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For a weakly perturbed circular binary planetesimal, the constant differential headwind does zero work averaged over the unperturbed orbit. Orbit-averaged drag work givesWith constant aerodynamic stopping times, $a(t)=a_0{e}^{-t/\tau }$. This averaging assumes drag is slow compared with the orbital period and does not appreciably distort the circular orbit within one period.

For a nearly circular stellar orbit of radius $a_b$ in gas moving at $(1-\eta )$ times the local circular speed, the averaged tangential gas drag on the centre of mass is $-\eta {\kappa }_b{v}_K$. Orbit-averaged drag work givesIf $m_2/m_1=ϵ\ll 1$ and aerodynamic stopping time scales as $m^k$, $0<k<1$, then $\tau /{\tau }_b\simeq \eta {ϵ}^k\ll 1$. Internal contraction can outpace stellar migration. Tightly gas-coupled drift requires solving radial and azimuthal motion together instead of this weak-drag approximation.

With $\hat{\mathbf{r}}=(\cos \theta ,\sin \theta )$ and $\hat{\mathbf{\theta }}=(-\sin \theta ,\cos \theta )$, project the relative gas drag to obtainThe specific orbital energy is $\epsilon =v^2/2-\mu /r=-\mu /(2a)$. Differentiation cancels the conservative gravitational work and gives $\dot{\epsilon }=\mathbf{v}\cdot \mathbf{f}$. For the circular orbit,Using $v^2=\mu /a$, the orbit-averaged drag work relation therefore gives the instantaneous leading-order changeThe differential headwind term can change sign during the orbit; the damping term always removes orbital energy.

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.