A planet has two satellite moons. Moon X has an orbital period of 2.13 days. Moon Y has an orbital period of about 3.53 days. Both moons have nearly circular orbits. Use Kepler's third law to find the distance of each satellite from the planet's center. The planet's mass is 2.0 10^26 kg.
Moon X km?
Moon Y km?
To find the distance of each satellite from the planet's center, we will use Kepler's third law. Kepler's third law states that the square of the orbital period of a satellite is directly proportional to the cube of its average distance from the center of the planet. Mathematically, it can be written as:
T^2 = k * r^3
Where T is the orbital period in seconds, r is the average distance from the center of the planet in meters, and k is a constant that depends on the gravitational force constant and the mass of the planet.
To solve for the distance of each satellite from the planet's center, we need to rearrange the equation:
r^3 = (T^2) / k
First, let's calculate the constant k. We can substitute the mass of the planet (2.0 × 10^26 kg) and the gravitational force constant (G) into the equation:
k = 4π^2 / G * M
Where G is the gravitational force constant (6.674 × 10^-11 m^3 kg^-1 s^-2) and M is the mass of the planet (2.0 × 10^26 kg).
Next, we can calculate the value of k:
k = 4π^2 / (6.674 × 10^-11 m^3 kg^-1 s^-2) * (2.0 × 10^26 kg)
Once we have the value of k, we can use the orbital periods of Moon X and Moon Y to solve for their distances.
For Moon X:
r_x^3 = (T_x^2) / k
where T_x is the orbital period of Moon X (2.13 days).
For Moon Y:
r_y^3 = (T_y^2) / k
where T_y is the orbital period of Moon Y (3.53 days).
To convert the orbital periods to seconds, we multiply them by 24 (hours per day), 60 (minutes per hour), and 60 (seconds per minute).
Finally, we can calculate the distances of Moon X and Moon Y from the planet's center by taking the cube root of the respective equations.
Let's perform the calculations step by step to find the distances of Moon X and Moon Y from the planet's center.