 Solution

For a circular binary star, the relative orbit has speed $v=\sqrt{GM/a}$. Its orbital angular momentum is thereforeLogarithmic differentiation, using $\dot{\mu }/\mu ={\dot{M}}_1/M_1+{\dot{M}}_2/M_2-\dot{M}/M$, givesFor conservative binary mass transfer, $\dot{M}=0$, ${\dot{J}}_{\mathrm{orb}}=0$, and ${\dot{M}}_2=-{\dot{M}}_1$. HenceSince a donor has ${\dot{M}}_1<0$, the orbit widens for $M_1<M_2$ and shrinks for $M_1>M_2$.

Write the Roche lobe radius as $R_L=aF(q)$, whereThenConservative transfer has $d\ln q/d\ln M_1=1+q$, so

The donor response $R_1\propto M_1^{-n}$ has ${\zeta }_{\ast }=d\ln R_1/d\ln M_1=-n$, whereasAfter $d\ln M_1<0$, overflow decreases only if ${\zeta }_{\ast }>{\zeta }_L$. The dynamical stability of binary mass transfer criterion is thusFor $q>1$, both terms make $n_{\mathrm{crit}}<0$, which is incompatible with $n>0$. Transfer from the more massive donor is therefore dynamically unstable: mass loss shrinks its Roche lobe while the donor expands. The resulting runaway commonly produces a common envelope, followed by envelope ejection into a tighter binary or by a stellar merger.

Solved by gpt-5.6-sol high.