Editing SSCpt1/Virial/FormFactors (section)

===Implication for Structural Form Factors===
On the other hand, our expressions for these two [[SSCpt1/Virial#Structural_Form_Factors|normalized energy components written in terms of the structural form factors]] are,
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<math>\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>
- \frac{3}{5} \chi^{-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}^2_M} \, ,
</math>
  </td>
</tr>

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  <td align="right">
<math>\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{4\pi n}{3} \cdot \chi^{-3/n}
\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)\frac{1}{\tilde\mathfrak{f}_M} \biggr]_\mathrm{eq}^{(n+1)/n}
\cdot \tilde\mathfrak{f}_A \, ,</math>
  </td>
</tr>
</table>
</div>
where, in equilibrium (see [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation|here]] and [[SSCpt1/Virial#Choices_Made_by_Other_Researchers|here]] for details),
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<math>\chi_\mathrm{eq} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{R_\mathrm{eq}}{R_\mathrm{Horedt}} \biggl\{ \frac{R_\mathrm{Horedt}}{R_\mathrm{norm}}\biggr\}</math>
  </td>
</tr>

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&nbsp;
  </td>
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<math>=</math>
  </td>
  <td align="left">
<math>\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)} 
\biggl\{ \biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{(n-1)/(n-3)}
\biggr\} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{M_\mathrm{limit}}{M_\mathrm{tot}} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\tilde\xi^2 \tilde\theta^'}{\xi_1^2 \theta^'_1} \biggr) \, ,</math>
  </td>
</tr>

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  <td align="right">
<math>\tilde\mathfrak{f}_M </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr) \, .</math>
  </td>
</tr>
</table>
</div>
Hence, we deduce that,
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<math>\tilde\mathfrak{f}_W </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math> - \frac{5}{3} \biggl[ \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]
\chi_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2} \cdot \tilde\mathfrak{f}^2_M 
</math>
  </td>
</tr>

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&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>\frac{5}{3} \biggl\{\biggl[ -\frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]\biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)} \biggr\}
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)} 
\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{[(n-1)-2(n-3)]/(n-3)} 
\cdot \biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr)^2 
</math>
  </td>
</tr>

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&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>3\cdot 5\biggl\{\biggl[ -\frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]\biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)} 
\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{(5-n)/(n-3)}\biggr\}
(-\tilde\theta^')^{[(1-n)+2(n-3)]/(n-3)} \tilde\xi^{[-(n-3)+2(1-n)]/(n-3)}
</math>
  </td>
</tr>

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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>3\cdot 5\biggl\{\biggl[ -\frac{W_\mathrm{grav}}{E_\mathrm{norm}} \biggr]\biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)} 
\biggl( \frac{\tilde\xi^2 \tilde\theta^'}{\xi_1^2 \theta^'_1}\biggr)^{(5-n)/(n-3)}\biggr\}
(-\tilde\theta^')^{(n-5)/(n-3)} \tilde\xi^{(5-3n)/(n-3)} \, .
</math>
  </td>
</tr>
</table>
</div>
If we now adopt the {{ VH74hereafter }} expression for the normalized gravitational potential energy, the product of terms inside the curly braces becomes,
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<math>
\biggl\{~~~\biggr\}_\mathrm{VH74}
</math>
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<math>=</math>
  </td>
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<math>
\frac{1}{(5-n)} 
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr] 
\biggl[ (-\tilde\xi^2 \tilde\theta^')_{\xi_1}^{(5-n)}  \cdot \frac{(n+1)^n}{4\pi} \biggr]^{1/(n-3)} 
\biggl[ \frac{4\pi}{(n+1)^n} \biggr]^{1/(n-3)} 
\biggl( \frac{\tilde\xi^2 \tilde\theta^'}{\xi_1^2 \theta^'_1}\biggr)^{(5-n)/(n-3)}
</math>
  </td>
</tr>

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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{(5-n)} 
\biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr] 
(-\tilde\xi^2 \tilde\theta^')^{(5-n)/(n-3)}  \, .
</math>
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</tr>
</table>
</div>
Therefore, 
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<math>\biggl[ \tilde\mathfrak{f}_W \biggr]_\mathrm{VH74}</math>
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  <td align="center">
<math>=</math>
  </td>
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<math>\frac{3\cdot 5}{(5-n)} \biggl[\tilde\xi^3 \tilde\theta^{n+1} + 3\tilde\xi^3 (\tilde\theta^')^2 - 3(-\tilde\xi^2 \tilde\theta^')\tilde\theta \biggr] 
\tilde\xi^{-5}  
</math>
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</tr>

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&nbsp;
  </td>
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<math>=</math>
  </td>
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<math>\frac{3\cdot 5}{(5-n)\tilde\xi^2} 
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] 
\, .
</math>
  </td>
</tr>
</table>
</div>

Now, from [[SSC/Virial/PolytropesEmbedded/FirstEffortAgain#PTtable|our earlier work]] we deduced that <math>\tilde\mathfrak{f}_A</math> is related to <math>\tilde\mathfrak{f}_W</math> via the relation,
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<math>\tilde\mathfrak{f}_A</math>
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  <td align="center">
<math>=</math>
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<math>\tilde\theta^{n+1} + \tilde\mathfrak{f}_W\biggl[ \frac{(n+1)}{3\cdot 5} \biggr] \tilde\xi^2 \, .</math>
  </td>
</tr>
</table>
</div>
Hence, we now have,
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<math>\biggl[ \tilde\mathfrak{f}_A \biggr]_\mathrm{VH74}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
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<math>\tilde\theta^{n+1} + \frac{(n+1)}{(5-n)} 
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] 
</math>
  </td>
</tr>

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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>\frac{1}{(5-n)} \biggl\{ 6\tilde\theta^{n+1} +  (n+1)
\biggl[3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \biggr\}
\, .
</math>
  </td>
</tr>
</table>
</div>

Building on the work of {{ VH74hereafter }}, we have, quite generally,

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<table border="1" align="center" cellpadding="5">
<tr>
<th align="center" colspan="1">
Structural Form Factors for <font color="red">Isolated</font> Polytropes
</th>
<th align="center" colspan="1">
Structural Form Factors for <font color="red">Pressure-Truncated</font> Polytropes
</th>
</tr>
<tr>
<td align="center">

<table border="0" cellpadding="5" align="center">
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  <td align="right">
<math>\mathfrak{f}_M</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ - \frac{3\theta^'}{\xi} \biggr]_{\xi_1} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathfrak{f}_W </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{3^2\cdot 5}{5-n} \biggl[ \frac{\theta^'}{\xi} \biggr]^2_{\xi_1} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathfrak{f}_A  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{3(n+1) }{(5-n)} ~\biggl[ \theta^' \biggr]^2_{\xi_1} 
</math>
  </td>
</tr>
</table>

</td>

<td align="center">

<table border="0" cellpadding="5" align="center">
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  <td align="right">
<math>\tilde\mathfrak{f}_M</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr) </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_W</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>\frac{3\cdot 5}{(5-n)\tilde\xi^2} 
\biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
\tilde\mathfrak{f}_A
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{(5-n)} \biggl\{ 6\tilde\theta^{n+1} +  (n+1)
\biggl[3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \biggr\}
</math>
  </td>
</tr>
</table>

</td>
</tr>
<tr>
  <td align="left" colspan="2">
We should point out that {{ LRS93bfull }} define a different set of dimensionless structure factors for ''isolated'' polytropic spheres  &#8212; <math>k_1</math> (their equation 2.9) is used in the determination of the internal energy; and <math>k_2</math> (their equation 2.10)  is used in the determination of the gravitational potential energy.
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<math>k_1</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{n(n+1)}{5-n} \biggr] \xi_1|\theta^'_1|</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>k_2</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{3}{5-n} \biggl[ \frac{4\pi |\theta^'_1|}{\xi_1} \biggr]^{1 / 3} </math>
  </td>
</tr>
</table>
</div>
Note that these are defined in the context of energy expressions wherein the central density, rather than the configuration's radius, serves as the principal parameter.  We note, as well, that for rotating configurations they define two additional dimensionless structure factors &#8212; <math>k_3</math> (their equation 3.17) is used in the determination of the rotational kinetic energy; and <math>\kappa_n</math> (their equation 3.14; also equation 7.4.9 of [<b>[[Appendix/References#ST83|<font color="red">ST83</font>]]</b>])  is used in the determination of the moment of inertia.
  </td>
</tr>
</table>
</div>

The singularity that arises when <math>n = 5</math> leads us to suspect that these general expressions fail in that one specific case.  Fortunately, as [[#Summary_.28n.3D5.29|we have shown in an accompanying discussion]], <math>\mathfrak{f}_W</math> and <math>\mathfrak{f}_A</math>, as well as <math>\mathfrak{f}_M</math>, can be determined by direct integration in this single case.