Statistics of gravitational microlensing magnification. II. Three-dimensional lens distribution

Abstract

In the first paper of this series, we studied the theory of gravitational microlensing for a planar distribution of point masses. In this second paper, we extend the analysis to a three-dimensional lens distribution. First we study the lensing properties of three-dimensional lens distributions by considering in detail the critical curves, the caustics, the illumination patterns, and the magnification cross sections σ(A) of multiplane configurations with two, three, and four point masses. For N* point masses that are widely separated in Lagrangian space (i.e., in projection), we find that there are ∼2 N* -1 critical curves in total, but that only ∼N* of these produce prominent caustic-induced features in σ(A) at moderate to high magnifications (A ≳ 2). In the case of a random distribution of point masses at low optical depth, we show that the multiplane lens equation near a point mass can be reduced to the single-plane equation of a point mass perturbed by weak shear. This allows us to calculate the caustic-induced feature in the macroimage magnification distribution P(A) as a weighted sum of the semianalytic feature derived in Paper I for a planar lens distribution. The resulting semianalytic caustic-induced feature is similar to the feature in the planar case, but it does not have any simple scaling properties, and it is shifted to higher magnification. The semianalytic distribution is compared with the results of previous numerical simulations for optical depth τ ≈0.1, and they are in better agreement than a similar comparison in the planar case. We explain this by estimating the fraction of caustics of individual lenses that merge with those of their neighbors. For τ = 0.1, the fraction is ≈20%, much less than the ≈55% for the planar case. In the three-dimensional case, a simple criterion for the low optical depth analysis to be valid is τ ≪ 0.4, though the comparison with numerical simulations indicates that the semianalytic distribution is a reasonable fit to P(A) for τ up to 0.2. © 1997. The American Astronomical Society. All rights reserved.