# Rotation matrix from J2000 to ITRF2008

I want to rotate a vector from the inertial J2000 frame into ITRF2008. I am using the NASA SPICE library, which provides a rotation matrix from J2000 to ITRF93 (they plan to upgrade the library to a modern ITRF in the near future but so far only ITRF93 is supported).

So my task is to then rotate from ITRF93 to ITRF2008. Given my vector $$V_{J2000}$$, I can easily compute $$V_{ITRF93} = R_{ITRF93}^{J2000} V_{J2000}$$ where the rotation matrix $$R_{ITRF93}^{J2000}$$ is given by the NASA SPICE library. Now, according to http://itrf.ign.fr/doc_ITRF/Transfo-ITRF2008_ITRFs.txt, the transformation from ITRF93 to ITRF2008 is given as follows: $$V_{ITRF93} = V_{ITRF2008} + T + M V_{ITRF2008}$$ where the vector $$T$$ and matrix $$M$$ are given in the link above. Rearranging this gives: $$V_{ITRF2008} = (I + M)^{-1} (V_{ITRF93} - T)$$ so this amounts to a translation by $$T$$ followed by a matrix multiplication. However, one issue is that the matrix $$I+M$$ is not orthogonal (according to the parameters in the link above). So this is not really a rotation, but a transformation followed by a scaling.

It seems to me there should be an orthogonal rotation matrix allowing me to go from ITRF93 to ITRF2008: $$V_{ITRF2008} = R_{ITRF2008}^{ITRF93} V_{ITRF93}$$ Does anyone know how to compute this rotation matrix?

My ultimate goal is to compute the spherical coordinate components of $$V_{ITRF2008}$$ (i.e. geocentric latitude, longitude and radius).