Understanding Doppler Shifts
Doppler shifting is caused by compression of the waves emitted from a source. For audio waves it is the physical objecting 'catching up' with waves emitted at a constant velocity (speed of sound) in front and 'leaving behind' the waves emitted backwards. Wikipedia has several articles to make this more understandable. For electromagnetic waves, it is actually time compression of the source, but it can be modeled as the same effect with the same equations.
We need to model the Doppler shifts of a radio signal in order to match real world data to an actual orbit (or position along a ground track of a known orbit).
Let's start simple. We have a road and there is an observer standing at some distance d from this road. A car is traveling at a constant velocity v. Let us assume that it is emitting engine noise at frequency f.
It is clear that this picture does not work for a satellite. A satellite is not traveling along a linear path, instead, a circle. Let's follow the same idea but close the track from a country road into a racetrack where the cars are traveling in a circle.
We can use the same variables, but now we can reference everyone against the center of the track instead of using the observer as the origin. The same car is traveling at the same constant velocity v on a track with a radius r while our observer is at distance rp from the center of the track, and an angle <math>\theta_p_0</math>.