Two planets A and B are moving in circular orbits around different stars. Planet A orbits a star of mass M with an orbital angular momentum per unit mass l_A . Planet B orbits a star of mass 8M with an orbital angular momentum per unit mass l_B = 2l_A . The ratio of the time periods of revolution of planet B to planet A , T_B T_A , is:

Step-by-step solution

The orbital angular momentum per unit mass is given by l = vr , where v is the orbital speed and r is the orbital radius. For a circular orbit, the orbital speed is v = GM r . Substituting v into the expression for l : l = GM r r = GMr Squaring both sides and solving for r gives: r = l^2 GM According to Kepler's third law, the time period of revolution is T = 2 r^3 GM . Substitute the expression for r into the time period formula: T = 2 ( l^2 GM )^3 GM = 2 l^6 G^4 M^4 = 2 l^3 G^2 M^2 This shows that the time period