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 Newtonian 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</math> |
<math>~\equiv</math> |
<math>~ -G \int\limits_V \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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we find that we may write the gravitational acceleration as,
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<math>~\vec{a}(\vec{x})</math> |
<math>~=</math> |
<math>~-\nabla_x \Phi \, .</math> |
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[BT87], p. 31, Eq. (2-5) |
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Drawn from Other Wiki Pages
It is clear, therefore, that Chandrasekhar uses the variable <math>~\vec{u}</math> instead of <math>~\vec{v}</math> to represent the inertial velocity field. More importantly, he adopts a different variable name and a different sign convention to represent the gravitational potential, specifically,
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<math>~ - \Phi = \mathfrak{B} </math> |
<math>~=</math> |
<math>~ G \int\limits_V \frac{\rho(\vec{x}^{~'})}{|\vec{x} - \vec{x}^{~'}|} d^3x^' \, .</math> |
Hence, care must be taken to ensure that the signs on various mathematical terms are internally consistent when mapping derivations and resulting expressions from [EFE] into this H_Book.
… which expresses simply the conservation of the angular momentum of the system. The symmetric part of the tensor expression gives what is generally referred to as (see [EFE] for details) the,
Tensor Virial Equation
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<math>~\frac{1}{2} \frac{d^2 I_{ij}}{dt^2}</math> |
<math>~=</math> |
<math>~2 \mathfrak{T}_{ij} + \mathfrak{W}_{ij} + \delta_{ij}\Pi \, ,</math> |
See Also
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© 2014 - 2021 by Joel E. Tohline |