Over a thousand new periodic orbits of a planar three-body system with unequal masses

Abstract

Abstract The three-body problem is common in astronomy, examples of which are the solar system, exoplanets, and stellar systems. Due to its chaotic characteristic, discovered by Poincaré, only three families of periodic three-body orbits were found in 300 years, until 2013 when Šuvakov and Dmitrašinović (2013, Phys. Rev. Lett., 110, 114301) found 13 new periodic orbits of a Newtonian planar three-body problem with equal mass. Recently, more than 600 new families of periodic orbits of triple systems with equal mass were found by Li and Liao (2017, Sci. China-Phys. Mech. Astron., 60, 129511). Here, we report 1349 new families of planar periodic orbits of the triple system where two bodies have the same mass and the other has a different mass. None of the families have ever been reported, except the famous “figure-eight” family. In particular, 1223 among these 1349 families are entirely new, i.e., with newly found “free group elements” that have been never reported, even for three-body systems with equal mass. It has been traditionally believed that triple systems are often unstable if they are non-hierarchical. However, all of our new periodic orbits are in non-hierarchical configurations, but many of them are either linearly or marginally stable. This might inspire the long-term astronomical observation of stable non-hierarchical triple systems in practice. In addition, using these new periodic orbits as initial guesses, new periodic orbits of triple systems with three unequal masses can be found by means of the continuation method, which is more general and thus should have practical meaning from an astronomical viewpoint.