We derive alternate and new closed-form analytic solutions for the non-equatorial eccentric bound trajectories, , , around a Kerr black hole by using the transformation . The application of the solutions is straightforward and numerically fast. We obtain and implement translation relations between energy and angular momentum of the particle, (, ), and eccentricity and inverse-latus rectum, (, ), for a given spin, , and Carter’s constant, , to write the trajectory completely in the (, , , ) parameter space. The bound orbit conditions are obtained and implemented to select the allowed combination of parameters (, , , ). We also derive specialized formulae for spherical and separatrix orbits. A study of the non-equatorial analog of the previously studied equatorial separatrix orbits is carried out where a homoclinic orbit asymptotes to an energetically bound spherical orbit. Such orbits simultaneously represent an eccentric orbit and an unstable spherical orbit, both of which share the same and values. We present exact expressions for and as functions of the radius of the corresponding unstable spherical orbit, , , and , and their trajectories, for () separatrix orbits; they are shown to reduce to the equatorial case. These formulae have applications to study the gravitational waveforms from extreme-mass ratio inspirals (EMRIs) using adiabatic progression following a sequence of Kerr geodesics, besides relativistic precession and phase space explorations. We obtain closed-form expressions of the fundamental frequencies of non-equatorial eccentric trajectories that are equivalent to the previously obtained quadrature forms and also numerically match with the equivalent formulae previously derived. We sketch non-equatorial eccentric, separatrix, zoom-whirl, and spherical orbits, and discuss their astrophysical applications.
Solutions for bound trajectories around a Kerr black hole]Astrophysically relevant bound trajectories around a Kerr black hole
Indian Institute of Astrophysics, Sarjapur Road, 2nd block Koramangala, Bangalore, 560034
E-mail: firstname.lastname@example.org, email@example.com
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.
It has now been established with observational evidence that black holes with masses ranging from - in X-ray binaries, to - in galactic nuclei, are ubiquitous. One among such important evidences is the recent detection of gravitational waves from the black hole binary merger [Abbott2016] and more such events are awaited to be detected by the planned LISA mission [Glampedakis2005]. One of the major objectives of the LISA mission is the detection of gravitational waves from EMRIs, most probably to be sourced from the compact objects spiralling and finally plunging onto the super-massive black hole (SMBH) in galactic nuclei. The dynamics of EMRIs is widely accepted as representative of test-particle motion, evolving adiabatically, in the spacetime of a rotating black hole. Understanding of such strong gravity regimes involves them using the Kerr metric [Kerr1963], which is a vacuum solution of Einstein’s equation for a rotating black hole. The study of trajectories around the black holes is critical to our understanding of the physical processes and their observational consequences [Narayan2005].
The trajectories in Kerr and Schwarzschild [Schwarzschild] geometries have been studied extensively. Some of these results are covered in a pioneering work by [C1983book] in an elegant manner. The key idea that the general trajectory in Kerr background can be expressed in terms of quadratures, was first given in [Carter1968]. In [Bardeen1972], the energy, , and angular momentum, , were expressed in terms of the circular orbit radius and the spin parameter ; the specific solution for the radius of the innermost stable circular orbit (ISCO) was also derived. The necessary conditions for bound geodesics for spherical orbits and the dragging of nodes along the direction of spin of a black hole was discussed [Wilkins1972]. The formulae have proved to be extremely useful in predicting observables in astrophysical applications like accretion around the black holes. For example, a general solution for a star in orbit around a rotating black hole was expressed in terms of quadratures [Vokrouhlicky1993] using the formulation given by [Carter1968]; the resulting integrals have been calculated numerically. The general expression in terms of quadratures for fundamental orbital frequencies , and , for a general eccentric orbit, were first derived by [Schmidt2002], where different cases for circular and equatorial orbits are also discussed but complete analytic trajectories were not calculated. An exact solution for non-spherical polar trajectories in Kerr geometry and an exact analytic expression for for eccentric orbits in the equatorial plane were derived [Kraniotis2007]. These were used to obtain the expressions for the periapsis advance and Lense-Thirring frequencies. The time-like geodesics were expressed in terms of quadratures involving hyper-elliptic, elliptic and Abelian integrals for Kerr and Kerr-(anti) de Sitter spacetimes including cosmological constant [Kraniotis2004] and applied in a semi-analytic treatment of Lense Thirring effect.
A time-like orbital parameter called Mino time [Mino2003] was introduced to decouple the and equations, which was then used to express a wider class of trajectory functions in terms of the orbital frequencies , and [Drasco2004]. These methods are applied to calculate closed-form solutions of the trajectories and their orbital frequencies [Fujita2009], using the roots of the effective potential. However, the solutions are expressed in terms of Mino time. The commensurability of radial and longitudinal frequencies, their resonance conditions for orbits in Kerr geometry, and their location in terms of spin and orbital parameter values were studied using numerical implementations of Carlson’s elliptic integrals [Brink2015]. Considering the problem of the precession of a test gyroscope in the equatorial plane of Kerr geometry, the analytic expressions to transform energy, angular momentum of the orbiting test particle, and spin of the black hole (, , ) to eccentricity, inverse-latus rectum of the bound orbit (, , ) parameters were obtained [Bini2016a]. The expressions for radial and orbital frequencies are obtained to the order for bound orbits and analytically for the marginally bound homoclinic orbits [Bini2016b]. The dynamical studies of an important family of Kerr orbits called separatrix or homoclinic orbits are important for computing the transition of spiralling to plunge in EMRIs emitting gravitational waves [Levin2009, Glampedakis2002]. The test particles (compact objects in this case) transit through an eccentric separatrix orbit in EMRIs, while progressing adiabatically, before they merge with the massive black hole.
In this paper, we study the generic bound trajectories, which are eccentric and inclined, around a Kerr black hole, and then we investigate the non- equatorial separatrix orbits as a special case. 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 Mino time, , which makes them numerically faster. We also implement the essential bound orbit conditions to choose the parameters (, , , ) of an allowed bound orbit, derived from the essential conditions on the parameters for the elliptic integrals involved in the trajectory solutions. We find that the space is more convenient for integrating the equations of motion as the turning points of the integrands are naturally specified in terms of the bound orbit conditions. The exact solutions for the trajectories are found in terms of not overly long expressions involving elliptic functions. We implement the translation formulae between (, ) and (, ) parameters that help us to express the trajectory solutions completely in the (, , , ) parameter space which we call the conic parameter space. We then study the case of non-equatorial separatrix trajectories in the conic parameter space. First, we write the essential equations for the important radii like , , and spherical light radius. Similar to the equatorial separatrix orbits, the non-equatorial separatrix or homoclinic trajectories asymptote to an energetically bound unstable spherical orbit, where the spherical orbit radius can vary between and . We show that the formulae for (, ) for the non-equatorial separatrix orbits can be expressed as functions of the radius of the corresponding spherical orbit, , , and , which also reduce to their equatorial counterpart [Levin2008] by implementing the limit . These formulae are obtained by using the expressions of and for the spherical orbits. Next, we derive the exact solutions for the non-equatorial separatrix trajectories by reducing our general eccentric trajectory solutions to this case. These solutions are important for investigating the behaviour of gravitational waveforms emitted by inspiralling and inclined test objects near non-equatorial separatrix trajectories in the case of EMRIs.
The ab-initio specification of the allowed geometry of bound orbits in the parameter space is crucial for the calculation of the orbital trajectories and its frequencies. These criteria are used in building, studying and sketching different types of trajectories around a Kerr black hole: for instance, spherical, non-equatorial eccentric, non-equatorial separatrix and zoom-whirl orbits, using our closed-form expressions for trajectories are constructed. We also derive closed-form analytic expressions for the fundamental frequencies of the general non-equatorial trajectories as functions of elliptic integrals around the Kerr black holes. We use a time-averaging method on the first-order equations of motion to derive these frequencies and show that our closed-form analytic expressions of frequencies match with the formulae given by [Schmidt2002] which were left in the quadrature form. We also reduce the general forms to the equatorial case, which is also a new form that is easier to implement and faster by a factor of .
This paper is organized as follows (see Fig. LABEL:concepts): in §LABEL:inofmttn, we review the basic equations describing motion around the Kerr black hole using Hamiltonian dynamics. In §LABEL:trans, we write the translation formulae from to parameter space. In §LABEL:analyticsoln, we derive the exact closed-form solutions for the trajectories by solving all involving integrals and writing them in terms of elliptic integrals, . In §LABEL:bndcnd, we give the essential bound orbit conditions on parameters applicable to the astrophysical situations. In §LABEL:equatorialtrajec, we reduce the analytic solutions to the case of equatorial plane. In §LABEL:ELforsphorbits, we derive the formulae for and for spherical orbits as a function of radius , , and . In §LABEL:emuseparatrix, we write the equations for the radii , , and spherical light radius. We then derive the exact expressions for and for the non-equatorial separatrix trajectories. In §LABEL:separatrixtrajec, we derive the trajectory solutions for the non-equatorial separatrix orbits. In §LABEL:trajec, we sketch and discuss various bound trajectories around the Kerr black hole. In §LABEL:frequencies, we derive the closed-form expressions of the fundamental frequencies in terms of elliptic integrals by the long time averaging method without using Mino time. In §LABEL:reduction, we conduct consistency checks by reducing the separatrix trajectories to the equatorial case, and also match the the azimuthal to polar frequency ratio, , with the spherical orbits case derived by [Wilkins1972]. We discuss possible applications of our trajectory solutions and frequency formulae in §LABEL:app. We summarize and conclude our results in §LABEL:summary and §LABEL:disc respectively.
|Boyer Lindquist coordinates|
|Time coordinate||Radial distance from the black hole|
|Polar angle||Azimuthal angle|
|Spin of the black hole|
|Common physical parameters|
|Horizon radius||Carter’s constant|
|Energy per unit rest mass of the test particle||z component of Angular momentum per unit|
|rest mass of the test particle|
|Generalized momentum for coordinate||Generalized momentum for coordinate|
|Generalized momentum for coordinate||=0 for photon orbits and for particle orbits|
|Radial effective potential for an eccentric test||Relativistic Hamiltonian for the geodesic motion|
|apastron distance ()||periastron distance ()|
|Third turning point of the test particle||Innermost turning point of the test particle|
|eccentricity parameter||inverse latus-rectum parameter|
|Integrals of motion|
|Terminology used for radial integrals|
|Terminology used for integrals|
|Spherical and separatrix orbits|
|radius of spherical orbit||radius of circular orbit|
|eccentricity of the separatrix orbits||inverse latus-rectum of the separatrix orbits|
|ISCO radius||Light radius|
|Azimuthal frequency||Radial frequency|
|Vertical oscillation frequency|