Difference between revisions of "User:Tohline/ThreeDimensionalConfigurations/ChallengesPt3"

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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 [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt2|''(Pt. 2)'']].  The focus here is on firming up our understanding of the relationships between various "tilted" Cartesian coordinate frames.
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.
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==Various Coordinate Frames==
==Various Coordinate Frames==


<table border="1" align="right" cellpadding="8">
===Riemann-Derived 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">
<tr><td align="center">
''Inertial Frame'' (green with subscript "0") <br />and ''Body Frame'' (black and unsubscripted).
''Inertial Frame'' (green with subscript "0") <br />and ''Body Frame'' (black and unsubscripted).
Line 15: Line 17:
</tr>
</tr>
<tr>
<tr>
   <td align="center">[[File:InertialAxes04.png|450px|Inertial and Body Frames]]</td>
   <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>
</tr>
</table>
</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,
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,
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</tr>
</tr>
</table>
</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.  The transformation from one frame to the other is accomplished via the relations provided in the table that follows.  For example, as viewed from the ''body'' frame, we have,
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">
<table border="0" cellpadding="5" align="center">


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</tr>
</tr>
</table>
</table>
<table border="1" align="center" width="50%" cellpadding="8"><tr><td align="left">
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>


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,
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,
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</tr>
</tr>
</table>
</table>
where,
<span  id="betagamma">where,</span>


<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">
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</table>
</table>


<table border="1" cellpadding="8" width="90%" align="center"><tr><td align="left">
<table border="1" cellpadding="8" width="90%" align="center">
<tr><td align="left" colspan="2">
<div align="center">'''Rotating-Frame Vorticity'''</div>
<div align="center">'''Rotating-Frame Vorticity'''</div>
</td>
</tr>
<tr>
<td align="center">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">


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</table>
</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>~\tan^{-1}[\zeta_2/\zeta_3] = 4.0013 ~\mathrm{rad} = 229.26^\circ</math>.
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">
<tr>
<td align="center" colspan="3"><font color="maroon">'''Off-Center Ellipse'''</font></td>
</tr>
<tr>
  <td align="right">
<math>~1</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~\biggl[\frac{x'}{x_\mathrm{max}} \biggr]^2 + \biggl[\frac{y' - y_c(z')}{y_\mathrm{max}} \biggr]^2 \, .</math>
  </td>
</tr>
</table>


<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>
</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">
<tr>
  <td align="right">
<math>~x'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~x_\mathrm{max}\cos(\dot\varphi t)</math>
  </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  <td align="right">
<math>~y' - y_c</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~y_\mathrm{max}\sin(\dot\varphi t) \, ,</math>
  </td>
</tr>


<tr>
  <td align="right">
<math>~\dot{x}'</math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<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 align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
  <td align="right">
<math>~\dot{y}' </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <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>
  </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">


<table border="1" align="center" cellpadding="8" width="80%">
<tr>
<tr>
   <td align="center" colspan="2">
  <td align="right">
Transformation Between ''Inertial'' Frame and ''Body'' Frame
<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>
   </td>
</tr>
</tr>
<tr>
</table>
<td align="left">
where, <math>~\beta</math> and <math>~\gamma</math> are as [[#betagamma|defined above]].  Also,
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~x</math>
<math>~\biggl[\frac{x_\mathrm{max}}{y_\mathrm{max}}  \biggr]^2</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 261: Line 368:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~x_0</math>
<math>~
\frac{a^2}{b^2 c^2}  (c^2\cos^2\theta + b^2\sin^2\theta)
= 1.05238  \, ,
</math>
   </td>
   </td>
</tr>
</tr>
Line 267: Line 377:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~y</math>
<math>~{\dot\varphi}^2 </math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 273: Line 383:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~y_0\cos\delta + z_0\sin\delta</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>
   </td>
   </td>
</tr>
</tr>
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<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~z</math>
<math>~y_c</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
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   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~z_0\cos\delta - y_0\sin\delta</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>
  </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>
   </td>
</tr>
</tr>
</table>
</table>


</td>
</td></tr></table>
<td align="left">
 
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">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~x_0</math>
<math>~\boldsymbol{u'}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 302: Line 466:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~x</math>
<math>~
\boldsymbol{\hat\imath'} \dot{x}'
+
\boldsymbol{\hat\jmath'} \dot{y}'
</math>
   </td>
   </td>
</tr>
</tr>
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<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~y_0</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
<math>~=</math>
<math>~~~\rightarrow~~</math>
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~y\cos\delta - z\sin\delta</math>
<math>~
\boldsymbol{\hat\imath} \dot{x}'
+
[\boldsymbol{\hat\jmath}\cos\theta + \boldsymbol{\hat{k}}\sin\theta] \dot{y}'
</math>
   </td>
   </td>
</tr>
</tr>
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<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~z_0</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
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   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~z\cos\delta + y\sin\delta</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>
   </td>
   </td>
</tr>
</tr>
</table>
</table>


</td>
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>
</tr>
</table>
</td></tr></table>
we have,
<table border="0" cellpadding="5" align="center">
<tr>
<tr>
<td align="left">
  <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">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\boldsymbol{\hat\imath}</math>
<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>
   <td align="center">
   <td align="center">
Line 345: Line 673:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\boldsymbol{{\hat\imath}_0}</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>
   </td>
   </td>
</tr>
</tr>
Line 351: Line 685:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\boldsymbol{\hat\jmath}</math>
&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>
   <td align="center">
   <td align="center">
Line 357: Line 707:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\boldsymbol{{\hat\jmath}_0}\cos\delta + \boldsymbol{\hat{k}_0}\sin\delta</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>
   </td>
   </td>
</tr>
</tr>
Line 363: Line 719:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\boldsymbol{\hat{k}}</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 369: Line 725:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\boldsymbol{\hat{k}_0}\cos\delta - \boldsymbol{{\hat\jmath}_0}\sin\delta</math>
<math>~
-x \biggl\{ \zeta_2\biggl[ \frac{c^2}{a^2 + c^2} \biggr]
\biggr\} \, .
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</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,


</td>
<td align="left">
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\boldsymbol{{\hat\imath}_0 }</math>
<math>~\boldsymbol{\hat\imath}\cdot \boldsymbol{u'}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 386: Line 744:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\boldsymbol{\hat\imath }</math>
<math>~
\dot\varphi \biggl( \frac{x_\mathrm{max}}{y_\mathrm{max}}\biggr)\biggl\{y_c - y\cos\theta - z\sin\theta\biggr\}</math>
   </td>
   </td>
</tr>
</tr>
Line 392: Line 751:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\boldsymbol{{\hat\jmath}_0 }</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 398: Line 757:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\boldsymbol{\hat\jmath }\cos\delta - \boldsymbol{\hat{k} }\sin\delta</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>
   </td>
   </td>
</tr>
</tr>
Line 404: Line 770:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\boldsymbol{{\hat{k}}_0 }</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 410: Line 776:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~\boldsymbol{\hat\jmath }\sin\delta + \boldsymbol{\hat{k} }\cos\delta</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>
  </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>
   </td>
</tr>
</tr>
</table>


</td>
<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>
</tr>
</table>
</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">


==Motivation==
<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>


===Where Are We Headed?===
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="5" align="center">
<tr>
<tr>
<td align="center" colspan="3"><font color="maroon">'''Off-Center Ellipse'''</font></td>
  <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>
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~1</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 435: Line 862:
   </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>~
-~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>
   </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,
 
<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">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~x'</math>
<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>
   <td align="center">
   <td align="center">
Line 450: Line 928:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~x_\mathrm{max}\cos(\dot\varphi t)</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>
   </td>
   </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
</tr>
 
<tr>
   <td align="right">
   <td align="right">
<math>~y' - y_0</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 460: Line 946:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~y_\mathrm{max}\sin(\dot\varphi t) \, ,</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>
   </td>
   </td>
</tr>
</tr>
Line 466: Line 958:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\dot{x}'</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 472: Line 964:
   </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>~
-~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>
   </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp;
</tr>
 
<tr>
   <td align="right">
   <td align="right">
<math>~\dot{y}' </math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 482: Line 982:
   </td>
   </td>
   <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 \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 492: Line 998:
* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt2|Construction Challenges (Pt. 2)]]
* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt2|Construction Challenges (Pt. 2)]]
* [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt3|Construction Challenges (Pt. 3)]]
* [[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:
* Related discussions of models viewed from a rotating reference frame:
** [[User:Tohline/PGE/RotatingFrame#Rotating_Reference_Frame|PGE]]
** [[User:Tohline/PGE/RotatingFrame#Rotating_Reference_Frame|PGE]]

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