Towards adiabatic waveforms for inspiral into Kerr black holes: A new model of the source for the time domain perturbation equation

Pranesh A. Sundararajan, Gaurav Khanna, and Scott A. Hughes

Phys. Rev. D 76, 104005 – Published 2 November 2007

Abstract

We revisit the problem of the emission of gravitational waves from a test mass orbiting and thus perturbing a Kerr black hole. The source term of the Teukolsky perturbation equation contains a Dirac delta function which represents a point particle. We present a technique to effectively model the delta function and its derivatives using as few as four points on a numerical grid. The source term is then incorporated into a code that evolves the Teukolsky equation in the time domain as a ( $2 + 1$ ) dimensional partial differential equation. The waveforms and energy fluxes are extracted far from the black hole. Our comparisons with earlier work show an order of magnitude gain in performance (speed) and numerical errors less than $1 %$ for a large fraction of parameter space. As a first application of this code, we analyze the effect of finite extraction radius on the energy fluxes. This paper is the first in a series whose goal is to develop adiabatic waveforms describing the inspiral of a small compact body into a massive Kerr black hole.

Received 5 March 2007

DOI:https://doi.org/10.1103/PhysRevD.76.104005

Authors & Affiliations

Pranesh A. Sundararajan¹, Gaurav Khanna², and Scott A. Hughes¹

¹Department of Physics and MIT Kavli Institute, MIT, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA
²Department of Physics, University of Massachusetts, Dartmouth, Massachusetts 02747, USA

Towards adiabatic waveforms for inspiral into Kerr black holes. II. Dynamical sources and generic orbits

Pranesh A. Sundararajan, Gaurav Khanna, Scott A. Hughes, and Steve Drasco

Phys. Rev. D 78, 024022 (2008)

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Vol. 76, Iss. 10 — 15 November 2007

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Images

Figure 1
Illustrations of quasinormal ringing for a black hole with $a / M = 0.9$ ; the $l = 2$ , $m = 0$ mode is shown here. Top panel: Evolution of the magnitude of the Teukolsky function, extracted at $r = 20 M$ , $θ = π / 2$ . We plot the time evolution of $\ln | Φ |$ at this position. Overplotted on this curve (dashed line) is a function $\propto \exp (- 0.078193 t / M)$ , demonstrating that we recover the expected decay law with a damping time $τ = 12.789 M$ . Bottom panel: Magnitude of the Fourier transform of $Φ (t)$ . Notice that it peaks at $ω = 0.41417 / M$ . These results for $ω$ and $τ$ are in excellent agreement with the expected values of $(ω M, M / τ) = (0.41, - 0.078)$ from Ref. 45 for quasinormal ringing of the $l = 2$ , $m = 0$ mode for $a = 0.9$ .Reuse & Permissions
Figure 2
The real part of the $m = 2$ mode of the Teukolsky function $Ψ$ as a function of time for a point particle of mass $μ / M = 0.01$ in a circular orbit of radius $r_{0} / M = 7.9456$ . These data were extracted in the equatorial plane ( $θ = π / 2$ ) at radius $R = 250 M$ . At this location the Teukolsky function is zero by construction until $t ≃ 250 M$ , at which point a spurious burst reaches the extraction radius. This burst is due to our unphysical initial conditions; it quickly propagates off the grid, leaving a reasonable physical solution for all time afterwards.Reuse & Permissions
Figure 3
The same as Fig. 2, but zooming in on the data for $t ≳ 350 M$ . Solid and dashed lines are the real and imaginary parts of $Ψ$ respectively. The Teukolsky function oscillates with a period of about $70 M$ ; since the source has a period of about $140 M$ , this is exactly what we expect for the $m = 2$ mode. We measure the total flux of energy carried by this mode to be $\dot{E} / μ^{2} = 1.708 \times 10^{- 4}$ , in agreement with previous results (see, e.g., Ref. 20).Reuse & Permissions
Figure 4
The same as Fig. 2, but now showing the data for a given moment in time ( $t = 312.5 M$ ) for a wide range of $r^{*}$ and $θ$ . Along with the outward propagating radiation packet visible at large radius, the nearly singular delta function source is clearly visible at the particle’s position.Reuse & Permissions
Figure 5
An illustration of our code’s convergence behavior. We show the fractional deviation in energy flux in the $| m | = 2$ mode, measured at $R = 250 M$ as a function of grid spacing. The grid is controlled by the integer $b$ using $δ r^{*} = 0.0625 \times 2^{- b / 4}$ , $δ θ = π / 30 \times 2^{- b / 4}$ , with $b \in [- 1, \dots, 4]$ . The upper data set is for fluxes measured from an orbit with $a = 0.8 M$ , $r_{0} = 5 M$ ; the lower set is for $a = 0.9 M$ , $r_{0} = 4.64 M$ . For each data set, the dotted line represents what we would expect for perfect second-order convergence (fit arbitrarily to the data for $b = 0$ ); the large dots represent our actual convergence data.Reuse & Permissions
Figure 6
A power law fit to our numerically extracted energy fluxes for the case $r_{0} / M = 10$ , $a / M = 0.99$ , $m = + 2$ . Numerical data is indicated by the dots; the curve is the best fit we obtain for the ansatz given by Eq. (5.1). For this case, the best fit parameters are $q = 7.45$ , $p = 2.06$ , ${\dot{E}}_{\infty} = 2.197 \times 10^{- 5}$ .Reuse & Permissions

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