On the Validity of the Coagulation Equation and the Nature of Runaway Growth

Abstract

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 Aij for gravitational interaction, with the power-law index of the mass-radius relation β=1/3 (for planetesimals) and 2/3 (for stars). We classify the mass dependence of Aij using the exponent λ for the merger between two particles of comparable mass, and the exponents μ and ν for the merger between a heavy particle and a light particle. For the two cases with λ≤1 and ν≤1, the mass spectrum evolves in an orderly fashion. For the remaining cases, which have ν>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 ν>1 cases are qualitatively different from the well-known example with Aij∝ij. For the latter case, which is in the range ν≤1 and λ>1, there is an analytic solution to the coagulation equation that is valid for a finite amount of time t0. We discuss a simplified merger problem that illustrates the qualitative differences in the solutions to the coagulation equation for the three classes of Aij. Our results strongly suggest that there are two types of runaway growth. For Aij with ν≤1 and λ>1, runaway growth starts at tcrit≈t0, the time at which the coagulation equation solution becomes invalid. For Aij with ν>1, which include all gravitational interaction cases expected to show runaway growth, the dependence of the time tcrit for the onset of runaway growth on the parameters of the problem is not yet well understood, but there are indications that tcrit (in units of 1/(n0A11)) may decrease slowly toward zero with increasing initial total number of particles n0. © 2000 Academic Press.