Vera Rubin's Discovery & The Dark Matter Mystery
In Newtonian mechanics and Keplerian orbits, the velocity of orbiting matter drops off as $1/\sqrt{r}$ at large distances from the galactic mass center ($v \propto r^{-1/2}$). However, spectroscopic observations of spiral galaxies performed by Vera Rubin and Kent Ford revealed that stellar velocity profiles remain conspicuously flat even far beyond the visible edge of luminous stellar disks.
To account for this discrepancy without altering general relativity, astrophysicists model a surrounding non-luminous Navarro-Frenk-White (NFW) Dark Matter Halo whose density distribution follows $\rho(r) = \frac{\rho_0}{\frac{r}{r_s}\left(1 + \frac{r}{r_s}\right)^2}$. This produces an enclosed mass $M(r) \propto r$ at moderate radii, balancing the $1/r$ gravitational force gradient to yield $v(r) \approx \text{const}$.