Figure 15

Figure 15: Lens map for the Schwarzschild metric. The observer is at $rO = 5m$ , the light sources are at $rS = 10m$ . $Θ$ is the colatitude on the observer’s sky and $Φ(Θ )$ is the angle swept out by the ray (see Figure 6). $Φ (Θ )$ was calculated with the help of Equation (87 ). $Θ$ is restricted by the opening angle $δ$ of the observer’s escape cone (see Figure 14). Rays with $Θ = δ$ asymptotically spiral towards the light sphere at $r = 3m$ . The first diagram (cf. [118], Figure 5) shows that $Φ (Θ )$ ranges from 0 to $∞$ if $Θ$ ranges from 0 to $δ$ . So there are infinitely many Einstein rings (dashed lines) whose angular radius approaches $δ$ . One can analytically prove [212, 246, 39] that the divergence of $Φ (Θ )$ for $Θ → δ$ is logarithmic. This is true whenever light rays approach an unstable light sphere [37]. The second diagram shows $Φ (Θ)$ over a logarithmic $Θ$ -axis. The graph of $Φ$ approaches a straight line which was called the “strong-field limit” by Bozza et al. [39, 37]. The picture illustrates that it is a good approximation for all light rays that make at least one full turn. The third diagram shows $cosΦ (Θ )$ over a logarithmic $Θ$ -axis. For every source position $0 < 𝜗 < π$ one can read the position of the images (dotted line). There are infinitely many, numbered by their order (89 ) that counts how often the light ray has crossed the axis. Images of odd order are on one side of the black hole, images of even order on the other. For the sources at $𝜗 = π$ and $𝜗 = 0$ one can read the positions of the Einstein rings.