Editing SSC/Virial/PolytropesEmbedded/SecondEffortAgain/Pt2 (section)

=====Parametric Relations=====
The other way is to determine the normalized mass and normalized radius individually through Stahler's pair of parametric relations.  Drawing partly from our [[SSC/Virial/PolytropesSummary#Detailed_Force-Balanced_Solution_2|above discussion]] and partly from a separate discussion where we provide a [[SSC/Structure/PolytropesEmbedded#Tabular_Summary_.28n.3D5.29|tabular summary of the properties of pressure-truncated <math>n=5</math> polytropes]], these are,
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<math>
~\mathcal{X}\biggr|_{n=5}
</math>
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<math>~=~</math>
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<math>
\biggl( \frac{5}{4\pi} \biggr)^{1/2} \tilde\xi \tilde\theta^{2} =
\biggl\{ \frac{3\cdot 5}{2^2 \pi} \biggl[ \frac{\tilde\xi^2/3}{(1+\tilde\xi^2/3)^{2}} \biggr] \biggr\}^{1/2} \, ,
</math>
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<math>
~\mathcal{Y}\biggr|_{n=5}
</math>
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<math>~=~</math>
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<math>
\biggl( \frac{5^3}{4\pi} \biggr)^{1/2} \tilde\theta (- \tilde\xi^2 \tilde\theta^') =
\biggl[  \biggl( \frac{3 \cdot 5^3}{2^2\pi} \biggr) \frac{(\tilde\xi^2/3)^3}{(1+\tilde\xi^2/3)^{4}} \biggr]^{1/2} \, .
</math>
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The entire sequence will be traversed by varying the Lane-Emden parameter, <math>~\tilde\xi</math>, from zero to infinity.  Using the first of these two expressions, we have determined, for example, that the point along the sequence corresponding to the maximum normalized equilibrium radius, <math>~\mathcal{X}_\mathrm{max}</math>, is associated with an embedded <math>~n=5</math> polytrope whose truncated, dimensionless Lane-Emden radius is,
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<math>
~\tilde\xi \biggr|_{\mathcal{X}_\mathrm{max}} = 3^{1/2} \, .
</math>
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Similarly, we have determined that the point along the sequence that corresponds to the maximum dimensionless mass, <math>~\mathcal{Y}_\mathrm{max}</math>, is associated with an embedded <math>~n=5</math> polytrope whose truncated, dimensionless Lane-Emden radius is, precisely,
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<math>
~\tilde\xi \biggr|_{\mathcal{Y}_\mathrm{max}} = 3 \, .
</math>
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Referring back to [[SSC/Structure/PolytropesEmbedded#Turning_Points|our review of turning points]] along equilibrium sequences and, especially, the work of {{ Kimura81bfull }}, we appreciate that the point that corresponds to the maximum mass, <math>~\mathcal{Y}_\mathrm{max}</math>, is the turning point that Kimura refers to as the "extremum in M<sub>1</sub>" along a p<sub>1</sub> sequence.  [[SSC/Structure/PolytropesEmbedded#Location_of_Pressure_Limit|As we have highlighted]], according to Kimura, this point should occur along the sequence where <math>~h_G=0</math>, that is, where the following condition applies:
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<math>~\frac{\tilde\theta^{n+1}}{(\tilde\theta^')^2} = \frac{(n-3)}{2}  \, .</math>
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For the specific case being studied here, namely, <math>~n = 5</math> polytropic configurations, we therefore expect from Kimura's work that <math>~[\tilde\theta^6/(\tilde\theta^')^2] = 1</math> at the "maximum mass" turning point.  Given that,
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<math>~\tilde\xi \biggr|_{\mathcal{Y}_\mathrm{max}} = 3</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>~\Rightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>\tilde\theta_{n=5} = \frac{1}{2}</math>&nbsp;&nbsp;&nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;&nbsp; <math>\tilde\theta_{n=5}^' = -\frac{1}{8} \, ,</math>
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we see that Kimura's condition holds and, hence, that our identification of the location along the sequence of the maximum mass matches Kimura's identification of the location of that turning point.

We appreciate, as well, that the point corresponding to the maximum normalized equilibrium radius, <math>\mathcal{X}_\mathrm{max}</math>, is the turning point that Kimura would reference as the "extremum in r<sub>1</sub>" along a p<sub>1</sub> sequence.  Following Kimura's analysis [[SSC/Structure/PolytropesEmbedded#TurningPointXmax|we have shown that this point occurs along the sequence where the following condition applies]]:
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<math>~\frac{\xi (-\theta^')}{\tilde\theta} = \frac{2}{(n-1)} \, ,</math>
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that is, for the specific case being studied here, we should expect <math>~[\tilde\xi (-\tilde\theta^')/\tilde\theta] = 1/2</math> at the "maximum radius" turning point.  Given that,
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<math>~\tilde\xi \biggr|_{\mathcal{X}_\mathrm{max}} = 3^{1/2}</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>~\Rightarrow</math>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
<math>\tilde\theta_{n=5} = 2^{-1/2}</math>&nbsp;&nbsp;&nbsp;&nbsp; and &nbsp;&nbsp;&nbsp;&nbsp; 
<math>\tilde\theta_{n=5}^' = -(2^3 \cdot 3)^{-1/2} \, ,</math>
</div>
we see that Kimura's condition holds and, hence, that our identification of the location along the sequence of the maximum radius matches Kimura's identification of the location of that turning point.
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