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Do geostationary satellites need to have the equator as the plane of rotation, and the earth's centre to be the centre of rotation? Can it rotate over, say, the Tropic of Cancer, focusing on a single city?

If not, in which case, we need a chain of geostationary satellites to do a function, why employ geostat satellites at all? Aren't there satellites at lower altitudes which could do a better job?

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Earth's center of mass must be at one of the two focus points of a satellite's elliptical orbit, or at the center of a circular orbit such as a geostationary orbit. One cannot orbit a certain latitude, except for the equator.

But there are clever alternatives for different purposes. Geosynchronous (as opposed to geostationary) orbits mean that the satellite will always be within sight from the same location on Earth, but it would move around and probably need antennas which continuously aim towards them. Japan is building its own regional GPS-system in this way.

There are also semi-synchronous orbits which put the satellite at the same place on the sky at the same time every day. This is used for some Earth observation purposes because the shadows on the ground remain the same for the same location on Earth everytime an image is taken.

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The GPS system in use by the United States are in medium Earth orbit (they orbit twice a sidereal day). –  LDC3 Aug 10 at 16:29
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@LDC3m Yes, the US uses about 24 satellites for global GPS coverage. Japan plans only 4 active satellites for regional coverage. –  LocalFluff Aug 10 at 17:17

Do geostationary satellites need to have the equator as the plane of rotation, and the earth's centre to be the centre of rotation?

To be stationary above a point, yes.

Can it rotate over, say, the Tropic of Cancer, focusing on a single city?

If the satellite's orbit touched the Tropic of Cancer, it would not be geostationary since the orbit about the center of the Earth would move the satellite north and south to reach the Tropic of Capricorn as well. Any geostationary satellite can be placed so that a designated city is always in line of sight, it doesn't need to be directly overhead.

If not, in which case, we need a chain of geostationary satellites to do a function, why employ geostat satellites at all?

Only one geostationary satellite is needed for some functions, such as TV broadcasts to the Eastern Seaboard.

Aren't there satellites at lower altitudes which could do a better job?

It depends on what the job is. For continuous broadcast and reception of signals, a geostationary satellite is best. For obtaining crop information, a low Earth orbit satellite might be better.

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"Any geostationary satellite can be placed so that a designated city is always in line of sight". So, I can do weather forecasting for Egypt or even Greenland using a single geostationary satellite? Consider that the az angle would be very tilted, and the distance would be too much. Way more than the ~35,000 km that it normally is. –  Raj Deshmukh Aug 10 at 15:56
    
If both locations can be seen at the same time, then only one satellite is needed. If not, then you will need 2 satellites. –  LDC3 Aug 10 at 15:59

You cannot point an antenna with a fixed orientation at a satellite that is not in a geostationary orbit. So to get satellite tv you'd not only need a bunch of satellites in orbit instead of one, but every receiver would need probably at least a couple of dishes on complex and highly accurate alt-az mounts.

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