194OAp J 92 . . 32IM 6 7 9 573 THE ASTROPHYSICALJOURNAL volume 92NOVEMBER1940number3 in theinteribr.Forthispurposeaccurateexpressionsforradiativeandconductive opacity arederived,bothforthenondegenerateenvelopeanddegeneratecoreof core theionizationisalmostcomplete(inenvelope,temperatureionization;in main partofthispaperisconcernedwiththecalculationtemperaturedistribution generacy theatomsareonlypartiallyionized,whereasbothinenvelopeand gated, anditisfoundthatinthetransitionregionbetweendegeneracynonde- by electronicconductioninthecore.Furthermore,freeelectrondensityisinvesti- stars. Itisshownthattheenergytransportchieflybyradiationinenvelopeand ing thechemicalcompositionofstaraswellpossibleerrorsinobserved if thetheoreticalvalueforradiusisassumed,honly8• 10cm.Thus,theenvelope most attentionispaidtothisstar.Theextensionofthenondegenerateenvelopeand the core,pressureionization). luminosity LandtheradiusR.ThedepthhofenvelopeSiriusBisabout6•10 B and40EridaniB;ofthetwoobservationalmaterialisbetterforSiriusB,sothat constitutes onlyaverysmallfractionofthetotalradius thestar(~10cm). consist exclusivelyof“Russellmixture.”Forapurehelium starhisevensmaller,and, the internaltemperaturearecalculatedforseveralwidelydifferentassumptionsregard- is almostindependentoftheassumedradius;moreover, degenerate coreispractical- cm, whentheobservedvaluesofR,LandmassMareusedstarisassumedto ly isothermal.Atthistemperatureandatthehighdensities of10to3•gm/cm low atCornellUniversity;preliminaryresultswerereported withH.A.Bethetothe which prevailinthecore,nuclearreactionsbetweenprotons andothernucleiwouldgo carbon andnitrogeninthesameabundanceasdomain-sequence stars,thehydrogen no appreciablenumberofprotonsispresentinsidewhite dwarfs. IfSiriusBcontains at averyrapidrate.Sincetheobservedluminosityissmall, wemustconcludethat American PhysicalSociety(cf.Phys.Rev.,55,681,1939,and 57,69,1940). o The centraltemperatureTofSiriusBis^15,000,000for pure Russellmixtureand © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem c The stateofmatterintheinteriorawhitedwarfstarisinvestigateddetail. Quantitative calculationsareperformedforthetwowell-knownwhitedwarfs,Sirius * Themajorpartofthisworkwasdonewhiletheauthor PresidentWhiteFel- AN INTERNATIONALREVIEWOFSPECTROSCOPYAND THE INTERNALTEMPERATUREOF WHITE DWARFSTARS* ASTRONOMICAL PHYSICS R. E.MARSHAK ABSTRACT 21 3 194OAp J 92 . . 32IM 6 8 -8 4 7 -3 6s _s 2 8 from main-sequencestarsofthesamemassintwofundamental itself theorypredictsthatthecarbon-cycleandproton-proton pose apureheliumstarwasinvestigatedforwhichrâî7•io°found;buteven other. Alowerreactionrateisobtainedforacentraltemperature;thispur- in main-sequencestarsbrighterthanthesun.For of twoprotonsismuchlessprobablethanfollowsfromwell-establishednucleartheory. preciably increasetheextentofnondegenerateenvelopeandthereforeradius is only5.7•10cm,ascomparedwithanobservedradiusof13.6cm.Thislarge cordance withChandrasekhar’stheoryofdegenerateconfigurations.Thetheoretical concentration Xcannotbemorethan2•io;andevenifCNarecompletely reaction giveaboutequalcontributionstotheluminosity. account fortheenergyevolutionoffainterstars.Forsun on theproton-protonreaction,usingFermiß-decaytheory, source ofenergymain-sequencestarshaveprovedquiteconclu- discrepancy isveryserious,sincetheobservationalvalueconsideredgoodandthere radius of40EridaniBagreeswellwiththeobservedradius,butSirius M, L,R,thattheenvelopehasadepthof3•10cm,centraltemperaturevalue at thistemperaturetheproton-protonreactionissofastthatXmustbelessthan10. Physical SocietyofLondon,1939. determined decayconstants,seemtoindicatethat’thisreactioncan sively thatthecarbon-cycleisresponsibleforenergyproduction dwarfs willbecomedarkobjects. of thewhitedwarfsisnotduetothermonuclearreactionsatall,butgravitationalcon- any lightnucleiotherthanheliumandhydrogenthat,moreover,thecombination fications ofthetheory(which,however,stillkeephydrogencontentsmall)willap- does notseemanywayoutfromthestandpointoftheory.Thus,noreasonablemodi- 30 •io°,thehydrogencontentanupperboundof8io~. absent, Xmustbelessthan2•io,becausetheprotonswillstillcombinewitheach 3 these circumstancestheradiusofstarisuniquelydeterminedbyitsmassinac- stars fainterthanthesunevidenceislessconclusive;calculations traction. Withthissourceofenergyitwilltakeatleast10yearsbeforethewhite the star.TheonlyfaintpossibilitywouldbetoassumethatSiriusBdoesnotcontain Gamow-Teller selectionrules,andtheaverageofasetempirically c H H H 123 2 a 3 1 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Recent investigations’*intothermonuclearprocessesasthe We must,therefore,concludethatnohydrogenispresentinwhitedwarfs.Under H.A.BetheandC.L.Critchfield,Phys.Rev.,54,248,1938. 3H. A.Bethe,Ap./.,92,118,1940. On theotherhand,itiswellknownthatwhitedwarfstarsdiffer H.A.BetheandR.E.Marshak,ReportsonProgressin Physics,6,1,London: H.A.Bethe,Phys.Rev.,55,434,1939. A furtherresultoftheseinvestigationsistheconclusionthatenergyproduction For 40EridaniBitturnsout,withtheassumptionofRussellmixtureandobserved 4 R.E.MarshakandH.A.Bethe,Phys.Rev.,56,210,1939. I. INTRODUCTION R. E.MARSHAK 194OAp J 92 . . 32IM 6 5 aspects :theyhaveextremelylowluminositiesandexceedinglyhigh been explainedintermsofthetheorydegeneratematter,andit production processotherthanthatoccurringinmain-sequencestars. has seemedlikelythattheirlowluminositiesareduetoanenergy- main-sequence stars.Thehighdensitiesinthewhitedwarfshave densities (asaresultoftheirverysmallradii),comparedwiththe moreover, todeterminethemodificationsintroducedintoChandra- marized insectionII.Similarly,thefree-electrondensity,pres- pressions havebeenderivedbytheauthor,andresultsaresum- way atallitwasnecessarytoobtainfairlyaccurateexpressionsfor finite temperatures,wehaveinvestigatedindetailthetemperature sekhar’s theoryofdegenerateconfigurationsbythepresence extreme degeneracybutalsoattheonsetofdegeneracy.Suchex- distribution intheinteriorofwhitedwarfs.Tomakeanyhead- the radiativeandconductiveopacitiesnotonlyunderconditionsof position, luminosity,andradius.InsectionVtheintegrationof face regionofthetwobest-knownwhitedwarfs,SiriusBand40 sure, andtheFermienergyarecalculatedfortransitionregion envelope arefoundunderdifferentassumptionsastochemicalcom- results giveninsectionIII. observational material,andsomeconclusions aredrawnregarding energy-production process.Theresultingtemperature-densitydis- Eridani B;boththetemperatureandextentofradiative white dwarfstars. equations ofstellarequilibriumiscarriedthroughtothecenter, and xi. tents andradiiofSiriusB40EridaniB. tribution isusedtodetermineupperboundsonthehydrogencon- the sourcesofenergy,radii,andevolutionary behaviorof quite independentofanyexplicitassumptionsconcerningthe (between theradiativeenvelopeanddegeneratecore), 5 6 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem In anattempttosettlethematterofenergyproductionand, In sectionIVthestarequationsareintegratedthroughsur- In thefinalsection(sec.VI)theoryiscomparedwith S.Chandrasekhar,IntroductiontotheTheoryofStellar Structure, esp.chaps,x R. E.Marshak,AnnalsofNewYorkAcademySciences, 1940. WHITE DWARFSTARS323 194OAp J 92 . . 32IM 6 8 5 57 is theconcentrationofRussellmixtureheavyelements,r found thatthefollowingexpressionisagoodapproximationfor formula forthedegenerateradiativeopacity.Inregion« Kr intheregions\(//kT«oand\f//kTo: eracy, and\f//kT^oholdsfortheintermediateregionofin- is cutdownconsiderably;ior\f//kT^—5,rattainsitsmaximum o, r^>>i,sothatthecontributionfrombound-freetransitions the guillotinefactor,and\[/isnegativeofFermienergy given byKramersformulacorrectedtheinsertionofa“guillo- ported notsolelybyradiativeprocessesbutalsoelectroniccon- lated valuesofrintheregion\p/kT«oforapureRussellmixture. free transitionscontribute.Table1inreference6givesthecalcu- value of196.5,andforlargernegativevalues\¡//kTonlythefree- cipient degeneracy).Equation(1)isvalidfor\¡//kTo;it The quantityTisthetemperature,kBoltzmannconstant,Xr tine factor.Thisformulamaybewritten In thelimit\[//kTequation(2)goesoverintousual 324 the electronsisdefinedas duction. IftheenergycurrentisQ,thermalconductivityAof (\p/kT ^>>oimpliesnondegeneracy,\f//kT
tween the strongly degenerate core and the nondegenerate surface. This is rather important because of the considerable uncertainty of the observational values of the radii of the white dwarfs. Then equation (22) has to be integrated subject to the condition that T = o when p = o. This can be done directly, provided we make the assumption that
can be taken equal to K at the point in question.18 We get
Kp - pp,.r. [1^]. to)
To obtain T and p as functions of r, we have to insert this expres- sion for Kpm equation (20) and get
dT _ i petf'GM , v dr - ~ 4 WÏ2 ’ ^24; with the condition T = o at r = equation (24) may be integrated to i GM _ T 4 m.2 ^ (25) h is the distance inward from the surface of the white dwarf and
Meff.
The latter quantity may be estimated fairly readily in the nonde- generate region and at the beginning of the degenerate region; farther in/equation (25) loses its usefulness. In any case, a change of radius of the star will simply change the depth h at which a certain temperature T occurs, in the ratio of R2.
18 For the justification of this assumption in the nondegenerate region see Ström» gren;7 as degeneracy sets in, appropriate corrections are easily made.
© American Astronomical Society • Provided by the NASA Astrophysics Data System 194OAp J 92 . . 32IM 19 calculations (Ap.J.,92,27,1940).Itturnsoutthattheagreement isremarkablygood, read hispaperpriortopublication. of interesttocompareourvaluesoíKwiththosegiven byP.M.Morse’sprecise to withinafewpercent.IamindebtedProfessorMorse fortheopportunityto dwarfs itwouldbenecessarytointegratesimultaneously theentire pute TandA,takingthesamevaluesforM,L,R(observed given by\f//kT=—3. ment withtheaccurateresultsgiveninTable2wasexcellent;for for SiriusB),Xr,andpasinthenumericalintegration.Theagree- to theradius,luminosity,andchemicalcomposition.Theseassump- and hforSiriusB40Eridaniunderdifferentassumptionsas tions were: For \p/kT<—3thediscrepancybecomesworse,sothatequations \p/kT >—3thediscrepancywasnevermorethan10percent. (a) SiriusB: (e) 40EridaniB: (d) SiriusB: (c) SiriusB: (b) SiriusB: 334 R (23) and(25)willnotbeappliedforstrongerdegeneracythanthat 19 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem The approximateformulae(23)and(25)wereemployedtocom- Intheregionÿ/kT>—3resultsarequitesensitive toK;itwastherefore In ordertoobtainanexactsolutionfortheinterior ofthewhite We obtaintheresultsgiveninTable3. For thepurposesofnextsectionwecalculatedvaluesp,T, R = = = ==:== M =Mobs.LLobs.RLtheor.^oXi(pure -ÆL ——ÜLobs.L2Lbs.R-^theor.XrI M Mohs.L~Tbs.R-^theor.XrI ■M- ~4/obs.LLohs.RRohs.XrI M —ÜLobs.Lbs.R-^theor.^XrI He 0 0 0 V. THEINTERIOROEWHITEDWARESTARS R. E.MARSHAK helium star) WHITE DWARF STARS 335 set of star equations, taking account of the variation in Mr, Lr, and r (which was not done in the surface region). The equations are9
dpo ß~i (26) dr dMr ^Trpr2 dr (27) dT _ 3 Kp L r (28) dr lÔTac T3 r2 dL r = 4irpr2€ . dr (29)
The energy production per gram-second, e, could be assumed to be given by some thermonuclear reaction such as the proton-proton reaction or the carbon-cycle, and one would adjust the hydrogen
TABLE 3
io7 4-5-I02 6.9-io6 107 2.5-103 9.1-io6 (a) IO7 6.0-io3 9.8- io6 IO7 10.7-io3 IO.I•IO6 9* 10a io3 6.9- io6 8- io6 io3 9.1-io6 (b) 7* ioi io3 9.8-io6 6* io6 io3 10.i•io6 6 2 (c) 8-io 5 io 5-io° (d) 6- io6 8-103 11-10° 5-107 i .4- io2 7-10° M. O- IO7 2.o-io3 II-io6
NOTE: The observed masses, luminosities, and radii were taken from a communication of Professor Chandra- sekhar (cf. also G. P. Kuiper, Ap.J., 88,472, 1938). The theoretical values were taken from the theory of degenerate configurations (cf. sec. V). content XH and the amount of Russell mixture Xr (determined by the helium content) so that the two boundary conditions, Mr = o,
Lr — o at r — o, would be satisfied simultaneously. This would be
© American Astronomical Society • Provided by the NASA Astrophysics Data System 194OAp J 92 . . 32IM 5 is settinginrapidlyandthetemperaturerisingveryslowly.The with cellent forbothSiriusBand40EridaniB. where This permitsustomakeagreatsimplificationasfirstapproxima- situation willbesimilarfor40EridaniB,althoughnotasfavorable. small distanceinwardfromthesurfaceofstarstrongdegeneracy sure, pc,andtofindthetemperaturedistributionwithoutmaking tion byneglectingthetemperature-dependenttermsingaspres- choice ofe.However,ananalysisTable2showsthatatavery tedious andwouldpresumablydependsensitivelyonthecorrect tion, andtheinitiallyassumedmass,radius,densitydistribu- of thetemperaturemaythenbetakenintoaccountasaperturba- relativistic andnonrelativisticdegeneracy)isgivenby equation ofstateadegenerategaselectrons(validforboth tion therebycorrected.Itwillturnoutthatthisprocedureisex- any explicitassumptionsregardingthesourceofenergy.Theeffects 336 R.E.MARSHAK the notation If wethenconsiderequations(26)and(27)togetherintroduce where yisthecentralvalueofy,itfoundthat thesetwoequations combine intothesingleone c © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Now itiswellknownthatwhenthetemperatureneglected 2l/-1 f(s) =s(2s—3)(s+i)3sinh5, l/3 82 S =I.OI•IO~^(pr7e) r =aco;y a =7.71•I°^;y*s+1,(31) c 2 22 co dœ = 6.01•iof(s),(30) i d (32) 194OAp J 92 . . 32IM 2 5 21 20 229 67 subject totheboundaryconditions=i,d
For 40 Eridani B, the variation of the temperature is not so slow. In this case we may approximate/(T) by a power of p, say, pm. The value of m will depend on the type of energy process; if thermo- nuclear reactions are considered, it will be higher for a reaction such as the carbon-cycle than for the proton-proton reaction. It will also depend on the variation of temperature with density and is deter- mined by the condition of self-consistency. Then we have24
For Sirius B a set of values for (Lr/L)^ was found from equation (37). We then integrated equation (28), taking account of the varia- tion of Tje in the outer region.25 It is found that the variation of T with p is about as T ^ pl/75. Even if the carbon-cycle is assumed, /(T) varies only as T20 ~ pI/4; for the proton combination the varia- tion would be only as T4 ~ pl/20. Thus, the choice ra = o is obviously a better one than, say, m = 1. This is so because m = 1 gives just about the same (T, p) dependence as m = o. To test the sensitivity of the results to inaccuracies in the luminosity, the temperature dis- tribution was calculated on the assumption L = 2Loh3^ Xr = 1, and 6 it is found that Tc = 18.4* io ° C. An uncertainty in the luminosity of a factor of 10 would produce an uncertainty in the central temper- ature of a factor of 2. For Kuiper’s value of the luminosity, the 24 To get an even better approximation, one could break up the range (o, a>i) into subregions, associating a slightly different (nonintegral) value of m with each subregion; this was not done in our calculation for reasons which will become apparent below. 2 s The quantity 7]e is different from 0.50 only for the first few points, so that it is justified to take r¡e = 0.50 for the star as a whole.
© American Astronomical Society • Provided by the NASA Astrophysics Data System 194OAp J 92 . . 32IM 9 60 9 of 1.25*io,and mination; thiswasnotthecaseforSiriusB. Then thetheoretical missible inviewoftheconsiderableuncertaintymassdeter- 0.40 Tfo,insteadofKuiper’svalue0.45Mq;thisismuchmore central temperatureofSiriusBisthus15.2-ioC,andfairly radius of40EridaniBis1.05•io,ascompared withKuiper’svalue convenient forthepurposesofChandrasekhar’stablesandisper- constant overalargerpartofthedegeneratecore.Thecomplete 340 R.E.MARSHAK temperature distributionisgiveninTable4. 32 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem For 40EridaniB,themasswastakenas7.95-iogm,i.e., 3-88. 3-87. 3.86. 3-85. 3-8.. 3 •7- 3.6.. 0.6.. O.4.. O. 2.. 0.0. 3-4-- 0. 8. 3.2.. 3.0. . 2.8.. 2.6.. 2.4.. 2.2.. 1.8.. i .6.. 1.4.. 1. 2.. 2.0. . i .0.. coi =3.60; s p =1.67•10; 6 Sirius B(rinUnitsofio) 0.00002 o.00009 0.00021 0.00036 0.00096 0.00220 0.00429 0.0199 0.0104 0.0333 0.0518 0.0765 0.109 0.436 0.930 0.980 0.151 0.203 0.268 0.346 0-537 0.752 0.850 0.644 i .000 W Pc) TABLE 4 6 pc =1.15•io. (Lr/Do 0.990 0.986 0.920 0.470 0.003 0.000 0.980 0.958 0-855 0.690 0.587 0.344 0.228 0.138 0.069 0.022 0.782 i .000 i .000 i .000 i .000 i .000 i .000 i .000 i .000 8 a =3.0•io; 1514 15.16 10.10 11.50 12.90 13.60 14.10 14-45 14.60 14-73 14-83 14.91 1511 15.18 15.20 15.20 15.20 15.20 15.20 14.98 15-03 1507 6.90 9.80 9.10 (40) 194OAp J 92 . . 32IM 31 1 for theproton-protonreactionand tion. Bothsolutionsgiveadependenceof T onpoftheform We haveintegratedequation(28)for40EridaniBinthesame responding toachoiceofw=oandm1forthe(L/i)distribu- manner asforSiriusBwithL=,2.1•10Xr1,subject T ~p/^anddifferverylittle.Since to theboundaryconditions rw oha © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Table 5containstheresultingtemperaturedistributioncor- 3-4°- 0.6.. 3-55- 3-5°- 3-45- 3-35- 3-3-. 3.2.. o. 2.. 0.0.. 0. 8. 0.4.. 3-0-- 2.4.. 2.2.. 1.8.. 2.8.. 2.6.. 2 .0.. 1 .6.. i.4-. 1. 2.. i .0.. 0.00012 0.0017 0.0042 0.0072 0.0100 0.0131 0.0216 0.0434 0.0732 0.160 0.218 o. 112 0.364 0.893 o. 286 0.450 0.542 0.637 0.731 0.818 0-951 0.987 i .000 6 40 EridaniB{TinUnitsofio) c 3*50 ii*io 3-55 WHITE DWARFSTARS for m- 0.997 0.987 0.999 0.978 0.909 0.944 o. 760 O.391 0.00414 0.000 1.000 0.849 0.654 O.525 0.267 O.154 0.0231 1.000 1.000 1.000 0.0729 1.000 1.000 a m ~p m -P**(41) TABLE 5 for w=i 0.997 0.990 0.994 0.975 0.940 0.902 0.566 0.008 0.000 0.820 0.128 0.042 a 1.000 o. 710 0.406 0.256 1.000 1.000 1.000 1.000 1.000 1.000 1.000 f 7* io T form=0 30.1 30.2 30.0 17-5 27.4 ii .0 14-5 19-5 21.0 21.8 24.0 25.0 25-9 26.7 27.9 28.3 28.6 29.2 28.9 29.7 29.9 29-5 7.0 T form=i 30.6 30.9 31.0 30.0 30.3 30.8 ii .0 14-5 17-5 21 .O 21.8 25.0 26.7 27.4 19-5 24.0 25-9 27.9 28.4 28.8 29.2 29.6 7.0 341 194OAp J 92 . . 32IM 265 2 222 6 1 for thecarbon-cycle,choicem=owouldbebetterfirst but ahighercentraltemperatureisattained—aboutthirtymillion temperature distributionvarymorein40EridaniBthanSiriusB, reaction andm=iforthesecond.Notonlydoesinternal 342 from theabovetemperatureanddensitydistributions,effectoftempera- degrees. Thisisduetothesmallermassdensitiesin40EridaniB. particle pressure,wewrite This maybedonebytreatingthetemperatureasaperturbationinequations ture inchangingtheinitiallyassumeddensitydistributionmustbecalculated. where wehavetaken(eq.[30])p=Bs*,C—4TGB.Neglectingtheheavy- (26) and(27).Combiningequations(27),weget Also the temperatureperturbationisa=4TrkT/inc4,A6.01.10,andm electron mass.Sincethedegeneratecoreofawhitedwarfispracticallyiso- perature-dependent partoftheelectrongaspressure.It turnsoutthattheheavy perturbed equation 15*1 o°C.Wemay,therefore,expandpaboutthepointZ=asfollows: thermal, wetakeaasconstantthroughoutthestar;e.g.,forSiriusB,Z= At anyrate,theorderofmagnitudewillbegivenbyourcalculation. particles contributeintheregionwheretemperature perturbation isveryslight. cle pressure(necessarilytemperaturedependent)contributes morethanthetem- Substituting theseexpressionsintoequation(42)andmakinguseoftheun- G 26 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Before anyconclusionscanbedrawnregardingthehydrogencontent,etc., Thisisjustified,ascanbeseenbyfindingthevalueofpfor whichtheheavy-parti- rJ/(5o)3[Af(Sn)] ?í[*í ]-1• • log s=+^. 0 AS12 6 Pg =K°)+-4a|^o(soi)/■ s =sS+^a, 2 Pg = r dr i d R. E.MARSHAK 2l/ ^ dr as(s +i) dpa dlog5 + W) 6 = -Cs, (42) (43) (44) 194OAp J 92 . . 32IM 2 2 2 6 2 2 equation (46)canbewritten If weexpresssintermsofChandrasekhar’sfunction,i.e., dr \r5 dz Í2 In theregionofnonrelativisticdegeneracy,weget dr r mass densityp=3.8*io(ifT]istakenas0.50). dz .2<¿2, relativistic degeneracy, The boundarybetweenthetworegionsisgivenbys=f,correspondingtoa If theexactexpressionisusedfor/^o),equation(45)somewhatcumbersome r dr to workwith;however,itisconsistentwithourapproximationtake 0 I d 0 e 0 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem With theindicatedsimplifications,equation(45)becomes,inregionof We getthefollowingequationforz={ds/àa)\ ü 35o<+,s2lo 2I/2 ~ Tr*'“^+/(-»)î8=~(>sjz r {íotó+i)+/'0o)s} 0 /(s) =2forrelativisticdegeneracy 0 f(s) =ffornonrelativisticdegeneracy 0 5 2 ^o\ dz\$_(dSo dr )Lo\^ s dr'r4H 3_ d?So_6_^0sCs; 0 2 s +i=y 0 WHITE DWARFSTARS z —rz]r=aú), 2 _ Fj$_/dso\d^Soi_¿£0] 2 2 + F{r)z=G{r),(48) [i ídSoVds¿^0] 2 [85^ \drJdr^rsl\' 0 25q \drj4^02rs Q dso dr ' rsdr4H 0 _8_ 3C5 3df (5q) dr \s 0 2 3^q ,±^£0] 4A 5odrJ ? * (45) (47) (46) 343 194OAp J 92 . . 32IM 28 where 344 R.E.MARSHAK and and equation(47)canbewritten where carried uptothepointco=1.9,wheref.At convenient andwillgiveacloseupperboundtotheeffectof thetemperatureperturba- equation (49)cannolongerbeused. Equation (49)wasthenintegrated,subjecttotheseboundary conditions,upto to theboundaryconditions2=0,dz/droatcoo.Theintegrationwas University ofRochester,forthecalculationfunctions F,G,A,B,C,D. tion. the pointco=3.67.Fromthistoboundary ofthestar(co!=3.727), this isequivalentto 28 27an 8 27 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem TheauthorisindebtedtoMr.RobertMannofthecollege ofartsandsciences, ThiscorrespondstoamassM=o.88üfod1.732 •10;thischoicewas We haveintegratedequation(48)forSiriusB,takingi/yl=0.2,subject G{r) = Fir) = 3C50 r(\ =iZl/^V_1^£îV4- r 2 ^ as2\á
With a knowledge of the z distribution, the modification in mass has been estimated as follows: Let p = p0 + Pi, where p0 is the density distribution cor- responding to T = o and is given by Chandrasekhar’s tables.5 The quantity pi is the perturbation term due to the finite temperature. Then p! = ^Bslza, and the change in mass ATf is
2 2 AM = ^pjdr = J r s 0zdr = AXM + A2M , where C3.67 3 2 5 AXM = 47iT$aa J œ slzdù) = 3.8 • 10 M , for T = 1.5 • io7°C
A2M is the uncertainty introduced in the region co = 3.67 to cOi = 3-73, and if -4 we take the total mass in this region then at most A2M = 1.5* 10 M. Hence, we may say AM 3-8 * 10 5 < < 1.9 • 10 4 .
For Sirius B, whose mass is just about equal to the sun’s mass, it is clear that a temperature perturbation of i5*io6° C—wfiich corresponds to the initial tem- perature—will lead to a AM'about one-hundredth of 1 per cent of the original mass. The theoretical radius must remain, and the initial temperature-density distribution may be used to gain an estimate of the concentration of reactants which must be present if the observed luminosity is assumed to be due to the occurrence of nuclear processes involving these reactants. A much more approximate calculation was performed for 40 Eridani B, in order to estimate the effect of the temperature in changing the unperturbed mass-density distribution. We took i/yl — 0.6, corresponding to a mass M = 0.04/M0, and for T the value 3o*io6°C throughout. It turns out that the change in mass is negligible again, i.e., AM = 3. io”3 M.
We may now compute the hydrogen content of Sirius B and of 40 Eridani B if we make an assumption regarding the source of energy. From eqúation (29) it follows that
fR L = 4w J per2dr . (50)
Suppose we were to assume that the carbon-cycle is responsible for the energy production in the white dwarfs (that is, that the concen-
© American Astronomical Society • Provided by the NASA Astrophysics Data System 194OAp J 92 . . 32IM 1 2930 2 30 31 3a 31 3 i nsurmountableobjectionsagainstanappreciableabundance ofHe*inthewhitedwarfs into equation(50),usingthetemperature-densitydistributions with Tmeasuredinmillionsofdegrees.Insertingthisexpression tration ofcarbonandnitrogenisthesameasinmain-sequencestars, found forSiriusBand40EridaniB,takingL=Z8.(9.7•10 proton reaction).Then white dwarfs,wehavealsocomputedthehydrogencontentincase ergs/sec forSiriusB,2.1•1040EridaniB),weget with Texpressedinmillionsofdegrees,andweget the onlyreactantsarehydrogennucleithemselves(i.e.,proton- say, ^ofipercentbyweight),thenweshouldhave 346 R.E.MARSHAK perature distributioninthewhitedwarfs,heconsidersonly thecarbon-cycleinmaking These valuescertainlygivethemaximumhydrogencontentunder Since itmightbetruethatcarbonandnitrogenareabsentinthe as apossibleimportantconstituentofthewhitedwarfs. However, therearealmost his estimates.Hisstatementsarethereforelessconclusive thanours,especiallyre- of astarconsistingheliumalone.Thislowersthetemperaturebe- garding theradiiofwhitedwarfs. of thewhitedwarfs.Inadditiontofactthathemakesnoattempt tocalculatethetem- any reasonableassumptionsregardingthechemicalcomposition (cf. W.A.Wildhack,Phys.Rev.,57,81,1940). Cornog (Phys.Rev.,56,379,1939),itmightbethoughtthat Heshouldbeinvestigated ob (i.e., anappreciableconcentrationofheavyelements). 2 1 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem 3° Inarecentpaper(M.N.,99,595,1939)Eddingtondiscussesthehydrogencontent However, wethoughtitworthwhiletoconsidertheextremecase 9 Cf.Tables4and5. 3 Inthiscalculationheavyheliumisimplied.viewof the workofAlvarezand 2I2r e =2.95•ioXpT-;, 8 Hr 7 s for SiriusBXh=2.2•10“; for 40EridaniBXh=3.6•io~. }2r for 40EridaniB....Xh=8.0•io“s. for SiriusBXh=2.4•io“, e =4•ioXpTe~;r^, H 194OAp J 92 . . 32IM 6 7 6 4 6 32 -8 3 7 1 of 10•io°C.Inthetransitionregionthereisaslightadditionaleffect in thesamedirectionduetogreaterdensityoffreeelectrons?7. tor of6(Z-/A=iinsteadasfortheRussellmixture).Thebound- ary temperatureobtainedbythemethodofsectionVis5•10instead cause theopacity,bothradiativeandconductive,isreducedbyafac- The contributionofthenucleitogaspressureisnotessential,for legitimate totakeXr=oandsetTje0.50indeducingthe half ofthevalueforRiissellmixture.Itisnecessarytoassume the higher77heavyisnearlycompensatedbylowertemperature. Xe =i.5*icrinordertogiveL¿obs-Thus,itwascertainly The centraltemperatureforSiriusBisthen7*io,i.e.,aboutone- proton reactionisassumedtoberesponsiblefortheenergyproduc- radius fromthemassandincalculatingtemperaturedistribu- pure heliumstars.Itismostplausibletoassumethatwhitedwarfs dwarfs aswell,inviewoftheirlowluminositiesandhighdensities. tion. This resultistrue,nomatterwhetherthecarbon-cycleorproton- Eridani Bmustbeverylow;thiswouldtruefortheotherwhite nitrogen. Thiswouldalmostcertainlyfollowifthewhitedwarfs tion, andremainstrueevenifthewhitedwarfsweretakentobe e contain someheavierelementsaswell,andamongthemcarbon are thelaststagesinevolutionofmain-sequencestars,sinceall stars inthemainsequence(exceptpossiblymuchlessmassive white dwarfsarenotthefinalstagesinevolutionof.main- main-sequence stars,thehydrogencontentofSirius Bwillbe2•10 reaction, thenumericalfactor4•io^hasbeenused.This differsfromequation(35) ply theenergy,evenwithobservedlowluminosities, onlyfora sequence stars,itislikelythatcarbonandnitrogenarepresent. than thesun)containaconsiderableamountofRussellmixtureand and of40EridaniB4•10“.Suchalowhydrogen contentwouldsup- consume onlyhydrogeninthecourseoftheirevolution.Evenif and H.A.Bethe,Phys.Rev.,56,210,1939);otherwise would bethreetimeslarger. n. 2,byafactorof10(cf.B.O.Grönblom,Phys.Rev.,56,508, 1939,andR.E.Marshak 2 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem We havefoundthatthehydrogencontentforSiriusBand40 Assuming, then,thesamecarbonandnitrogenconcentrationasin 3 Inmakingestimatesofthehydrogencontentonbasis oftheproton-proton WHITE DWARFSTARS347 VI. CONCLUSIONS 194OAp J 92 . . 32IM 3 23_s _s 811 5 fessor Kuiperhasalsokindlyinformedthewriterthatthere areatleasttwodistinct types ofwhitedwarfspectra:thetypewhichcontainsno traceofhydrogen,andthe from main-sequencestarsofthesamemass.Then thehydrogencon- type exemplifiedbySiriusB,whichexhibitsfairlystronghydrogen lines. tent, say,forastarofsolarmass,atthetime whengravitational further supportforhypothesis(2). explosions. Thesmallradiiwhichthewhitedwarfspossessfurnish main-sequence starsorbythecontractionoffragmentsgiant second alternativeseemsratherplausible,fromtheassumptionthat fusion throughadegenerategasisanextremelyslowprocess.The probably requiretoolongatimetoplaysignificantrole,sincedif- protons. (2)Theenergyproductionmaybeduelargelytogravita- observations ofthespectra.Then,eitherenergyproduction hydrogen contentcannotberesponsiblefortheenergyproduction. tion. be verymuchlargerthanintheinterior,asseemstoindicatedby observed frequencyofthewhitedwarfs,itseemsthatpresent E «6-io-Xergs/gm.ForSiriusBweobtain,since— contraction producesasmuchenergythethermonuclear processes the whitedwarfsareproducedeitherbygravitationalcontractionof could diffuseintotheinteriorandtherebyreplenishsourceof can takeplaceinthesurfacelayersthemselves,orhydrogen duction: (1)Thehydrogenconcentrationinthesurfacelayersmay ergs/gm-sec; therefore,thehydrogenwouldsupplyenergyfor For 40EridaniBwefindaboutthesametime,viz., very limitedtime.Theenergyevolvedinthetransformationofhydro- Since, ingeneral,suchshortlifetimeswouldbeincompatiblewiththe gen intoheliumisioergsperproton,andtherefore,altogether, 348 R.E.MARSHAK 2* 1er,E«1.2-ioergs/gm.TheluminosityofSiriusBis0.005 « 1.2•io^/o.oossec~8*ioyears,i.e.,onlyforaveryshorttime. H © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem Thus, itmightbesupposedthatthewhitedwarfshaveevolved The firstpossibilityinvolvingthediffusionofhydrogenwould 33 Cf.Öhman,Arkiv.f.Mat.Astron.ochFysik,Stockholm, 25B,no.21,1937.Pro- There areatleasttwoalternativepossibilitiesfortheenergypro- 196 1.2 •10X7-sec—3ioyears. -7 4 *io 0 J 0.026 194OAp J 92 . . 32IM 37 36 534 46-3 -4 35 9 38 2 10 fragments. larger thanChandrasekhar’scriticalmass,whichhavecontracted white dwarf(timebeforeitturnsintoadarkobject)willbegiven would thenarisefromthefurthergravitationalcontractionof gen muchmorerapidlythanstarsofsolarmass.Thewhitedwarfs large masshavealuminosityandhenceconsumetheirhydro- have originatedintheexplosionofmain-sequencestarsmasses by thedifferencebetweengravitationalenergywhichwhite gen contentatthetimeofexplosionwouldbesmall,sincestars greatly withoutdegeneracyhavingsetin.Presumably,thehydro- might benecessarytopostulatewithGamowthatthewhitedwarfs into helium)fromits35percentconcentrationinamain-sequence only theproton-protonreactionistakingplace.However,time hydrogen contentobtainedabove,evenwiththeassumptionthat is oftheorderio~.Thiscompatiblewithvalue sekhar) .5 E «S*ioergs,whilefor40EridaniB,AM/M~3*ioand where AM/Misessentiallythequantitycomputedinperturba- Phys. Rev.,55,718,1939);theresultisnotverysensitive to themass. tion calculation(cf.sec.V).ThusforSiriusB,AM/M«ioand dwarf hasatpresentandthatwhichitpossessesinacompletely stars notmuchearlierthanthat.Inviewofthisdifficulty,it great nebulaewereformedabout2•ioyearsagoandtheindividual From Chrandrasekhar’stablesitisseenthatAR/R«AM/M, of orderunity,andAi£isthedifferencebetweenradiusin concept oftheexpandinguniverseiscorrect—accordingtowhich star ofsolarmasstoasmallfraction1percentistheorder required forthehydrogentobéconsumed(throughitsconversion the completelydegeneratestateandradiusatfinitetemperature. degenerate state,namely,Eg=yGM/R-kR/R\7isaconstant io years.Thislongtimescalewouldbedifficulttoacceptifthe g 37 -5 38 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem If wegrantthatthermonuclearprocessesareabsent,thelifeofa Masseslargerthan6•65'nlMQcanneverbecomedegenerate (cf.Chandra- 34 ForthesunthiswouldhappenwhenRâîo.SRq,X ^ 10(cf.G.Gamow, Chandrasekhar,op.cit.,Table5,p.427. ^ Phys.Rev.,55,718,1939. h G. GamowandE.Teller,Phys.Rev.,55,654,1939. WHITE DWARFSTARS349 194OAp J 92 . . 32IM 8 3147 47 9 47 years. Theseresultsarequiteconsistentwiththedensityofwhite ergs/sec «3•ioyears.For40EridaniBtheminimumlifeis5 L «10ergs/sec,sothattheminimumlifeisioergs/10 it canbeshownthatascompletedegeneracyismorenearlyreached from thetemperature-densitydistributionswhichhavebeencalcu- Eg ~2-ioergs.Tobemoreaccurateweshouldalsoconsiderthe retical radiiseemtobeascertainthetheory ofdegeneratecon- dwarfs inspaceandwithanagefortheuniverseof~ioyears. the velocityofcontractiondiminishes),wehaveforSiriusB, Thus, theavailableenergyisgivenessentiallybygravitational We get radius—by morethanafactor2.Inviewofthe factthatthetheo- value oftheradiusSiriusBdisagreescompletely withtheobserved gible hydrogencontentisquitesatisfactory, while thetheoretical The agreementfoundfor40EridaniBonthe assumptionofnegli- source ofenergyinwhitedwarfstarsareadmittedlyinconclusive, continues toberadiatedatthepresentrate(thisisanupperlimit; ^thermal isduechieflytotheelectronsandequalfcTdV,where lated, thatE.,andEiarenegligiblecomparedtoEthermai-Now contribution andisaboutioergs.Ifitassumedthattheenergy 350 the situationwithregardtopredictedradii is muchmoredefinite. We canwrite able energy.ItcaneasilybeshownforSiriusBand40EridaniB, with thermal, radiative,andionizationcontributionstothetotalavail- v radoniz © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem 39 Ibid.,p.394. While theaboveargumentsforgravitationalcontractionas = 2I46 For 40EridaniB^thermal=i-S-io*?ergs. For SiriusB^thermal-5*oergs; Ethermal 1*7 _2l/39 5 =I.OI•IO(pr7e). 2 _ 7T^_p+i)*/* Tf)7je 21 R. E.MARSHAK me sm H i 63 ioa;y COi 2 TS(t)œdü) . 194OAp J 92 . . 32IM s 62 9 9 40 ples underlyingthecomputationofratehydrogencombina- limit of2•io~forthehydrogencontent.Unlessselectionrules would bedecreasedbyalargefactor^10)thereactionratecannot for ^-disintegrationareincorrect(inwhichcasetheprobability figurations (embodyingrelativisticeffects)andthephysicalprinci- are completelyabsent,theproton-protonreactiongivesanupper tions ofourassumptionscanleadtoanincreasedradiusforSiriusB. tion, itisworthwhiletoinquirewhetheranyreasonablemodifica- observed valueof1.4*10cm.Itmightbethoughtthatbecause ate configurationsthatR«0.6*iocm—tobecomparedwithan under veryhighpressures,is=1)andfromthetheoryofdegener- be wrongbymorethanafactorof10.Thenitmustfollowthat relativistic degeneracytheelectronswillpossesssufficientenergy pressure isnothighenoughinSiriusBfortheconversionofheavier lie =2(sinceonlyforhydrogen,incontrasttoallotherelements would onlyaggravatethesituation.Again,itmightbesaidthat neutrons whichareformedwillbecapturedbyothernuclei.The to combinewithprotonsandformneutrons;however,aneutron nuclei intoneutrons;furthermore,theformationofaneutroncore core willnotarise,becausesolittlehydrogenispresentthatthefew proximate agreementobtainedbetweentwo distinctmethodsof energy sources.Butthiswillnothelpmatters;theonlyconse- L =i,wasnotjustifiedifitistruethattheenergyproducedin the integrationofstarequationsnearsurface,assuming the envelopeandthatdegeneratecoredoesnotcontainany ly re-examinedtheobservationalmaterialfor SiriusB.Hecon- in error. The discrepancyinradiuswillremain.only'conclusionwecan generacy willsetinsooner,andtheenvelopestillbeverysmall. measurement oftheradius—by(T,T^f.)relationship andbythe draw, therefore,isthattheobservationsofradiusSiriusBare Einstein gravitationalredshift.Moreover,G. P. Kuiperhasrecent- quences willbethatthetemperaturegradientdecrease,de- dwarfs. r 3 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem In thecaseofSiriusB,evenifitistruethatcarbonandnitrogen We areveryreluctanttodrawsuchaconclusion, inviewoftheap- 4° ExceptforHe;buttheabundanceofmustbeinappreciable inthewhite WHITE DWARFSTARS351 194OAp J 92 . . 32IM 6 41 42 6A 43 6 value indicatedbythespectrum),togetagreementwiththeo- would havetobe80km/sec,insteadofthepresent20and error byafactorof2.Hepointsoutthattherelativisticredshift eludes thatitisextremelyimprobablethemeasuredradiusin 352 R.E.MARSHAK very likely.ThedubiousvalidityofrejectingtheGamow-Teller vision ofourpresentideasconcerningnuclearstructureandisnot is completelywrong—butthiswouldrequireafar-reachingre- obtained forapureheliumstarandminuteabundanceofCN, close, faintcompaniontoSiriusB,whichisatthesametime,awhite retical radius.Heconcludesthatonly“theadhocassumptionofa the effectivesurfacetemperature25,000°insteadofio,ooo°(the action willstillbefast.Theonlypossibilitywouldtomakethe temperature of15•io°C(Russellmixture).Withr~7°,asis dwarf, couldbringixtothevalue2.” selection rulesandreturningtotheoriginabFermiversionof additional assumptionthatthetheoryofproton-protonreaction the carbon-cyclewillbesufficientlyslow,butproton-protonre- It isobviousthatnoagreementcanbehopedforwithacentral of theproton-protonreactionbyalargefactor—afactorprobably ß-decay (whichasmentionedabovewoulddecreasetheprobability of /3-decaywhichisspindependentratherthanbyonenot. experimental resultsseemtobemorereadilyexplainedbyatheory experimental resultsontheßdisintegrationoflightnuclei.These gen inSiriusBandthereforealargerradius)isattestedtobyrecent sufficient topermitthepresenceofanappreciableamounthydro- ogous toHe=+2^He* ip+me~(in Teller theory.Sincethecombinationoftwoprotonstoforma reaction ispresumedtoprovidestrongconfirmationoftheGamow- publication. virtue oftheequalitylikeparticleforces), itwouldfollowthat deuteron withthesubsequentemissionofapositroniswhollyanal- In particular,thestrikinglylargeprobabilityofHeLi+e~ c e Gamow-Teller selectionrulesshouldbe,employed inthecomputa- 1 2 © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem But, ontheotherhand,theoryseemstobewellfounded. 4 TheauthorisindebtedtoProfessorKuiperforsending himhispaperpriorto 43 Cf.T.BjergeandK.J.Bröstom,Nature,138,400,1936. 4 Cf.B.O.Grönblom,op.cit.,p.509,andE.P.Wigner,Phys. Rev.,56,519,1939. 194OAp J 92 . . 32IM 6 present investigationhasatleastestablishedalmostbeyondques- bare possibilitythatthespinofHeisone,inwhichcaseGamow- ly besettledatpresent.Allsuchmodificationswoulddifficultto remaining experimentalevidencefortheGamow-Tellerrulescanbe Teller selectionruleswouldbeunnecessary.Whetherornotthe planation oftheradiusdiscrepancyforSiriusB. of nuclearphysicsandthattherereallyseemstobenosimpleex- reconcile withourpresent-dayknowledgeofnuclearforces.The explained bysimilarmodificationsinnucleartheorycannotdefinite- tion oftheproton-protonreactionrateinstars.However,thereisa script. HewouldalsoliketothankProfessorsS.Chandrasekharand tion thattheclaimofastrophysicsisindirectconflictwith the investigationandalsoforacriticalrevisionofentiremanu- Professor H.A.Betheformuchvaluablehelpatdifferentstagesin data andtables. G. P.Kuiperfortheirkindnessinsupplyingnecessaryastrophysical © American Astronomical Society •Provided bytheNASA Astrophysics DataSystem In conclusion,theauthorwouldliketoexpresshisgratitude University ofRochester Department ofPhysics May 1940 WHITE DWARFSTARS353