Modern planetary migration theory originated from the study of planetary rings (see Ward 1997). While the basic physical processes - density waves in quasi-Keplerian discs -are well studied, the application to the problem of forming planets in the nebula disc is not straightforward.
The basic problem that has to be solved to determine migration rates for a proto-giant planet orbiting in a nebula disc is the fluid dynamical analogue to the restricted three-body problem of celestial mechanics. In the classical problem of celestial mechanics the motion of a test particle is considered in the combined gravitational field of the Sun and a planet. For a proto-giant planet two modifications have to be made:
(i) a protoplanet, unlike a mature planet, is not well approximated by a point mass,
(ii) the test particles are replaced by a fluid with a finite pressure.
A protoplanet fills its Hill sphere and a considerable fraction of its mass is located at significant fractions of the Hill sphere (e.g. Mizuno 1980; Pecnik and Wuchterl 2005). Furthermore, the protoplanet builds up a significant contribution to the gas pressure at the Hill sphere. Typically, planet and nebula are in a mechanical equilibrium. This may only change when and if the planet collapses into the Hill sphere and does not rebound. Fluid dynamical calculations show that this is a non-trivial question that depends on the structure of the outer protoplanetary layers near the Hill sphere (Wuchterl 1995a). In consequence, the problem of a protoplanet in a nebula disc is not only a problem of gas motion in the gravitational potential of two centres but is controlled by the nebula gas flow and the largely hydrostatic equilibrium of the protoplanets themselves. The Hill spheres are filled by hydrostatic protoplanets at least up to the critical mass -20 • Earth, say - and by quasi-hydrostatic structures, typically up to 50-100 • Earth. In fact, strictly static solutions for protoplanets are published up to masses that closely approach that of Saturn (Wuchterl 1993). Static isothermal protoplanets may be found with masses comparable to Jupiter's (Pecnik and Wuchterl 2005). As a consequence, the protoplanetary migration problem is very far from the idealizations of essentially free gas motion in the potential of two point masses.
Because the problem is basically three-dimensional and the density structure of a protoplanet covers many orders of magnitude, additional approximations have to be made to solve the problem - either numerically or analytically. The basic analytical results (see Ward 1997) stem from solving the linearized fluid dynamical equations for power-law nebula surface densities and an approximate gravitational potential of the problem. The starting point is an unperturbed, quasi-Keplerian disc. The planet is approximated by an expansion of the perturbations induced by a point mass. The linear effect (spiral density waves launched in the disc) is deduced and the resulting torques of the waves on the planet are calculated, assuming that the waves dissipate by a break in the disc. If the waves (and the angular momentum carried) were to be reflected and return there would be no effect. This approach has at least two potential problems.
(i) The dense parts of the protoplanets, approximately in the inner half of the Hill sphere, that potentially carry a large fraction of momentum are treated as if there were no protoplanet - the density structure of the disc is assumed to be unperturbed by the protoplanetary structure even at the position of the planet's core, certainly throughout the Hill sphere. In that way the pressure inside the Hill sphere is dramatically underestimated. The Hill sphere effectively behaves like a hole in the idealized studies of the problem: the gravity of the protoplanet is introduced into the calculations, but the counteracting gas pressure of the static envelopes is omitted.
(ii) The unperturbed state needed for the linear analysis is an unperturbed Keplerian disc. But if a planet with finite mass is present, the unperturbed state is certainly not an axially symmetric disc and corrections have to be made at all azimuthal angles along the planet's orbit. The Keplerian disc in the presence of a protoplanet or an embryo is an artificial state that is found to decay in any non-linear calculation. It is certainly not a steady state, as would be required for a rigorous linear analysis. Therefore, the approach is not mathematically correct. It may turn out that the corrections are minor, as in the case of the , and the basic results hold despite considerable mathematical violence. But unlike in the Jeans case, where Bonnor-Ebert spheres show that there are indeed nearby static solutions, nothing similar is available for the planet-in-disc problem. In fact the respective steady flows are essentially unknown and it is questionable whether they exist at all in the fluid dynamical problem - they might always be a non-steady planetary wake trailing the planet. High resolution calculations (Koller and Li 2003;
Koller 2003) indeed show considerable vorticity and important effects on the torques at the corotation resonances with potentially important consequences for the migration rates.
Non-linear three-dimensional (and two-dimensional) hydrodynamic calculations of the problem are very challenging, both in terms of timescales and spatial scales. The 'atmospheric' structure of the protoplanet inside the Hill sphere can barely be resolved even in the highest resolution calculations (D'Angelo 2002, 2003, 2005) and the dynamics has to be done for simplified assumptions about the thermal structure and dynamical response of the nebula (usually locally isothermal or locally isentropic). The great value of these calculations is that they provide information about the complicated interaction of the planet with a nebula disc that can only be incompletely addressed by models with spherical symmetry that calculate the structure and energy budget of the proto-planet in great detail. The interaction regime between the outer protoplanetary envelope, inside half a Hill radius, and the unperturbed nebula disc, at five Hill radii, say, is only accessible by two- or three-dimensional calculations. Its nature is unknown owing to the lack of any reference flows, be they experimental or theoretical. This situation in my opinion is similar to that of the restricted three-body problem at the time when numerical integrations had just started.
Hence migration rates calculated numerically or analytically have to be considered preliminary and await confirmation by more complete studies of the problem. Agreement found between different investigations is within the very considerable assumptions outlined above and does not preclude considerable uncertainties in the migration rates of many orders of magnitude.
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