THE DETERMINATION OF THE CONSTANT
Mt3
Once a value for the constant k
has been established,
the mean sidereal periods of the three planets associated with
inverse-velocity
relation [3b] may be expressed as integral parts of an
exponential
function based on k = phi, the respective heliocentric
positions
of the planets in question, and the known sidereal period of Mercury (Mt
), i.e.,
[1] Mean Sidereal Period of Mercury
= Mt phi 0
[2] Mean Sidereal Period
of
Venus = Mt phi 2
[3] Mean Sidereal Period
of
Uranus = Mt phi 12
Because of the following velocity
expansions of Kepler's third law of
planetary motion:4
[4] Mean Velocity (Vr) = Mean
Distance (A)-1/2 = Mean Period (T)-1/3
[5] Inverse Velocity (Vi) =
Mean
Distance (A)1/2 = Mean Period (T)1/3
[6] Mean Distance (A) =
Inverse
Velocity(Vi)2 = Mean Period (T)2/3
inverse-velocity relation [3b]: ViVenus
- ViMercury
: Mean Velocity of Uranus (Vr) may expressed in terms of phi
and exponential Relations [1] through [6], i.e.,
[7] (Mt phi2)1/3
- Mt 1/3 (Mt phi12)-1/3
[7a] Mt 1/3phi2/3
- Mt 1/3 Mt-1/3phi-4
and then restated in terms of the
inverse velocity of Mercury since:
Mvi = Mt 1/3 ( Relation [5] ):
[8] Mvi(phi2/3
- 1) Mvi -1phi-4
[9] Mvi2(phi2/3
- 1) phi-4
[10] Mvi2 phi-4(
phi2/3 - 1)-1
But from Relation [6], Mvi2
is equivalent to the mean
distance of Mercury (Ma ), thus:
[11] Mean Distance of Mercury
= Mvi2 = Ma phi-4(phi2/3 -
1)-1
therefore a new phi-based value
for the mean sidereal period
of Mercury (Mt3) may be obtained from Relation [11]
and
the application of Kepler's Third Law:
[12] Mean Sidereal Period
of Mercury = Mt3 = phi-6(phi2/3-1)-3/2
= 0.2395640495... Years
RETURN
TO SECTION III