Analytic study of self-gravitating polytropic spheres with light rings

Ultra-compact objects describe horizonless solutions of the Einstein field equations which, like black-hole spacetimes, possess null circular geodesics (closed light rings). We study {\it analytically} the physical properties of spherically symmetric ultra-compact isotropic fluid spheres with a polytropic equation of state. It is shown that these spatially regular horizonless spacetimes are generally characterized by two light rings $\{r^{\text{inner}}_{\gamma},r^{\text{outer}}_{\gamma}\}$ with the property ${\cal C}(r^{\text{inner}}_{\gamma})\leq{\cal C}(r^{\text{outer}}_{\gamma})$, where ${\cal C}\equiv m(r)/r$ is the dimensionless compactness parameter of the self-gravitating matter configurations. In particular, we prove that, while black-hole spacetimes are characterized by the lower bound ${\cal C}(r^{\text{inner}}_{\gamma})\geq1/3$, horizonless ultra-compact objects may be characterized by the opposite dimensionless relation ${\cal C}(r^{\text{inner}}_{\gamma})\leq1/4$. Our results provide a simple analytical explanation for the interesting numerical results that have recently presented by Novotn\'y, Hlad\'ik, and Stuchl\'ik [Phys. Rev. D 95, 043009 (2017)].