Rung 1 · Eratosthenes, Alexandria, c. 240 BC
The size of the Earth
At noon on the summer solstice the Sun shone straight down a well in Syene. At the same moment in Alexandria, a vertical stick cast a shadow. Sunlight arrives in parallel rays, so the shadow's angle is the angle between the two cities measured at Earth's centre.
The two marked angles are alternate angles cut by one sunbeam, so they are equal. Stick heights are exaggerated.
- Fraction of a circle
- Circumference
- Circumference
The historical numbers (7.2° and 5,000 stadia) give a circumference of 250,000 stadia. How close that lands depends mostly on which stadion Eratosthenes meant, which historians still argue about. Try each one and watch the ledger.
Carries up: the Earth's radius sets the Moon's size in rung 2 and the Paris–Cayenne baseline for the Mars parallax in rung 3.
Rung 2 · Aristarchus and Hipparchus, 3rd–2nd century BC
The Moon, measured by Earth's shadow
During a lunar eclipse the Moon drifts through Earth's shadow at about one Moon-width per hour. Time the crossing and you learn how many Moons wide the shadow is. The shadow is a cone, and it narrows by exactly one Moon diameter on the way out, because the Sun and the Moon look the same size in our sky.
Side view, not to scale. The cone's edges converge at the Sun's angular size, about 0.5°.
- Earth diameter
- Earth ÷ Moon
- Moon diameter
- In Earth radii
Carries up: the Moon's distance is the short side of Aristarchus's triangle in rung 3. And the relation distance = size ÷ angle is the workhorse of every rung above.
Rung 3 · Aristarchus (c. 270 BC), Kepler (1619), Cassini and Richer (1672)
The Sun, and the Astronomical Unit
When the Moon is exactly half lit, the Sun–Moon–Earth angle is a right angle. Measure the angle between the Moon and the Sun in the sky, and the cosine gives the ratio of their distances. The method is sound, but near 90° it is ruthless.
Distance ratio on a log scale. A tenth of a degree near 90° is worth a factor of several.
- Sun ÷ Moon distance
- True ratio
The fix: one scale bar for Kepler's map
Kepler's third law (T² ∝ a³) maps the Solar System exactly, but only in AU. Mars's period of 1.881 years puts its orbit at 1.524 AU, and on the night of the 1672 opposition the map says Mars was 0.381 AU from Earth. One distance in kilometres scales the whole map. Cassini in Paris and Richer in Cayenne observed Mars at the same time and saw it shifted against the stars. Blink between their views and measure the shift.
Background stars are effectively infinitely far, so they don't move. Only Mars shifts. Tick marks are 5 arcseconds apart.
- Baseline (from your Earth)
- Distance to Mars
- Mars that night
- 0.381 AU
Carries up: the AU is the baseline for stellar parallax. Any error here is copied into the length of the parsec and into every distance above it.
Rung 4 · Bessel (1838), Hipparcos (1989), Gaia (2013)
Stellar parallax
Earth's orbit gives a baseline of 2 AU. Over a year, a nearby star traces a tiny ellipse against far-off background stars. The parallax angle p is half the shift, and the distance in parsecs is simply 1/p in arcseconds.
Left: the geometry, wildly exaggerated. Right: what the telescope sees, zoomed to fit. The fuzzy disk is the instrument's measurement error.
- Parallax p
- Signal ÷ error
- Instrument's reach (5σ)
- Distance
Carries up: parallax distances to thousands of nearby stars calibrate their true luminosities. That calibration is what rung 5 slides clusters against.
Rung 5 · Hertzsprung and Russell (1910s), main-sequence fitting
Standard candles and the main sequence
Light spreads over a sphere, so flux falls as 1/d². Plot nearby stars, whose parallax distances we know, by colour and true brightness, and most land on a narrow band: the main sequence. A distant cluster shows the same band, dimmed by its distance. Slide it up until it fits. The size of the slide is the distance.
Cyan: nearby stars calibrated by parallax (absolute magnitude M). Amber: the cluster, plotted as apparent magnitude minus your distance modulus. Brighter is up.
- Cluster vs calibrated band
- Dimming factor
- Distance
Older clusters (M67, NGC 188) have lost their hot, bright stars and grown a branch of red giants. Fit the band, not the giants. Interstellar dust is ignored here; in real work it dims and reddens every cluster a little.
Carries up: some Cepheids live in clusters like these. Your cluster distances set the zero point of the Leavitt law in rung 6.
Rung 6 · Henrietta Swan Leavitt (1912), Edwin Hubble (1923)
Cepheid variables
Cepheids pulse like clockwork, and the brighter ones pulse more slowly. Measure the period, read the true brightness off the Leavitt law, compare with how bright the star looks, and out comes the distance. Here are Hubble-style observations of a Cepheid in Andromeda, taken on scattered nights. Find its period by folding the data.
Top: brightness on each night it was observed. Bottom: the same points folded at your trial period. At the right period, the scatter collapses into one sawtooth.
- Fold quality
- Absolute mag M
- Mean apparent mag m
- Distance modulus
The Leavitt law. Dots are Cepheids in calibrated clusters, so they inherit your rung-5 distances. The dashed cyan line is the modern calibration.
Carries up: Cepheids in nearby galaxies that also hosted a Type Ia supernova set the supernova brightness scale in rung 7.
Rung 7 · Phillips (1993), Riess, Perlmutter and Schmidt (1998)
Type Ia supernovae
A white dwarf that nears 1.4 solar masses detonates and briefly shines like five billion Suns. They aren't perfectly identical, but brighter ones fade more slowly. Stretch a template light curve to fit, and the stretch tells you the true peak brightness.
A supernova in a distant galaxy. Fit the amber template to the points: stretch sets its width, peak sets its height.
- Fit (χ² per point)
- Decline in 15 days
- Standardised peak M
Forty nearby supernovae with Cepheid-calibrated distances. Raw peaks scatter widely; correcting for decline rate tightens them into a precise standard candle.
Carries up: supernova distances to galaxies hundreds of megaparsecs away are the x-axis of Hubble's law in rung 8.
Rung 8 · Slipher (1912), Hubble (1929)
Redshift and Hubble's law
Every element absorbs light at fixed wavelengths. In distant galaxies the same barcode appears, shifted red. Line up the lab wavelengths with each galaxy's dark lines to measure its redshift z, record it, and see velocity grow with distance.
Galaxy spectrum with absorption lines. The cyan ticks are lab wavelengths for calcium, hydrogen, magnesium and sodium, shifted by your z. Faint grey ticks show the unshifted lab positions.
- Hα would sit at
- Velocity cz
- SN distance
- Galaxies measured
Hubble diagram. The amber line is the best fit through the origin to the galaxies you have recorded. Hollow ticks on the axis are galaxies still waiting for a redshift.
Carries up: H₀ turns any redshift into a distance and sets the expansion rate used to integrate out to the edge in rung 9.
Rung 9 · Friedmann, Lemaître, and the cosmic microwave background
The edge of the observable universe
The universe is 13.8 billion years old, yet we see farther than 13.8 billion light years, because space kept stretching while the light travelled. Add up each step of the light's journey, each stretched by all the expansion that came after, and you get the distance to the source today.
Distance today versus redshift. The curve flattens toward a limit: the particle horizon, the farthest a signal could have come since the Big Bang.
- Dark energy ΩΛ
- Age of the universe
- Light travelled for
- Source distance today
The top of the ladder: a stick's shadow, an eclipse, a triangle across Earth's orbit, a pulsating star and a line in a spectrum, all multiplied together.
The summit
How your errors climbed the ladder
Every rung's ruler came from the rung below. The table shows where each of your numbers landed and how far it drifted from the modern value.
The Hubble tension
Real astronomers face the same problem at higher precision. The Cepheid-and-supernova ladder gives H₀ ≈ 73, while the cosmic microwave background, read through the standard model of cosmology, predicts about 67. The error bars don't overlap. Either a rung hides a subtle error, or the disagreement points to new physics.
Measurements of H₀ in km/s/Mpc with 1σ error bars. The gravitational-wave standard siren (GW170817) needs no ladder at all.