The Jeans’ criterion and the gravitational instability

Abstract

In this paper, we derive a Jeans’ criterion for the dynamical spherical collapse of a gravitational polytropic medium with constant γ. In opposition to the classical (static configuration) Jeans’ criterion where a threshold for stability appears, it is shown that the stability of the dynamical solution depends on the value of γ. For γ=4/3, the existence of a threshold is also obtained. However, for γ≠4/3, the system is always unstable with an instability growing according to a time-power, instead of the usual exponential law. In addition, two types of instabilities are obtained: the large wavelengths are unstable from the very beginning of the evolution, whereas the small ones, produce oscillations on a finite range of time and, finally, the system becomes unstable (oscillations stop). In this stage the pressure can be neglected. Since the collapse is unstable, we examine the fragmentation problem. The dynamical Jeans’ criterion suggests a cascade in which the mass M and the radius R of the pieces are connected by a relation of the type M∞R ².