STATISTICS OF GRAVITATIONAL MICROLENSING MAGNIFICATION .1. 2-DIMENSIONAL LENS DISTRIBUTION

The propagation of light from distant sources through a distribution o f 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 ex tend the analysis to a three-dimensional lens distribution. In the two -dimensional case, we derive the probability distribution of macroimag e magnification, P(A), at A -1 much greater than tau(2) for a low opti cal 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 sectio n sigma(A IS) of the point-mass plus weak-shear lens obeys a simple sc aling 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 obta in a caustic-induced feature in P(A) that also exhibits a simple scali ng property. This feature results in a 20% enhancement in P(A) at A ap proximate to 2/tau. In the low-magnification (A -1 much less than 1) l imit, the macroimage consists of a single bright primary image and a l arge number of faint secondary images formed close to each of the poin t masses. The magnifications of the primary and the secondary images c an be strongly correlated. Taking into account the correlations, we de rive P(A) for low magnification and find that P(A) has a peak of ampli tude approximate to 0.16/tau(2) at A -1 approximate to 0.84 tau(2). Th e low-magnification distribution matches smoothly the distribution for A -1 much greater than tau(2) in the overlapping regimes A -1 much gr eater than tau(2) and A much less than 1/tau. Finally, after a discuss ion of the correct normalization for P(A), we combine the results and obtain a practical semianalytic expression for the macroimage magnific ation distribution P(A). This semianalytic distribution is in qualitat ive agreement with the results of previous numerical simulations, but the latter show stronger caustic-induced features at moderate A (1.5 l ess than or similar to A less than or similar to 10) for tau as small as 0.1. We resolve this discrepancy by reexamining the criterion for l ow optical depth. A simple argument shows that the fraction of caustic s of individual lenses that merge with those of their neighbors is app roximately 1 - exp (-8 tau). For tau = 0.1, the fraction is surprising ly high: approximate to 55%. A simple criterion for the low optical de pth analysis to be valid is tau much less than 1/8, though the compari son with numerical simulations indicates that the semianalytic distrib ution is a reasonable fit to P(A) for tau up to 0.05.