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

TL;DR

This work analyzes the optical appearance of a Schwarzschild black hole illuminated by three geometrically thin accretion-disk models across varying inclinations, introducing a co-side/counter-side semi-disk decomposition to map disk emission to the observer plane. It develops photon-trajectory classifications and transfer functions that relate disk intersections to observer-plane coordinates, then computes observed intensities for optically thin and thick disks using distinct emission profiles. The key findings are that lensed emission regions expand on the co-side disk and contract on the counter-side with increasing inclination, while bright rings become more distorted; thick disks obscure partial rings and generally reduce image brightness. The results advance BH imaging studies and offer a new diagnostic pathway for testing gravitational theories through optical morphology, with potential extensions to more general disk geometries and rotating (Kerr) spacetimes.

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. 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.

Figures (18)

• Figure 1: The photon trajectories around a Schwarzschild BH. The black curves represent trajectories with impact parameters $b<b_c$, the red curve represents the photon trajectory corresponding to the critical impact parameter $b_c$, and the green curves depict trajectories with $b>b_c$. The black disk represents the BH.
• Figure 2: The schematic diagram of the observer and BH coordinate systems. The BH center is located at point $O$, and the observer is positioned at point $O'$. The viewing inclination is denoted by $\omega$. A photon emitted from point $P$ on the accretion disk arrives at point $P'$ on the observer's plane, undergoing a total azimuthal angle change $\angle POM \equiv \phi$, with $\eta$ representing the polar angle in the observer's plane.
• Figure 3: Schematic diagram of the co-side and counter-side semi-disks, where $O$ and $O'$ denote the positions of the BH and the observer, respectively. The dashed lines $NY"$ and $O'Y'$ represent the intersections of the plane $OO'Y'$ with the planes $XOY"$ and $X'O'Y'$ from Fig. \ref{['fig_yanshinew']}, respectively.
• Figure 4: Schematic of geometric relations in the photon trajectory plane, where $O$ is the BH position and $O'$ corresponds to the center of the observer’s plane. The blue dashed line represents the accretion disk, while the green dashed line indicates the plane $XOY"$.
• Figure 5: Light trajectories from a distant right-side observer with inclination $\omega$, intersecting the counter-side semi-disk (left) and co-side semi-disk (right) of a Schwarzschild BH. The black, orange, and red curves correspond to direct, lensed, and photon ring trajectories, respectively. The blue dashed curve depicts the accretion disk, while the green dashed line represents the $XOY"$ plane.