2 Mass transfer in close binaries driven by gravitational wave radiation

The rate of angular momentum loss ($\dot{J}$) of a binary system with a circular orbit due to gravitational wave radiation (GWR) is (Landau & Lishiftz 1971):

$$\left ( \frac {\dot{J}}{J} \right )_{\rm GWR} = -\frac{32}{5} \, \frac{G^3}{c^5} \frac{\,M \,m \,(M+m)}{a^{4}}\cdot$$ (1)

Here M and m are the masses of the two components and a is their orbital separation.

In a binary with stable mass transfer the change of the radius of the donor exactly matches the change of its Roche lobe. This condition combined with an approximate equation for the size of the Roche lobe (Paczynski 1967),

$$R_{\rm L } \approx 0.46 \, a \left(\frac{m}{ M + m}\right)^{1/3} \qquad \mbox{ for } m < 0.8 M,$$ (2)

may be used to derive the rate of mass transfer for a semi-detached binary in which the mass transfer is driven by GWR (Paczynnki 1967)

$$\frac{\dot{m}}{m} = \left ( \frac{\dot{J}}{J} \right )_{\rm... ...[\frac{\zeta (m)}{2} + \frac{5}{6} - \frac{m}{M} \right]^{-1}.$$ (3)

Here $\zeta (m)$ is the logarithmic derivative of the radius of the donor with respect to its mass ( $\zeta \equiv$ d $\ln r/$ d $\ln m$). For the mass transfer to be stable, the term in brackets must be positive, i.e.

$$q \equiv \frac{m}{M} < \frac{5}{6} + \frac{\zeta(m)}{2}\cdot$$ (4)

The mass transfer becomes dynamically unstable when this criterion is violated, probably causing the binary components to coalesce (Pringle & Webbing 1975;Tutukov & Yungelson 1979).