In a system such as the solar system where there is a concentration of mass near the center, the magnitude of that mass is determined as follows.

Let M be the central mass, m the mass of a satellite, r the radius of the satellite's orbit and G the gravitational constant. A balance of centrifugal force and gravitational attraction gives

mv²/r = GMm/r²

where v is the tangential velocity.

The mass of the satellite cancels out and

M = v²r/G

Thus if the tangential velocity and orbit radius are known then the value of the central mass M can be determined.

Consider a galactic cluster, a collection of fifty to a thousand galaxies.

The tangential velocity of galaxies on the periphery has been determined and thus the central mass can be computed for the cluster. The computed figure exceeds the estimated masses of all the galaxies in the cluster and hence astronomers conclude there must be some matter that is not seen. This hypothesized matter has been called dark matter.

The Distribution of Matter and Its Effect
on the Computation of Dark Matter

For some distributions of mass, such as in a sphere or a spherical shell, the gravitational attraction is the same as if the mass were concentrated at the center of the mass of the distribution. But for other distributions of mass this does not hold true.

Consider now the galaxies distributed throughout a thin disk and use a polar coordinate system. Galactic clusters are three dimensional but the limiting of the example to a two dimensional distribution does not affect the central point of the illustration. Likewise the galaxies in a cluster are not regularly spaced but this also does not affect the central point of the illustration.

An element at radius r and angle θ has xy coordinates of (r·cos(θ), r·sin(θ)) has a distance s from the point at R and angle 0 (xy coordinates (R, 0)) given by

s² = (R−r·cos(θ)² + (r·sin(θ))²
which reduces to
s² = R² + r² − 2rRcos(θ)

The radial component of the force is the important factor. Let φ be the angle between the radial line to (R, 0) and the point (r,, θ). This is the angle the force makes with the radial line to (R, 0). The cosine of the angle φ of the force is equal to (R−rcos θ)/s.

Galaxies are located at the nodes of a polar grid in which there are 19 angle lines separated by 2π/20 radians and the radial circles run from 0.1 to 0.9.

The radial attraction of the collection of the glaxies on a galaxy located at radial distance 1.0 is 6.115 times the attraction that would prevail if all of the mass of the galaxies were located at the center of the coordinate system. Thus there would seem to be dark matter which constitutes 83.65 percent of the mass of the system. This dark matter is of course spurious. It is notable that the figure accepted for the proportion of the universe which is dark matter is 83 per cent.

The center of gravitation of the system with respect to the point on the right of the system surrounded by the white circle is a point 0.4R from that point. This means the center of gravitation with respect to the point on the right is about 0.6 of the way from the center of the system to the point. This point is circled in yellow in the above diagram.

The ratio of the supposed dark matter to the actual matter will be sensitive to the spacing of the galaxies because the dominant source of the attraction is from the nearby galaxies.

(To be continued.)