Power-2 limb-darkening coefficients for the uvby, UBVRIJHK, SDSS ugriz, Gaia, Kepler, TESS, and CHEOPS photometric systems

II. PHOENIX spherically symmetric stellar atmosphere models^★

¹ Instituto de Astrofísica de Andalucía, CSIC, Apartado 3004, 18080 Granada, Spain
e-mail: claret@iaa.es
² Dept. Física Teórica y del Cosmos, Universidad de Granada, Campus de Fuentenueva s/n, 10871 Granada, Spain
³ Astrophysics Group, Keele University, Staffordshire ST5 5BG, UK

Received: 22 March 2023
Accepted: 6 April 2023

Abstract

Context. The phenomenon of limb-darkening is relevant to many topics in astrophysics, including the analysis of light curves of eclipsing binaries, optical interferometry, measurement of stellar diameters, line profiles of rotating stars, gravitational microlensing, and transits of extrasolar planets

Aims. Multiple parametric limb-darkening laws have been presented, and there are many available sources of theoretical limb-darkening coefficients (LDCs) calculated using stellar model atmospheres. The power-2 limb-darkening law allows a very good representation of theoretically predicted intensity profiles, but few LDCs are available for this law from spherically symmetric model atmospheres. We therefore present such coefficients in this work.

Methods. We computed LDCs for the space missions Gαiα, Kepler, TESS, and CHEOPS and for the passbands uυby, UBVRIJHK, and SDSS ugriz, using the PHOENIX-COND spherical models. We adopted two methods to characterise the truncation point, which sets the limb of the star: the first (M1) uses the point where the derivative dI(r)/dr is at its maximum – where I(r) is the specific intensity as a function of the normalised radius r – corresponding to µ_cri, and the second (M2) uses the midpoint between the point µ_cri and the point located at µ_cri–1. The LDCs were computed adopting the Levenberg–Marquardt least-squares minimisation method, with a resolution of 900 equally spaced µ points, and covering 823 model atmospheres for a solar metallicity, effective temperatures of 2300–12000 K, log g values from 0.0 to 6.0, and microturbulent velocities of 2 km s⁻¹. As our previous calculations of LDCs using spherical models included only 100 µ points, we also updated the calculations for the four-parameter law for the passbands listed above, and compared them with those from the power-2 law.

Results. Comparisons between the quality of the fits provided by the power-2 and four-parameter laws show that the latter presents a lower merit function, χ², than the former for both cases (M1 and M2). This is important when choosing the best approach for a particular science goal.