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We find that significant incompleteness in stellar number counts results in asignificant overestimate of the microlensing optical depth $\tau$ and eventrate per star per year $\Gamma$ toward the Galactic bulge from the first twoyears of the MOA-II survey. We find that the completeness in Red Clump Giant(RCG) counts $f_{\rm RC}$ decreases proportional to the galactic latitude $b$,as $f_{\rm RC}=(0.63\pm0.11)-(0.052\pm0.028)\times b$, ranging between 1 and0.7 at $b=-6^\circ\sim-1.5^\circ$. The previous measurements using all sourcesby Difference Image Analysis (DIA) by MACHO and MOA-I suffer the same bias. Onthe other hand, the measurements using a RCG sample by OGLE-II, MACHO and EROSwere free from this bias because they selected only the events associated withthe resolved stars. Thus, the incompleteness both in the number of events andstellar number count cancel out. We estimate $\tau$ and $\Gamma$ by correctingthis incompleteness. In the central fields with $|l|<5^\circ$, we find$\Gamma=[18.74\pm0.91]\times10^{-6}\exp[(0.53\pm0.05)(3-|b|)]$ star$^{-1}$yr$^{-1}$ and $\tau_{200}=[1.84\pm0.14]\times10^{-6}\exp[(0.44\pm0.07)(3-|b|)]$for the 427 events with $t_{\rm E}\leq200\,$days using all sources brighterthan $I_s\leq20$ mag. Our revised all-source $\tau$ measurements are about2-$\sigma$ smaller than the other all-source measurements and are consistentwith the RCG measurements within 1-$\sigma$. We conclude that the long-standingproblem on discrepancy between the high $\tau$ with all-source samples by DIAand low $\tau$ with RCG samples can probably be explained by the incompletenessof the stellar number count. A model fit to these measurements predicts$\Gamma=4.60\pm0.25\times10^{-5}$ star$^{-1}$ yr$^{-1}$ at $|b|\sim-1^\circ.4$and $-2^\circ.25<l<3^\circ.75$ for sources with $I<20$, where the future spacemission WFIRST will observe.

astrophysical journal/the astrophysical journal

Possible Solution of the long-standing discrepancy in the Microlensing  Optical Depth Toward the Galactic Bulge by correcting the stellar number  count