Special Cases: Supernova
Explosion and the Common Envelope

Supernova explosion in a binary is treated as an
instantaneous mass loss of the exploding star. We assume that an additional
kick velocity is imparted to young neutron stars due
to possible asymmetry of the collapse (see Section 5
for further discussion). In this case, the eccentricity and semi-major
axis of the binary after the explosion can be straightforwardly calculated
(Boersma, 1961[21]). We do this using
the following scheme.

First, velocities and locations of the components in orbit prior to
the explosion are calculated,

Then, the changing mass of the exploding star
is calculated and kick vwlocity
arbitrarily oriented in space is added to the orbital velocity,

Then, the transition to a new system barycenter frame is made (at this
point the spatial velocity of the new center of mass of the binary is computed),

Now in this new reference frame, the new total energy
and orbital angular momentum
are computed; if the new total energy is negative, then the new semimajor
axis a' and eccentricity e'
are calculated by using the new
and
; if the total energy is positive (that is, the binary is unbound) spatial
velocities of each component are calculated.

During the CE stage, an effective spiral-in of the
binary components occurs. This complicated process (first introduced by
Paczynsky, 1976[151]) is not fully understood
as yet, so we use the conventional energy consideration to find the binary
system parameters after the CE by introducing a parameter
measuring what fraction of the system's orbital energy (between the beginning
and the end of the spiralling-in process) is transformed into the binding
energy (gravitational minus thermal) of the ejected envelope. Thus

where
is the mass of the core of the mass losing star of initial mass
and radius
(which is simply a function of the initial separation
and the initial mass ratio
), and no substantial mass growth is assumed for the accretor (see, however,
Chevalier, 1993[31]). The lower
is, the closer the binary becomes after the CE stage. We will show below
that the evolutionary scenario allows values of
over a wide range,
-10.