How to find altitude AND orbital period?

<QUOTE>If I see a satellite as it rises over the horizon and watch it until it sets below the opposite horizon, what portion of the 360 degrees of its orbit have I seen?</QUOTE>

For low-altitude satellites, they usually are visible for anything from a few seconds (close to horizon) up to about 10 minutes or so (when passing overhead). Naturally, the exact interval of time depends on many things.

You should get a piece of paper and try it yourself. Draw two concentric circles, one being the surface and the other a circular orbit. Then pick a point on the surface and draw a line tangent to it. The places where the line cuts the orbit is the section of the orbit that will be seen from that point. How much of the orbit will be seen will depend on how "high" is the orbit.

If you want to measure its speed, and the satellite is emitting radio signals, and also if you know at which frequency it is emitting exactly, you can measure its velocity by listening to the frequency and measuring the Doppler shift. Kinda like this (fast forward to about minute 12 when the origin of GPS is discussed): http://www.ted.com/talks/steven_johnson_where_good...

Yes, there is a way. You need three sightings of the satellite, each with time, angle above horizon, and angle from north. Then you can combine them using Gauss' method.

However ... if the satellite is in low orbit (so that atmospheric drag is important), or if it changes its orbit during your observations, then Gauss's method will not work. Also, the method could be very sensitive to errors in the input data.

The whole method is complex, and involves concepts such as "Earth Centered Inertial" coordinates. It is best to use a commercial software package.