Gravitational Lensing

A massive lens bends light from a background source. Adjust mass, distances, and source offset to explore Einstein rings and arc pairs.

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Lensing weighs what never shines 🖖

The Einstein radius θ_E this tool computes depends on the lens mass and the three distances — and on nothing else. Luminosity does not appear. That makes lensing the rare mass measurement that never asks the mass to emit anything. Which matters, because most of it does not: Planck's 2018 fit puts cold dark matter at about 26.1% of the energy density against 4.9% for ordinary matter, a ratio above five to one. Weigh a foreground galaxy by its starlight and you miss most of it; weigh it by how far it bends a background quasar and you get the whole column, luminous or not. The first complete Einstein ring was resolved not in visible light but in the infrared — B1938+666, imaged with Hubble’s NICMOS camera and reported by King and colleagues in 1998.

A galaxy as a magnifying glass 🖖

Any concentration of mass curves the space around it, and light passing nearby follows that curve — so a foreground galaxy bends the light of a more distant object behind it. Instead of a single image you see two, an arc, or a complete ring, and each one is brightened. The heavier the lens and the closer the alignment, the larger the Einstein radius θ_E this tool computes. Astronomers exploit this as a free zoom lens to study galaxies too faint to see directly.

A lens that focuses on a line, not a point 🖖

A gravitational lens is a spectacularly bad optical instrument. A glass converging lens bends rays farther from its center the most, funnelling parallel light to a single focus; gravity does the opposite — the deflection α = 4GM/(c²b) grows as light passes closer to the mass. So parallel rays never meet at one point but pile up along a line stretching to infinity. That very 'flaw' is why lensed sources smear into rings and arcs instead of tidy dots.

Problem solved in full

A 10¹² solar-mass lens halfway to a source 1000 Mpc away 6 steps
A 10¹² solar-mass lens halfway to a source 1000 Mpc away. Find the Einstein radius, and then find what makes gravitational lensing worth the trouble.
1. One formula, and the geometry is entirely in the distance combination. Note that it is not the distance to the lens or to the source but the ratio D_LS/(D_LD_S) — a lens halfway is worth far more than one just in front of the source.
  
  $θ_{E} = \frac{4 GM}{c ^{2}} \cdot \frac{D _{L S}}{D _{L} D _{S}} M = 1 0^{12} M_{⊙}, D_{L} = 500, D_{S} = 1000, D_{L S} = 500 Mpc$
2. Take the mass factor first. 10¹² solar masses is a galaxy, and 4GM/c² is a length of about six thousand million million metres — which sounds enormous and is about half a light year.
  
  $\frac{4 GM}{c ^{2}} = \frac{4 ( 6.674 \times 1 0 ^{- 11} ) ( 1.989 \times 1 0 ^{42} )}{8.988 \times 1 0 ^{16}} = 5.908 \times 1 0^{15} m$
3. The distance factor is a reciprocal length, and with these three distances it collapses to one over 1000 Mpc exactly.
  
  $\frac{D _{L S}}{D _{L} D _{S}} = \frac{500}{500 \times 1000 Mpc} = \frac{1}{1000 Mpc} = 3.241 \times 1 0^{- 26} m^{- 1}$
4. Multiply, take the square root, and convert. A radian is 206,265 arcseconds, which is the same constant that defines the parsec.
  
  $θ_{E}^{2} = 1.915 \times 1 0^{- 10} ⟹ θ_{E} = 1.384 \times 1 0^{- 5} rad \times 206265 = 2.85 4^{''}$
5. Multiplying the angle by the lens distance gives the physical radius of the ring at the lens — the region of the galaxy whose mass is doing the bending.
  
  $R_{E} = θ_{E} D_{L} = 1.384 \times 1 0^{- 5} \times 500 Mpc = 6.92 kpc$
6. Finally, solve the same equation the other way round.
  
  $M = \frac{c ^{2}}{4 G} θ_{E}^{2} \frac{D _{L} D _{S}}{D _{L S}}$
Answer

2.854 arcseconds, and the payoff is the last line: you can invert it. Rearranged for M, the same equation turns a measured angle and two distances into a mass — and it never asks what the mass is made of. No light has to come from it, no orbit has to be timed, no assumption about temperature or ionisation enters. That is why lensing is the instrument of choice for weighing things that do not shine: cluster mass maps, and the measurements that made dark matter a quantity rather than an adjective. The figure shows the other half of the answer. Two rays leave the source, pass on opposite sides of the lens, and arrive together; spin the whole diagram about the dashed axis and the pair of images becomes a ring.

$θ_{E} = 2.85 4^{''}, R_{E} = 6.92 kpc$

References (4)

The dark-matter and baryon fractions quoted in the first block: N. Aghanim et al. (Planck Collaboration), "Planck 2018 results. VI. Cosmological parameters." Astronomy & Astrophysics 641, A6, 2020.
The first complete Einstein ring — imaged in the infrared, not in visible light: L. J. King, N. J. Jackson, R. D. Blandford et al., "A complete infrared Einstein ring in the gravitational lens system B1938+666." Monthly Notices of the Royal Astronomical Society 295, L41–L44, 1998.
Strong lensing as a mass measurement, reviewed: T. Treu, "Strong Lensing by Galaxies." Annual Review of Astronomy and Astrophysics 48, 87–125, 2010.
The deflection α = 4GM/(c²b) and the Einstein radius: P. Schneider, J. Ehlers and E. E. Falco, Gravitational Lenses, ch. 2 and 8. Springer, 1992. ISBN 978-3-540-66506-9.

By the 42Math team · Updated 2026-07-01

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

Einstein Ring - A 10¹² solar-mass galaxy at 500 Mpc lensing a source at 1000 gives θE = 2.854 arcseconds. With the source exactly behind it, the two images close into a complete ring.
Arc Pair - The same lens and distances as Einstein Ring, with the source moved 0.8 arcseconds off axis. θE is unchanged at 2.854, the ring breaks into images at +3.28 and -2.48, and the magnification falls from unbounded to 3.67.
Cluster Lens - A hundred times the mass buys ten times the Einstein radius, because θE goes as the square root of M: 29.87 arcseconds. The source sits at a hundredth of that and is magnified nearly a hundredfold.
Microlensing - One solar mass at 8 kpc gives θE = 0.71 milliarcseconds, four thousand times smaller than the galaxy ring. Nothing resolves that, which is why microlensing is seen as a brightening rather than as two images.