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=Spherically Symmetric Configurations Synopsis=
=Spherically Symmetric Configurations Synopsis (Using Style Sheet)=


{{LSU_HBook_header}}
{{LSU_HBook_header}}


==Structure==
===Tabular Overview===


==New Table Construction==
{| class="wikitable" style="margin: auto; color:black; width:85%;" border="1" cellpadding="12"
{| class="wikitable" style="margin: auto; color:black;" border="1" cellpadding="12"
|+ style="height:30px;" | <font size="+1">'''Spherically Symmetric Configurations that undergo Adiabatic Compression/Expansion'''</font> &#8212; adiabatic index, <math>~\gamma</math>
|+<font size="+1"><b>Spherically Symmetric Configurations that undergo Adiabatic Compression/Expansion</b></font> &#8212; adiabatic index, <math>~\gamma</math>
|-
|-
! colspan="2" |
! colspan="2" |
Line 50: Line 51:


|-  
|-  
! style="background-color:lightgreen;" colspan="2"|<font size="+1"><b>Equilibrium Structure</b></font>
! style="background-color:lightgreen;" colspan="2"|<b><font size="+1">Equilibrium Structure</font></b>
|-  
|-  
! style="text-allign:center;" width="50%" |<b>Detailed Force Balance</b>
! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">&#x2460;</font></b>&nbsp; &nbsp;<b>Detailed Force Balance</b>
! style="text-allign:center;" |<b>Free-Energy Analysis</b>
! style="text-align:center; background-color:lightblue" |<b><font color="maroon" size="+1">&#x2462;</font></b>&nbsp; &nbsp;<b>Free-Energy Identification of Equilibria</b>
|}
|-  
 
! style="vertical-align:top; text-align:left;" |Given a barotropic equation of state, <math>~P(\rho)</math>, solve the equation of
 
==Old Table Construction==
<table border="1" cellpadding="8" width="85%" align="center">
<tr>
  <td align="center" colspan="2" bgcolor="lightgreen">
<font size="+1"><b>Spherically Symmetric Configurations that undergo Adiabatic Compression/Expansion</b></font> &#8212; adiabatic index, <math>~\gamma</math>
  </td>
</tr>
<tr>
  <td align="center" colspan="2">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~dV = 4\pi r^2 dr</math>
  </td>
  <td align="center">
and 
  </td>
  <td align="left">
&nbsp;&nbsp;&nbsp;<math>~dM_r = \rho dV ~~~\Rightarrow ~~~M_r = 4\pi \int_0^r \rho r^2 dr</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~W_\mathrm{grav}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \int_0^R \biggl(\frac{GM_r}{r}\biggr) dM_r ~~ \propto ~~ R^{-1}</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~U_\mathrm{int}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{(\gamma -1)} \int_0^R 4\pi r^2 P dr ~~ \propto ~~ R^{3-3\gamma}</math>
  </td>
</tr>
</table>
  </td>
</tr>
<tr>
  <td align="center" colspan="2" bgcolor="lightgreen">
<font size="+1"><b>Equilibrium Structure</b></font>
  </td>
</tr>
<tr>
  <th align="center" width="50%">
Detailed Force Balance
  </th>
  <th align="center">
Free-Energy Analysis
  </th>
</tr>
 
<!--  BEGIN MAJOR 4th ROW -->
<tr>
<!--                                          FIRST COLUMN -->
  <td align="left">
<table border="0" cellpadding="5" align="left">
<tr>
  <td align="left">
Given a barotropic equation of state, <math>~P(\rho)</math>, solve the equation of
<div align="center">
<div align="center">
<font color="maroon"><b>Hydrostatic Balance</b></font><br />
<font color="maroon"><b>Hydrostatic Balance</b></font><br />
Line 130: Line 62:
</div>
</div>
for the radial density distribution, <math>~\rho(r)</math>.
for the radial density distribution, <math>~\rho(r)</math>.
</td>
! style="vertical-align:top; text-align:left;" rowspan="3"|The Free-Energy is,
</tr>
</table>
  </td>
<!--                                           THIRD COLUMN -->
  <td align="left" rowspan="3">
The Free-Energy is,
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
Line 157: Line 83:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~-a R^{-1} + bR^{3-3\gamma}+ cR^3 \, .</math>
<math>~-a \biggl(\frac{R}{R_0}\biggr)^{-1} + b\biggl(\frac{R}{R_0}\biggr)^{3-3\gamma}+ c\biggl(\frac{R}{R_0}\biggr)^3 \, .</math>
   </td>
   </td>
</tr>
</tr>
Line 165: Line 91:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\frac{d\mathfrak{G}}{dR}</math>
<math>~R_0 ~\frac{\partial\mathfrak{G}}{\partial R}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 171: Line 97:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~aR^{-2} +(3-3\gamma)bR^{2-3\gamma} + 3cR^2</math>
<math>~a\biggl(\frac{R}{R_0}\biggr)^{-2} +(3-3\gamma)b\biggl(\frac{R}{R_0}\biggr)^{2-3\gamma} + 3c\biggl(\frac{R}{R_0}\biggr)^2</math>
   </td>
   </td>
</tr>
</tr>
Line 182: Line 108:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\frac{1}{R}\biggl[ -W_\mathrm{grav} - 3(\gamma-1)U_\mathrm{int} + 3P_eV\biggr]</math>
<math>~\frac{R_0}{R}\biggl[ -W_\mathrm{grav} - 3(\gamma-1)U_\mathrm{int} + 3P_eV\biggr]</math>
   </td>
   </td>
</tr>
</tr>
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</tr>
</tr>
</table>
</table>
  </td>
|-
</tr>
! style="text-align:center; background-color:#ffff99;" |<b><font color="maroon" size="+1">&#x2461;</font></b>&nbsp; &nbsp;<b>Virial Equilibrium</b>
<!-- END MAJOR 4th ROW -->
|-
<tr>
! style="vertical-align:top; text-align:left;" |
  <th align="center">Virial Equilibrium</th>
</tr>
<tr>
  <td align="left">
<table border="0" cellpadding="5" align="left">
<tr>
  <td align="left">
Multiply the hydrostatic-balance equation through by <math>~rdV</math> and integrate over the volume:
Multiply the hydrostatic-balance equation through by <math>~rdV</math> and integrate over the volume:
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
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</tr>
</tr>
</table>
</table>
</td>
|}
</tr>
 
</table>
===Pointers to Relevant Chapters===
  </td>
 
</tr>
<!-- BACKGROUND MATERIAL -->
<tr>
 
  <td align="center" colspan="2" bgcolor="lightgreen">
<font size="+1" color="maroon"><b>&#x24EA; </b></font> Background Material:
<font size="+1"><b>Stability Analysis</b></font>
{| class="wikitable" style="margin: auto; color:black; width:100%;" border="0" cellpadding="5"
  </td>
|-
</tr>
! width="30px" style="text-align:right; vertical-align:top; "|&#x000B7;
<tr>
|[[User:Tohline/PGE#Principal_Governing_Equations|Principal Governing Equations]] (PGEs) in most general form being considered throughout this H_Book
  <th align="center" width="50%">
|-
Perturbation Theory
! width="30px" style="text-align:right; vertical-align:top; "|&#x000B7;
  </th>
|PGEs in a form that is relevant to a study of the ''Structure, Stability, &amp; Dynamics'' of [[User:Tohline/SphericallySymmetricConfigurations/PGE|spherically symmetric systems]]
  <th align="center">
|-
Free-Energy Analysis
! width="30px" style="text-align:right; vertical-align:top; "|&#x000B7;
  </th>
|[[User:Tohline/SR#Supplemental_Relations|Supplemental relations]] &#8212; see, especially, [[User:Tohline/SR#Barotropic_Structure|barotropic equations of state]]
</tr>
|}
<!-- BEGIN MAJOR STABILITY ROW -->
 
<tr>
<!-- DETAILED FORCE BALANCE -->
<!-- BEGIN 1ST LEFT STABILITY COLUMN -->
 
  <td align="left">
<font size="+1" color="maroon"><b>&#x2460; </b></font> Detailed Force Balance:
{| class="wikitable" style="margin: auto; color:black; width:100%;" border="0" cellpadding="5"
|-
! width="30px" style="text-align:right; vertical-align:top; "|&#x000B7;
|[[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29|Derivation of the equation of Hydrostatic Balance]], and a description of several standard strategies that are used to determine its solution &#8212; see, especially, what we refer to as [[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies#Technique_1|Technique 1]]
|}
 
<!-- VIRIAL EQUILIBRIUM -->
 
<font size="+1" color="maroon"><b>&#x2461; </b></font> Virial Equilibrium:
{| class="wikitable" style="margin: auto; color:black; width:100%;" border="0" cellpadding="5"
|-
! width="30px" style="text-align:right; vertical-align:top; "|&#x000B7;
|Formal derivation of the multi-dimensional, [[User:Tohline/VE#Second-Order_Tensor_Virial_Equations|2<sup>nd</sup>-order tensor virial equations]]
|-
! width="30px" style="text-align:right; vertical-align:top; "|&#x000B7;
|[[User:Tohline/VE#Scalar_Virial_Theorem|Scalar Virial Theorem]], as appropriate for spherically symmetric configurations
|-
! width="30px" style="text-align:right; vertical-align:top; "|&#x000B7;
|[[User:Tohline/VE#Generalization|Generalization]] of scalar virial theorem to include the bounding effects of a hot, tenuous external medium
|}
 
==Stability==
===Isolated &amp; Pressure-Truncated Configurations===
 
{| class="wikitable" style="margin: auto; color:black; width:85%;" border="1" cellpadding="12"
|-
! style="background-color:lightgreen;" colspan="2"|<font size="+1"><b>Stability Analysis: &nbsp; Applicable to Isolated & Pressure-Truncated Configurations</b></font>
|-
! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">&#x2463;</font></b>&nbsp; &nbsp;<b>Perturbation Theory</b>
! style="text-align:center; background-color:lightblue;" |<b><font color="maroon" size="+1">&#x2466;</font></b>&nbsp; &nbsp;<b>Free-Energy Analysis of Stability</b>
 
|-  
! style="vertical-align:top; text-align:left;" |
Given the radial profile of the density and pressure in the equilibrium configuration, solve the [[User:Tohline/SSC/VariationalPrinciple#Ledoux_and_Pekeris_.281941.29|eigenvalue problem defined]] by the,
Given the radial profile of the density and pressure in the equilibrium configuration, solve the [[User:Tohline/SSC/VariationalPrinciple#Ledoux_and_Pekeris_.281941.29|eigenvalue problem defined]] by the,
<div align="center">
<div align="center">
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   </td>
   </td>
</tr>
</tr>
<tr><td align="center" colspan="3">
[<b>[[User:Tohline/Appendix/References#P00|<font color="red">P00</font>]]</b>], Vol. II, &sect;3.7.1, p. 174, Eq. (3.145)
</td></tr>
</table>
</table>
</div>
</div>
to find one or more radially dependent, radial-displacement eigenvectors, <math>~x \equiv \delta r/r</math>, along with (the square of) the corresponding oscillation eigenfrequency, <math>~\omega^2</math>.
to find one or more radially dependent, radial-displacement eigenvectors, <math>~x \equiv \delta r/r</math>, along with (the square of) the corresponding oscillation eigenfrequency, <math>~\omega^2</math>.
  </td>
! style="vertical-align:top; text-align:left;" rowspan="5"|
<!-- END 1ST LEFT STABILITY COLUMN -->
<!--  BEGIN 1ST RIGHT STABILITY COLUMN -->
  <td align="left" rowspan="5">
The second derivative of the free-energy function is,
The second derivative of the free-energy function is,
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\frac{d^2 \mathfrak{G}}{dR^2}</math>
<math>~R_0^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 328: Line 280:
   <td align="left">
   <td align="left">
<math>~
<math>~
-2aR^{-3} + (3-3\gamma)(2-3\gamma)b R^{1-3\gamma} + 6cR
-2a\biggl(\frac{R}{R_0}\biggr)^{-3} + (3-3\gamma)(2-3\gamma)b \biggl(\frac{R}{R_0}\biggr)^{1-3\gamma} + 6c\biggl(\frac{R}{R_0}\biggr)
</math>
</math>
   </td>
   </td>
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   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\frac{1}{R^2}\biggl[
<math>~\biggl(\frac{R_0}{R} \biggr)^2\biggl[
2W_\mathrm{grav} - 3(\gamma-1)(2-3\gamma)U_\mathrm{int} + 6P_e V  
2W_\mathrm{grav} - 3(\gamma-1)(2-3\gamma)U_\mathrm{int} + 6P_e V  
\biggr] \, .
\biggr] \, .
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<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\Rightarrow~~~ R^2 \biggl[\frac{d^2\mathfrak{G}}{dR^2}\biggr]_\mathrm{equil}</math>
<math>~\Rightarrow~~~ R^2 \biggl[\frac{\partial^2\mathfrak{G}}{\partial R^2}\biggr]_\mathrm{equil}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
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</tr>
</tr>
</table>
</table>
Note the similarity with <b><font color="maroon" size="+1">&#x2465;</font></b>.
----
Alternatively, recalling that,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~3(\gamma - 1)U_\mathrm{int}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2S_\mathrm{therm} \, ,
</math>
   </td>
   </td>
<!--  END 1ST RIGHT STABILITY COLUMN -->
</tr>
</tr>
</table>
the conditions for virial equilibrium and stability, may be written respectively as,
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <th align="center" width="50%">
   <td align="right">
Variational Principle
<math>~3P_e V</math>
   </th>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ 2S_\mathrm{therm}+ W_\mathrm{grav} </math>
   </td>
</tr>
</tr>


<!--  BEGIN ANOTHER MAJOR STABILITY ROW -->
<tr>
<tr>
<!--  BEGIN 2ND LEFT STABILITY COLUMN -->
  <td align="right">
<math>~\Rightarrow~~~ R^2 \biggl[\frac{\partial^2\mathfrak{G}}{\partial R^2}\biggr]_\mathrm{equil}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
   <td align="left">
   <td align="left">
<math>~
2W_\mathrm{grav} - 2(2-3\gamma)S_\mathrm{therm} + 2 \biggl[ 2S_\mathrm{therm}+ W_\mathrm{grav}  \biggr]
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
4W_\mathrm{grav} + 6\gamma S_\mathrm{therm}  \, .
</math>
  </td>
</tr>
</table>
|-
! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">&#x2464;</font></b>&nbsp; &nbsp;<b>Variational Principle</b>
|-
! style="vertical-align:top; text-align:left;" |
Multiply the LAWE through by <math>~4\pi x dr</math>, and integrate over the volume of the configuration gives the,
Multiply the LAWE through by <math>~4\pi x dr</math>, and integrate over the volume of the configuration gives the,
<div align="center">
<div align="center">
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</div>
</div>


|-
! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">&#x2465;</font></b>&nbsp; &nbsp;<b>Approximation: &nbsp; Homologous Expansion/Contraction</b>
|-
! style="vertical-align:top; text-align:left;" |
If we ''guess'' that radial oscillations about the equilibrium state involve purely homologous expansion/contraction, then the radial-displacement eigenfunction is, <math>~x</math> = constant, and the governing variational relation gives,
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\omega^2 \int_0^R  r^2  dM_r</math>
  </td>
  <td align="center">
<math>~\leq</math>
  </td>
  <td align="left">
<math>~
(4- 3\gamma) W_\mathrm{grav}+ 3^2 \gamma  P_eV \, .
</math>
  </td>
</tr>
</table>
</div>
|}
===Bipolytropes===
{| class="wikitable" style="margin: auto; color:black; width:85%;" border="1" cellpadding="12"
|-
! style="background-color:lightgreen;" colspan="2"|<font size="+1"><b>Stability Analysis: &nbsp; Applicable to Bipolytropic Configurations</b></font>
|-
! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">&#x2467;</font></b>&nbsp; &nbsp;<b>Variational Principle</b>
! style="text-align:center; background-color:lightblue;" |<b><font color="maroon" size="+1">&#x2469;</font></b>&nbsp; &nbsp;<b>Free-Energy Analysis of Stability</b>
|-
! style="vertical-align:top; text-align:left;" |
<div align="center">
<font color="#770000">'''Governing Variational Relation'''</font><br />
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\biggl( \frac{2\pi}{3}\biggr)\sigma_c^2 \int_0^{R^*} (x r^*)^2  dM_r^* </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\gamma_c (\gamma_c-1) \int_0^{r^*_\mathrm{core}}  x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dU^*_\mathrm{int}
- (3\gamma_c - 4) \int_0^{r^*_\mathrm{core}}  x^2 dW^*_\mathrm{grav}
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ ~\gamma_e (\gamma_e-1) \int_{r^*_\mathrm{core}}^{R^*} x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dU^*_\mathrm{int}
- (3\gamma_e - 4)  \int_{r^*_\mathrm{core}}^{R^*} x^2 dW^*_\mathrm{grav}
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ ~3^2(\gamma_c - \gamma_e) x_i^2 P_i^* V_\mathrm{core}^*  \, .
</math>
  </td>
</tr>
</table>
</div>
! style="vertical-align:top; text-align:left;" rowspan="3"|
As we have detailed in an [[User:Tohline/SSC/BipolytropeGeneralization#Free_Energy_and_Its_Derivatives|accompanying discussion]], the first derivative of the relevant free-energy expression is,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~R ~\frac{\partial \mathfrak{G}}{\partial R}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2S_\mathrm{tot} + W_\mathrm{tot}
\, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~S_\mathrm{tot} \equiv S_\mathrm{core} + S_\mathrm{env}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; and &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~W_\mathrm{tot} \equiv W_\mathrm{core} + W_\mathrm{env} \, ;</math>
  </td>
</tr>
</table>
and the second derivative of that free-energy function is,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~R^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~2\biggl[
W_\mathrm{tot} + (3\gamma_c - 2) S_\mathrm{core} + (3\gamma_e-2)S_\mathrm{env}
\biggr] \, .
</math>
  </td>
</tr>
</table>
----
This stability criterion may be rewritten as,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\biggl[ R^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2} \biggr]_\mathrm{equil}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
<td align="left">
<math>~
2[(3\gamma_c -4) S_\mathrm{core}
+ (3\gamma_e -4) S_\mathrm{env} ] \, .
</math>
   </td>
   </td>
<!--  END 1ST LEFT STABILITY COLUMN -->
</tr>
</tr>
</table>
Hence, in bipolytropes, the marginally unstable equilibrium configuration (second derivative of free-energy set to zero) will be identified by the model that exhibits the ratio,
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <th align="center" width="50%">
   <td align="right">
Approximation: &nbsp; Homologous Expansion/Contraction
<math>~\frac{S_\mathrm{core}}{S_\mathrm{env}}</math>
   </th>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{(3\gamma_e - 4)}{(4 - 3\gamma_c)} \, .
</math>
   </td>
</tr>
</tr>
</table>


<!-- BEGIN ANOTHER MAJOR STABILITY ROW -->
See the [[User:Tohline/SSC/Stability/BiPolytropes#What_to_Expect_for_Equilibrium_Configurations|accompanying discussion]].
 
----
 
If &#8212; based for example on <b><font color="maroon" size="+1">&#x2466;</font></b> &#8212; we make the reasonable assumption that, in equilibrium, the statements,
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
<!-- BEGIN 2ND LEFT STABILITY COLUMN -->
  <td align="right">
<math>~2S_\mathrm{core} = 3P_i V_\mathrm{core} - W_\mathrm{core}</math>
  </td>
  <td align="center">
&nbsp; &nbsp; and &nbsp; &nbsp;
  </td>
  <td align="left">
<math>~2S_\mathrm{env} = - 3P_i V_\mathrm{core} - W_\mathrm{env} \, ,</math>
  </td>
</tr>
</table>
hold separately, then we satisfy the virial equilibrium condition, namely,
<table border="0" cellpadding="5" align="center">
 
<tr>
  <td align="right">
<math>~0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
   <td align="left">
   <td align="left">
<math>~2S_\mathrm{tot} + W_\mathrm{tot} \, ,</math>
  </td>
</tr>
</table>
and the second derivative of the relevant free-energy function can be rewritten as,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\biggl[ R^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2} \biggr]_\mathrm{equil}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
2(W_\mathrm{core} + W_\mathrm{env})
+ (3\gamma_c - 2) (3P_i V_\mathrm{core} - W_\mathrm{core})
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ (3\gamma_e-2)(-3P_i V_\mathrm{core} - W_\mathrm{env})
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
3^2 P_i V_\mathrm{core}(\gamma_c - \gamma_e)
+ (4-3\gamma_c ) W_\mathrm{core}
+ (4-3\gamma_e)W_\mathrm{env} \, .
</math>
  </td>
</tr>
</table>
Note the similarity with <b><font color="maroon" size="+1">&#x2468;</font></b> &#8212; temporarily, see [[User:Tohline/SSC/Stability/BiPolytropes#Revised_Free-Energy_Analysis|this discussion]].
|-
! style="text-align:center; background-color:#ffff99;" width="50%" |<b><font color="maroon" size="+1">&#x2468;</font></b>&nbsp; &nbsp;<b>Approximation: &nbsp; Homologous Expansion/Contraction</b>
|-
! style="vertical-align:top; text-align:left;" |
If we ''guess'' that radial oscillations about the equilibrium state involve purely homologous expansion/contraction, then the radial-displacement eigenfunction is, <math>~x</math> = constant, and the governing variational relation gives,
If we ''guess'' that radial oscillations about the equilibrium state involve purely homologous expansion/contraction, then the radial-displacement eigenfunction is, <math>~x</math> = constant, and the governing variational relation gives,


Line 508: Line 765:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\omega^2 \int_0^R  r^2  dM_r</math>  
<math>~\biggl( \frac{2\pi}{3}\biggr)\sigma_c^2 \int_0^R  r^2  dM_r</math>  
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\approx</math>
<math>~\leq</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~
(4- 3\gamma) W_\mathrm{grav}+ 3^2 \gamma   P_eV \, .
(4- 3\gamma_c) W_\mathrm{core}+ (4- 3\gamma_e) W_\mathrm{env}+ 3^2 (\gamma_c - \gamma_e)   P_i V_\mathrm{core} \, .
</math>
</math>
   </td>
   </td>
Line 521: Line 778:
</table>
</table>
</div>
</div>
  </td>
 
</tr>
|}
</table>


=See Also=
=See Also=

Latest revision as of 23:02, 4 February 2019


Spherically Symmetric Configurations Synopsis (Using Style Sheet)

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

Tabular Overview

Spherically Symmetric Configurations that undergo Adiabatic Compression/Expansion — adiabatic index, <math>~\gamma</math>

<math>~dV = 4\pi r^2 dr</math>

and

   <math>~dM_r = \rho dV ~~~\Rightarrow ~~~M_r = 4\pi \int_0^r \rho r^2 dr</math>

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

<math>~=</math>

<math>~- \int_0^R \biggl(\frac{GM_r}{r}\biggr) dM_r ~~ \propto ~~ R^{-1}</math>

<math>~U_\mathrm{int}</math>

<math>~=</math>

<math>~\frac{1}{(\gamma -1)} \int_0^R 4\pi r^2 P dr ~~ \propto ~~ R^{3-3\gamma}</math>

Equilibrium Structure
   Detailed Force Balance    Free-Energy Identification of Equilibria
Given a barotropic equation of state, <math>~P(\rho)</math>, solve the equation of

Hydrostatic Balance

LSU Key.png

<math>~\frac{dP}{dr} = - \frac{GM_r \rho}{r^2}</math>

for the radial density distribution, <math>~\rho(r)</math>.

The Free-Energy is,

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

<math>~=</math>

<math>~W_\mathrm{grav} + U_\mathrm{int} + P_eV</math>

 

<math>~=</math>

<math>~-a \biggl(\frac{R}{R_0}\biggr)^{-1} + b\biggl(\frac{R}{R_0}\biggr)^{3-3\gamma}+ c\biggl(\frac{R}{R_0}\biggr)^3 \, .</math>

Therefore, also,

<math>~R_0 ~\frac{\partial\mathfrak{G}}{\partial R}</math>

<math>~=</math>

<math>~a\biggl(\frac{R}{R_0}\biggr)^{-2} +(3-3\gamma)b\biggl(\frac{R}{R_0}\biggr)^{2-3\gamma} + 3c\biggl(\frac{R}{R_0}\biggr)^2</math>

 

<math>~=</math>

<math>~\frac{R_0}{R}\biggl[ -W_\mathrm{grav} - 3(\gamma-1)U_\mathrm{int} + 3P_eV\biggr]</math>

Equilibrium configurations exist at extrema of the free-energy function, that is, they are identified by setting <math>~d\mathfrak{G}/dR = 0</math>. Hence, equilibria are defined by the condition,

<math>~0</math>

<math>~=</math>

<math>~W_\mathrm{grav} + 3(\gamma-1)U_\mathrm{int} - 3P_eV\, .</math>

   Virial Equilibrium

Multiply the hydrostatic-balance equation through by <math>~rdV</math> and integrate over the volume:

<math>~0</math>

<math>~=</math>

<math>~-\int_0^R r\biggl(\frac{dP}{dr}\biggr)dV - \int_0^R r\biggl(\frac{GM_r \rho}{r^2}\biggr)dV</math>

 

<math>~=</math>

<math>~-\int_0^R 4\pi r^3 \biggl(\frac{dP}{dr}\biggr) dr - \int_0^R \biggl(\frac{GM_r}{r}\biggr)dM_r</math>

 

<math>~=</math>

<math>~-\int_0^R\biggl[ \frac{d}{dr}\biggl( 4\pi r^3P \biggr) - 12\pi r^2 P\biggr] dr + W_\mathrm{grav}</math>

 

<math>~=</math>

<math>~\int_0^R 3\biggl[ 4\pi r^2 P \biggr]dr - \int_0^R \biggl[ d(3PV)\biggr] + W_\mathrm{grav}</math>

 

<math>~=</math>

<math>~3(\gamma-1)U_\mathrm{int} + W_\mathrm{grav} - \biggl[ 3PV \biggr]_0^R \, .</math>

Pointers to Relevant Chapters

Background Material:

· Principal Governing Equations (PGEs) in most general form being considered throughout this H_Book
· PGEs in a form that is relevant to a study of the Structure, Stability, & Dynamics of spherically symmetric systems
· Supplemental relations — see, especially, barotropic equations of state


Detailed Force Balance:

· Derivation of the equation of Hydrostatic Balance, and a description of several standard strategies that are used to determine its solution — see, especially, what we refer to as Technique 1


Virial Equilibrium:

· Formal derivation of the multi-dimensional, 2nd-order tensor virial equations
· Scalar Virial Theorem, as appropriate for spherically symmetric configurations
· Generalization of scalar virial theorem to include the bounding effects of a hot, tenuous external medium

Stability

Isolated & Pressure-Truncated Configurations

Stability Analysis:   Applicable to Isolated & Pressure-Truncated Configurations
   Perturbation Theory    Free-Energy Analysis of Stability

Given the radial profile of the density and pressure in the equilibrium configuration, solve the eigenvalue problem defined by the,

LAWE:   Linear Adiabatic Wave (or Radial Pulsation) Equation

<math>~0</math>

<math>~=</math>

<math>~ \frac{d}{dr}\biggl[ r^4 \gamma P ~\frac{dx}{dr} \biggr] +\biggl[ \omega^2 \rho r^4 + (3\gamma - 4) r^3 \frac{dP}{dr} \biggr] x </math>

[P00], Vol. II, §3.7.1, p. 174, Eq. (3.145)

to find one or more radially dependent, radial-displacement eigenvectors, <math>~x \equiv \delta r/r</math>, along with (the square of) the corresponding oscillation eigenfrequency, <math>~\omega^2</math>.

The second derivative of the free-energy function is,

<math>~R_0^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2}</math>

<math>~=</math>

<math>~ -2a\biggl(\frac{R}{R_0}\biggr)^{-3} + (3-3\gamma)(2-3\gamma)b \biggl(\frac{R}{R_0}\biggr)^{1-3\gamma} + 6c\biggl(\frac{R}{R_0}\biggr) </math>

 

<math>~=</math>

<math>~\biggl(\frac{R_0}{R} \biggr)^2\biggl[ 2W_\mathrm{grav} - 3(\gamma-1)(2-3\gamma)U_\mathrm{int} + 6P_e V \biggr] \, . </math>

Evaluating this second derivative for an equilibrium configuration — that is by calling upon the (virial) equilibrium condition to set the value of the internal energy — we have,

<math>~3(\gamma-1)U_\mathrm{int}</math>

<math>~=</math>

<math>~3P_e V - W_\mathrm{grav} </math>

<math>~\Rightarrow~~~ R^2 \biggl[\frac{\partial^2\mathfrak{G}}{\partial R^2}\biggr]_\mathrm{equil}</math>

<math>~=</math>

<math>~2W_\mathrm{grav} - (2-3\gamma)\biggl[3P_e V - W_\mathrm{grav} \biggr] + 6P_e V </math>

 

<math>~=</math>

<math>~(4-3\gamma)W_\mathrm{grav} + 3^2\gamma P_e V \, . </math>

Note the similarity with .



Alternatively, recalling that,

<math>~3(\gamma - 1)U_\mathrm{int}</math>

<math>~=</math>

<math>~2S_\mathrm{therm} \, , </math>

the conditions for virial equilibrium and stability, may be written respectively as,

<math>~3P_e V</math>

<math>~=</math>

<math>~ 2S_\mathrm{therm}+ W_\mathrm{grav} </math>

<math>~\Rightarrow~~~ R^2 \biggl[\frac{\partial^2\mathfrak{G}}{\partial R^2}\biggr]_\mathrm{equil}</math>

<math>~=</math>

<math>~ 2W_\mathrm{grav} - 2(2-3\gamma)S_\mathrm{therm} + 2 \biggl[ 2S_\mathrm{therm}+ W_\mathrm{grav} \biggr] </math>

 

<math>~=</math>

<math>~ 4W_\mathrm{grav} + 6\gamma S_\mathrm{therm} \, . </math>


   Variational Principle

Multiply the LAWE through by <math>~4\pi x dr</math>, and integrate over the volume of the configuration gives the,

Governing Variational Relation

<math>~0</math>

<math>~=</math>

<math>~ \int_0^R 4\pi r^4 \gamma P \biggl(\frac{dx}{dr}\biggr)^2 dr - \int_0^R 4\pi (3\gamma - 4) r^3 x^2 \biggl( \frac{dP}{dr} \biggr) dr </math>

 

 

<math>~ - 4\pi \biggr[r^4 \gamma Px \biggl(\frac{dx}{dr}\biggr) \biggr]_0^R - \int_0^R 4\pi \omega^2 \rho r^4 x^2 dr \, . </math>

 

<math>~=</math>

<math>~ \int_0^R x^2 \biggl(\frac{d\ln x}{d\ln r}\biggr)^2 \gamma 4\pi r^2P dr - \int_0^R (3\gamma - 4)x^2 \biggl( - \frac{GM_r}{r} \biggr) 4\pi \rho r^2 dr </math>

 

 

<math>~ + \biggr[\gamma 4\pi r^3 Px^2 \biggl(-\frac{d\ln x}{d\ln r}\biggr) \biggr]_0^R - \int_0^R 4\pi \omega^2 \rho r^4 x^2 dr \, . </math>

Now, by setting <math>~(d\ln x/d\ln r)_{r=R} = -3</math>, we can ensure that the pressure fluctuation is zero and, hence, <math>~P = P_e</math> at the surface, in which case this relation becomes,

<math>~\omega^2</math>

<math>~=</math>

<math>~ \frac{\gamma (\gamma -1) \int_0^R x^2 \bigl(\frac{d\ln x}{d\ln r}\bigr)^2 dU_\mathrm{int} - \int_0^R (3\gamma - 4)x^2 dW_\mathrm{grav} + 3^2 \gamma x^2 P_eV}{ \int_0^R x^2 r^2 dM_r} </math>

   Approximation:   Homologous Expansion/Contraction

If we guess that radial oscillations about the equilibrium state involve purely homologous expansion/contraction, then the radial-displacement eigenfunction is, <math>~x</math> = constant, and the governing variational relation gives,

<math>~\omega^2 \int_0^R r^2 dM_r</math>

<math>~\leq</math>

<math>~ (4- 3\gamma) W_\mathrm{grav}+ 3^2 \gamma P_eV \, . </math>

Bipolytropes

Stability Analysis:   Applicable to Bipolytropic Configurations
   Variational Principle    Free-Energy Analysis of Stability

Governing Variational Relation

<math>~\biggl( \frac{2\pi}{3}\biggr)\sigma_c^2 \int_0^{R^*} (x r^*)^2 dM_r^* </math>

<math>~=</math>

<math>~ \gamma_c (\gamma_c-1) \int_0^{r^*_\mathrm{core}} x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dU^*_\mathrm{int} - (3\gamma_c - 4) \int_0^{r^*_\mathrm{core}} x^2 dW^*_\mathrm{grav} </math>

 

 

<math>~ + ~\gamma_e (\gamma_e-1) \int_{r^*_\mathrm{core}}^{R^*} x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dU^*_\mathrm{int} - (3\gamma_e - 4) \int_{r^*_\mathrm{core}}^{R^*} x^2 dW^*_\mathrm{grav} </math>

 

 

<math>~ + ~3^2(\gamma_c - \gamma_e) x_i^2 P_i^* V_\mathrm{core}^* \, . </math>

As we have detailed in an accompanying discussion, the first derivative of the relevant free-energy expression is,

<math>~R ~\frac{\partial \mathfrak{G}}{\partial R}</math>

<math>~=</math>

<math>~ 2S_\mathrm{tot} + W_\mathrm{tot} \, , </math>

where,

<math>~S_\mathrm{tot} \equiv S_\mathrm{core} + S_\mathrm{env}</math>

    and    

<math>~W_\mathrm{tot} \equiv W_\mathrm{core} + W_\mathrm{env} \, ;</math>

and the second derivative of that free-energy function is,

<math>~R^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2}</math>

<math>~=</math>

<math>~2\biggl[ W_\mathrm{tot} + (3\gamma_c - 2) S_\mathrm{core} + (3\gamma_e-2)S_\mathrm{env} \biggr] \, . </math>



This stability criterion may be rewritten as,

<math>~\biggl[ R^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2} \biggr]_\mathrm{equil}</math>

<math>~=</math>

<math>~ 2[(3\gamma_c -4) S_\mathrm{core} + (3\gamma_e -4) S_\mathrm{env} ] \, . </math>

Hence, in bipolytropes, the marginally unstable equilibrium configuration (second derivative of free-energy set to zero) will be identified by the model that exhibits the ratio,

<math>~\frac{S_\mathrm{core}}{S_\mathrm{env}}</math>

<math>~=</math>

<math>~ \frac{(3\gamma_e - 4)}{(4 - 3\gamma_c)} \, . </math>

See the accompanying discussion.


If — based for example on — we make the reasonable assumption that, in equilibrium, the statements,

<math>~2S_\mathrm{core} = 3P_i V_\mathrm{core} - W_\mathrm{core}</math>

    and    

<math>~2S_\mathrm{env} = - 3P_i V_\mathrm{core} - W_\mathrm{env} \, ,</math>

hold separately, then we satisfy the virial equilibrium condition, namely,

<math>~0</math>

<math>~=</math>

<math>~2S_\mathrm{tot} + W_\mathrm{tot} \, ,</math>

and the second derivative of the relevant free-energy function can be rewritten as,

<math>~\biggl[ R^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2} \biggr]_\mathrm{equil}</math>

<math>~=</math>

<math>~ 2(W_\mathrm{core} + W_\mathrm{env}) + (3\gamma_c - 2) (3P_i V_\mathrm{core} - W_\mathrm{core}) </math>

 

 

<math>~ + (3\gamma_e-2)(-3P_i V_\mathrm{core} - W_\mathrm{env}) </math>

 

<math>~=</math>

<math>~ 3^2 P_i V_\mathrm{core}(\gamma_c - \gamma_e) + (4-3\gamma_c ) W_\mathrm{core} + (4-3\gamma_e)W_\mathrm{env} \, . </math>

Note the similarity with — temporarily, see this discussion.

   Approximation:   Homologous Expansion/Contraction

If we guess that radial oscillations about the equilibrium state involve purely homologous expansion/contraction, then the radial-displacement eigenfunction is, <math>~x</math> = constant, and the governing variational relation gives,

<math>~\biggl( \frac{2\pi}{3}\biggr)\sigma_c^2 \int_0^R r^2 dM_r</math>

<math>~\leq</math>

<math>~ (4- 3\gamma_c) W_\mathrm{core}+ (4- 3\gamma_e) W_\mathrm{env}+ 3^2 (\gamma_c - \gamma_e) P_i V_\mathrm{core} \, . </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