Strong lensing cross sections for isothermal models - I. Finite source
  effects in the circular case
  The strong galaxy-galaxy lensing produces highly magnified and distorted
images of background galaxies in the form of arcs and Einstein rings.
Statistically, these effects are quantified, for example, in the number counts
of highly luminous sub-millimeter galaxies and of gravitational arcs. Two key
quantities to model these statistics are the magnification and the arc cross
sections. These are usually computed using either the circular infinitesimal
source approximation or ray-tracing simulations for sources of finite size. In
this work, we use an analytic solution for gravitational arcs to obtain these
cross sections as a function of image magnification and length-to-width ratio
in closed form, for finite sources. These analytical solutions provide simple
interpretations to the numerical results, can be employed to test the
computational codes, and can be used for fast a computation of the abundance of
distant sources and arcs. In this paper, the lens is modeled by a Singular
Isothermal Sphere, which is an excellent approximation to radial density
profile of Early-Type galaxies, and the sources are also axisymmetric. We
derive expressions for the geometrical properties of the images, such as the
area and several definitions of length and width. We obtain the magnification
cross section in exact form and derive a simple analytic approximation covering
the arc and Einstein ring regimes. The arc cross section is obtained down to
the formation of an Einstein ring and given in terms of elementary functions.
Perturbative expansions of these results are worked out, showing explicitly the
correction terms for finite sources.
