Towards modeling quasi-periodic oscillations of microquasars with oscillating slender tori
G. P. Mazur, F. H. Vincent, M. Johansson, E. Sramkova, G. Torok, P. Bakala, M. A. Abramowicz
AAstronomy & Astrophysics manuscript no. OscillatingTorus c (cid:13)
ESO 2018October 8, 2018
Towards modeling quasi-periodic oscillations of microquasars withoscillating slender tori
G. P. Mazur , , F. H. Vincent , M. Johansson , E. ˇSramkov´a , G. T¨or¨ok , P. Bakala , andM. A. Abramowicz , , N. Copernicus Astronomical Centree-mail: [email protected] Physics Department, Gothenburg University, SE-412-96 G¨oteborg, Swedene-mail: [email protected] Physics Department, Warsaw University,e-mail: [email protected] Chalmers University of Technologye-mail: [email protected] Physics Department, Silesian Universitye-mail: [email protected] e-mail: [email protected] e-mail: sram [email protected]
Received ; accepted
ABSTRACT
Context.
One of the often discussed models for high-frequency quasi-periodic oscillations of X-ray binaries is the oscil-lating torus model, which considers oscillation modes of slender accretion tori.
Aims.
Here, we aim at developing this model by considering the observable signature of an optically thick slender ac-cretion torus subject to simple periodic deformations.
Methods.
We compute light curves and power spectra of a slender accretion torus subject to simple periodic deforma-tions: vertical or radial translation, rotation, expansion, and shear.
Results.
We show that different types of deformations lead to very different
Fourier power spectra and therefore couldbe observationally distinguished.
Conclusions.
This work is a first step in a longer term study of the observable characteristics of the oscillating torusmodel. It gives promising perspectives on the possibility of constraining this model by studying the observed powerspectra of quasi-periodic oscillations.
Key words.
Accretion, accretion disks – Black hole physics – Relativistic processes
1. Introduction
Some microquasars exhibit high-frequency (millisecond)quasi-periodic oscillations (QPOs), which in this work al-ways refer to high-frequency oscillations; low-frequencyQPOs are not considered here. These high-frequency oscil-lations are characterized by a narrow peak in the sourcepower spectrum (see van der Klis 2004; Remillard &McClintock 2006, for a review). As the timescale of thisvariability is of the same order as the Keplerian orbitalperiod at the innermost stable circular orbit of the cen-tral black hole, it is highly probable that these phenomenaare linked with strong-field general relativistic effects. Theother different low-mass X-ray binaries that contain neu-tron stars display QPOs on these prominent timescales aswell. There is no consensus on the possible QPO origin anduniformity across different sources yet.A variety of models have been developed so far to ac-count for the high-frequency variability. Stella & Vietri(1999) and Stella et al. (1999) suggest that QPOs couldarise from the modulation of the X-ray flux by the pe-riastron precession and the Keplerian frequency of blobs of matter orbiting in an accretion disk around the cen-tral compact object. Also QPOs could be due to modu-lation of the X-ray flux by oscillations of a thin accretiondisk surrounding the central compact object (see Wagoner1999; Kato 2001). Fragile et al. (2001) propose that QPOsare due to the modulation of the X-ray flux caused by awarped accretion disk surrounding the central compact ob-ject. Pointing out the 3:2 ratio of some QPOs in differentsources, Abramowicz & Klu´zniak (2001) have proposed aresonance model in which these pairs of QPOs are due tothe beat between the Keplerian and epicyclic frequencies ofa particle orbiting around the central compact object.General relativistic ray tracing of radiating or irradiatedhot spots orbiting around black holes and neutron starswas founded at the turn of the millennium. For instance,Karas (1999) discusses already sophisticated numeric mod-eling of observable modulation from clumps distributedaround certain preferred circular orbits. Later, Schnittman& Bertschinger (2004) investigated in great detail predic-tions of a model of hot spot radiating isotropically on nearlycircular equatorial orbits. In their study, the hypothetic res-onance between the Keplerian and radial epicyclic frequen- a r X i v : . [ a s t r o - ph . H E ] M a y azur et al.: Oscillating tori for modeling QPOs cies gives rise to peaks in the modeled power spectrum.Tagger & Varni`ere (2006) advocate the fact that QPOs inmicroquasars are due to the triggering of a Rossby waveinstability in the accretion disk surrounding the centralcompact object. Ray-traced light curves have been recentlydeveloped for this model by Vincent et al. (2012). Finally,models involving oscillations of accretion tori have been de-veloped. As our work is a follow-up of these past studies,we present them in more details.The first study of a QPO model involving thick ac-cretion structure (tori) was developed by Rezzolla et al.(2003), who showed that p-mode oscillation of a numericallycomputed accretion torus can generate QPOs. Ray-tracedlight curves and power spectra of this model were derivedby Schnittman & Rezzolla (2006). The analysis of analyticalaccretion tori (for which all physical quantities are knownanalytically throughout the structure and at all times) as amodel for QPOs was initiated by Bursa et al. (2004), whoperformed simulations of ray-traced light curves and powerspectra of an optically thin oscillating slender torus. Thismodel took only into account simple vertical and radial si-nusoidal motion of a circular cross-section torus. To allow amore general treatment, a series of theoretical works werededicated to developing a proper model of general oscilla-tion modes of a slender or non-slender perfect-fluid hydro-dynamical accretion torus (Abramowicz et al. 2006; Blaeset al. 2006; Straub & ˇSr´amkov´a 2009). Considering hydro-dynamical perfect-fluid tori is interesting in terms of sim-plicity as all computations can be derived analytically andall physical quantities are known analytically at all times.The aim of this article is to strengthen the perfect-fluid oscillating slender torus model for QPOs by progress-ing towards determining its observable signature. Here, wepresent a first step towards this goal, which consists ofdetermining the observable signature of simple deforma-tions of a slender torus in the Schwarzschild metric. Wedo not use the mathematically well-defined oscillation for-malism derived by Blaes et al. (2006). We rather considerthe impact on the observables of elementary deformationsof the torus cross-section: translation, rotation, expansion,and shear. We simulate light curves and power spectra ofsuch tori subject to these simple deformations, taking intoaccount all relativistic effects that will affect radiation byusing a ray-tracing code. The focus of this article is thus todetermine the impact of the geometrical change of a slendertorus on the observed flux variation. We do not consider thephysics of the matter forming the torus.This model is extremely simple and does not claim togive a realistic view of the nature of QPOs. However, it doesallow determination in the simplest possible framework ofwhether different kinds of motions of a toroidal accretionstructure lead to potentially observable differences in thelight curves. This work thus makes it possible to go one stepfurther in the development of the oscillating torus model forQPOs.Section 2 describes the equilibrium slender torus, whileSection 3 describes its deformations in terms of translation,rotation, expansion, and shear. Section 4 shows the ray-traced light curves and power spectra of the deformed torus,and Section 5 gives conclusions and prospects.
2. Equilibrium of a slender accretion torus
This section derives the equations that describe the accre-tion torus at equilibrium. The spacetime is described bythe Schwarzschild metric in the Schwarzschild coordinates( t, r, θ, ϕ ), with geometrical units c = 1 = G and signature( − , + , + , +). The line element has then the following form:d s = g tt d t + g rr d r + g θθ d θ + g ϕϕ d ϕ = − (cid:18) − Mr (cid:19) d t + (cid:18) − Mr (cid:19) − d r + r (cid:0) d θ + sin θ d ϕ (cid:1) , (1)where M is the black hole mass. In the remaining article,we use units where this mass is 1.In this spacetime, we consider an axisymmetric, non-self-gravitating, perfect fluid, constant specific angular mo-mentum, which circularly orbits the accretion torus. Thistorus is assumed to be slender, meaning that its cross sec-tion is small as compared to its central radius. As the fluid follows circular geodesics, its 4-velocity can bewritten as u µ = A ( η µ + Ω ξ µ ) , (2)where Ω is the fluid’s angular velocity, and η µ and ξ µ are theKilling vectors associated with stationarity and axisymme-try, respectively. The constant A is given by imposing thenormalization of the 4-velocity, u µ u µ = − E and specific angular momentum L are definedas E = − η µ u µ = − u t (3) L = ξ µ u µ = u ϕ , which are geodesic constants of motion.The rescaled specific angular momentum (cid:96) is definedaccording to (cid:96) ≡ LE = − u ϕ u t . (4)We assume this rescaled specific angular momentum tobe constant throughout the torus: (cid:96) = (cid:96) = const . (5)The 4-acceleration along a given circular geodesic fol-lowed by the fluid is a µ = u ν ∇ ν u µ = − U ∂ µ U , (6)where U is the effective potential (see, e.g., Abramowiczet al. 2006): U = g tt + (cid:96) g ϕϕ . (7)The radial and vertical epicyclic frequencies for circularmotion are related to the second derivatives of this poten-tial: ω r = 12 E A g rr (cid:18) ∂ U ∂r (cid:19) ,ω θ = 12 E A g θθ (cid:18) ∂ U ∂θ (cid:19) . (8) In the Schwarzchild metric, these epicyclic frequenciesat the torus center are ω r = Ω K (cid:18) − r (cid:19) ,ω θ = Ω K (9)where a subscript 0 denotes here and in the rest of this arti-cle a quantity evaluated at the torus center. The Keplerianangular velocity is well known: Ω K = r − / .In the rest of this article, the torus central radius willbe fixed to the value r , satisfying the following condition ω θ ω r = 32 . (10)This choice is linked to our goal of applying the de-formed torus model to twin-peak QPOs. It results in r = 10 . . (11) Following Abramowicz et al. (2006), we define the surfaceof the torus by the locus of the zeros of a surface function f ( r, θ ). Introducing the following set of coordinates x = ( √ g rr ) (cid:18) r − r r (cid:19) , y = ( √ g θθ ) (cid:18) π/ − θr (cid:19) , (12)Abramowicz et al. (2006) show that the surface function isexpressed according to f = 1 − β (cid:0) ¯ ω r x + ¯ ω θ y (cid:1) , (13)where ¯ ω r = ω r Ω K , ¯ ω θ = ω θ Ω K . (14)the parameter β is related to the torus thickness, and thetorus being slender: β (cid:28) . (15)Abramowicz et al. (2006) also show that the x and y coordinates are of order β . We thus define a new set oforder-unity coordinates:¯ x = xβ , ¯ y = yβ , (16)and: f = 1 − (cid:0) ¯ ω r ¯ x + ¯ ω θ ¯ y (cid:1) . (17)The equilibrium slender torus has therefore an ellipticalcross section.
3. Deformations of a slender accretion torus
This section is devoted to determining the surface functionof the torus subject to simple deformations: translation,rotation, expansion, and shear. The 4-velocity of the per-turbed fluid is also given.
In this section, we consider various simple time-periodicdeformations of the torus cross-section. Our aim is to de-termine the new torus surface function corresponding tothese deformed states.All deformations boil down to performing transforma-tions on the (¯ x, ¯ y ) coordinates of the form:¯ x ( t ) = a ( t )¯ x (0) + a ( t )¯ y (0) + a ( t ) , (18)¯ y ( t ) = b ( t )¯ x (0) + b ( t )¯ y (0) + b ( t ) , where the a i and b i functions are simple trigonometric func-tions of time. Table 1 gives these functions for all defor-mations considered in this article as a function of a freeparameter λ and of the deformation’s pulsation ω . Thispulsation will have the same value for all the simulationsperformed in this article. We choose ω = Ω K . We note thatthis value is a choice as the deformations we consider arenot physically justified, but imposed for the simplicity ofthe model. We make here the simplest and most naturalchoice of considering the Keplerian pulsation.At this stage, the deformed torus we are considering hasthus only two degrees of freedom:- the torus thickness β parameter, with β (cid:28) λ , with typically λ ≈ β and λ will be chosen to satisfythe slender torus approximation for all times. The fluid 4-velocity in the deformed torus is given by u µ = u µ + δu µ , (19) u µ = A ( η µ + Ω K ξ µ )where u µ is the equilibrium 4-velocity already defined inEq. (2), which is assumed to be everywhere equal to itscentral value, in the slender torus limit.The value of the perturbation δu µ induced by the defor-mation is easily computed by using the coordinate trans-formations given in the previous Section. By writing a first-order expansion of ¯ x ( t + δt ) and ¯ y ( t + δt ) for a small timeincrement δt , it is straightforward to obtain the expressionsof d¯ x/ d t and d¯ y/ d t , which are themselves linearly relatedto d r/ d t and d θ/ d t .An example of this is the case of radial translation ofthe torus cross-section: u r = δu r (20) u θ = δu θ = 0 , with δu r = d r d t u t = λ ω r (cid:0) √ g rr (cid:1) cos( ωt ) u t (21)from Table 1.As the torus is assumed to have a constant rescaled an-gular momentum (cid:96) , the following relation holds between a a a b b b Radial translation 1 0 λ sin( ωt ) 0 1 0Vertical translation 1 0 0 0 1 λ sin( ωt )Rotation cos( ωt ) sin( ωt ) 0 − sin( ωt ) cos( ωt ) 0Expansion 1 + λ sin( ωt ) 0 0 0 1 + λ sin( ωt ) 0Radial simple shear 1 λ sin( ωt ) 0 0 1 0Vertical simple shear 1 0 0 λ sin( ωt ) 1 0Pure shear 1 + λ sin( ωt ) 0 0 0 [1 + λ sin( ωt )] − Table 1.
Functions a i and b i of Eq. 18 for all periodic deformations, λ is a free parameter and ω is the deformation’spulsation.the non-equilibrium temporal and azimuthal covariant com-ponents of the 4-velocity: u ϕ = − (cid:96) u t . (22)Adding the normalization condition u µ u µ = −
4. Light curves and power spectra of the deformedtorus
Light curves and power spectra of the deformed torus arecomputed by using the general relativistic ray-tracing code
GYOTO (Vincent et al. 2011). Null geodesics are integratedbackward in time from a distant observer’s screen to theoptically thick torus. The zero of the surface function isfound numerically along the integrated geodesic and thefluid 4-velocity is determined by using Eqs. (20). The spe-cific intensity emitted by the undeformed torus is chosento be constant throughout the surface. An image is thendefined as a map of specific intensity.Figure 1 shows the undeformed torus image as seen froman inclination of 45 ◦ and 85 ◦ . The two values of inclina-tion emphasize the bigger impact of the beaming effect athigh inclination, which strongly increases the dynamic ofthe image. Moreover, the apparent area of the primary andsecondary images vary, depending on inclination. The pri-mary image dominates clearly at small inclination, whereasthe secondary image’s apparent area becomes important athigh inclination. The value of β is chosen to ensure thatthe torus is indeed slender, i.e., that for all points insidethe torus | r − r | r (cid:28) , (cid:12)(cid:12)(cid:12) π − θ (cid:12)(cid:12)(cid:12) (cid:28) π . (23)We choose β = 0 .
05, a value that satisfies these con-straints well to be able to still abide by the slender torusapproximation even for the deformed (and, in particular,extended) torus. This value will be fixed for the remainderof this article.When the torus is deformed, two main effects will havean impact on the light curve: the change with time of pro-jected emitting area and the varying amplitude of relativis-tic effects as the torus surface is approaching or recedingfrom the black hole. To be able to compare the respectiveimpacts of these two effects on the various deformed tori,we require that: - the specific intensity emitted at the surface of the op-tically thick torus is inversely proportional to its crosssection. This will erase the effect of varying the flux bychanging the torus cross-section area for torus expan-sion (all other deformations leave the cross-section areaunchanged). Only the effect of changing the projectedemitting area will thus remain;- the ‘closest approach’ of the torus to the black hole, asdefined by the minimum value of the r coordinate, is thesame for all deformations. This ensures that the lightemitted by the torus experiences comparable generalrelativistic effects, whatever the deformation. Precisely,we require that at closest approach, ¯ x ≈ − . x = − . λ parameter on all deformations. We stress that λ is different for different deformations to allow a commonclosest approach for all deformations. Light curves are easily computed by calculating images ofthe deformed torus at various times. The time samplingwas chosen to be around 100 frames per Keplerian period,and images are 1000 × r = 10 . M is t K = 230 M , the time intervalbetween two frames is around δt = 2 M . The total numberof frames computed for one given light curve is N = 200,thus covering around two Keplerian periods. One point ona light curve is simply equal to the summation of all pixelsof a given frame (this boils down to summing intensity overall solid angles and thus to computing a flux).Power spectral densities are computed in the followingway. Let F ( t k ) be the flux value at time t k = kδt . The powerspectral density at frequency f k = k/ ( N δt ) is defined as thesquare of the modulus of the fast Fourier transform of thesignal, thus:
P SD ( f k ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N − (cid:88) j =0 F ( t k )exp(2 πijk/N ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (24) Figure 2a) displays the light curves obtained for all defor-mations of the slender torus, seen with an inclination pa-rameter of 5 ◦ , 45 ◦ , and 85 ◦ . The panels b) and c) of the Fig. 1.
Undeformed torus seen from an inclination of 45 ◦ (left) and 85 ◦ (right). The color bar, common to both panels,shows the ratio ∆ ≡ ( ν obs /ν em ) , where ν obs and ν em are the observed and emitted frequencies. This is the redshift factorthat modulates the observed intensity. At high inclination, the beaming effect (which makes the redshift factor brighteron the approaching side of the torus) is more pronounced and the apparent area of the secondary image is bigger.same figure show the power density spectra of all defor-mations. The spectra are presented in two distinct panelsfor visibility as the power associated with different defor-mations or the same deformation at different inclinationsvaries significantly.The biggest modulations (or biggest power densities)are obtained for deformations that lead to the biggestchange of the torus apparent area. These deformations de-pend on inclination, but always include expansion and oneor more shears.The variation of relativistic effects linked to the chang-ing torus location is less important than the change of ap-parent area. This is clearly demonstrated by noting that thesmallest modulation is always obtained for vertical transla-tion, which indeed leads to the smallest change of apparentarea.Whatever the inclination, expansion leads to a highpower density, while translations and rotation give lowpower densities. On the other hand, shears exhibit a veryinclination-dependent power density: radial and pure shearsgive high power for low inclination, while vertical shearsgive high power for high inclination. All these observablecharacteristics are directly linked to the change in apparentarea of the deformed torus at various inclinations.The time average of the light curves differs from onedeformation to the other. It is dictated by the average valueof the apparent area.The dominating frequency of the light curves dependsstrongly on the deformation. Power is balanced betweenone time and two times the Keplerian frequency ν K .Translations and expansion give a single-peak spectrumcentered on ν K . Rotation also gives a single-peak spectrum,but centered on 2 ν K , as all possible values of apparent ar-eas are covered after only half a Keplerian period in thiscase only. Shears give rise to a double-peak structure in thepower spectrum, with a different balance of power between ν K and 2 ν K for radial, vertical, and pure shear. Figure 2 shows that the power spectral density is an inter-esting probe of the deformation of a slender accretion torussurrounding a black hole. Different kinds of deformationslead to significantly different power spectra.It is interesting to note that the power density associ-ated with expansion and the dominant shears is always oneto two orders of magnitude higher than the power densityassociated with translations and rotation. This is due tothe highly different evolution of the torus apparent area forthese kinds of deformations.The single- or double-peak nature of the power spec-trum is also an interesting probe of the underlying kind ofdeformation. However, the very inclination-dependent as-pect of the shears power spectra, as opposed to the otherkind of deformations, presents a difficulty.Taking these two probes into account (maximum power,dominating frequency) it is possible to constrain the kind ofdeformation by studying the power spectrum. Knowing themaximum value of power density, it is possible to determinewhether the underlying deformation is within { expansion,shear } or { translation, rotation } . Then, as a function of thevalue of the dominating frequency, it is possible to deter-mine the kind of deformation: expansion, shear, translation,or rotation. But it is not straightforward to determine whatkind of translation or what kind of shear unless the incli-nation parameter is already constrained.However, this work is only a first step towards a moresophisticated treatment of the observable characteristics ofnon-equilibrium accretion slender tori. Therefore, it is notthe aim of this article to propose a way of constrainingthe motion of the accretion torus from the power densityspectrum properties.
5. Conclusions and perspectives
We compute light curves and power density spectraof slender optically thick accretion tori surrounding aSchwarzschild black hole, which are subject to simple de-formations (translations, rotation, expansion, and shears).
Fig. 2. a) Normalized light curves for torus deformations. b) Normalized power spectral densities. Set of deformationswith large maximal normalized values, ˜ P max (cid:38) .
5. b) Normalized power spectral densities. Complementary set ofdeformations with small maximal normalized values, ˜ P max (cid:46) .
25. We note that the power spectra are represented astriangular functions for readability only, although they are actually made of a succession of infinitely thin peaks.We show that the power spectrum can be used to constrainthe underlying kind of torus deformation.This is a first step towards the full study of the ob-servable characteristics of oscillating slender accretion tori.Our conviction is that combining simple deformations ofslender tori will allow realistic oscillations to be mimicked.From this perspective, our present result that it is pos-sible to differentiate various kinds of simple deformationsfrom the observed power spectra is promising. Future workwill determine whether this still applies when combinations of simple deformations are considered and whether thiscould lead to a practical tool for studying power spectraof high-frequency QPOs to constrain the underlying physi-cal model. We believe that such tests can finally bring im-portant outputs in the field of verifying the predictions ofgeneral relativity in the exploration of observational dataaccumulated so far. We also believe that outputs of sophis-ticated ray-tracing simulations could be key components ofthe proper interpretation of data received from planned fu- ture X-ray missions such as Large Observatory For X-rayTiming (Feroci et al. 2012).
Acknowledgements.
We acknowledge support from the Polish NCNUMO-2011/01/B/ST9/05439 grant, the Swedish VR grant, and theCzech grant GAˇCR 209/12/P740. We further acknowledge the projectCZ.1.07/2.3.00/20.0071 ‘Synergy’ supporting the international col-laboration of IF Opava. Computing was partly done using theDivision Informatique de l’Observatoire (DIO) HPC facilities fromObservatoire de Paris ( http://dio.obspm.fr/Calcul/ ). References