STATISTICS OF GRAVITATIONAL MICROLENSING MAGNIFICATION .3. 3-DIMENSIONAL LENS DISTRIBUTION

In the first paper of this series, we studied the theory of gravitatio nal microlensing for a planar distribution of point masses. In this se cond paper, we extend the analysis to a three-dimensional lens distrib ution. First we study the lensing properties of three-dimensional lens distributions by considering in detail the critical curves, the caust ics, the illumination patterns, and the magnification cross sections s igma(A) of multiplane configurations with two, three, and four point m asses. For N- point masses that are widely separated in Lagrangian sp ace (i.e., in projection), we find that there are similar to 2(N) - 1 critical curves in total, but that only similar to N- of these produ ce prominent caustic-induced features in sigma(A) at moderate to high magnifications (A greater than or similar to 2). In the case of a rand om distribution of point masses at low optical depth, we show that the multiplane lens equation near a point mass can be reduced to the sing le-plane equation of a point mass perturbed by weak shear. This allows us to calculate the caustic-induced feature in the macroimage magnifi cation distribution P(A) as a weighted sum of the semianalytic feature derived in Paper I for a planar lens distribution. The resulting semi analytic caustic-induced feature is similar to the feature in the plan ar case, but it does not have any simple scaling properties, and it is shifted to higher magnification. The semianalytic distribution is com pared with the results of previous numerical simulations for optical d epth tau approximate to 0.1, and they are in better agreement than a s imilar comparison in the planar case. We explain this by estimating th e fraction of caustics of individual lenses that merge with those of t heir neighbors. For tau = 0.1, the fraction is approximate to 20%, muc h less than the approximate to 55% for the planar case. In the three-d imensional case, a simple criterion for the low optical depth analysis to be valid is tau much less than 0.4, though the comparison with num erical simulations indicates that the semianalytic distribution is a r easonable fit to P(A) for tau up to 0.2.