Difference between revisions of "User:Tohline/2DStructure/ToroidalGreenFunction"

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NOTE: &nbsp; We have shifted to our "Ramblings" Appendix an [[User:Tohline/2DStructure/ToroidalCoordinates#Using_Toroidal_Coordinates_to_Determine_the_Gravitational_Potential|earlier version of this chapter]].


{{LSU_HBook_header}}
{{LSU_HBook_header}}


==Overview==


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:
Here we build upon our [[User:Tohline/AxisymmetricConfigurations/PoissonEq#Solving_the_.28Multi-dimensional.29_Poisson_Equation_Numerically|accompanying review]] of the types of numerical techniques that various astrophysics research groups have developed to solve for the Newtonian gravitational potential, <math>~\Phi(\vec{x})</math>, given a specified, three-dimensional mass distribution, <math>~\rho(\vec{x})</math>.  Our focus is on the use of toroidal coordinates to solve the ''integral'' formulation of the Poisson equation, namely,
 
 
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=See Also=
=See Also=

Revision as of 21:57, 15 June 2018

Using Toroidal Coordinates to Determine the Gravitational Potential

NOTE:   We have shifted to our "Ramblings" Appendix an earlier version of this chapter.

Whitworth's (1981) Isothermal Free-Energy Surface
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Here we build upon our accompanying review of the types of numerical techniques that various astrophysics research groups have developed to solve for the Newtonian gravitational potential, <math>~\Phi(\vec{x})</math>, given a specified, three-dimensional mass distribution, <math>~\rho(\vec{x})</math>. Our focus is on the use of toroidal coordinates to solve the integral formulation of the Poisson equation, namely,

<math>~ \Phi(\vec{x})</math>

<math>~=</math>

<math>~ -G \int \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .</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

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