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

Abstract

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 ${\kappa }_{\ast }$. Specifically, we find that at high magnifications, where the average total magnification of the macro-image is $<\mu >=|(1-{\kappa }_{\ast }{)}^2-{\gamma }^2{|}^{-1}\gg 1$, $<n>$ becomes correspondingly large, and is proportional to $<\mu >$. The ratio $<n>/<\mu >$ is largest near the line $\gamma =1-{\kappa }_{\ast }$ 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 $<n>\sim 1$.

As accepted for publication in ApJ. 11 pages, 4 figures, minor changes

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