M31 velocity vector

The M31 Velocity Vector.
I. Hubble Space Telescope Proper Motion Measurements
Sangmo Tony Sohn, Jay Anderson, and Roeland P. van der Marel
arXiv:1205.6863v1 [astro-ph.GA] 31 May 2012

Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218

tsohn@stsci.edu

ABSTRACT

We present the first proper motion measurements for the galaxy M31. We
obtained new V -band imaging data with the Hubble Space Telescope ACS/WFC
and the WFC3/UVIS instruments of three fields: a spheroid field near the minor
axis, an outer disk field along the major axis, and a field on the Giant Southern
Stream. The data provide 5–7 year time baselines with respect to pre-existing
deep first-epoch observations of the same fields. We measure the positions of
thousands of M31 stars and hundreds of compact background galaxies in each
field. High accuracy and robustness is achieved by building and fitting a unique
template for each individual object. The average proper motion for each field
is obtained from the average motion of the M31 stars between the epochs with
respect to the background galaxies. For the three fields, the observed proper mo-
tions (µW , µN ) are, in units of mas yr−1 , (−0.0458, −0.0376) ± (0.0165, 0.0154),
(−0.0533, −0.0104)±(0.0246, 0.0244), and (−0.0179, −0.0357)±(0.0278, 0.0272),
respectively. The ability to average over large numbers of objects and over the
three fields yields a final displacement accuracy of a few thousandths of a pixel,
corresponding to only 12 µas yr−1 . This is comparable to what has been achieved
for other Local Group galaxies using VLBA observations of water masers. Po-
tential systematic errors are controlled by an analysis strategy that corrects for
detector charge transfer inefficiency, and spatially- and time-dependent geometric
distortion and point-spread-function variations. The robustness of the proper-
motion measurements and uncertainties are supported by the fact that data from
different instruments, taken at different times and with different telescope orien-
tations, as well as measurements of different fields, all yield statistically consistent
results. Papers II and III of this series explore the implications of the new mea-
surements for our understanding of the history, future, and mass of the Local
Group.

Subject headings: proper motions — galaxies: individual (M31) — galaxies: kine-
matics and dynamics — Local Group

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1. Introduction

At a distance of ∼ 770 kpc (e.g., Freedman & Madore 1990), the Andromeda galaxy
M31 is the nearest giant spiral to the Milky Way. Together, these two galaxies dominate the
mass and dynamics of the Local Group. It is therefore of tremendous interest to determine
the velocity vector of M31 with respect to the Milky Way. While the line-of-sight velocity of
M31 is well-known from Doppler measurements, a determination of its proper motion (PM)
has so far remained elusive.
The PM of M31 has been sought for almost a century (e.g., Barnard 1917), but very
high astrometric accuracy is required to accomplish this. At the distance of M31, 100 km
s−1 corresponds to 0.027 mas yr−1 . This is well beyond the accuracy limit of current ground-
based optical observations even for long time baselines (e.g., > 40 year; e.g., Vieira et al.
2010). The two most accurate (∼ 0.01 mas yr−1 ) PM measurements in the Local Group come
from studies of water masers in M33 (Brunthaler et al. 2005) and IC 10 (Brunthaler et al.
2007). However, water masers in M31 have only recently been discovered (Darling 2011), and
it will take time until a sufficient baseline has been established to enable PM measurements.
Theoretical modeling of the local Universe has suggested that the Galactocentric rest-
frame velocity of M31 must have a tangential component Vtan ≤ 200 km s−1 . However, within
this limit many different orbits are still possible (Peebles et al. 2001). van der Marel & Guhathakurta
(2008) recently presented an estimate of M31’s transverse velocity based on indirect argu-
ments, using the known kinematics of satellite galaxies of M31 and the Local Group. Their
estimate implies that Vtan ≤ 56 km s−1 , at 68.3% confidence. In other words, M31 may be
moving directly (radially) towards the Milky Way. This might have drastic implications for
the future of the Milky Way, and even for the Sun itself (Cox & Loeb 2008). However, all
theoretical and indirect estimates of Vtan make assumptions about the structure and equi-
librium of the Local Group that may not hold true in our hierarchically-evolving universe.
Therefore, only a direct PM measurement will yield a robust determination of the M31
velocity vector.
The Hubble Space Telescope (HST) is an observing platform in space with unparalleled
astrometric capabilities. Whereas the absolute astrometric accuracy of HST is limited by the
accuracy of the Guide Star Catalog (∼ 0.2 arcsec for GSC2.3; Lasker et al. 2008), the current
onboard imagers of HST (e.g., ACS and WFC3) are capable of measuring relative positions
of multiple sources in a field to better than 0.5 mas. Therefore, it is possible to measure
very accurate absolute PMs of stars by measuring their displacement over time relative to
stationary background reference source(s) in the same field.
Quasi-stellar objects (QSOs) have conventionally served as reference sources in HST PM

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studies, due to their star-like appearance and luminous nature. Such a strategy has been
used in several studies for measuring absolute PMs of Milky Way satellite galaxies (e.g.,
Piatek et al. 2002, 2003, 2005, 2006, 2007; Kallivayalil et al. 2006a,b; Piatek et al. 2008).
However, the accuracies thus achieved are not sufficient at the distance of M31. This is due
in part to drawbacks of using QSOs. Not only must they be spectroscopically identified
in advance, but their distribution on the sky is sparse. For the small field of view of HST
imagers, these limits require the observer to find a QSO behind the field of interest. If one
exists, one must deliberately image the field containing the QSO at two or more different
epochs to measure the displacement of the stars relative to the QSO. Even then, the PM
uncertainty is limited by the positional accuracy of a single QSO.
Instead of using QSOs, one can use background galaxies as reference objects in HST
studies of absolute PMs. At the spatial resolutions of interest here, these sources are also
stationary due to their large distances. One great advantage of background galaxies is their
ubiquity: background galaxies are found in nearly every deep astronomical image. Moreover,
using many background galaxies can provide higher accuracy than using a single QSO. While
the individual positional accuracy is somewhat poorer for galaxies, their much larger number
√
provides an important N advantage in averaging. For example, Milone et al. (2006) and
Kalirai et al. (2007) used background galaxies to derive an accurate PM of the Galactic
globular cluster NGC 6397, at the distance of ∼ 3 kpc from the Sun. Similarly, Brown et al.
(2010) used background galaxies to determine the absolute PM of a hypervelocity star in
the Milky Way halo.
Unlike the situation with QSOs, galaxy positions are difficult to measure. One can
adopt a simple centroiding algorithm, but such algorithms have limited accuracy in the un-
dersampled images of HST’s wide-field detectors (Anderson & King 2000). Furthermore,
the resolved galaxies and the unresolved stars would be expected to have different centroid
biases, leading to spurious apparent PMs. As a better alternative, Mahmud & Anderson
(2008) developed a template-fitting method to measure accurate positions of resolved back-
ground galaxies. In this method, a tailor-made template is constructed for each individual
object, and the same template is used to measure a consistent position for that object in
each exposure. This is conceptually similar to the effective point spread function (ePSF) ap-
proach widely used for stars (Anderson & King 2000). Even for bright background galaxies,
the template-fitting method is able to measure positions at least twice as well as with simple
centroids (Mahmud & Anderson 2008).
In this paper we present the first proper motion measurements for the galaxy M31,
based on new HST ACS/WFC and the WFC3/UVIS imaging of three fields with deep pre-
existing data. We use the template-fitting method of Mahmud & Anderson (2008) as our

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basis, but make several innovations in the analysis to minimize systematic errors and ensure
robustness of the results. The final accuracy of only 12 µas yr−1 is significantly better than
what has been achieved by HST for other nearby galaxies, and is comparable to what has
been achieved for other Local Group galaxies using VLBA observations of water masers.
This paper is organized as follows. Section 2 describes the observations. Section 3
describes the analysis steps that lead to the PM measurements. Section 4 presents and
discusses the PM results thus obtained for the three target fields. Concluding remarks are
presented in Section 5.
This is the first paper in a series of three. In Paper II (van der Marel et al. 2012) we
derive the velocity vector of the M31 center of mass by correcting the PM results presented
here for the internal motions of stars within M31 and for the reflex solar motion. We then
combine the result with other estimates to determine the Galactocentric velocity and orbit
of M31, and use this to estimate the mass of the Local Group. In Paper III (van der Marel
et al. 2012b, in preparation) we use the results to study the future dynamical evolution of
the Local Group, and the expected merging of the Milky Way-M31-M33 System.

2. Data

2.1. First-Epoch Data

The data set we used for measuring the PM of M31 consists of HST images of three
fields obtained in two separate epochs. The fields are the SPHEROID, OUTERDISK,
and TIDALSTREAM fields of Brown et al. (2006) obtained under the science programs
GO-9453 and GO-10265 (PI: T. Brown). Figure 1 shows the location of the three M31 target
fields. The observation details for the first epoch data are given in Brown et al. (2006); field
coordinates and total exposure times are listed in their Table 1. In brief, the target fields
were observed between December 2002 and January 2005 in two filters (F606W and F814W)
with HST ACS/WFC to create color-magnitude diagrams (CMDs) that reach well below
the M31 main sequence turn-off. For each field, all exposures were obtained within 40 days
so they can be safely treated as a single epoch of data. Total per-field per-filter exposure
times ranged from 53-161 ksec. Individual images had exposure times ranging from 1,100
to 1,300 sec, and were dithered such that no two exposures placed a star on the same pixel.
In addition to the whole-pixel offsets, the dithering strategy also included sub-pixel shifts so
that every object was observed at a range of locations relative to the pixel boundaries; such
“pixel phase” coverage is critical for our program. The very deep first-epoch F606W data
were used by us for constructing super-sampled stacked images, identifying stars and galaxies,

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building templates for the template-fitting method, and providing reference positions of stars
and galaxies, as described in Section 3 below.

2.2. Second-Epoch Data

Our second-epoch data were observed between January 2010 and August 2010 using
ACS/WFC and WFC3/UVIS in F606W (HST program GO-11684, PI: van der Marel). We
designed our observations to minimize random and systematic errors with the following
strategies in mind. Pre-analysis of the first-epoch data revealed that the F606W filter pro-
vides slightly better astrometric handle on the background galaxies than the F814W filter,
so we chose to obtain second-epoch images only with F606W. Furthermore, we decided to
observe the three target fields with two different instruments instead of just one. This yields
comparable random error, but allows additional consistency checks on our PM results. Since
the purpose of the second-epoch observations was astrometry, and not deep photometry, the
total second-epoch exposure time was much less than in the first-epoch. We observed each
target field for two orbits (four half-orbit exposures) with WFC3/UVIS and one orbit (two
half-orbit exposures) with ACS/WFC. Individual images had exposure times ranging from
∼1,300 to 1,500 sec, slightly longer than in the first epoch, and were dithered in similar
fashion. The baselines between the two epochs are in the range of 5–7.5 years.
For the ACS observations, we used the same orientations and coordinates as used by
the first-epoch observations to maximize the overlapping area and so that any errors in the
static distortion solution would naturally cancel out. Due to different sets of guide stars
being used between the first- and second-epoch observations, the second-epoch ACS images
were slightly offset and rotated with respect to the first-epoch images. However, the offsets
are all within a negligible fraction of the ACS/WFC field of view (FOV), and the orientations
are within 0. 1 of each other.
◦

For the WFC3 observations, we pointed the telescope so that the resulting images
roughly share the same centers with the first epoch, but the WFC3 images were rotated
by roughly 45◦ with respect to the ACS images. This was to place the parallel ACS/WFC
fields overlapping with the first epoch parallel WFPC2 images, but the parallel fields will
not be discussed in this paper as they are not useful for astrometry at the accuracy that
we need. Because the FOV of WFC3/UVIS is 64% of that of the ACS/WFC, the observed
WFC3 images are fully contained within the ACS images despite the ∼ 45◦ difference in
their orientations. The details of our second-epoch data are summarized in Table 1.

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3. Analysis

3.1. Overview

Measurement of proper motions involves measuring the displacement between the po-
sition of an object at one time and its position at another. If this displacement can be
measured with respect to objects for which we know the absolute motion, then we can ob-
tain absolute PMs. In our case, we will measure the displacement of M31 stars with respect
to the background galaxies in the field to obtain an absolute PM. Our strategy is to first
align the stars in the first- and second-epoch images, restricting the alignment to those stars
confirmed to belong to M31. Then we measure the average displacement of the background
galaxies with respect to this moving frame of reference. The negative of this relative dis-
placement yields the mean absolute PM of the M31 stars. This measurement of the mean
does not require knowledge of the exact PM distribution of the M31 stars. This distribution
can be complex, because different structural components of M31 contribute to each field.
However, to transform the mean PM of the M31 stars in a field to an estimate of the M31
center of mass, one does need to construct a model for the internal kinematics of M31, which
is addressed in Paper II of this series.
The overall PM derivation process is summarized below.

1. We first create a high-resolution stacked image using the deep and well-dithered first-
epoch images for each target field using the distortion-corrected first exposure as the
reference frame.

2. Stars and galaxies are then identified from the stacked image for each field, and pho-
tometric measures and the CMD are used to specifically identify M31 reference stars.

3. We construct a template for each star and galaxy from the high-resolution image.

4. We then use the template to measure in a consistent fashion a position for each
star/galaxy in each individual exposure of each epoch.

5. We redefine the first-epoch reference frame using the template-based positions of the
stars, and determine the average first-epoch position of each galaxy in this frame.

6. We determine the template-measured positions of the stars in each of the six second-
epoch exposures for each field and use them to transform the template-measured po-
sitions of the galaxies into the first-epoch frame.

7. We then take the difference between the second- and first-epoch positions of the galaxies
to obtain the relative displacement of the galaxies with respect to the M31 stars.

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8. Finally, we multiply the relative displacements of the galaxies by −1 to obtain the mean
absolute displacement of the M31 stars, since in reality the galaxies are stationary and
the stars are moving. Division by the time baseline turns the displacements into PMs.

The following sections provide the details of each step.

3.2. Initial Processing of Individual Exposures

We downloaded all the required data in the form of flt.fits images from the STScI
HST archive. These images are already bias-subtracted, dark-subtracted, and flat-fielded
by the STScI data-reduction pipelines using the best available calibration data at the time
of data inquiry. Each multi-extension flt.fits file was converted to a single 4,096×4,096
image where the top chip scene is directly abutted to the bottom chip scene. This single-
format image allows a more efficient way of handling the data, and all computer software we
used throughout our analysis were specifically programmed to deal with this image format.
The HST detectors suffer from significant degradation in charge-transfer efficiency (CTE)
as time progresses. This impacts the astrometry of objects by effectively shifting the cen-
troid position of the flux distribution. This can be a particularly serious problem for our
second-epoch ACS/WFC data, taken almost 9 years after the installation of ACS. A pixel-
based CTE correction routine has recently been developed by Anderson & Bedin (2010) for
the ACS/WFC, and Brown et al. (2010) demonstrated that this correction works very well
for astrometric purposes. All first-epoch ACS/WFC images were processed through this
routine. For the second-epoch ACS/WFC images, we used a modified version of the CTE
correction code that also included a correction for X-CTE, in addition to the publicly avail-
able correction for Y -CTE. We found that the X-CTE correction made very little difference
to the positions measured in the images, compared to the other uncertainties in the results.
A CTE correction routine was not available for the WFC3/UVIS at the time of writing
this paper, so no correction was made to the WFC3 images. However, since at the time of the
observations, UVIS had spent only ∼ 10% as much time as ACS in the space environment, the
CTE degradation of our UVIS images is expected to be quite small. Although UVIS appears
to be losing charge-transfer efficiency faster than ACS did, the studies that show this have
focused on low-background images (e.g., Baggett et al. 2011). As our backgrounds are quite
high, we expect CTE losses to be very low for our images. Hence, the CTE degradation
should only have a minimal effect on our WFC3 PM measurements. A direct post-facto
consistency check of this is provided by comparison our ACS/WFC and WFC3/UVIS PMs
derived for the same fields (see Section 4.2 below).

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As the final step of the initial processing of individual images, we measured each star
using the img2xym WFC 9x10 program (Anderson & King 2006) on the ACS/WFC images,
and using a similar program on the WFC3/UVIS images. Both programs utilize library
PSFs to determine a position and a flux for each star. We will use these PSF-based star
positions to provide our initial handle on the coordinate transformations from one image to
another.

3.3. Distortion Corrections

The positions of objects on the HST detectors must be corrected for geometric distor-
tions before they can be used for any astrometric measurement. For the ACS/WFC detector,
the distortion-correction solutions by Anderson (2005) have been used in several astrometric
studies. These solutions are known to be better than 0.01 pixel or ∼0.5 mas, and were shown
to be stable between 2002 and 2007 (Anderson 2007; van der Marel et al. 2007). These avail-
able solutions were directly applied to the positions measured in our first-epoch ACS/WFC
images.
While reducing data for another program (GO-11677, PI: H. Richer), one of us (J.A.)
found that the ACS distortion solution appears to have changed by ∼ 0.005 pixel relative
to the solution before the HST Service Mission 4 (SM4). This may be related to a similar
variation of the PSF (Anderson et al., in prep.). It is unclear what caused these variations,
but they appear to be stable over time. We used the GO-11677 data set to develop a
correction to the pre-SM4 ACS/WFC distortion solution and applied it to the second-epoch
data. For the WFC3/UVIS data, we used the distortion corrections provided by Bellini et al.
(2011). These corrections are better than 0.008 pixel or ∼0.3 mas and appear to be stable
over the UVIS lifetime so far.
The accuracies of the distortion corrections are comparable to our final measured mo-
tions (as will be shown later). However, geometric distortion affects both star and galaxy
positions similarly, and hence drops out in a differential measurement to lowest order. This
is true in particular if each galaxy position is measured with respect to that of nearby stars
that fall on the same area of the detector. This is what we do in Section 3.8 below, to ensure
that higher order geometric distortion correction residuals do not affect our final results.

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3.4. Stacked Images

The deep and well-dithered first-epoch ACS/WFC data are well-suited for constructing
high-resolution stacked images. To do this, we first apply the distortion corrections (see
Section 3.3) to the positions of stars measured in Section 3.2. We then adopt the first
exposure of each field as the frame of reference, cross-identify stars in this exposure and
stars in the other exposures, and use their distortion-corrected positions to construct a six-
parameter least-squares linear transformation between the two frames. The six parameters
involve x-y translation, scale, rotation, and two components of skew. The need for these
linear transformations is discussed in Section 3.6.4 of Anderson & van der Marel (2010). We
use these transformations to convert the star positions measured in the various images into
the reference frame, giving us many estimates of the position of each star in the reference
frame. We average these positions to refine the reference-frame positions for all the stars,
and use these new average positions to improve the transformations (which had initially been
based on the positions as measured in the first frame itself). This procedure is iterated 2–3
times to improve the positions of both the stars and the transformations.
We use the star-based transformations to construct the stacked images. In order to get
better sampling, we super-sample the stacked image by a factor of 2 relative to the native
ACS/WFC pixel scale. The image-stacking process we used is similar to the commonly used
Drizzle algorithm (Fruchter & Hook 2002) with a point kernel, except that we included an
iterative procedure to regularize the sampling in a manner similar to iDrizzle (Fruchter
2011). The procedure involves no deconvolution, and as such the resulting image at every
point simply represents the flux that an actual flt image pixel would receive if it were
centered at that location in the frame. It is this property that allows us to construct empirical
templates (see Section 3.6.1) for our stars and galaxies. Figure 2 shows a 25′′ ×25′′ portion of
the 2× super-sampled stacked image for the SPHEROID field with M31 stars and background
galaxies identified, as described in the following sections.

3.5. Identifying Stars and Galaxies

All of our target fields are dominated by M31 stars, but there are also other sources, such
as foreground stars and background galaxies. Since our strategy for deriving the PM of M31
is to measure the displacement of the background galaxies relative to the co-moving frame
of reference defined by the M31 stars, it is important to accurately identify both background
galaxies and M31 stars before we proceed any further.

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3.5.1. Identifying M31 Reference Stars

The selection of M31 stars was carried out following the procedure below. For each
target field, we use the star list compiled in Section 3.4. Our initial star lists include only
sources that are found independently in a large number of first-epoch F606W exposures
(typically, > 35%); as such, the list is almost entirely free of cosmic rays and image defects.
There are, however, a large number of resolved sources, which could be either blended stars
or extended background galaxies. To identify them, we made use of the mean quality-
of-PSF-fit parameter (QFIT) reported by the img2xym WFC 9x10 program (see Figure 3a).
Selecting the objects with low values of QFIT allows us to filter out extended objects as well
as stars that are too close to other stars to provide accurate position measurements. To
ensure that all of the stars in the list are M31 members, we reduced the F814W images
and constructed a CMD (see Figure 3b). From this, we selected stars that lie roughly on
the M31 sequences. Although small in number, field stars in the Milky Way halo may be
included in our selection, but we believe that most of them will be filtered out in our next
step of selection. Finally, reference stars were selected based on their lack of motion with
respect to the other M31 stars. This was done by aligning the second-epoch star positions
with the first-epoch positions and iteratively rejecting the objects that have moved between
the epochs (see Figure 3c). For any given ACS/WFC target field, the M31 stars should
all be moving towards the same direction in space. Differential motions due to the internal
kinematics of M31 within a single ACS/WFC field should be negligible compared to the
observational uncertainties (see Paper II for details). We note that, in principle, better
detection and photometry of fainter stars can be achieved by measuring stars directly from
the stacked images as has been demonstrated by Brown et al. (2006). However, as our
main goal is doing astrometry, we are only interested in stars for which positions can be
reliably measured in individual exposures. Our final lists of M31 reference stars include
∼ 10, 000 stars for the SPHEROID and OUTERDISK fields, and ∼ 5, 000 stars for the
TIDALSTREAM field.

3.5.2. Identifying Background Galaxies

A quick visual inspection of the super-sampled stacked images of our target fields re-
veals that there are hundreds of background galaxies. We have already identified sources
using the img2xym WFC 9x10 program, but because that program is specifically designed
to measure stars, it neglects many of the extended sources. For this reason, SExtractor
(Bertin & Arnouts 1996) was separately run on the stacked images to detect and measure
extended sources. We generated a candidate list of galaxies by selecting sources from the

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SExtractor output mainly based on their MAG AUTO, CLASS STAR, and FLUX MAX parameters.
For each field, this candidate list included more than 1,000 sources, but we found that many
of the candidates are in fact multiple stars clustered together. We carefully identified sources
in the candidate list one by one, basing our judgment on their 2-d contours and 3-d surface
profiles. Whenever the identification was unclear, we excluded the source from our selection
to stay on the conservative side. We note that we desire the background galaxy list to be
as free of contamination by stars as possible because the PM measurements are sensitive to
the reference sources we choose to use. Finally, we considered only objects for which the
template-fitting method, described in the next section, yields position uncertainties of less
than 0.25 pixel in both X and Y . The final lists contain 368, 339, and 374 galaxies for the
SPHEROID, OUTERDISK, and TIDALSTREAM fields, respectively.

3.6. Measuring Positions of Objects with Templates

Our goal is to measure accurate PMs. The most crucial part of our analysis is therefore
to measure the positions of objects in the individual images as accurately as possible. Simple
methods such as fitting the objects with a two-dimensional Gaussian function or finding the
flux centroid are accurate only if the objects have well-sampled peaks. This is not the case
for most of the background galaxies in our target fields. To measure the positions of objects,
we therefore adopt the template-fitting method developed by Mahmud & Anderson (2008).

3.6.1. The Template-Fitting Method

The details and general diagnostics of the template-fitting method are documented
in Mahmud & Anderson (2008). Here we summarize only the basic concepts behind the
method. The distortion corrections in Section 3.3 and the linear transformations derived in
Section 3.4 allow us to associate a position (xr , yr )j in a given individual image j with a
position (xm , ym) in the stacked image, and vice versa. We are thus able to build a model
of what the galaxy or star should look like in an individual image by proper sampling of the
stacked image.
The goal is to measure in consistent fashion a position for each object (star or galaxy)
in each individual exposure. To do this, we construct a template model for each object, and
use it to measure that object in every exposure. The location within the template that we
will define to be its center is arbitrary, since in the end we care only about differences in
position. So we define the center of the brightest pixel in the stacked image to be the object

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center (or “handle” in the parlance of Mahmud & Anderson). With the center so defined,
we interpolate the stacked image at a super-sampled array of points, with the center at (0,0)
but with a pixel orientation and spacing that corresponds to the transformed exposure’s flt
coordinate system. The result is a template that can be interpolated at an array of locations
to tell us how much flux we would expect in an array of flt pixels for a presumed object
center. The template is custom-made for each individual exposure, based on the mapping
of that exposure’s pixel coordinate system into the master frame. The template centers are
all the same.
For each object in each exposure, we use the previously established transformations to
identify the location of the object in the exposure to within one pixel. We then evaluate
an array of trial locations for the center, covering an area of 1×1 pixels with a spacing of
0.01 pixel. At each trial location, interpolation of the template with a bicubic interpolation
algorithm tells us how much flux we should expect to measure in each pixel of the exposure.
We compare this model to the actual observed pixel values to obtain a quality-of-fit quantity,

Σpixels |OBS − MODELij |
qf itij = , (1)
Σpixels OBS

for each trial offset (i, j). The position of the galaxy or star then corresponds to the location
of the template center that gives the minimum qf itij .
In Section 3.2 we used empirical “library” PSF fits to the stars to define the transfor-
mations for construction of the stacked images. However, for the final PM measurements
we use the template procedure for both galaxies and stars. This minimizes any potential
for systematic errors due to differences in measurement techniques between different types
of objects. The sums in equation (1) were chosen to extend over the 5 × 5 pixel raster
centered on the object’s brightest pixel in the individual exposure. The size of this region
was deliberately chosen to be small, since most of the positional information for the stars
and compact galaxies of interest here is contained in the central core pixels.

3.6.2. First-Epoch ACS Data

We applied the template-fitting method to all stars and galaxies in the first-epoch
ACS/WFC data, to determine their positions in each exposure. To illustrate the quality of
the template-fitting method, we show in Figure 4 the best fit qf it (as defined in Section 3.6.1)
versus instrumental magnitude for the background galaxies in one of the first-epoch images.
The top row of Figure 5 shows the data, template fits, and residuals for three galaxies of
different brightness, and for one star.

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3.6.3. Second-Epoch ACS Data

The stacked images constructed in Section 3.4 represent the astronomical scene con-
volved with the average effective PSF appropriate for the first-epoch data. The effective PSF
represents the convolution of the instrumental PSF with the sensitivity profile of a pixel. The
astronomical scene for any given object should not change between our epochs of data, but
the effective PSF might change. This is certainly the case for the second-epoch WFC3/UVIS
data, discussed below, since both the instrumental PSF and the pixelization are different
than for the first-epoch ACS/WFC data. However, we also found that the effective PSF
for the second-epoch ACS/WFC data was not the same as for first-epoch ACS/WFC data.
Even though the pixelization remained the same, we found that a subtle change occurred in
the ACS PSF since SM4 (see Section 3.3).
It is important that any change in the effective PSF be explicitly accounted for in the
analysis. PSF changes can introduce centroid shifts, and these shifts can be different for
point sources and for extended sources. Our technique for measuring the M31 PM relies
on positional differences between stars and background galaxies. PSF changes can therefore
produce spurious PM results.
In order to deal with the differences in the effective PSF between epochs, we model
each object in a given second-epoch image as follows. First, we select bright and isolated
stars from the M31 reference star list compiled in Section 3.5.1 , and use them to derive a
7×7-pixel kernel that accounts for differences in effective PSF between the stacked image and
the individual exposure. Next, we interpolate the first-epoch template onto the flt pixel
grid of the second epoch image, as we did in Section 3.6.2. We then convolve the template
with the kernel derived in the earlier stage, and finally evaluate the goodness-of-fit quantity
qf it to find the object position that provides the minimum qf it. The middle row of Figure 5
shows the data, template fits, and residuals for three galaxies of different brightness, and
for one star, obtained with this procedure for one of the second-epoch ACS/WFC images
(jb4404vuq).
The complexity in this procedure lies in the determination of the kernel. Each kernel
has 49 free parameters. However, it is constrained by 5×5-pixel patches centered around all
the objects in each image. This is an over-determined problem that can be solved to find the
best-fitting kernel. We cast the problem into the form of a least-squares matrix equation,
and solved for the kernel using the observed pixel values for several thousand bright and
isolated stars distributed throughout the target fields. The solution was obtained through
singular value decomposition (Press et al. 1992). To fit the kernels, we first determined the
stellar positions of the bright and isolated stars without any kernel. Then we kept those
positions fixed, and optimized the kernels.

– 14 –

We ended up using a single kernel for each second-epoch image. We experimented
with using different kernels for different parts of each image (to deal with potential field-
dependent PSF variations), but found that this did not change the main results. In deriving
the kernel, we recognized that there is a degeneracy between shifting the kernel and shifting
the astronomical scene. We removed this degeneracy by constructing the pixel residuals that
went into the kernel with respect to the library-PSF-measured position (see Section 3.2) for
each star. This ensured that the kernel would reproduce on average the PSF-based positions
for the stars, and that it would properly account for changes in source shape due to known
PSF variations between epochs. Either way, the degeneracy does not affect absolute PMs
that are based on relative differences in position between stars and background galaxies.
The top panels of Figure 6 show the average of the derived normalized kernels for
the second-epoch ACS/WFC images. Slight asymmetries are evident, primarily in the X
direction. Without our kernel-based corrections, such asymmetric PSF differences would
bias astrometry.

3.6.4. Second-Epoch WFC3 Data

Our second epoch WFC3/UVIS images are rotated by ∼ 45 deg with respect to the
ACS images. Because we have many bright M31 stars in the field, we are able to derive six-
parameter linear transformations that relate the undistorted WFC3 positions to the stacked
image without difficulty.
To determine the positions of all objects in the field we used the same kernel-based
template-matching approach as for the second-epoch ACS data. In this case, the kernel
accounts not only for PSF variations, but also for variations in the PSF size and orientation
between epochs. The bottom row of Figure 5 shows the data, template fits, and residuals
for three galaxies of different brightness, and for one star, obtained with this procedure for
one of the second epoch WFC3/UVIS images (ib4401s1q).
The kernel size for the WFC3/UVIS data was chosen to be the same as for the ACS
data, and we used over 1,000 stars to derive the kernels. The bottom panels of Figure 6
show the average of the derived normalized kernels for the second-epoch WFC3/UVIS images.
There are slight asymmetries in both coordinate directions. This differs from the situation
for the second-epoch ACS/WFC data, because the WFC3/UVIS detector is rotated by ∼
45◦ in the HST focal plane with respect to the ACS/WFC detector. Also, the symmetric
part of the kernels differs between the second-epoch ACS/WFC images and the second-
epoch WFC3/UVIS images. This is because of the difference in PSF shapes between the

– 15 –

different detectors. Our kernel-based approach minimizes potential systematics arising from
measuring positions of stars and galaxies in images with different PSF characteristics.
Our PM measurement technique for each field compares the high-resolution stacked
image from the first-epoch to the individual exposures from the second epoch. The derived
kernels account for variations in the average PSF between the epochs, as well as exposure-
to-exposure PSF variations within the second epoch. We do not compare second-epoch
exposures to individual first-epoch exposures, but only to the high-resolution stacked image.
The stacked image is constructed from the first-epoch images, so on average, the PSF does
not change between the stacked image and the individual first-epoch images. Therefore, any
exposure-to-exposure PSF variations that may exist within the first epoch do not affect the
derived PMs. For this reason, there was no need to study or correct explicitly PSF variations
within the first epoch data set.

3.7. Reference-Frame Positions and Positional Uncertainties

The processes outlined above lead to accurate positions of stars and galaxies in each
individual exposure. We align the star positions between exposures using the same iterative
approach described in Section 3.4, but this time using the newly derived positions based
on the template-fitting method instead of the initial library-PSF-based positions. For each
exposure for a given field, this procedure yields a six-parameter linear-transformation with
respect to the first exposure in the first epoch for that field. For all objects (stars and
background galaxies) in all exposures, we apply the known geometric distortion solutions
and these linear-transformations, to obtain the position in the distortion-corrected frame of
the first exposure (i.e., j8f801abq, j92c01b4q, and j92c28ccq for SPHEROID, OUT-
ERDISK, and TIDALSTREAM, respectively). These distortion-corrected frames serve
as our reference frames.
In the first-epoch data we have many exposures. This yields multiple determinations
for the position of each star or galaxy in the reference frame. The RMS scatter among
these determinations provides a measure of the random positional uncertainty in a single
measurement for each object. Figure 7 shows the one-dimensional errors in positions for
M31 stars (upper panels) and background galaxies (lower panels) as a function of instru-
mental magnitude (defined as minstr = −2.5 log[electrons]) in the three separate target fields.
Brighter objects have a higher signal-to-noise ratio than fainter objects, and therefore have
more accurately determined positions. Stars are more compact than background galaxies,
and therefore also have more accurately determined positions. At fixed magnitude, the
scatter in positional accuracy is larger for the galaxies than for the stars. This is because

– 16 –

galaxies have a large variation in sizes and shapes, with the morphology also depending on
wavelength. The dependence of astrometric accuracy on size and shape was discussed in
Sohn, Anderson, & van der Marel (2010), and the dependence on wavelength was discussed
in Mahmud & Anderson (2008). At the bright end in Figure 7, the per-exposure accuracies
level off at 0.01–0.02 WFC pixels for both stars √ galaxies. When many measurements
and
are averaged, the positional errors decrease as 1/ N.

3.8. Proper Motions of Individual Objects

Using the above procedure,we determine for each object its average first-epoch position
in the reference frame. In a similar way, we measure template-based positions for each
second-epoch exposure, correct these positions for distortion, and transform them into the
reference frame using the template-measured star positions. We then compare these positions
with the first-epoch averages to estimate the PM displacement for that object. Division by
the time baseline of the observations then yields an estimate of the actual PM.
By construction, our method aligns the star fields between epochs. M31 stars will
therefore have zero PM on average. Figures 8 and 9 show the residual motion of each star
in X and Y (P MX and P MY ) as a function of chip coordinates for one of the second-epoch
ACS/WFC images, and for one of the second-epoch WFC3/UVIS images. These plots show
that the “average star” (given our selection criteria and brightness limits) is measured with
a precision of ∼ 0.05 pixel (but as shown in Figure 7, this is a strong function of brightness).
When averaging over an ensemble√ N stars in M exposures, the accuracy of the average PM
of
is in principle smaller by a factor NM . The stellar PMs show that there remain some small
systematic trends with position on the detector, at levels 0.02 pixel. These trends may be
related to small changes in the distortion solution between the first and second epochs, or
other low-level systematic effects that are not accounted for in our analysis. Such systematic
trends might affect PM results. It is therefore important that we correct for them. We note
that we found similar residual trends in all three target fields. This indicates that whatever
is causing the low-level systematic trends is related to the detector characteristics rather
than the internal kinematics of M31.
The background galaxies were not used to define any transformations between the first-
epoch and second-epoch data. Any motion between the galaxies and stars should therefore
show up as a displacement between the first- and second-epoch galaxy positions in the
reference frame. We measure these displacements in a two step process to remove any
systematic PM trends related to the position on the detector, such as those visible in Figures 8
and 9.

– 17 –

First, we calculate the difference between the first- and second-epoch positions for each
background galaxy in the reference frame. Then, for each galaxy, we compute the average
displacement of stars in the vicinity of the galaxy. We subtract this displacement from
the galaxy displacement. This “local correction” ensures that the displacement of each
background galaxy is measured only with respect to the M31 stars that fall on the same part
of the detector. This removes any remaining systematic PM residual associated with detector
position. Each local correction is constructed using stars of similar brightness (±1 mag in
the inner 5×5 pixels area) and within a 200 pixel region centered on the given background
galaxy. The matching of the stars and background galaxies in brightness was motivated
by the fact that some known detector effects (such as CTE), depend not only on detector
position, but also on the source brightness.
In the top panels of Figures 10 and 11, we show the X and Y proper motions (P MX and
P MY ) of each galaxy as a function of chip locations (X and Y ), for the same second-epoch
exposures as in Figures 8 and 9. There are many more stars than background galaxies in
each frame, and the positions of the stars are generally more accurately determined than
those of the galaxies (see Figure 7). The final PM error for each second-epoch exposure is
therefore dominated entirely by the astrometric accuracy for the background galaxies.

4. M31 Proper Motion

4.1. Final Results

To determine an average PM of the M31 stars in each field, we start with the measured
PM displacements of the background galaxies in the reference frame (including local correc-
tions as described above). For each of the 18 different second-epoch images, we calculate
the weighted average using the positional errors determined from the RMS among the first-
epoch measurements. Outliers were rejected in this process with an iterative 3-σ rejection
scheme applied to the 2-d PM distribution. The uncertainty of the average was computed
using the bootstrap method (Efron & Tibshirani 1993) with 10,000 bootstrap samples for
each case. This provides a rigorous error estimate, which is ultimately based on the actual
scatter between results inferred from different background galaxies. This naturally includes
any random errors due to photon-counting statistics, as well as any potentially remaining
systematic errors that lead to increased scatter (but any systematic errors that affect all
sources equally would not be quantified by this).
We then transformed the results to average PMs (and their associate errors) of M31
stars in the directions north and west, in units of mas yr−1 . To do this, we used the

– 18 –

orientation of the reference image (see Section 3.6.2) with respect to the sky (using the
FITS header keyword ORIENTAT), the time baseline, and the fact we are measuring the
relative displacement of the background galaxies with respect to the M31 stars in our target
fields. In Table 2, we tabulate the PM1 (µW , µN ) inferred from each second-epoch image
and the corresponding error, along with the number of background galaxies used for the PM
derivation. Figure 12 shows for each of the three target fields the PM estimates inferred
from the six independent second-epoch exposures (2 ACS/WFC + 4 WFC/UVIS).
The final estimate of the average PM of the M31 stars in each field is calculated by
taking the error-weighted mean of the six independent measurements listed in Table 2. The
results are shown in red in Figure 12, and they are tabulated in Table 3.
Figure 13 compares the final PM results for the three different M31 fields. The weighted
average of the results is shown in black. This weighted average is also listed in Table 3. The
figure shows results in physical units of km s−1 , instead of mas yr−1 . To transform the units,
we assumed a distance of 770 kpc (see references in van der Marel & Guhathakurta 2008),
so that 0.1 mas yr−1 = 365 km s−1 . Distance errors were not propagated in this conversion.
The final weighted average PM differs from zero at the ∼ 4.3-σ level. Therefore, we have
actually measured a motion, and we have not merely put an upper limit on any motion.

4.2. Consistency Checks

To have faith in the results, it is important to assess the internal consistency of the mea-
surements. There are many checks available for this, since we have performed measurements
using different exposures, with different instruments, and for different fields.
For a given field i (SPHEROID, OUTERDISK, or TIDALSTREAM), second-
epoch instrument j (ACS/WFC or WFC3/UVIS), and coordinate direction k (West or
North), we identify the set of l PM measurements µijkl (either four or two measurements,
depending on the instrument) with random errors ∆µijkl. There are a total of 36 measure-
ments (see Table 2). We combine the different measurements l to obtain the 12 weighted
averages µijk with random errors ∆µijk .
The quantity
2
µijkl − µijk
χ2
1 = (2)
ijkl
∆µijkl

1
µW is defined to be positive for an object moving towards the West, and therefore has the opposite sign
of the PM in the RA direction.

– 19 –

provides a measure of the extent to which different measurements for the same field and
with the same instrument agree to within the random errors. We find that χ2 = 26.2. In
1
absence of systematic errors, one expects χ2 to follow a χ2 probability distribution with
1
NDF = 36 − 12 = 24 degrees of freedom. The expectation value for such a distribution is
√
NDF , and the dispersion is ∼ 2NDF . Hence, the measurements for the same field with the
same instrument are statistically consistent with each other. This is also visually evident
from inspection of Figure 12, which shows furthermore that the agreement is least good for
the OUTERDISK field.
The quantity
(µi1k − µi2k )2
χ2 =
2 (3)
ik
∆µ2 + ∆µ2
i1k i2k

provides a measure of the extent to which measurements for the same field with different
instruments agree to within the random errors. We find that χ2 = 8.5. In this case NDF = 6,
2
so the measurements for the same field with the different instruments are also statistically
consistent with each other.
Finally, the quantity
2
µik − µk
χ2
3 = (4)
ik
∆µik
provides a measure of the extent to which measurements for different fields agree to within
the random errors. Here µik with random errors ∆µik are the weighted averages over all
exposures for given field (top three lines of Table 3). The µk are the weighted averages over
all fields (bottom line of Table 3)). We find that χ2 = 1.9. In this case NDF = 6 − 2 = 4, so
3
the measurements for different fields are also statistically consistent with each other. This
is also visually evident from inspection of Figure 13.
The fact that all measurements are statistically consistent indicates that there is no
additional scatter in the data that is unaccounted for by the random errors. This justifies
the use of weighted averages in combining the different measurements (which propagates
only random errors).

4.3. Evaluating Systematic Errors

The final weighted average PM corresponds to a displacement of 0.0074 ACS/WFC
pixel over the ∼ 7 year time baseline for the SPHEROID field. The per-coordinate error in
the displacement is only 0.0024 pixel. The displacement errors are even lower for the other
fields, which have shorter time baselines.

– 20 –

These displacements are very small, but they can nonetheless be accurately measured.
This can be understood based on first principles. A single source can be centroided to an
accuracy of order σ/(S/N), where σ is the Gaussian dispersion of the object size, and S/N is
the signal-to-noise ratio of the observation. For a compact source with S/N 100, this yields
uncertainties of 0.01–0.02 pixels, consistent with what we found in Figure 7. Therefore, even
a single bright compact source in a single exposure can provide an accuracy comparable to
the M31 PM displacement. Averaging over multiple sources in many exposures/fields reduces
the random uncertainties to thousandths of a pixel and hence allows a solid measurement of
the PM displacement.
The key to our robust measurement is, though, control of systematic errors, since those
are not guaranteed to decrease by averaging multiple measurements. We have paid careful
attention throughout our observation planning and subsequent analysis to identify possible
sources of systematic errors and adjusted our methodology to minimize them as following:

• The observations were obtained with the F606W filter, which has the best PSF and
distortion models for both ACS/WFC and WFC3/UVIS (see Section 2.2).

• The ACS/WFC images were obtained using the same telescope orientation and pointing
in both epochs (see Section 2.2). This allows any static astrometric residuals that
depend on the position on the detector (e.g., static geometric distortion solution errors)
to cancel out in the differential PM measurement.

• The ACS/WFC data were explicitly corrected for the effects of imperfect CTE, reduc-
ing any potential astrometric impact (see Section 3.2).

• We used different geometric distortion solutions for the different epochs of data, thus
minimizing any impact due to potential time-variations in the geometric distortion
solutions (see Section 3.3).

• We carefully selected our samples of M31 stars and background galaxies to minimize
any bias introduced by interlopers (see Section 3.5).

• We built a unique template for every individual source that was measured, thus avoid-
ing any ad hoc assumptions about the properties of PSF or galaxy shapes (see Sec-
tion 3.6.1).

• We fitted and accounted for PSF variations between different epochs, thereby mini-
mizing any potential PM biases (see Section 3.6.3).

– 21 –

• We measured the relative PM of each background galaxy with respect to only its
neighboring stars (the “local correction”), thus minimizing the impact of spurious
spatially varying PM residuals (see Section 3.8).

• The above local measurement was made with respect to only stars of similar brightness,
thus minimizing the impact of magnitude-dependent PM residuals, including CTE
effects (see Section 3.8).

• We obtained the final PM for each field by averaging over background galaxies at
many different locations on the detector. This minimizes the potential impact of local
peculiarities in detector properties or the astronomical scene (see Section 4.1). Our ap-
proach is significantly advantageous over PM studies that use only a single background
quasar in an image.

• We determined the random PM error for each exposure from the actual scatter between
results from different background galaxies. The resulting errors include all sources of
scatter, and not just those from photon-counting statistics (see Section 4.1).

• We used robust statistical measures with outlier rejection throughout our analysis,
thus minimizing the potential influence of individual erroneous measurements (e.g.,
Section 4.1).

• We showed that PM measurements with different instruments, taken at different times
and with different telescope orientations, as well as measurements of different fields,
all yield statistically consistent results (see Section 4.2). This rules out a large range
of possible residual systematic errors, including most possible astrometric residuals
specific to a given instrument or detector.

In summary, there is strong reason to believe that the PM measurements and the quoted
uncertainties are robust and free of the potential systematics considered in this section.

4.4. M31 motions: Other Contributions and Constraints

The results presented thus far do not directly measure the PM of the M31 center of
mass (COM). Instead, in every field we measure the sum of the internal kinematics of the
M31 stars and the COM motion. The rotation curve of M31 has an amplitude of ∼ 250
km s−1 (Corbelli et al. 2010). The contributions from internal kinematics are therefore not
necessarily negligible compared to the uncertainties in our measurements (see Figure 13).

– 22 –

In Paper II we model the internal kinematics explicitly, and we derive an unbiased
estimate for the COM PM. As it turns out, this estimate is not very different from the
weighted average presented in Table 3. We also show in Paper II that this estimate is
statistically consistent with the estimate for the transverse motion of M31 derived with the
independent methods of van der Marel & Guhathakurta (2008), based on the kinematics of
M31 and Local Group satellites. This is a further indication that any remaining systematic
uncertainties in our measurements are likely to be small.
The observed PMs presented here are heliocentric motions, i.e., not corrected for the
reflex solar motion in the Milky Way. This known reflex motion falls in the same quadrant
of PM space as our measurements (see van der Marel & Guhathakurta 2008). The actual
transverse motion of M31 with respect to the Milky Way is therefore closer to zero than what
we show in Figure 13. This provides another useful consistency check on our measurements,
since there are theoretical reasons to suspect that the transverse motion of M31 with respect
to the Milky Way should be small (e.g., Peebles et al. 2001). In fact, we show in Paper II
that the transverse motion implied by our data is consistent with zero, meaning that M31
is likely moving on a nearly direct radial orbit towards the Milky Way.

5. Conclusions

We have presented the first direct absolute PM measurements of three fields in M31,
our nearest giant companion galaxy in the Local Group. We used new second-epoch HST
data obtained with two different instruments, combined with very deep pre-existing first-
epoch data, spanning a time baseline of 5–7 years. With state-of-the-art analysis methods,
using background galaxies as stationary reference frame, we achieved a final PM accuracy of
∼ 12 µas yr−1 . This is comparable to the accuracies that have been achieved for other Local
Group galaxies using VLBA observations of water masers (Brunthaler et al. 2005, 2007).
Water masers were recently discovered in M31 (Darling 2011), but have yet to be used for a
PM determination. We have paid careful attention to control of systematic errors throughout
our analysis. A large range of consistency checks indicates that our PM measurements and
the quoted uncertainties are robust and free of unknown systematics. The new PM results
provide improved insights into the history, future, and mass of the Local Group. These
topics are explored in detail in Papers II and III of this series.
The techniques presented here for measuring the PM of stars with respect to background
galaxies are not only applicable to M31, but can be applied to a wide range of other problems.
For example, our group has ongoing HST observing programs to use the same techniques to
measure the PM of Leo I (GO-12270, PI: S. T. Sohn), the PM of dwarf galaxies near the

– 23 –

Local Group turn-around radius (GO-12273, PI: R. P. van der Marel), and the PM of stars
in the Sagittarius Stream (GO-12564, PI: R. P. van der Marel). More generally, any deep
wide-field space-based imager with a stable configuration, including future missions such as
WFIRST and EUCLID, might be able to use the techniques presented here to measure the
PMs of foreground objects from multiple epochs of data.
For M31 itself, it would be possible to improve the results presented here through
additional HST observations by, e.g., increasing the available number of fields, time baselines,
or signal-to-noise ratio per epoch. Meanwhile, water masers hold the potential to soon
provide PM measurements for individual sources in M31 (Darling 2011). They will yield
constraints on the transverse motion of the M31 center-of-mass that are independent from
those presented here and in Paper II. Furthermore, water masers might allow measurement
of additional effects, such as the M31 PM rotation, and the increase in M31’s apparent size
caused by its motion towards us.

Support for this work was provided by NASA through a grant for program GO-11684
from the Space Telescope Science Institute (STScI), which is operated by the Association of
Universities for Research in Astronomy (AURA), Inc., under NASA contract NAS5-26555.
The authors are grateful to Rachael Beaton, Gurtina Besla, Tom Brown, T. J. Cox, Mark
Fardal, and Raja Guhathakurta for contributing to the other papers in this series, and for
comments that helped improve the presentation of the present paper.
Facilities: HST (ACS/WFC; WFC3/UVIS).

– 24 –

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2007, ACS/ISR 2007-07, Calibration of ACS/WFC Absolute Scale and Rotation for
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van der Marel, R. P., Fardal, M., Besla, G., Beaton, R. L., Sohn, S. T., Anderson, J., Brown,
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This preprint was prepared with the AAS L TEX macros v5.2.
A

– 27 –

Fig. 1.— Figure taken from Brown et al. (2006). Appropriately scaled and rotated boxes
denote our three target ﬁelds (labeled). The underlying gray shading represents the density
of stars from the map of Ferguson et al. (2002). The ellipse marks the area within 30 kpc of
the galactic center in the inclined disk plane (labeled).

– 28 –

Fig. 2.— A 25′′ ×25′′ portion (1.5% of the total image area) of the stacked image for the
SPHEROID ﬁeld. Background galaxies used as positional references are enclosed by red
circles, while M31 stars that pass the selection criteria of Section 3.5.1 are marked with green
plus signs.

– 29 –

Fig. 3.— Selection of M31 stars in the SPHEROID ﬁeld based on (a) the quality-of-ﬁt
(QFIT) parameter, (b) the color-magnitude diagram, and (c) relative proper motion (divided
by the proper motion error for better scaling) with respect to the average proper motion. In
all three panels, blue points are stars that pass all three cuts, while red points are objects
that fail to pass at least one cut. Most objects in red located at the bright end of the color-
magnitude diagram were rejected because of their relative proper motions. These are likely
foreground stars.

– 30 –

Fig. 4.— Best fit qf it (as defined in Section 3.6.1) versus instrumental magnitude for the
background galaxies in one of the first-epoch images (j8f801caq). The quantity qf it is
a measure of the typical flux residual of the template-fit normalized by the total flux. It
therefore behaves as (S/N)−1 , so that brighter objects yield smaller values of qf it. The red
squares mark the objects shown in Figure 5.

– 31 –

Fig. 5.— Each row shows example results of the template-fitting method. Top row: a first-
epoch ACS/WFC exposure; middle row: a second-epoch ACS/WFC exposure; bottom row:
a second-epoch WFC3/UVIS exposure. Each row shows results for four different objects.
The first three objects (denoted GALAXY #1, #2, and #3), viewed from left to right, are
galaxies of decreasing brightness. The rightmost object (denoted STAR) is a star with similar
brightness to the second galaxy. Note that the bottom row images are rotated 45◦ clockwise
with respect to those in the top and middle rows because of the difference in orientations
between ACS and WFC3 observations (see Section 2.2). For each object in each image we
show three images: the observed pixels; the best-fit template; and the residual of the two.
The images show a 19 × 19 pixels patch for illustration, even though the template-fitting was
done using only the inner 5 × 5 pixels. The three galaxies in the top row are the same ones
for which the qf it value is indicated with a red square in Figure 4.

– 32 –

Fig. 6.— Grayscale images (left panels) and one-dimensional cuts (mid and right pan-
els) of the convolution kernels derived for the second-epoch ACS/WFC (top panels) and
WFC3/UVIS (bottom panels) data of the SPHEROID field. The one-dimensional cuts
show in green the central row (mid panels) and the central column (right panels) of the
kernel. Red squares show the same values in reverse order (equivalent to flipping the plots
about the center along the abscissa). Comparing the plots in different symbols (and col-
ors) provides a measure of asymmetries in the kernels, which could affect astrometry if left
uncorrected.

– 33 –

Fig. 7.— The total one-dimensional positional error per exposure as a function of instrumen-
tal magnitude, for M31 stars (upper panels) and background galaxies (lower panels). The
error is deﬁned as σ1−D = 1 (σX 2 + σY 2 ). Here σX and σY are the per-coordinate RMS
2
residuals with respect to the average, for the multiple ﬁrst-epoch measurements.

– 34 –

Fig. 8.— Displacements of individual stars (dark gray dots) versus detector location between
one of the second-epoch ACS/WFC images (jb4404vsq) of the SPHEROID field and the
average of the first-epoch images, plotted separately for X and Y positions. The units are
in native ACS/WFC pixels, and X and Y positions are in the reference frame. We also plot
the average displacements of stars and the RMS of the distribution for each 400-pixel bin in
red. The RMS is equivalent to the average uncertainty for an individual star. The 1-σ error
√
bars on the red data points equal RMS/ N , where N is the number of stars in the bin,
and these are smaller than the sizes of the points themselves. So while the displacements
average to zero by construction, there are statistically significant low-level trends indicative
of residual detector effects. We correct the measurements for these trends using the local
corrections described in Section 3.8.

– 35 –

Fig. 9.— As Figure 8, but now for one of the second-epoch WFC3/UVIS images (ib4401rsq)
of the SPHEROID ﬁeld.

– 36 –

Fig. 10.— Displacements of background galaxies versus detector location between one of the
second epoch ACS/WFC images (jb4404vsq) of the SPHEROID field and the average of
the first-epoch images, plotted separately for X and Y positions. The black points show the
relative displacements measured for different background galaxies. The weighted average for
all galaxies is shown as the red line; dashed red lines indicate the 1-σ error region around
√
the average. This region is smaller than the scatter between the points by a factor of ∼ N ,
where N is the number of background galaxies. The radius of each black point is proportional
to 1/∆, where ∆ is the PM measurement uncertainty for the particular background galaxy.
Hence, the area of each point is proportional to the weight a point receives in the final
weighted average. Symbols in green in the top left panel illustrate how the point size relates
to the PM uncertainty ∆. The units are in native ACS/WFC pixels, and X and Y positions
are in the reference frame.

– 37 –

0.2

0.1
PMX (pix)

0.0

-0.1

-0.2
0.2

0.1
PMY (pix)

0.0

-0.1

-0.2
0 1000 2000 3000 4000 0 1000 2000 3000 4000
X (pix) Y (pix)

Fig. 11.— As Figure 10, but now for one of the second-epoch WFC3/UVIS images
(ib4401rsq) of the SPHEROID ﬁeld.

– 38 –

Fig. 12.— Proper-motion results for the three target fields. Each black symbol with an
error bar indicates the weighted average PM of M31 stars in the given field, inferred from
a single second-epoch exposure as in Figures 10 and 11. Measurements using ACS/WFC
(open squares) and WFC3/UVIS (open triangles) are indicated with different symbols. The
solid red data point is the weighted average of the six separate measurements for each field.

– 39 –

M31 ALL 3 FIELDS
300

200

100
µN (km/s)

0

−100

−200

−300
−300 −200 −100 0 100 200 300
µW (km/s)

Fig. 13.— Average proper motions, converted to km s−1 using a distance to M31 of 770 kpc,
for each target ﬁeld (red closed circles). The error-weighted mean of the 3 ﬁelds is shown as
the black X mark.

– 40 –

Table 1: Description of Data for the Second Epoch Observations

Target Field Data set Detector PA V3a Exposure Time (s) ∆T b
jb4404vsq, jb4404vuq ACS/WFC 75.07 1289, 1289 7.10
SPHEROID
ib4401rsq, ib4401ruq, ib4401rxq, ib4401s1q WFC3/UVIS 255.14 1379, 1379, 1450, 1450 7.57
jb4405jgq, jb4405jiq ACS/WFC 247.17 1289, 1289 5.06
OUTERDISK
ib4402urq, ib4402uwq, ib4402vqq, ib4402vuq WFC3/UVIS 67.08 1420, 1420, 1491, 1491 5.55
jb4406muq, jb4406mwq ACS/WFC 21.85 1299, 1299 5.99
TIDALSTREAM
ib4403enq, ib4403epq, ib4403esq, ib4403ewq WFC3/UVIS 216.92 1379, 1379, 1450, 1450 5.47

a
The position angle of the HST V3 axis at the center of detector’s field of view.
b
Baseline between the first and second-epoch data in years.

Note. — All data in the second epoch were obtained with the F606W filter. Field coordinates and
descriptions of the first epoch-data are presented in Brown et al. (2006).

– 41 –

Table 2: Proper Motion Results for Individual Second Epoch Images
µW µN σµW σµN
Field Data Set (mas yr−1 ) (mas yr−1 ) (mas yr−1 ) (mas yr−1 ) Nused a
jb4404vsq −0.0839 −0.0212 0.0512 0.0327 308
jb4404vuq 0.0012 −0.0982 0.0297 0.0299 306
SPHEROID ib4401rsq −0.0085 0.0184 0.0606 0.0530 176
ib4401ruq −0.0564 −0.0379 0.0332 0.0442 176
ib4401rxq −0.0772 −0.0104 0.0660 0.0495 186
ib4401s1q −0.0895 −0.0126 0.0369 0.0339 172
jb4405jgq −0.0706 0.0085 0.0501 0.0504 310
jb4405jiq −0.1105 −0.0043 0.0547 0.0501 286
OUTERDISK ib4402urq −0.0123 0.0993 0.0679 0.0681 156
ib4402uwq −0.0668 −0.1609 0.0649 0.0759 152
ib4402vqq −0.0641 0.1353 0.0761 0.0757 161
ib4402vuq 0.0237 −0.1112 0.0589 0.0553 152
jb4406muq 0.0425 −0.0895 0.0574 0.0601 321
jb4406mwq 0.0104 0.0098 0.0564 0.0518 317
TIDALSTREAM ib4403enq −0.1116 −0.0572 0.0670 0.0752 185
ib4403epq −0.0792 −0.0358 0.0766 0.0812 196
ib4403esq −0.0445 −0.0474 0.0883 0.0731 175
ib4403ewq 0.0298 −0.0148 0.0794 0.0728 185

a
Number of background galaxies used for deriving the average proper motion.

– 42 –

Table 3: Final Proper Motion Results for the Three Target Fields

µW µN σµW σµN
Field (mas yr−1 ) (mas yr−1 ) (mas yr−1 ) (mas yr−1 )
SPHEROID −0.0458 −0.0376 0.0165 0.0154
OUTERDISK −0.0533 −0.0104 0.0246 0.0244
TIDALSTREAM −0.0179 −0.0357 0.0278 0.0272
weighted av.a −0.0422 −0.0309 0.0123 0.0117

a
Weighted average of the results for the three-target ﬁelds. This is not an unbiased estimate of the PM of
the M31 center-of-mass. It contains contributions also from the internal motions of stars in M31, which are
modeled and corrected in Paper II.

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