Mon. Not. R. Astron. Soc. 322, 785±799 (2001)
Forced nutations of a two-layer Earth model
Juan Getino1w and Jose M. FerraÂndiz2w
1
2
Grupo de MecaÂnica Celeste, Facultad de Ciencias, 47005 Valladolid, Spain
Depto. AnaÂlisis MatemaÂtico y MatemaÂtica Aplicada, Facultad de Ciencias, Universidad de Alicante, 03080 Alicante, Spain
Accepted 2000 October 30. Received 2000 October 3; in original form 2000 January 26
A B S T R AC T
In this paper we present a theory of the Earth rotation for a model composed of an inelastic
mantle and a liquid core, including the dissipation in the core±mantle boundary (CMB). The
main features of the theory are: (i) to be Hamiltonian, therefore the computation of some
complex inner torques can be avoided; (ii) to be self-consistent and non-dependent on a
previous rigid Earth theory, so there is no need to use transfer functions; (iii) to be analytical,
the solution being derived by perturbation methods. Numerical nutation series deduced from
the theory are compared with the IERS 96 empirical series, an accuracy better than 0.8 mas
in providing celestial ephemeris pole (CEP) offsets.
Key words: celestial mechanics ± Earth.
1
INTRODUCTION
In a previous Letter sent to this journal (Getino & FerraÂndiz 1999)
we introduced the first analytical nutation series that could be
referred to as accurate, since they kept the deviation in CEP
offsets with respect to 1996 IERS Conventions below 1 mas in the
time domain. These series were derived by using Hamiltonian
perturbation methods without relying on any specific geophysical
model, so that the procedure is more reminiscent of traditional
astronomy and celestial mechanics than of the usual modern
approach, which is closely dependent on geophysics. Once the
scientific community was promptly informed of the availability of
new analytical solutions to the nutations, reaching a similar level
of accuracy to recent series by Mathews et al. (1991), Dehant &
Defraigne (1997) and Schastok (1997), it seemed appropriate to
submit the theory used in their derivation, which constitutes the
main purpose of this paper.
Let us recall that we consider a rather simple Earth model
composed of two layers, the relative motion of which gives rise to
a dissipation at the mutual boundary. The mantle is assumed to be
deformed in an almost elastic way, the solution of the displacement vector being given by expressions like those of Takeuchi
(1951). A weak inelasticity is allowed and treated in the simplest
way through considerations of a uniform time delay in the
response to external perturbing torques. As for the core, it is
considered to be fluid, and its motion is referred to as a Tisserand
frame. The dissipation in the boundary is modelled as being
produced by a torque proportional to the differential angular
velocity between the two layers, gathering both the viscous and
electromagnetic coupling, like in Sasao, Okubo & Saito (1980). In
this framework, we define a suitable set of canonical variables that
generalizes the Andoyer one used by Kinoshita (1977), and then
w
E-mail: [email protected] (JG); [email protected] (JMF)
q 2001 RAS
apply Hori's perturbation methods (Hori 1966), after having
included the dissipation torques in the so-called `unperturbed'
solution. The process leads to analytical nutation series, depending
on several constant coefficients. The numerical values of these
coefficients cannot be computed from any well-established geophysical theory with accuracy enough to provide accurate values
for the nutation. Think for instance of the large error in the
computation of the dynamical ellipticity HD following preliminary
earth model (PREM) (Dziewonski & Anderson 1981) and 1066A
(Gilbert & Dziewonski 1975) models, pointed out by Dehant &
Capitaine (1997). The values of the parameters are thus obtained
by means of a generalized least-squares adjustment (Getino, MartõÂn
& Farto 1999) taking the IERS 96 series as a basic reference, so that
we could afford the computational overhead more easily, as
opposed to using a large set of observational data. The analytical
nutation series are specialized for those values of the relevant
parameters, producing a nutation series that has been named GF99-1.
Notice that their amplitudes are slightly different from the ones
published in the aforementioned Letter, because of a minor enhancement of the algorithms in the adjustment procedure, and inclusion
of the rate of general precession in longitude (see Section 8).
In the writing of this paper we have taken advantage of the
previous work done by the authors, where simpler Earth models
were worked out in detail to investigate nutations. Namely, the
mantle elasticity was extensively treated in Getino & FerraÂndiz
(1990, 1991a,b, 1994, 1995a) and Getino (1993), all of which used
Hamiltonian formalism; a Poincare model with a rigid mantle was
considered in Getino (1995a,b) without dissipation, and then in
Getino & FerraÂndiz (1997, 2000) with dissipation. Referring to
these papers, we have considerably shortened the mathematical
derivations and we expect that the main features of the theory are
easy to follow. Let us remark, for instance, that we do not need to
rely on any rigid Earth nutation series available in advance.
Instead, we arrive at a Hamiltonian function that gathers all the
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J. Getino and J. M. FerraÂndiz
contributions of the Earth components and dissipations considered
in our basic model. By specializing the parameters in a suitable
way, we can obtain the different particular, more restrictive
models previously studied following our approach, including the
rigid case of Kinoshita (1977).
Finally, let us point out that the study of other non-negligible
effects not included in this theory is currently under development.
For instance, the free nutations of a three-layer model, including a
solid inner core (SIC), have been worked out by Escapa et al.
(1999). Preliminary nutation series regarding the SIC effect were
presented at the 1998 JourneÂes (Getino & FerraÂndiz 1998). Short
periodic nutation terms have been also treated in the Hamiltonian
way, and the contributions coming from the triaxiality of the Earth
presented at the 1999 European Geological Society (EGS)
(FerraÂndiz et al. 1999). We also pursue work on oceanic and
atmospheric corrections, and we hope to be able to release an
extension of this theory, also self-consistent and more complete
and accurate, in the near future.
Note that s and s c are small quantities, of the order of 1026 rad;
in our first-order approximation we will neglect second-order
terms in these variables.
2.1
The effect of the elasticity of the mantle on the Earth rotation is
studied in detail in Getino & FerraÂndiz (1995a), which will be
referred to hereafter as Paper I. In this paper, the corresponding
deformation of the Earth is produced by two kind of perturbations:
tidal deformation, caused by the gravitational attraction of external
bodies (Moon and Sun), and rotational deformation, caused by the
rotation of the elastic body itself. As the rotational deformation
has only a secular effect, and here we are mainly interested in the
nutations, we limit ourself in this paper to the study of the tidal
deformation, which is produced by the perturbing tidal potential
(by unit mass)
Wt ˆ
2 T H E T W O - L AY E R S E A R T H M O D E L :
KINETIC ENERGY
The establishment of the problem follows similar guidelines to the
rigid mantle±liquid core model studied in Getino (1995a, 1995b)
and Getino & FerraÂndiz (1997), with a basic difference: the mantle
is no longer a rigid body, but an elastic one. Then, let OXYZ be a
non-rotating inertial reference frame, and Oxyz the frame of
principal axes of the Earth, rotating with an angular velocity v
with respect to the inertial frame. For the core we take a fixed
frame Oxc yc zc rotating with an angular velocity v c with respect to
OXYZ. Thus, we can write vc ˆ v ‡ dv; where dv is the angular
velocity of the core with respect to the mantle. The corresponding
tensors of inertia are Pm for the mantle, Pc for the core and
P ˆ Pm ‡ Pc for the total Earth. As usual, the field of velocities
of the liquid core is assumed to be composed of a dominant
uniform rotation and a small residual velocity due to the nonsphericity (Sasao et al. 1980). With this, if L, Lm and Lc are the
angular momenta of the total Earth, the mantle and the core
respectively, they satisfy
L ˆ Lm ‡ Lc ˆ Pm v ‡ Pc …v ‡ dv† ˆ Pv ‡ Pc dv:
…1†
Thus, it is known that the relative angular momentum of the small
residual velocity due to the effects of non-sphericity can be made
zero with an appropriate definition of the core rotation (Moritz
1982), or by taking the Tisserand axes as the core fixed frame
(Moritz 1984), as detailed in Getino (1995a). With these considerations, the kinetic energy is written as
1 t 21
T ˆ 12 …L 2 Lc †t P21
m …L 2 Lc † ‡ 2 Lc Pc Lc :
…2†
This expression is canonically formulated by means of a set of
canonical variables, l , m , n , L, M, N for the total Earth, and l c,
m c, n c, Lc, Mc, Nc for the core, with the help of the auxiliary
angles s , I, s c, Ic, as described in detail in Getino (1995b). Concretely, the angular momenta L and Lc are expressed in the form
1
1
0
0
K sin n
K c sin nc
C
C
B
B
C
C
B
…3†
LˆB
@ K cos n A; Lc ˆ @ 2K c cos nc A;
N ˆ M cos s
N c ˆ M c cos sc
where
K ˆ M sin s;
K c ˆ M c sin sc :
…4†
Tidal deformation and tensors of inertia
Gm* 2
r P2 …cos S†
r*3
with cos S ˆ
r r*
;
r r*
…5†
where G is the gravitational constant, r is the vector from the
origin to the point within the Earth where the potential is
evaluated, r is its modulus, and m*, r* and r* are the mass, the
vector from the origin (centre of the Earth) to the external
perturbing body (Moon, Sun), and its modulus, respectively. P2 is
the Legendre polynomial of second degree.
As a result of this perturbing potential, the elastic mantle, and
then (from the hypothesis of continuity) the fluid core, are
deformed, and the corresponding tensors of inertia are no longer
constants, as in the rigid case, but time-variable functions.
According to Paper I, the tensor of inertia of mantle and core
can be broken down into
Pm ˆ P0m ‡ Ptm ;
Pc ˆ P0c ‡ Ptc ;
…6†
where P0m and P0c are the constant tensors corresponding to the
rigid case (in the absence of deformation), which in the mantle
system are expressed by (Getino 1995b)
1
0
0
Am 0
C
B
C
P0m ˆ B
@ 0 Am 0 A;
0
0
Ac
B
P0c ˆ B
@ 0
0
0
0
Ac
0
Cm
0
1
C
0 C
A;
Cc
…7†
and Ptm and Ptc are the increments arising from the tidal
deformation. The elements tij of these tensors are function of time
through the positional coordinates of the perturbing bodies (Moon,
Sun), and their explicit expressions can be found in Paper I.
Nevertheless, in this paper we consider only the elements that
have influence in nutation, which are t13 and t23 in agreement with
Moritz (1982) and Sasao et al. (1980). Then, following Paper I,
these tensors can be written as
1
0
0
0 t13
C
B
Ptm ˆ Dtm B
0 t23 C
A;
@ 0
t13 t23 0
q 2001 RAS, MNRAS 322, 785±799
Forced nutations of a two-layer Earth model
0
0
B
t
tB
Pc ˆ Dc @ 0
t13
0
t13
1
0
C
t23 C
A;
t23
0
two terms
…8†
where the terms t13 and t23, depending on the position of the
perturbing body, are given by
3
a*
P12 …sin d† cos a;
t13 ˆ 22
r*
3
a*
t23 ˆ 22
P12 …sin d† sin a;
…9†
r*
a and d are the longitude and latitude of the perturbing body
referred to the mantle frame, a* is the semi-major axis of its orbit,
and P12 is the associated Legendre function. On the other hand, we
have the constant coefficients
Dtm ˆ
Gm* 2p
Im;
3
a* 15
Dtc ˆ
Gm* 2p
Ic;
3
a* 15
…10†
where Im and Ic depend respectively on the internal structure of the
mantle and core (Paper I and MartõÂn & Getino 1998). According
to these papers, these coefficients are of the order of 1036 (in c.g.s.
units).
2.2
Kinetic energy
From equations (2), (3), (6), (7) and (8) we can get the canonical
expression of the kinetic energy. The only point of this is to
compute the inverse of the tensors of the inertia. As the order of
magnitude of the ratio Dt =C is .1028, we can perform the
expansion (see Paper I)
…Pm;c †21 . …P0m;c †21 2 …P0m;c †21 Ptm;c …P0m;c †21 :
…11†
V ˆ V 0 ‡ V t;
T ˆ T 0 ‡ T t;
…12†
where the first term is the kinetic energy corresponding to the
rigid case
1
A
KK c
1
K 2 ‡ K 2c ‡
cos…n ‡ nc † ‡
T0 ˆ
Am
2Am
Ac
2Cm
C 2
…13†
N 2 2 2NN c ‡
Nc ;
Cc
with A ˆ Am ‡ Ac and C ˆ Cm ‡ Cc being the principal moments
of the total Earth. Obviously, T0 is the same expression as obtained
in Getino (1995b) for the rigid mantle±liquid core Earth model.
On the other hand, the term Tt represents the increment of the
kinetic energy due to the tidal deformation, and is expressed as
N 2 Nc t
D ‰K c …t13 sin nc 2 t23 cos nc †
Am C m m
V0
. 1028 ;
T0
Nc t
D K c …t13 sin nc 2 t23 cos nc †:
Ac Cc c
…16†
the term Vt will be neglected in this paper. Hence the potential
energy will be
3
Gm*
a*
V . V0 ˆ
…C
2
A†
P2 …sin d†:
…17†
3
r*
a*
3.1 Development of spherical functions
Now we need to transform the spherical functions which appear in
the perturbed terms of the kinetic energy and of the potential.
According to Paper I, and neglecting second-order terms, these
functions, expressed in terms of the canonical variables, simplify
to
3
X
a*
P2 …sin d† . 3 Bi cos Qi
r*
i
XX
2 3 sin s
Ci …t† cos…m 2 tQi †;
a*
r*
a*
r*
3
3
tˆ^1 i
P12 …sin d† cos a . 3
P12 …sin d† sin a . 3
XX
Ci …t† sin…m ‡ n 2 tQi †;
…18†
tˆ^1 i
XX
Ci …t† cos…m ‡ n 2 tQi †;
tˆ^1 i
where the functions Bi and Ci(t ) are
Bi ˆ 2 16 …3 cos2 I 2 1†A0i 2 12 sin 2I A1i 2 14 sin2 I A2i ;
Ci …t† ˆ 2 14 sin 2I A0i ‡ 4t sin I…1 ‡ t cos I†A2i
‡ 12 …1 ‡ t cos I†…21 ‡ 2t cos I†A1i :
…19†
Aji
The numerical values of the coefficients are given in Kinoshita
(1977), and updated in Kinoshita & Souchay (1990). As for the
argument Qi, we have
Qi ˆ m1 lM ‡ m2 lS ‡ m3 F ‡ m4 D ‡ m5 V;
with i ˆ …m1 ; m2 ; m3 ; m4 ; m5 †;
F ˆ lM ‡ gS ;
D ˆ lM ‡ gS ‡ hM 2 lS 2 gS 2 hS ;
where l, g and h are the Delaunay variables for the Moon (M) and
Sun (S).
…14†
3 POT ENTI AL ENERGY AND SPHERI CAL
FUNCTIONS
Following Paper I, the potential energy can also be separated into
q 2001 RAS, MNRAS 322, 785±799
Vt
. 10213 ;
T0
V ˆ h M 2 l;
2 K…t13 sin n ‡ t23 cos n†Š
2
…15†
where the term V0 corresponds to the rigid case (for an axially
symmetric Earth model), and Vt is the additional potential due to
the redistribution of mass caused by the tidal deformation. As the
orders of magnitude of these terms are
Then the final expression of the kinetic energy can be expressed as
Tt ˆ
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4
D I S S I PAT I O N I N T H E C M B
The dissipation in the core±mantle boundary (CMB) is also
included in our model. The problem has been undertaken in
Getino & FerraÂndiz (1995b, 1997), where the effect of the
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J. Getino and J. M. FerraÂndiz
dissipation on the free nutations is studied. As described in those
papers, following Sasao et al. (1980), the torque of the dissipative
forces including electromagnetic coupling and the effects of the
viscosity is expressed as
0
2R dv1 ‡ R 0 dv2
…20†
R
A
sin sc K cos…n ‡ nc † ‡ K c
Am
Ac
R*
C
R0
cos sc N 2
Nc 2
sin sc K sin…n ‡ nc †;
‡
Am
Cm
Cc
R*
C
Qnc ˆ
N2
Nc ;
Cm
Cc
R K
R0 K
A
QN c ˆ 2
: …21†
sin…n ‡ nc † ‡
cos…n ‡ nc † ‡
Am K c
Am K c
Ac
Qmc ˆ 2
According to the previous sections, the Hamiltonian of the system
(at the first order) is written as
…22†
where the terms T0, Tt and V0 are given respectively by equations
(13), (14) and (17), with the help of equation (18). In this
Hamiltonian we must include the effect of the dissipation, as
described below. Now we perform a first order analytical
integration of this Hamiltonian by using Hori's perturbation
method (1966), following the same procedure as in Kinoshita
(1977) for the rigid Earth, and as in Paper I for a simplified nonrigid case. This procedure is briefly described here.
First of all, the Hamiltonian (equation 22) is separated into an
unperturbed part, H0, corresponding to the free motion, and a
perturbed part H1, for the forced perturbations, in the form
H0 ˆ T 0
H1 ˆ T t ‡ V 0
:
A *2
K
Ac c
K*K c*
cos…n* ‡ nc*†
Am
1
C
N*2 2 2N*N c* ‡
N c*2 ;
‡
2Cm
Cc
K *2 ‡
‡
with k 00 ˆ 3
Gm*
…C 2 A†;
a*3
…25†
where the coefficient B0 ˆ B…00000† corresponds to the only
secular contribution of the expansion of P2(sin d ) (see Paper I).
Note that we have used asterisks to indicate the new variables
resulting from the canonical transformation. However, in the
following equations these asterisks will be omitted for the sake
of simplicity.
Finally, the generating function of the transformation is
…
…
W ˆ …H 1 2 H 1*† dt ˆ H 1 per dt;
…26†
where this integral is performed along the solution of the
unperturbed part. This solution, corresponding to the free motion
problem, is described in the next section.
5.1
Unperturbed solutions
The free motion problem is solved by means of the corresponding
equations of motion. In the presence of generalized forces, due to
the dissipative effects, these equations are of the form
5 H A M I LT O N I A N A N D F I R S T- O R D E R
I N T E G R AT I O N
H ˆ T 0 ‡ T t ‡ V 0;
1
2Am
H 1* ˆ V 0 sec ˆ k 00 B0
where R, R 0 and R* are coupling constants.
The dissipative torque is canonically formulated by means of
generalized forces, whose construction can be found in detail in
Getino & FerraÂndiz (1997). Here we limit ourselves to give the
expression of the necessary generalized forces for our approach.
These forces are
H ˆ H0 ‡ H1 !
H 0* ˆ
1
C
B
0
C
tc ˆ 2tm ˆ B
@ 2R dv2 2 R dv1 A;
2R*dv3
(
H 1* ˆ H 1 sec : According to Paper I, we have finally
…23†
q_ ˆ
­T 0
2 Qp ;
­p
p_ ˆ 2
­T 0
‡ Qq :
­q
…27†
Note that, since the effect of the deformations is included in the
perturbation, the system of the free motion is the same as in the
rigid mantle±liquid core model, solved in Getino & FerraÂndiz
(1997), where the readers can found a complete explanation. Here
we just summarize the main consequences.
First of all, for the third components of the angular velocities
we have that v3 ˆ constant ˆ V; and dv3 ˆ 0: Then, from
equations (1) and (3) we can deduce
N ˆ CV;
N c ˆ Cc V:
…28†
With these considerations, the free frequencies s 1 (Chandler
wobble, CW) and s 2 (free core nutation, FCN), characterizing the
free motion, are of the form
s1 ˆ m1 ;
s2 ˆ m2 ‡ id;
…29†
with
Then, we carry out a canonical transformation of the initial
Hamiltonian H into a new one, H*, which is easier to
integrate:
C2A
;
Am
A Cc 2 Ac
A
m2 ˆ 2V 1 ‡
‡ G0 ;
Am Ac
Ac
H ˆ H 0 ‡ H 1 ! H* ˆ H 0* ‡ H 1*;
dˆV
m1 ˆ V
…24†
by means of a generating function W. In the new Hamiltonian,
the unperturbed part is the same as in the old one, H 0* ˆ H 0 ;
including the effect of the dissipation, and for the new
disturbing term we take the secular part of H1, that is to say,
A
G;
Ac
…30†
where we have introduced the dimensionless constants of dissipation
Gˆ
R
;
VAm
G0 ˆ
R0
:
VAm
…31†
q 2001 RAS, MNRAS 322, 785±799
Forced nutations of a two-layer Earth model
These free frequencies are provided by the system of
equations
!
!
u_
u
ˆ iR
:
…32†
v_
v
In this system we have defined the variables
u ˆ K…sin n ‡ i cos n†; v ˆ K c …sin nc 2 i cos nc †;
and the matrix R is given by
0
1
r2
r1
Rˆ@
A A;
r 3 2 iVG r 4 ‡ iVG
Ac
…33†
…34†
whose coefficients are
Ac
A
r1 ˆ V
‡e
;
Am
Am
r 2 ˆ 2V
Cc 2 Ac
:
ec ˆ
Ac
…35†
…36†
…37†
6 E X P R E S S I O N O F T H E G E N E R AT I N G
FUNCTION
It has been said that the computation of the generating function (at
first order) for a non-rigid Earth model is not a trivial task (as it is
in the rigid case). A first attempt was used in Getino (1995b), by
means of a quite laborious method involving successive integrations by parts. Later, a more sophisticated technique was applied
in Getino & FerraÂndiz (2000), based on a complex formulation of
the problem. In this paper we present a new procedure, simpler
and easier to apply, which is in a sense a mixture of the two former
ones.
Auxiliary integrals
Let us begin by computing two integrals which will be very
useful in the development of this technique. These integrals are
q 2001 RAS, MNRAS 322, 785±799
Let P and Q be new integrals whose imaginary parts are
respectively I1 and I2, that is, I 1 ˆ Im{P}; I 2 ˆ Im{Q}; then
defined by
…
P ˆ iM sin se2in eih dt;
…
Q ˆ iM c sin sc einc eih dt;
u ˆ iM sin se2in ;
Once the solution of the unperturbed motion has been obtained,
we can construct the generating function according to equation
(26). However, in Getino (1995b) it was pointed out that the
presence of the liquid core makes the derivation of W much more
complicated than in the rigid case. The difficulties increase when
the effects of dissipation and tidal deformation are added.
Therefore we have decided to leave the calculation of W to the
next section.
6.1
…38†
…39†
…40†
On the other hand, with the help of equation (4) the variables u
and v in equation (33) can be expressed as
Another useful result, which will be used in the computation of
the generating function, is that, from the equations of motion
(equation 27), it is straightforward to derive at first order (Getino
1995b; Getino & FerraÂndiz 1997)
m_ ‡ n_ ˆ V:
…
I 2 ˆ M c sin sc cos…m ‡ n ‡ nc 2 tQi † dt:
h ˆ m ‡ n 2 tQ i :
where we have introduced the ellipticities
C2A
eˆ
;
A
written as
…
I 1 ˆ M sin s cos…m 2 tQi † dt;
where we have introduced the notation
A
…1 ‡ e†;
Am
Ac
0
…1 ‡ ec † ‡ G ;
r3 ˆ V
Am
A
A
r 4 ˆ 2V
…1 ‡ ec † ‡ G 0 ;
Am
Ac
789
v ˆ 2iM c sin sc einc :
…41†
From equations (39) and (41) the new integrals are written as
…
…
P ˆ ueih dt; Q ˆ 2 veih dt:
…42†
Now let us pay attention to the system of equations for the free
motion given by equation (32). This system can be transformed
into
!
„ ih !
„ ih !
P
_ dt
ue
ue dt
ˆ
iR
ˆ
iR
:
…43†
„ ih
„ ih
2Q
_ dt
ve
ve dt
The left hand side of equation (43) can be integrated by parts, and
we obtain
!
„ ih !
„ ih !
_ dt
ue
ue dt
ueih
2 inh „
ˆ
;
…44†
„ ih
ih
_ dt
ve
veih dt
ve
where, taking into account equation (37),
nh ˆ
dh
ˆ m_ ‡ n_ 2 tQ_i ˆ V 2 tni :
dt
…45†
From equations (43) and (44) we get
P
2Q
!
…R ‡ nh 1†a
ˆ
jR ‡ nh 1j
2iueih
2iveih
!
;
…46†
where the superscript a stands for the adjoint matrix, and 1 is the
unit matrix. As the eigenvalues of matrix R are precisely the free
frequencies s 1 and s 2 given by equations (29) and (30), taking
into account the expression of matrix R in equations (34) and (35),
we have that
jR ‡ nh 1j ˆ …nh ‡ s1 †…nh ‡ s2 †;
! 0
A
a
a
nh ‡ r 4 ‡ iGV
11
12
Ac
ˆ@
…R ‡ nh 1†a ˆ
a21 a22
2r3 ‡ iVG
2r2
1
A;
nh ‡ r 1
…47†
then, taking into account equation (41), the integrals P and Q are
J. Getino and J. M. FerraÂndiz
790
given by
P ˆ M sin sei…h2n†
F a2 ˆ
a11
2a12
‡ M c sin sc ei…h‡nc †
;
f 1 …f 2 ‡ i d†
f 1 …f 2 ‡ id†
2a21
a22
Q ˆ M sin sei…h2n†
‡ M c sin sc ei…h‡nc †
;
f 1 …f 2 ‡ id†
f 1 …f 2 ‡ id†
F b2 ˆ 2VG
…48†
with the help of the notation
f 2 ˆ nh ‡ m2 ˆ m2 ‡ V 2 tni :
…49†
Finally, taking the imaginary parts in equation (48), expressions
of the integrals I1 and I2 are obtained in the form
I 1 ˆ M sin s‰F a1 sin…h 2 n† ‡ F b1 cos…h 2 n†Š
‡
sin…h ‡ nc † ‡
F b2
cos…h ‡ nc †Š;
‡ M c sin sc ‰Ga2 sin…h ‡ nc † ‡ Gb2 cosh ‡ nc Š;
Ga1 ˆ
Gb1
…50†
…r 4 ‡ nh † f 2
;
f 1 … f 22 ‡ d 2 †
A f 2 2 r 4 2 nh
;
Ac f 1 … f 22 ‡ d 2 †
A r 1 ‡ nh
:
ˆ 2VG
Ac f 1 … f 22 ‡ d 2 †
A f 2 2 r 4 2 nh
;
F b1 ˆ VG
Ac
f 1 f 22
Ga1 ˆ
r3
;
f1 f2
Gb1 ˆ 2VG
A
r3
Ac
;
f 1 f 22
f2 ‡
…53†
Thus, the generating function can be divided into two parts
…
…
W ˆ H 1per dt ˆ …V 0per ‡ T t † dt ˆ W 0 ‡ W t ;
…54†
X
k0 X
Bi cos Qi 2 0
C i …t†‰M sin s cos…m 2 tQi †Š;
M tˆ^1
i±0
where the secular contribution corresponding to i ˆ 0 (see equation
25) has been excluded. By means of integral I1 (equations 38 and
50), W0 can be easily expressed as
W 0 ˆ W 00 ‡ W 01 ‡ W 02 ;
with
W 00 ˆ k 00
…51†
Notice that, d being different from zero, no exact resonance can
occur in those coefficients, as was explained in more detail in
Getino & FerraÂndiz (2000). Moreover, the small value of
parameter d for the Earth in comparison with the value of f2 for
any of the disturbing frequencies ni allows us to neglect d2, which
leads to the next simplified expressions
r 4 ‡ nh
F a1 ˆ
;
f1 f2
Generating function
…55†
A
r2
;
Ac f 1 … f 22 ‡ d 2 †
Gb2
6.2
V 0per ˆ k 00
r2 f 2
;
f 1 … f 22 ‡ d 2 †
…r 1 ‡ nh † f 2
ˆ
;
f 1 … f 22 ‡ d 2 †
which replace equation (51) to compute the forced nutations.
From equations (17) and (18), the periodic part of V0 is
A
r3
Ac
;
ˆ 2VG
f 1 … f 22 ‡ d 2 †
Ga2
…52†
„
6.2.1 Expression of W 0 ˆ V 0 per dt
r3 f 2
;
f 1 … f 22 ‡ d 2 †
F b2 ˆ 2VG
A r 1 ‡ nh
;
Ac f 1 f 22
where W0 is the contribution of the rigid part and Wt is due to the
tidal deformation. Let us compute each of these parts.
f2 ‡
F a2 ˆ
r 1 ‡ nh
;
f 1f 2
H 1per ˆ H 1 2 H 1 sec ˆ V 0 per ‡ T t :
where the following functions have been defined:
F b1 ˆ VG
A r2
;
Ac f 1 f 22
Following Section 5, the generating function is obtained by
integrating the periodic part of H1, that is
I 2 ˆ M sin s‰Ga1 sin…h 2 n† ‡ Gb1 cos…h 2 n†Š
F a1 ˆ
Ga2 ˆ
Gb2 ˆ 2VG
f 1 ˆ nh ‡ m1 ˆ m1 ‡ V 2 tni ;
M c sin sc ‰F a2
r2
;
f1 f2
X Bi
i±0
ni
…56†
sin Qi ;
ni ˆ
dQi
dt
XX
W 01 ˆ 2 k 00 sin s
C i …t†
i±0 tˆ^1
‰F a1
W 02 ˆ 2 k 00
sin…m 2 tQi † ‡ F b1 cos…m 2 tQi †Š;
XX
Mc
sin sc
Ci …t† ‰F a2 sin…h ‡ nc †
M
i±0 tˆ^1
‡ F b2 cos…h ‡ nc †Š:
…57†
„
6.2.2 Expression of W t ˆ T t dt
First of all, let us develop the expression of Tt given by equation
(14). From equations (9) and (18) we can write
X
t13 sin nc 2 t23 cos nc ˆ 6 Ci …t† cos…h~ ‡ nc †;
i;t
X
t13 sin n ‡ t23 cos n ˆ 26 C i …t† cos…h~ 2 n†;
…58†
i;t
q 2001 RAS, MNRAS 322, 785±799
Forced nutations of a two-layer Earth model
~ i is the argument of the spherical functions
where h~ ˆ m~ ‡ n~ 2 tQ
in (18). We use the symbol Ä to point out that this argument
corresponds to the coordinates of the perturbing body responsible
for the tidal deformation. This remark is necessary because the
derivatives leading to the canonical equation must be taken with
respect to the coordinates of the perturbed bodies. An analysis of
the distinction between perturbed and perturbing bodies is
accomplished in Paper I.
By means of equations (28) and (58), the tidal kinetic energy
can be finally expressed as
X
~
Tt ˆ
Ci …t†‰km
t M sin s cos…h 2 n†
i;t
c
~
‡ …km
t 2 kt †M c sin sc cos…h ‡ nc †Š;
…59†
where the dimensionless coefficients km
t concerning the deformation of the mantle and kct concerning the deformation of the core
are given by
km
t ˆ
6Dtm V
;
Am
kct ˆ
6Dtc V
:
Ac
…60†
Now we can integrate the expression (59) of Tt with the help of
integrals I1, I2, and the corresponding generating function can be
finally written as
W t ˆ W t1 ‡ W t2 ;
…61†
with
X
W t1 ˆ M sin s Ci …t† ‰Da1 sin…h~ 2 n† ‡ Db1 cos…h~ 2 n†Š;
W t2 ˆ M c sin sc
X
Ci …t† ‰Da2 sin…h~ ‡ nc † ‡ Db2 cos…h~ ‡ nc †Š;
i;t
…62†
where the functions Dpi are of the form
p
p
c p
Dpi ˆ km
t …F i ‡ Gi † 2 k t Gi ;
7
i ˆ 1; 2; p ˆ a; b:
…63†
F O R C E D N U TAT I O N S
The periodic perturbations, or forced nutations, are obtained
through the generating function by means of the well-known
relationships
D…m; n; l† ˆ
­W
;
­…M; N; L†
D…M; N; L† ˆ
2­W
:
­…m; n; l†
…64†
Next we compute the periodic perturbations of the fundamental
planes, that is to say, the plane perpendicular to the angular
momentum vector (or Andoyer plane), and the plane perpendicular to the figure axis of the Earth (figure plane or equatorial
plane).
7.1
Nutations of the Andoyer plane
The longitude of the node and the inclination of this plane are
given respectively by l and I ˆ cos21 …L=M†: The nutations
corresponding to these variables, known as Poisson terms, are
obtained through the equations (Kinoshita 1977)
21 ­W
1
­W
­W
; DI ˆ
Dl ˆ
:
…65†
2 cos I
­m
M sin I ­I
M sin I ­l
Neglecting second-order terms in small parameters, the contribution
q 2001 RAS, MNRAS 322, 785±799
of these nutations comes from the term W00 of the generating
function, which corresponds to the rigid perturbation. Using the
angles c ˆ 2l and 1 ˆ 2I (Kinoshita 1977), the Poisson terms
up to the first order are as follows (Getino 1995b):
X 1 ­ Bi sin Qi ;
Dc ˆ k0
sin I ­I ni
i±0
D1 ˆ k0
X …m5 † Bi
i±0
sin I ni
cos Qi ;
…66†
with the coefficient
k0 ˆ
k 00
3Gm* C 2 A
. 3
:
M
a* V C
…67†
The expressions in equation (66) are the same as those of
Kinoshita (1977) for the rigid Earth (with the simplification
A ˆ B†. That is, the presence of the liquid core and the effects of
elasticity and dissipation have no influence on the nutations of the
Andoyer plane, which agrees with the fact that the motion of the
angular momentum axis does not depend on the internal
constitution of the Earth (Moritz & Mueller 1987).
7.2 Nutations of the figure plane
The longitude of the node, cf ˆ 2lf ; and the inclination 1f ˆ
2I f ; of this plane are given up to the first order by (Kinoshita
1977)
lf ˆ l ‡
i;t
791
s
sin m;
sin I
I f ˆ I ‡ s cos m:
…68†
Following Kinoshita, the periodic perturbations of the increments
lf 2 l and I f 2 I; called Oppolzer terms, are given up to the first
order by
1
1
­W ­W
­W
D…lf 2 l† ˆ
sin m
2
‡ s cos m
;
­n
­m
­s
sin I M sin s
1
­W ­W
­W
2 s sin m
:
…69†
cos m
2
D…I f 2 I† ˆ
­n
­m
­s
M sin s
Let us study the contribution to the Oppolzer terms of each part
that forms the generating function.
7.2.1 Contribution of W0
Neglecting second order terms, the contribution of W0 arises from
the term W01. After a little arrangement, we obtain
2k0 X
C i …t† ‰F a1 t sin Qi 2 F b1 cos Qi Š;
sin I i;tˆ^1
X
D0 …1f 2 1† ˆ 2k0
Ci …t† ‰F a1 cos Qi ‡ F b1 t sin Qi Š:
D0 …cf 2 c† ˆ
…70†
i;tˆ^1
These expressions differ substantially from those of Kinoshita
(1977) for the rigid case: on the one hand, there are in (70) inphase and out-of-phase terms. Out-of-phase terms come from the
function F b1 ; which, according to equation (51), is produced by
the dissipation in the CMB (through the coefficients G and d). On
the other hand, amplitudes of in-phase terms, given by F a1 ; are
not the same as in the rigid case. As the term W01 corresponds to a
rigid mantle±liquid core model, with dissipation in the CMB and
without deformation, formulae (70) show the contribution of these
effects to the nutation of the figure plane.
792
J. Getino and J. M. FerraÂndiz
7.2.2 Contribution of Wt
This term is due to the tidal deformation. Neglecting second-order
terms, from equations (62) and (69) we get that the contribution is
given by term Wt1 in the form
21 X
Dt …cf 2 c† ˆ
C i …t†‰Da1 sin…h~ 2 m 2 n†
sin I i;tˆ^1
‡ Db1 cos…h~ 2 m 2 n†Š;
X
Ci …t†‰2Da1 cos…h~ 2 m 2 n†
Dt …1f 2 1† ˆ 2
i;tˆ^1
‡ Db1 sin…h~ 2 m 2 n†Š:
…71†
Once the derivatives of the equation of perturbation have been
performed, we can identify
h~ ; h ˆ m ‡ n 2 tQi ;
…72†
but this is the occasion to consider the effect of the inelasticity, as
explained in the next paragraph.
7.2.3 Effect of the inelasticity
The inelasticity of the mantle gives rise to a phase lag angle e in
the arguments of the nutation, related to the time of delay Dt
(Lambeck 1974; Mignard 1978). To take into account this effect
we just replace h by hin, where
h~in ; hin ˆ m ‡ n 2 tQi ‡ e:
…73†
Here we consider a frequency dependence of the phase lag in the
form
ei ˆ …V 2 tni †Dt;
…74†
Dt being the time delay. Introducing equation (74) into the
formulae in (71), the contribution of inelasticity to the Oppolzer
terms is then given by
21 X
Dt …cf 2 c† ˆ
C i …t†…E1 t sin Qi 2 E2 cos Qi †;
sin I i;tˆ^1
X
Dt …1f 2 1† ˆ 2
Ci …t†…E1 cos Qi ‡ E2 t sin Qi †;
…75†
i;tˆ^1
where, taking into account equation (63), functions E1 and E2 are
written as
E1 ˆ
a
km
t ‰2cos ei …F 1
‡
Ga1 †
‡
sin ei …F b1
‡
Gb1 †Š
‡ kct …cos ei Ga1 2 sin ei Gb1 †;
b
b
a
a
E2 ˆ km
t ‰2cos ei …F 1 ‡ G1 † 2 sin ei …F 1 ‡ G1 †Š
‡ kct …cos ei Gb1 ‡ sin ei Ga1 †:
7.3
…76†
Final expressions
Gathering together formulae (66), (70) and (75), the nutation
series in longitude, Dc f, and obliquity, D1 f, of the figure plane can
be collected as follows
X
out
Dcf ˆ
…Lin
i sin Qi ‡ Li cos Qi †;
i±0
X
out
D1f ˆ
…Oin
i cos Qi ‡ Oi sin Qi †;
i±0
…77†
in
where the amplitudes for longitude, Lin
i ; and obliquity, Oi ; of inphase terms are given by
X
k0 ­ Bi
tC i …t†
Lin
2
…k0 F a1 ‡ E1 †i;t ;
i ˆ
sin I ­I ni
sin I
tˆ^1
Oin
i ˆ k0
X
…m5 † Bi
2
Ci …t†…k0 F a1 ‡ E1 †i;t ;
sin I ni tˆ^1
…78†
and Oout
for out-of-phase
and the corresponding amplitudes Lout
i
i
terms are
X C i …t†
…k0 F b1 ‡ E2 †i;t ;
Lout
i ˆ
sin I
tˆ^1
X
tCi …t†…k0 F b1 ‡ E2 †i;t :
…79†
Oout
i ˆ 2
tˆ^1
8
E VA L UAT I O N O F T H E N U TAT I O N S E R I E S
We proceed in this section to a numerical computation of the
forced nutations given by equations (78) and (79). These
expressions depend on a set of parameters characterizing the
Earth model used, through coefficients ri in equation (35)
and E1,2, and the
appearing in definition of functions F a;b
1
coefficients for the deformation, Dm;c
t ; as well as the coefficients
k0M and k0S of the perturbing potential. It is well known that
present Earth models (PREM, 1066A) do not provide the required
accuracy in order to compare the obtained theoretical results with
the observational data, so that before any comparison we have to
perform an adjustment of the said parameters (Mathews et al.
1991; Mathews & Shapiro 1995; Getino & FerraÂndiz 2000).
8.1
Basic Earth Parameters (BEP)
Analytical formulae of the forced nutations (78), (79) can be
expressed as functions of the following set of free parameters
c
BEP ˆ {PCW ; PFCN ; Ac =Am ; G; Dt; …k0 †M ; …km
t †M ; …kt †M ; k S=M };
where PCW and PFCN are the periods corresponding to the free
frequencies m1 and m2 (30) respectively, that is to say,
21
A 21
A
; PFCN ˆ
…ec 2 G 0 †
;
…80†
PCW ˆ e
Am
Am
Ac/Am is the ratio between principal moments of core and mantle,
G is the coefficient of the dissipation (31), Dt is the time delay due
c
to the inelasticity (74), …k0 †M ; …km
t †M and …k t †M are respectively the
coefficients of the perturbing potential (67), the deformation of
mantle and the deformation of core (60) corresponding to the
Moon, and kS/M is a Sun to Moon ratio defined by
kS=M ˆ
…k0 †S
…km †
…kc †
ˆ mt S ˆ ct S :
…k0 †M …kt †M …kt †M
…81†
Finally we have taken the value V ˆ 7:292 12 1025 s21 from
Mathews et al. (1991), and for the obliquity I appearing in
equation (19), we take 10 ˆ 2I ˆ 84 381:412 arcsec from IERS96
(McCarthy 1996).
Note that the coefficients ri in equation (35) are expressed in
functions of the BEP as
r 1 ˆ V‰…Ac =Am † ‡ …PCW †21 Š;
r 2 ˆ 2V‰1 ‡ …Ac =Am † ‡ …PCW †21 Š;
q 2001 RAS, MNRAS 322, 785±799
Forced nutations of a two-layer Earth model
r3 ˆ
8.2
Table 1. (upper section) Values of BEP; (lower section) Comparison of
BEP with other sources.
…Ac =Am †
V‰1 ‡ …Ac =Am † ‡ …PFCN †21 Š;
1 ‡ …Ac =Am †
r 4 ˆ 2V‰1 ‡ …Ac =Am † ‡ …PFCN †21 Š:
…82†
General precession in longitude
The rate of the general precession in longitude depends on the
coefficients (k0)M and (k0)S of the perturbing potential, so that we
have to take this fact into account in order to get consistent values
for these parameters, and we have to include the corresponding
equation in the set of observational data. According to Souchay &
Kinoshita (1996), the required equation, neglecting second-order
terms, is as follows:
p 0A ˆ ‰…k0 †M M 0 ‡ …k0 †S S0 Š cos 10 ;
Parameter
q 2001 RAS, MNRAS 322, 785±799
Source
400.7 d
432.94 d
0.123 234
4.1 1026
4.67 min
7567.870 647 arcsec Jcy21
1906.852 345 arcsec Jcy21
4514.468 392 arcsec Jcy21
0.459 127
PFCN
432.94
429.2
433.8
This paper
Mathews & Shapiro (1995)
Dehant & Defraigne (1997)
Ac/Am
0.123 234
0.1284
0.1292
This paper
PREM
1066A
(k0)M
7567.8706 arcsec
7567.7216 arcsec
7567.8292 arcsec
7567.3199 arcsec
7567.3447 arcsec
7567.3461 arcsec
7567.3057 arcsec
This paper
Kinoshita (1977)
Seidelmann (1982)
Souchay et al. (1999)
Roosbeek & Dehant (1997)
Bretagnon et al. (1997)
Williams (1994)
Q
13 545
31 000
15 326
This paper
Defraigne, Dehant & Hinderer (1995)
Mathews & Shapiro (1995)
Dtm
4.75 1036
6.62 1036
This paper
MartõÂn & Getino (1998)
Dtc
1.45 1036
1.51 1036
This paper
MartõÂn & Getino 1998
Adjustment of parameters
With the previous considerations, the analytical series given by
equations (78) and (79), which provide nutations in longitude and
obliquity for in-phase and out-of-phase terms, are used to solve for
the parameters of BEP in order to fit the IERS96 series, which are
considered as observational data. Second-order effects, namely
orbital coupling and variation of nV (Kinoshita & Souchay 1990;
reviewed in Souchay et al. 1999), and geodesic nutation
(Fukushima 1991), have been added. Equation (83) for the rate
of general precession in longitude is also considered. Thus, as
mentioned in Section 1, values of BEP and the amplitudes of
nutation are slightly different from the ones obtained in Getino &
FerraÂndiz (1999).
The difficulty of the adjustment lies in the fact that the
dependence of the free parameters is not linear. To avoid this
problem we have used the method developed by Getino et al.
(1999). In the first attempt we tried to fit the nine parameters of
BEP. However, although the amplitudes of the nutations were in
good agreement with the IERS96 data, this process leads to a
physically meaningless value of PCW. Something similar appears
in Mathews & Shapiro (1995). For this reason we have decided to
fix this parameter to a value of 400.7 d (Mathews et al. 1991). In
fact, we have tested different values of PCW, but the evaluations of
the forced nutations show a negligible variation for reasonable
values of this parameter. The obtained values for the parameters
are listed in Table 1 (upper section).
Without entering into an astronomical or geophysical discussion
of this solution, let us remark that the values of the parameters are
close to the ones given by different authors or deduced from wellknown Earth models. A few of them are displayed in Table 1
(lower section).
Notice that, in order to compare the effects of dissipation, we
have introduced the so-called quality factor Q ˆ sRe =2sIm ;
which, referring to the FCN frequency and according to equation
(29), is given by jQj ˆ m2 =2d:
The larger relative discrepancy appears in the coefficient Dtm ;
concerning the elastic deformation of the mantle. We do not know
the exact reasons for it, but it might be related, at least, to the
Value
PCW
PFCN
Ac/Am
G
Dt
(k0)M
km
t †M
kct †M
kS/M
…83†
where M0 and S0 are coefficients of the lunisolar potential, with
values M 0 ˆ 0:496 3033 rad and S0 ˆ 0:500 2101 rad (Kinoshita
1977), and the part p 0 A of the general precession pA which must be
retained as the linear component in equation (83) is
p 0A ˆ 5040:6445 arcsec Jcy21 .
8.3
793
difficulty of deriving this coefficient from rheological parameters
of the Earth models, and the absence of oceans in our present
model.
8.4 Final results: GF99-1 nutation series
With the set of BEPs given in Table 1, we can evaluate the
amplitudes of the nutations. Numerical values are collected in the
GF99-1 nutation series shown in Table 2, in which deviations with
respect to IERS 96 are labelled as `errors'. Terms corresponding to
the often-called `geophysical nutations' have been displayed in
bold characters, to make the localization easier. As for in-phase
terms, almost all errors are less than 0.06 mas. Only two terms in
obliquity exceed that limit, being less than 0.12 mas, corresponding to the periods of 365.26 and 182.62 d. In longitude, the
exceptions appear for periods of 182.62 and 121.75 d, with errors
of less than 0.13 mas, and for 6798.36 d, with errors of less than
0.1 mas. As for out-of-phase terms, four terms in obliquity and
seven terms in longitude show errors greater than 0.1 mas; the
largest happen at the 13.66-d period, being less than 0.7 mas in
longitude and less than 0.3 mas in obliquity. Therefore, the
maximum magnitude of the errors with respect to the IERS 96 is
less than half of the corresponding maximum error in Schastok's
(1997) tables. In his series, out-of-phase errors in longitude reach
1.4 mas at the 6798.36- and 365.26-d periods.
On the other hand, from equations (78), (79) and (76), the
Oppolzer terms in longitude and obliquity can be expressed as a
794
J. Getino and J. M. FerraÂndiz
Table 2. GF99±1 Nutation Series; comparison with IERS conventions 1996 …unit ˆ mas†:
Period
26798.36
23399.18
23232.85
1615.74
1305.47
21095.17
2943.22
411.78
409.23
2388.27
386.00
365.26
365.22
2346.64
346.60
2329.79
212.32
205.89
2199.84
182.63
182.62
177.84
173.31
2169.00
131.67
121.75
119.61
117.54
91.31
35.03
234.85
234.67
32.76
232.61
31.96
231.81
231.66
29.80
29.53
229.26
27.78
27.67
27.55
227.44
227.33
27.32
27.09
26.98
226.88
25.62
25.42
23.94
23.86
23.77
22.47
215.91
215.39
14.80
14.77
214.73
14.63
14.19
14.19
14.16
13.81
13.78
213.75
13.66
13.63
13.61
Longitude
217206.184140
207.458877
2.332258
2.314213
4.592479
1.103214
.065830
2.440934
.073220
.100175
21.407335
147.574509
21.638617
21.271023
2.481350
2.088189
.405359
4.773408
2.575016
1.677202
21316.891443
12.824382
22.179951
.086066
.101242
251.808393
.363925
2.066682
21.586487
.074500
2.736946
2.058601
2.057783
2.198832
1.517024
215.698845
21.286221
.473491
2.403411
.095142
2.197623
6.314554
71.106876
25.795048
.139619
.056797
12.353064
2.044665
.405542
2.339895
2.073333
2.860831
.580684
2.076488
.129415
2.128090
2.435808
2.630791
6.339168
2.492853
.138422
2.067430
2.716260
2.067179
.218054
2.923345
2.228821
2227.777465
238.753729
2.585230
In-phase terms
Error
Obliquity
.092860
.029877
2.005258
2.003213
.002479
.001214
2.005169
2.035934
.004220
2.000824
2.001335
.036509
.055617
2.002023
2.004350
.001810
.000359
2.000591
.003983
.006202
.122557
.004382
2.001951
2.000933
2.013757
2.121393
.002925
2.000682
2.008487
2.000499
2.001946
.002398
.000216
.000167
.000024
.000154
2.000221
.002491
.018588
.001142
2.000623
.000554
2.011123
.001951
.000619
2.000202
.000064
.000665
.001542
2.001895
.000666
2.002168
.000684
.000511
.001415
2.000090
2.000808
.000208
.003168
.000146
2.000577
2.000430
2.002260
2.001179
.000054
.000345
2.000821
2.057465
2.001729
.000230
9205.382459
289.759664
2.000920
.135887
22.424761
.010527
2.035799
2.059328
2.034318
2.048777
.854512
7.271946
29.612759
.644085
.273942
.002771
2.219992
.047594
.302801
.013312
572.947812
26.903781
2.015626
2.045491
.000404
22.493688
2.195216
2.000196
.688081
2.039365
2.005236
.031632
.000447
.107435
2.800466
2.126624
.695332
2.004163
.003589
.000856
.085332
23.324039
2.690916
3.140949
2.060797
2.000557
25.333276
21.075856
.004062
.003611
.031650
21.234315
2.304851
.000888
2.055816
2.002334
2.008232
.327447
2.125134
.271520
2.059521
.001389
.307911
.034828
2.112963
2.062124
.126379
97.886220
20.072665
2.055665
Error
Longitude
.026459
2.012664
.000079
.004887
2.000761
.000527
.002200
2.004328
2.001318
.000222
2.002487
2.116053
2.021759
.002085
.002942
2.000228
.000007
2.000405
2.001198
2.000687
2.110187
2.006781
2.000626
2.001491
2.000595
.053688
2.001216
2.000196
.003081
2.000365
2.000236
2.000367
.000447
.000435
.000533
.000375
.001332
2.000163
.000589
.000856
.000332
2.001039
2.003916
2.000050
2.000797
2.000557
.000723
.000143
.000062
2.000388
.000650
.000684
.002148
.000888
2.000816
2.003334
.000767
.000447
2.000134
2.000479
.000478
.001389
.000911
2.000171
.001036
2.000124
.000379
.022220
2.003334
.000334
3.234908
2.069486
.000106
2.000133
.001749
2.001355
2.000197
2.046194
2.000758
2.004175
.010773
1.246573
.004897
.010140
.000144
.000223
.000654
2.000532
2.001025
2.000245
21.750250
.021819
.000329
.000165
2.000013
2.073708
.000650
.000007
2.002339
.000145
.000063
2.000110
2.000005
2.000371
.002989
.001637
2.002400
.000055
2.000048
2.000011
2.000325
.012561
.009560
2.010698
.000204
.000007
.020416
.004074
2.000056
2.000051
2.000121
.004786
.001166
2.000012
.000217
.000036
.000127
2.001314
.001942
2.000849
.000244
2.000021
2.001272
2.000140
.000457
.000969
2.000390
2.406622
2.081307
.000869
Out-of-phase terms
Error
Obliquity
2.410091
.001513
.000106
2.000133
.000749
2.000355
2.000197
2.011194
.000241
2.001175
.002773
.125573
2.001102
.004140
.000144
.000223
2.000345
.001467
2.000025
.000754
2.350250
.003819
2.000670
.000165
2.000013
2.019708
2.000349
.000007
2.000339
.000145
.001063
2.000110
2.000005
.000628
.001989
.019637
.001599
.001055
2.001048
2.000011
2.000325
.009561
.103560
.008301
.000204
.000007
.018416
.003074
2.001056
2.000051
2.000121
.004786
.001166
2.000012
.000217
.000036
.001127
2.001314
.016942
.001150
.000244
2.000021
2.002272
2.000140
.000457
.008969
.000609
2.675622
2.115307
.007869
1.450238
2.028495
.000007
2.000063
.001516
2.000276
2.000084
.018264
.000362
2.001669
2.005670
2.444451
.016243
.004174
2.000530
.000055
.000184
.002385
2.000320
.000886
2.625269
.005996
2.001167
.000052
.000056
2.026453
.000182
2.000037
2.000845
.000044
2.000419
2.000031
2.000032
2.000105
.000922
2.008936
2.000678
.000269
2.000229
.000054
2.000120
.003916
.040472
2.002977
.000073
.000032
.007565
.001272
.000230
2.000193
2.000045
.001778
.000367
2.000043
.000081
2.000072
2.000247
2.000432
.003600
2.000213
.000092
2.000038
2.000479
2.000046
.000151
.001659
2.000096
2.153338
2.026975
.001467
Error
2.102761
.000504
.000007
2.000063
.000516
2.000276
2.000084
.004264
.000362
2.000669
2.001670
2.246451
.004243
.002174
2.000530
.000055
.000184
2.000614
2.000320
2.000113
2.161269
.001996
2.000167
.000052
.000056
2.008453
.000182
2.000037
.000154
.000044
2.000419
2.000031
2.000032
2.000105
.000922
.000063
.001321
.000269
2.000229
.000054
2.000120
.004916
.001472
.005022
.000073
.000032
.007565
.001272
.000230
2.000193
2.000045
.001778
.000367
2.000043
.000081
2.000072
2.000247
2.000432
.000600
.000786
.000092
2.000038
2.000479
2.000046
.000151
.000659
2.000096
2.289338
2.058975
.000467
q 2001 RAS, MNRAS 322, 785±799
Forced nutations of a two-layer Earth model
795
Table 2 ± continued
Period
213.58
13.17
13.14
12.81
12.79
12.66
210.08
9.81
9.63
9.61
29.60
9.56
9.54
29.53
9.37
9.34
9.18
9.13
9.12
9.11
9.06
8.91
8.75
7.35
7.24
7.13
7.10
7.09
6.86
6.85
5.80
5.64
5.64
5.49
4.79
4.68
Longitude
2.066389
.759070
.081534
.644560
.102185
.092275
2.133491
2.282679
2.097163
.656869
2.048861
25.967315
21.020180
2.063922
2.288630
.166212
.157257
230.142810
25.162552
.333470
.114762
.248803
.093435
2.121337
2.265388
.058575
23.856892
2.663612
23.102781
2.534327
2.151729
2.132908
2.767538
2.289713
2.069112
2.109528
In-phase terms
Error
Obliquity
.001610
.002070
.000534
.001560
.001185
.001275
2.000491
2.000679
2.002163
2.000130
.001138
2.002315
.001819
.000077
2.001630
.001212
.000257
2.005810
.000447
2.000529
2.000237
.002803
2.000564
2.000337
2.001388
2.000424
2.002892
.000387
2.000781
2.001327
2.000729
.000091
.000461
2.000713
2.000112
2.001528
.036684
2.326099
2.042173
2.276834
2.052810
2.039627
2.003933
.121069
.049698
2.020333
.027443
2.554980
.521607
2.001995
.123550
2.071146
2.005103
12.898771
2.634562
2.010907
2.049104
2.106435
2.039961
.051769
.113204
2.002464
1.644716
.334615
1.322429
.268929
.064485
.066134
.326039
.123004
.029263
.046354
Error
Longitude
.000684
2.000099
2.000173
.000165
.001189
2.000627
.000066
2.000930
.000698
2.000333
.000443
.000980
2.000392
2.002995
.000550
.000853
2.000103
.002771
2.000437
.000092
2.000104
2.000435
.000038
2.000230
2.000795
2.003464
.001716
2.000384
2.000570
2.000070
2.001514
.000134
.001039
2.000995
.000263
2.000645
2.000113
.001361
.000171
.001160
.000215
.000166
.000062
2.000529
2.000210
.000321
2.000077
2.011218
2.002215
.000031
2.000544
.000313
.000080
2.057086
2.011262
.000172
.000217
.000473
.000178
2.000238
2.000522
.000038
2.007620
2.001487
2.006167
2.001202
2.000310
2.000305
2.001580
2.000599
2.000146
2.000233
Out-of-phase terms
Error
Obliquity
2.000113
.002361
.000171
.002160
.000215
.000166
.000062
2.001529
2.000210
.002321
2.000077
2.025218
2.004215
.000031
2.001544
.000313
.001080
2.134086
2.023262
.001172
.000217
.001473
.000178
2.000238
2.001522
.000038
2.022620
2.003487
2.018167
2.003202
2.001310
2.001305
2.005580
2.002599
2.000146
2.001233
2.000027
.000514
.000057
.000438
.000072
.000062
2.000075
2.000201
2.000072
.000371
2.000016
2.004279
2.000764
2.000036
2.000207
.000119
.000088
2.021812
2.003907
.000188
.000083
.000180
.000068
2.000091
2.000201
.000032
2.002938
2.000532
2.002381
2.000432
2.000120
2.000112
2.000615
2.000233
2.000057
2.000091
Error
2.000027
.000514
.000057
.000438
.000072
.000062
2.000075
2.000201
2.000072
.000371
2.000016
2.011279
2.001764
2.000036
2.000207
.000119
.000088
2.056812
2.011907
.000188
.000083
.000180
.000068
2.000091
2.000201
.000032
2.008938
2.001532
2.007381
2.001432
2.000120
2.000112
2.002615
2.001233
2.000057
2.000091
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.8
-0.6
1984
1986
1988
1990
1992
1994
1996
1984
1986
1988
1990
1992
1994
1996
Figure 1. Error in obliquity (in-phase and out-of-phase).
Figure 2. Error in longitude sin 1 (in-phase and out-of-phase).
sum of the contributions of different effects: the presence of the
liquid core (two-layer model, without deformation), plus the
effects due to the deformation of mantle and core. Note that, from
equations (35) and (51), the functions F a;b
1 are associated to the
mantle, and the functions Ga;b
concern the core, while the
1
c
coefficients km
t and kt correspond to deformation of mantle and
core respectively. Thus, deformation of the core comes from the
terms of the form kct Ga;b
1 in equation (76), while the effect of the
deformation of the core can be separated into `direct' effect,
a;b
resulting from terms in km
t F 1 ; and the `crossed' effect, for
m a;b
terms in kt G1 : Numerical values of these contributions
corresponding to the main nutations are listed in Table 3 for inphase terms and in Table 4 for out-of-phase terms.
To show the performance of the former series we present some
comparisons, in both the time and the frequency domain.
In the time domain, differences between CEP offsets provided
q 2001 RAS, MNRAS 322, 785±799
796
J. Getino and J. M. FerraÂndiz
Table 3. Oppolzer in-phase terms for nutations in longitude and obliquity: contribution of different effects …unit ˆ mas†:
Period
(days)
Two2layers
26798.36
23399.18
1305.47
21095.17
386.00
365.26
365.22
205.89
182.63
182.62
177.84
173.31
121.75
91.31
31.96
231.81
231.66
27.67
27.55
227.44
27.09
26.98
23.94
14.77
13.78
13.66
13.63
13.61
9.56
9.13
9.12
7.10
6.86
193.8532
24.0805
.2127
2.0251
.1747
54.2775
.7876
.7123
.2366
2110.7226
.9320
2.3010
25.0780
2.1697
.2237
21.8593
2.1122
.9792
8.4335
2.4587
1.8452
.3199
.4445
.7699
.3569
243.7126
28.0264
.3159
21.3548
27.0092
21.3131
21.0312
2.8466
Longitude
Def. mantle
Direct
Crossed
248.8511
1.0283
2.0536
.0063
2.0438
213.6504
2.1988
2.1794
2.0596
27.8962
2.2348
.0758
1.2793
.0427
2.0563
.4683
.0282
2.2467
22.1245
.1155
2.4648
2.0806
2.1119
2.1939
2.0899
11.0122
2.0220
2.0796
.3413
1.7658
.3308
.2597
.2132
48.0945
21.0137
.0525
2.0063
.0451
13.6782
.1832
.1797
.0597
226.0411
.2115
2.0759
21.1696
2.0382
.0412
2.4656
2.0414
.1743
2.1065
2.1840
.3461
.0565
.0808
.1874
.0864
26.6474
21.1276
.0764
2.1772
2.8975
2.1539
2.1163
2.0937
Def. core
Two2layers
2113.8638
2.3999
2.1244
.0150
2.1069
232.3832
2.4337
2.4255
2.1413
61.6524
2.5007
.1798
2.7690
.0905
2.0977
1.1024
.0980
2.4127
24.9873
.4357
2.8194
2.1339
2.1914
2.4437
2.2046
15.7378
2.6696
2.1809
.4197
2.1249
.3644
.2754
.2219
259.4885
1.5178
2.0469
.0258
.1625
18.1121
2.9175
.1254
.0359
53.4214
2.7152
2.0427
2.3069
.0752
2.1078
2.1548
.0704
2.4625
2.8811
.3047
2.7713
2.1505
2.1848
2.1718
2.0855
17.7563
3.4273
2.0767
.5425
2.8021
.5312
.4086
.3351
Obliquity
Def. mantle
Direct
Crossed
14.9913
2.3825
.0118
2.0065
2.0408
24.5534
.2310
2.0315
2.0090
213.4584
.1801
.0107
2.5811
2.0189
.0271
.0390
2.0177
.1165
.2220
2.0767
.1943
.0379
.0465
.0432
.0215
24.4732
2.8634
.0193
2.1366
2.7059
2.1338
2.1029
2.0844
214.7682
.3772
2.0115
.0064
.0405
4.5972
2.2256
.0338
.0099
12.7875
2.1734
2.0119
.5413
.0173
2.0226
.0078
.0215
2.0946
.0232
.0967
2.1507
2.0306
2.0351
2.0024
2.0013
2.8623
.5878
2.0012
.0758
.3837
.0783
.0494
.0398
Def. core
34.9636
2.8930
.0272
2.0151
2.0960
210.8840
.5342
2.0800
2.0235
230.2744
.4107
.0283
21.2815
2.0410
.0536
2.0186
2.0511
.2242
2.0550
2.2289
.3569
.0726
.0831
.0058
.0032
26.7766
21.3918
.0030
2.1796
2.9085
2.1854
2.1171
2.0943
0.2
0.6
0.4
0.1
0.2
0
0
-0.2
-0.1
-0.4
-0.2
-0.6
1984
1986
1988
1990
1992
1994
1996
1984
1986
1988
1990
1992
1994
1996
Figure 3. Error in obliquity (out-of-phase).
Figure 4. Error in obliquity (in-phase).
by IERS 96 and our work are plotted in Figs 1±6, where the
vertical units are mas and the horizontal units are years, starting
from 1984 and ending in 1996. Figs 1 and 2 include both in- and
out-of-phase components. It can be seen that the errors are kept
well below 1 mas, in fact they are lower than 0.7 mas. Figs 3 and 4
for obliquity, and Figs 5 and 6 for longitude, show how the largest
part of the error comes from the out-of-phase terms (,0.7 mas),
while the in-phase global errors are less than 0.25 mas. These plots
are useful to compare our series with series in Dehant & Defraigne
(1997), since they do not display nutation tables, but do display
several error curves. It can be seen in that paper that errors in the
out-of-phase component of the obliquity exceed 1.5 mas, and that
the same component of the longitude errors is twice as large,
reaching 4 mas (equivalent to 1.6 mas in CEP offset) ± see figs 8±
11 in that reference. Let us finally remark that our series do not
include oceanic corrections unlike the other two used in the
comparisons, in spite of providing double accuracy in the time
domain.
q 2001 RAS, MNRAS 322, 785±799
Forced nutations of a two-layer Earth model
797
Table 4. Oppolzer out-of-phase terms for nutations in longitude and obliquity: contribution of different effects
…unit ˆ mas†:
Period
(days)
26798.36
23399.18
1305.47
21095.17
386.00
365.26
365.22
205.89
182.63
182.62
177.84
173.31
121.75
91.31
31.96
231.81
231.66
27.67
27.55
227.44
27.09
26.98
23.94
14.77
13.78
13.66
13.63
13.61
9.56
9.13
9.12
7.10
6.86
Longitude
Def. mantle
Direct
Crossed
Two-layers
2.5867
2.0701
.0008
2.0014
.0390
4.4854
2.0567
.0087
.0023
2.4110
.0010
2.0027
2.0154
2.0004
.0001
.0022
.0002
.0006
.0086
.0008
.0013
.0002
.0002
.0004
.0001
2.0130
2.0021
.0001
2.0002
2.0011
2.0002
2.0001
2.0000
2.1801
.0024
2.0003
2.0001
2.0111
21.3944
.0158
2.0053
2.0016
.3905
2.0019
.0020
.0208
.0007
2.0010
.0095
.0006
2.0045
2.0432
.0027
2.0087
2.0015
2.0021
2.0038
2.0017
.2047
.0373
2.0015
.0061
.0318
.0059
.0045
.0037
Def. core
Two-layers
.2535
2.0040
.0009
.0000
2.0283
23.2438
.0613
2.0093
2.0026
21.3057
.0204
.0030
2.0562
2.0018
.0025
2.0029
2.0028
.0108
2.0117
2.0125
.0169
.0035
.0039
2.0005
2.0002
2.3175
2.0663
2.0002
2.0082
2.0413
2.0086
2.0051
2.0041
1.2920
2.0294
.0012
2.0003
2.0162
21.7594
.0313
2.0026
2.0006
2.2777
.0044
.0007
2.0080
2.0002
.0001
.0001
.0001
.0003
2.0003
.0004
.0005
.0001
.0001
2.0000
.0000
2.0056
2.0011
.0000
2.0001
2.0005
2.0001
2.0000
2.0000
.1646
2.0020
.0002
.0001
.0112
1.3977
2.0161
.0053
.0016
2.3532
.0014
2.0020
2.0186
2.0006
.0007
2.0096
2.0009
.0031
.0433
2.0042
.0064
.0010
.0015
.0038
.0017
2.1217
2.0201
.0015
2.0031
2.0158
2.0026
2.0019
2.0015
Obliquity
Def. Mantle
Direct
Crossed
.0472
2.0007
.0001
.0000
2.0047
2.5430
.0104
2.0015
2.0004
2.2326
.0035
.0004
2.0102
2.0003
.0005
.0004
2.0003
.0022
.0030
2.0016
.0037
.0007
.0008
.0007
.0003
2.0835
2.0162
.0003
2.0024
2.0127
2.0024
2.0017
2.0014
2.0426
.0006
2.0001
2.0000
.0047
.5450
2.0103
.0015
.0004
.2193
2.0034
2.0005
.0094
.0003
2.0004
.0005
.0004
2.0018
.0019
.0021
2.0028
2.0005
2.0006
.0000
.0000
.0533
.0111
.0000
.0013
.0069
.0014
.0008
.0006
Def. core
.1550
2.0019
.0002
.0001
.0105
1.3163
2.0152
.0050
.0015
2.3327
.0013
2.0019
2.0175
2.0006
.0007
2.0090
2.0008
.0029
.0408
2.0039
.0060
.0009
.0014
.0036
.0016
2.1146
2.0189
.0014
2.0029
2.0149
2.0024
2.0018
2.0015
0.6
0.15
0.4
0.1
0.2
0.05
0
0
-0.05
-0.2
-0.1
-0.4
-0.15
1984
1986
1988
1990
1992
1994
1984
1996
1986
1988
1990
1992
1994
1996
Figure 5. Error in longitude sin 1 (out-of-phase).
Figure 6. Error in longitude sin 1 (in-phase).
In the frequency domain it is convenient to express the results
as circular nutations, paying attention to the geophysical nutations.
The corresponding prograde and retrograde amplitudes for the
main nutation terms are listed in Table 5 (in-phase) and Table 6
(out-of-phase), together with the deviations with respect to
IERS96 (labelled as `error'). For in-phase terms the maximum
error is 0.08 mas for prograde amplitudes (semi-annual
period) and 0.05 mas for retrograde ones (annual period),
while concerning out-of-phase terms we found a maximum
deviation of 0.28 mas (13.66 d) in prograde and 0.15 mas (annual)
in retrograde.
q 2001 RAS, MNRAS 322, 785±799
9
C O N C L U S I O N S A N D F U T U R E V I S TA S
In the previous section we have shown that traditional (sometimes
called `astronomical') approaches to Earth rotation, based on
variational principles, are still valid to obtain accurate nutation
series, and even more accurate than the ones last published, which
798
J. Getino and J. M. FerraÂndiz
were based on `geophysical' approaches, at least in the time
domain and with reference to IERS 96 series, in spite of the
simplicity of the model.
It is natural to wonder about the extent to which the fitting of
Table 5. Prograde and retrograde amplitudes of nutation.
Comparison with IERS96 …unit ˆ mas† in-phase (main
terms).
Period
Prograde
Error
Retrograde
Error
26798.36
23399.18
365.26
365.22
182.62
121.75
27.55
13.66
13.63
9.13
21180.58
3.62
25.71
9.11
2548.39
221.55
14.49
294.25
217.74
212.44
20.032
0.000
0.065
0.022
0.080
20.051
20.000
20.022
0.001
20.002
28024.80
86.14
232.99
0.50
224.56
20.94
213.80
23.64
22.33
20.45
0.005
0.012
0.051
20.000
0.031
20.003
0.004
0.000
0.002
20.000
Table 6. Prograde and retrograde amplitudes of nutation.
Comparison with IERS96 …unit ˆ mas† out-of-phase (main
terms).
Period
Prograde
Error
Retrograde
Error
26798.36
23399.18
365.26
365.22
182.62
121.75
27.55
13.66
13.63
9.13
0.082
20.000
20.026
20.009
0.661
0.028
20.022
0.157
0.030
0.022
0.030
20.000
0.098
20.002
0.150
0.008
20.021
0.279
0.052
0.055
21.368
0.028
20.470
0.007
0.035
0.001
0.018
0.004
0.003
0.000
0.133
20.001
20.148
0.002
20.011
20.000
20.020
20.010
20.006
20.002
nutation series could benefit by including some noticeable but
neglected effects in the Earth model, as happened with oceanic
ones in our former approach. To get some insight into that
issue, we have computed modified nutation series (referred to
as GF99-1b) by adding the oceanic corrections computed by
Schastok (1997) to our analytical solution, and adjusting some
of the BEP again. The accuracy of the new series in the
frequency domain is remarkably improved. In Table 7 we
present a comparison of the main nutation terms with the most
accurate values computed by different authors, according to
table 4 in Dehant et al. (1999). It can be seen that GF99-1b
produces the best adjustment in the 18.6-yr period (with respect
to all the alternate solutions) whereas the worst term in our
series is the annual one. Let us say that in recent meetings
(JourneÂes 1999, Dresden; AGU Fall Meeting 1999, San
Francisco) we have reported on the results obtained by working
out the Hamiltonian theory for a three-layers Earth model, with
and without oceanic corrections ± the corrections were taken
from Schastock (1997) (based on an early Topex/Poseidon tide
model by Zahel 1995) or Dehant & Defraigne (1997) (based on
Chao et al. 1996). Further improvements occur, although the
comments are beyond the scope of this paper, that emphasize
the usefulness of the direct Hamiltonian approach to obtain
highly accurate nutation series independently of a previous
rigid-Earth solution, which is needed to compute transfer
functions in the other recent approaches.
AC K N O W L E D G M E N T S
This work has been partially supported by Spanish Projects
CICYT, Project No. ESP97±1816±C04±02, DGES, Project
No. PB95±696 and Junta de Castilla y LeoÂn Project No.
VA11/99.
Table 7. Main nutation terms with numerical oceanic corrections (taken from Dehant et al. 1999):
M. ˆ Mathews et al. (1998, 1999); D. D. ˆ Dehant & Defraigne (1997); S. ˆ Schastok (1997); GF991b ˆ This paper. Error ˆ absolute value of deviations with respect to IERS96 …unit ˆ mas†:
In-phase terms
Prograde
Retrograde
Value
Error
Value
Error
Period
Out-of-phase terms
Prograde
Retrograde
Value
Error
Value
Error
18.6 y
IERS96
M.
D. D.
S.
GF9921b
21180.55
21180.43
21180.48
21180.38
21180.57
0.12
0.07
0.17
0.02
28024.81
28024.77
28024.39
28024.67
28024.83
0.04
0.42
0.14
0.02
0.05
0.10
20.04
0.10
0.10
0.05
0.09
0.05
0.05
21.50
21.42
21.91
21.01
21.45
0.08
0.41
0.49
0.05
Annual
IERS96
M.
D. D.
S.
GF9921b
25.65
25.64
25.64
25.73
25.78
0.01
0.01
0.08
0.13
233.04
233.07
233.04
233.00
232.94
0.03
0.00
0.04
0.10
20.12
20.15
20.03
20.02
20.03
0.03
0.09
0.10
0.09
20.32
20.34
0.12
0.11
20.42
0.02
0.44
0.21
0.10
Semi2
annual
IERS96
M.
D. D.
S.
GF9921b
2548.47
2548.48
2548.40
2548.40
2548.44
0.01
0.07
0.07
0.03
224.59
224.56
224.55
224.55
224.56
0.03
0.04
0.04
0.03
0.51
0.51
0.45
0.49
0.59
0.00
0.06
0.02
0.08
0.05
0.04
0.05
0.07
0.05
0.01
0.00
0.02
0.00
13.66 d
IERS96
M.
D. D.
S.
GF9921b
294.22
294.19
294.26
294.22
294.23
0.03
0.04
0.00
0.01
23.64
23.65
23.64
23.64
23.64
0.01
0.00
0.00
0.00
20.12
20.12
20.15
20.17
20.15
0.00
0.03
0.05
0.03
0.01
0.01
0.01
0.02
0.02
0.00
0.00
0.01
0.01
q 2001 RAS, MNRAS 322, 785±799
Forced nutations of a two-layer Earth model
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