User:Tohline/Appendix/Ramblings/ConcentricEllipsodalT8Coordinates

Concentric Ellipsoidal (T8) Coordinates

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 (T8) 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.

Note that, in a separate but closely related discussion, we made attempts to define this coordinate system, numbering the trials up through "T7." In this "T7" effort, we were able to define a set of three, mutually orthogonal unit vectors that should work to define a fully three-dimensional, concentric ellipsoidal coordinate system. But we were unable to figure out what coordinate function, <math>~\lambda_3(x, y, z)</math>, was associated with the third unit vector. In addition, we found the <math>~\lambda_2</math> coordinate to be rather strange in that it was not oriented in a manner that resembled the classic spherical coordinate system. Here we begin by redefining the <math>~\lambda_2</math> coordinate such that its associated <math>~\hat{e}_3</math> unit vector lies parallel to the x-y plane.

Realigning the Second Coordinate

The first coordinate remains the same as before, namely,

<math>~\lambda_1^2</math>

<math>~=</math>

<math>~x^2 + q^2 y^2 + p^2 z^2 \, .</math>

This may be rewritten as,

<math>~1</math>

<math>~=</math>

<math>~\biggl( \frac{x}{a}\biggr)^2 + \biggl( \frac{y}{b}\biggr)^2 + \biggl(\frac{z}{c}\biggr)^2 \, ,</math>

where,

<math>~a = \lambda_1 \, ,</math>

<math>~b = \frac{\lambda_1}{q} \, ,</math>

<math>~c = \frac{\lambda_1}{p} \, .</math>

By specifying the value of <math>~z = z_0 < c</math>, as well as the value of <math>~\lambda_1</math>, we are picking a plane that lies parallel to, but a distance <math>~z_0</math> above, the equatorial plane. The elliptical curve that defines the intersection of the <math>~\lambda_1</math>-constant surface with this plane is defined by the expression,

<math>~\lambda_1^2 - p^2z_0^2</math>	<math>~=</math>	<math>~x^2 + q^2 y^2 </math>
<math>~\Rightarrow~~~1</math>	<math>~=</math>	<math>~\biggl( \frac{x}{a_{2D}}\biggr)^2 + \biggl( \frac{y}{b_{2D}}\biggr)^2 \, ,</math>

where,

<math>~a_{2D} = \biggl(\lambda_1^2 - p^2z_0^2 \biggr)^{1 / 2} \, ,</math>

<math>~b_{2D} = \frac{1}{q} \biggl(\lambda_1^2 - p^2z_0^2 \biggr)^{1 / 2} \, .</math>

At each point along this elliptic curve, the line that is tangent to the curve has a slope that can be determined by simply differentiating the equation that describes the curve, that is,

<math>~0</math>	<math>~=</math>	<math>~\frac{2x dx}{a_{2D}^2} + \frac{2y dy}{b_{2D}^2}</math>
<math>~\Rightarrow~~~\frac{dy}{dx}</math>	<math>~=</math>	<math>~- \frac{2x}{a_{2D}^2} \cdot \frac{b_{2D}^2}{2y} = - \frac{x}{q^2y} \, .</math>
<math>~\Rightarrow~~~\Delta y</math>	<math>~=</math>	<math>~- \biggl( \frac{x}{q^2y} \biggr)\Delta x \, .</math>

The unit vector that lies tangent to any point on this elliptical curve will be described by the expression,

<math>~\hat{e}_2</math>	<math>~=</math>	<math>~ \hat\imath~ \biggl\{ \frac{\Delta x}{[ (\Delta x)^2 + (\Delta y)^2 ]^{1 / 2}} \biggr\} + \hat\jmath~ \biggl\{ \frac{\Delta y}{[ (\Delta x)^2 + (\Delta y)^2 ]^{1 / 2}} \biggr\} </math>
	<math>~=</math>	<math>~ \hat\imath~ \biggl\{ \frac{1}{[ 1 + x^2/(q^4y^2) ]^{1 / 2}} \biggr\} - \hat\jmath~ \biggl\{ \frac{x/(q^2y)}{[ 1 + x^2/(q^4y^2) ]^{1 / 2}} \biggr\} </math>
	<math>~=</math>	<math>~ \hat\imath~ \biggl\{ \frac{q^2y}{[ x^2 + q^4y^2 ]^{1 / 2}} \biggr\} - \hat\jmath~ \biggl\{ \frac{x}{[ x^2 + q^4y^2 ]^{1 / 2}} \biggr\} \, .</math>

As we have discovered, the coordinate that gives rise to this unit vector is,

<math>~\lambda_2</math>

<math>~=</math>

<math>~\frac{x}{y^{1/q^2}} \, .</math>

Other properties that result from this definition of <math>~\lambda_2</math> are presented in the following table.

Direction Cosine Components for T8 Coordinates
<math>~n</math>	<math>~\lambda_n</math>	<math>~h_n</math>	<math>~\frac{\partial \lambda_n}{\partial x}</math>	<math>~\frac{\partial \lambda_n}{\partial y}</math>	<math>~\frac{\partial \lambda_n}{\partial z}</math>	<math>~\gamma_{n1}</math>	<math>~\gamma_{n2}</math>	<math>~\gamma_{n3}</math>
<math>~1</math>	<math>~(x^2 + q^2 y^2 + p^2 z^2)^{1 / 2} </math>	<math>~\lambda_1 \ell_{3D}</math>	<math>~\frac{x}{\lambda_1}</math>	<math>~\frac{q^2 y}{\lambda_1}</math>	<math>~\frac{p^2 z}{\lambda_1}</math>	<math>~(x) \ell_{3D}</math>	<math>~(q^2 y)\ell_{3D}</math>	<math>~(p^2z) \ell_{3D}</math>
<math>~2</math>	<math>~\frac{x}{ y^{1/q^2}}</math>	<math>~\frac{1}{\lambda_2}\biggl[\frac{x q^2 y }{(x^2 + q^4y^2)^{1 / 2}}\biggr] </math>	<math>~\frac{\lambda_2}{x}</math>	<math>~-\frac{\lambda_2}{q^2 y}</math>	<math>~0</math>	<math>~\frac{q^2 y }{(x^2 + q^4y^2)^{1 / 2}} </math>	<math>~- \frac{x }{(x^2 + q^4y^2)^{1 / 2}} </math>	<math>~0</math>
<math>~3</math>	---	---	---	---	---	---	---	---
<math>~\ell_{3D}</math> <math>~\equiv</math> <math>~(x^2 + q^4y^2 + p^4 z^2 )^{- 1 / 2} \, ,</math> <math>~\mathcal{D}</math> <math>~\equiv</math> <math>~(q^4 y^2 p^4 z^2 + x^2 p^4 z^2 + 4x^2q^4y^2)^{1 / 2} \, .</math>

The associated unit vector is, then,

<math>~\hat{e}_2</math>

<math>~=</math>

<math>~ \hat\imath~\biggl[ \frac{q^2 y }{(x^2 + q^4y^2)^{1 / 2}} \biggr] - \hat\jmath~\biggl[ \frac{x }{(x^2 + q^4y^2)^{1 / 2}} \biggr] \, . </math>

See Also

• Direction Cosines
• Trials up through T7 Coordinates

© 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