Long-Term Behavior of Planetesimals and the Formation of the Planets

On the origin of the solar system, we have investigated the longterm and long-range behaviors of an ensemble of planetesimals, which are rotating around the Sun, until they are finally captured by protoplanets within their Hill spheres. In this way, we have evaluated the growth time of the planets. We have considered, as an example, an initial state such that in the solar nebula with gas density of about 10- g a protoplanet with mass of the order of 1025 g and a great number of uniformly- populated planetesimals of smaller masses are rotating around the Sun. Generally, the Keplerian orbits of planetesimals are changed very gradually by a combination of three effects due, respectively, to the gas drag force, the mutual encounters, and the gravity of a protoplanet. For each of the effects, the rates of change of the three orbital elements, the semi-major axis, the eccentricity, and the inclination, have been calculated numerically and/or analytically. Combining these effects, we have evaluated the average rate of the long-term change of the semi- major axis as a stochastic process for an ensemble of planetesimals. We have found that protoplanets grow to the mass of the Earth and that of a Jupiter's core in periods of the order of lOT and 108 yr, respectively, and that these time scales are determined mainly by the rate of migration of planetesimals in regions far from the protoplanet. This migration is a result of the interplay of the gas drag force and the mutual encounters. Key words: Origin of the solar system; Planet formation; Restricted three-body problem; Stochastic process in the solar nebula.