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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
Rotation theory of celestial bodies: dynamical
relations of the pole motion and nutation
Yu. V. Barkin
Sternberg Astronomical Institute, Moscow, Russia
1. Problem
By construction Earth rotation theory the studies of the precession--nutation
motion and pole motion are considered usually separately. The reason of this
approach is obvious. The angle Q between the polar axis of inertia and axis of
rotation is small (of order 10 \Gamma6 ). The pole perturbations (of the different na­
ture) are very small to give big contribution to the nutation. But in any case we
must give evaluations of these possible correlation between pole motion, diurnal
rotation and space axis motion (precession and nutation). On the other hand,
we know about very high requirements to accuracy of the Earth rotation theory
(10 \Gamma5 \Gamma 10 \Gamma6 arcsec). If we take into account particularities of the pole motion by
construction of the nutation theory the analytical problem will be more compli­
cated as compared with classical approach. To study the above--mentioned effects
in the Earth rotation we use some new approaches to construct the Earth rotation
theory.
2. Force function. Unperturbed motion
The problem of constructing the analytical theory of the rotational motion
of the Earth takes one of the central places in celestial mechanics. Starting from
classical papers on this subject the equations of rotational motion in the oscu­
lating elements similar to that of Andoyer and angle­action variables have been
used (Laplace,1825; Poisson, 1827; Pontecoulant,1829; Tisserand,1891; Andoyer,
1923, and so on). But in constructing the force function of the problem in these
variables the authors used usually the approximate developments with additional
assumption about small eccentricities of the ellipsoid of inertia. Analytical solu­
tion of the problem was of restricted character. In (Kinoshita, 1977) in construct­
ing the Earth rotation theory similar simplifications were used (only the constant
of precession was presented in the explicit form in term of elliptic integrals).
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We have developed a more general theory to study the rotation of the de­
formable celestial bodies. The new effective forms of the canonical and non­
canonical equations in Andoyer and angle­action variables were suggested. These
variables were introduced on the basis of the integrable Euler­Chandler problem
for the attitude motion of an elastic body deformed by its own rotation. The
unperturbed motion with the elastic property taken into account is reduced to
the classical Euler­Poinsot problem with a specially changed principal moment of
inertia. The properties of the Euler­Chandler unperturbed motion of the Earth
are described in details (Chandler period, eccentricity of the pole trajectory, non­
uniform pole motion, etc.). The peculiarities of the Venus pole motion have been
studied as well.
3. Perturbations of the pole motion of the deformable isolated
body
We have studied the problem of rotation of the isolated deformable body
with a shell changeable in time. The components of the external shell tensor of
inertia are given as conditionally--periodic functions of time. The analytical for­
mulas for the secular and periodic perturbations have been obtained in Andoyer,
angle­action and classical Euler variables. Generally, these formulas are obtained
for arbitrary parameters of the considered unperturbed motion (for example for
arbitrary unperturbed value of the angle Q between the axis of the angular mo­
ment and the body pole axis). The analytical formulas for the precession constant
and the additive terms to the Chandler period and diurnal rotation due to the
gravitational attraction of the Moon and the Sun are obtained in terms of elliptic
functions and integrals. The perturbation theory of the Earth rotation has been
constructed in the elastic Andoyer and angle--action variables. Perturbations of
the Earth rotation due to the tidal and non­tidal variations of the Earth tensor
of inertia in gravitational field of the Moon and the Sun have been obtained.
4. Perturbations
Complete formulas for the first--order perturbations of the rotational motion
of an Earth's satellite moving in the gravitational field under the action of the
perturbing body (the Moon and the Sun) in the angle--action variables (for Euler
and Euler--Chandler unperturbed motions) have been derived. Amplitudes of all
perturbations of the first order were presented in terms of elliptic functions and
integrals of the action variables. It means that the analytical theory is applicable
for study of the attitude motion of the natural and artificial celestial bodies
with arbitrary dynamical structure and for arbitrary unperturbed pole motion
in accordance with Euler or Chandler--Euler problems. Perturbations in the pole
motion are calculated for the two initial values of the angle between polar axis
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of inertia and vector of the angular momentum Q = 0:12 00 and Q = 0:14 00 (these
values correspond to the real averaged pole trajectory). New nutation terms of
small amplitudes have been found. Results presented here are perspective for the
construction of the high--accuracy theory of the rotation of the Earth and other
celestial bodies with a non--ordinary unperturbed rotation (Venus, asteroids, etc.).
This work is supported by the grant of RFBR 02­05­64176 and grant of Spain for
2002--2003 Sabatico year at the Alicante university.
References
1. Getino J., Ferrandiz J. M. A Hamiltonian theory for an elastic Earth: elastic
energy of deformation, Celestial Mechanics, 1991, 51, 17--34.
2. Getino J., Ferrandiz J. M. A Hamiltonian theory for an elastic Earth: secular
acceleration, Celestial Mechanics, 1991, 52, 381--396.
3. Ferrandiz J. M., Barkin Yu. V., J. Getino, About Applications of Angle­
Action Variables in Rotation Dynamics of the Deformable Celestial Bodies,
In: (Eds. S. Ferraz­Mello, B. Morrando, J.­E. Arlot) Dynamics, ephemerides
and astrometry of the solar system. Proc. IAU Symposium No. 172 (Paris,
1995), 1996, 243--244.
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