Observable properties of orbits in exact bumpy spacetimes

We explore the properties of test-particle orbits in bumpy spacetimes—stationary, reflection-symmetric, asymptotically flat solutions of Einstein equations that have a non-Kerr (anomalous) higher-order multipole-moment structure but can be tuned arbitrarily close to the Kerr metric. Future detectors should observe gravitational waves generated during inspirals of compact objects into supermassive central bodies. If the central body deviates from the Kerr metric, this will manifest itself in the emitted waves. Here, we explore some of the features of orbits in non-Kerr spacetimes that might lead to observable signatures. As a basis for this analysis, we use a family of exact solutions proposed by Manko and Novikov which deviate from the Kerr metric in the quadrupole and higher moments, but we also compare our results to other work in the literature. We examine isolating integrals of the orbits and find that the majority of geodesic orbits have an approximate fourth constant of the motion (in addition to the energy, angular momentum, and rest mass) and the resulting orbits are triperiodic to high precision. We also find that this fourth integral can be lost for certain orbits in some oblately deformed Manko-Novikov spacetimes, leading to ergodic motion. However, compact objects will probably not end up on these chaotic orbits in nature. We compute the location of the innermost stable circular orbit (ISCO) and find that the behavior of an orbit in the approach to the ISCO can be qualitatively different depending on whether the location of the ISCO is determined by the onset of an instability in the radial or vertical direction. Finally, we compute periapsis and orbital-plane precessions for nearly circular and nearly equatorial orbits in both the strong and weak field, and discuss weak-field precessions for eccentric equatorial orbits.

Figure 1

Spacetime structure for $\chi =0.9$. The upper row shows zeros of $g_{tt}$ for $q=-1$ (left column), $q=0$ (middle column), and $q=1$ (right column). This defines the boundary of the ergoregion of the spacetime. The region with $g_{tt}>0$ is shaded. The bottom row shows points where $g_{\varphi \varphi }$ changes sign for the same values of $q$, and the region where $g_{\varphi \varphi }<0$ is shaded. This defines the region where closed timelike curves exist. The middle bottom panel is empty since there is no such region in the Kerr spacetime. The shape of the two boundaries is qualitatively the same for other values of $q$ with the same sign, although both regions grow as $|q|$ is increased.

Figure 2

The fractional errors in energy $E$ (solid line), angular momentum $L_z$ (dashed line), and the quantity $g_{\alpha \beta }{\dot{x}}^{\alpha }{\dot{x}}^{\beta }$ (dotted line) accumulated over 1700 orbits of a geodesic with $E=0.92$ and $L_z=2.5M$ in a spacetime with spin $\chi =0.9$ and anomalous quadrupole moment $q=0.95$.

Figure 3

Effective potential for geodesic motion around a Kerr black hole, with $E=0.95$, $L_z=3M$, and $\chi =0.9$. The curves indicate zeros of the effective potential. Allowed orbits are found in the small region around $\rho =0$, $z=0$ (rising and plunging orbits) or in the region containing $\rho =10$, $z=0$ (bound orbits).

Figure 4

Effective potential for geodesic motion around a bumpy black hole with $\chi =0.9$, $q=0.95$, $E=0.95$, and $L_z=3M$. The thick dotted curves indicate zeros of the effective potential. The trajectory of a typical geodesic in the outer region is shown by a thin curve. The regular pattern of self-intersections of the geodesic projection onto the $\rho -z$ plane indicates (nearly) regular dynamics.

Figure 5

Poincaré map showing $\mathrm{d}\rho /\mathrm{d}\tau$ vs $\rho$ for crossings of the $z=0$ plane for a sequence of orbits in the outer allowed region of the Kerr spacetime with $E=0.95$, $L_z=3M$, and $\chi =0.9$. The closed curves indicate the presence of a fourth isolating integral, which we know to be the Carter constant.

Figure 6

Poincaré map for a geodesic in the outer region of Fig. 4.

Figure 7

Poincaré map for a geodesic in the inner region of Fig. 4.

Figure 8

Absolute values of the Fourier transforms of $\rho (t)$ (solid line), $z(t)$ (dashed line), and the gravitational-wave component $h_{+}(t)$ (dotted line) in the frequency domain for an orbit in the outer region of Fig. 4. The frequency is displayed in units of $1/M$; the amplitude scaling is arbitrary.

Figure 9

Absolute values of the Fourier transforms of $\rho (t)$ (solid line), $z(t)$ (dashed line), and the gravitational-wave component $h_{+}(t)$ (dotted line) in the frequency domain for an orbit in the inner region of Fig. 4. The frequency is displayed in units of $1/M$; the amplitude scaling is arbitrary.

Figure 10

The evolution of the separation $\Delta \rho$ between the inner and outer bounded regions in the equatorial plane along an inspiral in a Manko-Novikov spacetime with $\chi =0.9$ and $q=0.95$. $\Delta \rho =0$ means that the two regions have merged and there is a single bounded region.

Figure 11

Properties of the equatorial ISCO in spacetimes with $\chi =0$, as a function of $q$. We show the $\rho$ coordinate of the ISCO (left panel) and the dimensionless frequency of the orbit at the ISCO (right panel). As described in the text, the ISCO radius has three branches, depending on whether it is determined by one of the two branches of radial instability or the branch of vertical instability. These branches are indicated separately in the diagram. For values of $q$ where all three branches are present, the dashed line denotes the OSCO and the dotted line denotes ${\tilde{\rho }}_{\mathrm{ISCO}}$ as discussed in the text. Allowed orbits lie above the curve in the left panel, and below the curve in the right panel.

Figure 12

As Fig. 11, but now for a spin of $\chi =0.9$. There are now two ISCO curves, one for prograde orbits and one for retrograde orbits. The allowed range of orbital frequencies is given by the region in between the two curves in the right-hand panel.

Figure 13

Periapsis precession $p_{\rho }$ versus azimuthal frequency ${\Omega }_{\varphi }$ for $\chi =0$ and various values of $q$.

Figure 14

Periapsis precession $p_{\rho }$ versus azimuthal frequency ${\Omega }_{\varphi }$ for $\chi =0.9$ and various values of $q$.

Figure 15

Orbital-plane precession $p_z$ versus azimuthal frequency ${\Omega }_{\varphi }$ for $\chi =0$ and various values of $q$. We do not show the case $q=0$ here, since there is no orbital-plane precession in Schwarzschild.

Figure 16

Orbital-plane precession $p_z$ versus azimuthal frequency ${\Omega }_{\varphi }$ for $\chi =0.9$ and various values of $q$.

Figure 17

Difference between periapsis precessions in a bumpy spacetime with $\chi =0$ and the Schwarzschild spacetime, $\Delta p_{\rho }({\Omega }_{\varphi },q)=p_{\rho }({\Omega }_{\varphi },q)-p_{\rho }({\Omega }_{\varphi },q=0)$.

Figure 18

Difference between periapsis precessions in a bumpy spacetime with $\chi =0.9$ and the Kerr spacetime with $\chi =0.9$, $\Delta p_{\rho }({\Omega }_{\varphi },q)=p_{\rho }({\Omega }_{\varphi },q)-p_{\rho }({\Omega }_{\varphi },q=0)$.

Figure 19

Difference between orbital-plane precessions in a bumpy spacetime with $\chi =0.9$ and the Kerr spacetime with $\chi =0.9$, $\Delta p_z({\Omega }_{\varphi },q)=p_z({\Omega }_{\varphi },q)-p_z({\Omega }_{\varphi },q=0)$.

Figure 20

Example of onset of chaos in the Newtonian quadrupole-octupole potential (a1). All plots are for orbits which start with $\dot{\rho }=0=z$, $\rho /M=3$ and have specific angular momentum $L_z=1.7M$. The left-hand panels are for energy $E=0.82$, while the right-hand panels have energy $E=0.81$. The top two plots show zeros of the effective potential, $V_{\mathrm{eff}}=0$, as defined by Eq. (a2), and the paths followed by the orbits in the $(\rho ,z)$ plane. The bottom two plots are Poincaré maps for crossings of the $z=0$ plane in each case.