Internal resonance in non-linear disk oscillations and the amplitude evolution of neutron star kilohertz QPOs

0.37 MB·English

by  J. Horak

Astronomy&Astrophysicsmanuscriptno.cross c ESO2009
(cid:13)
January20,2009
Internal resonance in non-linear disk oscillations and the
amplitude evolution of neutron star kilohertz QPOs
Jiˇr´ıHora´k1,Marek.A.Abramowicz2,3,WłodekKluz´niak4,3,PaolaRebusco5,andGabrielTo¨ro¨k6
9
0 1 AstronomicalInstituteoftheAcademyofSciences,Bocˇn´ıII1401/1a,141-31Praha4,CZ
0 2 DepartmentofPhysics,Go¨teborgUniversity,S-41296Go¨teborg,SE
2 3 CopernicusAstronomicalCentrePAN,Bartycka18,00-716Warsaw,PL
4 JohannesKeplerInstituteofAstronomy,ZielonaGo´raUniversity,ul.Lubuska2,65-265ZielonaGo´ra,PL
n 5 MITKavliInstituteforAstrophysicsandSpaceResearch,77MassachusettsAvenue,Cambridge,MA02139,USA
a 6 InstituteofPhysics,FacultyofPhilosophyandScience,SilesianUniversityinOpava,Bezrucˇovona´m.13,746-01Opava,CZ
J

0 Received...............2008;accepted...............2008
2
ABSTRACT

]
O Weexploresomepropertiesoftwinkilohertzquasiperiodicoscillations(QPOs)inasimpletoy-modelconsistingoftwooscillation
C modes coupled byageneral nonlinear force. Weexamine resonant effectsby slowlyvarying thevaluesof thetunable, and nearly
commensurable,eigenfrequencies.Thebehavioroftheactualoscillationfrequenciesandamplitudesduringaslowtransitionthrough
.
h the 3:2 resonance is examined in detail and it is shown that both are significantly affected by the nonlinearities in the governing
p equations. In particular, the amplitudes of oscillations reflect a resonant exchange of energy between the modes, as a result the
- initially weaker mode may become dominant after the transition. We note that a qualitatively similar behavior has been recently
o reportedinseveralneutronstarsourcesbyTo¨ro¨k(2008),whofoundthatthedifferenceofamplitudesinneutronstartwinpeakQPOs
r
changessignastheobservedfrequencyratiooftheQPOspassesthroughthevalue3:2.
t
s
a Keywords.X-rays:binaries—Stars:neutron—Accretion,accretiondisks
[

1
v 1. Introduction The appearance of rational ratios (mainly 3:2) be-
∼
6 tween the centroid of twin peak QPOs points at a non-
7 There is little doubt that kHz QPOs observed in low mass X- linear coupling between different modes of oscillations
0 ray binaries originate in the inner region of accretion disks of (Abramowicz&Kluz´niak 2001). The importance of a res-
3 the neutronstar or black hole sources. The very fact that these onant process was also pointed out by Psaltis&Norman
1. quasi-periodicmodulationsare observedin the X-rayflux, i.e., (2000); Kluz´niak&Abramowicz (2001); Titarchuk (2002);
0 in the most luminous part of the source spectrum—and at the Kluz´niaketal. (2004) and investigated in more detail, e.g.,
9 sametimeathighphotonenergies—impliesanorigininthein- in Abramowiczetal. (2003b); Rebusco (2004); Hora´ketal.
0 nerpartoftheaccretionflow,wheremostofthegravitationalpo- (2004);Leeetal.(2004);Hora´k&Karas(2006).
: tentialenergyoftheaccretingmatterisreleased.Time-of-flight
v Oneexampleofaninvestigationofresonancesbetweenspe-
argumentsalsoimplythatakHzmodulationofunbeamedradi-
i cificmodesinanaccretiondiskcanbefoundintheworkofKato
X ationmustoriginateina regionnotlargerthan 100km,with
∼ (2005b),whoproposedamodelinwhichthetwokHzQPOsare
anevensharperboundwhenthehighdegreeofcoherenceofthe
r duetoa nonlinearresonantinteractionbetweentheoscillations
a observedmodulationistakenintoaccount.
ofa thinaccretiondisk andan oscillatoryperturbationinduced
InthepresentpaperwefocusonkHzQPOsthatshowupin
bythe non-axisymmetricrotatingneutronstar. Both kHz oscil-
pairs,theso calledtwinpeakQPOs(seevanderKlis2000, for
lationsarelocalizedclosetoaresonantradius,wheretheinter-
areview),andweassumethattheycorrespondtooscillationsof
actionwiththeperturbationismostprominent.Thespatialprox-
anaccretion-diskstructure.Reproduciblefeaturesofthesehigh-
imityofthetwooscillationsmayfacilitateaninternalresonance
frequencyQPOsimplythattheyreflectsomefundamentalprop-
between them. As pointed out recently by Maoetal. (2008), a
ertiesofthesystems,suchastheirmassandspin.
resonancemayoccurbetweentwomodeswithnon-overlapping
Many models have been proposed, some involving or- wavefunctions,whenitismediatedbyatravelingwave.Inother
bital motions (Kluz´niaketal. 1990; Stella&Vietri 1999), oth- words,thetwooscillatorsinresonanceneednotbelocatedinthe
ers based on oscillations of the accretion disk (Wagoner same annulusof the accretiondisk. MorerecentlyKato (2008)
1999; Wagoneretal. 2001; Kato 2001, 2004a, 2005a,b, 2008; relaxedthelocalapproximationandanalyzedthenonlinearreso-
Pe´tri 2005; Pe´tri 2006) and of tori (Rezzollaetal. 2003b,a; nantexcitationofinertial-acousticdiskoscillationsindeformed
Bursaetal. 2004; Monteroetal. 2004, 2007; Blaesetal. 2006, disks(awarpeddiskoraone-armedpatternsymmetricwithre-
2007; Schnittman&Rezzolla 2006; Zanottietal. 2005; Hora´k specttotheequator).
2008).Mostofthesemodelsinvolvetwoormoreoscillatorymo-
Other QPO models involve vortices in the disk or mag-
tions,whichmayinteractinanon-linearmanner.
netic interactions. For example Tagger&Varnie`re (2006) sug-
gesttheformationoftwo orthreeRossby-wavevorticesinmi-
Sendoffprintrequeststo:[email protected] croquasardisks,whileLi&Narayan(2004)suggestedamodel
2 J.Hora´ketal.:InternalresonanceinneutronstarkilohertzQPOs
inwhichthetwoQPOsareidentifiedwithtwoRayleigh-Taylor
or Kelvin-Helmholtz unstable modes developed at the inter-
face between the disk and the magnetosphere.Similarly,in the 0.1 (A) (C) (B) (A)
Alfve´nwaveoscillationsmodel(Zhang2004;Zhangetal.2007)
thetwoQPOscorrespondtotheKeplerianorbitalfrequencyand y
g
the frequency of the Alfve´n waves excited at the same radius. er
n
Itislessclearwhethertheresonantinteractiondescribedinthis e
s
paperwouldapplytothosemodelsaswell. s
e
In this paper we consider some general properties of non- onl
linear resonance that may be applicable to many systems. We nsi0.05
e
specifyneitherthephysicalnatureoftheoscillators,northecou- m
plingsnecessaryforaresonancetooccur.InthecontextofkHz Di
QPOs,theoscillatorscouldbetakentobeeigenmodesoftheac-
cretiondisk or of an associated structure,perhapsonesalready
discussedintheextensivetheoreticalliteraturecitedabove.We
donotaddressthe crucialquestionofthe excitationanddamp- 0
ing of the modesin the turbulentdisk, particularlywhen MRI- 1.47 1.48 1.49 1.5
inducedturbulenceispresent(intheframeworkoftwocoupled Eigenfrequency ratio
oscillatorsthisissuewasadressedalreadybyVioetal.(2006)).
Eventually,one would hope to identify such resonances in nu-
Fig.1. Regions where a resonance is present (resonance
mericalmodelsofaccretionflow.Itisonlyrecentlythatinertial
‘tongues’) in the plane of the eigenfrequency ratio and the to-
modeshavestartedshowingup innumericalsimulations(Kato
talenergyofoscillations.Eachpointintheplanecorrespondsto
2004b;Brandenburg2005;Arrasetal.2006;Reynolds&Miller
onesingle(γ,ξ)-diskanddifferentregionstodifferenttopologies
2008; Machida&Matsumoto 2008). It may be premature to
ofthesedisks(seeFig.2).Regions(A),(B)and(C)correspond
lookforanumericalvalidationoftheideaofresonanceinaccre-
tozero,one,andtwofixedpointsinthe(γ,ξ)-disk.Theparame-
tiondisks,atleastuntiltwodistinctfrequenciesina3:2ratioare
tersofthesystemaregiveninthetext.
identifiedwithsomeregularityinthesimulations.Nevertheless,
numericalworkmayalreadybeofferingnewinsights.
Inwhatfollowswereviewsomegeneralpropertiesofinter- (e.g. Nayfeh&Mook 1979) we find the lowest order real dis-
nalresonanceandinvestigatetheamplitudeevolutionoftheos- placementsintheform:
cillationsbymodelingaslowpassageofthefrequenciesthrough
the 3:2 resonance.The resultspresentedin thispaper are more xℓ(t)= Aℓ(t)eiωℓt , xu(t)= Au(t)eiωut . (3)
relevant to neutron-star QPOs where the frequencies wander ℜh i ℜh i
overhundredsof hertz on the time-scale of severalhours,than The higher-order terms add only higher harmonics to the
toblackholeQPOs,whosefrequenciesaremorestable. quasiperiodicsignaldescribedbythesesolutions.Themaindif-
ferencewith respectto the solutionsof a linearsystem are that
thecomplexamplitudes
2. Mathematicsoftheinternalresonance
1 1
We model the QPOs using two coupled nonlinear oscillators Aℓ(t) aℓ(t)eiφℓ(t), Au(t) au(t)eiφu(t) (4)
≡ 2 ≡ 2
governedbytheequations
depend on time. We will refer to the actual frequencies of the
x¨ +ω2x = ω2f (x ,x ,x˙ ,x˙ ), (1) solution, ω⋆ and ω⋆, as the “observed frequencies.” They are
ℓ ℓ ℓ ℓ ℓ ℓ u ℓ u ℓ u
shifted with respect to the eigenfrequencies by the corrections
x¨u+ω2uxu = ω2ufu(xℓ,xu,x˙ℓ,x˙u). (2) ∆ω =φ˙ and∆ω =φ˙ .
ℓ ℓ u u
Weassumeenergyconservation,sothetworealamplitudes
TheoscillationsunderlyingtheupperandlowerQPOsareiden-
maybeparameterizedusingthefractionalenergyξ(t):
tifiedwiththedimensionlesssolutions x (t)and x (t).Thecou-
u ℓ
pling functions fℓ and fu are treated as a perturbation, and are E
further assumed to contain only nonlinear terms and to be in- a2ℓ =ξE, a2u =(1−ξ)ν, 0≤ξ ≤1, (5)
variantundertime inversion(e.g.,their Taylor expansionsstart
withthesecondorderanddonotcontainoddpowersofthetime where E a2 + νa2 is the total dimensionless energy and
derivatives x˙ , x˙ ).Thecoefficientsω > 0,andω > 0arethe ν is a con≡stantℓthat duepends on the properties of the system
ℓ u ℓ u
eigenfrequenciesofthetwooscillators. (Hora´k&Karas2006).Similarly,we introducethephasefunc-
In order to study the most relevant resonant effects, we tionγ(t)as
concentrate on the case when ω /ω is close to 3:2. We
u ℓ
chose this resonance as it is most prominent in the QPO data γ 2φu 3φℓ+σt. (6)
≡ −
(Abramowiczetal. 2003a; To¨ro¨ketal. 2008). The degree of
Thedifference2ω⋆ 3ω⋆expressedusingthefrequencycorrec-
closeness to the rational ratio is expressed by the detuning pa- u− ℓ
tionsis
rameterσ 2ω 3ω .
u ℓ
≡ −
Our analysis is applicable when the amplitudes of oscilla- 2ω⋆ 3ω⋆ =σ 3φ˙ +2φ˙ =γ˙. (7)
tions and the detuning parameter are small and are such that u − ℓ − ℓ u
a a ǫ 1 and σ ǫ2 respectively. In that case, per- Hence, the observedfrequenciesare commensurablewhenever
ℓ u
∼ ∼ ≪ ∼
turbationtechniquescan be used to find approximatesolutions γ˙ = 0,eveniftheeigenfrequenciesarenotexactlyso(thedevi-
oftheequations(1)and(2).Usingthemethodofmultiplescales ationisproportionaltoǫ 2).
J.Hora´ketal.:InternalresonanceinneutronstarkilohertzQPOs 3
Thedynamicsofthesystemisthengovernedbythetwofirst- Figure 1 shows these regions for the system described by the
orderordinarydifferentialequations fiducialsetofparameters,β=µ =µ =1,ν=9/4.
ℓ u
Someexamplesofthethreepossibletopologiesofthe(γ,ξ)-
1
ξ˙= βω (1 ξ)(ξE)3/2sinγ, (8) diskaregiveninFigure2.Allofthemcontaincirculatingtrajec-
u
8 − toriesforwhichγ(t)monotonicallyincreasesordecreasesinthe
γ˙=σ+ 14ωuE(cid:20)µℓξ+ µνu(1−ξ)+ 14β(3−5ξ)(ξE)1/2cosγ(cid:21),(9) fcualnlbraenegxep(ree.sgs.e,d0(≤wiγth<an2aπc)c.uTrhaecyraotfioupoftoobǫs3e)ravsedfrequencies
where β, µu and µℓ are as yet unspecified dimensionless con- ω⋆ 3 γ˙
stantscharacterizingagivensystem. ⋆ u = 1+ . (11)
R ≡ ω⋆ 2 2ω !
The values of the system parameters follow directly from ℓ u
thecoefficientsoftheTaylorexpansionsofthefunctions f and
ℓ Hence,thisratioisalwaysgreaterorsmallerthan3/2ifthesys-
f and depend on the particular physical model. For example,
u temevolvesalongthecirculatingtrajectorieswithincreasingor
Rebusco(2004)studiedsmallnonlinearepicyclicoscillationsin
decreasingγ,respectively.
nearly geodesic orbitsof test particles in Schwarzschildspace-
Inaddition,closer to the resonance,the (γ,ξ)-diskscontain
timeusingthemethodofmultiplescales.Later,Hora´k&Karas
a fixed point that corresponds to an extremum of the function
(2006) generalized her approach to motion in an arbitrary ax-
F(γ,ξ).Thispointisalwayssurroundedbylibratingorbitswith
isymmetric gravitational field. In both works the radial and
two turning points where γ˙ changes its sign. When the system
vertical oscillations of the particles are governed by equations
followsthesetrajectoriestheobservedfrequencyratiooscillates
of the form of (1) and (2), e.g., equations (14) and (15) in
around 3:2 . The regions of librating and circulating orbits are
Hora´k&Karas (2006). Moreover, the latter work contains ex-
dividedbyaseparatrix.Whenthe(γ,ξ)-diskcontainstwofixed
plicitformulaefortheconstantsβ,µ andµ intermsofderiva-
u ℓ points,thesecondpointisasaddlepoint,inwhichthetrajectory
tivesoftheeffectivepotential[seetherelationsaftertheirequa-
crossesitself.Consequently,ittakesinfinitetimeforthesystem
tion (68)]. In addition, a generalform of the equations (1) and
toreachthesaddlepointfromanyotherpointontheseparatrix.
(2) is suitable for studying resonance phenomena in different
physicalconditions. Hora´k (2008) examined nonlinear interac-
tion between two epicyclic modes of a slender accretion torus;
3. Slowpassagethroughtheresonance
sequation(6)inhispaperrepresentsaparticularexampleofthe
generalequationsdiscussedhere.
Letusnowsupposethattheeigenfrequenciesaretunableandare
Both,ξ(t)andγ(t),areslowlyvaryingfunctionsoftime,they subjecttosmallslowchanges.Morespecifically,weconsiderthe
describeaslowmodulationofamplitudesandphasesduringthe casewhentheeigenfrequenciesandcouplingsareslowlychang-
oscillations.Theamplitudemodulationoccursonthecharacter-
ingandthesystemgraduallypassesthroughtheresonance.We
istic time scale tmod ≡ ǫ−3tosc, with tosc ≡ ω−u1 being the char- concentrate on that part of this process in which the eigenfre-
acteristic timescale of the oscillations. The two amplitudes are quenciesare close to the rational3:2ratio. We assume thatthe
anticorrelatedandthevariationsofξcorrespondtoanexchange
total relative changes of the eigenfrequenciesare small (of the
ofenergybetweenthe modesthatkeepsthe totalenergyof the orderofǫ2),andthetimescaleforthischange,T,ismuchlonger
oscillations,E,constant.Sincetheoscillationsarenonlinear,the thananyothertimescaleconnectedtothenonlinearoscillations
observed frequencies are periodically modulated reflecting the
(inparticularT t ).
mod
modulationofthe amplitudes.InthecontextofQPOsthispro- ≫
Ingeneral,theoscillationsofthesystemarethesolutionsto
cess has already been discussed by Hora´ketal. (2004) in con-
equations(1)and(2),whereω andω aswellasthenonlinear
ℓ u
nectiontothecorrelationsbetweentheQPOfrequenciesandthe
functions f and f areknownfunctionsthatdependexplicitlyon
ℓ u
phaseofthenormal-branchoscillations(NBOs)Yuetal.(2001)
time.SincethecharacteristictimescaleoftheprocessT ismuch
The time-evolution of the system can be studied directly
longerthant ,theonlychangeintheequations(8)–(9)isthat
mod
from equations (8) and (9) or with the aid of another integral theeigenfrequenciesω andω andthecoefficientsβ,µ ,µ and
ℓ u ℓ u
ofmotion
νbecomeslowfunctionsoftime.Closetotheresonance,theos-
8σ µ cillationsaremostsensitivetothechangeofthedetuningparam-
F = ξ(cid:20)− E +µℓξ+ νu(2−ξ)+β(1−ξ)(ξE)1/2cosγ(cid:21). (10) eterσ(t)≡2ωu(t)−3ωℓ(t),whiletheyarepracticallyunaffected
bysmallchangesoftheotherquantities.Thisisbecausetheco-
Differentsolutionstrace differentcurves F(γ,ξ) = constin efficientsoftheothertermsin equations(8)and(9)arealways
the (γ,ξ)-disk. For given values of the parameters β, µ , µ , E multipliedbysmallquantities(oftheorderof ǫ 2 and ǫ 3).
ℓ u
∼ ∼
andσtheremayexistseveralfixedpointsinthisdisk,eachrep- We donotconsiderthe “singular”case whenthe two eigenfre-
resenting strictly periodic oscillations of the system. They cor- quenciesarenearlycorrelatedasω 1.5ω .Becausethechar-
u ℓ
≈
respondtooscillationswhoseamplitudesarenotmodulatedand acteristictimescaleofpassagethroughresonanceistakentobe
whose phases φ (t) and φ (t) are linear functions of time that much longer than that of the resonant modulation we may ne-
ℓ u
adjustthefrequenciesofoscillationstotheexact3:2ratio.The glecttimederivativesofσandE intheevolutionequations(8)
fixedpointsareeitherextremaorsaddlesofthefunctionsF(ξ,γ) and(9)thatdescribetheevolutionofthesystemonthetimescale
andtheirpositionsinthe(γ,ξ)-diskaregivenbytheconditions t .
mod
ξ˙ = γ˙ = 0 imposed on equations (8) and (9). Generally, their As the system passes through the resonance, the topol-
γ-coordinateequalskπwithkbeinganinteger. ogy of the (γ,ξ)-disk graduallychanges.The system trajectory
Thenumberoffixedpointsinthe(γ,ξ)-diskdependsonthe [γ(t),ξ(t)]doesnotstrictlyfollowthecurvesF(γ,ξ)=constbe-
energyE andonthedetuningparameterσ.Forafixedvalueof cause the ratio σ/E 1 is changing significantly.Indeed both
ω ,thelatterquantityisgivenbytheratiooftheeigenfrequen- quantitiesareoftheo∼rderofǫ2andthechangeofthetotalenergy
u
cies, ω /ω .Hence,itispossibletoindicateregionsinthe E is negligiblebecause the relative changeof ν is proportional
u ℓ
( ,ER) p≡lane for which different number of fixed points exists. toǫ2.Accordingly,inthefollowing,wetakeν=const.
R
4 J.Hora´ketal.:InternalresonanceinneutronstarkilohertzQPOs
(a) (b) (c)
1
=
ξ
P− P− P+
Fig.2.Threepossibletypesoftopologiesofthe(γ,ξ)-diskdenotedinFig.1asA,BandC(left,middleandright,respectively).The
polarangleandthedistancefromtheoriginrepresentthephasefunctionγandthefractionalenergyξ,respectively.Thevaluesof
thetotalenergyanddetuningparametercorrespondingtotheseexamplesare(σ/ω ,E) = (0.005,0.1),(0.025,0.1)and(0.02,0.1).
u
Eachcurvecorrespondstoasolutionofthegoverningequations(8)and(9).Thediskscorrespondingto(σ,E)fromtheresonance
tonguescontainboththelibrating(whiteregions),andcirculating(shadedregions)trajectories.ThefixedpointsP correspondtoa
saddlepointorextremaoftheintegralofmotionF(ξ,γ). ±
0.8 2
=
ξ
1.5
u.] 1
a.
[
e
c 0.5
n
e
r
γ =0 e
ff 0
di
e
ud −0.5
t
mpli −1
A
−1.5
−2
1.4 1.45 1.5 1.55 1.6
Observed frequency ratio
Fig.3.Behavioroftheresonantoscillationsofthesystemundergoingaslowsecularchangeofitseigenfrequencies.Thedetuning
parameter σ increases linearly with time so that the system gradually passes through the resonance. The left panel shows the
trajectoryof the system projectedinto the (γ,ξ)-disk. The rightpanel shows behaviorof the amplitude difference,δa = a a ,
ℓ u
(measuredinarbitraryunits)asafunctionoftheratiotheobservedfrequencies, ⋆ = ω⋆/ω⋆.Becauseoftheslowchangesi−nthe
R u ℓ
eigenfrequencies,the energyexchangebetweenmodesis notexactly symmetricandthe totalenergyis redistributedbetweenthe
modeswhenthesystempassedtheresonance.
The general effect of the resonance transition is illustrated Inallcasesthebehaviorofthesystemisqualitativelysimi-
by the example shown in Figure 3. The eigenfrequencies are lar:theinitialvalueofthedetuningparametercorrespondstoa
changedwithconstantrate,σ(t)=σ +σ˙ t.Theleftpanelshows point(σ,E)outsidetheresonancetongue.Hence,the(γ,ξ)-disk
i i
the evolution of the oscillations in the (γ,ξ)-disk and the right initiallycontainsonlycirculatingorbits.Oneofthemisclosely
panelshowstheobservablequantities(i.e.,theratioofobserved followed by the system (the outer “circles” in the left panel in
frequencies ⋆ versus the amplitude difference δa = a a ). Figure3).Theξ-coordinateslightlyoscillatesincorrespondence
ℓ u
These FigurRes are for σ = 0.1ω , σ˙ = 3 10 5ω2, w−here to a weak energyexchangebetween the modes. As the system
i − u × − u
ω istheupperfrequencyintheexactresonance.Thesystemis approaches the resonance, the magnitude of these oscillations
u
described by β = µ = µ = 5 and ν = 9/4, and the initial increases gradually. Then the system enters into the resonance
ℓ u
−
conditionsareξ ξ(0) = 0.5,γ γ(0) = 0.Thetotalenergy tongue and the topology of the (γ,ξ)-disk suddenly changes –
i i
≡ ≡
isfixedat E = 0.1.We alsotrieddifferentratesofpassageand thelibratingtrajectoriesappear.Whenthetrajectoryofthesys-
differentinitialconditions. temcrossestheseparatrixandenterstheregionoflibratingtra-
jectories, the energy exchange becomes asymmetric – i.e., the
J.Hora´ketal.:InternalresonanceinneutronstarkilohertzQPOs 5
inflowoftheenergyintooneofthemodesexceedstheoutflow. Table 1. Occurrenceof solutionwith frequency-lockingperiod
The system mayescape fromthisregionpassingthe separatrix insetsof200solutionsfordifferentvaluesofγ andβ. Ineach
i
again before it makes a single orbit (as in Fig. 3) or it can be settheinitialvalueξ isuniformlydistributedbetween0and1,
i
trappedin this regionfor longertime. A trappingin the region whiletheotherparametersarekeptconstant.
oflibratingorbitsisaccompaniedbyseveralchangesofthesign
ofγ˙,andthereforebychangesofthedirectionoforbitinginthe β=0 β=3 β=5 β=10
(γ,ξ)-disk. The ratio ⋆ of the observed frequenciesoscillates γ =0 0 44(22%) 64(32%) 89(48%)
around 1.5 [as followRs from equation (11)]. When the system γ i=π/2 0 47(24%) 60(30%) 95(48%)
i
leaves the resonance tongues and as it gets farther away from γi=π 0 43(22%) 60(30%) 88(44%)
them,theenergyexchangebetweenmodesbecomesmoresym-
metricandlessapparent.Thefinalfractionalenergyξ afterthe
f
closetoaratioofsmallintegernumbers.Amongthemostchar-
passageisgenerallydifferentfromtheinitialoneξ:theenergy
i
acteristic observable imprints of the internal resonance are (i)
isredistributedduetotheresonance.
Exchangeofenergybetweentheoscillations(ii)equalityofthe
We notethatthebehaviorofthe amplitudeswith respectto
qualityfactorsofoscillationsand(iii) lockingofthe frequency
the frequency ratio as shown in the right panel of Figure 3 is
ratio of oscillations.Our work was devotedto the mathematics
similartothatreportedforseveralatollsources(To¨ro¨k2008).
ofthefirstandthirdphenomena.
Anexampleofatrajectorytrappedintheregionofthelibrat-
SomemodelsofQPOs, which maysupportan internalres-
ing orbitsis shownin Figure 4. The parametersof the solution
onance of the type described in this work have been described
are σ = 0.2ω , σ˙ = 4 10 5ω2, β = µ = µ = 5 and
0 − u × − u ℓ − u in Section 1. In this work we explored only the 3:2 resonance
ν = 9/4.Theinitialconditionsare ξ = 0.9,γ = 0 andthe to-
i i whichistheonemostprominentinthedata(Abramowiczetal.
tal energy is E = 0.1. Obviously, the nonlinearities adjust the
2003a;To¨ro¨ketal.2008).Theexaminationofotherresonances
observedfrequenciessothattheirratiooscillatesaroundtheres-
maybedoneinananalogousway.
onancevalue3:2duringthetrappingperiod.
Assumingaslow(secular)changeoftheeigenfrequenciesof
We examined the stability of this process against changes
thetwo coupledoscillators,we demonstratedthatthe energyis
oftheinitialconditionsandotherparameterscharacterizingthe
redistributedbetweenthetwooscillatorswhenthesystempasses
passagethroughtheresonance.Theeffectsofchangingtheini-
througharesonance.Thiseffectcanbeseeninthechangeofthe
tial conditions are summarized in the left panel of Figure 5.
sign inthe amplitudedifferencewhentheratio ofthe eigenfre-
Again,the parametersof the systemwere setto the valuesβ =
quenciesratiocrossesarationalvalue,suchas3:2.Thisresultis
µ = µ = 5 and ν = 9/4 and the initial detuningparameter
a−ndℓtherauteofchangewereσ = 0.1ω ,σ˙ =3 10 5ω .The inqualitativeagreementwiththeobservationsforsomeneutron
0 − u × − u stars(To¨ro¨k2008).
solutionstrappedinthelibratingregionsandshowingforsome
A class of solutions exhibits a time interval during which
time an episode of what could be termed ‘frequency-locking’
a frequency-lockingof the two oscillations occurs. This corre-
are clearlyseparated fromthe others.These solutionsare quite
sponds to small oscillations of the actual frequencies around
commonforsmallξ .Inthatcasetheyareaccompaniedbysig-
i the ratio 3:2 over a (greater) range of eigenfrequency ratios
nificantexchangedenergy.Astheinitialfractionalenergyξ in-
i (Figure4).
creasestheybecomeratherexceptional.
Although different for the solutions with and without Acknowledgements. ThisworkwasstartedattheGo¨teborgUniversityandcon-
frequency-locking,theξ–ξ dependenceisfairlyindependentof tinuedatMPAGarching.JHisgreatfultoprof.ChengminZhangforusefuldis-
i f
theinitialphasefunctionγ asitcanbeseenfromtheleftpanel cussionsandacknowledgessupportofthegrantsGACˇR205/07/0052andGACˇR
i 205/06/P415,PRissupportedbythePappalardoFellowshipatMITandMAA
ofFigure5,wherethe solutionswithdifferentγ correspondto
i wassupportedbythePolishMinistryofEducationgrantN20300931/1466,WK
differentsymbols.Similarly,wealsocheckedthattherateofthe acknowledgespartialsupportthroughgrant1P03D00530andGTwassupported
transitionσ˙ hasno significantinfluenceonthe resultingξ –ξ bythegrantMSM4781305903.
i f
plot.
TherightpanelofFigure5showstheeffectofchangingthe
References
valueoftheparameterβ.Theredistributionofenergyinthesolu-
tionswithoutthe‘frequency-locking’periodbecomesmorepro- AbramowiczM.A.,Kluz´niakW.,2001,A&A,374,L19
nounced as the value of β increases. In addition, the solution AbramowiczM.A.,BulikT.,BursaM.,Kluz´niakW.,2003a,A&A,404,L21
AbramowiczM.A.,KarasV.,Kluz´niakW.,LeeW.H.,RebuscoP.,2003b,PASJ,
withthe‘frequencylocking’periodbecomemorefrequent(see
55,467
Table1).Thevalueβ = 0correspondstothecasewhenthereis
ArrasP.,BlaesO.,TurnerN.J.,2006,ApJ,645,L65
noresonantcouplingbetweenthetwooscillations,andξi = ξf. BlaesO.M.,ArrasP.,FragileP.C.,2006,MNRAS,369,1235
Ofcourse,thereareno‘frequency-locked’solutionsinthiscase. BlaesO.M.,Sˇra´mkova´E.,AbramowiczM.A.,Kluz´niakW.,TorkelssonU.,Aug.
2007,ApJ,665,642
The dependenceonthe remainingparameterscouldbe dis-
BrandenburgA.,2005,AstronomischeNachrichten,326,787
cussed in a similar way;however,it can be shownthatforsuf-
BursaM.,AbramowiczM.A.,KarasV.,Kluz´niakW.,2004,ApJ,617,L45
ficiently slow transitions of the resonance the whole dynamics Hora´kJ.,2008,A&A,486,1
depends only on a single parameter, βˆ = β√E/(µ µ /ν). Hora´kJ.,KarasV.,2006,A&A,451,377
ℓ u
− Hora´kJ.,AbramowiczM.A.,KarasV.,Kluz´niakW.,2004,PASJ,56,819
Therefore one does not expect any qualitatively new behavior
KatoS.,Oct.2001,PASJ,53,L37
astheotherparametersarechanged. KatoS.,Oct.2004a,PASJ,56,905
KatoS.,2005a,PASJ,57,679
KatoS.,2005b,PASJ,57,699
4. Discussionandconclusions KatoS.,2008,PASJ,60,111
KatoY.,Oct.2004b,PASJ,56,931
Kluz´niakW.,AbramowiczM.A.,2001,astro-ph/0105057
IfthetwinkHzQPOsinneutron-starscorrespondtotwophysi-
Kluz´niakW.,MichelsonP.,WagonerR.V.,1990,ApJ,358,538
caloscillatorsthathaveanonlinearcoupling,thetwoQPOsmay
Kluz´niakW.,AbramowiczM.A.,KatoS.,LeeW.H.,StergioulasN.,2004,ApJ,
exhibit resonance phenomena when the QPO frequencies are 603,L89
6 J.Hora´ketal.:InternalresonanceinneutronstarkilohertzQPOs
1 1.6
=
ξ
tio1.55
a
r
y
c
n
e
γ =0 u
q 1.5
e
r
f
d
e
v
r
e
s
b1.45
O
1.4
1.35 1.4 1.45 1.5 1.55 1.6
Eigenfrequency ratio
Fig.4.Exampleofatrajectorywithafrequency-lockingepisode.Left:Trajectoryprojectedontothe(γ,ξ)-disk.frequencylocking
correspondstotheredistributionoftheenergywhenthesystemistrappedintheregionofthelibratingorbits.Right:Thefrequency-
lockingeffectintheplotshowingratiooftheobservedfrequenciesversuseigenfrequencyratio.
1 1
FL FL
0.8 0.8
0.6 0.6
ξf ξf
0.4 0.4
γi=0 β=0
0.2 γi=π/2 0.2 β=3
γi=π β=10
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
ξi ξi
Fig.5.Energyexchangeduringtheforwardtransitionthroughthe3:2resonance(e.g.,fromthelowertothehighereigenfrequency
ratios). ξ and ξ are the fractionalenergyin the lower oscillation mode before and after the transition. Left: Dependence on the
i f
initial condition(differentsymbolsdenotedifferentinitial phases, γ = 0,π/2and π) for β = 5. Right: Dependenceon the value
i
of the parameter β (the initial phase is γ = 0). The branchescorrespondingto solutionswith (denotedby ‘FL’) and withoutthe
i
frequency-lockingperiodareclearlyseparated.Inthisplot,ξ issampleduniformly—anindicationoftheprobabilityoffindingthe
i
systemina‘frequency-locked’solutionisgivenbytherelativenumberofpointsontheFLbranch.
LeeW.H.,AbramowiczM.A.,Kluz´niakW.,2004,ApJ,603,L93 SchnittmanJ.D.,RezzollaL.,2006,ApJ,637,L113
LiL.X.,NarayanR.,2004,ApJ,601,414 StellaL.,VietriM.,Jan.1999,PhysicalReviewLetters,82,17
MachidaM.,MatsumotoR.,Jun.2008,PASJ,60,613 TaggerM.,Varnie`reP.,2006,ApJ,652,1457
MaoS.A.,PsaltisD.,MilsomJ.A.,2008,ArXiv:0805.0598 TitarchukL.,2002,ApJ,578,L71
MonteroP.J.,RezzollaL.,YoshidaS.,2004,MNRAS,354,1040 To¨ro¨kG.,2008,submittedtoA&A
MonteroP.J.,ZanottiO.,FontJ.A.,RezzollaL.,2007,MNRAS,378,1101 To¨ro¨kG.,AbramowiczM.A.,BakalaP.,etal.,2008,ActaAstronomica,58,15
NayfehA.H.,MookD.T.,1979,NonlinearOcillations,Wiley-Interscience,New vanderKlisM.,2000,ARA&A,38,717
York VioR.,RebuscoP.,AndreaniP.,MadsenH.,OvergaardR.V.,2006,A&A,452,
Pe´triJ.,2005,A&A,439,L27 383
Pe´triJ.,2006,Ap&SS,302,117 WagonerR.V.,1999,Phys.Rep.,311,259
PsaltisD.,NormanC.,2000,astro-ph/0001391 WagonerR.V.,SilbergleitA.S.,Ortega-Rodr´ıguezM.,2001,ApJ,559,L25
RebuscoP.,2004,PASJ,56,553 YuW.,vanderKlisM.,JonkerP.G.,2001,ApJ,559,L29
ReynoldsC.S.,MillerM.C.,2008,ArXiv:0805.2950 ZanottiO.,FontJ.A.,RezzollaL.,MonteroP.J.,2005,MNRAS,356,1371
RezzollaL.,YoshidaS.,MaccaroneT.J.,ZanottiO.,2003a,MNRAS,344,L37 ZhangC.,2004,A&A,423,401
RezzollaL.,YoshidaS.,ZanottiO.,2003b,MNRAS,344,978 Zhang C.M., Yin H.X., Zhao Y.H., Chang H.K., Song L.M., 2007,
J.Hora´ketal.:InternalresonanceinneutronstarkilohertzQPOs 7
AstronomischeNachrichten,328,491