THE SPHERICALLY SYMMETRIC GRAVITATIONAL COLLAPSE OF A CLUMP OF SOLIDS IN A GAS

Abstract

In the subject of planetesimal formation, several mechanisms have been identified that create dense particle clumps in the solar nebula. The present work is concerned with the gravitational collapse of such clumps, idealized as being spherically symmetric. Fully nonlinear simulations using the two-fluid model are carried out (almost) up to the time when a central density singularity forms. We refer to this as the collapse time. The end result of the study is a parametrization of the collapse time, in order that it may be compared with timescales for various disruptive effects to which clumps may be subject in a particular situation. An important effect that determines the collapse time is that as the clump compresses, it also compresses the gas due to drag. This increases gas pressure, which retards particle collapse and can lead to oscillation in the size and density of the clump. In the limit of particles perfectly coupled to the gas, the characteristic ratio of gravitational force to gas pressure becomes relevant and defines a two-phase Jeans parameter, Jt, which is the classical Jeans parameter with the speed of sound replaced by an effective wave speed in the coupled two-fluid medium. The parameter Jt remains useful even away from the perfect coupling limit because it makes the simulation results insensitive to the initial density ratio of particles to gas (Φ0) as a separate parameter. A simple ordinary differential equation model is developed. It takes the form of two coupled non-linear oscillators and reproduces key features of the simulations. Finally, a parametric study of the time to collapse is performed and a formula (fit to the simulations) is developed. In the incompressible limit Jt → 0, collapse time equals the self-sedimentation time, which is inversely proportional to the Stokes number. As Jt increases, the collapse time decreases with Jt and eventually becomes approximately equal to the dynamical time. Values of collapse time versus clump size are given for a minimum-mass solar nebula. Finally, the timescale of clump erosion due to turbulent strain is estimated.