Shadows of Kerr black holes with a Gaussian-distributed plasma in the polar direction

In this work, we study the shadows of Kerr black holes surrounded by an axisymmetric plasma, whose density takes a Gaussian distribution in the polar direction. Along the radial direction, we consider two models: In model A, the density of the plasma decays in a power law; in model B, the density obeys a logarithmic Gaussian distribution. Using the numerical backward ray-tracing method, we find that the size of the shadow is sensitive to the inclination angle of the observer due to the polar distribution of the density of the plasma. In particular, we pay special attention to model B and investigate the influence of the radial position of maximum density, the decay rate of the density toward the event horizon, and the opening angle of the plasma on the shape and size of the Kerr black hole shadows. The effects of the plasmas studied in this work can be qualitatively explained by taking the plasmas as convex lenses with the refractive index being less than 1.

Figure 1

Density contour map of a two-dimensional slice $\varphi =\text{const}$ for model A of the plasma with $N_{\max }=1$ and ${\xi }_{\theta }=\pi /9$. The horizontal and vertical coordinates are $\rho =r\sin \theta$ and $z=r\cos \theta$, respectively, where $\theta$ and $\varphi$ are polar and azimuthal angles, respectively, in the Boyer-Lindquist coordinate.

Figure 2

Density plots of the plasma for model B. The coordinates are the same as those in Fig. 1. $N_{\max }=1$, $\sigma =0.8$, $r_m=3$, and ${\xi }_{\theta }=\pi /9$.

Figure 3

A profile of a log-normal distribution with $r_m=3$ and $\sigma =0.512$. The width of the accretion is $d=8$.

Figure 4

A diagram of the Kerr black hole shadow curve, deformed by the plasma around the black hole. The shadow in this diagram is given with model B, and $k_B=0.8$, $r_m=5$, and $\sigma =0.18$.

Figure 5

The images of the Kerr black hole surrounded by plasma of model A. The spin is fixed at $a=0.998$ and $k_A=16$.

Figure 6

The variation of $\overline{R}/{\overline{R}}_{\text{Kerr}}$ with respect to the opening angle ${\xi }_{\theta }$. The spin is fixed at $a=0.998$ and $k_A=16$. Left: ${\theta }_o=\pi /2$; right: ${\theta }_o=17°$.

Figure 7

The images of the Kerr black hole surrounded by plasma of model A with different $k_A$. The inclination angle of the observer is fixed at ${\theta }_o=\pi /2$. The spin is fixed at $a=0.998$ and ${\xi }_{\theta }=0.36$.

Figure 8

The images of the Kerr black hole surrounded by plasma of model B for various $i$ and $r_m$. The inclination angle of the observer is ${\theta }_o=\pi /2$. The spin is $a=0.998$, and ${\xi }_{\theta }=0.36$ and $k_B=0.9$. The radial positions of maximum densities of each row from top to bottom are 2, 4, 6, 8, and 10, respectively.

Figure 9

The variation of $\overline{R}/{\overline{R}}_K$ with respect to $i$ and $r_m$. The inclination angle of the observer is fixed at ${\theta }_o=\pi /2$. The spin is fixed at $a=0.998$ and $k_B=0.8$.

Figure 10

Two diagrams of the light trajectories traveling on the slice of $\varphi =\text{const}$. For the sake of illustration qualitatively, we use circles instead of the actual isodensity lines of the plasma in Fig. 2 in the left plot. Since the plasma density becomes larger with the shrinking radius and the plasma’s refractive index is less than 1, the trajectory diverges after multiple refractions. In the right plot, we let the two isodensity lines have the same density $N_B/N_{\max }=0.1$ at a fixed $r_m$ with different $r_2$. Qualitatively, we can see that the photon would have a longer journey in the plasma of a wider plasma disk, which indicates it has a larger deflection angle when leaving the plasma.

Figure 11

A diagram of the convex lensing effect of plasma on the lights.

Figure 12

The images of the Kerr black hole surrounded by plasma of model B. The inclination angle of the observer is fixed at ${\theta }_o=\pi /2$, and the spin is fixed at $a=0.998$. $\sigma =0.265$ and $k_B=0.8$.

Figure 13

The images of the Kerr black hole surrounded by plasma of model B. The inclination angle of the observer is fixed at ${\theta }_o=17°$, and the spin is fixed at $a=0.998$. $\sigma =0.265$ and $k_B=0.8$.