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 ≈π times 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 20~\ per\ cent 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 30~\ per\ cent 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 20~\ per\ cent. It will soon become possible to robustly learn more about the space–time geometry of astrophysical BHs from such measurements.