How do you calculate the distance above the surface of the earth to geosynchronous orbit?
Answers
Answer:
The height of a geostationary orbit is calculated as the distance required to have an orbital period of 24 hours.
Explanation:
The force of gravity acting on a satellite is given by the formula
F=GMmr2.
Where G=6.67384m3kg⋅s2 is the gravitational constant, M=5.972⋅1024kg is the mass of the Earth, m is the mass of the satellite and ris the distance from the centre if the Earth to the satellite.
The centripetal force required to keep the satellite in orbit is given by the formula
F=m⋅r⋅ω2.
Where m and r are as above and
ω=2π24⋅60⋅60
is the angular velocity of the satellite in radians per second. The value of ω given is the angular velocity required to complete a full orbit, 2π radians, in 24 hours.
When a satellite is in orbit the gravitational force must equal the centripetal force which gives the formula
GMmr2=mrω2
The m cancels out and the formula can be rewritten as
r3=GMω2.
The distance is from the centre of the Earth so we need to subtract the radius of the Earth R=6,371,000m.
So the height of geostationary orbit h is given by the formula:
h=(GMω2)13−R
If the stated values of G, M, ω and R are put into the formula it gives a value of about 35,870,000 m.