# Transformation from Earth-centered inertial (ECI) to Geocentric (GEOC) satellite coordinates

I'm trying to follow the definition for converting Earth-centred inertial (ECI) coordinates (as output by SGP4) to Geocentric (GEOC) for later computation from this reference: Coordinate Systems for Space and Geophysical Application. However I can't find the alphaG value used in this equation:

$$\bar{X}=(\cos \alpha_G, \sin \alpha_G, 0)$$ $$\bar{Y}=\bar{Z} \times \bar{X}$$

I assume it's something based on the angle between the vernal equinox and Greenwich meridian but I'm not sure if I'm correct or how I would find this value. Any guidance/reference/advice welcome!

• The vernal equinox here is mean equinox, in a special way. The $\alpha_G$ is found from current time. – Deer Hunter Dec 4 '14 at 22:24
• @DeerHunter I've continued to research and I've found this definition: spenvis.oma.be/help/background/coortran/… Is this 'theta' the right value? – ThePlanMan Dec 4 '14 at 22:36
• The right idea, but the exact relationship with time is not the same. I ported SGP/SDP to C# maybe 5 years ago, so I don't remember the exact formula now, but you are on the right track. – Deer Hunter Dec 4 '14 at 22:42
• @DeerHunter thanks! I'll keep at it then! :D – ThePlanMan Dec 4 '14 at 22:50

The document you cited is fundamentally flawed. It has the z axes in the ECI and GEOC frames co-aligned. That's just wrong. It ignores precession and nutation.

The full theory is amazingly complex. You probably don't need that. (You would have graduate advisors who would have already pointed you to the necessary software if you did need the full theory.) You do however need to account for precession and nutation to some extent if you want even single precision accuracy.

For the full theory, you should read chapter 5 of IERS Technical Node 36, IERS Conventions (2010), which describes the transformation from the International Terrestrial Reference System (ITRS; what you call"ECI") and the Geocentric Celestial Reference System (GCRS; what you call "GEOC").

Software to compute the transformation matrix from ITRS to GCRS already exists. For that, I refer you to the International Astronomical Union's Standards Of Fundamental Astronomy (SOFA). Look at the left side of that page. There are cookbooks that instruct you on exactly how to use the relevant pieces of the SOFA software.

• SPICE is even easier to use, IMHO. – Deer Hunter Dec 5 '14 at 13:24
• Is the entire PDF wrong, or is it still useful for learning? Is there other books which have similar content but is correct. I'm a newbie to this area of study. – CMCDragonkai Aug 8 '16 at 14:38

Ignoring the effects of nutation and precession the conversion factor/angles can be determined ideally by projecting the angle between vernal equinox vector and the vector pointing the sun (Which is characteristic depending on time of the year and can also be determined by Mean difference of sidereal and solar time in angle multipled over number of days since spring equinox) over equitorial plane and adding an angular component of ( W(e)*T(u)).

W(e) being the angular velocity and T(u) being the universal time or time at greenwich.

However, Effects of Prescession and Nutation can also be introduced using Diurnal signature of Earth which is used by tracking stations to accurately transform the local coordinates to inertial. Here is an answer which gives information on Diurnal signature of the earth. I suppose that can be used to model Precession and Nutation effects in the conversion. Hope this Helps.