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 $T_{s.i.}=T|E{|}^{3/2}$, 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 $T_{s.i.}$ 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 $T_{s.i.}$, 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.

29 pages, 7 figures; v2: corrected stability, revised sections VI.B and VII; v3: published version, significantly shorter (large part of v2 moved to supplementary material)

References in corpus (5)

• Three Classes of Newtonian Three-Body Planar Periodic Orbits
• More than six hundreds new families of Newtonian periodic planar collisionless three-body orbits
• Topological Dependence of Kepler's Third Law for Collisionless Periodic Three-Body Orbits with Vanishing Angular Momentum and Equal Masses
• Angular momentum and topological dependence of Kepler's Third Law in the Broucke-Hadjidemetriou-Hénon family of periodic three-body orbits
• Periodic three-body orbits with vanishing angular momentum in the Jacobi-Poincare "strong" potential

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• A database of high precision trivial choreographies for the planar three-body problem
• Hundreds of new satellites of figure-eight orbit computed with high precision