Astrophysically relevant bound trajectories around a Kerr black hole
aa r X i v : . [ g r- q c ] F e b Astrophysically relevant bound tra jectories around aKerr black hole
Prerna Rana , and A. Mangalam , Indian Institute of Astrophysics, Sarjapur Road, 2nd block Koramangala,Bangalore, 560034, IndiaE-mail: [email protected] , [email protected] Abstract.
We derive alternate and new closed-form analytic solutions for the non-equatorialeccentric bound trajectories, { φ ( r, θ ) , t ( r, θ ) , r ( θ ) } , around a Kerr black hole byusing the transformation /r = µ (1 + e cos χ ) . The application of the solutions isstraightforward and numerically fast. We obtain and implement translation relationsbetween energy and angular momentum of the particle, ( E , L ), and eccentricity andinverse-latus rectum, ( e , µ ), for a given spin, a , and Carter’s constant, Q , to write thetrajectory completely in the ( e , µ , a , Q ) parameter space. The bound orbit conditionsare obtained and implemented to select the allowed combination of parameters ( e , µ , a , Q ). We also derive specialized formulae for equatorial, spherical and separatrixorbits. A study of the non-equatorial analog of the previously studied equatorialseparatrix orbits is carried out where a homoclinic orbit asymptotes to an energeticallybound spherical orbit. Such orbits simultaneously represent an eccentric orbit and anunstable spherical orbit, both of which share the same E and L values. We presentexact expressions for e and µ as functions of the radius of the corresponding unstablespherical orbit, r s , a , and Q , and their trajectories, for ( Q = 0 ) separatrix orbits;they are shown to reduce to the equatorial case. These formulae have applications tostudy the gravitational waveforms from extreme-mass ratio inspirals (EMRIs) usingadiabatic progression of a sequence of Kerr geodesics, besides relativistic precessionand phase space explorations. We obtain closed-form expressions of the fundamentalfrequencies of non-equatorial eccentric trajectories that are equivalent to the previouslyobtained quadrature forms and also numerically match with the equivalent formulaepreviously derived. We sketch non-equatorial eccentric, separatrix, zoom-whirl, andspherical orbits, and discuss their astrophysical applications.PACS numbers: 04.20.Jb, 04.70.-s, 04.70.Bw, 95.30.Sf, 97.60.Lf, 04.25.dg, 97.10.Gz,97.80.Jp, 98.62.Mw Submitted to:
Class. Quantum Grav.
1. Introduction
It has now been established with observational evidence that black holes with massesranging from - M ⊙ in X-ray binaries, to - M ⊙ in galactic nuclei, are ubiquitous. olutions for bound trajectories around a Kerr black hole E , and angular momentum, L , were expressed in terms of the circular orbit radius and the spin parameter a ; thespecific solution for the radius of the innermost stable circular orbit (ISCO) was alsoderived. The necessary conditions for bound geodesics for spherical orbits and thedragging of nodes along the direction of spin of a black hole was discussed [9]. Theformulae have proved to be extremely useful in predicting observables in astrophysicalapplications like accretion around the black holes. For example, a general solution for astar in orbit around a rotating black hole was expressed in terms of quadratures [10] usingthe formulation given by [7]; the resulting integrals have been calculated numerically.The general expression in terms of quadratures for fundamental orbital frequencies ν θ , ν φ and ν r , for a general eccentric orbit, were first derived by [11], where different casesfor circular and equatorial orbits are also discussed but complete analytic trajectorieswere not calculated. An exact solution for non-spherical polar trajectories in Kerrgeometry and an exact analytic expression for t ( r ) for eccentric orbits in the equatorialplane were derived [12]. These were used to obtain the expressions for the periapsisadvance and Lense-Thirring frequencies. The time-like geodesics were expressed interms of quadratures involving hyper-elliptic, elliptic and Abelian integrals for Kerrand Kerr-(anti) de Sitter spacetimes including cosmological constant [13] and appliedin a semi-analytic treatment of Lense Thirring effect.A time-like orbital parameter λ called Mino time [14] was introduced to decouple the r and θ equations, which was then used to express a wider class of trajectory functions interms of the orbital frequencies ν θ , ν φ and ν r [15]. These methods are applied to calculateclosed-form solutions of the trajectories and their orbital frequencies [16], using the rootsof the effective potential. However, the solutions are expressed in terms of Mino time.The commensurability of radial and longitudinal frequencies, their resonance conditionsfor orbits in Kerr geometry, and their location in terms of spin and orbital parametervalues were studied using numerical implementations of Carlson’s elliptic integrals [17].Considering the problem of the precession of a test gyroscope in the equatorial plane of olutions for bound trajectories around a Kerr black hole E , L , a ) to eccentricity, inverse-latusrectum of the bound orbit ( e , µ , a ) parameters were obtained [18]. The expressionsfor radial and orbital frequencies are obtained to the order e for bound orbits andanalytically for the marginally bound homoclinic orbits [19]. The dynamical studies ofan important family of Kerr orbits called separatrix or homoclinic orbits are importantfor computing the transition of spiralling to plunge in EMRIs emitting gravitationalwaves [20, 21]. The test particles (compact objects in this case) transit through aneccentric separatrix orbit in EMRIs, while progressing adiabatically, before they mergewith the massive black hole.This paper is an expanded version of the published article [23]. In this paper, westudy the generic bound trajectories, which are eccentric and inclined, around a Kerrblack hole, and then we investigate the non- equatorial separatrix orbits as a specialcase. We have solved the equations of motion and produce alternate and new closed-form solutions for the trajectories in terms of elliptic integrals without using Minotime, { φ ( r, θ ) , t ( r, θ ) , r ( θ ) or θ ( r ) } , which makes them numerically faster. We alsoimplement the essential bound orbit conditions to choose the parameters ( e , µ , a , Q )of an allowed bound orbit, derived from the essential conditions on the parameters forthe elliptic integrals involved in the trajectory solutions. We find that the e − µ spaceis more convenient for integrating the equations of motion as the turning points of theintegrands are naturally specified in terms of the bound orbit conditions. The exactsolutions for the trajectories are found in terms of not overly long expressions involvingelliptic functions. We implement the translation formulae between ( E , L ) and ( e , µ )parameters that help us to express the trajectory solutions completely in the ( e , µ , a , Q ) parameter space which we call the conic parameter space. We then study the case ofnon-equatorial separatrix trajectories in the conic parameter space. First, we write theessential equations for the important radii like innermost stable spherical orbit ( ISSO ),marginally bound spherical orbit (
M BSO ), and spherical light radius. Similar to theequatorial separatrix orbits, the non-equatorial separatrix or homoclinic trajectoriesasymptote to an energetically bound unstable spherical orbit, where the spherical orbitradius can vary between
M BSO and
ISSO . We show that the formulae for ( e , µ )for the non-equatorial separatrix orbits can be expressed as functions of the radius ofthe corresponding spherical orbit, r s , a , and Q , which also reduce to their equatorialcounterpart [20] by implementing the limit Q → . These formulae are obtained byusing the expressions of E and L for the spherical orbits. Next, we derive the exactsolutions for the non-equatorial separatrix trajectories by reducing our general eccentrictrajectory solutions to this case. These solutions are important for investigating thebehaviour of gravitational waveforms emitted by inspiralling and inclined test objectsnear non-equatorial separatrix trajectories in the case of EMRIs.The ab-initio specification of the allowed geometry of bound orbits in the parameterspace is crucial for the calculation of the orbital trajectories and its frequencies. Thesecriteria are used in building, studying and sketching different types of trajectories around olutions for bound trajectories around a Kerr black hole
4a Kerr black hole: for instance, spherical, non-equatorial eccentric, non-equatorialseparatrix and zoom-whirl orbits, using our closed-form expressions for trajectoriesare constructed. We also derive closed-form analytic expressions for the fundamentalfrequencies of the general non-equatorial trajectories as functions of elliptic integralsaround the Kerr black holes. We use a time-averaging method on the first-orderequations of motion to derive these frequencies and show that our closed-form analyticexpressions of frequencies match with the formulae given by [11] which were left inquadrature forms. We also reduce the general forms to the equatorial case, which is alsoa new form that is easier to implement and faster by a factor of ∼ .This paper is organized as follows (see Fig. 1): in §2, we review the basic equationsdescribing { r, θ, φ, t } motion around the Kerr black hole using Hamiltonian dynamics.In §2.1, we write the translation formulae from ( e, µ, a, Q ) to ( E, L, a, Q ) parameterspace. In §2.2, we derive the exact closed-form solutions for the trajectories by solving allinvolving integrals and writing them in terms of elliptic integrals, { φ ( r, θ ) , t ( r, θ ) , r ( θ )or θ ( r ) } . In §2.3, we give the essential bound orbit conditions on ( e, µ, a, Q ) parametersapplicable to the astrophysical situations. In §2.4, we reduce the analytic solutions to thecase of equatorial plane. In §3.1, we derive the formulae for E and L for spherical orbitsas a function of radius r s , a , and Q . In §3.2, we write the equations for the radii ISSO , M BSO , and spherical light radius. We then derive the exact expressions for e and µ for the non-equatorial separatrix trajectories. In §3.3, we derive the trajectory solutionsfor the non-equatorial separatrix orbits. In §4, we sketch and discuss various boundtrajectories around the Kerr black hole. In §5, we derive the closed-form expressionsof the fundamental frequencies in terms of elliptic integrals by the long time averagingmethod without using Mino time. In §6, we conduct consistency checks by reducingthe separatrix trajectories to the equatorial case, and also match the azimuthal to polarfrequency ratio, ν φ /ν θ , with the spherical orbits case derived by [9]. We discuss possibleapplications of our trajectory solutions and frequency formulae in §7. We summarizeand conclude our results in §8 and §9 respectively. In Table 1 a glossary of symbols isprovided.
2. Integrals of motion and bound orbits around Kerr black hole
In this section, we first set up the basic equations defining the integrals of motion of thegeneral eccentric orbit with Q = 0 around a Kerr black hole. We then write the formulaedefining the transformation from conic parameters ( e , µ ) to dynamical parameters ( E , L ) for bound orbits and also provide the conditions for the selection of the parameters ( e , µ , a , Q ) for the bound orbits. These results are essential for expressing the integrals ofmotion in ( e , µ , a , Q ) space for bound orbits. Finally, we derive and present an alternateand simple form of the analytic solutions for the integrals of motion in terms of standardelliptic integrals using the transformation /r = µ (1 + e cos χ ) . Such transformationslead to a compact and useful trajectory solution for the non-equatorial and eccentricorbits around a rotating black hole. Later, we reduce these results to a simpler form for olutions for bound trajectories around a Kerr black hole Boyer Lindquist coordinates t Time coordinate r Radial distance from the black hole θ Polar angle φ Azimuthal angle ρ r + a cos θ a Spin of the black hole
Common physical parameters u r τ Proper time r + Horizon radius Q Carter’s constant E Energy per unit rest mass of the test particle L z component of Angular momentum per unitrest mass of the test particle p t Generalized momentum for t coordinate p φ Generalized momentum for φ coordinate p r Generalized momentum for r coordinate m =0 for photon orbits and = 1 for particle orbits V eff Radial effective potential for an eccentric test H Relativistic Hamiltonian for the geodesic motionparticle trajectory r apastron distance ( = r a ) r periastron distance ( = r p ) r Third turning point of the test particle r Innermost turning point of the test particle e eccentricity parameter µ inverse latus-rectum parameter Integrals of motion χ defined by u = µ (1 + e cos χ ) ψ χ − π y e cos χ I Terminology used for radial integrals H Terminology used for θ integrals Spherical and separatrix orbits r s radius of spherical orbit r c radius of circular orbit e s eccentricity of the separatrix orbits µ s inverse latus-rectum of the separatrix orbits Z ISCO radius X Light radius
Fundamental frequencies ν φ Azimuthal frequency ν r Radial frequency ν θ Vertical oscillation frequency
Table 1.
Glossary of symbols used. §2 Kerr orbit dynamics inBoyer-Lindquist coordinates. §2.2
Exact solutions for the non-equatorialeccentric trajectories, Eqs. (12). §2.4
Reduction tothe equatorialplane, Eqs. (16). §4 Sketching varioustrajectories. §5 Fundamentalfrequencies fromthe long timeaverage method,Eqs. (37a, 37b, 37c). §6 Consistency checkof the results. §3.3
Deriving the exactsolutions for the non-equatorial separatrixtrajectories, Eqs.(34a, 34b, 34c). §3.2
Writing equations forvarious important radii,Eqs. (22-24), and deriving e s and µ s for the non-equatorial separatrixtrajectories, Eqs. (29). §2.1 Derivation oftranslationrelations between( E , L ) and ( e , µ )parameters, Eqs. (7). §2.3 Derivation ofuseful bound orbitconditions, Eqs. (14). §3.1
Derivation of theexpressions of E and L for the sphericalorbits, Eqs. (18). Figure 1.
The flow chart of the paper is shown with the sections labeled at top of thebox where the concept is presented. olutions for bound trajectories around a Kerr black hole M in the Boyer-Lindquist coordinates, x α = ( t, r, θ, φ ) , in geometrical units G = c = 1d s = − (cid:18) − rρ (cid:19) d t − ar sin θρ d φ d t + ρ r − r + a d r + ρ d θ + (cid:18) r + a + 2 ra sin θρ (cid:19) sin θ d φ , (1)where a = J/M is the specific angular momentum of the black hole and ρ = r + a cos θ . We have written the variables ρ, r , and t are in units of M . The relativisticand conservative Hamiltonian for the geodesic motion of a test particle in Kerr spacetime[7, 24, 25]: H = 12 g µν p µ p ν ≡ − m , = − ( r + a ) − ( r − r + a ) a sin θ r − r + a ) ρ p t − arρ ( r − r + a ) p t p φ + ( r − r + a ) − a sin θ r − r + a ) ρ sin θ p φ + ( r − r + a )2 ρ p r + 12 ρ p θ , (2)where m is the particle’s rest mass, p β are the conjugate momenta associated withparticle’s coordinates. To derive the complete set of constants of motion, a canonicaltransformation, ( q α , p β ) → ( Q α , P β ), can be found such that the Hamiltonian is cyclicand the new set of momenta, P β , are conserved along the world-line of the particle.A characteristic function is obtained to generate such transformation and Hamilton’sequations are used to obtain the first-order equations of motion [7, 24, 25]: m ρ d t d τ = r + a ( r + a − r ) (cid:2) E (cid:0) r + a (cid:1) − aL (cid:3) − a (cid:0) aE sin θ − L (cid:1) , (3a) m ρ d r d τ = ± √ R, (3b) m ρ d θ d τ = ± √ Θ , (3c) m ρ d φ d τ = a ( r + a − r ) (cid:2) E (cid:0) r + a (cid:1) − aL (cid:3) − aE + L sin θ , (3d)where R = (cid:2)(cid:0) r + a (cid:1) E − aL (cid:3) − (cid:0) r + a − r (cid:1) (cid:2) m r + ( L − aE ) + Q (cid:3) , (3e) Θ = Q − (cid:20)(cid:0) m − E (cid:1) a + L sin θ (cid:21) cos θ. (3f)We have written the variables ρ, r , and t in the units of M . The integrals of motion olutions for bound trajectories around a Kerr black hole τ − τ = Z rr r ′ dr ′ √ R + Z θθ a cos θ ′ d θ ′ √ Θ , (4a) φ − φ = − Z rr √ R ∂R∂L d r ′ − Z θθ √ Θ ∂ Θ ∂L d θ ′ = − I − H , (4b) t − t = 12 Z rr √ R ∂R∂E d r ′ + 12 Z θθ √ Θ ∂ Θ ∂E d θ ′ = 12 I + 12 H , (4c) Z rr d r ′ √ R = Z θθ d θ ′ √ Θ ⇒ I = H , (4d)where ∆ = r ′ − r ′ + a and I , I , H , H are integrals defined above and solved in§2.2.The equation for the radial motion around the Kerr black hole, Eq. (3b), can beexpressed in the form ( E − m )2 = m ρ r (cid:18) d r d τ (cid:19) + V eff ( r, a, E, L, Q ) , (5)where the term on the LHS represents the total energy, the first term on the RHSrepresents the radial kinetic energy and the second term on the RHS represents theradial effective potential given by V eff ( r, a, E, L, Q ) = − m r + L − a ( E − m ) + Q r − ( L − aE ) + Qr + a Q r . (6) E , L ) and ( e , µ ) We present the transformation of energy, angular momentum, and Carter’s constant ( E , L , Q ) space of the test particle to the eccentricity, inverse-latus rectum ( e , µ , Q ) spaceof its corresponding bound orbit. These relations can be derived if R ( r ) is factorizedand the periastron r p and apastron r a of the orbit are substituted as /µ (1 + e ) and /µ (1 − e ) respectively. Hence, the formulae connecting ( E , L ) and ( e , µ ) parametersfor bound orbits (a derivation of these formulae is given in Appendix A) are E ( e, µ, a, Q ) = h − µ (cid:0) − e (cid:1) (cid:0) µa Q − Q − x (cid:1) − µ (cid:0) − e (cid:1)i / , (7a)where x = L − aE and it can be written in terms of conic parameters as x ( e, µ, a, Q ) = − S − √ S − P R P , (7b)where P ( e, µ, a, Q ) = 14 a (cid:2)(cid:0) e (cid:1) µ − (cid:3) − µ (cid:0) − e (cid:1) , (7c) olutions for bound trajectories around a Kerr black hole S ( e, µ, a, Q ) = µ (cid:0) − e (cid:1) + µ (cid:0) − e (cid:1) (cid:0) µa Q − Q (cid:1) − a (cid:2)(cid:0) e (cid:1) µ − (cid:3) · (cid:20) µ − a − Q + a Qµ (cid:0) − e (cid:1) − µ (cid:0) e (cid:1) (cid:0) µa Q − Q (cid:1)(cid:21) , (7d) R ( e, µ, a, Q ) = 14 a (cid:20) µ − a − Q + a Qµ (cid:0) − e (cid:1) − µ (cid:0) e (cid:1) (cid:0) µa Q − Q (cid:1)(cid:21) . (7e)These expressions are used to derive analytic results for the integrals of motion, givenin §2.2, completely in the ( e , µ , a , Q ) parameter space. Next, we solve for the integrals of motion, i.e. Eqs. (4b-4d) and reduce them to a simpleform involving elliptic integrals. We first derive the expressions for the radial integrals I and I . We assume the starting point of the radial motion to be apastron point of thebound orbit, r = r a . The steps taken to obtain the reduced form of the radial integralsare as follows:(i) We make the substitution /r ′ = µ (1 + e cos χ ) and implement the method ofpartial fractions.(ii) Then make the substitutions, cos χ = 2 cos χ − and ψ = χ − π .(iii) Implement the variable transformation given by sin α = √ − m sin ψ √ − m sin ψ , where m isdefined by Eq. (9i).As a result the integrals of motion are expressed as functions of standard ellipticintegrals, given by I ( α, e, µ, a, Q ) = − [ C I ( α, e, µ, a, Q ) + C I ( α, e, µ, a, Q )] , (8a) I ( α, e, µ, a, Q ) = [ C I ( α, e, µ, a, Q ) + C I ( α, e, µ, a, Q ) + C I ( α, e, µ, a, Q ) + C I ( α, e, µ, a, Q )] , (8b) I ( α, e, µ, a, Q ) = 1 √ − m ( m + p ) (cid:20) m F (cid:0) α, k (cid:1) + p Π (cid:18) − ( p + m )1 − m , α, k (cid:19)(cid:21) , (8c) I ( α, e, µ, a, Q ) = 1 √ − m ( m + p ) (cid:20) m F (cid:0) α, k (cid:1) + p Π (cid:18) − ( p + m )1 − m , α, k (cid:19)(cid:21) , (8d) I ( α, e, µ, a, Q ) = 1 √ − m ( m + p ) (cid:2) m F (cid:0) α, k (cid:1) + 2 p m Π (cid:0) s , α, k (cid:1) + p I ( α, e, µ, a, Q ) (cid:3) , (8e) I ( α, e, µ, a, Q ) = 1 √ − m ( m + p ) (cid:2) m F (cid:0) α, k (cid:1) + p Π (cid:0) s , α, k (cid:1)(cid:3) , (8f) olutions for bound trajectories around a Kerr black hole I ( α, e, µ, a, Q ) = s sin α cos α p − k sin α − s ) ( k − s ) (cid:0) − s sin α (cid:1) − s − s ) ( k − s ) K (cid:0) α, k (cid:1) −
12 (1 − s ) F (cid:0) α, k (cid:1) + [ s − s (1 + k ) + 3 k ]2 (1 − s ) ( k − s ) Π (cid:0) s , α, k (cid:1) , (8g) I ( α, e, µ, a, Q ) = 2 µ (1 − e ) p C − A + √ B − AC F (cid:0) α, k (cid:1) , (8h)where C = 2 (1 − e ) µ [ La − xr + ] p ( A − B + C ) (1 − a ) ( a µ − a µe − r + ) , (9a) C = 2 (1 − e ) µ [ − La + 2 xr − ] p ( A − B + C ) (1 − a ) ( a µ − a µe − r − ) , (9b) C = 4 E (1 + e ) µ p ( A − B + C ) (1 − e ) , C = 8 E (1 + e ) p ( A − B + C ) , (9c) C = 4 a µ (1 − e ) ( − La + 2 Er − ) r − p ( A − B + C ) (1 − a ) ( a µ − a µe − r + ) , (9d) C = 4 aµ (1 − e ) (cid:0) − Lr − √ − a − Ear − + La (cid:1) r − p ( A − B + C ) (1 − a ) ( a µ − a µe − r − ) , (9e) A = Qa e µ (cid:0) − e (cid:1) , (9f) B = 2 eµ (cid:0) − e (cid:1) (cid:2) Qa µ − x − Q (cid:3) , (9g) C = µ (cid:0) − e (cid:1) (cid:2) µQa − x − Q (cid:3) + (cid:0) − E (cid:1) (cid:0) − e (cid:1) , (9h) n = 4 A A − B − √ B − AC , m = 4 A A − B + √ B − AC , (9i) k = n − m − m , s = − p − m − m , (9j) p = 2 e − e , p = 2 ea µa µ − a µe − r + , p = 2 ea µa µ − a µe − r − , (9k) x , = − B ± √ B − AC A , (9l)and where the variables E , L and x can be written as functions of ( e , µ , a , Q ) usingEqs. (7a-7e), which makes all the integrals to be only functions of ( e , µ , a , Q ). Thedefinition of the elliptic integrals involved, is given below [26]: F (cid:0) α, k (cid:1) = Z α d α p − k sin α , (10a) K (cid:0) α, k (cid:1) = Z α p − k sin α · d α, (10b) Π (cid:0) s , α, k (cid:1) = Z α d α (cid:0) − s sin α (cid:1) p − k sin α . (10c) olutions for bound trajectories around a Kerr black hole
10A complete derivation of these integrals is given in Appendix B. Next, to solve theintegrals H , H , and H of Eqs. (4b- 4d), we make the substitutions z = cos θ ′ and z = z − sin β [16] which reduces these integrals to H ( θ, θ , e, µ, a, Q ) = 2 Lz + 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) , (11a) 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) , (11b) 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) , (11c)where z ± = − P ′ ± p P ′ − Q ′ , P ′ = − L − Q − a (1 − E ) a (1 − E ) , Q ′ = Qa (1 − E ) . (11d)Hence, the equations of motion can now be written in short as φ − φ = 12 [ C I ( α, e, µ, a, Q ) + C I ( α, e, µ, a, Q ) − H ( θ, θ , e, µ, a, Q )] , (12a) t − t = 12 [ C I ( α, e, µ, a, Q ) + C I ( α, e, µ, a, Q ) + C I ( α, e, µ, a, Q )+ C I ( α, e, µ, a, Q ) + H ( θ, θ , e, µ, a, Q )] , (12b) I ( α, e, µ, a, Q ) = H ( θ, θ , e, µ, a, Q ) , (12c)where I , I , I , I , I , H , H and H are given by Eqs. (8c-8h; 11) respectively. Hence,all the integrals are written explicitly as functions of parameters ( e , µ , a , Q ) throughvariables α ( e, µ, a, Q ; χ ) and β ( e, µ, a, Q ; θ ) which are directly used to calculate ( r , θ , t )through Eqs. (12a-12c). The radial motion, which varies with the α , is assumed to havethe starting point at the apastron distance, r a or α = 0 , of the orbit and the startingpoint of the polar motion, β or θ , is an extra variable which can be chosen in the range { θ − , π − θ − } . This is tantamount to shifting the starting point of the motion in time oradjusting the initial value of the observed time, t .Once the initial points are fixed ( α = 0 , θ = θ ), Eqs. (12a) and (12b) are used tocalculate φ ( r, θ ) and t ( r, θ ) respectively, whereas Eq. (12c) gives r ( θ ) or θ ( r ) , whichcan be used to obtain t ( r ) or t ( θ ) and φ ( r ) or φ ( θ ) .The elegant alternate forms presented here help us to write useful and simplerexpressions of ( φ , t ) for the equatorial eccentric trajectories, as shown later in §2.4. olutions for bound trajectories around a Kerr black hole The bound orbit regions have been studied and divided in the ( E , L ) space accordingto the different types of possible r motion [27]. The most relevant astrophysical boundorbit region corresponds to the case where E < and there are four real roots of R ( r ) , r > r > r > r > , such that the bound orbit either exists between r and r or r and r , this has been defined as region III in the ( E , L ) plane by [27]. Since r and r are the outer most turning points of the effective potential, the bound orbit should existbetween these two in the astrophysical situations. We can implement this constraint inthe ( e , µ , a , Q ) space by imposing the condition k < on the parameter k used in theradial integrals in §2.2, where we have assumed that a bound orbit exists between r and r , which requires k < as an essential condition for the elliptic integrals to havereal values, Eqs. (10a-10c). This further implies n < (13a)where the substitution of Eq. (9i) in the above expression yields ( A + B + C ) > , (13b)and by using Eqs. (9f-9h) and (7a-7e) this implies (cid:2) µ a Q (1 + e ) + µ (cid:0) µa Q − x − Q (cid:1) (3 − e ) (1 + e ) + 1 (cid:3) > . (13c)Another necessary condition is that the periastron of the orbit r = 1 / [ µ (1 + e )] isoutside the horizon, which gives h µ (1 + e ) (cid:16) √ − a (cid:17)i < . (13d)Hence, the necessary and independent conditions for this region can be collectively givenas µ a Q (1 + e ) + µ (cid:0) µa Q − x − Q (cid:1) (3 − e ) (1 + e ) + 1 > , (14a) µ (1 + e ) (cid:16) √ − a (cid:17) < , (14b) E ( e, µ, a, Q ) < . (14c)There exists unstable bound orbits for E > specified as region IV in the ( E , L ) plane by [27], where the bound orbit exists between r and r . Such a situation isnot important from the astrophysical point of view, because the particle will follow abound trajectory between the outermost turning points, i.e. r and r , and hence theabove conditions, Eq. (14), together represent a necessary and sufficient condition forthe existence of bound orbits. olutions for bound trajectories around a Kerr black hole In this section, we apply the integrals of motion, Eqs. (12a, 12b), to the eccentricequatorial trajectories, where Q = 0 ( θ = π/ ). We show that the forms derived in§2.2 reduce to very compact expressions of ( φ , t ) involving trigonometric functions andelliptic integrals for the equatorial eccentric orbits. We implement the limit, Q → which leads to A → , Eq. (9f), and reduces the factors (1 + x ) , A (1 + x ) , using Eq.(9l), to (1 + x ) → (cid:18) − CB (cid:19) , and A (1 + x ) → − B, (15a)which gives A (1 + x ) (1 + x ) = A − B + C = µ (cid:0) − e (cid:1) (cid:2) − µ x (cid:0) − e − e (cid:1)(cid:3) , (15b)where the translation equation given by Eq. (7a) for Q = 0 is used to substitute for E .Also, m and n reduce to m = 2 BB − C = 4 µ ex [1 − µ x (3 − e − e )] , n = 4 AB B ( A − B ) + 2 AC = 0 . (15c)The substitution of these reduced expressions of m and n further simplifies the integrals I , I , I , and I , as shown in Appendix C, which finally yields the expressions forazimuthal angle and time coordinate for equatorial trajectories to be given by φ − φ = − I = a Π (cid:0) − p , ψ, m (cid:1) + b Π (cid:0) − p , ψ, m (cid:1) , (16a) t − t = 12 I = a I + b I + c I + d I , = a " p sin ψ cos ψ p − m sin ψ p ) ( m + p ) (cid:0) p sin ψ (cid:1) − F ( ψ, m )2 (1 + p ) + p K ( ψ, m )2 (1 + p ) ( m + p ) + Π (cid:0) − p , ψ, m (cid:1) (cid:26) a [ p + 2 p (1 + m ) + 3 m ]2 (1 + p ) ( m + p ) + b (cid:27) + c Π (cid:0) − p , ψ, m (cid:1) + d Π (cid:0) − p , ψ, m (cid:1) , (16b)where the substitution of Eq. (15b) into Eqs. (9a-9e) yields the reduced forms of theconstants given by a = C µ / [ La − xr + ] √ − a ( a µ − a µe − r + ) p − µ x (3 − e − e ) , (16c) b = C µ / [ − La + 2 xr − ] √ − a ( a µ − a µe − r − ) p − µ x (3 − e − e ) , (16d) a = C Eµ / (1 − e ) p − µ x (3 − e − e ) , (16e) b = C Eµ / (1 − e ) p − µ x (3 − e − e ) , (16f) olutions for bound trajectories around a Kerr black hole c = C a µ / ( − La + 2 Er − ) r − p [1 − µ x (3 − e − e )] (1 − a ) ( a µ − a µe − r + ) , (16g) d = C aµ / (cid:0) − Lr − √ − a − Er − a + La (cid:1) r − p [1 − µ x (3 − e − e )] (1 − a ) ( a µ − a µe − r − ) . (16h)The brackets of the factor [1 − µ x (3 − e − e )] in the expressions of c and d abovewere missing in the journal version [23], and has been corrected here. The correspondingfundamental frequency formulae for the equatorial trajectories are ν φ = c · [ φ ( ψ = π/ − φ ]2 πGM · [ t ( ψ = π/ − t ] , ν r = c GM · t r = c GM · [ t ( ψ = π/ − t ] . (17)These compact expressions, Eqs. (16, 17), for the equatorial eccentric trajectorieshave their importance in various astrophysical studies, in addition to, gyroscopeprecession and phase space studies.
3. Non-equatorial separatrix trajectories
The separatrix orbits have been studied for the equatorial plane around a Kerr black hole[28, 20]. They have been shown as homoclinic orbits which asymptote to an energeticallybound and unstable circular orbit. Here, we discuss the non-equatorial counterpart ofthese separatrix trajectories where these orbits asymptote to an energetically bound,unstable spherical orbit. These non-equatorial homoclinic trajectories are critical incalculating the evolution of test objects transiting from inspiral to plunge, which is notalways confined to the equatorial plane, as in EMRIs emitting gravitational radiation.In this section, we first deduce the expressions of E and L for the spherical orbitsas functions of the radius r s , and ( a , Q ). We then derive the exact expressions for theconic parameters ( e , µ ) for non-equatorial separatrix orbits as a function of the radiusof the corresponding spherical orbit, r s , and ( a , Q ). We also show that these formulaereduce to the equatorial case, previously derived in [20], when Q → is applied. Next,we derive the exact analytic expressions for the non-equatorial separatrix trajectoriesby reducing it from the general trajectory formulae, Eqs. (12a-12c). We find that inthis case, the radial part of the solutions can be reduced to a form that involves onlylogarithmic and trigonometric functions. Spherical orbits are the non-equatorial counterparts of circular orbits and set a crucialsignpost in the dynamical study of non-equatorial and separatrix trajectories. The exactexpressions for energy and angular momentum for the spherical orbits can be derived bysubstituting e = 0 and µ = 1 /r s , where r s is the radius of the orbit, in the expressions olutions for bound trajectories around a Kerr black hole Table 2.
This table summarizes all the integrals solved in §2.2, 2.4 to calculate theintegrals of motion in the Kerr geometry, where all the constants are defined in thetext.
Analytic solution of φ and t for Q = 0 φ − φ = ( C I + C I − H ) ; t − t = ( C I + C I + C I + C I + H ) I Z rr a √ R ∂R∂L d r ′ = − [ C I + C I ] I Z rr a √ R ∂R∂E d r ′ = [ C I + C I + C I + C I ] ,I Z ψ d ψ (cid:0) p sin ψ (cid:1) p − m sin ψ p − n sin ψ = 1 √ − m ( m + p ) (cid:20) m F (cid:0) α, k (cid:1) + p Π (cid:18) − p − m − m , α, k (cid:19)(cid:21) where sin α = √ − m sin ψ p − m sin ψ , ψ = χ − π and /r = µ (1 + e cos χ ) I Z ψ d ψ (cid:0) p sin ψ (cid:1) p − m sin ψ p − n sin ψ = 1 √ − m ( m + p ) (cid:20) m F (cid:0) α, k (cid:1) + p Π (cid:18) − p − m − m , α, k (cid:19)(cid:21) I Z ψ d ψ (cid:0) p sin ψ (cid:1) p − m sin ψ p − n sin ψ = 1 √ − m ( m + p ) (cid:2) m F (cid:0) α, k (cid:1) + 2 p m Π (cid:0) s , α, k (cid:1) + p I ( α, e, µ, a, Q ) (cid:3) ,I Z ψ d ψ (cid:0) p sin ψ (cid:1) p − m sin ψ p − n sin ψ = 1 √ − m ( m + p ) (cid:2) m F (cid:0) α, k (cid:1) + p Π (cid:0) s , α, k (cid:1)(cid:3) I Z ψ d ψ (cid:0) s sin ψ (cid:1) p − k sin ψ = s sin α cos α p − k sin α − s ) ( k − s ) (cid:0) − s sin α (cid:1) + [ s − s (1 + k ) + 3 k ]2 (1 − s ) ( k − s ) Π (cid:0) s , α, k (cid:1) −
12 (1 − s ) F (cid:0) α, k (cid:1) − s − s ) ( k − s ) K (cid:0) α, k (cid:1) H Z θθ √ Θ ∂ Θ ∂L d θ ′ = 2 Lz + a √ − E (cid:26) 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) − 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:27) H Z θθ √ Θ ∂ Θ ∂E d θ ′ = 2 Eaz + √ − E (cid:26) 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) − 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:27) . Analytic solution of φ and t for Q = 0 φ − φ = a Π (cid:0) − p , ψ, m (cid:1) + b Π (cid:0) − p , ψ, m (cid:1) t − t = a " p sin ψ cos ψ p − m sin ψ p ) ( m + p ) (cid:0) p sin ψ (cid:1) − F ( ψ, m )2 (1 + p ) + p K ( ψ, m )2 (1 + p ) ( m + p ) + d Π (cid:0) − p , ψ, m (cid:1) + c Π (cid:0) − p , ψ, m (cid:1) + Π (cid:0) − p , ψ, m (cid:1) (cid:26) a [ p + 2 p (1 + m ) + 3 m ]2 (1 + p ) ( m + p ) + b (cid:27) olutions for bound trajectories around a Kerr black hole E , L , and x given by Eqs. (7a-7e), which yields E = ( a Q + ( r s −
3) ( r s − r s − a r s [ r s (3 r s −
5) + Q ( r s ( r s −
4) + 5)] − a [ r s ( r s −
2) + a ] p a Q − r s Q ( r s −
3) + r s ) / r s (cid:2) r s ( r s − − a (cid:3) / , (18a) x = (cid:26) − a Q + r s ( r s −
3) [ r s − ( r s − Q ] + a r s ( r s + r s − Qr s + 8 Q ) − a [ r s ( r s −
2) + a ] p a Q − r s Q ( r s −
3) + r s (cid:27) / r / s (cid:2) r s ( r s − − a (cid:3) / , (18b)and L = x + aE. (18c)Similar formulae were derived in terms of inclination angle using an approximation in[29], whereas we have written the exact form in terms of the fundamental parametersand constant of motion Q . These formulae reduce to the energy and angular momentumformulae for circular orbits when Q = 0 is substituted [8]: E = r c − r c + a √ r c r c (cid:0) r c − r c + 2 a √ r c (cid:1) / , L = √ r c (cid:0) r c − a √ r c + a (cid:1) r c (cid:0) r c − r c + 2 a √ r c (cid:1) / . (19) Similar to the case of equatorial plane, the non-equatorial separatrix trajectories can beparametrized by the radius of unstable spherical orbits, r s , for a given combination of a and Q , where r s varies from MBSO to ISSO. The energy and angular momentum ofthe separatrix orbits can be determined by Eqs. (18a-18c) by varying r s between theextrema MBSO and ISSO radii. In the ( e , µ ) plane, these homoclinic orbits forms theboundary (other than e = 0 and e = 1 curves) of the allowed bound orbit region definedby Eq. (14) for a fixed a and Q ; see red curve in Fig. 2(a). The locus of this boundaryin the ( e , µ ) plane is obtained when equality is applied to the inequality Eq. (14a),which results in (cid:2) µ a Q (1 + e ) + µ (cid:0) µa Q − x − Q (cid:1) (3 − e ) (1 + e ) + 1 (cid:3) = 0 . (20)ISSO is a homoclinic orbit with e = 0 and MBSO is a homoclinic orbit with e = 1 ;hence the endpoints of the separatrix curve (red curve in Fig. 2(a)) represents the ISSOand MBSO radii. At these endpoints, the parameter µ takes values as described below: For ISSO , e = 0 for the homoclinic orbit gives µ = 2 r a r p r a = 1 r p = 1 r s . For MBSO , e = 1 for the homoclinic orbit gives µ = 1 + r p /r a r p = 12 r p = 12 r s . olutions for bound trajectories around a Kerr black hole μ ● S ● M (a) - - - - - - - - V e ff r s r r (b) Figure 2. (a) The shaded region depicts bound orbit region in the ( e , µ ) planedetermined by Eqs. (14) for a = 0 . and Q = 5 . The red boundary of the regionrepresents non-equatorial separatrix orbits with eccentricity of the orbit varying alongthe curve. The black dot represented by S corresponds to the ISSO with ( e = 0 , µ = 1 /r s ), whereas M represents the MBSO with ( e = 1 , µ = 1 / r s ); (b) The effectivepotential, Eq. (6), is shown for a non-equatorial separatrix orbit with E = 0 . , L = 2 . , a = 0 . , and Q = 5 , where the horizontal line represents the totalenergy given by (cid:0) E − (cid:1) / . The equations for ISSO and MBSO radii can be obtained using the equation ofseparatrix curve, Eq. (20), by plugging in ( e = 0 , µ = 1 /r s ) and ( e = 1 , µ = 1 / r s ) toderive ISSO and MBSO respectively (as shown in Appendix D). Hence, the equationsfor these radii are given by r s − r s − a r s + 36 r s + 8 a Qr s − a r s − a Qr s + 9 a r s − a Qr s +48 a Qr s + 16 a Q r s − a Qr s − a Q r s + 48 a Q r s − a Q = 0 , (22)for ISSO and r s − r s − a r s + 16 r s + 2 a Qr s − a r s − a Qr s + a r s − a Qr s +8 a Qr s + a Q r s − a Qr s − a Q r s + a Q = 0 . (23)for MBSO. The light radius for the spherical orbits can be obtained by equating thedenominator of Eq. (18a) to zero, so that E → ∞ , which has the well known form forthe equatorial light radius [8] given by X = 2 (cid:26) (cid:20)
23 arccos ( − a ) (cid:21)(cid:27) . (24)Fig. 3 shows the contours of these radii in the ( r s , a ) plane for various Q values.The effective potential diagram for the non-equatorial separatrix orbits showsdouble roots ( r = r ) of R ( r ) at the periastron of the eccentric orbit and it alsorepresents the spherical orbit radius, r s (see Fig 2(b)). One of the remaining two rootsof R ( r ) represents the apastron ( = r > r s ) of the eccentric orbit and the other innerroot ( = r < r s ) is not the part of bound trajectory. olutions for bound trajectories around a Kerr black hole - - r c a HorizonLight radiusMBCOISCO (a) - - r s a HorizonLight radiusMBSOISSO (b) - - r s a HorizonLight radiusMBSOISSO (c) - - r s a HorizonLight radiusMBSOISSO (d)
Figure 3.
The contours of different important radii around the Kerr black hole in the( r s , a ) plane for (a) Q = 0 , (b) Q = 5 , (c) Q = 10 , and (d) Q = 12 . Now, following a similar method used in [20], we derive the expressions for e and µ for separatrix orbits with Q = 0 . We write R ( r ) = 0 in the form u + a ′ u + b ′ u + c ′ u + d ′ = 0 , (25a)where u = 1 /r and a ′ = − x + Q ] a Q , b ′ = ( x + 2 aEx + a + Q ) a Q , c ′ = − a Q , d ′ = 1 − E a Q . (25b)For the separatrix orbits, Eq. (25a) can be written as ( u − u s ) · (cid:2) u − ( u + u ) u + u u (cid:3) = 0 , (26)where u s = 1 /r s , u = 1 /r apastron of the orbit, and u = 1 /r corresponds to theinner most root of R ( r ) . The comparison of u and constant term of the above equationwith those of Eq. (25a) further gives the expression u = 12 " − (cid:16) a ′ + 2 u s (cid:17) − s ( a ′ + 2 u s ) − d ′ u s . (27) olutions for bound trajectories around a Kerr black hole e s = u s − u u s + u , µ s = u s + u , (28)where the substitution of u and u s = 1 /r s yields e s = 4 + a ′ r s + q ( r s a ′ + 2) − d ′ r s − a ′ r s − q ( r s a ′ + 2) − d ′ r s , (29a) µ s = 14 r s (cid:20) − a ′ r s − q ( r s a ′ + 2) − d ′ r s (cid:21) ; (29b)since a homoclinic orbit has same energy and angular momentum of the unstablespherical orbit, as shown in Fig. 2(b); hence a ′ and d ′ can be rewritten using theformulae of E and L for the spherical orbits, Eqs. (18a-18c), to be a ′ = 2 (cid:26) a Q − r s ( r s −
3) [ r s − ( r s − Q ] − a r s ( r s + r s − Qr s + 8 Q )+ 2 a [ r s ( r s −
2) + a ] p a Q − r s Q ( r s −
3) + r s − Qr s (cid:2) r s ( r s − − a (cid:3) (cid:27) a Qr s (cid:2) r s ( r s − − a (cid:3) , (29c) d ′ = ( − a Q − ( r s −
3) ( r s − r s + a r s [ r s (3 r s −
5) + Q ( r s ( r s −
4) + 5)]+ r s (cid:2) r s ( r s − − a (cid:3) + 2 a [ r s ( r s −
2) + a ] p a Q − r s Q ( r s −
3) + r s ) a Qr s (cid:2) r s ( r s − − a (cid:3) . (29d)These expressions reduce to the ( e , µ ) formulae for the equatorial separatrix orbits(see Appendix E for the details) when the limit Q → is implemented, to the formspreviously derived by [20]: e s = − r c − r c − a + 8 a √ r c r c + a − r c , µ s = r c + a − r c r c (cid:0) √ r c − a (cid:1) . (30) In this section, we show the reduction of our general trajectory solutions, Eqs. 12,for the case of separatrix orbits with Q = 0 to simple expressions. The separatrix orhomoclinic orbits represent a curve in the ( e , µ ) plane for a fixed a and Q combination,Fig. 2, which is also the boundary of the bound orbit region defined by Eqs. (14). Thisseparatrix curve is defined by Eq. (20), which gives us the relation x + Q = 1 + 4 µ a Q (1 + e ) µ (3 − e ) (1 + e ) ; (31)this further reduces the expressions of A , B , C (Eqs. (9f-9h)) and correspondingly theexpressions of n and m to n = 1 or k = 1 , (32a) olutions for bound trajectories around a Kerr black hole Table 3.
This table summarizes the trajectory solution derived in §3.3 for the non-equatorial separatrix orbits.
Analytic solutions S = 1(1 + p ) " p p − ( p + m )ln s √ − m + p − ( p + m ) sin α √ − m − p − ( p + m ) sin α + ln (tan α + sec α ) √ − m S = 1(1 + p ) " p p − ( p + m )ln s √ − m + p − ( p + m ) sin α √ − m − p − ( p + m ) sin α + ln (tan α + sec α ) √ − m S = 1 √ − m ( m + p ) (cid:20) m ( m − p m + 2 p )(1 + p ) ln (tan α + sec α ) + p S + 2 p m (1 − m )(1 + p ) | s | tan − [ | s | sin α ] (cid:21) S = ln (tan α + sec α ) √ − m (1 + p ) + p √ − m ( m + p ) (1 + p ) | s | tan − [ | s | sin α ] S = 12 (1 − s ) " s sin α cos α (cid:0) − s sin α (cid:1) + 2 ln (tan α + sec α ) − s sin α + (cid:0) − s (cid:1) | s | tan − ( | s | sin α ) m = a Qµ e (1 + e ) (3 − e )[1 + 2 a ( − e ) Qµ ] . (32b)The integrals governing the vertical motion ( θ integrals) given by Eqs. (11a, 11b, 11c)retain their same form as they do not involve k = 1 , whereas, the radial integrals givenby Eqs. (8a-8h) reduce further, when k = 1 is substituted. The elliptic integrals reduceto forms involving trigonometric and logarithmic functions using the following identitiesgiven here Π (cid:0) q , α, (cid:1) = 11 − q " ln (tan α + sec α ) − q ln s q sin α − q sin α , where q > , q = 1 , = 11 − q (cid:2) ln (tan α + sec α ) + | q | tan − ( | q | sin α ) (cid:3) , where q < , (33a) F ( α,
1) = ln (tan α + sec α ) , (33b) K ( α,
1) = sin α. (33c)The final and simple expressions for the azimuthal angle, ( φ − φ ) , ( t − t ) , andthe equation relating r − θ motion for the non-equatorial separatrix trajectories (seeAppendix F for the derivation) are given by φ − φ = 12 ( p µ (1 + e ) (3 − e ) p e [1 + 2 a ( − e ) Qµ ] (1 − a ) (cid:20) [ La − xr + ]( a µ − a µe − r + ) S + [ − La + 2 xr − ]( a µ − a µe − r − ) S (cid:21) − H ) , (34a) olutions for bound trajectories around a Kerr black hole t − t = p (1 + e ) (3 − e ) p eµ [1 + 2 a ( − e ) Qµ ] ( Eµ (1 − e ) S + a µ ( − La + 2 Er − ) r − p (1 − a ) ( a µ − a µe − r + ) S + 2 E (1 − e ) S + aµ (cid:0) − Lr − √ − a − Ear − + La (cid:1) r − p (1 − a ) ( a µ − a µe − r − ) S ) + 12 H , (34b) µ (1 − e ) az + √ − E p C − A + √ B − AC ln (tan α + sec α ) = (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) . (34c)where integrals S − S are summarized in Table 3, and H ( θ, θ , e, µ, a, Q ) , H ( θ, θ , e, µ, a, Q ) are given by Eq. (11a), (11b) respectively.These expressions have their utility in evaluating the trajectory evolution ofinspiralling objects near the separatrix, and just before plunging, for extreme massratio inspirals (EMRIs) in gravitational wave astronomy [30, 31, 32].
4. Trajectories
The analytic solution of the integrals of motion presented in this paper in §2.2 providesa direct and exact recipe to study bound trajectories without involving numericalintegrations. These expressions have their utility in calculating extreme mass ratioinspirals (EMRIs) in gravitational wave astronomy, where numerical models consider anadiabatic progression through series of geodesics around a Kerr black hole [30, 31, 32].We now discuss various kinds of bound geodesics around Kerr black hole using ouranalytic solution for the integrals of motion. We use the translation formulae, Eqs. (7a-7e), to obtain the integrals of motion only in terms of ( e , µ , a , Q ) parameters. To sketchthe trajectories, we have chosen the starting point for the trajectories to be ( β = π/ , α = 0 ) as it follows from Eq. (4d). We use Eq. (12c) to calculate corresponding smallchange in θ or β with the small change in r or α and substitute corresponding ( r , θ ) or( α , β ) values in Eqs. (12a) and (12b) to calculate ( φ , t ).There are various possible kinds of bound orbits. Here, we take up the each caseand sketch these trajectories for different combinations of ( a , Q ), where the parametersvalues are tabulated in the Table 4. We take up slow rotating ( a = 0 . ) and fast rotatingblack hole situations ( a = 0 . or a = 0 . ), with both prograde and retrograde cases, forvarious Q values. The various features of these orbits are enumerated below:(i) Eccentric orbits : Figs. 5 and 6 represent eccentric bound prograde and retrogradetrajectories respectively, where the parameter values are depicted in the Table 4.The particle periodically oscillates between the periastron and the apastron, andis also bound between θ = arccos ( z − ) and θ = arccos ( − z − ) as shown in ( t - r )and ( t - θ ) plots in Figs. 5 and 6, whereas ( t - φ ) plots depict that φ varies between0 to π . We have fixed ( e , µ ) of the plotted trajectories and show the variationwith change in a and Q parameters. The motion of the trajectory increases in thevertical direction with increase in Q parameter. olutions for bound trajectories around a Kerr black hole Table 4.
This following table summarizes the values of conic parameters ( e , µ ) chosenin the listed orbit simulations to study eccentric, homoclinic and spherical orbits fordifferent ( a , Q ) combinations for both prograde and retrograde cases constructed usingEqs. (12). Type of Orbit
Inverse latus- Eccentricity Spin of Carter’s constant Varyingorbit rectum of the orbit of the orbit the black hole parameter µ e a Q
Eccentric orbits E1 0.1 0.6 0.2 3 a E2 0.1 0.6 0.8 3E3 0.1 0.6 0.2 8 a E4 0.1 0.6 0.8 8E5 0.1 0.6 -0.2 3 a E6 0.1 0.6 -0.8 3E7 0.1 0.6 -0.2 8 a E8 0.1 0.6 -0.8 8Homoclinic orbits H1 0.153 0.6 0.2 3 a and e H2 0.208 0.2 0.5 3H3 0.153 0.5 0.2 8 a H4 0.172 0.5 0.5 8H5 0.127 0.5 -0.2 3 a and e H6 0.127 0.2 -0.5 3H7 0.134 0.5 -0.2 8 a H8 0.123 0.5 -0.5 8Spherical orbits S1 0.222 0 0.5 3 a and Q S2 0.144 0 -0.5 8Zoom-whirl Z1 0.155 0.5 0.2 5 a Z2 0.226 0.5 0.8 5Z3 0.142 0.8 0.2 5 a Z4 0.212 0.8 0.8 5Z5 0.162 0.5 0.5 10 a Z6 0.179 0.5 0.8 10 (ii)
Homoclinic/Separatrix orbits : Homoclinic orbits are the separatrices betweeneccentric bound and plunge orbits, where the particle asymptotically approachesthe unstable spherical/circular orbit in both the distant past and the distantfuture. The energy and angular momentum of the orbiting particle simultaneouslycorrespond to a stable eccentric bound orbit and an unstable spherical/circularorbit. Separatrix orbits in the equatorial plane of a Kerr black hole are well studied,[20, 28, 33]. The homoclinic orbits form an important group in Kerr dynamics asthey represent the transition between inspiral and plunge orbits and hence, havetheir significance in the study of gravitational wave spectrum under the adiabaticapproximation. The homoclinic or separatrix orbits correspond to the boundaryof the region in ( e, µ, a, Q ) space, defined by Eq. (20). Separatrix orbits with Q = 0 also have similar features as the equatorial separatrix orbits, where the olutions for bound trajectories around a Kerr black hole a , increases the range of θ . The orbit initially followsan eccentric path and asymptotically approaches the periastron radius which alsocorresponds to the unstable spherical orbit radius as shown in ( t - r ) plots of Figs. 7and 8.(iii) Spherical orbits : Fig. 9 shows prograde and retrograde innermost stable sphericalorbits (ISSO), which are also the homoclinic orbits with e = 0 . All the sphericalstable orbits exist outside ISSO, whereas unstable spherical orbits are found betweenISSO and MBSO.(iv) Zoom-whirl orbits : Zoom whirl orbits are orbits where the particle takes a finitenumber of revolutions at the periastron before going back to the apastron, whichis an extreme form of the periastron precession. Their significance in gravitationalastronomy has been studied for the case of equatorial Kerr orbits [21]. Here, wediscuss zoom-whirl orbits with Q = 0 as shown in Fig. 10, where the particletakes finite revolutions with varying θ at the periastron before turning back to theapastron. We have chosen the value of µ very near to the separatrix, where usuallythe zoom whirl behavior is seen, for different values of ( e , a , Q ) combinations. Asexpected, the particle spends more time at the periastron, compared to the timetaken at apastron, to take a finite number of revolutions which is making the t − r plots appear flatter near the periastron, (see Fig. 10). We again see that theincrease in a increases the range of vertical motion of the orbit like for the eccentricorbits case. Homoclinic/Separatrix orbit family is the limiting case of the zoom-whirl orbit family where the particle takes infinite revolutions as it asymptotes tothe unstable spherical orbit.Now, we discuss how different kinds of orbits are distributed in the bound orbitregion in the ( e , µ ) plane defined by the Eq. (14) for a fixed combination of ( a , Q ).We fix a = 0 . and Q = 5 and show the shaded bound orbit region in Fig. 4, thatrepresents the eccentric orbits allowed. The black curve which is the boundary of theshaded region represents homoclinic or separatrix orbits. The curve defined by e = 0 represents all the spherical orbits with its end point at ISSO, which intersects with theseparatrix line. We fix e = 0 . depicted by the red curve in Fig. 4 and take differentvalues of µ , as depicted by the black dots on the red curve, and plot the correspondingtrajectories and study their corresponding behavior.We see from Figs. 11 and 12, that for a fixed e = 0 . , as µ is increased, the trajectoryshows zoom-whirl behavior as it gets closer to the separatrix or homoclinic orbit for thecorresponding e value. It can be seen in the t - r plot of Fig. 12(b) that the particle spendssome time at the periastron which clearly depicts the zoom-whirl behavior. Hence, itcan be said that zoom-whirl behavior is a near separatrix phenomenon and can occurat any eccentricity. olutions for bound trajectories around a Kerr black hole μ Figure 4.
The shaded region depicts the bound orbit region defined by Eq. (13c) inthe ( e , µ ) plane for a = 0 . and Q = 5 . The black curve represents the homoclinicorbits where the end points depict e = 0 and e = 1 homoclinic orbits correspondingto the ISSO and MBSO respectively. The red curve represents e = 0 . and we studyorbits with different µ values as depicted by the dots on this curve.
5. Fundamental frequencies
In this section, we derive the expressions for fundamental frequencies ( ν φ , ν r , ν θ ) interms of the integrals derived analytically in §2.2. We take a long time average of Eq.(4d) on both the sides so that lim T →∞ T Z rr dr √ R = lim T →∞ T Z θθ dθ √ Θ . (35a)As T → ∞ , there exists a large integer solutions, which can be found with arbitraryprecision, so that N r t r = N θ t θ = T , where N r and N θ are the number of radial andvertical oscillations; hence Eq. (35a) reduces to lim N r →∞ N r R r a r p dr √ R N r · t r = lim N θ →∞ N θ R π − θ − θ − dθ √ Θ N θ · t θ , (35b)where r p and r a are the periastron and apastron of the orbit and θ − correspondsto the starting point of the vertical oscillation, and where θ − = arccos( z − ) and π − θ − = − arccos( z − ) , which results in β varying from − π/ to π/ . Hence, usingEqs. (8h, 11c) we find ν θ ν r = R r a r p dr √ R R π − θ − θ − dθ √ Θ = a √ − E z + I (cid:0) π , e, µ, a, Q (cid:1) · F (cid:16) π , z − z (cid:17) , . (35c)The similar expression can also be derived using formulae given in [16]. Again, we takea long time average of Eq. (4c), so that lim T →∞ t − t T = lim T →∞ T (cid:20) Z rr √ R ∂R∂E d r ′ + 12 Z θθ √ Θ ∂ Θ ∂E d θ ′ (cid:21) , (36a) olutions for bound trajectories around a Kerr black hole (a) (b)(c) (d) Figure 5.
The figure shows prograde eccentric bound orbits (a) E1, (b) E2, (c) E3,and (d) E4 in the table 4, for various combinations of ( e , µ , a , Q ) satisfying Eq. (13c)and also presents the evolution of corresponding θ , φ and r with coordinate time, t . where using the same argument, again, of large possible integer solutions, so that N r t r = N θ t θ = T to find N r R r a r p √ R ∂R∂E d r ′ N r t r + 2 N θ R π − θ − θ − √ Θ ∂ Θ ∂E d θ ′ N θ t θ = ν r I + ν θ H , (36b)which gives ν r ( e, µ, a, Q ) = 1 I (cid:0) π , e, µ, a, Q (cid:1) + ν θ ν r H (cid:0) − π , π , e, µ, a, Q (cid:1) , (36c) ν θ ( e, µ, a, Q ) = 1 ν r ν θ I (cid:0) π , e, µ, a, Q (cid:1) + H (cid:0) − π , π , e, µ, a, Q (cid:1) . (36d) olutions for bound trajectories around a Kerr black hole (a) (b)(c) (d) Figure 6.
The figure shows retrograde eccentric bound orbits (a) E5, (b) E6, (c) E7,and (d) E8 in the table 4, for various combinations of ( e , µ , a , Q ) satisfying Eq. (13c)and also presents the evolution of corresponding θ , φ and r with coordinate time, t . The limits of integral I are α = { , π/ } , and that of H are β = { π/ , − π/ } . Thesubstitution of H (cid:0) − π , π , e, µ, a, Q (cid:1) and ν θ ν r from Eqs. (11b) and (35c) in the aboveequations give ν r ( e, µ, a, Q ) = F (cid:16) π , z − z (cid:17) (cid:2) I (cid:0) π , e, µ, a, Q (cid:1) + 2 a z EI (cid:0) π , e, µ, a, Q (cid:1)(cid:3) F (cid:16) π , z − z (cid:17) − a z EI (cid:0) π , e, µ, a, Q (cid:1) K (cid:16) π , z − z (cid:17) , (37a) olutions for bound trajectories around a Kerr black hole (a) (b)(c) (d) Figure 7.
The figure shows the prograde homoclinic orbits (a) H1, (b) H2, (c) H3,and (d) H4 in the table 4, for various combinations of ( e , µ , a , Q ) and also presentsthe evolution of corresponding θ , φ and r with coordinate time, t . ν θ ( e, µ, a, Q ) = a √ − E z + I (cid:0) π , e, µ, a, Q (cid:1) (cid:2) I (cid:0) π , e, µ, a, Q (cid:1) + 2 a z EI (cid:0) π , e, µ, a, Q (cid:1)(cid:3) F (cid:16) π , z − z (cid:17) − a z EI (cid:0) π , e, µ, a, Q (cid:1) K (cid:16) π , z − z (cid:17) . (37b)Similarly, taking the long time average of Eq. (4b) and the substitution of H and olutions for bound trajectories around a Kerr black hole (a) (b)(c) (d) Figure 8.
The figure shows the retrograde homoclinic orbits (a) H5, (b) H6, (c) H7,and (d) H8 in the Table 4, for various combinations of ( e , µ , a , Q ) and also presentsthe evolution of corresponding θ , φ and r with coordinate time, t . H from Eqs. (11a) and (11b) yields ν φ ( e, µ, a, Q ) = (cid:2) − I (cid:0) π , e, µ, a, Q (cid:1) − LI (cid:0) π , e, µ, a, Q (cid:1)(cid:3) F (cid:16) π , z − z (cid:17) + 2 LI (cid:0) π , e, µ, a, Q (cid:1) Π (cid:16) z − , π , z − z (cid:17) π (cid:2) I (cid:0) π , e, µ, a, Q (cid:1) + 2 a z EI (cid:0) π , e, µ, a, Q (cid:1)(cid:3) F (cid:16) π , z − z (cid:17) − a z EI (cid:0) π , e, µ, a, Q (cid:1) K (cid:16) π , z − z (cid:17) , (37c)where I , I , and I are given by Eqs. (8a)-(8g) and (8h). Hence, the fundamentalfrequencies are explicit functions of input parameters ( e , µ , a , Q ), which can be chosenusing the bound orbit conditions presented in §2.3. These frequency formulae also olutions for bound trajectories around a Kerr black hole (a)(b) Figure 9.
The figure shows the spherical orbits for (a) prograde, S1, (b) retrograde,S2, in the Table 4 along with the corresponding evolution of θ , φ and r with coordinatetime, t . Table 5.
This table summarizes the fundamental frequency formulae derived using thelong time average method in the Kerr geometry. The explicit expressions for integrals I , I and I are summarized in table 2. ν r ( e, µ, a, Q ) F (cid:18) π , z − z (cid:19) M (cid:26) [ I ( π ,e,µ,a,Q ) +2 EI ( π ,e,µ,a,Q )] F (cid:18) π , z − z (cid:19) − a z EI ( π ,e,µ,a,Q ) K (cid:18) π , z − z (cid:19)(cid:27) ν θ ( e, µ, a, Q ) a √ − E z + I ( π ,e,µ,a,Q ) M (cid:26) [ I ( π ,e,µ,a,Q ) +2 EI ( π ,e,µ,a,Q )] F (cid:18) π , z − z (cid:19) − a z EI ( π ,e,µ,a,Q ) K (cid:18) π , z − z (cid:19)(cid:27) ν φ ( e, µ, a, Q ) (cid:26) [ − I ( π ,e,µ,a,Q ) − LI ( π ,e,µ,a,Q )] F (cid:18) π , z − z (cid:19) +2 LI ( π ,e,µ,a,Q ) Π (cid:18) z − , π , z − z (cid:19)(cid:27) πM (cid:26) [ I ( π ,e,µ,a,Q ) +2 EI ( π ,e,µ,a,Q )] F (cid:18) π , z − z (cid:19) − a z EI ( π ,e,µ,a,Q ) K (cid:18) π , z − z (cid:19)(cid:27) match with the quadrature formulae derived in [11]; but here we have explicitly solvedthe integrals I , I and I in §2.2.
6. Consistency check with previous results
In order to verify our results, we have reduced our formulae for the non-equatorialseparatrix trajectories, Eqs. (34), to the case of equatorial separatrix orbits and foundthat they are consistent with earlier results derived in [20]. We also found that the olutions for bound trajectories around a Kerr black hole (a) (b)(c) (d)(e) (f) Figure 10.
The figure shows zoom whirl orbits (a) Z1, (b) Z2, (c) Z3, (d) Z4, (e) Z5,and (f) Z6 in the table 4, for various combinations of ( e , µ , a , Q ) satisfying Eq. (13c)and also presents the evolution of corresponding θ , φ and r with coordinate time, t . olutions for bound trajectories around a Kerr black hole (a) (b) Figure 11.
The figure shows the eccentric trajectories on the red curve of Fig. 4( e = 0 . , a = 0 . , Q = 5 ) for (a) µ = 0 . , and (b) µ = 0 . .(a) (b) Figure 12.
The figure shows the eccentric trajectories on the red curve of Fig. 4( e = 0 . , a = 0 . , Q = 5 ) for (a) µ = 0 . , and (b) µ = 0 . . We see that thetrajectory shown in (b) represent a zoom-whirl orbit. frequency ratio, ν φ /ν θ , from Eqs. (37b, 37c), reduce to the case of maximally rotatingblack hole, a = 1 , for spherical orbits previously derived in [9]. See Appendix G forthese derivations.
7. Applications
There are various important applications of our analytic solutions of the general non-equatorial trajectories and the fundamental frequencies for astrophysical studies as olutions for bound trajectories around a Kerr black hole
Gravitational waves : One of the crucial applications of our trajectory solution isthe case of gravitational waves from the extreme-mass ratio inspirals (EMRIs). Ouranalytic formulae are directly applicable for the frequency domain calculation ofthe gravitational waves using the Teukolsky formalism, [34] or Kludge scheme [35],and the orbits can be computed more accurately than the numerical calculations[36]. Also, the homoclinic orbits, which are the separatrix between plunge andbound geodesics [22, 28], have their importance to study the zoom-whirl behaviorof inspirals near separatrix [21, 37]. In this paper, we provide the analytic formulaefor eccentricity and inverse-latus rectum, ( e , µ ), for non-equatorial separatrix orbitswhich are crucial for the selection of these orbits for the study of gravitationalwaveforms in the Kerr geometry.(ii) Relativistic precession : The exact analytic formula for azimuthal angle, φ − φ , isuseful to find the precession of the orbits in the astrophysical systems like planets,black hole, and double pulsar systems. PSR J0737-3039 is one example of a doublepulsar system having two pulsars, PSR J0737-3039A and PSR J0737-3039B having23 ms [38] and 2.8 s [39] period respectively, which is useful to study the relativisticprecession phenomenon valid in a strong gravitational field. The periastron advancewas estimated in this source using the first PK parameter, ˙ ω [40]. Our exact analyticresults can be used to make a more accurate estimation of the relativistic advanceof the periastron in pulsar systems where one component is having a major spincontribution.(iii) Quasi-periodic oscillations (QPOs) : QPOs are broad peaks seen in the Fourierpower spectrum of the Neutron star X-ray binaries (NSXRB) and black hole X-ray binaries (BHXRB). The relativistic precession (RP) model was introduced [41]to explain the kHz QPOs in NSXRB and later applied to BHXRB [42]. The RPmodel can be used to calculate the black hole parameters assuming a circular oreccentric orbit is giving rise to a pair of observed high-frequency QPOs and asingular and nearly simultaneous corresponding Type-C QPO, [43, 44], where ourexact formulae for the fundamental frequencies are applicable.(iv)
Gyroscope precession : The calculation of precession of spin of a test gyroscopeis another application for the test of general relativity. In previous studies,approximate expressions were used for the fundamental frequencies as a seriesexpansion in terms of eccentricity up to order e around a Kerr black hole forthe stable bound orbits in the equatorial plane [18]. Our exact analytic results areuseful to estimate more accurate results which are useful to explain the reportedresults of geodetic drift rate and frame-dragging drift rate by the Gravity Probe B(GP-B) [45].(v) Phase space study : Study of dynamics of Kerr orbits by Poincaré maps is also welldiscussed [22]. Our closed-form solutions are directly applicable to the study of theextreme chaotic behavior of orbits like Zoom-whirl orbits, which are extreme forms olutions for bound trajectories around a Kerr black hole
8. Summary
The summary of this paper is given below:(i) We first translate the parameters ( E , L , a , Q ) to ( e , µ , a , Q ) using the translationformulae, Eqs. (7) to completely describe the trajectory solution in the ( e , µ , a , Q )space. We then select the allowed bound orbit by choosing the parameters ( e , µ , a , Q ) using the bound orbit conditions, Eqs. (14).(ii) We have derived the closed-form analytic solutions of the general eccentric trajec-tory in the Kerr geometry as function of elliptic integrals, { φ ( r, θ ) , t ( r, θ ) , r ( θ ) } ,Eqs. (12a12c). These trajectories around a Kerr black hole were previously derivedin terms of Mino time [16], λ , subject to the initial conditions on d r (0) / d λ and d θ (0) / d λ . The application of our trajectory solution to the various possible studiesis numerically faster and does not require any selection of initial conditions. Wechoose the starting point of the trajectory as the apastron of the orbit, r a or α = 0 ,and the initial polar angle, θ or β , is an extra parameter which can be arbitrarilychosen between maximum and minimum allowed θ range for a given Q . The inputvariables for plotting the trajectories are α and β which define the range of r and θ for a fixed combination of ( e , µ , a , Q ). These results are summarized in Table 2.(iii) We have derived the formulae for E and L for the spherical orbits as functions ofradius r s , a , and Q , given by Eqs. (18).(iv) We have derived the equations for ISSO, MBSO, and spherical light radius, Eqs.(22-24). The light radius derived is the same as that for the equatorial case.(v) We discussed the non-equatorial separatrix orbits, which asymptote to the unstablespherical radius sharing the same E and L values with the eccentric bound orbit.The radius of this unstable spherical radius for the separatrix orbit exists between M BSO and
ISSO . We write the exact forms for the eccentricity and inverse-latusrectum ( e s , µ s ) for the non-equatorial separatrix orbits as functions of r s , a , and Q , given by Eqs. (29).(vi) We use our general trajectory solutions to derive the equations of motions for non-equatorial separatrix orbits, given by Eqs. (34), and find that the radial part ofthe solutions can be completely reduced to the form containing only trigonometricand logarithmic functions. We also show the reduction of these trajectories to theequatorial case which is also a new and useful form and match the solutions withthe previously known result derived in [20]. Separatrix trajectories are essentialin the study of gravitational waves from EMRIs, where our analytic solutions aredirectly applicable. These results are summarized in Table 3.(vii) We discuss families of allowed bound orbit trajectories like non-equatorial eccentric,non-equatorial separatrix, zoom whirl, and spherical orbits around a rotating blackhole using our analytic solution for the trajectories. Homoclinic trajectories have olutions for bound trajectories around a Kerr black hole λ . We show that these expressions match with thosederived in [11] using Hamilton-Jacobi formulation, which were left in the quadratureform, and we have obtained a closed form using elliptic integrals. These expressionsare summarized in Table 5. We present the consistency of our trajectory solution byreducing it to the equatorial separatrix case and also show that the frequency ratio, ν φ /ν θ , matches with the standard expression derived [9] for the spherical orbits.The results include novel aspects given in (i) and (iii)-(vii), listed above, and alternatenew forms of the known formulae, given in the points (ii) and (viii) above. The equationsand tables providing these results are indicated in the points above.
9. Discussion and Conclusions
There are several notable results in the vast literature discussing various aspects ofdynamics in Kerr geometry such as the quadrature formulae for the trajectories ([6, 7]),circular orbit formulae [8], conditions for spherical orbits [9], expressions in terms ofquadratures for the oscillation frequencies [11], formulae for trajectories in terms ofquadratures for spherical polar motion [13], trajectories for non-spherical polar motion[12], and expressions for the trajectories and oscillation frequencies [16] in terms ofMino time [14]. Besides these key results there are other useful expressions reported forexample on separatrix orbits [20], and on eccentric equatorial bound orbits ([6, 18]).We discuss below the utility of the results in our paper:The recipe for calculating frequencies and trajectories by [16] is as follows: Theoperative equations are { φ ( λ ) , t ( λ ) , r ( λ ) } , Eqs. (6), (23)-(33), (35)-(45), which requirelinear combinations of many other equations. The analogy to r ( χ ) or χ ( r ) is r ( λ ) or λ ( r ) (Eqs. (26, 27)); the latter is non-trivial, whereas the former is simple. Given λ ,( φ, t ) are calculated subsequently inverting linear combinations of many other ellipticintegrals. We have numerically matched our frequency formulae with that given in [16]and we find that there is a minor typo in their expression of Γ below Eq. (20) in section3.3, where there is a factor of E/ missing in the term ( r − r )( r − r ) E ( k r ) . However,the correct factor has been applied to calculate the numbers in their Tables (1, 2, 3)given in [16]. By using the set of equations in [16] and comparing with our expressions,it is found that our calculation is easier to implement and numerically faster by ∼ ,in the equatorial case, for example.The novel results listed in (i) and (iii)-(vii) of the summary: translation conditionsof { E, L } → { e, µ } , bound orbit conditions, { E ( r s ) , L ( r s ) } , ISSO, MBSO, and light olutions for bound trajectories around a Kerr black hole Q = 0 separatrix curve in the e − µ plane besides providing the form of the trajectories. Usingthis, further studies can be carried for chaotic motion and study of gravitational wavesfrom zoom-whirl orbits which can be set-up by locating them near the separatrix locus,in the same spirit, as was done for the equatorial case [20, 21].The analytic results presented in this paper have direct applications in astrophysicsfor example, the study of non-equatorial separatrix orbits which has not been discussedbefore. They also help in understanding the highly eccentric behaviour of trajectoriesseen in numerical simulations [30] just before plunging onto the massive black hole inthe case of EMRIs which is possibly related to the eccentric and inclined homoclinicorbits, besides relativistic precession in other astrophysical systems like binary pulsarsand black holes, spin precession of gyroscopes around rotating black holes for the testof general relativity, and the study of chaotic orbits in the phase space. Appendix A. Deriving translation formulae between ( E , L ) and ( e , µ )parameters The turning points of radial motion around rotating black hole are derived from Eq.(25a), which can be factorized into two quadratics u + a ′ u + b ′ u + c ′ u + d ′ = (cid:16) u + a ′ u + b ′ (cid:17) (cid:16) u + a ′ u + b ′ (cid:17) , (A.1)and the comparison of coefficients on both sides yield the following relations a ′ + a ′ = a ′ , (A.2a) a ′ a ′ + b ′ + b ′ = b ′ , (A.2b) b ′ a ′ + b ′ a ′ = c ′ , (A.2c) b ′ b ′ = d ′ . (A.2d)Assuming that the first quadratic in Eq. (A.1) have the turning points of the orbit, u = µ (1 − e ) and u = µ (1 + e ) , implies that a ′ = − ( u + u ) = − µ and b ′ = µ (1 − e ) . Using Eqs. (A.2d) and (A.2a) to replace a ′ and b ′ in Eqs. (A.2b)and (A.2c) and substituting for a ′ = − ( u + u ) = − µ and b ′ = µ (1 − e ) yields − µ (cid:16) a ′ + 2 µ (cid:17) + µ (cid:0) − e (cid:1) + d ′ µ (1 − e ) = b ′ , (A.3a) µ (cid:0) − e (cid:1) (cid:16) a ′ + 2 µ (cid:17) − d ′ µ (1 − e ) = c ′ . (A.3b) olutions for bound trajectories around a Kerr black hole a ′ , b ′ , c ′ and d ′ and rearrangement of the terms in above equationsgives E ( e, µ, a, Q ) = 1 − µ (cid:0) − e (cid:1) (cid:0) µa Q − Q − x (cid:1) − µ (cid:0) − e (cid:1) , (A.4a) E = 12 ax (cid:20) − x − a + a Qµ (cid:0) − e (cid:1) − Q + 1 µ − (cid:0) e (cid:1) µ (cid:0) µa Q − Q − x (cid:1)(cid:21) , = C x + C x , (A.4b)where C = 12 a (cid:2)(cid:0) e (cid:1) µ − (cid:3) , (A.5a) C = 12 a (cid:20) µ − a − Q + a Qµ (cid:0) − e (cid:1) − µ (cid:0) e (cid:1) (cid:0) µa Q − Q (cid:1)(cid:21) . (A.5b)Next, the substitution of Eq. (A.4b) in Eq. (A.4a) gives x h C − µ (cid:0) − e (cid:1) i + x h µ (cid:0) − e (cid:1) + 2 C C + µ (cid:0) − e (cid:1) (cid:0) µa Q − Q (cid:1) − i + C = 0 , (A.6)which is further solved for x to obtain Eqs. (7b-7e). These relations also reduce to theequatorial case, Q = 0 , which was first derived in [18]. Appendix B. Solution of integrals I - I In this appendix, we show the solutions of the integrals given in §2.2. First, we derivethe radial integrals I − I , given by the Eqs. (8a-8h). We make the substitution /r ′ = µ (1 + e cos χ ) , which reduces the integrals to I = − µ (cid:0) − e (cid:1) Z χπ L − L − aE ) µy [1 − µy + a µ y ] p A cos χ + B cos χ + C d χ, (B.1a) I = 2 (1 − e ) µ Z χπ E + a Eµ y − a ( L − aE ) µ y y [1 − µy + a µ y ] p A cos χ + B cos χ + C d χ, (B.1b)where y = (1 + e cos χ ) . Further, we implement the partial fraction method to reducethe integrals to I = − − e ) µa Z χπ (cid:20) A y − y + + B y − y − (cid:21) p A cos χ + B cos χ + C d χ, (B.2a) I = 2 (1 − e ) µ a Z χπ (cid:20) A y + B y + C y − y + + D y − y − (cid:21) p A cos χ + B cos χ + C d χ, (B.2b) olutions for bound trajectories around a Kerr black hole y ± = r ± a µ , (B.3a) A = La µ − L − aE ) µr + √ − a , B = − La µ + 2 ( L − aE ) µr − √ − a , (B.3b) A = Ea µ , B = 2 Ea µ , C = a µ r − √ − a ( − La + 2 Er − ) , (B.3c) D = aµ r − √ − a (cid:16) − Lr − √ − a − Ear − + La (cid:17) . (B.3d)Next, we make the substitution, cos χ = 2 cos χ − and ψ = χ − π , which reducesthe integrals to I = − " C Z ψ d ψ (cid:0) p sin ψ (cid:1) p − m sin ψ p − n sin ψ + C Z ψ d ψ (cid:0) p sin ψ (cid:1) p − m sin ψ p − n sin ψ , = − [ C I + C I ] , (B.4a) I = " C Z ψ d ψ (cid:0) p sin ψ (cid:1) p − m sin ψ p − n sin ψ + C Z ψ d ψ (cid:0) p sin ψ (cid:1) p − m sin ψ p − n sin ψ + C Z ψ d ψ (cid:0) p sin ψ (cid:1) p − m sin ψ p − n sin ψ + C Z ψ d ψ (cid:0) p sin ψ (cid:1) p − m sin ψ p − n sin ψ , = C I + C I + C I + C I . (B.4b)where the constants, C - C , n , m , p , p , and p are defined by Eq. (9) in §2.2.First, we solve the integrals I , I or I which are of the form given by I a ≡ Z ψ d ψ (cid:0) p sin ψ (cid:1) p − m sin ψ p − n sin ψ , = Z ψ d ψ p − m sin ψ p − n sin ψ − p Z ψ sin ψ (cid:0) p sin ψ (cid:1) p − m sin ψ p − n sin ψ d ψ, = Z ψ d ψ p − m sin ψ p − n sin ψ + p m Z ψ − m sin ψ − (cid:0) p sin ψ (cid:1) p − m sin ψ p − n sin ψ d ψ, olutions for bound trajectories around a Kerr black hole I a = 1( p + m ) " m Z ψ d ψ p − m sin ψ p − n sin ψ + p Z ψ p − m sin ψ (cid:0) p sin ψ (cid:1) p − n sin ψ d ψ . (B.5)Now, the substitution given by sin α = √ − m sin ψ p − m sin ψ , (B.6)reduces the integrals in Eq. (B.5) to I a = 1 √ − m ( p + m ) (cid:20) m F (cid:18) α, n − m − m (cid:19) + p Π (cid:18) − p − m − m , α, n − m − m (cid:19)(cid:21) . (B.7)Hence, integrals given by Eqs. (8c, 8d, 8f) reduce to the forms given above. Next, wesolve for I , which is of the form I b ≡ Z ψ d ψ (cid:0) p sin ψ (cid:1) p − m sin ψ p − n sin ψ , = Z ψ p sin ψ − p sin ψ (cid:0) p sin ψ (cid:1) p − m sin ψ p − n sin ψ d ψ, = I a + p m Z ψ p − m sin ψ (cid:0) p sin ψ (cid:1) p − n sin ψ d ψ − p m I b ,I b = 1( m + p ) ( m I a + p "Z ψ p − m sin ψ (cid:0) p sin ψ (cid:1) p − n sin ψ d ψ − p Z ψ sin ψ p − m sin ψ (cid:0) p sin ψ (cid:1) p − n sin ψ d ψ , (B.8)where the substitution given by Eq. (B.6) reduces the second integral in the aboveequation to I b = 1( m + p ) (cid:20) m I a + p √ − m Π (cid:18) − p − m − m , α, n − m − m (cid:19) − p I c (cid:21) , (B.9)where I c ≡ Z ψ sin ψ p − m sin ψ (cid:0) p sin ψ (cid:1) p − n sin ψ d ψ = − n Z ψ (cid:0) − n sin ψ − (cid:1) p − m sin ψ (cid:0) p sin ψ (cid:1) p − n sin ψ d ψ,I c (cid:18) p n (cid:19) = 1 n " − Z ψ p − n sin ψ p − m sin ψ (cid:0) p sin ψ (cid:1) d ψ + Z ψ p − m sin ψ (cid:0) p sin ψ (cid:1) p − n sin ψ d ψ ; the substitution given by Eq. (B.6) reduces the above expression to I c = 1( n + p ) √ − m − Z α p − k sin α h p + m − m sin α i d α + Π (cid:18) − ( p + m )1 − m , α, k (cid:19) , (B.10) olutions for bound trajectories around a Kerr black hole p − k sin α gives I c = 1( n + p ) √ − m − Z α h p + m − m sin α i p − k sin α d α + Π (cid:18) − ( p + m )1 − m , α, k (cid:19) + k Z α sin α h p + m − m sin α i p − k sin α d α , (B.11)which can be further reduced to I c = 1 √ − m ( m + p ) (cid:26) Π (cid:18) − ( p + m )1 − m , α, n − m − m (cid:19) − I d (cid:27) , (B.12)where I d = Z α h p + m − m sin α i p − k sin α d α. (B.13)By defining integrals having a general form given by T n = Z d y (cid:0) h + g sin y (cid:1) n p − w sin y , (B.14)we use the following identity [26]: T n − = − g sin y cos y p − w sin y (cid:0) h + g sin y (cid:1) n − (2 n − w − (2 n −
3) [ g + 2 hg (1 + w ) + 3 h w ] T n − (2 n − w + 2 ( n −
2) [ g (1 + w ) + 3 hw ] T n − (2 n − w + 2 ( n − h ( g + h ) ( g + hw ) T n (2 n − w . (B.15)The integral I d has a form similar to T with (cid:16) h = 1 , g = p + m − m , w = k (cid:17) , which yields I d = T = 12 (1 + g ) ( g + k ) ( g sin α cos α p − k sin α (cid:0) g sin α (cid:1) + (cid:2) g + 2 g (cid:0) k (cid:1) + 3 k (cid:3) T − k T − ) , (B.16)where T − = Z α (cid:0) g sin α (cid:1) d α p − k sin α = Z α d α p − k sin α − gk Z α (cid:0) − k sin α − (cid:1) d α p − k sin α = (cid:16) gk (cid:17) F (cid:0) α, k (cid:1) − gk K (cid:0) α, k (cid:1) , (B.17)and T = Π (cid:0) − g, α, k (cid:1) , (B.18)were substituted. See Table 2 for the summary of the final expressions of I − I derivedusing the method given in this section. olutions for bound trajectories around a Kerr black hole Appendix C. Reduction to the equatorial plane ( Q = 0 ) In this appendix, we reduce the integrals of motion to the case of equatorial plane. Westart with the final expressions of I , I , I , and I (in §2.2) and take the limit Q → .As shown in §2.4, that for Q → , we have n → which gives k = − m − m . Now, wefirst reduce the expressions of I and I , Eqs. (8c, 8d) under the limit Q → . We usethe following identity (cf. [46], Eq. 160.02) to write Π (cid:0) α , ϕ, − k (cid:1) = k ′ h k F ( β , k ) + k ′ α Π ( α , β , k ) i(cid:16) α k ′ + k (cid:17) , (C.1)where sin β = p k sin ϕ p k sin ϕ , (C.2) α = α k ′ + k , k = k √ k and k ′ = k k , which reduces I and I to the forms I = 1 √ − m ( m + p ) (cid:20) m F (cid:0) α, k (cid:1) + p Π (cid:18) − ( p + m )1 − m , α, k (cid:19)(cid:21) = Π (cid:0) − p , ψ, m (cid:1) , (C.3) I = 1 √ − m ( m + p ) (cid:20) m F (cid:0) α, k (cid:1) + p Π (cid:18) − ( p + m )1 − m , α, k (cid:19)(cid:21) = Π (cid:0) − p , ψ, m (cid:1) . (C.4)The above expressions can be directly obtained if n = 0 is substituted in the definitionof I and I at the intermediate step, Eq. (B.4a). However, we aim to directly validatethe final forms of ( I − I ). Hence, we apply the reduced expressions of I and I to ( φ − φ ) for the equatorial plane to obtain Eq. (16a).Next, we reduce the coordinate time integral. The expression for I also reduces insimilar way to I and I by applying the identity, Eq. (C.1), to I = 1 √ − m ( m + p ) (cid:20) m F (cid:0) α, k (cid:1) + p Π (cid:18) − ( p + m )1 − m , α, k (cid:19)(cid:21) = Π (cid:0) − p , ψ, m (cid:1) , (C.5)which again can be directly obtained by substituting n = 0 in the definition of I in Eq. (B.4b). Next, we reduce the expression of I , Eq. (8e). We substitute for sin α = √ − m sin ψ p − m sin ψ , cos α = cos ψ p − m sin ψ , k = − m − m and s = − p + m − m intothe expression of I , Eq. (8g), and use the following identities (cf. [46], Eq. 160.02) F (cid:0) ϕ, − k (cid:1) = k ′ F (cid:0) β , k (cid:1) , (C.6) K (cid:0) ϕ, − k (cid:1) = 1 k ′ " K (cid:0) β , k (cid:1) − k sin β cos β p − k sin β , (C.7) olutions for bound trajectories around a Kerr black hole F ( α, k ) , K ( α, k ) , and Π ( s , α, k ) to F (cid:0) α, k (cid:1) = √ − m F (cid:0) ψ, m (cid:1) , (C.8) K (cid:0) α, k (cid:1) = 1 √ − m " K (cid:0) ψ, m (cid:1) − m sin ψ cos ψ p − m sin ψ , (C.9) Π (cid:0) s , α, k (cid:1) = √ − m p (cid:2)(cid:0) p + m (cid:1) Π (cid:0) − p , ψ, m (cid:1) − m F (cid:0) ψ, m (cid:1)(cid:3) . (C.10)The substitution of the above equations reduces I to I = ( ( p + m ) √ − m sin ψ cos ψ p (1 + p ) p − m sin ψ (cid:0) p sin ψ (cid:1) + ( p + m ) √ − m p (1 + p ) " K (cid:0) ψ, m (cid:1) − m sin ψ cos ψ p − m sin ψ + ( p − p m + 2 p − m ) √ − m p (1 + p ) (cid:2)(cid:0) p + m (cid:1) Π (cid:0) − p , ψ, m (cid:1) − m F (cid:0) ψ, m (cid:1)(cid:3) − (1 − m ) / F ( ψ, m )2 (1 + p ) ) . (C.11)Now, the substitution of I from the above equation and sin α , cos α , k , s into Eq.(8e) reduces I to I = p sin ψ cos ψ p − m sin ψ p ) ( m + p ) (cid:0) p sin ψ (cid:1) + [ p + 2 p (1 + m ) + 3 m ]2 (1 + p ) ( m + p ) Π (cid:0) − p , ψ, m (cid:1) −
12 (1 + p ) F (cid:0) ψ, m (cid:1) + p p ) ( m + p ) K (cid:0) ψ, m (cid:1) . (C.12)The above expression of I can also be directly obtained by substituting n = 0 in itsdefinition given in Eq. (B.4b) and applying the identity given by Eq. (B.15). Hence,the substitution of I , I , I and I from Eqs. (C.3), (C.4), (C.12) and (C.5) in Eq.(12b) gives the expression for coordinate time, t , which simplifies to Eq. (16b).See Table 2 for the final expressions of ( φ, t ) for the equatorial plane derived usingthe method given in this section. Appendix D. Innermost stable and marginally bound spherical radii
We discussed in §3.2 that ISSO and MBSO radii represent the end points of theseparatrix curve defined by Eq. (20). Hence, we use this to write equations for ISSOand MBSO.(i) For the case of ISSO, we substitute e = 0 and µ = 1 /r s into Eq. (20), which gives r s − r s (cid:0) Q + x (cid:1) + 4 a Q = 0 , (D.1)which further expands, by the substitution of x for spherical orbits, to r s − r s − a r s + 18 r s + 4 a Qr s − a r s − a Qr s + 24 a Qr s − a Q + 6 a (cid:0) a − r s + r s (cid:1) p a Q − Q ( r s − r s + r s = 0 . (D.2) olutions for bound trajectories around a Kerr black hole r s ,which factorizes to (cid:18) r s − r s − a r s + 36 r s + 8 a Qr s − a r s − a Qr s + 9 a r s − a Qr s + 48 a Qr s + 16 a Qr s − a Qr s − a Q r s + 48 a Q r s − a Q (cid:19) · (cid:0) r s − r s + 9 r s − a (cid:1) = 0 , (D.3)where the second factor corresponds to the light radius [8] and the equation forISSO is given by r s − r s − a r s + 36 r s + 8 a Qr s − a r s − a Qr s + 9 a r s − a Qr s +48 a Qr s + 16 a Q r s − a Qr s − a Q r s + 48 a Q r s − a Q = 0 . (D.4)For the equatorial plane, Q = 0 gives the following equation for the ISCO radius( Z ) Z − Z − a Z + 18 Z − a Z + 6 a (cid:0) a − Z + Z (cid:1) Z / = 0; (D.5)this equation can be factorized as Z / (cid:0) Z / − Z / − a (cid:1) (cid:0) Z − Z + 8 aZ / − a (cid:1) = 0 , (D.6)where the first bracket gives the solution for light radius and the second bracket, Z − Z + 8 aZ / − a = 0 , (D.7)gives the solution for ISCO [8] given by Z = n Z − [(3 − Z ) (3 + Z + 2 Z )] / o , (D.8a) Z = 1 + (cid:0) − a (cid:1) / h (1 + a ) / + (1 − a ) / i , (D.8b) Z = (cid:0) a + Z (cid:1) / . (D.8c)(ii) The condition for MBSO is derived by substituting e = 1 and µ = 1 / (2 Y ) in Eq.(20), which yields Y − Y (cid:0) Q + x (cid:1) + a Q = 0 , (D.9)where r a of the orbit reaches infinity and r s in the above equation corresponds tothe unstable radius. By substituting x for spherical orbits, the above equationreduces to Y − Y + 12 Y − a Y + a QY − a Y − a QY + 5 a QY − a Q, + 2 a (cid:0) a − Y + Y (cid:1) p a Q − Q ( Y − Y + Y = 0 . (D.10)The above expression expands to an equation of eleventh order in r s , whichfactorizes to (cid:18) Y − Y − a Y + 16 Y + 2 a QY − a Y − a QY + a Y − a QY + 8 a QY + a Q Y − a QY − a Q Y + a Q (cid:19) · (cid:0) X − X + 9 X − a (cid:1) = 0 , (D.11) olutions for bound trajectories around a Kerr black hole X ) [8] and the equationfor MBSO is given by Y − Y − a Y + 16 Y + 2 a QY − a Y − a QY + a Y − a QY +8 a QY + a Q Y − a QY − a Q Y + a Q = 0 . (D.12)In the equatorial plane, Q = 0 , this reduces to Y − Y + 12 Y − a Y − a Y + 2 a (cid:0) a − Y + Y (cid:1) Y / = 0 , (D.13)which gets further factorized to Y / (cid:0) Y + 2 Y / − a (cid:1) (cid:0) Y − Y / + a (cid:1) (cid:0) X / − X / − a (cid:1) = 0 , (D.14)where the last bracket in above equation gives the solution for light radius andthe first two brackets give retrograde and prograde solutions for marginally boundorbit in the equatorial plane [8]. According to the sign convention used in this paper( − < a < ), both retrograde and prograde cases are covered by the formula givenby Y = 2 − a + 2 √ − a. (D.15) Appendix E. Solution of ( e s , µ s ) in the equatorial separatrix case The expressions for eccentricity and inverse-latus rectum for the non-equatorialseparatrix orbits, given by Eqs. (29a) and (29b) can be written in the form e s = 2 u s h − a ′ − q ( a ′ + 2 u s ) − d ′ u s i − , (E.1) µ s = 14 " − a ′ − s ( a ′ + 2 u s ) − d ′ u s . (E.2)where the expressions for a ′ and d ′ are given by Eq. (25b). We solve for the factor inthe denominator of e s by substituting a ′ and d ′ and taking the limit Q → , we find " − a ′ − s ( a ′ + 2 u s ) − d ′ u s = (cid:20) x a Q + 1 a − x a Q (cid:18) Qx − u s a Qx − a Q u s x + E a Q u s x + O [ Q ] (cid:19)(cid:21) , (E.3) = u s + 12 x u s − E x u s . (E.4)Substitution of Eq. (E.4) in (E.1, E.2) gives e s = 2 u s x − E u s x + 1 − E , µ s = 14 (cid:20) u s + 1 − E x u s (cid:21) . (E.5) olutions for bound trajectories around a Kerr black hole x = ( L − aE ) and E from Eq. (19) into the above equations,we find e s = − r / s − r / s + 6 ar s − a r / s + 18 r / s − ar s − a r / s + 6 a r / s − r / s − ar s + a r / s + 6 r / s + 4 ar s − a r / s − a , = − (cid:16) r / s − r / s − a (cid:17) (cid:16) r s − r s + 8 ar / s − a (cid:17)(cid:16) r / s − r / s − a (cid:17) ( r s + a − r s ) = − r s − r s + 8 ar / s − a r s + a − r s . (E.6)and µ s = r s − r s − ar / s + a r s + 6 r s + 4 ar / s − a r s − a r / s r s (cid:16) r s − r s + a r s + a r s − ar / s + 4 ar / s − a r / s (cid:17) , = ( r s + a − r s ) (cid:16) r s − r s − ar / s (cid:17) r s ( r s + a − ar s / ) (cid:16) r s − r s − ar / s (cid:17) = r s + a − r s r s (cid:16) r / s − a (cid:17) , (E.7)which are the expressions in Eq. (30) given in §3.2. Appendix F. Reducing the radial integrals for the case of non-equatorialseparatrix trajectories
Here, we reduce the radial integrals in our general trajectory solutions presented in§2.2, for the case of separatrix orbits with Q = 0 . As shown in §3.3 that k = 1 for theseparatrix orbits, we use the identities given by Eq. (33) [26] to reduce the correspondingElliptic integrals.The substitution of these reduced form of the Elliptic integrals further reduces theintegrals of the form S and S , Eqs. (8c, 8d), to the general form S ≡ p ) " p p − ( p + m ) ln s √ − m + p − ( p + m ) sin α √ − m − p − ( p + m ) sin α + ln (tan α + sec α ) √ − m , (F.1)where p equals p and p for S and S respectively. Hence, from Eq. (8a) theexpression of I reduces in terms of S and S for the separatrix trajectories to I = − p µ (1 + e ) (3 − e ) p e [1 + 2 a ( − e ) Qµ ] (1 − a ) (cid:20) [ La − xr + ] S ( a µ − a µe − r + ) + [ − La + 2 xr − ] S ( a µ − a µe − r − ) (cid:21) . (F.2)Further, Eqs. (33) reduce the integrals S − S , Eqs. (8e-8h), to S = 1 √ − m ( m + p ) (cid:20) m ( m − p m + 2 p )(1 + p ) ln (tan α + sec α ) + p S + olutions for bound trajectories around a Kerr black hole p m (1 − m )(1 + p ) | s | tan − [ | s | sin α ] (cid:21) , (F.3) S = ln (tan α + sec α ) √ − m (1 + p ) + p √ − m ( m + p ) (1 + p ) | s | tan − [ | s | sin α ] , (F.4) S = 12 (1 − s ) " s sin α cos α (cid:0) − s sin α (cid:1) + 2 ln (tan α + sec α ) − s sin α + (cid:0) − s (cid:1) | s | tan − ( | s | sin α ) , (F.5) S = 2 µ (1 − e ) p C − A + √ B − AC ln (tan α + sec α ) . (F.6)Finally, the expression of I , Eq. (8b), reduces for the separatrix orbits in terms of S − S to I = 2 p (1 + e ) (3 − e ) p eµ [1 + 2 a ( − e ) Qµ ] ( Eµ (1 − e ) S + a µ ( − La + 2 Er − ) r − p (1 − a ) ( a µ − a µe − r + ) S + 2 E (1 − e ) S + aµ (cid:0) − Lr − √ − a − Ear − + La (cid:1) r − p (1 − a ) ( a µ − a µe − r − ) S ) . (F.7)See Table 3 for the summary of radial integrals for the separatrix trajectories. Appendix G. Derivations for the consistency check with the previousresults (i)
Equatorial separatrix orbits : Here, we show the reduction of the separatrixtrajectory formulae to the equatorial case. We substitute Q = 0 in the expressionof the azimuthal angle given by Eq. (34a), which yields φ − φ = 12 s µ (1 + e ) (3 − e ) e (1 − a ) (cid:20) [ La − xr + ]( a µ − a µe − r + ) I + [ − La + 2 xr − ]( a µ − a µe − r − ) I (cid:21) . (G.1)We saw that n = 0 for the equatorial orbits and m reduces to 1 for separatrixorbits, which gives α = 0 by definition, given by Eq. (B.6). Hence, we solve theintegrals in the form of ψ variable. First, we solve integrals I and I for separatrixorbits using I = Z ψ d ψ (cid:0) p sin ψ (cid:1) cos ψ , I = Z ψ d ψ (cid:0) p sin ψ (cid:1) cos ψ . (G.2) olutions for bound trajectories around a Kerr black hole p sin ψ = z for I and p sin ψ = z for I , and applying themethod of partial fractions, we find I = 1(1 + p ) [ p arctan ( p sin ψ ) + arctanh (sin ψ )] , (G.3a) I = 1(1 + p ) [ p arctan ( p sin ψ ) + arctanh (sin ψ )] . (G.3b)Hence, when the above equations for I and I are substituted into the expressionsfor azimuthal angle, Eq. (G.1), yields φ − φ = s µ (1 + e ) (3 − e ) e (1 − a ) (cid:26) ( La − xr + ) p · arctan ( p sin ψ )2 ( a µ + a µe − r + ) +( − La + 2 xr − ) p · arctan ( p sin ψ )2 ( a µ + a µe − r − ) + √ − a [ L − xµ (1 + e )] arctanh (sin ψ ) (cid:2) µ a (1 + e ) − µ (1 + e ) + 1 (cid:3) ) . (G.4)Now, plugging in Q = 0 , reduces the expression for coordinate time of the separatrixorbits, Eq. (34b), to t − t = s (1 + e ) (3 − e ) eµ ( Eµ (1 − e ) I + a µ ( − La + 2 Er − ) r − p (1 − a ) ( a µ − a µe − r + ) I + 2 E (1 − e ) I + aµ (cid:0) − Lr − √ − a − Ear − + La (cid:1) r − p (1 − a ) ( a µ − a µe − r − ) I ) . (G.5)And similarly, the integrals I and I reduce to I = Z ψ d ψ (cid:0) p sin ψ (cid:1) cos ψ = 1(1 + p ) " p (cid:0) p (cid:1) arctan ( p sin ψ ) + arctanh (sin ψ ) + p sin ψ (1 + p )2 (cid:0) p sin ψ (cid:1) , (G.6a) I = Z ψ d ψ (cid:0) p sin ψ (cid:1) cos ψ = 1(1 + p ) [ p arctan ( p sin ψ ) + arctanh (sin ψ )] . (G.6b)Hence, the substitution of I , I , I , I into Eq. (G.5) yields the expression forcoordinate time for equatorial separatrix orbits to be t − t = s (1 + e ) (3 − e ) eµ ( Ee sin ψµ (1 − e ) (cid:0) − e + 2 e sin ψ (cid:1) + E [3 − e + 4 µ (1 − e )] p · arctan ( p sin ψ )2 µ (1 − e ) (1 + e )+ aµr − √ − a ( − La + 2 Ear − ) p · arctan ( p sin ψ )( a µ + a µe − r + ) + (cid:0) − Lr − √ − a − Ear − + La (cid:1) · p · arctan ( p sin ψ )( a µ + a µe − r − ) EFERENCES + " E [1 + 2 µ (1 + e )] µ (1 + e ) + 2 µ [ − Laµ (1 + e ) + 2 E ] (cid:2) µ a (1 + e ) − µ (1 + e ) (cid:3) arctanh (sin ψ ) ) (G.7)We have numerically matched these expressions of the azimuthal angle andcoordinate time for equatorial separatrix orbits, Eqs. (G.4) and (G.7), with thosederived in [20].(ii) Spherical orbits :Now, we verify our frequency formulae with the spherical orbit case. From Eqs.(37c) and (37b), we have ν φ ν θ = 2 (" − I (cid:0) π , e, µ, , Q (cid:1) I (cid:0) π , e, µ, , Q (cid:1) − L F (cid:16) π , z − z (cid:17) + L · Π (cid:16) z − , π , z − z (cid:17)) π √ − E z + . (G.8)In [9], the azimuthal to polar motion frequency ratio, ν φ /ν θ , for maximally rotatingblack hole, a = 1 , was found to be ν φ ν θ = 2 n L · Π (cid:16) z − , π , z − z (cid:17) + ( P ∆ − − E ) F (cid:16) π , z − z (cid:17)o π √ − E z + , (G.9)where P = [ E ( r + 1) − L ] ( r − − E for a = 1 .For the case of spherical orbits, the limit e → reduces the ratio ( I /I ) to theratio of their integrands, which yields (cid:20) − I I − L (cid:21) = − E + [ E ( r + 1) − L ]( r − = (cid:0) P ∆ − − E (cid:1) ; (G.10)this reduces ν φ /ν θ from, Eq. (G.8) to Eq. (G.9), and establishes the consistency ofour frequency formulae with the spherical orbits case. References [1] B. P. Abbott et al. Observation of Gravitational Waves from a Binary Black Hole Merger.
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