Table Of ContentAstronomy&Astrophysicsmanuscriptno.short˙aph February2,2008
(DOI:willbeinsertedbyhandlater)
The orbital resonance model for twin peak kHz QPOs
Measuring the black hole spins in microquasars
MarekA.Abramowicz1,2,WłodekKluz´niak1,3,4,ZdeneˇkStuchl´ık5,andGabrielTo¨ro¨k1,5
4
1 UKAFFsupercomputerfacility,DepartmentofPhysicsandAstronomy,UniversityofLeicester,England
0
0 2 DepartmentofAstrophysics,ChalmersUniversityS-412-96Go¨teborg,Sweden, [email protected]
2 3 InstituteofAstronomy,ZielonaGo´raUniversityul.Lubuska2,PL-65-265ZielonaGo´ra,Poland
4 CopernicusAstronomicalCentre,Warszawa,Poland, [email protected]
n
5 DepartmentofPhysics,Bezrucˇovona´m.13,CZ-74601Opava,CzechRepublic, [email protected]
a
J
ReceivedNovember30,2003;acceptedDecember1,2003
2
2
Abstract. ManyGalacticblackholeandneutronstarsourcesinlowX-raymassbinariesshowQPOs(quasiperiodicoscilla-
1 tions)intheirobservedX-rayfluxes,i.e.,peaksintheFouriervariabilitypowerspectra.Pairsoftwinpeaksareobserved,and
v inblack-holesystemstheirfrequenciesν ,ν areinrationalratios.Forexample,inallfourmicroquasarswithtwinpeaks
upp down
4 observed,ν /ν =3/2.TherationalratioshavebeenpostulatedinamodelthatexplainedtwinpeakQPOsasanon-linear
upp down
6
resonancebetweenmodesofacretiondiskoscillations.ForthosemicroquasarswherethemassoftheX-raysourceisknown,
4
wedeterminetheblack-holespin,followingfromtheobservedQPOfrequencieswithinvariousmodelsofresonance.
1
0
Keywords. LMXRBblackholes–X-rayvariability–observations–theory
4
0
/
h 1. Introduction
p
- Observedpropertiesofthehighfrequency(fom∼1Hzto∼103Hz)quasiperiodicoscillations(QPOs)intheX-rayvariabilityof
o
neutronstarandblackholesourceshavebeenreviewede.g.byvanderKlis,(2000)andMcClintock&Remillard(2003).There
r
t isnogeneralagreementonaphysicalmechanismexcitingQPOs. TheresonancemodelproposedbyKluz´niak&Abramowicz
s
a (2000)recallstheseobservedfeaturesthatsharplyilluminatethephysicalnatureofthekHztwinpeakQPOs:
:
v
1. The Psaltis-Belloni-van der Klis-Mauche correlation, ν = 0.08ν , between low and high frequency for all QPOs in
i low high
X black hole, neutron stars and white dwarf sources (see Figure 1) proves that in general the QPOs phenomenon is due to
r accretiondiskoscillations,andnottomerekinematiceffects. Italsoprovesthatingeneral,QPOsfrequenciesdonotscale
a
withmass.However,QPOsinkilohertzrangeoftencomeastwinpeaksatfrequenciesν ,ν .ThetwinpeakkHzQPOs
upp down
inmicroquasarsseemstoscaleinverslywithmass,ν ∼ 1/M (seeFigure2).Thissuggestsastronggravityoriginofthis
upp
particularclassofaccretiondiskoscillations.
2. InallfourmicroquasarswithtwinpeakkHzQPOsdiscovered,ν /ν = 3/2 (Table1).Thissuggeststhatanon-linear
upp down
resonancemaybeatwork.
Withthisinmind,itisnaturaltolookforanexplanationoftwinpeakkHzQPOsbasedonstronggravity’soscillatonsinanon-
linearresonance.TheseareindeedtwofundamentalprinciplesoftheresonancemodelsuggestedbyKluz´niak&Abramowicz.
2. Thestronggravityoscillations
Considerablackhole1withthemass M andangularmomentumJ .Inthinblackholeaccretiondisks,matterspiralsdownthe
0 0
centralblackhole alongstreamlines thatare locatedalmoston theblack holeequatorialplane θ = θ = π/2,andthatlocally
0
differonlyslightlyfromafamilyofconcentriccirclesr=r =const.Thesmalldeviations,δr=r−r ,δθ=θ−θ aregoverned,
0 0 0
withaccuracytolinearterms,by
δr¨+ω2δr=δa , δθ¨+ω2δθ=δa . (1)
r r θ θ
Sendoffprintrequeststo:G.To¨ro¨k, [email protected]
1 WerescalemasswithM=GM /c2 =r ,andangularmomentumwitha= J c3/M3G2.WeuseBoyer-Lindquistcoordinates,t,r,θ,φ.
0 G 0
2 MarekA.Abramowiczetal.:TheorbitalresonancemodelfortwinpeakkHzQPOs
Table1.FrequenciesoftwinpeakkHzQPOsinmicroquasars
Source νupp[Hz] νdown[Hz] 2νupp/3νdown−1 Mass[M⊙]
(a)XTE1550–564 276 174 0.05747 8.4 —10.8
(a)GRO1655–40 450 300 0.00000 6.0 — 6.6
(a)GRS1915+105 168 113 0.00885 10.0 —18.0
(b)H 1743–322 240 160 0.00000 notmeasured
(a)FromMcClintock&Remillard(2003),(b)FromHomanetal.(2003).
6 8 M=1M0(cid:12) 12 14 16 18
2
2.7 - 500Hz
1655-40 - 450Hz
1 2.6 - 400Hz
- 350Hz
Hz℄) 0 Hz℄) 2.5 - 300Hz
[ [
(cid:23)low (cid:23)upp 2.4 1550-564 - 250Hz
( (
g -1 g
o o
l l
2.3 - 200Hz
-2
1915+105
2.2
-3
-2 -1 0 1 2 3 0.8 0.9 1 1.1 1.2 1.3
log((cid:23)high[Hz℄) log(M=M(cid:12))
Fig.1. (left) The ν = 0.08ν correlation for neutron star binaries (open circles), black hole binaries (filled circles), and
low high
whitedwarfsbinariesSSCyg(filleddiamonds),andVWHyi(opensquares).DataforLMXRBfromBelloni,Psaltis,&vander
Klis (2002)who discoveredthe correlation for these sources, and for CV from Mauche (2002)who found that it also applies
for CVs. See additionaldiscussion by Psaltis, & van der Klis (2002)and Woudt & Warner (2002).Figure is reproducedfrom
Mauche(2002)withpermissionoftheEditorsoftheAstrophysicalJournal.ThecorrelationprovesthattheQPOsphenomenon
mustbeduetoaccretiondisks’oscillationsandnottotheirkinematicslikee.g.Dopplermodulationoffluxesfromisolated‘hot
spots’:QPOsarewaves,notparticles.Indeed,itisnotpossiblethateverytimeaspotiscreatedatoneradiustoproduceν by
high
Dopplermodulation,anotherspotiscreatedatapreciselycorrelatedlargerradiustoproduceν = 0.08ν ,alsobyDoppler
low high
modulation. Abramowicz, Barret, Kluz´niak, Lee & Spiegel (2003) suggested that the low-high correlation in accretion disks
could occur severalreasons, well-knownin oceanography.On deep waters, in a train of travelling waves with high frequency
ν ,aboutevery9thwaveishigherduetotheside-band(Benjamin-Frei)instability.9thwavescausealowerfrequencyfeature
high
ν thatobviouslyobeysa correlation,ν = 0.11ν . On shallowbeaches,thelowerfrequencyfeaturesin travellingwaves
low low high
are caused by backflows. Both effects could operate in accretion disks, producingthe 1/0.08 ≈ 13 th wave and the low-high
correlation.
Fig.2. (right) The best fit for the frequency-massscaling for microquasars’kHz twin peak QPOs, νupp = 2793/M [Hz/M⊙],
adoptedfromMcClintock&Remillard(2003)whopointedoutthescaling.Thescalingprovesthatoscillationsarerelativistic,
i.e.theyoccurinstronggravity,ataradiusfixedintermsofthegravitationalradiusr =GM /c2.
G 0
Heredotdenotestimederivative.ForstrictlyKeplerian(i.e.free)motion,δa =0,δa =0,andtheaboveequationsdescribetwo
r θ
uncoupledharmonicoscillatorswiththeeigenfrequenciesω ,ω equal,inKerrgeometry(e.g.Nowak&Lehr,1999),
θ r
1/2
ω2 =Ω2 1−4ax−3/2+3a2x−2 , ω2 =Ω2 1−6x−1+8ax−3/2−3a2x−2 , Ω = GM0 x3/2+a −1, (2)
θ K r K K r3
(cid:16) (cid:17) (cid:16) (cid:17) G (cid:16) (cid:17)
where x = r/M.Thetwoepicyclicfrequencies,verticalν = ω /2πandradialν = ω /2π,areshowninFigure3togetherwith
θ θ r r
theKeplerianorbitalfrequencyν = Ω /2πforanonrotating(a = 0)blackhole,andforamoderatelyrotating(a = 0.8)black
K K
hole. Figures 4, and 5 show these frequecies in the whole range of a, from a maximally co-rotating black hole (a = 1) to a
maximallycounter-rotatingblackhole(a=−1).
MarekA.Abramowiczetal.:TheorbitalresonancemodelfortwinpeakkHzQPOs 3
280
?
rossat 6 a=0 1000 a=0:8
621Hz
240
800
200
℄ ℄
Hz 160 (cid:23)(cid:18)=(cid:23)K Hz 600
(cid:23)[ (cid:23)[ ? (cid:23)K
120
400
80 (cid:23)m(cid:0)a(cid:0)x(cid:0)(cid:0)(cid:18) (cid:23)(cid:18)
200
40 rh rph (cid:23)r rh rph (cid:23)m(cid:1)a(cid:1)x(cid:1)(cid:21)
(cid:23)r
0
2 4 6 8 10 12 14 1.5 2 2.5 3 3.5 4 4.5 5
r=M r=M
Fig.3. Orbital frequencyν , and the two epicyclic frequencies, radial ν , and vertical ν for Keplerian circular orbits around
K r θ
a 10M⊙ black hole. Such orbits are posible only for radii larger than the radius of the circular photon orbit rph. This limit
is labelled here and in other Figures by the subscript ph. Left panel for non-rotatingblack hole, right panel for a moderately
(a = 0.8) rotating one. In Newton’s theory with the 1/r potential, all the three frequencies are equal: ν = ν = ν . Strong
K θ r
Einstein’sgravitymakesν ≥ν >ν .
K θ r
2.1.Thehighestpossibleorbitalfrequency:ISCOandRISCO
InNewton’stheorywiththe−GM /rpotentialitis GM /r3/2 = ν = ν = ν ,butinthestronggravityofarotatingblackhole,
0 0 K r θ
ν > ν > ν .Theradialepicyclicfrequencyν hasamaximumataparticularcircularorbitwiththeradiusr > r ,depending
K θ r r ms
ontheblackholespin(Kato&Okazaki,1978),andgoestozeroatr ,themarginallystablecircularorbit.Theorbitisnowmost
ms
oftencalledISCO,whichisahandyshortname,andwesuggesttocallRISCOthemarginallyboundorbitatr .
mb
Location of the inner edge of accretion disk, and therefore also the value of the highest possible orbital frequency, depends
on disk’s radiative efficiency. Very thin disks with high radiative efficiency have their inner edges almost exactly at ISCO.
Correspondingly,theISCO orbitalfrequency,ν (r , isthehighestpossibleorbitalfrequencyforthindisks.However,asfirst
K ms)
proved analytically by Kozłowski, Jaroszyn´ski & Abramowicz (1978) and then confirmed in numerical simulations by many
authors, for accretion disks with low radiative efficiency such as radiative and ion tori, slim disks, and adafs, the orbits are
stabilizedbythepressuregradient,andtheinneredgegoesclosetoRISCO.Forsuchdisks,thehighestpossibleorbitalfrequency
isobviouslyhigherthantheorbitalfrequencyatISCO,ν (r )>ν (r ),asillustratedinFigure4(rightpanel).
K mb K ms
2.2.Stronggravity’sresonancesandtheinversemassscaling
Resonancesoccurwhentwoofthethreefrequenciesν ,ν ,ν ,areinrationalratios,e.g.ν /ν =n/m.Becauseallthreefrequen-
K r θ r θ
ciesν ,ν ,ν havethegeneralform,
K r θ
GM 1/2 M −1
ν= 0 f(x,a)=32293 0 f(x,a)[Hz], (3)
r3 M⊙!
withf(Gx,a)beinga dimensionlessfunction,anyparticularresonanceoccursatits ownresonanceradius x (a),andtherefore
n:m
each frequencyat resonancescales approximatelyas 1/M. Radii of the three particularresonances(3:1,2:1, 3:2) betweenthe
verticalandradialepicyclicfrequenciesareshowninFigure6.
3. Thenon-linearresonances
3.1.Importanceofthenon-lineareffects
TheeffectivepotentialU(r,θ;ℓ)fororbitalmotionofaparticlewithafixedangularmomentumℓ>ℓ hasaminimumatr (ℓ),
ms 0
correspondingto the locationof a stable circularorbit.Its Talyorexpansion(forsimplicity we write it onthe equatorialplane
θ=π/2),
1 ∂2U 1 ∂3U
U(r,ℓ)= (r−r )2+ (r−r )3+... (4)
2 ∂r2 ! 0 6 ∂r3 ! 0
0 0
containshigherthanquadraticterms,whichmeansthatsmalloscillationsaroundtheminimumatr−r aredescribedbynon-
0
lineardifferentialequations(e.g.Landau&Lifshitz,1976;Nayfeh&Mook,1979).Non-linearresonancesthatmaybeexcited
inthesenon-linearoscillationshaveseveralcharacteristicpropertiesthatcloselyresemblethoseobservedinQPOs:
4 MarekA.Abramowiczetal.:TheorbitalresonancemodelfortwinpeakkHzQPOs
1600 3.2
1400
3
1200
℄)
Hz℄ 1000 Hz 2.8 (cid:23)K(rmb)
[
[ (cid:23)
(cid:23) (
800 (cid:23)K(rph) log
2.6
600
(cid:23)K(rms)
400 a=(cid:0)1 2.4
a=1
1.5 2 2.5 3 3.5 4 4.5 0 0.2 0.4 0.6 0.8 1
r=M a
Fig.4. Left:The Keplerianorbitalfrequencyfor 10M⊙ mass black hole. Right: RISCO and ISCO frequenciesfor 10M⊙ mass
blackhole.TheaccretiondiskinneredgemustbelocatedbetweenRISCOandICSO,dependingondisk’sefficiency.
700
250
600
500 200
℄ 400 ℄
z z 150
H H
[ [
(cid:23) 300 (cid:23)
100
200
a=1
50
100
a=(cid:0)1
a=(cid:0)1
a=1
2 4 6 8 10 12 2 4 6 8 10 12
r=M r=M
Fig.5. Left: The verticalepicyclic frequencyfor 10M⊙ mass black hole. Right: the radial epicyclic frequencyfor 10M⊙ mass
blackhole.
1. Resonanceoccursinaregionwithafiniteoftenlargewidthδν.
2. Thefrequenciesofoscillationsν dependonamplitudeandforthisreasontheymaybetimedependentandmaydifferfrom
i
thefixedeigenvaluefrequenciesν ofthesystem,ν(t)=ν +δν(t).
i(0) i i(0) i
3. Combinationfrequencies,ν ±ν maybepresent.
i(0) k(0)
4. Subharmonicfrequenciesmaybepresent.
3.2.The3:2parametricresonance
Weshallstartwithanargumentappealingtophysicalintuitionandshowingthattheresonancenowdiscussedisaverynatural,
indeednecessary,consequenceofstronggravity.Inthindisks,randomfluctuationshaveδr≫δθ.Thus,δrδθisafirstorderterm
inδθandshouldbeincludedinthefirstorderequationforverticaloscillations(1).Theequationnowtakestheform,
δθ¨+ω2[1+hδr]δθ=δa , (5)
θ θ
wherehisaknownconstant.Thefirstorderequationforδrhasthesolutionδr= A cos(ω t).Insertingthisin(5)togetherwith
0 r
δa =0,onearrivesattheMathieuequation(A isabsorbedinh),
θ 0
δθ¨+ω2[1+h cos(ω t)]δθ=0, (6)
θ r
thatdescribestheparametricresonance.FromthetheoryoftheMathieuequationoneknownsthatwhen
ω ν 2
r = r = , n=1, 2, 3..., (7)
ω ν n
θ θ
MarekA.Abramowiczetal.:TheorbitalresonancemodelfortwinpeakkHzQPOs 5
16
280
a=0 14 3:2
240
12
200 2:1
3:1
℄ 10
z 160
H M
(cid:23)[ 120 2:1 r= 8 3:1
(cid:23)(cid:18)
6
3:2
80
4
40 (cid:23)r
2
0
5 6 7 8 9 10 11 0
-1 -0.5 0 0.5 1
r=M
a
Fig.6.Left:locationsof the threeresonances:the 3:2 parametric,and 2:1 and3:1 forcedfor Schwarzschildcase. Right:these
locationsdependingontheblackholespin.Alsoshownistheradiusr (dottedline)atwhichthestandardrelativisticShakura-
F
Sunyaevdisk(Page&Thorne,1974)emitslocallythemaximalflux,andtheradiusr (dashedline)correspondingtothepressure
C
centreofthemaximallythicktorus,i.e.thetoruswithconstantangularmomentumequaltotheKeplerianvalueatRISCO.
theparametricresonanceisexcited(Landau&Lifshitz,1976).Theresonanceisstrongestsforthesmallestpossiblevalueofn.
Becausenearblackholesν <ν ,thesmallestpossiblevalueforresonanceis n=3,whichmeansthat2ν =3ν .Thisexplains
r θ θ r
theobserved3:2ratio,because,obviously,
ν =ν , ν =ν , (8)
upp θ down r
Of course in real disks neither δr = A cos(ω t), nor δa = 0 exactly, but one may expect that because these equations are
0 r θ
approximately obeyed for thin disks, the parametric resonance will also be excited in realistic situations. And this is indeed
the case. The parametricresonanceof the typediscussed abovewas foundin numericalsimulationsof oscillationsin a nearly
Keplerian accretiondisk by Abramowicz et al. (2003).Their numericalresults were reproducedin an exact analytic solutions
firstbyRebusco(2003)andlaterconfirmedandgeneralisedbyHorak(2003).Theanalyticsolutionisaccurateuptothirdorder
terms in δr, δθ, and based on the methodof multiple scales (see e.g. Nayfeh& Mook,1976).Existence of the 3:2 parametric
resonanceis thereforea mathematicalporopertyof thin,nearlyKepleriandisks. Itwasfoundthatthe resonanceisexitedonly
inthenon-Kepleriancase,withsomeweakforcesδa , 0andδa , 0present.Theiroriginiscertainlyconnectedtostresses
θ r
(pressure,magneticfield,viscosity),butexactdetailsremaintobedetermined—atpresentδa andδa arenotcalculatedfrom
θ r
firstprinciplesbutdescribedbyanansatz2.
Theparametricresonanceoccursataparticularradiusr (a),determinedbythecondition3ω (r ,a)=2ω (r ,a)andequation
3:2 r 3:2 θ 3:2
(2). We show the function r (a) in Figure 6. In Figure 7 we fit the 3:2 resonance theoretically predicted frequencies to the
3:2
observationaldataforthethreemicroquasarswiththeknownmasses.
3.3.Theforced3:1and2:1resonances
A directresonantforcingofverticaloscillationsbytheradialonesthrougha pressurecoupling,andwithδa ∼ cos(ω t), was
r r
evidentinrecentnumericalsimulationsofoscillationsofaperfectfluidtorus(Lee,Abramowicz&Kluz´niak,2003).Thissupports
anidea(Abramowicz&Kluz´niak,2001)foranotherpossiblemodelforthetwinpeakkHzQPOs:aforcednon-linearoscillator,
δθ¨+ω2δθ+[nonlineartermsinδθ]=h(r)cos(ω t), ω =nω. (9)
θ r θ r
Obviously,thereisnovaluefornsuchthatω andω couldbeinthe3:2ratio.However,thenon-lineartermsalowthepresence
θ r
ofcombinationfrequenciesinaresonantsolutionsforδθ(t)(seee.g.Landau&Lifshitz,1974),
ω− =ωθ−ωr, ω+ =ωθ+ωr. (10)
2 Whilethelackofafullphysicalunderstandingisobviouslynotsatisfactory,theexperiencetellsthatsuchasituationisnotuncommonfor
non-linearsystems.Examplesareknownofmathematicallypossibleresonancescausingdamageinbridges,areoplanewingsetc.,forwhichno
specificphysicalexcitationmechanismcouldhavebeenpinneddown(Nayfeh&Mook,1976).
6 MarekA.Abramowiczetal.:TheorbitalresonancemodelfortwinpeakkHzQPOs
6 8 M1=M0(cid:12) 12 14 16 18
- 500Hz
1655-40 - 450Hz
2.6 - 400Hz
- 350Hz
Hz℄) 2.4 1550-564 -- 235000HHzz
[
(cid:23)upp - 200Hz
(
g
lo 2.2
- 0.994
1915+105
- 0.932
- 0.871
2 - 0.809 a
- 0.747
- 0.685
0.8 0.9 1 1.1 1.2 1.3
log(M=M(cid:12))
Fig.7.Fitofthefrequencyν =ν (M,a)predictedbythe3:2parametricresonancemodeltothefrequenciesobservedinthree
upp θ
microquasarswithknownmasses.Thespinparameteraisnotknownfromobservations,andthelinesa = constarecalculated
fromthemodel.Oneshouldnoticethatthededucedblackholespinsareratherhigh,whichisexpectedifmicroquasars’jetsare
asignatureoflargespinofthecentralblackhole.
Simplearthmeticshows(AbramowiczandKluz´niak,2001)thatthesecombinationfrequenciesmaybeinthe3:2ratioifandonly
ifn=2,orn=3andthatinthesetwocasestheobservedfrequenciesν =ω /2πandν =ω /2πareuniquelygiven
down down upp upp
by,
ωdown =ω− =2ωr, ωupp =ωθ =3ωr for n=3 forced epicyclic resonance ωθ =3ωr, (11)
ω =ω =3ω , ω =ω = 2ω for n=2 forced epicyclic resonance ω =2ω . (12)
upp + r down θ r θ r
WefitobservedQPOstothesepredictedbytheforcedepicyclic3:1and2:1resonancesinFigure8.
6 8 M1=M0(cid:12) 12 14 16 18 6 8 M1=0M(cid:12) 12 14 16 18
- 500Hz - 500Hz
1655-40 - 450Hz 1655-40 - 450Hz
2.6 - 400Hz 2.6 - 400Hz
- 350Hz - 350Hz
Hz℄) 2.4 1550-564 -- 235000HHzz Hz℄) 2.4 1550-564 -- 235000HHzz
(cid:23)[upp - 200Hz (cid:23)[upp - 200Hz
( (
log 2.2 log 2.2
- 0.608 - 0.440
1915+105 - 0.457 1915+105 - 0.269
- 0.305 - 0.099
2 - 0.154 a 2 - -0.071 a
- 0.002 - -0.242
- -0.150 - -0.412
0.8 0.9 1 1.1 1.2 1.3 0.8 0.9 1 1.1 1.2 1.3
log(M=M(cid:12)) log(M=M(cid:12))
Fig.8.Left:Fitofthepredictionsofthe3:1forcedresonancemodeltoobservations.Right:thesameforthe2:1forcedresonance.
Bothresonancesarebetweenverticalandradialepicyclicoscillations.
3.4.Keplerianresonances
The resonancesdiscussed so far were due to a couplingbetween epicyclicoscillations — radialand vertical.Such a coupling
existsinvarietyofrealisticphysicalsituations.Itismoredifficulttoimaginearealisticphysicalcouplingbetweenradialepicyclic
oscillations and orbital Keplerian oscillations. One possibility is connected to the fact that g-modes have pattern frequency
2πν = ±(ν ± mν ), and these can be in co-rotation resonance, i.e., with ν = ν (Kato, 2001). The case ν /ν = 3/2 is
m r K m K K r
excludedbyobservations(Figure9,leftpanel).Inaddition,co-rotationresonanceleadstodamping,andnotexcitation,ofmodes
(Kato,2002).
Anotherpossiblitythatmaycometo mindisbasedonthe followingidea(Spiegel,2003).When thepotentialvorticityiscon-
served,coherentvorticestendto formin pairswith oppositevorticity(Bracco,Provenzale,Spiegel& Yecko,1999).One may
imaginethatbecausethespatialdistancebetweenthetwostructuresthatoscillateswiththeepicyclicradialfrequency,depends
MarekA.Abramowiczetal.:TheorbitalresonancemodelfortwinpeakkHzQPOs 7
onthe velocityprofileofthe disk i.e. also onthe oscillationsoforbitalvelocity,a resonancebetweenthesetwo frequenciesis
possible.InFigure9weplotthepredictionoftheonlythreepossiblecasescorrespondingtoω /ω =3/2,
upp down
ω =3ω , ω =ω =2ω , ω =2ω , ω =ω = (1/3)ω , ω =ω , ω =ω = (2/3)ω . (13)
upp r down K r upp K down r K upp K down r K
6 8 M1=M0(cid:12) 12 14 16 18 6 8 M1=M0(cid:12) 12 14 16 18 6 8 M1=0M(cid:12) 12 14 16 18
- 500Hz - 500Hz - 500Hz
1655-40 - 450Hz 1655-40 - 450Hz 1655-40 - 450Hz
2.6 - 400Hz 2.6 - 400Hz 2.6 - 400Hz
- 350Hz - 350Hz - 350Hz
log((cid:23)[Hz℄)upp22..42 1550-564 --- 223050000HHHzzz log((cid:23)[Hz℄)upp22..42 1550-564 ---- 0223.050500050HHHzzz log((cid:23)[Hz℄)upp22..42 1550-564 ---- 0223.405040002HHHzzz
1915+105 1915+105 - 0.413 1915+105 -0.272
2 - 1.000 a 2 -- 00..123796 a 2 -- - 00..100619 a
- 0.002 - -0.240
- 0.790 - -0.134 - -0.410
0.8 0.9 1 1.1 1.2 1.3 0.8 0.9 1 1.1 1.2 1.3 0.8 0.9 1 1.1 1.2 1.3
log(M=M(cid:12)) log(M=M(cid:12)) log(M=M(cid:12))
Fig.9.From left:fit of the predictionsof the 3:2,3:1 and2:1 “Keplerian”resonanceto observations.Resonancesare between
radialepicyclicoscillationsandsomeoscillationsatKeplerianfrequency.
4. Conclusions
In Table 2 we summarize results of our estimates of the black hole spin for the three microquasars with known masses. The
uncertaintyinthespinestimateisduetouncertaintyinthepresentknowledgeoftheblackholemass.Inthisrespectwenotethat
inthecaseofGRO1655-40,thepossiblespinrangesgiveninTable2for3:1and2:1Keplerianresonancesare,respectively,0.45
–0.53,and0.31–0.42.TheyaremoreprecisethanthecorrespondingestimatesbyAbramowicz&Kluz´niak(2001),i.e.0.36–
0.67and0.2–0.6.TheimprovementintheaccuracyisduetotheimprovementintheknowledgeofthemassofGRO1655-40.
In2001themasswasknowntobeintherange5.5< M/M⊙ <7.9,whilein2003theaccuracyincreasedto6.0< M/M⊙ <6.6.
The resonance model is rather sensitive to observational constraints: the data already excludes the 3:2 Keplerian resonance
as a possible explanation of twin peak QPOs in case of two microquasars. All other resonannces discussed in the paper are
consistentwiththeexistingdata,butitisplausiblethatfutureobservationsmaynarrowdownthechoiceofaresonance.Future
developementsin accretion disk theory could also narrow down the choice. For example, it is often argued that the presence
ofrelativisticjetisasignatureoflargeblackholespin(BlandfordandZnajek1977,seehoweverFenderetal.2004).Because
microquasarsdohavejets,onewouldexceptthata≈1fortheirblackholes.Thisargument,ifproventrue,woulduniquelypoint
tothe3:2paramericresonance,astheonlypossiblechoice.
Different resonances occur at very different resonance radii. Figure 6 shows that accretion disk physics at these radii is also
verydiferent.The3:2parametricresonanceislocatedataveryouterpartoftheinnermostregionofthedisk,the2:1resonance
is located in the middle of this region, and the 3:1 resonance most close to the inner edge. Therefore, a physical excitation
mechanism(stillunknown)mustbeverydifferentfordifferentresonances.
If the resonace model is correct, and the kHz double peak QPOs are indeed due to non-linear strong gravity’s resonance, the
QPOs phenomenonwould havea fundamentalimportanceas a practicaltest of super-stronggravity.It couldalso be usefulin
severalastrophysicalapplicationsofblackholeastrophysics.Forexample,the1/M scalingofthetwinpeakQPOsfrequencies
withthe 3:2ratio(shownin Figure1),wasproposedbyAbramowicz,Kluz´niak,McClintock& Remillard(2003)asa method
forestimatingblackholemassesinAGNsandULXs,basedonMirabel&Rodr´ıguez(1998)analogybetweenmicroquasarsin
ourGalaxyanddistantquasars.Indeed,iftheanalogyisalsovalidforaccretiondiskoscillations,thendiscoveringinULXsthe
twin peak QPOs frequencieswith the 3:2 ratio,would resolvethe controversyabouttheir mass: if ULXs blackholeshavethe
samemassesasmicroquasars,thefrequencieswillbe∼100Hzasinmicroquasars.IfULXsblackholesare∼1000timesmore
massive,thefrequencieswillbe∼0.1Hzinstead.
Acknowledgements. WethankJ.McClintock,R.Remillard,andR.V.Wagonerforcomments.Mostoftheworkreportedherewasdoneatthe
UKAstrophysical FluidsFacility(UKAFF)andsupportedthroughtheEuropeanComissionAccesstoResearchInfrastructureactionofthe
ImprovingHumanPotentialProgram.Z.S.andG.T.weresupportedbytheCzechGACRgrants202/02/0735 and205/03/H144, andM.A.A.
andW.K.bythePolishKBNgrant2P03D01424.
References
AbramowiczM.A.,BulikT.,BursaM.,Kluz´niakW.,2003A&ALett.,404,L21
AbramowiczM.A.,ChenX.-M.,GranathM.,LasotaJ.-P.ApJ.,471,762
8 MarekA.Abramowiczetal.:TheorbitalresonancemodelfortwinpeakkHzQPOs
Table2.Summaryofangularmomentumestimatesforresonancemodels
Source 1550–564 1655–40 1915+105
Models
3:2[ν , ν] parametric +0.89—+0.99 +0.96—+0.99 +0.69—+0.99
θ r
2:1[ν , ν] forced +0.12—+0.42 +0.31—+0.42 -0.41—+0.44
θ r
3:1[ν , ν] forced +0.32—+0.59 +0.50—+0.59 -0.15—+0.61
θ r
3:2[ν , ν] “Keplerian” +0.79
K r
2:1[ν , ν] “Keplerian” +0.12—+0.43 +0.31—+0.42 -0.41—+0.44
K r
3:1[ν , ν] “Keplerian” +0.29—+0.54 +0.45—+0.53 -0.13—+0.55
K r
AbramowiczM.A.,CzernyB.,LasotaJ.-P.,SzuszkiewiczE.,1988ApJ.,332,646
AbramowiczM.A.,KarasV.,Kluz´niakW.LeeW.,RebuscoP.,2003Publ.Astr.Soc.Japan.,55,467
AbramowiczM.A.,Kluz´niakW.,2000,A&ALett.,374,L19
AbramowiczM.A.,Kluz´niakW.,AlmergrenG.J.E.,ThampanA.W.,WallinderF.,2002,Class.Q.Grav.,19,L57
AbramowiczM.A.,Kluz´niakW.,2003Gen.Rel.Grav.,35,69
AbramowiczM.A.,Kluz´niakW.,BarretD.,LeeW.,SpiegelE.,2003,(inpreparation)
BlandfordR.D,andZnajekR.L.1977.Non.Not.Roy.Astr.Soc.179,433-40
DeVilliersJ.-P.,HawleyJ.F.,KrolikJ.H.,2003astro-ph/0307260
FenderR.KinwahW.,JohnstonH.,TzioumisT.,JonkerP.,SpencerR.,vanderKlisM.,2004,Nature,427,222,2004(astro-ph/0401290)
HomanJ.,MillerJ.M.,WijnandsR.,SteeghsD.,BelloniT.,vanderKlisM.,LewinW.H.G.,2003,Atel16(http://integral.rssi.ru/atelmirror/)
IgumenshchevI.V.,NarayanR.,AbramowiczM.A.,2003,Ap.J.,592,1042
Jaroszyn´skiM.,AbramowiczM.A.,Paczyn´skiB.,1980,ActaAstronomica,30,1
KatoS.,2001Publ.Astron.Soc.Japan53,1
KatoS.,2001Publ.Astron.Soc.Japan53,L37
KatoS.FukueJ.,1980Publ.Astron.Soc.Japan32,377
KatoS.FukueJ.,1980Publ.Astron.Soc.Japan32,377
KatoS.,Fukue,J.,Mineshige,S.1998,Black-HoleAccretionDisks(Kyoto:KyotoUniversityPress)
Kluz´niakW.,AbramowiczM.A.,KatoS.,LeeW.H.,StergioulasN.,2003ApJ(inpress),astro-ph/0308035
Kluz´niakW.,AbramowiczM.A.,2000,Phys.Rev.Lett.(submitted),astro-ph/0105057
Kluz´niakW.,AbramowiczM.A.,2001,ActaPhys.Polon.,B32,3605
Kluz´niakW.,AbramowiczM.A.,2001,A&A(submitted),astro-ph/0203314
Kluz´niakW.,AbramowiczM.A.,2003,12thWorkshoponGeneralRelativityandGravitation,
(Tokyo:TokyoUniversityPress),astro-ph/0304345
KrolikJ.H.,HawleyJ.F.,2002ApJ.,573,754
LandauL.D.,LifshitzE.M.,1976Mechanics,3rdedition,(Oxford:PergamonPress)
LeeW.H.,AbramowiczM.A.,Kluz´niakW.,2003(inpreparation)
LiL.-X.,NarayanR.,2003,astro-ph/0310081
MaucheC.W.,2002,ApJ.,580,423
McCalumJ.E,RemillardR.A.,2003,astro-ph/0306213
MirabelI.F,andRodr´ıguezL.F.,1998,Nature392,673
Nowak,M.A.,Wagoner,R.V.1992,ApJ,393,697
NowakM.,LehrD.,1999,TheoryofBlackHoleAccretionDisks,
eds.AbramowiczM.A.,Bjo¨rnssonG.,PringleJ.E.,(Cambridge:CambridgeUniversityPress)
Nayfeh,A.H.,Mook,D.T.,1979,NonlinearOscillations,(NewYork:Wiley-Interscience)
Okazaki,A.T.,Kato,S.,Fukue,J.1987,Publ.Astron.Soc.Japan39,457
Perez,C.A.,Silbertgleit,A.S.,Wagoner,R.V.,Lehr,D.E.1997,ApJ,476,589
PsaltisD.,BelloniT.,vanderKlisM.,1999,ApJ.520,262
RebuscoP.2003,(inpreparation)
RezzollaL.,YoshidaS’i.,MaccaroneT.J.,ZanottiO.,2003Mon.Not.R.astr.Soc.,344,L37
Silbergleit,A.S.,Wagoner,R.V.,Ortega-Rodr´ıguez,M.2001,ApJ,548,335
SzuszkiewiczE.,MillerJ.C.,2001Mon.Not.R.astr.Soc.,328,36
vanderKlisM.,2000,Ann.Rev.ofA&A,38,717
Wagoner,R.V.1999,Phys.Rep.,311,259
Wagoner,R.V.,Silbergleit,A.S.,Ortega-Rodr´ıguez,M.2001,ApJ,559,L25
