# User:Tohline/VE/RiemannEllipsoids

(Difference between revisions)
 Revision as of 21:41, 3 August 2020 (view source)Tohline (Talk | contribs)← Older edit Revision as of 10:20, 4 August 2020 (view source)Tohline (Talk | contribs) (→Steady-State 2nd-Order Tensor Virial Equations)Newer edit → Line 8: Line 8: {{LSU_HBook_header}} {{LSU_HBook_header}} - ==Equilibrium Expressions== + Here we are only interested in determining the equilibrium conditions of uniform-density ellipsoids that have semi-axes, $~a_1, a_2, a_3$. - [[[User:Tohline/Appendix/References#EFE|EFE]] §11(b), p. 22] Under conditions of a stationary state, [the tensor virial equation] gives, + + ==General Coefficient Expressions== + + As has been detailed in an [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids#Gravitational_Potential|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:
- +
- $~2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij}$ + $+ ~A_1 +$ - $~=$ + $+ ~= +$ - $~- \delta_{ij}\Pi \, .$ + $~2\biggl(\frac{a_2}{a_1}\biggr)\biggl(\frac{a_3}{a_1}\biggr) + \biggl[ \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, , +$
- - - [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: -
- $~0$ + $+ ~A_3 +$ - $~=$ + $+ ~= +$ - $~ + [itex] - 2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} + \delta_{ij}\Pi + ~2\biggl(\frac{a_2}{a_1}\biggr) \biggl[ \frac{(a_2/a_1) \sin\theta - (a_3/a_1)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, , - + \Omega^2 I_{ij} - \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j dx + - \, . +$ [/itex]
[ [[User:Tohline/Appendix/References#EFE|EFE]], Chapter 2, §12, Eq. (64) ]
- - - - EFE (p. 57) also shows that … The potential energy tensor … for a homogeneous ellipsoid is given by -
- $~\frac{\mathfrak{W}_{ij}}{\pi G\rho}$ + $+ ~A_2 +$ - $~=$ + $+ ~= +$ - $~-2A_i I_{ij} \, ,$ + $~2 - (A_1+A_3) \, ,$
[ [[User:Tohline/Appendix/References#EFE|EFE]], Chapter 3, §22, Eq. (128) ]
- + - where + + where, $~F(\theta,k) and [itex]~E(\theta,k)$ are incomplete elliptic integrals of the first and second kind, respectively, with arguments, +
- $~I_{ij}$ + $~\theta = \cos^{-1} \biggl(\frac{a_3}{a_1} \biggr)$ - $~=$ +       and       - $~\tfrac{1}{5} Ma_i^2 \delta_{ij} \, ,$ + $~k = \biggl[\frac{1 - (a_2/a_1)^2}{1 - (a_3/a_1)^2} \biggr]^{1/2} \, .$
[ [[User:Tohline/Appendix/References#EFE|EFE]], Chapter 3, §22, Eq. (129) ]
[ [[User:Tohline/Appendix/References#EFE|EFE]], Chapter 3, §17, Eq. (32) ]
- + - + - is the moment of inertia tensor. + ==Adopted (Internal) Velocity Field== ==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: + 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: Line 125: Line 131: - + ==Equilibrium Expressions== - ==General Coefficient Expressions== + [[[User:Tohline/Appendix/References#EFE|EFE]] §11(b), p. 22] Under conditions of a stationary state, [the tensor virial equation] gives, - + - As has been detailed in an [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids#Gravitational_Potential|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: +
- +
- $+ [itex]~2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij}$ - ~A_1 + - [/itex] + - $+ [itex]~=$ - ~= + - [/itex] + - $~2\biggl(\frac{b}{a}\biggr)\biggl(\frac{c}{a}\biggr) + [itex]~- \delta_{ij}\Pi \, .$ - \biggl[  \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, , + - [/itex] +
+ + + [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: +
- $+ [itex]~0$ - ~A_3 + - [/itex] + - $+ [itex]~=$ - ~= + - [/itex] + - $+ [itex]~ - ~2\biggl(\frac{b}{a}\biggr) \biggl[ \frac{(b/a) \sin\theta - (c/a)E(\theta,k)}{(1-k^2) \sin^3\theta} \biggr] \, , + 2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} + \delta_{ij}\Pi + + \Omega^2 I_{ij} - \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j dx + \, .$ [/itex]
[ [[User:Tohline/Appendix/References#EFE|EFE]], Chapter 2, §12, Eq. (64) ]
+ + + + EFE (p. 57) also shows that … The potential energy tensor … for a homogeneous ellipsoid is given by +
- $+ [itex]~\frac{\mathfrak{W}_{ij}}{\pi G\rho}$ - ~A_2 + - [/itex] + - $+ [itex]~=$ - ~= + - [/itex] + - $~2 - (A_1+A_3) \, ,$ + $~-2A_i I_{ij} \, ,$
[ [[User:Tohline/Appendix/References#EFE|EFE]], Chapter 3, §22, Eq. (128) ]
- + - + where - where, $~F(\theta,k) and [itex]~E(\theta,k)$ are incomplete elliptic integrals of the first and second kind, respectively, with arguments, + -
+
- $~\theta = \cos^{-1} \biggl(\frac{c}{a} \biggr)$ + $~I_{ij}$ -       and       + $~=$ - $~k = \biggl[\frac{1 - (b/a)^2}{1 - (c/a)^2} \biggr]^{1/2} \, .$ + $~\tfrac{1}{5} Ma_i^2 \delta_{ij} \, ,$
[ [[User:Tohline/Appendix/References#EFE|EFE]], Chapter 3, §17, Eq. (32) ]
[ [[User:Tohline/Appendix/References#EFE|EFE]], Chapter 3, §22, Eq. (129) ]
- + - + + is the moment of inertia tensor. =Various Degrees of Simplification= =Various Degrees of Simplification=

# 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.

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) ]

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.

Describe …

Describe …

Describe …