The Long-Term Periodicities of the VEJ Spin-Orbit Coupling Model

The reader should be familiar with the contents of the
following paper before continuing with this post:

Wilson I.R.G., The Venus–Earth–Jupiter spin–orbit 
coupling model, Pattern Recogn. Phys., 1, 147–158,
2013

which can be freely downloaded at:
http://www.pattern-recogn-phys.net/1/147/2013/prp-1-147-2013.html

In this paper, Wilson (2013) constructs a Venus–Earth
–Jupiter spin–orbit coupling model from a combination
of the Venus–Earth–Jupiter tidal-torquing model and
the gear effect. The new model produces net tangential
torques that act upon the outer convective layers of the
Sun with periodicities that match many of the long-term
cycles that are found in the 10Be and 14C proxy records
of solar activity.

Wilson (2013) showed that there are 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:

– Tidal bulges are formed at the base of the convective
layers of the Sun by the periodical alignments of Venus
and the Earth.

– Jupiter applies a tangential gravitational torque to these
tidal bulges that either speed-up or slow-down parts of
the convective layer of the Sun.

– Jupiter’s net tangential torque increases the rotation rate
of the convective layers of the Sun for 11.07 yr (seven
Venus–Earth alignments lasting 11.19 yr) and then
decreases the rotation rate over the next 11.07 yr.

– The model produces periodic changes in rotation rate
of the convective layers of the Sun that result a 22.14 yr
(Hale-like) modulation of the solar activity cycle ( 14
Venus–Earth alignments lasting 22.38 yr).

– There is a long-term modulation of the net torque that
is equal to the mean time required for the 11.8622 yr
periodic change in Jupiter’s distance from the Sun to
realign with the 11.0683 yr tidal-torquing cycle of the
VEJ model.

The second way is via modulation of the VEJ 
tidal-torquing process via the gear effect:

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.

– This modulation is greatest whenever Saturn is in
quadrature with Jupiter. These periodic changes in the
modulation of the rotation rate vary over a 19.859 yr
period.

– The gear effect is most effective at the times when Venus
and the Earth are aligned on the same side of the Sun.

– There is a long-term modulation of the net torque that
has a period of 192.98 yr.

Note: The sidereal orbital periods used in this post are
those provided by:
http://nssdc.gsfc.nasa.gov/planetary/planetfact.html

 = sidereal orbital period of Venus = 0.615187(1) yrs

= sidereal orbital period of the Earth = 1.000000 yrs

= sidereal orbital period of Jupiter = 11.8617755(6) yrs

= sidereal orbital period of Saturn = 29.45663 yrs

= synodic period of Venus/Earth = 1.59866(5) yrs

= synodic period Jupiter/Saturn = 19.8585(3) yrs


The Physical Meaning for each of the Periodicities

The 22.136 Year Period of the VEJ Tidal-Torquing Model 

This is the time over which the angle between the nearest
VE tidal bulge (formed in the convective layers of the Sun)
and Jupiter moves from 0 to 180 degrees

Jupiter's net tangential torque increases the rotation rate of
the Sun's convective layers for the first 11.068 years and
then decreases the rotation rate for the remaining 11.068
years.

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 × 11.068 yr = 22.137 yr. This is essentially equal
to the mean length of the Hale magnetic sunspot cycle of
the Sun, which is 22.1 ± 2.0 yr (Wilson, 2011).

The 22.136 year period is simply half the realignment
time between Venus, the Earth and Jupiter (= 44.272
years) and it can be represented by the equation:

(Paul Vaughan - private communications).

The 165.42 year Modulation Period of the
Net Tangential Torque of Jupiter 

The 11.068 year period in the net tangential torque of
Jupiter acting upon the base of the Sun's convective
layer is modulated by the 11.862 year variation in the
mean distance of Jupiter from the Sun. This produces
a 165.42 year modulation in Jupiter's peak net tangential
torque given by:

The 193.02 year Modulation Period of the Gear Effect

The is the time required for the 22.137 yr periodicity of the
net tangential torque of Jupiter associated with the VEJ
Tidal-Torquing model to re-align with the 19.859 yr period
associated with the gear effect:

which can also be written as;

linking this modulation cycle to a multiple of the
period of time required for the planets Venus, the Earth,
Jupiter and Saturn to re-synchronize their orbits.

The 88 Year Gleissberg Cycle

The 88 year Gleissberg Cycle is a well identified
long-term periodicity that is seen in the level of solar
activity. The following equation shows that is merely
the synodic beat period between half the synodic
period of Jupiter/Saturn (= 9.9293 yrs) and seven
time the synodic period of Venus/Earth = 11.191 yrs.

Half the synodic period of Jupiter/Saturn is the time
between successive quadratures of Jupiter and Saturn
which is the main periodicity of the gear effect, while
seven times the synodic period of Venus/Earth
is the periodicity of the link between the VEJ
tidal-torquing model and the gear effect.

Of course, multiples of the Gleissberg period
correspond to long-term periodicities that were found
by McCracken et al. [2012]:

1 x 88.09 = 88.09 yrs --> 87.3 ± 0.4 yrs

4 x 88.09 = 352.36 yrs --> 350 ± 0.7 yrs

6 x 88.09 = 528.54 yrs --> 510 ± 15 yrs

8 x 88.09 = 704.72 yrs --> 708 ± 28 yrs

7 x 165.42 yrs = 6 x 193.02 yrs ≈ 1158 yrs

The following formula are direct consequence of
the above commensurablity:

The last equation links the orbital periods Venus and
the Earth to those of Jupiter and Saturn.