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 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 (mathcal{C}(r^{text {inner}}_{gamma })le mathcal{C}(r^{text {outer}}_{gamma })), where (mathcal{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 (mathcal{C}(r^{text {inner}}_{gamma })ge 1/3), horizonless ultra-compact objects may be characterized by the opposite dimensionless relation (mathcal{C}(r^{text {inner}}_{gamma })le 1/4). Our results provide a simple analytical explanation for the interesting numerical results that have recently presented by Novotný et al. (Phys Rev D 95:043009, 2017).