A protoplanet embedded in a gas disk perturbs the gas gravitationally, exciting trailing spiral density waves at its inner and outer Lindblad resonances. Each wave carries angular momentum away from the planet; because the outer disk (further from the star) responds slightly more strongly than the inner disk in a typical disk, the two torques don't cancel, leaving a small net differential Lindblad torque that is almost always negative — it drains angular momentum from the planet and makes it spiral inward. This is Type I migration, the regime for planets too light to open a gap in the disk (roughly below a few Earth masses to a Neptune mass, depending on disk viscosity).
The Tanaka, Takeuchi & Ward (2002) linear torque formula used here:
Γ_L = -(1.364 + 0.541 α) · (q/h)² · Σ_p r_p⁴ Ω_p²
q = M_planet / M_star, h = H/r (disk aspect ratio)
Σ_p ∝ r_p^(-α) (surface density power law)
Gas that co-orbits with the planet (the horseshoe region) also exchanges angular momentum through the corotation torque. In an isothermal disk this term is small and negative, but in a disk with a real entropy/temperature gradient it can become strongly positive and — for the right combination of mass and disk structure — overpower the Lindblad torque, temporarily halting or reversing migration. This "torque reversal" is one of the leading mechanisms proposed to keep growing rocky/icy cores from spiraling into their star before they can become gas-giant cores.
da/dt = 2 a Γ_tot / (M_planet a² Ω_p) (angular-momentum exchange)
τ_mig = a / |da/dt|
- Planet mass — sets q² in Γ_L; heavier planets torque the disk harder and migrate faster.
- Surface-density slope α — steeper disks (larger α) strengthen the differential Lindblad torque.
- Aspect ratio h — a thinner disk (smaller h) couples more strongly to the planet, shortening τ_mig.
- Corotation toggle — switches the reversal term on/off so you can see the bare Lindblad-only inward drift versus the more realistic, sometimes-outward net torque.
The spiral wake rendered in the disk is the same one- and two-armed pattern real hydrodynamic simulations show trailing a low-mass planet; the planet's orbit here integrates the migration rate directly, so its radius visibly decays (or holds/grows) as the torque balance changes.