## Why Phi: is the Moon a phi balloon? – part 2

Posted: November 9, 2018 by oldbrew in Astrophysics, moon, Phi
Tags: ,

Picking up from where we left off here

Three well-known aspects of lunar motion are:
Lunar declination – minimum and maximum degrees
Orbital parameters – perigee and apogee distances (from Earth)
Anomalistic month – minimum and maximum days

Standstill limits due to the lunar nodal cycle

‘The major standstill limit of the moon can be reached if the lunar node is near the vernal (or autumnal) point, and with the moon at its max. distance from the equator, equal to a declination at present days of 23.44° + 5.1454°= 28.59°.

The minor standstill limit of the moon can be reached if the lunar node is near the vernal (or autumnal) point, and with the moon at its min. distance from the equator, equal to a declination at present days of 23.44°- 5.1454° = 18.29°.’
http://iol.ie/~geniet/eng/moonperb.htm#nodes

28.59 / 18.29 = 1.5631492
4th root of 1.5631492 = 1.11815
This number leads to the key to the puzzle.

Note – 23.44° is the present day tilt of the Earth:
‘Earth’s obliquity oscillates between 22.1 and 24.5 degrees on a 41,000-year cycle; Earth’s mean obliquity is currently 23°26’12.6″ (or 23.43684°) and decreasing.’ – Wikipedia
So Earth is now near the mean of its min. and max. angles of tilt.

Perigee and Apogee

‘The Moon revolves around Earth in an elliptical orbit with a mean eccentricity of 0.0549. As a result, the Moon’s distance from Earth (center-to-center) varies with mean values of 363,396 km at perigee (closest) to 405,504 km at apogee (most distant).’
http://www.astropixels.com/ephemeris/moon/moonperap2001.html

405504 / 363396 = 1.1158735

Anomalistic month

‘The anomalistic month is defined as the time it takes the Moon to make one revolution around its orbit with respect to the perigee. The length of the mean anomalistic month as calculated for the year 2000 is 27.55455 days (27d 13h 18m 33s). However, the actual duration can vary by several days due to the gravitational perturbations of the Sun on the Moon’s eccentric orbit. […] The shortest anomalistic month is 24.629 days (2.925 days shorter than the mean) while the longest anomalistic month is 28.565 days (1.011 days longer than the mean).’ – astropixels.com

27.55455 / 24.629 = 1.1187847
(28.565 / 27.55455 = 1.0366708
1.0366708³ = 1.1140959)

Phi connection

Phi, or the golden ratio = (√5 + 1) / 2 = 1.618034
√5 / 2 = 1.118034 = Phi – 0.5

The lunar data results above all cluster round 1.118034, which is also the square root of 5/4.
Declination – minimum:geometric mean ratio is 5:4 approx, and g. mean:maximum ratio is 4:5 approx.
So the fourth root ratio mentioned earlier (standstill limits) is the result of two square roots.

1. oldbrew says:

Axial tilt: note that the obliquity varies only from about 22.0° to 24.5° – Berger 1976.
https://en.wikipedia.org/wiki/Axial_tilt#Long_term

24.5 / 22.0 = 1.113636 = 99.6% of 1.118034 (see post)
– – –
Using Fred Espenak’s lunar data for average current century apogee/perigee:
405408 / 362508 = 1.1183422
[Update: Horizontal parallax (H.P. = perigee HP/apogee HP) and Semi-Diameter (S.D.) give almost the same result as above]

2. oldbrew says:

The highest relative brightness at closest full moon perigee is given as:
1.301/1.163 = 1.1186586

See ‘Closest Full Moon Perigees (Super Moons)’ here:
http://www.astropixels.com/ephemeris/moon/fullperigee2001.html

This occurs next in 2034 (Nov. 25).

3. oldbrew says:

The ratio of the lunar draconic year to the Chandler wobble is about 1:1.1175² (just under 5:4).

As the ratio of 5:4 also relates to the standstills (see post*), maybe there’s a link there.

*Declination – minimum:geometric mean ratio is 5:4 approx, and g. mean:maximum ratio is 4:5 approx.

4. oldbrew says:

Thanks IW. I recalled your point that 5 Draconic years = ~4 Chandler wobbles.
Lunar Draconic Year (Yd) = 1/10th the orbital period of Jupiter / (Phi — 0.5)^2
…and J orbit = 10 Chandler wobbles, 2 J = 25 Draconic Years (both within a few days).

Re Jupiter, the ratio of J:nodal cycle is 1.1192^4 (= 1.5690335) :1.
That’s a 99.58% match to (5/4)² = 1.5625.

Anyway, these things all seem related to the variation of the tilt of the Earth, in number terms at least. Max./min. tilt = 1.114 approx.

5. tallbloke says:

I know we engineers use steam driven slide rules and the backs of fag packets rather than calculators, but is it worth mentioning that the Jupiter synodic period with the Earth-Moon system is 399 days, which is 1.09 years, which is around 98% of 1.118 years?

The beat period of 399 and 413 days is 32 years, which isn’t all that far off Ian’s 31 year lunar period and half the ~65 year ocean oscillation period.

6. tallbloke says:

Back in 2014 I said this to Ian

Ian: The question is why does it equal the evection oscillation period in ecliptic longitude.

As Paul shows, there is a link between the evection periods and the synodic and anomalistic months.
The 6 year period Paul derived necessarily fits with the contra-rotation of the lines of apse and nodes crossing each other every 3 years (Thanks Chaeremon).

The nature of evection from the farside link (which I gave Paul earlier too):
“Evection can be thought of as causing a slight reduction in the eccentricity of the lunar orbit around the times of the new moon and the full moon”
“Evection in latitude can be thought of as causing a slight increase in the inclination of the lunar orbit to the ecliptic at the times of the first and last quarter moons, and a slight decrease at the times of the new moon and the full moon.”

The 6 year period is three quarters of the Ea-Ve conjunction cycle, approximately half the Jupiter orbital period and approximately a third of the Jupiter-Saturn conjunction period. Coincidence? I doubt it. I think the Moon ‘fits in’ as best it can to the forces acting on it from other planets, and the evection represents the extent of its inability to balance all the forces. Something worth noting on that score is that the gravitational force felt by the Moon from Jupiter and Venus are very similar in magnitude. There has to be a reason for the squeezing and pulling that manifests itself in evection, and I suspect we might find it in JEV timings.

7. oldbrew says:

Let’s also mention that the ratio of the Jupiter-Saturn conjunction period to the mean solar Hale cycle (22.14 years) is 1.1145:1, very close to Earth’s min:max tilt ratio (> 99.9% match).

8. oldbrew says:

TB: ~6 years (2190.34 days) is the axial period of the lunar apsidal and nodal cycles, i.e. when the sum of their periods in proportion to that amount of time = 1.

The same period can be calculated using the full moon cycle and draconic year.
(FMC * DY) / (FMC – DY) = (LNC * LAC) / (LNC + LAC) = 2190.34 days
[Six tropical years = 2191.4531 days]

The small difference resolves after 1799 anomalistic years (= 300 * 2190.34 days) according to de Rop’s 1971 paper.
https://tallbloke.wordpress.com/2016/01/05/de-rops-long-term-lunar-cycle/

9. oldbrew says:

Re. The average of 14 x Synodic months = 413.4282446 days
and_______15 x anomalistic months = 413.318250 days
is approx Lunar Evective Cycle (LEC) = 413.3732473 days

7515 * 14 SM = 7516 LEC = 7517 * 15 AM = 7545 full moon cycles (503 * 15)
[1 LEC = 13 lunar evections]
7516 – 7515 = 1 (cycle?)

Tallbloke says: As Paul shows, there is a link between the evection periods and the synodic and anomalistic months.
– – –
7515 * 14 = 7014 * 15 SM [= 7515 * 14 SM]
No. of full moon cycles = AM – SM
Dividing totals by 15:
7517 AM – 7014 SM = 503 FMC

10. oldbrew says:

539 LEC = 539 * 413.3732473 days = 610 anomalistic years = 7545 SM (503 * 15)
539 * 14 = 7546 = 7545 (SM) + 1
(Dividing by 5: 1509 SM (503 * 3) = 122 anom. years)
– – –
The ratio of 539 LEC to 7516 LEC (previous comment) is the same as the ratio of synodic months to one full moon cycle.

11. oldbrew says:

tallbloke says: November 12, 2018 at 8:58 am

I know we engineers use steam driven slide rules and the backs of fag packets rather than calculators, but is it worth mentioning that the Jupiter synodic period with the Earth-Moon system is 399 days, which is 1.09 years, which is around 98% of 1.118 years?

Yes — 76 Jupiter-Earth conjunctions = 83 tropical years = 220/3 lunar evection cycles.
So 55 LEC (1/4 of 220) = 57 J-E (3/4 of 76) = 62.25 TY (3/4 of 83).

As 57 – 55 = 2, it could be said there’s a half-period of 62.25 TY / 2 = 31.125 TY.
Of course these periods are close to the 31/62 year lunar cycles in Ian Wilson’s studies.

12. oldbrew says:

Wiki: Matching synodic and anomalistic months

When tracking by counting cycles of 14 synodic months, a correction of 1 synodic month should take place after 18 cycles:

18×FC = 251×SM = 269×AM, not:
18×14 = 252×SM

The equality of 269 anomalistic months to 251 synodic months was already known to Chaldean astronomers

http://www.thefullwiki.org/Full_moon_cycle
– – –
The number of lunar evections in this formula would be 233 (13 * 18, -1):
233 LE + 18 = 251 SM (14 * 18, -1)
251 SM + 18 = 269 AM (15 * 18, -1) = 18 full moon cycles

233 is a Fibonacci number.
– – –
The ratio of lunar evection to solar rotation is about 1:1.119² (see earlier comments re this ratio).

Update – see: Why Phi? – a lunar evection model [16 Nov. 2018]
https://tallbloke.wordpress.com/2018/11/16/why-phi-a-lunar-evection-model/

13. oldbrew says: 