THEORY FOR GOES ORBIT
Q: Why are GOES 35,800 km from the center of the earth so that they are in geosynchronous orbit???
There are two relevant forces involved in this problem:
1. gravitational force of attraction between any two objects, given by:
2. centrifugal force an outward-directed force that normally balances the inward-directed centripital force, given by: . These forces are required to help maintain the circular trajectory of an object.
In our situation of a satellite in geosynchronous orbit, the outward-directed centrifugal force balances the inward-directed gravitational force. Hence, for a steady-state orbit, the force balance becomes:
Solving for vs, the tangential velocity of the satellite, from (1) yields:
Notice, that in (2), the mass of the satellite does not appear.
REALITY CHECK – what are we trying to solve for?????
OK, so what is vs?
The tangential velocity of the satellite (vs) is related to its orbital period, T through:
or or (3)
Eliminating between (3) and (2) gives:
Solving for the orbital period, T, gives:
OK, we still do not know r………but we’re getting closer. To find r, we still need to determine what T is…..
What is the constraint, in terms of angular velocity, on the satellite if it is to be in a geosynchronous orbit??????
Yes, where ws and we are the angular velocities of the satellite and earth, respectively.
The angular velocity (from basic physics) for the satellite is:
but from (3), recall that or (6)
Substituting (6) into (5) gives:
or solving for T, or (7)
recall that so (7) can be rewritten as: (8)
From (8), we now know the satellites orbital period, T.
By substituting (8) into (4) to eliminate T2 we get:
or solving for r yields: (9)
G = 6.67 x 10-11 Nm2kg-2
we = 7.29 x 10-5 rad s-1
Hence, substituting the above constants into (9) gives: