 Solution

Since the donor fills its Roche lobe, use $R_2=R_L$ and substitute its equilibrium mass-radius relation into the lobe approximation:Kepler's third law then givesso the Roche-lobe period-mass relation for a linear donor radius law isThe total binary mass cancels, rather than being assumed equal to the white-dwarf mass.

The orbital radii about the center of mass are $a_1=a{M}_2/M$ and $a_2=a{M}_1/M$. Summing the two orbital angular momenta givesWith transferred matter retained by the accretor, total mass is fixed. Logarithmic differentiation givesFor $\dot{J}=0$, the separation response is $\dot{a}/a=2(q-1){\dot{M}}_2/M_2$. Since $R_L\propto a{M}_2^{1/3}$ at fixed total mass,The second relation uses the supplied equilibrium donor sequence. As the donor loses mass, ${\dot{M}}_2<0$. If $q>4/3$, its Roche lobe shrinks fractionally faster than this donor radius, so the excess filling increases and drives more transfer: it is positive feedback. If $q<4/3$, infinitesimal loss instead reduces overfill on that assumed sequence, so some continuing driver is needed to retain contact.

There are two limits to the literal dynamical interpretation. The given initial range is $q<1$, so $q>4/3$ is an extrapolated instability test, not a state inside that range. Also, a true dynamical stability of binary mass transfer test needs the donor's adiabatic stellar radius response exponent ${\zeta }_{\mathrm{ad}}$, rather than its thermal-equilibrium exponent. In general,Thus $4/3$ is the result for the assumed ${\zeta }_{\ast }=1$ response. A fully convective idealized donor has ${\zeta }_{\mathrm{ad}}=-1/3$, giving a dynamical threshold $2/3$, already below $4/3$. The equilibrium radius law alone should not be used to assert a universal dynamical threshold.

One continuing driver is magnetic braking of a binary star. A magnetized wind carries away the donor's spin angular momentum; tides that maintain near-synchronous rotation replenish that spin from the orbit, removing orbital angular momentum. With an external $\dot{J}<0$, the lobe response becomesEquating this to the donor response proves the binary mass-transfer contact equation in this model:A negative torque therefore drives mass loss while maintaining contact for $q<4/3$, subject to the chosen donor response and braking assumptions.