Final answer:
Using Kepler's third law, we can determine the mass of the planet by knowing the period and radius of the satellite's orbit.
Step-by-step explanation:
Kepler's third law states that the square of the period of an orbiting object is directly proportional to the cube of its average distance from the center of the planet. We can use this law to find the mass of the planet:
Given: The satellite orbits once every 6 hours and 10 minutes, which is equivalent to 6.167 hours. The radius of the orbit is 9.44 × 10^7 m.
Using Kepler's third law, we can write:
T^2 = k * r^3
where T is the period of the orbit, r is the radius of the orbit, and k is a constant.
Solving for k:
k = T^2 / r^3
Plugging in the values:
k = (6.167^2) / (9.44 × 10^7)^3
k = 0.2552 / (6.77 × 10^21)
k ≈ 3.772 × 10^-23
Now, we can use the value of k to find the mass of the planet:
M = 4π^2r^3 / GT^2
where M is the mass of the planet, G is the gravitational constant, and π is a mathematical constant.
Plugging in the values:
M = (4π^2) * (9.44 × 10^7)^3 / (6.67430 × 10^-11) * (6.167^2)
M ≈ 1.024 × 10^24 kg
Therefore, the mass of the planet is approximately 1.024 × 10^24 kg.