Editing SSC/Structure/StahlerMassRadius (section)

==Comparison==
First, let's insert the definitions of the coefficients <math>~\mathcal{A}</math>, <math>~\mathcal{B}</math>, and <math>~\mathcal{C}</math> into the virial equilibrium expression, replacing, where necessary, the adiabatic exponent in favor of the polytropic index, using the relation, <math>~\gamma_g = (n+1)/n</math>.
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<math>~0</math>
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<math>~=</math>
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<math>~\mathcal{A} - \mathcal{B}\chi_\mathrm{eq}^{(n-3)/n}  +~ \mathcal{D}\chi_\mathrm{eq}^4</math>
  </td>
</tr>

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&nbsp;
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<math>~=</math>
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<math>~
\frac{1}{5} \cdot \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) 
\frac{1}{\tilde\mathfrak{f}_M} \biggr]^2 \cdot \tilde\mathfrak{f}_W 
~-~ \frac{4\pi}{3} 
\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 \biggl(\frac{R_\mathrm{eq}}{R_\mathrm{norm}}\biggr)^{(n-3)/n}  
+~ \biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}}\biggr) ^4
</math>
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Next, explicitly spelling out as well the definitions of our adopted normalization radius and normalization pressure &#8212; recognizing that <math>~P_\mathrm{norm} R_\mathrm{norm}^4 = GM^2_\mathrm{tot}</math> &#8212; and multiply the expression through by <math>[3GM_\mathrm{tot}^2/(4\pi)]</math>. 
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<math>~0</math>
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<math>~=</math>
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<math>~
\frac{3}{20\pi} \biggl( \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \biggr) GM_\mathrm{limit}^2  
~-~ GM_\mathrm{tot}^2
\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)  \biggr]_\mathrm{eq}^{(n+1)/n}
\biggl[ \frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{(n+1)/n}} \biggr] 
\biggl[ \biggl( \frac{K}{G} \biggr) M_\mathrm{tot}^{(1-n)/n} \biggr]  R_\mathrm{eq}^{(n-3)/n}
+~ P_e R_\mathrm{eq}^4
</math>
  </td>
</tr>

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&nbsp;
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<math>~=</math>
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<math>~
\frac{3}{20\pi} \biggl( \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \biggr) GM_\mathrm{limit}^2  
~-~ K
\biggl[ \frac{3}{4\pi} \cdot \frac{M_\mathrm{limit}}{\tilde\mathfrak{f}_M} \biggr]_\mathrm{eq}^{(n+1)/n} \mathfrak{f}_A
R_\mathrm{eq}^{(n-3)/n}
+~ P_e R_\mathrm{eq}^4 \, .
</math>
  </td>
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</table>
</div>
As has been pointed out in our separate, [[SSC/Virial/Polytropes#Solution_Expressed_in_Terms_of_K_and_M_.28Whitworth.27s_1981_Relation.29|more general discussion of the virial equilibrium of polytropes]], if we multiply this expression through by <math>~R_\mathrm{eq}^{-4}</math>, set all three structural form factors, <math>~\mathfrak{f}_i</math>, equal to unity, and replace <math>~M_\mathrm{limit}</math> with the notation, <math>~M_0</math>, the expression exactly matches the one presented as equation (5) of [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth], which reads:
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[[File:Whitworth1981Eq5.jpg|500px|center|Whitworth (1981, MNRAS, 195, 967)]]
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But I like this last version of our derived expression as well because it shows some resemblance to the mass-radius relationship presented by Stahler and highlighted above:  The first term on the left-hand-side is a constant times the square of the mass; the third term is a constant times the fourth power of the equilibrium radius; and the middle term shows a cross-product of the mass and radius (in our case, each is raised to a power other than unity). In an effort to make the comparison with Stahler even clearer, let's rewrite our expression in terms of the mass and equilibrium radius, normalized respectively to <math>~M_\mathrm{SWS}</math> and <math>~R_\mathrm{SWS}</math>.  
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<math>~0</math>
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<math>~=</math>
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<math>~
\frac{3G}{20\pi} \biggl( \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^2
\biggl[ \biggl( \frac{n+1}{nG} \biggr)^{3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \biggr]^2 
~+~ P_e \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^4 \biggl[ \biggl( \frac{n+1}{nG} \biggr)^{1/2} K_n^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \biggr]^4 
</math>
  </td>
</tr>

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&nbsp;
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&nbsp;
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<math>~
~-~ K \mathfrak{f}_A
\biggl[ \frac{3}{4\pi} \cdot \frac{1}{\tilde\mathfrak{f}_M} \biggr]^{(n+1)/n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n}
\biggl[ \biggl( \frac{n+1}{nG} \biggr)^{3/2} K_n^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \biggr]^{(n+1)/n}
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^{(n-3)/n} \biggl[ \biggl( \frac{n+1}{nG} \biggr)^{1/2} K_n^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \biggr]^{(n-3)/n}
</math>
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</tr>

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&nbsp;
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<math>~=</math>
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<math>~
\frac{3}{20\pi} \biggl( \frac{n+1}{n} \biggr)^{3} \biggl( \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^2
\biggl[  K_n^{4n} P_\mathrm{e}^{(3-n)} \biggr]^{1/(n+1)}  G^{-2}
~+~ \biggl( \frac{n+1}{n} \biggr)^{2} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^4 \biggl[ K_n^{4n} P_\mathrm{e}^{[(n+1)+2(1-n)]} \biggr]^{1/(n+1)} G^{-2}
</math>
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</tr>

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&nbsp;
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&nbsp;
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<math>~
~-~\mathfrak{f}_A
\biggl[ \frac{3}{4\pi} \cdot \frac{1}{\tilde\mathfrak{f}_M} \biggr]^{(n+1)/n} \biggl( \frac{n+1}{nG} \biggr)^{[3(n+1)+(n-3)]/2n} 
\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n}
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^{(n-3)/n} 
\biggl[ K^{1 + 2 + (n-3)/(n+1)} \biggr] \biggl[ P_\mathrm{e}^{(3-n) + (1-n)(n-3)/(n+1) } \biggr]^{1/(2n)}
</math>
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</tr>

<tr>
  <td align="right">
&nbsp;
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<math>~=</math>
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<math>~
\frac{3}{20\pi} \biggl( \frac{n+1}{n} \biggr)^{3} \biggl( \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^2
\biggl[  K_n^{4n} P_\mathrm{e}^{(3-n)} \biggr]^{1/(n+1)}  G^{-2}
~+~ \biggl( \frac{n+1}{n} \biggr)^{2} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^4 \biggl[ K_n^{4n} P_\mathrm{e}^{(3-n)} \biggr]^{1/(n+1)} G^{-2}
</math>
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</tr>

<tr>
  <td align="right">
&nbsp;
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&nbsp;
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<math>~
~-~\mathfrak{f}_A
\biggl[ \frac{3}{4\pi} \cdot \frac{1}{\tilde\mathfrak{f}_M} \biggr]^{(n+1)/n} \biggl( \frac{n+1}{n} \biggr)^{2} 
\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n}
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^{(n-3)/n} 
\biggl[ K^{4n/(n+1)} \biggr] \biggl[ P_\mathrm{e}^{(3-n)/(n+1) } \biggr] G^{-2}
</math>
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</tr>

<tr>
  <td align="right">
&nbsp;
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<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3}{20\pi} \biggl( \frac{n+1}{n} \biggr) \biggl( \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^2
~-~\mathfrak{f}_A
\biggl[ \frac{3}{4\pi} \cdot \frac{1}{\tilde\mathfrak{f}_M} \biggr]^{(n+1)/n} 
\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{SWS}} \biggr)^{(n+1)/n}
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^{(n-3)/n} 
~+~ \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}} \biggr)^4 
</math>
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</tr>
</table>
</div>

On 30 September 2014, J.E. Tohline showed that this expression is perfectly satisfied using Stahler's definitions of the normalized mass and normalized equilibrium radius along with Tohline's definitions of the structural form factors.  The middle term on the right-hand side includes the structural form factor, <math>~\mathfrak{f}_A</math>, which in turn is the sum of two pieces; the first piece of this form factor supplies the term that cancels the first term on the right-hand side of the equilibrium expression and the second piece cancels the third term on the right-hand side.  We have noticed that the first term on the right-hand side (via the factor, <math>~\mathfrak{f}_W</math>) and the first portion of the middle term (via the first term in the expression for <math>~\mathfrak{f}_A</math>), contain factors of <math>~(5-n)^{-1}</math>, which will cause these terms to blow up when considering truncated polytropes of index, <math>~n = 5</math>.  But this is precisely the case for which Stahler provides an analytic mass-radius relationship.  What happens to the virial expression if we multiply through by <math>~(5-n)</math>?