User:Tohline/VE/RiemannEllipsoids
Steady-State Tensor Virial Equations
By satisfying all nine components of the Second-Order Tensor Virial Equation, the entire set of Riemann S-Type Ellipsoids can be determined.
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General Coefficient Expressions
As has been detailed in an accompanying chapter, the gravitational potential anywhere inside or on the surface, <math>~(a_1,a_2,a_3) ~\leftrightarrow~(a,b,c)</math>, of an homogeneous ellipsoid may be given analytically in terms of the following three coefficient expressions:
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<math> ~A_1 </math> |
<math> ~= </math> |
<math>~2\biggl(\frac{b}{a}\biggr)\biggl(\frac{c}{a}\biggr) \biggl[ \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, , </math> |
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<math> ~A_3 </math> |
<math> ~= </math> |
<math> ~2\biggl(\frac{b}{a}\biggr) \biggl[ \frac{(b/a) \sin\theta - (c/a)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, , </math> |
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<math> ~A_2 </math> |
<math> ~= </math> |
<math>~2 - (A_1+A_3) \, ,</math> |
where, <math>~F(\theta,k)</math> and <math>~E(\theta,k)</math> are incomplete elliptic integrals of the first and second kind, respectively, with arguments,
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<math>~\theta = \cos^{-1} \biggl(\frac{c}{a} \biggr)</math> |
and |
<math>~k = \biggl[\frac{1 - (b/a)^2}{1 - (c/a)^2} \biggr]^{1/2} \, .</math> |
| [ EFE, Chapter 3, §17, Eq. (32) ] | ||
See Also
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© 2014 - 2021 by Joel E. Tohline |