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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.

For a point lens of mass $m$ at observer distance $D_l$ and source distance $D_s$ in a nearby Euclidean geometry, the Einstein radius in the lens plane is $R_E=\sqrt{4Gm{D}_l(D_s-D_l)/(c^2{D}_s)}$. Its angular version is ${\theta }_E=R_E/D_l$. This is the ring radius for perfect alignment and the characteristic transverse scale of a gravitational microlensing event. A crossing timescale is $t_E=R_E/v_{\perp }$, proportional to the square root of lens mass at fixed geometry and speed.

An aligned source behind an axisymmetric gravitational lens produces a circular image at an Einstein radius. Its angular radius satisfies ${\theta }_E^2=4G{M}_{\mathrm{projected}}(<D_d{\theta }_E)D_{ds}/(c^2{D}_d{D}_s)$. Exact alignment gives a ring; displaced sources generally give arcs or distinct images rather than a complete ring.

The microlensing optical depth is the probability, when small, that a source lies within one Einstein radius of a foreground lens. For point lenses and compact-object mass density ${\rho }_l$ in nearby Euclidean geometry,The individual lens mass cancels because $n_l={\rho }_l/m$ and $R_E^2\propto m$. Detection efficiencies and event durations still depend on individual masses and survey cadence.

Substituting $\eta =D_s\beta$ and $\xi =D_d\theta$ into the previous geometric relation verifiesFor a point mass gravitational lens, the given physical deflection yieldsAn aligned source has $\beta =0$, and therefore ${\theta }^2={\theta }_E^2$. Rotational symmetry turns these signed intersections into an Einstein ring, whose Einstein radius isFor an offset source, multiplying $\beta =\theta -{\theta }_E^2/\theta$ by $\theta$ gives a quadratic equation. Its two image positions areFor $\beta >0$, ${\theta }_{+}>0$ lies on the source side of the lens and ${\theta }_{-}<0$ lies on the opposite side.

A complete Einstein ring requires an axisymmetric gravitational lens and exact alignment of the source with the lens centre and observer: $\beta =0$. A finite aligned circular source gives a ring of finite width. The background source must lie behind the lens, so the lensing distance factor is positive.

With $\beta =0$ the thin gravitational lens equation givesThe Einstein radius here is angular, and the mass is projected inside the corresponding aperture. For an extended mass distribution the equation is implicit because the enclosed mass depends on ${\theta }_E$.

Using the supplied approximations, $D_{ds}/(D_s{D}_d)=1/(3\times 10^{25}\mathrm{m})$ and ${\theta }_E=3\times 10^{-4}$, the inferred mass isThusThis measures the projected aperture mass without requiring the singular isothermal sphere model used subsequently.

Objects near the derived cutoff are too faint for their absence in ordinary star counts alone to exclude a massive dark matter halo. The decisive test is gravitational microlensing: compact objects passing in front of background stars produce transient magnification even if they emit no detectable light. For a lens at distance $D_l$ and a source at $D_s$, the physical Einstein radius and event timescale areFor $D_l=10\mathrm{kpc}$, $D_s=50\mathrm{kpc}$ and $v_{\perp }=200\mathrm{km}{\mathrm{s}}^{-1}$, a $10^{-3}{M}_{\odot }$ lens gives $t_E\simeq 2.2$ days, within the regime tested by suitably sampled gravitational microlensing surveys. The microlensing optical depth isso at fixed compact-object mass density it is independent of the individual lens mass: reducing individual mass changes the event durations rather than hiding the total lensing probability.

EROS observations of the Magellanic Clouds found far fewer candidate events than a halo dominated by compact objects would predict and excluded such objects as the principal Galactic halo component over $6\times 10^{-8}<m/M_{\odot }<15$, under their tested halo models. Their often-quoted eight-percent bound is specifically near $0.4M_{\odot }$, not a uniform bound at every lens mass. A halo dominated by the approximately Jupiter-mass objects needed here is incompatible with those microlensing constraints. Deep optical and infrared counts also restrict a large population of luminous low-mass stars and brown dwarfs, while the cosmic baryon fraction provides a separate constraint on explaining all cosmological dark matter with baryonic objects.

The projected density $\Sigma (R)={\sigma }^2/(2GR)$ gives cylindrical mass $\pi {\sigma }^{2}R/G$, so the reduced deflection magnitude is the constant displayed angular Einstein radius. The signed thin gravitational lens equation is $\beta =\theta -{\theta }_E\mathrm{sgn}\theta$. A source with $0<\beta <{\theta }_E$ has two images; one with $\beta >{\theta }_E$ has one. The lensing convergence and lensing shear magnitude are both ${\theta }_E/(2|\theta |)$.