Why-Phi? – some Uranus-Neptune long-term resonances

Continuing our recent series of posts, with Uranus-Neptune conjunction data an obvious starting point for the table is where the difference between the number of Neptune orbits and U-N synods is 1.

647 U-N takes a long time (~110,900 years) but the accuracy of the whole number matches is very high.

Lucas no. (7 here) is fixed, and Fibonacci nos. follow the correct sequence (given their start no.).
Full Fib. series starts: 0,1,1,2,3,5,8,13,21…etc.
Multiplier: 0,1,1,2,3
Addition: 1,1,2,3,5

The Neptune orbits are multiples of 26 with the same Fibonacci adjustment:
Add 0,1,1,2,3 to the Neptune column numbers to get an exact multiple of 26 (which will be the pattern number in the last column).

Three Neptune orbits are ‘dropped’ in the period, since 26 * 26 = 676, but there are only 673 N orbits.

The corresponding number for Uranus is 6 dropped orbits (51*26, -1320), reflecting the fact that it completes nearly 2 orbits of the Sun per Neptune orbit.

To check the results, multiply the synods number by 374.4507 (degrees) then divide by the Neptune orbits. The result will always be very close to 360 degrees.

Another U-N Phi/Fibonacci connection can be found straight from their orbit ratio:
164.79132 (N) / 84.016846 (U) = 1.961408
1.961408 * 4 = 7.845632 = 2.6152106 = 34/13 (>99.993% match)

That’s very close to saying that the U:N ratio = 1: 3/4 of Phi².
The two orbit periods are very close to 84(U) and 165(N) years.
84 = 21*4 and 165 = 55*3.
Since 21 and 55 are Fibonacci numbers the ratio is confirmed.