A satellite is to be put into a circular low earth orbit at 149 km above surface. What orbital speed (km/s) is required?
The velocity required to maintain a circular orbit around the Earth may be computed from the following:
Vc = sqrt(µ/r)
where Vc is the circular orbital velocity in feet per second, µ (pronounced meuw as opposed to meow) is the gravitational constant of the earth, ~1.40766x10^16 ft.^3/sec.^2, and r is the distance from the center of the earth to the altitude in question in feet. Using 3963 miles for the radius of the earth, the orbital velocity required for a 250 miles high circular orbit would be Vc = 1.40766x10^16/[(3963+250)x5280] = 1.40766x10^16/22,244,640 = 25,155 fps. (17,147 mph.) Since velocity is inversely proportional to r, the higher you go, the smaller the required orbital velocity.
To calculate the orbital speed of a satellite in a circular orbit, we can use the following formula:
orbital speed (V) = √(GM/R)
where:
- G is the gravitational constant (approximately 6.674 × 10^-11 N(m/kg)^2)
- M is the mass of the celestial body being orbited (in this case, the Earth)
- R is the radius of the orbit, which is the sum of the radius of the Earth and the altitude of the satellite above the Earth's surface.
First, let's calculate R. The radius of the Earth is approximately 6,371 km. Adding the altitude of the satellite, which is 149 km, we get:
R = 6,371 km + 149 km = 6,520 km
Next, we need to know the mass of the Earth (M) in kilograms. The mass of the Earth is approximately 5.972 × 10^24 kg.
Substituting the values into the formula, we have:
V = √((6.674 × 10^-11 N(m/kg)^2) * (5.972 × 10^24 kg) / (6,520 km))
Before we calculate the final result, let's convert the radius to meters:
R = 6,520 km * 1,000 m/km = 6,520,000 m
Now, we can calculate the orbital speed:
V = √((6.674 × 10^-11 N(m/kg)^2) * (5.972 × 10^24 kg) / (6,520,000 m))
Simplifying the equation, we get:
V ≈ 7.903 km/s
Therefore, the orbital speed required for a satellite in a circular low Earth orbit at 149 km above the surface is approximately 7.903 km/s.