Part a) Part b) Part c) Part d) A distant star named α of mass M...

A As you centrifugal force = centripetal force GMm_x/r^2_x = m_x v^2/r_x rightarrow r_x = GM/V^2 1 Here V^2 = (2 p[i r_x/T)^2 = GM/4 pi^2 r^2_x T^2 - 1 2 r^3_x = GM/4 pi^2 T^2 = 6.67 times 10^-11 times 10^31/4 times pi^2 times (3.14159 times 10^7)^2 r^3_x = 1.6675 times 10^34 r_x = 2.5548 times 10^11 m (B) similarly r^3_y = GM/4 pi^2 T^2 3 Here T' = 9T = 9 times 3.14159 times 10^7 m = 6.67 times 10^-11 times 10^31 times 81 times (3.14159 times 10^7)^2/4 times (3.14159)^2 r^3_y = 135.0675 times 10^34 r_y = 1.1054 times 10^12 m (C) from eq 1 r_x = GM/V^2_x similarly r_y = GM/V^2_y So r_y/r_x = V^2_x/V^2_y implies V^2_x (r_y/r_x) times V^2_y Here r_y/r_x = 0.4 V_y = 2540 m/s \r\n V^2_x = 0.44 times (2540)^2 V^2_x = 2838704 V_x = 1684.84 m/s remember r_y/r_x = 0.44 means r_y < r_x (part d) g = GM/R^2 Here G = 6.61