Difference between revisions of "User:Tohline/SSC/FreeEnergy/PolytropesEmbedded"

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(→‎Free-Energy of Truncated Polytropes: Finished re-derivation of "case P" free-energy function)
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<math>~
<math>~
- \biggl[\frac{3}{5}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggr] \frac{GM^2}{R}  
- \biggl[\frac{3}{5}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggr] \frac{GM^2}{R}  
- \biggl[\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}}  \biggr] \frac{nKM^{(n+1)/n}}{R^{3/n}}
+ \biggl[\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}}  \biggr] \frac{nKM^{(n+1)/n}}{R^{3/n}}
+ \frac{4\pi}{3} \cdot P_e R^3  
+ \frac{4\pi}{3} \cdot P_e R^3  
\, ,</math>
\, ,</math>
Line 230: Line 230:
<math>
<math>
\mathfrak{G}_{K,M}^* \equiv \frac{\mathfrak{G}_{K,M}}{E_\mathrm{norm}} =  
\mathfrak{G}_{K,M}^* \equiv \frac{\mathfrak{G}_{K,M}}{E_\mathrm{norm}} =  
-3A\biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1} -~ nB \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-3/n}  
-3A\biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1} +~ nB \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-3/n}  
+~ \biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 \, ,
+~ \biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 \, ,
</math>
</math>
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<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~E_\mathrm{SWS} \equiv \frac{GM_\mathrm{SWS}^2}{R_\mathrm{SWS}}</math>
<math>~E_\mathrm{SWS} \equiv \biggl( \frac{n}{n+1} \biggr) \frac{GM_\mathrm{SWS}^2}{R_\mathrm{SWS}}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 316: Line 316:
   <td align="left">
   <td align="left">
<math>~
<math>~
\biggl( \frac{n+1}{n} \biggr)^{5/2} G^{-3/2}K^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]} \, .</math>
\biggl( \frac{n+1}{n} \biggr)^{3/2} G^{-3/2}K^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]} \, .</math>
   </td>
   </td>
</tr>
</tr>
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</div>
</div>


After implementing these normalizations, the expression that describes the P<sub>1</sub> Free-Energy surface is,
After implementing these normalizations &#8212; see our [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#Our_Case_M_Analysis|accompanying analysis]] for details &#8212; the expression that describes the P<sub>1</sub> Free-Energy surface is,
<div align="center">
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
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<math>~=</math>
<math>~=</math>
   </td>
   </td>
<!-- HIDE LONG RE-DERIVATION ...
   <td align="left">
   <td align="left">
<math>~
<math>~
\biggl\{\biggl( \frac{n+1}{n} \biggr)^{5/2} G^{-3/2}K^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]} \biggr\}^{-1}
\biggl\{\biggl( \frac{n+1}{n} \biggr)^{3/2} G^{-3/2}K^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]} \biggr\}^{-1}
</math>
</math>
   </td>
   </td>
Line 350: Line 351:
<math>~
<math>~
\times \biggl\{- \biggl[\frac{3}{5}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggr] \frac{GM^2}{R}  
\times \biggl\{- \biggl[\frac{3}{5}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggr] \frac{GM^2}{R}  
- \biggl[\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}}  \biggr] \frac{nKM^{(n+1)/n}}{R^{3/n}}
+ \biggl[\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}}  \biggr] \frac{nKM^{(n+1)/n}}{R^{3/n}}
+ \frac{4\pi}{3} \cdot P_e R^3  
+ \frac{4\pi}{3} \cdot P_e R^3  
\biggr\}
\biggr\}
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   <td align="left">
   <td align="left">
<math>~-
<math>~-
\biggl( \frac{n}{n+1} \biggr)^{5/2} G^{3/2}K^{-3n/(n+1)} P_\mathrm{e}^{(n-5)/[2(n+1)]}  
\biggl( \frac{n}{n+1} \biggr)^{3/2} G^{3/2}K^{-3n/(n+1)} P_\mathrm{e}^{(n-5)/[2(n+1)]}  
</math>
</math>
   </td>
   </td>
Line 381: Line 382:
<math>~
<math>~
\times \biggl\{\biggl[\frac{3}{5}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggr] \frac{GM^2}{R}  
\times \biggl\{\biggl[\frac{3}{5}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggr] \frac{GM^2}{R}  
+ \biggl[\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}}  \biggr] \frac{nKM^{(n+1)/n}}{R^{3/n}}
- \biggl[\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}}  \biggr] \frac{nKM^{(n+1)/n}}{R^{3/n}}
- \frac{4\pi}{3} \cdot P_e R^3  
- \frac{4\pi}{3} \cdot P_e R^3  
\biggr\}
\biggr\}
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   <td align="left">
   <td align="left">
<math>~-
<math>~-
\biggl( \frac{n}{n+1} \biggr)^{5/2} G^{3/2}K^{-3n/(n+1)} P_\mathrm{e}^{(n-5)/[2(n+1)]}  
\biggl( \frac{n}{n+1} \biggr)^{3/2} G^{3/2}K^{-3n/(n+1)} P_\mathrm{e}^{(n-5)/[2(n+1)]}  
</math>
</math>
   </td>
   </td>
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   <td align="left">
   <td align="left">
<math>~
<math>~
+ \biggl[n\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}}  \biggr] \biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl(\frac{R_\mathrm{SWS}}{R}\biggr)^{3/n}
- \biggl[n\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}}  \biggr] \biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl(\frac{R_\mathrm{SWS}}{R}\biggr)^{3/n}
\biggl[K M_\mathrm{SWS}^{(n+1)/n} R_\mathrm{SWS}^{-3/n}  \biggr]
\biggl[K M_\mathrm{SWS}^{(n+1)/n} R_\mathrm{SWS}^{-3/n}  \biggr]
</math>
</math>
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   <td align="left">
   <td align="left">
<math>~-
<math>~-
\biggl( \frac{n}{n+1} \biggr)^{5/2} G^{3/2}K^{-3n/(n+1)} P_\mathrm{e}^{(n-5)/[2(n+1)]}  
\biggl( \frac{n}{n+1} \biggr)^{3/2} G^{3/2}K^{-3n/(n+1)} P_\mathrm{e}^{(n-5)/[2(n+1)]}  
</math>
</math>
   </td>
   </td>
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   <td align="left">
   <td align="left">
<math>~
<math>~
+ \biggl[n\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}}  \biggr]  
- \biggl[n\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}}  \biggr]  
\biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl(\frac{R_\mathrm{SWS}}{R}\biggr)^{3/n}
\biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl(\frac{R_\mathrm{SWS}}{R}\biggr)^{3/n}
K \biggl[ \biggl( \frac{n+1}{nG} \biggr)^{3/2} K^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]}  \biggr]^{(n+1)/n} \biggl[ \biggl( \frac{n+1}{nG} \biggr)^{1/2} K^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]}  \biggr]^{-3/n}   
K \biggl[ \biggl( \frac{n+1}{nG} \biggr)^{3/2} K^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]}  \biggr]^{(n+1)/n} \biggl[ \biggl( \frac{n+1}{nG} \biggr)^{1/2} K^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]}  \biggr]^{-3/n}   
Line 521: Line 522:
   <td align="left">
   <td align="left">
<math>~-
<math>~-
\biggl( \frac{n}{n+1} \biggr)^{5/2} G^{3/2}K^{-3n/(n+1)} P_\mathrm{e}^{(n-5)/[2(n+1)]}  
\biggl( \frac{n}{n+1} \biggr)^{3/2} G^{3/2}K^{-3n/(n+1)} P_\mathrm{e}^{(n-5)/[2(n+1)]}  
</math>
</math>
   </td>
   </td>
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   <td align="left">
   <td align="left">
<math>~
<math>~
+ \biggl[n\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}}  \biggr]  
- \biggl[n\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}}  \biggr]  
\biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl(\frac{R_\mathrm{SWS}}{R}\biggr)^{3/n}
\biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl(\frac{R_\mathrm{SWS}}{R}\biggr)^{3/n}
K \biggl[ \biggl( \frac{n+1}{nG} \biggr)^{3(n+1)/(2n)} K^{2} P_\mathrm{e}^{(3-n)/(2n)}  \biggr]
K \biggl[ \biggl( \frac{n+1}{nG} \biggr)^{3(n+1)/(2n)} K^{2} P_\mathrm{e}^{(3-n)/(2n)}  \biggr]
Line 583: Line 584:
   <td align="left">
   <td align="left">
<math>~-
<math>~-
\biggl( \frac{n}{n+1} \biggr)^{5/2} G^{3/2}K^{-3n/(n+1)} P_\mathrm{e}^{(n-5)/[2(n+1)]}  
\biggl( \frac{n}{n+1} \biggr)^{3/2} G^{3/2}K^{-3n/(n+1)} P_\mathrm{e}^{(n-5)/[2(n+1)]}  
</math>
</math>
   </td>
   </td>
Line 612: Line 613:
   <td align="left">
   <td align="left">
<math>~
<math>~
+ \biggl[n\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}}  \biggr]  
- \biggl[n\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}}  \biggr]  
\biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl(\frac{R_\mathrm{SWS}}{R}\biggr)^{3/n}
\biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl(\frac{R_\mathrm{SWS}}{R}\biggr)^{3/n}
\biggl[ \biggl( \frac{n+1}{n} \biggr)^{3/2} G^{-3/2} K^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]}  \biggr]   
\biggl[ \biggl( \frac{n+1}{n} \biggr)^{3/2} G^{-3/2} K^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]}  \biggr]   
Line 643: Line 644:
   <td align="left">
   <td align="left">
<math>~-
<math>~-
\biggl[\frac{3}{5}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggr] \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2  
\biggl[\frac{3}{5}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggr] \biggl( \frac{n+1}{n} \biggr) \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2  
\biggl(\frac{R_\mathrm{SWS}}{R}\biggr)\biggr]
\biggl(\frac{R_\mathrm{SWS}}{R}\biggr)
</math>
</math>
   </td>
   </td>
Line 658: Line 659:
   <td align="left">
   <td align="left">
<math>~
<math>~
- \biggl[n\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}}  \biggr]  
+ \biggl[n\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}}  \biggr]  
\biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl(\frac{R_\mathrm{SWS}}{R}\biggr)^{3/n}
\biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl(\frac{R_\mathrm{SWS}}{R}\biggr)^{3/n}
\biggl( \frac{n}{n+1} \biggr)  
+ \frac{4\pi}{3} \cdot \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^3
+ \frac{4\pi}{3} \cdot \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^3 \biggl( \frac{n}{n+1} \biggr)  
</math>
  </td>
</tr>
 
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~-
\biggl[\frac{3}{5}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggl( \frac{n+1}{n} \biggr)\biggr] \biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2
\biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1}
</math>
  </td>
</tr>
 
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
&nbsp;
  </td>
  <td align="left">
<math>~
+ \biggl[n\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}}  \biggr]
\biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/n}
+ \frac{4\pi}{3} \cdot \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^3
</math>
</math>
   </td>
   </td>
</tr>
</tr>


<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
END SUPPRESSION OF LONG DERIVATION -->
  <td align="left">
<math>~- 3A\biggl( \frac{n+1}{n} \biggr)\biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1}
+ nB \biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/n}
+ \frac{4\pi}{3} \cdot \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^3
</math>
  </td>
</tr>
</table>
</table>
</div>
</div>

Revision as of 15:47, 13 July 2016

Free-Energy of Truncated Polytropes

Whitworth's (1981) Isothermal Free-Energy Surface
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In this case, the Gibbs-like free energy is given by the sum of three separate energies,

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

<math>~=</math>

<math>~W_\mathrm{grav} + \mathfrak{S}_\mathrm{therm} + P_eV</math>

 

<math>~=</math>

<math>~ - \biggl[\frac{3}{5}\cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \biggr] \frac{GM^2}{R} + \biggl[\biggl(\frac{3}{4\pi}\biggr)^{1/n} \frac{\tilde{\mathfrak{f}}_A}{\tilde{\mathfrak{\mathfrak{f}}}_M^{(n+1)/n}} \biggr] \frac{nKM^{(n+1)/n}}{R^{3/n}} + \frac{4\pi}{3} \cdot P_e R^3 \, ,</math>

where, as derived elsewhere,

Structural Form Factors for Pressure-Truncated Polytropes <math>~(n \ne 5)</math>

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

<math>~=</math>

<math>~ \biggl( - \frac{3\tilde\theta^'}{\tilde\xi} \biggr) </math>

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

<math>~=</math>

<math>\frac{3\cdot 5}{(5-n)\tilde\xi^2} \biggl[\tilde\theta^{n+1} + 3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] </math>

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

<math>~=</math>

<math>~\frac{1}{(5-n)} \biggl\{ 6\tilde\theta^{n+1} + (n+1) \biggl[3 (\tilde\theta^')^2 - \tilde\mathfrak{f}_M \tilde\theta \biggr] \biggr\} </math>

As we have shown separately, for the singular case of <math>~n = 5</math>,

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

<math>~=</math>

<math>~ ( 1 + \ell^2 )^{-3/2} </math>

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

<math>~=</math>

<math>~ \frac{5}{2^4} \cdot \ell^{-5} \biggl[ \ell \biggl( \ell^4 - \frac{8}{3}\ell^2 - 1 \biggr)(1 + \ell^2)^{-3} + \tan^{-1}(\ell ) \biggr] </math>

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

<math>~=</math>

<math>~ \frac{3}{2^3} \ell^{-3} [ \tan^{-1}(\ell ) + \ell (\ell^2-1) (1+\ell^2)^{-2} ] </math>

where, <math>~\ell \equiv \tilde\xi/\sqrt{3} </math>


In general, then, the warped free-energy surface drapes across a four-dimensional parameter "plane" such that,

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

<math>~=</math>

<math>~\mathfrak{G}(R, K, M, P_e) \, .</math>

In order to effectively visualize the structure of this free-energy surface, we will reduce the parameter space from four to two, in two separate ways: First, we will hold constant the parameter pair, <math>~(K,M)</math>; adopting Kimura's (1981b) nomenclature, we will refer to the resulting function, <math>~\mathfrak{G}_{K,M}(R,P_e)</math>, as an "M1 Free-Energy Surface" because the mass is being held constant. Second, we will hold constant the parameter pair, <math>~(K,P_e)</math>, and examine the resulting "P1 Free-Energy Surface," <math>~\mathfrak{G}_{K,P_e}(R,M)</math>.

The M1 Free-Energy Surface

It is useful to rewrite the free-energy function in terms of dimensionless parameters. Here we need to pick normalizations for energy, radius, and pressure that are expressed in terms of the gravitational constant, <math>~G</math>, and the two fixed parameters, <math>~K</math> and <math>~M</math>. We have chosen to use,

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

<math>~\equiv</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>~\equiv</math>

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

which, as is detailed in an accompanying discussion, are similar but not identical to the normalizations used by Horedt (1970) and by Whitworth (1981). The self-consistent energy normalization is,

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

<math>~\equiv</math>

<math>~P_\mathrm{norm} R^3_\mathrm{norm} \, .</math>

As we have demonstrated elsewhere, after implementing these normalizations, the expression that describes the M1 Free-Energy surface is,

<math> \mathfrak{G}_{K,M}^* \equiv \frac{\mathfrak{G}_{K,M}}{E_\mathrm{norm}} = -3A\biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-1} +~ nB \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^{-3/n} +~ \biggl( \frac{4\pi}{3} \biggr) \frac{P_e}{P_\mathrm{norm}} \biggl(\frac{R}{R_\mathrm{norm}}\biggr)^3 \, , </math>

where the constants,

<math>~A</math>

<math>~\equiv</math>

<math>\frac{1}{5} \cdot \frac{\tilde{\mathfrak{f}}_W}{\tilde{\mathfrak{f}}_M^2} \, ,</math>

<math>~B</math>

<math>~\equiv</math>

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

Given the polytropic index, <math>~n</math>, we expect to obtain a different M1 free-energy surface for each choice of the dimensionless truncation radius, <math>~\tilde\xi</math>; this choice will imply corresponding values for <math>~\tilde\theta</math> and <math>~\tilde\theta^'</math> and, hence also, corresponding (constant) values of the coefficients, <math>~A</math> and <math>~B</math>.


The P1 Free-Energy Surface

Again, it is useful to rewrite the free-energy function in terms of dimensionless parameters. But here we need to pick normalizations for energy, radius, and mass that are expressed in terms of the gravitational constant, <math>~G</math>, and the two fixed parameters, <math>~K</math> and <math>~P_e</math>. As is detailed in an accompanying discussion, we have chosen to use the normalizations defined by Stahler (1983), namely,

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

<math>~\equiv</math>

<math>~\biggl( \frac{n+1}{nG} \biggr)^{1/2} K^{n/(n+1)} P_\mathrm{e}^{(1-n)/[2(n+1)]} \, ,</math>

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

<math>~\equiv</math>

<math>~\biggl( \frac{n+1}{nG} \biggr)^{3/2} K^{2n/(n+1)} P_\mathrm{e}^{(3-n)/[2(n+1)]} \, ,</math>

The self-consistent energy normalization is,

<math>~E_\mathrm{SWS} \equiv \biggl( \frac{n}{n+1} \biggr) \frac{GM_\mathrm{SWS}^2}{R_\mathrm{SWS}}</math>

<math>~=</math>

<math>~ \biggl( \frac{n+1}{n} \biggr)^{3/2} G^{-3/2}K^{3n/(n+1)} P_\mathrm{e}^{(5-n)/[2(n+1)]} \, .</math>

After implementing these normalizations — see our accompanying analysis for details — the expression that describes the P1 Free-Energy surface is,

<math>~\mathfrak{G}_{K,P_e}^* \equiv \frac{\mathfrak{G}_{K,P_e}}{E_\mathrm{SWS}}</math>

<math>~=</math>

<math>~- 3A\biggl( \frac{n+1}{n} \biggr)\biggl( \frac{M}{M_\mathrm{SWS}}\biggr)^2 \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-1} + nB \biggl(\frac{M}{M_\mathrm{SWS}}\biggr)^{(n+1)/n} \biggl(\frac{R}{R_\mathrm{SWS}}\biggr)^{-3/n} + \frac{4\pi}{3} \cdot \biggl( \frac{R}{R_\mathrm{SWS}}\biggr)^3 </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