Exploring the role of accretion disk geometry in shaping black hole shadows

TL;DR

This paper investigates how geometrically thick accretion-disk geometries in Schwarzschild spacetime shape black hole shadows by decomposing the emitting region into radial one-dimensional segments and mapping their trajectories through transfer functions that capture direct, lensing, and photon-ring features. Across optically thin, optically thick, and partially optically thick regimes, it shows that disk thickness broadens the lensing ring and that the photon ring remains a narrow, boundary-defining feature, with its visibility governed by geometry and the first-intersection criterion in thick disks. A simple radiative-transfer model with a constant absorption coefficient $\oldsymbol{\chi}$ reveals a transition from thin to thick appearance around $\oldsymbol{\chi \sim (6M\psi_0)^{-1}}$, highlighting how absorption suppresses high-deflection paths. The framework provides a coherent, Lensing-ring–driven interpretation of high-resolution black hole images and can be extended to other spherically symmetric spacetimes or modified gravity scenarios.

Abstract

We study black hole imaging in the context of geometrically thick accretion disks in Schwarzschild spacetime. By decomposing the emitting region into a set of one-dimensional luminous segments, each characterized by its inclination angle and inner radius, we construct transfer functions that capture key image features-namely, the direct image, lensing ring, and photon ring. This approach allows a unified treatment of disk geometry and viewing angle. We explore three regimes: optically thin, optically thick, and partially optically thick disks. For optically thin flows, increasing the disk thickness (characterized by the half-opening angle $ψ_0$) broadens the lensing ring, gradually bridging the photon ring and the direct image. The photon ring remains narrow, but its position robustly defines the innermost edge of the lensing structure. In the optically thick case, image features are primarily determined by the first intersection of traced light rays with the disk, and we provide analytical criteria for the presence of lensing and photon rings based on the critical deflection angles. For partially optically thick disks, we adopt a simplified radiative transport model and find a critical absorption coefficient $χ\sim (6Mψ_0)^{-1}$ beyond which the image rapidly transitions from an optically thin- to thick-disk appearance. These results help clarify the respective roles of the photon and lensing rings across different disk configurations, and may offer a useful framework for interpreting future high-resolution black hole observations.

Figures (17)

• Figure 1: Schematic meridional cross-sectional view ($x-z$ plane) of a geometrically thick accretion disk. The disk is symmetric about the equatorial plane and extends outward from an inner radius $r_{\mathrm{in}}$. In general, the relative thickness is characterized by $h/r$. In this paper, we consider a simplified scenario where the thickness is described by an half-opening angle $\psi_0$ at $r_{\mathrm{in}}$.
• Figure 2: Coordinate systems for a photon trajectory (red solid curve) intersecting with a thin accretion disk. The Cartesian coordinates $(x, y, z)$ are related to the spatial spherical coordinates $(r, \theta, \phi)$ via the standard coordinate transformation. The observer $O'$ is located at a radius $r_{\rm obs} \gg M$ (in numerical calculations, we set $r_{\rm obs} = 10^5 M$). The accretion disk lies in the $z=0$ plane with an inclination angle $\theta_0$. The observer's local frame $(x', y', z')$ is obtained by first rotating the $(x, y, z)$ frame clockwise around the $y$-axis by an angle $\pi/2-\theta_0$, and then translating the origin to $O'$. From the observer's perspective, light rays emitted from points $P_1$ and $P_2$ appear to originate from a single point $P'$, whose projection onto the observer's image plane $y'$-$z'$ is denoted by $P$. The position of $P$ on the image plane is characterized by polar coordinates $(b, \alpha)$, where $b$ is the impact parameter and $\alpha$ is the angle between the projection of the photon trajectory onto the image plane and the $y'$-axis. $\beta$ is defined as the angle between the observer-black hole axis ($\overline{OO'}$) and the incident photon trajectory. $\varphi$ denotes the angle between the observer-black hole axis and the line connecting the black hole center to the photon emission point $P_1$.
• Figure 3: Sketch of one-dimensional luminous segments lying in the photon's orbital plane. The colorful thick solid lines depict representative emitting segments. Thin black lines illustrate representative photon trajectories, while the dashed line indicates the observer-black hole axis.
• Figure 4: The first three transfer functions for luminous segments at different inclination angles. Top left: Selected inclination angles. Top right, bottom left, and bottom right: First, second, and third transfer functions over a broad range of inclination angles, respectively.
• Figure 5: Observed flux from one-dimensional luminous segments with an inner radius of $r_{\rm in} = 6M$. The left panel corresponds to the optically thin case, while the right panel shows the optically thick case. These image lines correspond to projections of certain thin accretion disk structures onto the observer's image plane. Specifically, $\varphi_0 = 90^\circ$ corresponds to the $\alpha = {\pi}/{2}$ line (positive $z'-$axis) in the image of a face-on disk with inclination angle $\theta_0 = 0$; $\varphi_0 = 40^\circ$ and $\varphi_0 = 140^\circ$ correspond to the $\alpha = {3\pi}/{2}$(negative $z'-$axis) and $\alpha = {\pi}/{2}$ lines, respectively, in the image of a disk with inclination angle $\theta_0 = 50^\circ$. All disks assume $\kappa_{\rm ff} = 0.5$ and $\kappa_{\rm K} = 0.1$.