A Somewhat Complicated Explanation of Kepler’s Laws 77

Third Law: The Square of the Orbital Period of a

PlanetIs Directly Proportional to the Cube of the

Semimajor Axis of Its Orbit

This is the law that may be the most difficult to understand. In short, it means that the

length of the orbital period depends only on the semimajor axis of the orbit. Since it does

not depend on the eccentricity of the elliptical orbit, the period will be the same as long as

the semimajor axis is the same, regardless of whether the motion is circular or elliptical.

Incidentally, the semimajor axis is half of the major axis, which appeared in the descrip-

tion of the first law. In other words, it is the mean distance between the planet and the Sun.

Even before Kepler, scientists observed that planets with larger orbits took longer to

make one circuit (that is, their orbital period was larger). However, the mathematical rela-

tionship between that period and the semimajor axis of the orbit was not understood until

Kepler.

If we assume P is the orbital period in years and a is the semimajor axis (the mean

distance between a planet and the Sun) in astronomical units (or AUs, with 1 AU defined

as the distance between Earth and the Sun), then the following equation holds for Earth

(where a = 1 AU):

= 1

a

3

P

2

If we calculate this ratio for any planet in the solar system, we will get the following

table, which confirms Kepler’s Third Law.

SemiMajor Axis of a Planet’s Orbit and Orbital Period

Planet Semimajor axis

of orbit a (AUs)

a

3

Orbital period relative

to the fixed star’s P

(solar years)

P

2

a

3

/P

2

Mercury 0.3871 0.05800555 0.2409 0.05803281 0.9995

Venus 0.7233 0.37840372 0.6152 0.37847104 0.9998

Earth 1.0000 1 1.0000 1 1.0000

Mars 1.5237 3.53751592 1.8809 3.53778481 0.9999

Jupiter 5.2026 140.819017 11.8620 150.707044 1.0008

Saturn 9.5549 872.32524 29.4580 867.773764 1.0052

Uranus 19.2184 7098.25644 84.0220 7049.69648 1.0055

Neptune 30.1104 27299.1783 164.7740 27150.4711 1.0055

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