Editing SSC/Stability/InstabilityOnsetOverview (section)

===Turning Points along Sequences of Pressure-Truncated Polytropes===
Just as we have discussed above in the context of isothermal spheres, when <math>~n \ge 5</math>, equilibrium configurations of finite extent can be constructed by truncating the function, <math>~\Theta_H</math>, at some radius, <math>~0 < \tilde\xi < \infty</math> &#8212; in which case the surface density is finite and set by the value of <math>~\tilde\theta \equiv \Theta_H(\tilde\xi)</math> &#8212; and embedding them in a hot, tenuous medium that exerts an external pressure, <math>~P_e = K\rho_c^{(n+1)/n}\tilde\theta^{(n+1)/n}</math>, uniformly across the surface of the truncated sphere.  Polytropes having <math>~0 \le n <5</math> may similarly be truncated at any radius, <math>~0 < \tilde\xi < \xi_\mathrm{surf}</math>.  For any value of the index, <math>~n</math>, the internal structure of each such "pressure-truncated" polytrope is completely describable in terms of the same function, <math>~\Theta_H(\xi)</math>, that describes the structure of an isolated polytrope with the same index, except that the function becomes physically irrelevant beyond <math>~\tilde\xi</math>.  

<table border="0" cellpadding="8" align="right">
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  <th align="center">[[File:DataFileButton02.png|right|60px|file = Dropbox/WorkFolder/Wiki edits/EmbeddedPolytropes/CombinedSequences.xlsx --- worksheet = EqSeqCombined]]Figure 3: &nbsp;Equilibrium Sequences<br />of Pressure-Truncated Polytropes
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[[File:MassVsRadiusCombined02.png|350px|Equilibrium sequences of Pressure-Truncated Polytropes]]
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As is the case for pressure-truncated isothermal spheres, for each index, <math>~n</math>, a ''sequence'' of pressure-truncated polytropes is readily defined by varying the value of <math>~\tilde\xi</math> over the range,  <math>~0 < \tilde\xi < \infty</math> &#8212; or, as the case may be, over the range, <math>~0 < \tilde\xi < \xi_\mathrm{surf}</math>.  In an [[SSC/Structure/PolytropesEmbedded#Additional.2C_Numerically_Constructed_Polytropic_Configurations|accompanying discussion of the properties of such equilibrium configurations]], we have graphically displayed in a single diagram the mass-radius relation of sequences having n = 1, 2.5, 3, 3.05, 3.5, 5, and 6.   In Figure 3, shown here on the right, we have redrawn these mass-radius relations for the subset of sequences that have <math>~n \ge 3</math>, and have inserted as well the mass-radius relation for pressure-truncated isothermal spheres &#8212; copied from the [[#Fig1|middle panel of Figure 1, above]].  More specifically, adopting the mass and radius normalizations introduced by [[SSC/Structure/PolytropesEmbedded#Stahler.27s_Presentation|S. W. Stahler (1983)]], each Figure 3 ''polytropic'' sequence has been defined via the pair of parametric relations,
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<math>
~\frac{M}{M_\mathrm{SWS} }
</math>
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<math>~=~</math>
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<math>
\biggl( \frac{n^3 }{4\pi} \biggr)^{1 / 2}  \biggl[ \theta^{(n-3)/2} \xi^2 
\biggl| \frac{d\theta}{d\xi} \biggr| ~\biggr]_{\tilde\xi} \, ,
</math>
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<math>
~\frac{R}{R_\mathrm{SWS}}
</math>
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<math>~=~</math>
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<math>~\biggl(  \frac{n }{4\pi} \biggr)^{1 / 2} 
\biggl[ \xi \theta^{(n-1)/2} \biggr]_{\tilde\xi} \, .
</math>
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</table>
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In Figure 3, as in Figure 1, a small, yellow circular marker has been used to identify the configuration along each sequence for which the equilibrium radius reaches its maximum value; also, a small, green circular marker identifies the configuration along each sequence that has the maximum mass.  As we have reviewed in an [[SSC/Structure/PolytropesEmbedded#Other_Limits|accompanying discussion]], [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura (1981)] has shown that, along each such (polytropic) mass-radius sequence, the configuration associated with the "maximum radius" turning point (yellow marker) occurs precisely where,
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<math>~
\biggl[~ \frac{\xi}{\theta} \biggl|\frac{d\theta}{d\xi}\biggr|~ \biggr]_{\tilde\xi}
</math>
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<math>~=</math>
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<math>~\frac{2}{n-1} \, ,</math>
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<span id="MaximumMass">and the configuration associated with the "maximum mass" turning point (green marker) occurs precisely where,</span>
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<math>~
\biggl[ \frac{1}{\theta^{n+1}} \biggl(\frac{d\theta}{d\xi}\biggr)^2\biggr]_{\tilde\xi}
</math>
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<math>~=</math>
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<math>~\frac{2}{n-3} \, .</math>
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As we have [[SSC/Structure/PolytropesEmbedded#Horedt.27s_Derivation|highlighted elsewhere]], approximately a decade prior to Kimura's work, [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)] showed that this last criterion also precisely identifies the configuration that is associated with the <math>~P_e</math>-max turning point along polytropic pressure-volume sequences, analogous to the location of the uppermost green marker in the lefthand panel of our [[#Fig1|Figure 1]].  [[SSC/Structure/PolytropesEmbedded#Table3|Table 3 of an accompanying discussion]] gives the exact, analytically determined coordinate locations of the pair of turning points that arise along the <math>~n=5</math> sequence, as well as approximate, numerically determined coordinate locations of the "maximum radius" (yellow markers) and "maximum mass" (green markers) turning points that have been identified along the other pressure-truncated polytropic sequences displayed in [[#Turning_Points_along_Sequences_of_Pressure-Truncated_Polytropes|Figure 3]].  <span id="SpecificN5Reference">Because of its relevance</span> to our [[#n5Analytic|discussion of stability, below]], we note that, along the n = 5 sequence, the configuration that has the maximum radius (yellow marker) is truncated precisely at <math>~\tilde\xi = \sqrt{3}</math>, while the configuration that has the maximum mass (green marker) is truncated precisely at <math>~\tilde\xi = 3</math>.