By the term "Hallstatt" is called the cycle of about 2300 years. For the first time, I read about the existence of the 2300-year cycle in texts published on the Internet by Ray Tomes. There was probably also written the relationship 1/J-3/S+1/U+1/N (= 1/H).
The ratios of the synodic periods of the outer planets p1 = (U,N)/(J,S) and p2 = (U,N)/(S,U) are approximately p1 = 171/20 = 8.5 and p2 = 171/45 = 3.8. It offers to balance these to the ratio: (U,N):(S,U):(J,S) = 1: 1/4: 1/9. And here a series of relationships - dominated by the period (H) - follows. I thought it might be related to some unknown (hypothetical) planet and began to call it H (Hypos). Until recently (in 2018) I have read that it is customary to use the name "Hallstatt" for the period (so there is no need to change the abbreviation "H" ...).
Difference (L_{S}-L_{U}) a -(L_{S}-L_{N})
je d = L_{S}-L_{U}+L_{S}-L_{N}
= 2∙L_{S}-L_{U}-L_{N}
= 2∙(J,S)/S -(J,S)/U-(J,S)/N = (J,S)∙(2/S-1/U-1/N).
Deviation from full angle during (J,S): (1-d) = 1 - (J,S)∙(2/S-1/U-1/N).
During 1 year: h = (1-d)/(J,S) = 1/(J,S) - (2/S-1/U-1/N) = 1/J-1/S-2/S+1/U+1/N = 1/J-3/S+1/U+1/N.
So
In degrees:
d∙360° = 19.859∙(2/29.457-1/84.020-1/164.770)∙360° = 0.991433 ∙ 360° = 356.916° (1-d)∙360° = -3.084° (= 157.601° - 160.685°) h∙360° = -3.084° /19.859 years = 0.1553°/year. h = 0.1553/360 = 0.00043139 full angles / year. Period H: H = 1/h = 1/0.00043139 = 2318.1 years.
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).
(Similarly was world observed by ancient astronomers...)
Motion of an observer in the fictive space of synodical periods does not modulate orbital periods, but synodical periods.
Let H is period of stable resonance, 1/H=1/J-3/S+1/U+1/N (c. 2320 years). Observer moving with this period (in the world of combined synodical periods) will get the following values of ((J,S),(S,N)), ((J,S),(U,N)) and ((J,U),(U,N)):
1/((J,S),(S,N))-1/H = 1/J-2/S+1/N-1/H = 1/S-1/U = 1/(S,U) 1/((J,S),(U,N))-1/H = 1/J-1/S-1/U+1/N-1/H = 2/S-2/U = 2/(S,U) 1/((J,U),(U,N))-1/H = 1/J-2/U+1/N-1/H = 3/S-3/U = 3/(S,U)
Therefore for this observer it holds:
Resonant ratio of orbital periods of Uranus and Neptune is 1:2 (N/U =1.961);
period of inequality I = (U, N/2), approximately 4200 years.
Observer moving with period I gets periods of outer planets J',S',U',N':
1/J' = 1/J-2/N+1/U = 11.8953 years 1/S' = 1/S-2/N+1/U = 29.6636 years 1/U' = 1/U-2/N+1/U = 2/U-2/N = 85.722 years 1/N' = 1/N-2/N+1/U = 1/U-1/N = 171.444 years
For this observer N':U' is exactly 2/1. Ratio S'/J' is
approximately 5:2 and U'/S' approximately 3:1.
Period of inequality J-S: (S'/5,J'/2) = 2362 y and period of
inequality S-U: (U'/3,S'/1) = 778 y.
Value of period H (1/H = 1/J-3/S+1/U+1/N) remain the same: H = 2320 y.
Sidereal periods of outer planets fulfill equation:
For synodic periods: 1/H = 1/(J,S)-2/(S,U)-1/(U,N) 3/H = 4/(U,N)-1/(S,U) 5/H = 9/(U,N)-1/(J,S) 7/H = 4/(J,S)-9/(S,U) Generally m2/P-n2/Q = k/H, so P∙Q/(Q∙m2-P∙n2) = H/k. For comparison Bohr's quantization of atoms: 1/T = c∙R∙(1/m2-1/n2)
Our observer therefore realizes:
5/S'-2/J'=1/H (=5/S-2/J+3/U-6/N=1/J-3/S+1/U+1/N)
3/U'-1/S'=3/H (=5/U-4/N-1/S =3/J-9/S+3/U+3/N)
It holds:
1/H = 1/J- 3/S+1/U+1/N
3/H = -1/S+5/U-4/N
5/H = -1/J+1/S+9/U-9/N
7/H = 4/J-13/S+9/U
Values L=(3L_{J}-8L_{S})-(2L_{U}-7L_{N}), where L_{J},L_{S},L_{U},L_{N} are longitudes of planets in selected moments oscillates approximately around 120˚:
In conjunctions J-S is (3L_{J}-8L_{S})/5 = L_{J} = L_{S}, in conjunctions U-N is (2L_{U}-7L_{N})/5 = L_{U} = L_{N}.
Opposition J-UYear | 3 L_{J}[˚] | 8 L_{S} [˚] | 2 L_{U} [˚] | 7 L_{N} [˚] |
(3L_{J}-8L_{S})-(2L_{U}-7L_{N}) [˚] |
1810,46 | 124 | 230 | 83 | 309 | 254 – 134 = 120 |
1824,28 | 322 | 139 | 215 | 157 | 183 - 58 = 125 |
1838,09 | 141 | 51 | 334 | 7 | 90 – 327 = 123 |
1851,90 | 300 | 320 | 80 | 219 | 340 – 221 = 119 |
1865,70 | 98 | 229 | 185 | 73 | 229 – 113 = 116 |
1879,52 | 272 | 142 | 302 | 287 | 131 - 15 = 116 |
1893,33 | 108 | 49 | 72 | 142 | 58 – 290 = 128 |
1907,15 | 306 | 325 | 204 | 356 | 341 – 208 = 133 |
1920,97 | 127 | 230 | 325 | 208 | 257 – 116 = 141 |
1934,77 | 287 | 146 | 72 | 59 | 142 - 13 = 129 |
1948,58 | 84 | 49 | 176 | 268 | 35 – 269 = 127 |
1962,39 | 257 | 326 | 291 | 115 | 291 – 176 = 115 |
1976,21 | 91 | 230 | 61 | 323 | 221 - 97 = 123 |
1990,02 | 290 | 148 | 193 | 172 | 142 - 21 = 121 |
2003,84 | 113 | 52 | 315 | 22 | 61 – 293 = 128 |
2017,65 | 275 | 328 | 63 | 234 | 307 – 189 = 118 |
2031,45 | 71 | 233 | 167 | 88 | 198 - 80 = 118 |
2045,26 | 242 | 146 | 281 | 302 | 95 – 339 = 116 |
2059,08 | 74 | 55 | 50 | 157 | 19 – 253 = 126 |
2072,90 | 274 | 326 | 183 | 11 | 307 – 172 = 136 |
2086,71 | 98 | 238 | 306 | 223 | 220 - 83 = 138 |
Ian R.G. Wilson published a tidal model of Venus, Earth and Jupiter with a period of 11.07 years. He notices that the derived period of synchronization of these planets 575.52 years can be exactly a quarter of the Hallstatt cycle. At the same time he pointed out the possible connection with the moon's tidal tides, which show a significant period of 574.6 years.
While observing the alignment of planets V-E-J, it is actually possible to find the period H/4 - planetary configurations even show some kind of symmetry in time here. Symmetry centers appears in years:
111.5 AD, 687.1 AD, 1262.6 AD, 1838.2 AD
The following intervals appear between these data: (derived from observed Jupiter-Earth-Venus conjunctions with accuracy up to 2 degrees).
------ 111.5 AD 44.8, 65.6, 44.8, 44.8, 65.6, 89.5, 65.6, 155.1 years --------- 687.1 AD 155.1, 65.5, 89.5, 65.5, 89.5, 65.5, 44.8 years --------- 1262.6 AD 44.8, 65.5, 89.6, 65.5, 89.5, 65.6, 155.1 years --------- 1838.2 AD 155.0, 65.6 years
All intervals are multiples of approximately 11 years:
44.8 years = 4 * 11.20 years 65.6 years = 6 * 10.93 years 44.8 years = 4 * 11.20 years 44.8 years = 4 * 11.20 years 65.6 years = 6 * 10.93 years 89.5 years = 8 * 11.19 years 65.6 years = 6 * 10.93 years 155.1 years = 14 * 11.08 years
Here two periods take turns: one when "wins" average interval of the V-E conjunctions, from which come the periods 11.19-11.20 years and second with periods of the average interval of J-E conjunctions, fitting to 10.92-10.93 years. (Which one have to win is decided by speeds on elliptical orbits ...!?)
The average period from the observed H/4 = 575.55 years results to be 575.55 = 52 * 11.068 years.
At the breaks, Mars is emerging (in combination with V, E and J). With the period of 1151.1 years, Mars comes in line with the tides of Jupiter-Earth-Venus.
6.8.1262 9.3.1838In 1151.1 years Venus will make 1871 orbits (+ about 15 degrees extra) The Earth make 1151 orbits (+ approx. 15 degrees extra) and Mars approximately 612 orbits ... - Difference 1871- 1151 = 720 = 4 * 180, difference 1151-612 = 539 = 3 * 180 - 1 !?
Let us try to look for some period h that can have for inner planets meaning similar to period H of outer planets.
Period hLet h is period of stable resonance, 1/h = 1/M-4/V+2/E+1/R (c. 5.504 years). Observer moving with this period (in the world of combined synodical periods) will get the following values of ((M,V),(V,R)),((M,V),(E,R)) and ((M,E),(E,R)):
1/((M,V),(V,R))-1/h = 1/M-2/V+1/R-1/h = 2/V-2/E = 2/(V,E) 1/((M,V),(E,R))-1/h = 1/M-1/V-1/E+1/R-1/h = 3/V-3/E = 3/(V,E) 1/((M,E),(E,R))-1/h = 1/M-2/E+1/R-1/h = 4/V-4/E = 4/(V,E)
Therefore for this observer it holds: