User:Tohline/ThreeDimensionalConfigurations/EFE Energies
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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,
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<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,
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<math>~x</math> |
<math>~\equiv</math> |
<math>~\biggl[\frac{ab}{a^2 + b^2} \biggr]f \, ,</math> |
[ Paper I, Eq. (2.2) ]
or,
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<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,
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<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> |
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<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) ]
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<math>~\frac{5L}{M}</math> |
<math>~=</math> |
<math>~(a^2 + b^2)\Omega - 2ab\Lambda</math> |
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<math>~=</math> |
<math>~ (a^2 + b^2 + 2abx)\Omega \, ,</math> |
[ 1st expression — EFE, §53, Eq. (240) ]
[ 2nd expression — Paper I, Eq. (2.5) ]
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<math>~\frac{5C}{M}</math> |
<math>~=</math> |
<math>~(a^2 + b^2)\Lambda - 2ab\Omega</math> |
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<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
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© 2014 - 2021 by Joel E. Tohline |