The Einstein radius is the radius of an Einstein ring, and is a characteristic angle for gravitational lensingin general, as typical distances between images in gravitational lensing are of the order of the Einstein radius.
Derivation
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 α1 the total deflection by a point mass M is given by where By noting that, for small angles and with the angle expressed in radians, the point of nearest approach b1 at an angle θ1 for the lens L on a distance DL is given by, we can re-express the bending angle α1 as If we set θS as the angle at which one would see the source without the lens, and θ1 as the observed angle of the image of the source with respect to the lens, then one can see from the geometry of lensing that the vertical distance spanned by the angle θ1 at a distance DS 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 equal to, and rearranging, we get For a source right behind the lens,, and the lens equation for a point mass gives a characteristic value for θ1 that is called the Einstein angle, denoted θE. When θE is expressed in radians, and the lensing source is sufficiently far away, the Einstein Radius, denoted RE, is given by Putting and solving for θ1 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. 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 this radius could be as large as 100 arcsec. For a Gravitational microlensing event search for at galactic distances, 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 lowerray 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 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.
