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Most gravitational lenses have the standard configurations we illustrated in Section B.2. These configurations are easily understood in terms of the caustic structures generic to simple lens models. In this section we illustrate the origin of these basic geometries using simple mathematical examples. We build on the general outline of lensing theory from Part 1.

B.3.1. Some Nomenclature

Throughout this lecture we use comoving angular diameter distances (also known as proper motion distances) rather than the more familiar angular diameter distances because almost every equation in gravitational lensing becomes simpler. The distance between two redshifts i and j is

Equation 1 (B.1)

where OmegaM, OmegaLambda and Omegak = 1 - OmegaM - OmegaLambda are the present day matter density, cosmological constant and "curvature" density respectively, rH = c / H0 is the Hubble radius, and the function sinn(x) becomes sinh(x), x or sin(x) for open (Omegak > 0), flat (Omegak = 0) and closed (Omegak < 0) models (Carroll, Press & Turner [1992]). We use Dd, Ds and Dds for the distances from the observer to the lens, from the observer to the source and from the lens to the source. These distances are trivially related to the angular diameter distances, Dijang = Dij / (1 + zj), and luminosity distances, Dijlum = Dij(1 + zj). In a flat universe, one can simply add comoving angular diameter distances (Ds = Dd + Dds), which is not true of angular diameter distances. The comoving volume element is

Equation 2 (B.2)

for flat universes. We denote angles on the lens plane by vector{theta} = theta(coschi, sinchi) and angles on the source plane by vector{beta}. Physical lengths on the lens plane are vector{xi} = Ddang vector{theta}. The lensing potential, denoted by Psi(vector{theta}), satisfies the Poisson equation nabla2 Psi = 2kappa where kappa = Sigma / Sigmac is the surface density Sigma in units of the critical surface density Sigmac = c2(1 + zl) Ds / (4pi G Dd Dds). For a more detailed review of the basic physics, see Part 1.

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