Editing Galaxy rotation curve (section)

==Halo density profiles==
In order to accommodate a flat rotation curve, a density profile for a galaxy and its environs must be different than one that is centrally concentrated. Newton's version of [[Kepler's Third Law]] implies that the spherically symmetric, radial density profile {{math|''ρ''(''r'')}} is:
<math display="block">\rho(r) = \frac{v(r)^2 }{4 \pi G r^2}\left(1+2~ \frac{d\log v(r)}{d\log r}\right) </math>
where {{math|''v''(''r'')}} is the radial orbital velocity profile and {{math|''G''}} is the [[gravitational constant]]. This profile closely matches the expectations of a [[singular isothermal sphere profile]] where if {{math|''v''(''r'')}} is approximately constant then the density {{math|''ρ'' ∝ ''r''<sup>−2</sup>}} to some inner "core radius" where the density is then assumed constant. Observations do not comport with such a simple profile, as reported by Navarro, Frenk, and White in a seminal 1996 paper.<ref name="NAVARROETAL1996">{{cite journal |last1=Navarro |first1=J. F. |last2=Frenk |first2=C. S. |last3=White |first3=S. D. M. |year=1996 |title=The Structure of Cold Dark Matter Halos |journal=[[The Astrophysical Journal]] |volume=463 |pages=563–575 |arxiv=astro-ph/9508025 |bibcode=1996ApJ...462..563N |bibcode-access=free |doi=10.1086/177173 |doi-access=free}}</ref>

The authors then remarked that a "gently changing logarithmic slope" for a density profile function could also accommodate approximately flat rotation curves over large scales. They found the famous [[Navarro–Frenk–White profile]], which is consistent both with [[N-body simulations]] and observations given by
<math display="block">\rho (r) = \frac{\rho_0}{\frac{r}{R_s}\left(1+\frac{r}{R_s}\right)^2}</math>
where the central density, {{math|''ρ''<sub>0</sub>}}, and the scale radius, {{math|''R''<sub>''s''</sub>}}, are parameters that vary from halo to halo.<ref>{{Cite book |title=An Introduction to Modern Astrophysics |last2=Carroll |first2=Bradley W. |last1=Ostlie |first1=Dale A. |publisher=Cambridge University Press |year=2017 |pages=918 }}</ref> 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 better agreement with certain dark matter halo simulations.<ref>{{cite journal |last1=Merritt |first1=D. |last2=Graham |first2=A. |last3=Moore |first3=B. |last4=Diemand |first4=J. |last5=Terzić |first5=B. |year=2006 |title=Empirical Models for Dark Matter Halos. I. Nonparametric Construction of Density Profiles and Comparison with Parametric Models |journal=[[The Astronomical Journal]] |volume=132 |issue=6 |pages=2685–2700 |arxiv=astro-ph/0509417 |bibcode=2006AJ....132.2685M |bibcode-access=free |doi=10.1086/508988 |doi-access=free}}</ref><ref>{{cite journal |last1=Merritt |first1=D. |last2=Navarro |first2=J. F. |last3=Ludlow |first3=A. |last4=Jenkins |first4=A. |year=2005 |title=A Universal Density Profile for Dark and Luminous Matter? |journal=[[The Astrophysical Journal]] |volume=624 |issue=2 |pages=L85–L88 |arxiv=astro-ph/0502515 |bibcode=2005ApJ...624L..85M |bibcode-access=free |doi=10.1086/430636 |doi-access=free}}</ref>

Observations of orbit velocities in spiral galaxies suggest a mass structure according to:
<math display="block">v(r) = \left(r \, \frac{d\Phi}{dr}\right)^{1/2}</math>
with {{math|Φ}} the galaxy [[gravitational potential]].

Since observations of galaxy rotation do not match the distribution expected from application of Kepler's laws, they do not match the distribution of luminous matter.<ref name=Rubin1980/> This implies that spiral galaxies contain large amounts of dark matter or, alternatively, the existence of exotic physics in action on galactic scales. The additional invisible component becomes progressively more conspicuous in each galaxy at outer radii and among galaxies in the less luminous ones.{{clarify|date=August 2015}}

A popular interpretation of these observations is that about 26% of the mass of the Universe is composed of dark matter, a [[Hypothesis|hypothetical]] type of matter which does not emit or interact with [[electromagnetic radiation]]. Dark matter is believed to dominate the gravitational potential of galaxies and clusters of galaxies. Under this theory, galaxies are baryonic condensations of stars and gas (namely hydrogen and helium) that lie at the centers of much larger haloes of dark matter, affected by a gravitational instability caused by primordial density fluctuations.

Many cosmologists strive to understand the nature and the history of these ubiquitous dark haloes by investigating the properties of the galaxies they contain (i.e. their luminosities, kinematics, sizes, and morphologies). The measurement of the kinematics (their positions, velocities and accelerations) of the observable stars and gas has become a tool to investigate the nature of dark matter, as to its content and distribution relative to that of the various baryonic components of those galaxies.