A MNRAS 000, 1–14 (2015) Preprint 4 April 2019 Compiled using MNRAS L TEX style file v3.0 The white dwarf mass–radius relation and its dependence on the hydrogen envelope. Alejandra D. Romero,1⋆, S. O. Kepler1, S. R. G. Joyce2, G. R. Lauffer1 & A. H. C´orsico3,4 1Physics Institute, Universidade Federal do Rio Grande do Sul, Av. Bento Gon¸calves 9500, Brazil 2Dept. of Physics & Astronomy, University of Leicester, University Road, Leicester, LE1 7RH 3Facultad de Ciencias Astron´omicas y Geof´ısicas, Universidad Nacional de La Plata, La Plata 1900, Argentina 4CONICET, Consejo Nacional de Investigaciones Cientif´ıcas y T´ecnicas, Argentina Accepted XXX. Received YYY; in original form ZZZ ABSTRACT We present a study of the dependence of the mass–radius relation for DA white dwarf stars on the hydrogen envelope mass and the impact on the value of log g, and thus the determination of the stellar mass. We employ a set of full evolutionary carbon- oxygen core white dwarf sequences with white dwarf mass between 0.493 and 1.05M⊙. Computations of the pre-white dwarf evolution uncovers an intrinsic dependence of the maximum mass of the hydrogen envelope with stellar mass, i.e., it decreases when the total mass increases. We find that a reduction of the hydrogen envelope mass can lead to a reduction in the radius of the model of up to ∼ 12%. This translates directly into an increase in log g for a fixed stellar mass, that can reach up to 0.11 dex, mainly overestimating the determinations of stellar mass from atmospheric parameters. Fi- nally, we find a good agreement between the results from the theoretical mass–radius relation and observations from white dwarfs in binary systems. In particular, we find a −8 thin hydrogen mass of MH ∼ 2 × 10 M⊙, for 40 Eridani B, in agreement with previous determinations. For Sirius B, the spectroscopic mass is 4.3% lower than the dynamical mass. However, the values of mass and radius from gravitational redshift observations are compatible with the theoretical mass–radius relation for a thick hydrogen envelope −6 of MH = 2 × 10 M⊙. Key words: white dwarf stars — stellar evolution — binary stars 1 INTRODUCTION mines how much stellar material is returned to the interstel- lar medium, affecting the chemical evolution of the Galaxy. One of the results of Chandrasekhar’s theory for the struc- arXiv:1901.04644v2 [astro-ph.SR] 2 Apr 2019 ture of white dwarf stars is a dependence of the ra- Semi-empirical determinations of the mass–radius rela- dius with the stellar mass, known as the mass–radius tions can be obtained from atmospheric parameters, effec- relation. This relation is widely used in stellar astro- tive temperature and surface gravity, which combined with physics. It makes possible to estimate the stellar mass flux measurements and parallax, can lead to a determina- of white dwarf stars from spectroscopic temperatures and tion of the radius and stellar mass (Provencal et al. 1998; gravities, which in turn are used to determine the mass Holberg et al. 2012; B´edard et al. 2017). This technique distribution (see e.g., Koester et al. 1979; Bergeron et al. started with the works by Schmidt (1996) and Vauclair et al. 2001; Liebert et al. 2005; Holberg et al. 2012; Falcon et al. (1997), who used atmospheric parameters and trigonomet- 2012; Tremblay et al. 2017). In addition, a determination ric parallax measurements for 20 white dwarfs observed with of the white dwarf mass distribution contains information the Hypparcos satellite. Later, this technique was expanded about the star formation history and is directly related to include wide binary systems for which the primary has with the initial–final mass function (Catal´an et al. 2008; a precise parallax from Hipparcos (Provencal et al. 1998; Cummings et al. 2016; El-Badry et al. 2018) which deter- Holberg et al. 2012) and Gaia DR1 (Tremblay et al. 2017). However, this method is not completely independent of theo- retical models, since the determination of the radius depends ⋆ E-mail: [email protected] on the flux emitted at the surface of the star, that is based on c 2015 The Authors 2 Romero et al. the predictions of model atmospheres. In addition, the de- white dwarf cooling sequences employed are those from terminations of the effective temperature and surface grav- Romero et al. (2012, 2013, 2017), extracted from the full ity also rely on model atmospheres, usually through spec- evolutionary computations using the LPCODE evolutionary tral fitting, which can suffer from large uncertainties, up to code (Althaus et al. 2005; Renedo et al. 2010). The model ∼ 0.1 in log g and 1–10% in temperature (Joyce et al. 2018a; grid expands from ∼ 0.493M⊙ to 1.05M⊙ in white dwarf mass, Tremblay et al. 2019). where carbon-oxygen core white dwarfs are found. We also −3 For eclipsing binary systems, the mass and radius of consider a range in hydrogen envelope mass from ∼ 10 M∗ −10 the white dwarf component can be obtained from photo- to ∼ 5 × 10 M∗, depending on the stellar mass. metric observations of the eclipses and kinematic param- We compare our theoretical sequences with mass and eters, without relying on white dwarf model atmospheres, radius determinations for white dwarfs in binary systems, except for the determination of the effective temperature. in order to test the predictions of the theoretical mass– However, the specific configuration of eclipsing binaries im- radius relation and to measure the hydrogen content in the plies that they have most probably interacted in the past, as star, when possible. We consider four white dwarfs in as- common-envelope binaries (Tremblay et al. 2017). A sam- tromeric binaries – 40 Eridani B (Mason et al. 2017), Sirius ple of eclipsing binaries containing a white dwarf compo- B (Bond et al. 2017a), Procyon B (Bond et al. 2015) and nent applied to the study of the mass–radius relation can be Stein 2051 B (Sahu et al. 2017) – and a sample of 11 white found in Parsons et al. (2010, 2012a,b, 2017). In particular, dwarfs in detached eclipsing binaries (Parsons et al. 2017). Parsons et al. (2017) analysed a sample of 16 white dwarfs The paper is organized as follows. In section 2 we briefly in detached eclipsing binary and estimated their mass and describe the evolutionary cooling sequences used in our anal- radius up to a precision of 1–2 per cent. ysis. Section 3 is devoted to study the evolution of the hydro- Another method to test the mass–radius relation is to gen mass in the white dwarf cooling sequence for different rely on astrometric binaries with precise orbital parame- stellar masses. In section 4 we present an analysis on the de- ters, in particular a dynamical mass determination, and dis- pendence on the total radius of the white dwarf with the hy- tances. Examples of those systems are Sirius, 40 Eridani and drogen envelope mass and the possible impacts on the spec- Procyon, for which recent determinations of the dynamical troscopic stellar mass determinations. We also compare our masses based on detailed orbital parameters were reported theoretical cooling sequences with other model grids used in by Bond et al. (2017a), Mason et al. (2017) and Bond et al. the literature. Section 5 is devoted to present the compari- (2015), respectively. However, the radius of the white dwarf son between our theoretical models and the mass and radius component cannot be determined from orbital parameters obtained for white dwarfs in binary systems. Final remarks and other techniques are necessary to estimate this param- are presented in section 6. eters. In particular, for the systems mentioned above, the radius is estimated from the measured flux and precise par- allax, depending on model atmospheres. 2 COMPUTATIONAL DETAILS From evolutionary model computations for single stars 2.1 Input Physics it is known that the theoretical mass–radius relation de- pends systematically on effective temperature, core compo- The white dwarf cooling sequences employed in this work sition, helium abundance and hydrogen abundance in the are those from Romero et al. (2012, 2013, 2017), extracted case of DA white dwarf stars (Wood 1995; Fontaine et al. from full evolutionary computations calculated with the 2001; Renedo et al. 2010; Salaris et al. 2010; Romero et al. LPCODE evolutionary code. Details on the code can be 2015). Previous theoretical mass–radius relations (e.g. Wood found in Althaus et al. (2005, 2010); Renedo et al. (2010); 1995; Fontaine et al. 2001) have assumed a constant hy- Romero et al. (2015). LPCODE computes the evolution of drogen layer thickness which is applied to all models re- single stars with low and intermediate mass at the main gardless of progenitor and white dwarf mass, being typ- sequence, starting at the zero age main sequence, going −4 ically MH/M∗ = 10 (Iben & Tutukov 1984). However, through the hydrogen and helium burning stages, the ther- full evolutionary computations from Romero et al. (2012, mally pulsing and mass-loss stages on the AGB, to the white 2013) showed that the upper limit for the mass of the dwarf cooling evolution. Here we briefly mention the main hydrogen layer in DA white dwarf depends on the total input physics relevant for this work. mass of the remnant. The hydrogen content can vary from The LPCODE evolutionary code considers a simultaneous −3 MH /M∗ ∼ 10 , for white dwarf masses of ∼ 0.5M⊙, to treatment of non-instantaneous mixing and burning of ele- −6 MH /M∗ = 10 for massive white dwarfs with ∼ 1M⊙. In ments (Althaus et al. 2003). The nuclear network accounts addition, asteroseismological studies show strong evidence explicitly for 16 elements and 34 nuclear reactions, that in- of the existence of a hydrogen layer mass range in DA clude pp chain, CNO-cycle, helium burning and carbon ig- −9.5 −4 white dwarfs, within the range 10 < MH /M∗ < 10 , nition (Renedo et al.