Linear stability of periodic three-body orbits with zero angular momentum and topological dependence of Kepler's third law: a numerical test

Abstract

We test numerically the recently proposed linear relationship between the scale-invariant period $T_{\rm s.i.} = T |E|^{3/2}$, and the topology of an orbit, on several hundred planar Newtonian periodic three-body orbits. Here $T$ is the period of an orbit, $E$ is its energy, so that $T_{\rm s.i.}$ is the scale-invariant (s.i.) period, or, equivalently, the period at unit energy $|E| = 1$. All of these orbits have vanishing angular momentum and pass through a linear, equidistant configuration at least once. Such orbits are classified in ten algebraically well-defined sequences. Orbits in each sequence follow an approximate linear dependence of $T_{\rm s.i.}$, albeit with slightly different slopes and intercepts. The orbit with the shortest period in its sequence is called the "progenitor": six distinct orbits are the progenitors of these ten sequences. We have studied linear stability of these orbits, with the result that 21 orbits are linearly stable, which includes all of the progenitors. This is consistent with the Birkhoff-Lewis theorem, which implies existence of infinitely many periodic orbits for each stable progenitor, and in this way explains the existence and ensures infinite extension of each sequence.