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2.3 The Singular Isothermal Sphere (SIS)

Galaxies and clusters of galaxies that act as gravitational lenses can be approximated by singular isothermal spheres. It is then easy to relate an angular scaling parameter xiE, referred to as the Einstein radius, to the mass inside the corresponding light cone. The Einstein radius xiE corresponds to the ring image of a point source aligned exactly on the axis of the lens (Fig. 7). For a singular isothermal sphere the line of sight velocity dispersion is constant as well as the gradient of the projected potential. Therefore, the deviation angle is constant and xiE can be obtained from the one dimensional velocity dispersion sigma1D as :

Equation 10 (10)
Equation 10


Equation 11 (11)
Equation 11

The perfect alignment of the source on the optical axis of the deflector is quite unlikely, but with a small misalignment the observer will see two opposite arcs approximately located at the Einstein radius (Fig. 7).

In fact, the total mass inside the radius xiE mostly depends on the Einstein radius. More generally, the mass inside the cone of light limited by a large arc will be a robust parameter that does not depend strongly on the actual mass distribution. Therefore, observations of arcs give the possibility of quickly deriving the total gravitational mass and the mass to light ratio of a cluster core without any spectroscopic determination of the velocity dispersion. It is worth noting that for real clusters both methods of mass determination give results which do not differ by more than 10% in the cluster center (Mellier 1993, Mellier et al. 1994).

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