Full text of DOI:10.1134/1.1342447

Doklady Physics, Vol. 45, No. 12, 2000, pp. 672–675. Translated from Doklady Akademii Nauk, Vol. 375, No. 5, 2000, pp. 619–621.
Original Russian Text Copyright © 2000 by Konkina, Markov, Mikisha, Rykhlova.
ASTRONOMY, ASTROPHYSICS,
AND COSMOLOGY
On the Problem of Constructing Intermediate Trajectories
in the Theory of Elastic-Earth Rotation
around a Center of Mass
L. I. Konkina, Yu. G. Markov, A. M. Mikisha, and L. V. Rykhlova
Presented by Academician A.A. Boyarchuk February 21, 2000
Received November 21, 1999
The development of the exact theory of Earth rota- of the Earth’s own rotation around its axis, of the pre-
tion is a rather complicated mathematical problem and cession of the kinetic-moment vector for the deform-
requires the elimination of a number of simplifications
in the accepted theory [2]. In spite of the exceptionally
high level of modern observations, researchers have
failed to attain a complete understanding of such
dynamic effects as the free nutation of the Earth-rota-
tion axis and the variation of latitudes, both of which
are extremely necessary in constructing a highly accu-
rate theory of rotational motion for the deformable
Earth.
We attempted to study certain fine regularities in the
theory of Earth rotation around its center of mass from
more general positions, namely, translational–rota-
tional movement. As a starting theoretical model, we
used the intermediate two-body problem of the Earth–
Moon system, which made it possible to take into
account the barycentric distance. In this intermediate
motion, the Earth uniformly rotates and deforms under
the action of centrifugal forces of inertia and the lunar
gravitation field. The deformations are considered to
proceed quasi-statically (the inertia terms can be
ignored). In other words, the motion of the three-axis
able Earth, and of the evolution of the rotation-axis
inclination to the plane of the ecliptic. In such a system,
the natural separation of motions into fast and slow
motions takes place and these motions are described by
their corresponding parameters. Under certain condi-
tions, it is possible to isolate a set of slow parameters
(variables) whose rate of variation is asymptotically
slow (with respect to a certain small parameter), and the
evolutionary equations describing this variation are
separated from the remaining equations of the set [5].
Equations averaged over the fast variables for the trans-
lational–rotational motions of the Earth–Moon system
in the solar gravitational field are studied indepen-
dently. In the majority of cases, these equations turn out
to be a good approximation to the original equations for
a long (in the asymptotic sense) time interval. The
equations for the remaining variables form a fast com-
ponent of the Earth–Moon system and involve the evo-
lutionary-system variables as slowly varying parame-
ters. It should be noted that, from the standpoint of evo-
lutionary processes, the qualitative picture of the fast
elastic Earth as a whole around its center of mass can motions of the system is a background against which
be represented as the motion of a planet with an equi- the slow evolution of the orbital–rotational motion
librium configuration and “frozen” deformations. Fur-
thermore, on the basis of the intermediate model prob-
lem, it is of interest to consider the dynamics of evolu-
tionary processes, but already using perturbed motion
with allowance for dissipative factors and lunar–solar
perturbations.
In our opinion, when developing the theory of the
Earth’s rotation around its center of mass, an important
argument is the fact that the Earth–Moon dynamic sys-
tem is assigned to the class of systems with a slow evo-
lution in which it is possible to trace multistage
dynamic processes with various characteristic times.
Thus, it is possible to compare the characteristic times
occurs.
1. The choice of intermediate trajectories for the
Earth’s motion is based on the spatial variant of the
two-body (planet–satellite) problem and, namely, the
deformable-Earth–Moon system (the Moon is taken as
a mass point) and is analyzed from the positions of a
double planet. This automatically presumes the pres-
ence of a barycentre and allows for its position in sub-
sequent calculations. The model problem under consid-
eration is formulated as follows: let a deformable planet
(the Earth) and its satellite (the Moon) participate in the
mutual translational–rotational motion around their
common center of mass (barycentre). The satellite orbit
is inclined at an arbitrary angle to the planet’s equator.
The Earth is represented by a two-layer model with a
solid core and a viscoelastic mantle [1, 7], which are
individually continuous. We introduce the inertial sys-
Institute of Astronomy, Russian Academy of Sciences,
ul. Pyatnitskaya 48, Moscow, 109017 Russia
1028-3358/00/4512-0672$20.00 © 2000 MAIK “Nauka/Interperiodica”

ON THE PROBLEM OF CONSTRUCTING INTERMEDIATE TRAJECTORIES
673
tem of coordinates Cξ1ξ2ξ3 with the origin in the bary-
ξ3
center of the planetary system. Let G be the Earth’s
own kinetic moment and L be the orbital angular
momentum for the lunar center of mass C1 and the ter-
restrial center of mass C2 . The angular momentum K =
G + L of the entire planetary system is immobile in the
inertial space and coincides with the axis Cξ3 (see the
figure).
C1
K
G
L
R10
i > 0
i
The radii vectores for the points C1 and C2 in the
Cξ1ξ2ξ3 coordinate system are given in the form
C
R20
x3
h
Ri = RiR0i , Ri = ciR21, i = 1, 2,
ξ1
˜
ξ2
R0i = Γ3(h)Γ1(i)(cosϑ, sinϑ, 0), R20 = –R01,
h > 0
C2
(1)
–1
–1
˜
˜
Coordinate system and orientation of vectors.
c1 = m2m , c2 = m1m , m = m1 + m2.
Here, h, i, and ϑ are the ascending-node longitude, the
inclination, and the orbit true anomaly, respectively;
R21 = R21R021 is the radius vector drawn from the point
C2 to the point C1, so that R21 = R01 . The C2x1x2x3 Car-
tesian coordinate system is rigidly related to the solid
core of the planet. The axes of this system are directed
along the principal axes of inertia A, B, and C of the
planet. For this coordinate system, we may write out
allowance for the barycentric distance, they take the
form
J11[u]I22
˜
----------------------------------------------
A =
,
,
I22 + 3µ1R–3dJ211[u]
21
J22[u]I22
˜
----------------------------------------------
B =
(4)
I22 + 3µ1R–3dJ222[u]
21
J33[u]I22
O–1(t)R201 = (γ1, γ2, γ3),
O–1(t) = Γ3–1(ϕ1)Γ–11(δ2)Γ3–1(ϕ2)Γ–11(δ1)Γ3–1(ϕ3),
˜
----------------------------------------------
C =
,
I22 + 3µ1R–3dJ233[u]
(2)
21
2
2
2
˜
˜
d = cos h + cos isin h
2
2
2
2
2
2
where O(t) is the matrix specifying the passage from
the body axes to the inertial axes and is expressed in
˜
˜
+ cos δ1(sin h + cos icos h) + sin isin δ1,
˜
h = ϕ3 – h, µ1 = f m1.
Andoyer canonical variables: L, I2 = G , I3, ϕ1, ϕ2, and
ϕ3; and cosδ1 = I3/I2, cosδ2 = L/I2 [5, 6].
Here, f is the gravitation constant; Jii[u] = diag{J11[u],
J22[u], J33[u]} is the inertia-tensor components depen-
dent on the vector u of the elastic translation for the
deformed planet in the planetary coordinate system;
and J[0] = diag{A, B, C} for u = 0. Structural form (3)
coincides exactly with the traditional expression for the
Routh function of a perfectly rigid body in the Andoyer
variables and is a basis for introducing the action–angle
variables Ii and wi (i = 1, 2, 3). It should be noted that
the action–angle variables compose the Hamiltonian
part of the variables, whereas the generalized coordi-
nates (modal variables) describing the deformations of
the shell compose the Lagrangian part of the variables.
We describe the mutual orbital motion of mass cen-
ters in the Λ, H, ϑ, h Delone canonical variables, where
H is the projection of Λ onto the Cξ3-axis, Λ = L , and
cosi = H/Λ.
After a number of simple transformations and aver-
aging over the fast variables ϕ2 and ϑ, the Routh func-
tional of the intermediate problem is reduced to the
form (with an accuracy to an insignificant constant)
sin2 ϕ1 cos2 ϕ1
I22 – L2
---------------
2
L2
Thus, we introduce the following principal dynamic
parameters:
ᑬ =
--------------- + ---------------- + ------ + const, (3)
˜
˜
˜
A
B
2C
2
˜
˜ ˜
˜
2EC – I2
C(A – B)
κ2 =
,
λ2 = κ2
,
(5)
---------------------
---------------------
˜
˜
˜
2
where A , B , and C are the elastic-Earth principal
˜ ˜
˜
˜
A(B – C)
I2 – 2EA
moments of inertia modified under the action of the
centrifugal forces induced by the Earth’s rotation. With where E is the energy constant.
DOKLADY PHYSICS Vol. 45 No. 12
2000

674
KONKINA et al.
The relation between the action–angle variables and mediate motion (8) in the form (0 ≤ λ < 1)
the Andoyer variables is given by the following formu-
las [3, 4] (for brevity, we restrict ourselves by the case
0 ≤ λ < 1):
˜
˜
I
2 I2 – I1
2κ1 I2
2 A – C
2 + κ
----------------------------
,
-----------------
n
=
1 –
κ1 = 1 + κ2,
(10)
(2κ1 – 2 – κ ) .
w1
˜ ˜
κ1 AC
˜
˜
1 + κ2
I2
I2 – I1
--------------
2κ1I2
C – A
2
2I2
-------
πκ
π
2
2
2
I1 =
---------------- (λ2 + κ2)Π , κ , λ – λ K(λ) ,
---
-------------
n
=
w2
1 +
1 +
--
˜
˜
λ2 + κ2
C
κ1 A
I2 = I2, I3 = I3,
2. As an example, we consider the Chandler wobble
π F(ξ, λ)
-- ----------------
2K(λ)
--------------
π
of the Earth’s poles. The wobble is determined as a
motion of the rotation axis with respect to the figure
axis [8]:
w1 = ±
,
ξ = ±am(u, λ), u =
w1,
(6)
2
K(λ)
1
κ
w2 = ϕ2 ±
(1 + κ2)(κ2 + λ2)
--
I2λ
-----------------------------
Aω κ + λ
ωx
xp = ----- =
ω
cnu,
2
2
˜
π
2
F(ξ, λ)
----------------
K(λ)
2
× Π(ξ, κ2, λ) – Π , κ , λ
,
--
λI2 1 + κ2
ωy
yp = –----- = –
ω
-----------------------------
snu,
(11)
2
2
˜
Bω κ + λ
cotϕ1 = – 1 + κ2 tanξ, w3 = ϕ3.
2
2
˜
˜
I2
-------------------------
A
A
2
2
2
2
2
Here, K(λ), Π(π/2, κ2, λ) are the complete elliptic inte-
grals of the first and third kinds; and F(ξ, λ), Π(ξ, κ2,
λ) are the elliptic integrals of the first and third kinds.
-----
-----
ω =
λ + 2κ – λ 1 + 2κ sn u.
2
2
˜
˜
˜
C
C
A κ + λ
Here, snu and cnu are the Jacobi elliptic functions,
which are represented by the expansions in the Fourier
trigonometric series [3, 4]. In (11), according to the
accepted convention [8], yp is directed along the 90°
meridian of the western longitude.
In this case, Routh functional (3) can be written
out as
I22
C – A κ2
˜
˜
------
----------------------------------
ᑬ =
1 –
.
(7)
(λ2 + κ2)
˜
˜
2A
C
The period of the Chandler wobble is determined by
the following expression:
The general solution to the intermediate problem
with functional (7) has the form
2π
Ii = Ii(0), i = 1, 2, 3, w1 = nw t + w10,
T = -------.
(12)
1
n
W1
w2 = nw t + w20,
2
Formula (12) relates the period of the Chandler
wobble with the principal dynamic parameters intro-
duced by the authors: the energy constant, the kinetic-
moment modulus, and the Earth’s principal moments of
inertia.
˜
˜
I2(A – C)
πκ
(8)
-----------------------------------------------------------------------------
nw
=
,
1
˜ ˜
(1 + κ2)(κ2 + λ2)K(λ)
2AC
2
˜
˜
I2
A – CΠ(π/2, κ , λ)
--------------------------------------------
,
---
nw
=
1 –
2
˜
˜
K(λ)
C
A
The analytical expressions obtained in this study for
the moments of inertia of the Earth with allowance for
its elasticity and the translational type of its motion is a
basis for both constructing a highly accurate theory of
the Chandler wobble and for a comparison of this the-
ory with long-term astronomical observations and esti-
mates for the period of this vibration. It should be espe-
cially emphasized that, on the basis of the derived aver-
aged differential equations, we can trace the evolution
of both the pole vibrational motion and the moments of
where w10 and w20 are the initial values of the angular
variables.
Solving the first equation of set (6) with respect to λ
with an accuracy to λ2 inclusively, we obtain
2κ2(I2 – I1)
I2 1 + κ2
λ2 =
,
0 ≤ λ < 1.
(9)
---------------------------
Using (9), we can represent the frequencies of inter- inertia for the Earth.
DOKLADY PHYSICS Vol. 45
No. 12
2000

ON THE PROBLEM OF CONSTRUCTING INTERMEDIATE TRAJECTORIES
675
3. Yu. A. Sadov, Prikl. Mat. Mekh. 34, 962 (1970).
ACKNOWLEDGMENTS
4. Yu. V. Barkin, Astron. Astrophys. Trans. 17, 179 (1998).
This work was supported by the “Astronomy” State
Scientific and Applied Program, project no. 1.8.1.2.
5. V. G. Vil’ke, Analytical and Qualitative Methods in
Mechanics of Systems with an Infinite Number of
Degrees of Freedom (Mosk. Gos. Univ., Moscow, 1986).
6. Yu. G. Markov, Astron. Zh. 73, 748 (1996) [Astron. Rep.
40, 681 (1996)].
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Translated by V. Bukhanov
DOKLADY PHYSICS Vol. 45
No. 12
2000