More on Orbits, Part 4

With our polar coordinates (r, ν) in hand, there are a few more transformations.

The argument of latitude u is obtained by adding the argument of perigee ω to ν. Argument of perigee (or more generally, periapsis) is taken from the GPS almanac.

u = ν + ω

Next we convert to cartesian coordinates to get position in the orbital plane:

x' = r * cos(u)
y' = r * sin(u)

Now we need to account for rotation of the earth. The right ascension of the ascending node Ω₀ is given in the almanac, and it is valid at the beginning of the referenced GPS week, not at the time of applicability. So we add the rotation speed of the earth times the number of seconds since the start of the week.

Ω = Ω₀ - Ω_e(dot) * t

This will rotate our coordinates so we can get a longitude of the satellite relative to earth longitude. The quantity Ω_e(dot) is the rotation rate of the earth, 7.292115 x 10-5 radians per second.

With all these new quantities in hand, we'll use a rotation matrix to move the orbit into earth coordinates: latitude and longitude.