Popis: D:\binary\images\c_mevem.jpgSynchronization

Mercury and Venus

Venus action on Mercury

Venus is heavier than Mercury and revolves on a circular orbit. Mercury orbit is considerably eccentric.
After conjunction with Venus Mercury gets an additional impulse (Venus decreases moment of Mercury movement and it starts move like it was nearer to the Sun.) Before conjunction is Mercury on contrary decelerated.

Similar phenomena cause in case of satellites of Jupiter, Saturn and Uranus resonant motions. But Mercury and Venus do not show such a resonant binding. Conjunctions M-V appear also when Mercury is in aphelion, e.g. at end of year 1993 and near half of year 1999 (after c. 5.54 years ~ W/2, this cycle is not permanent,  it disappear after 9 or 10 occurrences replace by new, phase shifted cycle…):

(Interval) Conjunction M-V      Mercury in aphelion

-----------------------------       ------------------------

             1993.989                1993.986 1994.227

  ( 0.381)   1994.370                1994.468 1994.709

  ( 0.433)   1994.802                1994.950

  ...

             1998.718                1998.805 1999.046

  ( 0.433)   1999.150                1999.286

  ( 0.383)   1999.533                1999.528

Day on Mercury and Venus

Synchronization Mercury with Venus can be seen only in combination with rotation.
Solar day on Mercury lasts (M,Mr) = (87.969, 58.646) =175.938 days, on Venus (V,Vr) = (224.8008,-243.1)=116.796 days.

So it holds approximately:

 (M,Mr)/(V,Vr)= 3/2



Solar day on Venus is 1.5 multiple of solar day on Mercury. To keep periods exactly in this ratio some small changes are necessary, e.g. to shorten mean rotational period of Mercury by 2 hours.

Synchronization of inner planets

Beats of synodical periods

Beats of synodical periods of inner planets:

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

All periods of beats are parts of spectrum of period c. 1510-1513 years.

Eruptions of volcanoes

Two big volcanic explosions appeared at years of conjunctions V-E-R (interval c. 32 years):

         1777.313

( 6.447) 1783.761 Laki, Island 8.6. 1783 (až do r. 1784)

         …

         1809.289

( 6.462) 1815.751 Tambora, Sumbava 5-11.4. 1815

( 6.356) 1822.107

Conjunctions V-E-R in years 1783.8, 1815.8 and 1835.0 were very close dL<5°(step 1 day)

Eight-year period

Let us note that years of some big eruptions 1783, 1815, 1835, 1883 are congruent according to module of 4-years.

Synchronization of Galilean moons

Rotation of coordinates

Strong influences of Jovian satellites (Io,Europa,Ganymedes) cause computational difficulties (Wargentin, Lagrange, Laplace, Souillart). De Sitter (1918) solved this problem by movable co-ordinate system.

Let the system move with period T.  Then: (I',-E'/3,2/G') = 0 and I'= E'/2= G'/4;  where I'=(I,T), E'=(E,T), G'=(G,T) and T =(I,E/2)=(E,G/2).

        Solar (X,J)  Sidereal X   Modulated (X,T)

----------------------------------------------------

 I        1.769860      1.769138       1.762731 (1)

 E        3.554094      3.551181       3.525464 (2)

 G        7.166387      7.154553       7.050928 (4)

 K       16.753552     16.689018      16.135846

----------------------------------------------------

(I,E/2) 437.613       486.829          -

(E,G/2) 437.637       486.810          -

In the modulated system it holds: (I,E) = E,  (E,G) = G,   (I,G) = [I,E] = [E,G]/2; from relation 1/I-3/E+2/G follows:  (E, [I,G,K]) = 3∙(G,K).

Geometric axis [I,G,K]  move with regard to E with period, that is exact multiple of (G,K) (regardless of value K).

Other resonances

E.g.  4/I-9/E+5/K = 0 (38.1 days),  9/E-25/G+16/K = 0  (836.2 days)

Half-day period:  (I,K)  1.978915  =   4∙  0.49473 days, (E,K)  4.511072  =   9∙  0.50123 days,  (G,K) 12.523247  =  25∙  0.50093 days.
Synchronization with Jupiter:  1/I-3/E+5/G-7/K = 0 (107.65 days ? 9∙J)
Inclusion of Kallisto: 

1/I -3/E+5/G -7/K = 0

Popis: D:\binary\images\c_csync.jpgCoordination with (V,E)?

Conjunctions of all Galilean satellites of Jupiter appear with period 437.64 d (Meeus), i.e. 1.1982 years.
During this time Venus pass from the furthermost west to the furthermost east elongation (439-443 d).

Are Galilean moons synchronized with conjunctions of pair Venus-Earth?

V-E      |  conjunctions of all Galilean satellites

--------------------------------------------------

   1970.87   |   1970.89  1971.29  1971.69  1972.09

   1972.47   |   1972.49  1972.89  1973.29  1973.69

   1974.07   |   1974.09  1974.49  1974.89  1975.29

   1975.66   |   1975.69  1976.09  1976.49  1976.89

   1977.27   |   1977.28  1977.68  1978.08  1978.48

   1978.86   |   1978.88  1979.28  1979.68  1980.08

   1980.46   |   1980.48  1980.88  1981.28  1981.68

   1982.06   |   1982.08  1982.48 ...

Period 437.64 days corresponds approximately to:

Ø  3/4 synodic period Venus-Earth (3/4*(V,E)=3/4*583.92=437.94 days)

Ø  period of Chandler spiral motion of earth magnetic pole
(according to various authors it has value 1.17-1.21 y; e.g. Jeffreys: 1.202 y; Rudnik: 1.193 y, Fairbridge 1.1855 y,...)

Ø  period of nutation of earth axis (438 d)

Ø  J/10; tenth of orbital period of Jupiter (J/10=433.26 d)

Ø  16 sidereal lunations (16*27.3217=437.15 d)


Exact motion of Galilean satellites is based on observations of J.N.Delisle (1688-1768), A.Pingré (1711-1796), Delambre (1749-1822). There exist a large collection of data 1652-1982 (Jay H. Lieske, Jet Propulsion Laboratory).


Other indications of synchronization

Halley, Edmond, 1656-1742

Asteroids and comets 

The more eccentric orbit is, the more probable seems to be synchronization of body with certain orbital period (rather than with synodic period).

Halley comet (the periodic comet with mean orbital period CH= c. 76-77 years) is observed a few millennia. So, it possibly would synchronize with motion of other, greater bodies. Some theorists assume relation to period 13J/2 = 77.1 years (13/C-2/J-11/P=0).
From axial period J-S we get:

9∙[J,S]/2 = 9∙16.913/2 = 76.11 let

Comparison of longitude of axis J-S (La) with longitude of Halley comet in perihelion:


tp[year] Lj[°] Ls[°] La[°] Lh[°]
----------------------------------
 989,7 329 299 314 147.3
 1066,2 140 165 153 149.3
 1145,3 16 40 28 152.0
 1222,7 203 270 237 152.4
 1301,8 90 164 127 153.9
 1378,9 259 13 136 157.0
 1456,4 157.0
 1531,7 218 86 152 159.3
 1607,8 12 294 153 160.6
 1682,8 131 138 134 164.1
 1759,2 281 341 311 167.2
 1835,9 99 210 154 167.5
 1910,3 192 28 110 169.5
 1986,1 329 244 286 tp[year] Lj[°] Ls[°] La[°] Lh[°]          

----------------------------------

  -85,4   110 130   120   124.1

 -10,2   224 317   271   127.7

   66,1    23 183   103   128.0

  141,2   146   7    77   130.2

  218,3   318 239   279   131.3

  295,3   143  99   121   133.6

  374,1     9 333   171   136.4

  451,5                   137.5

  530,9    93  97    95   138.9

  607,2   242 305   273   141.3

  684,4    72 179   126   142.2

  760,4   214  15   114   144.0

  837,2    24 242   133   144.3

  912,6   155  80   117   145.7

Orbital period of asteroid Chiron is 50.42 years. From axial period J-S we get:


3∙[J,S] = 3∙16.913 = 50.74 let

Inner planets to axis Jupiter-Saturn

Let Vr is rotational period of Venus. Period (E,Vr) = (365.256,-243.01) = 291.85 days (9∙32.42 days) and (V,E)/2 = 291.96 days (9∙32.44 days),

so:

1/Vr-5/V+4/E=0

Let us consider observer moving with axial period [J,S].  He will register the following axial periods of pairs of inne planets:

([M,V], [J,S])= (126.43848, 6177.562)= 129.080 days=  4∙32.270 days

([V,E], [J,S])= (278.23510, 6177.562)= 291.357 days=  9∙32.373 days

([E,R], [J,S])= (476.93428, 6177.562)= 516.836 days= 16∙32.302 days

These periods are approximately n2 multiples of quanta c. 32.2-32.4 days. Their common period is c. 4646- 4662 days (12.720-12.764 let).

Gauss’s polygons

Motion of barycentre of system Sun-Jupiter-Saturn makes trefoil in the plane of motion. Similarly:

(U,N)/U = 2/1  -> one to two sides, (J,S)/J = 5/3  -> trefoil, (V,E)/V = 13/5  -> pentagon

Construction of regular n-gons with help of compass and ruler was know only for n=2j,3 a 5 (and combinations).

Gauss has shown, that it is possible to extend this with all Fermat’s numbers n=2k+1 (k=2t, tεN) i.e. n=17,257,65537,…

E.g.  in interval n=1..60 these n-gon constructions are possible:  n=2,3,4,5,6,8,10,12,15,16,17,20,24,30,32,34,40,48,51,60.

Let us note that approximately 4286 years /84.01 years = 51, i.e.:

 (U,N/2)/U   = 3∙17

Other relations

Let us consider stable resonances A= (V/3,-E/7,R/4) and B= (M,-V/5,E/4); i.e. (E,R):(V,R)=7:3 (2.33569), (V,E):(M,E)=5:1 (5.03913).

Composition of both relations makes stable resonance with beats (M,-V/2,-E/3,R/4):

1/M-2/V-3/E+4/R = 0

Metonic system

Cycles of Moon phases (Meton’s, Exeligm’s, Kallip’s,..)  makes multiples of 19 year period.

Highest beat of J-S contains (J/2,S/5)=883 years, (J,S/2)=60.95 years,  (J,S/3) = 57.01 years. 

S/1

S/2

S/3

S/4

S/5

S/6

S/7

S/8

J/1

19.859

60.947

57.013

19.422

11.705

8.376

6.522

5.340

J/2

7.426

9.929

14.978

30.474

883.27

28.507

14.487

9.711

It holds: (J,S/3)= (U,N)/3 i.e.

1/J-3/S-3/U+3/N = 0


Planetary interactions