Analytical Methods in Gravitational Microlensing

Abstract

We discuss analytical results dealing with photometric and astrometric gravitational microlensing. The first two sections concern approximation methods that allow us to get solutions of the general lens equation near fold caustics and cusp points up to any prescribed accuracy. Two methods of finding approximate solutions near the fold are worked out. The results are applied to derive new corrections to total amplifications of critical source images. Analytic expressions are obtained in case of the Gaussian, power-law, and limb-darkening extended source models; here we present the first nonzero corrections to the well-known linear caustic approximation. Possibilities to distinguish different source models in observations are discussed on the basis of statistical simulations of microlensed light curves. In the next section, we discuss astrometric microlensing effects in various cases of extended sources and extended lenses, including a simple model of weak statistical microlensing by extended dark matter clumps. Random walks of a distant source image microlensed by stochastic masses are estimated. We note that the bulk motion of foreground stars induces a small apparent rotation of the extragalactic reference frame. Compact analytical relations describing the statistics of such motions are presented.