Considering only the homogeneous part (M
=
U) of Eqs. (3.2) and (3.3) leads to free oscillations. The inhomogeneous solution gives the forced oscil-lation. In both cases, the oscillations can be related to the inertial or to the terrestrial system. A further criterion for the solution concerns the inertia tensor. For a rigid earth and neglecting internal mass shifts, this tensor is constant; this is not the case for a deformable earth.3.2 Coordinate systems
3.2.1 Definitions
Oscillations of axes. The oscillation of ~ with respect to the inertial space is called nutation. For the sake of convenience the effect is partitioned into the secular precession and the periodic nutation. The oscillation with respect to the terrestrial system is named polar motion. A simplified representation of polar motion is given by Fig. 3.2. The image of a mean position of ~ is denoted P in this polar plot of the unit sphere. The free oscillation results in a motion of the rotational axis along a circular cone, with its mean position as axis, and an aperture angle of about 0!'4 ~ 12 m. On the unit sphere, this motion is represented by a 6 m radius circle around P. The image of an instantaneous position of the free oscillating earth's rotational axis is denoted Ro. The period of the free motion amounts to about 430 days and is known as the Chandler period. The forced motion can also be described by a cone. In Fig. 3.2 this cone is mapped by the circle around the free position
Ro.
The radius of this circle is related to the tidal deformation and is approximately 0.5 m. The nearly diurnal period of the forced motion corresponds to the tesseral part of the tidal potential of second degree since the zonal and sectorial components have no influence.p
one day
Fig. 3.2. Polar motion of earth's rotational axis
26 3. Reference systems The respective motions of the angular momentum axis, which is within Of'OOl of the rotation axis, are very similar. The free motion of the angular mo-mentum axis deserves special attention because the forced motion can be removed by modeling the tidal attractions. The free polar motion is long-periodic and the free position in space is fixed since for M = !l the integration of Eq. (3.2) yields N = const. By the way, this result implies the law of conservation of angular momentum as long as no external forces are applied.
Because of the above mentioned properties, the angular momentum axis is appropriate to serve as a reference axis and the scientific community has named its free position in space Celestial Ephemeris Pole (CEP). A can-didate for serving as reference axis in the terrestrial system is the mean position of the rotational axis denoted by P, cf. Fig. 3.2. This position is called Conventional International Origin (CIO). For historical reasons, the CIO represents the mean position of ~ during the period 1900 until 1905.
Conventional Inertial System. By convention, the X~-axis is identical to the position of the angular momentum axis at a standard epoch denoted by J2000.0, cf. Sect. 3.3. The X~-axis points to the associated vernal equinox.
At present this equinox is realized kinematically by a set of fundamental stars, cf. Fricke et al. (1988). Since this system is defined conventionally and the practical realization does not necessarily coincide with the theoret-ical system, it is called Conventional Inertial Frame. Sometimes the term
"quasi-inertial" is used to point out that a geocentric system is not rigor-ously inertial because of the accelerated motion of the earth around the sun.
Conventional Terrestrial System. Again by convention, the X 3-axis is identi-cal to the mean position of the earth's rotational axis as defined by the CIO.
The X I-axis is associated with the mean Greenwich meridian. The realiza-tion of this system is named Convenrealiza-tional Terrestrial Frame and is defined by a set of terrestrial control stations serving as reference points, cf. for example Boucher and Altamimi (1989). Most of the reference stations are equipped with Satellite Laser Ranging (SLR) or Very Long Baseline Interferometry (VLBI) facilities.
Since 1987, GPS has used the World Geodetic System WGS-84 as a ref-erence, cf. Decker (1986). Associated with WGS-84 is a geocentric equipo-tential ellipsoid of revolution which is defined by the four parameters listed in Table 3.1. However, using the theory of the equipotential ellipsoid, nu-merical values for other parameters such as the geometric flattening (f = 1/298.2572221) or the semiminor axis (b = 6356752.314m) can be derived.
Note that the parameter values have been adopted from the Geodetic Ref-erence System 1980 (GRS-80) ellipsoid.
3.2 Coordinate systems
Table 3.1. Parameters of the WGS-84 ellipsoid Parameter and Value
a = 6378137 m J2 = 1082630 . 10-9
WE
=
7292115 . 10-11 rad . S-1Jt = 3986005 . 108m3 • s-2
x
z
Explanation
Semimajor axis of ellipsoid Zonal coefficient of second degree Angular velocity of the earth Earth's gravitational constant
R
-+---+-- Y
Fig. 3.3. Cartesian and ellipsoidal coordinates
27
A vector X in the terrestrial system can be represented by Cartesian coordinates X, Y, Z as well as by ellipsoidal coordinates <p, A, h, d. Fig. 3.3.
The rectangular coordinates are often called Earth-Centered-Earth-Fixed (ECEF) coordinates. The relation between the two sets of coordinates is given by, d. for example Heiskanen and Moritz (1967), p. 182,
(N
+
h) cos <p cos A(3.6) (N
+
h)cos<p sin A( !: N + h)
sin <pwhere <p, A, h are the ellipsoidal latitude, longitude, height, N is the radius of curvature in prime vertical, and a, b are the semimajor and semiminor axis of ellipsoid. More details on the transformation of Cartesian and ellipsoidal (i.e., geodetic) coordinates are provided in Sect. 10.2.1.
28 3. Reference systems 3.2.2 Transformations
General remarks. The transformation between the Conventional Inertial System (CIS) and the Conventional Terrestrial System (CTS) is performed by means of rotations. For an arbitrary vector ~ the transformation is given by
(3.7) with
RM rotation matrix for polar motion RS rotation matrix for sidereal time RN rotation matrix for nutation RP rotation matrix for precession.
The CIS, defined at the standard epoch J2000.0, is transformed into the in-stantaneous or true system at observation epoch by applying the corrections due to precession and nutation. The X~-axis of the true CIS represents the free position of the angular momentum axis and thus points to the CEP.
Rotating this system around the X~-axis and through the sidereal time by the matrix RS does not change the position of the CEP. Finally, the CEP is rotated into the CIO by RM which completes the transformation.
The rotation matrices in Eq. (3.7) are composed of the elementary matri-ces &{ a} describing a positive rotation of the coordinate system around the L-axis and through the angle a. As it may be verified from any textbook on vector analysis, the rotation matrices are given by
[
1 0m~a
]Rda}
=
0 cos a0 -sina cos a
[
cos a 0-Si~a
]R2{a} = 0 1 (3.8)
sin a 0 cos a [ cooa sin a 0
] .
Jb{a}
=
-si;a cos a 00 1
Note that the matrices given by Eq. (3.8) are consistent with right-handed coordinate systems. The rotation angle a has a positive sign for clockwise rotation as viewed from the origin to the positive L-axis.
Precession. A graphic representation of precession is given in Fig. 3.4. The position of the mean vernal equinox at the standard epoch to is denoted by
3.2 Coordinate systems 29
mean equator (t)
mean equator (to)
X~(t)
Fig. 3.4. Precession
Eo and the position at the observation epoch t is denoted by E. The pre-cession matrix RP is composed of three successive rotation matrices
cos z cos {} cos ( - sin zsin ( sin z cos {} cos (
+
cos zsin(sin {} cos (
- cos z cos {} sin ( - sin z cos ( - sin z cos {} sin (
+
cos zcos(- sin {} sin (
- cos z sin {}
- sin z sin {} (3.9) cos {}
with the precession parameters z, {}, (. These parameters are computed from the time series, cf. Nautical Almanac Office (1983), p. S19:
( =
2306!'2181 T+
0!'30188 T2+
0!'017998 T3 z = 2306!'2181 T+
1!'09468 T2+
0!'018203 T3{} =
2004!'3109 T - Of' 42665 T2 - 0!'041833 T3 .(3.10)
The parameter T represents the timespan expressed in Julian centuries of 36525 mean solar days between the standard epoch J2000.0 and the epoch of observation. To give a numerical example consider an observation epoch J1990.5 which corresponds to T = -0.095. With T and Eq. (3.10) the numerical values (
=
-219!'0880, z=
-219!'0809, and {}=
-190!'4134 are obtained. Substitution of these values into Eq. (3.9) gives the following30 3. Reference systems numerical precession matrix:
[ 0.999997318 RP
=
-0.002124301 -0.0009231500.002124301 0.000923150
1
0.999997744 -0.000000981 . -0.000000981 0.999999574
Nutation. A graphic representation of nutation is given in Fig. 3.5. The mean vernal equinox at the observation epoch is denoted by E and the true equinox by Et • The nutation matrix RN is composed of three successive rotation matrices where both the nutation in longitude t!.1jJ and the nutation in obliquity t!.E can be treated as differential quantities:
(3.11)
The mean obliquity of the ecliptic E has been determined, d. Nautical Al-manac Office (1983), p. 821, as
E
=
23°26'21!' 448 - 46!'8150 T - 0!'00059 T2+
0!'001813 T3 (3.12) where T is the same time factor as in Eq. (3.10). The nutation parameters t!.1jJ and t!.E are computed from the harmonic series:106 5
t!.1jJ
= L
ai sin(L
ej Ej}=
-17!'2 sin nm+ ...
i=l j=l
64 5 (3.13)
t!..s
= L
bi cos(L
ej Ej)=
9!'2 cosnm+ ...
i=l j=l
mean equator
~---II
b,[
Fig. 3.5. Nutation
3.2 Coordinate systems 31 The amplitudes ai, bi as well as the integer coefficients ej are tabulated for example in Nautical Almanac Office (1983), pp. S23-S26. The five funda-mental arguments Ej describe mean motions in the sun-earth-moon system.
The mean longitude Om of moon's ascending node is one of the arguments.
The moon's node retrogrades with a period of about 18.6 years and this period appears in the principal terms of the nutation series.
Sidereal time. The rotation matrix for sidereal time J1,.s is
(3.14) The computation of the apparent Greenwich sidereal time 00 is shown in the section on time systems, cf. Sect. 3.3.
The WGS-84 system is defined by a uniform angular velocity WE, cf.
Table 3.1. Consequently, instead of the apparent sidereal time the mean sidereal time must be used in the case of GPS for the rotation angle in Eq. (3.14).
Polar motion. To this point the instantaneous CEP has been obtained. The CEP must still be rotated into the CIa. This is achieved by means of the pole coordinates x p, yp which define the position of the CEP with respect to the CIa, cf. Fig. 3.6. The pole coordinates are determined by the International Earth Rotation Service (IERS) and are available upon request, cf. Feissel and McCarthy (1989). The rotation matrix for polar motion RM is given by
RM
=
R2{-xp} Rd-Yp}= [ ~
-xp
o
1 -YP xp1
.yp 1
(3.15)
y --.,.---"'*
CIa
Xp
CEP
ypx Fig. 3.6. Pole coordinates
32 3. Reference systems The rotation matrices RS and RM are often combined to form a single matrix RR for earth rotation:
(3.16) In the case of GPS the space-fixed coordinate system is already related to the CEP. Hence, RR is the only rotation matrix which must be applied for the transformation into the terrestrial system.