r7_a0.05_lz0.5

The results given here are for an orbit at r/M = 7, a/M = 0.05. 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 results include multipoles from l = 2 to l = 10.

Orbit 
quantities
Energy
0.9444671
z-component of 
angular momentum (Lz)
1.732474
Carter constant
9.125384
iota
60.16528 degrees
iota'
59.89819 degrees
Omega_phi
5.406504 × 10-2
Omega_theta
5.377557 × 10-2
 
Radiation 
reaction 
quantities
Energy flux to infinity
-3.947930 × 10-4
Energy flux down the horizon
-4.429323 × 10-7
Lz flux to infinity
-3.67665 × 10-3
Lz flux down the horizon
-2.716628 × 10-6
Rate of change of radius
-1.096448 × 10-1
Rate of change of Carter constant
-3.830044 × 10-2
Rate of change of inclination angle
+1.087463 × 10-5
 

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 by about 50% in this strong-field, small spin regime.

    Note in this case that the energy and angular momentum flux from the horizon is negative, unlike the positive flux in the a = 0.95 cases. This is because the ergosphere of such a slowly spinning black hole is far less effective at superradiantly scattering the gravitational waves --- most of the radiation just "falls down" the horizon.

    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) = 5.78 × 10-4 M-1. This difference is fairly small, so there are a large number of cycles in each peak.

    Here's a zoom on one of the peaks:

    The radiated energy has the following distribution:

    Notice that this is not as broad as in the case a = 0.95M. Also, there is a rather sharp break in the spectrum for omega > 0.5/M. Such a break is also evident when r = 100M and a = 0.05M.