= Kick-direction binary survival criterion
{title2=$\varepsilon'=\frac{v^2}{2}(1+2\alpha\cos\psi+\alpha^2-2\beta)$}
For an impulsive <supernova kick in a binary star> initially in a <circular Kepler orbit>, let $\beta$ be the remaining total mass fraction and $\alpha$ the kick speed divided by the original relative speed. The displayed post-kick <specific orbital energy> is negative for a bound pair and positive for escape. Over all directions, the velocity factor ranges from $(1-\alpha)^2$ to $(1+\alpha)^2$. Some direction leaves a bound pair exactly when $2\beta>(1-\alpha)^2$; some direction permits positive-energy escape exactly when $2\beta<(1+\alpha)^2$. Equalities are marginal <parabolic Kepler orbits>. The circular initial orbit and impulsive approximation are essential.
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