Hotspots and photon rings in spherically symmetric space-times

Future black hole (BH) imaging observations are expected to resolve finer features corresponding to higher order images of hotspots and of the horizon-scale accretion flow. In spherical space-times, the image order is determined by the number of half-loops executed by the photons that form it. Consecutive-order images arrive approximately after a delay time of ≈πtimes the BH shadow radius. The fractional diameters, widths, and flux-densities of consecutive-order images are exponentially demagnified by the lensing Lyapunov exponent, a characteristic of the space-time. The appearance of a simple point-sized hotspot when located at fixed spatial locations or in motion on circular orbits is investigated. The exact time delay between the appearance of its zeroth and first-order images agrees with our analytic estimate, which accounts for the observer inclination, with error for hotspots located about ≲ 5M from a Schwarzschild BH of mass M. Since M87∗ and Sgr A∗ host geometrically thick accretion flows, we also explore the variation in the diameters and widths of their first-order images with disc scale-height. Using a simple 'conical torus' model, for realistic morphologies, we estimate the first-order image diameter to deviate from that of the shadow by and its width to be ≲ 1.3M. Finally, the error in recovering the Schwarzschild lensing exponent (π), when using the diameters or the widths of the first and second-order images is estimated to be. It will soon become possible to robustly learn more about the space-time geometry of astrophysical BHs from such measurements.