Kepler’s Laws of Planetary Motion & Newton’s Law of Gravitation

In the early 1600s, Johannes Kepler proposed three laws of planetary motion as follows:

Kepler’s First law (Law of orbits)

Each planet moves around the sun in an elliptical orbit with the sun at one of its foci. An ellipse is a closed curve such that the sum of the distances from any point P on the curve to two fixed points (F1 , F2) remains constant.

Second law (Law of areas)

As the planet moves in its orbit, a line drawn from the sun to the planet sweeps out equal areas in equal intervals of time. Let PQS and RST be the areas swept by the line joining the planet and the sun in equal intervals of time. Kepler found that these areas are equal. Hence the speed of the planet around sun must be maximum at the perihelion position and minimum at the aphelion position.

Third law (Law of periods)

The squares of the periods of revolution of the planets are proportional to the cubes of their mean distances from the sun. If R is the mean distance of the planet from the sun and T is the period of its revolution the third law states that T² ∝ R³ . As per this law, the planets with the mean distances from the sun, their orbital periods and velocities are listed in the table.

A century later, Newton demonstrated that Kepler’s laws were the consequence of a simple force that exists between any two masses. Newton’s law of gravitation and laws of motion, provide the basis for the motion of planets and satellites.

Newton’s universal law of gravitation

Everybody in the universe attracts every other body with a force which is directly proportional to the product of the masses of the two bodies and inversely proportional to the square of the distance between them. If m₁ and m₂ are the masses of two bodies separated by a distance r, the force of attraction F between them is given by:

where G is the universal constant of gravitation. The value of G = 6.67 x10^-11 N m² Kg^-2 . The force of gravitation is directed along the line joining the two bodies. If m1 = m2 = 1 kg and r = 1 m then F = G. Thus the gravitational constant is equal to the force of attraction between two bodies each of mass 1 kg separated by a distance of 1 metre.