SATELLITE MOTION.

Geo-stationary orbit.

 Traveling on the circle orbit of radius r the satellite is acted upon by the force of the earth's gravity gmM/r², where g is the gravity constant, m is the mass of the satellite and M is the mass of the planet (earth in our case). According to second Newton law this force is equal to centripetal force mv²/r. From these two formulas we find the equation for the velocity of the satellite motion in the circle orbit:

v=(g M/r)^1/2

The period of the satellite revolution around the earth T_sat is equal to the length of the orbit 2pr divided by the velocity of the satellite motion v:

T_sat=2pr/v=2p (r³/gM)^1/2

If this orbit period Tsat is equal to the period of the globe spinning (about 24 hours), then the satellite will hang over the same equatorial region of the earth and such an orbit is called the geo-stationary orbit. The sky coordinates of the geo-stationary satellite are the same all the time and we can precisely orient/beam  on it the parabolic antenna of our receiver.

Circular orbits. "Iridium"

If the radius of the satellite orbit is less than the radius of the geo-stationary orbit, then the satellite will outstrip the spinning of the globe and we can not direct the satellite dish on it. Nevertheless, satellites on the low orbits provide more a powerful signal as compared to the satellites on the geo-stationary orbit, and using several satellites on the same orbit, permanent telecommunication can be achieved.

This principle was used in the telecommunication satellite system "Iridium", which uses 66 Low Earth Orbit satellites: 11 satellites in 6 orbits, as shown in the animation. Every satellite covers the area on the globe shown by a light spot. We can see in this figure that the spots overlap, covering the entire surface of the globe. This means that this satellite system provides continuous telephone communication at every point of the globe.

Ref. in Internet: Iridium, Wiki, FAQ About Iridium

Kepler's laws. The motion of satellites in elliptical orbits

First law: Orbits of all the planets are ellipses with the sun at one of the focal points.

Second law: Any planet moves by such a way that radius-vector from the sun to the planet covers in equal time intervals the equal areas.

Third law: T²/a³=const, where T is the period of the planet revolution, a is the length of the bigger axis of the orbit.

 The animation here shows the motion of a satellite in a highly elliptical orbit. We can see from this animation that in accordance with Kepler’s first law, the earth is situated at one of the focuses of the orbit and the satellite moves quicker at perigee (the point of the orbit that is nearest to the earth), than in apogee (the point of the orbit that is the most distant from the earth). Such highly elliptical orbits are, in fact, actually used for satellite telecommunication. Unlike the geo-stationary orbit, the satellites on elliptical orbits can "see" the poles of the earth. At apogee, the satellite moves slowly and hangs there for several hours. During these several hours, the satellite provides stable telecommunication, and then its place is taken by another satellite. Thus, several satellites in elliptical orbit are used to provide permanent telecommunication. (See Molniya system).