Algorithms for generating and analyzing black hole images based on the analytical equations of lightlike geodesics

The analytical orbit equation of the lightlike geodesics through any fixed spatial point in Schwarzschild spacetime is explicitly presented in terms of the impact parameter, and with the result, the algorithms for generating and analyzing the black hole images under various accretion models are constructed. For the static and infalling spherical shell models of optically thin accretion, the integrated intensity observed by a distant observer is derived based on the backward ray tracing method. 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, and then, the geometrical features and luminosity variations of these images are summarized. A notable feature of the black hole images for the infalling spherical shell model is that when the inner boundary of the spherical shell is sufficiently large, the doppler effects can strengthen the luminosity in the small adjacent region outside the shadow. The circular annulus models of optically and geometrically thin accretion are further explored. For a lightlike geodesic, the transfer functions establishing the connection between the emitting and receiving points are first provided in the algebraic manner, and the redshift factors in the static, infalling, and rotating circular annulus models are then deduced. With these results, in the three models, 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 furnished in Phys. Rev. D \textbf{100} (2019) 024018 are plotted. Finally, the geometrical features and the luminosity variations of these images are also summarized.