User:Tohline/Appendix/Ramblings/T1Coordinates
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Relationship Between HNM82 Models and T1 Coordinates
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>
One Special Case
For the specific case of <math>q^2 = 2</math> — that is, for <math>a_1/a_3 = \sqrt{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>.
HNM82
HNM82 derived a family of equilibrium models of rotationally flattened isothermal gas clouds whose gravitational potential is defined via the analytic expression,
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<math> \Phi </math> |
<math>=</math> |
<math> 2c_s^2 \ln\biggl[ \varpi^\gamma \cosh(\gamma\zeta) \biggr] </math> |
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<math>=</math> |
<math> 2c_s^2 \ln\biggl[ \frac{1}{2}(r+z)^\gamma + \frac{1}{2}(r-z)^\gamma \biggr] , </math> |
where <math>\gamma</math> is an index that identifies the specific member of the family of solutions, <math>c_s</math> is the isothermal sound speed, <math>r</math> is a spherical radius, that is,
<math> r \equiv ( \varpi^2 + z^2 )^{1/2} , </math>
and,
<math> \zeta \equiv \ln\biggl[ \frac{r + z}{\varpi} \biggr] = \sinh^{-1}\biggl(\frac{z}{\varpi}\biggr) . </math>
For completeness we note that these HNM82 models have a simple rotation profile in which <math>v_\varphi</math> is uniform in space, and the index <math>\gamma</math> is related to the uniform Mach number <math>\mathcal{M} \equiv v_\varphi/c_s</math> via the algebraic expression,
<math>\gamma = 1 + \frac{1}{2}\mathcal{M}^2</math> .
Immediately following their Eq. (3.4), HNM82 point out that "lines of gravity force which are perpendicular to the equi-potential surfaces are given by,"
<math> \frac{\varpi^{\gamma-1}}{\sinh[(\gamma-1)\zeta]} = \mathrm{constant} . </math> .
One Special Case
HNM82 also point out that, "in a special case <math>\gamma=2</math>, the equi-potential surfaces and the gravity lines are expressed simply as,"
<math> \varpi^2 + 2z^2 = \mathrm{constant} ~~~~\mathrm{and}~~~~ \frac{\varpi^2}{z} = \mathrm{constant} , </math>
respectively. This pair of HNM83 coordinate lines matches identically the T1 coordinate lines in the special case highlighted above, namely, in the case of <math>q^2 = 2</math>.
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