Editing SSCpt1/Virial (section)

====Normalizations====

=====Our Choices=====
It is appropriate for us to define some characteristic scales against which various physical parameters can be normalized &#8212; and, hence, their relative significance can be specified or measured &#8212; as the free energy of various systems is examined.  As the system size is varied in search of extrema in the free energy, we generally will hold constant the total system mass and the specific entropy of each fluid element.  (When isothermal rather than adiabatic variations are considered, the sound speed rather than the specific entropy will be held constant.)  Hence, following the lead of both {{ Horedt70full }} and {{ Whitworth81full }}, we will express the various characteristic scales in terms of the constants, <math>G, M_\mathrm{tot},</math> and the polytropic constant, <math>K.</math>  Specifically, we will normalize all length scales, pressures, energies, mass densities, and the square of the speed of sound by, respectively,
<div align="center">
<table border="1" align="center" cellpadding="5">
<tr><th align="center" colspan="2">
Adopted Normalizations
</th></tr>
<tr>
  <td align="center">
Adiabatic Cases
  </td>
  <td align="center">
Isothermal Case
<math>~(\gamma = 1; K = c_s^2)</math>
  </td>
</tr>
<tr><td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>R_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\biggl[ \biggl( \frac{G}{K} \biggr) M_\mathrm{tot}^{2-\gamma} \biggr]^{1/(4-3\gamma)} </math>
  </td>
</tr>

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

<tr>
  <td align="center" colspan="3">
----
  </td>
</tr>

<tr>
  <td align="right">
<math>E_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>P_\mathrm{norm} R_\mathrm{norm}^3 = 
\biggl[ KG^{3(1-\gamma)}M_\mathrm{tot}^{6-5\gamma} \biggr]^{1/(4-3\gamma)} </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\rho_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{3M_\mathrm{tot}}{4\pi R_\mathrm{norm}^3}
= \frac{3}{4\pi} \biggl[ \frac{K^3}{G^3 M_\mathrm{tot}^2} \biggr]^{1/(4-3\gamma )}  </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>c^2_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{P_\mathrm{norm}}{\rho_\mathrm{norm}}
= \frac{4\pi}{3} \biggl[ \frac{K}{(G^3 M_\mathrm{tot}^2)^{\gamma-1}} \biggr]^{1/(4-3\gamma )}  </math>
  </td>
</tr>
</table>

</td>

<td align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>R_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{G M_\mathrm{tot}}{c_s^2}   </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{c_s^8}{G^{3} M_\mathrm{tot}^{2}}   </math>
  </td>
</tr>

<tr>
  <td align="center" colspan="3">
----
  </td>
</tr>

<tr>
  <td align="right">
<math>E_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>M_\mathrm{tot} c_s^2 </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\rho_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{3}{4\pi} \biggl[ \frac{c_s^6}{G^3 M_\mathrm{tot}^2} \biggr]  </math>
  </td>
</tr>

<tr>
  <td align="right">
<math>c^2_\mathrm{norm}</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{4\pi}{3} \biggr) c_s^2 </math>
  </td>
</tr>
</table>

</td>
</tr>

<tr><td align="left" colspan="2">
Note that, given the above definitions, the following relations hold:
<div align="center">
<math>E_\mathrm{norm} = P_\mathrm{norm} R_\mathrm{norm}^3 = \frac{G M_\mathrm{tot}^2}{ R_\mathrm{norm}} 
= \biggl( \frac{3}{4\pi} \biggr) M_\mathrm{tot} c_\mathrm{norm}^2</math>
</div>
</td></tr>
</table>
</div>

It should be emphasized that, as we discuss how a configuration's free energy varies with its size, the variable <math>R_\mathrm{limit}</math> will be used to identify the configuration's size ''whether or not the system is in equilibrium,''  and the parameter,
<div align="center">
<math>\chi \equiv \frac{R_\mathrm{limit}}{R_\mathrm{norm}} \, ,</math>
</div>
will be used to identify the size as referenced to <math>R_\mathrm{norm}</math>.  When an equilibrium configuration is identified <math>(R_\mathrm{limit} \rightarrow R_\mathrm{eq})</math>, we will affix the subscript "eq," specifically,
<div align="center">
<math>\chi_\mathrm{eq} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \, .</math>
</div>

=====Choices Made by Other Researchers=====

As is detailed in a [[SSC/Structure/PolytropesEmbedded#General_Properties|related discussion]], our definitions of <math>R_\mathrm{norm}</math> and <math>P_\mathrm{norm}</math> are close, but not identical, to the scalings adopted by {{ Horedt70 }} and by {{ Whitworth81 }}. The following relations can be used to switch from our normalizations to theirs:
<div align="center">

<table border="0" cellpadding="5" align="center">
<tr>
  <td align="center">

<table border="0" cellpadding="5" align="center">
<tr><th colspan="3" align="center">[[SSC/Structure/PolytropesEmbedded|Hoerdt's (1970)]] Normalization</th><tr>

<tr>
  <td align="right">
<math>\biggl( \frac{R_\mathrm{Hoerdt}}{R_\mathrm{norm}} \biggr)^{4-3\gamma}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{(\gamma-1)}{\gamma} \biggl( 4\pi \biggr)^{\gamma-1}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\biggl( \frac{P_\mathrm{Hoerdt}}{P_\mathrm{norm}} \biggr)^{4-3\gamma}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[\frac{\gamma}{(\gamma-1)} \biggr]^{3\gamma} \biggl( \frac{1}{4\pi} \biggr)^{\gamma}</math>
  </td>
</tr>
</table>

  </td>
  <td align="center">
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
  </td>
  <td align="center">

<table border="0" cellpadding="5" align="center">
<tr><th colspan="3" align="center">[[SSC/Structure/PolytropesEmbedded|Whitworth's (1981)]] Normalization</th><tr>

<tr>
  <td align="right">
<math>\biggl( \frac{R_\mathrm{rf}}{R_\mathrm{norm}} \biggr)^{4-3\gamma}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{5\pi} \biggl( \frac{4\pi}{3} \biggr)^\gamma</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\biggl( \frac{P_\mathrm{rf}}{P_\mathrm{norm}} \biggr)^{4-3\gamma}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2^{-2(4+\gamma)} \biggl( \frac{3^4 \cdot 5^3}{\pi} \biggr)^\gamma</math>
  </td>
</tr>
</table>

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

It is also worth noting how the length-scale normalization that we are adopting here relates to the characteristic length scale, 
<div align="center">
<math>a_n \equiv \biggl[ \frac{1}{4\pi G} \biggl( \frac{H_c}{\rho_c} \biggr) \biggr]^{1/2} \, ,</math>
</div>
that has classically been adopted in the context of the [[SSC/Structure/Polytropes#Lane-Emden_Equation|Lane-Emden equation]], the solution of which provides a detailed description of the internal structure of spherical polytropes for a wide range of values of the polytropic index, {{ Template:Math/MP_PolytropicIndex }}.  
Recognizing that, via the [[SR#Barotropic_Structure|polytropic equation of state]], the pressure, density, and enthalpy of every element of fluid are related to one another via the expressions,
<div align="center">
<math>H\rho = (n+1)P</math> &nbsp;&nbsp;&nbsp;&nbsp;&hellip; and 
&hellip; &nbsp;&nbsp;&nbsp;&nbsp; <math>P = K_n\rho^{1+1/n} \, ,</math>
</div>
the specific enthalpy at the center of a polytropic sphere, <math>H_c/\rho_c</math>, can be rewritten in terms of {{ Template:Math/MP_PolytropicConstant }} and <math>\rho_c</math> to give,
<div align="center">
<math>~a_n = \biggl[ \frac{(n+1)K_n}{4\pi G} \rho_c^{(1/n) -1} \biggr]^{1/2} \, ,</math>
</div>
which is the definition of this classical length scale introduced by [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>] (see, specifically, his equation 10 on p. 87).  Switching from {{ Template:Math/MP_PolytropicIndex }} to the associated adiabatic exponent via the relation, <math>\gamma = 1+1/n ~~~\Rightarrow~~~ n = 1/(\gamma-1)</math>, we see that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl( \frac{a_n}{R_\mathrm{norm}} \biggr)^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\gamma}{\gamma-1} \biggr) \frac{K_n \rho_c^{(\gamma-2)}}{4\pi G}  \cdot \frac{1}{R_\mathrm{norm}^2}</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{4\pi}\biggl( \frac{\gamma}{\gamma-1} \biggr) \frac{K_n }{G}  \biggl( \frac{\rho_c}{\bar\rho} \biggr)^{(\gamma-2)} 
\biggl( \frac{3M_\mathrm{tot}}{4\pi R_\mathrm{eq}^3} \biggr)^{(\gamma-2)} 
\cdot \frac{1}{R_\mathrm{norm}^2} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{4\pi} \biggl( \frac{\gamma}{\gamma-1} \biggr) \biggl( \frac{3}{4\pi } \cdot \frac{\rho_c}{\bar\rho} \biggr)^{\gamma-2} 
\biggl[ \frac{K_n M_\mathrm{tot}^{\gamma-2} }{G}  \biggr]
\biggl( \frac{R_\mathrm{norm}}{R_\mathrm{eq}} \biggr)^{3{(\gamma-2)}} 
\cdot \frac{1}{R_\mathrm{norm}^{3\gamma-4}} 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{4\pi} \biggl( \frac{\gamma}{\gamma-1} \biggr) \biggl( \frac{3}{4\pi } \cdot \frac{\rho_c}{\bar\rho} \biggr)^{2-\gamma}
\chi_\mathrm{eq}^{6-3\gamma}
\biggl[ \frac{K_n M_\mathrm{tot}^{\gamma-2} }{G} \biggr]  
\cdot \biggl[ \biggl( \frac{G}{K} \biggr) M_\mathrm{tot}^{2-\gamma} \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{4\pi} \biggl( \frac{\gamma}{\gamma-1} \biggr) \biggl( \frac{3}{4\pi } \cdot \frac{\rho_c}{\bar\rho} \biggr)^{2-\gamma}
\chi_\mathrm{eq}^{6-3\gamma} \, .
</math>
  </td>
</tr>
</table>
</div>

Notice that, written in this manner, the scale length, <math>a_n</math>, cannot actually be determined unless the normalized equilibrium radius, <math>\chi_\mathrm{eq}</math>, is known.  We will encounter analogous situations whenever the free energy function is used to identify the physical parameters that define equilibrium configurations &#8212; key attributes of a system that should be held fixed as the system size (or some other order parameter) is varied cannot actually be evaluated until an extremum in the free energy is identified and the corresponding value of <math>\chi_\mathrm{eq}</math> is known.  Because solutions of the Lane-Emden equation directly provide detailed force-balance models of polytropic spheres, [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>] did not encounter this issue.  As we have [[SSC/Structure/Polytropes#Known_Analytic_Solutions|discussed elsewhere]], the equilibrium radius of a polytropic sphere is identified as the radial location, 
<div align="center">
<math>\xi_1 = \frac{R_\mathrm{eq}}{a_n} \, ,</math>
</div>
at which the Lane-Emden function, <math>\Theta_H(\xi)</math>, first goes to zero.  Bypassing the free-energy analysis and using knowledge of <math>\xi_1</math> to identify the equilibrium radius &#8212; specifically, setting,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\chi_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{R_\mathrm{eq}}{R_\mathrm{norm}} = \xi_1 \biggl(\frac{a_n}{R_\mathrm{norm}} \biggr) \, ,</math>
  </td>
</tr>
</table>
</div>

we can extend the above analysis to obtain,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl( \frac{a_n}{R_\mathrm{norm}} \biggr)^2</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{4\pi} \biggl( \frac{\gamma}{\gamma-1} \biggr) \biggl( \frac{4\pi }{3} \cdot \frac{\rho_c}{\bar\rho} \biggr)^{2-\gamma}
\biggl[ \xi_1 \biggl(\frac{a_n}{R_\mathrm{norm}} \biggr) \biggr]^{6-3\gamma}
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~~~\biggl( \frac{a_n}{R_\mathrm{norm}} \biggr)^{4-3\gamma}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>4\pi \biggl( \frac{\gamma-1}{\gamma} \biggr) \biggl( \frac{4\pi }{3} \cdot \frac{\rho_c}{\bar\rho} \cdot \xi_1^3\biggr)^{\gamma-2}
\, .
</math>
  </td>
</tr>
</table>
</div>