The Physics Classroom website says:
‘Kepler’s third law provides an accurate description of the period and distance for a planet’s orbits about the sun. Additionally, the same law that describes the T²/R³ ratio for the planets’ orbits about the sun also accurately describes the T²/R³ ratio for any satellite (whether a moon or a man-made satellite) about any planet. There is something much deeper to be found in this T²/R³ ratio – something that must relate to basic fundamental principles of motion.’
But is it really quite simple?
Take a few basic formulae for spheres:
Radius = r
Circumference = 2pi r
Area (flat surface of half sphere) = pi r²
Volume = (4pi r³) / 3
If we’re talking about planetary bodies:
Radius = distance from the Sun (aka ‘semi-major axis’)
Circumference = orbital distance (i.e. one ‘lap’ of the Sun)
Area = the area swept by one orbit (see graphic)
Volume = the volume of space needed to contain that orbit
The only variable is the radius. Pi and the numbers are constants so can be discarded for this exercise.
Therefore circumference varies as r, area varies as r², and volume varies as r³.
Variation = the difference between any two planetary bodies A and B.
So: volume variation / (circumference x area) variation = a constant (because r³ = r² x r).
For example, compare Venus and Earth:
Radius or SMA ratio = 0.723:1
Circumference ratio = 0.723²:1² = 0.522729:1
Volume ratio = 0.723³:1³ = 0.377933:1
Obviously: 0.377933 = 0.723 x 0.522729
Planetary data: http://nssdc.gsfc.nasa.gov/planetary/factsheet/
It also follows that:
(Circumference/area) ratio x orbit speed ratio = (orbit period ratio)²
[i.e. the ratio of any two planetary bodies]
So is it true that ‘There is something much deeper to be found in this T²/R³ ratio’?
Or is it quite straightforward?