Editing SSCpt1/Virial/FormFactors (section)

===Presentation===

{{ VH74full }} have provided analytic expressions for the gravitational potential energy and the internal energy &#8212; which they tag with the variable names, <math>~\Omega</math> and <math>~U</math>, respectively &#8212; that we can adopt in our effort to quantify the key structural form factors in the context of pressure-truncated polytropic spheres.  [The same expression for <math>~\Omega</math> is also effectively provided in &sect;1 of [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] through the definition of his coefficient, "A" (polytropic case).]
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<!-- 
[[Image:VialaHoredt1974.png|500px|center]]
Astronomy &amp; Astrophysics, 33: 195-202, (1974)<br />
POLYTROPIC SHEETS, CYLINDERS AND SPHERES WITH NEGATIVE INDEX<br />
Y. P. Viala &amp; Gp. Horedt<br />
-->
{{ VH74figure }}
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  <td align="right"><math>\Omega</math></td>
  <td align="center"><math>=</math></td>
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<math>
- G\int_0^M \frac{MdM}{r} = \frac{16\pi^2 G \rho_0^2 \alpha^5}{(5-n)} \biggl[
\mp \xi^3 \theta^{n+1} - 3\xi^3 (\theta')^2 - 3\xi^2 \theta (\theta')
\biggr] \, ,
</math>
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  <td align="right"><math>U</math></td>
  <td align="center"><math>=</math></td>
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<math>
\frac{1}{\gamma - 1}\int_V pdV = \frac{\alpha K \rho_0^{1 + 1/n}}{\gamma - 1} \int_0^\xi \theta^{n+1} 4\pi \alpha^2 \xi^2 d\xi
</math>
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  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
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<math>
\frac{\alpha K \rho_0^{1 + 1/n}}{\gamma - 1}  \cdot
\frac{4\pi \alpha^2(n+1)}{(5-n)}
\biggl[
\frac{2\xi^3 \theta^{n+1}}{n+1} \pm \xi^3 (\theta')^2 \pm \xi^2 \theta (\theta')
\biggr]_0^\xi \, .
</math>
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(the superior sign holds if <math>-1 < n < \infty</math>, the inferior if <math>-\infty < n < -1</math>)
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A couple of key equations drawn directly from {{ VH74 }} have been shown here.  As its title indicates, the paper includes discussion of &#8212; and accompanying equation derivations for &#8212; equilibrium self-gravitating, pressure-truncated, polytropic configurations having several different geometries:  planar sheets, axisymmetric cylinders, and spheres.  We have extracted derived expressions for the gravitational potential energy, <math>\Omega</math>, and the internal energy, <math>U</math>, that apply to spherically symmetric configurations only.  These authors also consider negative polytropic indexes; we are considering only values in the range, <math>0 \le n \le \infty</math>, so, as the accompanying parenthetical note indicates, when either <math>\pm</math> or <math>\mp</math> appears in an expression, we will pay attention only to the ''superior'' sign.
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Rewriting these two expressions to accommodate our parameter notations &#8212; recognizing, specifically, that <math>\alpha</math> is the [[SSC/Structure/Polytropes#Lane-Emden_Equation|familiar polytropic length scale]] (<math>a_n</math>; [[#Renormalization|expression provided below]]), <math>\rho_0</math> is the central density <math>(\rho_c)</math>, and <math>(\gamma - 1) = 1/n</math> &#8212; we have from {{ VH74 }},

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<math>\biggl[ W_\mathrm{grav} \biggr]_\mathrm{VH74}</math>
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<math>=</math>
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<math> -
\frac{(4\pi)^2}{(5-n)} \cdot G \rho_c^2 a_n^5
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr] \, ,
</math>
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<math>\biggl[ \mathfrak{S}_\mathrm{A} \biggr]_\mathrm{VH74}</math>
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<math>=</math>
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<math>
\frac{n(4\pi)^2}{3(5-n)} \cdot G \rho_c^2 a_n^5 
\biggl[\frac{6}{(n+1)} \cdot \tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr] \, .
</math>
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