In a previous paper1 I worked out the general form of Newton’s Law of Gravitation and applied it to the special case of a planet orbiting the sun. In this case Newton’s Law was modified by the factor
|1/(1 - v²/c²) �|
For the case of an object moving directly toward another object rather than orbiting, the genetal equation reduces to Newton’s Law multiplied by the factor
|(1 - v²/c²)|
This is exactly of the same form as Lorentz’s modification of Coulomb’s Law.
Before applying the new factor, it is important to realize that the galaxy cannot be represented as a pofnt mass; rather it should be represented as a flat disk. The Newtonian actraetion of a flat disk for a point mass has been worked out before², but will be repeated here.
In Figure I let the radius of the disk be r and let its surface density be s . I aim to find the attraction of the disk for a polnt mass located at P on the perpendicular line passing through the center of the disk. Let 0 be the origin of a system of polar ccordinates p and q, and let z be the distance along the line to the attracted location P.
Since pdpq is the area of an element in polar coordinates, the mass of such an element is
|dm = spdpdq||
The dietance of the element dm from P is
|R = (p² + z²)½||
and the attraction of the mass dm for the mass at P is
|- G dm = Gspdpdq|
|R² p² + z²|
and the component of the attraction along the exis is
|- Gspdpdq . z Gspdpdq|
|———— – = - ————||
|p² + z² R (p² + z²)3/2|
The total intensity of attraction of the disk for the point P mass is
|I = G s z||————–|
|o||o||(p² + z²)3/2|
|I = 2pGsz||————–|
|o||(p² + z²)3/2|
|I = 2pGs||————||——||
|(z² + r²)½||(z²)½|
Assuming z positive,
|I = 2pGs||———— - 1||
|(z² + r²)½|
Now with the modifying factor included, the acceleration of the point masa toward the disk is
|dv||[||z||]||[||1 - v²||]|
|— = 2pGs||———— - 1||—|
|dt||(z² + r²)½||c²|
|dv 2pGs||[||z||]||[||c² - v²||]|
|— = ——||———— - 1||
|dt c²||(z² + r²)½|
|dv = dv dz = v dv|
|— — — —||
|dt dz dt dz|
The crucial deduccion in Larson’s gravitational theory is that the gravitational force of any mass extends outwatd only a finite amount the gravitational force does not extend out to “infinity”, as commonly assumed. At the gravitational limit of the galaxy, which will be denoted by do, the attracted velocity of a mass is zero. This velocity becomes larger to the degree that the mass is loeated closer to the galaxy. Let the velocity be v at distance z. Then, separating the variables in equation 6 and integrating between the limits, the result is
c� - v�
(z² + r²)½
The outcome of this result is that
|v||[||- 4pGs||[(z² + r²)� - (do² + r²)½ + do - z]||
|– =||1 - e||——––||½||
For our galaxy the constants in the equation are as follows:
|G = 6.67 x 10-11 N - m²/kg²|
|c² = (3 x 108)² m²/s²|
|s = .2975 kg/m²|
|r = 4.626 x 1020 m|
|do = 2.177 x 1022 m|
With these values equation 9 becomes
|v||[||- 2.771 x 10-27[(z� + 2,140 x 10 41)� - 9.087 x 1013 - z]||]|
|– =||1 - e||½||
Speed in km/sec vs. distance ia kiloparsecs is plotted in the graph (Fig. 2). Great caution must be used in applying equation 10 to real masses:
- A globular cluster or a small galaxy associated with the Milky Way galaxy is not really a point mass; in fact, observation shows that the near side stars of such objects are attracted at the expense of the farside stars.
- Globular clusters are not falling directly toward the galactic center; rather they are orbiting.
- Small local galaxies are at a distance close to the gravitational limit of the galaxy — their veloclties aze difficult to measure and compare with theory.
Even so, the calculated velocities, neverthaless, agree in a very general vay with those observed for the local group of objects.
1. Satz, R.W. REClPROCITY Vol. IV, No. 2, p. 25, July 1974,
2. MacMillan, W., The Theory of the Potential (New York: McGraw Hill Book Company, 1930),pp. 15-16.