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A New, Simple Model for Black Hole High Frequency QPOs L. Rezzolla SISSA,InternationalSchoolforAdvancedStudies,Trieste,Italy INFN,SezionediTrieste,Trieste,Italy 4 0 Abstract. Observations of X-ray emissions from binary systems have always been considered important tools to test the 0 validity of General Relativity in strong-field regimes. The pairs and triplets of high frequency quasi-periodic oscillations 2 observedinbinariescontainingablackholecandidate,inparticular,havebeenproposedasameanstomeasuremoredirectly n theblackholepropertiessuchasitsmassandspin.NumerousmodelshavebeensuggestedovertheyearstoexplaintheQPOs a andtherichphenomenologyaccompanyingthem.Manyofthesemodelsrestonanumberofassumptionsandareattimesin J conflictwiththemostrecentobservations.Wehereproposeanew,simplemodelinwhichtheQPOsresultfrombasicp-mode oscillationsofanon-Kepleriandiscoffinitesize.WeshowthatwithinthisnewmodelallofthekeypropertiesoftheQPOs: 7 a)harmonicratiosoffrequenciesevenasthefrequencieschange;b)variationsintherelativestrengthofthefrequencieswith spectralenergydistributionandwithphotonenergy;c)smallandsystematicchangesinthefrequencies,canallbeexplained 1 simplygivenasinglereasonableassumption.1 v 8 7 0 A number of models have been proposed to explain resonancemodelsuggeststhataninitialturbulencespec- 1 thehighfrequencyquasi-periodicoscillations(HFQPOs) trumisamplifiedataradiuswherethe(radial)epicyclic 0 seen in accreting black hole systems [1, 2] and two of frequency is in resonance with the (azimuthal) orbital 4 these seem particularlypromisingin our view.The first frequency,withthetwofrequenciesbeingin(small)inte- 0 / modelproposed,thediscoseismicmodel[3],assertsthat gerratios.Theseannulitendtobeclosetotheblackhole h g-modes should become trapped in the potential well event horizon for the observed frequencies and, hence, p of a Keplerian (i.e. geometrically thin) disc in a Kerr generalrelativisticeffectscanbeimportant.Givenasuf- - o potential. The size of the region where the modes are ficientlyaccuratemassestimatefortheblackholeanda r trappeddependsonboththemassandthespinoftheac- properidentificationoftheratioofthefrequenciesinres- t s cretingblackhole.Additionalfrequenciesofoscillation onance,again,a blackhole spin canbe measuredfairly a should be expected from pressure modes and corruga- accurately.Itshouldbenotedthatgiventheobservedfre- : v tionmodes.Thepredictionsofthemodelarewellsum- quenciesof162and324Hz in GRS 1915+105itis not i X marisedbyKato[4].GivenpairsofhighfrequencyQPOs possible to producethe 40 and67Hz QPOs with a dis- and a proper identification of the frequencies with the coseismicmodelandthehigherfrequencyQPOswitha r a particular modes, one can measure both the black hole resonancemodelwhileretainingthesamevaluesforthe massandspintorelativelyhighaccuracy[5].Thediscov- blackholemassandspin[9]. eriesofthreesystemswheretheQPOsshowaharmonic A potential problem for both models is the observa- structurewithrelativelystrongpeaksseeninintegerra- tionalevidencefora frequencyjitter in the HFQPOs of tios[6,2]1:2,2:3,or1:2:3seemtocastsomedoubt XTE J1550-564, i.e. for small variations of about 10% upon this model. However, the model remains viable infrequenciesforabout15%ofthetime.Thisdifficulty fortheintermediatefrequencyQPOsinGRS1915+105, couldbeparticularlyseverefortheparametricresonance seenat67Hz[7]and40Hz[6].Thesecondmodelpro- model,whoserelevantfrequenciesconfinetheresonance posed, the parametric resonance model [8], asserts that to a narrow region in radial coordinates (This situation aharmonicrelationshipintheQPOsfrequenciescanbe is worsened for the 1:3 resonance for which the radial producedasaresultofresonancesandwasmotivatedby variationofthefrequenciesisevenmorerapid[2].). thesmallnumberintegerratiosofsomeoftheQPOsseen Wehereproposeanexplanationforthehighfrequency fromblackholescandidates.Inparticular,theparametric QPOs in black hole candidates that has distinct novel features and is based on a single assumption, namely that a non-Kepleriandisc orbits arounda rotatingblack 1 Thisworkwasincollaboration withS’i.Yoshida,T.J.Maccarone hole. More specifically, we propose a model based on andO.Zanotti the oscillation properties of a geometrically thick disc located in the vicinity of a Kerr black hole. A more a largeparameterspace in which boththe sizes and the detaileddiscussionofthismodelcanbefoundin[10]. physical properties (e.g. pressure and density profiles, In contrast to a Keplerian disc, a non-Keplerian disc distributions of specific angular momentum, polytropic (i.e.atorus)isconfinedtoafinite-sizeregiondetermined indices)arevariedconsiderably[12].Theeigenfunctions uniquelyby the distribution of the specific angularmo- and eigenfrequencies found in this way correspond to mentumandbythepressuregradients.Insimilaritywith global p-modes,havebeencomputedforthefundamen- a star, however,restoring forces can be used to classify tal mode of oscillation as well as for the first few over- differentmodesof oscillation in the disc. A first restor- tones, and all have been found to be in a harmonic se- ing force is the centrifugal force, which is responsible quence2:3:4:...towithinafewpercent. for inertial oscillationsof the orbitalmotion of the disc Figure 1 shows a typical example of the solution of and hencefor epicyclic oscillations.A second restoring theeigenvalueproblemforablackholewithmassM= forceisthegravitationalfieldinthedirectionverticalto 10M⊙ anditscomparisonwiththeobservationsofXTE theorbitalplaneandwhichwillproduceaharmonicos- J1550-564(othersourcescouldequallyhavebeenused). cillation across the equatorial plane if a portion of the In particular, we have plotted the value of the different discisperturbedintheverticaldirection.Athirdrestor- eigenfrequenciesfoundversusthe radial extensionL of ing force is provided by pressure gradients and the os- the torus, expressed in units of the gravitational radius cillationsproducedinthiswayarecloselyrelatedtothe r ≡GM/c2. The sequenceshave been calculated for a g sound waves propagating in a compressible fluid. In a distribution of specific angular momentum following a geometrically thick disc, the vertical and horizontalos- power-law[10], keepingthe position of maximumden- cillations are in general coupled and more than a sin- sity in the torus at r = 3.489, and for a black hole max glerestoringforcecaninterveneforthesamemode.Re- with dimensionlessspin parametera≡J/M2 =0.94to ferring the reader to [4] for a more detailed discussion, maximizethevalueforL.Indicatedwithasolidlineare it is here sufficient to remind that c modes are essen- thefundamentalfrequencies f, whilethefirstovertones tiallycontrolledbytheverticalgravitationalfield,thatg o are shown with a dashed line; each pointon the two 1 modesaremainlyregulatedbycentrifugalandpressure- linesrepresentsthenumericalsolutionoftheeigenvalue gradient forces, and that all of the restoring forces dis- problem. The asterisks represent the frequencies of the cussedaboveplayaroleinthecaseofpmodes.Itisalso HFQPOsdetectedinXTEJ1550-564at184and276Hz, usefultounderlinethatbecausewearehereinterestedin respectively.Note that the two plotted eigenfrequencies modes with a prevalent horizontal motion and frequen- are close to a 2:3 ratio over the full range of L con- ciesabovetheepicyclicone,weareessentiallyselecting sidered (this is shown in the inset) and that while they “inertial-acoustic” modes having centrifugal and pres- dependalsoonotherparametersintheproblem(e.g.the suregradientsasonlyrestoringforces.Hereafterwewill positionofr ,theangularmomentumdistribution,the max refertothesesimplyas pmodes. polytropic index), these dependences are very weak so Following a recent investigation of the dynamics of thatthefrequenciesdependeffectivelyonM,aandL.As perturbed relativistic tori [11], we have analysed the aresult,ifMandaareknown,thesolutionoftheeigen- globaloscillationpropertiesof such systemsand,more valueproblemcanbeusedtodeterminethedimensionof specifically, we have performed a perturbative analysis theoscillatingregionLaccurately. of axisymmetric modes of oscillation of relativistic tori Onthebasisoftheseresults,wesuggestthattheHFQ- in the Cowling approximation.The eigenvalueproblem POsobservedinblackholecandidatesystemscanbein- thatneedstobesolvedtoinvestigateconsistentlytheos- terpretedintermsof p-modeoscillationsofasmalltorus cillationpropertiesoffluidtoriissimplifiedconsiderably [10].Notethatallthepropertiesdiscussedsofardepend iftheverticalstructureisaccountedforbyanintegration simplyontheexistenceofnon-Kepleriandiscorbitingin intheverticaldirection.Doingsoremovesonespatialdi- thevicinityofablackhole.Suchaconfigurationcouldbe mensionfromtheproblemandyieldstosimpleordinary producedinavarietyofwaysandbasicallywheneveran differentialequations.Whileanapproximation,thissim- interveningprocess[e.g.largeviscosity,turbulence,hy- pler modelcapturesmost of the featuresrelevantto the drodynamical and magnetohydrodynamical (MHD) in- discussion of p-modes in relativistic tori and is a first- stabilities]modifiestheKepleriancharacteroftheflow. stepinaresearchfieldwhichisingreatpartunexplored. In what follows, we discuss how the predictions of We havesolvedtheeigenvalueproblemforanumber such a model can match the observed phenomenology of vertically-integratedrelativistic tori which have been andcanbeusedtoextractastrophysicalinformation. perturbedthroughthe introductionof arbitrarypressure (i) The harmonic relations between the HFQPO fre- perturbations. The tori are modelled as made of a per- quenciesinblackholecandidatesarenaturallyex- fect fluid obeying a polytropic equation of state for an plained within this model. In a sufficiently small ultrarelativisticgasofelectrons(otherpolytropicindices torus,infact,pmodeseffectivelybehaveastrapped havealsobeenconsidered).Theequilibriummodelsspan FIGURE1. p-mode frequencies for anon-Keplerian disc. Thehorizontal axisshows thenormalized radial dimensions of the torusL/rg whentheblackholehasamassM=10M⊙.Thefundamentalfrequencies f areindicatedwithasolidline,whilethe firstovertoneso1withadashedline.Thefilleddotindicatestheepicyclicfrequencyatrmax,whilethetwoasterisksshowthevalues oftheHFQPOsobservedinXTEJ1550-546.TheerrorbarsshowtheuncertaintyinthedeterminationofM. sound waves with allowed wavelengths that are rangewiththeharmonicstructurepreserved. l =(2/2)L,(3/2)L,(4/2)L,....Thefrequenciesof (v)Theobservedvariationsintherelativestrengthofthe these standing waves would be in an exact integer peakscanbeexplainedasavariationinthepertur- ratioonlyifthesoundspeedwereconstant.Inprac- bationsthetorusisexperiencing.(Thishasbeenre- tice this doesnothappen,butthe eigenfrequencies producedwithnumericalsimulations[11]).Further- foundareinasequenceverycloseto2:3:4. more, while the low frequencyovertonesare ener- (ii) Being global modes of oscillation, the same har- geticallyfavouredandthecorrespondingeigenfunc- monicityispresentatallradiiwithinthetorus.This tionspossesslessnodes,anynumberofharmonics removesthedifficultyencounteredintheresonance couldinprinciplebeobserved. modelandprovidesalsoalargerextentinradialco- (vi) The evidence that an overtone can be stronger ordinateswheretheemissivitycanbemodulated. thanthefundamentalintheharderX-raybands[1], (iii) Because the radial epicyclic frequency represents can be explained simply given that the overtone is the upperlimitforthedisc eigenfrequencies,these an oscillation preferentially of the innermost (and scalelike1/M.Thisisinagreementwiththeobser- hottest) part of the accretionflow (see [12] for the vations made of XTE J1550-564and GRO J1655- eigenfunctions). 40aslongasthespinsaresimilar[13].Ontheother A few remarks are worth making at this point. The hand,arathernarrowrangeofblackholespinshas firstoneisabouttheexistenceofanon-Keplerianorbital beensuggestedasapossibleexplanationforthenar- motion:thisisrequiredonlyveryclosetotheblackhole rowrangeofradio-to-X-rayfluxratiosintheGalac- (the inner edge of the torus can be in principle be lo- ticX-raybinarysystems[14]. catedatthemarginallyboundorbit)andbeyondthisre- (iv) The frequency jitter can be naturally interpreted gionthefluidmotioncanbeKeplerian.Stateddifferently, in terms of variations of the size of the oscillating the HFQPOs observed could be produced by the inner cavityL.Thefrequenciesmayalsodriftoveralarge parts of an otherwise standard, nearly-Keplerian, geo- metricallythinaccretiondisc,whereavarietyofphysical a result of the change in size of the torus, they should phenomenacanintroducepressure-gradients(3Dnumer- neverthelessappearinaharmonicratiotowithin5-10%. icalsimulationsinMHDseemtoindicatetheformation Bothoftheseobservationalrequirementsaresufficiently ofthesetorineartheblackhole[15]).Thesecondremark straightforwardtoassessandthereforeprovideasimple is about the stability of these tori to non-axisymmetric andeffectivewayoffalsifyingthismodel. oscillations. Itis well-knownthata stationary(i.e. non- Clearly, more work is needed to assess whether p- accreting) perfect fluid torus flowing in circular orbits modeoscillationsinasmall-sizetorusrepresentthecor- around a black hole is subject to a dynamical instabil- rect interpretation of the black hole QPO phenomenol- ity triggeredby non-axisymmetricperturbations[16]. It ogy. A first step in this direction will be made when a is less well-known, however, that the instability can be convincing evidence is provided that p modes can be suppressed if the flow is non-stationary. Stated differ- responsible for the modulatedX-ray emission in black- ently,a fluid torusaroundblack couldbe stable to non- hole spacetimes; work is now in progress in this direc- axisymmetricperturbationsifmass-accretiontakesplace tion[20].Weleaveourfinalremarkonthecomplexphe- [17,18,19].Becausethetoridiscussedhereareassumed nomenologyassociated to HFQPOs in BHCs to a well- tobetheterminalpartofstandardaccretiondiscs,weex- knownquote,whoseviewwefeeltoshare:“essentianon pectthemtobestabletonon-axisymmetricperturbations suntmultiplicandapraeternecessitatem”[21]. aslongasmagneticfieldsareunimportant. Note that this model also offers a simple way of ex- tracting astrophysically relevant information from the ACKNOWLEDGMENTS observationofHFQPOs.Werecallthatthefundamental p-modefrequencytendstotheradialepicyclicfrequency It is a pleasure to thank Shin’ichirou Yoshida, Tom k r2 atrmax inthelimitofavanishingtorussize,andthat MaccaroneandOlindoZanotti,mycollaboratorsinthis for a Kerr black hole this frequencyis a function of its research. Financial support has been provided by the massandspinonly.Exploitingthisproperty,afirstesti- MIUR and by the EU Network Programme (European mate of the blackhole spin andof the size ofthe oscil- RTNContractHPRN-CT-2000-00137). latingtoruscanbeobtainedoncethelowerfrequencyin theHFQPOsismeasuredaccurately.Inthiscase,infact, given a black hole candidate with measured mass M∗, REFERENCES a lower limit on the black hole spin a¯ can be deduced as the value of a at which the maximum epicyclic fre- 1. Strohmayer,T.,2001,ApJ,552,L49 quencyisequaltothelowerobservedHFQPOfrequency 2. Remillard,R.A.,Muno,M.P.,McClintock,J.E.&Orosz, [12]. Because for small tori L∼rmax, once a¯ has been J.A.,2002,ApJ,580,1030 determined,theradialpositionr¯ofthemaximumofthe 3. Nowak,M.A.&Wagoner,R.V.,1991,ApJ,378,656 epicyclic frequency provides an upper limit on the size 4. Kato,S.,2001,Publ.Astron.Soc.Japan,53,1 5. Wagoner,R.V.,Silberlgleit,A.S.&Ortega-Rodriguez, ofthetorus,thusprovidinganestimatedifficulttoobtain M.,2001,ApJ,559,L25 throughdirectobservations.Thesetwoestimatesshould 6. Strohmayer,T.,2001b,ApJ,554,L169 be consideredfirst approximationsonly and can be fur- 7. Morgan,E.H.,Remillard,R.A.&Greiner,J.,1997,ApJ, therrefinedthroughtheeigenvalueproblem. 482,993 Afinalcommentwillbereservedtothepossibilityof 8. Abramowicz,M.A.,&Kluzniak,W.,2001,A&A,374, findingsupporttothismodelfromtheobservationaldata. L19 9. Maccarone,T.J.,2002,MNRAS,336,1371 Therearenowextensiveobservationsoftheseblackhole 10. RezzollaL.,YoshidaS’i.,MaccaroneT.J.,ZanottiL, HFQPOs and interesting spectral and flux correlations 2003MNRAS,344,L37 are beginning to be found (see, for instance, [2], and 11. ZanottiO.,RezzollaL.,FontJ.A.,2003MNRAS,341,832 referencestherein).Theseobservationscouldbeusedto 12. RezzollaL.,YoshidaS’i.,ZanottiL,2003MNRAS,344, deducethepresenceofasmalltorusintheterminalpart 978 oftheaccretiondiscs. 13. Remillard,R.A.,Muno,M.,McClintcock,J.E.&Orosz, J., 2002, “New Views on Microquasars ”, Eds. P. Being based on a single assumption, our model for Duruchoux,Y.Fuchs&J.Rodriguez,p49 HFQPOs is simply constructed, but, equally simply, it 14. Fender,R.P.,2001,MNRAS,322,31 can be confuted. Two observational constraints, if not 15. DeVilliers,J.-P.,Hawley,J.F.,2003,ApJ,592,1060 met,willcastseriousdoubtsaboutthismodel.Thefirst 16. Papaloizou,J.C.B.,Pringle,J.E.1984,MNRAS208,721 one requires the lower HFQPO frequency to be always 17. Blaes,O.,1987,MNRAS,227,975 lessthanthemaximumpossibleepicyclicfrequencyfor 18. Blaes,O.,Hawley,J.F.,1988,ApJ,326,277 ablackholeofmassM∗,i.e.lowerHFQPOfrequency≤ 19. DeVilliers,J.-P.,Hawley,J.F.,2002,ApJ,577,866 max[k r(a=1,M∗)]. The second constraint, instead, re- 2201.. KOacrcaasm,,VT.,.,R1e4z9z6olElax,pLo.seittioala.u2r0e0a4,,Binolporgenpaaration quires that even if the HFQPOs frequencies change as