= Gas-driven migration of a binary centre of mass
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 $-\eta\kappa_bv_K$. <Orbit-averaged drag work> gives
$$
\dot a_b=-2\eta\kappa_ba_b,\qquad \tau_b=\frac1{2\eta\kappa_b}.
$$
If $m_2/m_1=\epsilon\ll1$ and <aerodynamic stopping time> scales as $m^k$, $0<k<1$, then $\tau/\tau_b\simeq\eta\epsilon^k\ll1$. Internal contraction can outpace stellar migration. Tightly gas-coupled drift requires solving radial and azimuthal motion together instead of this weak-drag approximation.
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