Editing SSC/Stability/Isothermal (section)

===Equilibrium Model===

====Review====
In an [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|accompanying discussion]], while reviewing the original derivations of {{ Ebert55 }} and {{ Bonnor56 }}, we have detailed the equilibrium properties of pressure-truncated isothermal spheres.  These properties have been expressed in terms of the ''isothermal Lane-Emden function'', <math>~\psi(\xi)</math>, which provides a solution to the governing,
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Isothermal Lane-Emden Equation <p></p>
{{ Math/EQ_SSLaneEmden02 }}
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A parallel presentation of these details can be found in &sect;2 &#8212; specifically, equations (2.4) through (2.10) &#8212; of {{ Yabushita68 }}.  Each of Yabushita's key mathematical expressions can be mapped to ours via the variable substitutions presented here in Table 1.

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<table border="1" align="center" cellpadding="5">
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  <td align="center" colspan="7">
<font size="+1"><b>Table 1:</b></font>&nbsp; Mapping Between Our Notation and that employed by {{ Yabushita68 }}
  </td>
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<tr>
  <td align="right">Yabushita's (1968) Notation:</td>
  <td align="center" width="8%"><math>~x</math></td>
  <td align="center" width="8%"><math>~\psi</math></td>
  <td align="center" width="8%"><math>~\mu</math></td>
  <td align="center" width="8%"><math>~M</math></td>
  <td align="center" width="8%"><math>~x_0</math></td>
  <td align="center" width="8%"><math>~p_0</math></td>
</tr>
<tr>
  <td align="right">[[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|Our Notation]]:</td>
  <td align="center"><math>~\xi</math></td>
  <td align="center"><math>~-\psi</math></td>
  <td align="center"><math>~\bar\mu</math></td>
  <td align="center"><math>~M_{\xi_e}</math></td>
  <td align="center"><math>~\xi_e</math></td>
  <td align="center"><math>~P_e</math></td>
</tr>
</table>
</div>

For example, given the system's sound speed, <math>~c_s</math>, and total mass, <math>~M_{\xi_e}</math>, the expression from [[SSC/Structure/BonnorEbert#Pressure|our presentation]] that shows how the bounding external pressure, <math>~P_e</math>, depends on the dimensionless Lane-Emden function, <math>~\psi</math>, is,
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  <td align="right">
<math>~P_e</math>
  </td>
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<math>~=</math>
  </td>
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<math>~\biggl( \frac{c_s^8}{4\pi G^3 M_{\xi_e}^2} \biggr) ~\xi_e^4 \biggl(\frac{d\psi}{d\xi}\biggr)^2_e e^{-\psi_e}</math>
  </td>
</tr>

<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \xi_e^2 \biggl(\frac{d\psi}{d\xi}\biggr)_e e^{-(1/2)\psi_e}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
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<math>~\frac{1}{c_s^4}\biggl[ G^3 M_{\xi_e}^2 ~(4\pi P_e)\biggr]^{1 / 2} \, ,</math>
  </td>
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</table>
</div>

which &#8212; see the boxed-in excerpt that follows &#8212; exactly matches equation (2.9) of {{ Yabushita68 }}, after recalling that the system's sound speed is related to its temperature via the relation,
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<math>c_s^2 = \frac{\Re T}{\bar{\mu}} \, .</math>
</div>

And, [[SSC/Structure/BonnorEbert#Radius|our expression]] for the truncated configuration's equilibrium radius is,
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  <td align="right">
<math>~R</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
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<math>~\frac{GM_{\xi_e}}{c_s^2} \biggl[ \xi \biggl(\frac{d\psi}{d\xi}\biggr) \biggr]_e^{-1}</math>
  </td>
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</table>
</div>
which &#8212; see the boxed-in excerpt that follows &#8212; matches equation (2.10) of {{ Yabushita68 }}.


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<table border="1" cellpadding="5" width="80%">
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Equations extracted<sup>&dagger;</sup> from p. 110 of <br />{{ Yabushita68figure }}
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  <td align="left">
<!-- [[File:Yabushita68Eqns.png|600px|center|Yabushita (1968)]] -->

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  <td align="right" width="50%">
<math>
x^2 \frac{d\psi}{dx} e^{1 /  2)\psi}\biggr|_{x=x_0}
</math>
  </td>
  <td align="center" width="5%"><math>=</math></td>
  <td align="left">
<math>
- \biggl(\frac{\mu}{\Re T}\biggr)^2 M G^{3 / 2} (4\pi p_0)^{1 / 2}
</math>
  </td>
  <td align="right" width="5%">(2.9)</td>
</tr>

<tr>
  <td align="right" width="50%">
<math>
R \equiv \ell_0 x_0
</math>
  </td>
  <td align="center" width="5%"><math>=</math></td>
  <td align="left">
<math>
- \frac{M G \mu}{\Re T} \biggl(x_0 \frac{d\psi}{dx}\biggr|_{x_0} \biggr)^{-1}
</math>
  </td>
  <td align="right" width="5%">(2.10)</td>
</tr>
</table>

  </td>
</tr>
<tr><td align="left">
<sup>&dagger;</sup>Layout of equations has been modified from the original publication.
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</table>
</div>

====P-V Diagram====

As we have discussed in a [[SSC/Structure/BonnorEbert#P-V_Diagram|separate chapter that focuses on the structural properties of pressure-truncated Isothermal spheres]], {{ Bonnor56 }} examined the ''sequence'' of equilibrium models that is generated by varying the truncation radius over the range, <math>0 < \xi_e < \infty</math>.  In a diagram that shows how <math>P_e(\xi_e)</math> varies with equilibrium volume, <math>V(\xi_e) \propto R^3</math>, Bonnor noticed that there is a pressure above which no equilibrium configurations exist.  The original P-V diagram published by [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor (1956)] has been reproduced here, in the left-hand panel of our Figure 1.  The right-hand panel of Figure 1 shows the same equilibrium sequence, as generated from our [[SSC/Structure/IsothermalSphere#Our_Numerical_Integration|numerical integration of the isothermal Lane-Emden equation]]; we have adopted [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|Whitworth's normalizations]], <math>P_\mathrm{rf}</math> and <math>V_\mathrm{rf}</math>.


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<b>Figure 1: &nbsp; Bonnor's P-V Diagram</b> (see [[SSC/Structure/BonnorEbert#Fig1|accompanying discussion]] for details)

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Figure extracted from p. 355 of <br />{{ Bonnor56figure }}
<!--
[http://adsabs.harvard.edu/abs/1956MNRAS.116..351B W. B. Bonnor (1956)]<p></p>
"''Boyle's Law and Gravitational Instability''"<p></p>
MNRAS, vol. 116, pp. 351 - 359 &copy; Royal Astronomical Society
-->
  </td>
  <td align="left">
The {{ Bonnor56 }} equilibrium sequence, but as generated from our [[SSC/Structure/IsothermalSphere#Our_Numerical_Integration|numerical integration of the isothermal Lane-Emden equation]]; in our plot, we have adopted normalizations, <math>P_\mathrm{rf}</math> and <math>V_\mathrm{rf}</math>, from {{ Whitworth81 }}.
  </td>
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[[File:Bonnor1956Fig1reproducedB.png|400px|center|Bonnor (1956, MNRAS, 116, 351)]]
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[[File:IsothermalEquilSequence.png|800px|center|Pressure-Truncated Isothermal Equilibrium Sequence]]
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</div>

As is [[SSC/Stability/Isothermal#From_the_Analysis_of_Taff_and_Van_Horn_.281974.29|discussed below]], in separate studies, {{ Yabushita68 }} and {{ TVH74 }} examined the lowest-order modes of radial oscillations that arise in pressure-truncated isothermal spheres.  In both studies, numerical techniques were used to solve the eigenvalue problem associated with the [[#IsothermalLAWE|isothermal LAWE]], as derived below.  The nine individual equilibrium models that were studied by {{ TVH74hereafter }} are identified by the nine small, filled circular markers along the sequence that has been displayed in the right-hand panel of our Figure 1; as labeled, they correspond to models with <math>~\xi_e</math> = 2, 3, 4, 5, 6, 7, 8, 9, and 10.  Earlier, Yabushita studied the oscillation modes of three of these same configurations; specifically, the models with <math>~\xi_e</math> = 6, 7, and 8, which straddle the location along the equilibrium sequence of the model associated with the pressure maximum (the [[SSC/FreeEnergy/EquilibriumSequenceInstabilities#Instabilities_Associated_with_Equilibrium_Sequence_Turning_Points|turning point]] labeled "A" in Bonnor's P-V diagram).

====Other Properties====
<span id="IsothermalVariables">Also, as has been summarized in our [[SSC/Structure/BonnorEbert#P-V_Diagram|accompanying discussion]] of the equilibrium properties of pressure-truncated isothermal spheres, we have,</span>

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<math>~r_0 </math>
  </td>
  <td align="center">
<math>~=</math>
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<math>~\biggl( \frac{c_s^2}{4\pi G \rho_c} \biggr)^{1/2} \xi \, ;</math>
  </td>
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  <td align="right">
<math>~P_0 = c_s^2 \rho_0 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~(c_s^2 \rho_c) e^{-\psi} \, ;</math>
  </td>
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<tr>
  <td align="right">
<math>~M_r  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{c_s^6}{4\pi G^3 \rho_c} \biggr)^{1/2} \biggl[ \xi^2 \frac{d\psi}{d\xi} \biggr]  \, .</math>
  </td>
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</table>
</div>

Hence, for isothermal configurations,
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<math>~g_0 \equiv \frac{GM_r}{r_0^2}</math>
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  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~G\biggl( \frac{c_s^6}{4\pi G^3 \rho_c} \biggr)^{1/2} \biggl[ \xi^2 \frac{d\psi}{d\xi} \biggr]
\biggl[ \biggl( \frac{c_s^2}{4\pi G \rho_c} \biggr)^{1/2} \xi\biggr]^{-2}</math>
  </td>
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  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~c_s^2  
\biggl( \frac{4\pi G \rho_c}{c_s^2} \biggr)^{1 / 2} 
\biggl( \frac{d\psi}{d\xi} \biggr) \, .
</math>
  </td>
</tr>
</table>
</div>