Dark Matter: Galaxy Rotation Curves and the Evidence for Invisible Mass

The Newtonian expectation vs the flat curve

For a test mass orbiting a spherically symmetric mass distribution M(r), Newtonian gravity predicts v(r) = √(GM(r)/r). Past the edge of a galaxy's visible disk, M(r) stops growing — almost all the light-emitting matter sits inside a well-defined radius — so v(r) should fall as 1/√r, exactly like the outer planets orbiting the Sun. In the 1970s, astronomer Vera Rubin, working with Kent Ford, measured rotation curves for dozens of spiral galaxies using a sensitive spectrograph and found the opposite: curves stayed flat, sometimes even rising, far beyond the visible disk. The only way to explain a flat curve is if M(r) keeps growing with radius — a vast reservoir of unseen mass in an extended halo.

v(r) = √(G·M(r)/r) → if v(r) = v₀ = constant (flat curve)
then M(r) ∝ r  ← mass grows linearly with radius, not bounded
ρ(r) ∝ 1/r²   ← isothermal-sphere density, first halo model

Rubin's work, building on Fritz Zwicky's 1933 study of the Coma galaxy cluster, turned dark matter from a footnote into a central problem of modern astrophysics. Observed rotation curves imply spiral galaxies contain roughly 5–10 times more mass in dark matter than in ordinary stars, gas and dust combined — independently confirmed by gravitational lensing and the cosmic microwave background.

The NFW halo profile

N-body cosmological simulations of structure formation (Navarro, Frenk & White, 1996) found that cold dark matter halos settle into a near-universal shape, the NFW profile: ρ(r) = ρ₀ / [(r/rₛ)(1+r/rₛ)²]. Near the centre (r ≪ rₛ) density diverges as 1/r — a steep "cuspy" core still debated against flatter "cored" profiles suggested by dwarf-galaxy observations (the core–cusp problem). Far out (r ≫ rₛ) density falls as 1/r³, steeper than the isothermal sphere, which is why real rotation curves gently decline at very large radii instead of staying perfectly flat forever. Stars and gas contribute only ~10–15% of a galaxy's mass; the dark halo supplies the remaining ~85–90% and extends several times further than the visible disk.

MOND: the rival explanation

Modified Newtonian Dynamics (MOND), proposed by Mordehai Milgrom in 1983, takes a different route: instead of adding unseen mass, it modifies gravity itself below an acceleration scale a₀ ≈ 1.2×10⁻¹⁰ m/s². MOND fits individual galaxy rotation curves impressively well with very few free parameters, but it struggles to explain cluster-scale gravitational lensing and the fine structure of the cosmic microwave background — both of which fit cold dark matter cleanly. Most cosmologists treat dark matter as the leading explanation, with MOND remaining an active minority research programme.

Frequently asked questions

Try it live

Everything above runs in your browser — open Dark Matter and compare the predicted Keplerian drop-off against the observed flat rotation curve as you reshape the dark matter halo. Nothing is installed, nothing is uploaded.