Difference between revisions of "User:Tohline/SSC/Virial/PolytropesEmbeddedOutline"

From VistrailsWiki
Jump to navigation Jump to search
(→‎The Physics: Finish establishing Whitworth's P_e(R_eq) relation, which is essentially the scalar virial theorem)
(→‎The Physics: Swap the order of a few subsections, putting discussion of Whitworth's work first)
Line 230: Line 230:
In the first case, the analysis reveals how <math>~R_\mathrm{eq}</math> varies with the applied external pressure and usually is displayed as a <math>~P_e(R_\mathrm{eq})</math> function.  The second case identifies a mass-radius relationship for the polytropic sequence under consideration and is usually displayed as a <math>~M_\mathrm{limit}(R_\mathrm{eq})</math> function.  (In the figure at the top of this page, a "Case M" mass-radius relation for pressure-truncated, <math>~n = 5</math> polytropic configurations is traced by the sequence of a dozen, small colored spherical dots that each reside at an extremum in the displayed free-energy function.)
In the first case, the analysis reveals how <math>~R_\mathrm{eq}</math> varies with the applied external pressure and usually is displayed as a <math>~P_e(R_\mathrm{eq})</math> function.  The second case identifies a mass-radius relationship for the polytropic sequence under consideration and is usually displayed as a <math>~M_\mathrm{limit}(R_\mathrm{eq})</math> function.  (In the figure at the top of this page, a "Case M" mass-radius relation for pressure-truncated, <math>~n = 5</math> polytropic configurations is traced by the sequence of a dozen, small colored spherical dots that each reside at an extremum in the displayed free-energy function.)


===Our Case P Analysis (raw)===
===Whitworth's (1981) Case P Analysis===
 
[http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth's (1981)] related <font color="red">'''Case P'''</font> analysis of pressure-truncated polytropic spheres produces the following governing free-energy function &#8212; referred to by Whitworth as the "global potential function":
====Coefficient Definitions====
<div align="center" id="WhitworthFreeEnergyExpression">
As has been both summarized and detailed in [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Free_Energy_Function_and_Virial_Theorem|an accompanying discussion]], our <font color="red">'''Case P'''</font> analysis has demonstrated that the following,
<div align="center" id="FreeEnergyExpression">
<font color="#770000">'''Algebraic Free-Energy Function'''</font><br />
 
<math>
<math>
\mathfrak{G}^* =  
\frac{2\mathcal{U}}{3M_0 K_1} =  
-3\mathcal{A} \chi^{-1} +~ n\mathcal{B} \chi^{-3/n} +~ \mathcal{D}\chi^3 \, ,
-\frac{3}{2} \biggl( \frac{R}{R_\mathrm{rf}}\biggr)^{-1} +~ \frac{2n}{3}\biggl( \frac{R}{R_\mathrm{rf}}\biggr)^{-3/n}  
+~ \frac{1}{6}\biggl(\frac{P_e}{P_\mathrm{rf}}\biggr) \biggl( \frac{R}{R_\mathrm{rf}}\biggr)^3 \, ;
</math>
</math>
</div>
</div>
properly governs the equilibrium structure and stability of pressure-truncated polytropic configurationsThis algebraic expression is identical to the [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#Overview|free-energy function given above]] if the following coefficient and variable substitutions are made:  
this expression is obtained from Whitworth's equation (10) after setting <math>~\delta_{1\eta} = 0</math>, that is, by choosing to ignore isothermal systems, and after setting <math>~\eta = (n+1)/n</math>, that is, after rewriting his adiabatic exponent <math>~(\eta)</math> in terms of the corresponding polytropic indexWhitworth's expression also is identical to the [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#Overview|free-energy function given above]] if the following coefficient and variable substitutions are made:  
 
<div align="center">
<div align="center">
<table border="1" cellpadding="10" align="center">
<table border="1" cellpadding="10" align="center">
<tr><td align="center">
<tr><td align="center">
Our <font color="red">Case P</font> Analysis
[http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth's (1981)] <font color="red">Case P</font> Analysis
</td></tr>
</td></tr>
<tr><td align="center">
<tr><td align="center">
Line 259: Line 257:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~3\mathcal{A}</math>
<math>~\frac{3}{2}</math>
   </td>
   </td>
</tr>
</tr>
Line 271: Line 269:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~n\mathcal{B} </math>
<math>~\frac{2n}{3}</math>
   </td>
   </td>
</tr>
</tr>
Line 283: Line 281:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\mathcal{D} \equiv \frac{4\pi}{3} \biggl(\frac{P_e}{P_\mathrm{norm}}\biggr)</math>
<math>~\frac{1}{6}\biggl(\frac{P_e}{P_\mathrm{rf}}\biggr)</math>
   </td>
   </td>
</tr>
</tr>
Line 295: Line 293:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\chi \equiv \frac{R}{R_\mathrm{norm}}</math>
<math>~\frac{R}{R_\mathrm{rf}}</math>
   </td>
   </td>
</tr>
</tr>
Line 307: Line 305:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\mathfrak{G}^{*} \equiv \frac{\mathfrak{G}}{E_\mathrm{norm}}</math>
<math>~\frac{2}{3} \biggl( \frac{\mathcal{U}}{\mathcal{U}_\mathrm{rf}} \biggr) </math>
   </td>
   </td>
</tr>
</tr>
Line 314: Line 312:
</table>
</table>
</div>
</div>
where (see an [[User:Tohline/SSC/Structure/PolytropesASIDE1#ASIDE:_Whitworth.27s_Scaling|accompanying ASIDE]]),
<div align="center">
<table border="0" cellpadding="5" align="center">


where &#8212; see, for example, our [[User:Tohline/SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Review|accompanying review]],
<tr>
<div align="center">
  <td align="right">
<table border="0" cellpadding="5">
<math>~R_\mathrm{rf}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>
\biggl[ \frac{\pi}{5^n} \biggl( \frac{2^2}{3}\biggr)^{(n+1)} K^{-n} G^{n} M_\mathrm{limit}^{n-1} \biggr]^{1/(n-3)}
</math>
  </td>
</tr>


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~R_\mathrm{norm}</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 327: Line 338:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\biggl[ \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1} \biggr]^{1/(n-3)} \, ,</math>
<math>~R_\mathrm{norm}
\biggl[ \frac{\pi}{5^n} \biggl( \frac{2^2}{3}\biggr)^{(n+1)} \biggr]^{1/(n-3)}
\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{(n-1)/(n-3)} \, ,
</math>
   </td>
   </td>
</tr>
</tr>
Line 333: Line 347:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~P_\mathrm{norm}</math>
<math>~P_\mathrm{rf}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\equiv</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\biggl[ \frac{K^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n-3)} \, , </math>
<math>~
\biggl[ 2^{-2(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n+1}
K^{4n} G^{-3(n+1)} M_\mathrm{limit}^{-2(n+1)}\biggr]^{1/(n-3)}
</math>
   </td>
   </td>
</tr>
</tr>
Line 345: Line 362:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~E_\mathrm{norm}</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 351: Line 368:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~  
<math>~P_\mathrm{norm}
\biggl[ K^n G^{-3}M_\mathrm{tot}^{n-5} \biggr]^{1/(n-3)} \, ,</math>
\biggl[ 2^{-2(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n+1}  
\biggr]^{1/(n-3)} \, ,
</math>
   </td>
   </td>
</tr>
</tr>


</table>
</div>
and, in terms of the [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Structural_Form_Factors|structural form factors]], <math>~\tilde\mathfrak{f}_M</math>, <math>~\tilde\mathfrak{f}_A</math>, and <math>~\tilde\mathfrak{f}_W</math>,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathcal{A}</math>
<math>~\mathcal{U}_\mathrm{rf} \equiv (M_0K_1)_\mathrm{Whitworth}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 370: Line 383:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>\frac{1}{5} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2
<math>~\biggl[ \biggl( \frac{3^4\cdot 5^3}{2^8\pi} \biggr) K^n G^{-3} M_\mathrm{limit}^{n-5}\biggr]^{1/(n-3)} </math>
\cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \, ,
</math>
   </td>
   </td>
</tr>
</tr>
Line 378: Line 389:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathcal{B}</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 384: Line 395:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>
<math>~E_\mathrm{norm} \biggl( \frac{3^4\cdot 5^3}{2^8\pi} \biggr)^{1/(n-3)}  
\biggl(\frac{3}{4\pi}\biggr)^{1/n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{(n+1)/n}
\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{(n-5)/(n-3)} \, .
\frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{(n+1)/n}} \, .
</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
</div>
</div>
Line 400: Line 409:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathcal{B} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{(n-3)/n} </math>
<math>~\frac{2}{3}\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{(n-3)/n} </math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 406: Line 415:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>\mathcal{A} + \frac{4\pi}{3} \biggl( \frac{P_\mathrm{e}}{P_\mathrm{norm}} \biggr)  
<math>\frac{1}{2} + \frac{1}{6} \biggl( \frac{P_\mathrm{e}}{P_\mathrm{rf}} \biggr)  
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4 \, ,
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^4 \, .
</math>
</math>
   </td>
   </td>
Line 414: Line 423:
</table>
</table>
</div>
</div>
which matches the [[User:Tohline/SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Nonrotating_Adiabatic_Configuration_Embedded_in_an_External_Medium|statement of virial equilibrium presented in our accompanying, more detailed analysis]].  A rearrangement of terms explicitly provides the desired <math>~P_e(R_\mathrm{eq})</math> function, namely,
A rearrangement of terms explicitly provides the desired <math>~P_e(R_\mathrm{eq})</math> function, namely,
<div align="center">
<div align="center">
<table border="0" cellpadding="5">
<table border="0" cellpadding="5">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~
<math>\biggl( \frac{P_\mathrm{e}}{P_\mathrm{rf}} \biggr)
\frac{P_\mathrm{e}}{P_\mathrm{norm}}  
</math>
</math>
   </td>
   </td>
Line 427: Line 435:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\frac{3}{4\pi}
<math>~\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{-4} \biggl[4\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{(n-3)/n} -3\biggr] \, ,</math>
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{-4}
  </td>
\biggl[ \mathcal{B} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{(n-3)/n} -\mathcal{A} \biggr] \, ,
</tr>
</math>
 
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{-3(n+1)/n} -3\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{-4}  \, ,</math>
   </td>
   </td>
</tr>
</tr>
Line 436: Line 453:
</table>
</table>
</div>
</div>
or (see the [[User:Tohline/SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Solution_Expressed_in_Terms_of_K_and_M_.28Whitworth.27s_1981_Relation.29|accompanying derivation]] for details),
or,
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
Line 448: Line 465:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~K \biggl( \frac{3M_\mathrm{limit}}{4\pi R_\mathrm{eq}^3} \biggr)^{(n+1)/n} \cdot \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{(n+1)/n}}
<math>~
- \biggl(\frac{3GM_\mathrm{limit}^2}{20\pi R_\mathrm{eq}^4} \biggr)\cdot \frac{\mathfrak{f}_W}{\mathfrak{f}^2_M} \, .</math>
4P_\mathrm{rf} R_\mathrm{rf}^{3(n+1)/n} R_\mathrm{eq}^{-3(n+1)/n} -3P_\mathrm{rf} R_\mathrm{rf}^4 R_\mathrm{eq}^{-4}  
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>


====Stability====
Similarly, according to the [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#Stability|above-derived stability criterion]], pressure-truncated polytropic configurations will only be stable if,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4 </math>
&nbsp;
  </td>
</td>
   <td align="center">
   <td align="center">
<math>~></math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>\frac{(n-3)\mathcal{A}}{4\pi(n+1)}  
<math>~
\biggl( \frac{P_e}{P_\mathrm{norm}} \biggr)^{-1} \, .
4R_\mathrm{eq}^{-3(n+1)/n}\biggl[ 2^{-2(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n+1}  
K^{4n} G^{-3(n+1)} M_\mathrm{limit}^{-2(n+1)}\biggr]^{1/(n-3)}
\biggl[ \frac{\pi}{5^n} \biggl( \frac{2^2}{3}\biggr)^{(n+1)} K^{-n} G^{n} M_\mathrm{limit}^{n-1} \biggr]^{3(n+1)/[n(n-3)]} 
</math>
</math>
   </td>
   </td>
</tr>
</tr>


</table>
</div>
Or, given that <math>~P_\mathrm{norm}R_\mathrm{norm}^4 = G M_\mathrm{tot}^2</math>, the criterion for stability may be written as,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~P_e </math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~></math>
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~
\frac{(n-3)\mathcal{A}}{4\pi(n+1)} \biggl( \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}^4} \biggr)  
-3R_\mathrm{eq}^{-4}\biggl[ 2^{-2(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n+1}
=\frac{(n-3)}{20\pi(n+1)} \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \biggl( \frac{GM_\mathrm{limit}^2}{R_\mathrm{eq}^4} \biggr)
K^{4n} G^{-3(n+1)} M_\mathrm{limit}^{-2(n+1)}\biggr]^{1/(n-3)}
\, .
\biggl[ \frac{\pi}{5^n} \biggl( \frac{2^2}{3}\biggr)^{(n+1)} K^{-n} G^{n} M_\mathrm{limit}^{n-1} \biggr]^{4/(n-3)
</math>
</math>
   </td>
   </td>
</tr>
</tr>


</table>
</div>


===Our Case P Analysis (compact)===
<tr>
As has been both summarized and detailed in [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Free_Energy_Function_and_Virial_Theorem|an accompanying discussion]], our <font color="red">'''Case P'''</font> analysis has demonstrated that the following,
  <td align="right">
<div align="center" id="RenormalizedFreeEnergyExpression">
&nbsp;
<font color="#770000">'''Renormalized Free-Energy Function'''</font><br />
</td>
 
  <td align="center">
<math>
<math>~=</math>
\mathfrak{G}^{**} \equiv \mathfrak{G}^* \biggl[ \frac{\mathcal{A}^3}{\mathcal{B}^n} \biggr]^{1/(n-3)} =
  </td>
-3 \Chi^{-1} +~ n\Chi^{-3/n} +~ \Pi_\mathrm{ad}\Chi^3 \, ,
  <td align="left">
<math>~
2^2 R_\mathrm{eq}^{-3(n+1)/n}\biggl[ 2^{-2n(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n(n+1)}  
K^{4n^2} G^{-3n(n+1)} M_\mathrm{limit}^{-2n(n+1)}\cdot
\frac{\pi^{3(n+1)}}{5^{3n(n+1)}} \biggl( \frac{2^2}{3}\biggr)^{3(n+1)(n+1)} K^{-3n(n+1)} G^{3n(n+1)} M_\mathrm{limit}^{3(n-1)(n+1)} \biggr]^{1/[n(n-3)]} 
</math>
</math>
</div>
  </td>
properly governs the equilibrium structure and stability of pressure-truncated polytropic configurations.  This expression is identical to the [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#Overview|free-energy function given above]] if the following coefficient and variable substitutions are made:
</tr>
<div align="center">
<table border="1" cellpadding="10" align="center">
<tr><td align="center">
Our <font color="red">Case P</font> Analysis
</td></tr>
<tr><td align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~a</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\rightarrow</math>
&nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~3</math>
<math>~
-3R_\mathrm{eq}^{-4}\biggl[ 2^{-2(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n+1}
K^{4n} G^{-3(n+1)} M_\mathrm{limit}^{-2(n+1)}\cdot
\frac{\pi^4}{5^{4n}} \biggl( \frac{2^2}{3}\biggr)^{4(n+1)} K^{-4n} G^{4n} M_\mathrm{limit}^{4(n-1)} \biggr]^{1/(n-3)} 
</math>
   </td>
   </td>
</tr>
</tr>
Line 531: Line 538:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~b</math>
&nbsp;
  </td>
</td>
   <td align="center">
   <td align="center">
<math>~\rightarrow</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~n</math>
<math>~
R_\mathrm{eq}^{-3(n+1)/n} \biggl( \frac{3}{2^2\pi}\biggr)^{(n+1)/n} K M_\mathrm{limit}^{(n+1)/n} 
- R_\mathrm{eq}^{-4} \biggl(\frac{3}{2^2\cdot 5\pi}\biggr) G M_\mathrm{limit}^2
</math>
   </td>
   </td>
</tr>
</tr>
Line 543: Line 553:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~c</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\rightarrow</math>
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\Pi_\mathrm{ad} \equiv \frac{P_e}{P_\mathrm{ad}}</math>
<math>~K \biggl( \frac{3M_\mathrm{limit}}{4\pi R_\mathrm{eq}^3} \biggr)^{(n+1)/n} 
- \frac{3GM_\mathrm{limit}^2}{20\pi R_\mathrm{eq}^4} \, .</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>


Recalling that <math>~\eta \leftrightarrow (n+1)/n</math>, it is clear that this <math>~P_e(R_\mathrm{eq})</math> relation exactly matches equation (5) of [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth], which reads:
<div align="center">
<table border="2">
<tr><td>
[[File:Whitworth1981Eq5.jpg|500px|center|Whitworth (1981, MNRAS, 195, 967)]]
</td></tr>
</table>
</div>
====Stability====
Similarly, according to the [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#Stability|above-derived stability criterion]], pressure-truncated polytropic configurations will only be stable if,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~x</math>
<math>~\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4 </math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\rightarrow</math>
<math>~></math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\Chi \equiv \frac{R}{R_\mathrm{ad}}</math>
<math>\frac{(n-3)\mathcal{A}}{4\pi(n+1)}
\biggl( \frac{P_e}{P_\mathrm{norm}} \biggr)^{-1} \, .
</math>
   </td>
   </td>
</tr>
</tr>


</table>
</div>
Or, given that <math>~P_\mathrm{norm}R_\mathrm{norm}^4 = G M_\mathrm{tot}^2</math>, the criterion for stability may be written as,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathcal{G}</math>
<math>~P_e </math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~\rightarrow</math>
<math>~></math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\mathfrak{G}^{**} </math>
<math>~
\frac{(n-3)\mathcal{A}}{4\pi(n+1)} \biggl( \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}^4} \biggr)
=\frac{(n-3)}{20\pi(n+1)} \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \biggl( \frac{GM_\mathrm{limit}^2}{R_\mathrm{eq}^4} \biggr)
\, .
</math>
   </td>
   </td>
</tr>
</tr>
</table>
 
</td></tr>
</table>
</table>
</div>
</div>
where,




===Whitworth's (1981) Case P Analysis===
 
[http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth's (1981)] related <font color="red">'''Case P'''</font> analysis of pressure-truncated polytropic spheres produces the following governing free-energy function &#8212; referred to by Whitworth as the "global potential function":
===Our Case P Analysis (raw)===
<div align="center" id="WhitworthFreeEnergyExpression">
 
====Coefficient Definitions====
As has been both summarized and detailed in [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Free_Energy_Function_and_Virial_Theorem|an accompanying discussion]], our <font color="red">'''Case P'''</font> analysis has demonstrated that the following,
<div align="center" id="FreeEnergyExpression">
<font color="#770000">'''Algebraic Free-Energy Function'''</font><br />
 
<math>
<math>
\frac{2\mathcal{U}}{3M_0 K_1} =  
\mathfrak{G}^* =  
-\frac{3}{2} \biggl( \frac{R}{R_\mathrm{rf}}\biggr)^{-1} +~ \frac{2n}{3}\biggl( \frac{R}{R_\mathrm{rf}}\biggr)^{-3/n}  
-3\mathcal{A} \chi^{-1} +~ n\mathcal{B} \chi^{-3/n} +~ \mathcal{D}\chi^3 \, ,
+~ \frac{1}{6}\biggl(\frac{P_e}{P_\mathrm{rf}}\biggr) \biggl( \frac{R}{R_\mathrm{rf}}\biggr)^3 \, ;
</math>
</math>
</div>
</div>
this expression is obtained from Whitworth's equation (10) after setting <math>~\delta_{1\eta} = 0</math>, that is, by choosing to ignore isothermal systems, and after setting <math>~\eta = (n+1)/n</math>, that is, after rewriting his adiabatic exponent <math>~(\eta)</math> in terms of the corresponding polytropic indexWhitworth's expression also is identical to the [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#Overview|free-energy function given above]] if the following coefficient and variable substitutions are made:  
properly governs the equilibrium structure and stability of pressure-truncated polytropic configurationsThis algebraic expression is identical to the [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#Overview|free-energy function given above]] if the following coefficient and variable substitutions are made:  
 
<div align="center">
<div align="center">
<table border="1" cellpadding="10" align="center">
<table border="1" cellpadding="10" align="center">
<tr><td align="center">
<tr><td align="center">
[http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth's (1981)] <font color="red">Case P</font> Analysis
Our <font color="red">Case P</font> Analysis
</td></tr>
</td></tr>
<tr><td align="center">
<tr><td align="center">
Line 610: Line 649:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\frac{3}{2}</math>
<math>~3\mathcal{A}</math>
   </td>
   </td>
</tr>
</tr>
Line 622: Line 661:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\frac{2n}{3}</math>
<math>~n\mathcal{B} </math>
   </td>
   </td>
</tr>
</tr>
Line 634: Line 673:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\frac{1}{6}\biggl(\frac{P_e}{P_\mathrm{rf}}\biggr)</math>
<math>~\mathcal{D} \equiv \frac{4\pi}{3} \biggl(\frac{P_e}{P_\mathrm{norm}}\biggr)</math>
   </td>
   </td>
</tr>
</tr>
Line 646: Line 685:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\frac{R}{R_\mathrm{rf}}</math>
<math>~\chi \equiv \frac{R}{R_\mathrm{norm}}</math>
   </td>
   </td>
</tr>
</tr>
Line 658: Line 697:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\frac{2}{3} \biggl( \frac{\mathcal{U}}{\mathcal{U}_\mathrm{rf}} \biggr) </math>
<math>~\mathfrak{G}^{*} \equiv \frac{\mathfrak{G}}{E_\mathrm{norm}}</math>
   </td>
   </td>
</tr>
</tr>
Line 665: Line 704:
</table>
</table>
</div>
</div>
where (see an [[User:Tohline/SSC/Structure/PolytropesASIDE1#ASIDE:_Whitworth.27s_Scaling|accompanying ASIDE]]),
 
where &#8212; see, for example, our [[User:Tohline/SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Review|accompanying review]],
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~R_\mathrm{rf}</math>
<math>~R_\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>~\biggl[ \biggl( \frac{G}{K} \biggr)^n M_\mathrm{tot}^{n-1} \biggr]^{1/(n-3)} \, ,</math>
\biggl[ \frac{\pi}{5^n} \biggl( \frac{2^2}{3}\biggr)^{(n+1)} K^{-n} G^{n} M_\mathrm{limit}^{n-1} \biggr]^{1/(n-3)}
</math>
   </td>
   </td>
</tr>
</tr>
Line 685: Line 723:
<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~P_\mathrm{norm}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 691: Line 729:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~R_\mathrm{norm}
<math>~\biggl[ \frac{K^{4n}}{G^{3(n+1)} M_\mathrm{tot}^{2(n+1)}} \biggr]^{1/(n-3)} \, , </math>
\biggl[ \frac{\pi}{5^n} \biggl( \frac{2^2}{3}\biggr)^{(n+1)} \biggr]^{1/(n-3)}
\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{(n-1)/(n-3)} \, ,
</math>
   </td>
   </td>
</tr>
</tr>
Line 700: Line 735:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~P_\mathrm{rf}</math>
<math>~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>~  
\biggl[ 2^{-2(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n+1}
\biggl[ K^n G^{-3}M_\mathrm{tot}^{n-5} \biggr]^{1/(n-3)} \, ,</math>
K^{4n} G^{-3(n+1)} M_\mathrm{limit}^{-2(n+1)}\biggr]^{1/(n-3)}
</math>
   </td>
   </td>
</tr>
</tr>


</table>
</div>
and, in terms of the [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Structural_Form_Factors|structural form factors]], <math>~\tilde\mathfrak{f}_M</math>, <math>~\tilde\mathfrak{f}_A</math>, and <math>~\tilde\mathfrak{f}_W</math>,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~\mathcal{A}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 721: Line 760:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~P_\mathrm{norm}  
<math>\frac{1}{5} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2
\biggl[ 2^{-2(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n+1}  
\cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \, ,
\biggr]^{1/(n-3)} \, ,
</math>
</math>
   </td>
   </td>
Line 730: Line 768:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mathcal{U}_\mathrm{rf} \equiv (M_0K_1)_\mathrm{Whitworth}</math>
<math>~\mathcal{B}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 736: Line 774:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\biggl[ \biggl( \frac{3^4\cdot 5^3}{2^8\pi} \biggr) K^n G^{-3} M_\mathrm{limit}^{n-5}\biggr]^{1/(n-3)} </math>
<math>
\biggl(\frac{3}{4\pi}\biggr)^{1/n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{(n+1)/n}
\frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{(n+1)/n}} \, .
</math>
   </td>
   </td>
</tr>
</tr>


<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~E_\mathrm{norm} \biggl( \frac{3^4\cdot 5^3}{2^8\pi} \biggr)^{1/(n-3)}
\biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{(n-5)/(n-3)} \, .
</math>
  </td>
</tr>
</table>
</table>
</div>
</div>
Line 762: Line 790:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\frac{2}{3}\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{(n-3)/n} </math>
<math>~\mathcal{B} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{(n-3)/n} </math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 768: Line 796:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>\frac{1}{2} + \frac{1}{6} \biggl( \frac{P_\mathrm{e}}{P_\mathrm{rf}} \biggr)  
<math>\mathcal{A} + \frac{4\pi}{3} \biggl( \frac{P_\mathrm{e}}{P_\mathrm{norm}} \biggr)  
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^4 \, .
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4 \, ,
</math>
</math>
   </td>
   </td>
Line 776: Line 804:
</table>
</table>
</div>
</div>
A rearrangement of terms explicitly provides the desired <math>~P_e(R_\mathrm{eq})</math> function, namely,
which matches the [[User:Tohline/SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Nonrotating_Adiabatic_Configuration_Embedded_in_an_External_Medium|statement of virial equilibrium presented in our accompanying, more detailed analysis]].  A rearrangement of terms explicitly provides the desired <math>~P_e(R_\mathrm{eq})</math> function, namely,
<div align="center">
<div align="center">
<table border="0" cellpadding="5">
<table border="0" cellpadding="5">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>\biggl( \frac{P_\mathrm{e}}{P_\mathrm{rf}} \biggr)
<math>~
\frac{P_\mathrm{e}}{P_\mathrm{norm}}  
</math>
</math>
   </td>
   </td>
Line 788: Line 817:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{-4} \biggl[4\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{(n-3)/n} -3\biggr] \, ,</math>
<math>~\frac{3}{4\pi}  
  </td>
\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{-4}
</tr>
\biggl[ \mathcal{B} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{(n-3)/n} -\mathcal{A} \biggr] \, ,
 
</math>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~4\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{-3(n+1)/n} -3\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{-4}  \, ,</math>
   </td>
   </td>
</tr>
</tr>
Line 806: Line 826:
</table>
</table>
</div>
</div>
or,
or (see the [[User:Tohline/SSC/Virial/PolytropesEmbedded/FirstEffortAgain#Solution_Expressed_in_Terms_of_K_and_M_.28Whitworth.27s_1981_Relation.29|accompanying derivation]] for details),
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
Line 818: Line 838:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~K \biggl( \frac{3M_\mathrm{limit}}{4\pi R_\mathrm{eq}^3} \biggr)^{(n+1)/n} \cdot \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{(n+1)/n}}
4P_\mathrm{rf} R_\mathrm{rf}^{3(n+1)/n} R_\mathrm{eq}^{-3(n+1)/n} -3P_\mathrm{rf} R_\mathrm{rf}^4 R_\mathrm{eq}^{-4}  
- \biggl(\frac{3GM_\mathrm{limit}^2}{20\pi R_\mathrm{eq}^4} \biggr)\cdot \frac{\mathfrak{f}_W}{\mathfrak{f}^2_M} \, .</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>
Notice that, when <math>~P_e \rightarrow 0</math>, this expression reduces to the solution we obtained for an isolated polytrope, expressed in terms of <math>~K</math> and <math>~M_\mathrm{limit}</math> (see the left-hand column of our [[User:Tohline/SSC/Virial/Polytropes#TwoPointsOfView|table titled "Two Points of View"]]).  Also notice that this equilibrium relation exactly matches the one derived by Whitworth &#8212; and rederived above &#8212; when all three structural form factors are set to unity.  This is as it should be because all of Whitworth's results were derived assuming uniform-density configurations.


====Stability====
Similarly, according to the [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#Stability|above-derived stability criterion]], pressure-truncated polytropic configurations will only be stable if,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \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{(n-3)\mathcal{A}}{4\pi(n+1)}  
4R_\mathrm{eq}^{-3(n+1)/n}\biggl[ 2^{-2(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n+1}
\biggl( \frac{P_e}{P_\mathrm{norm}} \biggr)^{-1} \, .
K^{4n} G^{-3(n+1)} M_\mathrm{limit}^{-2(n+1)}\biggr]^{1/(n-3)}
\biggl[ \frac{\pi}{5^n} \biggl( \frac{2^2}{3}\biggr)^{(n+1)} K^{-n} G^{n} M_\mathrm{limit}^{n-1} \biggr]^{3(n+1)/[n(n-3)]} 
</math>
</math>
   </td>
   </td>
</tr>
</tr>


</table>
</div>
Or, given that <math>~P_\mathrm{norm}R_\mathrm{norm}^4 = G M_\mathrm{tot}^2</math>, the criterion for stability may be written as,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~P_e </math>
   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp;
<math>~></math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~
-3R_\mathrm{eq}^{-4}\biggl[ 2^{-2(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n+1}
\frac{(n-3)\mathcal{A}}{4\pi(n+1)} \biggl( \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}^4} \biggr)  
K^{4n} G^{-3(n+1)} M_\mathrm{limit}^{-2(n+1)}\biggr]^{1/(n-3)}
=\frac{(n-3)}{20\pi(n+1)} \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \biggl( \frac{GM_\mathrm{limit}^2}{R_\mathrm{eq}^4} \biggr)
\biggl[ \frac{\pi}{5^n} \biggl( \frac{2^2}{3}\biggr)^{(n+1)} K^{-n} G^{n} M_\mathrm{limit}^{n-1} \biggr]^{4/(n-3)
\, .
</math>
</math>
   </td>
   </td>
</tr>
</tr>


</table>
</div>


<tr>
===Our Case P Analysis (compact)===
  <td align="right">
As has been both summarized and detailed in [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Free_Energy_Function_and_Virial_Theorem|an accompanying discussion]], our <font color="red">'''Case P'''</font> analysis has demonstrated that the following,
&nbsp;
<div align="center" id="RenormalizedFreeEnergyExpression">
</td>
<font color="#770000">'''Renormalized Free-Energy Function'''</font><br />
  <td align="center">
 
<math>~=</math>
<math>
  </td>
\mathfrak{G}^{**} \equiv \mathfrak{G}^* \biggl[ \frac{\mathcal{A}^3}{\mathcal{B}^n} \biggr]^{1/(n-3)} =
  <td align="left">
-3 \Chi^{-1} +~ n\Chi^{-3/n} +~ \Pi_\mathrm{ad}\Chi^3 \, ,
<math>~
2^2 R_\mathrm{eq}^{-3(n+1)/n}\biggl[ 2^{-2n(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n(n+1)}  
K^{4n^2} G^{-3n(n+1)} M_\mathrm{limit}^{-2n(n+1)}\cdot
\frac{\pi^{3(n+1)}}{5^{3n(n+1)}} \biggl( \frac{2^2}{3}\biggr)^{3(n+1)(n+1)} K^{-3n(n+1)} G^{3n(n+1)} M_\mathrm{limit}^{3(n-1)(n+1)} \biggr]^{1/[n(n-3)]} 
</math>
</math>
  </td>
</div>
</tr>
properly governs the equilibrium structure and stability of pressure-truncated polytropic configurations.  This expression is identical to the [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#Overview|free-energy function given above]] if the following coefficient and variable substitutions are made:
<div align="center">
<table border="1" cellpadding="10" align="center">
<tr><td align="center">
Our <font color="red">Case P</font> Analysis
</td></tr>
<tr><td align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~a</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
&nbsp;
<math>~\rightarrow</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~3</math>
-3R_\mathrm{eq}^{-4}\biggl[ 2^{-2(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n+1}
K^{4n} G^{-3(n+1)} M_\mathrm{limit}^{-2(n+1)}\cdot
\frac{\pi^4}{5^{4n}} \biggl( \frac{2^2}{3}\biggr)^{4(n+1)} K^{-4n} G^{4n} M_\mathrm{limit}^{4(n-1)} \biggr]^{1/(n-3)} 
</math>
   </td>
   </td>
</tr>
</tr>
Line 891: Line 923:
<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~b</math>
</td>
  </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\rightarrow</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~n</math>
R_\mathrm{eq}^{-3(n+1)/n} \biggl( \frac{3}{2^2\pi}\biggr)^{(n+1)/n} K M_\mathrm{limit}^{(n+1)/n} 
- R_\mathrm{eq}^{-4} \biggl(\frac{3}{2^2\cdot 5\pi}\biggr) G M_\mathrm{limit}^2
</math>
   </td>
   </td>
</tr>
</tr>
Line 906: Line 935:
<tr>
<tr>
   <td align="right">
   <td align="right">
&nbsp;
<math>~c</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~\rightarrow</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~K \biggl( \frac{3M_\mathrm{limit}}{4\pi R_\mathrm{eq}^3} \biggr)^{(n+1)/n} 
<math>~\Pi_\mathrm{ad} \equiv \frac{P_e}{P_\mathrm{ad}}</math>
- \frac{3GM_\mathrm{limit}^2}{20\pi R_\mathrm{eq}^4} \, .</math>
   </td>
   </td>
</tr>
</tr>
</table>
</div>


Recalling that <math>~\eta \leftrightarrow (n+1)/n</math>, it is clear that this <math>~P_e(R_\mathrm{eq})</math> relation exactly matches equation (5) of [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth], which reads:
<div align="center">
<table border="2">
<tr><td>
[[File:Whitworth1981Eq5.jpg|500px|center|Whitworth (1981, MNRAS, 195, 967)]]
</td></tr>
</table>
</div>
Notice that, when <math>~P_e \rightarrow 0</math>, this expression reduces to the solution we obtained for an isolated polytrope, expressed in terms of <math>~K</math> and <math>~M_\mathrm{limit}</math> (see the left-hand column of our [[User:Tohline/SSC/Virial/Polytropes#TwoPointsOfView|table titled "Two Points of View"]]).
====Stability====
Similarly, according to the [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#Stability|above-derived stability criterion]], pressure-truncated polytropic configurations will only be stable if,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4 </math>
<math>~x</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~></math>
<math>~\rightarrow</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>\frac{(n-3)\mathcal{A}}{4\pi(n+1)}
<math>~\Chi \equiv \frac{R}{R_\mathrm{ad}}</math>
\biggl( \frac{P_e}{P_\mathrm{norm}} \biggr)^{-1} \, .
</math>
   </td>
   </td>
</tr>
</tr>


</table>
</div>
Or, given that <math>~P_\mathrm{norm}R_\mathrm{norm}^4 = G M_\mathrm{tot}^2</math>, the criterion for stability may be written as,
<div align="center">
<table border="0" cellpadding="5">
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~P_e </math>
<math>~\mathcal{G}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~></math>
<math>~\rightarrow</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~
<math>~\mathfrak{G}^{**} </math>
\frac{(n-3)\mathcal{A}}{4\pi(n+1)} \biggl( \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}^4} \biggr)
=\frac{(n-3)}{20\pi(n+1)} \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \biggl( \frac{GM_\mathrm{limit}^2}{R_\mathrm{eq}^4} \biggr)
\, .
</math>
   </td>
   </td>
</tr>
</tr>
 
</table>
</td></tr>
</table>
</table>
</div>
</div>
 
where,





Revision as of 22:17, 7 February 2015


Virial Equilibrium of Embedded Polytropic Spheres

Free-Energy Surface

Click here for technical discussion of this Free-Energy surface

Overview

The free-energy function that is relevant to a discussion of the structure and stability of a pressure-truncated configuration having polytropic index, <math>~n</math>, has the form,

<math>~\mathcal{G}(x)</math>

<math>~=</math>

<math> -ax^{-1} +b x^{-3/n} + c x^3 \, , </math>

where <math>~x</math> identifies the size of the configuration. (As is explained more fully, below, the above figure displays a free-energy surface of this form for the case, <math>~n=5</math>.) If the coefficients, <math>~a, b</math>, and <math>~c</math>, are held constant while varying the configuration's size, we see that,

<math>~\frac{d\mathcal{G}}{dx}</math>

<math>~=</math>

<math> ax^{-2} - \frac{3b}{n}\cdot x^{-(3+n)/n} + 3c x^2 </math>

 

<math>~=</math>

<math> x^{-2} \biggl[ a - \frac{3b}{n}\cdot x^{(n-3)/n} + 3c x^4 \biggr] \, , </math>

and,

<math>~\frac{d^2\mathcal{G}}{dx^2}</math>

<math>~=</math>

<math> x^{-3} \biggl[ -2a + \frac{3(3+n)b}{n^2}\cdot x^{(n-3)/n} + 6c x^4 \biggr] \, . </math>

Equilibrium Configurations

The size, <math>~x_\mathrm{eq}</math>, of each equilibrium configuration is determined by setting, <math>d\mathcal{G}/dx = 0</math>. Hence, <math>~x_\mathrm{eq}</math> is given by the root(s) of the polynomial expression that is often referred to as the,

Scalar Virial Theorem

<math>~x^{(n-3)/n}_\mathrm{eq} </math>

<math>~=</math>

<math>\frac{n}{b} \biggl[\frac{a}{3} + c\cdot x^4_\mathrm{eq} \biggr] \, . </math>

(The equilibrium radii of <math>~n = 5</math> polytropic configurations having a variety of different masses are identified by the sequence of a dozen, small colored spherical dots in the above figure.)

Stability

The relative stability of each equilibrium configuration is determined by the sign of the second derivative of the free-energy function, evaluated at the specified equilibrium radius. Specifically, the systems being considered here are stable if the second derivative is positive, but they are unstable if the second derivative is negative. Evaluating the second derivative in this manner gives,

<math>~\biggl[ x^{3} \cdot \frac{d^2\mathcal{G}}{dx^2}\biggr]_\mathrm{eq}</math>

<math>~=</math>

<math> -2a + \frac{3(3+n)}{n} \biggl[\frac{a}{3} + c\cdot x^4_\mathrm{eq} \biggr] + 6c x^4_\mathrm{eq} </math>

 

<math>~=</math>

<math> -2a + \frac{(3+n)a}{n} + \frac{3(3+n)c}{n} \cdot x^4_\mathrm{eq} + 6c x^4_\mathrm{eq} </math>

 

<math>~=</math>

<math>\frac{9(n+1)c}{n}\cdot x^4_\mathrm{eq} - \frac{a(n-3)}{n} \, . </math>

Defining <math>~x_\mathrm{crit}</math> as the equilibrium radius at which this function goes to zero gives,

<math>~x_\mathrm{crit} </math>

<math>~\equiv</math>

<math> \biggl[ \frac{a(n-3)}{3^2c(n+1)} \biggr]^{1/4} \, . </math>

(The small red spherical dot in the above figure identifies the equilibrium configuration at <math>~x_\mathrm{crit} </math>.) We conclude, therefore, that pressure-truncated, equilibrium polytropic configurations having <math>~n > 3</math> are stable if,

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

<math>~></math>

<math> ~x_\mathrm{crit} \, , </math>

while they are unstable if,

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

<math>~<</math>

<math> ~x_\mathrm{crit} \, . </math>

The Physics

The above mathematical statements, ostensibly defining the free-energy function, the scalar virial theorem, and stability, cannot be interpreted in physical terms until the definitions of the various coefficients have been provided. In the discussion that follows, we will focus on sequences of equilibrium configurations that have a polytropic index <math>~n > 3</math> because, as has been foreshadowed in the above overview, such sequences include both stable and unstable equilbria and are therefore of considerable interest in an astrophysical context. Isothermal sequences — corresponding to <math>~n = \infty</math> — are of particular astrophysical interest; however, we will devote a great deal of attention to <math>~n=5</math> configurations because their structures can be defined entirely in terms of analytic expressions.


In order to determine the equilibrium radius, <math>~R_\mathrm{eq}</math>, of any pressure-truncated polytropic configuration, we must specify the configuration's mass, <math>~M_\mathrm{limit}</math>, its polytropic constant, <math>~K</math>, and the pressure, <math>~P_e</math>, of the external medium in which the configuration is embedded, and locate extrema in the resulting <math>~\mathfrak{G}(R)</math> function. Then, a sequence of equilibria can be identified if, for example:

  • Case P: <math>~P_e</math> is varied while holding <math>~K</math> and <math>~M_\mathrm{limit}</math> fixed; or
  • Case M: <math>~M_\mathrm{limit}</math> is varied while holding <math>~K</math> and <math>~P_e</math> fixed.

In the first case, the analysis reveals how <math>~R_\mathrm{eq}</math> varies with the applied external pressure and usually is displayed as a <math>~P_e(R_\mathrm{eq})</math> function. The second case identifies a mass-radius relationship for the polytropic sequence under consideration and is usually displayed as a <math>~M_\mathrm{limit}(R_\mathrm{eq})</math> function. (In the figure at the top of this page, a "Case M" mass-radius relation for pressure-truncated, <math>~n = 5</math> polytropic configurations is traced by the sequence of a dozen, small colored spherical dots that each reside at an extremum in the displayed free-energy function.)

Whitworth's (1981) Case P Analysis

Whitworth's (1981) related Case P analysis of pressure-truncated polytropic spheres produces the following governing free-energy function — referred to by Whitworth as the "global potential function":

<math> \frac{2\mathcal{U}}{3M_0 K_1} = -\frac{3}{2} \biggl( \frac{R}{R_\mathrm{rf}}\biggr)^{-1} +~ \frac{2n}{3}\biggl( \frac{R}{R_\mathrm{rf}}\biggr)^{-3/n} +~ \frac{1}{6}\biggl(\frac{P_e}{P_\mathrm{rf}}\biggr) \biggl( \frac{R}{R_\mathrm{rf}}\biggr)^3 \, ; </math>

this expression is obtained from Whitworth's equation (10) after setting <math>~\delta_{1\eta} = 0</math>, that is, by choosing to ignore isothermal systems, and after setting <math>~\eta = (n+1)/n</math>, that is, after rewriting his adiabatic exponent <math>~(\eta)</math> in terms of the corresponding polytropic index. Whitworth's expression also is identical to the free-energy function given above if the following coefficient and variable substitutions are made:

Whitworth's (1981) Case P Analysis

<math>~a</math>

<math>~\rightarrow</math>

<math>~\frac{3}{2}</math>

<math>~b</math>

<math>~\rightarrow</math>

<math>~\frac{2n}{3}</math>

<math>~c</math>

<math>~\rightarrow</math>

<math>~\frac{1}{6}\biggl(\frac{P_e}{P_\mathrm{rf}}\biggr)</math>

<math>~x</math>

<math>~\rightarrow</math>

<math>~\frac{R}{R_\mathrm{rf}}</math>

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

<math>~\rightarrow</math>

<math>~\frac{2}{3} \biggl( \frac{\mathcal{U}}{\mathcal{U}_\mathrm{rf}} \biggr) </math>

where (see an accompanying ASIDE),

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

<math>~\equiv</math>

<math> \biggl[ \frac{\pi}{5^n} \biggl( \frac{2^2}{3}\biggr)^{(n+1)} K^{-n} G^{n} M_\mathrm{limit}^{n-1} \biggr]^{1/(n-3)} </math>

 

<math>~=</math>

<math>~R_\mathrm{norm} \biggl[ \frac{\pi}{5^n} \biggl( \frac{2^2}{3}\biggr)^{(n+1)} \biggr]^{1/(n-3)} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{(n-1)/(n-3)} \, , </math>

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

<math>~\equiv</math>

<math>~ \biggl[ 2^{-2(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n+1} K^{4n} G^{-3(n+1)} M_\mathrm{limit}^{-2(n+1)}\biggr]^{1/(n-3)} </math>

 

<math>~=</math>

<math>~P_\mathrm{norm} \biggl[ 2^{-2(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n+1} \biggr]^{1/(n-3)} \, , </math>

<math>~\mathcal{U}_\mathrm{rf} \equiv (M_0K_1)_\mathrm{Whitworth}</math>

<math>~=</math>

<math>~\biggl[ \biggl( \frac{3^4\cdot 5^3}{2^8\pi} \biggr) K^n G^{-3} M_\mathrm{limit}^{n-5}\biggr]^{1/(n-3)} </math>

 

<math>~=</math>

<math>~E_\mathrm{norm} \biggl( \frac{3^4\cdot 5^3}{2^8\pi} \biggr)^{1/(n-3)} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{(n-5)/(n-3)} \, . </math>

Virial Equilibrium

Plugging these coefficient assignments into the above mathematical prescription of the virial theorem gives the following relationship between the applied external pressure and the resulting equilibrium radius of pressure-truncated polytropic configurations,

<math>~\frac{2}{3}\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{(n-3)/n} </math>

<math>~=</math>

<math>\frac{1}{2} + \frac{1}{6} \biggl( \frac{P_\mathrm{e}}{P_\mathrm{rf}} \biggr) \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^4 \, . </math>

A rearrangement of terms explicitly provides the desired <math>~P_e(R_\mathrm{eq})</math> function, namely,

<math>\biggl( \frac{P_\mathrm{e}}{P_\mathrm{rf}} \biggr) </math>

<math>~=</math>

<math>~\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{-4} \biggl[4\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{(n-3)/n} -3\biggr] \, ,</math>

 

<math>~=</math>

<math>~4\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{-3(n+1)/n} -3\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{rf}} \biggr)^{-4} \, ,</math>

or,

<math>~P_e </math>

<math>~=</math>

<math>~ 4P_\mathrm{rf} R_\mathrm{rf}^{3(n+1)/n} R_\mathrm{eq}^{-3(n+1)/n} -3P_\mathrm{rf} R_\mathrm{rf}^4 R_\mathrm{eq}^{-4} </math>

 

<math>~=</math>

<math>~ 4R_\mathrm{eq}^{-3(n+1)/n}\biggl[ 2^{-2(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n+1} K^{4n} G^{-3(n+1)} M_\mathrm{limit}^{-2(n+1)}\biggr]^{1/(n-3)} \biggl[ \frac{\pi}{5^n} \biggl( \frac{2^2}{3}\biggr)^{(n+1)} K^{-n} G^{n} M_\mathrm{limit}^{n-1} \biggr]^{3(n+1)/[n(n-3)]} </math>

 

 

<math>~ -3R_\mathrm{eq}^{-4}\biggl[ 2^{-2(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n+1} K^{4n} G^{-3(n+1)} M_\mathrm{limit}^{-2(n+1)}\biggr]^{1/(n-3)} \biggl[ \frac{\pi}{5^n} \biggl( \frac{2^2}{3}\biggr)^{(n+1)} K^{-n} G^{n} M_\mathrm{limit}^{n-1} \biggr]^{4/(n-3)} </math>

 

<math>~=</math>

<math>~ 2^2 R_\mathrm{eq}^{-3(n+1)/n}\biggl[ 2^{-2n(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n(n+1)} K^{4n^2} G^{-3n(n+1)} M_\mathrm{limit}^{-2n(n+1)}\cdot \frac{\pi^{3(n+1)}}{5^{3n(n+1)}} \biggl( \frac{2^2}{3}\biggr)^{3(n+1)(n+1)} K^{-3n(n+1)} G^{3n(n+1)} M_\mathrm{limit}^{3(n-1)(n+1)} \biggr]^{1/[n(n-3)]} </math>

 

 

<math>~ -3R_\mathrm{eq}^{-4}\biggl[ 2^{-2(5n+1)}\biggl( \frac{3^4\cdot 5^3}{\pi} \biggr)^{n+1} K^{4n} G^{-3(n+1)} M_\mathrm{limit}^{-2(n+1)}\cdot \frac{\pi^4}{5^{4n}} \biggl( \frac{2^2}{3}\biggr)^{4(n+1)} K^{-4n} G^{4n} M_\mathrm{limit}^{4(n-1)} \biggr]^{1/(n-3)} </math>

 

<math>~=</math>

<math>~ R_\mathrm{eq}^{-3(n+1)/n} \biggl( \frac{3}{2^2\pi}\biggr)^{(n+1)/n} K M_\mathrm{limit}^{(n+1)/n} - R_\mathrm{eq}^{-4} \biggl(\frac{3}{2^2\cdot 5\pi}\biggr) G M_\mathrm{limit}^2 </math>

 

<math>~=</math>

<math>~K \biggl( \frac{3M_\mathrm{limit}}{4\pi R_\mathrm{eq}^3} \biggr)^{(n+1)/n} - \frac{3GM_\mathrm{limit}^2}{20\pi R_\mathrm{eq}^4} \, .</math>

Recalling that <math>~\eta \leftrightarrow (n+1)/n</math>, it is clear that this <math>~P_e(R_\mathrm{eq})</math> relation exactly matches equation (5) of Whitworth, which reads:

Whitworth (1981, MNRAS, 195, 967)


Stability

Similarly, according to the above-derived stability criterion, pressure-truncated polytropic configurations will only be stable if,

<math>~\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4 </math>

<math>~></math>

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

Or, given that <math>~P_\mathrm{norm}R_\mathrm{norm}^4 = G M_\mathrm{tot}^2</math>, the criterion for stability may be written as,

<math>~P_e </math>

<math>~></math>

<math>~ \frac{(n-3)\mathcal{A}}{4\pi(n+1)} \biggl( \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}^4} \biggr) =\frac{(n-3)}{20\pi(n+1)} \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \biggl( \frac{GM_\mathrm{limit}^2}{R_\mathrm{eq}^4} \biggr) \, . </math>


Our Case P Analysis (raw)

Coefficient Definitions

As has been both summarized and detailed in an accompanying discussion, our Case P analysis has demonstrated that the following,

Algebraic Free-Energy Function

<math> \mathfrak{G}^* = -3\mathcal{A} \chi^{-1} +~ n\mathcal{B} \chi^{-3/n} +~ \mathcal{D}\chi^3 \, , </math>

properly governs the equilibrium structure and stability of pressure-truncated polytropic configurations. This algebraic expression is identical to the free-energy function given above if the following coefficient and variable substitutions are made:

Our Case P Analysis

<math>~a</math>

<math>~\rightarrow</math>

<math>~3\mathcal{A}</math>

<math>~b</math>

<math>~\rightarrow</math>

<math>~n\mathcal{B} </math>

<math>~c</math>

<math>~\rightarrow</math>

<math>~\mathcal{D} \equiv \frac{4\pi}{3} \biggl(\frac{P_e}{P_\mathrm{norm}}\biggr)</math>

<math>~x</math>

<math>~\rightarrow</math>

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

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

<math>~\rightarrow</math>

<math>~\mathfrak{G}^{*} \equiv \frac{\mathfrak{G}}{E_\mathrm{norm}}</math>

where — see, for example, our accompanying review,

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

<math>~=</math>

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

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

<math>~=</math>

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

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

<math>~=</math>

<math>~ \biggl[ K^n G^{-3}M_\mathrm{tot}^{n-5} \biggr]^{1/(n-3)} \, ,</math>

and, in terms of the structural form factors, <math>~\tilde\mathfrak{f}_M</math>, <math>~\tilde\mathfrak{f}_A</math>, and <math>~\tilde\mathfrak{f}_W</math>,

<math>~\mathcal{A}</math>

<math>~=</math>

<math>\frac{1}{5} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}} \biggr)^2 \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \, , </math>

<math>~\mathcal{B}</math>

<math>~=</math>

<math> \biggl(\frac{3}{4\pi}\biggr)^{1/n} \biggl( \frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr)^{(n+1)/n} \frac{\tilde\mathfrak{f}_A}{\tilde\mathfrak{f}_M^{(n+1)/n}} \, . </math>

Virial Equilibrium

Plugging these coefficient assignments into the above mathematical prescription of the virial theorem gives the following relationship between the applied external pressure and the resulting equilibrium radius of pressure-truncated polytropic configurations,

<math>~\mathcal{B} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{(n-3)/n} </math>

<math>~=</math>

<math>\mathcal{A} + \frac{4\pi}{3} \biggl( \frac{P_\mathrm{e}}{P_\mathrm{norm}} \biggr) \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4 \, , </math>

which matches the statement of virial equilibrium presented in our accompanying, more detailed analysis. A rearrangement of terms explicitly provides the desired <math>~P_e(R_\mathrm{eq})</math> function, namely,

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

<math>~=</math>

<math>~\frac{3}{4\pi} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{-4} \biggl[ \mathcal{B} \biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^{(n-3)/n} -\mathcal{A} \biggr] \, , </math>

or (see the accompanying derivation for details),

<math>~P_e </math>

<math>~=</math>

<math>~K \biggl( \frac{3M_\mathrm{limit}}{4\pi R_\mathrm{eq}^3} \biggr)^{(n+1)/n} \cdot \frac{\mathfrak{f}_A}{\mathfrak{f}_M^{(n+1)/n}} - \biggl(\frac{3GM_\mathrm{limit}^2}{20\pi R_\mathrm{eq}^4} \biggr)\cdot \frac{\mathfrak{f}_W}{\mathfrak{f}^2_M} \, .</math>

Notice that, when <math>~P_e \rightarrow 0</math>, this expression reduces to the solution we obtained for an isolated polytrope, expressed in terms of <math>~K</math> and <math>~M_\mathrm{limit}</math> (see the left-hand column of our table titled "Two Points of View"). Also notice that this equilibrium relation exactly matches the one derived by Whitworth — and rederived above — when all three structural form factors are set to unity. This is as it should be because all of Whitworth's results were derived assuming uniform-density configurations.


Stability

Similarly, according to the above-derived stability criterion, pressure-truncated polytropic configurations will only be stable if,

<math>~\biggl( \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \biggr)^4 </math>

<math>~></math>

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

Or, given that <math>~P_\mathrm{norm}R_\mathrm{norm}^4 = G M_\mathrm{tot}^2</math>, the criterion for stability may be written as,

<math>~P_e </math>

<math>~></math>

<math>~ \frac{(n-3)\mathcal{A}}{4\pi(n+1)} \biggl( \frac{GM_\mathrm{tot}^2}{R_\mathrm{eq}^4} \biggr) =\frac{(n-3)}{20\pi(n+1)} \cdot \frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_M^2} \biggl( \frac{GM_\mathrm{limit}^2}{R_\mathrm{eq}^4} \biggr) \, . </math>

Our Case P Analysis (compact)

As has been both summarized and detailed in an accompanying discussion, our Case P analysis has demonstrated that the following,

Renormalized Free-Energy Function

<math> \mathfrak{G}^{**} \equiv \mathfrak{G}^* \biggl[ \frac{\mathcal{A}^3}{\mathcal{B}^n} \biggr]^{1/(n-3)} = -3 \Chi^{-1} +~ n\Chi^{-3/n} +~ \Pi_\mathrm{ad}\Chi^3 \, , </math>

properly governs the equilibrium structure and stability of pressure-truncated polytropic configurations. This expression is identical to the free-energy function given above if the following coefficient and variable substitutions are made:

Our Case P Analysis

<math>~a</math>

<math>~\rightarrow</math>

<math>~3</math>

<math>~b</math>

<math>~\rightarrow</math>

<math>~n</math>

<math>~c</math>

<math>~\rightarrow</math>

<math>~\Pi_\mathrm{ad} \equiv \frac{P_e}{P_\mathrm{ad}}</math>

<math>~x</math>

<math>~\rightarrow</math>

<math>~\Chi \equiv \frac{R}{R_\mathrm{ad}}</math>

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

<math>~\rightarrow</math>

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

where,



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.
|   Go Home   |


Outline of Detailed Investigations Leading to Above Summary

First Effort

My first attempt to analytically define the free energy, and then the virial equilibrium, of pressure-truncated (embedded) polytropic spheres was developed as a direct extension of my description of the virial equilibrium of isolated polytropes. An important outcome of this "first effort" was the unveiling of analytic expressions for the key structural form factors, both for isolated polytropes and, separately, for pressure-truncated polytropic structures.

I am very confident that the form-factor expressions presented for isolated polytropes are all correct because they have been cross-checked with expressions for closely related "integral" parameters discussed by Chandrasekhar [C67]. Although the form-factor expressions derived for pressure-truncated polytropes make some sense — they look very similar to the ones presented for isolated polytropes and seem to behave properly for models which, based on detailed force-balanced analysis, are known to be in equilibrium — I have much less confidence that they are correct. A couple of strategies were developed in an effort to demonstrate the validity and utility of these more general form-factor expressions, resulting in the derivation of a concise virial equilibrium relation,

<math>\Pi_\mathrm{ad} = \chi_\mathrm{ad}^{-3\gamma} - \chi_\mathrm{ad}^{-4} \, ,</math>

that incorporates the newly defined normalization parameters, <math>~R_\mathrm{ad}</math> and <math>~P_\mathrm{ad}</math>. But subsequent derivations aimed at more conclusively demonstrating the correctness of the more general form-factor expressions were messy and got bogged down.

Second Effort

My second attempt to analytically define the free energy, and then the virial equilibrium, of pressure-truncated (embedded) polytropic spheres built upon my first effort and, for a couple of different polytropic indexes, focused on comparing the mass-radius relationship embodied in detailed force-balanced models against the mass-radius relationship implied by the virial theorem. A lot of reasonable results seem to have arisen from a discussion of models (done numerically using Excel) with <math>~n=4</math> polytropic index. And there are some nice aspects of models with an <math>~n=5</math> index, but these models raise some serious concerns related to the fact that two of our "derived" form-factor expressions involve division by the factor, <math>~(5-n)</math>, that is, division by zero.

Third Effort

In an attempt to answer the serious concern(s) raised during our first two efforts, we finally buckled down and performed the integrals necessary to determine expressions for key structural form factors in the cases where the internal structure is known analytically, specifically, for indexes <math>~n=5</math> and <math>~n=1</math>. The result is that the individual expressions derived by direct integration for <math>~\mathfrak{f}_W</math> and for <math>~\mathfrak{f}_A</math> do not match the general form-factor expressions that were rather cavalierly "derived" during our first effort. Oddly enough, as we discovered while fiddling around with the new results, the ratio of these form factors appears to be the same as before, namely,

<math>~\frac{\tilde\mathfrak{f}_W}{\tilde\mathfrak{f}_A - \tilde\theta^{n+1}}</math>

<math>~=</math>

<math>~ \biggl[ \frac{3\cdot 5}{(n+1) \tilde\xi^2 } \biggr] \, . </math>

It is worth noting that, as a result of this more thorough "third effort" examination, we have confirmed that the third key form factor,

<math>~\mathfrak{f}_M = \frac{\bar\rho}{\rho_c} = \biggl[- \frac{3\tilde\theta^'}{\tilde\xi}\biggr] \, ,</math>

which is the same as before and the same as for isolated polytropes. We also have determined that,

<math>~\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} = \biggl(\frac{\tilde\xi^2 \tilde\theta^'}{\xi_1 \theta^'_1} \biggr)\biggl[- \frac{\tilde\xi}{3\tilde\theta^'}\biggr] = - \frac{\tilde\xi^3 }{3\xi_1 \theta^'_1} \, , </math>

except in the case of <math>~n=5</math> structures, for which we have determined,

<math>~\biggl[\biggl(\frac{M_\mathrm{limit}}{M_\mathrm{tot}}\biggr) \frac{1}{\mathfrak{f}_M} \biggr]_{n=5} = \ell^3 = \biggl( \frac{\tilde\xi^2}{3} \biggr)^{3/2} \, . </math>

First Effort, Second Time Around

In an accompanying chapter, we reproduce the discussion associated with our "First Effort", as referenced above, but correct expressions for <math>~\mathfrak{f}_W</math> and <math>~\mathfrak{f}_A</math>, as identified in our "Third Effort" and, accordingly, re-derive various affected expressions that follow.

Second Effort, Second Time Around

In an accompanying chapter, we reproduce the discussion associated with our "Second Effort", as referenced above, but revise key sections to incorporate corrected expressions for the structural form factors.


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
|   H_Book Home   |   YouTube   |
Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS |
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