Using Toroidal Coordinates to Determine the Gravitational Potential
Overview
The set of 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 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 — and more commonly — as has been explicitly demonstrated in, respectively, Step 1 and 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:
| Table 1: Poisson Equation |
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| Integral Representation |
Differential Representation |
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<math>~ \Phi(\vec{x})</math>
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<math>~=</math>
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<math>~ -G \int \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .</math>
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<math>\nabla^2 \Phi = 4\pi G \rho</math>
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See Also
