Question

Calculate the mass of Saturn based on the orbit of Titan by means of Newton’s form of Kepler’s third law. Mass of Titan (M) = 1.35 ×10²³ kg; average  radius of the orbit of Titan (a) = 1222 × 10³ km; period of revolution of Titan (p) = 15.94 d; G  = 6.67 ×10^-11 Nm² /kg².

Answer 1

p² = (4π²/G(M + m))a³

Let m << M, M + m ≅ M

p² = (4π²/GM)a³

M = (4π²a³/Gp²)

p² = (15.94 x 24 x 60 x 60)² = 1.8967 x 10¹²

π² = (3.14)² = 9.8596

a³ = (1222 x 10⁶)³ = 1.824 x 10²⁷

M = ((4 x 9.8596 x 1.824 x 10²⁷)/(6.67 x 10^-1 x 1.8967 x 10¹²) = 5.686 x 10²⁶kg

M = 5.69 x 10²⁶kg