Difference between revisions of "User:Tohline/ThreeDimensionalConfigurations/EFE Energies"

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In addition to pulling from §53 of [[User:Tohline/Appendix/References#EFE|Chandrasekhar's EFE]], here, we lean heavily on the papers by [http://adsabs.harvard.edu/abs/1983ApJ...271..586W M. D. Weinberg & S. Tremaine (1983, ApJ, 271, 586)] (hereafter, WT83) and by [http://adsabs.harvard.edu/abs/1995ApJ...446..472C D. M. Christodoulou, D. Kazanas, I. Shlosman, & J. E. Tohline (1995, ApJ, 446, 472)] (hereafter, Paper I).
In addition to pulling from §53 of [[User:Tohline/Appendix/References#EFE|Chandrasekhar's EFE]], here, we lean heavily on the papers by [http://adsabs.harvard.edu/abs/1983ApJ...271..586W M. D. Weinberg & S. Tremaine (1983, ApJ, 271, 586)] (hereafter, WT83) and by [http://adsabs.harvard.edu/abs/1995ApJ...446..472C D. M. Christodoulou, D. Kazanas, I. Shlosman, & J. E. Tohline (1995, ApJ, 446, 472)] (hereafter, Paper I).


==Key Dimensionless Parameters==
==Sequence-Defining Dimensionless Parameters==


A Riemann sequence of ''S''-type ellipsoids is defined by the value of the dimensionless parameter,  
A Riemann sequence of ''S''-type ellipsoids is defined by the value of the dimensionless parameter,  
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   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\frac{1}{2}v^2 + \frac{1}{2}(a^2 + b^2)(\Lambda^2 + \Omega^2) - 2ab\Lambda\Omega - 2I \, ,</math>
<math>~\frac{1}{2}v^2 + \frac{1}{2}(a^2 + b^2)(\Lambda^2 + \Omega^2) - 2ab\Lambda\Omega - 2I </math>
  </td>
</tr>
 
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{\Omega^2}{2} [(a+bx)^2 + (b+ax)^2] - 2I \, ,</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>


[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">&sect;53, Eq. (239)</font> ]<br />
[ 1<sup>st</sup> expression &#8212; [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">&sect;53, Eq. (239)</font> ]<br />
[ [http://adsabs.harvard.edu/abs/1983ApJ...271..586W WT83], <font color="#00CC00">Eq. (5)</font> ]<br />
[ 2<sup>nd</sup> expression &#8212; [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I], <font color="#00CC00">Eq. (2.7)</font> ]<br />
[ [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I], <font color="#00CC00">Eq. (2.1)</font> ]<br />
</div>
</div>


<div align="center">
<table border="0" cellpadding="5" align="center">


==Relevant Energy Components==
<tr>
  <td align="right">
<math>~L</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{M}{5}\biggl[ (a^2 + b^2)\Omega - 2ab\Lambda\biggr]</math>
  </td>
</tr>


<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ (a^2 + b^2 + 2abx)\Omega \, ,</math>
  </td>
</tr>
</table>


As has been explicitly demonstrated in Chapter 3 of [[User:Tohline/Appendix/References#EFE|EFE]] and summarized in Table 2-2 (p. 57) of [[User:Tohline/Appendix/References#BT87|BT87]], for an homogeneous ellipsoid this volume integral can be evaluated analytically in closed form.  Specifically, at an internal point or on the surface of an homogeneous ellipsoid with semi-axes <math>~(x,y,z) = (a_1,a_2,a_3)</math>,
[ 1<sup>st</sup> expression &#8212; [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">&sect;53, Eq. (240)</font> ]<br />
[ 2<sup>nd</sup> expression &#8212; [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I], <font color="#00CC00">Eq. (2.5)</font> ]<br />
</div>


<div align="center">
<div align="center">
<math>
<table border="0" cellpadding="5" align="center">
~\Phi(\vec{x}) = -\pi G \rho \biggl[ I_\mathrm{BT} a_1^2 - \biggl(A_1 x^2 + A_2 y^2 +A_3 z^2 \biggr) \biggr],
 
</math><br />
<tr>
  <td align="right">
<math>~C</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\frac{M}{5}\biggl[ (a^2 + b^2)\Lambda - 2ab\Omega\biggr]</math>
  </td>
</tr>
 
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- [2ab + (a^2 + b^2)x ]\Omega \, ,</math>
  </td>
</tr>
</table>


[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, Eq. (40)</font><sup>1,2</sup> ]<br />
[ 1<sup>st</sup> expression &#8212; [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">&sect;53, Eq. (241)</font> ]<br />
[ [[User:Tohline/Appendix/References#BT87|BT87]], <font color="#00CC00">Chapter 2, Table 2-2</font> ]
[ 2<sup>nd</sup> expression &#8212; [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I], <font color="#00CC00">Eq. (2.6)</font> ]<br />
</div>
</div>
where,


=See Also=
=See Also=

Revision as of 02:35, 15 June 2016

Whitworth's (1981) Isothermal Free-Energy Surface
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Properties of Homogeneous Ellipsoids (2)

In addition to pulling from §53 of Chandrasekhar's EFE, here, we lean heavily on the papers by M. D. Weinberg & S. Tremaine (1983, ApJ, 271, 586) (hereafter, WT83) and by D. M. Christodoulou, D. Kazanas, I. Shlosman, & J. E. Tohline (1995, ApJ, 446, 472) (hereafter, Paper I).

Sequence-Defining Dimensionless Parameters

A Riemann sequence of S-type ellipsoids is defined by the value of the dimensionless parameter,

<math>~f</math>

<math>~\equiv</math>

<math>~\frac{\zeta}{\Omega} = </math> constant,

[ EFE, §48, Eq. (31) ]
[ WT83, Eq. (5) ]
[ Paper I, Eq. (2.1) ]

where, <math>~\zeta</math> is the system's vorticity as measured in a frame rotating with angular velocity, <math>~\Omega</math>. Alternatively, we can use the dimensionless parameter,

<math>~x</math>

<math>~\equiv</math>

<math>~\biggl[\frac{ab}{a^2 + b^2} \biggr]f \, ,</math>

[ Paper I, Eq. (2.2) ]

or,

<math>~\Lambda</math>

<math>~\equiv</math>

<math>~-\biggl[\frac{ab}{a^2 + b^2} \biggr] \Omega f \, .</math>

[ WT83, Eq. (4) ]


Conserved Quantities

Algebraic expressions for the conserved energy, <math>~E</math>, angular momentum, <math>~L</math>, and circulation, <math>~C</math>, are, respectively,

<math>~E</math>

<math>~=</math>

<math>~\frac{1}{2}v^2 + \frac{1}{2}(a^2 + b^2)(\Lambda^2 + \Omega^2) - 2ab\Lambda\Omega - 2I </math>

 

<math>~=</math>

<math>~\frac{\Omega^2}{2} [(a+bx)^2 + (b+ax)^2] - 2I \, ,</math>

[ 1st expression — EFE, §53, Eq. (239) ]
[ 2nd expression — Paper I, Eq. (2.7) ]

<math>~L</math>

<math>~=</math>

<math>~\frac{M}{5}\biggl[ (a^2 + b^2)\Omega - 2ab\Lambda\biggr]</math>

 

<math>~=</math>

<math>~ (a^2 + b^2 + 2abx)\Omega \, ,</math>

[ 1st expression — EFE, §53, Eq. (240) ]
[ 2nd expression — Paper I, Eq. (2.5) ]

<math>~C</math>

<math>~=</math>

<math>~\frac{M}{5}\biggl[ (a^2 + b^2)\Lambda - 2ab\Omega\biggr]</math>

 

<math>~=</math>

<math>~- [2ab + (a^2 + b^2)x ]\Omega \, ,</math>

[ 1st expression — EFE, §53, Eq. (241) ]
[ 2nd expression — Paper I, Eq. (2.6) ]

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

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