Editing SSC/FreeEnergy/PolytropesEmbedded/Pt3C (section)

=Nonstandard Examination=
In our introductory remarks, [[#Free-Energy_of_Bipolytropes|above]], we said the warped free-energy surface drapes across a five-dimensional parameter "plane" such that,
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<math>~\mathfrak{G}_{51}</math>
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<math>~=</math>
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<math>~\mathfrak{G}(R, K_c, M_\mathrm{tot}, q, \nu) \, .</math>
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From a more pragmatic point of view, we should have said that the "five-one" free-energy surface drapes across a five-dimensional parameter "plane" such that,
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<math>~\mathfrak{G}_{51}</math>
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  <td align="center">
<math>~=</math>
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<math>~\mathfrak{G}(R, K_c, M_\mathrm{tot}, \ell_i, \tfrac{\mu_e}{\mu_c}) \, .</math>
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In our initial, standard examination of the structure of this warped free-energy surface, we held  three parameters fixed &#8212; namely, <math>~K_c, M_\mathrm{tot}, \tfrac{\mu_e}{\mu_c}</math> &#8212; and plotted <math>~\mathfrak{G}_{51}(\ell_i, \Chi\equiv R/R_\mathrm{eq})</math>.  Now, let's fix <math>~\Chi = 1</math> and plot <math>~\mathfrak{G}_{51}(\ell_i, \tfrac{\mu_e}{\mu_c})</math>.  The following plot shows how a portion of the <math>~(\ell_i, \mu_e/\mu_c)</math> grid maps onto the traditional <math>~(q, \nu)</math> plane.  The numerical labels identify lines of constant <math>~\xi_i = \sqrt{3}\ell_i</math> &#8212; 7 (light green), 9 (pink), and 12 (orange) &#8212; and lines of constant <math>~\mu_e/\mu_c</math> &#8212; 0.330 (purple), 0.315 (dark green), and 0.305 (white/blue).

[[File:GridOnNuQplot.png|center|500px|xi-ell grid drawn on q-nu grid]]