Gravitational Microlensing near Caustics. II. Cusps

We present a rigorous, detailed study of the generic, quantitative propertiesof gravitational microlensing near cusp catastrophes. We derive explicitformulas for the total magnification and centroid of the images created forsources outside, on, and inside the cusped caustic. We obtain results on howthe image magnifications scale with respect to separation from the cuspedcaustic for arbitrary source positions. Along the axis of symmetry of the cusp,the total magnification $\mu$ scales as $\mu \propto u^{-1}$, where $u$ is thedistance of the source from the cusp, whereas perpendicular to this axis, $\mu\propto u^{-2/3}$. When the source passes through a point $\bu_0$ on a fold arcabutting the cusp, the image centroid has a jump discontinuity; we present aformula for the size of the jump in terms of the local derivatives of the lenspotential. The total magnifications for a small extended source located bothon, and perpendicular to, the axis of symmetry are also derived, for bothuniform and limb darkened surface brightness profiles. We find that thedifference in magnification between a finite and point source is $< 5%$ forseparations of $> 2.5$ source radii from the cusp point, while the effect oflimb-darkening is $< 1%$ in the same range. Our predictions for the astrometricand photometric behavior of both pointlike and finite sources passing near acusp are illustrated and verified using numerical simulations of thecusp-crossing Galactic binary-lens event MACHO-1997-BUL-28. We discuss severalpossible applications of our results to Galactic binary lenses and quasarmicrolensing.Comment: 52 pages, 12 figures. Revised version, minor changes. Accepted to ApJ, to appear in the November 20, 2002 issue (v580). Companion paper to astro-ph/011253