User:Tohline/DarkMatter/UniformSphere
Force Exerted by a Uniform-Density Shell or Sphere
|
|---|
| | Tiled Menu | Tables of Content | Banner Video | Tohline Home Page | |
Tohline 1982
General Derivation from Notes Dated 29 November 1982
If the force per unit mass exerted at the position, <math>~\vec{r}</math>, from a single point mass, <math>~m</math>, is given by,
|
<math>~\vec{F}</math> |
<math>~=</math> |
<math>~- \biggl( \frac{G^'m}{r} \biggr) \frac{\vec{r}}{r} \, ,</math> |
then the force per unit mass exerted at <math>~\vec{x}</math> by a continuous mass distribution, whose mass density is defined by the function <math>~\rho(\vec{x}^')</math>, is,
|
<math>~\vec{F}(\vec{x})</math> |
<math>~=</math> |
<math>~- \int G^' \rho(\vec{x}^') \biggl[ \frac{\vec{x}^' - \vec{x}}{| \vec{x}^' - \vec{x} |^2} \biggr] d^3x^' \, .</math> |
This central force can also be expressed in terms of the gradient of a scalar potential, <math>~\Phi(\vec{x})</math>, specifically,
|
<math>~\vec{F}(\vec{x})</math> |
<math>~=</math> |
<math>~- \vec\nabla\Phi(\vec{x}) \, ,</math> |
where,
|
<math>~\Phi(\vec{x}) </math> |
<math>~=</math> |
<math>~ \int G^' \rho(\vec{x}^') \ln | \vec{x}^' - \vec{x} | d^3x^' \, .</math> |
For a spherically symmetric mass distribution, <math>~\rho(r^')</math>, the magnitude of the force that is directed along the radial vector, <math>~\vec{r}^'</math>, and measured from the center of the mass distribution can be expressed as the following single integral over <math>~r^'</math>:
|
<math>~F(r) \equiv \vec{F}\cdot \frac{\vec{r}}{r} </math> |
<math>~=</math> |
<math>~ -2\pi G^' \int\limits_{R_1}^{R_2} \rho(r^') (r^')^2 \biggl[\frac{1}{r} + \frac{1}{2r^2 r^'} \biggl( r^2 - {r^'}^2 \biggr) \ln\biggl( \frac{r^' + r}{|r^' - r|} \biggr) \biggr] dr^' \, .</math> |
This integral can be completed analytically if <math>~\rho(r^') = \rho_0</math>, that is, for a uniform-density mass distribution. Independent of whether the limits of integration, <math>~R_1</math> and <math>~R_2</math>, are less than or greater than <math>~r</math>, the integral gives,
|
<math>~F(r) </math> |
<math>~=</math> |
<math>~ - \frac{3G^'}{8r} \biggl( \frac{4\pi}{3}\rho_0 \biggr) \biggl\{ \biggl( R_2^3 - R_1^3 \biggr) + r^2 \biggl(R_2 - R_1\biggr) </math> |
|
|
|
<math>~ + r^3 \biggl[ \frac{1}{2} + \frac{1}{2}\biggl( \frac{R_1}{r} \biggr)^4 - \biggl( \frac{R_1}{r} \biggr)^2\biggr] \ln\biggl( \frac{R_1 + r}{|R_1 - r|} \biggr) </math> |
|
|
|
<math>~ - r^3 \biggl[ \frac{1}{2} + \frac{1}{2}\biggl( \frac{R_2}{r} \biggr)^4 - \biggl( \frac{R_2}{r} \biggr)^2\biggr] \ln\biggl( \frac{R_2 + r}{|R_2 - r|} \biggr) \biggr\} \, .</math> |
If the position, <math>~r</math>, is located outside of a uniform-density sphere, then <math>~R_1 = 0</math> and <math>~R_2 < r</math>, so the aggregate acceleration becomes,
|
<math>~F(r)_\mathrm{out} </math> |
<math>~=</math> |
<math>~ - \frac{3G^'}{8r} \biggl( \frac{4\pi}{3}\rho_0 \biggr) \biggl\{ R_2^3 + r^2 R_2 - r^3 \biggl[ \frac{1}{2} + \frac{1}{2}\biggl( \frac{R_2}{r} \biggr)^4 - \biggl( \frac{R_2}{r} \biggr)^2\biggr] \ln\biggl( \frac{r+R_2}{r- R_2} \biggr) \biggr\} </math> |
|
|
<math>~=</math> |
<math>~ - \frac{G^' M(R_2)}{r} \biggl\{ 1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{R_2}{r} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1} \biggr\} \, , </math> |
where, <math>~M(R_2) \equiv 4\pi \rho_0 R_2^3/3</math>. If the position, <math>~r</math>, is located interior to a uniform-density shell, then <math>~r < R_1 < R_2</math> and the aggregate acceleration is,
|
<math>~F(r)_\mathrm{shell} </math> |
<math>~=</math> |
<math>~ - \frac{4\pi}{3} G^' \rho_0 R_2 r \biggl\{1 - \frac{R_1}{R_2} - 3 \sum_{n=1}^{\infty} \biggl[ \biggl( \frac{r}{R_2} \biggr)^{2n} - \frac{R_1}{R_2} \biggl( \frac{r}{R_1} \biggr)^{2n}\biggr] \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1} \biggr\} \, . </math> |
If <math>~r</math> is inside a uniform-density sphere, then <math>~R_1 = 0</math> and <math>~ r < R_2</math>, so the aggregate acceleration is,
|
<math>~F(r)_\mathrm{in} </math> |
<math>~=</math> |
<math>~ - \frac{4\pi}{3} G^' \rho_0 R_2 r \biggl\{ 1 - 3 \sum_{n=1}^{\infty} \biggl( \frac{r}{R_2} \biggr)^{2n} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1} \biggr\} \, . </math> |
Limiting Cases
Some limiting cases are of interest for the uniform sphere, i.e., when <math>~R_1 = 0</math>. First, notice that (Gradshteyn & Ryzhik 1965, formula 0.141-2),
|
<math>~\sum_{n=1}^{\infty} \biggl[(2n-1)(2n+1)(2n+3) \biggr]^{-1}</math> |
<math>~=</math> |
<math>~ \frac{1}{12} \, .</math> |
Sitting on the Surface: Therefore, when <math>~r = R_2</math> — that is, on the surface of the uniform-density sphere,
|
<math>~F</math> |
<math>~=</math> |
<math>~ - \frac{3 G^' M(R_2)}{4R_2} \, .</math> |
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 <math>~4/3</math> of the sphere's radius away.
Well Inside the Surface: When <math>~r \ll R_2</math>,
|
<math>~F(r)_\mathrm{in}</math> |
<math>~\approx</math> |
<math>~ - \frac{G^' M(R_2)}{R_2} \biggl( \frac{r}{R_2} \biggr) \, ,</math> |
that is, the acceleration grows linearly with <math>~r</math>, as in any harmonic potential.
Well Outside the Sphere: When <math>~r \gg R_2</math>,
|
<math>~F(r)_\mathrm{out}</math> |
<math>~\approx</math> |
<math>~ - \frac{G^' M(R_2)}{r} \, ,</math> |
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
|
|
|---|
|
© 2014 - 2021 by Joel E. Tohline |