Context. Systematic theoretical calculations of Doppler beaming factors are scarce in the literature, particularly in the case of white dwarfs. Additionally, there are no specific calculations for the limb-darkening coefficients of 3D white dwarf models.
Aims. The objective of this research is to provide the astronomical community with Doppler beaming calculations for a wide range of effective temperatures, local gravities, and hydrogen/metal content for white dwarfs as well as stars on both the main sequence and the giant branch. In addition, we present the theoretical calculations of the limb-darkening coefficients for 3D white dwarf models for the first time.
Methods. We computed Doppler beaming factors for DA, DB, and DBA white dwarf models, as well as for main sequence and giant stars covering the transmission curves of the Sloan, UBVRI, HiPERCAM, Kepler, TESS, and Gaia photometric systems. The calculations of the limb-darkening coefficients for 3D models were carried out using the least-squares method for these photometric systems.
Results. The input physics of the white dwarf models for which we have computed the Doppler beaming factors are: chemical compositions log [H/He] = −10.0 (DB), −2.0 (DBA), and He/H = 0 (DA), with log g varying between 5.0 and 9.5 and effective temperatures in the range 3750–100 000 K. The beaming factors were also calculated assuming non-local thermodynamic equilibrium for the case of DA white dwarfs with Teff > 40 000 K. For the mixing-length parameters, we adopted ML2/α = 0.8 (DA case) and 1.25 (DB and DBA). The Doppler beaming factors for main sequence and giant stars were computed using the ATLAS9 version, characterized by metallicities ranging from [−2.5, 0.2] solar abundances, with log g varying between 0 and 5.0 and effective temperatures between 3500 and 50 000 K. The adopted microturbulent velocity for these models was 2.0 km s−1. The limb-darkening coefficients were computed for three-dimensional DA and DB white dwarf models calculated with the CO5BOLD radiation-hydrodynamics code. The parameter range covered by the three-dimensional DA models spans log g values between 7.0 and 9.0 and Teff between 6000 and 15 000 K, while He/H = 0. The three-dimensional DB models cover a similar parameter range of log g between 7.5 and 9.0 and Teff between 12 000 and 34 000 K, while logH/He = −10.0. We adopted six laws for the computation of the limb-darkening coefficients: linear, quadratic, square root, logarithmic, power-2, and a general one with four coefficients.
Conclusions. The beaming factor calculations, which use realistic models of stellar atmospheres, show that the black body approximation is not accurate, particularly for the filters u, u′, U, g, g′, and B. The black body approach is only valid for high effective temperatures and/or long effective wavelengths. Therefore, for more accurate analyses of light curves, we recommend the use of the beaming factors presented in this paper. Concerning limb-darkening, the distribution of specific intensities for 3D models indicates that, in general, these models are less bright toward the limb than their 1D counterparts, which implies steeper profiles. To describe these intensities better, we recommend the use of the four-term law (also for 1D models) given the level of precision that is being achieved with Earth-based instruments and space missions such as Kepler and TESS (and PLATO in the future).
First discovery of an ultra-cool white dwarf benchmark in common proper motion with an M dwarf