The White Dwarf Mass–Radius Relation and Its Dependence on The

arXiv:1901.04644v2 [astro-ph.SR] 2 Apr 2019 MNRAS umnse l 2016 al. et Cummings mass the determine to used mass e.g., astro- are stellar and (see turn temperatures stellar the distribution in spectroscopic ra- estimate in mass–radius from which the stars to gravities, used the dwarf of possible white widely as makes dependence of known is It a mass, relation physics. is stellar This stars the relation. dwarf struc- with the white for dius theory of Chandrasekhar’s of ture results the of One INTRODUCTION 1 ⋆ related directly ( function is mass and initial–ﬁnal history information the formation contains with star distribution the mass dwarf about white the of 2012 2001

.H Córsico H. A. & ljnr .Romero, D. Alejandra dependence envelope. its hydrogen and the relation on mass–radius dwarf white The cetdXX eevdYY noiia omZZZ form original in YYY; Received XXX. Accepted c 1 2 3 4 hsc nttt,Uiesdd eea oRoGad oSu do Grande Rio do Federal Universidade Institute, Physics et fPyis&Atooy nvriyo ecse,Univ Leicester, of University Astronomy, & Physics of Dept. autdd inisAto´mcsyGeof´ısicas, Astronómicas Univers y Ciencias de Facultad OIE,CneoNcoa eIvsiainsCientif´ıca Investigaciones de Nacional Consejo CONICET, -al [email protected] E-mail: 05TeAuthors The 2015 ; ; ibr ta.2005 al. et Liebert rmlye l 2017 al. et Tremblay 000 , 1 – 14 21)Pern pi 09Cmie sn NA L MNRAS using Compiled 2019 April 4 Preprint (2015) ; ose ta.1979 al. et Koester lBdye l 2018 al. et El-Badry ; 3 , obr ta.2012 al. et Holberg 4 .I diin determination a addition, In ). xgncr ht wr eune ihwiedafms between mass dwarf ev white full with of set sequences a dwarf employ We white mass. core stellar oxygen the of determination the thus ABSTRACT wr tr ntehdoe neoems n h mato h v the on impact the and mass envelope hydrogen the on stars dwarf eainadosrain rmwiedaf nbnr ytm.I p In theor systems. the of binary from mass in hydrogen results dwarfs thin the white between from atmospher observations agreement and from good relation mass a find stellar we of nally, determinations the overestimating edt euto nterdu ftemdlo pto up of model the hydroge of the radius of the reduction de in a it reduction that i.e., a mass, find to We stellar lead intrins increases. with envelope mass an hydrogen total uncovers the the evolution of mass dwarf maximum pre-white the the of Computations noa nraein increase an into r optbewt h hoeia asrdu eainfrathick a for r relation gravitational mass–radius from of theoretical radius the and with th mass compatible lower are 4.3% of values is the mass However, spectroscopic mass. the B, Sirius For determinations. e words: Key 1 ⋆ M .O Kepler O. S. , epeetasuyo h eedneo h asrdu relation mass–radius the of dependence the of study a present We H aa´ ne l 2008 al. Catalán et = 2 ; × egrne al. et Bergeron hc deter- which ) ; 10 acne al. et Falcon ht wr tr tla vlto iaystars binary — evolution stellar — stars dwarf white − 6 M ⊙ log . ddNcoa eL lt,L lt 90 Argentina 1900, Plata La Plata, La de Nacional idad g Técnicas, Argentina y s M 1 riyRa,Lietr E 7RH LE1 Leicester, Road, ersity ,A.BnoGncle 50 Brazil Gon¸calves 9500, Bento Av. l, o xdselrms,ta a ec pt .1dx mainly dex, 0.11 to up reach can that mass, stellar fixed a for ; .R .Joyce G. R. S. , H ∼ 2 ino h aisadselrms ( mass stellar determina- and a to radius lead the can of parallax, tion with and combined effec- which measurements gravity, parameters, flux surface atmospheric and from temperature obtained tive be can tions Galaxy. the of evolution chemical interstel the the to affecting returned medium, is lar material stellar much how mines obr ta.2012 al. et Holberg ( by works the with 2012 started al. et Holberg nteflxeitda h ufc ftesa,ta sbsdon based is that star, the of surface the at depen emitted radius flux the the of on determination the theo- since of models, independent retical completely not is method this However, i aalxmaueet o 0wiedaf bevdwith observed dwarfs white the 20 for measurements parallax ric oicuewd iayssesfrwihtepiayhas primary the from which parallax for precise systems binary a wide include to × 1997 10 Hypparcos eieprcldtriain ftems–aisrela- mass–radius the of determinations Semi-empirical − ,woue topei aaeesadtrigonomet- and parameters atmospheric used who ), 8 M ⊙ o 0EiaiB nareetwt previous with agreement in B, Eridani 40 for , aelt.Ltr hstcnqewsexpanded was technique this Later, satellite. 2 and ) .R Lauffer R. G. , ; ´ dr ta.2017 al. Bédard et Gaia Schmidt Hipparcos ∼ 12% R ( DR1 hstasae directly translates This . ( 1996 ( rmlye l 2017 al. et Tremblay rvna ta.1998 al. et Provencal neoems can mass envelope n rvna ta.1998 al. et Provencal dhf observations edshift riua,w n a find we articular, ltoaycarbon- olutionary yrgnenvelope hydrogen and ) tclmass–radius etical .9 and 0.493 cprmtr.Fi- parameters. ic ntedynamical the an .Ti technique This ). leof alue cdpnec of dependence ic A 1 T E o Awhite DA for tl l v3.0 file style X acare al. et Vauclair rae when creases log 1 . g 05 and , M ds ⊙ ). - . ; ; 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 parallax, 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 depends 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. 2010). −6.3 with an average of MH /M∗ ∼ 10 (Fontaine & Brassard We consider the occurrence of extra-mixing beyond 2008; Castanheira & Kepler 2009; Romero et al. 2012). The each convective boundary following the prescription of mass of the hydrogen layer is an important factor, since Herwig et al. (1997), except for the thermally pulsating the mass–radius relation varies by 1-15 per cent, depend- AGB phase. We treated the extra–mixing as a time– ing on the white dwarf mass and temperature, whether a dependent diffusion process, assuming that the mixing veloc- −4 −10 thick (10 M∗) ora thin(10 M∗) hydrogen layer is assumed ities decay exponentially beyond each convective boundary. (Tremblay et al. 2017). The diffusion coefficient is given by DEM = D0 exp(2z/ fHP), In this work we study the dependence of the mass– where HP is the pressure scaleheight at the convective bound- radius relation with the mass of the hydrogen layer. The ary, D0 is the diffusion coefficient of unstable regions close

MNRAS 000, 1–14 (2015) The mass-radius relation 3 to the convective boundary, z is the geometric distance Romero et al. (2012, 2013, 2017). The values of stellar mass from the edge of the convective boundary, and f describes of our model grid are listed in table 1, along with the hydro- the efficiency, and was set to f = 0.016 (see Romero et al. gen and helium content as predicted by single stellar evo- 2015, for details). Mass loss episodes follow the prescrip- lution, for an effective temperature of ∼ 12 000 K. The cen- tion from Schröder & Cuntz (2005) during the core helium tral abundance of carbon and oxygen for each mass is also burning and the red giant branch phases, and the pre- listed. Note that the value of the hydrogen content listed in scription of Vassiliadis & Wood (1993) during the AGB and table 1 is the maximum possible value, since a larger hydro- thermally pulsating AGB phases (De Gerónimo et al. 2017, gen mass will trigger nuclear reactions, consuming all the 2018). During the white dwarf evolution, we considered the exceeding material (see section 3). The upper limit for the distinct physical processes that modify the inner chemical possible hydrogen content shows a strong dependence on the −4 profile. In particular, element diffusion strongly affects the stellar mass. It ranges from 1.6 × 10 M⊙ for M∗ = 0.493M⊙ −6 −4 chemical composition profile throughout the outer layers. In- to 1.5 × 10 M⊙ for M∗ = 1.050M⊙, with a value ∼ 10 M∗ deed, our sequences develop a pure hydrogen envelope with for the averaged-mass sequence of M∗ = 0.60M⊙, for effective increasing thickness as evolution proceeds. Our treatment temperatures near the beginning of the ZZ Ceti instabil- of time dependent diffusion is based on the multicompo- ity strip. The helium abundance also shows a dependence nent gas treatment presented in Burgers (1969). We con- with the stellar mass, decreasing monotonically with the in- sider gravitational settling and thermal and chemical diffu- crease of the stellar mass. In particular, the sequence with 3 4 12 13 14 16 sion of H, He, He, C, C, N and O (Althaus et al. 1.05M⊙ was obtained by artificially scaling the stellar mass 2003). To account for convection process in the interior of from the 0.976M⊙ sequence at high effective temperatures the star, we adopted the mixing length theory, in its ML2 fla- (see Romero et al. 2013, for details). Since no residual he- vor, with the free parameter α = 1.61 (Tassoul et al. 1990) lium burning is present in the cooling sequence the helium during the evolution previous to the white dwarf cooling content does not change, and both sequences present similar curve, and α = 1 during the white dwarf evolution. Last, helium content. we considered the chemical rehomogenization of the inner Uncertainties related to the physical processes occurring carbon-oxygen profile induced by Rayleigh-Taylor instabili- during the AGB stage, lead to uncertainties in the amount ties following Salaris et al. (1997). of hydrogen remaining on the envelope of a white dwarf For the white dwarf stage, the input physics of the star. For instance, the hydrogen mass can be reduced to code includes the equation of state of Segretain et al. (1994) a factor of two as a result of the carbon enrichment of the for the high density regime complemented with an up- envelope due to third dredge–up episodes at the thermally dated version of the equation of state of Magni & Mazzitelli pulsing AGB phase (Althaus et al. 2015). Also, the hydro- (1979) for the low density regime. Other physical ingredi- gen envelope mass depends on the initial metallicity of the ents considered in LPCODE are the radiative opacities from progenitor, being a factor of 2 thicker when the metallic- the OPAL opacity project (Iglesias & Rogers 1996) supple- ity decreasses from Z = 0.01 to Z = 0.001 (Renedo et al. mented at low temperatures with the molecular opacities 2010; Romero et al. 2015). However, the mass loss rate dur- of Alexander & Ferguson (1994). Conductive opacities are ing the AGB and planetary nebula stages will not strongly those from Cassisi et al. (2007), and the neutrino emission impact the amount of hydrogen left on the white dwarf rates are taken from Itoh et al. (1996) and Haft et al. (1994). (Althaus et al. 2015). Cool white dwarf stars are expected to crystallize as In order to compute cooling sequences with different val- a result of strong Coulomb interactions in their very dense ues of the thickness of the hydrogen envelope, in particular interior (van Horn 1968). In the process two additional en- thinner than the value expected by the burning limit, 1H was ergy sources, i.e., the release of latent heat and the release of replaced with 4He at the bottom of the hydrogen envelope gravitational energy associated with changes in the chemical (see Romero et al. 2012, 2013, for details). This procedure is composition of the carbon–oxygen profile induced by crystal- done at high effective temperatures (& 90 000K), so the tran- lization (Garcia-Berro et al. 1988; Winget et al. 2009), are sitory effects caused by the artificial procedure are quickly considered self-consistently and locally coupled to the full set washed out. The values of hydrogen content as a function of of equations of stellar evolution. The chemical redistribution the stellar mass are depicted in Figure 1. The thick red line due to phase separation has been considered following the connects the values of the maximum value of MH predicted procedure described in Montgomery & Winget (1999) and by our stellar evolution computations. Salaris et al. (1997). To assess the enhancement of oxygen in the crystallized core we used the azeotropic-type formu- lation of Horowitz et al. (2010). 3 THE EVOLUTION OF THE HYDROGEN CONTENT 2.2 Model Grid After the end of the TP-AGB stage, during the post-AGB The DA white dwarf cooling sequences considered in this evolution at nearly constant luminosity, simple models of work are the result of full evolutionary computations of pro- the white dwarf progenitors show that CNO cycle reactions genitor stars with stellar masses between 0.95 and 6.6M⊙ reduce the hydrogen content below a critical value. If the at the zero age main sequence. The initial metallicity was star has a white dwarf mass of 0.6 M⊙, the value of the −4 set to Z = 0.01. As a result, the stellar mass range in critical hydrogen mass is ∼ 2 × 10 M⊙ (Iben 1982, 1984; the cooling sequence expands from ∼ 0.493M⊙ to 1.05M⊙, Iben & Renzini 1983). Residual nuclear burning will reduce −4 where carbon-oxygen core white dwarfs are found. These se- the hydrogen mass in the surface layers to ∼ 10 M⊙. These quences were presented in the works of Renedo et al. (2010); values change with white dwarf mass when the evolution

MNRAS 000, 1–14 (2015) 4 Romero et al.

10 -2.5 post-AGB 9 WD-maxTeff -3 WD-12000 K 8 -3.5 ) * )

7 * /M H /M -4 H 6 -log(M log(M -4.5 5 -5

4 -5.5

3 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 -6 M*/Msun 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 M*/Msun Figure 1. Grid of DA WD evolutionary sequences considered in Figure 2. Hydrogen mass as a function of the white dwarf stellar this work in the M∗/M⊙ vs. − log(MH /M∗) plane. Each circle corresponds to a sequence of models representative of white dwarf mass for three point during the evolution: during the nearly con- ∼ stars characterized by a given stellar mass and hydrogen envelope stant luminosity stage at the post-AGB at Teff 10 000 K (post- mass. The envelope mass is measured at an effective temperature AGB), at the point of maximum effective temperature when the of 12 000 K. The red line connects the sequences with the maxi- star enters the cooling sequence (WD-maxTeff ) and on the cooling ∼ mum values for the thickness of the hydrogen envelope, predicted curve at Teff 12 000K (WD-12000K). evolutionary computations. by our evolutionary computations.

mass occurs during the post-AGB evolution, for all stellar Table 1. The main characteristics of our set of DA white dwarf M ∼ 0.6M⊙ models. The stellar mass at the white dwarf stage is listed in masses, due to CNO shell–burning. For a WD the ∼ × −4 column 1. Also listed are the hydrogen mass at 12 000 K (column hydrogen content is 2.8 10 M⊙ at 10 000 K on the post- −4 2), the helium mass (column 3) and the central abundances of AGB stage, and it reduces to ∼ 1.1 × 10 M⊙ when the star carbon (column 4) and oxygen (column 5). enters the white dwarf cooling curve. Residual burning at the cooling curve can further reduce the hydrogen envelope by a factor of ∼ 2 (see figure M⋆/M⊙ − log(MH/M⋆) − log(MHe/M⋆) XC XO 2). Figures 3 and 4 show the temporal evolution of the hy- 0.493 3.50 1.08 0.268 0.720 drogen content for sequences with white dwarf stellar mass 0.525 3.62 1.31 0.278 0.709 0.609 and 0.998 M⊙, respectively, during the cooling curve. 0.548 3.74 1.38 0.290 0.697 Also shown are the luminosity given by the nuclear burning 0.570 3.82 1.46 0.301 0.696 of hydrogen due to the CNO bi-cycle and the pp chain as 0.593 3.93 1.62 0.283 0.704 0.609 4.02 1.61 0.264 0.723 a function of the logarithm of the cooling time in years, 0.632 4.25 1.76 0.234 0.755 measured from the point of maximum effective tempera- 0.660 4.26 1.92 0.258 0.730 ture when the star enters the white dwarf cooling curve. 0.705 4.45 2.12 0.326 0.661 As can be seen, the hydrogen burning is reducing the hy- 0.721 4.50 2.14 0.328 0.659 drogen envelope mass during the early evolutionary phases 0.770 4.70 2.23 0.332 0.655 of the white dwarf stage. At the beginning of the cooling 0.800 4.84 2.33 0.339 0.648 sequence, the 0.609 M⊙ sequence has a hydrogen content of 0.837 5.00 2.50 0.347 0.640 −5 MH/M⊙ = 8.6 × 10 , which is reduced due to residual hydro- 0.878 5.07 2.59 0.367 0.611 −5 gen burning to a constant value of MH/M⊙ = 5.8 × 10 after 0.917 5.41 2.88 0.378 0.609 1.76 Gyr. As can be seen from figure 3, that the CNO bi-cycle 0.949 5.51 2.92 0.373 0.614 0.976 5.68 2.96 0.374 0.613 dominates de energy production due to hydrogen burning ∼ 0.998 5.70 3.11 0.358 0.629 for the first 110 Myr of the cooling sequences. Hydrogen- 1.024 5.74 3.25 0.356 0.631 burning due to the pp chain lasts longer, causing a small 1.050 5.84 2.96 0.374 0.613 reduction in the hydrogen content of the model. Once the hydrogen content decreases below a certain threshold the pressure at the bottom of the envelope is not large enough to support further nuclear reactions. After no residual nu- previous and during the cooling curve are computed consis- clear burning is present in the star, the main energy source tently. In figure 2 we show the hydrogen mass as a function of the white dwarf is the release of gravothermal energy. of the white dwarf mass for three points during the evolu- A similar scenario is found for the more massive sequence tion: the point with Teff ∼ 10 000 K during the post-AGB shown in figure 4. The star enters the cooling sequence with −6 stage, previous to the white dwarf phase (solid line), the an hydrogen content of MH/M⊙ = 7.3 × 10 , 10 times smaller point of maximum effective temperature when the star en- than the hydrogen content in the 0.609 M⊙ model at the ters the cooling curve (dashed line) and at Teff ∼ 12 000K same stage. After 580 Myrs the hydrogen content reaches −6 on the white dwarf cooling curve (dot-dashed line). As can a somewhat constant value of MH/M⊙ = 1.97 × 10 . In this be seen from figure 2, the major reduction of the hydrogen case, because of the higher temperature at the base of the

MNRAS 000, 1–14 (2015) The mass-radius relation 5

4 0.609 -3.86 3

-3.88 2

1 -3.9 0 -3.92 ) ) sun H -1 -3.94

log(M -2 log(L/L

-3.96 -3

-3.98 -4 log(M /M*) H -5 -4 CNO pp -6 -4.02 -7 1 2 3 4 5 6 7 8 9 10 log(t) [yr] Figure 5. Evolution in the H-R diagram for a sequence of 1M⊙ in the white dwarf cooling sequence with an artiﬁcially increased −4 Figure 3. Temporal evolution of the hydrogen content MH (in hydrogen envelope of MH ∼ 10 M∗ at the begining of the cooling units of M∗) and the ratio of hydrogen nuclear burning to surface curve. This scenario sumilates the possible acretion of material luminosity for CNO bi-cycle and pp chain, for a white dwarf se- due to interaction with a binary companion. The color bar indi- quence with stellar mass 0.609 M⊙. The time corresponds to the cates the amount of hydrogen left in the envelope of the star. Note cooling time in yr measured from the point of maximum eﬀective that, if residual thermonuclear burning is considered, the amount −6 temperature at the beginning of the cooling curve. of hydrogen decreases and reaches a value of ∼ 2.5×10 M∗, similar to that obtained from single stellar evolution.

3 -5.15 0.998 -5.2 2

-5.25 1

-5.3 0 on the distance to the companion, a certain amount of mass -5.35 can be added on top of the white dwarf. However, if the -1 ) hydrogen content exceeds the limiting value for nuclear re- ) sun H -5.4 actions, the H–shell at the bottom of the hydrogen envelope -2 -5.45 can be activated and the additional hydrogen is consumed, log(M log(L/L leading to an equilibrium mass. To explore this scenario, -5.5 -3 we computed the cooling evolution of a 1M⊙ white dwarf -5.55 -4 sequence with a thick hydrogen layer. To simulate the ac- creted material we artificially increased the hydrogen enve- -5.6 log(M /M ) -5 H * lope mass at high effective temperatures, near the beginning CNO -5.65 of the cooling sequence. For a white dwarf with 1M⊙ the re- pp -6 -5.7 maining hydrogen content predicted by single stellar evolu- = −6 -7 tion is MH /M∗ 2.1 × 10 (see Table. 1). Thus, we increased 1 2 3 4 5 6 7 8 9 10 −4 M /M∗ = log(t) [yr] the hydrogen mass to a factor 100, H 10 , and computed the following evolution considering possible sources of Figure 4. Same as figure 3 but for a sequence with stellar mass nuclear burning. Figure 5 shows the evolution in the H–R 0.998 M⊙. The time corresponds to the cooling time in yr. diagram for this sequence. The color bar indicates the hydrogen envelope mass in a logarithmic scale. As expected, the hydrogen burning shell at the bottom of the hydrogen enve- envelope, the contribution to the energy production from lope is active again, consuming the excess in the hydrogen the CNO bi-cycle is five orders of magnitude larger than the content. The residual burning makes the model move to the contribution from the proton-proton chain at the beginning high luminosity and low temperature region of the H–R dia- of the cooling sequence, being the dominant source while gram, similar to what happens in the low mass regime that residual nuclear reactions are still active. produces pre-ELM white dwarf stars (Althaus et al. 2013; Istrate et al. 2014). Once the hydrogen content is reduced 3.1 Massive white dwarf with ”thick” hydrogen below the limiting value for nuclear burning, the star settles onto the cooling sequence one more time with a hydrogen envelope −6 mass of ∼ 2.5 × 10 M∗, similar to the value obtained from The hydrogen content in the envelope of a white dwarf can single evolution computations. Therefore, the final hydrogen increase due to accretion from a companion star. Depending content would not change significantly due to accretion.

MNRAS 000, 1–14 (2015) 6 Romero et al.

2 7.2 Table 2. Radius (in R⊙) and surface gravity (g in cm/s ) ex- Thick tracted from theoretical cooling sequences, for 40 000 K (columns -4.5 2 and 3), and 20 000 K (columns 4 and 5), for the stellar masses 0.493 -5.3 7.4 -6.4 presented in Figure 6. -7.4 -9.3 7.6 0.493 R(40kK) log g R(20kK) log g

thick 0.019137 7.5683 0.016157 7.7153 7.8 0.609 -5.3 0.017661 7.6380 0.015266 7.7646 -6.4 0.017222 7.6586 0.014986 7.7793

log (g) -7.4 0.017017 7.6690 0.014927 7.7828 8 0.705 -9.3 0.016826 7.6788 0.014814 7.7894

8.2 0.609 R(40kK) log g R(20kK) log g 0.837 thick 0.014794 7.8825 0.013306 7.9747 8.4 -5.3 0.014181 7.9206 0.012982 7.9973 -6.4 0.013899 7.9368 0.012796 8.0086 0.997 -7.4 0.013766 7.9451 0.012723 8.0136 8.6 -9.3 0.013658 7.9520 0.012662 8.0177

50000 40000 30000 20000 10000 0.705 R(40kK) log g R(20kK) log g Teff [K] thick 0.0125432 8.0891 0.0116590 8.1526 Figure 6. Cooling tracks for different hydrogen envelopes in the -5.3 0.0122555 8.1093 0.0114827 8.1658 Teff −log g plane. The thickness of the hydrogen envelope decreases -6.4 0.0120894 8.1211 0.0113798 8.1737 from top to bottom, and it is labelled in the figure in log(MH/M∗). -7.4 0.0120037 8.1273 0.0113274 8.1777 The stellar mass for each group of sequences is indicated. -9.3 0.0119206 8.1334 0.0112778 8.1815

0.837 R(40kK) log g R(20kK) log g 4 RADIUS AND HYDROGEN CONTENT thick 0.0103397 8.3315 0.0098452 8.3744 As it was mentioned, the mass-radius relation depends on -5.3 0.0110280 8.3365 0.0098002 8.3781 the amount of hydrogen and helium present in the star. In -6.4 0.0101528 8.3474 0.0097275 8.3846 particular, the hydrogen envelope, although it contains a -7.4 0.0100880 8.3530 0.0096864 8.3882 very small amount of mass, appears to be the dominant pa- -9.3 0.0100311 8.3579 0.0096513 8.3914 rameter in the determination of the radius for DA white 0.998 R(40kK) log g R(20kK) log g dwarfs (Tremblay et al. 2017). Thus a reduction of the hydrogen, and/or helium content, will lead to a smaller radius thick 0.008221 8.6082 0.007945 8.6377 and thus to an increase in the surface gravity. This effect is -6.4 0.008124 8.6172 0.007886 8.6430 depicted in figure 6, where we show white dwarf evolution- -7.4 0.008084 8.6215 0.007860 8.6458 -9.3 0.008050 8.6251 0.007837 8.6484 ary sequences in the log g − Teff plane for stellar masses from 0.493 to 0.998M⊙. The black line indicates the sequence with the thickest hydrogen envelope, as predicted by single stellar evolution. As expected, the surface gravity increases when log g ∼ 0.038 the hydrogen content decreases for a given stellar mass at spectroscopic fits in is dex (Liebert et al. 2005; a given temperature. The effect is less important for higher Barstow et al. 2005), it is possible to estimate the hydro- ∼ 0.7M⊙ stellar masses, since they form with thinner hydrogen en- gen layer for stellar masses lower than , but not for velopes. For instance, the maximum mass for the hydrogen higher stellar masses, unless the uncertainties are reduced to less than 0.011 dex in log g. content in a 0.998 M⊙ sequence is 100 times thinner than that corresponding to a 0.6 M⊙ (see table. 1 for details). The radius and log g for the stellar masses are listed in Table 2 4.1 Comparison with other theoretical models for effective temperatures of 40 000 and 20 000 K. Comparing the results for the sequences with the thick- In the literature we can find a few grids of theoretical white est envelope with those having the thinnest hydrogen enve- dwarf cooling sequences. Wood (1995) computed DA white lope of the grid, the reduction in the total radius is between dwarf cooling sequences to study the white dwarf luminos- 8-12% for a stellar mass of 0.493 M⊙, 5-8% for stellar mass ity function of our Galaxy. He considered a stratified model 0.609 M⊙ and almost negligible, 1-2%, for a model with 0.998 with various central compositions, from pure C to pure O. M⊙, within the ranges of hydrogen mass considered in this The computations start as polytropes and the early evolu- work. However, this reduction in the total radius has a strong tion is characterized by a contraction phase at a constant 2 impact in the surface gravity value, specially for high effec- luminosity of ∼ 10 L⊙, before entering the cooling curve tive temperatures, being 0.11 dex, 0.08 dex and 0.02 dex in (Winget et al. 1987). The growing degeneracy in the core log g for stellar masses 0.493, 0.609 and 0.998 M⊙, respec- halts the contraction and the surface temperature reaches tively, for Teff = 40 000 K. For an effective temperature of a maximum of Teff ≥ 100 000 K. In particular, the models 20 000 K, de increase in log g is 0.08, 0.07 and 0.011 dex, re- presented in Wood (1995) have a fixed hydrogen content of −4 spectively. Considering that the real mean uncertainty from ∼ 10 M∗.

MNRAS 000, 1–14 (2015) The mass-radius relation 7

Another widely used set of white dwarf sequences are 1 7.4 those computed by the Montreal group , and published in Fontaine et al. (2001) Fontaine et al. (2001). The first set of models for DA white LPCODE dwarfs had a C pure core and a helium and hydrogen content −2 −4 7.6 of 10 M∗ and 10 M∗, respectively, for sequences with stellar mass in the range between 0.2 and 1.3 M⊙. Latter, additional 0.50 models with a C/O = 50/50 where computed with a helium 7.8 −2 content 10 M∗ and two different values for the hydrogen −4 −10 mass, being 10 M∗ (”thick”) and 10 M∗ (”thin”). 0.60 Finally, Salaris et al. (2010) presented a set of white 8 dwarf cooling sequences with stellar masses between 0.54 log(g) and 1 M⊙. For each white dwarf mass an initial model 8.2 0.70 2 was converged at L ∼ 10 L⊙, considering a chemical composition profile taken from pre-white dwarf computations, 0.80 specifically at the first thermal pulse (Salaris et al. 2000). 8.4 −4 The hydrogen and helium mass are set to be MH = 10 M∗ −2 0.90 and MHe = 10 M∗, respectively for all stellar masses, as in 8.6 Fontaine et al. (2001). In all cases, the authors assume a constant thickness of 1.00 the hydrogen layer for all stellar masses. In order to keep 8.8 60000 50000 40000 30000 20000 10000 the hydrogen mass at a strictly constant value, no residual T [K] thermonuclear burning has been included in the calculations. eff

In particular, Salaris et al. (2010) stated that H–burning at Figure 7. Comparison in the Teff − log g plane of the theoretical the bottom of the hydrogen envelope is negligible in all but cooling sequences computed with LPCODE (solid lines) and those −4 the more massive models. In addition, the cooling sequences extracted from Fontaine et al. (2001) with MH = 10 M∗ (dashed presented in those works do not compute the post-AGB and lines). The labels indicates the stellar mass of the sequences. In planetary nebula stages, crucial to determine the hydrogen order to match the stellar mass values from Fontaine et al. (2001), envelope mass at the white dwarf stage. we interpolated the cooling sequences within our model grid (see As we show in section 2.2, the maximum mass of hydro- Table 1). gen left on top of a white dwarf model depends on the stellar mass. In particular, the hydrogen envelope mass is larger sources at the cooling sequence, can lead to overestimated −4 than 10 M∗ for stellar masses lower than ∼ 0.6M⊙, while se- or underestimated spectroscopic masses. quences with white dwarf masses larger than ∼ 0.6M⊙ show −4 thinner hydrogen envelopes, with masses below 10 M∗. The extension of the hydrogen envelope will impact the total ra- 5 MEASURING THE HYDROGEN MASS dius of the star and consequently the value of log g. For instance, the hydrogen envelope mass for a ∼ 1M⊙ white dwarf In this section we estimate the hydrogen mass content of a −6 model is ∼ 10 M∗, 100 times thinner than the fixed value selected sample of white dwarf stars in binary systems. The considered in previous works, leading to a ∼ 5% decrease in mass and radius for the objects considered in our analysis the stellar radius for that stellar masses and radii. were taken from the literature and were estimated using dif- In figure 7 we compare our canonical sequences, those ferent techniques, which are in principle, independent of the with the thickest hydrogen envelope obtained from sin- theoretical mass–radius relation. First we consider two well gle stellar evolution, to the theoretical cooling sequences studied members of astrometric binary systems, 40 Eridani −4 from Fontaine et al. (2001) with MH = 10 M∗, in the B and Sirius B. Aditionally, we consider two non-DA white Teff − log g plane. Similar results are found when we compare dwarfs, Procyon B and Stein 2051 B. Finally, we analyse the to the cooling sequences from Wood (1995) and Salaris et al. results obtained from a sample of eleven detached eclipsing (2010). To match the stellar mass values from Fontaine et al. binaries. We do not consider the data fully based on spectro- (2001), we interpolated the cooling sequences within our scopic techniques since the uncertainties in the atmospheric model grid. Note that for a stellar mass 0.6M⊙ the model parameters, specially in log g, is too large to estimate the computed with the LPCODE perfectly overlaps with the model hydrogen envelope mass (Joyce et al. 2018a). In each case computed by Fontaine et al. (2001). This is a consequence of we analyse the results and the uncertainties and how they the value of the hydrogen envelope mass, which for this stel- impact the determination of the hydrogen layer mass. −4 lar mass is nearly ∼ 10 M∗ in both cases. For stellar masses below 0.6 M⊙, in this case 0.5 M⊙, the cooling sequence com- 5.1 Astrometric binaries puted with LPCODE shows a lower log g – larger radius – than that from Fontaine et al. (2001), while the opposite happens Astrometric binaries are an important tool to test the mass- for sequences with stellar masses larger than 0.6 M⊙. Thus, radius relation, since independent determinations of the stel- models with fixed hydrogen envelopes, that do not consider lar mass can be obtained from the dynamical parameters the pre-white dwarf evolution and/or the residual burning of the binary system and accurate distances. The determination of the radius, on the other hand, is not model- independent, since it depends on the value of the flux emit- 1 http://www.astro.umontreal.ca/∼bergeron/CoolingModels/ ted at the surface itself. We consider that, for these systems,

MNRAS 000, 1–14 (2015) 8 Romero et al. the dynamical parameters and distances are precise enough 8.02 to set constrains not only on the theoretical mass–radius re- 0.609 lation but on the hydrogen content. In this section we use 0.593 8 0.570 the observational determinations of mass and radius for four -4.28 -5.34 white dwarfs in astrometric binary systems. -6.33 -7.34 7.98 -8.32 5.1.1 40 Eridani B -9.33 0.6-04 0.6-10 40 Eridani B was for many years reported as a low mass 7.96 white dwarf with a stellar mass of ∼ 0.4M⊙. Recently, Mason et al. (2017) determined a dynamical mass of 0.573 ± log (g) 0.018M⊙ using observations of the orbit covering a longer 7.94 period of time and the updated Hipparcos parallax. Lat- ter, Bond et al. (2017b) determined the atmospheric parameters of this star using spectroscopy and obtained an effec- 7.92 tive temperature of 17 200 ± 110 K and a surface gravity of log g = 7.957 ± 0.020. In addition, these authors determined 7.9 the radius of 40 Eridani B using photometric observations 40 Eri B combined with the distance (see for details Bédard et al. 17500 17400 17300 17200 17100 17000 16900 16800 2017), being R = 0.01308 ± 0.00020R⊙. Teff [K] From the spectroscopic parameters derived by Bond et al. (2017b), we computed the stellar mass for Figure 8. Location of 40 Eridani B in the Teff − log g plane. Solid 40 Eridani B from evolutionary tracks in the Teff − log g lines correspond to thick envelope sequences, those with the larger plane for different values of the hydrogen envelope thickness. amount of hydrogen allowed by single stellar evolution. Thinner In Figure 8 we depict the location of 40 Eridani B along envelopes are depicted with different lines (see inset in the fig- with theoretical white dwarf cooling tracks computed with ure). The labels associated to the sequences with different thin log(M /M∗) LPCODE, with different hydrogen envelope thickness ranging envelopes correspond to the value of H . Different col- −4 −10 ors indicate different stellar masses (in solar units). Green and from 10 M∗ to 2 × 10 M∗, and stellar mass between 0.570 blue lines correspond to sequences with 0.570 and 0.593 M⊙, re- and 0.609 M⊙. We also plotted two cooling sequences from spectively, while the black solid line correspond to a sequences −4 Fontaine et al. (2001) with 0.6M⊙ for thick (10 M∗) and with 0.609M⊙. The magenta lines corresponds to sequences from −10 thin (10 M∗) hydrogen envelopes. Fontaine et al. (2001) for a stellar mass 0.6M⊙ with thick (0.6–04) If we consider the uncertainties in the atmospheric pa- and thin (0.6–10) hydrogen envelopes. rameters, specifically in log g, the stellar mass varies when the different hydrogen envelope thickness is taken into account (Tremblay et al. 2017). From figure 8 we note that Another way to estimate the hydrogen mass in 40 Eri- the spectroscopic stellar mass is higher if we consider thick dani B is by comparing the radius and dynamical mass to envelope tracks, those with the thickest hydrogen envelope the theoretical mass–radius relation. This is shown in fig- allowed by our models of single stellar evolution, and it de- ure 9, where we depict the mass–radius relations for six creases for thinner hydrogen envelopes. We computed the different envelope thickness. The solid black curve corre- stellar mass for each envelope thickness. The results are spond to the mass–radius relation for thick envelope se- listed in Table 3, along with the corresponding hydrogen quences. The thinnest hydrogen envelope in the model grid ∼− mass in solar units, the determinations of the dynamical is log(MH/M∗) 9.33. From figure 9, we see that the obser- mass (Shipman et al. 1997; Mason et al. 2017) and the spec- vations of 40 Eridani B are in agreement with a thin enve- troscopic mass (Bond et al. 2017b). We notice that the spec- lope solution. Specifically, the upper limit for the hydrogen troscopic stellar mass varies from 0.594M⊙, for the thick en- mass is the same as the one obtained using the value of the velope set of tracks, to 0.571M⊙, ∼ 4% lower, for the thinnest dynamical mass and the spectroscopic parameters (Figure value. Note that the values for the spectroscopic mass for 8). the two thinnest envelopes are the same, implying that this The cooling age for 40 Eridani B, for a stellar mass of ± ∼ parameter is not sensitive to the hydrogen envelope once (0.573 0.0011)M⊙, is 145 Myrs. Considering the Initial– −9 it is thinner than 2.79 × 10 M⊙. The spectroscopic mass to–final mass relation from Romero et al. (2015) for solar ± that better matches the value of the dynamical mass from metallicity, we estimate a progenitor mass of 1.53 0.11M⊙ ± Mason et al. (2017) is the one characterized by an hydrogen and a total age of 2.68 0.43 Gyr for 40 Eridani B. −8 envelope of MH = 2.63×10 M⊙. If we consider 1 σ uncertain- −6 ties, the hydrogen mass is between MH = 2.63 × 10 M⊙ and −10 5.1.2 Sirius B MH = 2.67 × 10 M⊙. Thus, we conclude that the hydrogen envelope for 40 Eridani B should be thinner than the value Sirius B is the brightest and nearest of all white dwarfs, lo- predicted by single stellar evolution. This result is consis- cated at 2.65 pc. Combining the information from the or- tent with previous works (Holberg et al. 2012; Bédard et al. bital parameters and parallax, Bond et al. (2017a) deter- 2017; Bond et al. 2017b). In particular, Bond et al. (2017b) mined a dynamical mass of MB = 1.018 ± 0.011M⊙ for Sir- found a spectroscopic mass somewhat lower than the values ius B. In addition, Bond et al. (2017a) reported spectro- presented in this work, but they also found a thin envelope scopic atmospheric parameters of Teff = 25 369 ± 46 K and −10 for 40 Eridani B, consistent with MH = 10 M⊙. log g = 8.591±0.016 and a radius of R = 0.008098±0.000046R⊙ .

MNRAS 000, 1–14 (2015) The mass-radius relation 9

Table 3. Stellar mass determinations for diﬀerent hydrogen envelope layers, considering the spectroscopic atmospheric parameters for 40 Eridani B (left) and Sirius B (right). Also listed are the dynamical mass (Mason et al. 2017) and the spectroscopic mass derived by Bond et al. (2017b). Dynamical masses and other spectroscopic determinations are listed in the last three rows. Note: Values of the stellar mass determined by the techniques: (1) dynamical mass, (2) fully spectroscopical, (3) gravitational redshift.

MH /M⊙ Mass [M⊙] MH /M⊙ Mass [M⊙] 6.87 × 10−5 0.594 ± 0.010 ······ 3.07 × 10−5 0.589 ± 0.010 ······ 2.64 × 10−6 0.580 ± 0.010 2.02 × 10−6 0.974 ± 0.013 2.67 × 10−7 0.575 ± 0.010 3.16 × 10−7 0.970 ± 0.015 2.63 × 10−8 0.573 ± 0.011 3.66 × 10−8 0.968 ± 0.014 2.79 × 10−9 0.571 ± 0.011 4.00 × 10−9 0.967 ± 0.014 2.65 × 10−10 0.571 ± 0.011 4.87 × 10−10 0.966 ± 0.014

Shipman et al. (1997)(1) 0.501 ± 0.011 Bond et al. (2017a)(1) 1.018 ± 0.011 Mason et al. (2017)(1) 0.573 ± 0.018 Barstow et al. (2005)(2) 0.978 ± 0.005 Bond et al. (2017b)(2) 0.565 ± 0.031 Joyce et al. (2018b)(3) 1.017 ± 0.025

0.016 (2001). The spectroscopic mass, determined using our evo- Thick lutionary tracks, results in ∼ 0.974M⊙ for the sequences -5.3 0.015 -6.4 with the thickest envelope, 4.3 % lower than the dynami- -7.4 -8.4 cal mass, in agreement with previous determinations of the -9.3 spectrocopic mass (Barstow et al. 2005; Holberg et al. 2012; 0.014 B´edard et al. 2017; Joyce et al. 2018a). The spectroscopic

sun stellar mass for each hydrogen envelope mass, computed us-

R/R ing the LPCODE cooling tracks is listed in table 3. Also listed, 0.013 are stellar mass determinations from observations. 40 Eri B Also from Fig. 10, while the cooling track from 0.012 Fontaine et al. (2001) for thin hydrogen envelope (1.0-10) overlaps with the cooling sequences of LPCODE with the same mass, the track with thick hydrogen envelope (1.0-04, 10−4) 0.011 0.5 0.55 0.6 0.65 0.7 overlaps with the LPCODE tracks with stellar mass of 0.973M⊙. M/M sun We can easily explain the difference in log g with the different total hydrogen mass in the models, since, for our models, Figure 9. Location of 40 Eridani B in the mass–radius plane, the thickest hydrogen envelope mass for a ∼ 1M⊙ white dwarf where the values correspond to the dynamical mass (Mason et al. −6 ∼ 2 × 10 M∗ 2017) and radius (Bond et al. 2017b). The curves are the mass– is , two orders of magnitude thinner than the value adopted by Fontaine et al. (2001) (see section 4.1). We radius relation for an effective temperature of Teff = 17 200 K, corresponding to the spectroscopic temperature of 40 Eridani B. compute an additional sequence with 0.988M⊙ and thick hy- −4 Each curve is characterized by a value of MH /M∗. The black solid drogen envelope of ∼ 10 M∗, labelled as 0998-thick in figure line (”Thick”) corresponds to the maximum hydrogen content al- 10. We use the same technique described in section 3.1 but, lowed by single stellar evolution. we turned off all hydrogen nuclear reactions, to keep its hydrogen content fixed. Note that with an hydrogen envelope ∼ 100 times more massive our model with 0.998M⊙ is able to Note that, the surface gravity calculated from radius and dy- nearly reproduce the spectroscopic surface gravity for Sirius namical mass is log g = 8.629 ± 0.007, not compatible with B. Although the hydrogen content is the dominant factor, the spectroscopic value within 2 σ (Bond et al. 2017a). More additional discrepancies in the surface gravities between the recently, Joyce et al. (2018b) determine the mass for Sir- thick envelope models can be explained with the difference −2 ius B using the effect of gravitational redshift and a ra- in the helium content, being arbitrarily set to 10 M∗ for the models from Fontaine et al. (2001), and being set by stellar dius from the flux and the parallax, and obtained values −3.1 evolution to 10 M∗ for the models computed with LPCODE. of M∗ = 1.017 ± 0.025M⊙ and R∗ = 0.00803 ± 0.00011R⊙, in agreement with those of Bond et al. (2017a). Note that, the uncertainties associated to the spectro- Using the spectroscopic parameters reported in scopic determinations of the atmospheric parameters in the Bond et al. (2017a) we proceed to estimate the stellar mass literature correspond to internal errors and could be as large of Sirius B. In Figure 10 we depict the location of Sirius B as 1.2 % in effective temperature, and 0.038 dex in log g in the Teff − log g plane. Cooling tracks for stellar masses in (Barstow et al. 2005; Liebert et al. 2005). In the case of Sir- the range of 0.949 − 1.024M⊙ are color-coded for each stellar ius B, Joyce et al. (2018a) computed the atmospheric pa- mass and the values are indicated for each group. The solid rameters using different spectra for HST and found a dis- lines correspond to thick envelope sequences, while thinner persion for log g of 0.05 dex, leading to spectroscopic stel- −6 envelopes, i.e. with MH/M∗ < 2 × 10 , are depicted with dif- lar masses between 0.874 and 0.962 M⊙. With this criteria, ferent lines, with increasing log g when MH decreases. The the uncertainties in the spectroscopic mass are three times figure includes cooling curves with 1M⊙ from Fontaine et al. larger than the ones considered by Bond et al. (2017a). In

MNRAS 000, 1–14 (2015) 10 Romero et al.

8.7 0.0087 Fontaine01-thick Joyce18b-Gaia Thick 1.024 Joyce18b-Hipp 8.68 Thin - 9.33 Bond17a MESA-1.019 Bastow05 0.0084 This work 8.66 MESA-1.012 0.998 8.64 sun 0.0081 R/R 1.0-10 M +R 8.62 dy 0.998-Thick 0.0078

log (g) 8.6 0.976 Spec Sirius B 8.58 1.0-04 0.0075 0.949 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 M/M 8.56 sun Figure 11. Same as figure 9 but for Sirius B and an effective 8.54 temperature of Teff = 25 369 K. The points show the location of Sirius B the spectroscopic mass obtain in this work combined with the ra- 8.52 dius from Joyce et al. (2018b) (circle), the dynamical mass from 25600 25400 25200 25000 Bond et al. (2017a) (square) and the gravitational redshift mass T [K] eff from Joyce et al. (2018b) considering parallax from Hipparcos (full triangle) and Gaia DR2 (open triangle). Figure 10. Location of Sirius B in the Teff − log g plane, using the spectroscopic determinations (Spec) and the dynamical mass combined with the radius (Mdy+R) from Bond et al. (2017a). The lines correspond to theoretical white dwarf sequences, that are scopic mass computed in this work combined with the radius color-coded in stellar mass: black lines for 0.949M⊙, red lines for from Joyce et al. (2018b). 0.976M⊙, blue lines for 0.998M⊙ and magenta lines for 1.024M⊙. As expected, the spectroscopic mass and the dynami- Solid lines correspond to canonical sequences. Thinner envelopes cal mass from Bond et al. (2017a) do not agree within the are depicted with different lines, being thinner as the log g in- uncertainties in 1 σ. However, the results from Joyce et al. creases. The violet lines correspond to the cooling tracks from (2018b) are compatible with our theoretical mass – radius Fontaine et al. (2001) for a stellar mass of 1.0M⊙ with thick (1.0- relation, within 1 σ for a hydrogen envelope with MH ∼ 04) and thin (1.0-10) hydrogen envelopes. Just for comparison, we −6 2 × 10 M∗. Note that, the mass and radius for Sirius B from included two tracks computed with the MESA code with stellar Joyce et al. (2018b) are also in agreement with the “thick” masses 1.012 and 1.019 M⊙ from Lauffer et al. (2018). The thick = −4 brown line labelled ”0.998-Thick” correspond to a cooling track envelope, with MH 10 M∗, sequences from (Fontaine et al. computed with LPCODE having thicker hydrogen envelopes than 2001). the canonical value (see text for details). A thicker envelope could perhaps be expected if Sirius B had accretion episodes after the residual nuclear burning . has turned off. Considering that the Sirius system is a vi- sual binary with a period of ∼ 50 yr (Bond et al. 2017a), this scenario can be disregarded. The accretion rate of hy- addition, the uncertainties presented by Bond et al. (2017a) −17 drogen from the interstellar medium is less than 10 M⊙/yr correspond to uncorrelated internal uncertainties of the fit- (Dupuis et al. 1993; Koester & Kepler 2015), too low to ting, even though the orbital parameters and stellar masses −4 build a thick hydrogen envelope of ∼ 10 M∗. In any case, are correlated. By computing the uncertainties using a sim- as it was shown in section 3.1, the increase of the hydro- ple error propagation statistics, we obtain an uncertainty gen content will trigger nuclear burning at the base of the ∼ 48% larger for the mass of Sirius B, implying that the −6 envelope, reducing the hydrogen mass to ∼ 10 M∗. quoted uncertainties could be underestimated. Davis et al. (2011) determined the structure parameters In Figure 11 we compare the observational parameters for Sirius A using photometry and spectroscopy combined for Sirius B with our theoretical models using the mass– with parallax, to be R = 1.7144 ± 0.009R⊙, T = 9845 ± 64 radius relation. The different lines correspond to theoret- K and L = 24.74 ± 0.70L⊙. Considering the uncertainties in ical mass–radius relations for an effective temperature of mass and metallicity, we estimate an age between 205 − 245 T ff = 25 369 K. The solid black line corresponds to the e Myr. With a cooling age of 115 ± 6 Myr, the stellar mass of sequences with the thickest hydrogen envelope allowed by +0.47 the progenitor of Sirius B is 5.11 M⊙, in agreement with single stellar evolution, computed with LPCODE, while the −0.28 the value obtained by Bond et al. (2017a) and Liebert et al. solid magenta line correspond to the thinnest envelope, with −9.33 (2005). MH ∼ 10 M∗. We also show the theoretical mass–radius relation from Fontaine et al. (2001) with hydrogen envelope −4 mass M ∼ 10 M∗ as a dashed line. We include the grav- H 5.1.3 Binaries with non-DA white dwarf components itational redshift mass from Joyce et al. (2018b) obtained using parallaxes from Hiparcos (full triangle) and Gaia DR2 In this section we briefly consider two non-DA white dwarfs (open triangle), from Bond et al. (2017a) (full square) and in binary systems: Procyon B and Stein 2051 B. Procyon from Barstow et al. (2005) (red diamond). With a black cir- B is a DQZ white dwarf with an effective temperature of cle, we show the result obtained by considering the spectro- 7740 ± 50 K (Provencal et al. 2002) in a binary system with

MNRAS 000, 1–14 (2015) The mass-radius relation 11

0.014 the eﬀective temperature of the white dwarf component is H-Thick H-Thin determine using spectroscopy. We selected the objects with He stellar masses larger than ∼ 0.5M⊙, that are covered by our 0.013 Procyon B Stein2051 B model grid. The selected sample is depicted in ﬁgure 13, were we compare the observations extracted from Parsons et al. 0.012 (2017) to the theoretical mass–radius relation. The solid

sun (dashed) lines correspond to the theoretical mass–radius re- − R/R 10 lations for canonical (MH/M⊙ = 10 ) models for effective 0.011 temperatures from 60 000 K to 10 000 K, from top to bottom, in steps of 10 000 K (see figure for details). From this figure we see a very good agreement between models and 0.01 observations, being also consistent in effective temperature. T = 7450 K eff Next we proceed to measure the hydrogen content in the 0.009 selected sample of eleven white dwarfs. The sample is listed 0.5 0.55 0.6 0.65 0.7 0.75 0.8 M/Msun in table 4, along with the effective temperature, stellar mass and radius extracted from Parsons et al. (2017). For each Figure 12. Same as figure 9 but for the non-DA white dwarfs object we compare the observed mass and radius with our Procyon B and Stein 2051 B, where the observations are those theoretical mass–radius relation, considering different thick- from Bond et al. (2015) and Sahu et al. (2017), respectively. The ness of the hydrogen envelope. From the selected sample, curves are the mass–radius relation for an effective tempera- only five objects, GK Vir, NN Ser, J0138-0016, J0121+1744 ture Teff = 7450 K, for thick (solid balck line) and thin (MH = −9 10 M∗, red point-dashed line) and DB white dwarf models from and J1123-1155, show uncertainties small enough to mea- Althaus et al. (2009) (green dashed line). sure the hydrogen envelope mass, within our model grid. The remaining objects are consistent with the theoretical mass–radius relation but the uncertainties are too large to a slightly evolved subgiant of spectral type F5 IV-V. The constrain the mass of the hydrogen content. The results for Procyon system was analysed by Bond et al. (2015) using the five objects are depicted in figure 14, while the values precise relative astrometry for HST observations combined for the hydrogen envelopes are listed in the last column of with ground base observations and parallax. For Procyon B, table 4. From figure 14 it shows that all five objects have the dynamical mass resulted in M∗ = 0.593±0.006M⊙ and the a canonical hydrogen envelope, i,e., the maximum amount radius, determined using flux and parallax measurements, of hydrogen as predicted by stellar evolution theory. This is was 0.01232 ± 0.00032R⊙. expected given the mass range of the objects, for which the Stein 2051 B, is a DC white dwarf with Teff = 7122 ± hydrogen envelope is intrinsically thicker. Also, note that the 181 K, in a binary system with a main sequence companion larger differences between the theoretical mass–radius rela- of spectral type M4. The stellar mass was determined by tions for different hydrogen envelope mass occurs for low Sahu et al. (2017) using astrometric microlensing, being M = stellar masses and higher effective temperatures, as shown 0.75±0.051M⊙, while the radius of R∗ = 0.0114±0.0004R⊙ was in figure 6. determine using photometry and parallax measurements. Figure 12 shows the position of Procyon B and Stein 2051 B as compared to the theoretical mass–radius relations 6 CONCLUSIONS for Teff = 7450 K. The red point-dashed line corresponds = −9 to thin hydrogen envelope models, with MH 10 M∗ while In this work we studied the mass–radius relation for white the green dashed line corresponds to DB white dwarf mod- dwarf stars and its dependence with the hydrogen envelope els from Althaus et al. (2009). We also included the mass- mass. In particular, how the extension of the hydrogen en- LP- radius relation for thick envelope models computed with velope affects the radius, and the surface gravity, which di- CODE , those with the thickest hydrogen envelope allowed by rectly impacts the calculation of the stellar mass using at- stellar evolution. Considering the uncertainties reported by mospheric parameters, i.e., the spectroscopic mass. Bond et al. (2015), Procyon B is in very good agreement We find that, comparing the sequences with the thickest with our theoretical models for thin H-envelope, as it was envelope with those having the thinnest hydrogen envelope found by Bond et al. (2015, 2017b), but also is in agree- in our model grid, the reduction in the radius is around 8- ment with the theoretical mass–radius relation for DB white 12%, 5-8% and 1-2% for stellar masses of 0.493 M⊙, 0.609 dwarfs. The results for Stein 2051 B are not that conclusive M⊙ and 0.998 M⊙, respectively. As expected the differences since the uncertainties are too large, but are still consistent are larger for models with lower stellar mass since, for these with the theoretical models. objects, the maximum hydrogen envelope left on top of a white dwarf star is thicker, according to single stellar evolution theory. The reduction of the stellar radius translates 5.2 Eclipsing binaries directly into an increase in log g for a fixed stellar mass, Parsons et al. (2017) presented mass and radius determina- that can reach up to 0.11 dex, for low mass and high effec- tions for 16 white dwarfs in detached eclipsing binaries with tive temperatures. Considering that the mean uncertainty in low mass main sequence stars companions and combined log g is 0.038 dex, then it is possible to measure the hydrogen them with 10 previous measurements to test the theoretical envelope mass. mass–radius relation. The mass and radius are estimated In addition, the maximum hydrogen mass allowed by from the eclipses and radial velocity measurements, while stellar evolution theory is mass dependent, being thinner

MNRAS 000, 1–14 (2015) 12 Romero et al.

60000 0.024 50000 40000 30000 0.022 20000 10000 CSS 21357 0.02 GK Vir NN Ser 0.018 QS Vir J0024+1745

sun J0138-0016 0.016 J0314+0206 R/R J0121+1744 J1123-1155 0.014 J1307+2156 V471 Tau 0.012

0.01

0.008

0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 M/Msun

Figure 13. The white dwarf Mass-Radius relation. Solid lines correspond to thick envelope models, while dashed lines correspond to −10 sequences with MH /M∗ = 10 . From top to bottom, the solid lines are color coded, corresponding to eﬀective temperatures of 60 000 K (black), 50 000 K (blue), 40 000 K (red), 30 000 K (green), 20 000 K (violet), 10 000 K (orange). The data point are the sample extracted from Parsons et al. (2017) and indicated with diﬀerent symbols that are color coded as well.

Table 4. White dwarf stars in detached eclipsing binaries analysed in ﬁgure 14. For each object we list the eﬀective temperature, stellar mass and radius (columns 2, 3 and 4) from Parsons et al. (2017) and the hydrogen envelope mass determined in this work.

obj ID Teﬀ (K) M(M⊙) R(R⊙) MH /M∗ CSS 21357 15909 ± 285 0.6579 ± 0.0097 0.01221 ± 0.00046 5.8 × 10−5 − 4.6 × 10−10 GK Vir 50000 ± 673 0.5618 ± 0.0142 0.01700 ± 0.00030 (1.67 ± 0.30) × 10−4 NN Ser 63000 ± 3000 0.5354 ± 0.0117 0.02080 ± 0.00020 (2.18 ± 0.23) × 10−4 QS Vir 14220 ± 350 0.7816 ± 0.0130 0.01068 ± 0.00007 1.8 × 10−5 − 4.6 × 10−10 J0024+1745 8272 ± 580 0.5340 ± 0.0090 0.01398 ± 0.00070 2.2 × 10−4 − 2.6 × 10−10 J0138-0016 3570 ± 100 0.5290 ± 0.0100 0.01310 ± 0.00030 2.3 × 10−4 − 5.8 × 10−10 J0314+0206 46783 ± 7706 0.5967 ± 0.0088 0.01597 ± 0.00022 (1.25 ± 0.01) × 10−4 J0121+1744 10644 ± 1721 0.5338 ± 0.0038 0.01401 ± 0.00032 (2.21 ± 0.73) × 10−4 J1123-1155 10210 ± 87 0.6050 ± 0.0079 0.01278 ± 0.00037 (1.02 ± 0.10) × 10−4 J1307+2156 8500 ± 500 0.6098 ± 0.0031 0.01207 ± 0.00061 9.6 × 10−5 − 1.4 × 10−10 V471 Tau 34500 ± 1000 0.8400 ± 0.0500 0.01070 ± 0.00070 9.8 × 10−6 − 4.8 × 10−10

−4 than MH = 10 M∗, for white dwarf masses larger than 0.6M⊙, sure the mass of the hydrogen envelope. We analyse a sam- and thicker for masses below 0.6M⊙. Thus, considering a ple of white dwarf in astrometric and eclipsing binaries, for −4 hydrogen envelope of MH = 10 M∗ for all stellar masses, which it is possible to determine the mass and radius inde- can lead to overestimated or underestimated spectroscopic pendently of the theoretical models. We consider four white masses. dwarfs in astrometric binaries with very well determined or- The hydrogen envelope mass is the dominant factor in- bital parameters and a sample of eleven white dwarf stars fluencing the value of the radius. The central composition in detached eclipsing binary systems. Our main results are leads to less than 1% difference in the radius, as shown by the following. Tremblay et al. (2017). For the helium content, the radius • For 40 Eridani B, we find a spectroscopic mass of can be reduced by ∼ 1.1% if the helium mass is reduced by (0.573 ± 0.0011)M⊙ and a hydrogen envelope mass of MH ∼ more than a factor of 10. −8 2.63 × 10 M⊙. This result is in agreement with previous de- We also use the mass–radius relation as a tool to mea- terminations, pointing to a thin hydrogen envelope solution.

MNRAS 000, 1–14 (2015) The mass-radius relation 13

∼ Thick at least 50% or the difference is due to the fitting method 0.02 -5 and/or current atmospheric models. -6 • Observations of both non-DA white dwarfs, Procyon -7 B and Stein 2051 B, are consistent with a thin hydrogen −9 0.016 -9 envelope (MH . 10 ) as found by Bond et al. (2017b), but GK Vir also with the pure He theoretical models from Althaus et al. 0.012 (2009). • For a sample of 11 white dwarfs in detached eclipsing NN Ser binaries we found a good agreement between the theoretical 0.02 mass–radius relation and the observations. For five objects, in the low mass range (. 0.6M⊙), we measured the hydro- 0.016 gen mass and found thick hydrogen envelopes in all cases. For the remaining objects the uncertainties are too large to 0.012 constrain the hydrogen envelope mass, but the observations are in agreement with the theoretical mass–radius relation. J0121+1744 In general the mass–radius relation computed using our 0.014 models is in good agreement with the observations. For some sun objects we were able to constrain the hydrogen envelope mass given the lower uncertainties in the observed mass and R/R 0.012 radius. However, for most objects uncertainties are still too large. High mass white dwarf models show that these stars −6 0.01 are born with hydrogen envelopes of ∼ 10 M∗ or thinner. 0.018 J1123-1155 Thus the challenge of constrain the hydrogen mass is higher since the difference in radius is 1-2% within the hydrogen mass range allowed by our model grid. 0.015 Finally, we emphasize that the maximum hydrogen content left on top of a white dwarf is mass dependent, when 0.012 the evolution of the white dwarf progenitor in computed consistently. In particular, the hydrogen envelope is thinner J0314+0206 than the canonical value for stellar masses larger than 0.6M⊙ 0.014 and thicker for stellar masses below that value. Not taking into account this dependence can lead to a overestimation 0.012 of the stellar mass when the determination is based on spectroscopy, i.e., using the atmospheric parameters log g and effective temperature. 0.01 0.5 0.55 0.6 0.65 0.7 ACKNOWLEDGEMENTS M/Msun We thank our anonymous referee for the constructive com- Figure 14. Comparision with the observational values for mass and radius for five white dwarf in eclipsing binaries presented in ments and suggestions. ADR, SOK and GRL had finan- Parsons et al. (2017), with the theoretical mass-radius relations cial support from CNPq and PRONEX-FAPERGS/CNPq for different masses of the hydrogen envelope. Each panel corre- (Brazil). SRGJ acknowledges support from the Science and sponds to a single object. The hydrogen envelope for each curve Technology Facilities Council (STFC, UK). AHC had finan- is color-coded (see inset in the figure). cial support by AGENCIA through the Programa de Mod- ernización Tecnológica BID 1728/OC-AR, and by the PIP 112-200801-00940 grant from CONICET. Special thanks to Leandro Althaus for computing the model described in figure The cooling age for 40 Eridani B is ∼ 145 Myr, with a total 5 and for the very useful comments on the manuscript. This age of 2.68±0.43 Gyr and a progenitor mass of 1.53±0.11M⊙. research has made use of NASA Astrophysics Data System. • For Sirius B, we find a spectrocopic stellar mass of (0.974 ± 0.013M⊙ in agreement with previous determinations (Barstow et al. 2005). In addition, the gravitational redshift REFERENCES mass from Joyce et al. (2018b) as compared with our theoretical mass–radius relation, are in agreement within 1 σ, Alexander, D. R., & Ferguson, J. W. 1994, ApJ, 437, 879 −6 Althaus, L. G., Serenelli, A. M., Córsico, A. H., & Montgomery, considering a thick hydrogen envelope of MH = 2.02 × 10 . M. H. 2003, A&A, 404, 593 The cooling age for Sirius B is 115 ± 6 Myr, leading to a +0.47 Althaus, L. G., Serenelli, A. M., Panei, J. A., et al. 2005, A&A, 5.11 M⊙ stellar mass of the progenitor of −0.28 . As compared 435, 631 to the dynamical mass, the spectrocopic value is 4.3% lower Althaus, L. G., Panei, J. A., Miller Bertolami, M. M., et al. 2009, than that obtained by Bond et al. (2017a), and not compat- ApJ, 704, 1605 ible within the uncertainties. We conclude that, either the Althaus, L. G., Córsico, A. H., Bischoff-Kim, A., et al. 2010, ApJ, uncertainties in the dynamical mass are underestimated by 717, 897

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