Table Of ContentDifficulties withthe QPOs ResonanceModel
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0
Paola Rebusco
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2 Kavli Institute forAstrophysics and Space Research, MIT, Cambridge, MA 02139, USA
n
a ”Toknow that weknow what weknow,
J and to know that wedonot know what wedonot know,
3 that is trueknowledge.”
2
Nicolaus Copernicus
]
h
p
- Abstract
o
r
t Highfrequencyquasi-periodicoscillations (HFQPOs)havebeendetectedinmicroquasars andneutronstarsystems.
s
Theresonancemodelsuggested byKlu´zniak &Abramowicz (2000,2001) explainstwin QPOsas twoweaklycoupled
a
[ nonlinear resonant epicyclic modes in the accretion disk. Although this model successfully explains many features
of theobserved QPOs, it still faces difficulties and shortcomings. Here we summarize theaspects of thetheory that
1
remain a puzzleandwe brieflydiscuss likely developments.
v
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Keywords: accretion, general relativity, X-rays:binaries,nonlinear resonance, QPOs
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3
.
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1. Introduction in strong fields. Many models have been pro-
0
8 posed in order to understand HFQPOs: some of
0 Manygalacticblackholeandneutronstarsources them involve orbital motions (e.g., Kluzniaketal.,
v: in low-mass X-ray binaries show quasi-periodic 1990; Stella &Vietri, 1998; Lamb& Miller, 2003;
i variability in their observed X-ray fluxes. Some of Schnittman &Bertschinger, 2004), others consider
X
the quasi-periodic oscillations (QPOs) are of the accretion disk oscillations (e.g., Wagoneretal.,
r
a order of few hundred Hz (hence the name high fre- 2001;Kato,2001; Rezzollaet al.,2003;Kato,2007;
quency)andoftencomeinpairsoftwinpeaksinthe Tassev&Bertschinger, 2007). None is definitive.
Fourier power spectra (see vanderKlis& Murdin, Klu´zniak&Abramowicz(2000,2001)proposedthat
2000;Remillard& McClintock,2006,fora review). certainpropertiesofHFQPOsinneutronstarscould
High frequency QPOs draw much interest be- be explainedby resonantmotions ofaccretingfluid
cause their frequencies lie in the range of orbital in strong gravity, and predicted that in black hole
frequencies few Schwarzschild radii outside the systemstwoHFQPOsinafrequencyratioof1:2,1:3,
central source. Furthermore for different sources n:m, could be detected. Abramowicz&Klu´zniak
they scale with 1/M, where M is the mass of the (2001) observed that the newly discovered HFQ-
central compact object (McClintock &Remillard, POs in GRO J 1655-05(Strohmayer, 2001) were in
2006). These two characteristics make QPOs at- a 3 : 2 ratio, and proposed a more specific model
tractive tools to possibly test General Relativity of resonance. This idea guided successive studies
(e.g., Abramowiczetal., 2003b; Rebusco, 2004;
1 Supported by the Pappalardo Postdoctoral Fellowship in Hora´ketal., 2004; Klu´zniaketal., 2004) aimed
Physics at MIT. at clarifying and expanding the theory. Wlodek
Preprintsubmitted toElsevier 4February2008
Klu´zniak talked about the accomplishments of the has a second pair (Strohmayer, 2001) in a 5:3 fre-
model (see the contribution in the same Proceed- quency ratio(Kluzniak& Abramowicz,2002).
ings),herewe focusonits limits. Contrarily to what it may seem at first sight, all
these features are naturally explained in the res-
2. High Frequency QPOs and Nonlinear onance model. Indeed the absence of a peak may
Resonances be due to the relativistic modulation of the emit-
ted light (Bursaetal., 2004). Moreover although
the strongest resonance is expected to occur when
The fundamental features of the nonlinear reso-
the frequencies are in 3 : 2 ratio, lower order reso-
nancemodelarethat:
nances (such as 5 : 3) and subharmonics can still
be excited and become visible (Abramowiczetal.,
– HFQPOs arise from a nonlinear resonance in ac-
2003b; Rebusco, 2004; Hora´ketal., 2004). In the
cretiondisks inGeneralRelativity.
resonance picture the vicinity of the observedratio
tocommensurateratioindicatesthatthefrequency
– The frequencies of nonlinear oscillations depend
correctionsaresmall.
on the amplitudes of the same oscillations. As a
∗ The real problems of the model are more substan-
consequence the observed frequencies (ν ) may
tial: how are the QPOs excited? How are they
differ from the eigenfrequencies (ν0) and can
∗ coupled? In a preliminary toy-model based on the
vary in time (ν = ν0+∆ν(t)). We refer to this
ideaofAbramowicz&Klu´zniak(2001),themotion
well-known nonlinear phenomenon as frequency
of a single particle on a perturbed geodesic was
correction.
analyzed. In this simple model an ad-hoc parame-
ter wasadded(Abramowiczetal.,2003b;Rebusco,
– The two frequencies of resonant modes are ap-
2004), that would provide the energy to increase
proximately in the ratio of small integers, most
theamplitudesoftheresonantoscillations.Thisap-
likelyinthe ratio3:2.
proach works well mathematically, but the physics
remains unexplained. A more detailed fluid model
3. Difficultieswiththe NonlinearResonance
of the disk should be developed in order to jus-
Model
tify this excitation mechanism and to understand
whetherforexampleaformofdiskinstability(e.g.,
All models of QPOs are essentially dynami-
Papaloizou&Pringle, 1984, or magnetic instabil-
cal models, that miss any emission mechanisms
ity)couldarousethe oscillations.
and any connections to the spectral states of the
Accretion disks are highly turbulent. Vio etal.
sources. The only seminal steps in this direc-
(2006) investigated the possibility that stochastic
tion have been done by Bursaetal. (2004) and
turbulence could excite and feed the resonance. To
Schnittman& Bertschinger(2004), who considered
this purpose a white noise term was added to the
general relativistic ray-tracing from an oscillating
vertical component of the perturbed geodesic and
slim torus and an orbiting hot-spot respectively.In
itwasfoundthatasmallnoisecouldindeedtrigger
what follows we concentrate on the difficulties in
the epicyclic resonance (see figure 2). This result
thepuredynamics,sincetheradiationproblemstill
opens an interesting perspective. Still a more real-
hasto be addressedbyallthe models.
istic scenario,that takes into accountcolorednoise
HFQPOs from black holes are stable (for many
andstochasticforcingofadisk(ratherthanasingle
years), while neutron star QPOs are variable and
particle)shouldbe evaluated.
present a rich phenomenology: consequently the
Finally in the alpha of the QPOs resonance theory
twoclassesofsystems willbe treatedseparately.
it was always mentioned that gravity and pressure
would provide the coupling between the resonant
3.1. Difficulties in Black Hole Systems modes. However it was recently estimated that in
thin disks and slender tori such a coupling is too
Four microquasars display pairs of HFQPOs, weak to actually transfer energy between epicyclic
whose centroids ratio is consistent with 3 : 2 modes in a reasonably short time (Jiˇr´ı Hora´k,
(Remillard&McClintock, 2006, and see Figure 1). private communication). This difficulty holds in
Sometimes the twin peaks are observed simultane- neutron star systems as well. One possibility could
ously,mostoftenonlyoneispresent.GRS1915+150 be to consider the coupling of epicyclic modes in
2
Fig.1.Lower(νdown)versusupper(νupp)frequenciesinobservedtwinpeakHFQPOs.Thefourmicroquasarslieonthelower
left part of the plot, on the line with slope 3:2 . Neutron star systems (all the others) cross the 3:2 line, but at different
times of observation maybe ina differentposition andratio(courtesy of MichalBursa).
different geometric configurations. Still this may modes arise, with some modes in 3 : 2 ratio. One
originate new dilemmas: for instance, in a thick of the challenges consists to identify the simulated
torus the epicyclic oscillations (if not damped) do modes with analytic solutions and to tag if there is
notsimplyscalewiththemassofthecentralobject, anyresonances.
but depend strongly on thermodynamical quanti- We conclude this section on microquasars pointing
ties. Moreover Sr´amkov´aetal. (2007) showed that atanotherpuzzle:Toroketal.(2006)reportedthat
inasymmetricnewtonianslender/thicktorusthere the spin estimates from the resonance model do
is no pressure coupling between epicyclic modes. not match the estimates obtained by fitting spec-
Non-axisymmetrymayplay animportantrole. tral continua. This is a puzzle that any relativistic
Another option would be to consider non-epicyclic QPOsmodelmustaddress.
modes, but one should understand why two non-
epicyclic modes are more important than others.
3.2. Difficulties in NeutronStarSystems
Although much has been done, the excitation and
coupling of modes still needs to be developed and
a close interaction between analytic and numerical As we have seen there is no general agreement
work is fundamental. Simulated axisymmetric tori aboutthenatureofHFQPOsinblackholesystems.
slightlyoffequilibriumdonotincontrovertiblypro- Forneutronstarsthesituationisevenmorecomplex
duce QPOs : they need to be ”kicked” (Lee etal., and the scientific community is torn between those
2004; Zanottietal., 2005; Blaesetal., 2006). Once who think that HFQPOs in neutron star systems
the modes are excited (via velocity, density per- have the same originas in microquasars,and those
turbations or periodic forcing), then a spectrum of who sustain that it is a different phenomenon. In
the scenario proposed by Abramowicz&Klu´zniak
3
Fig. 2. Spectra (upper panels) and phase diagrams (lower panels) for radial (ρ) and vertical (z) perturbed geodesics. The
displacements are in units of Schwarzschild radii, the frequencies are scaled to kHz (assuming a central mass M of 2 M⊙).
The left plots are noise free, while the right have a small noise in the z direction. Clearly the amplitudes of oscillation are
much greater in presence of stochastic forcing. Note that if turbulence becomes too strong, then any periodicity is lost and
the motionbecomes chaotic. Seesection 3.1 andthe original paper (Vioet al., 2006)fora moreextensive explanation.
(2001)HFQPOshavethe samerootinbothclasses 2005; Abramowiczet al., 2007). The divergence
ofsources,with someimportantcaveats. fromtheexactratioandthepossibilityofmorethan
Twin HFQPOs are observed in accreting neutron one ratios (as suggestedby Bellonietal., 2005) are
stars: however both peaks can be in a different po- naturalconsequencesofthenonlinearnatureofthe
sition at a different time (e.g., vander Klis, 2005). oscillators(e.g.,Landau& Lifshitz,1969)
Thefrequencyshiftcanbehigherthanafactorof2 The bright Sco X-1 is often taken as a paradigm
(seeFigure1).Theratioofthetwocentroidsissome- for neutron star sources. Different detections lie
times in the vicinity of 3 : 2, other times it differs on a line with a slope close, but not equal to
from the exact ratio (up to ∼ 10%). Bellonietal. 3 : 2 (vanderKliset al., 1997). Abramowiczetal.
(2005) argue that in neutron star sources there is (2003b) and Rebusco (2004) reproduced part of
not a preferred fixed ratio (as Abramowiczetal., this line, by considering the frequencies and ra-
2003a, claim) and the occurrence of ratios close to tio shifts as frequencies corrections (∆ν(t)) due
3:2is biased. to nonlinear resonance. This result is qualitatively
In the resonance model these QPOs are still inter- interesting, but the toy-model they adopt is too
pretedasresonantepicyclicmodes,withamplitudes simple.Moreoverinthenonlineartheorythelowest
larger than in the case of black holes. The neutron orderfrequencycorrectionsscalewiththesquareof
starsurfaceplaysanimportantrole:theX-rayemis- the amplitude of the perturbation. Shifts as large
sion is indeed modulated at the boundary layer, as those observed would mean so high amplitudes
where the accreted matter hits the star (Hora´k, thatthe adoptedexpansionwouldfail.
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HFQPOs in neutron star sources may be related
to the spin of the central object. In the very first
detections the difference of the frequencies of the
two peaks was found to be consistent with being
constant and equal to the spin of the neutron star.
When more sources were observed, they were di-
vided into two classes: slow (ν < 400 Hz) and
spin
fastrotators,inwhichthedifferencewasequaltothe
spin or half of it respectively (Muno etal., 2001).
FromthestudyofScoX-1Abramowicz& Klu´zniak
(2004) pointed that HFQPOs have ”little to do
with the rotation of a stellar surface or any mag-
netic field structure anchored in the star”. Very
recently M´endez& Belloni (2007) re-examined all
the available data and found that the difference of
HFQPOsisapproximatelyconstantbutnotrelated
to the spin and this issue is currently under debate
(Strohmayeretal.,2007;Barretetal.,2007).Mod-
Fig.3.Measuredqualityfactorofthelower(dots)andupper
els of HFQPOs became more or less popular each
(squares) HFQPOs of 4U 1636-536. The solid and dashed
time that a correlation or not with the spin was lines show the estimates done withthe toy model proposed
found (see M´endez&Belloni, 2007, for a review inBarretet al.(2006), fromwhichthis figureis taken.
apropos). The original structure of the resonance
modelisunrelatedtothespinofthecentralobject. 4. Conclusions
It was however subsequently proposed (Lee etal.,
2004) that in neutron star sources the resonance Thenonlinearresonancemodelforhighfrequency
could be excited by direct forcing of the rotating QPOs encounters many internal difficulties. This
star, leading to frequency differences equal to the maymarkthe endofthe modelorthe beginning of
spinorhalfofit.Itisworthnoticingthatthispossi- a new era.
bility is not structural in the resonance model, but In our opinion the presence of twin peaks in (al-
is a variation of it. On the other hand this brings most)commensurateratioandthedetectionofsub-
back the problem of the excitation of the modes, harmonicsstillstronglysupportthe ideathatweak
likein the caseofblack holesystems. nonlinear resonance is a fundamental ingredient in
Eventuallyanotheraspectsremainstobeexplained, the physicsofHFQPOs.
thatisthebehaviorofthequalityfactorofHFQPOs It is like smelling chocolate in a birthday cake: the
(Q=ν/FWHM),thataccountsforthecoherenceof smell is the unmistakable signature of the presence
the oscillations. Barretetal. (2006) reported that ofchocolate.Howevertheother(mathematicaland
the quality factor of the lower HFQPOs is higher physical)ingredients,theirrelativeimportanceand
thanthatoftheupper.Furthermore,whiletheQof a draft of the recipe (mechanisms) are needed to
the lower frequencies increases with the frequency knowwhichcakeitis.
up to a break frequency and then drops, the Q of
the upper frequenciesincreasessteadily(see Figure
3). These measurements suggest that in different
sources there is a common physical mechanism not
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