Finite-source and finite-lens effects in astrometric microlensing
  The aim of this paper is to study the astrometric trajectory of microlensing
events with an extended lens and/or source. We consider not only a dark lens
but also a luminous lens as well. We find that the discontinuous finite-lens
trajectories given by Takahashi (2003) will become continuous in the
finite-source regime. The point lens (source) approximation alone gives an
under (over)estimation of the astrometric signal when the size of the lens and
source are not negligible. While the finiteness of the source is revealed when
the lens transits the surface of the source, the finite-lens signal is most
prominent when the lens is very close to the source. Astrometric microlensing
towards the Galactic bulge, Small Magellanic Cloud and M31 are discussed, which
indicate that the finite-lens effect is beyond the detection limit of current
instruments. Nevertheless, it is possible to distinguish between self-lensing
and halo lensing through a (non-)detection of the astrometric ellipse. We also
consider the case where the lens is luminous itself, as has been observed where
a lensing event was followed up with the Hubble Space Telescope. We show that
the astrometric signal will be reduced in a luminous-lens scenario. The
physical properties of the event, such as the lens-source flux ratio, the size
of the lens and source nevertheless can be derived by fitting the astrometric
trajectory.
