Editing Apps/SMS (section)

===Mass Normalization===
Now, according to {{ BAC84hereafter }} (see their equation 8), when the total pressure is written in polytropic form &#8212; specifically, if we set,
<div align="center">
<math>P = K\rho^{(1+1/n_p)} </math>
</div>

&#8212; the mass-scaling for relativistic configurations will depend on <math>~G</math>, <math>~c</math>, <math>~K</math>, and <math>~n_p</math> via the expression,

<div align="center">
<math>~M_u = K^{n_p/2} G^{-3/2} c^{3-n_p} = \biggl( \frac{K}{G}\biggr)^{3/2} \biggl(\frac{K}{c^2}\biggr)^{(n_p-3)/2} \, .</math>
</div>

<!-- COMMENT OUT SMALL SECTION
It is convenient to rewrite this expression in the form,

<div align="center">
<math>~M_u = M_\mathrm{norm} \biggl(\frac{K}{c^2}\biggr)^{(n_p-3)/2} \, ,</math>
</div>

and to determine, first, an expression for the mass-normalization when <math>~n_p = 3</math>, namely,
<div align="center">
<math>~M_\mathrm{norm} \equiv \biggl( \frac{K}{G}\biggr)^{3/2} .</math>
</div>
-->

====Polytropic Index Equals 3====
Referencing our [[SSC/Structure/BiPolytropes/Analytic1.53#HighlightedExpressions|separate discussion of Milne's (1930) work]], when <math>~n_p = 3</math>, the polytropic constant is related to the relevant set of physical parameters via the relation,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~K_{3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \biggl( \frac{\Re}{\bar\mu}\biggr)^4 \biggl(\frac{1-\beta}{\beta^4}\biggr) \frac{3}{a_\mathrm{rad}} \biggr]^{1/3} \, .</math>
  </td>
</tr>
</table>
</div>
Adopting the {{ BAC84hereafter }} terminology, this means that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl(\frac{K_{3}}{G}\biggr)^3</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{\Re}{\bar\mu}\biggr)^4 \biggl[\frac{1-\beta}{\beta^4}\biggr] \frac{3}{G^3 a_\mathrm{rad}} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{k Y_T}{m_B}\biggr)^4 \biggl[\sigma^4(1+\sigma^{-1})^3\biggr] \frac{3}{G^3 a_\mathrm{rad}} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ M_\mathrm{norm} \equiv \biggl(\frac{K_{3}}{G}\biggr)^{3/2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(1+\sigma^{-1})^{3/2} \biggl( \frac{k Y_T}{m_B}\biggr)^2  \biggl(\frac{3}{G^3 a_\mathrm{rad}}\biggr)^{1/2} \sigma^2 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\biggl(1+\frac{3}{2\sigma} \biggr) \biggl( \frac{k Y_T}{m_B}\biggr)^2  \biggl(\frac{3}{G^3 a_\mathrm{rad}}\biggr)^{1/2} \sigma^2 \, .</math>
  </td>
</tr>
</table>
</div>
When radiation pressure significantly dominates over gas pressure &#8212; that is, in the limit <math>~\sigma \gg 1</math> &#8212; the leading factor is approximately unity, in which case we see that this expression for <math>~M_\mathrm{norm}</math> exactly matches the expression for <math>~M_{u,3}</math> given by equation (10) of {{ BAC84hereafter }}.

====Polytropic Index Slightly Less Than 3====
More generally, equating the two expressions for the total pressure and drawing (twice) on the expression for <math>~\sigma</math> [[#Ratio_of_Radiation_Pressure_to_Gas_Pressure|provided above]], we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~K\rho^{(1 + 1/n_p)}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~Y_T n k T + \frac{a_\mathrm{rad}}{3} T^4 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a_\mathrm{rad}}{3} (1+\sigma^{-1})T^4 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{a_\mathrm{rad}}{3} (1+\sigma^{-1})\biggl[ \frac{3Y_T n k \sigma}{a_\mathrm{rad}} \biggr]^{4/3} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{3}{a_\mathrm{rad}} \biggr)^{1/3}(1+\sigma^{-1})\biggl[ Y_T n k \sigma \biggr]^{4/3} \, .</math>
  </td>
</tr>
</table>
</div>

Now, from [[#GammaApprox|above]] we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~1 + \frac{1}{n_p} = \Gamma</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\frac{4}{3} + \frac{1}{6\sigma} \, ,</math>
  </td>
</tr>
</table>
</div>

so the lefthand-side of this last expression can be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~K\rho^{(1+1/n_p)}</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~K\rho^{(4/3+1/6\sigma)} = K(m_B n)^{4/3} \rho^{1/6\sigma} \, .</math>
  </td>
</tr>
</table>
</div>
This means that, for any <math>~\sigma \gg 1</math>,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~K </math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{3}{a_\mathrm{rad}} \biggr)^{1/3} \biggl( \frac{Y_T  k \sigma}{m_B} \biggr)^{4/3} (1+\sigma^{-1}) \rho^{-1/6\sigma} \, .</math>
  </td>
</tr>
</table>
</div>
This matches exactly expression (7) in {{ BAC84hereafter }}.  Again from [[#GammaApprox|above]] &#8212; and continuing to assume <math>~\sigma \gg 1</math> &#8212; we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~1 + \frac{1}{n_p} \approx \frac{4}{3} + \frac{1}{6\sigma} </math>
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow ~~~~</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\frac{1}{n_p} \approx \frac{1}{3}\biggl(1 + \frac{1}{2\sigma}\biggr) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow ~~~~</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~n_p \approx 3\biggl(1 + \frac{1}{2\sigma}\biggr)^{-1} \approx 3\biggl(1 - \frac{1}{2\sigma}\biggr)</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp; &nbsp; &nbsp; <math>~~~~\Rightarrow ~~~~</math>&nbsp; &nbsp; &nbsp; 
  </td>
  <td align="left">
<math>~\frac{(n_p-3)}{2} \approx - \frac{3}{4\sigma} \, .</math>
  </td>
</tr>
</table>
</div>

Hence, when the polytropic index is slightly less than 3, the mass normalization is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~M_u</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{K}{G}\biggr)^{3/2} \biggl(\frac{K}{c^2}\biggr)^{-3/4\sigma}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{1}{G} \biggl( \frac{3}{a_\mathrm{rad}} \biggr)^{1/3}
\biggl( \frac{Y_T  k \sigma}{m_B} \biggr)^{4/3} (1+\sigma^{-1}) \rho^{-1/6\sigma} \biggr]^{3/2} 
\biggl[\frac{1}{c^2} \biggl( \frac{3}{a_\mathrm{rad}} \biggr)^{1/3}
\biggl( \frac{Y_T  k \sigma}{m_B} \biggr)^{4/3} (1+\sigma^{-1}) \rho^{-1/6\sigma} \biggr]^{-3/4\sigma} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\biggl( \frac{3}{G^3 a_\mathrm{rad}} \biggr)^{1/2}
\biggl( \frac{Y_T  k }{m_B} \biggr)^{2} (1+\sigma^{-1})^{3/2} \sigma^2 \biggr] \rho^{-1/4\sigma} 
\biggl[\frac{1}{c^2} \biggl( \frac{3}{a_\mathrm{rad}} \biggr)^{1/3}
\biggl( \frac{Y_T  k \sigma}{m_B} \biggr)^{4/3} (1+\sigma^{-1}) \rho^{-1/6\sigma} \biggr]^{-3/4\sigma} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[M_\mathrm{norm} \biggr]  
\biggl\{\frac{1}{c^2} \biggl( \frac{3}{a_\mathrm{rad}} \biggr)^{1/3}
\biggl( \frac{Y_T  k \sigma}{m_B} \biggr)^{4/3} (m_B n)^{1/3} \biggl[(1+\sigma^{-1}) \rho^{-1/6\sigma} \biggr] \biggr\}^{-3/4\sigma} </math>
  </td>
</tr>
</table>
</div>
Drawing again from the definition of <math>~\sigma</math> [[#Ratio_of_Radiation_Pressure_to_Gas_Pressure|provided above]], we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl(\frac{3}{a_\mathrm{rad}}\biggr)^{1/3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~T (\sigma Y_T n k)^{-1/3} \, ,</math>
  </td>
</tr>
</table>
</div>
so this last relation can be rewritten as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~M_u</math>
  </td>
  <td align="center">
<math>~\approx</math>
  </td>
  <td align="left">
<math>~
M_\mathrm{norm} \cdot f \biggl[\frac{m_B c^2}{T \sigma Y_T k} \biggr]^{3/4\sigma} 
\approx f \cdot M_{u,3} \biggl(1+\frac{3}{2\sigma} \biggr) \biggl[\frac{m_B c^2}{T \sigma Y_T k} \biggr]^{3/4\sigma} 
\, ,</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<math>f \equiv \biggl[(1+\sigma^{-1})^{-1}\rho^{1/6\sigma} \biggr]^{3/4\sigma} 
\approx \biggl( 1 - \frac{3}{4\sigma^2}\biggr) (n m_B)^{1/8\sigma^2}\, ,</math>
</div>
which certainly is close to unity when <math>~\sigma \gg 1</math>.  After setting <math>~f=1</math>, this last expression for <math>~M_u</math> exactly matches the expression presented as equation (9) in {{ BAC84hereafter }}.