Archive for the ‘Maths’ Category


Encylopaedia Britannica on the Metonic cycle:

Metonic cycle, in chronology, a period of 19 years in which there are 235 lunations, or synodic months, after which the Moon’s phases recur on the same days of the solar year, or year of the seasons. The cycle was discovered by Meton (fl. 432 bc), an Athenian astronomer.

Calendar Wiki’s opening paragraphs on the Metonic cycle say:

The Metonic cycle or Enneadecaeteris in astronomy and calendar studies is a particular approximate common multiple of the year (specifically, the seasonal i.e. tropical year) and the synodic month. Nineteen tropical years differ from 235 synodic months by about 2 hours. The Metonic cycle’s error is one full day every 219 years, or 12.4 parts per million.

19 tropical years = 6939.602 days
235 synodic months = 6939.688 days

It is helpful to recognize that this is an approximation of reality.

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When Pi is not 3.14 – by PBS America

Posted: March 28, 2020 by oldbrew in Maths

A simple practical demo here…

Wikipedia explains: The arc length of the cycloid.

[Credit: Zorgit @ Wikipedia]

Taxicab geometry – Wikipedia


Wikipedia says:

Dansgaard–Oeschger events (often abbreviated D–O events) are rapid climate fluctuations that occurred 25 times during the last glacial period. Some scientists say that the events occur quasi-periodically with a recurrence time being a multiple of 1,470 years, but this is debated. —

The 25 occurrences of 1470 years are represented in this synodic chart posted in the comments of our 2018 blog post:
Possible origin of Dansgaard-Oeschger abrupt climate events.

Re. the ‘debate’, let’s take a line from this paper:
On the 1470-year pacing of Dansgaard-Oeschger warm events
Michael Schulz
First published: 01 May 2002
Citations: 99
‘a fundamental pacing period of ~1470 years seems to control the timing of the onset of the Dansgaard-Oeschger events.’

Another study: Timing of abrupt climate change: A precise clock
Stefan Rahmstorf
First published: 21 May 2003

An analysis of the GISP2 ice core record from Greenland reveals that abrupt climate events appear to be paced by a 1,470-year cycle with a period that is probably stable to within a few percent; with 95% confidence the period is maintained to better than 12% over at least 23 cycles. This highly precise clock points to an origin outside the Earth system; oscillatory modes within the Earth system can be expected to be far more irregular in period.

[bold added]

However, researchers often admit defeat when looking for a viable mechanism to explain its regularity, or just say there isn’t one to date.

Kepler’s trigon – the orientation of consecutive Jupiter-Saturn synodic periods, showing the repeating triangular shape (trigon).


Returning to the synodics chart, a relevant number doesn’t appear in it. The Jupiter-Saturn conjunction of 19.865~ years is an important period in the solar system, and it returns to almost the same position after every three occurrences, as Johannes Kepler noted with his ‘trigon’, centuries ago.

We can work out the rate of movement per conjunction in degrees:
360 – ((360 / S) * J-S) = 117.147 degrees
(360 / 117.147) * J-S = 61.046482y (‘JS-360’)
[Data: https://ssd.jpl.nasa.gov/?planet_phys_par ]

Then, from the chart:
1470*25 / ‘JS-360’ = 602.00029
Check: (602*360) / 117.147 = 1849.983 (1850 J-S, see chart)
Since ‘JS-360’ is almost exactly a whole number (602), the Jupiter-Saturn conjunction should be in its original position at the end of the 25 D-O cycles.

Adding 602 to the orbits of each planet = multiples of 25:
223(N) + 602 = 825 (25*33) = 1850-1025(S-N)
[33 = 74-41]
1248(S) + 602 = 1850 (25*74)
3098(J) + 602 = 3700 (25*74*2)

Another way to get multiples of 25:
Add 2 to each orbit number (see chart), and subtract 2 from 602.

More on the 602 number:
602 = 14*43
14*61.046482y = 854.651y
43 J-S = 854.197y
These two results are only about half a year apart, and we find:
43*43 = 1849 J-S
Add 1 = 1850 J-S completing the 25 D-O cycle.

43*61.046482y = 2625 years (2624.9987)
1470:2625 = 14:25 ratio
1470*25 = 2625*14 (hence 602 of ‘JS-360’ = 14*43)

Obliquity note:
28 D-O = 41160 years, a fair match to the expected 41 kyr period.
One paper refers to a fit between D-O and obliquity.
Others support the notion of a link — possibly a topic for another post.
(28*25*1470 = 1,029,000 years)

Example of a 1470 year period from Arnholm’s solar simulator — click on image to enlarge:

Showing Neptune, Jupiter, Saturn and Earth.
* * *
Another one — Jupiter, Neptune, Saturn

Kepler-90 Planets Orbit Close to Their Star [credit: NASA/AMES]


In part 1 we looked at the inner four planets: b,c,i and d. Here in part 2 we’ll look at the outer four: e,f,g and h – with a dash of d included.

The largest planet in the system is h, the outermost of the eight so far found, and it’s about the same size as Jupiter. It’s ‘an exoplanet orbiting within the habitable zone of the early G-type main sequence star Kepler-90’, says Wikipedia. However, ‘it is a gas giant with no solid surface’, so probably no aliens lurking there.

It wasn’t that easy to find synodic patterns of interest, but here we have two examples, both involving planet h.

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Golden rectangle: Fibonacci spiral


Unusually, the eight planets in the Kepler-90 system were found using machine learning. “It’s very possible that Kepler-90 has even more planets that we don’t know about yet,” NASA astronomer Andrew Vanderburg said.
– – –
From Wikipedia’s Near resonances section on exoplanet Kepler-90:

“Kepler-90’s eight known planets all have periods that are close to being in integer ratio relationships with other planets’ periods; that is, they are close to being in orbital resonance.

The period ratios b:c, c:i and i:d are close to 4:5, 3:5 and 1:4, respectively (4: 4.977, 3: 4.97 and 1: 4.13) and d, e, f, g and h are close to a 2:3:4:7:11 period ratio (2: 3.078: 4.182: 7.051: 11.102; also 7: 11.021).

f, g and h are also close to a 3:5:8 period ratio (3: 5.058: 7.964). Relevant to systems like this and that of Kepler-36, calculations suggest that the presence of an outer gas giant planet facilitates the formation of closely packed resonances among inner super-Earths.”
– – –
Let’s look at it another way i.e. at the synodic periods rather than the orbit ratios, as these tend to deliver more clear-cut results, starting with a model for the first four planets: b,c,i and d, which we’ll call the inner planets. Their orbits of the star are in a range of 7-60 days.

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Jupiter-Saturn-Earth orbits chart


This was just about to go live when a new idea involving the Sun cropped up, now added to the original. The source data is from NASA JPL as usual.

From our 2015 de Vries post we saw that the 2503 year period, which the numbers were based on, consisted of 85 Saturn and 211 Jupiter orbits [see chart on the right].

Taking Saturn’s orbit period, and using JPL’s planetary data we find:
10755.7 days * 85 = 914234.5 days

The lunar year is 13 lunar orbits of Earth:
27.321582 days * 13 = 355.18056 days

914234.5 / 355.18056 = 2573.9992 (2574) = 13 * 198 lunar years

Number of beats of Saturn and the lunar year = 2574 – 85 = 2489 in 2503 years.
2503 – 2489 = 14
Number of Jose cycles in 2503 years = 14 (= 126 Jupiter-Saturn conjunctions, i.e. 9 J-S * 14).

Therefore the difference per Jose cycle between ‘Saturn-lunar year’ beats and Earth years is exactly one.

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Poster from the NASA Exoplanets Exploration Program’s Exoplanet Travel Bureau [credit: NASA/JPL-CalTech]


Before we start – ‘Pulsar planets are planets that are found orbiting pulsars, or rapidly rotating neutron stars.’

Wikipedia tells us:
‘PSR B1257+12, previously designated PSR 1257+12, […] is a pulsar located 2,300 light-years from the Sun in the constellation of Virgo. It is also named Lich, after a powerful, fictional undead creature of the same name.

The pulsar has a planetary system with three known planets, named “Draugr” (PSR B1257+12 b or PSR B1257+12 A), “Poltergeist” (PSR B1257+12 c, or PSR B1257+12 B) and “Phobetor” (PSR B1257+12 d, or PSR B1257+12 C), respectively.

They were both the first extrasolar planets and the first pulsar planets to be discovered; B and C in 1992 and A in 1994.

A is the lowest-mass planet yet discovered by any observational technique, with somewhat less than twice the mass of Earth’s moon.’

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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)

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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.

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Image credit: naturalnavigator.com


The contention here is that in the time taken for 14 lunar nodal cycles, the difference between the number of Saros eclipse cycles and lunar apsidal cycles (i.e the number of ‘beats’ of those two periods) is exactly 15.

Since 15-14 = 1, this period of 260.585 tropical years might itself be considered a cycle. It is just over 9 Inex eclipse cycles (260.5 years) of 358 synodic months each, by definition.

Although it’s hard to find references to ~260 years as a possible climate and/or planetary period, there are a few for the half period i.e. 130 years, for example here.

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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.

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Image credit: interactivestars.com


In 2015 this post discussed long-term lunar precession from an apsidal, or anomalistic, standpoint.

We saw that all the numbers related to an exact number (339) of Metonic cycles (19 tropical years each, as discussed below).

Here we show the equivalent from a nodal, or draconic, standpoint.

Again, all the numbers relate to an exact number (337 this time) of Metonic cycles.

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Credit: NASA’s Goddard Space Flight Center


First the report, then a brief Talkshop analysis.

NASA’s Transiting Exoplanet Survey Satellite (TESS) has discovered a world between the sizes of Mars and Earth orbiting a bright, cool, nearby star, reports MessageToEagle.com.

The planet, called L 98-59b, marks the tiniest discovered by TESS to date.

Two other worlds orbit the same star.

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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.

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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).

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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).

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Lunar evections and the Saros cycle

Posted: May 7, 2019 by oldbrew in Maths, moon, solar system dynamics
Tags:

Credit: Matthew Zimmerman @ English Wikipedia


The Saros cycle can be used to predict eclipses of the Sun and Moon, and is usually defined as 223 lunar synodic months, or about 11 days over 18 years.

But there are a few other lunar-related periods which can used to arrive at 223.

One Saros cycle can be said to be the difference between the number of:
— anomalistic months and full moon cycles (239 – 16)
— draconic months and draconic years (242 – 19)
— tropical months and tropical years (241 – 18)

That may be fairly well known, but then there are the lunar evections.

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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.

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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.

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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.

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