The Mean Number of Extra Microimage Pairs for Macrolensed Quasars

When a gravitationally lensed source crosses a caustic, a pair of images iscreated or destroyed. We calculate the mean number of such pairs ofmicro-images $$ for a given macro-image of a gravitationally lensed pointsource, due to microlensing by the stars of the lensing galaxy. This quantitywas calculated by Wambsganss, Witt & Schneider (1992) for the case of zeroexternal shear, $\gamma=0$, at the location of the macro-image. Since inrealistic lens models a non-zero shear is expected to be induced by the lensinggalaxy, we extend this calculation to a general value of $\gamma$. We find acomplex behavior of $$ as a function of $\gamma$ and the normalized surfacemass density in stars $\kappa_*$. Specifically, we find that at highmagnifications, where the average total magnification of the macro-image is$<\mu>=|(1-\kappa_*)^2-\gamma^2|^{-1}\gg 1$, $$ becomes correspondinglylarge, and is proportional to $<\mu>$. The ratio $/<\mu>$ is largest nearthe line $\gamma=1-\kappa_*$ where the magnification $<\mu>$ becomes infinite,and its maximal value is 0.306. We compare our semi-analytic results for $$to the results of numerical simulations and find good agreement. We find thatthe probability distribution for the number of extra image pairs is reasonablydescribed by a Poisson distribution with a mean value of $$, and that thewidth of the macro-image magnification distribution tends to be largest for$\sim 1$.Comment: As accepted for publication in ApJ. 11 pages, 4 figures, minor change