Editing SSC/Structure/BiPolytropes/MurphyUVplane (section)

==Chandrasekhar's U and V Functions==
As presented by {{ Murphy83a }}, most of the development and analysis of this model was conducted within the framework of what is commonly referred to in the astrophysics community as the "U-V" plane.  Specifically in the context of the model's <math>~n_c=1</math> core, this pair of referenced functions is:
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<math>U_{1E} \equiv \xi \theta \biggl(- \frac{d\theta}{d\xi}\biggr)^{-1}</math>
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<math>=</math>
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<math>\sin\xi\biggl[\frac{1}{\xi^{2}} (\sin\xi - \xi\cos\xi)\biggr]^{-1}</math>
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&nbsp;
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<math>=</math>
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<math>\frac{\xi^2}{(1 - \xi\cot\xi)} \, ;</math>
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<math>(n_c+1) V_{1E} \equiv  (n_c+1)\frac{\xi}{ \theta} \biggl(- \frac{d\theta}{d\xi}\biggr)</math>
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<math>=</math>
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<math>\frac{2\xi^2}{\sin\xi} \biggl[ \frac{1}{\xi^{2}} (\sin\xi - \xi\cos\xi) \biggr]</math>
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&nbsp;
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<math>=</math>
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<math>2(1 - \xi\cot\xi) \, .</math>
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Correspondingly, in the context of the model's <math>n_e=5</math> envelope, the pair of referenced functions is:
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<math>U_{5F} \equiv \xi \phi^5 \biggl(- \frac{d\phi}{d\xi}\biggr)^{-1}</math>
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<math>=</math>
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<math>
\frac{B^{-4}\xi \sin^5\Delta}{\xi^{5/2}\{3-2\sin^2\Delta\}^{5/2}}
\biggl[ \frac{2\xi^{3/2}(3-2\sin^2\Delta)^{3/2}}{3\sin\Delta - 2\sin^3\Delta -3\cos\Delta }  \biggr]
</math>
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&nbsp;
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<math>=</math>
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<math>
\frac{2B^{-4}\sin^5\Delta}{[3-2\sin^2\Delta][3\sin\Delta - 2\sin^3\Delta -3\cos\Delta ]}
</math>
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&nbsp;
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<math>=</math>
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<math>
\frac{-2B^{-4}\sin^5\Delta}{[2+\cos(2\Delta)][3\cos\Delta - \frac{3}{2}\sin\Delta - \frac{1}{2}\sin(3\Delta) ]} \, ;
</math>
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<math>(n_e+1) V_{5F} \equiv (n_e+1) \frac{\xi}{ \phi} \biggl(- \frac{d\phi}{d\xi}\biggr)</math>
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<math>=</math>
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<math>\frac{6\xi^{3/2}\{3-2\sin^2\Delta\}^{1/2}} {\sin\Delta}
\frac{[3\sin\Delta - 2\sin^3\Delta -3\cos\Delta ] }{2\xi^{3/2}(3-2\sin^2\Delta)^{3/2}} 
</math>
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&nbsp;
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<math>=</math>
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<math>\frac{3[3\sin\Delta - 2\sin^3\Delta -3\cos\Delta ] }{\sin\Delta (3-2\sin^2\Delta)}
</math>
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&nbsp;
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<math>=</math>
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<math>\frac{-6[3\cos\Delta - \frac{3}{2}\sin\Delta - \frac{1}{2}\sin(3\Delta)  ] }{2\sin\Delta [2+\cos(2\Delta)]} \, .
</math>
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In an effort to demonstrate correspondence with the published work of {{ Murphy83a }}, we have reproduced his expressions for these governing U-V functions in the following boxed-in image.  

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Expressions for U-V functions extracted from p. 176 of &hellip;<br />
{{ Murphy83afigure }} 
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<!-- [[File:Murphy1983UVfunctionsBoth.png|center|700px|U and V Functions from Murphy (1983)]]-->
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<math>U_{1E} </math>
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<math>=</math>
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<math>-~\xi^2 \tan\xi/(\xi - \tan\xi) \, ,</math>
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<math>(n+1)V_{1E} </math>
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<math>=</math>
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<math>-~2(\xi - \tan\xi)/\tan\xi \, ,</math>
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<math>U_{5F}</math>
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<math>=</math>
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<math>-~2\sin^5(\ln\sqrt \xi)/\{ [ 2 + \cos(\ln\xi) ] [3 \cos(\ln \sqrt \xi) </math>
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&nbsp;
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&nbsp;
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<math>-~\tfrac{3}{2} \sin(\ln \sqrt \xi) - \tfrac{1}{2}\sin(3\ln\sqrt \xi) ] \} \, , </math>
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<math>(n+1) V_{5F}</math>
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<math>=</math>
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<math>-~6[3\cos(\ln\sqrt \xi) - \tfrac{3}{2}\sin(\ln\sqrt \xi) - \tfrac{1}{2}\sin(3 \ln \sqrt \xi) ] / </math>
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&nbsp;
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&nbsp;
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<math>\{ [2 \sin(\ln \sqrt\xi)] [ 2 + \cos(\ln \xi)] \} \, . </math>
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The match between our expressions and those presented by Murphy becomes clear upon recognizing that, in our notation, 
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<math>\Delta </math>
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<math>=</math>
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<math>\ln\sqrt{A\xi} </math>
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<math>\Rightarrow ~~~~ 2\Delta </math>
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<math>=</math>
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<math>2 \ln\sqrt{A\xi} = \ln(A\xi) </math>
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and, &nbsp; &nbsp; &nbsp; 
<math>3\Delta </math>
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<math>=</math>
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<math>3\ln[\sqrt{A\xi}] \, ;</math>
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and, in laying out these function definitions, Murphy has implicitly assumed that the two scaling coefficients, <math>A</math> and <math>B</math>, are unity.  

<font color="red">'''CAUTION:'''</font>  Presented in this fashion &#8212; that is, by using <math>\xi</math> to represent the dimensionless radial coordinate in all four expressions &#8212; Murphy's expressions seem to imply that the independent variable defining the radial coordinate in the bipolytrope's core is the same as the one that defines the radial coordinate in the structure's envelope.  In general, this will not be the case, so we have explicitly used a different independent variable, <math>\eta</math>, to mark the envelope's radial coordinate in our expressions.  It is clear from other elements of his published derivation that Murphy understood this distinction but, as is explained more fully below, errors in his final model specifications may have resulted from not explicitly differentiating between this variable notation.