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Put $ϵ=m_2/m_1\ll 1$ and $t_{s2}/t_{s1}={ϵ}^k$, with $0<k<1$. The gas-driven contraction of a planetesimal binary and gas-driven migration of a binary centre of mass times obeyThereforeThe low-mass secondary has the shorter aerodynamic stopping time and chiefly controls internal damping, whereas the primary chiefly controls centre of mass drag. The pair can contract substantially, potentially merging, before its stellar orbital radius changes appreciably. This conclusion is restricted to a pair that remains bound and within the adopted linear-drag regime.

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.