Cosmic Distance Ladder (2D)

Rung 1: parallax

As the Earth moves around the Sun, a nearby star shifts against the distant background by an angle p (the parallax), where 1 AU subtends p at distance d. With p in arcseconds and d in parsecs:

The measurement has an error σ_p (Gaia reaches about 0.02 mas for bright stars). The page treats a parallax as usable when p > 3σ_p, which limits this rung to d < 1/(3σ_p) (about 17 kpc for 0.02 mas, though with 10 % errors the useful range is shorter).

Rung 2: standard candles

If the true luminosity of a star is known, its apparent magnitude gives the distance through the distance modulus:

Classical Cepheids pulsate with a period tied to luminosity (Leavitt law, V band): M ≈ −2.43 (log₁₀P − 1) − 4.05, with an intrinsic scatter of about 0.1 mag. Parallaxes of nearby Cepheids calibrate the relation, which then reaches galaxies out to about 40 Mpc.

Rung 3: supernovae and the Hubble flow

Type Ia supernovae reach M ≈ −19.3 after standardisation (scatter about 0.15 mag) and are visible across billions of parsecs. For galaxies far enough that peculiar motions are small, the recession velocity follows Hubble's law:

The page simulates N supernovae at random distances 30-1500 Mpc, adds 0.15 mag brightness scatter and 250 km/s of peculiar velocity, estimates each distance from its apparent magnitude and fits a line through the origin: H₀ = Σv d / Σd².

Honest simplifications

• Euclidean Hubble law with no cosmological K-correction, extinction, time dilation of the light curve or curvature: it is accurate only for z below about 0.1 (d below about 400 Mpc); the high-distance part of the plot therefore uses the linear law as a model, not as a claim about the real Universe.
• The sky-shift panel draws the star's schematic distance not to scale and the real parallax ellipse is the true value times a fixed 0.5 ratio for the vertical axis (ecliptic latitude about 30 degrees).
• Parallax noise is a deterministic pseudo-random draw; Cepheid and supernova scatter come from a seeded generator.
• The real "Hubble tension" (about 67 versus 73 km/s/Mpc between early- and late-Universe methods) is not modelled; the preset only shows how a different true H₀ is recovered.

Checks run in the page

(1) The distance modulus inverts exactly over 28 distances from 1 pc to 5 Gpc. (2) The half-angle between sightlines from opposite sides of the orbit, computed with arctangents, equals 1000/d mas. (3) The fitted H₀ recovers the input within 3 %.

• Press Nearby star: parallax is large and its estimate is accurate; move the distance slider outward and watch the red error disc swallow the shift.
• Increase the parallax error to 1 mas: the parallax rung shrinks to about 300 pc.
• Press LMC Cepheid: parallax no longer works but the Cepheid rung does; change the period and see the absolute magnitude move on the Leavitt law.
• Raise the supernova scatter to 0.4 mag: the Hubble diagram widens and the fitted H₀ gets a bigger error bar.
• Press Hubble tension: H0 = 73 and compare the fit with the input line.