Difference between revisions of "User:Tohline/Appendix/Mathematics/ToroidalSynopsis01"

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==Basics==
Here we attempt to bring together &#8212; in as succinct a manner as possible &#8212; [[User:Tohline/2DStructure/ToroidalCoordinates#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential|our approach]] and [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong's (1973)]  approach to determining the gravitational potential of an axisymmetric, uniform-density torus that has a major radius, <math>~R</math>, and a minor, cross-sectional radius, <math>~d</math>.  The relevant toroidal coordinate system is one based on an ''anchor ring'' of major radius,  
Here we attempt to bring together &#8212; in as succinct a manner as possible &#8212; [[User:Tohline/2DStructure/ToroidalCoordinates#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential|our approach]] and [http://adsabs.harvard.edu/abs/1973AnPhy..77..279W C.-Y. Wong's (1973)]  approach to determining the gravitational potential of an axisymmetric, uniform-density torus that has a major radius, <math>~R</math>, and a minor, cross-sectional radius, <math>~d</math>.  The relevant toroidal coordinate system is one based on an ''anchor ring'' of major radius,  
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   <td align="left">
<math>~2\pi^2 Rd^2 \, .</math>
<math>~2\pi^2 Rd^2 \, .</math>
  </td>
</tr>
</table>
</div>
==Exploration==
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~(\varpi + a)  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ [(z-Z_0)^2 - r_1^2]^{1 / 2}</math>
  </td>
  <td align="center">&nbsp; &nbsp; and, &nbsp; &nbsp;</td>
  <td align="right">
<math>~(\varpi^' + a)  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ [(z^'-Z_0)^2 - (r_1^')^2]^{1 / 2}</math>
  </td>
</tr>
</table>
</div>
<div align="center">
<table border="0" cellpadding="5" align="center">
<tr>
  <td align="right">
<math>~\Rightarrow ~~~ (\varpi  - \varpi^')  </math>
  </td>
  <td align="center">
<math>~=</math>
  </td>
  <td align="left">
<math>~ [(z-Z_0)^2 - r_1^2]^{1 / 2} - [(z^'-Z_0)^2 - (r_1^')^2]^{1 / 2}</math>
   </td>
   </td>
</tr>
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Revision as of 04:53, 3 June 2018

Synopsis of Toroidal Coordinate Approach

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

Here we attempt to bring together — in as succinct a manner as possible — our approach and C.-Y. Wong's (1973) approach to determining the gravitational potential of an axisymmetric, uniform-density torus that has a major radius, <math>~R</math>, and a minor, cross-sectional radius, <math>~d</math>. The relevant toroidal coordinate system is one based on an anchor ring of major radius,

<math>~a^2 \equiv R^2 - d^2 \, .</math>

If the meridional-plane location of the anchor ring — as written in cylindrical coordinates — is, <math>~(\varpi, z) = (a,Z_0)</math>, then the preferred toroidal-coordinate system has meridional-plane coordinates, <math>~(\eta, \theta)</math>, defined such that,

<math>~\eta</math>

<math>~=</math>

<math>~\ln\biggl(\frac{r_1}{r_2} \biggr) \, ,</math>

    and,    

<math>~\cos\theta</math>

<math>~=</math>

<math>~\frac{(r_1^2 + r_2^2 - 4a^2)}{2r_1 r_2} \, ,</math>

where,

<math>~r_1^2 </math>

<math>~\equiv</math>

<math>~(\varpi + a)^2 + (z-Z_0)^2 \, ,</math>

    and,    

<math>~r_2^2 </math>

<math>~\equiv</math>

<math>~(\varpi - a)^2 + (z-Z_0)^2 \, ,</math>

and <math>~\theta</math> has the same sign as <math>~(z-Z_0)</math>. Mapping the other direction, we have,

<math>~\varpi</math>

<math>~=</math>

<math>~\frac{a \sinh\eta }{(\cosh\eta - \cos\theta)} \, ,</math>

    and,    

<math>~z-Z_0</math>

<math>~=</math>

<math>~\frac{a \sin\theta}{(\cosh\eta - \cos\theta)} \, .</math>

The three-dimensional differential volume element is,

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

<math>~=</math>

<math>\varpi d\varpi ~dz ~d\psi</math>

<math>~=</math>

<math>~\biggl[ \frac{a^3\sinh\eta}{(\cosh\eta - \cos\theta)^3} \biggr] d\eta~ d\theta~ d\psi \, .</math>

Note that, if <math>~\eta_0</math> identifies the surface of the uniform-density torus, then,

<math>~\cosh\eta_0</math>

<math>~=</math>

<math>~\frac{R}{d} \, ,</math>

     

<math>~\sinh\eta_0</math>

<math>~=</math>

<math>~\frac{a}{d} \, ,</math>

    and,    

<math>~\coth\eta_0</math>

<math>~=</math>

<math>~\frac{R}{a} \, ;</math>

and when the integral over the volume element is completed — that is, over all <math>~\psi</math>, over all <math>~\theta</math>, and over the "radial" interval, <math>~\eta_0 \le \eta \le \infty</math> — the resulting volume is,

<math>~V</math>

<math>~=</math>

<math>~\frac{2\pi^2 \cosh\eta_0}{\sinh^3\eta_0}</math>

<math>~=</math>

<math>~2\pi^2 Rd^2 \, .</math>

Exploration

<math>~(\varpi + a) </math>

<math>~=</math>

<math>~ [(z-Z_0)^2 - r_1^2]^{1 / 2}</math>

    and,    

<math>~(\varpi^' + a) </math>

<math>~=</math>

<math>~ [(z^'-Z_0)^2 - (r_1^')^2]^{1 / 2}</math>

<math>~\Rightarrow ~~~ (\varpi - \varpi^') </math>

<math>~=</math>

<math>~ [(z-Z_0)^2 - r_1^2]^{1 / 2} - [(z^'-Z_0)^2 - (r_1^')^2]^{1 / 2}</math>

See Also

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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