The Einstein radius is the radius of an Einstein ring, and is a characteristic angle for gravitational lensing in general, as typical distances between images in gravitational lensing are of the order of the Einstein radius.
In the following derivation of the Einstein radius, we will assume that all of mass M of the lensing galaxy L is concentrated in the center of the galaxy.
For a point mass the deflection can be calculated and is one of the classical tests of general relativity. For small angles ñ<sub>1</sub> the total deflection by a point mass M is given (see Schwarzschild metric) by
where
By noting that, for small angles and with the angle expressed in radians, the point of nearest approach b<sub>1</sub> at an angle ø<sub>1</sub> for the lens L on a distance D<sub>L</sub> is given by , we can re-express the bending angle ñ<sub>1</sub> as
If we set ø<sub>S</sub> as the angle at which one would see the source without the lens (which is generally not observable), and ø<sub>1</sub> as the observed angle of the image of the source with respect to the lens, then one can see from the geometry of lensing (counting distances in the source plane) that the vertical distance spanned by the angle ø<sub>1</sub> at a distance D<sub>S</sub> is the same as the sum of the two vertical distances and . This gives the lens equation
which can be rearranged to give
By setting (eq. 1) equal to (eq. 2), and rearranging, we get
For a source right behind the lens, , the lens equation for a point mass gives a characteristic value for ø<sub>1</sub> that is called the Einstein angle, denoted ø<sub>E</sub>. When ø<sub>E</sub> is expressed in radians, and the lensing source is sufficiently far away, the Einstein Radius, denoted R<sub>E</sub>, is given by
Putting and solving for ø<sub>1</sub> gives
The Einstein angle for a point mass provides a convenient linear scale to make dimensionless lensing variables. In terms of the Einstein angle, the lens equation for a point mass becomes
Substituting for the constants gives
In the latter form, the mass is expressed in solar masses ( and the distances in Gigaparsec (Gpc). The Einstein radius is most prominent for a lens typically halfway between the source and the observer.
For a dense cluster with mass at a distance of 1 Gigaparsec (1 Gpc) this radius could be as large as 100 arcsec (called macrolensing). For a Gravitational microlensing event (with masses of order ) search for at galactic distances (say ), the typical Einstein radius would be of order milli-arcseconds. Consequently, separate images in microlensing events are impossible to observe with current techniques.
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 ñ the positions ø<sub>I</sub>(ø<sub>S</sub>) of the images can then be calculated. For small deflections this mapping is one-to-one and consists of distortions of the observed positions which are invertible. This is called weak lensing. 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.