Editing DarkMatter/CK2015 (section)

==Building Equilibrium Models with Flat Rotation Curves==

It finally dawned on me that CK15 are pointing out &#8212; what has been obvious (Ha!) to me for years &#8212; that if you want to build an equilibrium, self-gravitating configuration in which all of the equatorial-plane motions are perfectly circular, the model result will be ''very'' different if you're dealing with a gaseous configuration than if you're dealing with a stellar-dynamic system.  

* In stellar-dynamic (or ''aged'' planetary!) systems, you're pretty much stuck with "Keplerian" velocity profiles; if you want to generate a flat rotation curve, you'd better modify gravity!  
* But in gas-dynamical systems, radial pressure gradients can help you out big time!  Indeed, you can ''a priori'' choose almost any rotation profile that you like and feel confident that you can build an equilibrium, axisymmetric configuration with that profile; the density/pressure profile will readily bend to your wishes and, as CK15 emphasize, the gravitational potential will happily tag along.

In a chapter of my online H_Book, I constructed a table that lists a wide variety of "[[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|simple rotation profiles]]" that have been ''chosen'' by different researchers over the years as they investigated the equilibrium and/or stability properties of a wide variety of self-gravitation ''gaseous'' configurations.  So we've all "known" this for years!  But, it just so happens that v = constant was almost never the choice!  The paper by {{ HNM82full }} &#8212; see  [[Apps/HayashiNaritaMiyama82#Rotationally_Flattened_Isothermal_Structures|our related chapter discussion]] &#8212; presents one exception; and that paper was exceptional because the authors were actually able to find an ''analytic'' solution.  

Why has everyone (before CK15) chosen profiles that are not v-constant?  I think the answer is that, since the equations allow you to specify virtually any profile ''a priori'', everyone looked for a '''physically justified''' reason for picking a particular profile.  Observations gave us no help because, after all, you generally can't see ''inside'' of a rotationally flattened gaseous object.  Favorite choices were:  Uniform rotation (a natural result of viscosity); uniform specific angular momentum (because that is the power-law limit that is marginally stable against the axisymmetric Raleigh instability); and near-Keplerian (because that gives flat disks in which pressure is not very important).  There is also the so-called n' rotation profile used by <!-- {{ OB73full }} --> [http://adsabs.harvard.edu/abs/1973ApJ...180..171O Ostriker &amp; Bodenheimer (1973)] &#8212; see, for example, their &sect;IIIa, which references equilibrium configurations with (n = 3/2, n' = 0, 1, 3/2).  This example means:  build a model with an n = 3/2 equation of state, but with an angular momentum profile that is the same as in a ''uniformly-rotating'' n = 0 sphere, or n = 1 sphere, or n = 3/2 sphere.  Why should anyone try a v-constant profile?  Until CK15(!), there was no physical justification for picking v-constant.  (Except, of course, for {{ HNM82 }}, who picked this ''strange'' profile because it allowed them to uncover an analytic description of the underlying structure.)  

So, CK15 think they understand why nature picks v-constant profiles in gaseous disks.  There argument is, first of all, that '''observations''' are showing us that nature likes this particular profile.  It just so happens that galaxy disks are the only gaseous structures that have allowed astronomers to "see inside" the gaseous structure; and nature is saying, I am not so interested in uniform rotation, or uniform specific angular momentum, or even near-Keplerian!  I like flat rotation curves!  CK15 think that they know why flat rotation curves are preferred; and they are sufficiently confident in this physical reasoning to ''predict'' that young planetary disks also will reveal flat rotation curves!  I wish them luck with this prediction!

''VERY IMPORTANT NOTE'':  It is very important to appreciate that the best observations that show flat rotation curves in galaxies come from measurements of radiation from ''gas'', and not from the underlying stars.  (See comments on ''stellar'' dynamic systems below.)  The radio observations are measuring doppler shifts in neutral hydrogen (HI).  Even Vera Rubin's optical measurements come from the emission lines in HII regions; while stars are associated with (and actually are responsible for heating) these regions, they are very young and have just been born from the gaseous (flat rotation-curve) structure.