The angular radius of an Einstein ring

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The angular radius of an Einstein ring (also called the Einstein angle or Einstein radius in angular terms, commonly denoted θ_E or θ₁) is the angular size (as seen from Earth) of the perfectly circular image formed when a distant point-like source is exactly aligned behind a point-mass gravitational lens (such as a star or compact galaxy), with perfect symmetry.

This concept originates from Albert Einstein's 1936 paper "Lens-Like Action of a Star by the Deviation of Light in the Gravitational Field" (published in Science, vol. 84, p. 506), where he revisited gravitational light deflection in the context of lensing by a star.

Earlier ideas of ring-like images appeared in Orest Chwolson's 1924 work, but Einstein provided the relativistic calculation and formula still used today.

Definition

In the case of perfect alignment (source angular position β = 0 relative to the lens), light rays are deflected symmetrically around the lens, producing a ring rather than discrete images. The angular radius of this ring is the characteristic scale of the lensing effect.

For a point-mass lens of mass M, the original/standard expression for the angular radius θ_E (in radians) is:

$$ \theta_E = \sqrt{ \frac{4GM}{c^2} \cdot \frac{D_{LS}}{D_L \, D_S} } $$

where:

θ_E is the angular Einstein radius (in radians; typically converted to arcseconds for observations),

G is the gravitational constant,

M is the mass of the lensing object,

c is the speed of light,

D_L is the angular-diameter distance from observer to lens,

D_S is the angular-diameter distance from observer to source,

D_LS is the angular-diameter distance from lens to source.

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