Three-body_orbits_Correlations

We analyze the world set of data for planar periodic three-body orbits with zero angular momentum that pass through a symmetrical collinear configuration (syzygy), and find correlations between scale-invariant parameters as the ratio m3 m of the central m3 and the equal m masses, the topological quantifier Lf (the number of letters in the free-group F2(a, b) word describing the topology of an orbit, as defined in Refs. [1, 5]), the scale-invariant kinematic variables, such as the scale-invariant period T|E| 3/2 , or the scale-invariant angular momentum L|E| 1/2 , on one hand, and the distribution of marginally stable orbits, on another. We find a distribution that falls within a compact non-zero measure domain with clearly defined, apparently smooth boundaries in the space spanned by the topological index Lf and the mass ratio m3 m . On this basis, we predict an increase of the number of linearly stable orbits, as the mass ratio m3 m falls below 0.5. We also predict the (approximate) domain of these orbits’ initial conditions. Some consequences for astronomy/astrophysics are briefly discussed.