Statistics and Universality in Simplified Models of Planetary Formation

In this paper, we modify Laskar's simplified model of planetary evolution and accretion [J. Laskar, Phys. Rev. Lett, vol 84, p 3240 (2000)] to account for the full conservation of the total angular momentum of the system, and extend it to incorporate an accretion probability that depends on the mass and relative velocity of the colliding particles. We present statistical results for the mass and eccentricity of the planets formed, in terms of their semi-major axes, for a large number of realisations of different versions of the model. In particular, we find that by combining the mass-dependent accretion probability and the velocity-selection mechanism, the planets formed display a systematic occurrence at specific locations. By introducing properly scaled variables, our results are universal with respect to the total angular momentum of the system, the mass of the planetesimal disc, and the mass of the central star.