Self-Lensing By A Stellar Disk

I derive a general expression for the optical depth $\tau$ for gravitational lensing of stars in a disk by Massive Compact Objects (Machos) in the same disk. For the more restricted case where the disk is self-gravitating and the stars and Machos have the same distribution function, I find $\tau = 2\VEV{v^2}/c^2\sec^2 i$ where $\VEV{v^2}$ is the mass-weighted vertical velocity dispersion, and $i$ is the angle of inclination. This result does not depend on any assumptions about the velocity distribution. As an example, if stars within the bar of the Large Magellanic Cloud (LMC) account for the observed optical depth $\tau\sim 8\times 10^{-8}$ as has recently been suggested, then $v\gsim 60\,\kms$. This is substantially larger than the measured dispersions of known LMC populations.