Compute the orbital radius of an earth satellite that has an equatorial orbit and always remains above a fixed point P on the earth's surface. Communication satellites have such "geosynchronous" orbits (see figure). These satellites are used to relay radio and television signals around the world.
How do you determine the altitude at which a satellite must fly in order to complete one orbit in the same time period that it takes the earth to make one complete rotation?
The force exerted by the earth on the satellite derives from
...................................................F = GMm/r^2
where G = the universal gravitational constant, M = the mass of the earth, m = the mass of the satellite and r = the radius of the satellite from the center of the earth.
GM = µ = 1.407974x10^16 = the earth's gravitational constant.
The centripetal force required to hold the satellite in orbit derives from F = mV^2/r.
Since the two forces must be equal, mV^2/r = µm/r^2 or V^2 = µ/r.
The circumference of the orbit is C = 2Pir.
A geosycnchronous orbit is one with a period equal to the earth's rotational period, which, contrary to popular belief, requires 23hr-56min-4.09sec. to rotate 360º, not 24 hours. Therefore, the time to complete one orbit is 23.93446944 hours or 86,164 seconds
Squaring both sides, 4Pi^2r^2 = 86164^2
But V^2 = µ/r
Therefore, 4Pi^2r^2/(µ/r) = 86164^2 or r^3 = 86164^2µ/4Pi^2
Thus, r^3 = 86164^2(1.407974x10^16)/4Pi^2 = 2.647808686x10^24
Therefore, r = 138,344,596 feet. = 26,201.6 miles.
Subtracting the earth's radius of 3963 miles, the altitude for a geosynchronous satellite is ~22,238 miles.
Oh, so we're talking about satellites that never want to miss their favorite spot on Earth! How considerate of them! Now, to calculate the orbital radius of a geosynchronous satellite, we need some information. Do you happen to know the radius of the Earth, or the period of time it takes for the Earth to complete one rotation?
To calculate the orbital radius of a geosynchronous satellite above a fixed point on the Earth's surface, we need to consider two main factors: the Earth's radius and the orbital period of the satellite.
1. Earth's radius:
The average radius of the Earth is roughly 6,371 kilometers (or 3,959 miles). This value represents the distance from the center of the Earth to its surface.
2. Orbital period:
A geosynchronous satellite completes one orbit around the Earth in the same amount of time it takes for the Earth to complete one rotation on its axis. This period is approximately 24 hours, or 86,400 seconds.
Using the orbital period, we can calculate the orbital radius using the formula:
Orbital radius = (G * M * T^2) / (4π^2)^(1/3)
where G is the gravitational constant (6.674 × 10^-11 m^3/kg/s^2), M is the mass of the Earth (5.972 × 10^24 kg), and T is the orbital period in seconds.
Plugging in the values:
Orbital radius = (6.674 × 10^-11 * 5.972 × 10^24 * (86,400)^2) / (4π^2)^(1/3)
Simplifying the expression:
Orbital radius ≈ 42,164 km (kilometers) or 26,199 miles
Therefore, the orbital radius of a geosynchronous satellite above a fixed point on the Earth's surface is approximately 42,164 kilometers (or 26,199 miles).
To compute the orbital radius of a geosynchronous satellite that remains above a fixed point on Earth's surface, we need to consider the following information:
1. Rotation period of the Earth: The time it takes for the Earth to complete one full rotation on its axis is approximately 24 hours.
2. Equatorial orbit: The satellite orbits in the plane of the Earth's equator.
3. Geosynchronous orbit: The satellite's orbital period matches the rotation period of the Earth, resulting in the satellite always remaining above a fixed point P on the Earth's surface.
To calculate the orbital radius, we can start with the formula for the orbital period of a satellite, given by:
T = 2π√(R³/GM)
where:
T is the orbital period (24 hours in this case),
R is the orbital radius we want to find,
G is the gravitational constant (approximately 6.67430 x 10^(-11) m³/kg/s²),
M is the mass of the Earth (approximately 5.97219 x 10^24 kg).
Since the orbital period T is 24 hours, we can substitute these values into the equation and solve for R:
24 = 2π√(R³/GM)
To isolate R, we can square both sides, divide by 4π², and multiply by GM:
R³ = (GM/4π²) * 24²
Now, substitute the known values and calculate:
R³ ≈ ((6.67430 x 10^(-11) m³/kg/s²) * (5.97219 x 10^24 kg)) / (4π²) * (24²)
R³ ≈ (3.986 x 10^(14) m³/s²) / (4π²) * (24²)
R³ ≈ (3.986 x 10^(14) m³/s²) / (4π²) * (576)
R³ ≈ 421641.6 x 10^(14) m³/s²
Now, take the cube root to find R:
R ≈ ∛(421641.6 x 10^(14) m³/s²)
R ≈ ∛(421641.6) x 10^(4) m
R ≈ 421.634 x 10^4 m
Therefore, the orbital radius of the geosynchronous satellite is approximately 421,634 kilometers.