Galaxy Rotation Curves
Rotation and mass curves
Rotation curve showing velocity versus radius for the currently selected galaxy preset.
Core equations
Dark matter evidence appears when the observed $V(R)$ stays flat out where the visible-matter-only curve is already declining.
Quadrature composition
$$V_{\rm total}(R)=\sqrt{V_{\rm bulge}^2+V_{\rm disk}^2+V_{\rm halo}^2}$$
Dark matter inference
$$M_{\rm dark}(<R)=\frac{V_{\rm obs}^2R}{G}-M_{\rm vis}(<R)$$
Visible-matter benchmark
$$V_{\rm Kep}(R)\propto R^{-1/2} \quad (R\gg R_d)$$
MOND comparison
$$V_{\rm MOND}\approx\left(G\,M_{\rm vis}\,a_0\right)^{1/4}$$
Dark matter $\neq$ dark energy
Dark matter is non-luminous mass that clumps in galaxy halos and clusters. Dark energy is a separate phenomenon — it drives the accelerating expansion of the universe. They share the word "dark" but are unrelated physics.
How to read these panels
Build a galaxy mass model and compare the observed flat rotation curve to what the visible matter alone can produce.
Galaxy schematic. Face-on pedagogical view. Published curves are inclination-corrected intrinsic $V(R)$. Rotation and mass curves. Compare the total curve, the visible-matter-only curve, the individual components, and an optional MOND prediction.
What to notice
- When the dark halo is removed, the rotation curve falls off as $R^{-1/2}$ at large radii, approaching the Keplerian limit you expect far outside the visible mass. Closer in the disk is not a point mass, so the curve is not Keplerian there.
- Flat rotation curves at large $R$ require extra mass beyond what we see — the dark halo component supplies this.
- The dark-to-visible mass ratio grows with radius: dark matter dominates the outskirts of galaxies.
Model notes
- The halo uses an NFW (Navarro-Frenk-White) density profile; the disk uses an exponential surface density; the bulge uses a Hernquist profile.
- Component velocities add in quadrature because the gravitational potentials superpose: $V^2 = V_{\rm bulge}^2 + V_{\rm disk}^2 + V_{\rm halo}^2$. The disk term is the exact Freeman solution for an exponential surface density, not a spherical approximation to it.
- The MOND curve uses the simple interpolating function $\mu(x) = x/(1+x)$, so $a = \tfrac{1}{2}a_N\left[1 + \sqrt{1 + 4a_0/a_N}\right]$: Newtonian at small $R$ where $a_N \gg a_0$, and approaching the deep-MOND limit $V^4 \to G\,M\,a_0$ at large $R$. Its baryonic acceleration $a_N = V_{\rm baryons}^2/R$ comes from the same exponential disk and bulge as the Newtonian curve, so the two lines differ only by the interpolation.
- Rotation curves are one line of evidence for dark matter. Cluster dynamics, gravitational lensing (Bullet Cluster), and CMB anisotropies all indicate most matter is non-luminous.
Explore further
- Doppler Shift: The 21-cm readout in this demo uses the same Doppler formula. Open Doppler Shift
- Spectral Lines: See why hydrogen emits at specific wavelengths, including the 21-cm spin-flip line. Open Spectral Lines