Validity of the Statistical Coagulation Equation and Runaway Growth of Protoplanets

Abstract

The coagulation equation has been widely used to describe various kinds of accretion processes. Owing to its statistical property, however, the coagulation equation has a serious defect in some cases, i.e., sometimes the total mass of a system is not conserved. The aim of the present study is to find the necessary and sufficient conditions under which the ordinary coagulation equation is valid physically. On the basis of the stochastic viewpoint, H. Tanaka and K. Nakazawa (1993, J. Geomag. Geoelectr. 45, 361-381) derived the stochastic coagulation equation which describes exactly the accretion process. They also obtained analytic solutions to the stochastic coagulation equation for three kinds of the bilinear coalescence rates (i.e., A ij = 1, i + j , and i × j). By comparing these analytical solutions with those to the ordinary coagulation equation, we examine the conditions under which the ordinary coagulation equation is valid. The results are summarized as follows: (i) In the cases of A ij = 1 and i + j, the coagulation equation remains valid before bodies grow to the mass comparable to the total mass, N. (ii) In the case of A ij = i × j, the coagulation equation is valid until the stage in which bodies with mass comparable to or larger than N 2/3 appear. Additionally, we calculate the time-variation of the masses of the largest two bodies in a system using the solutions to the stochastic coagulation equation. As a result, we can combine the above conditions (i) and (ii) by a single statement: the ordinary coagulation equation is valid before the stages where the runaway growth starts. This conclusion is probably useful not only for cases of these limited coalescence rates but also for more general cases.