Consider a binary system on a circular orbit, consisting of stars of masses . We shall assume that the star explodes suddenly and symmetrically (thus, receiving no "kick" due to the mass ejection) and its remaining mass is . The following dynamics of the system are determined by the conservation of energy and angular momentum after the explosion. From energy conservation we get:
where is the reduced mass, is the stars' relative velocity and are the semi-major axis' before and after the explosion correspondingly. The L.H.S. is the gravitational + kinetic energy right after the explosion, and the R.H.S. is the Keplerian energy of the resulting orbit. From this equation we get that:
Plugging in and defining (where is the ejected mass and is the total final mass of the two stars), we have:
From angular conservation, we have:
where, again, the L.H.S. is the angular momentum right after the explosion and the R.H.S. is the Keplerian angular momentum of the resulting orbit. Squaring this equation and plugging in we get:
Using the ratio for the semi major axis in the energy equation, we get:
It follows that that for symmetrical explosions, the resulting eccentricity follows the simple relation: . In particular there is no bound orbit when , i.e when the system losses more than half its original mass (as shown in Maximum mass loss in a supernova that leaves a bound binary).