Editing SSC/Virial/Polytropes/Pt2 (section)

====Renormalization====

=====Grunt Work=====

Returning to the dimensionless form of the virial expression and multiplying through by <math>~[-\chi_\mathrm{eq}/(3D)]</math>, we obtain,
<div align="center">
<math>
\chi_\mathrm{eq}^4 = \frac{B}{D} \chi_\mathrm{eq}^{4-3\gamma_g} - \frac{A}{D} \, ,
</math>
</div>
or, after plugging in [[SSCpt1/Virial#Gathering_it_all_Together|definitions of the coefficients]], <math>~A</math>, <math>~B</math>, and <math>~D</math>, and rewriting <math>~\chi_\mathrm{eq}</math> explicitly as <math>~R_\mathrm{eq}/R_\mathrm{norm}</math>,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math> 
\biggl(\frac{3}{4\pi}\biggr)^{\gamma_g} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{\gamma_g}
\biggl( \frac{P_e}{P_\mathrm{norm}} \biggr)^{-1} \frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{\gamma_g}}
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{4-3\gamma_g} - 
\frac{3}{20\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr)^{-1} 
\frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \, .</math>
  </td>
</tr>
</table>
</div>
This relation can be written in a more physically concise form, as follows.  First, normalize <math>~P_e</math> to a new pressure scale &#8212; call it <math>~P_\mathrm{ad}</math> &#8212; and multiply through by <math>~(R_\mathrm{norm}/R_\mathrm{ad})^4</math> in order to normalizing <math>~R_\mathrm{eq}</math> to a new length scale,<math>~R_\mathrm{ad}</math>: 
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{ad}} \biggr)^4  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math> 
\biggl(\frac{3}{4\pi}\biggr)^{\gamma_g} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{\gamma_g}
\biggl( \frac{P_\mathrm{norm}}{P_\mathrm{ad}} \biggr) \biggl( \frac{P_e}{P_\mathrm{ad}} \biggr)^{-1} \frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{\gamma_g}}
\biggl( \frac{R_\mathrm{norm}}{R_\mathrm{ad}} \biggr)^{3\gamma_g}\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{ad}} \biggr)^{4-3\gamma_g} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math> 
~- 
\frac{3}{20\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2} \biggl( \frac{P_\mathrm{norm}}{P_\mathrm{ad}} \biggr)
\frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \biggl( \frac{P_e}{P_\mathrm{ad}} \biggr)^{-1} \biggl( \frac{R_\mathrm{norm}}{R_\mathrm{ad}} \biggr)^{4} \, ,
</math>
  </td>
</tr>
</table>
</div>
or,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\chi_\mathrm{ad}^4  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math> 
\biggl(\frac{3}{4\pi}\biggr)^{\gamma_g} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{\gamma_g}
\biggl( \frac{P_\mathrm{norm}}{P_\mathrm{ad}} \biggr) \biggl( \frac{R_\mathrm{norm}}{R_\mathrm{ad}} \biggr)^{3\gamma_g} 
\frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{\gamma_g}}\cdot \frac{\chi_\mathrm{ad}^{4-3\gamma_g} }{\Pi_\mathrm{ad}} - 
\frac{3}{20\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2}
\biggl( \frac{P_\mathrm{norm}}{P_\mathrm{ad}} \biggr) \biggl( \frac{R_\mathrm{norm}}{R_\mathrm{ad}} \biggr)^{4}
\frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \cdot \frac{1}{\Pi_\mathrm{ad}}   \, ,</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\chi_\mathrm{ad}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{ad}} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Pi_\mathrm{ad}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{P_\mathrm{e}}{P_\mathrm{ad}} \, .</math>
  </td>
</tr>
</table>
</div>
By demanding that the leading coefficients of both terms on the right-hand-side of the expression are simultaneously unity &#8212; that is, by demanding that,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\biggl(\frac{3}{4\pi}\biggr)^{\gamma_g} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{\gamma_g}
\biggl( \frac{P_\mathrm{norm}}{P_\mathrm{ad}} \biggr) \biggl( \frac{R_\mathrm{norm}}{R_\mathrm{ad}} \biggr)^{3\gamma_g} 
\frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{\gamma_g}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 \, ,</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{3}{20\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2}
\biggl( \frac{P_\mathrm{norm}}{P_\mathrm{ad}} \biggr) \biggl( \frac{R_\mathrm{norm}}{R_\mathrm{ad}} \biggr)^{4}
\frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 \, ,</math>
  </td>
</tr>
</table>
</div>
we obtain the expressions for <math>~R_\mathrm{ad}/R_\mathrm{norm}</math> and <math>~P_\mathrm{ad}/P_\mathrm{norm}</math> as shown in the following table.
<div align="center">
<table border="1" align="center" cellpadding="5">
<tr><th align="center" colspan="1">
Renormalization for Adiabatic (''ad'') Systems
</th></tr>
<tr><td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\frac{R_\mathrm{ad}}{R_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ \frac{1}{5} \biggl( \frac{4\pi}{3} \biggr)^{\gamma_g-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2-\gamma_g}
\frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_A \cdot \tilde\mathfrak{f}_M^{2-\gamma_g}} \biggr]^{1/(4-3\gamma_g)} </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[
\frac{1}{5^n} \biggl( \frac{4\pi}{3}\biggr)\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{n-1}
\frac{\tilde\mathfrak{f}_W^n}{\tilde\mathfrak{f}_A^n \cdot \tilde\mathfrak{f}_M^{n-1}}
\biggr]^{1/(n-3)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\frac{P_\mathrm{ad}}{P_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~ \biggl[ \tilde\mathfrak{f}_A^4 \cdot \biggl( \frac{3\cdot  5^3}{4\pi} \cdot 
\frac{\tilde\mathfrak{f}_M^2}{\tilde\mathfrak{f}_W^3} \biggr)^{\gamma_g} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2\gamma_g} \biggr]^{1/(4-3\gamma)}  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[
\tilde\mathfrak{f}_A^{4n} \biggl(\frac{3\cdot 5^3}{4\pi} \biggr)^{n+1} \biggl( \frac{\tilde\mathfrak{f}_M^2}{\tilde\mathfrak{f}_w^3} \biggr)^{n+1}
\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2(n+1)}
\biggr]^{1/(n-3)}
</math>
  </td>
</tr>
</table>

</td>
</tr>
</table>
</div>
 
Using these new normalizations, we arrive at the desired, concise virial equilibrium relation, namely,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\Pi_\mathrm{ad} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\chi_\mathrm{ad}^{-3\gamma_g} - \chi_\mathrm{ad}^{-4} \, ,</math>
  </td>
</tr>
</table>
</div>
or,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\chi_\mathrm{ad}^{4-3\gamma_g} - \Pi_\mathrm{ad} \chi_\mathrm{ad}^4</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 \, .</math>
  </td>
</tr>
</table>
</div>


For the sake of completeness, we should develop expressions for both <math>~\chi_\mathrm{ad}</math> and <math>~\Pi_\mathrm{ad}</math> that are entirely in terms of the Lane-Emden function, <math>~\tilde\theta</math>, its derivative, <math>~\tilde\theta'</math>, and the associated dimensionless radial coordinate, <math>\tilde\xi</math>, at which the function and its derivative are to be evaluated.  (Adopting a unified notation, we will set <math>~\gamma_g \rightarrow (n+1)/n</math>.)

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

<tr>
  <td align="right">
<math>~\chi_\mathrm{ad} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{ad}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{Horedt}} \cdot \frac{R_\mathrm{Horedt}}{R_\mathrm{norm}} \cdot \frac{R_\mathrm{norm}}{R_\mathrm{ad}}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;  
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
r_a \cdot \biggl[ \frac{4\pi}{(n+1)^n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{n-1} \biggr]^{1/(n-3)} \cdot 
\biggl[ 5^n \biggl( \frac{3}{4\pi} \biggr) \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{1-n}
\frac{\tilde\mathfrak{f}_A^n}{\tilde\mathfrak{f}_W^n \cdot \tilde\mathfrak{f}_M^{1-n}}
\biggr]^{1/(n-3)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~\chi_\mathrm{ad}^{n-3} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
3 r_a^{n-3} \cdot \biggl[ \frac{5}{(n+1)} \cdot
\frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_W} \biggr]^n \tilde\mathfrak{f}_M^{n-1} \, .
</math>
  </td>
</tr>
</table>
</div>

Inserting the functional expression for <math>r_a</math> from {{ Horedt70 }}, and our structural form factors, <math>\tilde\mathfrak{f}_i</math>, gives,

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

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

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

<tr>
  <td align="right">
<math>~\Rightarrow ~~~~ \chi_\mathrm{ad}^{(n-3)/n}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{(5-n)}{3(n+1)} \cdot (-\tilde\theta')^{-2} \biggl[ \frac{3(n+1)}{(5-n)} (-\tilde\theta')^2 + \theta^{n+1} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~1 + 
\frac{(5-n)}{3(n+1)} \cdot \frac{\tilde\theta^{n+1}}{(-\tilde\theta')^2} \, .
</math>
  </td>
</tr>
</table>
</div>
And,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\Pi_\mathrm{ad} \equiv \frac{P_e}{P_\mathrm{ad}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{P_e}{P_\mathrm{Horedt}} \cdot \frac{P_\mathrm{Horedt}}{P_\mathrm{norm}} \cdot \frac{P_\mathrm{norm}}{P_\mathrm{ad}}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
p_a \cdot \biggl[ \frac{(n+1)^3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{-2}\biggr]^{(n+1)/(n-3)} \cdot 
\biggl[ \tilde\mathfrak{f}_A^{-4n} \cdot \biggl( \frac{4\pi}{3\cdot  5^3} \cdot 
\frac{\tilde\mathfrak{f}_W^3}{\tilde\mathfrak{f}_M^2} \biggr)^{(n+1)} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^{2(n+1)} 
\biggr]^{1/(n-3)}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow~~~\Pi_\mathrm{ad}^{n-3} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
p_a^{n-3}~ \tilde\mathfrak{f}_A^{-4n} \cdot \biggl[ 
\frac{(n+1)^3}{3\cdot  5^3} \biggl( \frac{\tilde\mathfrak{f}_W^3}{\tilde\mathfrak{f}_M^2} \biggr)
\biggr]^{n+1} \, .
</math>
  </td>
</tr>
</table>
</div>

Inserting the functional expressions for <math>~p_a</math> from {{ Horedt70 }}, and our structural form factors, <math>\tilde\mathfrak{f}_i</math>, gives,
<div align="center">
<table border="0" cellpadding="5" align="center">

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

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

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\tilde\theta^{(n+1)(n-3)}
\biggl[ \frac{3(n+1) }{(5-n)} ~( \tilde\theta^' )^2  + \tilde\theta^{n+1} \biggr]^{-4n} 
~\biggl[ \frac{3(n+1)}{(5-n)} ( \tilde\theta' )^{2} \biggr]^{3(n+1)} \, .
</math>
  </td>
</tr>
</table>
</div>
(Not a particularly simple or transparent expression!)

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

<tr>
  <td align="right">
<math>~a_\mathrm{ad}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\frac{3(n+1) }{(5-n)} ~(\tilde\theta^' )^2 \, ,</math> &nbsp;&nbsp;&nbsp; and,
  </td>
</tr>

<tr>
  <td align="right">
<math>~b_\mathrm{ad}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\tilde\theta^{n+1} \, ,</math>
  </td>
</tr>
</table>
</div>
the expressions for the dimensionless equilibrium radius and the dimensionless external pressure may be written as, respectively,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>~\chi_\mathrm{ad}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ 1 + \frac{b_\mathrm{ad}}{a_\mathrm{ad}} \biggr]^{n/(n-3)} \, ,
</math> &nbsp;&nbsp;&nbsp; and,
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Pi_\mathrm{ad} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
b_\mathrm{ad} \biggl[ \frac{a_\mathrm{ad}^{3(n+1)} }{( a_\mathrm{ad} + b_\mathrm{ad} )^{4n} } \biggr]^{1/(n-3)}
=
\frac{b_\mathrm{ad}}{a_\mathrm{ad}} \biggl[ 1 + \frac{b_\mathrm{ad}}{a_\mathrm{ad}} \biggr]^{-4n/(n-3)}
\, .
</math>
  </td>
</tr>
</table>
</div>

Using these expressions, it is easy to demonstrate that the virial equilibrium relation is satisfied, namely,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\Pi_\mathrm{ad} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\chi_\mathrm{ad}^{-3(n+1)/n} - \chi_\mathrm{ad}^{-4} \, .</math>
  </td>
</tr>
</table>
</div>

{{ SGFworkInProgress }}