Free motion around black holes with discs or rings: Between integrability and chaos. VI. The Melnikov method

Motivated by black holes surrounded by accretion structures, we consider in this series static and axially symmetric black holes “perturbed” gravitationally as being encircled by a thin disc or a ring. In previous papers, we employed several different methods to detect, classify, and evaluate chaos which can occur, due to the presence of the additional source, in timelike geodesic motion. Here we apply the Melnikov-integral method, which is able to recognize how stable and unstable manifolds behave along the perturbed homoclinic orbit. Since the method standardly works for systems with 1 degree of freedom, we first suggest its modification applicable to 2 degrees of freedom (which is our case), starting from a suitable canonical transformation of the corresponding Hamiltonian. The Melnikov function reveals that, after the perturbation, the asymptotic manifolds tend to split and intersect, consistent with the chaos found by other methods in previous papers.

Figure 1

Examples of homoclinic orbits (top row) of the Nowak-Wagoner pseudopotential, (middle row) of the Schwarzschild space-time, and (bottom row) of the Reissner-Nordström space-time. In the left column, radial shapes of the effective potentials are drawn in red with the homoclinic-orbit energy levels indicated in green, while in the right column, spatial shapes of these orbits are shown in green (the unstable circular orbits which they have as both past and future asymptotes are also plotted in black, but are barely visible, even though we only draw the $-3\pi \le \vartheta \le 3\pi$ part of the homoclinic orbits). For easy comparison, we select orbits with the same maximal radius (apocenter) of $r=17.6M$; this corresponds to $\ell =2.6M$ for Nowak-Wagoner, to $\ell =3.669M$ for Schwarzschild, and to $\ell =3.091M$ for Reissner-Nordström.

Figure 2

Examples of the Melnikov-function behavior. (Left plot) $M({\vartheta }_0)$ for the extreme Reissner-Nordström black hole encircled by a light Majumdar-Papapetrou (MP) ring at $b=15M$, for orbital parameters $\mathcal{E}=0.9428$, $\ell =3M$, ${\ell }_{\varphi }=0$. (Right plot) $M({\vartheta }_0)$ for the pseudo-Newtonian (Nowak-Wagoner) Schwarzschild-type potential encircled by a light inverted first Morgan-Morgan (MM) disc at $b=6M$, for orbital parameters $\mathcal{E}=-0.0510$, $\ell =2.5M$, ${\ell }_{\varphi }=M$. The sinelike shape is in general more frequent than the more complicated behavior shown in the right plot.

Figure 3

Dependence of the Melnikov function on angular momentum, illustrated on the pseudo-Newtonian (Nowak-Wagoner) Schwarzschild black hole surrounded by a light inverted first MM disc. (Left plot) Dependence of $M({\vartheta }_0)$ on total angular momentum $\ell$ for a disc at $b=10M$ and for ${\ell }_{\varphi }=M$. The wave-shape amplitude grows with increasing $\ell$; the shown curves correspond to $\ell =2.5M$, $2.55M$, $2.6M$, $2.7M$, and $2.8M$ (the respective orbital energies are $\mathcal{E}=-0.05098$, $-0.04521$, $-0.03839$, $-0.02167$, $-0.00068$). (Right plot) Dependence of $M({\vartheta }_0)$ on ${\ell }_{\varphi }$ for a disc at $b=20M$, for $\ell =2.6M$ and $\mathcal{E}=-0.03839$. The wave-shape amplitude increases with decreasing ${\ell }_{\varphi }$ (and fixed $\ell$); the shown curves correspond to ${\ell }_{\varphi }=2.6M$ [this yields $M({\vartheta }_0)=0$], $2.5M$, $2.3M$, $2M$, $1.5M$, $1M$, and 0.

Figure 4

Dependence of the Melnikov function on the external-source Weyl radius $b$. (Left plot) For the pseudo-Newtonian (Nowak-Wagoner) Schwarzschild black hole surrounded by a light inverted first MM disc, with orbital parameters $\mathcal{E}=-0.05098$, $\ell =2.5M$, ${\ell }_{\varphi }=M$. Line coloring: $b=M$, blue (almost coincides with the axis); $b=5M$, black; $b=6M$, red; $b=7M$, yellow; $b=10M$, light blue; $b=15M$, green; $b=25M$, violet. (Right plot) Similar plot for the extreme Reissner-Nordström black hole surrounded by a light MP ring, with orbital parameters $\mathcal{E}=0.9428$, $\ell =3M$, ${\ell }_{\varphi }=M$. Line coloring: $b=0$, brown (coincides with the axis); $b=10M$, blue; $b=12M$, red; $b=15M$, green; $b=20M$, yellow. [Note that $b=0$ means that the MP-ring Weyl radius is the same as that of the horizon, which for the extreme Reissner-Nordström (RN) black hole is a viable option.]

Figure 5

Two examples of a discontinuity of the Melnikov function which may appear at the location of the external source. They have been obtained for two different configurations of the extreme Reissner-Nordström black hole encircled by the MP ring. (See the main text for a comment on this feature.)

Figure 6

Equatorial Poincaré section of a typical orbit lying close to a separatrix of an extreme Reissner-Nordström black hole (of mass $M$), (top row) before and (bottom row) after a perturbation due to a MP ring with mass $\mathcal{M}=0.01M$. The orbit has constants of motion $\mathcal{E}=0.942809$, $\ell =M$. The left column shows the entire orbit, while the right column shows details of the vicinity of the corresponding unstable circular orbit.

Figure 7

A detailed zoom (even more detailed on the right-hand side) of the equatorial Poincaré diagram of three different orbits (having $\mathcal{E}=0.942809$, $\ell =M$) passing close to the unstable periodic orbit of the system of extreme RN black hole perturbed by the MP ring with $\mathcal{M}=0.01M$. The orbits are distinguished by colors.