Optical appearance of Schwarzschild black holes with optically thin and thick accretion disks at various inclination angles

In this paper, we systematically investigate the optical appearance of a Schwarzschild black hole illuminated by three geometrically thin accretion disk models under varying observational inclination angles. Based on the geometric relationship between the black hole and observer, we first divide the accretion disk into co-side and counter-side semi-disks. We then analyze light ray trajectories, and calculate the total number of orbits and transfer functions for both semi-disks. The results reveal distinct inclination-dependence of lensed regions on different semi-disks: as inclination increases, the lensed region contracts for the counter-side semi-disk while expanding for the co-side one. Furthermore, through explicit specification of the emission profiles of the three models, we present optical images for both optically thin and thick disk scenarios at different inclinations. The results demonstrate that: (i) the bright rings in all three models become progressively compressed and deviate from circularity as inclination increases; (ii) for thick disks, partial rings are obscured and the overall intensity is lower than thin disks. These results may advance our understanding of general black hole imaging processes and provide a new approach to test gravitational theories through optical morphology studies.