14.17:
Kepler’s Third Law of Planetary Motion
Consider a planet orbiting the Sun in an elliptical orbit. If the planet takes time T to complete one orbit around the Sun, its angular velocity is expressed as two pi divided by T.
The centripetal force acting on the orbiting planet is expressed as the products of its mass, square of its angular velocity, and its average distance from the Sun.
Substituting for omega, the centripetal force equation can be obtained in terms of the planet's orbital period and the orbit's semi-major axis.
The Sun's gravitational force provides the centripetal force for the planet to orbit around the Sun. Substituting and rearranging the terms, square of orbital period equals the product of a constant term and the cube of the orbit's semi-major axis.
Therefore, the square of a planet's orbital period is proportional to the cube of the orbit's semi-major axis. This is popularly known as Kepler's third law of planetary motion. The proportionality constant, which involves the gravitational constant and the Sun's mass, was derived later when Newton stated his law of gravitation.
14.17:
Kepler’s Third Law of Planetary Motion
In the early 17th century, German astronomer and mathematician Johannes Kepler postulated three laws for the motion of planets in the solar system. In 1909, he formulated his first two laws based on the observations of his forebears, Nikolaus Copernicus and Tycho Brahe. However, in 1918, he published his third law of planetary motion, which gives a precise mathematical relationship between a planet's average distance from the Sun and the amount of time it takes to revolve around the Sun. It states that
The proportionality constant for this law was derived much later, after Newton established the universal law of gravitation. However, Kepler formatted this third law of planetary motion based on the observation of Galilean moons orbiting Jupiter.
This law established the ratios of the average distances of each planet from the Sun, to the Earth's average distance from the Sun, as shown in Table 1.
Table 1: Ratio of Planetary Distances from the Sun Compared to the Earth-Sun Distance.
| Material | Planetary Distance Ratio |
| Mercury | 0.387 |
| Venus | 0.723 |
| Earth | 1.000 |
| Mars | 1.520 |
| Jupiter | 5.200 |
| Saturn | 9.570 |
| Uranus | 19.170 |
| Neptune | 30.180 |
The actual value for the distance of the Earth from the Sun was calculated for the first time during a transit of Venus using the parallax method. Using this value of the astronomical unit, the actual distances to all other planets were obtained. This law is very useful to estimate the distances of all the satellites around the Jovian planets, just by observing their time periods.
This text is adapted from Openstax, University Physics Volume 1, Section 13.5 Kepler’s Laws of Motion.