Black hole shadows, photon rings, and lensing rings

The presence of a bright ``photon ring'' surrounding a dark ``black hole shadow'' has been discussed as an important feature of the observational appearance of emission originating near a black hole. We clarify the meaning and relevance of these heuristics with analytic calculations and numerical toy models. The standard usage of the term ``shadow'' describes the appearance of a black hole illuminated from all directions, including from behind the observer. A backlit black hole casts a somewhat larger shadow. Neither ``shadow'' heuristic is particularly relevant to understanding the appearance of emission originating near the black hole, where the emission profile and gravitational redshift play the dominant roles in determining the observed size of the central dark area. A photon ring results from light rays that orbit around the black hole in the near-field region before escaping to infinity, where they arrive near a ring-shaped ``critical curve'' on the image plane. Although the brightness can become arbitrarily large near this critical curve in the case of optically thin emitting matter near the black hole, we show that the enhancement is only logarithmic, and hence is of no relevance to present observations. For optically thin emission from a geometrically thin or thick disk, photons that make only a fraction of an orbit will generically give rise to a much wider ``lensing ring,'' which is a demagnified image of the back of the disk, superimposed on top of the direct emission. For nearly face-on viewing, the lensing ring is centered at a radius $\ensuremath{\sim}5%$ larger than the photon ring and, depending on the details of the emission, its width is $\ensuremath{\sim}0.5--1M$ (where $M$ is the mass of the black hole). It can be relatively brighter by a factor of 2--3, as compared to the surrounding parts of the image, and thus could provide a significant feature in high-resolution images. Nevertheless, the characteristic features of the observed image are dominated by the location and properties of the emitting matter near the black hole. We comment on the recent M87* Event Horizon Telescope observations and mass measurement.