Editing SSC/Structure/PolytropesEmbedded/n1 (section)

====Overlap with Whitworth's Presentation====
The solid green curve in the two top panels of Figure 1 shows how <math>R_\mathrm{eq}</math> varies with the applied external pressure <math>P_e</math> for this pressure-bounded <math>~n=1</math> model sequence.  In the top-right panel, following the lead of [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth] (1981, MNRAS, 195, 967) &#8212; for clarification, read the [[SSC/Structure/PolytropesASIDE1|accompanying ASIDE]] &#8212; these two quantities have been respectively normalized (or, "referenced") to,
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R_\mathrm{rf}\biggr|_\mathrm{n=1} \equiv \biggl( \frac{3^2 \cdot 5}{2^4 \pi} \biggr)^{1/2} \biggl(\frac{K}{G}\biggr)^{1/2} ~~~\Rightarrow ~~~ \frac{R_\mathrm{eq}}{R_\mathrm{rf}} = \biggl( \frac{2^3}{3^2 \cdot 5} \biggr)^{1/2} \xi_e \, ;
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and,
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P_\mathrm{rf}\biggr|_\mathrm{n=1} \equiv \frac{2^6 \pi}{3^4 \cdot 5^3}  \biggl(\frac{G^3 M^2}{K^2}\biggr) ~~~\Rightarrow ~~~ \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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Note that this pair of mathematical expressions has been recorded to the immediate right of Whitworth's name in our [[SSC/Structure/PolytropesEmbedded#n1Summary|<math>~n=1</math> summary table]].
In the top-left panel of Figure 1, the solid green curve shows the identical sequence, but plotted as <math>~\log(p_a)</math> versus <math>~log(r_a)</math>, for easier comparison with Horedt's work.  The pair of mathematical expressions defining <math>~r_a(\xi_e)</math> and <math>~p_a(\xi_e)</math> has been recorded to the immediate right of Horedt's name in the same [[SSC/Structure/PolytropesEmbedded#n1Summary|summary table]].


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'''Figure 1:''' <font color="darkblue">
Equilibrium R-P Diagram &#8212; Referred to by [http://adsabs.harvard.edu/abs/1981PASJ...33..273K Kimura (1981)] as an "M<sub>1</sub> Sequence"
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All of the plots shown in this figure illustrate how the equilibrium radius of a pressure-bounded polytrope varies with the applied external pressure.  In the right-hand column, the log-log plots display a normalized <math>~P_e</math> along the horizontal axis and a normalized <math>~R_\mathrm{eq}</math> along the horizontal axis; in the left-hand column, these axes are flipped, and a different normalization is used.  One primary intent of all the diagrams is to show that, for polytropic sequences having <math>~n > 3</math> (or, equivalently, sequences having <math>\gamma_g \equiv 1 + 1/n < 4/3),</math> no equilibrium models exist above some limiting external pressure.
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[[File:HoredtPlot2.png|250px|center|To be compared with Horedt (1970)]]
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[[File:WhitworthPlot2.png|250px|center|To be compared with Whitworth (1981)]]
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[[File:Horedt_PRdiagram0.png|250px|center|Horedt (1970) Figure 1]]
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[[File:WhitworthFig1bCopy.jpg|300px|center|Whitworth (1981) Figure 1b]]
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[[File:Horedt_EmbeddedPolytrope.png|300px|center|Horedt (1970) Title Page]] 
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[[File:Whitworth1981TitlePage0.png|200px|center|Whitworth (1981) Title Page]]
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''Bottom Left'' [reproduction of Figure 1 from [http://adsabs.harvard.edu/abs/1970MNRAS.151...81H Horedt (1970)]]:  All three displayed sequences &#8212; <math>~n=4</math> (<math>~\gamma_g = 1.25</math>), <math>~n=5</math> (<math>~\gamma_g = 1.20</math>), and <math>~n=\infty</math> (<math>~\gamma_g = 1</math>, hence, isothermal) &#8212; exhibit an upper limit for the bounding pressure.  Each sequence displays two segments &#8212; a solid segment and a dashed segment &#8212; indicating that, below the maximum allowed value of <math>~P_e</math>, it is possible to construct two (or more) equilibrium configurations; models lying along the solid segment of each displayed curve are expected to be dynamically stable while models lying along the dashed segments are unstable.

''Bottom Right'' [reproduction of Figure 1b from [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth (1981)]]:  Model sequences are shown for five different effective adiabatic indexes &#8212; <math>~\gamma_g = 1/3,~ 2/3,~ 1,~ 4/3,</math> and <math>~ 5/3</math> &#8212; corresponding, respectively, to polytropic indexes <math>~n = -2/3, -1/3, \infty, ~3/2, </math> and <math>~3</math>.  The three sequences having <math>~\gamma_g < 4/3</math> exhibit an upper limit for the bounding pressure.  Both the stable (solid) curve segment and the unstable (dashed) curve segment are drawn for the isothermal <math>~(\gamma_g = 1)</math> sequence, which is also displayed (as the <math>~n=\infty</math> sequence) in Horedt's diagram.

''Top'':  Plots that we have generated for direct comparison with Horedt's diagram (''left'') and with Whitworth's diagram (''right'').  Both plots display only the two sequences that are analytically prescribable:   <math>~n=1</math> (<math>~\gamma_g = 2</math>) and <math>~n=5</math> (<math>~\gamma_g = 1.20</math>).  Along the <math>~n=1</math> (green) sequence, stable equilibrium models can be constructed for all values of <math>~P_e</math>.  Along the <math>~n=5</math> sequence, equilibrium models only exist for values of <math>~P_e</math> less than the critical value, <math>~P_\mathrm{max} = (2^5\cdot 3^9/5^9) P_\mathrm{rf} = (3^{12}/2^{24}) P_\mathrm{Horedt}</math>; below this critical pressure, the sequence has two branches denoted by blue diamonds (stable models) and red squares (unstable models).
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