Tides come from the fact that different parts of a system do not
fall in exactly the same way in a non-uniform gravity field. In the
case of a protoplanetary disk perturbed by an orbiting, prograde
protoplanet, the protoplanet tides raise a wake in the disk which
causes the orbital elements of the planet to change over time. The
most spectacular result of this process is a change in the
protoplanet's semi-major axis, which can decrease by orders of
magnitude on timescales shorter than the disk lifetime. This drift
in the semi-major axis is called planetary migration, and is the
most important aspect of planet–disk interactions. In this chapter,
we first describe how the planet and disk exchange angular momentum
and energy at the Lindblad and corotation resonances. Next we
review the various types of planetary migration that have so far
been contemplated: type I migration, which corresponds to low-mass
planets (less than a few Earth masses) triggering a linear disk
response; type II migration, which corresponds to massive planets
(typically at least one Jupiter mass) that open up a gap in the
disk; “runaway” or type III migration, which corresponds to
sub-giant planets that orbit in massive disks; and stochastic or
diffusive migration, which is the migration mode of low- or
intermediate-mass planets embedded in turbulent disks. Third, we
discuss questions linked to the planet eccentricity, in particular
how the eccentricity is affected by the planet–disk interaction.
Fourth, we discuss the various numerical schemes that have been used
to describe planet–disk interactions. We discuss their strengths
and weaknesses, and list the results that numerical simulations have
achieved over the past decade.