Editing SSC/Stability/BiPolytropes/51Models (section)

===Fixed Core Mass===
Initially, our normalization was based on [[SSC/Structure/BiPolytropes/Analytic51#Normalization|holding <math>K_c</math> and the central density <math>(\rho_0)</math> constant]].  Specifically,

<table border="0" cellpadding="3" align="center">
<tr>
  <td align="right">
<math>\rho^*</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{\rho}{\rho_0}</math>
  </td>

  <td align="center">; &nbsp;&nbsp;&nbsp;</td>

  <td align="right">
<math>r^*</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{r}{[K_c^{1/2}/(G^{1/2}\rho_0^{2/5})]}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P^*</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{P}{K_c\rho_0^{6/5}}</math>
  </td>

  <td align="center">; &nbsp;&nbsp;&nbsp;</td>

  <td align="right">
<math>M_r^*</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{M_r}{[K_c^{3/2}/(G^{3/2}\rho_0^{1/5})]}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>H^*</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{H}{K_c\rho_0^{1/5}}</math>
  </td>

  <td align="center">. &nbsp;&nbsp;&nbsp;</td>

  <td align="right" colspan="3">
&nbsp;
  </td>
</tr>

</table>
We also have explored a [[SSC/Structure/BiPolytropes/51RenormaizePart2#Basic_Equilibrium_Structure|"new normalization"]] based on holding <math>K_c</math> and <math>M_\mathrm{tot}</math> constant.  Here we want to perform a Bonnor-Ebert-type analysis, examining how <math>P_i</math> varies with radius if we hold <math>K_c</math> and the ''core mass'' constant along an equilibrium sequence.  According to our initial normalization &#8212; see, for example, [[SSC/Structure/BiPolytropes/Analytic51Renormalize#Step_4:_Throughout_the_core|here]] &#8212; we can write,

<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
<math>M_\mathrm{core}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c^3}{G^3 \rho_0^{2/5} } \biggr]^{1/2} \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} ( \xi_i \theta_i )^3</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow ~~~ \rho_0^{1 / 5}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c^3}{G^3 M_\mathrm{core}^2 } \biggr]^{1/2} \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} ( \xi_i \theta_i )^3</math>
  </td>
</tr>
</table>

Therefore, from the analytic profiles that describe the core, we have,

<table border="0" cellpadding="3" align="center">

<tr>
  <td align="right">
<math>\rho_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_0 \theta_i^5</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\rho_0 \theta_i^5</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>K_c \rho_0^{6/5} \theta_i^6</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>K_c \rho_0^{6/5} \theta_i^6</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>r</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c}{G\rho_0^{4/5}} \biggr]^{1/2} \biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c}{G\rho_0^{4/5}} \biggr]^{1/2} \biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi_i</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>M_i</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c^3}{G^3 \rho_0^{2/5} } \biggr]^{1/2} \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} ( \xi_i \theta_i)^3 </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \frac{K_c^3}{G^3 \rho_0^{2/5} } \biggr]^{1/2} \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} ( \xi_i \theta_i)^3 </math>
  </td>
</tr>
</table>

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

<tr>
  <td align="right"><math>\rho_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\rho_0 \theta_i^5
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl\{
\biggl[ \frac{K_c^3}{G^3 M_\mathrm{core}^2 } \biggr]^{1/2} \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} ( \xi_i \theta_i )^3
\biggr\}^5 \theta_i^5
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{K_c^3}{G^3 M_\mathrm{core}^2 } \biggr]^{5/2} \biggl( \frac{2\cdot 3}{\pi } \biggr)^{5/2} \xi_i^{15} \theta_i^{20} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>P_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
K_c \rho_0^{6/5} \theta_i^6
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
K_c\biggl\{
\biggl[ \frac{K_c^3}{G^3 M_\mathrm{core}^2 } \biggr]^{1/2} \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} ( \xi_i \theta_i )^3
\biggr\}^6 \theta_i^6
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{K_c^{10}}{G^9 M_\mathrm{core}^6 } \biggr] \biggl( \frac{2\cdot 3}{\pi } \biggr)^{3} \xi_i^{18} \theta_i^{24} \, , 
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>r_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{K_c}{G\rho_0^{4/5}} \biggr]^{1/2} \biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi_i
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{K_c}{G}\biggr]^{1/2} \biggl\{ 
\biggl[ \frac{K_c^3}{G^3 M_\mathrm{core}^2 } \biggr]^{1/2} \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} ( \xi_i \theta_i )^3
\biggr\}^{-2} \biggl(\frac{3}{2\pi}\biggr)^{1/2} \xi_i
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{G}{K_c}\biggr]^{5/2} M_\mathrm{core}^{-1} 
\biggl(\frac{\pi}{2^3 3}\biggr)^{1/2} \xi_i^{-5} \theta_i^{-6}
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>\Rightarrow ~~~ \mathrm{volume}~=\biggl(\frac{2^2\pi}{3}\biggr)r_i^3</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl(\frac{2^2\pi}{3}\biggr)\biggl\{
\biggl[ \frac{G}{K_c}\biggr]^{5/2} M_\mathrm{core}^{-1} 
\biggl(\frac{\pi}{2^3 3}\biggr)^{1/2} \xi_i^{-5} \theta_i^{-6}
\biggr\}^3
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{G}{K_c}\biggr]^{15/2} M_\mathrm{core}^{-3} 
\biggl(\frac{\pi}{2 \cdot 3}\biggr)^{5/2}
\xi_i^{-15} \theta_i^{-18} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right"><math>M_i</math></td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{K_c^3}{G^3 \rho_0^{2/5} } \biggr]^{1/2} \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} ( \xi_i \theta_i)^3 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
\biggl[ \frac{K_c^3}{G^3 } \biggr]^{1/2} \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1 / 2}
\biggl\{
\biggl[ \frac{K_c^3}{G^3 M_\mathrm{core}^2 } \biggr]^{1/2} \biggl( \frac{2\cdot 3}{\pi } \biggr)^{1/2} ( \xi_i \theta_i )^3
\biggr\}^{-1} ( \xi_i \theta_i)^3 
</math>
  </td>
</tr>

<tr>
  <td align="right">&nbsp;</td>
  <td align="center"><math>=</math></td>
  <td align="left">
<math>
M_\mathrm{core} \, .
</math>
  </td>
</tr>
</table>

Immediately below we reproduce [[SSC/Structure/PolytropesEmbedded#Fig3|Figure 3 from our accompanying discussion of ''embedded (pressure-truncated) polytropes'' having <math>n=5</math>]].  Notice that frame (a) contains a plot that displays our "yet another normalization" expressions for <math>P_i</math> vs. volume.

<div align="center" id="Fig3">
<table border="1" align="center" cellpadding="8" width="1050px">
<tr>
  <td align="center" colspan="6">
Equilibrium Sequences of Pressure-Truncated, n = 5 Polytropic Spheres<br />(viewed from several different astrophysical perspectives)
  </td>
</tr>
<tr>
  <td align="center"><font color="black" size="+2">&#x25CF;</font></td><td align="center"><math>~\xi_e</math></td>
  <td align="center" width="300px"><sup>&dagger;</sup>External Pressure vs. Volume<br /><font size="-1">(Fixed Mass)</font></td>
  <td align="center" width="300px">Mass vs. Radius<br /><font size="-1">(Fixed External Pressure)</font></td>
  <td align="center" width="300px"><sup>&Dagger;</sup>Mass vs. Central Density<br /><font size="-1">(Fixed External Pressure)</font></td>
  <td align="center" width="300px">Mass vs. Central Density<br /><font size="-1">(Fixed Radius)</font></td>
</tr>
<tr>
  <td align="center" colspan="1"><font color="yellow" size="+2">&#x25CF;</font></td>  <td align="center" colspan="1">&radic;3</td>
  <td align="center" colspan="1" rowspan="4">(a)<br />
[[File:N5Sequence01B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
  </td>
  <td align="center" colspan="1" rowspan="4">(b)<br />
[[File:N5Sequence02B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
  </td>
  <td align="center" colspan="1" rowspan="4">(c)<br />
[[File:N5Sequence03B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
  </td>
  <td align="center" colspan="1" rowspan="4">(d)<br />
[[File:N5Sequence04B.png|300px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
  </td>
</tr>
<tr>
  <td align="center" colspan="1"><font color="darkgreen" size="+2">&#x25CF;</font></td>  <td align="center" colspan="1">3</td>
</tr>
<tr>
  <td align="center" colspan="1"><font color="purple" size="+2">&#x25CF;</font></td>  <td align="center" colspan="1">&radic;15</td>
</tr>
<tr>
  <td align="center" colspan="1"><font color="red" size="+2">&#x25CF;</font></td>  <td align="center" colspan="1">9.01</td>
</tr>
<tr>
  <td align="center" colspan="2">&nbsp;</td>
  <td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^3 \biggl[ \xi^{18} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-12} \biggr]_\tilde\xi</math><br /> vs. <br />
<math>\biggl(\frac{\pi}{2\cdot 3}\biggr)^{5/2} \biggl[ \xi^{-15} \biggl(1 + \frac{\xi^2}{3} \biggr)^{9}\biggr]_\tilde\xi</math>
</td>
  <td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2}\biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. 
<br /> <math>\biggl(\frac{3}{2\pi}\biggr)^{1 / 2} \biggl[ \xi \biggl(1 + \frac{\xi^2}{3} \biggr)^{-1} \biggr]_\tilde\xi</math></td>
  <td align="center" colspan="1"><math>\biggl(\frac{2\cdot 3}{\pi}\biggr)^{1 / 2} \biggl[ \xi^{3} \biggl(1 + \frac{\xi^2}{3} \biggr)^{-2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \biggl(1 + \frac{\xi^2}{3} \biggr)^{5/2}\biggr]_\tilde\xi</math>
  </td>
  <td align="center" colspan="1"><math>\biggl[ \frac{2^3\cdot 3}{\pi} \biggr]^{1 / 4} \biggl[  \xi^{5/2}\biggl(1 + \frac{\xi^2}{3}\biggr)^{-3 / 2}\biggr]_\tilde\xi</math> <br /> vs. <br /> <math>\biggl[ \frac{3}{2\pi} \biggr]^{5 / 4} \tilde\xi^{5 / 2}</math>
  </td>
</tr>
</table>
</div>