The Mean Number of Extra Micro-Image Pairs for Macro-Lensed Quasars

When a gravitationally lensed source crosses a caustic, a pair of images is created or destroyed. We calculate the mean number of such pairs of micro-images $<n>$ for a given macro-image of a gravitationally lensed point source, due to microlensing by the stars of the lensing galaxy. This quantity was calculated by Wambsganss, Witt & Schneider (1992) for the case of zero external shear, $\gamma=0$, at the location of the macro-image. Since in realistic lens models a non-zero shear is expected to be induced by the lensing galaxy, we extend this calculation to a general value of $\gamma$. We find a complex behavior of $<n>$ as a function of $\gamma$ and the normalized surface mass density in stars $\kappa_*$. Specifically, we find that at high magnifications, where the average total magnification of the macro-image is $<\mu>=|(1-\kappa_*)^2-\gamma^2|^{-1}\gg 1$, $<n>$ becomes correspondingly large, and is proportional to $<\mu>$. The ratio $<n>/<\mu>$ is largest near the line $\gamma=1-\kappa_*$ where the magnification $<\mu>$ becomes infinite, and its maximal value is 0.306. We compare our semi-analytic results for $<n>$ to the results of numerical simulations and find good agreement. We find that the probability distribution for the number of extra image pairs is reasonably described by a Poisson distribution with a mean value of $<n>$, and that the width of the macro-image magnification distribution tends to be largest for $<n>\sim 1$.