A Stability Timescale for Non-Hierarchical Three-Body Systems

The gravitational three-body problem is a fundamental problem in physics and has significant applications to astronomy. Three-body configurations are often considered stable as long the system is hierarchical; that is, the two orbital distances are well-separated. However, instability, which is often associated with significant energy exchange between orbits, takes time to develop. Assuming two massive objects in a circular orbit and a test particle in an eccentric orbit, we develop an analytical formula estimating the time it takes for the test particle's orbital energy to change by an order of itself. We show its consistency with results from N-body simulations. For eccentric orbits in particular, the instability is primarily driven not by close encounters of the test particle with one of the other bodies, but by the fundamental susceptibility of eccentric orbits to exchange energy at their periapsis. Motivated by recent suggestions that the galactic center may host an intermediate-mass black hole (IMBH) as a companion to the massive black hole Sgr A*, we use our timescale to explore the parameter space that could harbor an IMBH for the lifetime of the S-cluster of stars surrounding Sgr A*. Furthermore, we show that the orbit of an S-star can be stable for long timescales in the presence of other orbital crossing stars, thus suggesting that the S-cluster may be stable for the lifetimes of its member stars.