Strong gravitational lensing

Strong gravitational lensing produces multiple resolved images, arcs, or rings when a sufficiently massive foreground lens lies close to the source's line of sight.

For a thin lens, relative arrival times follow the difference of $\frac{1}{2}|\theta -\beta {|}^2-\psi (\theta )$ between images, multiplied by $(1+z_l)D_l{D}_s/(c{D}_{ls})$. Both geometry and gravitational potential contribute; an additive delay common to all images is unobservable.

The distance combination $D_{\Delta t}=(1+z_l)D_l{D}_s/D_{ls}$ converts a dimensionless thin-lens delay difference into an observed time delay. Inference from a variable lensed source also requires a mass model and control of external convergence.

The two resolved quasar images of 0957+561 provide an observational question: are they separate sources, or different light paths from one source? In May 1979, spectrophotometry used the McDonald Observatory 2.7 m reflecting telescope, the UVITS spectrograph, and an intensified dissector scanner at the f/9 focus of the Cassegrain reflector. A 600-groove-per-millimetre diffraction grating produced 4100-7100 ångström scans sampled every 2 ångströms, with about 7 ångström resolution over most of that range. Two spectrograph slits, about 52 arcseconds apart, recorded object and sky background simultaneously for subtraction. Interleaving observations of the two images reduced instrumental and atmospheric gain changes.^[ref]

The readout was received light versus wavelength for each image. Closely matching absorption line redshifts supported a common source and absorber; differences in continua and line strengths motivated differential interstellar extinction. Spectral agreement supplied evidence for gravitational lensing, rather than proving the point-mass gravitational lens model. The authors discussed a physical double and discrepant optical and radio light ratios. Their note added in proof attributed the redder spectrum to foreground galaxy contamination and emphasized an extended mass distribution.^[ref]

A critical curve is a locus in the image plane where the Jacobian matrix of the thin gravitational lens equation becomes singular. Its formal point-source lensing magnification diverges, while finite source size regularizes the measured flux. Its mapped locus in the source plane is a gravitational-lensing caustic.

The local and mean lensing convergence obey $2\kappa =2\bar{\kappa }+r{\bar{\kappa }}^{\prime }$. Hence the radial derivative of the lens mapping is $1-{\alpha }^{\prime }(r)=1-2\kappa +\bar{\kappa }$. Its zeros are radial gravitational-lensing critical curves, where radial stretching diverges and images merge on crossing the mapped gravitational-lensing caustic.

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.

A caustic is the source-plane image of a gravitational-lensing critical curve. Crossing a generic fold caustic creates or destroys a pair of images of opposite parity. For a circular lens, a tangential critical circle maps to an aligned-source point, while a radial critical circle can map to a finite source-plane circle. Critical curves and caustics consequently occupy different planes and must not be identified as the same angular locus.

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.