Lunar-planetary links to the Lucas sequence – part 3 and summary

Posted: April 22, 2019 by oldbrew in Fibonacci, Lucas, moon, solar system dynamics
Tags: ,

Earth from the Moon [image credit: NASA]

Part 3

To recap, the Lucas series starts: 2, 1, 3, 4, 7, 11, 18, 29 … (adding the last two numbers each time to find the next number in the series).

Note: for clarity, the three parts of this mini-series should be read in order (links below).

Since Part 1 showed that 7 Jupiter-Saturn conjunctions (J-S) = 11 * 13 lunar tropical years (LTY), and from Part 2 we know that 363 LTY = 353 Earth tropical years (TY), these numbers of occurrences can be integrated by applying another multiple of 13:
363 = 3*11*11 LTY
therefore
353 * 13 TY = 3*11*11*13 LTY = 3*7*11 J-S

7 and 11 are Lucas numbers.
13 is a Fibonacci number.
3 belongs to both series.

One further step would be to multiply by 6.
(3*7*11 J-S) * 6 = 7*11*18 J-S = 126 J-S * 11
7,11 and 18 are Lucas numbers.

Jupiter-Saturn-Earth orbits chart

Since the total is a multiple of 126 J-S, we can now refer back to the ‘de Vries cycle’ chart (right) from Part 2, which shows that 85 Saturn = 211 Jupiter orbits = 126 J-S (= 211 – 85).

So 11 of those long periods (of ~2503 years each) corresponds to 353*13*6 TY.
2503*11 = ~27533 sidereal years
353*13*6 = 27534 tropical years exactly
(a sidereal year is very slightly longer than a tropical year)

Cross-check:
27534 TY * 365.24219 d = 10,056,578 days
85*11 Saturn @ 10755.7 d = 10,056,579 days

Summary

Part 1 Shows the Lucas number links between the Lunar tropical year, the Chandler wobble and the Venus rotation period.

Part 2 Relates Jupiter-Saturn conjunctions via Lucas numbers to the periods listed in Part 1.

Part 3 Relates Earth’s tropical year to all the periods listed in Parts 1 and 2, showing further links to the Lucas number series.

1. tallbloke says:

This is very impressive work.
Can De Ropp’s period be worked in too?

2. oldbrew says:

Hi TB. Re de Rop:

The difference between 6 lunar wobbles and 37 lunar tidal years (=13*37 lunar orbits) is very small, about a third of a day in almost 36 years.

37 * 124 = 4588 LTY
13 * 353 = 4589 LTY [see Part 3 post, above]

That may be as close as we can get with that idea.
– – –
PS I found a clear link between the Jupiter-Earth synodics and the Lucas series 😎

3. Paul Vaughan says:

Primorial Golden Triangle Split Primer

(2310)*(30030) / (2310 – 30030) = 2502.5
(2310)*(30030) / (2310 + 30030) = 2145
(2310)*(30030) / ( (2310 + 30030) / 2 ) = 4290

(2310)*(2502.5) / ( (2310 + 2502.5) / 2 ) = 2402.4

=
The concrete details of a given construction may be messy, but if the construction satisfies a universal property, one can forget all those details: all there is to know about the construct is already contained in the universal property. Proofs often become short and elegant if the universal property is used rather than the concrete details.
[…]
Universal properties occur everywhere in mathematics. By understanding their abstract properties, one obtains information about all these constructions and can avoid repeating the same analysis for each individual instance.
=
https://en.wikipedia.org/wiki/Universal_property

4. tallbloke says:

Paul, I looked up primorial but can’t see how it relates to your numbers above. Any extra hints?

5. […] Lunar-planetary links to the Lucas sequence – part 3 and summary […]

6. Paul Vaughan says: