If correct, this hypothesis would be a major paradigm change in our understanding of how the Sun works, although the seeming lack of influence from any of the other planets with significant gravitational pulls on the Sun, relative to the Earth and Jupiter, is suspicious.
The average strength of the gravitational pull of the planets on the Sun, normalized so that Earth's pull on the Sun is equal to one, to three significant digits, is as follows:
* Jupiter 11.7
* Venus 1.56
* Saturn 1.04
* Mercury 0.369
* Mars 0.0463
* Uranus 0.0396
* Neptune 0.0188
* Pluto 0.00000140
Given this, one would expect Venus, Saturn, and Mercury's orbits to have non-negligible effects as well.
Venus has an almost perfectly circular orbit, so that might explain a lack of a sunspot cycle effect from its gravitational pull, but this is not true of Mercury or Saturn.
Mercury's 88 day period and strongly elliptical orbit ought to be very measurable in the data as well in this hypothesis even if its small size reduces the magnitude of its impact.
Saturn's 29.4 year orbit and moderate elliptical orbit also ought to be discernible, but is long enough that the small sample size of Saturn's orbits in a three hundred year old data set whose quality declines in the older data could reduce the statistical significance of this signal.
The sunspot number record covers over three centuries.These numbers measure the activity of the Sun. This activity follows the solar cycle of about eleven years.
In the dynamo-theory, the interaction between differential rotation and convection produces the solar magnetic field. On the surface of Sun, this field concentrates to the sunspots. The dynamo-theory predicts that the period, the amplitude and the phase of the solar cycle are stochastic.
Here we show that the solar cycle is deterministic, and connected to the orbital motions of the Earth and Jupiter. This planetary-influence theory allows us to model the whole sunspot record, as well as the near past and the near future of sunspot numbers. We may never be able to predict the exact times of exceptionally strong solar flares, like the catastrophic Carrington event in September 1859, but we can estimate when such events are more probable. Our results also indicate that during the next decades the Sun will no longer help us to cope with the climate change. The inability to find predictability in some phenomenon does not prove that this phenomenon itself is stochastic.
Lauri Jetsu, "Sunspot cycles are connected to the Earth and Jupiter" arXiv:2311.08317 (November 14, 2014).
A 2022 paper includes Venus as well. There is also a 1975 paper purporting to rule out this relationship (and a 2022 paper as well) with a 2022 rebuttal (which is related to the 2022 paper).
Skimming the literature, it does seem that more accurate modeling of the shape of planetary orbits, the actual locations of planets on those orbits, and inclusion of more planets, does produce reasonably good fits to the Sun spot data, although it isn't a conclusive result.