  The imaging by the Event Horizon Telescope (EHT) of the supermassive central
objects at the heart of the M87 and Milky Way (Sgr A$^\star$) galaxies, has
marked the first step into peering at the photon rings and central brightness
depression that characterize the optical appearance of black holes surrounded
by an accretion disk. Recently, Vagnozzi et. al. [S.~Vagnozzi, \textit{et al.}
arXiv:2205.07787 [gr-qc]] used the claim by the EHT that the size of the {\it
shadow} of Sgr A$^\star$ can be inferred by calibrated measurements of the
bright ring enclosing it, to constrain a large number of spherically symmetric
space-time geometries. In this work we use this result to study some features
of the first and second photon rings of a restricted pool of such geometries in
thin accretion disk settings. The emission profile of the latter is described
by calling upon three analytic samples belonging to the family introduced by
Gralla, Lupsasca and Marrone, in order to characterize such photon rings using
the Lyapunov exponent of nearly bound orbits and discuss its correlation with
the luminosity extinction rate between the first and second photon rings. We
finally elaborate on the chances of using such photon rings as observational
discriminators of alternative black hole geometries using very long baseline
interferometry.
