How may one calculate the escape velocity of a binary system at a given radius from the COM?

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? Many thanks in advance for any help.

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.