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Register a new userRotationally and tidally distorted compact stars A theoretical approach to the gravity-darkening exponents for white dwarfs
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A. Claret
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To the best of our knowledge, there are no specific calculations of gravity-darkening exponents for white dwarfs in the literature. On the other hand, the number of known eclipsing binaries whose components are tidally and/or rotationally distorted white dwarfs is increasing year on year. Our main objective is to present the first theoretical approaches to the problem of the distribution of temperatures on the surfaces of compact stars distorted by rotation and/or tides in order to compare with relevant observational data. We find discrepancies between the gravity-darkening exponents calculated with our methods and the predictions of the von Zeipel theorem, particularly in the cases of cold white dwarfs; although the discrepancy also applies to higher effective temperatures under determined physical conditions. We find physical connections between the gravity-darkening exponents calculated using our modified method of triangles strategy with the convective efficiency (defined here as the ratio of the convective to the total flux). A connection between the entropy and the gravity-darkening coefficients is also found: variations of the former cause changes in the way the temperature is distributed on distorted stellar surfaces. On the other hand, we have generalised the von Zeipel theorem for the case of hot white dwarfs. Such a generalisation allows us to predict that, under certain circumstances, the value of the gravity-darkening exponent may be smaller than 1.0, even in the case of high effective temperatures.
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We identify two new tidally distorted white dwarfs (WDs), SDSS J174140.49+652638.7 and J211921.96-001825.8 (hereafter J1741 and J2119). Both stars are extremely low mass (ELM, < 0.2 Msun) WDs in short-period, detached binary systems. High-speed photometric observations obtained at the McDonald Observatory reveal ellipsoidal variations and Doppler beaming in both systems; J1741, with a minimum companion mass of 1.1 Msun, has one of the strongest Doppler beaming signals ever observed in a binary system (0.59 pm 0.06% amplitude). We use the observed ellipsoidal variations to constrain the radius of each WD. For J1741, the stars radius must exceed 0.074 Rsun. For J2119, the radius exceeds 0.10 Rsun. These indirect radius measurements are comparable to the radius measurements for the bloated WD companions to A-stars found by the Kepler spacecraft, and they constitute some of the largest radii inferred for any WD. Surprisingly, J1741 also appears to show a 0.23 pm 0.06% reflection effect, and we discuss possible sources for this excess heating. Both J1741 and J2119 are strong gravitational wave sources, and the time-of-minimum of the ellipsoidal variations can be used to detect the orbital period decay. This may be possible on a timescale of a decade or less.
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). 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 $y(lambda)$ 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 $left(partial{ln I_o(lambda)}/{partial{ln g}}right)_{T_{rm eff}}$. 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), $-$2.0 (DBA) and He/H = 0 (DA), with log g varying between 5.0 and 9.5 and effective temperatures between 3750 K-100,000 K. For effective temperatures higher than 40,000 K, the models were also computed adopting nonlocal thermal equilibirum (DA). The adopted mixing-length parameters are ML2/$alpha = 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 T$_{rm eff}$ can be performed upon request.
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. 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).
We present the discovery of an unusual, tidally-distorted extremely low mass white dwarf (WD) with nearly solar metallicity. Radial velocity measurements confirm that this is a compact binary with an orbital period of 2.6975 hrs and a velocity semi-amplitude of K = 108.7 km/s. Analysis of the hydrogen Balmer lines yields an effective temperature of Teff = 8380 K and a surface gravity of log g = 6.21 that in turn indicate a mass of M = 0.16 Msol and a cooling age of 4.2 Gyr. In addition, a detailed analysis of the observed metal lines yields abundances of log Mg/H = -3.90, log Ca/H = -5.80, log Ti/H = -6.10, log Cr/H = -5.60, and log Fe/H = -4.50, similar to the sun. We see no evidence of a debris disk from which these metals would be accreted though the possibility cannot entirely be ruled out. Other potential mechanisms to explain the presence of heavy elements are discussed. Finally, we expect this system to ultimately undergo unstable mass transfer and merge to form a ~0.3-0.6 Msol WD in a few Gyr.
Observational evidence of white dwarf planetary systems is dominated by the remains of exo-asteroids through accreted metals, debris discs, and orbiting planetesimals. However, exo-planets in these systems play crucial roles as perturbing agents, and can themselves be perturbed close to the white dwarf Roche radius. Here, we illustrate a procedure for computing the tidal interaction between a white dwarf and a near-spherical solid planet. This method determines the planets inward and/or outward drift, and whether the planet will reach the Roche radius and be destroyed. We avoid constant tidal lag formulations and instead employ the self-consistent secular Darwin-Kaula expansions from Bou{e} & Efroimsky (2019), which feature an arbitrary frequency dependence on the quality functions. We adopt wide ranges of dynamic viscosities and spin rates for the planet in order to straddle many possible outcomes, and provide a foundation for the future study of individual systems with known or assumed rheologies. We find that: (i) massive Super-Earths are destroyed more readily than minor planets (such as the ones orbiting WD 1145+017 and SDSS J1228+1040), (ii) low-viscosity planets are destroyed more easily than high-viscosity planets, and (iii) the boundary between survival and destruction is likely to be fractal and chaotic.
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