Figure 17: Luminosity distance and ellipticity (image distortion) in
the Schwarzschild spacetime. The data
are the same as in Figures 15 and 16. If point sources of equal bolometric luminosity are distributed at
, the plotted function
gives
their magnitude on the observer’s sky, modulo
an additive constant . For
the calculation of one
needs and (see Figure 16), and the
general relations (41)
and (48). This procedure
follows [84] (cf. [85, 119]). For source
and observer at large radius, related calculations can also be found in [211, 246, 201, 337]. Einstein
rings have magnitude in the
ray-optical treatment. For a light
source not on the axis, the image of order is fainter than the
image of order by magnitudes,
see [211, 246]. (This is
strictly true in the “strong-field
limit”, or “strong-bending limit”, which is explained in the
caption of Figure 15.) The above picture
is similar to Figure 6
in [246]. Note that it
refers to point sources and not to a radiating
spherical surface of
constant surface brightness; by Equation (54), the latter would
show a constant intensity. The lower part of the diagram
illustrates image distortion in terms
of .
Clearly, is infinite at
each Einstein ring. The double-logarithmic representation shows that beyond the second
Einstein ring all images are extremely elongated in the tangential direction, . Image distortion in the
Schwarzschild spacetime is also treated in [85, 121, 120], an
approximation formula is derived in [240].
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