 Solution

For weak gas drag, evaluate the work on the unperturbed circular orbit and average over one orbital period. The gas velocity changes negligibly during that period and $⟨\cos \theta ⟩=0$. Hence gas-driven contraction of a planetesimal binary obeysWith constant aerodynamic stopping times,This is the secular, orbit-averaged separation. It assumes the perturbations are small enough for the orbit to remain nearly circular; a large differential headwind can excite eccentricity or disrupt a weakly bound pair instead.