Quasi-periodic oscillatory motion of particles orbiting a distorted deformed compact object
QQuasi-periodic oscillatory motion of particles orbiting a distorted deformed compact object
Shokoufe Faraji ∗ and Audrey Trova † University of Bremen, Center of Applied Space Technology and Microgravity (ZARM), 28359 Germany (Dated: February 14, 2021)This work explores the dynamic properties of test particles surrounding a distorted, deformed compact object.The astrophysical motivation was to choose such background, which could constitute a more reasonable modelof a real situation that arises in the vicinity of compact objects with the possibility of having parameters asthe extra physical degrees of freedom. This can facilitate associating observational data with astrophysicalsystems. This work’s main goal is to study the dynamic regime of motion and quasi-periodic oscillation in thisbackground, depending on di ff erent parameters of the system. Also, we exercise the resonant phenomena of theradial and vertical oscillations at their observed quasi-periodic oscillations frequency ratio of 3 : 2. I. INTRODUCTION
Quasi-periodic oscillations (QPOs) of X-ray power spec-tral density have been observed at low (Hz) and high (kHz)frequencies in some observations, and they were discoveredin the eighties [1]. They were also detected in several BlackHole candidates [2]. Quasi-period oscillations have also beenobserved in supermassive black hole light curves [3–8]. Also,recently the source went through Burst Alert Telescope (BAT)onboard Swift [9, 10]. QPOs, peak features in the X-ray ob-served from stellar-mass BHs and neutron stars, are likely toarise from quite near the compact object itself and exhibit fre-quencies that scale inversely with the black hole mass, andallow us to probe and study the nature of accretion in highlycurved space-time [11–19].One of the first QPO models is the Relativistic PrecessionModel (RPM), which identifies the twin-peak QPO frequen-cies with two frequencies, namely the Keplerian and the peri-astron frequencies. In the past years, the RPM is served to ex-plain the twin-peak QPOs in several LMXBs [20–23]. How-ever, this model has some di ffi culties explaining relativelylarge observed high frequencies QPO amplitudes and inferredexistence of preferred orbits. To modify this model, the con-cept of orbital resonance models was proposed [24, 25]. Re-garding this modification, the high frequencies QPOs (two-picks) are considered as the resonances between oscillationmodes of the accreted fluid - the well-known ratio 3 : 2epicyclic resonance model - identify the resonant frequencieswith frequencies of radial and vertical epicyclic axisymmet-ric modes of disc oscillations. The correlation is a cost ofresonant corrections to these frequencies [26, 27]. While thesecret of this 3 : 2 ratio has still not been clearly revealed.Nevertheless, the oscillations occur only in certain states ofluminosity, and hardness [2, 28, 29], and this phenomenon isnot universal [28]. For a review of quasi-periodic oscillationsobservation and theory, see for example [30, 31].In this paper, we are motivated by the partial success ofthe above models. By referring to these models, we assumethe QPOs are caused by the fundamental epicyclic frequenciesassociated with the orbital motion of the matter in the accre- ∗ [email protected] † [email protected] tion disk also their combinations. In fact, with the other kindof resonance with ratios given by small integral numbers like1 : 1 , , one can also explain observed QPOs frequencieswith the same 3 : 2 ratio [22, 24, 32].On the one hand, the properties of these fundamental fre-quencies, as mentioned above, have been extensively studiedin the case of particles motion in the di ff erent backgrounds[33–43] among many others. On the other hand, still there areopen questions in this area.We present our work in the background of a distorted, de-formed compact object which is static and axisymmetric. Thisbackground is the simplest generalization of the so-called q-metric up to quadrupole moments [44]. This metric has twoparameters, aside from the central object’s mass, namely dis-tortion parameter and deformation parameter, which are notindependent of each other. Besides, in the absence of oneor other, one can recover either the q-metric or distortedSchwarzschild metric [45]. From a dynamical point of view,these parameters can be seen as perturbation parameters of theSchwarzschild spacetime. We explain this metric briefly inSection II. In this respect, the first static and axially symmet-ric solution with arbitrary quadrupole moment are describedby [46]. Then [47] introduced a static solution with arbitraryquadrupole in prolate spheroidal coordinates. Later, an equiv-alent form of this metric was found by Zipoy and Voorhees[48, 49] known as γ -metric or σ -metric, and later on, by rep-resenting this metric in terms of a new parameter q, is knownas q-metric [50]. This area of study has been discussed ex-tensively in the literature and generalized in many respects[51–56], among many others.There are several motives to study the circular motion ofparticles in this background. In the relativistic astrophysicalstudy, it is assumed that astrophysical compact objects aredescribed by the Schwarzschild or Kerr space-times. How-ever, besides these setups, others can imitate a black hole’sproperties, such as the electromagnetic signature [57]. It isalso possible that some astrophysical observations may notbe fitted within the general theory of relativity by using theSchwarzschild or Kerr metric [58, 59], like as the mentionedratio of QPOs. Also, the astrophysical systems are not al-ways isolated as they surrounded by di ff erent kinds of matterand radiation. In addition, while in the more realistic sce-nario, the rotation should take into account; however, it hasbeen shown the possibility of observed resonant oscillationsdirectly when they occur in the inner parts of accretion flow, a r X i v : . [ a s t r o - ph . H E ] F e b even if the source of radiation is steady and perfectly axisym-metric [21]. Another motivation to choose such background isto constitute a reasonable model of a real situation that arisesin the vicinity of this compact object, where it is not alwaysisolated, with the possibility of analytic analysis through ex-ercising parameters of the model where can be treated as thedegrees of freedom of the system.In this paper, we also explore the dynamics of test particlesaround a distorted, deformed compact object. This discussioncan approximate a diluted astrophysical plasma’s complex dy-namics, where they can be located around the system.The paper’s organization is as follows: Section II presentsthe background object and a study of the motion of test parti-cles in this background. While Section III explains epicyclicfrequencies and stable circular geodesics. The parametric res-onances present in Section IV. Finally, the conclusions aresummarized in Section V.Throughout this work, we use the signature ( − , + , + , + ) andgeometrized unit system G = = c , otherwise specified there.Latin indices run from 1 to 3, while Greek ones take valuesfrom 0 to 3. II. SPACE-TIME OF DISTORTED DEFORMED COMPACTOBJECT
There is a space-time which is the simplest asymptoticallyflat solution of Einstein equation with quadrupole moment. Inthis respect, the first static and axially symmetric solution witharbitrary quadrupole moment were described in [46]. Then[47] introduced a static solution with arbitrary quadrupole inprolate spheroidal coordinates. Later, Zipoy and Voorhees[48, 49] found an equivalent transformation that leads to asimple solution which can be treated analytically and knownas γ -metric or σ -metric, and later on, with representing it interms of a new parameter is known as q-metric [50].Now in this work, we consider generalized q-metric, whichhas q-metric as the seed metric and considers the existence ofa static and axially symmetric external distribution of matterin its vicinity. By its construction, this metric is only validlocally [45, 60]. In [61] the metric and its circular geodesicsare studied. The metric has this formd s = − (cid:32) x − x + (cid:33) (1 + α ) e ψ dt + M ( x − e − ψ (cid:32) x + x − (cid:33) (1 + α ) (cid:32) x − x − y (cid:33) α (2 + α ) e γ (cid:32) dx x − + dy − y (cid:33) + (1 − y ) d φ (cid:35) , (1)where t ∈ ( −∞ , + ∞ ), x ∈ (1 , + ∞ ), y ∈ [ − , φ ∈ [0 , π ).Where M is a parameter that can be identified as the mass ofthe body generating the field, which is expressed in the dimen-sion of length. The function ˆ ψ plays the role of gravitationalpotential, and the function ˆ γ is obtained by an integration ofthe explicit form of the function ˆ ψ . These are given by ˆ ψ = − β (cid:104) − x y + x + y − (cid:105) , (2)ˆ γ = − x β (1 − y ) + β x − − y )( − x y + x + y − . (3)This metric contains three free parameters, namely the totalmass, quadrupole moments α deformation parameter, and β distortion parameter, which are taken to be relatively smalland connected to the q-metric and the presence of externalmass distribution, respectively. In the case of β =
0, thisturns to the mentioned q-metric, and in the case of α = β = ff ects to these parameters as new physical de-grees of freedom. Also, as we have some freedom in definingthese momentum variables, we look for ones that minimizecomputational time and numerical errors. Also, the circulargeodesics in this background studied in [44].The relation between the prolate spheroidal coordinates( t , x , y , φ ), and the Schwarzschild coordinates ( t , r , θ, φ ) isgiven by x = rM − , y = cos θ. (4)In addition, the related kinematic quantities in this work,specific energy, angular momentum and Keplerian orbital fre-quency, respectively, read as E = e − ˆ ψ (cid:115)(cid:32) x − x + (cid:33) + α x − Sx − S , (5) L = ± e − ˆ ψ (1 + x ) (cid:115)(cid:32) x + x − (cid:33) α Sx − S , (6) Ω = e ψ (cid:32) x − x + (cid:33) α (cid:115) x − x + Sx − S . (7)Where S : = + α + β x − β x . (8) A. Dynamic of particle in the equatorial plane
The e ff ective potential in the equatorial plane is given bythis relation [44] V E ff = (cid:32) x − x + (cid:33) ( α + e ψ (cid:15) + L e ψ M ( x + (cid:32) x − x + (cid:33) α . (9) FIG. 1. The domain of existence of the circular orbits in the equato-rial plane in the ( x , α )-plane. The lightlike orbits are located on theblack curve, 2 S − x =
0. The timelike orbits’ positions are form-ing the blue area, which is bound by the locus of the null geodesics2 S − x = S = x , β ) is plane. The lightlike orbits are located on the blackcurve, 2 S − x =
0. The timelike orbits’ positions are forming the bluearea, which is bound by the locus of the null geodesics 2 S − x = S = Regarding, the domain of existence of the circular orbits inthe equatorial is plotted in Figures 1 and 2. In Figure 1 thisdomain is plotted in the ( x , α )-plan in terms of the distortedparameter β , while in Figure 2 this region is plotted in the( x , β )-plane for a range of values of the deformation parameter α . The particle motion is limited by the energetic boundariesgiven by E : = E = V e ff . By analysing the e ff ective potential,we can have general properties of the dynamics of a particle inthis background. In fact, possible types of orbits, in general,dependent on the parameters of (cid:15) , E , L , α and β . Analytically,dependent on the number of positive real zeros and the sign of E − (cid:15) , one obtains di ff erent types of trajectories. Here we onlydiscussed bounded timelikes trajectory as we are interestedin studying oscillation of particle for a small perturbation of the orbit. In general, test particles’ motion can be chaotic inthis background for some combinations of parameters α and β . Here we focus on bounded orbits. Due to the behaviour ofthe e ff ective potential V e ff , one can distinguish four di ff erenttypes of the energetic boundary related to having or lackingof initial conditions on both boundaries. Figure 3 correspondsto the existence of both initial conditions on inner and outerboundaries, where particles trapped in some region forminga toroidal shape around the central object. In this Figure thetrajectories for some choices of the parameters are plotted. III. EPICYCLIC FREQUENCIES AND STABILITY OFCIRCULAR GEODESICS
In fact, in the accretion disk processes, a variety of os-cillatory motions are expected. Indeed, circular and quasi-circular orbits seem to be crucial from the point of view ofaccretion processes. In the study of the relativistic accretiondisk, three frequencies are relevant. The Keplerian orbital fre-quency ν K = Ω π , radial frequency ν x = ω x π and the vertical fre-quency ν y = ω y π . A resonance between these frequencies canbe a source of quasi-periodic oscillations that leads to chaoticand quasi-periodic variability in X-ray fluxes observations inmany galactic objects.The Relativistic Precession Model (RPM) is one of themodels of study QPO as mentioned earlier. In this model itassumes this is caused by the epicyclic frequencies associ-ated with the quasi-Keplerian motion in the accretion disks.In RPM the upper frequency is defined as the Keplerian fre-quency ν U = ν K and the lower frequency is defined as theperiastron frequency i.e. ν p : = ν L = ± ( ν K − ν x ). Their correla-tions are obtained by varying the radius of the associated cir-cular orbit in a reasonable range. Within this framework, it isusually assumed that the variable component of the observedX-ray signal places in a bright localized spot or blob orbit-ing the compact object on a slightly eccentric orbit. There-fore because of the relativistic e ff ects, the observed radiationis supposed to be periodically modulated.In this section, we explain the stability of circular motion.In this spacetime circular motion in equatorial plane, and therelation between parameters extensively studied in [44]. Theequation of motion for a test particle is the geodesic equation d x µ ds + Γ µνρ dx ν ds dx ρ ds = . (10)To adapt this equation for this mentioned background, weneed to replace all necessary Christo ff el symbols [44], alsosubstitute ( x = x , y =
0) as we are in the equatorial plane.To describe the more general class of orbits slightly deviatedfrom the circular geodesics in the equatorial plane x µ , we canuse the di ff eomorphism x (cid:48) µ = x µ + ξ µ , and write down geodesicequation (10) for this perturbation. By considering terms upto linear order in ξ µ we obtain [62], d ξ µ dt + γ µη d ξ η dt + ξ η ∂ η U µ = , (11) FIG. 3. Timelike geodesic for di ff erent pairs of ( α, β ). The trajectories in the ( r , φ ) section and in the complete 3D are plotted. In the firstcolumn, both configurations E = .
90 and L =
25. In the second column, both configurations have E = .
94 and L =
22. In the centralcolumn, both configurations have E = .
93 and L =
12. In the fourth column, both configurations have E = .
93 and L =
15. In the last one,both configurations have E = .
96 and L = where γ µη = (cid:104) Γ µηδ u δ ( u ) − (cid:105) y = , (12) U µ = (cid:104) γ µη u η ( u ) − (cid:105) y = . (13)Here the 4-velocity for the circular orbits in the equatorialplane is taken as u µ = u (1 , , , Ω ). Then integration of theequation (11) for the t and φ components leads to d ξ η dt + γ ην ξ ν = , (14) d ξ x dt + ω x ξ x = , (15) d ξ y dt + ω x ξ y = , (16) where in the first equation η can be taken t , or φ ; and ω x = ∂ x U x − γ x η γ η x , (17) ω y = ∂ y U y . (18)This equation system describes the free radial phase and verti-cal oscillations of a particle around the circular geodesics. Foran alternative definition of the epicyclic harmonic motion, see[63]. The sign of frequencies ω x and ω y determine the dy-namic, so that we have either a stable circular orbits, or even atiny perturbation can make a strong deviation from the unper-turbed path.In the Schwarzschild black ground, these frequencies inspheroidal coordinates are given by FIG. 4. Stability of the timelike circular orbits in the ( x , α ). Timelikecircular orbits exist in the light blue area. This region is bounded bythe thick dark line (2 S − x = w x =
0. Thearea depicted by the blue-grey region shows the domain of stability.In the chosen range, w y > ω y = x + , (19) ω x = x + (cid:32) − x + (cid:33) . (20)The stability of the circular orbits is determined by the radialepicyclic frequency because the vertical frequency coincideswith the orbital frequency ω y = Ω . As it is seen from the equa-tions (19) the vertical epicyclic frequency is a monotonicallydecreasing function of x , and we have ω x < ω y = Ω , also FIG. 5. Stability of the timelike circular orbits in the ( x , β )-plane.Timelike circular orbit exists in the light blue area. This region isbounded by the thick dark line (2 S − x =
0) and the dashed line ( S = w y = w x =
0. Theblue-grey region shows the stability domain with respect to chosenparameters. there exists a periapsis shift for bounded quasi-elliptic trajec-tory implying the e ff ect of relativistic precession that changingthe radius of the orbit [64]. Indeed, this ordering between thefrequencies contributes to the possible resonances that mayhave in a given background. The behaviour of the frequencieshelps us to distinguish possible trajectories around a stablecircular orbit.These epicyclic frequencies in the background of a dis-torted, deformed compact object are written as, w x = Ω e − γ (1 − / x ) − α (2 + α ) ( x − (cid:34) S − x )( S − x ) − ( x − S (1 + α + β x ) (cid:35) , (21) w y = Ω e − γ (1 − / x ) − α (2 + α ) S (1 + α + β x ) , (22)where S is given by the relation (8). Note that these frequen-cies are measured concerning the proper time of a comovingobserver. The signs of these fundamental frequencies providea natural condition of having the valid domain of existence ofcircular and quasi-circular orbits. We explore these frequen-cies and the valid region more perspicaciously on Figures.In Figure 4 and 5, the region of the stability of the timelikecircular geodesics is plotted. This region are explored in Fig-ure 4 in the ( x , α )-plane for di ff erent values of the distortionparameter β and in Figure 5 in the ( x , β )-plane for di ff erentvalues of the deformation parameter α . In Figure 4 as the signof radial frequency suggests outside of the red curve and inthe blue region, the timelike geodesics are unstable for radialperturbations and only are stable for the vertical perturbations.In the Figure 5, both the curves w y = w x = ff ect of parameter α is more profound ratherthan β . In fact, increasing β tends to shrink the range of α forallowing to have stability.The interesting situation in this background, contrary to theSchwarzschild case, is the possibility of various ordering situ-ations that arises among frequencies. For analysing the orderof magnitude of these frequencies, we start with, w x w y = S (2 S − x )( S − x )( x − + α + β x ) − β and α are not in-dependent of each other; namely, fixing one of them restrictsanother one’s domain. For example, for a small enough pa-rameter α , we have this range of orders for parameter β • α < −O (10 − ) < β < O (10 − ) − O (10 − ) • α > −O (10 − ) < β < O (10 − ) − O (10 − )Using this analysis, it turns out that for a given α and β >
0, wealways have w x < w y . However, for β < S (2 S − x )( S − x ) and regarding its sign we havethree di ff erent orderings. These various cases are investigatedin Figures 6 and 7, where we compare the three frequenciesby plotting the curves of ω x , ω y and Ω .In Figure 6 the di ff erent epicyclic frequencies in the ( x , α )-plane for di ff erent values and signs of the distortion parameter β are plotted. Also, the w y = Ω , w x = Ω and w x = w y curves are specified. Moreover, on the left panel, the darkarea means w x > Ω and on the right panel w x > w y . We illustrated both in one panel to be easy to compare and analyzethe behaviour of di ff erent regions. In Figure 7 the di ff erentepicyclic frequencies are also presented but in the ( x , β )-planefor di ff erent values of the deformation parameter α .By considering the Figures 6 and 7, we can extract interest-ing information about the order of the epicyclic frequencies inthis background, which strongly influenced by the parametersand their signs and both α and β parameters have a crucial rolein the order of the frequencies, as we see the lines are inter-secting each other multiple times in both planes of discussion.According to these figures, one can discuss di ff erent pos-sibilities. In Figure 6 in the ( x , α )-plane, if we are interestedin the order of the frequencies above the red line where w x and w y are both positives, we see that the ordering is goingto be di ff erent after intersections. One appears when the redcurve crossing the orange curve and the other one when thethree curves yellow, pink and orange cross each other, mean-ing where Ω = w y = w x .Thus, a first possibility appears: Above the red curve,where there is no intersection point, only we have w y > Ω .On the contrary, if a crossing point between the red and theorange appears, both orderings w y > Ω and w y < Ω arepossible. Following that, a second behaviour appears when w y > Ω and w y < Ω are possible: appearing the secondcrossing point makes the order among the three frequencieseven more varies. In the following, we give the order for dif-ferent cases concerning di ff erent situations. In the last row ofFigure 6, we can extract three di ff erent regions: • from the red curve to the pink curve, the order is thefollowing w x < Ω < w y . • from the pink line to the yellow line, Ω < w x < w y . • from the yellow line to the top of the box, Ω < w y < w x .For β ≤
0, as it has been seen also from Figures Several dif-ferent regions can appear: • above the red curve and below the orange, three regionsare present:- below the pink and the yellow, we get the follow-ing order w x < w y < Ω .- above the yellow and below the pink (small area)we get w y < w x < Ω .- above the yellow and the pink, w y < Ω < w x . • also, above the red curve and above, the orange one,three regions appear:- below the pink and the yellow, we get the follow-ing order w x < Ω < w y .- above the pink and below the yellow (small area)we get Ω < w x < w y .- above the yellow and the pink, Ω < w y < w x .One can produce the same analysis by using Figure 7. Wecan note the presence of the two crossing points in the( x , β ) − plane. The analysis of ordering among the di ff erentfrequencies will be the same as the previous one. IV. PARAMETRIC RESONANCES
Before the twin peak, HF QPOs have been discovered inmicroquasars; this existence and ratio 3 : 2 have pointed outin [65]. Also, in [66] authors suggested that the ratios shouldbe rational due to the resonances in quasi-Keplerian accretiondisks. Apparently, this fact is well supported by observations.Also, this 3 : 2 ratio as the ratio of ( ν U : ν L ) is seen most oftenin the twin HF QPOs in the LMXB containing Microquasars.In this section we study this phenomena by means of para-metric resonance [67], and we identify the upper and lowerfrequencies ( ν U , ν L ) with ( ν x , ν y ) or ( ν y , ν x ). In this respect weexplore this fact by standard procedure through the Mathieu’sequation. This equation is a linear second-order ODE, whichdi ff ers from the one corresponding to a harmonic oscillator inthe existence of a periodic and sinusoidal forcing of the sti ff -ness coe ffi cient as f ( t ) = f + f cos( ω x t ). The equation thenis given by d ξ y dt + ω y (cid:104) + ω y h cos( ω x t ) (cid:105) ξ y = h = f f (cid:29) h , and frequency ν x = ω x π . Thenatural, unexcited period is ν y = ω y π .It is well known that this set up performs free oscillationaround the stable equilibrium case. While, if the sti ff ness termcontains the parametric excitation, i.e. f (cid:44)
0, the motioncan stay bounded, which is referred to as stable or the motionbecomes unbounded, where this case is referred to as unstable(see for example [68, 69]).The resonance excitation arises for special values of fre-quencies. In contrast to the standard resonance epicyclicmodel, the oscillating test particles in this background allowboth frequency ratios ν y : ν x = ν y : ν x = ff erent combinationsof parameters α and β , also it is possible to have other ratioswhich can be relevant in other observed data like in other twinfrequencies observed in the Microquasar GRS 1915 +
105 (seefor example [70]). In Figure 8 and 9, the resonance ratio w x : w y = w x : w y = β , in the possible rangeof α , the 2 : 3 resonance ration is not possible.In addition, we can identify the frequencies upper and lowerfrequencies ( ν U , ν L ) with di ff erent combinations of ν x and ν y which can also reproduce the ratio of 3 : 2, as mentioned ear-lier. In Figure 10, the epicyclic frequencies ratio at the maxi-mum of ω y is depicted. We see that the curves take their min- imum at di ff erent radius depending on the choice of β wherethey are plotted with respect to α . We see that in all cases, al-most the positive and negative values of β take their maximumat the same radios; however, a further analysis reveal that thisradio is smaller for negative values of this parameter. Also, themaximums in all parameters depend on the ratio; for example,we see that as this ratio becomes larger, the maximum hap-pens in smaller radios. This means that the resonance is notmonotonic after some distance from the central object, whichdepends on the combination of parameters in this background.Of course, considering rotation can modify the radial pro-files of the vertical and radial frequencies, which is the subjectof future works. V. SUMMARY AND CONCLUSION
In this work, we studied the dynamics of particles and thequasi-periodic oscillation by studying fundamental frequen-cies of the circular motion around a deformed compact ob-ject up to the quadrupole. This background has q-metricas the seed metric while considering a distribution of mat-ter in its vicinity. This metric is static and axisymmetricup to quadrupoles also in the external surrounding matterand contains two parameters: distortion parameter β and de-formation parameter α . This background was briefly ex-plained in Section II. These two parameters’ dependency re-flects into motion and epicyclic frequencies of particles thatcost strong deviation from the correspondence quantities inthe Schwarzschild case. In this respect, one can exploredi ff erent orderings among fundamental frequencies and var-ious possibilities to reproduce the ratio of 3 : 2 via di ff er-ent combinations of parameters which is not the case in eitherSchwarzschild or in q-metric.Interestingly, also it is possible from observational data toassign some restrictions on the parameters in this metric. Inthis regard, also the construction of a thin accretion disk canlimit having some combinations of these parameters and mod-ify the metric in this sense [71].A further step of this work can be considering rotation thatleads to modifying the radial profiles of the vertical and radialfrequencies, which is the subject of our future work. Also, themagnetic field can serve as a fundamental input in this sys-tem to model more real astrophysical systems. One can alsoextend this work from a single particle to a complex systemlike accretion disks. It is also of some interest to apply thesemodels as the initial conditions in the numerical simulationsand test their ability to account for observable constraints ofastrophysical systems. ACKNOWLEDGEMENTS
The authors thank the research training group GRK 1620,”Models of Gravity”, funded by the German Research Founda-tion (DFG).
FIG. 6. The di ff erent epicyclic frequencies in the ( x , α )-plane are examined. On all the plots, the yellow line depicts w x = w y , the pink lineshows w x = Ω , and the orange line is w y = Ω . Moreover, the light blue area corresponds, as in the Figure 4 and 5, to the area where circularorbits exist (bounded by the thick black line 2 S − x = w x > Ω . In the second row, the darker blue area satisfies w x > w y . Byconsidering the first and second rows together, one can have di ff erent ordering for these frequencies.FIG. 7. The di ff erent epicyclic frequencies in the ( x , β )-plane are examined. As in the Figures 4 and 5, the red line shows w x = w x and w y are positive. On all the plots, the yellow line depicts w x = w y , the pink line shows w x = Ω , and the orange lineis w y = Ω . Furthermore, the light blue area corresponds as in the Figure 4 and 5 to the area where circular orbits exist (bounded by the thickblack line 2 S − x = S = w x > Ω . In the second row, the darker blue area satisfies w x > w y . Byconsidering the first and second rows together, one can have di ff erent ordering for these frequencies.[1] M. van der Klis. Millisecond Oscillations in X-ray Binaries. Annu. Rev. Astron. Astrophys. , 38:717–760, January 2000.[2] Je ff rey E. McClintock and Ronald A. Remillard. Black holebinaries , volume 39, pages 157–213. Oxford, 2006.[3] Marek Gierli´nski, Matthew Middleton, Martin Ward, and ChrisDone. A periodicity of ˜1hour in X-ray emission from the activegalaxy RE J1034 + Nature (London) , 455(7211):369–371, September 2008.[4] Dacheng Lin, Jimmy A. Irwin, Olivier Godet, Natalie A. Webb,and Didier Barret. A ˜3.8 hr Periodicity from an Ultrasoft Ac-tive Galactic Nucleus Candidate.
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FIG. 8. The light blue area depicts the region of existence of thecircular orbits. This region is bounded by the thick black curve (2 S − x =
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Astrophys. J. , 585(2):665–676, March2003.[8] Krista Lynne Smith, Richard F. Mushotzky, Patricia T. Boyd,and Robert V. Wagoner. Evidence for an Optical Low- FIG. 9. The light blue area depicts the region of existence of thecircular orbits. This region is bounded by the thick black curve (2 S − x =
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