User:Tohline/DarkMatter/UniformSphere

Force Exerted by a Uniform-Density Shell or Sphere

Tohline's Derivations Circa 1983

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>

See Also

• Stabilizing a Cold Disk with a 1/r Force Law
• Does Gravity Exhibit a 1/r Force on the Scale of Galaxies?

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation