  We consider a family of isolated inhomogeneous steady states to the
gravitational Vlasov-Poisson system with a point mass at the centre. They are
parametrised by the polytropic index $k>1/2$, so that the phase space density
of the steady state is $C^1$ at the vacuum boundary if and only if $k>1$. We
prove the following sharp dichotomy result: if $k>1$ the linear perturbations
Landau damp and if $1/2< k\le1$ they do not.
  The above dichotomy is a new phenomenon and highlights the importance of
steady state regularity at the vacuum boundary in the discussion of long-time
behaviour of the perturbations. Our proof of (nonquantitative) gravitational
relaxation around steady states with $k>1$ is the first such result for the
gravitational Vlasov-Poisson system. The key step in the proof is to show that
no embedded eigenvalues exist in the essential spectrum of the linearised
system.
