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

Astrophysics Group, Keele University, Keele, Staffordshire ST5 5BG, UK
e-mail: p.maxted@keele.ac.uk

Received: 2 November 2018
Accepted: 30 November 2018

Abstract

Context. The power-2 law, I_λ( μ) = 1 − c(1−μ^α), 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.

Aims. Our aim is to derive an analytic approximation that can be used to quickly and accurately calculate light curves of transiting exoplanets using the power-2 limb-darkening law.

Methods. 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.

Results. 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; ∼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.

Conclusions. The qpower2 algorithm can be used to efficiently and accurately analyse large numbers of high-precision transit light curves using Monte Carlo methods.