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CYCLE SUBJECTS / ASTRONOMY / LADMA ->
Vladimir Ladma
Axial period of two periods P, Q is period:
[P,Q] = 2/(1/P+1/Q)= 2*P*Q/(Q+P).
We designate axial period with brackets [].
For any periods A,B,C and constant k it holds:
Axial period is period, after which the axis of angle P-S-Q align with its original position; S is the point around which motion of bodies P,Q happens (centre of gravity).
At time of planetary conjunction P-Q, line S-P is aligned with line S-Q.
During time t the first line cover angle t/P (*360°), meanwhile
the second line angle t/Q (*360°).
Axis of these lines to the original line then contains angle:
t/Q+(t/P-t/Q)/2 = t/Q+t/(2P)-t/(2Q) = t *(1/(2P)+1/(2Q)) (*360°).
So, during time [P,Q] axis runs through full angle 360°.
We can imagine axial period (in contrast to synodic period) as an orbital
period; (and we can, as the case may be, compare it with other orbital periods).
Gravity centre of (three) bodies moves (around S) approximately with axial
period.
[M,V]= 0.3461696 y (126.43848 d) [M,E]= 0.3881986 y (141.78956 d) [M,R]= 0.4270134 y (155.96664 d) [V,E]= 0.7617662 y (278.23510 d) [V,R]= 0.9271408 y (338.63814 d) [E,R]= 1.3057750 y (476.93428 d)Pair axial periods of outer planets:
[J,S]= 16.9132418 y ( 6177.562 d) [J,U]= 20.7889842 y ( 7593.176 d) [J,N]= 22.1307494 y ( 8083.256 d) [S,U]= 43.6210092 y (15932.574 d) [S,N]= 49.9791756 y (18254.894 d) [U,N]=111.2908942 y (40649.000 d)
Configuration of more bodies is axially symmetric, when certain axes are identified into one axis. In case of even number of bodies such situation occurs, when all axes of selected pairs of bodies align. Alignment of two axes can be at angle 0° or 180°, period of alignments is half of synodic period of axes.
Four outer planets of solar system have symmetrical configurations of three types:
I/ JS-UN
([J,S],[U,N])/2 = (16.9132418,111.2908942)/2 = 19.94423/2 let = 9.97212 years
Date (Difference) Julian date (Math. date) 1905 May 5 AD ( 10.38741) #2416970.5 (1905.34664) 1915 Apr 7 AD ( 9.92197) #2420594.5 (1915.26883) 1925 Jan 25 AD ( 9.80424) #2424175.5 (1925.07328) 1935 Mar 4 AD ( 10.10267) #2427865.5 (1935.17617) 1944 Oct 7 AD ( 9.59617) #2431370.5 (1944.77254) 1954 Jul 21 AD ( 9.78508) #2434944.5 (1954.55782) 1964 Dec 21 AD ( 10.42026) #2438750.5 (1964.97831) 1974 Dec 28 AD ( 10.01780) #2442409.5 (1974.99632) 1985 Jan 8 AD ( 10.03149) #2446073.5 (1985.02802) 1995 Mar 25 AD ( 10.20671) #2449801.5 (1995.23494) 2004 Sep 29 AD ( 9.51677) #2453277.5 (2004.75192) 2014 Jun 10 AD ( 9.69473) #2456818.5 (2014.44685) 2024 Aug 30 AD ( 10.22313) #2460552.5 (2024.67021) 2034 Jul 20 AD ( 9.88638) #2464163.5 (2034.55680) 2044 Aug 31 AD ( 10.11636) #2467858.5 (2044.67337) 2055 Jan 16 AD ( 10.37645) #2471648.5 (2055.05005) 2064 Sep 19 AD ( 9.67556) #2475182.5 (2064.72582) 2074 Jul 31 AD ( 9.86174) #2478784.5 (2074.58777) 2084 Sep 26 AD ( 10.15743) #2482494.5 (2084.74541) 2094 Jun 20 AD ( 9.73032) #2486048.5 (2094.47594)
II/ JU-SN
([J,U],[S,N])/2 = (20.7889842, 49.9791756)/2 = 35.59471/2 let = 17.79735 years
Date (Difference) Julian date (Math. date) 1910 Dec 11 AD ( 17.42642) #2419016.5 (1910.94841) 1929 Feb 19 AD ( 18.19302) #2425661.5 (1929.14181) 1947 Jul 31 AD ( 18.44216) #2432397.5 (1947.58437) 1964 May 23 AD ( 16.81314) #2438538.5 (1964.39787) 1982 Aug 26 AD ( 18.25873) #2445207.5 (1982.65699) 1999 Oct 14 AD ( 17.13347) #2451465.5 (1999.79082) 2018 May 6 AD ( 18.55989) #2458244.5 (2018.35111) 2036 Feb 23 AD ( 17.80151) #2464746.5 (2036.15300) 2053 Jun 13 AD ( 17.30322) #2471066.5 (2053.45658) 2071 Jun 4 AD ( 17.97399) #2477631.5 (2071.43096) 2089 Mar 8 AD ( 17.76044) #2484118.5 (2089.19177)
III/ JN-SU
<[>([J,N],[S,U])/2 = (22.1307494, 43.6210092)/2 = 44.92108/2 let = 22.46054 yearsDate (Difference) Julian date (Math. date) 1911 Nov 28 AD ( 21.81246) #2419368.5 (1911.91215) 1933 Aug 3 AD ( 21.68104) #2427287.5 (1933.59365) 1956 May 22 AD ( 22.80082) #2435615.5 (1956.39496) 1979 Jul 6 AD ( 23.12115) #2444060.5 (1979.51661) 2001 Apr 30 AD ( 21.81793) #2452029.5 (2001.33500) 2023 May 29 AD ( 22.07803) #2460093.5 (2023.41350) 2046 Jul 13 AD ( 23.12389) #2468539.5 (2046.53789) 2069 Oct 9 AD ( 23.24162) #2477028.5 (2069.78000) 2091 Sep 13 AD ( 21.92745) #2485037.5 (2091.70791)
Mentioned examples are from the interval 1900-2100.
More (interval 0-2250).
Axis of angle Jupiter-Sun-Neptune moves with period:
[J,N] = [11.861983,164.770132] = 22.1307494 y ( 8083.256 d)
Axis of angle Saturn-Sun-Uranus with period:
[S,U] = [29.457159, 84.020473] = 43.6210092 y (15932.574 d)
These axes align with mean period:
([J,N],[S,U]) = (22.1307494, 43.6210092) = 44.92108 y = 2*22.46054 y = 4*11.23027 y
In the years 1540-1950, solar minima appear when axes contain angle
near to 0° or 90°.
year (diff.) extreme year (diff.) extreme -------------------------------------- 1518.3 (-1.3) MAX 1523.8 (+0.2) min 1529.2 (-1.2) MAX -------------------------------------- break of regularity 1541.1 (+1.9) min 1547.0 ( 0.0) MAX 1551.9 (+2.1) min 1558.3 (-0.3) MAX 1563.8 (?) - 1570.2 (+0.8) MAX 1574.8 (?) - 1580.6 (+0.4) MAX 1585.5 (-1.5) min 1591.9 (+1.1) MAX 1597.2 (-0.8) min 1603.0 (+1.0) MAX 1608.0 (+1.0) min 1613.3 (+0.7) MAX 1619.4 (+1.0) min 1625.5 (-0.5) MAX 1631.3 (+1.7) min 1636.5 (?) - 1642.3 (?) - 1647.7 (+2.3) MAX 1654.3 (+0.7) min 1659.5 (+1.5) MAX 1665.2 (+0.8) min 1669.8 (?) - 1676.0 (?) - 1681.2 ( 0.0) MAX 1687.6 (+1.9) min 1692.3 (+1.7) MAX 1698.0 ( 0.0) min 1703.3 (+2.2) MAX 1709.8 (+2.2) min 1715.5 (+2.2) MAX 1721.3 (+2.2) min 1726.5 (+1.0) MAX 1731.9 (+2.1) min 1738.0 (+0.7) MAX 1743.8 (+1.2) min 1749.4 (+0.9) MAX 1754.1 (+1.1) min 1760.0 (+1.5) MAX 1764.9 (+1.6) min 1771.7 (-2.0) MAX 1776.6 (+1.8) MAX? 1782.6 (?) - 1787.3 (+0.8) MAX? 1793.7 (?) - 1799.3 (-1.0) min 1805.9 (-0.7) MAX 1810.8 (-0.2) min 1816.5 (-0.1) MAX 1821.9 (+1.4) min 1828.1 (+1.8) MAX 1833.7 (+0.2) min 1839.0 (-1.8) MAX 1844.3 (-0.8) min 1849.2 (-1.1) MAX 1855.5 (+0.5) min 1861.0 (-0.9) MAX 1867.1 (+0.1) min 1871.8 (-1.2) MAX 1877.9 (+1.0) min 1883.1 (+0.8) MAX 1890.1 (-0.5) min 1895.1 (-1.0) MAX 1901.1 (+0.6) min 1905.8 (+1.2) MAX 1912.0 (+1.6) min 1917.5 (+0.1) MAX 1923.5 (+0.1) min 1928.5 (-0.1) MAX 1933.6 (+0.2) min 1939.3 (-2.1) MAX 1944.9 (-0.7) min 1951.2 -------------------------------------- break of regularity 1956.4 (+1.5) MAX 1962.2 (+2.3) min 1967.2 (?) - 1974.0 (?) - 1979.5 ( 0.0) MAX 1985.6 (+0.9) min 1990.3 (-0.3) MAX 1996.2 (+0.3) minThis regularity is more notably disturbed only in the years 1770-1790.
------ 1944.0: 78.50° solar minimum -- 1945.0: 86.5° --- 1946.0: 85.4° ------- 1947.0: 77.4° ----------- 1948.0: 69.4° --------------- 1949.0: 61.4° ------------------- 1950.0: 53.4° ----------------------- 1951.0: 45.4° -------- ------------------- 1952.0: 37.3° | --------------- 1953.0: 29.3° | ----------- 1954.0: 21.3° | ------- 1955.0: 13.3° | --- 1956.0: 5.3° V -- 1957.0: 2.6° ------ 1958.0: 10.6° SOLAR MAXIMUM ---------- 1959.0: 18.7° -------------- 1960.0: 26.7° ------------------ 1961.0: 34.7° ---------------------- 1962.0: 42.7° -------------------- 1963.0: 50.7° ---------------- 1964.0: 58.7° ------------ 1965.0: 66.7° solar minimum -------- 1966.0: 74.8° ---- 1967.0: 82.8°
Let Lj,Ls,Lu,Ln be longitudes of outer planets. When angle (Ls-Lj)
is equal to angle (Lu-Ln), then axis of bodies S,N aligns with axis
of bodies J,U.
These configurations repeat with mean period:
([J,U],[S,N]) = 35.5948 y,
i.e. period of cca 3 orbital periods of Jupiter (3*11.8620=35.5860).
During period 1500-2050 axes always align (0°, 180°) after Jupiter passing through pericentre or apocentre (usually within 1 year).
J,perihelion| (dif) Ls-Lj=Lu-Ln | J,aphelion
------------------------------------------------------------
1500.95 | (17.51) 1501.71 (18.22) 1519.93 | 1518.74
1536.54 | (17.91) 1537.84 (17.12) 1554.95 | 1554.33
1572.13 | (18.06) 1573.02 (18.19) 1591.21 | 1589.91
1607.72 | (17.54) 1608.75 (18.38) 1627.13 | 1625.50
1643.31 | (16.63) 1643.75 (18.43) 1662.19 | 1661.09
1678.89 | (17.75) 1679.93 (17.96) 1697.89 | 1696.67
1714.49 | (17.78) 1715.67 (17.19) 1732.86 | 1732.26
1750.08 | (18.20) 1751.06 (18.42) 1769.48 | 1767.84
1785.66 | (17.14) 1786.62 (18.24) 1804.86 | 1803.43
1821.25 | (16.84) 1821.70 (18.57) 1840.27 | 1839.02
1856.85 | (17.83) 1858.10 (17.62) 1875.72 | 1874.60
1892.43 | (17.79) 1893.50 (17.45) 1910.95 | 1910.19
1928.02 | (18.18) 1929.13 (18.46) 1947.59 | 1945.77
1963.61 | (16.81) 1964.40 (18.26) 1982.66 | 1981.36
1999.20 | (17.14) 1999.79 (18.58) 2018.37 | 2016.95
2034.79 | (17.78) 2036.15 (17.32) 2053.47 | 2052.53
Invasion of glaciers in the Alps; data of Bruckner cycle:
J,perihelion| Invasion of glaciers |J,aphelion|Bruckner|
------------------------------------------------------------
1500.95 | | 1518.74 | |
1536.54 | | 1554.33 | |
1572.13 | | 1589.91 | |
1607.72 | 1600-1610 | 1625.50 | |
1643.31 | 1632-1644 | 1661.09 | |
1678.89 | 1664-1685 | 1696.67 | |
1714.49 | 1712-1716 1734-1743 | 1732.26 | 1736 |
1750.08 | 1767-1770 | 1767.84 | 1771 |
1785.66 | 1787-1790 | 1803.43 | 1806 |
1821.25 | 1814-1825 | 1839.02 | 1841 |
1856.85 | 1845-1856 | 1874.60 | 1871 |
1892.43 | 1880-1894 | 1910.19 | |
1928.02 | 1914-1924 | 1945.77 | |
1963.61 | | 1981.36 | |
1999.20 | | 2016.95 | |
2034.79 | | 2052.53 | |
Let us assume, the both mentioned hypothetical patterns (solar activity, synchronization with Jupiter) holds.
So,
([J,U],[S,N]) = 3*J,
and ([J,N],[S,U]) = 4*W.
(1/J+1/N)-(1/S+1/U) = 1/(4W),
(1/J+1/U)-(1/S+1/N) = 1/(3J).
(3/J+3/N)-(3/S+3/U) = 3/(4W),
(3/J+3/U)-(3/S+3/N) = 3/(3J).
Then
3/(4W) = 2/J-3/S
W= (3J/8,S/4) = 3/4*14.97834
W= 3/4(J/2,S/3)= 11.23375 y
2W =(3J/4,S/2) = 22.4675 y
4W = (3J/2,S) = 44.9350 y
(W= [J,212.1 y], i.e. approximately [J,B/2], where B is Babylonian period (427 y)).
