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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
where v is the tangential velocity.
The mass of the satellite cancels out and
Thus if the tangential velocity and orbit radius are known then the value of the central mass M can be determined.
The tangential velocity of stars on the periphery have been determined and thus the central mass has be computed for galaxies. The computed figure exceeds the estimated masses of all the stars in the galaxies and hence astronomers conclude there must be some matter that is not seen. This hypothesized matter has been called 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 a the center of the mass. But for other distributions of mass this does not hold true.
Consider three point masses of magnitude m arrayed in a straight line with a separation distance r between them. The open circle denotes the center of mass of the two masses on the right.
The force of gravitation F due to the center mass and one end mass on the other end mass is
On the other hand if the other two masses are considered concentrated at their center of mass which is a distance (3/2)r from the end mass (shown as the open circle in the above diagram). The computed force on end mass would then be
The tangential velocity of the system would be based upon F_{1} but the computed force based upon the presumption that the atracting masses are concentrated at their center of mass is F_{2}. There would appear to be some missing mass. The proportion missing would be determined by the ratio of F_{1} to F_{2}; i.e. (5/4)/(8/9)=45/32=1.40625. That is to say, the computation would indicate that there should to 40.625 percent more mass than is observed in the system. This dark matter is nonexistent; its apparent existence is due to the error of presuming that distributed masses can replaced with one mass all concentrated at the center of mass of the distributed masses.
The major part of the attraction in the example comes from the closest mass; i.e., 80 percent. In the concentrated version the mass of that closest mass accounts for only 50 percent of the attraction. The point at which the two masses could be concentrated and have the same gravitational effect as the original distribution, their gravitational center, is at distance of 1.264r.
Let us push the example a bit further. Suppose there are five equal masses arrayed in a straight line with a distance r between them.
The computations of the forces are as follows:
The ratio of F_{1} to F_{2} is 4.4488, indicating that the supposed dark matter constitutes 77.52 percent of the mass of the system. The center of gravitation of the four masses for the mass at the left is at distance of approximately 1.676r from it.
Now consider the mass distributed throughout a thin disk and use a polar coordinate system. An element at radius r and angle θ has xy coordinates of (rcos θ, rsin θ) has a distance s from the point at R and angle 0 (xy coordinates (R, 0)) given by
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.
Stars 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 stars on a star located at radial distance 1.0 is 6.115 times the attraction that would prevail if all of the stars of the collection 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.
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 stars because the dominant source of the attraction is from the nearby stars.
(To be continued.)
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