Analytical solutions of bound timelike geodesic orbits in Kerr spacetime

We present two methods for integrating forced geodesic equations in the Kerr spacetime. The methods can accommodate arbitrary forces. As a test case, we compute inspirals caused by a simple drag force, mimicking motion in the presence of gas. We verify that both methods give the same results for this simple force. We find that drag generally causes eccentricity to increase throughout the inspiral. This is a relativistic effect qualitatively opposite to what is seen in gravitationalradiation-driven inspirals, and similar to what others have observed in hydrodynamic simulations of gaseous binaries. We provide an analytic explanation by deriving the leading order relativistic correction to the Newtonian dynamics. If observed, an increasing eccentricity would thus provide clear evidence that the inspiral was occurring in a non-vacuum environment.Our two methods are especially useful for evolving orbits in the adiabatic regime. Both use the method of osculating orbits, in which each point on the orbit is characterized by the parameters of the geodesic with the same instantaneous position and velocity. Both methods describe the orbit in terms of the geodesic energy, axial angular momentum, Carter constant, azimuthal phase, and two angular variables that increase monotonically and are relativistic generalizations of the eccentric anomaly. The two methods differ in their treatment of the orbital phases and the representation of the force. In the first method, the geodesic phase and phase constant are evolved together as a single orbital phase parameter, and the force is expressed in terms of its components on the Kinnersley orthonormal tetrad. In the second method, the phase constants of the geodesic motion are evolved separately and the force is expressed in terms of its Boyer-Lindquist components. This second approach is a direct generalization of earlier work by Pound and Poisson [1] for planar forces in a Schwarzschild background.

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“…In particular, an action-angle formulation of the Kerr solution exists [19], in which the equations take the form…”

Section: Action-angle Formulationmentioning

confidence: 99%

Forced motion near black holes

et al. 2011

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“…Later on, the motion of test particles was extensively investigated in Kerr spacetimes where the circular geodesics has also been examined [19,20,21,22]. Recently, in [23], analytic solutions of the bound timelike geodesics of test particles in Kerr spacetime have been presented in terms of elliptic integrals using Mino time. In [24] and [25], the geodesic equations are analytically solved in the background of Schwarzschild-(anti) de Sitter spacetimes, where the solutions are expressed in terms of Kleinian sigma functions.…”

Section: Introductionmentioning

confidence: 99%

Motion of the charged test particles in Kerr-Newman-Taub-NUT spacetime and analytical solutions

2016

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In this work, we study the motion of charged test particles in KerrNewman-Taub-NUT spacetime. We analyze the angular and the radial parts of the orbit equations and examine the possible orbit types. We also investigate the spherical orbits and their stabilities. Furthermore, we obtain the analytical solutions of the equations of motion and express them in terms of Jacobian and Weierstrass elliptic functions. Finally, we discuss the observables of the bound motion and calculate the perihelion shift and Lense-Thirring effect for the bound orbits.

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“…Since the whole method is implemented using arbitrary-precision arithmetic and uses a numerical implementation [55][56][57] of the analytical series solution to the Teukolsky equation devised by Mano, Suzuki, and Takasugi [58,59], individual modes can be solved to almost any desired accuracy. The limiting step in the accuracy of this method comes from fitting for the large-l tail of the mode sum.…”

Section: The Teukolsky-mst-cck Methodsmentioning

confidence: 99%

Numerical computation of the effective-one-body potentialqusing self-force results

Akçay

Meent

2016

Phys. Rev. D

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The effective-one-body theory (EOB) describes the conservative dynamics of compact binary systems in terms of an effective Hamiltonian approach. The Hamiltonian for moderately eccentric motion of two non-spinning compact objects in the extreme mass-ratio limit is given in terms of three potentials: a(v),d(v), q(v). By generalizing the first law of mechanics for (non-spinning) black hole binaries to eccentric orbits, [A. Le Tiec Phys. Rev. D92, 084021 (2015)] recently obtained new expressions ford(v) and q(v) in terms of quantities that can be readily computed using the gravitational self-force approach. Using these expressions we present a new computation of the EOB potential q(v) by combining results from two independent numerical self-force codes. We determine q(v) for inverse binary separations in the range 1/1200 ≤ v 1/6. Our computation thus provides the first-ever strong-field results for q(v). We also obtaind(v) in our entire domain to a fractional accuracy of 10 −8 . We find that our results are compatible with the known post-Newtonian expansions ford(v) and q(v) in the weak field, and agree with previous (less accurate) numerical results ford(v) in the strong field.

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Analytical solutions of bound timelike geodesic orbits in Kerr spacetime

Cited by 163 publications

References 37 publications

Forced motion near black holes

Forced motion near black holes

Motion of the charged test particles in Kerr-Newman-Taub-NUT spacetime and analytical solutions

Numerical computation of the effective-one-body potentialqusing self-force results

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