where a is the semi-major axis and H the thickness of the disk. Mp is the mass of the planet and o is the surface density of the disk. For a planet of one Earth mass at one AU, ti « 105 yr. This time is shorter than the lifetime of disks of gas or the time for the formation of planets.
When the planet is more massive (Type II migration), it creates an empty zone in the disk of gas, the width of which is about twice the Hill radius (see Sect. 5.4.3). In this case, the timescale of the migration depends on the viscosity of the disk, but no longer depends on the mass of the planet:
Unlike Type I migration, here the planet migrates at the same time as the disk, and not relative to the disk. If a/H = 10 and 1/a = 1000, tii amounts to about 5000 years at a distance of one AU, which is, like Type I migration, far shorter than the time for the formation of the planets and the lifetime of gaseous disks.
The transition between the two types of migration occurs when the Hill's radius of the planet exceeds the scale height of the disk, i.e., for a planetary mass Mp such that:
For a/H = 10 and a solar-mass star, the transition occurs at 3 MJ. Numerous numerical studies have shown that this mechanism can account for the observations. Several problems persist, however. For certain configurations, migration takes place towards the exterior of the system. In some cases, the evolution of orbital momentum may also be a chance affair, which does not follow a systematic course.
Interaction between protoplanets and a disk of planetesimals has been suggested to explain the outward migration of Uranus and Neptune. However, to explain the decrease of several AU that is required by the orbits of observed exoplanets, a very massive disk would be required, amounting to about 10 per cent of the stellar mass. In addition, this mechanism may also result in an increase in the semi-major axes, as in the case of the Solar System.
Another mechanism is interaction with a stellar companion (the Kozai' mechanism). If the system initially includes a planet in an inclined orbit, perturbations by the stellar companion may cause the planet's orbit to evolve towards a smaller, less inclined orbit with high eccentricity. If the planet's periapsis comes close enough to the star, its orbit is re-circularized through tidal effects. This theory requires the discovery of a mechanism to explain a planet that is initially situated in an extremely inclined orbit.
Interaction between planets may also lead to migration. If a system contains two planets whose separation is less than:
where ai is the orbital radius of the planet of mass Mi, one possible evolution is an inwards migration of the inner planet, with an increase in its eccentricity. The
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