Skip to main content

Testing the three-body stability limit at very long time

Presentation #104.04 in the session “Dynamical Stability in Exoplanet Systems”.

Published onJun 01, 2021

Testing the three-body stability limit at very long time

According to Harrington (1972, CeMec 6, 322 ), the stability of a hierarchical three-body system depends crucially on a single parameter *Q*, the pericenter distance of the outer orbit in units of the semi-major axis of the inner orbit. The critical value of this parameter, at which the stability is guaranteed, is usually called the stability limit. Harrington found that the limit is 3.5 for direct orbits and 2.75 for retrograde orbits. Later work has shown that other orbital parameters as well as masses of the bodies are also important, see e.g Mylläri et al. 2018 (MNRAS 476, 830). The latter paper also showed that it is important to specify the length of time for which the stability is required. In some problems the length of time *N*, measured in the number of revolutions in the outer orbit, has to be very long. For that purpose, we have modified the derivation of the stability formula in such a way that it includes *N*, and that it also allows us the find a limiting value at infinite time. The new expression is *Q* = *A* (*N *^{1/2} *f*)^{1/(}^{b}^{ + log }^{N}^{)} where the function *f* (typically *f* ~1) depends on the masses and orbital elements, while *A* and *b* are parameters which we have determined numerically for *N* < 10^{6}, *A* = 2.5, *b* = 5.3. At the limit, *N* → infinity, (*N* ^{1/2} *f*)^{1/(}^{b}^{ + log }^{N}^{)} → 10^{1/2} . In this work we have carried numerical simulations with *N* going up to 10^{7} , using different values of the masses and the orbital elements. We find that generally the parameter dependence disappears, as expected, and we confirm the most simple result for the three-body problem at very long time, *Q* = 2.5 10^{1/2} . Our limit is more than twice the value derived by Harrington. Putting *N* ~ 10–20, as Harrington did, our values are in agreement.