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 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.

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.

The $k_R=+1$ orbital resonance of an $m$-fold perturbation of a galactic disk. For near-circular stellar orbits, ${\Omega }_R$ is the radial epicyclic frequency. The harmonic changes radial action and azimuthal angular momentum in the ratio $\Delta J_R/\Delta J_{\phi }=1/m$, so an orbit gaining angular momentum gains random radial motion.

The $k_R=-1$ orbital resonance of an $m$-fold perturbation of a galactic disk. For near-circular stellar orbits, ${\Omega }_R$ is the radial epicyclic frequency. The harmonic changes radial action and azimuthal angular momentum in the ratio $\Delta J_R/\Delta J_{\phi }=-1/m$, so an orbit losing angular momentum gains random radial motion.