Editing SSCpt1/Virial/FormFactors/Pt3 (section)

==Fiddling Around==
NOTE (from Tohline on 17 March 2015):  Chronologically, this "Fiddling Around" subsection was developed before our discovery of the {{ VH74hereafter }} derivations.  It put us on track toward the correct development of general expressions for the structural form factors that are applicable to pressure-truncated polytropic spheres.  But this subsection's conclusions are superseded by the {{ VH74hereafter }} work.

In this subsection, for simplicity, we will omit the "tilde" over the variable <math>\xi</math>.  In the case of <math>n=1</math> structures,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\theta^{n+1}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
\biggl( \frac{\sin\xi}{\xi}\biggr)^2 
= \frac{1}{2\xi^2} \biggl[ 1 - \cos(2\xi) \biggr] 
= \frac{1}{2^2\xi^3} \biggl[ 2\xi - 2\xi\cos(2\xi) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~\mathfrak{f}_A - \theta^{n+1}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{2^2\xi^3}  \biggl[6\xi  - 3\sin(2\xi ) \biggr] 
- \frac{1}{2^2\xi^3} \biggl[ 2\xi - 2\xi\cos(2\xi) \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{2^2\xi^3}  \biggl[4\xi  - 3\sin(2\xi ) + 2\xi\cos(2\xi) \biggr] \, .
</math>
  </td>
</tr>

</table>
</div>
But, we also have shown that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl( \frac{2^3 \xi^5}{3\cdot 5} \biggr) \mathfrak{f}_W</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
\biggl[ 4\xi - 3\sin(2\xi) + 2\xi \cos(2\xi ) \biggr] \, .
</math>
  </td>
</tr>

</table>
</div>
Hence, we see that,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

</table>
</div>

Similarly, in the case of <math>n = 5</math> structures,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\theta^{n+1}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
(1 + \ell^2)^{-3}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~\biggl( \frac{2^3}{3} \ell^{3}  \biggr) \biggl[ \mathfrak{f}_A - \theta^{n+1} \biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
[ \tan^{-1}(\ell ) + \ell (\ell^4-1) (1+\ell^2)^{-3} ]  - \biggl( \frac{2^3}{3} \ell^{3}  \biggr) (1 + \ell^2)^{-3}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\tan^{-1}(\ell ) + \ell \biggl(\ell^4-\frac{8}{3}\ell^2 - 1 \biggr) (1+\ell^2)^{-3} \, .
</math>
  </td>
</tr>

</table>
</div>
But, we also have shown that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl( \frac{2^4}{5} \cdot \ell^{5} \biggr) \mathfrak{f}_W</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
\ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell )  \, .
</math>
  </td>
</tr>

</table>
</div>
Hence, we see that,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
<math>\Rightarrow ~~~~\biggl( \frac{2\xi^2}{5} \biggr) \mathfrak{f}_W</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>  
\mathfrak{f}_A - \theta^{n+1} \, .
</math>
  </td>
</tr>

</table>
</div>
This is pretty amazing!  Both examples produce almost exactly the same relationship between the two structural form factors, <math>\mathfrak{f}_A</math> and <math>\mathfrak{f}_W</math>.  I think that we are well on our way toward nailing down the generic, analytic relationship and, in turn, a generally applicable mass-radius relationship for pressure-truncated polytropic configurations.

Okay &hellip; here is the final piece of information.  In the case of isolated polytropes, we know that the correct expressions for the structural form factors are as summarized in the following table:
<div align="center">
<table border="1" align="center" cellpadding="5">
<tr><th align="center" colspan="1">
Structural Form Factors for <font color="red">Isolated</font> Polytropes
</th></tr>
<tr><td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\tilde\mathfrak{f}_M</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ - \frac{3\Theta^'}{\xi} \biggr]_{\tilde\xi} </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{3^2\cdot 5}{5-n} \biggl[ \frac{\Theta^'}{\xi} \biggr]^2_{\tilde\xi} </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{3(n+1) }{(5-n)} ~\biggl[ \Theta^' \biggr]^2_{\tilde\xi} 
</math>
  </td>
</tr>
</table>

</td>
</tr>
</table>
</div>
We notice, from this, that the ratio,

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{(n+1)\tilde\xi^2 }{3\cdot 5} \, .
</math>
  </td>
</tr>
</table>
Even in the case of the two pressure-truncated polytropes, analyzed above, this ratio proves to give the correct prefactor on <math>\mathfrak{f}_W</math>.  So we ''suspect'' that the universal relationship between the two form factors is,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

</table>
</div>