= Einstein angular radius
{c}
{title2=$\theta_E=\sqrt{4GM D_{ls}/(c^2D_lD_s)}$}
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 $\boldsymbol0=(1-\theta_E^2/r^2)\boldsymbol\theta$. 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.
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