Question

Titan is the largest moon of Saturn. It orbits Saturn at a distance of 1.22 x 10^6 km with a period of 16days. Using these data, estimate the mass of Saturn.​

a) 4.5 x 10^26 kg
b) 5.7 x 10^26 kg
c) 6.9 x 10^26 kg
d) 8.2 x 10^26 kg

Answer 1

To estimate the mass of Saturn, Kepler's third law is used where the square of Titan's orbital period is proportional to the cube of its orbital distance from Saturn, allowing us to calculate Saturn's mass and select the correct option.

Step-by-step explanation:

To estimate the mass of Saturn using Titan's orbit data, we apply Kepler's third law of planetary motion. According to this law, the square of the orbital period of a moon is directly proportional to the cube of the semi-major axis of its orbit, with the proportionality constant being related to the mass of the central body, in this case, Saturn.

Given that Titan orbits Saturn with a period (T) of 16 days and at a distance (r) of 1.22 x 106 km, we can use the formula derived from Kepler's law:

T2 = (4π2/GMSaturn)r3

Where G is the gravitational constant, MSaturn is the mass of Saturn, and r is the radius of Titan's orbit.

Rearranging the formula and solving for MSaturn, we get:

MSaturn = (4π2r3)/(GT2)

Inserting the known values and solving for Saturn's mass provides us with an estimate. Comparing our result with the given options, we choose the one that matches our calculation.