
 7  &  115.53  & 1272.89  & 17.759  &  0.009  & 16.865  &  0.007  &  4 \nl
 8  &  788.36  &  302.76  & 17.776  &  0.031  & 16.698  &  0.018  &  3 \nl
 9  &  355.20  & 1389.03  & 17.814  &  0.010  & 16.834  &  0.010  &  4 \nl
%10  &  154.98  & 1420.89  & 17.817  &  0.016  & 16.968  &  0.010  &  3
%
\enddata
%
\tablecomments{This table is presented in its entirety in the
electronic edition of the Astronomical Journal.  A portion is shown
here for guidance regarding its form and content.}
%
\end{deluxetable}

%TAB04

\begin{deluxetable}{crrrrrrc}
\tablewidth{0pt}
\tablecaption{Photometry of Sgr from Shallow Frames \label{tbl-4}}
\tablehead{
\colhead{ID$_{\rm SL95}$}                       & 
\colhead{$X_{\rm pix}$}         &
\colhead{$Y_{\rm pix}$}         &
\colhead{$V$}                   & 
\colhead{$\sigma_V$}            &
\colhead{$I$}                   &
\colhead{$\sigma_I$}            &
\colhead{$N_{\rm obs}$}
}
\startdata
% & 
 1  &  580.46  &  205.14  & 12.913  &  0.001  & 11.699  &  0.003  &  4 \nl
 2  &  436.71  &  698.96  & 13.992  &  0.001  & 12.609  &  0.003  &  4 \nl
 3  &  321.17  &  170.12  & 14.074  &  0.001  & 12.749  &  0.002  &  4 \nl
 4  &  699.23  &  764.98  & 14.098  &  0.003  & 13.069  &  0.006  &  4 \nl
 5  &  118.78  &  300.19  & 14.239  &  0.002  & 13.255  &  0.005  &  4 \nl
 6  &  971.21  &   92.99  & 14.310  &  0.001  & 12.844  &  0.003  &  4 \nl
 7  &  674.70  &  166.16  & 14.412  &  0.001  & 13.499  &  0.003  &  4 \nl
 8  &  238.29  &  439.99  & 14.424  &  0.001  & 13.094  &  0.001  &  4 \nl
 9  &  384.45  &  343.89  & 14.620  &  0.001  & 13.493  &  0.002  &  4 \nl
%10  &  340.02  &  640.82  & 14.724  &  0.001  & 13.820  &  0.006  &  3
%
\enddata
%
\tablecomments{This table is presented in its entirety in the
electronic edition of the Astronomical Journal.  A portion is shown
here for guidance regarding its form and content.}
%
\tablecomments{This data is identical to that in Table 2 of SL95,
except that the $XY$ coordinates have been transformed to the 
system defined by Figure 3.}
%
\end{deluxetable}

%TAB05

\begin{deluxetable}{crrrrrrc}
\tablewidth{0pt}
\tablecaption{Photometry of Sgr from Deep Frames \label{tbl-5}}
\tablehead{
\colhead{ID}                    & 
\colhead{$X_{\rm pix}$}         &
\colhead{$Y_{\rm pix}$}         &
\colhead{$V$}                   & 
\colhead{$\sigma_V$}            &
\colhead{$I$}                   &
\colhead{$\sigma_I$}            &
\colhead{$N_{\rm obs}$}
}
\startdata
% & 
 1  & 1052.15  &  693.69  & 17.763  &  0.023  & 16.928  &  0.007  &  5 \nl
 2  &  424.85  & 1019.85  & 17.809  &  0.007  & 16.922  &  0.010  &  4 \nl
 3  &  746.79  &  972.15  & 17.837  &  0.016  & 16.917  &  0.008  &  5 \nl
 4  &  728.44  &  576.48  & 17.846  &  0.018  & 16.953  &  0.007  &  5 \nl
 5  &  919.97  &  286.07  & 17.847  &  0.005  & 17.001  &  0.009  &  5 \nl
 6  &  643.08  & 1013.63  & 17.848  &  0.010  & 16.948  &  0.005  &  5 \nl
 7  &  224.65  &  932.54  & 17.853  &  0.007  & 17.011  &  0.015  &  5 \nl
 8  &  393.06  &  469.94  & 17.887  &  0.004  & 17.069  &  0.007  &  6 \nl
 9  &  783.77  &  332.38  & 17.888  &  0.005  & 17.068  &  0.008  &  6 \nl
%10  &  314.42  &  989.99  & 17.890  &  0.008  & 17.029  &  0.009  &  5
%
\enddata
%
\tablecomments{This table is presented in its entirety in the
electronic edition of the Astronomical Journal.  A portion is shown
here for guidance regarding its form and content.}
%
\end{deluxetable}

%TAB06

\begin{deluxetable}{crrrrrrc}
\tablewidth{0pt}
\tablecaption{Photometry of the Control Field \label{tbl-6}}
\tablehead{
\colhead{ID}                    & 
\colhead{$X_{\rm pix}$}         &
\colhead{$Y_{\rm pix}$}         &
\colhead{$V$}                   & 
\colhead{$\sigma_V$}            &
\colhead{$I$}                   &
\colhead{$\sigma_I$}            &
\colhead{$N_{\rm obs}$}
}
\startdata
% & 
 1  &  276.87  &  157.31  & 17.840  &  0.004  & 17.088  &  0.009  &  3 \nl
 2  &  347.96  & 1292.79  & 17.924  &  0.015  & 16.994  &  0.004  &  4 \nl
 3  & 1517.80  &  260.91  & 17.929  &  0.004  & 16.763  &  0.023  &  3 \nl
 4  &  187.02  &  836.07  & 17.940  &  0.007  & 17.196  &  0.006  &  3 \nl
 5  & 1148.93  & 1108.48  & 17.942  &  0.014  & 17.074  &  0.009  &  3 \nl
 6  &  737.02  & 1199.04  & 17.947  &  0.036  & 16.749  &  0.009  &  3 \nl
 7  &  929.88  &  143.74  & 17.960  &  0.008  & 17.188  &  0.009  &  3 \nl
 8  & 1469.66  & 1287.70  & 17.993  &  0.010  & 17.163  &  0.009  &  4 \nl
 9  &  610.79  & 1027.83  & 17.997  &  0.018  & 16.778  &  0.013  &  4 \nl
%10  &  686.76  &  145.92  & 17.999  &  0.009  & 17.120  &  0.001  &  3
%
\enddata
%
\tablecomments{This table is presented in its entirety in the
electronic edition of the Astronomical Journal.  A portion is shown
here for guidance regarding its form and content.}
%
%\tablecomments{Equatorial coordinates are derived from the Digitized
%Sky Survey for three stars:  (1) ID=???, $\alpha_{J2000} = N^{\rm
%h} N^{\rm m} N^{\rm s}$, $\delta_{J2000} = N^{\circ} N^{\prime}
%N^{\prime\prime}$;   (2) ID=???, $\alpha_{J2000} = N^{\rm
%h} N^{\rm m} N^{\rm s}$, $\delta_{J2000} = N^{\circ} N^{\prime}
%N^{\prime\prime}$;   (3) ID=???, $\alpha_{J2000} = N^{\rm
%h} N^{\rm m} N^{\rm s}$, $\delta_{J2000} = N^{\circ} N^{\prime}
%N^{\prime\prime}$.}
%
\end{deluxetable}

%TAB07

\begin{deluxetable}{lcrrrrrc}
\tablewidth{0pt}
\tablecaption{Observing Runs for Variable Star Photometry \label{tbl-7}}
\tablehead{
\colhead{Run}               & 
\colhead{Dates}               &
\colhead{Telescope}      & 
\colhead{CCD}   &
\colhead{FOV}   &
\colhead{$N_{\rm V}$}            & 
\colhead{$N_{\rm I}$} &
\colhead{Symbol}
}
\startdata
%
1 & 1994/06/17--18 & CTIO 0.9-m & Tek1k \#1 &  6.8 &  6 & 6 & {\em triangle} \nl
2 & 1994/06/26--27 & CTIO 0.9-m & Tek2k \#3 & 13.6 &  7 & 7 & {\em square}  \nl
3 & 1994/10/13--17 & CTIO 0.9-m & Tek2k \#4 & 13.6 &  7 & 0 & {\em pentagon} \nl
4 & 1995/06/28--29 & CTIO 4.0-m & Tek2k \#4 & 14.6 & 11 & 5 & {\em circle} \nl
5 & 1996/06/13     & CTIO 0.9-m & Tek2k \#3 & 13.6 &  2 & 2 & {\em cross} 
%
\enddata
\end{deluxetable}

%TAB08

\begin{deluxetable}{rrrcccl}
\tablewidth{0pt}
\tablecaption{Mean Photometry of Suspected Variable Stars \label{tbl-8}}
\tablehead{
\colhead{Star}               & 
\colhead{$X_{\rm pix}$}      &
\colhead{$Y_{\rm pix}$}      & 
\colhead{${\overline{V}}$}   &
\colhead{${\overline{V-I}}$} & 
\colhead{$\Delta V_{\rm min}^{\rm max}$} &
\colhead{Comment\tablenotemark{a}}
}
\startdata
%
V20 &  949.3 & 1055.0 & 18.34 & 0.22 & 0.11 & not variable \nl %%T19
V22 &  345.6 &  773.7 & 16.52 & 1.31 & 0.05 & not variable \nl %%T11
V27 &   15.6 & 1275.3 & 17.74 & 0.81 & 0.14 & not variable \nl %%T23 = vfe.27a
V53 &  552.8 &  666.9 & 18.13 & 0.86 & 0.37 & not variable \nl %%T07 = vfe.53a
V67 &  889.1 &  809.6 & 17.99 & 0.68 & 0.90 & variable \nl % E08
V69 & 1364.3 &   42.4 & 17.58 & 0.82 & 1.10 & variable \nl % T01
V72 & 1070.5 &  668.7 & 15.93 & 1.13 & 0.04 & not variable \nl %%T02
V73 &  876.1 &  844.9 & 18.08 & 0.66 & 0.62 & variable \nl % T04 = vfe.73b
V75 &  912.1 &  851.4 & 17.94 & 0.92 & 1.26 & variable \nl % T03 = vfe.75a
V79 &  625.9 &  880.9 & 18.10 & 0.47 & 0.67 & variable \nl % T16 = vfe.79a
V81 &  756.9 &  912.3 & 17.98 & 0.98 & 0.42 & variable \nl % T25 = vfe.81c
SV1 &  657.7 &  718.0 & 18.59 & 0.35 & 0.93 & variable?  \nl %% A27
SV2 &  746.3 &  908.9 & 18.27 & 0.93 & 1.45 & possible EA \nl % A28  EA w/ P=2.11765 other poss too
SV3 &  803.6 &  828.3 & 17.81 & 1.07 & 1.27 & variable?   \nl  % A33 
%SV4 & 1261.5 & 1522.5 & 17.53 & 0.93 & 2.13 & possible EA \nl % B02
%EA w/ P=0.40978 others poss too -- SV4 == V91 Algol
%
\enddata
\tablenotetext{a}{See Appendix for more information.}
\end{deluxetable}

%TAB09

\begin{deluxetable}{rrrrlrrrrrccccrrl}
\tablewidth{0pt}
\tablecaption{Photometric Properties of Short Period Variable Stars \label{tbl-9}}
\tablehead{
\colhead{Star}               & 
\colhead{$X_{\rm pix}$}      &
\colhead{$Y_{\rm pix}$}      & 
\colhead{Per}                &
\colhead{Typ}                &
\colhead{$\langle V \rangle$}        &
\colhead{$\langle I \rangle$}        &
\colhead{$V_{\rm max}$}              &
\colhead{$V_{\rm min}$}              &
\colhead{$E_{\rm max}$}              &
\colhead{$N_{\rm obs}$}              &
\colhead{$N_{\rm tmp}$}              &
\colhead{$rms$}             &
\colhead{$(V-I)_{\rm min}$}          &
\colhead{$\epsilon (V-I)_{\rm min}$} &
\colhead{$N_{\rm min}$}              &
\colhead{Note}
}
\startdata
%
V01 &  747.1 &  998.3 &  1.34769 & Cep  & 17.257 & 16.589 & 16.795 & 17.776 &  640.34534 & 
 31 &  7 &  0.075 &  0.827 &  0.026 &  2 & * \nl  %B04
V02 &  938.3 &  796.5 &  1.09461 & Cep? & 17.292 & 16.738 & 16.604 & 17.884 &  641.18047 & 
 32 &  6 &  0.030 &  0.721 & 99.999 &  1 & * \nl  %B05
V03 & 1145.4 &  778.8 &  0.57536 & RRab & 18.201 & 17.525 & 17.632 & 18.529 &  641.62160 & 
 26 &  1 &  0.022 &  0.752 & 99.999 &  1 &  \nl  %A22
V04 & 1452.0 &  685.6 &  0.48011 & RRab & 18.189 & 17.522 & 17.507 & 18.618 &  641.28792 & 
 26 &  3 &  0.029 &  0.808 &  0.005 &  3 &  \nl  %A06
V05 &  837.8 &  504.6 &  0.57911 & RRab & 18.112 & 17.487 & 17.494 & 18.490 &  641.27471 & 
 32 &  3 &  0.025 &  0.711 &  0.033 &  3 &  \nl  %A11
V06 &  308.0 & 1279.4 &  0.53290 & RRab & 18.104 & 17.467 & 17.436 & 18.521 &  641.34148 & 
 25 &  3 &  0.038 &  0.701 &  0.017 &  3 &  \nl  %A02
V07 &  360.2 &  922.7 &  0.59261 & RRab & 18.206 & 17.506 & 17.753 & 18.468 &  641.18662 & 
 32 &  3 &  0.017 &  0.785 &  0.016 &  3 &  \nl  %A15
V12 &  165.4 &  278.1 &  0.31985 & RRc  & 16.628 & 16.042 & 16.388 & 16.869 &  641.50453 & 
 27 &  7 &  0.036 &  0.651 &  0.017 &  2 &  \nl  %B07
V14 & 1210.2 & 1371.1 &  0.70489 & RRab?& 18.023 & 88.888 & 17.446 & 18.372 &  641.51657 & 
  9 &  3 &  0.029 &  1.045 & 99.999 &  1 &  * \nl  %14a
V15 &  582.5 & 1101.7 &  0.58686 & RRab & 18.229 & 17.506 & 17.800 & 18.481 &  641.11748 & 
 32 &  5 &  0.054 &  0.787 &  0.009 &  4 &  \nl  %A34
V23 & 1165.9 & 1635.8 &  0.51767 & RRab & 18.230 & 17.634 & 17.535 & 18.647 &  641.51703 & 
 20 &  1 &  0.046 &  0.623 & 99.999 &  1 &  \nl  %A17
V28 & 1103.8 &  971.5 &  0.51304 & RRab & 18.176 & 17.567 & 17.408 & 18.648 &  641.25888 & 
 27 &  1 &  0.021 &  0.743 &  0.014 &  4 &  \nl  %A14
V29 &  625.6 &  507.9 &  0.58970 & RRab & 18.239 & 17.561 & 17.726 & 18.530 &  641.45177 & 
 31 &  1 &  0.043 &  0.784 &  0.020 &  7 &  \nl  %V29
V30 &  915.0 &  813.2 &  0.57388 & RRab & 18.234 & 17.597 & 17.715 & 18.540 &  641.40869 & 
 31 &  3 &  0.038 &  0.761 &  0.021 &  4 &  \nl  %A13
V31 &  573.3 &  576.2 &  0.64661 & RRab & 18.192 & 17.490 & 17.969 & 18.354 &  641.29377 & 
 32 &  6 &  0.037 &  0.746 &  0.014 &  7 &  \nl  %E18
V32 &  878.9 &  394.2 &  0.52132 & RRab & 18.134 & 17.538 & 17.331 & 18.633 &  641.51681 & 
 32 &  1 &  0.022 &  0.754 &  0.010 & 11 &  \nl  %A03
V33 &  492.7 &  983.2 &  0.49108 & RRab & 18.236 & 17.655 & 17.448 & 18.723 &  641.48930 & 
 30 &  1 &  0.036 &  0.689 &  0.065 &  3 &  \nl  %A05
V34 &  375.3 &  680.5 &  0.50141 & RRab?& 18.227 & 17.617 & 17.390 & 18.753 &  641.14426 & 
 32 &  1 &  0.105 &  0.747 &  0.013 &  9 & * \nl  %A01
V35 &  851.9 &  619.9 &  0.52672 & RRab & 18.213 & 17.651 & 17.492 & 18.649 &  641.58894 & 
 32 &  1 &  0.024 &  0.753 &  0.012 &  5 &  \nl  %E04
V36 &  853.8 & 1106.0 &  0.59906 & RRab & 18.158 & 17.439 & 17.827 & 18.345 &  641.26755 & 
 32 &  5 &  0.034 &  0.762 &  0.023 &  3 &  \nl  %A44
V37 &  632.5 &  908.9 &  0.62743 & RRab & 18.220 & 17.568 & 17.934 & 18.433 &  641.36121 & 
 31 &  6 &  0.031 &  0.750 &  0.028 &  3 &  \nl  %V37
V38 &  812.4 &  653.0 &  0.61111 & RRab & 18.057 & 17.347 & 17.741 & 18.272 &  641.61304 & 
 31 &  2 &  0.028 &  0.783 &  0.019 &  5 &  \nl  %E23
V39 &  581.0 &  575.5 &  0.59962 & RRab & 18.246 & 17.543 & 17.858 & 18.471 &  641.02917 & 
 32 &  5 &  0.039 &  0.768 &  0.016 &  5 &  \nl  %A35
V40 &  473.3 &  688.9 &  0.58637 & RRab & 18.235 & 17.543 & 17.805 & 18.538 &  641.12260 & 
 31 &  2 &  0.040 &  0.739 &  0.010 &  4 &  \nl  %A21
V41 &  839.9 & 1105.0 &  0.61768 & RRab & 18.096 & 17.427 & 17.542 & 18.502 &  641.12561 & 
 32 &  2 &  0.023 &  0.784 &  0.006 &  8 &  \nl  %E06
V42 &  863.4 &  972.2 &  0.32643 & RRc  & 18.185 & 17.676 & 17.925 & 18.449 &  641.31181 & 
 32 &  7 &  0.027 &  0.547 &  0.038 &  5 &  \nl  %A47
V43 &  847.5 &  456.4 &  0.59284 & RRab & 18.145 & 17.487 & 17.585 & 18.467 &  641.14792 & 
 32 &  1 &  0.042 &  0.795 &  0.038 &  2 &  \nl  %A09
V44 &  544.3 &  839.0 &  0.61779 & RRab & 18.235 & 17.544 & 17.991 & 18.415 &  640.28031 & 
 29 &  6 &  0.036 &  0.730 &  0.011 &  6 &  \nl  %V44
V45 &  485.1 & 1087.4 &  0.48885 & RRab & 18.199 & 17.603 & 17.658 & 18.531 &  641.57369 & 
 30 &  5 &  0.038 &  0.729 &  0.011 &  4 &  \nl  %A26
V46 &  639.2 &  726.8 &  0.60473 & RRab & 18.094 & 17.437 & 17.735 & 18.300 &  641.31119 & 
 29 &  5 &  0.033 &  0.751 & 99.999 &  1 &  \nl  %V46
V47 &  948.5 &  740.8 &  0.50694 & RRab?& 18.174 & 17.553 & 17.642 & 18.476 &  641.55164 & 
 31 &  1 &  0.054 &  0.678 &  0.016 &  8 &  \nl  %A08
V48 &  636.6 & 1399.1 &  0.68702 & RRab & 18.104 & 17.383 & 17.714 & 18.405 &  641.10221 & 
 27 &  6 &  0.015 &  0.855 &  0.069 &  6 &  \nl  %A07
V49 &  417.6 &  588.6 &  0.33128 & RRc  & 18.034 & 17.575 & 17.831 & 18.233 &  641.46954 & 
 32 &  7 &  0.025 &  0.503 &  0.025 &  3 &  \nl  %A43
V50 &  876.2 & 1048.5 &  0.56152 & RRab & 18.122 & 17.445 & 17.842 & 18.310 &  641.62160 & 
 32 &  2 &  0.030 &  0.782 &  0.083 &  3 &  \nl  %V50
V51 & 1201.4 & 1311.3 &  0.60545 & RRab & 18.205 & 17.484 & 17.810 & 18.434 &  641.55984 & 
 26 &  5 &  0.022 &  0.784 &  0.028 &  2 &  \nl  %A38
V52 &  620.0 & 1023.2 &  0.64083 & RRab & 18.208 & 17.486 & 17.966 & 18.386 &  641.05639 & 
 28 &  6 &  0.023 &  0.767 &  0.013 &  8 &  \nl  %V52
V54 & 1476.8 &  533.6 &  0.57039 & RRab & 18.199 & 17.505 & 17.854 & 18.396 &  641.62160 & 
 26 &  5 &  0.022 &  0.753 &  0.015 &  2 &  \nl  %E14
V55 &  266.0 & 1159.5 &  0.42606 & RRab & 18.586 & 17.948 & 18.000 & 19.054 &  641.26159 & 
 26 &  4 &  0.028 &  0.744 &  0.034 &  4 &  \nl  %A10
V56 &  427.9 &   44.8 &  0.36766 & RRc  & 18.132 & 17.506 & 17.926 & 18.335 &  641.32895 & 
 24 &  7 &  0.021 &  0.703 &  0.018 &  4 &  \nl  %E02
V57 &  673.9 & 1486.7 &  0.54663 & RRab & 18.189 & 17.578 & 17.512 & 18.593 &  641.29705 & 
 24 &  1 &  0.025 &  0.733 &  0.016 &  2 &  \nl  %A04
V58 & 1330.4 &  993.8 &  0.54866 & RRab & 18.220 & 17.523 & 17.733 & 18.506 &  641.45426 & 
 26 &  3 &  0.031 &  0.748 &  0.023 &  2 &  \nl  %A45
V59 &  136.2 &  324.0 &  0.59942 & RRab & 18.169 & 17.459 & 17.724 & 18.415 &  641.19541 & 
 27 &  1 &  0.016 &  0.774 & 99.999 &  1 &  \nl  %A39
V60 &  165.9 &  205.9 &  0.57585 & RRab & 18.247 & 17.519 & 17.706 & 18.556 &  641.38262 & 
 27 &  1 &  0.028 &  0.837 &  0.059 &  2 &  \nl  %A19
V61 &  973.1 &  708.6 &  0.60392 & RRab & 18.177 & 17.469 & 17.767 & 18.403 &  641.08681 & 
 28 &  1 &  0.043 &  0.692 &  0.033 &  2 &  \nl  %A37
V62 &  944.8 &  593.0 &  0.61355 & RRab & 18.273 & 17.578 & 17.962 & 18.448 &  641.47128 & 
 32 &  5 &  0.029 &  0.761 &  0.009 &  7 &  \nl  %E10
V63 &  422.6 &  727.2 &  0.59011 & RRab?& 18.265 & 17.618 & 17.850 & 18.501 &  641.16485 & 
 32 &  3 &  0.051 &  0.735 &  0.013 &  5 & * \nl  %A30
V65 & 1122.7 & 1363.8 &  0.56599 & RRab & 17.590 & 16.880 & 17.129 & 17.942 &  641.47331 & 
 25 &  4 &  0.040 &  0.762 &  0.011 &  4 &  \nl  %B03
V74 &  413.2 & 1080.0 &  0.28892 & RRc  & 18.119 & 17.654 & 17.877 & 18.391 &  641.48003 & 
 32 &  8 &  0.021 &  0.534 &  0.008 &  5 &  \nl  %V74
V76 &  677.0 &  578.1 &  0.70702 & RRab & 18.083 & 17.390 & 17.739 & 18.344 &  641.42010 & 
 31 &  6 &  0.025 &  0.754 &  0.012 & 10 & * \nl  %A46
V77 &  628.8 &  556.7 &  0.57691 & RRab?& 18.162 & 17.533 & 17.639 & 18.459 &  641.38815 & 
 29 &  1 &  0.055 &  0.792 &  0.022 &  5 & * \nl  %E28
V78 &  704.0 &  981.5 &  0.40200 & RRc? & 17.953 & 17.370 & 17.764 & 18.138 &  641.40613 & 
 31 &  7 &  0.039 &  0.588 &  0.107 &  2 &  \nl  %V78
V80 &  675.0 &  936.9 &  0.63089 & ab?  & 18.141 & 17.442 & 17.875 & 18.284 &  641.51687 & 
 31 &  3 &  0.041 &  0.709 &  0.023 &  4 & * \nl  %V80
V82 &  625.7 &  704.4 &  0.58719 & RRab?& 18.159 & 17.484 & 17.715 & 18.405 &  641.62160 & 
 29 &  1 &  0.055 &  0.775 &  0.028 &  5 & * \nl  %82b
V83 &   86.4 &  513.5 &  0.57835 & RRab & 18.190 & 17.502 & 17.742 & 18.456 &  641.45735 & 
 26 &  5 &  0.038 &  0.694 &  0.050 &  3 &  \nl  %A12
V84 &  613.0 &  788.0 &  0.59972 & RRab & 18.019 & 17.446 & 17.345 & 18.421 &  641.20839 & 
 29 &  1 &  0.044 &  0.816 &  0.070 &  6 &  \nl  %A18
V85 &  519.6 &  763.2 &  0.52046 & RRab?& 18.135 & 17.499 & 17.635 & 18.428 &  641.27914 & 
 33 &  3 &  0.052 &  0.723 &  0.029 &  7 & * \nl  %A20
V86 &  620.3 &  963.3 &  0.58367 & RRab & 18.147 & 17.487 & 17.634 & 18.519 &  641.18793 & 
 31 &  2 &  0.047 &  0.764 &  0.012 &  4 &  \nl  %A23
V87 &  667.0 & 1045.0 &  0.55512 & RRab & 18.211 & 17.545 & 17.722 & 18.497 &  641.36791 & 
 32 &  3 &  0.048 &  0.799 &  0.020 &  3 &  \nl  %A25
V88 & 1148.1 &  459.5 &  0.48578 & RRab?& 18.187 & 17.559 & 17.581 & 18.540 &  640.56880 & 
 20 &  1 &  0.075 &  0.699 &  0.022 &  6 & * \nl  %A31
V89 & 1218.1 &  591.6 &  0.61083 & RRab & 18.255 & 17.563 & 17.730 & 18.553 &  641.51654 & 
 21 &  1 &  0.037 &  0.769 &  0.005 &  2 &  \nl  %A36
V90 & 1086.8 &  950.4 &  0.61307 & RRab & 18.165 & 17.460 & 17.812 & 18.426 &  641.29177 & 
 30 &  4 &  0.022 &  0.786 &  0.021 &  3 &  \nl  %A50
V91 & 1261.4 & 1522.5 &  0.40978 & EA?  & 17.373 & 16.907 & 17.137 & 19.787 &  641.51670 & 
 13 & 10 &  0.047 &  1.103 & 99.999 &  1 &  \nl  %B02
V92 &  744.9 &  887.4 &  0.48470 & RRab & 17.879 & 17.117 & 17.240 & 18.272 &  641.62160 & 
 25 &  3 &  0.062 &  0.780 &  0.233 &  5 &  \nl  %B09
V93 & 1157.9 &  771.3 &  0.55809 & RRab & 17.475 & 16.746 & 17.077 & 17.692 &  641.17122 & 
 25 &  1 &  0.017 &  0.811 &  0.002 &  3 &  \nl  %B10
V94 & 1299.0 & 1022.3 &  0.64440 & RRab & 18.230 & 17.467 & 18.024 & 18.375 &  641.39542 & 
 26 &  4 &  0.023 &  0.804 &  0.021 &  3 &  \nl  %E07
V95 &  559.8 &  745.3 &  0.55838 & RRab & 18.007 & 17.212 & 17.499 & 18.293 &  640.25483 & 
 24 &  1 &  0.034 &  0.860 &  0.037 &  4 &  \nl  %E15
V96 &  -57.4 & 1444.5 &  0.55511 & RRab & 18.206 & 17.487 & 17.692 & 18.496 &  641.22414 & 
 20 &  1 &  0.039 &  0.792 &  0.047 &  2 &  \nl  %E16
V97 &  835.1 &  875.0 &  0.33673 & RRc  & 18.131 & 17.640 & 17.905 & 18.357 &  640.53505 & 
 29 &  7 &  0.042 &  0.591 &  0.031 &  7 &  \nl  %E26
V98 &  662.3 & 1027.8 &  0.62435 & RRab & 18.177 & 17.470 & 17.876 & 18.403 &  641.14585 & 
 30 &  6 &  0.031 &  0.758 &  0.017 &  5 &  \nl  %E27
V99 & 1153.6 &  626.0 &  0.34769 & EW?  & 18.337 & 17.546 & 18.208 & 18.584 &  641.29129 & 
 25 &  9 &  0.043 &  0.800 &  0.028 &  3 & * \nl  %E29
V100 &  818.9 &  772.8 &  0.96473 & RRab?& 18.223 & 17.590 & 17.787 & 18.566 &  640.59600 &
  16 &  6 &  0.074 &  0.748 &  0.072 &  3 & *     %E30
%
\enddata
\end{deluxetable}

%TAB10

\begin{deluxetable}{rrrcccl}
\tablewidth{0pt}
\tablecaption{Mean Photometry of Long Period Variable Stars \label{tbl-10}}
\tablehead{
\colhead{Star}               & 
\colhead{$X_{\rm pix}$}      &
\colhead{$Y_{\rm pix}$}      & 
\colhead{${\overline{V}}$}   &
\colhead{${\overline{V-I}}$} & 
\colhead{$\Delta V_{\rm min}^{\rm max}$} &
\colhead{Comment}
}
\startdata
%
V25 &  897.2 &  657.7 & 15.39 & 1.81 & 0.83 & -- \nl %C01
V66 & 1217.5 & 1338.6 & 16.24 & 3.20 & 0.27 & -- \nl %T02
V71 &  970.4 &  734.0 & 14.95 & 1.63 & 0.28 & -- \nl %C07
%
V101 &  716.6 &  212.2 & 16.54 & 2.79 & 0.70 & -- \nl %C04
V102 &  103.9 & 1133.4 & 16.95 & 3.02 & 0.34 & -- \nl %D01
V103 &  712.3 &  621.8 & 17.74 & 4.65 & 1.12 & -- \nl %D02
V104 & 1173.6 &  677.2 & 15.98 & 2.52 & 0.53 & -- \nl %D03
V105 &  817.0 &  531.7 & 16.08 & 2.54 & 0.27 & -- \nl %D05 low ampl?
V106 &  118.0 &  758.7 & 15.56 & 2.30 & 0.22 & -- \nl %T05  
%
V107 &  720.1 &  854.2 & 16.22 & 1.40 & 1.25 & crowded, probable LPV \nl %D06
V108 & 1196.0 & 1356.0 & 16.12 & 2.70 & 0.20 & low amplitude, possible LPV \nl
       %T07  0.058   LPV? (Vamp ~ 0.2mag)
V109 & 1009.4 & 1114.5 & 15.27 & 1.56 & 0.12 & low amplitude, possible LPV \nl
       %T11  0.011   max Vamp~0.1 mag, MAYBE
V110 &  798.9 & 1138.7 & 15.19 & 1.69 & 0.12 & low amplitude, possible LPV \nl 
       %T12  0.011   max Vamp~0.1 mag, MAYBE
V111 &  773.4 &  740.2 & 15.39 & 1.71 & 0.67 & crowded, probable LPV \nl
       %T13  0.034   LPV?, some scatt w/in run, but more var run-run
V112 &  754.6 &  766.4 & 15.33 & 1.81 & 0.60 & crowded, probable LPV \nl
       %T18  0.081   probably LPV, 1 multi-hit, only 12 V points 
%
V113 &   43.7 &  524.4 & 16.12 & 2.47 & 0.12 & low amplitude, possible LPV \nl 
       %T03 Vamp~0.1
V114 &  671.6 &  471.4 & 15.35 & 1.65 & 0.07 & low amplitude, possible LPV \nl
       %T08  0.013   max Vamp~0.05 mag, MAYBE 
V115 &  791.8 &  691.2 & 15.17 & 1.72 & 0.09 & low amplitude, possible LPV 
       %T10  0.009   max Vamp~0.05 mag, MAYBE
%
\enddata
\end{deluxetable}

%TAB11

\begin{deluxetable}{rrrcccl}
\tablewidth{0pt}
\tablecaption{Mean Photometry of Faint Variable Stars \label{tbl-11}}
\tablehead{
\colhead{Star}               & 
\colhead{$X_{\rm pix}$\tablenotemark{a}}      &
\colhead{$Y_{\rm pix}$\tablenotemark{a}}      & 
\colhead{${\overline{V}}$}   &
\colhead{${\overline{V-I}}$} & 
\colhead{$\Delta V_{\rm min}^{\rm max}$} &
\colhead{Comment}
}
\startdata
%
V116 & 1169.7 &  989.3 & 19.43 & 0.42 & 0.59 & binary   \nl %%X12
V117 & 1141.7 & 1313.4 & 19.03 & 0.94 & 0.31 & binary   \nl %%X20
Sgr1 &  378.9 &  555.5 & 18.25 & 0.88 & 0.46 & RR Lyrae \nl %%F05
Sgr2 &  652.0 &  858.8 & 20.38 & 0.62 & 1.31 & binary   \nl %%F08
%
\enddata
\tablenotetext{a}{Coordinates for V116 and V117 refer to Figure 1,
while coordinates for Sgr1 and Sgr2 refer to Figure 3.}
\end{deluxetable}

%TAB12

\begin{deluxetable}{lrcccccc}
\tablewidth{0pt}
\tablecaption{Time-Series Photometry for All Variable Stars \label{tbl-12}}
\tablehead{
\colhead{Star}               & 
\colhead{Time}      &
\colhead{Phase}      & 
\colhead{$V$}   &
\colhead{$\sigma_V$} & 
\colhead{$I$} &
\colhead{$\sigma_I$} & 
\colhead{Code\tablenotemark{a}}
}
\startdata
%
V01 &  520.88660 &  0.289 &  17.321 &   0.010 &  16.570 &   0.013 &  33 \nl %B04
V01 &  520.89368 &  0.294 &  17.335 &   0.009 &  16.623 &   0.014 &  33 \nl %B04
V01 &  520.89972 &  0.298 &  17.361 &   0.011 &  16.607 &   0.013 &  30 \nl %B04
V01 &  521.66339 &  0.865 &  17.119 &   0.012 &  16.595 &   0.012 &  33 \nl %B04
V01 &  521.67090 &  0.871 &  17.114 &   0.011 &  16.522 &   0.009 &  33 \nl %B04
V01 &  521.67688 &  0.875 &  17.056 &   0.012 &  16.527 &   0.013 &  33 \nl %B04
V01 &  529.63379 &  0.779 &  17.453 &   0.014 &  16.745 &   0.020 &  40 \nl %B04
%V01 &  529.71240 &  0.838 &  17.290 &   0.027 &  16.631 &   0.014 & 40 \nl %B04
%
%~... &   ...~~~~ &   ...  &   ...   &   ...   &   ...   &   ...   & ... \nl
%
%V01 &  897.86609 &  0.012 &  16.807 &   0.009 &  16.312 &   0.009 &  93 \nl %B04
%V01 &  897.88843 &  0.029 &  16.805 &   0.010 &  16.308 &   0.023 &  93 \nl %B04
%V01 & 1278.67260 &  9.999 &  17.649 &   0.018 &  16.856 &   0.023 &  41 \nl %B04
%V01 & 1278.67780 &  9.999 &  17.654 &   0.021 &  16.855 &   0.021 &  41 \nl %B04
%V02 &  520.88660 &  0.187 &  17.098 &   0.010 &  16.585 &   0.014 &  33 \nl %B05
%V02 &  520.89368 &  0.193 &  17.109 &   0.009 &  16.589 &   0.014 &  33 \nl %B05
%V02 &  520.89972 &  0.199 &  17.122 &   0.009 &  16.597 &   0.015 &  30 \nl %B05
%V02 &  521.66339 &  0.896 &  17.226 &   0.009 &  16.709 &   0.012 &  33   %B05
%
\enddata
%
\tablecomments{This table is presented in its entirety in the
electronic edition of the Astronomical Journal.  A portion is
shown here for guidance regarding its form and content.}
%
%\medskip
\tablenotetext{a}{Code corresponding to the symbols in Figures 8 and 9.
See footnote 6 for an explanation.}
%\tablenotetext{a}{Code corresponding to the symbols in Figures NN-MM.
%This is a SuperMongo point type.  The first digit indicates the symbol
%shape and run number (see Table NN): 3 = {\em triangle} = Run1, 
%4 = {\em square} = Run2, 5 = {\em pentagon} = Run3, 9 = {\em circle} =
%Run4.  The second digit indicates whether the seeing was better or
%worse than the median seeing (3 or 1, respectively).  The exception
%to this rule is code=41, which indicates Run5.}
% 
\end{deluxetable}

%TAB13

\begin{deluxetable}{cccc}
%\tablewidth{34pc}
\tablewidth{0pt}
\tablecaption{Observed Cluster Parameters \label{tbl-13}}
\tablehead{
\colhead{Cluster} & 
\colhead{$V({\rm HB})$}   & 
\colhead{$[Fe/H]$}   & 
\colhead{$E(V-I)$} 
}
\startdata
%
M92  &   $15.10 \pm 0.03$\tablenotemark{a} & 
$-2.24 \pm 0.08$\tablenotemark{b}  &  0.03\tablenotemark{c} \nl
%
M54  &   $18.17 \pm 0.02$\tablenotemark{d} & 
$-1.79 \pm 0.08$\tablenotemark{d}  &  0.18\tablenotemark{d} \nl
%
M5   &   $15.09 \pm 0.02$\tablenotemark{e} & 
$-1.40 \pm 0.06$\tablenotemark{b}  &  0.04\tablenotemark{c}
%
\enddata
\tablenotetext{a}{Carney et al. (1992).}
\tablenotetext{b}{Zinn \& West (1984).}
\tablenotetext{c}{Harris (1996).}
\tablenotetext{d}{This paper.}
\tablenotetext{e}{Sandquist et al. (1992).}
\end{deluxetable}

%TAB14

\begin{deluxetable}{cccccc}
%\tablewidth{34pc}
\tablecaption{Cluster Ages from the SGB \label{tbl-14}}
\tablewidth{0pt}
\tablehead{
\colhead{Cluster} & 
\colhead{$[Fe/H]$}   & 
\colhead{$V({\rm HB})$}   & 
\colhead{$V({\rm SGB})$}   & 
\colhead{$\Delta V$}  &
\colhead{Age} 
}
\startdata
%
M92   & $-2.24 \pm 0.08$\tablenotemark{a} & 
$15.10 \pm 0.03$\tablenotemark{c} & $17.89 \pm 0.03$ & 
$2.79 \pm 0.04$ &  $14.5 \pm 0.5$ \nl
%
Ter 8 & $-2.00 \pm 0.10$\tablenotemark{a} & 
$17.95 \pm 0.05$\tablenotemark{d} & $20.77 \pm 0.02$ & 
$2.82 \pm 0.05$ &  $15.5 \pm 0.8$ \nl
%
Arp 2 & $-1.84 \pm 0.09$\tablenotemark{a} & 
$18.23 \pm 0.05$\tablenotemark{e} & $20.94 \pm 0.03$ & 
$2.71 \pm 0.06$ &  $14.1 \pm 0.9$ \nl
%
M54   & $-1.79 \pm 0.08$\tablenotemark{b} & 
$18.17 \pm 0.02$\tablenotemark{b} & $20.99 \pm 0.02$ & 
$2.82 \pm 0.03$ &  $15.7 \pm 0.5$ \nl
%
M5    & $-1.40 \pm 0.06$\tablenotemark{a} & 
$15.09 \pm 0.02$\tablenotemark{f} & $17.87 \pm 0.03$ &
$2.78 \pm 0.04$ &  $14.5 \pm 0.7$ \nl
%
Ter 7 & $-0.82 \pm 0.15$\tablenotemark{a} & 
$17.76 \pm 0.07$\tablenotemark{e} & $20.44 \pm 0.04$ & 
$2.68 \pm 0.08$ &  $ 9.3 \pm 1.8$
%
\enddata
\tablenotetext{a}{Zinn \& West (1984).}
\tablenotetext{b}{This paper.}
\tablenotetext{c}{Carney et al. (1992).}
\tablenotetext{d}{Montegriffo et al. (1998).}
\tablenotetext{e}{Sarajedini \& Layden (1997).}
\tablenotetext{f}{Sandquist et al. (1992).}
%M92 (Carney et al. 1992), Ter 8 (Montegriffo et
%al.1998), Arp 2 (Sarajedini \& Layden 1997), M54 (this paper), 
%M5 (Sandquist et al. 1992), Ter 7 (Sarajedini \& Layden 1997).}
\end{deluxetable}

%TAB15

\begin{deluxetable}{ccc}
%\tablewidth{34pc}
\tablewidth{0pt}
\tablecaption{Final Metallicities and Ages \label{tbl-15}}
\tablehead{
\colhead{Population} & 
\colhead{$[Fe/H]$}   & 
\colhead{Age}
}
\startdata
%
Old Field       & $-1.3 \pm 0.2$        & $10.5 \pm 1.5$ \nl
Intermed Field  & $-0.7 \pm 0.2$        & $5.0 \pm 1.0$ \nl
\medskip
Young Field     & $-0.4 \pm 0.3$        & 0.5 -- 3.0 \nl
% & & \nl
Ter 8           & $-2.00 \pm 0.10$      & $14.5 \pm 0.8$ \nl
Arp2            & $-1.84 \pm 0.09$      & $13.1 \pm 0.9$ \nl
M54             & $-1.79 \pm 0.08$      & $14.7 \pm 0.5$ \nl
Ter 7           & $-0.82 \pm 0.15$      & $ 8.3 \pm 1.8$ 
%
\enddata
\end{deluxetable}

%TAB16

\begin{deluxetable}{lcrrrr}
%\tablewidth{34pc}
\tablewidth{0pt}
\tablecaption{Red Giant Star Counts \label{tbl-16}}
\tablehead{
\colhead{Field}        & 
\colhead{Range}        &
\colhead{Area}         & 
\colhead{Z-Poor}   &
\colhead{Mid-Z}    &
\colhead{Z-Rich}   
}
\startdata
%
M54      & 100-620 pix & 60.4 & 227  &  33  &  43  \nl
M54      & $>$620 pix  & 61.4 &  42  &  21  &  26  \nl
Sgr      & all         & 67.9 &  11  &  17  &  12  \nl
MUSKKK   & all         &225.6 &  30  &   6  &   6
%
\enddata
\end{deluxetable}

%TAB17

\begin{deluxetable}{lcrrrr}
%\tablewidth{34pc}
\tablewidth{0pt}
\tablecaption{MSTO Star Counts \label{tbl-17}}
\tablehead{
\colhead{Field}        & 
\colhead{Range}        &
\colhead{Area}         & 
\colhead{1.0 Gyr}    &
\colhead{2.5 Gyr}    
%\colhead{Z-Rich}   
}
\startdata
%
M54      & 300--650 pix& 53.7 &  64  & 189  \nl
M54      & $>$650 pix  & 55.2 &  35  & 102  \nl
Sgr      & all         & 67.9 &  44  &  88  \nl
Control  & all         &131.5 &   4  &  91  \nl
MUSKKK   & all         &225.6 &  10  & 106 
%
\enddata
\end{deluxetable}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\clearpage

\begin{center}
\bf{Figure Captions}
\end{center}

\bigskip

\figcaption{A 30 second $V$-band exposure of M54.  The field is 11.1
arcmin on a side.  North is to the right and East is up.  The
coordinate system employed in Tables 2 and 3 correspond to this
figure, with ($X$, $Y$) = (1, 1) at the lower-left and (1551, 1551) at
the upper-right.  The cluster center is at ($X$, $Y$) = (726, 806)
pixels.  Three stars from Table 2 are marked with their ID numbers
(37, 46, 49), and three stars from Table 3 are also marked (15, 51,
70).  A FITS-format image of this figure is available on request from
ACL. \label{fig1}}
% FIG_IMHS/m54_publF.gif

\figcaption{Differences between the shallow M54 4.0-m photometry and
the M54 photometry of SL95, in the sense (4.0-m -- SL95).  Panels (a)
and (b) show the magnitude and color differences as a function of $V$
magnitude.  Panels (c) and (d) show the differences as a function of
$V-I$ color.  Panels (e)--(h) show the differences as a function of
$X$ and $Y$ position in the 4.0-m coordinate system.  All stars
appearing in both photometry sets were used, regardless of the size of
their estimated errors. \label{fig2}}

\figcaption{A 600 second $V$-band exposure of our Sgr field.  The
image is 8.2 arcmin on a side.  North is to the right and East is up.
The coordinate system employed in Tables 4 and 5 correspond to this
figure, with ($X$, $Y$) = (1, 1) at the lower-left and (1150, 1150) at
the upper-right.  Three stars from Table 4 are marked with their ID
numbers (20, 49, 68), and three stars from Table 5 are also marked (5,
12, 86).  A FITS-format image of this figure is available on request
from ACL. \label{fig3}}
% FIG_IMHS/sgr_publF.gif

\figcaption{Differences between the present Sgr 4.0-m photometry and
the Sgr photometry of SL95, in the sense (4.0-m -- SL95).  The panels
are as for Figure 2.  \label{fig4}}
%All stars appearing in both photometry sets were
%used, regardless of the size of their estimated errors. \label{fig4}}
% FIG3/fig3.eps

\figcaption{The CMDs of (a) the Sgr field, and (b) the M54 field.  Deep
and shallow photometry sets have been spliced together as described in
the text.  Only stars with errors in $V$ and $V-I$ less than 0.10 and
0.14, respectively are shown. \label{fig5}}

\figcaption{The CMDs from (a) the Sgr field, and (b) our control field.
Only stars with errors in $V$ and $V-I$ less than 0.10 and 0.14,
respectively, are shown.  In (c), a subset of the stars from (b) have been
subtracted statistically from those in (a), resulting in a clearer
view of the Sgr sequences.  In (d), a subset of the control sample
of MUSKKK has been subtracted from the stars in (a), providing a
complementary view of Sgr. \label{fig6}}
% FIG_CMDS/cmdsubtr.sgr_cntrl.eps

\figcaption{The CMDs from (a) all the stars in the M54 field, (b) the Sgr
field, and (c) stars in the M54 field with radii between 300 and 600 pixels.
Only stars with errors in $V$ and $V-I$ less than 0.10 and 0.14,
respectively, are shown.  In (d), the stars from (b) have been
subtracted statistically from those in (c), resulting in a clearer
view of the M54 sequences alone -- note that some traces of the
youngest Sgr population remain. \label{fig7}}
% FIG_CMDS/cmdsubtr.m54_sgr.eps

\figcaption{$V$-band light curves for 71 short-period variables,
listed by variable star designation.  The points represent
observations and the curve represents the best-fit template.  The
point symbols are described in footnote 6.  In each panel, the
magnitude range is 1.7 mag and the minor tick marks are separated by
0.1 mag, except in the case of V91. \label{fig8}}
% FIGS/lcs{1,2,3}.eps

\figcaption{Magnitude as a function of time for the long period variable
V25.  The time axis is broken into the five observing runs as marked.
The top set of plots are for $V$ magnitude, and the bottom set are for
$I$ magnitude. \label{fig9}}
%FIG_LPV/mag_time.lpv_V25.sm

\figcaption{Finder charts for each variable star in our field of view,
listed by variable star designation.  Each finder is 20 arcsec on a
side, and is oriented with North to the top and East to the left.
\label{fig10}}
% FIG_IMHS/vars_final.gif

\figcaption{The CMD from stars in Table 2 with errors $\sigma_V < 0.10$
and $\sigma_{V-I} < 0.14$, and located more than 150 pix from the
cluster center.  The mean magnitudes and colors of the variable stars
are marked as follows: ab-type RR Lyrae ({\em circles}), c-type RR
Lyrae ({\em triangles}), Cepheids ({\em stars}), eclipsing binaries
({\em squares}).  For the RR
%({\em asterisks}), long period variables ({\em squares}).  For the RR
Lyrae, filled and open symbols indicate stars with template fits
having rms values less than and greater than 0.033 mag, respectively.
\label{fig11}}
%For the long period variables, the open squares indicate stars listed
%as ``probable'' or ``possible'' LPV stars in Table 10.
% FIG_CMDS/cmd_vars2.eps

\figcaption{The cumulative radial distributions of blue HB stars (BHB),
RR Lyrae stars (RRL), red clump stars (RC), and ``anonymous red HB''
(AHB) stars in our field of view.  The AHB was defined as all stars
with $18.17 < V < 18.33$ and $0.75 < V-I < 0.90$. 
%This definition was
%based on the CMD in Figure 6a of SL95, upon which the AHB was first
%identified. 
SL95 showed that the BHB stars belong to M54, while the
RC stars belong to Sgr. \label{fig12}}
%FIGS/rad_cumdist.eps

\figcaption{The $V$-band amplitudes of our RR Lyrae stars are plotted
as a function of their logarithmic periods.  Symbols are as for Figure
11, except that squares indicate the four field RRL.  Symbols
overstruck with a {\em cross} lie more than 500 pix from the cluster
center.  The solid lines show the period-amplitude relations for M3
and M9 derived by Clement \& Shelton (1999), and the dotted line is
the M3 relation shifted to the position expected for RRab in a
population with $[Fe/H] \approx -1.1$ (Sandage 1982). \label{fig13}}


\figcaption{The field-subtracted CMD of M54 from Figure 7d.  The
curves are the fiducial sequences of M5 ($[Fe/H] = -1.40$, solid
curve) and M92 ($[Fe/H] = -2.24$, dashed curve) from
Johnson \& Bolte (1998), shifted to M54 as described in
the text. \label{fig14}}
%[m54_fids.eps]

\figcaption{The observed luminosity functions at $(V-I)_g - 0.21$ mag
for the indicated clusters, expressed as a difference from $V({\rm
HB})$.  The curves are Gaussian fits to the SGB peaks; the maximum of
each curve is used to estimate the age of the cluster. \label{fig15}}

\figcaption{(a) The field-subtracted Sgr CMD from Figure 6d.  The
curves are Bertelli et al. (1994) isochrones for $[Fe/H] = -1.3$ dex
and ages (from left to right) 0.004, 1.0, 2.0, 4.0, 6.3, 7.9, 10.0,
12.0, and 13.8 Gyr.  (b) The same as in panel (a) but the isochrones
are for $[Fe/H] = -0.7$ dex.  (c) The same as (a) but the isochrones
are for $[Fe/H] = -0.4$ dex.  (d) The field-subtracted M54 CMD from
Figure 7d.  The lower set of isochrones are for $[Fe/H] = -1.7$ and
ages 10.0, 12.0, 13.8, and 15.8 Gyr.  The upper set of isochrones are
for $[Fe/H] = -0.4$ and ages 2.0, 2.5, and 3.2 Gyr. \label{fig16}}
%[cmd4_iso.eps]

\figcaption{The field-subtracted CMD of Sgr from Figure 6d.  (a) The
curves are the fiducial sequences of M5 ($[Fe/H] = -1.40$, solid
curve) and M92 ($[Fe/H] = -2.24$, dashed curve) from Johnson \& Bolte
(1998). (b) The curves are the fiducials from M67 ($[Fe/H] = 0.00$,
solid curve, Montgomery et al. 1993) and NGC 2204 ($[Fe/H] = -0.34$,
dashed curve, Kassis et al. 1997). The fiducials were shifted to Sgr
as described in the text. \label{fig17}}
%[sgr\_fids.eps]

\figcaption{The age-metallicity relation for Sgr.  The values for the
clusters (circles) and field components (squares) are taken from
Table 15. (b) The data are repeated with the age-metallicity relation
predicted by the closed-box chemical enrichment model described 
in \S 9.  \label{fig18}}

\figcaption{Projected stellar density of RGB stars as a function of
radial distance from M54 in arcmin.  (a) Stars in the metal-poor box
defined by the RGB of M54, (b) stars in the intermediate-metallicity
box, $[Fe/H] \approx -1.3$, and (c) stars in the metal-rich box,
$[Fe/H] \approx -0.5$ dex.  The MUSKKK control sample point is located
at $R = 19$ for convenience.  \label{fig19}}

\figcaption{Stellar density of MSTO stars as a function of radial
distance from M54 in arcmin.  (a) Stars in the ``1 Gyr population''
box defined by the plume of young, blue Sgr stars, and (b) stars in
the ``2.5 Gyr population'' described in \S 9.  The control fields are
placed at $R \approx 19$ (the open circle represents the
MUSKKK field).  \label{fig20}}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\appendix

\section{Details on Individual Variable Stars}

{\bf V1} is listed in \markcite{hsh73}SH73 as a Cepheid with P=1.34956
days.  Our period, P=1.34769 days, is close to this value, and the
mean magnitude is about 0.9 mag brighter than the cluster RR Lyrae.
These properties are consistent with the Period-Luminosity relation
for a Population II Cepheid pulsating in the first-overtone
(\markcite{nem94}Nemec et al. 1994).  The star's position 3.2 arcmin
from the cluster center suggests it is a member of M54.

{\bf V2} is listed in \markcite{hsh73}SH73 with P=0.5111 days.  Our
period search found no strong minima below P=0.8 days, and produced an
excellent light curve at P=1.09461.  The light curve properties are
similar to the Anomalous Cepheids in the Sextans dwarf spheroidal
galaxy found by \markcite{mfk95}Mateo et al. (1995b).  The mean
magnitude of this star is about 0.9 mag brighter than the cluster RR
Lyrae, in rough agreement with the Period-Luminosity relation for an
Anomalous Cepheid pulsating in the fundamental mode
(\markcite{nem94}Nemec et al. 1994).  The star's position 3.5 arcmin
from the cluster center suggests it is a member of M54.

{\bf V14} appears to be variable in our data set, but is present in
only a few of our frames.  We folded the existing data by a series of
periods around the \markcite{hsh73}SH73 value of P=0.6892, and obtained a best fit at
P=0.70489 days.  The small number of observations makes the period and
mean photometry derived for this star quite uncertain.  The $XY$
position in \markcite{hsh73}SH73 is in error; the variable star marked in \markcite{rn59}RN59 is
correct.

{\bf V20} appears not to be variable, based on the star marked on the
\markcite{rn59}RN59 finder.

{\bf V21} could not be identified on our images.  The \markcite{rn59}RN59 finder
chart position disagrees with the $XY$ position listed in \markcite{hsh73}SH73.  Five
candidate stars in the regions in question were checked for
variability, but no significant variations were detected.  A note in
\markcite{hsh73}SH73 questions the variability of this star; we support this
conclusion.

{\bf V22} appears not to be variable, based on the star marked on the
\markcite{rn59}RN59 finder.

{\bf V26} could not be identified on our images.  The \markcite{rn59}RN59 finder
chart position disagrees with the $XY$ position listed in \markcite{hsh73}SH73.  Four
candidate stars in the regions in question were checked for
variability, but nothing significant was detected.

{\bf V27} appears not to be variable, based on the star marked on the
\markcite{rn59}RN59 finder.  However, the stellar image is badly blended.  \markcite{hsh73}SH73 lists
this star with a single magnitude (rather than with a maximum and
minimum), and the note ``med'' appears in the magnitude comments.

{\bf V34} light curve has large scatter; period and mean photometry
are consequently uncertain.

{\bf V47:} light curve has large scatter; period and mean photometry
are consequently uncertain.

{\bf V53} appears not to be variable, though it was identified
positively with the star marked on the \markcite{rn59}RN59 finder.

{\bf V63:} light curve has large scatter; period and mean photometry
are consequently uncertain.  Could also be an RRc with P=0.39339 days.

{\bf V67} was identified positively from the \markcite{rn59}RN59 finder chart, and
appears to have real variations, but we could not determine a period
for this star.
%E08

{\bf V69} is blended our our images, and the resulting photometric
confusion makes it difficult to tell whether this star is truly
variable.  

{\bf V72} does not vary significantly in our data set.  The
identification was made using the \markcite{rn59}RN59 finder.  \markcite{hsh73}SH73 lists it as a
possible eclipsing binary; there may have been no eclipses during our
observations.

{\bf V73} could not be identified on our images.  The \markcite{rn59}RN59 finder
chart position disagreed with the $XY$ position listed in \markcite{hsh73}SH73.  Five
candidate stars in the regions in question were checked for
variability.  Possible variations were detected in the star marked in
Figure 10, but no acceptable periods were found.
%73b

{\bf V75} could not be confirmed as a variable in our data set,
since its image is badly blended with that of another star.

{\bf V76} is marked on the \markcite{rn59}RN59 finder as a close pair of stars,
neither of which appears to be variable.  However, we detected an
ab-type RR Lyrae 3 arcmin East of the marked star, and suspect that
the \markcite{rn59}RN59 finder is in error (indeed, \markcite{rn59}RN59 classified V76 as an RR
Lyrae).  We have assigned this RR Lyrae to V76.
%A46
 
{\bf V77:} light curve has large scatter; period and mean photometry
are consequently uncertain.

{\bf V79} could not be identified positively on the \markcite{rn59}RN59 finder.
Two candidate stars in the region in question were checked for
variability.  Definite variations were detected in the star marked in
Figure 10, but no acceptable periods were found.
%79a

{\bf V80:} crowded!  Identification from \markcite{rn59}RN59 finder
uncertain, but variability is certain in the star marked on Figure 10,
and a reasonable period was determined, despite the relatively large
scatter in the resulting light curve.  This star may exhibit the
Blazhko effect.
%80b

{\bf V81:} crowded!  Identification from \markcite{rn59}RN59 finder
not possible.  Five candidate stars in the region in question were
checked for variability.  Possible variations were detected in the
star marked in Figure 10, but no acceptable periods were found.  No
magnitude or period information were listed in \markcite{hsh73}SH73
%81c

{\bf V82:} light curve has large scatter; period and mean photometry
are consequently uncertain.

{\bf V88:} light curve has large scatter; period and mean photometry
are consequently uncertain.

{\bf V99} is a possible W UMa contact binary.  The light curve has large
scatter; period and mean photometry are consequently uncertain.

{\bf V100:} light curve has large scatter and relatively poor phase
coverage; period and mean photometry are consequently uncertain.


{\bf SV1} is a new suspected variable.  Could not determine period.

{\bf SV2} is a new suspected variable.  Obtained a reasonable Algol light
curve at P=2.11765 days, but too few data points to be certain.

{\bf SV3} is a new suspected variable.  Could not determine period.

{\bf SV4} is a new suspected variable.  Obtained a reasonable Algol light
curve at P=0.40978 days, but too few data points to be certain.



\end{document}
