Editing SSCpt1/Virial (section)

===Expressions for Various Energy Terms===
We begin, here, by deriving an expression for each of the terms in the free-energy function as appropriate for spherically symmetric systems.  In deriving each expression, we keep in mind two issues:  First, for a given size system a determination of each term's total contribution to the free energy generally will involve integration through the entire volume of the configuration, effectively "summing up" the differential mass in each radial shell,
<div align="center">
<math>
dm = \rho(\vec{x}) d^3x = 4\pi \rho(r) r^2 dr \, ,
</math>
</div>
weighted by some specific energy expression.  Second, each term must be formulated in such a way that it is clear how the energy contribution depends on the overall system size.

====Volume Integrals====
We note, first, that the mass enclosed within each interior radius, <math>r</math>, is
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>M_r(r) = \int\limits_V dm</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\int_0^r  4\pi r^2 \rho dr  \, .</math>
  </td>
</tr>
</table>
</div>
Hence, if the volume of the configuration extends out to a radius denoted by <math>R_\mathrm{limit}</math>, the configuration mass is,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>M_\mathrm{limit}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\int_0^{R_\mathrm{limit}}  4\pi r^2 \rho dr  \, .</math>
  </td>
</tr>
</table>
</div>

<table align="center" border="1" width="80%" cellpadding="8">
<tr><td align="left">
NOTE:  The following considerations have led us to formally draw a distinction between <math>M_\mathrm{limit}</math> and the "total" mass, <math>M_\mathrm{tot}</math>, that we use (see below) for normalization.  

<font color="maroon"><b>Isolated Polytropes</b></font>:  For [[SSC/Virial/Polytropes#Isolated_Nonrotating_Adiabatic_Configuration|isolated polytropes]], the limit of integration, <math>R_\mathrm{limit}</math>, will be the natural edge of the configuration, where the pressure and mass-density drop to zero.  In this case, <math>M_\mathrm{limit}</math> quite naturally corresponds to the total mass of the configuration.  

<font color="maroon"><b>Pressure-Truncated Polytropes</b></font>:  But, a [[SSC/Virial/Polytropes#Nonrotating_Adiabatic_Configuration_Embedded_in_an_External_Medium|configuration embedded in an external medium]] of pressure, <math>P_e</math>, will have a (pressure-truncated) surface whose radius, <math>R_\mathrm{limit}</math>, corresponds to the radial location at which the configuration's internal pressure drops to a value that equals <math>P_e</math>.  In this case as well, one might choose to refer to <math>M_\mathrm{limit}</math> as the total mass; on the other hand, it might be more useful to distinguish <math>M_\mathrm{limit}</math> from <math>M_\mathrm{tot}</math>, continuing to rely on <math>M_\mathrm{tot}</math> to represent the mass of the corresponding ''isolated'' polytrope.  

<font color="maroon"><b>BiPolytropes</b></font>:  When discussing [[SSC/BipolytropeGeneralizationVersion2#Bipolytrope_Generalization|bipolytropes]], the limit of integration, <math>R_\mathrm{limit}</math>, will naturally refer to the radial location that defines the outer edge of the configuration's "core" and, at the same time, identifies the radial "interface" between the bipolytrope's core and its envelope.  In this case, <math>M_\mathrm{limit}</math> corresponds to the mass of the core rather than to the total mass of the bipolytropic configuration.
</td></tr>
</table>

<font color="red">Confinement by External Pressure:</font>  For spherically symmetric configurations, the energy term due to confinement by an external pressure can be expressed, simply, in terms of the configuration's radius, <math>R_\mathrm{limit}</math>, as,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>P_e V</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>P_e \int_0^{R_\mathrm{limit}}  4\pi r^2 dr  = \frac{4\pi}{3} P_e R_\mathrm{limit}^3 \, .</math>
  </td>
</tr>
</table>
</div>

<font color="red">Gravitational Potential Energy:</font>  From our discussion of the [[VE#Scalar_Virial_Theorem|scalar virial theorem]] &#8212; see, specifically, the reference to Equation (18), on p. 18 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] &#8212; the gravitational potential energy is given by the expression,
<div align="center">
<math>
W_\mathrm{grav} = - \int\limits_V \rho x_i \frac{\partial\Phi}{\partial x_i} d^3 x
= - \int\limits_V \vec{r} \cdot \nabla\Phi dm = - \int_0^{R_\mathrm{limit}} \biggl( r \frac{d\Phi}{dr} \biggr) dm \, .
</math>
</div>
For spherically symmetric systems, the

<div align="center">
<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

{{ Template:Math/EQ_Poisson01 }}
</div>
becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\frac{1}{r^2} \frac{d}{dr} \biggl( r^2 \frac{d\Phi}{dr} \biggr) </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>4\pi G \rho(r) \, , </math>
  </td>
</tr>
</table>
</div>
which implies,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>r^2 \frac{d\Phi}{dr} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\int_0^r 4\pi G \rho(r) r^2 dr = GM_r(r) \, .</math>
  </td>
</tr>
</table>
</div>
<span id="Wgrav">Hence</span> &#8212; see, also, p. 64, Equation (12) of [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>] &#8212; the desired expression for 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) dm = - \int_0^{R_\mathrm{limit}} \frac{G}{r}\biggl[\int_0^r 4\pi r^2 \rho dr \biggr] 4\pi r^2 \rho dr \, .</math>
  </td>
</tr>
</table>
</div>


<div id="AlternateGravPotEnergy">
<table border="1" align="center" width="80%" cellpadding="8">
<tr><td align="left">
Also, as pointed out by [<b>[[Appendix/References#C67|<font color="red">C67</font>]]</b>] &#8212; see p. 64, Equation (16) &#8212; it may sometimes prove advantageous to recognize that, if a spherically symmetric system is in hydrostatic balance, an alternate expression for the total gravitational potential energy is,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>W_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>+ \frac{1}{2} \int_0^{R_\mathrm{limit}} \Phi(r) dm \, .</math>
  </td>
</tr>
</table>
</div>
</td></tr>
</table>
</div>

<div>
<font color="red">Rotational Kinetic Energy:</font>  We will also consider a system that is rotating with a specified [[AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|''simple'' angular velocity profile]], <math>\dot\varphi(\varpi)</math>, in which case, from our discussion of the [[VE#Scalar_Virial_Theorem|scalar virial theorem]] &#8212; see, specifically, the reference to Equation (8), on p. 16 of [<b>[[Appendix/References#EFE|<font color="red">EFE</font>]]</b>] &#8212; the (ordered) kinetic energy,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>T_\mathrm{kin}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{2} \int\limits_V \rho |\vec{v} |^2 d^3x = \frac{1}{2} \int\limits_V |\vec{v} |^2 dm  \, ,</math>
  </td>
</tr>
</table>
</div>
is entirely rotational kinetic energy, specifically,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>T_\mathrm{kin} = T_\mathrm{rot}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{2} \int\int\int \dot\varphi^2 \varpi^2 dm
= \frac{1}{2} \int_0^{R_\mathrm{limit}} \dot\varphi^2 \varpi^2   \int_{-\sqrt{{R_\mathrm{limit}}^2 - \varpi^2}}^{\sqrt{{R_\mathrm{limit}}^2 - \varpi^2}}  \rho(r(\varpi,z)) 2\pi \varpi d\varpi dz\, .</math>
  </td>
</tr>
</table>
</div>

<font color="red">Reservoir of Thermodynamic Energy:</font>  As has been explained in [[VE#Reservoir_of_Thermodynamic_Energy|our introductory discussion of the Gibbs-like free energy]], formulation of an expression for the reservoir of thermodynamic energy, <math>\mathfrak{S}_\mathrm{therm}</math>, depends on whether the system is expected to evolve adiabatically or isothermally.  For [[VE#Isothermal_Systems|isothermal systems]],
<div align="center" id="Reservoir">
<math>
\mathfrak{S}_\mathrm{therm} ~~\rightarrow ~~\mathfrak{S}_I 
= + \int\limits_V c_s^2  \ln \biggl(\frac{\rho}{\rho_\mathrm{norm}}\biggr) dm
= c_s^2  \int_0^{R_\mathrm{limit}} \ln \biggl(\frac{\rho}{\rho_\mathrm{norm}}\biggr) 4\pi r^2 \rho dr \, ,
</math>
</div>
where, <math>c_s</math> is the isothermal sound speed and <math>\rho_\mathrm{norm}</math> is a (as yet unspecified) reference mass density; while, for [[VE#Adiabatic_Systems|adiabatic systems]],
<div align="center">
<math>
\mathfrak{S}_\mathrm{therm} ~~\rightarrow ~~ \mathfrak{S}_A 
= + \int\limits_V  \frac{1}{({\gamma_g}-1)} \biggl( \frac{P}{\rho} \biggr) dm
= \frac{1}{({\gamma_g}-1)}  \int_0^{R_\mathrm{limit}}  4\pi r^2 P dr 
 \, ,</math>
</div>
where, <math>P(r)</math> is the system's pressure distribution and <math>\gamma_g</math> is the specified adiabatic index.

====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>

====Implementation====
=====Normalize=====
We will now judiciously introduce our adopted normalizations into the [[#Expressions_for_Various_Energy_Terms|above-defined free-energy term expressions]], using asterisks to denote dimensionless variables that have been accordingly normalized; for example, 
<div align="center">
<math>
r^* \equiv \frac{r}{R_\mathrm{norm}} \, , ~~~~~~ P^* \equiv \frac{P}{P_\mathrm{norm}} \, , ~~~~~~ </math>
&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; <math>\rho^* \equiv \frac{\rho}{\rho_\mathrm{norm}} \, .
</math>
</div>

<font color="red">Normalized Mass:</font>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>M_r(r^*)  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
R_\mathrm{norm}^3 \rho_\mathrm{norm} \int_0^{r^*}  4\pi (r^*)^2 \rho^* dr^* 
= M_\mathrm{tot} \int_0^{r^*}  3(r^*)^2 \rho^* dr^*  \, .
</math>
  </td>
</tr>
</table>
</div>

<font color="red">Confinement by External Pressure (Normalized Volume):</font>  
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>P_e V</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>E_\mathrm{norm} \biggl[ \frac{4\pi}{3} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) 
\biggl(\frac{R_\mathrm{limit}}{R_\mathrm{norm}}\biggr)^3 \biggr] \, .</math>
  </td>
</tr>
</table>
</div>

<font color="red">Normalized Gravitational Potential Energy:</font>  
<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^{\chi=R_\mathrm{limit}^*} \biggl[\frac{M_r(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^{\chi = R_\mathrm{limit}^*} 3\biggl[\frac{M_r(r^*)}{M_\mathrm{tot}} \biggr]  r^* \rho^* dr^* \, .
</math>
  </td>
</tr>
</table>
</div>

<font color="red">Normalized Reservoir of Thermodynamic Energy:</font>  
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\mathfrak{S}_I</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>E_\mathrm{norm}  \int_0^{\chi=R_\mathrm{limit}^*} 3 \ln (\rho^*) (r^*)^2 \rho^* dr^* \, ,</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\mathfrak{S}_A</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{E_\mathrm{norm}}{({\gamma_g}-1)}  \int_0^{\chi=R_\mathrm{limit}^*}  4\pi (r^*)^2 P^* dr^* \, .</math>
  </td>
</tr>
</table>
</div>

<font color="red">Normalized Rotational Kinetic Energy:</font>  
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>T_\mathrm{rot}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\pi \dot\varphi_c^2 R_\mathrm{norm}^5 \rho_\mathrm{norm} 
\int_0^{\chi=R_\mathrm{limit}^*} \biggl[ \frac{\dot\varphi^2}{\dot\varphi_c^2} \biggr] (\varpi^*)^3  d\varpi^*
\int_{-\sqrt{\chi^2 - (\varpi^*)^2}}^{\sqrt{\chi^2 - (\varpi^*)^2}}  (\rho^*)  dz^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{5^2\pi}{2^2} \biggr) \biggl[ \frac{J^2 R_\mathrm{norm} \rho_\mathrm{norm}}{M_\mathrm{tot}^2} \biggr] \chi_\mathrm{eq}^{-4}
\int_0^{\chi=R_\mathrm{limit}^*} \biggl[ \frac{\dot\varphi^2}{\dot\varphi_c^2} \biggr] (\varpi^*)^3  d\varpi^*
\int_{-\sqrt{\chi^2 - (\varpi^*)^2}}^{\sqrt{\chi^2 - (\varpi^*)^2}}  (\rho^*)  dz^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{3\cdot 5^2}{2^4} \biggr) \biggl[  \frac{J^2}{M_\mathrm{tot}} \biggl(\frac{E_\mathrm{norm} }{G M_\mathrm{tot}^2 }\biggr)^2 \biggr] \chi_\mathrm{eq}^{-4}
\int_0^{\chi=R_\mathrm{limit}^*} \biggl[ \frac{\dot\varphi^2}{\dot\varphi_c^2} \biggr] (\varpi^*)^3  d\varpi^*
\int_{-\sqrt{\chi^2 - (\varpi^*)^2}}^{\sqrt{\chi^2 - (\varpi^*)^2}}  (\rho^*)  dz^*
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>E_\mathrm{norm}
\biggl( \frac{3^2\cdot 5^2}{2^6 \pi} \biggr) \biggl[  \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4}  \biggr] \chi_\mathrm{eq}^{-4}
\int_0^{\chi=R_\mathrm{limit}^*} \biggl[ \frac{\dot\varphi^2}{\dot\varphi_c^2} \biggr] (\varpi^*)^3  d\varpi^*
\int_{-\sqrt{\chi^2 - (\varpi^*)^2}}^{\sqrt{\chi^2 - (\varpi^*)^2}}  (\rho^*)  dz^* \, ,
</math>
  </td>
</tr>
</table>
</div>
where, 
<div align="center">
<math>\dot\varphi_c \equiv \frac{5J}{2M_\mathrm{tot} R_\mathrm{eq}^2} = 
\frac{5}{2} \biggl[ \frac{J}{M_\mathrm{tot} R_\mathrm{norm}^2} \biggr] \chi_\mathrm{eq}^{-2} \, ,</math>
</div>
is a characteristic rotation frequency in the equilibrium configuration whose value is set once the system's total angular momentum, <math>J</math>, is specified.

=====Separate Time &amp; Space=====
Our intent is to vary the size of the configuration <math>(R_\mathrm{limit})</math> while holding the (properly normalized) internal structural profile fixed, so let's separate the spatial integral over the (fixed) structural profile from the time-varying configuration size.  Making use of the dimensionless ''internal'' coordinates,
<div align="center">
<math>x \equiv \frac{r}{R_\mathrm{limit}} \, ,~~~~w \equiv \frac{\varpi}{R_\mathrm{limit}} \, ,
~~~~\zeta \equiv \frac{z}{R_\mathrm{limit}} \, ,
</math>
</div>
that always run from zero to one, we have,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>r^*</math>
  </td>
  <td align="center">
<math>\rightarrow~</math>
  </td>
  <td align="left">
<math>
x \biggl( \frac{R_\mathrm{limit}}{R_\mathrm{norm}} \biggr) = x \chi \, ;
</math>
&nbsp;&nbsp;&nbsp;and, likewise, &nbsp;&nbsp;&nbsp;
<math>
~~~~\varpi^* ~\rightarrow~ w \chi \, ;
~~~~z^* ~\rightarrow~ \zeta \chi \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\rho^*</math>
  </td>
  <td align="center">
<math>\rightarrow~</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\rho(x)}{\bar\rho} \biggr] \biggl( \frac{\bar\rho}{\rho_\mathrm{norm}} \biggr)
= \biggl[ \frac{\rho(x)}{\bar\rho} \biggr] \biggl( \frac{M_\mathrm{limit}/R_\mathrm{limit}^3}{M_\mathrm{tot}/R_\mathrm{norm}^3} \biggr)
= \biggl[ \frac{\rho(x)}{\bar\rho} \biggr] \biggl(  \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \chi^{-3}  
= \frac{\rho_c}{\bar\rho} \biggl[ \frac{\rho(x)}{\rho_c} \biggr] \biggl(  \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \chi^{-3} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>P^*</math>
  </td>
  <td align="center">
<math>\rightarrow~</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{P(x)}{P_c} \biggr] \biggl( \frac{P_c}{P_\mathrm{norm}} \biggr)
= \biggl[ \frac{P(x)}{P_c} \biggr] \biggl( \frac{K\rho_c^\gamma}{P_\mathrm{norm}} \biggr)
= \biggl[ \frac{P(x)}{P_c} \biggr] \biggl( \frac{\rho_c}{\bar\rho} \biggr)^\gamma 
\biggl[ \frac{(3M_\mathrm{limit}/4\pi R_\mathrm{limit}^3)^\gamma}{K^{-1}P_\mathrm{norm}} \biggr]
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;&nbsp;&nbsp;&nbsp;
  </td>
  <td align="left">
<math>
= \biggl[ \frac{P(x)}{P_c} \biggr] \biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]^\gamma 
\biggl[ \frac{K M_\mathrm{tot}^\gamma}{P_\mathrm{norm} R_\mathrm{norm}^{3\gamma}} \biggr] \biggl(  \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^\gamma
\biggl( \frac{R_\mathrm{limit}}{R_\mathrm{norm}}  \biggr)^{-3\gamma}
= \biggl[ \frac{P(x)}{P_c} \biggr] \biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]^\gamma \biggl(  \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^\gamma
\chi^{-3\gamma} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{\dot\varphi}{\dot\varphi_c}</math>
  </td>
  <td align="center">
<math>\rightarrow~</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\dot\varphi(w)}{\dot\varphi_\mathrm{limit}} \biggr] \biggl( \frac{\dot\varphi_\mathrm{limit}}{\dot\varphi_c}\biggr)
= \biggl[ \frac{\dot\varphi(w)}{\dot\varphi_\mathrm{limit}} \biggr] \biggl( \frac{R_\mathrm{limit}}{R_\mathrm{eq}}\biggr)^{-2}
= \biggl[ \frac{\dot\varphi(w)}{\dot\varphi_\mathrm{limit}} \biggr] \chi_\mathrm{eq}^{2} \chi^{-2} \, .
</math>
  </td>
</tr>
</table>
</div>

=====Summary of Normalized Expressions=====
Hence, our normalized expressions become,
<div align="center">
<table border="1" cellpadding="8">
<tr><th align="center">
Normalized Expressions
</th></tr>
<tr><td align="center">

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

<tr>
  <td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot}}  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{x}  3x^2 \biggl[ \frac{\rho(x)}{\rho_c} \biggr]  dx \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{P_e V}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{4\pi}{3} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr)  \chi^3 \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>
- \chi^{-1} \biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
\int_0^{1} 3x \biggl[\frac{M_r(x)}{M_\mathrm{tot}} \biggr]  \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
 <td align="left">
<math>
- \frac{3}{5} \chi^{-1} \biggl( \frac{\rho_c}{\bar\rho} \biggr)^2_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2
\int_0^{1} 5x \biggl\{\int_0^{x}  3x^2 \biggl[ \frac{\rho(x)}{\rho_c} \biggr]  dx\biggr\}  \biggl[ \frac{\rho(x)}{\rho_c} \biggr] dx \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{4\pi}{3({\gamma_g}-1)} \cdot \chi^{3-3\gamma}
\biggl\{ \biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]_\mathrm{eq}^{\gamma} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^\gamma
\int_0^{1}  3x^2 \biggl[ \frac{P(x)}{P_c} \biggr]  dx \biggr\} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{\mathfrak{S}_I}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\int_0^{1} 
\biggl\{ \ln \biggl[ \frac{\rho(x)}{\bar\rho} \biggr] -3\ln \biggl[ \frac{R_\mathrm{edge}}{R_\mathrm{norm}} \biggr]  \biggr\} 
3 x^2 \biggl[ \frac{\rho(x)}{\bar\rho} \biggr] dx </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-3 \ln \chi  + \mathrm{constant} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{T_\mathrm{rot}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\chi^{-2}
\biggl( \frac{3^2\cdot 5^2}{2^6 \pi} \biggr) \biggl[  \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr] \biggl( \frac{\rho_c}{\bar\rho}  \biggr)_\mathrm{eq} 
\int_0^{1} \biggl[ \frac{\dot\varphi(w)}{\dot\varphi_\mathrm{edge}} \biggr]^2 w^3  dw
\int_{-\sqrt{1 - w^2}}^{\sqrt{1 - w^2}}  \biggl[ \frac{\rho(w,\zeta)}{\rho_c} \biggr]   d\zeta \, .
</math>
  </td>
</tr>
</table>

</td></tr>
<tr><td align="left>
<font color="red">NOTE to self (21 September 2014)<b></b></font>:  The expressions for <math>\mathfrak{S}_I</math> and <math>T_\mathrm{rot}</math> may not properly account for the ratio, <math>M_\mathrm{limit}/M_\mathrm{tot}</math>.
</td></tr>
</table>
</div>


It should be emphasized that the coefficient involving the density ratio, <math>(\rho_c/\bar\rho)</math>, that lies outside of the integral in most of these expressions depends only on the internal structure, and not the overall size, of the configuration.  It can therefore be evaluated at any time.  We usually will choose to evaluate this coefficient in an equilibrium state, that is, when <math>R_\mathrm{limit} \rightarrow R_\mathrm{eq}</math>.  Accordingly, the subscript "eq" has been attached to this coefficient.  The inverse of this density ratio can be obtained from the integral expression for <math>M_r</math> by recognizing that <math>M_r \rightarrow M_\mathrm{limit}</math> when the upper limit on the integral <math>x \rightarrow 1</math>.  Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\biggl(\frac{\rho_c}{\bar\rho} \biggr)^{-1}_\mathrm{eq}  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\int_0^{1}  3x^2 \biggl[ \frac{\rho(x)}{\rho_c} \biggr]_\mathrm{eq}  dx \, .</math>
  </td>
</tr>
</table>
</div>
This coefficient also may be rewritten in terms of the central pressure in the equilibrium state; specifically, using a sequence of steps similar to the ones that were used, above, in rewriting <math>P^*</math>, we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>
\biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]_\mathrm{eq}^{\gamma} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^\gamma
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl[ \biggl( \frac{P_c}{P_\mathrm{norm}} \biggr) \chi^{3\gamma} \biggr]_\mathrm{eq} \, .</math>
  </td>
</tr>
</table>
</div>

=====Looking Ahead to Bipolytropes=====
<div id="BiPolytrope">
<table border="1" align="center" width="90%" cellpadding="20">
<tr><td align="left">
<b><font color="purple">ASIDE:</font></b> When we discuss the free energy of bipolytropic configurations, we will need to divide the expression for <math>\mathfrak{S}_A/E_\mathrm{norm}</math> into two parts &#8212; one accounting for the reservoir of thermodynamic energy in the bipolytrope's "core" and one accounting for the reservoir of thermodynamic energy in the bipolytrope's "envelope."  It is useful to develop this two-part expression here, while the definition of <math>\mathfrak{S}_A</math> is fresh in our minds and to show how the two-part expression reduces to the simpler expression for <math>\mathfrak{S}_A/E_\mathrm{norm}</math>, just derived, when there is no distinction drawn between the properties of the core and the envelope. 

In what follows, we will use the subscript ''core'' (or "c") when referencing physical properties of the bipolytrope's core and the subscript ''env'' (or "e") for the envelope; and, as above, we will use <math>x \equiv r/R_\mathrm{edge}</math> to denote the dimensionless radial location within a configuration of radius, <math>R_\mathrm{edge}</math>.  The dimensionless radial coordinate, <math>q \equiv x_i = r_i/R_\mathrm{edge}</math>, will identify the radial ''interface'' where the core meets the envelope; that is, <math>q</math> will identify both the outer edge of the core and the inner edge of the envelope.  In general, separate expressions will define the run of pressure through the core and through the envelope.  We can assume that, for the core, the pressure drops monotonically from a value of <math>P_0</math> at the center of the configuration according to an expression of the form,
<div align="center">
<math>P_\mathrm{core}(x) = P_0 [1 - p_c(x)]</math> &nbsp; &nbsp;&nbsp; for &nbsp; &nbsp;&nbsp; <math>0 \leq x \leq q \, ,</math>
</div>
and that, for the envelope, the pressure drops monotonically from a value of <math>P_{ie}</math> at the interface according to an expression of the form,
<div align="center">
<math>P_\mathrm{env}(x) = P_{ie} [1 - p_e(x)]</math> &nbsp; &nbsp;&nbsp; for &nbsp; &nbsp;&nbsp; <math>q \leq x \leq 1 \, ,</math>
</div>
where <math>p_c(x)</math> and <math>p_e(x)</math> are both dimensionless functions that will depend on the equations of state that are chosen for the core and envelope, respectively.  By prescription, the pressure in the envelope must drop to zero at the surface of the bipolytropic configuration, hence, we should expect that <math>p_e(1) = 1</math>.  Furthermore, by prescription, the pressure in the core will drop to a value, <math>P_{ic}</math>, at the interface, so we can write,
<div align="center">
<math>P_{ic} = P_0 [1 - p_c(q)] \, .</math>
</div>

In equilibrium &#8212; that is, when <math>R_\mathrm{edge} = R_\mathrm{eq}</math> &#8212; we will demand that the pressure at the interface be the same, whether it is referenced in the core or in the envelope, that is, we will demand that <math>P_{ic} = P_{ie} \, .</math>  It will therefore prove to be strategically advantageous to rewrite the expression for the run of pressure through the core in terms of the pressure at the interface rather than in terms of the central pressure; specifically,
<div align="center">
<math>P_\mathrm{core}(x) = P_{ic} \biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr] \, .</math> 
</div>
Referencing these prescriptions for <math>P_\mathrm{core}(x)</math> and <math>P_\mathrm{env}(x)</math>, the two-part expression for the reservoir of thermodynamic energy is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{({\gamma_c}-1)}  \int_0^{r_i/R_\mathrm{norm}}  4\pi (r^*)^2 P^*_\mathrm{core} dr^*
+ \frac{1}{({\gamma_e}-1)}  \int_{r_i/R_\mathrm{norm}}^\chi  4\pi (r^*)^2 P^*_\mathrm{env} dr^* 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi \chi^3 }{({\gamma_c}-1)} \biggl[ \frac{P_{ic}}{P_\mathrm{norm}} \biggr] \int_0^q  \biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr]  x^2 dx
+ \frac{4\pi \chi^3 }{({\gamma_e}-1)} \biggl[ \frac{P_{ie}}{P_\mathrm{norm}} \biggr]  \int_q^1  \biggl[1 - p_e(x) \biggr]  x^2 dx \, .
</math>
  </td>
</tr>
</table>
</div>
As is implied by the subscripts on the adiabatic exponents that appear in the leading factor of each of the two terms, we are assuming that, as the bipolytropic system expands or contracts, the thermodynamic properties of the material in the envelope will vary as prescribed by an adiabat of index, <math>\gamma_e</math>, while the thermodynamic properties of material in the core will vary as prescribed by a, generally different, adiabat of index, <math>\gamma_c</math>.  Therefore, as the radius of the bipolytropic configuration, <math>R_\mathrm{edge}</math>, is varied, the density of each fluid element will vary and, in the core, the pressure of each fluid element will vary as <math>P \propto \rho^{\gamma_c}</math> while, in the envelope, the pressure of each fluid element will vary as <math>P \propto \rho^{\gamma_e}</math>.  If we furthermore assume that the mass in the core and the mass in the envelope remain constant during a phase of contraction or expansion, the density of each fluid element will vary as <math>R_\mathrm{edge}^{-3}</math>, whether the material is associated with the core or with the envelope.  Therefore, using the subscript, "eq," to identify the value of thermodynamic quantities when the system is in an equilibrium state and, accordingly, <math>R_\mathrm{edge} = R_\mathrm{eq}</math>, we can write,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\biggl[ \frac{P}{P_\mathrm{eq}} \biggr]_\mathrm{core}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\rho}{\rho_\mathrm{eq}} \biggr)^{\gamma_c} = \biggl( \frac{R_\mathrm{edge}}{R_\mathrm{eq}} \biggr)^{-3\gamma_c} \, ,</math>
  </td>
</tr>
</table>
</div>
and,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\biggl[ \frac{P}{P_\mathrm{eq}} \biggr]_\mathrm{env}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\biggl( \frac{\rho}{\rho_\mathrm{eq}} \biggr)^{\gamma_e} = \biggl( \frac{R_\mathrm{edge}}{R_\mathrm{eq}} \biggr)^{-3\gamma_e} \, .</math>
  </td>
</tr>
</table>
</div>
In particular, for any <math>R_\mathrm{edge}</math>, material associated with the core that lies at the interface will have a pressure given by the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>P_{ic}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(P_{ic})_\mathrm{eq} \biggl( \frac{R_\mathrm{edge}}{R_\mathrm{eq}} \biggr)^{-3\gamma_c} 
= (P_{ic})_\mathrm{eq} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{+3\gamma_c}\biggl( \frac{R_\mathrm{edge}}{R_\mathrm{norm}} \biggr)^{-3\gamma_c} 
= (P_{ic})_\mathrm{eq} \chi_\mathrm{eq}^{+3\gamma_c} \chi^{-3\gamma_c} 
\, ,</math>
  </td>
</tr>
</table>
</div>
while material associated with the envelope that lies at the interface will have a pressure given by the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>P_{ie}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
(P_{ie})_\mathrm{eq} \biggl( \frac{R_\mathrm{edge}}{R_\mathrm{eq}} \biggr)^{-3\gamma_e} 
= (P_{ie})_\mathrm{eq} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{+3\gamma_e}\biggl( \frac{R_\mathrm{edge}}{R_\mathrm{norm}} \biggr)^{-3\gamma_e} 
= (P_{ie})_\mathrm{eq} \chi_\mathrm{eq}^{+3\gamma_e} \chi^{-3\gamma_e} 
\, .</math>
  </td>
</tr>
</table>
</div>
Hence,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi }{({\gamma_c}-1)}  \biggl[ \frac{P_{ic} \chi^{3\gamma_c}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi^{3-3\gamma_c} 
\int_0^q  \biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr]  x^2 dx
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>
+ ~\frac{4\pi }{({\gamma_e}-1)}  \biggl[ \frac{P_{ie} \chi^{3\gamma_e}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi^{3-3\gamma_e} 
\int_q^1  \biggl[1 - p_e(x) \biggr]  x^2 dx \, .
</math>
  </td>
</tr>
</table>
</div>
----


Now, let's see how this expression simplifies if <math>P_{ie} = P_{ic}</math> and <math>\gamma_e = \gamma_c</math> and, hence, the properties of the envelope are indistinguishable from the properties of the core.  We note, first, that in this limit, <math>P_\mathrm{core}(x)</math> and <math>P_\mathrm{env}(x)</math> must be identical functions of <math>x</math>, that is, it must be the case that <math>p_e(x)</math> is related to <math>p_c(x)</math> via the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>1 - p_e(x) </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1 - p_c(x)}{1-p_c(q)} \, .</math>
  </td>
</tr>
</table>
</div>

We therefore obtain,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi }{({\gamma_c}-1)}  \biggl[ \frac{P_{ic} \chi^{3\gamma_c}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi^{3-3\gamma_c} 
\biggl\{ \int_0^q  \biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr]  x^2 dx + \int_q^1  \biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr]  x^2 dx \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi }{({\gamma_c}-1)}  \biggl[ \frac{P_0 \chi^{3\gamma_c}}{P_\mathrm{norm}} \biggr]_\mathrm{eq} \chi^{3-3\gamma_c} 
\biggl\{ \int_0^1  \biggl[1 - p_c(x)\biggr]  x^2 dx  \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi }{({\gamma_g}-1)} \cdot \chi^{3-3\gamma}
\biggl\{ \biggl[ \biggl(\frac{3}{4\pi} \biggr) \frac{\rho_c}{\bar\rho} \biggr]_\mathrm{eq}^{\gamma}
\int_0^{1}  \biggl[ \frac{P(x)}{P_c} \biggr] x^2 dx \biggr\} \, ,
</math>
  </td>
</tr>
</table>
</div>
as desired.

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

====Idealized Configuration====
(For simplicity throughout this subsection, we will assume that the mass enclosed within the configuration's limiting radius, <math>M_\mathrm{limit}</math>, equals the normalization mass, <math>M_\mathrm{tot}</math>.)  In the idealized situation of a configuration that has uniform density, <math>\rho(x) = \rho_c</math> &#8212; and, hence, the density ratio <math>\rho_c/\bar\rho = 1</math> &#8212; the mass interior to each radius is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{M_r(x)}{M_\mathrm{tot} }  </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\int_0^{x}  3x^2  dx = x^3 \, ,</math>
  </td>
</tr>
</table>
</div>
and the normalized gravitational potential energy is,
<div align="center">
<table border="0" cellpadding="8" 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>
- \frac{3}{5} \chi^{-1} \int_0^{1} 5x \biggl\{ x^3\biggr\}  dx = -\frac{3}{5} \chi^{-1} \, .
</math>
  </td>
</tr>
</table>
</div>

If, in addition, the configuration is uniformly rotating with angular velocity, <math>\dot\varphi = \dot\varphi_\mathrm{edge}</math>, and has uniform pressure, <math>P_c</math>, evaluation of the ordered kinetic energy and thermodynamic energy integrals yields,
<div align="center">
<table border="0" cellpadding="8" align="center">

<tr>
  <td align="right">
<math>\frac{T_\mathrm{rot}}{E_\mathrm{norm} }</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>2\chi^{-2}
\biggl( \frac{3^2\cdot 5^2}{2^6\pi} \biggr) \biggl[  \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr] 
\int_0^{1}  w^3  dw
\int_{0}^{\sqrt{1 - w^2}}   d\zeta 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\chi^{-2}
\biggl( \frac{3^2\cdot 5^2}{2^5\pi} \biggr) \biggl[  \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr] 
\int_0^1 w^3 (1-w^2)^{1/2} dw  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\chi^{-2}
\biggl( \frac{3^2\cdot 5^2}{2^5\pi} \biggr)\biggl[  \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr]  
\biggl[ -\frac{1}{15} (1-w^2)^{3/2} (3w^2 +2) \biggr]_0^1  
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\chi^{-2}
\biggl( \frac{3\cdot 5}{2^4 \pi} \biggr) \biggl[  \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr]  
\, , 
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{4\pi }{3({\gamma_g}-1)} \cdot \chi^{3-3\gamma}
\biggl\{ \biggl(\frac{3}{4\pi} \biggr)^{\gamma}\int_0^{1}  3x^2  dx \biggr\} 
= \frac{1}{({\gamma_g}-1)} \biggl(\frac{3}{4\pi} \biggr)^{\gamma-1} \chi^{3-3\gamma}
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{\mathfrak{S}_I}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>-3 \ln \chi  + \mathrm{constant} \, ,
</math>
  </td>
</tr>
</table>
</div>
where the various dimensionless integration variables are, <math>x \equiv (r/R)</math>, <math>\zeta \equiv (z/R)</math>, and <math>w \equiv (\varpi/R)</math>.

====Structural Form Factors====
Keeping in mind the expressions that arise in the case of our just-defined, idealized configuration, in more realistic cases we generally will write each energy term as follows:
<div align="center">
<table border="0" cellpadding="8" 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>
- \frac{3}{5} \chi^{-1} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}^2_M} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{T_\mathrm{rot}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math> 
\biggl( \frac{3\cdot 5}{2^4 \pi} \biggr)\chi^{-2} \biggl[  \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr] \biggl( \frac{\rho_c}{\bar\rho}  \biggr)_\mathrm{eq} \cdot \frac{\mathfrak{f}_T}{\mathfrak{f}_M} \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{4\pi}{3({\gamma_g}-1)} \cdot \chi^{3-3\gamma}
\biggl[ \frac{3}{4\pi} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \biggr]_\mathrm{eq}^{\gamma}
\cdot \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{\gamma}}  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{4\pi}{3({\gamma_g}-1)} \cdot \chi^{3-3\gamma}
\biggl[ \biggl( \frac{P_c}{P_\mathrm{norm}} \biggr)\chi^{3\gamma} \biggr]_\mathrm{eq} \cdot \mathfrak{f}_A \, ,</math>
  </td>
</tr>
</table>
</div>

<span id="FormFactors">where the dimensionless form factors, <math>\mathfrak{f}_i</math>, which are assumed to be independent of the overall configuration size and will each usually of order unity, are</span>,

<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>\mathfrak{f}_M </math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\int_0^1  3\biggl[ \frac{\rho(x)}{\rho_c}\biggr] x^2 dx = \biggl( \frac{\bar\rho}{\rho_c} \biggr)_\mathrm{eq} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathfrak{f}_W</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>3\cdot 5 \int_0^1 \biggl\{ \int_0^x  \biggl[ \frac{\rho(x)}{\rho_c}\biggr] x^2 dx \biggr\}  \biggl[ \frac{\rho(x)}{\rho_c}\biggr] x dx\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathfrak{f}_T</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{15}{2} \int_0^1 \biggl[ \frac{\dot\varphi(w)}{\dot\varphi_\mathrm{edge}} \biggr]^2 w^3 dw   \int_0^{\sqrt{1 - w^2}}  
\biggl[ \frac{\rho(w,\zeta)}{\rho_c} \biggr] d\zeta\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\mathfrak{f}_A</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\int_0^1 3\biggl[ \frac{P(x)}{P_c}\biggr]  x^2 dx \, .</math>
  </td>
</tr>

</table>
</div>
In each case, the "idealized" energy expression is retrieved if/when the relevant form factor, <math>\mathfrak{f}_i</math>, is set to unity.

====Some Detailed Examples====

In an [[SSCpt1/Virial/FormFactors#Structural_Form_Factors|accompanying discussion]], we derive detailed expressions for a selected subset of the above structural form factors and corresponding energy terms in the case of spherically symmetric configurations that obey an {{ Template:Math/MP_PolytropicIndex }} = 5 or an {{ Template:Math/MP_PolytropicIndex }} = 1 polytropic equation of state.  The hope is that this will illustrate, in a clear and helpful manner, how the task of calculating form factors is to be carried out, in practice; and, in particular, to provide one nontrivial example for which analytic expressions are derivable.  This should help with the task of debugging numerical algorithms that are designed to calculate structural form factors for more general cases, which cannot be derived analytically.  The limits of integration will be specified in a general enough fashion that the resulting expressions can be applied, not only to the structures of ''isolated'' polytropes, but to [[SSC/Virial/PolytropesSummary#Further_Evaluation_of_n_.3D_5_Polytropic_Structures|''pressure-truncated'' polytropes]] that are embedded in a hot, tenuous external medium and to the [[SSC/Structure/BiPolytropes/Analytic51#Free_Energy|cores of bipolytropes]].