An Effective 1-D Model of Planetary Condensation A. Balaˇ z, A. Beli´ c and A. Bogojevi´ c Institute of Physics P.O.B. 57, Belgrade 11001, Yugoslavia Abstract An effective 1-D model of planetary condensation through gravitational accretion is presented. The formation of planetary systems is is simulated starting from as many as N = 10 10 initial particles, and the spontaneous appearance of two distinct types of condensate is investigated. The two types of condensate, light and heavy, are distinguished by the way they scale with changes of N . The positions, mass and spin of the heavy condensates (planets) are found to be in very good agreement with Solar system data. 1 Introduction There has been a large increase of the amount of available data on extra-solar planetary systems [1], and an even more dramatic increase is expected in the near future [2]. For this reason the long standing problem of the detailed understanding of planet formation [3] has received a lot of attention lately [4, 5]. The ultimate goal of such studies is to assess the likelihood of the formation of Earth-like planets. The exponential increase in computing power during the last two decades has made numerical simulation the most promising tool to achieve this goal. However, the brute force numerical simulation of the formation of whole planetary systems is still impossible due to the large number of initial particles (N  10 6 ) needed to resolve the resulting planets with reasonable accuracy. In order to deal with this problem we have constructed an effective model of planetary accretion. The simplifications in the model have made it effectively one dimensional, which not only makes gravitational condensation more transparent, but also makes possible the derivation of certain analytical results. 2 The Model Our effective planetary condensation model starts from a given planar distribution of N initial particles, all of the same mass and with no spin. The particles have a uniform angular distribution, while the radial distribution is given by ρ(r), which determines the initial conditions. The dynamics is modeled by two independent processes—free propagation of particles and instanta- neous interactions. Between interactions all particles move on circular trajectories according to Kepler’s laws. The only interaction allowed is the merging of two particles into one. The merging happens if the two particles satisfy an interaction criterion given bellow. The result of the merging of two bodies with masses m 1 and m 2 , at positions  r 1 and  r 2 , and with spins S 1 and S 2 , is a new body with mass m 1 + m 2 , position  R, and spin S = S 1 + S 2 + L 1 + L 2 − L. L 1 , L 2 and L are respectively the orbital angular momenta of the first, second and final particle. The point of joining follows from energy conservation (once we neglect heating due to accretion, energies due to the particles spin, as well as the potential energies between pairs of condensing particles). We find m 1 + m 2 R = m 1 r 1 + m 2 r 2 . (1) The interaction criterion chosen is quite natural. We impose F △t  |△ p |, where F is a mean value of the gravitational force between the bodies during the collision and △t ∼ |△ r |/|△v | is a characteristic 1