Black hole images under spherical-shell and circular-annulus accretion models in Schwarzschild spacetime: a semianalytical approach

In the static and infalling spherical-shell models of optically thin accretion on Schwarzschild black hole, the formulas for the integrated intensities observed by a distant observer are derived, and by taking the monochromatic emission pattern with a $1/r^{2}$ radial profile as example, the black hole images for the spherical shell with different boundaries are plotted. For these BH images, the geometric and luminosity features are summarized, and the qualitative explanations of the luminosity variations between the static and infalling spherical-shell models are provided. A notable feature of the black hole image in the infalling spherical-shell model is that when the inner boundary of the spherical shell is far from the bound photon orbit, the observed luminosity near the exterior of the shadow is enhanced. The circular-annulus models of optically and geometrically thin accretion on Schwarzschild black hole are further explored. For a lightlike geodesic, the analytical forms of the transfer functions working for all impact parameter values are first given, and the redshift factors in the static, infalling, and rotating circular-annulus models are then deduced. With these results, in the three situations, the formulas for the integrated intensities observed by a distant observer viewing the circular annulus at an inclination angle are derived, and the corresponding black hole images for each emission pattern provided in Phys. Rev. D \textbf{100} (2019) 024018 are plotted. Finally, for the BH images of arbitrary order, the geometric and luminosity features are also summarized, and the qualitative explanations of the luminosity variations between different CA models are also given.