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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, ~\vec{r}, from a single point mass, ~m, is given by,

~\vec{F}

~=

~- \biggl( \frac{G^'m}{r} \biggr) \frac{\vec{r}}{r} \, ,

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

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

~=

~- \int G^' \rho(\vec{x}^') \biggl[ \frac{\vec{x}^' - \vec{x}}{| \vec{x}^' - \vec{x} |^2} \biggr] d^3x^' \, .

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

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

~=

~- \vec\nabla\Phi(\vec{x}) \, ,

where,

~\Phi(\vec{x})

~=

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

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

~F(r) \equiv \vec{F}\cdot \frac{\vec{r}}{r}

~=

~ -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^' 
\, .

This integral can be completed analytically if ~\rho(r^') = \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^'}{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)

 

 

~ +
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)

 

 

~ -  
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\} \, .

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{out}

~=

~ 
- \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\}

 

~=

~ 
- \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\} \, ,

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{shell}

~=

~ 
- \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\} \, .


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

~F(r)_\mathrm{in}

~=

~ 
- \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\} \, .

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}^{\infty} \biggl[(2n-1)(2n+1)(2n+3)  \biggr]^{-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{3 G^' 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{in}

~\approx

~ - \frac{G^' M(R_2)}{R_2}  \biggl( \frac{r}{R_2} \biggr) \, ,

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{out}

~\approx

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

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

See Also


Whitworth's (1981) Isothermal Free-Energy Surface

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