On Apollo missions to the Moon, the command module orbited at an altitude of 150 above the lunar surface.
How long did it take for the command module to complete one orbit?
To determine the time it took for the command module to complete one orbit around the Moon during the Apollo missions, we need to know the orbital period of the command module. The orbital period is the time it takes for an object to complete one full orbit around a celestial body.
During the Apollo missions, the command module orbited at an altitude of 150 miles above the lunar surface. The orbital period of an object depends on its altitude and the mass of the celestial body it is orbiting. In this case, the celestial body is the Moon, which has a mass of approximately 7.35 x 10^22 kilograms.
To calculate the orbital period of the command module, we can use Kepler's Third Law of planetary motion, which states that the square of the orbital period (T) is proportional to the cube of the semi-major axis (a) of the orbit. The semi-major axis is the average distance between the object and the center of the celestial body.
In this case, the altitude of the command module is the sum of the radius of the Moon (1,738.14 kilometers) and the altitude above the lunar surface (which we convert to kilometers). So the semi-major axis (a) can be calculated as follows:
a = radius of the Moon + altitude above the lunar surface
a = 1,738.14 km + (150 miles * 1.60934 km/mile)
a = 1,738.14 km + 241.401 km
a = 1,979.541 km
Once we have the semi-major axis (a), we can calculate the orbital period (T) using the equation:
T^2 = (4π^2 / GM) * a^3
Where G is the gravitational constant (6.67430 × 10^-11 m^3 kg^-1 s^-2) and M is the mass of the Moon (7.35 x 10^22 kg). We convert the values from kilometers to meters for consistency.
T^2 = (4π^2 / (6.67430 × 10^-11 m^3 kg^-1 s^-2 * 7.35 x 10^22 kg)) * (1,979,541 m)^3
After performing the calculation, we find:
T^2 ≈ 239,083,654 seconds^2
Taking the square root of both sides gives us:
T ≈ 15,465 seconds
Therefore, it took approximately 15,465 seconds or 4 hours and 17 minutes for the command module to complete one orbit around the Moon during the Apollo missions.