Let the period of revolution of a planet at a distance R from a star be T. Prove that if it was at a distance of 2R from the star, its period of revolation will be `sqrt8T`.

Explanation

`T= (2pi)/(sqrt(GM)) r^(3//2), ` where T = period of revolution of a planet around the sun, M= mass of the sun, G = gravitational constant and r = radius of the orbit assumed to be circular = distance of the planet from the sun.
For ` r= R and T = T_(1)`
` T_(1) = (2pi)/(sqrt(GM)) R^(3//2)`
From r = 2 R and T = `T_(2)`
` T_(2) = (2pi)/(sqrt(GM)) (2R)^(3//2) = (2pi)/(sqrt(GM)) R^(3//2) xx 2^(3//2)= T_(1) xx 2^(3//2)`
` T_(2) = T_(1) sqrt(8) = sqrt(8)T`