Einstein radius

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 α₁ 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 b₁ at an angle θ₁ for the lens L on a distance D_L is given by b₁ = θ₁D_L, we can re-express the bending angle α₁ as

If we set θ_S as the angle at which one would see the source without the lens (which is generally not observable), 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 (counting distances in the source plane) that the vertical distance spanned by the angle θ₁ at a distance D_S is the same as the sum of the two vertical distances θ_SD_S and α₁D_LS. 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, θ_S = 0, and the lens equation for a point mass gives a characteristic value for θ₁ that is called the Einstein radius, denoted θ_E. Putting θ_S = 0 and solving for θ₁ 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 (M_☉ and the distances in Gigaparsec (Gpc). The Einstein radius is most prominent for a lens typically halfway between the source and the observer.

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