Editing SSC/Structure/PolytropesEmbedded/Other (section)

====Kimura's Derivation====

Appreciating that Kimura uses the subscript "1," rather than a tilde, to identify equilibrium parameter values, the requisite expression is equation (22) from [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura's "Paper II,"] as displayed in the following boxed image:   
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Excerpts (edited) from [http://adsabs.harvard.edu/abs/1981PASJ...33..299K Kimura (1981b)] 
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[[File:KimuraEq00.png|500px|center|Kimura (1981b) Expressions]]
<!-- [[Image:AAAwaiting01.png|500px|center|Kimura (1981b) Expressions]] -->
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where,
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[[File:KimuraEq01.png|500px|center|Kimura (1981b) Expressions]] 
<!-- [[Image:AAAwaiting01.png|500px|center|Kimura (1981b) Expressions]] -->
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Drawing on the additional parameter and variable definitions provided in our [[SSC/Structure/PolytropesEmbedded#Kimura.27s_Presentation|discussion of Kimura's presentation, above]], we can rewrite this key expression as,
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<math>~\frac{R_\mathrm{eq}}{P_e} \cdot \frac{dP_e}{dR_\mathrm{eq}}\biggr|_M ~~\rightarrow ~~ \frac{d\ln{p_1}}{d \ln{r_1}}</math>
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<math>~=</math>
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<math>~
\frac{v_G \cdot h_G}{k_G} \, ,</math>
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where,
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<math>~v_G</math>
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<math>~=</math>
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<math>~\frac{2}{[1-2(n+1)^{-1}]} =\frac{ 2(n+1)}{n-1} \, ,
</math>
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<math>~u_G</math>
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<math>~=</math>
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<math>~(3-1)-\biggl[\frac{1}{1-2(n+1)^{-1}} \biggr] = 2-\frac{(n+1)}{(n-1)} = \frac{(n-3)}{(n-1)} \, ,
</math>
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<math>~h_G</math>
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<math>~=</math>
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<math>~
\frac{1}{u_G} \biggl[ \frac{\zeta \theta^n}{\phi^'} \biggr]_1 - \frac{1}{v_G} \biggl[ \frac{\zeta \phi^'}{\theta} \biggr]_1
=
\frac{(n-1)}{(n-3)} \biggl[ \frac{\tilde\xi \tilde\theta^n}{-\tilde\theta^'} \biggr] - 
\frac{(n-1)}{2(n+1)} \biggl[ \frac{(n+1)\tilde\xi (-\tilde\theta^')}{\tilde\theta} \biggr]
</math>
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&nbsp;
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<math>~=</math>
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<math>~ \frac{(n-1)\tilde\xi}{2(n+1)(n-3)\tilde\theta (-\tilde\theta^')}
\biggl\{ 2(n+1) \tilde\theta^{n+1} + (3-n) (n+1) (-\tilde\theta^')^2 \biggr\} \, ,
</math>
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<math>~k_G</math>
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<math>~=</math>
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<math>~1-
\frac{1}{u_G} \biggl[ \frac{\zeta \theta^n}{\phi^'} \biggr]_1 
=1-
\frac{(n-1)}{(n-3)} \biggl[ \frac{\tilde\xi \tilde\theta^n}{-\tilde\theta^'} \biggr] 
=
\frac{1}{ (n-3) (-\tilde\theta^') } \biggl\{ (n-3)(- \tilde\theta^') - (n-1) \tilde\xi \tilde\theta^n \biggr\}
</math>
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&nbsp;
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<math>~=</math>
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<math>~ 
\frac{1}{ (n+1)(n-3) (-\tilde\theta^')^2 } \biggl\{ (n-3)(n+1) (-\tilde\theta^')^2 
- (n-1)\tilde\xi [(n+1) \tilde\theta^n (-\tilde\theta^')] \biggr\} \, .
</math>
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&nbsp;
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<math>~=</math>
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<math>~ 
\frac{-1}{ (n+1)(n-3) (-\tilde\theta^')^2 } \biggl\{ (3-n)(n+1) (-\tilde\theta^')^2 
+ (1-n)\tilde\xi [(n+1) \tilde\theta^n (\tilde\theta^')] \biggr\} \, .
</math>
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Hence, from Kimura's work we find,
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<math>~\frac{R_\mathrm{eq}}{P_e} \cdot \frac{dP_e}{dR_\mathrm{eq}}\biggr|_M </math>
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<math>~=</math>
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<math>~
\frac{(n+1)\tilde\xi \tilde\theta^'}{\tilde\theta}
\biggl\{ \frac{2(n+1) \tilde\theta^{n+1} + (3-n) (n+1) (\tilde\theta^')^2}{(3-n)(n+1) (\tilde\theta^')^2 
+ (1-n)\tilde\xi [(n+1) \tilde\theta^n \tilde\theta^'] } \biggr\} \, .
</math>
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Appreciating that <math>~f^' = [(n+1)\tilde\theta^n \tilde\theta^']</math>, we see that the expression inside the curly braces here matches exactly the expression inside the curly braces that was obtained through Horedt's derivation, as it should!  The prefactor is different in the two expressions only because Kimura's result is for a logarithmic derivative whereas Horedt's derivation is not; the ratio of the two prefactors is, simply, the ratio,
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<math>~\frac{R_\mathrm{eq}/R_\mathrm{Horedt}}{P_e/P_\mathrm{Horedt}} </math>
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<math>~=</math>
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<math>~\frac{\tilde\xi}{\tilde\theta_n^{n+1}}\cdot ( -\tilde\xi^2 \tilde\theta' )^{[(1-n)-2(n+1)]/(n-3)} 
</math>
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&nbsp;
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<math>~=</math>
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<math>~\frac{\tilde\xi}{\tilde\theta_n^{n+1}}\cdot ( -\tilde\xi^2 \tilde\theta' )^{-(3n+1)/(n-3)} \, .
</math>
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In a [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#KimuraApplication|separate discussion]], specifically focused on the <math>~n=5</math> mass-radius relationship, we show how Kimura's analysis of turning points can be usefully applied.