Editing SSC/Structure/PolytropesEmbedded/n1 (section)

===Tabular Summary (n=1) ===
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Table 1: &nbsp;Properties of <math>~n=1</math> Polytropes Embedded in an External Medium of Pressure <math>~P_e</math> 
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(and, accordingly, truncated at radius <math>~\xi_e</math>)
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<math>
~\theta_1 = \frac{\sin\xi_e}{\xi_e} 
</math>
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&nbsp; &nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp; &nbsp;
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<math>
~\frac{d\theta_1}{d\xi} \biggr|_{\xi_e} = \frac{\cos\xi_e}{\xi_e} - \frac{\sin\xi_e}{\xi_e^2}
</math>
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[http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)]
<br>for<br>
fixed <math>~(M,K_n)</math>
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<math>
~r_a = \frac{R_\mathrm{eq}}{R_\mathrm{Horedt}} = \xi_e
</math>
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<math>
~p_a = \frac{P_e}{P_\mathrm{Horedt}} = \biggl[ \frac{\sin\xi_e}{\xi_e(\sin\xi_e - \xi_e \cos\xi_e )} \biggr]^2
</math>
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[http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)]
<br>for<br>
fixed <math>~(M,K_n)</math>
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\frac{R_\mathrm{eq}}{R_\mathrm{rf}} = \biggl( \frac{2^3}{3^2 \cdot 5} \biggr)^{1/2} \xi_e
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<math>
\frac{P_e}{P_\mathrm{rf}} = \biggl( \frac{3^4 \cdot 5^3}{2^7} \biggr) \biggl[ \frac{\sin\xi_e}{\xi_e(\sin\xi_e - \xi_e \cos\xi_e )} \biggr]^2
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[http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler (1983)]
<br>for<br>
fixed <math>~(P_e,K_n)</math>
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\frac{R_\mathrm{eq}}{R_\mathrm{SWS}} = (4\pi)^{-1/2} \xi_e
</math>
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<math>
\frac{M}{M_\mathrm{SWS}} = (4\pi)^{-1/2} \biggl[ \frac{\xi_e(\sin\xi_e - \xi_e \cos\xi_e )}{\sin\xi_e} \biggr]
</math>
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NOTE:  None of the analytic expressions for the dimensionless radius, pressure, or mass presented in this table explicitly appear in the referenced articles by Horedt, by Whitworth, or by Stahler but, as is discussed fully above, they are straightforwardly derivable from the more general relations that appear in these papers.  
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