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In last week's post, the idealistic case of gravitational lensing producing a perfect Einstein ring has been analysed. In this issue we will still keep it simple, but move the source slightly off the direct line to the deflector galaxy, while keeping it in the same observer-deflector plane. Figure 1 shows the resultant image for small angles, again highly exaggerated for clarity.
Figure 1: 
The angle alpha is obtained from last week's φ and α0 and the (new) offset angle β, shown here in the negative direction.(a)
In this particular scenario, the positive side do not have an image, because the deflector galaxy is taken as "solid" and hence not transparent. It's easy to spot that when the offset angle β=0, then α± reduces to ±α0.
Like in the past, let's stick in some values and obtain a feel for the magnitudes. Using the same lens and distances as last week (α0 = 1 arcsec, Dd=1 Gly and Ds=2 Gly) and taking β- = -2 arcsec, the lens-to-image angle comes out as: α- = -2.41 arcsec. (Hint, since eq. 3 contains only angles, one can work directly in arcsecs.)
Real gravitational lenses seldom produce such smooth Einstein rings, because the lens mass is normally not evenly distributed. In such cases, multiple images of distant objects are formed, either as arcs of incomplete Einstein rings, or in the case of quasars, multiple images. Shown below is a particularly good example of an "Einstein Cross" (G2237+030) formed by a lensed quasar.(b)
Figure 2: 
Quasars are the highly energetic cores of remote active galaxies and the most luminous objects in the universe, capable of radiating over a trillion times as much energy as the Sun from a region little larger than the Solar System.
Before gravitational lensing was well understood, astronomers were puzzled by the multiple images of essentially the same objects (with identical spectral properties). Today gravity's lenses are valuable tools in astronomy.
Black holes are also efficient gravitational lenses, but due to their relatively compact size, the rings and multiple images are not directly resolvable from Earth. However, when a black hole happens to wander across our line of sight to a distant star, the lensing effect may temporarily brighten the star's image by a factor 10 or more.
Such events are known as "micro-lensing" and are used to estimate the population of dark baryonic matter in our own galaxy.(c) It is also used to search for planets around visible stars - this is done by studying the variations of the star's light curve during a micro-lensing event.
This concludes this very brief summary of gravitational lensing. The floor is open for discussion, questions or other other inputs.
Notes:
(a) All units are SI and positive angles are counter-clockwise, as usual. Equations adapted from: George K. Francis et al.: http://new.math.uiuc.edu/superball/pb.pdf)
(b) Image credit: http://www.daviddarling.info/encyclopedia/E/Einstein_Cross.html
(c) Do not confuse "dark baryonic matter" with exotic "cold dark matter" of the non-baryonic variety, which presumably makes up ~20% of the mass of the universe. Baryonic matter is "ordinary" stuff (black holes, brown dwarfs etc.) that do not radiate enough in any e.m. band to be visible, making up some 10-20% of the total matter energy, Ωm. See the Friedmann Equation in Relativity 4 Engineers for a discussion on how Ω is made up.
-J
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