Editing SSC/Structure/PolytropesEmbedded/n1 (section)

====Overlap with Stahler's Presentation====
We can invert the above expression for <math>~P_e(K,M)</math> to obtain the following expression for <math>~M(K,P_e)</math>:
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<math>~M= K \biggl[\frac{2}{\pi} \cdot \frac{P_e}{G^3} \biggr]^{1/2} \biggl[ \frac{\xi_e(\sin\xi_e - \xi_e \cos\xi_e )}{\sin\xi_e} \biggr]</math> .
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If, following Stahler's lead, we normalize this expression by <math>~M_\mathrm{SWS}</math> (evaluated for <math>~n=1</math>) and we normalize the above expression for <math>~R_\mathrm{eq}</math> by <math>~R_\mathrm{SWS}</math> (evaluated for <math>~n=1</math>), we obtain,
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<math>
\frac{M}{M_\mathrm{SWS}}
</math>
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<math>~=~</math>
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<math>
K \biggl[\frac{2}{\pi} \cdot \frac{P_e}{G^3} \biggr]^{1/2} \biggl[ \frac{\xi_e(\sin\xi_e - \xi_e \cos\xi_e )}{\sin\xi_e} \biggr]
\biggl[ \biggl( \frac{G}{2} \biggr)^{3/2} K^{-1} P_\mathrm{ex}^{-1/2}  \biggr]
</math>
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&nbsp;
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<math>~=~</math>
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<math>
(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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<math>
\frac{R_\mathrm{eq}}{R_\mathrm{SWS}}
</math>
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<math>~=~</math>
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<math>
\biggl[ \frac{K}{2\pi G} \biggr]^{1/2} \xi_e \biggl[ \frac{G}{2K} \biggr]^{1/2} =
(4\pi)^{-1/2} \xi_e \, .
</math>
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<span id="Stahler1983Fig17">
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'''Figure 2:''' <font color="darkblue">Equilibrium Mass-Radius Diagram </font> 
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[[File:Stahler1983TitlePage0.png|300px|center|Stahler (1983) Title Page]]
<!-- [[Image:AAAwaiting01.png|300px|center|Stahler (1983) Title Page]] -->
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''Top:'' A slightly edited reproduction of Figure 17 in association with Appendix B of  [http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler] (1983, ApJ, 268, 165).  Stahler's figure caption reads, in part, "Mass-radius relation for bounded polytropes (schematic).  Each curve is labeled by the appropriate value or range" of {{Math/MP_PolytropicIndex}} &hellip; "As the cloud density increases from unity, all curves leave the origin with the same slope &hellip;"  


''Bottom:''  Curves depict the exact, analytically derived mass-radius relationship for truncated <math>~n = 1</math> (purple squares) and <math>~n = 5</math> (blue diamonds) polytropes that are embedded in an external medium of pressure <math>~P_e</math>; the relevant mathematical expressions are presented to the immediate right of Stahler's name in, respectively, our [[SSC/Structure/PolytropesEmbedded#n1Summary|<math>~n=1</math> summary table]] and our [[SSC/Structure/PolytropesEmbedded#n5Summary|<math>~n=5</math> summary table]].  As the dimensionless truncation radius, <math>~\xi_e</math>, increases steadily from zero, both curves exhibit very similar behavior up to <math>~M_n \equiv M/M_\mathrm{SWS} \approx 0.5</math>; thereafter the normalized mass and normalized radius continue to steadily increase along the <math>~n = 1</math> sequence, but the <math>~n = 5</math> sequence eventually bends back on itself, returning to the origin as <math>~\xi_e \rightarrow \infty</math>.  


''Comparison:'' The monotonic <math>P-R</math> behavior of the analytically derived solution for {{Math/MP_PolytropicIndex}} = 1 <math>(\gamma_g = 2)</math>, shown above, is consistent with the behavior of the numerically derived solutions presented by Whitworth for slightly lower values of <math>\gamma_g</math> = 5/3 and 4/3.  The analytically derived solution for {{Math/MP_PolytropicIndex}} = 5 <math>(\gamma_g = 6/5)</math> shows that, above some limiting pressure, no equilibrium configuration exists; this is consistent with the behavior of the numerically derived solutions presented by Whitworth for all values of <math>\gamma_g < 4/3 \, .</math>
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[[File:Stahler_MRdiagram1.png|300px|center|Stahler (1983) Figure 17 (edited)]]
<!-- [[Image:AAAwaiting01.png|300px|center|Stahler (1983) Figure 17 (edited)]] -->
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[[File:Stahler1983Comparison.png|300px|center|To be compared with Stahler (1983)]]
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