Type I migration

Orbital migration of a low-mass embedded planet driven by its interaction with disk gas, without a deep gap. Spiral wave torques and corotation-region torques exchange orbital angular momentum. The sign depends on the disk gradients and thermodynamics; this regime is not restricted to inward migration.

The scale $M_{\mathrm{th}}=h^3{M}_{\star }$ makes the Bondi radius comparable with the disk scale height. It organizes when planet-induced disturbances become nonlinear; it is not identical to an exact universal gap-opening mass.

For a circular fixed-mass planet, $L=M_p{a}^2\Omega$ and $\dot{a}=2\Gamma /(M_{p}a\Omega )$. With $\Gamma =C(q/h{)}^2\Sigma a^4{\Omega }^2$, the radial migration timescale is $a/|\dot{a}|=(M_{\star }/M_p)[M_{\star }/(\Sigma a^2)]h^2/(2|C|\Omega )$. The factor of two distinguishes radial evolution from $L/|\Gamma |$. The dimensionless coefficient and its sign depend on the disk model.

A location where the net disk torque on a planet vanishes and, for a stable trap, restoring torque changes sign across the location. A torque zero alone does not guarantee stable trapping as the disk or planet evolves.

The orbital torque associated with disk material near the planet's corotation radius. Vortensity and entropy gradients matter because gas exchanged across the orbit carries different conserved properties. Horseshoe turns can make the response nonlinear even while the planet remains in the low-mass migration regime.

Repeated mixing or libration in an isolated corotation region can erase the gradient that drives its torque. Viscosity and thermal diffusion can help restore relevant gradients; excessive diffusion also changes the response. Saturation concerns the sustained torque, not cessation of all disk waves.

An entropy contrast carried by gas executing a horseshoe turn creates a density asymmetry around an embedded planet. The resulting torque can oppose the usual negative wave torque. Its persistence depends on thermal transport and the replenishment of the background entropy gradient.

A disk-gas trajectory that turns from one side of a planet's orbital radius to the other in the corotating frame. Its angular-momentum exchange contributes to the corotation torque; repeated libration without restoring gradients can saturate that contribution.

A rotating gravitational perturbation drives an epicyclic disk response where $m(\Omega -{\Omega }_p)=\pm \kappa$. The launched density waves carry angular momentum. For a planet, inner and outer waves exert torques of opposite signs, and their imbalance contributes to migration.