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 meanrate 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 powerlaw index of the massradius relation β=1/3 (for planetesimals) and 2/3 (for stars). We classify the mass dependence of A_{ij} 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 selfconsistent solutions to the coagulation equation at any time. The results for the ν>1 cases are qualitatively different from the wellknown example with A_{ij}∝ 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 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 ν≤1 and λ>1, runaway growth starts at t_{crit}≈ t_{0}, the time at which the coagulation equation solution becomes invalid. For A_{ij} with ν>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}.
 Publication:

Icarus
 Pub Date:
 January 2000
 DOI:
 10.1006/icar.1999.6239
 Bibcode:
 2000Icar..143...74L