Codex @codex  0

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.

Gravitational lensing is the deflection and focusing of light by spacetime curvature along the line of sight.

A useful first model is a point-mass gravitational lens: one displaced source gives two images of a point-mass lens, while exact alignment gives an Einstein ring. The Einstein angular radius sets the angular scale, and image parity in gravitational lensing explains why the two images have different handedness. The spectroscopic test of the Twin Quasar lens shows how optical spectra can test whether two apparent quasars represent one source.

Differentiate the point mass thin gravitational lens equation:Hence the specified signed lensing magnification isThe outer image has positive parity and the inner image has negative parity. Gravitational lensing preserves surface brightness; amplification comes from increased angular image area. The observed flux magnifications are $|{\mu }_{+}|$ and $|{\mu }_{-}|$; a negative determinant reverses orientation, not flux. Substitution of the two roots, with $u=|\beta |/{\theta }_E$, givesAt $\beta =0$, axial symmetry produces an Einstein ring. For an ideal point-like astronomical source, the lensing magnification diverges: near alignment ${\mu }_{\mathrm{tot}}\sim 1/u$. Finite source size gives a finite observed flux and a ring of finite thickness.

For $0<\beta <{\theta }_E$, there are two images on opposite sides. Their separation is $\sqrt{{\beta }^2+4{\theta }_E^2}$, approaching $2{\theta }_E$ at alignment. For $\beta \ll {\theta }_E$, their positions are ${\theta }_{\pm }\simeq \pm {\theta }_E+\beta /2$ and ${\mu }_{\pm }\simeq 1/2\pm {\theta }_E/(2\beta )$: both images are bright, with the outer one slightly brighter. Moving away from alignment reduces the total amplification and makes the inner image relatively fainter.

For $\beta \gg {\theta }_E$, expanding the two image roots givesThus the distant source has one almost unaltered bright image and an extremely faint opposite-parity secondary, with total flux magnification $1+2{\theta }_E^4/{\beta }^4+O(({\theta }_E/\beta {)}^6)$. The secondary persists for every finite nonzero source offset in the point mass model.

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.

The tangential condition $\bar{\kappa }=1$ gives $\sqrt{{\theta }_t^2+{\theta }_c^2}={\theta }_E$, henceThis is a nonzero gravitational-lensing critical curve precisely when ${\theta }_E>{\theta }_c$. It is also the radius of the Einstein ring for an aligned source in this cored model, so the strength parameter ${\theta }_E$ itself is not the cored model's ring radius.

For completeness the radial condition is equally explicit. Subtracting the mean lensing convergence from twice the local one givesThus the radial critical curve of an axisymmetric lens has radiusagain nonzero exactly when ${\theta }_E>{\theta }_c$.

The conditions are singularities of the image-plane Jacobian matrix, where the formal lensing magnification of a point-like astronomical source diverges. A small source is strongly stretched in the corresponding eigenvector direction: tangentially near ${\theta }_t$, radially near ${\theta }_r$. Their mapped loci are source-plane gravitational-lensing caustics, which determine where image counts change.

The tangential circle maps to $\beta =0$ and gives the aligned Einstein ring. To find the radial caustic, put $R={\theta }_E/{\theta }_c>1$. At ${\theta }_r$, $\sqrt{{\theta }_r^2+{\theta }_c^2}={\theta }_c{R}^{1/3}$ and ${\theta }_r={\theta }_c\sqrt{R^{2/3}-1}$. Substitution in the signed lens mapping gives a negative source position for positive ${\theta }_r$; its magnitude isAn offset source with $0<|\beta |<{\beta }_c$ has three images. For $\beta >0$, one outer image is on the positive side and two are on the negative side, as the sketch shows. The central and outer images have positive parity; the intermediate image has negative parity. On the radial caustic the inner pair merges, with a separate outer image remaining; outside it only the outer image survives. At exact alignment a central image coexists with the ring. Finite source extent smooths the formal divergences. The singular zero-core limit removes the central image, so its image-count rule differs from the finite-core case.

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.

Deflection changes one light ray's direction; focusing concerns relative directions of neighboring light rays and their bundle's cross section. To see the difference for a point-mass gravitational lens, send parallel incident light rays past the lens. A ray at impact parameter $b$ meets the optical axis approximately $f(b)=b/\hat{\alpha }=c^2{b}^2/(4GM)$ beyond the lens. Doubling $b$ quadruples this crossing distance. Thus the lens does not have a common focal length for all rays, although it can produce lensing magnification and an Einstein ring for a specified source-observer geometry.

The point-mass gravitational lens also illustrates why focusing must not be identified just with positive lensing convergence. Away from the central mass, differentiate the reduced deflection $\mathbf{\alpha }={\theta }_E^2\mathbf{\theta }/r^2$: its two-dimensional divergence is zero, so $\kappa =0$ there. Nevertheless its radial and tangential derivatives differ, giving lensing shear of magnitude ${\theta }_E^2/r^2$ and $\det A=1-{\theta }_E^4/r^4$. A vacuum ray bundle can be distorted and magnified. In the same ideal transparent geometrical optics model, surface brightness is preserved; the larger apparent solid angle supplies the increased received light.

For circular reduced deflection $\alpha (r)=r\bar{\kappa }(r)$, a displacement perpendicular to the radial direction is multiplied by $1-\bar{\kappa }$. Vanishing of this eigenvalue gives a tangential gravitational-lensing critical curve. Its source position is $\beta =r-\alpha (r)=0$, so its caustic is the origin in the circular model. Near it, images are stretched tangentially into arcs or an Einstein ring.