Circular and noncircular nearly horizon-skimming orbits in Kerr spacetimes

We have performed a detailed analysis of orbital motion in the vicinity of a nearly extremal Kerr black hole. For very rapidly rotating black holes—spin parameter $a\equiv J/M>0.9524M$—we have found a class of very strong-field eccentric orbits whose orbital angular momentum $L_z$ increases with the orbit’s inclination with respect to the equatorial plane, while keeping latus rectum and eccentricity fixed. This behavior is in contrast with Newtonian intuition, and is in fact opposite to the normal behavior of black hole orbits. Such behavior was noted previously for circular orbits; since it only applies to orbits very close to the black hole, they were named “nearly horizon-skimming orbits.” Our current analysis generalizes this result, mapping out the full generic (inclined and eccentric) family of nearly horizon-skimming orbits. The earlier work on circular orbits reported that, under gravitational radiation emission, nearly horizon-skimming orbits exhibit unusual inspiral, tending to evolve to smaller orbit inclination, toward prograde equatorial configuration. Normal orbits, by contrast, always demonstrate slowly growing orbit inclination—orbits evolve toward the retrograde equatorial configuration. Using up-to-date Teukolsky-based fluxes, we have concluded that the earlier result was incorrect—all circular orbits, including nearly horizon-skimming ones, exhibit growing orbit inclination under radiative backreaction. Using kludge fluxes based on a Post-Newtonian expansion corrected with fits to circular and to equatorial Teukolsky-based fluxes, we argue that the inclination grows also for eccentric nearly horizon-skimming orbits. We also find that the inclination change is, in any case, very small. As such, we conclude that these orbits are not likely to have a clear and peculiar imprint on the gravitational waveforms expected to be measured by the space-based detector LISA.

Figure 1

Left panel: Inclination angles ${\theta }_{\mathrm{inc}}$ for which bound stable orbits exist for a black hole with spin $a=0.998M$. The allowed range for ${\theta }_{\mathrm{inc}}$ goes from ${\theta }_{\mathrm{inc}}=0$ to the curve corresponding to the eccentricity under consideration, ${\theta }_{\mathrm{inc}}={\theta }_{\mathrm{inc}}^{\max }$. Right panel: Same as the left panel, but for an extremal black hole, $a=M$. Note that in this case ${\theta }_{\mathrm{inc}}^{\max }$ never reaches zero.

Figure 2

Left panel: Noncircular nearly horizon-skimming orbits for $a=0.998M$. The heavy solid line indicates the separatrix between stable and unstable orbits for equatorial orbits ($\iota ={\theta }_{\mathrm{inc}}=0$). All orbits above and to the left of this line are unstable. The dashed-dotted line (green in the online color version) bounds the region of the $(p,e)$-plane where $\partial L_z/\partial {\theta }_{\mathrm{inc}}>0$ for all allowed inclination angles (“Region A”). All orbits between this line and the separatrix belong to Region A. The dotted line (red in the online color version) bounds the region $(L_z{)}_{\mathrm{most}\mathrm{bound}}<(L_z{)}_{\mathrm{least}\mathrm{bound}}$ (“Region B”). Note that B includes A. The dashed line (blue in the online color version) bounds the region where $\partial L_z/\partial {\theta }_{\mathrm{inc}}>0$ for at least one inclination angle (“Region C”); note that C includes B. All three of these regions are candidate generalizations of the notion of nearly horizon-skimming orbits. Right panel: Same as the left panel, but for the extreme spin case, $a=M$. In this case the separatrix between stable and unstable equatorial orbits is given by the line $p/M=1+e$.