qpower2 – a fast and accurate algorithm for the computation of exoplanet transit light curves with the power-2 limb-darkening law. (arXiv:1812.01606v1 [astro-ph.EP])

<a href="http://arxiv.org/find/astro-ph/1/au:+Maxted_P/0/1/0/all/0/1">P. F. L. Maxted</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Gill_S/0/1/0/all/0/1">S. Gill</a> (Keele University, UK)

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.

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.

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