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:

or                                                                 (1)

Solving for vs, the tangential velocity of the satellite, from (1) yields:

(2)

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:

(4)

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:

(5)

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)

We know:

G = 6.67 x 10-11 Nm2kg-2

me = 5.97 x 1024 kg

we = 7.29 x 10-5 rad s-1

Hence, substituting the above constants into (9) gives: