Statistics of Gravitational Microlensing Magnification. I. Two‐dimensional Lens Distribution

(Abridged) In this paper we refine the theory of microlensing for a planardistribution of point masses. We derive the macroimage magnificationdistribution P(A) at high magnification (A-1 >> tau^2) for a low optical depth(tau << 1) lens distribution by modeling the illumination pattern as asuperposition of the patterns due to individual ``point mass plus weak shear''lenses. We show that a point mass plus weak shear lens produces an astroid-shaped caustic and that the magnification cross-section obeys a simple scalingproperty. By convolving this cross-section with the shear distribution, weobtain a caustic-induced feature in P(A) which also exhibits a simple scalingproperty. This feature results in a 20% enhancement in P(A) at A approx 2/tau.In the low magnification (A-1 << 1) limit, the macroimage consists of a brightprimary image and a large number of faint secondary images formed close to eachof the point masses. Taking into account the correlations between the primaryand secondary images, we derive P(A) for low A. The low-A distribution has apeak of amplitude ~ 1/tau^2 at A-1 ~ tau^2 and matches smoothly to the high-Adistribution. We combine the high- and low-A results and obtain a practicalsemi-analytic expression for P(A). This semi-analytic distribution is inqualitative agreement with previous numerical results, but the latter showstronger caustic-induced features at moderate A for tau as small as 0.1. Weresolve this discrepancy by re-examining the criterion for low optical depth. Asimple argument shows that the fraction of caustics of individual lenses thatmerge with those of their neighbors is approx 1-exp(-8 tau). For tau=0.1, thefraction is surprisingly high: approx 55%. For the purpose of computing P(A) inthe manner we did, low optical depth corresponds to tau << 1/8.Comment: 35 pages, including 6 figures; uses AASTeX v4.0 macros; submitted to Ap