Asri-unix.737
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utcsrgv!utzoo!decvax!ucbvax!ARPAVAX:C70:sri-unix!David.Smith@CMU-10A
Tue Feb 9 19:31:04 1982
Orbital mechanix, please!
In fact, higher orbits are slower any way you want to measure. But
due to the gravity well, they are still at a higher energy level.
Consider:
f = GMm/r^2
where f is the force exerted between two bodies, G is the gravitational
constant, M is the mass of the primary, m the mass of the satellite,
and r the radius between centers. Assume M >> m.
For a circular orbit, we use
f = ma (force, mass, acceleration)
a = rw^2 (acceleration, radius, angular velocity)
w = v/r (angular velocity, tangential velocity, radius)
Equating gravitational force on the satellite with the force required to
keep it in circular orbit,
GMm/r^2 = mrw^2 = mv^2/r
which produces
w = sqrt( GM/r^3 )
v = sqrt( GM/r )
which clearly shows angular and tangential velocity dropping as radius
rises.
Low earth satellites travel at nearly 18,000 mph; geosynchronous
satellites travel at around 6,000 mph; the moon travels at around
2,000 mph with respect to earth's center.
Consider the task of moving a satellite from LEO to GEO. (I'll
pull a few numbers out of my hat because I'm lazy, but the exact
numbers aren't the point.) Starting at 18,000 mph at 150 miles
up, you burn the rockets to accelerate it to 22000 mph. With
more than circular velocity, the satellite climbs, trading speed
for altitude, until it reaches apogee at geosynchronous height
(around 23,000 miles) with 2000 mph. Since circular velocity
there is 6000 mph, the satellite will drop back. It falls until
it reaches 22000 mph at its next perigee, 150 miles up. At next
apogee (23,000 miles, 2,000 mph), you burn the engines again to
raise the speed to 6,000 mph. This raises the perigee to put the
satellite into synchronous orbit. The satellite has gone from
a 18,000 mph circular orbit to a 6,000 mph one purely by firing
its rockets to increase speed.
- David Smith
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