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Force Exerted by a Uniform-Density Shell or Sphere

Whitworth's (1981) Isothermal Free-Energy Surface
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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>~- \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>~- \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\nabla\Phi(\vec{x}) \, ,</math>


<math>~\Phi(\vec{x}) </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>~ -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>~ - \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>~ - \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>~ - \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>~ - \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>~ - \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>~ \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>~ - \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>~ - \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>~ - \frac{G^' M(R_2)}{r} \, ,</math>

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

See Also

Whitworth's (1981) Isothermal Free-Energy Surface

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