More on Orbits

Awhile back, I wrote about my orbit simulator. This has been on my mind again, as I'm dealing with some issues at work that have to do with GPS receivers. GPS is useful to synchronize widely dispersed systems such as cellular base stations. It not only provides location, but also very accurate timing. In fact, it is the timing that allows you to triangulate your exact position.

My concern is in predicting which satellites are in view at any given time. There are some websites that track satellite orbits and predict passes over your location, but I'm looking for something more specific to what I want. How to go about this? It turns out that a lot of the same equations I used in my orbit simulator come into play for predicting locations of real world satellites.

The first good reference is the ICD-GPS-200C document which is pretty easy to find on the web. The equations starting in Table 20-IV are what we need.

Since the satellite orbits are perturbed by various influences, the GPS almanac is updated daily. The orbits are generally stable, but will have some small variance from day to day. Almanacs are available here.

The first equation I'll deal with is the mean motion, n.

n = sqrt(GMe / a³)

GMe is 3.986 x 10¹⁴ and relates to the gravitational constant and the mass of the earth. The semi-major axis a has to do with the size of the elliptical orbit. The almanac gives the square root of a in the line that starts with SQRT(A).

The mean motion relates to the inverse of the orbital period and gives the number of radians per second that a given satellite is moving around its orbit. This will be important for finding the position at a given time, because the almanac just gives a snapshot of where it was, or will be, at some reference time.

That is all for this post. The next step is finding the position in orbit at a particular time.