Editing SSC/SynopsisStyleSheet (section)

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

{| class="Synopsis1A" style="margin: auto; color:black; width:85%;" 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>
|-
! colspan="2" |

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<math>dV = 4\pi r^2 dr</math> 
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and   
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&nbsp;&nbsp;&nbsp;<math>dM_r = \rho dV ~~~\Rightarrow ~~~M_r = 4\pi \int_0^r \rho r^2 dr</math>
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<math>W_\mathrm{grav}</math>
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<math>=</math>
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<math>- \int_0^R \biggl(\frac{GM_r}{r}\biggr) dM_r ~~ \propto ~~ R^{-1}</math>
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<math>U_\mathrm{int}</math>
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<math>=</math>
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<math>\frac{1}{(\gamma -1)} \int_0^R 4\pi r^2 P dr ~~ \propto ~~ R^{3-3\gamma}</math>
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|- 
! style="background-color:lightgreen;" colspan="2"|<b><font size="+1">Equilibrium Structure</font></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-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
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<font color="maroon"><b>Hydrostatic Balance</b></font><br />
{{ Math/EQ_SShydrostaticBalance01 }}
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for the radial density distribution, <math>\rho(r)</math>.
! style="vertical-align:top; text-align:left;" rowspan="3"|The Free-Energy is,
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<math>\mathfrak{G}</math>
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<math>=</math>
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<math>W_\mathrm{grav} + U_\mathrm{int} + P_eV</math>
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&nbsp;
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<math>=</math>
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<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>
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Therefore, also,  
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<math>R_0 ~\frac{\partial\mathfrak{G}}{\partial R}</math>
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<math>=</math>
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<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>
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&nbsp;
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<math>=</math>
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<math>\frac{R_0}{R}\biggl[ -W_\mathrm{grav} - 3(\gamma-1)U_\mathrm{int} + 3P_eV\biggr]</math>
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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,
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<math>0</math>
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<math>=</math>
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<math>W_\mathrm{grav} + 3(\gamma-1)U_\mathrm{int} - 3P_eV\, .</math>
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|- 
! style="text-align:center; background-color:#ffff99;" |<b><font color="maroon" size="+1">&#x2461;</font></b>&nbsp; &nbsp;<b>Virial Equilibrium</b>
|-
! style="vertical-align:top; text-align:left;" |
Multiply the hydrostatic-balance equation through by <math>rdV</math> and integrate over the volume:
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<math>0</math>
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<math>=</math>
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<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>
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&nbsp;
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<math>=</math>
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<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>
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&nbsp;
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<math>=</math>
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<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>
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&nbsp;
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<math>=</math>
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<math>\int_0^R 3\biggl[ 4\pi r^2 P \biggr]dr - \int_0^R \biggl[ d(3PV)\biggr] + W_\mathrm{grav}</math>
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&nbsp;
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<math>=</math>
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<math>3(\gamma-1)U_\mathrm{int}  + W_\mathrm{grav} - \biggl[ 3PV \biggr]_0^R \, .</math>
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|}

===Pointers to Relevant Chapters===

<!-- BACKGROUND MATERIAL -->

<font size="+1" color="maroon"><b>&#x24EA; </b></font> Background Material:
{| class="Synopsis1B" style="margin: auto; color:black; width:100%;" border="0" cellpadding="5"
|-
! width="30px" style="text-align:right; vertical-align:top; "|&#x000B7; 
|[[PGE#Principal_Governing_Equations|Principal Governing Equations]] (PGEs) in most general form being considered throughout this H_Book
|-
! width="30px" style="text-align:right; vertical-align:top; "|&#x000B7; 
|PGEs in a form that is relevant to a study of the ''Structure, Stability, &amp; Dynamics'' of [[SSCpt1/PGE|spherically symmetric systems]]
|-
! width="30px" style="text-align:right; vertical-align:top; "|&#x000B7; 
|[[SR#Supplemental_Relations|Supplemental relations]] &#8212; see, especially, [[SR#Barotropic_Structure|barotropic equations of state]]
|}

<!-- DETAILED FORCE BALANCE -->

<font size="+1" color="maroon"><b>&#x2460; </b></font> Detailed Force Balance:
{| class="Synopsis1C" style="margin: auto; color:black; width:100%;" border="0" cellpadding="5"
|-
! width="30px" style="text-align:right; vertical-align:top; "|&#x000B7; 
|[[SSCpt2/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 [[SSCpt2/SolutionStrategies#Technique_1|Technique 1]]
|}

<!-- VIRIAL EQUILIBRIUM -->

<font size="+1" color="maroon"><b>&#x2461; </b></font> Virial Equilibrium:
{| class="Synopsis1D" 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, [[VE#Second-Order_Tensor_Virial_Equations|2<sup>nd</sup>-order tensor virial equations]] 
|-
! width="30px" style="text-align:right; vertical-align:top; "|&#x000B7; 
|[[VE#Scalar_Virial_Theorem|Scalar Virial Theorem]], as appropriate for spherically symmetric configurations
|-
! width="30px" style="text-align:right; vertical-align:top; "|&#x000B7; 
|[[VE#Generalization|Generalization]] of scalar virial theorem to include the bounding effects of a hot, tenuous external medium
|}