Editing SSC/Structure/PolytropesASIDE1

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=ASIDE: Whitworth's Scaling=

This provides details in support of our associated discussion of [[SSC/Structure/PolytropesEmbedded|embedded polytropic spheres]].

In his study of the "global gravitational stability [of] one-dimensional polytropes,"  [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth] (1981, MNRAS, 195, 967) normalizes (or "references") various derived mathematical expressions for configuration radii, <math>R</math>, and for the pressure exerted by an external bounding medium, <math>P_\mathrm{ex}</math>, to quantities he refers to as, respectively, <math>R_\mathrm{rf}</math> and <math>P_\mathrm{rf}</math>.  The paragraph from his paper in which these two reference quantities are defined is shown here: 
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<table border="2">
<tr><td>
[[File:WhitworthScalingText.png|600px|center|Whitworth (1981, MNRAS, 195, 967)]]
</td></tr>
</table>
</div>
In order to map Whitworth's terminology to ours, we note, first, that he uses <math>M_0</math> to represent the spherical configuration's total mass, which we refer to simply as <math>M</math>; and his parameter <math>\eta</math> is related to our {{Math/MP_PolytropicIndex}} via the relation,
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<math>\eta = 1 + \frac{1}{n} \, .</math>
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Hence, Whitworth writes the polytropic equation of state as,
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<math>P = K_\eta \rho^\eta \, ,</math>
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whereas, using our standard notation, this same key relation is written as,
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{{Math/EQ_Polytrope01}} ;
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and his parameter <math>K_\eta</math> is identical to our {{Math/MP_PolytropicConstant}}.  

According to the second (bottom) expression identified by the red outlined box drawn above,
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<math>
P_\mathrm{rf} = \frac{3^4 5^3}{2^{10} \pi} \biggl( \frac{K_1^4}{G^3 M^2} \biggr) \, ,
</math>
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and inverting the expression inside the green outlined box gives,
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<math>
K_1 = \biggl[ K_n (4 P_\mathrm{rf})^{\eta - 1} \biggr]^{1/\eta} \, .
</math>
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Hence,
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<math>
P_\mathrm{rf} = \frac{3^4 5^3}{2^{10} \pi} \biggl( \frac{1}{G^3 M^2} \biggr)\biggl[ K_n (4 P_\mathrm{rf})^{\eta - 1} \biggr]^{4/\eta} \, ,
</math>
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or, gathering all factors of <math>P_\mathrm{rf}</math> to the left-hand side,
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<math>
P_\mathrm{rf}^{(4-3\eta)} = 2^{-2(4+\eta)} \biggl( \frac{3^4 5^3}{\pi} \biggr)^\eta \biggl[ \frac{K_n^4}{G^{3\eta} M^{2\eta}} \biggr] \, .
</math>
</div>
Analogously, according to the first (top) expression identified inside the red outlined box,
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<math>
R_\mathrm{rf} = \frac{2^2 GM}{3\cdot 5 K_1} = 2^{2/\eta} \biggl( \frac{GM}{3\cdot 5}\biggr) K_n^{-1/\eta}  P_\mathrm{rf}^{(1-\eta)/\eta} 
~~~~\Rightarrow~~~~ R_\mathrm{rf}^\eta = \frac{2^{2}}{K_n} \biggl( \frac{GM}{3\cdot 5}\biggr)^\eta P_\mathrm{rf}^{(1-\eta)} \, .
</math>
</div>

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<table border="1" width="90%">
  <tr>
  <td colspan="4" align="center">'''Examples'''</td>
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<tr>
  <td align="center">
{{Math/MP_PolytropicIndex}}
  </td>
  <td align="center">
<math>\eta = 1+1/n</math>
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  <td align="center">
<math>P_\mathrm{rf}</math>
  </td>
  <td align="center">
<math>R_\mathrm{rf}</math>
  </td>
</tr>

<tr>
  <td align="center">
1
  </td>
  <td align="center">
2
  </td>
  <td align="center">
<math>\frac{2^{6}\pi}{3^4 5^3}  \biggl[ \frac{G^{3} M^{2} }{K^2}\biggr]</math>
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  <td align="center">
<math>\biggl[  \frac{3^2 5}{2^4 \pi} \biggl( \frac{K}{G} \biggr) \biggr]^{1/2}</math>
  </td>
</tr>

<tr>
  <td align="center">
5
  </td>
  <td align="center">
6/5
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  <td align="center">
<math>\frac{3^{12} 5^{9}}{2^{26} \pi^3} \biggl[ \frac{K^{10}}{G^9 M^6} \biggr]</math>
  </td>
  <td align="center">
<math>\biggl[ \frac{2^{12} \pi}{3^6 5^5} \biggl( \frac{G^5 M^4}{K^5} \biggr) \biggr]^{1/2}</math>
  </td>
</tr>

<tr>
  <td align="center">
<math>\infty</math>
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  <td align="center">
1
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  <td align="center">
<math> \frac{3^4 5^3}{2^{10}\pi}  \biggl[ \frac{K^4}{G^{3} M^{2} }\biggr]</math>
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  <td align="center">
<math>\frac{2^2GM}{3\cdot 5 K}</math>
  </td>
</tr>

</table>
</div>

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