On the validity of the coagulation equation and the nature of runaway growth

The coagulation equation, which is widely used for modeling growth in planet formation and other astrophysical problems, is the mean-rate equation that describes the evolution of the mass spectrum of a collection of particles due to successive mergers. A numerical code that can yield accurate solutions to the coagulation equation with a reasonable number of mass bins is developed, and it is used to study the properties of solutions to the coagulation equation. We consider limiting cases of the merger rate coefficient A(ij) for gravitational interaction, with the power-law index of the mass-radius relation beta = 1/3 (for planetesimals) and 2/3 (for stars). We classify the mass dependence of A(ij) using the exponent lambda for the merger between two particles of comparable mass, and the exponents mu and nu for the merger between a heavy particle and a light particle. For the two cases with nu less than or equal to 1 and lambda less than or equal to 1, the mass spectrum evolves in an orderly fashion. For the remaining cases, which have nu > 1, we find strong numerical and analytical evidence that there are no self-consistent solutions to the coagulation equation at any time. The results for the nu > 1 cases are qualitatively different from the well-known example with A(ij) proportional to ij. For the latter case, which is in the range nu less than or equal to and lambda > 1, there is an analytic solution to the coagulation equation that is valid for a finite amount of time t(0). We discuss a simplified merger problem that illustrates the qualitative differences in the solutions to the coagulation equation for the three classes of A(ij). Our results strongly suggest that there are two types of runaway growth. For A(ij) with nu less than or equal to 1 and lambda > 1, runaway growth starts at t(crit) approximate to t(0), the time at which the coagulation equation solution becomes invalid. For A(ij) with nu > 1, which include all gravitational interaction cases expected to show runaway growth, the dependence of the time t(crit) for the onset of runaway growth on the parameters of the problem is not yet well understood, but there are indications that t(crit) (in units of 1/(n(0)A(11))) may decrease slowly toward zero with increasing initial total number of particles n(0). (C) 2000 Academic Press.