Action of the Moon

Moon spend more than two days (in average L²/(E-L), i.e.. 2.209 days) to catch up length, that was meantime made by the Earth. (This period of lunar phases differs from the third of the mean orbital period of Mercury only by several hours M/3 =29.323 days). Ø 29.530588 days: synodic moon (E,L), so called "lunation", change of phases (new moon, full moon)

• 29.530588 days: synodic moon (E,L), so called "lunation", change of phases (new moon, full moon)

If a year divides to 12 equal months, then one month has c. 30.5 days. Approximately the same period is observed in beats (1:4) of orbital periods of Martian satellites Phobos (0.31891 days) and Deimos (1.26244 days).

• 30.43685 days: E_t/12 = 365.2422/12 (tropical)
• 30.43803 days: E/12 = 365.25636/12 (sidereal)
• 30.50036 days: (Phobos,Deimos/4) = (0.31891,1.26244/4)

Sometimes it appears to be more favourable to divide year to 13 months. Orbital period of Venus is approximately 8/13 of our year. For observer on the Earth, every (V,E), i.e. c. 583.92 days, Venus passes Earth (in direction of Earth motion). Moon passes this place (against Venus direction) every c. 29.53 days. So, with regard to Venus, orbital period of Moon is little bit shorter.

• In space of synodical periods we have: ((E,L),-(V,E)) = (29.53, 583.92) = 28.11 days.
• Resonant period of atmospheric oscillation is c. 10.5 hours (Feynman). Let us note, that (2*10.5, 24.0) = 168 hour = 7 days (one week).
• Cycle of 28-day was observed in solar corona (Archenhold, 1938).
• One of the Mayan calendars (Tun-Uc) was sequence of 28 day months (natural months). Computationl year or bat cycle (Tzotz tun) was made of thirteen of such months: 364 days = 4∙91 days = 13∙28 days.

• 31.80 days: evection of Lunar motion (31.81)

Cycle of evection 1.132 years = 413.46 days = 2*206.73 days. (E,Y) = (1.00,8.5413) = 1.1326 years.

Period of solar eclipses 18.03 years.

• 18 years: Norwegian icebergs (Rekstad, 1924)
• 18.0 years: (J/2, 8.85 years)
• 18.03 years: (6585 days) Saros, 223 lunations
• 18.0303 years: Lunar tidal cycle  18.028292-18.031030 years (Wood)
• 18.031 years: saros (Moon eclipses) (6585.33 d, 9483583 min)
• 18.04 years: 16*1.1274 years; 3*6.01 years
• 18.058 years: (5*(E,Moon),J/30) = (147.653, 144.42) = 5*3.6116 years
• 18.14 years: (S,U)*2/5
• 18.02874743: saros

Large tidal floods 1959.991, 1978.021.

Nodal line of lunar orbit (intersection of orbital plains of Moon and Earth) moves against Earth motion (with period c. NL=-18.6 years).

• 4.7448 years: 2J/5

• 9.27 years: abundance, river freeze cycle, U-mg values
• 9.275 years: cycle of solar motion (Landscheidt)

• 18.31 years: N/9, 4/S-16/U
• 18.33 years: tree rings (Illinois)
• 18.506 years: 26 tzolkin = 26*0.71177 years
• 18.56 years: (E,J)*17=6.549*17/6=17*1.092 years (18.53 years: J*25/16)

• 18.595 years: ((J,S),lunar modulation) = (19.855 y, 586 y)
• 18.59953 years: (Julian years) sidereal revolution of the lunar nodes (6793.477 d)
• 18.604 years: nutation (Fairbridge,1997)
• 18.6040 years: Moon synergic cycle (Wood)
• 18.605 years: retrograd motion of Moon nodes (19.35 st/year), rotation of the E changes, nutatopn of E

(18.605,19.85931) = 294.57 y = 883.71 /3 y; (18.605/2, 8.8743)= 192.79 y

• 18.61 y (18.6-18.7 y) nutace, (19.86,18.69) -> 318 years
• 18.611 years: ((S/5,J/2)/2, 3*J/2)
• 18.6134 years: (18.05694-18.06574) the Lunar nodal cycle (Fairbridge,1997)

31 year period of eclipses.

• 31.007 years (3*31.007≈5*18.61 years) Combination of Saros and Evection.
• 31 years: Moon – variation of perigee.
• 31.008 years: the Lunar perigee-syzygy cycle (Fairbridge, Sanders, Wood)

Mayan period of eclipses - 46 tzolkin.

46∙z= 11960 days (32.7447 years). Corresponds to 405 lunations (405∙29.5306 days).

It reminds Douglass period of tree rings (32.8 years). Period of beats (R/3,V) = 32.82 years.
Three periods of Jupiter modulated by Babylonian period gives: [3∙J, B]= [35.586,427.1]= 32.85 years.

Examples of solar eclipses in the Middle and the South America:

All data in Mayan tables of eclipses do not need to coincide with the solar or lunar eclipses (in the given area). Mayan could register Moon near nodes (i.e. intersections of lunar orbit with ecliptic) in the time of main phases of the Moon (full moon, new moon).

E.g. Moon near nodes (accuracy 30°):

    Difference   Mat.date   Date
    ( 176.60000) (1997.75) 1997 Sep 28 AD
    ( 178.10000) (1998.24) 1998 Mar 26 AD
    ( 176.00000) (1998.72) 1998 Sep 18 AD
    ( 148.80000) (1999.13) 1999 Feb 13 AD
    ( 176.40000) (1999.61) 1999 Aug  9 AD

    Difference   Mat.date   Date
    (1033.90000) (1991.53) 1991 Jul 11 AD
    ( 176.90000) (1992.02) 1992 Jan  4 AD
    ( 177.80000) (1992.50) 1992 Jun 29 AD
    (1033.00000) (1995.33) 1995 Apr 28 AD
    ( 177.60000) (1995.82) 1995 Oct 23 AD
    (1211.10000) (1999.13) 1999 Feb 15 AD
    ( 176.20000) (1999.62) 1999 Aug 10 AD

• 11 years Tritos – eclipse
• 29 years Inex – eclipse

Let us consider motion with period P, having synodic period related to Merkury the same as synodic period of Venus related to Earth, i.e. it has to hold:

 (M,P) = (V,E)

i.e. [P,V] = [M,E], resp. 1/P = 1/M-1/V+1/E.

Hence P = 103.55 days = 207.10 days/2. Beats (205.9, 207.1) makes c. 35540 days (97.3 years).

"Semestral year" makes period of 12 lunations (two semesters). Mercury completes 4 orbits around the Sun during this period.

• 351.86 days: 4*87.965 days;
• 354.36706 days: 12*(L,E) = 12*29.530588 days

• 8.826 years: 4/9*(J,S)
• 8.85 years: Lunar Perigee (apsid line), (c. 3232 d) (27.32154,27.55455) = 3230.9 d
• 8.85 years: 3231.5 d (tropical), 3232.61 d (sidereal)
• 8.8743 years: Lunar Perigee (40.567 dg/y)(?)

Period of rotation of lunar perigee is approximately equal to quarter of Bruckner’s period.

• 9.5 years: transit of Mercury across solar disc
• 19 years: Metonic cycle (Meton,Chong), lunar phases
• 19 years: 235 lunations; Chang; 235*29.53059=6939.689 days; (235=19*12+7)
• 19*365.2422=6939.602 d (error 1 day/219 years), lunar phases fall to given day of the year.
• 19.00 years: (6940 days) Saros, 235 lunations, (10 Inex - 15 Saros), beats (365.2425/12, 29.530589) -> 2.7153 years = 19 years/7
• 19.043 years: (U,N)/9=171.39/9 years
• 19.053 years: (18.61,800.8)= 3*6.3509 years
• 19.2177 years: 27 tzolkin = 27*0.71177 years
• 37.4 years: cycle in tree rings (giant sequoia)
• 37.724 years: 53 tzolkin
• 37.784 years: Mayan lunar eclipse -> 11959 days
• 38.8237 years: 2*(J,12*J)
• 38.824 years: 2* [19.71, 1284]
• 74.3 years: 4*18.57 years; 222.6 years/3; 520 years/7;
• 74.7 years: free libration of the Moon, libration W (Moshier)

Let us consider ratio J/(E,R)= 50/9. Period J and period (E,R) are multiples of quantum c. 86.65 days. Four such quanta gives eclipse year.

Period Ey is approximate fraction of Jovian period and period of conjunctions Earth-Mars:

• 346.607148 days: J∙2/25
• 346.620031 days: ekliptic year
• 346.638272 days: (E,R)∙4/9

Axes of triads of inner planets move with these periods:

  [M,V,E] = 161.67475 days   [M,V,R] = 173.67527 days
  [M,E,R] = 192.78896 days   [V,E,R] = 347.06907 days

Also

• 347.1 days: [V,E,R]*3 = 115.67*3 (note: (M,E)= 115.87 days)
• 347.61 days: 2∙[M,V,R]

Let us consider motion with period Q, having synodic period regarding to Venus the same as is the synodic period of Earth and Mars, i.e. it has to hold:

 (Q,V) = (E,R) 

i.e. [Q,R] = [V,E], resp. 1/Q = 1/V+1/E-1/R.

Hence Q = 174.44340 days. Beats (173.31, 174.44) make c. 26750 days (73.2 years).

A phenomenon called El-Nino occurs during warming of waters of the eastern Pacific Ocean, phenomenon La Niña during cooling. Because it shows that both phenomena significantly affect the global climate system (western or eastern wind direction ...), their more detailed monitoring was initiated. By comparing of the air pressure at sea level so called Oscillation index is calculated (negative values ​​for the El-Nino positive for La Niña).

For fundamental period of oscillation is considered so. Landsberg's approximately 2.2-year (26 months) period.

• Period 2.2 years was also seen in a stream of neutrinos.
• Conjunctions Earth - Mars with a period (E, R) = 2,135 years
• Cycles 4.2 to 4.4 years and 8.8 years (spectral analysis of oscillations) recall period Y / Y and 2.
• Period c. 26.8 lunations was mentioned as economic cycle (Finex U.S. Dollar Index).
• Also double cycle conjunction with E-J; period 2 * (E, J) = 2 * 1,092 = 2,184 years.

Natural run of oscillations is disturbed by eruptions of volcanoes, e.g. eruption of El Chichon (April 1982) caused a short-term cooling, eruption Mount Pinatubo (1991) marked cooling of the globe.

Let us call period (E,J), i.e. approximately 400 days (398-402 days), oscillation year. The period coincides with 13.5 lunations (does Jupiter force Earth and Moon to synchronize?).

• 1.091479 years: 13.5*(E,L) = 13.5*29.530588 days = 398.66294 days
• 1.092085 years: (E,J) = (1.0000174, 11.861983) = 398.88404 days

The following beats observed in the system of inner planets are approximately multiples of oscillation year.

• (M,E/4) = (0.2408467, 1.0000174/4)= 6.57513 years=  6*1.095855 years
• (V,R/3) = (0.6151973,1.8808480/3)= 32.81964 years= 30*1.093988 years

Mean period 2.17 years has been associated to c. 1/5 of Wolf's cycle. Landscheidt's model of synchronization of climatic oscillations (El Niňo,..) with solar activity assume shift c. 0.382 W, where W is Wolf cycle (c. 11.1 y). (Theodor Landscheidt connect the number 0.382 with golden section.) Let us note, that: Y/(2W) = 8.54/22.2 = 0.385.

The following schema brings data of conjunctions of Earth (E), Mars (R) and line of lunar nodes (L_n) on the left and data of solar activity extremes (Smin, Smax) on the right.

Differences between Hale minima make 10*(E,R) = 21.35 years, in observed interval (1900-2000). The same does not hold for a wider interval (cycles become longer).

Solar minima and E-L_n-R

 E-Ln-R             Smin    Smax
--------------------------------------------
1894.7     -            -        1894.1
-         1899.0        1901.7  -
1903.2     -            -        1907.0
-         1907.5        -        -
1911.8     -            1913.6  -           Hale's minimum
--------------------------------------------
-         1916.0        -        1917.6
1920.3     -            1923.6  -
-         1924.6        -        1928.4
1928.9     -            -        -
-         1933.1        1933.8  -           Hale's minimum
--------------------------------------------
1937.4     -            -        1937.4
-         1941.7        1944.2  -
1946.0     -            -        1947.5
-         1950.2        -        -
1954.5     -            1954.3  -            Hale's minimum
--------------------------------------------
-         1958.8        -        1957.9
1963.0     -            1964.9  -
-         1967.3        -        1968.9
1971.6     -            -        -
-         1975.9        1976.5  -            Hale's minimum
--------------------------------------------
1980.1     -            -        1979.9
-         1984.4        1986.8  -
1988.7     -            -        1989.6
-         1993.0     -        -
1997.2     -            1996.4  -            Hale's minimum
--------------------------------------------