Question

Suppose we have made a model of the Solar system scaled down in the ratio $\eta$ but of materials of the same mean density as the actual materials of the planets and the Sun. How will the orbital periods of revolution of planetary models change in this case?

Answer 1

$T=\frac{2\pi R^{\frac{3}{2}}}{\sqrt{\gamma m_s}}$
If the distances are scaled down, $R^{\frac{3}{2}}$ decreases by a factor ${\eta }^{\frac{3}{2}}$ and so does $m_s$. Hence $T$ does not change.