Difference between revisions of "User:Tohline/SphericallySymmetricConfigurations/Virial"

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(→‎Energy Content for a System of a Given Size and Internal Structure: Correct various expressions, mostly involving f_M)
 
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__FORCETOC__
__FORCETOC__


=Virial Equilibrium=
=Virial Equilibrium of Spherically Symmetric Configurations=
{{LSU_HBook_header}}
{{LSU_HBook_header}}
==Free Energy Expression==
==Free Energy Expression==
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<div align="center">
<div align="center">
<math>
<math>
\mathfrak{G} = W + \mathfrak{W}_\mathrm{therm} + T_\mathrm{rot} + P_e V + \cdots
\mathfrak{G} = W_\mathrm{grav} + \mathfrak{S}_\mathrm{therm} + T_\mathrm{kin} + P_e V + \cdots
</math>
</math>
</div>  
</div>  
Here, we have explicitly included the gravitational potential energy, <math>~W</math>, the rotational kinetic energy, <math>~T_\mathrm{rot}</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{W}_\mathrm{therm}</math>, the reservoir of thermodynamic energy that is available to perform work as the system expands or contracts.
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, [[User:Tohline/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===
===Expressions for Various Energy Terms===
In deriving useful expressions for each of the terms in the Gibbs-like free-energy expression, we need to consider two issues:  First, for a given size system, a determination of each term's total contribution to the free energy may involve integration through the entire volume of the configuration.  Second, each term must be formulated in such a way that it is clear how the energy contribution depends on the system size.
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">
====Energy Content for a System of a Given Size and Internal Structure====
<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.


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</math>, as,
====Volume Integrals====
We note, first, that the mass enclosed within each interior radius, <math>~r</math>, is
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~P_e V</math>
<math>~M_r(r) = \int\limits_V dm</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
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   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\frac{4\pi}{3} P_e R^3 \, .</math>
<math>~ \int_0^r  4\pi r^2 \rho dr  \, .</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
 
Hence, if the volume of the configuration extends out to a radius denoted by <math>~R_\mathrm{limit}</math>, the configuration mass is,
For a spherically symmetric system with a given density distribution, <math>~\rho(r)</math>, the mass enclosed within each interior radius, <math>~r</math>, is
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~M_r(r)</math>
<math>~M_\mathrm{limit}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
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   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ \int_0^r 4\pi r^2 \rho dr  \, ,</math>
<math>~ \int_0^{R_\mathrm{limit}} 4\pi r^2 \rho dr  \, .</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
and the total mass is,
 
<table align="center" border="1" width="65%" 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 [[User:Tohline/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 [[User:Tohline/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 [[User:Tohline/SSC/BipolytropeGeneralization_Version2#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">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~M_\mathrm{tot}</math>
<math>~P_e V</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
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   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ \int_0^R 4\pi r^2 \rho dr  \, .</math>
<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>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
As pointed out by, for example, [[User:Tohline/Appendix/References|Chandrasekhar [C67]]] &#8212; see p. 64, Equation (12) &#8212; the total gravitational potential energy is, therefore,
 
<font color="red">Gravitational Potential Energy:</font>  From our discussion of the [[User:Tohline/VE#Scalar_Virial_Theorem|scalar virial theorem]] &#8212; see, specifically, the reference to Equation (18), on p. 18 of [<b>[[User:Tohline/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 />
 
{{User:Tohline/Math/EQ_Poisson01}}
</div>
becomes,
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~W</math>
<math>~\frac{1}{r^2} \frac{d}{dr} \biggl( r^2 \frac{d\Phi}{dr} \biggr) </math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 78: Line 112:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ - \int_0^R \biggl( \frac{GM_r}{r} \biggr) dm = - \int_0^R \frac{G}{r}\biggl[\int_0^r 4\pi r^2 \rho dr \biggr] 4\pi r^2 \rho dr \, .</math>
<math>~4\pi G \rho(r) \, , </math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
 
which implies,
<table border="1" align="center" width="90%" cellpadding="20">
<div align="center">
<tr><td align="left">
<table border="0" cellpadding="5" align="center">
Also, as pointed out by [[User:Tohline/Appendix/References|Chandrasekhar [C67]]] &#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>
<tr>
   <td align="right">
   <td align="right">
<math>~W</math>
<math>~r^2 \frac{d\Phi}{dr} </math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 97: Line 128:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ - \frac{1}{2} \int_0^R \Phi(r) dm \, .</math>
<math>~\int_0^r 4\pi G \rho(r) r^2 dr = GM_r(r) \, .</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
</td></tr>
<span id="Wgrav">Hence</span> &#8212; see, also, p. 64, Equation (12) of [<b>[[User:Tohline/Appendix/References#C67|<font color="red">C67</font>]]</b>] &#8212; the desired expression for the gravitational potential energy is,
</table>
 
If, in addition, this system is rotating with a specified [[User:Tohline/AxisymmetricConfigurations/SolutionStrategies#Simple_Rotation_Profile_and_Centrifugal_Potential|''simple'' angular velocity profile]], <math>~\dot\varphi(\varpi)</math>, the total rotational kinetic energy content is,
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~T_\mathrm{rot}</math>
<math>~W_\mathrm{grav}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 117: Line 144:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ \frac{1}{2} \int\int\int \dot\varphi^2 \varpi^2 dm
<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>
= \frac{1}{2} \int_0^R \dot\varphi^2 \varpi^2  \int_{-\sqrt{R^2 - \varpi^2}}^{\sqrt{R^2 - \varpi^2}}  \rho(r(\varpi,z)) 2\pi \varpi d\varpi dz\, .</math>
   </td>
   </td>
</tr>
</tr>
Line 124: Line 150:
</div>
</div>


As has been explained in [[User:Tohline/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{W}_\mathrm{therm}</math>, depends on whether the system is expected to evolve adiabatically or isothermally.  For [[User:Tohline/VE#Isothermal_System|isothermal systems]],
<div align="center">
<math>\mathfrak{W}_I = c_s^2  \int_0^R \ln \biggl(\frac{\rho}{\rho_0}\biggr) 4\pi r^2 \rho dr \, ,</math>
</div>
where, <math>~c_s</math> is the isothermal sound speed; while, for [[User:Tohline/VE#Adiabatic_Systems|adiabatic systems]],
<div align="center">
<math>\mathfrak{W}_A = \frac{1}{({\gamma_g}-1)}  \int_0^R  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.


<div align="center">
<div id="AlternateGravPotEnergy">
<table border="1" cellpadding="8" align="center" width="90%">
<table border="1" align="center" width="90%" cellpadding="20">
<tr>
  <th align="center">
Idealized Spherical Configuration
  </th>
</tr>
<tr><td align="left">
<tr><td align="left">
In the idealized situation of a configuration that has uniform density, <math>~\rho_c</math>, has uniform pressure, <math>~P_c</math>, and is uniformly rotating with angular velocity, <math>~\dot\varphi_c</math>, evaluation of the integrals yields,
Also, as pointed out by [<b>[[User:Tohline/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">
<div align="center">
<table border="0" cellpadding="8" align="center">
<table border="0" cellpadding="5">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~W</math>
<math>~W_\mathrm{grav}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 154: Line 165:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ - \biggl( \frac{2^4 \pi^2}{3} \biggr) G\rho_c^2 R^5 \int_0^1  x^4 dx
<math>~ + \frac{1}{2} \int_0^{R_\mathrm{limit}} \Phi(r) dm \, .</math>
= - \frac{3}{5} \biggl( \frac{GM_\mathrm{tot}^2}{R} \biggr) \, ,</math>
   </td>
   </td>
</tr>
</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 [[User:Tohline/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 [[User:Tohline/VE#Scalar_Virial_Theorem|scalar virial theorem]] &#8212; see, specifically, the reference to Equation (8), on p. 16 of [[User:Tohline/Appendix/References#EFE|EFE]] &#8212; the (ordered) kinetic energy,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~T_\mathrm{rot}</math>
<math>~T_\mathrm{kin}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 167: Line 187:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ 2\pi R^5 \rho_c \dot\varphi^2_c  \int_0^1 w^3 dw \int_0^{\sqrt{1 - w^2}}  d\zeta
<math>~ \frac{1}{2} \int\limits_V \rho |\vec{v} |^2 d^3x = \frac{1}{2} \int\limits_V |\vec{v} |^2 dm \, ,</math>
= 2\pi R^5 \rho_c \dot\varphi^2_c \int_0^1 w^3 (1-w^2)^{1/2} dw </math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
is entirely rotational kinetic energy, specifically,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~T_\mathrm{kin} = T_\mathrm{rot}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 180: Line 204:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ 2\pi R^5 \rho_c \dot\varphi^2_c  \biggl[ -\frac{1}{15} (1-w^2)^{3/2} (3w^2 +2) \biggr]_0^1
<math>~ \frac{1}{2} \int\int\int \dot\varphi^2 \varpi^2 dm
= \frac{4\pi}{15} R^5 \rho_c \dot\varphi^2_c
= \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>
= \frac{1}{5} M_\mathrm{tot} R^2 \dot\varphi^2_c \, ,</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>


<font color="red">Reservoir of Thermodynamic Energy:</font>  As has been explained in [[User:Tohline/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 [[User:Tohline/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 [[User:Tohline/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 [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] and [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)], 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>
<tr>
   <td align="right">
   <td align="center">
<math>~\mathfrak{W}_I</math>
Adiabatic Cases
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
Isothermal Case
  </td>
<math>~(\gamma = 1; K = c_s^2)</math>
  <td align="left">
<math>~ c_s^2 \ln \biggl(\frac{\rho_c}{\rho_0}\biggr) 4\pi R^3 \rho_c \int_0^1 x^2 dx
= c_s^2 M_\mathrm{tot} \ln \biggl(\frac{\rho_c}{\rho_0}\biggr) \, ,</math>
   </td>
   </td>
</tr>
</tr>
<tr><td align="center">


<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathfrak{W}_A</math>
<math>~R_\mathrm{norm}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\frac{4\pi R^3 P_c}{({\gamma_g}-1)}  \int_0^1  x^2 dx =  \frac{M_\mathrm{tot}}{({\gamma_g}-1)} \frac{P_c}{\rho_c} \, ,</math>
<math>~\biggl[ \biggl( \frac{G}{K} \biggr) M_\mathrm{tot}^{2-\gamma} \biggr]^{1/(4-3\gamma)} </math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
where, <math>~M_\mathrm{tot} = (4\pi \rho_c R^3/3)</math>, and 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>.
</td></tr>
</table>
</div>


Keeping in mind the expressions that arise in the case of an idealized spherical configuration, we generally will write the expression for the total mass as,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~M_\mathrm{tot}</math>
<math>~P_\mathrm{norm}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ \frac{4\pi}{3} R^3 \rho_c \cdot \mathfrak{f}_M ~~~~~ \biggl(\Rightarrow ~ \mathfrak{f}_M = \frac{\bar\rho}{\rho_c} \biggr) \, ,</math>
<math>~\biggl[ \frac{K^4}{G^{3\gamma} M_\mathrm{tot}^{2\gamma}} \biggr]^{1/(4-3\gamma)</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>


and we generally will write each energy term as follows:
<div align="center">
<table border="0" cellpadding="8" align="center">
<tr>
<tr>
  <td align="right">
   <td align="center" colspan="3">
<math>~W</math>
----
  </td>
   <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ - \frac{3}{5} \biggl( \frac{GM_\mathrm{tot}^2}{R} \biggr) \cdot \frac{\mathfrak{f}_W}{\mathfrak{f}_M^2} \, ,</math>
   </td>
   </td>
</tr>
</tr>
Line 251: Line 282:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~T_\mathrm{tot}</math>
<math>~E_\mathrm{norm}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ \frac{1}{2} \biggl( \frac{2}{5} M_\mathrm{tot} R^2 \biggr) \dot\varphi^2_c \cdot \frac{\mathfrak{f}_T }{\mathfrak{f}_M} \, ,</math>
<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>
   </td>
</tr>
</tr>
Line 263: Line 295:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathfrak{W}_I</math>
<math>~\rho_\mathrm{norm}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ M_\mathrm{tot} c_s^2 \biggl[\ln\biggl( \frac{\rho_c}{\rho_0} \biggr)  + \frac{\mathfrak{f}_I }{\mathfrak{f}_M} \biggl] \, ,</math>
<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>
   </td>
</tr>
</tr>
Line 275: Line 308:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathfrak{W}_A</math>
<math>~c^2_\mathrm{norm}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ \frac{M_\mathrm{tot}}{({\gamma_g}-1)} \frac{P_c}{\rho_c} \cdot \frac{\mathfrak{f}_A}{\mathfrak{f}_M} \, ,</math>
<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>
   </td>
</tr>
</tr>
</table>
</table>
</div>


<span id="FormFactors">where the dimensionless form factors, <math>~\mathfrak{f}_i</math> &#8212; each usually of order unity &#8212; are</span>,
</td>
 
<td align="center">


<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathfrak{f}_M = \frac{\bar\rho}{\rho_c}</math>
<math>~R_\mathrm{norm}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ 3\int_0^1  \biggl[ \frac{\rho(x)}{\rho_c}\biggr] x^2 dx  \, ,</math>
<math>~\frac{G M_\mathrm{tot}}{c_s^2</math>
   </td>
   </td>
</tr>
</tr>
Line 305: Line 339:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathfrak{f}_W</math>
<math>~P_\mathrm{norm}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 311: Line 345:
   </td>
   </td>
   <td align="left">
   <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>
<math>~\frac{c_s^8}{G^{3} M_\mathrm{tot}^{2}}   </math>
  </td>
</tr>
 
<tr>
  <td align="center" colspan="3">
----
   </td>
   </td>
</tr>
</tr>
Line 317: Line 357:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathfrak{f}_T</math>
<math>~E_\mathrm{norm}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 323: Line 363:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ \frac{15}{2} \int_0^1 \biggl[ \frac{\dot\varphi(w)}{\dot\varphi_c} \biggr]^2 w^3 dw  \int_0^{\sqrt{1 - w^2}} 
<math>~ M_\mathrm{tot} c_s^2 </math>
\biggl[ \frac{\rho(w,\zeta)}{\rho_c} \biggr] d\zeta\, ,</math>
   </td>
   </td>
</tr>
</tr>
Line 330: Line 369:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathfrak{f}_I</math>
<math>~\rho_\mathrm{norm}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 336: Line 375:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ 3\int_0^1 \biggl[ \frac{\rho(x)}{\rho_c}\biggr]  \ln \biggl[ \frac{\rho(x)}{\rho_c}\biggr]  x^2 dx \, ,</math>
<math>
\frac{3}{4\pi} \biggl[ \frac{c_s^6}{G^3 M_\mathrm{tot}^2} \biggr]  </math>
   </td>
   </td>
</tr>
</tr>
Line 342: Line 382:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathfrak{f}_A</math>
<math>~c^2_\mathrm{norm}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 348: Line 388:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ 3\int_0^1 \biggl[ \frac{P(x)}{P_c}\biggr]  x^2 dx \, .</math>
<math>~
\biggl( \frac{4\pi}{3} \biggr) c_s^2 </math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</td>
</tr>
<tr><th 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>
</th></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>
</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.


====Dependence on Size====
=====Choices Made by Other Researchers=====


Variation with size:
As is detailed in a [[User:Tohline/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 [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] and by  [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)]. The following relations can be used to switch from our normalizations to theirs:
<div align="center">
<div align="center">
<table border="0" cellpadding="8" 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">[[User:Tohline/SSC/Structure/PolytropesEmbedded#Horedt.27s_Presentation|Hoerdt's (1970)]] Normalization</th><tr>
 
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~W</math>
<math>~\biggl( \frac{R_\mathrm{Hoerdt}}{R_\mathrm{norm}} \biggr)^{4-3\gamma}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 368: Line 436:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ - \frac{3}{5} \biggl( \frac{GM_\mathrm{tot}^2}{R_0} \biggr) \biggl( \frac{R}{R_0} \biggr)^{-1} \cdot \mathfrak{f}_W \, ,</math>
<math>~ \frac{(\gamma-1)}{\gamma} \biggl( 4\pi \biggr)^{\gamma-1}</math>
   </td>
   </td>
</tr>
</tr>
Line 374: Line 442:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~T_\mathrm{tot}</math>
<math>~\biggl( \frac{P_\mathrm{Hoerdt}}{P_\mathrm{norm}} \biggr)^{4-3\gamma}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 380: Line 448:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ \frac{5}{4} \frac{J^2}{MR_0^2} \biggl( \frac{R}{R_0} \biggr)^{-2} \cdot \mathfrak{f}_T \, ,</math>
<math>~ \biggl[\frac{\gamma}{(\gamma-1)} \biggr]^{3\gamma} \biggl( \frac{1}{4\pi} \biggr)^{\gamma}</math>
   </td>
   </td>
</tr>
</tr>
</table>


<tr>
  <td align="right">
<math>~\mathfrak{W}_I</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="center">
<math>~ M_\mathrm{tot} c_s^2 \biggl[\mathfrak{f}_I -3\ln\biggl( \frac{R}{R_0} \biggr) \cdot \mathfrak{f}_M \biggl] \, ,</math>
 
  </td>
<table border="0" cellpadding="5" align="center">
</tr>
<tr><th colspan="3" align="center">[[User:Tohline/SSC/Structure/PolytropesEmbedded#Whitworth.27s_Presentation|Whitworth's (1981)]] Normalization</th><tr>


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathfrak{W}_A</math>
<math>~\biggl( \frac{R_\mathrm{rf}}{R_\mathrm{norm}} \biggr)^{4-3\gamma}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 404: Line 470:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~ \frac{M_\mathrm{tot}}{({\gamma_g}-1)} \biggl[ K\rho_c^{\gamma_g-1}\biggr] \cdot \mathfrak{f}_A
<math>~ \frac{1}{5\pi} \biggl( \frac{4\pi}{3} \biggr)^\gamma</math>
= \frac{M_\mathrm{tot} K}{({\gamma_g}-1)} \biggl[ \rho_0\biggr]^{\gamma_g-1}  
\biggl( \frac{\rho_c}{\rho_0} \biggr)^{\gamma_g-1} \cdot \mathfrak{f}_A
</math>
   </td>
   </td>
</tr>
</tr>
Line 413: Line 476:
<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~\biggl( \frac{P_\mathrm{rf}}{P_\mathrm{norm}} \biggr)^{4-3\gamma}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 419: Line 482:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>\frac{M_\mathrm{tot} K}{({\gamma_g}-1)} \biggl[ \frac{3M_\mathrm{tot}}{4\pi R_0^3} \biggr]^{\gamma_g-1}
<math>~ 2^{-2(4+\gamma)} \biggl( \frac{3^4 \cdot 5^3}{\pi} \biggr)^\gamma</math>
\biggl( \frac{R}{R_0} \biggr)^{-3(\gamma_g-1)} \cdot \mathfrak{f}_A
  </td>
\, .</math>
</tr>
</table>
 
   </td>
   </td>
</tr>
</tr>
Line 427: Line 492:
</div>
</div>


 
It is also worth noting how the length-scale normalization that we are adopting here relates to the characteristic length scale,  
 
===Gathering it all Together===
Gathering all of the terms together we find that, to within an additive constant, the expression for the free energy is,  
<div align="center">
<div align="center">
<math>
<math>~a_n \equiv \biggl[ \frac{1}{4\pi G} \biggl( \frac{H_c}{\rho_c} \biggr) \biggr]^{1/2} \, ,</math>
\mathfrak{G} = -A\biggl( \frac{R}{R_0} \biggr)^{-1} +~ (1-\delta_{1\gamma_g})B\biggl( \frac{R}{R_0} \biggr)^{-3(\gamma_g-1)} -~ \delta_{1\gamma_g} B_I \ln \biggl( \frac{R}{R_0} \biggr) +~ C \biggl( \frac{R}{R_0} \biggr)^{-2} +~ D\biggl( \frac{R}{R_0} \biggr)^3 \, ,
</math>
</div>
</div>
where, <math>R_0</math> is an, as yet unspecified, scale length,
that has classically been adopted in the context of the [[User:Tohline/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, <math>~n</math>
Recognizing that, via the [[User:Tohline/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">
<div align="center">
<table border="0" cellpadding="5">
<math>~H\rho = (n+1)P</math> &nbsp;&nbsp;&nbsp;&nbsp;&hellip; and
&hellip; &nbsp;&nbsp;&nbsp;&nbsp; <math>P = K\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 <math>~K</math> and <math>~\rho_c</math> to give,
<div align="center">
<math>~a_n = \biggl[ \frac{(n+1)K}{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>[[User:Tohline/Appendix/References#C67|<font color="red">C67</font>]]</b>] (see, specifically, his equation 10 on p. 87).  Switching from <math>~n</math> 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>
<tr>
   <td align="right">
   <td align="right">
<math>~A</math>
<math>~\biggl( \frac{a_n}{R_\mathrm{norm}} \biggr)^2</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\equiv</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>\frac{3}{5} \frac{GM_\mathrm{tot} ^2}{R_0} \cdot \mathfrak{f}_W \, ,</math>
<math>~\biggl( \frac{\gamma}{\gamma-1} \biggr) \frac{K \rho_c^{(\gamma-2)}}{4\pi G} \cdot \frac{1}{R_\mathrm{norm}^2}</math>
   </td>
   </td>
</tr>
</tr>
Line 453: Line 524:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~B</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\equiv</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~\frac{1}{4\pi}\biggl( \frac{\gamma}{\gamma-1} \biggr) \frac{K }{G} \biggl( \frac{\rho_c}{\bar\rho} \biggr)^{(\gamma-2)}  
\frac{K M_\mathrm{tot} }{(\gamma_g-1)} \biggl( \frac{3M_\mathrm{tot} }{4\pi R_0^3} \biggr)^{\gamma_g - 1} \cdot \mathfrak{f}_A
\biggl( \frac{3M_\mathrm{tot}}{4\pi R_\mathrm{eq}^3} \biggr)^{(\gamma-2)}  
= \frac{\bar{c_s}^2 M_\mathrm{tot} }{(\gamma_g - 1)} \cdot \mathfrak{f}_A \, ,
\cdot \frac{1}{R_\mathrm{norm}^2}
</math>
</math>
   </td>
   </td>
Line 468: Line 539:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~B_I</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\equiv</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<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}
3c_s^2 M_\mathrm{tot}  \cdot \mathfrak{f}_M \, ,
\biggl[ \frac{K 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>
</math>
   </td>
   </td>
Line 482: Line 555:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~C</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\equiv</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<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}
\frac{5J^2}{4M_\mathrm{tot} R_0^2} \cdot \mathfrak{f}_T \, ,
\chi_\mathrm{eq}^{6-3\gamma}
\biggl[ \frac{K M_\mathrm{tot}^{\gamma-2} }{G} \biggr] 
\cdot \biggl[ \biggl( \frac{G}{K} \biggr) M_\mathrm{tot}^{2-\gamma} \biggr]
</math>
</math>
   </td>
   </td>
Line 496: Line 571:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~D</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\equiv</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<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}
\frac{4}{3} \pi R_0^3 P_e \, .
\chi_\mathrm{eq}^{6-3\gamma} \, .
</math>
</math>
   </td>
   </td>
Line 510: Line 585:
</div>
</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 <math>~K</math>) &#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>~R</math>) for a given choice of <math>~\gamma_g</math>.
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>[[User:Tohline/Appendix/References#C67|<font color="red">C67</font>]]</b>] did not encounter this issue. As we have [[User:Tohline/SSC/Structure/Polytropes#Known_Analytic_Solutions|discussed elsewhere]], the equilibrium radius of a polytropic sphere is identified as the radial location,
 
<div align="center">
==Visual Representation==
<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">
<div align="center">
<table border="2" cellpadding="8">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="center" colspan="2">
   <td align="right">
'''Figure 1:''' <font color="darkblue">Free Energy Surface </font>  
<math>~\chi_\mathrm{eq}</math>
  </td>
  <td align="center">
<math>~=</math>
   </td>
   </td>
</tr>
   <td align="left">
<tr>
<math>~\frac{R_\mathrm{eq}}{R_\mathrm{norm}} = \xi_1 \biggl(\frac{a_n}{R_\mathrm{norm}} \biggr) \, ,</math>
   <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 [[User:Tohline/SphericallySymmetricConfigurations/Virial#Bounded_Isothermal|further discussion]], below) &#8212; is, specifically,
<div align="center">
<font size="-1">
<math>
\frac{\mathfrak{G}}{3Mc_s^2} = 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}/(3Mc_s^2)</math>, increases upward.
</td>
  <td align="center" bgcolor="black">
[[File:3DFreeEnergy.jpg|350px|center|Free Energy Surface]]
   </td>
   </td>
</tr>
</tr>
Line 541: Line 606:
</div>
</div>


==Energy Extrema==
we can extend the above analysis to obtain,
As is illustrated in [[User:Tohline/SphericallySymmetricConfigurations/Virial#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. 
<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>


For a given choice of the set of physical parameters <math>M</math>, <math>K</math>, <math>J</math>, <math>P_e</math>, and <math>\gamma_g</math>, extrema occur wherever,
<tr>
<div align="center">
  <td align="right">
<math>
<math>\Rightarrow~~~~~\biggl( \frac{a_n}{R_\mathrm{norm}} \biggr)^{4-3\gamma}</math>
\frac{d\mathfrak{G}}{dR} = 0 \, .
  </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>
</math>
  </td>
</tr>
</table>
</div>
</div>
For the free energy function identified above,  
 
====Implementation====
=====Normalize=====
We will now judiciously introduce our adopted normalizations into the [[User:Tohline/SphericallySymmetricConfigurations/Virial#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">
<div align="center">
<math>
<math>
\frac{d\mathfrak{G}}{dR} = \frac{1}{R_0} \biggl[ A\chi^{-2} +~ (1-\delta_{1\gamma_g})~3(1 - \gamma_g) B\chi^{2 -3\gamma_g} -~ \delta_{1\gamma_g} B_I \chi^{-1} ~ -2C \chi^{-3} +~ 3D\chi^2 \biggr] \, .
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>
</math>
</div>
</div>
where,
 
<font color="red">Normalized Mass:</font>
<div align="center">
<div align="center">
<math>\chi \equiv \frac{R}{R_0}  \, .</math>
<table border="0" cellpadding="5" align="center">
</div>
So <math>\chi_\mathrm{eq} \equiv (R/R_0)_\mathrm{eq}</math> is obtained from the real root(s) of the equation,
<div align="center">
<math>
A \chi^{-2} +~ (1-\delta_{1\gamma_g})~3(1 - \gamma_g) B\chi^{2 -3\gamma_g} -~ \delta_{1\gamma_g} B_I \chi^{-1} ~ -2C \chi^{-3} +~ 3D\chi^2 = 0 \, ,
</math>
</div>
or, equivalently, from the roots of the equation,
<div align="center">
<math>
2C \chi^{-2}  + ~ (1-\delta_{1\gamma_g})~3(\gamma_g-1) B\chi^{3 -3\gamma_g} +~ \delta_{1\gamma_g} B_I ~-~A\chi^{-1}  -~ 3D\chi^3 = 0 \, .
</math>
</div>
As a definition of equilibrium states, this last expression is also the well-known scalar virial equation, derivable from the first moment of the equation of motion.  A more recognizable expression can be obtained by replacing each of the terms by the energy contents that they represent:
<div align="center">
<math>
2(T_\mathrm{rot} + S) + W - 3P_e V = 0 \, .
</math>
</div>
In this expression, <math>S</math> is the thermal energy content of the configuration; the relationship between <math>S</math> and the configuration's total internal energy, <math>U</math>, is provided in our [[User:Tohline/VE#Adiabatic|associated derivation of both the adiabatic and isothermal free energy functions]].


=Examples=
<tr>
==Isolated, Nonrotating Configuration==
  <td align="right">
For a nonrotating configuration <math>(C=J=0)</math> that is not influenced by the effects of a bounding external medium <math>(D=P_e = 0)</math>, the statement of virial equilibrium is,
<math>~M_r(r^*) </math>
<div align="center">
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
<math>
(1-\delta_{1\gamma_g})~3(\gamma_g-1) B\chi^{3 -3\gamma_g} +~ \delta_{1\gamma_g} B_I ~-~A\chi^{-1}   = 0 \, .
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>
</math>
  </td>
</tr>
</table>
</div>
</div>


===Isothermal===
<font color="red">Confinement by External Pressure (Normalized Volume):</font>
For isothermal configurations <math>(\delta_{1\gamma_g} = 1)</math>, one and only one equilibrium state arises where,
<div align="center">
<div align="center">
<math>
<table border="0" cellpadding="5" align="center">
B_I = A\chi^{-1} \, ,
<tr>
</math>
  <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>
</div>
that is,
 
<font color="red">Normalized Gravitational Potential Energy:</font> 
<div align="center">
<div align="center">
<math>
<table border="0" cellpadding="5" align="center">
R_\mathrm{eq} = R_0 \chi_\mathrm{eq} = \frac{A}{B_I}\cdot R_0 = \frac{GM}{5c_s^2} \, .
</math>
</div>


===Adiabatic===
<tr>
For adiabatic configurations <math>(\delta_{1\gamma_g} = 0)</math>, one equilibrium state exists for each value of <math>\gamma_g</math> and it occurs where,
  <td align="right">
<div align="center">
<math>~W_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
<math>
3(\gamma_g-1) B\chi^{3 -3\gamma_g} = A\chi^{-1} \, ,
- 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>
</math>
</div>
  </td>
that is, where,
 
<div align="center">
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
<math>
R_\mathrm{eq} = R_0 \chi_\mathrm{eq} = \biggl[ \frac{3(\gamma_g-1) B}{A} \cdot R_0^{(3\gamma_g-4)} \biggr]^{1/(3\gamma_g-4)} = \biggl[ 5\biggl( \frac{3}{4\pi} \biggr)^{\gamma_g-1} \cdot \frac{KM^{(\gamma_g-2)}}{G} \biggr]^{1/(3\gamma_g-4)} \, .
- E_\mathrm{norm} \int_0^{\chi = R_\mathrm{limit}^*} 3\biggl[\frac{M_r(r^*)}{M_\mathrm{tot}} \biggr]  r^* \rho^* dr^* \, .
</math>
</math>
  </td>
</tr>
</table>
</div>
</div>
Accordingly, the equilibrium mass-radius relationship for adiabatic configurations of a given specific entropy is,
 
<font color="red">Normalized Reservoir of Thermodynamic Energy:</font> 
<div align="center">
<div align="center">
<math>
<table border="0" cellpadding="5" align="center">
M^{(\gamma_g - 2)} \propto R_\mathrm{eq}^{(3\gamma_g -4)} \, .
<tr>
</math>
  <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>
</div>
Notice that, for <math>\gamma_g=2</math>, the equilibrium radius depends only on the specific entropy of the gas and is independent of the configuration's mass.  Conversely, notice that, for <math>\gamma_g = 4/3</math>, the mass of the configuration is independent of the radius.  For <math>\gamma_g</math> &#x3E; <math> 2</math> or <math>\gamma_g </math>&#x3C; <math>4/3</math>, configurations with larger mass (but the same specific entropy) have larger equilibrium radii.  However, for <math>\gamma_g</math> in the range, <math>2</math> &#x3E; <math>\gamma_g </math> &#x3E; <math>4/3</math>, configurations with larger mass have smaller equilibrium radii.  Note that the result obtained for the isothermal configuration could have been obtained by setting <math>\gamma_g = 1</math> in this adiabatic solution, because <math>K = c_s^2</math> when  <math>\gamma_g = 1</math>.
and,
 
It is also instructive to write the coefficient <math>B</math> in terms of the average sound speed as defined above.  In this case,
<div align="center">
<div align="center">
<math>
<table border="0" cellpadding="5" align="center">
R_\mathrm{eq} = R_0 \biggl[ \frac{GM}{5 \bar{c_s}^2 R_0} \biggr]^{1/(4- 3\gamma_g)} \, ,
<tr>
</math>
  <td align="right">
</div>
<math>~\mathfrak{S}_A</math>
so the equilibrium radius of an isolated, nonrotating, uniform density, adiabatic sphere is,
  </td>
<div align="center">
  <td align="center">
<math>
<math>~=</math>
R_\mathrm{eq} = R_0 = \frac{GM}{5 \bar{c_s}^2 } \, .
  </td>
</math>
  <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>
</div>


==Nonrotating Configuration Embedded in an External Medium==
<font color="red">Normalized Rotational Kinetic Energy:</font>
For a nonrotating configuration <math>(C=J=0)</math> that is embedded in, and is influenced by the pressure <math>P_e</math> of, an external medium, the statement of virial equilibrium is,
<div align="center">
<div align="center">
<math>
<table border="0" cellpadding="5" align="center">
(1-\delta_{1\gamma_g})~3(\gamma_g-1) B\chi^{3 -3\gamma_g} +~ \delta_{1\gamma_g} B_I ~-~A\chi^{-1}  -~ 3D\chi^3 = 0 \, .
</math>
</div>


===Bounded Isothermal===
<tr>
For isothermal configurations <math>(\delta_{1\gamma_g} = 1)</math>, we deduce that equilibrium states exist at radii given by the roots of the equation,
  <td align="right">
<div align="center">
<math>~T_\mathrm{rot}</math>
<math>
  </td>
B_I ~-~A\chi^{-1}  -~ 3D\chi^3 = 0 \, .
  <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>
</math>
</div>
  </td>
</tr>


====Bonnor's (1956) Equivalent Relation====
<tr>
Inserting the expressions for the coefficients <math>B_I</math>, <math>A</math>, and <math>D</math> gives,
  <td align="right">
<div align="center">
&nbsp;
<math>
  </td>
3Mc_s^2 ~- \frac{3}{5} \frac{GM^2}{R} = 3 P_e \biggl( \frac{4\pi}{3} R^3\biggr) \, ,
  <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>
</math>
</div>
  </td>
or, because the volume <math>V = (4\pi R^3/3)</math> for a spherical configuration, we can write,
</tr>
<div align="center">
<math>
3P_e V = 3Mc_s^2 ~- \frac{3}{5} \biggl( \frac{4\pi}{3} \biggr)^{1/3} \frac{GM^2}{V^{1/3}}  \, .
</math>
</div>
It is instructive to compare this expression for a self-gravitating, isothermal equilibrium sphere to the one that was presented in 1956 by [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor] (1956, MNRAS, 116, 351) as equation (1.2) in a paper titled, "Boyle's Law and Gravitational Instability":
<div align="center">
<table border="2">
<tr><td>
[[File:Bonnor1951Eq1.2.jpg|600px|center|Bonnor (1956, MNRAS, 116, 351)]]
</td></tr>
</table>
</div>
Once we realize that, for an isothermal configuration, twice the thermal energy content, <math>2S</math>, can be written as <math>(3NkT)</math> just as well as via the product, <math>(3Mc_s^2)</math>, we see that our expression is identical to Bonnor's if we set the prefactor on Bonnor's last term, <math>\alpha = (4\pi/3)^{1/3}/5</math>.  (Indeed, later on the first page of his paper, Bonnor points out that this is the appropriate value for <math>\alpha</math> when considering a uniform density sphere.)


====P-V Diagram====
<tr>
Returning to the dimensionless form of this expression and multiplying through by <math>[-\chi/(3D)]</math>, we obtain,
  <td align="right">
<div align="center">
&nbsp;
<math>
  </td>
\chi^4 - \frac{B_I}{3D} \chi + \frac{A}{3D} = 0 \, .
  <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>
</math>
</div>
  </td>
Now, taking a cue from the solution presented above for an isolated isothermal configuration, we choose to set the previously unspecified scale factor, <math>R_0</math>, to, 
</tr>
<div align="center">
 
<math>
<tr>
R_0 = \frac{GM}{5c_s^2} \, ,
  <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>
</math>
  </td>
</tr>
</table>
</div>
</div>
in which case <math>B_I = A</math>, and the quartic equation governing the radii of equilibrium states becomes, simply,
where,  
<div align="center">
<div align="center">
<math>
<math>\dot\varphi_c \equiv \frac{5J}{2M_\mathrm{tot} R_\mathrm{eq}^2} =
\chi^4 - \frac{\chi}{\Pi} + \frac{1}{\Pi} = 0 \, ,
\frac{5}{2} \biggl[ \frac{J}{M_\mathrm{tot} R_\mathrm{norm}^2} \biggr] \chi_\mathrm{eq}^{-2} \, ,</math>
</math>
</div>
</div>
where,
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">
<div align="center">
<math>
<math>~x \equiv \frac{r}{R_\mathrm{limit}} \, ,~~~~w \equiv \frac{\varpi}{R_\mathrm{limit}} \, ,
\Pi \equiv \frac{3D}{B_I} = \frac{4\pi R_0^3 P_e}{3Mc_s^2} = \frac{4\pi P_e G^3 M^2}{3\cdot 5^3 c_s^8} \, .
~~~~\zeta \equiv \frac{z}{R_\mathrm{limit}} \, ,
</math>
</math>
</div>
</div>
For a given choice of <math>P_e</math> and <math>c_s</math>, <math>\Pi^{1/2}</math> can represent a dimensionless mass, in which case,
that always run from zero to one, we have,
<div align="center">
<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>
<math>
M = \Pi^{1/2} \biggl( \frac{3\cdot 5^3}{2^2\pi}\biggr)^{1/2} \biggl( \frac{c_s^8}{P_e G^3} \biggr)^{1/2} \, .
~x \biggl( \frac{R_\mathrm{limit}}{R_\mathrm{norm}} \biggr) = x \chi \, ;
</math>
</math>
</div>
&nbsp;&nbsp;&nbsp;and, likewise, &nbsp;&nbsp;&nbsp;
Alternatively, for a given choice of configuration mass and sound speed, this parameter, <math>\Pi</math>, can be viewed as a dimensionless external pressure; or, for a given choice of <math>M</math> and <math>P_e</math>, <math>\Pi^{-1/8}</math> can represent a dimensionless sound speed.  In most of what follows we will view <math>\Pi</math> as a dimensionless external pressure. 
 
The above quartic equation can be rearranged immediately to give the external pressure that is required to obtain a particular configuration radius, namely,
<div align="center">
<math>
<math>
\Pi = \frac{(\chi - 1)}{\chi^4} \, .
~~~~\varpi^* ~\rightarrow~ w \chi \, ;
~~~~z^* ~\rightarrow~ \zeta \chi \, ;
</math>
</math>
</div>
  </td>
The resulting behavior is shown by the black curve in Figure 2.
</tr>


<div align="center">
<table border="2" cellpadding="8">
<tr>
<tr>
   <td align="center" colspan="2">
   <td align="right">
'''Figure 2:''' <font color="darkblue">Equilibrium Isothermal P-V Diagram </font>  
<math>~\rho^*</math>
  </td>
  <td align="center">
<math>~\rightarrow~</math>
   </td>
   </td>
</tr>
   <td align="left">
<tr>
   <td valign="top" width=450 rowspan="1">
The black curve traces out the function,
<div align="center">
<math>
<math>
\Pi = (\chi - 1)/\chi^4 \, ,
\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>
</math>
</div>
and shows the dimensionless external pressure, <math>\Pi</math>, that is required to construct a nonrotating, self-gravitating, isothermal sphere with an equilibrium radius <math>\chi</math>.  The pressure becomes negative at radii <math>\chi < 1</math>, hence the solution in this regime is unphysical.
[[User:Tohline/SphericallySymmetricConfigurations/Virial#Visual_Representation|Figure 1]] displays the free energy surface that "lies above" the two-dimensional parameter space (<math>1.2 < \chi < 1.51</math>; <math>0.103 < \Pi < 0.104</math>) that is identified here by the thin, red rectangle.
</td>
  <td align="center" bgcolor="white">
[[File:Bonnor1956Fig1.jpg|450px|center|Equilibrium P-R Diagram]]
   </td>
   </td>
</tr>
</tr>


</table>
<tr>
</div>
In the absence of self-gravity (''i.e.,'' <math>A=0</math>), the product of the external pressure and the volume should be constant.  The corresponding relation, <math>\Pi = \chi^{-3}</math>, is shown by the blue dashed curve in the figure.  As the figure illustrates, when gravity is included the P-V relationship pulls away from the PV = constant curve at sufficiently small volumes.  Indeed, the curve turns over at a finite pressure, <math>\Pi_\mathrm{max}</math>, and for every value of <math>\Pi < \Pi_\mathrm{max}</math> a second, more compact equilibrium configuration appears.  The location of <math>\Pi_\mathrm{max}</math> along the curve is identified by setting <math>\partial\Pi/\partial\chi = 0</math>, that is, it occurs where,
<div align="center">
<math>
\frac{\partial\Pi}{\partial\chi} = -4 \chi^{-5}(\chi - 1) + \chi^{-4} = 0 \, ,
</math>
 
<math>
\Rightarrow ~~~~~ \chi = \frac{2^2}{3} \approx 1.333333 \, .
</math>
</div>
<span id="BonnorEbertMass">Hence,</span>
<div align="center">
<math>\Pi_\mathrm{max} = \biggl( \frac{2^2}{3} \biggr)^{-4} \biggl( \frac{2^2}{3}-1 \biggr) = \frac{3^3}{2^8} \approx 0.105469\, ;</math>
</div>
therefore, from above,
<div align="center">
<math>
M_\mathrm{max} = \biggl( \frac{3^4\cdot 5^3}{2^{10}\pi}\biggr)^{1/2} \biggl( \frac{c_s^8}{P_e G^3} \biggr)^{1/2}
\approx 1.77408 \biggl( \frac{c_s^8}{P_e G^3} \biggr)^{1/2} \, .
</math>
</div>
 
====Quartic Solution====
In the above <math>P-V</math> diagram discussion, we rearranged the quartic equation governing equilibrium configurations to give <math>\Pi</math> for any chosen value of <math>\chi</math>.  Alternatively, the four roots of the quartic equation &#8212; <math>\chi_1</math>, <math>\chi_2</math>, <math>\chi_3</math> and <math>\chi_4</math> in the presentation that follows &#8212; will identify the radii at which a spherical configuration will be in equilibrium for any choice of the external pressure, <math>\Pi</math>, assuming the roots are real. 
<div align="center">
<table border="1" cellpadding="10" bgcolor="darkblue">
<tr>
  <td align="center" bgcolor="lightblue">
Roots of the quartic equation:  <math>\chi^4 - \chi \Pi^{-1}+ \Pi^{-1} = 0 </math>
  </td>
</tr>
 
<tr><td>
 
<div align="center">
<table border="0" cellpadding="3">
<tr>
   <td align="right">
   <td align="right">
<math>\chi_1</math>
<math>~P^*</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>=</math>
<math>~\rightarrow~</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>
+\frac{1}{2} y_r^{1/2} + \frac{1}{2} D_q \, ;
\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>
</math>
   </td>
   </td>
Line 795: Line 897:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>\chi_2</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>=</math>
&nbsp;&nbsp;&nbsp;&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>
+\frac{1}{2} y_r^{1/2} - \frac{1}{2} D_q \, ;
= \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>
</math>
   </td>
   </td>
Line 809: Line 915:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>\chi_3</math>
<math>~\frac{\dot\varphi}{\dot\varphi_c}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>=</math>
<math>~\rightarrow~</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>
-\frac{1}{2} y_r^{1/2} + \frac{1}{2} E_q \, ;
\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>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>


<tr>
=====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">
   <td align="right">
<math>\chi_4</math>
<math>~\frac{M_r(x)}{M_\mathrm{tot}}  </math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>=</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~ \biggl( \frac{\rho_c}{\bar\rho} \biggr)_\mathrm{eq} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)
-\frac{1}{2} y_r^{1/2} - \frac{1}{2} E_q \, ,
\int_0^{x}  3x^2 \biggl[ \frac{\rho(x)}{\rho_c} \biggr]  dx \, ,</math>
</math>
   </td>
   </td>
</tr>
</tr>


</table>
<tr>
</div>
  <td align="right">
where,
<math>~\frac{P_e V}{E_\mathrm{norm}}</math>
<div align="center">
  </td>
<table border="0" cellpadding="3">
  <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>
<tr>
   <td align="right">
   <td align="right">
<math>D_q</math>
<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>\equiv</math>
<math>~=</math>
   </td>
   </td>
  <td align="left">
<td align="left">
<math>
<math>
y_r^{1/2} \biggl[ \frac{2}{\Pi} y_r^{-3/2} - 1 \biggr]^{1/2}  \, ,
- \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>
</math>
   </td>
   </td>
Line 857: Line 984:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>E_q</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>\equiv</math>
<math>~=</math>
   </td>
   </td>
  <td align="left">
<td align="left">
<math>
<math>
y_r^{1/2} \biggl[ - \frac{2}{\Pi} y_r^{-3/2} - 1 \biggr]^{1/2}  \, ,
- \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>
</math>
   </td>
   </td>
</tr>
</tr>


</table>
<tr>
</div>
  <td align="right">
and,
<math>~\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
<div align="center">
  </td>
<math>
  <td align="center">
y_r \equiv \biggl( \frac{1}{2\Pi^2} \biggr)^{1/3} \biggl\{ \biggl[ 1 + \sqrt{1-\frac{2^8}{3^3}\Pi} \biggr]^{1/3} + \biggl[ 1 - \sqrt{1-\frac{2^8}{3^3}\Pi} \biggr]^{1/3} \biggr\} \, ,
<math>~=</math>
</math>
  </td>
</div>
  <td align="left">
is the real root of the cubic equation,
<math>~\frac{4\pi}{3({\gamma_g}-1)} \cdot \chi^{3-3\gamma}
<div align="center">
\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
<math>
\int_0^{1} 3x^2 \biggl[ \frac{P(x)}{P_c} \biggr]  dx \biggr\} \, ,</math>
y^3 - \frac{4y}{\Pi} - \frac{1}{\Pi^{2}} = 0 \, .
  </td>
</math>
</tr>
</div>
 
</td></tr>
</table>
</div>
 
Because <math>\Pi</math> must be positive in physically realistic solutions, we conclude that the two roots involving <math>E_q</math> &#8212; that is, <math>\chi_3</math> and <math>\chi_4</math> &#8212; are imaginary and, hence, unphysical.  The other two roots  &#8212; <math>\chi_1</math> and <math>\chi_2</math> &#8212; will be real only if the arguments inside the radicals in the expression for <math>y_r</math> are positive.  That is, <math>\chi_1</math> and <math>\chi_2</math> will be real only for values of the dimensionless external pressure,
<div align="center">
<math>\Pi \leq \Pi_\mathrm{max} \equiv \frac{3^3}{2^8} \, .</math>
</div>
This is the same upper limit on the external pressure that was derived above, via a different approach.
 
When combined, a plot of <math>\chi_1</math> versus <math>\Pi</math> and <math>\chi_2</math> versus <math>\Pi</math> will reproduce the solid black curve shown in Figure 2, but with the axes flipped. The top-right quadrant of Figure 3 presents such a plot, but in logarithmic units along both axes; also <math>\Pi</math> is normalized to <math>\Pi_\mathrm{max}</math> and <math>\chi</math> is normalized to the equilibrium radius <math>(4/3)</math> at that pressure. This is the manner in which [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth] (1981, MNRAS, 195, 967) chose to present this result for uniform-density, spherical isothermal <math>(\gamma_\mathrm{g}=1)</math> configurations.  Our solid and dashed curve segments &#8212; identifying, respectively, the <math>\chi_1(\Pi)</math> and <math>\chi_2(\Pi)</math> solutions to the above quadratic equation &#8212; precisely match the solid and dashed curve segments labeled "1" in Whitworth's Figure 1a (replicated here in the bottom-right quadrant of Figure 3).


<div align="center">
<table border="2" cellpadding="8">
<tr>
<tr>
   <td align="center" colspan="2">
   <td align="right">
'''Figure 3:''' <font color="darkblue">Equilibrium R-P Diagram </font>  
<math>~\frac{\mathfrak{S}_I}{E_\mathrm{norm}}</math>
   </td>
   </td>
</tr>
  <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>
<tr>
   <td valign="top" width=450 rowspan="2">
   <td align="right">
''Top:'' The solid curve traces the function <math>\chi_1(\Pi)</math> and the dashed curve traces the function <math>\chi_2(\Pi)</math>, where <math>\chi_1</math> and <math>\chi_2</math> are the two real roots of the quartic equation,
&nbsp;
<div align="center">
  </td>
<math>
  <td align="center">
\chi^4 - \frac{\chi}{\Pi} + \frac{1}{\Pi} = 0 \, .
<math>~=</math>
  </td>
  <td align="left">
<math>~-3 \ln \chi + \mathrm{constant} \, ,
</math>
</math>
</div>
Logarithmic units are used along both axes; <math>\Pi</math> is normalized to <math>\Pi_\mathrm{max}</math>; and <math>\chi</math> is normalized to the equilibrium radius <math>(4/3)</math> at <math>\Pi_\mathrm{max}</math>.
''Bottom:'' A reproduction of Figure 1a from [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth] (1981, MNRAS, 195, 967).  The solid and dashed segments of the curve labeled "1" identify the equilibrium radii, <math>R_\mathrm{eq}</math>, that result from embedding a uniform-density, isothermal <math>(\gamma_\mathrm{g} = 1)</math> gas cloud in an external medium of pressure <math>P_\mathrm{ex}</math>. 
''Comparison:'' The curve shown above that traces out <math>\chi_1(\Pi)</math> and <math>\chi_2(\Pi)</math> should be identical to the "Whitworth" curve labeled "1".
</td>
  <td align="center" bgcolor="white">
[[File:WhitworthLogFig1a_norm.jpg|450px|center|To be compared with Whitworth (1981)]]
   </td>
   </td>
</tr>
</tr>
<tr>
<tr>
   <td align="center" bgcolor="white">
  <td align="right">
[[File:WhitworthFig1aCopy.jpg|450px|center|Whitworth (1981) Figure 1a]]
<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>
   </td>
</tr>
</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 ratio of M_limit to M_tot.]
</td></tr>
</table>
</table>
</div>
</div>


===Bounded Adiabatic===
 
For adiabatic configurations <math>(\delta_{1\gamma_g} = 0)</math>, equilibrium states exist at radii given by the roots of the following expression:
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">
<div align="center">
<math>
<table border="0" cellpadding="5" align="center">
3(\gamma_g-1) B\chi^{3 -3\gamma_g} ~-~A\chi^{-1}  -~ 3D\chi^3 = 0 \, .
</math>
</div>


====Whitworth's (1981) Equivalent Relation====
<tr>
This is precisely the same condition that derives from setting equation (3) to zero in [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth's] (1981, MNRAS, 195, 967) discussion of the "global gravitational stability for one-dimensional polytropes."  The overlap with Whitworth's narative is perhaps clearer after introducing the algebraic expressions for the coefficients <math>A</math>, <math>B</math>, and <math>D</math>, dividing the equation through by <math>(3\chi^3 V_0) = (4\pi R^3)</math>, and rewriting <math>R</math> as <math>R_\mathrm{eq}</math> to obtain,
  <td align="right">
<div align="center">
<math>~\biggl(\frac{\rho_c}{\bar\rho} \biggr)^{-1}_\mathrm{eq} </math>
<math>
  </td>
P_e = K \biggl( \frac{3M}{4\pi R_\mathrm{eq}^3} \biggr)^{\gamma_g} - \biggl( \frac{3GM^2}{20\pi R_\mathrm{eq}^4} \biggr) \, .
  <td align="center">
</math>
<math>~=</math>
</div>
  </td>
This exactly matches equation (5) of [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth], which reads:
  <td align="left">
<div align="center">
<math>~ \int_0^{1}  3x^2 \biggl[ \frac{\rho(x)}{\rho_c} \biggr]_\mathrm{eq}  dx \, .</math>
<table border="2">
  </td>
<tr><td>
</tr>
[[File:Whitworth1981Eq5.jpg|500px|center|Whitworth (1981, MNRAS, 195, 967)]]
</td></tr>
</table>
</table>
</div>
</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,
====P-V Diagram====
Returning to the dimensionless form of this expression and multiplying through by <math>[-\chi/(3D)]</math>, we obtain,
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>
<math>
\chi^4 - (\gamma_g - 1)\frac{B}{D} \chi^{4-3\gamma_g} + \frac{A}{3D} = 0 \, .
\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>
</math>
</div>
  </td>
Writing the coefficient, <math>B</math>, in terms of the average sound speed and setting the radial scale factor equal to the equilibrium radius of an isolated adiabatic sphere, that is, setting,
  <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">
<div align="center">
<math>
<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>
R_0 = \frac{GM}{5\bar{c_s}^2} \, ,
</math>
</div>
</div>
the equation governing the radii of ''adiabatic'' equilibrium states becomes,  
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">
<div align="center">
<math>
<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>
\chi^4 - \frac{1}{\Pi_a} \chi^{(4-3\gamma_g)} + \frac{1}{\Pi_a} = 0 \, ,
</math>
</div>
</div>
where,
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">
<div align="center">
<math>
<math>~P_{ic} = P_0 [1 - p_c(q)] \, .</math>
\Pi_a \equiv \frac{4\pi P_e G^3 M^2}{3\cdot 5^3 \bar{c_s}^8} \, .
</math>
</div>
</div>
As in the isothermal case, for a given choice of configuration mass and sound speed, this parameter, <math>\Pi_a</math>, can be viewed as a dimensionless external pressure.  Alternatively, for a given choice of <math>P_e</math> and <math>\bar{c_s}</math>, <math>\Pi_a^{1/2}</math> can represent a dimensionless mass; or, for a given choice of <math>M</math> and <math>P_e</math>, <math>\Pi_a^{-1/8}</math> can represent a dimensionless sound speed.  Here we will view it as a dimensionless external pressure. 


Unlike the isothermal case, for an arbitrary value of the adiabatic exponent, <math>\gamma_g</math>, it isn't possible to invert this equation to obtain an analytic expression for <math>\chi</math> as a function of <math>\Pi_a</math>. But we can straightforwardly solve for <math>\Pi_a</math> as a function of <math>\chi</math>.  The solution is,
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">
<div align="center">
<math>
<math>~P_\mathrm{core}(x) = P_{ic} \biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr] \, .</math>  
\Pi_a = \frac{\chi^{(4- 3\gamma_g)} - 1}{\chi^4} \, .
</math>
</div>
</div>
For physically relevant solutions, both <math>\chi</math> and <math>\Pi_a</math> must be nonnegative.  Hence, as is illustrated by the curves in Figure 4, the physically allowable range of equilibrium radii is,
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">
<div align="center">
<math>
<table border="0" cellpadding="5" align="center">
1 \le \chi \le \infty \, ~~~~~\mathrm{for}~ \gamma_g < 4/3 \, ;
</math>


<math>
<tr>
0 < \chi \le 1 \, ~~~~~~\mathrm{for}~ \gamma_g > 4/3 \, .
  <td align="right">
</math>
<math>~\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
</div>
  </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>


<div align="center">
<table border="2" cellpadding="8">
<tr>
<tr>
   <td align="center" colspan="2">
   <td align="right">
'''Figure 4:''' <font color="darkblue">Equilibrium Adiabatic P-V Diagram </font>  
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
   </td>
   </td>
</tr>
   <td align="left">
<tr>
   <td valign="top" width=450 rowspan="1">
The curves trace out the function,
<div align="center">
<math>
<math>
\Pi_a = (\chi^{4-3\gamma_g} - 1)/\chi^4 \, ,
\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>
</math>
</div>
for six different values of <math>\gamma_g</math> (<math>2, ~5/3, ~7/5, ~6/5, ~1, ~2/3</math>, as labeled)  and show the dimensionless external pressure, <math>\Pi_a</math>, that is required to construct a nonrotating, self-gravitating, uniform density, adiabatic sphere with an equilibrium radius <math>\chi</math>.  The mathematical solution becomes unphysical wherever the pressure becomes negative.
The solid red curve, drawn for the case <math>\gamma_g = 1</math>, is identical to the solid black (isothermal) curve displayed above in Figure 1.
</td>
  <td align="center" bgcolor="white">
[[File:AdabaticBoundedSpheres_Virial.jpg|450px|center|Equilibrium Adiabatic P-R Diagram]]
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</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,
Each of the <math>\Pi_a(\chi)</math> curves drawn in Figure 4 exhibits an extremumIn each case this extremum occurs at a configuration radius, <math>\chi_\mathrm{extreme}</math>, given by,
<div align="center">
<div align="center">
<math>
<table border="0" cellpadding="5" align="center">
\frac{\partial\Pi_a}{\partial\chi} = 0 \, ,
<tr>
</math>
  <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>
</div>
that is, where,
and,
<div align="center">
<div align="center">
<math>
<table border="0" cellpadding="5" align="center">
4 - 3\gamma_g \chi^{4-3\gamma_g} = 0 ~~~~\Rightarrow ~~~~~ \chi_\mathrm{extreme} = \biggl[ \frac{4}{3\gamma_g} \biggr]^{1/(4-3\gamma_g)} \, .
<tr>
</math>
  <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>
</div>
For each value of <math>\gamma_g</math>, the corresponding dimensionless pressure is,
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">
<div align="center">
<math>
<table border="0" cellpadding="5" align="center">
\Pi_a \biggr|_\mathrm{extreme} = \biggl(\frac{4}{3\gamma} - 1 \biggr) \biggl[ \frac{3\gamma_g}{4} \biggr]^{4/(4-3\gamma_g)} \, .
<tr>
</math>
  <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>
</div>
 
while material associated with the envelope that lies at the interface will have a pressure given by the relation,
Note, first, that for <math>\gamma_g > 4/3</math>, an equilibrium configuration with a positive radius can be constructed for all physically realistic &#8212; that is, for all positive &#8212; values of <math>\Pi_a</math>.  Also, consistent with the behavior of the curves shown in Figure 4, the extremum arises in the regime of physically relevant &#8212; ''i.e.,'' positive &#8212; pressures only for values of <math>\gamma_g < 4/3</math>; and in each case it represents a ''maximum'' limiting pressure.
 
====Maximum Mass====
=====Isothermal=====
When <math>\gamma_a = 1</math> we retrieve the maximum pressure that was derived above for the special case of an isothermal configuration, namely,
<div align="center">
<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>
<math>
\Pi_\mathrm{max} = \Pi_a\biggr|_\mathrm{extreme}^{(\gamma_g = 1)} = \frac{3^3}{2^8} \, .
(P_{ie})_\mathrm{eq} \biggl( \frac{R_\mathrm{edge}}{R_\mathrm{eq}} \biggr)^{-3\gamma_e}
</math>
= (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>
</div>
This translates into a maximum mass for a pressure-bounded isothermal configuration of,
Hence,
<div align="center">
<div align="center">
<math>M_\mathrm{max} = \Pi_\mathrm{max}^{1/2}  \biggl(\frac{3\cdot 5^3}{2^2\pi} \biggr)^{1/2} \biggl( \frac{c_s^8}{G^3 P_e} \biggr)^{1/2}
<table border="0" cellpadding="5" align="center">
= \biggl(\frac{3^4\cdot 5^3}{2^{10}\pi} \biggr)^{1/2} \biggl( \frac{c_s^8}{G^3 P_e} \biggr)^{1/2} \, .</math>
</div>


=====<math>n=5</math> Polytropic=====
<tr>
When <math>\gamma_a = 6/5</math> &#8212; which corresponds to an <math>n=5</math> polytropic configuration &#8212; we obtain,
  <td align="right">
<div align="center">
<math>~\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
<math>
\Pi_\mathrm{max} = \Pi_a\biggr|_\mathrm{extreme}^{(\gamma_g = 6/5)}  
\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( \frac{3^{18}}{2^{10}\cdot 5^{10}} \biggr) \, ,
\int_0^q  \biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr]  x^2 dx
</math>
</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>
</div>
which corresponds to a maximum mass for pressure-bounded <math>n=5</math> polytropic configurations of,
----
<div align="center">
<math>M_\mathrm{max} = \Pi_\mathrm{max}^{1/2}  \biggl(\frac{3\cdot 5^3}{2^2\pi} \biggr)^{1/2} \biggl( \frac{\bar{c_s}^8}{G^3 P_e} \biggr)^{1/2}
= \biggl(\frac{3^{19}}{2^{12}\cdot 5^{7}\pi} \biggr)^{1/2} \biggl( \frac{\bar{c_s}^8}{G^3 P_e} \biggr)^{1/2} \, .</math>
</div>
This result can be compared to [[User:Tohline/SSC/Structure/LimitingMasses#Bounded_Isothermal_Sphere_.26_Bonnor-Ebert_Mass|other determinations of the Bonnor-Ebert mass limit]].


{{LSU_WorkInProgress}}


=BiPolytrope=
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,
[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.
<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 will adopt the following approach:
We therefore obtain,
* 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">
<div align="center">
<math>\xi \equiv \frac{r}{r_i}</math>.
<table border="0" cellpadding="5" align="center">
</div>
Then, <math>\xi_i = 1</math> and <math>\xi_s \equiv R/r_i</math>.


==Expressions for Mass==
<tr>
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, the mass of the core, the mass of the envelope, and the total mass are, respectively,
<div align="center">
<math>M_\mathrm{core} = \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]</math>
&nbsp;&nbsp;<math>\Rightarrow</math> &nbsp;&nbsp;
<math>\frac{\rho_c}{\rho_0} = \frac{M_\mathrm{core}}{M_0} \biggl( \frac{r_i}{R_0}\biggr)^{-3}</math>  ;
 
<math>M_\mathrm{env} = \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]</math>
&nbsp;&nbsp;<math>\Rightarrow</math> &nbsp;&nbsp; 
<math>\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> ;
 
<math>M_\mathrm{tot} = \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> ;
 
</div>
where, <math>M_0 \equiv 4\pi \rho_0 R_0^3/3</math>.  Letting <math>\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math> &#8212;
which also means, <math>M_\mathrm{env}/M_\mathrm{tot} = (1-\nu) </math> &#8212; 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{(1-\nu)}{\nu (\xi_s^3 - 1)} </math> ,
</div>
and,
<div align="center">
<math>\nu (\xi_s^3 - 1) \biggl( \frac{\rho_e}{\rho_c} \biggr)  =  (1-\nu) </math>
&nbsp;&nbsp; <math>\Rightarrow</math> &nbsp;&nbsp; <math>\nu = \biggl[ 1 + \biggl( \frac{\rho_e}{\rho_c} \biggr) (\xi_s^3 - 1) \biggr]^{-1}</math> .
</div>
 
Following the work of [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Sch&ouml;nberg &amp; Chandrasekhar (1942)] &#8212; see [[User:Tohline/SSC/Structure/LimitingMasses#Sch.C3.B6nberg-Chandrasekhar_Mass|our accompanying discussion]]  &#8212; we are seeking equilibrium configurations in the <math>\nu - q</math> plane where,
<table align="center" border="0" cellpadding="10">
<tr>
   <td align="right">
   <td align="right">
<math>\nu</math>
<math>~\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>\equiv</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>\frac{M_\mathrm{core}}{M_\mathrm{tot}} </math>, &nbsp; &nbsp;&nbsp; (as also defined here)
<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>
   </td>
</tr>
</tr>
Line 1,148: Line 1,306:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>q</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>\equiv</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>\frac{r_i}{R} = \frac{1}{\xi_s}</math> .
<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>
   </td>
</tr>
</tr>
</table>
 
We also will be examining the stability of configurations, looking for extrema in the free energy plane where <math>\nu</math> is allowed to vary while holding <math>q</math> fixed.  According to the above relations, this can be accomplished by varying the ratio, <math>\rho_e/\rho_c</math>.  Notice that this can also be viewed as follows:  Changing the ratio of densities, <math>\rho_e/\rho_c</math>, while holding <math>q</math> fixed will cause <math>\nu</math> to vary.  If we want to impose this perturbation on a configuration of constant total mass, then an additional contraint on the choice of <math>\rho_e</math> and <math>\rho_c</math> is,
<div align="center">
<math>\rho_c|_0  + \rho_e|_0 (\xi_s^3 - 1) = \rho_c|_\mathrm{new}  + \rho_e|_\mathrm{new}(\xi_s^3 - 1) </math>
 
<math>\Rightarrow</math> &nbsp; &nbsp;
<math>\rho_c|_0  - \rho_c|_\mathrm{new}  =  \rho_e|_\mathrm{new}(\xi_s^3 - 1) - \rho_e|_0 (\xi_s^3 - 1) </math>
 
<math>\Rightarrow</math> &nbsp; &nbsp;
<math>\rho_c|_0 ( 1 - f_c ) =  \rho_e|_0 (\xi_s^3 - 1) \biggl[ f_e - 1 \biggr]</math>
 
<math>\Rightarrow</math> &nbsp; &nbsp;
<math>f_e  = 1 + \biggl[ \frac{( 1 - f_c )}{ (\xi_s^3 - 1) } \biggr] \biggl( \frac{\rho_c}{\rho_e} \biggr)_0 \, ,</math>
</div>
 
where, <math>f_c \equiv \rho_c|_\mathrm{new}/ \rho_c|_0</math> and <math>f_e \equiv \rho_e|_\mathrm{new}/ \rho_e|_0</math>.  This also means that, for a given ''initial''
choice of the density ratio <math>(\rho_e/\rho_c)</math> and the factor, <math>f_c</math>, by which you want the core density to increase or decrease, the new density ratio will be,
<table align="center" border="0" cellpadding="10">
 
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>
&nbsp;
\biggl( \frac{\rho_e}{\rho_c} \biggr)_\mathrm{new}
  </td>
</math>
  <td align="center">
<math>~=</math>
   </td>
   </td>
  <td align="center"><math>=</math></td>
   <td align="left">
   <td align="left">
<math>
<math>
\frac{f_e}{f_c} \biggl( \frac{\rho_e}{\rho_c} \biggr)_0
\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>
</math>
   </td>
   </td>
</tr>
</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>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~\frac{M_r(x)}{M_\mathrm{tot} }  </math>
  </td>
  <td align="center">
<math>~=</math>
   </td>
   </td>
  <td align="center"><math>=</math></td>
   <td align="left">
   <td align="left">
<math>
<math>~ \int_0^{x} 3x^2  dx = x^3 \, ,</math>
\frac{1}{f_c} \biggl\{ \biggl( \frac{\rho_e} {\rho_c}\biggr)_0 + \biggl[ \frac{( 1 - f_c )}{ (\xi_s^3 - 1) } \biggr] \biggr\} \, .
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
 
</div>
 
and the normalized gravitational potential energy is,
A variation in <math>\nu</math> will imply that the mass contained in the core will vary, since <math>\nu = M_\mathrm{core}/M_\mathrm{tot}</math>.  The particular algebraic relation makes sense because, a decrease in the ratio <math>\rho_e/\rho_c</math> will mean that <math>\nu</math> increases, which also means that a relatively larger fraction of the mass is in the core.
 
==Energy Expressions==
 
The gravitational potential energy of the bipolytropic configuration is obtained by integrating over the following differential energy contribution,
<div align="center">
<div align="center">
<math>dW = - \biggl( \frac{GM_r}{r} \biggr) dm</math> .
<table border="0" cellpadding="8" align="center">
</div>
 
Hence,
<table border="0" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>W = W_\mathrm{core} + W_\mathrm{env}</math>
<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm} }</math>
   </td>
   </td>
   <td align="left">
   <td align="center">
<math>~=</math>
  </td>
<td align="left">
<math>
<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\}
- \frac{3}{5} \chi^{-1} \int_0^{1} 5x \biggl\{ x^3\biggr\} dx = -\frac{3}{5} \chi^{-1} \, .
</math>
</math>
   </td>
   </td>
</tr>
</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>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~\frac{T_\mathrm{rot}}{E_\mathrm{norm} }</math>
  </td>
  <td align="center">
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~ 2\chi^{-2}
= - 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
\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_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\}
\int_0^{1} w^3 dw
</math>
\int_{0}^{\sqrt{1 - w^2}}  d\zeta
</math>
   </td>
   </td>
</tr>
</tr>
Line 1,242: Line 1,403:
   <td align="right">
   <td align="right">
&nbsp;
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~ \chi^{-2}
= - \frac{3GM^2_\mathrm{core}}{r_i} \biggl\{ \int_0^\xi^4 d\xi
\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_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\}
\int_0^1 w^3 (1-w^2)^{1/2} dw 
</math>
</math>
   </td>
   </td>
Line 1,254: Line 1,418:
   <td align="right">
   <td align="right">
&nbsp;
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>\chi^{-2}
= - \frac{3GM^2_\mathrm{core}}{r_i} \biggl\{ \frac{1}{5}  
\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{\rho_e}{\rho_c} \biggr) \int_1^{\xi_s} \xi d\xi
\biggl[ -\frac{1}{15} (1-w^2)^{3/2} (3w^2 +2) \biggr]_0^1 
+ \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \int_1^{\xi_s} (\xi^3 - 1) \xi d\xi \biggr\}
</math>
</math>
   </td>
   </td>
Line 1,267: Line 1,433:
   <td align="right">
   <td align="right">
&nbsp;
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~ \chi^{-2}
= - \biggl( \frac{GM^2_\mathrm{tot}}{R} \biggr) 3\nu^2 \xi_s \biggl\{ \frac{1}{5}  
\biggl( \frac{3\cdot 5}{2^4 \pi} \biggr) \biggl\frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr]
+ \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>
</math>
   </td>
   </td>
</tr>
</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>.
The internal energy of the bipolytropic configuration is,
<table border="0" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>
<math>~\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
U = U_\mathrm{core} + U_\mathrm{env}
  </td>
</math>
  <td align="center">
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>=
<math>~\frac{4\pi }{3({\gamma_g}-1)} \cdot \chi^{3-3\gamma}
\frac{2}{3}\biggl\{ \frac{S_\mathrm{core}}{(\gamma_c-1)} + \frac{S_\mathrm{env}}{(\gamma_e-1)} \biggr\}  
\biggl\{ \biggl(\frac{3}{4\pi} \biggr)^{\gamma}\int_0^{1} 3x^2  dx \biggr\}
</math>
= \frac{1}{({\gamma_g}-1)} \biggl(\frac{3}{4\pi} \biggr)^{\gamma-1} \chi^{3-3\gamma}
\, ,</math>
   </td>
   </td>
</tr>
</tr>
Line 1,299: Line 1,462:
<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~\frac{\mathfrak{S}_I}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>=
<math>~-3 \ln \chi + \mathrm{constant} \, ,
\biggl\{ n_c M_\mathrm{core} K_c \rho_c^{1/n_c} + n_e M_\mathrm{env} K_e \rho_e^{1/n_e}  \biggr\} \, .
</math>
</math>
   </td>
   </td>
</tr>
</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>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
   </td>
   </td>
  <td align="left">
<td align="left">
<math>=
<math>
M_\mathrm{tot} \biggl\{ n_c \nu K_c \rho_c^{1/n_c} + n_e (1-\nu) K_e \rho_e^{1/n_e} \biggr\} \, .
- \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>
</math>
   </td>
   </td>
Line 1,321: Line 1,497:
<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~\frac{T_\mathrm{rot}}{E_\mathrm{norm}}</math>
  </td>
  <td align="center">
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>=
<math>~
M_\mathrm{tot} \biggl\{ n_c \nu K_c \biggl[ \biggl( \frac{3M_\mathrm{tot}}{4\pi} \biggr) \nu r_i^{-3} \biggr]^{1/n_c} +
\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} \, ,
n_e (1-\nu) K_e \biggl[ \biggl( \frac{3M_\mathrm{tot}}{4\pi} \biggr) (1-\nu)(\xi_s^3-1)^{-1} r_i^{-3} \biggr]^{1/n_e} \biggr\} \, .
</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
Now, try to write this in such a way that the pressure in the envelope equals the pressure in the core (uniform pressure configuration) even as the overall radius of the configuration is varied in order to search for the equilibrium configuration.  That is, set
<table align="center"  border="0" cellpadding="8">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>K_c \rho_c^{1+1/n_c} </math>
<math>~\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
=
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>K_e \rho_e^{1+1/n_e}</math>
<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>
   </td>
</tr>
</tr>
Line 1,350: Line 1,525:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>\Rightarrow ~~~~~~~~ \frac{K_e}{K_c}</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
=
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>\frac{\rho_c^{1+1/n_c}}{\rho_e^{1+1/n_e}}</math>
<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>
   </td>
</tr>
</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>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~\mathfrak{f}_M </math>
   </td>
   </td>
   <td align="center">
   <td align="center">
=
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>\biggl[ \biggl( \frac{3M_\mathrm{tot}}{4\pi} \biggr) \nu r_i^{-3} \biggr]^{1+1/n_c} \biggl[ \biggl( \frac{3M_\mathrm{tot}}{4\pi}  
<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>
\biggr) (1-\nu)(\xi_s^3-1)^{-1} r_i^{-3} \biggr]^{-1-1/n_e} \, .</math>
   </td>
   </td>
</tr>
</tr>
</table>
Hence, the expression for the internal energy becomes,
<table border="0" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>
<math>~\mathfrak{f}_W</math>
U
  </td>
</math>
  <td align="center">
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>=
<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>
K_c M_\mathrm{tot} \biggl\{ n_c \nu \biggl[ \biggl( \frac{3M_\mathrm{tot}}{4\pi} \biggr) \nu r_i^{-3} \biggr]^{1/n_c} +
n_e (1-\nu) \biggl[ \biggl( \frac{3M_\mathrm{tot}}{4\pi} \biggr) \nu r_i^{-3} \biggr]^{1+1/n_c} \biggl[ \biggl( \frac{3M_\mathrm{tot}}{4\pi}
\biggr) (1-\nu)(\xi_s^3-1)^{-1} r_i^{-3} \biggr]^{-1}
\biggr\} \, .
</math>
   </td>
   </td>
</tr>
</tr>
Line 1,395: Line 1,568:
<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;  </td>
<math>~\mathfrak{f}_T</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
   <td align="left">
   <td align="left">
<math>=
<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}
K_c M_\mathrm{tot} \nu \biggl[ \biggl( \frac{3M_\mathrm{tot}}{4\pi} \biggr) \nu r_i^{-3} \biggr]^{1/n_c}  
\biggl[ \frac{\rho(w,\zeta)}{\rho_c} \biggr] d\zeta\, ,</math>
\biggl\{ n_c  + n_e (\xi_s^3-1) \biggr\}
</math>
   </td>
   </td>
</tr>
</tr>
Line 1,406: Line 1,581:
<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;  </td>
<math>~\mathfrak{f}_A</math>
  <td align="left">
<math>=
K_c M_\mathrm{tot} \biggl( \frac{3M_\mathrm{tot}}{4\pi} \biggr)^{1/n_c} R^{-3/n_c}
[ n_c  + n_e (\xi_s^3-1) ] \nu^{1+1/n_c} \xi_s^{3/n_c}
</math>
   </td>
   </td>
</tr>
   <td align="center">
</table>
<math>~\equiv</math>
 
==Virial Analysis==
Employing the above derived expressions, the free energy may be written as,
<div align="center">
<math>
\mathfrak{G} = W + U  = - A\biggl( \frac{R}{R_0} \biggr)^{-1} + B \biggl( \frac{R}{R_0} \biggr)^{-3/n_c} \, ,
</math>
</div>
where, <math>R_0</math> is a, as yet unspecified, scale length, and,
 
<table border="0" align="center">
 
<tr>
  <td align="right">
<math>
A
</math> 
  </td>
   <td align="center">
<math>=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~ \int_0^1 3\biggl[ \frac{P(x)}{P_c}\biggr]  x^2 dx \, .</math>
\biggl( \frac{3GM^2_\mathrm{tot}}{5R_0} \biggr) \nu^2 \xi_s \biggl\{ 1  
+ \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (\xi_s^2 - 1)
+ \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ (\xi_s^5 - 1) - \frac{5}{2}(\xi_s^2-1) \biggr] \biggr\} \, ,
</math>
   </td>
   </td>
</tr>
</tr>


<!-- (August 2015) DELETE NEXT EQUATION, AS DEFINITION IN TERMS OF AVERAGE PRESSURE IS LIKELY NOT CORRECT ...
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>B</math>
<math>~\mathfrak{f}_A</math>
</td>
  </td>
   <td align="center">
   <td align="center">
<math>=</math>
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~ \int_0^1 3\biggl[ \frac{P(x)}{P_c}\biggr]  x^2 dx = \biggl( \frac{\bar{P}}{P_c} \biggr)_\mathrm{eq} \, .</math>
n_cK_c M_\mathrm{tot}  \biggl( \frac{3M_\mathrm{tot}}{4\pi R_0^3} \biggr)^{1/n_c}
\biggl\{ \biggl[ 1  + \frac{n_e}{n_c} (\xi_s^3-1) \biggr] \nu^{1+1/n_c} \xi_s^{3/n_c} \biggr\} \, .
</math>
   </td>
   </td>
</tr>
</tr>
END DELETION -->


<tr>
</table>
   <td align="right">
</div>
&nbsp;
In each case, the "idealized" energy expression is retrieved if/when the relevant form factor, <math>~\mathfrak{f}_i</math>, is set to unity.
</td>
 
====Some Detailed Examples====
 
In an [[User:Tohline/SSC/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 <math>~n=5</math> or an <math>~n=1</math>  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 debug numerical algorithms that are designed to calculate structural form factors for more general cases that 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 [[User:Tohline/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 [[User:Tohline/SSC/Structure/BiPolytropes/Analytic5_1#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">
   <td align="center">
<math>=</math>
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<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>
n_c M_\mathrm{tot} \bar{c_s}^2 \biggl\{ \biggl[ 1  + \frac{n_e}{n_c} (\xi_s^3-1) \biggr] \nu^{1+1/n_c} \xi_s^{3/n_c} \biggr\} \, .
</math>
   </td>
   </td>
</tr>
</tr>
</table>
These should be used in conjunction with the relations derived above, namely,
<table align="center" border="0" cellpadding="10">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>
<math>~B</math>
\biggl( \frac{\rho_e}{\rho_c} \biggr)
  </td>
</math>
  <td align="center">
<math>~\equiv</math>
   </td>
   </td>
  <td align="center"><math>=</math></td>
   <td align="left">
   <td align="left">
<math>
<math>
\frac{1}{f_c} \biggl\{ \biggl( \frac{\rho_e} {\rho_c}\biggr)_0 + \biggl[ \frac{( 1 - f_c )}{ (\xi_s^3 - 1) } \biggr] \biggr\} \, ,
\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>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
 
and,
<div align="center">
<math>
\nu = \biggl[ 1 + \biggl( \frac{\rho_e}{\rho_c} \biggr) (\xi_s^3 - 1) \biggr]^{-1} \, ,
</math>
</div>
where, we recall,
<div align="center">
<math>f_c \equiv \frac{\rho_c}{\rho_c|_0}</math> &nbsp; &nbsp; and &nbsp; &nbsp; <math>\xi_s \equiv \frac{R}{r_i}</math>&nbsp; .
</div>
 
===Scenarios===
 
Generally, we will define a bipolytropic configuration by specifying:
<ul>
<li>The polytropic index of the core, <math>n_c</math>, and the envelope <math>n_e</math> &nbsp; ;
<li>The total mass, <math>M_\mathrm{tot}</math>&nbsp; ;
<li>The specific entropy of the core material, via the specification of the polytropic constant of the core, <math>K_c</math>&nbsp; .
</ul>
Next, we will choose:
<ul>
<li>The fractional radius of the core, <math>r_i/R = 1/\xi_s</math> &nbsp; ;
<li>The initial ratio of density in the envelope to density in the core, <math>(\rho_e/\rho_c)_0</math> &nbsp; ; also set <math>f_c = 1</math>, indicating that
<math>(\rho_e/\rho_c) = (\rho_e/\rho_c)_0</math>.
</ul>
With these values in hand, we can determine:
<ul>
<li>The ratio of the core mass to total mass, <math>\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math> &nbsp; ;
<li>The free-energy coefficients, <math>A</math> and <math>B</math> &nbsp; ;
</ul>
and the free energy <math>\mathfrak{G}</math> can be evaluated for a wide variety of choices of configuration radii, <math>R</math>.
 
We retrieve the expression for the single polytrope by setting <math>n_e = n_c</math> and <math>(\rho_e/\rho_c)_0 = 1</math>, for any choice of <math>\xi_s</math>.
 
===Equilibria===
 
The radii of equilibrium configurations, <math>R_\mathrm{eq}</math>, are identified by the condition,
<div align="center">
<math>\frac{\partial \mathfrak{G}}{\partial R} = 0 \, .</math>
</div>
As in the case of the single polytrope, this condition is satisfied if we set <math>R_0 = R_\mathrm{eq}</math> and,
<div align="center">
<math>
\frac{An_c}{3B} = 1 \, .
</math>
</div>
This implies,
<table border="0" cellpadding="8" align="center">
 
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>
&nbsp;
\frac{5\bar{c_s}^2 R_\mathrm{eq}}{GM_\mathrm{tot}}
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
<math>~=</math>
=
</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>
\nu^{-1/n_c} \xi_s^{2-3/n_c} \biggl\{ 1 + \frac{5}{2} \biggl(\frac{\rho_e}{\rho_c} \biggr) (\xi_s^2 - 1) + \biggl(\frac{\rho_e}{\rho_c} \biggr)^2
\frac{4\pi}{3}  
\biggl[ (\xi_s^5 - 1) - \frac{5}{2}(\xi_s^2 - 1) \biggr] \biggr\}  
\biggl[ \biggl( \frac{P_c}{P_\mathrm{norm}} \biggr)\chi^{3\gamma} \biggr]_\mathrm{eq}  
\biggl[ 1 + \frac{n_e}{n_c}( \xi_s^3 - 1 ) \biggr]^{-1} \, ,
\cdot \mathfrak{f}_A \, ,
</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
where,
<div align="center">
<math>
\frac{\rho_e}{\rho_c} = \biggl(\frac{1}{\nu} - 1 \biggr) (\xi_s^3 - 1)^{-1} \, ,
</math>
</div>
and
<div align="center">
<math>
\xi_s^{-3} \le \nu \le 1 \, .
</math>
</div>
Finally, the value of the free energy in each equilibrium configuration is,
<table border="0" cellpadding="8" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>
<math>~C</math>
\mathfrak{G}\biggr|_{R_\mathrm{eq}}
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
<math>~\equiv</math>
=
</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>
B - A
\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>
</math>
   </td>
   </td>
Line 1,596: Line 1,684:
<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~D</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>
<math>~\equiv</math>
=
</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>
n_c M_\mathrm{tot} \bar{c_s}^2 \biggl\{ \biggl[ 1  + \frac{n_e}{n_c} (\xi_s^3-1) \biggr] \nu^{1+1/n_c} \xi_s^{3/n_c} \biggr\}
\biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} \, .
-
\biggl( \frac{3GM^2_\mathrm{tot}}{5R_\mathrm{eq}} \biggr) \nu^2 \xi_s \biggl\{ 1
+ \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (\xi_s^2 - 1)
+ \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ (\xi_s^5 - 1) - \frac{5}{2}(\xi_s^2-1) \biggr] \biggr\}
</math>
</math>
   </td>
   </td>
</tr>
</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 <math>~K</math>) &#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>
<tr>
   <td align="right">
   <td align="center" colspan="2">
<math>
'''Figure 1:''' <font color="darkblue">Free Energy Surface </font>  
\biggl( M_\mathrm{tot} \bar{c_s}^2 \biggr)^{-1} \mathfrak{G}\biggr|_{R_\mathrm{eq}}
</math>
   </td>
   </td>
   <td align="center">
</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 [[User:Tohline/SphericallySymmetricConfigurations/Virial#Bounded_Isothermal|further discussion]], below) &#8212; is, specifically,
<div align="center">
<font size="-1">
<math>
<math>
=
\mathfrak{G}^* = 3000\biggl[ - \frac{1}{\chi} - \ln\chi + \frac{\Pi}{3}\chi^3 + 0.9558 \biggr] \, .
</math>
  </td>
  <td align="left">
<math>
n_c  \biggl\{ \biggl[ 1  + \frac{n_e}{n_c} (\xi_s^3-1) \biggr] \nu^{1+1/n_c} \xi_s^{3/n_c} \biggr\}  
-  
\biggl( \frac{3GM_\mathrm{tot}}{5\bar{c_s}^2 R_\mathrm{eq}} \biggr) \nu^2 \xi_s \biggl\{ 1
+ \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (\xi_s^2 - 1)
+ \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ (\xi_s^5 - 1) - \frac{5}{2}(\xi_s^2-1) \biggr] \biggr\}
</math>
</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>
   </td>
</tr>
</tr>
</table>
</div>


<tr>
==Energy Extrema==
  <td align="right">
As is illustrated in [[User:Tohline/SphericallySymmetricConfigurations/Virial#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. 
&nbsp;
 
  </td>
For a given choice of the set of physical parameters <math>~M</math>, <math>~K</math>, <math>~J</math>, <math>~P_e</math>, and <math>~\gamma_g</math>, extrema occur wherever,
  <td align="center">
<div align="center">
<math>
<math>
=
\frac{d\mathfrak{G^*}}{d\chi} = 0 \, .
</math>
</math>
  </td>
</div>
  <td align="left">
For the free energy function identified above,
<div align="center">
<math>
<math>
n_c  \biggl\{ \biggl[ 1  + \frac{n_e}{n_c} (\xi_s^3-1) \biggr] \nu^{1+1/n_c} \xi_s^{3/n_c} \biggr\}  
\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 \, ,
3 \nu^2 \xi_s \biggl[ 1 + \frac{n_e}{n_c}( \xi_s^3 - 1 ) \biggr] \nu^{1/n_c} \xi_s^{3/n_c-2}
</math>
</math>
  </td>
</div>
</tr>
<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">
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>
<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>
</math>
  </td>
</div>
  <td align="left">
<!-- COMMENT OUT THIS SECTION
As a definition of equilibrium states, this last expression is also the well-known scalar virial equation, derivable from the first moment of the equation of motion.  A more recognizable expression can be obtained by replacing each of the terms by the energy contents that they represent:
<div align="center">
<math>
<math>
n_c \nu^{1 + 1/n_c} \xi_s^{3/n_c} \biggl[ 1  + \frac{n_e}{n_c} (\xi_s^3-1) \biggr]
2(T_\mathrm{rot} + S) + W - 3P_e V = 0 \, .
\biggl\{ 1 - \frac{3 \nu}{n_c \xi_s}
\biggr\}
</math>
</math>
  </td>
</div>
</tr>
In this expression, <math>S</math> is the thermal energy content of the configuration; the relationship between <math>S</math> and the configuration's total internal energy, <math>U</math>, is provided in our [[User:Tohline/VE#Adiabatic|associated derivation of both the adiabatic and isothermal free energy functions]].
END OF COMMENT -->
 
 
<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 [http://adsabs.harvard.edu/abs/1976ApJ...208..113W Weber (1976)] &#8212; ''Oscillation and Collapse of Interstellar Clouds'' &#8212; and the work of [http://adsabs.harvard.edu/abs/1985ApJ...292..181T Tohline (1985)] &#8212; ''Star Formation:  Phase Transition, not Jeans Instability'' &#8212; which, in turn draws upon [http://adsabs.harvard.edu/abs/1981ApJ...248..717T Tohline (1981)].  In preparation for that discussion, we will go ahead and show how [http://adsabs.harvard.edu/abs/1985ApJ...292..181T Tohline's (1985)] 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 [http://adsabs.harvard.edu/abs/1976ApJ...208..113W Weber's (1976)] "energy integral" &#8212; his equation (B3) &#8212; relates to our dimensionless free-energy function.
 
----
 
 
<table border="1" cellpadding="5" align="center">
<tr><td>
[[Image:Tohline1985_Eq9.png|500px|center]]
</td></tr>
</table>
</table>


==Virial Analysis (2nd Try) ==
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 [http://adsabs.harvard.edu/abs/1985ApJ...292..181T Tohline (1985)] was only concerned with isothermal systems; then, because [http://adsabs.harvard.edu/abs/1985ApJ...292..181T Tohline (1985)] normalizes each energy term by
Let's go back to a more general expression for the internal energy, namely,
<div align="center">
<table border="0" 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>
<tr>
   <td align="right">
   <td align="right">
<math>
<math>~0</math>
U = U_\mathrm{core} + U_\mathrm{env}
  </td>
</math>
  <td align="center">
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>=
<math>\frac{3E_\mathrm{norm}}{E^*}\biggl[~A\chi_\mathrm{eq}^{-1} ~- \biggl( \frac{2C}{3}\biggr) \chi_\mathrm{eq}^{-2}  
M_\mathrm{tot} \biggl\{ n_c \nu K_c \biggl[ \biggl( \frac{3M_\mathrm{tot}}{4\pi} \biggr) \nu r_i^{-3} \biggr]^{1/n_c} +
~  +~ D\chi_\mathrm{eq}^3 - ~ B_I \biggr] \, .</math>
n_e (1-\nu) K_e \biggl[ \biggl( \frac{3M_\mathrm{tot}}{4\pi} \biggr) (1-\nu)(\xi_s^3-1)^{-1} r_i^{-3} \biggr]^{1/n_e} \biggr\}
</math>
   </td>
   </td>
</tr>
</tr>
 
</table>
<tr>
</div>
Next, because [http://adsabs.harvard.edu/abs/1985ApJ...292..181T Tohline (1985)] 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 [http://adsabs.harvard.edu/abs/1981ApJ...248..717T Tohline (1981)], 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">
   <td align="right">
&nbsp;
<math>~A</math>
  </td>
  <td align="center">
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>=
<math>\frac{1}{5} \biggl( \frac{\sin^{-1}e}{e} \biggr) \, ,</math>
M_\mathrm{tot} \biggl\{ n_c \nu K_c \biggl[ \biggl( \frac{3M_\mathrm{tot}}{4\pi R_0^3} \biggr) \nu \xi_s^{3} \biggr]^{1/n_c} \biggl(\frac{R}{R_0}\biggr)^{-3/n_c} +
n_e (1-\nu) K_e \biggl[ \biggl( \frac{3M_\mathrm{tot}}{4\pi R_0^3} \biggr) (1-\nu)(\xi_s^3-1)^{-1} \xi_s^{3} \biggr]^{1/n_e} \biggl(\frac{R}{R_0}\biggr)^{-3/n_e}  \biggr\} \, .
</math>
   </td>
   </td>
</tr>
</tr>
</table>
Ultimately, we will relate <math>K_e</math> to <math>K_c</math> by demanding that initially the pressure is identical in both layers. 
As derived earlier, this will be accomplished via the expression,
<table align="center"  border="0" cellpadding="8">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>\frac{K_e}{K_c}</math>
<math>~B_I</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
=
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>\biggl[ \frac{\rho_c^{1+1/n_c}}{\rho_e^{1+1/n_e}} \biggr]_0 \, .</math>
<math>
1 \, ,
</math>
   </td>
   </td>
</tr>
</tr>
</table>
The free energy may now be written as,
<div align="center">
<math>
\mathfrak{G} = W + U  = - A\biggl( \frac{R}{R_0} \biggr)^{-1} + B_c \biggl( \frac{R}{R_0} \biggr)^{-3/n_c} + B_e \biggl( \frac{R}{R_0} \biggr)^{-3/n_e} \, ,
</math>
</div>
where, <math>R_0</math> is a, as yet unspecified, scale length, and,
<table border="0" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>
<math>~C</math>
A
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>=</math>
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>
\biggl( \frac{3GM^2_\mathrm{tot}}{5R_0} \biggr) \nu^2 \xi_s \biggl\{ 1
\frac{3\cdot 5}{2^4 \pi} \biggl[ \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr
+ \frac{5}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (\xi_s^2 - 1)
= \biggl( \frac{3^2}{5^2} \biggr) \frac{E_\mathrm{norm}}{E^*} \, ,
+ \biggl( \frac{\rho_e}{\rho_c} \biggr)^2 \biggl[ (\xi_s^5 - 1) - \frac{5}{2}(\xi_s^2-1) \biggr] \biggr\} \, ,
</math>
</math>
   </td>
   </td>
Line 1,750: Line 1,852:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>B_c</math>
<math>~D</math>
</td>
  </td>
   <td align="center">
   <td align="center">
<math>=</math>
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>
n_c K_c M_\mathrm{tot}  \biggl( \frac{3M_\mathrm{tot}}{4\pi R_0^3} \biggr)^{1/n_c} \nu^{1+1/n_c} \xi_s^{3/n_c} \, ,
\biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} (1-e^2)^{1/2} \, .
</math>
</math>
   </td>
   </td>
</tr>
</tr>
 
</table>
</div>
Inserting these coefficient definitions, our "free energy extrema" relation becomes,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>B_e</math>
<math>~0</math>
</td>
  </td>
   <td align="center">
   <td align="center">
<math>=</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>\frac{3E_\mathrm{norm}}{E^*}
n_e K_e M_\mathrm{tot} \biggl( \frac{3M_\mathrm{tot}}{4\pi R_0^3} \biggr)^{1/n_e} (1 - \nu)^{1+1/n_e} \xi_s^{3/n_e} (\xi_s^3 - 1)^{-1/n_e} \, .
\biggl[~\frac{1}{5} \biggl( \frac{\sin^{-1}e}{e} \biggr) \chi_\mathrm{eq}^{-1}
</math>
~- \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>
   </td>
</tr>
</tr>
</table>
</table>
These should be used in conjunction with the relations derived above, namely,
</div>
<table align="center" border="0" cellpadding="10">
Next we need to appreciate that [http://adsabs.harvard.edu/abs/1985ApJ...292..181T Tohline (1985)] 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>
<tr>
   <td align="right">
   <td align="right">
<math>
<math>~\beta \equiv \frac{T_\mathrm{rot}}{|W_\mathrm{grav}|}</math>
\biggl( \frac{\rho_e}{\rho_c} \biggr)
  </td>
</math>
  <td align="center">
<math>~=</math>
   </td>
   </td>
  <td align="center"><math>=</math></td>
   <td align="left">
   <td align="left">
<math>
<math>~\frac{C\chi^{-2}}{3A \chi^{-1}} = \biggl( \frac{3}{5} \biggr) \frac{E_\mathrm{norm}}{E^*}
\frac{1}{f_c} \biggl\{ \biggl( \frac{\rho_e} {\rho_c}\biggr)_0 + \biggl[ \frac{( 1 - f_c )}{ (\xi_s^3 - 1) } \biggr] \biggr\} \, ,
\biggl( \frac{\sin^{-1}e}{e} \biggr)^{-1} \chi^{-1}</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
 
and,
<tr>
<div align="center">
  <td align="right">
<math>\Rightarrow~~~~\chi_\mathrm{eq}^{-1} </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
<math>
\nu = \biggl[ 1 + \biggl( \frac{\rho_e}{\rho_c} \biggr) (\xi_s^3 - 1) \biggr]^{-1} \, ,
\biggl( \frac{5}{3} \biggr) \frac{E^*}{E_\mathrm{norm}}  
</math>
\biggl( \frac{\sin^{-1}e}{e} \biggr)\beta_\mathrm{eq} \, .
</div>
where, we recall,
<div align="center">
<math>f_c \equiv \frac{\rho_c}{\rho_c|_0}</math> &nbsp; &nbsp; and &nbsp; &nbsp; <math>\xi_s \equiv \frac{R}{r_i}</math>&nbsp; .
</div>
 
===Equilibria===
Now let's evaluate the variation of the free energy with <math>R</math>.
<div align="center">
<math>
R_0 \cdot \frac{\partial \mathfrak{G}}{\partial R}\biggr|_{A, B_c, B_e} = A \biggl(\frac{R}{R_0}\biggr)^{-2} - \frac{3B_c}{n_c} \biggl(\frac{R}{R_0}\biggr)^{-1-3/n_c}
- \frac{3B_e}{n_e} \biggl(\frac{R}{R_0}\biggr)^{-1-3/n_e} \, .
</math>
</math>
  </td>
</tr>
</table>
</div>
</div>
Equilibria are defined by setting this first-derivative of <math>\mathfrak{G}</math> to zero. So, letting <math>R_0 \equiv R_\mathrm{equil}</math>,
Hence, our expression for the "free energy extrema" becomes,
we see that the configuration is in equilibrium when <math>R = R_0</math>, which means,
<div align="center">
<div align="center">
<math>
<table border="0" cellpadding="5" align="center">
n_c n_e A = 3(n_e B_c + n_c B_e) \, .
</math>
</div>
Now let's evaluate the second derivative at the equilibrium radius where <math>R = R_0</math>:
<table border="0" cellpadding="8" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>
<math>~0</math>
R_0^2 \cdot \frac{\partial^2 \mathfrak{G}}{\partial R^2}\biggr|_{A, B_c, B_e}
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>=</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>
-2 A \biggl(\frac{R}{R_0}\biggr)^{-3}
\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{3B_c}{n_c} \biggl( 1+\frac{3}{n_c} \biggr) \biggl(\frac{R}{R_0}\biggr)^{-2-3/n_c}
~  +~ \frac{4\pi P_e}{P_\mathrm{norm}} (1-e^2)^{1/2}  
+ \frac{3B_e}{n_e} \biggl( 1+\frac{3}{n_e} \biggr) \biggl(\frac{R}{R_0}\biggr)^{-2-3/n_e}  
\biggl[ \biggl( \frac{3^3}{5^3} \biggr) \biggl( \frac{E_\mathrm{norm}}{E^*} \biggr)^4
</math>
\biggl( \frac{\sin^{-1}e}{e} \biggr)^{-3}\biggr] \beta_\mathrm{eq}^{-3}
- ~ \frac{3E_\mathrm{norm}}{E^*}  </math>
   </td>
   </td>
</tr>
</tr>
Line 1,842: Line 1,940:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>
&nbsp;
n_c n_e R_0^2 \cdot \frac{\partial^2 \mathfrak{G}}{\partial R^2}\biggr|_{A, B_c, B_e}
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>=</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>  
<math>
3n_e B_c \biggl( 1+\frac{3}{n_c} \biggr) + 3n_c B_e \biggl( 1+\frac{3}{n_e} \biggr) - 2n_c n_e A
2 \biggl\{
</math>
\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>
   </td>
</tr>
</tr>
</table>
</div>
Now,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<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>
   <td align="center">
   <td align="center">
<math>=</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>  
<math>~\frac{2\pi \cdot 3^3}{5^3} \biggl[ \frac{P_e}{(E^*)^4} \biggr] ( GM_\mathrm{tot}^2)^3</math>
3n_e B_c \biggl( 1+\frac{3}{n_c} \biggr) + 3n_c B_e \biggl( 1+\frac{3}{n_e} \biggr) - 6(n_e B_c + n_c B_e)
</math>
   </td>
   </td>
</tr>
</tr>
Line 1,872: Line 1,976:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>
&nbsp;
(n_c n_e)^2 R_0^2 \cdot \frac{\partial^2 \mathfrak{G}}{\partial R^2}\biggr|_{A, B_c, B_e}
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>=</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>  
<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>
3n_e^2 B_c (3 -n_c ) + 3n_c^2 B_e ( 3 - n_e) \, .
</math>
   </td>
   </td>
</tr>
</tr>
</table>
Let's relate <math>B_e</math> to <math>B_c</math>.
<table border="0" cellpadding="8" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>B_c</math> 
&nbsp;
</td>
  </td>
   <td align="center">
   <td align="center">
<math>=</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>
n_c K_c M_\mathrm{tot}  \biggl( \frac{3M_\mathrm{tot}}{4\pi R_0^3} \biggr)^{1/n_c} \nu^{1+1/n_c} \xi_s^{3/n_c}  
~\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>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
which is the definition of the coefficient "<math>~k</math>" that is provided as equation (7) of [http://adsabs.harvard.edu/abs/1985ApJ...292..181T Tohline (1985)].  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>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<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>
   <td align="center">
   <td align="center">
<math>=</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>0 \, .</math>
n_c K_c M_\mathrm{tot}  \biggl[ \frac{\rho_c|_0 }{\nu \xi_s^3} \biggr]^{1/n_c} \nu^{1+1/n_c} \xi_s^{3/n_c}
= n_c K_c M_\mathrm{tot}  \biggl[ \rho_c|_0 \biggr]^{1/n_c} \nu
= \biggl[ \frac{n_c \nu M_\mathrm{tot}}{\rho_c|_0} \biggr] K_c \biggl( \rho_c \biggr)_0^{1+1/n_c}
</math>
   </td>
   </td>
</tr>
</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 [http://adsabs.harvard.edu/abs/1985ApJ...292..181T Tohline (1985)] 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 [http://adsabs.harvard.edu/abs/1985ApJ...292..181T Tohline's (1985)] 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>
<tr>
   <td align="right">
   <td align="right">
<math>B_e</math>
<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>
   <td align="center">
   <td align="center">
<math>=</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>0 \, .</math>
  </td>
</tr>
</table>
</div>
----
<table border="1" cellpadding="5" align="center">
<tr><td>
<!-- [[Image:Weber1976_EqB3.png|500px|center]] -->
[[Image:AAAwaiting01.png|500px|center]]
</td></tr>
</table>
Plugging the same set of modified leading coefficients into our derived expression for the free energy becomes,
<div align="center">
<math>
<math>
n_e K_e M_\mathrm{tot} \biggl( \frac{3M_\mathrm{tot}}{4\pi R_0^3} \biggr)^{1/n_e} (1 - \nu)^{1+1/n_e} \xi_s^{3/n_e} (\xi_s^3 - 1)^{-1/n_e}  
\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>
</math>
  </td>
</div>
</tr>
Now, recognize that,
<div align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~\chi</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>=</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~\alpha \biggl( \frac{R_0}{R_\mathrm{norm}} \biggr) = \biggl( \frac{2^2}{3\cdot 5} \biggr) \alpha \, ,</math>
n_e K_e M_\mathrm{tot}  \biggl[ \frac{(\xi_s^3 - 1) \rho_e|_0 }{(1-\nu) \xi_s^3} \biggr]^{1/n_e} (1 - \nu)^{1+1/n_e} \xi_s^{3/n_e} (\xi_s^3 - 1)^{-1/n_e}
   </td>
= n_e K_e M_\mathrm{tot}  \biggl[ \rho_e|_0 \biggr]^{1/n_e} (1 - \nu)
</math>
   </td>
</tr>
</tr>


<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~(1 - e^2)^{1/2}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>=</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~\frac{Z}{R} = \frac{\gamma}{\alpha} \, ,</math>
\biggl[ \frac{n_e (1 - \nu) M_\mathrm{tot}}{\rho_e|_0} \biggr]  K_e \biggl( \rho_e \biggr)_0^{1+1/n_e}
</math>
   </td>
   </td>
</tr>
</tr>
Line 1,964: Line 2,098:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>\Rightarrow ~~~~ B_e</math>
<math>~\frac{P_e}{P_\mathrm{norm}}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>=</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<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}
B_c \biggl[ \frac{n_e (1 - \nu) }{n_c \nu } \biggr]  \biggl( \frac{\rho_e}{\rho_c}\biggr)_0^{-1} =
\, ,</math>
B_c \biggl( \frac{n_e}{n_c} \biggr) (\xi_s^3 -1) \, .
</math>
   </td>
   </td>
</tr>
</tr>
</table>
Hence,
<table border="0" cellpadding="8" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>
<math>~\frac{3\cdot 5}{2^4 \pi} \biggl[ \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr]</math>
(n_c n_e)^2 R_0^2 \cdot \frac{\partial^2 \mathfrak{G}}{\partial R^2}\biggr|_{A, B_c, B_e}
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>=</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>  
<math>~\frac{1}{3} \biggl( \frac{2}{5} J_\mathrm{Weber} \biggr)^2
3n_eB_c \biggl\{ n_e (3 -n_c ) + n_c ( 3 - n_e)(\xi_s^3 -1) \biggr\} \, .
\, ,</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
 
</div>
This second derivative becomes negative and, hence, the configuration becomes unstable, when,
where, for axisymmetric configurations (set <math>~\beta=\alpha</math> in [http://adsabs.harvard.edu/abs/1976ApJ...208..113W Weber's (1976)] equation 12),
<div align="center">
<div align="center">
<math>
<math>J_\mathrm{Weber} \equiv \alpha^2 \Omega = \biggl( \frac{R}{R_0} \biggr)^2 (\dot\varphi_c t_0)^2 \, .</math>
\xi_s < \xi_\mathrm{crit} \equiv \biggl[ 1 + \frac{n_e(n_c-3)}{n_c(3-n_e)} \biggr]^{1/3} \, ,
</math>
</div>
</div>
that is, when,
Hence, our expression for the free energy may be written as,
<div align="center">
<div align="center">
<math>
<table border="0" cellpadding="5" align="center">
q \equiv \frac{1}{\xi_s} > \biggl[ 1 + \frac{n_e(n_c-3)}{n_c(3-n_e)} \biggr]^{-1/3} \, .
</math>
</div>


Some example values of <math>q_\mathrm{crit}</math> are provided in the following table.
<table border="1" cellpadding="8" align="center">
<tr>
<tr>
   <th align="center" colspan="6">
   <td align="right">
Table 1:  Values of <math>q_\mathrm{crit}</math> and <math>\nu_\mathrm{crit}</math> for various bipolytropes
<math>~\mathfrak{G}^*</math>
  </th>
</tr>
 
<tr>
  <th align="center" rowspan="2">
<math>n_c</math>
  </th>
  <th align="center" rowspan="2">
<math>n_e</math>
  </th>
  <th align="center" rowspan="2">
<math>\xi_\mathrm{crit}^3</math>
  </th>
  <th align="center" rowspan="2">
<math>q_\mathrm{crit}</math>
  </th>
 
  <th align="center" colspan="2">
<math>\nu_\mathrm{crit}</math>
  </th> 
</tr>
 
<tr>
  <th align="center">
<math>\frac{\rho_e}{\rho_c} = 1</math>
  </th>
  <th align="center">
<math>\frac{\rho_e}{\rho_c} = \frac{1}{2}</math>
  </th>
</tr>
 
<tr>
  <td align="center">
<math>5</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>1</math>
<math>~=</math>
   </td>
   </td>
   <td align="center">
   <td align="left">
<math>\frac{6}{5}</math>
<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>
   </td>
   <td align="center">
</tr>
<math>0.941</math>
 
<tr>
   <td align="right">
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>\frac{5}{6}</math>
<math>~=</math>
   </td>
   </td>
 
   <td align="left">
   <td align="center">
<math>
<math>\frac{10}{11}</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>
   </td>
</tr>
</tr>


<tr>
<tr>
   <td align="center">
   <td align="right">
<math>5</math>
<math>\Rightarrow~~~~\frac{4}{3} \mathfrak{G}^*</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>\frac{3}{2}</math>
<math>~=</math>
   </td>
   </td>
   <td align="center">
   <td align="left">
<math>\frac{7}{5}</math>
<math>
  </td>
J^2_\mathrm{Weber} \alpha^{-2}
  <td align="center">
-~ \frac{4}{3} \ln \chi^3 +~ \frac{1}{3} \alpha^2 \gamma \biggl[ P_\mathrm{ext} \biggr]_\mathrm{Weber}
<math>0.894</math>
- 3 \biggl( \frac{\sin^{-1}e}{e} \biggr) \alpha^{-1} \, .
</math>
   </td>
   </td>
</tr>
</table>
</div>
The right-hand-side of this expression exactly matches [http://adsabs.harvard.edu/abs/1976ApJ...208..113W Weber's (1976)] "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 align="center">
<math>\frac{5}{7}</math>
  </td>


  <td align="center">
</td></tr>
<math>\frac{10}{12}</math>
</table>
  </td>
</div>
</tr>


<tr>
=Examples=
  <td align="center">
* Polytropic Spheres
<math>\gg 3</math>
** [[User:Tohline/SSC/Virial/Polytropes#Isolated.2C_Nonrotating_Configuration|Isolated, Nonrotating Configuration]]
  </td>
** [[User:Tohline/SSC/Virial/Polytropes#Nonrotating_Configuration_Embedded_in_an_External_Medium|Nonrotating Configuration Embedded in an External Medium]]
  <td align="center">
<math>\frac{3}{2}</math>
  </td>
  <td align="center">
<math>2</math>
  </td>
  <td align="center">
<math>0.794</math>
  </td>


  <td align="center">
* Isothermal Spheres
<math>\frac{1}{2}</math>
** [[User:Tohline/SSC/Virial/Isothermal#Isolated.2C_Nonrotating_Configuration|Isolated, Nonrotating Configuration]]
  </td>
** [[User:Tohline/SSC/Virial/Isothermal#Nonrotating_Configuration_Embedded_in_an_External_Medium|Nonrotating Configuration Embedded in an External Medium]]


  <td align="center">
<math>\frac{2}{3}</math>
  </td>
</tr>
</table>


The associated critical value of the mass ratio, <math>\nu_\mathrm{crit}</math>, depends on the selection of the chosen density ratio via the relation,
<div align="center">
<math>
\nu_\mathrm{crit} = \biggl[1 + \biggl(\frac{\rho_e}{\rho_c}\biggr)(\xi_\mathrm{crit}^3 - 1) \biggr]^{-1} \, .
</math>
</div>


{{LSU_WorkInProgress}}


=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.] 


==Virial Analysis (3rd Try) ==
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>.


As in the <math>2^\mathrm{nd}</math> try, above, the free energy may be written as,
Use the dimensionless radius,
<div align="center">
<div align="center">
<math>
<math>\xi \equiv \frac{r}{r_i}</math>.
\mathfrak{G} = W + U  = - A \chi^{-1} + B_c \chi^{-3/n_c} + B_e \chi^{-3/n_e} \, ,
</math>
</div>
</div>
where, <math>\chi \equiv R/R_0</math>.
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">
<div align="center">
<math>
<math>M_\xi = \frac{4\pi}{3} \rho_c r_i^3 \xi^3</math> .
\frac{\partial\mathfrak{G}}{\partial \chi} = A \chi^{-2} -\frac{3}{n_c} B_c \chi^{-(1+3/n_c)} -\frac{3}{n_e} B_e \chi^{-(1+3/n_e)} \, ;
</div>
</math>
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,


<math>
\frac{\partial^2\mathfrak{G}}{\partial \chi^2} = -2 A \chi^{-3} + \frac{3}{n_c} \biggl(1+\frac{3}{n_c}\biggr) B_c \chi^{-(2+3/n_c)}
+ \frac{3}{n_e} \biggl(1+\frac{3}{n_e}\biggr) B_e \chi^{-(2+3/n_e)} \, .
</math>
</div>
We obtain the equilibrium radius, <math>\chi_E</math>, when <math>\partial\mathfrak{G}/\partial\chi = 0</math>.  Hence,
the relation governing the equilibrium radius is,
<div align="center">
<div align="center">
<table border="0" cellpadding="5">
<table border="0" cellpadding="5" align="center">
 
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>
<math>~M_\mathrm{core}</math>
A \chi_E^{-2}  
  </td>
</math>
  <td align="center">
<math>~=</math>
   </td>
   </td>
  <td align="center"><math>=</math></td>
   <td align="left">
   <td align="left">
<math>
<math>
\frac{3}{n_c} B_c \chi_E^{-(1+3/n_c)} +\frac{3}{n_e} B_e \chi_e^{-(1+3/n_e)}
\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>
</math>
   </td>
   </td>
Line 2,167: Line 2,252:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>
<math>~M_\mathrm{env}</math>
\Rightarrow ~~~~~ \alpha
  </td>
</math>
  <td align="center">
<math>~=</math>
   </td>
   </td>
  <td align="center"><math>=</math></td>
   <td align="left">
   <td align="left">
<math>
<math>
\chi_E^{1-3/n_c} +\beta \chi_E^{1-3/n_e}
\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>
</math>
   </td>
   </td>
</tr>
</tr>


</table>
<tr>
</div>
   <td align="right">
where,
<math>~M_\mathrm{tot}</math>
<div align="center">
<math>\alpha \equiv \frac{n_c A}{3B_c} \, ;</math>
 
<math>\beta \equiv \frac{n_c B_e}{n_e B_c} \, .</math>
</div>
Note that for the isothermal case (<math>n_c = \infty</math> in the above exponents),
<div align="center">
<math>\alpha \equiv \frac{A}{B_c} \, ;</math>
 
<math>\beta \equiv \frac{3 B_e}{n_e B_c} \, .</math>
</div>
 
And at this equilibrium radius, the second derivative of the free energy has the value,
<div align="center">
<table border="0" cellpadding="5">
<tr>
   <td align="right">
<math>
\chi_E^3 \biggl( \frac{n_c}{3B_c} \biggr) \frac{\partial^2\mathfrak{G}}{\partial \chi^2} \biggr|_E
</math>
   </td>
   </td>
   <td align="right">
   <td align="center">
<math>
<math>~=</math>
=
</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>
-2 \alpha + \biggl(1+\frac{3}{n_c}\biggr) \chi_E^{1-3/n_c}
\frac{4\pi}{3} r_i^3 \biggl[\rho_c  + \rho_e(\xi_s^3 - 1) \biggr]
+ \biggl(1+\frac{3}{n_e}\biggr) \beta \chi_E^{1-3/n_e} \, ,
= 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>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
which, when combined with the condition for equilibrium gives,
 
<div align="center">
Following the work of [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Sch&ouml;nberg &amp; Chandrasekhar (1942)] &#8212; see [[User:Tohline/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 border="0" cellpadding="5">
<table align="center" border="0" cellpadding="10">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>
<math>~\nu</math>
\chi_E^3 \biggl( \frac{n_c}{3B_c} \biggr) \frac{\partial^2\mathfrak{G}}{\partial \chi^2} \biggr|_E
</math>
   </td>
   </td>
   <td align="right">
   <td align="center">
<math>
<math>~\equiv</math>
=
</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~\frac{M_\mathrm{core}}{M_\mathrm{tot}} \, ,</math>
-2 \alpha  + \biggl(1+\frac{3}{n_e}\biggr) \beta \chi_E^{1-3/n_e}  
+ \biggl(1+\frac{3}{n_c}\biggr) \biggl[  \alpha  - \beta \chi_E^{1-3/n_e} \biggr]
</math>
   </td>
   </td>
</tr>
</tr>
Line 2,242: Line 2,300:
<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~q</math>
   </td>
   </td>
   <td align="right">
   <td align="center">
<math>
<math>~\equiv</math>
=
</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~\frac{r_i}{R} = \frac{1}{\xi_s} \, .</math>
\biggl(\frac{3}{n_c} - 1\biggr) \alpha  + 3\beta \biggl(\frac{1}{n_e} - \frac{1}{n_c}\biggr) \chi_E^{1-3/n_e} 
</math>
   </td>
   </td>
</tr>
</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>
<tr>
   <td align="right">
   <td align="right">
<math>~W_\mathrm{grav} = W_\mathrm{core} + W_\mathrm{env}</math>
  </td>
  <td align="left">
<math>
<math>
\Rightarrow ~~~~~ \chi_E^3 \biggl( \frac{n_e n_c^2}{3B_c} \biggr) \frac{\partial^2\mathfrak{G}}{\partial \chi^2} \biggr|_E
= - 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>
</math>
   </td>
   </td>
</tr>
<tr>
   <td align="right">
   <td align="right">
<math>
&nbsp;
=
</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>
n_e (3- n_c) \alpha  + 3\beta (n_c - n_e) \chi_E^{1-3/n_e}
= - 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>
</math>
   </td>
   </td>
</tr>
</tr>


</table>
<tr>
</div>
  <td align="right">
Finally, the equilibrium configuration is stable as long as this second derivative is positive, that is, for,
&nbsp;
<div align="center">
  </td>
  <td align="left">
<math>
<math>
\chi_E^{3/n_e-1} \frac{3\beta (n_c - n_e) }{n_e (n_c- 3) \alpha} = \frac{3\beta (1- n_e/n_c) }{n_e (1- 3/n_c) \alpha} \, .
= - \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>
</math>
</div>
  </td>
</tr>


===Isothermal Core===
<tr>
If the core is isothermal, we set <math>n_c = \infty</math>, in which case stability occurs for,
  <td align="right">
<div align="center">
&nbsp;
  </td>
  <td align="left">
<math>
<math>
\chi_E^{3/n_e-1} \frac{3\beta}{n_e \alpha} \, .
= - \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>
</math>
</div>
  </td>
</tr>


====&nbsp; &nbsp;&nbsp; Envelope with <math>n=3/2</math>====
<tr>
If we choose an <math>n_e = 3/2</math> envelope, we obtain stability for,
  <td align="right">
<div align="center">
&nbsp;
  </td>
  <td align="left">
<math>
<math>
\chi_E <  \frac{2\beta}{\alpha}\, .
= - \frac{3GM^2_\mathrm{tot}}{R} \biggl( \frac{M_\mathrm{core}}{M_\mathrm{tot}} \biggr)^2 \xi_s \biggl\{ \frac{1}{5}
</math>
+ \frac{1}{2} \biggl( \frac{\rho_e}{\rho_c} \biggr) (\xi_s^2 - 1)
</div>
+ \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\}
In this case, the equilibrium radius condition is,
<div align="center">
<math>
\chi_E^2 - \alpha \chi_E + \beta =0
</math>
</math>
  </td>
</tr>


<math>
</table>
\Rightarrow ~~~~ \chi_E = \frac{1}{2}\biggl[\alpha \pm \biggl(  \alpha^2 -4\beta \biggr)^{1/2}  \biggr] =
\frac{\alpha}{2}\biggl[1 \pm \biggl(  1 -\frac{4\beta}{\alpha^2} \biggr)^{1/2}  \biggr]
</math>
 
</div>


 
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)], we will rewrite this expression for the gravitational potential energy as,
====&nbsp; &nbsp;&nbsp; Envelope with <math>n=1</math>====
If, instead, we choose an <math>n_e = 1</math> envelope, we obtain stability for,
<div align="center">
<div align="center">
<math>
<table border="0" cellpadding="5" align="center">
\chi_E <  \sqrt{\frac{3\beta}{\alpha} }\, .
<tr>
</math>
  <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>
</div>
In this case, the equilibrium radius condition is,
where,
<div align="center">
<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>
<math>
\alpha = \chi_E + \beta \chi_E^{-2} \, ,
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>
</math>
  </td>
</tr>


<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>
<math>
\Rightarrow ~~~~ \chi_E^3 - \alpha \chi_E^2 + \beta = 0 \, .
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>
</math>
  </td>
</tr>
</table>
</div>
=See Also=
<ul>
<li>[[User:Tohline/SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>
</ul>


</div>




{{LSU_HBook_footer}}
{{LSU_HBook_footer}}

Latest revision as of 01:11, 30 May 2017


Virial Equilibrium of Spherically Symmetric Configurations

Whitworth's (1981) Isothermal Free-Energy Surface
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Free Energy Expression

Review

As has been 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,

<math> \mathfrak{G} = W_\mathrm{grav} + \mathfrak{S}_\mathrm{therm} + T_\mathrm{kin} + P_e V + \cdots </math>

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, 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,

<math> dm = \rho(\vec{x}) d^3x = 4\pi \rho(r) r^2 dr \, , </math>

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

<math>~M_r(r) = \int\limits_V dm</math>

<math>~=</math>

<math>~ \int_0^r 4\pi r^2 \rho dr \, .</math>

Hence, if the volume of the configuration extends out to a radius denoted by <math>~R_\mathrm{limit}</math>, the configuration mass is,

<math>~M_\mathrm{limit}</math>

<math>~=</math>

<math>~ \int_0^{R_\mathrm{limit}} 4\pi r^2 \rho dr \, .</math>

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.

Isolated Polytropes: For 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.

Pressure-Truncated Polytropes: But, a 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.

BiPolytropes: When discussing 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.


Confinement by External Pressure: 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,

<math>~P_e V</math>

<math>~=</math>

<math>~P_e \int_0^{R_\mathrm{limit}} 4\pi r^2 dr = \frac{4\pi}{3} P_e R_\mathrm{limit}^3 \, .</math>

Gravitational Potential Energy: From our discussion of the scalar virial theorem — see, specifically, the reference to Equation (18), on p. 18 of [EFE] — the gravitational potential energy is given by the expression,

<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>

For spherically symmetric systems, the

Poisson Equation

LSU Key.png

<math>\nabla^2 \Phi = 4\pi G \rho</math>

becomes,

<math>~\frac{1}{r^2} \frac{d}{dr} \biggl( r^2 \frac{d\Phi}{dr} \biggr) </math>

<math>~=</math>

<math>~4\pi G \rho(r) \, , </math>

which implies,

<math>~r^2 \frac{d\Phi}{dr} </math>

<math>~=</math>

<math>~\int_0^r 4\pi G \rho(r) r^2 dr = GM_r(r) \, .</math>

Hence — see, also, p. 64, Equation (12) of [C67] — the desired expression for the gravitational potential energy is,

<math>~W_\mathrm{grav}</math>

<math>~=</math>

<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>


Also, as pointed out by [C67] — see p. 64, Equation (16) — 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,

<math>~W_\mathrm{grav}</math>

<math>~=</math>

<math>~ + \frac{1}{2} \int_0^{R_\mathrm{limit}} \Phi(r) dm \, .</math>


Rotational Kinetic Energy: We will also consider a system that is rotating with a specified simple angular velocity profile, <math>~\dot\varphi(\varpi)</math>, in which case, from our discussion of the scalar virial theorem — see, specifically, the reference to Equation (8), on p. 16 of EFE — the (ordered) kinetic energy,

<math>~T_\mathrm{kin}</math>

<math>~=</math>

<math>~ \frac{1}{2} \int\limits_V \rho |\vec{v} |^2 d^3x = \frac{1}{2} \int\limits_V |\vec{v} |^2 dm \, ,</math>

is entirely rotational kinetic energy, specifically,

<math>~T_\mathrm{kin} = T_\mathrm{rot}</math>

<math>~=</math>

<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>

Reservoir of Thermodynamic Energy: As has been explained in 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 isothermal systems,

<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>

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 adiabatic systems,

<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>

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 — and, hence, their relative significance can be specified or measured — 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 Horedt (1970) and Whitworth (1981), 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,

Adopted Normalizations

Adiabatic Cases

Isothermal Case <math>~(\gamma = 1; K = c_s^2)</math>

<math>~R_\mathrm{norm}</math>

<math>~\equiv</math>

<math>~\biggl[ \biggl( \frac{G}{K} \biggr) M_\mathrm{tot}^{2-\gamma} \biggr]^{1/(4-3\gamma)} </math>

<math>~P_\mathrm{norm}</math>

<math>~\equiv</math>

<math>~\biggl[ \frac{K^4}{G^{3\gamma} M_\mathrm{tot}^{2\gamma}} \biggr]^{1/(4-3\gamma)} </math>


<math>~E_\mathrm{norm}</math>

<math>~\equiv</math>

<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>

<math>~\rho_\mathrm{norm}</math>

<math>~\equiv</math>

<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>

<math>~c^2_\mathrm{norm}</math>

<math>~\equiv</math>

<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>

<math>~R_\mathrm{norm}</math>

<math>~\equiv</math>

<math>~\frac{G M_\mathrm{tot}}{c_s^2} </math>

<math>~P_\mathrm{norm}</math>

<math>~\equiv</math>

<math>~\frac{c_s^8}{G^{3} M_\mathrm{tot}^{2}} </math>


<math>~E_\mathrm{norm}</math>

<math>~\equiv</math>

<math>~ M_\mathrm{tot} c_s^2 </math>

<math>~\rho_\mathrm{norm}</math>

<math>~\equiv</math>

<math> \frac{3}{4\pi} \biggl[ \frac{c_s^6}{G^3 M_\mathrm{tot}^2} \biggr] </math>

<math>~c^2_\mathrm{norm}</math>

<math>~\equiv</math>

<math>~ \biggl( \frac{4\pi}{3} \biggr) c_s^2 </math>

Note that, given the above definitions, the following relations hold:

<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>

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,

<math>~\chi \equiv \frac{R_\mathrm{limit}}{R_\mathrm{norm}} \, ,</math>

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,

<math>~\chi_\mathrm{eq} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \, .</math>

Choices Made by Other Researchers

As is detailed in a 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 Horedt (1970) and by Whitworth (1981). The following relations can be used to switch from our normalizations to theirs:

Hoerdt's (1970) Normalization

<math>~\biggl( \frac{R_\mathrm{Hoerdt}}{R_\mathrm{norm}} \biggr)^{4-3\gamma}</math>

<math>~=</math>

<math>~ \frac{(\gamma-1)}{\gamma} \biggl( 4\pi \biggr)^{\gamma-1}</math>

<math>~\biggl( \frac{P_\mathrm{Hoerdt}}{P_\mathrm{norm}} \biggr)^{4-3\gamma}</math>

<math>~=</math>

<math>~ \biggl[\frac{\gamma}{(\gamma-1)} \biggr]^{3\gamma} \biggl( \frac{1}{4\pi} \biggr)^{\gamma}</math>

     

Whitworth's (1981) Normalization

<math>~\biggl( \frac{R_\mathrm{rf}}{R_\mathrm{norm}} \biggr)^{4-3\gamma}</math>

<math>~=</math>

<math>~ \frac{1}{5\pi} \biggl( \frac{4\pi}{3} \biggr)^\gamma</math>

<math>~\biggl( \frac{P_\mathrm{rf}}{P_\mathrm{norm}} \biggr)^{4-3\gamma}</math>

<math>~=</math>

<math>~ 2^{-2(4+\gamma)} \biggl( \frac{3^4 \cdot 5^3}{\pi} \biggr)^\gamma</math>

It is also worth noting how the length-scale normalization that we are adopting here relates to the characteristic length scale,

<math>~a_n \equiv \biggl[ \frac{1}{4\pi G} \biggl( \frac{H_c}{\rho_c} \biggr) \biggr]^{1/2} \, ,</math>

that has classically been adopted in the context of the 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, <math>~n</math>. Recognizing that, via the polytropic equation of state, the pressure, density, and enthalpy of every element of fluid are related to one another via the expressions,

<math>~H\rho = (n+1)P</math>     … and …      <math>P = K\rho^{1+1/n} \, ,</math>

the specific enthalpy at the center of a polytropic sphere, <math>~H_c/\rho_c</math>, can be rewritten in terms of <math>~K</math> and <math>~\rho_c</math> to give,

<math>~a_n = \biggl[ \frac{(n+1)K}{4\pi G} \rho_c^{(1/n) -1} \biggr]^{1/2} \, ,</math>

which is the definition of this classical length scale introduced by [C67] (see, specifically, his equation 10 on p. 87). Switching from <math>~n</math> to the associated adiabatic exponent via the relation, <math>~\gamma = 1+1/n ~~~\Rightarrow~~~ n = 1/(\gamma-1)</math>, we see that,

<math>~\biggl( \frac{a_n}{R_\mathrm{norm}} \biggr)^2</math>

<math>~=</math>

<math>~\biggl( \frac{\gamma}{\gamma-1} \biggr) \frac{K \rho_c^{(\gamma-2)}}{4\pi G} \cdot \frac{1}{R_\mathrm{norm}^2}</math>

 

<math>~=</math>

<math>~\frac{1}{4\pi}\biggl( \frac{\gamma}{\gamma-1} \biggr) \frac{K }{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>

 

<math>~=</math>

<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 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>

 

<math>~=</math>

<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 M_\mathrm{tot}^{\gamma-2} }{G} \biggr] \cdot \biggl[ \biggl( \frac{G}{K} \biggr) M_\mathrm{tot}^{2-\gamma} \biggr] </math>

 

<math>~=</math>

<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>

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 — 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, [C67] did not encounter this issue. As we have discussed elsewhere, the equilibrium radius of a polytropic sphere is identified as the radial location,

<math>~\xi_1 = \frac{R_\mathrm{eq}}{a_n} \, ,</math>

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 — specifically, setting,

<math>~\chi_\mathrm{eq}</math>

<math>~=</math>

<math>~\frac{R_\mathrm{eq}}{R_\mathrm{norm}} = \xi_1 \biggl(\frac{a_n}{R_\mathrm{norm}} \biggr) \, ,</math>

we can extend the above analysis to obtain,

<math>~\biggl( \frac{a_n}{R_\mathrm{norm}} \biggr)^2</math>

<math>~=</math>

<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>

<math>\Rightarrow~~~~~\biggl( \frac{a_n}{R_\mathrm{norm}} \biggr)^{4-3\gamma}</math>

<math>~=</math>

<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>

Implementation

Normalize

We will now judiciously introduce our adopted normalizations into the above-defined free-energy term expressions, using asterisks to denote dimensionless variables that have been accordingly normalized; for example,

<math> r^* \equiv \frac{r}{R_\mathrm{norm}} \, , ~~~~~~ P^* \equiv \frac{P}{P_\mathrm{norm}} \, , ~~~~~~ </math>         and       <math>\rho^* \equiv \frac{\rho}{\rho_\mathrm{norm}} \, . </math>

Normalized Mass:

<math>~M_r(r^*) </math>

<math>~=</math>

<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>

Confinement by External Pressure (Normalized Volume):

<math>~P_e V</math>

<math>~=</math>

<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>

Normalized Gravitational Potential Energy:

<math>~W_\mathrm{grav}</math>

<math>~=</math>

<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>

 

<math>~=</math>

<math> - E_\mathrm{norm} \int_0^{\chi = R_\mathrm{limit}^*} 3\biggl[\frac{M_r(r^*)}{M_\mathrm{tot}} \biggr] r^* \rho^* dr^* \, . </math>

Normalized Reservoir of Thermodynamic Energy:

<math>~\mathfrak{S}_I</math>

<math>~=</math>

<math>~E_\mathrm{norm} \int_0^{\chi=R_\mathrm{limit}^*} 3 \ln (\rho^*) (r^*)^2 \rho^* dr^* \, ,</math>

and,

<math>~\mathfrak{S}_A</math>

<math>~=</math>

<math>~\frac{E_\mathrm{norm}}{({\gamma_g}-1)} \int_0^{\chi=R_\mathrm{limit}^*} 4\pi (r^*)^2 P^* dr^* \, .</math>

Normalized Rotational Kinetic Energy:

<math>~T_\mathrm{rot}</math>

<math>~=</math>

<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>

 

<math>~=</math>

<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>

 

<math>~=</math>

<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>

 

<math>~=</math>

<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>

where,

<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>

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 & 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,

<math>~x \equiv \frac{r}{R_\mathrm{limit}} \, ,~~~~w \equiv \frac{\varpi}{R_\mathrm{limit}} \, , ~~~~\zeta \equiv \frac{z}{R_\mathrm{limit}} \, , </math>

that always run from zero to one, we have,

<math>~r^*</math>

<math>~\rightarrow~</math>

<math> ~x \biggl( \frac{R_\mathrm{limit}}{R_\mathrm{norm}} \biggr) = x \chi \, ; </math>    and, likewise,     <math> ~~~~\varpi^* ~\rightarrow~ w \chi \, ; ~~~~z^* ~\rightarrow~ \zeta \chi \, ; </math>

<math>~\rho^*</math>

<math>~\rightarrow~</math>

<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>

<math>~P^*</math>

<math>~\rightarrow~</math>

<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>

 

    

<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>

<math>~\frac{\dot\varphi}{\dot\varphi_c}</math>

<math>~\rightarrow~</math>

<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>

Summary of Normalized Expressions

Hence, our normalized expressions become,

Normalized Expressions

<math>~\frac{M_r(x)}{M_\mathrm{tot}} </math>

<math>~=</math>

<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>

<math>~\frac{P_e V}{E_\mathrm{norm}}</math>

<math>~=</math>

<math>~ \frac{4\pi}{3} \biggl( \frac{P_e}{P_\mathrm{norm}} \biggr) \chi^3 \, ,</math>

<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>

<math>~=</math>

<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>

 

<math>~=</math>

<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>

<math>~\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>

<math>~=</math>

<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>

<math>~\frac{\mathfrak{S}_I}{E_\mathrm{norm}}</math>

<math>~=</math>

<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>

 

<math>~=</math>

<math>~-3 \ln \chi + \mathrm{constant} \, , </math>

<math>~\frac{T_\mathrm{rot}}{E_\mathrm{norm}}</math>

<math>~=</math>

<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>

[NOTE to self (21 September 2014): The expressions for <math>~\mathfrak{S}_I</math> and <math>~T_\mathrm{rot}</math> may not properly account for ratio of M_limit to M_tot.]


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,

<math>~\biggl(\frac{\rho_c}{\bar\rho} \biggr)^{-1}_\mathrm{eq} </math>

<math>~=</math>

<math>~ \int_0^{1} 3x^2 \biggl[ \frac{\rho(x)}{\rho_c} \biggr]_\mathrm{eq} dx \, .</math>

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,

<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>

<math>~=</math>

<math>~\biggl[ \biggl( \frac{P_c}{P_\mathrm{norm}} \biggr) \chi^{3\gamma} \biggr]_\mathrm{eq} \, .</math>

Looking Ahead to Bipolytropes

ASIDE: 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 — 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,

<math>~P_\mathrm{core}(x) = P_0 [1 - p_c(x)]</math>      for      <math>~0 \leq x \leq q \, ,</math>

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,

<math>~P_\mathrm{env}(x) = P_{ie} [1 - p_e(x)]</math>      for      <math>~q \leq x \leq 1 \, ,</math>

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,

<math>~P_{ic} = P_0 [1 - p_c(q)] \, .</math>

In equilibrium — that is, when <math>~R_\mathrm{edge} = R_\mathrm{eq}</math> — 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,

<math>~P_\mathrm{core}(x) = P_{ic} \biggl[\frac{1 - p_c(x)}{1-p_c(q)} \biggr] \, .</math>

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,

<math>~\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>

<math>~=</math>

<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>

 

<math>~=</math>

<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>

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,

<math>~\biggl[ \frac{P}{P_\mathrm{eq}} \biggr]_\mathrm{core}</math>

<math>~=</math>

<math>~\biggl( \frac{\rho}{\rho_\mathrm{eq}} \biggr)^{\gamma_c} = \biggl( \frac{R_\mathrm{edge}}{R_\mathrm{eq}} \biggr)^{-3\gamma_c} \, ,</math>

and,

<math>~\biggl[ \frac{P}{P_\mathrm{eq}} \biggr]_\mathrm{env}</math>

<math>~=</math>

<math>~\biggl( \frac{\rho}{\rho_\mathrm{eq}} \biggr)^{\gamma_e} = \biggl( \frac{R_\mathrm{edge}}{R_\mathrm{eq}} \biggr)^{-3\gamma_e} \, .</math>

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,

<math>~P_{ic}</math>

<math>~=</math>

<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>

while material associated with the envelope that lies at the interface will have a pressure given by the relation,

<math>~P_{ie}</math>

<math>~=</math>

<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>

Hence,

<math>~\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>

<math>~=</math>

<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>

 

 

<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>



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,

<math>~1 - p_e(x) </math>

<math>~=</math>

<math>~\frac{1 - p_c(x)}{1-p_c(q)} \, .</math>

We therefore obtain,

<math>~\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>

<math>~=</math>

<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>

 

<math>~=</math>

<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>

 

<math>~=</math>

<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>

as desired.

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> — and, hence, the density ratio <math>~\rho_c/\bar\rho = 1</math> — the mass interior to each radius is,

<math>~\frac{M_r(x)}{M_\mathrm{tot} } </math>

<math>~=</math>

<math>~ \int_0^{x} 3x^2 dx = x^3 \, ,</math>

and the normalized gravitational potential energy is,

<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm} }</math>

<math>~=</math>

<math> - \frac{3}{5} \chi^{-1} \int_0^{1} 5x \biggl\{ x^3\biggr\} dx = -\frac{3}{5} \chi^{-1} \, . </math>

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,

<math>~\frac{T_\mathrm{rot}}{E_\mathrm{norm} }</math>

<math>~=</math>

<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>

 

<math>~=</math>

<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>

 

<math>~=</math>

<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>

 

<math>~=</math>

<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>

<math>~\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>

<math>~=</math>

<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>

<math>~\frac{\mathfrak{S}_I}{E_\mathrm{norm}}</math>

<math>~=</math>

<math>~-3 \ln \chi + \mathrm{constant} \, , </math>

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:

<math>~\frac{W_\mathrm{grav}}{E_\mathrm{norm}}</math>

<math>~=</math>

<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>

<math>~\frac{T_\mathrm{rot}}{E_\mathrm{norm}}</math>

<math>~=</math>

<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>

<math>~\frac{\mathfrak{S}_A}{E_\mathrm{norm}}</math>

<math>~=</math>

<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>

 

<math>~=</math>

<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>

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,

<math>~\mathfrak{f}_M </math>

<math>~\equiv</math>

<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>

<math>~\mathfrak{f}_W</math>

<math>~\equiv</math>

<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>

<math>~\mathfrak{f}_T</math>

<math>~\equiv</math>

<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>

<math>~\mathfrak{f}_A</math>

<math>~\equiv</math>

<math>~ \int_0^1 3\biggl[ \frac{P(x)}{P_c}\biggr] x^2 dx \, .</math>

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 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 <math>~n=5</math> or an <math>~n=1</math> 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 debug numerical algorithms that are designed to calculate structural form factors for more general cases that 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 pressure-truncated polytropes that are embedded in a hot, tenuous external medium and to the 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,

<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>

where,

<math>~A</math>

<math>~\equiv</math>

<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>

<math>~B</math>

<math>~\equiv</math>

<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>

 

<math>~=</math>

<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>

<math>~C</math>

<math>~\equiv</math>

<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>

<math>~D</math>

<math>~\equiv</math>

<math> \biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} \, . </math>

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 <math>~K</math>) — or, in the isothermal case, sound speed (<math>~c_s</math>) — 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

Figure 1: Free Energy Surface

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 — derived from the above expression for <math>\mathfrak{G}^*</math>, with <math>~\gamma_\mathrm{g} = 1</math> and <math>~C=0</math> (see further discussion, below) — is, specifically,

<math> \mathfrak{G}^* = 3000\biggl[ - \frac{1}{\chi} - \ln\chi + \frac{\Pi}{3}\chi^3 + 0.9558 \biggr] \, . </math>

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.

Free Energy Surface

Energy Extrema

As is illustrated in 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>, <math>~K</math>, <math>~J</math>, <math>~P_e</math>, and <math>~\gamma_g</math>, extrema occur wherever,

<math> \frac{d\mathfrak{G^*}}{d\chi} = 0 \, . </math>

For the free energy function identified above,

<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>

so <math>\chi_\mathrm{eq} \equiv R_\mathrm{eq}/R_\mathrm{norm}</math> is obtained from the real root(s) of the equation,

<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>


ASIDE: When we discuss the equilibrium of isothermal, rotating configurations that are immersed in an external medium, we will draw on the work of Weber (1976)Oscillation and Collapse of Interstellar Clouds — and the work of Tohline (1985)Star Formation: Phase Transition, not Jeans Instability — which, in turn draws upon Tohline (1981). In preparation for that discussion, we will go ahead and show how Tohline's (1985) statement of virial equilibrium — his equation (9) — is the same as the equation that defines free energy extrema that has been derived here; and we will show how Weber's (1976) "energy integral" — his equation (B3) — relates to our dimensionless free-energy function.



Tohline1985 Eq9.png

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 Tohline (1985) was only concerned with isothermal systems; then, because Tohline (1985) normalizes each energy term by

<math>~E^* \equiv \biggl( \frac{2^2 \cdot 3^2}{5^3} \biggr) \frac{G^2 M_\mathrm{tot}^5}{J^2} \, ,</math>

instead of by our <math>~E_\mathrm{norm}</math>, we need to multiply our equation through by the ratio,

<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>

With these three modifications, our "free energy extrema" relation becomes,

<math>~0</math>

<math>~=</math>

<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>

Next, because Tohline (1985) 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 Tohline (1981), the leading coefficients of two of our energy terms should be modified to include a factor involving the configuration's eccentricity,

<math>e \equiv \biggl( 1 - \frac{Z_\mathrm{eq}^2}{R_\mathrm{eq}^2} \biggr)^{1/2} \, ,</math>

in order to account for rotational flattening. Properly adjusted, the four coefficients are,

<math>~A</math>

<math>~\equiv</math>

<math>\frac{1}{5} \biggl( \frac{\sin^{-1}e}{e} \biggr) \, ,</math>

<math>~B_I</math>

<math>~\equiv</math>

<math> 1 \, , </math>

<math>~C</math>

<math>~\equiv</math>

<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>

<math>~D</math>

<math>~\equiv</math>

<math> \biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} (1-e^2)^{1/2} \, . </math>

Inserting these coefficient definitions, our "free energy extrema" relation becomes,

<math>~0</math>

<math>~=</math>

<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>

Next we need to appreciate that Tohline (1985) 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,

<math>~\beta \equiv \frac{T_\mathrm{rot}}{|W_\mathrm{grav}|}</math>

<math>~=</math>

<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>

<math>\Rightarrow~~~~\chi_\mathrm{eq}^{-1} </math>

<math>~=</math>

<math> \biggl( \frac{5}{3} \biggr) \frac{E^*}{E_\mathrm{norm}} \biggl( \frac{\sin^{-1}e}{e} \biggr)\beta_\mathrm{eq} \, . </math>

Hence, our expression for the "free energy extrema" becomes,

<math>~0</math>

<math>~=</math>

<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>

 

<math>~=</math>

<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>

Now,

<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>

<math>~=</math>

<math>~\frac{2\pi \cdot 3^3}{5^3} \biggl[ \frac{P_e}{(E^*)^4} \biggr] ( GM_\mathrm{tot}^2)^3</math>

 

<math>~=</math>

<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>

 

<math>~=</math>

<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>

which is the definition of the coefficient "<math>~k</math>" that is provided as equation (7) of Tohline (1985). Hence, dropping the factor of two out front, our expression for "free energy extrema" becomes,

<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>

<math>~=</math>

<math>0 \, .</math>

Finally, realizing that the square of the sound speed is related to our <math>~c_\mathrm{norm}^2</math> via the relation [note that Tohline (1985) uses <math>~a^2</math> in place of <math>~c_s^2</math>],

<math>~c_s^2 = \biggl( \frac{3}{4\pi} \biggr) c_\mathrm{norm}^2 \, ,</math>

it is clear that this last form of our "free energy extrema" expression is identical to Tohline's (1985) virial equilibrium equation (9), which appears in print in a simpler but also more cryptic form as,

<math> \beta_\mathrm{eq} \biggl( \frac{\sin^{-1}e}{e} \biggr)^2 \biggl( \frac{1}{2} - \beta_\mathrm{eq} \biggr) + kV^* - F_s^* </math>

<math>~=</math>

<math>0 \, .</math>




AAAwaiting01.png

Plugging the same set of modified leading coefficients into our derived expression for the free energy becomes,

<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>

Now, recognize that,

<math>~\chi</math>

<math>~=</math>

<math>~\alpha \biggl( \frac{R_0}{R_\mathrm{norm}} \biggr) = \biggl( \frac{2^2}{3\cdot 5} \biggr) \alpha \, ,</math>

<math>~(1 - e^2)^{1/2}</math>

<math>~=</math>

<math>~\frac{Z}{R} = \frac{\gamma}{\alpha} \, ,</math>

<math>~\frac{P_e}{P_\mathrm{norm}}</math>

<math>~=</math>

<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>

<math>~\frac{3\cdot 5}{2^4 \pi} \biggl[ \frac{J^2 c_\mathrm{norm}^2}{G^2 M_\mathrm{tot}^4} \biggr]</math>

<math>~=</math>

<math>~\frac{1}{3} \biggl( \frac{2}{5} J_\mathrm{Weber} \biggr)^2 \, ,</math>

where, for axisymmetric configurations (set <math>~\beta=\alpha</math> in Weber's (1976) equation 12),

<math>J_\mathrm{Weber} \equiv \alpha^2 \Omega = \biggl( \frac{R}{R_0} \biggr)^2 (\dot\varphi_c t_0)^2 \, .</math>

Hence, our expression for the free energy may be written as,

<math>~\mathfrak{G}^*</math>

<math>~=</math>

<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>

 

<math>~=</math>

<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>

<math>\Rightarrow~~~~\frac{4}{3} \mathfrak{G}^*</math>

<math>~=</math>

<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>

The right-hand-side of this expression exactly matches Weber's (1976) "energy integral" for oblate-spheroidal configurations — see his equation (B3) for the case, <math>~e > 0</math> — 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.]


Examples



Work-in-progress.png

Material that appears after this point in our presentation is under development and therefore
may contain incorrect mathematical equations and/or physical misinterpretations.
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BiPolytrope

[Following a discussion that Tohline had with Kundan Kadam on 3 July 2013, 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,

<math>\xi \equiv \frac{r}{r_i}</math>.

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,

<math>M_\xi = \frac{4\pi}{3} \rho_c r_i^3 \xi^3</math> .

The expression for the mass interior to any position within the envelope, <math>1 \le \xi \le \xi_s</math>, is,

<math>M_\xi = \frac{4\pi}{3} r_i^3 \biggl[\rho_c + \rho_e(\xi^3 - 1) \biggr]</math> .

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,

<math>~M_\mathrm{core}</math>

<math>~=</math>

<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>

<math>~M_\mathrm{env}</math>

<math>~=</math>

<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>

<math>~M_\mathrm{tot}</math>

<math>~=</math>

<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>

Following the work of Schönberg & Chandrasekhar (1942) — see our accompanying discussion — we will discuss bipolytropic equilibrium configurations in the context of a <math>~\nu - q</math> plane where,

<math>~\nu</math>

<math>~\equiv</math>

<math>~\frac{M_\mathrm{core}}{M_\mathrm{tot}} \, ,</math>

<math>~q</math>

<math>~\equiv</math>

<math>~\frac{r_i}{R} = \frac{1}{\xi_s} \, .</math>

With this in mind we can write,

<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> ,

and,

<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>

Energy Expressions

The gravitational potential energy of the bipolytropic configuration is obtained by integrating over the following differential energy contribution,

<math>dW_\mathrm{grav} = - \biggl( \frac{GM_r}{r} \biggr) dm</math> .

Hence,

<math>~W_\mathrm{grav} = W_\mathrm{core} + W_\mathrm{env}</math>

<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>

 

<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>

 

<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>

 

<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>

 

<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>

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 — <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 Schönberg & Chandrasekhar (1942), we will rewrite this expression for the gravitational potential energy as,

<math>~W_\mathrm{grav}</math>

<math>~=</math>

<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>

where,

<math>~f\biggl(q, \frac{\rho_e}{\rho_c} \biggr)</math>

<math>~\equiv</math>

<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>

 

<math>~=</math>

<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>


See Also


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation