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=Virial Equilibrium of Spherically Symmetric Configurations=
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<font size="-1">[[H_BookTiledMenu#Spherically_Symmetric_Configurations|<b>Free-Energy<br />of<br />Spherical<br />Systems</b>]]</font>
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==Free Energy Expression==
===Review===
As has been [[VE#Free_Energy_Expression|introduced elsewhere in a more general context]], associated with any isolated, self-gravitating, gaseous configuration we can identify a total Gibbs-like free energy, <math>\mathfrak{G}</math>, given by the sum of the relevant contributions to the total energy of the configuration,
<div align="center">
<math>
\mathfrak{G} = W_\mathrm{grav} + \mathfrak{S}_\mathrm{therm} + T_\mathrm{kin} + P_e V + \cdots
</math>
</div> 
Here, we have explicitly included the gravitational potential energy, <math>W_\mathrm{grav}</math>, the ordered kinetic energy, <math>T_\mathrm{kin}</math>, a term that accounts for surface effects if the configuration of volume <math>V</math> is embedded in an external medium of pressure <math>P_e,</math> and <math>\mathfrak{S}_\mathrm{therm}</math>, the reservoir of thermodynamic energy that is available to perform work as the system expands or contracts.  A mathematical expression encapsulating the physical definition of each of these energy terms, in full three-dimensional generality, [[VE#Free_Energy_Expression|can be found in our introductory discussion]] of the scalar virial theorem and the free-energy function.

===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]].

===Gathering it All Together===
Gathering all of the terms together we find that, to within an additive constant, the expression for the normalized free energy is, 
<div align="center">
<math>
\mathfrak{G}^* \equiv \frac{\mathfrak{G}}{E_\mathrm{norm}} = 
-3A\chi^{-1} -~ \frac{(1-\delta_{1\gamma_g})}{(1-\gamma_g)} B \chi^{3-3\gamma_g} -~ \delta_{1\gamma_g} 3\ln \chi 
+~ C \chi^{-2} +~ D\chi^3 \, ,
</math>
</div>
where, 
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>A</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{1}{5} \cdot \biggl[ \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr) \frac{1}{\mathfrak{f}_M} \biggr]^2 \cdot \mathfrak{f}_W \, ,</math>
  </td>
</tr>

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

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi}{3} 
\biggl[ \biggl( \frac{P_c}{P_\mathrm{norm}} \biggr)\chi^{3\gamma} \biggr]_\mathrm{eq} 
\cdot \mathfrak{f}_A \, ,
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>C</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{3\cdot 5}{2^4 \pi} \biggl[ \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr] \cdot \frac{\mathfrak{f}_T}{\mathfrak{f}_M} \, ,
</math>
  </td>
</tr>

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

Once the pressure exerted by the external medium (<math>P_e</math>), and the configuration's mass (<math>M_\mathrm{tot}</math>), angular momentum (<math>J</math>), and specific entropy (via {{ Template:Math/MP_PolytropicConstant }}) &#8212; or, in the isothermal case, sound speed (<math>c_s</math>) &#8212;  have been specified, the values of all of the coefficients are known and the above algebraic expression for <math>\mathfrak{G}^*</math> describes how the free energy of the configuration will vary with the configuration's size (<math>\chi</math>) for a given choice of <math>\gamma_g</math>.

==Visual Representation==
<div align="center">
<table border="2" cellpadding="8">
<tr>
  <td align="center" colspan="2">
'''Figure 1:''' <font color="darkblue">Free Energy Surface </font> 
  </td>
</tr>
<tr>
  <td valign="top" width=450>
This segment of the free energy "surface" shows how the free energy varies as the size of the configuration and the applied external pressure are varied, while all other relevant physical attributes are held fixed.  

The plotted function &#8212; derived from the above expression for <math>\mathfrak{G}^*</math>, with <math>\gamma_\mathrm{g} = 1</math> and <math>C=0</math> (see [[SSCpt1/Virial#Examples|further discussion]], below) &#8212; is, specifically,
<div align="center">
<font size="-1">
<math>
\mathfrak{G}^* = 3000\biggl[ - \frac{1}{\chi} - \ln\chi + \frac{\Pi}{3}\chi^3 + 0.9558 \biggr] \, .
</math>
</font>
</div>
As shown, the size of the configuration <math>(\chi)</math> increases to the right from <math>1.2</math> to <math>1.51</math>; the dimensionless external pressure <math>(\Pi)</math> increases into the screen from <math>0.103</math> to <math>0.104</math>; and the dimensionless free energy, <math>\mathfrak{G}^*</math>, increases upward.
 </td>
  <td align="center" bgcolor="black">
[[File:3DFreeEnergy.jpg|350px|center|Free Energy Surface]]
  </td>
</tr>
</table>
</div>

==Energy Extrema==
As is illustrated in [[#Visual_Representation|Figure 1]], the free energy surface generally will exhibit multiple local minima and local maxima, and may also possess one or more points of inflection. The locations along the energy surface where these special points arise identify equilibrium states, and the associated values of <math>(R/R_0)_\mathrm{eq}</math> give the radii of the equilibrium configurations.  

For a given choice of the set of physical parameters <math>M</math>, {{ Template:Math/MP_PolytropicConstant }}, <math>J</math>, <math>P_e</math>, and <math>\gamma_g</math>, extrema occur wherever,
<div align="center">
<math>
\frac{d\mathfrak{G^*}}{d\chi} = 0 \, .
</math>
</div>
For the free energy function identified above, 
<div align="center">
<math>
\frac{d\mathfrak{G^*}}{d\chi}  = 
3A\chi^{-2} -~ (1-\delta_{1\gamma_g})~3 B\chi^{2 -3\gamma_g} -~ \delta_{1\gamma_g} 3\chi^{-1} ~ -2C \chi^{-3} +~ 3D\chi^2  \, ,
</math>
</div>
<span id="GeneralVirial">so <math>\chi_\mathrm{eq} \equiv R_\mathrm{eq}/R_\mathrm{norm}</math> is obtained from the real root(s) of the equation,</span>
<div align="center">
<math>
2C \chi_\mathrm{eq}^{-2}  + ~ (1-\delta_{1\gamma_g})~3 B\chi_\mathrm{eq}^{3 -3\gamma_g} +~ \delta_{1\gamma_g} 3 ~
-~3A\chi_\mathrm{eq}^{-1}  -~ 3D\chi_\mathrm{eq}^3 = 0 \, .
</math>
</div>

<div id="Tohline85">
<table border="1" align="center" width="90%" cellpadding="20">
<tr><td align="left">
<b><font color="purple">ASIDE:</font></b> When we discuss the equilibrium of isothermal, rotating configurations that are immersed in an external medium, we will draw on the work of {{ Weber76full }} and the work of {{ Tohline85full }} which, in turn, draws upon {{ Tohline81full }}.  In preparation for that discussion, we will go ahead and show how the {{ Tohline85 }} statement of virial equilibrium &#8212; his equation (9) &#8212; is the same as the equation that defines free energy extrema that has been derived here; and we will show how the {{ Weber76 }} "energy integral" &#8212; his equation (B3) &#8212; relates to our dimensionless free-energy function.

----


<table border="1" cellpadding="10" align="center" width="75%">
<tr><td align="center">
<!-- [[Image:Tohline1985_Eq9.png|500px|center]] -->
<!--
The Astrophysical Journal, 292: 181-187, 1985 May 1<br />
STAR FORMATION: PHASE TRANSITION, NOT JEANS INSTABILITY<br />
Joel E. Tohline<br />
-->
{{ Tohline85figure }}<br />&nbsp;<br />
<math>\beta(\sin^{-1}e/e)^2 [\tfrac{1}{2} - \beta ] + kV^* - F_s^* = 0 \, .</math>&nbsp; &nbsp; &nbsp; &nbsp;(9)
</td></tr>
</table>


First, in order to match sign conventions, we need to multiply our "free energy extrema" equation through by minus one; second, we should set <math>\delta_{1\gamma_g} = 1</math> because {{ Tohline85 }} was only concerned with isothermal systems; then, because {{ Tohline85 }} normalizes each energy term by
<div align="center">
<math>E^* \equiv \biggl( \frac{2^2 \cdot 3^2}{5^3} \biggr) \frac{G^2 M_\mathrm{tot}^5}{J^2} \, ,</math>
</div>
instead of by our <math>E_\mathrm{norm}</math>, we need to multiply our equation through by the ratio,
<div align="center">
<math>\frac{E_\mathrm{norm}}{E^*} = \biggl( \frac{5^3}{2^4 \cdot 3\pi} \biggr) \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \, .</math>
</div>
With these three modifications, our "free energy extrema" relation becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{3E_\mathrm{norm}}{E^*}\biggl[~A\chi_\mathrm{eq}^{-1} ~- \biggl( \frac{2C}{3}\biggr) \chi_\mathrm{eq}^{-2}    
~  +~ D\chi_\mathrm{eq}^3 - ~ B_I \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
Next, because {{ Tohline85 }} considered only uniform-density configurations, all of the dimensionless filling factors can be set to unity in the definitions of the leading coefficients of all of our energy terms; but, following {{ Tohline81 }}, the leading coefficients of two of our energy terms should be modified to include a factor involving the configuration's eccentricity,
<div align="center">
<math>e \equiv \biggl( 1 - \frac{Z_\mathrm{eq}^2}{R_\mathrm{eq}^2} \biggr)^{1/2} \, ,</math>
</div>
in order to account for rotational flattening.  Properly adjusted, the four coefficients are,
<div align="center">
<table border="0" cellpadding="5">
<tr>
  <td align="right">
<math>A</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{1}{5} \biggl( \frac{\sin^{-1}e}{e} \biggr) \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>B_I</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
1 \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>C</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\frac{3\cdot 5}{2^4 \pi} \biggl[ \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr]  
= \biggl( \frac{3^2}{5^2} \biggr) \frac{E_\mathrm{norm}}{E^*}  \, ,
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>D</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} (1-e^2)^{1/2} \, .
</math>
  </td>
</tr>
</table>
</div>
Inserting these coefficient definitions, our "free energy extrema" relation becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{3E_\mathrm{norm}}{E^*}
\biggl[~\frac{1}{5} \biggl( \frac{\sin^{-1}e}{e} \biggr) \chi_\mathrm{eq}^{-1} 
~- \frac{E_\mathrm{norm}}{E^*} \biggl( \frac{2\cdot 3}{5^2} \biggr)  \chi_\mathrm{eq}^{-2}    
~  +~ \biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} (1-e^2)^{1/2} \chi_\mathrm{eq}^3 - ~ 1 \biggr] \, .</math>
  </td>
</tr>
</table>
</div>
Next we need to appreciate that {{ Tohline85 }} adopted the dimensionless parameter, <math>\beta \equiv T_\mathrm{rot}/|W_\mathrm{grav}|</math>, instead of the normalized radius, <math>\chi</math>, as the order parameter that is varied when searching for extrema in the free-energy function.  So, in our equation that defines "free energy extrema" we need to replace <math>\chi_\mathrm{eq}</math> with <math>\beta_\mathrm{eq}</math>, using the relation,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\beta \equiv \frac{T_\mathrm{rot}}{|W_\mathrm{grav}|}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{C\chi^{-2}}{3A \chi^{-1}} = \biggl( \frac{3}{5} \biggr) \frac{E_\mathrm{norm}}{E^*} 
\biggl( \frac{\sin^{-1}e}{e} \biggr)^{-1} \chi^{-1}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~~\chi_\mathrm{eq}^{-1} </math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{5}{3} \biggr) \frac{E^*}{E_\mathrm{norm}} 
\biggl( \frac{\sin^{-1}e}{e} \biggr)\beta_\mathrm{eq} \, .
</math>
  </td>
</tr>
</table>
</div>
Hence, our expression for the "free energy extrema" becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>0</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{\sin^{-1}e}{e} \biggr)^2 \beta_\mathrm{eq} ~- 2\biggl( \frac{\sin^{-1}e}{e} \biggr)^2 \beta_\mathrm{eq}^{2}    
~  +~ \frac{4\pi P_e}{P_\mathrm{norm}} (1-e^2)^{1/2} 
\biggl[ \biggl( \frac{3^3}{5^3} \biggr) \biggl( \frac{E_\mathrm{norm}}{E^*} \biggr)^4
\biggl( \frac{\sin^{-1}e}{e} \biggr)^{-3}\biggr] \beta_\mathrm{eq}^{-3}
- ~ \frac{3E_\mathrm{norm}}{E^*}  </math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
2 \biggl\{
\beta_\mathrm{eq} \biggl( \frac{\sin^{-1}e}{e} \biggr)^2 \biggl( \frac{1}{2} - \beta_\mathrm{eq} \biggr)    
~  + \frac{2\pi \cdot 3^3}{5^3} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \biggl( \frac{E_\mathrm{norm}}{E^*} \biggr)^4
\biggl[ 
\beta_\mathrm{eq}^{-3}\biggl( \frac{\sin^{-1}e}{e} \biggr)^{-3} (1-e^2)^{1/2}\biggr] 
- ~ \biggl( \frac{3}{2} \biggr) \frac{5^3}{2^4 \cdot 3 \pi} \biggl[ \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr]
\biggr\} \, .</math>
  </td>
</tr>
</table>
</div>
Now,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\frac{2\pi \cdot 3^3}{5^3} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \biggl( \frac{E_\mathrm{norm}}{E^*} \biggr)^4</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{2\pi \cdot 3^3}{5^3} \biggl[ \frac{P_e}{(E^*)^4} \biggr] ( GM_\mathrm{tot}^2)^3</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{2\pi \cdot 3^3}{5^3} \biggl[\biggl( \frac{2^2 \cdot 3^2}{5^3} \biggr) \frac{G^2 M_\mathrm{tot}^5}{J^2} \biggr]^{-4} ( P_e G^3 M_\mathrm{tot}^6)</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\pi\biggl( \frac{5^{9}}{2^7 \cdot 3^5} \biggr) \frac{J^8 P_e }{G^5 M_\mathrm{tot}^{14}}  
= \frac{10 \pi}{3} \biggl( \frac{5^{2}}{2^2 \cdot 3} \biggr)^4 \frac{J^8 P_e }{G^5 M_\mathrm{tot}^{14}} \, ,
</math>
  </td>
</tr>
</table>
</div>
which is the definition of the coefficient "<math>k</math>" that is provided as equation (7) of {{ Tohline85 }}.  Hence, dropping the factor of two out front, our expression for "free energy extrema" becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\beta_\mathrm{eq} \biggl( \frac{\sin^{-1}e}{e} \biggr)^2 \biggl( \frac{1}{2} - \beta_\mathrm{eq} \biggr)    
~  + \frac{10 \pi}{3} \biggl( \frac{5^{2}}{2^2 \cdot 3} \biggr)^4 \frac{J^8 P_e }{G^5 M_\mathrm{tot}^{14}} 
\biggl[ 
\beta_\mathrm{eq}^{-3}\biggl( \frac{\sin^{-1}e}{e} \biggr)^{-3} (1-e^2)^{1/2}\biggr] 
- ~ \frac{3}{4\pi} \biggr( \frac{5^3}{2^3 \cdot 3} \biggr) \biggl[ \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr]
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0 \, .</math>
  </td>
</tr>
</table>
</div>
Finally, realizing that the square of the sound speed is related to our <math>c_\mathrm{norm}^2</math> via the relation [note that {{ Tohline85 }} uses <math>a^2</math> in place of <math>c_s^2</math>],
<div align="center">
<math>c_s^2 = \biggl( \frac{3}{4\pi} \biggr) c_\mathrm{norm}^2 \, ,</math>
</div>
it is clear that this last form of our "free energy extrema" expression is identical to the {{ Tohline85 }} virial equilibrium equation (9), which appears in print in a simpler but also more cryptic form as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>
\beta_\mathrm{eq} \biggl( \frac{\sin^{-1}e}{e} \biggr)^2 \biggl( \frac{1}{2} - \beta_\mathrm{eq} \biggr)    + kV^* - F_s^*
</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>0 \, .</math>
  </td>
</tr>
</table>
</div>


----


<table border="1" cellpadding="10" align="center" width="75%">
<tr><td align="center">
<!-- [[Image:Weber1976_EqB3.png|500px|center]] -->
<!--
The Astrophysical Journal, 208: 113-126, 1976 August 15<br />
OSCILLATION AND COLLAPSE OF INTERSTELLAR CLOUDS<br />
Stephen V. Weber<br />
-->
{{ Weber76figure }}<br />&nbsp;<br />
<math>E = \dot{a}^2 + \dot\gamma^2/2 + J^2/\alpha^2 
- \tfrac{4}{3} \log \alpha^2 \gamma + \tfrac{1}{3}\alpha^2 \gamma P_\mathrm{ext} -
\begin{cases}\biggl(\frac{3}{\alpha e}\biggr) \sin^{-1}e, & e > 0, \\ \biggl( \frac{3}{2\alpha e} \biggr) \log Q , & e<0 \, . \end{cases}
</math>&nbsp; &nbsp; &nbsp; &nbsp;(B3)
</td></tr>
</table>


Plugging the same set of modified leading coefficients into our derived expression for the free energy gives,
<div align="center">
<math>
\mathfrak{G}^* = \frac{3\cdot 5}{2^4 \pi} \biggl[ \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr] \chi^{-2} 
 -~ 3 \ln \chi +~ \biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} (1-e^2)^{1/2} \chi^3 
- \frac{3}{5} \biggl( \frac{\sin^{-1}e}{e} \biggr) \chi^{-1}\, .
</math>
</div>
Now, recognize that,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\chi</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\alpha \biggl( \frac{R_0}{R_\mathrm{norm}} \biggr) = \biggl( \frac{2^2}{3\cdot 5} \biggr) \alpha \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>(1 - e^2)^{1/2}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{Z}{R} = \frac{\gamma}{\alpha} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{P_e}{P_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{P_e}{P_0} \cdot \frac{P_0}{P_\mathrm{norm}} = \frac{3^4 \cdot 5^3}{2^{10} \pi} \biggl[ P_\mathrm{ext} \biggr]_\mathrm{Weber} 
\, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\frac{3\cdot 5}{2^4 \pi} \biggl[ \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr]</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>\frac{1}{3} \biggl( \frac{2}{5} J_\mathrm{Weber} \biggr)^2
\, ,</math>
  </td>
</tr>
</table>
</div>
where, for axisymmetric configurations (set <math>\beta=\alpha</math> in equation 12 of {{ Weber76 }}),
<div align="center">
<math>J_\mathrm{Weber} \equiv \alpha^2 \Omega = \biggl( \frac{R}{R_0} \biggr)^2 (\dot\varphi_c t_0)^2 \, .</math>
</div>
Hence, our expression for the free energy may be written as,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>\mathfrak{G}^*</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{1}{3} \biggl( \frac{2}{5} J_\mathrm{Weber}\biggr)^2 \biggl( \frac{3\cdot 5}{2^2} \biggr)^2 \alpha^{-2} 
 -~ 3 \ln \chi +~ \biggl( \frac{4\pi}{3} \biggr) \frac{3^4 \cdot 5^3}{2^{10} \pi} \biggl[ P_\mathrm{ext} \biggr]_\mathrm{Weber} \biggl( \frac{2^2}{3\cdot 5} \biggr)^3 \alpha^2 \gamma
- \frac{3}{5} \biggl( \frac{\sin^{-1}e}{e} \biggr) \biggl( \frac{3\cdot 5}{2^2} \biggr) \alpha^{-1}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\biggl( \frac{3}{2^2} \biggr)  J^2_\mathrm{Weber} \alpha^{-2} 
 -~ \ln \chi^3 +~ \frac{1}{2^{2} } \alpha^2 \gamma \biggl[ P_\mathrm{ext} \biggr]_\mathrm{Weber}  
- \frac{3^2}{2^2} \biggl( \frac{\sin^{-1}e}{e} \biggr) \alpha^{-1} \, .
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>\Rightarrow~~~~\frac{4}{3} \mathfrak{G}^*</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
J^2_\mathrm{Weber} \alpha^{-2} 
 -~ \frac{4}{3} \ln \chi^3 +~ \frac{1}{3} \alpha^2 \gamma \biggl[ P_\mathrm{ext} \biggr]_\mathrm{Weber}  
- 3 \biggl( \frac{\sin^{-1}e}{e} \biggr) \alpha^{-1} \, .
</math>
  </td>
</tr>
</table>
</div>
The right-hand-side of this expression exactly matches the {{ Weber76 }} "energy integral" for oblate-spheroidal configurations &#8212; see his equation (B3) for the case, <math>e > 0</math> &#8212; except that Weber's energy integral includes an additional pair of terms (<math>{\dot\alpha}^2 + {\dot\gamma}^2/2</math>) to account for kinetic energy associated with the overall collapse or expansion of the configuration.  [NOTE:  The logarithmic term ultimately needs to be <math>\ln \alpha^2\gamma</math> instead of <math>\ln\chi^3</math> in order to reflect an oblate-spheroidal, rather than spherical, volume.  This term also needs to be fixed in the above discussion of Tohline's work.]
</td></tr>
</table>
</div>

=Examples=
* Polytropic Spheres
** [[SSC/Virial/Polytropes#Isolated.2C_Nonrotating_Configuration|Isolated, Nonrotating Configuration]]
** [[SSC/Virial/Polytropes#Nonrotating_Configuration_Embedded_in_an_External_Medium|Nonrotating Configuration Embedded in an External Medium]]

* Isothermal Spheres
** [[SSC/Virial/Isothermal#Isolated.2C_Nonrotating_Configuration|Isolated, Nonrotating Configuration]]
** [[SSC/Virial/Isothermal#Nonrotating_Configuration_Embedded_in_an_External_Medium|Nonrotating Configuration Embedded in an External Medium]]


{{ SGFworkInProgress }}

=BiPolytrope=
[Following a discussion that Tohline had with Kundan Kadam on <font color="red">3 July 2013</font>, we have decided to carry out a virial equilibrium and stability analysis of nonrotating bipolytropes.]  

We will adopt the following approach:
* Properties of the core <math>\cdots</math> 
** Uniform density, <math>\rho_c</math>;
** Polytropic constant, <math>K_c</math>, and polytropic index, <math>n_c</math>;
** Surface of the core at <math>r_i</math>;
* Properties of the envelope <math>\cdots</math> 
** Uniform density, <math>\rho_e</math>;
** Polytropic constant, <math>K_e</math>, and polytropic index, <math>n_e</math>;
** Base of the core at <math>r_i</math> and surface at <math>R</math>.

Use the dimensionless radius,
<div align="center">
<math>\xi \equiv \frac{r}{r_i}</math>.
</div>
Then, <math>\xi_i = 1</math> and <math>\xi_s \equiv R/r_i</math>.

==Expressions for Mass==
Inside the core, the expression for the mass interior to any radius, <math>0 \le \xi \le 1</math>, is,
<div align="center">
<math>M_\xi = \frac{4\pi}{3} \rho_c r_i^3 \xi^3</math> .
</div>
The expression for the mass interior to any position within the envelope, <math>1 \le \xi \le \xi_s</math>, is,
<div align="center">
<math>M_\xi = \frac{4\pi}{3} r_i^3 \biggl[\rho_c  + \rho_e(\xi^3 - 1) \biggr]</math> .
</div>
Hence, in terms of a reference mass, <math>~M_0 \equiv 4\pi \rho_0 R_0^3/3</math>, the mass of the core, the mass of the envelope, and the total mass are, respectively,

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

<tr>
  <td align="right">
<math>M_\mathrm{core}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi}{3} \rho_c r_i^3 
= M_0 \biggl[ \frac{\rho_c}{\rho_0} \biggl( \frac{r_i}{R_0}\biggr)^3 \biggr] 
~~~~~\Rightarrow~~~~~ 
\frac{\rho_c}{\rho_0} = \frac{M_\mathrm{core}}{M_0} \biggl( \frac{r_i}{R_0}\biggr)^{-3} \, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>M_\mathrm{env}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi}{3} r_i^3 \biggl[\rho_e (\xi_s^3 - 1) \biggr] =
M_0 (\xi_s^3 - 1) \biggl[ \frac{\rho_e}{\rho_0} \biggl( \frac{r_i}{R_0}\biggr)^3 \biggr] 
~~~~~\Rightarrow~~~~~   
\frac{\rho_e}{\rho_0} = \frac{M_\mathrm{env}}{M_0} \biggl( \frac{r_i}{R_0}\biggr)^{-3} (\xi_s^3 - 1)^{-1}\, ;
</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>M_\mathrm{tot}</math>
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
\frac{4\pi}{3} r_i^3 \biggl[\rho_c  + \rho_e(\xi_s^3 - 1) \biggr] 
= M_0 \biggl( \frac{\rho_c}{\rho_0} \biggr) \biggl( \frac{r_i}{R_0}\biggr)^3  \biggl[ 1 + \frac{\rho_e}{\rho_c} (\xi_s^3 - 1) \biggr]  \, .
</math>
  </td>
</tr>
</table>
</div>

Following the work of [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Sch&ouml;nberg &amp; Chandrasekhar (1942, ApJ, 96, 1615)] &#8212; see [[SSC/Structure/LimitingMasses#Sch.C3.B6nberg-Chandrasekhar_Mass|our accompanying discussion]]  &#8212; we will discuss bipolytropic equilibrium configurations in the context of a <math>\nu - q</math> plane where,
<table align="center" border="0" cellpadding="10">
<tr>
  <td align="right">
<math>\nu</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{M_\mathrm{core}}{M_\mathrm{tot}} \, ,</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>q</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>\frac{r_i}{R} = \frac{1}{\xi_s} \, .</math>
  </td>
</tr>
</table>

With this in mind we can write,
<div align="center">
<math>\frac{\rho_e}{\rho_c} =  \frac{M_\mathrm{env}}{M_\mathrm{core}} (\xi_s^3 - 1)^{-1} 
=  \frac{q^3 (1-\nu)}{\nu (1-q^3)} </math> ,
</div>
and,
<div align="center">
<math>\nu \biggl(\frac{1-q^3}{q^3}\biggr) \biggl( \frac{\rho_e}{\rho_c} \biggr)  =  (1-\nu)
~~~~~\Rightarrow~~~~~ 
\nu = \biggl[ 1 + \biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl(\frac{1-q^3}{q^3}\biggr) \biggr]^{-1} \, .</math>
</div>

==Energy Expressions==

The gravitational potential energy of the bipolytropic configuration is obtained by integrating over the following differential energy contribution,
<div align="center">
<math>dW_\mathrm{grav} = - \biggl( \frac{GM_r}{r} \biggr) dm</math> .
</div>

Hence,
<table border="0" align="center">

<tr>
  <td align="right">
<math>W_\mathrm{grav} = W_\mathrm{core} + W_\mathrm{env}</math>
  </td>
  <td align="left">
<math>
= - G \biggl\{ \int_0^{r_i} \biggl( \frac{M_r}{r} \biggr) 4\pi r^2 \rho_c dr + \int^R_{r_i} \biggl( \frac{M_r}{r} \biggr) 4\pi r^2 \rho_e dr \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="left">
<math>
= - G \biggl\{ \int_0^1 \biggl( \frac{4\pi }{3} \rho_c r_i^3 \xi^3 \biggr) 4\pi r_i^2 \rho_c \xi d\xi 
+ \int_1^{\xi_s} \frac{4\pi}{3} \rho_c r_i^3 \biggl[ 1 + \frac{\rho_e}{\rho_c}(\xi^3 - 1) \biggr] 4\pi r_i^2 \rho_e \xi d\xi \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="left">
<math>
= - \frac{3GM^2_\mathrm{core}}{r_i} \biggl\{ \int_0^1  \xi^4 d\xi 
+ \int_1^{\xi_s} \biggl[ 1 + \frac{\rho_e}{\rho_c}(\xi^3 - 1) \biggr] \biggl( \frac{\rho_e}{\rho_c} \biggr) \xi d\xi \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="left">
<math>
= - \frac{3GM^2_\mathrm{core}}{r_i} \biggl\{ \frac{1}{5} 
+ \biggl( \frac{\rho_e}{\rho_c} \biggr) \int_1^{\xi_s} \xi d\xi 
+ \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \int_1^{\xi_s}  (\xi^3 - 1) \xi d\xi \biggr\}
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="left">
<math>
= - \frac{3GM^2_\mathrm{tot}}{R} \biggl( \frac{M_\mathrm{core}}{M_\mathrm{tot}} \biggr)^2 \xi_s \biggl\{ \frac{1}{5} 
+ \frac{1}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (\xi_s^2 - 1)
+ \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ \frac{1}{5}(\xi_s^5 - 1) - \frac{1}{2}(\xi_s^2-1) \biggr] \biggr\}
</math>
  </td>
</tr>

</table>

I like the form of this expression.  The leading term, which scales as <math>R^{-1}</math>, encapsulates the behavior of the gravitational potential energy for a given choice of the internal structure, namely, a given choice of <math>\xi_s</math>, <math>\nu</math>, and density ratio <math>(\rho_e/\rho_c)</math>.  Actually, only two internal structural parameters need to be specified &#8212; <math>\xi_s</math> and <math>f_c</math>; from these two, the expressions shown above allow the determination of both <math>(\rho_e/\rho_c)</math> and <math>\nu</math>.  Keeping in mind our desire to discuss the properties of bipolytropes in the context of the <math>\nu - q</math> plane introduced by [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Sch&ouml;nberg &amp; Chandrasekhar (1942, ApJ, 96, 1615)], we will rewrite this expression for the gravitational potential energy as,
<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>- \frac{3}{5} \biggl( \frac{GM_\mathrm{tot}^2}{R} \biggr) \frac{\nu^2}{q} \cdot f\biggl(q, \frac{\rho_e}{\rho_c} \biggr) \, ,</math>
  </td>
</tr>
</table>
</div>
where,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
  <td align="right">
<math>f\biggl(q, \frac{\rho_e}{\rho_c} \biggr)</math>
  </td>
  <td align="center">
<math>\equiv</math>
  </td>
  <td align="left">
<math>
1 + \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl(\frac{1}{q^2} - 1 \biggr)
+ \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ \biggl( \frac{1}{q^5} - 1 \biggr) - \frac{5}{2}\biggl(\frac{1}{q^2} - 1 \biggr) \biggr] 
</math>
  </td>
</tr>

<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>=</math>
  </td>
  <td align="left">
<math>
1 + \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) \frac{1}{q^5} \biggl[ (q^3- q^5 )
+ \biggl( \frac{\rho_e}{\rho_c} \biggr) \biggl( \frac{2}{5} -q^3 + \frac{3}{5}q^5\biggr) \biggr] \, .
</math>
  </td>
</tr>
</table>
</div>

=See Also=
<ul>
<li>[[SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>
<li>[[SSC/Index|Spherically Symmetric Configurations (SSC) Index]]</li>
</ul>


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