How did Newton guess the inverse-square rule?

There has always been a great interest in the motion of planets. By the 16th century, a lot of data on the motion of planets had been collected by many astronomers. Based on these data, Johannes Kepler derived three laws that govern the planets’ motion. These are called Kepler’s laws.

These are:

Fig – Kepler’s first law of Planetary motion

Fig – Kepler’s second law of Planetary motion

Fig – Kepler’s third law of Planetary motion

  1. The orbit of a planet is an ellipse with the Sun at one of the foci, as shown in the figure given below. In this figure
    is the position of the Sun.
  2. The line joining the planet and the Sun sweep equal areas in equal intervals of time.
  3. The cube of the mean distance of a planet from the Sun is proportional to the square of its orbital period T. Or, r3/T2 = constant.

It is important to note that Kepler could not give a theory to explain the motion of planets. It was Newton who showed that the cause of the planetary motion is the gravitational force that the Sun exerts on them. Newton used the third law of Kepler to calculate the gravitational force of attraction. The gravitational force of the earth is weakened by distance. A simple argument goes like this. We can assume that the planetary orbits are circular. Suppose the orbital velocity is v and the radius of the orbit is r. Then the force acting on an orbiting planet is given by F ∝ v 2/r.
If T denotes the period, then v = 2πr / T,
so that v 2 ∝ r2 / T2
.We can rewrite this as v2 ∝ (1/r) × ( r3/T2 ). Since r3/T2 is constant by Kepler’s third law, we have v 2 ∝ 1/r. Combining
this with F ∝ v 2/ r, we get, F ∝ 1/ r2

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