Equilibrium states in galactic dynamics can be described as stationary solutions of the Vlasov–Poisson system, which is the non-relativistic case, or of the Vlasov–Einstein system, which is the relativistic case. To obtain spherically symmetric stationary solutions the distribution function of the particles (stars) on phase space is taken to be a function Φ(E, L) of the particle energy and angular momentum. We give a new condition on Φ which guarantees that the resulting steady state has finite mass and compact support both for the non-relativistic and the relativistic case. The condition is local in the sense that only the asymptotic behaviour of Φ for E → E0 needs to be prescribed, where E0 is a cut-off energy above which no particles exist.