Analytic solutions to the maximum and average exoplanet transit depth for common stellar limb darkening laws

Abstract

Context. The depth of an exoplanetary transit in the light curve of a distant star is commonly approximated as the squared planet-to-star radius ratio, (Rp/Rs)2. Stellar limb darkening, however, can result in significantly deeper transits. An analytic solution would be worthwhile to illustrate the principles of the problem and predict the actual transit signal required for the planning of transit observations with certain signal-to-noise requirements without the need of computer-based transit simulations. Aims. We calculate the overshoot of the mid-transit depth caused by stellar limb darkening compared to the (Rp/Rs)2 estimate for arbitrary transit impact parameters. In turn, this allows us to compute the true planet-to-star radius ratio from the transit depth for a given parameterization of a limb darkening law and for a known transit impact parameter. Methods. We compute the maximum emerging specific stellar intensity covered by the planet in transit and derive analytic solutions for the transit depth overshoot. Solutions are presented for the linear, quadratic, square-root, logarithmic, and nonlinear stellar limb darkening with arbitrary transit impact parameters. We also derive formulae to calculate the average intensity along the transit chord, which allows us to estimate the actual transit depth (and therefore Rp∕Rs) from the mean in-transit flux. Results. The transit depth overshoot of exoplanets compared to the (Rp/Rs)2 estimate increases from about 15% for main-sequence stars of spectral type A to roughly 20% for sun-like stars and some 30% for K and M stars. The error in our analytical solutions for Rp∕Rs from the small planet approximation is orders of magnitude smaller than the uncertainties arising from typical noise in real light curves and from the uncertain limb darkening. Conclusions. Our equations can be used to predict with high accuracy the expected transit depth of extrasolar planets. The actual planet radius can be calculated from the measured transit depth or from the mean in-transit flux if the stellar limb darkening can be properly parameterized and if the transit impact parameter is known. Light curve fitting is not required.