Three-body equations of motion in successive post-Newtonian approximations

TL;DR

The authors address the three-body figure-eight problem in General Relativity by deriving the ADM-gauge Hamiltonian up to 2PN and solving the conservative equations of motion to high precision, confirming that $H$, ${\bf P}$, and ${\bf L}$ are well conserved during evolution. They develop 1PN and 2PN initial data for equal-mass bodies, revealing how the required momenta scale with system size via a scaling parameter $\lambda$ and providing explicit fitting formulas for $|{\bf p}_3|$ as a function of $\lambda$. The results show that relativistic corrections become significant around distances of order $100M$ and that Newtonian dynamics is a good approximation only for much larger separations; near smaller separations, gravitational radiation would drive collapse, absent in the conservative PN truncation. The methods yield both insight into the PN regime of the three-body problem and practical initial data for subsequent full nonlinear numerical simulations of relativistic triple systems.

Abstract

There are periodic solutions to the equal-mass three-body (and N-body) problem in Newtonian gravity. The figure-eight solution is one of them. In this paper, we discuss its solution in the first and second post-Newtonian approximations to General Relativity. To do so we derive the canonical equations of motion in the ADM gauge from the three-body Hamiltonian. We then integrate those equations numerically, showing that quantities such as the energy, linear and angular momenta are conserved down to numerical error. We also study the scaling of the initial parameters with the physical size of the triple system. In this way we can assess when general relativistic results are important and we determine that this occur for distances of the order of 100M, with M the total mass of the system. For distances much closer than those, presumably the system would completely collapse due to gravitational radiation. This sets up a natural cut-off to Newtonian N-body simulations. The method can also be used to dynamically provide initial parameters for subsequent full nonlinear numerical simulations.

Figures (9)

• Figure 1: The relative error of the Hamiltonian constraint in the Lagrangian approximation. This figure is derived by using the orbit in Imai:2007gn.
• Figure 2: The relative error of the Hamiltonian constraint evaluate by using the orbit derived from the Hamiltonian formalism.
• Figure 3: Figure-eight motions. We show $\lambda=1$ (solid line) $\lambda=10$ (dashed line) and $\lambda=100$ (dotted line).
• Figure 4: $\lambda$-$|{\bf p}_3|$ relation with points obtained numerically.
• Figure 5: The relative error of the fitting.