Editing SSC/Structure/BiPolytropes/Analytic15 (section)

==Step 5: Interface Conditions==

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&nbsp;
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Setting <math>~n_c=1</math> and <math>~n_e=5~~~~~\Rightarrow</math>
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<math>~\frac{\rho_e}{\rho_0}</math>
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<math>~=</math>
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<math>~\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta^{n_c}_i \phi_i^{-n_e}</math>
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<math>~=</math>
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<math>~\biggl( \frac{\mu_e}{\mu_c} \biggr) \theta_i \phi_i^{-5}</math>
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<math>~\biggl( \frac{K_e}{K_c} \biggr) </math>
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<math>~=</math>
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<math>~\rho_0^{1/n_c - 1/n_e}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-(1+1/n_e)} \theta^{1 - n_c/n_e}_i</math>
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<math>~=</math>
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<math>~\biggl[\rho_0^{4}\biggl( \frac{\mu_e}{\mu_c} \biggr)^{-6} \theta^{4}_i\biggr]^{1/5}</math>
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<math>~\frac{\eta_i}{\xi_i}</math>
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<math>~=</math>
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<math>~\biggl[ \frac{n_c + 1}{n_e+1} \biggr]^{1/2} \biggl( \frac{\mu_e}{\mu_c}\biggr) \theta_i^{(n_c-1)/2} \phi_i^{(1-n_e)/2}</math>
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<math>~=</math>
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<math>~\biggl( \frac{1}{3} \biggr)^{1/2} \biggl( \frac{\mu_e}{\mu_c}\biggr) \phi_i^{-2}</math>
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<math>~\biggl( \frac{d\phi}{d\eta} \biggr)_i</math>
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<math>~=</math>
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<math>~\biggl[ \frac{n_c + 1}{n_e + 1} \biggr]^{1/2} \theta_i^{- (n_c + 1)/2} \phi_i^{(n_e+1)/2} \biggl( \frac{d\theta}{d\xi} \biggr)_i</math>
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<math>~=</math>
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<math>~\biggl( \frac{1}{3} \biggr)^{1/2} \theta_i^{- 1} \biggl( \frac{d\theta}{d\xi} \biggr)_i \phi_i^3</math>
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<span id="Alternative"><font color="red">Alternative:</font>  In our</span> [[SSC/Structure/BiPolytropes#UVplane|introductory description of how to build a bipolytropic structure]], we pointed out that, instead of employing these last two fitting conditions, Chandrasekhar [[Appendix/References#C67|[<b><font color="red">C67</font></b>]]] found it useful to employ, instead, the ratio of the <math>3^\mathrm{rd}</math> to <math>4^\mathrm{th}</math> expressions, which in the present case produces,
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<math>
\frac{\eta_i \phi_i^{5}}{(d\phi/d\eta)_i} = \frac{\xi_i \theta_i}{(d\theta/d\xi)_i} \biggl( \frac{\mu_e}{\mu_c}\biggr) \, ,
</math>
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and the product of the <math>3^\mathrm{rd}</math> and <math>4^\mathrm{th}</math> expressions, which in the present case generates,
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<math>
\frac{3\eta_i (d\phi/d\eta)_i}{ \phi_i } = \frac{\xi_i (d\theta/d\xi)_i}{ \theta_i } \biggl( \frac{\mu_e}{\mu_c}\biggr) \, .
</math>
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In what follows we will sometimes refer to the first of these two expressions as Chandrasekhar's "U-constraint" and we will sometimes refer to the second as Chandrasekhar's "V-constraint."  As is explained in [[SSC/Structure/BiPolytropes/MurphyUVplane#Chandrasekhar's_U_and_V_Functions|an accompanying discussion]], {{ Murphy83a }} followed Chandrasekhar's lead and extracted fitting conditions from this last pair of expressions.  In seeking the most compact analytic solution, we have found it advantageous to invoke our standard <math>3^\mathrm{rd}</math> fitting expression in tandem with the Chandrasekhar's V-constraint.