Linear stability analysis of symmetric periodic simultaneous binary collision orbits in the planar pairwise symmetric four-body problem

Abstract

We apply the symmetry reduction method of Roberts to numerically analyze the linear stability of a one-parameter family of symmetric periodic orbits with regularizable simultaneous binary collisions in the planar pairwise symmetric four-body problem with a mass m ∈ (0, 1] as the parameter. This reduces the linear stability analysis to the computation of two eigenvalues of a 3 × 3 matrix for each m ∈ (0, 1] obtained from numerical integration of the linearized regularized equations along only the first one-eighth of each regularized periodic orbit. The results are that the family of symmetric periodic orbits with regularizable simultaneous binary collisions changes its linear stability type several times as m varies over (0, 1], with linear instability for m close or equal to 0.01, and linear stability for m close or equal to 1.

Stability of the Rhomboidal Symmetric-Mass Orbit

“…The orbit was shown to be linearly stable in [4]. It was later shown that this orbit could be extended to symmetric masses in [2] (see also [3]), and linear stability for this extension was shown for an interval of certain mass ratios in [1].…”

“…Different choices of Y 0 will give different properties of the monodromy matrix. It is also worth noting that Y 0 is independent of the value of m for this orbit, which is not always true (see [1]. )…”

“…In a separate study of the rhomboidal four-body problem with unequal masses, Waldvogel [22] notes that "sufficiently simple systems may bear the chance of permitting theoretical advances," and identifies the rhomboidal configuration as one such system. Indeed, much of the analysis performed in this paper is far simpler than that of [1].…”

Understanding the Dynamics of Collision and Near-Collision Motions in the N-Body Problem

“…How close will 2012 DA14 pass by Earth? A mere 17000 miles (27000 km) 4 . In cosmic terms, this close shave of 2012 DA14 with Earth in 2013 is a near-collision motion.…”

“…Bakker, Ouyang, Yan and Simmons [3] then numerically extended this non-collinear periodic simultaneous binary collision solution to unequal masses 0 < m < 1. In 2012, Bakker, Mancuso, and Simmons [4] have numerically determined that the noncollinear periodic simultaneous binary collision solution is spectrally stable when 0.199 < m < 0.264 and 0.538 < m ≤ 1 and is linearly unstable for the remaining values of m. Long-term numerical integrations of the regularized equations done by Bakker, Ouyang, Yan, and Simmons [3] suggest instability when 0.199 < m < 0.264 and stability when 0.538 < m ≤ 1 in the sense of Definition 1. For these latter values of m could the near-collision solutions in the PPS4BP that look like the non-collinear periodic simultaneous binary collision solution be collision-free and bounded for all time?…”