VisTrails Home

User:Tohline/VE/RiemannEllipsoids

From VisTrailsWiki

(Difference between revisions)
Jump to: navigation, search
Line 203: Line 203:
</table>
</table>
</div>
</div>
 +
 +
=Various Degrees of Simplification=
 +
 +
==Riemann S-Type Ellipsoids==
 +
Describe &hellip;
 +
 +
==Jacobi and Dedekind Ellipsoids==
 +
Describe &hellip;
 +
 +
==Maclaurin Spheroids==
 +
 +
Describe &hellip;
 +
=See Also=
=See Also=

Revision as of 21:41, 3 August 2020


Contents

Steady-State 2nd-Order Tensor Virial Equations

By satisfying all six — not necessarily unique — components of the Second-Order Tensor Virial Equation, the entire set of Riemann Ellipsoids can be determined.

Whitworth's (1981) Isothermal Free-Energy Surface
|   Tiled Menu   |   Tables of Content   |  Banner Video   |  Tohline Home Page   |

Equilibrium Expressions

[EFE §11(b), p. 22] Under conditions of a stationary state, [the tensor virial equation] gives,

~2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij}

~=

~- \delta_{ij}\Pi \, .

[This] provides six integral relations which must obtain whenever the conditions are stationary.

When viewing the (generally ellipsoidal) configuration from a rotating frame of reference, the 2nd-order TVE takes on the more general form:

~0

~=

~
2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} + \delta_{ij}\Pi 
+ \Omega^2 I_{ij} - \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j dx
\, .

[ EFE, Chapter 2, §12, Eq. (64) ]

EFE (p. 57) also shows that … The potential energy tensor … for a homogeneous ellipsoid is given by

~\frac{\mathfrak{W}_{ij}}{\pi G\rho}

~=

~-2A_i I_{ij} \, ,

[ EFE, Chapter 3, §22, Eq. (128) ]

where

~I_{ij}

~=

~\tfrac{1}{5} Ma_i^2 \delta_{ij} \, ,

[ EFE, Chapter 3, §22, Eq. (129) ]

is the moment of inertia tensor.

Adopted (Internal) Velocity Field

EFE (p. 130) states that … The kinematical requirement, that the motion ~(\vec{u}), associated with ~\vec{\zeta}, preserves the ellipsoidal boundary, leads to the following expressions for its components:

~u_1

~=

~- \biggl[ \frac{a_1^2}{a_1^2 + a_2^2}\biggr] \zeta_3 x_2 + \biggl[ \frac{a_1^2}{a_1^2+a_3^2}\biggr] \zeta_2 x_3 \, ,

~u_2

~=

~- \biggl[ \frac{a_2^2}{a_2^2 + a_3^2}\biggr] \zeta_1 x_3 + \biggl[ \frac{a_2^2}{a_2^2+a_1^2}\biggr] \zeta_3 x_1 \, ,

~u_3

~=

~- \biggl[ \frac{a_3^2}{a_3^2 + a_1^2}\biggr] \zeta_2 x_1 + \biggl[ \frac{a_3^2}{a_3^2+a_2^2}\biggr] \zeta_1 x_2 \, .

[ EFE, Chapter 7, §47, Eq. (1) ]


General Coefficient Expressions

As has been detailed in an accompanying chapter, the gravitational potential anywhere inside or on the surface, ~(a_1,a_2,a_3) ~\leftrightarrow~(a,b,c), of an homogeneous ellipsoid may be given analytically in terms of the following three coefficient expressions:


~A_1


~=

~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] \, ,


~A_3


~=


~2\biggl(\frac{b}{a}\biggr) \biggl[  \frac{(b/a) \sin\theta - (c/a)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, ,


~A_2


~=

~2 - (A_1+A_3) \, ,

where, ~F(\theta,k) and ~E(\theta,k) are incomplete elliptic integrals of the first and second kind, respectively, with arguments,

~\theta = \cos^{-1} \biggl(\frac{c}{a} \biggr)

      and      

~k = \biggl[\frac{1 - (b/a)^2}{1 - (c/a)^2} \biggr]^{1/2} \, .

[ EFE, Chapter 3, §17, Eq. (32) ]

Various Degrees of Simplification

Riemann S-Type Ellipsoids

Describe …

Jacobi and Dedekind Ellipsoids

Describe …

Maclaurin Spheroids

Describe …


See Also


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2020 by Joel E. Tohline
|   H_Book Home   |   YouTube   |
Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS |

Personal tools