Redefining Kepler's 3rd Law of Planetary Motion As One of Orbital Motion
Kepler's 3rd law of motion is a simple use of proportions and proportions are scalable, having no units.
I propose this law can be used for moons as well as planets.
Its current definition does not make this clear.
Its current definition of Kepler's 3rd Law is simple:
" The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit. "
It can be expressed by the formula: P^2 = R^3
where P is the period in years and R is the radius in AU.
I find no description of this equation explaining the values are ratios not values with specific units being ignored.
This equation is NOT actually year^2 = AU^3 because mixing these inconsistent units is invalid.
The trigger for this post is this equation apparently fails at using units correctly, but despite an apparent mistake the equation works.
The solution is a better description of this equation and its use.
The values in the equation are ratios so they actually have no units.
All of the major bodies in the solar system rotating around the Sun have consistent ellipses around the barycenter.
The simplest ratio to use in the equation is basing the ratio values on Earth's orbit.
This simple equation has unit-less ratios using the proportions of the planet's orbit to Earth's orbit.
This equation can be expressed in a slightly different format:
PF^2 = RF^3
where PF is the Period Factor and RF is the Radius Factor.
PF is the scalar factor relative to the baseline period with the ratio of two values in whatever units are used to describe the period. The values from each planet could be years, days, hours, or anything consistent. This is a ratio where its common units are critical.
The ratios require unit consistency but, as a result, the equation uses values with no units.
RF is the scalar factor relative to the baseline radius from a ratio in whatever units are used to describe the radius. The values from each planet could be AU, km, or anything consistent. This factor is a ratio.
Currently these factors are the ratio for each planet to Earth's values as the baseline.
That is why the numbers of years and of AU work. This equation can also use ratio values from a different basis than Earth's orbit parameters.
If the baseline were Mercury's orbit then the PF is the planet's ratio to Mercury's period and the RF is the ratio to Mercury's orbit radius.
The planetary data:
Mercury =0.3871 AU, 0.2409 yr
Venus = 0.7233 AU, 0.6152 yr
Earth = 1AU, 1yr
Each factor is a ratio from a body's value divided by basline value. Currently we use Earth for the baseline for planets.
For Mercury based on Earth, PF = 0.2409AU / 1 AU, RF= 0.1871 yr / 1 yr or 0.1871 with no units
For Mercury based on Venus, PF = 0.391. RF= 0.5352
These values based on Venus for Mercury conform to the P^2 = R^3 expectation.
As required, there are no units with the ratio values entered in the equation.
All of the planets could have their factor based on their ratio to another planet because their ellipses are consistent.
The interesting application of this new definition of Krpler's 3rd Law is its application to systems of moons around a primary.
Jupiter has a number of large moons:
1 Io, 2 Europa, 3 Ganymede, 4 Callisto, 5 Amalthea, 6 Himalia, 7 Elara
If the moons other than Europa have their PF and RF based on the orbit of Eropa. this squared=cubed equation applies to all these moons. Europa can have its ratios based on Io.
In these cases of distant moons the deviation is small but expected given the approximate values available.
This approach using ratios works for the moons around Mars, Jupiter, Saturn, Uranus, Neptune, and Pluto.
In the case of Saturn's moons, the orbit of Janus can be the baseline for most moons. Janus can use Mimas for its baseline.
An issue with Saturn's moon collection is after the main moons there are more moons with higher eccentricities.
As Wikipedia describes: Saturn has 24 regular satellites in prograde orbits not inclined. The remaining 58 are irregular satellites with a mix of prograde and retrograde.
In the case of moons around Uranus, the orbit of Miranda can be the baseline for most moons. Miranda can use Ophelia for its baseline. Uranus also has irregular satellites.
In the case of moons around Neptune, the orbit of Naiad can be the baseline for most moons. Naiad can use Thalassa for its baseline.
In the case of moons around Pluto, the orbit of Styx can be the baseline for most moons. Styx can use Charon for its baseline.
In the case of only 2 moons around Mars, the orbit of one is the baseline for the other. The equation still applies for both.
The equation's deviation from equality can increase as a moon's orbit eccentricity gets far from a circle. Some of the distant moons around planets have rather high eccentricity values and often very long periods.
This simple equation involves ratios and is not sensitive to the units used for the baseline messurements.
This observation suggests planets and their moons share a common mechanism for their orbit characteristics around their primary, whether the primary is the Sun or a planet.
On January 29 I posted "Solar System Synchronicity" about the consistent conformance to Kepler's 3rd Law. This post is somewhat a sequel.
I expect the consistency between planet ellipses and moon ellipses is important for understanding these systems of disparate masses..
The giant planets are probably a mix of original larger partners with smaller satellites being captured later. At the end of this process of a diverse collection, Kepler's 3rd law still describes the orbits.
The new tentative definition of the 3rd Law of Orbital Motion:
A planet or moon is part of a collection of similar bodies in rotation around a primary. Kepler defined these ellipses in the other laws.
The orbital period of a body is expressed as a ratio to the period of another body in the collection.
The semi-major axis of a body is expressed as a ratio to the axis of another body in the collection.
The new distinction:
The square of the orbital period of a body (from a ratio) is directly proportional to the cube of the semi-major axis of its orbit (from a ratio).
For all bodies in the solar system rotating around the Sun, the Earth is the usual body serving as the ratio reference, using the body's orbit in years for the period and the axis in AU for those ratios.
In a collection of moons where some have high eccentricities there is less consistency across all bodies in the system.
Perhaps this new description of Kepler's 3rd Law is not clear enough.
This is an equation's description and its correct application.
Perhaps, this change cannot be described both completely and briefly.
If the moons are not relevant, then this description change addresses only the apparent problem with units in the equation.
I just don't know whether anyone realized Kepler's 3rd law needs this change in definition for the consistency of its application to both planets and moons.
Kepler published his laws of planetary motion around 1619.
Over 400 years, if anyone has noted Kepler's 3rd law is using ratios, neither Brittanica nor Wikipedia mentions it.
After the original post, I added
In any system of bodies rotating around a primary, The value for the PF is proportional among all the bodies just as the RF is proportional among all the bodies in their respective orbits.
In other words, the astronomical unit and the year unit we use are proportional among the bodies in the system. In our Solar system we can use our AU and our year because the Earth is in orbit with others, while in other systems these two reference values for the equation ratios are proportional in the particular system.
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