Why Phi? – Jupiter-Earth harmonics and the Lucas series

Jupiter – the dominant planet in the solar system

The aim here is to show a Lucas number based pattern in seven rows of synodic data.
There’s also a Fibonacci number element to this, as shown below.
The results can be linked back to an earlier post on planetary harmonics (see below).

The nearest Lucas number equation leading to the Jupiter orbit period in years is:
76/7 + 1 = 11.857142 (1, 7 and 76 are Lucas numbers).
The actual orbit period is 11.862615 years (> 99.95% match).
[Planetary data source]

It turns out that 7 Jupiter orbits take slightly over 83 years, while 76 Jupiter-Earth (J-E) synodic conjunctions take almost exactly 83 years. One J-E synod occurs every 1.09206 years. (83/76 = 1.0921052).

In the interest of clarity, whole numbers have been used below but very slight variations do exist.

The data table in question is on the right.

Lucas series starts: 2,1,3,4,7,11,18,29,47,76 (etc: add any two consecutive numbers for next one). [2 is not used here – see footnotes].
Fibonacci series starts: 0,1,1,2,3,5,8,13,21,34 (etc: add any two consecutive numbers for next one).

Explanation:
On the first row, 1 (Lucas no.) Jupiter orbit times the fixed multiplier (7) = 7.
Add the Fibonacci number (0 on the first row), sum of 7+0 = 7.
The number of Earth orbits (years) during 7 Jupiter orbits is 83.
(J-E is always the difference between the number of J and E orbits).

Repeat for the other six lines, converting the number of Jupiter orbits into years to obtain Earth orbits and J-E. On the last row, 7*29 plus 8 = 211 J orbits.

Analysis:
The Jupiter numbers form their own series, and are 7 times the Lucas number in the first column, plus the next (ascending) Fibonacci number.
Any two consecutive Jupiter orbit numbers add up to the next number on the list.
Or, the difference between any two consecutive numbers equals the previous number.

>>> The Earth orbits (years) are always 12 times the Jupiter number, less the Lucas number.
Example (second row): 12 * 22 = 264, minus 3 = 261.

>>> The J-E synods are always 11 times the Jupiter number, less the Lucas number.
Example (second row): 11 * 22 = 242, minus 3 = 239.

>>> As the Jupiter orbit numbers increase, their ‘neighbour’ ratios converge on the golden ratio (Phi).
Example (last two rows): 211/131 = 1.610687 (> 99.954% of Phi).

So the contention is that the table does contain clear numerical patterns.

Footnotes:
7 and 11 are Lucas numbers.
12 is the nearest whole number of years to one Jupiter orbit.
The Lucas series starts (here) from 1 to ensure an incremental progression.
The Fibonacci series starts from zero.

Jupiter-Saturn-Earth orbits chart

Planetary theory:
The last row of the table relates to this chart from an earlier blog post:
Why Phi? – Jupiter, Saturn and the de Vries cycle

The chart shows the same 211 Jupiter orbits, 2503 Earth years and 2292 J-E conjunctions that appear in the last row of the earlier data table (above). The data ties in with 85 Saturn orbits and 126 Jupiter-Saturn conjunctions.

Readers may wish to refer to the link for detailed discussion of the data in this chart, including the de Vries cycle connection.
– – –
Also from the Talkshop:
Lunar-planetary links to the Lucas sequence – part 3 and summary

Why Phi? – some Earth-Mars orbital harmonics