 Type I migration timescale from disk torque

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 $\Omega =\sqrt{G{M}_{\star }/a^3}$. Suppose the torque on the planet iswhere the dimensionless coefficient $C$ can have either sign and depends on the disk model.

The planet's orbital angular momentum is $L=M_p{a}^2\Omega =M_p\sqrt{G{M}_{\star }a}$. Its derivative gives $\dot{L}=(M_{p}a\Omega /2)\dot{a}=\Gamma$, hencePositive $\Gamma$ 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.