Runaway Heating By R-modes of Neutron Stars in Low

arXiv:astro-ph/9810471v129Oct1998RunawayHeatingByR-ModesofNeutronStarsinLowMassX-rayBinaries.YuriLevinTheoreticalAstrophysics,CaliforniaInstituteofTechnology,Pasadena,California91125Received;accepted–2–ABSTRACTRecentlyAnderssonet.al.,andBildstenhaveindependentlysuggestedthatanr-modeinstabilitymightberesponsibleforstallingtheneutron-starspin-upinstronglyaccreting,LowMassX-rayBinaries(LMXBs).Weshowthatifthisdoesoccur,thentherearetwopossibilitiesfortheresultingneutron-starevolution:Ifther-modedampingisadecreasingfunctionoftemperature,thenthestarundergoesacyclicevolution:(i)accretionalspin-uptriggerstheinstabilityneartheobservedmaximumspinrate;(ii)ther-modesbecomehighlyexcitedthroughgravitational-radiation-reaction,andinafractionofayear(0.13yrsinaparticularmodelthatwehaveconsidered)theyviscouslyheatthestaruptoT∼2.5×109K;(iii)r-modegravitational-radiation-reactionthenspinsthestardownintspindown≃0.08(ffinal/130Hz)−6yrstoalimitingrotationalfrequencyffinal,whoseexactvaluedependsonthenotfullyunderstoodmechanismsofr-modedamping;(iv)ther-modeinstabilityshutsoff;(v)theneutronstarslowlycoolsandisspunupbyaccretionfor∼5×106yrs,untilitonceagainreachestheinstabilitypoint,closingthecycle.Theshortnessoftheepochofr-modeactivitymakesitunlikelythatr-modesarecurrentlyexcitedintheneutronstarofanygalacticLMXBs,andunlikelythatadvancedLIGOinterferometerswillseegravitationalwavesfromextragalacticLMXBs.Nevertheless,thiscyclicevolutioncouldberesponsibleforkeepingtherotationalfrequencieswithintheobservedLMXBfrequencyrange.If,ontheotherhand,ther-modedampingistemperatureindependent,thenasteadystatewithconstantangularvelocityandTcore≃4×108Kisreached,inwhichr-modeviscousheatingisbalancedbyneutrinocoolingandaccretionalspin-uptorqueisbalancedbygravitational-radiation-reactionspin-downtorque.–3–Inthiscase(asBildstenandAnderssonet.al.haveshown)theneutronstarsinLMXBscouldbepotentialsourcesofperiodicgravitationalwaves,detectablebyenhancedLIGOinterferometers.–4–1.IntroductionMostoftherapidlyaccretingneutronstarsinLowMassX-rayBinaries(LMXBs)areobservedtorotateinastrikinglynarrowrangeoffrequencies—from260Hzto330Hz(seee.g.VanderKlis1997).Anaturalexplanationforthiscouldbesomemechanismwhichpreventsfurtherneutron-starspin-uponcetherotationalfrequencyissufficientlyhigh.Recentlyseveralsuchmechanismswereproposed:WhiteandZhang(1997)suggestedthatmagneticbrakingcouldberesponsibleforhaltingthespin-up;thisideawillnotbediscussedhere.Bildsten(1998)pointedoutthat,becausegravitationalradiationreactionisasharplyincreasingfunctionofrotationalfrequency,itmighthaltthespin-ip.Inhisoriginalmanuscript,Bildstenidentifiedonemechanismfortriggeringthenecessarygravitationalwaves:lateraldensityvariationscausedbytemperaturedependenceofelectroncapturereactions.Whilehismanuscriptwasbeingrefereed,Bildstenlearnedofthediscoverythatanr-modeinstability,drivenbygravitationalradiation,canbeverystronginspinningneutronstars(Andersson1998,FriedmanandMorsink1998,Lindblom,OwenandMorsink1998,Andersson,KokkotasandSchutz1998,Owenetal1998);andther-modeexpertslearnedofBildsten’sgravitational-waveideaforsaturatingLMXBspinup.Bothgroupsindependentlysawtheconnection:Bildsten(1998)andAndersson,KokkotasandStergioulas(1998)proposedthatther-modeinstabilitycouldprovideenoughgravitational-radiationreactiontohalttheLMXBspinup.Inthisletterweexaminetheconsequencesofthisproposal.Ourconclusionsdependcruciallyonwhetherthedissipationofther-modesdecreaseswithtemperature(asisthecase,e.g.,whenshearviscositydominatesther-modedamping),orinsteadistemperature-independent(asisthecasewhen,e.g.,themutualfrictionofprotonandneutronsuperfluidsdominatesthedamping).Intheformercase(Section2ofthispaper)wefindthattheneutronstarwillundergoaspin-up—heating—spin-down—cooling–5–cycle;inthelattercase(Section3)itwillprobablysettledowntoastableequilibriumstatewithaninternalneutron-startemperatureofabout4×108K.2.“Viscous”r-modedampingLetusconsiderfirstthecasewhendissipationisadecreasingfunctionoftemperature.Weshowthat,ifsomer-modesbecomeunstableinaneutronstarspunupbyaccretion,thentheyheatuptheneutronstarthroughshearviscousity.Astheneutronstarheatsup,ther-modesbecomemoreunstable.Athermo-gravitationalrunawaytakesplace,inwhichther-modeamplitudegrows,asaresultofthisgrowththestar’stemperaturerises,thedissipationbecomesweakerandtheinstabilitybecomesstronger.Withinafractionofayearther-modes’gravitationalradiationreactionspinsthestardowntoarotationfrequencywhichisclosetotheminimumofthecriticalstabilitycurve(probablyaround100−150Hz,buttheexactvaluedependsonpoorlyunderstooddissipationmechanisms—seebelow),withafinaltemperatureofabout2×109K.Theinstabilitythenshutsoffandthestarbeginsaseveral-million-yearepochofneutrinocoolingandaccretionalspinup,leadingbacktotheoriginalinstabilitypoint.Fig.1showsatypicalevolutionarytrajectoryA→B→C→D→Boftheneutronstarinthelog(T8)−˜Ωplane,whereT8isthetemperatureofthestar’scoremeasuredinunitsof108K,and˜Ω=Ω/√πG¯ρ.HereΩistheangularvelocityoftheneutronstarand¯ρisit’smeandensity.TheportionA→Bofthecurverepresentstheaccretionalspin-upofthe