Galaxy Rotation Curve - Halo Density Profiles

Halo Density Profiles

In order to accommodate a flat rotation curve, a density profile for galactic environs must be different than one that is centrally concentrated. Newton's version of Kepler's Third Law states that the radial density profile ρ(r) equals

where v(r) is the radial orbital velocity profile and G is the gravitational constant. This profile closely matches the expectations of a singular isothermal sphere profile where if v(r) is approximately constant then the density to some inner "core radius" where the density leveled off to a constant. Observations did not comport with such a simple profile as reported by Navarro, Frenk, and White in a seminal 1996 paper:

If more massive halos were indeed associated with faster rotating disks and so with brighter galaxies, a correlation would be expected between the luminosity of binary galaxies and the relative velocity of their components. Similarly, there should be a correlation between the velocity of a satellite galaxy relative to its primary and the rotation velocity of the primary's disk. No such correlations are apparent in existing data.

The authors then remarked, as did a few others before them, that a "gently changing logarithmic slope" for a density profile could also accommodate approximately flat rotation curves over large scales. They wrote down the famous Navarro–Frenk–White profile which is consistent both with N-body simulations and observations given by

where the central density, ρ₀, and the scale radius, R_s, are parameters that vary from halo to halo. In part because the slope of the density profile diverges at the center, other alternative profiles have been proposed, for example, the Einasto profile which has exhibited as good or better agreement with certain dark matter halo simulations.