Statistics of gravitational microlensing magnification. I. Two-dimensional lens distribution

The propagation of light from distant sources through a distribution of clumpy matter, acting as point-mass lenses, produces multiple images that contribute to the total brightness of the observed macroimages. In this paper, we refine the theory of gravitational microlensing for a planar distribution of point masses. In an accompanying paper, we extend the analysis to a three-dimensional lens distribution. In the two-dimensional case, we derive the probability distribution of macroimage magnification, P(A), at A -1 ≫ τ 2 for a low optical depth lens distribution by modeling the illumination pattern as a superposition of the patterns due to individual "point-mass plus weak-shear" lenses. A point-mass lens perturbed by weak shear S produces an astroid-shaped caustic. We show that the magnification cross section σ(A / S) of the point-mass plus weak-shear lens obeys a simple scaling property, and we provide a useful analytic approximation for the cross section. By convolving this cross section with the probability distribution of the shear due to the neighboring point masses, we obtain a caustic-induced feature in P(A) that also exhibits a simple scaling property. This feature results in a 20% enhancement in P(A) at A ≈ 2/τ. In the low-magnification (A - 1 ≪ 1) limit, the macroimage consists of a single bright primary image and a large number of faint secondary images formed close to each of the point masses. The magnifications of the primary and the secondary images can be strongly correlated. Taking into account the correlations, we derive P(A) for low magnification and find that P(A) has a peak of amplitude ≈0.16/τ 2 at A - 1 ≈ 0.84τ 2. The low-magnification distribution matches smoothly the distribution for A - 1 ≫ τ 2 in the overlapping regimes A - 1 ≫ τ 2 and A ≪ 1/τ. Finally, after a discussion of the correct normalization for P(A), we combine the results and obtain a practical semianalytic expression for the macroimage magnification distribution P(A). This semianalytic distribution is in qualitative agreement with the results of previous numerical simulations, but the latter show stronger caustic-induced features at moderate A (1.5 ≲ A ≲ 10) for τ as small as 0.1. We resolve this discrepancy by reexamining the criterion for low optical depth. A simple argument shows that the fraction of caustics of individual lenses that merge with those of their neighbors is approximately 1 - exp (-8τ). For τ = 0.1, the fraction is surprisingly high: ≈ 55%. A simple criterion for the low optical depth analysis to be valid is τ ≪1/8, though the comparison with numerical simulations indicates that the semianalytic distribution is a reasonable fit to P(A) for τ up to 0.05. © 1997. The American Astronomical Society. All rights reserved.link_to_subscribed_fulltex