Conjunctions, inequalities

Introduction

We will use - from the three ways of expressing dates - mainly "the mathematical" and "the historical" way.

historical      astronomical    mathematical
-------------------------------------------
1.4.2005 AD       2005(.25)         2005.25
1.4.   1 AD          1(.25)            1.25
1.4.   1 BC          0(.25)            0.25
1.4.   2 BC         -1(.25)           -0.75
1.4.2005 BC      -2004(.25)        -2003.75

Synodic period

Periods of motion

Basic orbital periods of bodies are given with regard to stars. These are so called sidereal periods.

In the next computations we will use mean periods according to VSOP87 (Bretagnon, Variations Seculaires des Orbites Planetaires).

J  11.8620 years   ( 4332.59 days)
S  29.457159 years (10759.23 days)
U  84.020473 years (30688.48 days)
N 164.770132 years (60182.29 days)

M   0.2408467 years ( 87.96926 days)
V   0.6151973 years (224.70080 days)
E   1.0000174 years (365.25636 days)
R   1.8808480 years (686.97973 days)

Conjunction is a close apparition (join, alignment) of two or more bodies. For simplicity we will consider only alignment of bodies in plane perpendicular to approximate plane of planetary motion (i.e. conjunction in longitude). Exact alignments in one line are sporadic and they are studied in connection with other phenomena (eclipses of Sun and Moon, transits of Mercury and Venus through solar disc…)

If bodies are observed from the Sun, we speak about heliocentric conjunction (conjunction with the Sun).
E.g. if bodies ordered Sun-Venus-Earth-Mars are in a straight line, we say Venus-Earth, Earth-Mars, Venus-Mars and Venus-Earth-Mars are in conjunction (seen from the Sun).

Practical astronomy understands by conjunction usually geocentric conjunction. If Sun-Venus-Earth-Mars are aligned, it is said, Venus is in conjunction and Mars in opposition (with the Sun, seen from the Earth).

Synodic period of two bodies

Mean period, with which (helio)centric conjunctions of two bodies repeat, is called synodic period.
Synodic (relative) period of two periods P,Q is period:

 (P,Q) = 1/(1/P-1/Q)= P∙Q/(Q-P) 

We designate synodic period with round brackets ().

For any periods A,B and constant k it holds:

In practical astronomy it is implied, that one of the period is orbital period of the Earth. E.g. synodic period of Jupiter is determined to be c. 399 days. It is period of Jupiter with regard to Earth: (E,J) = (365.256,4332.59) = 398.9 days.
Se synodickou period se postupně rozvírá (a pak zase přivírá) úhel P-S-Q; S is centrum, bod (center of gravity systems) okolo kterého tělesa P,Q obíhají.
With synodic period the angle P-S-Q gradually opens (and then closes); S is the point (centre), around which motion of bodies P,Q happens.

(J,N)= 12.7822 years ( 4668.69 days)
(J,U)= 13.8120 years ( 5044.81 days)
(J,S)= 19.8589 years ( 7253.45 days)
(S,N)= 35.8699 years (13101.47 days)
(S,U)= 45.3602 years (16567.82 days)
(U,N)=171.4443 years (62620.01 days)

(M,R)= 0.276217 years (100.8882 days)
(M,E)= 0.317255 years (115.8775 days)
(M,V)= 0.395801 years (144.5662 days)
(V,R)= 0.914227 years (333.9215 days)
(V,E)= 1.598690 years (583.9214 days)
(E,R)= 2.135349 years (779.9361 days)

Synodic day

Synodic day is rotational period Tr measured with regard to orbital period T, i.e. synodic period (Tr,T).

In case of planet of the Sun we speak about “solar day”, e.g. if Earth as rotational period Tr =1 stellar (sidereal) day, is its solar day equal to (1.0, 365.256) = 1.0027 stellar days. We divide solar day to 24 hours.

Reversing of polarity

Let us have a number of small periods Pi: (P0, P1,..., Pn), and greater period Q. If periods Pi have common multiple P, which is approximately equal to Q, it could appear, all the system has period P.
But evidently it is not true. Deviations of periods (P-Q) will gradually accumulate; during synodic period (P,Q).

Such accumulation of deviations can appear as transformation of cycle P, as change of its polarity.

E.g. periods 3,4 and 13 years makes common multiple c. 12-13 years. It can be well approximated by period P=12 years. But after a longer time we will register beats with period c. (12,13)=156 years.

In solar system periods (U,N)= 171.44 years and 9∙(J,S)=178.730 differ by more than 7 years. Though they are usually covered by so called 180-years period. Deviations of periods oscilates with period c. (9∙(J,S),(U,N))=(178.7, 171.4)=4200 years.

I.Charvatova has found period c. 4400 years (resp. its aliquots 2200 years and 1100 years) in motion of the Solar system gravity centre. According to I.Charvatova is the basic interval made of c. 55 conjunctions (J,S), i.e. 1100 years. Observed deviations of motional characteristics are in turns positive and negative (intervals in years): (-2200,-1100) +; (-1100,0) -; (0,+1100) +; (+1100,+2200) -.

Distortion

Mentioned definition of conjunctions is not physically correct, it is only geometrical construction. We neglect finite speed of light as well as time-space relations and so on.

E.g. to compare of conjunctions V-E (period 1.6 years) and conjunctions J-N (period 12.8 years), we have to consider space distance about 29 AU. (Light run this distance c. 4.3∙10¹² m during c. 14300 seconds, i.e. c. 4 hours…).

Venus-Earth interaction

Internal cycle

By „internal cycle” we mean cycle of inner planets with period I=6.4 years.

Period I

Resonance 3:4:7 of synodical periods Venus, Earth and Mars:

• 6.394758 years: 4*(V,E) = 4*1.5986896 years.
• 6.399591 years: 7*(V,R) = 7*0.9142273 years.
• 6.406046 years: 3*(E,R) = 3*2.1353487 years.
• 6.391093 years: (J,N)/2 = 12.7821869 years /2.
• Mayan cycle 9 tzolkin: 9*z= 2340 days= 6.4 years, a combination of three main numbers 9 (lords of nights), 13 and 20.
• Stable resonance (J,S/3,-U,-N/3)= 2340.8 days =6.41 years.

Period 6.4 years

Indication of regularity

Conjunctions Venus-Earth-Mars repeats approximately every 6.4 years. During 4 conjunctions V-E, 3 conjunctions E-R appear.

This motion repeats regularly like clockwork.

 V-E        E-R
 ---------  --------
 1886.140   1886.181
 1887.728
            1888.281
 1889.335     
            1890.406
 1890.929

 V-E        E-R
 ---------  --------
 1892.528   1892.593
 1894.132
            1894.806
 1895.720
            1896.949
 1897.330

  V-E       E-R
 ---------  --------
 1898.923   1899.055
 1900.520
            1901.147
 1902.127     
            1903.241
 1903.715

 V-E        E-R
 ---------  --------
 1905.322   1905.352
 ....         ....

If conjunctions could be heard, the sound would remind a drummer that play 3 triplets during 4 beats (beats are conjunctions Venus-Earth).

• 6.394758 y: 4*(V,E) = 4*1.5986896 years
• 6.399591 y: 7*(V,R) = 7*0.9142273 years
• 6.406046 y: 3*(E,R) = 3*2.1353487 years

(E,R)/(V,E)= 779.9361 days/583.9214 days= 2.1353487 years/1.5986896 years= 1.335687.

  (E,R)/(V,E) ~  4/3 

Missing loop?

V-E         E-R 
----------- ----------- 
2039.6081   2040.0078 
2041.2152 
            2042.1051 
2042.8032 
            2044.1968 
2044.4104 
----------- ----------- 
2046.0066   2046.2968 
2047.6000 
            2048.4269 
2049.2099 
            2050.6200 
2050.7979 
----------- ----------- 
2052.4023   2052.8267 
...          

After c. 150 years regularity of conjunctions V-E-R quite disappear (see outline on the right):

Entire cycle repeats with period c. 300 years.

Ratio (E,R)/(V,E) is not 4/3 (192/144), but rather 191/143. After c. 300 years one conjunction V-E and one E-R "drops out": 191∙ 1.5986896 years = 305.350 years, 143∙ 2.1353487 years = 305.355 years

So: (E,R)/(V,E) = (4∙48-1)/(3∙48-1); 48∙((V,E),(E,R)) = 48∙ 6.361133 years = 305.334 years

Mayan evaluate period: 140*(E,R)= 140*780 days = 109200 days (299 years=23*13 years), or 187*(V,E).
In this case: (E,R)/(V,E) = (4*47-1)/(3*47-1).

Likewise: (4∙(E,R))/((V,E),(E,R)) = (4∙12-1)/(3∙12-1) or R/Ey=(2∙54-1)/54=107/54, see Ecliptic year.

Line configurations

Instants, when configuration Earth-Venus-Mars makes a line, repeat also (like conjunctions) approximately with period 6.4 years. Besides, another structure appears here:

Mars stays four times (three periods R) at the same place. Later (after 6.4 years from the beginning) the same pattern starts again (phase shift F, c. 250-300 days).
Let: 3*R + F = 6.4 y. Then F = 6.4 - 3*1.880848 = 0.757 years = 276.7 days.

 Date        (Interval) Time
-----------------------------------
 1885 Mar 29 (1.88364) 1885.24820
 1887 Feb  9 (1.86721) 1887.11545
 1889 Jan  3 (1.90007) 1889.01556
 1889 Oct  1 (0.74196) 1889.75754
 1891 Aug 23 (1.89185) 1891.64943
 1893 Jul  8 (1.87543) 1893.52490
 1895 Jun  5 (1.90828) 1895.43322
 1896 Feb 10 (0.68446) 1896.11770
 1897 Dec 29 (1.88364) 1898.00138
 1899 Nov 17 (1.88364) 1899.88506
 1901 Oct  9 (1.89185) 1901.77696
 1902 Jul 31 (0.80767) 1902.58464
 1904 Jun 18 (1.88364) 1904.46832
 ...

Every c. 300 years one period R "vanishes". Complete cycle 6.4 y vanishes after c. 1020 y (300 y*6.4/1.88)?

Period 12.8 years

Mayan period 13 tun

Period 13*18*20 days = 4680 days (12.81 years).

13 Mayan tuns = Double period of conjunctions V-E-R:

• 8*(V,E) = 12.789516 years = 2* 6.394758 years
• 6*(E,R) = 12.812088 years = 2* 6.406044 years
• Mayan cycle 18 tzolkin: 18*z= 4680 days = 12.81 years
• ([J,N],U)= 12.74 years
• (J,N)= 4668.7 days= 12.78 years
• (U/2,S/3)= 12.81 years,...
• 12.74 years: ([J,N],U)
• 12.743 years: (2.2066,1.8809)
• 12.74682 years: ze simulation motion center of gravity známých planet
• 12.76 years: solar spectrum (Charvátová)
• 12.79100 years: simulation motion center of gravity známých planet
• 12.79891 years 14 conjunctions V-R (beats 9867.3 years)
• 12.8 years významná period ve Slunečním spektru (Charvátová)
• 12.81184 years: 18 tzolkin = 18*0.71177 years
• 12.81475 years: (2/U,3/S)
• 25.6237 years: 36 tzolkin = 36*0.71177 years

Mars in perihelion

Because frac(6.4/1.88)= 2/5, observer of Mars with period 6.4 years realizes, that Mars makes pentagon like as conjunctions V-E, frac(1.6/1.0) = 3/5. Oppositions V-E, Mars at perihelion:

 Rozdíl   Datum     Lv   Le    Lr    Lv-Le 
-------------------------------------------- 
 ( 31.98) 1715.86:  47.6 228.2 326.0 -180.6 
 ( 31.98) 1747.84:  38.7 219.9 326.6 -181.1 
 ( 31.98) 1779.81:  29.9 211.5 327.2 -181.6 
 ( 31.98) 1811.79:  21.2 203.2 327.8 -182.0 
 ( 31.97) 1843.76:  12.1 194.2 328.2 -182.1 
 ( 31.98) 1875.74:   3.5 185.8 328.8 -182.3 
 ( 31.98) 1907.72: 354.9 177.4 329.4  177.5 
 ( 31.98) 1939.69: 346.3 168.9 330.0  177.4 
 ( 31.98) 1971.67: 337.8 160.5 330.6  177.3

View from Mars

Seen from Mars: Venus moves forwards with period (V,R/3)=32.819643 y, meanwhile Earth moves backwards with period (E,R/2)= 15.780949 y.

Resultant motion makes period: [32.819643, 15.780949] = 10.656765 years. Five periods (E,R): 5*(E,R) = 10.676744 years.
Equality of these periods gives unstable resonance: 5/V - 6/E - 4/R = 0 (1139 years)

Note: modulation (10.656765, 302.4347) = 11.046 years.

Other periods

Period 256 years

Mayan period. 13 Mayan katuns make 256.3 years.

Triple conjunctions J-S (observed from the Earth) returns after 257-258 years; e.g. 333, 411, 452, 710, 967, 1008, 1306, 1425 AD.

Incorporation of Mars

Resonance of beats

Integer ratios of pairs Mercury-Earth (E/M=4/1) and Venus-Mars (R/V=3/1) lead to beats

• R1 = (E/4,M) = (1.0000/4, 0.2408) = 6.575 years,
• R₂ = (R/3,V) = (1.8808/3, 0.6152) = 32.82 years,

 1/M-5/V-4/E+15/R = 0 

Inequalities

Period of inequality

Two bodies P and Q repeat their positions (e.g. in conjunction at the same place), if q periods P is equal to p periods Q, so if: q*P = p*Q, i.e. P/Q=p/q, where p,q are whole numbers.

Let

 q/Q -p/P = 1/I 

Period I is called period of inequality (or inequality period, inequality):

 I = (Q/q,P/p) = P∙Q/(q∙P-p∙Q) 

Usually I is on order of greater then P and Q (I>>P, I>>Q).

The place, where planets repeat their positions move with period I.

Big trigon

Conjunction of planets J and S appears on an average every 19.859 years. During this time Jupiter get approximately:
(J,S)/J *360° = 1.67416*360 = 360+242.698°. And Saturn approximately (J,S)/S *360° = 0.67416*360 = 242.698°.

Because 240°=(2/3)*360°, conjunction places make equilateral triangle, so called "big trigon". During 19.859 years this trigon moves by
((J,S)/S - 2/3)*360° = 242.698-240 = 2.698°.

After 120°/2.698 * 19.859 y, i.e. c. 900 years (great inequality) the second apex appear at the starting point, and after c. 1800 years the third apex. The whole triangle returns to its original position after c. 2700 years.

If there was no rotation, conjunction line would be oriented in the same direction every 3*(J,S). So, also after 42*(J,S), 45*(J,S) and 48*(J,S). During this period (c. 900 years) trigon takes approximately 120° forward. Therefore conjunction line is oriented in the same direction every c. 43, 46, and 49 conjunctions.
43∙(J,S) = 853.9 years, 46∙(J,S) = 913.5 years, 49∙(J,S) = 973.1 years

Change of planetary speed

J.H.Lambert has noted, that mean speed of Saturn increased compared to speed from Galileo’s measurements. This deviation was later make clear by Laplace with help of effect of small denominators.

Babylonian period

Babylonian period is distinct period of motion of inner planets. It makes approximately 427 years.

Introduction

Ray Tomes included Babylonian period into his schemes (see  Theoretical Cycles Periods ):

427.0   213.5  106.8   53.38  26.69

142.3   71.17  35.59  17.79   8.897  4.448  2.224

        1. 415-425 years
        2. 305-314 years
        3. 260-280 years, affects moisture
        4. 177-227 years, almost all, very pronounced,
              an intensity cycle, affects temperature
              (median 202 years)
        5. 154-157 years, a length cycle, 13 Jovian years
        6. 143-148 years, a length cycle, 13 average cycles
        7. 104-105 years, half of the 200-year cycle
        8.  85- 90 years
        9.  78- 79 years, the Gleissberg cycle
        10.  63- 67 years, 1834-1901, (1954-?)
        11.  57- 59 years
        12.  51- 52 years, 1783-1834, 1901-1954
        13.  43- 45 years
    

E-Moon-R-J-S

It holds approximately: B/2 = 25*Y = 100*(E,R) = 225*(E,Ln) = 18*J, so

 2*(E,R)/J = Y/2J = 9/25 

Simulation of motion of R-J-S makes patterns with period 853.9 years.

Great inequality

Great inequality (J/2,S/5) makes c. 2*B=72*J=29*S=400*(E,R). Let us assume I= 2∙B = 72∙J = 29∙S' = 43∙(J,S').
Then using J=11.8620 we get: I=854.06 years, B=427.03 years, S' = 29.450441 years. Synodic period (J,S') = 19.861925 years.

If value of great inequality J-S was just value I= 2*B= 854.06 years, then derived periods differ from the Bretagnon’s periods by ratio:
S/S'=29.457158/29.450441 = 1.000228; (J,S')/(J,S)=19.861925/19.8588709= 1.000154.

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 non-uniform motion of bodies) in irregular intervals.

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

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.

      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
    

       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
    

      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).

Long-term changes

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

• by planets Jupiter and Saturn itself (elliptical orbits, great inequality)
• by planets Uranus and Neptune (perturbations).

• Mean cycles of these changes are:

• cycle of 60-61 years (59.58 y, with phase shift, see above)
• cycle of 85.5 years (85.72 y = (U,N)/2 = 171.44/2 y)

Let us try to put these two cycles together.
With regard to phase shifts, we will write one function for each interval (c. 800-1000 years).
The exact instants of shifts are not clear - perhaps they even do not exist - one cycle fades while the other one become stronger:

       Interval        Function
       -----------------------------------------------------------------
       (-2000,-1000)   |Ap∙sin(2π(t-1111)/P)+Aq∙sin(2π(t-1157)/Q)|
       (-1000,    0)   |Ap∙sin(2π(t-1131)/P)+Aq∙sin(2π(t-1157)/Q)|
       (    0, 1000)   |Ap∙sin(2π(t-1151)/P)+Aq∙sin(2π(t-1157)/Q)|
       ( 1000, 2000)   |Ap∙sin(2π(t-1171)/P)+Aq∙sin(2π(t-1157)/Q)|
       ( 2000, ... )   |Ap∙sin(2π(t-1191)/P)+Aq∙sin(2π(t-1157)/Q)|
       ....
    

Here P=3*(J,S)=59.58 y, Q=(U,N)/2=85.72 y, Ap,Aq are constants and t is time [years]. Values 1111, 1131, 1151, 1171 AD follows c. 20 years phase shifts.
The value 1157 AD was selected 20 years after conjunction (U,N).

Egyptian minimum (-1300,-1200), Homér's minimum (-800,-700)

Greek minimum (-450,-350)

Middle age minimum (650,705)

Middle age maximum (1120,1280), Wolf's minimum (1280,1340)

Sporer's minimum (1400,1500), Maunder's minimum (1645,1715), minimum (1870,1930), end of millenium maximum (1930-2030)

Comments:

Maunder's minimum started about 1650 AD. An analogous "zero-point" was e.g. at 800 AD, i.e. 850 years (2*B, i.e. 12 Bruckner's cycles) before Maunder's.

• Comparison with Schove's maxima:
• 175-200, (290-300?), 350-375, 500-510, 530, 565, 585, (675?), 745,765,
• (830-840?), 920-930, 960-980, (1100-1150), (1200), 1360-1370,
• 1530-1580, 1720-1740, 1770-1790, (1840-1870?), 1940-1990,...
• Diagrams fail to make clear minimum 1800-1820 AD.
• About 1820 conjunction J-S (1821.76), U-N (1821.01) (and barycentric loop) occured at the same time; J was in perihelion (1821.37).
• G.Sporer called attention to a possible low activity in years 1887-1889.

Other data:

Synchronization of deviations

Both mentioned cycles coincide with Babylonian period c. 427 years: 61.0 ∙7 = 427.0 years, 85.5 ∙5 = 427.5 years.

Derived periods

[(J,S/2), (U,N)/2]= [60.95, 85.72]= 35.6 years (Bruckner's period),

((J,S/2), (U,N)/2)= (60.95, 85.72)= 210.9 years (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 years, F1=19.859 years 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).

Prediction

For years 2000-2700 AD.

There would be next "Maunder's minima" around years 2150,  2350 and 2500?!.

Other relations

Conjunctions E-R

Conjunctions E-R  appears  close to geometrical axis J-S with period B/4. E.g. (with accuracy 5°) in years 1636.3,1743.2,1850.0 or 1869.1 and 1975.9.

It holds:

 ([J,S]/2, Y) = 2∙B 

i.e. [J,S]/2, 4∙(E,R))= (16.9132418/2, 4∙2.1353487)= (8.45662, 8.54140)= 852.04 years= 2∙426.02 years.

Outer planets

Lt us assume two relations:

• synchronization of axes of outer planets with Jupiter

 ([J,U],[S,N]) = 3∙J 

• synchronization of axes of outer planets with solar cycle

 ([J,N],[S,U]) = 4∙W 

From equations (1/J+1/U)-(1/S+1/N) = 1/(3J) and (1/J+1/N)-(1/S+1/U) = 1/(4W) it follows W= 3/4(J/2,S/3)= 11.23375 years = [J,212.1 years].

And hence:

 W= 3/4(J/2,S/3)= [J,B/2] 

(But in comparison with the mean period of Solar cycle Ŵ=11.0-11.1 years is value 11.23 years too high).

Derived periods

Period 71 years

Babylonian counted also with the cycle B/6 = 6*J = 71.17 years (see Earthquakes).

65 oscillatory years give c. 71 years, i.e. 6 orbital period Jupitera.

1.091854 years: 9/2*(E,R/2)/65 = 9/2*(0.999979,1.880711/2)/65 = 70.97052/65 = 9/2*15.7712270/65

• 70 years: geological cycle (mud of Lake Saks)
• 70 years: cycle from tree rings (Japanese cedar tree)
• 70.9856 years: 65 oscillation years, 65*(E,J)=65*1.0920855 (tropical)
• 71.1391 years: 6*J (tropical), 71.1769 years: 6*J (anomalistic) =100 tzolkin
• 71.1785 years: B/6 = 427.07 years/6
• 71.1861 years: 4627.1/65 (925.4/13)
• 71.74 years: 2*(S,N) =2*35.87 years
• 73 years: geological cycle (banded clays)

Period B/2

Extrapolation of low solar activity from 1800-1830 to 1980-2010 went wrong, 180 years cycle failed. But in this case also 320 years cycle go wrong, because of Maunder’s minimum  in years  1660-1690.

In Schove’s dates  periods of high   (SSS,SS,S) and low (WW,W) solar activity seems to change with period more than 180 years. E.g. progress with step 205 years (about half of Babylonian period)  we get e.g. years: 350, 555, 760, 965, 1170, 1375, 1580, 1785, 1990.  Nearby (±40 years)  of these dates  relatively high maxima appear. (Tidal action of planet J-V-E cannot make such differences…!?)

Period 284 years

Simulation of motion of inner planets (with Bretagnon’s data) establishes especially part 2/3 of Babylonian period, i.e. period c. 284 years

Period 1025 years

I have noted the possibility of phase shifts of conjunction cycles while reading texts (paragraph about period 1020-1030 years) of climatologist Timo Niroma:

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

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

Period G.L.Siscoe 1025 years: 5125/5;

minima v Eddyho diagramu Solar activity (-9625,-8600,-7575,-6550,-5525,-4500,-3475, -2450,-1425, -400, +625, 1650, 2675, 3700)

Conjunctions Uranus-Neptune

Data in the following table progress in the rough with Babylonian period 427 years. On the left are instants of conjunctions E-Ln-R and instants of Jupiter in perihelion.

  E-Ln-R      Jp        U-N
 ---------------------------------
 -2021.55  -2022.63
 -1594.48  -1595.57 * (-1605.74)
 -1167.41  -1168.50 
  -740.34   -741.44 * ( -745.20)
  -313.26   -314.37
   113.81    112.69 * ( 111.19)
   540.88    539.76
   967.94    966.83 * ( 965.47)
  1395.01   1393.90
  1822.08   1820.97 * ( 1821.01)
  2249.15   2248.04
  2676.22   2675.11 * ...

Possible correlations with conjunctions Uranus-Neptune are marked on the right.

If mean period (U,N) was 170.813 years, then it would be equal to 20* Y= 80* (E,R)= 240 z. It would have integer ratio to Babylonian period (2:5), to 1025 years period of polar lights G.L.Siscoe (1:6) and also to Mayan period M=5125 years (1:30).

The following data (with period M/5 = 5125/5 = 1025 y) seams to appear near to minima on Eddy diagram of solar activity:
-9625,-8600,-7575,-6550,-5525,-4500,-3475, -2450,-1425, -400, 625, 1650, (2675), (3700).

These data (with period M/3 = 5125/3 = 1708 y) remind great eruptions of volcanoes:
-5049 ?, -3341 Avelino, -1633 Thera, 75 Vesuvius, 1783 Laki, 3491 ?.

Laplace's cycle

Cycle of approximately 900-years (840-960 years?) created by the so-called great inequality of the planets Jupiter and Saturn. G. Beutler in the book Methods of Celestial Mechanics gives period 890 years, prof. A.E.Roy writes in the book Orbital motion about 900-year oscillation. Last extremes of Laplace's cycle was approximately in years 1560 (minimum) and 2000 (maximum).

Synchronization of perihelia

Mayan calendar round (52 years) and computation of tuns (1-18) makes period 9 Aztec centuries i.e. 9*104 = 936 years.

With the same period perihelia of Jupiter and Saturn synchronize. Value (J_a/2,S_a/5) computed from anomalistic period J_a,S_a  makes c. 938-939 years.

Mean Day Motion

The values nP=360/P are generally called the Mean Angular Velocities (MAV), see Angular velocity.
If period P is given in days, we speake about Mean Day Motion (MDM).

If in given time planet run just with mean velocity, its motion is sometimes called "true motion" (=mean motion).
If a planet is in state of true (mean) motion, its actual period is the mean period.
Actual velocity of motion deviates from velocity of true (mean) motion.

Precision

Precision of antique astronomical instruments was not better than c. 10’ (0.167 dg) - in comparison with that precision of values in age of Edmund Halley achieved 10”.
However the precision of some results in antique was very high, e.g. Hipparchos determined MDM of the Earth with the precision 0.0435” (0.000012 dg). Values of MDM given in the Almagest (by Claudius Ptolemaios) lead - after correction of precession - to the following MDM for Jupiter and Saturn: nJ=299,104581“ nS=120,422528“. These values corresponds to periods J=3600/299,104581 * 360/365,25 = 11.862923 years and S=3600/120,422528 * 360/365,25 = 29.465040 years.
Bretagnon theory (VSOP) makes these MDM values: <nJ>=299.1283“ and <nS>=120.4547“, i.e. periods : J=11.861983 years, S=29.457159 years.
For year 2000 NASA provides these actual MDM values: nJ=299.1124“ and nS=120.4943“.

The period (I) of the Jupiter –Saturn inequality period gets from ratio S:J near to resonance 5:2: I(J,S) = 1/(5/S-2/J) . Because the denominator is small, the resulting value strongly depends on precision of mean orbital periods of Jupiter and Saturn. From Bretagnon data I(11.861983, 29.457158) = 883.3 years, from Ptolemaio’s data I(11.862923, 29.465040) = 909.0 years.

                     J  S     A                          B                 C   
                1/   0  0    0.59954649739               0.00000000000     0.00000000000
                2/   2 -5    0.00573506125               1.44396306420     7.11354700080
                3/   1 -2    0.00062308554               3.41857056095   103.09277421860 

                     J  S     A                          B                 C   
                17/  1  0    0.00001824700               5.72883078185     529.69096509460 

Energy

Some scientists consider about connection of Laplace's cycle with observed millennial cycle of solar activity.
When Jupiter get - for its own accerelation - some extra energy from the Sun, then some lost of the solar energy would be observed. Years in which Q take the values -1: -9000,-8120,-7240,-6360,-5480,-4600,-3720,-2840,-1960,-1080,-200,680,1560, 2440,…
The year 1560 corresponds to so called Suess' minimum.

There was oposition of Uranus against other outer planets in years 691 and 1561 (difference 870 years):

Solar rotation

Synodical rotation period of the Sun seen from the Earth was estimated to be c.27.275 days.
In some years before (~Maunder minimum) it was lower: