Prog. Theor. Phys. Vol. 121 No. 4 (2009) pp. 843-874
An Efficient Numerical Method for Computing Gravitational Waves Induced by a Particle Moving on Eccentric Inclined Orbits around a Kerr Black Hole
1Theoretical Physics, Raman Research Institute,
Bangalore 560 080, India
2Department of Earth and Space Science, Graduate School of Science,
Osaka University, Toyonaka 560-0043, Japan
(Received January 27, 2009)
We develop a numerical code to compute
gravitational waves induced by a particle moving on
eccentric inclined orbits around a Kerr black hole.
For such systems, the black hole perturbation method is applicable.
The gravitational waves can be evaluated by solving the
Teukolsky equation with a point like source term, which is computed from
the stress-energy tensor of a test particle moving on generic bound
In our previous papers, we computed the homogeneous solutions of the
Teukolsky equation using a formalism developed by Mano, Suzuki and
Takasugi and showed that we could
compute gravitational waves efficiently and very accurately
in the case of circular orbits on the equatorial plane.
Here, we apply this method to eccentric inclined orbits.
The geodesics around a Kerr black hole have three constants of motion:
energy, angular momentum and the Carter constant.
We compute the rates of change of the Carter
constant as well as those of energy and angular momentum.
This is the first time that the rate of change of the Carter constant has been
We also treat the case of highly eccentric orbits with e = 0.9.
To confirm the accuracy of our codes, several tests are performed.
We find that the accuracy is only limited by the truncation of
ℓ-, k- and n-modes, where ℓ is the index of
the spin-weighted spheroidal harmonics, and n and k are the
harmonics of the radial and polar motion, respectively.
When we set the maximum of ℓ to 20, we obtain
a relative accuracy of 10-5 even in the highly eccentric case of
e = 0.9. The accuracy is better for lower eccentricity.
Our numerical code is expected to be useful for computing
templates of the extreme mass ratio inspirals, which is one of the
main targets of the Laser Interferometer Space Antenna (LISA).
Subject Index :
DOI : 10.1143/PTP.121.843
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