  We present a new system of equations that fully characterizes adiabatic,
radial perturbations of perfect fluid stars within the theory of general
relativity. The properties of the system are discussed, and, provided that the
equilibrium spacetime verifies some general regularity conditions, analytical
solutions for the perturbation variables are found. As illustrative examples,
the results are applied to study perturbations of selected classical exact
spacetimes, and the first oscillation eigenfrequencies are computed. Exploiting
the new formalism, we derive an upper bound for the maximum compactness of
stable, perfect fluid stars, which is equation-of-state-agnostic and
significantly smaller than the Buchdahl bound.
