This is an investigation of the relative error involved in replacing the force due to distributed objects with the force with the force those objects would have concentrated at the center of their distribution. The analysis has relevance for phenomena at the nuclear and atomic level all the way up to the galactic cluster level.

There is a wonderful theorem in mathematical physics to the effect that a uniform charge on a spherical surface has the same effect for an inverse distance squared force as that charge concentrated at the center of the sphere. This theorem extends to a charge distributed uniformly throughout a spherical ball. This theorem however does not apply if the charged object has a shape other than a sphere, such as a disk. It also does not apply is the charge on the spherical shape is not uniform. This in particular means that the theorem does not apply even if the shape is spherical but the charge (or mass) is located only at specific points on the spherical surface.

Here is a depiction of a galaxy.

The gravitational attraction of the collection of stars was computed for the star on the right just outside of the collection. The radial attraction of the collection of stars on a star located at that radial distance is 6.115 times the attraction that would prevail if all of the stars of the collection were located at the center of the collection. That is because the force on the star is dominated by the attraction of the nearby stars.

The center of gravitation of the system with respect to the star on the right of the system surrounded by the white circle is the point is circled in yellow in the above diagram. Thus, despite the circular symmetry of the collection, the force due to the collection is greatly different from that which would prevail if the total mass of the collection were concentrated at the center of the collection.

Additionally, the theorem does not apply if the force is not an inverse distance squared force.

What the analysis deals with is the circumstances in which the theorem holds approximately and asymptotically holds exactly.

The Simplest Case

Consider first two equal objects located at ±σ distances from the origin.

For an inverse distance squared (IDS) force the force experienced per unit charge at a distance x from the origin is

F(x) = 1/(x+σ)² + 1/(x+σ)²

The force that would prevail if both objects located at the origin is

F₀(x) = 2/x²

The ratio of the true force F to the approximation F₀(x) can be put into the form

F(x)/F₀(x) = (1/2)[1/(1+(σ/x)² + 1/(1−(σ/x)²]

The important aspect of this formula is that the ratio F(x)/Fsub>0(x) is a function solely of (σ/x).

Note that the absolute error ΔF is equal to F(x)−F₀(x) and therefore

ΔF/F₀(x) = F(x)/F₀(x) − 1

Furthermore

ΔF/F(x) = ΔF/(F₀(x)+ΔF)
which upon division of the numerator
and denominator by F₀(x) gives
ΔF/F(x) = = (ΔF/F₀(x))/(1 + ΔF/F₀(x))

Thus the value of F(x)/F₀(x) establishes the value of the relative error ΔF/F(x).

As the ratio (σ/x) goes to zero the ratio F(x)/F₀(x) goes to 1.0 and the relative error goes to zero.

If (σ/x)=0.1 then

F(x)/F₀(x) = (1/2)[1/1.01 + 1/0.99] = 1.030507091

Therefore the relative error is about 3 percent. If (σ/x)=0.01 then

F(x)/F₀(x) = (1/2)[1/1.0001 + 1/0.9999] = 1.00030005

The relative error is then about 0.03 of 1 percent.

The width of our Milky Way Galaxy is about 100,000 light years. The distances between galaxies in a galactic cluster are 20 to 40 times that much. The maximum value of (σ/x) is then 1/40=0.025 and therefore the error involved in assuming the gravitational between galaxies is the same as if all their masses were concentrated at their centers is less than 0.2 of 1 percent. On the other hand, for a point just 10 percent beyond the edge of the galaxy the error could be as much as 100 percent.

Generalization

Consider now two charges such that the midpoint of the line between them is at the origin and that line is at an angle of θ with respect to the x-axis.

The distances from the charges to a point a distance x from the origin on the x-axis are given by

y₁² = σ²sin²(θ) + (x − σcos(θ))²
and
y₂² = σ²sin²(θ) + (x + σcos(θ))²

These reduce to

y₁² = x² − 2xσcos(θ) + σ²
and
y₂² = x² + 2xσcos(θ) + σ²

This means that the force is given by

F(x) = 1/(x² − 2xσcos(θ) + σ²) + 1/(x² + 2xσcos(θ) + σ²)

The ratio of this force to F₀(x)=2/x² can be put into the form

F(x)F₀(x) = (1/2)[1/(1 −2(σ/x)cos(θ + (σ/x)²) + 1/(1 +2(σ/x)cos(θ + (σ/x)²)]

Again the error is a function of (σ/x) and the error goes to zero as (σ/x) goes to zero.

For a symmetrical collection the above result could be applied pair by pair. The error involved in substituting the the force which would prevail if the collection of objects were concentrated at the center of the collection is definitely less than that which would prevail based upon the maximum extent of the collection relative to the distance. The overall error would be determined by the average distance of the objects from the center compared to the distance at which the force is being observed.

A Force that is not Strictly
an Inverse Distance Squared Law

Consider a force of the following form

F(x) = exp(−x)/x²

Two objects located at ±σ away from the origin would exert the following force at a point a distance of x from the origin on the x-axis

F(x) = exp(−(x+σ))/(x+σ)² + exp(−(x−σ))/(x−σ)²
which reduces to
F(x) = exp(−x)[exp(−σ)/(x+σ)² + exp(σ)/(x−σ)²

The force that would be exerted if both objects were located at the origin is

F₀(x) = 2exp(−x)/x²

The ratio of the force can be put into the form

F(x)/F₀(x) = (1/2)[exp(−σ)/(1+(σ/x))² + exp(σ)/(1−(σ/x))²

In this case the ratio of forces is not solely a function of (σ/x) and it does not go to zero as (σ/x) goes to zero. For the case of σ=1 and (σ/x)=0.1 the ratio is about 1.83 and the relative error is about 45 percent.

Conclusions

For inverse square law forces the relative error in replacing the force at a distance x from a collection of objects with the force which would prevail if all of the objects were located at the center of the collection depends upon the ratio of the average radius σ of the collection to the distance x. The relative error involved goes to zero as the ratio (σ/x) goes to zero. This does not hold for forces that do not have a strictly inverse distance squared law.