Isofrequency pairing of geodesic orbits in Kerr geometry

Bound geodesic orbits around a Kerr black hole can be parametrized by three constants of the motion: the (specific) orbital energy, angular momentum, and Carter constant. Generically, each orbit also has associated with it three frequencies, related to the radial, longitudinal, and (mean) azimuthal motions. Here, we note the curious fact that these two ways of characterizing bound geodesics are not in a one-to-one correspondence. While the former uniquely specifies an orbit up to initial conditions, the latter does not: there is a (strong-field) region of the parameter space in which pairs of physically distinct orbits can have the same three frequencies. In each such isofrequency pair, the two orbits exhibit the same rate of periastron precession and the same rate of Lense-Thirring precession of the orbital plane, and (in a certain sense) they remain “synchronized” in phase.

Figure 1

The $({\Omega }_{\phi },e)$ parameter space for bound geodesic orbits in Schwarzschild geometry. Bound orbits are confined to the region right of the curve marked separatrix. Thin (blue) curves are contour lines of constant ${\Omega }_r$. The marginal contour line ${\Omega }_r=0$ is shown as a thick (red) line. ${\Omega }_r$ takes its greatest value at the point marked $c$, representing a (slightly perturbed) circular orbit of radius $r_c=8M$. The dotted (black) line shows the curve along which the Jacobian matrix of the transformation $(p,e)\leftrightarrow ({\Omega }_r,{\Omega }_{\phi })$ becomes singular. The singular curve intersects the $e=0$ axis at $b$, corresponding to a circular orbit of radius $r_b=(39+\sqrt{145})M/8\simeq 6.3802M$. Any vertical (${\Omega }_{\phi }=\mathrm{const}$) line left of $b$ intersects some ${\Omega }_r=\mathrm{const}$ contours twice. Each pair of intersections identifies a pair of isofrequency orbits; a sample pair is marked in the plot. Each and every orbit between the separatrix and the singular curve has an isofrequency dual between the singular curve and the dashed (green) curve representing circular-orbit duals (COD). The COD is the locus of all orbits dual to circular orbits of radius $r$ with $r_i=6M<r<r_b$.

Figure 2

Orbital trajectories in the equatorial plane of a Schwarzschild black hole for a sample pair of isofrequency orbits. The motion is counterclockwise, and the black hole is drawn to scale. Orbit 1 (red, round markers) has parameters $(p_1,e_1)=(6.255,0.05)$, and orbit 2 (blue, square markers) has parameters $(p_2,e_2)\simeq (6.718788076,0.3522488173)$. Both share the same orbital frequencies, $({\tilde{\Omega }}_r,{\tilde{\Omega }}_{\phi })\simeq (0.01257801,0.06426083)$. The orbital period of both orbits is $T_r\simeq 499.535318M$, and each accumulates $\Delta \phi \simeq 32.100669$ radians during that period. Both orbits start at their periastron marker 0 along the radial line $P1$. Each successive marker shows the orbital phase after a time period of $n\times T_r/8$, where $n$ is the marker number. At $T_r/2$ (marker 4), both orbits are synchronized again at their apastra along the line $A1$. When each test body has completed one orbit (marker 8), they are again synchronized at their periastra along the line $P2$. Both orbits have precessed by the same amount over their common radial period.

Figure 3

Evolution of $r(t)$ and $\phi (t)-{\Omega }_{\phi }t$ for the isofrequency pair shown in Fig. 2. Both radial and azimuthal motions are “phase-synchronized” on average.

Figure 4

The $({\Omega }_{\phi },e)$ parameter space for bound equatorial geodesic orbits in Kerr geometry with $a=0.5M$. Compare with Fig. 1. The relevant features are as in the Schwarzschild case, and the existence of isofrequency pairing below the COD is similarly evident. We indicate a sample pair with $(p,e)=(4.915656,0.45)$ and (4.62288270, 0.26313140), both having frequencies $({\Omega }_{\phi },{\Omega }_r)=(0.112675037,0.01291945)$. Labelled points on the horizontal axis correspond to circular orbits of radii (left to right) $r_w\simeq 3.8994M$ (whirl radius of marginally bound marginally stable orbit; orbit of highest azimuthal frequency), $r_i\simeq 4.2330M$ (ISCO), $r_b\simeq 4.5039M$ (outermost orbit in an isofrequency pair), and $r_c\simeq 5.7628M$ (orbit of highest radial frequency, $M{\Omega }_r\simeq 0.03312$).

Figure 5

Range of isofrequency pairing (shaded area), as a function of the black hole spin, for orbits in the equatorial plane. The large-spin portion of the plot is shown separately in an inset for clarity. Isofrequency orbits are confined to the strong-field frequency regime ${\Omega }_{\phi }>{\Omega }_{\phi }^{\min }$. The frequency ${\Omega }_{\phi }^{\max }$ is the highest attainable by any bound orbit (at given $M$, $a$), corresponding to the whirl frequency of the marginally bound, marginally stable orbit with $\mathcal{E}=1$ (which is also the azimuthal frequency of the “unstable” circular orbit with that energy).

Figure 6

Illustration of isofrequency pairing in triperiodic orbits. Thin solid (blue) lines are contours of constant ${\Omega }_r$ in the $(e,{\Omega }_{\theta })$ plane, for inclined eccentric orbits with fixed ${\tilde{\Omega }}_{\phi }=0.14$. Here, $a=0.7M$. (The “empty” lower-right corner of the diagram lies outside the parameter space of bound orbits). Dashed vertical lines are sample ${\Omega }_{\theta }=\mathrm{const}$ contours, along each of which we indicate a pair of isofrequency orbits. The parameters of these three pairs are given in Table  (sample pairs 1, 2 and 3, from right to left). All essential features are as in Figs. 1, 4. Similar contour maps can be obtained for other values of ${\tilde{\Omega }}_{\phi }$ and $a$.

Figure 7

A sample pair of triperiodic isofrequency orbits for $a=0.7M$. The orbits depicted correspond to sample pair 3 from Table  (also the leftmost pair in Fig. 6), with orbit 1 shown on the left and orbit 2 shown on the right. The top row shows the motion in the $(x,y)$ plane, and the bottom row shows the motion in the $(x,z)$ plane, where $x=r\cos \phi \sin \theta /M$, $y=r\sin \phi \sin \theta /M$, and $z=r\cos \theta /M$. The black hole is shown to scale. In both orbits, the motion begins at $t=0=\lambda$ at periastron, with $\phi =0$ and $\theta =\pi /2$. In integrating the geodesic equations, we used the method of Drasco and Hughes [7], which avoids numerical difficulties near the orbital turning points. We show the portion of the orbits between $\lambda =0$ and $\lambda =30M^{-1}$.

Figure 8

Evolution of $r(t)$, $\phi (t)-{\Omega }_{\phi }t$, and $\cos [{\theta }_1(t)]-\cos [{\theta }_2(t)]$ for sample pair 3 of Table  and Fig. 7. Both orbits begin at $t=0$ at periastron with $\phi =0$ and $\theta =\pi /2$. Triperiodic isofrequency orbits are synchronized only in a long-time average sense. Periastra are reached only approximately at the same time (as a closer inspection of the upper panel would reveal), but the time differences should average to zero over a long time. The same applies to the average azimuthal motion (middle panel, where a close inspection reveals that the azimuthal phases of the two orbits are not in precise agreement at the periastra) and to the motion in $\theta$ (lower panel). In the latter case, we show the difference between the two longitudinal phases, which remains quasiperiodic. It would have not remained quasiperiodic had the two orbits not been in an isofrequency pair.