🌌 Rotation Curve of a Spiral Galaxy

If only visible mass M(r) existed, orbital speed would fall off with distance, like planets around the Sun: v_Kepler(r) = √(G·M_vis / r) But real spiral galaxies keep spinning at nearly constant speed far past the visible disk. Adding an invisible halo whose mass grows roughly linearly with radius, ρ(r) ≈ ρ₀ / (1 + (r/r_s)²), reproduces the flat curve: v_obs(r) = √(G·(M_vis+M_halo) / r) ≈ const The gap between the two curves on the graph is the observational evidence for dark matter — mass we can't see but whose gravity we can measure.