Connecting the 208 Year de Vries Cycle with the Earth-Moon-Venus System

     UPDATED & CORRECTED 23/08/2013

     Direct instrumental observations of the Sun since 1610 have shown that the level of sunspot activity on the Sun has a mean periodicity of 22.3 years, known as the Hale cycle. In addition, these observations of the Sun have shown that there are longer-term periodicities present in the level of solar activity.

     One of the most prominent long-term cycles that have been identified is the ~210 year de Vries (Suess) cycle. However, because of the limited time over which instrumental observations have been available, the confirmation of the de Vries cycle [1] has required the use of proxies such as de-trended δC14 from tree rings [2,3], Be10 levels in the GRIP ice cores [4,5,6], and dust profiles in GISP2 ice cores [7].  These proxy observations have indicated that:

a) the de Vries cycle amplitude varies with a period of about 2200 years [6]. In other words, its appearance is intermittent in nature.

b)  the largest amplitude of the de Vries cycle are found near Hallstatt cycle minima centered at 8,200, 5,500, 2,500 and 800 B.P .[6]

c) grand solar minima occur preferentially at minima of the Hallstatt cycle that are characterized by large de Vries cycle amplitudes [6].

d) the cycle length is somewhere in the range 205 - 210 years, with the more precise estimates being in the range 207-208 years.

     Abreu et al. (2012) [5] have identify a 208 year period in a 9400 year reconstruction of the solar modulation potential that is derived from C14 and Be10 observations taken from ice cores. The solar modulation potential is thought to be a good indicator the strength of the solar magnetic field that is responsible for the deflection of cosmic ray, and so a good proxy of the overall level of past solar magnetic activity. Abreu et al. (2012) [5] also show that there is a 208 year period in the planetary induced torques that could act upon any asymmetric structure in the boundary layer known as the solar tachocline. These authors propose that it these planetary induced torques that could be responsible for modulating the long-term solar magnetic activity on the Sun.

      Abreu et al. (2012) [5] do not identify the specific physical mechanism that is responsible for producing the 208 year period in the planetary torques, although it is reasonable to assume that it must be linked in some way with the synodic interactions between orbital period of Jupiter [the main source of planetary torque] and one or more of the other planets.

     However, it can be shown that there is a natural 208 year periodicity associated with the position of the Earth in its orbit when it is observed at intervals separated by half the precession cycle of the Lunar line-of-apse, in a reference frame that is fixed with respect to the stars.

     The following diagram shows the angle that the Earth in its orbit about the Sun forms with a fixed direction in a sidereal reference frame, at time steps of half the precession period of the lunar line-of-apse (= 4.42558131 sidereal years for 2000.0). This angle is plotted as a function of time measured in sidereal years.

[Notes:
A: The lunar line-of-apse is a line passing through the centre of the Earth that connects the perigee and apogee of the lunar orbit.
B: The lunar line-of-apse precesses about the Earth once every 8.85116364 sidereal years, when measured with respect to the fixed stars.
C: The reference direction in the sidereal frame that was used (as T = 0 years) is that of the Earth on January 1st 2000.0 at 00:00 UT.
D: The following values for the Anomalistic month = 27.554549878-(0.00000001039*T) and the Sidereal month = 27.321661547+(0.000000001857*T) were used for all calculations, where T is the number of sidereal years since 2000.0.]

     The above diagram clearly shows that there is a natural 208 year periodicity in the alignment of the Earth with respect to the fixed with the stars when it is observed every half precession cycle of the Lunar line-of-apse. It also shows that the 208 year periodicity in alignment slowly drifts out synchronization over a period ~ 500 years. Hence, this Earth/Lunar alignment pattern exhibits two characteristics that mimic those of the de Vries cycle, namely a periodicity of 208 years and slow loss of synchronicity on millennial time scales.

     An alternative way to display the 208 year Earth/Lunar cycle that is based upon ephemeris data rather than extrapolated mean lunar orbital data is shown in the following diagram [8]. 

     This diagram shows the number of hours perigee is away from New or Full Moon, when the Moon is at perigee between the 27th of December and the 10th of January, plotted against time in years. 

   

  The limits that have been placed upon the dates of perigee are designed to restrict the observational window to +/- seven days either side of a nominal fixed date in the seasonal calender (in this case the 3rd of January which roughly corresponds to modern day Perihelion). In essence, they are restricting the observational window to +/- a quarter of a lunar orbit either side of a point in the Earth's orbit that is "fixed" with respect to the stars.

[Skip down to ******* bar if you want to avoid the following details]

     In this diagram, you see long diagonal bands from the upper left to the lower right of the diagram. The dots of the same colour in any given long diagonal band lie inside the +/- seven day calender window. Each dot is separated from its immediate neighbor (of the same colour) by almost exactly 31 years. If you move down and to the right along the long diagonal bands, from a dot of one colour, to a dot of the next colour that has the same date in the seasonal calender, you jump 106 years.

     There are three separate colour groupings in this diagram. The first grouping, vertically from the top to the bottom of the diagram, is green, red and black. The second is yellow, blue, and brown and the third is orange and purple. As you move vertically down from one colour to the next in a given colour grouping, you advance by almost exactly four years e.g. if you pick a set of green dots near the top of the diagram, the set of red dots immediately below it are shifted forward in time by four years, and the set black dots immediately below the red dots are shifted forward by a further four years.

****************

     The important point to note is that the symbols in this diagram form long diagonal lines or waves that are separated by almost exactly 208 years [see the red spacing bars in the above diagram that link points that are separated by 208 years].

      Detailed investigations show that paired points in the above diagram that are separated by 208 years, occur almost exactly 2 1/3 days apart in the seasonal calendar. This accounts for the change between the observed time of perigee and the time of New/Full Moon that occur between the paired points separated by 208 years (i.e. it accounts for the slope of the red line and, hence, the tilt of the diagonal waves). 

     It turns out that the 2 1/3 day slippage backward in the seasonal dates of the alignments between New/Full Moon and the lunar perigee every 208 years, corresponds to a westward slippage of ~ 40 arc seconds per year. This close to the westward drift of the equinoxes by 50.3 arc seconds per year that is caused by the precession of the equinox. 

     So, what it is telling us is that if we correct the above diagram for the effects of the precession of equinoxes (i.e. correct for the drift between our co-ordinate frame and the fixed stars) we get a Earth/Lunar repetition cycle for the position of the Earth in its orbit (with respect to the stars) of 208 years.    

IMPORTANT IMPLICATIONS FOR THE INTEGRATED EARTH/MOON/VENUS SYSTEM  

     Interestingly, the Earth/Venus pentagram alignment pattern resets itself with respect to the Sun and the fixed stars once every 149.5 VE alignments (of 1.59866 years) = 238.9996251 years ( with an error of 0.134964 degrees).

[see the updated and corrected blog post at:
http://astroclimateconnection.blogspot.com.au/2013/08/the-vej-tidal-torquing-model-can.html  ]

     Similarly, the relative position of the Moon in its orbit about the Earth compared to the Lunar line-of-apse reset themselves with respect to the Sun and the fixed stars almost exactly once every 31.0 sidereal years (actually closer to 31.0 sidereal years + 2 days). This comes about because:

31.0 sidereal years _________= 11322.94725312 days
383.5 synodic lunar months ___= 11324.980825 days
411.0 anomalistic lunar months _= 11324.92000 days
27.5 Full Moon Cycles _______= 11324.071833 days

     This means that that if you have a New Moon at closest perigee, 31.00 sidereal years (+ 2 days) later, you will have a Full Moon at closest perigee, on almost the same day of the calender year.

     Now, it seems quiet remarkable that:

a) The position of the Earth in its orbit, as seen once every half precession cycle of the Lunar line-of-apse (= 4.42558131 sidereal years for 2000.0), resets itself with respect to the stars once every 208.0 sidereal years.

b) The relative position of the Moon in its orbit about the Earth compared to the Lunar line-of-apse reset themselves with respect to the Sun and the fixed stars almost exactly once every 31.0 sidereal years.

c) 208.0 sidereal years + 31.0 sidereal years = 239.0 sidereal years 

and that

d) the Earth/Venus pentagram alignment pattern resets itself with respect to the Sun and the fixed stars once every 149.5 VE alignments (of 1.59866 years) = 238.9996251 years.

     One is left with the feeling that this is more than just a coincidence.     

References

1. Rogers, M. L., Richards, M.T. and Richards, D. St. P. (2006), Long-term variability in the length of the solar cycle, preprint, arXiv: astro-ph/0606426v3
2. Peristykh, A.N. and Damon, P.E. (2003) Persistence of the gleissberg 88-year solar cycle over the last ~12,00 years: Evidence from cosmogenic isotopes. Journal of Geophysical Research 108, 1003.
3. Stuiver, M. and Braziunas, T.F. (1993) Sun, ocean, climate and atmospheric CO2: An evaluation of causal and spectral relationships. Holocene 3, 289-305
4. Wagner, G., Beer, J., Masarik, J., Muscheler, R., Kubik, P. W., Mende, W., Laj, C., Raisbeck, G.M. and Yiou, F., (2001), Presence of the Solar deVries cycle (≈205 years) during the last ice age, Geophysical Research Letters 28 (2), 303-306
5. Abreu J. A., Beer J., Ferriz-Mas A., McCracken K.G., and Steinhilber F., (2012), Is there a planetary influence on solar activity?, A&A 548, A88.
6. Steinhilber F., et al., (2012), 9,400 years of cosmic radiation and solar activity from ice cores and tree rings, PNAS,  vol. 109, no. 16, 5967–5971
7. Ram, M. and Stolz, M. R. (1999) Possible solar influences on the dust profile of the GISP2 ice core from Central Greenland, Geophysical Research Letters, 26 (8), 1043-1046

8. Lunar Perigee and Apogee Calculator: http://www.fourmilab.ch/earthview/pacalc.html

The VEJ Tidal Torquing Model can explain many of the long-term changes in the level of solar activity.

 II. The 2300 year Hallstatt Cycle (*)
Updated and Corrected 23/08/2103

     It has long been recognized that there is a prominent 208 year de Vries (or Suess) cycle in the level of solar activity. Its appearance, however, is intermittent. Careful analysis of the Be10 and C14 ice-core records show that the de Vries cycle is most prominent during epochs that are separated by about 2300 years (Vasiliev and Dergachev, 2002).  This longer modulation period in the level of solar activity is known as the Hallstatt cycle ( Vitinsky et al., 1986; Damon and Sonett, 1991; Vasiliev and Dergachev, 2002).  

Summary:  

1. The VEJ tidal-Torquing model proposes that tidal bulges are formed in the base of the convective layers of the Sun by the periodically alignments of Venus and the Earth, and that it is the gravitational torque that are applied to these bulges by Jupiter that are responsible for the long-term modulation of the Solar activity cycle. Hence, it needs to be shown that the torques that are applied by Jupiter to these tidally induced bulges, naturally exhibit a 2300 year Hallstatt-like cycle.  

2. The length of the 243 year Venus transit cycle is set by the time it takes for the Earth-Venus-Sun line to re-align itself with one of the "fixed" nodes of Venus' orbit. Hence, the length of the transit cycle is determined by time it takes for the five-pointed star Venus-Earth alignment pattern to re-establish rotational symmetry plus the time it then takes for the Venus-Earth alignments to return to the "fixed" node [Note: "fixed" in this case means, roughly the same position with respect to the fixed stars].   

3. A similar strategy to that used to determine the length of the Venus transit cycle is then applied to determine the time required to precisely re-align the Jupiter torque cycle. The full length of this torque cycle includes both the time required for the realignment of the orbital position of Jupiter with respect to the Venus-Earth alignment pattern and the time required for Jupiter to re-establish its rotational symmetry pattern with respect to the fixed stars. 

4. It is shown that the Jupiter torque cycle naturally exhibits a 2302 year Hallstatt-like cycle.

(*) Note that most of the values used in this blog post are stated to four decimal places. This is not being done to claim that the values have a precision to this level of accuracy but solely for the purposes of delaying the curtailment of the number of decimal places until the end of the calculations. In addition, it is important to note that the calculation done here are just a preliminary attempt to explain why the VEJ Tidal-Torquing model produces changes in planetary torque acting upon the Sun that exhibit a Hallstatt-like cycle. A detailed analysis of ephemeris data will have to be done before these preliminary results can be confirmed.
   

The 243 year Venus Transit Cycle

Venus-Earth Alignments in a Reference Frame That is Fixed with Respect to the Stars.

     The following diagram shows five consecutive alignments of Venus and the Earth following the alignment of 2004. Each inferior conjunction of the Earth and Venus (i.e. VE alignment) is separated from the previous one by the Venus-Earth synodic cycle i.e. 1.59866 years. This means that, on average, the Earth-Venus-Sun line moves by 144.4824 degrees in retrograde direction, once every VE alignment. Hence, E-V-S line returns to almost the same orientation with respect the stars after five VE alignments or eight Earth (sidereal) years [actually 7.9933 years].

     The above figure shows that after five VE alignments (i.e 7.9933 years), the E-V-S line falls short from completing one full orbit of the Sun with respect to the stars by [(360-(360*(7.9933 - 7.0000))) =] 2.412 degrees. Hence, the E-V-S line slowly revolves about the Sun, taking 150 EV alignments (= 239.7990 years) to move backwards [clockwise in the above diagram] by one point in the five pointed star or pentagram pattern. [Note: the actual movement is 72.36 degrees over the 239.7990 years while the mean spacing between each point on the five pointed star is 72.2412 degrees]. 

     Hence, the 239.7990 year realignment symmetry for the VE alignments naturally produces a 243 year repetition time between the transits of Venus in front of the Sun. The reason for this is that the star point that is aligned with the South Node of Venus' orbit (i.e. the one pointing out of the figure above) moves one to the left after 239.7990 years, and so two extra VE alignments are required (along lines 4 and 5 in the above diagram) on top of the 150 VE aligns in 239.7990 years. This means that it takes 152 VE alignments = 242.9963 ~ 243 years before the Earth and Venus re-align near the South Node of Venus' orbit,  again.

[CORRECTION - (Thanks to Ulric Lyons)]
The precise re-alignment period for the VE Pentagram is actually 149.5 VE alignments of 1.59866 sidereal years = 238.9996251 sidereal years. However, since, a half VE alignment cannot give a transit of Venus across the Sun - the precise alignment occurs with Venus on the far side of the Sun. This means that transits of Venus in front of the Sun repeat at either 147 VE alignments = 235.0029759 sidereal years or 152 VE alignments = 242.9962744 sidereal years, with the latter being favored in recent times.      
   
     Technically, a complete repetition cycle of the pentagram pattern in VE alignments requires that E-V-S line revolves backwards by two star points (i.e. 144.4824 degrees) on five separate occasions. Hence, it takes 299 EV alignments (= 477.99934 years) to rotate backwards by two pentagonal star points and 1495 VE alignments (= 2389.9967 ~ 2390 years) to move backwards through the full Venus-Earth alignment pentagram pattern. 

     Hence, the VE alignment pentagram has a 2390 year Hallstatt-like symmetry re-alignment cycle with respect to the fixed stars. There raises the possibility that this 2390 year cycle could play a role in modulating any long term cycles that exist in the torque being applied by Jupiter to the VE tidal bulge. 

     However, to look for these long-term periodicities in Jupiter's torque, we need to investigate how Jupiter moves, with respect to the periodic VE tidal bulge in the convective layers of the Sun. This requires us to look at the motion of Jupiter in both a fixed frame with respect to the stars and a frame that is revolving about the Sun at the same rate as the periodic VE alignments.

Jupiter in a Reference Frame that is Fixed with Respect to the Stars

     The diagram immediately below shows the orbital position of Jupiter, starting at (0,1), every 0.79933 years, over a period of 35.9699 years [i.e. just over three orbits of the Sun]. It is clear from this diagram that, in a reference frame that is fixed with respect to the stars,  the symmetry pattern perfectly re-aligns after moves roughly 24.26 degrees in a clockwise (pro-grade) direction. It takes Jupiter 71.9397 years (i.e. just over six orbits of the Sun or 45 VE aligns) to move 23.30 degrees in a clockwise (pro-grade) direction, to approach with one degree of producing a re-alignment of rotational symmetry.

Jupiter in a Reference Frame that is Rotating with the Earth-Venus-Sun Line 

The Movement of Jupiter with Respect to the Tidal-Bulge that is Induced in the Convective Layers of the Sun by Periodic Alignments of Venus and the Earth.

     The slow revolution of the Earth-Venus-Sun alignment axis can be removed provided you place yourself in a framework that rotates by 215.5176 degrees in a pro-grade direction [with respect to the fixed stars] once every 1.59866 years. In this rotating framework, Jupiter moves in a pro-grade direction (with respect to the Earth-Venus-Sun line) by 12.9993 degrees per [inferior conjunction] VE alignment.

     The following diagram shows the position of Jupiter every VE alignment (i.e. 1.59866 years) in reference frame that is rotating with the Earth-Venus-Sun alignment line. This keeps the Earth and Venus at the 12:00 o'clock position in this diagram whenever the number of VE aligns is even and at the 6:00 o'clock position whenever the number of VE aligns is odd. In contrast, Jupiter starts out at JO and moves 12.9993 degrees every 1.59866 years, taking 11.07 years to move exactly 90 degrees in the clockwise (pro-grade) direction and 11.19 years to the position marked J7.    

    Also shown on this diagram is the position of Jupiter after 27, 28 and 29 VE alignments. This tells us that Jupiter completes exactly one orbit in the VE reference frame once every 44.28 years (= 11.07 years x 4), with the nearest VE alignment taking place at 28 VE alignments (= 44.7625 years) when Jupiter has moved 3.9796 degrees past realignment with its original position at JO.

     The following table shows how Jupiter advances by one orbit + 3.9796 degrees every 28 VE alignments until the alignment of Jupiter with the Earth-Venus-Sun line progresses forward by   
13 orbits in the VE reference frame plus 51.7345 degrees. This angle (see * in table) is almost exactly equal to the angle moved by Jupiter in 4 VE aligns (i.e. 4 x 12.99927 degrees = 51.9971 degrees). 
   
VE_multiple______Angle of______Orbits_+__Degrees   
of 12.9993_______Jupiter______________________
degrees

____ 28_________363.9796_______1__+___3.9796
____56_________727.9592_______2__+___7.9592
____84________1091.9387_______3__+__11.9387
___112________1455.9183_______4__+__15.9183
___140________1819.8979_______5__+__19.8979
___168________2183.8775_______6__+__23.8775
___196________2547.8571_______7__+__27.8571
___224________2911.8366_______8__+__31.8366
___252________3275.8162_______9__+__35.8162
___280________3639.7958______10__+__39.7958
___308________4003.7754______11__+__43.7754
___336________4367.7550______12__+__47.7550
___364________4731.7345______13__+__51.7345__*

This means that Jupiter returns to almost exact re-alignment with the Earth-Venus-Sun line after:

(364 - 4) VE aligns = 360 VE aligns = 575.5176 years 

[i.e. 12.9993 orbits of Jupiter in a retro-grade direction in the VE reference frame, falling 0.2625 degrees short of exactly 13 full orbits]

Re-aligning the Movement of Jupiter in the Rotating VE Reference Frame with its Movement in the Reference Frame that is Fixed with the Stars

    The following diagram shows the precise alignments Jupiter with the Earth-Venus-Sun line at
575.5176 years (360 VE aligns) and 1151.0352 years (720 VE aligns) in a frame of reference that is fixed with respect to the stars. Jupiter lags behind the VE alignments by 0.2654 degrees and 0.5251 degrees, respectively. 

     The next diagram (directly below) shows the precise alignments of Jupiter with the Earth-Venus-Sun line at 1726.5528 years (1080 VE aligns) and 2302.0704 years (1440 VE aligns) in a frame of reference that is fixed with respect to the stars. Jupiter lags behind the VE alignments by 0.7876 degrees and 1.0502 degrees, respectively. 

     The important point to note is that after four precise Jupiter alignments of 575.5176 years (= 2302.0704 years), the position of Jupiter advances from its initial position at JO (see the third diagram in this blog post) by 24.2983 degrees. This angle is almost exactly the same as 24.26 degrees of rotation that is required to produce a re-alignment of the rotational symmetry of Jupiter, in the reference frame that is fixed with respect to the stars.

     Hence, the period of time required for Jupiter to precisely re-align with the Earth-Venus-Sun line in a reference frame that is fixed with respect to the stars is 2302 years. This is the Hallstatt-like cycle that is naturally found in the planetary configurations that are driving the VEJ Tidal-Torquing model for solar activity.  

References

Damon, P.E. and Sonett, C.P., 1991, “Solar and terrestrial components of the atmospheric 14C variation spectrum”, in The Sun in Time, (Eds.) Sonett, C.P., Giampapa, M.S., Matthews, M.S., pp. 360–388, University of Arizona Press, Tucson.

Vasiliev, S.S. and Dergachev, V.A., 2002, “The ∼2400-year cycle in atmospheric radiocarbon concentration: bispectrum of ¹⁴C data over the last 8000 years”, Ann. Geophys., 20, 115–120.
http://www.ann-geophys.net/20/115/2002/

Vitinsky, Y.I., Kopecky, M. and Kuklin, G.V., 1986, Statistics of Sunspot Activity (in Russian), Nauka, Moscow