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MNRAS000,1–8(2017) Preprint10January2017 CompiledusingMNRASLATEXstylefilev3.0 Constraining Effective Temperature, Mass and Radius of Hot White Dwarfs Elvis do A. Soares(cid:63) Instituto de F´ısica, Universidade Federal do Rio de Janeiro, C.P. 68528, Rio de Janeiro 21945-970, RJ, Brazil AcceptedXXX.ReceivedYYY;inoriginalformZZZ 7 1 ABSTRACT 0 Byintroducingasimplifiedtransportmodelofouterlayersofwhitedwarfswederivean 2 analytical semi-empirical relation which constrains effective temperature-mass-radius n forwhitedwarfs.Thisrelationisusedtoclassifyrecentdataofwhitedwarfsaccording a to their time evolution in non-accretion process of cooling. This formula permit us J to study the population map of white dwarfs in the central temperature and mass 9 plane, and discuss the relation with the ignition temperature for C-O material. Our effective temperature-mass-radius relation provide a quick method to estimate the ] mass of newly observed white dwarfs from their spectral measurements of effective R temperature and superficial gravity. S . Key words: stars:evolution–stars:atmospheres–whitedwarfs–radiativetransfer h p - o r t 1 INTRODUCTION 10 000 K) white dwarfs (WDs) has increased enormously s a mainly due to the Sloan Digital Sky Survey (SDSS; Eisen- White dwarfs (WD) are frequently treated in the zero tem- [ steinetal.2006).Follow-uphighqualityground-basedspec- perature approximation of highly degenerate electron gas troscopy of survey objects yield large samples of hot WDs 1 pressure,whichprovideanadequatedescriptionofthestel- withprecisemeasurementsofeffectivetemperaturesandsu- v lar structure as a whole. On the other hand, in the area perficial gravities. More than 10,000 spectroscopically iden- 5 closetothesurfaceweknowthattheobservedwhitedwarfs tified white dwarfs with determined effective temperatures 9 haveconsiderablehigheffectivetemperatures(T )ranging 2 from5,000Ktoover100,000K,losingtheirtheremffalenergy (Teff) and superficial gravities (logg) have been detected to 2 date(Kepleretal.2007;Kleinmanetal.2013),givingusthe emittingradiation(e.g.,Koester&Chanmugam1990).The 0 opportunity to explore the white dwarf mass distribution, externallayers,althoughtheamountofmasscontainedthere . which ultimately provides insights into mass-loss processes 1 issmall,determinethethermalevolutionofthewholewhite duringstellarevolutionandthemassbudgetoftheGalaxy. 0 dwarfs, while the bulk degenerate electrons keep the star’s 7 Beyond that, from the astrophysical point of view, hot core essentially isothermal due to its high thermal conduc- 1 WDs are important: (i) to elucidate the evolutionary links tivityandhotterthanthecrust.Theradiativeopacityinthe : between WDs and their pre-white dwarf progenitors, i.e., v outermost layers prevents the white dwarf to cool quickly. whethertheyarefromtheasymptoticgiantbranch(AGB), i Comparison between observations and models requires the X the extended horizontal branch, stellar mergers, or binary corrections from the finite temperature effects to the white r dwarf stellar structure. evolution. (ii) to understand their roles in the process of a chemical evolution of the Galaxy, because white dwarf pro- Many works have been developed, since the discovery genitors lose their outer layers which are carbon, nitrogen, ofthemaximummassofidealwhitedwarfs(WD)byChan- and oxygen rich at the top of the asymptotic giant branch drasekhar (1931), in the field of finite temperature correc- (AGB) (iii) to improve our knowledge of type Ia supernova tions to the degenerate equation of state (EoS) (e.g., Mar- events, with important underlying implications for cosmol- shak 1940; Hubbard & Wagner 1970; de Carvalho et al. ogy (e.g., Hillebrandt et al. 2013). 2014).However,ascommentedinBoshkayevetal.(2016),a systematicanalysisusingempiricalmass-radiusrelationsob- ThemaingoalofthisworkistousethedatafromSloan tainedfromthespectroscopicorphotometricmeasurements Digital Sky Survey (SDSS) Data Release 7 to constrain the ofmassesandradiiisstillneededtounderstandtheprecise effective temperature, mass and radius of hot white dwarfs structure and the dynamics of time evolution of WDs. inasimpleanalyticalrelation,introducingaverysimplified Moreover, the total number of observed hot (T > thermaltransportmodeloftheouterlayers.Thepresentpa- eff perisorganizedasfollows.Firstweintroducetheextended model for outer radiative layers in section 2. Based on this (cid:63) E-mail:[email protected] model, we derive an analytic relation among the effective (cid:13)c 2017TheAuthors 2 E. do A. Soares temperature, mass and radius containing two parameters, idealgasEoSandtheKramersopacity,κ=κ ρT−3.5.With 0 which related to the the transport properties of the outer this we have layers. In section 3 by using the SDSS-DR7 data, we de- (cid:18) 2 4ac4πGM µ (cid:19) termine these two parameters as function of WD’s mass. In ρ2 = T6.5, (4) 8.5 3 κ L N k 0 A B section4usingthusobtainedsemi-empiricalmass-radiusrelation for WD for each effective temperatures, we discuss integrating the Eq.(3) from P =0 when T =0. The Eq.(4) theimplicationsoftheseconstraintsforcentraltemperature isawell-knownresult,ascanbeseeninShapiro&Teukolsky andignitionofthenuclearmaterialinsideWDinsection5. (1983). Insection6weuse our results toestimate masses andradii Using the Eq.(4), which is valid for any r inside the for others observed hot white dwarfs, as in the Gaia DR1. outerlayer,wecaneliminateρfromEq.(1)andintegratethis In section 7, we discuss our results from physical point of equationfromaneffectiveradius(regionwherethephotons view and perspectives for the further study. decouplefromsurface)wherethetemperatureistheeffective temperature to the external radius of the WD where the temperature is zero, we get (cid:18) (cid:19) (cid:18) (cid:19) 1 µ GM R 2 EXTENDED MODEL FOR OUTER T = −1 . (5) eff 4.25N k R R RADIATIVE LAYERS OF WHITE DWARFS A B eff It is reasonable to assume that the effective radius can ThehighlydegeneratedelectrongasinsideaWDprovidesa berelatedwiththeChandrasekharradius,R =ξR (M), eff ch highthermalconductivityasaresultofthelargemeanfree withξ∼1,sincetheeffectiveradiusshouldbeverycloseto pathofthedegenerateelectronsinthefilledFermisea(e.g., the core surface. Then Weissetal.2004).Suchhighthermalconductivitytogether withthelackofnuclearreactionsdonotallowlargetemper- (cid:18) M (cid:19)(cid:18) R (cid:19)−1(cid:20)(cid:18) R (cid:19) (cid:21) T =(588,862 K)µ −1 , aturegradients,leadingtoanalmostuniformtemperaturein eff M R ξR (M) (cid:12) ⊕ ch theWDinterior.Ontheotherhand,inthedomaincloseto (6) its surface, the density ρ decreases and the matter becomes quickly non-degenerate. Then, the dominant heat transfer whereµandξareparameterstobedetermined.Forsimplic- is the radiative one (and a little convection), and the heat itywefurtherusetheanalyticalapproximatedexpressionfor conductionbecomesmuchsmallerifcomparedtothedegen- theradiusofidealwhitedwarfsRch(M)givenbyNauenberg erate electron gas. Therefore, we expect that the structure (1972) as of a WD can be modeled as an isothermal core covered by non-degenerate surface layers which isolates the degenerate 2.45354 (cid:18) M (cid:19)−1/3(cid:34) (cid:18) M (cid:19)4/3(cid:35)1/2 R (M)= R 1− core from the outer space (e.g., Kippenhahn et al. 2012). ch µ ⊕ M M e ch ch To exploit the above image, let us introduce a simple (7) approach to describe the energy transfer mechanism in the outer layers of a white dwarf. The main simplification con- with M = 5.816 M /µ2 and µ is the mean molecular ch (cid:12) e e sistsinattributingtheoutermostlayersastheregionrespon- weight per electron. sible by the thermal regulation of the white dwarf and the TheEq.(6)determinestheeffectivetemperaturesofthe core responsible for the mechanical regulation of the stellar white dwarf stars as a function of their masses and their structure, i.e., the core is in a hydrostatic equilibrium and radii. This relation is semi-empirical because the correction the outer layers is in a stationary state of radiative energy parameters µ and ξ are fitted by data, but it is physically transfer. The outer layer region starts where the degener- based on the model of transport phenomena in the outer atematterbecomesnon-degenerate(idealgas)matter.The layers of the white dwarf stars. thermal gradient and the hydrostatic equilibrium are main- tained by 3 SEMI-EMPIRICAL RELATION AMONG dT 3 κρ L =− r (1) MASS, RADIUS AND EFFECTIVE dr 4acT34πr2 TEMPERATURE and The hot WD are composed mainly by carbon (C) and oxy- dP GM =− rρ, (2) gen(O),buttheirobservedspectrashowusthattheiratmo- dr r2 sphere is dominated by hydrogen (H) or helium (He), with whereM =(cid:82)rρ4πr2dr andL =(cid:82)r(cid:15)4πr2dr.Dividingone the dominant element almost thousand times more abun- r 0 r 0 equation by the other, we can write dant than the other elements in that location. For these dP 16πacG1 (cid:18)M (cid:19)(cid:18) L (cid:19)M reasons,weexpectthenµe =2forthecoreofbothDA-WD = r T3 (3) and DB-WD due to their internal composition, but the pa- dT 3 κ M L L r rametersµandξofDA-WDandDB-WDmustbedifferent. whereM andLarerespectivelythemassandluminosityof thestar.Thisequationcanbeintegratedwiththehypothesis 3.1 Data Set and Procedure thattheouterlayersaretoothintocontributetothemass, i.e., M ≈ M and there is no energy generation ((cid:15) = 0) in Using the data from the Sloan Digital Sky Survey (SDSS) r these layers, i.e., L ≈L. By supposition, the material here DataRelease7,Kleinmanetal.(2013)reported42,154spec- r is a non-degenerate andfully ionizedgas,sowe canuse the troscopically confirmed white dwarf stars. From the 14,120 MNRAS000,1–8(2017) Constraining Effective Temperature, Mass and Radius of Hot White Dwarfs 3 clean DAs classified by the spectra, we use the 2,216 stars 1.5 ionized4He with S/N ≥ 15 and T ≥ 13,000 K. Of the 923 stars eff ) which they classified as clean DBs, we use the 140 stars M 1 ( with S/N ≥15 and T ≥16,000 K. µ ThemassesoftheeffidentifiedcleanDAandDBstarsare 0.5 ionized1H calculated from the effective temperature T and superfi- eff 1.1 cial gravity g values obtained by spectra. These relations are based on full evolutionary calculation of hydrogen-rich M) 1 Reff =Rch DA white dwarfs and hydrogen-deficient DB white dwarfs, ξ( as discussed in that paper. These evolutionary sequences 0.9 constituteacompleteandhomogeneousgridofwhitedwarf 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 models that captures the core features of progenitor evolution,inparticulartheinternalchemicalstructuresexpected Mass (solar mass) in the different types of white dwarf stars. Thereforewewillnotconsiderthedatauncertaintiesto Figure1.ParametersµandξasafunctionofmassforDA-WD. fit the parameters µ and ξ, assuming that such cannot be Thetoppanelshowstworeferencesforµasdottedlines,apure divided in systematical uncertainties from the stellar evo- ionizedHegasandapureionizedHgas.Inthebottompanelwe lution simulations, and observational uncertainties coming representthecasewhereReff=Rch asdottedline. fromthespectra.Ofcourse,thisproceduremustberevised in the future when the systematical uncertainties were rec- ognized. 99 Then,toadjusttheparametersµandξwechooseanar- 0.85 0.7 0.6 0.55 M =0.5 ) M rowrangeofmassesaroundameanvalueM withawidthof K 88 (cid:12) 0.001M(cid:12) sufficienttodeterminetheparametersforasingle 40 0.475 value of mass, because white dwarf stars with same mass 1 77 ( must evolve similarly. e ur 66 t a 0.45 er 55 3.2 DA white dwarfs mp 0.425 0.4 e 44 Forthehydrogen-richDAwhitedwarfs,themainbehaviorof T thecoefficientsµ(M)andξ(M)canbefittedbythesimplest ve 33 forms cti e ff 22  E 0.48(0.02) M/M(cid:12) <0.448 11 µ(M)= 4.2(0.2)MM −1.4(0.1) 0.448≤M/M(cid:12) ≤0.503 00..55 11 11..55 22 22..55 33 33..55 44 44..55 55 0.78(0.01)(cid:12)MM +0.32(0.01) M/M(cid:12) >0.503 Radius (earth radius) (cid:12) (8) and Figure 2. Semi-empirical radius-effective temperature relation M for DA-WD with different masses. The orange circles represent ξ(M)=0.984(0.002)−0.021(0.003) , (9) some values of mass and the blue lines are their correspondent M (cid:12) fits.ThelightgraycirclesaretheavailabledataforDA-WDfrom according with the Figure 1, and the uncertainties are rep- theSDSS-DR7. resented in the parentheses. A transition from a pure hydrogen composition for a 0.4 M . Bellow this mass value, the parameters cannot be hydrogen-heliummixtureintheouterlayersispresentedby (cid:12) adjusted because the low statistics of the data. the parameter µ(M) in the top panel of Fig.1. This outer layer composition transition is marked by an core composition transition. In fact, WD with mass below 0.452 M(cid:12) are 3.3 DB white dwarfs helium-core white dwarf stars (Althaus et al. 2009a) and For the hydrogen-deficient DB white dwarfs, the main be- WD with mass above 0.452 M are carbon-oxygen white (cid:12) havior of the coefficients µ(M) and ξ(M) can be fitted by dwarf stars (Althaus et al. 2005). the simplest forms TheFigure2illustratestheradius-effectivetemperature of DA-WD with different masses using the Eq.(6) and the µ(M)=1.25(0.03)−0.59(0.05) M , (10) parameters µ(M) and ξ(M) given by Eqs.(8) and (9) for M(cid:12) several values of WD masses (blue lines). The orange cir- clesrepresentthecorrespondentdataforthesemassvalues. ξ(M)=0.92(0.03)+0.02(0.05) M , (11) The light gray circles are the available data for DA-WD in M (cid:12) the SDSS-DR7. Our analytic lines are not displayed for all according with the Figure 3. data but there is a excellent agreement between the semi- For hydrogen-deficient white dwarf stars with stellar empirical relation and the DA-WD data for masses above mass values from 0.515 to 0.870 M (Althaus et al. 2009b) (cid:12) MNRAS000,1–8(2017) 4 E. do A. Soares 1.5 ionized4He 4 SEMI-EMPIRICAL MASS-RADIUS RELATION ) M 1 ( Themass-radiusrelationisafundamentalingredienttoun- µ 0.5 ionized1H derstand the physics of white dwarfs. The first mass-radius relationgivenbyHamada&Salpeter(1961)assumedazero 1.1 temperature fully degenerate core. Finite temperature cor- M) 1 Reff =Rch rectionstoCandOnuclearmaterialandthenon-degenerate ( outer layers of He and H were included by Althaus et al. ξ 0.9 (2010).Recently,Holbergetal.(2012)constraintthedegen- erate mass-radius relation with the observations, but there 0.5 0.6 0.7 0.8 is a doubt about the favored models to estimate the mass andradiusofWD,using”thick”Henvelopesor”thin”Hen- Mass (solar mass) velopes.Therefore,themass-radiusrelationofwhitedwarfs is not greatly constrained by observations. Figure3.ParametersµandξasafunctionofmassforDB-WD. OurconstrainingrelationEq.(6)canbeinvertedtogive Thetoppanelshowstworeferencesforµasdottedlines,apure ionizedHegasandapureionizedHgas.Inthebottompanelwe representthecasewhereReff=Rch asdottedline. (cid:34) 1 (cid:18)T (cid:19)(cid:18) M (cid:19)−1(cid:18)ξR (M)(cid:19)(cid:35)−1 R=ξR (M) 1− eff ch , ch µ T M R 0 (cid:12) ⊕ (12) which provides a simple analytical mass-radius relation of K) 5 0.820.740.650.590.520.47 white dwarfs, with T = 588,862 K. The knowledge about 0 40 M =0.30 the parameters µ and ξ are the only physical ingredients to 1 M be added from the observations. ( (cid:12) e 4 For DA-WD we can use the parameters µ and ξ found r tu in Section 3.2. The Figure 5 represents this mass-radius re- a r lationofDA-WDwithdifferenteffectivetemperatures.The e p 3 data are represented in narrow ranges of effective tempera- m e turewithwidthof1000K,symbolizedastheorangecircles. T The blue lines are the correspondent mass-radius relation e tiv 2 obtained from Eq.(6), where T4 = T/(104 K). The Chan- c drasekhar mass-radius relation for CO ideal white dwarf e ff E is represented for comparison. We can note an ideal white dwarf behavior to white dwarf with mass above 1 M . 1 (cid:12) 1 1.5 2 2.5 3 ThisbehaviorcomesfromthedenominatorinEq.(12)when the temperature parcel becomes smaller than the Chan- Radius (earth radius) drasekharradiusparcel.Amass-radiusrelationforDBwhite dwarfs can be obtained similarly, getting the parameters of Section 3.3, and the result is presented in Figure 6. Figure 4. Semi-empirical radius-effective temperature relation A recent paper from Tremblay et al. (2016) reports a forDB-WDwithdifferentmasses.Theredcirclesrepresentsome sample of white dwarf parallaxes, including 4 directly ob- values of mass and the green lines are their correspondent fits. ThelightgraycirclesaretheavailabledataforDB-WDfromthe served DA-WD and other wide binaries WD. This data set SDSS-DR7. can be combined with spectroscopic atmospheric parameters, as effective temperature and superficial gravity, to study the mass-radius relationship. Using the data from that paper, we can reproduce the there is not core composition transition, and consequently estimatedmassesandradiifromtheGaia-DR1andcompare theparameterµ(M)variesslowlyinthisrangeofmassval- with our semi-empirical mass-radius relation, as illustrated ues. in Fig.7. The purple solid circles are the directly observed Alike the case of DA-WD, the Figure 4 illustrates DA-WD,alsoidentifiedinTable3,andtheyellowopencir- the radius-effective temperature of DB-WD with different clesarethewidebinariesDA-WD.Oursemi-empiricalmass- massesusingtheEq.(6)andtheparametersµ(M)andξ(M) radius relation is represented as the blue lines for different given in this section. The red circles represent some mass effectivetemperatures.Sinceourmass-radiusrelationisob- values for which the semi-empirical relation is calculated, tained from the SDSS-DR7 and agreed very well with the representedbythegreenlines.Thelightgraycirclesarethe directly observed DA-WD from Gaia, we suggest that the availabledataforDB-WDintheSDSS-DR7.Thereisagain wide binaries data be reviewed in future analysis. a good agreement between the semi-empirical relation and There is a observed sample of eclipsing white-dwarfs the DB-WD data despite the low statistics. Note that the wherethederivationofbothmassandradiusisindependent. very isolated point for M =0.3M is stay also well on our In Table 1, we calculate the radius R (the last column) for (cid:12) curve. the given T and M = M which is to be compared eff eclipse MNRAS000,1–8(2017) Constraining Effective Temperature, Mass and Radius of Hot White Dwarfs 5 5 3 1.3 3.0 6.0 T4=9.0 1.3 3.0 6.0 T4=9.0 ) 4 ) us us 2.5 di di a a r 3 r h h rt rt 2 a a e e Rch ( 2 ( us Rch us di di 1.5 a a R 1 R 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0.2 0.4 0.6 0.8 1 1.2 Mass (solar mass) Mass (solar mass) Figure 5.Semi-empiricalmass-radiusrelationforDA-WDwith Figure 7.Semi-empiricalmass-radiusrelationforDA-WDwith differenteffectivetemperatures.Theorangecirclesrepresentsome differenteffectivetemperatures(bluelines).Thepurplesolidcir- values of effective temperature and the blue lines are their cor- cles correspond to the directly observed DA-WD and the yellow respondent fits. The light gray circles are the available data for open circles are the DA-WD observed in wide binaries from the DA-WDfromtheSDSS-DR7.Thedashedlinecorrespondstothe Gaia DR1. The dashed line is the Chandrasekhar mass-radius Chandrasekharmodelforidealwhitedwarfsstars. relationforidealWD. given by 5 T (M) (cid:18) M (cid:19)(cid:18)ξR (M)(cid:19)−1 lim =µ ch . (13) 1.63.0 T4=4.5 T M R 0 (cid:12) ⊕ s) 4 Using this definition for the limiting temperature, the u di radiusofthehotwhitedwarfiswrittenasR∝(Tlim−T)−1. ra 3 InFig.8,weshowTlim asfunctionofmassM.Thelight h gray circles are the effective temperature distribution as a t ar function of mass. The blue region represents the forbidden e ( 2 regionfortheeffectivetemperatureofhotDA-WD.Infact, us Rch there is not a single point in the bulk of the region, which adi indicatesthatthistemperaturelimitexhibitsaphysicalbe- R 1 havior of data, although be a mathematical limit of our model. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 5 CENTRAL TEMPERATURE AND Mass (solar mass) NUCLEAR IGNITION Although astronomical observables are stellar atmospheric Figure 6.Semi-empiricalmass-radiusrelationforDB-WDwith quantities,thestellarinteriorquantitiesareofgreatimpor- different effective temperatures. The red circles represent some tancetoastrophysics.Forinstance,thecentraltemperature values of effective temperature and the green lines are their cor- andthecentraldensitydeterminethechemicalevolutionof respondent fits. The light gray circles are the available data for the star and its nuclear energy generation. DB-WDfromtheSDSS-DR7.Thedashedlinecorrespondstothe Therelationbetweeneffective temperatureandcentral Chandrasekharmodelforidealwhitedwarfsstars. temperatureisgivenbyKoester(1976)intheapproximated form to the observed radius R (the fourth column) . We T4 eclipse eff =2.05×10−10T2.56 (14) find that our semi-empirical mass-radius relation is in good g c agreement with observations. wheregisthesuperficialgravity.Thisrelationwasobtained by fitting the data from simulations and give us a good estimate of the central temperature inside WD stars. 4.1 Effective Temperature Limit We can estimate the central temperature of WD-data ThestarradiusR,fromEq.(12),becomesinfinitewhenthe from SDSS-DR7, using their radii, mass and effective tem- effective temperature is equal to the limiting value, T , perature.InFig.9werepresentthesedataasthegraycircles. lim MNRAS000,1–8(2017) 6 E. do A. Soares Table 1.TheestimatedradiifortheobservedDAwhitedwarfstarsfromEclipsingBinaries. Name Teff (K) Meclipse (M(cid:12)) Reclipse (R⊕) R(R⊕) CSS41177A 22,500(60) 0.378(0.023) 2.425(0.045) 2.64(0.21) NNSer 63,000(3000) 0.535(0.012) 2.27(0.02) 2.31(0.12) SDSS0857+0342 37,400(400) 0.514(0.049) 2.69(0.09) 2.11(0.48) SDSSJ1212-0123 17,710(40) 0.439(0.050) 1.83(0.01) 2.07(0.28) GKVir 50,000(670) 0.562(0.014) 1.85(0.03) 1.91(0.07) QSVir 14,220(350) 0.781(0.013) 1.165(0.008) 1.14(0.02) V471Tau 34,500(1000) 0.840(0.050) 1.17(0.08) 1.10(0.08) 4K) 1100 vin) εO+O=εν MM(cid:12) =0.4 0.45 0.475 0.50.55 0 el 1e( 88 (K 109 εC+C=εν 00..67 r e tu ur 0.8 a t er 66 ra mp pe 108 m e T e T 44 e tiv ral c t e n 107 Eff 22 Ce 00..22 00..44 00..66 00..88 11 11..22 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Mass (solar mass) Effective Temperature (105 Kelvin) Figure 8. Effective temperature limit as a function of mass for Figure9.Centraltemperatureasafunctionoftheeffectivetem- DA-WD(blueline).Theblueregionrepresentstheforbiddenre- peratureforDA-WDwithdifferentmasses(bluelines).Thegray gion of the effective temperature to the correspondent mass of circlesrepresentthecentraltemperaturetoeachDA-WDstarin DA-WD.ThelightgraycirclesaretheavailabledataforDA-WD the available data from the SDSS-DR7, and the orange circles fromtheSDSS-DR7.Thedotlinecorrespondstothetemperature arethehighestcentraltemperatureforWDfromthesedata.The thresholdforidealwhitedwarfsstars. dashed lines are the ignition curves for carbon and oxygen fu- sions. The dotted line is our effective temperature threshold for hotwhitedwarfs. Using our semi-empirical mass-radius relation, we can obtain the central temperature of WD using just their effective temperatures and their masses, since their radii can gen ignition lines as the dashed lines, using the Eq.(14) to be calculated with this relation. The central temperature estimate the correspondent effective temperature. of DA-WDs as a function of their effective temperature is InTable2wepresentsixDA-WDwithcentraltemper- plotted for different mass values in Fig.9, as blue lines. ature (the last column) above 99% of the carbon ignition The ignition of the nuclear material inside the white temperature,representedbytheorangecirclesinFig.9.The dwarf is determined by the balance between nuclear energy star 200646.50-124410.9 is a special case because it is the generation rate and local heat losses. We consider the case only one WD above the carbon ignition line. The fact that where the heat losses are mainly caused by neutrino emis- thisWDdidnotbecomeasupernovacanbeunderstoodby sion,whichisappropriateinwhitedwarfs,e.g.,formodeling its internal composition, i.e., if there is more oxygen then Type Ia supernova events (Hillebrandt & Niemeyer 2000). carboninitscore,weexpectnoignitioninsidethisWD.We The ignition temperature for the material depends on the suggest that more careful observations and more detailed centraldensityofthestar.Weusethefittingformulaforig- simulations must be directed to the modeling of this WD nitiontemperatureforcarbonandoxygenfusionsasafunc- star. tion of mass density, the Eq.(A.1) in Potekhin & Chabrier (2012). The central densities for white dwarfs with different masses are calculated using the hydrostatic equilibrium 6 ESTIMATING MASSES equation with the EoS of degenerate electrons, i.e., hydro- staticequilibriumforidealwhitedwarfs.Thisassumptionis One of the great achievements in white dwarf research has enough a posteriori because the temperature corrections to beenthecapacitytomeasuretheeffectivetemperaturesand the EoS are important just above the ignition temperature superficial gravities. In particular, the spectroscopic tech- for oxygen. In Figure 9 we represent both carbon and oxy- niquedevelopedbyBergeronetal.(1992)foranalyzingthe MNRAS000,1–8(2017) Constraining Effective Temperature, Mass and Radius of Hot White Dwarfs 7 Table 2.ThehighestestimatedcentraltemperaturedeterminedfortheobservedDAwhitedwarfstarsfromSDSS-DR7. Name M (M(cid:12)) R(R⊕) Teff (K) logTc (K) 200646.50-124410.9 0.539(0.017) 3.959(0.546) 99,018(2529) 8.84(0.05) 091442.70+041455.9 0.538(0.037) 3.445(0.692) 85,714(5102) 8.70(0.08) 113303.70+290223.0 0.466(0.012) 4.375(0.577) 73,149(2867) 8.69(0.05) 102624.05+091554.8 0.573(0.021) 2.957(0.335) 92,989(3088) 8.68(0.05) 224653.73-094834.5 0.553(0.019) 3.077(0.313) 87,805(2600) 8.67(0.04) 080403.06+083030.8 0.526(0.041) 3.406(0.799) 82,219(6036) 8.67(0.10) Balmer line of hydrogen in (DA) white dwarfs has become Table 3. The estimated masses for the directly observed DA the standard method for measuring the effective tempera- whitedwarfstarsfromtheGaia-DR1. ture and surface gravity of these stars which represent 80% of the white dwarf population. In addition to being infre- Name Teff(K) logg(cm/s2) MGaia(M ) M (M ) quentthantheirhydrogen-lineDAcounterparts,thehotter (cid:12) (cid:12) 0232+035 66,950(1440) 7.40(0.07) 0.490(0.113) 0.518(0.013) DB stars are characterized by an optical spectrum where 1314+293 56,800(1250) 7.89(0.07) 0.516(0.096) 0.644(0.028) theneutralheliumtransitionsexhibitlittlesensitivitytoef- 1647+591 12,510(200) 8.34(0.05) 0.860(0.103) 0.807(0.031) fective temperature, as discussed in Bergeron et al. (2011). 2117+539 14,680(240) 7.91(0.05) 0.573(0.071) 0.561(0.025) The mass-radius relation is fundamental to compute white dwarf masses from these accurate measurements. Usingoursemi-empiricalmass-radiusrelation,Eq.(12), 7 DISCUSSION AND CONCLUSIONS we can obtain the superficial gravity g as a function of the Inthispaper,introducingaverysimplemodelfortheouter massM andtheeffectivetemperatureT ofthewhitedwarf eff layerofhotWDsweanalyzedtheSDSS-DR7andderiveda star according simple, analytic semi-phenomenological relation among effective temperature, mass and radius of hot white dwarfs, the Eq.(6). GM Fromthisrelation,weobservethattherearetwoessen- g(M,T )= , (15) eff R(M,T )2 tial differences between hydrogen-rich DA white dwarf and eff hydrogen-deficient DB white dwarf: their outer layer composition and their effective temperature range. AsdiscussedinSion(2011),theDA-WDaremucheas- which we can be numerically invert to obtain the mass of ier to classify because the Balmer lines of hydrogen across the WD as a function of T and logg. eff a wide range of effective temperature T , from 4,000 up to eff As a test case, the most recent measurements of Sirius 120,000 K and higher, whereas DB-WD exhibit He I lines B from Holberg et al. (1998) can be used to determine its but with a lower effective temperature range, from 12,000 mass. Sirius B is a hydrogen-rich DA-WD whose the effec- to 45,000 K. Confirming the different effective temperature tivetemperatureis24,790(100)Kandthesurfacegravityis ranges for DA and DB. logg = 8.57(0.06). Then, using the Eq.(15), we can obtain The difference in the outer layer composition was pre- themassvalueofM =0.960(0.035)M(cid:12).Thisresultisclose sented by the parameter µ of both DA and DB, and it in- to the refined estimates of the mass M = 1.034(0.026) M(cid:12) dicates that the parameter is closely related to the mean using other method of measurement, as the Hipparcos par- molecular weight of this region, whose the information give allax method. us clues about the chemical composition of the material. Another example would be the PG0948+534 reported The parameter ξ must be related to the Rosseland by Preval & Barstow (2016) as currently one of the hottest optical-depth mentioned in Baschek et al. (1991) and it de- DAwhitedwarfstars.Theauthorswereabletomeasurethe pends on the chemical composition of the material, due to Teff andthelogg forthisWD,findingTeff =110,000Kand theopacityofthematerial.Thefactofξ(cid:46)1showsthatthe logg=7.58.ForthecaseofPG0948+534,wefindthemass region responsible for the photon emission is essentially in valueofM =0.640M(cid:12),thatcorroboratesthehypothesisof the border of the core described by the degenerate electron DA hot white dwarfs with mass between 0.5−0.7 M(cid:12) are gas,suggestingthatasmallportionofthissurfaceismelted the hottest observed DA stars, as can be seen in Figure 9. into the outer layer. The best test is to compare our mass estimates with Our result permits us to obtain a mass-radius relation, direct mass measurements by independent methods, such theEq.(12),andestimatesofradiiforWDsforknownmass as those presented in Gaia-DR1 by Tremblay et al. (2016). andtemperaturewithothermethods.Furthermore,ourfor- As discussed in the Section 4, the data of directly observed mula exhibits a mathematical limit to the effective temper- DA-WD are in better agreement with our model than the ature, and curiously there is not a single white dwarf star dataofwidebinariesWD.InTable3,weestimatethemass inthebulkoftheforbiddenregionimposedbythislimit.A (the last column) using the atmospheric measurements of furtherstudytounderstandtheexistenceofsuchalimiting effective temperature and superficial gravity for these WD, temperature is required. byEq.(15),andcomparewiththeobservedmassM (the Thecentraltemperaturecanbeevaluatedusingthere- Gaia fourth column). lationbetweeneffective temperature andsuperficialgravity MNRAS000,1–8(2017) 8 E. do A. Soares derivedfromKoester(1976)usingnumericalmodelsofWD. Holberg J. B., Oswalt T. D., Barstow M. A., 2012, The Astro- ThedatafromSDSS-DR7presentsixDA-WDwithcentral nomicalJournal,143,68 temperaturesveryclosetothecarbonignitiontemperature. HubbardW.B.,WagnerR.L.,1970,TheAstrophysicalJournal, There is only one DA-WD with central temperature above 159,93 the ignition temperature. If our analytic expression reflects KeplerS. O., KleinmanS. J., Nitta A., Koester D.,Castanheira B. G., Giovannini O., Costa a. F. M., Althaus L. G., 2007, thephysicalsystematicscorrectly,wemaythinkofthepos- MonthlyNoticesoftheRoyalAstronomicalSociety,375,1315 sibility that the core of this WD is composed by oxygen Kippenhahn R., Weigert A., Weiss A., 2012, Stellar Structure instead of carbon. Numerical simulations and future obser- and Evolution, 2 edn. Astronomy and Astrophysics Library, vations are required for the better understanding whether Springer-VerlagBerlinHeidelberg such WD is a possible Type Ia supernova progenitor. Kleinman S. J., et al., 2013, The Astrophysical Journal Supple- The mass-radius relation obtained in this work allows mentSeries,204,5 ustoobtainmassestimatesfromatmosphericmeasurements KoesterD.,1976,AstronomyandAstrophysics,52,415 ofeffectivetemperatureandsuperficialgravity.Weusethis Koester D., Chanmugam G., 1990, Reports on Progress in method to estimate masses of the well known Sirius B and Physics,53,837 other DA-WD from the recent Gaia-DR1. Although they MarshakR.E.,1940,TheAstrophysicalJournal,92,321 NauenbergM.,1972,TheAstrophysicalJournal,175,417 are distinct methods, our mass evaluations are in good ac- Potekhin A. Y., Chabrier G., 2012, Astronomy & Astrophysics, cordance with the masses measured by Gaia, considering 538,A115 theiruncertainties.Thisresultconfirmourrelation,Eq.(6), PrevalS.P.,BarstowM.A.,2016,e-printarXiv:1610.01677,pp1– as a great constraining for effective temperature, mass and 6 radius of hot white dwarfs. ShapiroS.L.,TeukolskyS.A.,1983,BlackHoles,WhiteDwarfs and Neutron Stars: The Physics of Compact Objects, 1 edn. Wiley-VCH SionE.M.,2011,inHoardD.D.W.,ed.,,WhiteDwarfAtmo- ACKNOWLEDGEMENTS spheresandCircumstellarEnvironments.Wiley-VCHVerlag GmbH & Co. KGaA, Weinheim, Germany, Chapt. 1, pp 1– The author acknowledges the members of ICE group of the 23, doi:10.1002/9783527636570.ch1, http://dx.doi.org/10. Institute of Physics for fruitful discussions and comments. 1002/9783527636570.ch1 In particular, the author would like to thank Profs. T. Ko- TremblayP.E.,etal.,2016,e-printarXiv:1611.00629,pp1–13 damaandJ.R.T.deMelloNetoforreadingthemanuscript Weiss A., Hillebrandt W., Thomas H.-C., Ritter H., 2004, Cox andtheirsuggestions.Thisworkisfinanciallysupportedby andGiuli’sPrinciplesofStellarStructure,2edn.Advancesin CNPq. Astronomy&Astrophysics,CambridgeScientificPublishers de Carvalho S. M., Rotondo M., Rueda J. 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