Gravitational Microlensing near Caustics. I. Folds

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Abstract
We study the local behavior of gravitational lensing near fold catastrophes. Using a generic form for the lensing map near a fold, we determine the observable properties of the lensed images, focusing on the case in which the individual images are unresolved, i.e., microlensing. Allowing for images not associated with the fold, we derive analytic expressions for the photometric and astrometric behavior near a generic fold caustic. We show how this form reduces to the more familiar linear caustic, which lenses a nearby source into two images that have equal magnification, opposite parity, and are equidistant from the critical curve. In this case, the simplicity and high degree of symmetry allow for the derivation of semianalytic expressions for the photometric and astrometric deviations in the presence of finite sources with arbitrary surface brightness proles. We use our results to derive some basic properties of astrometric microlensing near folds; in particular, we predict, for finite sources with uniform and limb-darkening profiles, the detailed shape of the astrometric curve as the source crosses a fold. We find that the astrometric effects of limb darkening will be difficult to detect with the currently planned accuracy of the Space Interferometry Mission for Galactic bulge sources; however, this also implies that astrometric measurements of other parameters, such as the size of the source, should not be compromised by an unknown amount of limb darkening. We verify our results by numerically calculating the expected astrometric shift for the photometrically well-covered Galactic binary lensing event OGLE-1999-BUL-23, finding excellent agreement with our analytic expressions. Our results can be applied to any lensing system with fold caustics, including Galactic binary lenses and quasar microlensing.