  An important requirement for the Space Interferometry Mission (SIM) is to
carry out precision astrometry in crowded fields. This capability is crucial,
for example, to accurately measure proper motions of bright stars in nearby
galaxies. From such measurements, one can obtain distance estimates, explore
the dynamics of these systems and measure the mass of the Milky Way itself. In
this paper we investigate errors introduced by confusion, i.e. the presence of
objects other than the targeted star in the SIM field of view (FOV). Using
existing HST images of fields in M31, the LMC and the Galaxy we simulate the
background within the SIM FOV and estimate the errors in the measured position
of the target star. Our simulations account for the error contribution from
photon statistics. We also study the effects of pointing imperfections when a
field is revisited which result in errors in the measured proper motion. We use
the simulations to explore the measurement accuracy of several SIM key programs
which will require crowded field astrometry. In M31, the error in the absolute
position of the targets could be significant for all but the brightest targets.
Our results also indicate that in the case of the brightest targets in M31, and
for all likely target magnitudes in the other cases, confusion-induced proper
motion errors are well within the SIM requirements. However, targets which vary
in flux between measurements can be susceptible to enhanced proper motion
errors. We also find that for an on-source integration time of one hour, photon
noise is larger than or of comparable magnitude to the confusion-induced
position error for bright targets and dominates over proper motion error in
most cases.
