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Origin of the Poisson Equation

In deriving the,

Poisson Equation

\nabla^2 \Phi = 4\pi G \rho

we will follow closely the presentation found in §2.1 of [BT87].


Whitworth's (1981) Isothermal Free-Energy Surface
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According to Isaac Newton's inverse-square law of gravitation, the acceleration, ~\vec{a}(\vec{x}), felt at any point in space, ~\vec{x}, due to the gravitational attraction of a distribution of mass, ~\rho(\vec{x}), is obtained by integrating over the accelerations exerted by each small mass element, ~\rho(\vec{x}^{~'}) d^3x', as follows:

~\vec{a}(\vec{x})

~=

~
\int \biggl[\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3}\biggr] G\rho(\vec{x}^{~'}) d^3 x' \, ,

[BT87], p. 31, Eq. (2-2)

where, ~G is the universal gravitational constant.

Step 1

In the astrophysics literature, it is customary to adopt the following definition of the,

Scalar Gravitational Potential

~ \Phi(\vec{x})

~\equiv

~ -G \int \frac{\rho(\vec{x}^{~'})}{|\vec{x}^{~'} - \vec{x}|} d^3x^' \, .

[BT87], p. 31, Eq. (2-3)
[EFE], §10, p. 17, Eq. (11)
[T78], §4.2, p. 77, Eq. (12)

(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, ~|\vec{x}^{~'} - \vec{x}|^{-1}, with respect to ~\vec{x} is,

~\nabla_x \biggl[ \frac{1}{|\vec{x}^{~'} - \vec{x}|} \biggr]

~=

~
\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3} \, ,

[BT87], p. 31, Eq. (2-4)

and given that, in the above expression for the gravitational acceleration, the integration is taken over the volume that is identified by the primed ~(\vec{x}~{'}), rather than the unprimed ~(\vec{x}), coordinate system, we find that we may write the gravitational acceleration as,

~\vec{a}(\vec{x})

~=

~\int G\rho(\vec{x}^{~'})  \nabla_x \biggl[ \frac{1}{|\vec{x}^{~'} - \vec{x}|} \biggr]d^3 x'

 

~=

~ \nabla_x \biggl\{ G \int \biggl[ \frac{\rho(\vec{x}^{~'}) }{|\vec{x}^{~'} - \vec{x}|} \biggr]d^3 x'\biggr\}

 

~=

~-\nabla_x \Phi \, .

[BT87], p. 31, Eq. (2-5)

Step 2

Next, we realize that the divergence of the gravitational acceleration takes the form,

~\nabla_x \cdot \vec{a}(\vec{x})

~=

~
\nabla_x \cdot \int \biggl[\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3}\biggr] G\rho(\vec{x}^{~'}) d^3 x'

 

~=

~
\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' \, .

[BT87], p. 31, Eq. (2-6)

Examining the expression inside the curly braces, we find that,

~\nabla_x \cdot \biggl[\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3}\biggr]

~=

~
- \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]

(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 ~|\vec{x}^{~'} - \vec{x}|^1, whereas it should be ~|\vec{x}^{~'} - \vec{x}|^3 as written here.) When ~(\vec{x}^{~'} - \vec{x}) \ne 0, we may cancel the factor ~|\vec{x}^{~'} - \vec{x}|^2 from top and bottom of the last term in this equation to conclude that,

~\nabla_x \cdot \biggl[\frac{\vec{x}^{~'} - \vec{x}}{|\vec{x}^{~'} - \vec{x}|^3}\biggr] = 0

      when,      

~
(\vec{x}^{~'} \ne \vec{x}) \, .

[BT87], p. 31, Eq. (2-8)

Therefore, any contribution to the integral must come from the point ~\vec{x}^{~'} = \vec{x}, 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 ~\rho(\vec{x}~{'}) = \rho(\vec{x}) 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 ~\vec{x}^{~'} = \vec{x} and, in turn, this surface integral may be written in terms of an integral over the solid angle, ~d^2\Omega, to give:

~\nabla_x \cdot \vec{a}(\vec{x})

~=

~
-G\rho(\vec{x}) \int d^2\Omega

 

~=

~
-4\pi G\rho(\vec{x}) \, .

[BT87], p. 32, Eq. (2-9b)

Step 3

Finally, combining the results of Step 1 and Step 2 gives the desired,

Poisson Equation

\nabla^2 \Phi = 4\pi G \rho

which serves as one of the principal governing equations in our examination of the Structure, Stability, & Dynamics of Self-Gravitating Fluids.

See Also

Whitworth's (1981) Isothermal Free-Energy Surface

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