2. Runaway velocities from symmetric SNe
Consider a circular pre-SN orbit consisting of a helium star with mass (the progenitor of the black hole) and a companion star with mass m. Assume that the helium star explodes in a symmetric SN during which an amount of mass, is ejected instantaneously and decouples gravitationally from the system. If the binary will remain bound. The post-SN eccentricity, period and orbital separation are given by Bhattacharya & van den Heuvel Bhattacharya and van den Heuvel (1991)
where we define
and subscripts i denote the pre-SN system. Since the observed black hole binaries all have short orbital periods ( days) tidal forces act to re-circularize the post-SN orbit. The parameters of the re-circularized orbit are given by
And similarly . Here we have ignored the effects of the impact of the ejected shell on the companion star and assume there is no mass loss or transfer during the re-circularization phase. From conservation of momentum one finds an expression for the resulting runaway velocity (recoil) of the system
For convenience this equation can be expressed as
If we know the masses of the stellar components and the orbital period after the re-circularization () we can calculate from the observed runaway velocity, . However, we observe mass-transferring binaries which might have evolved due to loss of angular momentum by gravitational radiation or magnetic braking before the mass transfer started and/or might have transferred already a significant amount of mass from the donor to the black hole. Before applying Eq. (7) to the observed systems we have to correct for these effects.
Also, one has to check whether the binary before the SN would be detached, i.e. that both stars do not fill their Roche lobes at the moment the SN explodes.
© European Southern Observatory (ESO) 1999
Online publication: December 2, 1999