Rigorous "Rich Argument" in Microlensing Parallax

Abstract

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.