Why Phi – the Fibonacci resonances of the TOI-451 exoplanets

NASA’s exoplanet hunter (TESS)
[image credit: MIT]

This three-planet system has orbit periods ranging from under two to over sixteen days, obviously another very compact group. Their star is slightly smaller and less powerful than our Sun.

Planets b and c are a fraction of Jupiter’s size, but planet d is vast with a radius of over four Jupiters, or about 45 Earth radii.

Why Phi? – resonances of exoplanets LHS 1140 b and c

Wikipedia says:
LHS 1140 is a red dwarf in the constellation of Cetus…The star is over 5 billion years old and has 15% of the mass of the Sun. LHS 1140’s rotational period is 130 days…LHS 1140 is known to have two confirmed rocky planets orbiting it, and a third candidate planet not yet confirmed.

Planet b was in the media spotlight in 2017:
LHS 1140b: Potentially Habitable Super-Earth Found Orbiting Nearby Red Dwarf – Sci-News.

“This is the most exciting exoplanet I’ve seen in the past decade,” said Dr. Jason Dittmann, an astronomer at the Harvard-Smithsonian Center for Astrophysics and lead author of the Nature paper.
. . .
“The LHS 1140 system might prove to be an even more important target for the future characterization of planets in the habitable zone than Proxima b or TRAPPIST-1,” concluded co-authors Dr. Xavier Delfosse and Dr. Xavier Bonfils, both at the CNRS and IPAG in Grenoble, France.

Friday Fibonacci Fun

Someone make this with cheese and tortilla chips so I can have a fibonacho pic.twitter.com/FyAir9GUgJ

— jayynecobb (@JayyneCobb) April 10, 2020

Such a cleverly made 12 second clip. Fullscreen it and enjoy

YZ Ceti update: resonant echoes of Pluto’s moons

There’s been a data update for the three planet system of star YZ Ceti, which featured in our 2018 post: Why Phi? – resonant exoplanets of star YZ Ceti. According to NASA the third planet YZ Ceti d is a ‘super Earth’, about 1.14 times the mass of our planet.

The paper:
‘The CARMENES search for exoplanets around M dwarfs.
Characterization of the nearby ultra-compact multiplanetary system YZ Ceti’
(Submitted on 5 Feb 2020)

With an additional 229 radial velocity measurements obtained since the discovery publication, we reanalyze the YZ Ceti system and resolve the alias issues.

Are Earth’s obliquity and axial precession in a long-term 5:8 ratio?

Earth’s tilt moves back and forth between about 22 and 24.5 degrees

If there is a mean ratio of 5:8 it would be linked to the known variation of Earth’s tilt, which in turn causes variation in the precession and obliquity periods.

Encyclopedia Britannica’s definition says:
Precession of the equinoxes, motion of the equinoxes along the ecliptic (the plane of Earth’s orbit) caused by the cyclic precession of Earth’s axis of rotation…The projection onto the sky of Earth’s axis of rotation results in two notable points at opposite directions: the north and south celestial poles. Because of precession, these points trace out circles on the sky.

(Axial precession is another term for ‘precession of the equinoxes’).

Our 2016 unified precession post started with this quote from Wikipedia (bolds added):
Because of apsidal precession the Earth’s argument of periapsis slowly increases; it takes about 112000 years for the ellipse to revolve once relative to the fixed stars. The Earth’s polar axis, and hence the solstices and equinoxes, precess with a period of about 26000 years in relation to the fixed stars. These two forms of ‘precession’ combine so that it takes about 21000 years for the ellipse to revolve once relative to the vernal equinox, that is, for the perihelion to return to the same date (given a calendar that tracks the seasons perfectly).

Three linked precessions

In units of 1,000 years:
21 * (16/3) = 112
112 * (3/13) = 25.846~ (near 26)
25.846~ * (13/16) = 21
That was the number theory of the ‘unified precession’ post, i.e. a 3:13:8*2 ratio.

Where might the obliquity period, known to be somewhere near 41,000 years, fit into that?

Referring to the chart (above, right) and converting decimals to whole numbers:
AY – SY = 328 = 109*3, +1
SY – TY = 1417 = 109*13
AY – TY = 1745 (328 + 1417) = 109*16, +1
[327:1417:1744 = 3:13:16]

So that supports the number theory.

Starting out, I just updated the chart to include an entirely theoretical obliquity period of 8/5 times axial precession, linking it to the other known cycles as suggested by my 2016 comment to the unified precession post, here.

That post was a follow-up to: Why Phi? – some Moon-Earth interactions, which showed how:
The period of 6441 tropical years (6440.75 sidereal years) is one quarter of the Earth’s ‘precession of the equinox’.
Multiplying by 4: 25764 tropical years = 25763 sidereal years.
The difference of 1 is due to precession.

[NB Wikipedia quotes 25772 years (‘disputed – discuss’) for this precession cycle, but as it’s not a fixed number the question is: what is the mean period? Earth is currently around the mid-point of the tilt variation, moving towards minimum tilt i.e a shorter precession period. Astronoo says 25765 years.]

But then I came across two things: a paper by EPJ van den Heuvel, cited in Wikipedia, and another entry in Wikipedia (see below), that together suggested viable alternative numbers but with the same 5:8 ratio.

On the Precession as a Cause of Pleistocene Variations of the Atlantic Ocean Water Temperatures
— E. P. J. van den Heuvel (1965)

From the summary:
‘The Fourier spectrum (Fig. 8) shows two significant main periods, P1 = 40000 years and P2 = 12825 years*. The first period agrees well with the period of the oscillations of the obliquity of the ecliptic. The second period corresponds very well with the half precession period.’
[*But the specific periods found were: 42857, 39474 and 12825 years]

From Wikipedia – Axial tilt – long term (Wikipedia):
‘For the past 5 million years, Earth’s obliquity has varied between 22° 2′ 33″ and 24° 30′ 16″, with a mean period of 41,040 years. This cycle is a combination of precession and the largest term in the motion of the ecliptic.’

41040:12825 = 16:5 exactly. Since 12825 is the half precession period, the full period ratio is 8:5 as in the chart, but with slightly different numbers.

If this is correct, the 25764y period in the chart would need adjusting by a factor of 225/226:
25764 * (225/226) = 25650 = 2 * 12825

The Wikipedia obliquity period of 41040 years is divisible by 19, so is an exact number of Metonic cycles (2160), as is the revised axial precession of 25650 years (1350). So the alternative period equals a reduction of 6 Metonic cycles of axial precession. The idea of a role for the Moon in Earth’s obliquity has been put forward before.

Of course 225/226 represents less than half a percent of correction, so could be argued to be negligible.
– – –
Now something else has turned up, written around the same time as two Talkshop posts already referred to:
The Secret of the Long Count, by John Martineau

In the ‘Long Count’ section of the article the writer also puts forward an argument for a (mean) 5:8 ratio of obliquity and axial (equinoctial) precession, using some historical context (see below).

So at least one other person has been thinking along the same lines. Note that 2,3,5,8 and 13 are Fibonacci numbers.

– – –
The Secret of the Long Count

In the summer of 2012 I visited Carnac, accompanied by Geoff Stray. Howard Crowhurst runs an annual midsummer conference there and we had been invited to speak at the 2012-themed event. Halfway through his presentation, Crowhurst was describing his hunches surrounding megalithic awareness of the 41,000-year cycle, when he casually mentioned a startling fact:

The 41,000-year cycle very precisely consisted of eight Mayan Suns.

I did a double take. Eight suns, but five made precession! Startled, I cornered Geoff Stray. He had already come across the eight Suns figure for the obliquity cycle, but not realised the significance of 5:8, while Howard Crowhurst had been unaware of the fact that five Suns gave a value for Precession. We had cracked it.

One Mayan Sun is 5,125 years.

Five Suns give the Precessional Cycle

5 x 5125 = 25,625 years (current value 25,700 years, 75 years out)

Eight Suns give the Earth’s Obliquity Cycle.

8 x 5125 = 41,000 years (current value 41,040 years, 40 years out)

Five and eight! The two long cycles that most affect the Earth relate as 5:8 and are both encoded by the Long Count. The Maya must have known. No wonder they drew so many pictures of jawbones. Five and eight! The same two numbers displayed by human teeth are the same two numbers as those used by the plants all around us, and these are the same two numbers that connect us with our closest neighbour Venus, and the same two numbers that relate the two long cycles that affect Earth-bound astronomy.

[emphasis by the author]

From: The Secret of the Long Count, by John Martineau

One of the closest exoplanet pairs to the 3:2 mean motion resonance: K2-19b and c

Artist’s impression of the Kepler telescope [credit: Wikipedia]

So said researchers in their 2015 study which had that title. Then a third planet was seen.

In the abstract they say:

Methods. Our search through two separate pipelines led to the independent discovery of K2-19b and c, a two-planet system of Neptune-sized objects (4.2 and 7.2 R⊕), orbiting a K dwarf extremely close to the 3:2 mean motion resonance. The two planets each show transits, sometimes simultaneously owing to their proximity to resonance and the alignment of conjunctions.

NASA quip: ‘Honey I Shrunk the Planetary System’

The Kepler-42 system as compared to the Jovian system [credit: NASA/JPL-Caltech]

The headline was NASA’s joke about both the size and the short orbit periods (all less than two days) of the three planets in the Kepler-42 system.

The discovery of this system dates back to 2012, but there don’t seem to be any numbers on resonant periods, so we’ll supply some now.

Wikipedia says:
‘Kepler-42, formerly known as KOI-961, is a red dwarf located in the constellation Cygnus and approximately 131 light years from the Sun. It has three known extrasolar planets, all of which are smaller than Earth in radius, and likely also in mass.’

‘On 10 January 2012, using the Kepler Space Telescope three transiting planets were discovered in orbit around Kepler-42. These planets’ radii range from approximately those of Mars to Venus. The Kepler-42 system is only the second known system containing planets of Earth’s radius or smaller (the first was the Kepler-20 system). These planets’ orbits are also compact, making the system (whose host star itself has a radius comparable to those of some hot Jupiters) resemble the moon systems of giant planets such as Jupiter or Saturn more than it does the Solar System.’

The three planets in order of distance from their star (nearest first) are c,b and d. They all have very short orbit periods ranging from under half a day to less than two days, and the star has only 13% of the power of our Sun.

A nearby binary star system with four planets

Pairs or multiple systems of stars which orbit their common center of mass. If we can measure and understand their orbital motion, we can estimate the stellar masses.

Relatively nearby, that is…

‘Upsilon Andromedae is located fairly close to the Solar System… (44 light years). Upsilon Andromedae A has an apparent magnitude of +4.09, making it visible to the naked eye even under moderately light-polluted skies, about 10 degrees east of the Andromeda Galaxy.’ – Wikipedia

The larger of the binary stars is ups_And A, which has 4 planets orbiting it: b,c,d and e.

The information on this star system was recently updated, so let’s have a look.

TESS discovers habitable zone planet in GJ 357 system

Credits: NASA’s Goddard Space Flight Center/Chris Smith

Following the report we analyse the orbital data for evidence of resonances.

A planet discovered by NASA’s TESS has pointed the way to additional worlds orbiting the same star, one of which is located in the star’s habitable zone, reports SciTechDaily.

If made of rock, this planet may be around twice Earth’s size.

The new worlds orbit a star named GJ 357, an M-type dwarf about one-third the Sun’s mass and size and about 40% cooler that our star. The system is located 31 light-years away in the constellation Hydra.

In February, TESS cameras caught the star dimming slightly every 3.9 days, revealing the presence of a transiting exoplanet — a world beyond our solar system — that passes across the face of its star during every orbit and briefly dims the star’s light.

Why Phi? – exoplanetary resonances of HD 40307

Moons of Pluto

This one may have slipped through the net, so to speak. The link to Pluto is explained below.

Star HD 40307 has six planets orbiting between 7 and 198 days, but here the focus will be on the outer three: e, f and g. These were reported in 2012 (whereas b, c, and d were found in 2008).

However, it seems the resonances described below have been overlooked, if lack of related internet search results can be relied on.

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.