= Solution
Treat the <stars> as <point masses> for the orbital motion, neglecting their spin <angular momenta>. Let $M=M_1+M_2$. Their distances from the <centre of mass> are $a_1=aM_2/M$ and $a_2=aM_1/M$. The sum of their orbital <angular momenta> is
$$
\begin{aligned}
J&=(M_1a_1^2+M_2a_2^2)\Omega\\
&=\frac{M_1M_2}{M}a^2\left(\frac{GM}{a^3}\right)^{1/2}.
\end{aligned}
$$
Thus the <circular-binary orbital angular momentum> is
$$
\boxed{J=\frac{M_1M_2}{M_1+M_2}\sqrt{G(M_1+M_2)a}.}
$$
For <conservative binary mass transfer>, both $M$ and $J$ are fixed. In terms of the <binary mass ratio> $q=M_1/M_2$, $M_1=Mq/(1+q)$ and $M_2=M/(1+q)$, hence
$$
J^2=GM^3a\frac{q^2}{(1+q)^4},\qquad\boxed{a(1+q)^{-4}q^2=C,\quad C=\frac{J^2}{GM^3}.}
$$
<Mass conservation> also gives $dM_2=-dM_1$, so
$$
\frac{d\log M_2}{d\log M_1}=-q,\qquad\frac{d\log q}{d\log M_1}=1+q.
$$
Differentiating $J\propto M_1M_2a^{1/2}$ at fixed $M,J$ gives $d\log a/d\log M_1=2(q-1)$. Therefore the stipulated <Roche lobe> law has the <Roche-lobe radius response exponent>
$$
\zeta_L=\frac{d\log R_L}{d\log M_1}=\frac3{10}(1+q)+2(q-1)=\frac{23q-17}{10}.
$$
Define the overfill by $\Delta=\log(R_1/R_L)$. On a time scale short compared with the slow change of $K$, the donor's response is $d\log R_1=\beta\,d\log M_1$. Thus
$$
d\Delta=(\beta-\zeta_L)d\log M_1.
$$
During <mass> loss $d\log M_1<0$. If $\zeta_L>\beta$, <mass> loss increases the overfill and therefore drives more <mass> loss: this is positive feedback, giving more rapid <Roche-lobe overflow>. If $\zeta_L<\beta$, <mass> loss decreases the overfill and can balance the slow expansion due to <stellar evolution>. More explicitly, with $k_{\mathrm{ev}}=d\log K/dt>0$, contact requires
$$
0=\dot\Delta=k_{\mathrm{ev}}+(\beta-\zeta_L)\frac{\dot M_1}{M_1}.
$$
A slow negative transfer rate can satisfy this only on the stable side $\beta>\zeta_L$. The rapid-transfer condition is therefore
$$
\boxed{q>\frac{17+10\beta}{23}.}
$$
Its time scale depends on the physical donor response represented by $\beta$; it need not be a dynamical time scale if $\beta$ is a thermal-equilibrium mass-radius exponent.
For <Roche-lobe overflow feedback with a constant wind torque factor>, retain <mass conservation> to the stated approximation but allow the prescribed angular-momentum loss. Since $d\log J=f\,d\log M_1$, differentiation gives
$$
f=1-q+\frac12\frac{d\log a}{d\log M_1},\qquad\frac{d\log a}{d\log M_1}=2(q-1+f).
$$
The new <Roche-lobe radius response exponent> is
$$
\zeta_L=\frac3{10}(1+q)+2(q-1+f)=\frac{23q-17+20f}{10}.
$$
The same overfill calculation gives
$$
\boxed{q>\frac{17-20f+10\beta}{23}.}
$$
A loss of <angular momentum> has $f>0$ because $\dot M_1<0$, and it lowers the critical ratio by making the orbit contract more strongly.
For the specified solar-mass donor exponent $\beta=1/2$, the answer is
$$
\boxed{q>\frac{22-20f}{23},\qquad q>0.}
$$
Without the wind torque this becomes \b[$q>22/23\simeq0.957$], so donors slightly less massive than their companions can already lie on the rapid-transfer side of this model. For $0<f<11/10$ the lower threshold is $(22-20f)/23$, and for $f\geq11/10$ every positive $q$ satisfies the strict inequality. There is no finite upper bound on $q$ from this criterion. Since the paper does not specify $f$, the wind case has a conditional range rather than a unique numerical cutoff.
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