A planet is revolving in a circular orbit with a time period T around the center of a star solely under the gravity of the star. Suppose the distance between the star and the planet is halved. The individual radii of the star and the planet are also halved, keeping their uniform mass densities unchanged. What will be the time period of the new orbit of the planet?

Step-by-step solution

The time period of a planet revolving in a circular orbit around a star (considering the two-body system) is given by: T = 2 r^3 G(M_s + M_p) where r is the distance between their centers, M_s is the mass of the star, and M_p is the mass of the planet. The mass of a spherical body with radius R and uniform density is M = 4 3 R^3 . When the radii of both the star and the planet are halved while keeping their densities unchanged, their new masses become: M_s' = 4 3 ( R_s 2 )^3 _s = M_s 8 M_p' = 4 3 ( R_p 2 )^3 _p = M