Difference between revisions of "User:Tohline/ThreeDimensionalConfigurations/EFE Energies"
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=Properties of Homogeneous Ellipsoids (2)= | =Properties of Homogeneous Ellipsoids (2)= | ||
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). | |||
==Terminology== | |||
A Riemann sequence of ''S''-type ellipsoids is defined by the value of the dimensionless parameter, | |||
<div align="center"> | |||
<table border="0" cellpadding="5" align="center"> | |||
<tr> | |||
<td align="right"> | |||
<math>~f</math> | |||
</td> | |||
<td align="center"> | |||
<math>~\equiv</math> | |||
</td> | |||
<td align="left"> | |||
<math>~\frac{\zeta}{\Omega} = </math> constant, | |||
</td> | |||
</tr> | |||
</table> | |||
[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">§48, Eq. (31)</font> ]<br /> | |||
[ [http://adsabs.harvard.edu/abs/1983ApJ...271..586W WT83], <font color="#00CC00">Eq. (5)</font> ]<br /> | |||
[ [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I], <font color="#00CC00">Eq. (2.1)</font> ]<br /> | |||
</div> | |||
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, | |||
<div align="center"> | |||
<table border="0" cellpadding="5" align="center"> | |||
<tr> | |||
<td align="right"> | |||
<math>~x</math> | |||
</td> | |||
<td align="center"> | |||
<math>~\equiv</math> | |||
</td> | |||
<td align="left"> | |||
<math>~\biggl[\frac{ab}{a^2 + b^2} \biggr]f \, ,</math> | |||
</td> | |||
</tr> | |||
</table> | |||
[ [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I], <font color="#00CC00">Eq. (2.2)</font> ]<br /> | |||
</div> | |||
or, | |||
<div align="center"> | |||
<table border="0" cellpadding="5" align="center"> | |||
<tr> | |||
<td align="right"> | |||
<math>~\Lambda</math> | |||
</td> | |||
<td align="center"> | |||
<math>~\equiv</math> | |||
</td> | |||
<td align="left"> | |||
<math>~-\biggl[\frac{ab}{a^2 + b^2} \biggr] \Omega f \, .</math> | |||
</td> | |||
</tr> | |||
</table> | |||
[ [http://adsabs.harvard.edu/abs/1983ApJ...271..586W WT83], <font color="#00CC00">Eq. (4)</font> ]<br /> | |||
</div> | |||
==Relevant Energy Components== | ==Relevant Energy Components== | ||
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where, | where, | ||
=See Also= | =See Also= | ||
Revision as of 01:18, 15 June 2016
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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).
Terminology
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) ]
Relevant Energy Components
As has been explicitly demonstrated in Chapter 3 of EFE and summarized in Table 2-2 (p. 57) of 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>,
<math>
~\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>
[ EFE, Chapter 3, Eq. (40)1,2 ]
[ BT87, Chapter 2, Table 2-2 ]
where,
See Also
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