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

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 uνby, 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 μcri, and the second (M2) uses the midpoint between the point μcri and the point located at μcn-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, X2, than the former for both cases (M1 and M2). This is important when choosing the best approach for a particular science goal.