Compute rotation matrix / quaternion ITRF--> ICRF at a given time or in real time

updated : June 2023

Compute time series of the Earth orientation matrix / quaternion 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 / quaternion
output for one line : Modified Julian Date in TAI ; civil date in UTC ; matrix elements a11, a12, a13, a21, a22, a23, a31, a32, a33 ; quaternion elements q1, q2, q3, q4

Select :
        Matrix from ITRF to ICRF
        Matrix from Celestial Intermediate Frame (CIF) to ICRF (precession-nutation)
        Matrix from Terrestrial Intermediate Frame (TIF) to CIF (Earth angle rotation matrix)
        Matrix from ITRF to Terrestrial Intermediate Frame (TIF)
        Matrix from ITRF to CIF (diurnal rotation + polar motion)

Projection of the geographic axis on the equatorial plane J2000.0 OXY (unit : arcsecond)


N.B.: by taking 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 Reference Frame (ITRF) 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 10-5 rad/s)   without precession-nutation

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

Figure axes for the real Earth and a rigid Earth (Kinoshita Souchay model) projected on the equatorial plane J2000.0 OXY (unit : arcsecond)

Transformation from the international terrestrial reference system (ITRF) to the international celestial reference frame (ICRF): transformation coordinate and associated quaternion. Compution of Celestial Pole coordinates (X,Y), Earth rotation angle, Celestial Intermediate Origin locator, Terrestrial Intermediate Origin locator are performed by FORTRAN programs of the SOFA IAU library, consistent with the IAU 2000/2006 resolutions.

  • The transformation is derived through rotation quaternions (Bizouard C. and Cheng Y.T., 2022, to be published). The matrix is calculated from the quaternion expressing the rotation from ITRF to ICRF

  • from 1962 to the current week the transformation can include the whole EOP set of the IERS combined series C04 (daily step, 0hUTC), as well as diurnal and semi-diurnal effects produced by ocean tides, which amount about 0.5 mas (milliarcsecond). Accuracy is 0.05 mas corresponding to 5 10-10 rad. Bewteen two dates at 0hUTC the EOP are interpolated according to a cubic spline.

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

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

  • The last digit (14) of the matrix/quaternion elements is equivalent to an angular uncertainty of 2 10-6mas. or a time uncertainty of 0.14 nano seconds (in reason of the diurnal rotation).

  • Components of the instantaneous rotation vector are deduced from the quaternion 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), Paul-Eric Pottie (SYRTE, France, November 2021) for their worthful comments, which allow us to improve/correct the computation of the Earth rotation matrix.