3.6 Lens mapping

In the vicinity of an arbitrary point, the lens mapping as shown in Equation (7View Equation) can be described by its Jacobian matrix π’œ:
( ) ( ) ∂-βƒ—β ( ∂-αi(βƒ—πœƒ-)) ( ∂2-ψ(βƒ—πœƒ)) π’œ = βƒ— = δij − ∂πœƒ = δij − ∂πœƒ ∂πœƒ . (29 ) ∂ πœƒ j i j
Here we made use of the fact (see [26Jump To The Next Citation Point164Jump To The Next Citation Point]), that the deflection angle can be expressed as the gradient of an effective two-dimensional scalar potential ψ: βƒ— ∇ πœƒψ = βƒ—α, where
DLS 2 ∫ ψ(βƒ—πœƒ) = D--D-- c2 Φ (βƒ—r)dz, (30 ) L S
and Φ(βƒ—r) is the Newtonian potential of the lens.

The determinant of the Jacobian π’œ is the inverse of the magnification:

1 μ = ------. (31 ) det π’œ
Let us define
2 -∂-ψ--- ψij = ∂πœƒi∂πœƒj. (32 )
The Laplacian of the effective potential ψ is twice the convergence:
ψ11 + ψ22 = 2κ = trψij. (33 )
With the definitions of the components of the external shear γ,
βƒ— 1- βƒ— βƒ— γ1(πœƒ) = 2(ψ11 − ψ22) = γ(πœƒ) cos[2φ (πœƒ)], (34 )
and
γ2(βƒ—πœƒ) = ψ12 = ψ21 = γ(βƒ—πœƒ) sin[2φ (βƒ—πœƒ)] (35 )
(where the angle φ reflects the direction of the shear-inducing tidal force relative to the coordinate system), the Jacobian matrix can be written
( ) ( ) ( ) 1 − κ − γ1 − γ2 1 0 cos2φ sin 2φ π’œ = |( − γ 1 − κ + γ |) = (1 − κ)|( 0 1 |) − γ |( sin 2φ − cos 2φ |) . (36 ) 2 1
The magnification can now be expressed as a function of the local convergence κ and the local shear γ:
−1 -----1------- μ = (detπ’œ ) = (1 − κ )2 − γ2 . (37 )
Locations at which det A = 0 have formally infinite magnification. They are called critical curves in the lens plane. The corresponding locations in the source plane are the caustics. For spherically symmetric mass distributions, the critical curves are circles. For a point lens, the caustic degenerates into a point. For elliptical lenses or spherically symmetric lenses plus external shear, the caustics can consist of cusps and folds. In Figure 4View Image the caustics and critical curves for an elliptical lens with a finite core are displayed.
View Image

Figure 4: The critical curves (upper panel) and caustics (lower panel) for an elliptical lens. The numbers in the right panels identify regions in the source plane that correspond to 1, 3 or 5 images, respectively. The smooth lines in the right hand panel are called fold caustics; the tips at which in the inner curve two fold caustics connect are called cusp caustics.

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