On the Microlensing Optical Depth of the Galactic Bar

The microlensing probability (optical depth $\tau$) toward the Galactic center carries information about the mass distribution of the Galactic bulge/bar, so can be used to constrain the very uncertain shape parameters of the bar. We find $tau$ depends on the bar mass, radial profile, angle, axis scale lengths and boxyness by a few simple analytical formulae, which shows: (1) $\tau$ is proportional to the mass of the bar, $M$. (2) $\tau$ falls along the minor axis with a strong gradient. (3) An oblate bulge can have more optical depth than a triaxial bar if the bar angle $\alpha>45$ degress. (4) $\tau$ is the largest if the angle $\alpha$ and the axis ratio $y_0/x_0$ conspires so that $y_0/x_0=\tan \alpha$. (5) At a fixed field on the minor axis but away from the center, boxy bars with a flat density profile tend to give a larger optical depth than ellipsoidal bars with a steep profile. (6) Main sequence sources should have a significantly lower (20-50\% lower) optical depth than red clump giants if main sequence stars are not observed as deep as the bright clump giants. An application to four COBE-constrained models (Dwek et al. 1994) shows most models produce optical depth $2\sigma$ lower than MACHO and OGLE observed values even with both a massive bar $2.8\times 10^{10} M_\odot$ and a full disk. The high $\tau$ argues for a massive ($> 2\times 10^{10}M_\odot$) boxy bar with $y_0/x0\approx \tan\alpha$ and $\alpha<20$ deg and with a flat radial profile up to corotation.