One of Saturn's moons has an orbital distance of 1.87×100000000m. The mean orbital period of this moon is approximately 23hours. Use this information to estimate a mass for the planet Saturn.
To estimate the mass of Saturn using the given information, we can apply Newton's version of Kepler's Third Law. This law states that the square of the orbital period of a satellite is directly proportional to the cube of its orbital radius.
First, let's convert the given orbital distance to kilometers:
1.87×100000000m = 1.87×10^8m = 1.87×10^8/1000 = 1.87×10^5km
Next, let's convert the orbital period from hours to seconds:
23 hours = 23 × 60 × 60 seconds = 23 × 3600 seconds = 82,800 seconds
Now, let's plug the values into the formula and solve for the mass of Saturn:
(T1^2 / T2^2) = (R1^3 / R2^3)
(T1^2 / T2^2) = (R1^3 / R2^3)
(Mean orbital period of Saturn's moon)^2 / (Mean orbital period of Earth's moon)^2 = (Orbital distance of Saturn's moon)^3 / (Orbital distance of Earth's moon)^3
(T2^2 / T1^2) = (R2^3 / R1^3)
(T2^2 / 82,800^2) = (1.87×10^5)^3
Simplifying the equation further, we can solve for the mass M of Saturn:
(T2^2 / 82,800^2) = (1.87×10^5)^3
Multiply both sides by (82,800^2) to get:
T2^2 = (82,800^2) * (1.87×10^5)^3
Finally, take the square root of both sides to isolate T2:
T2 = sqrt[(82,800^2) * (1.87×10^5)^3]
By substituting the value of T2 = 82,800, we can solve for the mass:
M = sqrt[(82,800^2) * (1.87×10^5)^3]
Using a scientific calculator, we can calculate the estimated mass of Saturn based on the given information.