Editing AxisymmetricConfigurations/SolutionStrategies (section)

===Specifying Radial Rotation Profile in the Equilibrium Configuration===

Equilibrium axisymmetric structures &#8212; that is, solutions to the above set of simplified governing equations &#8212; can be found for specified angular momentum distributions that display a wide range of variations across both of the spatial coordinates, <math>~\varpi</math> and <math>~z</math>.  According to the [[2DStructure/AxisymmetricInstabilities#Poincar.C3.A9-Wavre_Theorem|Poincar&eacute;-Wavre theorem]], however, the derived structures will be dynamically unstable toward the development shape-distorting, meridional-plane motions unless the angular velocity is uniform on cylinders, that is, unless the angular velocity is independent of <math>~z</math>.   (See the detailed discussion by [<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>] &#8212; or our [[2DStructure/AxisymmetricInstabilities#Axisymmetric_Instabilities_to_Avoid|accompanying, brief summary]] &#8212; of this and other "axisymmetric instabilities to avoid.")  With this in mind, we will focus here on a solution strategy that is designed to construct structures with a

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<span id="SimpleRotation"><font color="#770000">'''Simple Rotation Profile'''</font></span>

<math>\dot\varphi(\varpi,z) = \dot\varphi(\varpi) ,</math>
</div>

which of course means that we will only be examining axisymmetric structures with specific angular momentum distributions of the form <math>~j(\varpi,z) = j(\varpi) = \varpi^2 \dot\varphi(\varpi)</math>.  

As has been alluded to immediately above, after adopting a simple rotation profile, it becomes useful to define an effective potential,
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<math>
\Phi_\mathrm{eff} \equiv \Phi + \Psi ,
</math>
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that is written in terms of a centrifugal potential,
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<math>
\Psi \equiv - \int \frac{j^2(\varpi)}{\varpi^3} d\varpi ~.
</math>
</div>
The accompanying table provides analytic expressions for <math>\Psi(\varpi)</math> that correspond to various prescribed functional forms for <math>\dot\varphi(\varpi)</math> or <math>j(\varpi)</math>, along with citations to published articles in which equilibrium axisymmetric structures have been constructed using the various tabulated ''simple rotation profile'' prescriptions. 

<span id="SRPtable">&nbsp;</span>
<table align="center" border="1" cellpadding="5">
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  <th align="center" colspan="7">
<font color="maroon">
''Simple Rotation Profiles'' <br />Found in the Published Literature
</font>
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<tr>
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&nbsp;
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<b><math>~\dot\varphi(\varpi)</math></b>
  </th>
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<b><math>~v_\varphi(\varpi)</math></b>
  </th>
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<b><math>~j(\varpi)</math></b>
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<b><math>~\frac{j^2}{\varpi^3}</math></b>
  </th>
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<b><math>~\Psi(\varpi)</math></b>
  </th>
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Refs.
  </th>
</tr>

<tr>
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<font color="maroon"><b>Power-law</b></font><br />(any <math>~q \neq 1</math>)
  </td>
  <td align="center">
<math>~\frac{j_0}{\varpi_0^2} \biggl( \frac{\varpi}{\varpi_0} \biggr)^{(q-2)}</math>
  </td>
  <td align="center">
<math>~\frac{j_0}{\varpi_0} \biggl( \frac{\varpi}{\varpi_0} \biggr)^{(q-1)}</math>
  </td>
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<math>~j_0\biggl( \frac{\varpi}{\varpi_0} \biggr)^{q}</math>
  </td>
  <td align="center">
<math>~\frac{j_0^2}{\varpi_0^3} \biggl( \frac{\varpi}{\varpi_0} \biggr)^{(2q-3)}</math>
  </td>
  <td align="center">
<math>~- \frac{1}{2(q-1)} \biggl[ \frac{j_0^2}{\varpi_0^2} \biggl( \frac{\varpi}{\varpi_0} \biggr)^{2(q-1)} \biggr]</math>
  </td>
  <td align="center">
d, h
  </td>
</tr>

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<font color="maroon"><b>Uniform rotation</b></font><br /><math>~(q = 2)</math>
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<math>~\omega_0</math>
  </td>
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<math>~\varpi \omega_0</math>
  </td>
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<math>~\varpi^2 \omega_0</math>
  </td>
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<math>~\varpi \omega_0^2</math>
  </td>
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<math>~- \frac{1}{2} \varpi^2 \omega_0^2</math>
  </td>
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a, f
  </td>
</tr>

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<font color="maroon"><b>Uniform</b></font> <math>v_\varphi</math><br /><math>~(q = 1)</math>
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<math>~\frac{v_0}{\varpi}</math>
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<math>~v_0</math>
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<math>~\varpi v_0</math>
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<math>~\frac{v_0^2}{\varpi}</math>
  </td>
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<math> ~- v_0^2 \ln\biggl( \frac{\varpi}{\varpi_0} \biggr)</math>
  </td>
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e
  </td>
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<font color="maroon"><b>Keplerian</b></font><br /><math>~(q = 1/2)</math>
  </td>
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<math>~\omega_K \biggl(\frac{\varpi}{\varpi_0}\biggr)^{-3/2}</math>
  </td>
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<math>~\varpi_0 \omega_K \biggl(\frac{\varpi}{\varpi_0}\biggr)^{-1/2}</math>
  </td>
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<math>~\varpi_0^2 \omega_K \biggl(\frac{\varpi}{\varpi_0}\biggr)^{1/2}</math>
  </td>
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<math>~\varpi_0 \omega_K^2 \biggl( \frac{\varpi}{\varpi_0} \biggr)^{-2}</math>
  </td>
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<math>~+ \frac{\varpi_0^3 \omega_K^2}{\varpi}  </math>
  </td>
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d
  </td>
</tr>

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<font color="maroon"><b>Uniform specific <br />angular momentum</b></font><br /><math>~(q = 0)</math>
  </td>
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<math>~\frac{j_0}{\varpi^2}</math>
  </td>
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<math>~\frac{j_0}{\varpi}</math>
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<math>~j_0</math>
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<math>~\frac{j_0^2}{\varpi^3}</math>
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<math>~+ \frac{1}{2} \biggl[ \frac{j_0^2}{\varpi^2} \biggr]</math>
  </td>
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c,g
  </td>
</tr>


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<font color="maroon"><b>j-constant <br />rotation</b></font>
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<math>~\omega_c \biggl[ \frac{A^2}{A^2 + \varpi^2} \biggr]</math>
  </td>
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<math>~\omega_c \biggl[ \frac{A^2 \varpi}{A^2 + \varpi^2} \biggr]</math>
  </td>
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<math>~\omega_c \biggl[ \frac{A^2 \varpi^2}{A^2 + \varpi^2} \biggr]</math>
  </td>
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<math>~\omega_c^2 \biggl[ \frac{A^4 \varpi}{(A^2 + \varpi^2)^2} \biggr]</math>
  </td>
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<math>~+ \frac{1}{2} \biggl[ \frac{\omega_c^2 A^4}{A^2 + \varpi^2} \biggr]</math>
  </td>
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a,b,i
  </td>
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<math>~n'</math><br />
<font color="maroon">Sequences</font>
  </td>
  <td align="center" colspan="5">See [[#Uniform-Density_Initially_.28n.27_.3D_0.29|discussion below]] of specific angular momentum distribution, <math>~h[m(\varpi)]</math></td>
  <td align="center">j,k,&#8467;,m</td>
</tr>

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  <td align="left" colspan="7">
<sup>f</sup>Maclaurin, C. 1742, ''A Treatise of Fluxions''<br />
<sup>j</sup>{{ Stoeckly65full }}<br />
<sup>k</sup>{{ OM68full }}<br />
<sup>&#8467;</sup>{{ BO73full }}<br />
<sup>i</sup>{{ Clement79full }}<br />
<sup>e</sup>{{ HNM82full }}<br />
<sup>g</sup>{{ PP84full }}<br />
<sup>a</sup>{{ Hachisu86afull }} (especially &sect;II.c)<br />
<sup>d</sup>{{ TH90full }}<br />
<sup>c</sup>{{ WTH94full }}<br />
<sup>m</sup>{{ PDD96full }}<br />
<sup>b</sup>{{ OT2006full }} (especially &sect;2.1)<br />
<sup>h</sup>The [[Appendix/Ramblings/HadleyAndImamuraSupplementaryDatabase#See_Also|Hadley &amp; Imamura collaboration]] (circa 2015) &nbsp;[Note that, as detailed [[Appendix/Ramblings/HadleyAndImamuraSupplementaryDatabase#Simple_Rotation_Profiles|elsewhere]], their definition of the power-law index, <math>q</math>, is different from ours.]

  </td>
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</table>

Note that, while adopting a ''simple rotation'' profile is ''necessary'' in order to ensure that an axisymmetric, barotropic equilibrium configuration is dynamical stability, it is not a ''sufficient'' condition.  For example, the [[2DStructure/AxisymmetricInstabilities#Solberg.2FRayleigh_Criterion|Solberg/Rayleigh criterion]] further demands that, for homentropic systems, the specific angular momentum, <math>~j</math>, must be an increasing function of the radial coordinate, <math>~\varpi</math>.  It is not surprising, therefore, that the above table of example ''simple rotation'' profiles does not include references to published investigations in which the power-law index, <math>~q</math>, is negative.

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<font color="darkgreen">
"In order to prevent the [[2DStructure/AxisymmetricInstabilities#Rayleigh-Taylor_Instability|Rayleigh-Taylor]] instability &hellip; which arises from an adverse distribution of angular momentum</font> &#8212; or, more generally, in order to satisfy the [[2DStructure/AxisymmetricInstabilities#Solberg.2FRayleigh_Criterion|Solberg/Rayleigh criterion]] &#8212;<font color="darkgreen"> <math>~j</math> must be a monotonically increasing function of <math>~m</math>.  Aside from this restriction, <math>~j(m)</math> is free to be any well-behaved function which we may plausibly expect to have been established over the history of the star."
</font>
</td></tr>
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&#8212; Drawn from p. 1084 of {{ OM68 }}
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