Semirelativistic approximation to gravitational radiation from encounters with nonspinning black holes

The capture of compact bodies by black holes in galactic nuclei is an important prospective source for low frequency gravitational wave detectors, such as the planned Laser Interferometer Space Antenna. This paper calculates, using a semirelativistic approximation, the total energy and angular momentum lost to gravitational radiation by compact bodies on very high eccentricity orbits passing close to a supermassive, nonspinning black hole; these quantities determine the characteristics of the orbital evolution necessary to estimate the capture rate. The semirelativistic approximation improves upon treatments which use orbits at Newtonian order and quadrupolar radiation emission, and matches well onto accurate Teukolsky simulations for low eccentricity orbits. Formulas are presented for the semirelativistic energy and angular momentum fluxes as a function of general orbital parameters.

Figure 1

Radial gravitational potential for a zoom-whirl orbit. The dashed line corresponds to the energy of the orbit. The orbit oscillates in the region where the potential (solid curve) lies below the energy line. If the energy is too high and the orbit passes inside the maximum of the radial potential ($r_{\mathrm{UCO}}$), the particle plunges into the black hole.

Figure 2

Sample gravitational waveform (top) from the zoom-whirl orbit indicated in Fig. 1. We show the plus polarization of the gravitational wave as a function of time. The radiation is emitted predominantly in a high frequency burst during the whirl phase of the orbit. For comparison, the waveform from the Keplerian orbit with the same parameters is shown in the bottom diagram.

Figure 3

Comparison between the semirelativistic results and Peters and Mathews as a function of periapse for orbits with fixed geodesic eccentricity $e=0.99$. The solid lines are the results from the semirelativistic approximation discussed here. The dashed and dotted lines are the Peters and Mathews results with geodesic parameters and Keplerian parameters, respectively. We use a logarithmic vertical scale and plot the absolute value of the energy (upper plot) and angular momentum (lower plot) fluxes.

Figure 4

Comparison between accurate Teukolsky results and various approximations, for orbits with fixed eccentricity $e=0.5$ and a variety of periapses. The plots show the ratio of the flux computed using a given approximation to the flux obtained from the Teukolsky calculation. The upper plot shows the ratio of the energy fluxes, while the lower plot is the ratio of the angular momentum fluxes. In both plots, the solid lines are the semirelativistic results. The dashed and dotted lines are for Peters and Mathews, evaluated with geodesic parameters and Keplerian parameters, respectively. The latter lines cut off at small periapse since there are no corresponding Keplerian orbits in that region.

Figure 5

Sample inspiral trajectories in the $(r_p,e)$ plane. The solid lines illustrate inspiral trajectories. The dotted line marks the separatrix and all points to the left of this line are plunging orbits. The dashed line is the locus of points with $de/dt=0$. In the region between this line and the separatrix, $de/dt>0$ on all trajectories.

Figure 6

Ratio of the energy content of kludge gravitational waves to the change in energy of the source particle orbit, relative to the energy at plunge. This is shown as a function of time until plunge (in unite of $M^2/m$) for eccentricities at plunge $e_{\mathrm{pl}}=0.1$, $0.2\dots 0.9$ (from lowermost curve to uppermost).

Figure 7

Error using fitting functions to approximate the analytic expressions for the energy (upper plot) and angular momentum loss (lower plot). In each plot, the absolute percentage error in the fit is shown for fitting functions of various orders, $N=2,\cdots ,6$.

Figure 8

“Hyperbolic” orbits (i.e., orbits with $E>1$) that are captured after one close encounter with the black hole, for various mass ratios. Orbits whose energy and angular momentum place them above and to the right of the line for a given mass ratio remain unbound and are not captured. The line which begins at bottom left and curves around to top right in the figure is the separatrix line separating unstable plunging orbits (to its left) from stable orbits.