Gravity and limb-darkening coefficients for compact stars : DA, DB, and DBA eclipsing white dwarfs

Context. The distribution of the specific intensity over the stellar disk is an essential tool for modeling the light curves in eclipsing binaries, planetary transits, and stellar diameters through interferometric techniques, line profiles in rotating stars, gravitational microlensing, etc. However, the available theoretical calculations are mostly restricted to stars on the main sequence or the giant branch, and very few calculations are available for compact stars. Aims. The main objective of the present work is to extend these investigations by computing the gravity and limb-darkening coefficients for white dwarf atmosphere models with hydrogen, helium, or mixed compositions (types DA, DB, and DBA). Methods. We computed gravity and limb-darkening coefficients for DA, DB, and DBA white dwarfs atmosphere models, covering the transmission curves of the Sloan, UBVRI, Kepler, TESS, and Gaia photometric systems. Specific calculations for the HiPERCAM instrument were also carried out. For all calculations of the limb-darkening coefficients we used the least-squares method. Concerning the effects of tidal and rotational distortions, we also computed for the first time the gravity-darkening coefficients y(λ) for white dwarfs using the same models of stellar atmospheres as in the case of limb-darkening. A more general differential equation was introduced to derive these quantities, including the partial derivative (∂ln Io(λ)/∂ln g)Teff. Results. Six laws were adopted to describe the specific intensity distribution: linear, quadratic, square root, logarithmic, power-2, and a more general one with four coefficients. The computations are presented for the 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 between 3750 and 100 000 K. For effective temperatures higher than 40 000 K, the models were also computed adopting nonlocal thermal equilibirum (DA). The adopted mixing-length parameters are ML2/α = 0.8 (DA case) and 1.25 (DB and DBA). The results are presented in the form of 112 tables. Additional calculations, such as for other photometric systems and/or different values of log[H/He], log g, and Teff can be performed upon request

Doppler beaming factors for white dwarfs, main sequence stars, and giant stars. Limb-darkening coefficients for 3D (DA and DB) white dwarf models

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 (NLTE) 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 15000 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 log H/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)

Calibration of the mixing-length theory for structures of helium-dominated atmosphere white dwarfs

We perform a calibration of the mixing-length parameter at the bottom boundary of the convection zone for helium-dominated atmospheres of white dwarfs. This calibration is based on a grid of 3D DB (pure-helium) and DBA (helium-dominated with traces of hydrogen) model atmospheres computed with the co5bold radiation-hydrodynamics code, and a grid of 1D DB and DBA envelope structures. The 3D models span a parameter space of hydrogen-to-helium abundances in the range −10.0 ≤ log (H/He) ≤−2.0, surface gravities in the range 7.5 ≤ log g ≤ 9.0, and effective temperatures in the range 12 000 K ≲ Teff ≲ 34 000 K. The 1D envelopes cover a similar atmospheric parameter range, but are also calculated with different values of the mixing-length parameter, namely 0.4 ≤ ML2/α ≤ 1.4. The calibration is performed based on two definitions of the bottom boundary of the convection zone: the Schwarzschild and the zero convective flux boundaries. Thus, our calibration is relevant for applications involving the bulk properties of the convection zone including its total mass, which excludes the spectroscopic technique. Overall, the calibrated ML2/α is smaller than what is commonly used in evolutionary models and theoretical determinations of the blue edge of the instability strip for pulsating DB and DBA stars. With calibrated ML2/α we are able to deduce more accurate convection zone sizes needed for studies of planetary debris mixing and dredge-up of carbon from the core. We highlight this by calculating examples of metal-rich 3D DBAZ models and finding their convection zone masses. Mixing-length calibration represents the first step of in-depth investigations of convective overshoot in white dwarfs with helium-dominated atmospheres

Horizontal spreading of planetary debris accreted by white dwarfs

White dwarfs with metal-polluted atmospheres have been studied widely in the context of the accretion of rocky debris from evolved planetary systems. One open question is the geometry of accretion and how material arrives and mixes in the white dwarf surface layers. Using the three-dimensional (3D) radiation hydrodynamics code CO5BOLD, we present the first transport coefficients in degenerate star atmospheres that describe the advection–diffusion of a passive scalar across the surface plane. We couple newly derived horizontal diffusion coefficients with previously published vertical diffusion coefficients to provide theoretical constraints on surface spreading of metals in white dwarfs. Our grid of 3D simulations probes the vast majority of the parameter space of convective white dwarfs, with pure-hydrogen atmospheres in the effective temperature range of 6000–18 000 K and pure-helium atmospheres in the range of 12 000–34 000 K. Our results suggest that warm hydrogen-rich atmospheres (DA; ≳13000 K) and helium-rich atmospheres (DB and DBA; ≳30000 K) are unable to efficiently spread the accreted metals across their surface, regardless of the time dependence of accretion. This result may be at odds with the current non-detection of surface abundance variations in white dwarfs with debris discs. For cooler hydrogen- and helium-rich atmospheres, we predict a largely homogeneous distribution of metals across the surface within a vertical diffusion time-scale. This is typically less than 0.1 per cent of disc lifetime estimates, a quantity that is revisited in this paper using the overshoot results. These results have relevance for studies of the bulk composition of evolved planetary systems and models of accretion disc physics

Phase-Resolved Spectroscopy of Gaia14aae: Line Emission From Near the White Dwarf Surface

AM CVn binaries are a class of ultracompact, hydrogen-deficient binaries, each consisting of a white dwarf accreting helium-dominated material from a degenerate or semi-degenerate donor star. Of the 56 known systems, only Gaia14aae undergoes complete eclipses of its central white dwarf, allowing the parameters of its stellar components to be tightly constrained. Here, we present phase-resolved optical spectroscopy of Gaia14aae. We use the spectra to test the assumption that the narrow emission feature known as the `central spike' traces the motion of the central white dwarf. We measure a central spike velocity amplitude of $13.8 \pm 3.2$ km/s, which agrees at the 1 $\sigma$ level with the predicted value of $17.6 \pm 1.0$ km/s based on eclipse-derived system parameters. The orbital phase offset of the central spike from its expected position is $4 \pm 15$ $^\circ$, consistent with 0 $^\circ$. Doppler maps of the He I lines in Gaia14aae show two accretion disc bright spots, as seen in many AM CVn systems. The formation mechanism for the second spot remains unclear. We detect no hydrogen in the system, but we estimate a 3 $\sigma$ limit on H$\alpha$ emission with an equivalent width of -1.14 \AA. Our detection of nitrogen and oxygen with no corresponding detection of carbon, in conjunction with evidence from recent studies, mildly favours a formation channel in which Gaia14aae is descended from a cataclysmic variable with a significantly evolved donor.Comment: 16 pages, accepted by MNRA

Phase-resolved spectroscopy of Gaia14aae: line emission from near the white dwarf surface

AM CVn binaries are a class of ultracompact, hydrogen-deficient binaries, each consisting of a white dwarf accreting helium-dominated material from a degenerate or semi-degenerate donor star. Of the 56 known systems, only Gaia14aae undergoes complete eclipses of its central white dwarf, allowing the parameters of its stellar components to be tightly constrained. Here, we present phase-resolved optical spectroscopy of Gaia14aae. We use the spectra to test the assumption that the narrow emission feature known as the ‘central spike’ traces the motion of the central white dwarf. We measure a central spike velocity amplitude of 13.8 ± 3.2 km s^(−1), which agrees at the 1σ level with the predicted value of 17.6 ± 1.0 km s^(−1) based on eclipse-derived system parameters. The orbital phase offset of the central spike from its expected position is 4 ± 15°, consistent with 0°. Doppler maps of the HeI lines in Gaia14aae show two accretion disc bright-spots, as seen in many AM CVn systems. The formation mechanism for the second spot remains unclear. We detect no hydrogen in the system, but we estimate a 3σ limit on H α emission with an equivalent width of −1.14 Å. Our detection of nitrogen and oxygen with no corresponding detection of carbon, in conjunction with evidence from recent studies, mildly favours a formation channel in which Gaia14aae is descended from a cataclysmic variable with a significantly evolved donor