The polytropic hydrodynamic vortex describes an effective $(2+1)$-dimensional acoustic spacetime with an inner reflecting boundary at $r=r_{ \text{ c}}$. This physical system, like the spinning Kerr black hole, possesses an ergoregion of radius $r_{ \text{ e}}$ and an inner non-pointlike curvature singularity of radius $r_{ \text{ s}}$. Interestingly, the fundamental ratio $r_{ \text{ e}}/r_{ \text{ s}}$ which characterizes the effective geometry is determined solely by the dimensionless polytropic index $N_{ \text{ p}}$ of the circulating fluid. It has recently been proved that, in the $N_{ \text{ p}}=0$ case, the effective acoustic spacetime is characterized by an { \it infinite} countable set of reflecting surface radii, $\{ r_{ \text{ c}}(N_{ \text{ p}};n)\}^{ n=\infty}_{ n=1}$, that can support static (marginally-stable) sound modes. In the present paper we use { \it analytical} techniques in order to explore the physical properties of the polytropic hydrodynamic vortex in the $N_{ \text{ p}}>0$ regime. In particular, we prove that in this physical regime, the effective acoustic spacetime is characterized by a { \it finite} discrete set of reflecting surface radii, $\{ r_{ \text{ c}}(N_{ \text{ p}},m;n)\}^{ n=N_{ \text{ max}}}_{ n=1}$, that can support the marginally-stable static sound modes (here $m$ is the azimuthal harmonic index of the acoustic perturbation field). Interestingly, it is proved analytically that the dimensionless outermost supporting radius $r^{ \text{ max}}_{ \text{ c}}/r_{ \text{ e}}$, which marks the onset of superradiant instabilities in the polytropic hydrodynamic vortex, increases monotonically with increasing values of the integer harmonic index $m$ and decreasing values of the dimensionless polytropic index $N_{ \text{ p}}$.
Categories: gr-qc astro-ph.HE hep-th