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(Create the ASIDE to explain Whitworth's (1981) scaling of pressure-bounded polytropes)
 
(→‎ASIDE: Whitworth's Scaling: Discuss Whitworth's normalizations)
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=ASIDE: Whitworth's Scaling=
=ASIDE: Whitworth's Scaling=
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:
<div align="center">
<table border="2">
<tr><td>
[[File:WhitworthScalingText.jpg|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 {{User:Tohline/Math/MP_PolytropicIndex}} via the relation,
<div align="center">
<math>\eta = 1 + \frac{1}{n} \, .</math>
</div>
Hence, Whitworth writes the polytropic equation of state as,
<div align="center">
<math>P = K_\eta \rho^\eta \, ,</math>
</div>
whereas, using our standard notation, this same key relation is written as,
<div align="center">
{{User:Tohline/Math/EQ_Polytrope01}} ;
</div>
and his parameter <math>K_\eta</math> is identical to our {{User:Tohline/Math/MP_PolytropicConstant}}. 


===Bounded Adiabatic===
According to the second (bottom) expression identified by the red outlined box drawn above,
For adiabatic configurations <math>(\delta_{1\gamma_g} = 0)</math>, equilibrium states exist at radii given by the roots of the following expression:
<div align="center">
<math>
P_\mathrm{rf} = \frac{3^4 5^3}{2^{10} \pi} \biggl( \frac{K_1^4}{G^3 M^2} \biggr) \, ,
</math>
</div>
and inverting the expression inside the green outlined box gives,
<div align="center">
<div align="center">
<math>
<math>
3(\gamma_g-1) B\chi^{3 -3\gamma_g} ~-~A\chi^{-1} -~ 3D\chi^3 = 0 \, .
K_1 = \biggl[ K_n (4 P_\mathrm{rf})^{\eta - 1} \biggr]^{1/\eta} \, .
</math>
</math>
</div>
</div>
This is precisely the same condition that derives from setting equation (3) to zero in [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth's] (1981, MNRAS, 195, 967) discussion of the "global gravitational stability for one-dimensional polyropes."  The overlap with Whitworth's narative is perhaps clearer after introducing the algebraic expressions for the coefficients <math>A</math>, <math>B</math>, and <math>D</math>, dividing the equation through by <math>(3\chi^3 V_0) = (4\pi R^3)</math>, and rewriting <math>R</math> as <math>R_\mathrm{eq}</math> to obtain,
Hence,
<div align="center">
<div align="center">
<math>
<math>
P_e = K \biggl( \frac{3M}{4\pi R_\mathrm{eq}^3} \biggr)^{\gamma_g} - \biggl( \frac{3GM^2}{20\pi R_\mathrm{eq}^4} \biggr) \, .
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>
</math>
</div>
</div>
This exactly matches equation (5) of [http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth], which reads:
or, gathering all factors of <math>P_\mathrm{rf}</math> to the left-hand side,
<div align="center">
<div align="center">
<table border="2">
<math>
<tr><td>
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] \, .
[[File:WhitworthScalingText.jpg|500px|center|Whitworth (1981, MNRAS, 195, 967)]]
</math>
</td></tr>
</div>
Analogously, according to the first (top) expression identified inside the red outlined box,
<div align="center">
<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>
 
<div align="center">
<table border="1" width="90%">
  <tr>
  <td colspan="4" align="center">'''Examples'''</td>
  </tr>
<tr>
  <td align="center">
{{User:Tohline/Math/MP_PolytropicIndex}}
  </td>
  <td align="center">
<math>\eta = 1+1/n</math>
  </td>
  <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>
  </td>
  <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
  </td>
  <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>
  </td>
  <td align="center">
1
  </td>
  <td align="center">
<math> \frac{3^4 5^3}{2^{10}\pi}  \biggl[ \frac{K^4}{G^{3} M^{2} }\biggr]</math>
  </td>
  <td align="center">
<math>R_\mathrm{rf} = \frac{2^2GM}{3\cdot 5 K}</math>
  </td>
</tr>
 
</table>
</table>
</div>
</div>
Ideally we would like to invert this equation to obtain an analytic expression for the configuration's equilibrium radius in terms of the physical parameters, <math>M</math>, <math>K</math>, and <math>P_e</math>.  However, this cannot be accomplished for an arbitrary value of the adiabatic exponent, <math>\gamma_g</math>.
 




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Revision as of 23:02, 20 October 2012

Whitworth's (1981) Isothermal Free-Energy Surface
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ASIDE: Whitworth's Scaling

In his study of the "global gravitational stability [of] one-dimensional polytropes," 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:

Whitworth (1981, MNRAS, 195, 967)

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>~n</math> via the relation,

<math>\eta = 1 + \frac{1}{n} \, .</math>

Hence, Whitworth writes the polytropic equation of state as,

<math>P = K_\eta \rho^\eta \, ,</math>

whereas, using our standard notation, this same key relation is written as,

<math>~P = K_\mathrm{n} \rho^{1+1/n}</math> ;

and his parameter <math>K_\eta</math> is identical to our <math>~K_\mathrm{n}</math>.

According to the second (bottom) expression identified by the red outlined box drawn above,

<math> P_\mathrm{rf} = \frac{3^4 5^3}{2^{10} \pi} \biggl( \frac{K_1^4}{G^3 M^2} \biggr) \, , </math>

and inverting the expression inside the green outlined box gives,

<math> K_1 = \biggl[ K_n (4 P_\mathrm{rf})^{\eta - 1} \biggr]^{1/\eta} \, . </math>

Hence,

<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>

or, gathering all factors of <math>P_\mathrm{rf}</math> to the left-hand side,

<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>

Analogously, according to the first (top) expression identified inside the red outlined box,

<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>

Examples

<math>~n</math>

<math>\eta = 1+1/n</math>

<math>P_\mathrm{rf}</math>

<math>R_\mathrm{rf}</math>

1

2

<math>\frac{2^{6}\pi}{3^4 5^3} \biggl[ \frac{G^{3} M^{2} }{K^2}\biggr]</math>

<math>\biggl[ \frac{3^2 5}{2^4 \pi} \biggl( \frac{K}{G} \biggr) \biggr]^{1/2}</math>

5

6/5

<math>\frac{3^{12} 5^{9}}{2^{26} \pi^3} \biggl[ \frac{K^{10}}{G^9 M^6} \biggr]</math>

<math>\biggl[ \frac{2^{12} \pi}{3^6 5^5} \biggl( \frac{G^5 M^4}{K^5} \biggr) \biggr]^{1/2}</math>

<math>\infty</math>

1

<math> \frac{3^4 5^3}{2^{10}\pi} \biggl[ \frac{K^4}{G^{3} M^{2} }\biggr]</math>

<math>R_\mathrm{rf} = \frac{2^2GM}{3\cdot 5 K}</math>


Whitworth's (1981) Isothermal Free-Energy Surface

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