r7_a0.95_lz0.5

The results given here are for an orbit at r/M = 7, a/M = 0.95. To make the orbit inclined, I started with the z-component of angular momentum needed to make a prograde, equatorial orbit and cut that angular momentum in half. I then solved for the energy and Carter constant needed to produce a stable circular orbit. The following table summarizes the resultant orbital characteristics, and the characteristic effects of radiation reaction on that orbit. These multipoles include results from l = 2 to l = 12.

Orbit 
quantities
Energy
0.9372461
z-component of 
angular momentum (Lz)
1.475696
Carter constant
7.988996
iota
62.43110 degrees
iota'
57.71486 degrees
Omega_phi
5.42369 × 10-2
Omega_theta
4.953899 × 10-2
 
Radiation 
reaction 
quantities
Energy flux to infinity
-3.398686 × 10-4
Energy flux down the horizon
+1.022124 × 10-6
Lz flux to infinity
-3.362842 × 10-3
Lz flux down the horizon
+4.219116 × 10-5
Rate of change of radius
-4.657365 × 10-2
Rate of change of Carter constant
-3.125239 × 10-2
Rate of change of inclination angle
+1.207253 × 10-4
 

All of these quantities are in non-dimensional units. To get to physical units, apply the following rules:

  • Multiply the energy by mu (where mu is the mass of the orbiting particle); multiply the angular momentum by mu M (where M is the black hole mass); multiply the Carter constant by mu2 M2.

  • Divide the frequencies by M.

  • Multiply rate of change of energy by (mu/M)2; multiply rate of change of angular momentum by mu2/M; multiply rate of change of Carter constant by mu3.

  • Multiply rate of change of radius by (mu/M).

  • Multiply rate of change of inclination angle by (mu/M2).

    Note that the inclination angle increases, as predicted by Fintan Ryan [Phys. Rev. D 52, R3159 (1995)]. Fintan's formula overpredicts the rate, though, by roughly a factor of three in this strong-field, large spin regime (Fintan's results are valid only in the weak field, and are most relevant to small spins).

    Note also that the energy and angular momentum flux from the horizon is positive, i.e. the particle's energy and angular momentum increase due to radiation falling into the horizon. This is due to superradiant scattering: radiation is scattered by the hole's ergosphere, absorbing some of the energy stored in the spin of the spacetime, and then gives a "kick" to the orbiting particle.

    Here's what the waveform looks like at this point in the particle's evolution:

    The time axis is for a 107 solar mass black hole. This waveform is observed in the hole's equatorial plane. The blue lines are h+, the red lines h×. The low-frequency amplitude modulation is due to Lense-Thirring precession, i.e. the dragging of the orbit's nodes by the black hole's spin. The frequency of this modulation is 2 × (Omegaphi - Omegatheta) = 9.40 × 10-3 M-1. Since this difference is relatively large in this strong-field region the orbital precession is rather rapid, and there aren't too many cycles in each peak. Note also that there is a lot of high-frequency structure in this waveform (eg, the sharp wiggles in the plus polarization near t = 4 hrs). This is because for such a close orbit of a rapidly rotating black hole many harmonics contribute to the waveform.

    The radiated energy has the following distribution:

    Note that this spectrum is relatively broad: a lot of high harmonics radiate out for strong-field orbits of rapidly rotating holes.