Derivation of Newton’s law of Gravitation from Kepler’s law

Newton’s law of gravitation proposed by Sir Issac Newton in 1687 stated that the force of attraction between two objects is directly proportional to the product of masses of the two objects and is inversely proportional to the square of the distance between the two. Newton’s law of gravitation and Kepler’s law of planetary motion are closely related. They state that a force keeps the planets revolving while staying fixed in their orbits. Kepler’s law states that the planets revolve around the sun in their fixed elliptical orbits. The shape of their orbits and their staying fixed in those respective orbits is because of the force of attraction between the sun and the planets.

The derivation of newton’s law of gravitation from Kepler’s law :

Suppose that the mass of the sun is M and that of the plant is m. That planet is revolving around the sun which revolves around the sun in an orbit of radius x. The constant angular velocity is ω. Also, suppose that T is the time period of its revolution around the sun.

The constant angular velocity ω can be written as –

ω = 2π/T … (1) 

The centripetal force which is acting on the planet for its circular motion is:

F = mrω2 

F = mr(2π/T)2

F = 4π2mr/R2

Now, looking at Kepler’s third law we can say –

T² α r³ (Or) T² = Kr³

Here, K is the constant of proportionality. The constant of proportionality is said to be the constant value of the ratio between the two proportional quantities.

F = 4π2mr/Kr3

F = 4π2/K×mr2 … (2)

F ∝ m/r2 (∵ 4π2/K is a constant) … (3)

The centripetal force here is provided by the gravitational force of attraction between the sun and the earth. Newton said that the force of attraction between the planet and the sun is mutual. The force F derived has a relation of direct proportionality with the mass of the planet. Also, it is directly proportional to the mass of the sun. Hence,

4π2/K ∝ M 

(Or) 4π2/K = GM … (4)

K is the constant of proportionality.

If we substitute equation (4) in the equation (2), what we get is –

F = GMm/r2

This is Newton’s law of gravitation. Thus, by following these steps, we can give the derivation of Newton’s law of gravitation from Kepler’s law.

It is easy to get the derivation of Newton’s law of gravitation from Kepler’s law if we follow the steps accurately and use the correct formulas.

The conclusions Newton came to on the basis of Kepler’s law of planetary motion –

There were certain things that Newton deduced after analysing Kepler’s law. They were :

• It is the centripetal force that acts on the planet due to the sun. This centripetal force is directed towards the sun.

• The force which acts on the planet due to the sun is directly proportional to the product of the masses of both the objects respectively.

• The force which acts on the planet due to the sun is inversely proportional to the square of the distance between the objects. It must be kept in mind that the distance between the two is measured from the centres of both the objects.

Importance of Newton’s law of gravitation

The law of gravitation given by Newton in 1687 has many uses in physics. Some of them are:

• It can be used to determine the trajectory of the astronomical objects and also in measuring their motions.

• It explains the rotation of all the planets around the sun in their fixed elliptical orbits.

• We can measure and detect the force of attraction between any two planets or objects in the universe.

• It helps in keeping the things on the surface of the earth in balance. Imagine if you threw a ball at the sky and never came back? The gravity doesn’t let that happen.

• It is used in the derivation of many other equations in physics and also in solving various problems in physics using it.

Importance of Kepler’s law 

The three laws of planetary motion proposed by Kepler helped the scientists to get their heads clear about many things. The laws stated various facts about the things related to the planets and their motions which helped in breaking of the prevailing myths regarding them. Some of the important things related to Kepler’s laws are:

• The first law proposed by him helped the people to know that the planets revolve around the sun in elliptical orbits.

• The second law proposed by him said that the motion of planets is proportionally faster in their orbits when the distance between them and the sun decreases.

• His third law states that the square of the time period of the revolution of the planet around the sun is directly proportional to the cube of its semi-major axis.

• Kepler’s law helped Newton to give his law of gravitation.

Conclusion

Both Newton’s law of gravitation and Kepler’s law are of great significance in physics. It is very easy to get the derivation of Newton’s law of gravitation from Kepler’s law. If one follows the steps accurately and also pays attention to the formulas used, it is an easy thing to do. Newton’s law of gravitation derived from Kepler’s law can further be used to derive other physical equations and laws.