Editing SSC/Structure/BiPolytropes/Analytic15 (section)

===First Constraint===
Calling upon Chandrasekhar's V-constraint, as just defined above &#8212; see also [[SSC/Structure/BiPolytropes/MurphyUVplane#Chandrasekhar.27s_U_and_V_Functions|our accompanying discussion]] for elaboration on [http://adsabs.harvard.edu/abs/1983PASAu...5..175M Murphy's (1983)] "V<sub>5F</sub>" and "V<sub>1E</sub>" function notations &#8212; one fitting condition at the interface is,
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<math>~- \frac{2\xi_i }{ 3\theta_i } \biggl( \frac{d\theta}{d\xi} \biggr)_i \biggl( \frac{\mu_e}{\mu_c}\biggr)</math>
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<math>~=</math>
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<math>~2\biggl[ \frac{\eta (- d\phi/d\eta)}{\phi} \biggr]_i </math>
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&nbsp;
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<math>~=</math>
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<math>~\frac{[3\sin\Delta_i - 2\sin^3\Delta_i -3\cos\Delta_i ] }{\sin\Delta_i (3-2\sin^2\Delta_i)}</math>
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&nbsp;
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<math>~=</math>
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<math>~\frac{3 - 2\sin^2\Delta_i -3\cot\Delta_i}{(3-2\sin^2\Delta_i)} \, .</math>
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The left-hand side of this expression is inherently positive over the physically viable radial coordinate range, <math>~0 \ge \xi_i \ge \pi</math> and its value is known once the radial coordinate of the edge of the core has been specified.  So, defining the interface parameter,
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<math>~
\kappa_i \equiv - \frac{2\theta_i^' \xi_i}{3\theta_i} \biggl( \frac{\mu_e}{\mu_c} \biggr)
\, ,</math>
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we will recast the first constraint into, what will henceforth be referred to as, the
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<font color="#770000">'''Key Nonlinear Interface Relation'''</font>
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<math>\kappa_i</math>
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<math>=</math>
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<math>\frac{3 - 2\sin^2\Delta_i -3\cot\Delta_i}{(3-2\sin^2\Delta_i)} \, .</math>
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In a [[SSC/Structure/BiPolytropes/Analytic15/Pt2#Analytic_Solution_of_Key_Interface_Relation|separate subsection (Part II) of this chapter]], we present a closed-form analytic solution, <math>\Delta_i(\kappa_i)</math>, to this nonlinear equation.


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===Second Constraint (Earliest Try)===

====Obtained from Third Interface Condition====
Now, our [[SSC/Structure/BiPolytropes/Analytic15#Step_5:_Interface_Conditions|3<sup>rd</sup> interface condition, as detailed above]], states that,
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<math>~\frac{\eta_i}{\xi_i}</math>
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<math>~=</math>
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<math>~3^{-1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr) \phi_i^{-2}</math>
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&nbsp;
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<math>~=</math>
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<math>~3^{-1/2} \biggl( \frac{\mu_e}{\mu_c} \biggr)  \biggl[ \frac{\eta^{1/2} (3-2\sin^2\Delta_i)^{1/2}}{B^{-1}\sin\Delta_i} \biggr]^{2} </math>
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<math>~\Rightarrow ~~~~ B^2</math>
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<math>~=</math>
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<math>~\frac{\sqrt{3}}{\xi_i} \biggl( \frac{\mu_e}{\mu_c} \biggr)^{-1}  \biggl( \frac{3}{\sin^2\Delta_i} - 2\biggr)^{-1} \, .  </math>
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