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(The Six Off-Diagonal Elements)
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For example &#8212; as is explicitly illustrated on p. 130 of EFE &#8212; for <math>~i=2</math> and <math>~j=3</math>,
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<math>~0</math>
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<math>~=</math>
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<math>~
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2 \mathfrak{T}_{23} - \Omega_2\Omega_3 I_{33} + 2\Omega_1 \cancelto{0}{\int_V \rho u_3x_3 dx}
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- 2\Omega_3 \int_V \rho u_1x_3 dx \, ,
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whereas for <math>~i=3</math> and <math>~j=2</math>,
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<math>~0</math>
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<math>~=</math>
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<math>~
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2 \mathfrak{T}_{ij} - \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j dx
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\, ,
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where, in both cases, we have acknowledged that, given the above specific prescription for the internal velocity field, <math>~\vec{u}</math>,
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<math>~\int_V \rho u_i x_j dx</math>
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<math>~=</math>
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<math>~0</math>
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  <td align="right">&nbsp; &nbsp; &nbsp; if  &nbsp; &nbsp;<math>~i = j \, .</math>
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Revision as of 13:34, 4 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
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Here we are only interested in determining the equilibrium conditions of uniform-density ellipsoids that have semi-axes, ~a_1, a_2, a_3.

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), of an homogeneous ellipsoid may be given analytically in terms of the following three coefficient expressions:


~A_1


~=

~2\biggl(\frac{a_2}{a_1}\biggr)\biggl(\frac{a_3}{a_1}\biggr)
\biggl[  \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,


~A_3


~=


~2\biggl(\frac{a_2}{a_1}\biggr) \biggl[  \frac{(a_2/a_1) \sin\theta - (a_3/a_1)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{a_3}{a_1} \biggr)

      and      

~k = \biggl[\frac{1 - (a_2/a_1)^2}{1 - (a_3/a_1)^2} \biggr]^{1/2} \, .

[ EFE, Chapter 3, §17, Eq. (32) ]

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) ]

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.

The Three Diagonal Elements

For ~i = j = 1, we have,

~0

~=

~
2 \mathfrak{T}_{11} + \mathfrak{W}_{11} + \Pi 
+ \Omega^2 I_{11} - \Omega_1\Omega_k I_{k1} + 2\epsilon_{1lm}\Omega_m \int_V \rho u_lx_1 dx

 

~=

~
2 \mathfrak{T}_{11} + \mathfrak{W}_{11} + \Pi + \Omega^2 I_{11} 
- \Omega_1^2I_{11} 
+ 2 \Omega_3 \int_V \rho u_2x_1 dx
- 2\Omega_2 \int_V \rho u_3x_1 dx

 

~=

~
2 \mathfrak{T}_{11} + \mathfrak{W}_{11} + \Pi 
+( \Omega_2^2 + \Omega_3^2) I_{11} 
+ 2 \Omega_3 \int_V \rho u_2x_1 dx
- 2\Omega_2 \int_V \rho u_3x_1 dx

Similarly, for ~i = j = 2,

~0

~=

~
2 \mathfrak{T}_{22} + \mathfrak{W}_{22} + \Pi 
+ \Omega^2 I_{22} - \Omega_2\Omega_k I_{k2} + 2\epsilon_{2lm}\Omega_m \int_V \rho u_lx_2 dx

 

~=

~
2 \mathfrak{T}_{22} + \mathfrak{W}_{22} + \Pi 
+ (\Omega_1^2 + \Omega_3^2) I_{22} + 2\Omega_1 \int_V \rho u_3x_2 dx
- 2\Omega_3 \int_V \rho u_1x_2 dx

and, for ~i=j=3,

~0

~=

~
2 \mathfrak{T}_{33} + \mathfrak{W}_{33} + \Pi 
+ \Omega^2 I_{33} - \Omega_3\Omega_k I_{k3} + 2\epsilon_{3lm}\Omega_m \int_V \rho u_lx_3 dx

 

~=

~
2 \mathfrak{T}_{33} + \mathfrak{W}_{33} + \Pi 
+ (\Omega_1^2 + \Omega_2^2) I_{33}  + 2\Omega_2 \int_V \rho u_1x_3 dx
- 2\Omega_1 \int_V \rho u_2 x_3 dx

The Six Off-Diagonal Elements

Notice that the off-diagonal components of both ~I_{ij} and ~\mathfrak{W}_{ij} are zero. Hence, the equilibrium expression that is dictated by each off-diagonal component of the 2nd-order TVE is,

~0

~=

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

For example — as is explicitly illustrated on p. 130 of EFE — for ~i=2 and ~j=3,

~0

~=

~
2 \mathfrak{T}_{23} - \Omega_2\Omega_3 I_{33} + 2\Omega_1 \cancelto{0}{\int_V \rho u_3x_3 dx}
- 2\Omega_3 \int_V \rho u_1x_3 dx \, ,

whereas for ~i=3 and ~j=2,

~0

~=

~
2 \mathfrak{T}_{ij} - \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j dx
\, ,

where, in both cases, we have acknowledged that, given the above specific prescription for the internal velocity field, ~\vec{u},

~\int_V \rho u_i x_j dx

~=

~0

      if    ~i = j \, .

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
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