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

Abstract: 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. Subsequently, we analyze their differences in photon orbit distributions, transfer functions, and received intensities. Our results reveal distinct inclination-dependent behaviors between the two semi-disks: when the inclination approaches $\pi/2$, the co-side semi-disk develops richer ring structures with the lensed emission dominating the total brightness. 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.