User:Tohline/SR/PoissonOrigin
Origin of the Poisson Equation
In deriving the,
Poisson Equation
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<math>\nabla^2 \Phi = 4\pi G \rho</math> |
we will follow closely the presentation found in §2.1 of [BT87].
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According to Isaac Newton's inverse-square law of gravitation, the acceleration, <math>~\vec{a}(\vec{x})</math>, felt at any point in space, <math>~\vec{x}</math>, due to the gravitational attraction of a distribution of mass, <math>~\rho(\vec{x})</math>, is obtained by integrating over the accelerations exerted by each small mass element, <math>~\rho(\vec{x}^{~'}) d^3x'</math>, as follows:
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<math>~\vec{a}(\vec{x})</math> |
<math>~=</math> |
<math>~ \int \biggl[\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3}\biggr] G\rho(\vec{x}^{~'}) d^3 x' \, , </math> |
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[BT87], p. 31, Eq. (2-2) |
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where, <math>~G</math> is the universal gravitational constant.
Now, in the astrophysics literature, it is customary to adopt the following definition of the,
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Scalar Gravitational Potential |
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<math>~ \Phi(\vec{x})</math> |
<math>~\equiv</math> |
<math>~ -G \int \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .</math> |
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[BT87], p. 31, Eq. (2-3) |
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(Note: As we have detailed in a separate discussion, throughout [EFE] Chandrasekhar adopts a different sign convention as well as a different variable name to represent the gravitational potential.) Recognizing that the gradient of the function, <math>~|\vec{x}^{~'} - \vec{x}|^{-1}</math>, with respect to <math>~\vec{x}</math> is,
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<math>~\nabla_x \biggl[ \frac{1}{|\vec{x}^{~'} - \vec{x}|} \biggr]</math> |
<math>~=</math> |
<math>~ \frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3} \, , </math> |
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[BT87], p. 31, Eq. (2-4) |
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and given that, in the above expression for the gravitational acceleration, the integration is taken over the volume that is identified by the primed <math>~(\vec{x}~{'})</math>, rather than the unprimed <math>~(\vec{x})</math>, coordinate system, we find that we may write the gravitational acceleration as,
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<math>~\vec{a}(\vec{x})</math> |
<math>~=</math> |
<math>~\int G\rho(\vec{x}^{~'}) \nabla_x \biggl[ \frac{1}{|\vec{x}^{~'} - \vec{x}|} \biggr]d^3 x' </math> |
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<math>~=</math> |
<math>~ \nabla_x \biggl\{ G \int \biggl[ \frac{\rho(\vec{x}^{~'}) }{|\vec{x}^{~'} - \vec{x}|} \biggr]d^3 x'\biggr\}</math> |
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<math>~=</math> |
<math>~-\nabla_x \Phi \, .</math> |
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[BT87], p. 31, Eq. (2-5) |
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See Also
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© 2014 - 2021 by Joel E. Tohline |