 Solution

Let $\mathbf{V}$ be the centre of mass velocity and $\mathbf{v}$ the binary relative velocity. Summing the two linear gas drag forces givesThe last term averages to zero over the internal circular orbit. On the nearly circular stellar orbit, $\mathbf{V}\simeq v_K\hat{\mathbf{\theta }}$ and ${\mathbf{v}}_g=(1-\eta )v_K\hat{\mathbf{\theta }}$, so the averaged tangential acceleration is $-\eta {\kappa }_b{v}_K$.

With specific orbital energy $-G{M}_{\star }/(2a_b)$, the orbit-averaged drag work is $-\eta {\kappa }_b{v}_K^2$, where $v_K^2=G{M}_{\star }/a_b$. Thus gas-driven migration of a binary centre of mass givesThis uses the weak-drag, nearly Keplerian approximation implicit in applying the circular-orbit work relation. Strongly gas-coupled orbits require a coupled radial-azimuthal drift solution.