 Natal kick binding criterion

For an initially circular binary losing mass instantaneously, let $f$ be the retained total mass fraction and $q$ the kick speed in units of the initial relative orbital speed. A kick exactly parallel or opposite to the exploding star's velocity leaves negative relative orbital energy precisely for the title inequality. Equality is marginal escape. For an opposite kick $q=1$, the relative velocity and angular momentum vanish: negative energy then produces a radial collision, rather than a persistent finite-pericentre binary.