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(Steady-State 2nd-Order Tensor Virial Equations)
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{{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, <math>~a_1, a_2, a_3</math>.
-
[<b>[[User:Tohline/Appendix/References#EFE|<font color="red">EFE</font>]]</b> &sect;11(b), p. 22] <font color="#007700">Under conditions of a stationary state, [the tensor virial equation] gives,</font>
+
 
 +
==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, <math>~(a_1,a_2,a_3)</math>, of an homogeneous ellipsoid may be given analytically in terms of the following three coefficient expressions:
<div align="center">
<div align="center">
-
<table border="0" cellpadding="5" align="center">
+
<table align="center" border=0 cellpadding="3">
<tr>
<tr>
   <td align="right">
   <td align="right">
-
<math>~2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} </math>
+
<math>
 +
~A_1
 +
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
-
<math>~=</math>
+
<math>
 +
~=
 +
</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
-
<math>~- \delta_{ij}\Pi \, .</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>
   </td>
   </td>
</tr>
</tr>
-
</table>
 
-
</div>
 
-
<font color="#007700">[This] provides six integral relations which must obtain whenever the conditions are stationary</font>.
 
-
When viewing the (generally ellipsoidal) configuration from a rotating frame of reference, the 2<sup>nd</sup>-order TVE takes on the more general form:
 
-
<table border="0" cellpadding="5" align="center">
 
<tr>
<tr>
   <td align="right">
   <td align="right">
-
<math>~0</math>
+
<math>
 +
~A_3
 +
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
-
<math>~=</math>
+
<math>
 +
~=
 +
</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
-
<math>~
+
<math>
-
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
+
-
\, .
+
</math>
</math>
   </td>
   </td>
</tr>
</tr>
-
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 2, &sect;12, Eq. (64)</font> ]</td></tr>
 
-
</table>
 
-
 
-
EFE (p. 57) also shows that &hellip; <font color="#007700">The potential energy tensor &hellip; for a homogeneous ellipsoid is given by</font>
 
-
<table border="0" cellpadding="5" align="center">
 
<tr>
<tr>
   <td align="right">
   <td align="right">
-
<math>~\frac{\mathfrak{W}_{ij}}{\pi G\rho}</math>
+
<math>
 +
~A_2
 +
</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
-
<math>~=</math>
+
<math>
 +
~=
 +
</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
-
<math>~-2A_i I_{ij} \, ,</math>
+
<math>~2 - (A_1+A_3) \, ,</math>
   </td>
   </td>
</tr>
</tr>
-
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, &sect;22, Eq. (128)</font> ]</td></tr>
+
 
</table>
</table>
-
<font color="#007700">where</font>
+
</div>
 +
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,
 +
<div align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">
-
<math>~I_{ij}</math>
+
<math>~\theta = \cos^{-1} \biggl(\frac{a_3}{a_1} \biggr)</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
-
<math>~=</math>
+
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;
   </td>
   </td>
   <td align="left">
   <td align="left">
-
<math>~\tfrac{1}{5} Ma_i^2 \delta_{ij} \, ,</math>
+
<math>~k = \biggl[\frac{1 - (a_2/a_1)^2}{1 - (a_3/a_1)^2} \biggr]^{1/2} \, .</math>
   </td>
   </td>
</tr>
</tr>
-
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, &sect;22, Eq. (129)</font> ]</td></tr>
+
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, &sect;17, Eq. (32)</font> ]</td></tr>
</table>
</table>
-
 
+
</div>
-
<font color="#007700">is the moment of inertia tensor.</font>
+
==Adopted (Internal) Velocity Field==
==Adopted (Internal) Velocity Field==
-
EFE (p. 130) states that &hellip; <font color="#007700">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:</font>
+
EFE (p. 130) states that the &hellip; <font color="#007700">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:</font>
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
Line 125: Line 131:
</table>
</table>
-
 
+
==Equilibrium Expressions==
-
==General Coefficient Expressions==
+
[<b>[[User:Tohline/Appendix/References#EFE|<font color="red">EFE</font>]]</b> &sect;11(b), p. 22] <font color="#007700">Under conditions of a stationary state, [the tensor virial equation] gives,</font>
-
 
+
-
As has been detailed in an [[User:Tohline/ThreeDimensionalConfigurations/HomogeneousEllipsoids#Gravitational_Potential|accompanying chapter]], the gravitational potential anywhere inside or on the surface, <math>~(a_1,a_2,a_3) ~\leftrightarrow~(a,b,c)</math>, of an homogeneous ellipsoid may be given analytically in terms of the following three coefficient expressions:
+
<div align="center">
<div align="center">
-
<table align="center" border=0 cellpadding="3">
+
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">
-
<math>
+
<math>~2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} </math>
-
~A_1
+
-
</math>
+
   </td>
   </td>
   <td align="center">
   <td align="center">
-
<math>
+
<math>~=</math>
-
~=
+
-
</math>
+
   </td>
   </td>
   <td align="left">
   <td align="left">
-
<math>~2\biggl(\frac{b}{a}\biggr)\biggl(\frac{c}{a}\biggr)
+
<math>~- \delta_{ij}\Pi \, .</math>
-
\biggl[  \frac{F(\theta,k) - E(\theta,k)}{k^2 \sin^3\theta} \biggr] \, ,
+
-
</math>
+
   </td>
   </td>
</tr>
</tr>
 +
</table>
 +
</div>
 +
<font color="#007700">[This] provides six integral relations which must obtain whenever the conditions are stationary</font>.
 +
When viewing the (generally ellipsoidal) configuration from a rotating frame of reference, the 2<sup>nd</sup>-order TVE takes on the more general form:
 +
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">
-
<math>
+
<math>~0</math>
-
~A_3
+
-
</math>
+
   </td>
   </td>
   <td align="center">
   <td align="center">
-
<math>
+
<math>~=</math>
-
~=
+
-
</math>
+
   </td>
   </td>
   <td align="left">
   <td align="left">
-
<math>
+
<math>~
-
~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
 +
\, .
</math>
</math>
   </td>
   </td>
</tr>
</tr>
 +
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 2, &sect;12, Eq. (64)</font> ]</td></tr>
 +
</table>
 +
 +
EFE (p. 57) also shows that &hellip; <font color="#007700">The potential energy tensor &hellip; for a homogeneous ellipsoid is given by</font>
 +
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">
-
<math>
+
<math>~\frac{\mathfrak{W}_{ij}}{\pi G\rho}</math>
-
~A_2
+
-
</math>
+
   </td>
   </td>
   <td align="center">
   <td align="center">
-
<math>
+
<math>~=</math>
-
~=
+
-
</math>
+
   </td>
   </td>
   <td align="left">
   <td align="left">
-
<math>~2 - (A_1+A_3) \, ,</math>
+
<math>~-2A_i I_{ij} \, ,</math>
   </td>
   </td>
</tr>
</tr>
-
 
+
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, &sect;22, Eq. (128)</font> ]</td></tr>
</table>
</table>
-
</div>
+
<font color="#007700">where</font>
-
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,
+
-
<div align="center">
+
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
   <td align="right">
   <td align="right">
-
<math>~\theta = \cos^{-1} \biggl(\frac{c}{a} \biggr)</math>
+
<math>~I_{ij}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
-
&nbsp; &nbsp; &nbsp; and &nbsp; &nbsp; &nbsp;
+
<math>~=</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
-
<math>~k = \biggl[\frac{1 - (b/a)^2}{1 - (c/a)^2} \biggr]^{1/2} \, .</math>
+
<math>~\tfrac{1}{5} Ma_i^2 \delta_{ij} \, ,</math>
   </td>
   </td>
</tr>
</tr>
-
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, &sect;17, Eq. (32)</font> ]</td></tr>
+
<tr><td align="center" colspan="3">[ [[User:Tohline/Appendix/References#EFE|EFE]], <font color="#00CC00">Chapter 3, &sect;22, Eq. (129)</font> ]</td></tr>
</table>
</table>
-
</div>
+
 
 +
<font color="#007700">is the moment of inertia tensor.</font>
=Various Degrees of Simplification=
=Various Degrees of Simplification=

Revision as of 10:20, 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.

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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