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,

_{} 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

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 *T ^{2}*

_{} or solving for *r*
yields: _{} (9)

We
know:

*G* = 6.67 x 10^{-11} Nm^{2}kg^{-2}

*w** _{e}* = 7.29 x 10

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