Photon ring signatures as tests for alternative spherically symmetric geometries with thin accretion disks

The imaging by the Event Horizon Telescope (EHT) of the supermassive central objects at the heart of the M87 and Milky Way (Sgr A∗) galaxies, has marked the first step into peering towards the shadow and photon ring structures that characterize the optical appearance of black holes. Recently, Vagnozzi et. al. [S. Vagnozzi, et al. arXiv:2205.07787 [gr-qc]], used the claim by the EHT that the size of the shadow of Sgr A∗ can be inferred by calibrated measurements of the bright photon ring, to constrain a large number of spacetime geometries. In this work we use this observation to study the first and second photon rings of a restricted pool of such geometries in thin accretion disk settings, since they are expected to be decreasingly dependent on the assumptions of the astrophysical accretion flow. The latter is described by calling upon three analytic intensity profiles introduced by Gralla, Lupsasca and Marrone in order to study the properties of such photon rings in relation to the Lyapunov exponents of nearly bound orbits, and discuss its correlation with the extinction rate between the first and second photon rings in such emission models. We finally elaborate on the chances of using such photon rings as observational discriminators of alternative black hole geometries using very long baseline interferometry.