Why Phi? – the Saturn-Uranus connection

Distances not to scale.

This is an easy data table to interpret.

The Uranus orbits are all Fibonacci numbers, and the synodic conjunctions are all a 3* multiple of Fibonacci numbers.
[Fibonacci series starts: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …etc.]

In addition, the difference between the two is always a Lucas number. And that’s it for Saturn-Uranus, which would make for a very short blog post.

But it’s possible to go further.

Note that the first row in the table above starts with 13 Uranus orbits.

The time period for that is 1092.2189 years, which is almost the same as 1000 Jupiter-Earth conjunctions (1092.06 years).

In the second table, a multiplier of 13 is used to get closer to whole numbers of J-E conjunctions, then by rounding the results to the nearest whole number another pattern emerges.

For each Fibonacci multiple of 1000 J-E synods, a Fibonacci number can be added to get an exact match to the (Fibonacci) number of Uranus orbits in the first column.

In fact two more rows in the table would bring us to 55*13 Uranus orbits, which featured in an earlier blog post:
Why Phi? – a long-term Jupiter-Saturn-Uranus model

On the right is the chart from that post.

The number of J-E conjunctions for that time period would be 55008 (1000*55,+8), continuing the Fibonacci progression.
55008 J-E = 55*13 U = 715 Uranus orbits

This is the long period (60,072 years) discussed in that blog post.
55008 * 1.09206 years (J-E) = 60072.036 y.
60072 – 55008 = 5064 Jupiter orbits