Action of the Moon

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Lunar period

Moon orbit

With regard to stars - Moon has sidereal period L:

With regard to nodal line - Moon orbits with draconic period Ln:

With regard to vernal point Lt:

Moon came near to the Earth every c. 27.6 days::

Rotation of the Earth changes also with this period (changes are caused by Moon on eccentric orbit).

The Sun completes rotation with regard to Earth after c. (25.38, 365.25) = 27.27 days. Interval 27.2-27.4 days coincides with period of Moon orbiting (draconic, tropical and sidereal) and also with period of moon libration in latitude.

(Contitiograms were based on "cycle of emotion" length of 27 days)

Lunar phases

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)

Calendar month

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).

Natural month

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.

Evection

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

Periods of eclipses

Lunar year of eclipses

"Lunar year of eclipses" (eclipse year) is period with what nodal line of Moon points to Sun.

Node line moves against the earth motion (with period c. 18.6 years), so lunar year is a bit shorter than solar year, it makes approximately (1.0,-18.6)=0.949 years, i.e. 346.6 days. (We consider orientation of nodal line. Otherwise nodal line points to Sun every 346.6/2=173.3 days).

inclination of lunar orbit with period c. 173.3 days (346.6 days).
Chaldean period Saros

Period of solar eclipses 18.03 years.

Large tidal floods 1959.991, 1978.021.

Lunar line of nodes

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

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

Wood's period

31 year period of eclipses.

Mayan period

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:

Date 1

Date 2

Interval

 7.10.804

 6. 7.837

837.541-804.794 = 32.747 years

26. 9.805

25. 6.838

838.511-805.766 = 32.745 years

10. 2.826

 9.11.858

858.886-826.141 = 32.745 years

25. 5.830

21. 2.863

863.170-830.426 = 32.744 years

Lunar crossing of ecliptic

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
Higher accuracy:
    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

Other periods

Derived periods

Semester

Period 6-ti lunations, i.e. c. 177 days (0.49 years). Zákryty Venuše Měsícem occurs with mean period 177 days.

Change of eccentricity
Eccentricity of lunar orbit changes with period c. 205.9 days.

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).

Lunar semestral year

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

Turn of lunar perigee

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

Meton's calendar cycle

Eclipse year

c_eyear Eclipse year is period, with which line of lunar nodes points to the Sun. Line of nodes circle opposite direction to motion of Earth (with period c. 18.6 y). Eclipse year is little bit shorter than solar year, it takes approximately:
Ey = (1.0,-18.6134) = 0.94901 y, 346.62 days.
Both ends of the line points to the Sun with period c. 346.6/2=173.3 days. Period 173.3 days is observed also in changes of inclination of he moon orbit (Meeus).

Influence of Jupiter

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:

Axes of inner planets

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

Conjunction with Venus

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).

Period Y

Conjunctions Earth-Mars

Nodal line of the Moon (Ln) points to the Earth (E) and Mars (R) with period:

 Y = 4∙(E,R) = 9∙(E,Ln

i.e. 8.5413 years (c. 3120 days).

Conjunction E-Ln-R

Conjunctions E-Ln-R:
1916.03 (January)                             2027.06 (January)
1924.61 (August)      1975.85 (November)      2035.61 (August)
1933.15 (February)    1984.35 (May)           2044.14 (February)
1941.66 (September)   1992.92 (December)      2052.69 (September)
1950.23 (March)       2001.44 (June)            ....
1958.75 (October)     2009.99 (December-January)
1967.27 (April)       2018.57 (July)

After 13 of such periods (111.04 y) cycle returns to the same season. (To observe this phenomenon, calendar with 13 months would be better.)

Period 111.050 y was derived by Wood as period of lunar tides. Period 112 y (4*28 y) was used by so called "weather calendars" or "centennial calendars". Axial period [U,N]=111.2909 years. With approximately 111 years period also axis of the angle of the planets Uranus and Neptune moves: [U-N] = 111.2909 years. Eclipse year is synchronized with conjunctions Earth-Mars with period: Y = 4∙(E,R) = 9∙(E,Ln), i.e. 8.5413 years (c. 3120 days).

Conjunctions E-Ln-R (outline)

Climatic oscillations

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.

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.

Oscillation year

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?).

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

Correlation with solar minima

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 (Ln) 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-Ln-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
--------------------------------------------

Outline of relations

Lunar periods

Name Mark Value Symbol
Sidereal month L 27.3217 days (l)
Drakonic month Ln 27.2122 days F
Anomalistic month Lp 27.5546 days (l)
Synodic month Ls=(L,E) 29.53059 days D
Earth orbit E 365.2596 days l'
Line of apsides P 3232.1 days
Nodal period Ω=(Ln,L) 6798.383 days (18.6 years) Ω
Some selected relations: Related to Mayan calendar: