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Thayer Watkins
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Particle Systems Under an
Inverse Distance Squared Force

There is a wonderful theorem of mathematical physics which says that the effect of a mass uniformly distributed over a spherical surface is the same as if the mass were concentrated at the the center of the sphere at all points outside of the sphere. This theorem is easily extended to a mass uniformly distributed over a spherical ball. However the theorem does not extend to mass not uniformly distributed over a spherical surface or over a non-spherical surface or solid. In particular. it does apply to a system of discrete particles.

What is true is that as the distance from the center of mass of the particles increases without bound the effect asymptotically approaches the the effect the particles would have if they were concentrated at the center of mass of the particles. An equivalent form is that as the scale of the distribution relative to the distance from the center of mass goes to zero the effect asymptotically approaches the effect the particle would have if they were concentrated at the center of mass.

The Effect as the Distance from the
Center of Mass Increases Without Bound

Let i, yi, zi, mi} be the position coordinates and mass of the i-th particle, with the origin being at the center of mass of the particles. This latter means that

Σmixi = 0, Σmiyi = 0, Σmizi = 0

Consider a point on the x-axis at a distance X from the origin; i.e., a point at coordinates (X, 0, 0). The distance from the i-th particle to the point (X, 0, 0) is

si = [(X-xi)² + yi² + zi²]½

Note that si can be expressed as

si = X[(1-xi/X)² + (yi/X)² + (zi/X)²]½

The magnitude of the force at (X, 0,0) due to the system of particles is

ΣGmi/si²

where G is the gravitational constant.

The component of the force at (X, 0, 0) directed along the x-axis is the sum of the forces due to each particle multiplied times the cosine of the angle ψi between that force and the x-axis. That cosine is equal to

cos(ψi) = (X−xi)/si
or, equivalently,
cos(ψi) = (1−xi/X)/[(1-xi/X)² + (yi/X)² + (zi/X)²]½

Note that the limit of cos(ψi) as X→∞ is 1.

Therefore the force directed along the x-axis at (X, 0, 0) is

Fx = Σ (Gmi)/(X²[(1-xi/X)² + (yi/X)² + (zi/X)²])cos(ψi)

If the mass of the particles were concentrated at the origin the force (X, 0, 0) would be

F0 = G(Σmj)/X²

and it would be directed along the x-axis.

The ratio of the two forces is then

Fx/F0 = Σ(mi/Σmj)cos(ψi)/[(1-xi/X)² + (yi/X)² + (zi/X)²]

The limit of this ratio at X increases without bound is

limX→∞ (Fx/F0) = (Σmi/Σmi)(1)/(1-0-0) = 1

The Limit as the Scale of the Particle Arrangement
to the Distance from the Origin Goes to Zero

Now consider the coordinates being given by

xi = Rricos(θi)
yi = Rrisin(θi)cos(φi)
zi = Rrisin(θi)sin(φi)

where R is the scale of the system as measured by the maximum radius of a sphere centered on the center of mass and enclosing all the particles and ri≤1 for all i. The above transformation of coordinates is based upon a spherical coordinate system in which the x-axis coincides with the axis of the sphere and θ is a latitude angle. The angle φ corresponds to longitude, the angle in the yz plane between the radial for the point and the z-axis.

Now

cos(ψi) = (1−(R/X)ricos(θi)

The limit of cos(ψi) as (R/X) goes to zero is 1 for all i.

Let ζi be defined as

ζi = [(1-(R/X)ricos(θi)² + [(R/X)risin(θi)cos(φi)]² + [(R/X)risin(θi)sin(φi)]²]

Note that

limR/X → 0 ζi = 1
for all i

The ratio Fx/F0 is now

Fx/F0 = Σ(mi/Σmj)cos(ψi)/ζi

Therefore

limR/X → 0(Fx/F0) = Σ(mi/Σmj)·1/1 = (Σmi/Σmj) = 1

Thus the effect of a distributed system of particles asymptotically approaches the effect that would prevail if all of the mass were concentrated at the center of mass of the system as the ratio of the scale of the system to the distance from the center of mass approaches zero.

Some Empirical Comparisons

The radius of our Milky Way galaxy is about 50,000 light years. The typical distance between galaxies in a galactic cluster is about forty times greater. Thus R/X for the distances between galaxies in a galactic cluster is about 1/40=0.025.

The radius of a Uranium nucleus is about 7.5 fermi (f). The radius of the first electron orbit is about 53,000 f. Thus at the distance of the electrons from the nucleus the ratio R/X is 1.4×10-4. Thus considering the electrostatic charge as being concentrated at the center of the nucleus is a good approximation for distances comparable to the electrons.

(To be continued.)


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