Galaxy Rotation Curve & Dark Matter Lab
Stars far out in a spiral galaxy orbit just as fast as stars near the bright center, which is not what the visible mass predicts. Tune the visible disk and a dark matter halo, then watch the total rotation curve flatten and match the observed points only when unseen mass is added.
Guided Experiment: Can you fit the flat rotation curve without any dark matter?
Set the dark halo velocity vInf to 0 so the galaxy has only visible matter. Predict whether you can match the observed flat rotation curve, which stays near 150 km/s out to 30 kpc, by adjusting only the visible mass and the visible scale radius.
Write your hypothesis in the Lab Report panel, then click Next.
Controls
The total mass of stars and gas in the bulge and disk. Visible matter alone gives a Keplerian decline beyond the bright disk.
The flat outer velocity that the dark matter halo settles to. Set vInf to 0 for no dark matter, and watch the outer curve fall off.
The core size of the pseudo-isothermal halo. A larger core makes the halo rise to its flat velocity more gradually.
Rotation curve
Galaxy structure
Rotation curve and dark matter
Good fit
The total curve tracks the observed points to an RMS of 5.1 km/s. A modest visible disk plus an extended dark halo holds the outer velocity flat at about 149 km/s.
Fit quality
Good fit
RMS residual 5.1 km/s
Outer dark matter fraction
81%
halo share of mass at 30 kpc
Peak velocity
153 km/s
highest point on the total curve
Velocity at 30 kpc
149 km/s
total curve, outer galaxy
Outer curve shape
Flat
stays near the peak
Visible only at 30 kpc
65 km/s
what stars and gas alone would give
The total velocity combines visible matter and the dark halo in quadrature, vTotal = √(vVis² + vHalo²). At 30 kpc the visible matter alone would give about 65 km/s and declining, but the observed curve stays near 149 km/s. That gap between the flat curve and the falling visible contribution is the dark matter.
Data Table
(0 rows)| # | Visible mass (1e10 Msun) | Scale radius (kpc) | vInf (km/s) | Core radius (kpc) | RMS residual (km/s) | Dark fraction (%) | Fit |
|---|
Reference Guide
Orbital Velocity and the Keplerian Expectation
A star on a circular orbit balances gravity against its orbital motion, so its speed depends on the mass enclosed inside its orbit. In astronomer units the orbital velocity is v = √(G M(r) / r).
- Inside the bright disk the enclosed mass rises, so the velocity rises.
- Beyond almost all the visible mass, M(r) stops growing.
- Then v falls off as 1 over √r, the Keplerian decline.
This is the same falloff that planets show in the Solar System, where almost all the mass is in the Sun. Visible matter alone predicts the same shape for a galaxy.
The Observed Flat Rotation Curve
Real spiral galaxies do not show the Keplerian decline. Measured rotation curves rise near the center and then stay flat, holding a roughly constant velocity far past the edge of the visible disk.
v(r) ≈ constant in the outer galaxy
A flat curve means the enclosed mass keeps growing in proportion to radius, M ∝ r, even where there is almost no visible light. Something massive and unseen must extend far beyond the stars.
The Dark Matter Halo Model
The lab adds a pseudo-isothermal dark matter halo on top of the visible disk. The halo velocity starts near zero at the center and rises to a flat asymptotic value vInf in the outer galaxy.
vHalo(r) = vInf √(1 − (rc / r) arctan(r / rc))
The visible and dark contributions add in quadrature, vTotal = √(vVis² + vHalo²). The halo keeps the outer curve flat by supplying mass that grows with radius, just what a flat rotation curve requires.
Dark Matter Fraction and the Evidence
At each radius the dark matter share of the dynamical mass is the halo velocity squared over the total velocity squared.
dark fraction = vHalo² / (vVis² + vHalo²)
In the outer galaxy this fraction climbs toward 1, so most of the mass is dark. Vera Rubin and Kent Ford measured flat rotation curves in spiral galaxies in the 1970s and showed that the outer regions orbit far faster than the visible mass can explain. Galaxy rotation curves remain a key piece of evidence for dark matter.
