Stability of Encounterless Spherical Stellar Systems

We show that a sufficient condition for stability for the spherical modes of a stellar system is that the distribution function $F\left(\epsilon \right)$ decrease with the energy $\epsilon$. For this demonstration we approximate the function $F\left(\epsilon \right)$ by a succession of steps corresponding to a multiple "water-bag" model. The total energy is shown to be an absolute minimum: The second variation $\delta W$ of the energy is always strictly positive and there is no marginal mode.