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> |
or,
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<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,
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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>~\rightarrow</math> |
<math>~\cancelto{0}{\frac{1}{2}v^2} + \frac{1}{2} [(a+bx)^2 + (b+ax)^2]\Omega^2 - 2I \, ,</math> |
[ 1st expression — EFE, §53, Eq. (239) ]
[ 2nd expression — Paper I, Eq. (2.7) ]
where — see an accompanying discussion for the definitions of <math>~A_1</math>, <math>~A_2</math>, and <math>~A_3</math>,
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<math>~I</math> |
<math>~=</math> |
<math>~A_1a^2 + A_2b^2 + A_3c^2 \, ;</math> |
[ 1st expression — EFE, §53, Eq. (239) ]
[ 2nd expression — Paper I, Eq. (2.8) ]
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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) ]
Note that, based on the units chosen in Paper I, <math>~M = 5</math>, and <math>~abc = 15/4</math>.
Free Energy Surface(s)
Consider a self-gravitating ellipsoid having the following properties:
- Semi-axis lengths, <math>~(x,y,z)_\mathrm{surface} = (a,b,c)</math>, and corresponding volume, <math>~4\pi/(3abc)</math> ; and consider only the situations <math>0 \le b/a \le 1</math> and <math>0 \le c/a \le 1</math> ;
- Total mass, <math>~M</math> ;
- Uniform density, <math>~\rho = (3 M)/(4\pi abc) </math> ;
- Figure is spinning about its c axis with angular velocity, <math>~\Omega</math> ;
- Internal, steady-state flow exhibiting the following characteristics:
- No vertical (z) motion;
- Elliptical (x-y plane) streamlines everywhere having an ellipticity that matches that of the overall figure, that is, <math>~e = (1-b^2/a^2)^{1/2}</math> ;
- The velocity components, <math>~v_x</math> and <math>~v_y</math>, are linear in the coordinate and, overall, characterized by the magnitude of the vorticity, <math>~\zeta</math> .
Such a configuration is uniquely specified by the choice of six key parameters: <math>~a</math>, <math>~b</math>, <math>~c</math>, <math>~M</math>, <math>~\Omega</math>, and <math>~\zeta</math> .
We are interested, here, in examining how the free energy of such a system will vary as it is allowed to "evolve" as an incompressible fluid — i.e., holding <math>~\rho</math> fixed — through different ellipsoidal shapes while conserving its total mass. Following Paper I, we choose to set <math>~M = 5</math> — which removes mass from the list of unspecified key parameters — and we choose to set <math>~\rho = \pi^{-1}</math>, which is then reflected in a specification of the semi-axis, <math>~c</math>, in terms of the other two semi-axes, namely,
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<math>~c</math> |
<math>~=</math> |
<math>~\frac{3M}{4\pi(ab)\rho} = \frac{15}{4ab} \, .</math> |
Moving forward, then, a unique ellipsoidal configuration is identified via the specification of four, rather than six, key parameters — <math>~a</math>, <math>~b</math>, <math>~\Omega</math>, and <math>~\zeta</math> — and the free energy of that configuration is given by the expression,
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<math>~E</math> |
<math>~=</math> |
<math>~\frac{1}{2} [(a+bx)^2 + (b+ax)^2]\Omega^2 - 2I \, ,</math> |
where,
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<math>~x</math> |
<math>~\equiv</math> |
<math>~\biggl[\frac{ab}{a^2 + b^2} \biggr]\frac{\zeta}{\Omega} \, ,</math> |
and,
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<math>~I</math> |
<math>~=</math> |
<math>~A_1a^2 + A_2b^2 + A_3\biggl(\frac{15}{4ab}\biggr)^2 \, ,</math> |
[ 1st expression — EFE, §53, Eq. (239) ]
[ 2nd expression — Paper I, Eq. (2.8) ]
— see an accompanying discussion for the definitions of the ellipticity-dependent coefficients, <math>~A_1(e)</math>, <math>~A_2(e)</math>, and <math>~A_3(e)</math>.
See Also
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© 2014 - 2021 by Joel E. Tohline |