Migration of giant planets in a time-dependent planetesimal accretion disc

In this paper, we further develop the model for the migration of planets introduced in Del Popolo et al. (2001). We first model the protoplanetary nebula as a time-dependent accretion disc and find self-similar solutions to the equations of the accretion disc that give to us explicit formulas for the spatial structure and the temporal evolution of the nebula. These equations are then used to obtain the migration rate of the planet in the planetesimal disc and to study how the migration rate depends on the disc mass, on its time evolution and on some values of the dimensionless viscosity parameter alpha. We find that planets that are embedded in planetesimal discs, having total mass of 10^{-4}-0.1 M_{\odot}, can migrate inward a large distance for low values of alpha (e.g., alpha \simeq 10^{-3}-10^{-2}) and/or large disc mass and can survive only if the inner disc is truncated or because of tidal interaction with the star. Orbits with larger $a$ are obtained for smaller value of the disc mass and/or for larger values of alpha. This model may explain several orbital features of the recently discovered giant planets orbiting nearby stars.