Hotspots and Photon Rings in Schwarzschild Black Hole Spacetimes

Future black hole (BH) imaging observations with better sensitivity and angular resolution are expected to resolve finer features corresponding to higher-order images of both hotspots that are produced in the accretion flow, as well as of the entire emitting region. In spherically symmetric spacetimes, the image order is determined by the maximum number of half-loops executed around the BH by the photons that form it. Due to the additional half-loop, consecutive-order images arrive after a delay time of approximately π times the BH shadow radius. Furthermore, the deviation of the diameters from that of the shadow, widths, and flux-densities of consecutive-order images are exponentially demagnified by the lensing Lyapunov exponent, a characteristic of the spacetime. We compare the exact time delay between the appearance of the zeroth and first-order images of a hotspot to our best analytic estimate and find an error ≲ 50% for hotspot locations within ≈ 10M from a Schwarzschild BH of mass M. We also explore the possibilities for inferring kinetic properties of hotspots, and also our inclination, from future BH movies. Furthermore, since the targets of such observations host geometrically-thick accretion flows (also jets), we obtain simple theoretical estimates of the variation in the diameters and widths of their first-order images, for varying disk scale-heights. We find realistically that the deviation of the former from that of the shadow is ≲ 30% and that the latter remain ≲ 1.3M. Finally, we estimate the error in recovering the lensing exponent, when using the first and second-order images, to be ≲ 20%. This provides further evidence that future observations could yield new and independent estimates of the shadow size and, in principle, of the lensing exponent, allowing us to robustly learn about the spacetimes of astrophysical BHs.