Evolution of planetesimal discs and planets migration

In this paper, we further develop the model for the migration of planets introduced in Del Popolo, Gambera and Ercan, and extended to time-dependent planetesimal accretion disks in Del Popolo and Eksi. More precisely, the assumption of Del Popolo and Eksi that the surface density in planetesimals is proportional to that of gas is released. Indeed, the evolution of the radial distribution of solids is governed by many processes: gas-solid coupling, coagulation, sedimentation, evaporation/condensation, so that the distribution of planetesimals emerging from a turbulent disk does not necessarily reflect that of gas. In order to describe this evolution we use a method developed in Stepinski and Valageas which, using a series of simplifying assumptions, is able to simultaneously follow the evolution of gas and solid particles for up to $10^7 {\rm yr}$. Then, the distribution of planetesimals obtained after $10^7 {\rm yr}$ is used to study the migration rate of a giant planet through the migration model introduced in \cite{DP1}. This allows us to investigate the dependence of the migration rate on the disk mass, on its time evolution and on the value of the dimensionless viscosity parameter alpha. We find that in the case of disks having a total mass of $10^{-3}-10^{-1} M_{\odot}$, and $10^{-4}<\alpha<10^{-1}$, planets can migrate inward over a large distance while if $M_{\rm d}<10^{-3} M_{\odot}$ the planets remain almost at their initial position for $\alpha>10^{-3}$ and only in the case $\alpha<10^{-3}$ the planets move to a minimum value of orbital radius of $\simeq 2 {\rm AU}$. Moreover, the observed distribution of planets in the period range 0-20 days can be easily obtained from our model.