A satellite with an orbital period of exactly 24.0 h is always positioned over the same spot on Earth. This is known as a geosynchronous orbit. Television, communication, and weather satellites use geosynchronous orbits. At what distance would a satellite have to orbit Earth in order to have a geosynchronous orbit?
set the force of gravity equal to centripetal force.
GMm/r^2=mw^2*r
where w=2pi/period (set period to seconds in a day)
solve for r
The time it takes a satellite to orbit the earth, its orbital period, can be calculated from
T = 2(Pi)sqrt[a^3/µ]
where T is the orbital period in seconds, Pi = 3.1416, a = the semi-major axis of an elliptical orbit = (rp+ra)/2 where rp = the perigee (closest) radius and ra = the apogee (farthest) radius from the center of the earth, µ = the earth's gravitational constant = 1.407974x10^16 ft.^3/sec.^2. In the case of a circular orbit, a = r, the radius of the orbit.
The geostationary orbit is one where a spacecraft or satellite appears to hover over a fixed point on the Earth's surface. There is only one geostationary orbit in contrast to there being many geosynchronous orbits. What is the difference you ask? A geosycnchronous orbit is one with a period equal to the earth's rotational period, which, contrary to popular belief, is 23hr-56min-4.09sec., not 24 hours. Thus, the required altltude providing this period is ~22,238.64 miles, or ~35,787.875 kilometers.
To determine the distance at which a satellite would have to orbit Earth in order to achieve a geosynchronous orbit, we can use Kepler's third law of planetary motion.
Kepler's third law states that the square of the period of revolution (T) of a satellite is proportional to the cube of its average distance from the center of the planet (r).
Mathematically, this can be represented as:
T^2 = k * r^3
Where T is the period of revolution in seconds, r is the distance from the center of Earth to the satellite in meters, and k is a constant.
In this case, the satellite has an orbital period of 24.0 hours, which is equivalent to 24.0 * 60 * 60 = 86,400 seconds.
To find the distance (r), we need to solve for r in the equation above.
Plugging in the values:
86,400^2 = k * r^3
We can now solve for r by rearranging the equation:
r^3 = (86,400^2) / k
Taking the cube root of both sides:
r = (86,400^2)^(1/3) / k^(1/3)
The constant k depends on the gravitational constant (G), the mass of Earth (M), and the period of Earth's rotation (T).
By substituting the values:
k = (G * M * T^2) / (4 * π^2)
The gravitational constant (G) is approximately 6.67430 × 10^(-11) m^3 kg^(-1) s^(-2), the mass of Earth (M) is approximately 5.972 × 10^(24) kg, and the period of Earth's rotation (T) is 24 hours.
Calculating the constant (k):
k = (6.67430 × 10^(-11) * 5.972 × 10^(24) * (24 * 60 * 60)^2) / (4 * π^2)
Substituting this value into the equation for r:
r = (86,400^2)^(1/3) / ((6.67430 × 10^(-11) * 5.972 × 10^(24) * (24 * 60 * 60)^2 / (4 * π^2))^(1/3)
Evaluating this expression will give us the distance at which the satellite needs to orbit Earth in order to achieve a geosynchronous orbit.
To determine the distance at which a satellite would have a geosynchronous orbit, we need to understand the concept of geosynchronous orbit and how it relates to the Earth's rotation.
A geosynchronous orbit refers to a specific orbital period of a satellite, which is exactly 24 hours. This means the satellite takes 24 hours to complete one orbit around the Earth, allowing it to remain stationary relative to a fixed point on Earth's surface.
Since the Earth completes one full rotation on its axis in approximately 24 hours, a geosynchronous satellite's orbital period matches the Earth's rotation period. As a result, the satellite appears to stay fixed in the sky while moving at the same speed and direction as the Earth's rotation.
To calculate the distance at which a satellite would have a geosynchronous orbit, we can use the formula for orbital period. The orbital period (T) of a satellite is related to the distance from the center of the Earth (r) by the following formula:
T = 2π√(r³/GM)
Where:
T = Orbital period (in seconds)
π = Pi, approximately 3.14159
r = Distance between the satellite and the center of the Earth (in meters)
G = Gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2)
M = Mass of the Earth (approximately 5.972 × 10^24 kg)
In the case of a geosynchronous orbit, the orbital period (T) is 24 hours, which can be converted to seconds as follows:
T = 24 hours × 60 minutes/hour × 60 seconds/minute = 86,400 seconds
Now, we can rearrange the formula to solve for the distance (r):
r = (GMT²/4π²)^(1/3)
Substituting the values:
r = ((6.67430 × 10^-11 m^3 kg^-1 s^-2) × (5.972 × 10^24 kg) × (86,400 seconds)² / (4π²))^(1/3)
Calculating this equation will give us the distance at which a satellite would have a geosynchronous orbit.
r ≈ 42,164 kilometers
Therefore, a satellite would need to orbit at a distance of approximately 42,164 kilometers from the center of the Earth to have a geosynchronous orbit.