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Vladimir Ladma

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Conjunction

Conjunction is a close apparition (join, alignment) of two bodies, as seen from the Sun.
Mentioned definition of conjunctions is not exact, it does not comprise definition of "close apparition" of bodies. We neglect finite speed of light and deformations of time-space relations.

E.g. if bodies ordered Sun-Venus-Earth-Mars are on straight line, we say Venus-Earth, Earth-Mars, Venus-Mars and Venus-Earth-Mars are in conjunction (i.e. in conjunction with the Sun, heliocentric conjunction).
Practical astronomy use geocentric definition of conjunction. If Sun-Venus-Earth-Mars are aligned, it is said, Venus is in conjunction, but Mars in opposition (with the Sun, seen from the Earth).

Synodic period

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 parentheses ().

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

(A,B) = -(B,A)
(k*A,k*B) = k*(A,B)
((A,M),(B,M)) = (A,B).

It is often implied, one period is orbital period of Earth.
E.g. synodic period of Jupiter is determined to be c. 399 days. It means period of Jupiter with regard to Earth:
(E,J) = (365.256,4332.59) = 398.9 days.

With synodic period the angle P-S-Q gradually opens (and then closes); S is the point, around which motion of bodies P,Q happens (centre of gravity).

Synodic day

Rotational (sidereal) period of planet is given with regard to stars. With regard to centre, around which the planet move, it takes one "synodic day" (in case of Sun, so called solar day).
Synodic day is synodic period (Tr,T), i.e. rotational period Tr measured with regard to orbital period T.

Action of synodical periods

Let us assume fictive space, where only synodical periods can be perceived (motion of bodies is hidden). E.g. we can hear a crack (AB) during conjunction of two bodies (A and B); during conjunction of three bodies (A,B,C) three cracks (AB, AC, and BC).
Ancient astronomers observed world similarly...

Space, at which synodic periods act, is determined only by periods P and Q and does not have any direct reflection in motion around centre. It is no good to compare synodical and orbital periods.
The case of synchronization synodical and orbital periods is exemption to this rule, see Synchronization with Jupiter

Examples

Pair synodical periods of inner planets:

(M,R)=  0.2762169 y (100.8882 days)
(M,E)=  0.3172552 y (115.8775 days)
(M,V)=  0.3958007 y (144.5662 days)
(V,R)=  0.9142273 y (333.9215 days)
(V,E)=  1.5986896 y (583.9214 days)
(E,R)=  2.1353487 y (779.9361 days)

Pair synodical periods of outer planets:

(J,N)= 12.7821869 y ( 4668.69 days)
(J,U)= 13.8119497 y ( 5044.81 days)
(J,S)= 19.8588709 y ( 7253.45 days)
(S,N)= 35.8698787 y (13101.47 days)
(S,U)= 45.3602286 y (16567.82 days)
(U,N)=171.4442476 y (62620.01 days)

Congruences of solar flares

Let us compute reminders of division by synodical periods of inner planets for each particular date of solare flare.
E.g. in years 1957-1964 we get:

          (M,R)    (M,E)    (M,V)    (V,R)    (V,E)   (E,R)
          100.9 d  115.9 d  144.6 d  333.9 d  583.9 d  779.9 d
----------------------------------------------------------------
1957.792:  79.3     33.1     40.7    168.4      5.7    385.3
1957.958:  39.0     93.7    101.4    229.0     66.3    	445.9
1959.042:  31.4     26.1     63.6    291.0    462.2     61.9
1959.542:  12.2     92.9    101.6    139.7     60.9    244.6
1960.625:   4.2     24.9     63.5    201.4    456.5    640.1
1961.458:   5.8     97.4     78.6    171.7    176.8    164.4
1962.708:  58.8     90.4    101.5    294.3     49.4    621.0
1963.375: 100.7    102.3     55.9    204.0    293.1     84.7
1963.708:  20.5    108.1     33.0    325.7    414.7    206.3

Flares 1957.958, 1959.542 and 1962.708 occured approximately at the same phase of conjunction cycle M-V-E.

Multiple conjunctions

Let us compute for each selected period T how exactly it contains integer multiples of all synodic periods. We will sum numbers g=f (pro f<0.5), resp. g= 1-f (for f>0.5), where
f = frac(T/(P,Q))
We will select only such periods, which have inaccuracy ∑g lower then specified limit.

Inner planets
∑g<0.2 (step 0.005 y, T<100 y):

Number of conjunctions  T [y] (∑g) Interval T [y]
(  23, 20, 16, 7, 4, 3)  6.350 (0.18)  6.345- 6.355
( 116,101, 81,35,20,15) 32.040 (0.15) 32.035-32.050
( 139,121, 97,42,24,18) 38.395 (0.07) 38.380-38.405
( 255,222,178,77,44,33) 70.435 (0.17) 70.430-70.435
( 278,242,194,84,48,36) 76.785 (0.12) 76.775-76.795
...

∑g<0.15 (step 0.005 y, T<900 y):

T [y](∑g)
247.770(0.10), 254.120(0.14), 286.165(0.05), 292.510(0.14),
324.555(0.04), 362.950(0.09), 533.935(0.14), 540.280(0.15),
572.325(0.08), 578.675(0.11), 610.715(0.04), 617.065(0.14),
649.110(0.08), 687.505(0.13), 858.490(0.12), 864.835(0.10)
...

Outer planets
∑g<0.5 (step 0.05 y, T<900 y):

Number of conjunctions  T [y] (∑g) Interval T [y]
(  9, 13, 14,  4,  5, 1) 178.95 (0.16) 177.50- 181.00
( 16, 23, 25,  7,  9, 2) 317.75 (0.44) 317.45- 319.55
( 18, 26, 28,  8, 10, 2) 357.90 (0.33) 357.05- 359.70
( 25, 36, 39, 11, 14, 3) 498.50 (0.40) 496.70- 499.10
( 27, 39, 42, 12, 15, 3) 536.85 (0.49) 536.85- 537.30
( 34, 49, 53, 15, 19, 4) 677.45 (0.39) 676.15- 678.20
( 43, 62, 67, 19, 24, 5) 856.40 (0.38) 855.60- 857.20

∑g<0.5 (step 0.05 y, T<6000 y):

T [y] (∑g)
1367.65 (0.46), 1546.65 (0.38), 1725.60 (0.39), 1866.20 (0.43),
1905.15 (0.44), 2045.15 (0.26), 2085.15 (0.49), 2224.10 (0.10),
2403.05 (0.07), 2582.00 (0.23), 2721.60 (0.42), 2760.95 (0.40),
2901.55 (0.40), 3080.50 (0.39), 3259.50 (0.43), 3770.70 (0.39),
3949.70 (0.37), 4128.65 (0.38), 4269.25 (0.36), 4309.25 (0.41),
4448.20 (0.19), 4627.15 (0.03), 4806.10 (0.14), 4944.85 (0.47),
4985.05 (0.30), 5125.65 (0.42), 5164.00 (0.47), 5304.60 (0.41),
5483.50 (0.40), 5662.55 (0.50), 5994.80 (0.48), .	..

Notes:

period c. 180 y is often cited in connection with motion of Solar system gravity centre;
period c. 500 y is supported by synodic period (Neptune,Pluto);
period c. 855 y is made from two Babylonian periods (427 y);
periods c. 676 y, 2582 y tj.130*(J,S) and 5125 y are Mayan periods;
period c. 2045 y is double of G.L.Siscoe period (1020-1030 y) found in polar lights data;
period 4627.15 y has on the interval (0-6000 y) smallest inaccuracy (0.03); at present it is assumed to be the most accurate period of multiple conjunctions (so called All-Planets Synod). (It consists of two subperiods 2224.10 +2403.05 y;
4627/25 = 2221/12 = 2406/13 = 185.1 y: period of resonance 1/J-3/S+2/N,
((U,N),H), see Inverse motion).

Period 180 years

Period 180 years is the smallest common multiple of selected orbital, synodical and axial periods of outer planet. But period (U,N)= 171.44 years does not fit in the schema, it differs from 9*(J,S)=178.730 by more than 7 years.

 1.178.730 9*(J,S), (U,N)?
 2. 89.365
 3. 59.577 3*(J,S)
 4. 44.682 (S,U)
 5. 35.746 (S,N), Bruckner period
 6. 29.788 S
 7. 25.533 [S,N]/2
 8. 22.341 [S,U], ([J,N],[S,U])/2, Hale period
 9. 19.859 (J,S)
10. 17.873 ([J,U],[S,N])/2, 3/2 J
11. 16.248
12. 14.894 S/2
13. 13.748 (J,U)
14. 12.766 (J,N), (conjunctions of inner planets)
15. 11.915 J
16. 11.171 [J,N]/2, Wolf period
17. 10.514 [J,U]/2
18.  9.929 (J,S)/2, ([J,S],[U,N])/2
19.  9.407
20.  8.936 ([J,U],[S,N])/2
21.  8.511 [J,S]/2

Switching of cycles

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).

Mentioned period of 180 years is the case of this phenomenon. All the system of outer planets does not have this period. Synodic period of Uranus and Neptune rather differ from 180 years:
(U,N)= 171.4 y.
Deviations accumulate with period c. (9*(J,S),(U,N))=(178.7, 171.4)= 4200 years.

I.Charvatova has found period c. 4400 years, and its aliquotes 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 parentheses, years): (-2200,-1100) +; (-1100,0) -; (0,+1100) +; (+1100,+2200) -;