We explore the sensitivity of two-point-mass gravitational microlensing to the extended nature of the source star, as well as the related sensitivity to its limb darkening. We demonstrate that the sensitive region, usually considered to be limited to a source-diameter-wide band along the caustic, is strongly expanded near cusps, most prominently along their outer axis. In the case of multi-component caustics, facing cusps may form a region with a non-negligible extended-source effect spanning the gap between them. We demonstrate that for smaller sources the size of the sensitive region extending from a cusp measured in units of source radii increases, scaling as the inverse cube root of the radius. We study the extent of different sensitivity contours and show that for a microlensed Galactic bulge giant the probability of encountering at least a 1% extended-source effect is higher than the probability of caustic crossing by 40-60% when averaged over a typical range of lens-component separations, with the actual value depending on the mass ratio of the components. We derive analytical expressions for the extended-source effect and chromaticity for a source positioned off the caustic. These formulae are more generally applicable to any gravitational lens with a sufficiently small source. Using exactly computed amplifications we test the often used linear-fold caustic approximation and show that it may lead to errors on the level of a few percent even in near-ideal caustic-crossing events. Finally, we discuss several interesting cases of observed binary and planetary microlensing events and point out the importance of our results for the measurement of stellar limb darkening from microlensing light curves.

Bennett et al.

Muraki et al.

Among the 25 planetary systems detected up to now by gravitational microlensing, there are two cases of a star with two planets, and two cases of a binary star with a planet. Other, yet undetected types of triple lenses include triple stars or stars with a planet with a moon. The analysis and interpretation of such events is hindered by the lack of understanding of essential characteristics of triple lenses, such as their critical curves and caustics. We present here analytical and numerical methods for mapping the critical-curve topology and caustic cusp number in the parameter space of n-point-mass lenses. We apply the methods to the analysis of four symmetric triple-lens models, and obtain altogether 9 different critical-curve topologies and 32 caustic structures. While these results include various generic types, they represent just a subset of all possible triplelens critical curves and caustics. Using the analyzed models, we demonstrate interesting features of triple lenses that do not occur in two-point-mass lenses. We show an example of a lens that cannot be described by the Chang-Refsdal model in the wide limit. In the close limit we demonstrate unusual structures of primary and secondary caustic loops, and explain the conditions for their occurrence. In the planetary limit we find that the presence of a planet may lead to a whole sequence of additional caustic metamorphoses. We show that a pair of planets may change the structure of the primary caustic even when placed far from their resonant position at the Einstein radius.

Observations of microlensing transit events can be used to measure the limb darkening of the lensed star. We discuss the advantages and drawbacks of several microlensing light curve inversion methods. The method of choice in this work is inversion by means of decomposition of the stellar surface brightness profile. We construct an ideal basis by principal component analysis of brightness profiles obtained from model atmosphere calculations. Limb darkening approximations using such a basis are superior to those using standard power-law limb darkening laws. We perform a full analysis of simulated single-lens microlensing transit events including a detailed error analysis of the method. In realistic events with a low impact parameter the brightness profile of the source can be recovered with a relative accuracy of 2% from the center of the source disk to 0.9 of the disk radius. We show that in the particular case of the observed MACHO Alert 95-30 event the intrinsic complex variability of the lensed red giant hinders efforts to recover its surface features.Comment: 47 pages, 11 figures - accepted by The Astrophysical Journa

We explore the sensitivity of limb-darkening coefficients computed from stellar atmosphere models to different least-squares fitting methods. We demonstrate that conventional methods are strongly biased to fitting the stellar limb. Our suggested method of fitting by minimizing the radially integrated squared residual yields improved fits with better flux conservation. The differences of the obtained coefficients from commonly used values are observationally significant. We show that the new values are in better agreement with solar limb-darkening measurements, as well as with coefficients reported from analyses of eclipsing binary light curves.

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