Do Jupiter and Saturn Have Anything to Do with Sunspots?

The conjunction of Jupiter and Saturn reminded me of an argument I had with someone about the solar cycle, the
way that the number of spots on the sun's surface oscillate over time. There are roughly 11 years
between peaks of activity, the "solar maxima", and the reason may not be well-understood.

The argument began because a few details were stuck in my brain:

1. the length of time it takes Jupiter to travel around the sun, 4333 earth days or about 12 earth years,
2. the way that our oceans are pulled by very distant objects, the moon, Sun and also Jupiter (to a much smaller extent),
3. the chemical and nuclear reactions powering the sun had not reached an equilibrium, even after billions of years.

Was it possible there was a connection? Could Jupiter be causing tides on the Sun?

The other side of the argument pointed out that Jupiter takes 12
years to orbit Sun and the solar cycle is much closer to 11. Case closed.

One night while slipping in and out of a dream state, I started imagining an orrery and realized that
while Jupiter takes 12 years to orbit, its relationship with Saturn repeats itself more often because
Saturn is also moving at the same time. Saturn's year is 10765 earth days or about 26 earth years.

Thanks to the magic of
open source software and the generosity of Chris A. Jager, I was able to build out my own
digital orrery to simulate the planets.

Tidal calculations can be elaborate and complex. I came up with the most basic of calculations that
summed the gravitational effects of eight planets on the sun:

First, the simulator converted the distance
between the planets and the sun and squared it. This isn't constant because the planets don't move
in circular orbits. Then it multiplied by the planet's weight.

Second, the simulator added the gravitational pull in the two major directions of the plane (x,y) --
with one trick.

Observers of Earth tides know that the oceans rise higher during the so-called spring tides, which
occur when the moon and the sun are either on the same side of Earth (new moon) or on exact
opposite sides (full moon). The water flows both toward and away from the gravitational mass,
an effect that is still hard for me to understand with any intuition. So capture this effect,
I measured the alignment of the planets by flipping the gravitational force vector by 180 degrees
if it was on one side of the plane. It's like putting it in a mathematical mirror.

The enhanced orrery can graph the gravitational pull according to this simple formula. (See
the function computeGravity in T6_createLayeredCanvas for exact details.)

How much time lies in between peaks? Roughly 11 years and a few months. Incidentally, a bit of deeper research
shows that these relationships have been explored before. While much of the basic literature suggests that reasons
behind the solar cycle are a mystery, others have calculated the orbits and noticed the relationship.
Here are just a few pointers:

• In L'Astronomie of November 1925, M. Jean Malburet found a relationship between the sun-spot activity and the orbits of Jupiter, Venus and Earth.
  Hazel Marie Losh at
  the Mt. Wilson observatory examined Malburet's work but noted skeptically, "There's little variation in the actual tidal forces, even when the interval between a syzygy, Earth-Jupiter, and a syzygy, Venus-Jupiter , vaaries from zero to twenty days. ... It seems entirely improbable that this small variation should control sun-spot activity...."
• Nicola Scafetta reported in May 2012
  suggested that the relatively gentle tidal forces, may stimulate some kind of resonant nuclear reaction
  that amplifies the effect. It focuses on the "spring tides of Saturn and Jupiter", in other words when
  then two planets are 90 degrees apart.
• In October 2016,
  researchers at the Helmholtz-Zentrum Dresden-Rossendorf focused on the
  alignment of Venus, Earth and Jupiter, something that happens every 11.07 years.

Clearly there is plenty of exploration to be done. The simulator below uses all eight
planets and does not focus on any particular alignment.
You're welcome to use the code to explore other configurations and weightings.

Some questions I have are:

• The distance between the planets and the sun seem to affect the size of the maxima and minima. Do
  these changes correspond to any larger solar cycles?
• Are there better ways to compute the tidal forces?
• Should the simulator take into account the length of the Sun's own rotation (roughly 27 earth days on average)?
• Could this be useful in predicting the amount of solar activity (aurorae, spots, radiation storms)?