# Itan's orbital radius is about 1222000 km and its orbital period is just under 16 days. From this information, estimate the mass of Saturn (in kg) to 2 sig figs. Use "e-notation" to report your answer.

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## Answers

Answer:[tex]5.67\cdot 10^{26} kg[/tex]Explanation:For a satellite in orbit around a planet, the gravitational force between the planet and the satellite is equal to the centripetal force that keeps the satellite in circular motion.So, we can write:[tex]\frac{GMm}{r^2}=m\frac{v^2}{r}[/tex]whereG is the gravitational constantM is the mass of the planetm is the mass of the satelliter is the orbital radius of the satellitev is the speed of the satelliteThe equation can be rewritten as[tex]M=\frac{v^2r}{G}[/tex]Also, we can write the orbital speed as the ratio between the length of the orbit (circumference of the orbit) and orbital period, T:[tex]v=\frac{2\pi r}{T}[/tex]Substituting into the equation for M,[tex]M=\frac{4\pi^2 r^3}{GT^2}[/tex]Here we have:[tex]r=1,222,000 km = 1.222\cdot 10^6 km = 1.222\cdot 10^9 m[/tex] is the orbital radius[tex]T=16 d \cdot 86400 s/d=1.38\cdot 10^6 s[/tex] is the orbital periodSubstituting, we find the mass of the Saturn:[tex]M=\frac{4\pi^2 (1.222\cdot 10^9)^3}{(6.67\cdot 10^{-11})(1.38\cdot 10^6)^2}=5.67\cdot 10^{26} kg[/tex]