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.1) |
where _{M}, _{} and _{k} = 1 - _{M} - _{} are the present day matter density, cosmological constant and "curvature" density respectively, r_{H} = c / H_{0} is the Hubble radius, and the function sinn(x) becomes sinh(x), x or sin(x) for open (_{k} > 0), flat (_{k} = 0) and closed (_{k} < 0) models (Carroll, Press & Turner [1992]). We use D_{d}, D_{s} and D_{ds} 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, D_{ij}^{ang} = D_{ij} / (1 + z_{j}), and luminosity distances, D_{ij}^{lum} = D_{ij}(1 + z_{j}). In a flat universe, one can simply add comoving angular diameter distances (D_{s} = D_{d} + D_{ds}), which is not true of angular diameter distances. The comoving volume element is
(B.2) |
for flat universes. We denote angles on the lens plane by = (cos, sin) and angles on the source plane by . Physical lengths on the lens plane are = D_{d}^{ang} . The lensing potential, denoted by (), satisfies the Poisson equation ^{2} = 2 where = / _{c} is the surface density in units of the critical surface density _{c} = c^{2}(1 + z_{l}) D_{s} / (4 G D_{d} D_{ds}). For a more detailed review of the basic physics, see Part 1.