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E. Agol, A. Farmer, K. Mandel (Caltech)
We will prove a gravitational lensing theorem: the magnification of a large source of uniform brightness by a small foreground point-mass lens is M=1+(2RE2-RL2)/RS2, where RS is the radius of the source and RE and RL are the Einstein radius and size of the lens projected into the source plane; this provides an accurate approximation to the exact magnification for RL,RE << RS. Remarkably, this result is independent of the shape of the source or position of the lens (except near the edges). We show that this formula can be generalized to include limb-darkening of a circular source by simply inserting the surface-brightness at the position of the foreground object (divided by the average surface-brightness of the star). We use this theorem to compute transit lightcurves in binaries for which the foreground star or planet has a size and Einstein radius much smaller than the background star. White dwarf stars in binaries with semi-major axes of 0.1 AU have a size comparable to their Einstein radii. Thus, white dwarfs orbiting main-sequence stars can show either brightening or eclipse. Since white dwarfs are similar in size to terrestrial planets, we predict how many white dwarf-main sequence binaries might be found in transit searches for terrestrial type planets such as Kepler and Eddington. We estimate that dozens of systems may be found, although the precise number will depend on the properties of low-level variability of the main-sequence stars in these binaries. We also apply these results to planetary transits (RE~0), demonstrating that this simple approximation accurately describes the exact limb-darkened transit lightcurves. Using the HST lightcurve of HD 209458 from Brown et al. (2001), the ratio of the planet radius to stellar radius is measured to be Rp/R*=0.1207±0.0003 , marginalized over the limb-darkening.
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The author(s) of this abstract have provided an email address for comments about the abstract: agol@tapir.caltech.edu
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Bulletin of the American Astronomical Society, 34, #4
© 2002. The American Astronomical Soceity.