 Solution

Put $m=m_1+m_2$, $\mu =Gm$, $n=\sqrt{\mu /a^3}$, $v=na$, and $\theta =nt$. Work instantaneously in the inertial frame in which the centre of mass is at rest. The unperturbed circular relative orbit hasThe individual velocities are ${\mathbf{v}}_1=-(m_2/m)\mathbf{v}$ and ${\mathbf{v}}_2=(m_1/m)\mathbf{v}$. Linear gas drag gives ${\mathbf{F}}_i/m_i=-({\mathbf{v}}_i-v_g\hat{\mathbf{y}})/t_i$, where $t_i=t_{si}$ are the aerodynamic stopping times. DefineThen the gas drag on a planetesimal binary isEquivalently, $\mathbf{f}=-\kappa \mathbf{v}+\Delta v_g\hat{\mathbf{y}}$. If the centre of mass moves, $v_g\hat{\mathbf{y}}$ here is its relative gas velocity, held approximately fixed during one short binary orbit.