Editing SSC/Virial/PolytropesSummary (section)

=Serious Concern=

==Statement of Concern==
Throughout our discussion of embedded (pressure-truncated) polytropes &#8212; both on this "summary" page and in an [[User:Tohline/SSC/Virial/Polytropes#Virial_Equilibrium_of_Adiabatic_Spheres|accompanying chapter]] where critical background derivations are presented &#8212;  we have used expressions for the structural form factors that include an overall leading factor of <math>~(5-n)^{-1}</math>.  For clarity, the form factors that we have used [[User:Tohline/SSC/Virial/Polytropes#Summary|for ''isolated'' polytropes]] is reprinted on the lefthand side of the following table while the ones that we have used [[User:Tohline/SSC/Virial/Polytropes#PTtable|for ''pressure-truncated'' polytropes]] is reprinted on the righthand side of the 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>
  <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">
<tr>
  <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">
<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}  + \tilde\Theta^{n+1}
</math>
  </td>
</tr>
</table>

  </td>
</tr>
</table>
</div>
This factor seemed destined to become a nuisance in the specific case of <math>~n=5</math> polytropic structures.  But we did not let its appearance in these expressions deter us from using a free-energy analysis to study the equilibrium and stability of spherical polytropes because, after all, the factor of <math>~(5-n)^{-1}</math> appears in [[User:Tohline/Appendix/References|Chandrasekhar's [C67]]] expression for the gravitational potential energy of ''isolated'' polytropes &#8212; see his Equation (90), p. 101. In retrospect, its appearance in the structural form factors for ''isolated'' polytropes did not prove to be a problem because, via a free-energy and virial theorem analysis, the [[User:Tohline/SSC/Virial/Polytropes#Isolated_Nonrotating_Adiabatic_Configuration|derived expression for the configuration's equilibrium radius]] depends on the ratio of <math>~f_W</math> to <math>~f_A</math>, so the awkward factor of <math>~(5-n)^{-1}</math> cancels out.

However, in our discussion of ''pressure-truncated'' <math>~n=5</math> polytropic structures, the factor of <math>~(5-n)^{-1}</math> did not conveniently cancel out at the appropriate time and we were forced to carry out some logical contortions [[User:Tohline/SSC/Virial/PolytropesSummary#Plotting_the_Virial_Theorem_Relation|as we tried to compare the mass-radius relation obtained from the virial theorem]] to Stahler's mass-radius relation, which was derived from detailed force-balance arguments.  This leads us to seriously question whether our, rather casually derived, expressions for the structural form factors in ''pressure-truncated'' polytropes are correct.

==Further Evaluation of n = 5 Polytropic Structures==
Throughout most of this subsection, we will adopt the shorthand notation,
<table align="center" border="1" cellpadding="10">
<tr><td align="center">
<math>~\ell \equiv \frac{\tilde\xi}{\sqrt{3}} ~~~~~\Rightarrow ~~~~~ \ell^2 = \frac{\tilde\xi^2}{3} \, .</math>
</td></tr>
</table>
This will not only simplify the appearance of some expressions, it will facilitate direct comparison with an expression for the free-energy coefficient, <math>~\mathcal{A}</math>, that has been derived in a [[User:Tohline/SSC/Virial/FormFactors#Structural_Form_Factors|companion chapter]] following a different train of logic and with an expression for the normalized gravitational potential energy that has been derived via a brute-force integration in association with our [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#Free_Energy|discussion of bipolytropic configurations]] where the variable, <math>~x_i</math>, has the same definition as <math>~\ell</math>.  

===Free-Energy Expression===

From our [[User:Tohline/SphericallySymmetricConfigurations/Virial#Free_Energy_Expression|general review of the topic]], to within an additive constant, the free-energy of a nonrotating, pressure-truncated polytrope comes from the sum of three principal energy terms, namely,
<div align="center">
<math>
\mathfrak{G} = W_\mathrm{grav} + \mathfrak{S}_\mathrm{therm} + P_e V \, .
</math>
</div> 
Furthermore, as has been shown in our extended [[User:Tohline/SphericallySymmetricConfigurations/Virial#Gathering_it_all_Together|introductory discussion of free energy]], the corresponding ''normalized'' free energy (applied to <math>~n=5</math> or <math>~\gamma_g = 6/5</math> configurations) is, 
<div align="center">
<math>
\mathfrak{G}^* \equiv \frac{\mathfrak{G}}{E_\mathrm{norm}} = 
-3A\chi^{-1} + 5B \chi^{-3/5} +~ D\chi^3 \, ,
</math>
</div>
where, 
<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<math>~\chi</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>\frac{R}{R_\mathrm{norm}}  \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~A</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>\frac{1}{5} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~B</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\biggl(\frac{3}{4\pi} \biggr)^{1/5} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{6/5}  \cdot \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{6/5}} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~D</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} \, .
</math>
  </td>
</tr>
</table>
</div>

Also note that the relevant normalizations are,
<div align="center">
<math>~E_\mathrm{norm} \equiv \biggl( \frac{K^5}{G^3} \biggr)^{1/2} \, ;</math>&nbsp; &nbsp; &nbsp; &nbsp;
<math>~R_\mathrm{norm} \equiv \biggl( \frac{G^5 M_\mathrm{tot}^4}{K^5} \biggr)^{1/2} \, ;</math>&nbsp; &nbsp; &nbsp; &nbsp;
<math>~P_\mathrm{norm} \equiv \frac{K^{10}}{G^9 M_\mathrm{tot}^{6}}  \, .</math>&nbsp; &nbsp; &nbsp; &nbsp;
</div>

===Virial Theorem===
The traditional expression for the virial theorem in this context is,
<div align="center">
<math>
~2S_\mathrm{therm}^* + W_\mathrm{grav}^* - \frac{3P_e V}{E_\mathrm{norm}} = 0 \, .
</math>
</div>
From our [[User:Tohline/VE#Adiabatic_Systems|introductory discussion of the thermodynamic energy reservoir]], we know that, for <math>~\gamma_g=6/5</math> configurations,
<div align="center">
<math>
~S_\mathrm{therm} = \frac{3}{2}(\gamma_g-1) \mathfrak{S}_\mathrm{therm} = \frac{3}{10} \mathfrak{S}_\mathrm{therm}\, .
</math>
</div>
So, making this substitution and recognizing that <math>~E_\mathrm{norm} = P_\mathrm{norm}R_\mathrm{norm}^3</math>, the (normalized) virial theorem expression becomes,
<div align="center">
<math>
~\frac{3}{5}\mathfrak{S}_\mathrm{therm}^* + W_\mathrm{grav}^* - \frac{3P_e V}{P_\mathrm{norm}R_\mathrm{norm}^3} = 0 \, .
</math>
</div>

Furthermore, by comparing terms in the first free-energy expression, above, with the second (normalized) free-energy expression, we see that,
<div align="center">
<math>~W_\mathrm{grav}^* \equiv \frac{W_\mathrm{grav}}{E_\mathrm{norm}} \rightarrow -3A\chi^{-1} \, ;</math>&nbsp; &nbsp; &nbsp; &nbsp;
<math>~\mathfrak{S}_\mathrm{therm}^* \equiv \frac{\mathfrak{S}_\mathrm{therm}}{E_\mathrm{norm}} \rightarrow 5B\chi^{-3/5} \, ;</math>&nbsp; &nbsp; &nbsp; &nbsp;
<math>~\frac{P_e V}{E_\mathrm{norm}} \rightarrow D\chi^{3}  \, .</math>&nbsp; &nbsp; &nbsp; &nbsp;
</div>
Hence, the normalized virial theorem may be written as,
<div align="center">
<math>
3B \chi^{-3/5} -3A\chi^{-1}  -~ 3D\chi^3 =0\, .
</math>
</div>

For sake of consistency, let's check this by holding the coefficients <math>~A</math>, <math>~B</math>, and <math>~D</math> constant and setting <math>~d\mathfrak{G}^*/d\chi</math> equal to zero:
<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<math>~\frac{d\mathfrak{G}^*}{d\chi} = - 3B \chi^{-8/5} + 3A\chi^{-2}  +~ 3D\chi^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0  </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ - \chi^{-1} \biggl[ 3B \chi^{-3/5} - 3A\chi^{-1} -~ 3D\chi^3 \biggr] </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0  \, .</math>
  </td>
</tr>
</table>
</div>
This, in turn, implies that the expression inside the square brackets sums to zero, which identically matches the (normalized) traditional virial theorem expression.  Excellent!

===Borrowing from Bipolytrope Discussion===
In an accompanying chapter that presents the [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1|detailed force-balanced models of <math>~(n_c, n_e) = (5, 1)</math> bipolytropes]] we explicitly show that, for configurations with the correct equilibrium radius, the virial theorem is satisfied.  In the case of bipolytropes, which are not embedded in an external medium, the relevant normalized virial theorem states that,
<div align="center">
<math>
~(2{S}_\mathrm{therm}^* + W_\mathrm{grav}^*)_\mathrm{core} + (2{S}_\mathrm{therm}^* + W_\mathrm{grav}^*)_\mathrm{env} = 0 \, .
</math>
</div>

In the bipolytrope, the (truncated) <math>~n=5</math> core is confined by an <math>~n=1</math> envelope; in addition to demanding that the relevant virial theorem be satisfied, there is also a constraint that the pressure at the inner edge of the envelope be equal to the pressure at the (truncated) outer edge of the core.  As we have just discussed, for a (truncated) <math>~n=5</math> polytrope that is confined by a hot, tenuous external medium instead of by an enveloping envelope, the relevant normalized virial theorem is,
<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<math>~(2{S}_\mathrm{therm}^* + W_\mathrm{grav}^*)_\mathrm{core} - \frac{3P_e V_\mathrm{eq}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~0  </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ (2{S}_\mathrm{therm}^* + W_\mathrm{grav}^*)_\mathrm{core} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{3P_e }{(K^5/G^3)^{1/2}} \biggl( \frac{4\pi}{3} R_\mathrm{eq}^3 \biggr) </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
 </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi \biggl( \frac{2\cdot 3}{5} \biggr)^{3/2}  \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}}\biggr)^3 \, ,</math>
  </td>
</tr>
</table>
</div>
where,  [[User:Tohline/SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|as discussed/defined in an accompanying chapter of this H_Book]], we have adopted the normalization radius, <math>~R_\mathrm{SWS}</math>, first introduced by [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Steven W. Stahler (1983)].  For <math>~n=5</math> configurations, its definition is,
<div align="center">
<math>
R_\mathrm{SWS}\biggr|_{n=5} = \biggl( \frac{2\cdot 3}{5} \biggr)^{1/2} \biggl[ \frac{(K^{5}/G^3)^{1/2}}{P_\mathrm{e}}\biggr]^{1/3} \, .
</math>
</div>
As has [[User:Tohline/SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|also been discussed in the accompanying chapter]], we can deduce from Stahler's detailed force-balanced models that the equilibrium radius of embedded, <math>~n=5</math> polytropes is given in terms of the dimensionless, ''truncated'' Lane-Emden radius, <math>~\tilde\xi</math> &#8212; and our corresponding variable, <math>\ell</math> &#8212; by the expression,
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} }\biggr|_{n=5}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{5}{4\pi} \biggr)^{1/2} \tilde\xi \tilde\theta^2 
= \biggl( \frac{5}{4\pi} \biggr)^{1/2} \tilde\xi \biggl(1 + \frac{\tilde\xi^2}{3}\biggr)^{-1} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{5\cdot 3}{2^2\pi} \biggr)^{1/2} \ell (1 + \ell^2)^{-1} \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, upon careful evaluation of the thermal energy and gravitational potential energy of truncated <math>~n=5</math> polytropes, we should find that,
<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<math>~(2{S}_\mathrm{therm}^* + W_\mathrm{grav}^*)_\mathrm{core} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\pi \biggl( \frac{2\cdot 3}{5} \biggr)^{3/2}  \biggl[ \biggl( \frac{5\cdot 3}{2^2\pi} \biggr)^{1/2} \ell (1 + \ell^2)^{-1}
\biggr]^3 </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{2\cdot 3^6}{\pi} \biggr)^{1/2} \ell^3 (1 + \ell^2)^{-3} \, .
</math>
  </td>
</tr>
</table>
</div>
Well, it just so happens that, in our [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#twoSplusWcore|accompanying chapter that presents the detailed force-balanced models of <math>~(n_c, n_e) = (5, 1)</math> bipolytropes]], we explicitly carried out the volume integrals defining these two key components of the free energy expression with the results being,
<div align="center">
<table border="0" cellpadding="4">

<tr>
  <td align="right">
<math>~(2S^* + W^*)_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{2\cdot 3^6}{\pi} \biggr)^{1/2} \biggl[ x_i ^3 (1 + x_i^2)^{-3} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>
Realizing that the variable, <math>~x_i</math>, in that context is the same as <math>~\ell</math>, in the present context, we see that the two separately derived results are identical to one another.

===Determining Expressions for Free-Energy Coefficients===
We should be able to convert the separately derived expression for <math>~W_\mathrm{grav}^*</math> into an expression for the free-energy coefficient, <math>~A</math>, in equilibrium configurations.  As [[User:Tohline/SSC/Virial/PolytropesSummary#Virial_Theorem|noted above]], for a fixed value of <math>~A</math>,
<div align="center">
<math>~W_\mathrm{grav}^* ~~\rightarrow ~~ -3A\chi^{-1} \, .</math>
</div>
Therefore, in an equilibrium configuration, we can write,
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
~W_\mathrm{grav}^*
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-3A\chi_\mathrm{eq}^{-1} = -3A \biggl( \frac{R_\mathrm{norm}}{R_\mathrm{eq}}\biggr)
= -3A \biggl( \frac{R_\mathrm{SWS}}{R_\mathrm{eq}}\biggr)\biggl( \frac{R_\mathrm{norm}}{R_\mathrm{SWS}}\biggr)</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~\Rightarrow ~~~ A
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
-\frac{1}{3} W_\mathrm{grav}^* \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{SWS}}\biggr)\biggl( \frac{R_\mathrm{SWS}}{R_\mathrm{norm}}\biggr) \, .</math>
  </td>
</tr>
</table>
</div>
Now, from immediately above, we know that,
<div align="center">
<table border="0" cellpadding="4">

<tr>
  <td align="right">
<math>
~\frac{R_\mathrm{eq}}{R_\mathrm{SWS} }\biggr|_{n=5}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{5\cdot 3}{2^2\pi} \biggr)^{1/2} \ell (1 + \ell^2)^{-1}  \, ;
</math>
  </td>
</tr>
</table>
</div>
and, from our [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#Free_Energy|accompanying discussion of the free-energy of bipolytropic configurations]], we know that,
<div align="center">
<table border="0" cellpadding="4">

<tr>
  <td align="right">
<math>~W^*_\mathrm{core}</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ - \biggl( \frac{3^8}{2^5\pi } \biggr)^{1/2} 
\biggl[ x_i\biggl(x_i^4 - \frac{8}{3}x_i^2 - 1\biggr) (1 + x_i^2)^{-3}  + \tan^{-1}(x_i) \biggr] \, . </math>
  </td>
</tr>
</table>
</div>
So, again realizing that <math>~x_i</math> and <math>~\ell</math> are interchangeable, we have,
<div align="center">
<table border="0" cellpadding="4">

<tr>
  <td align="right">
<math>~A</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ \frac{1}{3}
\biggl( \frac{3^8}{2^5\pi } \biggr)^{1/2} \biggl( \frac{5\cdot 3}{2^2\pi} \biggr)^{1/2} \ell (1 + \ell^2)^{-1}
\biggl[ \ell \biggl(\ell^4 - \frac{8}{3}\ell^2 - 1\biggr) (1 + \ell^2)^{-3}  + \tan^{-1}(\ell) \biggr] 
\biggl( \frac{R_\mathrm{SWS}}{R_\mathrm{norm}}\biggr)
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ 
\biggl( \frac{3^7 \cdot 5}{2^7\pi^2 } \biggr)^{1/2} \ell (1 + \ell^2)^{-1}
\biggl[ \ell \biggl(\ell^4 - \frac{8}{3}\ell^2 - 1\biggr) (1 + \ell^2)^{-3}  + \tan^{-1}(\ell) \biggr] 
\biggl( \frac{R_\mathrm{SWS}}{R_\mathrm{norm}}\biggr) \, .
</math>
  </td>
</tr>
</table>
</div>
Finally, we need to determine an expression for the ratio, <math>~R_\mathrm{SWS}/R_\mathrm{norm}</math>.  Drawing the definition of <math>~R_\mathrm{norm}</math> from [[User:Tohline/SphericallySymmetricConfigurations/Virial#Normalizations|our introductory chapter on the virial equilibrium]], we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~R_\mathrm{norm}\biggr|_{\gamma = 6/5}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \biggl( \frac{G}{K}\biggr) M_\mathrm{tot}^{2-6/5} \biggr]^{1/(4-18/5)}
= \biggl[ \biggl( \frac{G}{K}\biggr) M_\mathrm{tot}^{4/5} \biggr]^{5/2}
= \biggl( \frac{G}{K}\biggr)^{5/2} M_\mathrm{tot}^{2} \, .
</math>
  </td>
</tr>
</table>
</div>
From our [[User:Tohline/SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|tabular summary of Stahler's derived mass &amp; radius relationships for truncated, <math>~n=5</math> polytropes]] we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~M_\mathrm{limit}^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
M_\mathrm{SWS}^2 \biggl[ \biggl( \frac{3\cdot 5^3}{2^2\pi} \biggr) \ell^6 (1+\ell^2)^{-4} \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>
In addition, from our [[User:Tohline/SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|review of Stahler's defined normalizations]], we see that,
<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<math>~M_\mathrm{SWS}^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2^3 \cdot 3^3}{5^3} \biggr) G^{-3} K^{10/3} P_e^{-1/3} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
and, &nbsp;&nbsp;&nbsp;<math>~R_\mathrm{SWS}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2 \cdot 3}{5} \biggr)^{1/2} G^{-1/2} K^{5/6} P_e^{-1/3} \, ,
</math>
  </td>
</tr>
</table>
</div>
which, when combined to cancel <math>~P_e</math> gives,
<div align="center">
<table border="0" cellpadding="5">

<tr>
  <td align="right">
<math>~M_\mathrm{SWS}^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2^6 \cdot 3^6}{5^6} \biggr)^{1/2} G^{-3} K^{10/3} 
\biggl( \frac{5}{2 \cdot 3} \biggr)^{1/2} G^{1/2} K^{-5/6} R_\mathrm{SWS} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2^5 \cdot 3^5}{5^5} \biggr)^{1/2}\biggl( \frac{K}{G}\biggr)^{5/2} R_\mathrm{SWS} \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~M_\mathrm{limit}^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2^5 \cdot 3^5}{5^5} \biggr)^{1/2}\biggl( \frac{K}{G}\biggr)^{5/2}  
\biggl[ \biggl( \frac{3\cdot 5^3}{2^2\pi} \biggr) \ell^6 (1+\ell^2)^{-4} \biggr] R_\mathrm{SWS}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \frac{1}{R_\mathrm{SWS}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2\cdot 3^7 \cdot 5}{\pi^2} \biggr)^{1/2}\biggl( \frac{K}{G}\biggr)^{5/2}  
\biggl[ \ell^6 (1+\ell^2)^{-4} \biggr] \frac{1}{M_\mathrm{limit}^2} \, .
</math>
  </td>
</tr>
</table>
</div>
In combination with the expression for <math>~R_\mathrm{norm}</math>, then, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{R_\mathrm{norm}}{R_\mathrm{SWS}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl( \frac{2\cdot 3^7 \cdot 5}{\pi^2} \biggr)^{1/2} 
\biggl[ \ell^6 (1+\ell^2)^{-4} \biggr] \biggl(\frac{M_\mathrm{tot}}{M_\mathrm{limit}}\biggr)^{2} \, ,
</math>
  </td>
</tr>
</table>
</div>
which means that, for truncated <math>~n=5</math> polytropes, the expression for the free-energy coefficient is,
<div align="center">
<table border="0" cellpadding="4">

<tr>
  <td align="right">
<math>~A</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ 
\biggl( \frac{3^7 \cdot 5}{2^7\pi^2 } \biggr)^{1/2} \biggl( \frac{\pi^2}{2\cdot 3^7 \cdot 5} \biggr)^{1/2}
\ell (1 + \ell^2)^{-1} \cdot \ell^{-6} (1+\ell^2)^{4}  
\biggl[ \ell \biggl(\ell^4 - \frac{8}{3}\ell^2 - 1\biggr) (1 + \ell^2)^{-3}  + \tan^{-1}(\ell) \biggr] 
\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ 
2^{-4} \ell^{-5} (1+\ell^2)^{3}  
\biggl[ \ell \biggl(\ell^4 - \frac{8}{3}\ell^2 - 1\biggr) (1 + \ell^2)^{-3}  + \tan^{-1}(\ell) \biggr] 
\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{2} \, .
</math>
  </td>
</tr>
</table>
</div>
Finally, drawing from our [[User:Tohline/SSC/Virial/FormFactors#Gravitational_Potential_Energy|accompanying derivation of expressions for the structural form factors in this case]], we know that,
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>~\frac{M_\mathrm{limit}}{M_\mathrm{tot} } </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
\ell^3 (1+\ell^2)^{-3/2} \, ,
</math>
  </td>
</tr>
</table>
</div>
which gives,
<div align="center">
<table border="0" cellpadding="4">

<tr>
  <td align="right">
<math>~A</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>~ 
\frac{\ell}{2^{4} }  
\biggl[ \ell \biggl(\ell^4 - \frac{8}{3}\ell^2 - 1\biggr) (1 + \ell^2)^{-3}  + \tan^{-1}(\ell) \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

This exactly matches the expression for the free-energy coefficient, <math>~\mathcal{A}</math>, that we derived separately in conjunction with our [[User:Tohline/SSC/Virial/FormFactors#Gravitational_Potential_Energy|derivation of expressions for the structural form factors]].

==Take Care Comparing Gravitational Potential Energies==
Does this derived relation for the coefficient, <math>~A</math>, make sense?  Well, we've derived the relation by comparing two separate expressions for the gravitational potential energy that were normalized in slightly different ways, so the leading numerical coefficient may not be correct.  We need to repeat the derivation, checking the relative normalizations carefully.  But before doing this, let's determine what we ''expected'' the relation to be, based on the expressions for the structural form factors that we have been using.

From the [[User:Tohline/SSC/Virial/PolytropesSummary#Serious_Concern|lead-in paragraphs of this subsection]], we have previously assumed that,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>~\mathcal{A}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>\frac{1}{5} \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} 
= \frac{1}{5} \cdot \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \biggl\{ \frac{3^2\cdot 5}{5-n} \biggl[ \frac{\tilde\theta^'}{\tilde\xi} \biggr]^2 \biggr\}
\biggl\{ \biggl[ - \frac{3\tilde\theta^'}{\tilde\xi} \biggr] \biggr\}^{-2} = \frac{1}{(5-n)} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \, .
</math>
  </td>
</tr>

</table>
</div>
According to the line of reasoning presented above, the coefficient, <math>~A</math>, is related to the gravitational potential energy via the expression,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~W_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-3A\chi^{-1} E_\mathrm{norm} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-3A \biggl( \frac{K^5}{G^3} \biggr)^{1/2} \biggl( \frac{G^5 M_\mathrm{tot}^4}{K^5} \biggr)^{1/2} \frac{1}{R}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-3A \biggl(\frac{GM_\mathrm{tot}^2}{R} \biggr) \, .</math>
  </td>
</tr>
</table>
</div>
From our [[User:Tohline/SphericallySymmetricConfigurations/Virial#Expressions_for_Various_Energy_Terms|introductory layout of the free-energy function for polytropes]] &#8212; see, also, p. 64, Equation (12) of [[User:Tohline/Appendix/References|Chandrasekhar [C67]]] &#8212; the gravitational potential energy is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~W_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \int_0^{R_\mathrm{limit}} \biggl( \frac{GM_r}{r} \biggr) 4\pi r^2 \rho dr \, ,</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~M_r</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\int_0^r 4\pi r^2 \rho dr \, .</math>
  </td>
</tr>
</table>
</div>
Now, independent of the chosen normalization, if we use <math>~M_\mathrm{tot}</math> to represent the total mass of an ''isolated'' <math>~n=5</math> polytrope, then from [[User:Tohline/SSC/Structure/PolytropesEmbedded#Review_2|an earlier review]], we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~M_\mathrm{tot}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggr[ \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr]^{1/2}  \rho_c^{-1/5} \, ,</math>
  </td>
</tr>
</table>
</div>
and we can write, in terms of the Lane-Emden dimensionless radius, <math>~\xi</math>,
<div align="center">
<math>
\frac{M_r}{M_\mathrm{tot}} = \xi^3 (3 + \xi^2)^{-3/2}  \, .
</math> 
</div>
===Virial Chapter===
Now, in our discussion of the virial equilibrium of embedded polytropes, we used the normalizations specified above and wrote,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~W_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
- 4\pi GM_\mathrm{tot} R_\mathrm{norm}^2 \rho_\mathrm{norm} \int_0^{R_\mathrm{limit}/R_\mathrm{norm}} \biggl[\frac{M_r}{M_\mathrm{tot}} \biggr]  r^* \rho^* dr^* 
</math>
  </td>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
- E_\mathrm{norm} \int_0^{R_\mathrm{limit}/R_\mathrm{norm}} 3\biggl[\frac{M_r}{M_\mathrm{tot}} \biggr]  r^* \rho^* dr^* \, .
</math>
  </td>
</tr>
</table>
</div>
We can replace <math>~r^* \equiv r/R_\mathrm{norm}</math> with <math>~\xi \equiv r/a_5</math> by recognizing that, 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~M_\mathrm{tot}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(  \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr)^{1/2} \rho_c^{-1/5} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \rho_c^{-2/5} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(  \frac{\pi G^3}{2\cdot 3^4 K^3} \biggr)  M_\mathrm{tot}^2 \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~a_5</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(  \frac{3K}{2\pi G} \biggr)^{1/2} \rho_c^{-2/5} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(  \frac{3K}{2\pi G} \biggr)^{1/2} \biggl(  \frac{\pi G^3}{2\cdot 3^4 K^3} \biggr)  M_\mathrm{tot}^2
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(  \frac{\pi}{2^3\cdot 3^7} \biggr)^{1/2}  \biggl(  \frac{G}{K} \biggr)^{5/2} M_\mathrm{tot}^2 \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~R_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(  \frac{G}{K} \biggr)^{5/2} M_\mathrm{tot}^{2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~ \frac{a_5}{R_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(  \frac{\pi}{2^3\cdot 3^7} \biggr)^{1/2}  \, .
</math>
  </td>
</tr>
</table>
</div>
Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~r^*</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(  \frac{\pi}{2^3\cdot 3^7} \biggr)^{1/2} \xi \, .
</math>
  </td>
</tr>
</table>
</div>

Also,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3}{4\pi} \biggl[ \frac{K}{G} \biggr]^{15/2} M_\mathrm{tot}^{-5}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3}{4\pi} \biggl[ \frac{K}{G} \biggr]^{15/2} \biggl[ \biggl(  \frac{2\cdot 3^4 K^3}{\pi G^3} \biggr)^{1/2} \rho_c^{-1/5}\biggr]^{-5}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3}{4\pi} \biggl[ \frac{K}{G} \biggr]^{15/2} \biggl(  \frac{\pi G^3}{2\cdot 3^4 K^3} \biggr)^{5/2} \rho_c
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(\frac{3^2}{2^4\pi^2} \biggr)^{1/2} \biggl(  \frac{\pi^5 }{2^5 \cdot 3^{20}} \biggr)^{1/2} \rho_c
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(  \frac{\pi^3 }{2^9 \cdot 3^{18}} \biggr)^{1/2} \rho_c \, .
</math>
  </td>
</tr>
</table>
</div>
Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\rho^* \equiv \frac{\rho}{\rho_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(  \frac{2^9 \cdot 3^{18}}{\pi^3 } \biggr)^{1/2} \frac{\rho}{\rho_c} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl(  \frac{2^9 \cdot 3^{18}}{\pi^3 } \biggr)^{1/2} \biggl[ 1 + \frac{\xi^2}{3} \biggr]^{-5/2} \, .
</math>
  </td>
</tr>
</table>
</div>
So the energy integral becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm} }</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
- 3 \biggl(  \frac{\pi}{2^3\cdot 3^7} \biggr) \biggl(  \frac{2^9 \cdot 3^{18}}{\pi^3 } \biggr)^{1/2} 
\int_0^{\tilde\xi} \biggl[\xi^3 (3 + \xi^2)^{-3/2}  \biggr]  \xi \biggl[ 1 + \frac{\xi^2}{3} \biggr]^{-5/2} d\xi 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
- \biggl(  \frac{2^3 \cdot 3^{6}}{\pi } \biggr)^{1/2} 
\int_0^{\tilde\xi} \xi^4 \biggl[ 1 + \frac{\xi^2}{3} \biggr]^{-4} d\xi \, .
</math>
  </td>
</tr>
</table>
</div>
This needs to be compared with the <math>~W_\mathrm{grav}^*</math> integral that we previously have handled in [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#Expression_for_Free_Energy|the chapter discussing bipolytrope models]].