qpower2 - a fast and accurate algorithm for the computation of exoplanet transit light curves with the power-2 limb-darkening law

Abstract

The power-2 law, \(I_{\lambda}(\mu)=1- c (1 - \mu^{\alpha})\), accurately represents the limb-darkening profile for cool stars. It has been implemented in a few transit models to-date using numerical integration but there is as-yet no implementation of the power-2 law in analytic form that is generally available. An algorithm to implement the power-2 law is derived using a combination of an approximation to the required integral and a Taylor expansion of the power-2 law. The accuracy of stellar and planetary radii derived by fitting transit light curves with this approximation is tested using light curves computed by numerical integration of limb-darkening profiles from 3D stellar model atmospheres. Our algorithm (qpower2) is accurate to about 100 ppm for broad-band optical light curves of systems with a star-planet radius ratio p = 0.1. The implementation requires less than 40 lines of python code so can run extremely fast on graphical processing units (GPUs; \(\sim\) 1 million models per second for the analysis of 1000 data points). Least-squares fits to simulated light curves show that the star and planet radius are recovered to better than 1% for p < 0.2. The qpower2 algorithm can be used to efficiently and accurately analyse large numbers of high-precision transit light curves using Monte Carlo methods.