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 microimages [n] for a given macroimage of a gravitationally lensed point source due to microlensing by the stars of the lensing galaxy. This quantity was calculated by Wambsganss, Witt, and Schneider in 1992 for the case of zero external shear, gamma = 0, at the location of the macroimage. Since in realistic lens models a nonzero 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 macroimage is [mu] = \(1 - kappa(*))(2) -gamma(2)\(-1) >>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 semianalytic results for [n] with the results of numerical simulations and find good agreement. We find that the probability distribution for the number of extra microimage pairs is reasonably described by a Poisson distribution with a mean value of [n] and that the width of the macroimage magnification distribution tends to be largest for [n] similar to 1.