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 revovution of a planet around the Sun, M = mass 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, T=T_(1)`
`therefore T_(1)=(2pi)/(sqrt(GM))R^(3//2)`
For `r=2R, T=T_(2)`
`therefore T_(2)=(2pi)/(sqrt(GM))(2R)^(3//2)=(2pi)/(sqrt(GM))R^(3//2)x2^(3//2)=T_(1)2^(3//2)`
`therefore T_(2)=T_(1)sqrt8=sqrt8T.`