Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.stsci.edu/~marel/psdir/boeker_sim_SPIE02.ps Äàòà èçìåíåíèÿ: Sun Sep 21 19:36:04 2003 Äàòà èíäåêñèðîâàíèÿ: Sat Dec 22 03:02:43 2007 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 31

Measuring Stellar Proper Motions in Crowded Fields with
SIM
Torsten B˜oker a , Ronald J. Allen, Jayadev Rajagopal, and Roeland P. van der Marel
STScI, 3700 San Martin Drive, Baltimore, MD 21218
a On assignment from the Space Telescope Operations Division of the European Space Agency.
ABSTRACT
Astrometry in crowded fields is an important component of the science program of the Space Interferometry
Mission (SIM). Resolving multiple point sources within the SIM beam, or imaging of complicated, extended
source structures requires a (large) number of interferometer baselines. As the spacecraft design keeps evolving,
the impact on various key projects needs to be studied. In this paper, we discuss the capabilities of the latest
SIM design (with only two baselines available for science measurements) for measuring stellar proper motions
in crowded fields. Using the nucleus of the Andromeda Galaxy (M 31) as a case study, we quantify the roll
angle increment needed to enable such measurements with the reduced SIM baseline set. In particular, we
demonstrate that SIM can measure Keplerian motion of luminous stars around the # 3 â 10 7 M# black hole in
M 31, provided that the spacecraft roll angle can be chosen in increments of around 4 # or smaller.
Keywords: Interferometry: missions (SIM), synthesis imaging, Galaxies: individual (M31)
1. INTRODUCTION
The goals and objectives of astrometry in crowded stellar fields are the same as those for isolated stars: to
measure as accurately as possible the positions and proper motions of the stars. Unfortunately, attaining the
highest astrometric accuracy of SIM requires that the stars are isolated, and special techniques for observation
and data analysis will be required in crowded fields. However, some of the most interesting environments in
astrophysics involve crowded stellar fields. For example, the central regions of many galaxies apparently harbor
black holes, and these objects appear to be more massive in the larger, more evolved galaxies. The study of the
stellar nuclei of galaxies can therefore further our understanding of galaxy evolution with time and in particular,
the interplay between active galactic nuclei (AGN) and the stars in their immediate vicinity.
The Space Interferometry Mission (SIM) has the potential to make important contributions along this line.
With a 10 m long baseline, it promises at least a fourfold improvement in spatial resolution over the Hubble
Space Telescope. SIM has experienced considerable design changes over the last few years. The original SIM
layout had seven siderostats distributed along a 10m long truss, providing evenly spaced baselines between 1 m
and 10 m. However, cost­cutting measures and the desire to optimize the astrometric accuracy of SIM resulted
in a design which only provides a single 10 m baseline for the science interferometer, and a 9 m baseline for the
guide interferometer. The guide interferometer can probably be used for observations of the science target which
will certainly be useful for synthesis imaging. Nevertheless, the new design drastically reduces the number of
interferometric baselines available for synthesis imaging observation, and will also adversely a#ect the accuracy
of astrometric measurements in crowded star fields. In addition, the SIM specifications with respect to its ability
to obtain accurate measurements at di#erent roll angles have not yet been finalized. The roll angle increment
is an important parameter for the accuracy of separating the various fringe components from individual source
components within the field of view.
The purpose of this paper is to quantify the impact of the SIM design changes on a particular science
program, namely the study of stellar proper motions around the massive black hole in the Andromeda galaxy
(M 31). In § 2, we describe the nature of our SIM science project, and detail the implementation of the M 31
source model that is the subject of this study. We discuss some aspects of our simulations in § 3. In § 4, we
present our results, in particular the inferred requirements for SIM roll angle control.
Send correspondence to T. B˜oker (boeker@stsci.edu).

2. STELLAR PROPER MOTIONS AROUND MASSIVE DARK OBJECTS: A SIM
SCIENCE PROGRAM
Observational evidence for massive dark objects (MDOs) in galactic nuclei has been gathered for more than
two decades. The MDO is commonly assumed to be a supermassive black hole (BH), but only in a few cases
have plausible alternatives to a BH been ruled out. Recently, the focus has changed from individual galaxies
to demographic studies (see, e.g., Richstone et al. 1998, Ho 2001). The current paradigm holds that BHs are
common in galaxy nuclei, and that their masses scale roughly linearly with the bulge mass of the host galaxy.
Observational constraints on the presence of BHs in galaxies can be derived either from stellar or gas
kinematics. The most compelling evidence so far was obtained from stellar proper motions in the Milky Way
galaxy (Eckart & Genzel 1997; Ghez et al. 1998), and water maser gas in NGC4258 (Miyoshi et al. 1995).
In general, stars are better suited for probing the gravitational potential than viscous gaseous matter, because
they can be considered collisionless test particles in most cases. However, stellar motions are also extremely
di#cult to trace observationally, because of the small angular scales involved. While stellar dynamical evidence
for MDOs has been obtained for several galaxies from integrated light measurements with the Hubble Space
Telescope (HST; e.g., van der Marel et al. 1997), the Milky Way is the only case in which stellar proper motions
have been directly observed. These observations have essentially ruled out alternatives to a single BH in the
center of the Milky Way.
The unprecedented astrometric accuracy of SIM o#ers the unique opportunity to measure for the first time
the stellar proper motions in the nuclei of external galaxies. Multi­baseline imaging measurements with SIM
can yield a resolution of about 8 milliarcseconds (mas) at 600 nm. However, the quality of the restored images
is not easily predictable because their signal­to­noise ratio depends on many parameters, the most important
ones being the (u,v)­coverage and the structure of the source. Therefore, detailed modeling for each individual
target is required to optimize the observing strategy and to predict the quality of the reconstruction. For these
reasons, we have developed IMsim, a software package to simulate the synthesis imaging mode of SIM (B˜oker
& Allen 1999a).
Figure 1. The central 3.5 ## of M 31 in a greyscale representation that emphasizes B­band and UV light (Brown et
al. 1998). The circle denotes a 1.2 ## diameter aperture, close to the expected SIM field of view. The SIM imaging­mode
resolution will be # 0.7% of the size of the circle.

2.1. The Andromeda Galaxy
The Andromeda Galaxy (M 31) is the nearest giant spiral galaxy, and its nucleus can therefore be studied at
higher resolution than for any comparable galaxy. Interestingly, the sub­arcsec nuclear structure of M31 has
turned out be a considerable puzzle, which indicates that there remains much to be learned about galactic
nuclei in general. HST images of M 31 have revealed a double nucleus (Lauer et al. 1993). Integrated light
spectroscopy of absorption features from the ground and with HST indicate that one of the two nuclei hosts
an MDO of # 3 â 10 7 M# (e.g., Kormendy & Bender 1999; Statler et al. 1999). This nucleus is bluer than its
surroundings and contains UV­bright stars with apparent magnitudes up to m V # 21. It is hence believed to
be a young star cluster surrounding a BH (Brown et al. 1998; Lauer et al. 1998). The second nucleus has been
postulated to be the result of crowding of stars at the apocenters of their orbits in an eccentric disk (Tremaine
1995). We expect that SIM measurements of stellar proper motions will for the first time provide a detailed
understanding of the nuclear structure of M 31. Fig. 1 shows a greyscale image of the central 3.5 ## from Brown et
al. (1998), with the SIM field­of­view superposed. The observed line­of­sight velocity dispersion at the position
of the central star cluster, measured at the # 0.1 ## resolution of HST, is # = 440 km s -1 (Statler et al. 1999).
The velocity dispersion in the plane of the sky is a factor of # 2 larger, which --- at the M 31 distance of 770 kpc
--- corresponds to about 0.8 mas proper motion over the SIM mission lifetime of five years. This is equivalent to
10% of the SIM beam width. Because the velocity dispersion around a BH increases towards the center as 1/ # r,
stars at radii around the SIM resolution limit of 10mas should therefore have several mas of proper motion over
the SIM mission lifetime. Moreover, about 30% of the stars will have velocities larger than #, because of the
Maxwellian velocity distribution. We have demonstrated in the past that the original SIM design was easily
capable of measuring such motions (B˜oker et al. 1999b). We show below that similar results with the new SIM
design are feasible only if the roll angle increment can be chosen freely and with high accuracy.
2.2. Source Model
Our input source model of the nuclear cluster in M 31 for the simulations is the same as described in B˜oker
et al (1999b). The surface brightness within r = 0.15 ## is # 13.7 mag/arcsec 2 in V (Lauer et al. 1998). We
assume a SIM throughput of 33% and interferometer elements of 30 cm diameter, and model the central 0.3 ##
of the SIM field of view. We have constructed a population of stars with a luminosity function according to the
Salpeter (1955) initial mass function (IMF). We conservatively assume a uniform background surface brightness
of µ V = 15.5 on which we superpose randomly placed stars drawn from the luminosity function until the surface
brightness observed by Lauer et al. (1998) is reached. For the particular model discussed here (Figure 2), the
SIM field of view contains 8 stars with m v = 21, 16 stars with m v = 21.5, 30 stars with m v = 22, and 50
stars with m v = 22. The source model is calculated in five channel maps of 100 nm width between 500 nm
and 1000 nm. We assume blackbody spectra for the stars according to their spectral type. The success of the
M 31 program will depend on the magnitude of the individual stars, a quantity that is not well constrained even
from the best available data. V­band magnitudes of M V = -3.5 (corresponding to m V = 21 at the distance of
M 31) are typical for bright giants (Luminosity Class II) around spectral type A0. If the cluster is very young,
it could contain supergiant stars which are brighter than M V = -3.5. It should be noted, however, that the
relatively smooth surface brightness distribution found in the HST WFPC2 images of Lauer et al. (1998) is not
very supportive of this possibility, because much brighter stars would stand out above the background.
3. SIMULATIONS
It is intuitively clear that the loss of short baselines in the new SIM design, and the associated ``gaps'' in the
(u,v)­coverage will prevent accurate recovery of source structure at arbitrary spatial frequencies. However,
crowded stellar fields are (more or less by definition) dominated by point sources, and hence one may hope to
recover much of their morphology with a (u,v)­coverage that is heavily weighted towards the longest baselines.
Indeed, if the object under investigation is known to contain only point sources, then it is advantageous to
only use the highest spatial frequencies of the aperture. This fact is routinely exploited in aperture­masking
observations with large ground­based telescopes.
With only the longest baselines available, the roll angle increment has to be reduced significantly in order
to disentangle the numerous fringe components of the individual stars. This is demonstrated in Fig. 3 which

Figure 2. Left: input source model for the nuclear star cluster in M 31. The stellar magnitudes range from 21 to 22.5
mag in V. Right: reconstructed image after 1500 CLEAN iterations. The axes are labeled in units of grid pixels. Each
pixel has a size of 5 mas, the field is therefore # 0.3 ## wide. Contour levels are 10%, 20%, ..., 100% of the peak surface
brightness. This simulation used both the 9 m and 10 m baselines, a roll angle increment of 1 # , and a total on­source
integration time of 20 h.
compares the Point Spread Function (PSF) for two (u,v)­coverages obtained with di#erent roll angle increments.
Clearly, the ``noisy'' PSF obtained with roll angle steps of 8 # will allow accurate imaging of point sources only
with exceptionally high signal­to­noise data.
The requirement of fine roll angle sampling causes a new complication for our simulation code. Past versions
of IMsim have relied on Fast Fourier Transforms (FFT) for computationally e#cient simulations. The use of
FFTs requires that the source model as well as the sampled baselines lie on a regular grid of coordinates. In
the case of the original SIM design with its complete baseline set, relatively large roll angle increments were
su#cient for adequate (u,v)­coverage, and a simple ``nearest­neighbor'' approach for placing visibility data on
the grid did not cause significant errors.
However, this is no longer true for the simulations discussed here. The small roll angle increments necessary
to recover large numbers of point sources with very few baselines require extremely fine sampling in order to
avoid ``gridding'' errors, and the resulting large gridsizes would cause computation times to exceed reasonable
limits. For this reason, we have developed a version of IMsim that uses a Discrete Fourier Transform (DFT)
to sample the complex visibilities of the source at the actual baseline coordinates. This method ( which better
resembles the way ``real'' data are analyzed), avoids gridding errors, but is somewhat more costly in computing
time than an FFT for a comparable field size.
In addition to these technical aspects, our simulations are based on the following conditions and assumptions:
. We consider photon statistics, but no instrumental imperfections. In particular, the noise distribution
does not account for possible systematic calibration errors of the interferometer, which will degrade the
accuracy of the fringe phase and amplitude measurements. While the latest version of IMsim has the
ability to incorporate various models for the distribution of such instrumental errors, not enough is known
at this point about the SIM characteristics to make meaningful assumptions in this regard.
. SIM allows scientific use of the guide interferometer with its 9 m baseline.

Figure 3. Point spread function achieved by using both the 9 m and 10 m baselines in spectral synthesis mode for two
cases of roll angle increments: 1 # (left) and 8 # (right).
. SIM measures dispersed fringes in five 100nm­wide channels between 500nm and 1000nm. By combining
the complex visibilities in spectral synthesis mode, one e#ectively improves the (u,v)­coverage.
. We allow a total (on­source) integration time of 20 h for each simulated observation, which is evenly
distributed between all baselines. For example, in the case of 1 # roll angle increment and two baselines,
each measurement is 100 s long (360 â 2 â 100 s = 20 h), while for a 2 # roll angle increments, the time per
measurement is 200 s, etc.
4. RESULTS AND DISCUSSION
In order to quantify the accuracy of stellar positions in the synthesized images, we take the following approach.
After the CLEAN residuals have been added to the CLEANed image, we extract one­dimensional surface
brightness ``cuts'' through those recovered sources that are not obviously a#ected by close neighbors. Especially
for the sparser (u,v)­coverages, only the brightest stars are recovered with reasonable signal­to­noise ratios, and
we have therefore concentrated on those eight stars with m v = 21. We then fit simple Gaussians to the stellar
profiles, and compare the position of the best­fit Gaussian to the ``true'' position as known from the input
model. The positional errors derived in this way have a distribution centered on zero, their standard deviation
is a measure of the typical astrometric uncertainty. The error in a proper motion measurement, derived from
simple di#erencing of two position measurements, will be larger by a factor of # 2.
Figure 4 summarizes the positional accuracy as a function of the roll angle increment. As expected, the
positional errors increase as the (u,v)­coverage becomes sparser, and the experiment quickly becomes unfeasible
if the roll­angle increment exceeds about 4 # . Note also the significant loss of accuracy when only the 10m­baseline
is used (diamond symbol).
We believe that our analysis is conservative in a number of ways. For one, more sophisticated deconvolution
methods that make use of available data at low spatial frequencies ­ for example, from HST images ­ will help
to reduce the impact of the missing short SIM baselines. In addition, the simplistic one­dimensional Gauss
fits to the stellar positions can only provide a rough (under)estimate of the accuracy. Multiple component,
two­dimensional fitting routines will certainly improve the fidelity of position measurements. Nevertheless, it is
clear from the analysis presented here that proper motion studies in the nucleus of M 31 are testing the limits of

the capabilities of the ``new'' SIM mission. Whether this study (and others in similarly crowded fields) remain
viable SIM projects will mostly depend on the magnitude and nature of the systematic calibration errors, and
the ability to control the spacecraft roll angle.
Figure 4. Astrometric accuracy of stellar positions in the M 31 nucleus as a function of SIM roll angle increment. The
horizontal line indicates the requirement for measuring positions with an accuracy that is at least three times smaller
than the expected stellar proper motions after 5 years. Asterisks are for use of both the 9 m and 10 m baselines, the
diamond is for use of the 10 m baseline only.
REFERENCES
1. B˜oker, T. & Allen, R. J. 1999a, ApJS, 125, 123
2. B˜oker, T., Allen, R. J., & Rajagopal, J. 1999b, in Optical and IR Interferometry from Ground and Space,
Unwin, S. & Stachnik, R. eds., ASP Conf. 194, 219
3. Brown, T. M., Ferguson, H. C., Stanford, S. A., & Deharveng, J.­M. 1998, ApJ, 504, 113
4. Eckart, A. & Genzel, R. 1997, MNRAS, 284, 576
5. Ghez, A. M. et al. 1998, ApJ, 509, 678
6. Ho, L. C. 2001, ApJ, 564, 120
7. Kormendy, J. & Bender, R. 1999, ApJ, 522, 772
8. Lauer, T. R. et al. 1993, AJ, 106, 1436
9. Lauer, T. R., Faber, S. M., Ajhar, E. A., Grillmair, C. J., & Holtzman, J. A. 1998, AJ, 116, 2263
10. Miyoshi, M., et al. 1995, Nature, 373, 127
11. Richstone, D. O. et al. 1998, Nature, 395, A14
12. Salpeter, E. 1955, ApJ, 121, 161
13. Statler, T. S., King, I. R., Crane, P., & Jedrzejewski, R. I. 1999, AJ, 117, 894
14. Tremaine, S. 1995, AJ, 110, 628
15. van der Marel, R. P., de Zeeuw, P. T., Rix, H.­W., & Quinlan, G. D. 1997, Nature, 385, 610