Saturday, July 22, 2017

Pro Tip: Tin-Foil Hat Alarmists Tilt Your Heads 21 Degrees to the Left!!

The New York Times is trying to create a global-warming scare campaign by claiming that the large increases in the amounts of CO2 in the Earth's atmosphere in 2015 and 2016 are unprecedented and that they [possibly] signal an ominous change in the balance between the natural sources and sinks of carbon in the environment. The reporter Justin Gillis mentions that unusually high increases in CO2 in 2015 and 2016 may have something to do with the effects of the 2015/16 El Nino, though he quickly downplays this explanation because it detracts from the scare campaign that he trying to promote.

This post shows that most of what is being reported in the NYT on this topic is FAKE NEWS


Carbon in Atmosphere Is Rising, Even as Emissions Stabilize
Justin Gillis June 26th, 2017 NYT

The reader can read an extract from the article at the end of this post and the full article at:



Now let's look at the real facts.


Here is the plot that shows the annual increase in CO2 [measured in parts per million - ppm] recorded at Mauna Loa (Hawaii) from 1959 and 2016.  A least-squares line-of-best-fit is superimposed upon the the data (red line). Visible on the right-hand side of the plot is the so-called "unprecedented" peak in annual CO2 increase for the years 2015 and 2016. 

Ref: NOAA Earth Systems Research Laboratory (ESRL) Global Monitoring Division - Trends in Atmospheric Carbon Dioxide - Annual Mean Growth Rate Mauna Loa Hawaii:  https://www.esrl.noaa.gov/gmd/ccgg/trends/gr.html

It is immediately evident that the claim that level of CO2 in "unprecedented" is false because it does not take into account fact that there is more than one factor that is contributing to the annual change in CO2. Firstly, there is a slow increase in magnitude of the annual change in CO2 that is roughly linear with time. Secondly, there are short-term fluctuations in the annual change in CO2 that take place over a year or two. 

If the red line in the figure above is used to remove the long-term changes in the annual increase in CO2 [This is the equivalent of the tin-foil hat alarmists tilting their heads 21 degrees to the left], you get the curve for the de-trended annual change in CO2 (measured in ppm) that is displayed at the top the following figure.   


What this curve shows is that the recent increase in the annual change in CO2 in 2015 and 2016 are no unprecedented when compared to earlier increases, provided allowance is made for a slowly increasing linear rise in the annual change in CO2 over the last 57 years.

A second curve is displayed below the top curve of this figure that shows the NINO3.4 sea-surface temperatures (SST) anomalies. These SST anomalies can be used to determine when El Nino events are occurring in the Pacific Ocean. The timing of these El Nino events are highlighted in red.

Ref: The Month Nino3.4 SST Anomaly from NOAA: 
http://www.cpc.ncep.noaa.gov/data/indices/sstoi.indices

A comparison between the top and middle curve in the above figure, clearly shows that on every occasion between 1959 and 2016 when there has been an El Nino event, it has been accompanied by a corresponding increase in the annual change in CO2, with two exception. The two exceptions are the El Nino events in 1964 and 1991.

The bottom curve in the above figure shows the stratospheric aerosol optical depth at 550 nm. This index is excellent indicator of recent major volcanic eruptions that have taken place in the tropical region of the planet. There are three main eruptions that are evident in this time series i.e. the eruptions in 1964 (Agung), 1983 (El Chichon) and 1991 (Mt. Pinatubo). Each of these eruptions injected massive amounts of aerosols (mainly sulphur dioxide) into the stratosphere that led to the significant cooling of global atmospheric and (to a lesser extent) oceanic temperatures over following 1 - 2 years.

Ref: Stratospheric aerosol optical depth at 550 nm used in GISS climate simulations

https://data.giss.nasa.gov/modelforce/strataer/tau.line_2012.12.txt

A comparison of this curve with the two above it clearly shows that the cooling associated with the two largest volcanic eruptions (Agung - 1964 and Mt. Pinatubo - 1991)  completely reversed any increases in the annual change in CO2 caused by the El Nino events of 1964 and 1982/83.

Hence it is reasonable to conclude that:

a) virtually all of the variance in the annual change in CO2 on time scales
    less than about 2 years can be attributed to the El Nino/La Nina ENSO
    weather cycle.

b) the increases in the annual change in CO2 in 2015 and 2016 were just
      what you would expect given the major 2015/16 El Nino event i.e. these
      were in no way extraordinary.  

c) there is an underlying long-term increase in the annual change in
     atmospheric CO2 concentration that could be attributed to human CO2
    emissions.

One thing that is clear from the two figure above, is that there is a natural source/sink of CO2 that can be associated with the EL Nino/La Nina ENSO phenomenon that is significantly contributing to the short time scale (< 2 years) increase in the annual change in CO2. 

Another factor that the NYT article doesn't take into account is that atmospheric concentration of CO2 (measured in ppm) has increase from about 316 ppm in 1959 to about 404 ppm in 2016. 
     


This means that if we are to make a valid comparison between annual increases in atmospheric CO2 over the full 57 year period we must allow for the significant change in atmospheric CO2 concentration of this time period.

In this vane, the following plot shows the annual percentage increase in atmospheric CO2 at Mauna Loa Hawaii between 1959 and 2016. This plot re-emphasizes the point made earlier that [in terms of percentage change] the increases in the annual change in CO2 are not out of the ordinary compared to previous EL Nino events .       

  
It also shows that the long-term increase in the annual change in CO2 is probably not taking place in a smooth linear fashion, as would be expected if human CO2 emissions were the primary contributor, but is in fact occurring in distinct step-like increases that last for about 18 to 20 years [note the steps at 1978 and 1997/98 El Nino events.

This step-like increases are also seen in the world's sea surface temperatures, as the following graph from the WUWT article by Bob Tisdale shows:

Ref: https://wattsupwiththat.com/2011/03/11/tisdale-on-enso-step-changes-in-rss-global-temperature-data/  



This plot shows the volcano-adjusted RSS TLT (Temperature - Lower Troposphere) anomaly for points north of 20 degrees North between 1979 and 2011. 

A comparison of this figure with the next figure above shows that the long-term percentage increase in the annual change in atmospheric CO2 appears to be showing the same step-like increase as the lower troposphere temperatures - strong suggesting that that there may be sources and sinks of CO2 that are NOT only associated with human emissions of CO2 but which are (also) most likely related in some way to these atmospheric temperature changes.

If this is true then this blows the whole human-induced, CO2-driven climate change model out of the water, strongly implying that human emissions only play a minor role. This would be agreement with the work of Bob Tisdale and Dr. Murray Salby.

The reader is referred to an excellent set of article by Bob Tisdale on this topic:

What Causes Sea Surface Temperature (SST) To Rise?


and a video by Dr. Salby at:

New video: Dr. Murry Salby – Control of Atmospheric CO2


  
The extract from the New York Times' article:

Justin Gillis reports that:

"For more than two years, the monitoring station here [in Tasmania] , along with its counterparts across the world, has been flashing a warning: The excess carbon dioxide scorching the planet rose at the highest rate on record in 2015 and 2016. A slightly slower but still unusual rate of increase has continued into 2017.

Scientists are concerned about the cause of the rapid rises because, in one of the most hopeful signs since the global climate crisis became widely understood in the 1980's, the amount of carbon dioxide that people are pumping into the air seems to have stabilized in recent years, at least judging from the data that countries compile on their own emissions. 

That raises a conundrum: If the amount of the gas that people are putting out has stopped rising, how can the amount that stays in the air be going up faster than ever? Does it mean the natural sponges that have been absorbing carbon dioxide are now changing?

To me, it’s a warning,” said Josep G. Canadell, an Australian climate scientist who runs the Global Carbon Project, a collaboration among several countries to monitor emissions trends. Scientists have spent decades measuring what was happening to all of the carbon dioxide that was produced when people burned coal, oil and natural gas. They established that less than half of the gas was remaining in the atmosphere and warming the planet. The rest was being absorbed by the ocean and the land surface, in roughly equal amounts.

In essence, these natural sponges were doing humanity a huge service by disposing of much of its gaseous waste. But as emissions have risen higher and higher, it has been unclear how much longer the natural sponges will be able to keep up."

and he further reports that:

"Many of them suspect an El Niño climate pattern that spanned those two years, one of the strongest on record, may have caused the faster-than-usual rise in carbon dioxide, by drying out large parts of the tropics. The drying contributed to huge fires in Indonesia in late 2015 that sent a pulse of carbon dioxide into the atmosphere. Past 
El Niños have also produced rapid increases in the gas, though not as large as the recent ones.

Yet scientists are not entirely certain that the El Niño was the main culprit; the idea cannot explain why a high rate of increase in carbon dioxide has continued into 2017, even though the El Niño ended early last year."

Thursday, April 20, 2017

I Need Some Help to Solve an Interesting Lunar Puzzle

The Conundrum

[N.B. A Full Moon Cycle (FMC = 411.78443025 days epoch J2000.0) is the time required for the Sun (as seen from the Earth) to complete one revolution with respect to the Perigee of the Lunar orbit. In other words, it is the time between repeat occurrences of the Perigee end of the lunar line-of-apse pointing at the Sun. The value for the length of the FMC is equal to the synodic product of the Synodic (lunar phase) month (= 29.530588853 days) with the anomalistic month (= 27.554549878 days) for Epoch J2000.0. The anomalist month is the time required for the Moon to return to the Perigee of the lunar orbit.]  

[N.B. The Lunar Anomalistic Cycle (LAC = 3232.60544062 days = 8.85023717 sidereal years) is the time required for the lunar line-of-apse to precess once around the sky with respect to the stars. This corresponds to a 0.11136528O per day movement of the lunar line-of-apse in a pro-grade (clockwise) direction. The value for the length of the LAC is equal to the synodic product of the anomalistic month (= 27.554549878 days) with the sidereal lunar month (= 27.231661547 days) for Epoch J2000.0. The sidereal month is the time required for the Moon to rotate once around its orbit with respect to the stars.]  

The diagram below shows the Perigee of the lunar orbit pointing at the Sun at 0.0 days. In addition, the diagram shows the Perigee of the lunar orbit once again pointing at the Sun after one Full Moon Cycle (FMC) = 411.78443025 days. It takes more than 1.0 sidereal year (= 365.256363004 days) for the Perigee to realign with the Sun because of the slow pro-grade (clockwise) precession of the lunar line-of-apse once every 8.85023717 sidereal years.




1.0 FMC falls short of 15 anomalistic months (= 413.31824817 days) by 1.53381792 days (= 1.5117449198O). During these 1.5117449198 days the Perigee end of the lunar line-of-apse rotates by 0.17081406in a prograde direction, producing an overall movement of the line-of-apse (red line) of 1.34093086O (= 1.5117449198O – 0.17081406O) with respect to the Earth-Sun line (blue line).

if we let:

  DT  =  (15 anomalistic months -- FMC) = (413.31824817 -- 411.78443025) days 
         = 1.53381792 days
      S = the angular revolution (in degrees) of the Earth about the Sun over DT days.
          = 1.5117449198 degrees
      L = the orbital precession (in degrees) of the lunar line-of-apse over DT days.
          = 0.1708140574 degrees
then  

      D = S -- L = angle between the lunar line-of-apse and the Earth-Sun line after DT days.
          = 1.3409308624 degrees

we find that if we take the incremental angle between the lunar line-of-apse and the Earth-Sun line over DT days (= 1.3409308624 degrees) and divide it by the 360 degrees of movement of the angle between the lunar line-of-apse and the Earth-Sun line that has occurred over the previous FMC, it effectively has the same value as the incremental number of days between 15 anomalistic months and 1.0 FMC (=1.5338172 days) divided by 1.0 FMC i.e.    

     (S -- L) / 360 degrees = (15 anomalistic months - FMC) / FMC = 0.0037248080            (1)

While this is not remarkable, what is remarkable, however, is that both of these fractions (whether they be measured in degrees or days) are precisely equal to the cumulative annual precession of the Perihelion of the Earth's orbit (measured in days) over a period of 1.0 FMC!

= (11.723"/3600) deg. per yr x (365.256363004 days / 360 deg.) x (411.78443025 / 365.256363004)
= 0.0037248062 days per 1.127384686 sidereal years.

N.B. The current value for the precession of the Perihelion of the Earth's orbit is 11.615 arc seconds per year. However it is increasing and will achieve a value of 11.723 arc seconds in roughly 2490 A.D.   

Here's the Rub [updated 25/04/2017]

1. The cumulative precession of the Earth's orbit over a period of 1.0 FMC has the dimensions of days per year!

I must be missing something. Why the strange units of days per 1.127384686 sidereal years? Can anyone help me understand why I get these weird dimensionless units? [updates 22/04/2017]

2.  The increase in angle between the lunar line-of-apse and the Earth-Sun line as you move from 1.0 FMC to 15 anomalistic months (= 1.34093086 degrees) seems to be almost precisely equal to the FULL annual precession of the Perihelion of the Earth's orbit (= 11.723 arc seconds per year) PER DAY accumulate over 1.0 FMC i.e.

[(11.723 / 3600) deg per YEAR] x 411.78443025 days = 1.34093024 degrees

The question is, what angular motion associated with the movements in Sun-Earth-Moon system can cause the ANNUAL precession of the Perihelion of the Earth's orbit to accumulate DAILY?

I am not aware of any mechanism that would produce a motion like this and I would appreciate if anyone could solve this interesting lunar puzzle for me!

Is it something to to do with the interaction between the mean and true anomalies of the Earth's orbit and the Moon's orbit? Could motion of the Earth and moon about the common centre-of-gravity have and effect? What about the effects of 18.6 year nutation of the Earth's rotation axis?

THANKS IN ADVANCE 




Tuesday, December 20, 2016

A Direct Connection Between the Venus, Earth and Jupiter Tidal-Torquing Cycles (a Proposed Driver of Solar Activity) and the Long-Term Strength of Perigean Spring Tides



IMPORTANT CLAIMS:

1. The Synodic (phase) cycle of the Moon precisely re-synchronises with the times of the Extreme Perigean Spring tides (EPST) once every 574.60 topical years.

2. Jupiter precisely re-synchronises itself in a frame of reference that is rotating with the Earth-Venus-Sun line once every 575.52 tropical years.

3. The orientation of Jupiter to the Earth-Venus-Sun line produce the tidal torques that act upon the base of the convective layers of the Sun which are thought to be responsible for the periodic changes in the level of magnetic activity on the surface of the Sun (i.e. the Solar Cycle).

4. 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 4 x 575.52 = 2302 years. This is the Hallstatt cycle that is intimately associated with the planetary configurations that are driving the VEJ Tidal-Torquing model for solar activity.

5. Hence, the repetition period for strongest of the Extreme Perigean Spring tides appears to match that of the planetary tidal-torquing forces that are thought to be responsible for driving the Solar sunspot cycle.

6. The Venus–Earth–Jupiter (VEJ) tidal-torquing model is based on the idea that the planet that applies the dominant gravitational force upon the outer convective layers of the Sun is Jupiter, and after Jupiter, the planets that apply the dominant tidal forces upon the outer convective layers of the Sun are Venus and the Earth. Periodic alignments of Venus and the Earth on the same or opposite sides of the Sun, once every 0.7993 sidereal Earth years, produce temporary tidal bulges on the opposite sides of the Sun’s surface. Whenever these temporary tidal bulges occur, Jupiter’s gravitational force tugs upon the tidally induced asymmetries and either slows down or speeds-up the rotation rate of plasma near the base of the convective layers of the Sun. The VEJ tidal-torquing model proposes that it is the variations in the rotation rate of the plasma in Sun’s lower convective layer, produced by the torque applied by Jupiter upon the periodic Venus–Earth (VE) tidal bulges that modulate the Babcock–Leighton solar dynamo. Hence, the model asserts that it is the modulating effects of the planetary tidal-torquing that are primarily responsible for the observed long-term changes in the overall level of solar activity.

What makes this simple VEJ tidal-torquing model so intriguing is the time period over which the Jupiter’s gravitational force speeds up and slows down the rotation rate of the Sun’s outer layers. Jupiter’s movement of 13.00 deg. per 1.5987 yr with respect to closest tidal bulge means that Jupiter will increase the rotation speed of the lower layers of the Sun's convective zone for 11.07 yrs. This is almost the same amount of time as to average length of the Schwabe sunspot cycle (11.1 ± 1.2 yrs, Wilson, 2011).

In addition, for the next 11.07 yrs, Jupiter will start to lag behind the closest tidal bulge by 13.00 deg. every 1.5987 yrs, and so its gravitational force will pull on the tidal bulges in such a way as to slow down the rotation rate of the outer convective layers of the Sun. Hence, the basic unit of change in the Sun’s rotation rate (i.e. an increase followed by a decrease in rotation rate) is 2 × 11.07 = 22.14 yrs. This is essentially equal to the mean length of the Hale magnetic sunspot cycle of the Sun, which is 22.1 ± 2.0 yrs (Wilson, 2011).

It is important to note that, the actual torques that are applied by Jupiter to the temporary tidal bulges induced by alignments of Venus and the Earth, vary in-phase with the observed 11 year sunspot cycle.

ARTICLE:

A. The 574.6 Year Cycle in Extreme Perigean Spring Tides


     Extreme Perigean Spring tides (EPST) occur when a New moon occurs at time when the Perigee of the lunar orbit points directly at the Sun or when a Full Moon occurs when the Perigee of the lunar orbit points directly away from the Sun (the latter are often called Extreme Super Moons). Figure 1 shows a schematic diagram an EPST occurring at a New Moon.

Figure 1.      




     EPST at New Moon re-occur once every 18 Full Moon Cycles (FMC) = 20.2937 tropical years (where one tropical year = 365.242189 days).

[Note: A FMC (= 411.78443029 days epoch J2000.0) is the the time required for the Sun (as seen from the Earth) to complete one revolution with respect to the Perigee of the Lunar orbit. In other words, it is the time between repeat occurrences of the Perigee of the lunar orbit pointing at the Sun. This value for the FMC is based upon a Synodic month = 29.53058885 days, an anomalistic month = 27.55454988 days - Epoch J2000.0.]  

     Figure 2, below, has as its initial starting point (T = 0.0 tropical years), a New Moon taking place at the precise time that the Perigee of the lunar orbit points directly at the Sun. In addition, this figure shows the number of days to (negative values on the y-axis) or from (positive values on the y-axis) a New/Full Moon for each of the EPST's that occur over the next 618.4 tropical years.

     Figure 2 shows that the point representing the New Moon at T = 0.0 tropical years is part of a triplet of points with the other two points occurring at  (-1.1274 tropical years, 1.64 days) and (+1.1274 tropical years, -1.64 days). Hence, a point starting at (-1.1274 tropical years, 1.64 days), reaches the x-axis at (573.4727 topical years, 0.0 days), leading to an overall repetition cycle of 574.600 tropical years.
   
Figure 2.



B. The Venus–Earth–Jupiter (VEJ) Tidal-Torquing Model 
     (A Proposed Driver of Solar Activity)


Quote from: Wilson, I.R.G.: The Venus–Earth–Jupiter spin–orbit coupling modelPattern Recogn. Phys., 1, 147-158

     "The Venus–Earth–Jupiter (VEJ) tidal-torquing model is based on the idea that the planet that applies the dominant gravitational force upon the outer convective layers of the Sun is Jupiter, and after Jupiter, the planets that apply the dominant tidal forces upon the outer convective layers of the Sun are Venus and the Earth. Periodic alignments of Venus and the Earth on the same or opposite sides of the Sun, once every 0.7993 sidereal Earth years, produces temporary tidal bulges on the opposite sides of the Sun’s surface (Fig. 3 – red ellipse). 

Figure 3.


      Whenever these temporary tidal bulges occur, Jupiter’s gravitational force tugs upon the tidally induced asymmetries and either slows down or speeds-up the rotation rate of plasma near the base of the convective layers of the Sun. The VEJ tidal-torquing model proposes that it is the variations in the rotation rate of the plasma in Sun’s lower convective layer, produced by the torque applied by Jupiter upon the periodic Venus–Earth (VE) tidal bulges that modulate the Babcock–Leighton solar dynamo. Hence, the model asserts that it is the modulating effects of the planetary tidal-torquing that are primarily responsible for the observed long-term changes in the overall level of solar activity. 

      It is important to note that tidal bulges will be induced in the surface layers of the Sun when Venus and the Earth are aligned on the same side of the Sun (inferior conjunction), as well as when Venus and the Earth are aligned on opposite sides of the Sun (superior conjunction). This means that whenever the gravitational force of Jupiter increases/decreases the tangential rotation rate of the surface layer of the Sun at inferior conjunctions of the Earth and Venus, there will be a decrease/increase the tangential rotation rates by almost the same amount at the subsequent superior conjunction. 

     Intuitively, one might expect that the tangential torques of Jupiter at adjacent inferior and superior conjunctions should cancel each other out. However, this is not the case because of a peculiar property of the timing and positions of Venus– Earth alignments. Each inferior conjunction of the Earth and Venus (i.e. VE alignment) is separated from the previous one by the Venus–Earth synodic period (i.e. 1.5987 yr). This means that, on average, the Earth–Venus–Sun line moves by 144.482 degrees in the retrograde direction, once every VE alignment. Hence, the Earth–Venus–Sun line returns to almost the same orientation with respect to the stars after five VE alignments of almost exactly eight Earth (sidereal) years (actually 7.9933 yr). Thus, the position of the VE alignments trace out a five pointed star or pentagram once every 7.9933 yr that falls short of completing one full orbit of the Sun with respect to the stars by (360−(360×(7.9933− 7.0000))) = 2.412 degrees (fig. 4). 

Figure 4.




     In essence, the relative fixed orbital longitudes of the VE alignments means that, if we add together the tangential torque produced by Jupiter at one superior conjunction, with the tangential torque produced by Jupiter at the subsequent inferior conjunction, the net tangential torque is in a pro-grade/retrograde direction if the torque at the inferior conjunction is greater/less than that of the torque at the superior conjunction.

Figure 5.





     What makes this simple tidal-torquing model so intriguing is the time period over which the Jupiter’s gravitational force speeds up and slows down the rotation rate of the Sun’s outer layers. 

     Figure 5 shows Jupiter, Earth and Venus initially aligned on the same side of the Sun (position 0). In this configuration, Jupiter does not apply any tangential torque upon the tidal bulges (the position of the near-side bulge is shown by the black 0 just above the Sun’s surface). Each of the planets, 1.5987 yrs later, moves to their respective position 1's. At this time, Jupiter has moved 13.00 deg. ahead of the far-side tidal bulge (marked by the red 1 just above the Sun’s surface) and the component of its gravitational force that is tangential to the Sun’s surface tugs on the tidal bulges, slightly increasing the rotation rate of the Sun’s outer layers.

     After a second 1.5987 yrs, each of the planets moves to their respective position 2's. Now, Jupiter has moved 26.00 deg. ahead of the near-side tidal bulge (marked by the black 2 just above the Sun’s surface), increasing Sun’s rotation rate by roughly twice the amount that occurred at the last alignment. This pattern continues with Jupiter getting 13.00 deg. further ahead of the nearest tidal bulge, every 1.5987 yrs. Eventually, Jupiter will get 90 deg. ahead of the closest tidal bulge and it will no longer exert a net torque on these bulges that is tangential to the Sun’s surface and so it will stop increasing the Sun’s rotation rate. 

      Interestingly, Jupiter’s movement of 13.00 deg. per 1.5987 yr with respect to closest tidal bulge means that Jupiter will get 90 deg. ahead of the closest tidal bulge in 11.07 yrs. This is almost the same amount of time as to average length of the Schwabe sunspot cycle (11.1 ± 1.2 yrs, Wilson, 2011). In addition, for the next 11.07 yrs, Jupiter will start to lag behind the closest tidal bulge by 13.00 deg. every 1.5987 yrs, and so its gravitational force will pull on the tidal bulges in such a way as to slow down the rotation rate of the outer convective layers of the Sun. Hence, the basic unit of change in the Sun’s rotation rate (i.e. an increase followed by a decrease in rotation rate) is 2 × 11.07 = 22.14 yrs. This is essentially equal to the mean length of the Hale magnetic sunspot cycle of the Sun, which is 22.1 ± 2.0 yrs (Wilson, 2011)." 


C. The 575.52 year Realignment Cycle for 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.

     Figure 6 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 (at roughly 91 degrees).

Figure 6.
 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 = 575.52 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]


APPENDIX
The Planetary Connection to the 2300 Hallstatt Cycle

      It has long been recognised that there is a prominent 208 year de Vries (or Suess) cycle in the level of solar activity. The following blog post shows that the there is a 208.0 year de Vries cycle in the alignment between the times that Perigee of the Lunar orbit points directly at the Sun and the Earth's seasons provided that you measure the alignment in a reference frame that is fixed with respect to the Perihelion of the Earth's orbit:   

http://astroclimateconnection.blogspot.com.au/2016/05/there-is-natural-208-year-de-vries-like.html   

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., 1986Damon and Sonett, 1991Vasiliev and Dergachev, 2002).
  
Jupiter in a Reference Frame that is Fixed with Respect to the Stars

     Figure_A1 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.

Figure_A1


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

    Figure_A2 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.

Figure_A2





  Figure_A3 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.

Figure_A3


  The important point to note is that after four precise Jupiter alignments of 575.5176 years (i.e. 4 x 575.5176 = 2302.07 years), the position of Jupiter advances from its initial position at JO (see figure 6 and figure_A1 above) 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