High-accuracy statistical simulation of planetary accretion: I. Test of the accuracy by comparison with the solution to the stochastic coagulation equation

Abstract

The object of this series of studies is to develop a highly accurate statistical code for describing the planetary accumulation process. In the present paper, as a first step, we check the validity of the method proposed by Wetherill and Stewart (1989) by comparing the results obtained by their method with the analytical solution to the stochastic coagulation equation (or to a well-evaluated numerical solution). As the collisional probability A_{i j} between bodies with masses of im₁ and j_m1 (m₁ being the unit mass), we consider the two cases: one is A_i j∝ i × j and another is A_{i j} ∝ min(i, j )(i 1/3 + j^1/3)(i + j ). In both cases, it is known that runaway growth occurs. The latter case corresponds to a simplified model of the planetesimal accumulation. We assumed that a collision of two bodies leads to their coalescence. Wetherill and Stewart’s method contains some parameters controlling the practical numerical computation. Among these, two parameters are important: the mass division parameter δ, which determines the mass ratio of the adjacent mass batches, and the time division parameter ∈, which controls the size of a time step in numerical integration. Through a number of numerical simulations for the case of A_{i j} = i × j, we find that when δ ≤ 1.6 and ∈ ≤ 0.03 the numerical simulation can reproduce the analytical solution within a certain level of accuracy independently of the size of the body system. For the case of the planetesimal accumulation, it is shown that the simulation with δ ≤ 1.3 and ∈ ≤ 0.04 can describe precisely runaway growth. Because the accumulation process is stochastic, in order to obtain reliable mean values it is necessary to take the ensemble mean of the numerical results obtained with different random number generators. It is also found that the number of simulations, N_c, demanded to obtain the reliable mean value is about 500 and does not strongly depend on the functional form of A_{i j}. From the viewpoint of the numerical handling, the above value of δ(≤ 1.3) and N_c(∼500) are reasonable and, hence, we conclude that the numerical method proposed by Wetherill and Stewart is a valid and useful method for describing the planetary accumulation process. The real planetary accumulation process is more complex since it is coupled with the velocity evolution of the planetesimals. In the subsequent paper, we will complete the high-accuracy statistical code which simulate the accumulation process coupled with the velocity evolution and test the accuracy of the code by comparing with the results of N -body simulation.