# Shadow of a Schwarzschild black hole surrounded by a Bach-Weyl ring

## TL;DR

This paper investigates how a Bach-Weyl ring influences the shadow of a Schwarzschild black hole, revealing chaotic photon dynamics, fractal structures, and the dependence of shadow shape on system parameters.

## Contribution

It introduces the study of black hole shadows affected by a Bach-Weyl ring, highlighting non-integrable photon motion and chaotic lensing effects.

## Key findings

- Chaotic photon trajectories cause complex shadow shapes.
- Fractal structures emerge in the black hole shadow.
- Shadow size and shape depend on ring mass and radius.

## Abstract

We have studied the shadows of a Schwarzschild black hole surrounded by a Bach-Weyl ring through the backward ray-tracing method. The presence of Bach-Weyl ring leads to that the photon dynamical system is non-integrable and then chaos would appear in the photon motion, which affects sharply the black hole shadow. The size and shape the black hole shadow depend on the black hole parameter, the Bach-Weyl ring mass and the Weyl radius between black hole and ring. Some self-similar fractal structures also appear in the black hole shadow, which originates from the chaotic lensing. We also study the change of the image of Bach-Weyl ring with the ring mass and the Weyl radius. Finally, we analyze the invariant manifolds of Lyapunov orbits near the fixed points and discuss further the formation of the shadow of a Schwarzschild black hole with Bach-Weyl ring.