User:Tohline/SR/PoissonOrigin
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Origin of the Poisson Equation
In deriving the,
we will follow closely the presentation found in §2.1 of [BT87].
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According to Isaac Newton's inversesquare law of gravitation, the acceleration, , felt at any point in space, , due to the gravitational attraction of a distribution of mass, , is obtained by integrating over the accelerations exerted by each small mass element, , as follows:



[BT87], p. 31, Eq. (22) 
where, is the universal gravitational constant.
Step 1
In the astrophysics literature, it is customary to adopt the following definition of the,
Scalar Gravitational Potential 




[BT87], p. 31, Eq. (23) 
(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, , with respect to is,



[BT87], p. 31, Eq. (24) 
and given that, in the above expression for the gravitational acceleration, the integration is taken over the volume that is identified by the primed , rather than the unprimed , coordinate system, we find that we may write the gravitational acceleration as,









[BT87], p. 31, Eq. (25) 
Step 2
Next, we realize that the divergence of the gravitational acceleration takes the form,






[BT87], p. 31, Eq. (26) 
Examining the expression inside the curly braces, we find that,



(Note: Ostensibly, this last expression is the same as equation 27 of [BT87], but apparently there is a typesetting error in the BT87 publication. As printed, the denominator of the first term on the righthand side is , whereas it should be as written here.) When , we may cancel the factor from top and bottom of the last term in this equation to conclude that,

when, 

[BT87], p. 31, Eq. (28) 
Therefore, any contribution to the integral must come from the point , 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 out of the integral. Via the divergence theorem (for details, see appendix 1.B — specifically, equation 1B42 — of [BT87]), the remaining volume integral may be converted into a surface integral over the small volume centered on the point and, in turn, this surface integral may be written in terms of an integral over the solid angle, , to give:






[BT87], p. 32, Eq. (29b) 
Step 3
Finally, combining the results of Step 1 and Step 2 gives the desired,
which serves as one of the principal governing equations in our examination of the Structure, Stability, & Dynamics of SelfGravitating Fluids.
See Also
© 2014  2019 by Joel E. Tohline 