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``Ecology'' of Magnetic Rotators Evolutionary Scenarios of Binary Semi-major axis change

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.

1. First, velocities and locations of the components in orbit prior to the explosion are calculated,
2. Then, the changing mass of the exploding star is calculated and kick vwlocity arbitrarily oriented in space is added to the orbital velocity,
3. 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),
4. 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.

``Ecology'' of Magnetic Rotators Evolutionary Scenarios of Binary Semi-major axis change