THE GRAVITATIONAL ATTRACTION OF THE GALAXY

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

(1)

The dietance of the element dm from P is

R = (p² + z²)½

(2)

and the attraction of the mass dm for the mass at P is

- G dm   =  Gspdpdq
—       ————	(3)
R²        p² + z²

and the component of the attraction along the exis is

- Gspdpdq   .    z          Gspdpdq
————      –   =  - ————	(4)
p² + z²           R         (p² + z²)3/2

The total intensity of attraction of the disk for the point P mass is

	ò	n	ò	2p	pdpdq
I = G s z					————–
		o		o	(p² + z²)3/2

 

	ò	r	pdp
I = 2pGsz			————–
		o	(p² + z²)3/2

 

	[	z                 -	z	]
I = 2pGs		————	——			(5)
		(z² + r²)½	(z²)½

Assuming z positive,

	[	z	]
I = 2pGs		———— -  1			(5¹)
		(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						(6)
dt          c²		(z² + r²)½

 

dv  =  dv   dz  =   v  dv
—      —   —          —	(7)
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 d_o, 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

ò	v	vdv	ò	z	2pGs	zdz	-	ò	z	2pGsdz
		————— =			———–	————				————	(8)
	o	c� - v�		do	c²	(z² + r²)½			do	c²

The outcome of this result is that

v	[		- 4pGs	[(z² + r²)� - (do² + r²)½ + do - z]			½	]
– =		1 - e	——––						½	(9)
c			c²

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		(10)
c

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:

1. 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.
2. Globular clusters are not falling directly toward the galactic center; rather they are orbiting.
3. 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.

References:

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.