If a galaxy's mass were dominated by its visible stars and gas, orbital speed should fall off with radius the way planets do around the Sun (Keplerian decline). Real spiral galaxies instead show flat rotation curves: stars far from the center orbit almost as fast as ones near the middle. The only way to reconcile this with Newtonian gravity is extra, unseen mass — an extended dark-matter halo — whose enclosed mass keeps growing roughly linearly with radius.
v(r) = sqrt( G · M(<r) / r )
Keplerian (light-only): M(<r) → const ⇒ v(r) ∝ 1/√r
Flat curve (+ halo): M(<r) ∝ r ⇒ v(r) → constant
- Dark-matter halo mass — scales the invisible mass envelope; at 0 the curve falls off Keplerian, at high values it flattens hard.
- Visible disk mass — scales the star/gas mass that we can actually see and would predict alone.
- Time speed — speeds up or slows down the orbital animation.
- Rotation-curve toggle — shows/hides the live velocity-vs-radius graph overlaying observed (cyan) against Keplerian-predicted (dim) curves.
Vera Rubin's 1970s measurements of exactly this flatness in spiral galaxies like M31 provided some of the first strong evidence for dark matter.