and Cosmology

2. The Milky Way as a Galaxy
68
face brightness (or specific intensity) is preserved. The
flux of a source is given as the product of surface brightness
and solid angle. Since the former of the two factors
is unchanged by light deflection, but the solid angle
changes, the observed flux of the source is modified. If
S 0 is the flux of the unlensed source and S the flux of
an image of the source, then
μ := S S 0
= ω ω s
(2.82)
describes the change in flux that is caused by a magnification
(or a diminution) of the image of a source. Obviously,
the magnification is a purely geometrical effect.
Magnification for “Small” Sources. For sources and
images that are much smaller than the characteristic
scale of the lens, the magnification μ is given by the
differential area distortion of the lens mapping (2.77),
( )∣ μ =
∂β ∣∣∣
−1
( )∣
∣ det ≡
∂θ ∣ det ∂βi ∣∣∣
−1
∂θ j
. (2.83)
have magnification either larger or less than unity, depending
on y. The magnification of the two images is
illustrated in Fig. 2.24, while Fig. 2.25 shows the magnification
for several different source positions y. For
y ≫ 1, one has μ + 1andμ − ∼ 0, from which we
draw the following conclusion: if the source and lens
are not sufficiently well aligned, the secondary image is
strongly demagnified and the primary image has magnification
very close to unity. For this reason, situations
with y ≫ 1 are of little relevance since then essentially
only one image is observed which has about the same
flux as the unlensed source.
For y → 0, the two magnifications diverge,
μ ± →∞. The reason for this is purely geometric: in
this case, out of a zero-dimensional point source a onedimensional
image, the Einstein ring, is formed. This
divergence is not physical, of course, since infinite magnifications
do not occur in reality. The magnifications
remain finite even for y = 0, for two reasons. First, real
sources have a finite extent, and for these the magnifi-
Hence for small sources, the ratio of solid angles of the
lensed image and the unlensed source is described by
the determinant of the local Jacobi matrix. 11
The magnification can therefore be calculated for
each individual image of the source, and the total magnification
of a source, given by the ratio of the sum of
the fluxes of the individual images and the flux of the
unlensed source, is the sum of the magnifications for
the individual images.
Magnification for the Point-Mass Lens. For a pointmass
lens, the magnifications for the two images (2.81)
are
μ ± = 1 4
(
√
y
√
y2 + 4 + y2 + 4
± 2
y
)
. (2.84)
From this it follows that for the “+”-image μ + > 1for
all source positions y =|y|, whereas the “−”-image can
11 The determinant in (2.83) is a generalization of the derivative in one
spatial dimension to higher dimensional mappings. Consider a scalar
mapping y = y(x); through this mapping, a “small” interval Δx is
mapped onto a small interval Δy, whereΔy ≈ (dy/dx) Δx. The
Jacobian determinant occurring in (2.83) generalizes this result for
a two-dimensional mapping from the lens plane to the source plane.
Fig. 2.24. Illustration of the lens mapping by a point mass M.
The unlensed source S and the two images I 1 and I 2 of the
lensed source are shown. We see that the two images have
a solid angle different from the unlensed source, and they also
have a different shape. The dashed circle shows the Einstein
radius of the lens