Exact solutions that describe boson or fermion
stars have been found only numerically (in 3 + 1 dimensions). For
this reason there is no boson star model for which the lightlike
geodesics could be studied analytically. Numerical studies of
lensing have been carried out by Dabrowski and Schunck [70] for a transparent
spherically symmetric static maximal boson star, and by Bilić,
Nikolić, and Viollier [30
] for a transparent
spherically symmetric static maximal fermion star. For the case of
a fermion-fermion star (two components) see [171]. In all three articles
the authors use the “almost exact lens map” of Virbhadra and Ellis
(see Section 4.3) which is valid for observer and
light source in the asymptotic region and almost aligned. Dabrowski and Schunck [70
] also discuss how
the alignment assumption can be dropped. The lensing features found
in [70] for the boson star
and in [30] for the fermion
star have several similarities. In both cases, there is a
tangential caustic and a radial caustic (recall Figure 8
for terminology). A
(point) source on the tangential caustic (i.e., on the axis of
symmetry through the observer) is seen as a (1-dimenional) Einstein
ring plus a (point) image in the center. If the (point) source is
moved away from the axis the Einstein ring breaks into two (point)
images, so there are three images altogether. Two of them merge and
vanish if the radial caustic is crossed. So the qualitative lensing
features are quite different from a Schwarzschild black hole with
(theoretically) infinitely many images (see Section 5.1). The essential difference is
that in the case of a boson or fermion star there are no circular
lightlike geodesics towards which light rays could asymptotically
spiral.