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 v_s, 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 v_s?

 

The tangential velocity of the satellite (v_s) 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 w_s and w_e 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

          m_e = 5.97 x 10²⁴ kg

          w_e = 7.29 x 10^-5 rad s^-1

 

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

 

R = 35,786 km for GOES

 

There it is……