Supernova kick in a binary star

Asymmetric ejecta or neutrino emission can impart a natal kick to a compact remnant. The impulsive change in relative velocity and total mass can alter the binary's semimajor axis and eccentricity or unbind the system.

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.

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.

A disrupted binary star with positive post-impulse specific orbital energy ${\epsilon }^{\prime }$ has relative speed $V=\sqrt{2{\epsilon }^{\prime }}$ at infinite separation, because the Newtonian gravitational potential energy tends to zero. With the notation of the kick-direction binary survival criterion, $V=v\sqrt{1+2\alpha \cos \psi +{\alpha }^2-2\beta }$. This is not a systemic recoil velocity. A marginal parabolic Kepler orbit has $V=0$ at infinity.