= Solution
For a fixed satellite mass on an adiabatically changing <circular orbit>, $L_s=M_s\sqrt{GM_*r_0}=M_sr_0^2\Omega$. The satellite <torque> is minus the disc <torque>. Hence
$$
\frac12M_sr_0\Omega\frac{dr_0}{dt}=-\Gamma,\qquad
\frac{dr_0}{dt}=-\frac{8\chi}{9}\,\beta q\frac{\Sigma_0r_0^2}{M_*}\frac{r_0\Omega}{(H/r_0)^2}.
$$
The <circular-orbit migration rate from disk torque>, in the normalization supplied by the question, is
$$
\boxed{\frac{dr_0}{dt}=-\frac49\,\beta q\frac{\Sigma_0r_0^2}{M_*}\frac{r_0\Omega}{(H/r_0)^2}.}
$$
It is inward for a positive <mass density> slope and outward for a negative one. The complete-flyby normalization makes its magnitude four times larger, with the same direction. This is a local <torque> estimate; the hypothesis of a nearly circular slowly migrating orbit is needed when using the derivative of the circular <angular momentum>.
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