User:Tohline/DarkMatter/UniformSphere
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Force Exerted by a UniformDensity Shell or Sphere
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Tohline 1982
General Derivation from Notes Dated 29 November 1982
If the force per unit mass exerted at the position, , from a single point mass, , is given by,



then the force per unit mass exerted at by a continuous mass distribution, whose mass density is defined by the function , is,



This central force can also be expressed in terms of the gradient of a scalar potential, , specifically,



where,



For a spherically symmetric mass distribution, , the magnitude of the force that is directed along the radial vector, , and measured from the center of the mass distribution can be expressed as the following single integral over :



This integral can be completed analytically if , that is, for a uniformdensity mass distribution. Independent of whether the limits of integration, and , are less than or greater than , the integral gives,









If the position, , is located outside of a uniformdensity sphere, then and , so the aggregate acceleration becomes,






where, . If the position, , is located interior to a uniformdensity shell, then and the aggregate acceleration is,



If is inside a uniformdensity sphere, then and , so the aggregate acceleration is,



Limiting Cases
Some limiting cases are of interest for the uniform sphere, i.e., when . First, notice that (Gradshteyn & Ryzhik 1965, formula 0.1412),



Sitting on the Surface: Therefore, when — that is, on the surface of the uniformdensity sphere,



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 of the sphere's radius away.
Well Inside the Surface: When ,



that is, the acceleration grows linearly with , as in any harmonic potential.
Well Outside the Sphere: When ,



which is in line with the adopted pointmass 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) — NonNewtonian forces and the invisible mass problem
 Sanders (2014) — A Historical Perspective on Modified Newtonian Dynamics
© 2014  2020 by Joel E. Tohline 