  For stationary, barotropic fluids in Newtonian gravity we give simple
criteria on the equation of state and the "law of motion" which guarantee
finite or infinite extent of the fluid region (providing a priori estimates for
the corresponding stationary Newton-Euler system). Under more restrictive
conditions, we can also exclude the presence of "hollow" configurations. Our
main result, which does not assume axial symmetry, uses the virial theorem as
the key ingredient and generalises a known result in the static case. In the
axially symmetric case stronger results are obtained and examples are
discussed.
