**Appendix
**

The derivation of two equations

**
A.
****
The Hubble constant anomaly.**

This derivation of the Hubble constant equation rests on three assumptions. The first is that the universe is a regularly expanding sphere. The second is that light travels in straight lines. The third is that universe follows Euclidean geometry.

Let the velocity of an object at the edge of a universe be I. If the radius of the universe is R then the mean increase in the radius over the previous second is I/2. From Euclidean geometry of similar figures the following ratios apply (Figure 1)

Equation A1.
I/2R* *= i/2r
= v/2D

Where i is the velocity of an object that is r distance along that radius and v is the velocity of a galaxy that is D distance from earth. The mean volume of the universe during that next second is

Equation A2
Volume = (4/3)^{3}

Equation A3 I/2 = vR/2D

Substituting in Equation A2 and expanding the equation, but discarding the third and fourth components of the expansion as being too small to be significant when D is very large, results in

Equation A4
Volume = (4/3)^{3}

^{
}
=
(4/3)^{3}
+ 3R^{2}Rv/2D)

For a previous time, t, the volume was

Equation A5
Volume =
(4/3)^{3}(1-t3v/2D)

When the volume was insignificantly small compared with the present size of the universe, that is effectively zero, at the beginning of time

Equation A6
0 = (4/3)^{3
}(1-t3v/2D)

= 1-t3v/2D

That is

Equation A7 t = 2D/3v = 2/3 x Ho

Equation A8 3v/D = 2/t

But t is the
age of the universe when the light from a distant galaxy that we now
receive, was emitted. It has
taken D_{t} billions of years
for that light to reach us.
(See figure A1.) That is

Equation
A9
3v/D = 2/(Age -D_{t })

When v is in fractions of the velocity of light and D
is in billions of light years converting
this to km.s^{-1}Mpc^{-1} results in

Equation A10
Ho = 3.26 x 200/(Age-D_{t})

It follows that the Hubble value should increase with distance, albeit that when that distance is less than a thousand Mega parsecs the effect is so small as not to be noticeable.

**B. The Relativity correction**

The standard equation defining the effective magnitude
has four components, Equation 4. Each of these four components is time
sensitive and each needs
correcting by use of Special relativity’s basic equation, (Equation 7). The
correction needs to be in the log format using the log base applicable to
Magnitudes (i.e. log x 2.5). The result is

Equation B1.
Effective Mag = observed Mag - A_{x} – K-
a(s-1)-10(log(1/(1-*v*_{c}^{2}* _{
}*))

^{0.5}

Where *v*_{c}*
*is the velocity expressed as a fraction of the velocity of light. When
this is applied to determining the logarithm of distance ,(Equation 3) and
simplifying this becomes

Equation B2.
Log distance (Mpc) = (Effective Mag -
5.3)/5 + log(1/(1-*
v*_{c}^{2})

For the determination of the Hubble constant v/D where
v is in kilometres per second, the velocity also must be corrected for
relativity’s distortion of time i.e.

Equation
B3
v = observed v x (1/(1-*
v*_{c}^{2}))^{0.5}

This simplifies further to become Equation 5