A geometric origin for quasi-periodic oscillations in black hole X-ray binaries
DDraft version October 27, 2020
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A Geometric Origin for Quasi-periodic Oscillations in Black Hole X-ray Binaries
Prerna Rana and A. Mangalam Indian Institute of Astrophysics, Sarjapur Road, 2nd Block Koramangala, Bangalore-560034 (Received ; Revised ; Accepted )
Submitted to ApJABSTRACTWe expand the relativistic precession model to include nonequatorial and eccentrictrajectories and apply it to quasi-periodic oscillations (QPOs) in black hole X-ray bi-naries (BHXRBs) and associate their frequencies with the fundamental frequencies ofthe general case of nonequatorial (with Carter’s constant, Q (cid:54) = 0 ) and eccentric ( e (cid:54) = 0 )particle trajectories, around a Kerr black hole. We study cases with either two or threesimultaneous QPOs and extract the parameters { e , r p , a , Q }, where r p is the perias-tron distance of the orbit, and a is the spin of the black hole. We find that the orbitswith [ Q = 0 − should have e (cid:46) . and r p ∼ − for the observed range of QPOfrequencies, where a ∈ [0 , , and that the spherical trajectories { e = 0 , Q (cid:54) = 0 } with Q = 2 − should have r s ∼ − . We find nonequatorial eccentric solutions for bothM82 X-1 and GROJ 1655-40. We see that these trajectories, when taken together, spana torus region and give rise to a strong QPO signal. For two simultaneous QPO cases,we found equatorial eccentric orbit solutions for XTEJ 1550-564, 4U 1630-47, and GRS1915+105, and spherical orbit solutions for BHXRBs M82 X-1 and XTEJ 1550-564. Wealso show that the eccentric orbit solution fits the Psaltis-Belloni-Klis correlation ob-served in BHXRB GROJ 1655-40. Our analysis of the fluid flow in the relativistic diskedge suggests that instabilities cause QPOs to originate in the torus region. We alsopresent some useful formulae for trajectories and frequencies of spherical and equatorialeccentric orbits. Keywords:
Astrophysical black holes; Stellar mass black holes; Kerr black holes; Accre-tion; General relativity; X-ray binary stars; Geodesics INTRODUCTIONBlack hole X-ray binaries (BHXRBs) are systems with a primary black hole gravitationally boundto a nondegenerate companion star. These systems display transient behavior exhibiting high X-ray
Corresponding author: A. [email protected]@iiap.res.in a r X i v : . [ a s t r o - ph . H E ] O c t Rana and Mangalam 2020 luminosities ( L X ∼ erg s − ) during the outburst state, lasting from a few days to many months,followed by a long quiescent state ( L X ∼ erg s − ) (Remillard et al. 2006). The triggering of theseX-ray outbursts has been modeled as an instability arising in the accretion disk when the accretionrate is not adequate for the continuous flow of matter to the black hole, and when a critical surfacedensity is reached (Dubus et al. 2001). However, the disk instability model has not been able toexplain the outbursts of much shorter or longer time-scales; for example, BHXRB GRS 1915+105has shown a high X-ray luminosity state for more than 10 yr (Fender & Belloni 2004). During theoutburst phase, the X-ray intensity shows rapid variations with timescales ranging from millisecondsto a few seconds, which are most likely to arise in the proximity to the black hole ( r ∼ r I , where r I represents the innermost stable spherical orbit (ISSO)). The power density spectrum (PDS) of theX-ray intensity, which is commonly used to probe this fast variability, exhibits distinct features calledquasi-periodic oscillations (QPOs) during the outburst period with their peak frequency, ν , rangingfrom 0.01 to 450 Hz (Remillard et al. 2006; Belloni & Stella 2014). QPOs can be distinguishedfrom other broad features of the PDS by their high-quality factor ν / FWHM (cid:38) . Hence, the studyof properties and origin of QPOs in BHXRBs is crucial to understanding the properties of inneraccretion flow close to the black hole, where general relativistic effects are ascendant.QPOs in BHXRBs are categorized as low-frequency QPOs (LFQPOs) with ν < Hz, which areagain classified as type A, B, or C based on their various properties, and high-frequency QPOs(HFQPOs) with ν > Hz (Motta 2016). These different types of QPOs are also known to showa remarkable association with various spectral states during the outburst phase (Fender et al. 2004;Remillard et al. 2006; Fender & Belloni 2012; Motta 2016). The launch of the Rossi X-ray TimingExplorer (RXTE) in 1995 with its high sensitivity significantly increased the detection of BHXRBs,and made it possible to detect HFQPOs in their PDS in the late 1990s (Belloni & Stella 2014),for example, the detection of 300 Hz and 450 Hz QPOs in GROJ 1655-40 (Remillard et al. 1999b;Strohmayer 2001a); QPOs in the range 102-284 Hz, at 188 Hz, 249-276 Hz and near 183 Hz, 283Hz in XTEJ 1550-564 (Homan et al. 2001; Miller et al. 2001; Remillard et al. 2002); 67 Hz, 40 Hz,and 170 Hz in GRS 1915+105 (Morgan et al. 1997; Strohmayer 2001b; Belloni et al. 2006); 250Hz in XTEJ 1650-500 (Homan et al. 2003); 240 Hz and 160 Hz in H1743-322 (Homan et al. 2005;Remillard et al. 2006); and more. Some of these HFQPOs have been detected simultaneously alongwith their peak frequencies showing nearly 3:2 or 5:3 ratios, indicating a resonance phenomenon(Remillard et al. 2006; Belloni & Stella 2014). There is also an interesting case of BHXRB GROJ1655-40 which showed three QPOs simultaneously − two HFQPOs and one type C LFQPO (Mottaet al. 2014a). Understanding of the origin of HFQPOs and their simultaneity has been the primefocus of the observational studies as well as the theoretical models.The study of general relativistic effects is important for a theoretical understanding of the origin ofQPOs and their connection with various spectral states during the X-ray outburst, as these signalsappear to emanate very close to the black hole. Several existing models, based on the instabilitiesin the accretion disk and other geometrical effects, which attempt to explain the origin of LFQPOsand HFQPOs. Most of these models assume that the disk inhomogeneities orbiting in the innermostregions of the accretion disk are the cause of high variability in the X-ray flux, resulting in QPOs inthe PDS. A widely accepted model among them is the relativistic precession model (RPM) (Stella &Vietri 1999; Stella et al. 1999), which ascribes two simultaneous HFQPOs to the azimuthal, ν φ , andperiastron precession frequencies, ( ν φ − ν r ) , and a third simultaneous type C LFQPO to the nodal Geometric origin for QPOs ( ν φ − ν θ ) , of a self-emitting blob of matter in the accretion disk. The RPMhas been applied to the cases of BHXRBs GROJ 1655-40 (Motta et al. 2014a) and XTEJ 1550-564(Motta et al. 2014b) to estimate the spin parameter and mass of the black hole, where they assumedthe precession frequencies of nearly circular particle trajectories in the accretion disk around a Kerrblack hole. Recently, in contrast with the localized assumption of the RPM, the most frequentlydetected type C QPOs in BHXRBs have been modeled as the Lense − Thirring frequency of a radiallyextended thick torus precessing as a rigid body (Ingram et al. 2009; Ingram & Done 2011, 2012).This model describes the increase in type C QPO frequency with the hard to soft spectral transitionduring outburst as coincident with the decrease in outer radius of the torus and also shows that themaximum type C QPO frequency should be close to 10 −
30 Hz (Motta et al. 2018). Other modelswhich concentrate on the 3:2 or 5:3 resonance phenomena of simultaneous HFQPOs under the regimeof particle approach; for instance, the nonlinear resonance models (Kato 2004, 2008; Török et al. 2005,2011) which explain the phenomenon of simultaneous HFQPOs as an excitation due to the nonlinearresonant coupling between the oscillations within the accretion disk. One such nonlinear resonancephenomenon is the parametric resonance between radial, ν r , and vertical, ν θ , oscillation frequenciesof particles in the accretion disk (Abramowicz et al. 2003). Another explanation of HFQPOs is basedon the Keplerian and radial frequencies of the deformation of the clumps of matter that is due tothe simulated tidal interactions in the accretion disk (Germanà et al. 2009). A recent model involvesthe study of (magneto)hydrodynamic instabilities, for example, in particular, to understand the 3:2resonance of HFQPOs using the general relativistic and ray-tracing simulations (Tagger & Varnière2006; Varniere et al. 2019).The RPM takes into account of the fundamental phenomenon of relativistic precession, which isdominant and inevitable in the strong-field regime around a black hole. Although the emissionmechanism for the production of QPOs with strong rms ( ∼
20 %) is hitherto unknown, it explainssome important observational relations, for example, the Psaltis − Belloni − Klis (PBK) (Psaltis et al.1999), which is a positive correlation between the HFQPOs and the LFQPOs in different BHXRBs.In a few other BHXRBs, the characteristic frequency of a broad feature (not a QPO) in the PDSduring the hard state shows the same correlation with the LFQPOs. This correlation has beenexplained using the RPM as a variation of the radius of origin around the Kerr black hole, tracingthe QPO frequency.In this paper, we expand the RPM from a restricted study of circular orbits and explore the fun-damental frequency range of the nonequatorial eccentric, equatorial eccentric, and spherical particletrajectories around a Kerr black hole and associate them with the properties of QPOs. We call this asthe generalized RPM (GRPM). The general trajectory solutions around a Kerr black hole and theircorresponding fundamental frequencies have been extensively studied before (Schmidt 2002; Fujita& Hikida 2009; Rana & Mangalam 2019a,b). The existence of nonequatorial eccentric, equatorialeccentric, and spherical orbits near a rotating black hole is tangible, and hence the relativistic pre-cession of these orbits can also be included in the model for the emission of QPOs. The quadratureform of the general trajectory solution { φ , θ , r , t } around a Kerr black hole (Carter 1968) and thecorresponding fundamental frequencies { ν φ , ν r , ν θ } (Schmidt 2002) are well known. Later, the com-plete analytic form for the trajectories and the fundamental frequencies was derived in terms of theMino time (Mino 2003) and the standard elliptic integrals (Fujita & Hikida 2009). More recently, amore compact, analytic, and numerically faster form was derived, in terms of the standard elliptic Rana and Mangalam 2020 integrals, for the particle trajectory solutions and their fundamental frequencies was derived (Rana &Mangalam 2019a,b). We use these analytic formulae for the fundamental frequencies via the GRPMfor the periastron and nodal precession of nonequatorial eccentric, equatorial eccentric, and sphericaltrajectories around a Kerr black hole to associate them with the detected QPO frequencies. TheRPM was previously predicted for circular { e = 0 , Q = 0 } orbits (Stella & Vietri 1999; Stella et al.1999). We now include { e (cid:54) = 0 , Q (cid:54) = 0 } orbits in this paradigm and test the more general model inthis paper. Finally, we show that the eccentric trajectory solution also satisfies the PBK correlationfor the case of BHXRB GROJ 1655-40. Non-equatorial eccentricorbits, eQ ; §2.1, §3.2.1 e (cid:54) = Q (cid:54) = Spherical orbits, Q ;§2.2, §3.2.2 e =0 and Q (cid:54) = Circular orbits, e =0 and Q =0,Previously applied to QPOs(Motta et al. 2014a,b). Equatorial eccentricorbits, e ; §2.1, §3.2.1 e (cid:54) = Q =0. Figure 1.
Flowchart of various Kerr orbits (with the nomenclature used here of nonequatorial eccentric( eQ ), spherical ( Q ), eccentric equatorial ( e ), and circular ( ) orbits) studied to explore QPO frequenciesusing the GRPM in various sections of this paper, where the most specialized case of circular orbits waspreviously studied (Motta et al. 2014a,b). Clearly, the GRPM is valid strictly only when e (cid:54) = 0 . The paper is structured as follows. We first motivate the association of fundamental frequencies ofthe general eccentric and spherical trajectories with the QPOs in BHXRBs assuming the GRPM in§2.1 and §2.2; see Figure 1 for the terminology used for eQ (general case), Q (spherical), e (eccentricequatorial), and (circular orbits). We then take up the cases of BHXRBs M82 X-1, GROJ 1655-40, XTEJ 1550-564, 4U 1630-47, and GRS 1915+105, where HFQPOs have been discovered before.We discuss their observation history in Appendix D, and we discuss observations of each BHXRBthat we use for our analysis in §3.1. Using the observed QPO frequencies in these BHXRBs, wecalculate the corresponding orbital parameters. The method for the parameter estimation and itscorresponding errors are discussed in §3.2 and in Appendix E. We discuss the results for generaleccentric trajectories in §3.2.1, and those corresponding to the spherical orbit in §3.2.2. We alsoshow in §4 that the PBK correlation is well explained by the eccentric trajectory solutions found inthe case of BHXRB GROJ 1655-40. In §5, we compare our model with another model for the fluidflow in the general-relativistic thin accretion disk. We finally discuss and conclude our results in §6. Geometric origin for QPOs Introduction, §1 GRPM, §2 Model justifica-tion and assump-tions, §2.1, 2.2Appendix AFormulae for ver-tical oscillations( e (cid:54) = 0 , Q = 0 ) PARAMETERS,§3:2 QPOs search3 QPOs searchAppendix BSpecializedformulae forspherical orbits( e = 0 , Q (cid:54) = 0 ) Sources descrip-tion and solutionsearch strategy,§3.1, Appendix DAppendix CReduction of e and Q fre-quency formulaeto orbits Method and sta-tistical analysiscommon and spe-cific to 2 QPOsand 3 QPOs,§3.2, Appendix EResults:Eccentric or-bits, §3.2.1;Spherical or-bits, §3.2.2PBK corre-lation, §4Comparisonwith the GRmodel for fluidflow in a thinaccretion disk, §5Discussion,caveats, andconclusions, §6 Figure 2.
Concept flowchart of the paper.
A glossary of symbols used in this article is given in Table 1, and a concept flowchart of the paper isgiven in Figure 1.
Table 1.
Glossary of Symbols Used.
Common physical parameters c Speed of light G Gravitational constant M • Mass of the black hole a Spin of the black hole scaled by (cid:0) GM • /c (cid:1) M M • /M (cid:12) Orbital parameters E Energy per unit rest mass of the L z z component of Angular momentumparticle, scaled by mc per unit rest mass of the particle,scaled by ( GM • /c ) L Angular momentum per unit rest Q Carter’s constant scaled by (cid:0) GM • /c (cid:1) mass of the particle, scaled by ( GM • /c ) r a Apastron distance of the orbit r p Periastron distance of the orbitscaled by (cid:0) GM • /c (cid:1) scaled by (cid:0) GM • /c (cid:1) e Eccentricity parameter µ Inverse latus-rectum parameter r s Radius of spherical orbit scaled r I ISSO radius scaled by (cid:0) GM • /c (cid:1) by (cid:0) GM • /c (cid:1) Fundamental frequencies ν φ Azimuthal frequency ν np Nodal precession frequency, (cid:0) ν φ − ν θ (cid:1) ν θ Vertical oscillation frequency ν pp Periastron precession frequency, (cid:0) ν φ − ν r (cid:1) ν r Radial frequency ν Centroid frequency of the QPO ¯ ν Frequency scaled by the factor (cid:0) c /GM • (cid:1) Probability analysis for estimating parameter errors P Probability density (space) P Normalized probability density (space) N Normalization factor J l Jacobian of transformation from frequencyto parameter space
Rana and Mangalam 2020 (a) (b)
Figure 3.
Generalized relativistic precession phenomenon for Q (cid:54) = 0 , near a black hole (BH) at the center,rotating anticlockwise, where Ω pp represents the periastron precession and Ω np represents the nodal precessionfrequency. The initial point of the trajectory is indicated by point A, from where the particle follows aneccentric trajectory before completing one (a) radial or (b) vertical oscillation to reach point B. The particlesweeps an extra ∆ φ azimuthal angle during one (a) radial or (b) vertical oscillation because the azimuthalmotion is faster than the radial or vertical motion causing the periastron or nodal precession.2. GENERALIZED RELATIVISTIC PRECESSION MODEL (GRPM)The relativistic precession is a phenomenon that is due to strong gravity near a rotating black hole,and its consequence for QPOs originating very close to the black hole is studied. We motivate theassociation of QPOs in BHXRBs with the fundamental frequencies of general nonequatorial boundparticle trajectories around a Kerr black hole through the GRPM. Figure 3 shows the periastron andnodal precession of an eccentric particle trajectory near the equatorial plane of a rotating black hole.We suggest that the instabilities in the inner region close to the rotating black hole might providea radiating plasma cloud (it could be a blob or a torus with the collection of such trajectoriesdegenerate in the parameter space) with enough energy and angular momentum to attain an eccentric( e (cid:54) = 0 ) trajectory, or a nonequatorial trajectory ( Q (cid:54) = 0 , Carter’s constant, Carter (1968)), or bothsimultaneously ( e (cid:54) = 0 , Q (cid:54) = 0 ). The Carter’s constant can be roughly interpreted as representativeof the residual of the angular momentum in the x − y plane, Q ∝ L − L z , so we have Q = 0 for the equatorial orbits where L = L z . We first try to find the suitable range for the parameters, { e, r p , a, Q } , of these orbits that produce the fundamental frequencies to compare with the observedrange of QPO frequencies in BHXRBs, where r p represents the periastron point of the orbit and a represents the spin of the black hole. We divide our study of the trajectories into three categories(see Figure 1), where a particle follows one of these:1. A nonequatorial eccentric trajectory ( e (cid:54) = 0 , Q (cid:54) = 0 ) called eQ .2. An equatorial eccentric trajectory ( e (cid:54) = 0 , Q = 0 ) called e . Geometric origin for QPOs
73. A nonequatorial and noneccentric, also called a spherical trajectory ( e = 0 , Q (cid:54) = 0 ), called Q .We are using dimensionless parameters ( G = c = M • = 1 ) as the convention in this article for sim-plicity, so that r p → r p / ( GM • /c ) , r a → r a / ( GM • /c ) , a → J/ ( GM • /c ) , and Q → Q/ ( GM • /c ) ,where J is the angular momentum and M • is the mass of the black hole, and r a is the apastron pointof the bound orbit, while e = ( r a − r p ) / ( r a + r p ) , the eccentricity parameter, is dimensionless bydefinition (see Table 1). We also define another mass parameter M = M • /M (cid:12) scaled by solar massfor convenience. The most general nonequatorial trajectory ( eQ ) around a Kerr black hole comprisesof periastron precession in the orbital plane, superimposed on the precession of the orbital planeabout the spin axis of the rotating black hole. Figure 4 shows one such trajectory around a Kerrblack hole centered at the origin. (a) (b) Figure 4.
Example of eQ trajectory with parameters { e = 0 . , r p = 5 . , a = 0 . , Q = 5 } around a Kerrblack hole at the origin, with its spin pointing in the positive z -direction: (a) shows the side view of the orbitrepresenting the nodal precession phenomenon of the orbital plane about the spin axis of the black hole; (b)top view of the orbit showing the periastron precession phenomenon. There are a variety of bound Kerr orbits, for example, nonequatorial eccentric, separatrix, zoom-whirl, and spherical orbits, that have been systematically studied before [e.g. Rana & Mangalam(2019a,b) and references within]. Hence, here we first discuss the distribution of these orbits in theparameter space and then isolate the most plausible type of orbits, which should give us the observedrange of QPO frequencies assuming the GRPM. A complete description of various types of trajectoriesis given in Table 2, where MBSO(MBCO) is the marginally bound spherical (circular) orbit, and ISCOis the innermost stable circular orbit. These bound orbits are distributed in particular regions in theparameter space and into different parameter ranges for different types of orbits. In Figure 5, weshow how this distribution belongs in different regions in the ( r , a ) plane, where r = R/R g representsdistance from the black hole, and R g = ( GM • /c ) . These regions are separated by important radii,which are shown as various curves for the equatorial ( Q = 0 ) and nonequatorial ( Q = 4 ) trajectoriesin Figure 5, where we see that the (un)stable bound orbits are found in regions 1, 2, and 3. Region4 is beyond the light radius, which extends down to the horizon radius [ r + = (cid:0) √ − a (cid:1) ], wherebound particle orbits are not present, which means any particle in this region would plunge intothe black hole, and region 5 is inside the horizon surface. Hence, we restrict our exploration search Rana and Mangalam 2020 of suitable parameters for required QPO frequencies to the regions 1 and 2, where stable circular(spherical), equatorial (nonequatorial) eccentric, zoom-whirl, and separatrix orbits are found.
Table 2.
Various Types of Trajectories around a Kerr Black Hole with Their Description and the Region in the ( r , a )Plane Where They Are Found, as Shown in Figure 5. Type of Orbit or Radius Description Region or Curve a Eccentric (1), eQ or e • Stable eccentric bound orbits. 1 and 2Separatrix (1), (2), eQ or e • They are the intermediate case between bound and 2plunge orbits, while their periastron points correspondto an unstable spherical (or circular) orbit, where aparticle reaches asymptotically. • The eccentricity of a separatrix orbit increases as itsperiastron moves closer to the black hole for a given a . • The r p of a separatrix orbit with a given eccentricitydefines the innermost radial limit for the eccentric boundorbits having the same eccentricity.Zoom-whirl (1), (3), eQ or e • Represent an extreme form of the periastron 1 and 2precession in the strong-field regime. • A particle spends enough time near the periastronto make finite spherical (or circular) revolutions beforezooming out to the apastron point. • Found near and outside the separatrices.Stable spherical (circular) (1), Q ( ) • Have a constant radius with the precession of 1orbital plane partially spanning the surface of a spherearound the black hole. • Found outside ISSO (ISCO).Unstable spherical (circular) (1), Q ( ) • Have a constant radius like stable spherical 2 and 3(circular) orbits. • Found outside MBSO (MBCO).ISSO (ISCO) (1), Q ( ) • Innermost stable spherical (circular) orbit. Black curve • Defined by Equation (22) of Rana & Mangalam (2019b).MBSO (MBCO) (1), Q ( ) • Marginally bound spherical (circular) orbit. Blue curve • Defined by Equation (23) of Rana & Mangalam (2019b).Light radius (1), Q or • Only a photon orbit can exist at this radius. Green curve • Defined by Equation (24) of Rana & Mangalam (2019b). • Innermost boundary for the unstable spherical(circular) particle orbits. a The regions for e and orbits are shown in Figure 5(a), whereas eQ or Q orbits are shown in Figure 5(b). References —(1) Rana & Mangalam (2019a,b); (2) Levin & Perez-Giz (2009), Perez-Giz & Levin (2009); (3)Glampedakis &Kennefick (2002).
These bound orbits can also be shown as a region in the ( e , µ ) space, which is defined as e = r a − r p r a + r p , µ = r a + r p r a r p , (1) Geometric origin for QPOs a HorizonLight radiusMBCOISCO (a) a HorizonLight radiusMBSOISSO (b)
Figure 5.
Important radii: the ISCO (ISSO), MBCO (MBSO), light radius, and the horizon. These radiiseparate various kinds of orbits outside a Kerr black hole in the ( r , a ) plane, indicated by different regionsthat are depicted by numbers, for (a) the equatorial orbits with Q = 0 , and (b) nonequatorial orbits with Q = 4 . where r a is the apastron point of the orbit. This bound orbit region is shown as a shaded region inFigure 6. The condition for these bound orbits is given by (Rana & Mangalam 2019a,b) (cid:2) µ a Q (1 + e ) + µ (cid:0) µa Q − x − Q (cid:1) (3 − e ) (1 + e ) + 1 (cid:3) ≥ , (2)where µ can also be written as µ = 1 / [ r p (1 + e )] , where the equality sign corresponds to the separatrixtrajectories. This bound orbit region shown in Figure 6 only includes regions 1 and 2 of the ( r p , a )plane shown in Figure 5. The RPM has been applied to two cases of BHXRBs, assuming the μ ● S ● M (a) - - - - - - - - V e ff r r = r = r (b) - - V e ff r r = r (c) Figure 6. (a) The shaded region represents all possible bound orbits in the ( e , µ ) plane for { a = 0 . , Q = 5 },where S depicts the ISSO and M depicts the MBSO radius, and the red curve represents separatrix orbits(see Rana & Mangalam (2019a), Figure (2a)); the corresponding effective potential diagrams are shown as afunction of r for (b) ISSO and (c) MBSO, where the horizontal black curve represents (cid:0) E − (cid:1) / and thevertical red curve represents the horizon radius, and { r , r , r , r } are four roots of the effective potential,which are also the turning points of a trajectory, and where r = ∞ for MBSO. precession of nearly circular orbits (negligible eccentricity ) in the equatorial plane of a Kerr black as there is no periastron precession for e = 0 . Rana and Mangalam 2020 hole (Motta et al. 2014a,b). In general, the observed range of HFQPOs in BHXRBs is 40-500 Hz,whereas that of type C LFQPOs is 10 mHz to 30 Hz (Remillard et al. 2006; Belloni & Stella 2014).The formulae for fundamental particle frequencies of nearly circular and equatorial orbits are givenby Bardeen et al. (1972) and Wilkins (1972); see Appendix C for the derivation of these formulaefrom the general frequency formulae of e (Equation (5)) and Q (Equation (7)) orbits: ν φ ( r, a ) = c πGM • r / + a ) , ¯ ν φ ( r, a ) = ν φ ( c /GM • ) = 12 π ( r / + a ) , (3a) ν r ( r, a ) = ν φ (cid:18) − r − a r + 8 ar / (cid:19) / , ¯ ν r ( r, a ) = ν r ( c /GM • ) , (3b) ν θ ( r, a ) = ν φ (cid:18) a r − ar / (cid:19) / , ¯ ν θ ( r, a ) = ν θ ( c /GM • ) , (3c)where { ¯ ν φ , ¯ ν r , ¯ ν θ } are the dimensionless frequencies, where we use the convention a > for theprograde and a < for the retrograde orbits in this article. Using these formulae and assumingthe RPM, it was retrodicted for BHXRB GROJ 1655-40 and XTEJ 1550-564 that these signalsoriginated very close to and outside the ISCO radius, at nearly r = 5 . ± . for GROJ 1655-40and r = 5 . ± . for XTE J1550-564 (Motta et al. 2014a,b). We show that the expected QPOfrequency range associated with the orbits in the RPM { ν φ , ν pp ≡ ( ν φ − ν r ) , ν np ≡ ( ν φ − ν θ ) } is valid for a wide range of r , where ν φ , ν pp , and ν np correspond to the HFQPO-1, HFQPO-2, andtype C LFQPO, respectively . To illustrate this, we present a mass-independent model of thesefrequencies. In Table 3, we have shown the observed range of the HFQPO and LFQPO frequenciesin BHXRBs along with a typical range in dimensionless values { ¯ ν φ , ¯ ν pp , ¯ ν np }, obtained by scalingthe observed frequencies of HFQPOs in BHXRBs using the corresponding known value of the blackhole mass M ∼ − (given in Table 5). For a BHXRB, the typical frequency range of the type CQPOs is 10 mHz to 30 Hz, and we have scaled this frequency range with M = 10 (a typical massvalue for BHXRB) to obtain the dimensionless frequency range. This provides an expected range ofthe geometrical orbital parameters independent of the black hole mass that implies largely a rangeof r p . Figure 7 shows the contours of ¯ ν φ , ¯ ν pp , and ¯ ν np for the orbits, using Equations (3a − r , a ) plane outside the ISCO radius (region 1 of Figure 5(a)). We find the following:1. The expected range of simultaneous QPO frequencies corresponds to a wide range of r ∼ − for the orbits, which is typically the inner region of the accretion disk.2. The simultaneous QPOs, if associated with the orbits, should originate very near to theISCO radius.3. We expect much higher QPO frequency values { ¯ ν φ (cid:38) . , ¯ ν pp (cid:38) . , ¯ ν np (cid:38) . } for the orbits near the ISCO radius for a (cid:38) . , as seen in Figure 7, which are outside the observedQPO frequency range.Now, with this, we can explore the frequency range of the nonequatorial eccentric, equatorialeccentric, and spherical orbits using a similar approach assuming the GRPM in the regions 1 and 2of Figure 5 (shaded region of Figure 6). where pp and np represent the periastron and nodal precession frequencies, respectively. Geometric origin for QPOs Table 3.
Summary of the Observed QPO Frequency Range in BHXRBs and Their CorrespondingDimensionless Values Derived from Data Given in Table 5.
Type of QPO QPOs in the Observed QPO Frequency Dimensionless Frequency RangeRPM and GRPM Range in Hz ¯ ν = ν · − / (cid:0) c /GM • (cid:1) HFQPO-1 ν φ −
500 2 − HFQPO-2 ν pp −
300 1 − Type C LFQPO ν np − −
30 10 − − a (a) a (b) a (c) Figure 7.
Dimensionless frequency contours are shown for circular and equatorial trajectories ( ), usingEquations (3a − r , a ) plane outside the ISCO radius, which is indicated by a thick black contouras also depicted in Figure 5, for HFQPOs (a) ¯ ν φ , (b) ¯ ν pp , and type C LFQPO (c) ¯ ν np assuming the RPM. Nonequatorial and Equatorial Eccentric Orbits: eQ and e We first discuss the useful formulae of the fundamental frequencies for the nonequatorial andequatorial eccentric particle trajectories derived in Rana & Mangalam (2019a,b). Later, we use theseformulae to explore the required frequency range for QPOs in BHXRBs, based on the GRPM, anddetermine the corresponding parameter range { e , r p , a , Q } associated with these trajectories.As shown in Figure 4, the orbital plane of a nonequatorial eccentric trajectory oscillates with respectto the spin axis of the black hole, along with the phenomenon of periastron precession taking placein the orbital plane. A complete analytic trajectory solution and the fundamental frequencies forsuch trajectories around a Kerr black hole were derived in terms of { e , µ , a , Q } parameters (Rana &Mangalam 2019a,b), where µ is the inverse latus rectum of the orbit, and it can be written in termsof { e, r p } as µ = 1 / [ r p (1 + e )] . The expressions of dimensionless fundamental frequencies for thesetrajectories are given by (Rana & Mangalam 2019a,b) ¯ ν φ ( e, r p , a, Q ) = [ − I ( e, r p , a, Q ) − L z I ( e, r p , a, Q )] F (cid:16) π , z − z (cid:17) + 2 L z I ( e, r p , a, Q ) Π (cid:16) z − , π , z − z (cid:17) π (cid:110) [ I ( e, r p , a, Q ) + 2 a z EI ( e, r p , a, Q )] F (cid:16) π , z − z (cid:17) − a z EI ( e, r p , a, Q ) K (cid:16) π , z − z (cid:17)(cid:111) , (4a)2 Rana and Mangalam 2020 ¯ ν r ( e, r p , a, Q ) = F (cid:16) π , z − z (cid:17)(cid:110) [ I ( e, r p , a, Q ) + 2 a z EI ( e, r p , a, Q )] F (cid:16) π , z − z (cid:17) − a z EI ( e, r p , a, Q ) K (cid:16) π , z − z (cid:17)(cid:111) , (4b) ¯ ν θ ( e, r p , a, Q ) = a √ − E z + I ( e, r p , a, Q )2 (cid:110) [ I ( e, r p , a, Q ) + 2 a z EI ( e, r p , a, Q )] F (cid:16) π , z − z (cid:17) − a z EI ( e, r p , a, Q ) K (cid:16) π , z − z (cid:17)(cid:111) , (4c)where L z is the z -component of a particle’s angular momentum and E is its energy per unit restmass, which can be explicitly expressed as the functions of { e , µ , a , Q } parameters [see Equations(5a − I ( e, r p , a, Q ) , I ( e, r p , a, Q ) , and I ( e, r p , a, Q ) are theradial integrals of motion given in their simplest analytic forms, along with the constants involved, byEquations (6a − − − F ( ϕ, p ) , K ( ϕ, p ) ,and Π ( q , ϕ, p ) used in Equations (4a − e ), the expressions for the azimuthal and radialfundamental frequencies can be further reduced to a form simpler than Equations (4a − ¯ ν φ ( e, r p , a ) = a Π (cid:0) − p , π , m (cid:1) + b Π (cid:0) − p , π , m (cid:1) π a ( p ) (cid:20) − F (cid:0) π , m (cid:1) + p K ( π ,m )( m + p ) (cid:21) + Π (cid:0) − p , π , m (cid:1) (cid:20) a [ p +2 p ( m ) +3 m ] ( p )( m + p ) + b (cid:21) + c Π (cid:0) − p , π , m (cid:1) + d Π (cid:0) − p , π , m (cid:1) , (5a) ¯ ν r ( e, r p , a ) = 12 a ( p ) (cid:20) − F (cid:0) π , m (cid:1) + p K ( π ,m )( m + p ) (cid:21) + Π (cid:0) − p , π , m (cid:1) (cid:20) a [ p +2 p ( m ) +3 m ] ( p )( m + p ) + b (cid:21) + c Π (cid:0) − p , π , m (cid:1) + d Π (cid:0) − p , π , m (cid:1) , (5b) ¯ ν θ ( e, r p , a ) = 2¯ ν r ( e, µ, a ) µ / (cid:112) ( x + a + 2 aEx ) · F (cid:16) π , m (cid:17) π [1 − µ x (3 − e − e )] / , (5c)where x = ( L z − aE ) , and { p , p , p } are given by Equation (7k) of Rana & Mangalam (2019a),while m is given by Equation (13c) of Rana & Mangalam (2019a), for the e orbits. See AppendixA for the derivation of Equation (5c), which is a novel reduced form for ¯ ν θ .Now, we use these frequency formulae, Equations (4a − e , r p , a , Q } for eQ and Equations (5a − e trajectories to find { e , r p , a } toretrodict the observed range of QPOs in BHXRBs, which is provided in Table 3. In Figures 8 − Geometric origin for QPOs r p a - - - - - - (a) r p a - - - - - - (b) r p a - - - - - (c) r p a - - - - - - (d) r p a - - - - - (e) r p a - - - - - (f) Figure 8.
The contours of δ φ ( e, r p , a, Q ) are shown in the ( r p , a ) plane for eccentric orbits around aKerr black hole, where the parameter combinations are (a) { e = 0 . , Q = 0 }, (b) { e = 0 . , Q = 2 }, (c){ e = 0 . , Q = 4 }, (d) { e = 0 . , Q = 0 }, (e) { e = 0 . , Q = 2 }, and (f) { e = 0 . , Q = 4 }. we have shown the variation of the quantities δ φ ( e, r p , a, Q ) = [¯ ν φ ( e, r p , a, Q ) − ¯ ν φ ( e = 0 , r p , a, Q = 0)]¯ ν φ ( e = 0 , r p , a, Q = 0) , (6a) δ pp ( e, r p , a, Q ) = [¯ ν pp ( e, r p , a, Q ) − ¯ ν pp ( e = 0 , r p , a, Q = 0)]¯ ν pp ( e = 0 , r p , a, Q = 0) , (6b) δ np ( e, r p , a, Q ) = [¯ ν np ( e, r p , a, Q ) − ¯ ν np ( e = 0 , r p , a, Q = 0)]¯ ν np ( e = 0 , r p , a, Q = 0) , (6c)in the ( r p , a ) plane for combinations of e = {0.25, 0.5} and Q = {0, 2, 4}. These quantities providea fractional deviation between frequencies of general eccentric orbits and circular orbits for the samespin and periastron radius. For this comparison, we have calculated the frequency corresponding toa circular orbit at the same radius, r p = r , for a fixed value of parameter a . Hence, the deviation, δ , between frequencies defined in this manner is dominated by the parameters e and Q . Also, thesedeviations are shown only in the region where ¯ ν φ ( e, r p , a, Q ) , ¯ ν pp ( e, r p , a, Q ) , and ¯ ν np ( e, r p , a, Q ) arein the range of QPO frequencies allowed by the observations, as provided in Table 3. Hence, theseplots together give us the information of deviation of frequencies from circularity, as the e and Q parameters are varied, along with information on the range of ( e, r p , a, Q ) for general eccentric orbits4 Rana and Mangalam 2020 r p a - - - - - - - (a) r p a - - - - - - - (b) r p a - - - - - - - (c) r p a - - - - - - - (d) r p a - - - - - - - (e) r p a - - - - - - - (f) Figure 9.
The contours of δ pp ( e, r p , a, Q ) are shown in the ( r p , a ) plane for eccentric orbits around aKerr black hole, where the parameter combinations are (a) { e = 0 . , Q = 0 }, (b) { e = 0 . , Q = 2 }, (c){ e = 0 . , Q = 4 }, (d) { e = 0 . , Q = 0 }, (e) { e = 0 . , Q = 2 }, and (f) { e = 0 . , Q = 4 }. allowed by the observed range of QPO frequency. The 3:2 and 5:3 ratios of the simultaneous HFQPOs,seen in a few BHXRBs, are also a remarkable phenomenon that we need to fathom; for example, 300Hz and 450 Hz HFQPOs were seen in GROJ 1655-40 (Remillard et al. 1999b; Strohmayer 2001a), and240 Hz and 160 Hz HFQPOs in H1743-322 (Homan et al. 2005; Remillard et al. 2006). Assuming theGRPM, this ratio is given by ν φ /ν pp = ¯ ν φ / ¯ ν pp , which is a dimensionless quantity. The contours of thisratio are shown in Figure 11 for the six combinations from the set e = { . , . } , Q = { , , } . Theblue contours in Figures 8 −
11 represent the ISSO radius, and black contours represent the MBSOradius as also indicated in Figure 5, whereas the magenta color contours represent the separatrixorbits, given by the equality in Equation (2), defining the innermost limit for r p of an eccentric orbitwith a given e .A summary of the results is given below:1. A novel and reduced form of ¯ ν θ ( e, r p , a ) for e trajectories, given by Equation (5c), is derivedin Appendix A.2. Assuming the GRPM, (non)equatorial eccentric trajectories with small to moderate eccentric-ities, e (cid:46) . , with Q ∼ − also generate the expected range of QPO frequencies, { ¯ ν φ , ¯ ν pp , ¯ ν np }, in BHXRBs, as shown in Table 3. We have not taken very high values for the Q Geometric origin for QPOs r p a - - - - - - (a) r p a - - - - - - (b) r p a - - - - (c) r p a - - - - - - (d) r p a - - - - - - (e) r p a - - - - - - (f) Figure 10.
The contours of δ np ( e, r p , a, Q ) are shown in the ( r p , a ) plane for eccentric orbits around aKerr black hole, where the parameter combinations are (a) { e = 0 . , Q = 0 }, (b) { e = 0 . , Q = 2 }, (c){ e = 0 . , Q = 4 }, (d) { e = 0 . , Q = 0 }, (e) { e = 0 . , Q = 2 }, and (f) { e = 0 . , Q = 4 }. parameter, as the particle oscillation is expected to be close to the equatorial plane in typicalBHXRB scenarios.3. The effective r p ranges that produce the required QPO frequency ranges are ∆ r p ∼ − for ¯ ν φ , ∆ r p ∼ − for ¯ ν pp , and ∆ r p ∼ − for ¯ ν np , where a varies from 0 to 1. While these ∆ r p values are strongly dependent on e , they are only weakly dependent on the Q parameter.The frequency ¯ ν np (see Figure 10) increases with a , which implies that we expect to find hightype C LFQPO values (nearly ¯ ν np ∼ . ) for the black holes with high spin.4. As e increases, the allowed region shifts close to the black hole. In other words, we expect(non)equatorial eccentric orbits close to the black hole to create the allowed frequency range,whereas circular orbits at comparatively larger radius cater to the same frequency range (seeFigure 7). This is consistent with the finding that the GRPM favors slightly eccentric andstrongly relativistic orbits. We also see that as e increases, the frequencies deviate and decreasefrom corresponding circular orbit frequencies; for example, ¯ ν φ decreases by 30% for e = 0 . to 60% for e = 0 . (see Figure 8), ¯ ν pp decreases by 40% for e = 0 . to 79% for e = 0 . (seeFigure 9), and ¯ ν np decreases by 40% for e = 0 . to 80% for e = 0 . (see Figure 10).5. The dependence of these frequencies on Q is very weak. Although the change is comparativelysmall, we see that these frequencies increase with Q . For example, the maximum increase in ¯ ν φ Rana and Mangalam 2020 r p a (a) r p a (b) r p a (c) r p a (d) r p a (e) r p a (f) Figure 11.
The HFQPO frequency ratio, ¯ ν φ / ¯ ν pp , contours are shown for eccentric orbits around a Kerrblack hole in the ( r p , a ) plane, assuming the GRPM, where the parameter combinations are (a) { e = 0 . , Q = 0 }, (b) { e = 0 . , Q = 2 }, (c) { e = 0 . , Q = 4 }, (d) { e = 0 . , Q = 0 }, (e) { e = 0 . , Q = 2 }, and (f){ e = 0 . , Q = 4 }. is ∼
3% (see Figure 8) and ∼
10% for ¯ ν pp (see Figure 9), whereas it is ∼
3% for ¯ ν np (see Figure10) as Q changes from to . Even for high Q values, say Q ∼ , the change in frequenciesis of the same order.6. Expectedly, the associated frequencies increase as the r p of a trajectory decreases for a given{ e, a, Q }, where r p of an eccentric trajectory is limited by the corresponding separatrix orbit,having the same { e , a , Q } values.7. As shown in Figure 11, the 3:2 or 5:3 ratios of HFQPOs originate in the region very closeto the separatrix orbits, which is between MBSO and ISSO radii corresponding to typically ∆ r p ∼ − ; this range is dependent on a since r p decreases as a increases. The frequency ratiocontours shift close to the black hole as e is increased, whereas these contours move toward large r p as Q is increased. This indicates that nonequatorial orbits show a 3:2 or 5:3 ratio of HFQPOfrequencies farther away from the black hole than the equatorial orbits, and eccentric orbitshave such ratios comparatively closer to the black hole than the circular orbits. Therefore, eQ and orbits close to the black hole can account for these ratios, as e and Q have cancelingeffects. 2.2. Spherical Orbits: Q Geometric origin for QPOs eQ trajectories, the spherical orbits ( Q ) are also specific to the rotating black holes.They are the orbits with a constant radius, r s , where the orbital plane precesses on a sphere aboutthe spin axis of the black hole. Similar to the ISCO and MBCO radii for circular orbits, ISSO andMBSO radii exist for the spherical orbits that are functions of the a and Q parameters (Rana &Mangalam 2019a,b). We explore the ranges of parameters, { r s , a , Q }, for spherical orbits allowedby the observed frequency range of QPOs (see Table 3). The fundamental frequency formulae forthe spherical orbits reduce to the form given by (see Appendix B for a derivation: Equations (B14),(B15b), and (B16c)) ¯ ν φ ( r s , a, Q ) = (cid:26)(cid:20) − (2 L z r s − L z r s − r s aE )∆ − L z (cid:21) F (cid:16) π , z − z (cid:17) + L z · Π (cid:16) z − , π , z − z (cid:17)(cid:27) π (cid:26)(cid:20) [ E ( a r s + r s + 2 a r s ) − L z ar s ]∆ + a z E (cid:21) F (cid:16) π , z − z (cid:17) − a z E · K (cid:16) π , z − z (cid:17)(cid:27) , (7a) ¯ ν r ( r s , a, Q ) = (cid:112) r s (1 − E ) + (3 Qa − x r s − Qr s ) · F (cid:16) π , z − z (cid:17) πr s (cid:26)(cid:20) [ E ( a r s + r s + 2 a r s ) − L z ar s ]∆ + a z E (cid:21) F (cid:16) π , z − z (cid:17) − a z E · K (cid:16) π , z − z (cid:17)(cid:27) , (7b) ¯ ν θ ( r s , a, Q ) = a √ − E z + (cid:26)(cid:20) [ E ( a r s + r s + 2 a r s ) − L z ar s ]∆ + a z E (cid:21) F (cid:16) π , z − z (cid:17) − a z E · K (cid:16) π , z − z (cid:17)(cid:27) , (7c)where ∆ = r s + a − r s , and z ± are given by Equation (9d) of Rana & Mangalam (2019a). InFigure 12, we show the contours of the quantities δ φ ( r s , a, Q ) = [¯ ν φ ( r s , a, Q ) − ¯ ν φ ( r s , a, Q = 0)]¯ ν φ ( r s , a, Q = 0) , (8a) δ pp ( r s , a, Q ) = [¯ ν pp ( r s , a, Q ) − ¯ ν pp ( r s , a, Q = 0)]¯ ν pp ( r s , a, Q = 0) , (8b) δ np ( r s , a, Q ) = [¯ ν np ( r s , a, Q ) − ¯ ν np ( r s , a, Q = 0)]¯ ν np ( r s , a, Q = 0) , (8c)for QPOs in the ( r s , a ) plane for spherical orbits with Q = { , } assuming the GRPM, usingEquations (7a − φ ( r s , a, Q ) , t ( r s , a, Q ) }, given by Equa-tion (B12), and the fundamental frequencies { ¯ ν φ ( r s , a, Q ) , ¯ ν r ( r s , a, Q ) , ¯ ν θ ( r s , a, Q ) }, given byEquation (7), for spherical trajectories are derived in Appendix B.2. Assuming the GRPM, we see that the spherical orbits with Q ∼ − are in the expected rangeof QPO frequencies for BHXRBs. The allowed range of r s to source the QPOs is typically ∼ − for ¯ ν φ (see Figures 12(a) and 12(b)), ∼ − for ¯ ν pp (see Figures 12(c) and 12(d)),and ∼ − for ¯ ν np (see Figures 12(e) and 12(f)), where a varies from 0 to 1.8 Rana and Mangalam 2020 r s a (a) r s a (b) r s a (c) r s a (d) r s a (e) r s a (f) Figure 12.
The contours of δ φ ( r s , a, Q ) are shown for (a) Q =2, (b) Q =4; δ pp ( r s , a, Q ) for (c) Q =2, (d) Q =4; and δ np ( r s , a, Q ) for (e) Q =2, (f) Q =4 in the ( r s , a ) plane for the spherical orbits around a Kerr blackhole.
3. The frequencies change weakly with Q . The maximum changes in frequencies are ∼ −
3% for ¯ ν φ , ∼ −
23% for ¯ ν pp , and ∼ −
8% for ¯ ν np as Q changes from 2 to 4 for the spherical orbits.The associated frequencies increase as r s decreases for a given { a , Q }.4. We see from Figure 13 that the 3:2 or 5:3 ratio of HFQPOs, ¯ ν φ / ¯ ν pp , for spherical orbits shouldemanate in the region r s ∼ − for Q =2 and r s ∼ . − . for Q =4. The ranges of r s arealso dependent on a , where r s for a given ratio contour decreases as a increases. PARAMETER ESTIMATION OF ORBITS IN BLACK HOLE SYSTEMSWITH OBSERVED QPOSNext, we take up a few cases of black hole systems that are known to have shown either two or threesimultaneous QPOs in their PDS, and we extract the parameter values of the nonequatorial eccentric( eQ ), equatorial eccentric ( e ), and the spherical orbits ( Q ) corresponding to the observed QPOfrequencies using our GRPM. The solution for a given GRPM class ( eQ , Q , e ) being attempted hereis based on balancing the knowns (number of simultaneous frequencies, two or three) with the numberof unknown parameters { e, r p , a, Q } (see Table 4 illustrating this criterion). For the three frequencycases (M82 X-1 and GROJ 1655-40), we have to either supply a from available data or deduce thisusing a procedure that involves minimizing χ in the unknown parameter volume. For the geometricstudy of orbits that is of importance here, we have taken the view that the best approximation to a isto be determined first, and then the solution vector { e, r p , Q } (which is crucial for the orbital shape) Geometric origin for QPOs r s a (a) r s a (b) Figure 13.
The HFQPO frequency ratio, ¯ ν φ / ¯ ν pp , contours are shown for the spherical orbits around a Kerrblack hole in the ( r s , a ) plane, assuming the GRPM for (a) Q = 2 and (b) Q = 4 . Table 4.
Various GRPMs, Their Corresponding Unknown Parameters, and Underlying Assumptionsfor BHXRBs with Three and Two Simultaneous QPOs.
BHXRBs with Three QPOs
GRPM Model Parameters Number of Parameters Number of Observed QPOs eQ { e , r p , a , Q } 4 3 a e { e , r p , a } 3 3 Q { a , r s , Q } 3 3 BHXRBs with Two QPOs e { e , r p } 2 2 b Q { r s , Q } 2 2 b Note — a need to supply a from the best fit of χ ; b a is fixed from the available data (see Table 5). for the peak probability is found. We have taken slightly different approaches for the two sources asexact solutions are found only in one of the two sources (M82 X-1), where we minimize χ in the a dimension to isolate a . In the other case where no exact solution vector is found (GROJ 1655-40),and where it is computationally expensive to explore the full four-dimensional parameter volumeof { e, r p , Q, a } in a fine-grained manner, we have only done a primary coarse-grained search to find a sufficiently accurately and then proceeded to determine the unknown parameters { e, r p , Q } by afine-grid search. The two QPO frequency cases (XTEJ 1550-564, 4U 1630-47, and GRS 1915+105)are searched by direct fine-grid computations assuming a from available data (see Table 4 and 5).We describe our parameter search criteria below:1. For BHXRBs with three simultaneous QPOs, that is, M82 X-1 and GROJ 1655-40 (see Table5), since a type C LFQPO is also present, which corresponds to the nodal oscillation frequency( ν np ), we search for all eQ , e , and Q orbit solutions. We use Equations (4a − − ν φ , ν pp , ν np } and find the parameters { e , r p , a } of eQ and e orbits for M82 X-1 and GROJ 1655-40. Next, we calculate the most probable spin of the0 Rana and Mangalam 2020 black hole to estimate { e , r p , Q } of the orbit. Similarly, we study the Q orbits as solutionsto the QPOs using Equations (7a − r s , a , Q } for these BHXRBs.Hence, the parameters searched for these cases are eQ and e , {M = fixed from observations , e, r p , a, Q = { , , , , }} ,Q , {M = fixed from observations , e = 0 , r s , a, Q } . (9a)2. For BHXRBs with two simultaneous QPOs, that is, XTEJ 1550-564, 4U 1630-47, and GRS1915+105 (see Table 5), we expect that the solutions are likely to be equatorial as the LFQPO,or ν np oscillation, is absent (this is consistent with no large-amplitude nodal oscillations andstrictly equatorial orbits). Hence, we search for e solutions using Equations (5a − ν φ , ν pp } to find { e , r p } of the orbit. However, we also check for the Q orbital solution in thesesystems and estimate the parameters { r s , Q } using { ν φ , ν pp }. Hence, the parameters searchedfor in these cases are e , { ( M , a ) = fixed from observations , e, r p , Q = 0 } ,Q , { ( M , a ) = fixed from observations , e = 0 , r s , Q } . (9b)We have summarized the history of black hole systems considered here in Appendix D. In §3.1, wesummarize the observations related to QPO detection, mass, and spin estimation and the parameterswe estimated for each source. In §3.2, we explain the method used to estimate the parameters ofthese orbits and corresponding errors and then present the results for the (non)equatorial eccentricorbits in §3.2.1, and spherical orbits in §3.2.2.3.1. Source Selection
Here we summarize the QPO observations of the black hole systems that we have selected toimplement the GRPM for the general eccentric and spherical trajectories. We have chosen caseswhere either two or three simultaneous QPOs have been detected, which are as follows:1. M82 X-1: We use the HF-analog QPOs of M82 X-1 along with the other detected LFQPOs(Pasham & Strohmayer 2013a) to estimate the parameters { e , r p , a } of the eQ and e trajecto-ries, where the QPOs are created, by varying Q in the range − using the GRPM. Next, usingthese results, we calculate the most probable value of a to estimate the remaining parameters{ e , r p , Q }, using three simultaneous QPO frequencies, in §3.2.1. In our analysis, we have as-sumed the mass of the black hole to be M = 428 (Pasham et al. 2014). We also search for the Q orbit solution and estimate the corresponding parameters { r s , a , Q } assuming the GRPMin §3.2.2. In this paper, we have assumed that the LFQPOs are simultaneous with 3.32 ± ± M = 428 , e , r p , a , Q } (see Equation (9a)).2. GROJ 1655-40: We use three simultaneous frequencies detected, 441 ± ± ± eQ and e trajectoriesassuming the GRPM in §3.2.1. We also explore a Q trajectory solution. For this BHXRB, we Geometric origin for QPOs M = 5 . (Beer & Podsiadlowski2002). We did not find any Q orbit solution for this BHXRB. Hence, we explore the parameterspace of { M = 5 . , e , r p , a , Q } (see Equation (9a)).3. XTEJ 1550-564: We use the simultaneous frequencies, 268 ± ± e , r p } of the orbit assuming the e orbit in §3.2.1. We alsoestimate the parameters { r s , Q } of the Q orbit using these QPO frequencies in §3.2.2. Weassumed that the mass of the black hole is M = 9 . , as estimated using the optical spectro-photometric observations (Orosz et al. 2011), and that the spin of the black hole is a = 0 . (Miller & Miller 2015), estimated from the disk continuum spectrum. Hence, we explore theparameter space of { M = 9 . , a = 0 . , e , r p } for e orbits and { M = 9 . , a = 0 . , r s , Q }for Q orbits (see Equation (9b)).4. 4U 1630-47: We use the twin HFQPOs at 179.3 ± ± e orbits to find the parame-ters { e , r p } in §3.2.1. We assumed the mass of the black hole to be M = 10 , calculated from thescaling of the photon index of the Comptonized spectral component with the LFQPOs (Seifinaet al. 2014). We fixed the spin of the black hole to a = 0 . , as previously estimated fromthe fit to the reflection spectrum using NuSTAR observations (King et al. 2014). We did notfind a Q orbit solution for this BHXRB. Hence, we explore the solution space of { M = 10 , a = 0 . , e , r p } for the e orbit (see Equation (9b)).5. GRS 1915+105: We take simultaneous HFQPOs at 69.2 ± ± e orbits using the GRPM and calculate the corresponding parameters { e , r p } in §3.2.1. We fixed the mass of the black hole to M = 10 . , estimated using the near-infraredspectroscopic observations (Steeghs et al. 2013). We assumed the spin of the black hole to be a = 0 . , calculated by fitting to the disk reflection spectrum using NuSTAR observations(Miller et al. 2013). We did not find a Q orbit solution for this BHXRB. Hence, we explorethe solution space of { M = 10 . , a = 0 . , e , r p } for the e orbit (see Equation (9b)).We have summarized the BHXRB data in the Table 5 along with the frequencies of detected QPOs,and previously known values of mass and spin of the black hole, along with their references.3.2. Method Used and Results
We apply the GRPM to associate the fundamental frequencies of eQ , e , and Q orbits with QPOs.In Appendix E, we describe a generic procedure that we have used to estimate errors in the orbitalparameters. A flowchart of this method is provided in Figure 14. Next, we summarize the resultscorresponding to the eQ and e models in §3.2.1 and the Q model in §3.2.2.3.2.1. Nonequatorial and Equatorial Eccentric Orbits ( eQ and e ) We have taken the cases of five BHXRBs, known to have either three or two simultaneous detectionsof QPOs in their observations, to study the eccentric and nonequatorial trajectories as solutions tothe QPOs assuming the GRPM. Here we summarize the results for the cases of three and twosimultaneous QPOs separately, as discussed below:1.
Three simultaneous QPOs : In our analysis, varying the dimensionless parameter Q = { , , , , } ∝ ( L − L z ) gives us various trajectory solutions having different { e , r p , a , Q }2 Rana and Mangalam 2020
Joint Probability density P ( ν ) = (cid:81) i P i ( ν i ) , i =1 to l ,where l =3 for M82 X-1 and GROJ 1655-40,and l =2 for XTEJ 1550-564, 4U 1630-47, and GRS 1915+105.JacobianFind J l using Equation (E29), where i, j =1 to l .Eccentric orbits - x i ’s = { e , r p , a } for l =3, and x i ’s = { e , r p } for l =2.Spherical orbits - x i ’s = { r s , a , Q } for l =3, and x i ’s = { r s , Q } for l =2.Exact solutionsFind the exact solutions for x j ’s using the frequency formu-lae, Equations (4), (5) and (7). Fix M or { M , a } to the pre-viously known values for l =3 and l =2 respectively, see Table 5.Probability density and normalization factorChoose appropriate range for x j ’s near exact solution and resolu-tions ∆ x j ’s to find P ([ x ]) and N using Equations (E30) and (E31a).For the case with no exact solution, for example GROJ 1655-40, wechoose complete range of parameters to calculate P ([ x ]) and N .Normalized probability densityFind P ([ x ]) using Equation (E31b), which is P ( x , x , x ) for l =3 and P ( x , x ) for l =2.Integrated profilesIntegrate P ([ x ]) to obtain the profile in each dimension:Eccentric orbits - { P ( e ) , P ( r p ) , P ( a ) } for l =3, { P ( e ) , P ( r p ) } for l =2.Spherical orbits - { P ( r s ) , P ( Q ) , P ( a ) } for l =3, { P ( r s ) , P ( Q ) } for l =2.Three simultaneous QPOsMultiple trajectory solutions were estimated for M82 X-1 andGROJ 1655-40 with varying spin. We choose the most probablevalue of a and estimate the exact solutions { e , r p , Q }, prob-ability density profiles { P ( e ) , P ( r p ) , P ( Q ) }, and the cor-responding errors using the same procedure described above. Figure 14.
Flowchart of the method used to estimate the orbital solutions for QPOs and correspondingerrors.
Geometric origin for QPOs Table 5.
Summary of Existing BHXRBs That Exhibit Either Three or Two Simultaneous QPOs.
S.No. BHXRB ν (Hz) ν (Hz) ν (Hz) M a ModelClasses
1. M82 X-1 5.07 ± (a) ± (a) (cid:0) . ± . × − (cid:1) (b) ± (a) - eQ , e , Q
2. GROJ 1655-40 441 ± (c) ± (c) ± (c) ± (d) - eQ , e , Q
3. XTE J1550-564 268 ± (e) ± (e) - 9.1 ± (f) +0 . − .
45 (g) e , Q
4. 4U 1630-47 179.3 ± (h) ± (h) - 10 ± (i) +0 . − .
014 (j) e , Q
5. GRS 1915+105 69.2 ± (k) ± (k) - 10.1 ± (l) ± (m) e , Q Note —The first two rows represent the cases having twin HFQPOs with simultaneous type-C QPO. The remaining rowsshow the cases of BHXRB having only twin HFQPOs. The columns show the source name, QPO frequencies, and previouslymeasured mass through optical, infra-red or X-ray observations, previously known spin of the black hole measured by fit tothe Fe K α line or to the continuum spectrum (for 1 and 2 we calculate the parameter a from our method), and the class ofGRPM applied to estimate the parameters. The references are indicated by lower case letters (a-m). References —(a) Pasham et al. (2014), (b) Pasham & Strohmayer (2013a), (c) Motta et al. (2014a), (d) Beer & Podsiadlowski(2002), (e) Miller et al. (2001), (f) Orosz et al. (2011), (g) Miller & Miller (2015), (h) Klein-Wolt et al. (2004), (i) Seifinaet al. (2014), (j) King et al. (2014), (k) Strohmayer (2001b), (l) Steeghs et al. (2013), (m) Miller et al. (2013). combinations. We first find the exact solutions for the parameters { e , r p , a }, given in the Table6, by equating the centroid frequencies of three simultaneous QPOs (see Table 5) to { ν φ , ν pp , ν np } for each value of Q = { , , , , } using our analytic formulae (Equations (4a − e , r p , a } using the method discussed in Appendix E (seeFigure 14) for each value of Q . The results of fits to the integrated profiles { P ( e ) , P ( r p ) , P ( a ) } are summarized in the Table 6. Since the spin of the black hole is not expected to vary,we estimate the most probable spin value for these black holes and then estimate the orbitalparameters { e , r p , Q } and their corresponding errors again using the same method discussedin Appendix E (see Figure 14). The results for each case are as follows:• M82 X-1 : In this case, we find that the (non)equatorial trajectories with small to moderateeccentricities e ∼ − r p = − a = − Q between 0 and 4.Starting with these exact solutions, the most probable value of the spin is found first. InFig 15(a), we show the spin variation in the parameter solutions for QPOs as a functionof Q . Next, to estimate the most probable value of the spin, we minimize the function χ a = (cid:88) i ( a − a i ) σ a i , (10a)which gives a = (cid:80) i (cid:0) a i /σ a i (cid:1)(cid:80) i (cid:0) /σ a i (cid:1) , (10b)4 Rana and Mangalam 2020 a (a) a (b) Figure 15.
The figures show σ errors in the spin parameters for various Q values for exact solutions of (a)M82 X-1, and corresponding to the peak of the probability distributions for (b) GROJ 1655-40, as given inthe Table 6. The upper and lower dashed curves correspond to the limits of the calculated errors. Althougheach Q value corresponds to a different spin of the black hole, the calculated values, and corresponding errorsare within a narrow band which puts a sharp and reasonable constraint on the spin of the black hole. where i = 1 − corresponds to six probable solutions for a , and the σ i values are thecorresponding 1 σ errors, where five of these are given in Table 6, and the remainingone corresponds to the spherical orbit solution found for M82 X-1 given in Table 9. Byincluding these six solutions, we have spanned the complete ( e , Q ) parameter space,which is bounded by e and Q solutions. This gives us the most probable spin valueof a = 0 . . Hence, we fix the spin of the black hole to this most probable estimateand then calculate the remaining parameters { e , r p , Q } and corresponding errors usingthe method given in Appendix E and Figure 14. We find the exact solution for QPOs at{ e = 0 . , r p = 4 . , Q = 2 . } calculated by equating centroid QPO frequenciesto { ν φ , ν pp , ν np } while fixing a = 0 . . The probability density distribution profiles{ P ( e ) , P ( r p ) , P ( Q ) }, along with their model fit, and the probability contours in theparameter plane { e , Q }, { r p , e }, and { Q , r p } are shown in Figure 16. The results ofthe model fit to the integrated profiles are summarized in the Table 7. The correspondingerrors are quoted with respect to the exact solution of the parameters, which slightly differfrom the peak of the integrated profiles { P ( e ) , P ( r p ) , P ( Q ) }, as expected (see Figure16).• GROJ 1655-40 : For this case, we did not find the exact solution for the parameters { e , r p , a } when the centroid frequencies of QPOs, Table 5, are equated to { ν φ , ν pp , ν np }. However,we generate the probability density profiles P ( e ) , P ( r p ) , and P ( a ) for each value of Q between 0 and 4. The results of fits for these profiles are summarized in Table 6. Wefound that the probability density peaks near very small eccentricities e ∼ . − . forvarious values of Q , whereas r p ranges between . and . and a ranges between . and . ; see Table 6. The change in the value of the spin of the black hole as a functionof Q is shown in Figure 15(b) for GROJ 1655-40. Next, we find the most probable valueof the spin for this BHXRB. Since we did not find any exact solution for the parametersby equating centroid frequencies of QPOs to the frequency formulae, we calculated the χ function given by χ = ( ν φ − ν ) σ + ( ν pp − ν ) σ + ( ν np − ν ) σ , (11) Geometric origin for QPOs Q + % % % (a) r p e + % % % (b) r p + % % % (c) ( e ) (d) r p ( r p ) (e) ( Q ) (f) Figure 16.
The probability contours in the parameter planes are shown in (a) { e , Q }, (b) { r p , e }, and (c){ Q , r p }, where the + sign marks the exact solution for the parameters for QPOs in M82 X-1 with a = 0 . .The probability density profiles are shown in (d) P ( e ) , (e) P ( r p ) , and (f) P ( Q ) , where the black pointsrepresent integrated probability densities and the blue curves are their model fit. The dashed vertical linesenclose a region with 68.2% probability, and the solid vertical line marks the peak of the profiles. in the four-dimensional parameter space { e , r p , a , Q } using Equations (4a) − (4c) for{ ν φ , ν pp , ν np }, and we numerically found the minimum χ = 2 . for the parametercombination { e = 0 . , r p = 5 . , a = 0 . , Q = 0 }. This is a primary coarse-grainedsearch to find a viable solution of a . Next, we assume a = 0 . corresponding to theminimum χ to calculate the final solution for the parameters { e , r p , Q }, which are the keyparameters for the geometric study, using the more accurate fine-grid method describedin Appendix E and Figure 14. We find that the probability density peaks near { e = 0 . , r p = 5 . , Q = 0 }. The results of fitting to the { P ( e ) , P ( r p ) , P ( Q ) } profiles aresummarized in the Table 7, whereas these profiles with their model fit and the probabilitycontours in the parameter plane { e , Q }, { r p , e }, and { Q , r p } are shown in Figure 17.Hence, we conclude for both M82 X-1 and GROJ 1655-40 that (non)equatorial trajectories(both eQ and e ) with small or moderate eccentricities in the region very close to the blackhole are the solutions for the observed QPOs assuming our GRPM. A self-emitting blob ofmatter close to a Kerr black hole can have enough energy and angular momentum to attain aneccentric and nonequatorial trajectory. These results are also consistent with the conclusionsmade in §2.1 that the trajectories having small to moderate eccentricities with Q = 0 − arealso possible solutions for the observed range of QPO frequencies in BHXRBs.The errors in QPO frequencies cause to a distribution in the solution space { e , r p , Q } assolutions using our GRPM, as shown in Figures 16 and 17. We take various combinations ofthese parameters within the range of 1 σ errors, as summarized in Table 7, as any such parameter6 Rana and Mangalam 2020 Q + % % % (a) r p e + % % % (b) r p + % % % (c) ( e ) (d) r p ( r p ) (e) ( Q ) (f) Figure 17.
The probability contours in the parameter planes are shown in (a) { e , Q }, (b) { r p , e }, and (c){ Q , r p }, where the + sign marks the peak of the probability density for GROJ 1655-40 with a = 0 . . Theprobability density profiles are shown in (d) P ( e ) , (e) P ( r p ) , and (f) P ( Q ) , where the dashed verticallines enclose a region with 68.2% probability and the solid vertical line marks the peak of the profiles. combination is a probable solution for the frequencies within the width of QPOs observed inthe power spectrum. In Figure 18, we have plotted together the trajectories for these parametercombinations for both BHXRBs M82 X-1 and GROJ 1655-40. Each trajectory has differentparameter values { e , r p , Q } and is indicated by a different color, where we fixed the spin ofthe black hole to a = 0 . for M82 X-1 and a = 0 . for GROJ 1655-40. Hence, thesetrajectories, having fundamental frequencies very close to each other and within the width ofthe QPO, together simulate the strong rms of the observed QPOs. The trajectories togetherspan a torus in the region . − . for M82 X-1 and . − . for GROJ 1655-40, whichshould be the emission region for QPOs, where we expect precession frequencies of both the eQ and e trajectories. The ISCO radius is ∼ for both the cases of BHXRB. We suggest that thesimultaneous HFQPO and LFQPO emission should be from a region that is close to the inneredge of the accretion disk ( r in ), where both eQ and e trajectories span a torus; the disk edgecould be a source of blobs that are generating QPOs, as we will argue later in §5. In contrast,a rigid body precession model is invoked by some authors (Ingram et al. 2009; Ingram & Done2011, 2012), where Lense − Thirring precession of a rigid torus is suggested as the origin of thetype C QPOs. Here, instead of the rigid precession of a solid torus, we propose that a collectiveprecession of various trajectories, spanning a torus region, explains the origin of HFQPOs andLFQPOs simultaneously. We argue that HFQPOs originate when r in comes in very close to theblack hole at some point during the outburst (the soft state). In the hard state, r in is fartherout, and the type C QPO is more frequent and it is more prone to the vertical oscillations Geometric origin for QPOs (a) (b) Figure 18.
The figures show various trajectories together having parameter combinations { e , r p , Q } withinthe estimated range of 1 σ errors, as tabulated in the Table 7, for (a) M82 X-1 and (b) GROJ 1655-40. Thespin of the black hole is fixed to the most probable estimates, which are a = 0 . for M82 X-1 and a = 0 . for GROJ 1655-40. Each color corresponds to a different parameter combination, where { e = 0 . − . ; r p = 4 . − ; Q = 1 − } for M82 X-1 and { e = 0 . − . ; r p = 5 . − . ; Q = 0 − . } for GROJ1655-40. ( ν np ). This scenario explains the increase in the frequency of type C QPOs with a decrease in r in , while the spectrum transits from the hard to soft state.2. Two simultaneous QPOs : We have considered only equatorial eccentric trajectories, Q =0, forthese BHXRBs, as we can estimate only two parameters of the orbit corresponding to twosimultaneous QPOs. First, we find the exact solutions for the parameters { e , r p }, summarizedin Table 8, by equating the centroid frequencies of two simultaneous QPOs (see Table 5) to{ ν φ , ν pp } using our analytic formulae for Q =0, Equations (5a) and (5b). Then we calculate theerrors in the parameters { e , r p } using the method discussed in Appendix E (see Figure 14).The results are summarized in Table 8. These results are described below:• XTEJ 1550-564 : We find that an equatorial trajectory with eccentricity e = 0 . with r p = 4 . (see Table 8) as a solution for the observed QPO frequencies in XTEJ 1550-564.The calculated probability density profiles in e and r p dimensions, P ( e ) and P ( r p ) , werefound to be skew symmetric and were fit by an interpolating function. The correspondingerrors were obtained by taking the integrated probability of 68.2% about the peak value ofthe probability density distributions. The quoted errors are calculated with respect to theexact solution of the parameters, which slightly deviates from the peak of the integratedprofiles { P ( e ) and P ( r p ) }; see Figure 19 and Table 8. These profiles, correspondingmodel fit, and the probability contours in the ( e , r p ) plane are shown in Figure 19.8 Rana and Mangalam 2020 T a b l e . Su mm a r y o f R e s u l t s C o rr e s p o nd i n g t o ( N o n ) e q u a t o r i a l E cce n tr i c S o l u t i o n s ( e Q a nd e ) f o r B HX R B s M X - nd G R O J - . B H X R B Q e R a n g e R e s o l u t i o n E x a c t M o d e l F i t r p R a n g e R e s o l u t i o n E x a c t M o d e l F i t a R a n g e R e s o l u t i o n E x a c t M o d e l F i t ∆ e S o l u t i o n t o P ( e ) ∆ r p S o l u t i o n t o P (cid:0) r p (cid:1) ∆ a S o l u t i o n t o P ( a ) e r p a M X - . − . . . . + . − . . − . . . . + . − . . − . . . . ± . . − . . . . + . − . . − . . . + . − . . − . . . . ± . . − . . . . + . − . . − . . . . + . − . . − . . . . ± . . − . . . . + . − . . − . . . . + . − . . − . . . . ± .
009 4 . − . . . . + . − . . − . . . . + . − . . − . . . . ± . G R O J - − . . - . + . − . . − . . - . + . − . . − . . - . ± .
003 1 − . . - . + . − . . − . . - . + . − . . − . . - . ± .
003 2 − . . - . + . − . . − . . - . + . − . . − . . - . ± .
003 3 − . . - . + . − . . − . . - . + . − . . − . . - . ± .
003 4 − . . - . + . − . . − . . - . + . − . . − . . - . ± . N o t e — T h ec o l u m n s d e s c r i b e t h e r a n g e o f p a r a m e t e r v o l u m ec o n s i d e r e d f o r { e , r p , a } w i t h a c h o s e n r e s o l u t i o n t o c a l c u l a t e t h e n o r m a li ze dp r o b a b ili t y d e n s i t y a t e a c hp o i n t i n s i d e t h e p a r a m e t e r v o l u m e u s i n g E q u a t i o n ( E b ) , t h ee x a c t s o l u t i o n s f o r { e , r p , a } c a l c u l a t e du s i n g E q u a t i o n s ( ) − ( c ) , a nd t h e r e s u l t s o f t h e m o d e l fi tt o P ( e ) , P (cid:0) r p (cid:1) , a nd P ( a ) , f o r e a c h v a l u e o f Q b e t w ee n nd . Geometric origin for QPOs T a b l e . Su mm a r y o f R e s u l t s f o r { e , r p , Q } P a r a m e t e r S o l u t i o n s a nd C o rr e s p o nd i n g E rr o r s f o r Q P O s i n B HX R B s M X - nd G R O J - . B H X R B e R a n g e R e s o l u t i o n E x a c t M o d e l F i t r p R a n g e R e s o l u t i o n E x a c t M o d e l F i t Q R a n g e R e s o l u t i o n E x a c t M o d e l F i t ∆ e S o l u t i o n t o P ( e ) ∆ r p S o l u t i o n t o P (cid:0) r p (cid:1) ∆ Q S o l u t i o n t o P ( Q ) e r p Q M X - . − . . . . + . − . . − . . . . + . − . − . . . + . − . G R O J - − . . - . + . − . . − . . - . + . − . − . - + . N o t e — T h ec o l u m n s d e s c r i b e t h e r a n g e o f p a r a m e t e r v o l u m e t a k e n f o r { e , r p , Q } , a nd t h ec h o s e n r e s o l u t i o n t o c a l c u l a t e t h e n o r m a li ze dp r o b a b ili t y d e n s i t y a t e a c hp o i n t i n s i d e t h e p a r a m e t e r v o l u m e , t h ee x a c t s o l u t i o n s , a nd t h e r e s u l t s o f t h e m o d e l fi tt o t h e i n t e g r a t e dp r o fi l e s . T h e s p i n o f t h e b l a c k h o l e i s fi x e d t o t h e m o s t p r o b a b l ee s t i m a t e s , w h i c h a r e a = . f o r M X - nd a = . f o r G R O J - . Rana and Mangalam 2020
Table 8.
Summary of Results Corresponding to the Equatorial Eccentric Orbit Solutions for BHXRBs XTEJ1550-564, 4U 1630-47, and GRS 1915+105.
BHXRB e Range Resolution Exact Model Fit r p Range Resolution Exact Model Fit ∆ e Solution to P ( e ) ∆ r p Solution to P ( r p ) e r p XTEJ 1550-564 . − . +0 . − . . − . +0 . − .
4U 1630-47 . − . +0 . − . − . +0 . − . GRS 1915+105 . − . ± . − +0 . − . Note — The columns describe the parameter range considered for { e , r p }, its resolution, the exact solutions for { e , r p }calculated using { ν φ , ν pp } for XTEJ 1550-564 and GRS 1915+105, and using { ν φ , ν np } for 4U 1630-47 using Equations (5a),(5b), and (5c), and results of the model fit to P ( e ) and P ( r p ) . •
4U 1630-47 : We found an exact solution at { e = 0 . , r p = 2 . } (see Table 8) byequating { ν φ , ν np } instead of { ν φ , ν pp } to the centroid QPO frequencies. This might bebecause the QPO with a lower frequency of ∼ Hz (see Table 5) is too small to be anHFQPO. The calculated probability density profiles in the e and r p dimensions, the corre-sponding model fit, and the probability contours in the ( e , r p ) plane are shown in Figure20. In this case, too, we see that P ( e ) and P ( r p ) profiles are skew, such that the inte-grated probability is 68.2% about the peak value of the probability density distributions,and the errors are quoted with respect to the exact solution of the parameters, whichslightly deviates from the peak of the integrated profiles P ( e ) and P ( r p ) (see Figure 20and Table 8). We see that a highly eccentric orbit is found as the most probable solution.• GRS 1915+105 : We found an exact solution at { e = 0 . , r p = 1 . }; see Table 8.We find a highly eccentric equatorial trajectory as the most probable solution that cangive the observed QPO frequencies in GRS 1915+105. This result is similar to the case of4U 1630-47, which leads us to observe that a black hole with a high spin value prefers ahighly eccentric orbit solution to simultaneous QPOs. The calculated probability densityprofiles P ( e ) and P ( r p ) are well fit by the Gaussian. The corresponding model fit andthe probability contours in the ( e , r p ) plane are shown in Figure 21.Hence, we conclude that for XTEJ 1550-564, 4U 1630-47, and GRS 1915+105, the e model inthe region r p = 1 . − . are the probable cause of the observed QPOs in the power spectrum.We found high eccentricity values for the orbits as solutions for QPOs in the cases of BHXRB4U 1630-47 and GRS 1915+105, and this indicates that black holes with very high spin valuesprefer highly eccentric orbits in the QPO solutions.We show all of the eccentric trajectory solutions together for both Q = 0 and Q (cid:54) = 0 in Figure 22 inthe ( r p , a ) plane along with the radii ISCO (ISSO), MBCO (MBSO), light radius, and the horizon. Geometric origin for QPOs ( e ) (a) r p e + % % (b) r p + % % % (c) r p ( r p ) (d) Figure 19.
The integrated density profiles of BHXRB XTEJ 1550-564 are shown in (a) P ( e ) and (d) P ( r p ) , where the dashed vertical lines enclose a region with 68.2% probability, and the solid vertical linecorresponds to the peak of the profiles. The probability contours of the parameter solution are shown in the(b) ( r p , e ) and (c) ( e , r p ) planes, where the + sign marks the exact solution. We see that the calculated eccentric orbit solutions are found in region 1 of the ( r p , a ) plane (asdefined in Figure 5) and near ISCO for Q = 0 in the cases of BHXRB 4U 1630-47, GROJ 1655-40,and GRS 1915+105. The trajectory solutions are found in region 2 near ISCO for XTEJ 1550-564( Q = 0 ) and near ISSO for M82 X-1 ( Q = 2 . ; as defined in Figure 5). These results are alsoconsistent with the results discussed in §2.1, except that very high e values are found for trajectories inBHXRB 4U 1630-47 and GRS 1915+105. Hence, we conclude that the eccentric trajectory solutionswith Q = 0 and Q (cid:54) = 0 for the observed QPOs in BHXRBs are found either in the region 1 or region2 of the ( r p , a ) plane but close to the ISCO (ISSO) curve; we call this radius as R . As all theseorbit solutions are distributed near R , it is expected that this radius is very close to the inner edgeradius, r in , of the circular accretion disk, which could also be a source of blobs that are generatingthese QPOs. The torus region, shown in Figure 18, spans a part of regions 1 and 2 near R , whichcan be represented as (cid:0) R − ∆ (cid:1) , where ∆ i represents a small deviation from R (which need not bethe center point of the torus in this scenario). This means that the orbits near R are induced bythe instabilities in the inner flow to be (non)equatorial and eccentric.3.2.2. Spherical Orbits
Here we summarize the results of associating the spherical orbits around a Kerr black hole withQPOs in BHXRBs. We limited this study to the cases of BHXRBs M82 X-1 and XTEJ 1550-564, aswe found the exact solutions for the parameters { r s , a , Q } or { r s , a } for only these two BHXRBs whenwe solved { ν φ = ν , ν pp = ν , ν np = ν } for M82 X-1 and { ν φ = ν , ν pp = ν } for XTEJ 1550-5642 Rana and Mangalam 2020 ( e ) (a) r p e + % % (b) r p + % % % (c) r p ( r p ) (d) Figure 20.
The integrated density profiles are shown in (a) P ( e ) and (d) P ( r p ) for BHXRB 4U 1630-47,where the dashed vertical lines enclose a region with 68.2% probability, and the solid vertical line correspondsto the peak of the profiles. The probability contours of the parameter solution are shown in the (b) ( r p , e )and (c) ( e , r p ) planes, where the + sign marks the exact solution. using Equations (7a − M82 X-1 : We found the exact solution for a spherical orbit at { r s = 6 . , a = 0 . , Q =6 . } for M82 X-1. The spherical trajectory with these parameter values is shown in Figure23(a). The calculated probability density profiles and the model fit are shown in Figure 24. The P ( r s ) and P ( Q ) profiles were found to be skew symmetric, and the integrated probabilityis 68.2% about the peak of the probability density distribution between the error bars, while P ( a ) is well fit by a Gaussian. We see that the spin of the black hole is also found very close tothe spin solutions estimated in §3.2.1. We conclude that along with the eQ trajectories havingmoderate eccentricities, as discussed in §3.2.1, a spherical trajectory ( Q ) at r s = 6 . with Q = 6 . is also a viable solution that can produce the observed QPO frequencies in M82X-1. The corresponding spin estimate a = 0 . ± . was utilized in §3.2.1 using Equation(10b) to calculate the most probable value of the spin for M82 X-1.• XTEJ 1550-564 : A spherical trajectory solution was found at r s = 5 . and Q = 2 . forBHXRB XTEJ 1550-564 that is shown in Figure 23(b), and the calculated probability densityprofiles, the Gaussian model fit, and the probability contours in the { r s , Q } plane are shownin Figure 25. So, along with an e trajectory, as discussed in §3.2.1, a Q orbit is also a viablecandidate for the observed QPOs in the temporal power spectrum of XTEJ 1550-564. Geometric origin for QPOs ( e ) (a) r p e + % % (b) r p + % % % (c) r p ( r p ) (d) Figure 21.
The integrated density profiles are shown in (a) P ( e ) and (d) P ( r p ) for BHXRB GRS1915+105, where the dashed vertical lines enclose a region with 68.2% probability, and the solid vertical linecorresponds to the peak of the profiles. The probability contours of the parameter solution are shown in the(b) ( r p , e ) and (c) ( e , r p ) planes, where the + sign marks the exact solution. r p a HorizonLight radiusMBCOISCO (a) r p a HorizonLight radiusMBSOISSO (b)
Figure 22.
Equatorial eccentric orbit solutions for QPOs observed in BHXRBs GROJ 1655-40 (purple),XTEJ 1550-564 (cyan), 4U 1630-47 (brown), and GRS 1915+105 (orange) for (a) Q = 0 ; and (b) thenonequatorial eccentric orbit solution for BHXRB M82 X-1 (magenta) for Q = 2 . . Rana and Mangalam 2020 (a) (b)
Figure 23.
Spherical trajectories corresponding to the exact solutions calculated for (a) M82 X-1 at { r s =6 . , a = 0 . , Q = 6 . } and for (b) XTEJ 1550-564 at { r s = 5 . , a = 0 . , Q = 2 . }, as alsoprovided in Table 9. r s ( r s ) (a) ( a ) (b) ( Q ) (c) Figure 24.
Probability density profiles in { r s , a , Q } dimensions for M82 X-1: (a) P ( r s ) , (b) P ( a ) ,and (c) P ( Q ) . The black points represent normalized probability density profiles generated using themethod described in §3.2, and the blue curves are the model fit, and the results are summarized in Table9. The errors for the P ( r s ) and P ( Q ) profiles are obtained such that the integrated probability betweenthe vertical dashed curves is 68.2%, whereas the vertical thick curves correspond to the peak value of thereduced probability density distributions. Geometric origin for QPOs T a b l e . Su mm a r y o f R e s u l t s C o rr e s p o nd i n g t o t h e Sph e r i c a l O r b i t S o l u t i o n s f o r B HX R B s M X - nd X T E J - . B H X R B r s R a n g e R e s o l u t i o n E x a c t M o d e l F i t Q R a n g e R e s o l u t i o n E x a c t M o d e l F i t a R a n g e R e s o l u t i o n E x a c t M o d e l F i t ∆ r s S o l u t i o n t o P ( r s ) ∆ Q S o l u t i o n t o P ( Q ) ∆ a S o l u t i o n t o P ( a ) r s Q a M X - . − . . . . + . − . − . . . + . − . . − . . . . ± . X T E J - − . . . ± . . − . . . + . − . ---- N o t e — T h ec o l u m n s d e s c r i b e t h e r a n g e o f p a r a m e t e r v o l u m ec o n s i d e r e d f o r { r s , a , Q }a nd i t s r e s o l u t i o n t o c a l c u l a t e t h e n o r m a li ze dp r o b a b ili t y d e n s i t y u s i n g E q u a t i o n ( E b ) , t h ee x a c t s o l u t i o n s f o r { r s , a , Q } c a l c u l a t e du s i n g E q u a t i o n s ( ) − ( c ) , t h e v a l u e o f p a r a m e t e r s c o rr e s p o nd i n g t o t h e p e a k o f t h e i n t e g r a t e dp r o fi l e s i n { r s , a , Q } , a nd r e s u l t s o f t h e m o d e l fi tt o P ( r s ) , P ( Q ) , a nd P ( a ) . Rana and Mangalam 2020 r s ( r s ) (a) r s + % % (b) r s Q + % % % (c) ( Q ) (d) Figure 25.
The integrated density profiles are shown in (a) P ( r s ) and (d) P ( Q ) for the spherical orbitsolution of BHXRB XTEJ 1550-564, where the dashed vertical lines enclose a region with 68.2% probability,and the solid vertical line corresponds to the peak of the profiles. The probability contours of the parametersolution are shown in the (b) ( Q , r s ) and (c) ( r s , Q ) planes, where the + sign marks the exact solution. We found that the spherical trajectories are also possible solutions for QPOs in BHXRBs M82 X-1( a = 0 . , Q = 6 . , r s = 6 . , r I = 5 . ) and XTEJ 1550-564 ( a = 0 . , Q = 2 . , r s = 5 . , r I = 4 . ). This indicates that the spherical trajectory solutions are in region 1 of the ( r , a ) plane,as defined in Figure 5; for both BHXRBs, and they are very close to the ISSO radius, r I . Theseresults are also consistent with the results discussed in §2.2, where the QPO-generating region isclose to the ISSO curve in the ( r , a ) plane. For the case of M82 X-1, the spherical trajectory solutionhas a different value of spin compared to the ones estimated in §3.2.1, but it is very close to theother estimates given in Table 6. This value of spin, together with other results in Table 6, is usedto estimate the most probable value of spin of the black hole for M82 X-1, which is a = 0 . . Wealso see that a low eccentric trajectory prefers a high Q value and vice versa, as seen from the resultsshown in Table 6. As the Q value of the orbit is increased, the eccentricity of the trajectory solutiondecreases for both BHXRBs M82 X-1 and GROJ 1655-40. This trend is also followed here: for thespherical orbit ( e = 0 ), Q ∼ is found as a solution for M82 X-1 and Q ∼ . for XTEJ 1550-564,whereas a moderately eccentric trajectory solution was found with Q = 0 for XTEJ 1550-564; seeTable 8.We conclude that various kinds of Kerr orbits, for example, spherical { e = 0 , Q (cid:54) = 0 }, equatorialeccentric { e (cid:54) = 0 , Q = 0 }, and nonequatorial eccentric { e (cid:54) = 0 , Q (cid:54) = 0 }, are also viable solutions forQPOs in BHXRBs. Hence, such trajectories with similar fundamental frequencies can together givea strong QPO signal in the temporal power spectrum. THE PBK CORRELATION
Geometric origin for QPOs ν np ( Hz ) ν ϕ ( H z ) (a) ν np ( Hz ) ν pp ( H z ) (b) ν np ( Hz ) ν pp ( H z ) (c) ν np ( Hz ) ν np ( H z ) (d) Figure 26.
The PBK correlation is shown for BHXRB GROJ 1655-40 as previously observed [data pointsare from Motta et al. (2014a)]. The observed correlation is in good agreement with the frequencies of the e solution estimated, where { e = 0 . , a = 0 . , Q = 0 , M = 5 . }, for GROJ 1655-40 in §3.2.1, where(a) ν φ , (b) ν pp in the low-frequency range, (c) ν pp in the high-frequency range, and (d) ν np are shown. Theblue, black, and red data points represent the L u , L l , and L LF components of the PDS, respectively. Themagenta curves show the theoretical values of frequencies. A tight correlation between the frequencies of two components in the PDS of various sources,including black hole and neutron star X-ray binaries, was discovered (Psaltis et al. 1999). Such acorrelation among various variability components of the PDS in both types of sources suggests acommon and important emission mechanism for these signals. This correlation is either betweentwo QPOs, an LFQPO and either of the two HFQPOs, or it is between an LFQPO and high-frequency broadband noise components. We adopt the definition of Belloni et al. (2002) for thesevariability components: L LF for LFQPO, and L l and L u for lower and upper HFQPOs or broad noisecomponents. A systematic study of 571 RXTE observations was carried out for BHXRB GROJ 1655-40 between 1996 March and 2005 October (Motta et al. 2014a), and they also found such correlationbetween the type C QPOs and high-frequency QPOs and broadband components (either L l or L u ; seeTables 1 and 2 and Figure 5 of Motta et al. (2014a)). In this study, they calculated mass, spin of theblack hole, and the radius at which QPOs originated { M = 5 . , a = 0 . , r = 5 . } (Motta et al.2014a) using { L u = ν φ , L l = ν pp , L LF = ν np }, assuming that circular equatorial orbits are the originof three simultaneous QPOs in the RPM ( model as defined in Figure 1). Using the estimatedvalues of M and a , they fit the PBK correlation of variability components in GROJ 1655-40 byvarying r .Here we apply the e model solution calculated in §3.2.1 assuming { L u = ν φ , L l = ν pp , L LF = ν np },using the observation ID having three simultaneous QPOs detected in GROJ 1655-40 (shown in Table8 Rana and Mangalam 2020 r p ν ( H z ) ν ϕ ν pp (a) r p ν np ( H z ) (b) Figure 27.
The frequencies (a) ν φ and ν pp , (b) ν np are shown as function of r p , for the e solution vector{ e = 0 . , a = 0 . , Q = 0 , M = 5 . }. Table 10.
Nonequatorial Eccentric Orbit ( eQ ) So-lutions for L l and L LF Components Detected inRXTE Observations of GROJ 1655-40 (Motta et al.2014a), Where the First Row Corresponds to theObservation ID with Three Simultaneous QPOs. L LF L l r p e Q (Hz) (Hz)17.3 298 5.25 0.071 00.106 3.3 29.179 0.077 24.4230.117 3.9 28.228 0.083 33.9030.123 4 27.758 0.083 33.3920.128 4 27.389 0.083 32.6420.11 3.5 28.818 0.082 33.6220.115 3.7 28.392 0.083 34.0280.128 4.2 27.389 0.083 33.0100.157 4.8 25.576 0.083 30.9641.333 29 12.464 0.079 10.9210.46 12 17.826 0.085 22.343 Note — The mass of the black hole was fixed to M = 5 . andspin was fixed to a = 0 . . M = 5 . (Beer & Podsiadlowski2002) and the spin of the black hole to the most probable value, a = 0 . , estimated by minimizingthe χ function, given by Equation (11). We fix e and Q to the values estimated by the fine-gridmethod { e = 0 . , Q = 0 } and vary r p to calculate the frequencies. In Figure 26, we show thecorrelations of the frequencies corresponding to the parameters { e = 0 . , a = 0 . , Q = 0 },which are in good agreement with the PBK correlation. In Figure 27, these frequencies are shownas functions of r p . We see that the data points for L u components fit very well (see Figure 26(a)),whereas the L l components show a good fit in the high-frequency range [see Figures 26(b),26(c)].The L LF components also show good agreement with the eccentric orbit solution (see Figure 26(d)).Thirty-four L l and L LF components which were detected simultaneously in the same observationID [see Table 1 of Motta et al. (2014a)]. To calculate r p , we first solve for L LF = ν np for the solution Geometric origin for QPOs e = 0 . , Q = 0 , a = 0 . , M = 5 . }; this locates the r p , where oscillations are present, toa good approximation. Using these r p values, we simultaneously solve { ν pp = L l , ν np = L LF } usingthe centroid frequencies of these components and estimate the exact solutions for parameters { e , Q }with { a = 0 . , M = 5 . }. In 10 out of 34 cases, we found low-eccentricity eQ solutions for thesePDS components, where the calculated parameters are shown in Table 10. We find orbits with high Q values at large r p (this is expected as Q ∝ L − L z ) as solutions for these PDS components. Thisexercise confirms the existence of eQ in addition to e solutions for QPOs. GAS FLOW NEAR ISSO (ISCO)In this section, we discuss our torus picture of eccentric trajectories, and we examine the model offluid flow in the general-relativistic thin disk around a Kerr black hole (Penna et al. 2012; Mohan& Mangalam 2014) with the aim of finding a source of the e , eQ , and Q trajectories. In thismodel, the region around the rotating black hole was divided into various regimes: (1) the plungeregion between the ISCO radius and black hole horizon dominated by gas pressure and electronscattering based opacity, (2) the edge region at and very near to the ISCO radius dominated by gaspressure and electron scattering based opacity, (3) the inner region outside the edge region with smallradii comparable to ISCO dominated by radiation pressure and electron scattering based opacity, (4)the middle region outside the inner region where gas pressure again dominates over the radiationpressure and electron scattering based opacity, (5) the outer region far from the black hole horizonand outside the middle region dominated by gas pressure and electron scattering based opacity. Theanalytic forms for the important quantities like flux of radiant energy, F , temperature, T , and radialvelocity in the locally nonrotating frame, β r , were given for these different regions (as functions of r , a , viscosity, α , accretion rate, ˙ m = ˙ M • / ˙ M Edd , and M • ) where nonzero stresses were incorporated atthe inner edge of the disk in this model (Penna et al. 2012). Also, the expression for quality factor Q φ ( r, a, β r ) was derived for ν φ QPO frequencies in the equatorial plane, which is given by [Mohan &Mangalam (2014), typo fixed in Equation (10)] Q φ ( r, a, β r ) = −√ A πβ r ∆ r / (cid:34) − ( A Ω − ar ) Σ ∆ (cid:35) − / , (12)where A = ( r + a ) − a ∆ sin θ , ∆ = r + a − r , Σ = r + a cos θ , and Ω = 1 / (cid:0) r / + a (cid:1) , andwhere θ = π/ is assumed in Equation (12). Using this formula, one can obtain the quality factor ofthe QPO in various regions close to the black hole by substituting the β r of the corresponding regionas defined above. The expressions for β r in the edge and inner regions are given by (Equations (12),(13) of Mohan & Mangalam (2014)) β r,edge = − . × − α / m − / ˙ m / r − / B / C − / D / Φ − / , (13a) β r,inner = − . α ˙ m r − / A B − C − / D − / S − Φ , (13b)where m = M • / M (cid:12) , C = 1 − r − + 2 ar − / [there is a typo in the expression of C , Equation(A4c), in Penna et al. (2012)]; and A , B , D , S , and Φ are given in Penna et al. (2012) (EquationsA4(a), (b), (d), (o) and (3.6)).0 Rana and Mangalam 2020
In Figure 28(a) and 28(b), we have shown the contours for β r and Q φ for the edge region in the ( r , a ) plane, and the p gas /p rad ratio as a function of r in Figure 28(c). One can discern the transitionfrom the inner to edge region by the sudden increase of the p gas /p rad ratio, as seen in Figure 28(c),which is given by [Penna et al. (2012), Equation (3.7g)] p gas p rad = 1 . × − m − / α − / ˙ m − r / A − / B / DS / Φ − . (14) a - - - - - - - - (a) a (b) p ga s / p r ad Edgeregion Innerregion (c)
Figure 28.
Contours of (a) β r and (b) Q φ in the ( r , a ) plane in the edge region of the general-relativisticthin disk, and (c) p gas /p rad as a function of r with a = 0 . (where the dotted vertical curve corresponds toISCO and the solid vertical curve corresponds to r when p gas /p rad = 1 ). We have fixed { α = 0 . , m = 1 , ˙ m = 0 . }. Table 11.
Ranges of r , Pressure Ratio, p gas /p rad , Quality Factor, Q φ , and Radial Velocity, β r , in the Edge andInner Regions of Fluid Flow in the Relativistic Thin Accretion Disk around a Kerr Black Hole (Penna et al.2012; Mohan & Mangalam 2014), Where We Have Fixed { m = 1 , α = 0 . } for BHXRBs. Region ( a = 0 . , ˙ m = 0 .
1) ( a = 0 . , ˙ m = 0 .
1) ( a = 0 . , ˙ m = 0 .
3) ( a = 0 . , ˙ m = 0 . (cid:0) r, p gas /p rad , β r , Q φ (cid:1) (cid:0) r, p gas /p rad , β r , Q φ (cid:1) (cid:0) r, p gas /p rad , β r , Q φ (cid:1) (cid:0) r, p gas /p rad , β r , Q φ (cid:1) Edge . − .
93 4 . − .
87 4 . − .
25 4 . − . . − .
84 1 . − .
41 1 . − .
921 1 . − . - (2 . − . × − - (3 . − . × − - (1 . − . × − - (1 . − . × − . − .
12 1019 . − .
07 694 . − .
59 773 . − . Inner . − .
22 4 . − .
81 5 . − . . − . . − .
998 0 . − .
999 0 . − .
998 0 . − . - (1 . − . × − - (1 . − . × − - (1 . − . × − - (1 . − . × − . − .
85 501 . − . . − .
26 55 . − . In Table 11, we give the range of { r , Q φ , β r , p gas /p rad } for the edge and inner regions for differentcombinations of a and ˙ m , fixing { m = 1 , α = 0 . } for BHXRBs, with a low accretion rate ( ˙ m (cid:39) . ) Geometric origin for QPOs ˙ m (cid:39) . ) corresponding to thesoft spectral state of BHXRBs. We see a sharp rise in p gas /p rad values in the edge region in Figure28(c). The ranges of Q φ in both the edge and inner regions are very high compared to those observedin BHXRBs ( Q φ = 5 − ). We, therefore, suggest that the QPOs are coming from a region veryclose to and inside ISCO; we identify this with the torus region, consisting of geodesics (Penna et al.2012), and hence Q φ is different. This is also supported by the observation that the edge-flow-sourcedgeodesics populate the torus region obtained here for M82 X-1 ( r = 4 . − . ) and GROJ 1655-40( r = 5 . − . ); see Figure 18. Specifically, the sharp pressure ratio gradient suggests that theedge region can be a launchpad for the instabilities that then oscillate with fundamental frequencies,causing geodesic flows in the torus region inside ISCO ( r < r ISCO ), where the fluid motion is close toHamiltonian flow. A further understanding of this proposal (or conjecture) can be gained by carryingout a detailed model or simulation of the GRMHD flow in the edge region. DISCUSSION, CAVEATS, AND CONCLUSIONSThe QPOs in BHXRBs have been an important probe for comprehending the inner accretion flowclose to the rotating black hole. Many theoretical models have been proposed in the past to explainits origin and in particular LFQPOs and HFQPOs (Kato 2004; Török et al. 2005; Tagger & Varnière2006; Germanà et al. 2009; Ingram et al. 2009; Ingram & Done 2011, 2012). These various modelshave been able to explain different properties of QPOs. For example, one of these models attributesthe HFQPOs to the Rossby instability under the general relativistic regime (Tagger & Varnière 2006),whereas another model attributes type C QPOs to the Lense − Thirring precession of a rigid torus ofmatter around a Kerr black hole (Ingram et al. 2009; Ingram & Done 2011, 2012). Although thesemodels can explain either LFQPOs or HFQPOs, they do not explain the simultaneity of these QPOs,as previously observed in BHXRB GROJ 1655-40 (Motta et al. 2014a). The RPM, which is basedon the geometric phenomenon of the relativistic precession of particle trajectories, explains thesesimultaneous QPOs as { ν φ , ( ν φ − ν r ) , ( ν φ − ν θ ) } of a self-emitting blob of matter (or instability) ina bound orbit near a Kerr black hole. We have extended the RPM for QPOs in BHXRBs to studyand associate the fundamental frequencies of the bound particle trajectories near a Kerr black hole,which are eQ , e , and Q solutions with the frequencies of QPOs. We call this as the generalizedRPM (GRPM). Recently, novel and compact analytic forms for the trajectories around a Kerr blackhole and their fundamental frequencies were derived (Rana & Mangalam 2019a,b). We applied theseformulae to the GRPM to extract the QPO frequencies. Graphical examples of these trajectoriesaround a Kerr black hole are shown in Figures 18, 23, and 29. A summary of these results is givenin Table 12.We add the following caveats and conclusions:1. Novel and useful formulae : We have derived novel forms for the spherical trajectory solu-tions { φ ( r s , a, Q ) , t ( r s , a, Q ) }, given by Equation (B12), and their fundamental frequencies{ ¯ ν φ ( r s , a, Q ) , ¯ ν r ( r s , a, Q ) , ¯ ν θ ( r s , a, Q ) }, given by Equation (7). A reduced form of the verti-cal oscillation frequency, ¯ ν θ ( e, r p , a ) given by Equation (5c), for equatorial eccentric orbits isalso derived in Appendix A. These new and compact formulae are useful for various theoret-ical studies of Kerr orbits, besides other astrophysical applications (e.g., Rana & Mangalam(2020)).2 Rana and Mangalam 2020
PRODUCED BY AN AUTODESK STUDENT VERSION P R O DUC E D B Y AN AU T O D ES K S T UD E N T VE R S I O N PRODUCED BY AN AUTODESK STUDENT VERSION P R O DUC E D B Y AN AU T O D ES K S T UD E N T VE R S I O N Figure 29.
The cartoon shows a geometric model explaining the region of origin of QPOs assuming themore general nonequatorial eccentric trajectories in the GRPM, where the torus extent is R − ∆ (and toruswidth ∆ r = ∆ + ∆ ). Table 12.
Summary of Orbital Solutions Found for QPOs Observed in Five BHXRBs Using the GRPM in ThisArticle, and the Corresponding Region of the ( r p , a ) Plane Where QPOs Originate. BHXRB Number Model e r p a Q MBSO ISCO ISSO Region inof QPOs Class ( r p , a ) Plane M82 X-1 3 eQ . +0 . − . . +0 . − . .
299 2 . +1 . − . Q +0 . − . ± +2 . − . eQ . +0 . − . . +0 . − . .
283 0 +0 . - 5.039 - 1XTE J1550-564 2 e +0 . − . +0 . − . Q ± +1 . − . e +0 . − . +0 . − . e ± +0 . − . Orbital solutions : The fundamental frequencies of the eQ , e , and Q trajectories are in therange of QPO signals observed in BHXRBs, so these are viable solutions for explaining theobserved QPOs in BHXRBs M82 X-1, GROJ 1655-40, XTEJ 1550-564, 4U 1630-47, and GRS1915+105 in the GRPM paradigm. We see that these trajectory solutions are found in eitherregion 1 or 2 of the ( r , a ) plane, as defined in Figure 5, and shown in Figure 22. The values of Geometric origin for QPOs Q solution. Asummary of these parameter solutions and corresponding MBSO, ISCO, and ISSO radii for allBHXRBs is given in Table 12.3. Trajectories in the torus : We found trajectories, having different parameter combinationswithin the estimated range of errors in the orbital parameters and having fundamental fre-quencies within the width of the observed QPOs, as solutions for QPOs in BHXRBs M82X-1 and GROJ 1655-40. We also found that the distinct parameter solutions found for thesecases follow a trend that, as the eccentricity of the orbit decreases, the Q value increases fora given QPO frequency pair. This behavior can also be understood from Figures 8 −
10, wherethe frequencies increase as Q increases, but decrease as e increases for a given r p . This im-plies that to obtain the degenerate parameter solutions for the same set of frequencies, a loweccentricity ⇐⇒ high Q trend is expected. We also found that these trajectories span a torusregion near the Kerr black hole, as shown in Figure 18, which together give rise to the samepeaks in the power spectrum. This should also explain the strong rms seen for the HFQPOsand type C LFQPOs. Another possibility of a rigidly precessing torus was suggested (Ingramet al. 2009; Ingram & Done 2011, 2012); our proposal consists of a nonprecessing torus, whichincludes all viable solutions of the GRPM: eQ , e , and Q trajectories.4. Torus region : The emission of simultaneous QPOs is expected from a region where differenttrajectories having similar fundamental frequencies span a torus, as shown in Figure 18 andthey can together show a strong peak in the power spectrum. The inner radius of the circularaccretion disk is expected to be close to this torus region in such a scenario. In Figure 29, wedepict this geometric model where the emission region of the simultaneous QPOs is shown asa torus region close to the inner edge of the accretion disk. This torus region is expected to beoutside the MBSO radius, and the ISSO radius is expected to be in between the torus regionfor the eccentric orbit solutions, as observed in the case of M82 X-1. The torus region can berepresented as R − ∆ , where R is an e = 0 orbit (ISCO or ISSO) and ∆ i represents the regionvery close to R . The width of the torus region in this model is given by ∆ r = (∆ + ∆ ) . Allof the orbit solutions are found to be distributed near R ; hence, it is expected that this radiuscorresponds to the inner edge radius, r in , of the circular accretion disk. This torus region existsin region 1 and(or) 2 near the R radius. Due to the instabilities in the inner flow, we arguethat the nearly e orbits near the R radius transcend to eQ orbits. Based on the geometry ofthe orbits and the emission region, we plan to build a detailed GRMHD-based model to expandon the GRPM paradigm. More cases of BHXRBs with three simultaneous QPOs, if detectedin the future, will help us test our models.5. Highly eccentric solutions : For QPOs in BHXRBs 4U 1630-47 and GRS 1915+105, we foundhighly eccentric e solutions. This indicates that black holes with high spin values prefer highlyeccentric trajectories as solutions to the QPOs. This behavior can also be understood from4 Rana and Mangalam 2020
Figures 8 −
10, where we see that for black holes with very high spins, the QPOs originate veryclose to the black hole, and the solution contours move close to the black hole as e increases.This implies that more eccentric orbits are preferred for a given frequency pair of QPOs for ablack hole with very high spin. We do not find any spherical orbit solution for QPOs in thesetwo BHXRBs, which confirms that the orbital solution is purely equatorial, but such highlyeccentric solutions are unlikely. We expect more and better estimates of the orbital solutionsin the future if a more precise estimate of the spin is available, or if three simultaneous QPOsare discovered in BHXRBs 4U 1630-47 and GRS 1915+105. For the case that we studied inthis paper of 4U 1630-47, the lower frequency of the QPO pair probably has a different originthan the high-frequency feature suggested by Klein-Wolt et al. (2004). However, even in sucha scenario, the frequency range of this QPO still implies an origin near the torus region in ourmodel. There was also another pair of QPOs observed in 4U 1630-47 (Klein-Wolt et al. 2004),for which there was no exact solution found in the orbital parameter space.6. Nonequatorial solutions : In the case of BHXRBs M82 X-1 and XTE 1550-564, we found both eQ ( e for XTE 1550-564) and Q solutions, and the spin determinations are slightly differentfor the two different types of trajectory solutions. These solutions were found close to andoutside their corresponding ISSO radii. The mass of the black hole in case of M82 X-1 wasfixed to the intermediate-mass black hole (IMBH) range, M = 428 , because the QPOs observedin the low-frequency range (3 − M , if found in the IMBH range, will notsignificantly change the result. However, if, in the future, a more reliable and precise estimateplaces it in the stellar-mass range, then the outcome from the GRPM will be dramaticallydifferent. The QPOs observed in XTE 1550-564 by Miller et al. (2001) were later shown to bethe result of the data averaging by Motta et al. (2014b), where the same QPO moved up in thefrequency, appearing as a distinct QPO. As in the case of 4U 1630-47, the range of this QPOfrequency still implies an origin near the torus region.7. Spectral states : We suggest that HFQPOs originate when r in comes very close (nearISCO/ISSO) to the black hole during the soft spectral state of the outburst. When r in isfarther out as in the hard state, the resulting type C QPO frequency is of the order of milli-hertz. As a type C QPO occurs more frequently and is prone to the vertical oscillations, theincrease in its frequency is explained as an increase in ν np when r in decreases, with the spectraltransition from the hard to soft state.8. Circularity : The RPM was previously applied to understand the QPOs observed in BHXRBsGROJ 1655-40 and XTEJ 1550-564 (Motta et al. 2014a,b) using the fundamental frequenciesof orbits. We have found an eQ solution for GROJ 1655-40 very close to an equatorial orbithaving a very small eccentricity e ∼ . (see Table 12), which is in a very close agreementwith the solution found by Motta et al. (2014a), where their estimated mass of the black hole, M = 5 . , is also very close to our assumption, M = 5 . (see Table 5). Our most probablespin estimated for GROJ 1655-40, a = 0 . , is almost the same as found by Motta et al.(2014a), a ∼ . , but our solution provides a more precise estimation of e and Q values whileconfirming a near orbit solution as assumed by Motta et al. (2014a). For the case of XTEJ Geometric origin for QPOs M = 9 . by Motta et al. (2014b)as also in our model. Our assumption for the spin, a = 0 . +0 . − . (Orosz et al. 2011), is alsonearly the same as the value estimated by Motta et al. (2014b); but our model gives the e and Q solutions for XTEJ 1550-564, having moderate e = 0 . +0 . − . and Q = 2 . +1 . − . values,respectively (see Table 12). This indicates that the assumption of circularity is not alwaysvalid.9. Solution degeneracy : To study the impact of the GRPM (with nonzero e and Q ), we haveexplored the behavior of { δ φ , δ pp , δ np }( e , r p , a , Q ) as defined in Equation (6) as deviationsfrom the behavior (circularity). We find that the frequencies are strongly dependent on e but not so much on Q (see Figures 8 − e , r p , a , Q }, called the isofrequency pairs, for a given combination ofQPO frequencies. This degeneracy is a known behavior of trajectories around a Kerr blackhole (Warburton et al. 2013), where different combinations of { E , L z , Q } can have the sameset { ν φ , ν r , ν θ } for a fixed a . An evidence of this degeneracy is also seen in Figures 8 − δ φ , δ pp , δ np }( e , r p , a , Q ) have multiple solutions; that is, for a given δ value, there are different combinations of { e , Q } that have distinct contours on the ( r p , a )plane. Unlike RPM, the mass of the black hole is always assumed from the previous estimatesin the GRPM, which is a valid assumption because the underlying physics or behavior of theKerr orbits is independent of M • . The GRPM, along with the statistical method (Figure 14,Appendix E) that is applied, provides a more precise estimation of the spin of the black hole.10. Frequency ratio : The 3:2 and 5:3 ratios of the simultaneous HFQPOs are a phenomenon ob-served in a few cases of BHXRBs: 300 and 450 Hz HFQPOs in GROJ 1655-40 (Remillardet al. 1999b; Strohmayer 2001a), 240 and 160 Hz HFQPOs in H1743-322 (Homan et al. 2005;Remillard et al. 2006). Such claims, other than the case of GROJ 1655-40, are probably not real(Belloni et al. 2012). Hence, the possibility of such ratios is still causes skepticism. However,if true, the GRPM suggests that the origin of these ratios is very close to the torus region and r in .11. The PBK correlation : In §4, we show that the e solution { e = 0 . , a = 0 . , Q = 0 , M = 5 . }, estimated using a fine-grid method in §3.2.1, fits the PBK correlation that waspreviously observed in BHXRB GROJ 1655-40 (Motta et al. 2014a). This fit is shown in Figure26. We also found that 10 observation IDs, where L l and L LF (broad frequency components)were detected simultaneously (Motta et al. 2014a), show low-eccentricity eQ solutions, wherethe calculated parameters are shown in Table 10. The calculated Q values are consistent withlarge r p and small e values. This exercise suggests that eQ solutions for QPOs are viable.12. Probing the disk edge with a GR fluid model : We study a model of fluid flow in the general-relativistic thin accretion disk (Penna et al. 2012; Mohan & Mangalam 2014). We find thatthe disk edge flows into a torus region containing Hamiltonian geodesics that was obtained forM82 X-1 ( r = 4 . − . ) and GROJ 1655-40 ( r = 5 . − . ). Specifically, the sharp gradientof the p gas /p rad pressure ratio, seen in Figure 28(c), suggests that the edge region is a launchpad for the instabilities that orbit with fundamental frequencies of the geodesics in the edgeand geodesic regions, which then follow the geodesics inside the torus region and also close tothe edge region, where Hamiltonian dynamics is applicable, that is built into the GRPM. The6 Rana and Mangalam 2020 range of { r , Q φ , β r , p gas /p rad } for the edge and inner regions for different combinations of a and ˙ m , fixing { m = 1 , α = 0 . }, is given in Table 11, and the contours of β r and Q φ in the ( r , a )plane for the edge region are shown in Figures 28(a) and 28(b). The ranges of Q φ (tuned to ∆ ν ,the width of the observed QPO), which is defined by orbits in the torus which was providedby observed frequency centroids, in both the edge and inner regions are very high comparedto those observed in BHXRBs ( Q φ = 5 − ). We are suggesting that the QPOs originatein the geodesic region. We also see that the edge is adjacent to the torus region (consistingof geodesics) found for M82 X-1 and GROJ 1655-40, implying that the QPOs are originatingfrom geodesics close to the edge region. Hence, the particle and gas dynamics models togetherjustify the scenario sketched in Figure 29, of a unified fluid-particle picture that is the following:the source of the particles in the torus are dynamical instabilities of plasma blobs ejected fromthe edge region. These blobs have zero α and therefore obey the Hamiltonian dynamics. Theclue that the torus region physically overlaps with the edge and geodesic regions is a subjectof future detailed GRMHD models (and simulations).13. Isofrequency combinations : In the cases with three simultaneous QPOs, once a is fixed (tothe most probable value or the previously estimated value), it is easy to predict the remainingparameters { e , r p , Q } using three QPO frequencies. In the case of M82 X-1 and GROJ 1655-40, when a was fixed to the most probable value (Table 7), we obtained a single solution for{ e , r p , Q } and their errors { ∆ e , ∆ r p , ∆ Q }, where this range of parameters spans the torusregion based on the GRPM. However, there is a finite possibility (Warburton et al. 2013) thatdistinct solutions for the { e , r p , Q } triad are obtained for the same triple QPO frequency set,subject to the bound orbit conditions: ≤ e < , Q ≥ , and Equation (2). This completelydepends on the values of the QPO frequency set that are further subject to the constraints ofbound orbit conditions. In the cases where only two simultaneous QPOs exist, it is difficultto predict whether an e = 0 orbit will be preferred over an e > orbit, or a Q = 0 orbit willbe preferred over a Q > orbit, or vice versa. This will be clear when more cases of threesimultaneous QPOs are found and whether they yield distinct solution sets for { e , r p , a , Q },thereby indicating if the torus region at the disk edge is indeed the geometric origin of QPOs.From our numerical experiment, we find a distinct exact solution for { e , r p , Q } for the threesimultaneous QPOs case, where a was fixed to the most probable value. The RPM restrictsthe search to { e = 0 , Q = 0 } orbital solutions, while the GRPM expands it to more generalbut astrophysically possible { e (cid:54) = 0 , Q (cid:54) = 0 } solutions and thereby subsumes the RPM withinits framework. Hence, the GRPM provides more realistic orbit solutions around a Kerr blackhole that are outside the scope of the RPM, thus giving more impetus to probes of physicalmodels of the origin of QPOs.14. Caveats : The results predicted by the GRPM are subject to the veracity of the observed datathat are inputs to our model. For example, in the case of 4U 1630-47 and GRS 1915+105,very highly eccentric orbit solutions obtained by the GRPM are unlikely; this implies that veryhigh spin values in these cases are probably unreliable. Similarly, if M82 X-1 does not hostan IMBH but a stellar-mass black hole or a neutron star, then the results predicted by theGRPM will change drastically. Also, for 4U 1630-47 and XTEJ 1550-564, where the inputfrequencies of QPOs are not very reliable (Klein-Wolt et al. 2004; Motta et al. 2014b), asdiscussed before, the results obtained by the GRPM might not be physically meaningful. As
Geometric origin for QPOs
Future work : In the near future, we expect suitable observational results from the currentlyoperative Indian X-ray satellite, AstroSat, and from future missions, such as eXTP, which isproposed to have instruments with much higher sensitivity for fast variations and X-ray timing.If simultaneous QPO signals are observed from these missions, we expect to test our GRPMfurther.We would like to thank the anonymous referee for detailed and insightful suggestions that haveimproved our paper significantly. We acknowledge the DST SERB grant No. CRG 2018/003415for financial support. We would like to thank Dr. Prashanth Mohan for his useful suggestions andcomments. We would like to thank Saikat Das for helping us with Figure 3 and 29. We acknowledgethe use and support of the IIA-HPC facility.APPENDIX A. VERTICAL OSCILLATION FREQUENCY FOR ECCENTRIC ORBITS ABOUTEQUATORIAL PLANE WITH Q = 0 Here, we derive the θ oscillation frequency for the equatorial eccentric orbits about the equatorialplane. Using Equations (4b) and (4c), we can write ¯ ν θ ¯ ν r = a √ − E z + I ( e, µ, a, Q )2 F (cid:16) π , z − z (cid:17) , (A1a)where the substitution of I ( e, µ, a, Q ) from Equation (6h) of Rana & Mangalam (2019a) into theabove equation yields ¯ ν θ ¯ ν r = µ (1 − e ) a √ − E z + F (cid:16) π , k (cid:17)(cid:112) C − A + √ B − ACF (cid:16) π , z − z (cid:17) . (A1b)By the substitution of A , B , and C using Equations (7f − Q = 0 for the equatorial orbits, we find (cid:113) C − A + √ B − AC = µ / (cid:0) − e (cid:1) (cid:2) − µ x (cid:0) − e − e (cid:1)(cid:3) / . (A2a)Also, from Equation (9d) of Rana & Mangalam (2019a), we see that z − = 0 , z + = (cid:112) L z + a (1 − E ) a (cid:112) (1 − E ) = √ x + a + 2 aExa (cid:112) (1 − E ) , (A2b)for Q = 0 , which implies that F (cid:18) π , z − z (cid:19) = π . (A2c)8 Rana and Mangalam 2020
Hence, Equations (A1b) − (A2c) together reduce ¯ ν θ / ¯ ν r for equatorial orbits to ¯ ν θ ¯ ν r = 2 µ / √ x + a + 2 aEx · F (cid:16) π , k (cid:17) π [1 − µ x (3 − e − e )] / . (A3)We see from Equations (7f − k = ( n − m ) / (1 − m ) can bewritten in terms of A , B , and C as k = 2 √ B − AC (cid:0) − A + C + √ B − AC (cid:1) , (A4)where the substitution of A , B , and C for Q = 0 gives k = m = 4 ex µ [1 − µ x (3 − e − e )] . (A5)Hence, we can write ¯ ν θ for the equatorial orbits as ¯ ν θ ( e, µ, a ) = 2¯ ν r ( e, µ, a ) µ / (cid:112) ( x + a + 2 aEx ) · F (cid:16) π , k (cid:17) π [1 − µ x (3 − e − e )] / , (A6)where ¯ ν r ( e, µ, a ) is given by Equation (5b) and k is given by Equation (A5). B. TRAJECTORY AND FREQUENCY FORMULAE FOR SPHERICAL ORBITS1. Azimuthal angle and coordinate time: The integrals of motion for a general nonequatorialtrajectory of a particle with rest mass m around a Kerr black hole have been derived usingthe Hamilton − Jacobi method, in terms of the Boyer − Lindquist coordinates ( r , φ , θ , t ) (Carter1968; Schmidt 2002): φ − φ = − (cid:90) rr √ R ∂R∂L z d r (cid:48) − (cid:90) θθ √ Θ ∂ Θ ∂L z d θ (cid:48) = − I − H , (B7a) t − t = 12 (cid:90) rr √ R ∂R∂E d r (cid:48) + 12 (cid:90) θθ √ Θ ∂ Θ ∂E d θ (cid:48) = 12 I + 12 H , (B7b) (cid:90) rr d r (cid:48) √ R = (cid:90) θθ d θ (cid:48) √ Θ ⇒ I = H , (B7c)where R and Θ are given by R = (cid:104)(cid:16) r (cid:48) + a (cid:17) E − aL z (cid:105) − ∆ (cid:104) r (cid:48) + ( L z − aE ) + Q (cid:105) , (B7d) Θ = Q − (cid:20)(cid:0) − E (cid:1) a + L z sin θ (cid:48) (cid:21) cos θ (cid:48) . (B7e)We have from Equation (B7c) that d r (cid:48) √ R = d θ (cid:48) √ Θ ; (B8)
Geometric origin for QPOs φ − φ = − (cid:20) ∂R∂L z H + H (cid:21) , t − t = 12 (cid:20) ∂R∂E H + H (cid:21) . (B9)Since r = r s is constant for the spherical orbits, the expressions of ∂R∂L z and ∂R∂E can bewritten as ∂R∂L z = 2 (2 L z r s − L z r s − r s aE )∆ , ∂R∂E = 2 [ E ( a r s + r s + 2 a r s ) − L z ar s ]∆ , (B10)and the integrals H , H , and H have been previously derived to be (Fujita & Hikida 2009;Rana & Mangalam 2019a) H ( θ, θ , e, µ, a, Q ) = 2 L z z + a √ − E (cid:26) F (cid:18) arcsin (cid:18) cos θ z − (cid:19) , z − z (cid:19) − F (cid:18) arcsin (cid:18) cos θz − (cid:19) , z − z (cid:19) +Π (cid:18) z − , arcsin (cid:18) cos θz − (cid:19) , z − z (cid:19) − Π (cid:18) z − , arcsin (cid:18) cos θ z − (cid:19) , z − z (cid:19)(cid:27) , (B11a) H ( θ, θ , e, µ, a, Q ) = 2 Eaz + √ − E (cid:26) K (cid:18) arcsin (cid:18) cos θz − (cid:19) , z − z (cid:19) − F (cid:18) arcsin (cid:18) cos θz − (cid:19) , z − z (cid:19) − K (cid:18) arcsin (cid:18) cos θ z − (cid:19) , z − z (cid:19) + F (cid:18) arcsin (cid:18) cos θ z − (cid:19) , z − z (cid:19)(cid:27) , (B11b) H ( θ, θ , e, µ, a, Q ) = 1 a √ − E z + (cid:26) F (cid:18) arcsin (cid:18) cos θ z − (cid:19) , z − z (cid:19) − F (cid:18) arcsin (cid:18) cos θz − (cid:19) , z − z (cid:19)(cid:27) , (B11c)where z ± are given by Equation (9d) of Rana & Mangalam (2019a). Hence, the substitutionof Equations (B10) and (B11) into Equation (B9) yields the expressions of ( φ − φ , t − t ) forthe spherical orbits, given by φ − φ = 1 a √ − E z + (cid:26) ( a L z − aEr s )∆ (cid:20) F (cid:18) arcsin (cid:18) cos θz − (cid:19) , z − z (cid:19) − F (cid:18) arcsin (cid:18) cos θ z − (cid:19) , z − z (cid:19)(cid:21) − L z (cid:20) Π (cid:18) z − , arcsin (cid:18) cos θz − (cid:19) , z − z (cid:19) − Π (cid:18) z − , arcsin (cid:18) cos θ z − (cid:19) , z − z (cid:19)(cid:21)(cid:27) , (B12a) t − t = 1 a √ − E z + (cid:26) Ea z (cid:20) K (cid:18) arcsin (cid:18) cos θz − (cid:19) , z − z (cid:19) − K (cid:18) arcsin (cid:18) cos θ z − (cid:19) , z − z (cid:19)(cid:21) + (cid:20) F (cid:18) arcsin (cid:18) cos θ z − (cid:19) , z − z (cid:19) − F (cid:18) arcsin (cid:18) cos θz − (cid:19) , z − z (cid:19)(cid:21) · (cid:20) Ea z + E ( a r s + r s + 2 a r s ) − L z ar s ∆ (cid:21)(cid:27) . (B12b)0 Rana and Mangalam 2020
2. Fundamental frequencies: The closed forms for fundamental frequencies associated with thenonequatorial eccentric bound trajectories have been previously derived (Schmidt 2002; Rana& Mangalam 2019a) and are given by Equations (4a) − (4c). We first reduce the commondenominator of these expressions to the case of spherical orbits. If we take I ( e, µ, a, Q ) commonfrom the denominator, it gives (cid:20)(cid:0) I + 2 a z EI (cid:1) F (cid:18) π , z − z (cid:19) − a z EI K (cid:18) π , z − z (cid:19)(cid:21) = I (cid:20)(cid:18) I I + 2 a z E (cid:19) F (cid:18) π , z − z (cid:19) − a z EK (cid:18) π , z − z (cid:19)(cid:21) , (B13)where by definition I /I = ∂R∂E for spherical orbits, which is given by Equation (B10). Hence,Equations (B13), (B10), and (4c) combine to give the vertical oscillation frequency for thespherical orbits: ¯ ν θ ( r s , a, Q ) = a √ − E z + (cid:26)(cid:20) [ E ( a r s + r s + 2 a r s ) − L z ar s ]∆ + a z E (cid:21) F (cid:16) π , z − z (cid:17) − a z EK (cid:16) π , z − z (cid:17)(cid:27) . (B14)Next, using Equation (B13), the azimuthal frequency, Equation (4a), can be written as ¯ ν φ ( r s , a, Q ) = (cid:26)(cid:20) − I I − L z (cid:21) F (cid:16) π , z − z (cid:17) + 2 L z · Π (cid:16) z − , π , z − z (cid:17)(cid:27) π (cid:26)(cid:20) [ E ( a r s + r s + 2 a r s ) − L z ar s ]∆ + a z E (cid:21) F (cid:16) π , z − z (cid:17) − a z EK (cid:16) π , z − z (cid:17)(cid:27) , (B15a)where I /I = ∂R∂L z , which is given by Equation (B10). Hence, the azimuthal frequency forthe spherical orbits is given by ¯ ν φ ( r s , a, Q ) = (cid:26)(cid:20) − (2 L z r s − L z r s − r s aE )∆ − L z (cid:21) F (cid:16) π , z − z (cid:17) + L z · Π (cid:16) z − , π , z − z (cid:17)(cid:27) π (cid:26)(cid:20) [ E ( a r s + r s + 2 a r s ) − L z ar s ]∆ + a z E (cid:21) F (cid:16) π , z − z (cid:17) − a z EK (cid:16) π , z − z (cid:17)(cid:27) . (B15b)Similarly, the radial oscillation frequency, Equation (4b), can be written for the spherical orbitsby using Equation (B13) as ¯ ν r ( r s , a, Q ) = F (cid:16) π , z − z (cid:17) I (cid:26)(cid:20) [ E ( a r s + r s + 2 a r s ) − L z ar s ]∆ + a z E (cid:21) F (cid:16) π , z − z (cid:17) − a z EK (cid:16) π , z − z (cid:17)(cid:27) , (B16a)where, for spherical orbits the integral I reduces to a constant, as shown below. Geometric origin for QPOs k , Equation (A4), reduces to zero because A = B = 0 (Equations(7f), (g) of Rana & Mangalam (2019a)) for spherical orbits ( e = 0 ). Hence, I ( e = 0 , µ, a, Q ) (Equation (6h) of Rana & Mangalam (2019a)) reduces to I = 2 µ √ C F (cid:16) π , k = 0 (cid:17) = πr s (cid:112) r s (1 − E ) + (3 Qa − x r s − Qr s ) . (B16b)Hence, the radial oscillation frequency for spherical orbits reduces to ¯ ν r ( r s , a, Q ) = (cid:112) r s (1 − E ) + (3 Qa − x r s − Qr s ) · F (cid:16) π , z − z (cid:17) πr s (cid:26)(cid:20) [ E ( a r s + r s + 2 a r s ) − L z ar s ]∆ + a z E (cid:21) F (cid:16) π , z − z (cid:17) − a z EK (cid:16) π , z − z (cid:17)(cid:27) . (B16c) C. REDUCTION OF FREQUENCY FORMULAE TO THE EQUATORIAL CIRCULAR CASEHere, we reduce the fundamental frequency formulae to the known case of equatorial circular orbits( ). We show this reduction from both the equatorial eccentric ( e ) and the spherical ( Q ) orbitsbelow:1. Reduction from e orbits : We see that for circular orbits ( e = 0 ), the expressions of m , p , p , and p (Equations (7i), (k) of Rana & Mangalam (2019a)) reduce to m = p = p = p = 0 . (C17)We first make the subtitution m = 0 in Equations (5a) − (5c), which gives ¯ ν φ = a Π (cid:0) − p , π , (cid:1) + b Π (cid:0) − p , π , (cid:1) π (cid:26) Π (cid:0) − p , π , (cid:1) (cid:20) a ( p +2 ) ( p ) + b (cid:21) + c Π (cid:0) − p , π , (cid:1) + d Π (cid:0) − p , π , (cid:1)(cid:27) , (C18a) ¯ ν r = 12 (cid:26) Π (cid:0) − p , π , (cid:1) (cid:20) a ( p +2 ) ( p ) + b (cid:21) + c Π (cid:0) − p , π , (cid:1) + d Π (cid:0) − p , π , (cid:1)(cid:27) , (C18b) ¯ ν θ = ¯ ν r µ / (cid:112) ( x + a + 2 aEx ) (cid:112) − µ x . (C18c)Next, the substitution of p = p = p = 0 in Equation (C18) yields ¯ ν φ = a + b π ( a + b + c + d ) , (C19a) ¯ ν r = 1 π ( a + b + c + d ) , (C19b) ¯ ν θ = ¯ ν r µ / (cid:112) ( x + a + 2 aEx ) (cid:112) − µ x . (C19c)2 Rana and Mangalam 2020
By substituting e = 0 in Equation (16) of Rana & Mangalam (2019b), we find that a + b = 2 µ / ( L z − xµ ) (cid:112) − µ x (1 − µ + a µ ) , (C20a) a + b + c + d = 2 ( E + Ea µ − axµ ) µ / (cid:112) − µ x (1 − µ + a µ ) . (C20b)Now, by substituting Equation (C20) in Equation (C19), we get ¯ ν φ = µ ( L z − xµ )2 π ( E + Ea µ − axµ ) , (C21a) ¯ ν r = µ / (cid:112) − µ x (1 − µ + a µ )2 π ( E + Ea µ − axµ ) , (C21b) ¯ ν θ = µ (1 − µ + a µ ) (cid:112) ( x + a + 2 aEx )2 π ( E + Ea µ − axµ ) . (C21c)The expressions of E , L z , and x for orbits are given by (Bardeen et al. 1972) E = (cid:0) r c − r c + a √ r c (cid:1) r c (cid:0) r c − r c + 2 a √ r c (cid:1) / , (C22a) L z = √ r c (cid:0) r c + a − a √ r c (cid:1) r c (cid:0) r c − r c + 2 a √ r c (cid:1) / , (C22b) x = r c (cid:16) r / c − a (cid:17)(cid:0) r c − r c + 2 a √ r c (cid:1) / , (C22c)where r c is the radius of the circular orbit. These expressions can be also be obtained bysubstituting { e = 0 , Q = 0 , µ = 1 /r c } in the more general expressions given by Equation (5)of Rana & Mangalam (2019a). Finally, by substituting E , L z , x , and µ = 1 /r c from Equation(C22) into Equation (C21), we recover the frequency formulae for orbits: ¯ ν φ = 12 π (cid:16) r / c + a (cid:17) , (C23a) ¯ ν r = ¯ ν φ (cid:18) − r c − a r c + 8 ar / c (cid:19) / , (C23b) ¯ ν θ = ¯ ν φ (cid:18) a r c − ar / c (cid:19) / , (C23c)as given by Equation (3).2. Reduction from Q orbits : We find that for circular orbits ( Q = 0 ), the expressions of z ± [Equation (9d) of Rana & Mangalam (2019a)] reduce to z − = 0 , z + = (cid:112) L z + a (1 − E ) a √ − E . (C24) Geometric origin for QPOs Q orbits, Equation (7), yields ¯ ν φ = ( − L z r c + L z r c + 2 r c aE )2 π [ E ( a r c + r c + 2 a r c ) − L z ar c ] , (C25a) ¯ ν r = (cid:112) r c (1 − E ) − x r c ∆2 πr c [ E ( a r c + r c + 2 a r c ) − L z ar c ] , (C25b) ¯ ν θ = (cid:112) L z + a (1 − E )∆2 π [ E ( a r c + r c + 2 a r c ) − L z ar c ] . (C25c)Using the expressions of E , L z , and x from Equation (C22), we find that (cid:2) E (cid:0) a r c + r c + 2 a r c (cid:1) − L z ar c (cid:3) = r / c ∆ (cid:16) r / c + a (cid:17)(cid:16) r c − r c + 2 ar / c (cid:17) / , (C26a) (cid:0) − L z r c + L z r c + 2 r c aE (cid:1) = r / c ∆ (cid:16) r c − r c + 2 ar / c (cid:17) / , (C26b) (cid:112) r c (1 − E ) − x r c = r / c (cid:16) r c − r c − a + 8 ar / c (cid:17) / (cid:16) r c − r c + 2 ar / c (cid:17) / , (C26c) (cid:112) L z + a (1 − E ) = (cid:113) r c + 3 a r c − ar / c (cid:16) r c − r c + 2 ar / c (cid:17) / . (C26d)Finally, substituting these factors, given by Equation (C26), in Equation (C25), we recoverthe expressions for orbits, which are given by ¯ ν φ = 12 π (cid:16) r / c + a (cid:17) , (C27a) ¯ ν r = ¯ ν φ (cid:18) − r c − a r c + 8 ar / c (cid:19) / , (C27b) ¯ ν θ = ¯ ν φ (cid:18) a r c − ar / c (cid:19) / , (C27c)as given in Equation (3). D. SOURCE HISTORYWe summarize the history of each BHXRB below:4
Rana and Mangalam 2020
1. M82 X-1: This is the brightest X-ray source in the M82 galaxy. This source is thought to harboran intermediate-mass black hole because of its very high X-ray luminosity, average 2 −
10 keVluminosity ∼ × erg s − , and variability characteristics (Patruno et al. 2006; Casella et al.2008; Pasham & Strohmayer 2013b), although other models claim that it might contain a blackhole of mass ∼ M (cid:12) (Okajima et al. 2006). However, the discovery of twin-peak and stableQPOs at 3.32 ± ± ± M (cid:12) (Pasham et al. 2014), making it an intermediate-massblack hole system.2. GROJ 1655-40: This is one among the few BHXRBs in the Milky Way galaxy whose BH massis known with good precision through the dynamical studies of the infrared and optical ob-servations during the quiescent state (Beer & Podsiadlowski 2002). GROJ 1655-40 is also oneof the BHXRBs known to produce relativistic radio jets having a double-lobed radio structure(Mirabel & Rodríguez 1994). The first detection of two simultaneous HFQPOs near ∼ and Hz in GROJ 1655-40 was reported by Strohmayer (2001a). The detection of 300HzQPO was reported in BHXRB GROJ 1655-40 (Remillard et al. 1999b), and later the detec-tion of a simultaneous 450Hz QPO along with 300Hz in the same observations was confirmed(Strohmayer 2001a). A systematic study of the LFQPOs and HFQPOs in 571 RXTE obser-vations taken between the years 1996 and 2005 was carried out by Motta et al. (2014a), whodetected three simultaneous QPOs (two HFQPOs and one LFQPO) at 441 ± ± ± − (3c) assuming the RPM (Motta et al. 2014a).3. XTEJ 1550-564: This BHXRB was first detected by ASM/RXTE on 1998 September 7. Sincethen, it has undergone four X-ray outbursts between the years 1998 and 2002 as observedby RXTE, among which the 1998 September to 1999 May outburst was the most luminousone. XTEJ 1550-564 is also among the few BHXRBs that have shown HFQPOs; for example,QPOs with frequencies in the range 185 −
237 Hz were detected during the 1998-1999 outburst(Remillard et al. 1999a; Homan et al. 2001). After a quiescent period of a few months, XTEJ1550-564 again underwent a short X-ray outburst in the period 2000 April to May followinga fast rise and an exponential decay of the X-ray luminosity. The simultaneous occurrence oftwo HFQPOs at 268 ± ± ∼
183 Hz along with a simultaneous type C LFQPO at ∼
13 Hz and type B LFQPOat ∼ ∼ ◦ − ◦ (Kuulkers et al. 1998). This source is oneamong the few BHXRBs to show HFQPOs during its 1998 outburst in the frequency range Geometric origin for QPOs ∼ ∼ ∼
180 Hz, and then a decrease in QPO frequencies was observed duringthe decay of the outburst.5. GRS 1915+105: This BHXRB is known to be a very bright system during the whole RXTEperiod, showing its peculiar behavior and have also shown superluminal radio outflows (Mirabel& Rodríguez 1994). This is also the first BHXRB to show an HFQPO at a characteristicconstant frequency of ∼
67 Hz (Morgan et al. 1997) in the RXTE observations taken during1996 April to May. Later, simultaneous ∼
67 Hz and ∼
40 Hz QPOs were discovered in theRXTE observations taken during 1997 July and November (Strohmayer 2001b). A systematicstudy of all RXTE observations of GRS 1915+105 discovered 51 observations that showeddetection of HFQPOs, out of which 48 observations showed the centroid frequency of QPOsin the range 63 −
71 Hz (Belloni & Altamirano 2013a). Another pair of simultaneous HFQPOswas also discovered at ∼
34 Hz and ∼
68 Hz (Belloni & Altamirano 2013b). E. METHOD FOR ERRORS ESTIMATION OF THE ORBITAL PARAMETERSHere, we describe a generic procedure that we have used to estimate errors in the orbital parameters.A flowchart of this method is provided in Figure 14.1. We assume that the QPO frequencies, ν , ν and ν , are Gaussian distributed with mean valuesat ν , ν , and ν (with ν > ν > ν ), which are equal to the observed QPO centroidfrequencies (see Table 5). For BHXRBs with two simultaneous QPOs, we only have ν and ν .The joint probability density distribution of these frequencies will be given by P ( ν ) = l (cid:89) i =1 P i ( ν i ) , (E28a)where l = 3 and l = 2 for BHXRBs with three and two simultaneous QPOs, respectively. Here, P i ( ν i ) represents the Gaussian distribution of the i th QPO frequency, given by P i ( ν i ) = 1 (cid:112) πσ i exp (cid:34) − ( ν i − ν i ) σ i (cid:35) . (E28b)2. We find the Jacobian of the transformation from frequency to orbital parameter space usingthe formulae of fundamental frequencies, which are given by J l = ∂ν i ∂x j ; J l = J , , J , , (E29a)where { i, j } = l and x j represent the orbital parameters, and J is given by J = ∂ν ∂x ∂ν ∂x ∂ν ∂x ∂ν ∂x ∂ν ∂x ∂ν ∂x ∂ν ∂x ∂ν ∂x ∂ν ∂x and J = (cid:34) ∂ν ∂x ∂ν ∂x ∂ν ∂x ∂ν ∂x (cid:35) . (E29b)6 Rana and Mangalam 2020
For general eccentric trajectories ( Q (cid:54) = 0 ), which are implemented for BHXRBs with threeQPOs, we have { x , x , x } = { e , r p , a }, whereas for equatorial eccentric trajectories ( Q =0 ), implemented for BHXRBs with two QPOs, we have { x , x } = { e , r p }. Similarly, for thespherical orbit case, these parameters are { x , x , x } = { r s , Q , a } or { x , x } = { r s , Q }. TheJacobian is completely expressible in terms of the standard elliptic integrals and can be easilyevaluated from Equation (E29) and using the frequency formulae, Equations (4), (5), and (7),where ν = ν φ , ν = ( ν φ − ν r ) , and ν = ( ν φ − ν θ ) according to the RPM and GRPM. Theanalytic expressions for the elements of the Jacobian are too long to reproduce here, but theyare used to make our computations faster.3. Next, we write the probability density distribution in the parameter space given by P ([ x ]) = P ( ν ) |J l | , (E30)where [ x ] represent the set of parameters { x , x , x } for l = 3 and { x , x } for l = 2 , and { ν , ν , ν } or { ν , ν } are substituted in terms of parameters using our analytic formulae.We take Q = { , , , , } for the general { e , Q } trajectory solutions that are implemented forthe sources M82 X-1 and GROJ 1655-40. For each fixed value of Q , we find the correspondingprobability density distribution in the parameter space using Equation (E30).4. We calculate the exact solutions for parameters by solving ν φ = ν , ν pp = ν , and ν np = ν using Equations (4a − − − M for l = 3 ,and both M and a for l = 2 to the previous values; see Table 5. We find 1 σ errors inthe parameters by taking an appropriate parameter volume around the exact solution, andwe generate sets of parameter combinations with resolution ∆ x j in this volume. The chosenparameter range, exact solutions, and corresponding resolutions are summarized in Tables 6, 8,and 9. We then calculate the probability density using Equation (E30), for all of the generatedparameter combinations and normalize the probability density by the normalization factor N = (cid:80) k P ([ x ] k ) ∆ V k V , ∆ V k = l (cid:89) j =1 ∆ x j,k , V = (cid:88) k ∆ V k , (E31a)where k varies from 1 to the number of total parameter combinations taken in the parametervolume, and [ x ] k is the k th combination of the parameters in the parameter volume. Hence,the normalized probability density is given by P ([ x ]) = P ([ x ]) N . (E31b)5. The allowed parameter combinations for the bound orbits are governed by the condition Equa-tion (2). For a spherical orbit, we have e = 0 . Hence, we ensure that the parameters ( e , r p , a , Q ) for eccentric and ( r s , a , Q ) for spherical trajectories follow the bound orbit condition. Ifany parameter combination does not obey the bound orbit condition, then P ([ x ]) is taken tobe zero at that point in the parameter volume. Geometric origin for QPOs P ([ x ]) , Equation (E31b), in two dimen-sions to obtain the profile in the remaining third dimension of the parameters for BHXRBswith three simultaneous QPOs, and similarly by integrating in one dimension for the two QPOcases, we obtain the profile in the other dimension. So we finally obtain the one-dimensionaldistributions P ( e ) , P ( r p ) , and P ( a ) .7. Finally, we fit the normalized probability density profiles in each of the parameter dimensionsto find the corresponding mean values, and quoted errors are obtained such that they containa probability of 68.2% about the peak value of the probability density. The results of these fitsare given in Tables 6, 8, and 9.8. For BHXRBs M82 X-1 and GROJ 1655-40, we find various orbital solutions showing varying{ a , Q } values. As the spin of the black hole should be fixed, we choose the most probable valueof a , and then we estimate the remaining parameters { e , r p , Q }, their profiles { P ( e ) , P ( r p ) , P ( Q ) }, and the corresponding errors using the same procedure given above in steps 1 to 6,where the orbital parameters are now given by { x , x , x } = { e , r p , Q }.9. Although we have made accurate calculations described above, to obtain a rough and quickestimate of the errors, we may use the following procedure. Assuming that the probabilitydensity is Gaussian distributed independently in e , r p and a parameters, the normalized jointprobability density distribution is given by P ( e, r p , a ) = 1(2 π ) / σ e σ r p σ a exp (cid:40) − (cid:34)(cid:18) e − e σ e (cid:19) + (cid:18) r p − r p σ r p (cid:19) + (cid:18) a − a σ a (cid:19) (cid:35)(cid:41) , (E32a)where the distribution is centered at the exact solution ( e , r p , a ), and σ e , σ r p , and σ a arethe corresponding 1 σ errors, derived using the method described above. The total probabilitycontained in a volume V in ( e, r p , a ) space is given by p = 1(2 π ) / σ e σ r p σ a (cid:90) (cid:90) (cid:90) V exp (cid:40) − (cid:34)(cid:18) e − e σ e (cid:19) + (cid:18) r p − r p σ r p (cid:19) + (cid:18) a − a σ a (cid:19) (cid:35)(cid:41) d e · d r p · d a ; (E32b)so that the total probability p inside an ellipsoid in ( e , r p , a ) space specified by (cid:34)(cid:18) e − e σ e (cid:19) + (cid:18) r p − r p σ r p (cid:19) + (cid:18) a − a σ a (cid:19) (cid:35) = s , (E32c)is given by p = (cid:114) π (cid:90) s exp (cid:18) − s (cid:19) s d s = 2 √ π γ (cid:18) , s (cid:19) , (E32d)where γ (cid:18) , s (cid:19) is the incomplete gamma function.8 Rana and Mangalam 2020
Similarly, for two QPO cases, the joint probability density distribution can be written as P ( e, r p ) = 12 πσ e σ r p exp (cid:40) − (cid:34)(cid:18) e − e σ e (cid:19) + (cid:18) r p − r p σ r p (cid:19) (cid:35)(cid:41) . (E33a)The total probability contained in a surface S in ( e, r p ) space is given by p = 12 πσ e σ r p (cid:90) (cid:90) S exp (cid:40) − (cid:34)(cid:18) e − e σ e (cid:19) + (cid:18) r p − r p σ r p (cid:19) (cid:35)(cid:41) d e · d r p . (E33b)The total probability inside an ellipse, specified by (cid:34)(cid:18) e − e σ e (cid:19) + (cid:18) r p − r p σ r p (cid:19) (cid:35) = s , (E33c)is given by p = (cid:90) s exp (cid:18) − s (cid:19) s d s = 1 − exp (cid:18) − s (cid:19) . (E33d)For a given p , we can calculate s and s , and hence evaluate the error ellipsoid corresponding s s s Figure 30.
Figure showing s and s as a function of probability p given by Equations (E32d) and (E33d). to p . s and s are shown as a functions of p in Figure 30. This can be used to get roughestimates of the error distribution of the parameters. However, we calculate them exactly in§3.2. REFERENCES Abramowicz, M. A., Karas, V., Kluzniak, W., Lee,W. H., & Rebusco, P. 2003, Publications of theAstronomical Society of Japan, 55, 467,doi: 10.1093/pasj/55.2.467 Bardeen, J. M., Press, W. H., & Teukolsky, S. A.1972, ApJ, 178, 347, doi: 10.1086/151796Beer, M. E., & Podsiadlowski, P. 2002, MNRAS,331, 351, doi: 10.1046/j.1365-8711.2002.05189.x