Using Toroidal Coordinates to Determine the Gravitational Potential

The detailed derivations and associated scratch-work that support the summary discussion of this chapter can be found under the Appendix/Ramblings category of this H_Book.

Preamble

As I have studied the structure and analyzed the stability of (both self-gravitating and non-self-gravitating) toroidal configurations over the years, I have often wondered whether it might be useful to examine such systems mathematically using a toroidal — or at least a toroidal-like — coordinate system. Is it possible, for example, to build an equilibrium torus for which the density distribution is one-dimensional as viewed from a well-chosen toroidal-like system of coordinates?

I should begin by clarifying my terminology. In volume II (p. 666) of their treatise on Methods of Theoretical Physics, Morse & Feshbach (1953; hereafter MF53) define an orthogonal toroidal coordinate system in which the Laplacian is separable.¹ (See details, below.) It is only this system that I will refer to as the toroidal coordinate system; all other functions that trace out toroidal surfaces but that don't conform precisely to Morse & Feshbach's coordinate system will be referred to as toroidal-like.

I became particularly interested in this idea while working with Howard Cohl (when he was an LSU graduate student). Howie's dissertation research uncovered a Compact Cylindrical Greens Function technique for evaluating Newtonian potentials of rotationally flattened (especially axisymmetric) configurations.^2,3 The technique involves a multipole expansion in terms of half-integer-degree Legendre functions of the <math>2^\mathrm{nd}</math> kind — see NIST digital library discussion — where, if I recall correctly, the argument of this special function (or its inverse) seemed to resemble the radial coordinate of Morse & Feshbach's orthogonal toroidal coordinate system — see more on this, below.

Statement of the Problem

Expression for the Axisymmetric Potential

Cohl & Tohline (1999; hereafter CT99) derive an expression for the Newtonian gravitational potential in terms of a Compact Cylindrical Green's Function expansion. They show — see, for example, their equation (31) — that when expressed in terms of cylindrical coordinates, the potential at any meridional location, <math>\varpi = R_*</math> and <math>~Z = Z_*</math>, due to an axisymmetric mass distribution, <math>~\rho(\varpi, Z)</math>, is

where,

<math> q_0 = \int_\Sigma \varpi^{1/2} Q_{-1/2}(\Chi) \rho(\varpi, Z) d\sigma, </math>

<math>~d\sigma = d\varpi dZ</math> is a differential area element in the meridional plane, and the dimensionless argument (the modulus) of the special function, <math>~Q_{-1/2}</math>, is,

<math> \Chi \equiv \frac{R_*^2 + \varpi^2 + (Z_* - Z)^2}{2R_* \varpi} . </math>

Next, following the lead of CT99, we note that according to the Abramowitz & Stegun (1965),

where, the function <math>~K(\mu)</math> is the complete elliptical integral of the first kind and, for our particular problem,

<math>~\mu^2</math>	<math>~\equiv</math>	<math>~2(1+\Chi)^{-1}</math>
	<math>~=</math>	<math>~ 2\biggl[ 1+\frac{R_^2 + \varpi^2 + (Z_ - Z)^2}{2R_* \varpi}\biggr]^{-1} </math>
	<math>~=</math>	<math>~ \biggl[\frac{4R_\varpi}{(R_ + \varpi)^2 + (Z_* - Z)^2} \biggr] \, . </math>

Hence, we can write,

<math> q_0 = \int\int \varpi^{1/2} \mu K(\mu) \rho(\varpi, Z) d\varpi dZ \, . </math>

As has been explained in an accompanying set of notes, this is precisely the same expression for the gravitational potential that A. Trova, J.-M. Huré and F. Hersant (2012; MNRAS, 424, 2635) used in their study of the potential of self-gravitating, axisymmetric discs.

Our objective, here, is to examine whether or not it might be advantageous to transform this expression to one in which the double integral is performed on a toroidal, rather than a cylindrical, coordinate system.

Chosen Test Mass Distribution

For purposes of illustration, we will follow the lead of Trova, Huré & Hersant (2012) — see the left-hand panel of the following figure ensemble — and seek to determine the gravitational potential, both inside and outside, of a uniform-density, equatorial-plane torus whose (pink) meridional cross-section is exactly a circle. More specifically, as illustrated in our Figure 1 — see the right-hand panel of the following figure ensemble — at all azimuthal angles, a cross-section through the (pink) torus is prescribed by the familiar algebraic expression for an off-center circle, namely,

<math>~(\varpi_t - \varpi)^2 + Z^2</math>

Everywhere inside this toroidal surface we set <math>~\rho(\varpi, Z) = \rho_0</math>, that is, the density is uniform with the value, <math>~\rho_0</math>.

Figure 4 extracted without modification from p. 2640 of Trova, Huré & Hersant (2012)

"The Potential of Discs from a 'Mean Green Function' "

Monthly Notices of the Royal Astronomical Society, vol. 424, pp. 2635-2645 © RAS

Our Figure 1

Notice that another off-center circle — this one colored purple, rather than pink — appears in our Figure 1 diagram. In the discussion that follows, it will be used to represent the meridional-plane cross-section of one axisymmetric surface in an MF53 toroidal-coordinate system. Here we simply point out that this "surface" is also prescribed by an algebraic expression for an off-center circle, namely,

<math>~(R_0 - \varpi)^2 + (Z_0 - Z)^2</math>

Toroidal Coordinates

Properties

Here we highlight certain properties and features of the MF53 toroidal coordinate system; more details can be found in a related set of our online notes. Most importantly in the context of our discussion, if (at all azimuthal angles) the origin of the toroidal coordinate system is placed at the cylindrical-coordinate location, <math>~(a, Z_0), </math> the pair of orthogonal coordinates, <math>~(\xi_1, \xi_2)</math>, is related to the cylindrical coordinate pair, <math>~(\varpi, Z)</math>, via the expressions,

<math> ~\frac{\varpi}{a} </math>	<math> ~= </math>	<math> ~\frac{(\xi_1^2 - 1)^{1/2}}{\xi_1 - \xi_2} \, , </math>
<math> ~\frac{(Z_0 - Z)}{a} </math>	<math> ~= </math>	<math> ~\pm~\frac{(1-\xi_2^2)^{1/2}}{\xi_1 - \xi_2} \, . </math>

An off-center circle — such as the black circle depicted in our Figure 1 diagram — is generated If a value of the "radial" coordinate, <math>~\xi_1</math>, is chosen from within the range,

and held fixed while the "angular" coordinate, <math>~\xi_2</math>, is varied over the range,

Hereafter, we will refer to this <math>~\xi_1</math> = constant circle as a "<math>\xi_1</math>-circle." A <math>\xi_1</math>-circle of radius zero and, hence, the origin of the toroidal coordinate system is associated with the upper limiting value of the radial coordinate, namely, <math>~\xi_1 = \infty</math>; as the value of <math>~\xi_1</math> is decreased monotonically, the radius of the circle (for example, the black circle of radius, <math>~r_0</math>, in our Figure 1) steadily grows; and the radius of the (black) circle becomes infinite at the radial coordinate's other limiting value, <math>~\xi_1 = 1</math>.

In the <math>~Z = Z_0</math> plane, the location of the inner and outer edges of the toroidal-coordinate surface are determined by setting <math>~\xi_2 = -1</math> (inner) and <math>~\xi_2 = +1</math> (outer). Hence,

<math> ~\biggl(\frac{\varpi}{a}\biggr)_\mathrm{inner} </math>	<math>~=</math>	<math> ~\frac{(\xi_1^2 - 1)^{1/2}}{\xi_1 +1} = \biggl[\frac{(\xi_1 - 1)}{(\xi_1 + 1)} \biggr]^{1/2} \, , </math>
<math> ~\biggl(\frac{\varpi}{a}\biggr)_\mathrm{outer} </math>	<math>~=</math>	<math> ~\frac{(\xi_1^2 - 1)^{1/2}}{\xi_1 - 1} = \biggl[\frac{(\xi_1 + 1)}{(\xi_1 - 1)} \biggr]^{1/2} \, . </math>

Hence, also, the (cylindrical) radial location of the "center" of each toroidal-coordinate surface — labeled <math>~R_0</math> in our Figure 1 — is given by the expression,

<math> R_0 = \frac{a}{2} \biggl[ \biggl(\frac{\varpi}{a}\biggr)_\mathrm{outer} + \biggl(\frac{\varpi}{a}\biggr)_\mathrm{inner} \biggr] = \frac{a\xi_1}{(\xi_1^2 - 1)^{1/2}} \, , </math>

and the surface's cross-sectional radius — labeled <math>~r_0</math> in our Figure 1 — is given by the expression,

<math> r_0 = \frac{a}{2} \biggl[ \biggl(\frac{\varpi}{a}\biggr)_\mathrm{outer} - \biggl(\frac{\varpi}{a}\biggr)_\mathrm{inner} \biggr] = \frac{a}{(\xi_1^2 - 1)^{1/2}} \, . </math>

This last expression quantifies, and its simplicity reinforces, our earlier statement; that is, as the value of <math>~\xi_1</math> is decreased monotonically, the radius of the circle, <math>~r_0</math>, steadily grows. The next-to-last expression makes it clear, as well, that <math>~R_0</math> gets larger and, therefore, the location of the center of a <math>\xi_1</math>-circle shifts farther away from the symmetry axis as the value of <math>~\xi_1</math> is decreased. Notice that, for any off-center circle, the ratio of these to lengths gives the value of the toroidal-coordinate system's dimensionless "radial" coordinate, that is,

<math>~\biggl[\frac{a\xi_1}{(\xi_1^2 - 1)^{1/2}}\biggr] \biggl[ \frac{(\xi_1^2 - 1)^{1/2}}{a}\biggr] = \xi_1 \, .</math>

Notice, furthermore, that there is a particular combination of these two lengths that is independent of <math>~\xi_1</math>, namely,

<math>~r_0 \biggl[\biggl( \frac{R_0}{r_0} \biggr)^2 - 1 \biggr]^{1/2} </math>

<math>~\frac{a}{(\xi_1^2 - 1)^{1/2}} \biggl[\xi_1^2 - 1 \biggr]^{1/2} = a \, .</math>

This is a manner in which one can determine the radial position, <math>~a</math>, of the origin of the toroidal coordinate system that could legitimately be associated with any particular off-center circle, such as the black circle drawn in our Figure 1.

Connection With the Physical Problem

Earlier, we stated that the purple, off-center circle displayed in Figure 1 is prescribed by the algebraic expression,

<math>~(R_0 - \varpi)^2 + (Z_0 - Z)^2</math>

Let's plug in the "toroidal-coordinate" expressions for each parameter that appears on the left-hand side of this relation and see whether, after simplification, it reduces to the right-hand side.

LHS	<math>~=</math>	<math>~(R_0 - \varpi)^2 + (Z_0 - Z)^2</math>
	<math>~=</math>	<math>~\biggl[ \frac{a\xi_1}{(\xi_1^2 - 1)^{1/2}} - \frac{a(\xi_1^2 - 1)^{1/2}}{\xi_1 - \xi_2} \biggr]^2 + \biggl[\pm~\frac{a(1-\xi_2^2)^{1/2}}{\xi_1 - \xi_2}\biggr]^2</math>
	<math>~=</math>	<math>~\frac{a^2}{(\xi_1^2 - 1)} \biggl\{ \biggl[ \xi_1 - \frac{(\xi_1^2 - 1)}{\xi_1 - \xi_2} \biggr]^2 + \biggl[\frac{(1-\xi_2^2)^{1/2}(\xi_1^2 - 1)^{1/2} }{\xi_1 - \xi_2}\biggr]^2 \biggr\} </math>
	<math>~=</math>	<math>~\frac{a^2}{(\xi_1^2 - 1)(\xi_1-\xi_2)^2} \biggl\{ \biggl[ \xi_1(\xi_1-\xi_2) - (\xi_1^2 - 1) \biggr]^2 + (1-\xi_2^2)(\xi_1^2 - 1) \biggr\} </math>
	<math>~=</math>	<math>~\frac{a^2}{(\xi_1^2 - 1)(\xi_1-\xi_2)^2} \biggl[ ( 1-\xi_1\xi_2 )^2 + (\xi_1^2 - 1 -\xi_1^2\xi_2^2 + \xi_2^2) \biggr] </math>
	<math>~=</math>	<math>~\frac{a^2}{(\xi_1^2 - 1)(\xi_1-\xi_2)^2} \biggl[ \xi_1^2 -2\xi_1\xi_2 + \xi_2^2 \biggr] </math>
	<math>~=</math>	<math>~\frac{a^2}{(\xi_1^2 - 1)} \, . </math>

This, indeed, equals the right-hand side of the relation, which is, <math>~r_0^2</math>. It all nicely checks out!

Next, taking a hint from the EUREKA! moment recorded in our accompanying notes, let's rewrite the function <math>~\Chi</math> in terms of toroidal rather than cylindrical coordinates, where <math>~\Chi</math> is the argument of the special function, <math>~Q_{-1/2}</math>, that appears in the above definition of <math>~q_0</math>. More specifically, let's assume that the coordinate location at which the gravitational potential is to be evaluated, <math>~(R_*, Z_*)</math>, is taken to be the cylindrical-coordinate location of the origin of the toroidal coordinate system, <math>~(a, Z_0)</math>. Given this association, we can write,

<math>~\Chi </math>	<math>~\equiv</math>	<math>~\frac{R_^2 + \varpi^2 + (Z_ - Z)^2}{2R_* \varpi}</math>
	<math>~=</math>	<math>~\frac{a^2 + \varpi^2 + (Z_0 - Z)^2}{2a \varpi}</math>
	<math>~=</math>	<math>~\frac{1 + (\varpi/a)^2 + [(Z_0 - Z)/a]^2}{2(\varpi/a) }</math>
<math>~\Rightarrow~~~~ \Chi^2 </math>	<math>~=</math>	<math>~\biggl[ 2 \biggl( \frac{\varpi}{a} \biggr) \biggr]^{-2} \biggl\{ 1 + \biggl(\frac{\varpi}{a} \biggr)^2 + \biggl[\frac{(Z_0 - Z)}{a} \biggr]^2 \biggr\}^2 </math>
	<math>~=</math>	<math>~\biggl[ \frac{2(\xi_1^2 - 1)^{1/2}}{\xi_1 - \xi_2} \biggr]^{-2} \biggl\{ 1 + \biggl[\frac{(\xi_1^2 - 1)^{1/2}}{\xi_1 - \xi_2} \biggr]^2 + \biggl[\pm~\frac{(1-\xi_2^2)^{1/2}}{\xi_1 - \xi_2} \biggr]^2 \biggr\}^2 </math>
	<math>~=</math>	<math>~\frac{(\xi_1 - \xi_2)^2}{4(\xi_1^2 - 1)} \biggl[ 1 + \frac{(\xi_1^2 - 1)}{(\xi_1 - \xi_2)^2} + \frac{(1-\xi_2^2)}{(\xi_1 - \xi_2)^2} \biggr]^2 </math>
	<math>~=</math>	<math>~\frac{1}{4(\xi_1^2 - 1)(\xi_1 - \xi_2)^2} \biggl[ (\xi_1 - \xi_2)^2 + (\xi_1^2 - 1) + (1-\xi_2^2) \biggr]^2 </math>
	<math>~=</math>	<math>~\frac{[ 2\xi_1(\xi_1 - \xi_2 ) ]^2}{4(\xi_1^2 - 1)(\xi_1 - \xi_2)^2} </math>
	<math>~=</math>	<math>~\frac{\xi_1^2}{\xi_1^2 - 1} \, . </math>

Hence, when the function, <math>~q_0</math>, is rewritten in terms of the elliptic integral of the first kind, <math>~K(\mu)</math>, the modulus of <math>~K</math> can be written as,

<math>~\mu^2</math>	<math>~=</math>	<math>~2[1+\Chi]^{-1}</math>
	<math>~=</math>	<math>~2\biggl[1+\frac{\xi_1}{(\xi_1^2 - 1)^{1/2}} \biggr]^{-1}</math>
	<math>~=</math>	<math>~\frac{2(\xi_1^2 - 1)^{1/2}}{(\xi_1^2 - 1)^{1/2}+\xi_1} \, .</math>

This is the key result motivating the use of a toroidal coordinate system to evaluate the gravitational potential: When expressed in an appropriately defined toroidal coordinate system, the modulus of the special function is a function of one, rather than two, spatial coordinates. This gives some hope that the integral over the second (angular) coordinate, <math>~\xi_2</math>, can be completed analytically, giving rise to an expression for the gravitational potential whose evaluation only requires numerical integration over a single (radial) coordinate, <math>~\xi_1</math>.

Finally, drawing on discussion in our accompanying set of notes, we recognize that, expressed in terms of toroidal coordinates, the differential area element in the meridional plane is,

<math>~d\sigma</math>

<math>~ a^2 \biggl[ \frac{d\xi_1}{(\xi_1 - \xi_2)(\xi_1^2 - 1)^{1/2}} \biggr] \biggl[ \frac{d\xi_2}{(\xi_1 - \xi_2)(1-\xi_2^2)^{1/2}} \biggr] \, . </math>

Putting everything together, then, the (indefinite) integral expression for <math>~q_0</math>, expressed in terms of toroidal-coordinates, is,

<math>~q_0</math>	<math>~=</math>	<math>~ \int \int \biggl[ \frac{a(\xi_1^2 - 1)^{1/2}}{\xi_1 - \xi_2} \biggr]^{1/2} \mu K(\mu) \rho(\xi_1, \xi_2) \biggl[ \frac{a}{(\xi_1 - \xi_2)(\xi_1^2 - 1)^{1/2}} \biggr] \biggl[ \frac{a}{(\xi_1 - \xi_2)(1-\xi_2^2)^{1/2}} \biggr] d\xi_1 d\xi_2 </math>
	<math>~=</math>	<math>~a^{5/2} \int (\xi_1^2 - 1)^{-1/4} \mu K(\mu) d\xi_1 \int \rho(\xi_1, \xi_2) \biggl[ \frac{d\xi_2}{(\xi_1 - \xi_2)^{5/2}(1-\xi_2^2)^{1/2}} \biggr] </math>
	<math>~=</math>	<math>~2^{1/2} a^{5/2} \int [ (\xi_1^2 - 1)^{1/2}+\xi_1 ]^{-1/2} K(\mu) d\xi_1 \int \rho(\xi_1, \xi_2) \biggl[ \frac{d\xi_2}{(\xi_1 - \xi_2)^{5/2}(1-\xi_2^2)^{1/2}} \biggr] \, . </math>

Making the substitution,

<math>~\xi_2~ \rightarrow ~ \sin\theta</math> <math>~\Rightarrow</math> <math>~d\xi_2~ \rightarrow ~ \cos\theta ~d\theta</math> ,

and,

<math>~\xi_1~ \rightarrow ~ \cosh x</math> <math>~\Rightarrow</math> <math>~d\xi_1~ \rightarrow ~ \sinh x ~dx</math> ,

gives,

<math>~2^{1/2} a^{5/2} \int\limits_{x_\mathrm{min}}^{x_\mathrm{max}} \frac{K(\mu) \sinh x ~dx}{( \sinh x+\cosh x )^{1/2}} \int\limits_{\theta_\mathrm{min}}^{\theta_\mathrm{max}} \rho(\xi_1, \theta) \biggl[ \frac{d\theta}{(\xi_1 - \sin\theta)^{5/2}} \biggr] \, , </math>

where, written in terms of <math>~x</math>,

<math>~ \biggl[\frac{2\sinh x}{\sinh x+\cosh x}\biggr]^{1/2} </math>

and where we have now explicitly introduced four parameters to set definite limits on the nested pair of integrations.

Perform Angular Integration Over Test Mass Distribution

While, in principle, the pair of nested integrals must be carried out over all of space using the limits,

<math>~\xi_1|_\mathrm{min} = 1 \, , ~\xi_1|_\mathrm{max} = \infty \, , ~\theta_\mathrm{min} = - \tfrac{\pi}{2} \, ,</math> and <math>~\theta_\mathrm{max} = + \tfrac{\pi}{2} \, ,</math>

in practice, the limits should be set to reflect the volume boundaries of the mass distribution that is responsible for generating the gravitational potential. Here we present analytic expressions for these limits in the case of the (pink) toroidal test mass distribution specified above; details regarding the derivation of these expressions can be found in our accompanying set of notes.

Identifying Limits of Integration

Figure 2

The animation shown here in Figure 2 builds upon the configuration displayed in our Figure 1, above. It shows a meridional cross-section through the selected (pink) uniform-density, toroidal mass distribution, whose geometric properties are fully determined by specifying values for <math>~\varpi_t</math> and <math>~r_t</math>. (For the example illustrated in Figure 2, we have specified the same values used by Trova, Huré & Hersant (2012) to construct their Figure 4, as reprinted above; namely, <math>~\varpi_t = 3/4</math> and <math>~r_t = 1/4</math>.) Throughout the Figure 2 animation sequence, these two parameters have been fixed — thereby fixing the properties of the (pink) torus — and, in addition, we have fixed the location of the origin of a toroidal coordinate system — identified by the red-filled circular dot. We explicitly associate this coordinate-system origin with the (cylindrical) coordinates of the point in space at which we choose to evaluate the gravitational potential, namely, <math>~(R_*, Z_*) = (a, Z_0)</math>. [For illustration purposes, in Figure 2 we have set <math>~(a, Z_0) = (1/3, 3/4)</math>.]

While the values of the four primary model parameters <math>~(a, Z_0, \varpi_t, r_t)</math> are held fixed, Figure 2 depicts in a quantitatively precise manner how the size of a <math>~\xi_1</math>= constant toroidal surface (the off-center circle traced by the sequence of black dots) varies as the value of the radial coordinate, <math>~\xi_1</math>, is varied. In each frame of the animation sequence, the value of <math>~\xi_1</math> that was used to define the (black) <math>\xi_1</math>-circle is printed in the lower-right corner of the image; additional quantitative details associated with each animation frame can be obtained from the table titled, "Example 2" in our accompanying notes.

As the value of <math>~\xi_1</math> is varied from large values (small black circles) to smaller values (larger black circles), there is a maximum value, <math>~\xi_1|_\mathrm{max}</math>, at which the <math>\xi_1</math>-circle first makes contact with the (pink) equatorial-plane torus, and there is a minimum value, <math>~\xi_1|_\mathrm{min}</math>, at which it makes its final contact. These are the limiting values of the toroidal radial coordinate to be used in the integration that produces <math>~q_0</math>. At all values within the parameter range,

<math>~\xi_1|_\mathrm{max} > \xi_1 > ~\xi_1|_\mathrm{min} \, ,</math>

the <math>\xi_1</math>-circle intersects the surface of the torus in two locations, defined by two different values of the associated angular coordinate, <math>~\xi_2</math> — see, for example, the coordinates listed in the table associated with example 2, in our accompanying notes. For each relevant value of <math>~\xi_1</math>, these are the limiting values of the toroidal angular coordinate to be used in the integration that produces <math>~q_0</math>. It should be realized that, at the first and final points of contact, the two values of <math>~\xi_2</math> will be degenerate.

Notice in the animation that, while the origin of the selected toroidal coordinate system (the filled red dot) remains fixed, the center of the <math>\xi_1</math>-circle does not remain fixed. In order to highlight this behavior, the location of the center of the <math>\xi_1</math>-circle has been marked by a filled, light-blue square and, in keeping with the earlier Figure 1 diagram, a vertical, light-blue line connects this center to the equatorial plane of the cylindrical coordinate system.

Associated Analytic Expressions

We define the following terms that are functions only of the four principal model parameters, <math>~(a, Z_0, \varpi_t, r_t)</math>, and therefore can be treated as constants while carrying out the pair of nested integrals that determine <math>~q_0</math>:

<math>~\kappa</math>	<math>~\equiv</math>	<math>~ Z_0^2 + a^2 - (\varpi_t^2 - r_t^2) = \frac{5^2}{2^4\cdot 3^2} \approx 0.17361111\, ; </math>
<math>~C</math>	<math>~\equiv</math>	<math>~1 + \biggl( \frac{2Z_0}{\kappa}\biggr)^2 ( \varpi_t^2 - r_t^2) = \frac{17 \cdot 1409}{5^4} \approx 38.3248 \, ; </math>
<math>~\beta_\pm</math>	<math>~\equiv</math>	<math>~ - \frac{\kappa}{2} \biggl[ \frac{\varpi_t \mp r_t \sqrt{C}}{(\varpi_t + r_t)(\varpi_t - r_t)} \biggr] = -~\frac{5^2}{2^6\cdot 3}\biggl[ 1\mp \sqrt{ \frac{17\cdot 1409}{3^2\cdot 5^4}} \biggr] \, . </math>

Then,

<math>\xi_1|_\mathrm{max} = \biggl[1-\biggl( \frac{a}{\varpi_t-\beta_+} \biggr)^2 \biggr]^{-1/2} \approx 1.1927843</math> and <math>\xi_1|_\mathrm{min} = \biggl[1-\biggl( \frac{a}{\varpi_t-\beta_-} \biggr)^2 \biggr]^{-1/2} \approx 1.0449467\, ,</math>

which implies,

<math>x_\mathrm{max} = \tanh^{-1}\biggl( \frac{a}{\varpi_t-\beta_+} \biggr) \approx 0.000 </math> and <math>x_\mathrm{min} = \tanh^{-1}\biggl( \frac{a}{\varpi_t-\beta_-} \biggr) \approx 0.000 \, .</math>

Also let,

<math>~\xi_1 \rightarrow \cosh x \, ,</math> in which case, <math>~(\xi_1^2 - 1)^{1/2} \rightarrow \sinh x</math> and, <math>~(1-\xi_1^{-2})^{-1/2} = \frac{\xi_1}{(\xi_1^2 - 1)^{1/2}} \rightarrow \coth x\, ,</math>

and define,

<math>~A(\xi_1)</math>	<math>~\equiv</math>	<math>~1 - \biggl[ \frac{\varpi_t}{Z_0} - \frac{a}{Z_0}(1 - \xi_1^{-2})^{-1/2}\biggr]^2</math>
	<math>~=</math>	<math>~1 + \frac{a}{Z_0} \biggl[ \coth x - \frac{\varpi_t}{a} \biggr]^2 \, ;</math>
<math>~B(\xi_1)</math>	<math>~\equiv</math>	<math>~(1 - \xi_1^{-2})^{-1/2} - \frac{\varpi_t}{a}\biggl[1 - \biggl( \frac{2aZ_0}{\kappa} \biggr) \biggr]</math>
	<math>~=</math>	<math>~\coth x - \frac{\varpi_t}{a} + \biggl( \frac{2\varpi_t Z_0}{\kappa} \biggr) \, . </math>

Then,

<math>~\xi_2\biggr|_\pm</math>

<math>~ \cosh x - \sinh x \biggl( \frac{2aZ_0}{\kappa}\cdot \frac{A}{B} \biggr) \biggl[1 \pm \sqrt{1-\frac{AC}{B^2}}\biggr]^{-1} \, . </math>

References

Morse, P.M. & Feshmach, H. 1953, Methods of Theoretical Physics — Volumes I and II
Cohl, H.S. & Tohline, J.E. 1999, ApJ, 527, 86-101
Cohl, H.S., Rau, A.R.P., Tohline, J.E., Browne, D.A., Cazes, J.E. & Barnes, E.I. 2001, Phys. Rev. A, 64, 052509

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