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%          IAU SYMPOSIUM 172 (July 1995)
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%            J. Getino and P. Martin
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\begin{document}
\author{Juan Getino\thanks{Grupo de Mec\'anica Celeste, Departamento de Matem\'a
tica Aplicada Fundamental,
Facultad de Ciencias, Universidad de Valladolid, E-47005 Valladolid, Spain.}
\and
Pablo Mart\'{\i}n\thanks{Grupo de Mec\'anica Celeste, Departamento de Matem\'ati
ca Aplicada a la
Ingenier\'{\i}a, E.T.S. de Ingenieros In\-dus\-tria\-les, Universidad de Vallado
lid,
E-47011 Valladolid, Spain.}}
\title{Influence of the liquid and solid core in the tidal variation of geopotential}
\maketitle
\begin{abstract}
The accuracy levels achieved in obtaining and dealing with satellite data
means that the coefficients of the development of the Earth's gravitational
potential can no longer be considered as constants, but as functions of the
time. The most studied variation, both theoretically and experimentally, is
that of the $J_2$, which, on the other hand, produces the greatest indirect
effect on the observations as they are closely related to the variation of
the main moment of inertia and of the universal time, the latter reaching
more than 2 ms. The LAGEOS data analysis allowed Yoder {\sl et al.}, already
in 1983, not only to identify and improve the predicted values of the
periodical variations of $J_2$ of tidal origin, but also a secular decrease
associated with post glacial phenomena and other unexplained effects.

However, the knowledge of the effect of other zonal or tesseral harmonics
has shown less progress, and its experimental calculation presents greater
problems although it is usually acknowledged that the uncertainty in the
values of the low order harmonics should be diminished if global models with
real centimeter or subcentimeter accuracy are to be obtained.

In previous papers, Ferr\'andiz and Getino (1993) and Ferr\'andiz, Getino
and Caballero (1993), the problem of the analytical prediction of the tidal
variations of zonal or tesseral harmonics for the first terms of the
geopotential is undertaken. Later, in Getino and Ferr\'andiz (1994)
analytical expressions for any order $n$ are obtained. The process followed
is essentially similar to that used by the authors in their Hamiltonian
theory for the Earth's rotation, in such a way that the solution is
calculated directly in terms of integrals of functions that include the
displacement vector components obtained starting from a selected Earth
model, so the result depends above all on experimental data of a geophysical
kind. Notice that the study is limited to the contribution of the mantle.

This paper is an extension of the previous ones, where the effect of the
deformation of the liquid outer core (LOC) and solid inner core (SIC) is
taken into account. We perform a numerical integration of the elasticity
differential equations in order to obtain the necessary parameters to
compute the effect of the tidal deformation of the whole Earth. Numerical
tables showing the periodic variation of coefficients are given, as well as
a new prediction for $\Delta UT1.$ For $\delta J_2$ and $\delta J_{22}$ the
amplitudes reach such a magnitude that both two variations should not be
ignored in studies involving the analysis of highly precise satellite
tracking data.
\end{abstract}
\noindent
{\bf Key words}: Geopotential, Solid tides, Elastic Earth.
\begin{thebibliography}{9}
\bibitem{}  Ferr\'andiz, J.M. and Getino, J.: 1993, ``On the tidal variation
of geopotential'', {\it Celes. Mech.}, {\bf 57}, 279--292.

\bibitem{}  Ferr\'andiz, J.M., Getino, J. and Caballero, R.: 1993,
``Accurate analytical predictions of the tidal variations of the
geopotential coefficients''. In {\it AAS/GSFC International Symposium on
Space Flight Dynamics}, April 1993, {\bf 2}, AAS 93-312.

\bibitem{}  Getino, J., Ferr\'andiz, J.M.: 1994, ``General expressions of the
tidal variations of geopotential coefficients'', in {\it Dynamics and
Astrometry of Natural and Artificial Celestial Bodies}, Kurzynska, Barlier,
Seidelmann and Wytrzyszczak eds., Astronomical Observatory of A. Mickiewic
University, Poznan, 341--346.

\bibitem{}  Yoder, C.F. et al.: 1983, ``Secular variation of the Earth's
gravitational harmonic $J_2$ coefficient from LAGEOS and nontidal
acceleration of Earth rotation'', {\it Nature}, {\bf 303}, 757--762.
\end{thebibliography}
\end{document}
