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Apologies if this is a bit general, even some pointers in the right direction would be great (reading material etc.)

Given an object of mass $m_3$ travelling outward at speed $v$ from the centre of mass of a binary star system of two stars of masses $m_1$ and $m_2$ at angle $\theta$ from the $m_1$ end of the major axis of the elliptical orbit of the binary system is there an inequality that determines if the body of mass $m_3$ will escape the binary system?

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Many thanks in advance for any help.

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For the sake of simplicity assume the binaries' motion to be circular. Then you can use the formalism of the circular restricted 3-body problem (CR3BP) to model the motion of the test particle $m_3$. Your Lagrangian will be time-independent and the conserved quantity (Jacobi constant) can be evaluated at infinity to give an equation for conditions of escape, in the same way you would find a particle under the influence of two others to have an escape velocity of $v = (2Gm_1/r_1 + 2Gm_2/r_2)^{1/2}$ by taking the total energy to be 0 at infinity.

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