Comparison of the power-2 limb-darkening law from the STAGGER-grid to Kepler light curves of transiting exoplanets

Abstract

Context. Inaccurate limb-darkening models can be a significant source of error in the analysis of the light curves for transiting exoplanet and eclipsing binary star systems, particularly for high-precision light curves at optical wavelengths. The power-2 limbdarkening law, I λ (µ) = 1 − c (1 − µ α ), has recently been proposed as a good compromise between complexity and precision in the treatment of limb-darkening. Aims. My aim is to develop a practical implementation of the power-2 limb-darkening law and to quantify the accuracy of this implementation. Methods. I have used synthetic spectra based on the 3D stellar atmosphere models from the Stagger-grid to compute the limbdarkening for several passbands (UBVRI, CHEOPS, TESS, Kepler, etc.). The parameters of the power-2 limb-darkening laws are optimized using a least-squares fit to a simulated light curve computed directly from the tabulated I λ (µ) values. I use the transformed parameters h 1 = 1 − c (1 − 2 −α ) and h 2 = c2 −α to directly compare these optimized limb-darkening parameters to the limb darkening measured from Kepler light curves of 16 transiting exoplanet systems. Results. The posterior probability distributions (PPDs) of the transformed parameters h 1 and h 2 resulting from the light curve analysis are found to be much less strongly correlated than the PPDs for c and α. The agreement between the computed and observed values of (h 1 , h 2 ) is generally very good but there are significant differences between the observed and computed values for Kepler-17, the only star in the sample that shows significant variability between the eclipses due to magnetic activity (star spots).Conclusions. The tabulation of h 1 and h 2 provided here can be used to accurately model the light curves of transiting exoplanets. I also provide estimates of the priors that should be applied to transformed parameters h 1 and h 2 based on my analysis of the Kepler light curves of 16 stars with transiting exoplanets.

The ExoTETHyS Package: Tools for Exoplanetary Transits around Host Stars

“…At wavelengths shorter than 1 µm the two-coefficient laws are unreliable for exoplanet spectroscopy, so that the claret-4 law must be preferred. These conclusions are in agreement with previous studies based on both simulated and real data (Espinoza & Jordán 2016;Morello et al 2017;Morello 2018;Maxted 2018). Figure 7 shows that, for a fixed transit geometry, the peak-to-peak of light-curve residuals is roughly proportional to the weighted-r QS RMS of stellar intensity residuals.…”

NGTS and WASP photometric recovery of a single-transit candidate from TESS

“…We fitted the decorrelated limb-darkening parameters h 1 & h 2 from Eqn. 1 & 2 of Maxted (2018). Following the suggestion by Maxted (2018), Gaussian priors were centred on interpolated values of h 1 and h 2 (from Tab.…”

Note on the Power-2 Limb-darkening Law

“…(3) Maxted (2018) further stated that "These definitions impose the constraints h 1 < 1 and h 1 + h 2 ≤ 1 that are required so that the flux is positive at all points on the stellar disc." Unfortunately, the inequality highlighted in bold does not, in fact, satisfy that condition.…”

“…Some areas correspond to intensity distributions that may not be realized in actual stellar atmospheres. For example, using the 3D staggergrid (Magic et al 2015) of stellar atmosphere models for cool stars, Maxted (2018) finds no cases outside the ranges 0.37 < h 1 < 0.87 or 0.25 < h 2 < 0.78 for unspotted stars. This corresponds to a significantly smaller region than the unit square in the (q 1 , q 2 ) plane, and a greatly reduced area in the (c, α) plane.…”

“…The dots show 1000 uniform samples randomly chosen from the (q1, q2) unit square, and then cast into the Maxted and Hestroffer regions. The green colored dots correspond to the more likely pairs of coefficients, based on stellar atmosphere models as found in Maxted (2018).…”