ACCEPTEDBYTHEASTROPHYSICALJOURNALLETTERS PreprinttypesetusingLATEXstyleemulateapjv.10/09/06 THELATITUDEOFTYPEIX-RAYBURSTIGNITIONONRAPIDLYROTATINGNEUTRONSTARS RANDALLL.COOPERANDRAMESHNARAYAN Harvard-SmithsonianCenterforAstrophysics,60GardenStreet,Cambridge,MA02138 AcceptedbyTHEASTROPHYSICALJOURNALLETTERS ABSTRACT We investigate the latitude at which type I X-ray bursts are ignited on rapidly rotating accreting neutron ˙ stars. Wefindthat,forawiderangeofaccretionratesM,ignitionoccurspreferentiallyattheequator,inaccord ˙ ˙ withtheworkofSpitkovskyetal. However,forarangeofM belowthecriticalM abovewhichburstscease, ˙ ignition occurs preferentially at higher latitudes. The range of M over which nonequatorial ignition occurs 7 is an increasingfunctionof the neutronstar spin frequency. These findingshave significantimplicationsfor 0 thermonuclearflamepropagation,andtheymayexplainwhyoscillationsduringtheburstrisearedetectedpre- 0 2 dominantlywhentheaccretionrateishigh. TheyalsosupportthesuggestionofBhattacharyya&Strohmayer thatnon-photosphericradiusexpansiondouble-peakedburstsandtheunusualharmoniccontentofoscillations n duringtheriseofsomeburstsresultfromignitionatorneararotationalpole. a J Subjectheadings:accretion,accretiondisks—stars: neutron—X-rays:binaries—X-rays:bursts 4 2 1. INTRODUCTION (Fryxell&Woosley 1982; Bildsten 1995; Spitkovskyetal. 2002,hereafterSLU02).Time-resolvedspectroscopyandob- 1 Type I X-ray bursts are thermonuclear explosions servations of large amplitude oscillations during the rise of v that occur on the surfaces of accreting neutron stars in some type I X-ray bursts from rapidly rotatingneutron stars 8 low-mass X-ray binaries (LMXBs) (Babushkinaetal. 0 1975; Grindlay&Heise 1975; Grindlayetal. 1976; supportthisconclusion(Strohmayeretal.1997,1998). 7 Belianetal. 1976; Woosley&Taam 1976; Joss 1977; SLU02 investigated the dynamics of localized ignition on 1 Maraschi&Cavaliere1977;Lamb&Lamb1977,1978),and a rapidly rotating neutron star. They found that the latitude 0 at which localized ignition occurs is of great importance in they are triggered by unstable hydrogen or helium burning 7 determiningthe burningfrontpropagationspeed. Itcanalso (for reviews, see Cumming 2004; Strohmayer&Bildsten 0 affect the stability and nature of zonal flows during the de- 2006). The physics of type I X-ray bursts is generally well / cay phase of bursts (Cumming 2005), the harmonic content h understood, and detailed time-dependent one-dimensional ofburstriseoscillations(Bhattacharyya&Strohmayer2005), p models (e.g., Woosleyetal. 2004; Fiskeretal. 2006) have - beenrathersuccessfulatreproducingthegrosscharacteristics and the light curves of non-photospheric radius expansion o (PRE)bursts(Bhattacharyya&Strohmayer2006a,b).SLU02 of bursts, such as their fast rise times of ∼1 s, durationsof r assert that, due to the reduction of the effective gravita- t ∼10–100s,andrecurrencetimesofafewhourstodays. s tional acceleration, ignition is likely to occur at the equator Nearly all type I X-ray burst models are one-dimensional a of a rapidly rotating neutron star. However, recent observa- : and thus implicitly assume that matter accretes spherically v tionssuggestotherwise. Bhattacharyya&Strohmayer(2005, onto a nonrotating neutron star. Therefore, ignition occurs i 2006a,b)arguethattheharmoniccontentofburstriseoscilla- X simultaneouslyovertheentirestellarsurface. However,these tionsandnon-PREdouble-peakedburstsfromtheLMXB4U assumptions are clearly inapplicable to the vast majority of r 1636–536,whichharborsa neutronstar with spin frequency a accreting neutron stars. The Rossi X-Ray Timing Explorer ν=581Hz,requireignitiontooftentimesoccuratornearthe hasdetectedhighlysinusoidaloscillationswithfrequenciesof rotationalpole,atleastinthissource. 45–1122Hzduringburstsin17LMXBs(Strohmayer&Bild- InthisLetter,weevaluatetheoreticallythelatitudeatwhich sten 2006 and references therein; Bhattacharyyaetal. 2006; typeIX-rayburstsonrapidlyrotatingneutronstarsaremost Bhattacharyya 2006; Kaaretetal. 2006). It is thought that probably ignited. In §2 we outline the physics that governs theburstoscillationfrequencycorrespondstotheneutronstar the latitude of type I X-ray burst ignition and determine the spin frequency (e.g. Strohmayer&Markwardt 1999), which ignitionlatitudeasafunctionoftheglobalaccretionrate. We implies that many neutron stars that exhibit bursts are in discussourresultsandconcludein§3. fact rapidly rotating. Indeed, Munoetal. (2001, 2004) have shownthatthepropertiesofburstsdependsensitivelyonthe 2. IGNITIONLATITUDE neutron star rotation rate, and so rotation must be consid- Inthissection,wefirstoutlinethebasicphysicsthatdeter- eredintheoreticalburstmodels. Furthermore,sincethetime minesthe latitudeoftypeI X-rayburstignitionona rapidly required to accumulate a sufficient amount of fuel to trig- rotatingneutronstar using a simple model. We then use the ger a burst is much greater than the duration of the result- global linear stability analysis of Cooper&Narayan (2005) ing burst, it is highly unlikely that all of the accreted fuel tocarryoutmoredetailedcalculations. overtheentireneutronstarsurfacewillignitesimultaneously (Joss 1978; Shara 1982). Thus, contrary to the assumptions 2.1. BasicPhysics ofmosttheoreticalmodels,ignitionalmostsurelyoccursata point,andtheresultingthermonuclearflamesubsequentlyen- Usingtheone-zoneburstignitionmodelofBildsten(1998), gulfs the whole stellar surface in ∼1 s, the burst rise time we expand upon the work of SLU02 to determine the most probablelatitudeofburstignitiononthesurfaceofarapidly Electronicaddress:[email protected],[email protected] rotating neutron star. The reader is encouraged to refer to 2 Cooper&Narayan these worksforfurtherdetails. We neglectgeneralrelativis- p /2 tic correctionsthroughoutfor clarity. FollowingSLU02, we assumethatatalltimespriortoignitiontheaccretedplasma is in hydrostatic equilibrium and is at rest in the corotating frame(e.g.Inogamov&Sunyaev1999).Wepresumethatthe accretedmatterspreadsinsuchawayastominimizethegrav- n = 600 Hz itational potential energy of the accreted layer. The base of n p /4 n = 300 Hz g theaccretedlayeristhereforeanequipotentialsurface,andso i the pressure at the base of the accreted layer is the same at l every latitude (e.g., Clayton 1983). Hydrostatic equilibrium thus implies that P =Σ (λ)g (λ) is independent of lati- layer eff tude,wherePisthepressureatthebaseoftheaccretedlayer, Σ isthecolumndepth,g istheeffectivegravitationalac- 0.0 layer eff celeration,andλ≡π/2- θisthelatitude.Thus,Σ ∝g- 1, layer eff 0.0 0.2 0.4 0.6 0.8 1.0 andso . . Σ˙ ∝g- 1, (1) M/M (p /2) eff crit whereΣ˙ =Σ˙(λ)isthelocalmassaccretionrateperunitarea. FIG.1.—Plotshowsλign,thelatitudeatwhichtypeIX-rayburstsigniteon arapidlyrotatingneutronstar,derivedfromaone-zoneburstmodelfortwo Adecpctohrdatinwghtiochehqeulaiutimonig(2n0it)esoΣf iBgnilvdasrteiens(a1s998), the column sgtleolblaarlraoctcarteiotinonfreraqtueesnMc˙i,esigνn.itMion=o1c.c4uMrs⊙aatnthdeReq=u1a0tokrm(λ.F=o0r)a.wHiodweeravnegr,efoofr ˙ Σign∝Σ˙- 1/5ge- f2f/5. (2) aatrhainggheeroflaMtitundeeasr.tThehecrriatnicgaeloraftMe˙aobvoevrewwhhicichhnbounresqtsuacteoarsiea,liiggnniittiioonnooccccuurrss is∝ν2. Combiningequations(1)and(2)givestheignitiontime asa functionoflatitude where ν, M, and R are the spin frequency, mass, and radius t (λ)≡Σ /Σ˙ ∝g4/5. (3) oftheneutronstar,respectively.Thus,foraneutronstarwith ign ign eff spin frequencyν =600Hz, type I X-ray bursts will ignite at ˙ ˙ Thustign,thetimeittakestoaccreteacriticalamountoffuel theequatorforM/Mcrit(π/2).0.89andoffoftheequatorfor to trigger a type I X-ray burst, is an increasing function of ˙ ˙ 0.89.M/M (π/2)<1.Withinthelatterrangeofaccretion crit g . Duetocentrifugalacceleration,g islowestattheequa- eff eff rates,thelatitudeatwhichignitionoccursisgivenby torofarapidlyrotatingneutronstar,whichmeansthatt (λ) ign ˙ ˙ is a minimum at the equator. Therefore, ignition will occur 0, M<M (0), crit preferentiallyattheequator(SLU02). neHutorownevsetra,rnduocelesarnobturanlwinagysontritghgeesrurafabcuersotf. anBoatchcrtehteinog- λign= arccos"ννKr1- M˙M˙crcitr(itπ(λ/)2) 2/3#, M˙crit(0)<M˙ <M˙crit(π/2), (cid:16) (cid:17) retical models (e.g., Fujimotoetal. 1981; Paczyn´ski 1983; (8) Bildsten 1998) and observations(e.g. Cornelisseetal. 2003; whereνK is the Keplerian frequency. Figure 1 shows a plot ˙ Gallowayetal.2006)suggestthatthereisalocalcriticalac- ofλign asafunctionofM. WeseethattypeIX-rayburstig- cretion rate Σ˙ above which nuclear burning is stable, and nitiononarapidlyrotatingneutronstaroccurspreferentially thus bursts docrnitot occur. By equation (1), Σ˙ is related to neartheequatorforawiderangeofM˙. However,forasmall theglobalcriticalaccretionrateM˙ suchthat crit rangeof M˙ near M˙crit, ignition occursat higherlatitudes be- cause nuclear burning becomes stable near the equator. As Σ˙ ∝g- 1M˙ , (4) equation(7)illustrates,therangeofaccretionratesoverwhich crit eff crit ˙ ˙ nonequatorialignitionoccursincreasesdramaticallywithν. whereM =M (λ)isdefinedtobethecriticalglobalaccre- crit crit tion rate abovewhichnuclearburningisstable atlatitude λ. 2.2. GlobalLinearStabilityAnalysis Accordingtoequation(24)ofBildsten(1998), We now use the global linear stability analysis of Σ˙ ∝g1/2, (5) Cooper&Narayan (2005), which is an expanded and im- crit eff provedversionofthemodelofNarayan&Heyl(2003),tode- andso M˙ ∝g3/2. (6) terminethe latitude of type I X-ray burstignitionon rapidly crit eff rotating neutron stars. We assume that matter accretes at a ˙ This implies that, since g is lowest at the equator and in- global rate M onto a rapidly rotating neutron star of mass eff creases towards the rotational pole, there exists a range of M =1.4M⊙, radius R=10 km, and spin frequency ν =650 accretion rates for which nuclear burning is stable near the Hz,anditspreadsoverthestellarsurfaceinthesamemanner equatorandunstablenearthepole. Consequently,foragiven asdescribedin§2.1. We setthecompositionoftheaccreted ˙ λ andcorrespondingcriticalaccretionrateM (λ ),nuclear mattertobethatoftheSun,suchthatattheneutronstarsur- 0 crit 0 burning is stable for all λ<λ and unstable for all λ>λ . face the hydrogen mass fraction X =0.7, helium mass frac- 0 0 Clearly, at these accretion rates, bursts will be triggered off tionY =0.28, CNO mass fraction ZCNO =0.016, and heavy the equator. For M˙ >M˙ (π/2), no bursts are triggered at elementfractionZ=0.004,whereZ referstoallmetalsother crit anylatitude. Takingthefunctionalderivativeofequation(6) thanCNO. ˙ gives We plottheignitiontimetign asa functionofM forλ=0, ˙ ˙ π/4, and π/2 in Figure 2. For M .0.16M , t (λ) is a δM˙crit ≈0.11 ν 2 1.4M⊙ R 3, (7) minimum at the equator, and so bursts igniteEdpdrefigenrentially M˙ (π/2) 600Hz M 10km near the equator, in agreement with both SLU02 and equa- crit (cid:16) (cid:17) (cid:18) (cid:19)(cid:18) (cid:19) BurstIgnitionLatitude 3 delayed mixed bursts, see Cooper&Narayan 2006). In ef- 0.8 fect,thedelayedmixedburstregimesignificantlyextendsthe ˙ 0.7 range of M over which nonequatorial ignition occurs. The global linear stability analysis and the two-zone model of 0.6 l = p /2 Cooper&Narayan(2006)agreebetterwithobservationsthan s) l = p /4 allothercurrentburstmodels, butthisagreementholdsonly ay 0.5 l = 0 inatime-averagedsense(e.g.Gallowayetal.2006). Bythis d (n 0.4 wemeanthat,whileobservationsimplyt˙hatthe˙meanign˙ition g timeht iisanincreasingfunctionofM forM&0.15M , ti ign Edd 0.3 which is in accord with the results shown in Figure 2, the ignitiontimet measuredbetween pairs of bursts observed ign 0.2 in natureexhibitssignificantdeviationsrelativeto the mean. Ourmodelscannotaccountforthis behavior. Since the pre- 0.1 dictedburstignitionlatitudeismodel-dependent,andnocur- 0.10 0.15 0.20 0.25 0.30 0.35 . . rentmodelcansuccessfullyreproducethechaoticbehaviorof M/M bursts, we have little confidence in the accuracy of our pre- Edd FIG. 2.— Plot showstign(λ), the type I X-ray burst ignition timescale, dictedignitionlatitudesλign. However,alltheoreticalmodels atthreedifferentlatitudesλforaneutronstarwithspinfrequencyν=650 wouldpredictthatburstsignitepreferentiallyneartheequator ˙ ˙ ˙ ˙ Hz, M=1.4M⊙, andR=10km. Foraccretion rates M.0.16MEdd,tign at low valuesof M and off of the equatorat higherM. This islowestattheequator, whichmeansthatburstsigniteattheequator. For resultisveryrobust. ˙ ˙ ˙ 0.16.M/MEdd .0.19, the burstignition latitude increases withM. For Recent observations have generated a renewed interest in 0.19.M˙/M˙Edd.0.3,althoughnuclearburningisunstableatallλ˙,ig˙nition the latitude of type I X-ray burst ignition. First, non-PRE occurspreferentiallynearthepolewheretignislowest.For0.3.M/MEdd. double peaked bursts have been observed in a few LMXBs 0.34,nuclearburningbecomesstableatlowlatitudes,andignitionstilloccurs preferentiallynearthepole. (e.g. Sztajnoetal. 1985; Penninxetal. 1989; Kuulkersetal. tion (3). Here, M˙ =8.3×1017 gs- 1 denotes the mass ac- 2002). Bhattacharyya&Strohmayer(2006a,b)arguethatig- Edd nition near a rotational pole can explain the non-PRE dou- cretionrate at whichthe accretionluminosityis equalto the ˙ ˙ ble peakedburstsobservedin 4U 1636–536,whichcontains Eddingtonlimit. For 0.16.M/M .0.19, λ gradually Edd ign a neutron star with ν = 581 Hz. Their simple model of ˙ increaseswithincreasingM, againinaccordwiththeresults nonequatorial ignition and subsequent thermonuclear flame of §2.1. However, Figure 2 shows a new regime of bursts propagation and temporary stalling near the equator quali- ˙ ˙ for M & 0.19M , the regime of “delayed mixed bursts,” tatively reproduces both the light curves and spectral pro- Edd which was first identified by Narayan&Heyl (2003). De- files of such bursts. Second, although burst oscillations de- layedmixedburstsaremixedhydrogenandheliumburststrig- tectedinboththeriseanddecayareusuallyquitesinusoidal, gered by unstable helium burning, and they are precededby Bhattacharyya&Strohmayer(2005)reportevidenceforsub- a long period of stable nuclear burning. Using their global stantialharmoniccontentintheoscillationsduringtheriseof linear stability analysis, Narayan & Heyl found that, within aburstfrom4U1636–536.Theyagainsuggestthatnonequa- this range of accretion rates, (1) the ignition timescale is an torialignitionandsubsequentflamepropagationcan explain ˙ increasing function of M (which is consistent with obser- theobservations. Theseauthorsacknowledgedthatnonequa- vations, e.g., vanParadijsetal. 1979, 1988; Cornelisseetal. torialignitionisatoddswiththeworkofSLU02,whoargued ˙ 2003; Remillardetal. 2006; Gallowayetal. 2006), and (2) forequatorialignitionatallM. Ourresultthatnonequatorial the criticalglobalaccretionrate abovewhich delayedmixed ignition is more likely when the accretion rate is high natu- burstsoccurisanincreasingfunctionofthegravitationalac- rallyreconcilesthisdiscrepancy. ˙ ˙ ˙ celeration g . Thus, for the range 0.19. M/M .0.3, That nonequatorial ignition is preferred at higher M may eff Edd althoughnuclear burningis unstableat all latitudes, ignition explain why oscillations in the rising phase of some bursts occurs preferentially near the pole where t is lowest. For onrapidlyrotatingneutronstarsoccurpredominantlyonthe ign 0.3.M˙/M˙ .0.34, nuclear burningbecomescompletely banana branch, i.e. when the inferred accretion rate is high Edd stableatlowlatitudes,andignitionagainoccurspreferentially (Strohmayer&Bildsten2006).Thiscanbeunderstoodasfol- nearthepole. ForM˙/M˙ &0.34,nuclearburningisstable lows. SLU02 find thatburningfrontspropagatemuchfaster Edd ˙ atalllatitudes,andnoburstsoccur. neartheequatorthannearthepoles. IfM islowandignition occursneartheequator,theflamewillspreadinlongitudeona 3. DISCUSSIONANDCONCLUSIONS timescalethatismuchshorterthantheburstrisetimescaleand Theresultsofboththesimpleone-zonemodelofBildsten quickly produce an axisymmetric belt. An observer would (1998) and the more detailed global linear stability analysis detectnooscillationsduringtherisingphasebecausethereis ˙ ofCooper&Narayan(2005)suggestthatburstsigniteonthe no longitudinalasymmetry. On the other hand, if M is high equatoratlowaccretionratesandoffoftheequatorathigher and ignition occurs close to, but not directly at, a rotational accretion rates. The global linear stability analysis predicts pole, the flame willspread in the longitudinaldirectionon a that nonequatorial ignition occurs over a much larger range timescalethatiscomparabletotheburstrisetimescale. This of M˙ – nearly 50% for ν =650 Hz versus only about 10% slowlongitudinalspreadwouldcreateanon-axisymmetrichot in the one-zonemodel– and that ignition should occur near spot and hence oscillations. We note that oscillations in the the pole for most of this range. The differences in the re- decay phase of some bursts are again observed only when a sults of the two models are due to the delayed mixed burst systemisonthebananabranch.However,burstoscillationsin regime,whichispredictedinthegloballinearstabilityanal- thedecayphasearenotwellunderstoodtheoretically,andwe ysis but not in the one-zone model (for an explanation of arehesitanttosuggestaconnectionbetweentheseoscillations 4 Cooper&Narayan andnonequatorialignition. ignitionismorelikelytooccurwithinthisconfinementregion We stress that our results apply only to rapidly rotating, than at a specific latitude. Although some bursts observed weakly-magnetic neutron stars. Rapid rotation induces a from the accreting millisecond pulsar SAX J1808.4–3658 stronglatitudinaldependenceontheeffectivegravitationalac- exhibit timing features that Bhattacharyya&Strohmayer celerationwhichrestrictsburstignitiontoapreferredlatitude. (2006c) suggest originate from midlatitudinal ignition, we Foranonrotating(orslowlyrotating)neutronstar, allphysi- speculatethatitisduetomagneticconfinement. calquantitiesareindependentofλ(ornearlyso),andignition couldoccuratanylatitude.Thus,oscillationsduringtheburst ˙ risecouldoccuratanyMforslowlyrotatingneutronstars. WethankJoshGrindlayfordiscussionsthatmotivatedthis Strong magnetic fields may channel and confine accreted investigationandtherefereeforusefulcomments. Thiswork mattertosomeregionoftheneutronstarsurface.Inthiscase, wassupportedbyNASAgrantNNG04GL38G. REFERENCES Babushkina, O. P., Bratolyubova-Tsulukidze, L. S., Kudryavtsev, M. I., Kuulkers,E.,Homan,J.,vanderKlis,M.,Lewin,W.H.G.,&Méndez,M. Melioranskiy, A. S., Savenko, I. A., & Yushkov, B. 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