  The \lq\lq effective geometry" formalism is used to study the perturbations
of a perfect barotropic Newtonian self-gravitating rotating and compressible
fluid coupled with gravitational backreaction. The case of a uniformly rotating
polytrope with index $n=1$ is investigated, due to its analytical tractability.
Special attention is devoted to the geometrical properties of the underlying
background acoustic metric, focusing in particular on null geodesics as well as
on the analog light cone structure.
