The period of Mars' rotation is 24 hours, 37 minutes, and 23 seconds. At what altitude above Mars would a "Mars-stationary" satellite orbit?
A Mars-stationary satellite would orbit at an altitude of approximately 3,436 kilometers above the surface of Mars.
To determine the altitude of a "Mars-stationary" satellite, we need to consider the rotation period of Mars and its gravitational pull.
The term "Mars-stationary" suggests that the satellite would stay fixed in relation to the Martian surface, appearing to hover above one specific point. Since Mars' rotation period is given as 24 hours, 37 minutes, and 23 seconds, we can equate this to one Martian day.
To calculate the altitude, we'll use the concept of the synchronous orbit. In a synchronous orbit, a satellite orbits a celestial body at a specific altitude and matches the body's rotation, effectively appearing stationary from the surface.
To find the altitude, we can use the formula:
Altitude = (G * M * T^2 / 4π^2)^(1/3) - R,
where G is the gravitational constant, M is the mass of Mars, T is the rotation period of Mars, and R is the radius of Mars.
First, we need to gather the necessary data:
- The rotation period of Mars is 24 hours, 37 minutes, and 23 seconds.
- The gravitational constant (G) is approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2.
- The mass of Mars (M) is approximately 6.39 × 10^23 kg.
- The radius of Mars (R) is approximately 3,389.5 kilometers (km) or 3,389,500 meters (m).
Now we can plug in the values and calculate the altitude:
Altitude = (6.67430 × 10^-11 * 6.39 × 10^23 * (24 * 3600 + 37 * 60 + 23)^2 / (4 * π^2))^(1/3) - 3,389,500 m.
Evaluating this equation will give us the approximate altitude above Mars at which a "Mars-stationary" satellite would orbit.
To determine the altitude above Mars at which a "Mars-stationary" satellite would orbit, we need to use Kepler's third law and the period of rotation of Mars.
1. Convert the period of rotation into hours:
24 hours + (37 minutes / 60 minutes per hour) + (23 seconds / 3600 seconds per hour) = 24.6231 hours
2. Use the formula for the period of a satellite to derive the altitude:
T = 2π√(r³/GM)
T = period of rotation of Mars (in hours)
r = radius of Mars plus altitude of the satellite (in kilometers)
G = gravitational constant (6.67430 × 10^-20 km³/(kg·s²))
M = mass of Mars (6.39 × 10^23 kg)
3. Rearrange the formula to solve for the altitude, r:
r = [(T² * GM) / (4π²)]^(1/3) - radius of Mars
4. Plug in the values and calculate the altitude:
r = [(24.6231² * (6.67430 × 10^-20) * (6.39 × 10^23)) / (4π²)]^(1/3) - (3,389.5 km)
r ≈ 20,400 km
Therefore, a "Mars-stationary" satellite would orbit at an altitude of approximately 20,400 kilometers above Mars.