Self-Gravitating, Incompressible (Dyson-Wong) Tori

Much of the introductory material of this chapter has been drawn from the paper by Tohline & Hachisu (1990) titled, The Breakup of Self-Gravitating Rings, Tori, and Accretion Disks.

Overview

In his pioneering work, F. W. Dyson (1893, Philosophical Transactions of the Royal Society of London. A., 184, 43 - 95) and (1893, Philosophical Transactions of the Royal Society of London. A., 184, 1041 - 1106) used analytic techniques to determine the approximate equilibrium structure of axisymmetric, uniformly rotating, incompressible tori. C.-Y. Wong (1974, ApJ, 190, 675 - 694) extended Dyson's work, using numerical techniques to obtain more accurate — but still approximate — equilibrium structures for incompressible tori having solid body rotation. Since then, Y. Eriguchi & D. Sugimoto (1981, Progress of Theoretical Physics, 65, 1870 - 1875) and I. Hachisu, J. E. Tohline & Y. Eriguchi (1987, ApJ, 323, 592 - 613) have mapped out the full sequence of Dyson-Wong tori, beginning from a bifurcation point on the Maclaurin spheroid sequence.

Thin Ring Approximation

MacMillan (1930)

In §102 of the book titled, The Theory of the Potential, W. D. MacMillan (1958; originally, 1930) derives an analytic expression for the gravitational potential of a uniform, infinitesimally thin, circular "hoop" of radius, <math>~a</math>; as shown, immediately below, the hoop is labeled, <math>~H</math>, in his Figures 60 and 61.

Figures 60 & 61 extracted without modification from, respectively, p. 195 & 196 of MacMillan (1958)

The Theory of the Potential, New York: McGraw-Hill

In setting up this problem, MacMillan (1958) says (verbatim text is typeset in a dark green font), Let <math>~P</math> be any point in space not in <math>~H</math>. From <math>~P</math> drop the perpendicular <math>~PQ = z</math> to the plane of the hoop. Draw the diameter of the circle <math>~BOA</math> which, extended, passes through <math>~Q</math>. Let <math>~m</math> be any point on the circle, and draw

Evidently <math>~\rho_1</math> and <math>~\rho_2</math> are the minimum and maximum values of <math>~\rho</math> as the point <math>~m</math> runs around the circle. If the angle <math>~mOA</math> is represented by <math>~2\omega</math>, the arc element is <math>~ds = 2ad\omega</math>, and — after multiplying MacMillan's §102, equation (1) through by "minus <math>~G</math>" — the expression for the gravitational potential is

<math>~- 2Ga\sigma \int_0^\pi \frac{d\omega}{\rho} = - \biggl( \frac{GM}{\pi} \biggr) \int_0^\pi \frac{d\omega}{\rho} \, ,</math>

where, <math>~\sigma</math> is the (uniform) linear mass density around the hoop, hence, the total mass of the hoop is <math>~M = 2\pi a \sigma</math>.

Referring further to MacMillan's Figure 60 — digitally reproduced, above — if the lengths <math>~mQ</math> and <math>~OQ</math> are represented by <math>~h</math> and <math>~r</math>, respectively, then

<math>~\rho_1^2</math>	<math>~=</math>	<math>~(r-a)^2 + z^2 \, ,</math>
<math>~\rho_2^2</math>	<math>~=</math>	<math>~(r+a)^2 + z^2 \, ,</math>
<math>~\rho^2</math>	<math>~=</math>	<math>~z^2 + h^2 </math>
	<math>~=</math>	<math>~z^2 + r^2 + a^2 - 2ar\cos(2\omega) \, .</math>

Following MacMillan (1958) (p. 196), the expression for <math>~\rho^2</math> can further be written,

<math>~\rho^2</math>	<math>~=</math>	<math>~ (r^2 + a^2 + z^2)(\cos^2\omega + \sin^2\omega) - 2ar(\cos^2\omega - \sin^2\omega) </math>
	<math>~=</math>	<math>~ [(r-a)^2 + z^2]\cos^2\omega + [(r+a)^2 + z^2]\sin^2\omega </math>
	<math>~=</math>	<math>~ \rho_1^2 \cos^2\omega + \rho_2^2\sin^2\omega \, . </math>

Hence, the expression for the potential becomes,

<math>~- \biggl( \frac{2GM}{\pi} \biggr) \int_0^{\pi/2} \frac{d\omega}{[\rho_1^2 \cos^2\omega + \rho_2^2\sin^2\omega]^{1 / 2}} \, .</math>

As MacMillan (1958) argues, … this expression shows that <math>~V</math> is symmetric in <math>~\rho_1</math> and <math>~\rho_2</math>, for if <math>~\omega</math> is replaced by <math>~(\tfrac{\pi}{2} - \psi)</math> it becomes

<math>~- \biggl( \frac{2GM}{\pi} \biggr) \int_0^{\pi/2} \frac{d\psi}{[\rho_1^2 \sin^2\psi + \rho_2^2\cos^2\psi]^{1 / 2}} \, ,</math>

and therefore

Along the axis of the hoop, <math>~\rho_1 = \rho_2</math>, and if <math>~\rho_a</math> is their common value, it is seen at once that the value of the potential along this axis <math>~V_a</math> is,

Furthermore, according to MacMillan (1958), … the function <math>~V(\rho_1, \rho_2)</math> is homogeneous of degree "- 1" in <math>~\rho_1</math> and <math>~\rho_2</math>. Therefore, <math>~\rho_1 V</math> is homogeneous of degree zero and depends only upon the ratio <math>~\rho_1/\rho_2</math>. With this in mind, let's rewrite the expression for the potential in the form,

<math>~\rho_1V</math>	<math>~=</math>	<math>~- \frac{2GM}{\pi} \int_0^{\pi/2} \biggl[\sin^2\psi + \biggl( \frac{\rho_2^2}{\rho_1^2} \biggr)\cos^2\psi \biggr]^{-1 / 2} d\psi </math>
	<math>~=</math>	<math>~- \frac{2GM}{\pi} \biggl( \frac{\rho_1}{\rho_2} \biggr) \int_0^{\pi/2} \biggl[\biggl( \frac{\rho_1^2}{\rho_2^2} \biggr)\sin^2\psi + \biggl( 1 - \sin^2\psi \biggr)\biggr]^{-1 / 2} d\psi </math>
	<math>~=</math>	<math>~- \frac{2GM}{\pi} \biggl( \frac{\rho_1}{\rho_2} \biggr) \int_0^{\pi/2} \biggl[1 - \biggl(1 - \frac{\rho_1^2}{\rho_2^2} \biggr)\sin^2\psi \biggr]^{-1 / 2} d\psi \,, .</math>

In addition to the hoop, <math>~H</math>, Figure 61 in §102 of MacMillan (1958) — digitally reproduced, above — displays a curve in the meridional plane of the hoop for which the ratio,

where <math>~c</math> is a constant. As MacMillan (1958) argues, the displayed curve is a circle because this equation is the equation of a circle in bipolar coordinates; and this circle … divides the line <math>~BCAD</math> harmonically, since by this last equation,

It is clear, therefore, that at every point along this meridional circle, the potential is given by the expression,

<math>~V</math>	<math>~=</math>	<math>~- \frac{2GMc}{\pi \rho_1} \int_0^{\pi/2} \frac{d\psi}{ \sqrt{1 - k^2 \sin^2\psi }} </math>
	<math>~=</math>	<math>~- \biggl[ \frac{2GMc}{\pi \rho_1} \biggr] K(k^2) \, ,</math>
MacMillan (1958), §102, Eq. (5)

where, <math>~K(k^2)</math> is the complete elliptic integral of the first kind for the modulus,

whose value always lies between zero and unity.

Again, referencing MacMillan's Figure 61, we note that when the potential is being evaluated at point <math>~D</math>, we can write,

and

Geometrically Thick Tori with Circular Cross-Section

Wong (1973, 1974)

In a paper titled, Toroidal Figures of Equilibrium, C.-Y. Wong (1974, ApJ, 190, 675 - 694) remarks that a "detailed analysis of toroidal figure of equilibrium has not received much attention since the last century. Previous work on this problem was carried out by":

Poincaré (1885a, C. R. Acad. Sci., 100, 346), (1885b, Bull. Astr., 2, 109), (1885c, Bull. Astr. 2, 405). — references copied from paper by Wong (1974)
S. Kowalewsky (1885, Astronomische Nachrichten, 111, 37) — Zusätze und Bemerkungen zu Laplace's Untersuchung über die Gestalt der Saturnsringe
F. W. Dyson (1893, Philosophical Transaction of the Royal Society London. A., 184, 43 - 95) — The Potential of an Anchor Ring. Part I.
1. In this paper, Dyson derives the gravitational potential exterior to the ring mass distribution
F. W. Dyson (1893, Philosophical Transaction of the Royal Society London. A., 184, 1041 - 1106) — The Potential of an Anchor Ring. Part II.
1. In this paper, Dyson derives the gravitational potential inside the ring mass distribution

Wong argues that a "reexamination of the toroidal figures of equilibrium is … necessary, because in all the previous analyses the physical quantities are expanded as a power series of the inverse of the aspect ratio. Such an expansion breaks down in the interesting region of small aspect ratios where one wishes to observe the transition between the Maclaurin sequence to the toroidal sequence. Furthermore, the classical solutions … can only treat small perturbations from a circular meridian …"

Principal Simplification: Following Poincaré, Dyson, and Kowalewsky, Wong confines his analysis to toroidal structures that have (a) uniform and incompressible mass distribution, and throughout which (b) the angular velocity is assumed to be independent of positions.

It is worth pointing out that Wong pursued this astrophysically relevant research problem at a time when, apparently, the principal focus of his work was nuclear physics. We suspect this is the case because, (a) his byline lists Oak Ridge National Laboratory as his employer; (b) in the acknowledgement section of his paper, Wong states that he "is indebted to Professor J. A. Wheeler who either consciously or unconsciously introduced the author to the subject matter with his toroidal geons and toroidal nuclei;" and Wong references and draws upon a paper that he published one year earlier — specifically, C.-Y. Wong (1973, Annals of Physics, 77, 279 - 353) — titled, Toroidal and Spherical Bubble Nuclei.

Introducing Toroidal Coordinates

Figure 1 extracted without modification from C.-Y. Wong (1973)
"Toroidal and Spherical Bubble Nuclei'"

Annals of Physics, vol. 77, pp. 279 - 353 © Elsevier Science

Wong (1973, Annals of Physics, 77, p. 284)

C.-Y. Wong (1973) introduces the toroidal coordinate system <math>~(\eta, \theta, \psi)</math> as follows (direct quotes from the article are displayed here in a dark green font). Referencing the figure — shown here on the right — that has been extracted without modification from the article, the surfaces of constant <math>~\eta</math> are generated by rotating a circle about the axis of symmetry, the <math>~z</math>-axis. These surfaces are toroidal surfaces. A toroidal surface of coordinate <math>~\eta</math> can be characterized by a "major radius" <math>~R</math> and a "minor radius" <math>~d</math> … The quantity <math>~\eta</math> varies from zero to infinity. The larger the value of <math>~\eta</math>, the smaller is the "minor radius" <math>~d</math>; when <math>~\eta</math> approaches infinity, the two-dimensional toroidal surface degenerates into a 1-dimensional circle with a radius <math>~a</math>. [Note that, otherwise, <math>~R</math> (the location of the center of the circular cross-section of the torus) does not coincide with <math>~a</math> (the location of the off-axis "origin" of the toroidal coordinate system).] The surfaces of constant <math>~\theta</math> are spherical bowls. The coordinate <math>~\theta</math> is defined in such a way that points above the x-y plane are characterized by positive values of <math>~\theta</math> while points below the x-y plane by negative values of <math>~\theta</math>. Thus — as is the case for a traditional spherical coordinate system — <math>~\theta</math> varies from <math>~- \pi</math> to <math>~+\pi</math>. As is also the case for a spherical coordinate system, the surfaces of constant <math>~\psi</math> are half planes through the axis of symmetry. The coordinate <math>~\psi</math> varies from <math>~0</math> to <math>~2\pi</math>.

Given a toroidal surface of major radius <math>~R</math> and minor radius <math>~d</math>, the parameter <math>~a</math> is defined such that,

<math>~a^2</math>	<math>~\equiv</math>	<math>~R^2 - d^2 \, ,</math>
Wong (1973), Eq. (2.8)

and the corresponding "radial" coordinate location <math>~\eta_0</math> of the relevant toroidal surface is,

<math>~\eta_0</math>	<math>~=</math>	<math>~\cosh^{-1}\biggl(\frac{R}{d}\biggr) \, .</math>
Wong (1973), Eq. (2.9)

Alternatively, given <math>~\eta_0</math> and the value of the parameter <math>~a</math>, we have,

<math>~R</math>	<math>~=</math>	<math>~a \coth\eta_0 \, ,</math>
<math>~d</math>	<math>~=</math>	<math>~\frac{a}{\sinh\eta_0} \, .</math>
Wong (1973), Eqs. (2.10) & (2.11)

Hence, the aspect ratio is,

<math>~\frac{R}{d}</math>	<math>~=</math>	<math>~\cosh\eta_0 \, .</math>
Wong (1973), Eq. (2.12)

Given the value of the scale-length, <math>~a</math>, the relationship between toroidal coordinates and Cartesian coordinates is [see equations 2.1 - 2.3 of Wong (1973)],

<math>~x</math>	<math>~=</math>	<math>~\frac{a \sinh\eta \cos\psi}{(\cosh\eta - \cos\theta)} \, ,</math>
<math>~y</math>	<math>~=</math>	<math>~\frac{a \sinh\eta \sin\psi}{(\cosh\eta - \cos\theta)} \, ,</math>
<math>~z</math>	<math>~=</math>	<math>~\frac{a \sin\theta}{(\cosh\eta - \cos\theta)} \, ;</math>

or, mapping the other direction [see equations 2.13 - 2.15 of Wong (1973),

<math>~\eta</math>	<math>~=</math>	<math>~\ln\biggl(\frac{r_1}{r_2} \biggr) \, ,</math>
<math>~\cos\theta</math>	<math>~=</math>	<math>~\frac{(r_1^2 + r_2^2 - 4a^2)}{2r_1 r_2} \, ,</math>
<math>~\tan\psi</math>	<math>~=</math>	<math>~\frac{y}{x} \, ,</math>

where,

<math>~r_1^2 </math>	<math>~\equiv</math>	<math>~[(x^2 + y^2)^{1 / 2} + a]^2 + z^2 \, ,</math>
<math>~r_2^2 </math>	<math>~\equiv</math>	<math>~[(x^2 + y^2)^{1 / 2} - a]^2 + z^2 \, ,</math>

and <math>~\theta</math> has the same sign as <math>~z</math>. Drawing from equations (2.7), (2.17) and (2.18) of Wong (1973), we see that the volume, <math>~V</math>, of a torus that is bounded by surface <math>~\eta_s</math> is,

<math>~\frac{V}{a^3} = \frac{1}{a^3} \iiint\limits_{\eta_s}d^3 r</math>

<math>~\iiint\limits_{\eta_s} \biggl[ \frac{\sinh\eta}{(\cosh\eta - \cos\theta)^3} \biggr] d\eta~ d\theta~ d\psi~ = \frac{2\pi^2\cosh{\eta_s}}{\sinh^3\eta_s} \, .</math>

If <math>~\eta_s \rightarrow \eta_0</math> then, in terms of the major and the minor radii of the torus, the volume is,

<math>~V</math>	<math>~=</math>	<math>~2\pi^2 Rd^2 \, .</math>
Wong (1973), Eq. (2.19)

If such a torus has a uniform density, <math>~\rho_0</math>, throughout, and a total charge (mass), <math>~q</math>, then the charge (mass) and density will be related through the toroidal-coordinate expression (see Wong's equation 2.51),

Also, as Wong (1973) points out (see his equation 2.50), in this case the density distribution may be written as,

<math>~\rho(\eta^', \theta^', \psi^')</math>

<math>~\rho_0 \Theta(\upsilon) \, ,</math>

where, the argument <math>~\upsilon \equiv (\eta - \eta_s)</math>, and <math>~\Theta(\upsilon)</math> is the step function defined by,

<math>~\Theta(\upsilon)</math>	<math>~=</math>	<math>~0</math>	for <math>~\upsilon < 0 \, ,</math>
<math>~\Theta(\upsilon)</math>	<math>~=</math>	<math>~1</math>	for <math>~\upsilon \ge 0 \, .</math>

The Coulomb Potential

As Wong (1973) reminds us, the Coulomb potential, <math>~U({\vec{r}}~')</math>, at a point <math>~{\vec{r}}~'</math> due to an arbitrary charge distribution, <math>~\rho({\vec{r}})</math>, is,

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

Referencing, for example, equation (3) of Cohl and Tohline (1999), we see that if we let <math>~\rho({\vec{r}})</math> represent a mass distribution instead of a charge distribution, this identical expression will give the Newtonian gravitational potential if we simply multiply through by (conventionally, the negative of) the gravitational constant, <math>~G</math>.

From the above expression for the differential volume element in toroidal coordinates, the right-hand side of this expression for the potential becomes,

<math>~U(\eta^',\theta^',\psi^')</math>	<math>~=</math>	<math>~\rho_0 a^3 \iiint \frac{1}{\|~\vec{r} - {\vec{r}}^{~'} ~\|} \biggl[ \frac{\Theta(\upsilon) \sinh\eta}{(\cosh\eta - \cos\theta)^3} \biggr] d\eta~ d\theta~ d\psi~ \, .</math>
Wong (1973), Eq. (2.52)

Next, Wong (1973) points out that in toroidal coordinates the Green's function is,

<math>~\frac{1}{\|~\vec{r} - {\vec{r}}^{~'} ~\|} </math>	<math>~=</math>	<math>~ \frac{1}{\pi a} \biggl[ (\cosh\eta - \cos\theta)(\cosh \eta^' - \cos\theta^') \biggr]^{1 /2 } \sum\limits_{m,n} (-1)^m \epsilon_m \epsilon_n ~\frac{\Gamma(n-m+\tfrac{1}{2})}{\Gamma(n + m + \tfrac{1}{2})} </math>
		<math>~ \times \cos[m(\psi - \psi^')][\cos[n(\theta - \theta^')] ~\begin{cases}P^m_{n-1 / 2}(\cosh\eta) ~Q^m_{n-1 / 2}(\cosh\eta^') ~~~\eta^' > \eta \\P^m_{n-1 / 2}(\cosh\eta^') ~Q^m_{n-1 / 2}(\cosh\eta)~~~\eta^' < \eta \end{cases}\, , </math>
Wong (1973), Eq. (2.53)

where, <math>~P^m_{n-1 / 2}, Q^m_{n-1 / 2}</math> are "Legendre functions of the first and second kind with order <math>~n - \tfrac{1}{2}</math> and degree <math>~m</math> (toroidal harmonics)," and <math>~\epsilon_m</math> is the Neumann factor, that is, <math>~\epsilon_0 = 1</math> and <math>~\epsilon_m = 2</math> for all <math>~m \ge 1</math>. After plugging this Green's function into the expression for the potential, then integrating over the azimuthal angle — which is permitted, here, because the density distribution, <math>~\rho(\vec{r})</math>, is assumed to be axisymmetric — Wong (1973) obtains,

<math>~U(\eta^',\theta^')</math>	<math>~=</math>	<math>~ 2\rho_0 a^2 (\cosh \eta^' - \cos \theta^')^{1 / 2} \sum\limits_n \epsilon_n \int_{\eta_0}^\infty d\eta \int_{-\pi}^{\pi} \biggl[\frac{\cos[n(\theta - \theta^')]}{(\cosh\eta - \cos\theta)^{5 / 2}} \biggr]d\theta </math>
		<math>~ \times \sinh\eta ~\begin{cases}P_{n-1 / 2}(\cosh\eta) ~Q_{n-1 / 2}(\cosh\eta^') ~~~\eta^' > \eta \\P_{n-1 / 2}(\cosh\eta^') ~Q_{n-1 / 2}(\cosh\eta)~~~\eta^' < \eta \end{cases}\, </math>
Wong (1973), Eq. (2.55)

which is valid for any azimuthal angle, <math>~\psi^'</math>. Notice that the step function, <math>~\Theta(\upsilon)</math>, no longer explicitly appears in this expression for the Coulomb (or gravitational) potential; it has been used to establish the specific limits on the "radial" coordinate integration. Next, he completes the integration over the angle, <math>~\theta</math>, to obtain,

<math>~U(\eta^',\theta^')</math>	<math>~=</math>	<math>~ \frac{2^{9 / 2}\rho_0 a^2}{3} (\cosh \eta^' - \cos \theta^')^{1 / 2} \sum\limits_n \epsilon_n \cos(n\theta^') \int_{\eta_0}^\infty d\eta \biggl[ \frac{Q^2_{n-1 / 2}(\cosh\eta)}{\sinh\eta} \biggr] </math>
		<math>~ \times \begin{cases}P_{n-1 / 2}(\cosh\eta) ~Q_{n-1 / 2}(\cosh\eta^') ~~~\eta^' > \eta \\P_{n-1 / 2}(\cosh\eta^') ~Q_{n-1 / 2}(\cosh\eta)~~~\eta^' < \eta \end{cases}\, . </math>
Wong (1973), Eq. (2.57)

TO BE DONE:

As we have suggested in an accompanying discussion, this expression should match our separate "special case" evaluation of the potential of a uniform-density torus if this expression is evaluated at <math>~\eta^' = +\infty</math>, and all values of the polar angle, <math>~\theta^'</math>.

Finally, Wong (1973) was able to complete the integration over the radial coordinate, <math>~\eta</math> to obtain an expression for the potential — most generally, his equation (2.59) — at all interior as well as all exterior coordinate positions, <math>~(\eta^', \theta^')</math>. Here, we will present only the interior solution, which is:

<math>~U(\eta^',\theta^')\biggr\|_{\mathrm{for}~\eta^' \ge \eta_0}</math>	<math>~=</math>	<math>~ \frac{2^{3 / 2}}{3\pi^2} \biggl(\frac{q}{a}\biggr) \frac{\sinh^3 \eta_0}{\cosh\eta_0} \biggl\{ - \frac{3\pi^2}{2^{5/ 2}} \biggl[ \frac{\sinh^2\eta^'}{(\cosh \eta^' - \cos \theta^')^2} \biggr] +~ (\cosh \eta^' - \cos \theta^')^{1 / 2} </math>
		<math>~ \times \sum\limits_{n=0}^\infty \epsilon_n \cos(n\theta^') Q_{n-1 / 2}(\cosh\eta^') B_n(\cosh\eta_0) \biggr\} \, , </math>
Wong (1973), Eq. (2.65)

where,

<math>~B_n(\cosh\eta_0)</math>	<math>~\equiv</math>	<math>~ (n+\tfrac{1}{2})P_{n+1/2} (\cosh\eta_0)Q^2_{n-1/2} (\cosh\eta_0) - (n-\tfrac{3}{2})P_{n-1/2} (\cosh\eta_0)Q^2_{n+1/2} (\cosh\eta_0) \, . </math>
Wong (1973), Eq. (2.62)

Fukushima (2016)

Setup

T. Fukushima (2016, AJ, 152, id. 35, 31 pp.) has also used zonal toroidal harmonics to examine the gravitational field external to ring-like objects. But, while continuing to associate the parameter, <math>~a</math>, with the equatorial-plane radius of the central ring, and to reference the same polar <math>~(\theta)</math> and azimuthal <math>~(\psi)</math> angles as Wong (1973), Fukushima uses a meridional-plane radial coordinate, <math>~u</math>, that, in practice, maps to Wong's (1973) coordinate, <math>~\eta</math>, via the expression,

Figure 1 extracted without modification from T. Fukushima (2016)
"Zonal Toroidal Harmonic Expansions
of External Gravitational Fields for Ring-like Objects'"

<math>~u \leftrightarrow \cosh\eta \, .</math>

Fukushima also introduces a set of coordinate-dependent functions — <math>~\nu(u), s(\theta), c(\theta),</math> and <math>D(u,\theta) \equiv (u - c)</math> — that, in effect, allow his derivations to be presented in a relatively compact form. Table 1, below, details the relationships between this set of functions and the coordinates/functions used by Wong as well as the coordinates/functions preferred by Morse & Feshbach (1953) that we have adopted in an associated discussion. Then, for example, the mapping between Fukushima's adopted set of meridional-plane coordinates and cylindrical coordinates <math>~(\varpi,z)</math> is:

<math>~\frac{\varpi}{a} = \frac{\nu}{D}</math>

and

Using toroidal coordinates, a meridional-plane circle describing a cross-section through the torus is defined by setting <math>~u</math> = constant. The relevant relation can be obtained by combining this pair of expressions to eliminate <math>~\theta</math>. From the first, we see that,

<math>~\frac{\varpi}{a}</math>	<math>~=</math>	<math>~\frac{\nu}{(u - \cos\theta)}</math>
<math>~\Rightarrow u - \cos\theta </math>	<math>~=</math>	<math>~\frac{a\nu}{\varpi}</math>
<math>~\Rightarrow \cos^2\theta </math>	<math>~=</math>	<math>~\biggl( u - \frac{a\nu}{\varpi^2} \biggr)^2\, .</math>

Using the second to replace <math>~D</math> in the first, we obtain,

<math>~\frac{\varpi}{a} \biggl( \frac{a}{z}\biggr)</math>	<math>~=</math>	<math>~\frac{\nu}{\sin\theta}</math>
<math>~\Rightarrow \sin\theta </math>	<math>~=</math>	<math>~\frac{z\nu}{\varpi}</math>
<math>~\Rightarrow \cos^2\theta </math>	<math>~=</math>	<math>~1- \frac{z^2\nu^2}{\varpi^2} \, .</math>

Together, this pair of relations implies,

<math>~ \biggl( u - \frac{a\nu}{\varpi} \biggr)^2 </math>	<math>~=</math>	<math>~ 1- \frac{z^2\nu^2}{\varpi^2} </math>
<math>~\Rightarrow \biggl( u - \frac{a\nu}{\varpi} \biggr)^2 + \frac{z^2\nu^2}{\varpi^2} </math>	<math>~=</math>	<math>~ 1 </math>
<math>~\Rightarrow \frac{\varpi^2}{\nu^2}\biggl( u - \frac{a\nu}{\varpi} \biggr)^2 + z^2 </math>	<math>~=</math>	<math>~ \frac{\varpi^2}{\nu^2} </math>
<math>~\Rightarrow z^2</math>	<math>~=</math>	<math>~ \frac{\varpi^2}{\nu^2} - \biggl( \frac{\varpi u}{\nu} -a \biggr)^2 </math>
	<math>~=</math>	<math>~ \frac{\varpi^2}{\nu^2} - \biggl( \frac{\varpi^2 u^2}{\nu^2} - \frac{2a\varpi u}{\nu} + a^2 \biggr) </math>
	<math>~=</math>	<math>~ \frac{\varpi^2}{\nu^2}\biggl[ 1 - u^2 \biggr] + \frac{2a\varpi u}{\nu} - a^2 \, . </math>

Now, given that (see Table 1), <math>~\nu^2 = (u^2-1)</math>, this last expression can be rewritten as,

<math>~z^2</math>	<math>~=</math>	<math>~ - \varpi^2 + \frac{2a\varpi u}{\nu} + \frac{a^2}{\nu^2} \biggl[ 1 - u^2 \biggr] </math>
	<math>~=</math>	<math>~ \frac{a^2}{\nu^2} - \varpi^2 + \frac{2a\varpi u}{\nu} - \frac{a^2u^2}{\nu^2} </math>
	<math>~=</math>	<math>~ \frac{a^2}{\nu^2} - \biggl[\varpi - \frac{au}{\nu} \biggr]^2 </math>
<math>~\Rightarrow \biggl[\varpi - \frac{au}{\nu} \biggr]^2 + z^2 </math>	<math>~=</math>	<math>~ \frac{a^2}{\nu^2} </math>
Fukushima (2016), Eq. (27)

This last expression not only matches equation (27) in Fukushima (2016) but, as has been pointed out in an accompanying discussion of the paper by Trova, et al. (2012), its form matches the familiar algebraic expression for an off-axis circle. The circle — which, here, is associated with a meridional cross-section of the <math>~u</math> = constant torus — has a (cross-sectional) radius,

and its center is shifted a distance,

<math>~\varpi_t = \frac{au}{\nu} \, ,</math>

away from the symmetry <math>~(z)</math> axis. Via his equations (26) and (25), Fukushima (2016) labels these key geometric lengths as, respectively, <math>~R_R</math> and <math>~R_C</math>. We should point out that the other key geometric length, <math>~a</math> — which defines the size of the central ring as depicted in Figure 1 of Fukushima (2016) — is related to <math>~\varpi_t</math> and <math>~r_t</math> via the expression,

<math>~a^2 = \varpi_t^2 - r_t^2 \, .</math>

And it should be emphasized that <math>~a</math> is always smaller than <math>~\varpi_t = R_C</math>, except in the limit of <math>~u \rightarrow \infty \Rightarrow u/\nu = 1</math>, in which case the two lengths are the same.

TABLE 1: Mapping Between Coordinate-Dependent Functions
Fukushima (2016)	<math>~\leftrightarrow</math>	Wong (1973)	<math>~\leftrightarrow</math>	MF53
<math>~u</math>	<math>~\leftrightarrow</math>	<math>~\cosh\eta</math>	<math>~\leftrightarrow</math>	<math>~\xi_1</math>
<math>~\nu(u) = (u^2 - 1)^{1 / 2}</math>	<math>~\leftrightarrow</math>	<math>~\sinh\eta</math>	<math>~\leftrightarrow</math>	<math>~(\xi_1^2 - 1)^{1 / 2}</math>
<math>~s(\theta) = \sin\theta</math>	<math>~\leftrightarrow</math>	<math>~ \sin\theta</math>	<math>~\leftrightarrow</math>	<math>~ (1-\xi_2^2)^{1 / 2}</math>
<math>~c(\theta) = \cos\theta</math>	<math>~\leftrightarrow</math>	<math>~ \cos\theta</math>	<math>~\leftrightarrow</math>	<math>~ \xi_2</math>
<math>~D(u,\theta) = u - c(\theta)</math>	<math>~\leftrightarrow</math>	<math>~\cosh\eta - \cos\theta</math>	<math>~\leftrightarrow</math>	<math>~\xi_1 - \xi_2</math>

Letting <math>~P</math> mark the external point at which the gravitational potential is to be evaluated — see his Figure 1, a digital replica of which is shown immediately above, on the left — Fukushima (2016) uses <math>~q</math> to represent the shortest distance between <math>~P</math> and the central ring, and uses <math>~p</math> to represent the greatest distance between <math>~P</math> and the central ring. It is straightforward to show that,

<math>~p</math>	<math>~=</math>	<math>~\biggl[(\varpi+a)^2 + z^2\biggr]^{1 / 2}</math>	<math>~=</math>	<math>~a\biggl[ \frac{2(u+\nu)}{D}\biggr]^{1 / 2} \, ,</math>
Fukushima (2016), Eqs. (21) & (23)

and,

<math>~q</math>	<math>~=</math>	<math>~\biggl[(\varpi-a)^2 + z^2\biggr]^{1 / 2}</math>	<math>~=</math>	<math>~a\biggl[ \frac{2(u-\nu)}{D}\biggr]^{1 / 2} \, .</math>
Fukushima (2016), Eqs. (22) & (24)

It is worth detailing how the cylindrical-coordinate expression for <math>~p</math> is converted into the stated expression in terms of Fukushima's coordinates. (Transformation of <math>~q</math> is done in an analogous fashion.) For <math>~p</math>, we have,

<math>~\frac{p^2}{a^2}</math>	<math>~=</math>	<math>~ \biggl( \frac{\varpi}{a} + 1\biggr)^2 + \frac{z^2}{a^2} </math>
	<math>~=</math>	<math>~ \biggl( \frac{\nu}{D} + 1\biggr)^2 + \frac{s^2}{D^2} </math>
	<math>~=</math>	<math>~ \frac{2\nu}{D} + \frac{\nu^2}{D^2} + 1 + \frac{s^2}{D^2} </math>
	<math>~=</math>	<math>~ \frac{2\nu}{D} + \frac{1}{D^2} \biggl[ (u^2-1) + (u-c)^2 + s^2\biggr] </math>
	<math>~=</math>	<math>~ \frac{2\nu}{D} + \frac{1}{D^2} \biggl[ u^2-1 + u^2 - 2uc + c^2 + s^2\biggr] </math>
	<math>~=</math>	<math>~ \frac{2\nu}{D} + \frac{2u}{D^2} \biggl[ u - c\biggr] </math>
	<math>~=</math>	<math>~ \frac{2(\nu + u)}{D} \, , </math>

where, along the way, we have used the fact that <math>~(s^2 + c^2) = 1</math> and, again, recognized that, <math>~\nu^2 = (u^2-1)</math>.

Uniform-Density, Circular Torus

In our accompanying statement of this problem, we have written,

<math>~- \frac{2G}{R_*^{1 / 2}} \int\int \varpi^{1 / 2} \mu K(\mu) \rho(\varpi, Z) d\varpi dZ \, ,</math>

where, <math>~K(\mu)</math> is the complete elliptic integral of the first kind, and,

<math>~ \biggl[\frac{4R_*\varpi}{(R_* + \varpi)^2 + (Z_* - Z)^2} \biggr] \, . </math>

Fukushima (2016) tackles this same problem in §5.2 of his article. If, to minimize confusion, we replace his asterisk-superscript notation with primes, Fukushima's expression for the external gravitational potential is (see his equations 138 - 142),

<math>~\Phi(R,z)\biggr\|_\mathrm{external}</math>	<math>~=</math>	<math>~ -4G\rho_0 R \int_{R1}^{R2} \biggl\{ \int_{-z(R^')}^{z(R^')}\biggl[ \frac{K(m^')}{ \sqrt{(R+R^')^2 + (z - z^')^2 }} \biggr] dz^' \biggr\} dR^' </math>
	<math>~=</math>	<math>~ -4G\rho_0 R \int_{R1}^{R2} \biggl\{ \int_{-z(R^')}^{z(R^')}\biggl[ K(m^') \biggl( \frac{m^'}{4RR^'} \biggr)^{1 / 2}\biggr] dz^' \biggr\} dR^' </math>
	<math>~=</math>	<math>~ -2G\rho_0 R^{1 / 2} \int_{R1}^{R2} \biggl\{ \int_{-z(R^')}^{z(R^')}\biggl[ K(m^') \biggl( \frac{m^'}{R^'} \biggr)^{1 / 2}\biggr] dz^' \biggr\} dR^' \, , </math>

where,

<math>~\equiv</math>

and the integration limit,

It appears as though the definition of the elliptic integral argument, <math>~m^' \leftrightarrow \mu^2</math>, is identical in the two cases, but the leading radial factor does not appear to be the same — in one case it is, <math>~(\varpi/R_*)^{1 / 2}</math>, while in the other case it is, <math>~(R/R^')^{1 / 2}</math>. This discrepancy needs to be resolved!

User:Tohline/Apps/DysonWongTori

Contents

Self-Gravitating, Incompressible (Dyson-Wong) Tori

Overview

Thin Ring Approximation

MacMillan (1930)

Geometrically Thick Tori with Circular Cross-Section

Wong (1973, 1974)

Introducing Toroidal Coordinates

The Coulomb Potential

Fukushima (2016)

Setup

Uniform-Density, Circular Torus

See Also

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