The growth of planetary embryos: Orderly, runaway, or oligarchic?

We consider the growth of a protoplanetary embryo embedded in a planetesimal disk. We take into account the dynamical evolution of the disk caused by (1) planetesimal-planetesimal interactions, which increase random motions and smooth gradients in the disk, and (2) gravitational scattering of planetesimals by the embryo, which tends to heat up the disk locally and repels planetesimals away. The embryo's growth is self-consistently coupled to the planetesimal disk dynamics. We demonstrate that the details of the evolution depend on only two dimensionless parameters incorporating all the physical characteristics of the problem: the ratio of the physical radius to the Hill radius of any solid body in the disk (which is usually a small number), and the number of planetesimals inside an annulus of the disk with width equal to the planetesimal Hill radius (which is usually large). We derive simple scaling laws describing embryo-disk evolution for different sets of these parameters. The results of exploration in the framework of our model of several situations typical for the protosolar nebula can be summarized as follows: Initially, the planetesimal disk dynamics is not affected by the presence of the embryo, and the growth of the embryo's mass proceeds very rapidly in the runaway regime. Later, when the embryo starts being dynamically important its accretion slows down, similar to the oligarchic growth picture. The scenario of orderly growth originally suggested by Safronov is never realized in our calculations; the scenario of runaway growth suggested by Wetherill & Stewart is only realized for a limited range in mass. The slow character of the planetesimal accretion in the oligarchic stage of the embryo's accumulation leads to a considerable increase of the protoplanetary formation timescale compared with that following from a simple runaway accretion picture valid for homogeneous planetesimal disks.