Circular motion and gravity
Working anything out about orbits involves two basic ideas: circular
motion and gravity. For something to go in a circle it needs a centripetal
force. This could be the tension in the string tied to a bucket of water
as you swing it around your head, or it could be the friction on the
tyres of a car as it corners.
For a satellite the centripetal force is provided by the Earth's gravitational
attraction. So the starting point for any calculation is:
R = total distance between the Earth and satellite
G = gravitational constant
M = mass of the Earth
m = mass of satellite
v = speed of satellite
Another useful starting point is to look at the acceleration due to
gravity (or the force per unit mass - otherwise known as gravitational
field strength) where you are. Even when they are in orbit, the satellites
are accelerated by gravity:
where g' is the gravitational field strength at that orbital
height - which is much less as you move further away.
For example, for a geostationary satellite:
The acceleration due to gravity decreases as the satellite's orbit
gets bigger. This is why they can afford to go more slowly without risking
being pulled back down to Earth.
Wherever a satellite is orbiting, the centre of its orbit has to be
the centre of the Earth; this defines the direction of the force exerted
by the Earth.
Lose weight - go to the equator
You weigh less when standing on the equator than you do when standing
on the poles. Firstly the radius of the Earth north-south is less than
east-west (the Earth is an oblate spheroid), so you are nearer to the
centre of the earth. Secondly you are slightly 'thrown out' from the
Earth by its spin. In your frame of reference you experience a centrifugal
force which is away from the centre of your circular motion. This is
the same as the feeling of being thrown outwards on spinning fairground
Astronauts on the orbiting space station are weightless. This is not
because there is no pull of gravity on them. It is because the floor
of the space station does not push back upon them - they experience
no reaction force. They are in free fall as the space station is in
free fall. In fact they train for weightlessness by going up in a large
training jet which then free falls. The astronauts and the jet fall
at the same rate (accelerating at 9.8m/s2 as they are near to the Earth).
The astronauts never catch up with the jet so it never exerts a reaction
force on them - this is how they feel weightless.