User:Tohline/VE/RiemannEllipsoids

Steady-State 2^nd-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, <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 2^nd-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 2^nd-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) ]

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

<math>~\int_V \rho u_i x_j dx</math>	<math>~=</math>	<math>~0</math>	if    <math>~i = j \, .</math>
[ EFE, Chapter 7, §47, Eq. (5) ]

Various Degrees of Simplification

Riemann S-Type Ellipsoids

Describe …

Jacobi and Dedekind Ellipsoids

Describe …

Maclaurin Spheroids

Describe …

See Also

• Properties of Maclaurin Spheroids
• Excerpts from Maclaurin's (1742) A Treatise of Fluxions

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation