A Self-Consistent Model for a Full Cycle of Recurrent Novae -- Wind Mass Loss Rate and X-Ray Luminosity

0.76 MB·

by  Mariko Kato

TOAPPEARINTHEASTROPHYSICALJOURNAL
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
ASELF-CONSISTENTMODELFORAFULLCYCLEOFRECURRENTNOVAE–WINDMASSLOSSRATEAND
X-RAYLUMINOSITY
MARIKOKATO
DepartmentofAstronomy,KeioUniversity,Hiyoshi,Yokohama223-8521,Japan;
HIDEYUKISAIO
AstronomicalInstitute,GraduateSchoolofScience,TohokuUniversity,Sendai,980-8578,Japan
7
1 AND
0 IZUMIHACHISU
2 DepartmentofEarthScienceandAstronomy,CollegeofArtsandSciences,TheUniversityofTokyo,3-8-1Komaba,Meguro-ku,Tokyo153-8902,Japan
toappearintheAstrophysicalJournal
n
a
ABSTRACT
J
An unexpectedlyslow evolutionin the pre-optical-maximumphase was suggested in the very shortrecur-
7
renceperiodnovaM31N2008-12a. Toobtainreasonablenovalightcurveswehaveimprovedourcalculation

methodbyconsistentlycombiningopticallythickwindsolutionsofhydrogen-richenvelopeswithwhitedwarf
]
E (WD)structurescalculatedbyaHenyey-typeevolutioncode. Thewindmasslossrateisproperlydetermined
H with high accuracy. We have calculated light curve models for 1.2 and 1.38 M⊙ WDs with mass accretion
ratescorrespondingtorecurrenceperiodsof10and1yr,respectively.Theoutburstlasts590/29daysinwhich
.
h thepre-optical-maximumphaseis82/16days,for1.2/1.38M ,respectively.Opticallythickwindsstartatthe
⊙
p endofX-rayflashandceaseatthebeginningofsupersoftX-rayphase. WealsopresentsupersoftX-raylight
- curvesincludingapromptX-rayflashandlatersupersoftX-rayphase.
o
r Subjectheadings:nova,cataclysmicvariables–stars: individual(M31N2008-12a)–whitedwarfs–X-rays:
t binaries
s
a
[
1. INTRODUCTION tic observationalplans (Moriietal. 2016; Katoetal. 2015,

1 2016). Katoetal. (2016)showedthatoutburstsofrecurrent
A nova is a thermonuclear runaway event on a mass-
v novaeare so weak thattheir pre-optical-maximumevolution
accretingwhitedwarf(WD)(Nariaietal. 1980;Iben 1982;
5 couldbeveryslow. TheyreportedthenondetectionofX-ray
Prialnik 1986; Sionetal. 1979; Sparksetal. 1978). The-
2 fluxduring8dayspriortothe2015outburstofM31N2008-
oretical light curves of nova outbursts have been calculated
8 12a.TheyestimatedthattheX-rayflashhadalreadyoccurred
mainlybased ona steady-stateapproximation(i.e., optically
1 long (9- 16days) before the optical maximum. Such unex-
thickwindtheory:Kato&Hachisu 1994)withsuccessfulre-
0
pectedslowevolutionoftheveryearlyphase,inspite ofthe
. production of characteristic properties, such as multiwave-
1 lengthlightcurvesincludingsupersoftX-rayphase,ofanum- veryfastopticaldecline,indicatesthefactthatwedonotfully
0 understandnovaoutburstsyet.
ber of classical novae (Hachisu&Kato 2006, 2007, 2010,
7 Recurrentnovaeofveryshortrecurrenceperiods(P <10
2014,2015,2016a,b;Hachisuetal. 2008;Katoetal. 2009). rec
1 yr) were recently reported: 0.5 or 1 yr for M31N 2008-12a
In spite of these successfulresults, this methodis inapplica-
: (Darnleyetal. 2014, 2015, 2016; Henzeetal. 2014, 2015;
v ble when a steady-state is not a good approximation. Such
Tangetal. 2014; Darnleyetal. 2016), 5 yr for M31N
i failuresoccurforveryfastevolutionsinrecurrentnovaelike
X 1963-09c(Williamsetal. 2015),∼6yrforLMCN1968-12a
U Sco and RS Oph and pre-maximum-opticalphases of all
(Darnleyetal. 2016), and 10 yr for U Sco (e.g., Schaefer
r kinds of novae. Thus, we need to take a different approach
a 2010). Recurrent novae are binaries harboring a massive
tofollowtheentireevolutionofshortperiodrecurrentnovae,
white dwarf (WD). Short recurrence period novae occur in
whichweaiminthepresentwork.
Multicycle nova outbursts have been calculated with verymassiveWDsofMWD&1.2M⊙withveryhighmassac-
Henyey-type evolution codes. These codes, however, meet cretionratesofM˙acc&1.5×10- 7M⊙yr- 1(Prialnik&Kovetz
numerical difficulties when the nova envelope expands to a 1995; Wolfetal. 2013a,b; Tangetal. 2014; Katoetal.
giant size. To continue numerical calculation beyond the 2014). Such massive WDs are considered to be candidates
extended stage, various authors have adopted various mass for Type Ia supernova (SN Ia) progenitors (Hachisuetal.
loss schemes and approximations (Prialnik&Kovetz 1995; 1999a,b; Hachisu&Kato 2001; Hachisuetal. 2010;
Kovetz 1998; Denissenkovetal. 2013; Idanetal. 2013; Han&Podsiadlowski 2004; Li&vandenHeuvel 1997;
Wolfetal. 2013a,b; Katoetal. 2014, 2015; Tangetal. Kato&Hachisu 2012). SNe Ia playveryimportantrolesin
2014). These works, however, paid little attention in repro- astrophysicsasastandardcandleformeasuringcosmological
ducing reliable nova light curves, so there is still room for distances (Perlmutteretal. 1999; Riess 1998) and as main
improvement. producersofirongroupelementsinthechemicalevolutionof
Recently, X-ray flash of a nova, a brief X-ray brightening galaxies(Kobayashietal. 1998). However,theirimmediate
beforetheopticalmaximum,cameintoattentionwithrealis- pre-explosion progenitors remain unclear (e.g., Maozetal.
2014; Pagnotta&Schaefer 2014). Thus, studies of very
[email protected] shortrecurrenceperiodnovaeareveryimportant.
2 Katoetal.
Ouraimofthisworkistoestablishaself-consistentnumer- start of the calculation. In the 1.38 M⊙ model, we restarted
icalmethodfor modelinga fullcycle of novaoutbursts. We the accretion immediately after the mass loss stopped. This
combine the structure of optically thick winds with a model isbecauseweadoptinternalmodelsofKatoetal. (2016). In
obtained by our Henyey-type code at each phase of a nova, the 1.2 M⊙ model, we restarted accretion when the outburst
which gives a mass loss rate and surface values of tempera- isalmostfinished,tocomparethedurationofthesupersoftX-
tureandluminositythatarerequiredtocalculatemultiwave- ray phase with thatobtained from the static sequence which
length light curves. In order to obtain successful fittings in doesnotincludemass-accretion(seeSection3.3).
˙
ohuigrhfimrstasast-taecmcprettwioenhraavteeocfhoMs˙eancca=12..20M×⊙10W- 7DMw⊙ithyra- 1vberey- casTehbeyvcaalusee.sTohfethdeetpaialrsaomfettheersse, Mpa0r,amR0e,tearnsdarRe1exaprelacinheodseinn
cause the nova explosion is weak and expansion is slow, so Section5.2.
the envelope structure is close to that of steady-state enve-
lope. We have also calculated wind structures fitted to the 2.2. WindMassLoss
evolutionmodelsofa1.38M⊙ WDwith1yrrecurrencepe- Shortlyaftertheonsetofunstablenuclearburning,theen-
riodpublishedinKatoetal. (2016). Section2describesour
velopeexpandsandopticallythickwindsareaccelerated.The
numericalmethod, and Section 3 presents our results on the
occurrenceofwindsisdetectedatthephotosphereusingthe
1.2M⊙ WDindetail. The1.38M⊙ WDmodelisbrieflyde- BC1 surfaceboundaryconditioninKato&Hachisu (1994).
scribed in Section 4. Discussion and conclusions follow in
Toobtainenvelopestructureswithwinds,wesolvetheequa-
Sections5and6,respectively.
tionsofmotion,masscontinuity,radiativeenergytransferby
diffusion,andenergyconservationassumingsteady-statecon-
2. NUMERICALMETHOD
ditionsandsphericalsymmetry(Kato&Hachisu 1994). The
WecalculatedWDstructuresfromthecenteroftheWDup
boundaryconditionsof the wind solution are set at the pho-
tothephotosphere. We adopttheopticallythickwinds(e.g.,
tosphere and critical point (=sonic point) (Kato&Hachisu
Kato 1983,1985;Kato&Hachisu 1994)asthemainmech- 1994),whichdeterminestwovalues,i.e.,thetemperatureand
anism of mass loss during the nova outburst. The optically
radius of the critical point, (T , R ). We calculated the op-
cr cr
thickwindshavebeenappliedtoanumberofnovaeandsuc-
tically thick wind solutions down to a chosen inner bound-
cessfully reproduced light curves and characteristic proper- ary,i.e.,wherethetemperaturerisestoaspecifiedvalue(e.g.
ties(e.g.,Hachisu&Kato 2006, 2007;Hachisuetal. 2008; T =T)andobtaintheenvelopemassofthewind(Mwind)be-
Hachisu&Kato 2010, 2014, 2015, 2016a,b; Katoetal. f env
tween the photosphereand the pointof T =T. In short, for
2009). The interior structures of WDs are calculated with a f
a chosen temperature (T =T) a wind solution is character-
Henyey code (Section 2.1). When the optically thick winds f
ized by T and R . The other quantities are automatically
occur the structure of mass-losing envelope is calculated as cr cr
calculated,such asthe massloss rate, windvelocity,and lo-
describedinSection2.2. WeusetheUV planeforourfitting
calluminosity(L ). IngeneralL isthesumofradiativeand
processofouterandinnerstructures(Section2.3)intheway r r
convective luminosities. In our calculation convection does
asdescribedinSection2.4.
notoccuratthematchingpointandL isalwaysequaltothe
r
radiativeluminosity.
2.1. Time-dependentCalculation
Evolutionmodelsofmass-accretingWDsarecalculatedby 2.3. FittingintheUV Plane
using the same Henyey-type code as in Katoetal. (2014,
In the fitting procedure we use the UV plane
2016) and Hachisuetal. (2016). Accretion energy outside
(Chandrasekhar 1939; Schwarzschild 1958; Hayashietal.
thephotosphereisnotincluded. Wehavecalculatedmulticy-
1962),whereU andV arethehomologyinvariantsdefinedas
clenovaoutburstsuntilthecharacteristicpropertiesofflashes
becomeunchanged.Theevolutionisstartedwiththeequilib- dlnM 4πr3ρ
riummodelforagivenmassaccretionrate(Katoetal. 2014), U ≡ r = , (2)
dlnr M
inwhichtheWDcoreisthermallyinasteadystate. Thus,the r
WDcoretemperatureisnotafreeparameter. and
Inalaterphaseofashellflash,thephotosphericradius(as dlnP GM ρ
V ≡- = r , (3)
wellastheluminosity)increasestothepointwheretheouter
dlnr rP
boundary condition of our evolution code fails because the
densitybecomestoosmallatT ∼2×105K(attheFeopacity whereMr isthemasswithintheradiusr.
In general, the fitting point of the wind solution (U , V )
peak, Iglesias&Rogers 1996). To avoid the numericaldif- w w
moves downward in the UV plane if we increase the wind
ficulty, we start mass loss, which hinders the growth of the
radius, when the photospheric radius Rn at time-step n ex- masslossrate(i.e.,increaseTcrorRcr). Ontheotherhand,the
ph fittingpointofHenyey-codesolutions(U,V)movesupward
ceedsaradiusR . ThemasslossrateforthenextstepM˙n+1 is f f
0 evol fora largermasslossrate. Thus,we findauniquematching
calculatedas
pointforthesamewindmasslossrateandenvelopemassin
M˙n+1 =M˙n exp[10(logRn - logRn- 1)]. (1) theUV plane. MatchingU andV at Tf guarantees the con-
evol evol ph ph tinuation of the mechanical structure at the fitting point so
AtthebeginningofmasslossM˙n isreplacedwithM˙ (<0), that the radius and density are automatically matched there
evol 0
whichisanarbitraryparametertogiveaninitialmasslossrate (seeEquations(2)and(3)). Inaddition,asthereisnoenergy
atRph=R0. Thisequationgivesarapidincrease(decrease)in source/sinkaroundthefittingplacethelocalluminosityLr is
˙ automaticallyfitted.
|M | when the photospheric radius is increasing (decreas-
evol
ing).Equation(1)isuseduntilR becomessmallerthanara-
˙ ph 2.4. FittingProcedure
diusR ,thenM isswitchedtotheaccretionrategivenatthe
1 evol
RecurrentNovaModel 3
Inthewindphaseweneedseveralstepsofiterationuntilwe
obtainafinalresultinwhichthewindmasslossisconsistently TABLE1
takenintoaccount.Ourfittingprocedureisasfollows. SUMMARYOFRECURRENTNOVAMODELS
(1) First, we calculate nova outbursts with the Henyey code
Subjecta units 1.2 M⊙ 1.38 M⊙
assumingatime-variablemasslossrateuntiltheshellflashes
reachalimitcycle.Weusethelastonecycleinthefollowing M˙acc 10- 7M⊙yr- 1 2.0 1.6
procedure. Prec yr 9.9 0.95
logTcore K 7.93 8.03
(2)WhenthemasslossoccursontheHenyeycode,wecalcu- Macc 10- 7M⊙ 17 1.4
lateopticallythickwindsolutionswithaspecifiedparameters, Mig 10- 7M⊙ 26 2.0
(Tcr,Rcr),downtothefittingplaceT =Tf. Ifthefittingcondi- Lmnuacx 106L⊙ 2.2 3.9
tion(Uw,Vw)=(Uf,Vf)isnotsatisfied,wechangeTcr andRcr logTnmucax K 8.11 8.23
untiltheconditionissatisfied. Weneedtypically20iterations tnovab day 590 29
donebyhand. ttwX-infldashc ddaayy 113100 109.77
(3)IftheenvelopemassMewnivnd ofthewindsolutiondoesnot tSSSc day 440 7.8
match to that of the interior solution we change the fitting trised day 82 16
place T and repeat the same process until the two envelope tdecaye day 510 13
f
masses are sufficiently close to each other. A few to several aTheWDmassandmassaccretionratearegivenparameters
iterationsareneededfortheprocess. andrecurrenceperiodandotherquantitiesaretheresultsof
calculation.
w(4i)thWaewreeplllamceadtcthheedowutienrdpsaorltu(tTion<.TTfh)eonf,twheeeovbotlauitnioancmonotdine-l bcdduurraattiioonnooffththeeflXas-hrawyitflhasLhpha>nd1S0S4LS⊙p.hase: LX(0.3- 1.0
uousstructure from the WD core up to the photospherethat keV)>103L⊙for1.2M⊙,but>104L⊙for1.38M⊙.
consistently includes the optically thick wind at a specified d thetimeelapsedbetweenthetimeofthepeaknuclearlu-
time during the wind phase. The photospheric temperature, minosityandthetimeofthepeakmasslossrate.
velocity, and wind mass loss rate etc. are uniquely deter- eopticaldecaytimecalculatedastdecay=tnova- trise.
mined.
thatwestopmasslossatlogT (K)=5.60).Theresultantmass
(5)Weneedtodosuchaprocedure(2)-(4)foreachtimestep. ph
lossratesareshownlaterinFigure7.
However,ourwindphasecontains∼10,000timesteps,which
Table1summarizestheresultsofourcalculations.TheWD
isfarbeyondthecapacityofmanualhandlings. Forthisrea-
massandmass-accretionratearegivenparameters. Theoth-
son,weselectabout30epochsandapplytheiterationproce-
ers are the results of calculation: the recurrence period P ,
dure(2)-(4)byhand.Thus,weobtainstellarstructuremodels rec
temperature at the WD center T , mass accreted after the
foronecycleofnovaoutburst. core
previousoutburst until the beginning of the currentoutburst
(6) The mass loss rates of the optically thick winds, which
(=masstransferredfromthe companion)M , ignitionmass
variestimetotime,isgenerallydifferentfromthoseassumed acc
M which is the envelope mass at the epoch of maximum
instep(1).Werepeatedtheprocess(1)-(5)assumingdifferent ig
nuclearluminosityLmax, maximumtemperatureattheepoch
mass loss rates M˙0 in Equation(1) until the calculated wind ofmaximumnuclearnluucminosityTmax,novaoutburstduration
mass loss rates well match to the assumed values. It takes nuc
severaliterations. tnova forLph>104L⊙,durationofthewindphasetwind,dura-
Amongtheaboveprocess,themostdifficultprocessisstep tionsoftheX-rayflashtX- flash,andSSSphasetSSS,risingtime
t which is the time elapsed between the time of the peak
(1). A Henyey-type code often meets numerical difficulties rise
nuclear luminosity and the time of the peak mass loss rate,
withoutassuminglargemasslossrateswhentheenvelopeex- decaying time t =t - t . Here, the durations of the
pands. If we assume too large mass loss rates, however, the decay nova rise
X-rayflashandSSSphasearedefinedasthetimespanwhen
procedure(1)-(6)doesnotconverge.
Notethatthefittingpointbetweenthewindandtheinterior LX>104L⊙for1.38M⊙followedbyKatoetal. (2016),but
should be carefully chosen especially in the first trial. The LX >103 L⊙ for 1.2 M⊙ because of faint X-ray luminosity,
where L means the luminosity in a band of 0.3- 1.0 keV.
fittingpointshouldbedeepenoughtobeinsensitivetotreat- X
Notethattheoutburstdurationt isdifferentfromthesum-
ments in the surface region, especially when we assume in- nova
adequatemass lossratesin the trialprocess. Thispointalso mation of the three phases, tX- flash+twind+tSSS, because the
definitionisdifferent.
shouldbeshallowenoughtoensurethetime-dependentterm
Figure1showsthelastcycleofour1.2M WDmodel.We
ǫ (gravitationalenergyreleaserateperunitmass: see equa- ⊙
tigon(1)inHachisuetal. 2016)isnegligible. stoppedthemassaccretion(M˙acc=2.0×10- 7M⊙yr- 1)when
Our procedure (1)-(6) currently needs many iterations by the luminosity increases to logLph/L⊙ = 4.32 (logTph (K)=
hand,andthus,takesmuchhuman-time. Forthisreason,we 5.85), and resumed mass accretion after the flash when the
havecalculatedonly1.2 and1.38M⊙ models. We certainly luminositydecreasestologLph/L⊙=4.0(logTph (K)=5.83).
needtoimproveourfittingprocedure. TheperiodofmassaccretionisdepictedinFigure1(b). The
recurrenceperiodisobtainedtobe9.9yr.
3. 1.2M⊙MODEL Figure1(a)showsthephotosphericluminosity,L ,nuclear
ph
We have calculated two models of 1.2 and 1.38 M . The burningenergyrelease rate integratedfromthe centerof the
⊙
1.38M modelarealmostthesameasthosepartlypublished WDtothesurface,L ,andtotalgravitationalenergyrelease
⊙ nuc
in Katoetal. (2016). Thus, we first describe the 1.2 M⊙ rate,LG,definedastheintegrationofǫgfromthecenterofthe
modelandthen summarizeforthe 1.38M resultsfocusing WD to the photosphere (equation (3) in Katoetal. 2016).
⊙
onthedifference.Forthemasslossparametersofthe1.2M⊙ ThesupersoftX-ray(0.3- 1.0 keV) luminosityLX is calcu-
modeldiscussed in Section 2.1 we adoptlogR0/R⊙ =- 1.25 latedassumingtheblackbodyemissionofphotospherictem-
(correspondingto the case that we start mass loss at logT perature T . The lower panel (b) shows the change of the
ph ph
(K)=5.54)andlogR1/R⊙=- 1.36(correspondingtothecase photospherictemperatureTphandradiusRph.
4 Katoetal.
FIG.2.—CloseupviewofaveryearlyphaseoftheshellflashinFigure
1:thenuclearburningluminosity(blackline),Lnuc,photosphericluminosity
(thick red line), Lph, and gravitational energy release rate (blue line), LG.
Mostofthenuclearluminosityisabsorbedintheburningshellasexpressed
bylargenegativevaluesofLG.Asaresult,thephotosphericluminosity,Lph,
ismuchsmallerthanLnuc.Thethinredlinedenotes10timesthephotospheric
luminosity,Lph×10.
itsmaximumatpointB.Astheenvelopeexpands,thephoto-
FIG.1.—Temporalchangeinonecycleofanovaoutburstona1.2M⊙
WDwithamassaccretionrateof2×10- 7 M⊙ yr- 1. (a)Thephotospheric spherictemperaturedecreasesandopticallythickwindsstart
luminosityLph(redsolidline),integratedfluxofnuclearburningLnuc(black toblowatpointC.Thewindscontinueuntilitreachespoint
dashedline),integratedgravitationalenergyreleaserateLG(bluedottedline), G. At point F the photospheric temperature attains a mini-
andsupersoft(0.3-1.0keV)X-rayflux(solidorangeline). (b)Thephoto-
mum. After that the photospheric radius decreases and the
pspehrieordicotfemacpcereratitounreisTpinhd(ibclaatcekdsboylidthleinheo)raiznodnrtaaldibulsueRlpihne(r.edsolidline).The temperatureTph increaseswith time. The envelopemass de-
creasesowingtowindmasslossandhydrogenburning.
A closeup view in a very early phase is shown in Figure
Figure3(b)showsacloseupviewofthesametrack. There
2. ThenuclearenergyreleaseratereachesLnuc=2.2×106L⊙ isazigzagpart(D-E-F)arisingfromthechangeofchemical
at maximum. This energy is mostly absorbed in the lower
composition in the surface layers. Just after the thermonu-
partoftheburningregionasshownbyalargenegativevalue
clearrunawaysetsin,convectiondevelopsthroughouttheen-
of L (<0). Thus, only a very small part of L is trans-
G nuc velope up close to the photosphere. In the uppermost layer
portedoutwarduptothephotosphere. Asaresult,thephoto-
where the convection never reaches, the chemical composi-
spheric luminosity L does not exceed the Eddingtonlumi-
ph tion remains to be the same as that of the accreted matter,
nosity. Thesepropertiesarethesameasthosereportedinthe
i.e., X = 0.7, Y = 0.28 and Z = 0.02. In the region where
1.38 M⊙ model (Figure 2 in Katoetal. 2016). Comparing theconvectionreachedthehydrogenmass-fractiondecreased
the two models in Table 1, we find that Lmax of the 1.2 M
nuc ⊙ to X ∼0.6. Figure 4 shows distributionsof hydrogenmass-
modelisalmostahalfofthe1.38M⊙model,i.e.,theoutburst fraction in the envelope at selected stages, in which all the
isveryweak.
layerwithX=0.7havebeenblownoffinthewindalittleafter
Theabsorbedenergy(L <0)isreleasedinthelaterphase
G stageE.ThegradientofX resultsintheopacitygradient(for
of the outburst. As shown in Figure 1(a), L (>0) reaches
G larger X, opacity is larger owing to electron scattering) that
about 10 % of L during the outburst. After the shell flash
ph causesadditionalaccelerationofthewinds. Thus,the winds
ends,L quicklydecreases,butL slowlydecreasesbecause
nuc ph blowrelativelystrongeruntiltheregionwithX =0.7iscom-
themass-accretionresumesandL contributestoL . During
G ph pletely blown off. As the chemicalcomposition distribution
thequiescentphaseLph≈LGisalmostconstant(logLph/L⊙& approachesuniformwindsbecomeweakandthephotospheric
2.4)asshowninFigure1(a).
temperatureriseswithtime.Astheshellflashgoesontheen-
velope expands and the photospheric temperature decreases
3.1. H-RDiagram
again. Thus, the track in the H-R diagram has a zigzag ex-
Figure3 showsthe trackin the H-R diagramfor thesame cursionbeforeitreachespointFinFigure3(b).Theenvelope
periodinFigure1. Duringthe quiescentphasetheaccreting masscontinuouslydecreasesthroughthesestages. Notethat
WD model stays around point A. After the unstable nuclear thedistributionofhydrogenabundanceX inFigure4isthatof
burning sets in, the bolometric luminosity quickly increases theevolutionmodel. InwindsolutionweassumedX-profile
whereasthe photosphericradiusstaysalmostconstant(from mimictothecorrespondingevolutionmodel.
pointAtoB).ThenuclearenergygenerationrateL reaches Thedottedand dashedlinesin Figure3(b)are the steady-
nuc
RecurrentNovaModel 5
FIG.3.—EvolutiontrackofthemodelinFigure1. Filledcirclesonthe FIG.4.— Runsofthehydrogenmassfraction,X,intheWDenvelopesat
trackcorrespondtothecharacteristic stages. A:minimumphotosphericlu-
selectedstages. (a)X distributionintheentirehydrogen-richenvelope. (b)
minosity,Lph=Lmphin. B:maximumnuclearluminosityLnuc=Lmnuacx. C:opti- Closeupviewofthesurfaceregionofpanel(a). Inthebothpanels,theleft
callythickwindsstart. D:firstlocalmaximumexpansion. E:theuppermost bound(M- Mr=0)correspondstothephotosphere.Theaccretedmatterhas
hydrogen-richenvelopeofX=0.7isblownoff. F:maximumexpansionof ahydrogencontentofX=0.7initially. Theenvelopemassdecreasesowing
thephotosphere. G:opticallythickwindscease. H:maximumphotospheric towindmasslossandnuclearburning.Eachstageisindicatedbesidetheline.
temperatureTph=Tpmhax. (a)Thebluearrowsindicate thedirectionofevo- TheoutermostX=0.7layerisblownoffjustafterstageE.Theconvective
lution. Thegreensolidlinesindicate thelinesoflogRph/R⊙=- 1and- 2 regionisindicatedbythehorizontallinesintheupperpartofpanel(a),which
(left).(b)Closeupview.ThethinsurfacelayerofX=0.7isblownoffinthe becomesnarroweranddisappearsjustafterstageE.Thematchingpointof
opticallythickwindsthroughC-D-E.Afterwardtheenvelopecompositionis theouterwindsolutionwiththeinnerstructureisindicatedbyfilledcircles.
almostuniformatX∼0.6.Thedottedanddashedlinesindicatesteady-state
sequenceswiththechemicalcompositionofX=0.6,Y=0.38andZ=0.02, maximumexpansionofthephotosphere(epochF).Thepho-
andX=0.7,Y=0.28andZ=0.02,respectively(seeSection3). tosphere reaches logRph (cm)=10.61 (0.58 R⊙). This figure
state/static sequences that represent the nova decay phase alsoshowsthedistributionofthelocalluminosityLr andthe
(Kato 1999; Kato&Hachisu 1994). The dashed line is localEddingtonluminositydefinedby
for a 1.2 M⊙ WD with the uniform chemical composition 4πcGM
of X = 0.7, Y = 0.28 and Z = 0.02 while the dotted is for L = WD, (4)
Edd
X =0.6, Y =0.38 and Z =0.02. The WD radius, which is κ
neededinobtainingsteady-statemodels,i.e.,theradiusatthe where κ is the opacity. We adopt OPAL opacity
bottomofthehydrogen-richenvelope,istakenfromourevo- (Iglesias&Rogers 1996). The local Eddington luminosity
lutioncalculationaslogR/R⊙=- 2.22. has a small dip corresponding to a small peak in the OPAL
Inthedecayphase(stagesF,G,andlater)theevolutionpath opacityowingtoionizedOandNe(logT (K)∼6.2–6.3),and
is veryclose to thatof the uppersteady-state sequence(dot- alargedecreasecorrespondingtoalargeFepeakatlogT (K)
ted line). In this phase, the envelope composition is almost ∼5.2(seeFigure6ofKatoetal. 2016).Thewindisacceler-
uniform at X = 0.6, Y = 0.38, and Z = 0.02, and the enve- atedwheretheopacityincreasesoutward. Thewindvelocity
lopestructureisclosetothatofthesteady-state(stageFtoG) barelyexceedstheescapevelocityvesc=p2GMWD/r.
andofhydrostaticbalance(afterstage G)as wellasthermal The dotted lines show the interior structure of the model
equilibrium. Thus, the nova evolves along with the steady- calculated with the Henyey code in step (1) in Section 2.4.
state/staticsequence. The connecting point with the wind solution is depicted by
In the rising phase, on the other hand, the envelope does filled circles. To demonstratethe smoothfitting, we plotthe
notyetreachthermalbalancebutapproachesthebalancewith modelstructurebeyondtheconnectingpoint.Itshowsagood
time. TheenvelopestillhasanX =0.7layeronitstop,sothe agreementwiththesteadystatesolutionnotonlyonthecon-
evolutionpathisclosetothelowerdashedline. nectingpointbutalsointhewideoutsideregion.
Figure6 demonstrateshow the optically thickwinds’qui-
3.2. Envelopestructure
etly’ceaseatstageG.Inthedecayphaseofthenovaoutburst,
Figure5showstheenvelopestructuresofthewindsolution thephotospherictemperatureincreaseswithtimeandthedip
(solid lines) and Henyey code solution (dotted lines) at the of L becomes shallower. The solid line depicts the enve-
Edd
6 Katoetal.
FIG.5.—Internalstructureatthemaximumexpansionofthephotosphere FIG.6.— SameasFigure5,butjustbefore(solid)andat(dotted)stageG.
(pointFinFigure3). Weplot,toptobottom,theescapevelocityvesc(thin ThewindbecomesweakercomparedwiththatatstageFinFigure5. The
solidblackline),windvelocityv(solidblackline),temperatureT (redline), local super-Eddington region becomes much narrower. At stage G, when
densityρ(blueline), localluminosityLr (blackline), andlocalEddington theopticallythickwindstops(dottedline),thereisnolocalsuper-Eddington
luminosityLEdd(redline). Thesolidlinesindicatetheopticallythickwind region.Noconvectionoccurs.
solution, whereasthedottedlinesshowtheinnerstructureobtained bythe
Henyeycode. ThelocalEddingtonluminosity,shownonlyfromtheoutside
decreases) toward stage F, the wind mass loss rate increases
ofthefittingpoint(filled circle), becomes smaller thantheradiative lumi-
nosityforlogr(cm)>10.1. Theinteriorsolutionisplottedbeyondthefit- with time. After stage F the mass loss rate decreases as the
ting point todemonstrate that the inner andouter solutions are veryclose starevolvesbackfromstageFtoGintheH-Rdiagram(Fig-
toeach other. Theopencircles denotethecritical point ofthewindsolu- ure 3). The wind stopped at stage G when the SSS phase
tion(Kato&Hachisu 1994). Convectiveregionhasdisappearedbeforethis
begins. In this way the X-ray turn on/off is closely related
stage.
totheoccurrenceoftheopticallythickwinds. Themassloss
lope structure just before stage G. The wind is still acceler- rateincreases(decreases)asthephotospherictemperaturede-
ated, buttheaccelerationregion,whereL >L , isgetting creases (increases) which will be shown later (Figure 12 in
r Edd
narrower.ThisregiondisappearsatstageG(dottedline).The Section5.1).
resemblance of the two structures demonstrates that the op- A sequence of static and steady-state solutions is also
tically thick winds gradually weaken and stop quietly. This showninFigure7(a)forcomparison. Thedottedredlinede-
propertywasalreadypointedoutinthesteady-statesequences pictstheX-raylightcurvecalculatedfromthemodelshown
(Kato&Hachisu 1994)thattheopticallythickwindsquietly by the dotted line in Figure 3. The rightmost point corre-
ceaseswhenthephotospherictemperaturerisestobeyondthe spondstotheepochofextinctionofhydrogenburning. This
Fepeak. X-ray light curve shows a good agreement with the evolu-
tional one but the termination point comes earlier. This dif-
3.3. X-rayLightCurveandWindMassLoss ference is explained as follows. The evolution time of the
equilibriumsequenceiscalculatedfromthemassdecreasing
In Figure 7 the upper panel (a) shows the bolometric and
rateowingtonuclearburning.Ontheotherhand,inthetime-
supersoft X-ray luminosities during the outburst, while the
dependentcalculation,theevolutiontimeisinfluencedbythe
lower panel (b) shows its closeup view with wind mass loss
gravitational energy release at the bottom of the envelope,
rates. Theoverallone-cycleviewisalreadyshowninFigure
i.e.,theevolutiontimeisdeterminedbythemassdecreasing
1. ThefirstbrightX-rayphaseis theX-rayflash thatoccurs
rateowingtonuclearburningandadditionalenergysourceof
just after the onset of thermonuclear runaway (Katoetal.
gravitational energy release rate L . This additional energy
2016). ThetinyX-raypeakshortlyaftertheX-rayflashcor- G
source, amounts about 10% of the photospheric luminosity
respondstothesmallleftwardexcursiontowardstageEinthe
(Figure1),lengthenstheSSSlifetime.Consideringthiseffect
H-Rdiagram(Figure3). ThesupersoftX-rayphasebeginsat
we can conclude that the steady-state sequence is consistent
stageGandendsshortlyafterstageH.
withthetime-dependentcalculation.
Thepanel(b)alsoshowsthetemporalchangeofthewind
mass loss rate which is closely related to the change in the
X-ray light curve. The X-ray flash ends when the envelope 4. 1.38M⊙MODEL
expands and the optically thick winds starts at stage C. As Our multicycle shell flash calculation of a 1.38 M⊙ WD
the photospheric radius increases (photospheric temperature with M˙acc =1.6×10- 7 M⊙ yr- 1 was published (Katoetal.
RecurrentNovaModel 7
FIG.7.—(a)Temporalchangeofsupersoft(0.3-1.0keV)X-rayluminosity
WLXD(bmlaocdkelli.nTe)haenddopttheodtoresdphleinriecdluenmoitneosstihtyeLXp-hra(syolliigdhrtecdulrinvee)ooffsttheead1y.2-sMta⊙te mFasIsGa.c8c.r—etioSnamraeteaosft1h.o6s×ei1n0-F7igMur⊙e1y,r-b1u.tforthe1.38M⊙ modelwitha
sequence, in which the wind stops at the small filled circle and hydrogen
nuclear burning stopsatthesmallopen circle (seesection 3.3). Epochof
stageHisindicated.(b)Closeupviewofthefirsthalfpartoftheupperpanel
˙
withthemasslossrateofopticallythickwindMwind(redline).Dottedpurple
˙
lineistheassumedmasslossrateMevolinprocess(1)inSection2.4.Stages
B,C,E,F,andGareindicated.
2016),butitincludedonlytheearlyphaseoftheX-rayflash.
UsingthemethodofcalculationdiscussedinSection2.4we
improvedthe fitting in the wind mass loss phase. We adopt
logR0/R⊙ =- 1.00 and logR1/R⊙ =- 1.15. The recurrence
periodis0.95yr. Thecharacteristicpropertiesofthismodel
aresummarizedinTable1. Here,wereportourresultsfocus-
ingonthedifferencefromthe1.2M case.
⊙
Figure 8 shows the last one cycle of our calculation. The
outburst duration (Lph > 104 L⊙) is 29 days. In the inter-
pulse phase, the gravitational energy release rate L domi-
G
natesthephotosphericluminosityL . Theenergybudgetin
ph
the early phase, corresponding to Figure 2, is already pub-
lished in Katoetal. (2016). These characteristic properties
aresameasthoseinthe1.2M WDmodel.
⊙
Figure 9(a) shows one cycle of nova outbursts in the H-R
diagram. Acloseupviewisshowninpanel(b). Similarlyto
the1.2M model(Figure3),stagesC-D-Ecorrespondtothe
⊙
periodthat the uppermostlayer with X =0.7, of mass 1.6×
10- 9 M ,areblownoffinthewind. Afterthat,thechemical
⊙
composition in the envelope is almost uniform at X =0.58,
Y =0.4andZ=0.02,andtheenvelopeevolvesalongwiththe
lineofsteady-state/staticsequence.
Figure 10 shows the internal structure at stage F. The ve- FIG.9.— SameasthoseinFigure3,butforthe1.38M⊙model.Inpanel
(b),thedottedanddashedlinesindicatethesequencesofsteady-state/static
locityquicklyrises(aroundlogr(cm)∼10.0)wherethelocal
solutionsofthechemicalcompositionofX=0.55, Y =0.43,andZ=0.02,
Eddingtonluminosityquicklydecreasescorrespondingtothe andX=0.7, Y =0.28,andZ=0.02. Inthesesequences,theWDradiusis
Feopacitypeak.Theenvelopemassatthisstageisassmallas assumedtobelogR/R⊙=- 2.57(takenfromourevolutioncalculation). See
1.1×10- 7M⊙ whichisnotenoughtoexpandtoagiantsize, themaintextformoredetails.
sothephotosphericradiusisassmallaslogR (cm)=10.52
ph
8 Katoetal.
FIG.10.—SameasthoseinFigure5,butforthe1.38M⊙model.Convec- FIG.11.— (a)Thenuclear burningluminosity, Lnuc (black line), photo-
tiveregionhasdisappearedbeforethisstage. spheric luminosity, Lph (redline), rateofenergy release owingtogravita-
((K0.)4=85R.1⊙0).,andthesurfacetemperatureisratherhighatlogTph t(iuopnpaelrcbolnuteralcintieo)ni,nLthGe(olouwtbeurrsbtlupehalsineeo)f,tahned1p.3h8otMos⊙phmeroicdetle.m(bp)erTahtuereX,-Trapyh
luminosityforthe0.3-1.0keVband,LX (blackline),andwindmassloss
To demonstrate a good matching between the interior and ˙ ˙
rateMwind(redline).ThemasslossrateMevolinitiallyassumedinstep(1)in
windsolutions, the interiorstructureoutsidethe fittingpoint Section2.4isshownbythedottedblueline.Anothertrialiterationisadded:
˙ ˙
isalsoplotted;thisouterpartwasreplacedwiththewindso- Mevol(dottedgreenline)andMwind(solidgreenline).Forthesetwolinesthe
lutioninstep(4)inSection2.4. Thedottedlineisveryclose timeisnormalizedtofitthewindphaseoftheredlinemodel. SeeSection
5.2formoredetails.
to the steady-state wind solution, indicating that the interior
structureissmoothlyconnectedtothewindsolution.
Figure11(a) showsthetemporalchangesin L , L , and
ph nuc
L as well as the change of T . The supersoft X-ray light
G ph
curveshowninpanel(b)iscalculatedfromL andT . The
ph ph
X-rayflash(whenLX>104 L⊙)lasts0.77daysandtheSSS
phase 7.8 days. Between them there is a faint X-ray peak
correspondingtostageE.
Thismodelmaybecomparedwiththeobservationalprop-
erties of the 1 yr recurrence period nova M31N 2008-12a.
Katoetal. (2015) estimated the WD mass of M31N 2008-
12a to be 1.38 M⊙ from the SSS duration that lasts about 8
days(Henzeetal. 2014, 2015; Darnleyetal. 2016). In the
presentpaper,wedonotconsidertheopticallightcurvesbe-
cause the relation between the optical brightness and wind
masslossrateinrecurrentnovaeisuncertain,whichremains
to be a future work. If we assume that the optical magni-
tude peaks at stage F, i.e., when the wind mass loss rate is
maximum, the X-ray turn-ontime is 4 daysafter the optical
maximum,whichisroughlyconsistentwithM31N2008-12a
(5.9±0.5 days: Henzeetal. 2015), considering the insuffi-
cientfittingqualityofour1.38M⊙ model. Detailedcompar-
ison with the observationis beyondthe scope of the present
paper. FIG.12.— Masslossratesofopticallythickwindsagainstthephotospheric
temperatureforthe1.2and1.38M⊙models.Theopenstarsymbolsindicate
5. DISCUSSION fromstageCtoFinFigures3and9,whereasfilledcirclesindicateatand
afterstageF.Thethinlinesdepicttherelationobtainedfromthesteady-state
5.1. WindMassLossRate
sequencesfor1.2and1.38M⊙ withthechemicalcompositionofX =0.6,
Figure 12 shows the mass loss rate of the optically thick Y=0.38,andZ=0.02(lowerblackline)andX=0.55,Y=0.43,andZ=0.02
(upperredline),respectively.
windscalculatedinthefittingprocess(Section2.4)againstthe
RecurrentNovaModel 9
photospherictemperatureT . Themasslossrateincreasesas 5.3. Comparisonwithotherworks
ph
T decreases. Intheextendedenvelopestheaccelerationre-
ph In the present work we calculated nova evolution models
gion (where the opacity increases outward) locates deep be-
withveryfinemasszoningsandshorttimesteps. Asaresults,
low the photosphere where the density is large (see Figures
we are able to followthe temporalchangeof the wind mass
5 and 6), which results in a large mass loss rate. The mass
loss, including the gradual increase and decrease, as in Fig-
lossratealsoweaklydependsontheWDmass. The1.38M⊙ ures7and11. Wealsoshowedhowtheupperenvelopewith
modelshowsaslightlyhighermasslossratesbecausethesur-
X =0.7isblownoff(Figure4andzigzagtracksinFigures3
facegravityisstrongerinmoremassiveWDs.
and9).
˙
This figurealso shows the two Mwind - Tph relations(solid Manytime-dependentcalculationsonnovaehavebeenpre-
lines) obtained from the steady-state sequences of 1.2 and sentedso far, butnoneof themshoweda zigzagbehaviorin
1.38 M⊙, correspondingto the upper sequence in each H-R the H-R diagram. For example, Prialnik&Kovetz (1995)
diagram (Figure 3(b) for 1.2 M⊙ and Figure 9(b) for 1.38 calculatednovamulticycleoutburstsforvariousWDmasses,
M⊙). The dependence on the chemical composition is very ofwhichlightcurvesarerecentlypublishedbyHillmanetal.
weak, so the solar composition sequences are omitted. The (2014). Theirlightcurvesshownozigzagbehaviorlikeours.
masslossratesobtainedfromourtime-dependentcalculations The differencemightarise fromthe differencein zoningsof
agreeverywellwiththoseofsteady-statesequencesthrough- theenvelope,becauseifthemasszoningdidnotresolvewell
outthewindphase,includingthezigzagpath,C-D-E-F-G. thesuperficialX =0.7region,themodelwouldbeunableto
˙
ThisagreementoftheMwind-Tphrelationsisreasonablebe- follow the temporal change of the chemical composition of
causetheinteriorstructureofourtime-dependentcalculation thesuperficiallayers.Thisreasoningmaybesupportedbythe
showsa goodagreementwith the steady state solution. The description“convectionhadalreadyreachedthesurfaceofthe
envelope is accelerated where the opacity rapidly increases star” (Section 2.1 Prialnik&Kovetz 1995). In addition, in
outwardatT ∼2×105K,sothatsimilarenvelopestructures their nova outbursts, the photospheric temperature suddenly
result in similar photospheric temperatures and wind mass- jumpstotheminimumvalue,andstaythere,andjumpsback.
lossrates. Thiscorrespondsto,inourFigure3,thatthestarjumpsfrom
stageCtoFinstantaneouslyandjumpsbacktostageG.Such
5.2. DetailsofNumericalCalculation abehaviorcouldoccuriflargetimestepswereadopted,skip-
InSection2.1wedescribedthemasslossrateM˙ which ping the gradual increase/decrease periods of the mass loss
evol
isinitiallyassumed. Afterseveraltime-consumingiterations, rates.
we found that better results are obtained if we choose R to Kovetz (1998)presentedanalgorithmtoplaceanoptically
0
be close to those wherethe opticallythick wind stopsin the thickwind solutionontopof a hydrostaticstellar configura-
steady-statesequence(Kato&Hachisu 1994),i.e.,whenthe tion.Thecalculationshowsnoindicationofthezigzagshape.
˙ Thisisbecausetheadoptedsmallnumberofmasszones(less
photospherictemperatureis close to thatwhere M =0 in
wind
than 50) is not sufficientto treatthe very thin X =0.7 layer.
Figure12.
˙ Despitethedetaileddescriptionofthealgorithm,noinforma-
Figure 7(b) shows M (dotted purple line) and resultant
evol tionontheconnectingpointisgiven.Inthe1.0M model,the
windmass-lossrate(solidredline)inthe1.2M⊙model.The windbeginsatlogT (K)=5.36andceasesatlog⊙T (K)=5.5,
resultant wind mass loss rates are somewhat larger than the eff eff
which is roughly consistent with our model with the OPAL
assumedvalue(19%atthetimeofmaximumwindmassloss)
opacity.Attheoccurrenceofthewindthemasslossratesud-
buttraceswelltheglobalchangeincludingearlysmallpeakin
denlyjumpsfromzerotothe maximumvaluefollowedbya
theepisodeofsurfacecompositionchange(seeSection3.1).
gradualdecrease. Thegradualdecreaseisconsistentwithour
This agreement is satisfactorily well, considering numerical
calculation.
difficultiesinstep(1).
˙ Idanetal. (2013) adopted Kovetz’ (1998) mass loss
Figure 11(b) shows1.38 M⊙ modelsfor two sets of Mevol schemewithsufficientnumberofmasszones(>8000),how-
(dotted lines) and correspondingresultant (solid lines) mass
ever,thepaperdiscussesinteriorstructuresindetailwithlittle
˙ ˙
loss rates Mwind. Blue dotted line show Mevol obtained with descriptionofthesurfaceregion.
logR0/R⊙=- 1.0andlogR1/R⊙=- 1.15,whichresultsinthe In the previous paper (Katoetal. 2015) we presented a
wind mass loss rate shown by the solid red line. The 25% multi-wavelengthlight curvemodelof 1.38M⊙ WD forthe
matching is obtained at the maximum mass loss rate (of the 1 yr recurrenceperiodnovaM31N2008-12a. We adopteda
redline)butthereis a shortperiodof a largediscrepancyof similarmethodofnumericalcalculationtothepresentwork,
factor2justbeforethewindsstop. but connected the interior structure at a much deeper place
˙
Dotted green line shows Mevol obtained adopting with the wind solutions. One of the reason why we chose
logR0/R⊙ = - 1.5 and logR1/R⊙ = - 1.3. In this case suchfittingpointwasthatournumericalcalculationwiththe
˙
M has no large enhancementin the final wind phase, but Henyey code did not converge unless large mass loss rates
evol
a sharp narrow peak in the very early stage. The resultant wereassumed,andthefittingplacewaschosenforthestruc-
windmasslossrates, M˙ , denotedbythe solidgreenline, turetobealmostindependentoftheassumedmasslossrate.
wind
however, are much small in the later phase. To summarize, However,insuchadeepenvelopewithalargermasslossrate,
˙ the gravitational energy release rate per unit mass ǫ is not
withasmallerM (dottedblueline)wegetaslightlylarger g
evol
˙ ˙ negligible,sothefittingofsteady-statesolutionswiththein-
M except the final peak, and with a larger M (dotted
wind ˙ evol terioryieldedmediocremodels. Theresultantmasslossrates
green line), we get a much smaller M . We suppose that
wind were about 3 times larger than the present work. Thus, the
a well converged model should have wind mass loss rates
windphasewasshorterbya factorof∼3. Thisroughlyex-
between them, which may be closer to the red line model
plainsthedifferencebetweenthepreviousestimateofadura-
thanthesolidgreenline. Therefore,weadoptedtheredline
tionof5daysandourpresentestimateof19days.
model.
10 Katoetal.
Inthepresentworkwehaveimprovedthenumericaltech- the maximum expansionof the photosphere, its temperature
nique and adoptmuch thinner atmosphereand mass zoning, isashighas105Kinthebothcases.
andconnecttotheinteriorstructureataplacewhereǫgisneg- 3.WeobtainedazigzagtrackintheH-Rdiagramcorrespond-
ligible. Asaresult,matchingofinnerandoutersolutionsare ingtotheepochthattheverysurfacezoneofX=0.7isblown
sufficientlywell as shownin Figures5 and 10. Our 1.2 M⊙ off in the wind, uncoveringthe region mixed by convection
modelshowsverygoodconvergenceofiterationprocess(Sec- during the runaway, which therefore has a lower H fraction
tion 2.4) throughout the wind phase that support our fitting correspondinglydifferentopacity.
procedure. Our 1.38 M⊙ modelshows a better convergence 4. Our 1.38 M⊙ model is roughly consistent with the
excepta shortperiodjust beforethe SSS phase. This model timescaleofthe1yrrecurrenceperiodnovaM31N2008-12a.
mustbeimprovedfurtherinthefuture. 5. The relation between the wind mass loss rate and photo-
spherictemperaturein ourevolutionmodelagreeswellwith
6. CONCLUSIONS those of steady-state/static sequence. The envelope struc-
ture at each stage are also very similar to those of steady-
Ourmainresultsaresummarizedasfollows.
state/staticsequenceinthewideregionexceptthedeepinte-
1. We present a new calculation procedure to follow nova
riorwherethegravitationalenergyreleaserateperunitmass
outbursts in which the optically thick wind is consistently
ǫ isnotnegligible.
introduced. We have found that steady-state wind solutions g
smoothlymatchwiththeinteriorcalculatedwiththeHenyey-
typecode. MK and IH express our gratitude to T. Iijima and Astro-
2. Wecalculatedonecyclenovaoutburston1.2and1.38M nomical Observatory of Padova (Asiago) for warm hospi-
⊙
WDs with the mass-accretion rates corresponding to the re- tality during which we initiated the present work. We also
currenceperiodsofP =10and1yr,respectively.Thewind thank the anonymous referee for useful comments that im-
rec
mass loss rate is accurately obtained in the new calculation proved the manuscript. This research has been supported
procedure. The X-ray flash ends when the optically thick in partby Grants-in-Aidfor Scientific Research (15K05026,
windsbegin. TheSSSphasestartswhenthewindscease. At 16K05289)oftheJapanSocietyforthePromotionofScience.
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