One of Saturn's moons has an orbital distance of 1.87 x 108 m. The mean orbital period of this moon is approximately 23 hours. Use this information to estimate a mass for the planet Saturn.
The mass of Saturn cannot be estimated from this information alone. To calculate the mass of Saturn, you would need to know the gravitational force between Saturn and the moon, as well as the moon's velocity.
To estimate the mass of Saturn using the given information, we can apply Newton's version of Kepler's Third Law:
T² = ((4π²)/(G(M + m))) × r³
Where:
T = Orbital period of the moon (in seconds)
G = Gravitational constant (6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻²)
M = Mass of Saturn
r = Orbital distance of the moon
Converting the orbital period of the moon from hours to seconds:
23 hours = 23 × 60 × 60 seconds = 82,800 seconds
Plugging in the given values:
(82,800)² = ((4π²)/(6.674 × 10⁻¹¹(M + m))) × (1.87 × 10⁸)³
Simplifying:
(6.86424 × 10⁹) = ((4π²)/(6.674 × 10⁻¹¹(M + m))) × (6.3286 × 10²⁴)
Cancelling common factors:
(1.02804 × 10²) = ((4π²)/(M + m)) × (1.66875 × 10¹⁴)
Rearranging the equation to solve for M:
M + m = ((4π²)/(1.02804 × 10²)) × (1.66875 × 10¹⁴)
M + m = (1.02918 × 10¹⁴)
Assuming the mass of the moon is negligible in comparison to Saturn, we can consider m to be approximately zero.
Therefore, the estimated mass of Saturn (M) is:
M = 1.02918 × 10¹⁴ kg
Hence, the estimated mass of Saturn is approximately 1.02918 × 10¹⁴ kg.
To estimate the mass of a planet using the orbital distance and period of one of its moons, we can use Kepler's Third Law of Planetary Motion. The law states that the square of the orbital period of a moon (T) is proportional to the cube of its orbital distance (r) from the planet:
T^2 ∝ r^3
We can rewrite the equation as:
T^2 = k * r^3
Where k is a constant.
Let's plug in the known values:
T^2 = (23 hours)^2
r^3 = (1.87 x 10^8 m)^3
Now we can solve for k:
(23 hours)^2 = k * (1.87 x 10^8 m)^3
Solving for k gives us:
k = (23 hours)^2 / (1.87 x 10^8 m)^3
Next, we need to find the value of k in the appropriate units. We convert the hours to seconds and the meters to kilometers to have consistent units:
k = (23 hours)^2 / (1.87 x 10^8 m)^3
k = (23 x 3600 s)^2 / (1.87 x 10^5 km)^3
Now we can substitute the values back into the equation to find k:
k ≈ (23 x 3600 s)^2 / (1.87 x 10^5 km)^3
After calculating the value of k, we can use it to estimate the mass of Saturn. However, we need one more piece of information to complete the estimation. We require the value of the gravitational constant, G, which is approximately 6.6743 x 10^-11 m^3 kg^-1 s^-2. With G and the estimated value of k, we can calculate the mass of Saturn.