197 5ApJ. . .200. .145W present paperisconcernedwiththeformationof present. periodic X-raysourceswhicharebasedonneutron where theobservationaldatamostdirectlyapply.The eccentric, orbit(seealso Colgate 1970;Sutantyo mass losscanleaveaneutron starinabound,albeit who showedthatsupernovamodels involvingmoderate to acertainextentbyMcCluskey andKondo(1971), half themassisejectedimpulsively,aquestion that abinarysystemwillbedisruptedifmorethan fills itsRochelobeandislosingmattertotheneutron were putforthexplainingthesystemintermsofa tion withthestarHZHerbyLiller(1972),models that futuremaybeevenmorebizarrethanthe give importantcluestothefutureevolutionandahint systems. Weshallseethatthestudyofpasthistory neutron starsbysupernovaexplosionsinclosebinary work inthisareaisdevotedtothepresentphysical is happeningtothemnow?WhattheirfateMost binary systemsimmediatelyraisesintriguingquestions and Pines1973).Similarmodelsareenvisagedforthe star (Forman,Jones,andLiller1972;Lamb,Pethick, rotating neutronstarorbitingacompanionwhich state ofsuchpostulatedneutronstarssincethatis about theirevolution.Howweretheyformed?What The AstrophysicalJournal,200:145-157,1975August15 stars inclosebinaryorbits.Thisquestionwasanswered raised astotheself-consistencyofmodelsfor a supernovaexplosion.Sincesimpledynamicsdictates et al.1972;DavidsonandOstriker1973). of suchsystemscoupledwithsomethecurrentdata other knownperiodicX-raysource,CenX-3(Schreier Her X-lbyTananbaumetal.(1972)anditsassocia- © 1975.TheAmericanAstronomicalSociety.Allrightsreserved.PrintedinU.S.A. 1974a, b\Cheng1974). The notionthatneutronstarsmayexistinclose Neutron starformationisgenerallyassociatedwith Soon afterthediscoveryofperiodicX-raysource © American Astronomical Society • Provided by theNASA Astrophysics Data System the systemandratiooffinaltoinitialsemimajoraxesdependontwosixparameters, two basictypesofmodels:oneinwhichthecompanionisa“normal”star(polytrope,n=3) to agoodapproximationtheeffectofimpingingblastwavewhichstripsandablatesmass discussed. depending onthelevelofapproximation.Resultsforcalculationseccentricityimpartedto from thecompanioncanbeexpressedintermsofasingleparameter.Theeccentricityimpartedto and oneinwhichthecompanionisa“red-giant”phase. orbit musthaveafiniteeccentricity.TherelevanceofthisresulttoX-raybinarysystemsis an initiallycircularsystemarepresentedbymeansofgraphs,tables,andsimplefittingformulaefor Subject headings:binaries—neutronstarssupernovae We concludethatneutronstarscanbeformedinclosebinarysystemsbuttheresulting The generalproblemofasupernovaexplosioninbinarystarsystemisstudied.Weshowthat Center forAstrophysics,HarvardCollegeObservatoryandSmithsonianAstrophysical I. INTRODUCTION J. CraigWheeler,M.Lecar,andChristopherF.McKee Received 1974December23;revised1975February17 SUPERNOVAE INBINARYSYSTEMS ABSTRACT mass strippingandablation fromthecompanion.We different assumptions(see§ III). geometry issomewhatdifferent thanthatofCheng eccentricity imparted.Ourtreatmentofthestripping cross sectionofthecompanionstarandhence from theouterportions.Thelattereffectreduces (1974), andthecalculation oftheablationrestson important respects.McCluskeyandKondo(1971) tricity intotheorbitsofstarsifsystemisnot, to HerX-landCenX-3hasbeenpublishedelsewhere A preliminaryversionofourresultsastheypertain same wayaswedobutneglectthestrippingofmass and Sutantyo(1974a,b)treatablationinmuchthe collision andablationwillresultinanimpulsetothe may servetoablatematterfromthestar(Colgate from thesystem,whichwillinduceacertaineccen- The firstdirecteffectoftheexplosionismassloss dissipation inclosebinarysystemswillalsobetreated easily accessibleandasgenerallyapplicablepossible. different typesofexplosionsundervariousconditions. star, therebyincreasingtheeccentricityoforbit. which collidesinelasticallywillbethermalizedand by theejecta.Second,someofenergymatter mass maybedirectlyblownfromthecompanionstar interaction ofthesupernovaejectawithcompanion in fact,disruptedentirely.Othereffectsconcernthe as farthedynamicsofbinaryareconcerned. separately (Lecar,McKee,andWheeler1975). referred toasWML).Therelatedquestionoftidal supernova explosionsinbinarysystems,including star. Wetreatthreeprimaryeffectshere.First,some (Wheeler, McKee,andLecar1974,hereinafter We haveattemptedtopresentourresultssoasbe 1970). Lastly,momentumtransferredtothestarby The nexttwosectionspresent thecalculationof We assumethatthesupernovaisanimpulsiveevent We havedoneamoregeneralstudyoftheeffects Our workdiffersfromrelatedpapersinseveral 146 WHEELER, LECAR, AND McKEE Vol. 200 show that to a good approximation this calculation incident mass to Mc is termed the incident mass depends on only one (geometry-dependent) parameter. fraction, The parametrization of supernova models is given in § IV, and the results are given in § V in terms of the r - 1 ^sn R2 ln 2 (8) eccentricity which is imparted to various binary sys- ~ 4 Mc «o ' tems by different types of supernovae. A final dis- cussion and the application to X-ray sources is The condition that matter at x is ejected from the star presented in § VI. is that î;(x) exceed the escape velocity ves(x); from equations (6)-(8) this condition can be written II. MASS STRIPPING FROM COMPANION To estimate the mass directly stripped from the companion star, we first divide the companion into concentric cylinders of radius b and area Inbclb, with The parameter Fin depends only on geometry and axes along the line joining the centers of the two stars. masses whereas T, strictly speaking, depends on the Treating the incident ejecta as a plane slab and the distribution of matter in the star through ves. In cylinders as independent, we then determine if the practice, however, ves is only weakly dependent on momentum incident on each of the cylinders is suffi- position within the regions to be stripped or ablated cient to accelerate it to the escape velocity (cf. Spitzer in the cases we have considered; hence, by using a and Saslaw 1966). To describe the structure of a star typical (i.e., surface) value for ves we can regard T, as of radius Æ in cylindrical coordinates, we define defined in equation (9), as essentially a “geometric” parameter. We thus adopt the quantity T* as the basic x = blR, £ = z/R , (1) parameter describing the stripping and ablation of mass from the star; it is simply the ratio of the 1 available momentum of the supernova ejecta to the and let Mc'(x) be the mass of the companion per unit area at x. One readily finds momentum the companion would have if it moved at a velocity ver As we shall see, if T » 1, the companion Mc'(x) = 2/> R0 (x) , (2) is entirely disrupted. The assumption is made that the c 1 main part of the star remains stationary while the where p is the central density of the star and outer layers are stripped off, so our model is not likely c to be accurate when T » 1. In calculating the stripping we have used «w = j- pWie + x^)]dè. o) 112 Pc J 0 *;es = [2GMc(x)IRx] (10) The mass interior to x is as an approximation for the escape velocity. The critical impact parameter xcrit at which matter Mc(x) = 47tR*pc2(x)9 (4) is just barely ejected from the star is determined by where evaluating equation (9) at the lower limit of the inequality. In solving for xcrit we have accounted for the variation in T through equation (10) merely 02(x) = i*x'^ix’W . (5) Jo because it was easy to do so numerically. The error made had we replaced Y by its constant surface value Note that Mc(l) = Mc is the total mass of the com- would have been negligible in comparison with the panion so that approximation inherent in assuming ves to be a func- tion of x only. In what follows we regard Y as inde- M^ix) pendent of x in accord with the previous discussion. Mc'(x) = 2 (6) 27rR a>2(l) The corresponding mass fraction stripped from the star is In our model, the velocity v(x) imparted to the shell at is determined by conservation of momentum: ¿WpOF) = 1 - Mc(xcrit)lMc. (11)
[Msn' + Mc\x)]v(x) = Msn'i;sn , (7) The basic geometry is illustrated schematically in Figure 1. 2 where MSN' = MSN/47ra0 is the mass per unit area of In Figure 2 we have evaluated xcrit and Fstrlp for an the ejecta, the velocity of the ejecta, and a0 the n = 3 polytrope, which is appropriate for the massive, initial separation of the two stars. The ratio of the relatively unevolved, star one might expect for the companion at the time of the supernova explosion. 1 For comparison, we have also constructed a 10 M0 In WML this mass was denoted Mp since the primary is “red giant” model with a small core containing unambiguously distinguished from the compact remnant. In 18 percent of the mass and an envelope of radius the present context which includes the state prior to the 14 _9 3 explosion, identification of the primary is ambiguous whereas 10 cm and constant density p = 4 x 10 gcm“ ; identification of the companion to the exploding star is not. this model is illustrated in Figure 3.
© American Astronomical Society • Provided by the NASA Astrophysics Data System 197 5ApJ. . .200. .145W 2 4a crlt [ polytrope of«=3impliesanapproximatelinear The peakintheintegrandofequation(12)occursatj¡Rx\ follows fromequations(4),(5),and(11)whereby relation betweenTandFintheouterregions.This corresponding massisassumedtobeejected. cylindrical geometryofthecompanionstarincludingouter tonically rapidlydecreasingfunctionofx.Thevalue\o/ x ~0.2,sofor^0.5theintegrandisamono-Pin=t^sn^snI—I>(16) Fabiate isablated.Arrowsindicatethedirectioninwhich portions fromwhichmassfractioniWpisstrippedand material, withtheimpactingportiondenotedbyF,and explosion showingtheincidentplanewaveofsupernova No. 1,1975 Folate, areshownasafunction of the“geometrical”parameterT(seetext)formodelinwhich thecompanionhasstruc- ture ofapolytropeindex«= 3. striv in strip The steeplydecreasingdensitydistributionforthe Fig. 1.—Schematicrepresentationoftheeffect Fig. 2.—Thecriticalradiusbeyond whichmassisstripped,xt,andthefractionslost to strippingandablating,Ft, cri srip © American Astronomical Society • Provided by theNASA Astrophysics Data System *2(1) -$(*)c 2 *2(1) -3 -2 -I 0+1 *0)L 2 SUPERNOVAE INBINARYSYSTEMS x^>(x)dx . 1 (12) LOG y on constant densityenvelopeofmassM,Fcanagain 0.1 (cf.Table1).Forthe“redgiant”modelwitha where wefindtheproportionalityconstant be evaluatedsemianalyticallywithgoodaccuracy: the lowerlimitx;so,toafirstapproximation, the incidentmassfractionFandhence(foragiven of theintegralisthusdominatedbyvalueat ^sn/^bs) ^AssumingthatRislessthantheradius directly strippedawayisstruckbytheejecta,astrong single variableM/-sNandfind minimize theupperlimitonFwithrespectto (Paczynski 1971).SinceM<.,onecan of theRochelobeRimplies momentum tothestar.Theincidentis will causemattertobeablatedandimpart shock willbedrivenintothestar;resultingheating estrip crit in cpre in SNpreSN L Finally, wenotethatanupperlimitcanbeplacedon When thatpartofthecompanionwhichisnot -^strip OC^hi(^crit)^'ï%^crit)5*^crit a — <~0.38+0.2log(M/M.)(14) 0 cpreSN III. IMPULSETOTHECOMPANION (12 ^ =(^)-^h)',(13a) 813b '■•“»“(ii'*')- <> a) Discussion F <0.08.(15) ln 147 197 5ApJ. . .200. .145W 2 6 9314 point thatthesoundspeedcwerecomparableto pressure behindtheshockisP~pv,whichincreases to becomparablethebindingenergyofstar. with densityp;whentheshockpenetratessofarinto between ahigh-velocityslabofgas(theejecta)and nova ejectaandthecompanionstarinsomewhatmore mass lossandimpulsewouldbeconsiderablyreduced. rupted entirelyorbegivenalargeimpulse.Onthe escape velocityv,thenthestarcouldeitherbedis- is distributed.Ifalltheshockedgaswereheatedto The importantquestionthenbecomeshowtheenergy and itispossiblefortheinterceptedenergy, the staratavelocitysomewhatgreaterthanv.The idealize theproblemasone-dimensionalcollision we shallneglectthesphericalgeometryofstarand detail (cf.Colgate1970;Cheng1974).Forsimplicity reverse shockwillpropagatebackintotheejecta(cf. the starthatthispressureexceedsinejecta,a ejecta initiallyactlikeapistonanddriveshockinto (the star).Sincethedensityofejectagreatly smaller amountofmatterinwhichc>v,thenthe other hand,ifmostoftheenergyweredepositedina exceeds thatoftheouteratmospherestar, stationary slabofgaswithastrongdensitygradient amount ofmasscomparabletothatintheejecta, high velocityandisanalogous totheejecta;region the encounterwithdensitygradientmovesata analytic solutionobtainedbyWhitham(1958)forthe This decelerationisdescribedbytheapproximate as itrunsfartherintotheincreasingdensityofstar. ‘piston” willslowdownandtheshockdecelerate McKee 1974).Oncetheforwardshockhassweptupan “core” andaconstantdensityenvelopewithp=4x10"gcm~R10cm. effectively infiniteinextent(see Fig.4).Hefoundthat key differencebetweenour problemandthatof 148 Whitham isthatheassumed theshockedgasis with thedensitygradientis analogous tothestar.The strong shockwhichpassesfromauniformmediuminto slightly differentproblemofthepropagationa one withadensitygradient.Thegasshockedpriorto s es SN ses Let usconsidertheinteractionbetweensuper- Fig. 3.—Sameas2butforthemodelinwhichcompanionistakentobea“redgiant”starof10M©with1.8 © American Astronomical Society • Provided by theNASA Astrophysics Data System WHEELER, LECAR,ANDMcKEE ejecta isexplicitlytakenintoaccount andmomentumiscon- shock intostar,(a)Cheng’s(1974)modelisequivalentto patterned afterthatofColgate (1970), thefiniteextentof effectively infiniteinextentbeforereachingthedensitygradient the shockhasbeenpropagatinginauniformgaswhichis Whitham’s (1958)solutionforthepropagationofashockinto served ;asaresulttheshockdoes notpropagateasfarintothe shock intothestarisoverestimated, (b)Inourmodel,whichis (the star).Asaresultthemomentum availablefordrivingthe an increasingdensitygradient.Inthismodelitisassumedthat star asinCheng’smodel. Fig. 4.—Distinctiofibetweenmodelsforpropagationof Vol. 200 No. 1, 1975 SUPERNOVAE IN BINARY SYSTEMS 149 ß the shock velocity *;sh varies as p~ , where ß = 0.24 When the shock has penetrated a mass fraction for a gas with a ratio of specific heats y = 5/3. The F(x, 0 represented by a normalized depth £ in a 2 0 52 pressure behind the shock varies as pvsh ~ p - ; the cylindrical shell at a normalized cylindrical radius x, resulting pressure gradient acts to decelerate the gas momentum conservation gives shocked previously to a higher velocity. Our problem will begin to deviate from his when the reverse shock Msn'vsn = [Msn' + Mc'(x)F(x, OH*, 0 , (17) (which was not included in his analysis) gets through where primes denote quantities per unit area, M '(x) the ejecta and a rarefaction wave is reflected back into c is given in equation (2), and v(x9 0 is the mean the shocked gas. This wave will propagate at the sound velocity of the shocked portion of the shell and the velocity in a gas which has been compressed by a corresponding incident ejecta. Note that the energy factor ^4 and so will overtake the forward shock not -2 per unit mass is proportional to (1 + MC'F/MSN') too long after the latter has swept up an amount of and hence is concentrated in the outer layers ; the mass comparable to that of the ejecta; this will result possibility that the star will be disrupted seems in a weakening of the forward shock relative to that unlikely. given by Whitham’s theory. The rarefaction wave The condition required for the matter to escape the accelerates the shocked stellar material away from the star is that its thermal energy exceed the binding companion ; the outermost material has been shocked energy, which can be expressed as to cs ~ and escapes freely from the star, but at some point in the star the thermal energy becomes too t;2(x,a>^s2. (18) small for the matter to escape from the star. Two approaches have previously been used to Only a small error is made by applying this criterion determine the amount of matter ablated and the in the frame in which the companion is at rest before corresponding impulse given to the star: Colgate it is struck by the ejecta. (1970) applied conservation of momentum and The mass fraction which escapes, Fes, is that at assumed that the fluid velocity just behind the shock which v(x, i) equals ves which, as before, we approxi- is equal to the mean velocity of all the shocked stellar mate as a constant, and so from equation (17) we get material; and Cheng (1974) numerically integrated the equations of motion in a manner which is essentially equivalent to the analytic solution of Whitham de- scribed above. Since Cheng’s analysis is equivalent to From equation (17) we see that the condition that the Whitham’s, it neglects the finite extent of the piston velocity v(x) imparted to the whole shell be just the and the corresponding effects of the rarefaction wave; escape velocity is just the condition that FeS(xcrit) = 1. the result is that overall momentum is not conserved. Our description of stripping and ablation are thus Cheng’s estimate thus provides an approximate upper compatible in the sense that equation (7) is just a limit to the velocity of the shocked material as shown special case of equation (17). schematically in Figure 4. The momentum transfer to the star can then be The total specific energy of a mass element written as immediately behind the shock will be E = Etherm&i + iv2. Both Colgate and Cheng assume that the internal Fe3 2 2 ll2 Ap' = M/ f (v - ves ) dF + 2Msn’vsn , (20) thermal energy initially imparted will become available Jo as kinetic energy as an element of matter adiabatically expands from the star; the work done by one part of where the second term expresses the impulse due to the reflection of the incident ejecta. For the usual case in the star on another serves only to lower the velocity of 2 a given mass element as more mass is swept up. The which (vSN¡ves) » 1, the integral gives , velocity attained on expansion is then V(2£thermai) Ap' = M 'i; (1 + In 2v lv ) , (21) which is numerically equivalent to the initial velocity sn sn SN es imparted in the case of a strong shock for which which is equal to Colgate’s result except for the ^thermal := The picture is then that a given element factor 2 in the logarithm which arises from our use of 2 2 112 of shocked matter “remembers” its initial velocity (v — ves ) rather than v as the integrand of equation via its internal energy even though its actual velocity (20). decreases as the shock continues to sweep up more The mass ablated from the companion star is matter; hence the matter is eventually ablated at its /*-*crit initial velocity. 2 -^ablate = -K 2ttxM '{x)F {x)dx . (22) Jo c eB b) Derivation The ablated mass fraction Fablate = Mablate/Mc can Since Colgate’s analysis is consistent with momen- then be determined with the aid of equation (19): tum conservation, it may give a more reasonable estimate of the ablated mass and momentum than <23) Cheng’s analysis. To determine the momentum and — mass of the ablated material, we generalize Colgate’s or 2 work to a certain extent. Fablzte x Txorlt , (24)
© American Astronomical Society • Provided by the NASA Astrophysics Data System 197 5ApJ. . .200. .145W equivalent ofthebindingenergyremnant. where Mistheinitialtotalmassofbinarysystem and Ai=Mp_sN-^snMisthemass system : induce aneccentricityeininitiallycircularorbit. weaken theshock. have neglectedsucheffectsasradiationdiffusionout impacted onthecompanion,butlarge-scaleinhomo- the ejectawerestillofsmallscalewhen within theslab.Thepresentanalysismightstillbe factors willplayanincreasinglyimportantrole.We Let €bethefractionofmasslostfrombinary geneities mightappreciablyalterthepicture.Alsowe have alsoassumedthattheejectaarehomogeneous the orderofradiuscompanion,geometrical possibly importantcorrectionsshouldbementioned. reasonable inanaveragesenseifinhomogeneities geometry throughout;butasseparationsbecomeof We havealreadydiscussedtheomissionofdetails of theincidentshockfrontwhichwouldtendto where polytrope andamodelredgiant,respectively.Note mental radialvelocity that FágateapproacheszeroatlargeYbecausewehave of therarefactionwave.Wehaveassumedaplane assumed thatstrippingoccursfirst. labiate asafunctionofYforthecasesan«=3 or since v£constant.Figures2and3alsogive T BErerem 150 es The suddenmasslossandradialimpulsewill This analysisisrathercrude,andsomeofthe The impulsetothecompanionresultsinanincre- © American Astronomical Society • Provided by theNASA Astrophysics Data System € =(M+MF)[M,(28) snbeejectcT [1ln = VuFX?+_(^N/t>es)], alnCTi(26) 3 5 3 3 2 4 7 2 10 0.170.730.29 ~1 indeterminate 0.01 0.78 ~10- 0.0060.013.6x10" 0.1 0.64 0.01 0.040.051.8xlO" 0.2 0.58 0.02 0.070.095.9x10~ 0.4 0.52 0.04 0.110.151.7x10' 0.001 0.87 ~10- 7.6x10~-05.8lO" 4 0.290.350.34 0.69 1.2 1.. 0.440.100.19 0.29 5.1x10" 2 0.370.190.28 0.47 0.28 5 0.260.430.34 0.77 2.2 Eccentricity DeterminationParametersasaFunctionofY (seetext)foraCompanionwith IV. INDUCEDECCENTRICITY ^ejeot ^Fg^rip Vr =(25) M{\-F^’ ce A Reject ■Xcrit FstripFt>iteject F ae the StructureofaPolytropewithn=3 WHEELER, LECAR,ANDMcKEE TABLE 1 2 2 Pipjvof =2fMaIRandY. then basicallydeterminedbytwoparameters, being considered.Withtheseapproximations,vlvis given inTables1and2forthetwotypesofmodels The functionVandotherparametersofinterestare « =3polytropemodels,and/ä6fortheredgiant. which isapproximatelyconstant:/æ3-5forthe c0T and where quite good:takingvlv»1,onefinds expressed intermsofM,_Afand¿^0*0, eccentricity isdeterminedbyeandvlv;canbe represented byan«=3polytropeoraredgiant.The stars inthesystem,providedcompanioncanbe binary orbitfollowinganexplosiveeventinoneofthe evaluate theeccentricityandsemimajoraxisofa some approximations.Thefirstapproximationis (see eq.[26]). and (vlv)isdeterminedbyv,[vY mass fractioneexceeds[1—fa/vo)]/!. r0 and or happen concurrently,onehas SNes cvreSNrem r0 r0SNes The expressionforv[vmaybesimplifiedbymaking The systemisdisrupted(û->oo,e1)iftheejected Using theresultspresentedabove,onecannow When boththemasslossandradialimpulse r0 2 FOT) EE[Tx/(1-F)], criteieot aw 2 2 (s)-fe)’ J =(1-)(1e) a 1-2e(vlv) € 0r / =1+In(2i>/t>), SNes 2 * +(Vr/Vo) 2 (1 -e) 1 - Vol. 200 (34) (33) (32) (31) (30) (29) 197 5ApJ. . .200. .145W 2 2 2 2A 2 17 the moredetailedcalculationstowithinabout10per- parameter estimateoftheeccentricity.Onecansafely two fairlyextrememodels(models9and12ofWML). values aslong(vjv)»e.Forvl^and equation (37a)andtheapproximatione£;const.= equation (29)isalsoarelativelysmalleffect.Useof disregard thevariationofewithRlainmostcases estimate ofetogive,respectively,athree-ortwo- numerical coefficientwaschosenasameanbetween generally appreciablylessthanunity.Theparticular approximation rationalizedwiththenotionthatwhile the strippingandablation. models asthedetailedcalculations, inviewofthe estimate fortheeccentricity ofthen=3polytrope equation (37b)mayactually giveaboutasgoodan included. Althoughitscrudenessismoreobvious, M ~,thecontributionoftoemustbe cent fore~1andmuchbettermoremoderate M/M inequation(29)willreproducetheresultsof significantly toe.Thevariationofthedenominatorin such conditionswhereFMnolongercontributes because (vlv)dominateseinequation(29)until Equation (37a)or(37b)canthenbeusedwithan duces (vlv)fromthemoredetailedcalculationsto appropriate valueof/,thisapproximateformrepro- simplified physicalmodelwe haveadoptedtoestimate ^sW^es issignificantlygreaterthanunity,|M/M with Y^0.5).Equation(37b)isanobviouslycrude within 10percentformostcasesofinterest(i.e.,those Equation (37a)representsamoredirectlyaccessible of (Rla)regardedasoneparameter.Withthe “two-parameter” formfor(v[v)withthecoefficient we have Then with have No. 1,1975SUPERNOVAEINBINARYSYSTEMS151 r0 0 snbeBE snt eiectc r0 r0 SNC 0 r0 V x0.09Y-for^0.5;henceinthatregimewe A fittoF(Y)forthepolytropewith«=3gives © American Astronomical Society • Provided by theNASA Astrophysics Data System 2A ^ 0.35(Rla). 0 /^ Q—vpi.7 \v/ MR 0. 0t vp ^-^SN%N/ ? 4 ~ 4Mv\a) 0.01 -1-01xIO' 0.1 -1-00.100.01 0.2 0.98~00.190.055 ces0 0.4 0.950.0250.360.390.35 1.0 0.60.450.360.813.6 4 Mv ces Eccentricity DeterminationParametersasaFunctionofT(seetext)for ^SN Companion witha“RedGiant”Structure •^-crit FstripFabiate-RejectV 2.4 (37b) (37a) (36) (35) TABLE 2 _1 _1 _1 -1 those basedonironcoresasfoundinmassivestars prove tohavemoredowithdetailsofouterenvelope in aclosebinarysystemwheremasstransferhas found inintermediate-massstars(~4-8M),and categories :thosebasedondegeneratecarboncoresas Arnett 1973;Tammann1973). curves andspectralproperties,phenomenawhichmay fundamentally differentiatedbytheiropticallight Type IIsupernovaeislessclearsincethelatterare explosion mechanismbeitthermonuclear(cf.Wheeler, may bedifferentiatedfromlow-mass,high-velocity root ofthemassenvelope. in whichcasethevelocityscalesinverselyassquare with amoremassivebutlesstightlyboundenvelope, have avelocity~10,000kmsunlessitinteracted teristic (escape)velocityof~10,000kms.Asmall few MeVpernucleoncorrespondingtoacharac- The bindingenergyatthesurfaceofsuchacoreis be significantlylessthan a Chandrasekharmass mass ofthecorewhichgenerates aneutronstarcannot rounded byalessdenseenvelopeofcarbonand hydrogen layer.Theironcorecouldbebareorsur- such asimpleschemereallycharacterizesTypeIand Wilson 1971),orapulsarmechanism(cf.Ostrikerand rounding thedegeneratecoreinwhichexplosion supernovae merelybythemassofenvelopesur- amount ofmassejectedfromsuchacorewouldthen stood intermsofamodelwhichtheexplosion have highermass(~solarmasses)butlowervelocities km s);andTypeII,whicharegenerallybelievedto have lowmass(~0.1M©)andhighvelocity(~10,000 categories: TypeI,whicharegenerallythoughtto In anycaseevolutionarycalculationsindicatethatthe (> 10Mo).Thecarboncorecouldbebare,particularly structure thanwithgrossenergetics(cf.Falkand Gunn 1970;BodenheimerandOstriker1974).Whether Buchler, andBarkat1973),neutrinotransport(cf. occurs. Thisfollowsroughlyindependentlyofthe sets alowerlimitontheejected massM. occurred, oritcouldbesurroundedbyadistended originates insomesortofelectrondegeneratecore. (~5000 kms).Thissituationcanbecrudelyunder- oxygen andperhapsadistendedhydrogenenvelope. 1973). Giventhemassof starwhichexplodes,this 0 ~1.4M (Paczynski1970; ArnettandSchramm SN 0 Thus perhapshigh-mass,low-velocitysupernovae Theoretical supernovamodelsfallintotwobroad Observationally, supernovaefallintotwobroad V. SUPERNOVAMODELS 152 WHEELER, LECAR, AND McKEE Calculations by Wilson (1971) and Wheeler and Wilson (1975) indicate that one cannot rule out the possibility of neutron star formation without mass ejection. Even in the case of direct collapse to form a neutron star, however, the binding energy must decrease, accompanied by an effective loss of rest mass in the form of neutrinos. We have accounted for this effective mass loss by fitting the following simple formula to the binding energy for the neutron star models of Baym, Pethick, and Sutherland (1971):
^frem — (Afpre.SN)
x MsN 2 [i - 0,13^^ ~Q j ] , (38) where MYem and Mpre_sN are the proper masses of the remnant and the presupernova object, respectively. This gravitational mass defect is not important unless the actual mass ejected is small, but it does provide a finite lower limit to the mass loss in any process of neutron star formation. To parametrize the supernova explosions, we have selected two basic presupernova models. One is a bare core of M = 1.4 M© and the other a core of 4M©. The latter is envisaged as having an inner core of 1.4 M© and is patterned after the work of Arnett and Schramm (1973). Such a presupernova configuration has been invoked by van den Heuvel and Heise (1972) in discussing the binary evolution leading to the production of compact X-ray sources. While cores of M > 4 M© are surely plausible, there is some sugges- Fig. 5.—The eccentricity e is plotted as a function of the tion of a gap between 1.4M© and 4M© cores for ratio of the radius of the companion, R, to the initial separa- evolutionary reasons, and hence we have not con- tion a0 for three models: one corresponding to Her X-l (see Table 3); one corresponding to Cen X-3 (see Table 4); and one sidered presupernovae of intermediate masses. for which the companion is a “red giant” model with M = We have chosen a range of velocities for the super- 10 M©, Moore = 1.8 M©, envelope density /> = 4 x 10“9 nova ejecta from 2000 to 20,000 km s-1. The lower g cm-3, and = 1014 cm (see Table 6). The solid lines repre- limit is suggested from the maximum filament veloci- sent the complete calculations discussed in the text. The dashed -1 lines represent the results for the approximation given in ties of about 1500 km s in the Crab Nebula (cf. eq. (44), as discussed in § IV. The dot-dashed lines represent a Trimble 1968), with certain license taken to allow for minimum case where only the impulse from the incident the inefficiency of the supposed initial shock accelera- momentum is taken into account, thereby neglecting reflection tion. The upper limit is characteristic of the maximum of the ejecta and ablation. For reference, the present values of mass motion velocities observed (Minkowski 1968) Rla0 corresponding to the models illustrated for Her X-l and Cen X-3 are ~0.17 and ~0.21, respectively. These values are and is about the escape velocity from a 1.4 M© white probably not those existing at the time of the explosion because dwarf. of the effect of the explosion and the subsequent evolution of We assume the supernova explosion to be spherically the system inferred from the present low eccentricity (see text symmetric. Any asymmetries would tend to increase for discussion). the eccentricity imparted (see De Loore et al. 1974). incident momentum, omitting both reflection of the VI. APPLICATION supernova ejecta and ablation from the companion. Figure 5 gives the eccentricity as a function of Rla0 A second model corresponds to Cen X-3 where for several models of interest. One model is of the type again we take parameters from WML (their model 12), pertinent to Her X-l as discussed by WML (their M — 15.6 M©, M . = 4 M©, Mg^- = 3.7 A/© c pre SN 11 model 9). We take M = 2.09 M©, M e_s = 1-4 M©, (yielding MBe ~ 0, e ^ 0.19), R = 3.2 x 10 cm, c pr N 1 Msn = 0.1 M© (yielding MBE = 0.15 M© and e ^ and vSN = 5000 km s^ . The approximation of equa- 11 -1 0.072), i? = 1.1 x 10 cm, and t;SN = 20,000 km s . tion (39) and incident impulse only are again given for Also shown for comparison is the approximation comparison. A third model has Mpre.sN = 1.4 M©, -1 Msn = 0.1 M©, and = 20,000 km s , but the e2 = f!j-0.35(W- (39) companion is taken to be the massive “red giant” with 14 Mc = 10 M© and R = 10 cm. The increase in eccentricity as a0 decreases as shown and the case where the impulse is due only to the in Figure 5 is primarily due to the impulse given the
© American Astronomical Society • Provided by the NASA Astrophysics Data System 1 co CM o o O' LO ^r LO ¡s ft
TABLE 3
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