Irodov Problem 1.204

As seen from problem 1.202 the time period of a planet is given by where R and r are the distance of the planet at its extremities from the sun and Ms is the sun's mass. Now scaling down the entire solar system by a factor of will have two effect, i) R and r will scale by a factor and ii) the radii of all the planets will shrink by a factor and hence their masses will shrink by a factor (since their volumes will shrink by a factor and their average densities are the same). Thus, the expression will remain unchanged, in other words the time periods of the planets will remain unchanged.