User:Tohline/Apps/DysonPotential

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Dyson (1893)

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

In his pioneering work, F. W. Dyson (1893a, Philosophical Transactions of the Royal Society of London. A., 184, 43 - 95) and (1893b, 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.

External Potential

His Derived Expression

(See an accompanying Ramblings Chapter for additional derivation details.) On p. 62 of Dyson (1893a), we find the following approximate expression for the potential at point "P", anywhere exterior to an anchor ring:

Anchor Ring Schematic

Caption: Anchor ring schematic, adapted from figure near the top of §2 (on p. 47) of Dyson (1893a)

Equation image extracted without modification from p. 62 of Dyson (1893a)

The Potential of an Anchor Ring, Phil. Trans. Royal Soc. London. A., Vol. 184

The Potential Exterior to an Anchor Ring

In Dyson's expression, the leading factor of <math>~F</math> is the complete elliptic integral of the first kind, namely,

<math>~F = F(\mu)</math>

<math>~\equiv</math>

<math>~\int_0^{\pi/2} \frac{d\phi}{\sqrt{1 - \mu^2 \sin^2\phi}} \, ,</math>

where, <math>~\mu \equiv (R_1 - R)/(R_1 + R)</math>. Similarly, <math>~E = E(\mu)</math> is the complete elliptic integral of the second kind.

Comparison With Thin Ring Approximation

In the limit of <math>~a/c \rightarrow 0</math>, Dyson's expression gives,

<math>~V_\mathrm{Dyson}</math>

<math>~=</math>

<math>~\frac{4K(\mu)}{R+R_1} \, ,</math>

where we have acknowledged that, in the twenty-first century, the complete elliptic integral of the first kind is more customarily represented by the letter, <math>~K</math>. In a separate discussion, we have shown that the gravitational potential of an infinitesimally thin ring is given precisely by the expression,

<math>~\biggl[ \frac{\pi}{GM}\biggr] \Phi_\mathrm{TR}</math>

<math>~=</math>

<math>~- \frac{2K(k)}{R_1} \, ,</math>

where, <math>~k \equiv [1-(R/R_1)^2]^{1 / 2}</math>. Is Dyson's expression identical to this one when <math>~a/c = 0</math> ?

Proof

Taking a queue from our accompanying discussion of toroidal coordinates, if we adopt the variable notation,

<math>~\eta \equiv \ln\biggl(\frac{R_1}{R}\biggr) \, ,</math>

then we can write,

<math>~\cosh\eta = \frac{1}{2}\biggl[e^\eta + e^{-\eta}\biggr]</math>

<math>~=</math>

<math>~\frac{R^2 + R_1^2}{2RR_1} \, ,</math>

which implies that,

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

<math>~=</math>

<math>~\biggl[ 1 - \biggl(\frac{R}{R_1}\biggr)^2 \biggr]^{1 / 2} \, .</math>

This is the definition of the parameter, <math>~k</math>, in the expression for <math>~\Phi_\mathrm{TR}</math>. Now, if we employ the Descending Landen Transformation for the complete elliptic integral of the first kind, we can make the substitution,

<math>~K(k)</math>

<math>~=</math>

<math>~ (1 + k_1)K(k_1) \, , </math>

      where,      

<math>~k_1</math>

<math>~\equiv</math>

<math>~ \frac{1-\sqrt{1-k^2}}{1+\sqrt{1-k^2}} \, . </math>

But notice that, <math>~\sqrt{1-k^2} = e^{-\eta}</math>, in which case,

<math>~k_1 </math>

<math>~=</math>

<math>~ \frac{1-e^{-\eta}}{1+e^{-\eta}} </math>

<math>~=</math>

<math>~ \frac{1-R/R_1}{1+R/R_1} </math>

<math>~=</math>

<math>~ \frac{R_1-R}{R_1+R} \, , </math>

which is the definition of the parameter, <math>~\mu</math>, in the expression for <math>~V_\mathrm{Dyson}</math>. Hence, we can write,

<math>~\biggl[ \frac{\pi}{GM}\biggr] \Phi_\mathrm{TR}</math>

<math>~=</math>

<math>~- \frac{2}{R_1} \biggl[(1+k_1)K(k_1) \biggr] </math>

 

<math>~=</math>

<math>~- \frac{2K(\mu)}{R_1} \biggl[1+\frac{R_1-R}{R_1+R} \biggr] </math>

 

<math>~=</math>

<math>~- \frac{4K(\mu)}{R_1+R} \, .</math>

Aside from the adopted sign convention, this is indeed precisely the expression given by <math>~V_\mathrm{Dyson}</math> when <math>~a/c = 0</math> .

Evaluation

Dyson's Figures

In his effort to illustrate the behavior of equipotential contours in the space exterior to various anchor rings, Dyson evaluated his expression for the potential up through <math>~\mathcal{O}(\tfrac{a^2}{c^2})</math>; that is, he evaluated the function,

<math>~V_2 \equiv V_\mathrm{Dyson}\biggr|_{\mathcal{O}(a^2/c^2)}</math>

<math>~=</math>

<math>~ \frac{4K(\mu)}{R+R_1}\biggl[1 - \frac{1}{8}\biggl(\frac{a^2}{c^2}\biggr) \cos^2\biggl( \frac{\psi}{2}\biggr)\biggr] + \frac{(R + R_1)E(\mu)}{RR_1}\biggl[\frac{1}{8}\biggl(\frac{a^2}{c^2}\biggr) \cos\psi \biggr] \, . </math>

Figures 1 - 6 from Dyson (1893a) — replicated immediately below — show his resulting set of contours for six cases: Tori (anchor rings) having aspect ratios of <math>~a/c = 0, 1/5, 2/5, 3/5, 4/5, 1</math>. Click on an image to view the contour plot at higher resolution. In what follows we present results from our own evaluation of this "V2" function for the single case of an anchor ring having <math>~a/c = 2/5</math>.


Figures 1 - 6 extracted without modification from pp. 63-66 of F. W. Dyson (1893)

The Potential of an Anchor Ring, Phil. Trans. Royal Soc. London. A., Vol. 184

The Potential Exterior to an Anchor Ring; R/d = infinity
The Potential Exterior to an Anchor Ring; R/d = 5
The Potential Exterior to an Anchor Ring; R/d = 2.5
The Potential Exterior to an Anchor Ring; R/d = 1.667
The Potential Exterior to an Anchor Ring; R/d = 1.25
The Potential Exterior to an Anchor Ring; R/d = 1

Our Attempt to Replicate

First, let's test the accuracy of Dyson's (1893a) "series expansion" expression for the elliptic integrals, <math>~K(\mu)</math> and <math>~E(\mu)</math>; in the following table, the high-precision evaluations labeled "Numerical Recipes" have been drawn from the tabulated data that is provided in our accompanying discussion of incomplete elliptic integrals. According to, for example, Wikipedia, the relevant series-expansion expressions are:

<math>~K(\mu)</math>

<math>~=</math>

<math>~ \frac{\pi}{2} \biggl\{ 1 + \biggl[\frac{1}{2}\biggr]^2\mu^2 + \biggl[ \frac{1\cdot 3}{2\cdot 4}\biggr]^2\mu^4 + \biggl[ \frac{5\cdot 3\cdot 1}{6\cdot 4\cdot 2} \biggr]^2 \mu^6 + \cdots + \biggl[ \frac{(2n-1)!!}{(2n)!!}\biggr]^2 \mu^{2n} +\cdots \biggr\} \, ; </math>

<math>~E(\mu)</math>

<math>~=</math>

<math>~ \frac{\pi}{2} \biggl\{ 1 ~-~ \biggl[\frac{1}{2}\biggr]^2\frac{\mu^2}{1} ~-~ \biggl[ \frac{1\cdot 3}{2\cdot 4}\biggr]^2 \frac{\mu^4}{3} ~- ~\biggl[ \frac{5\cdot 3\cdot 1}{6\cdot 4\cdot 2} \biggr]^2 \frac{\mu^6}{5} ~- ~\cdots ~- ~\biggl[ \frac{(2n-1)!!}{(2n)!!}\biggr]^2 \frac{\mu^{2n}}{2n-1} ~-~ \cdots \biggr\} \, . </math>

These expressions — up through <math>~\mathcal{O}(\mu^4)</math> — can be found in the middle of p. 58 of Dyson (1893a). We strongly suspect that, in constructing the equipotential contours shown in his figures 1-6, Dyson used expressions for <math>~K(\mu)</math> and <math>~E(\mu)</math> that were more accurate than this. For example, we found it necessary to include terms up through <math>~\mathcal{O}(\mu^{10})</math> in order to match to three digits accuracy the potential contour values and coordinate locations reported by Dyson.

<math>~\mu</math> Numerical Recipes Series expansion up through <math>~\mathcal{O}(\mu^4)</math> Series expansion up through <math>~\mathcal{O}(\mu^{10})</math>
<math>~K(\mu)</math> <math>~E(\mu)</math> <math>~K(\mu)</math> <math>~E(\mu)</math> <math>K(\mu)~</math> <math>~E(\mu)</math>
0.34202014 1.62002589 1.52379921 1.6198 1.5239 1.6200263 1.5237989
0.57357644 1.73124518 1.43229097 1.7239 1.4336 1.73124518 1.43230
0.76604444 1.93558110 1.30553909 1.8773 1.3150 1.93558109 1.3061
0.90630779 2.30878680 1.16382796 2.042 1.199 2.308784 1.1700
0.98480775 3.15338525 1.04011440 2.16 1.12 3.150 1.069

We actually used the "descending Landen transformation" to evaluate <math>~K(\mu)</math> through <math>~\mathcal{O}(\mu^{10})</math>.


For <math>~c=1</math> and a specification of the ratio, <math>~a/c</math>, take the following steps to map out an equipotential curve that has <math>~V_2 = V_0</math>:

  • Choose a value of <math>~R \ge a</math>
    • Guess a value of <math>~(c-R) \le R_1 \le (c+R) ~~~\Rightarrow ~~~ \varpi = (R_1^2 - R^2)/(4c)</math>     and,     <math>~z = \pm \sqrt{ R_1^2 - (c+\varpi)^2}</math>
    • Set <math>~ \cos\psi = (R_1^2 + R^2 - 4c^2)/(2RR_1)</math>
    • Evaluate the function, <math>~V_2</math>
    • If <math>~V_2 \ne V_0</math> to the desired accuracy, loop back up and guess another value of <math>~R_1</math>
  • If <math>~V_2 = V_0</math> to the desired accuracy, save the coordinate location, <math>~(\varpi,z)</math>, and loop back up to pick another value of <math>~R</math>


The Potential Exterior to an Anchor Ring; R/d = 2.5
   Compare with Dyson

Tabulated Data

As the data in the following table documents, we have been able to construct equipotential contours that agree with Dyson, not only qualitatively, but quantitatively. For example:

  • The dark green contour has been designed to touch the surface of the torus precisely where its outermost edge cuts through the equatorial plane <math>~(\varpi,z) = (1.4,0)</math>. This means that <math>~R = 0.4</math> and <math>~R_1 = 2.4</math>. (These four coordinate values are highlighted in pink in the second major column of the table.) When we plugged these values of <math>~R</math> and <math>~R_1</math> into Dyson's expression for <math>~V_2</math>, we determined that the value of the potential at this point on the torus surface is 0.8551 — see the yellow-highlighted heading of the second major table column. Compare this to the value of 0.855 that Dyson has printed just below the Figure 3 x-axis where a fiducial identifies the coordinate, <math>~\varpi = 1.4</math>. As has been catalogued at the bottom of table column #2, we have found that this dark-green contour touches the vertical axis at the coordinate location, <math>~(\varpi,z) = (0,0.572)</math>, for which, <math>~R_1 = R = 1.1518</math>.
  • By design — see the coordinate values highlighted in pink in table column #1 — our outermost (pink) contour touches the equatorial plane at <math>~(\varpi,z) = (1.5,0) ~\Rightarrow ~ (R,R_1) = (0.5,2.5)</math>. When we plugged these values of <math>~R</math> and <math>~R_1</math> into Dyson's expression for <math>~V_2</math>, we determined that the value of the potential at this point outside the torus is 0.7737 — see the yellow-highlighted heading of table column #1. Compare this to the value of 0.777 that Dyson has printed just below the Figure 3 x-axis where a fiducial identifies the coordinate, <math>~\varpi = 1.5</math>. As has been catalogued at the bottom of table column #1, we have found that this pink contour touches the vertical axis at the coordinate location, <math>~(\varpi,z) = (0,0.794)</math>, for which, <math>~R_1 = R = 1.2766</math>.
  • Similarly, we have constructed contours that intersect the equatorial plane at the fiducials marking <math>~\varpi = 0.0</math> (red curve & table column #5), <math>~\varpi = 0.2</math> (light-green curve & table column #6), and <math>~\varpi = 0.4</math> (light-blue curve & table column #7). According to our calculations, they correspond, respectively, to values of the potential, <math>~V_2 = 0.9800</math> (Dyson's corresponding fiducial label is 0.980), <math>~V_2 = 0.9896</math> (Dyson's corresponding fiducial label is 0.990), and <math>~V_2 = 1.0212</math> (Dyson's corresponding fiducial label is 1.021).
  • Finally, we constructed two contours (blue and orange) by initially specifying the value of the potential, rather than specifying the coordinate values <math>~(R,R_1)</math>. We used the values of the potential that Dyson associated with the fiducials along the vertical axis at <math>~(\varpi,z) = (0.0,0.4)</math> and at <math>~(\varpi,z) = (0.0,0.2)</math>: Respectively, <math>~V_2 = 0.912</math> — blue contour detailed in our table column #3 — and <math>~V_2 = 0.961</math>— orange contour detailed in our table column #4. We determined that these two contour curves intersected the vertical axis at, respectively, <math>~(\varpi,z) = (0.0, 0.402)</math> and <math>~(\varpi,z) = (0.0, 0.204)</math>, that is, at coordinate locations that were nearly identical to the locations labeled by Dyson.


Coordinates of Points that Trace Seven Different Equipotential Contours External to the Anchor Ring With <math>~c/a = 5/2</math>

Column #1 Column #2 Column #3 Column #4 Column #5 Column #6 Column #7
V2 = 0.7737

R

R1

<math>~\varpi</math>

z

0.5000

2.5000

1.500

0.000

0.5005

2.4990

1.499

0.043

0.504

2.4889

1.485

0.137

0.510

2.4720

1.463

0.215

0.520

2.4445

1.426

0.298

0.530

2.4177

1.391

0.358

0.550

2.3665

1.324

0.444

0.580

2.2940

1.232

0.532

0.610

2.2265

1.146

0.592

0.640

2.1632

1.067

0.636

0.700

2.0465

0.925

0.696

0.800

1.8745

0.718

0.749

0.9000

1.7240

0.541

0.774

1.000

1.5890

0.381

0.786

1.100

1.4670

0.236

0.791

1.2000

1.3558

0.100

0.793

1.277

1.2766

0.000

0.794



V2 = 0.8551

R

R1

<math>~\varpi</math>

z

0.400

2.4000

1.400

0.000

0.405

2.3830

1.379

0.144

0.410

2.3668

1.358

0.199

0.425

2.3190

1.299

0.302

0.450

2.2458

1.210

0.398

0.480

2.1655

1.115

0.466

0.520

2.0690

1.003

0.520

0.570

1.9610

0.880

0.557

0.620

1.8635

0.772

0.577

0.700

1.7240

0.621

0.588

0.800

1.5712

0.457

0.588

0.900

1.4360

0.313

0.581

1.000

1.3147

0.182

0.575

1.100

1.2050

0.061

0.572

1.1518

1.1518

0.000

0.572


V2 = 0.9120

R

R1

<math>~\varpi</math>

z

1.0776

1.0776

0.000

0.402

1.000

1.1582

0.085

0.404

0.950

1.2135

0.143

0.409

0.900

1.2715

0.202

0.416

0.800

1.3979

0.328

0.435

0.700

1.5401

0.470

0.458

0.600

1.7040

0.636

0.477

0.550

1.7970

0.732

0.480

0.500

1.8998

0.840

0.474

0.475

1.9560

0.900

0.464

0.440

2.0410

0.993

0.440

0.400

2.1510

1.117

0.383


V2 = 0.9610

R

R1

<math>~\varpi</math>

z

1.0206

1.0206

0.000

0.204

0.9500

1.0937

0.073

0.210

0.900

1.1488

0.127

0.221

0.800

1.2685

0.242

0.257

0.700

1.4030

0.370

0.304

0.600

1.5572

0.516

0.355

0.550

1.6440

0.600

0.378

0.500

1.7395

0.694

0.395

0.450

1.8462

0.801

0.404

0.410

1.9690

0.929

0.394



V2 = 0.9800

R

R1

<math>~\varpi</math>

z

1.0000

1.0000

0.000

0.000

0.900

1.1053

0.103

0.072

0.800

1.2225

0.214

0.147

0.700

1.3543

0.336

0.222

0.600

1.5050

0.476

0.293

0.550

1.5897

0.556

0.325

0.500

1.6827

0.645

0.352

0.450

1.7865

0.747

0.372

0.400

1.9050

0.867

0.377


V2 = 0.9896

R

R1

<math>~\varpi</math>

z

0.8000

1.2000

0.200

0.000

0.7950

1.2062

0.206

0.034

0.780

1.2248

0.223

0.068

0.760

1.2503

0.246

0.099

0.730

1.2895

0.282

0.134

0.700

1.3305

0.320

0.166

0.650

1.4022

0.386

0.213

0.600

1.4796

0.457

0.256

0.550

1.5633

0.535

0.294

0.500

1.6552

0.622

0.328

0.450

1.7573

0.721

0.353

0.400

1.8737

0.838

0.366



V2 = 1.0212

R

R1

<math>~\varpi</math>

z

0.6000

1.4000

0.400

0.000

0.5950

1.4078

0.407

0.048

0.580

1.4315

0.428

0.097

0.570

1.4477

0.443

0.120

0.540

1.4978

0.488

0.171

0.500

1.5688

0.553

0.224

0.450

1.6663

0.644

0.275

0.400

1.7767

0.749

0.312

Intermediate Step

Objective

As has been reprinted above, on p. 62 of Dyson's Part I we find his power-series expression for the external potential, namely,

<math>~\frac{\pi V_\mathrm{Dyson}}{GM} \biggr|_{\mathcal{O}(a^4/c^4)}</math>

<math>~=</math>

<math>~ \frac{4K(\mu)}{R+R_1}\biggl\{ 1 ~-~ \frac{1}{8}\biggl(\frac{a^2}{c^2}\biggr) \cos^2\biggl( \frac{\psi}{2}\biggr) - \frac{1}{768}\biggl(\frac{a}{c}\biggr)^4 \biggl[ 5 ~+~ 8\cos\psi ~-~ \cos^2\psi ~-~ 4\cos^3\psi ~-~ \frac{4c^2}{RR_1} \cos2\psi \biggr] \biggr\} </math>

 

 

<math>~ + \frac{(R + R_1)E(\mu)}{RR_1}\biggl\{ \frac{1}{8}\biggl(\frac{a}{c}\biggr)^2 \cos\psi ~-~\frac{1}{192} \biggl(\frac{a}{c}\biggr)^4 \biggl[ 2\cos^2\psi ~-~4\cos\psi ~+~ \frac{2c^2}{RR_1}\cos2\psi \biggr] \biggr\} \, , </math>

where — as in the context of toroidal coordinates — we occasionally will make the substitution, <math>~e^\eta = R_1/R</math>, and therefore,

<math>~\mu</math>

<math>~\equiv</math>

<math>~\frac{R_1 - R}{R_1+R} = \frac{e^\eta - 1}{e^\eta + 1} \, . </math>

In order to facilitate matching boundary conditions at the surface of the torus, between the exterior and interior expressions for the gravitational potential, Dyson rewrites this Part I expression for the external potential and — explicitly evaluating it on the torus surface — sets, <math>~R = a</math>. Specifically, on p. 1049 of Dyson's Part II we find equation (6), which reads,

<math>~\frac{V}{2\pi a^2}</math>

<math>~=</math>

<math>~ \ln\biggl(\frac{8c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr) \biggl[\ln\biggl(\frac{8c}{a}\biggr) - \frac{5}{4}\biggr] \cos\chi + \biggl\{ \frac{1}{16} \biggl[ \ln\biggl(\frac{8c}{a}\biggr) - \frac{5}{2} \biggr] + \frac{3}{16} \biggl[\ln\biggl(\frac{8c}{a}\biggr) +\frac{17}{36} - \frac{72}{36}\biggr]\cos2\chi\biggr\}\biggl(\frac{a^2}{c^2}\biggr) </math>

 

 

<math>~ + \biggl\{ \frac{3}{32}\biggl[ \ln\biggl(\frac{8c}{a}\biggr) - \frac{25}{12}\biggr]\cos\chi + \frac{5}{64}\biggl[ \ln\biggl(\frac{8c}{a}\biggr)+\frac{7}{24} - \frac{48}{24}\biggr]\cos3\chi \biggr\} \biggl(\frac{a^3}{c^3}\biggr) </math>

 

 

<math>~ + \biggl\{ \frac{9}{256}\biggl[ \ln\biggl(\frac{8c}{a}\biggr) - 2\biggr] + \frac{7}{128}\biggl[ \ln\biggl(\frac{8c}{a}\biggr) - \frac{19}{168} - 2\biggr]\cos2\chi + \frac{35}{1024} \biggl[ \ln\biggl(\frac{8c}{a}\biggr) - 2 + \frac{19}{120}\biggr]\cos4\chi \biggr\} \biggl(\frac{a^4}{c^4}\biggr) ~+~\cdots </math>

In order to obtain this alternate power-series expression, Dyson …

  • Expresses angular variations in terms of the angle, <math>~\chi</math>, instead of the angle, <math>~\psi</math>; these two angles are identified in the above schematic.
  • Employs power-series expansions of both elliptic integral functions, <math>~K(\mu)</math> and <math>~E(\mu)</math>.
  • Uses the binomial theorem to develop a number of other power-series expressions.

In what follows we will attempt to demonstrate that this second (Part II, equation 6) expression is identical to the first.

The Ratio R1/c

Note that, via the law of cosines,

<math>~R_1^2</math>

<math>~=</math>

<math>~(2c)^2 + R^2 - 4Rc\cos\chi</math>

<math>~\Rightarrow ~~~\biggl(\frac{R_1}{c}\biggr)^2</math>

<math>~=</math>

<math>~4 + \biggl( \frac{R}{c}\biggr)^2 - 4\biggl(\frac{R}{c}\biggr)\cos\chi</math>

At the surface of the torus, where <math>~R=a</math>, we therefore have,

<math>~\frac{R_1}{c}</math>

<math>~=</math>

<math>~2\biggl[ 1 - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr]^{1 / 2} \, .</math>

Low Order

Employing the binomial theorem, we can write,

<math>~\biggl(\frac{R_1}{c}\biggr)^{-1}</math>

<math>~=</math>

<math>~\frac{1}{2}\biggl[ 1 - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr]^{-1 / 2} </math>

 

<math>~\approx</math>

<math>~\frac{1}{2}\biggl\{ 1 - \frac{1}{2} \biggl[ - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr] + \frac{3}{8}\biggl[ - \biggl(\frac{a}{c}\biggr)\cos\chi + \cancelto{0}{\frac{1}{4}\biggl( \frac{a}{c}\biggr)^2} \biggr]^2\biggr\}</math>

 

<math>~\approx</math>

<math>~\frac{1}{2}\biggl\{ 1 + \frac{1}{2} \biggl(\frac{a}{c}\biggr)\cos\chi - \frac{1}{8}\biggl( \frac{a}{c}\biggr)^2 + \frac{3}{8}\biggl(\frac{a}{c}\biggr)^2\cos^2\chi \biggr\}</math>

 

<math>~\approx</math>

<math>~ \frac{1}{2}\biggl\{ 1 + \frac{1}{2} \biggl(\frac{a}{c}\biggr)\cos\chi +\biggl( \frac{a}{c}\biggr)^2 \biggl[ \frac{3}{8}\cos^2\chi -\frac{1}{8}\biggr] \biggr\} </math>

<math>~\Rightarrow~~~ 1 + \frac{a}{c} \biggl(\frac{R_1}{c}\biggr)^{-1}</math>

<math>~\approx</math>

<math>~1 + \frac{1}{2}\biggl( \frac{a}{c}\biggr) + \frac{1}{4} \biggl(\frac{a}{c}\biggr)^2\cos\chi </math>

<math>~\Rightarrow~~~ \biggl[ 1 + \frac{a}{c} \biggl(\frac{R_1}{c}\biggr)^{-1} \biggr]^{-1}</math>

<math>~\approx</math>

<math>~1 - \biggl[ \frac{1}{2}\biggl( \frac{a}{c}\biggr) + \frac{1}{4} \biggl(\frac{a}{c}\biggr)^2\cos\chi \biggr] + \biggl[ \frac{1}{2}\biggl( \frac{a}{c}\biggr) + \frac{1}{4} \biggl(\frac{a}{c}\biggr)^2\cos\chi \biggr]^2 </math>

 

<math>~\approx</math>

<math>~ 1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) + \frac{1}{4} \biggl( \frac{a}{c}\biggr)^2 (1 - \cos\chi) </math>

Higher Order

Adopting the shorthand notation,

<math>~\gamma \equiv \frac{1}{2}\biggl(\frac{R_1}{c}\biggr) \, ,</math>      and       <math>~b \equiv - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \, ,</math>

and employing the binomial theorem, we can write,

<math>~\gamma = \biggl[ 1 + b \biggr]^{1 / 2}</math>

<math>~=</math>

<math>~ 1 + \frac{1}{2}b - \frac{1}{2^3}b^2 + \frac{1}{2^4}b^3 - \frac{3\cdot 5}{2^7\cdot 3}b^4 + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ 1 + \frac{1}{2}\biggl[ - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr] - \frac{1}{2^3}\biggl[ - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr]^2 </math>

 

 

<math>~ + \frac{1}{2^4}\biggl[ - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr]^3 - \frac{3\cdot 5}{2^7\cdot 3}\biggl[ - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr]^4 + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ 1 ~-~ \frac{1}{2} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 - \frac{1}{2^3}\biggl[\biggl(\frac{a}{c}\biggr)^2\cos^2\chi - \frac{1}{2}\biggl( \frac{a}{c}\biggr)^3 \cos\chi + \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^4 \biggr] </math>

 

 

<math>~ + \frac{1}{2^4}\biggl[ - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr]\biggl[\biggl(\frac{a}{c}\biggr)^2\cos^2\chi - \frac{1}{2}\biggl( \frac{a}{c}\biggr)^3 \cos\chi \biggr] - \frac{3\cdot 5}{2^7\cdot 3}\biggl[ \biggl(\frac{a}{c}\biggr)^4\cos^4\chi \biggr] + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ 1 ~-~ \frac{1}{2} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 -~\frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2\cos^2\chi ~+~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 \cos\chi ~-~ \frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 </math>

 

 

<math>~ -~\frac{1}{2^4} \biggl(\frac{a}{c}\biggr)^3\cos^3\chi ~+~ \frac{1}{2^5}\biggl( \frac{a}{c}\biggr)^4 \cos^2\chi + \frac{1}{2^6} \biggl(\frac{a}{c}\biggr)^4\cos^2\chi ~-~ \frac{3\cdot 5}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^4\cos^4\chi ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ 1 ~-~ \frac{1}{2} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (1-\cos^2\chi) +~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (\cos\chi - \cos^3\chi) +~\frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 \biggl[~-~ 1 ~+~ 6 \cos^2\chi ~-~ 5 \cos^4\chi \biggr] ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, . </math>

Also, we have,

<math>~\frac{1}{\gamma} = 2\biggl(\frac{R_1}{c}\biggr)^{-1} = \biggl[ 1 + b \biggr]^{-1 / 2}</math>

<math>~=</math>

<math>~ 1 -\frac{1}{2}b + \frac{3}{2^3}b^2 - \frac{3\cdot 5}{2^4\cdot 3}b^3 + \frac{3\cdot 5\cdot 7}{2^7\cdot 3}b^4 + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~1 - \frac{1}{2}\biggl[- \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2\biggr] + \frac{3}{2^3}\biggl[- \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2\biggr]^2 </math>

 

 

<math>~ - \frac{3\cdot 5}{2^4\cdot 3}\biggl[- \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2\biggr]^3 + \frac{3\cdot 5\cdot 7}{2^7\cdot 3}\biggl[- \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2\biggr]^4 + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr)</math>

 

<math>~=</math>

<math>~1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi - \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 + \frac{3}{2^3}\biggl[\biggl(\frac{a}{c}\biggr)^2\cos^2\chi ~-~ \frac{1}{2} \biggl(\frac{a}{c}\biggr)^3\cos\chi ~+~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^4\biggr] </math>

 

 

<math>~ - \frac{3\cdot 5}{2^4\cdot 3}\biggl[- \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2\biggr] \biggl[\biggl(\frac{a}{c}\biggr)^2\cos^2\chi ~-~ \frac{1}{2} \biggl(\frac{a}{c}\biggr)^3\cos\chi \biggr] + \frac{3\cdot 5\cdot 7}{2^7\cdot 3}\biggl[\biggl(\frac{a}{c}\biggr)^4\cos^4\chi \biggr] + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr)</math>

 

<math>~=</math>

<math>~1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi - \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 + \frac{3}{2^3}\biggl(\frac{a}{c}\biggr)^2\cos^2\chi ~-~ \frac{3}{2^4}\biggl(\frac{a}{c}\biggr)^3\cos\chi ~+~ \frac{3}{2^7}\biggl( \frac{a}{c}\biggr)^4 </math>

 

 

<math>~ + \frac{3\cdot 5}{2^4\cdot 3} \biggl[\biggl(\frac{a}{c}\biggr)^3\cos^3\chi ~-~ \frac{1}{2} \biggl(\frac{a}{c}\biggr)^4\cos^2\chi \biggr] - \frac{3\cdot 5}{2^6\cdot 3} \biggl[\biggl(\frac{a}{c}\biggr)^4\cos^2\chi\biggr] + \frac{3\cdot 5\cdot 7}{2^7\cdot 3}\biggl[\biggl(\frac{a}{c}\biggr)^4\cos^4\chi \biggr] + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr)</math>

 

<math>~=</math>

<math>~1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 \biggl[ 3\cos^2\chi - 1 \biggr] + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3\biggl[ 5\cos^3\chi ~-~ 3\cos\chi \biggr] </math>

 

 

<math>~ ~+~ \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 \biggl[ 3 ~-~ 30 \cos^2\chi ~+~ 35 \cos^4\chi \biggr] ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, . </math>

Hence,

<math>~1~+~\biggl( \frac{a}{c}\biggr)\biggl(\frac{R_1}{c}\biggr)^{-1} </math>

<math>~=</math>

<math>~1 ~+~\frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 \biggl[ 3\cos^2\chi - 1 \biggr] ~+~ \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^4\biggl[ 5\cos^3\chi ~-~ 3\cos\chi \biggr] ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, . </math>

And, adopting the shorthand notation,

<math>~d \equiv \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 \biggl[ 3\cos^2\chi - 1 \biggr] ~+~ \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^4\biggl[ 5\cos^3\chi ~-~ 3\cos\chi \biggr]

\, ,</math>

we have,

<math>~\biggl[1~+~\biggl( \frac{a}{c}\biggr)\biggl(\frac{R_1}{c}\biggr)^{-1}\biggr]^{-1} </math>

<math>~=</math>

<math>~ 1 -d + d^2 - d^3 + d^4 ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~-~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 \biggl[ 3\cos^2\chi - 1 \biggr] ~-~ \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^4\biggl[ 5\cos^3\chi ~-~ 3\cos\chi \biggr] </math>

 

 

<math>~+~ \biggl\{ \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 \biggl[ 3\cos^2\chi - 1 \biggr] \biggr\} \biggl\{ \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 \biggl[ 3\cos^2\chi - 1 \biggr] \biggr\} </math>

 

 

<math>~-~ \biggl[ \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi \biggr] \biggl[ \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi \biggr]^2 ~+~\frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^4 ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~-~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 \biggl[ 3\cos^2\chi - 1 \biggr] ~-~ \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^4\biggl[ 5\cos^3\chi ~-~ 3\cos\chi \biggr] </math>

 

 

<math>~+~ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggl\{ \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 \biggl[ 3\cos^2\chi - 1 \biggr] \biggr\} </math>

 

 

<math>~+~ \biggl\{\frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi \biggr\} \biggl\{ \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi \biggr\} ~+~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 \biggl[ 3\cos^2\chi - 1 \biggr] \biggl\{ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggr\} </math>

 

 

<math>~-~ \biggl[ \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi \biggr] \biggl[ \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi \biggr]^2 ~+~\frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^4 ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~-~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 \biggl[ 3\cos^2\chi - 1 \biggr] ~-~ \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^4\biggl[ 5\cos^3\chi ~-~ 3\cos\chi \biggr] </math>

 

 

<math>~+~ \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2 ~+~ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^3\cos\chi ~+~ \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^4 \biggl[ 3\cos^2\chi - 1 \biggr] ~+~ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^3\cos\chi +\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^4\cos^2\chi ~+~\frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^4 \biggl[ 3\cos^2\chi - 1 \biggr] </math>

 

 

<math>~-~ \biggl[ \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2 ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^3\cos\chi ~+~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^4\cos^2\chi \biggr] \biggl[ \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi \biggr] ~+~\frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^4 ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ 1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2(1-\cos\chi ) ~+~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 \biggl[ 2\cos\chi ~+~2 (\cos\chi -1) ~-~ 2( 3\cos^2\chi - 1 ) \biggr] </math>

 

 

<math> ~+~ \frac{1}{2^5} \biggl(\frac{a}{c}\biggr)^4 \biggl[ ( 3\cos^2\chi - 1 ) + 2\cos^2\chi ~+~( 3\cos^2\chi - 1 ) ~-~2 \cos\chi ~-~ 4 \cos\chi ~+~2 ~-~ ( 5\cos^3\chi ~-~ 3\cos\chi ) \biggr] ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ 1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2(1-\cos\chi ) ~+~ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^3 ( 2\cos\chi ~-~ 3\cos^2\chi ) </math>

 

 

<math> ~+~ \frac{1}{2^5} \biggl(\frac{a}{c}\biggr)^4 ( -~9 \cos\chi ~+~8\cos^2\chi ~-~ 5\cos^3\chi ) ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, . </math>

Relationship Between Angles

Drawing on the Law of Cosines, as above, we can state that on the torus surface,

<math>~R_1^2</math>

<math>~=</math>

<math>~(2c)^2 + a^2 - 4ac\cos\chi</math>

Alternatively, applying the Law of Cosines to the angle, <math>~\psi</math>, we have,

<math>~(2c)^2</math>

<math>~=</math>

<math>~R_1^2 + a^2 - 2aR_1\cos\psi</math>

<math>~\Rightarrow ~~~\cos\psi</math>

<math>~=</math>

<math>~\frac{R_1^2 + a^2 - 4c^2}{2aR_1} \, .</math>

Therefore, anywhere along the surface of the torus, we can switch from one of these angles to the other via the relation,

<math>~\cos\psi</math>

<math>~=</math>

<math>~\frac{1}{2a}\biggl[ 4c^2 + a^2 - 4ac\cos\chi + a^2 - 4c^2\biggr] \biggl[4c^2 + a^2 - 4ac\cos\chi\biggr]^{-1 / 2} </math>

 

<math>~=</math>

<math>~- \biggl[\cos\chi ~-~ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggr] \biggl[1 - \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi\biggr]^{-1 / 2} \, .</math>

Cosine ψ Expansion

Employing the binomial theorem, we therefore can write,

<math>~\cos\psi</math>

<math>~=</math>

<math>~- \biggl[\cos\chi ~-~ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggr] \biggl\{ 1 - \frac{1}{2} \biggl[- \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr] + \frac{3}{8}\biggl[- \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr]^2 </math>

 

 

<math>~ - \frac{5}{2^4}\biggl[- \biggl(\frac{a}{c}\biggr)\biggr]^3 + \frac{5\cdot 7}{2^7}\biggl[- \biggl(\frac{a}{c}\biggr)\biggr]^4 + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \biggr\} </math>

 

<math>~=</math>

<math>~- \biggl[\cos\chi ~-~ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggr] \biggl\{ 1 ~+~ \frac{1}{2} \biggl(\frac{a}{c}\biggr) ~-~ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 \cos\chi ~+~ \frac{3}{8}\biggl[ \biggl(\frac{a}{c}\biggr)^2 ~-~ \frac{1}{4}\biggl(\frac{a}{c}\biggr)^3 \cos\chi ~+~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^4 \cos^2\chi \biggr] </math>

 

 

<math>~ +~ \frac{5}{2^4} \biggl(\frac{a}{c}\biggr)^3 ~+~ \frac{5\cdot 7}{2^7} \biggl(\frac{a}{c}\biggr)^4 ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \biggr\} </math>

 

<math>~=</math>

<math>~- \biggl[\cos\chi ~-~ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggr] \biggl\{ 1 ~+~ \frac{1}{2} \biggl(\frac{a}{c}\biggr) ~+~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)^2\biggl[ 3 ~-~ \cos\chi \biggr] +~ \frac{1}{2^5} \biggl(\frac{a}{c}\biggr)^3 \biggl[ 10 ~-~ 3 \cos\chi \biggr]~+~ \frac{1}{2^7} \biggl(\frac{a}{c}\biggr)^4 \biggl[3 \cos^2\chi ~+~ 5\cdot 7 \biggr] ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \biggr\} </math>

<math>~\Rightarrow ~~~ \cos\psi \biggr|_{\mathcal{O}(a^2/c^2)}</math>

<math>~=</math>

<math>~- \biggl[\cos\chi ~-~ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggr] \biggl\{ 1 ~+~ \frac{1}{2} \biggl(\frac{a}{c}\biggr) ~+~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)^2\biggl[ 3 ~-~ \cos\chi \biggr]

\biggr\} 

</math>

 

<math>~=</math>

<math>~ - \cos\chi \biggl\{ 1 ~+~ \frac{1}{2} \biggl(\frac{a}{c}\biggr) ~+~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)^2\biggl[ 3 ~-~ \cos\chi \biggr]

\biggr\} 

+ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggl\{ 1 ~+~ \frac{1}{2} \biggl(\frac{a}{c}\biggr)

\biggr\} 

</math>

 

<math>~=</math>

<math>~ -\cos\chi ~+~ \frac{1}{2} \biggl(\frac{a}{c}\biggr)(1-\cos\chi) ~+~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)^2\biggl[2 - 3\cos\chi + \cos^2\chi \biggr] \, . </math>

Cosine-Squared Expansion

Letting,

<math>~b \equiv \biggl[- \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr] \, ,</math>

via the binomial theorem we have,

<math>~\cos^2\psi</math>

<math>~=</math>

<math>~\biggl[\cos\chi ~-~ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggr]^2 \biggl[1 - \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi\biggr]^{-1 } </math>

 

<math>~=</math>

<math>~\biggl[\cos\chi ~-~ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggr]^2 \biggl\{ 1 - b + b^2 - b^3 + b^4 - \mathcal{O}(b^5) \biggr\} </math>

 

<math>~=</math>

<math>~\biggl[\cos\chi ~-~ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggr]^2 \biggl\{ 1 - \biggl[- \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr] + \biggl[- \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr]^2 </math>

 

 

<math>~ - \biggl[- \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr]^3 + \biggl[- \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr]^4 + \mathcal{O}\biggl( \frac{a^5}{c^5} \biggr) \biggr\} </math>

 

<math>~=</math>

<math>~\biggl[\cos\chi ~-~ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggr]^2 \biggl\{ 1 + \biggl[\biggl(\frac{a}{c}\biggr) - \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr] + \biggl[\biggl(\frac{a}{c}\biggr)^2 - \frac{1}{2}\biggl(\frac{a}{c}\biggr)^3 \cos\chi + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^4 \cos^2\chi \biggr] </math>

 

 

<math>~ + \biggl[\biggl(\frac{a}{c}\biggr) ~-~ \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr]\biggl[\biggl(\frac{a}{c}\biggr)^2 - \frac{1}{2}\biggl(\frac{a}{c}\biggr)^3 \cos\chi + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^4 \cos^2\chi \biggr] </math>

 

 

<math>~ + \biggl[\biggl(\frac{a}{c}\biggr)^2 - \frac{1}{2}\biggl(\frac{a}{c}\biggr)^3 \cos\chi + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^4 \cos^2\chi \biggr]^2 + \mathcal{O}\biggl( \frac{a^5}{c^5} \biggr) \biggr\} </math>

 

<math>~=</math>

<math>~\biggl[\cos\chi ~-~ \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggr]^2 \biggl\{ 1 + \biggl(\frac{a}{c}\biggr) - \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi + \biggl(\frac{a}{c}\biggr)^2 - \frac{1}{2}\biggl(\frac{a}{c}\biggr)^3 \cos\chi + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^4 \cos^2\chi </math>

 

 

<math>~ ~+~ \biggl(\frac{a}{c}\biggr)^3 ~-~ \frac{1}{2}\biggl(\frac{a}{c}\biggr)^4 \cos\chi ~-~ \frac{1}{4}\biggl(\frac{a}{c}\biggr)^4 \cos\chi + \biggl(\frac{a}{c}\biggr)^4 + \mathcal{O}\biggl( \frac{a^5}{c^5} \biggr) \biggr\} </math>

 

<math>~=</math>

<math>~ \biggl[\cos^2\chi ~-~ \biggl( \frac{a}{c}\biggr)\cos\chi ~+~ \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2 \biggr] \biggl\{1 ~+~ \biggl(\frac{a}{c}\biggr) ~+~ \biggl(\frac{a}{c}\biggr)^2\biggl[1 ~-~ \frac{1}{4} \cos\chi \biggr] ~+~ \biggl(\frac{a}{c}\biggr)^3 \biggl[1~-~ \frac{1}{2} \cos\chi \biggr] </math>

 

 

<math>~ + \biggl(\frac{a}{c}\biggr)^4\biggl[1 ~+~ \frac{1}{2^4} \cos^2\chi ~-~ \frac{3}{4} \cos\chi \biggr] + \mathcal{O}\biggl( \frac{a^5}{c^5} \biggr) \biggr\} </math>

 

<math>~=</math>

<math>~ \cos^2\chi \biggl\{1 ~+~ \biggl(\frac{a}{c}\biggr) ~+~ \biggl(\frac{a}{c}\biggr)^2\biggl[1 ~-~ \frac{1}{4} \cos\chi \biggr] ~+~ \biggl(\frac{a}{c}\biggr)^3 \biggl[1~-~ \frac{1}{2} \cos\chi \biggr] + \biggl(\frac{a}{c}\biggr)^4\biggl[1 ~+~ \frac{1}{2^4} \cos^2\chi ~-~ \frac{3}{4} \cos\chi \biggr] \biggr\} </math>

 

 

<math> ~-~ \biggl( \frac{a}{c}\biggr)\cos\chi \biggl\{1 ~+~ \biggl(\frac{a}{c}\biggr) ~+~ \biggl(\frac{a}{c}\biggr)^2\biggl[1 ~-~ \frac{1}{4} \cos\chi \biggr] ~+~ \biggl(\frac{a}{c}\biggr)^3 \biggl[1~-~ \frac{1}{2} \cos\chi \biggr] \biggr\} </math>

 

 

<math>~+~ \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2 \biggl\{1 ~+~ \biggl(\frac{a}{c}\biggr) ~+~ \biggl(\frac{a}{c}\biggr)^2\biggl[1 ~-~ \frac{1}{4} \cos\chi \biggr] \biggr\} ~+~ \mathcal{O}\biggl( \frac{a^5}{c^5} \biggr) </math>

 

<math>~=</math>

<math>~ \cos^2\chi ~+~ \biggl(\frac{a}{c}\biggr)\biggl[ \cos^2\chi ~-~\cos\chi \biggr] ~+~ \biggl(\frac{a}{c}\biggr)^2\biggl[ \cos^2\chi ~-~ \frac{1}{4} \cos^3\chi ~-~ \cos\chi ~+~\frac{1}{2^2}\biggr] </math>

 

 

<math> ~+~ \biggl(\frac{a}{c}\biggr)^3\biggl\{ \cos^2\chi \biggl[1~-~ \frac{1}{2} \cos\chi \biggr] ~-~ \cos\chi \biggl[1 ~-~ \frac{1}{4} \cos\chi \biggr]~+~\frac{1}{2^2} \biggr\} </math>

 

 

<math> ~+~\biggl(\frac{a}{c}\biggr)^4 \biggl\{ \cos^2\chi \biggl[1 ~+~ \frac{1}{2^4} \cos^2\chi ~-~ \frac{3}{4} \cos\chi \biggr] ~-~ \cos\chi \biggl[1~-~ \frac{1}{2} \cos\chi \biggr]~+~\frac{1}{2^2} \biggl[1 ~-~ \frac{1}{4} \cos\chi \biggr] \biggr\} ~+~ \mathcal{O}\biggl( \frac{a^5}{c^5} \biggr) </math>

 

<math>~=</math>

<math>~ \cos^2\chi ~+~ \biggl(\frac{a}{c}\biggr)\biggl[ \cos^2\chi ~-~\cos\chi \biggr] ~+~ \biggl(\frac{a}{c}\biggr)^2\biggl[ \frac{1}{2^2} ~-~ \cos\chi~+~ \cos^2\chi ~-~ \frac{1}{4} \cos^3\chi \biggr] </math>

 

 

<math> ~+~ \biggl(\frac{a}{c}\biggr)^3\biggl[ \frac{1}{2^2}~-~ \cos\chi ~+~ \frac{5}{4} \cos^2\chi ~-~ \frac{1}{2} \cos^3\chi \biggr] ~+~\biggl(\frac{a}{c}\biggr)^4 \biggl[ \frac{1}{2^2} ~-~ \frac{17}{2^4} \cos\chi ~+~ \frac{3}{2} \cos^2\chi ~-~ \frac{3}{4} \cos^3\chi ~+~\frac{1}{2^4} \cos^4\chi \biggr] ~+~ \mathcal{O}\biggl( \frac{a^5}{c^5} \biggr) </math>

Cosine-Cubed Expansion

Again, letting,

<math>~b \equiv \biggl[- \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr] \, ,</math>

via the binomial theorem we have,

<math>~\cos^3\psi</math>

<math>~=</math>

<math>~\biggl[\frac{1}{2}\biggl( \frac{a}{c}\biggr) ~-~ \cos\chi \biggr]^3 \biggl[1 - \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi\biggr]^{-3 / 2 } </math>

 

<math>~=</math>

<math>~\biggl[\frac{1}{2}\biggl( \frac{a}{c}\biggr) ~-~ \cos\chi \biggr]^3 \biggl\{ 1 -\frac{3}{2}\biggl[ b \biggr] + \frac{3\cdot 5}{2^3} \biggl[ b \biggr]^2 - \frac{3\cdot 5\cdot 7}{2^4\cdot 3}\biggl[ b \biggr]^3 + \frac{3\cdot 5\cdot 7\cdot 9}{2^7\cdot 3}\biggl[ b \biggr]^4 + \mathcal{O}(b^5) \biggr\} </math>

 

<math>~=</math>

<math>~\biggl[\frac{1}{2}\biggl( \frac{a}{c}\biggr) ~-~ \cos\chi \biggr]^3 \biggl\{ 1 -\frac{3}{2}\biggl[ - \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr] + \frac{3\cdot 5}{2^3} \biggl[ - \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr]^2 </math>

 

 

<math>~ - \frac{5\cdot 7}{2^4}\biggl[ - \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr]^3 + \frac{5\cdot 7\cdot 9}{2^7}\biggl[ - \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr]^4 + \mathcal{O}(b^5) \biggr\} </math>

 

<math>~=</math>

<math>~\biggl[\frac{1}{2}\biggl( \frac{a}{c}\biggr) ~-~ \cos\chi \biggr]^3 \biggl\{ 1 + \frac{3}{2}\biggl(\frac{a}{c}\biggr) - \frac{3}{2^3}\biggl(\frac{a}{c}\biggr)^2 \cos\chi + \frac{3\cdot 5}{2^3} \biggl[ \biggl(\frac{a}{c}\biggr)^2 - \frac{1}{2} \biggl(\frac{a}{c}\biggr)^3 \cos\chi + \frac{1}{2^4} \biggl(\frac{a}{c}\biggr)^4 \cos^2\chi \biggr] </math>

 

 

<math>~ - \frac{5\cdot 7}{2^4}\biggl[ - \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \cos\chi \biggr] \biggl[ \biggl(\frac{a}{c}\biggr)^2 - \frac{1}{2} \biggl(\frac{a}{c}\biggr)^3 \cos\chi + \frac{1}{2^4} \biggl(\frac{a}{c}\biggr)^4 \cos^2\chi \biggr] </math>

 

 

<math>~ + \frac{5\cdot 7\cdot 9}{2^7}\biggl[ \biggl(\frac{a}{c}\biggr)^2 - \frac{1}{2} \biggl(\frac{a}{c}\biggr)^3 \cos\chi + \frac{1}{2^4} \biggl(\frac{a}{c}\biggr)^4 \cos^2\chi \biggr]^2 + \mathcal{O}(b^5) \biggr\} </math>

 

<math>~=</math>

<math>~\biggl[\frac{1}{2}\biggl( \frac{a}{c}\biggr) ~-~ \cos\chi \biggr]^3 \biggl\{ 1 + \frac{3}{2}\biggl(\frac{a}{c}\biggr) - \frac{3}{2^3}\biggl(\frac{a}{c}\biggr)^2 \cos\chi + \frac{3\cdot 5}{2^3} \biggl[ \biggl(\frac{a}{c}\biggr)^2 - \frac{1}{2} \biggl(\frac{a}{c}\biggr)^3 \cos\chi + \frac{1}{2^4} \biggl(\frac{a}{c}\biggr)^4 \cos^2\chi \biggr] </math>

 

 

<math>~ + \frac{5\cdot 7}{2^4} \biggl(\frac{a}{c}\biggr) \biggl[ \biggl(\frac{a}{c}\biggr)^2 - \frac{1}{2} \biggl(\frac{a}{c}\biggr)^3 \cos\chi \biggr] - \frac{5\cdot 7}{2^6}\biggl(\frac{a}{c}\biggr)^4 \cos\chi + \frac{3^2\cdot 5\cdot 7}{2^7}\biggl(\frac{a}{c}\biggr)^4 + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \biggr\} </math>

 

<math>~=</math>

<math>~\biggl[ \cos^2\chi ~-~\biggl( \frac{a}{c}\biggr)\cos\chi + \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2 \biggr] \biggl[\frac{1}{2}\biggl( \frac{a}{c}\biggr) ~-~ \cos\chi \biggr] \biggl\{ 1 + \frac{3}{2}\biggl(\frac{a}{c}\biggr) - \frac{3}{2^3}\biggl(\frac{a}{c}\biggr)^2 \cos\chi + \frac{3\cdot 5}{2^3}\biggl(\frac{a}{c}\biggr)^2 </math>

 

 

<math>~ - \frac{3\cdot 5}{2^4}\biggl(\frac{a}{c}\biggr)^3 \cos\chi + \frac{3\cdot 5}{2^7} \biggl(\frac{a}{c}\biggr)^4 \cos^2\chi + \frac{5\cdot 7}{2^4} \biggl(\frac{a}{c}\biggr)^3 - \frac{5\cdot 7}{2^5} \biggl(\frac{a}{c}\biggr)^4 \cos\chi - \frac{5\cdot 7}{2^6}\biggl(\frac{a}{c}\biggr)^4 \cos\chi + \frac{3^2\cdot 5\cdot 7}{2^7}\biggl(\frac{a}{c}\biggr)^4 + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \biggr\} </math>

 

<math>~=</math>

<math>~\biggl\{ -\cos^3\chi + \frac{3}{2}\biggl( \frac{a}{c}\biggr) \cos^2\chi - \frac{3}{2^2}\biggl( \frac{a}{c}\biggr)^2\cos\chi + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^3 \biggr\} </math>

 

 

<math>~\times \biggl\{ 1 + \frac{3}{2}\biggl(\frac{a}{c}\biggr) + \frac{3}{2^3} \biggl(\frac{a}{c}\biggr)^2 \biggl[5 - \cos\chi \biggr] + \frac{5}{2^4} \biggl(\frac{a}{c}\biggr)^3 \biggl[ 7 - 3\cos\chi \biggr] + \frac{5}{2^7} \biggl(\frac{a}{c}\biggr)^4 \biggl[ 3\cos^2\chi - 2^2\cdot 7 \cos\chi - 2\cdot 7 \cos\chi + 3^2\cdot 7 \biggr] + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \biggr\} </math>

 

<math>~=</math>

<math>~-\cos^3\chi \biggl\{ 1 + \frac{3}{2}\biggl(\frac{a}{c}\biggr) + \frac{3}{2^3} \biggl(\frac{a}{c}\biggr)^2 \biggl[5 - \cos\chi \biggr] + \frac{5}{2^4} \biggl(\frac{a}{c}\biggr)^3 \biggl[ 7 - 3\cos\chi \biggr] + \frac{5}{2^7} \biggl(\frac{a}{c}\biggr)^4 \biggl[ 3\cos^2\chi - 2^2\cdot 7 \cos\chi - 2\cdot 7 \cos\chi + 3^2\cdot 7 \biggr] \biggr\} </math>

 

 

<math>~+ \frac{3}{2}\biggl( \frac{a}{c}\biggr) \cos^2\chi \biggl\{ 1 + \frac{3}{2}\biggl(\frac{a}{c}\biggr) + \frac{3}{2^3} \biggl(\frac{a}{c}\biggr)^2 \biggl[5 - \cos\chi \biggr] + \frac{5}{2^4} \biggl(\frac{a}{c}\biggr)^3 \biggl[ 7 - 3\cos\chi \biggr] \biggr\} </math>

 

 

<math>~ - \frac{3}{2^2}\biggl( \frac{a}{c}\biggr)^2\cos\chi \biggl\{ 1 + \frac{3}{2}\biggl(\frac{a}{c}\biggr) + \frac{3}{2^3} \biggl(\frac{a}{c}\biggr)^2 \biggl[5 - \cos\chi \biggr] \biggr\} + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^3 \biggl\{ 1 + \frac{3}{2}\biggl(\frac{a}{c}\biggr) \biggr\} + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~-\cos^3\chi \biggl\{ 1 + \frac{3}{2}\biggl(\frac{a}{c}\biggr) + \frac{3}{2^3} \biggl(\frac{a}{c}\biggr)^2 \biggl[5 - \cos\chi \biggr] + \frac{5}{2^4} \biggl(\frac{a}{c}\biggr)^3 \biggl[ 7 - 3\cos\chi \biggr] + \frac{5}{2^7} \biggl(\frac{a}{c}\biggr)^4 \biggl[ 3\cos^2\chi - 2^2\cdot 7 \cos\chi - 2\cdot 7 \cos\chi + 3^2\cdot 7 \biggr] \biggr\} </math>

 

 

<math>~+ \frac{3}{2} \cos^2\chi \biggl\{ \biggl( \frac{a}{c}\biggr) + \frac{3}{2}\biggl(\frac{a}{c}\biggr)^2 + \frac{3}{2^3} \biggl(\frac{a}{c}\biggr)^3 \biggl[5 - \cos\chi \biggr] + \frac{5}{2^4} \biggl(\frac{a}{c}\biggr)^4 \biggl[ 7 - 3\cos\chi \biggr] \biggr\} </math>

 

 

<math>~ - \frac{3}{2^2} \cos\chi \biggl\{ \biggl( \frac{a}{c}\biggr)^2 + \frac{3}{2}\biggl(\frac{a}{c}\biggr)^3 + \frac{3}{2^3} \biggl(\frac{a}{c}\biggr)^4 \biggl[5 - \cos\chi \biggr] \biggr\} + \frac{1}{2^3} \biggl( \frac{a}{c}\biggr)^3 + \frac{3}{2^4}\biggl(\frac{a}{c}\biggr)^4 </math>

 

 

<math>~ + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

Coefficients of Elliptic Integrals

Rewriting the external potential, as provided in the above-stated objective, and evaluating it at the torus surface,

<math>~\frac{\pi V_\mathrm{Dyson}}{GM} \biggr|_{\mathcal{O}(a^4/c^4)}</math>

<math>~=</math>

<math>~ \frac{4K(\mu)}{a+R_1}\biggl\{ t_K \biggr\} + \frac{(a + R_1)E(\mu)}{aR_1}\biggl\{ t_E \biggr\} </math>

 

<math>~=</math>

<math>~ \frac{4K(\mu)}{c} \biggl(\frac{R_1}{c}\biggr)^{-1} \biggl[1 + \biggl(\frac{a}{c}\biggr) \biggl(\frac{R_1}{c}\biggr)^{-1} \biggr]^{-1}\biggl\{ t_K \biggr\} + \frac{E(\mu) }{a}\biggl[1 + \biggl(\frac{a}{c}\biggr) \biggl(\frac{R_1}{c}\biggr)^{-1} \biggr] \biggl\{ t_E \biggr\} \, , </math>

where,

<math>~t_K</math>

<math>~\equiv</math>

<math>~ 1 ~-~ \frac{1}{8}\biggl(\frac{a^2}{c^2}\biggr) \cos^2\biggl( \frac{\psi}{2}\biggr) - \frac{1}{768}\biggl(\frac{a}{c}\biggr)^4 \biggl[ 5 ~+~ 8\cos\psi ~-~ \cos^2\psi ~-~ 4\cos^3\psi ~-~ \frac{4c^2}{RR_1} \cos2\psi \biggr] \, , </math>

and,

<math>~t_E</math>

<math>~\equiv</math>

<math>~ \frac{1}{8}\biggl(\frac{a}{c}\biggr)^2 \cos\psi ~-~\frac{1}{192} \biggl(\frac{a}{c}\biggr)^4 \biggl[ 2\cos^2\psi ~-~4\cos\psi ~+~ \frac{2c^2}{RR_1}\cos2\psi \biggr] \, . </math>

Given our derived power-series expressions for various trigonometric functions, these coefficients can be rewritten as,

<math>~t_K</math>

<math>~=</math>

<math>~ 1 ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1 + \cos\psi) + \frac{1}{2^6\cdot 3}\biggl(\frac{a}{c}\biggr)^3 \biggl(\frac{R_1}{c}\biggr)^{-1}(2\cos^2\psi - 1) - \frac{1}{2^8\cdot 3}\biggl(\frac{a}{c}\biggr)^4 \biggl[ 5 ~+~ 8\cos\psi ~-~ \cos^2\psi ~-~ 4\cos^3\psi \biggr] </math>

 

<math>~=</math>

<math>~ 1 ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 \biggl\{ 1 -\cos\chi ~+~ \frac{1}{2} \biggl(\frac{a}{c}\biggr)(1-\cos\chi) ~+~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)^2\biggl[2 - 3\cos\chi + \cos^2\chi \biggr]

\biggr\}

</math>

 

 

<math>~ + \frac{1}{2^6\cdot 3}\biggl(\frac{a}{c}\biggr)^3 \biggl(\frac{R_1}{c}\biggr)^{-1}\biggl\{ 2 \biggl[ \cos^2\chi ~+~ \biggl(\frac{a}{c}\biggr)\biggl( \cos^2\chi ~-~\cos\chi \biggr) \biggr] - 1 \biggr\} </math>

 

 

<math>~ - \frac{1}{2^8\cdot 3}\biggl(\frac{a}{c}\biggr)^4 \biggl\{ 5 ~-~ 8\cos\chi ~-~ \cos^2\chi ~+~ 4\cos^3\chi \biggr\} + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ 1 ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 \biggl[ 1 -\cos\chi \biggr] ~-~ \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^3 \biggl[ 1-\cos\chi \biggr] ~-~ \frac{1}{2^7}\biggl(\frac{a}{c}\biggr)^4 \biggl[2 - 3\cos\chi + \cos^2\chi \biggr] </math>

 

 

<math>~ + \frac{1}{2^6\cdot 3} \biggl(\frac{R_1}{c}\biggr)^{-1}\biggl\{ \biggl(\frac{a}{c}\biggr)^3\biggl(2 \cos^2\chi - 1 \biggr) ~+~ 2\biggl(\frac{a}{c}\biggr)^4\biggl( \cos^2\chi ~-~\cos\chi \biggr) \biggr\} </math>

 

 

<math>~ - \frac{1}{2^8\cdot 3}\biggl(\frac{a}{c}\biggr)^4 \biggl\{ 5 ~-~ 8\cos\chi ~-~ \cos^2\chi ~+~ 4\cos^3\chi \biggr\} + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ 1 ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 \biggl[ 1 -\cos\chi \biggr] + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^3\biggl\{ \biggl(\frac{R_1}{c}\biggr)^{-1} \biggl(2 \cos^2\chi - 1 \biggr) ~-~ 2\cdot 3 \biggl( 1-\cos\chi \biggr) \biggr\} </math>

 

 

<math>~ + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl\{ 2^3 \biggl(\frac{R_1}{c}\biggr)^{-1} \biggl( \cos^2\chi ~-~\cos\chi \biggr) - \biggl[ 5 ~-~ 8\cos\chi ~-~ \cos^2\chi ~+~ 4\cos^3\chi \biggr] ~-~ 2\cdot 3\biggl[2 - 3\cos\chi + \cos^2\chi \biggr] \biggr\} + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ 1 ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 ( 1 -\cos\chi ) + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^3\biggl[ \biggl(\frac{R_1}{c}\biggr)^{-1} (2 \cos^2\chi - 1 ) -6~+~6\cos\chi \biggr] </math>

 

 

<math>~ + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl[ 2^3 \biggl(\frac{R_1}{c}\biggr)^{-1} ( \cos^2\chi ~-~\cos\chi ) ~-17 + 26\cos\chi -5 \cos^2\chi ~-~ 4\cos^3\chi \biggr] + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, ; </math>

and,

<math>~t_E</math>

<math>~=</math>

<math>~ \frac{1}{8}\biggl(\frac{a}{c}\biggr)^2 \cos\psi ~-~\frac{1}{2^5\cdot 3} \biggl(\frac{a}{c}\biggr)^3 \biggl(\frac{R_1}{c}\biggr)^{-1} \biggl\{ 2 \biggl[\cos^2\psi\biggr] - 1\biggr\} ~-~\frac{1}{2^5\cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl[ \cos^2\psi ~-~2\cos\psi \biggr] </math>

 

<math>~=</math>

<math>~ \frac{1}{8}\biggl(\frac{a}{c}\biggr)^2 \biggl\{ -\cos\chi ~+~ \frac{1}{2} \biggl(\frac{a}{c}\biggr)(1-\cos\chi) ~+~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)^2\biggl[2 - 3\cos\chi + \cos^2\chi \biggr] \biggr\} </math>

 

 

<math>~ ~-~\frac{1}{2^5\cdot 3} \biggl(\frac{a}{c}\biggr)^3 \biggl(\frac{R_1}{c}\biggr)^{-1} \biggl\{ 2 \biggl[\cos^2\chi ~+~ \biggl(\frac{a}{c}\biggr)\biggl( \cos^2\chi ~-~\cos\chi \biggr)\biggr] - 1\biggr\} ~-~\frac{1}{2^5\cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl[\cos^2\chi ~+~2\cos\chi \biggr] + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ -\frac{1}{2^3} \biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^4} \biggl(\frac{a}{c}\biggr)^3(1-\cos\chi) ~+~ \frac{1}{2^6} \biggl(\frac{a}{c}\biggr)^4\biggl[2 - 3\cos\chi + \cos^2\chi \biggr] </math>

 

 

<math>~ ~-~\frac{1}{2^5\cdot 3} \biggl(\frac{a}{c}\biggr)^3 \biggl(\frac{R_1}{c}\biggr)^{-1} \biggl[ 2\cos^2\chi ~+~ 2\biggl(\frac{a}{c}\biggr)\biggl( \cos^2\chi ~-~\cos\chi \biggr) - 1\biggr] ~-~\frac{1}{2^5\cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl( \cos^2\chi ~+~2\cos\chi \biggr) + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ -\frac{1}{2^3} \biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^4} \biggl(\frac{a}{c}\biggr)^3(1-\cos\chi) ~-~\frac{1}{2^5\cdot 3} \biggl(\frac{a}{c}\biggr)^3 \biggl(\frac{R_1}{c}\biggr)^{-1} \biggl( 2\cos^2\chi - 1\biggr) </math>

 

 

<math>~ ~-~\frac{1}{2^4\cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl(\frac{R_1}{c}\biggr)^{-1} \biggl( \cos^2\chi ~-~\cos\chi \biggr) ~+~ \frac{1}{2^6 \cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl[ 6 - 13\cos\chi + \cos^2\chi \biggr] + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ -\frac{1}{2^3} \biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^5\cdot 3} \biggl(\frac{a}{c}\biggr)^3 \biggl[ 6(1-\cos\chi) ~-~\biggl(\frac{R_1}{c}\biggr)^{-1} ( 2\cos^2\chi - 1 ) \biggr] </math>

 

 

<math>~ ~+~ \frac{1}{2^6 \cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl[ ( 6 - 13\cos\chi + \cos^2\chi ) ~-~4 \biggl(\frac{R_1}{c}\biggr)^{-1} ( \cos^2\chi ~-~\cos\chi ) \biggr] + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, . </math>

Now, inserting to the appropriate order the above expression for the ratio, <math>~R_1/c</math> — namely,

<math>~\biggl(\frac{R_1}{c}\biggr)^{-1}</math>

<math>~=</math>

<math>~ \frac{1}{2} + \frac{1}{2^2} \biggl(\frac{a}{c}\biggr)\cos\chi + \mathcal{O}\biggl(\frac{a^2}{c^2}\biggr) \, , </math>

we have,

<math>~t_K</math>

<math>~=</math>

<math>~ 1 ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 ( 1 -\cos\chi ) + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^3 \biggl\{ \biggl[ \frac{1}{2} + \frac{1}{2^2} \biggl(\frac{a}{c}\biggr)\cos\chi \biggr] (2 \cos^2\chi - 1 ) -6~+~6\cos\chi \biggr\} </math>

 

 

<math>~ + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl\{ 2^2 ( \cos^2\chi ~-~\cos\chi ) ~-17 + 26\cos\chi -5 \cos^2\chi ~-~ 4\cos^3\chi \biggr\} + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ 1 ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 ( 1 -\cos\chi ) + \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( -13~+~12\cos\chi +2 \cos^2\chi ) + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 (2 \cos^3\chi - \cos\chi ) </math>

 

 

<math>~ + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 (-17 + 22\cos\chi - \cos^2\chi ~-~ 4\cos^3\chi ) + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ 1 ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 ( 1 -\cos\chi ) + \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( -13~+~12\cos\chi +2 \cos^2\chi ) </math>

 

 

<math>~ + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 (-17 + 21\cos\chi - \cos^2\chi ~-~ 2\cos^3\chi ) + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, ; </math>

and,

<math>~t_E</math>

<math>~=</math>

<math>~ -\frac{1}{2^3} \biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^5\cdot 3} \biggl(\frac{a}{c}\biggr)^3 \biggl\{ 6(1-\cos\chi) ~+~\biggl[ \frac{1}{2} + \frac{1}{2^2} \biggl(\frac{a}{c}\biggr)\cos\chi \biggr] (1- 2\cos^2\chi ) \biggr\} </math>

 

 

<math>~ ~+~ \frac{1}{2^6 \cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl\{ ( 6 - 13\cos\chi + \cos^2\chi ) ~+~2 ( \cos\chi - \cos^2\chi) \biggr\} + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ -\frac{1}{2^3} \biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 13- 12\cos\chi - 2\cos^2\chi ) ~+~ \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^4 (\cos\chi - 2\cos^3\chi ) </math>

 

 

<math>~ ~+~ \frac{1}{2^6 \cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( 6 - 11\cos\chi - \cos^2\chi ) + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ -\frac{1}{2^3} \biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 13- 12\cos\chi - 2\cos^2\chi ) </math>

 

 

<math>~ ~+~ \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( 12 -21 \cos\chi - 2\cos^2\chi - 2\cos^3\chi ) + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, . </math>

Alternate "Small" Argument of Elliptic Integrals

Defining the "small parameter,"

<math>~k'</math>

<math>~\equiv</math>

<math>~ \sqrt{1-\mu^2} </math>

 

<math>~=</math>

<math>~ \biggl[1 - \biggl( \frac{e^\eta - 1}{e^\eta + 1}\biggr)^2 \biggr]^{1 / 2} </math>

 

<math>~=</math>

<math>~ \biggl[\frac{(e^\eta + 1)^2 - (e^\eta - 1)^2 }{(e^\eta + 1)^2} \biggr]^{1 / 2} </math>

 

<math>~=</math>

<math>~ \biggl[\frac{(e^{2\eta} + 2e^\eta+ 1)- (e^{2\eta} - 2e^\eta + 1) }{(e^\eta + 1)^2} \biggr]^{1 / 2} </math>

 

<math>~=</math>

<math>~ \biggl[\frac{4e^\eta }{(e^\eta + 1)^2} \biggr]^{1 / 2} = \biggl[\frac{4e^{-\eta} }{(1 + e^{-\eta})^2} \biggr]^{1 / 2} </math>

 

<math>~=</math>

<math>~2\biggl( \frac{R}{R_1}\biggr)^{1 / 2} \biggl[1 + \frac{R}{R_1} \biggr]^{-1} \, . </math>

At the surface of the torus, where <math>~R=a</math>, we therefore have,

<math>~k'</math>

<math>~=</math>

<math>~ 2\biggl( \frac{a}{c}\biggr)^{1 / 2}\biggl( \frac{R_1}{c}\biggr)^{-1 / 2} \biggl[1 + \frac{a}{c}\cdot \frac{c}{R_1} \biggr]^{-1} </math>

<math>~\Rightarrow ~~~ \frac{4}{k'}</math>

<math>~=</math>

<math>~2\biggl( \frac{a}{c}\biggr)^{- 1 / 2}\biggl( \frac{R_1}{c}\biggr)^{1 / 2} \biggl[1 + \frac{a}{c}\cdot \frac{c}{R_1} \biggr] </math>

 

<math>~=</math>

<math>~2\biggl( \frac{c}{a}\biggr)^{1 / 2} \biggl[ 4 + \biggl( \frac{a}{c}\biggr)^2 - 4\biggl(\frac{a}{c}\biggr)\cos\chi \biggr]^{1 / 4} \biggl\{ 1 + \frac{a}{c}\cdot \biggl[ 4 + \biggl( \frac{a}{c}\biggr)^2 - 4\biggl(\frac{a}{c}\biggr)\cos\chi \biggr]^{-1 / 2} \biggr\} </math>

 

<math>~=</math>

<math>~\biggl( \frac{2^3c}{a}\biggr)^{1 / 2} \biggl[ 1 - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr]^{1 / 4} \biggl\{ 1 + \frac{a}{2c}\cdot \biggl[ 1 - \biggl(\frac{a}{c}\biggr)\cos\chi + \biggl( \frac{a}{2c}\biggr)^2 \biggr]^{-1 / 2} \biggr\} </math>

<math>~\Rightarrow ~~~ \ln \frac{4}{k'}</math>

<math>~=</math>

<math>~\frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{4}\ln\biggl[ 1 - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr] + \ln\biggl\{ 1 + \frac{a}{2c}\cdot \biggl[ 1 - \biggl(\frac{a}{c}\biggr)\cos\chi + \biggl( \frac{a}{2c}\biggr)^2 \biggr]^{-1 / 2} \biggr\} </math>

 

<math>~\approx</math>

<math>~ \frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{4}\ln\biggl[ 1 - \biggl(\frac{a}{c}\biggr)\cos\chi \biggr] + \ln\biggl[ 1 + \frac{a}{2c} \biggr] </math>

 

<math>~\approx</math>

<math>~ \frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{a}{2c}\biggl[ 1 - \frac{1}{2} \cos\chi \biggr] </math>

Alternatively, if — as above — we adopt the shorthand notation, <math>~\gamma \equiv R_1/(2c)</math>, we can write,

Summary

<math>~\gamma</math>

<math>~=</math>

<math>~ 1 ~-~ \frac{1}{2} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (1-\cos^2\chi) +~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (\cos\chi - \cos^3\chi) +~\frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 \biggl[~-~ 1 ~+~ 6 \cos^2\chi ~-~ 5 \cos^4\chi \biggr] ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, ; </math>

<math>~\frac{1}{\gamma}</math>

<math>~=</math>

<math>~ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 \biggl[ 3\cos^2\chi - 1 \biggr] + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3\biggl[ 5\cos^3\chi ~-~ 3\cos\chi \biggr] ~+~ \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 \biggl[ 3 ~-~ 30 \cos^2\chi ~+~ 35 \cos^4\chi \biggr] ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, . </math>

<math>~k'</math>

<math>~=</math>

<math>~ \biggl( \frac{2a}{c}\biggr)^{1 / 2} \gamma^{-1 / 2} \biggl[1 + \biggl(\frac{a}{c}\biggr) \frac{1}{2\gamma} \biggr]^{-1} </math>

 

<math>~=</math>

<math>~ \biggl( \frac{2a}{c}\biggr)^{1 / 2} \gamma^{-1 / 2} \biggl[1 + \biggl(\frac{a}{c}\biggr) \frac{1}{\gamma} + \biggl(\frac{a}{c}\biggr)^2 \frac{1}{4\gamma^2} \biggr]^{-1 / 2} </math>

 

<math>~=</math>

<math>~ \biggl( \frac{2a}{c}\biggr)^{1 / 2} \biggl[\gamma + \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \frac{1}{\gamma} \biggr]^{-1 / 2} </math>

<math>~\Rightarrow ~~~(k')^{2m}</math>

<math>~=</math>

<math>~ \biggl( \frac{2a}{c}\biggr)^{m } \biggl[\gamma + \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \frac{1}{\gamma} \biggr]^{-m } = \biggl( \frac{2a}{c}\biggr)^{m } \Gamma^{-m} </math>

<math>~\Rightarrow ~~~ \frac{4}{k'}</math>

<math>~=</math>

<math>~ \biggl( \frac{2^3c}{a}\biggr)^{1 / 2} \biggl[\gamma + \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \frac{1}{\gamma} \biggr]^{1 / 2} = \biggl( \frac{2^3c}{a}\biggr)^{1 / 2} \Gamma^{1 / 2} </math>

<math>~\Rightarrow ~~~\ln \frac{4}{k'}</math>

<math>~=</math>

<math>~ \frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\ln \Gamma \, , </math>

where,

<math>~\Gamma</math>

<math>~\equiv</math>

<math>~ \gamma + \biggl(\frac{a}{c}\biggr) + \frac{1}{4}\biggl(\frac{a}{c}\biggr)^2 \frac{1}{\gamma} </math>

 

<math>~=</math>

<math>~ 1 ~-~ \frac{1}{2} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (1-\cos^2\chi) +~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (\cos\chi - \cos^3\chi) +~\frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 \biggl[~-~ 1 ~+~ 6 \cos^2\chi ~-~ 5 \cos^4\chi \biggr] ~+~ \biggl(\frac{a}{c}\biggr) </math>

 

 

<math>~ +~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2 \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 \biggl[ 3\cos^2\chi - 1 \biggr] + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3\biggl[ 5\cos^3\chi ~-~ 3\cos\chi \biggr] ~+~ \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 \biggl[ 3 ~-~ 30 \cos^2\chi ~+~ 35 \cos^4\chi \biggr] \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ 1 + \biggl(\frac{a}{c}\biggr)(1~-~ \frac{1}{2} \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (1-\cos^2\chi) +~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (\cos\chi - \cos^3\chi) +~\frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 (-~ 1 ~+~ 6 \cos^2\chi ~-~ 5 \cos^4\chi ) </math>

 

 

<math>~ +~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2 + \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^3\cos\chi + \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^4 ( 3\cos^2\chi - 1 ) ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ 1 + \biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) +~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (3\cos\chi - \cos^3\chi) +~\frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 (- 5 ~+~ 18 \cos^2\chi ~-~ 5 \cos^4\chi ) ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, . </math>

Now, if we adopt the shorthand notation,

<math>~g \equiv \biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) +~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (3\cos\chi - \cos^3\chi) +~\frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 (- 5 ~+~ 18 \cos^2\chi ~-~ 5 \cos^4\chi ) ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, , </math>

we also have,

<math>~\ln\Gamma = \ln (1 + g)</math>

<math>~=</math>

<math>~ g - \frac{1}{2}g^2 + \frac{1}{3}g^3 - \frac{1}{4}g^4 ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ \biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) +~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (3\cos\chi - \cos^3\chi) +~\frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 (- 5 ~+~ 18 \cos^2\chi ~-~ 5 \cos^4\chi ) </math>

 

 

<math>~ - \frac{1}{2}\biggl[ \biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) +~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (3\cos\chi - \cos^3\chi) \biggr]^2 </math>

 

 

<math>~ + \frac{1}{3}\biggl[ \biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) \biggr]^3 - \frac{1}{4}\biggl[ \biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi ) \biggr]^4 ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ \biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) +~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (3\cos\chi - \cos^3\chi) +~\frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 (- 5 ~+~ 18 \cos^2\chi ~-~ 5 \cos^4\chi ) </math>

 

 

<math>~ - \frac{1}{2} \biggl\{ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2 (2~-~ \cos\chi )^2 + \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (3-\cos^2\chi) (2~-~ \cos\chi ) +~ \frac{1}{2^5}\biggl( \frac{a}{c}\biggr)^4 (3\cos\chi - \cos^3\chi)(2~-~ \cos\chi ) </math>

 

 

<math>~ ~+~ \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^3 (3-\cos^2\chi) (1~-~ \frac{1}{2} \cos\chi ) ~+~ \frac{1}{2^6}\biggl( \frac{a}{c}\biggr)^4 (3-\cos^2\chi) (3-\cos^2\chi) ~+~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^4 (3\cos\chi - \cos^3\chi)(1~-~ \frac{1}{2} \cos\chi ) \biggr\} </math>

 

 

<math>~ + \frac{1}{3}\biggl[ \biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) \biggr] \biggl[ \biggl(\frac{a}{c}\biggr)^2 (1~-~ \frac{1}{2} \cos\chi )^2 ~+~ \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^3 (1~-~ \frac{1}{2} \cos\chi ) (3-\cos^2\chi) \biggr] - \frac{1}{2^6} \biggl(\frac{a}{c}\biggr)^4(2~-~ \cos\chi )^4 ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) ~-~\frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 (2~-~ \cos\chi )^2 </math>

 

 

<math>~ +~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (3\cos\chi - \cos^3\chi) ~-~ \frac{1}{2^5}\biggl( \frac{a}{c}\biggr)^3 (3-\cos^2\chi) (2~-~ \cos\chi ) ~-~ \frac{1}{2^5}\biggl( \frac{a}{c}\biggr)^3 (3-\cos^2\chi) (2~-~ \cos\chi ) ~+~\frac{1}{2^3\cdot 3}\biggl(\frac{a}{c}\biggr)^3 (2~-~ \cos\chi )^2 (2~-~\cos\chi ) </math>

 

 

<math>~ ~-~ \frac{1}{2^6}\biggl( \frac{a}{c}\biggr)^4 (3\cos\chi - \cos^3\chi)(2~-~ \cos\chi ) ~-~ \frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 (3-\cos^2\chi) (3-\cos^2\chi) ~-~ \frac{1}{2^6}\biggl( \frac{a}{c}\biggr)^4 (3\cos\chi - \cos^3\chi)(2~-~ \cos\chi ) </math>

 

 

<math>~ ~+~ \frac{1}{2^4\cdot 3}\biggl( \frac{a}{c}\biggr)^4 (2~-~ \cos\chi ) (3-\cos^2\chi)(2~-~ \cos\chi ) +\frac{1}{2^5\cdot 3}\biggl( \frac{a}{c}\biggr)^4 (3-\cos^2\chi) (2~-~ \cos\chi )^2 +~\frac{1}{2^7}\biggl( \frac{a}{c}\biggr)^4 (- 5 ~+~ 18 \cos^2\chi ~-~ 5 \cos^4\chi ) - \frac{1}{2^6} \biggl(\frac{a}{c}\biggr)^4(2~-~ \cos\chi )^4 ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

And,

<math>~~(k')^{2}</math>

<math>~=</math>

<math>~ \biggl( \frac{2a}{c}\biggr) \Gamma^{-1} </math>

 

<math>~=</math>

<math>~ \biggl( \frac{2a}{c}\biggr) (1+g)^{-1} </math>

 

<math>~=</math>

<math>~ \biggl( \frac{2a}{c}\biggr) \biggl\{1 ~-~ g ~+~ g^2 ~-~ g^3 \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ \biggl( \frac{2a}{c}\biggr) \biggl\{1 ~-~ \biggl[ \biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) +~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3 (3\cos\chi - \cos^3\chi) \biggr] </math>

 

 

<math>~ ~+~ \biggl[\biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) \biggr]^2 ~-~ \biggl[ \biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi )

\biggr]^3 

\biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ \biggl( \frac{2a}{c}\biggr) \biggl\{1 ~-~ \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2\biggl[ (3-\cos^2\chi) ~+~2(2~-~ \cos\chi )^2 \biggr] </math>

 

 

<math>~ +~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3\biggl[ (3\cos\chi - \cos^3\chi) ~+~ (2~-~ \cos\chi )(3-\cos^2\chi) ~-~ (2~-~ \cos\chi )^3 \biggr] \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

And,

<math>~~(k')^{4}</math>

<math>~=</math>

<math>~ \biggl( \frac{2a}{c}\biggr)^2 \Gamma^{-2} </math>

 

<math>~=</math>

<math>~ \biggl( \frac{2a}{c}\biggr)^2 (1+g)^{-2} </math>

 

<math>~=</math>

<math>~ \biggl( \frac{2a}{c}\biggr)^2 \biggl\{1 ~-~ 2g ~+~ 3g^2 \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ \biggl( \frac{2a}{c}\biggr)^2 \biggl\{1 ~-~ 2\biggl[ \biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) \biggr] ~+~ 3\biggl[ \biggl(\frac{a}{c}\biggr) (1~-~ \frac{1}{2} \cos\chi ) \biggr]^2 \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ \biggl( \frac{2a}{c}\biggr)^2 \biggl\{1 -~\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi ) ~+~ \frac{1}{2^2} \biggl(\frac{a}{c}\biggr)^2\biggl[ 3(2~-~ \cos\chi )^2 ~-~ (3-\cos^2\chi) \biggr] \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ 4\biggl( \frac{a}{c}\biggr)^2 -~\biggl(\frac{a}{c}\biggr)^3 (8~-~ 4\cos\chi ) ~+~ \biggl(\frac{a}{c}\biggr)^4 ( 9 - 12\cos\chi + 4\cos^2\chi ) ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

Elliptic Integral Expressions

Hence, drawing upon the series expansion for the complete elliptic integral of the first kind that can be found, for example, as eq. (8.113.3) in the Fourth Edition of Gradshteyn & Ryzhik (1965), we can write,

<math>~K(\mu)</math>

<math>~=</math>

<math>~ \ln \frac{4}{k^'} + \frac{1}{2^2}\biggl( \ln\frac{4}{k^'} - \frac{2}{1\cdot 2} \biggr){k'}^2 + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 \biggl( \ln\frac{4}{k^'} - \frac{2}{1\cdot 2} - \frac{2}{3\cdot 4} \biggr){k'}^4 </math>

 

 

<math>~ + \biggl( \frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\biggr)^2 \biggl( \ln\frac{4}{k^'} - \frac{2}{1\cdot 2} - \frac{2}{3\cdot 4} - \frac{2}{5\cdot 6} \biggr){k'}^6 + \cdots </math>

 

<math>~=</math>

<math>~ \ln \frac{4}{k^'} + \frac{1}{2^2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2 + \frac{3^2}{2^6} \biggl( \ln\frac{4}{k^'} - \frac{7}{6} \biggr){k'}^4 + \frac{5^2}{2^8} \biggl( \ln\frac{4}{k^'} - \frac{37}{30} \biggr){k'}^6 + \cdots </math>

Now, we recognize that,

<math>~\biggl(\frac{a}{2}\biggr) \frac{4K(\mu)}{R_1+a}</math>

<math>~=</math>

<math>~ \biggl(\frac{a}{2c}\biggr) 4K(\mu) \biggl[\frac{R_1}{c} + \frac{a}{c} \biggr]^{-1} </math>

 

<math>~=</math>

<math>~ \biggl(\frac{a}{c}\biggr) 2K(\mu) \biggl(\frac{R_1}{c}\biggr)^{-1}\biggl[1 + \frac{a}{c} \biggl(\frac{R_1}{c}\biggr)^{-1}\biggr]^{-1} </math>

 

<math>~\approx</math>

<math>~ \biggl(\frac{a}{c}\biggr) \biggl(\frac{R_1}{c}\biggr)^{-1}\biggl[1 + \frac{a}{c} \biggl(\frac{R_1}{c}\biggr)^{-1}\biggr]^{-1}\biggl\{ 2\ln \frac{4}{k^'} + \frac{1}{2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2 \biggr\} </math>


Also, drawing upon the series expansion for the complete elliptic integral of the second kind that can be found, for example, as eq. (8.114.3) in the Fourth Edition of Gradshteyn & Ryzhik (1965), we can write,

<math>~E(\mu)</math>

<math>~=</math>

<math>~ 1 ~+~ \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{1\cdot 2}\biggr)(k')^2 ~+~ \frac{1^2\cdot 3}{2^2\cdot 4}\biggl( \ln \frac{4}{k'} - \frac{2}{1\cdot 2} - \frac{1}{3\cdot 4}\biggr)(k')^4 ~+~ \frac{1^2\cdot 3^2\cdot 5}{2^2\cdot 4^2\cdot 6}\biggl( \ln \frac{4}{k'} - \frac{2}{1\cdot 2} - \frac{2}{3\cdot 4} - \frac{1}{5\cdot 6}\biggr)(k')^6 ~+~ \cdots </math>

 

<math>~=</math>

<math>~ 1 ~+~ \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{2}\biggr)(k')^2 ~+~ \frac{3}{2^4}\biggl( \ln \frac{4}{k'} - 1 - \frac{1}{3\cdot 4}\biggr)(k')^4 ~+~ \frac{3^2\cdot 5}{2^7\cdot 3}\biggl( \ln \frac{4}{k'} - 1 - \frac{1}{2\cdot 3} - \frac{1}{2\cdot 3\cdot 5}\biggr)(k')^6 ~+~ \cdots </math>

<math>~\Rightarrow ~~~ \biggl(\frac{a}{2}\biggr) \frac{(R_1+R)E(\mu)}{RR_1}</math>

<math>~\approx</math>

<math>~ \frac{1}{2} \biggl[ 1 + \frac{a}{c}\biggl( \frac{R_1}{c}\biggr)^{-1} \biggr] \biggl\{ 1 + \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{2}\biggr)(k')^2 \biggr\} </math>

External Potential at Torus Surface

Initial Low Resolution

Hence,

<math>~\biggl(\frac{a}{2}\biggr)V_2 </math>

<math>~\approx</math>

<math>~ \biggl[1 - \frac{1}{8}\biggl(\frac{a^2}{c^2}\biggr) \cos^2\biggl( \frac{\psi}{2}\biggr)\biggr] \biggl(\frac{a}{c}\biggr) \biggl(\frac{R_1}{c}\biggr)^{-1}\biggl[1 + \frac{a}{c} \biggl(\frac{R_1}{c}\biggr)^{-1}\biggr]^{-1}\biggl\{ 2\ln \frac{4}{k^'} + \frac{1}{2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2 \biggr\} </math>

 

 

<math>~ + \biggl[\frac{1}{8}\biggl(\frac{a^2}{c^2}\biggr) \cos\psi \biggr] \frac{1}{2} \biggl[ 1 + \frac{a}{c}\biggl( \frac{R_1}{c}\biggr)^{-1} \biggr] \biggl\{ 1 + \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{2}\biggr)(k')^2 \biggr\} </math>

<math>~\Rightarrow ~~~ cV_2 </math>

<math>~\approx</math>

<math>~ \biggl[1 - \cancelto{0}{\frac{1}{8}\biggl(\frac{a^2}{c^2}\biggr) \cos^2\biggl( \frac{\psi}{2}\biggr)}\biggr] \biggl(\frac{R_1}{c}\biggr)^{-1}\biggl[1 + \frac{a}{c} \biggl(\frac{R_1}{c}\biggr)^{-1}\biggr]^{-1}\biggl\{ 4\ln \frac{4}{k^'} + \biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2 \biggr\} </math>

 

 

<math>~ + \biggl[\frac{1}{8}\biggl(\frac{a}{c}\biggr) \cos\psi \biggr] \biggl[ 1 + \frac{a}{c}\biggl( \frac{R_1}{c}\biggr)^{-1} \biggr] \biggl\{ 1 + \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{2}\biggr)(k')^2 \biggr\} \, . </math>

Hence,

<math>~\Rightarrow ~~~ cV_2 </math>

<math>~\approx</math>

<math>~ \biggl(\frac{R_1}{c}\biggr)^{-1}\biggl[1 + \frac{a}{c} \biggl(\frac{R_1}{c}\biggr)^{-1}\biggr]^{-1}\biggl\{ 4\ln \frac{4}{k^'} + \biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2 \biggr\} </math>

 

 

<math>~ + \biggl[\frac{1}{8}\biggl(\frac{a}{c}\biggr) \cos\psi \biggr] \biggl[ 1 + \frac{a}{c}\biggl( \frac{R_1}{c}\biggr)^{-1} \biggr] \biggl\{ 1 + \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{2}\biggr)(k')^2 \biggr\} </math>

 

<math>~\approx</math>

<math>~ \biggl\{ 1 + \frac{1}{2} \biggl(\frac{a}{c}\biggr)\cos\chi +\biggl( \frac{a}{c}\biggr)^2 \biggl[ \frac{3}{8}\cos^2\chi -\frac{1}{8}\biggr] \biggr\} \biggl[1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) + \frac{1}{4} \biggl( \frac{a}{c}\biggr)^2 (1 - \cos\chi)\biggr] \biggl\{ 2\ln \frac{4}{k^'} + \frac{1}{2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2 \biggr\} </math>

 

 

<math>~ + \frac{1}{8}\biggl(\frac{a}{c}\biggr) \cos\psi \biggl[ 1 + \frac{1}{2}\biggl( \frac{a}{c}\biggr) + \frac{1}{4} \biggl(\frac{a}{c}\biggr)^2\cos\chi \biggr] \biggl\{ 1 + \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{2}\biggr)(k')^2 \biggr\} \, . </math>

To order <math>~(a/c)^1</math>, this gives,

<math>~\Rightarrow ~~~ cV_2 </math>

<math>~\approx</math>

<math>~ 2\ln \frac{4}{k^'} + \frac{1}{2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2 + \biggl(\frac{a}{c}\biggr)(\cos\chi -1) \ln \frac{4}{k^'} + \frac{1}{8}\biggl(\frac{a}{c}\biggr) \cos\psi </math>

 

<math>~\approx</math>

<math>~ 2\ln \frac{4}{k^'} + \biggl(\frac{a}{c}\biggr) \biggl\{ - 1 + \cos\chi \ln \frac{4}{k^'} + \frac{1}{8} \cos\psi \biggr\} </math>

 

<math>~\approx</math>

<math>~ 2\ln \frac{4}{k^'} + \biggl(\frac{a}{c}\biggr) \biggl\{ - 1 + \cos\chi \biggl[ \ln \frac{4}{k^'} - \frac{1}{8} \biggr] \biggr\} </math>

 

<math>~\approx</math>

<math>~ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{a}{c}\biggl[ 1- \frac{1}{2} \cos\chi \biggr] + \biggl(\frac{a}{c}\biggr) \biggl\{ \frac{1}{2} \cos\chi \biggl[\ln\biggl( \frac{2^3c}{a}\biggr) - \frac{1}{4} \biggr] -1 \biggr\} </math>

 

<math>~\approx</math>

<math>~ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{a}{c} \biggl\{ 1- \frac{1}{2} \cos\chi + \frac{1}{2} \cos\chi \biggl[\ln\biggl( \frac{2^3c}{a}\biggr) - \frac{1}{4} \biggr] -1 \biggr\} </math>

 

<math>~\approx</math>

<math>~ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr) \biggl[\ln\biggl( \frac{2^3c}{a}\biggr) - \frac{5}{4} \biggr] \cos\chi \, . </math>



We are trying to match equation (6) in Dyson's (1893b) "Part II", that is,

<math>~\frac{V}{2\pi a^2}</math>

<math>~=</math>

<math>~ \ln\biggl(\frac{8c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr) \biggl[\ln\biggl(\frac{8c}{a}\biggr) - \frac{5}{4}\biggr] \cos\chi + \biggl\{ \frac{1}{16} \biggl[ \ln\biggl(\frac{8c}{a}\biggr) - \frac{5}{2} \biggr] + \frac{3}{16} \biggl[\ln\biggl(\frac{8c}{a}\biggr) +\frac{17}{36} - \frac{72}{36}\biggr]\cos2\chi\biggr\}\biggl(\frac{a^2}{c^2}\biggr) </math>

 

 

<math>~ + \biggl\{ \frac{3}{32}\biggl[ \ln\biggl(\frac{8c}{a}\biggr) - \frac{25}{12}\biggr]\cos\chi + \frac{5}{64}\biggl[ \ln\biggl(\frac{8c}{a}\biggr)+\frac{7}{24} - \frac{48}{24}\biggr]\cos3\chi \biggr\} \biggl(\frac{a^3}{c^3}\biggr) </math>

 

 

<math>~ + \biggl\{ \frac{9}{256}\biggl[ \ln\biggl(\frac{8c}{a}\biggr) - 2\biggr] + \frac{7}{128}\biggl[ \ln\biggl(\frac{8c}{a}\biggr) - \frac{19}{168} - 2\biggr]\cos2\chi + \frac{35}{1024} \biggl[ \ln\biggl(\frac{8c}{a}\biggr) - 2 + \frac{19}{120}\biggr]\cos4\chi \biggr\} \biggl(\frac{a^4}{c^4}\biggr) ~+~\cdots </math>

High Resolution

Summary

<math>~2\biggl(\frac{R_1}{c}\biggr)^{-1}</math>

<math>~=</math>

<math>~ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 \biggl[ 3\cos^2\chi - 1 \biggr] + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3\biggl[ 5\cos^3\chi ~-~ 3\cos\chi \biggr] ~+~ \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 \biggl[ 3 ~-~ 30 \cos^2\chi ~+~ 35 \cos^4\chi \biggr] ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, . </math>

<math>~\biggl[1~+~\biggl( \frac{a}{c}\biggr)\biggl(\frac{R_1}{c}\biggr)^{-1}\biggr]^{-1} </math>

<math>~=</math>

<math>~ 1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2(1-\cos\chi ) ~+~ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^3 ( 2\cos\chi ~-~ 3\cos^2\chi ) ~+~ \frac{1}{2^5} \biggl(\frac{a}{c}\biggr)^4 ( -~9 \cos\chi ~+~8\cos^2\chi ~-~ 5\cos^3\chi ) ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, . </math>

<math>~t_K</math>

<math>~=</math>

<math>~ 1 ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 ( 1 -\cos\chi ) + \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( -13~+~12\cos\chi +2 \cos^2\chi ) + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 (-17 + 21\cos\chi - \cos^2\chi ~-~ 2\cos^3\chi ) + \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, . </math>

<math>~1~+~\biggl( \frac{a}{c}\biggr)\biggl(\frac{R_1}{c}\biggr)^{-1} </math>

<math>~=</math>

<math>~ 1 ~+~\frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 ( 3\cos^2\chi - 1 ) ~+~ \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^4 ( 5\cos^3\chi ~-~ 3\cos\chi ) ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) \, . </math>

<math>~t_E</math>

<math>~=</math>

<math>~ -\frac{1}{2^3} \biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 13- 12\cos\chi - 2\cos^2\chi ) ~+~ \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( 12 -21 \cos\chi - 2\cos^2\chi - 2\cos^3\chi ) + f_{E5}\biggl(\frac{a}{c}\biggr)^5 + \mathcal{O}\biggl(\frac{a^6}{c^6}\biggr) \, . </math>


<math>~\frac{\pi V_\mathrm{Dyson}}{GM/c} </math>

<math>~=</math>

<math>~ 2K(\mu) \biggl[~2\biggl(\frac{R_1}{c}\biggr)^{-1}\biggr] \biggl[1 + \biggl(\frac{a}{c}\biggr) \biggl(\frac{R_1}{c}\biggr)^{-1} \biggr]^{-1}\biggl\{ t_K \biggr\} + E(\mu) \biggl[1 + \biggl(\frac{a}{c}\biggr) \biggl(\frac{R_1}{c}\biggr)^{-1} \biggr] \biggl\{\biggl(\frac{c}{a}\biggr) t_E \biggr\} </math>

 

<math>~=</math>

<math>~ 2K(\mu) \biggl\{ ~2\biggl(\frac{R_1}{c}\biggr)^{-1}\biggr\} </math>

 

 

<math>~ \times \biggl\{ \biggl[1 + \biggl(\frac{a}{c}\biggr) \biggl(\frac{R_1}{c}\biggr)^{-1} \biggr]^{-1} \biggr\} </math>

 

 

<math>~ \times \biggl\{ t_K \biggr\} </math>

 

 

<math>~ + E(\mu) \biggl\{ 1 + \biggl(\frac{a}{c}\biggr) \biggl(\frac{R_1}{c}\biggr)^{-1} \biggr\} </math>

 

 

<math>~ \times \biggl\{\biggl(\frac{c}{a}\biggr) t_E \biggr\} </math>

 

<math>~=</math>

<math>~ 2K(\mu) \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 \biggl[ 3\cos^2\chi - 1 \biggr] + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3\biggl[ 5\cos^3\chi ~-~ 3\cos\chi \biggr] ~+~ \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 \biggl[ 3 ~-~ 30 \cos^2\chi ~+~ 35 \cos^4\chi \biggr] \biggr\} </math>

 

 

<math>~ \times \biggl\{ 1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2(1-\cos\chi ) ~+~ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^3 ( 2\cos\chi ~-~ 3\cos^2\chi ) ~+~ \frac{1}{2^5} \biggl(\frac{a}{c}\biggr)^4 ( -~9 \cos\chi ~+~8\cos^2\chi ~-~ 5\cos^3\chi ) \biggr\} </math>

 

 

<math>~ \times \biggl\{ 1 ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 ( 1 -\cos\chi ) + \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( -13~+~12\cos\chi +2 \cos^2\chi ) + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 (-17 + 21\cos\chi - \cos^2\chi ~-~ 2\cos^3\chi ) \biggr\} </math>

 

 

<math>~ + E(\mu) \biggl\{ 1 ~+~\frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 \biggl[ 3\cos^2\chi - 1 \biggr] ~+~ \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^4\biggl[ 5\cos^3\chi ~-~ 3\cos\chi \biggr] \biggr\} </math>

 

 

<math>~ \times \biggl\{-\frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi ~+~ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 12\cos\chi - 2\cos^2\chi ) ~+~ \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 12 -21 \cos\chi - 2\cos^2\chi - 2\cos^3\chi ) + f_{E5}\biggl(\frac{a}{c}\biggr)^4 \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ 2K(\mu) \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 \biggl[ 3\cos^2\chi - 1 \biggr] + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3\biggl[ 5\cos^3\chi ~-~ 3\cos\chi \biggr] ~+~ \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 \biggl[ 3 ~-~ 30 \cos^2\chi ~+~ 35 \cos^4\chi \biggr] \biggr\} </math>

 

 

<math>~ \times \biggl\{ 1 ~-~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 ( 1 -\cos\chi ) + \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( -13~+~12\cos\chi +2 \cos^2\chi ) + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 (-17 + 21\cos\chi - \cos^2\chi ~-~ 2\cos^3\chi ) </math>

 

 

<math>~ - \frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^3 ( 1 -\cos\chi ) - \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4( -13~+~12\cos\chi +2 \cos^2\chi ) +\frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2(1-\cos\chi ) - \frac{1}{2^6}\biggl(\frac{a}{c}\biggr)^4(1-\cos\chi )^2 </math>

 

 

<math>~ ~+~ \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^3 ( 2\cos\chi ~-~ 3\cos^2\chi ) ~+~ \frac{1}{2^5} \biggl(\frac{a}{c}\biggr)^4 (-~9 \cos\chi ~+~8\cos^2\chi ~-~ 5\cos^3\chi ) \biggr\} </math>

 

 

<math>~ + E(\mu) \biggl\{ -\frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi \biggl[ 1 ~+~\frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi ~+~ \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 ( 3\cos^2\chi - 1 ) \biggr] </math>

 

 

<math>~ + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 12\cos\chi - 2\cos^2\chi ) \biggl[ 1 ~+~\frac{1}{2}\biggl( \frac{a}{c}\biggr) ~+~ \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2\cos\chi \biggr] </math>

 

 

<math>~ + \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 12 -21 \cos\chi - 2\cos^2\chi - 2\cos^3\chi ) \biggl[ 1 ~+~\frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggr] + f_{E5}\biggl(\frac{a}{c}\biggr)^4 \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ 2K(\mu) \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 \biggl[ 3\cos^2\chi - 1 \biggr] + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3\biggl[ 5\cos^3\chi ~-~ 3\cos\chi \biggr] ~+~ \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 \biggl[ 3 ~-~ 30 \cos^2\chi ~+~ 35 \cos^4\chi \biggr] \biggr\} </math>

 

 

<math>~ \times \biggl\{ 1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) +\frac{3}{2^4}\biggl(\frac{a}{c}\biggr)^2(1-\cos\chi ) + \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 \biggl[ ( -13~+~12\cos\chi +2 \cos^2\chi ) ~+~ 12 ( 1 -\cos\chi ) ~+~ 48 ( 2\cos\chi ~-~ 3\cos^2\chi ) \biggr] </math>

 

 

<math>~ + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl[ (-17 + 21\cos\chi - \cos^2\chi ~-~ 2\cos^3\chi ) - ( -13~+~12\cos\chi +2 \cos^2\chi ) - 12 (1-\cos\chi )^2 ~+~ 24 (-~9 \cos\chi ~+~8\cos^2\chi ~-~ 5\cos^3\chi ) \biggr] \biggr\} </math>

 

 

<math>~ + E(\mu) \biggl\{ ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi ~ -\frac{1}{2^4} \biggl(\frac{a}{c}\biggr)^2 \cos\chi ~ -\frac{1}{2^5} \biggl(\frac{a}{c}\biggr)^3 \cos^2\chi ~ -\frac{1}{2^7} \biggl(\frac{a}{c}\biggr)^4 ( 3\cos^3\chi - \cos\chi ) </math>

 

 

<math>~ + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 12\cos\chi - 2\cos^2\chi ) ~+~\frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 13- 12\cos\chi - 2\cos^2\chi ) ~+~ \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( 13- 12\cos\chi - 2\cos^2\chi )\cos\chi </math>

 

 

<math>~ + \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 12 -21 \cos\chi - 2\cos^2\chi - 2\cos^3\chi ) ~+~\frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( 12 -21 \cos\chi - 2\cos^2\chi - 2\cos^3\chi ) + f_{E5}\biggl(\frac{a}{c}\biggr)^4 \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ 2K(\mu) \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 ( 3\cos^2\chi - 1 ) + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 ( 5\cos^3\chi ~-~ 3\cos\chi ) ~+~ \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 ( 3~-~ 30 \cos^2\chi~+~ 35 \cos^4\chi ) \biggr\} </math>

 

 

<math>~ \times \biggl\{ 1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) +\frac{3}{2^4}\biggl(\frac{a}{c}\biggr)^2(1-\cos\chi ) + \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 \biggl[ ( -13~+~12\cos\chi +2 \cos^2\chi ) ~+~ ( 12 -12\cos\chi ) ~+~ ( 96\cos\chi ~-~ 144\cos^2\chi ) \biggr] </math>

 

 

<math>~ + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl[ (-17 + 21\cos\chi - \cos^2\chi ~-~ 2\cos^3\chi ) + ( 13~-~12\cos\chi -2 \cos^2\chi ) + (-12 + 24\cos\chi - 12\cos^2\chi) ~+~ (-~216 \cos\chi ~+~192\cos^2\chi ~-~ 120\cos^3\chi ) \biggr] \biggr\} </math>

 

 

<math>~ + E(\mu) \biggl\{ ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2\biggl[ ( 13- 12\cos\chi - 2\cos^2\chi ) ~ -12 \cos\chi \biggr] </math>

 

 

<math>~ ~+~\frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 \biggl[ ( 13- 12\cos\chi - 2\cos^2\chi ) + ( 12 -21 \cos\chi - 2\cos^2\chi - 2\cos^3\chi ) ~ -~12 \cos^2\chi \biggr] </math>

 

 

<math>~ ~+~ \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl[ ( 13\cos\chi - 12\cos^2\chi - 2\cos^3\chi ) ~+~( 12 -21 \cos\chi - 2\cos^2\chi - 2\cos^3\chi ) ~ -~6 ( 3\cos^3\chi - \cos\chi ) + 2^8\cdot 3 f_{E5} \biggr] \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

That is,

<math>~\frac{\pi V_\mathrm{Dyson}}{GM/c} </math>

<math>~=</math>

<math>~ 2K(\mu) \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 ( 3\cos^2\chi - 1 ) + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 ( 5\cos^3\chi ~-~ 3\cos\chi ) ~+~ \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 ( 3~-~ 30 \cos^2\chi~+~ 35 \cos^4\chi ) \biggr\} </math>

 

 

<math>~ \times \biggl\{ 1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) +\frac{3}{2^4}\biggl(\frac{a}{c}\biggr)^2(1-\cos\chi ) + \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( -1 ~+~ 96\cos\chi~-~ 142\cos^2\chi ) + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 (-16 ~-~183\cos\chi ~+~177 \cos^2\chi ~-~ 122\cos^3\chi ) \biggr\} </math>

 

 

<math>~ + E(\mu) \biggl\{ ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi - 2\cos^2\chi ) ~+~\frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 25- 33\cos\chi - 16\cos^2\chi - 2\cos^3\chi ) </math>

 

 

<math>~ ~+~ \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( 12 -2 \cos\chi - 14\cos^2\chi - 22\cos^3\chi + 2^8\cdot 3 f_{E5} ) \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ 2K(\mu) \biggl\{ 1 - \frac{1}{2}\biggl( \frac{a}{c}\biggr) +\frac{3}{2^4}\biggl(\frac{a}{c}\biggr)^2(1-\cos\chi ) + \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( -1 ~+~ 96\cos\chi~-~ 142\cos^2\chi ) + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 (-16 ~-~183\cos\chi ~+~177 \cos^2\chi ~-~ 122\cos^3\chi ) </math>

 

 

<math>~ + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\cos\chi - \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2\cos\chi +\frac{3}{2^5}\biggl(\frac{a}{c}\biggr)^3(\cos\chi-\cos^2\chi ) + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( - \cos\chi ~+~ 96\cos^2\chi~-~ 142\cos^3\chi ) </math>

 

 

<math>~ + \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 ( 3\cos^2\chi - 1 ) + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 (1- 3\cos^2\chi ) ~+~\frac{3}{2^7}\biggl(\frac{a}{c}\biggr)^4(1-\cos\chi ) ( 3\cos^2\chi - 1 ) </math>

 

 

<math>~ + \frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^3 ( 5\cos^3\chi ~-~ 3\cos\chi ) ~+~ \frac{1}{2^5}\biggl(\frac{a}{c}\biggr)^4 (3\cos\chi ~-~5\cos^3\chi ) + \frac{1}{2^7} \biggl( \frac{a}{c}\biggr)^4 ( 3~-~ 30 \cos^2\chi~+~ 35 \cos^4\chi ) \biggr\} </math>

 

 

<math>~ + E(\mu) \biggl\{ ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi - 2\cos^2\chi ) ~+~\frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 25- 33\cos\chi - 16\cos^2\chi - 2\cos^3\chi ) </math>

 

 

<math>~ ~+~ \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( 12 -2 \cos\chi - 14\cos^2\chi - 22\cos^3\chi + 2^8\cdot 3 f_{E5} ) \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ 2K(\mu) \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) +\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 \biggl[3(1-\cos\chi ) - 4 \cos\chi + 2 ( 3\cos^2\chi - 1 )\biggr] </math>

 

 

<math>~ + \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 \biggl[ 36 (\cos\chi-\cos^2\chi ) + 24 (1 ~-~ 3\cos\chi - 3\cos^2\chi +5\cos^3\chi ) + ( -1 ~+~ 96\cos\chi~-~ 142\cos^2\chi ) \biggr] </math>

 

 

<math>~ + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 \biggl[ (-16 ~-~183\cos\chi ~+~177 \cos^2\chi ~-~ 122\cos^3\chi ) ~+~18(1-\cos\chi ) ( 3\cos^2\chi - 1 ) + ( - \cos\chi ~+~ 96\cos^2\chi~-~ 142\cos^3\chi ) ~+~ 24 (3\cos\chi ~-~5\cos^3\chi ) + 6 ( 3~-~ 30 \cos^2\chi~+~ 35 \cos^4\chi ) \biggr] \biggr\} </math>

 

 

<math>~ + E(\mu) \biggl\{ ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi - 2\cos^2\chi ) ~+~\frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 25- 33\cos\chi - 16\cos^2\chi - 2\cos^3\chi ) </math>

 

 

<math>~ ~+~ \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( 12 -2 \cos\chi - 14\cos^2\chi - 22\cos^3\chi + 2^8\cdot 3 f_{E5} ) \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

 

<math>~=</math>

<math>~ 2K(\mu) \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) +\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi + 6\cos^2\chi ) + \frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 (23 ~+~ 60\cos\chi - 250\cos^2\chi +120\cos^3\chi ) </math>

 

 

<math>~ + \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( -16 ~-~94\cos\chi ~+~147 \cos^2\chi ~-~ 194\cos^3\chi + 210 \cos^4\chi ) \biggr\} </math>

 

 

<math>~ + E(\mu) \biggl\{ ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi - 2\cos^2\chi ) ~+~\frac{1}{2^7\cdot 3} \biggl(\frac{a}{c}\biggr)^3 ( 25- 33\cos\chi - 16\cos^2\chi - 2\cos^3\chi ) </math>

 

 

<math>~ ~+~ \frac{1}{2^8\cdot 3} \biggl(\frac{a}{c}\biggr)^4 ( 12 -2 \cos\chi - 14\cos^2\chi - 22\cos^3\chi + 2^8\cdot 3 f_{E5} ) \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

Insert Expressions for K and E

Summary

<math>~k'</math>

<math>~=</math>

<math>~ 2\biggl( \frac{a}{c}\biggr)^{1 / 2}\biggl( \frac{R_1}{c}\biggr)^{-1 / 2} \biggl[1 + \frac{a}{c}\cdot \biggl( \frac{R_1}{c} \biggr)^{-1} \biggr]^{-1} </math>

<math>~\Rightarrow ~~~ \frac{4}{k'}</math>

<math>~=</math>

<math>~2\biggl( \frac{a}{c}\biggr)^{- 1 / 2}\biggl( \frac{R_1}{c}\biggr)^{1 / 2} \biggl[1 + \frac{a}{c}\cdot \biggl( \frac{R_1}{c} \biggr)^{-1} \biggr] </math>

 

<math>~=</math>

<math>~2\biggl( \frac{c}{a}\biggr)^{1 / 2} \biggl[ 4 + \biggl( \frac{a}{c}\biggr)^2 - 4\biggl(\frac{a}{c}\biggr)\cos\chi \biggr]^{1 / 4} \biggl\{ 1 + \frac{a}{c}\cdot \biggl[ 4 + \biggl( \frac{a}{c}\biggr)^2 - 4\biggl(\frac{a}{c}\biggr)\cos\chi \biggr]^{-1 / 2} \biggr\} </math>

 

<math>~=</math>

<math>~\biggl( \frac{2^3c}{a}\biggr)^{1 / 2} \biggl[ 1 - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr]^{1 / 4} \biggl\{ 1 + \frac{a}{2c}\cdot \biggl[ 1 - \biggl(\frac{a}{c}\biggr)\cos\chi + \biggl( \frac{a}{2c}\biggr)^2 \biggr]^{-1 / 2} \biggr\} </math>

<math>~\Rightarrow ~~~ \ln \frac{4}{k'}</math>

<math>~=</math>

<math>~\frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{4}\ln\biggl[ 1 - \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{4}\biggl( \frac{a}{c}\biggr)^2 \biggr] + \ln\biggl\{ 1 + \frac{a}{2c}\cdot \biggl[ 1 - \biggl(\frac{a}{c}\biggr)\cos\chi + \biggl( \frac{a}{2c}\biggr)^2 \biggr]^{-1 / 2} \biggr\} </math>

 

<math>~\approx</math>

<math>~ \frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{4}\ln\biggl[ 1 - \biggl(\frac{a}{c}\biggr)\cos\chi \biggr] + \ln\biggl[ 1 + \frac{a}{2c} \biggr] </math>

 

<math>~\approx</math>

<math>~ \frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{a}{2c}\biggl[ 1 - \frac{1}{2} \cos\chi \biggr] </math>


Remember that (see above),

<math>~K(\mu)</math>

<math>~=</math>

<math>~ \ln \frac{4}{k^'} + \frac{1}{2^2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2 + \frac{3^2}{2^6} \biggl( \ln\frac{4}{k^'} - \frac{7}{6} \biggr){k'}^4 + \frac{5^2}{2^8} \biggl( \ln\frac{4}{k^'} - \frac{37}{30} \biggr){k'}^6 + \cdots </math>

And (see above),

<math>~E(\mu)</math>

<math>~=</math>

<math>~ 1 ~+~ \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{2}\biggr)(k')^2 ~+~ \frac{3}{2^4}\biggl( \ln \frac{4}{k'} - 1 - \frac{1}{3\cdot 4}\biggr)(k')^4 ~+~ \frac{3^2\cdot 5}{2^7\cdot 3}\biggl( \ln \frac{4}{k'} - 1 - \frac{1}{2\cdot 3} - \frac{1}{2\cdot 3\cdot 5}\biggr)(k')^6 ~+~ \cdots </math>


We are trying to match equation (6) in Dyson's (1893b) "Part II", that is,

<math>~\frac{V}{2\pi a^2}</math>

<math>~=</math>

<math>~ \ln\biggl(\frac{8c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr) \biggl[\ln\biggl(\frac{8c}{a}\biggr) - \frac{5}{4}\biggr] \cos\chi + \biggl\{ \frac{1}{16} \biggl[ \ln\biggl(\frac{8c}{a}\biggr) - \frac{5}{2} \biggr] + \frac{3}{16} \biggl[\ln\biggl(\frac{8c}{a}\biggr) +\frac{17}{36} - \frac{72}{36}\biggr]\cos2\chi\biggr\}\biggl(\frac{a^2}{c^2}\biggr) </math>

 

 

<math>~ + \biggl\{ \frac{3}{32}\biggl[ \ln\biggl(\frac{8c}{a}\biggr) - \frac{25}{12}\biggr]\cos\chi + \frac{5}{64}\biggl[ \ln\biggl(\frac{8c}{a}\biggr)+\frac{7}{24} - \frac{48}{24}\biggr]\cos3\chi \biggr\} \biggl(\frac{a^3}{c^3}\biggr) </math>

 

 

<math>~ + \biggl\{ \frac{9}{256}\biggl[ \ln\biggl(\frac{8c}{a}\biggr) - 2\biggr] + \frac{7}{128}\biggl[ \ln\biggl(\frac{8c}{a}\biggr) - \frac{19}{168} - 2\biggr]\cos2\chi + \frac{35}{1024} \biggl[ \ln\biggl(\frac{8c}{a}\biggr) - 2 + \frac{19}{120}\biggr]\cos4\chi \biggr\} \biggl(\frac{a^4}{c^4}\biggr) ~+~\cdots </math>


To First Order

<math>~~(k')^{2}</math>

<math>~=</math>

<math>~ \biggl( \frac{2a}{c}\biggr) \biggl\{1 ~-~ \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2\biggl[ (3-\cos^2\chi) ~+~2(2~-~ \cos\chi )^2 \biggr] </math>

 

 

<math>~ +~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^3\biggl[ (3\cos\chi - \cos^3\chi) ~+~ (2~-~ \cos\chi )(3-\cos^2\chi) ~-~ (2~-~ \cos\chi )^3 \biggr] \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^5}{c^5}\biggr) </math>

<math>~\ln \frac{4}{k'}</math>

<math>~=</math>

<math>~ \frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\ln \Gamma </math>

 

<math>~=</math>

<math>~ \frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\biggl\{ \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2 (3-\cos^2\chi) ~-~\frac{1}{2^3}\biggl(\frac{a}{c}\biggr)^2 (2~-~ \cos\chi )^2 ~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) \biggr\} </math>

Hence,

<math>~\frac{\pi V_\mathrm{Dyson}}{GM/c} </math>

<math>~=</math>

<math>~ 2K(\mu) \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) \biggr\} + E(\mu) \biggl\{ ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^2}{c^2}\biggr) </math>

 

<math>~\approx</math>

<math>~ \biggl[ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) \biggr] \biggl\{ 2K(\mu) \biggr\} ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi \biggl\{~E(\mu) \biggr\} </math>

 

<math>~\approx</math>

<math>~ \biggl[ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) \biggr] \biggl\{ 2\ln \frac{4}{k'} + \frac{1}{2}\biggl[ \ln\frac{4}{k'} - 1 \biggr] k'^2 \biggr\} ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi \biggl\{~1 + \frac{1}{2}\cancelto{0}{\biggl[ \ln\frac{4}{k'} - \frac{1}{2} \biggr] k'^2}\biggr\} </math>

 

<math>~\approx</math>

<math>~ \biggl\{ 2\ln \frac{4}{k'} + \frac{1}{2}\biggl[ \ln\frac{4}{k'} - 1 \biggr] \biggl(\frac{2a}{c}\biggr) \biggr\} + \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) \biggl\{ 2\ln \frac{4}{k'} +\frac{1}{2} \cancelto{0}{\biggl[ \ln\frac{4}{k'} - 1 \biggr] k'^2} \biggr\} ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi \biggl\{~1 + \frac{1}{2}\cancelto{0}{\biggl[ \ln\frac{4}{k'} - \frac{1}{2} \biggr] k'^2}\biggr\} </math>

 

<math>~\approx</math>

<math>~ 2\ln \frac{4}{k'} + \biggl(\frac{a}{c}\biggr)\biggl[ \ln\frac{4}{k'} - 1 \biggr] + \biggl(\frac{a}{c}\biggr)(\cos\chi - 1) \biggl\{ \ln \frac{4}{k'} \biggr\} ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi </math>

 

<math>~\approx</math>

<math>~ 2\ln \frac{4}{k'} + \biggl(\frac{a}{c}\biggr)\biggl[ \ln\frac{4}{k'} - 1 + (\cos\chi - 1) \biggl( \ln \frac{4}{k'} \biggr) ~-~ \frac{1}{2^3} \cos\chi \biggr] </math>

 

<math>~\approx</math>

<math>~ 2\ln \frac{4}{k'} + \biggl(\frac{a}{c}\biggr)\biggl[ - 1 + \cos\chi \biggl( \ln \frac{4}{k'} \biggr) ~-~ \frac{1}{2^3} \cos\chi \biggr] </math>

 

<math>~\approx</math>

<math>~ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr) \biggl\{ (2~-~ \cos\chi ) - 2 + \cos\chi \biggl[ \ln\biggl(\frac{2^3c}{a}\biggr)\biggr] ~-~ \frac{1}{4} \cos\chi \biggr\} </math>

 

<math>~\approx</math>

<math>~ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr) \biggl[ \ln\biggl(\frac{2^3c}{a}\biggr) ~-~ \frac{5}{4} \biggr]\cos\chi </math>

To Second Order

<math>~\frac{\pi V_\mathrm{Dyson}}{GM/c} </math>

<math>~=</math>

<math>~ \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) +\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi + 6\cos^2\chi ) \biggr\} \biggl\{ 2K(\mu) \biggr\} </math>

 

 

<math>~ + \biggl\{ ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi - 2\cos^2\chi ) \biggr\} \biggl\{ E(\mu) \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) </math>

 

<math>~=</math>

<math>~ \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) +\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi + 6\cos^2\chi ) \biggr\} \biggl\{ 2\ln \frac{4}{k^'} + \frac{1}{2}\biggl( \ln\frac{4}{k^'} - 1 \biggr){k'}^2 + \frac{3^2}{2^5} \biggl( \ln\frac{4}{k^'} - \frac{7}{6} \biggr){k'}^4 \biggr\} </math>

 

 

<math>~ + \biggl\{ ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi - 2\cos^2\chi ) \biggr\} \biggl\{ 1 ~+~ \frac{1}{2}\biggl( \ln \frac{4}{k'} - \frac{1}{2}\biggr)(k')^2 \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) </math>

 

<math>~=</math>

<math>~ \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) +\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi + 6\cos^2\chi ) \biggr\} \biggl\{ 2\biggl[\frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\ln \Gamma\biggr] + \frac{1}{2}\biggl[ \frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\ln \Gamma - 1 \biggr]{k'}^2 + \frac{3^2}{2^5} \biggl[ \frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\ln \Gamma - \frac{7}{6} \biggr] {k'}^4 \biggr\} </math>

 

 

<math>~ + \biggl\{ ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi - 2\cos^2\chi ) \biggr\} \biggl\{ 1 ~+~ \frac{1}{2}\biggl[ \frac{1}{2} \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\ln \Gamma - \frac{1}{2}\biggr](k')^2 \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) </math>

 

<math>~=</math>

<math>~ \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) +\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi + 6\cos^2\chi ) \biggr\} \biggl\{ \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) + \ln \Gamma\biggr] + \frac{1}{2^2}\biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) + \ln \Gamma - 2 \biggr] \biggl( \frac{2a}{c}\biggr) \biggl[ 1 ~-~ \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi )\biggr] + \frac{3}{2^6} \biggl[ 3 \ln\biggl( \frac{2^3c}{a}\biggr) + \cancelto{0}{3\ln \Gamma} - 7 \biggr] 4\biggl( \frac{a}{c}\biggr)^2 \biggr\} </math>

 

 

<math>~ + \biggl\{ ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi - 2\cos^2\chi ) \biggr\} \biggl\{ 1 ~+~ \frac{1}{2^2}\biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) + \cancelto{0}{\ln \Gamma} - 1 \biggr] \biggl( \frac{2a}{c}\biggr) \biggl[ 1 ~-~\cancelto{0}{ \frac{1}{2}\biggl(\frac{a}{c}\biggr)} (2~-~ \cos\chi )\biggr] \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) \, , </math>

where,

<math>~\ln\Gamma</math>

<math>~=</math>

<math>~ \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2( -1 ~+~ 4\cos\chi -2\cos^2\chi ) ~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) \, . </math>

Hence,

<math>~\frac{\pi V_\mathrm{Dyson}}{GM/c} </math>

<math>~=</math>

<math>~ \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) +\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi + 6\cos^2\chi ) \biggr\} \biggl\{ \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2( -1 ~+~ 4\cos\chi -2\cos^2\chi ) \biggr] </math>

 

 

<math>~ + \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) - 2 + \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi ) \biggr] \biggl[ 1 ~-~ \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi )\biggr] + \frac{3}{2^4}\biggl( \frac{a}{c}\biggr)^2 \biggl[ 3 \ln\biggl( \frac{2^3c}{a}\biggr) - 7 \biggr] \biggr\} </math>

 

 

<math>~ + \biggl\{ ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi - 2\cos^2\chi ) \biggr\} \biggl\{ 1 ~+~ \frac{1}{2}\biggl( \frac{a}{c}\biggr)\biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) - 1 \biggr] \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) </math>

 

<math>~=</math>

<math>~ \biggl\{ 1 + \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) +\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi + 6\cos^2\chi ) \biggr\} \biggl\{ \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi ) + \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2( -1 ~+~ 4\cos\chi -2\cos^2\chi ) \biggr] </math>

 

 

<math>~ + \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) - 2 \biggr] + \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2 (2~-~ \cos\chi ) - \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2 \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) - 2 \biggr] (2~-~ \cos\chi ) + \frac{3}{2^4}\biggl( \frac{a}{c}\biggr)^2 \biggl[ 3 \ln\biggl( \frac{2^3c}{a}\biggr) - 7 \biggr] \biggr\} </math>

 

 

<math>~ ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi ~+~ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi - 2\cos^2\chi ) ~-~ \frac{1}{2^4} \biggl(\frac{a}{c}\biggr)^2\cos\chi \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) - 1 \biggr] ~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) </math>

 

<math>~=</math>

<math>~ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr) (2~-~ \cos\chi ) ~-~ \frac{1}{2^3} \biggl(\frac{a}{c}\biggr)\cos\chi + \frac{1}{2}\biggl( \frac{a}{c}\biggr) \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) - 2 \biggr] + \frac{1}{2}\biggl(\frac{a}{c}\biggr)(\cos\chi - 1) \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) \biggr] </math>

 

 

<math>~ ~+~ \frac{1}{2^3}\biggl( \frac{a}{c}\biggr)^2( -1 ~+~ 4\cos\chi -2\cos^2\chi ) + \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2 (2~-~ \cos\chi ) - \frac{1}{2^2}\biggl( \frac{a}{c}\biggr)^2 \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) - 2 \biggr] (2~-~ \cos\chi ) + \frac{3}{2^4}\biggl( \frac{a}{c}\biggr)^2 \biggl[ 3 \ln\biggl( \frac{2^3c}{a}\biggr) - 7 \biggr] </math>

 

 

<math>~ + \frac{1}{2^2}\biggl(\frac{a}{c}\biggr)^2(\cos\chi - 1) (2~-~ \cos\chi ) +\frac{1}{2^4}\biggl(\frac{a}{c}\biggr)^2 (1-7\cos\chi + 6\cos^2\chi ) \ln\biggl( \frac{2^3c}{a}\biggr) </math>

 

 

<math>~ ~+~ \frac{1}{2^6\cdot 3} \biggl(\frac{a}{c}\biggr)^2 ( 13- 24\cos\chi - 2\cos^2\chi ) ~-~ \frac{1}{2^4} \biggl(\frac{a}{c}\biggr)^2\cos\chi \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) - 1 \biggr] ~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) </math>

 

<math>~=</math>

<math>~ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)\biggl\{ 4(2~-~ \cos\chi ) ~-~ \cos\chi + \biggl[ 4 \ln\biggl( \frac{2^3c}{a}\biggr) - 8 \biggr] + \biggl[ 4\ln\biggl( \frac{2^3c}{a}\biggr) \biggr] (\cos\chi - 1) \biggr\} </math>

 

 

<math>~~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{ 24( -1 ~+~ 4\cos\chi -2\cos^2\chi ) + 48 (2~-~ \cos\chi ) - 48\biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) - 2 \biggr] (2~-~ \cos\chi ) + 36\biggl[ 3 \ln\biggl( \frac{2^3c}{a}\biggr) - 7 \biggr] </math>

 

 

<math>~ + 48(\cos\chi - 1) (2~-~ \cos\chi ) ~+~12 (1-7\cos\chi + 6\cos^2\chi ) \ln\biggl( \frac{2^3c}{a}\biggr) ~+~ ( 13- 24\cos\chi - 2\cos^2\chi ) ~-~12\cos\chi \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) - 1 \biggr] \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) </math>

 

<math>~=</math>

<math>~ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2^3}\biggl(\frac{a}{c}\biggr)\biggl\{ 4(2~-~ \cos\chi ) ~-~ \cos\chi + \biggl[ 4 \ln\biggl( \frac{2^3c}{a}\biggr) - 8 \biggr] + \biggl[ 4\ln\biggl( \frac{2^3c}{a}\biggr) \biggr] (\cos\chi - 1) \biggr\} </math>

 

 

<math>~~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{ (-24 + 96\cos\chi-48\cos^2\chi) + (96~-~ 48\cos\chi ) ~+~ \biggl[2- \ln\biggl( \frac{2^3c}{a}\biggr) \biggr] (96~-~ 48 \cos\chi ) + \biggl[ 108 \ln\biggl( \frac{2^3c}{a}\biggr) - 252 \biggr] </math>

 

 

<math>~ + 48(3\cos\chi - \cos^2\chi - 2 ) ~+~ (12 - 84\cos\chi + 72\cos^2\chi ) \ln\biggl( \frac{2^3c}{a}\biggr) ~+~ ( 13- 24\cos\chi - 2\cos^2\chi ) ~+~12\cos\chi ~-~12\cos\chi \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) - \biggr] \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) </math>

 

<math>~=</math>

<math>~ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) ~-~ \frac{5}{4}\biggr]\cos\chi </math>

 

 

<math>~~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{ -24 ~+~ 96\cos\chi -48\cos^2\chi + 96~-~ 48\cos\chi + 192 - 252 -96\cos\chi ~+~ ( 13- 12\cos\chi - 2\cos^2\chi ) + (144\cos\chi - 48\cos^2\chi -96 ) </math>

 

 

<math>~ +36 \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr)\biggr]\cos\chi ~+~(12 - 84\cos\chi + 72\cos^2\chi -96 + 108) \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr)\biggr] \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) </math>

 

<math>~=</math>

<math>~ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) ~-~ \frac{5}{4}\biggr]\cos\chi </math>

 

 

<math>~~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{ -71 ~+~84\cos\chi -98\cos^2\chi ~+~(24 - 48\cos\chi + 72\cos^2\chi ) \biggl[ \ln\biggl( \frac{2^3c}{a}\biggr)\biggr] \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) </math>

 

<math>~=</math>

<math>~ \ln\biggl( \frac{2^3c}{a}\biggr) + \frac{1}{2}\biggl(\frac{a}{c}\biggr)\biggl[ \ln\biggl( \frac{2^3c}{a}\biggr) ~-~ \frac{5}{4}\biggr]\cos\chi ~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{ -71 ~+~84\cos\chi -98\cos^2\chi ~+~24\ln\biggl( \frac{2^3c}{a}\biggr)(1 - 2\cos\chi + 3\cos^2\chi ) \biggr\} ~+~ \mathcal{O}\biggl(\frac{a^3}{c^3}\biggr) </math>

In an effort to compare this expression with equation (6) from Dyson's (1893b) "Part II", we should make the substitutions,

<math>~\ln\biggl(\frac{2^3c}{a}\biggr) \rightarrow (\lambda +2)</math>       and       <math>~2\cos^2\chi \rightarrow 1 + \cos2\chi \, .</math>

This means,

<math>~\frac{\pi V_\mathrm{Dyson}}{GM/c}\biggr|_{\mathcal{O}(a^2/c^2)} </math>

<math>~=</math>

<math>~ ~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{ -71 ~+~84\cos\chi -49(1+\cos2\chi ) ~+~24(\lambda + 2)(1 - 2\cos\chi ) ~+~36(\lambda+2)(1 + \cos2\chi ) \biggr\} </math>

 

<math>~=</math>

<math>~ ~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{ -71 ~+~84\cos\chi -49 -49 \cos2\chi ~+~24(\lambda + 2 -2\lambda \cos\chi - 4\cos\chi) ~+~36(\lambda+2 +\lambda\cos 2\chi + 2\cos 2\chi) \biggr\} </math>

 

<math>~=</math>

<math>~ ~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{ -71 ~+~84\cos\chi -49 -49 \cos2\chi ~+~24\lambda + 48 -48\lambda \cos\chi - 96\cos\chi ~+~36\lambda+72 +36\lambda\cos 2\chi + 72\cos 2\chi \biggr\} </math>

 

<math>~=</math>

<math>~ ~+~ \frac{1}{2^6\cdot 3} \biggl( \frac{a}{c}\biggr)^2\biggl\{ 60\lambda -48\lambda \cos\chi - 12\cos\chi +36\lambda\cos 2\chi + 23\cos 2\chi \biggr\} </math>

 

<math>~=</math>

<math>~ ~+~ \biggl( \frac{a}{c}\biggr)^2\biggl\{ \frac{5\lambda}{16} - \frac{(4\lambda + 1)}{16}~\cos\chi +\frac{3(\lambda+\tfrac{23}{36})}{16}\cos 2\chi \biggr\} \, . </math>

This expression differs from the 2nd-order term in Dyson's equation (6) by the amount,

<math>~\Delta \biggr|_{\mathcal{O}(a^2/c^2)} </math>

<math>~=</math>

<math>~ \biggl( \frac{a}{c}\biggr)^2\biggl\{ \frac{5\lambda}{16} - \frac{(4\lambda + 1)}{16}~\cos\chi +\frac{3(\lambda+\tfrac{23}{36})}{16}\cos 2\chi \biggr\} - \biggl(\frac{a}{c}\biggr)^2 \biggl\{ \frac{\lambda - \frac{1}{2}}{16} + \frac{3(\lambda + \frac{17}{36})}{16}\cos2\chi \biggr\} </math>

 

<math>~=</math>

<math>~ \frac{1}{16\cdot 12}\biggl( \frac{a}{c}\biggr)^2\biggl\{ 60\lambda - (48\lambda + 12)~\cos\chi +(36\lambda+23)\cos 2\chi \biggr\} - \frac{1}{16\cdot 12}\biggl(\frac{a}{c}\biggr)^2 \biggl\{ 12\lambda - 6 + (36\lambda + 17) \cos2\chi \biggr\} </math>

 

<math>~=</math>

<math>~ \frac{1}{16\cdot 12}\biggl( \frac{a}{c}\biggr)^2\biggl\{ 48\lambda + 6 - (48\lambda + 12)~\cos\chi +(6)\cos 2\chi \biggr\} </math>

 

<math>~=</math>

<math>~ \frac{1}{2^5}\biggl( \frac{a}{c}\biggr)^2\biggl\{ (\cos 2\chi -1)- (8\lambda + 2)~(1+\cos\chi) \biggr\} </math>

 

<math>~=</math>

<math>~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^2\biggl\{ (\cos\chi - 1)- (4\lambda + 1)~ \biggr\} (1+\cos\chi) </math>

 

<math>~=</math>

<math>~ \frac{1}{2^4}\biggl( \frac{a}{c}\biggr)^2 (\cos\chi - 2- 4\lambda ) (1+\cos\chi) </math>

Interior Potential

In equation (9) on p. 1050 of his "Part II", Dyson (1893b) presents the following power-series expression for the gravitational potential at points inside the torus:

<math>~V</math>

<math>~=</math>

<math>~ 2\pi a^2 \biggl\{ L + \frac{1}{2}\biggl( 1 - \frac{R^2}{a^2}\biggr) + \frac{a}{c}\biggl[ \frac{(L-1)}{2}\biggl(\frac{R}{a}\biggr) - \frac{R^3}{8a^3} \biggr]\cos\chi </math>

 

 

<math>~+~ \frac{a^2}{c^2}\biggl[ - ~\frac{(L - \tfrac{1}{4})}{16} ~+~ \frac{(L-1)}{8} \biggl(\frac{R^2}{a^2} \biggr) ~-~ \frac{3}{64} \biggl( \frac{R^4}{a^4}\biggr) ~+~ \frac{3(L-\tfrac{5}{4})}{16} \biggl(\frac{R^2}{a^2}\biggr) \cos 2\chi ~-~\frac{5}{96} \biggl( \frac{R^4}{a^4}\biggr) \cos 2\chi \biggr] ~+~ \cdots \biggr\} </math>

where,

<math>~L</math>

<math>~\equiv</math>

<math>~\ln\biggl(\frac{8c}{a}\biggr) \, .</math>

Note that, for the example illustrated above, <math>~a/c = 2/5</math> and, hence, <math>~L = \ln(20) = 2.99573</math>. Therefore, at any point on the surface of this example torus,

<math>~\frac{V}{2\pi a^2}</math>

<math>~\approx</math>

<math>~ \ln(20) + \frac{2}{5}\biggl[ \frac{\ln(20)-1}{2} - \frac{1}{8} \biggr]\cos\chi </math>

 

 

<math>~+~ \frac{2^2}{5^2}\biggl[ - ~\frac{4\ln(20) - 1}{64} ~+~ \frac{\ln(20)-1}{8} ~-~ \frac{3}{64} ~+~ \frac{12\ln(20)-15}{64} \cos 2\chi ~-~\frac{10}{3\cdot 64} \cos 2\chi \biggr] </math>

 

<math>~=</math>

<math>~ \ln(20) + \frac{2}{40}\biggl[ 4\ln(20)- 5 \biggr]\cos\chi ~+~ \frac{2^2}{2^6\cdot 5^2}\biggl\{ - ~[4\ln(20) - 1] ~+~ [8\ln(20)-8] ~-~ 3 ~+~ [12\ln(20)-15] \cos 2\chi ~-~\biggl(\frac{10}{3} \biggr) \cos 2\chi \biggr\} </math>

 

<math>~=</math>

<math>~ \ln(20) + \frac{1}{20}\biggl[ 4\ln(20)- 5 \biggr]\cos\chi ~+~ \frac{1}{2^4\cdot 5^2}\biggl\{ 4\ln(20)-10 ~+~ \frac{1}{3}\biggl[ 36\ln(20)-55 \biggr] \cos 2\chi \biggr\} </math>


Red Contour

Let's construct an equipotential contour that extends the red contour into the interior region. Let's begin by evaluating Dyson's interior potential expression at the coordinate location where the red contour touches the surface of the torus. According to Column #5, this point on the surface has coordinates, <math>~(\varpi,z) = (0.867, 0.377)</math>; or, equivalently, <math>~R = a = 0.4</math> and,

<math>~\cos\chi</math>

<math>~=</math>

<math>~ \frac{1-\varpi}{R} = 0.3325 </math>

<math>~\Rightarrow ~~~ \chi</math>

<math>~=</math>

<math>~ \cos^{-1}(0.3325) = 1.2318 \, . </math>

Hence, for this specific point on the torus surface, we find,

<math>~\frac{V}{2\pi a^2}</math>

<math>~\approx</math>

<math>~ 3 + \biggl[\frac{7}{20}\biggr]\cos\chi ~+~ \frac{1}{2^4\cdot 5^2}\biggl\{ 2 ~+~ \biggl[ \frac{53}{3} \biggr] \cos 2\chi \biggr\} = 3.0870 \, . </math>

Contour
(column #)
<math>~V_2</math> (external) Surface <math>~\varpi</math> <math>~\chi</math> (radians)
blue (3) 0.9120 1.117 1.8676
orange (4) 0.961 0.929 1.392
red (5) 0.9800 0.867 1.2318
light green (6) 0.9800 0.838 1.1538
light blue (7) 1.0212 0.749 0.8925

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

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