Donor-wind Roche-lobe response (${\zeta }_L=(1-f){\zeta }_0+f{\zeta }_1$)

Under donor-wind angular-momentum loss and the Roche lobe approximation $R_L=0.46a(M_d/M{)}^{1/3}$, the Roche-lobe radius response exponent is affine in retained fraction: ${\zeta }_L=(1-f){\zeta }_0+f{\zeta }_1$, where $q=M_d/M_a$, ${\zeta }_0=1/3-4q/[3(1+q)]$ and ${\zeta }_1=2q-5/3$. Differentiate $\log R_L=\log a+(\log M_d-\log M)/3+\mathrm{constant}$ and the circular-binary orbital angular momentum to obtain this expression.

A donor star with stellar radius response exponent ${\zeta }_{\ast }$ can maintain exact contact in the donor-wind Roche-lobe response model for some retained fraction $0\le f\le 1$ precisely when ${\zeta }_{\ast }$ lies between ${\zeta }_0$ and ${\zeta }_1$. This follows because the attainable Roche-lobe radius response exponents form that closed interval. If its endpoints coincide, all fractions work when ${\zeta }_{\ast }$ equals the common value and none work otherwise. With negative donor mass-loss rate, $d\log (R_d/R_L)/dt=({\zeta }_{\ast }-{\zeta }_L){\dot{M}}_d/M_d$ distinguishes detachment from increasing overfill. Dynamical stability of binary mass transfer requires the adiabatic donor response.