Difference between revisions of "User:Tohline/AxisymmetricConfigurations/SolvingPE"

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You have arrived at this page from our [[User:Tohline/H_BookTiledMenu#Axisymmetric_Equilibrium_Structures|''Tiled Menu'']] by clicking on the chapter title that is also referenced in the panel of the following table that is colored light blue.  You may proceed directly to that chapter by clicking (again) on the same chapter title, as it appears in the table.  However, we have brought you to this intermediate page in order to bring to your attention that there are a number of additional chapters that have a strong thematic connection to the chapter you have selected.  The common thread is the "Key Equation" presented in the top panel of the table.
You have arrived at this page from our [[User:Tohline/H_BookTiledMenu#Axisymmetric_Equilibrium_Structures|''Tiled Menu'']] by clicking on the chapter title that is also referenced in the panel of the following table that is colored light blue.  You may proceed directly to that chapter by clicking (again) on the same chapter title, as it appears in the table.  However, we have brought you to this intermediate page in order to bring to your attention that there are a number of additional chapters that have a strong thematic connection to the chapter you have selected.  The common thread is the "Key Equation" presented in the top panel of the table.


{{LSU_HBook_header}}
{{LSU_HBook_header}}
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<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\Phi(R_*,Z_*)</math>
<math>~\Phi(\varpi,z)\biggr|_\mathrm{axisym}</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
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   <td align="left">
   <td align="left">
<math>~
<math>~
- \frac{G}{\pi} \iint\limits_\mathrm{config} \biggl[ \frac{\mu}{(R_* \varpi)^{1 / 2}} \biggr] K(\mu) \rho(\varpi, z) 2\pi \varpi d\varpi dz \, ,</math>
- \frac{G}{\pi} \iint\limits_\mathrm{config} \biggl[ \frac{\mu}{(\varpi~ \varpi^')^{1 / 2}} \biggr] K(\mu) \rho(\varpi^', z^') 2\pi \varpi^'~ d\varpi^' dz^' \, ,</math>
   </td>
   </td>
</tr>
</tr>
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<tr>
<tr>
   <td align="right">
   <td align="right">
<math>~\mu^2</math>
<math>~\mu</math>
   </td>
   </td>
   <td align="center">
   <td align="center">
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   <td align="left">
   <td align="left">
<math>~
<math>~
\biggl[\frac{4R_*\varpi}{(R_* + \varpi)^2 + (Z_* - z)^2} \biggr] \, .
\biggl[\frac{4\varpi~\varpi^'}{(\varpi + \varpi^')^2 + (z - z^')^2} \biggr]^{1 / 2}
</math>
</math>
   </td>
   </td>

Revision as of 18:07, 29 July 2018

Common Theme: Determining the Gravitational Potential for Axisymmetric Mass Distributions

You have arrived at this page from our Tiled Menu by clicking on the chapter title that is also referenced in the panel of the following table that is colored light blue. You may proceed directly to that chapter by clicking (again) on the same chapter title, as it appears in the table. However, we have brought you to this intermediate page in order to bring to your attention that there are a number of additional chapters that have a strong thematic connection to the chapter you have selected. The common thread is the "Key Equation" presented in the top panel of the table.

Whitworth's (1981) Isothermal Free-Energy Surface
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<math>~\Phi(\varpi,z)\biggr|_\mathrm{axisym}</math>

<math>~=</math>

<math>~ - \frac{G}{\pi} \iint\limits_\mathrm{config} \biggl[ \frac{\mu}{(\varpi~ \varpi^')^{1 / 2}} \biggr] K(\mu) \rho(\varpi^', z^') 2\pi \varpi^'~ d\varpi^' dz^' \, ,</math>

where,

<math>~\mu</math>

<math>~=</math>

<math>~ \biggl[\frac{4\varpi~\varpi^'}{(\varpi + \varpi^')^2 + (z - z^')^2} \biggr]^{1 / 2} </math>

Dyson-Wong
Tori
(Thin Ring Approximation)

Dyson-Wong Tori
(Thin Ring Approximation)

Solving the
Poisson Equation

Solving the Poisson Equation You have arrived at this page from our Tiled Menu by clicking on the chapter title that is also referenced in the panel of the following table that is colored light blue. You may proceed directly to that chapter by clicking (again) on the same chapter title, as it appears in the table. However, we have brought you to this intermediate page in order to bring to your attention that there are a number of additional chapters that have a strong thematic connection to the chapter you have selected. The common thread is the "Key Equation" presented in the top panel of the table.

Using
Toroidal Coordinates
to Determine the
Gravitational
Potential

Using Toroidal Coordinates to Determine the Gravitational Potential

Wong's
(1973)
Analytic Potential

Wong's (1973) Analytic Potential

Trova, Huré & Hersant (2012)

Related Discussions

Whitworth's (1981) Isothermal Free-Energy Surface

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