= Type I migration timescale from disk torque
{c}
{title2=$t_a=a/|\dot a|$}
For a fixed-<mass> <planet> in a <circular orbit> of radius $a$, let $q=M_p/M_\star$ be the planet-to-star <mass ratio>, $\Sigma$ the local <surface density>, and $h=H/a$ the <disk aspect ratio> defined by <disk scale height> $H$. The local <angular velocity> is $\Omega=\sqrt{GM_\star/a^3}$. Suppose the <torque> on the <planet> is
$$
\Gamma=C(q/h)^2\Sigma a^4\Omega^2,
$$
where the dimensionless coefficient $C$ can have either sign and depends on the disk model.
The planet's orbital <angular momentum> is $L=M_pa^2\Omega=M_p\sqrt{GM_\star a}$. Its derivative gives $\dot L=(M_pa\Omega/2)\dot a=\Gamma$, hence
$$
t_a=\frac a{|\dot a|}=\frac1{2|C|\Omega}\frac{M_\star}{M_p}\frac{M_\star}{\Sigma a^2}h^2.
$$
Positive $\Gamma$ in this convention produces outward <planetary migration>; negative $\Gamma$ produces inward <planetary migration>. The <torque> exerted on the disk has the opposite sign. The radial evolution timescale is half $L/|\Gamma|$, because $L\propto\sqrt a$. This expression assumes the low-<mass> <Type I migration> regime and the stated torque law, rather than a universal direction or rate.
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