Apogee = position furthest away from Earth. Earth. Perihelion = position closest to the sun. Moon. Perigee = position closest to Earth. Sun. Aphelion = position furthest away from the sun. (Eccentricities greatly exaggerated!)
Lunar evection has been described as the solar perturbation of the lunar orbit.
One lunar evection is the beat period of the synodic month and the full moon cycle. The result is that it should average about 31.811938 days (45809.19 minutes).
Comparing synodic months (SM), anomalistic months (AM), and lunar evections (LE) with the full moon cycle (FMC) we find:
1 FMC = 13.944335 SM
1 FMC = 13.944335 + 1 = 14.944335 AM
1 FMC = 13.944335 – 1 = 12.944335 LE
Since 0.944335 * 18 = 16.9983 = 99.99% of 17, and 18 – 17 = 1, we can say for our model:
18 FMC = 233 LE (18*13, -1) = 251 SM (18*14, -1) = 269 AM (18*15, -1)
See: 3 – Matching synodic and anomalistic months.
Noting that 233 is a Fibonacci number, if we multiply by 8:
144 FMC (18*8) = 233*8 LE (for model purposes)
8, 144 and 233 are all Fibonacci (F) numbers, and 144 and 233 are consecutive F numbers i.e.:
233 / 144 = Phi
Therefore: FMC / 8 LE = Phi, or the golden ratio
Cross-check: 12.944335 / 8 = 1.6180418 (Phi = 1.618034)
This means that for any Fibonacci number (F1) of full moon cycles, the number of lunar evections in that period will be F2 * 8, where F2 is the next number in the Fibonacci series. The accuracy of the result increases as the numbers increase, which is the nature of the Fibonacci series.
An example would be the period of 55 full moon cycles, which is close to both 62 tropical years and 7 lunar apsidal cycles (55 + 7 = 62). Using F1 = F2 * 8:
55 FMC = 22648.143 days
The next Fibonacci number after 55 is 89.
89 * 8 LE = 22650.1 days
Accuracy of match is > 99.991%.
Update: 12.944335 / 8 = 1.6180418 = Phi
12.944335 / (12.944335 – 8) = 2.6180133 = Phi²
This is self-similarity, a feature of the golden ratio and fractals.
12.944335 * (12.944335 – 8) = 8²
Therefore 8 is the geometric mean of the two sides of the equation.
In this graphic, 8 plugs into a and (12.944335 – 8) plugs into b, so a + b = 12.944335
[Wikipedia – Line segments in the golden ratio: The golden ratio (phi) represented as a line divided into two segments a and b, such that the entire line is to the longer a segment as the a segment is to the shorter b segment.]
Related: Predicting the Start of the Next El Niño Event — by Ian Wilson.
oldbrew,
This is a brilliant discovery! Well done!!
The questions now become:
a) What is so special about 8 lunar evection cycles (i.e. 254.495529 days)?
B) What is 8*LE physically related to?
It turns out that
(8*LE) / SM =(8 + (1/phi*))
__________= 8 + 0.6180309
__________= 8.6180309 (actual value of 8 + (1/phi) = 8.6180339)
If you use SM = Synodic month = 29.5305889 days and LE = 31.8119411 days.
Note:
[A check of the above formula between 8*LE and SM can be done by substituting in the definition of LE = (FMC*SM) / (FMC – SM)___which, after a bit of manipulation gives you FMC / 8*LE = phi ]
Substituting:
8*LE = (8 + 1/phi) SM into
FMC / (8*LE) = phi
gives us:
FMC / SM = (8*phi) + 1
If you use:
FMC = 411.78443025 days
SM = 29.530588853 days
you get a tentative phi* = 1.618042 which is not that far off real phi = 1.6180339
I don’t know if this kicks the can any further down the road but it is a start.
oldbrew,
A minor point. Another way of writing___________8 * LE = (8 + (1/phi)) SM
is_______________________________________8 * phi * LE = ((8 * phi) + 1) * SM
And just to throw a stick into a hornet’s nest –
You can show that (using either days or years as your units):
18 LEC – 18 FMC = SM
and
18 LEC – 18 = 2 * CW = QBO
where
SM = Synodic month = 29.5305889 days
LEC = 14 x SM = 413.4282446 days = Lunar Evection Cycle
CW = The Chandler Wobble
QBO = The Quisi-Biennial Oscillation
Note: I have a slightly different definition of the lunar evective month (LE) which uses 14 * SM
instead of the FMC to define the evective month.
The Lunar Evection Cycle (LEC = 14 SynM = 413.4282446 days = 1.13189 sidereal years) is the time it takes for the lunar evective month to re-align with the lunar synodic month.
This realignment effectively takes place when 13 LE = 14 SM, which sets the length of the LE of (14/13) x SM = 31.802173 days.
The length of the LEC is determined by the beat period between the EM and the synodic month
(SynM),
(𝟏 / 𝐒𝐌) − (𝟏 / LE) = (𝟏 / 𝐋𝐄𝐂)
I am not sure which definition is correct ?!?
I had a link that quoted it in minutes, which was the same number I found. I’ll dig it out…here it is:
General relativity theory explains the Shnoll effect and makes possible forecasting earthquakes and weather cataclysms
D Rabounski, L Borissova – Progress in Physics, 2014 – books.google.com
• 23 hours, 56 min= 1436 min (stellar day); • 365 days, 6 hours, 9 min= 525969 min (stellar year);
• 24 hours, 50 min= 1490 min (lunar day); • 27 days, 7 hours, 43 min= 39343 min (lunar month);
• 31 days, 19 hours, 29 min= 45809 min (period of the lunar evection)
– – –
It’s the first item if you put ‘lunar evection’ in the Google scholar search box.
Re.: you get a tentative phi* = 1.618042 which is not that far off real phi = 1.6180339
Yes, 1.618042 * 8 = 12.944336 (see post). Doing that as a division (12.944335 / 8) was when the penny really dropped for me!
– – –
Re.: What is so special about 8 lunar evection cycles (i.e. 254.495529 days)?
Don’t know, apart from being 1/Phi of a full moon cycle – which is itself phi-related (e.g. 55/7 is 3 times 55/21 = Phi²).
Update: which makes a lunar apsidal cycle around Phi³ * 3 * 8 LE.
There are three types of lunar libration:
1- Libration in longitude is a consequence of the Moon’s orbit around Earth being somewhat eccentric, so that the Moon’s rotation sometimes leads and sometimes lags its orbital position.
2- Libration in latitude is a consequence of the Moon’s axis of rotation being slightly inclined to the normal to the plane of its orbit around Earth. Its origin is analogous to the way in which the seasons arise from Earth’s revolution about the Sun. Also significant is the fact that the Moon’s orbit is inclined to the plane of the ecliptic by a little more than 5°. As it is the Sun which illuminates the Moon – and both the Sun and the Earth are always located in the plane of the ecliptic – the Moon is sometimes illuminated from above and sometimes from below, allowing us to see some of the lunar surface beyond the poles.
3- Diurnal libration is a small daily oscillation due to the Earth’s rotation, which carries an observer first to one side and then to the other side of the straight line joining Earth’s center to the Moon’s center, allowing the observer to look first around one side of the Moon and then around the other. This is because the observer is on the surface of the Earth, not at its centre.
What’s 0.99335??
Sorry. What is the 0.944335 appearing above?
wert – it’s just the remainder after subtracting the whole number.
Given that (when units are expressed in years):
18 LEC – 18 FMC = SM____________(1)
and
18 LEC – 18 = 2 * CW = QBO________(2)
where
SM = Synodic month = 29.5305889 days
LEC = 14 x SM = 413.4282446 days = Lunar Evection Cycle
CW = The Chandler Wobble
QBO = The Quisi-Biennial Oscillation
FMC = 411.78443025 days
(1) means that (when units are expresed in years):
18 * (14 * SM) – 18 FMC = SM or 18 FMC = 251 SM
251 SM = 7412.177802 days
18 FMC = 7412.119745 days
error = 0.0581 days = 1.39 hours.
(2) means that (when units are expressed in years):
252 SM – 18 = 2 * CW = QBO
18 FMC + SM – 18 = 2 * CW = QBO
18 FMC – 18 = 2 * CW – SM = QBO – SM
And how about this little gem?
https://astroclimateconnection.blogspot.com/search?updated-max=2017-04-20T21:45:00-07:00&max-results=5&start=10&by-date=false
This should be a little better!
Using astroclimatelink’s figures, LEC in days and FMC in days, 266.05~ FMC = 1 + 265.00000~ LEC = 44 Inex – 54 Saros.
The full moon cycle has the same ratio to the apsidal cycle as the QBO does to the nodal cycle, i.e. about (3 * Phi²):1.
https://tallbloke.wordpress.com/2017/01/08/why-phi-a-lunar-ratios-model/
Re: 251 SM = 7412.177802 days
18 FMC = 7412.119745 days
error = 0.0581 days = 1.39 hours
[Reply] The ‘error’ is because: 0.944335 * 18 = 16.9983 = 99.99% of 17 [see post]
So the 233 LE: 251 SM: 269 AM all need the same slight correction (250.9983 etc.)
The 18 FMC still applies as the difference between the numbers is unchanged.
– – –
Re. Inex:
1 Inex = 28.945 tropical years (358 synodic months)
28.945 / 18 = 1.6080555 TY (99.383% of Phi)
[Reply] That’s because 18 and 29 are Lucas numbers, which are closely related to Fibonacci numbers. (Lunar evections per full moon cycle came in just below 13 = Fib. no.).
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/lucasNbs.html
521 is a Lucas number.
521 / 18 = 28.9444(4 recurring), marginally less than 1 Inex.
Re: The questions now become:
a) What is so special about 8 lunar evection cycles (i.e. 254.495529 days)?
[Reply] Maybe it’s just self-similarity:
FMC / 254.495529 days = 1.6180418 = Phi
FMC – 254.495529 days = 157.28891 days
254.495529 days / 157.28891 days = 1.6180131 = Phi = 8 / (12.944335 – 8)
And finally: 12.944335 / (12.944335 – 8) = 2.6180133 = Phi²
oldbrew,
You are correct in pointing out that the evective month is defined in terms of the beat period between the synodic (phase) month and the FMC.
Lunar evection is the change in the eccentricity (i.e. egg shape) of the lunar orbit caused by the fact that, between Full and New moon, the Moon is falling towards the Sun, and so its orbital speed is increasing (i.e moving faster and faster than its normal speed),
and between Full and New moon,and between New and Full moon, it is climbing out of the Sun’s gravitational pull, and so its orbital speed is decreasing (i.e. moving slower and slower than its normal speed). This affects the shape of the lunar orbit because a faster orbital speed means that the Moon moves slightly closer to the Earth, and a slower orbital speed means that it moves slightly further away.When the Perigee of the lunar orbit points towards the Sun (i.e. the pointy end of the egg-shaped lunar orbit is directly between the Earth and the Sun), the lunar orbit is pushed in most just prior to New Moon and pulled out the most just prior to Full Moon. Hence, an alignment of the Perigee of the lunar orbit with syzygy (i.e. an alignment of the Earth, Moon, and Sun at New Moon) produces a stretching of the egg shape of the lunar orbit or increase in lunar orbital eccentricity.
[typo corrected: “, and between New and Full moon”]
oldbrew,
You are correct in pointing out that the evective month is defined in terms of the beat period between the synodic (phase) month and the FMC.
Lunar evection is the change in the eccentricity (i.e. egg shape) of the lunar orbit caused by the fact that, between Full and New moon, the Moon is falling towards the Sun, and so its orbital speed is increasing (i.e moving faster and faster than its normal speed), and between New and Full moon, it is climbing out of the Sun’s gravitational pull, and so its orbital speed is decreasing (i.e. moving slower and slower than its normal speed). This affects the shape of the lunar orbit because a faster orbital speed means that the Moon moves slightly closer to the Earth, and a slower orbital speed means that it moves slightly further away.
When the Perigee of the lunar orbit points towards the Sun (i.e. the pointy end of the egg-shaped lunar orbit is directly between the Earth and the Sun), the lunar orbit is pushed in most just prior to New Moon and pulled out the most just prior to Full Moon. Hence, an alignment of the Perigee of the lunar orbit with syzygy (i.e. an alignment of the Earth, Moon, and Sun at New Moon) produces a stretching of the egg shape of the lunar orbit or increase in lunar orbital eccentricity.
The ratio of the lunar evection to solar rotation is about 4:5.
LE * 4/5 = 25.45 days
Average solar rotation = 25.38 days
https://svs.gsfc.nasa.gov/4623
– – –
Note: the axial period of the Metonic cycle and the lunar rotation/orbit period is 25.42~ days.
(25.42d / 19 Trop. Yrs.) + (25.42d / 27.321582d) = 1
Metonic = 19 TY = 254 lunar months = 273 (254 + 19) axials.
273 = 21 * 13 (Fibonacci numbers).
I’m so drunk, just had surgery on some troublesome tooth, but the headache I got from reading this has neutralised the pain. Enough to have a chat.
There is a ratio connection between orbital bodies, I think the subject is very important, I have already posted here a possible ratio for a tenth planet, (awhile back) if we can check these ratios I provided and relate them into time and space coordinates for orbital bodies, we will have somewhere to look, it will be a step forward.
Like I said, this subject is important, Neptune was discovered in a similar way…
I have some issues with the math provided, It does not account for time and distance variations, although saying that, it is probable as in likely that the lunar orbit will strike each beat on its orbit than not.
btw can you please stop being satirical with your post headlines, no one gets it, it’s a very complex subject that maybe 10 people on the planet would understand, so the “inside joke” is basically falling on deaf ears and comes across as weird.
lol I’m going to shut up now. good luck 😉
baz – it’s true that we use mean values in the calcs. Anomalistic month for example can vary by +/- 2-3 days.
https://eclipse.gsfc.nasa.gov/SEhelp/moonorbit.html#anomalistic
Re: B) What is 8*LE physically related to?
[Reply] 12.944335 * (12.944335 – 8) = 8²
Therefore 8 is the geometric mean of the two sides of the equation [see update to the post re. self-similarity].
https://en.wikipedia.org/wiki/Geometric_mean
– – –
Re: It turns out that
(8*LE) / SM =(8 + (1/phi*))
__________= 8 + 0.6180309
__________= 8.6180309 (actual value of 8 + (1/phi) = 8.6180339)
[Reply] Yes, it must do because in one full moon cycle no. of SM = no. of LE, + 1.
So the corresponding difference in 0.618~ of one FMC has to be 0.618~ of one SM.
Perigee/Apogee. These are the points when the Moon, due to its non-circular orbit, is closest and furthest (respectively) to the Earth. The Moon moves at its greatest speed when it is at perigee and at its slowest when furthest from the Earth at apogee. The gravitational pull of the Moon is much stronger at perigee than at apogee.
The apogee/perigee points (the line of apsides that connects them) are not fixed along the ecliptic, but move slowly forward along the ecliptic over a nine year period.
Lunar Speed. In addition, this line of apsides also fluctuates backwards and forwards in the ecliptic slightly with a period of 31.81 days. This is due to the eccentricity of the Moon’s orbit, and this fluctuation is called evection. The resulting effect is that the Moon speeds up and slows down at different rates in the four weeks from one perigee to the next. [bold added]
http://www.astrologysoftware.com/resources/articles/lphenom.asp
This 31.81 days period matches our result. It may well be that it’s ‘due to the eccentricity of the Moon’s orbit’, but it’s the beat period of the full moon cycle and the synodic month [see blog post].
For a basic lunar model, we could have (see chart below re. abbreviations):
828 AM = 766 SM = 711 LE (~99.14% true)
Then:
828 AM – 766 SM = 62 TY
766 SM – 711 LE = 55 FMC
62 TY – 55 FMC = 7 LAC (~99.986% true)
But the only correct numbers here are 7 LAC and 766 SM.
All the others need a small adjustment, which was mostly explained using this chart from another post: Lunar precession update
https://tallbloke.wordpress.com/2017/10/15/lunar-precession-update/
Here the numbers are all 7 short of a multiple of 104, except LAC and SM (see link above for details).
Lunar evection is not included, but the number would be 73951, which is SM – FMC (79664 – 5713).
73951 = 104 * 711, plus 7 — so LE is 7 over, rather than 7 short like the others.
This is logical because if FMC were not 7 short (of 5720 = 104 * 55), LE would have to be 7 less.
NB the total period is a quarter of the Earth’s axial precession i.e. 90 degrees.
To put it another way, for every 104 (13*8) lunar apsidal cycles:
1 lunar evection is ‘gained’ at the expense of 1 each of full moon cycles, anomalistic months, tropical months and tropical years which are ‘lost’.
The anomalistic month is the beat period of lunar evection in longitude (= 31.81194 days) and half the synodic month (29.5305889 days / 2).
The draconic month is the beat period of lunar evection in latitude (= 32.280776 days) and half the synodic month (29.5305889 days / 2).
A new post will explain this in more detail [link below].
[…] oldbrew on Why Phi? – a lunar evect… […]
Re. What is so special about 8 lunar evection cycles (i.e. 254.495529 days)?
You need 233 of those (i.e. 8 LE) to equal 144 (18*8) full moon cycles.
8, 144 and 233 are Fibonacci numbers.
233/144 = Phi
From Wikipedia:
(22639.55 + 4586.45) / (22639.55 – 4586.45) = 3/2 (99.46% true)
Actual ratio = 279:185 (279:186 = 3:2)
The same result can be obtained when comparing the mean Jupiter-Uranus conjunction period (13.812904 years) to the solar inertial motion cycle aka José cycle (~178.8 years).
Just substitute J-U for the lunar evection and the José cycle for the full moon cycle.
Then 144 Jose (SIM) cycles = 233 * 8 J-U (> 99.991% true).
8 J-U = 110.50323 years
110.50323 * Phi = 178.79798 years
8 J-U is the geometric mean of 178.79798y and (178.79798 – 110.50323) y.
That means the ratio of 8 J-U to the whole cycle = the ratio of the cycle remainder to 8 J-U (both 1:~Phi).
– – –
Footnote: 144 José cycles is only about 25 years short of the Earth’s current axial precession period of ~25,770 years. The period is known to be declining in the current epoch.
===
Related: see the paper by Gerry Pease (link here)…
https://tallbloke.wordpress.com/2016/12/01/gerry-pease-links-improved-and-updated-solar-planetary-paper/