In general, the loss of matter and radiation is non-spherical, so that the remnant of the supernova explosion (neutron star or black hole) acquires some recoil velocity called kick velocity . In a binary, the kick velocity should be added to the orbital velocity of the pre-supernova star.
The usual treatment proceeds as follows. Let us consider a pre-SN binary with initial masses and . The stars move in a circular orbit with orbital separation and relative velocity . The star explodes leaving a compact remnant of mass . The total mass of the binary decreases by the amount . It is usually assumed that the compact star acquires some additional velocity (kick velocity) (see detailed discussion in Section 3.4). Unless the binary is disrupted, it will end up in a new orbit with eccentricity , major semi axis , and angle between the orbital planes before and after the explosion. In general, the new barycentre will also receive some velocity, but we neglect this motion. The goal is to evaluate the parameters , , and .
It is convenient to work in an instantaneous reference frame centered on right at the time of explosion. The -axis is the line from to , the -axis points in the direction of , and the -axis is perpendicular to the orbital plane. In this frame, the pre-SN relative velocity is , where (see Equation (13)). The initial total orbital momentum is . The explosion is considered to be instantaneous. Right after the explosion, the position vector of the exploded star has not changed: . However, other quantities have changed: and , where is the kick velocity and is the reduced mass of the system after explosion. The parameters and are being found from equating the total energy and the absolute value of orbital momentum of the initial circular orbit to those of the resulting elliptical orbit (see Equations (10, 14, 12)):
which results in
The condition for disruption of the binary system depends on the absolute value of the final velocity, and on the parameter . The binary disrupts if its total energy defined by the left-hand-side of Equation (38) becomes non-negative or, equivalently, if its eccentricity defined by Equation (41) becomes . From either of these requirements one derives the condition for disruption:
Since and , one can write Equation (43) in the form
So far, we have considered an originally circular orbit. If the pre-SN star moves in an originally eccentric orbit, the condition of disruption of the system under symmetric explosion reads
where is the distance between the components at the moment of explosion.
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