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).
Terminology
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) ]
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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