Final answer:
To find the average distance from Titan to the center of Saturn, Kepler's Third Law can be applied where the formula a^3 = (G · M · T^2)/(4π^2) is used, with T representing Titan's orbital period and M being Saturn's mass.
Step-by-step explanation:
The student's question is related to the use of Kepler's Third Law to find the average distance from Saturn's moon Titan to the center of Saturn, based on its orbital period. To solve this problem, we apply Kepler's Third Law of planetary motion, which states that the square of the orbital period (T) of a planet is directly proportional to the cube of the semi-major axis (a) of its orbit, where T is the orbital period and a is the average distance from the center of the planet to the moon.
The formula derived from Kepler's Third Law is:
T^2 = (4π^2)/(G(M+m)) · a^3
Where:
- T is the orbital period
- G is the gravitational constant (6.674×10^-11 N·m^2/kg^2)
- M is the mass of the planet (Saturn in this case)
- m is the mass of the moon (Titan)
- a is the average distance from the center of the planet to the moon
Since the mass of Titan is much smaller compared to the mass of Saturn, it can be neglected in the formula. Thus, we have:
T^2 ≈ (4π^2)/(G · M) · a^3
Rearranging the formula to solve for a, we get:
a^3 = (G · M · T^2)/(4π^2)
Substituting the given values (T = 15.9 days, M = 5.68×10^23 kg), and converting T to seconds, we can calculate the value of a. After solving, we can find the average distance corresponding to one of the given options.