DarkMatter/UniformSphere

Force Exerted by a Uniform-Density Shell or Sphere

Tohline 1982

General Derivation from Notes Dated 29 November 1982

If the force per unit mass exerted at the position, $\overrightarrow{r}$, from a single point mass, $m$, is given by,

$\overrightarrow{F}$

$=$

$-\left(\frac{G^{\text{'}}m}{r}\right)\frac{\overrightarrow{r}}{r},$

then the force per unit mass exerted at $\overrightarrow{x}$ by a continuous mass distribution, whose mass density is defined by the function $\rho ({\overrightarrow{x}}^{\text{'}})$, is,

$\overrightarrow{F}(\overrightarrow{x})$

$=$

$-\int G^{\text{'}}\rho ({\overrightarrow{x}}^{\text{'}})\left[\frac{{\overrightarrow{x}}^{\text{'}}-\overrightarrow{x}}{|{\overrightarrow{x}}^{\text{'}}-\overrightarrow{x}{|}^2}\right]d^3{x}^{\text{'}}.$

This central force can also be expressed in terms of the gradient of a scalar potential, $\mathrm{\Phi }(\overrightarrow{x})$, specifically,

$\overrightarrow{F}(\overrightarrow{x})$

$=$

$-\overrightarrow{\mathrm{\nabla }}\mathrm{\Phi }(\overrightarrow{x}),$

where,

$\mathrm{\Phi }(\overrightarrow{x})$

$=$

$\int G^{\text{'}}\rho ({\overrightarrow{x}}^{\text{'}})\ln |{\overrightarrow{x}}^{\text{'}}-\overrightarrow{x}|d^3{x}^{\text{'}}.$

For a spherically symmetric mass distribution, $\rho (r^{\text{'}})$, the magnitude of the force that is directed along the radial vector, ${\overrightarrow{r}}^{\text{'}}$, and measured from the center of the mass distribution can be expressed as the following single integral over $r^{\text{'}}$:

$F(r)\equiv \overrightarrow{F}\cdot \frac{\overrightarrow{r}}{r}$

$=$

$-2\pi G^{\text{'}}\underset{R_1}{\overset{R_2}{\int }}\rho (r^{\text{'}})(r^{\text{'}}{)}^2[\frac{1}{r}+\frac{1}{2r^2{r}^{\text{'}}}(r^2-{r^{\text{'}}}^2)\ln \left(\frac{r^{\text{'}}+r}{|r^{\text{'}}-r|}\right)]d{r}^{\text{'}}.$

This integral can be completed analytically if $\rho (r^{\text{'}})={\rho }_0$, that is, for a uniform-density mass distribution. Independent of whether the limits of integration, $R_1$ and $R_2$, are less than or greater than $r$, the integral gives,

$F(r)$	$=$	$-\frac{3G^{\text{'}}}{8r}\left(\frac{4\pi }{3}{\rho }_0\right)\left\{\right(R_2^3-R_1^3)+r^2(R_2-R_1)$
		$+r^3[\frac{1}{2}+\frac{1}{2}(\frac{R_1}{r}{)}^4-\left(\frac{R_1}{r}{)}^2\right]\ln \left(\frac{R_1+r}{|R_1-r|}\right)$
		$-r^3[\frac{1}{2}+\frac{1}{2}(\frac{R_2}{r}{)}^4-\left(\frac{R_2}{r}{)}^2\right]\ln \left(\frac{R_2+r}{|R_2-r|}\right)\}.$

If the position, $r$, is located outside of a uniform-density sphere, then $R_1=0$ and $R_2<r$, so the aggregate acceleration becomes,

$F(r{)}_{\mathrm{o}\mathrm{u}\mathrm{t}}$	$=$	$-\frac{3G^{\text{'}}}{8r}\left(\frac{4\pi }{3}{\rho }_0\right)\{R_2^3+r^2{R}_2-r^3[\frac{1}{2}+\frac{1}{2}(\frac{R_2}{r}{)}^4-(\frac{R_2}{r}{)}^2]\ln (\frac{r+R_2}{r-R_2}\left)\right\}$
	$=$	$-\frac{G^{\text{'}}M(R_2)}{r}\{1-3\sum _{n=1}^{\mathrm{\infty }}(\frac{R_2}{r}{)}^{2n}[(2n-1)(2n+1)(2n+3){]}^{-1}\},$

where, $M(R_2)\equiv 4\pi {\rho }_0{R}_2^3/3$. If the position, $r$, is located interior to a uniform-density shell, then $r<R_1<R_2$ and the aggregate acceleration is,

$F(r{)}_{\mathrm{s}\mathrm{h}\mathrm{e}\mathrm{l}\mathrm{l}}$

$=$

$-\frac{4\pi }{3}{G}^{\text{'}}{\rho }_0{R}_{2}r\{1-\frac{R_1}{R_2}-3\sum _{n=1}^{\mathrm{\infty }}[(\frac{r}{R_2}{)}^{2n}-\frac{R_1}{R_2}(\frac{r}{R_1}{)}^{2n}\left]\right[(2n-1)(2n+1)(2n+3){]}^{-1}\}.$

If $r$ is inside a uniform-density sphere, then $R_1=0$ and $r<R_2$, so the aggregate acceleration is,

$F(r{)}_{\mathrm{i}\mathrm{n}}$

$=$

$-\frac{4\pi }{3}{G}^{\text{'}}{\rho }_0{R}_{2}r\{1-3\sum _{n=1}^{\mathrm{\infty }}(\frac{r}{R_2}{)}^{2n}[(2n-1)(2n+1)(2n+3){]}^{-1}\}.$

Limiting Cases

Some limiting cases are of interest for the uniform sphere, i.e., when $R_1=0$. First, notice that (Gradshteyn & Ryzhik 1965, formula 0.141-2),

$\sum _{n=1}^{\mathrm{\infty }}[(2n-1)(2n+1)(2n+3){]}^{-1}$

$=$

$\frac{1}{12}.$

Sitting on the Surface: Therefore, when $r=R_2$ — that is, on the surface of the uniform-density sphere,

$F$

$=$

$-\frac{3G^{\text{'}}M(R_2)}{4R_2}.$

So the force acts as though the mass is all concentrated at a point, not at the center of the sphere, but at a distance $4/3$ of the sphere's radius away.

Well Inside the Surface: When $r\ll R_2$,

$F(r{)}_{\mathrm{i}\mathrm{n}}$

$\approx$

$-\frac{G^{\text{'}}M(R_2)}{R_2}\left(\frac{r}{R_2}\right),$

that is, the acceleration grows linearly with $r$, as in any harmonic potential.

Well Outside the Sphere: When $r\gg R_2$,

$F(r{)}_{\mathrm{o}\mathrm{u}\mathrm{t}}$

$\approx$

$-\frac{G^{\text{'}}M(R_2)}{r},$

which is in line with the adopted point-mass specification.

See Also

• Finzi (1963) — On the Validity of Newton's Law at a Long Distance
• Notes from Beatrice Tinsley dated July 3, 1978
• Stabilizing a Cold Disk with a 1/r Force Law
• Does Gravity Exhibit a 1/r Force on the Scale of Galaxies?
• Kuhn & Kruglyak (1987) — Non-Newtonian forces and the invisible mass problem
• Sanders (2014) — A Historical Perspective on Modified Newtonian Dynamics