Rigorous "Rich Argument" in Microlensing Parallax
  I show that when the observables $(\vec \pi_{{\rm E}},t_{{\rm
E}},\theta_{{\rm E}},\pi_s,\vec \mu_s)$ are well measured up to a discrete
degeneracy in the microlensing parallax vector $\vec \pi_{{\rm E}}$, the
relative likelihood of the different solutions can be written in closed form
$P_i = K H_i B_i$, where $H_i$ is the number of stars (potential lenses) having
the mass and kinematics of the inferred parameters of solution $i$ and $B_i$ is
an additional factor that is formally derived from the Jacobian of the
transformation from Galactic to microlensing parameters. The Jacobian term
$B_i$ constitutes an explicit evaluation of the ``Rich Argument'', i.e., that
there is an extra geometric factor disfavoring large-parallax solutions in
addition to the reduced frequency of lenses given by $H_i$. Here $t_{{\rm E}}$
is the Einstein timescale, $\theta_{{\rm E}}$ is the angular Einstein radius,
and $(\pi_s,\vec \mu_s)$ are the (parallax, proper motion) of the microlensed
source. I also discuss how this analytic expression degrades in the presence of
finite errors in the measured observables.
