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=Challenges Constructing Ellipsoidal-Like Configurations (Pt. 2)=
=Challenges Constructing Ellipsoidal-Like Configurations (Pt. 3)=


This chapter extends the accompanying chapters titled, [[User:Tohline/ThreeDimensionalConfigurations/Challenges|''Construction Challenges (Pt. 1)'']] and [[.  The focus here is on an SCF technique that will incorporate specification of a Lagrangian flow-flied.
This chapter extends the accompanying chapters titled, [[User:Tohline/ThreeDimensionalConfigurations/Challenges|''Construction Challenges (Pt. 1)'']] and [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt2|''(Pt. 2)'']].  The focus here is on firming up our understanding of the relationships between various "tilted" Cartesian coordinate frames.


{{LSU_HBook_header}}
{{LSU_HBook_header}}


==Motivation==
==Various Coordinate Frames==


===Where Are We Headed?===
===Riemann-Derived Expressions===
In a [[User:Tohline/ThreeDimensionalConfigurations/RiemannTypeI#Try_Again|separate discussion]], we have shown that, as viewed from a frame that "tumbles" with the (purple) body of a Type 1 Riemann ellipsoid, each Lagrangian fluid element moves along an elliptical path in a plane that is tipped by an angle <math>~\theta</math> about the x-axis of the body.  (See the yellow-dotted orbits in Figure panels 1a and 1b below).  As viewed from the (primed) coordinates associated with this tipped plane, by definition, z' = 0 and dz'/dt = 0, and the planar orbit is defined by the expression for an,
<table border="0" cellpadding="10" align="right" width="30%"><tr><td align="center">
<table border="1" align="center" cellpadding="8">
<tr><td align="center">
''Inertial Frame'' (green with subscript "0") <br />and ''Body Frame'' (black and unsubscripted).
</td>
</tr>
<tr>
  <td align="center">[[File:InertialAxes05.png|400px|Inertial and Body Frames]]</td>
</tr>
<tr><td align="center">
For our chosen [[User:Tohline/ThreeDimensionalConfigurations/RiemannTypeI#Case_I|Example Type I Ellipsoid]], we have, <math>~\Omega_2 = 0.3639</math> and <math>~\Omega_3 = 0.6633</math>, in which case, <math>~\Omega_0 = 0.7566</math> and <math>~\delta = 0.5018 ~\mathrm{rad} = 28.75^\circ</math>.
</td>
</tr>
</table>
</td></tr></table>
 
The purple (ellipsoidal) configuration is spinning with frequency, <math>~\Omega_0</math> about the <math>~z_0</math>-axis of the "inertial frame," as illustrated; that is,
<table border="0" cellpadding="5" align="center">
 
<tr>
  <td align="right">
<math>~\boldsymbol\Omega</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\boldsymbol{\hat{k}_0}\Omega_0 \, .</math>
  </td>
</tr>
</table>
Also as illustrated, the "body frame," which is attached to and aligned with the principal axes of the purple ellipsoid, is tilted at an angle, <math>~\delta</math>, with respect to the inertial frame.  Hence, as viewed from the ''body'' frame, we have,
<table border="0" cellpadding="5" align="center">
 
<tr>
  <td align="right">
<math>~\boldsymbol\Omega</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[ \boldsymbol{\hat\jmath }\sin\delta + \boldsymbol{\hat{k} }\cos\delta \biggr]\Omega_0 \, .</math>
  </td>
</tr>
</table>
 
Now, adhering to the notation used by [[User:Tohline/Appendix/References#EFE|[<font color="red">EFE</font>] ]] &#8212; see, for example, the first paragraph of &sect;51 (p. 156) &#8212; we should write,
<table border="0" cellpadding="5" align="center">
 
<tr>
  <td align="right">
<math>~\boldsymbol\Omega</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\boldsymbol{\hat\jmath }\Omega_2 + \boldsymbol{\hat{k} }\Omega_3
~~~~~\Rightarrow ~~~ \Omega_2 = \Omega_0\sin\delta
</math>&nbsp; &nbsp; and, &nbsp; &nbsp;
<math>~\Omega_3 = \Omega_0\cos\delta \, .</math>
  </td>
</tr>
</table>
This means that,
<table border="0" cellpadding="5" align="center">
 
<tr>
  <td align="right">
<math>~\Omega_0</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[\Omega_2^2 + \Omega_3^2 \biggr]^{1 / 2}
</math>&nbsp; &nbsp; and, &nbsp; &nbsp;
<math>~\delta = \tan^{-1}\biggl[ \frac{\Omega_2}{\Omega_3} \biggr] \, .</math>
  </td>
</tr>
</table>
 
As we have summarized in an [[User:Tohline/ThreeDimensionalConfigurations/RiemannTypeI#EFEvelocities|accompanying discussion]] of Riemann Type 1 ellipsoids, [[User:Tohline/Appendix/References#EFE|[<font color="red">EFE</font>] ]]  provides an expression for the velocity vector of each fluid element, given its  instantaneous ''body''-coordinate position (x, y, z) = (x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>) &#8212; see his Eq. (154), Chapter 7, &sect;51 (p. 156).  As viewed from the rotating frame of reference, the three component expressions are,
 
<table border="0" cellpadding="5" align="center">
 
<tr>
  <td align="right">
<math>~\dot{x} = u_1 = \boldsymbol{\hat\imath} \cdot \boldsymbol{u}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 y - \biggl(\frac{a}{c}\biggr)^2 \beta \Omega_2 z</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 z \, ,</math>
  </td>
</tr>
 
<tr>
  <td align="right">
<math>~\dot{y} = u_2 = \boldsymbol{\hat\jmath} \cdot \boldsymbol{u}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \gamma \Omega_3 x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~+\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3 x \, ,</math>
  </td>
</tr>
 
<tr>
  <td align="right">
<math>~\dot{z} = u_3 = \boldsymbol{\hat{k}} \cdot \boldsymbol{u}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~+ \beta \Omega_2 x</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~- \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 x \, ,</math>
  </td>
</tr>
</table>
<span  id="betagamma">where,</span>
 
<table border="0" cellpadding="5" align="center">
 
<tr>
  <td align="right">
<math>~\beta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \frac{\zeta_2}{\Omega_2}
</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
  <td align="right">
<math>~\gamma</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{\zeta_3}{\Omega_3} \, .
</math>
  </td>
</tr>
</table>
 
<table border="1" cellpadding="8" width="90%" align="center">
<tr><td align="left" colspan="2">
<div align="center">'''Rotating-Frame Vorticity'''</div>
</td>
</tr>
<tr>
<td align="center">
<table border="0" cellpadding="5" align="center">
 
<tr>
  <td align="right">
<math>~\boldsymbol{\zeta} \equiv \boldsymbol{\nabla \times}\bold{u}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath} \biggl[ \frac{\partial \dot{z} }{\partial y} - \frac{\partial \dot{y}}{\partial z} \biggr]
+ \boldsymbol{\hat\jmath} \biggl[ \frac{\partial \dot{x}}{\partial z} - \frac{\partial \dot{z}}{\partial x} \biggr]
+ \bold{\hat{k}} \biggl[ \frac{\partial \dot{y}}{\partial x} - \frac{\partial \dot{x}}{\partial y} \biggr]
</math>
  </td>
</tr>
 
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\jmath} \biggl\{
\biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 + \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2
\biggr\}
+ \bold{\hat{k}} \biggl\{
\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3 + \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3
\biggr\}
</math>
  </td>
</tr>
 
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\jmath} ~\zeta_2
+ \bold{\hat{k}} ~\zeta_3 \, .
</math>
  </td>
</tr>
</table>
 
For our chosen [[User:Tohline/ThreeDimensionalConfigurations/RiemannTypeI#Case_I|Example Type I Ellipsoid]], we have, <math>~\zeta_2 = -2.2794</math> and <math>~\Omega_3 = -1.9637</math>, in which case, <math>~\zeta_\mathrm{rot} = (\zeta_2^2 + \zeta_3^2)^{1 / 2} = 2.2794</math> and <math>~\xi \equiv \tan^{-1}[\zeta_2/\zeta_3] = 4.0013 ~\mathrm{rad} = 229.26^\circ</math>.
  </td>
  <td align="center">
[[File:VorticityAxis04.png|350px|center|Vorticity Axis]]
  </td>
</tr>
 
</table>
 
===Tipped Orbit Planes===
 
====Summary====
In a [[User:Tohline/ThreeDimensionalConfigurations/RiemannTypeI#Try_Again|separate discussion]], we have shown that, as viewed from a frame that "tumbles" with the (purple) body of a Type 1 Riemann ellipsoid, each Lagrangian fluid element moves along an elliptical path in a plane that is tipped by an angle <math>~\theta</math> about the x-axis of the body.  As viewed from the (primed) coordinates associated with this tipped plane, by definition, z' = constant and dz'/dt = 0, and the planar orbit is defined by the expression for an,
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
Line 23: Line 268:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\biggl(\frac{x'}{x_\mathrm{max}} \biggr)^2 + \biggl(\frac{y' - y_0}{y_\mathrm{max}} \biggr)^2 \, .</math>
<math>~\biggl[\frac{x'}{x_\mathrm{max}} \biggr]^2 + \biggl[\frac{y' - y_c(z')}{y_\mathrm{max}} \biggr]^2 \, .</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
As a function of time, the x'-y' coordinates and associated velocity components of each Lagrangian fluid element are given by the expressions,
 
<table border="0" cellpadding="10" align="right" width="30%"><tr><td align="center">
<table border="1" align="center" cellpadding="8">
<tr><td align="center">
''Tipped Orbit Frame'' (yellow, primed) <br />
</td>
</tr>
<tr>
  <td align="center">[[File:TippedAxes03.png|350px|Tipped Orbital Planes]]</td>
</tr>
<tr><td align="center">
Given that b/a = 1.25 and c/a = 0.4703 for our chosen [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt2#Example_Equilibrium_Model|Example Type I Ellipsoid]], we find that, <math>~\theta = - 0.3320 ~\mathrm{rad} = -19.02^\circ</math>.
</td>
</tr>
</table>
</td></tr></table>
Notice that the offset, <math>~y_c</math>, is a function of the tipped plane's vertical coordinate, <math>~z'</math>.  As a function of time, the x'-y' coordinates and associated velocity components of each Lagrangian fluid element are given by the expressions,
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">


Line 42: Line 303:
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
   <td align="right">
   <td align="right">
<math>~y' - y_0</math>
<math>~y' - y_c</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 60: Line 321:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~- x_\mathrm{max}~ \dot\varphi \cdot \sin(\dot\varphi t) = (y_0 - y') \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr] \dot\varphi </math>
<math>~- x_\mathrm{max}~ \dot\varphi \cdot \sin(\dot\varphi t) = (y_c - y') \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr] \dot\varphi </math>
   </td>
   </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
Line 71: Line 332:
   <td align="left">
   <td align="left">
<math>~y_\mathrm{max}~\dot\varphi \cdot \cos(\dot\varphi t) = x' \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr] \dot\varphi \, .</math>
<math>~y_\mathrm{max}~\dot\varphi \cdot \cos(\dot\varphi t) = x' \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr] \dot\varphi \, .</math>
  </td>
</tr>
</table>
As has been summarized in an [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt2#Try_Tipped_Plane_Again|accompanying discussion]], we have determined that (numerical value given for our chosen example Type I ellipsoid),
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\tan\theta</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
- \frac{\zeta_2}{\zeta_3} \biggl[ \frac{a^2 + b^2}{a^2 + c^2} \biggr]  \frac{c^2}{b^2}
=
- \frac{\beta \Omega_2}{\gamma \Omega_3} 
=
-0.34479\, ,
</math>
  </td>
</tr>
</table>
where, <math>~\beta</math> and <math>~\gamma</math> are as [[#betagamma|defined above]].  Also,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\biggl[\frac{x_\mathrm{max}}{y_\mathrm{max}}  \biggr]^2</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{a^2}{b^2 c^2}  (c^2\cos^2\theta + b^2\sin^2\theta)
= 1.05238  \, ,
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~{\dot\varphi}^2 </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\zeta_3^2\biggl[ \frac{b^2}{a^2 + b^2} \biggr]^2
\biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]^2 \biggl[1 + \tan^2\theta \biggr]
= 1.68818\, ,
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~y_c</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~+ \frac{z' b^2 \tan\theta}{c^2 \cos^2\theta + b^2\sin^2\theta}
=
+z' \tan\theta  \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}}  \biggr]^2 \frac{a^2}{c^2}
=
\biggl( \frac{z'}{ \cos\theta }\biggr)(-1.40038)
\, .</math>
  </td>
</tr>
</table>
Note that this last expression has been obtained by making the substitutions, <math>~y_0 \rightarrow y_c</math> and <math>~z_0 \rightarrow -z'/\cos\theta</math>, in the [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt2#OffCenter|accompanying derivation's expression]] for <math>~y_0</math>.
====Demonstration====
In order to transform a vector from the "tipped orbit" frame (primed coordinates) to the "body" frame (unprimed), we use the following mappings of the three unit vectors:
<table border="1" align="center" width="40%" cellpadding="8"><tr><td align="left">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\boldsymbol{\hat\imath'}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\boldsymbol{\hat\imath} \, ,</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\boldsymbol{\hat\jmath'}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~\boldsymbol{\hat\jmath}\cos\theta + \boldsymbol{\hat{k}}\sin\theta \, ,</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\boldsymbol{\hat{k}'}</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~-\boldsymbol{\hat\jmath}\sin\theta + \boldsymbol{\hat{k}}\cos\theta \, .</math>
  </td>
</tr>
</table>
</td></tr></table>
Given that, by design in our "tipped orbit" frame, there is no vertical motion &#8212; that is, <math>~\dot{z}' = 0</math> &#8212; mapping the (primed coordinate) velocity to the body (unprimed) coordinate is particularly straightforward.  Specifically,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\boldsymbol{u'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath'} \dot{x}'
+
\boldsymbol{\hat\jmath'} \dot{y}'
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~~~\rightarrow~~</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath} \dot{x}'
+
[\boldsymbol{\hat\jmath}\cos\theta + \boldsymbol{\hat{k}}\sin\theta] \dot{y}'
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath} \biggl\{
(y_c - y') \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr) \dot\varphi
\biggr\}
+
[\boldsymbol{\hat\jmath}\cos\theta + \boldsymbol{\hat{k}}\sin\theta] \biggl\{
x' \biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr) \dot\varphi
\biggr\} \, .
</math>
  </td>
</tr>
</table>
Recognizing, [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt2#Tipped_Orbital_Plane|as before]], that the relevant coordinate mapping is,
<table border="1" align="center" width="40%" cellpadding="8"><tr><td align="left">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~x'</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~x \, ,</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~y'</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~y\cos\theta + z\sin\theta \, ,</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~z'</math>
  </td>
  <td align="center">
<math>~\rightarrow</math>
  </td>
  <td align="left">
<math>~z\cos\theta - y\sin\theta \, ,</math>
  </td>
</tr>
</table>
</td></tr></table>
we have,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\boldsymbol{u'}</math>
  </td>
  <td align="center">
<math>~~~\rightarrow~~~</math>
  </td>
  <td align="left">
<math>~
\boldsymbol{\hat\imath}  \dot\varphi \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)\biggl\{y_c - y\cos\theta - z\sin\theta\biggr\}
+
\boldsymbol{\hat\jmath}  \dot\varphi
\biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr)
\biggr\{ x\cos\theta \biggr\}
+
\boldsymbol{\hat{k}}  \dot\varphi
\biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr)
\biggr\{ x\sin\theta \biggr\} \, ,
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~y_c</math>
  </td>
  <td align="center">
<math>~~~\rightarrow~~~</math>
  </td>
  <td align="left">
<math>~
+[z\cos\theta - y\sin\theta] \tan\theta  \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}}  \biggr]^2 \frac{a^2}{c^2}
\, .</math>
  </td>
</tr>
</table>
Written in terms of the "body" frame coordinates, therefore, the 2<sup>nd</sup> and 3<sup>rd</sup> components of this velocity vector are, respectively:
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\boldsymbol{\hat\jmath}\cdot \boldsymbol{u'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x \dot\varphi
\biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr)
\cos\theta
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x \biggl\{ \zeta_3^2\biggl[ \frac{b^2}{a^2 + b^2} \biggr]^2
\biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]^2 \biggl[1 + \tan^2\theta \biggr]
\biggr\}^{1 / 2}
\biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr)
\cos\theta
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x \biggl\{ \zeta_3 \biggl[ \frac{b^2}{a^2 + b^2} \biggr]
\biggr\} \, ,
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\boldsymbol{\hat{k}}\cdot \boldsymbol{u'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x \dot\varphi
\biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr)
\sin\theta
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x \biggl\{ \zeta_3^2\biggl[ \frac{b^2}{a^2 + b^2} \biggr]^2
\biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]^2 \biggl[1 + \tan^2\theta \biggr]
\biggr\}^{1 / 2}
\biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr)
\sin\theta
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x \biggl\{ \zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr]
\biggr\}
\tan\theta
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
x \biggl\{ \zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr]
\biggr\}
\biggl\{
- \frac{\zeta_2}{\zeta_3} \biggl[ \frac{a^2 + b^2}{a^2 + c^2} \biggr]  \frac{c^2}{b^2}
\biggr\}
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-x \biggl\{ \zeta_2\biggl[ \frac{c^2}{a^2 + c^2} \biggr]
\biggr\} \, .
</math>
  </td>
</tr>
</table>
These expressions perfectly match the body-coordinate expressions derived by Riemann (see [[#Riemann-Derived_Expressions|above]]) for, respectively, <math>~\dot{y}</math> and <math>~\dot{z}</math>.  The 1<sup>st</sup> component is,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\boldsymbol{\hat\imath}\cdot \boldsymbol{u'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\dot\varphi \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)\biggl\{y_c - y\cos\theta - z\sin\theta\biggr\}</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl\{ \zeta_3^2\biggl[ \frac{b^2}{a^2 + b^2} \biggr]^2
\biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]^2 \biggl[1 + \tan^2\theta \biggr]
\biggr\}^{1 / 2}
\biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)
\biggl\{y_c
- y\cos\theta - z\sin\theta\biggr\}
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr]
\biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2
\biggl\{\frac{y_c}{\cos\theta}
- y - z\tan\theta\biggr\}
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2 \biggl\{
\zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{y_c}{\cos\theta}
-~y\cdot \zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr]
+~ z\cdot \zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{\zeta_2}{\zeta_3} \biggl[ \frac{a^2 + b^2}{a^2 + c^2} \biggr]  \frac{c^2}{b^2}
\biggr\}
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2 \biggl\{
\zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{y_c}{\cos\theta}
-~y\cdot \zeta_3\biggl[ \frac{a^2}{a^2 + b^2} \biggr] \frac{b^2}{a^2}
+~ z\cdot \zeta_2\biggl[ \frac{a^2}{a^2 + c^2} \biggr]  \frac{c^2}{a^2}
\biggr\} \, .
</math>
  </td>
</tr>
</table>
So, implementing the mapping of <math>~y_c</math>, the first term inside the curly braces becomes,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{y_c}{\cos\theta}</math>
  </td>
  <td align="center">
<math>~~~\rightarrow~~~</math>
  </td>
  <td align="left">
<math>~
\frac{\zeta_3}{\cos\theta}\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \biggl\{
+[z\cos\theta - y\sin\theta] \tan\theta  \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}}  \biggr]^2 \frac{a^2}{c^2}
\biggr\}
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\zeta_3 \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}}  \biggr]^2 \frac{a^2}{c^2} \biggl\{ -y\tan^2\theta  \biggr\}
+
\zeta_3 \biggl[ \frac{b^2}{a^2 + b^2} \biggr]\tan\theta \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}}  \biggr]^2 \frac{a^2}{c^2}  \biggl\{ z \biggr\}
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~y \cdot \zeta_3 \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}}  \biggr]^2 \frac{a^2}{c^2} \biggl\{ \tan^2\theta  \biggr\}
-
z \cdot \zeta_2  \biggl[ \frac{c^2 }{a^2 + c^2} \biggr]  \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}}  \biggr]^2 \frac{a^2}{c^2}  \biggl\{ \tan^2\theta \biggr\}
</math>
  </td>
</tr>
<tr>
  <td align="right">
<math>~\Rightarrow ~~~ \biggl(\frac{x_\mathrm{max}}{y_\mathrm{max}}  \biggr)^2 \zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{y_c}{\cos\theta}</math>
  </td>
  <td align="center">
<math>~~~\rightarrow~~~</math>
  </td>
  <td align="left">
<math>~
-~y \cdot \zeta_3 \biggl[ \frac{b^2}{a^2 + b^2} \biggr]  \frac{a^2}{c^2} \biggl\{ \tan^2\theta  \biggr\} -
z \cdot \zeta_2  \biggl[ \frac{c^2 }{a^2 + c^2} \biggr]  \frac{a^2}{c^2}  \biggl\{ \tan^2\theta \biggr\}
</math>
  </td>
</tr>
</table>
<div align="left">
<math>
\biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr]^2~=~\frac{a^2}{b^2c^2}  (c^2\cos^2\theta + b^2\sin^2\theta)
</math>
<math>
\biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr]^2 \biggl[ 1 + \tan^2\theta \biggr]~=~\frac{a^2}{b^2c^2}  (c^2 + b^2\tan^2\theta)
</math>
</div>
Therefore,
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\boldsymbol{\hat\imath}\cdot \boldsymbol{u'}</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~y \cdot \zeta_3 \biggl[ \frac{b^2}{a^2 + b^2} \biggr]  \frac{a^2}{c^2} \biggl\{ \tan^2\theta  \biggr\} -
z \cdot \zeta_2  \biggl[ \frac{c^2 }{a^2 + c^2} \biggr]  \frac{a^2}{c^2}  \biggl\{ \tan^2\theta \biggr\}
~+~
\biggl\{
z\cdot \zeta_2\biggl[ \frac{c^2}{a^2 + c^2} \biggr] 
-~y\cdot \zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr]
\biggr\} \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~y \cdot \zeta_3 \biggl[ \frac{b^2}{a^2 + b^2} \biggr]  \frac{a^2}{c^2} \biggl\{ \tan^2\theta  \biggr\}
-~y\cdot \zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2
~+~
z\cdot \zeta_2\biggl[ \frac{c^2}{a^2 + c^2} \biggr]  \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2
-~z \cdot \zeta_2  \biggl[ \frac{c^2 }{a^2 + c^2} \biggr]  \frac{a^2}{c^2}  \biggl\{ \tan^2\theta \biggr\}
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~y \cdot \zeta_3 \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \biggl\{ \frac{b^2}{c^2} \cdot \tan^2\theta
+\frac{b^2}{a^2} \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2 \biggr\}
~+~
z\cdot \zeta_2\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \biggl\{  \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2
-~ \frac{a^2}{c^2}  \cdot \tan^2\theta \biggr\}
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~y \cdot \zeta_3 \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \biggl\{ \frac{b^2}{c^2} \cdot \tan^2\theta
+ \frac{1}{c^2}  (c^2\cos^2\theta + b^2\sin^2\theta) \biggr\}
~+~
z\cdot \zeta_2\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \biggl\{  \frac{a^2}{b^2c^2}  (c^2\cos^2\theta + b^2\sin^2\theta)
-~ \frac{a^2}{c^2}  \cdot \tan^2\theta \biggr\}
</math>
  </td>
</tr>
<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
-~y \cdot \zeta_3 \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \frac{1}{c^2\cos^2\theta}\biggl\{b^2 \sin^2\theta
+ (c^2\cos^2\theta + b^2\sin^2\theta)\cos^2\theta \biggr\}
~+~
z\cdot \zeta_2\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \biggl\{  \frac{a^2}{b^2c^2}  (c^2\cos^2\theta + b^2\sin^2\theta)
-~ \frac{a^2}{c^2}  \cdot \tan^2\theta \biggr\}
</math>
   </td>
   </td>
</tr>
</tr>
Line 76: Line 994:


=See Also=
=See Also=
* [[User:Tohline/ThreeDimensionalConfigurations/Challenges|Construction Challenges (Pt. 1)]]
* [[User:Tohline/ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids|Riemann Type 1 Ellipsoids]]
* [[User:Tohline/ThreeDimensionalConfigurations/Challenges#Challenges_Constructing_Ellipsoidal-Like_Configurations|Construction Challenges (Pt. 1)]]
* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt2|Construction Challenges (Pt. 2)]]
* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt2|Construction Challenges (Pt. 2)]]
* [[User:Tohline/ThreeDimensionalConfigurations/RiemannTypeI#Riemann_Type_1_Ellipsoids|Riemann Type 1 Ellipsoids]]
* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt3|Construction Challenges (Pt. 3)]]
* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt4|Construction Challenges (Pt. 4)]]
* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt5|Construction Challenges (Pt. 5)]]
* Related discussions of models viewed from a rotating reference frame:
** [[User:Tohline/PGE/RotatingFrame#Rotating_Reference_Frame|PGE]]
** <font color="red"><b>NOTE to Eric Hirschmann &amp; David Neilsen...  </b></font>I have moved the earlier contents of this page to a new Wiki location called [[User:Tohline/Apps/RiemannEllipsoids_Compressible|Compressible Riemann Ellipsoids]].




{{LSU_HBook_footer}}
{{LSU_HBook_footer}}

Latest revision as of 21:51, 11 May 2021

Challenges Constructing Ellipsoidal-Like Configurations (Pt. 3)

This chapter extends the accompanying chapters titled, Construction Challenges (Pt. 1) and (Pt. 2). The focus here is on firming up our understanding of the relationships between various "tilted" Cartesian coordinate frames.

Whitworth's (1981) Isothermal Free-Energy Surface
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Various Coordinate Frames

Riemann-Derived Expressions

Inertial Frame (green with subscript "0")
and Body Frame (black and unsubscripted).

Inertial and Body Frames

For our chosen Example Type I Ellipsoid, we have, <math>~\Omega_2 = 0.3639</math> and <math>~\Omega_3 = 0.6633</math>, in which case, <math>~\Omega_0 = 0.7566</math> and <math>~\delta = 0.5018 ~\mathrm{rad} = 28.75^\circ</math>.

The purple (ellipsoidal) configuration is spinning with frequency, <math>~\Omega_0</math> about the <math>~z_0</math>-axis of the "inertial frame," as illustrated; that is,

<math>~\boldsymbol\Omega</math>

<math>~=</math>

<math>~\boldsymbol{\hat{k}_0}\Omega_0 \, .</math>

Also as illustrated, the "body frame," which is attached to and aligned with the principal axes of the purple ellipsoid, is tilted at an angle, <math>~\delta</math>, with respect to the inertial frame. Hence, as viewed from the body frame, we have,

<math>~\boldsymbol\Omega</math>

<math>~=</math>

<math>~\biggl[ \boldsymbol{\hat\jmath }\sin\delta + \boldsymbol{\hat{k} }\cos\delta \biggr]\Omega_0 \, .</math>

Now, adhering to the notation used by [EFE] — see, for example, the first paragraph of §51 (p. 156) — we should write,

<math>~\boldsymbol\Omega</math>

<math>~=</math>

<math>~\boldsymbol{\hat\jmath }\Omega_2 + \boldsymbol{\hat{k} }\Omega_3 ~~~~~\Rightarrow ~~~ \Omega_2 = \Omega_0\sin\delta </math>    and,     <math>~\Omega_3 = \Omega_0\cos\delta \, .</math>

This means that,

<math>~\Omega_0</math>

<math>~=</math>

<math>~ \biggl[\Omega_2^2 + \Omega_3^2 \biggr]^{1 / 2} </math>    and,     <math>~\delta = \tan^{-1}\biggl[ \frac{\Omega_2}{\Omega_3} \biggr] \, .</math>

As we have summarized in an accompanying discussion of Riemann Type 1 ellipsoids, [EFE] provides an expression for the velocity vector of each fluid element, given its instantaneous body-coordinate position (x, y, z) = (x1, x2, x3) — see his Eq. (154), Chapter 7, §51 (p. 156). As viewed from the rotating frame of reference, the three component expressions are,

<math>~\dot{x} = u_1 = \boldsymbol{\hat\imath} \cdot \boldsymbol{u}</math>

<math>~=</math>

<math>~\biggl(\frac{a}{b}\biggr)^2 \gamma \Omega_3 y - \biggl(\frac{a}{c}\biggr)^2 \beta \Omega_2 z</math>

<math>~=</math>

<math>~- \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 y + \biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 z \, ,</math>

<math>~\dot{y} = u_2 = \boldsymbol{\hat\jmath} \cdot \boldsymbol{u}</math>

<math>~=</math>

<math>~- \gamma \Omega_3 x</math>

<math>~=</math>

<math>~+\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3 x \, ,</math>

<math>~\dot{z} = u_3 = \boldsymbol{\hat{k}} \cdot \boldsymbol{u}</math>

<math>~=</math>

<math>~+ \beta \Omega_2 x</math>

<math>~=</math>

<math>~- \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 x \, ,</math>

where,

<math>~\beta</math>

<math>~=</math>

<math>~ - \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \frac{\zeta_2}{\Omega_2} </math>

      and,      

<math>~\gamma</math>

<math>~=</math>

<math>~ - \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{\zeta_3}{\Omega_3} \, . </math>

Rotating-Frame Vorticity

<math>~\boldsymbol{\zeta} \equiv \boldsymbol{\nabla \times}\bold{u}</math>

<math>~=</math>

<math>~ \boldsymbol{\hat\imath} \biggl[ \frac{\partial \dot{z} }{\partial y} - \frac{\partial \dot{y}}{\partial z} \biggr] + \boldsymbol{\hat\jmath} \biggl[ \frac{\partial \dot{x}}{\partial z} - \frac{\partial \dot{z}}{\partial x} \biggr] + \bold{\hat{k}} \biggl[ \frac{\partial \dot{y}}{\partial x} - \frac{\partial \dot{x}}{\partial y} \biggr] </math>

 

<math>~=</math>

<math>~ \boldsymbol{\hat\jmath} \biggl\{ \biggl[ \frac{a^2}{a^2 + c^2} \biggr] \zeta_2 + \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \zeta_2 \biggr\} + \bold{\hat{k}} \biggl\{ \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \zeta_3 + \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \zeta_3 \biggr\} </math>

 

<math>~=</math>

<math>~ \boldsymbol{\hat\jmath} ~\zeta_2 + \bold{\hat{k}} ~\zeta_3 \, . </math>

For our chosen Example Type I Ellipsoid, we have, <math>~\zeta_2 = -2.2794</math> and <math>~\Omega_3 = -1.9637</math>, in which case, <math>~\zeta_\mathrm{rot} = (\zeta_2^2 + \zeta_3^2)^{1 / 2} = 2.2794</math> and <math>~\xi \equiv \tan^{-1}[\zeta_2/\zeta_3] = 4.0013 ~\mathrm{rad} = 229.26^\circ</math>.

Vorticity Axis

Tipped Orbit Planes

Summary

In a separate discussion, we have shown that, as viewed from a frame that "tumbles" with the (purple) body of a Type 1 Riemann ellipsoid, each Lagrangian fluid element moves along an elliptical path in a plane that is tipped by an angle <math>~\theta</math> about the x-axis of the body. As viewed from the (primed) coordinates associated with this tipped plane, by definition, z' = constant and dz'/dt = 0, and the planar orbit is defined by the expression for an,

Off-Center Ellipse

<math>~1</math>

<math>~=</math>

<math>~\biggl[\frac{x'}{x_\mathrm{max}} \biggr]^2 + \biggl[\frac{y' - y_c(z')}{y_\mathrm{max}} \biggr]^2 \, .</math>

Tipped Orbit Frame (yellow, primed)

Tipped Orbital Planes

Given that b/a = 1.25 and c/a = 0.4703 for our chosen Example Type I Ellipsoid, we find that, <math>~\theta = - 0.3320 ~\mathrm{rad} = -19.02^\circ</math>.

Notice that the offset, <math>~y_c</math>, is a function of the tipped plane's vertical coordinate, <math>~z'</math>. As a function of time, the x'-y' coordinates and associated velocity components of each Lagrangian fluid element are given by the expressions,

<math>~x'</math>

<math>~=</math>

<math>~x_\mathrm{max}\cos(\dot\varphi t)</math>

      and,      

<math>~y' - y_c</math>

<math>~=</math>

<math>~y_\mathrm{max}\sin(\dot\varphi t) \, ,</math>

<math>~\dot{x}'</math>

<math>~=</math>

<math>~- x_\mathrm{max}~ \dot\varphi \cdot \sin(\dot\varphi t) = (y_c - y') \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr] \dot\varphi </math>

      and,      

<math>~\dot{y}' </math>

<math>~=</math>

<math>~y_\mathrm{max}~\dot\varphi \cdot \cos(\dot\varphi t) = x' \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr] \dot\varphi \, .</math>

As has been summarized in an accompanying discussion, we have determined that (numerical value given for our chosen example Type I ellipsoid),

<math>~\tan\theta</math>

<math>~=</math>

<math>~ - \frac{\zeta_2}{\zeta_3} \biggl[ \frac{a^2 + b^2}{a^2 + c^2} \biggr] \frac{c^2}{b^2} = - \frac{\beta \Omega_2}{\gamma \Omega_3} = -0.34479\, , </math>

where, <math>~\beta</math> and <math>~\gamma</math> are as defined above. Also,

<math>~\biggl[\frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]^2</math>

<math>~=</math>

<math>~ \frac{a^2}{b^2 c^2} (c^2\cos^2\theta + b^2\sin^2\theta) = 1.05238 \, , </math>

<math>~{\dot\varphi}^2 </math>

<math>~=</math>

<math>~ \zeta_3^2\biggl[ \frac{b^2}{a^2 + b^2} \biggr]^2 \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]^2 \biggl[1 + \tan^2\theta \biggr] = 1.68818\, , </math>

<math>~y_c</math>

<math>~=</math>

<math>~+ \frac{z' b^2 \tan\theta}{c^2 \cos^2\theta + b^2\sin^2\theta} = +z' \tan\theta \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]^2 \frac{a^2}{c^2} = \biggl( \frac{z'}{ \cos\theta }\biggr)(-1.40038) \, .</math>

Note that this last expression has been obtained by making the substitutions, <math>~y_0 \rightarrow y_c</math> and <math>~z_0 \rightarrow -z'/\cos\theta</math>, in the accompanying derivation's expression for <math>~y_0</math>.

Demonstration

In order to transform a vector from the "tipped orbit" frame (primed coordinates) to the "body" frame (unprimed), we use the following mappings of the three unit vectors:

<math>~\boldsymbol{\hat\imath'}</math>

<math>~\rightarrow</math>

<math>~\boldsymbol{\hat\imath} \, ,</math>

<math>~\boldsymbol{\hat\jmath'}</math>

<math>~\rightarrow</math>

<math>~\boldsymbol{\hat\jmath}\cos\theta + \boldsymbol{\hat{k}}\sin\theta \, ,</math>

<math>~\boldsymbol{\hat{k}'}</math>

<math>~\rightarrow</math>

<math>~-\boldsymbol{\hat\jmath}\sin\theta + \boldsymbol{\hat{k}}\cos\theta \, .</math>

Given that, by design in our "tipped orbit" frame, there is no vertical motion — that is, <math>~\dot{z}' = 0</math> — mapping the (primed coordinate) velocity to the body (unprimed) coordinate is particularly straightforward. Specifically,

<math>~\boldsymbol{u'}</math>

<math>~=</math>

<math>~ \boldsymbol{\hat\imath'} \dot{x}' + \boldsymbol{\hat\jmath'} \dot{y}' </math>

 

<math>~~~\rightarrow~~</math>

<math>~ \boldsymbol{\hat\imath} \dot{x}' + [\boldsymbol{\hat\jmath}\cos\theta + \boldsymbol{\hat{k}}\sin\theta] \dot{y}' </math>

 

<math>~=</math>

<math>~ \boldsymbol{\hat\imath} \biggl\{ (y_c - y') \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr) \dot\varphi \biggr\} + [\boldsymbol{\hat\jmath}\cos\theta + \boldsymbol{\hat{k}}\sin\theta] \biggl\{ x' \biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr) \dot\varphi \biggr\} \, . </math>

Recognizing, as before, that the relevant coordinate mapping is,

<math>~x'</math>

<math>~\rightarrow</math>

<math>~x \, ,</math>

<math>~y'</math>

<math>~\rightarrow</math>

<math>~y\cos\theta + z\sin\theta \, ,</math>

<math>~z'</math>

<math>~\rightarrow</math>

<math>~z\cos\theta - y\sin\theta \, ,</math>

we have,

<math>~\boldsymbol{u'}</math>

<math>~~~\rightarrow~~~</math>

<math>~ \boldsymbol{\hat\imath} \dot\varphi \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)\biggl\{y_c - y\cos\theta - z\sin\theta\biggr\} + \boldsymbol{\hat\jmath} \dot\varphi \biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr) \biggr\{ x\cos\theta \biggr\} + \boldsymbol{\hat{k}} \dot\varphi \biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr) \biggr\{ x\sin\theta \biggr\} \, , </math>

where,

<math>~y_c</math>

<math>~~~\rightarrow~~~</math>

<math>~ +[z\cos\theta - y\sin\theta] \tan\theta \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]^2 \frac{a^2}{c^2} \, .</math>

Written in terms of the "body" frame coordinates, therefore, the 2nd and 3rd components of this velocity vector are, respectively:

<math>~\boldsymbol{\hat\jmath}\cdot \boldsymbol{u'}</math>

<math>~=</math>

<math>~ x \dot\varphi \biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr) \cos\theta </math>

 

<math>~=</math>

<math>~ x \biggl\{ \zeta_3^2\biggl[ \frac{b^2}{a^2 + b^2} \biggr]^2 \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]^2 \biggl[1 + \tan^2\theta \biggr] \biggr\}^{1 / 2} \biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr) \cos\theta </math>

 

<math>~=</math>

<math>~ x \biggl\{ \zeta_3 \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \biggr\} \, , </math>

<math>~\boldsymbol{\hat{k}}\cdot \boldsymbol{u'}</math>

<math>~=</math>

<math>~ x \dot\varphi \biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr) \sin\theta </math>

 

<math>~=</math>

<math>~ x \biggl\{ \zeta_3^2\biggl[ \frac{b^2}{a^2 + b^2} \biggr]^2 \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]^2 \biggl[1 + \tan^2\theta \biggr] \biggr\}^{1 / 2} \biggl( \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr) \sin\theta </math>

 

<math>~=</math>

<math>~ x \biggl\{ \zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \biggr\} \tan\theta </math>

 

<math>~=</math>

<math>~ x \biggl\{ \zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \biggr\} \biggl\{ - \frac{\zeta_2}{\zeta_3} \biggl[ \frac{a^2 + b^2}{a^2 + c^2} \biggr] \frac{c^2}{b^2} \biggr\} </math>

 

<math>~=</math>

<math>~ -x \biggl\{ \zeta_2\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \biggr\} \, . </math>

These expressions perfectly match the body-coordinate expressions derived by Riemann (see above) for, respectively, <math>~\dot{y}</math> and <math>~\dot{z}</math>. The 1st component is,

<math>~\boldsymbol{\hat\imath}\cdot \boldsymbol{u'}</math>

<math>~=</math>

<math>~ \dot\varphi \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)\biggl\{y_c - y\cos\theta - z\sin\theta\biggr\}</math>

 

<math>~=</math>

<math>~ \biggl\{ \zeta_3^2\biggl[ \frac{b^2}{a^2 + b^2} \biggr]^2 \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]^2 \biggl[1 + \tan^2\theta \biggr] \biggr\}^{1 / 2} \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr) \biggl\{y_c - y\cos\theta - z\sin\theta\biggr\} </math>

 

<math>~=</math>

<math>~ \zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2 \biggl\{\frac{y_c}{\cos\theta} - y - z\tan\theta\biggr\} </math>

 

<math>~=</math>

<math>~\biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2 \biggl\{ \zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{y_c}{\cos\theta} -~y\cdot \zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr] +~ z\cdot \zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{\zeta_2}{\zeta_3} \biggl[ \frac{a^2 + b^2}{a^2 + c^2} \biggr] \frac{c^2}{b^2} \biggr\} </math>

 

<math>~=</math>

<math>~\biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2 \biggl\{ \zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{y_c}{\cos\theta} -~y\cdot \zeta_3\biggl[ \frac{a^2}{a^2 + b^2} \biggr] \frac{b^2}{a^2} +~ z\cdot \zeta_2\biggl[ \frac{a^2}{a^2 + c^2} \biggr] \frac{c^2}{a^2} \biggr\} \, . </math>

So, implementing the mapping of <math>~y_c</math>, the first term inside the curly braces becomes,

<math>~\zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{y_c}{\cos\theta}</math>

<math>~~~\rightarrow~~~</math>

<math>~ \frac{\zeta_3}{\cos\theta}\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \biggl\{ +[z\cos\theta - y\sin\theta] \tan\theta \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]^2 \frac{a^2}{c^2} \biggr\} </math>

 

<math>~=</math>

<math>~ \zeta_3 \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]^2 \frac{a^2}{c^2} \biggl\{ -y\tan^2\theta \biggr\} + \zeta_3 \biggl[ \frac{b^2}{a^2 + b^2} \biggr]\tan\theta \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]^2 \frac{a^2}{c^2} \biggl\{ z \biggr\} </math>

 

<math>~=</math>

<math>~ -~y \cdot \zeta_3 \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]^2 \frac{a^2}{c^2} \biggl\{ \tan^2\theta \biggr\} - z \cdot \zeta_2 \biggl[ \frac{c^2 }{a^2 + c^2} \biggr] \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}} \biggr]^2 \frac{a^2}{c^2} \biggl\{ \tan^2\theta \biggr\} </math>

<math>~\Rightarrow ~~~ \biggl(\frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr)^2 \zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{y_c}{\cos\theta}</math>

<math>~~~\rightarrow~~~</math>

<math>~ -~y \cdot \zeta_3 \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{a^2}{c^2} \biggl\{ \tan^2\theta \biggr\} - z \cdot \zeta_2 \biggl[ \frac{c^2 }{a^2 + c^2} \biggr] \frac{a^2}{c^2} \biggl\{ \tan^2\theta \biggr\} </math>

<math> \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr]^2~=~\frac{a^2}{b^2c^2} (c^2\cos^2\theta + b^2\sin^2\theta) </math>

<math> \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr]^2 \biggl[ 1 + \tan^2\theta \biggr]~=~\frac{a^2}{b^2c^2} (c^2 + b^2\tan^2\theta) </math>

Therefore,

<math>~\boldsymbol{\hat\imath}\cdot \boldsymbol{u'}</math>

<math>~=</math>

<math>~ -~y \cdot \zeta_3 \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{a^2}{c^2} \biggl\{ \tan^2\theta \biggr\} - z \cdot \zeta_2 \biggl[ \frac{c^2 }{a^2 + c^2} \biggr] \frac{a^2}{c^2} \biggl\{ \tan^2\theta \biggr\} ~+~ \biggl\{ z\cdot \zeta_2\biggl[ \frac{c^2}{a^2 + c^2} \biggr] -~y\cdot \zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \biggr\} \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2 </math>

 

<math>~=</math>

<math>~ -~y \cdot \zeta_3 \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{a^2}{c^2} \biggl\{ \tan^2\theta \biggr\} -~y\cdot \zeta_3\biggl[ \frac{b^2}{a^2 + b^2} \biggr] \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2 ~+~ z\cdot \zeta_2\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2 -~z \cdot \zeta_2 \biggl[ \frac{c^2 }{a^2 + c^2} \biggr] \frac{a^2}{c^2} \biggl\{ \tan^2\theta \biggr\} </math>

 

<math>~=</math>

<math>~ -~y \cdot \zeta_3 \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \biggl\{ \frac{b^2}{c^2} \cdot \tan^2\theta +\frac{b^2}{a^2} \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2 \biggr\} ~+~ z\cdot \zeta_2\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \biggl\{ \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)^2 -~ \frac{a^2}{c^2} \cdot \tan^2\theta \biggr\} </math>

 

<math>~=</math>

<math>~ -~y \cdot \zeta_3 \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \biggl\{ \frac{b^2}{c^2} \cdot \tan^2\theta + \frac{1}{c^2} (c^2\cos^2\theta + b^2\sin^2\theta) \biggr\} ~+~ z\cdot \zeta_2\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \biggl\{ \frac{a^2}{b^2c^2} (c^2\cos^2\theta + b^2\sin^2\theta) -~ \frac{a^2}{c^2} \cdot \tan^2\theta \biggr\} </math>

 

<math>~=</math>

<math>~ -~y \cdot \zeta_3 \biggl[ \frac{a^2}{a^2 + b^2} \biggr] \frac{1}{c^2\cos^2\theta}\biggl\{b^2 \sin^2\theta + (c^2\cos^2\theta + b^2\sin^2\theta)\cos^2\theta \biggr\} ~+~ z\cdot \zeta_2\biggl[ \frac{c^2}{a^2 + c^2} \biggr] \biggl\{ \frac{a^2}{b^2c^2} (c^2\cos^2\theta + b^2\sin^2\theta) -~ \frac{a^2}{c^2} \cdot \tan^2\theta \biggr\} </math>

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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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