Dynamical environments of relativistic binaries: The phenomenon of resonance shifting

In this article, we explore both numerically and analytically how the dynamical environments of mildly relativistic binaries evolve with increasing the general relativity factor $\gamma$ (the normalized inverse of the binary size measured in the units of the gravitational radius corresponding to the total mass of the system). Analytically, we reveal a phenomenon of the relativistic shifting of mean-motion resonances: on increasing $\gamma$, the resonances between the test particle and the central binary shift, due to the relativistic variation of the mean motions of the primary and secondary binaries and the relativistic advance of the tertiary’s pericenter. To exhibit the circumbinary dynamics globally, we numerically integrate equations of the circumbinary motion of a test particle and construct relevant scans of the maximum Lyapunov exponents and stability diagrams in the “pericentric distance–eccentricity” plane of initial conditions. In these scans and diagrams, regular and chaotic domains are identified straightforwardly. Our analytical and numerical estimates of the shift size are in a good agreement. Prospects for identification of the revealed effect in astronomical observations are discussed.

Figure 1

The rate of apsidal precession of the particle’s orbit around a single primary, as a function of the relativistic factor $\gamma$. For all panels, we set the particle’s pericentric distance $q=3$. The eccentricities: (a) $e=0.1$, (b) $e=0.2$, (c) $e=0.3$. The dots represent the results of our numerical simulations, and the dashed straight line is the theoretical $\omega =2\pi \frac{{\omega }_{\mathrm{E}}}{n}$, where $\frac{{\omega }_{\mathrm{E}}}{n}$ is given by Eq. (8).

Figure 2

The same as in Fig. 1, but at $e=0.01$ (all panels) and $q=1.5$ [panel (a)], $q=2.0$ [panel (b)], and $q=2.2$ [panel (c)].

Figure 3

Scans of the maximum Lyapunov exponent along the $q$ axis at $e=0.01$. The black curve corresponds to $\gamma ={10}^{-7}$, and the red one to $\gamma ={10}^{-2}$. The red arrows indicate the resonance shift direction.

Figure 4

Upper panels: “$q\text{−}e$” stability diagrams at fixed $\mu =0.1$. Chaotic and regular zones are shown in red and blue, respectively. (a) $\gamma ={10}^{-50}\sim 0$; (b) $\gamma ={10}^{-3}$; (c) $\gamma ={10}^{-2}$. Panels below: The corresponding histograms of computed values of Lyapunov exponents; see the text for details.

Figure 5

The stability diagram $\gamma$ vs $q$ in the neighborhood of the $3/1$ resonance. Chaotic and regular zones are shown in red and blue, respectively. The theoretical curve is white dashed.

Figure 6

The normalized difference $(\omega -{\omega }_{\mathrm{E}})/{\omega }_{\mathrm{E}}$ as a function of $\gamma$. The numerical code is used to compute $\omega$, and ${\omega }_{\mathrm{E}}$ is calculated by Eq. (8). The black and red curves correspond to $q=1.95$ and $q=2.05$, respectively; $e=0.01$.