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Stability of Broucke's isosceles orbit

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  • 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 $ 0.595 < m_1 < 0.715 $, whereas the two-degrees-of-freedom orbit is linearly stable for a much wider interval.

    Mathematics Subject Classification: Primary: 70F16; Secondary: 37N05, 37J25.

    Citation:

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  • Figure 1.  The coordinates for Broucke's Orbit

    Figure 2.  Broucke's orbit as it evolves in time. One-quarter period is shown. The remainder of the orbit is obtained by a symmetric extension

    Figure 3.  A plot of the value of $ Q_4(0) $. The value of $ m_1 $ is plotted on the horizontal axis

    Figure 4.  A plot of the numerically-computed upper-left entry $ k_{11} $ from the matrix $ K $ in Equation 11 (vertical) against $ m_1 $ (horizontal)

    Figure 5.  A vertically rescaled plot of the graph from Figure 4

    Figure 6.  A plot of the value of $ e $ from Equation (11) corresponding to stability in the 2DF setting

    Figure 7.  A vertically rescaled plot of the value of $ e $. The curve appears to be asymptotic to the line $ y = 1 $ as $ m_1 \to 0^- $

    Figure 8.  A plot of the second non-trivial eigenvalue of $ K $

    Figure 9.  A vertically rescaled plot of the value of the second non-trivial eigenvalue of $ K $. For the values of $ m_1 \in [0.595, 0.715] $, this second eigenvalue lies within the interval $ [-1, 1] $

    Figure 10.  Both eigenvalue plots superimposed. The two graphs cross at roughly $ m_1 = 0.71 $, where the difference between the two is numerically $ 0.000429 $

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