= 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 $\langle\cos\theta\rangle=0$. Hence <gas-driven contraction of a planetesimal binary> obeys
$$
\langle\dot a\rangle=-2\kappa a=-\frac a\tau,
\qquad
\boxed{\tau=\frac{m_1+m_2}{2(m_1/t_{s2}+m_2/t_{s1})}.}
$$
With constant <aerodynamic stopping times>,
$$
\boxed{a(t)=a_0e^{-t/\tau}.}
$$
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.
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