San José State University |
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Its Supposed Existence Stems from a Few Naive Mistakes in Analysis |
The standard line of analysis purporting to demonstrate the existence of dark matter goes as follows. The velocities of some component of a system, such as stars in a galaxy or galaxies in a galactic cluster, are measured. The gravitational attractions required to keep these components in their orbits are computed. The mass that would be required at the center of the system to sustain such attractions is computed. That mass is then compared with known mass of the system. If the required mass is greater than the known mass the difference is attributed to the existence of dark matter.
The mistake of this is in saying that the attraction at a point due to spatially distributed masses is the same as if all the masses were concentrated at the center of mass. There is a wonderful theorem in mathematical physics which says that if a mass is uniformly distributed over a sphere its effect is the same as if all the mass were concentrated at the center of the sphere. This theorem extends to spherical shells but it does not extend to other geometrid configurations such as disks and cylinders.
The uniformity of the distribution is essential. It doesn't extend to distributions of point-like particles. Thus so-called dark matter is a spurious creation of of erroneous analysis.
Now consider a spatial structure of star masses distributed throughout a thin disk described using a polar coordinate system. Any structure would work just as well. Some simple ones are shown in the Appendix. This one has the advantage of simplicity for ease of computation. It also generally resembles the structure of a galaxy with a concentration of mass in the center.
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
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−r·cos 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 0.1R to 0.9R.
The point of interest is the star located on the right side of the disk. Its radial distance to the center is R.
The radial attraction of the collection of stars on that star located to the right of the galaxy 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 in order to duplicate that attraction with a mass located at the center of the coordinate that mass would have to be 6.115 times the total mass of the galaxy. There would thus 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 is surrounded by the white circle. It is a point 0.4R from the center of the star on the right of the galaxy . 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.
There is a different center of gravity for each different star of the galaxy.
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.
Another phenomenon that is purported to be evidence of dark matter is galactic lensing. Such lensing is evidence of matter in a galaxy not specifically dark matter.
(To be continued.)
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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.