PGE/PoissonOrigin

Origin of the Poisson Equation

Poisson

We will follow closely the presentation found in §2.1 of [BT87] in deriving the,

Poisson Equation

${\mathrm{\nabla }}^2\mathrm{\Phi }=4\pi G\rho$

 

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

$\overrightarrow{a}(\overrightarrow{x})$	$=$	$\int \left[\frac{{\overrightarrow{x}}^{{}^{\prime }}-\overrightarrow{x}}{|{\overrightarrow{x}}^{{}^{\prime }}-\overrightarrow{x}{|}^3}\right]G\rho ({\overrightarrow{x}}^{{}^{\prime }})d^3{x}^{\prime },$
[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
$\mathrm{\Phi }(\overrightarrow{x})$	$\equiv$	$-G\int \frac{\rho ({\overrightarrow{x}}^{{}^{\prime }})}{|{\overrightarrow{x}}^{{}^{\prime }}-\overrightarrow{x}|}{d}^3{x}^{\text{'}}.$
[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, $|{\overrightarrow{x}}^{{}^{\prime }}-\overrightarrow{x}{|}^{-1}$, with respect to $\overrightarrow{x}$ is,

${\mathrm{\nabla }}_x\left[\frac{1}{|{\overrightarrow{x}}^{{}^{\prime }}-\overrightarrow{x}|}\right]$	$=$	$\frac{{\overrightarrow{x}}^{{}^{\prime }}-\overrightarrow{x}}{|{\overrightarrow{x}}^{{}^{\prime }}-\overrightarrow{x}{|}^3},$
[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 $(\overrightarrow{x}\text{'})$, rather than the unprimed $(\overrightarrow{x})$, coordinate system, we find that we may write the gravitational acceleration as,

$\overrightarrow{a}(\overrightarrow{x})$	$=$	$\int G\rho ({\overrightarrow{x}}^{{}^{\prime }}){\mathrm{\nabla }}_x\left[\frac{1}{|{\overrightarrow{x}}^{{}^{\prime }}-\overrightarrow{x}|}\right]d^3{x}^{\prime }$
	$=$	${\mathrm{\nabla }}_x\{G\int [\frac{\rho ({\overrightarrow{x}}^{{}^{\prime }})}{|{\overrightarrow{x}}^{{}^{\prime }}-\overrightarrow{x}|}\left]d^3{x}^{\prime }\right\}$
	$=$	$-{\mathrm{\nabla }}_x\mathrm{\Phi }.$
[BT87], p. 31, Eq. (2-5)

Step 2

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

${\mathrm{\nabla }}_x\cdot \overrightarrow{a}(\overrightarrow{x})$	$=$	${\mathrm{\nabla }}_x\cdot \int \left[\frac{{\overrightarrow{x}}^{{}^{\prime }}-\overrightarrow{x}}{|{\overrightarrow{x}}^{{}^{\prime }}-\overrightarrow{x}{|}^3}\right]G\rho ({\overrightarrow{x}}^{{}^{\prime }})d^3{x}^{\prime }$
	$=$	$\int G\rho ({\overrightarrow{x}}^{{}^{\prime }})\{{\mathrm{\nabla }}_x\cdot [\frac{{\overrightarrow{x}}^{{}^{\prime }}-\overrightarrow{x}}{|{\overrightarrow{x}}^{{}^{\prime }}-\overrightarrow{x}{|}^3}\left]\right\}{d}^3{x}^{\prime }.$
[BT87], p. 31, Eq. (2-6)

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

${\mathrm{\nabla }}_x\cdot \left[\frac{{\overrightarrow{x}}^{{}^{\prime }}-\overrightarrow{x}}{|{\overrightarrow{x}}^{{}^{\prime }}-\overrightarrow{x}{|}^3}\right]$

$=$

$-\frac{3}{|{\overrightarrow{x}}^{{}^{\prime }}-\overrightarrow{x}{|}^3}+3\left[\frac{({\overrightarrow{x}}^{{}^{\prime }}-\overrightarrow{x})\cdot ({\overrightarrow{x}}^{{}^{\prime }}-\overrightarrow{x})}{|{\overrightarrow{x}}^{{}^{\prime }}-\overrightarrow{x}{|}^5}\right]$

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 $|{\overrightarrow{x}}^{{}^{\prime }}-\overrightarrow{x}{|}^1$, whereas it should be $|{\overrightarrow{x}}^{{}^{\prime }}-\overrightarrow{x}{|}^3$ as written here. In an Errata to BT87, the authors have identified this error along with its correction as the first among a list of innocuous errors.

When $({\overrightarrow{x}}^{{}^{\prime }}-\overrightarrow{x})\ne 0$, we may cancel the factor $|{\overrightarrow{x}}^{{}^{\prime }}-\overrightarrow{x}{|}^2$ from top and bottom of the last term in this equation to conclude that,

${\mathrm{\nabla }}_x\cdot \left[\frac{{\overrightarrow{x}}^{{}^{\prime }}-\overrightarrow{x}}{|{\overrightarrow{x}}^{{}^{\prime }}-\overrightarrow{x}{|}^3}\right]=0$	when,	$({\overrightarrow{x}}^{{}^{\prime }}\ne \overrightarrow{x}).$
[BT87], p. 31, Eq. (2-8)

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

${\mathrm{\nabla }}_x\cdot \overrightarrow{a}(\overrightarrow{x})$	$=$	$-G\rho (\overrightarrow{x})\int d^2\mathrm{\Omega }$
	$=$	$-4\pi G\rho (\overrightarrow{x}).$
[BT87], p. 32, Eq. (2-9b)

Step 3

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

Poisson Equation

${\mathrm{\nabla }}^2\mathrm{\Phi }=4\pi G\rho$

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

See Also

• Ulrich D. Jentschura & Jonathan Sapirstein (April, 2018), arXiv:1801.10224v2 [math-ph], Green Function of the Poisson Equation: $D=2,3,4$
• Mark Viola (April 2021) Generalizations of Poisson's Equation -- Math Stack Exchange

  "… we find the Green function for Poisson's equation, ${\mathrm{\nabla }}^2{G}_0(\overrightarrow{x}|\overrightarrow{y})=-\delta (\overrightarrow{x}-\overrightarrow{y})$ is given by " $G_0(\overrightarrow{x}|\overrightarrow{y})$ $=$ $\frac{\mathrm{\Gamma }(n/2-1)}{4{\pi }^{n/2}|\overrightarrow{x}-\overrightarrow{y}{|}^{n-2}},$ where $\delta (\overrightarrow{x})$ is the Dirac Delta. Hence, when the right-hand-side source function is spatially extended, … the solution of the Poisson's equation ${\mathrm{\nabla }}^{2}u(\overrightarrow{x})=p(\overrightarrow{x})$ can be written as $u(\overrightarrow{x})$ $=$ $\int p(\overrightarrow{x}+\overrightarrow{y})\left[\frac{\mathrm{\Gamma }(n/2-1)}{4{\pi }^{n/2}|\overrightarrow{y}{|}^{n-2}}\right]d^n\overrightarrow{y}.$ Example 1 [$n=3,\mathrm{\Gamma }(n/2-1)={\pi }^{1/2}$]: $u(\overrightarrow{x})$ $=$ $\int \left[\frac{p(\overrightarrow{x}+\overrightarrow{y})}{4\pi |\overrightarrow{y}|}\right]d^3\overrightarrow{y};$ Example 2 [$n=4,\mathrm{\Gamma }(n/2-1)=1$]: $u(\overrightarrow{x})$ $=$ $\int \left[\frac{p(\overrightarrow{x}+\overrightarrow{y})}{4{\pi }^2|\overrightarrow{y}{|}^2}\right]d^4\overrightarrow{y};$ Example 3 [$n=5,\mathrm{\Gamma }(n/2-1)=\sqrt{\pi }/2$]: $u(\overrightarrow{x})$ $=$ $\int \left[\frac{p(\overrightarrow{x}+\overrightarrow{y})}{8{\pi }^2|\overrightarrow{y}{|}^3}\right]d^5\overrightarrow{y}.$