The Schwarzschild lens model is the simplest gravitational lens model. It treats the lensing object as a point mass in the lens plane, and always produces two images (although one image may be highly demagnified).

For the Schwarzschild lens, the deflection angle, , is:

where

*M*is the lens mass, is the impact parameter in the lens plane,*G*is the gravitational constant and*c*is the speed of light. The closer a light ray passes to the lens, the greater the deflection.If we use the dimensionless gravitational lens equation:

*y = x - α(x)*relating positions in the source plane,

*y*and lens plane*x*, the deflection angle for the Schwarzschild lens reduces to:*α(x) = 1/x*This equation can be inverted to give a simple quadratic equation, and hence (the two image locations) as a function of

*y*. We now have enough information to determine:- The location of the source, if we know the location of one or both of the images, using
*y = x - 1/x*; and - The location of both images, if we know the location of the source, using . The corresponding gravitational magnifications of the two images are .

For a point source which is directly in line with the observer and a Schwarzschild lens, the image will be a ring with radius:

called the Einstein ring or Einstein radius. The distances,

*D*, are angular diameter distances between the [O]bserver, [L]ens and [S]ource planes._{ij}

- The location of the source, if we know the location of one or both of the images, using