User:Tohline/Appendix/Ramblings/T1Coordinates

Relationship Between T1 Coordinates and the Equilibrium Models of HNM82

Preamble

In the mid-1990s I invested time trying to gain a better understanding of the "<math>3^\mathrm{rd}</math> Integral of Motion" that is discussed especially in the context of galaxy dynamics. For example, BT87 discuss the behavior of particle orbits in static potentials in which equipotential contours are nested oblate spheroidal surfaces with uniform eccentricity. This set of equipotential contours does not conform to, and therefore cannot be defined by, the traditional Oblate Spheroidal Coordinate system — detailed, for example, in MF53 — because in that traditional coordinate system surfaces of constant <math>\xi_1</math> are confocal rather than concentric oblate spheroids. In an effort to uncover a closed-form mathematical prescription for the "<math>3^\mathrm{rd}</math> integral" in this case, I developed an orthogonal coordinate system in which surfaces of constant <math>\xi_1</math> are concentric oblate spheroids. The properties of this T1 coordinate system are detailed in the Appendix of the original version of this H_Book. Here are two relevant links:

• Original presentation of T1 Coordinates; (This page of shtml text may only be viewable using Internet Explorer.)
• PDF-formatted presentation of T1 Coordinates

I have just realized (in May, 2010) that there is a connection between this T1 Coordinate system and the equipotential contours that arise from at least one of the equilibrium models of the axisymmetric structure of rotationally flattened isothermal gas clouds presented by Hayashi, Narita & Miyama (1982; hereafter HNM82). What follows is a discussion of this connection.

T1 Coordinates

A spheroidal surface with semi-axes <math>a_1</math> & <math>a_3</math> specified, respectively, along the <math>\varpi</math> and <math>z</math> axes will be defined by the expression,

<math> \biggl(\frac{\varpi}{a_1}\biggr)^2 + \biggl(\frac{z}{a_3}\biggr)^2 = 1 . </math>

Hence, a coordinate system that perfectly overlays a set of concentric spheroidal surfaces should have a radial <math>\xi_1</math> coordinate of the form,

<math> \xi_1 \equiv \biggl[ z^2 + \biggl( \frac{\varpi}{q}\biggr)^2 \biggr]^{1/2} , </math>

where the degree of flattening of the concentric surfaces is specified by the (constant) coefficient, <math>q \equiv (a_1/a_3)</math>. (The spheroidal surfaces will be oblate if <math>q > 1</math> and prolate if <math>q < 1</math>.)

A complementary meridional-plane angular coordinate that is everywhere orthogonal to this radial coordinate is,

<math> \xi_2 \equiv \tan^{-1}\biggl[ \frac{\varpi}{z^{1/q^2}} \biggr] . </math>

These are the essential elements of the so-called T1 Coordinate system. Because we are only dealing here with axisymmetric configurations, the third coordinate, which is everywhere orthogonal to the first two, is the familiar azimuthal coordinate,

<math> \xi_3 \equiv \tan^{-1}\biggl[ \frac{y}{x} \biggr] . </math>

Notice that, for the specific case of <math>q^2 = 1/2</math>, we will find that

<math> 2z^2 + \varpi^2 = \mathrm{constant} , </math>

along surfaces of constant <math>\xi_1</math>; and we will find that

<math> \frac{\varpi^2}{z} = \mathrm{constant} , </math>

along surfaces of constant <math>\xi_2</math>.

 

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