Estimation of the Earth’s tensor of inertia from recent geodetic and astronomical data
The transformation of the second-degree harmonic coefficients C2m and S2m in the case of a finite commutative rotation was derived instead of the traditional Lambeck’s approach based on an infinitesimal rotation.
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Інститут геофізики ім. С.I. Субботіна НАН України
2009
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| Цитувати: | Estimation of the Earth’s tensor of inertia from recent geodetic and astronomical data / A.N. Marchenko, N.P. Yarema // Геодинаміка. — 2009. — № 1(8). — С. 24-43. — Бібліогр.: 39 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-185342025-02-09T22:33:04Z Estimation of the Earth’s tensor of inertia from recent geodetic and astronomical data Визначення тензора інерції Землі за сучасними даними астрономії та геодезії Marchenko, A.N. Yarema, N.P. Геодезія The transformation of the second-degree harmonic coefficients C2m and S2m in the case of a finite commutative rotation was derived instead of the traditional Lambeck’s approach based on an infinitesimal rotation. 2009 Article Estimation of the Earth’s tensor of inertia from recent geodetic and astronomical data / A.N. Marchenko, N.P. Yarema // Геодинаміка. — 2009. — № 1(8). — С. 24-43. — Бібліогр.: 39 назв. — англ. 1992-142X https://nasplib.isofts.kiev.ua/handle/123456789/18534 en Геодинаміка application/pdf Інститут геофізики ім. С.I. Субботіна НАН України |
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English |
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Геодезія Геодезія |
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Геодезія Геодезія Marchenko, A.N. Yarema, N.P. Estimation of the Earth’s tensor of inertia from recent geodetic and astronomical data Геодинаміка |
| description |
The transformation of the second-degree harmonic coefficients C2m and S2m in the case of a finite commutative rotation was derived instead of the traditional Lambeck’s approach based on an infinitesimal rotation. |
| format |
Article |
| author |
Marchenko, A.N. Yarema, N.P. |
| author_facet |
Marchenko, A.N. Yarema, N.P. |
| author_sort |
Marchenko, A.N. |
| title |
Estimation of the Earth’s tensor of inertia from recent geodetic and astronomical data |
| title_short |
Estimation of the Earth’s tensor of inertia from recent geodetic and astronomical data |
| title_full |
Estimation of the Earth’s tensor of inertia from recent geodetic and astronomical data |
| title_fullStr |
Estimation of the Earth’s tensor of inertia from recent geodetic and astronomical data |
| title_full_unstemmed |
Estimation of the Earth’s tensor of inertia from recent geodetic and astronomical data |
| title_sort |
estimation of the earth’s tensor of inertia from recent geodetic and astronomical data |
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Інститут геофізики ім. С.I. Субботіна НАН України |
| publishDate |
2009 |
| topic_facet |
Геодезія |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/18534 |
| citation_txt |
Estimation of the Earth’s tensor of inertia from recent geodetic and astronomical data / A.N. Marchenko, N.P. Yarema // Геодинаміка. — 2009. — № 1(8). — С. 24-43. — Бібліогр.: 39 назв. — англ. |
| series |
Геодинаміка |
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AT marchenkoan estimationoftheearthstensorofinertiafromrecentgeodeticandastronomicaldata AT yaremanp estimationoftheearthstensorofinertiafromrecentgeodeticandastronomicaldata AT marchenkoan viznačennâtenzoraínercíízemlízasučasnimidanimiastronomíítageodezíí AT yaremanp viznačennâtenzoraínercíízemlízasučasnimidanimiastronomíítageodezíí |
| first_indexed |
2025-12-01T10:54:03Z |
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| fulltext |
1(8)/2009
24 © A.N. Marchenko, N.P. Yarema, 2009
521.21/22 A.N. Marchenko, N.P. Yarema
ESTIMATION OF THE EARTH’S TENSOR OF INERTIA FROM RECENT
GEODETIC AND ASTRONOMICAL DATA
The transformation of the second-degree harmonic coefficients mC2 and mS2 in the case of a finite
commutative rotation was derived instead of the traditional Lambeck’s approach based on an infinitesimal
rotation. The modified Lambeck’s formulae avoid uncertainty in the deviatoric part of inertia tensor and allow
simple transformation of the 2nd-degree harmonic coefficients and zonal coefficients of an arbitrary degree
(including their temporal changes) via orthogonal matrixes. These formulae together with exact solution of the
eigenvalue-eigenvector problem are applied to determine static components and accuracy of the Earth’s tensor of
inertia from the adjustment in the principal axes system of mC2 , mS2 from recent four gravity field models
(EGM2008, GGM03S, ITG-GRACE03S, and EIGEN-GL04S1) and eight values HD of the dynamical ellipticity
all reduced to the common MHB2000 precession constant at the epoch J2000. The second solution contains the
same parameters based on these four sets of mm SC 22 , and only one HD from the MHB2000 model and
corresponds better to the IERS Conventions 2003 and latest gravity field determinations. Two solutions for static
components consist of the adjusted five 2nd-degree harmonic coefficients related to the IERS reference pole
given by the conventional mean pole coordinates at the epoch 2000 (IERS Conventions 2003), the orientation of
principal axes in this system, the principal moments (A, B, C) of inertia, and other associated parameters. The
evolution with time of the above-mentioned static parameters was estimated in the principal axes system from
the GRACE time series of )(2 tC m , )(2 tS m derived in five different centers of analysis over the time interval
from 2002 to 2008. Special attention is given to the direct computation of temporally varying principal axes and
moments of inertia based on )(2 tC m , )(2 tS m and the estimation of their mean values together with periodic
constituents on given time-period. Stability of the positions of the equatorial inertia axes ( A , B ) and the angle
between two quadrupole axes located in the plane of the axes A and C of inertia is found. The estimated
longitude A of the principal axis A as the parameter of the Earth’s triaxiality in the precession-nutation theory
and 2J precession rate Ap of the precession constant are recommended for the Earth’s rotation theory.
Additionally to some permanent constituents periodic components at seasonal and shorter time scale were
evaluated.
Key words: the earth’s inertia tensor; principal axes and moments of inertia; Lambeck’s approach.
Introduction
Estimation of the Earth’s fundamental
parameters including elements of the tensor of
inertia is the traditional area of interest of the IAG
[Bursa, 1995; Groten, 2000; Groten 2004]. Suitable
solutions for the Earth’s principal moments of
inertia (A, B, C), principal axes ( A , B , C ), and
other fundamental constants were obtained in
[Marchenko, Schwintzer, 2003; Marchenko, 2007]
from the adjustment (in the principal axes system)
at one chosen epoch of several sets of the second
degree harmonic coefficients mC2 , mS2 of the
Earth’s gravity models all referred to different
epochs with a spacing of 18 years in between and
values of the dynamical ellipticity DH . Derived
from GRACE observations recent gravity field
models give more accurate solutions for the time-
dependent coefficients )(2 tC m , )(2 tS m . In
addition, latest determinations of the dynamical
ellipticity DH are based on the non-rigid Earth’s
rotation theory including the MHB2000 precession-
nutation model [Mathews et al., 2002] estimated
from VLBI observations during the time-period of
20 years, adopted by the IAU, and recommended by
the IERS Conventions 2003 [McCarthy and Petit,
2004]. After the launch of CHAMP and GRACE
satellites the combination of new gravity field
models, Earth’s orientation series, and geophysical
fluids data have led to a number of important
contributions with the treatment of )(tHH DD
and )(2 tC m , )(2 tS m as the sum of constant and
variable (secular or/and periodic) parts caused by
mass redistribution within the Earth’s system
[Marchenko and Schwintzer, 2001; Bourda and
Capitaine, 2004; Chen et al., 2005; Fernández,
2007; Gross et al., 2007]. The consistency of such
investigations and the modeling of the time
evolution require additionally to the consistent set
of fundamental constants more precise theories to
determine the dynamic figure of the Earth, the
orientation of the principal axes in the Earth’s-fixed
system and its evolution with time from geodetic
)(2 tC m , )(2 tS m and astronomical )(tH D
parameters.
This study aims to derive more accurate
expressions for the transformation of the second-
degree coefficients and zonal coefficients of an
arbitrary degree through a finite commutative
rotation instead of the most widely used
25
approximate Lambeck’s approach based on an
infinitesimal rotation [Lambeck, 1971; Reigber,
1981]. The modified Lambeck’s formulae for polar
coordinates considered at the sphere avoid
uncertainty in the deviatoric part of inertia tensor in
comparison with the usual planar approximation
and allow simple reduction of the 2nd-degree
harmonic coefficients and zonal coefficients of an
arbitrary degree together with their temporal
changes to the figure axis C . On the other hand,
various solutions of the coefficients mC2 , mS2
transformed in the ( A , B , C ) system and DH -
estimates (expressed through (A, B, C)-values)
represent initial information for the determination
of the principal moments (A, B, C) via simultaneous
adjustment by iterations providing in this way their
agreement with different sets of geodetic and
astronomical constants [Marchenko and Schwin-
tzer, 2003]. The last approach is analyzed
additionally to select initial values for iterations,
which can be slightly differed from the mean
moment of inertia of a homogeneous planet.
In contrast to the previous papers [Marchenko
and Schwintzer, 2003; Marchenko, 2007] the fully
normalized coefficients mC2 , mS2 are selected
from the recent four gravity field models
EGM2008, GGM03S, ITG-GRACE03S, and
EIGEN-GL04S1 constructed in different centers of
analysis, based on different data sets, and referred
to various epochs with a spacing of 5 years in
between. The secular change in the 2nd-degree
zonal coefficient 1-11
20 yr101.1628C is
adopted for these gravity fields together with the
simple linear model for 21C , 21S represented by
the mean pole’s drift with the reference mean pole
coordinates 405.0)( 0txp , 735.0)( 0ty p at
the epoch t0=2000 according to the IERS
Conventions 2003 [McCarthy and Petit, 2004]. It
has to be pointed out that mC2 , mS2 of the
conventional solution EGM96 given at epoch 1986
(IERS Conventions 2003) were replaced by mC2 ,
mS2 of the new gravity field model EGM2008
based on surface gravity data only [Pavlis et al.,
2008] and referred to epoch J2000 with 21C , 21S
selected in agreement with this epoch [Pavlis,
2008]. The Earth’s fundamental parameters were
estimated from the weighted least squares
adjustment of the new set of mC2 , mS2 of four
gravity field models and eight values HD of the
dynamical ellipticity [Williams, 1994; Souchay and
Kinoshita, 1996; Hartmann et al., 1997; Bretagnon
et al., 1998; Roosbeek and Dehant, 1998; Mathews
et al., 2002; Fukushima, 2003; Capitaine et al.,
2003] all reduced to the common value
yr/550.287922Ap of the MHB2000
precession constant at epoch J2000.
Because the modified Lambeck’s approach
allows simple transformation of mm SC 22 , via
orthogonal matrixes based on a finite commutative
rotation the corresponding formulae were applied in
the adjustment of the geodetic-only parameters
mm SC 22 , of the four gravity field models to the
IERS reference pole. Hence, the solution for static
components consists of the adjusted mm SC 22 , -
coefficients related to the reference IERS pole at
the epoch 2000, the orientation of principal axes in
this system, the principal moments of inertia (A, B,
C) of the Earth, HD, the coefficients in the Eulerian
dynamical equations, and other associated values.
Another solution contains the same parameters
based on these four sets of mm SC 22 , and only one
HD from the MHB2000 theory recommended by the
IERS Conventions 2003. In this way the second
solution for the time-independent principal
moments of inertia and other associated parameters
as a by-product of this adjustment at epoch
corresponds better to the frequently used IERS
Conventions 2003 and latest gravity field
determinations instead of the old conventional
model EGM96.
Secular changes of dynamical ellipticity DH
and precession constant were estimated via 20C
temporal variation preliminary transformed via
modified Lambeck’s formulae to the figure axis
C . These estimates were compared with other
results. Temporally varying components of the
tensor of inertia were found from adjusted value of
the dynamical ellipticity HD, the secular variation
DH , and the GRACE time series of )(2 tC m ,
)(2 tS m derived in five different centers of analysis
on the period from 2002 to 2008: 1) CNES-GRGS;
2) CSR Release 04; 3) GFZ Release 04; 4) JPL
Release 04.1; 5) ITG-GRACE03S. Special attention
is given not only to the direct computation of
temporally varying principal axes and moments of
inertia based on these time series of )(2 tC m ,
)(2 tS m but to the estimation of their mean values
and periodic components on given time-period from
time-frequency analysis at seasonal and shorter
time scale. As a result, additionally to some
permanent constituents of discussed parameters as
mean values at mean epoch their periodic stable
changes were also detected.
Transformation of 2nd degree harmonic
coefficients based on the Lambeck’s approach
Simultaneous adjustment of appropriate sets of
the harmonic coefficients ( mm SC 22 , ) to the
adopted reference pole based on the standard
1(8)/2009
26
approach [Lambeck, 1971; Reigber, 1981] was
considered in [Marchenko and Schwintzer, 2003]
by means of the equation:
Zxy gPg , (1)
where the matrix xyP depends only on the
coordinates pp yx , of the mean pole at chosen
epoch including also order 2 terms; the vector
T
2222212120 ,,,, SCSCCg , (2a)
(hereafter the symbol T denotes transposition) of
the fully normalized second degree coefficients
mC2 and mS2 , adopted in the Earth body-fixed
frame XYZ, shall be denoted by
T
2222212120 ,,,, BABAAZg , (2b)
if given in the coordinate system ZYX , which
is close to XYZ but with a difference in the
orientation of the third axis with Z-Z' being equal to
the mean pole coordinates.
According to [Lambeck, 1971] the pole
coordinates pp yx , are connected in the planar
approximation with the so-called amplitude p and
azimuth p as
ppPx cos , ppPy sin , (3)
that leads to the expressions for pp , in the
following form
22
ppp yx ,
p
p
p x
y
tan . (4)
To avoid the planar approximation (3) and the
corresponding non-orthogonal matrix xyP we will
consider the angles p, p and pp yx , at the unit
sphere for further determination pp , from the
solution of associated spherical triangles. It is easy
to verify that after some simple algebra the
following relationships are valid
ppp yx 22 tantantan , (5 )
p
p
p x
y
tan
tan
tan , (5 )
ppPx costantan , (6 )
ppPy sintantan , (6 )
pp
pp
p
yx
yx
22 sinsin1
coscos
cos , (7)
which give exact expressions for the polar
coordinates p, p. Eqs. (5 – 7) will get a special
importance for similar to Eq. (1) transformation,
where the non-orthogonal matrix xyP will replace
by some orthogonal matrix R , which is
depended on the polar coordinates p, p adopted
now in spherical approximation.
Thus, we will consider a transformation of the
coefficients ( mm SC 22 , ), defined in the coordinate
system ( ZYX ,, ), into the coordinate system
ZYX , which is obtained by a certain finite
rotation of the XYZ – system around the origin.
Hence, the potential V2 of the 2nd degree may be
written in the following forms
HrrT
5
2
2 2
1
r
GMaPV XYZ system (8a)
rHr T
5
2
2 2
1
r
GMaPV X Y Z system (8b)
where
202121
21202222
21222022
521515
1551515
1515515
CSC
SCCS
CSCC
H ,(9a)
202121
21202222
21222022
521515
1551515
1515515
ABA
BAAB
ABAA
H .(9b)
The matrices H and H are defined in the
geocentric coordinate systems ( ZYX ,, ) and
( ZYX ), respectively, representing the
deviatoric part of inertia tensor; the vectors Tr and
Tr contain the Cartesian coordinates of the current
point P in these systems. GM is the product of the
gravitational constant G and the planet’s mass
M ; a is the semimajor axis of the ellipsoid of
revolution; r is the distance from the origin of a
coordinate system to the current point P.
It should be pointed out that the rotation of the
system XYZ around the origin can be expressed via
the three matrixes of elementary rotations )( 11R ,
)( 22R , )( 33R . According to [Madelund,
1957] there are only two kinds of commutative
rotations. First one is an infinitesimal rotation.
Second one is a finite rotation about the fixed axis.
An infinitesimal rotation was considered in
[Marchenko and Schwintzer, 2003] for the
adjustment of mm SC 22 , -coefficients. To resolve a
possible ambiguity for various sequences of finite
rotations we will use this second type of a
commutative rotation with the following
transformation of the coordinate vector
PPPPP
PPPPPPPP
PPPPPPP
cossinsinsincos
sinsincos)cos1(cos)1(coscossin
sincos)1(coscossin1)1(coscos
2
2
Q , (10)
27
rQr , (11)
is the rotation matrix depended on the polar
coordinates of the axis Z in the system XYZ: p is
the polar distance of the axis Z and p is the
longitude of this axis defined by the Eqs. (5–7).
It is easy to verify that the matrix Q can be
constructed in the following way
)()()( 323 PPP RRRQ , (12)
where
)cos(0)sin(
010
)sin(0)cos(
)(
22
22
22R , (13a)
100
0)cos()sin(
0)sin()cos(
)( 33
33
33R , (13b)
by means of the rotation about the angles
P2 and P3 around the nodes line of
the XYZ and X Y Z systems. Clearly, the inverse
transformation reads
rRRRrQr )()()( 323
T
PPP , (14)
due to the orthogonality of the rotation matrix Q.
By inserting (11)and (14) into (8) we get
rQHQr TT
5
2
2 2
1
r
GMaPV , (15a)
rQHQr TT
5
2
2 2
1
r
GMaPV (15b)
Eq. (15a) represents now the potential V2 with
reference to the X Y Z system and the harmonic
coefficients mm SC 22 , given in the XYZ system. Eq.
(15b) describes the potential V2 in the XYZ system
with the harmonic coefficients mm BA 22 , related to
the X Y Z system.
It has to be noted that the tesseral coefficients
)(21 IERSC and )(21 IERSS related to the IERS
reference pole are based on the [Lambeck, 1971;
Reigber, 1981] formulae
pp ySxCCIERSC 22222021 )3()( , (16 )
pp xSyCCIERSS 22222021 )3()( , (16 )
used also in the approximate form
pxCIERSC 2021 3)( , (17 )
pyCIERSS 2021 3)( . (17 )
Thus, Eq. (16) is recommended by IERS
Conventions 2003 [MacCarthy and Petit, 2004] for
the computation of )(21 IERSC , )(21 IERSS . But
Lambeck’s standard approach may be developed to
the expressions for all 2nd degree coefficients
2/)(3)( 22
222020 pp yxCCIERSC
pppp yxSyxC 22
22
20 32/)( , (18)
3/)()( 22
202222 pp yxCCIERSC , (19 )
3/2)( 202222 pp yxCSIERSS , (19 )
and we can verify Eqs. (16 – 17) by considering the
characteristic equation of the matrices H (or H)
and deriving the first invariant )(Trace1 HI for
new harmonic coefficients )(22 IERSCA mm ,
)(22 IERSSB mm through Eqs. (16 – 19). Of
course, the equality 01I is satisfied by Eqs. (9)
trivially for arbitrary sets of mm SC 22 , or
mm BA 22 , . Nevertheless, after some easy algebra
we may get using Eqs. (18 – 19):
2220
2
1 155)( CCxIERSI p
222220
2 152155 SyxCCy ppp , (20)
as a rule non-zero value in Eq. (20), if the planar
approximation [Eqs. (16 – 19)] was used.
For example, the application of Eq. (20) to the
conventional EGM96 gravity model leads to
14
1 102.0)(IERSI instead of the trivial case
and we note again that 0)(1 IERSI can be
obtained only by the direct computation of the first
invariant based on Eq. (9). Hence, Eq. (20) allows
us to demonstrate a level of accuracy of the planar
approximation. Transformation in Eqs. (15) via the
matrix Q represents here an exception, because all
mC2 , mS2 or mm BA 22 , are results of the
commutative orthogonal rotation that always gives
zero value of QHH Trace()(Trace1I
0)(Trace) HQT .
Thus, in contrast to the Lambeck’s formulae in
planar approximation, the transformation (15) of V2
from XYZ to X Y Z system by applying the matrix
Q makes available to keep the first invariant I1 of
the deviatoric part H of inertia tensor. In this case
all elements of the matrix H, expressed through
mC2 , mS2 , or all elements of the matrix H ,
expressed via mm BA 22 , , are connected by the
commutative orthogonal rotation that leads to
01I in both cases. If the non-orthogonal matrix
xyP is used instead of the matrix Q, we get the
non-zero first invariant 0)(Trace1 HI
1510 .
Basic relationships for the adjustment of 2nd
degree harmonic coefficients to adopted reference
pole
Let us now consider the vector Zg consisting
of the harmonic coefficients mm BA 22 , in the
X Y Z system and taking into account Eq. (15b) we
find the following auxiliary matrix
1(8)/2009
28
)()( 33 PP RHRH . (21)
After simple manipulations in Eq. (21), we
come to the possibility of direct transformation of
the vector Zg to some vector g
T
2222212120 ,,,, SCSCC of harmonic coeffici-
ents
ZP gRg )( ,
PP
PP
PP
PP
P
2cos2sin000
2sin2cos000
00cossin0
00sincos0
00001
)(R
,(22)
where )( PR is the (5x5)-orthogonal matrix of
rotation about the angle P. Making our
manipulations in the same manner we can get some
new auxiliary matrix
)()( 22 PP RHRH , (23)
or the auxiliary vector
T
2222212120 ,,,, SCSCCg
of harmonic coefficients
gRg )( P , (24)
where
PP
PPP
PP
P
P
P
PPP
P
cos0sin00
0
4
3
4
2cos0
2
2sin
4
3
4
2cos3
sin0cos00
0
2
2sin02cos
2
2sin3
0
4
3
4
2cos30
2
2sin3
4
1
4
2cos3
)(R
,
(25)
is the (5x5)-orthogonal matrix of rotation about the
angle P. Taking into consideration Eq. (14) finally
we come to the following transformation of the
vector Zg given in the X Y Z system, to the vector
T
2222212120 ,,,, SCSCCg adopted in the XYZ
system
gRg )( P (26 )
ZPPP gRRRg )()()( . (26 )
Then taking into account some properties of
these orthogonal matrixes, the inverse
transformation from the vector g (XYZ system) to
the vector Zg (X Y Z system) reads
gRRRg )()()( PPPZ . (27)
Eq. (27) can be considered as the observational
equations for further adjustment of different sets of
the 2nd degree harmonic coefficients Zg to the
IERS reference pole fixed by the conventional
mean pole coordinates. Additional conditions for
the harmonic coefficients 02121 BA can be
obtained from Eq. (27), if the axis Z will coincide
with the figure axis C.
The harmonic coefficients of the degree n=2 can
be derived from Eq. (26) and represented now in
the matrix form
ZZ
rrrrr
rrrrr
rrrrr
rrrrr
rrrrr
ggRg
5545352515
4544342414
3534332313
2524232212
1514131211
, (28)
with the elements (29), (30). Then, according to Eq.
(27) the inverse transformation admits the
representation with the orthogonal matrix TR
obtained by the transposition of the orthogonal
matrix R in Eq. (28) with elements given by
Eqs. (29 – 30):
,cossincos2
,sincos
,2/)1cossincos4(
,sincos
),cos2(cossinsin
,sincoscos
),1cos2(sinsin
,sincos
),cos(sincos
,sincoscos
,sinsincos3
,2/)sin3(
,cossinsin3
,cossincos3
,2/)1cos3(
2
3
22
55
1
2
345
22
3
22
44
535
3
2
34
2
22
33
3
2
25
424
223
2
2
2
22
2
15
1
2
14
13
12
2
11
PPP
PP
PPP
PP
PPPP
PPP
PPP
PP
PPP
PPP
PPP
P
PPP
PPP
P
ur
uur
ur
ur
ur
ur
ur
ur
ur
ur
r
ur
r
r
r
(29)
Where
.1sincos2cos2
,sin2cos
,1cos
,1cos2
,1cos2
22
5
2
14
3
2
2
2
1
PPP
PP
P
P
P
u
uu
u
u
u
(30)
gRg T
Z . (31)
The last relationship together with Eqs. (29
30) will be considered as basic equation for the
adjustment to the adopted IERS reference pole of
different 2nd degree harmonic coefficients Zg
chosen as observations according to various gravity
field models.
In particular, making further manipulations, it is
easy to verify that the degree n zonal harmonic
coefficients in these two coordinate systems can be
formed as
29
n
m
PnmPnmPnm
m
n PmBmAC
0
0 )(cos~sincos)1( , (32a)
n
m
pnmpnmpnmn PmSmCA
0
0 )(cos~)sincos( , (32b)
where )(cos~
PnmP are A. Schmidt’s quasi-
normalized by the factor )!(
)!(
0 )2( mn
mn
m
associated Legendre functions of the first kind
( 0m is the Kronecker delta). If m=0 these
functions coincide with )(cos PnmP . If m>0 we
have for the fully normalized Legendre functions
)(cos PnmP the following relationship:
)(cos~12)(cos PnmPnm PnP .
Then we will split up the matrix (18) onto two
parts
RRR constP )( , (33)
00000
0
4
300
4
3
00000
00000
0
4
300
4
1
constR , (34)
PP
PPP
PP
P
P
P
PPP
cos0sin00
0
4
2cos0
2
2sin
4
2cos3
sin0cos00
0
2
2sin02cos
2
2sin3
0
4
2cos30
2
2sin3
4
2cos3
R
,
(35)
that leads to extracting in Eqs. (26 27) some
constant terms, longitude – only terms, and
longitude – polar distance terms. The constant
terms exist in the expressions for 20C , 22C , 22S
coefficients only.
If the coefficients mC2 and mS2 are given, Eq.
(28) to Eq. (30) can be applied to compute mA2
and mB2 related to the axis Z . 21A and 21B then
read
2225222421232122201221 SrCrSrCrCrA ,(36a)
2235223421332123201321 SrCrSrCrCrB ,(36b)
where the harmonic coefficients 21A and 21B must
be zero by definition, if the axis Z and the figure
axis C are coinciding at t0. By this, Eqs. (36) give
a tool to test whether gravity field models are
referred to a common axis C .
Transformation of 2nd degree harmonic
coefficients from initial to principle axes
coordinate system
Assuming our initial information consisting of
the vector g (Eq. (2a)) of 2nd-degree coefficients
and their variance-covariance matrix, we will use
for the transformation of ( mm SC 22 , ) to the
principal axes system the exact closed solution of
the eigenvalue problem with accuracy estimation by
rigorous error propagation. Let us give briefly
according to [Marchenko and Schwintzer, 2003;
Marchenko, 2003] the corresponding closed
expressions for the transformation of ( mm SC 22 , ),
defined in an adopted Earth’s-fixed coordinate
system (X, Y, Z), to the vector
T
2220 0,,0,0,~ AAg of the two nonzero
harmonic coefficients 20A , 22A in the coordinate
system of the Earth’s principal axes of inertia
( A , B , C ). The potential V2 of the second degree
may be written in the following way
rHr ~~~
2
15 T
5
2
2 r
GMaPV ,(37 )
Hrr T
5
2
2 2
15
r
GMaPV (37 )
where the deviatoric matrix H is defined by Eq.
(9a) and
3
200
0
3
0
00
3
~
20
20
22
20
22
A
AA
AA
H (38)
The matrix H~ is adopted in the system of prin-
cipal axes of inertia ( A , B , C ); the vector T~r
contains the Cartesian coordinates of the current
point P in this system; ( 2220 , AA ) are fully
normalized harmonic coefficients in the Earth’s
principal axes of inertia system ( A , B , C ).
The computation of the harmonic coefficients
20A , 22A requires a transformation of the matrix H
[Eq. (9a)] into the diagonal form H~ [Eq. (38)].
Solving the eigenvalue problem for the
corresponding deviatoric tensor (Eq. (9a)) in the
case of the given quadratic form HrrT we get
eigenvalues i in the following non-linear form
[Marchenko and Schwintzer, 2003]:
33
~
sin
3
~
sin
33
~
sin
3
2 2
3
2
1 k
, (39)
where the auxiliary angle ~ is expressed by means
1(8)/2009
30
of the invariants 22 kI and 3I :
3
2
31
2
33sin~
k
I
,
2
~
2
,
( 0Trace1 HI ), (40)
with
2
0
2
22
2
20
2
2
2
222
m
mm AASCIk , (41)
2
22
2
22
2
21
2
21
20
3
20
3 22
333
2det SCSCCCI H +
)~det(2 222121
2
21
2
2122 HSSCSCC . (42)
Here the 2nd degree variance 2k and 3I
represent the invariant characteristics of the gravity
field, which are independent of linear
transformations of the coordinate system (X, Y, Z).
Thus, Eqs. (39) to (42) provide the computation
of the harmonic coefficients ( 2220 , AA ) in the
principle axes coordinate system via the simple
expressions
2
3 3
20A ,
2
21
22A . (43)
The matrix H~ can be used also in the following
way
CBA
BCA
ACB
200
020
002
3
2
1
00
00
00
15~15H , (44)
where A, B, and C are the Earth‘s principal
moments of inertia normalized by the factor
2/1 Ma . As a result, if the eigenvalues i are
found, we come after some easy algebra to the
following relationships for differences between
these normalized moments of inertia
223
152 AAB , 20
22 5
3
15 AAAC , (45 )
20
22 5
3
15 AABC , (45 )
represented by means of the harmonic coefficients
( 2220 , AA ) in the principal axes system. Similarly,
these differences can be expressed also through
parameters of the Earth’s gravitational quadrupole
[Marchenko, 1979; Marchenko, 1998]:
20
22
2
2
2 5
3
15~ AA
Ma
MMAC , (46)
2
~
sin
2
~cos1 2
AC
BC
, (47 )
2022
2022
3
33~cos
AA
AA
, (47 )
where 2M is the moment of the quadrupole and ~
is the angle between two quadrupole axes, located
in the plane of the axes A and C . The parameter
~cos of the Earth’s triaxiality as the cosine of an
angle has a bounded range of variation,
1~cos1 , and enables us via Eqs. (46, 47)
to obtain “limiting” relationships between the
principal moments of inertia, 2nd degree
harmonic coefficients in the principal axes
system 2/0 2022 AA , and the polar pf and
equatorial ef flattenings pe ff0 [Marchen-
ko, 1979].
The estimation of the normalized principal
moments of inertia can be obtained now by
involving the dynamical ellipticity DH :
DH
AC 205
with
C
A
C
BACH D
205
2
2
(48)
Substitution of Eq. (48) into Eq. (45) gives
DH
AAAA 2022
20
5
3
155 , (49 )
DH
AAAB 2022
20
5
3
155 . (49 )
Therefore, with DH known, the computation
of the polar moment of inertia (normalized by the
factor 2/1 Ma ), DHAC /5 20 , the trace
)(Tr I :
m
D
I
H
ACBA 3325)ITr( 20 ,(50)
of the Earth’s tensor of inertia I considered in the
principal axes
C
B
A
00
00
00
I , (51)
and functions of the principal moments of inertia
( , , - dynamical flattenings) used in the
integration of the Eulerian dynamical equations
[Bretagnon et al., 1998; Hartmann et al., 1999]:
C
AB
B
AC
A
BC ,, , (52)
are straightforward, if the fully normalized
harmonic coefficients, 20A , 22A are computed
through Eqs. (39 – 43). Then the orientation of the
31
principal axes in the XYZ frame is based on the
exact solution of eigenvector problem, using mC2 ,
mS2 only without the dynamical ellipticity HD
[Marchenko and Schwintzer, 2003].
Estimation of the Earth’s fundamental parameters
in the principal axes coordinate system
The harmonic coefficients of 2nd degree and their
temporal variations are selected from the following
four gravity field models derived in various centers
of analysis: three solutions resulting from satellite
tracking data and GRACE observations for
different time-periods, GGM03S [Tapley et al.,
2007], ITG-GRACE03S [Mayer-Gürr, 2007], and
EIGEN-GL04S1 [Förste et al., 2008], and one
gravity field model of high resolution, EGM2008
(Pavlis et al., 2008), based on surface gravimetry
only. The time variable coefficients in these models
are referred to different epochs with a spacing of 5
years in between. Among these models the
harmonic coefficients of ITG-GRACE03S have
non-calibrated errors, which were multiplied on the
factor 10 according to the recommendation of
[Mayer-Gürr, 2008]. To be consistent, the
following transformations were applied to values
given (after reductions) in Table 1: (a) prediction of
)(2 tC m , )(2 tS m for a common epoch 2000, (b)
reduction of 20C to a common permanent tide
system, and (c) scaling of these coefficients to
common values of GM=398600.4415 km3/s2 and
a=6378136.49 m.
Table 1.
Geodetic parameters in the zero-frequency system (GM=398600.4415 km3/s2; a=6378136.49 m;
epoch: t0=2000; 405.0px , 735.0py ).
Model 6
20 10C 6
21 10C 6
21 10S 6
22 10C 6
22 10S
EGM2008 -484.16928852
0.000007
-0.00020662
0.000007
0.00138441
0.000007
2.43938343
0.000007
-1.40027362
0.000007
ITG-GRACE03 -484.16928857
0.000006
-0.00026548
0.000006
0.00147539
0.000006
2.43938345
0.000006
-1.40027368
0.000006
GGM03S -484.16929290
0.000047
-0.00020659
0.000008
0.00138442
0.000008
2.43934997
0.000008
-1.40029646
0.000008
EIGEN-GL04S1 -484.16944263
0.000025
-0.00024172
0.000016
0.00137671
0.000016
2.43936442
0.000017
-1.40028586
0.000017
For the transformation of 20C from the tide-free
system fC20 to the zero-frequency tide system ZC20
the following relation was used :
5/3.0103.1108 -8
2020 -CC fZ . (53)
The IERS Conventions 2003 recommends the
simple linear model representing the mean pole’s
drift as
)(
)(
)(
)(
)(
0
0
0
0
0 tt
ty
tx
ty
tx
y
x
p
p
p
p
p
p , (54)
where 405.0)( 0txp , 735.0)( 0ty p are the
mean pole coordinate at the reference epoch
t0=2000; /yr][0.00083)( 0txp ,
/yr][0.00395)( 0ty p are the secular variations
in )(),( 00 tytx pp valid in the vicinity of t0. The
linear model (54) can be applied only for the
transformation of the harmonic coefficients 21C
and 21S caused by a linear drift of the mean pole
[Eq. (17)], involving into the temporal variations
21C and 21S :
21002121 )()()( CtttCtC , (55 )
21002121 )()()( StttStS , (55 )
1-11
02021 yr100.337)(3 txCC p , (56 )
1-11
02021 yr101.606)(3 tyCS p , (56 )
because for other coefficients we get from Eq. (54)
0222220 SCC . Additionally to Eqs. (55 –
56) we will take into account the non-tidal secular
drift in the zonal coefficient
20002020 )()()( CtttCtC , (57 )
1-11
20 yr101.1628C . (57 )
In order to determine the Earth’s normalized
principal moments of inertia CBA ,, we use Eqs.
(48 49). Table 2 lists eight estimations of DH
and the values of the underlying precession
constant Ap . The first five DH were discussed in
Dehant et al. (1999) as ‘the best values to be used’
in the rigid nutation theory in the year 1999.
Another three solutions for the dynamical flattening
correspond to the non-rigid Earth’s rotation theory
including the MHB2000 precession-nutation model
[Mathews et al., 2002] estimated from VLBI
observations during the time-period of 20 years,
adopted by the IAU, and recommended by the
IERS Conventions 2003 [McCarthy and Petit,
2004]. The value DH by [Krasinsky, 2006] has a
large deviation from other determinations DH and
1(8)/2009
32
for this reason was omitted. For only three selected
DH accuracy estimates are found in the literature.
From the initial values of the dynamical
ellipticity DH given in Table 2 (also assumed to
refer to J2000) seven values differ in the adopted
according to IERS Conventions 2003 (IAU2000
Precession-Nutation model) precession constant
yr/50.2879225Ap . To transform the
associated quantities from different Ap to the
common value yr/50.2879225Ap the
differential relationship of Souchay and Kinoshita
(1996) was used
AA
A
D
D dpdp
p
HdH 7104947.6 , (58)
where Adp is expressed in arcseconds per Julian
century and we get the values DH given as
‘transformed HD to the MHB2000 precession
constant’ in Table 2. Eqs. (48 – 50) reflect a direct
dependence of A, B, C, and of the mean moment of
inertia 3/)(Tr ImI
Table 2.
Determinations of the dynamical ellipticity HD
Reference
Initial value of the
precession constant
pA [ /yr], J2000
Initial value of the
dynamical ellipticity HD
Transformed HD to the
MHB2000 precession
constant
yr/50.2879225Ap
Williams, 1994 50.287700 0.0032737634 0.003273777851
Souchay and Kinoshita,
1996
50.287700 0.0032737548 0.003273769251
Hartmann et al., 1999 50.288200 0.003273792489 0.003273774466
Bretagnon et al., 1998 50.287700 0.003273766818
0.000000000023 0.003273781269
Roosbeek and Dehant,
1998
50.287700 0.0032737674 0.003273781851
Mathews et. Al., 2002,
(MHB2000)
50.2879225
±0.000018
0.0032737949
±0.0000000012 0.003273794900
Fukushima, 2003 50.287955
±0.000003
0.0032737804
±0.0000000003 0.003273778289
Capitaine et al., 2003 50.28796195 0.00327379448 0.003273791918
The parameter 3/152 22AAB is also slightly
depending on the adopted permanent tide system
because 20C enters into the computation of the
coefficient 22A through Eq. (43). The indirect effect
of the permanent tide may either be included in the
20C -coefficient (zero-frequency tide system) or
excluded (tide-free system). It is assumed that the
DH values are related to the zero-frequency tide
system [Bursa, 1995; Groten, 2000].
With given variance-covariance matrices of
20A , 22A , the Earth’s principal moments of inertia
CBA ,, are determined from a weighted least-
squares adjustment of the astronomical and
geodetic parameters, all referred to a common
permanent tide system and one epoch 2000. As
‘observations’ generally are taken (a) the eight
values for DH (Table 2) and (b) the four sets of
20A , 22A in the principal axes system, computed
from the coefficients given in Table 1 by applying
Eq. (43). Using Eq. (45) and Eq. (48) we get the
over-determined system of non-linear observation
equations
,)(
152
3
,)2(
52
1
,
2
2
)(
22
)(
22
)(
20
)(
20
)()(
jj
jj
i
H
i
D
AAB
ACBA
H
C
BAC
(59)
with respect to the normalized principal moments
(A, B, C). )(i
DH (i=1,2,..k), )(
20
jA , and )(
22
jA
(j=1,2,..l) are treated as observations with being
an error component. For k values of )(i
DH and for l
sets of degree 2 harmonic coefficients )(
20
jA ,
)(
22
jA of l gravity field models we get according to
[Marchenko and Schwintzer, 2003] the system of
(k+2l) observation equations,
)(
)(
)(
)(
22
)(
20
)(2
0
00
00
22
20
0
152
3
152
3
5
1
52
1
52
1
22
1
2
1
j
A
j
A
j
H
j
j
i
D D
A
A
H
C
B
AC
BA
CC ,(60)
where 000 ,, CBA are some approximate values of
CBA ,, ; 0)()(
D
j
D
j
D HHH ,
33
0
20
)(
20
)(
20 AAA jj , 0
22
)(
22
)(
22 AAA jj ; and
CBA ,, are the corrections provided by the
solution of the normal equation system following
from Eq. (60) through iterations.
A number of iterations depends on the initial
values A0, B0, C0 in Eq. (60). Traditional
characteristic for such an adjustment of
astronomical and geodetic parameters is a high
close to +1 correlation between the solved
parameters, the three moments of inertia.
Nevertheless, the selection of the value Im=0.4 of
the mean moment of inertia of a homogeneous
planet as initial values for A0=B0=C0=0.4 leads also
to the convergence process but requires about 10
iterations. Finally in zero approximation were
adopted A0=B0=0.3 and C0=0.35. Usually with the
last amounts of A0, B0, and C0 it is enough to make
4 – 5 iterations. For each of the 8 values HD an
identical standard deviation 8H = 0.799 10-8
derived from the scattering about the mean value
was assumed for the weighting in the subsequent
adjustment by applying weights two times greater
for the last three values HD from Table 2,
corresponding to the non-rigid rotation theory, than
for other HD.
Table 3.
Results of the simultaneous adjustment of the astronomical DH and geodetic 20A , 22A parameters (zero-frequency-
tide system; GM=398600.4415 km3/s2; a=6378136.49 m, epoch: 2000)
Parameter S1: 8 HD + 4 gravity field models S2: 1 HD + 4 gravity field models
Solved
A 0.329612131 0.00000073 0.329611131 0.00000019
B 0.329619393 0.00000073 0.329618393 0.00000019
C 0.330698397 0.00000073 0.330697398 0.00000019
Derived
mI 0.329976640 0.00000073 0.329975641 0.00000019
DH 0.0032737850 0.0000000072 0.0032737949 0.0000000019
610)( AC 1086.266646 0.000049 1086.266646 0.000049
610)( BC 1079.004263 0.000049 1079.004263 0.000049
610)( AB 7.262383 0.000043 7.262383 0.000043
ABC /)( (3273.5575 0.072) 10-6 (3273.5674 0.019) 10-6
BAC /)( (3295.5180 0.073) 10-6 (3295.5280 0.019) 10-6
CAB /)( (21.9607 0.001) 10-6 (21.9608 0.0001) 10-6
6
20 10A 484.1692942 0.000009 484.1692942 0.000012
6
22 10A 2.8127085 0.000013 2.8127085 0.000017
1/f 298.256508 0.000008 298.256508 0.000008
1/fe 91434.77 0.4 91434.77 0.6
The variance-covariance matrices of ( 20A ,
22A )-sets are also taken into account. RMS
differences before and after iterations are equal to
0.05 and 0.6·10-8, respectively. Simultaneous
adjustment of the eight values of )(i
DH and four
models of the 2nd degree harmonic coefficients,
taken from the Table 1 and transformed to the
principal moments systems ( 20A , 22A ) is given in
the first column of Table 3 as the solution S1. The
second solution S2 represents the adjustment of
only one DH from the MHB2000 theory and four
sets of the same harmonic coefficients from Table
1.
Thus, two solutions, computed for the epoch
2000, are derived from two combinations of eight
(S1) and one (S2) values of )(i
DH plus the 2nd
degree harmonics of the gravity field models
EGM2008, GGM03S, ITG-GRACE03S, and
EIGEN-GL04S1. Apart from the solved
parameters, the other fundamental parameters of the
Earth derived from the three moments of inertia are
given in Table 3 together with their accuracy
estimates from error propagation [Marchenko and
Schwintzer, 2003]. Better accuracy of the S2
solution reflects a level of agreement of geodetic
parameters since only one HD was adopted in this
case. In general both sets of parameters from Table
3 have small differences on the level of accuracy
estimates. Nevertheless, the second solution S2
corresponds better to the frequently used IERS
Conventions 2003 and latest gravity field models
instead of the conventional EGM96.
1(8)/2009
34
Time-independent constituent adjusted to the
IERS reference pole
Let us now will examine values of 21A and
21B which must be zero by definition, if the axis Z
and the figure axis C are coinciding at t0. Eqs. (36)
give a good opportunity to test whether the adopted
here gravity field models are referred to a common
axis C .
Table 4 lists the obtained differences about zero
for adopted 405.0px and 735.0py (taken
from IERS Conventions 2003 at epoch 2000) and
leads to the conclusion that the reference systems of
considered models do not exactly match. We get
differences up to one order greater than the standard
deviations given in Table 1 for 2121, SC . However
these differences are smaller than the same values
in [Marchenko and Schwintzer, 2003] given for old
gravity field models.
To avoid the differences in Table 4 when fixing
a unique figure axis C we determine one set of the
coefficients mC2 and mS2 at epoch 2000 from a
least squares adjustment of the given six sets,
taking into account their variance-covariance
matrices and the two natural conditions for the left-
hand sides of Eqs. (36): 21A = 21B = 0.
For l adopted gravity models we initially
compute the harmonic coefficients )(
2
j
mA , )(
2
j
mB
(j=1,2,...l) treated further as observations.
Table 4.
Harmonic coefficient 21A and 21B at t0=2000
based on Eqs. (36) and adopted 405.0px and
735.0py (from IERS Conventions 2003)
Parameter EGM2008 ITG-
GRACE03 GGM03S EIGEN-
GL04S1
A21··1010 0.160 -0.429 -0.160 -0.191
B21··1010 -0.632 0.278 -0.632 -0.709
Applying Eq. (31) we get the observation
equations in the linear form
)(
5
)(
4
)(
3
)(
2
)(
1
)(
21
)(
22
)(
21
)(
21
)(
20
21
22
21
21
20
T
j
j
j
j
j
j
j
j
j
j
B
A
B
A
A
S
C
S
C
C
R , (61)
with the 5 unknown elements of the vector
T
2222212120 ,,,, SCSCCg ; )( j
i are error
components.
Table 5.
Results of a simultaneous adjustment of the mC2 , 22S parameters to the IERS reference pole fixed by the mean
pole coordinates 405.0px , 735.0py at epoch 2000 (zero-frequency-tide system;
GM=398600.4415 km3/s2; a=6378136.49 m)
Parameter 4 models: EGM2008, ITG-GRACE03S,
GGM03S, EIGEN-GL04S1
2 models:
EGM2008, ITG-GRACE03S
6
20 10C -484.16929419 0.000020 -484.169288549 0.000023
6
21 10C -0.00022261 3.1 10-11 -0.00022261 4.0 10-11
6
21 10S 0.00144761 6.6 10-11 0.00144761 7.7 10-11
6
22 10C 2.43937396 0.000016 2.439383442 0.000022
6
22 10S -1.40028032 0.000017 -1.40027366 0.000022
The orthogonal matrix TR of this system depends
only through Eqs. (3-7) on the mean pole
405.0px and 735.0py at epoch 2000. The
vector g results from the solution of the normal
system following from Eq. (61) with the two
additional conditions, i.e. zero left hand sides in Eqs.
(36).Taking for all (l=4) gravity models the
harmonic coefficients )(
2
j
mA and )(
2
j
mB in the
ZYX frame as observations, we get in this way
our basic set of adjusted mC2 , mS2 - coefficients
to IERS Reference pole (at the epoch 2000), given
in Table 5 in the first column. Second solution from
Table 5 (l=2, second column) based on the two
EGM2008 and ITG-GRACE03S gravity field
models was developed in the same manner only for
the comparison of adjusted to the IERS 2003 pole
sets of mC2 , mS2 and corresponding accuracy
estimates. Note that initial set of harmonic
coefficients for the construction of the EGM2008
model was taken from the ITG-GRACE03S
solution and small differences between mC2 , mS2
in Table 1 (excluding 21C and 21S of EGM2008
adopted according to the IERS Conventions 2003)
reflect the influence of ‘the inclusion of the surface
gravity data into the least-squares adjustment’.
35
Both sets of these coefficients from Table 5
restore exactly the adopted mean pole coordinates
405.0px and 735.0py , if inserted into the
following expressions based on Eqs. (5 – 7):
2
22
2
22
2
20
2122212220
3
)3(
SCC
SSCCCx p , (62 )
2
22
2
22
2
20
2122212220
3
)3(
SCC
CSSCCy p . (62 )
Applying the exact equations to the first set of
adjusted mC2 , mS2 , the orientation of the principal
axes A , B , and C are computed for each
Table 6.
Spherical coordinates of the principal axes and their accuracy (epoch 2000)
Gravity field
model
Lat. A
[degree]
Lon. A
[degree]
Lat. B
[degree]
Lon. B
[degree]
Lat. C
[degree]
Lon. C
[degree]
EGM2008 -0.000038
0.0000005
345.0715
0.0001
0.000088
0.000005
75.0715
0.0001
89.999904
0.0000005
278.3486
0.2885
ITG-
GRACE03S
-0.000043
0.0000004
345.0715
0.0001
0.000093
0.0000004
75.0715
0.0001
89.999897
0.0000004
280.053074
0.2328
GGM03S -0.000038
0.0000005
345.0711
0.0001
0.000088
0.0000005
75.0711
0.0001
89.999904
0.0000005
278.3476
0.3180
EIGEN-GL04S1 -0.000040
0.000001
345.0713
0.0002
0.000087
0.000001
75.0713
0.0002
89.999904
0.000001
279.8118
0.6604
Adjusted mC2 ,
mS2 (4 models)
-0.000040
0.3 10-9
345.0714
0.0002
0.000092
0.1 10-9
75.0714
0.0002
89.999900
0.6 10-11
278.6014
0.2 10-5
individual gravity field model and for the adjusted
second-degree coefficients. The results are given in
spherical coordinates in Table 6 and for the axis C
also in polar coordinates (Table 7).
Table 7.
Polar coordinates of the principal axis C
and their accuracy (epoch 2000)
Gravity field model Cx [0.001"] Cy [0.001"]
EGM2008 50.1 1.7 341.4 1.8
ITG-GRACE03S 64.5 1.5 363.8 1.6
GGM03S 50.1 1.9 341.4 1.9
EIGEN-GL04S1 58.7 4.0 339.5 4.0
Adjusted mC2 , mS2
(Table 5, 4 models)
54.0
0.110-4
357.0
0.210-4
It should be pointed out, that such ‘high’
accuracy of 21C , 21S in Table 5 are result from the
application of the mentioned conditions
02121 BA [Eqs. (36)]. Accuracy of xc and yc
in Table 7 for the adjusted harmonic coefficients
mC2 , mS2 and accuracy of the corresponding
latitudes of the principal axes A , B , and C
(Table 6) again reflect the mentioned influence of
conditions 02121 BA which were initially
introduced via adjustment to the adopted IERS
reference pole fixed at epoch 2000 by the mean
pole coordinates in Eq. (36).
After transformation of adjusted mC2 , mS2
based on four models to the principal axes system
in view of accuracy estimation we get comparable
numerical values with the coefficients 20A , 22A
(S1) of Table 3. Hence, their combination with the
adjusted 0000000072.0500.00327378DH
gives similar values for other parameters of the
solution S1 in Table 3. Therefore, the first columns
of Table 3 and Table 5 can be considered as one
consistent set of the Earth’s fundamental parameters
at epoch 2000 given in the principal axes and the
Earth’s-fixed systems, respectively. In comparison
with previous results [Marchenko and Schwintzer,
2003] based on Eq. (1) we get generally slightly
better accordance between the adjustment of
astronomical and geodetic constants and the separate
adjustment of the 2nd harmonic coefficients only to
the IERS reference pole. But differences between
adjusted mC2 , mS2 based on Eqs. (1 – 3) and Eqs.
(61) have values about 10-15 that corresponds to the
non-zero )(Trace1 HI in the case of the
traditional Lambeck’s approach [Eqs. (1 – 3)].
Earth’s time-dependent parameters from GRACE
The time-dependent 2nd-degree harmonic
coefficients )(2 tC m , )(2 tS m were taken from the
International Center for Global Earth’s Models of
the IAG and extracted for the following GRACE
time series: CNES-GRGS, CSR Release 04, GFZ
Release 04, JPL Release 04.1, and ITG-
GRACE03S time-dependent solution [Mayer-Gürr,
2007]. These mC2 , mS2 with a step size from 10
days (CNES-GRGS) to one month (other solutions)
1(8)/2009
36
were applied to the direct computation of
temporally evolving components of the Earth’s
inertia tensor and other associated parameters on
the time period from 2002.3 to 2008.5 years.
To be consistent the following transformations
were used to values )(2 tC m , )(2 tS m as in the case
of time-independent constituent: (1) reduction of
20C to a common zero-frequency tide system (Eq.
(53)), and (2) scaling of these coefficients to
common values of GM=398600.4415 km3/s2 and
a=6378136.49 m. Taking into account the adjusted
dynamical ellipticity 500.00327378DH , all
parameters listed in Table 3, Table 6, and Table 7
were determined now as time-dependent for each
related moment of time according to these four
solutions on the total period from 2002.3 to 2008.5
years.
Because of a great number of various
parameters computed for each moment of time t we
give only their evolution for the axes A , B , and
C of inertia. For other illustrations it is sufficient
to give only mean values of some time-dependent
quantities obtained by averaging their instant values
on the given time-period from 2002.3 to 2008.5
years. Fig. 1 and Fig.2 show temporal changes from
GRACE of longitudes of the axes A , B , and C .
Table 8 demonstrates mean longitudes of these
axes and mean values of the angle ~ (Eq. (47))
between two quadrupole axes, located in the plane
of the axes A and C of inertia. Table 9 lists
obtained average values of polar coordinates of the
figure axis C for the same period related to the
corresponding mean epochs about 2005.
A comparison of each initial )(2 tC m , )(2 tS m
taken from various centers of analysis leads to the
conclusion about systematic differences existing
C B
A
Fig. 1. Longitude of the axes A , B , C of
inertia from CNES-GRGS ( ), CSR ( ), GFZ
( ), JPL ( ), and ITG-GRACE03S ( ) time
series for the period from 2002.3
to 2008.5 years
Fig. 2. Longitude of the axis A of inertia from
CNES-GRGS ( ), CSR ( ), GFZ ( ), JPL ( ),
and ITG-GRACE03S ( ) time series for the
period from 2002.3 to 2008.5 years
Table 8.
Mean longitudes of the principal axes A , C and
mean values of the angle
~
between two
quadrupole axes (Eq. (47)) (period from 2002.3
to 2008.5 years)
Mean
values
Longitude
A
Longitude
C
Angle ~
[Eq. 47]
CNES-
GRGS
345.0714
0.00005
281.0880
0.16
170.61988
0.000008
CSR
Release 04
345.0711
0.00002
279.4887
0.09
170.61988
0.000003
GFZ
Release 04
345.0712
0.00001
280.5852
0.05
170.61988
0.000002
JPL
Release 04.1
345.0709
0.00001
278.7286
0.02
170.61985
0.000001
ITG-
GRACE03
345.0715
0.00006
280.0541
0.23
170.61986
0.000010
Table 9.
Mean coordinates of the figure axis C
for the period from 2002.3 to 2008.5 years
Gravity field
model
Mean
epoch
[year]
Cx
[0.001"]
Cy
[0.001"]
CNES-
GRGS 2005.46 70.2
1.0
358.5
1.0
CSR
Release 04 2005.48 56.2
0.5
336.6
0.5
GFZ
Release 04 2005.72 61.3
0.3
328.4
0.3
JPL
Release 04.1 2005.44 52.7
0.2
343.0
0.2
ITG-
GRACE03 2004.96 64.5
1.5
363.8
1.6
in these series. Fig. 2, Table 8, and Table 9 reflect
these probable systematic trends in five deter-
37
minations of the time-dependent coefficients
)(2 tC m , )(2 tS m . Nevertheless some derived
parameters illustrated by Table 8 and Fig. 1 are
generally permanent taking into account accuracy
estimation of their static part. In contrast to the
evident temporal change of the figure axis C (Fig.
1) we get a remarkable stability in time of the
position of the inertia axes A and B derived from
GRACE (Fig. 1). Processing of the CHAMP
quarterly solutions [Reigber et al., 2003] for
)(2 tC m , )(2 tS m (period from 2000.9 to 2003.4
years) produces the same conclusion about stability
of the axis A (and B ) with the mean longitude
E0706.345A . In addition, we get a similar
accordance with previous results [Marchenko, 2007]
based on such GRACE time series as CSR Release
01, GFZ Release 03, and JPL Release 02, which
allow the same general conclusion excluding small
differences in relation to values from Table 8. It has
to be pointed out, that the direction of the principal
axis A is considered in the precession-nutation
theory [Bretagnon et al., 1998; Roosbeek and
Dehant, 1998] as the parameter of the Earth’s
triaxiality or the longitude A of the major axis of
the equatorial ellipse. Thus, the adjusted to the
IERS reference pole at the epoch 2000 numerical
value 0002.0E0714.345A (Table 6) in
terms of accuracy estimation agrees perfectly with
those from Table 8 and may be recommended for
the Earth’s rotation theory: A
0002.0W9286.14 . But a most stable value
represents another parameter of the Earth’s
triaxiality, o6199.170~ , the angle between the
quadrupole axes.
If orientation of the Earth’s principal axes of
inertia, the angle ~ , and some other parameters
depend only on the )(2 tC m , )(2 tS m coefficients,
the determination of temporal changes of the
Earth’s tensor of inertia requires according to Eqs.
(48 – 49) the dynamical ellipticity HD. To compute
the principal moments of inertia A, B, and C (Eqs.
(48 – 49)) from adjusted 500.00327378DH
(related to J2000) and GRACE )(2 tC m , )(2 tS m at
each given moment t, which is different from the
standard epoch 2000, an additional correction H to
HD should be applied. Special study of the )(2 tC m ,
)(2 tS m GRACE series led to a non-stable
determination of the secular variation of 22A adopted
finally as 022A . Therefore, we assume the non-
tidal variation C in the moment of inertia C as a
function of 20C only [Yoder et al., 1983] as zonal
forces do not change the revolution shape of the body'
[Melchior, 1978] and come to 2/CBA
from the condition for the trace const)(Tr I
[Rochester and Smylie, 1974]. By this we get from
Eqs. (48) and (49) the secular change of HD :
)
3
21(
20
20
DDD HH
A
A
H , (63 )
)
3
21(
3
)Trace(5
20
20
220 DDD HH
C
C
C
AH I , (63 )
if secular variations in different coordinate system
are equal 2020 CA . To verify this equality we
will use Eq. (32b) written for the time-dependent
harmonic coefficients )(tCnm , )(tSnm , and 0nA .
Differentiation of 0nA , )(tCnm , )(tSnm in Eq.
(32b) with respect to time t gives
n
m
CnmCnmCnmn PmSmCA
0
0 )(cos~)sincos( , (64)
the equation for the reduction of the given
mm SC 22 , to the unknown 20A through the polar
coordinates C and C of the figure axis C
which are considered in Eq. (64) as time-
independent and known at fixed epoch. With 20C ,
21C , 21S taken from Eqs. (56 – 57),
02222 SC , and the position C and C of the
axis C supposed to coincide with the mean pole
coordinates at epoch 2000 (IERS Conventions
2003), we get from Eq. (64) the estimation
20
11
20 yr/110162795.1 CA which is
slightly differed from 20C (Eq. (57)) on the smaller
value 16105.0 than accuracy estimates of
20C and other temporal variations.
That is why we neglect this correction and get
numerically 111 yr108453.7DH with 20C
taken from Eq. (57). This amounts to
)( 0ttHH D for the reduction of HD from the
year t0=2000 to each moment t related to )(2 tC m ,
)(2 tS m of GRACE time series. (Note that
10101.7H for the reduction of HD from the
year 2000 to 2009). Then, applying for parameters
connected with 1-11
20 yr101.1628C the
following linear dependence
)()( 0ttFtF , (65)
where dt
tdFF )( and t0 is chosen reference epoch,
we give in Table 10 using Eq. (65) the
corresponding estimates of different secular
changes according to (Marchenko, 2007).
It has to be pointed out that similar estimates of
secular changes 111 yr1086.7DH and
38
Table 10.
Secular changes in some astronomical and geodetic parameters corresponding to the secular drift in the
coefficient 2020 CA (t0=2000) [Marchenko, 2007]
Parameter )()( 0ttFtF F
2020 CA )( 02020 ttAA 20A =1.1628 10-11 [1/yr]
DH )(
3
)Trace(5 0220 tt
C
AH D
I
DH = 7.8453 10-11[1/yr]
Ap )( 0tt
p
HHp
A
D
DA
Ap = 0.0121 [ /cy2]
A )(
3
5
0
20 ttAA
A =0.8667 10-11 [1/yr]
B )(
3
5
0
20 ttAB
B =0.8667 10-11 [1/yr]
C )(
3
52
0
20 ttAC
C = 1.7334 10-11 [1/yr]
A
BC
)(
3
35
02
20 tt
A
ABCA = 7.8970 10-11 [1/yr]
B
AC
)(
3
35
02
20 tt
B
BACA = 7.8968 10-11 [1/yr]
C
AB
)(
3
5
02
20 tt
C
ABA =5.7552 10-16 [1/yr]
f )(
2
53
0
20 ttAf
f = 3.9001 10-11 [1/yr]
Table 11.
Mean values of the principal moments of inertia A, B, and C from the GRACE series of )(2 tC m , )(2 tS m ,
adjusted 500.00327378DH , and 111 yr108453.7DH (for the period from 2002.3 to 2008.5 years)
Mean values Mean epoch
[year]
Principal
moment A
Principal
moment B
Principal
moment C
CNES-GRGS 2005.46 0.32961228 0.32961954 0.33069855
CSR Release 04 2005.48 0.32961220 0.32961946 0.33069846
GFZ Release 04 2005.72 0.32961215 0.32961941 0.33069841
JPL Release 04.1 2005.44 0.32961220 0.32961947 0.33069847
ITG-GRACE03 2004.96 0.32961217 0.32961943 0.33069843
111 yr104.7DH were found under the same
condition to conserve changes in the trace Trace(I)
of inertial tensor by [Marchenko and Schwintzer,
2003] and [Bourda and Capitaine, 2004]
respectively. Small differences in all DH -values
are explained by the application of various sets of
chosen constants entering in Eq. (63).
Among parameters from Table 10 all secular
changes have the same order as variation 20A
excluding and Ap . According to [Marchenko,
2007] the variation Ap was called by the 2J
precession rate with the estimated range (–11.6 to –
16.8) 10-3 [ /centuries2], which is depended on the
adopted 202 5CJ having the value ‘about
0.7% classical acceleration induced by ecliptic
motion and two orders of magnitude larger than
tidally induced accelerations’. Williams’ 2J
precession rate Ap = –0.014 [ /cy2] was based on
the old determinations of the variation 2J .
Nevertheless his estimation given in 1994 agrees
well with those from Table 10. Because the derived
value 012.0Ap [ /cy2] was based on Eq. (58)
and the secular variation 1-11
20 yr101.1628C
adopted for recent gravity field models this
parameter also may be recommended for the
Earth’s rotation theory.
39
Table 11 illustrates mean values of the principal
moments of inertia A, B, and C derived from the
)(2 tC m , )(2 tS m GRACE series, the adjusted
dynamical ellipticity 500.00327378DH , and
the secular change 111 yr108453.7DH by
averaging the instant values A(t), B(t), and C(t) on
given time-period. Taking into account the previous
results by [22] based on such GRACE time series as
CSR Release 01, GFZ Release 03, and JPL Release
02, the comparison of the GRACE only principal
moments of inertia from Table 11 with the adjusted
quantities A, B, and C given in Table 3 leads to a good
agreement in terms of accuracy estimation in all cases
of )(2 tC m , )(2 tS m GRACE series. Nevertheless,
only secular variations in the 2nd degree time-
dependent GRACE coefficients are not sufficient
for the description of )(2 tC m , )(2 tS m -changes.
For example, another representation for the series
of )(2 tC m , )(2 tS m was introduced by adopting
Table 12.
Contribution of nearly annual time variations of time-dependent parameters given in percentages
to common periodic changes
Parameter CNES-
GRGS
CSR-
r104
GFZ-
r104
JPL-
r104.1
ITG-
GRACE03
Parameters in the principal axes system
A 54% 24% 33% 39% 45%
B 54% 25% 33% 39% 46%
C 54% 22% 33% 39% 48%
20A 54% 23% 33% 38% 50%
22A 41% 37% 44% 85% 32%
~ 42% 40% 48% 85% 51%
Longitudes of the principal axes A and C
A 51% 59% 45% 5%
C 72% 24% 41% 60% 38%
the model of secular, annual, and semi-annual
periodic variations, based on the EIGEN-GL04S
static gravity field model and the GRACE 10-days
solutions [Lemoine et al., 2007]. Taking into
account that time-dependent parameters from Table
8 and Table 11 depend on )(2 tC m , )(2 tS m -
coefficients, these are then analyzed after removing
a linear trend for the detection of basic periods
derived from a spectral analysis using the following
model
i
i
i
i tt
P
AttFFtF )(2cos)()( 000
, (66)
for time-dependent function )(tF with the
simultaneous determination of all components iA ,
i , and iP of an oscillation, including periods iP .
As a result, close to annual and semi-annual terms
among estimated periods were observed with
common contributions more than 50% in all
determinations. Table 12 reflects the contribution of
nearly annual variations only into common periodic
changes, which are different for various centers of
analysis. Thus, although exist some basic part of
discussed parameters given in Table 8 and Table 11
we detect their small deviations having annual,
semi-annual, and other terms. On the other hand,
mean values of these parameters agreed well with
their ‘static’ values from Table 3 and can be
considered as some permanent constituents given at
the corresponding mean epochs.
Conclusions
In order to avoid uncertainty in the deviatoric
part H of inertia tensor the transformation of the
second-degree harmonic coefficients mm SC 22 , was
developed especially for the case of a finite
commutative rotation via modified Lambeck’s
formulae applied to polar coordinates considered at
the sphere. The modified Lambeck’s approach
allows simple transformation of the 2nd-degree
harmonic coefficients and zonal coefficients of an
arbitrary degree (including their temporal changes)
via orthogonal matrixes. This transformation was
used in the two individual adjustments of the
geodetic only parameters mm SC 22 , of four gravity
field models adopted in the Earth’s-fixed system to
the IERS reference pole given by the conventional
mean pole coordinates 405.0px and
735.0py at epoch 2000 (IERS Conventions
2003). The same sets of mm SC 22 , -coefficients
together with eight values of the dynamical
ellipticity HD all reduced to the common MHB2000
precession constant yr/50.2879225Ap were
used in the two general adjustments given in the
1(8)/2009
40
principal axes system with respect to the Earth’s
principal moments of inertia A, B, and C. Results of
the first adjustment of geodetic and astronomical
‘constants’ represent one set (S1) of consistent
parameters given in Table 3, Table 5, and Table 6 at
one chosen epoch J2000 as time-independent
constituent of the orientation of principal axes in
the Earth’s-fixed system, the principal moments (A,
B, C) of inertia, HD, coefficients in the Eulerian
dynamical equations, and other associated
parameters. The second solution contains the same
parameters based on the same four sets of
mm SC 22 , and only one HD from the MHB2000
model and corresponds better to the frequently used
IERS Conventions 2003 and latest gravity field
determinations.
Time-dependent components of the Earth’s
tensor of inertia were found from the time-dependent
)(2 tC m , )(2 tS m GRACE time series of the
following five solutions: CNES-GRGS; CSR Release
04; GFZ Release 04; JPL Release 04.1; ITG-
GRACE03S. The condition A= B= C/2 to
conserve Trace(I) of the inertia tensor when
changing the dynamical ellipticity DH from the
reference epoch t0=2000 to a current moment of
time t was applied via variation DH for the
estimation of DH , 2J precession constant rate,
and other parameters. These estimations are based
on the modified Lambeck’s approach and derived
closed expression for the reduction of the mm SC 22 ,
secular variations related to the standard Earth-
fixed system to the unknown 20A related to the
figure axis C through the polar coordinates C
and C of the axis C , which were fixed for the
epoch 2000. Estimation of 20A
yr/110162795.1 11 secular variation leads to a
slightly different from 20C value. It has to be
pointed out that mean values of the principal
moments A, B, and C of inertia given at the epoch
about 2005 based only on the )(2 tC m , )(2 tS m
GRACE series, DH , and DH agree well with the
adjusted quantities A, B, and C at the epoch J2000.
A stability in time of the position of the axes A
and B of inertia and the angle ~ between two
quadrupole axes, located in the plane of the axes
A and C , was observed from the time-dependent
)(2 tC m , )(2 tS m GRACE time series. Since the
longitude A of the principal axis A is considered
in the nutation theory as the parameter of the
Earth’s triaxiality, the estimated value
0002.0W9286.14A can be recommended
for the Earth’s rotation theory together with the 2J
precession rate 012.0Ap [ /cy2] of the
precession constant Ap . Nevertheless, periodic
components at seasonal and shorter time scale were
evaluated for the detection of basic periods derived
from a spectral analysis. As a result, nearly annual
and semi-annual terms among estimated periods
were observed with common contributions more
than 50% in all determinations. Hence, although
exist some permanent constituents of discussed
parameters (as mean values at mean epoch) their
small deviations have also stable terms with about
annual and semi-annual periods, which are different
for various centers of analysis.
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