Suppose that the gravitational force varies inversely as the nth power of distance. Then, the times period of a planet in circular orbit of radius R around the sun will be proportional to

If the gravitational force varies inversely with the n-th power of the distance, the relationship between the gravitational force and the orbital radius changes accordingly. For a planet in a circular orbit, the centripetal force required for circular motion is provided by this gravitational force. The balance of these forces determines the planet’s orbital velocity.

The time period of the orbit depends on the radius of the orbit and the orbital velocity. By analyzing this relationship under the modified gravitational law, it can be shown that the time period of the planet’s orbit is proportional to R⁽ⁿ ⁺ ¹⁾/². This reflects the dependence of orbital dynamics on the nature of the gravitational force.