Editing SSC/Structure/BiPolytropes/Analytic1.53/Pt3 (section)

===Step 4:  Throughout the core (0 &le; &xi; &le; &xi;<sub>i</sub>)===
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Specify:  <math>~K_c</math> and <math>~\rho_0 ~\Rightarrow</math>
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&nbsp;
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<math>~\rho</math>
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<math>~=</math>
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<math>~\rho_0 \theta^{n_c}</math>
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<math>~=</math>
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<math>~\rho_0 \theta^{3/2}</math>
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<math>~P</math>
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<math>~=</math>
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<math>~K_c \rho_0^{1+1/n_c} \theta^{n_c + 1}</math>
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<math>~=</math>
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<math>~K_c \rho_0^{5/3} \theta^{5/2}</math>
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<math>~r</math>
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<math>~=</math>
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<math>~\biggl[ \frac{(n_c + 1)K_c}{4\pi G} \biggr]^{1/2} \rho_0^{(1-n_c)/(2n_c)} \xi</math>
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<math>~=</math>
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<math>~\biggl[ \frac{5K_c}{8\pi G} \biggr]^{1/2} \rho_0^{-1/6} \xi</math>
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<math>~M_r</math>
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<math>~=</math>
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<math>~4\pi \biggl[ \frac{(n_c + 1)K_c}{4\pi G} \biggr]^{3/2} \rho_0^{(3-n_c)/(2n_c)} \biggl(-\xi^2 \frac{d\theta}{d\xi} \biggr)</math>
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<math>~=</math>
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<math>~\frac{1}{2^2(2 \pi )^{1/2}}  \biggl[ \frac{5K_c}{G} \biggr]^{3/2} 
\rho_0^{1/2} \biggl(-\xi^2 \frac{d\theta}{d\xi} \biggr)</math>
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By comparison, the expressions that {{ Milne30 }} derived for the run of <math>~\rho</math>, <math>~r</math>, and <math>~M_r</math> throughout the core are presented in his paper as, respectively, equations (90), (88), and (87).   In an effort to facilitate this comparison, Milne's expressions &#8212; which also specifically apply to the outer edge of the core, whose identity is associated with primed variable names in Milne's notation &#8212; are reprinted as extracted equations in the following boxed-in table. 

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Equations extracted<sup>&dagger;</sup> from <br />
{{ Milne30figure }}
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<!-- [[File:CoreRelations01.png|500px|center|Milne (1930)]] -->
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    <td align="right"><math>\rho</math></td>
    <td align="center" width="3%"><math>=</math></td>
    <td align="left"><math>\lambda_2 \psi^{3 / 2}</math></td>
    <td align="right" width="5%">(90)</td>
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    <td align="right"><math>r^'</math></td>
    <td align="center" width="3%"><math>=</math></td>
    <td align="left"><math>\eta^' \biggl( \frac{5K}{8\pi G\beta} \biggr)^{1 / 2} \lambda_2^{-1 / 6}</math></td>
    <td align="right" width="5%">(88)</td>
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    <td align="right"><math>M(r^')</math></td>
    <td align="center" width="3%"><math>=</math></td>
    <td align="left"><math>
-~\frac{1}{4(2\pi)^{1 / 2}} \biggl( \frac{5K}{G\beta} \biggr)^{3 / 2}\lambda_2^{1 / 2} (\eta^')^2 \biggl(\frac{d\psi}{d\eta}\biggr)_{\eta = \eta^'}
</math></td>
    <td align="right" width="5%">(87)</td>
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<tr><td align="left"><sup>&dagger;</sup>Equations displayed here, with presentation order &amp; layout modified from the original publication.</td></tr>
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It is clear that the agreement between our derivation and Milne's is exact, once it is realized that Milne has used <math>~\psi(\eta)</math> to represent the Lane_Emden function for the <math>~n_c = \tfrac{3}{2}</math> core, whereas we have represented this function by <math>~\theta(\xi)</math>; and Milne has identified the configuration's central density as <math>~\lambda_2</math>, whereas we have used the notation, <math>~\rho_0</math>.