Einstein radius

When θE is expressed in radians, and the lensing source is sufficiently far away, the Einstein Radius, denoted RE, is given by Putting θS = 0 and solving for θ1 gives The Einstein angle for a point mass provides a convenient linear scale to make dimensionless lensing variables.

For a dense cluster with mass Mc ≈ 10×1015 M☉ at a distance of 1 Gigaparsec (1 Gpc) this radius could be as large as 100 arcsec (called macrolensing).

Likewise, for the lower ray of light reaching the observer from below the lens, we have and and thus The argument above can be extended for lenses which have a distributed mass, rather than a point mass, by using a different expression for the bend angle α the positions θI(θS) of the images can then be calculated.

For large deflections one can have multiple images and a non-invertible mapping: this is called strong lensing.

Note that in order for a distributed mass to result in an Einstein ring, it must be axially symmetric.