Galaxy Rotation - 2D Top-Down Disc and Rotation Curve

For a circular orbit the gravitational pull of the mass inside radius r supplies the centripetal acceleration, so v(r)^2 = G M(<r) / r for a spherical distribution. Three components add their squared speeds: v^2 = v_disc^2 + v_bulge^2 + v_halo^2.

Disc: an exponential surface density Sigma = Sigma0 exp(-R/Rd) with the exact thin-disc result (Freeman 1970) v^2 = 4 pi G Sigma0 Rd y^2 [I0(y)K0(y) - I1(y)K1(y)], y = R/2Rd, using Bessel functions. Bulge: a Hernquist sphere, v^2 = G M r/(r + a)^2. Halo: an NFW profile, M(<r) = 4 pi rho0 rs^3 [ln(1 + x) - x/(1 + x)], x = r/rs.

Visible matter alone gives a curve that peaks and then falls nearly as r^(-1/2) (Keplerian). A dark halo keeps it flat, as radio observations of real spiral galaxies show. The orbital period T = 2 pi r / v grows with radius, so inner stars lap outer ones and spiral patterns wind up.

Simplifications: the mass numbers are round illustrative values (disc 5, bulge 1 and halo scale mass 30 in units of 10^10 solar masses), the halo is spherical and not fitted to a specific galaxy, and the arms are only a density pattern carried by the stars. The 3D page uses an empirical speed law; this page uses Newtonian mass models, so the shapes agree but the numbers are not identical. The mock observations are the default model with 6 percent scatter, not real data.

• Choose Visible matter only: the curve falls after the peak, and the outer tracer ring barely moves.
• Drag the halo strength up and down at fixed disc mass and watch the flat part rise.
• Turn on Mock observations and tune the halo until the curve passes through the points.