THE DETERMINATION OF THE CONSTANT Mt3

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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 (M_t ), i.e.,

[1]Mean Sidereal Period of Mercury = M_t phi ⁰
[2]Mean Sidereal Period of Venus = M_t phi ²
[3]Mean Sidereal Period of Uranus = M_t 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]: Vi_Venus - Vi_Mercury : Mean Velocity of Uranus (Vr) may expressed in terms of phi and exponential Relations [1] through [6], i.e.,

[7] (M_t phi²)^1/3 - M_t ^1/3 (M_t phi¹²)^-1/3

[7a] M_t ^1/3phi^2/3 - M_t ^1/3 M_t^-1/3phi^-4

and then restated in terms of the inverse velocity of Mercury since: Mvi = M_t ^1/3 ( Relation [5] ):

[8] Mvi(phi^2/3 - 1) Mvi ^-1phi^-4

[9] Mvi²(phi^2/3 - 1) phi^-4

[10]Mvi² phi^-4( phi^2/3 - 1)^-1

But from Relation [6], Mvi² is equivalent to the mean distance of Mercury (M_a ), thus:

[11] Mean Distance of Mercury = Mvi² = M_a phi^-4(phi^2/3 - 1)^-1

therefore a new phi-based value for the mean sidereal period of Mercury (M_t3) may be obtained from Relation [11] and the application of Kepler's Third Law:

[12] Mean Sidereal Period of Mercury = M_t3 = phi^-6(phi^2/3-1)^-3/2 = 0.2395640495... Years

RETURN TO SECTION III