Past exam of the astrophysics course of the University of Oxford / 2018 / 6 / c

For a kick parallel or antiparallel to star 1's original velocity, the immediate post-explosion relative speed is $|v_0\pm v^{\pm }|$. ThusA positive-energy, hyperbolically escaping pair therefore satisfies preciselyThe absolute value is essential when a sufficiently large opposite kick reverses the relative tangential velocity.

Marginal case. Strictly speaking, a pair is gravitationally bound only for $E^{\prime }<0$. If unbound includes marginal escape, replace the printed strict inequality by $\ge$. For a concrete counterexample to an unrestricted “if and only if” with $>$, take no kick and $\Delta M=M_0/2$. Both sides equal one and the energy is zero: this is not a bound ellipse, although the printed strict inequality is false.