Geoscience Reference
In-Depth Information
{
x
p
,
y
p
}
services and parameterized by the set of pole coordinates
. The prerequisite
derivations have been published by Brzezi nski (
1992
) or Gross (
1992
).
After finding the kinematical relationship between the TRS-based quantities
{
, a link to the dynamical equations of rotation, such as Eq. (
30
),
needs to be established, i.e. geophysical excitation and geodetic observation of Earth
rotation have to be connected properly. This task appears to be straightforward since
Eq. (
30
) incorporates the polar motion of Earth's instantaneous rotation axis. How-
ever, that equation has been deduced theoretically, based on the dynamical definition
of a reference system (the Tisserand mean mantle frame). Hence, it has to be asked if
it is legitimate to study polar motion with respect to the TRS by using formulae valid
in the Tisserand mean mantle frame. Chao (
1984
) considered the evoked inconsis-
tencies in the frequency band of the Chandler Wobble and found the resultant errors
to be negligible. Moreover, current realizations of the TRS consider the concept of
the mean crust (cf. Altamimi et al.
2003
) and thus may provide an approximation to
a Tisserand frame of the mantle. The validity of such an assumption is supported by
the fact that motions in the mantle on time scales up to decades are concentrated in
the upper layer of the astenosphere (A. Brzezi nski, personal communication).
The considerations now closely follow Gross (
1992
), who assumes all celestial
motions of the CIP to be perfectly accounted for by an appropriate precession-
nutation model. The celestial reference is thus a modified (intermediate) form of the
GCRS, rotated by the matrix
Q
x
p
,
y
p
}
and
{
m
1
,
m
2
}
, with its pole represented by the CIP. The TRS is
rotating with respect to this GCRS and the motion is characterized by the angular
velocity vector
(
t
)
ω
i
, where the subscript
i
labels quantities given in the inertial system.
If the origins of both the TRS and GCRS coincide, the transformation of position
vectors from one system to the other can be accomplished by aid of a time-dependent
rotation matrix
x
=
Rx
i
(64)
ω
=
R
ω
i
.
(65)
The velocity of a point that rests with respect to the GCRS follows from the derivative
of Eq. (
64
)
=
Rx
i
=
RR
T
x
x
˙
RR
T
≡
C
.
(66)
Considering that
x
can be equivalently calculated from
˙
˙
=−
ω
×
x
x
the components of the rotation vector
ω
can be associated with an antisymmetric
matrix (Gross
1992
)
