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.
Step 1
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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Step 2
Next, we realize that the divergence of the gravitational acceleration takes the form,
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<math>~\nabla_x \cdot \vec{a}(\vec{x})</math> |
<math>~=</math> |
<math>~ \nabla_x \cdot \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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<math>~=</math> |
<math>~ \int G\rho(\vec{x}^{~'}) \biggl\{ \nabla_x \cdot \biggl[\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3}\biggr] \biggr\} d^3 x' \, . </math> |
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[BT87], p. 31, Eq. (2-6) |
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Examining the expression inside the curly braces, we find that,
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<math>~\nabla_x \cdot \biggl[\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3}\biggr] </math> |
<math>~=</math> |
<math>~ - \frac{3}{|\vec{x}^{~'} - \vec{x}|^3} + 3 \biggl[ \frac{ (\vec{x}^{~'} - \vec{x}) \cdot (\vec{x}^{~'} - \vec{x}) }{|\vec{x}^{~'} - \vec{x}|^5}\biggr] </math> |
(Note: Ostensibly, this last expression is the same as equation 2-7 of [BT87], but apparently there is a typesetting error in the BT87 publication. As printed, the denominator of the first term on the right-hand side is <math>~|\vec{x}^{~'} - \vec{x}|^1</math>, whereas it should be <math>~|\vec{x}^{~'} - \vec{x}|^3</math> as written here.) When <math>~(\vec{x}^{~'} - \vec{x}) \ne 0</math>, we may cancel the factor <math>~|\vec{x}^{~'} - \vec{x}|^2</math> from top and bottom of the last term in this equation to conclude that,
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<math>~\nabla_x \cdot \biggl[\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3}\biggr] = 0</math> |
when, |
<math>~ (\vec{x}^{~'} \ne \vec{x}) \, . </math> |
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[BT87], p. 31, Eq. (2-8) |
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Therefore, any contribution to the integral must come from the point <math>~\vec{x}^{~'} = \vec{x}</math>, and we may restrict the volume of integration to a small sphere … centered on this point. Since, for a sufficiently small sphere, the density will be almost constant through this volume, we can take <math>~\rho(\vec{x}~{'}) = \rho(\vec{x})</math> out of the integral. Via the divergence theorem (for details, see appendix 1.B — specifically, equation 1B-42 — of [BT87]), the remaining volume integral may be converted into a surface integral over the small volume centered on the point <math>~\vec{x}^{~'} = \vec{x}</math> and, in turn, this surface integral may be written in terms of an integral over the solid angle, <math>~d^2\Omega</math>, to give:
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<math>~\nabla_x \cdot \vec{a}(\vec{x})</math> |
<math>~=</math> |
<math>~ -G\rho(\vec{x}) \int d^2\Omega </math> |
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<math>~=</math> |
<math>~ -4\pi G\rho(\vec{x}) \, . </math> |
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[BT87], p. 32, Eq. (2-9b) |
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Step 3
Finally, combining the results of Step 1 and Step 2 gives the desired,
Poisson Equation
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<math>\nabla^2 \Phi = 4\pi G \rho</math> |
which serves as one of the principal governing equations in our examination of the structure, stability, and dynamics of self-gravitating fluids.
See Also
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© 2014 - 2021 by Joel E. Tohline |