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Polytropic & Isothermal Tori

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

Here we will focus on the analysis of the structure self-gravitating tori that are composed of compressible — specifically, polytropic and isothermal — fluids as presented in a series of papers by Jeremiah P. Ostriker:

I believe that much, if not all, of this material was drawn from Ostriker's doctoral dissertation research at the University of Chicago (and Yerkes Observatory) under the guidance of S. Chandrasekhar.


Coordinate System

In §IIa of Paper II, Ostriker defines a set of orthogonal coordinates, <math>~(r,\phi,\theta)</math>, that is related to the traditional Cartesian coordinate system, <math>~(x,y,z)</math>, via the relations,

<math>~x</math>

<math>~=</math>

<math>~(R+r\cos\phi)\cos\theta \, ,</math>

<math>~y</math>

<math>~=</math>

<math>~(R+r\cos\phi)\sin\theta \, ,</math>

<math>~z</math>

<math>~=</math>

<math>~r\sin\phi \, .</math>

As Ostriker states, "The coordinate <math>~r</math> is the distance from a reference circle of radius <math>~R</math> (later chosen to be the major radius of the ring) …" The angle, <math>~\theta</math>, plays the role of the azimuthal angle, as is familiar in both cylindrical and spherical coordinates, while, here, <math>~\phi</math> is a meridional-plane polar angle measured counterclockwise from the equatorial plane. For axisymmetric systems, there will be no dependence on the azimuthal angle, so the pair of relevant coordinates in the meridional plane are,

<math>~\varpi \equiv (x^2+y^2)^{1 / 2}</math>

<math>~=</math>

<math>~R+r\cos\phi \, ,</math>

    and,    

<math>~z</math>

<math>~=</math>

<math>~r\sin\phi \, .</math>

Figure 1 extracted without modification from p. 1077 of J. P. Ostriker (1964; Paper II)

"The Equilibrium of Self-Gravitating Rings"

ApJ, vol. 140, pp. 1067-1087 © American Astronomical Society

Figure 1 from Ostriker (1964) Paper II

For later reference, we note that (see eq. 3 of Paper II) the corresponding line element is,

<math>~\delta s^2</math>

<math>~=</math>

<math>~ \delta r^2 + r^2 \delta\phi^2 + (R+r\cos\phi)^2\delta\theta^2 \, , </math>

which means that the relevant scale factors for the adopted coordinate system, <math>~(r,\phi,\theta)</math>, are

<math>~h_1 = 1 \, ,</math>       <math>~h_2 = r \, ,</math>       <math>~h_3 = (R+r\cos\phi) \, ,</math>

and the relevant differential volume element is,

<math>~d^3 x</math>

<math>~=</math>

<math>~h_1 h_2 h_3 dr d\phi d\theta = r(R+r\cos\phi) dr d\phi d\theta\, . </math>


Second Attempt

Single Offset Circle

Now an off-center circle whose major and minor radii are, respectively, <math>~(\varpi_0,d)</math>, will be described by the expression,

<math>~d^2</math>

<math>~=</math>

<math>~ (\varpi - \varpi_0)^2 + z^2 \, . </math>

where both <math>~d</math> and <math>~\varpi_0</math> are held constant while mapping out the variation of <math>~z</math> with <math>~\varpi</math>. If we acknowledge that, in general, <math>~\varpi_0 \ne R_\mathrm{JPO}</math>, then we know how <math>~r</math> varies with <math>~\phi</math> via the relation,

<math>~d^2</math>

<math>~=</math>

<math>~ \biggl[ R_\mathrm{JPO} + r\cos\phi - \varpi_0\biggr]^2 + r^2\sin^2\phi </math>

 

<math>~=</math>

<math>~ (R_\mathrm{JPO}-\varpi_0)^2 + 2\biggl[ (R_\mathrm{JPO}-\varpi_0) r\cos\phi \biggr] +r^2 </math>

<math>~\Rightarrow ~~~ 0 </math>

<math>~=</math>

<math>~ r^2 + 2r\biggl[ (R_\mathrm{JPO}-\varpi_0) \cos\phi \biggr] + \biggl[(R_\mathrm{JPO}-\varpi_0)^2 - d^2\biggr] </math>

<math>~\Rightarrow ~~~ r </math>

<math>~=</math>

<math>~ \frac{1}{2}\biggl\{ - 2\biggl[ (R_\mathrm{JPO}-\varpi_0) \cos\phi \biggr] \pm \sqrt{ 4\biggl[ (R_\mathrm{JPO}-\varpi_0) \cos\phi \biggr]^2 - 4\biggl[(R_\mathrm{JPO}-\varpi_0)^2 - d^2\biggr] } \biggr\} </math>

 

<math>~=</math>

<math>~ \frac{1}{2}\biggl\{ 2\biggl[ (\varpi_0 - R_\mathrm{JPO}) \cos\phi \biggr] \pm \sqrt{ 4\biggl[ (\varpi_0 - R_\mathrm{JPO}) \cos\phi \biggr]^2 - 4\biggl[(\varpi_0 - R_\mathrm{JPO})^2 - d^2\biggr] } \biggr\} </math>

<math>~\Rightarrow~~~ \frac{r}{ (\varpi_0 - R_\mathrm{JPO}) }</math>

<math>~=</math>

<math>~ \cos\phi \pm \sqrt{ \cos^2\phi - 1 + d^2 (\varpi_0 - R_\mathrm{JPO})^{-2} } </math>

 

<math>~=</math>

<math>~ \cos\phi \pm \sqrt{ d^2 (\varpi_0 - R_\mathrm{JPO})^{-2}-\sin^2\phi } </math>

In order to align this expression with the terminology (and variable labels) that we use in the context of a toroidal coordinate system, we associate the radius of the anchor ring as <math>~R_\mathrm{JPO}\leftrightarrow a</math>, and we associate the major radius of each circular torus as <math>~\varpi_0 \leftrightarrow R_0</math>. We therefore have,

<math>~\frac{r}{ (R_0-a) }</math>

<math>~=</math>

<math>~ \cos\phi \pm \sqrt{ d^2 (R_0-a)^{-2}-\sin^2\phi } </math>

<math>~\Rightarrow ~~~ \frac{r}{a}</math>

<math>~=</math>

<math>~\biggl(\frac{R_0}{a}-1 \biggr) \biggl[ \cos\phi \pm \sqrt{ \biggl(\frac{d}{a}\biggr)^2 \biggl(\frac{R_0}{a}-1 \biggr)^{-2}-\sin^2\phi } \biggr] </math>

and, the coordinates of points along the surface of the torus <math>~(\varpi,z)</math> are provided by the expressions,

<math>~\varpi</math>

<math>~=</math>

<math>~ a + (R_0 - a)\cos\phi \biggl[ \cos\phi \pm \sqrt{ d^2 (R_0 - a)^{-2}-\sin^2\phi } \biggr] </math>

<math>~z</math>

<math>~=</math>

<math>~ (R_0 - a)\sin\phi \biggl[ \cos\phi \pm \sqrt{ d^2 (R_0 - a)^{-2}-\sin^2\phi } \biggr] </math>

We have tested this pair of expressions using Excel and have successfully demonstrated that they do, indeed, trace out a circle of radius, <math>~d</math>, whose center is offset from the symmetry axis by a distance, <math>~R_0</math>.

Set of Circles Whose Offset Increases With Circle Diameter

A set of nested off-center circles will be described by allowing <math>~R_0 = R_0(d)</math>, that is, by having the off-set distance, <math>~R_0</math>, vary with the size of the circle, <math>~d</math>. The above prescription for the normalized "coordinate" <math>~r/a</math> will work for any prescribed <math>~R_0(d)</math> function.

But a particular <math>~R_0(d)</math> function is demanded if we want this derived prescription to represent the behavior of toroidal coordinates. In a toroidal coordinate system, a specification of the value of the "radial" coordinate, <math>~\eta</math>, automatically dictates the ratio <math>~R_0/d</math>; but we are not at liberty to separately define the value of the difference, <math>~(R_0 - d)</math>. Instead, we must enforce the toroidal-coordinate relation,

<math>~a^2</math>

<math>~=</math>

<math>~R_0^2 - d^2</math>

<math>~\Rightarrow~~~ \frac{R_0}{a}-1</math>

<math>~=</math>

<math>~\biggl[ 1 + \delta^2\biggr]^{1 / 2} -1 \, ,</math>

where we have adopted the shorthand notation, <math>~\delta\equiv d/a</math>. Hence,

<math>~\frac{r}{a}</math>

<math>~=</math>

<math>~[ \sqrt{1+\delta^2} -1 ] \{ \cos\phi \pm [\delta^2 ( \sqrt{1+\delta^2} -1 )^{-2}-\sin^2\phi ]^{1 / 2} \} </math>

Now, in a toroidal coordinate system, there is a similar "radial" coordinate, <math>~\eta</math>, whose value varies with distance from the anchor ring of radius, <math>~a</math>. Its value depends on both <math>~R_0</math> and <math>~d</math> via the relation,

<math>~R_0 = d\cosh\eta \, .</math>

This means that,

<math>~\cosh\eta</math>

<math>~=</math>

<math>~\frac{1}{\delta}\biggl(\frac{R_0}{a}\biggr) = \frac{\sqrt{1+\delta^2}}{\delta} </math>

<math>~\Rightarrow~~~ \delta^2 \cosh^2\eta</math>

<math>~=</math>

<math>~1 + \delta^2</math>

<math>~\Rightarrow~~~ \delta^2 </math>

<math>~=</math>

<math>~\frac{1}{\cosh^2\eta - 1} = \frac{1}{\sinh^2\eta} </math>

<math>~\Rightarrow~~~ \sqrt{1 + \delta^2} </math>

<math>~=</math>

<math>~\biggl[1 + \frac{1}{\sinh^2\eta} \biggr]^{1 / 2} = \coth\eta \, ,</math>

which also means that,

<math>~\frac{r}{a}</math>

<math>~=</math>

<math>~[ \coth\eta -1 ] \biggl\{ \cos\phi \pm \biggl[ ( \cosh\eta -\sinh\eta )^{-2} -\sin^2\phi \biggr]^{1 / 2} \biggr\} \, . </math>

Case of Small Offset

Another way to look at this issue is to go back to the expression,

<math>~d^2</math>

<math>~=</math>

<math>~ (R_\mathrm{JPO}-\varpi_0)^2 + 2\biggl[ (R_\mathrm{JPO}-\varpi_0) r\cos\phi \biggr] +r^2 </math>

<math>~\Rightarrow ~~~ \delta^2</math>

<math>~=</math>

<math>~\biggl(\frac{r}{a}\biggr)^2 + \frac{r}{a}\biggl[ 2\biggl(1 - \frac{R_0}{a}\biggr)\biggr] \cos\phi + \biggl(1 - \frac{R_0}{a}\biggr)^2 </math>

and assume that, while still dependent on the radial coordinate, the dimensionless offset is small. That is, assume that,

<math>~\Delta(\delta) \equiv 1 - \frac{R_0(\delta)}{a} \ll 1 \, .</math>

In this case, we can write,

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

<math>~\approx</math>

<math>~\biggl(\frac{r}{a}\biggr)^2 + 2\Delta(\delta) \biggl( \frac{r}{a} \biggr) \cos\phi +\cancelto{0}{\Delta^2(\delta)} \, . </math>

And differentiating both sides of the expression with respect to <math>~r/a</math> gives,

<math>~0 </math>

<math>~\approx</math>

<math>~2\biggl(\frac{r}{a}\biggr) + 2\Delta(\delta) \cos\phi</math>

COMMENT by Tohline (15 August 2018): I'm not sure that this is leading where I had hoped. I am gearing up to draw a comparison between these last expressions and eq. (74) in Ostriker's (1973) Paper II.


Gravitational Potential

Potential of a Thin Hoop

In §IIb of his Paper II, Ostriker (1973) derives an expression for the gravitational potential of a torus in the Thin Ring approximation, beginning specifically with the integral form of the Poisson equation that is widely referred to in the astrophysics community as an expression for the,

Scalar Gravitational Potential

<math>~ \Phi(\vec{x})</math>

<math>~\equiv</math>

<math>~ -G \iiint \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .</math>

[BT87], p. 31, Eq. (2-3)
[EFE], §10, p. 17, Eq. (11)
[T78], §4.2, p. 77, Eq. (12)

(Note:   Consistent with the usage favored by his doctoral dissertation advisor in [EFE], throughout his collection of 1973 papers Ostriker adopts a different sign convention as well as a different variable name to represent the gravitational potential.) Employing Ostriker's adopted coordinate system, and recognizing that (see his eq. 21), "the distance between the point of integration <math>~(0,0,\theta^')</math> and the point of observation <math>~(r,\phi,0)</math>" is,

<math>~|\vec{x}^{~'} - \vec{x}| = [4R(R+r\cos\phi) \sin^2(\tfrac{1}{2}\theta^') + r^2]^{1 / 2} \, ,</math>

this expression for the gravitational potential becomes,

<math>~ \Phi(r,\phi)</math>

<math>~=</math>

<math>~ -G \int \frac{\rho(\vec{x}^{~'})}{[4R(R+r\cos\phi) \sin^2(\tfrac{1}{2}\theta^') + r^2]^{1 / 2} } d^3x^' </math>

See Also

The following quotes have been taken from Petroff & Horatschek (2008):

§1:   "The problem of the self-gravitating ring captured the interest of such renowned scientists as Kowalewsky (1885), Poincaré (1885a,b,c) and Dyson (1892, 1893). Each of them tackled the problem of an axially symmetric, homogeneous ring in equilibrium by expanding it about the thin ring limit. In particular, Dyson provided a solution to fourth order in the parameter <math>~\sigma = a/b</math>, where <math>~a = r_t</math> provides a measure for the radius of the cross-section of the ring and <math>~b = \varpi_t</math> the distance of the cross-section's centre of mass from the axis of rotation."

§7:   "In their work on homogeneous rings, Poincaré and Kowalewsky, whose results disagreed to first order, both had made mistakes as Dyson has shown. His result to fourth order is also erroneous as we point out in Appendix B."

  1. Shortly after their equation (3.2), Marcus, Press & Teukolsky make the following statement: "… we know that an equilibrium incompressible configuration must rotate uniformly on cylinders (the famous "Poincaré-Wavre" theorem, cf. Tassoul 1977, &Sect;4.3) …"


 

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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