User:Tohline/ThreeDimensionalConfigurations/EFE Energies

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>

[ EFE, §48, Eq. (40) ]
[ Paper I, Eq. (2.2) ]

or,

<math>~\Lambda</math>

<math>~\equiv</math>

<math>~-\biggl[\frac{ab}{a^2 + b^2} \biggr] \Omega f = -\Omega x \, .</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>~\rightarrow</math>	<math>~\cancelto{0}{\frac{1}{2}v^2} + \frac{1}{2} [(a+bx)^2 + (b+ax)^2]\Omega^2 - 2I \, ,</math>

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

where,

<math>~I</math>

<math>~=</math>

<math>~A_1a^2 + A_2b^2 + A_3c^2 \, ;</math>

[ 1^st expression — EFE, §53, Eq. (239) ]
[ 2^nd expression — Paper I, Eq. (2.8) ]

<math>~\frac{5L}{M}</math>	<math>~=</math>	<math>~(a^2 + b^2)\Omega - 2ab\Lambda</math>
	<math>~=</math>	<math>~ (a^2 + b^2 + 2abx)\Omega \, ;</math>

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

<math>~\frac{5C}{M}</math>	<math>~=</math>	<math>~(a^2 + b^2)\Lambda - 2ab\Omega</math>
	<math>~=</math>	<math>~- [2ab + (a^2 + b^2)x ]\Omega \, .</math>

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

Note that, based on the units chosen in Paper I, <math>~M = 5</math>, and <math>~abc = 15/4</math>.

See Also

• Properties of Maclaurin Spheroids
• Excerpts from Maclaurin's (1742) A Treatise of Fluxions

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation