Two planets A and B, having masses m₁ and m₂, move around the sun in circular orbits of radii r₁ and r₂ respectively. If the angular momentum of A is L and that of B is 3L, the ratio of the time periods (T_A)/(T_B) is

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(4) 1/(27)((m₂)/(m₁))³. For a circular orbit v=√(GM)/r, so the angular momentum about the sun is L=mvr=m√GMr. Therefore (L_A)/(L_B)=1/3=(m₁)/(m₂)√(r₁)/(r₂), which gives (r₁)/(r₂)=1/9((m₂)/(m₁))². Kepler's third law gives (T_A)/(T_B)=((r₁)/(r₂))^3/2. Substituting, (T_A)/(T_B)=1/(27)((m₂)/(m₁))³.

Two planets A and B, having masses m₁ and m₂, move around the sun in circular orbits of radii r₁ and r₂ respectively. If the angular momentum of A is L and that of B is 3L, the ratio of the time periods (T_A)/(T_B) is

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