Einstein angular radius (${\theta }_E=\sqrt{4GM{D}_{ls}/(c^2{D}_l{D}_s)}$)

The Einstein angular radius is the angular form of the Einstein radius: $R_E=D_l{\theta }_E$ in the lens plane. Its displayed value applies to a point-mass gravitational lens with the angular diameter distances defined above. It depends on source distance as well as lens mass and position. At fixed distances, multiplying $M$ by four doubles ${\theta }_E$.

At exact alignment, the vector thin gravitational lens equation is $0=(1-{\theta }_E^2/r^2)\mathbf{\theta }$. For a nonzero image direction it requires $r={\theta }_E$ and leaves the direction unrestricted: all azimuths form an Einstein ring. The signed equation's roots $\theta =\pm {\theta }_E$ are only a diameter cut through that circle. A source of finite angular size makes an annulus or arcs with finite received light; the infinite point-source lensing magnification at alignment is an idealization. An Einstein ring expresses rotational symmetry and source alignment, not a circular arrangement of emitting stars.