Compute orientation matrix ITRF--> ICRF at a given time or in real time

updated : January 2008

N.B: bug in the program related to leap second has been corrected, however some suplementary check have to be done before to confirm the validity of the computation of the rotation matrix from ITRF to ICRF. Thank you for any coments.

Compute time series of the Earth orientation matrix or derived parameters between two dates:

Time scale for civil gregorian date : UTC         Step in seconds

First date: year   month   day     hour   min   second
 Last date: year   month   day     hour   min   second

Include combined EOP C04 (1962 - today + 6 months prediction) when available :
celestial pole offsets   UT1 - UTC   Polar motion

Include diurnal and semidiurnal variations produced by ocean tides (IERS conventions 2000)


Orientation matrix
output for one line : Modified Julian Date in TAI ; civil date in UTC ; matrix elements a11, a12, a13, a21, a22, a23, a31, a32, a33
Projection of the geographic axis on the equatorial plane J2000.0 OXY (unit : arcsecond)


N.B.: by taking UT1-UTC and the polar motion to zero one obtains the equatorial celestial coordinates of the Celestial Intermediate Pole

Components (ω123) of the instantaneous rotation vector ω:

in the international terrestrial frame Oxyz
in the local frame Ox'y'z' : Oz'=vertical (either geographic or astronomical), Ox' tangent to the meridian, towards South, Oy' in the horizontal plane, directly orthogonal to Ox':
Latitude (degree)   longitude (degree)
in the International Celestial Reference Frame (ICRF) OXYZ
      in nanorad/s   1/Ω,ω2/Ω) in arcsec (Ω=7.292115 e-5 rad/s)

  do not draw
  draw (date,ω1)
  draw (date,ω2)
  draw (date,ω3)
  draw (ω12)  

Precession-nutation matrix

Transformation coordinate M from the international terrestrial reference system (ITRF) to the international celestial reference frame (ICRF) : Celestial coordinates ( X Y Z) = M x Terrestrial coordinates ( x y z ) - We use the FORTRAN CODE of the VLBI analysis Software GLORIA developped by Anne-Marie Gontier (Paris Observatory), and compatible with the IAU 2000 resolutions

  • from 1962 to the current week the matrix can include the whole EOP set of the IERS combined series C04 (daily step, 0hUTC, time resolution for such EOP fluctuations is about 6 days), as well as diurnal and semi-diurnal effects produced by ocean tides, which amount about 0.001". Accuracy is 0.0001", corresponding to 5 10-10 rad. In turn the time resolution of the matrix can reach approximatively 6 hours. Bewteen two dates at 0hUTC the EOP are interpolated according to a cubic spline.

  • from the current week to 6 months after the matrix is computed from the prediction of the polar motion and UT1-UTC.

  • outside the range [1962, current week+ 6 months prediction] the rotation matrix is restricted to the diurnal rotation and the precession-nutation model IAU 2000.

  • The last digit (12) of the matrix element corresponds to the pico radian level. Corresponding time resolution for the matrix variation is 14 nano seconds.

  • Components of the instantaneous rotation vector are deduced from the orientation matrix and its time derivative.

    We are grateful to J.C. Marty (CNES, France), Markus Nitshke, Dr. Feipeng Zhang (GFZ, Germany), Luca Cerri (CNES, France), Dr. Jim Beaupre (Teledyne Solutions, USA), Hervé Manche (IMCCE, Observatoire de Paris), Josef Slavicek (Prague, January 2008) for their worthful comments, which allow us to improve/correct the computation of the Earth rotation matrix.