Galaxy Rotation Curves and the Case for Dark Matter

A curve that refuses to fall

Point a telescope at a spiral galaxy, measure the Doppler shift of hydrogen gas or stars at increasing distance from the centre, and plot orbital speed against radius. Newtonian gravity makes a firm prediction: once you are past the bulk of the visible mass, speed should fall off as v(r) ∝ 1/√r, the same Keplerian decline that governs the planets around the Sun. That is not what galaxies do. From roughly the edge of the visible disk outward, v(r) stays flat — often within 10% of its peak value all the way to the last measurable gas cloud, tens of kiloparsecs out.

live demo · orbital speed vs radius, Newtonian prediction vs observed● LIVE

Vera Rubin and Kent Ford established this decisively in the 1970s and 1980s, extending earlier hints from Fritz Zwicky's 1933 study of the Coma cluster. Rubin measured rotation curves for dozens of spiral galaxies using an optical spectrograph and found the same flat shape every time, regardless of a galaxy's size or luminosity. The visible stars and gas — the part that emits light — account for only a fraction of the gravity needed to hold those outer orbits together at that speed.

Doing the accounting

The orbital speed of a test mass on a circular orbit is fixed by how much mass sits inside its radius: v(r) = √(G M(r) / r). Model a galaxy as three components — a central bulge, a thin stellar disk with an exponential surface-brightness profile, and gas — and each contributes its own v(r), added in quadrature. Sum them and the composite curve peaks near the edge of the optical disk and then falls, exactly like the planets. The observed curve does not fall. Something else is contributing mass at large radius that emits no light at all.

The NFW halo

The standard fix is a roughly spherical halo of dark matter enveloping the visible disk, with far more mass than the stars and gas combined. Cosmological N-body simulations of cold dark matter consistently produce halos with a characteristic density profile, named for Navarro, Frenk and White (1996):

ρ(r) = ρ0 / [ (r/rs) · (1 + r/rs)² ]

ρ0 = characteristic density,  rs = scale radius
inner region  (r << rs):  ρ ∝ 1/r        — cuspy
outer region  (r >> rs):  ρ ∝ 1/r³       — steep falloff

Integrating this profile to get enclosed mass M(r) and feeding it back into v(r) = √(GM(r)/r) produces a rotation-curve contribution that keeps rising gently and then flattens over a very wide range of radii — precisely the missing piece. Add the halo's contribution in quadrature to the disk and bulge and the composite curve matches the data: it climbs through the bulge-dominated centre, has a shoulder where the disk peaks, and then stays flat because the halo's growing enclosed mass compensates for the growing radius.

Why not just fix gravity?

The main rival explanation, Modified Newtonian Dynamics (MOND), replaces the halo with a change to the force law below a tiny acceleration threshold a0 ≈ 1.2×10⁻¹⁰ m/s², and it fits individual rotation curves impressively well with one free parameter per galaxy. Where it struggles is everything else dark matter also explains: the Bullet Cluster's separated mass and light, the cosmic microwave background's acoustic peaks, and large-scale structure formation. The dark-matter halo picture is the one that survives all of these tests simultaneously, which is why it remains the working model, even though the particle itself has never been directly detected.

What the demo is doing

The simulation builds v(r) from disk, bulge and NFW-halo mass profiles that you can dial independently, then overlays the pure-Newtonian, stars-only prediction so the gap is visible directly. Turn the halo mass to zero and you get the Keplerian fall-off Rubin never saw; dial in a realistic halo mass fraction — dark matter outweighs baryonic matter roughly 5 to 1 in the modern cosmological budget — and the curve goes flat, matching real 21 cm and optical rotation-curve surveys.

Frequently asked questions

Why does a flat rotation curve mean there is extra mass?

Orbital speed on a circular orbit depends only on the mass enclosed inside that radius: v = sqrt(GM(r)/r). If the visible light traces all the mass, v should fall as 1/sqrt(r) once you are past the bright disk, the same way planetary speeds fall with distance from the Sun. A flat curve instead means M(r) keeps growing with radius, which requires mass that keeps extending outward without emitting light.

Is dark matter the only explanation for flat rotation curves?

It is the dominant one because it also explains gravitational lensing by clusters, the separation of mass and light in colliding clusters like the Bullet Cluster, and the pattern of fluctuations in the cosmic microwave background. Modified gravity theories such as MOND fit individual rotation curves well but have a harder time with those other independent lines of evidence.

Why does the NFW profile fall off as 1/r cubed at large radius?

It comes from how cold, collisionless dark matter clumps under gravity in cosmological simulations: particles falling into a growing halo settle into an orbit-averaged density that steepens with radius. The r cubed falloff keeps the total halo mass finite while still extending the enclosed mass far beyond the visible disk, which is exactly what is needed to flatten the rotation curve.

Try it live

Everything above runs in your browser — open Galaxy Rotation Curves and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.

▶ Open Galaxy Rotation Curves simulation

Galaxy Rotation Curves: The Case for Dark Matter