The conceptual basis for Hubble’s Law in the Reciprocal System has been discussed by Mr. Larson in a number of his works. This paper will present some additional mathematical details.
Hubble’s Law is commonly written as
|v = Hr||
where v is the velocity of a distant galaxy, in km/sec, r is the distance to the galaxy, in Mpc, and H is Hubble’s constant, in km sec-1 Mpc-1 . In differential form, the equation is
|dr/dt = Hr||
However, as shown in Larson’s The Structure of the Physical Universe,¹ the recession starts at the gravitational limit of our galaxy, denoted by ro. Thus the correct expression is
|dr/dt = H (r - ro)||
Clearly the velocity is zero when r = ro.
The equation is a first order linear differential equation² and can be easily solved for r. The result if
|r = ro + (ri - ro)eHt||
where ri is the initial position of the external galaxy.
Of great interest is the determination of Hubble’s constant from first principles. According to the Structure, the ratio of effective to total gravitational units is 1/156.4444. At the distance a galaxy recedes at the speed of light, the effective gravitational force drops betow the value of unity and vanishes. Thus
|1/156.4444 · MG/r1² =1||
(equation 159 of Structure). Solving for r1, the limiting distance, yields
|r1 = MG½ /12.51||
(equation 160 of Structure).
Putting this vatue of r1 in Hubble’s Law, one can solve for the constant:
|H = c/r1-ro » c/r1||
where c is the velocity of light.
The value of the constant thus depends on the value of the mass of the Galaxy, MG. According to reference three, this is
|MG = 2.587 * 1041 kg||
With this value the constant is
|h = 114.522 · km/ sec Mpc||
However, according to Sandage the value of H is
|H = 75.0 · km/sec Mpc||
This implies that the actual value of the mass of the Galaxy is
|MG = 6.032 * 1041||(11)|
or 2.33 times that estimated.
It seems to me that Hubble’s constant has been more accurately determined than the mass of the Galaxy. Thus the analysis leads to the conclusion that the mass of our Galaxy is greater than supposed—probably because of a white dwarf galactic core that still remains difficult to observe.
- Dewey B. Larson The Structure of the Physical Universe, First Edition (Portland, Oregon: North Pacific Publishers, 1959.
- Richard Bronson , Modern Introductor Differential Equations (New York: McGraw-Hill Book Company, 1973.
- Martin Harwit, Astrophysical Concepts (New York: John Wiley & Sons, 1973).