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

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[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">&sect;48, Eq. (40)</font> ]<br />
[ [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I], <font color="#00CC00">Eq. (2.2)</font> ]<br />
[ [http://adsabs.harvard.edu/abs/1995ApJ...446..472C Paper I], <font color="#00CC00">Eq. (2.2)</font> ]<br />
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<math>~-\biggl[\frac{ab}{a^2 + b^2} \biggr] \Omega f \, .</math>
<math>~-\biggl[\frac{ab}{a^2 + b^2} \biggr] \Omega f = -\Omega x \, .</math>
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[ [http://adsabs.harvard.edu/abs/1983ApJ...271..586W WT83], <font color="#00CC00">Eq. (4)</font> ]<br />
[ [http://adsabs.harvard.edu/abs/1983ApJ...271..586W WT83], <font color="#00CC00">Eq. (4)</font> ]<br />
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==Conserved Quantities==
==Conserved Quantities==

Revision as of 02:56, 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>

[ 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>~=</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>~\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>

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

[ 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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