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
we = 7.29 x 10-5
rad s-1
Hence,
substituting the above constants into (9) gives: