Kepler's 3rd Law of Planetary Motion

"A change to the orbital period (p) of a planet, squared, is proportional to a change to its semi-major axis (a), cubed."

The 3rd Law of Kepler is commonly referred to as the ‘Harmonic Law’, and is usually expressed mathematically as follows:


Of the sum total of all the components defining the orbital ellipse of each planet, Kepler’s 3rd law details a most specific relationship between the semi-major axis of each planetary body, and its orbital period; such that a precise change to either one must result in a precise change to the other, in accordance with a power relationship of 3 to 2.

The semi-major axis, as already noted, is the mean distance between any given planet and the sun; the mathematical average between its distance at aphelion and at perihelion. The orbital period is the total time required by a planet to complete one full orbit about the sun, equating thus to the area of the ellipse.

Diagram 1: In the slight elliptical orbit above, the planet is at its mean distance from the sun when at position M1 (pictured) or M2. The actual mean distance or semi-major axis itself is denoted by the line connecting the sun at the focal point to the planet at M1. The whole area of the ellipse denoted by P equates to one complete orbital period. (Point P specifies the perihelion and point A the aphelion).

The orbital configurations of the planets as they stand prove the truth of Kepler’s 3rd law. This is not evident however if just any one planet is examined in isolation. At least two planets must be evaluated together to demonstrate that the law is indeed valid in its expressed form.

For this reason, Kepler’s 3rd Law itself is usually converted into the form of a mathematical equation, allowing the details of two elliptical orbits to be compared together. An evaluation of any two planets within the solar system employing the equation form of this law readily confirms its validity, enabling one thus to understand how it is capable of unifying the whole solar system around the sun, as the primary central body:

By inputting values into this equation actually representative of the semi-major axis and orbital periods of any two planets within the solar system, one can see exactly how Kepler's law captures a real relationship between the two planets. An example, making use of the Earth and Mars will serve to illustrate this, by solving the harmonic equation for one unknown value, using three known values.

In this instance, the orbital period of the Earth, the semi-major axis of the Earth, and the orbital period of Mars will be chosen as the three known values. The remaining unknown to be calculated shall be the semi-major axis of Mars.

According to modern observations, the real magnitudes for each of the three chosen known values are as follows [1]:

   Earth Orbital Period
      = 365.2421897 Solar days (24 hours)

   Earth Semi-Major Axis
      = 81801193.26 Ideal Geographical Miles (1 IGM = 6000 ft)

   Mars Orbital Period
      = 686.9297110 Solar days (24 hours)

Substituting the above values into the stated equation, one is thus able to determine the semi-major axis of Mars (a) as follows:

One need only compare the calculated answer using Kepler’s 3rd law, with a known value for the semi-major axis of Mars to see that they are close [2]:

124638661.47 IGM

The actual observation of Mars does confirm then that the answer is indeed correct, and the expressed power relations of the law are physically valid.

It is important to realise though that this is not something that is confined just to these two planetary bodies. Any two planets could be used in this equation and it would prove true. Kepler’s law in effect unites the planets of the whole solar system, revealing that the total time taken to complete one orbit about the sun determines the semi-major axis of a planet.

Thus, in the above example, if Mars, as a result of some outside agency, were to alter its orbital period so that it was completed in only 500 solar days instead of 686.9297110, then the semi-major axis of Mars would itself be forced to change in order to maintain a stable orbit. In such an instance Mars would be required to move nearer to the sun. The actual determination of its new semi-major axis could be obtained by essentially judging the old orbit against the new. Thus, the ratio of 500 / 686.9297110 could be input into the above equation. The eventual answer produced for ‘a’ would then reveal the semi-major axis of Mars that would accompany a 500 day orbital period. Kepler’s 3rd law thus embodies the idea of a principled relationship existent between two very specific planetary features i.e. orbital period and semi-major axis.


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References

[1] P. Kenneth Seidelmann (Editor) Explanatory Supplement to the (1992) Astronomical Almanac. University Science Books
The values used for the example calculation are referenced and derived from the above noted publication as follows:

i) Earth Tropical Year = 365.2421897 days (p.698)

ii) Earth Semi-major Axis is taken to be the product of the following:
1 astronomical unit of length = 149597870000 metres (p.700)
Mean distance of the Earth from Sun = 1.0000010178 AU (p.700) Therefore: (149597870000 / 1000) × 1.0000010178 = 149598022.26071 kilometres And: Converted into Ideal Geographical miles:
149598022.26071 × 0.6213711922 = 92955901.44289 Statute Miles 92955901.44289 × 0.88 = 81801193.26975 IGM

iii) Mars Orbital Period is derived from the following:
Tropical Period of Mars in Julian years = 1.88071105 (p.704)
1 Julian year = 365.25 days (p.730) Therefore: 1.88071105 × 365.25 = 686.9297110125 days

[2] Kenneth Seidelmann (Editor) 1992 Ibid

Mean distance of Mars (from Sun) in AU given as 1.5236793419 (p.704)
Therefore: 1.5236793419 × (149597870 × 0.6213711922 × 0.88) = 1.5236793419 × 81801110.012 = 124638661.47 IGM