Twin-peak quasiperiodic oscillations as an internal resonance

0.27 MB·English

by  J. Horak

Astronomy&Astrophysicsmanuscriptno.4039horka3 (cid:13)c ESO2008
February5,2008
Twin-peak quasiperiodic oscillations as an internal resonance
(ResearchNote)
6 J.Hora´kandV.Karas
0
0
2 AstronomicalInstitute,AcademyofSciences,Bocˇn´ıII,CZ-14131Prague,CzechRepublic

n
a Received12August2005;accepted30December2005
J

3 ABSTRACT

1 Aims.Twointer-relatedpeaksoccurinhigh-frequencypowerspectraofX-raylightcurvesofseveralblack-holecandidates.Wefurtherexplore
v theideathatanon-linearresonancemechanism,operatinginstrong-gravityregime,isresponsibleforthesequasi-periodicoscillations(QPOs).
3
Methods. By extending the multiple-scales analysis of Rebusco, we construct two-dimensional phase-space sections, which enable us
5
to identify different topologies governing the system and to follow evolutionary tracks of the twin peaks. This suggests that the original
0
(Abramowicz & Kluz´niak) parametric-resonance scheme can be viewed as an ingenuous account of the QPOs model with an internal
1
resonance.
0
6 Results.We show an example of internal resonance in a system with up to two critical points, and we describe a general technique that
0 permitstotreatothercasesinasystematicalmanner.Aseparatrixdividesthephase-spacesectionsintoregionsofdifferenttopology:insidethe
/ librationregiontheevolutionarytracksbringtheobservedtwin-peakfrequenciestoanexactrationalratio,whereasinthecirculationregion
h
theobservedfrequenciesremainoffresonance.Ourschemepredictsthepowershouldcyclicallybeexchangedbetweenthetwooscillations.
p
Likewise the high-frequency QPOsin neutron-star binaries, also in black-hole sources one expects, as ageneral property of the non-linear
-
o model,thatslightdetuningpushesthetwin-peakfrequenciesoutofsharpresonance.
r
t
s
a
Keywords.Accretion,accretion-discs–Blackholephysics–Quasi-periodicoscillations
:
v
i
X
1. Introduction should prefer the 3:2 ratio; it also suggests this could be
r
a understood if a non-linear coupling mechanism operates in
Twin peaks occur in high-frequency (∼ 50–450 Hz) power a black-hole accretion disc, where strong-gravity effects are
spectraofX-ray(∼2–60keV)lightcurvesofseveralblack-hole essential. Indeed, especially in those black-hole candidates
candidates (see van der Klis 2006; McClintock & Remillard where the high-frequencyQPOs have been reported,they oc-
2003 for recent reviews of observational properties and the- cur very close to the ratio of small integer numbers, 3:2 in
oretical interpretations). This transient phenomenon seems to particular(Miller etal. 2001; Strohmayer2001; Homanet al.
be connected with the kilohertz quasi-periodic oscillations 2004;Remillard etal. 2005;Maccarone& Schnittman2005).
(QPOs) in neutron star sources, of which more examples are Nowadays, the original account can be viewed as a na¨ıve
known (about the tens at present). The nature of black hole modelwiththeinternalresonance.Variousrealizationsofthis
high-frequencyQPOsremainspuzzlingdespitevarietyofmod- scheme have been examined in terms of accretion disc/torus
els proposedin the literature. In neutron-starlow-mass X-ray oscillations (e.g. Abramowicz et al. 2003; Bursa et al. 2004;
binariesthesetwinQPOsareknowntooccuroftensimultane- Kato2004;Li&Narayan2004;Schnittman&Rezzolla2005;
ously and they can be highly coherent (Q & 102; e.g. Barret Zanottietal.2005).
etal.2005),slowlywanderinginfrequenciesbetweendifferent
So far the “right” model has not yet been identified.
observations, whereas in black-hole candidates the QPO co-
However,ithasbeenrecognizedthatfruitfulknowledgeabout
herencyappearstobelower(Q ∼ 2–10)andthepresenceofa
commonpropertiesofhigh-frequencyQPOscanbegainedby
pairpopsuponlywhenacollectionofobservationsiscarefully
investigating a very general resonance scheme, which likely
analyzed.
governs matter near a compact accreting body. To this aim,
In Abramowicz & Kluz´niak (2001) and Kluz´niak & Abramowicz et al. (2003) examined the epicyclic resonances
Abramowicz (2001) an idea of accretion disc resonance was inanearly-geodesicmotioninstronggravity.Rebusco(2004)
proposed,whichnaturallyincorporatespairsoffrequenciesoc- andHora´k(2004),byemployingthemethodofmultiplescales
curring in a ratio of small integer numbers.This scheme pre- (Nayfeh & Mook 1979), have demonstrated that the 3:2 res-
dictstheobservedfrequencyratiosinblack-holeQPOsources onance is indeed the most prominent one near horizon of a
2 J.Hora´k&V.Karas:Twin-peakquasiperiodicoscillationsasaninternalresonance(RN)
central black hole. Only certain frequency combinations are Forthecentralfieldweassumetheform
allowed,dependingonsymmetrieswhichthesystemexhibits,
GM
andonlysomeoftheallowedcombinationshavechancetogive Φs(r)=− , (6)
r˜
risetoastrongresonance.
Inthepresentpaperwepursuethisapproachfurtherandwe where we set r˜ = r or r˜ = r − RS (to adopt the Newtonian
findtracksthatanaxiallysymmetricsystemwithtwodegrees orthepseudo-Newtonianapproximations;RS ≡ 2GM/c2).We
of freedom,near resonance,shouldfollow in the plane of en- assume a circular ring (mass m, radius a) as a source of the
ergy(ofthe oscillations)versusradius(wherethe oscillations perturbingpotential,
takeplace).Weshowdifferenttopologiesofthephasespacein
2GmK(k)
thewaythatcloselyresemblesthemethodofdisturbingfunc- Φr(r,θ)=− π B1/2, (7)
tion,familiarfromthestudiesoftheevolutionofmeanorbital
where K(k) is a complete elliptical integral of the first kind,
elementsincelestialmechanics(Kozai1962;Lidov1962).The
B(r,θ) ≡ r2+a2+2arsinθ,k(r,θ) ≡4arsinθ/B(r).
analogyis veryilluminatingand it providesa systematic way
ofdistinguishingtopologicallydifferentstatesofthesystem.In Atthispointa remarkis necessaryregardingthe interpre-
tation of the potential (5)–(7): we conceive it as a toy-model
particular,one can discriminate regionsof phase space where
forstrong-gravityeffectsandinternalresonancesthatweseek
the observed frequencyratio fluctuates around an exact ratio-
inthesystemofablackholeandanaccretiondisc,buttheori-
nalnumberfromthoseregionswherethisratioremainsalways
gin of the perturbing potential Φ(r,θ) is not supposed to be
outsidetheresonance.Ourmodelsuggeststhatevenblack-hole r
thegravitationalfieldoftheaccretiondiscitself.Ofcoursethe
twinQPOsshouldvaryinfrequencyandtheyshouldnotstay
innerdiscisnotself-gravitatinginblackholebinariesandthe
at a firmly fixed frequencyratio, albeit the expected variation
ringisnotintroducedherewiththeaimofrepresentingtheac-
isverysmall–certainlylessthanwhathasbeenfrequentlyre-
cretion flow gravity. What we imagine is that hydrodynamic
portedinneutronstarbinariesandwhatcanbetestedwiththe
and magnetic forces are producing qualitative effects, which
dataavailableatpresent.
canbecapturedbytheringpotentialinourequations(1)–(2).
Ontheotherhand,ourapproachisrathergeneralanditisworth
2. Aconservativesystemwithtwodegreesof
rememberingthatthesameformalismcanbesuccessfullyap-
freedom
plied also in systems where the disc self-gravity plays a non-
negligiblerole.
2.1.Non-lineartermsinthegoverningequations
Also in general relativity, weakly perturbed (i.e. nearly-
Letusconsideranoscillatorysystemwithtwodegreesoffree- geodesic) motion of gas elements orbiting around a
dom, which is described by coupled differential equations of Schwarzschildblackholecanbedescribedbyeffectivepoten-
theform tialthatcontainsasphericaltermarisingfromthegravitational
δ¨ρ+ω2 δρ = f (δρ,δθ,δ˙ρ,δ˙θ), (1) fieldofthecentralblackhole,plusaperturbingterm,whichwe
r ρ
assume axially symmetric. Naturally, the interpretationof the
δ¨θ+ω2 δθ = f (δρ,δθ,δ˙ρ,δ˙θ). (2)
θ θ perturbingtermismorecomplicatedifonewouldliketoderive
We assume that the right-hand-side functions are nonlinear its particular form from an exact solution of Einstein’s equa-
(theirTaylorexpansionsstart withthe secondorder),andthat tions.We donotwantto enterintocomplicationsofa special
theyareinvariantunderreflectionoftime.Clearly,theseequa- modelhere(butseeLetelier2003;Karasetal.2004;Semera´k
tionsincludethecaseofanearlycircularmotionunderthein- 2004andreferencescitedthereinforareviewandexamplesof
fluenceofaperturbingforce:δρandδθaresmallperturbations spacetimes that contain a black hole and a gravitating ring in
oftheposition,whereasω (r)andω (r)haveameaningofthe generalrelativity).Also,onemayaskwhethertheadoptedap-
r θ
radialandtheverticalepicyclicfrequenciesalongthecircular proachcancomprehendframe-draggingeffectsofKerrspace-
orbitr=r ,θ=π/2: timewhiletheblackholerotationisconsideredasperturbation
0
to the spherical field of a static non-rotatingblack hole; even
∂2U 1 ∂2U
ω2 = , ω2 = , (3) if a general answer to that question would be affirmative, the
r ∂r2 θ r2 ∂θ2 !
0 Kerr metric has rather special properties concerning the inte-
wheretheeffectivepotentialis grabilityofthegeodesicmotionandtheformofnon-sphericity,
whichquicklydecayswithradius.Ourcurrentopinionissuch
ℓ2
U(r,θ)≡Φ(r,θ)+ , (4) thattherotation-relatedeffectsofKerrmetriccannotbeviewed
2r2sin2θ
as the originof the perturbationrequiredfor black-holehigh-
Φ(r,θ) is the gravitational potential, for which axial symme- frequencytwinQPOs.
try and staticity will be assumed. These assumptions make Theangularmomentumofatestparticleorbitingthecentre
our system qualitativelydifferent from modelsrequiringnon- onanequatorialcircularorbitis
axisymmetricperturbations.
As a generic example, let the total gravitational field be ℓ≡r3 ∂Φ =GMr r2 +µr(r+a)E(k0)+(r−a)K(k0) , (8)
givenas a superpositionofthe centralpotentialofa spherical ∂r! "r˜2 π(r2−a2) #
star,Φ (r),plusanaxisymmetrictermΦ(r,θ),
s r whereE(k)isacompleteellipticalintegralsofthesecondkind
Φ(r,θ)=Φ (r)+Φ(r,θ). (5) andtheright-hand-sidetermsareevaluatedattheorbitradius,
s r
J.Hora´k&V.Karas:Twin-peakquasiperiodicoscillationsasaninternalresonance(RN) 3
2 2
u u
stable stable n n
s s
1.5 stable 1.5 table table stable
n
u
a) a)
l / l(K 1 l / l(K 1 stable
0.5 0.5
u
n
s
ta
b
le
0 0
0 0.5 1 1.5 2 0 0.5 1 1.5 2
r / a r / a
Fig.1.Angularmomentumℓ(r)(thickline)ofatestparticleonacircularorbitinthecombinedgravitationalfieldofaspherical
bodyandaring.Themassoftheringrelativetothecentralmassisµ=0.1andtheparticlemassissettounity.Leftpanel:thecase
ofNewtoniancentralfield.Rightpanel:thepseudo-Newtoniancase.Radiushasbeenscaledwithrespecttotheringradius(here,
a = 9R ), and the angularmomentumhasbeen scaled by the valueof Keplerian angularmomentumℓ (a). Keplerianangular
S K
momentuminthecentralfieldisalsoplotted(thinline).CircularorbitsareRayleighunstableandtheepicyclicapproximationis
inadequateinregionswheretheangularmomentumdecreaseswithradius.
5 5
4 4
3 3
ωθ ωθ
ω , r ω , r
2 2
1 1
0 0
0 0.5 1 1.5 2 0 0.5 1 1.5 2
r / a r / a
Fig.2.Thebehaviourofepicyclicfrequencies(thicklines;ω –top,ω –bottom)providesbasiccluestotheoriginofoscillations
θ r
andthe effectsexpectedto occurin stronggravity.Left/rightpanelsreferto the Newtonian/pseudo-Newtoniancases, asin the
previousfigure.Unitsoftheordinatesarescaledby GM/a3tomakefrequenciesdimensionless.Thefrequenciesinabsenceof
thering(µ=0)arealsoindicatedforreference(thinplines;ω (r)=ω (r)intheNewtoniancase).
θ r
k ≡k(r ,π/2),µ≡m/M.Figure1capturesatypicalcurveofthe Thedifferencebetweenthe radialepicyclicfrequencyandthe
0 0
angularmomentumasafunctionofradius.Thecorresponding orbitalfrequencygivestheshiftofpericentre.Thedifferenceof
orbitalfrequencyis vertical epicyclic and orbital frequenciesgives the nodal pre-
cession.
GM r2 (r+a)E(k )+(r−a)K(k )
Ω2(r)= +µr 0 0 . (9) Internalresonancescanoccurinthesystem(1)–(2).Inor-
r3 "r˜2 π(r2−a2) # dertocapturethisphenomenon,wecarryoutamultiple-scales
Equation(3)givestheepicyclicfrequencies, expansion(Nayfeh&Mook1979)intheform
GM r2(3r˜−2r) 2µ (r−a)2K(k )−a2E(k ) 4 4
ω2(r)= + 0 0 ,(10) δρ(t,ǫ)= ǫnρ (T ), δθ(t,ǫ)= ǫnθ (T ), (12)
r r3 " r˜3 π r2(r−a)2(r+a) # n µ n µ
Xn=1 Xn=1
GM r2 2µ E(k )
ω2(r)= + 0 . (11) whereT =ǫµtaretreatedasindependenttimescales.Wewill
θ r3 "r˜2 π (r−a)2(r+a)# µ
terminatethe expansionatthe fourthorder(µ = 0,1,2,3;the
Theepicyclicfrequenciesasfunctionsofradiusareplottedin number of time scales is the same as the order at which the
Figure 2 (we will drop the index “0” for the sake of brevity). expansionsaretruncated).
4 J.Hora´k&V.Karas:Twin-peakquasiperiodicoscillationsasaninternalresonance(RN)
Timederivativestakeaformofexpansions Table 1. Possible resonancesand secular terms in the second
orderofapproximation.Onlyregularseculartermsarepresent
d 4 d2 4 4 ifωθ andωr areoutsideresonance,asindicatedintheleftcol-
= ǫµD , = ǫµ+νD D , (13) umn. Subsequent lines refer to a system in 1:2 and 2:1 reso-
dt µ dt2 µ ν
Xµ=0 Xµ=0Xν=0 nances, respectively. In each record correspondingto the res-
onances, the first/second row gives an expression for secular
where Dµ ≡ ∂/∂Tµ. The method tackles the governingequa- terms in radial/vertical oscillations. To simplify our notation
tionsintheirgeneralform. weintroducedΛ ≡C(ρ)andK ≡C(θ).
α α α α
As a specific example we can adopt an explicit form de-
scribingtheorbitalmotion,
ω :ω Secularterms
1∂U θ r
δρ¨+ω2δρ=(1+δρ)δθ˙2− −ω2δρ , (14)
r "r ∂r r #
Outside −2iω D A
r 1 ρ
δθ¨+ω2δθ=−2 δρ˙δθ˙ − 1 ∂U −ω2δθ . (15) resonance −2iωθD1Abθ
θ 1+δρ "(1+δρ)2 ∂θ θ #
b
1:2 −2iω D A , K A A
r 1 ρ 1001 −ρ θ
Because equations (14)–(15) are conservative, the growth of
−2iω D A , Λ A2
energyinonemodemustbebalancedbytheenergylossinthe θ 1bθ 0200bρ b
othermode.Close toresonanceradii,wherethetwoepicyclic b b
2:1 −2iω D A , K A2
frequenciesareinratioofsmallintegers,theperiodicexchange r 1 ρ 0002 θ
−2iω D A , Λ A A
ofenergyshouldoccurinamorepronouncedrate.Becauseam- θ 1bθ 0110bρ −θ
plitudesoftheoscillationsareconnectedwitheccentricityand b bb
inclination,thesolutionalternatesbetweenaninclined,almost
circular trajectory at certain stages, and an eccentric, almost
2.2.Solvabilityconstraints
equatorialcase at some other time. We remarkthat, in a non-
linearsystem,theeigenfrequenciesω ,ω areexpectedtodif- Inthefirstorder,weobtainequationsdescribingtwoindepen-
r θ
ferfromtheobserved(i.e.predicted)frequencies,whichcanbe dentharmonicoscillators,
revealede.g.byFourieranalysisofdatatimeseries.Relevance
D2+ω2 ρ =0, D2+ω2 θ =0. (17)
of this fact for QPOs was first recognized by Abramowicz et 0 r 1 0 θ 1
(cid:16) (cid:17) (cid:16) (cid:17)
al. (2003) and Rebusco (2004), when they discussed a model Thesolutionscanbeexpressedintheform
for Sco X-1. It was further employed by Hora´k et al. (2004),
ρ =A +A , θ = A +A , (18)
whosuggestedthataconnectionshouldexistbetweenthehigh- 1 ρ −ρ 1 θ −θ
frequencyQPOsandnormal-branchoscillations. wherebwe dbenote Ax ≡bAxeibωxT0, A−x = A∗xe−iωxT0 with x = ρ
Weexpandtheeffectivepotentialderivativesandthefunc- or θ. The complexamplitudes A generallydependon slower
x
b b
tions f and f intoTaylorseries,uptothe fourthorderabout scales,T ,T andT .
ρ θ 1 2 3
acircularorbit.Theexpansionprovidesmanynonlinearterms Analgorithmicnatureofthemultiple-scalesmethodallows
containingvariousderivatives ustodeterminetheformofsolvabilityconditionsinaconserva-
tivesystemwithtwodegreesoffreedom.Theconditionsarise
∂i+jU byeliminatingthetermsthataresecularinthefastestvariable,
u ≡ . (16)
ij ∂ri∂θj! T0.Thisconstraintisimposed,becauseotherwisethesolutions
[r0,π/2] intheformofserieswouldnotconvergeuniformly.Infact,the
reason why the same number of scales is required as the or-
By imposing constraints on the potential and its derivatives,
derofapproximationintheexpansion(12) isthatonesecular
weidentifyresonancesthatareexpectedinaparticularsystem
termgetseliminatedineachorder,andthereforethefunctions
(andwerejectthoseresonancesthatcannotberealized).Inthe
A (T )aredeterminedbythesamenumberofsolvabilitycon-
next subsection, formulation of these constraints is still kept x µ
ditionsasthenumberoftheirvariables.
completely general, valid for the arbitrary form of U. Only
Inthesecondorder,thetermsproportionaltoǫ2 intheex-
later, in subsection 2.3, we come to our original motivation
pandedleft-handsideofthegoverningequations(1)–(2)are
fromtheorbitalmotionaroundablackholeandweemploythe
symmetryofthepotentialU.Weconsiderthecaseofpotential δ¨x+ω2x = D2+ω2 x +2iω D A −2iω D A . (19)
symmetricwithrespecttotheequatorialplane.Thisimpliesthe x 2 0 x 2 x 1 x x 1 −x
h i (cid:16) (cid:17)
conditionu = 0,k ∈ N, somewhatreducingthenumber Ontheright-handside,thesecond-ordbertermsresubltfromthe
i(2k+1)
oftermsintheexpansions. expansion of the nonlinearity fx(δρ,δθ,δ˙ρ,δ˙θ) with D0ρ1 and
D θ inplaceofδ˙ρandδ˙θ,respectively.Theycanbeexpressed
Amplitudesoftheoscillationsarecharacterizedbyasmall 0 1
aslinearcombinationsofquadratictermsconstructedfromA
parameter: δρ ∼ ǫ, δθ ∼ ǫ. We impose the solvability con- ±ρ
straints and seek a solution in the form (12). To this aim, we andA±θ: b
fidersrtswofriateppthroexeimxpaltiicoint.formoftheseconstraintsindifferentor- hfx(δbρ,δθ,δ˙ρ,δ˙θ)i2 =|Xα|=2Cα(x)Aα−1ρAαρ2Aα−3θAαθ4, (20)
b b b b
J.Hora´k&V.Karas:Twin-peakquasiperiodicoscillationsasaninternalresonance(RN) 5
whereα = (α ,...,α )and|α| = α +...+α .Theconstants Table2.Possibleresonancesandseculartermsinthethirdor-
1 4 1 4
C(x) aregivenbyangularfrequenciesofω andbycoefficients derofapproximation.Individualrecordshaveasimilarmean-
α x
oftheTaylorexpansionof f .Equatingtheright-handsidesof ingasinTab.1.
x
eqs.(19)–(20)wefind
D2+ω2 x = −2iω D A +2iω D A
0 x 2 x 1 x x 1 −x ω :ω Secularterms
(cid:16) (cid:17) θ r
+ C(x)bAα1Aα2Aα3Abα4. (21)
α −ρ ρ −θ θ
|Xα|=2 Outside 2iω D A , K |A |2A , K |A2|A
b b b b r 2 ρ 1200 ρ ρ 0111 θ ρ
The rhs of eq. (21) contains one secular term independently resonance 2iω D Ab, Λ |A |2Ab, Λ |A2|Ab
θ 2 θ 1101 ρ θ 0012 θ θ
of the eigenfrequencies ω and ω . However, additional sec-
r θ
1:3 2iω D Ab, K |A |2Ab, K |A2|Ab, K A3
ular terms may appear in the resonance. For example, when r 2 ρ 1200 ρ ρ 0111 θ ρ 0030 θ
ωr ≈ 2ωθ, the terms proportional to A2θ in the ρ-equation 2iωθD2Abθ, Λ1101|Aρ|2Abθ, Λ0012|A2θ|Abθ, Λ0120AbρA2−θ
(x → ρ)and A A intheθ-equation(x → θ)becomesecular
ρ −θ b 1:1 2iω D Ab, K |A |2Ab, K |A2|Ab, K b|Ab|2A ,
andtheyshouldbealsoincludedtothesolvabilityconditions. r 2 ρ 1200 ρ ρ 0111 θ ρ 1110 ρ θ
Thesimilarsitbuabtionhappenswhenω ≈ω /2.Theseareinter- K0012|Aθb|2Aθ, K0210A2ρbA−θ, K1002A−bρA2θ b
r θ 2iω D A , Λ |A |2A , Λ |A2|A , Λ |A |2A ,
nalresonances,whichshowaqualitativelydifferentbehaviour: θ 2 θb 1101 ρbbθ 0012 bθ θb 2100 ρ ρ
thecorrespondingtermsaresecularonlyforspecial(resonant) Λ |Ab|2A , Λ1002Ab A2, Λ Ab2A b
0021 θ θ −ρ θ 0210 ρ −θ
combinationsof ω and ω , contrary to the terms that appear
always and are refrerred toθas regular secular terms. Possible 3:1 2iωrD2Aρ,bK1200|Aρ|2bAρ,bK0111|A2θ|bAρb, K2001A2−ρAθ
resonancesin the second orderof approximationand the cor- 2iωθD2Abθ, Λ1101|Aρ|2Abθ, Λ0012|A2θ|Abθ, Λ0300Ab3ρ b
respondingsecular terms in eq. (21) are listed in Table 1. Let b b b b
usassume,foramoment,thatthesystemisfarfromanyreso-
nance.Then eliminate the terms that are secular independently of ω , ω .
r θ
Theresultingsolvabilityconditionstaketheform
D A =0. (22)
1 x
i
Thefrequenciesandamplitudesareconstantandthebehaviour D A =− K |A |2+K |A |2 A , (27)
2 ρ 2ω 1200 ρ 0111 θ ρ
ofthesystemisalmostidenticalaswhatonefindsinthelinear r h i
i
approximation. The only difference is the presence of higher D A =− Λ |A |2+Λ |A |2 A . (28)
harmonicsoscillatingwithfrequencies2ωr,2ωθ and|ωr±ωθ|. 2 θ 2ωθ h 1101 ρ 0012 θ i θ
They are given by a particular solution of equation (21) after Aparticularsolutionofeq.(26)isgivenbyalinearcombina-
eliminatingthesecularterm, tionofcubictermsconstructedfromA andA ,
±ρ ±θ
x2 =|Xα|=2Qα(x)Abα−1ρAbαρ2Abα−3θAbαθ4, (23) x3 =|Xα|=3Q(α3,x)Aα−1ρAαρ2Aα−3θAαθ4, b b (29)
b b b b
Under the assumption of time-reflection invariance, constants
Q(x) arerealandtheirrelationtoconstantsC(x) becomesobvi- whereallcoefficientsQ(α3,x)arenowreal.
α α
Finally,inthefourthorderoftheapproximation,
ousbysubstitutingx intoequation(21).Wefind
2
Q(x) = Ck(xlm)n . (24) hδ¨x+ω2xxi4 =(cid:16)D20+ω2x(cid:17)x3+2D3D0x1+2D0D2x2. (30)
klmn ω2x−[(k−l)ωr+(m−n)ωθ]2 TheoperatorD0D2actson x2,givenbyeq.(23).Theresulting
formisfoundbyemployingthesolvabilityconditions(27)and
In the thirdorder,the discussion is analogousin manyre-
spects. The terms proportionalto ǫ3, which appear on the lhs (28):
ofthegoverningequations,aregivenby
2D0D2x2 =ω2x Jα(x)Aα−1ρAαρ2Aα−3θAαθ4, (31)
δ¨x+ω2x = D2+ω2 x +2iω D A −2iω D A . (25) |Xα|=4
x 3 0 x 3 x 2 x x 2 −x b b b b
h i (cid:16) (cid:17)
The terms containing D1x1 and D1x2bvanish in cobnsequence whereJα(x)arerealconstants.Byexpandingtherhswearriveat
of the solvability condition (22). The rhs contains now cubic thegoverningequation
termsoftheTaylorexpansion.Weobtain
D2+ω2 x = −2iω D A +2iω D A
0 x 4 x 3 x x 3 −x
D2+ω2 x = −2iω D A +2iω D A h i
0 x 3 x 2 x x 2 −x + C(x)bAα1Aα2Aα3Abα4, (32)
(cid:16) (cid:17) α −ρ ρ −θ θ
+ C(x)bAα1Aα2Aα3Abα4, (26) |Xα|=4
α −ρ ρ −θ θ b b b b
|Xα|=3
b b b b withC(x) realconstants.Onlyoneseculartermindependentof
α
where constants C(x) are real. The secular terms are summa- ω and ω appears on the rhs: −2iω D A . The sum contains
α r θ x 3 x
rizedin Table2 togetherwithresonancespossibleinthethird onlytermsthatbecomesecularneararesonance.Theseterms
b
order of approximation. Again, far from any resonance we andthesolvabilityconditionsarelistedinTable3.
6 J.Hora´k&V.Karas:Twin-peakquasiperiodicoscillationsasaninternalresonance(RN)
Table3.Possibleresonancesinthefourthorderofapproxima- ω :ω = 1:2, 1:1, 3:2 and 1:4. These are all possible combi-
θ r
tion. nations that may occur within the given order of approxima-
tion(thefirstthreecaseswereoriginallyidentifiedbyRebusco
2004,althoughshedoesnotmentionthefourthpossiblecom-
ω :ω Secularterms bination).A generalargumentof non-linearanalysis suggests
θ r
thatthedominantresonancesarethoseoneswhichcorrespond
Outside 2iωrD3Aρ to ratios of small natural numbers, although not every con-
resonance 2iωθD3Abθ ceivable resonant combination comes up in a given physical
1:4 2iωrD3Abρ, K0004A4θ system. Indeed, it appears that the 3:2 ratio is the most im-
2iω D A , Λ A A3 portantcase whenthe high-frequencyQPO pairsare debated,
θ 3bθ 0103bρ θ
however, the true role of this resonance has not yet been un-
b bb
2:3 2iω D A , K A A3 derstood; see also the discussion in Bursa (2005) and Lasota
r 3 ρ 0130 ρ −θ
2iω D A , Λ A2A2 (2005).Proceedingfurthertohigher-ordertermsoftheexpan-
θ 3bθ 0220bρb−θ
sions revealseven moreresonances,butthese are expectedto
b bb
3:2 2iω D A , K A2 A2 be very weak (recently various kinds of weird combinations
r 3 ρ 2002 −ρ θ
2iω D A , Λ A3A havebeenexaminedbyTo¨ro¨ketal.2005). Hereafterwecon-
θ 3bθ 0310bρ b−θ
centrateonthefirstthreecombinations.
b bb
4:1 2iω D A , K A3A
r 3 ρ 0301 ρ θ
2iω D A , Λ A4
θ 3bθ 0400bρb Thecaseof1:2resonance
b
Thesolvabilityconditionstaketheform(cp.Tab.1andHora´k
2004)
Anotablefeatureofinternalresonancesk:listhatkω and
r i
lω need not be infinitesimally close to each other, as might D A = − K A2, (37)
θ 1 ρ 2ω 0002 θ
be expected from the linear analysis. Consider, for example, r
b i b
an internal resonance 1:2, i.e. ωθ ≈ 2ωr. By eliminating the D1Aθ = −2ω Λ0110AρA−θ,. (38)
secularterms,weobtainsolvabilityconditions(seeTab.1) θ
b b b
Thecoefficientsoftheresonanttermsaregivenby
−2iω D A +K A A = 0, (33)
r 1 ρ 1001 −ρ θ
u u
−2iωθbD1Aθ+Λ0b200Ab2ρ = 0. (34) K0002 =−ω2θ − 2r12, Λ0110 =−2ω2θ − r12, (39)
0 0
In each of thebse equatiobns the first term is regular, while the satisfyingamutualrelation2K =Λ .
0002 0110
second term is nearly secular (resonant) one. The solvability
conditions give us the long-term behaviour of the amplitudes
Thecaseof1:1resonance
andphasesofoscillations.Supposenowthatthesystemdeparts
fromthesharpratiobysmall(first-order)deviationsωθ =2ωr+ Thesolvabilityconditionsforthefirstorder,D A = D A =0,
1 ρ 1 θ
ǫσ,whereσisthedetuningparameter.Thetermsproportional
implythatthe complexamplitudesA and A dependonlyon
toA A andA2stillremainsecularwithrespecttothevariable ρ θ b b
−ρ θ ρ the secondscale T . The 1:1 (ω ≈ ω ) resonanceis the only
2 r θ
T0.bThbiscanbbedemonstratedfromA−ρAθ = A∗ρAθei(ωθ−ωr)T0 = epicyclic resonance of the system bwith reflbection symmetry,
A∗ρAθeiσT1eiωrT0.SimilarrelationholdbsfobrA2ρ. which occursin the thirdorderof approximation.The depen-
denceonT impliesaslowerbehaviour.Thesolvabilitycondi-
2
b
tionsare
2.3.Solutionofthesolvabilityconstraints
i
2
soBibdyteacsionomfrpetahlaretiinToganysthloefor-creoxfepuffiannccidteieondntssgwoρviit(ehTrnsj)ianmagneedqpuoθawi(tTieorjns)sot(hf1aǫ)t–o(cn2a)nb,owbthee D2Abρ = −+K2ω01r1(cid:20)1K|A1θ20|20A(cid:12)(cid:12)(cid:12)Aρρ+(cid:12)(cid:12)(cid:12) KAb1ρ002A−ρA2θ , (40)
(cid:21)
solvedsuccessively.Afterrearrangingtoa“canonical”form,
i b 2 b b
D A = − Λ A A
hD20+ω2riρn = X KijklAi−ρAρjAk−θAlθ, (35) 2bθ +Λ2ωθ(cid:20)|A11|021A(cid:12)(cid:12)(cid:12) ρ+(cid:12)(cid:12)(cid:12) Λbθ A2A (41)
D2+ω2 θ = Λ Abi AbjAbk Abl, (36) 0012 θ θ 0210 ρ −θ
0 θ n ijkl −ρ ρ −θ θ (cid:21)
h i X b b b
where n is the order of abpprobximbatbion. In this way we iden- andthecoefficientsoftheresonanttermsaregivenby
tify constants Kijkl and Λijkl. The studied gravitational poten- 5u2
tialissymmetricwithrespecttotheequatorialplane,therefore, K = r2 30 − 1u , (42)
1200 06ω 2 40
tdheerivseartiiveses(3o5f)t–h(e36e)ffceacntinvoetpcootnentatiinaltwerimthsrpersoppeocrtttioonθa.lHtoenocded, 1  θ 2u2
K = −10ω2+ 12 −3u
contrarytoageneralcase,onlyspecificresonancesoccurhere: 0111 3 θ r2ω2 22
 0 θ
J.Hora´k&V.Karas:Twin-peakquasiperiodicoscillationsasaninternalresonance(RN) 7
8 3u
−6r u +u + 30 , (43)
0 30 12r ω2 
1 6u2 0 θ  u u
K = −6ω2+ 12 −3u −2r u − 12 30 , (44)
1002 6 θ r2ω2 22 0 30 ω2 
u 7u 0 θ5u2 10 θ 
Λ = − 04 − 12 + 12 + ω2, (45)
0012 2r2 6r 6r2ω2 3 θ
0 0 0 θ
Λ = K , (46)
0210 1002
Λ = K . (47)
1101 0111
Thecaseof3:2resonance
Thesolvabilityconditionsinvolveboththethirdandthefourth
orders.Hence,theamplitudesA ,A arefunctionsofbothtime
ρ θ
scales T and T . The elimination of regular secular terms in
3 4
thethirdorder(3ω ≈2ω ;seeTab.2)gives
r θ
Fig.3. Comparison between the analytical constraint
D A = − i K A 2A +K |A |2A , (48) E(aρ,aθ) = const (an ellipse), derived in multiple-scales
2 ρ 2ωr (cid:20) 1200(cid:12) ρ(cid:12) ρ 0111 θ ρ(cid:21) approximation, and the corresponding exact (numerical)
D Ab = − i Λ (cid:12)(cid:12)A (cid:12)(cid:12)2Ab +Λ |A |2Ab . (49) solution. The curly curve is the numerical solution of the
2 θ 2ωθ (cid:20) 1101(cid:12) ρ(cid:12) θ 0012 θ θ(cid:21) oscillationamplitudesaρ(t),aθ(t).Theagreementbetweenthe
b (cid:12)(cid:12) (cid:12)(cid:12) b b twocurvesdemonstratesthataccuracyoftheapproximationis
withthecoefficients satisfactoryovertheentiretimespan.Seethetextfordetails.
15u2
K1200 = r02 8ω30 − 12u40, (50) thesolvabilityconditionsadopttheexplicitform
θ
K = 1 −15ω2+ 9u12 −4u 2iωrD3Aρ = K2002 A⋆ρ 2A2θ ei(σ˜2T2+σ˜3T3), (58)
Λ00011121 = −4−21ur80402r0+u3106θ34+59r4u02urω21102222θω−2θr101+1563uω32u2θ0r2102,+ 11365ω2θ, ((5512)) F222iiiiωωωnarθθlDDDly223,AAAbρθθy===su(cid:20)(cid:20)ΛΛKb0s113t121i000tu10A(cid:16)t(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)3ρiAAnAρρg⋆θ(cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)22ae++ip(σΛK˜o2Tl00a012+r1121σ˜f3||oAATr3θθm)||,22(cid:21)(cid:21)oAAfθρc.,omplex amplitu(((d665e109s))),
Λ1101 = K0111. (53) Aρ ≡ 12a˜ρeiφρ andAθ ≡ 21a˜θeiφθ,wegetasetofeightequations
governingthelong-termbehaviourofphasesandamplitudes:
Theeliminationoftheresonanttermsgivesthesolvabilitycon-
ditioninthefourthorder(Tab.3), D2a˜ρ=0, D2a˜θ =0, (62)
K Λ
D3Aρ =−2ωi K2002A2−ρA2θ, D3Aθ =−2ωi Λ0310A3ρA−θ, (54) D3a˜ρ= 1620ω0r2a˜2ρa˜2θsinγ, D3a˜θ =−1603ω1θ0a˜3ρa˜θsinγ, (63)
r θ 1
whebretheresonantcboeffibcientsarbe b b D2φρ=−8ωr hK1200a˜2ρ+K0111a˜2θi,
1
K2002 =−1156ω2θ + 2372 ur102 + 16345 ru02ω2122θ − 214238 ru03ω3124θ DD2φφθ==−−K8ω20θ02hΛa˜11a˜021ac˜2ρos+γΛ, 001D2a˜φ2θi,=−Λ0310 a˜3cosγ, ((6654))
−9u + 27 u12u22 − 27r u + 81u212u30 3 ρ 16ωr ρ θ 3 θ 16ωθ ρ
8 22 16 r0ω2θ 16 0 30 64 r0ω4θ where the phase function has been introduced as γ(T2,T3) ≡
−σ T −σ T −3φ +2φ . The amplitudesa˜ and a˜ of the
− 9 r0u22u30 − 81 r02u302 − 81 r0u12u230 osc2illa2tions3va3ryslowρly,becθausetheydependonρlyontθhethird
16 ω2θ 256 ω2θ 512 ω4θ time-scale T3. Phases φr and φθ of the oscillations evolve on
r u u
−1r u − 9 r2u − 9 0 12 40, (55) bothtimescalesT2andT3.
4 0 30 64 0 40 128 ω2
θ
Λ = 2K . (56) 3. Thesystemevolutionnearthe3:2resonance
0310 3 2002
Byintroducingthedetuningparameter, 3.1.Theintegralsofmotion
ω Thecaseof3:2resonanceisparticularlyrelevantforthehigh-
σ≡3 r −2=ǫ2σ˜ +ǫ3σ˜ , (57)
ω 2 3 frequencyQPOs,bothonobservationalandtheoreticalgrounds
θ
8 J.Hora´k&V.Karas:Twin-peakquasiperiodicoscillationsasaninternalresonance(RN)
u1 u2 r3:2 r3:2
0.2
0.15
F(u)
0.02 Type B
0.1
u) 0.05 G(u) Type C
F(2
u), 0 ε
F(1 -0.05 0.01
-0.1
-F(u) Type A
-0.15
-0.2
0 0.2 0.4 0.6 0.8 1 0
u 0.5 0.55 0.6 0.65 0.7
r / a
Fig.4.ThefunctionsF ≡ F(u),F ≡ −F(u),andG(u)from
1 2
eq.(81).Thesystemevolutionislimitedtotheintervalhu1,u2i, Fig.5. The 3:2 inner resonancesin the gravitationalfield of a
wherethecondition|F(u)|≥|G(u)|issatisfied. sphericalpseudo-Newtonianstarandaring(a=9R ,µ=0.1).
S
Theregionsofdifferentphase-planetopologyareidentifiedin
(r,E)-plane.Threetypescanbedistinguishedaccordingtothe
(cf. Abramowicz & Kluz´niak 2001; Kluz´niak et al. 2004 for
numberofcriticalpoints:A–nocriticalpoint(thesystemisfar
argumentsin favour of 3:2 ratio in high-frequencyQPOs and
fromresonance);B–onecriticalpoint;C–twocriticalpoints.
forfurtherreferences).We thereforediscussthiscaseinmore
detail,however,similardiscussioncouldbepresentedalsofor
other resonances (Hora´k 2005). Reintroducing single physi- Then the oscillations are described by two equations for ξ(t)
cal time t, the equations for the second and the third order andγ(t),
can be combinedtogether.Time derivativesare then givenby
ξ˙= 1βω ξ2 1−ξ2 E3/2sinγ, (71)
d/dt=ǫ2D2+ǫ3D3.Amplitudesandphasesoftheoscillations 16 θ
(cid:16) (cid:17)
aregovernedbyequations γ˙=−σω + 1ω E µ ξ2
θ 4 θ r
a˙r = 214βωra2ra2θsinγ, (66) +94µθ 1−ξ2 h+ 41βξ 3−5ξ2 E1/2cosγ , (72)
a˙ = − 1βω a3a sinγ, (67) (cid:16) (cid:17) (cid:16) (cid:17) i
θ 16 θ r θ whichsatisfytheidentity
ω a
γ˙ = −σωθ+ 4θ (cid:20)µra2r +µθa2θ + 2r (cid:16)αa2θ −βa2r(cid:17)cosγ(cid:21), (68) ξ˙dγ−γ˙dξ=0. (73)
where a = ǫa˜ , a = ǫa˜ and µ , µ and β are defined by Substituting for ξ˙ and γ˙ from eqs. (71)–(72), eq. (73) implies
ρ ρ θ θ r θ
relations Λ = 2K = βω2, K −Λ = ω2µ , and that
0310 3 2002 θ 1200 1101 r r
K −Λ =ω2µ .Theamplitudesandphasesarenotmutu-
0111 0012 θ θ F ≡ 8 1−ξ2 σ+E µ ξ4− 4µ 1−ξ2 2
allyindependent;theequations(66)–(67)implythatthequan- r 9 θ
(cid:16) (cid:17) (cid:20) (cid:16) (cid:17) (cid:21)
tity
+βE3/2ξ3 1−ξ2 cos γ (74)
E=a2+ 9a2 =const (69) (cid:16) (cid:17)
ρ 4 θ is a secondintegralof motion.Fora givenvalueofenergyE,
remains conserved during the system evolution. Clearly, E is thesystemfollowsF = constcurves.Intheotherwords,pro-
proportional to the total energy of the oscillations. The exis- jectionofthesolutiononto(γ,ξ)-planesatisfies
tence of this integral is a general property of a conservative
F(γ,ξ)=const. (75)
system.Naturally,itisnotlimitedtotheparticularformofthe
perturbingpotential(7), which we considerhere, and it holds This allowsus to constructtwo-dimensionalphase-spacesec-
in Newtonian,pseudo-Newtonianas well as general-relativity tionsinwhichthesystemevolutiontakesplace.
versions of the equations of motion (the pseudo-Newtonian
casewasexamined,indetail,byAbramowiczetal.2003,and
3.2.Stationarypointsandthephase-planetopology
Hora´k 2004). One can watch the accuracyto which E is con-
servedinordertoverifytheanalyticalapproachagainsttheex- Stationary points are given by the condition ξ˙ = γ˙ = 0.
actnumericalsolution(we showsuchcomparisonin Figure3 According to eq. (71), γ-coordinate of these points satisfies
forthepseudo-NewtoniancaseofAbramowiczetal.2003). sinγ = 0, and therefore γ = kπ with k being an integer.
Equations(66)–(67)canbemergedinasingleequationby Substituting γ˙ = 0 and cosγ = ±1 in eq. (72), we find a cu-
introducingthefollowingparameterization: bicequation,
a2 =ξ2E, a2 = 4 1−ξ2 E. (70) −4σ+ µ ξ2+ 4µ 1−ξ2 E±βξ 3−5ξ2 E3/2 =0. (76)
ρ θ 9 r 9 θ
(cid:16) (cid:17) h (cid:16) (cid:17)i (cid:16) (cid:17)
J.Hora´k&V.Karas:Twin-peakquasiperiodicoscillationsasaninternalresonance(RN) 9
The solutiondeterminesξ-coordinateof the stationarypoints. whereT canberoughlyapproximatedas
In the case of small oscillations (E ≪ 1), the solution can be
approximatedbykeepingonlytermsupto thelinearoneinE 16π
T ∼ E−3/2. (86)
ineq.(76).Weobtain βω
θ
9σ¯ −µ
ξ2 = θ , (77) Notice that this time-scale is longer than the period of indi-
94µr−µθ vidualoscillationsanditsayshowfastthesystemswapsitself
between the radialand verticaloscillation modes. The simple
where σ¯ ≡ σ/E. The first correctionto this solution is of the
estimate(86)isquitepreciseinmostpartsofthephasespace,
order of E1/2. Deviations between ξ-coordinatesof stationary
although it becomes inaccurate near stationary points, where
pointsatoddandevenmultiplesofπareofthesameorder.
therateofenergyexchangeslowsdown.
The solution (77) lies within the allowed range provided
Finally,weillustrateourresultswithasimplecase,which
that 1µ ≶ σ¯ ≶ 1µ withthedenominatorD ≡ 9µ −µ ≶ 0,
4 r 9 θ 4 r θ wealreadyintroducedatthebeginningofthepaper(sect.2.1):
respectively.ThiscanbeexpressedintermsofenergyE:given
the gravitational field generated by a pseudo-Newtonian star
thedetuningparameterσ,stationarypointsappearinthe(γ,ξ)
andanarrowcircularring.Theresonantconditionω /ω =3/2
planeiftheenergyofoscillationssatisfies θ r
is now fulfilled at three different radii, two of them lying be-
σ σ tweenthestarandthering,andthethirdoneoutsidethering.
9 ≶E≶4 for D≷0. (78)
µ µ The resonances occurring at first two radii are called the in-
θ r
ner resonances, whereas the latter is the outer resonance (not
The examination of phase-plane topology near critical
to be confused with ‘internal resonance’, which they are all).
pointsleadstotheequation
We restrict ourselves to the inner resonances, for which we
∂ξ˙ ∂γ˙ ∂ξ˙ ∂γ˙ find σ, µr, µθ and β as functions of r. For a fixed radius, in-
−λ −λ − =0 (79) equalities(78)giveustheenergyrangeoftheoscillations.The
∂ξ ! ∂γ ! ∂γ ∂ξ
result is shown in Figure 5, where we identify three different
foreigenvaluesλofthesystemoflinearizedequations(71)and phase-planetopologiesin the(r,E)-section.Thesecanbedis-
(72).Evaluatingthepartialderivativesatthecriticalpointand tinguished by the number of critical points and the shape of
keepingonlythetermsofthelowestorderinE3/2,weobtain separatrices.ThetopologychangeisevidentinFigure6,where
two-dimensionalplots are constructed for the integral of mo-
λ2 =± 1ω2βξ3 1−ξ2 DE5/2. (80) tionF(ξ,γ).
72 θ
(cid:16) (cid:17)
Examiningthe sign of λ2 demonstratesthatthe criticalpoints
3.4.Frequenciesoftheresonantoscillations
ofcentraltopologyalternatewiththoseofsaddletopology.
Equations(63)–(65)givetheshiftofactual(observed)frequen-
3.3.Thetimedependence cies of oscillations, ω∗r and ω∗θ, with respect to the eigenfre-
quenciesω andω :
r θ
The equation for ξ(t) can be derived by combining eqs. (71)
and(74).Eliminatingcosγ,wearriveattherelation ω∗ =ω +φ˙ , ω∗ =ω +φ˙ . (87)
r r r θ θ θ
Ku˙2 = F2(u)−G2(u), (81)
Theserelationscanbecombinedtofind
whereweintroducedanewvariableu(t)≡ξ2.TheconstantK
andfunctionsF(u)andG(u)aredefinedby 2ω∗−3ω∗ =2ω −3ω +2φ˙ −3φ˙ =γ˙. (88)
θ r θ r θ r
1 8 2 Theobservedfrequenciesareinexact3:2ratioif(andonlyif)
K ≡ , (82)
E3/2 ωθβ! thetime-derivativeofthephasefunctionγvanishes.Animme-
F(u) ≡ u3/2(1−u) (83) diate implicationfor the frequenciesof stationary oscillations
with constant amplitudes is that they stay in exact 3:2 ratio,
1
G(u) ≡ βE3/2 F −8σ(1−u)−µrEu2+ 49µθE(1−u)2 . (84) even if the eigenfrequenciesmay depart from it. Outside sta-
h i tionary points, it is evident from Fig. 6 that γ˙ = 0 represents
The motion is allowed only for u˙2 ≥ 0, and so the condition turningpointsonlibrationtracks(thoseones,whichareencir-
±F(u) = G(u)givesustwoturningpoints,u andu ,between cledbytheseparatrixcurve).Hence,eq.(88)discriminatesbe-
1 2
which the evolutionary path oscillates. The functions ±F(u) tween librating and circulating trajectories in the (γ,ξ)-plane.
andG(u)areplottedinFigure4. Circulatingtrajectoriesspanthefullrangeof−π ≤ γ < πand
Theperiodofenergyexchangecanbefoundbyintegrating they donotcontainany turningpoint;γ˙ remainsnonzeroand
eq.(81): thetwinfrequenciesnevercrosstheexact3:2ratiointheregion
ofcirculation.Ontheotherhand,therearetwopointsγ˙ =0on
16 u2 du eachlibratingtrajectory.Insuchstatetheratioofobservedfre-
T = E−3/2 , (85)
βωθ Zu1 F2(u)−G2(u) quenciesslowlyfluctuatesabout3:2.
p
10 J.Hora´k&V.Karas:Twin-peakquasiperiodicoscillationsasaninternalresonance(RN)
1 1 1
0.8 0.95 0.8
0.6 0.9 0.6
ξ ξ ξ
0.4 0.85 0.4
0.2 0.8 0.2
0 0.75 0
-π -π/2 0 π/2 π -π -π/2 0 π/2 π -π -π/2 0 π/2 π
γ γ γ
Fig.6.Differenttopologiesofthephase-spacesectionsareshownin(ξ,γ)-plane.Thesystemevolutionfollowsthecontourlines
F(ξ,γ)=const.TheirshapedeterminesapossiblerangeofthefrequencychangeoftheQPOs.Separatrixcurve(thickline;cases
B and C) dividesthe regionsof circulatingtracksfromthe regionsof libration.Parametersand notationas in previousfigure:
A(left;r=0.6a,E=0.02),B(middle;r=0.65a,E=0.02),C(right;r=0.55a,E=0.05).
4. Conclusions modelsincludingnon-gravitationalforces(seePe´tri2005fora
recentexpositionoftheproblemandforreferences).Asanext
We have discussed the resonance scheme for high-frequency
step towards an astrophysically realistic scheme, one should
QPOs via multiple-scales analysis, assuming an axisymmet-
take dissipativeand non-potentialforcesinto account,as well
ricconservativesystemwithtwodegreesoffreedom.Thisap-
as non-axisymmetricperturbations. These will allow our sys-
proachprovidesausefulinsightintogeneralpropertiesthatare
tem to migrate across contours in the phase-plane and to un-
common to different conceivable mechanisms driving the os-
dergo transitions when crossing separatrices. Such additional
cillations, although it does not address the question how the
terms could also supplementthe influence of externalforcing
observedsignalisactuallyformedandmodulated.Inoursce-
and initiate the oscillations of the system. The internal reso-
nario, amplitudes and phases of the oscillations are mutually
nancewouldthendefinetheactualfrequenciesthatareexcited;
connectedand theyfollowtracksin the phase space with dis-
thiswaythestronggravityunmasksitself.
tinct topologies. The particular form was assumed to couple
theoscillationmodesviathenon-sphericaltermsinthegravi- Acknowledgements. VK appreciates fruitful discussions with par-
tationalfieldofa ring.We considerthisto bea toy-modelfor ticipants of the Aspen Center for Physics workshop ‘Revealing
rather generalbehaviour,which should take place in any sys- Black Holes’, and both of us thank for the hospitality of
temgovernedbyequationsoftype(1)–(2). NORDITA (Copenhagen). We gratefully acknowledge the financial
support from the Czech Science Foundation (refs. 205/06/P415 and
WeassumedtheNewtonian(orthepseudo-Newtonian)de-
202/06/0041)andtheGrantAgencyoftheAcademyofSciences(ref.
scription of the central gravitational field with a perturbation
IAA300030510)thathavebeenhelpingusatdifferentstagesofthepa-
byanalignedringasanexample.Theadoptedformisnotes-
perpreparation. TheAstronomical Institutehasbeenoperatedunder
sentialforgeneralconclusions.Infact,equations(1)–(2)cover theprojectAV0Z10030501.
alsothenearly-geodesicmotionaroundaSchwarzschildblack
hole.Comparedwiththepseudo-Newtoniancase, generalrel-
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geodesic motion, and hence prevent the occurrence of inter- Stars,Vol.4/5,eds.S.Hled´ık&Z.Stuchl´ık(SilesianUniversity,
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Hora´kJ.,2005,Thesis(CharlesUniversity,Prague)
nismlaunchingandmaintainingtheoscillationsremainsunan-
Hora´kJ.,AbramowiczM.A.,KarasV.,Kluz´niakW.,2004,PASJ,56,
swered.
819
Variousoptionsforthegeneralisationofourschemecould KarasV.,Hure´J.-M.,Semera´kO.,2004,CQG,21,R1
bemotivatedbypapersofotherauthorswhoproposedspecific KatoS.,2004,PASJ,56,905