EARTH MOTION IN SPACE AND REFERENCE SYSTEM TRANSFORMATIONS |
Before defining the CCRS and CTRS, the earth's motion in space needs to
be briefly reviewed (further details can be found in, for example, SEEBER, 1993). Due to the torques
exerted primarily by the gravitational attraction of the moon and sun (and
secondarily by other planets), the equator and the ecliptic of the Celestial Sphere precess and nutate. Precession,
with a period of approximately 26000 years, consists of two components:
luni-solar and planetary precession. The luni-solar effect causes a slow
westerly circular motion of the pole of the equator relative to the pole
of the ecliptic. The attraction of the planets causes an eastward motion
of the equinox by about 12" (arcseconds) per century and a decrease
of the obliquity by about 47" per century. Precession causes the equinox
to move along with the equator by about 50" per century. Nutation
is a short-period, irregular motion of the pole with a period ranging from
14 days to 18.6 years, and has a maximum amplitude of about 20". Both
are described by the motion of the equator and equinox with respect to the
fixed equator and equinox of a given epoch, as shown in the Figure below.
In order to relate the CCRS to the CTRS, the parameters of polar motion
and earth rotation are also needed. The motion of the true celestial pole
with respect to the pole of a conventionally selected earth-fixed (terrestrial)
reference frame is known as Polar Motion. The movement
of the true pole is described by means of two parameters, xp
and yp. Another parameter is the angle between the true equinox-of-date
and the Greenwich meridian, known as the Greenwich Apparent Sidereal
Time, denoted by 
g. The transformation from a space-fixed reference frame
to an earth-fixed reference frame is accounted for by the parameter g.
The terrestrial reference system transformed from a celestial system at
true-of-date is known as the Instantaneous Terrestrial
Reference System (ITRS). Therefore, in order to link the true-of-date
celestial system and the CTRS, the earth orientation between these two system
coordinate axes is defined by the values of xp, yp,
and g, also known collectively as the Earth
Orientation Parameters.
Figure 1. Precession and Nutation. (Adapted from TORGE,1980)
The CCRS is a right-handed coordinate system with its origin at the centre of mass of the earth. It is defined by the mean equator and equinox of J2000.0. (The standard epoch J2000.0 is 12hr on January 1, 2000.) The mean equator and equinox is the fictitious equator and equinox, derived after removing the effects of nutation. In other words, applying nutation to the mean equator and equinox of a given date yields the instantaneous or true equator and equinox of that date. If the satellite ephemeris (or station positions) are given in terms of the coordinate system which is defined by the true (or mean) equator and equinox of some reference date (not J2000.0), the reference system is known as the True-of-Date Reference System (or Mean-of-Date Reference System), or simply as the TDRS (or MDRS). It should be noted that the MDRS differs from the CCRS by the variations in time of the directions of the earth's spin axis. This variation is described by the effect of precession.
The CTRS is geocentric, right-handed, and orthogonal; the z-axis of this system is aligned with the mean pole of the earth; the x-axis is in the equatorial plane pointing towards the Greenwich meridian. For historical reasons the mean position of the true celestial pole during the period between 1900 and 1905 has been adopted as the mean pole of the earth, designated as the Conventional International Origin (CIO) or Conventional Terrestrial Pole (CTP).
The transformation between the CCRS and CTRS is therefore accounted for by the four transformation components: precession, nutation, earth rotation, and polar motion.
(1) Transformation from the CCRS to MDRS (in terms of Precession) is defined by:
| r' ( to ) = Pr r | (1.2-6) |
where
Pr = R3
( -ZA ) . R2 (
A ) . R3 ( - 
A ) |
(1.2-7) |
and
| r | is the position vector ( x , y , z ), expressed as CCRS coordinates |
| r'( to ) | is the position vector ( x' , y' , z' ), expressed as MDRS coordinates with respect to reference epoch time to, |
R1( ),R2(),R3() |
are the rotation matrice for anticlockwise rotations through an angle about
the x-,y-, and z-axis respectively, and |
| ZA,A,A |
are the equatorial precession angles (SEEBER, 1993) |
(2) Transformation from the MDRS to TDRS (in terms of Nutation) is defined by:
| r'' = N r' | (1.2-8) |
where
N=R1(- - ).R3(- ).R1() |
(1.2-9) |
and
| r'' | is the position vector (x'', y'', z''), expressed as TDRS coordinates, |
|
is the obliquity of the ecliptic, the angle between the equator and the ecliptic, |
|
is the nutation in obliquity, and |
|
is the nutation in longitude. |
(3) Transformation from the TDRS to ITRS (in terms of Earth Rotation) is given by:
| r''' = E r'' | (1.2-10) |
where
| E = R3(g) |
(1.2-11) |
and r''' is the position vector (x''' y''', z'''), expressed as ITRS coordinates.
(4) Transformation from the ITRS to CTRS (in terms of Polar Motion) is given by:
| re = Pm r''' | (1.2-12) |
where
| Pm = R2(-xp).R1(-yp) | (1.2-13) |
and re is the position vector (xe, ye, ze), expressed as CCRS coordinates.
The complete transformation from the CCRS to the CTRS therefore is:
| re = Pm E.N.Pr.r | (1.2-14) |
The reverse transformation from the CTRS to the CCRS is given by:
| r = (Pm E.N.Pr)T re | (1.2-15) |
The various component parameters required to evaluate the Polar Motion Pm, Earth Rotation E, Nutation N, and Precession Pr matrices are described in detail in SEEBER (1993). The transformation process between the various coordinate systems is summarised in the Figure below.
Figure 2. Coordinate system transformations.
Note, the ITRS is still not very useful for the most precise applications
because stations on the surface of the earth move even with respect to this
"earth-fixed" reference system due to geodynamic phenomena, the
most important of which is plate tectonic motion. The ITRS
as defined at some instant in time is the basis of the International
Terrestrial Reference Frame (ITRF
-- section 2.1.4).
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© Chris Rizos, SNAP-UNSW, 1999