Galaxy rotation curve

Remarks

Check of the formula: on a circular orbit the centripetal acceleration $v_c^2 / R$ equals the inward radial force per unit mass $\partial \Phi / \partial R$. The potential is linked to the mass density by Poisson's equation $\nabla^2 \Phi = 4 \pi G \rho$.

Example: for a spherical mass distribution with mass $M(r)$ inside radius $r$, Newton's law of gravitation gives $\partial \Phi / \partial r = G M(r)/r^2$, so $v_c(r)^2 = G M(r)/r$. For a point mass $M$ this gives the Keplerian curve $v_c = \sqrt{GM/r}$. For a density $\rho(r) = v_0^2 / (4 \pi G r^2)$ one gets $M(r) = v_0^2 r / G$ and the flat curve $v_c = v_0$.

Non-example: in a potential with $\partial \Phi / \partial R < 0$ at some radius the net force there points outward, so no circular orbit exists and $v_c$ is not defined at that radius.

In practice the rotation curve is measured from Doppler shifts of the gas, often the 21 cm line of neutral hydrogen.