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How to perform the conversion of (r,v) vector from ECEF to ECI (J2000) coordinate frame?

What I tried

I multiplied the matrix ([cosA -sinA 0] [sinA cosA 0] [0 0 1]) by the state vector, where A is Greenwitch mean sidereal time.

This equation shows that the Z axis doesn't change. However, I have the ephemeris of a real satellite in both coordinate frames, and there Z axis differs! Where is the issue?

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  • $\begingroup$ If I understand correctly, your question is equivalent to: in the J2000 frame, is the line from the Earth's center to the Earth's North Pole coincident with the z axis. The answer, as @DavidHammen notes, is no, though it's pretty close since we're only 18 years out from 2000. If you just need it for a fixed time, you can cheat and use HORIZONS. If you want the formula for the North Pole's J2000 direction, that's a bit trickier. $\endgroup$
    – user21
    Commented Mar 16, 2018 at 16:37

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A reference frame that has its Z axis fixed with respect to the Earth's rotation axis is not inertial. The Earth's rotation axis undergoes precession and nutation, and on a shorter term, undergoes polar motion as well.

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  • $\begingroup$ What is the method then? $\endgroup$
    – Tarlan Mammadzada
    Commented Mar 16, 2018 at 4:07
  • $\begingroup$ @TarlanMammadzada -- You need to get ephemeris data to find the change in the axis of the earths rotation since the epoch of the ECI frame, and apply that change to the result of the calculation you describe in the question. $\endgroup$
    – antlersoft
    Commented Mar 16, 2018 at 15:53

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