Relativistic Pythagorean three-body problem

We study the influence of relativity on the chaotic properties and dynamical outcomes of an unstable triple system; the Pythagorean three-body problem. To this end, we extend the brutus N-body code to include post-Newtonian pairwise terms up to 2.5 order, and the first order Taylor expansion to the Einstein-Infeld-Hoffmann equations of motion. The degree to which our system is relativistic depends on the scaling of the total mass (the unit size was 1 parsec). Using the brutus method of convergence, we test for time-reversibility in the conservative regime, and demonstrate that we are able to obtain definitive solutions to the relativistic three-body problem. It is also confirmed that the minimal required numerical accuracy for a successful time-reversibility test correlates with the amplification factor of an initial perturbation, as was found previously for the Newtonian case. When we take into account dissipative effects through gravitational wave emission, we find that the duration of the resonance, and the amount of exponential growth of small perturbations depend on the mass scaling. For a unit mass $\le 10{\mathrm{M}}_{\odot }$, the system behavior is indistinguishable from Newton’s equations of motion, and the resonance always ends in a binary and one escaping body. For a mass scaling up to ${10}^7{\mathrm{M}}_{\odot }$, relativity gradually becomes more prominent, but the majority of the systems still dissolve in a single body and an isolated binary. The first mergers start to appear for a mass of $\sim {10}^5{\mathrm{M}}_{\odot }$, and between ${10}^7{\mathrm{M}}_{\odot }$ and ${10}^9{\mathrm{M}}_{\odot }$ all systems end prematurely in a merger. These mergers are preceded by a gravitational wave driven in-spiral. For a mass scaling $\ge {10}^9{\mathrm{M}}_{\odot }$, all systems result in a gravitational wave merger upon the first close encounter. Relativistic three-body encounters thus provide an efficient pathway for resolving the final parsec problem. The onset of mergers at the characteristic mass scale of ${10}^7{\mathrm{M}}_{\odot }$ potentially leaves an imprint in the mass function of supermassive black holes.

Figure 1

Three solutions to the relativistic Pythagorean problem that end in a merger between two objects. Per row, we show the system’s global chaotic behavior in the left panel, and a zoom-in of the merger of two components in the right panel. The zoomed-in region is visualized by a small black box in the left panel, while the right panels are centralized at the center of mass of the merger progenitors.

Figure 2

Exponential growth of a small initial perturbation, $\delta$, as a function of time in the relativistic Pythagorean problem. Each curve was obtained with a different value of $\zeta$. A time reversible evolution, and thus a symmetric curve, was achieved for the specified values of $\epsilon$ and $L_w$.

Figure 3

Dependence of lifetime of the Pythagorean triple system on the strength of relativistic effects. The results in dimensionless quantities are given in the bottom panel, while the same results in physical units are given in the top panel.

Figure 4

Orbital elements of the ejected binary black holes. The dimensionless speed of light, $\zeta$, is a proxy for the total mass (decreasing $\zeta$ corresponds to increasing mass). The crosses denote binaries which merge within a Hubble time.

Figure 5

Amplification factor as a function of amplification time. Here the amplification time, $T_A$, is defined to be the minimum between the lifetime, or the moment at which the phase space separation reached $\delta =0.1$. We fit an analytical model with a constant Lyapunov time scale, $\log A=T_A/{\tau }_L+\gamma$, with ${\tau }_L=4.3\pm 0.2$ (see grey dashed curve).

Figure 6

Numerical accuracy required to achieve a time-reversible solution as a function of amplification factor. The linear fit has a slope of $-1.23\pm 0.09$, which confirms the anti-correlation between numerical accuracy and amplification factor. Deeper into the relativistic regime (smaller values of $\zeta$), the triples show less chaotic behavior (smaller amplification factors).