Stability of Broucke's Isosceles Orbit

Abstract

We extend the result of Yan to Broucke's isosceles orbit with masses $m_1$, $m_1$, and $m_2$ with $2m_1 + m_2 = 3$. Under suitable changes of variables, isolated binary collisions between the two mass $m_1$ particles are regularizable. We analytically extend a method of Roberts to perform linear stability analysis in this setting. Linear stability is reduced to computing three entries of a $4 \times 4$ matrix related to the monodromy matrix. Additionally, it is shown that the four-degrees-of-freedom setting has a two-degrees-of-freedom invariant set, and linear stability results in the subset comes `for free' from the calculation in the full space. The final numerical analysis shows that the four-degrees-of-freedom orbit is linearly unstable except for the interval $0.555 < m_1 < 0.730$, whereas the two-degrees-of-freedom orbit is linearly stable for a much wider interval.