Dark Matter in the Universe by John N. Bahcall

Author:John N. Bahcall [Bahcall, John N.]
Language: eng
Format: epub
ISBN: 9789812388407
Google: OS-NvaVJxqsC
Published: 2011-01-12T15:05:20+00:00

Gravitational Lenses

121

Fig. 10.

Bending angle diagram and critical curves of an isothermal sphere with a core. The lens now has two critical curves C+ and C− located at θ+ and θ− in the lens plane. The C+ critical line is still due to the degenerate caustic at the origin of the source plane associated with the Einstein ring image. The C− critical line is located at the point where the line is tangent to the bending angle curve. At this point, the image becomes extended in the radial direction leading to two bright image straddling the C− critical line. For sources inside the caustic associated with C−, there are three images, and for sources outside, there is only one image.

3.5. Elliptical Potentials

If we introduce some ellipticity into the potential, the lens can generate five images, and the five image cross section exceeds the three image cross section at moderate amplification. Highly amplified images are no longer opposed when the mean amplification of the two brightest images ¯

A12 −1, where is the ellipticity of the potential. A simple argument shows that this is a general result. The time delay between two opposed images can be estimated using a Taylor expansion as ∆t ∼ Dθ2+/ ¯

A12

where θ+ is the radius of the C+ critical line. If the potential has ellipticity , an azimuthal perturbation ∆φ ∼ Dθ2+ will be imposed on the arrival time surface near the critical line. When ∆t ∼ ∆φ, the opposed images will be displaced leading to a five image geometry with the brightest two images allied. Numerical simulations verify this rule for a range of potentials. In general this means that highly amplified images will not have large separations on the sky; instead, pairs of bright images will closely straddle critical lines. The total cross section for producing an image with mean magnification of the two brightest images ¯

A12 > A is

σ( ¯

A12 > A) ∼ S+

(3.12)

A2

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