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for a Set of Particles Attracting Another Particle
Let the particles of the set be labeled 1 to N and the particle being attracted by them be labeled the zeroeth particle. Let the masses and coordinates be given as mi and (xi, yi, zi), respectively.
The magnitude of the force F on the zeroeth particle is given by
The force is a vector and has a direction as well as the above magnitude. The center of gravitation is a point on the line of the force F a distance R such that
where M is the sum of the masses.
Then R is given by
If the masses are imagined to be located at some point more distant from the zerorth particle than the center of gravity the magnitude of the mass MDM required to have the attraction on the zeroeth particle then
where S is the distance from the zeroeth particle to the selected point.
The difference (MDM−M) would be called the Dark Matter. It is totally spurious. The selected point might be the center of mass, but there is no real justification for concentrating the gravitational mass at the center of mass.
For a computational illustration see Spuriosity of So-called Dark Matter
The concept of dark matter is totally spurious. It arose from the analytical error of thinking there is some justification for comparing actual attractions with the attraction that would come from concentrating the mass of a system at the center of mass. The only point where there would be a justification for concentrating the mass is the center of gravitation for each particle. For these there is no dark matter.
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