Planetary migration

Planetary migration changes a planet's orbital semimajor axis through torques from a disc or exchanges of energy and angular momentum with other bodies, often followed by tidal dissipation. Short-period giant planets, eccentric progenitors, resonant architectures, and stellar obliquity provide complementary constraints rather than one unique migration diagnostic.

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.

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.

A planet transfers angular momentum to disk material through its spiral wakes, tending to push gas away from its orbital annulus. A persistent deep gap requires this clearing to compete with pressure and disk transport. A gap can reduce local supply without completely shutting off gas streams or fixing the final planet mass.

For a fixed mass on a Keplerian circular orbit, $L_s=M_s\sqrt{G{M}_{\ast }{r}_0}$ has derivative $M_s{r}_0\Omega /2$. The torque on the satellite is minus the torque $\Gamma$ on the disk. This gives the displayed drift law. In the density-slope satellite torque model, ${\dot{r}}_0=-(8\chi /9)\beta q({\Sigma }_0{r}_0^2/M_{\ast })r_0\Omega (H/r_0{)}^{-2}$.

The stellar obliquity is the angle between a star's spin and a planet's orbital angular momentum. The Rossiter-McLaughlin effect constrains the angle $\lambda$ between the sky projections of those two axes, rather than a component of $\psi$. If the orbital and stellar-spin inclinations to the line of sight are $i_{\mathrm{orb}}$ and $i_{\ast }$, then $\cos \psi =\cos i_{\ast }\cos i_{\mathrm{orb}}+\sin i_{\ast }\sin i_{\mathrm{orb}}\cos \lambda$; $\lambda$ alone generally does not determine $\psi$. Large misalignments constrain formation and planetary migration, but can also reflect a tilted birth disc.

A transiting body hides different velocity regions of a rotating stellar disc and causes an anomalous measured Doppler shift. Its time dependence constrains the sky-projected spin-orbit angle, which is not generally the full stellar obliquity.