Editing SSC/Virial/Polytropes/Pt2 (section)

====Contrast with Detailed Force-Balanced Solution====
As has just been demonstrated, the virial theorem provides a mathematical expression that allows us to ''relate'' the equilibrium radius of a configuration to the applied external pressure, once the configuration's mass and either its specific entropy or central pressure are specified.  In contrast to this, as has been [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres|discussed in detail in another chapter]], {{ Horedt70full }},   {{ Whitworth81 }} and {{ Stahler83full }} have each derived separate analytic expressions for <math>~R_\mathrm{eq}</math> and <math>~P_e</math> &#8212; given in terms of the Lane-Emden function, <math>~\Theta</math>, and its radial derivative &#8212; without demonstrating how the equilibrium radius and external pressure directly relate to one another.  That is to say, solution of the detailed force-balanced equations provides a pair of equilibrium expressions that are parametrically related to one another through the Lane-Emden function.  For example &#8212; see our [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation|related discussion for more details]] &#8212; {{ Horedt70 }} derives the following set of parametric equations relating the configuration's dimensionless radius, <math>~r_a</math>, to a specified dimensionless bounding pressure, <math>~p_a</math>:
<div align="center">
<table border="0" cellpadding="3">

<tr>
  <td align="right">
<math>
~r_a \equiv \frac{R_\mathrm{eq}}{R_\mathrm{Horedt}}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>
~p_a \equiv \frac{P_\mathrm{e}}{P_\mathrm{Horedt}}
</math>
  </td>
  <td align="center">
<math>~=~</math>
  </td>
  <td align="left">
<math>
\tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)} \, ,
</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="3">

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

<tr>
  <td align="right">
<math>
~P_\mathrm{Horedt}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
K^{4n/(n-3)}\biggl[ \frac{(n+1)^3}{4\pi G^3 M_\mathrm{limit}^2} \biggr]^{(n+1)/(n-3)}  \, .
</math>
  </td>
</tr>
</table>
</div>

It is important to appreciate that, in the expressions for <math>~r_a</math> and <math>~p_a</math>, the tilde indicates that the Lane-Emden function and its derivative are to be evaluated, not at the radial coordinate, <math>~\xi_1</math>, that is traditionally associated with the "first zero" of the Lane-Emden function and therefore with the surface of the ''isolated polytrope,'' but at the radial coordinate, <math>~\tilde\xi</math>, where the internal pressure of the isolated polytrope equals <math>~P_e</math> and at which the ''embedded'' polytrope is to be truncated.  The coordinate, <math>~\tilde\xi</math>, therefore identifies the surface of the embedded &#8212; or, pressure-truncated &#8212; polytrope.  We also have taken the liberty of attaching the subscript "limit" to <math>~M</math> in both defining relations because it is clear that {{ Horedt70 }} intended for the normalization mass to be the mass of the pressure-truncated object, not the mass of the associated ''isolated'' (and untruncated) polytrope.  In anticipation of further derivations, below, we note here the ratio of the {{ Horedt70 }} normalization parameters to ours, assuming <math>~\gamma = (n+1)/n</math>:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{R_\mathrm{Horedt}}{R_\mathrm{norm}} \biggr)^{n-3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl[ \frac{4\pi}{(n+1)^n}\biggl( \frac{G}{K} \biggr)^n M_\mathrm{limit}^{n-1} \biggr]
\biggl[ \biggl( \frac{K}{G}\biggr)^n M_\mathrm{tot}^{1-n}\biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\frac{4\pi}{(n+1)^n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{n-1} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\biggl( \frac{P_\mathrm{Horedt}}{P_\mathrm{norm}} \biggr)^{n-3}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
K^{4n}\biggl[ \frac{(n+1)^3}{4\pi G^3 M_\mathrm{limit}^2} \biggr]^{n+1} \biggl[ \frac{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}}{K^{4n}} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 
\biggl[ \frac{(n+1)^3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2}\biggr]^{n+1} \, .
</math>
  </td>
</tr>
</table>
</div>

Next, we demonstrate that this pair of parametric relations satisfies the virial theorem and, in so doing, demonstrate how <math>~r_a</math> and <math>~p_a</math> may be ''directly'' related to each other.  Given that the normalization radius and normalization pressure chosen by {{ Horedt70 }} are defined in terms of <math>~K</math> and <math>~M_\mathrm{limit}</math>, we begin with the [[SSC/Virial/Polytropes#Solution_Expressed_in_Terms_of_K_and_M_.28Whitworth.27s_1981_Relation.2|virial theorem derived above]] in terms of <math>~K</math> and <math>~M_\mathrm{limit}</math>, setting <math>~\gamma_g = (n+1)/n</math>.

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

<tr>
  <td align="right">
<math>~P_e </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~K \biggl( \frac{3M_\mathrm{limit}}{4\pi R_\mathrm{eq}^3} \biggr)^{(n+1)/n} \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{(n+1)/n}} 
- \biggl(\frac{3GM_\mathrm{limit}^2}{20\pi R_\mathrm{eq}^4} \biggr) \frac{\mathfrak{f}_W}{\mathfrak{f}^2_M} \, .</math>
  </td>
</tr>
</table>
</div>

After setting <math>~R_\mathrm{eq} = r_a R_\mathrm{Horedt} </math>, a bit of algebraic manipulation shows that the first term on the right-hand side of the virial equilibrium expression becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
K \biggl( \frac{3M_\mathrm{limit}}{4\pi R_\mathrm{eq}^3} \biggr)^{(n+1)/n} \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{(n+1)/n}}
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
r_a^{-3(n+1)/n} \mathfrak{f}_A \biggl[ \biggl( \frac{3}{\mathfrak{f}_M} \biggr)^{n-3} \frac{(n+1)^{3n}}{(4\pi)^n}\biggr]^{(n+1)/[n(n-3)]}
[K^{4n} G^{-3(n+1)} M_\mathrm{limit}^{-2(n+1)} ]^{1/(n-3)} \, ,
</math>
  </td>
</tr>
</table>
</div>
while the second term on the right-hand side becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\biggl(\frac{3GM_\mathrm{limit}^2}{20\pi R_\mathrm{eq}^4} \biggr) \frac{\mathfrak{f}_W}{\mathfrak{f}^2_M} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3}{5} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} \cdot r_a^{-4} (4\pi)^{-(n+1)/(n-3)} (n+1)^{4n/(n-3)}~
[K^{4n} G^{-3(n+1)} M_\mathrm{limit}^{-2(n+1)} ]^{1/(n-3)} \, .
</math>
  </td>
</tr>
</table>
</div>
But, using Horedt's expression for <math>~P_e</math>, the left-hand side of the virial equilibrium equation becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~P_e = p_a P_\mathrm{Horedt}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
p_a~(4\pi)^{-(n+1)/(n-3)} ~(n+1)^{3(n+1)/(n-3)}
[K^{4n} G^{-3(n+1)} M_\mathrm{limit}^{-2(n+1)} ]^{1/(n-3)} \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, the statement of virial equilibrium is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
p_a~
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggr\{ r_a^{-3(n+1)/n} \mathfrak{f}_A \biggl[ \biggl( \frac{3}{\mathfrak{f}_M} \biggr)^{n-3} \frac{(n+1)^{3n}}{(4\pi)^n}\biggr]^{(n+1)/[n(n-3)]}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~~
- \frac{3}{5} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} \cdot r_a^{-4} (4\pi)^{-(n+1)/(n-3)} (n+1)^{4n/(n-3)} \biggr\}(4\pi)^{(n+1)/(n-3)} ~(n+1)^{-3(n+1)/(n-3)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathfrak{f}_A \biggl( \frac{3}{\mathfrak{f}_M \cdot r_a^3} \biggr)^{(n+1)/n} 
- \frac{3(n+1)}{5} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} \cdot r_a^{-4} \, ;
</math>
  </td>
</tr>
</table>
</div>

or, multiplying through by <math>~r_a^4</math> and rearranging terms,

<div align="center">
<table border="1" align="center" cellpadding="8">
<tr><td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~
\mathfrak{f}_A \biggl( \frac{3}{\mathfrak{f}_M} \biggr)^{(n+1)/n} r_a^{(n-3)/n} - p_a r_a^4
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3(n+1)}{5} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} \, .
</math>
  </td>
</tr>
</table>

</td></tr>
</table>
</div>

Now, {{ Horedt70 }} has given analytic expressions for <math>~r_a</math> and <math>~p_a</math> in terms of the Lane-Emden function and its first derivative.  The question is, what should the expressions for our structural form factors be in order for this virial expression to hold true for all pressure-truncated polytropic structures?  As has been [[SSC/Virial/Polytropes#Summary|summarized above]], in the case of an ''isolated'' polytrope, whose surface is located at <math>~\xi_1</math> and whose global properties are defined by evaluation of the Lane-Emden function at <math>~\xi_1</math>, we know that (see the [[SSC/Virial/Polytropes#Summary|above summary]]), 

<div align="center">
<table border="0" 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>~\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>
</tr>
</table>
</div>
These same expressions may or may not work for pressure-truncated polytropes, even if the evaluation radius is shifted from <math>~\xi_1</math> to <math>~\tilde\xi</math>.  Let's see &hellip;

<font color="red"><b>
January 13, 2015:
</b></font> 
As is noted in our [[SSC/Virial/PolytropesEmbeddedOutline#Third_Effort|accompanying outline of work]], I no longer believe that <math>~\mathfrak{f}_W</math> and <math>~\mathfrak{f}_A</math> have the same expressions as in isolated polytropes.  Hence, all of the material that follows is suspect and needs to be reworked.

{{ SGFworkInProgress }}

Inserting the expressions for <math>r_a</math> and <math>p_a</math>, as provided by {{ Horedt70 }}, into the virial equilibrium expression, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~
\frac{3(n+1)}{5} \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathfrak{f}_A \biggl( \frac{3}{\mathfrak{f}_M} \biggr)^{(n+1)/n} [ \tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)}  ]^{(n-3)/n}
~-~ 
\tilde\theta_n^{n+1}( -\tilde\xi^2 \tilde\theta' )^{2(n+1)/(n-3)} [ \tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)}  ]^{4}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathfrak{f}_A \biggl( \frac{3}{\mathfrak{f}_M} \biggr)^{(n+1)/n} [ \tilde\xi ( -\tilde\xi^2 \tilde\theta' )^{(1-n)/(n-3)}  ]^{(n-3)/n}
~-~ 
\tilde\theta_n^{n+1} \tilde\xi^4 [( -\tilde\xi^2 \tilde\theta' )^{2(n+1)+4(1-n)}  ]^{1/(n-3)} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathfrak{f}_A \biggl( \frac{3}{\mathfrak{f}_M} \biggr)^{(n+1)/n} [ \tilde\xi^{(n-3)} ( -\tilde\xi^2 \tilde\theta' )^{(1-n)}  ]^{1/n}
~-~
\tilde\theta_n^{n+1} \tilde\xi^4 [( -\tilde\xi^2 \tilde\theta' )^{-2}  ]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\mathfrak{f}_A \biggl( \frac{3}{\mathfrak{f}_M} \biggr)^{(n+1)/n}  \tilde\xi^{-(n+1)/n} ( -\tilde\theta' )^{(1-n)/n}  
~-~
\frac{\tilde\theta_n^{n+1} }{( -\tilde\theta' )^{2} } \, .
</math>
  </td>
</tr>

</table>
</div>
If we assume that both of the structural form factors, <math>~\mathfrak{f}_W</math> and <math>~\mathfrak{f}_M</math>, have the same functional expressions as in the case of isolated polytropes (but evaluated at <math>~\tilde\xi</math> instead of at <math>~\xi_1</math>), the virial relation further reduces to the form,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
 \frac{\mathfrak{f}_A}{(- \tilde\theta' )^2} - \frac{3(n+1)}{5-n} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~
\Rightarrow~~~\mathfrak{f}_A 
</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{3(n+1)}{5-n} (- \tilde\theta' )^2 + \tilde\theta_n^{n+1} \, .
</math>
  </td>
</tr>

</table>
</div>
This all seems to make a great deal of sense.  Only the structural parameter that is derived from an integral over the pressure distribution, <math>~\mathfrak{f}_A</math>, gets modified when the polytropic configuration is truncated.   Notice, as well, that the term that has been added to the definition of <math>~\mathfrak{f}_A</math> naturally goes to zero in the limit of <math>~\tilde\xi \rightarrow \xi_1</math>, that is, for an isolated polytrope.  We should definitely go back to the original definitions of all three structural parameters and prove that this is the case.  But, in the meantime, here is the summary:

<div align="center" id="PTtable">
<table border="1" align="center" cellpadding="5">
<tr><th align="center" colspan="1">
<font size="+1" color="red"><b>WRONG!!</b></font>&nbsp; For the correct form-factor expressions, go [[SSC/FreeEnergy/PolytropesEmbedded#Free-Energy_of_Truncated_Polytropes|here]].
</th></tr>
<tr><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>~\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>

<font size="+1" color="red"><b>WRONG!!</b></font>&nbsp; For the correct form-factor expressions, go [[SSC/FreeEnergy/PolytropesEmbedded#Free-Energy_of_Truncated_Polytropes|here]].

Notice that, in an effort to differentiate them from their counterparts developed earlier for "isolated" polytropes, we have affixed a tilde to each of these three form-factors, <math>~\mathfrak{f}_i</math>.