User:Tohline/VE/RiemannEllipsoids
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.
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Here we are only interested in determining the equilibrium conditions of uniform-density ellipsoids that have semi-axes, <math>~a_1, a_2, a_3</math>.
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)</math>, of an homogeneous ellipsoid may be given analytically in terms of the following three coefficient expressions:
<math> ~A_1 </math> |
<math> ~= </math> |
<math>~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] \, , </math> |
<math> ~A_3 </math> |
<math> ~= </math> |
<math> ~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] \, , </math> |
<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,
<math>~\theta = \cos^{-1} \biggl(\frac{a_3}{a_1} \biggr)</math> |
and |
<math>~k = \biggl[\frac{1 - (a_2/a_1)^2}{1 - (a_3/a_1)^2} \biggr]^{1/2} \, .</math> |
[ EFE, Chapter 3, §17, Eq. (32) ] |
Adopted (Internal) Velocity Field
EFE (p. 130) states that the … kinematical requirement, that the motion <math>~(\vec{u})</math>, associated with <math>~\vec{\zeta}</math>, preserves the ellipsoidal boundary, leads to the following expressions for its components:
<math>~u_1</math> |
<math>~=</math> |
<math>~- \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 \, ,</math> |
<math>~u_2</math> |
<math>~=</math> |
<math>~- \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 \, ,</math> |
<math>~u_3</math> |
<math>~=</math> |
<math>~- \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 \, .</math> |
[ EFE, Chapter 7, §47, Eq. (1) ] |
Equilibrium Expressions
[EFE §11(b), p. 22] Under conditions of a stationary state, [the tensor virial equation] gives,
<math>~2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} </math> |
<math>~=</math> |
<math>~- \delta_{ij}\Pi \, .</math> |
[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:
<math>~0</math> |
<math>~=</math> |
<math>~ 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 \, . </math> |
[ EFE, Chapter 2, §12, Eq. (64) ] |
EFE (p. 57) also shows that … The potential energy tensor … for a homogeneous ellipsoid is given by
<math>~\frac{\mathfrak{W}_{ij}}{\pi G\rho}</math> |
<math>~=</math> |
<math>~-2A_i I_{ij} \, ,</math> |
[ EFE, Chapter 3, §22, Eq. (128) ] |
where
<math>~I_{ij}</math> |
<math>~=</math> |
<math>~\tfrac{1}{5} Ma_i^2 \delta_{ij} \, ,</math> |
[ EFE, Chapter 3, §22, Eq. (129) ] |
is the moment of inertia tensor.
The Three Diagonal Elements
For <math>~i = j = 1</math>, we have,
<math>~0</math> |
<math>~=</math> |
<math>~ 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 </math> |
|
<math>~=</math> |
<math>~ 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 </math> |
|
<math>~=</math> |
<math>~ 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 </math> |
Similarly, for <math>~i = j = 2</math>,
<math>~0</math> |
<math>~=</math> |
<math>~ 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 </math> |
|
<math>~=</math> |
<math>~ 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 </math> |
and, for <math>~i=j=3</math>,
<math>~0</math> |
<math>~=</math> |
<math>~ 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 </math> |
|
<math>~=</math> |
<math>~ 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 </math> |
The Six Off-Diagonal Elements
Notice that the off-diagonal components of both <math>~I_{ij}</math> and <math>~\mathfrak{W}_{ij}</math> are zero. Hence, the equilibrium expression that is dictated by each off-diagonal component of the 2nd-order TVE is,
<math>~0</math> |
<math>~=</math> |
<math>~ 2 \mathfrak{T}_{ij} - \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j dx \, . </math> |
For example — as is explicitly illustrated on p. 130 of EFE — for <math>~i=2</math> and <math>~j=3</math>,
<math>~0</math> |
<math>~=</math> |
<math>~ 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 \, , </math> |
|
[ EFE, Chapter 7, §47, Eq. (3) ] |
whereas for <math>~i=3</math> and <math>~j=2</math>,
<math>~0</math> |
<math>~=</math> |
<math>~ 2 \mathfrak{T}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \int_V \rho u_1x_2 dx - 2\Omega_1 \cancelto{0}{\int_V \rho u_2 x_2 dx} \, . </math> |
|
[ EFE, Chapter 7, §47, Eq. (4) ] |
Given our adoption of a uniform-density configuration whose surface has a precisely ellipsoidal shape and, along with it, our adoption of the above specific prescription for the internal velocity field, <math>~\vec{u}</math>, we recognize that,
This has allowed us to set to zero one of the integrals in each of these last two expressions. In what follows, we will benefit from recognizing, as well, that,
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Adding this pair of governing expressions we obtain,
<math>~0</math> |
<math>~=</math> |
<math>~ \biggl[ 2 \mathfrak{T}_{23} - \Omega_2\Omega_3 I_{33} - 2\Omega_3 \int_V \rho u_1x_3 dx \biggr] + \biggl[2 \mathfrak{T}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \int_V \rho u_1x_2 dx \biggr] </math> |
|
<math>~=</math> |
<math>~4 \mathfrak{T}_{23} - \Omega_2\Omega_3(I_{22}+ I_{33} ) + 2 \int_V \rho u_1 (\Omega_2 x_2 - \Omega_3 x_3) dx \, ; </math> |
[ EFE, Chapter 7, §47, Eq. (6) ] |
and subtracting the pair gives,
<math>~0</math> |
<math>~=</math> |
<math>~ \biggl[ 2 \mathfrak{T}_{23} - \Omega_2\Omega_3 I_{33} - 2\Omega_3 \int_V \rho u_1x_3 dx \biggr] - \biggl[2 \mathfrak{T}_{32} - \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \int_V \rho u_1x_2 dx \biggr] </math> |
|
<math>~=</math> |
<math>~ \Omega_2\Omega_3 (I_{22} - I_{33} ) - 2 \int_V \rho u_1 ( \Omega_2 x_2 + \Omega_3 x_3) dx \, . </math> |
[ EFE, Chapter 7, §47, Eq. (7) ] |
Various Degrees of Simplification
Riemann S-Type Ellipsoids
Describe …
Jacobi and Dedekind Ellipsoids
Describe …
Maclaurin Spheroids
Describe …
Appendices: Various Integrals Over Ellipsoid Volume
Throughout this set of appendices, we work with a uniform-density ellipsoid whose surface is defined by the expression,
<math>~1</math> |
<math>~=</math> |
<math>~ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} \, . </math> |
Appendix A: Volume
Here we seek to find the volume of the ellipsoid via the Cartesian integral expression,
<math>~V</math> |
<math>~=</math> |
<math>~ \iiint dx ~dy ~dz \, . </math> |
Preliminaries
First, we will integrate over <math>~x</math> and specify the integration limits via the expression,
<math>~x_\ell</math> |
<math>~\equiv</math> |
<math>~ a\biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} \, ; </math> |
second, we will integrate over <math>~z</math> and specify the integration limits via the expression,
<math>~z_\ell</math> |
<math>~\equiv</math> |
<math>~ c\biggl[ 1 - \frac{y^2}{b^2} \biggr]^{1 / 2} \, ; </math> |
third, we will integrate over <math>~y</math> and set the limits of integration as <math>~\pm b</math>.
Carry Out the Integration
Following thestrategy that has just been outlined, we have,
<math>~V</math> |
<math>~=</math> |
<math>~ \iint dy ~dz \int_{-x_\ell}^{+x_\ell} dx = \iint dy ~dz \biggl[ x \biggr]_{-x_\ell}^{+x_\ell} = 2\int dy \int x_\ell ~dz </math> |
|
<math>~=</math> |
<math>~ 2a\int dy \int \biggl[ 1 - \frac{y^2}{b^2} - \frac{z^2}{c^2} \biggr]^{1 / 2} dz = \frac{2a}{c} \int dy \int_{-z_\ell}^{+z_\ell} \biggl[ z_\ell^2- z^2 \biggr]^{1 / 2} dz </math> |
|
<math>~=</math> |
<math>~ \frac{2a}{c} \int \frac{dy}{2} \biggl[ z\sqrt{ z_\ell^2- z^2 } + z_\ell^2 \sin^{-1} \biggl( \frac{z}{|z_\ell |} \biggr) \biggr]_{-z_\ell}^{+z_\ell} </math> |
|
<math>~=</math> |
<math>~ \frac{2a}{c} \int \biggl[ z_\ell \cancelto{0}{\sqrt{ z_\ell^2- z_\ell^2 }} + z_\ell^2 \sin^{-1} \biggl(1\biggr) \biggr] dy = \frac{2a}{c} \int \biggl[ \frac{\pi}{2} z_\ell^2 \biggr] dy </math> |
|
<math>~=</math> |
<math>~ \pi a c \int_{-b}^{+b} \biggl( 1 - \frac{y^2}{b^2} \biggr) dy = \pi a c \biggl[ y - \frac{y^3}{3b^2} \biggr]_{-b}^{+b} </math> |
|
<math>~=</math> |
<math>~ \frac{4\pi}{3} \cdot a b c\, . </math> |
Appendix B: Coriolis Component u1x2
<math>~\iiint [u_1 y] ~dx ~dy ~dz</math> |
<math>~=</math> |
<math>~ \iiint \biggl\{ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 z \biggr\} y ~dx ~dy ~dz </math> |
|
<math>~=</math> |
<math>~ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \iiint y^2 ~dx ~dy ~dz + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \iiint yz ~dx ~dy ~dz </math> |
|
<math>~=</math> |
<math>~ - \biggl[ \frac{a^2}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \int dz \int_{-x_\ell}^{+x_\ell} dx + \biggl[ \frac{a^2}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \int z ~dz \int_{-x_\ell}^{+x_\ell} dx </math> |
|
<math>~=</math> |
<math>~ - \biggl[ \frac{2a^2}{a^2 + b^2}\biggr]\zeta_3 \int y^2 dy \int x_\ell dz + \biggl[ \frac{2a^2}{a^2+c^2}\biggr] \zeta_2 \int y ~dy \int z~x_\ell ~dz </math> |
See Also
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© 2014 - 2021 by Joel E. Tohline |