Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 317 / 4

For a circular binary with fixed total mass $M=M_1+M_2$, the orbital angular momentum isConservative binary mass transfer keeps $J$ and $M$ fixed, so $a\propto (M_1{M}_2{)}^{-2}$. Since $M_1=Mq/(1+q)$ and $M_2=M/(1+q)$,

Conservative transfer gives $d\log M_2/d\log M_1=-q$ and hence $d\log q/d\log M_1=1+q$. It follows thatand, using $R_L\simeq 0.4a{q}^{2/9}$,

Let $f=\log (R_1/R_L)$. While the system is detached, $M_1$ is constant, soDuring Roche-lobe overflow, $\dot{\log M_1}=-f/{\tau }_{\mathrm{dyn}}$, whileThereforeIf $\beta >\alpha$, the stable fixed point isThe corresponding mass-transfer rate isso the donor overfills its Roche lobe by only a tiny amount while losing mass on the slow nuclear timescale.

If $\beta <\alpha$, overflow is unstable. Starting at contact time $t_0$ with $f(t_0)=0$,The overflow and mass-loss rate grow exponentially on a dynamical timescale, leading toward unstable mass transfer or a common-envelope phase.