Hotspots and Photon Rings in Spherically-Symmetric Spacetimes

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 spacetimes, 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 $\approx\pi$ 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 spacetime. 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 $\lesssim 20\%$ error for hotspots located about $\lesssim 5M$ from a Schwarzschild BH of mass $M$. Since M87$^\star$ and Sgr A$^\star$ host geometrically-thick accretion flows, we also explore the variation in the diameters and widths of their first-order images with disk 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 $\lesssim 30\%$ and its width to be $\lesssim 1.3M$. Finally, the error in recovering the Schwarzschild lensing exponent ($\pi$), when using the diameters or the widths of the first and second-order images is estimated to be $\lesssim 20\%$. It will soon become possible to robustly learn more about the spacetime geometry of astrophysical BHs from such measurements.