Planetary migration in protoplanetary disks

In the current paper, we further develop the model for the migration of planets introduced in Del Popolo et al. (2001) and extended to time-dependent accretion discs in Del Popolo and Eksi (2002). We use a method developed by Stepinski and Valageas (1996, 1997), that is able to simultaneously follow the evolution of gas and solid particles for up to $10^7 {\rm yr}$. The disc model is coupled to the migration model introduced in Del Popolo et al. (2001) in order to obtain the migration rate of the planet in the planetesimal disc. We find that in the case of discs having total mass of $10^{-3}-0.1 M_{\odot}$, and $0.1<\alpha<0.0001$, planets can migrate inward a large distance while if $M<10^{-3} M_{\odot}$ the planets remain almost in their initial position for $0.1<\alpha<0.01$ and only in the case $\alpha<0.001$ the planets move to a minimum value of orbital radius of $\simeq 2 {\rm AU}$. The model gives a good description of the observed distribution of planets in the period range 0-20 days.