User:Tohline/Appendix/Ramblings/Bordeaux
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- | =Université de Bordeaux= | + | =Université de Bordeaux (Part 1)= |
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Through a research collaboration at the [https://www.u-bordeaux.com Université de Bordeaux], [https://ui.adsabs.harvard.edu/abs/2019MNRAS.487.4504B/abstract B. Basillais & J. -M. Huré (2019), MNRAS, 487, 4504-4509] have published a paper titled, ''Rigidly Rotating, Incompressible Spheroid-Ring Systems: New Bifurcations, Critical Rotations, and Degenerate States.'' | Through a research collaboration at the [https://www.u-bordeaux.com Université de Bordeaux], [https://ui.adsabs.harvard.edu/abs/2019MNRAS.487.4504B/abstract B. Basillais & J. -M. Huré (2019), MNRAS, 487, 4504-4509] have published a paper titled, ''Rigidly Rotating, Incompressible Spheroid-Ring Systems: New Bifurcations, Critical Rotations, and Degenerate States.'' | ||
+ | We discuss this topic in a [[User:Tohline/Appendix/Ramblings/BordeauxSequences#Spheroid-Ring_Systems|separate, accompanying chapter]]. | ||
==Exterior Gravitational Potential of Toroids== | ==Exterior Gravitational Potential of Toroids== | ||
[https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract J. -M. Huré, B. Basillais, V. Karas, A. Trova, & O. Semerák (2020), MNRAS, 494, 5825-5838] have published a paper titled, ''The Exterior Gravitational Potential of Toroids.'' Here we examine how their work relates to the published work by [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong (1973, Annals of Physics, 77, 279)], which we have separately [[User:Tohline/Apps/Wong1973Potential#Wong.27s_.281973.29_Analytic_Potential|discussed in detail]]. | [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract J. -M. Huré, B. Basillais, V. Karas, A. Trova, & O. Semerák (2020), MNRAS, 494, 5825-5838] have published a paper titled, ''The Exterior Gravitational Potential of Toroids.'' Here we examine how their work relates to the published work by [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong (1973, Annals of Physics, 77, 279)], which we have separately [[User:Tohline/Apps/Wong1973Potential#Wong.27s_.281973.29_Analytic_Potential|discussed in detail]]. | ||
- | === | + | ===Our Presentation of Wong's (1973) Result=== |
- | + | ||
- | + | ||
- | + | ||
+ | <table border="1" cellpadding="8" align="center" width="80%"> | ||
+ | <tr><td align="center">'''Summary:''' First three terms in [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong's (1973)] expression for the gravitational potential at any point, P(ϖ, z), outside of a uniform-density torus.</td></tr> | ||
+ | <tr><td align="left"> | ||
+ | |||
+ | [[File:WongTorusIllustration02.png|500px|center|Wong diagram]] | ||
+ | |||
+ | ---- | ||
+ | |||
+ | <table border="0" cellpadding="5" align="center"> | ||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
- | <math>~\ | + | <math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W0}(\varpi,z)\biggr|_\mathrm{exterior}</math> |
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
- | <math>~ | + | <math>~=</math> |
</td> | </td> | ||
<td align="left"> | <td align="left"> | ||
<math>~ | <math>~ | ||
- | \ | + | - \biggl( \frac{2^{3} }{3\pi^3} \biggr) |
+ | \Upsilon_{W0}(\eta_0) \biggl\{ | ||
+ | \frac{a}{ r_1 } \cdot \boldsymbol{K}(k) \biggr\}\, , | ||
</math> | </math> | ||
</td> | </td> | ||
</tr> | </tr> | ||
- | |||
- | |||
- | |||
- | |||
- | |||
- | |||
- | |||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
- | <math>~\ | + | <math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W1}(\varpi,z)\biggr|_\mathrm{exterior}</math> |
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
Line 47: | Line 49: | ||
<td align="left"> | <td align="left"> | ||
<math>~ | <math>~ | ||
- | - \frac{ | + | - \biggl( \frac{2^{3} }{3\pi^3} \biggr) |
+ | \Upsilon_{W1}(\eta_0) \times \cos\theta | ||
+ | \biggl\{ \frac{a}{r_2} \cdot | ||
+ | \boldsymbol{E}(k) \biggr\} \, , | ||
</math> | </math> | ||
</td> | </td> | ||
</tr> | </tr> | ||
- | |||
- | |||
- | |||
- | |||
- | |||
- | |||
- | |||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
- | <math>~\frac{ | + | <math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W2}(\varpi, z)\biggr|_\mathrm{exterior}</math> |
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
Line 68: | Line 66: | ||
<td align="left"> | <td align="left"> | ||
<math>~ | <math>~ | ||
- | - \frac{ | + | - \biggl( \frac{2^{3} }{3\pi^3} \biggr)\Upsilon_{W2}(\eta_0) |
- | [\ | + | \times \cos(2\theta) |
- | \biggr\} \, | + | \biggl\{ |
+ | \biggl[ \frac{r_1^2 + r_2^2}{2r_1 r_2} \biggr] \frac{a}{r_2} \cdot \boldsymbol{E}(k) | ||
+ | - | ||
+ | \frac{a}{r_1} \cdot \boldsymbol{K}(k) | ||
+ | \biggr\} \, , | ||
</math> | </math> | ||
</td> | </td> | ||
Line 76: | Line 78: | ||
</table> | </table> | ||
- | + | where, once the major ( R ) and minor ( d ) radii of the torus — as well as the vertical location of its equatorial plane (Z<sub>0</sub>) — have been specified, we have, | |
- | </ | + | |
- | + | ||
- | + | ||
<table border="0" cellpadding="5" align="center"> | <table border="0" cellpadding="5" align="center"> | ||
<tr> | <tr> | ||
<td align="right"> | <td align="right"> | ||
- | <math>~ | + | <math>~a^2 </math> |
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
Line 91: | Line 90: | ||
<td align="left"> | <td align="left"> | ||
<math>~ | <math>~ | ||
- | + | R^2 - d^2</math> | |
+ | and, | ||
+ | <math>~\cosh\eta_0 \equiv \frac{R}{d} ~~~~\Rightarrow ~~~\sinh\eta_0 = \frac{a}{d} | ||
+ | \, , | ||
</math> | </math> | ||
</td> | </td> | ||
- | < | + | </tr> |
+ | |||
+ | <tr> | ||
<td align="right"> | <td align="right"> | ||
- | <math>~\ | + | <math>~r_1^2</math> |
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\equiv</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~(\varpi + a)^2 + (z - Z_0)^2 \, ,</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~r_2^2</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\equiv</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~(\varpi - a)^2 + (z - Z_0)^2 \, ,</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\cos\theta</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\equiv</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\biggl[\frac{ r_1^2 + r_2^2 - 4a^2}{2r_1 r_2} \biggr] \, ,</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~k</math> | ||
</td> | </td> | ||
<td align="center"> | <td align="center"> | ||
Line 103: | Line 143: | ||
<td align="left"> | <td align="left"> | ||
<math>~ | <math>~ | ||
- | \biggl[ ( | + | \biggl[ \frac{r_1^2 - r_2^2}{r_1^2} \biggr]^{1 / 2} |
+ | = \biggl[ \frac{4a\varpi}{r_1^2} \biggr]^{1 / 2} | ||
+ | = \biggl[ \frac{4a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \biggr]^{1 / 2} | ||
+ | \, . | ||
</math> | </math> | ||
</td> | </td> | ||
Line 109: | Line 152: | ||
</table> | </table> | ||
- | + | ---- | |
- | + | ||
- | === | + | <table border="0" cellpadding="5" align="center"> |
+ | <tr> | ||
+ | <td align="left" colspan="2"> </td> | ||
+ | <td align="left" colspan="1">Leading Coefficient Expressions …</td> | ||
+ | <td align="right" colspan="1" width="30%">… evaluated for: </td> | ||
+ | <td align="center" colspan="1"><math>~\frac{R}{d} = \cosh\eta_0 = 3</math> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Upsilon_{W0}(\eta_0)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\equiv</math> | ||
+ | </td> | ||
+ | <td align="left" colspan="2"> | ||
+ | <math>~ | ||
+ | \biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] \biggl\{ | ||
+ | K(k_0)\cdot K(k_0) [ \cosh\eta_0(1 - \cosh\eta_0) ] | ||
+ | + 2K(k_0)\cdot E(k_0) [ \cosh^2\eta_0 + 1 ] | ||
+ | - E(k_0)\cdot E(k_0) [ \cosh\eta_0(1 + \cosh\eta_0) ] | ||
+ | \biggr\} \, , | ||
+ | </math> | ||
+ | </td> | ||
+ | <td align="center"><font color="red">7.134677</font></td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Upsilon_{W1}(\eta_0)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\equiv</math> | ||
+ | </td> | ||
+ | <td align="left" colspan="2"> | ||
+ | <math>~\biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] | ||
+ | \biggl\{ | ||
+ | K(k_0)\cdot K(k_0) [\cosh\eta_0(1 - \cosh\eta_0)] | ||
+ | +~2K(k_0)\cdot E(k_0) [(3\cosh^2\eta_0 - 1)] | ||
+ | -~5 E(k_0)\cdot E(k_0) [\cosh\eta_0(1+\cosh\eta_0)] | ||
+ | \biggr\} | ||
+ | \, , | ||
+ | </math> | ||
+ | </td> | ||
+ | <td align="center"><font color="red">0.130324</font></td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Upsilon_{W2}(\eta_0)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\equiv</math> | ||
+ | </td> | ||
+ | <td align="left" colspan="2"> | ||
+ | <math>~ | ||
+ | \frac{2^{3 / 2}}{3^2} \biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] \biggl\{ | ||
+ | K ( k_0 ) \cdot K(k_0) \cdot \cosh\eta_0(1 - \cosh\eta_0) [ 16\cosh^2\eta_0 - 13] | ||
+ | + 2 K ( k_0 ) \cdot E(k_0) [ 16\cosh^4\eta_0 -13\cosh^2\eta_0 + 3 ] | ||
+ | </math> | ||
+ | </td> | ||
+ | <td align="center"> </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left" colspan="2"> | ||
+ | <math>~ | ||
+ | -~E(k_0) \cdot E(k_0) \cdot \cosh\eta_0 (1 + \cosh\eta_0) [ 3 +16\cosh^2\eta_0 ] | ||
+ | \biggr\} \, , | ||
+ | </math> | ||
+ | </td> | ||
+ | <td align="center"><font color="red">0.003153</font></td> | ||
+ | </tr> | ||
+ | <tr><td align="left" colspan="5">where,</td></tr> | ||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~k_0</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\equiv</math> | ||
+ | </td> | ||
+ | <td align="left" colspan="2"> | ||
+ | <math>~ | ||
+ | \biggl[ \frac{2}{\cosh\eta_0 + 1} \biggr]^{1 / 2} \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | <td align="center"><font color="red">0.707106781</font></td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | NOTE: In evaluating these "leading coefficient expressions" for the case, <math>~R/d = 3</math>, we have used the complete elliptic integral evaluations, '''K'''(k<sub>0</sub>) = <font color="red">1.854074677</font> and '''E'''(k<sub>0</sub>) = <font color="red">1.350643881</font>. | ||
+ | </td></tr> | ||
+ | </table> | ||
====Setup==== | ====Setup==== | ||
From our [[User:Tohline/Apps/Wong1973Potential#D0andCn|accompanying discussion of Wong's (1973) derivation]], the exterior potential is given by the expression, | From our [[User:Tohline/Apps/Wong1973Potential#D0andCn|accompanying discussion of Wong's (1973) derivation]], the exterior potential is given by the expression, | ||
+ | <div align="center"> | ||
<table border="0" cellpadding="5" align="center"> | <table border="0" cellpadding="5" align="center"> | ||
Line 134: | Line 274: | ||
</tr> | </tr> | ||
</table> | </table> | ||
- | + | [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], §II.D, p. 294, Eqs. (2.59) & (2.61) | |
+ | </div> | ||
where, | where, | ||
+ | <div align="center"> | ||
<table border="0" cellpadding="5" align="center"> | <table border="0" cellpadding="5" align="center"> | ||
Line 168: | Line 310: | ||
</tr> | </tr> | ||
</table> | </table> | ||
+ | [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)], §II.D, p. 294, Eq. (2.63) | ||
+ | </div> | ||
and where, in terms of the major ( R ) and minor ( d ) radii of the torus — or their ratio, ε ≡ d/R, | and where, in terms of the major ( R ) and minor ( d ) radii of the torus — or their ratio, ε ≡ d/R, | ||
<table border="0" cellpadding="5" align="center"> | <table border="0" cellpadding="5" align="center"> | ||
Line 1,148: | Line 1,292: | ||
</table> | </table> | ||
- | + | <span id="Qrecurrence"> </span> | |
<table border="1" align="center" width="80%" cellpadding="10"> | <table border="1" align="center" width="80%" cellpadding="10"> | ||
<tr><td align="left"> | <tr><td align="left"> | ||
Line 1,252: | Line 1,396: | ||
<td align="left"> | <td align="left"> | ||
<math>~ | <math>~ | ||
- | -~\frac{1}{2^2}\biggl\{ z k_0~K ( k_0 ) | + | -~\frac{1}{2^2} |
+ | \biggl\{ z k_0~K ( k_0 ) | ||
~-~(z^2+3) \biggl[ \frac{2}{(z-1)(z^2-1)} \biggr]^{1 / 2} E(k_0)\biggr\} | ~-~(z^2+3) \biggl[ \frac{2}{(z-1)(z^2-1)} \biggr]^{1 / 2} E(k_0)\biggr\} | ||
</math> | </math> | ||
Line 1,294: | Line 1,439: | ||
<td align="left"> | <td align="left"> | ||
<math>~ | <math>~ | ||
- | 5 Q^{2}_{- \tfrac{1}{2}}(z_0)-4z Q_{+\tfrac{1}{2}}^{2}(z_0) \ | + | 5 Q^{2}_{- \tfrac{1}{2}}(z_0)-4z Q_{+\tfrac{1}{2}}^{2}(z_0) |
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 5 \biggl\{ [2^3(z-1)(z^2-1)]^{-1 / 2} [4zE(k_0) - (z-1)K(k_0)] \biggr\} | ||
+ | + z \biggl\{ z k_0~K ( k_0 ) | ||
+ | ~-~(z^2+3) \biggl[ \frac{2}{(z-1)(z^2-1)} \biggr]^{1 / 2} E(k_0)\biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 2^{1 / 2}[ (z-1)(z^2-1) ]^{- 1 / 2} \biggl\{ [5 z] | ||
+ | ~-~z (z^2+3) \biggr\} E(k_0) | ||
+ | + \biggl\{ z^2 k_0~ | ||
+ | - [(z-1)(z^2-1)]^{-1 / 2} [ 2^{-3 / 2} \cdot 5 (z-1)] \biggr\}K(k_0) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 2^{-3 / 2}(z+1)^{-1 / 2} [4 z^2 - 5 ]K(k_0) | ||
+ | -~2^{1 / 2}[ (z-1)(z^2-1) ]^{- 1 / 2} (z^2 - 2)z E(k_0) | ||
</math> | </math> | ||
</td> | </td> | ||
Line 1,576: | Line 1,769: | ||
<td align="left"> | <td align="left"> | ||
<math>~ | <math>~ | ||
+ | \biggl[ \frac{2(3z^2 - 1)}{(z^2-1)} \biggr]K(k_0)\cdot E(k_0) | ||
-~\biggl[ \frac{z}{(z+1)} \biggr] K(k_0)\cdot K(k_0) | -~\biggl[ \frac{z}{(z+1)} \biggr] K(k_0)\cdot K(k_0) | ||
-~\biggl[ \frac{ 5z}{(z-1)} \biggr] E(k_0)\cdot E(k_0) | -~\biggl[ \frac{ 5z}{(z-1)} \biggr] E(k_0)\cdot E(k_0) | ||
- | + | </math> | |
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Rightarrow ~~~(z_0^2-1)C_1(z_0)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 2(3z^2 - 1) K(k_0)\cdot E(k_0) | ||
+ | -~z_0(z_0-1) K(k_0)\cdot K(k_0) | ||
+ | -~5z_0(z_0+1) E(k_0)\cdot E(k_0) \, . | ||
</math> | </math> | ||
</td> | </td> | ||
</tr> | </tr> | ||
</table> | </table> | ||
+ | |||
+ | Hence, we have, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W1}(\eta,\theta)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - \frac{2^{3} a}{3\pi^3} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] C_1(\cosh\eta_0) \biggl[\frac{ r_1^2 + r_2^2 - 4a^2}{2r_1 r_2^2} \biggr] | ||
+ | \boldsymbol{E}(k) \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | ====Third (n = 2) Term==== | ||
+ | |||
+ | =====Part A===== | ||
+ | The third (n = 2) term in [[User:Tohline/Apps/Wong1973Potential#D0andCn|Wong's (1973) expression for the exterior potential]] is, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W2}(\eta,\theta)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | -D_0 | ||
+ | (\cosh\eta - \cos\theta)^{1 / 2} \cdot 2 \cos(2\theta) \cdot C_2(\cosh\eta_0)P_{+\frac{3}{2}}(\cosh\eta) \, , | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | where, <math>~D_0</math> is the same as [[#Setup|above]], and, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~C_2(\cosh\eta_0)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\equiv</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\tfrac{5}{2}Q_{+\frac{5}{2}}(\cosh \eta_0) Q_{+\frac{3}{2}}^2(\cosh \eta_0) | ||
+ | - \tfrac{1}{2} Q_{+\frac{3}{2}}(\cosh \eta_0)~Q^2_{+ \frac{5}{2}}(\cosh \eta_0) \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | <table border="1" align="center" width="80%" cellpadding="10"> | ||
+ | <tr><td align="left"> | ||
+ | In order to evaluate <math>~C_2(z)</math>, we will need the following pair of expressions in addition to the ones already used: | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~Q^0_{m-\tfrac{1}{2}}</math> [[User:Tohline/Appendix/Equation_templates#Example_Recurrence_Relations|recurrence]] with m = 3, gives: | ||
+ | <math>~Q_{+\tfrac{5}{2}}(z_0)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\frac{8}{5} z~Q_{+\tfrac{3}{2}}(z_0) - \frac{3}{5} Q_{+\tfrac{1}{2}}(z_0)</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Rightarrow~~~15Q_{+\tfrac{5}{2}}(z_0)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~8 z~\biggl[ (4z^2 - 1 )k_0 K(k_0) - 4 z[2(z+1)]^{1 / 2} E(k_0) \biggr] | ||
+ | - | ||
+ | 9 \biggl[ z~k_0 K(k_0) - [2(z+1)]^{1 / 2} E(k_0) \biggr]</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | z~k_0 K(k_0) \biggl[ 8(4z^2 - 1 ) - 9 \biggr] | ||
+ | + | ||
+ | [2(z+1)]^{1 / 2} E(k_0) \biggl[-32z^2 + 9 \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | z~k_0 K(k_0) [ 32z^2 - 17 ] | ||
+ | + | ||
+ | [2(z+1)]^{1 / 2} E(k_0) [9 -32z^2 ] \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | Hence, <math>~Q_{+\frac{5}{2}}(3)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~0.002080867 \, .</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | And, setting m = 2 in the [[#Qrecurrence|above recurrence relation for]] <math>~Q^2_{m+\frac{1}{2}}(z)</math> gives, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\biggl[ (m - \tfrac{3}{2})Q^{2}_{m+\tfrac{1}{2}} (z) \biggr]_{m=2}</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl[ (2m)z Q_{m-\tfrac{1}{2}}^{2}(z) - (m + \tfrac{3}{2})Q^{2}_{m - \tfrac{3}{2}}(z) \biggr]_{m=2} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Rightarrow ~~~ Q^{2}_{+\tfrac{5}{2}} (z) </math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 8z Q_{+\tfrac{3}{2}}^{2}(z) - 7 Q^{2}_{+ \tfrac{1}{2}}(z) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 8z \biggl[ 5 Q^{2}_{- \tfrac{1}{2}}(z_0)-4z Q_{+\tfrac{1}{2}}^{2}(z_0) \biggr] - 7 Q^{2}_{+ \tfrac{1}{2}}(z) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 40z Q^{2}_{- \tfrac{1}{2}}(z_0) | ||
+ | - [32z^2 +7]Q_{+\tfrac{1}{2}}^{2}(z_0) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 40z \biggl\{ | ||
+ | [2^3(z-1)(z^2-1)]^{-1 / 2} [4zE(k_0) - (z-1)K(k_0)] | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + \frac{[32z^2 +7]}{4} \biggl\{ | ||
+ | z k_0~K ( k_0 ) | ||
+ | ~-~(z^2+3) \biggl[ \frac{2}{(z-1)(z^2-1)} \biggr]^{1 / 2} E(k_0) | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Rightarrow ~~~ 4Q^{2}_{+\tfrac{5}{2}} (z) </math> </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 2^5\cdot 5z \biggl\{ 2^{1 / 2} | ||
+ | [(z-1)(z^2-1)]^{-1 / 2} [zE(k_0) ] | ||
+ | - | ||
+ | 2^{-3 / 2}[(z-1)(z^2-1)]^{-1 / 2} [(z-1)K(k_0)] | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + [32z^2 +7] \biggl\{ | ||
+ | z k_0~K ( k_0 ) | ||
+ | ~-~2^{1 / 2}(z^2+3) [ (z-1)(z^2-1) ]^{-1 / 2} E(k_0) | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl\{ | ||
+ | 2^{11 / 2}\cdot 5 [z^2 ] - 2^{1 / 2} [32z^2 +7] (z^2+3) | ||
+ | \biggr\}[(z-1)(z^2-1)]^{-1 / 2} E(k_0) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | -~2^{7 / 2}\cdot 5 \biggl\{ [(z-1)(z^2-1)]^{-1 / 2} (z-1) \biggr\} z K(k_0) | ||
+ | + [32z^2 +7] \biggl\{ 2^{1 / 2}[z + 1]^{-1 / 2} \biggr\} z K ( k_0 ) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 2^{1 / 2}\biggl\{ 32z^2 - 33 \biggr\} z [z + 1]^{-1 / 2} K ( k_0 ) | ||
+ | -~2^{1 / 2} | ||
+ | \biggl\{ | ||
+ | 32z^4 - 57 z^2 + 21 | ||
+ | \biggr\}[(z-1)(z^2-1)]^{-1 / 2} E(k_0) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 2^{1 / 2}[z + 1]^{-1 / 2} [z-1]^{-1 }\biggl[ (32z^2 - 33) z (z-1) K ( k_0 ) | ||
+ | -~(32z^4 - 57 z^2 + 21)E(k_0) \biggr] | ||
+ | \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | Hence, <math>~Q^2_{+\frac{5}{2}}(3)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~0.03377378 \, .</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | </td></tr></table> | ||
+ | |||
+ | =====Part B===== | ||
+ | |||
+ | Let's evaluate <math>~C_2(z)</math> specifically for the case where <math>~z = \cosh\eta_0 = 3</math>, using the already separately evaluated values of the four relevant toroidal functions. We find, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~2C_2(3)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~5Q_{+\frac{5}{2}}(3) Q_{+\frac{3}{2}}^2(3) | ||
+ | - Q_{+\frac{3}{2}}(3)~Q^2_{+ \frac{5}{2}}(3) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 5\cdot ( 0.002080867 ) \times ( 0.132453829 ) - ( 0.014544576 ) \times (0.03377378 ) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 8.868687\times 10^{-4} \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | Next, let's develop a consolidated expression for <math>~C_2(z_0)</math> that replaces all the toroidal functions with complete elliptic integrals of the first and second kind. | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~2C_2(z_0)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~5Q_{+\frac{5}{2}}(z_0) Q_{+\frac{3}{2}}^2(z_0) | ||
+ | - Q_{+\frac{3}{2}}(z_0)~Q^2_{+ \frac{5}{2}}(z_0) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \frac{1}{3}\biggl\{ | ||
+ | z~k_0 K(k_0) [ 32z^2 - 17 ] | ||
+ | + | ||
+ | [2(z+1)]^{1 / 2} E(k_0) [9 -32z^2 ] | ||
+ | \biggr\} | ||
+ | \times \biggl\{ | ||
+ | 2^{-3 / 2}(z+1)^{-1 / 2} [4 z^2 - 5 ]K(k_0) | ||
+ | -~2^{1 / 2}[ (z-1)(z^2-1) ]^{- 1 / 2} (z^2 - 2)z E(k_0) | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - \frac{1}{2^2\cdot 3} | ||
+ | \biggl\{ (4z^2 - 1 )k_0 K(k_0) - 4 z[2(z+1)]^{1 / 2} E(k_0) \biggr\} | ||
+ | \times \biggl\{ | ||
+ | 2^{1 / 2}[z + 1]^{-1 / 2} [z-1]^{-1 }\biggl[ (32z^2 - 33) z (z-1) K ( k_0 ) | ||
+ | -~(32z^4 - 57 z^2 + 21)E(k_0) \biggr] | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Rightarrow ~~~ 2^{2} \cdot 3 (z^2-1) C_2(z_0)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl\{ | ||
+ | K(k_0) z[ 32z^2 - 17 ] | ||
+ | + | ||
+ | (z+1) E(k_0) [9 -32z^2 ] | ||
+ | \biggr\} | ||
+ | \times \biggl\{ | ||
+ | (z-1) [4 z^2 - 5 ]K(k_0) | ||
+ | -~4 (z^2 - 2)z E(k_0) | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - ~ | ||
+ | \biggl\{ (4z^2 - 1 ) K(k_0) - 4 z(z+1) E(k_0) \biggr\} | ||
+ | \times \biggl\{ | ||
+ | (32z^2 - 33) z (z-1) K ( k_0 ) | ||
+ | -~(32z^4 - 57 z^2 + 21)E(k_0) | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl\{ | ||
+ | (z-1)[ 32z^2 - 17 ] [4 z^2 - 5 ]z K(k_0) \cdot K(k_0) | ||
+ | -~4 (z^2 - 2)z^2 [ 32z^2 - 17 ] K(k_0) \cdot E(k_0) | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + \biggl\{ | ||
+ | (z-1) (z+1) [9 -32z^2 ] [4 z^2 - 5 ]K(k_0) \cdot E(k_0) | ||
+ | -~4 (z^2 - 2)z (z+1) [9 -32z^2 ] E(k_0) \cdot E(k_0) | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + ~ | ||
+ | \biggl\{ | ||
+ | (32z^4 - 57 z^2 + 21)(4z^2 - 1 ) K(k_0) \cdot E(k_0) | ||
+ | -~(32z^2 - 33) z (z-1)(4z^2 - 1 ) K ( k_0 ) \cdot K(k_0) | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + ~ | ||
+ | \biggl\{ | ||
+ | 4 z(z+1)(32z^2 - 33) z (z-1) K ( k_0 ) \cdot E(k_0) | ||
+ | -~4 z(z+1)(32z^4 - 57 z^2 + 21)E(k_0) \cdot E(k_0) | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~(z-1)\biggl\{ | ||
+ | \biggl[ | ||
+ | ( 32z^2 - 17 ) (4 z^2 - 5 )z \biggr] | ||
+ | -~\biggl[ (32z^2 - 33) z (4z^2 - 1 ) \biggr] \biggr\} K ( k_0 ) \cdot K(k_0) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + \biggl\{ \biggl[ | ||
+ | (z-1) (z+1) (9 -32z^2 ) (4 z^2 - 5 )\biggr] | ||
+ | -~\biggl[ 4 (z^2 - 2)z^2 ( 32z^2 - 17 ) \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + ~ | ||
+ | \biggl[ | ||
+ | (32z^4 - 57 z^2 + 21)(4z^2 - 1 ) \biggr] | ||
+ | + ~ \biggl[ 4 z(z+1)(32z^2 - 33) z (z-1)\biggr]\biggr\} K ( k_0 ) \cdot E(k_0) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | -~2z(z+1) \biggl\{ \biggl[ 2 (32z^4 - 57 z^2 + 21) \biggr] | ||
+ | +~2\biggl[ (z^2 - 2) (9 -32z^2 )\biggr] \biggr\} E(k_0) \cdot E(k_0) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | z(z-1)\biggl\{[ 52 - 64z^2 ] \biggr\} K ( k_0 ) \cdot K(k_0) | ||
+ | -~4z(z+1) \biggl\{ [ 3 +16z^2 ]\biggr\} E(k_0) \cdot E(k_0) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + \biggl\{ | ||
+ | \biggl[ 5(32z^4 - 41z^2 + 9 ) \biggr] | ||
+ | - \biggl[ (32z^4 - 57 z^2 + 21)\biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + ~ | ||
+ | 4z^2\biggl[ (32z^4 - 57 z^2 + 21) | ||
+ | + (32z^4 - 65z^2 + 33) + (-32z^4 + 41z^2 -9 ) +~( -32z^4 + 81z^2 - 34 ) | ||
+ | \biggr]\biggr\} K ( k_0 ) \cdot E(k_0) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 4z(z-1)\biggl\{ 13 - 16z^2 \biggr\} K ( k_0 ) \cdot K(k_0) | ||
+ | -~4z(z+1) \biggl\{ [ 3 +16z^2 ]\biggr\} E(k_0) \cdot E(k_0) | ||
+ | + 8\biggl\{ | ||
+ | 16z^4 -13z^2 + 3 \biggr\} K ( k_0 ) \cdot E(k_0) \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | Finally, let's evaluate this consolidated expression for the specific case of <math>~z_0 = \cosh\eta_0 = 3</math>, remembering that in this specific case <math>~k_0 = 2^{-1 / 2}</math>, <math>~K(k_0) = 1.854074677</math>, and <math>~E(k_0) = 1.350643881</math>. We find, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~2C_2(z_0)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | [2 \cdot 3 (z^2-1) ]^{-1} \biggl\{ | ||
+ | 4z(z-1) [ 13 - 16z^2 ] K ( k_0 ) \cdot K(k_0) | ||
+ | -~4z(z+1) [ 3 +16z^2 ] E(k_0) \cdot E(k_0) | ||
+ | + 8[ | ||
+ | 16z^4 -13z^2 + 3 ] K ( k_0 ) \cdot E(k_0) | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | [48 ]^{-1} \biggl\{ | ||
+ | -24[ 131 ] K ( k_0 ) \cdot K(k_0) | ||
+ | -~48 [ 147] E(k_0) \cdot E(k_0) | ||
+ | + 8[ 1182 ] K ( k_0 ) \cdot E(k_0) | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 8.8708 \times 10^{-4} \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | <font color="red">This matches the numerically evaluated expression, from above (6/30/2020)</font>. There is a tremendous amount of cancellation between the three key terms in this expression, so the match is only to three significant digits.</tr> | ||
+ | |||
+ | =====Part C===== | ||
+ | |||
+ | Next … | ||
+ | |||
+ | <table border="1" cellpadding="8" align="center" width="60%"> | ||
+ | <tr><td align="left"> | ||
+ | <div align="center">'''Useful Relations from Above'''</div> | ||
+ | |||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\cosh\eta</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\frac{r_1^2 + r_2^2}{2r_1 r_2} \, ;</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\sinh\eta</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\frac{r_1^2 - r_2^2}{2r_1 r_2} \, ;</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\varpi</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\frac{r_1^2 - r_2^2}{2a} \, ;</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\cosh\eta - \cos\theta</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\frac{2a^2}{r_1 r_2} \, ;</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~ \cos\theta</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\frac{r_1^2 + r_2^2-4a^2}{2r_1 r_2} \, ;</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\frac{2}{\coth\eta + 1}</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\frac{4a\varpi}{r_1^2} \, .</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | </td></tr> | ||
+ | </table> | ||
+ | |||
+ | |||
+ | Now, from our tabulation of [[User:Tohline/Appendix/Equation_templates#Example_Recurrence_Relations|example recurrence relations]], we see that, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~ P_{+\frac{3}{2}}(\cosh\eta)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\frac{4}{3} \cdot \cosh\eta ~P_{+\frac{1}{2}}(\cosh\eta) - \frac{1}{3} ~ P_{-\frac{1}{2}}(\cosh\eta) </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\frac{4}{3} \cdot \cosh\eta \biggl[ \frac{\sqrt{2}}{\pi} (\sinh\eta)^{+1 / 2} k^{-1} \boldsymbol{E}(k) \biggr] | ||
+ | - | ||
+ | \frac{1}{3} ~ \biggl[ \frac{\sqrt{2}}{\pi}~ (\sinh\eta)^{-1 / 2} ~k \boldsymbol{K}(k)\biggr]</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\frac{2^{1 / 2}}{3\pi} | ||
+ | \biggl[ 4 \cosh\eta (\sinh\eta)^{+1 / 2} k^{-1} \boldsymbol{E}(k) | ||
+ | - | ||
+ | (\sinh\eta)^{-1 / 2} ~k \boldsymbol{K}(k)\biggr] | ||
+ | \, ,</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | where, as above, | ||
+ | <div align="center"> | ||
+ | <math>~ | ||
+ | k \equiv \biggl[ \frac{2}{\coth\eta + 1} \biggr]^{1 / 2} | ||
+ | = | ||
+ | \biggl[ \frac{4a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \biggr]^{1 / 2} | ||
+ | = | ||
+ | \biggl[ \frac{4a\varpi}{r_1^2} \biggr]^{1 / 2} \, . | ||
+ | </math> | ||
+ | </div> | ||
+ | So we have, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W2}(\eta,\theta)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | -\frac{2^{5/2} }{3\pi^2} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] | ||
+ | C_2(\cosh\eta_0)\cos(2\theta) | ||
+ | \biggl\{ (\cosh\eta - \cos\theta)^{1 / 2}P_{+\frac{3}{2}}(\cosh\eta) \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | -\frac{2^{3} }{3^2\pi^3} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] | ||
+ | C_2(\cosh\eta_0)\cos(2\theta)\biggl( \frac{2a^2}{r_1 r_2}\biggr)^{1 / 2} | ||
+ | \biggl\{ | ||
+ | 4 \cosh\eta (\sinh\eta)^{+1 / 2} k^{-1} \boldsymbol{E}(k) | ||
+ | - | ||
+ | (\sinh\eta)^{-1 / 2} ~k \boldsymbol{K}(k) | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | -\frac{2^{3} }{3^2\pi^3} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] | ||
+ | C_2(\cosh\eta_0)\cos(2\theta)\biggl( \frac{2a^2}{r_1 r_2}\biggr)^{1 / 2} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \times | ||
+ | \biggl\{ | ||
+ | 4 \biggl[ \frac{r_1^2 + r_2^2}{2r_1 r_2} \biggr] \biggl[\frac{r_1^2 - r_2^2}{2r_1 r_2} \biggr]^{+1 / 2} \biggl[ \frac{4a}{r_1^2} \biggl( \frac{r_1^2 - r_2^2}{2a} \biggr)\biggr]^{-1 / 2} \boldsymbol{E}(k) | ||
+ | - | ||
+ | \biggl[ \frac{r_1^2 - r_2^2}{2r_1 r_2} \biggr]^{-1 / 2} ~\biggl[ \frac{4a}{r_1^2}\biggl( \frac{r_1^2 - r_2^2}{2a} \biggr) \biggr]^{1 / 2} \boldsymbol{K}(k) | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | -\frac{2^{9/2} }{3^2\pi^3} \biggl[ \frac{\sinh^3\eta_0}{\cosh\eta_0}\biggr] | ||
+ | C_2(\cosh\eta_0)\cos(2\theta) | ||
+ | \times | ||
+ | \biggl\{ | ||
+ | \biggl[ \frac{r_1^2 + r_2^2}{2r_1 r_2} \biggr] \frac{a}{r_2} \cdot \boldsymbol{E}(k) | ||
+ | - | ||
+ | \frac{a}{r_1} \cdot \boldsymbol{K}(k) | ||
+ | \biggr\} \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | Finally, inserting the expression for <math>~\sinh^2\eta_0~ C_2(\cosh\eta_0) = (z_0^2-1)C_2(z_0)</math> that we have derived, above, gives, | ||
+ | |||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W2}(\eta,\theta)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | -\frac{2^{9/2} }{3^3\pi^3} \biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] | ||
+ | \times \cos(2\theta) | ||
+ | \biggl\{ | ||
+ | \biggl[ \frac{r_1^2 + r_2^2}{2r_1 r_2} \biggr] \frac{a}{r_2} \cdot \boldsymbol{E}(k) | ||
+ | - | ||
+ | \frac{a}{r_1} \cdot \boldsymbol{K}(k) | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \times \biggl\{ | ||
+ | z(z-1) [ 13 - 16z^2 ] K ( k_0 ) \cdot K(k_0) | ||
+ | -~z(z+1) [ 3 +16z^2 ] E(k_0) \cdot E(k_0) | ||
+ | + 2 [ 16z^4 -13z^2 + 3 ] K ( k_0 ) \cdot E(k_0) | ||
+ | \biggr\} \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | ====Summary==== | ||
+ | |||
+ | Once the major ( R ) and minor ( d ) radii of the torus have been specified, two key model parameters can be immediately determined, namely, | ||
+ | <div align="center"> | ||
+ | <math>~a^2 \equiv R^2 - d^2\, ,</math> and, <math>~\cosh\eta_0 \equiv \frac{R}{d} \, ,</math> | ||
+ | </div> | ||
+ | in which case also, <math>~\sinh\eta_0 = a/d \, .</math> Once the mass-density ( ρ<sub>0</sub> ) of the torus has been specified, the torus mass is given by the expression, | ||
+ | <div align="center"> | ||
+ | <math>~M = 2\pi^2 \rho_0 d^2 R \, .</math> | ||
+ | </div> | ||
+ | In addition to the principal pair of meridional-plane coordinates, <math>~(\varpi, z)</math>, it is useful to define the pair of distances, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~r_1^2</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\equiv</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~(\varpi + a)^2 + (z - Z_0)^2 \, ,</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~r_2^2</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\equiv</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~(\varpi - a)^2 + (z - Z_0)^2 \, ,</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | where, the equatorial plane of the torus is located at <math>~z = Z_0</math>. As we have shown above, the leading (n = 0) term in Wong's (1973) series-expression for the gravitational potential anywhere outside the torus is, | ||
+ | |||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W0}(\varpi,z)\biggr|_\mathrm{exterior}</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - \frac{2^{3} }{3\pi^3} | ||
+ | \biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] | ||
+ | \frac{a}{ r_1 } \cdot \boldsymbol{K}(k) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \times \biggl\{ | ||
+ | K(k_0)\cdot K(k_0) [ \cosh\eta_0(1 - \cosh\eta_0) ] | ||
+ | + 2K(k_0)\cdot E(k_0) [ \cosh^2\eta_0 + 1 ] | ||
+ | - E(k_0)\cdot E(k_0) [ \cosh\eta_0(1 + \cosh\eta_0) ] | ||
+ | \biggr\} \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | where, the two distinctly different arguments — one with, and one without a zero subscript — of the complete elliptic-integral functions are, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~k</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\equiv</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl[ \frac{2}{\coth\eta + 1} \biggr]^{1 / 2} | ||
+ | = | ||
+ | \biggl[ \frac{4a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \biggr]^{1 / 2} | ||
+ | = \biggl[ \frac{4a\varpi}{r_1^2} \biggr]^{1 / 2} | ||
+ | = \biggl[ \frac{r_1^2 - r_2^2}{r_1^2} \biggr]^{1 / 2} \, , | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~k_0</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\equiv</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl[ \frac{2}{\cosh\eta_0 + 1} \biggr]^{1 / 2} \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | As we also have shown above, the second (n = 1) term in Wong's (1973) series-expression for the exterior gravitational potential is, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W1}(\varpi,z)\biggr|_\mathrm{exterior}</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - \frac{2^{3} }{3\pi^3} \biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] \biggl[\frac{ r_1^2 + r_2^2 - 4a^2}{2r_1 r_2} \biggr] \frac{a}{r_2} \cdot | ||
+ | \boldsymbol{E}(k) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\times | ||
+ | \biggl\{ | ||
+ | K(k_0)\cdot K(k_0) [\cosh\eta_0(1 - \cosh\eta_0)] | ||
+ | +~2K(k_0)\cdot E(k_0) [(3\cosh^2\eta_0 - 1)] | ||
+ | -~5 E(k_0)\cdot E(k_0) [\cosh\eta_0(1+\cosh\eta_0)] | ||
+ | \biggr\} | ||
+ | \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | Note that a transformation from the <math>~(r_1, r_2)</math> coordinate pair to the toroidal-coordinate pair <math>~(\eta, \theta)</math> includes the expression, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\cos\theta</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \frac{r_1^2 + r_2^2 - 4a^2}{2r_1 r_2} \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | So this (n = 1) term's explicit dependence on "cos(nθ)" is clear. Finally, the third (n = 2) term in Wong's (1973) series-expression for the exterior gravitational potential is, | ||
+ | |||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W2}(\eta,\theta)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | -\frac{2^{3} }{3\pi^3} \biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] | ||
+ | \times \cos(2\theta) | ||
+ | \biggl\{ | ||
+ | \biggl[ \frac{r_1^2 + r_2^2}{2r_1 r_2} \biggr] \frac{a}{r_2} \cdot \boldsymbol{E}(k) | ||
+ | - | ||
+ | \frac{a}{r_1} \cdot \boldsymbol{K}(k) | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \times \frac{2^{3 / 2}}{3^2}\biggl\{ | ||
+ | K ( k_0 ) \cdot K(k_0) \cdot \cosh\eta_0(1 - \cosh\eta_0) [ 16\cosh^2\eta_0 - 13] | ||
+ | + 2 K ( k_0 ) \cdot E(k_0) [ 16\cosh^4\eta_0 -13\cosh^2\eta_0 + 3 ] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | -~E(k_0) \cdot E(k_0) \cdot \cosh\eta_0 (1 + \cosh\eta_0) [ 3 +16\cosh^2\eta_0 ] | ||
+ | \biggr\} \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | ===The Huré, ''et al'' (2020) Presentation=== | ||
+ | |||
+ | {{LSU_WorkInProgress}} | ||
+ | |||
+ | ====Notation==== | ||
+ | |||
+ | In [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Huré, et al. (2020)], the major and minor radii of the torus surface ("shell") are labeled, respectively, R<sub>c</sub> and b, and their ratio is denoted, | ||
+ | <div align="center"> | ||
+ | <math>~e \equiv \frac{b}{R_c} \, .</math> | ||
+ | |||
+ | [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Huré, et al. (2020)], §2, p. 5826, Eq. (1) | ||
+ | </div> | ||
+ | The authors work in cylindrical coordinates, <math>~(R, Z)</math>, whereas we refer to this same coordinate-pair as, <math>~(\varpi_W, z_W)</math>. The quantity, | ||
+ | <div align="center"> | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Delta^2</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\equiv</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | [R + (R_c + b\cos\theta_H)]^2 + [Z - b\sin\theta_H]^2 \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Huré, et al. (2020)], §2, p. 5826, Eqs. (5) & (7) | ||
+ | </div> | ||
+ | |||
+ | We have affixed the subscript "H" to their meridional-plane angle, θ, to clarify that it has a different coordinate-base definition from the meridional-plane angle, θ, that appears in our above discussion of [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong's (1973)] work. In their paper, the subscript "0" is used in the case of an infinitesimally thin hoop <math>~(b \rightarrow 0)</math>, that is to say, | ||
+ | <div align="center"> | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Delta_0^2</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | [R + R_c]^2 + Z^2 \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Huré, et al. (2020)], §3, p. 5827, Eq. (13) | ||
+ | </div> | ||
+ | |||
+ | Generally, the argument (modulus) of the complete elliptic integral functions is, | ||
+ | <div align="center"> | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~k_H</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \frac{2}{\Delta}\biggl[ R (R_c + b\cos\theta_H) \biggr]^{1 / 2} \, , | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Huré, et al. (2020)], §2, p. 5826, Eq. (4) | ||
+ | </div> | ||
+ | and, as stated in the first sentence of their §3, reference may also be made to the ''complementary modulus'', | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~k'_H</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\equiv</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~[1 - k_H^2]^{1 / 2} \, .</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | (Again, we have affixed the subscript "H" in order to differentiate from the modulus employed by [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W Wong (1973)].) And in the case of an infinitesimally thin hoop <math>~(b\rightarrow 0)</math>, | ||
+ | <div align="center"> | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~[k^2_H]_0</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \frac{4R R_c}{\Delta_0^2} \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Huré, et al. (2020)], §3, p. 5827, Eq. (12) | ||
+ | </div> | ||
+ | |||
+ | ====Key Finding==== | ||
+ | On an initial reading, it appears as though the most relevant section of the [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Huré, et al. (2020)] paper is §8 titled, ''The Solid Torus.'' They write the gravitational potential in terms of the series expansion, | ||
+ | <div align="center"> | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Psi_\mathrm{grav}(\vec{r})</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\approx</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \Psi_0 + \sum\limits_{n=1}^N \Psi_n \, , | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Huré, et al. (2020)], §7, p. 5831, Eq. (42) | ||
+ | </div> | ||
+ | where, after setting <math>~M_\mathrm{tot} = 2\pi^2\rho_0 b^2 R_c </math> and acknowledging that <math>~V_{0,0} = 1 \, ,</math> we can write, | ||
+ | <div align="center"> | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Psi_0 </math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - \frac{GM_\mathrm{tot}}{r} \biggl[ \frac{r}{\Delta_0} \cdot \frac{2}{\pi} \boldsymbol{K}([k_H]_0) \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Huré, et al. (2020)], §8, p. 5832, Eqs. (52) & (53) | ||
+ | </div> | ||
+ | and, | ||
+ | <div align="center"> | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\frac{1}{e^2} \biggl[ \Psi_1 + \Psi_2 \biggr]</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - \frac{G\pi \rho_0 R_c b^2}{4 (k'_H)^2 \Delta_0^3} \biggl\{ | ||
+ | [\Delta_0^2 - 2R_c(R_c + R)]\boldsymbol{E}(k_H) - (k'_H)^2 \Delta_0^2 \boldsymbol{K}(k_H) | ||
+ | \biggr\} \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Huré, et al. (2020)], §8, p. 5832, Eq. (54) | ||
+ | </div> | ||
+ | |||
+ | Rewriting this last expression in a form that can more readily be compared with Wong's work, we obtain, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\frac{2^3\pi}{e^2} \biggl[ \frac{ \Psi_1 + \Psi_2 }{GM} \biggr]</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - \frac{ 1 }{(k'_H)^2 \Delta_0^3}\biggl\{ | ||
+ | [\Delta_0^2 - 2R_c(R_c + R)]\boldsymbol{E}(k_H) - (k'_H)^2 \Delta_0^2 \boldsymbol{K}(k_H) | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - \biggl[ \frac{ \Delta_0^2 - 2R_c(R_c + R)}{(1 - k_H^2) \Delta_0^2} \biggr] \frac{\boldsymbol{E}(k_H)}{\Delta_0} + \frac{\boldsymbol{K}(k_H)}{\Delta_0} \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | <span id="Step01">Hence, also,</span> | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~ \frac{ \Psi_0 }{GM} + \biggl[ \frac{ \Psi_1 + \Psi_2 }{GM} \biggr]</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~- \frac{2}{\pi}\biggl\{ | ||
+ | \frac{\boldsymbol{K}([k_H]_0)}{\Delta_0} | ||
+ | \biggr\} + | ||
+ | \frac{e^2}{2^3\pi}\biggl\{ | ||
+ | \frac{\boldsymbol{K}(k_H)}{\Delta_0} | ||
+ | - \biggl[ \frac{ \Delta_0^2 - 2R_c(R_c + R)}{(1 - k_H^2) \Delta_0^2} \biggr] \frac{\boldsymbol{E}(k_H)}{\Delta_0} | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~- \frac{2}{\pi \Delta_0}\biggl\{ | ||
+ | \boldsymbol{K}([k_H]_0) | ||
+ | - | ||
+ | \frac{e^2}{2^4} \cdot \boldsymbol{K}(k_H) \biggr\} | ||
+ | - \frac{2}{\pi\Delta_0} \cdot \frac{e^2}{2^4}\biggl\{\biggl[ \frac{ \Delta_0^2 - 2R_c(R_c + R)}{(1 - k_H^2) \Delta_0^2} \biggr] | ||
+ | \biggr\} \boldsymbol{E}(k_H) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Rightarrow ~~~- \frac{\pi \Delta_0}{2} \biggl[ \frac{ \Psi_0 + \Psi_1 + \Psi_2 }{GM} \biggr] </math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \boldsymbol{K}([k_H]_0) | ||
+ | - | ||
+ | \frac{e^2}{2^4} \cdot \boldsymbol{K}(k_H) | ||
+ | + \frac{e^2}{2^4} \biggl[ \frac{ \Delta_0^2 - 2R_c(R_c + R)}{(1 - k_H^2) \Delta_0^2} \biggr] \boldsymbol{E}(k_H) \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | ===Compare First Terms=== | ||
+ | |||
+ | Rewriting the first term in the [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Huré, et al. (2020)] series expression for the potential, we have, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\frac{\Psi_0}{GM} </math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - \frac{2}{\pi} \biggl\{ \frac{\boldsymbol{K}([k_H]_0) }{[ (\varpi_W + R_c)^2 + z_W^2]^{1 / 2}} \biggr\} \, , | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | where, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~[k_H]_0</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl[ \frac{4\varpi_W R_c}{\Delta_0^2} \biggr]^{1 / 2} | ||
+ | = | ||
+ | \biggl\{ \frac{4\varpi_W R_c}{ (\varpi_W + R_c)^2 + z_W^2} \biggr\}^{1 / 2} \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | For comparison, the first term in Wong's expression is, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\frac{\Phi_\mathrm{W0}}{GM} </math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - \biggl( \frac{2^{3} }{3\pi^3} \biggr) | ||
+ | \Upsilon_{W0}(\eta_0) \biggl\{ | ||
+ | \frac{\boldsymbol{K}(k) }{ r_1 } \biggr\}\, , | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | where, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~a^2 </math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\equiv</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | R^2 - d^2 ~~~\Rightarrow ~~~ a = R_c(1 - e^2)^{1 / 2} \, , | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~r_1^2</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\equiv</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\biggl[ \varpi + R_c (1 - e^2 )^{1 / 2}\biggr]^2 + \biggl[z - Z_0 \biggr]^2 \, ,</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~k</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\equiv</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl\{ \frac{4\varpi R_c(1-e^2)^{1 / 2}}{[\varpi + R_c(1-e^2)^{1 / 2}]^2 + [z - Z_0]^2} \biggr\}^{1 / 2} | ||
+ | \, , | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Upsilon_{W0}(\eta_0)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \frac{\sinh\eta_0}{\cosh\eta_0}\biggl\{ | ||
+ | K(k_0)\cdot K(k_0) [ \cosh\eta_0(1 - \cosh\eta_0) ] | ||
+ | + 2K(k_0)\cdot E(k_0) [ \cosh^2\eta_0 + 1 ] | ||
+ | - E(k_0)\cdot E(k_0) [ \cosh\eta_0(1 + \cosh\eta_0) ] | ||
+ | \biggr\} \, , | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \frac{(1-e^2)^{1 / 2}}{e^2} \biggl\{ | ||
+ | - K(k_0)\cdot K(k_0) (1-e) | ||
+ | + 2K(k_0)\cdot E(k_0) (1+e^2) | ||
+ | - E(k_0)\cdot E(k_0) (1+e) | ||
+ | \biggr\} \, , | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~k_0</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl[ \frac{2}{1+1/e} \biggr]^{1 / 2} | ||
+ | = | ||
+ | \biggl[ \frac{2e}{1+e} \biggr]^{1 / 2} \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | This expression is correct for any value of the aspect ratio, <math>~e</math>. But let's set <math>~Z_0 = 0</math> — as Huré, et al. (2020) have done — then see how the expression simplifies for an infinitesimally thin hoop, that is, if we let <math>~e \rightarrow 0</math>. First we note that, | ||
+ | |||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~k\biggr|_{e\rightarrow 0}</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl\{ \frac{4\varpi R_c}{[\varpi + R_c]^2 + z^2} \biggr\}^{1 / 2} | ||
+ | \, , | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | so in this limit the modulus of the complete elliptic integral of the first kind becomes identical to the modulus used by Huré, et al. (2020), <math>~[k_H]_0</math>. Next, we note that, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~r_1\biggr|_{e\rightarrow 0}</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~[( \varpi + R_c )^2 + z^2]^{1 / 2} \, .</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | As a result, we can write, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\frac{\Phi_\mathrm{W0}}{GM} \biggr|_{e\rightarrow 0}</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \frac{\Psi_0}{GM} \cdot | ||
+ | \biggl[ | ||
+ | \biggl( \frac{2^{2} }{3\pi^2} \biggr) | ||
+ | \Upsilon_{W0}(\eta_0) | ||
+ | \biggr]_{e\rightarrow 0} \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | Now let's evaluate the coefficient, <math>~\Upsilon_{W0}</math>, in the limit of <math>~e \rightarrow 0</math>. | ||
+ | |||
+ | |||
+ | <table align="center" border="1" width="100%" cellpadding="8"><tr><td align="left"> | ||
+ | |||
+ | <div align="center"> | ||
+ | <math>~\Upsilon_{W0}</math>, in the limit of <math>~e \rightarrow 0</math>. | ||
+ | </div> | ||
+ | First, drawing from our [[User:Tohline/Apps/Wong1973Potential#Phase_0C|separate examination of the behavior of complete elliptic integral functions]], we appreciate that, | ||
+ | |||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\frac{2^2}{\pi^2} \biggl[K(k_0) \cdot E(k_0) \biggr]</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 1 ~+~\frac{1}{2^5} ~k_0^4 | ||
+ | ~+~\frac{1}{2^5} ~ k_0^6 | ||
+ | + \mathcal{O}(k_0^{8}) \, , | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\frac{2^2}{\pi^2} \biggl[K(k_0) \cdot K(k_0) \biggr]</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 1 + \frac{1}{2} k_0^2 | ||
+ | + \frac{11}{2^5} ~k_0^4 | ||
+ | + \frac{17}{2^6} ~ k_0^6 | ||
+ | + \mathcal{O}(k_0^{8}) \, , | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\frac{2^2}{\pi^2} \biggl[E(k_0) \cdot E(k_0) \biggr]</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 1 - \frac{1}{2} ~k_0^2 ~-~ \frac{1}{2^5} ~ k_0^4 ~-~\frac{1}{2^6} ~ k_0^6 | ||
+ | + \mathcal{O}(k_0^{8}) \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | Next, employing the [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial expansion]], we find that, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~k_0^2</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~2e(1+e)^{-1}</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~2e( 1 - e +e^2 - e^3 + e^4 - e^5 + \cdots ) \, ;</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | and, | ||
+ | |||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~k_0^4</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~4e^2(1+e)^{-2}</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~4e^2( 1 - 2e + 3e^2 - 4e^3 + 5e^4 - 6e^5 + \cdots ) \, .</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | Hence, we have, | ||
+ | |||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\frac{2^2}{\pi^2} \biggl[ \frac{e^2}{(1 - e^2)^{1 / 2}} \biggr] \Upsilon_{W0}(\eta_0)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - (1-e) \biggl[ 1 + \frac{1}{2} k_0^2 | ||
+ | + \frac{11}{2^5} ~k_0^4 | ||
+ | + \mathcal{O}(k_0^{6}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + 2 (1+e^2) \biggl[ 1 ~+~\frac{1}{2^5} ~k_0^4 | ||
+ | + \mathcal{O}(k_0^{6}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - (1+e) \biggl[ 1 - \frac{1}{2} ~k_0^2 ~-~ \frac{1}{2^5} ~ k_0^4 ~ | ||
+ | + \mathcal{O}(k_0^{6}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - (1-e) \biggl[ 1 + e ( 1 - e +e^2 - e^3 + e^4 - e^5 + \cdots ) | ||
+ | + \frac{11}{2^3} \cdot ~e^2( 1 - 2e + 3e^2 - 4e^3 + 5e^4 - 6e^5 + \cdots ) | ||
+ | + \mathcal{O}(e^{3}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + 2 (1+e^2) \biggl[ 1 ~+~\frac{1}{2^3} \cdot~e^2( 1 - 2e + 3e^2 - 4e^3 + 5e^4 - 6e^5 + \cdots ) | ||
+ | + \mathcal{O}(e^{3}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - (1+e) \biggl[ 1 - e ~( 1 - e +e^2 - e^3 + e^4 - e^5 + \cdots ) ~ | ||
+ | -~ \frac{1}{2^3} \cdot~ e^2( 1 - 2e + 3e^2 - 4e^3 + 5e^4 - 6e^5 + \cdots ) ~ | ||
+ | + \mathcal{O}(e^{3}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - (1-e) \biggl[ 1 + e ( 1 - e ) | ||
+ | + \frac{11}{2^3} \cdot ~e^2 | ||
+ | \biggr] | ||
+ | + 2 (1+e^2) \biggl[ 1 ~+~\frac{1}{2^3} \cdot~e^2 | ||
+ | \biggr] | ||
+ | - (1+e) \biggl[ 1 - e ~( 1 - e ) ~ | ||
+ | -~ \frac{1}{2^3} \cdot~ e^2 ~ | ||
+ | \biggr] | ||
+ | + \mathcal{O}(e^{3}) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - \biggl[ 1 + e ( 1 - e ) | ||
+ | + \frac{11}{2^3} \cdot ~e^2 | ||
+ | \biggr] | ||
+ | +e \biggl[ 1 + e | ||
+ | \biggr] | ||
+ | + 2 \biggl[ 1 ~+~\frac{1}{2^3} \cdot~e^2 | ||
+ | \biggr] | ||
+ | + 2 e^2 | ||
+ | - \biggl[ 1 - e ~( 1 - e ) ~ | ||
+ | -~ \frac{1}{2^3} \cdot~ e^2 ~ | ||
+ | \biggr] | ||
+ | - e \biggl[ 1 - e | ||
+ | \biggr] | ||
+ | + \mathcal{O}(e^{3}) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | -1 - e + e^2 | ||
+ | - \frac{11}{2^3} \cdot ~e^2 | ||
+ | + e + e^2 | ||
+ | + 2 ~+~\frac{1}{2^2} \cdot~e^2 | ||
+ | + 2 e^2 | ||
+ | -1 + e - e^2 ~ | ||
+ | +~ \frac{1}{2^3} \cdot~ e^2 ~ | ||
+ | - e +e^2 | ||
+ | + \mathcal{O}(e^{3}) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\frac{e^2}{2^3} | ||
+ | \biggl[ 2^5 | ||
+ | - 11~ | ||
+ | ~+~3 \biggr]~ | ||
+ | + \mathcal{O}(e^{3}) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 3e^2 | ||
+ | + \mathcal{O}(e^{3}) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Rightarrow ~~~ \frac{2^2}{3\pi^2} \Upsilon_{W0}(\eta_0)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | [1 + \mathcal{O}(e^{1})]\cdot (1 - e^2)^{1 / 2} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Rightarrow ~~~ \biggl[ \frac{2^2}{3\pi^2} \Upsilon_{W0}(\eta_0) \biggr]_{e\rightarrow 0}</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 1 \, .</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | </td></tr></table> | ||
+ | |||
+ | Given that, | ||
+ | |||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~ \biggl[ \frac{2^2}{3\pi^2} \Upsilon_{W0}(\eta_0) \biggr]_{e\rightarrow 0}</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 1 \, ,</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | we conclude that, | ||
+ | |||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\frac{\Phi_\mathrm{W0}}{GM} \biggr|_{e\rightarrow 0}</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \frac{\Psi_0}{GM} \, , | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | that is, we conclude that <math>~\Psi_0</math> matches <math>~\Phi_{W0}</math> in the limit of, <math>~e\rightarrow 0</math>. | ||
+ | |||
+ | ===Go to Higher Order=== | ||
+ | |||
+ | Let's keep higher order terms in Wong's n = 0 component, and let's examine contributions to the same order that come from Wong's n = 1 and (if necessary) n = 2 components. | ||
+ | |||
+ | <span id="Step02">First, note that,</span> | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \Delta_0 \biggl( \frac{2^{2} }{3\pi^2} \biggr) | ||
+ | \Upsilon_{W0}(\eta_0) \biggl\{ | ||
+ | \frac{\boldsymbol{K}(k) }{ r_1 } \biggr\}\, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | |||
+ | ====Keeping Higher Order in Wong's First Component==== | ||
+ | |||
+ | |||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\frac{2^2}{\pi^2} \biggl[K(k_0) \cdot E(k_0) \biggr]</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 1 ~+~\frac{1}{2^5} ~k_0^4 | ||
+ | ~+~\frac{1}{2^5} ~ k_0^6 | ||
+ | ~+~\frac{231}{2^{13}} ~ k_0^8 | ||
+ | + \mathcal{O}(k_0^{10}) \, , | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\frac{2^2}{\pi^2} \biggl[K(k_0) \cdot K(k_0) \biggr]</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 1 + \frac{1}{2} k_0^2 | ||
+ | + \frac{11}{2^5} ~k_0^4 | ||
+ | + \frac{17}{2^6} ~ k_0^6 | ||
+ | + \frac{1787}{2^{13}} ~k^8 | ||
+ | + \mathcal{O}(k_0^{10}) | ||
+ | \, , | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\frac{2^2}{\pi^2} \biggl[E(k_0) \cdot E(k_0) \biggr]</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 1 - \frac{1}{2} ~k_0^2 ~-~ \frac{1}{2^5} ~ k_0^4 | ||
+ | ~-~\frac{1}{2^6} ~ k_0^6 | ||
+ | ~-~\frac{77}{2^{13}} ~ k_0^8 | ||
+ | + \mathcal{O}(k_0^{10}) \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | <table border="1" width="80%" cellpadding="5" align="center"> | ||
+ | <tr><td align="center">'''Add One Additional Term'''</td></tr> | ||
+ | <tr><td align="center"> | ||
+ | |||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\frac{2^2}{\pi^2} \biggl[K(k) \cdot E(k) \biggr]</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl\{ | ||
+ | 1 + \biggl( \frac{1}{2} \biggr)^2k^2 | ||
+ | + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4 | ||
+ | + \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6 | ||
+ | + \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 k^8 | ||
+ | + \cdots | ||
+ | + \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 k^{2n} | ||
+ | + \cdots | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \times~ | ||
+ | \biggl\{ | ||
+ | 1 - \frac{1}{2^2} ~k^2 | ||
+ | - \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4 | ||
+ | - \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5} | ||
+ | - \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 \frac{k^8}{7} | ||
+ | ~-~ \cdots | ||
+ | \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 \frac{k^{2n}}{2n-1} | ||
+ | ~-~ \cdots | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl\{ | ||
+ | 1 - \frac{1}{2^2} ~k^2 | ||
+ | - \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4 | ||
+ | - \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5} | ||
+ | - \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 \frac{k^8}{7} | ||
+ | \biggr\} | ||
+ | ~+~ | ||
+ | \biggl\{ | ||
+ | \biggl( \frac{1}{2} \biggr)^2k^2 | ||
+ | \biggr\} | ||
+ | \times~ | ||
+ | \biggl\{ | ||
+ | 1 - \frac{1}{2^2} ~k^2 | ||
+ | - \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4 | ||
+ | - \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5} | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~+~ | ||
+ | \biggl\{ | ||
+ | \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4 | ||
+ | \biggr\} | ||
+ | \times~ | ||
+ | \biggl\{ | ||
+ | 1 - \frac{1}{2^2} ~k^2 | ||
+ | - \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4 | ||
+ | \biggr\} | ||
+ | ~+~ | ||
+ | \biggl\{ | ||
+ | \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6 | ||
+ | \biggr\}\times | ||
+ | \biggl\{1 - \frac{1}{2^2} ~k^2 | ||
+ | \biggr\} | ||
+ | + \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 k^8 | ||
+ | + \mathcal{O}(k^{10}) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl\{ | ||
+ | 1 - \frac{1}{2^2} ~k^2 | ||
+ | - \frac{3}{2^6}~ k^4 | ||
+ | - \biggl(\frac{5}{2^8}\biggr)~k^6 | ||
+ | - \biggl( \frac{5^2 \cdot 7}{2^{14}}\biggr)~k^8 | ||
+ | \biggr\} | ||
+ | ~+~ | ||
+ | \biggl\{ | ||
+ | \biggl( \frac{1}{2^2} \biggr)k^2 | ||
+ | - \frac{1}{2^4} ~k^4 | ||
+ | - \frac{3}{2^8}~ k^6 | ||
+ | -\frac{5}{2^{10}} ~k^8 | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | ~+~ | ||
+ | \biggl( \frac{3^2}{2^6}\biggr) k^4 | ||
+ | ~-~ | ||
+ | \biggl( \frac{3^2}{2^8}\biggr) k^6 | ||
+ | ~-~ \biggl(\frac{3^3}{2^{12}}\biggr) ~k^8 | ||
+ | ~+~ | ||
+ | \biggl( \frac{5^2}{2^8}\biggr) k^6 | ||
+ | ~-~\biggl(\frac{5^2}{2^{10}}\biggr) ~k^8 | ||
+ | ~+~\biggl(\frac{5^2\cdot 7^2}{2^{14}}\biggr) ~k^8 | ||
+ | + \mathcal{O}(k^{10}) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 1 | ||
+ | ~+~\biggl[ \frac{1}{2^2} | ||
+ | - \frac{1}{2^2} \biggr] ~k^2 | ||
+ | ~+~\biggl[ \frac{3^2}{2^6} | ||
+ | - \frac{3}{2^6} | ||
+ | - \frac{1}{2^4} \biggr]~k^4 | ||
+ | ~+~\biggl[ \frac{5^2}{2^8} | ||
+ | - \frac{5}{2^8} | ||
+ | - \frac{3}{2^8} | ||
+ | ~-~\frac{3^2}{2^8} \biggr]~ k^6 | ||
+ | + \biggl[ | ||
+ | \biggl(\frac{5^2\cdot 7^2}{2^{14}}\biggr) - \biggl(\frac{5^2}{2^{10}}\biggr) - \biggl(\frac{3^3}{2^{12}}\biggr) - \frac{5}{2^{10}} - \biggl( \frac{5^2 \cdot 7}{2^{14}}\biggr) | ||
+ | \biggr]~k^8 | ||
+ | + \mathcal{O}(k^{10}) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 1 ~+~\frac{1}{2^5} ~k^4 | ||
+ | ~+~\frac{1}{2^5} ~ k^6 | ||
+ | ~+~\frac{231}{2^{13}} ~ k^8 | ||
+ | + \mathcal{O}(k^{10}) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\frac{2^2}{\pi^2} \biggl[K(k) \cdot K(k) \biggr]</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl\{ | ||
+ | 1 + \biggl( \frac{1}{2} \biggr)^2k^2 | ||
+ | + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4 | ||
+ | + \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6 | ||
+ | + \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 k^8 | ||
+ | + \cdots | ||
+ | + \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 k^{2n} | ||
+ | + \cdots | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \times~ | ||
+ | \biggl\{ | ||
+ | 1 + \biggl( \frac{1}{2} \biggr)^2k^2 | ||
+ | + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4 | ||
+ | + \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6 | ||
+ | + \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 k^8 | ||
+ | + \cdots | ||
+ | + \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 k^{2n} | ||
+ | + \cdots | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl\{ | ||
+ | 1 | ||
+ | + \frac{1}{2^2} k^2 | ||
+ | + \frac{3^2}{2^6} k^4 | ||
+ | + \frac{5^2}{2^8} k^6 | ||
+ | + \frac{5^2 \cdot 7^2}{2^{14}} k^8 | ||
+ | \biggr\} | ||
+ | \times~ | ||
+ | \biggl\{ | ||
+ | 1 | ||
+ | + \frac{1}{2^2} k^2 | ||
+ | + \frac{3^2}{2^6} k^4 | ||
+ | + \frac{5^2}{2^8} k^6 | ||
+ | + \frac{5^2 \cdot 7^2}{2^{14}} k^8 | ||
+ | \biggr\} | ||
+ | + \mathcal{O}(k^{10}) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl\{ | ||
+ | 1 | ||
+ | + \frac{1}{2^2} k^2 | ||
+ | + \frac{3^2}{2^6} k^4 | ||
+ | + \frac{5^2}{2^8} k^6 | ||
+ | + \frac{5^2 \cdot 7^2}{2^{14}} k^8 | ||
+ | \biggr\} | ||
+ | ~+~\frac{1}{2^2} k^2 | ||
+ | \biggl\{ | ||
+ | 1 | ||
+ | + \frac{1}{2^2} k^2 | ||
+ | + \frac{3^2}{2^6} k^4 | ||
+ | + \frac{5^2}{2^8} k^6 | ||
+ | \biggr\} | ||
+ | ~+~\frac{3^2}{2^6} k^4 | ||
+ | \biggl\{ | ||
+ | 1 | ||
+ | + \frac{1}{2^2} k^2 | ||
+ | + \frac{3^2}{2^6} k^4 | ||
+ | \biggr\} | ||
+ | ~+~ | ||
+ | \biggl\{ | ||
+ | \frac{5^2}{2^8} k^6 | ||
+ | \biggr\} | ||
+ | \biggl\{ | ||
+ | 1 | ||
+ | + \frac{1}{2^2} k^2 | ||
+ | \biggr\} | ||
+ | ~+~ | ||
+ | \biggl\{ | ||
+ | \frac{5^2 \cdot 7^2}{2^{14}} k^8 | ||
+ | \biggr\} | ||
+ | + \mathcal{O}(k^{10}) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl\{ | ||
+ | 1 + \frac{1}{2^2} k^2 | ||
+ | + \frac{3^2}{2^6} k^4 | ||
+ | + \frac{5^2}{2^8} k^6 | ||
+ | + \frac{5^2 \cdot 7^2}{2^{14}} k^8 | ||
+ | \biggr\} | ||
+ | ~+~ | ||
+ | \biggl\{ | ||
+ | \frac{1}{2^2} k^2 | ||
+ | + \frac{1}{2^4} k^4 | ||
+ | + \frac{3^2}{2^8} k^6 | ||
+ | + \frac{5^2}{2^{10}} k^8 | ||
+ | \biggr\} | ||
+ | ~+~ | ||
+ | \biggl\{ | ||
+ | \frac{3^2}{2^6} k^4 | ||
+ | ~+~\frac{3^2}{2^8} k^6 | ||
+ | ~+~\frac{3^4}{2^{12}} k^8 | ||
+ | \biggr\} | ||
+ | ~+~ | ||
+ | \biggl\{ | ||
+ | \frac{5^2}{2^8} k^6 | ||
+ | ~+~\frac{5^2}{2^{10}} k^8 | ||
+ | \biggr\} | ||
+ | ~+~ | ||
+ | \biggl\{ | ||
+ | \frac{5^2 \cdot 7^2}{2^{14}} k^8 | ||
+ | \biggr\} | ||
+ | + \mathcal{O}(k^{10}) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 1 + \biggl[ \frac{1}{2^2} | ||
+ | ~+~ \frac{1}{2^2} \biggr] ~k^2 | ||
+ | + \biggl[ \frac{3^2}{2^6} | ||
+ | + \frac{1}{2^4} | ||
+ | ~+~\frac{3^2}{2^6} \biggr]~k^4 | ||
+ | + \biggl[ \frac{5^2}{2^8} | ||
+ | + \frac{3^2}{2^8} | ||
+ | ~+~\frac{3^2}{2^8} | ||
+ | ~+~\frac{5^2}{2^8} \biggr]~ k^6 | ||
+ | ~+~\biggl[ | ||
+ | \frac{5^2 \cdot 7^2}{2^{14}}+ \frac{5^2}{2^{10}} ~+~\frac{3^4}{2^{12}} ~+~\frac{5^2}{2^{10}} ~+~\frac{5^2 \cdot 7^2}{2^{14}} | ||
+ | \biggr]k^8 | ||
+ | + \mathcal{O}(k^{10}) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 1 + \frac{1}{2} k^2 | ||
+ | + \frac{11}{2^5} ~k^4 | ||
+ | + \frac{17}{2^6} ~ k^6 | ||
+ | + \frac{1787}{2^{13}} ~k^{8} | ||
+ | + \mathcal{O}(k^{10}) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | |||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\frac{2^2}{\pi^2} \biggl[E(k) \cdot E(k) \biggr]</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl\{ | ||
+ | 1 - \frac{1}{2^2} ~k^2 | ||
+ | - \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4 | ||
+ | - \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5} | ||
+ | - \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 \frac{k^8}{7} | ||
+ | ~-~ \cdots | ||
+ | \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 \frac{k^{2n}}{2n-1} | ||
+ | ~-~ \cdots | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \times~ | ||
+ | \biggl\{ | ||
+ | 1 - \frac{1}{2^2} ~k^2 | ||
+ | - \frac{1^2\cdot 3}{2^2\cdot 4^2}~ k^4 | ||
+ | - \biggl(\frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2~\frac{ k^6 }{5} | ||
+ | - \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 \frac{k^8}{7} | ||
+ | ~-~ \cdots | ||
+ | \biggl[ \frac{(2n-1)!!}{2^n n!} \biggr]^2 \frac{k^{2n}}{2n-1} | ||
+ | ~-~ \cdots | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl\{ | ||
+ | 1 - \frac{1}{2^2} ~k^2 | ||
+ | - \frac{3}{2^6}~ k^4 | ||
+ | - \frac{5}{2^8}~ k^6 | ||
+ | - \frac{5^2\cdot 7}{2^{14}} ~k^8 | ||
+ | \biggr\} | ||
+ | \times | ||
+ | \biggl\{ | ||
+ | 1 - \frac{1}{2^2} ~k^2 | ||
+ | - \frac{3}{2^6}~ k^4 | ||
+ | - \frac{5}{2^8}~ k^6 | ||
+ | - \frac{5^2\cdot 7}{2^{14}} ~k^8 | ||
+ | \biggr\} | ||
+ | + \mathcal{O}(k^{10}) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl\{ | ||
+ | 1 - \frac{1}{2^2} ~k^2 | ||
+ | - \frac{3}{2^6}~ k^4 | ||
+ | - \frac{5}{2^8}~ k^6 | ||
+ | - \frac{5^2\cdot 7}{2^{14}} ~k^8 | ||
+ | \biggr\} | ||
+ | ~+~ | ||
+ | \biggl\{- \frac{1}{2^2} ~k^2 | ||
+ | \biggr\} | ||
+ | \times | ||
+ | \biggl\{ | ||
+ | 1 - \frac{1}{2^2} ~k^2 | ||
+ | - \frac{3}{2^6}~ k^4 | ||
+ | - \frac{5}{2^8}~ k^6 | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | ~+~ | ||
+ | \biggl\{ | ||
+ | - \frac{3}{2^6}~ k^4 | ||
+ | \biggr\} | ||
+ | \times | ||
+ | \biggl\{ | ||
+ | 1 - \frac{1}{2^2} ~k^2 | ||
+ | - \frac{3}{2^6}~ k^4 | ||
+ | \biggr\} | ||
+ | ~+~ | ||
+ | \biggl\{ | ||
+ | - \frac{5}{2^8}~ k^6 | ||
+ | \biggr\} | ||
+ | \times | ||
+ | \biggl\{ | ||
+ | 1 - \frac{1}{2^2} ~k^2 | ||
+ | \biggr\} | ||
+ | ~+~ | ||
+ | \biggl\{ | ||
+ | - \frac{5^2\cdot 7}{2^{14}}k^8 | ||
+ | \biggr\} | ||
+ | + \mathcal{O}(k^{10}) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | |||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl\{ | ||
+ | 1 - \frac{1}{2^2} ~k^2 | ||
+ | - \frac{3}{2^6}~ k^4 | ||
+ | - \frac{5}{2^8}~ k^6 | ||
+ | - \frac{5^2\cdot 7}{2^{14}} ~k^8 | ||
+ | \biggr\} | ||
+ | ~+~ | ||
+ | \biggl\{ | ||
+ | - \frac{1}{2^2} ~k^2 + \frac{1}{2^4} ~k^4 | ||
+ | + \frac{3}{2^8}~ k^6 | ||
+ | + \frac{5}{2^{10}}~k^8 | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | ~+~ | ||
+ | \biggl\{ | ||
+ | ~-~ \frac{3}{2^6}~ k^4 | ||
+ | ~+~ \frac{3}{2^8}~ k^6 | ||
+ | + \frac{3^2}{2^{12}}~k^8 | ||
+ | \biggr\} | ||
+ | ~+~ | ||
+ | \biggl\{ | ||
+ | ~-~\frac{5}{2^8}~ k^6 | ||
+ | + \frac{5}{2^{10}}~k^8 | ||
+ | \biggr\} | ||
+ | ~+~ | ||
+ | \biggl\{ | ||
+ | ~-~ \frac{5^2\cdot 7}{2^{14}}k^8 | ||
+ | \biggr\} | ||
+ | + \mathcal{O}(k^{10}) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 1 + \biggl[ - \frac{1}{2^2}- \frac{1}{2^2} \biggr]k^2 | ||
+ | + \biggl[ - \frac{3}{2^6}+ \frac{1}{2^4}~-~ \frac{3}{2^6} \biggr] k^4 | ||
+ | + \biggl[ - \frac{5}{2^8}+ \frac{3}{2^8}~+~ \frac{3}{2^8}~-~\frac{5}{2^8} \biggr] k^6 | ||
+ | + \biggl[ - \frac{5^2\cdot 7}{2^{14}} + \frac{5}{2^{10}}+ \frac{3^2}{2^{12}}+ \frac{5}{2^{10}}~-~ \frac{5^2\cdot 7}{2^{14}} \biggr] k^8 | ||
+ | + \mathcal{O}(k^{10}) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 1 + \biggl[ - \frac{2}{2^2} \biggr]k^2 | ||
+ | + \biggl[ - \frac{3}{2^5}+ \frac{2}{2^5} \biggr] k^4 | ||
+ | + \biggl[ - \frac{5}{2^7}+ \frac{3}{2^7} \biggr] k^6 | ||
+ | + \biggl[ - \frac{5^2\cdot 7}{2^{13}} + \frac{5\cdot 2^4}{2^{13}}+ \frac{2 \cdot 3^2}{2^{13}}\biggr] k^8 | ||
+ | + \mathcal{O}(k^{10}) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 1 - \frac{1}{2} ~k^2 ~-~ \frac{1}{2^5} ~ k^4 ~-~\frac{1}{2^6} ~ k^6 ~-~\frac{77}{2^{13}} ~ k^8 | ||
+ | + \mathcal{O}(k^{10}) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | </td></tr> | ||
+ | </table> | ||
+ | |||
+ | |||
+ | Next, employing the [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial expansion]], we find that, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~k_0^2</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~2e(1+e)^{-1}</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~2e( 1 - e +e^2 - e^3 + e^4 - e^5 + \cdots ) \, ;</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~k_0^4</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~4e^2(1+e)^{-2}</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~4e^2( 1 - 2e + 3e^2 - 4e^3 + 5e^4 - 6e^5 + \cdots ) \, ;</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~k_0^6</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~2^3e^3(1+e)^{-3}</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~2^3e^3( 1 - 3e + 6e^2 - 10e^3 + 15e^4 - 21e^5 + \cdots ) \, ;</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~k_0^8</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~2^4e^4(1+e)^{-4}</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~2^4e^4(1 - 4e + 10 e^2 - 20e^3 + 35e^4 - 56e^5 + \cdots) \, .</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | <span id="Step03">Hence, we have,</span> | ||
+ | |||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\frac{2^2}{\pi^2} \biggl[ \frac{e^2}{(1 - e^2)^{1 / 2}} \biggr] \Upsilon_{W0}(\eta_0)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - (1-e) \biggl[ 1 + \frac{1}{2} k_0^2 | ||
+ | + \frac{11}{2^5} ~k_0^4 | ||
+ | + \frac{17}{2^6} ~ k_0^6 | ||
+ | + \frac{1787}{2^{13}} ~k_0^8 | ||
+ | + \mathcal{O}(k_0^{10}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + 2 (1+e^2) \biggl[ 1 ~+~\frac{1}{2^5} ~k_0^4 | ||
+ | ~+~\frac{1}{2^5} ~ k_0^6 | ||
+ | ~+~ \frac{231}{2^{13}}~k_0^8 | ||
+ | + \mathcal{O}(k_0^{10}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - (1+e) \biggl[ 1 - \frac{1}{2} ~k_0^2 ~-~ \frac{1}{2^5} ~ k_0^4 ~ | ||
+ | ~-~\frac{1}{2^6} ~ k_0^6 | ||
+ | ~-~\frac{77}{2^{13}}~k_0^8 | ||
+ | + \mathcal{O}(k_0^{10}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - (1-e) \biggl[ 1 + e ( 1 - e +e^2 -e^3 ) | ||
+ | + \frac{11}{2^3} \cdot ~e^2( 1 - 2e + 3e^2) | ||
+ | + \frac{17}{2^6} ~\cdot 2^3 e^3 ( 1 - 3e) | ||
+ | + \frac{1787}{2^{13}} \cdot 2^4e^4 | ||
+ | + \mathcal{O}(e^{5}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + 2 (1+e^2) \biggl[ 1 ~+~\frac{1}{2^5} \cdot~4e^2( 1 - 2e + 3e^2) | ||
+ | ~+~\frac{1}{2^5} ~ 2^3e^3 (1-3e) | ||
+ | ~+~\frac{231}{2^{13}} \cdot 2^4e^4 | ||
+ | + \mathcal{O}(e^{5}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - (1+e) \biggl[ 1 - e ~( 1 - e +e^2 -e^3) ~ | ||
+ | -~ \frac{1}{2^5} \cdot~ 4e^2( 1 - 2e +3e^2) ~ | ||
+ | ~-~\frac{1}{2^6} ~ 2^3e^3(1 - 3e) | ||
+ | ~-~\frac{77}{2^{13}}~2^4e^4 | ||
+ | + \mathcal{O}(e^{5}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - 2^{-9}(1-e) \biggl[ 512 (1+ e - e^2 +e^3 -e^4 ) | ||
+ | + 704 \cdot ~( e^2 - 2e^3 + 3e^4) | ||
+ | + 1088 ~\cdot ( e^3 - 3e^4) | ||
+ | + 1787 \cdot e^4 | ||
+ | + \mathcal{O}(e^{5}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + (1+e^2) 2^{-9}\biggl[ 1024 | ||
+ | ~+~128 \cdot~( e^2 - 2e^3 + 3e^4) | ||
+ | ~+~256 ~ (e^3-3e^4) | ||
+ | ~+~462 \cdot e^4 | ||
+ | + \mathcal{O}(e^{5}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - 2^{-9}(1+e) \biggl[ 512 ~(1 -e + e^2 -e^3 + e^4) ~ | ||
+ | -~ 64 \cdot~ ( e^2 - 2e^3 +3e^4) ~ | ||
+ | ~-~64 ~ (e^3 - 3e^4) | ||
+ | ~-~77~e^4 | ||
+ | + \mathcal{O}(e^{5}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - 2^{-9}(1-e) \biggl[ | ||
+ | 512 + 512e + 192e^2 + e^3(512 - 1408 + 1088) + e^4(704-512 - 3264 + 1787) | ||
+ | + \mathcal{O}(e^{5}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + 2^{-9}(1+e^2) \biggl[ | ||
+ | 1024 + 128e^2 + e^3(-256 + 256 ) + e^4(384 -768 + 462) | ||
+ | + \mathcal{O}(e^{5}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - 2^{-9}(1+e) \biggl[ | ||
+ | 512 - 512e + e^2(512 - 64 ) + e^3(-512 +128 -64 ) + e^4(512 - 192 - 192 - 77) | ||
+ | + \mathcal{O}(e^{5}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - 2^{-9}(1-e) \biggl[ | ||
+ | 512 + 512e + 192e^2 + 192e^3 - 1285 e^4 | ||
+ | + \mathcal{O}(e^{5}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + 2^{-9}(1+e^2) \biggl[ | ||
+ | 1024 + 128e^2 + 78e^4 | ||
+ | + \mathcal{O}(e^{5}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - 2^{-9}(1+e) \biggl[ | ||
+ | 512 - 512e + 448e^2 - 448 e^3 + 51e^4 | ||
+ | + \mathcal{O}(e^{5}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 2^{-9}\biggl[ | ||
+ | (- 192+ 128 - 448)e^2 | ||
+ | + (- 192 + 448) e^3 | ||
+ | + (1285 + 78- 51)e^4 | ||
+ | + \mathcal{O}(e^{5}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + 2^{-9} e \biggl[ | ||
+ | 1024e + (192- 448)e^2 | ||
+ | + (192+ 448) e^3 | ||
+ | + \mathcal{O}(e^{4}) | ||
+ | \biggr] | ||
+ | + 2^{-9} e^2 \biggl[ | ||
+ | 1024 + 128e^2 | ||
+ | + \mathcal{O}(e^{3}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 2^{-9} \biggl[ | ||
+ | (- 192+ 128 - 448)e^2 + 2048 e^2 | ||
+ | + (1285 + 78- 51)e^4 | ||
+ | + (192+ 448) e^4 | ||
+ | + 128e^4 | ||
+ | + \mathcal{O}(e^{5}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 2^{-9} \biggl[ | ||
+ | 1536e^2 | ||
+ | + 2080e^4 | ||
+ | + \mathcal{O}(e^{5}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 3e^2 | ||
+ | + \biggl[ \frac{5\cdot 13}{2^4}\biggr] e^4 | ||
+ | + \mathcal{O}(e^{5}) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Rightarrow ~~~ \frac{2^2}{3\pi^2} \Upsilon_{W0}(\eta_0)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl[1 + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2 + \mathcal{O}(e^{3}) \biggr]\cdot (1 - e^2)^{1 / 2} \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | Hence, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | [1 + \mathcal{O}(e^{2})]\cdot (1 - e^2)^{1 / 2} | ||
+ | \biggl\{ | ||
+ | \frac{\boldsymbol{K}(k) }{ r_1 } \biggr\}\Delta_0 \, , | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | or, more precisely, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl[1 + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2 + \mathcal{O}(e^{3}) \biggr]\cdot (1 - e^2)^{1 / 2} | ||
+ | \biggl\{ | ||
+ | \frac{\boldsymbol{K}(k) }{ r_1 } \biggr\}\Delta_0 \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | ====Next Factors==== | ||
+ | |||
+ | |||
+ | Now, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Delta_0^2</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | (\varpi_W + R_c)^2 + z_W^2 \, , | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~r_1^2</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\biggl[ \varpi_W + R_c (1 - e^2 )^{1 / 2}\biggr]^2 + z_W^2 \, ,</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Rightarrow ~~~ r_1^2 - \Delta_0^2</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\biggl[ \varpi_W + R_c (1 - e^2 )^{1 / 2}\biggr]^2 + z_W^2 - \biggl[ (\varpi_W + R_c)^2 + z_W^2 \biggr]</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\varpi_W^2 + 2\varpi_W R_c (1 - e^2 )^{1 / 2} + R_c^2 (1 - e^2 ) + z_W^2 - [\varpi_W^2 + 2\varpi_W R_c + R_c^2 + z_W^2 ]</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~2\varpi_W R_c [(1 - e^2 )^{1 / 2} - 1] -e^2 R_c^2 \, . </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | ---- | ||
+ | |||
+ | Again, drawing from the [[User:Tohline/Appendix/Ramblings/PowerSeriesExpressions#Binomial|binomial theorem]], we have, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~(1 -e^2)^{1 / 2}</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 1 - \frac{1}{2}e^2 + \biggl[\frac{ \tfrac{1}{2}(-\tfrac{1}{2}) }{ 2 } \biggr]e^4 - \biggl[ \frac{ \tfrac{1}{2}(-\tfrac{1}{2} )(-\tfrac{3}{2} ) }{ 3! } \biggr]e^6 | ||
+ | + \biggl[ \frac{ \tfrac{1}{2} (-\tfrac{1}{2})(-\tfrac{3}{2})(-\tfrac{5}{2}) }{ 4! } \biggr]e^8 + \cdots | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 1 - \frac{1}{2}e^2 - \frac{1}{2^3} e^4 - \frac{1}{2^4}e^6 - \frac{5}{2^7} e^8 - \mathcal{O}(e^{10}) \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | ---- | ||
+ | |||
+ | <div align="center" id="Step04"> | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Rightarrow ~~~ r_1^2 - \Delta_0^2</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\varpi_W R_c \biggl[- e^2 - \frac{1}{2^2} e^4 - \frac{1}{2^3}e^6 - \frac{5}{2^6} e^8 - \mathcal{O}(e^{10}) \biggr] -e^2 R_c^2 </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Rightarrow ~~~\frac{ r_1^2}{\Delta_0^2} </math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~1 | ||
+ | -e^2 \biggl[ \frac{R_c(R_c + \varpi_W) }{\Delta_0^2}\biggr] | ||
+ | - \frac{\varpi_W R_c}{\Delta_0^2} \biggl[\frac{1}{2^2} e^4 + \frac{1}{2^3}e^6 + \frac{5}{2^6} e^8 + \mathcal{O}(e^{10}) \biggr] </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | </div> | ||
+ | |||
+ | |||
+ | |||
+ | ====Now Work on Elliptic Integral Expressions==== | ||
+ | |||
+ | |||
+ | From a [[User:Tohline/2DStructure/ToroidalGreenFunction#Series_Expansions|separate discussion]] we can draw the series expansion of <math>~\boldsymbol{K}(k)</math>, specifically, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\frac{2K(k)}{\pi}</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 1 + \biggl( \frac{1}{2} \biggr)^2k^2 | ||
+ | + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4 | ||
+ | + \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6 | ||
+ | + \cdots | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | where, | ||
+ | |||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\frac{k^2}{4}</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\equiv</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \frac{1}{4} \biggl[ \frac{r_1^2 - r_2^2}{r_1^2} \biggr] | ||
+ | = \biggl[ \frac{a\varpi}{r_1^2} \biggr] | ||
+ | = \biggl[ \frac{a\varpi}{(\varpi + a)^2 + (z - Z_0)^2} \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \frac{\varpi_W R_c(1-e^2)^{1 / 2}}{[ \varpi_W + R_c(1-e^2)^{1 / 2} ]^2 + z_W^2} \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | Also, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\frac{2K(k_H)}{\pi}</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 1 + \biggl( \frac{1}{2} \biggr)^2k_H^2 | ||
+ | + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k_H^4 | ||
+ | + \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k_H^6 | ||
+ | + \cdots | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | where, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\frac{k_H^2}{4}</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \frac{1}{\Delta^2}\biggl[ R (R_c + b\cos\theta_H) \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \frac{\varpi_W (R_c + b\cos\theta_H)}{[\varpi_W + (R_c + b\cos\theta_H)]^2 + [z_W - b\sin\theta_H]^2} \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | What we want to do is write <math>~K(k)</math> in terms of <math>~K(k_H)</math>. Let's try … | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\frac{2K(k)}{\pi}</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\frac{2K(k_H)}{\pi} + \delta_K \, ,</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | |||
+ | where, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\delta_K</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\equiv</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\frac{2K(k)}{\pi} - \frac{2K(k_H)}{\pi} </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\biggl\{1 + \frac{k^2}{4} | ||
+ | + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4 | ||
+ | + \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6 | ||
+ | + \cdots | ||
+ | \biggr\} | ||
+ | - \biggl\{1 + \frac{k_H^2}{4} | ||
+ | + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k_H^4 | ||
+ | + \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k_H^6 | ||
+ | + \cdots | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\approx</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\biggl\{1 + \frac{k^2}{4} | ||
+ | \biggr\} | ||
+ | - \biggl\{1 + \frac{k_H^2}{4} | ||
+ | \biggr\} | ||
+ | = \frac{k^2}{4} - \frac{k_H^2}{4} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \frac{\varpi_W R_c(1-e^2)^{1 / 2}}{[ \varpi_W + R_c(1-e^2)^{1 / 2} ]^2 + z_W^2} | ||
+ | - | ||
+ | \frac{\varpi_W (R_c + b\cos\theta_H)}{[\varpi_W + (R_c + b\cos\theta_H)]^2 + [z_W - b\sin\theta_H]^2} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl\{ \varpi_W R_c(1-e^2)^{1 / 2} \biggr\} \biggl\{\varpi_W^2 +R_c^2 + z_W^2 + 2\varpi_W R_c(1-e^2)^{1 / 2} - R_c^2 e^2 \biggr\}^{-1} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - | ||
+ | \biggl\{ \varpi_W R_c (1 + e\cos\theta_H) \biggr\} \biggl\{[\varpi_W + R_c(1 + e\cos\theta_H)]^2 + [z_W - R_c e\sin\theta_H]^2 \biggr\}^{-1} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\approx</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl\{ \varpi_W R_c \biggl[ 1-\frac{1}{2} e^2 + \mathcal{O}(e^4)\biggr] \biggr\} \biggl\{\varpi_W^2 +R_c^2 + z_W^2 + 2\varpi_W R_c\biggl[ 1-\frac{1}{2} e^2 + \mathcal{O}(e^4)\biggr] - R_c^2 e^2 \biggr\}^{-1} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - | ||
+ | \biggl\{ \varpi_W R_c \biggl[ 1 + e\cos\theta_H \biggr] \biggr\} | ||
+ | \biggl\{\varpi_W^2 + 2\varpi_W R_c(1+e\cos\theta_H) | ||
+ | + R_c^2(1 + 2e\cos\theta_H + e^2\cos^2\theta_H) + z_W^2 - 2 z_W R_c e\sin\theta_H + R_c^2 e^2 \sin\theta^2_H \biggr\}^{-1} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\approx</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl\{\varpi_W R_c - e^2 \biggl[ \frac{\varpi_W R_c}{2} \biggr] + \mathcal{O}(e^4) \biggr\} | ||
+ | \biggl\{ [ (\varpi_W +R_c)^2 + z_W^2 ] - e^2 \biggl[ \varpi_W R_c + R_c^2 \biggr] + \mathcal{O}(e^4) \biggr\}^{-1} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - | ||
+ | \biggl\{ \varpi_W R_c + e \biggl[ \varpi_W R_c \cos\theta_H \biggr]\biggr\} | ||
+ | \biggl\{ [ (\varpi_W^2 + R_c)^2 + z_W^2 ] | ||
+ | + 2R_c e(R_c \cos\theta_H+ \varpi_W \cos\theta_H - z_W \sin\theta_H) + R_c^2 e^2 \biggr\}^{-1} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\approx</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl\{ \frac{\varpi_W R_c}{ [ (\varpi_W +R_c)^2 + z_W^2 ] } \biggl[1 - \frac{1}{2}e^2 \biggr] \biggr\} | ||
+ | \biggl\{ 1 - e^2 \biggl[ \frac{ \varpi_W R_c + R_c^2 }{ (\varpi_W +R_c)^2 + z_W^2 } \biggr] \biggr\}^{-1} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | -~ | ||
+ | \biggl\{ \frac{ \varpi_W R_c }{ [(\varpi_W^2 + R_c)^2 + z_W^2 ] } \biggl[ 1 + e\cos\theta_H \biggr] \biggr\} | ||
+ | \biggl\{ 1 | ||
+ | + e\biggl[ \frac{2R_c (R_c \cos\theta_H+ \varpi_W \cos\theta_H - z_W \sin\theta_H)}{ (\varpi_W^2 + R_c)^2 + z_W^2 }\biggr] | ||
+ | + e^2 \biggl[ \frac{R_c^2 }{ (\varpi_W^2 + R_c)^2 + z_W^2 } \biggr] \biggr\}^{-1} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\approx</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \frac{\varpi_W R_c}{ [ (\varpi_W +R_c)^2 + z_W^2 ] } \biggl[1 - \frac{1}{2}e^2 \biggr] | ||
+ | \biggl\{ 1 + e^2 \biggl[ \frac{ \varpi_W R_c + R_c^2 }{ (\varpi_W +R_c)^2 + z_W^2 } \biggr] \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | -~\frac{ \varpi_W R_c }{ [(\varpi_W^2 + R_c)^2 + z_W^2 ] } \biggl[ 1 + e\cos\theta_H \biggr] | ||
+ | \biggl\{ 1 | ||
+ | - e\biggl[ \frac{2R_c (R_c \cos\theta_H+ \varpi_W \cos\theta_H - z_W \sin\theta_H)}{ (\varpi_W^2 + R_c)^2 + z_W^2 }\biggr] | ||
+ | - e^2 \biggl[ \frac{R_c^2 }{ (\varpi_W^2 + R_c)^2 + z_W^2 } \biggr] \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\approx</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | -~\frac{ e\cdot \varpi_W R_c }{ [(\varpi_W^2 + R_c)^2 + z_W^2 ] } | ||
+ | \biggl[ \cos\theta_H - \frac{2R_c (R_c \cos\theta_H+ \varpi_W \cos\theta_H - z_W \sin\theta_H)}{ (\varpi_W^2 + R_c)^2 + z_W^2 }\biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | +~\frac{e^2 \cdot \varpi_W R_c}{ [ (\varpi_W +R_c)^2 + z_W^2 ] } | ||
+ | \biggl\{ \biggl[ \frac{ \varpi_W R_c + R_c^2 }{ (\varpi_W +R_c)^2 + z_W^2 } - \frac{1}{2} \biggr] | ||
+ | - \biggl[ \frac{R_c^2 }{ (\varpi_W^2 + R_c)^2 + z_W^2 } \biggr] | ||
+ | - \cos\theta_H \biggl[ \frac{2R_c (R_c \cos\theta_H+ \varpi_W \cos\theta_H - z_W \sin\theta_H)}{ (\varpi_W^2 + R_c)^2 + z_W^2 }\biggr] | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\approx</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | -~\frac{ e\cdot \varpi_W R_c }{ [(\varpi_W^2 + R_c)^2 + z_W^2 ] [ (\varpi_W^2 + R_c)^2 + z_W^2 ] } | ||
+ | \biggl[ \cos\theta_H [(\varpi_W^2 + R_c)^2 + z_W^2 ] -2R_c (R_c \cos\theta_H+ \varpi_W \cos\theta_H - z_W \sin\theta_H) \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | +~\frac{\tfrac{1}{2}e^2 \cdot \varpi_W R_c}{ [ (\varpi_W +R_c)^2 + z_W^2 ][ (\varpi_W^2 + R_c)^2 + z_W^2 ] } | ||
+ | \biggl\{ 2 (\varpi_W R_c + R_c^2 ) - [ (\varpi_W^2 + R_c)^2 + z_W^2 ] | ||
+ | - 2R_c^2 | ||
+ | - 4R_c\cos\theta_H (R_c \cos\theta_H+ \varpi_W \cos\theta_H - z_W \sin\theta_H) | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\approx</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | -~\frac{ e\cdot \varpi_W R_c }{ \Delta_0^2 } | ||
+ | \biggl[ \cos\theta_H - \frac{2R_c (R_c \cos\theta_H+ \varpi_W \cos\theta_H - z_W \sin\theta_H)}{\Delta_0^2} \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | +~\frac{ e^2 \cdot \varpi_W R_c}{2 \Delta_0^4 } | ||
+ | \biggl\{ 2 (\varpi_W R_c + R_c^2 ) - [ (\varpi_W^2 + R_c)^2 + z_W^2 ] | ||
+ | - 2R_c^2 | ||
+ | - 4R_c\cos\theta_H (R_c \cos\theta_H+ \varpi_W \cos\theta_H - z_W \sin\theta_H) | ||
+ | \biggr\} \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | |||
+ | Let's subtract <math>~K([k_H]_0)</math> from the potential expression. But first, let's adopt the shorthand notation … | ||
+ | |||
+ | <table border="1" width="80%" align="center" cellpadding="8"><tr><td align="left"> | ||
+ | Given that, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl\{ | ||
+ | \boldsymbol{K}(k) \biggr\} \frac{\Delta_0}{r_1}~\biggl( \frac{2^2}{3\pi^2} \biggr) \Upsilon_{W0}(\eta_0) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\biggl\{ \boldsymbol{K}(k) \biggr\} \biggl[\frac{r_1^2 }{ \Delta_0^2 } \biggr]^{-1 / 2} | ||
+ | \biggl[1 + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2 + \mathcal{O}(e^{3}) \biggr]\cdot (1 - e^2)^{1 / 2} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\biggl\{ \boldsymbol{K}(k) \biggr\} \biggl\{1 + \frac{e^2}{2} \biggl[ \frac{R_c(R_c + \varpi_W)}{\Delta_0^2}\biggr] + \mathcal{O}(e^4)\biggr\} | ||
+ | \biggl[1 + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2 + \mathcal{O}(e^{3}) \biggr]\cdot \biggl[ 1 - \frac{1}{2}e^2 - \mathcal{O}(e^{4}) \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | let's define the variable, <math>~\mathcal{A}</math>, such that, | ||
+ | |||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl\{\boldsymbol{K}(k)\biggr\} \{1 + e^2\mathcal{A}\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Rightarrow ~~~ \{ 1 + e^2 \cdot \mathcal{A} \}</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\biggl\{1 + \frac{e^2}{2} \biggl[ \frac{R_c(R_c + \varpi_W)}{\Delta_0^2}\biggr] + \mathcal{O}(e^4)\biggr\} | ||
+ | \biggl[1 + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2 + \mathcal{O}(e^{3}) \biggr]\cdot \biggl[ 1 - \frac{1}{2}e^2 - \mathcal{O}(e^{4}) \biggr]</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\approx</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\biggl\{1 + \frac{e^2}{2} \biggl[ \frac{R_c(R_c + \varpi_W)}{\Delta_0^2}\biggr] \biggr\} | ||
+ | \biggl[1 + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2 \biggr] \biggl[ 1 - \frac{1}{2}e^2 \biggr]</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\approx</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\biggl\{1 + \frac{e^2}{2} \biggl[ \frac{R_c(R_c + \varpi_W)}{\Delta_0^2}\biggr] + \biggl( \frac{5\cdot 13}{2^4\cdot 3}\biggr) e^2 - \frac{1}{2}e^2 \biggr\}</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Rightarrow ~~~ 2\mathcal{A}</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\approx</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\biggl[ \frac{R_c(R_c + \varpi_W)}{\Delta_0^2}\biggr] + \biggl( \frac{41}{2^3\cdot 3}\biggr) \, . </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | </td></tr></table> | ||
+ | |||
+ | |||
+ | We can therefore write, | ||
+ | |||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] - K([k_H]_0)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\approx</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~- K([k_H]_0) + | ||
+ | \biggl\{ K(k_H) + \frac{\pi}{2} \cdot \delta_K\biggr\} | ||
+ | \{ 1 + e^2 \cdot \mathcal{A} \} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\approx</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~- K([k_H]_0) | ||
+ | + | ||
+ | K(k_H) | ||
+ | \{ 1 + e^2 \cdot \mathcal{A} \} | ||
+ | + | ||
+ | \frac{\pi}{2} \cdot \delta_K \, , | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | where we should keep in mind that <math>~\delta_k</math> is <math>~\mathcal{O}(e^1)</math>. So, let's examine the piece, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\frac{2}{\pi} \biggl[ K(k_H) - K([k_H]_0) \biggr]</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl\{1 + \frac{k_H^2}{4} | ||
+ | + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k_H^4 | ||
+ | + \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k_H^6 | ||
+ | + \cdots | ||
+ | \biggr\} - | ||
+ | \biggl\{1 + \frac{k_H^2}{4} | ||
+ | + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k_H^4 | ||
+ | + \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k_H^6 | ||
+ | + \cdots | ||
+ | \biggr\}_{e\rightarrow 0} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\approx</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \frac{k_H^2}{4} - \biggl[ \frac{k_H^2}{4} \biggr]_{e\rightarrow 0} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl[ \frac{\varpi_W R_c (1 + e\cos\theta_H)}{[\varpi_W + R_c (1 + e\cos\theta_H)]^2 + [z_W - R_c e\sin\theta_H]^2} \biggr] | ||
+ | - | ||
+ | \biggl[ \frac{\varpi_W R_c (1 + e\cos\theta_H)}{[\varpi_W + R_c (1 + e\cos\theta_H)]^2 + [z_W - R_c e\sin\theta_H]^2} \biggr]_{e\rightarrow 0} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\varpi_W R_c (1 + e\cos\theta_H) | ||
+ | \biggl\{ [\varpi_W + R_c (1 + e\cos\theta_H)]^2 + [z_W - R_c e\sin\theta_H]^2 \biggr\}^{-1} | ||
+ | - | ||
+ | \biggl[ \frac{\varpi_W R_c }{(\varpi_W + R_c )^2 + z_W^2} \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\varpi_W R_c (1 + e\cos\theta_H) | ||
+ | \biggl\{ | ||
+ | \varpi_W^2 + 2\varpi_W R_c(1 + e\cos\theta_H) + R_c^2(1 + 2e\cos\theta_H + e^2\cos^2\theta_H) + z_W^2 -2z_W R_c e\sin\theta_H + R_c^2 e^2\sin^2\theta_H | ||
+ | \biggr\}^{-1} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - | ||
+ | \biggl[ \frac{\varpi_W R_c }{(\varpi_W + R_c )^2 + z_W^2} \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | -\biggl[ \frac{\varpi_W R_c }{(\varpi_W + R_c )^2 + z_W^2} \biggr] | ||
+ | + \biggl[ \frac{ \varpi_W R_c (1 + e\cos\theta_H)}{ (\varpi_W + R_c)^2 + z_W^2 } \biggr] | ||
+ | \biggl\{1 | ||
+ | + \frac{ e [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H] }{(\varpi_W + R_c)^2 + z_W^2} | ||
+ | + \frac{e^2 [R_c^2\cos^2\theta_H + R_c^2 \sin^2\theta_H ] }{(\varpi_W + R_c)^2 + z_W^2} | ||
+ | \biggr\}^{-1} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\approx</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | -\biggl[ \frac{\varpi_W R_c }{(\varpi_W + R_c )^2 + z_W^2} \biggr] | ||
+ | +\biggl[ \frac{ \varpi_W R_c (1 + e\cos\theta_H)}{ (\varpi_W + R_c)^2 + z_W^2 } \biggr] | ||
+ | \biggl\{1 | ||
+ | - \frac{ e [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H] }{(\varpi_W + R_c)^2 + z_W^2} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - \frac{e^2 [R_c^2\cos^2\theta_H + R_c^2 \sin^2\theta_H ] }{(\varpi_W + R_c)^2 + z_W^2} | ||
+ | + \frac{ e^2 [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]^2 }{ [(\varpi_W + R_c)^2 + z_W^2]^2} | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\approx</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl[ \frac{ \varpi_W R_c}{ (\varpi_W + R_c)^2 + z_W^2 } \biggr] | ||
+ | \biggl\{ | ||
+ | - \frac{ e [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H] }{(\varpi_W + R_c)^2 + z_W^2} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - \frac{e^2 [R_c^2\cos^2\theta_H + R_c^2 \sin^2\theta_H ] }{(\varpi_W + R_c)^2 + z_W^2} | ||
+ | + \frac{ e^2 [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]^2 }{ [(\varpi_W + R_c)^2 + z_W^2]^2} | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | +\biggl[ \frac{ \varpi_W R_c}{ (\varpi_W + R_c)^2 + z_W^2 } \biggr] | ||
+ | \biggl\{ \frac{e\cos\theta_H [(\varpi_W + R_c)^2 + z_W^2] -e^2\cos\theta_H [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H] }{(\varpi_W + R_c)^2 + z_W^2} | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\approx</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl[ \frac{ \varpi_W R_c}{ \Delta_0^2 } \biggr] | ||
+ | \biggl\{ | ||
+ | e\cos\theta_H | ||
+ | - \frac{ eR_c\cos\theta_H [2\varpi_W + 2 R_c - 2z_W \tan\theta_H] }{ \Delta_0^2} | ||
+ | - \frac{e^2 R_c^2 }{ \Delta_0^2} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - \frac{e^2\cos\theta_H [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H] }{ \Delta_0^2} | ||
+ | + \frac{ e^2 [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]^2 }{ \Delta_0^4} | ||
+ | \biggr\} \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | Now we have, | ||
+ | |||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] - K([k_H]_0)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\approx</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~- K([k_H]_0) | ||
+ | + | ||
+ | K(k_H) | ||
+ | \{ 1 + e^2 \cdot \mathcal{A} \} | ||
+ | + | ||
+ | \frac{\pi}{2} \cdot \delta_K | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Rightarrow ~~~ - \Delta_0 \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\approx</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \frac{2}{\pi} K([k_H]_0) | ||
+ | + \frac{2}{\pi} \biggl[ K(k_H) - K([k_H]_0) \biggr] | ||
+ | + | ||
+ | \delta_K | ||
+ | + | ||
+ | \frac{2}{\pi} K(k_H) e^2 \cdot \mathcal{A} \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | But, as we have just demonstrated, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\frac{2}{\pi} \biggl[ K(k_H) - K([k_H]_0) \biggr]+ \delta_K | ||
+ | </math> | ||
+ | <td align="center"> | ||
+ | <math>~\approx</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl[ \frac{ \varpi_W R_c}{ \Delta_0^2 } \biggr] | ||
+ | \biggl\{ | ||
+ | e\cos\theta_H | ||
+ | - \frac{ eR_c\cos\theta_H [2\varpi_W + 2 R_c - 2z_W \tan\theta_H] }{ \Delta_0^2} | ||
+ | - \frac{e^2 R_c^2 }{ \Delta_0^2} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - \frac{e^2\cos\theta_H [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H] }{ \Delta_0^2} | ||
+ | + \frac{ e^2 [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]^2 }{ \Delta_0^4} | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | -~\frac{ e\cdot \varpi_W R_c }{ \Delta_0^2 } | ||
+ | \biggl[ \cos\theta_H - \frac{2R_c (R_c \cos\theta_H+ \varpi_W \cos\theta_H - z_W \sin\theta_H)}{\Delta_0^2} \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | +~\frac{ e^2 \cdot \varpi_W R_c}{2 \Delta_0^4 } | ||
+ | \biggl\{ 2 (\varpi_W R_c + R_c^2 ) - [ (\varpi_W^2 + R_c)^2 + z_W^2 ] | ||
+ | - 2R_c^2 | ||
+ | - 4R_c\cos\theta_H (R_c \cos\theta_H+ \varpi_W \cos\theta_H - z_W \sin\theta_H) | ||
+ | \biggr\} \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | <td align="center"> | ||
+ | <math>~\approx</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl[ \frac{e^2 \cdot \varpi_W R_c}{ \Delta_0^4 } \biggr] | ||
+ | \biggl\{ | ||
+ | - R_c^2 - \cos\theta_H [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | +~(\varpi_W R_c + R_c^2 ) - \frac{1}{2}[ (\varpi_W^2 + R_c)^2 + z_W^2 ] | ||
+ | - R_c^2 | ||
+ | - 2R_c\cos\theta_H (R_c \cos\theta_H+ \varpi_W \cos\theta_H - z_W \sin\theta_H) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + \frac{ [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]^2 }{ \Delta_0^2} | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr><td align="center" colspan="3"><font color="red">TEMPORARY BREAK HERE</font></td></tr> | ||
+ | </table> | ||
+ | |||
+ | Hence, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~ - \frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\approx</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | K([k_H]_0) | ||
+ | + | ||
+ | K(k_H) e^2 \cdot \mathcal{A} | ||
+ | +~\frac{\pi}{2} \biggl[ \frac{e^2 \cdot \varpi_W R_c}{ \Delta_0^4 } \biggr] | ||
+ | \biggl\{ | ||
+ | - R_c^2 - \cos\theta_H [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | +~(\varpi_W R_c + R_c^2 ) - \frac{1}{2}[ (\varpi_W^2 + R_c)^2 + z_W^2 ] | ||
+ | - R_c^2 | ||
+ | - 2R_c\cos\theta_H (R_c \cos\theta_H+ \varpi_W \cos\theta_H - z_W \sin\theta_H) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + \frac{ [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]^2 }{ \Delta_0^2} | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | ===Include Second Wong Term=== | ||
+ | |||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\biggl( \frac{a}{GM} \biggr) \Phi_\mathrm{W1}(\varpi,z)\biggr|_\mathrm{exterior}</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - \biggl( \frac{2^{3} }{3\pi^3} \biggr) | ||
+ | \Upsilon_{W1}(\eta_0) \times \cos\theta | ||
+ | \biggl\{ \frac{a}{r_2} \cdot | ||
+ | \boldsymbol{E}(k) \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Rightarrow ~~~ - \frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W1}}{GM} \biggr] </math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl( \frac{2^{2} }{3\pi^2} \biggr) | ||
+ | \Upsilon_{W1}(\eta_0) \times \cos\theta | ||
+ | \biggl\{ \frac{\Delta_0}{r_2} \cdot | ||
+ | \boldsymbol{E}(k) \biggr\} \, ; | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | ====Leading (Upsilon) Coefficient==== | ||
+ | |||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Upsilon_{W1}(\eta_0)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\equiv</math> | ||
+ | </td> | ||
+ | <td align="left" colspan="2"> | ||
+ | <math>~\biggl[ \frac{\sinh\eta_0}{\cosh\eta_0}\biggr] | ||
+ | \biggl\{ | ||
+ | K(k_0)\cdot K(k_0) [\cosh\eta_0(1 - \cosh\eta_0)] | ||
+ | +~2K(k_0)\cdot E(k_0) [(3\cosh^2\eta_0 - 1)] | ||
+ | -~5 E(k_0)\cdot E(k_0) [\cosh\eta_0(1+\cosh\eta_0)] | ||
+ | \biggr\} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Rightarrow~~~ \biggl[ \frac{e^2}{ (1 - e^2)^{1 / 2}} \biggr] \Upsilon_{W1}(\eta_0)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left" colspan="2"> | ||
+ | <math>~ | ||
+ | - (1-e)K(k_0)\cdot K(k_0) | ||
+ | +~2(3-e^2)K(k_0)\cdot E(k_0) | ||
+ | -~5(1+e) E(k_0)\cdot E(k_0) \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Rightarrow ~~~ \frac{2^2}{\pi^2} \biggl[ \frac{e^2}{(1 - e^2)^{1 / 2}} \biggr] \Upsilon_{W1}(\eta_0)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - (1-e) \biggl[ 1 + \frac{1}{2} k_0^2 | ||
+ | + \frac{11}{2^5} ~k_0^4 | ||
+ | + \frac{17}{2^6} ~ k_0^6 | ||
+ | + \frac{1787}{2^{13}} ~k_0^8 | ||
+ | + \mathcal{O}(k_0^{10}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + 2 (3-e^2) \biggl[ 1 ~+~\frac{1}{2^5} ~k_0^4 | ||
+ | ~+~\frac{1}{2^5} ~ k_0^6 | ||
+ | ~+~ \frac{231}{2^{13}}~k_0^8 | ||
+ | + \mathcal{O}(k_0^{10}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - 5(1+e) \biggl[ 1 | ||
+ | - ~\frac{1}{2} ~k_0^2 ~ | ||
+ | -~ \frac{1}{2^5} ~ k_0^4 ~ | ||
+ | ~-~\frac{1}{2^6} ~ k_0^6 | ||
+ | ~-~\frac{77}{2^{13}}~k_0^8 | ||
+ | + \mathcal{O}(k_0^{10}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - (1-e) \biggl[ 1 + \frac{1}{2} \cdot 2e( 1 - e +e^2 - e^3 ) | ||
+ | + \frac{11}{2^5} ~\cdot 4e^2( 1 - 2e + 3e^2 ) | ||
+ | + \frac{17}{2^6} ~ \cdot 2^3e^3( 1 - 3e ) | ||
+ | + \frac{1787}{2^{13}} ~\cdot 2^4e^4 | ||
+ | + \mathcal{O}(e^{5}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + 2 (3-e^2) \biggl[ 1 ~+~\frac{1}{2^5} ~\cdot 4e^2( 1 - 2e + 3e^2 ) | ||
+ | ~+~\frac{1}{2^5} ~ \cdot 2^3e^3( 1 - 3e ) | ||
+ | ~+~ \frac{231}{2^{13}}~\cdot 2^4e^4 | ||
+ | + \mathcal{O}(e^{5}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - 5(1+e) \biggl[ 1 | ||
+ | - ~\frac{1}{2} ~\cdot 2e( 1 - e +e^2 - e^3 ) ~ | ||
+ | -~ \frac{1}{2^5} ~ \cdot 4e^2( 1 - 2e + 3e^2 ) | ||
+ | ~-~\frac{1}{2^6} ~ \cdot 2^3e^3( 1 - 3e ) | ||
+ | ~-~\frac{77}{2^{13}}~\cdot 2^4e^4 | ||
+ | + \mathcal{O}(e^{5}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - 2^{-9}(1-e) \biggl[ 2^9 + 2^9e( 1 - e +e^2 - e^3 ) | ||
+ | + 2^6 \cdot 11 ~\cdot e^2( 1 - 2e + 3e^2 ) | ||
+ | + 2^6\cdot 17 ~ \cdot e^3( 1 - 3e ) | ||
+ | + 1787 ~\cdot e^4 | ||
+ | + \mathcal{O}(e^{5}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + 2^{-9} (6- 2e^2) \biggl[ 2^9 ~+~2^6~\cdot e^2( 1 - 2e + 3e^2 ) | ||
+ | ~+~2^7 \cdot e^3( 1 - 3e ) | ||
+ | ~+~ 231~\cdot e^4 | ||
+ | + \mathcal{O}(e^{5}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - 2^{-9}(5+ 5e) \biggl[ 2^9 | ||
+ | - ~2^9 \cdot e( 1 - e +e^2 - e^3 ) ~ | ||
+ | -~ 2^6 \cdot e^2( 1 - 2e + 3e^2 ) | ||
+ | ~-~2^6 \cdot e^3( 1 - 3e ) | ||
+ | ~-~77~\cdot e^4 | ||
+ | + \mathcal{O}(e^{5}) | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Rightarrow ~~~\frac{2^{11}}{\pi^2} \biggl[ \frac{e^2}{(1 - e^2)^{1 / 2}} \biggr] \Upsilon_{W1}(\eta_0)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - 2^9 - 2^9e( 1 - e +e^2 - e^3 ) | ||
+ | - 2^6 \cdot 11 ~\cdot e^2( 1 - 2e + 3e^2 ) | ||
+ | - 2^6\cdot 17 ~ \cdot e^3( 1 - 3e ) | ||
+ | - 1787 ~\cdot e^4 | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + 2^9e + 2^9e^2( 1 - e +e^2 ) | ||
+ | + 2^6 \cdot 11 ~\cdot e^3( 1 - 2e ) | ||
+ | + 2^6\cdot 17 ~ \cdot e^4 | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + 6\biggl[ 2^9 ~+~2^6~\cdot e^2( 1 - 2e + 3e^2 ) | ||
+ | ~+~2^7 \cdot e^3( 1 - 3e ) | ||
+ | ~+~ 231~\cdot e^4 | ||
+ | \biggr] | ||
+ | -2^{10}e^2 ~-~2^7~\cdot e^4 | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + 5 \biggl[ -2^9 | ||
+ | + ~2^9 \cdot e( 1 - e +e^2 - e^3 ) ~ | ||
+ | +~ 2^6 \cdot e^2( 1 - 2e + 3e^2 ) | ||
+ | ~+~2^6 \cdot e^3( 1 - 3e ) | ||
+ | ~+~77~\cdot e^4 | ||
+ | \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | +~5 \biggl[ -2^9e | ||
+ | + ~2^9 \cdot e^2( 1 - e +e^2 ) ~ | ||
+ | +~ 2^6 \cdot e^3( 1 - 2e ) | ||
+ | ~+~2^6 \cdot e^4 | ||
+ | \biggr] | ||
+ | + \mathcal{O}(e^{5}) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 2^9( e^2 - e^3 + e^4 ) | ||
+ | - 2^6 \cdot 11 ~( e^2 - 2e^3 + 3e^4 ) | ||
+ | - 2^6\cdot 17 ~( e^3 - 3e^4 ) | ||
+ | - 1787 ~e^4 | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | +2^9 ( e^2 - e^3 +e^4 ) | ||
+ | + 2^6 \cdot 11 ( e^3 - 2e^4 ) | ||
+ | + 2^6\cdot 17 ~ e^4 | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | +~ 3\cdot 2^7~( e^2 - 2e^3 + 3e^4 ) | ||
+ | ~+~3\cdot 2^8 ( e^3 - 3e^4 ) | ||
+ | ~+~ 2\cdot 3^2 \cdot 7\cdot 11~\cdot e^4 | ||
+ | -2^{10}e^2 ~-~2^7~\cdot e^4 | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + 5\cdot 2^9 ( - e^2 +e^3 - e^4 ) ~ | ||
+ | +~ 5\cdot 2^6 ( e^2 - 2e^3 + 3e^4 ) | ||
+ | ~+~5\cdot 2^6 ( e^3 - 3e^4 ) | ||
+ | ~+~5\cdot 77~e^4 | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | +5\cdot 2^9 ( e^2 - e^3 +e^4 ) ~ | ||
+ | +~ 5\cdot 2^6 ( e^3 - 2e^4 ) | ||
+ | ~+~5\cdot 2^6 e^4 | ||
+ | + \mathcal{O}(e^{5}) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | e^2 [ 2^9 - 2^6\cdot 11 + 2^9 + 3\cdot 2^7 -2^{10} -5\cdot 2^9 + 5\cdot 2^6 + 5\cdot 2^9 ] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + e^3 [ -2^9 - 2^7\cdot 11 - 2^6\cdot 17 - 2^9 +2^6\cdot 11 - 3\cdot 2^8+3\cdot 2^8 + 5\cdot 2^9-5\cdot 2^7 +5\cdot 2^6 -5\cdot 2^9 + 5\cdot 2^6] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + e^4 [ 2^9 - 3\cdot 11\cdot 2^6+3\cdot 17\cdot 2^6 - 1787 + 2^9 - 11\cdot 2^7 + 17\cdot 2^6 + 3^2\cdot 2^7 - 3^2\cdot 2^8+ 2\cdot 3^2\cdot 7\cdot 11 - 2^7 | ||
+ | - 5\cdot 2^9 + 3\cdot 5\cdot 2^6 - 3\cdot 5\cdot 2^6 + 5\cdot 7\cdot 11 + 5\cdot 2^9 - 5\cdot 2^7 +5\cdot 2^6] | ||
+ | + \mathcal{O}(e^{5}) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | e^2 [ - 2^6\cdot 11 + 3\cdot 2^7 + 5\cdot 2^6 ] | ||
+ | + e^3 [ -2^{10} - 2^7\cdot 11 - 2^6\cdot 17 +2^6\cdot 11 ] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + e^4 [ 2^{10}~-~ 1787~+~ 2\cdot 3^2\cdot 7\cdot 11~+~ 5\cdot 7\cdot 11 ~+~ 2^6\cdot (3\cdot 17 - 3\cdot 11 - 22 + 17 + 18 - 36 - 2 - 10 + 5) ] | ||
+ | + \mathcal{O}(e^{5}) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 2^6 e^2 [ 0 ] ~- 2^8 \cdot 11 e^3 + e^4 [ 2^{10}~-~ 1787~+~ 7\cdot 11\cdot 23 ~-~ 2^8\cdot 3 ] | ||
+ | + \mathcal{O}(e^{5}) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - 2^8 \cdot 11 e^3 + 2^4\cdot 3\cdot 5e^4 | ||
+ | + \mathcal{O}(e^{5}) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Rightarrow ~~~\frac{2^{2}}{\pi^2} \biggl[ \frac{e^2}{(1 - e^2)^{1 / 2}} \biggr] \Upsilon_{W1}(\eta_0)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | - \frac{11}{2} e^3 + \frac{3\cdot 5}{2^5} e^4 | ||
+ | + \mathcal{O}(e^{5}) \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | </table> | ||
+ | |||
+ | <table border="1" cellpadding="8" align="center" width="90%"> | ||
+ | <tr> | ||
+ | <td align="center" bgcolor="black"><font color="white">'''Floating Comparison Summary'''</font></td> | ||
+ | </tr> | ||
+ | <tr><td align="left"> | ||
+ | As [[#Step01|shown above]], the first three terms of the [https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5825H/abstract Huré, et al. (2020)] series expression may be written as, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~- \frac{\pi \Delta_0}{2} \biggl[ \frac{ \Psi_0 + \Psi_1 + \Psi_2 }{GM} \biggr] </math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \boldsymbol{K}([k_H]_0) | ||
+ | - | ||
+ | \frac{e^2}{2^4} \cdot \boldsymbol{K}(k_H) | ||
+ | + \frac{e^2}{2^4} \biggl[ \frac{ \Delta_0^2 - 2R_c(R_c + R)}{(1 - k_H^2) \Delta_0^2} \biggr] \boldsymbol{E}(k_H) \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | Let's see how it compares to the first term of Wong's (1973) expression which, as [[#Step02|shown separately above]], can be written in the form, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \Delta_0 \biggl( \frac{2^{2} }{3\pi^2} \biggr) | ||
+ | \Upsilon_{W0}(\eta_0) \biggl\{ | ||
+ | \frac{\boldsymbol{K}(k) }{ r_1 } \biggr\}\, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | ---- | ||
+ | First, as [[#Step03|shown above]], | ||
+ | |||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\frac{2^2}{3\pi^2} \Upsilon_{W0}(\eta_0)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl[1 + \biggl(\frac{5\cdot 13}{2^4\cdot 3} \biggr)e^2 + \mathcal{O}(e^{3}) \biggr]\cdot (1 - e^2)^{1 / 2} \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | Note that, in order to determine the functional form of the <math>~\mathcal{O}(e^{2})</math> term in this expression, we will have to include <math>~k_0^8</math> terms in the various expressions for products of elliptic integrals. Second, [[#Step04|we have shown that]], | ||
+ | |||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\frac{ r_1^2}{\Delta_0^2} </math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~1 | ||
+ | -e^2 \biggl[ \frac{R_c(R_c + \varpi_W) }{\Delta_0^2}\biggr] | ||
+ | - \frac{\varpi_W R_c}{\Delta_0^2} \biggl[\frac{1}{2^2} e^4 + \frac{1}{2^3}e^6 + \frac{5}{2^6} e^8 + \mathcal{O}(e^{10}) \biggr] </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Rightarrow ~~~ \frac{\Delta_0}{r_1} </math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\approx</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | 1 + e^2 \biggl[ \frac{R_c(R_c + \varpi_W) }{2\Delta_0^2}\biggr] | ||
+ | \, ,</math> and we are defining <math>~\delta_K</math> such that, | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~K(k)</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~K(k_H) + \frac{\pi}{2} \cdot \delta_K \, .</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | ---- | ||
+ | |||
+ | Hence, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~-\frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W0}}{GM} \biggr] </math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\approx</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl\{ K(k_H) + \frac{\pi}{2} \cdot \delta_K\biggr\} | ||
+ | \biggl\{ 1 + e^2 \biggl[ \frac{R_c(R_c + \varpi_W) }{2\Delta_0^2}\biggr] \biggr\}\biggl[1 + \biggl(\frac{5\cdot 13}{2^4\cdot 3} \biggr)e^2 + \mathcal{O}(e^{3}) \biggr]\cdot (1 - e^2)^{1 / 2} | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\approx</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \boldsymbol{K}([k_H]_0) | ||
+ | + | ||
+ | \boldsymbol{K}(k_H) e^2 \cdot \mathcal{A} | ||
+ | +~\frac{\pi}{2} \biggl[ \frac{e^2 \cdot \varpi_W R_c}{ \Delta_0^4 } \biggr] | ||
+ | \biggl\{ | ||
+ | - R_c^2 - \cos\theta_H [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | +~(\varpi_W R_c + R_c^2 ) - \frac{1}{2}[ (\varpi_W^2 + R_c)^2 + z_W^2 ] | ||
+ | - R_c^2 | ||
+ | - 2R_c\cos\theta_H (R_c \cos\theta_H+ \varpi_W \cos\theta_H - z_W \sin\theta_H) | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | + \frac{ [2\varpi_W R_c \cos\theta_H + 2 R_c^2 \cos\theta_H - 2z_W R_c \sin\theta_H]^2 }{ \Delta_0^2} | ||
+ | \biggr\} \, , | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | and, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\mathcal{A}</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~\approx</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\biggl[ \frac{R_c(R_c + \varpi_W)}{2\Delta_0^2}\biggr] + \biggl( \frac{41}{2^4\cdot 3}\biggr) \, . </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | ---- | ||
+ | |||
+ | Second, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~- \frac{\pi \Delta_0}{2} \biggl[ \frac{\Phi_\mathrm{W1}}{GM} \biggr] </math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \boldsymbol{E}(k) | ||
+ | \biggl\{ \frac{\Delta_0 \cdot \cos\theta}{r_2} \biggr\} | ||
+ | \biggl[ \biggl( \frac{2^{2} }{3\pi^2} \biggr) \Upsilon_{W1}(\eta_0) \biggr] | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | \biggl\{ \boldsymbol{E}(k_H) + \frac{\pi}{2}\cdot \delta_E \biggr\} | ||
+ | \biggl\{ \frac{\Delta_0 \cdot \cos\theta}{r_2} \biggr\} | ||
+ | \biggl[- \biggl(\frac{11}{2\cdot 3} \biggr) e + \biggl(\frac{5}{2^5}\biggr) e^2 + \mathcal{O}(e^{5}) \biggr]\cdot (1 - e^2)^{1 / 2} \, . | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | </td></tr> | ||
+ | </table> | ||
+ | |||
+ | ====Geometric Factor==== | ||
+ | |||
+ | By definition, | ||
+ | |||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\Delta_0^2</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~ | ||
+ | (\varpi_W + R_c)^2 + z_W^2 \, , | ||
+ | </math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~r_1^2</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\biggl[ \varpi_W + R_c (1 - e^2 )^{1 / 2}\biggr]^2 + z_W^2 \, ,</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~r_2^2</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\biggl[ \varpi_W - R_c (1 - e^2 )^{1 / 2}\biggr]^2 + z_W^2 \, ,</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~\cos^2\theta</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~\biggl[ \frac{r_1^2 + r_2^2 - 4R_c^2(1-e^2)}{2r_1 r_2} \biggr]^2 \, .</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | Hence, | ||
+ | <table border="0" cellpadding="5" align="center"> | ||
+ | |||
+ | <tr> | ||
+ | <td align="right"> | ||
+ | <math>~r_2^2 - \Delta_0^2 \cdot \cos^2\theta</math> | ||
+ | </td> | ||
+ | <td align="center"> | ||
+ | <math>~=</math> | ||
+ | </td> | ||
+ | <td align="left"> | ||
+ | <math>~</math> | ||
+ | </td> | ||
+ | </tr> | ||
+ | </table> | ||
+ | |||
+ | =See Also= | ||
+ | <ul> | ||
+ | <li>Université de Bordeaux (Part 2): [[User:Tohline/Appendix/Ramblings/BordeauxSequences|Spheroid-Ring Sequences]]</li> | ||
+ | <li>Université de Bordeaux (Part 3): [[User:Tohline/Appendix/Ramblings/BordeauxPostDefense|Discussions Following Dissertation Defense]]</li> | ||
+ | </ul> | ||
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Current revision as of 17:05, 16 December 2020
Contents |
Université de Bordeaux (Part 1)
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Spheroid-Ring Systems
Through a research collaboration at the Université de Bordeaux, B. Basillais & J. -M. Huré (2019), MNRAS, 487, 4504-4509 have published a paper titled, Rigidly Rotating, Incompressible Spheroid-Ring Systems: New Bifurcations, Critical Rotations, and Degenerate States.
We discuss this topic in a separate, accompanying chapter.
Exterior Gravitational Potential of Toroids
J. -M. Huré, B. Basillais, V. Karas, A. Trova, & O. Semerák (2020), MNRAS, 494, 5825-5838 have published a paper titled, The Exterior Gravitational Potential of Toroids. Here we examine how their work relates to the published work by C.-Y. Wong (1973, Annals of Physics, 77, 279), which we have separately discussed in detail.
Our Presentation of Wong's (1973) Result
Summary: First three terms in Wong's (1973) expression for the gravitational potential at any point, P(ϖ, z), outside of a uniform-density torus. | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
where, once the major ( R ) and minor ( d ) radii of the torus — as well as the vertical location of its equatorial plane (Z0) — have been specified, we have,
NOTE: In evaluating these "leading coefficient expressions" for the case, |
Setup
From our accompanying discussion of Wong's (1973) derivation, the exterior potential is given by the expression,
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Wong (1973), §II.D, p. 294, Eqs. (2.59) & (2.61)
where,
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Wong (1973), §II.D, p. 294, Eq. (2.63)
and where, in terms of the major ( R ) and minor ( d ) radii of the torus — or their ratio, ε ≡ d/R,
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These expressions incorporate a number of basic elements of a toroidal coordinate system. In what follows, we will also make use of the following relations:
Once the primary scale factor,
Given that (sin2θ + cos2θ) = 1, we have,
We deduce as well that,
Given the definitions,
we can use the transformations,
Or we can use the transformations,
Additional potentially useful relations can be found in an accompanying chapter wherein we present a variety of basic elements of a toroidal coordinate system. |
Leading (n = 0) Term
Wong's Expression
Now, from our separate derivation we have,
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And if we make the function-argument substitution, , in the "Key Equation,"
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Abramowitz & Stegun (1995), p. 337, eq. (8.13.3) |
we can write,
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where, from above, we recognize that,
So, the leading (n = 0) term gives,
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Thin-Ring Evaluation of C0
In an accompanying discussion of the thin-ring approximation, we showed that as
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Hence, in this limit we can write,
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More General Evaluation of C0
NOTE of CAUTION: In our above evaluation of the toroidal function, |