Where do I find ECEF coordinates of a satellite?

Question

I am taking a university course named 'satellite navigation' as a part of my master degree in aerospace engineering. For the exam we are required to elaborate the coordinates of a LEO satellite (it really does not depend which one). For this reason, I need the coordinates (the state vector, at least the position) in the Earth Centered - Earth Fixed ECEF frame. However, I cannot find any useful resource/archive.

I would really appreciate some help :)

Answer 1

Where do I find ECEF coordinates of a satellite?

You don't "find it." That information is not published. You'll have to compute it rather than find it.

What is published for many satellites orbiting the Earth is what are called Two Line Elements (TLEs, for short). There are many questions on this site that ask about TLEs. These questions are so common that there is a tag for them: two-line-elements.

A Two Line Element set is designed for use with the Simplified General Perturbations #4 (SGP4) algorithm. The output of this algorithm includes position and velocity in True Equator / Mean Equinox (TEME) of epoch coordinates. One of the inputs to this algorithm is the epoch time.

Note that

• TLEs are low fidelity; the target is single precision floating point accuracy,
• TLEs are not intended for use beyond a week or two of the TLE's epoch time, and
• The outputs of the SGP4 algorithm are position and velocity in TEME of epoch (or perhaps TEME of date; there's a subtle difference between the two).

This means a fairly simple series of calculations can be used to convert the SGP4 output to ECEF:

1. Optional: Calculate UT1 at the TLE epoch time and at the observation time.
   This is optional because (a) It's not clear whether the TLE epoch time is in UT1 or UTC, and (b) TLEs are not intended for sub-second usage.
2. Calculate the TEME of epoch coordinates of the vehicle using the TLE for the vehicle.
3. Calculate GMST at the TLE epoch time. This will directly yield the transformation matrix between TEME (of epoch) and ECEF.
4. Calculate the time difference in seconds (UT1 or UTC) between the TLE epoch time and observation time. Multiply by $2\pi/86400$. (Alternatively, you may want to use 86400 plus the excess length of day for the epoch time instead of 86400, but once again, SGP4 is not intended for sub-second usage.) This will directly yield the transformation matrix between TEME (of epoch) of epoch and TEME (of observation).
5. Apply the transformations from steps #2 and #3 to the results of step #4.