Statement: - According to this law the square of time period of a planet is ∝ to the cube of the semi-major axis of its orbit.

Suppose earth is revolving around the sun then the square of the time period (time taken to complete one revolution around sun) is ∝ to the cube of the semi major axis.

It is known as Law of Periods as it is dependent on the time period of planets.

Derivation of 3^rd Law: assumption: The path of the planet is circular.

Let m=mass of planet

M= mass of sun

According to Newton’s Law of Gravitation:

F= GMm/r²

F_c=mv²/r

where

F_c =centripetal force which helps the planet to move around sun in elliptical order.

F = Fc

GMm/r²= mv²/r where r=radius of the circle

GM/r= v² (1)

v= 2 πr/T

Squaring both the sides the above equation

v²=4 π²r²/T²

putting the value (1)

GM/r=4 πr²/T²

T²= (4 π² r³/GM) where (4 π²/GM) = constant

T²=r³ (In ellipse semi-major axis is same as radius of the circle)