Microlensing by Stars in the Disk of M31

The optical depth to microlensing toward M31 due to known stars in the disk of M31 itself is $\tau\sim 2\times 10^{-7}e^{-r/d}$ where $d$ is the disk scale length and $r$ is the distance along the major axis. Thus, there can be significant lensing toward the M31 disk even if M31 contains no dark compact objects. The optical depth has a strong dependence on azimuthal angle: at fixed radius $\tau\propto[1+ (h/d)\tan i\cos\phi]^{-2}$ where $h$ is the scale height of the disk, $i=75^\circ$ is the inclination of M31, and $\phi$ is the azimuthal angle relative to the near minor axis. By measuring the optical depth as a function of radial and azimuthal position, it is possible to estimate $h$ and $d$ for the mass of the M31 disk, and so determine whether the disk light traces disk mass. Ground-based observations in $0.\hskip-2pt''5$ seeing of 0.8 $\rm deg^2$ once per week could yield $\sim 3$ events per year. With an ambitious space-based project, it would be possible to to observe $\sim 80$ events per year. If lensing events were dominated by the M31 spheroid rather than the disk, the event rate would be higher and the spheroid's parameters could be measured. The pattern of optical depths differs substantially for a disk and a spheroid.