Einstein Ring

The idea: a lens with no glass

Light always takes the path that extremises travel time. Near a mass, spacetime is curved, so the "straight line" a photon follows is bent toward the mass — exactly as a glass lens bends light by slowing it down. A galaxy is therefore a lens, just one made of gravity rather than glass. Put a bright background source directly behind that gravitational lens, and the source's light reaches you from every direction around the lens at once. Those directions trace out a circle on the sky: the Einstein ring.

The crucial ingredient is alignment. A gravitational lens is not a single focal point but a line of focus stretching back from the lens — there is no single plane where everything comes to a crisp image, which is why an ordinary star makes a hopeless camera lens. But when the source, the lens, and your eye are nearly colinear, the symmetry of the situation forces the image into a ring. Nudge the alignment off-axis and the ring breaks: first into two unequal arcs on opposite sides of the lens, then, as you move further, into a single dominant arc plus a faint counter-image. The complete, unbroken ring is the rare and beautiful limit of near-perfect alignment.

How much gravity bends light

The whole phenomenon rests on one number from general relativity: the deflection angle of a light ray grazing a point mass M at impact parameter b. Newtonian "corpuscle" reasoning gives half the right answer; the full relativistic result, which Eddington confirmed at the 1919 solar eclipse, is

α̂ = 4GM / (c² b)

For light grazing the Sun's limb (M = 1 M☉, b = R☉) this is 1.75 arcseconds — the famous Eddington measurement. The factor of 4GM/c² recurs everywhere in lensing; note that 2GM/c² is the Schwarzschild radius, so the deflection is twice the Schwarzschild radius divided by the impact parameter. For an extended lens like a galaxy, you add up the deflection from all the mass, and what matters is the projected mass — the mass seen in the two-dimensional plane of the sky, integrated along the line of sight.

The Einstein radius

Demand perfect colinear alignment (source angle β = 0 in the lens equation) and the bending of light closes the ring at one characteristic angular radius, the Einstein radius:

θ_E = √[ (4GM / c²) · (D_LS / (D_L · D_S)) ]

Here M is the projected mass enclosed inside the ring, and D_L, D_S, D_LS are the angular-diameter distances to the lens, to the source, and from the lens to the source. (In an expanding universe D_LS is not simply D_S − D_L; the distances must be computed in the cosmological metric.) The lens equation that this comes from is

β = θ − (D_LS / D_S) · α̂(θ)        (the lens equation)
θ_E² = (4GM/c²)(D_LS / D_L D_S)      (ring radius for β = 0)

Two features make this equation a workhorse of cosmology. First, θ_E depends on the mass and on geometry only — there is no dependence on what the mass is made of, how it shines, or its dynamical state. Second, you can invert it to read the mass straight off the sky:

M(<θ_E) = (c² / 4G) · θ_E² · (D_L D_S / D_LS)

Measure the ring's angular radius, plug in the distances (from redshifts plus a cosmology), and out comes the projected mass inside the ring — a measurement that, unusually for astronomy, makes no assumption about stellar populations, gas content, or virial equilibrium.

Putting in real numbers

For a singular isothermal sphere — a good first model for a galaxy halo with flat rotation curve velocity dispersion σ — the Einstein radius takes a clean closed form:

θ_E = 4π (σ/c)² (D_LS / D_S)
    ≈ 1.4″ × (σ / 220 km s⁻¹)² × (D_LS / D_S)

A Milky-Way-like galaxy with σ ≈ 220 km/s, lensing a source at twice its distance (so D_LS/D_S ≈ 0.5), produces θ_E ≈ 0.7 arcsec — the typical size of the galaxy–galaxy rings found by the Sloan Lens ACS (SLACS) survey. The table below scales across the lens-mass ladder:

Lens	Mass inside ring	θ_E (typical)	Physical R_E	Found with
Single foreground star	~1 M☉	~1 mas	~few AU	Microlensing light curve
Massive star / stellar lens	~30 M☉	~5 mas	~tens of AU	Astrometric microlensing
Dwarf galaxy	~10⁹ M☉	~0.1″	~0.5 kpc	HST / JWST
L* galaxy (SLACS)	~3 × 10¹¹ M☉	~1″	~5 kpc	HST, SLACS / BELLS
Group / giant elliptical	~10¹³ M☉	~3–5″	~20 kpc	HST, ground 8 m
Galaxy cluster	~10¹⁴–10¹⁵ M☉	~10–50″	~100–300 kpc	HST Frontier Fields, Euclid

For scale, the full Moon spans about 1800 arcseconds, so even a cluster's 30″ ring is sixty times smaller than the Moon, and the galaxy-scale 1″ rings are nearly two thousand times smaller — comfortably beyond ground-based seeing of ~1″ without adaptive optics, which is why nearly every clean ring image comes from Hubble, JWST, or interferometers like ALMA and the VLA.

Why rings are also magnifiers

Lensing conserves surface brightness but not flux: it stretches the source's image over more solid angle, so the total light you collect goes up. The total magnification of a point source at the centre of a symmetric lens formally diverges, and a perfect ring is the brightest possible lensing configuration. Real sources have finite size, so the magnification is large but finite — typically factors of 5 to 50 for galaxy-scale rings, and up to hundreds near the critical curves of clusters.

This turns Einstein rings and arcs into cosmic telescopes. The most distant, faint galaxies we can study in detail are almost all gravitationally magnified: the lens does the light-gathering that no human-built telescope can match. JWST's deepest spectra of redshift-9-to-13 galaxies, and ALMA's resolution of star-forming clumps in the z = 3.04 ring SDP.81 down to ~50–100 parsec scales, both exploit the magnification that comes free with the ring.

How perfect must the alignment be?

A complete ring requires the source to lie within roughly an Einstein radius of perfect colinearity, projected back to the source plane. Because θ_E is an arcsecond-scale angle for a galaxy lens, and because the source must additionally be physically extended enough to wrap around, complete unbroken rings are rare; partial rings, arcs, double images, and quads are far more common. The probability that a random distant galaxy is strongly lensed at all is of order one in a thousand to one in ten thousand, which is why blind ring searches comb millions of galaxies.

Source–lens alignment	Lens shape	Image configuration	Example
Near-perfect, β ≈ 0	Circular	Complete Einstein ring	SDP.81; the "Molten Ring" (GAL-CLUS-022058s)
Small offset	Circular	Two arcs / partial ring	SLACS partial rings
Source inside inner caustic	Elliptical	Four images (Einstein cross)	Q2237+0305 (Einstein Cross)
Source on a fold caustic	Elliptical	Bright arc + counter-image	Cluster giant arcs
Large offset	Any	Two images, one faint	Doubly imaged quasars

Famous rings and what they taught us

• MG 1131+0456. The first Einstein ring ever imaged, by Jacqueline Hewitt, Bernard Burke and colleagues with the VLA in 1988 — a radio ring about 1.75″ across produced by a foreground galaxy lensing a background radio source.
• SDP.81, the "ALMA ring." A z = 3.04 dusty star-forming galaxy lensed by a z = 0.30 elliptical. ALMA's 2014 Long Baseline Campaign resolved it at ~30 milliarcsecond resolution, mapping molecular gas and dust clumps in the source down to ~50–100 pc and revealing a possible dwarf-galaxy substructure in the lens halo.
• The "Molten Ring" / GAL-CLUS-022058s. One of the largest and most complete known Einstein rings, imaged by Hubble — a striking near-360° ring around a luminous elliptical in a galaxy cluster.
• The Einstein Cross, Q2237+0305. Not a ring but its elliptical-lens cousin: a single z = 1.7 quasar imaged four times around a z = 0.04 foreground galaxy nucleus. Decades of microlensing variability in its four images probe the structure of the quasar's accretion disk.
• JWST and Euclid harvests. JWST imaging has produced exquisitely sharp rings (e.g. the "JWST ring" around a foreground elliptical), and ESA's Euclid mission is expected to discover of order 100,000 strong lenses across its 14,000-square-degree survey, transforming rings from rare curiosities into a statistical population.

What rings are used for

• Weighing dark matter. Because θ_E gives the total projected mass with no luminosity assumption, comparing the lensing mass with the stellar mass inferred from the galaxy's light directly reveals the dark-matter content within the ring radius. Galaxy-scale lenses typically need a dark-matter contribution of order half the projected mass within ~5 kpc.
• Mapping the inner mass profile. The exact shape and thickness of a ring, plus the positions of any extra images, constrain whether the lens has a cored or cuspy density profile — a direct test of cold-dark-matter predictions on galaxy scales.
• Detecting dark substructure. Tiny distortions and surface-brightness anomalies in a ring betray clumps of dark matter (subhalos of ~10⁷–10⁹ M☉) that emit no light at all. SDP.81 and the lens SDSS J0946 ("the Jackpot") have both yielded such detections.
• Measuring the Hubble constant. For variable sources (lensed quasars, lensed supernovae like the multiply imaged SN Refsdal), the time delays between images depend on H_0. Time-delay cosmography (H0LiCOW, TDCOSMO) yields H_0 to a few percent, independent of the distance ladder.
• Magnifying the early universe. Rings and arcs act as natural telescopes, delivering magnified, sometimes spatially resolved views of galaxies that would otherwise be unreachably faint at redshifts beyond 9.

Common misconceptions and edge cases

• "The ring is a real object." It is not. There is one background source; the ring is its distorted, multiplied image. Move the observer or the source and the ring reshapes or vanishes — it is a projection effect, not a physical structure around the lens.
• "You need a black hole to make a ring." Any sufficient projected mass works. Almost all known rings are made by ordinary galaxies and clusters, where dark matter and starlight do the bending; black holes are far too compact and low in mass to make a resolvable galaxy-scale ring.
• "The Einstein radius is the size of the lens." No — θ_E is an angle set jointly by the enclosed mass and the lens–source–observer geometry. Put the same galaxy in front of a more distant source and θ_E grows, because D_LS/D_S rises; the galaxy hasn't changed.
• "Microlensing makes tiny rings you can see." A star lensing a star does form a ring, but at the sub-milliarcsecond scale of stellar lensing it is utterly unresolvable as a circle. You see only a smooth, symmetric brightening and fading of the background star — the photometric signature that OGLE, MOA, and Roman exploit to find planets.
• "Lensing changes the source's colour or spectrum." Gravitational deflection is achromatic — it bends all wavelengths identically (unlike a glass lens). A colour gradient across a ring reflects the source's own structure, or differential dust in the lens, not the lensing itself.
• "Einstein discovered it and expected to see rings." The geometry predates Einstein's 1936 note (Chwolson described it in 1924), and Einstein explicitly doubted it could ever be observed — he was thinking of star-on-star lensing. Zwicky's 1937 insight that galaxies are the right lenses is what made rings a real observational programme.