= Solution
In a close <binary star>, the rotating effective gravitational potential defines two <Roche lobes> meeting at the <inner Lagrange point>. Stellar expansion or orbital shrinkage can bring the donor into <Roche-lobe overflow>. Gas then flows through the <inner Lagrange point>; its <angular momentum> can form an <accretion disk> around the companion. <wind mass transfer in a binary star> is also possible without Roche-lobe filling. For a circular, synchronized binary, the <Eggleton Roche-lobe radius approximation> is
$$
\frac{R_L}{a}\simeq\frac{0.49q^{2/3}}{0.6q^{2/3}+\ln(1+q^{1/3})},\qquad q=\frac{M_d}{M_a}.
$$
This describes the lobe geometry, not the donor's dynamical response.<Approximations to the radii of Roche lobes by Peter P. Eggleton>[Printed p. 368, equation 2]{ref}
<conservative binary mass transfer> has fixed total <mass> and <orbital angular momentum>, neglecting stellar spin. Using $J=M_dM_a\sqrt{Ga/(M_d+M_a)}$ gives
$$
\frac{\dot a}{a}=-2\dot M_d\left(\frac1{M_d}-\frac1{M_a}\right).
$$
Since $\dot M_d<0$, transfer from the more massive donor initially shrinks the orbit; after <binary mass ratio> reversal it expands. Stability depends on both stars and their changing lobes. Let $\zeta_*=d\ln R_d/d\ln M_d$ and $\zeta_L=d\ln R_L/d\ln M_d$. A small <mass> loss reduces the relative overfill if \b[$\zeta_*>\zeta_L$]. Use the adiabatic donor response for dynamical stability, and its thermal response for slower adjustment; an equilibrium mass-radius law alone cannot decide both.
Stable transfer can proceed on nuclear or thermal timescales, while dynamically unstable overflow may create a <common envelope>, orbital spiral-in and either envelope ejection or a merger. Escaping gas, <gravitational radiation> and <magnetic braking> make evolution nonconservative by changing <orbital angular momentum>. <accretion> onto a <compact stellar object> releases <power> of order $GM_a\dot M/R_a$, producing bright <X-ray binaries>; radiation-driven outflows can limit retained <mass>. Eclipses, <radial velocities> and disk emission constrain the geometry and <masses>, but inclination, irradiation and assumptions about <mass> or angular-momentum loss affect the inference. \b[The essential feedback is the donor's radius relative to its evolving <Roche lobe>.]
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