Missing loop

• Conjunctions Venus-Earth-Mars
• Disturbance of regularity
• Disappearance of cycle
• Beats of synodical periods
• Line Earth-Venus-Mars
• Period 32 years

Conjunctions Venus-Earth-Mars

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

This motion regularly repeats like clockwork.

V-E           E-R
---------- -----------
1886.140   1886.181
1887.728
           1888.281
1889.335
           1890.406
1890.929
---------- -----------
1892.528   1892.593
1894.132
           1894.806
1895.720
           1896.949
1897.330
---------- -----------
1898.923   1899.055
1900.520
           1901.147
1902.127
           1903.241
1903.715
---------- -----------
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 y
6.399591 y: 7*(V,R) = 7*0.9142273 y
6.406046 y: 3*(E,R) = 3*2.1353487 y

(E,R)/(V,E)= 779.9361 d/583.9214 d= 2.1353487 y/1.5986896 y= 1.335687

Integer ratios are not exact also in other cases. Cycles in the solar system are usually modulated by other cycles, see Resonances.

But what is cause of this inaccuracy?

Disturbance of regularity

After c. 150 years regularity of conjunctions V-E-R quite disappear:

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
...

Entire cycle repeats with period c. 300 years.

Disappearance of cycle

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 "drop out":
191* 1.5986896 y = 305.350
143* 2.1353487 y = 305.355
So: (E,R)/(V,E) = (4*48-1)/(3*48-1)

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

Beats of synodical periods

Beats of synodical periods of inner planets:
(computed as fractions of period ((V,E),(E,R))= 6.361133 years)

 Synodic period [years (days)]         Divisor   Beats [years]
 -------------------------------------------------------------
((V,E),(E,R)) = 6.3611330 (2323.40382)  1.0000(  1) -
(E,R)         = 2.1353487 ( 779.93610)  2.9790(  3) 302.4347
(V,E)         = 1.5986896 ( 583.92137)  3.9790(  4) 302.4348
(M,R)         = 0.2762169 ( 100.88821) 23.0295( 23) 215.7181
(V,R)         = 0.9142273 ( 333.92152)  6.9579(  7) 151.2174
((M,R),(V,E)) = 0.3339086 ( 121.96012) 19.0505( 19) 125.9101
(M,E)         = 0.3172552 ( 115.87748) 20.0505( 20) 125.9100
(M,V)         = 0.3958007 ( 144.56622) 16.0716( 16)  88.8993
((M,E),(E,R)) = 0.3726159 ( 136.09797) 17.0716( 17)  88.8993
((M,V),(V,E)) = 0.5260357 ( 192.13455) 12.0926( 12)  68.7041
((M,V),(E,R)) = 0.4858576 ( 177.45949) 13.0926( 13)  68.7041
((M,V),(V,R)) = 0.6979809 ( 254.93752)  9.1136(  9)  55.9858

(M=0.240847, V=0.615197, E=1.000017, R=1.880848)

All periods of beats are parts of spectrum of period P, P is c. 1500 years:

302.4347 * 5 = 1512.1 years
215.7181 * 7 = 1510.0 years
151.2174 *10 = 1512.2 years
125.9101 *12 = 1510.9 years
88.8993  *17 = 1511.3 years
68.7041  *22 = 1511.5 years
55.9858  *27 = 1511.6 years

Let us consider stable resonances A= (V/3,-E/7,R/4) and B= (M,-V/5,E/4).
From conjunction of both relations we get C=(M,-V/2,-E/3,R/4). It holds aproximately A/B = 302.44/40.86 = 37/5 (C is c. 36 years). Common multiple is 5*A= 37*B= 42*C= P.

Line Earth-Venus-Mars

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

 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
....

Mars stays three times (with period R) at the same place. Later (after 6.4 years) the same picture appears again (with phase shift F, c. 250-300 days).
Let: 3*R + F = 6.4 y. Then F = 6.4 - 3*1.880848 = 0.757 let = 276.7 days.

It seems, every c. 300 years one period R "vanish".
Complete cycle 6.4 y vanish during c. 1020 y (300 y*6.4/1.88).

Period 32 years

Conjunctions V-E repeat their positions with regard to Earth in every 8 years, conjunctions V-E-R repeat with period 6.4 years. So, conjunctions V-E-R appear at the same place with regard to Earth in every 32 years.

32 y = 3*10.67 y
32 y = 4* 8.00 y
32 y = 5* 6.40 y

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 resonance:
5/V - 6/E - 4/R = 0 (1139 y)
(This resonance is unstable: 5-6-4 != 0)

Observer of Mars with period 6.4 years realizes: Mars makes pentagon
(frac(6.4/1.88) = 2/5) - like as conjunctions V-E (during 8-year period).