Difference between revisions of "User:Tohline/Appendix/Ramblings/ConcentricEllipsodalCoordinates"

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


<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\biggl[ \biggl( \frac{x}{\lambda_1} \biggr)^2
+ \biggl( \frac{q^2y}{\lambda_1} \biggr)^2
+ \biggl(\frac{p^2 z}{\lambda_1} \biggr)^2 \biggr]^{1 / 2}
</math>
  </td>
</tr>


<tr>
  <td align="right">
&nbsp;
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~
\frac{1}{\lambda_1 \ell_{3D}}
</math>
  </td>
</tr>
</table>
where,
<table border="0" cellpadding="5" align="center">


The properly normalized
<tr>
  <td align="right">
<math>~\ell_{3D}</math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~\biggl[ x^2 + q^4y^2 + p^4 z^2 \biggr]^{- 1 / 2} \, .</math>
  </td>
</tr>
</table>


Next, we appreciate that the vector that is normal to theWhat is the expression for the unit vector normal to the surface at <math>~(x, y, z)</math> when written in terms of Cartesian unit vectors?
It is therefore clear that the ''properly normalized'' normal unit vector that should be associated with any <math>~\lambda_1</math> = constant ellipsoidal surface is,
<table border="0" cellpadding="5" align="center">


Well, to start with we know that <math>~\lambda_1^2</math> is constant across the entire surface, so at any point on this specified surface we must find,
<tr>
  <td align="right">
<math>~\hat{e}_1 </math>
  </td>
  <td align="center">
<math>~\equiv</math>
  </td>
  <td align="left">
<math>~
\frac{ \bold\hat{n} }{ [ \bold{\hat{n}} \cdot \bold{\hat{n}} ]^{1 / 2} }
=
\hat\imath (x \ell_{3D}) + \hat\jmath (q^2y \ell_{3D}) + \hat\jmath (p^2 z \ell_{3D}) \, .
</math>
  </td>
</tr>
</table>
From our [[User:Tohline/Appendix/Ramblings/DirectionCosines#Scale_Factors|accompanying discussion of direction cosines]], it is clear, as well, that the scale factor associated with the <math>~\lambda_1</math> coordinate is,
<table border="0" cellpadding="5" align="center">
<table border="0" cellpadding="5" align="center">


<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~0</math>
<math>~h_1^2</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
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   </td>
   </td>
   <td align="left">
   <td align="left">
<math>~2x dx + 2q^2y dy + 2p^2z dz \, .</math>
<math>~\lambda_1^2 \ell_{3D}^2 \, .</math>
  </td>
</tr>
</table>
We can also fill in the top line of our direction-cosines table, namely,
 
 
<table border="1" cellpadding="8" align="center" width="60%">
<tr>
  <td align="center" colspan="4">
'''Direction Cosines for T6 Coordinates'''
<br />
<math>~\gamma_{ni} = h_n \biggl( \frac{\partial \lambda_n}{\partial x_i}\biggr)</math>
  </td>
</tr>
<tr>
  <td align="center" width="10%"><math>~n</math></td>
  <td align="center" colspan="3"><math>~i = x, y, z</math>
</tr>
<tr>
  <td align="center"><math>~1</math></td>
  <td align="center">&nbsp;<br />
<math>~x\ell_{3D}</math><br />&nbsp;
  <td align="center"><math>~q^2 y \ell_{3D}</math>
  <td align="center"><math>~p^2 z \ell_{3D}</math>
</tr>
<tr>
  <td align="center"><math>~2</math></td>
  <td align="center">
&nbsp;<br />
---
<br />&nbsp;
  <td align="center">
&nbsp;<br />
---
<br />&nbsp;
  <td align="center">
&nbsp;<br />
---
<br />&nbsp;
  </td>
</tr>
<tr>
  <td align="center"><math>~3</math></td>
  <td align="center">
&nbsp;<br />
---
<br />&nbsp;
  </td>
  <td align="center">
&nbsp;<br />
---
<br />&nbsp;
  </td>
  <td align="center">
&nbsp;<br />
---
<br />&nbsp;
   </td>
   </td>
</tr>
</tr>

Revision as of 17:54, 27 October 2020

Concentric Ellipsoidal (T6) Coordinates

Whitworth's (1981) Isothermal Free-Energy Surface
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Background

Building on our general introduction to Direction Cosines in the context of orthogonal curvilinear coordinate systems, and on our previous development of T3 (concentric oblate-spheroidal) and T5 (concentric elliptic) coordinate systems, here we explore the creation of a concentric ellipsoidal (T6) coordinate system. This is motivated by our desire to construct a fully analytically prescribable model of a nonuniform-density ellipsoidal configuration that is an analog to Riemann S-Type ellipsoids.

Orthogonal Coordinates

We start by defining a "radial" coordinate whose values identify various concentric ellipsoidal shells,

<math>~\lambda_1</math>

<math>~\equiv</math>

<math>~(x^2 + q^2 y^2 + p^2 z^2)^{1 / 2} \, .</math>

When <math>~\lambda_1 = a</math>, we obtain the standard definition of an ellipsoidal surface, it being understood that, <math>~q^2 = a^2/b^2</math> and <math>~p^2 = a^2/c^2</math>. (We will assume that <math>~a > b > c</math>, that is, <math>~p^2 > q^2 > 1</math>.)

A vector, <math>~\bold{\hat{n}}</math>, that is normal to the <math>~\lambda_1</math> = constant surface is given by the gradient of the function,

<math>~F(x, y, z)</math>

<math>~\equiv</math>

<math>~(x^2 + q^2 y^2 + p^2 z^2)^{1 / 2} - \lambda_1 \, .</math>

In Cartesian coordinates, this means,

<math>~\bold{\hat{n}}(x, y, z)</math>

<math>~=</math>

<math>~ \hat\imath \biggl( \frac{\partial F}{\partial x} \biggr) + \hat\jmath \biggl( \frac{\partial F}{\partial y} \biggr) + \hat{k} \biggl( \frac{\partial F}{\partial z} \biggr) </math>

 

<math>~=</math>

<math>~ \hat\imath \biggl[ x(x^2 + q^2 y^2 + p^2 z^2)^{- 1 / 2} \biggr] + \hat\jmath \biggl[ q^2y(x^2 + q^2 y^2 + p^2 z^2)^{- 1 / 2} \biggr] + \hat\jmath \biggl[ p^2 z(x^2 + q^2 y^2 + p^2 z^2)^{- 1 / 2} \biggr] </math>

 

<math>~=</math>

<math>~ \hat\imath \biggl( \frac{x}{\lambda_1} \biggr) + \hat\jmath \biggl( \frac{q^2y}{\lambda_1} \biggr) + \hat\jmath \biggl(\frac{p^2 z}{\lambda_1} \biggr) \, , </math>

where it is understood that this expression is only to be evaluated at points, <math>~(x, y, z)</math>, that lie on the selected <math>~\lambda_1</math> surface — that is, at points for which the function, <math>~F(x,y,z) = 0</math>. The length of this normal vector is given by the expression,

<math>~[ \bold{\hat{n}} \cdot \bold{\hat{n}} ]^{1 / 2}</math>

<math>~=</math>

<math>~ \biggl[ \biggl( \frac{\partial F}{\partial x} \biggr)^2 + \biggl( \frac{\partial F}{\partial y} \biggr)^2 + \biggl( \frac{\partial F}{\partial z} \biggr)^2 \biggr]^{1 / 2} </math>

 

<math>~=</math>

<math>~ \biggl[ \biggl( \frac{x}{\lambda_1} \biggr)^2 + \biggl( \frac{q^2y}{\lambda_1} \biggr)^2 + \biggl(\frac{p^2 z}{\lambda_1} \biggr)^2 \biggr]^{1 / 2} </math>

 

<math>~=</math>

<math>~ \frac{1}{\lambda_1 \ell_{3D}} </math>

where,

<math>~\ell_{3D}</math>

<math>~\equiv</math>

<math>~\biggl[ x^2 + q^4y^2 + p^4 z^2 \biggr]^{- 1 / 2} \, .</math>

It is therefore clear that the properly normalized normal unit vector that should be associated with any <math>~\lambda_1</math> = constant ellipsoidal surface is,

<math>~\hat{e}_1 </math>

<math>~\equiv</math>

<math>~ \frac{ \bold\hat{n} }{ [ \bold{\hat{n}} \cdot \bold{\hat{n}} ]^{1 / 2} } = \hat\imath (x \ell_{3D}) + \hat\jmath (q^2y \ell_{3D}) + \hat\jmath (p^2 z \ell_{3D}) \, . </math>

From our accompanying discussion of direction cosines, it is clear, as well, that the scale factor associated with the <math>~\lambda_1</math> coordinate is,

<math>~h_1^2</math>

<math>~=</math>

<math>~\lambda_1^2 \ell_{3D}^2 \, .</math>

We can also fill in the top line of our direction-cosines table, namely,


Direction Cosines for T6 Coordinates
<math>~\gamma_{ni} = h_n \biggl( \frac{\partial \lambda_n}{\partial x_i}\biggr)</math>

<math>~n</math> <math>~i = x, y, z</math>
<math>~1</math>  

<math>~x\ell_{3D}</math>
 

<math>~q^2 y \ell_{3D}</math> <math>~p^2 z \ell_{3D}</math>
<math>~2</math>

 
---
 

 
---
 

 
---
 

<math>~3</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