Lucas resonances of the three planet system TOI-1749

Artist’s impression of an exoplanetary system [credit: NASA]

The three exoplanets of the TOI-1749 system are labelled b, c and d. A year ago, an article in the Astronomical Journal on this system by a group of scientists noted a near 2:1 orbital ratio of planets c and d, but made no reference to synodic periods.

Now, with newer data from exoplanet.eu, we analyse the synodics. The mean period between alignments of two planets with their star is as follows:
b-c = 5.0989527 days
c-d = 8.922796
b-d = 3.2447389

From that, the ratios can be obtained:
7 b-c = 35.692668 days
4 c-d = 35.691184
11 b-d = 35.692127

4:7:11 coincides with the Lucas number series which is closely related to the Fibonacci series and the golden ratio.

Resonances of the Kepler-184 three-planet system

There doesn’t seem to be any online discussion of this planetary system, first seen in 2014 – but it turns out be interesting anyway.

This is a Lucas series set-up, the planets being b, c, and d in order of proximity to the star.

Starting with the orbits:
19 b = 203.006394 days
10 c = 203.03005
7 d = 203.1565
(data: exoplanet.eu)

Lucas resonances in six-planet Kepler-20 system

In 2011, astronomers were saying:
“We’ve crossed a threshold: For the first time, we’ve been able to detect planets smaller than the Earth around another star.”

The planets in question were Kepler-20 e and Kepler-20 f.

In the end six planets were detected: b,e,c,f,d, and g (in order of proximity to their star). Orbit periods range from about 9.38 to 63.55 days, all the planets being closer to the star than Mercury is to the Sun.

A NASA article had the title: Kepler-20, An Unusual Planetary System — referring to the alternate large/small sizes of the planets.

Why Phi? – exoplanetary resonances of Kepler-102

Kepler Space Telescope [credit: NASA]

Star Kepler-102 has five known planets, lettered b,c,d,e,f. These all have short-period orbits between 5 and 28 days. Going directly to the orbit period numbers we find:
345 b = 1824.0012 d
258 c = 1824.4263 d
177 d = 1825.1709 d
113 e = 1824.4629 d
(for comparison: about 1-2 days short of 5 Earth years)

For the purposes of this post planet f (the furthest of the five from its star) is excluded, except to say that in terms of conjunctions 8 e-f = 11 d-e. Now let’s look for some resonances of the inner four planets.

Why Phi? – Jupiter-Venus harmonics and the Fibonacci/Lucas series

Jupiter – the dominant planet in the solar system

The aim here is to show a Lucas number based pattern in five rows of synodic data, then add in a note on Mercury as well.

There’s also a strong Fibonacci number element to this, as shown below.

The results can be linked back to earlier posts on planetary harmonics involving the Lucas and Fibonacci series (use ‘search this site’ box on our home page).

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).

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.

Why Phi? – more Jupiter-Saturn orbital harmonics

We’re now looking for a pattern arising from the Jupiter-Saturn synodic conjunctions and the orbit periods.

Focussing on the numbers of Jupiter orbits that are equal, or nearly equal, to an exact number of Saturn orbits (years), a pattern can be found by first subtracting the number of conjunctions from the number of Saturn orbits.

Why Phi? – some Earth-Mars orbital harmonics

A simple pattern emerges when looking at the Earth-Mars synodic conjunctions.

Focussing on the numbers of Mars orbits that are equal, or almost equal, to an exact number of Earth orbits (years), the pattern can be found by subtracting the number of conjunctions from the number of Mars orbits.

The difference between the two sets of numbers follows the Fibonacci series, which is strongly related to the golden ratio.

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).

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

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 2 showed that 7 Jupiter-Saturn conjunctions (J-S) = 11 * 13 lunar tropical years (LTY), and from Part 1 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.

Lunar-planetary links to the Lucas sequence – part 2

Image credit: Wikipedia

In Part 1 the period of time in question was 13 lunar tropical years (LTY). Here we show how this relates to the Jupiter-Saturn conjunction and other significant periods.

13 LTY = 169 * 27.321582 days (lunar orbit period) = 4617.3473 days
1 Jupiter-Saturn conjunction = 19.865036 years * 365.25636 days = 7255.8307 days
[planetary data source]

This gives a ratio of exactly 11:7 as follows:
11 * 4617.3473 = 50790.82 days
7 * 7255.8307 = 50790.814 days
7 and 11 are Lucas numbers.

Jupiter-Saturn-Earth orbits chart

Multiplying by 3 (which is both a Fibonacci and a Lucas number), the results from Part 1 can now be ‘plugged in’ to the chart on the right from a previous blog post, which is based on multiples of 21 (3 * 7) Jupiter-Saturn conjunctions, as the chart on the right shows.

Quoting from that post:
The synodic periods all occur in multiples of six, and one sixth of 2503 years is 417.1666 years which is 21 J-S, 382 J-E, 403 S-E and two de Vries cycles.

Updating that, the matching periods now are:
2 de Vries cycles
21 Jupiter-Saturn conjunctions (3 * 7)
382 Jupiter-Earth conjunctions
403 Saturn-Earth conjunctions (13 * 31)
352 Chandler wobbles (11 * 32)
429 Lunar tropical years (11 * 39 or 13 * 33)
627 Venus rotations (11 * 57 or 19 * 33)

Therefore: 31 Saturn-Earth = 33 Lunar tropical years.

Quoting from Part 1 of the post:
353 Earth tropical years (ETY) = 363 Lunar tropical years = 10 beats

363 LTY = 33 * 11
Therefore: 363 LTY = 31 * 11 (341) Saturn-Earth conjunctions (= synodic periods).

Quoting from another earlier post – Sidorenkov and the lunar or tidal year:
4719 LM = 128930.54 days  [note: 4719 = 363 * 13 i.e. 363 LTY]
4366 SM = 128930.55 days
4727 CR = 128930.34 days
5080 SSR = 128930.40 days
353 TY = 128930.49 days
(see post re. abbreviations).

NASA’s Saturn Fact Sheet says re. Saturn-Earth:
Synodic period (days) 378.09

TY = tropical years
128930.49 days / 341 S-E = 378.09527 days
This ties Saturn to Sidorenkov’s 353 year period, which is therefore 11/13ths of 21 J-S.
Also: 11/13ths of 429 LTY = 363 LTY.

Footnote:
In the graphic the full period is 126 J-S, described as 6 * 21.
It could also be described as 7 * 18, which are Lucas numbers.

Lunar-planetary links to the Lucas sequence – part 1

The Lucas spiral, made with quarter-arcs, is a good approximation of the golden spiral when its terms are large [credit: Wikipedia]

Here we show numerical connections between the Moon, the Earth and Venus. These will be carried forward into part 2 of the post. The focus is on the smaller Lucas numbers (3-18).

Wikipedia says: The Lucas sequence has the same recursive relationship as the Fibonacci sequence, where each term is the sum of the two previous terms, but with different starting values.

A look at the numbers:
19 Venus rotations = 169 (13²) lunar rotations
Lunar tropical year = 13 lunar rotations / orbits (1 rotation = 1 orbit)
So: 19 Venus rotations = 13 Lunar tropical years
(13 is a Fibonacci number. The Lunar tropical year is derived from the nearest whole number of lunar orbits to one Earth orbit.)

169 * 27.321582 = 4617.3473 days (Data source)
19 * 243.018 = 4617.342 days (Data source)

Now we bring in the Chandler wobble:
13*3 = 39
39 Lunar tropical years = 32 Chandler wobbles
19*3 = 57

Referring to the chart on the right:
7 and 18 are Lucas numbers.
This theme will continue in part 2 of the post.

(32 + 57 = 89 axial, and 89 is a Fibonacci number. In 1/89th of the period the sum of CW and Ve(r) occurrences is 1).

Re. the period of the Chandler wobble:
39 LTY / 32 CW = (169 * 3 * 27.321582) / 32 = 432.8763 days

Or, if we say 27 Chandler wobbles = 32 Earth tropical years:
(365.24219 * 32) / 27 = 432.8796 days

The two results are almost identical (Wikipedia rounds it to 433 days).

Note:
353 Earth tropical years (ETY) = 363 Lunar tropical years = 10 beats
1 beat = 35.3 ETY which is linked to the Chandler Wobble
See: Sidorenkov – THE CHANDLER WOBBLE OF THE POLES AND ITS AMPLITUDE MODULATION

These numbers also feed into part 2 of the post, with more planetary links.