A periodic table for black hole orbits

Understanding the dynamics around rotating black holes is imperative to the success of future gravitational wave observatories. Although integrable in principle, test-particle orbits in the Kerr spacetime can also be elaborate, and while they have been studied extensively, classifying their general properties has been a challenge. This is the first in a series of papers that adopts a dynamical systems approach to the study of Kerr orbits, beginning with equatorial orbits. We define a taxonomy of orbits that hinges on a correspondence between periodic orbits and rational numbers. The taxonomy defines the entire dynamics, including aperiodic motion, since every orbit is in or near the periodic set. A remarkable implication of this periodic orbit taxonomy is that the simple precessing ellipse familiar from planetary orbits is not allowed in the strong-field regime. Instead, eccentric orbits trace out precessions of multileaf clovers in the final stages of inspiral. Furthermore, for any black hole, there is some point in the strong-field regime past which zoom-whirl behavior becomes unavoidable. Finally, we sketch the potential application of the taxonomy to problems of astrophysical interest, in particular its utility for computationally intensive gravitational wave calculations.

Figure 1

This figure is a preview of Fig. 12. Rows 1 and 3 show exactly periodic orbits. Rows 2 and 4 show nearby aperiodic orbits.

Figure 2

Left panel: Half of the $(z=2,w=0,v=1)$ periodic orbit. Right panel: The full $(z=2,w=0,v=1)$ closed orbit. The orbit has $a=0$, $L=3.980393$, and $E=0.973101$.

Figure 3

Left panel: Half of the $(z=2,w=1,v=1)$ periodic orbit. Right panel: The full $(z=2,w=1,v=1)$ closed orbit. The orbital parameters are $a=0$, $L=3.718679$, and $E=0.966555$.

Figure 4

Four-leafed orbits with no whirls. Left panel: Leaves are traced out in sequential order for the $(z=4,w=1,v=1)$ closed orbit. Right panel: Leaves are traced out of order for the $(z=4,w=1,v=3)$ closed orbit. The orbital parameters are $a=0$, $L=3.834058$ for both. The energy of the leftmost orbit is $E=0.979032$, and the energy of the rightmost orbit is $E=0.979842$.

Figure 5

Left panel: The full $(z=1,w=1,v=0)$ closed orbit. The orbital parameters are $a=0$, $L=2.714326$, and $E=0.932703$. Right panel: The full $(z=1,w=2,v=0)$ closed orbit. The orbital parameters are $a=0$, $L=3.535534$, and $E=0.948491$.

Figure 6

Left panel: The closed $(z=3,w=1,v=1)$ orbit. Right panel: The closed $(z=300,w=1,v=103)$ orbit looks like a precession of the $(z=3,w=1,v=1)$ orbit. Only six of the 300 leaves are shown. The first segment from apastron to apastron is emphasized in bold. The orbital parameters are $a=0$, $L=3.834058$ for both. The leftmost orbit has energy $E=0.979304$, and the rightmost orbit has energy $E=0.979331$.

Figure 7

The effective radial potential $V_{\mathrm{eff}}(r)$ for very large $r$ (plotted on a $\log r$ scale) in the vicinity of solar system values. The horizontal line at $V_{\mathrm{eff}}=1/2$ corresponds to particle energy $E=1$.

Figure 8

Schwarzschild orbits with values of $L$ just above $L_{\mathrm{IBCO}}$. The vertical axes in (b) and (c) are $w+v/z=(\Delta {\phi }_r/2\pi -1)=q$.

Figure 9

Schwarzschild orbits with $L_{\mathrm{ISCO}}<L<L_{\mathrm{IBCO}}$. The vertical axes in (b) and (c) are $w+v/z=(\Delta {\phi }_r/2\pi -1)=q$.

Figure 10

A homoclinic orbit $(1,\infty ,0)$ for $L_{\mathrm{ISCO}}<L<L_{\mathrm{IBCO}}$. The orbital parameters are $a=0$, $L=3.636619$, and $E=0.958373$.

Figure 11

All $z=1$, 2, 3 orbits with $w=0$ for the first column, $w=1$ for the middle column, and $w=2$ for the last column. All orbits lie on the $L=3.9$ line of Fig. 9. Orbits increase in energy from top to bottom and left to right. The first radial cycle is emphasized in bold for each orbit. Notice the first and second entries in the first column are blank, indicating the inaccessibility of the (1, 0, 0) and (3, 0, 1) orbits.

Figure 12

A series of $w=1$ orbits for the $L=3.9$ line of Fig. 9. Orbits increase in energy from top to bottom and left to right. All $z=1$, 2, 3, 4 orbits are shown. Also shown are randomly selected high $z$ orbits. Notice that the high $z$ orbits look like precessions of the energetically closest low $z$ orbits.

Figure 13

Prograde orbits around a spinning black hole with $a=0.995$. The lines indicate increasing $L$ from left to right through the values $L=1.57$, 1.61, 1.82, 2, 2.1.

Figure 14

A series of $w=3$ and $w=4$ orbits for the $L=1.82$ line of Fig. 13. Orbits increase in energy from top to bottom and left to right. All $z=1$, 2, 3, 4 orbits are shown.

Figure 15

A series of $w=3$ orbits for the $L=2$ line of Fig. 13. Orbits increase in energy from top to bottom and left to right. All $z=1$, 2, 3, 4 orbits are shown. Also shown are randomly selected high $z$ orbits. Notice that the high $z$ orbits look like precessions of the energetically closest low $z$ orbits.

Figure 16

Left panel: ${\omega }_{\phi }$ versus the radii of stable circular orbits. Right panel: ${\omega }_r$ versus the radii of stable circular orbits. Increasing from right to left, the spin values are $a=0$, 0.5, 0.8, 0.95, 0.995, 0.998. All orbits are prograde. Note that, in the left panel, the curves for different values of $a$ nearly overlap and are hard to distinguish.

Figure 17

The ratio ${\omega }_{\phi }/{\omega }_r$, which equals $1+w+v/z$, as a function of the radii of stable circular orbits. Increasing from right to left, the spin values are $a=0$, 0.5, 0.8, 0.95, 0.995, 0.998. All orbits are prograde.