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

=====Discussion=====
Notice that if the quadratic equation is used to map out the mass-radius relationship, the parameter, <math>~\tilde\xi</math>, never explicitly enters the discussion.  Instead, a radius <math>~0 \le \mathcal{X} \le \mathcal{X}_\mathrm{max}</math> is specified and the ''two'' equilibrium masses associated with <math>~\mathcal{X}</math> &#8212; call them, <math>~\mathcal{Y}_+</math> and <math>~\mathcal{Y}_-</math> &#8212; are determined.  (The values of the two masses are degenerate at both limiting values of <math>~\mathcal{X}</math>.)  If the pair of parametric relations is used, instead, only ''one'' value of the mass is obtained for each specified value of <math>~\tilde\xi</math>.  As <math>~\tilde\xi</math> is increased from <math>~0</math> to <math>~\sqrt{3}</math>, <math>~\mathcal{X}</math> increases monotonically from <math>~0</math> to <math>~\mathcal{X}_\mathrm{max}</math> and the corresponding mass is (only) <math>~\mathcal{Y}_-</math>; that is, as <math>~\tilde\xi</math> is increased from <math>~0</math> to <math>~\sqrt{3}</math>, we move away from the origin in a counter-clockwise direction along the lower segment (colored orange in the [[#MRplot|above figure]]) of the plotted equlibrium sequence.  Then, as <math>~\tilde\xi</math> is increased from <math>~\sqrt{3}</math> to <math>~\infty</math>, we continue to move in a counter-clockwise direction along the equilibrium sequence, but now along the upper segment (colored green in the [[#MRplot|above figure]]) of the sequence, back to the origin; that is to say, <math>~\mathcal{X}</math> steadily decreases from <math>~\mathcal{X}_\mathrm{max}</math> back to <math>~0</math> and this time the relevant associated mass is the positive root of the quadratic relation, <math>~\mathcal{Y}_+</math>. 

Clearly, then, each value of <math>~\mathcal{X}</math> is associated with two different values of the parametric parameter, <math>~\tilde\xi</math>.  By inverting the <math>~\mathcal{X}(\tilde\xi)</math> parametric expression we see that, the two values of <math>~\tilde\xi</math> associated with a given equilibrium radius are,
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<math>~\tilde\xi_\pm</math>
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<math>~=</math>
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<math>~\biggl\{\frac{3}{\alpha} \biggl[ 1 \pm \sqrt{1 - \alpha^2} \biggr]  \biggr\}^{1/2} \, ,</math>
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</tr>
</table>
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where,
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<math>~\alpha</math>
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<math>~\equiv</math>
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<math>~\frac{(\mathcal{X}/\mathcal{X}_\mathrm{xmax})^2}{2-(\mathcal{X}/\mathcal{X}_\mathrm{xmax})^2} \, .</math>
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</table>
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We note as well that, for a given equilibrium radius, <math>~\mathcal{X}</math>, the ''ratio'' of the two mass solutions is given by a very simple expression, namely,
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  <td align="right">
<math>~\frac{\mathcal{Y}_-}{\mathcal{Y}_+} = \frac{\tilde\xi_-^2}{3}</math>
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; or &nbsp; &nbsp; &nbsp; &nbsp; 
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  <td align="left">
<math>~\frac{\mathcal{Y}_+}{\mathcal{Y}_-} = \frac{\tilde\xi_+^2}{3} \, .</math>
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</tr>
</table>
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This implies, as well, that,
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<math>~\tilde\xi_+ \cdot \tilde\xi_-</math>
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<math>~=</math>
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<math>~3\, .</math>
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