The Mean Number of Extra Microimage Pairs for Macrolensed 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, \γ=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 \γ. We find a complex behavior of $<n>$ as a function of \γ and the normalized surface mass density in stars \κ_*. Specifically, we find that at high magnifications, where the average total magnification of the macro-image is <\μ>=|(1-\κ_*)²-\γ²|⁻¹\ 1, $<n>$ becomes correspondingly large, and is proportional to <\μ>. The ratio <n>/<\μ> is largest near the line \γ=1-\κ_* where the magnification <\μ> 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>\~ 1.