Nuclear-driven Roche-overfill relaxation ($\dot{x}=t_{\mathrm{nuc}}^{-1}-A(q)x/(f{t}_{\mathrm{dyn}})$)

Let $x=\log (R/R_L)$, $q=M_1/M_2$, ${\dot{M}}_2=-f{\dot{M}}_1=M_{1}x/t_{\mathrm{dyn}}$, and $d\log R/dt=n{\dot{M}}_1/M_1+t_{\mathrm{nuc}}^{-1}$. Under the binary period invariant for a constant wind angular-momentum factor and $R_L=0.46a(M_1/M{)}^{1/3}$, the restoring coefficient is $A(q)=n+5/3-2fq-2(3\lambda +1)(1-f)q/[3(1+q)]$. Locally freezing the masses gives an attracting overfill $x_{\ast }=f{t}_{\mathrm{dyn}}/[A(q)t_{\mathrm{nuc}}]$ precisely when $A(q)>0$. If $A$ is of order one and $t_{\mathrm{dyn}}\ll t_{\mathrm{nuc}}$, transfer is driven slowly by nuclear expansion. A negative $A$ gives positive feedback on the short overfill timescale; detailed outcomes then exceed this linear contact model.