Editing AxisymmetricConfigurations/SolutionStrategies (section)

====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>