﻿ Solar activity - Resonance

# Solar activity - Resonance

You're reading an English translation of the original Czech page.
 A.M.Molchanov, Russian mathematician. He devoted himself to mathematical modeling in biology, ecology, genetics, etc. He assumed resonance structure of the solar system. The development of the planet's leads to resonances and orbital periods in the ratio of small integers. He suggested some relationships for both the sun and the sun system, as well as for the systems of Jupiter and Saturn.

## Resonance of orbital periods

### Introduction

Three Jupiter and three Uranus moons respect the so-called Laplace resonance, which limits the possibility of multiple conjunctions. It is a certain principle of minimal interaction - which ensures balance.

Therefore, resonance does not generally only represent a disturbing element (which can break objects, destroy bridges, etc.), but it can also be a thing to ensure synchronization and stability.

There is a simple argument regarding the moon of Jupiter / Uranus: they are synchronized thanks to the tidal forces. According to A.M.Molchanov (1965, 1968), however, the bodies can be synchronized even when operating by very weak forces. Molchanov considers "small dissipative forces", in some newer theories also other hypotheses were proposed (M. B. Gubaidullin, 2015, see also N. Scafetta, 2013).

According to the behavior of Jupiter / Uranus moons, generalization offers: multiple conjunctions of bodies are not desirable in the Solar System - there is a tendency for them to appear only minimally, not to appear exactly or to be somehow balanced.

Small (tidal) effects can enhance the natural oscillation of the solar atmosphere and cause great changes.

### Resonances of the inner planets

J.J.Condon and R.R.Schmidt at work (1975) derive a resonance relationship in the form Lt = 3 * Lv -5 * Le + 2 * Lj where Lv, Le and Lj are the longitude of Venus, Earth and Jupiter.

From the comparison of the synodic periods of the planets, ((E, J) / 5, (V, J) / 3) = 2 · W. Then 1 / (2W) = 3 / V-5 / E + 2 / J, and 2W = 22.13505 years, ie, W = 11.06753 years .
Maximum solar activity occurs approximately when (for the longitude of Venus, Earth and Jupiter):

#### 3LV - 5LE + 2LJ = kπ

But that does not always apply ...

### Wilson's refinement

Here we derive:

#### 41LV - 69LE + 28LJ = kπ

A period of 22,386 years is based on 41/69 (K.Takahashi, 1967).

### Maya

For 2W = 22.3855 years, Jupiter will go back by 40.6 dg, ie, if we observe the system with a 2W period, we will sense that Jupiter is orbiting the period with the period about 198.4 years. If he really moved, he would meet Saturn every 25.65 years, ie 4 * I or 3 * Y, where I is the period of the inner planets = 6.4 years and Y is 8.54 years. The difference between 25.65 and 22.39 years is 3.26 years.   (52 Tun / 72 Almanac Cycle 18720 d, Calendar Round 52 Haab / 73 Almanac Cycle 18980 d).

### Molchanov's Resonances

Molchanov's Resonance: [M, -V, -E / 2, -R] [V, -R / 3, -S] [E, -R, 2, J, / 2] [U, -U / 7], [U, -N / 2], [U, -P / 3] (plus their linear combination ...)

### Resonance with period H

In the repetition of the climatic periods, there is a so-called Hallstatt cycle (H = 2300 years)      which correlates with the resonance of orbital periods of outer planets.

The difference of 2403.05 years and 2224.10 years (with the sum of 4627 years) is 178.95 years,      but the common divisor is the period 185.1 years (+ -0.2 years)      which can be modeled by resonance 1 / J-3 / S + 2 / N (or ((U, N), H)).

## Balancing of Jupiter

Line of conjunctions V-E-J move (backward) with period: P= (W,J) = (11.06753, 11.861983) = 165.25 years, i.e. approximately with orbital period of Neptune.
Axial period Jupiter-Neptun [J,N] = 22.13075 y = 2*11.06538 years coincides with period of Solar activity.
Some authors (Ray Tomes,…) connect solar period with these two bodies ( Jupiter, Neptune).
(In combination with resonance of inner planet it leads to 6/V-10/E+3/J-1/N = 0, i.e. unstable resonance).

Importance of Jupiter (with a view to the weight) is clear. Neptune might acquire its influence by the distance from the sun, see, for example value (M * R2 (moment of inertia):

 M [1024 kg] a [109 m] M*a2 [1039kg*m²] Jupiter 1899,0 778,6 1151,2 Saturn 568,0 1433,5 1167,2 Uran 86,8 2872,5 716,2 Neptun 102,0 4495,1 2061,0.

### Resonance

Also period of resonance 1/W = 1/J + 1/U – 1/N correlates by value W = 11.094 years with the mean period of the Solar cycle. Let us write it to form of stable resonance: 1/W - 1/J + 1/N - 1/U = 0.

Resonance maxima occur, when it holds (for longitudes of Jupiter, Jupiter perihelion, Uranus and Neptune):

#### Lj – Ljp + Lu - Ln = π+2kπ

Resonance minima occur, when it holds (for longitudes of Jupiter, Jupiter perihelion, Uranus and Neptune):

#### Lj – Ljp + Lu - Ln = 2kπ

This resonance binds the period of solar activity W with orbital periods of planets Jupiter, Neptune, Uranus. Saturn's here seemingly outside game - however it is bound by resonance of outer planets to Jupiter perihelion (viz Influence of Jupiter)!?

The resonance arise as a sum of the two delta angles ∆J + ∆UN where:

• ∆J = Lj-Ljp (Jupiter with regard to its perihelion)
• ∆UN = Lu-Ln (Uranus with regard to Neptune).
 n Mo ∆J ∆UN n Mo ∆J ∆UN n Mo ∆J ∆UN n Mo ∆J ∆UN -30 1416,7 316 224 -14 1594,4 306 234 2 1772,0 296 244 18 1949,5 283 257 -29 1428,0 296 244 -13 1605,6 286 254 3 1783,1 273 267 19 1960,5 257 283 -28 1439,2 275 265 -12 1616,6 262 278 4 1794,1 246 294 20 1971,4 229 311 -27 1450,2 251 289 -11 1627,6 235 305 5 1805,0 218 322 21 1982,4 203 337 -26 1461,2 224 316 -10 1638,5 207 333 6 1815,9 192 348 22 1993,4 180 0 -25 1472,0 197 343 -9 1649,4 182 358 7 1826,9 168 312 23 2004,5 159 21 -24 1483,0 172 8 -8 1660,5 160 20 8 1838,1 149 31 24 2015,7 140 40 -23 1494,1 150 30 -7 1671,6 140 40 9 1849,3 130 50 25 2026,8 120 60 -22 1505,3 130 50 -6 1682,8 120 60 10 1860,4 109 71 26 2037,9 96 84 -21 1516,4 110 70 -5 1694,0 99 81 11 1871,5 85 95 27 2049,0 70 110 -20 1527,5 88 92 -4 1705,0 74 106 12 1882,5 58 122 28 2060,0 42 137 -19 1538,6 63 117 -3 1716,0 47 133 13 1893,5 31 149 29 2071,1 18 162 -18 1549,6 36 144 -2 1727,1 21 159 14 1904,6 7 173 30 2082,3 356 184 -17 1560,7 10 170 -1 1738,2 357 183 15 1915,9 346 194 31 2093,6 336 204 -16 1571,9 346 194 0 1749,5 336 204 16 1927,2 326 213 32 2104,8 316 224 -15 1583,7 326 214 1 1760,8 316 224 17 1938,4 306 234 33 2116 294 246
Simplified diagram of solar activity power follows.
Observed maximum so (according to axes direction) can be of 4 distinct types,
see e.g. I.1727.5 (140.0), II.1870.6 (140.5), III. 1989.6 (150.1), IV. 1957.9 (201.3) :

## Maxima of Solar activity

### Comparison of extremes

In the following overview these maxima are compared:
• maximum of Solar activity MS
• maximum of Wood's resonance of inner planets MI: 3LV - 5LE + 2LJ = kπ
• maximum of resonance of balancing the outer planets MO: LJ - LN + LU = LJP + π (+2kπ)
• maximum of resonance of geometrical axes MA: (LJ - LS) + (LN - LU) = kπ/2

 N WS WI WO WA WI-WS Wo-WS Wo-WI -23 1492 1494,6 1494,1 1490,4 2,6 2,1 -0,5 -22 1505 1505,8 1505,2 1501,6 0,8 0,2 -0,6 -21 1519,0 1517,3 1516,4 1513,1 -1,7 -2,6 -0,9 -20 1528,0 1527,5 1527,5 1523,7 -0,5 -0,5 0,0 -19 1539,0 1539,1 1538,6 1535,6 0,1 -0,4 -0,5 -18 1548,0 1549,6 1549,6 1546,9 1,6 1,6 0,0 P-maximum -17 1558,0 1560,8 1560,7 1558,1 2,8 2,7 -0,1 -16 1572,0 1572,5 1571,9 1570,1 0,5 -0,1 -0,6 -15 1581,0 1583,6 1583,7 1580,4 2,6 2,7 0,1 -14 1591,0 1594,6 1594,4 1591,7 3,6 3,4 -0,2 -13 1604,0 1606,2 1605,6 1602,8 2,2 1,6 -0,6 1607 - Kepler's spot -12 1615,5 1616,8 1613,1 1613,0 1,3 1,1 -0,2 1618 - Simon Mair's (Marius) spots -11 1626,0 1627,8 1627,6 1625,3 1,8 1,6 -0,2 -10 1639,5 1638,8 1638,5 1636,3 -0,7 -1,0 -0,3
 n WS WI WO WA WI-WS WO-WS WO-WI -9 1649,0 1649,9 1649,4 1647,5 0,9 0,4 -0,5 -8 1660,0 1660,8 1660,5 1659,2 0,8 0,5 -0,3 1660 - Hevelius spots, 1661 - Boyle and Picard sunspot group -7 1675,0 1672,2 1671,6 1669,5 -2,8 -3,4 -0,6 1671 (August) - Pickard spot, Hevelius spot, 1676 - Pickard extreme, Maunder -6 1685,0 1682,5 1682,8 1681,0 -2,5 -2,2 0,3 1684 - Pickard extreme, Cassini, Kirch -5 1693,0 1693,5 1694,0 1692,2 0,5 1,0 0,5 -4 1705,5 1705,2 1705,0 1703,1 -0,3 -0,5 -0,2 -3 1718,2 1716,0 1716,0 1715,2 -2,2 -2,2 0,0 1716 - visible aurora, 1714-15 - P-maximum, end of Maunder period -2 1727,5 1727,4 1727,1 1726,3 -0,1 -0,4 -0,3 -1 1738,7 1738,5 1738,2 1737,8 -0,2 -0,5 -0,3 0 1750,3 1750,1 1749,5 1749,2 -0,2 -0,8 -0,6 1752 - second maximum 1 1761,5 1761,3 1760,8 1759,7 -0,2 -0,7 -0,5 2 1769,7 1771,8 1772,0 1771,4 2,1 2,3 0,2 1774 - second maximum 3 1778,4 1782,8 1783,1 1782,3 4,4 4,7 0,3 4 1788,1 1794,5 1794,1 1793,4 6,4 6,0 -0,4
 n WS WI WO WA WI-WS WO-WS Wo-WI 5 1805,2 1805,5 1805,0 1805,7 0,3 -0,2 -0,5 6 1816,4 1815,6 1815,9 1816,3 -0,8 -0,5 0,3 7 1829,9 1827,2 1826,9 1827,7 -2,7 -3,0 -0,3 8 1837,2 1838,3 1838,1 1838,7 1,1 0,9 -0,2 9 1848,1 1849,2 1849,3 1848,9 1,1 1,2 0,1 1847-48 - Giant sunspots 10 1860,1 1860,2 1860,4 1860,7 0,1 0,3 0,2 1859 - proton event (Carrington), 1864 - second maximum, 1859-60 - Giant sunspots 11 1870,6 1871,3 1871,5 1871,6 0,7 0,9 0,2 1870-71 - Giant sunspots, 1870 - proton event (Young) 12 1883,9 1881,7 1882,5 1882,9 -2,2 -1,6 -0,8 1882 - P-maximum, 1882 - Giant sunspot, 1882 - proton events (Maunder) 13 1894,1 1893,5 1893,5 1894,9 -0,6 -0,6 0,0 1892-93 - Giant sunspots,  1892 -  proton events (Rudeaux) 14 1907,0 1904,5 1904,7 1905,7 -2,5 -2,3 -0,2 1905,07,08 - Giant sunspots, 1908 - proton event 15 1917,6 1916,1 1915,9 1917,3 -1,5 -1,7 -0,2 1917 - Giant sunspot, 1917 - proton event 16 1928,4 1926,7 1927,2 1928,4 -1,7 -1,2 -0,5 1926,29 - Giant sunspots, 1926 - proton event 17 1937,4 1937,9 1938,4 1939,1 0,5 1,0 0,5 1935,37-42 - Giant sunspots, 1938 - proton event 18 1947,5 1949,5 1949,6 1951,1 2,0 2,1 0,1 1946 - proton event, 1949 - proton event
 n WS WI WO WA WI-WS WO-WS WO-WI 19 1957,9 1960,5 1960,5 1962,1 2,6 2,6 0,0 1958 - the highest level since Galileo's observations (1610), 1956,59-61 - proton events 20 1968,9 1972,2 1971,4 1973,8 3,3 2,5 -0,8 1969 - proton event, 1972 - extreme flare, 1972 - second maximum 21 1979,9 1983,2 1982,4 1985,5 3,3 2,5 -0,8 1981 - extreme flare 22 1989,6 1993,2 1993,4 1996,1 3,6 3,8 0,2 1990 - the second highest level since Galileo's observations (1610) 23 2000,5 2004,3 2004,6 2007,8 3,8 4,1 0,3 24 ? 2015,3 2015,7 2018,3 ? ? 0,4 25 ? 2026,1 2026,8 2028,9 ? ? 0,7 26 ? 2036,6 2037,9 2040,9 ? ? 1,3 27 ? 2048,1 2049,0 2051,5 ? ? 0,9 2050-51 - P-maximum 28 ? 2059,5 2060,0 2063,4 ? ? 0,5 29 ? 2070,6 2071,1 2074,7 ? ? 0,5

## Subsequent considerations

### Regularities

Theoretical extremes (maxima as well as minima) of solar activity take place about with the same period as the actual extremes. The biggest differences in maximum are observed in years 1583-1606, 1675-1685, 1778- 1788, 1829, 1958-2000. The biggest differences in minimum are observed in years 1553,1619, 1784, 1902-1924,1996.

Justin Flynn has noticed certain regularities of these deviations (personal communication).

Consider the following chart:

### Rule of alternating deviations

Let us divide the Uranus-Neptune conjuction cycle to following time-periods, and to write departures taken from the column Mo-MS of the tables above:

```    Time-      Departures
period     sign  maxima   minima
------------------------------------
1543-1586  -
1586-1628  +
1628-1671  -
1671-1714  +
1714-1757  -  1718-1761  1745-1765
1757-1800  +  1769-1788  1777-1799
1800-1842  -  1805-1830  1810-1832
1842-1885  +  1838-1871  1844
1885-1928  -  1884-1928  1856-1943
1928-1971  +  1938-2004  1954-2008
1971-2014  -  ?????????
2014-2057  +
```

For the period 1714 -1971 all seems to work quite well. But this rule (like the rule from the paragraph Coincidence of geometrical axes) fails - and again roughly after year 1960!?)

### Correction of extremes WO

Let dUN = Neptune.Longitude - Uranus.Longitude, dJJa = Jupiter.Longitude - Jupiter.LongPerihelion - 180 and we look for extremes where dUN + correction = dJJa (with a given precission, e.g. 5 dg).

The following function was derived from the previous paragraph: f = Cos(2*dUN) * abs(Cos(2*dUN)) = sign(Cos(2*dUN)) * Cos2(2*dUN) with a correction angle equal to k * f. For value of the constant k = 120 we get the following table:

```    Diff    Theory     Jp    J    S    U    N  f(dUN)  Corrected   Observation
------------------------------------------------------------------------------
11,28   1749,39 -  15  346  231  322  118  +0.38   1750,73     1750.3  *
1757
11,25   1760,65 -  14  326  358    6  143   0.00   1760,72     1761.5  *
11,20   1771,84 +  14  305  146   51  167  -0.30   1770,79     1769.7  *
11,09   1782,93 +  13  281  276   98  191  -0.87   1779,56     1778.4  *
11,01   1793,94 +  14  256   46  148  215  -0.70   1791,19     1788.1  ?
1800
10,92   1804,86 -  15  229  191  200  238  +0.06   1805,13     1805.2  *
10,92   1815,79 -  15  203  314  249  262  +0.98   1819,56     1816.4  ?
11,01   1826,79 -  14  179   92  296  286  +0.73   1829.80     1829.9  *
11,14   1837,94 +  14  159  233  340  310  +0.21   1838,80 	   1837.2  .
1842
11,17   1849,11 +  14  140  359   24  334  -0.02   1849,04     1848.1  *
11,17   1860,28 +  14  119  147   69  359  -0.48   1858,55     1860,1  *
11,12   1871,39 +  15   97  277  118   24  -0.99   1868,05     1870,6  ?
10,98   1882,37 -  15   69   48  169   48  -0.29   1881,41     1883,9  ?
1885
11,01   1893,38 -  14   41  193  220   72  +0.25   1894,28     1894,1  *
11,14   1904,52 -  14   17  318  270   97  +1.00   1908,13     1907,0  *
11,23   1915,75 -  14  356  101  315  122  +0.66   1918,04 	   1917,6  *
11,34   1927,08 -  16  338  242  360  147  +0.14   1927,54     1928,4  *
1928
11,23   1938,31 +  16  317   10   44  171  -0.08   1938,11     1937,4  *
11,12   1949,42 +  14  294  157   91  195  -0.63   1947,03 	   1947,5  *
10,98   1960,40 +  14  267  284  141  218  -0.99   1956,54     1957,9  *
10,90   1971,30 +  14  239   55  192  242  -0.03   1971,21     1968,9  ?
1971
10,95   1982,25 +  15  214  199  242  265  +0.68   1984,90     1979,9  ??
11,03   1993,29 +  16  191  323  289  289  +0.94   1997,03     1989,6  ??
11,12   2004,40 +  15  170  105  334  313  +0.47   2006,28 	   2000,5  ??
11,14   2015,55    14  150  243   17  338  +0.78   2015,70             ?
2014
```

Rows with "acceptable" results are marked by asterisks.

## Other resonances

### Listing

Resonances observed above are not only resonances producing cycles with period close to the period of solar activity. For example (J/1,-S/2,+U/4,-N/3) = 21.8 years while (J/1,-S/3,+U/4,-N/2) = 55.8 years. Also (S/1,+U/3,-N/4) = 22.0 years, therefore e.g. cos(Ls+3*Lu-4*Ln) oscillates with 11-years period etc.

```    J  S  U  N        k*W          M
------------------------------------
0  3  1 -4        11,2       -971,5
1  0  2 -3        11,1      -1671,2
2 -3  3 -2        11,1      -5971,4
0  1  3 -4        22,0       7807,8
1 -1 -1  1        22,5      -1376,5
0  1 -1  0        45,4      -1724,2
1 -2  0  1        44,5      -6717,5
0  0  2 -2        85,7       2829,8
0  1 -3  2        96,3      -1072,9
1 -3  3 -1        82,7       1275,0
```

### Resonance with inclusion of Saturn

Now let us consider resonance with period 44.5 years according to row:

``` J  S  U  N        k*W          M
------------------------------------
1 -2  0  1        44,5      -6717,5
```

After rewriting to the form:

#### LJ - 2* LS + LN = LJP + k*π/2

we get new values WN (we will compare them to extremes WO):

```Spacing     WN     WO
0,00   1605,503	1605,6
9,09   1614,593	1613,1
14,84   1629,433	1627,6
8,27   1637,701	1638,5
10,82   1648,516	1649,4
13,03   1661,548	1660,5
8,10   1669,652	1671,6
14,26   1683,917	1682,8
9,67   1693,582	1694,0
9,75   1703,329	1705,0
13,61   1716,936	1716,0
9,42   1726,355	1727,1
11,80   1738,155	1738,2
12,21   1750,366	1749,5
9,34   1759,702	1760,8
13,53   1773,228	1772,0
9,91   1783,139	1783,1
```
```Spacing   WN     WO
9,75   1792,886	1794,1
14,35   1807,232	1805,0
8,19   1815,419	1815,9
12,70   1828,123	1826,9
11,14   1839,266	1838,1
8,27   1847,534	1849,3
14,59   1862,127	1860,4
9,25   1871,382	1871,5

10,16   1881,539	1882,5
13,28   1894,818	1893,5
9,34   1904,154	1904,7
12,54   1916,694	1915,9
11,47   1928,166	1927,2
9,75   1937,913	1938,4
13,61   1951,520	1949,6
9,58   1961,103	1960,5
```
```
Spacing   WN     WO
10,98   1972,082	1971,4
13,11   1985,196	1982,4
8,27   1993,465	1993,4
14,10   2007,565	2004,6

9,58   2017,148	2015,7
8,52   2025,663	2026,8
14,76   2040,420	2037,9
8,76   2049,181	2049,0
```
• Good agreement between the extremes WN and WO exists due to resonance [J/3, -S/8, -U, N/5] (it is derived from the difference of the terms for WN and WO as a function of planetary longitudes...). Leading period of this resonance is (J/3, S/8) = 53.555 years (ie. B/8, divisor of Babylonian period B).
• But - whether all the periods have to be measured relative to Jupiter's perihelion, or to other points (e.g. each planet to its perihelion etc.) - is not clear.
• The values of the spacing (i.e. the distance of two neighboring extremes) appear to be bi-modal, similar to the real solar cycles. But do not correspond to the observed intervals.

## Other observations

### Significant deviation

Interval of years 1761 - 1805 (with a centre around y. 1783) may be important. There would be solar-activity extremes expected on years marked by # (odd rows), but real activity was observed rather on years marked by asterisks:

```Theory      Jp    J    S    U    N    Observation
-------------------------------------------------------
# 1760,65 ***  14  326  358    6  143    1761.5
1765,96      14  136   66   27  154
# 1771,85 ***  14  305  146   51  167    1769.7
1777,08 ***  13  115  211   73  179    1778.4
# 1782,94      13  281  276   98  191
1788,14 ***  13   90  334  122  202    1788.1
# 1793,94      14  256   46  148  215
1799,14      14   63  117  173  226
# 1804,87 ***  15  229  191  200  238    1805.2
```

There is certain symmetry or perpendicularity, alignment of outer planets on the year. 1783 (see image of planetary position - Jupiter perihelion is on the right side of the image...!?)