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^inner_γ, r^outer_γ} with the property C(r^inner_γ) ≤ C(r^outer_γ), where C ≡ 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 C(r^inner_γ) ≥ 1/3, horizonless ultra-compact objects may be characterized by the opposite dimensionless relation C(r^inner_γ) ≤ 1/4. Our results provide a simple analytical explanation for the interesting numerical results that have recently presented by Novotny et al. (Phys Rev D 95:043009, 2017). (orig.)