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 moon from the planet's center, we can use Kepler's third law, which relates the orbital period and distance of a satellite to the mass of the planet. Kepler's third law can be stated as:
T^2 = (4π^2 / G * M) * r^3
where:
T is the orbital period,
G is the gravitational constant (approximately 6.674 × 10^-11 m^3/kg/s^2),
M is the mass of the planet, and
r is the distance from the planet's center to the satellite.
To determine the distance of each satellite moon, we need to rearrange the equation to solve for r:
r = (T^2 * G * M / 4π^2)^(1/3)
Given:
Mass of the planet (M) = 2.0 × 10^26 kg
Orbital period of Moon X (T_X) = 2.13 days
Orbital period of Moon Y (T_Y) = 3.53 days
Let's calculate the distance for each moon.
For Moon X:
Convert the orbital period of Moon X to seconds:
T_X = 2.13 days × 24 hours/day × 60 minutes/hour × 60 seconds/minute = 183,888 seconds
Substitute the values into the equation:
r_X = (183,888^2 * 6.674 × 10^-11 * 2.0 × 10^26 / (4π^2))^(1/3)
Calculating the value:
r_X ≈ 3.50 × 10^8 km
Therefore, the distance of Moon X from the planet's center is approximately 3.50 × 10^8 km.
For Moon Y:
Convert the orbital period of Moon Y to seconds:
T_Y = 3.53 days × 24 hours/day × 60 minutes/hour × 60 seconds/minute = 304,032 seconds
Substitute the values into the equation:
r_Y = (304,032^2 * 6.674 × 10^-11 * 2.0 × 10^26 / (4π^2))^(1/3)
Calculating the value:
r_Y ≈ 4.16 × 10^8 km
Therefore, the distance of Moon Y from the planet's center is approximately 4.16 × 10^8 km.