Revision as of 03:28, 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.

Solving the Poisson Equation

Using Toroidal Coordinates to Determine the Gravitational Potential

Wong's (1973) Analytic Potential

Dyson-Wong Tori
(Thin Ring Approximation)

Trova, Huré & Hersant (2012)

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-=Solving the (Multi-dimensional) Poisson Equation Numerically=
+=Common Theme: Determining the Gravitational Potential for Axisymmetric Mass Distributions=
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-==Overview==
+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.
-The set of [[User:Tohline/PGE#Principal_Governing_Equations|Principal Governing Equations]] that serves as the foundation of our study of the structure, stability, and dynamical evolution of self-gravitating fluids contains an equation of motion (the ''Euler'' equation) that includes an acceleration due to local gradients in the (Newtonian) gravitational potential, <math>~\Phi</math>.  As has been pointed out in an [[User:Tohline/SR/PoissonOrigin#Origin_of_the_Poisson_Equation|accompanying chapter that discusses the origin of the Poisson equation]], the mathematical definition of this acceleration is fundamentally drawn from Isaac Newton's inverse-square law of gravitation, but takes into account that our fluid systems are not ensembles of point-mass sources but, rather, are represented by a continuous ''distribution'' of mass via the function, <math>~\rho(\vec{x},t)</math>.  As indicated, in our study, <math>~\rho</math> may depend on time as well as space.  The acceleration felt at any point in space may be obtained by integrating over the accelerations exerted by each differential mass element.  Alternatively &#8212; and more commonly &#8212; as has been explicitly demonstrated in, respectively, [[User:Tohline/SR/PoissonOrigin#Step_1|Step 1]] and [[User:Tohline/SR/PoissonOrigin#Step_3|Step 3]] of the same accompanying chapter, at any point in time the spatial variation of the gravitational potential, <math>~\Phi(\vec{x})</math>, is determined from <math>~\rho(\vec{x})</math> via either an ''integral'' or a ''differential'' equation as follows:
-==Common Theme==
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 [[User:Tohline/AxisymmetricConfigurations/PoissonEq#Solving_the_.28Multi-dimensional.29_Poisson_Equation_Numerically|Solving the Poisson Equation]]
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Revision as of 03:28, 29 July 2018

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Common Theme: Determining the Gravitational Potential for Axisymmetric Mass Distributions

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