An energy principle for the stability of charged dust cloud

The stability of a charged dust cloud embedded in a quasineutral plasma background is studied using an energy principle. In this equilibrium, the electric fields arising due to the inhomogeneous dust distribution within the plasma background balances its self-gravity. The perturbed equation for the normal modes is obtained in terms of the Lagrangian displacement of the dust fluid $ξ⃗$ with a Hermitian operator. Hence, the square of the eigenvalue is real. The total potential energy (gravitational and electrostatic) integral is expressed as a quadratic in $ξ⃗$⁠. The compression due to the electrostatic field (or equivalently the acoustic modes) and the gravity due to the plasma background is shown to make a stabilizing contribution, while the gravity of the dust makes a destabilizing contribution to the potential energy integral. The stabilizing contribution due to the plasma background, which is missed in analysis based on Jean’s swindle, is a new effect and appears in our stability analysis due to a proper treatment of the equilibrium. This stability theory is applied to the study of the gravitational stability of one-dimensional spherically symmetric dust clouds. The equation for the radial oscillations is shown to be of the Sturm-Liouville form; hence the stability of the fundamental mode decides the overall stability of the configuration. A simple trial function is used to obtain the eigenvalue of the fundamental mode. The low dust density branch of the equilibrium is shown to be robustly stable mainly due to the electric field compression. At high dust densities, the charge reduction effects weaken the field compression. This destabilizes the fundamental mode and the over all configuration. A spherically symmetric collapse of the dust cloud sets in as the eigenvalue of the fundamental mode crosses zero. The radial oscillations of the low density dust cloud, the collapse at high densities, and the resulting mass limit has been predicted earlier and seen in recent numerical simulations.