  The hydrodynamic evolution of self-gravitating gaseous stars is governed by
the Euler-Poisson equations. We study the structure of the linear approximation
of barotropic perturbations around spherically symmetric equilibria based on
functional analytic tools. In contrast to folklore, we show that the spectrum
of the linearized operator for general perturbations is not of the
Sturm-Liouville type unless the perturbations are restricted. In particular, we
prove that it is of the Sturm-Liouville type for irrotational perturbations.
