The Gear Effect + the VEJ Tidal Torquing Model = The VEJ Spin-Orbit Coupling Model

             

 I. The Gear Effect

     

     Golfers use a physical principle called the Gear Effect to either slice or hook a golf ball off the tee.Figure 1 shows how the Gear Effect works.

     If the golf ball hits the (curved) face of the club off-centre, it applies a force (horizontal black arrow) to the club which induces a clock-wise rotation of the club head (green arrow) about its centre-of-mass (bottom right yellow circle with cross-hairs). The resultant rotation of the face of the the club head (red arrow) applies a side-ways force to the golf ball at the point of contact, producing an anti-clockwise rotation (blue arrow) of the golf ball (Note: The ball will roll from the toe (top) towards the centre of the club face). This particular application of the Gear Effect produces a hook shot.  

 figure 1

[In relation to the following arguments, it is important to note that rotational motion of the more massive golf club head will be considerably smaller than the less massive golf ball. In addition, it is also important to note that the centre-of-mass of the golf ball is independent from the centre-of-mass of the club head.]

     The purpose of this article is to show how the Gear Effect can be combined with the VEJ Tidal Torquing model to produce a Spin-Orbit Coupling model that links the rotation rate of the outer layers of the Sun to the Sun's motion about the centre-of-mass of the solar system (CMSS).

     In order to understand how the Gear Effect can be combined with the VEJ Tidal Torquing model, however, we must first show how the orbital motions of the Jovian planets determine the Sun's motion about the CMSS (often called the Solar Inertial Motion or SIM) and then discuss the Quadrature Effect.

II. The Solar Inertial Motion

Reference: Wilson et al. [2008]

     Given the fact that the Sun is over 1000 times the mass of Jupiter, it is often assumed that the CMSS is located at the centre of the Sun. In fact, the centre of the Sun moves about the CMSS in a series of complex spirals with the distance between the two varying from 0.01 to 2.19 solar radii (Jose 1965). This motion is the result of the gravitational forces of the Jovian planets tugging on the Sun.

     Jose (1965) quantified the motion of the Sun about the CMSS and showed that the time rate of change of the Sun’s angular momentum about the instantaneous centre of curvature = dP/dT , or torque, varies in a quasi-sinusoidal manner similar to the variation seen in the solar sunspot number. In fact, Jose (1965) found that the temporal agreement between variations in dP/dT and the solar sunspot number were so good that it strongly hinted that there was a connection between the planetary induced torques acting on the Sun and sunspot activity. However, he did not fully explain how this connection worked.

 figure 2

Figure 2: This shows a typical orbit for the Sun about the CM of the Solar System, with the position of the Sun marked by an ‘X’ at the times when Jupiter and Saturn are in opposition (1), first quadrature (2), conjunction (3), second quadrature (4), and opposition (5) [see Notes 1 and 2 below].

     Figure 2 shows a typical orbit of the Sun about the CMSS. It is not the simple ellipse that you would expect if gravitational effects of Jupiter dominated the Sun’s motion. The Sun’s orbit about the CMSS deviates from an ellipse primarily because of the added influence of Saturn. Obviously, when Jupiter is at inferior conjunction as seen from Saturn, i.e. the planets are on the same side of the Sun (see note 1), the Sun will be at its greatest distance from the CMSS; when Jupiter is at superior conjunction, as seen from Saturn, i.e. the planets are on opposite sides of the Sun (see note 2), the Sun will be closest to the CMSS. Similarly, when the planets are in quadrature, the Sun’s distance from the CMSS will be roughly the same and somewhere in between these two extremes.

This point is highlighted in figure 2 where we have marked a set of sequential events concerning the orbits of Jupiter and Saturn along the Sun’s orbit about the CMSS. Jupiter and Saturn start in opposition at (1), first quadrature at (2), conjunction at (3), second quadrature at (4) and finally back to opposition at (5).

[Note 1:  In an inferior conjunction, the superior planet (Saturn) is in opposition’to the Sun, as seen from the inferior planet (Jupiter), and so we will refer to this as Jupiter and Saturn being in opposition.]
[Note 2: When Jupiter is at superior conjunction as seen from Saturn, we will refer to this as Jupiter and Saturn being in conjunction.]

     The net effect of adding the gravitational influence of Saturn to that of Jupiter upon the Sun’s orbit about the CMSS is as follows:

a) The times at which the Sun experiences maximum torque (dP/dT ) as it moves around the CMSS, corresponds very closely with the times of quadrature for Jupiter and Saturn (Jose 1965) i.e. points (2) and (4) in figure 2.

b) Similarly, the times at which the torque acting on the Sun is zero (this also the time at which the torque acting on the Sun is most rapidly changing) correspond very closely with the times of opposition and conjunction of Jupiter and Saturn, i.e. points (1), (3), and (5) in Figure 2.

[Note: For the purposes of the following argument, we are limiting ourselves to a solar system that has only four planets, Venus, the Earth, Jupiter and Saturn, all moving in circular orbits at their mean distances from the Sun]

III. The Quadrature Effect

     Every 9.9±1.0 yr, the planet Saturn is in quadrature with the planet Jupiter (i.e. the angle between Saturn and Jupiter, as seen from the Sun, is 90 degrees).

 figure 3

     Figure 3 shows the orbital configuration of a quadrature of Jupiter and Saturn when Saturn follows Jupiter in its orbit. Referring to this diagram, we see that Saturn drags the CM of the Sun, Jupiter, Saturn system (CMSJSa) off the line joining the planet Jupiter to the Sun. As a result, the gravitational force of the Sun acting upon Jupiter speeds up its orbital motion about the CMSJSa. At the same time, the gravitational force of Jupiter acting on the Sun slows down the orbital speed of the Sun about the CMSJSa.

     However, the reverse is true at the next quadrature, when Saturn precedes Jupiter in its orbit. In this planetary configuration, the mutual force of gravitation between the Sun and Jupiter slows down Jupiter’s orbital motion about the CMSJSa and speeds up the Sun’s orbital motion about the CMSJSa. Hence, the Sun’s orbital speed about the CMSJSa (as well as the CMSS) should periodically decrease and then increase as you move from one quadrature to next (Jose 1965). The curves published by Jose (1965) showing the motion of the Sun about the CMSS can be used to directly measure the speed of the Sun along its orbital path.

 figure 4

     Figure 4 shows the speed of the Sun along its orbit, between the oppositions of Jupiter and Saturn in 1842.2 and 1861.9. Superimposed on this figure are symbols showing the syzygies (i.e. alignments) and quadratures of Jupiter and Saturn. This figure clearly shows that our prediction about the Sun’s orbital speed about the CMSS is indeed correct. In this plot, we see that the speed of the Sun almost halves (from ∼16 to 8 ms^(−1)) over the period from 1842 to 1850, roughly centred on the time of the first quadrature in 1846.5. And then after reaching a minimum near conjunction in 1851.8, the speed almost doubles (from ∼ 8 to 15 ms^(−1)) over the period from 1850 to 1860, again roughly centred on the time of quadrature in 1856.9.

SKIP DOWN TO SECTION IV IF YOU WANT TO AVOID DETAILS 

     It is important to note that it is not just the speed of the Sun about the CMSS that changes between oppositions but also the Sun’s orbital radius about the CMSS as well. During the eight-year time period between 1842 and 1850, for example, the Sun’s orbital radius about the CMSS changed from ∼2 solar radii to almost zero. This means that there was an overall decrease in the Sun’s angular momentum about the CMSS of ∼4.5×10^(40) Nms. Similarly, between 1850 and 1860, the Sun’s orbital radius about the CMSS increased from zero to 1.5 solar radii, resulting in an increase of the Sun’s angular momentum about the CMSS of ∼3.2×10^(40) Nms. Thus, the torque acting on the Sun about the CMSS starts out at zero at opposition, reaches a minimum value at the first quadrature (when Saturn follows Jupiter), returns to
zero at the following conjunction, reaches a maximum at the second quadrature (when Saturn precedes Jupiter), and the finally returns to zero when the planets return to opposition. Variations in the torque of this nature produces a strong breaking of the Sun’s orbital motion about the CMSS near the time of first quadrature, accompanied by a significant decrease in the Sun’s angular momentum. This is followed by a strong acceleration of the Sun’s orbital motion about the CMSS near the time of the second quadrature,
accompanied by a comparable increase in the Sun’s angular momentum.

     Published plots of the torque (dP/dT ) acting on the Sun, where the torque is measured about the instantaneous centre-of-curvature of the Sun’s orbit about the CMSS (Jose 1965), show that this is in fact what happens to the Sun. Jose’s (1965) plots show the torque varying in a quasi-sinusoidal manner, starting out at zero at opposition, reaching a minimum at the first quadrature, returning to zero at the following conjunction, reaching a maximum at the second quadrature and finally returning to zero at the next opposition. The average time taken for this cycle to repeat itself is simply set by the synodic period of Jupiter
and Saturn, i.e. 19.858 yr.

 figure 5

     Figure 5 shows a plot of dP/dT , derived from data in the paper by Jose (1965), for one cycle between 1901.6 and 1920.9 (solid line). Superimpose on this plot (dashed line) is a sinusoidal function with a period equal to the time between the consecutive oppositions at 1901.8 and 1921.8 (i.e. 20.0 yr) and an amplitude chosen to match of the first minimum.

     The curves in figure 5 show that it is a reasonable first approximation to say that the Sun’s orbital motion around the CMSS (as well as the CMSJSa) is being driving by a torque (dP/dT ) that is varying sinusoidally with a period of ~ 19.9 yr. However, there are other weaker perturbing influences that are advancing or retarding the times for the maxima, minima and zero points of dP/dT, compared to the cardinal planetary configurations. These weaker perturbations are primarily caused by the combined gravitational influences of Neptune and Uranus (Fairbridge and Shirley 1987). However, for the purposes of the following arguments, we will ignore their effects upon the motion of the Sun in this article.
                                                                                                   

IV. Differentiating Between the Quadrature Effect to the Gear Effect

     In figure 6 below, we re-plot figure 3 with the terrestrial planet Venus preceding Jupiter in its orbit. As with figure 3, figure 6 shows the situation where Saturn and Jupiter are in quadrature, with Saturn following Jupiter. In these circumstances, the Quadrature Effect ensures that the Sun's anti-clockwise motion about the CMSJSa  will be slowed by the gravitational force of Jupiter.

      If the Sun's speed about the SMSJSa slows down between one opposition and the following quadrature (as shown in figure 6 below), then the same must be true for the terrestrial planets, since their orbital motion is, for all intents and purposes, constrained to move about the centre-of-mass of the Sun rather than the CMSJSa. The red arrows in figure 6 represent the decrease in speed of the Sun and Venus as they revolve in an anti-clockwise direction about the SMSJSa. This decrease in speed is shared by both the Sun and Venus so that the two bodies effectively moves as one, maintaining their orientation and spacing.

      Hence, the Quadrature Effect should have little or no effect upon the  the VEJ Tidal-Torquing model and it should not be able to modulate the rotation rate of the outer layers of the Sun. This is true simply because there is no (significant) change in the relative positions of the Sun and Venus.

THE QUADRATURE EFFECT

figure 6

     Now, imagine that the Jupiter-Sun-Saturn system (with its own CM = CMSJSa) is like the golf club head in the Gear Effect and that the planet Venus (also with its own CM) is like the golf ball.

    In figure 7, shown below, we see that Venus applies a gravitational torque to the Jupiter-Sun-Saturn system that forces this system to reduce its orbital velocities about the CMSJSa (red arrow). In terms of the Gear Effect analogy, this is the equivalent of the club head rotating in a clock-wise direction about its centre-of-mass.

                                                              THE GEAR EFFECT

figure 7

[Note: Of course, the size of the velocity reduction in the Jupiter-Sun-Saturn system would be minute compared to that caused by the Quadrature Effect primarily because the combined mass of the Sun, Jupiter and Saturn is orders of magnitude larger than that of Venus.]

     In like manner, in figure 7, we see that the Jupiter-Sun-Saturn system applies a gravitational torque to Venus that speeds up the motion of Venus about the CMSJSa (dark curved arrow emanating from Venus) .

Hence, there are three critical points that we need to note about the Gear Effect:

a) Unlike the Quadrature Effect, the torques involved in the Gear Effect try to change the orientation and spacing between the Sun and Venus e.g. in relation to the specific case shown above, even though these gravitational torques are very minute, they produce a net anti-clockwise rotation of the Sun and Venus about their mutual center-of-mass (yellow cross). In terms of the Gear Effect analogy, this is the equivalent of the golf ball rotating in a counter-clock-wise direction.

[It is the offset between the CMSJSa and the centre-of-mass of the Sun-Venus system that is crucial for producing the net anti-clockwise rotation of the Sun and Venus about their mutual centre-of-mass.]    

b) Even though the net gravitational torques try to produce an anti-clockwise rotation of the Sun and Venus about their mutual center-of-mass, some of the resulting angular momentum will probably end up changing the rotation rates of both Venus and the outer layers of the Sun.

c) Given the minute nature of the torques applied and velocity changes involved, it is obvious that the effects of the Gear Effect will be greatest at the times when Venus and the Earth are aligned on the same side of the Sun. At these times, the Jupiter-Sun-Saturn system (at quadrature) would experience the greatest gravitational force from the Terrestrial planets and the centre-of-mass of the aligned Sun-Venus-Earth system would be furthest from the centre of the Sun.

Thus, the Gear Effect should have an effect upon the VEJ Tidal-Torquing model and it should be able to modulate the changes in rotation rate of the outer layers of the Sun that are being caused by the VEJ tidal-torquing.

V. The VEJ Spin-Orbit Coupling Model

     There appear to be at least two ways that the Jovian and Terrestrial planets can influence bulk motions in the convective layers of the Sun. 

The first is via the VEJ Tidal Torquing Process

a) Tidal bulges are formed in the convective layers of the Sun by the periodical alignments of Venus and the Earth.

b) Jupiter applies a gravitational torque to these tidal bulges that either speed up and slow down the outer convective layers of the Sun.  

c) Jupiter's net torque increases the rotation rate of the surface layers of the Sun for seven Venus-Earth alignments (lasting 11.19 years) and then decreases the rotation rate over the next seven Venus-Earth alignments (also lasting 11.19 years). 

d) The model produces periodic changes in rotation rate of the outer convective layers of the Sun that are responsible for the 22.38 year Hale-like modulation of the Solar activity cycle. 

The Second is via modulation of the VEJ Tidal-Torquing Process via the the Gear Effect 

a) The Gear Effect modulates the changes in rotation rate of the outer convective layers of the Sun that are being driven by the VEJ tidal-torquing effect. 

b) This modulation is greatest whenever Saturn is is quadrature with Jupiter. These periodic changes in the modulation of the rotation rate increase and decrease over a 19.859 year period. 

c) The Gear Effect is most effective at the times when Venus and the Earth are aligned on the same side of the Sun.

     Hence, it makes sense to combine the VEJ Tidal-Torquing Model with the Gear Effect to produce a new model called the VEJ Spin-Orbit Coupling Model. This new model is called a spin orbit coupling model for the simple reason that its net outcome is to produce link between changes in the rotation rate of the outer convective layers of the Sun (SPIN) [mostly likely near the Sun's equatorial regions)] and changes in the Sun's motion about the CMSS (ORBIT).  

     Finally, it needs to be pointed out that the combined bulk motions in the outer layers of the Sun that are produced by the VEJ Tidal Torquing Model and the Gear Effect, should exhibit a long term periodicity that are multiples of the synodic product of basic periodicities for two process i.e. 22.38 years and 19.859 years, respectively. Hence:

(22.38 x 19.859) / (22.38 - 19.859) =  176.30 years.   

These multiples include:

176.30 / 2 = 88.15 years  - Gleissberg Cycle  

2 x 176.30 = 352.6 years

3 x 176.30 = 528.9 years

4 x 176.30 = 705.2 years

as well as

1151.0- 176.30 = 974.7 - Eddy Period

The 176.30 year period is subject modulation by the period of the Jupiter-realignment cycle for the VEJ Tidal-Torquing Model of 1151.0 years. This produces the 208 year de Vries Cycle.
http://astroclimateconnection.blogspot.com.au/2013/08/the-vej-tidal-torquing-model-can.html

(1151.0 x 176.30) / (1151.0 - 176.30) = 208.2 years  - de Vries Cycle

All these are very close to the periods that are found by McCracken et al. [2013] using two 9400 year long  Be10 records from the Arctic and the Antarctic and a similar length C14 record.

SPIN-ORBIT MODEL_____McCracken et al.______Cycle____
______(years)________________(years)___________________

______88. 2_________________87.3±0.4               Gleissberg
_____208.2__________________208±2.4               de Vries
_____352.6__________________350±7
_____528.9__________________510±15
_____705.2__________________708±28
_____974.7__________________976±53                 Eddy
____2302.1_________________2310±304               Hallstatt(*)

(*) See: http://astroclimateconnection.blogspot.com.au/2013/08/the-vej-tidal-torquing-model-can.html

References

1. Fairbridge, R.W. & Shirley, J. H., 1987, Prolonged Minima and the 179-yr Cycle of the Solar Inertial Motion, Sol. Phys. 110, 191-210

2. Jose, P.D.: 1965, Sun’s motion and sunspots. Astron. J. 70, 193 – 200.

3. McCracken, K.G., Beer, J., Steinhilber, F, and Abreu, J.: 2013, A phenomenological study of the

cosmic ray variations over the past 9400 years, and their implications regarding solar activity and the solar dynamo, Sol. Phys., Volume 286, Issue 2, pp. 609 – 627

4. Wilson, I.R.G., Carter, B.D., and Waite, I.A., 2008, Does a Spin-Orbit Coupling Between the Sun and the Jovian Planets Govern the Solar Cycle?, Publications of the Astronomical Society of Australia, 2008, 25, 85 – 93. http://www.publish.csiro.au/paper/AS06018.htm