Editing AxisymmetricConfigurations/SolutionStrategies (section)

=Axisymmetric Configurations (Solution Strategies)=
<!--
Equilibrium, axisymmetric '''structures''' are obtained by searching for time-independent, steady-state solutions to the [[AxisymmetricConfigurations/PGE#Axisymmetric_Configurations_.28Part_I.29|identified set of simplified governing equations]].  We begin by writing each governing equation in Eulerian form and setting all partial time-derivatives to zero:

<div align="center">
<span id="Continuity"><font color="#770000">'''Equation of Continuity'''</font></span>

<math>\cancelto{0}{\frac{\partial\rho}{\partial t}} + \frac{1}{\varpi} \frac{\partial}{\partial\varpi} \biggl[ \rho \varpi \dot\varpi \biggr] 
+ \frac{\partial}{\partial z} \biggl[ \rho \dot{z} \biggr] = 0 </math><br />


<span id="PGE:Euler">The Two Relevant Components of the<br />
<font color="#770000">'''Euler Equation'''</font>
</span>
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~
\cancelto{0}{\frac{\partial \dot\varpi}{\partial t}} + \biggl[ \dot\varpi \frac{\partial \dot\varpi}{\partial\varpi} \biggr] + 
\biggl[ \dot{z} \frac{\partial \dot\varpi}{\partial z} \biggr]  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] + \frac{j^2}{\varpi^3}  
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~
\cancelto{0}{\frac{\partial \dot{z}}{\partial t}} + \biggl[ \dot\varpi \frac{\partial \dot{z}}{\partial\varpi} \biggr] + 
\biggl[ \dot{z} \frac{\partial \dot{z}}{\partial z} \biggr]
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr] 
</math>
  </td>
</tr>
</table>

<span id="PGE:AdiabaticFirstLaw">Adiabatic Form of the<br />
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />
<math>~
\biggl\{\cancel{\frac{\partial \epsilon}{\partial t}} + \biggl[ \dot\varpi \frac{\partial \epsilon}{\partial\varpi} \biggr] + \biggl[ \dot{z} \frac{\partial \epsilon}{\partial z} \biggr]\biggr\} +
P \biggl\{\cancel{\frac{\partial }{\partial t}\biggl(\frac{1}{\rho}\biggr)} + 
\biggl[ \dot\varpi \frac{\partial }{\partial\varpi}\biggl(\frac{1}{\rho}\biggr) \biggr] + 
\biggl[ \dot{z} \frac{\partial }{\partial z}\biggl(\frac{1}{\rho}\biggr) \biggr] \biggr\} = 0
</math>


<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

<math>
\frac{1}{\varpi} \frac{\partial }{\partial\varpi} \biggl[ \varpi \frac{\partial \Phi}{\partial\varpi} \biggr] + \frac{\partial^2 \Phi}{\partial z^2} = 4\pi G \rho . 
</math><br />
</div>


The steady-state flow field that will be adopted to satisfy both an axisymmetric geometry and the time-independent constraint is, <math>~\vec{v} = \hat{e}_\varphi (\varpi \dot\varphi)</math>.  That is, <math>~\dot\varpi = \dot{z} = 0</math> but, in general, <math>~\dot\varphi</math> is not zero and can be an arbitrary function of <math>~\varpi</math> and <math>~z</math>, that is, <math>~\dot\varphi = \dot\varphi(\varpi,z)</math>.  We will seek solutions to the above set of coupled equations for various chosen spatial distributions of the angular velocity <math>~\dot\varphi(\varpi,z)</math>, or of the specific angular momentum, <math>~j(\varpi,z) = \varpi^2 \dot\varphi(\varpi,z)</math>.


<span id="2DgoverningEquations">After setting the radial and vertical velocities to zero,</span> we see that the <math>1^\mathrm{st}</math> (continuity) and <math>4^\mathrm{th}</math> (first law of thermodynamics) equations are trivially satisfied while the <math>2^\mathrm{nd}</math> &amp; <math>3^\mathrm{rd}</math> (Euler) and <math>5^\mathrm{th}</math> (Poisson) give, respectively,

<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~
\biggl[ \frac{1}{\rho}\frac{\partial P}{\partial\varpi} + \frac{\partial \Phi}{\partial\varpi}\biggr] - \frac{j^2}{\varpi^3}  
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~
\biggl[ \frac{1}{\rho}\frac{\partial P}{\partial z} + \frac{\partial \Phi}{\partial z} \biggr] 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~
\frac{1}{\varpi} \frac{\partial }{\partial\varpi} \biggl[ \varpi \frac{\partial \Phi}{\partial\varpi} \biggr] + \frac{\partial^2 \Phi}{\partial z^2}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi G \rho \, .</math>
  </td>
</tr>
</table>

</div>

As has been outlined in our discussion of [[SR#Time-Independent_Problems|supplemental relations for time-independent problems]], in the context of this H_Book we will close this set of equations by specifying a structural, barotropic relationship between {{Math/VAR_Pressure01}} and {{Math/VAR_Density01}}. 

==Solution Strategy==
-->

==Lagrangian versus Eulerian Representation==
In our overarching specification of the set of [[PGE#Principal_Governing_Equations|''Principal Governing Equations'']], we have included a,

<div align="center">
<span id="ConservingMomentum:Lagrangian"><font color="#770000">'''Lagrangian Representation'''</font></span><br />
of the Euler Equation,

{{Math/EQ_Euler01}}

[<b>[[Appendix/References#BLRY07|<font color="red">BLRY07</font>]]</b>], p. 13, Eq. (1.55)
</div>

When seeking a solution to the set of governing equations that describes a steady-state equilibrium configuration &#8212; as has already been suggested in our [[PGE/Euler#Eulerian_Representation|accompanying discussion of "other forms of the Euler equation"]] &#8212; it is preferable to start from an,
<div align="center">
<span id="ConservingMomentum:Eulerian"><font color="#770000">'''Eulerian Representation'''</font></span><br />
of the Euler Equation,

{{Math/EQ_Euler02}}
</div>


because steady-state configurations are identified by setting the ''partial'' time derivative, rather than the ''total'' time derivative, to zero.  Notice that if the objective is to find an equilibrium configuration in which the fluid velocity is not zero &#8212; consider, for example, a configuration that is rotating &#8212; then throughout the configuration, the velocity field must be taken into account, in addition to the gradient in the gravitational potential, when determining the pressure distribution.  Specifically, for steady-state flows, the required relationship is,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \frac{1}{\rho} \nabla P</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \nabla \Phi - (\vec{v} \cdot \nabla) \vec{v} \, .</math>
  </td>
</tr>
</table>

As we also have [[PGE/Euler#in_terms_of_the_vorticity:|mentioned elsewhere]], by drawing upon a relevant [https://en.wikipedia.org/wiki/Vector_calculus_identities#Dot_product_rule dot product rule vector identity], this expression can be rewritten in terms of the fluid vorticity, <math>~\vec\zeta \equiv \nabla\times\vec{v}</math>, as,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \frac{1}{\rho} \nabla P</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \nabla \biggl[ \Phi + \frac{1}{2}\vec{v}\cdot \vec{v} \biggr] -  \vec\zeta \times \vec{v} \, .</math>
  </td>
</tr>
</table>

<span id="CentrifugalPotential">In certain astrophysically relevant situations &#8212; such as the adoption of any one of the ''simple rotation'' profiles identified immediately below &#8212; the nonlinear velocity term involving the "convective operator" can be rewritten in terms of the gradient of a scalar (centrifugal) potential, that is,</span>
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(\vec{v} \cdot \nabla) \vec{v}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\nabla \Psi \, .</math>
  </td>
</tr>
</table>
In such cases, the condition required to obtain a steady-state equilibrium configuration is given by the considerably simpler mathematical relation,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~ \frac{1}{\rho} \nabla P</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \nabla \biggl[ \Phi + \Psi \biggr]  \, .</math>
  </td>
</tr>
</table>

In the subsection of this chapter (below) titled, [[#Double_Check_Vector_Identities|''Double Check Vector Identities,'']] we explicitly demonstrate for four separate "simple rotation profiles"  that these three separate steady-state balance expressions do indeed generate identical mathematical relations.

==Simple Rotation Profile and Centrifugal Potential==
{| class="AxisymmetricConfigurations" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
|- 
! style="height: 150px; width: 150px; background-color:white;" |[[H_BookTiledMenu#Two-Dimensional_Configurations_.28Axisymmetric.29|<b>''Simple''<br />Rotation<br />Profiles</b>]]
|}

<table border="0" cellpadding="3" align="center" width="60%">
<tr><td align="left">
<font color="darkgreen">
"&hellip; A necessary and sufficient condition for <math>\dot{\varphi}</math> &hellip; to be independent of <math>z</math> is that the surfaces of constant pressure coincide with the surfaces of constant density, i.e., that P be a function of &rho; only."</font>  In this case, a centrifugal potential, <math>\Psi</math>, can be defined &#8212; see the integral expression provided below &#8212; and it "<font color="darkgreen">is also a function of <math>\rho</math> only &hellip;  When <math>\Psi</math> exists, the equations of state and of energy conservation may be thought of as determining the form of the P-&rho; relationship. Hence, by prescribing a P-&rho; relationship, one avoids the complications of those further equations.  This affects a major simplification of the formal problem of constructing rotating configurations.  This procedure will, of course, be inadequate for certain objectives &hellip;"
</font>
</td></tr>
<tr><td align="right">
&#8212; Drawn from p. 466 of {{ Lebovitz67_XXXIV }} 
</td></tr></table>
===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

<div align="center">
<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,
<div align="center">
<math>
\Phi_\mathrm{eff} \equiv \Phi + \Psi ,
</math>
</div>
that is written in terms of a centrifugal potential,
<div align="center">
<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">
<tr>
  <th align="center" colspan="7">
<font color="maroon">
''Simple Rotation Profiles'' <br />Found in the Published Literature
</font>
  </th>
</tr>

<tr>
  <td align="center">
&nbsp;
  </td>
  <th align="center">
<b><math>~\dot\varphi(\varpi)</math></b>
  </th>
  <th align="center">
<b><math>~v_\varphi(\varpi)</math></b>
  </th>
  <th align="center">
<b><math>~j(\varpi)</math></b>
  </th>
  <th align="center">
<b><math>~\frac{j^2}{\varpi^3}</math></b>
  </th>
  <th align="center">
<b><math>~\Psi(\varpi)</math></b>
  </th>
  <th align="center">
Refs.
  </th>
</tr>

<tr>
  <td align="center">
<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>
  <td align="center">
<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>

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

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

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

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


<tr>
  <td align="center">
<font color="maroon"><b>j-constant <br />rotation</b></font>
  </td>
  <td align="center">
<math>~\omega_c \biggl[ \frac{A^2}{A^2 + \varpi^2} \biggr]</math>
  </td>
  <td align="center">
<math>~\omega_c \biggl[ \frac{A^2 \varpi}{A^2 + \varpi^2} \biggr]</math>
  </td>
  <td align="center">
<math>~\omega_c \biggl[ \frac{A^2 \varpi^2}{A^2 + \varpi^2} \biggr]</math>
  </td>
  <td align="center">
<math>~\omega_c^2 \biggl[ \frac{A^4 \varpi}{(A^2 + \varpi^2)^2} \biggr]</math>
  </td>
  <td align="center">
<math>~+ \frac{1}{2} \biggl[ \frac{\omega_c^2 A^4}{A^2 + \varpi^2} \biggr]</math>
  </td>
  <td align="center">
a,b,i
  </td>
</tr>

<tr>
  <td align="center">
<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>

<tr>
  <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>
</tr>
</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.

<table border="0" cellpadding="3" align="center" width="60%">
<tr><td align="left">
<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>
<tr><td align="right">
&#8212; Drawn from p. 1084 of {{ OM68 }}
</td></tr></table>

===Prescribing Mass-Dependent Rotation Profile Based on an Initial Spherical Configuration===

Each of the ''simple rotation profiles'' listed in Table 1 has been defined by specifying the radial distribution of the specific angular momentum, <math>j(\varpi)</math>, ''in the rotationally flattened equilibrium configuration.''  Here we follow the lead of {{ Stoeckly65full }}, of {{ BO73full }} and of {{ MPT77full }} and, instead, present rotation profiles that are defined by specifying the function, <math>j(m_\varpi)</math>, where <math>m_\varpi</math> is a function describing how the fractional mass enclosed inside <math>\varpi</math> varies with <math>\varpi</math>.

To better clarify what is meant by the function, <math>m_\varpi</math>, consider a configuration (not necessarily in equilibrium) that is spherically symmetric and that exhibits an &#8212; as yet unspecified &#8212; mass-density profile, <math>\rho(r)</math>.  The [[SSCpt2/SolutionStrategies#Solution_Strategies|mass enclosed within each spherical radius]] is,
<div align="center">
<math>M_r = \int_0^r 4\pi r^2 \rho( r ) dr \, ,</math>
</div>
and, if the radius of the configuration is <math>R</math>, then the configuration's total mass is,
<div align="center">
<math>M = \int_0^R 4\pi r^2 \rho( r ) dr \, .</math>
</div>

In contrast, the mass enclosed within each ''cylindrical'' radius, <math>\varpi</math>, is
<div align="center">
<math>M_\varpi = 2\pi \int_0^\varpi \varpi d\varpi \int_0^{\sqrt{R^2 - \varpi^2}} \rho( r ) 2dz \, ,</math>
</div>
where it is understood that the argument of the density function is, <math>r = \sqrt{\varpi^2 + z^2} </math>.  

<span id="Example1">'''Example #1''':</span>  If the configuration has a uniform density, <math>\rho_0</math>, then its total mass is, <math>M = 4\pi \rho_0 R^3/3</math>, and

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>M_\varpi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4\pi \rho_0 \int_0^\varpi \varpi [R^2 - \varpi^2]^{1 / 2}d\varpi 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi}{3} \rho_0 \biggl[R^3 - (R^2 - \varpi^2)^{3 / 2} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>M -
\frac{4\pi}{3} \rho_0 \biggl[(R^2 - \varpi^2)^{3 / 2} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~m_\varpi \equiv \frac{M_\varpi}{M}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>1 -
\biggl[1 - \frac{\varpi^2}{R^2}\biggr]^{3 / 2} \, .
</math>
  </td>
</tr>
</table>
</div>

'''Example #2''':  If the spherically symmetric configuration has a density profile given by the function,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\rho(r)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_0 \biggl[\frac{\sin (\pi r/R)}{\pi r/R}  \biggr] \, ,</math>
  </td>
</tr>
</table>
</div>

then [[SSC/Structure/Polytropes#n_.3D_1_Polytrope|its total mass]] is, <math>M = 4 \rho_0 R^3/\pi</math>, and

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>M_\varpi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4\pi \rho_0\int_0^\varpi \varpi d\varpi \int_0^{\sqrt{R^2 - \varpi^2}} 
\biggl\{ \frac{\sin (\pi \sqrt{\varpi^2 + z^2} /R)}{\pi \sqrt{\varpi^2 + z^2} /R}  \biggr\} dz
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4 \rho_0 R^3\int_0^\chi \chi d\chi \int_0^{\sqrt{1 - \chi^2}} 
\biggl\{ \frac{\sin (\pi \sqrt{\chi^2 + \zeta^2} )}{\sqrt{\chi^2 + \zeta^2}}  \biggr\} d\zeta
</math>
  </td>
</tr>
</table>
</div>

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>M_\varpi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4\pi \rho_0 \biggl\{ \int_{\sqrt{R^2 - \varpi^2}}^R dz 
\int_0^\sqrt{R^2-z^2} \biggl[ \frac{\sin (\pi \sqrt{\varpi^2 + z^2} /R)}{\pi \sqrt{\varpi^2 + z^2} /R}  \biggr] \varpi d\varpi
+ \int_0^{\sqrt{R^2 - \varpi^2}} dz 
\int_0^\varpi \biggl[ \frac{\sin (\pi \sqrt{\varpi^2 + z^2} /R)}{\pi \sqrt{\varpi^2 + z^2} /R}  \biggr] \varpi d\varpi
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4 \rho_0 R^3 \biggl\{ \int_{\sqrt{1 - \chi^2}}^1 d\zeta 
\int_0^\sqrt{1-\zeta^2} \biggl[ \frac{\sin (\pi \sqrt{\chi^2 + \zeta^2})}{ \sqrt{\chi^2 + \zeta^2}}  \biggr] \chi d\chi
+ \int_0^{\sqrt{1 - \chi^2}} d\zeta 
\int_0^\chi \biggl[ \frac{\sin (\pi \sqrt{\chi^2 + \zeta^2})}{ \sqrt{\chi^2 + \zeta^2}}  \biggr] \chi d\chi
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4 \rho_0 R^3 \biggl\{ \int_{\sqrt{1 - \chi^2}}^1 
\biggl[  - \frac{ \cos(\pi\sqrt{\zeta^2 + \chi^2})}{\pi}    \biggr]_0^\sqrt{1-\zeta^2} d\zeta
+ \int_0^{\sqrt{1 - \chi^2}} 
\biggl[ - \frac{ \cos(\pi\sqrt{\zeta^2 + \chi^2})}{\pi}  \biggr]_0^\chi  d\zeta
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4 \rho_0 R^3}{\pi} \biggl\{ \int_{\sqrt{1 - \chi^2}}^1 
\biggl[  - \cos(\pi) + \cos(\pi\zeta)  \biggr] d\zeta
+ \int_0^{\sqrt{1 - \chi^2}} 
\biggl[ \cos(\pi\zeta ) - \cos(\pi\sqrt{\zeta^2 + \chi^2}) \biggr]  d\zeta
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4 \rho_0 R^3}{\pi} \biggl\{ \int_{\sqrt{1 - \chi^2}}^1 d\zeta
+  \int_0^1 \cos(\pi\zeta)   d\zeta
- \int_0^{\sqrt{1 - \chi^2}} 
\cos(\pi\sqrt{\zeta^2 + \chi^2})  d\zeta
\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4 \rho_0 R^3}{\pi} \biggl\{ \biggl[ z \biggr]_{\sqrt{1 - \chi^2}}^1
+  \frac{1}{\pi} \int_0^\pi \cos(u)   du
- \int_0^{\sqrt{1 - \chi^2}} 
\cos(\pi\sqrt{\zeta^2 + \chi^2})  d\zeta
\biggr\}
</math>
  </td>
</tr>
</table>
</div>

====Uniform-Density Initially (n' = 0)====

Drawing directly from &sect;IIc of {{ Stoeckly65 }}, <font color="orange">&hellip; consider a large, gaseous mass, initially a homogeneous sphere of mass <math>M</math> and angular momentum <math>J</math> rotating as a solid body, and suppose it contracts in such a way that cylindrical surfaces remain cylindrical and each such surface retains its angular momentum.  Let <math>\rho_0</math>, <math>R_0</math>, and <math>\dot\varphi_0</math> denote the initial density, radius, and angular velocity of the</font> [initially unstable configuration]<font color="orange">, <math>\varpi_0(\varpi)</math> the initial radius of the surface now at radius <math>\varpi</math>, and <math>M_\varpi(\varpi)</math> the mass inside this surface.  The conditions on the contraction are then</font>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>M - M_\varpi(\varpi)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
4\pi \rho_0 \int_{\varpi_0(\varpi)}^{R_0} \biggl[ \biggl(R_0^2 - (\varpi_0^')^2\biggr) \biggr]^{1 / 2} \varpi_0^' d\varpi_0^' \, ,
</math>
  </td>
</tr>
</table>
</div>
<font color="orange">and</font>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\dot\varphi(\varpi) \varpi^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\dot\varphi_0 [\varpi_0(\varpi)]^2 \, .</math>
  </td>
</tr>
</table>
</div>
<font color="orange">By integrating, eliminating <math>\varpi_0(\varpi)</math> between these equations, and eliminating <math>\rho_0</math>, <math>R_0</math>, and <math>\dot\varphi_0</math> in favor of <math>M</math> and <math>J</math>, one finds the relation of <math>\dot\varphi(\varpi)</math> to the mass distribution to be</font>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\dot\varphi(\varpi)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{5J}{2M\varpi^2}\biggl\{ 1 - [1 - m(\varpi) ]^{2 / 3} \biggr\} \, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ Stoeckly65 }}, &sect;II.c, eq. (12)
  </td>
</tr>
</table>
</div>
where, the mass fraction,
<div align="center">
<math>m(\varpi) \equiv \frac{M_\varpi(\varpi)}{M} \, .</math>
</div>

As noted, this is equation (12) of {{ Stoeckly65 }}; it also appears, for example, as equation (45) in {{ OM68 }}, as equation (12) in {{ BO70full }}, and in the sentence that follows equation (3) in {{ BO73 }}.  As Stoeckly points out, the angular momentum distribution implied by this functional form of <math>~\dot\varphi</math> satisfies the [[2DStructure/AxisymmetricInstabilities#Solberg.2FRayleigh_Criterion|Solberg/Rayleigh stability criterion]] &#8212; that is,
<div align="center">
<math>\frac{dj^2}{d\varpi} > 0 </math>
</div>
&#8212; initially, and also in the final equilibrium configuration because every cylindrical surface conserves specific angular momentum and the surfaces do not reorder themselves.

<table border="1" width="80%" align="center" cellpadding="5"><tr><td align="left">
We should be able to obtain the identical result by extending [[#Example1|Example 1]] above.  Attaching the subscript "0" to <math>\varpi</math> in order to acknowledge that, here, the initial configuration is a uniform-density sphere (n' = 0), our derivation gives,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>m_\varpi \equiv \frac{M_\varpi}{M}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>1 -
\biggl[1 - \frac{\varpi_0^2}{R^2}\biggr]^{3 / 2} \, ,
</math>
  </td>
</tr>
</table>
from which we see that,
<table border="0" align="center" cellpadding="5">
<tr>
  <td align="right">
<math>\frac{\varpi_0^2}{R^2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1 - \biggl[1 - m_\varpi \biggr]^{2 / 3} \, .
</math>
  </td>
</tr>
</table>
Now, the total angular momentum, <math>J</math>, of this initial configuration &#8212; a uniformly rotating <math>(\dot\varphi_0)</math>, uniform-density sphere &#8212; is,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>J = I{\dot\varphi}_0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{2}{5}MR^2{\dot\varphi}_0
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ {\dot\varphi}_0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{5J}{2MR^2} \, ,
</math>
  </td>
</tr>
</table>
in which case, the specific angular momentum of each fluid element &#8212; which is conserved as the configuration contracts or expands &#8212; is given by the expression,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>\dot\varphi \varpi^2 = {\dot\varphi}_0 \varpi_0^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{5J}{2MR^2} \cdot \varpi_0^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{5J}{2M} \biggl\{ 1 - \biggl[1 - m_\varpi \biggr]^{2 / 3}\biggr\} \, .
</math>
  </td>
</tr>
</table>
Q.E.D.
</td></tr></table>

Now, just as the fraction of the configuration's mass that lies ''interior to'' radial position, <math>\varpi</math>, is detailed by the function, <math>m_\varpi</math>, let's use <math>\ell_\varpi</math> to detail what fraction of the configuration's angular momentum lies ''interior to'' <math>m_\varpi</math>.  We have,
<table border="0" align="center" cellpadding="5">

<tr>
  <td align="right">
<math>J \ell_\varpi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\int_0^{m_\varpi} (\dot\varphi \varpi^2) M \cdot dm_\varpi
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \ell_\varpi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{5}{2} \int_0^{m_\varpi} \biggl\{ 1 - \biggl[1 - m_\varpi \biggr]^{2 / 3}\biggr\} dm_\varpi
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \frac{2}{5} \cdot \ell_\varpi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\int_0^{m_\varpi} dm_\varpi
-
\int_0^{m_\varpi} \biggl[1 - m_\varpi \biggr]^{2 / 3}dm_\varpi
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
m_\varpi
+
\biggl[
\frac{3}{5}\biggl(1 - m_\varpi\biggr)^{5/3}
\biggr]_0^{m_\varpi}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
m_\varpi
+
\frac{3}{5}\biggl(1 - m_\varpi\biggr)^{5/3}
-\frac{3}{5}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
-\biggl(1 - m_\varpi\biggr)
+
\frac{3}{5}\biggl(1 - m_\varpi\biggr)^{5/3}
+ \biggl(1-\frac{3}{5}\biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow \ell_\varpi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1 - \frac{5}{2}\biggl(1 - m_\varpi\biggr)
+
\frac{3}{2}\biggl(1 - m_\varpi\biggr)^{5/3} \, .
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ MPT77 }}, &sect;IV.a, eq. (4.3)
  </td>
</tr>
</table>

====Centrally Condensed Initially (n' > 0)====

<!--
Here, following [http://adsabs.harvard.edu/abs/1973ApJ...180..159B Bodenheimer &amp; Ostriker (1973)], we introduce an approach to specifying a wider range of physically reasonable angular momentum distributions; text that appears in an dark green font has been taken ''verbatim'' from this foundational paper.
-->
In &sect;III.d (pp. 1084 - 1086) of {{ OM68 }}, we find the following relations:
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>h(m) \equiv \biggl[\frac{M}{J}\biggr] j(m)</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
a_1 + a_2(1-m)^{\alpha_2} + a_3(1-m)^{\alpha_3} \, ,
</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="3">
{{ OM68 }}, &sect;III.d, Eq. (50)<br />
{{ OB68 }}, p. 1090, Eq. (6)<br />
{{ BO73 }}, &sect;II, Eq. (4)<br />
[<b>[[Appendix/References#T78|<font color="red">T78</font>]]</b>], &sect;10.4 (p. 254), Eq. (44)<br />
{{ PDD96 }}, &sect;2.1, Figure 1
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{1}{\alpha_2} = q_1</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{2\beta - \alpha \beta(2n+5)}{\alpha \beta(2n+5) - (2n + 3)} \, ,
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>\frac{1}{\alpha_3} = q_2</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{2n+3}{2} \, ,
</math>
  </td>
  <td align="center" colspan="4">&nbsp;</td>
</tr>

<tr>
  <td align="right">
<math>b_1</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{\alpha (q_2 + 1) - 1}{\alpha (q_2 - q_1)} \, ,
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>b_2</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{ 1 - \alpha (q_1+1)}{\alpha (q_2 - q_1)} \, ,
</math>
  </td>
  <td align="center" colspan="4">&nbsp;</td>
</tr>

<tr>
  <td align="right">
<math>a_1</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
b_1(q_1+1) + b_2(q_2+1) \, ,
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>a_2</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
-b_1(q_1+1) \, ,
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>a_3</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
- b_2(q_2+1) \, .
</math>
  </td>
</tr>

</table>

Ostriker &amp; Mark claim that the analytical expression for <math>\dot\varphi (\varpi) = j[m(\varpi)]/\varpi^2</math> that was derived by {{ Stoeckly65 }} for a uniform-density sphere, is retrieved by setting, <math>(n, \alpha, \beta) = (0, \tfrac{2}{5}, \tfrac{3}{2}) \, .</math>  Let's see &hellip;
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\lim_{n\rightarrow 0} \biggl[ q_1 \biggr]</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\lim_{n\rightarrow 0} \biggl[ \frac{-\tfrac{6n}{5} }{-\tfrac{4n}{5}}\biggr] = + \frac{3}{2} \, ;</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~q_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
+\frac{3}{2} \, ;
</math>
  </td>
<td align="center" colspan="3">&nbsp;</td>
</tr>

<tr>
  <td align="right">
<math>~b_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{\tfrac{2}{5} (\tfrac{3}{2} + 1) - 1}{\tfrac{2}{5} (\tfrac{3}{2} - \tfrac{2}{3})} 
=
\frac{0}{\tfrac{1}{3}} =0 \, ;
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~b_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{ 1 - \tfrac{2}{5} (\tfrac{2}{3}+1)}{\tfrac{2}{5} (\tfrac{3}{2} - \tfrac{2}{3})} 
=
\frac{ \tfrac{1}{3}}{\tfrac{1}{3} } = 1 \, ;
</math>
  </td>
<td align="center" colspan="3">&nbsp;</td>
</tr>

<tr>
  <td align="right">
<math>~a_1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\tfrac{5}{2}  \, ;
</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp;</td>
  <td align="right">
<math>~a_2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
0 \, ;
</math>
  </td>
  <td align="right">
<math>~a_3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \tfrac{5}{2} \, .
</math>
  </td>
</tr>
</table>

This implies,
<table border="1" cellpadding="10" align="center"><tr><td align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~h(m)\biggr|_{n' = ~0} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{5}{2}\biggl[1 - (1-m)^{2/3} \biggr] \, .
</math>
  </td>
</tr>
</table>
</td></tr></table>

Q. E. D.

In addition, from p. 163 (Table 1) of {{ BO73 }} we find the following table of coefficient values.

<table border="1" cellpadding="8" align="center">
<tr>
  <td align="center" colspan="6">
<b>Coefficients for <math>~h(m)</math> Expression</b><br />
[from K. Braly's (1969) unpublished undergraduate thesis, Princeton University]
</td>
<td align="center">
Figure &amp; caption extracted from p. 715 of<br />{{ PDD96figure }}<br />&copy; American Astronomical Society
</td>
</tr>
<tr>
  <td align="center"><math>~n^'</math></td>
  <td align="center"><math>~a_1</math></td>
  <td align="center"><math>~a_2</math></td>
  <td align="center"><math>~a_3</math></td>
  <td align="center"><math>~\alpha_2 = \frac{1}{q_1}</math></td>
  <td align="center"><math>~\alpha_3 = \frac{1}{q_2}</math></td>
  <td align="center" rowspan="10">[[File:PickettDurisenDavis96Fig1.png|400px]]</td>
</tr>
<tr>
  <td align="center">0</td>
  <td align="center">+2.5</td>
  <td align="center"><math>~\cdots</math></td>
  <td align="center">-2.5</td>
  <td align="center"><math>~\cdots</math></td>
  <td align="center"><math>~\tfrac{2}{3}</math></td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{1}{2}</math></td>
  <td align="center">+3.068133</td>
  <td align="center">+0.203667</td>
  <td align="center">-3.271800</td>
  <td align="center">+0.801297</td>
  <td align="center"><math>~\tfrac{1}{2}</math></td>
</tr>
<tr>
  <td align="center">1</td>
  <td align="center">+3.825819</td>
  <td align="center">+0.857311</td>
  <td align="center">-4.68313</td>
  <td align="center">+0.650981</td>
  <td align="center"><math>~\tfrac{2}{5}</math></td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{3}{2}</math></td>
  <td align="center">+4.887588</td>
  <td align="center">+2.345310</td>
  <td align="center">-7.232898</td>
  <td align="center">+0.525816</td>
  <td align="center"><math>~\tfrac{1}{3}</math></td>
</tr>
<tr>
  <td align="center">2</td>
  <td align="center">+6.457897</td>
  <td align="center">+6.018111</td>
  <td align="center">-12.476007</td>
  <td align="center">+0.417472</td>
  <td align="center"><math>~\tfrac{2}{7}</math></td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{5}{2}</math></td>
  <td align="center">+8.944150</td>
  <td align="center">+18.234305</td>
  <td align="center">-27.178455</td>
  <td align="center">+0.321459</td>
  <td align="center"><math>~\tfrac{1}{4}</math></td>
</tr>
<tr>
  <td align="center">3</td>
  <td align="center">+13.270061</td>
  <td align="center">+163.26149</td>
  <td align="center">-176.53154</td>
  <td align="center">+0.235287</td>
  <td align="center"><math>~\tfrac{2}{9}</math></td>
</tr>
<tr>
  <td align="center" colspan="6">
<b>Coefficients for <math>~h(m)</math> Expression</b><br />
used by {{ OB68 }}, p. 1090, Eq. (6)
</td>
</tr>
<tr>
  <td align="center"><math>~\tfrac{3}{2}</math></td>
  <td align="center">+4.8239</td>
  <td align="center">+1.8744</td>
  <td align="center">-6.6983</td>
  <td align="center">+0.5622</td>
  <td align="center"><math>~\tfrac{1}{3}</math></td>
</tr>
</table>

==Double Check Vector Identities==

Let's plug a few different [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|simple rotation profiles]] into the Euler equation, using a cylindrical-coordinate base to demonstrate that the three expressions are identical, namely, that
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(\vec{v} \cdot \nabla) \vec{v}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\vec\zeta \times \vec{v} + \frac{1}{2}\nabla (v^2)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\nabla \Psi \, .</math>
  </td>
</tr>
</table>

===Uniform Rotation===
In the case of uniform rotation, we have,
<div align="center">
<math>~\vec{v} = \hat{e}_\varphi (v_\varphi) = \hat{e}_\varphi (\varpi \omega_0) ~~~\Rightarrow~~~ \frac{j^2}{\varpi^3} = \frac{(\varpi v_\varphi)^2}{\varpi^3} = \frac{(\varpi^2\omega_0)^2}{\varpi^3} = \varpi \omega_0^2\, ,</math>
</div>
where, <math>~\omega_0</math> is independent of radial position.  This also means that,
<div align="center">
<math>
\Psi \equiv - \int \frac{j^2(\varpi)}{\varpi^3} d\varpi = - \frac{1}{2} \varpi^2 \omega_0^2~;
</math>
</div>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\vec\zeta = \nabla \times \vec{v}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat{e}_\varpi \biggl[ -\cancel{ \frac{\partial v_\varphi}{\partial z} }\biggr] + \hat{e}_z \biggl[ \frac{1}{\varpi} \frac{\partial (\varpi v_\varphi)}{\partial \varpi} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat{e}_z \biggl[ \frac{1}{\varpi} \frac{\partial (\varpi^2 \omega_0 )}{\partial \varpi} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat{e}_z ( 2\omega_0 )
</math>
  </td>
</tr>
</table>

[A] &nbsp; Hence,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(\vec{v} \cdot \nabla) \vec{v}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\hat{e}_\varpi \biggl[ - \frac{v_\varphi \cdot v_\varphi}{\varpi} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\hat{e}_\varpi \biggl[ - \frac{(\varpi \omega_0)\cdot (\varpi \omega_0)}{\varpi} \biggr] = - \hat{e}_\varpi (\varpi \omega_0^2) \, .</math>
  </td>
</tr>
</table>

[B] &nbsp; Alternatively,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\vec\zeta \times \vec{v} + \frac{1}{2}\nabla (v^2)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\hat{e}_z ( 2\omega_0 ) \times \hat{e}_\varphi (\varpi \omega_0) + \hat{e}_\varpi \frac{1}{2} \biggl[ \frac{\partial}{\partial\varpi} (\varpi^2 \omega_0^2) \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ \hat{e}_\varpi  \biggl\{ -( 2\omega_0 ) (\varpi \omega_0) + (\varpi \omega_0^2)  \biggr\} = - \hat{e}_\varpi (\varpi \omega_0^2) \, .</math>
  </td>
</tr>
</table>

[C] &nbsp; Or,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\nabla \Psi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\hat{e}_\varpi \biggl[- \frac{1}{2} \frac{\partial}{\partial\varpi} (\varpi^2 \omega_0^2) \biggr] = - \hat{e}_\varpi (\varpi \omega_0^2) \, .</math>
  </td>
</tr>
</table>
This demonstrates that, in the case of uniform angular velocity, all three expressions are identical.

===Power Law===
In the case of a power-law expression, we have,
<div align="center">
<math>~\vec{v} = \hat{e}_\varphi (v_\varphi) = \hat{e}_\varphi \biggl[ \frac{j_0}{\varpi_0^2} \biggl( \frac{\varpi}{\varpi_0}\biggr)^{(q-1)} \biggr] 
~~~\Rightarrow~~~ \frac{j^2}{\varpi^3} = \biggl[ \frac{j_0^2}{\varpi_0^3} \biggl( \frac{\varpi}{\varpi_0}\biggr)^{(2q-3)} \biggr] \, ,</math>
</div>
where, <math>~j_0</math> and <math>~\varpi_0</math> are both independent of radial position.  This also means that,
<div align="center">
<math>
\Psi \equiv - \int \frac{j^2(\varpi)}{\varpi^3} d\varpi = - \frac{1}{2(q-1)} \biggl[ \frac{j_0^2}{\varpi_0^2} \biggl( \frac{\varpi}{\varpi_0}\biggr)^{2(q-1)} \biggr]~;
</math>
</div>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\vec\zeta = \nabla \times \vec{v}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat{e}_\varpi \biggl[ -\cancel{ \frac{\partial v_\varphi}{\partial z} }\biggr] + \hat{e}_z \biggl[ \frac{1}{\varpi} \frac{\partial (\varpi v_\varphi)}{\partial \varpi} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat{e}_z~ \frac{1}{\varpi} \frac{\partial }{\partial \varpi} \biggl[ \frac{j_0}{\varpi_0} \biggl( \frac{\varpi}{\varpi_0}\biggr)^{q} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat{e}_z~ \frac{q}{\varpi} \biggl[ \frac{j_0}{\varpi_0^{q+1}} ( \varpi)^{q-1} \biggr] 
=
\hat{e}_z~ q \biggl[ \frac{j_0}{\varpi_0^{3}} \biggl( \frac{\varpi}{\varpi_0} \biggr)^{q-2} \biggr]\, .
</math>
  </td>
</tr>
</table>

[D] &nbsp; Hence,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(\vec{v} \cdot \nabla) \vec{v}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\hat{e}_\varpi \biggl[ - \frac{v_\varphi \cdot v_\varphi}{\varpi} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \hat{e}_\varpi \frac{1}{\varpi} \biggl[ \frac{j_0^2}{\varpi_0^4} \biggl( \frac{\varpi}{\varpi_0}\biggr)^{2(q-1)} \biggr]  
= - \hat{e}_\varpi \biggl[ \frac{j_0^2}{\varpi_0^5} \biggl( \frac{\varpi}{\varpi_0}\biggr)^{(2q-3)} \biggr]\, .</math>
  </td>
</tr>
</table>

[E] &nbsp; Alternatively,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\vec\zeta \times \vec{v} + \frac{1}{2}\nabla (v^2)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\hat{e}_z~ q \biggl[ \frac{j_0}{\varpi_0^{3}} \biggl( \frac{\varpi}{\varpi_0} \biggr)^{q-2} \biggr] \times 
\hat{e}_\varphi \biggl[ \frac{j_0}{\varpi_0^2} \biggl( \frac{\varpi}{\varpi_0}\biggr)^{(q-1)} \biggr] 
+ \hat{e}_\varpi \frac{1}{2}  \frac{\partial}{\partial\varpi} \biggl[ \frac{j_0^2}{\varpi_0^4} \biggl( \frac{\varpi}{\varpi_0}\biggr)^{(2q-2)} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\hat{e}_\varpi~ q \biggl[ \frac{j_0^2}{\varpi_0^{5}} \biggl( \frac{\varpi}{\varpi_0} \biggr)^{2q-3} \biggr]  
+ \hat{e}_\varpi(q-1)  \biggl[ \frac{j_0^2}{\varpi_0^5} \biggl( \frac{\varpi}{\varpi_0}\biggr)^{(2q-3)} \biggr]</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-\hat{e}_\varpi~ \biggl[ \frac{j_0^2}{\varpi_0^{5}} \biggl( \frac{\varpi}{\varpi_0} \biggr)^{2q-3} \biggr]  \, .
</math>
  </td>
</tr>
</table>

[F] &nbsp; Or,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\nabla \Psi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\hat{e}_\varpi\frac{\partial}{\partial\varpi} \biggl\{- \frac{1}{2(q-1)} \biggl[ \frac{j_0^2}{\varpi_0^2} \biggl( \frac{\varpi}{\varpi_0}\biggr)^{2(q-1)} \biggr] \biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- ~\hat{e}_\varpi\frac{\partial}{\partial\varpi} \biggl[ \frac{j_0^2}{\varpi_0^3} \biggl( \frac{\varpi}{\varpi_0}\biggr)^{2q-3} \biggr] </math>
  </td>
</tr>
</table>
This demonstrates that, in the case of power-law angular velocity profile, all three expressions are identical.

===Uniform v<sub>&#x03C6;</sub>===
In the case of a uniform <math>~v_\varphi</math> (i.e., a flat rotation curve), we have,
<div align="center">
<math>~\vec{v} = \hat{e}_\varphi (v_\varphi) = \hat{e}_\varphi v_0 
~~~\Rightarrow~~~ \frac{j^2}{\varpi^3} =  \frac{v_0^2}{\varpi} \, ,</math>
</div>
where, <math>~v_0</math> is independent of radial position.  This also means that,
<div align="center">
<math>
\Psi \equiv - \int \frac{j^2(\varpi)}{\varpi^3} d\varpi = - v_0^2 \ln \biggl( \frac{\varpi}{\varpi_0} \biggr)~;
</math>
</div>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\vec\zeta = \nabla \times \vec{v}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat{e}_\varpi \biggl[ -\cancel{ \frac{\partial v_\varphi}{\partial z} }\biggr] + \hat{e}_z \biggl[ \frac{1}{\varpi} \frac{\partial (\varpi v_\varphi)}{\partial \varpi} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat{e}_z \biggl( \frac{v_0}{\varpi} \biggr) \, .
</math>
  </td>
</tr>
</table>

[G] &nbsp; Hence,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(\vec{v} \cdot \nabla) \vec{v}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\hat{e}_\varpi \biggl[ - \frac{v_\varphi \cdot v_\varphi}{\varpi} \biggr] = -~\hat{e}_\varpi \biggl[ \frac{v_0^2}{\varpi} \biggr] \, .</math>
  </td>
</tr>
</table>

[H] &nbsp; Alternatively,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\vec\zeta \times \vec{v} + \frac{1}{2}\nabla (v^2)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat{e}_z \biggl( \frac{v_0}{\varpi} \biggr) \times \hat{e}_\varphi v_0
+ \hat{e}_\varpi~ \frac{1}{2} \frac{\partial}{\partial \varpi} (v_0^2)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\hat{e}_\varpi \biggl( \frac{v_0^2}{\varpi} \biggr) \, .
</math>
  </td>
</tr>
</table>

[I] &nbsp; Or,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\nabla \Psi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\hat{e}_\varpi\frac{\partial}{\partial\varpi} \biggl\{- v_0^2 \ln \biggl( \frac{\varpi}{\varpi_0} \biggr)\biggr\}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~ \hat{e}_\varpi v_0^2  \biggl(\frac{\varpi}{\varpi_0} \biggr)^{-1} \frac{1}{\varpi_0}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-~ \hat{e}_\varpi \biggl( \frac{v_0^2}{\varpi} \biggr)  \, .</math>
  </td>
</tr>
</table>
This demonstrates that, in the case of a constant <math>~v_\varphi</math> profile, all three expressions are identical.

===j-Constant Rotation===
In the case of so-called j-constant rotation, we have,
<div align="center">
<math>~\vec{v} = \hat{e}_\varphi (v_\varphi) = \hat{e}_\varphi ~\omega_c \biggl[ \frac{A^2\varpi}{A^2 + \varpi^2}\biggr] 
~~~\Rightarrow~~~ \frac{j^2}{\varpi^3} = \frac{(\varpi v_\varphi)^2}{\varpi^3} = 
\frac{\omega_c^2}{\varpi} \biggl[ \frac{A^2\varpi}{A^2 + \varpi^2}\biggr]^2 =
\biggl[ \frac{\omega_c^2 A^4\varpi}{(A^2 + \varpi^2)^2}\biggr]
\, ,
</math>
</div>

where, <math>~\omega_c</math>, and the characteristic length, <math>~A</math>, are both independent of radial position.  This also means that,

<div align="center">
<math>
\Psi \equiv - \int \frac{j^2(\varpi)}{\varpi^3} d\varpi =  +\frac{1}{2}\biggl[ \frac{\omega_c^2 A^4}{(A^2 + \varpi^2)}\biggr]~;
</math>
</div>
and,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\vec\zeta = \nabla \times \vec{v}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat{e}_\varpi \biggl[ -\cancel{ \frac{\partial v_\varphi}{\partial z} }\biggr] + \hat{e}_z \biggl[ \frac{1}{\varpi} \frac{\partial (\varpi v_\varphi)}{\partial \varpi} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat{e}_z \biggl\{ \frac{\omega_c}{\varpi} \frac{\partial }{\partial \varpi}  \biggl[ \frac{A^2\varpi^2}{A^2 + \varpi^2}\biggr]\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat{e}_z~ \frac{\omega_c}{\varpi}  \biggl\{ \biggl[ 2A^2\varpi(A^2 + \varpi^2)^{-1} \biggr] - \biggl[ 2A^2\varpi^3(A^2 + \varpi^2)^{-2} \biggr]\biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat{e}_z~  \biggl[2\omega_c A^4 (A^2 + \varpi^2)^{-2} \biggr] \, .
</math>
  </td>
</tr>
</table>

[J] &nbsp; Hence,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~(\vec{v} \cdot \nabla) \vec{v}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\hat{e}_\varpi \biggl[ - \frac{v_\varphi \cdot v_\varphi}{\varpi} \biggr] </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~\hat{e}_\varpi \frac{\omega_c^2}{\varpi} \biggl[ \frac{A^2\varpi}{A^2 + \varpi^2}\biggr]^2 
=
-~\hat{e}_\varpi \biggl[ \frac{\omega_c^2A^4 \varpi}{(A^2 + \varpi^2)^2} \biggr] \, .
</math>
  </td>
</tr>
</table>

[K] &nbsp; Alternatively,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\vec\zeta \times \vec{v} + \frac{1}{2}\nabla (v^2)</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\hat{e}_z~  \biggl[2\omega_c A^4 (A^2 + \varpi^2)^{-2} \biggr] \times \hat{e}_\varphi ~\omega_c \biggl[ \frac{A^2\varpi}{A^2 + \varpi^2}\biggr] + \frac{1}{2} \hat{e}_\varpi \frac{\partial}{\partial \varpi}\biggl[ \omega_c^2 A^4\varpi^2 (A^2 + \varpi^2)^{-2}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \hat{e}_\varpi ~  \biggl[ \frac{2\omega_c^2 A^6 \varpi }{(A^2 + \varpi^2)^{3}} \biggr] + \hat{e}_\varpi \biggl[  \omega_c^2 A^4\varpi (A^2 + \varpi^2)^{-2} - 2 \omega_c^2 A^4\varpi^3 (A^2 + \varpi^2)^{-3}\biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
 \hat{e}_\varpi \biggl[ \frac{ \omega_c^2 A^4\varpi }{ (A^2 + \varpi^2)^{2}} - \frac{2 \omega_c^2 A^4\varpi^3}{ (A^2 + \varpi^2)^{3} }  
- \frac{2\omega_c^2 A^6 \varpi }{(A^2 + \varpi^2)^{3}} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
 \hat{e}_\varpi \biggl[  (A^2 + \varpi^2) - 2 \varpi^2  - 2A^2 \biggr] \frac{ \omega_c^2 A^4\varpi }{ (A^2 + \varpi^2)^{3}} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~ \hat{e}_\varpi \biggl[ \frac{ \omega_c^2 A^4\varpi }{ (A^2 + \varpi^2)^{2}} \biggr] \, .
</math>
  </td>
</tr>
</table>

[L] &nbsp; Or,
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\nabla \Psi</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\hat{e}_\varpi ~ \frac{1}{2} \frac{\partial}{\partial \varpi}\biggl[ \omega_c^2 A^4 (A^2 + \varpi^2)^{-1} \biggr] 
= - \hat{e}_\varpi  \biggl[ \frac{ \omega_c^2 A^4 \varpi }{ (A^2 + \varpi^2)^{2}} \biggr] \, .</math>
  </td>
</tr>
</table>
This demonstrates that, in the case of a j-constant rotation profile, all three expressions are identical.