Linear Stability of Elliptic Relative Equilibria of Restricted Four-body Problem

Abstract

In this paper, we consider the linear stability of the elliptic relative equilibria of the restricted 4-body problems where the three primaries form a Lagrangian triangle. By reduction, the linearized Poincaré map is decomposed to the essential part, the Keplerian part and the elliptic Lagrangian part where the last two parts have been studied in literature. The linear stability of the essential part depends on the masses parameters $\alpha$, $\beta$ with $\alpha \ge \beta >0$ and the eccentricity $e\in [0,1)$. Via $\om$-Maslov index theory and linear differential operator theory, we obtain the full bifurcation diagram of linearly stable and unstable regions with respect to $\alpha$, $\beta$ and $e$. Especially, two linearly stable sub-regions are found.

arXiv comment76 pages, 6 figures. arXiv admin note: text overlap with arXiv:1206.6162 by other authors

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Cited by in corpus (2)

• Linear Instability of Elliptic Rhombus Solutions to the Planar Four-body Problem
• Linear stability of the elliptic relative equilibria for the restricted N-body problem: two special cases