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Vladimir Ladma
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Overlapping of cycles

Separation of conjunctions

Synodic period is mean period of repetition of conjunctions. In reality conjunctions occur (in consequence of elliptical orbits and nonuniform motion of bodies) in irregular intervals.

E.g. conjunctions J-S, during years 1940-2000 (in parenthesis intervals):
1940.85, ( 20.41), 1961.26, ( 20.01), 1981.28, ( 19.15) and 2000.43.

Overlapping of cycles of conjunctions J-S

Intervals of conjunctions Jupiter-Saturn repeats in triads, i.e. with mean period c. 60 years (Chinese astrological cycle),
3*(J,S) = 3*19.859 y = 59.577 y.

In every 900 years (great inequality) phase shift appears and new sequence of triads begins.

E.g. maximum separation c. 20.5 years appears before conjunctions in years 750.03, 1723.11 and 2636.65.

I. 750.03 

  0: ( 20.49)  750.03 ( 19.59)  769.62 ( 19.47)  789.08
  1: ( 20.48)  809.56 ( 19.69)  829.25 ( 19.38)  848.63
  2: ( 20.48)  869.11 ( 19.77)  888.88 ( 19.32)  908.20
  3: ( 20.45)  928.65 ( 19.86)  948.51 ( 19.25)  967.76
  4: ( 20.42)  988.18 ( 19.96) 1008.14 ( 19.19) 1027.34
  5: ( 20.38) 1047.72 ( 20.04) 1067.76 ( 19.15) 1086.91
  6: ( 20.34) 1107.26 ( 20.11) 1127.37 ( 19.14) 1146.50
  7: ( 20.29) 1166.79 ( 20.18) 1186.97 ( 19.12) 1206.09
  8: ( 20.22) 1226.31 ( 20.26) 1246.57 ( 19.12) 1265.70
  9: ( 20.15) 1285.85 ( 20.31) 1306.16 ( 19.15) 1325.32
 10: ( 20.07) 1345.39 ( 20.36) 1365.74 ( 19.19) 1384.93
 11: ( 19.97) 1404.91 ( 20.41) 1425.32 ( 19.25) 1444.57
 12: ( 19.88) 1464.44 ( 20.45) 1484.90 ( 19.30) 1504.20
 13: ( 19.78) 1523.98 ( 20.48) 1544.46 ( 19.37) 1563.83
 14: ( 19.70) 1583.53 ( 20.48) 1604.01 ( 19.47) 1623.47
 15: ( 19.59) 1643.06 ( 20.49) 1663.56 ( 19.55) 1683.11

II. 1723.11 

  0: ( 19.51) 1702.61 ( 20.49) 1723.11 ( 19.64) 1742.75
  1: ( 19.41) 1762.16 ( 20.49) 1782.66 ( 19.73) 1802.38
  2: ( 19.34) 1821.73 ( 20.47) 1842.19 ( 19.84) 1862.03
  3: ( 19.26) 1881.29 ( 20.44) 1901.73 ( 19.92) 1921.65
  4: ( 19.21) 1940.85 ( 20.41) 1961.26 ( 20.01) 1981.28
  5: ( 19.15) 2000.43 ( 20.37) 2020.80 ( 20.08) 2040.88
  6: ( 19.14) 2060.02 ( 20.31) 2080.34 ( 20.16) 2100.50
  7: ( 19.11) 2119.61 ( 20.25) 2139.86 ( 20.23) 2160.09
  8: ( 19.12) 2179.22 ( 20.18) 2199.39 ( 20.30) 2219.70
  9: ( 19.12) 2238.82 ( 20.11) 2258.93 ( 20.34) 2279.27
 10: ( 19.16) 2298.44 ( 20.01) 2318.45 ( 20.40) 2338.85
 11: ( 19.22) 2358.07 ( 19.92) 2377.99 ( 20.44) 2398.43
 12: ( 19.27) 2417.70 ( 19.82) 2437.52 ( 20.47) 2457.99
 13: ( 19.34) 2477.33 ( 19.73) 2497.06 ( 20.49) 2517.55
 14: ( 19.43) 2536.98 ( 19.63) 2556.61 ( 20.49) 2577.10

III. 2636.65 

  0: ( 19.51) 2596.61 ( 19.53) 2616.15 ( 20.51) 2636.65
  1: ( 19.60) 2656.26 ( 19.45) 2675.71 ( 20.49) 2696.20
  ....

Phase of new cycle is always shifted by on an average (J,S)=19.859 y with regard to the previous cycle.
Phase shifts cause, that the mean cycle of conjunctions (with regard to intervals of separation) seems to be (statistically,...) little bit longer, approximately 19.859*(n+1)/n, where n is c. 42-48.
Triads then lasts c. 61 years, i.e. approximately (J,S/2)=60.95 years (1/7 of Babylonian period 427 years).

Secular changes of solar activity

Greatest deviations from mean intervals of conjunctions of planet Jupiter and Saturn are caused:

Mean cycles of these changes are:

Let us try to put these two cycles together.
With regard to phase shifts, we will write one function for each interval (cca 900 y):

 Interval	        Function
 ----------------------------------------------------------------------
 (-2050,-975)       Ap*sin(2π*(t-1111)/P)+Aq*sin(2π*(t-1137)/Q)
 (-1025,  50)       Ap*sin(2π*(t-1131)/P)+Aq*sin(2π*(t-1137)/Q)
 (    0,1075)       Ap*sin(2π*(t-1151)/P)+Aq*sin(2π*(t-1137)/Q)
 ( 1025,2100)       Ap*sin(2π*(t-1171)/P)+Aq*sin(2π*(t-1137)/Q)
 ....

Here P=3*(J,S)=59.58 y, Q=(U,N)/2=85.72 y, Ap,Aq are constants and t time [years]. Years 1111, 1131, 1151, 1171 follows c. 20 years phase shifts, 1137 is (an arbitrary) year of conjunction (U,N).

Composed functions would make clear some secular changes of solar activity.
E.g. solar minima: Egyptian (-1400,-1200), Homer (-800,-630), middleage (650,705), Wolf (1270,1340), Sporer (1400,1500), Maunder (1645,1715),...

Functions have zero-points e.g. in years:

P-function:  .., 1409, 1469, 1528, 1588, 1648, 1707, 1767, 1826, 1886, 1946, 2005
Q-function:  .., 1393,   1479,   1564,   1650,   1735,    1821,    1906,   1992

Beginning of Maunder minimum corresponds to zero-point 1648-1650. An analogous zero-point was in years 794-795, i.e. 854 years (2*B) before the Maunder one.

Period 1025 years

More notable extremes of solar activity seems to appear every c. 1025 years, i.e. 6*(U,N), 12B/5.

Timo Niroma, "Sunspots: Sunspot cycles and supercycles and their
tentative causes":

"The auroral data of G. L. Siscoe of the years 450-1700 (Rev. Geophysics and Space
Physics 18, 1980) give another chance to try to calculate a value for the 1000-
year cycle. The lowest superminimum (smoothed) between 450 and 1450 appeared from
620 to 680. It precedes the lowest superminimum of this millennium, the Maunder
Minimum in 1640-1700 by 1020 years. The next superminimum after this pre-Maunder
is in 780-800, which apparently corresponds to the superminimum in 1800-1820 both
by duration and relative height with a 1020 year delay. The third superminimum in
the Siscoe data is in 850-880 corresponding to the superminimum in 1880-1920 about
1030 years later. The Siscoe supermaxima in 740-770, 820-850, and 900-930
correspond to supermaxima beginning 1030, 1010, and 1050 years later, so that a
supercycle of 1020-1030 years in average length is rather apparent."
(See above - overlapping of cycles).

Synchronization of observed deviations

Both mentioned cycles coincides with Babylonian period cca 427 years:

Derived periods:
[(J,S/2), (U,N)/2]= [60.95, 85.72]= 35.6 let (Bruckner period),
((J,S/2), (U,N)/2)= (60.95, 85.72)= 210.9 let (half of Babylonian period).

Are these cycles synchronized?

Let us assume, it holds:
P=a1*P1+F1 = a2*P2+F2,
where a1=14, P1=59.577 y, F1=19.859 let and a2=5, P2= 171.44 y, F2=?

Then P = 14*59.577 + 1*19.859 = 43*19.859 = 853.94 y (2*B) and phase shift
F2 = P- 5*171.44 = -3.27 y.
If F2= 0, then (U,N) = 2*B/5 = 2*36*J/5 = 427.031/5=170.813 y. And hence Neptunian period: N = ((U,N),U)= (170.813, 84.020)= 165.358 y. (Deviation 0.4 % from Bretagnon mean period 164.770 y).