Question

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.

Answer 1

Answer:

$5.67\cdot 10^{26} kg$

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:

$\frac{GMm}{r^2}=m\frac{v^2}{r}$

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

$M=\frac{v^2r}{G}$

Also, we can write the orbital speed as the ratio between the length of the orbit (circumference of the orbit) and orbital period, T:

$v=\frac{2\pi r}{T}$

Substituting into the equation for M,

$M=\frac{4\pi^2 r^3}{GT^2}$

Here we have:

$r=1,222,000 km = 1.222\cdot 10^6 km = 1.222\cdot 10^9 m$ is the orbital radius

$T=16 d \cdot 86400 s/d=1.38\cdot 10^6 s$ is the orbital period

Substituting, we find the mass of the Saturn:

$M=\frac{4\pi^2 (1.222\cdot 10^9)^3}{(6.67\cdot 10^{-11})(1.38\cdot 10^6)^2}=5.67\cdot 10^{26} kg$