Inspiral into Kerr black holes

Background

This section shows results for the inspiral of bodies that are ``circular'': limiting to constant Boyer-Lindquist radius in the absence of radiation reaction, but spiralling through a sequence of orbits of gradually changing radius and inclination when radiative effects are included. The formalism used to compute gravitational waves and backreaction on these circular orbits is presented here:

S. A. Hughes, Phys. Rev. D 61, 084004 (2000).

(Some of the results that were incorporated into that paper can be found here.) I next developed techniques for taking that formalism and actually computing gravitational waveforms and quasi-circular inspiral trajectories. These techniques are described (along with major results) in the following paper:

S. A. Hughes, Phys. Rev. D 64, 064004 (2001).

Results

The signals we hope to measure with gravitational-wave astronomy are best described as sounds. Here, I present what a small body spiraling into a Kerr black hole would sound like for a couple of interesting cases. In order for the sounds to fall into the frequency band of the human ear, I had to tweak the masses from what one would expect to encounter in nature. In reality, we expect to measure waves from binaries consisting of ``small'' compact bodies (neutron stars or black holes of 5 - 100 solar masses) spiraling into holes of 10⁵ - 10⁶ solar masses or so. The results presented here correspond to spiral-in to a hole with 130 solar masses (for a = 0.998 M) and 30 solar masses (a = 0.3594 M). The mass ratio in both cases is 1:10000.

Radiative evolution of orbits around a black hole with a/M = 0.998: the inspiral trajectories that the orbits follow, and sound files (.au and .wav format) of the gravitational waveforms that they generate.

Radiative evolution of orbits around a black hole with a/M = 0.3594: the inspiral trajectories that the orbits follow, and sound files (.au and .wav format) of the gravitational waveforms that they generate.

I found that it is necessary to interpolate the radiation reaction data using two dimensional cubic spline interpolation; linear interpolation noticeable affects the gravitational-wave phasing. To illustrate this, compare the following two sounds. The both correspond to inspiral in the equatorial plane (iota = 0 degrees) of a black hole with a = 0.95 M. Only the l = 2 data is included; this is enough to make the illustrate the point.

Waveform generated using spline interpolation (.au format only).

Waveform generated using linear interpolation (.au format only).

In the second case, you should be able to hear quite easily the points at which the inspiraling body crosses a grid domain; it sounds like a car shifting gears.

Generic Kerr orbits

Here are some early results concerning the ``generic'' orbits of Kerr black holes: orbits that are inclined and eccentric. Here are some simple visualizations of the motion of a body on such an orbit. (These visualizations assume that light doesn't couple to gravity --- I have not traced photon geodesics to see how such an orbit would actually ``look''.) Here is a body orbiting a black hole with tilt angle iota = 62.5 degrees, with eccentricity e = 0.62, and with semi-latus rectum (consult your favorite textbook on orbital dynamics for a definition) p = 6.32 M. This body comes in to Boyer-Lindquist radius r = 3.9 M, and goes out to r = 16.7 M. The black hole is not shown; it would be a sphere of radius M (it has maximal spin) at the center of both images. TECHNICAL POINT: in these animations, the frames of the movie are even steps in the orbiting body's proper time.

Here is the ``appearance'' down the hole's spin axis.

Here is the ``appearance'' in the hole's equatorial plane.

Next, a film for a closer orbit: tilt angle iota = 13.83 degrees, eccentricity e = 0.52, and semi-latus rectum p = 1.73 M. This body comes in to Boyer-Lindquist radius r = 1.14 M, and goes as out to r = 3.58 M. The black hole has spin a = 0.998 M. NOTE: in this animation the frames are evenly spaced in Boyer-Lindquist time (yes, the code is evolving...), so that this more nearly corresponds to what an observer at infinity would ``see''. This orbit was selected by my collaborator Teviet Creighton, who is busily finding lots of goofy orbital features for us to ponder.

Here is the ``appearance'' in the hole's equatorial plane.

Generic Kerr inspirals, kludgy results

Current efforts focus on understanding inspirals and gravitational waveforms from inspiral through a sequence of generic orbits. Codes to implement rigorous strong field radiation reaction in this case aren't yet available, so we're proceeding with a crude approximation for now. Briefly, we assume (a) that the orbit remains at constant inclination angle [as rigorous studies of circular orbits have shown is likely], (b) that on short timescales the motion is well-described by the geodesic equations, and (c) that on long timescales the orbital energy and angular momentum change as mandated by weak-field formulae. (c) in particular is surely wrong, but is likely to be close enough to right that it will be useful.

A code for implementing this is now working well enough that Teviet and I are able to get first results out. Here are some sounds for inspiral into a hole with a = 0.95M.

Initial p = 5M, initial e = 0.2, inclination angle = 25 degrees.
The plus polarization (.au format).
The plus polarization (.wav format).

Initial p = 5M, initial e = 0.7, inclination angle = 25 degrees.
The plus polarization (.au format).
The plus polarization (.wav format).

Initial p = 5M, initial e = 0.95, inclination angle = 25 degrees.
The plus polarization (.au format).
The plus polarization (.wav format).

Initial p = 5M, initial e = 0.95, inclination angle = 60 degrees.
The plus polarization (.au format).
The plus polarization (.wav format).

All of these sounds are for observation down the hole's spin axis. The mass ratio is M/mu = 3500; the total mass is set in such a way that the whole wave lies nicely in the human audio band.

Here are some sounds for inspiral into a hole with a = 0.998M.

Initial p = 2.2M, initial e = 0.95, inclination angle = 20 degrees.
The plus polarization (.au format).
The plus polarization (.wav format).

Initial p = 2.5M, initial e = 0.97, inclination angle = 20 degrees.
The plus polarization (.au format).
The plus polarization (.wav format).

All sounds are for observation down the hole's spin axis. The mass ratio is M/mu = 15000; the total mass is set in such a way that the whole wave lies nicely in the human audio band.

The kludge used to make these sounds is particularly crude. A paper by myself, Kostas Glampedakis, and Daniel Kennefick will appear soon that improves on this calculation and carefully codifies an easy-to-use kludge that can be used to develop approximate inspiral waveforms. These approximate waveforms will be used to explore LISA data analysis issues while more rigorous codes and computations are developed.

Last modified 8 April 2002.