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(Created page with '__FORCETOC__ =Virial Equilibrium of Pressure-Truncated Polytropes= Here we will draw heavily from an [[User:Tohline/SSC/FreeEnergy/PolytropesEmbedded#Pressure-Truncated_Polytrop…')
 
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Here we will draw heavily from an [[User:Tohline/SSC/FreeEnergy/PolytropesEmbedded#Pressure-Truncated_Polytropes|accompanying ''Free Energy Synopsis'']].
Here we will draw heavily from an [[User:Tohline/SSC/FreeEnergy/PolytropesEmbedded#Pressure-Truncated_Polytropes|accompanying ''Free Energy Synopsis'']].


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{{LSU_HBook_header}}
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</tr>
</tr>
</table>
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and,


<table border="0" cellpadding="5" align="center">


<tr>
  <td align="right">
<math>~\frac{d^2\mathfrak{G}^*}{dx^2}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{1}{x^3} \biggl[ \biggl( \frac{3b}{n} \biggr)\biggl(\frac{n+3}{n}\biggr) x^{(n-3)/n} + 6c x^4 - 2a  \biggr]  \, .</math>
  </td>
</tr>
</table>


=See Also=
=See Also=

Revision as of 18:04, 7 February 2019

Virial Equilibrium of Pressure-Truncated Polytropes

Here we will draw heavily from an accompanying Free Energy Synopsis.


Whitworth's (1981) Isothermal Free-Energy Surface
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In the context of spherically symmetric, pressure-truncated polytropic configurations, the relevant free-energy expression is,

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

<math>~=</math>

<math>~W_\mathrm{grav} + U_\mathrm{int} + P_e V \, .</math>

When rewritten in a suitably dimensionless form — see two useful alternatives, below — this expression becomes,

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

<math>~=</math>

<math>~- a x^{-1} + bx^{-3/n} + c x^3 \, ,</math>

where <math>~x</math> is the configuration's dimensionless radius and <math>~a</math>, <math>~b</math>, and <math>~c</math> are constants. We therefore have,

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

<math>~=</math>

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

and,

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

<math>~=</math>

<math>~\frac{1}{x^3} \biggl[ \biggl( \frac{3b}{n} \biggr)\biggl(\frac{n+3}{n}\biggr) x^{(n-3)/n} + 6c x^4 - 2a \biggr] \, .</math>

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

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