The first part of this post found a statistical relation between the planetary period of rotation [in days] and the diameter [in kilometres] for the Gas Giants of the Solar System.
However, describing this statistical relationship in more detail is problematical because of our limited understanding of the Gas Giants.
There are various observations, theories, models and calculations in the published literature regarding the Gas Giants. Unfortunately [for me], I find it impossible to separate fact from fiction [speculation] in the literature. This is especially true when the author liberally quotes from other sources that may [or may not] be equally speculative.
Wikipedia [for example] generally provides plenty of introductory bluster that can easily be mistaken for “fact” while their more detailed “small print” is littered with speculation, uncertainty and doubt.
Jupiter is thought to consist of a dense core with a mixture of elements, a surrounding layer of liquid metallic hydrogen with some helium, and an outer layer predominantly of molecular hydrogen.
Beyond this basic outline, there is still considerable uncertainty.
The core is often described as rocky, but its detailed composition is unknown, as are the properties of materials at the temperatures and pressures of those depths
Standard planetary models suggest that the interior of Saturn is similar to that of Jupiter, having a small rocky core surrounded by hydrogen and helium with trace amounts of various volatiles.
While the model considered above is reasonably standard, it is not unique; other models also satisfy observations. For instance, if substantial amounts of hydrogen and rocky material are mixed in the ice mantle, the total mass of ices in the interior will be lower, and, correspondingly, the total mass of rocks and hydrogen will be higher. Presently available data does not allow science to determine which model is correct.
Neptune’s internal structure resembles that of Uranus. Its atmosphere forms about 5% to 10% of its mass and extends perhaps 10% to 20% of the way towards the core, where it reaches pressures of about 10 GPa. Increasing concentrations of methane, ammonia and water are found in the lower regions of the atmosphere.
Wikipedia is fairly precise regarding the measurement of planetary radius for the Gas Giants. Unfortunately, we don’t know precisely what they have measured. It’s definitely not a “rocky core” because there is no certainty regarding their existence. Wikipedia is also fairly precise regarding the measurement of surface temperature and surface gravity. Unfortunately, we don’t know precisely which surfaces have been measured. Again, all I know is that it’s definitely not the “core”.
From the perspective of “atmospheric corotation” it is impossible to know whether the outer observed surfaces [of the Gas Giants] are corotating with the central “core” of the planets.
Therefore, at this stage, I will assume that the figures used in this analysis are general proxies that represent [for the Gas Giants]:
1) The diameter of the “corotating atmosphere”.
2) The rotational period of the “corotating atmosphere”
The initial analysis of “diameter versus rotation” returned 0.9921 for “R Squared”.
Using these very limited data items it is possible to compare the “rotational period” with other attributes that are derived by calculation, such as: radius, circumference, rotation surface velocity, circular surface area and spherical surface area.
Subsequent analysis of these calculated [derived] values returned 0.9988 for “R Squared” when the “circular surface area” is plotted against the “rotation surface velocity”.
Interestingly, this suggests that the unknown “attribute” of the Solar System that may be driving planetary rotation [of the Gas Giants] is associated with their “circular surface area” [visual profile].
Using the derived formula it is possible to calculate a generalised view of the relationship between the “Corotational Radius” and the ”Corotational Period” of the planets in the Solar System.
The next step in the analysis is to determine whether these generalised formulae have any predictive ability when applied to the Terrestrial Planets that have corotating atmospheres.
This analysis has followed “the data” – not the glimpse of Phi that appeared in the initial analysis.
However, for anyone looking for an elegant Phi approximation then here is one possibility: