Equatorial timelike circular orbits around generic ultracompact objects

For a stationary, axisymmetric, asymptotically flat, ultra-compact [$i.e.$ containing light-rings (LRs)] object, with $\mathbb{Z}_2$ north-south symmetry fixing an equatorial plane, we establish that the structure of timelike circular orbits (TCOs) in vicinity LRs, for either rotation direction, depends exclusively on radial stability LRs. Thus, unstable LR delimits region TCOs (no TCOs) radially above (below) it; stable below (above) it. Corollaries are discussed both horizonless objects and black holes. We illustrate these results variety exotic stars examples non-Kerr holes, which also compute efficiency associated converting gravitational energy into radiation by material particle falling under adiabatic sequence TCOs. most studied, it is possible to obtain efficiencies larger than maximal Kerr $i.e.$ $42\%$.