LIGHT / MASSThe mystery
An interactive astronomy presentation

Weighing a galaxy
with light

A ring in the sky. A line of algebra.
And a lot of missing mass.

Hubble field showing the Cosmic Horseshoe as a blue ring around a gold galaxy
Hubble observation · ESA/Hubble & NASA
01 / The mechanism

Mass bends light

More mass bends it more. Passing farther away bends it less.

lens galaxyincoming lightbAngles exaggerated
bend = 4GM / c²b

Try doubling mass, then doubling distance.

02 / The geometry

A whole circle of paths reaches us

When source, lens and telescope line up, we see an Einstein ring.

Real Hubble image. Gold: your simulated ring.
ESA/Hubble & NASA
Gold paths

Light that lands on Earth.

Blue paths

Light that misses us.

Drag the scene to rotate. Slide to zero mass to straighten the rays.

Illustrative geometry. Bending exaggerated. Ring radius scales with √mass.

03 / The relationship

Four times the mass. Twice the radius.

The ring grows with the square root of mass.

Reference ringYour ring
radius² ∝ mass

Source and lens distances stay fixed.

04 / The evidence

The Cosmic Horseshoe

A real ring around SDSS J1148+1930, discovered in 2007.

Enlarged Hubble view of the blue Cosmic Horseshoe around its central foreground galaxy5.0″ radius
Hubble crop · ESA/Hubble & NASA. Overlay illustrates the published scale.
Lens redshift0.444
Source redshift2.379
Observed radius≈ 5.0″

″ means arcseconds: 1/3,600 of a degree. A 5″ radius makes a 10″ diameter.

Belokurov et al. 2007
05 / The distances

Redshift gives us the geometry

Expansion stretches light. A cosmology converts that stretch into distances.

EarthLensSource

Lens redshift stays at 0.444. Mpc means megaparsec, about 3.26 million light-years.

Angular-diameter distances. Dls ≠ Ds − Dl. Diagram not to scale.

06 / The formula with real numbers

The numbers go straight into the formula

Follow each color from the symbol to its value. Change an input to recalculate.

c² / 4G
Constants of nature
(2.998 × 10⁸)²
÷ (4 × 6.674 × 10⁻¹¹)
= 3.37 × 10²⁶ kg/m
θ²
Radius, then square it
DlDs / Dls
Distances, then metres

Detailed model: 5.02 ± 0.09 trillion Suns · Dye et al. 2008

07 / The prediction

The stars predict a smaller ring

We measured a 5″ radius. How large a ring would the stars alone produce?

Gold: observed radius 5.0″

Real Hubble photograph · ESA/Hubble & NASA. Rings are illustrative model overlays.

Simplified enclosed-mass comparison. A full prediction also needs the stars’ radial distribution. The two percentages represent different stellar-population models.

Spiniello et al. 2011
08 / The inference

Most of the gravitating mass is unseen

The light accounts for only part of what gravity weighs.

Blue halo: inferred mass, not a photograph

Real Hubble photograph · ESA/Hubble & NASA. Rings are illustrative model overlays.

Gold: stars (17%) · Blue: additional mass · Outline: lensing total

Dark matter is the standard explanation. Lensing weighs total mass; it does not identify a particle.

09 / A useful challenge

Could a small measuring error close the gap?

Change the radius while keeping the geometry and stellar estimate fixed.

Dashed: 5″ observed. Blue: stellar estimate.

Real Hubble photograph · ESA/Hubble & NASA. Rings are illustrative model overlays.

mass / original mass = (radius / 5″)²

A sensitivity experiment, not the published uncertainty or a test of every systematic effect.

10 / Optional bonus

Even simulations face a lensing puzzle

A different comparison: observed cluster lenses versus dark-matter simulations.

Hubble image of MACS J1206 with an artist's blue dark matter overlay
MACS J1206. Blue is an artist’s dark-matter overlay, not visible matter. NASA/ESA + CLASH team, Caminha, Meneghetti, Natarajan & Kornmesser

Lensing probability, normalized

1× reference

The 2020 study found >10× the predicted efficiency of small galaxy lenses inside 11 clusters.

This is a probability comparison, not a 10× bend angle. Historical result; interpretation depends on the models.

Meneghetti et al. 2020
11 / The takeaway

Light lets us weigh what we cannot see

Assuming general relativity, the ring gives total mass. Starlight gives the stars’ share.

Same distances. Twice the ring radius.
How much mass?

Choose an answer.

Follow the evidence

Belokurov 2007 Discovery, ring size, redshiftsDye 2008 Detailed mass modelSpiniello 2011 Stellar fractions and assumptionsMeneghetti 2020 Cluster lensing comparison

Flat cosmology: H₀ = 70 km/s/Mpc, Ωm = 0.3, ΩΛ = 0.7. This presentation reproduces a historical example.

12 / An alternative explanation

Could gravity work differently?

Modified gravity changes the link between visible matter and its gravitational effects.

Orbital speed (km/s)0200400550Distance from galaxy center (kpc)

Gold: Newtonian, visible mass only
Blue: MOND, same visible mass

Illustrative spherical galaxy, 10¹¹ Suns. Curves are calculations, not observations of the Horseshoe.

MOND changes dynamics at tiny accelerations. It explains many galaxy rotation patterns without adding a dark halo.

At low acceleration: g ≈ √(gN a₀)

For our lensing story: predicting bent light requires a relativistic theory, such as TeVeS. MOND alone does not supply a new Horseshoe ring prediction. Galaxy clusters and cosmology remain difficult tests.

gN: pull from visible matter under Newtonian gravity.
a₀ ≈ 1.2 × 10⁻¹⁰ m/s². Demo uses a smooth transition between regimes.

Famaey & McGaugh 2012 · Bekenstein 2004