Einstein Radius - Derivation

Derivation

In the following derivation of the Einstein radius, we will assume that all of mass of the lensing galaxy 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 the total deflection by a point mass is given (see Schwarzschild metric) by

where

is the impact parameter (the distance of nearest approach of the lightbeam to the center of mass)

is the gravitational constant,

is the speed of light.

By noting that, for small angles and with the angle expressed in radians, the point of nearest approach b at an angle for the lens on a distance is given by, we can re-express the bending angle as

(eq. 1)

If we set as the angle at which one would see the source without the lens (which is generally not observable), and 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 at a distance is the same as the sum of the two vertical distances plus . This gives the lens equation,

,

which can be rearranged to give

(eq. 2)

By setting (eq. 1) equal to (eq. 2), and rearranging, we get

For a source right behind the lens, and the lens equation for a point mass gives a characteristic value for that is called the Einstein radius, denoted . Putting and solving for gives

The Einstein radius for a point mass provides a convenient linear scale to make dimensionless lensing variables. In terms of the Einstein radius, 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 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.

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 . The positions 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.