Editing SSC/Stability/Isothermal (section)

===Behavior of First Logarithmic Derivative===

<span id="LogarithmicDerivative">For the record,</span> let's develop an expression for the logarithmic derivative of this displacement function.
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<math>\frac{d\ln x}{d\ln \xi} = \frac{\xi}{x} \cdot \frac{dx}{d\xi}</math>
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<math>=</math>
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<math>
\biggl(\frac{3-n}{n-1}\biggr) \biggl[ 1 
+ \frac{n(\theta^')^2 }{\theta^{n+1}} + \frac{3\theta^' }{\xi \theta^{n}} \biggr] 
\biggl[ 1 + \biggl(\frac{n-3}{n-1}\biggr) \frac{\theta^' }{\xi \theta^{n}}  \biggr]^{-1} 
</math>
  </td>
</tr>

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&nbsp;
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<math>=</math>
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<math>
- (n-3) \biggl[ 1 
+ \frac{n(\theta^')^2 }{\theta^{n+1}} + \frac{3\theta^' }{\xi \theta^{n}} \biggr] 
\biggl[ (n-1) + (n-3) \frac{\theta^' }{\xi \theta^{n}}  \biggr]^{-1} 
</math>
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</tr>

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<math>\Rightarrow ~~~ \biggl[ (n-1) + (n-3) \frac{\theta^' }{\xi \theta^{n}}  \biggr] \frac{d\ln x}{d\ln \xi} </math>
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<math>=</math>
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<math>
- (n-3) \biggl[ 1 
+ \frac{n(\theta^')^2 }{\theta^{n+1}} + \frac{3\theta^' }{\xi \theta^{n}} \biggr]  
</math>
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<math>\Rightarrow ~~~ \biggl[ (n-1)  \biggr] \frac{d\ln x}{d\ln \xi} </math>
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<math>=</math>
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<math>
- (n-3) \biggl\{ \biggl[ 1 
+ \frac{n(\theta^')^2 }{\theta^{n+1}} + \frac{3\theta^' }{\xi \theta^{n}} \biggr]  
+ \biggl[ \frac{\theta^' }{\xi \theta^{n}}  \biggr] \frac{d\ln x}{d\ln \xi} \biggr\}
</math>
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</tr>

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&nbsp;
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<math>=</math>
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<math>
- (n-3) \biggl\{ \biggl[ 1 
+ \frac{n(\theta^')^2 }{\theta^{n+1}}  \biggr]  
+ \biggl[ \frac{\theta^' }{\xi \theta^{n}}  \biggr] \biggl[3 + \frac{d\ln x}{d\ln \xi}\biggr] \biggr\}
</math>
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Now, as we have [[#From_Yabushita.27s_.281992.29_Analysis|discussed above]], {{ Yabushita92 }} has argued that for pressure-truncated configurations, the appropriate surface boundary condition should be,
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<math>\frac{d\ln x}{d\ln\xi} = -3 \, .</math>
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If we adopt this surface boundary condition, then this just-derived expression becomes,

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<math>3 (n-1)   </math>
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<math>=</math>
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<math>
(n-3) \biggl[ 1 
+ \frac{n(\theta^')^2 }{\theta^{n+1}}  \biggr]  
</math>
  </td>
</tr>

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<math>\Rightarrow~~~
(n-3) \biggl[ \frac{n(\theta^')^2 }{\theta^{n+1}}  \biggr]
</math>
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<math>=</math>
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<math>
2n 
</math>
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<math>\Rightarrow~~~
\frac{(\theta^')^2 }{\theta^{n+1}} 
</math>
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<math>=</math>
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<math>
\frac{2 }{(n-3) } \, .
</math>
  </td>
</tr>
</table>
</div>
According to {{ Horedt70 }} &#8212; see our [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Derivation|accompanying discussion]] &#8212; this is precisely the functional condition that pinpoints the <math>~P_e</math>-max turning point(s) along the equilibrium sequence in a <math>~P-V</math> diagram, for all pressure-truncated, polytropic spheres with <math>~n \ge 3</math>.  And, according to {{ Kimura81b }} &#8212; again, see our [[SSC/Structure/PolytropesEmbedded#Other_Limits|accompanying discussion]] &#8212; this is precisely the functional condition that pinpoints the maximum mass turning point(s) along equlibrium sequences as displayed in a mass-radius diagram.

<!-- CHRISTY OR COX BOUNDARY CONDITIONS ...
On the other hand, as we have [[SSC/Perturbations#ChristyCox|discussed separately]] &#8212; see also, [[#From_Yabushita.27s_.281992.29_Analysis|above]] &#8212; in order to align with the separate derivations presented by [http://adsabs.harvard.edu/abs/1966ARA%26A...4..353C Christy (1965)] and [http://adsabs.harvard.edu/abs/1967IAUS...28....3C Cox (1967)]  &#8212; the corresponding boundary condition at the surface is, 
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<math>~\frac{dh}{dx}</math>
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<math>~=</math>
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<math>~- \frac{h}{x} \biggr[  3 - \frac{4}{\gamma}  + \cancelto{0}{\frac{x s^2}{\gamma q}} \biggr]</math>
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  <td align="left" colspan="2">
&nbsp;
  </td>
</tr>

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&nbsp;
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<math>~=</math>
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<math>~\frac{n-3}{n+1} \biggl(\frac{h}{x} \biggr) \, .</math>
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  <td align="left">
&nbsp; &nbsp; &nbsp; &nbsp; at &nbsp; &nbsp; &nbsp; &nbsp; 
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<math>~x = x_0 \, .</math>
  </td>
</tr>
</table>
</div>
-->