Editing SSC/Structure/BonnorEbert (section)

===Whitworth's Normalization===
[http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth] (1981, MNRAS, 195, 967) examined the behavior of model sequences that result from embedding polytropes with various effective adiabatic indexes <math>~(\gamma_g = 1/3,~ 2/3,~ 1,~ 4/3,~ 5/3)</math> in an external medium of pressure <math>~P_e</math>.  Following Ebert's lead, in his Figure 1b, Whitworth showed how the size of the configuration varies with <math>~P_e</math> along each sequence.  A reproduction of Whitworth's Figure 1b has been displayed [[SSC/Structure/PolytropesEmbedded#WhitworthFig1b|in our separate discussion of pressure-bounded polytropic configurations that have analytic prescriptions]], but we have again displayed the figure here, in the right-most panel of our Figure 2, because Whitworth included pressure-bounded isothermal configurations <math>~(\gamma_g = 1)</math> among his models.  The red arrow identifies the isothermal sequence.  Whitworth's Figure 1b effectively serves as an extension of Ebert's Figure 2, shown above in the left-most panel of our Figure 2.  In order to assist comparison with Ebert's figure, this time we have rotated and flipped Whitworth's figure relative to its original orientation because Whitworth chose to plot <math>~R</math> versus <math>~P_e</math> whereas Ebert plotted <math>~P_e</math> versus <math>~R</math>.

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'''Figure 3:  Whitworth's P-R Diagrams'''
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[[File:WhitworthFig1bCopy.jpg|400px|center|Whitworth (1981) Figure 1b]]
<!-- [[Image:AAAwaiting01.png|250px|center|Ebert (1955) Figure 2]] -->
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'''Pertaining to Reprinted Figure''' <br />(both on the top-left and on the right)
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<tr><td align="right" width="25%">Author (year):</td><td align="left">[http://adsabs.harvard.edu/abs/1981MNRAS.195..967W A. Whitworth (1981)]</td></tr>
<tr><td align="right">Article Title.:</td><td align="left">''Global Gravitational Stability for One-Dimensional Polytropes''</td></tr>
<tr><td align="right">Journal Title:</td><td align="left">'''Monthly Notices of the Royal Astronomical Society'''</td></tr>
<tr><td align="right">Volume &amp; Pages:</td><td align="left">'''195''', pp. 967 - 977</td></tr>
<tr><td align="right">Figure No.:</td><td align="left">1b (p. 971)</td></tr>
<tr><td align="right">Copyright:</td><td align="left">[https://ras.ac.uk Royal Astronomical Society] &#8212;</td></tr>
<tr><td align="right">&nbsp;</td><td align="left">Permission to reuse this figure has been granted implicitly as [[Appendix/Images#MNRAS|summarized here]] and as [https://academic.oup.com/mnras/pages/rights_and_new_business_development detailed by the publisher here].</td></tr>
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----
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<table border="0" cellpadding="2" align="center" width="90%"><tr><td align="center">'''Pertaining to Reprinted Figure''' (on the right)</td></tr>
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Same as reprinted figure on the top-left, except image has been flipped, both vertically and horizontally, so that the axes correspond with the axes of the diagrams presented above in our Figure 2a; a red arrow (our modification) points to the most relevant (isothermal) equilibrium sequence.
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<p></p>
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<table border="0" cellpadding="2" align="center" width="90%"><tr><td align="center">'''Pertaining to Our Original Figure''' (on the bottom-left)</td></tr>
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Caption:  In an effort to demonstrate more explicitly how Whitworth's 1981 work ties back to the original work on isolated isothermal spheres that was published by Emden in 1907, the blue curve displays a plot of <math>~P_e/P_\mathrm{rf}</math> versus <math>~V/V_\mathrm{rf}</math> that we have constructed using data drawn directly from [[SSC/Structure/IsothermalSphere#Emden.27s_Numerical_Solution|Table 14 of Emden (1907)]].
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[[File:WhitworthFig1bFlippedClipped.png|300px|center|Whitworth (1981) Figure 1b]]
<!-- [[Image:AAAwaiting01.png|250px|center|Whitworth (1981) Figure 1b]] -->
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[[File:WhitworthFig1bUsingEmdenData.png|300px|center|P-R Diagram Using Emden's (1907) data]]
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It should be noted that the scaling adopted along both axes in Whitworth's Figure 1b is different from the scaling used by Ebert.  As is detailed in the [[SSC/Structure/PolytropesASIDE1|accompanying ASIDE]], Whitworth "referenced" <math>P_e</math> and <math>R</math> to, respectively,
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<math>~P_\mathrm{rf} = \biggl( \frac{3^4\cdot 5^3}{2^8}  \biggr) P_0 \, ,</math>
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<math>
~R_\mathrm{rf} = \biggl( \frac{2^2}{3\cdot 5} \biggr)  R_0 \, .
</math>
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In generating the right-hand panel of our Figure 1, we have adopted Whitworth's scaling.  That is to say, we have plotted <math>~P_e/P_\mathrm{rf}</math> versus <math>~V/V_\mathrm{rf}</math>, where,
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<math>~\frac{P_e}{P_\mathrm{rf}}</math>
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<math>~=</math>
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<math>~\frac{P_e}{P_0} \cdot \frac{P_0}{P_\mathrm{rf}} = \biggl( \frac{2^8}{3^4\cdot 5^3}  \biggr)\xi_e^4 \biggl(\frac{d\psi}{d\xi}\biggr)^2_e e^{-\psi_e} \, ,</math>
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and,
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<math>~\frac{V}{V_\mathrm{rf}}</math>
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<math>~=</math>
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<math>~\frac{V}{V_0} \biggl( \frac{R_0}{R_\mathrm{rf}}\biggr)^{3} = \biggl( \frac{3\cdot 5}{2^2} \biggr)^3 \biggl[ \xi \biggl(\frac{d\psi}{d\xi}\biggr) \biggr]_e^{-3} \, .</math>
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