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

From VistrailsWiki
Jump to navigation Jump to search
 
(23 intermediate revisions by the same user not shown)
Line 1: Line 1:
__FORCETOC__<!--  will force the creation of a Table of Contents -->
__FORCETOC__<!--  will force the creation of a Table of Contents -->
<!-- __NOTOC__ will force TOC off -->
<!-- __NOTOC__ will force TOC off -->
=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.
Line 272: Line 272:
</tr>
</tr>
</table>
</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>
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,
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 319: Line 335:
</tr>
</tr>
</table>
</table>
As has been summarized in an [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt2#Try_Tipped_Plane_Again|accompanying discussion]], we have determined that,
 
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="0" cellpadding="5" align="center">


Line 333: Line 350:
- \frac{\zeta_2}{\zeta_3} \biggl[ \frac{a^2 + b^2}{a^2 + c^2} \biggr]  \frac{c^2}{b^2}  
- \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}  \, ,
- \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>
</math>
   </td>
   </td>
Line 344: Line 585:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\beta</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 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>
   <td align="center">
   <td align="center">
Line 351: Line 708:
   <td align="left">
   <td align="left">
<math>~
<math>~
- \biggl[ \frac{c^2}{a^2 + c^2} \biggr] \frac{\zeta_2}{\Omega_2}
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>
   </td>
   </td>
<td align="center">&nbsp; &nbsp; &nbsp; and, &nbsp; &nbsp; &nbsp; </td>
</tr>
 
<tr>
   <td align="right">
   <td align="right">
<math>~\gamma</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 363: Line 726:
   <td align="left">
   <td align="left">
<math>~
<math>~
- \biggl[ \frac{b^2}{a^2 + b^2} \biggr] \frac{\zeta_3}{\Omega_3} \, .
-x \biggl\{ \zeta_2\biggl[ \frac{c^2}{a^2 + c^2} \biggr]  
\biggr\} \, .
</math>
</math>
   </td>
   </td>
</tr>
</tr>
</table>
</table>
Also,
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">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\biggl[\frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]^2</math>
<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>
   <td align="center">
   <td align="center">
<math>~=</math>
<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>
   <td align="left">
   <td align="left">
<math>~
<math>~
\frac{a^2}{b^2 c^2}  (c^2\cos^2\theta + b^2\sin^2\theta)  \, ,
\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>
   </td>
   </td>
Line 387: Line 840:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~{\dot\varphi}^2 </math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 394: Line 847:
   <td align="left">
   <td align="left">
<math>~
<math>~
\zeta_3^2\biggl[ \frac{b^2}{a^2 + b^2} \biggr]^2
\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\}
\biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr]^2 \biggl[1 + \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>
   </td>
   </td>
Line 402: Line 856:
<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~y_c</math>
&nbsp;
   </td>
   </td>
   <td align="center">
   <td align="center">
Line 408: Line 862:
   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~- \frac{z' b^2 \sin\theta\cos\theta}{c^2 \cos^2\theta + b^2\sin^2\theta}  
<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' \sin\theta \cos\theta \biggl[\frac{y_\mathrm{max}}{x_\mathrm{max}}  \biggr]^2 \frac{a^2}{c^2}
-
\, .</math>
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>


====Demonstration====
<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>
</tr>
</table>


=See Also=
=See Also=
Line 423: 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
|   Tiled Menu   |   Tables of Content   |  Banner Video   |  Tohline Home Page   |

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