Collapsar Uncorking and Jet Eruption in Gamma-Ray Bursts

Microlensing surveys derive the microlensing optical depth toward various directions such as the Galactic center from the distribution of observed Einstein radius crossing times. I show that the formula that is being used is invalid for "exotic" lensing events. The corrected formula is derived. The "parallax" effect (Earth motion) requires no correction. Corrections for the finite sizes of sources and wide binary lenses are small (typically less than 1%), except for blended events. Corrections for intermediate-type binaries such as MACHO LMC-9 can be substantial.