Figure 15: Lens map for the Schwarzschild
metric. The observer is at , the light sources are at . is the
colatitude on the observer’s sky and is the angle swept out
by the ray (see Figure 6).
was calculated with the help
of Equation (87). is
restricted by the opening angle
of the observer’s escape cone
(see Figure 14). Rays with asymptotically spiral
towards the light sphere at . The first diagram (cf. [119], Figure 5)
shows that ranges from 0
to if ranges from 0 to . So there are infinitely many
Einstein rings (dashed lines) whose
angular radius approaches . One can analytically prove [211, 246, 39] that
the divergence of for is logarithmic. This is true
whenever light rays approach an unstable light sphere [37]. The second
diagram shows over a
logarithmic -axis. The
graph of 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 over a logarithmic -axis. For every source
position 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
and one can read the positions of
the Einstein rings.
|