The Thermal Evolution following a Superburst on an Accreting Neutron Star

0.18 MB·English

by  Andrew Cumming

DRAFTVERSIONFEBRUARY2,2008
PreprinttypesetusingLATEXstyleemulateapjv.10/10/03
THETHERMALEVOLUTIONFOLLOWINGASUPERBURSTONANACCRETINGNEUTRONSTAR
ANDREWCUMMING1 ANDJAREDMACBETH
DepartmentofAstronomyandAstrophysics,UniversityofCalifornia,SantaCruz,CA95064;[email protected]
DraftversionFebruary2,2008
ABSTRACT
SuperburstsareveryenergeticTypeIX-rayburstsdiscoveredinrecentyearsbylongtermmonitoringofX-ray
bursters, and believed to be due to unstable ignition of carbon in the deep ocean of the neutron star. In this
Letter, we follow the thermal evolution of the surface layers as they cool following the burst. The resulting
lightcurvesagreeverywellwithobservationsforlayermassesintherange1025–1026gexpectedfromignition
4 calculations, and for an energyrelease &1017 erg per gram duringthe flash. We show that at late times the
0
coolingfluxfromthelayerdecaysasapowerlawF t- 4/3,givingtimescalesforquenchingofnormalType
0
∝
2 I bursting of weeks, in good agreement with observationallimits. We show that simultaneousmodelling of
superburstlightcurvesandquenchingtimespromisestoconstrainboththethicknessofthefuellayerandthe
n
energydeposited.
a
J Subjectheadings:accretion,accretiondisks-X-rays:bursts-stars:neutron

5
1
1. INTRODUCTION calmixing.Wedonotcalculateignitionconditions,butrather

treattheamountofenergydepositedandthethicknessofthe
1 Type I X-ray bursts from accreting neutron stars in low
fuellayerasfreeparameters2. In§2, we describeourcalcu-
v mass X-ray binaries involve unstable thermonuclear burn-
lationsof the subsequentthermalevolutionof the layer, and
7 ing of accreted hydrogen (H) and helium (He) (Lewin, van
presentasimpleanalyticmodelwhichhelpstounderstandthe
1 Paradijs,&Taam1995).Inthelastfewyears,longtermmon-
3 itoringofX-rayburstersbyBeppoSAXandtheRossiX-Ray numericalresults. At late times, the flux evolvesas a power
1 TimingExplorer(RXTE)hasrevealedanewclassofveryen- lawintimeratherthantheexponentialdecayfoundbyCB01
0 ergeticTypeIX-raybursts,nowknownas"superbursts"(see foraone-zonemodel.In§3,weusethelongtermfluxevolu-
4 tionofthelayertopredictthetimescaleofquenchingofType
Strohmayer & Bildsten 2003; Kuulkers 2003 for reviews).
0 The 1042 erg energies and several hour durations of super- Iburstsafterthesuperburst,andcomparetoobservations.
/
h burstsare100–1000timesgreaterthanusualTypeIbursts. In
2. TIMEEVOLUTIONOFTHESUPERBURST
p addition,theyarerare:sofar8havebeenseenfrom7sources,
- with recurrencetimes notwell-constrained, but estimated as Afterthefuelburns,thecoolingofthelayerisdescribedby
o
1 year (Kuulkers 2002; in ’t Zand et al. 2003; Wijnands theentropyequation
r ∼
t 2001),insteadofhourstodaysforusualTypeIbursts.
s ∂T 1∂F
:a ignTihtieoncuorfrecnatrbpoicntuartediesntshiatitessuρperb1u0rs8t–s1a0r9egdcume-to3.uHnsytdabrole- cP ∂t =- ǫν- ρ ∂r (1)
v genandheliumburnatρ 105–1∼06gcm- 3viatherp-process
Xi (Wallace&Woosley1981∼),producingchieflyheavyelements wheretheheatfluxisF =- K(∂T/∂r),andǫν istheneutrino
energy loss rate. The layer remains in hydrostatic balance,
beyondtheirongroup(includingnucleiasmassiveasA=104;
r inwhichcase ausefulindependentcoordinateisthe column
a Schatzetal. 2001),butwithsomeresidualcarbon(massfrac- depthyintothestar(units: gcm- 2),wheredy=- ρdr,giving
tionX 0.01–0.1)(Schatzetal. 2003). Cumming&Bild-
C ∼ apressureP=gy. Thesurfacegravityisg=(GM/R2)(1+z),
sten (2001)(hereafterCB01) showed that this small amount
where 1+z=(1- 2GM/Rc2)- 1/2 is the gravitationalredshift
of carboncan ignite unstablyoncethe massof the ash layer
reaches 1025 g(seealsoStrohmayer&Brown2002). This factor. In this paper, we assume M=1.4M⊙ and R=10km,
∼ givingz=0.31andg =g/1014cms- 2=2.45.
fits well with observedsuperburstenergiesforX 0.1 and 14
C
≈ Tofindthetemperatureprofilejustafterthefuelburns,we
anenergyreleasefromthenuclearburningof1MeVpernu-
depositanenergyE =E 1017ergg- 1throughoutthelayer.
cleon. The low thermalconductivityofthe rp-processashes nuc 17
Sincecarbonburningtoirongives 1018 ergg- 1,weexpect
givesalargetemperaturegradientandignitionattherequired
≈
mass (CB01). The heavy nuclei may also photodisintegrate E17 =1 to correspond to XC 0.05–0.1, depending on how
≈
much energy is contributed by photodisintegration (Schatz
to iron groupduringthe flash, enhancingthe nuclear energy
et al. 2003). At each depth, we calculate the temperature
release (Schatz, Bildsten, & Cumming2003). Thereforesu-
perburstsofferanopportunitytostudytherp-processashes. of the layer, Tf, from TTifcPdT = Enuc, where Ti the initial
Previous authors used one-zone models to estimate the temperature. FollowingRCB01, a simple analytic estimate of
time-dependence of the flash (CB01; Strohmayer & Brown T is as follows. The electrons are degenerateand relativis-
f
2002). In this paper, we present the first multi-zone models tic for ρ&107 gcm- 3, giving ρY =1.5 108 gcm- 3 P3/4,
of the cooling phase of superbursts. Unlike normal Type I e × 26
bursts,thetimetoburnthefuelismuchlessthantheconvec- Fermi energyEF =2.7 MeV P216/4, and pressure scale height
tive turnover time. We therefore assume that the fuel burns
2 ThisisasimilarapproachtoEichler&Cheng(1989)whostudiedthe
locallyandinstantaneouslyinplace,withoutsignificantverti-
thermalresponseofaneutronstartoenergydepositionatdifferentdepths.
However, the transient events they consider are less energetic than super-
1HubbleFellow bursts.
2
H=y/ρ=67mP1/4Y /g ,whereP=P 1026ergcm- 3. The
26 e 14 26
heatcapacityc isdeterminedmainlybytheelectrons,c
P P
π2(Y k /m )(k T/E )=2.6 107 ergg- 1K- 1 TY P- 1/4. In≈-
e B p B F × 9 e 26
tegratingandassumingT T ,wefindtheconvectivelysta-
i f
bletemperatureprofileT =≪3.6 109KE1/2P1/8Y- 1/8,insen-
f × 17 26 e
sitivetodepthanddependingmainlyonE .
17
Our thermal evolution code uses the method of lines, in
whichtherighthandside ofequation(1)is differencedover
a spatial grid, and the resulting set of ordinary differen-
tial equations integrated using a stiff integrator. We choose
a uniform grid in sinh- 1(logy/y ), which concentrates grid
b
points around the base of the layer y , resolving the initial
b
temperature discontinuity. We place the outer boundary at
y=108gcm- 2,andsetflux T4there;attheinnerboundary,
typicallyy 1014 gcm- 2,w∝eassumevanishingflux. Weas-
sumethelay≈erisheatedbeforetheflashbya1021ergcm- 2s- 1
flux from the crust, and that all the fuel burns to 56Fe. Our
results are not sensitive to the details of the grid, or bound-
ary conditions(fortimes longerthan the thermaltime at the
top zone). We calculate the equation of state, opacity, neu-
trino emissivity, and heat capacity as described by Schatz et
al.(2003).
Figure1showstemperatureandfluxprofiles10minutes,1
FIG.1.—Temperatureandfluxprofilesforyb=1012gcm- 2,E17=1,and hour,1day,and10daysafterignitionforamodelwithE =1
tan=d1100mdianyuste(slo(nsgo-liddaslihneeds)l,in1ehs)ouarft(edroitgtenditiloinne.s)I,n1thdeaylo(wsheorrpt-adnaeslh,etdhelitne=s)1, andyb=1012 gcm- 2. Figure2showsaseriesoflightc1u7rves
hour(1day)fluxprofileisshownscaledbyafactorof30(300). fordifferenty andE . Atearlytimes, astheouterpartsof
b 17
the layer thermally adjust, the radiative flux dependsmostly
onE . Atlatetimes,afterthecoolingwavereachesthebase
17
of the layer, the flux depends mostly on y , and falls off as
b
a power law F t- 4/3. Figure 3 shows the cumulative en-
ergyreleasefor∝y =1012 and1013 gcm- 2 andE =1–3. In
b 17
the first few hours, the energy released from the surface is
1042 ergs, the exact value being mainly sensitive to E ,
17
≈
rather than depth. A significant fraction of the heat is ini-
tially conductedinwardsandreleased ona longertimescale,
aspointedoutbyStrohmayer&Brown(2002).
Thephysicalreasonforthelate-timepowerlawfluxdecay
is that as time evolves, the peak of the temperature profile
moves to greater depths where the thermal timescale to the
surface is longer3 (see Fig. 1). The simplest analytic model
isaslabwithconstantthermaldiffusivityD,whosetempera-
tureis perturbedclose to the surface, forexampleata depth
x=a (where x=0 is the surface). For a delta-function per-
turbation initially, the temperature evolution is given by the
Green’sfunction4
1 ax x2+a2
T(x,t)= sinh exp - (2)
FIG. 2.—Fluxfromthesurfaceagainsttimefordifferent valuesforthe √πDt (cid:16)2Dt(cid:17) (cid:18) 4Dt (cid:19)
energyreleaseE17andlayerthicknessyb(ingcm- 2). and the surface flux is F (∂T/∂x) t- 3/2exp(- τ/t),
x=0
∝ ∝
where τ is the thermal time at the initial heating depth
τ = 4a2/D. For an initial "top hat" temperature pro-
file T(x < a) = 1, T(x > a) = 0, the surface flux is F
(τ/t)1/2 1- exp(- τ/t) . Fort <τ, before the cooling wav∝e
reachest(cid:2)hebaseofthe(cid:3)layer,F t- 1/2;fort>τ,thesolution
∝
3Asimilarproblemisohmicdecayofcrustalmagneticfields,wherepower
lawdecayisalsoexpected(Sang&Chanmugam1987;Urpin,Chanmugam,
&Sang1994)
4 Asimplewaytoobtainthisresultistoapplythemethodofimagesto
theGreen’sfunctionforanunboundeddomainT(x,t)∝t- 1/2exp - x2/4Dt .
FIG. 3.—Cumulative energyreleasedfromthesurface asafunction of Eichler&Cheng(1989)deriveasimilarresultforapowerlawd(cid:0)ependenc(cid:1)e
time,foryb=1012 (solidlines)and1013 gcm- 2 (dottedlines)andineach ofconductivityondepth,whichalsoshowsself-similarbehavioratlatetimes
case(bottomtotop)E17=1,2,and3. (seeLyubarsky,Eichler,&Thompson2002forarecentapplicationtocooling
ofSGR1900+14afteranoutburst).
3
isindependentoftheinitialtemperatureprofile,andF t- 3/2.
∝
The numerical results show a similar behavior, although
with different power law indices. The relevant timescale in
thiscaseist =H2/D,whereD=K/ρc . Theelectroncon-
cool P
ductivity is K =π2n k2T/3m ν , where m =E /c2, and ν
e B ⋆ c ⋆ F c
istheelectroncollisionfrequency. Whenelectronsdominate
the heat capacity, the thermal diffusivity takes the particu-
larly simple temperature-independentform D=c2/3ν . For
c
electron-ioncollisions, ν =9.3 1016 s- 1 P1/4 Z2/A Λ /Y
c × 26 h i ei e
(e.g.seeAppendixofSchatzetal.1999),giving
Y Z2/A Λ g - 5/4
t =3.8hrsy3/4 eh i ei 14 (3)
cool 12 (cid:18) 6 (cid:19)(cid:16)2.45(cid:17)
(see also eq. [10]ofCB01), where we insertthe appropriate
numbersfor56Fecomposition. Thesimple“tophat”solution
forconstantconductivitymotivatesa fitto the numericalso-
lutions,
F =0.2t- 0.2E7/4 1- exp - 0.63t4/3E- 5/4t- 1.13 , (4)
25 hr 17 cool 17 hr
h (cid:16) (cid:17)i FIG. 4.—Observedandpredicted quenchingtimescales ofnormalType
Iburstsfollowingasuperburst. Weshowthepredictedquenchingtimefor
wherethr=t/1hour. Fort>tcool,F25=0.13E117/2(t/tcool)- 4/3. yb=1012 (solidlines)and1013 gcm- 2 (dottedlines)asafunctionofboth
The transition from F t- 0.2 to F t- 4/3 occurs when thecriticalfluxneededtostabilizeH/HeburningFcrit,andtheaccretionrate
t/t E- 1.1. ∝ ∝ (givenintermsofFcrit byeq.[5]). Foreach value ofyb, weshowcurves
Ecoqoula≈tion17(4)fitsthenumericalresultstobetterthanafactor for(bottomtotop)E17=1˙,2and3. Theobservedupperorlowerlimitson
tquench,andtheestimatedM,aretakenfromTable1ofKuulkers2003,with
oftwo formodelswithoutsubstantialneutrinoemission. As anupdatedvaluefor4U1636-53(Kuulkers,privatecommunication).
emphasised by Strohmayer & Brown (2002), neutrino cool-
ing is important for large carbon fractions: it depresses the
flux at t 5–10 hours for the models with E = 2 and 3,
17
y=1013 g≈cm- 2 in Figure 2. Whenever neutrinos dominate powerlawdecayusingsuperbursttails,althoughthisdepends
the cooling, the peak temperatureis largeenoughthat emis- uponbeingabletosubtractouttheunderlyingaccretionlumi-
sionisbypairannhilation. Agoodfittotheneutrinoenergy nosity,F 0.1(M˙/0.1M˙ ),inareliableway.
lossrateisǫν≈104ergg- 1s- 1T912y1- 23/2,givingacoolingtime Anothaecrcr,w25a≈y to probe the lEadtde-time cooling is to use the
tpdνeor=maticunPraTete/frǫwoνhm=ene2qt.ν5ua×<tio1tcn0oo1(l22,)sorgyiw15v2/e4hTse9nt-ν1E0≈.17I3n>0s0e2r.ht3irnsyg1-E23t1/-h725e0..pNeeauktrtienmos- r“(feerqom.ugme.anrKtkchhauebeusdlleuk”ep)oresbfrobs2rue0rr0vtsqt3aut)teien.ocmhnCp≈Bothr0aa1wtrieplTyerykospspteaofbsoiIelldilzobewtuhsriasnttthgsethdtHehise/cHaopseopulbepinauergrrnb(fliunaurrgsxet.
Anestimateofthecriticalstabilizingflux,F ,isasfollows.
crit
3. COMPARISONTOOBSERVATIONS The condition for temperature fluctuations to grow, and un-
The cooling curves in Figure 2 compare well with ob- stableHeignitiontooccur,isνǫ3α=ηǫcool (Fushiki&Lamb
served lightcurves, including a rapid initial decay on hour 1987), where ǫ3α is the triple alpha (3α) energy production
timescales, followed by an extendedtail of emission (as ob- rate, ǫcool is a local approximationto the coolingrate, and ν
served following some superbursts, e.g. KS 1731-260, Ku- andη aretherespectivetemperaturesensitivities. Foralarge
ulkersetal. 2002; Ser X-1, Cornelisse etal. 2002). We will fluxfrombelow,theHeburnsstablybeforereachingthisig-
present a detailed comparison with the observed superburst nitioncondition,atadepthwherethetimetoaccumulatethe
lightcurvesinafuturepaper. Theinitialdecayfromthepeak layerequalstheHeburningtime,y/m˙ =YQ3α/ǫ3α, whereY
dependsmostly on E , and so it should be possible to con- istheHemassfraction,m˙ isthelocalaccretionrateperunit
17
strain the amount of fuel consumed in the superburst. Our area,andQ3α=5.84 1017ergg- 1=0.606MeVpernucleon
×
models do not resolve the peak itself, since this depends on is the 3α energy release. At the transition from unstable to
the detailsof how the burningpropagatesoutto the surface; stable burning, both criteria are satisfied at the base of the
however,forE17&2–3,thefluxexceedstheEddingtonflux, H/He layer. Using the first condition to eliminate ǫ3α from
FtiEodnd,=fo3r×tim1e0s2c5aelergscomfm- 2ins-u1t/es(1. S+uXp)e,rbwuhresrtepeXakisluthmeinHosfirtaiecs- t6h.e2sec1o0n22d,eragndcmw-r2itsi-n1g(m˙ǫc/omo˙l≈)F(Y/y/,0g.3iv)(eνs/F4η=).νSm˙oQm3eαoYf/tηhi=s
Edd
×
are generally less than the Eddington luminosity (Kuulkers flux is provided by hot CNO burning of accreted H, F
H
2003), implying E . 2. The one exception is the super- ǫ y=5.8 1021 ergcm- 2s- 1y (Z/0.01) (Cumming & Bild≈-
17 H 8
burstfrom4U1820-30,whichshoweddramaticphotospheric sten2000;×Zisthemetallicity);theremainderisF =F- F .
crit H
radius expansion lasting for several minutes (Strohmayer & Thisestimateagreeswellwithamoredetailedcalculationus-
Brown 2002). This is consistent with the proposal that this ing the ignition models of Cumming & Bildsten (2000), in
source, which accretesandburnsHe rich material, produces whichwefindFcrit m˙Q3α 0.7MeVperaccretednucleon,
≈ ˙ ≈
largequantitiesofcarbon(Strohmayer&Brown2002;Cum- almostindependentofM. Therefore,
ming 2003a). The transition to the late-time power law oc-
curs after t ≈4 h E1-71.11y132/4 (eq. [5]), which correspondsto Fcrit,22≈6(m˙/m˙Edd) (5)
F 0.13E2 . It may therefore be possible to measure the (see also Paczynski 1983a; Bildsten 1995). Equation (5) in
25 ≈ 17
4
thelimitt t gives constrainboth the thicknessof the fuellayer and the energy
cool
≫
deposited,particularlywhencombinedwithmodelsofnormal
tquench=38tcoolFc-ri3t/,242E137/8=6daysy132/4Fc-ri3t/,242E137/8, (6) TypeIburstsfromthesamesource(Cumming2003a,b).
Thereis still muchto bedonein termsof theory. Perhaps
whichgivest in termsof the thicknessof the layer and
quench the most important issues are the physics of the rise (which
energyrelease.
setstheinitialconditionforoursimulations),andproduction
Figure 5 compares the predicted and observed quenching
ofthefuel. Importantcluesto thefirstarethe observedpre-
times. Theobservationsoft andaccretionrates(usedto
quench cursorstosuperbursts,whichmaybenormalTypeIburstsig-
findF fromeq.[5])aretakenfromKuulkers(2003)(except
crit nitedbythesuperburst.Recentprogresshasbeenmadeonthe
for4U 1636-53,which hasa revisedupperlimitof 23days,
second,withindicationsfromboththeory(Schatzetal.2003;
Kuulkers private communication). The observationsare up-
Woosleyetal.2003)andobservations(in’tZandetal.2003)
peror lowerlimits only: nonetheless, the generalagreement
that stable burning may be required to produce enough car-
is very good and supports the quenching picture suggested
bon to power superbursts. A self-consistent model of H/He
by CB01. There is much to learn from a careful compari-
burning,followedbyaccumulationand ignitionof the ashes
sonof superburstlightcurvesandthe correspondingquench-
mayrequireabetterunderstandingofthetransitionfromun-
ingtimes,separatelyconstrainingbothE andy .
17 b stable to stable burning observed in normal Type I bursting
(e.g.Cornelisseetal.2003).
4. SUMMARYANDCONCLUSIONS
Wehavepresentedthefirstmulti-zonemodelsofthecool-
ing phase of superbursts. The flux decay is notexponential,
butpower-law(eq.[5]). Fort <t , wheret isthecool-
cool cool
ing time at the base of the layer, the flux depends mostly We thank P. Arras, E. Brown, R. Cornelisse, E. Kuulkers,
on the energy release E17, and is insensitive to depth: the G.Ushomirsky,S.Woosley,andJ.in’tZandforusefulcom-
inwards travelling cooling wave does not yet know that the ments and discussions. AC is supported by NASA through
layer has a finite thickness. For t >tcool, the flux decays as Hubble Fellowship grant HF-01138 awarded by the Space
apowerlawF t- 4/3,independentoftheinitialtemperature Telescope Science Institute, which is operated by the Asso-
∝
profile. The power law decay at late times gives predicted ciation of Universities for Research in Astronomy, Inc., for
Type I burst quenching times of weeks (eq. [6]), consistent NASA,undercontractNAS5-26555. JMacknowledgessup-
withobservationallimits. Futurecomparisonsofbothsuper- portfromDOEgrantNo.DE-FC02-01ER41176totheSuper-
burstlightcurvesandquenchingtimeswithobservationswill novaScienceCenter/UCSC.
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