Limb darkening in spherical stellar atmospheres

¹ Argelander Institute for Astronomy, University of Bonn, Auf dem Huegel 71, 53121 Bonn, Germany
e-mail: hneilson@astro.uni-bonn.de
² Department of Chemical and Physical Sciences, University of Toronto Mississauga, 3359 Mississauga Road N., Mississauaga, Ontario, L5L 1C6, Canada
e-mail: lester@astro.utoronto.ca
³ Department of Astronomy & Astrophysics, University of Toronto, 50 St. George Street, Toronto, Ontario, M5S 3H4, Canada

Received: 1 February 2011
Accepted: 11 April 2011

Abstract

Context. Stellar limb darkening, I(μ = cosθ), is an important constraint for microlensing, eclipsing binary, planetary transit, and interferometric observations, but is generally treated as a parameterized curve, such as a linear-plus-square-root law. Many analyses assume limb-darkening coefficients computed from model stellar atmospheres. However, previous studies, using I(μ) from plane-parallel models, have found that fits to the flux-normalized curves pass through a fixed point, a common μ location on the stellar disk, for all values of T_eff, log g and wavelength.

Aims. We study this fixed μ-point to determine if it is a property of the model stellar atmospheres or a property of the limb-darkening laws. Furthermore, we use this limb-darkening law as a tool to probe properties of stellar atmospheres for comparison to limb-darkening observations.

Methods. Intensities computed with plane-parallel and spherically-symmetric Atlas models (characterized by the three fundamental parameters L_⋆, M_⋆ and R_⋆) are used to reexamine the existence of the fixed μ-point for the parametrized curves.

Results. We find that the intensities from our spherical models do not have a fixed point, although the curves do have a minimum spread at a μ-value similar to the parametrized curves. We also find that the parametrized curves have two fixed points, μ₁ and μ₂, although μ₂ is so close to the edge of the disk that it is missed using plane-parallel atmospheres. We also find that the spherically-symmetric models appear to agree better with published microlensing observations relative to plane-parallel models.

Conclusions. The intensity fixed point results from the choice of the parametrization used to represent the limb darkening and from the correlation of the coefficients of the parametrization, which is a consequence of their dependence on the angular moments of the intensity. For spherical atmospheres, the coefficients depend on the three fundamental parameters of the atmospheres, meaning that limb-darkening laws contain information about stellar atmospheres. This suggests that limb-darkening parameterizations fit with spherically-symmetric model atmospheres are powerful tools for comparing to observations of red giant stars.