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]
where
G is the gravitational constant
M is the mass of the planet
m is the mass of the satellite
r is the orbital radius of the satellite
v is the speed of the satellite
The 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 period
Substituting, 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]
