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

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<math>~[(\varpi + a)^2 + (z-Z_0)^2 \, ,</math>
<math>~(\varpi + a)^2 + (z-Z_0)^2 \, ,</math>
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<math>~[(\varpi - a)^2 + (z-Z_0)^2 \, ,</math>
<math>~(\varpi - a)^2 + (z-Z_0)^2 \, ,</math>
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and <math>~\theta</math> has the same sign as <math>~(z-Z_0)</math>.
and <math>~\theta</math> has the same sign as <math>~(z-Z_0)</math>. Note that, if <math>~\eta_0</math> identifies the surface of the uniform-density torus, then,
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<math>~\cosh\eta_0</math>
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<math>~=</math>
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<math>~\frac{R}{d} \, ,</math>
  </td>
  <td align="center">&nbsp; &nbsp; &nbsp;</td>
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<math>~\sinh\eta_0</math>
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<math>~=</math>
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<math>~\frac{a}{d} \, ,</math>
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  <td align="center">&nbsp; &nbsp; and &nbsp; &nbsp;</td>
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<math>~\coth\eta_0</math>
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  <td align="center">
<math>~=</math>
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<math>~\frac{R}{a} \, .</math>
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  <td align="center">&nbsp; &nbsp; &nbsp;</td>
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=See Also=
=See Also=

Revision as of 00:45, 3 June 2018

Synopsis of Toroidal Coordinate Approach

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

<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>

<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>. 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>

     

See Also

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation