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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
Analytical theories of the motion of the planets and
of the rotation of the rigid Earth
P. Bretagnon
Institut de m'ecanique c'eleste, Observatoire de Paris, Paris, France
1. Planetary theories
The improvement of the techniques of observation and the needs for the space­
craft navigation make necessary the construction of high precision solutions for
the motion of the planets. From VSOP82 and VSOP87 solutions (Bretagnon,
1982, Bretagnon and Francou,1988), we undertook the construction of analytical
theories of the motion of planets at the 10 \Gamma10 (0.02 mas) level. If the quality
of the observations of the giant planets does not require such a precision, such
a level is necessary for Mercure, Venus, the Earth and Mars and is reached for
the first three planets. It is not the case for Mars because of the uncertainty of
the masses of a great number of disturbing asteroids. The intrinsic quality of the
resolution of the equations is such as the error on the position of Mars for fixed
values of the mass of the asteroids is about 500 meters (20 \Theta 10 \Gamma10 radian) over
one century and of 20 meters (10 \Gamma10 radian) over 10 years. But, as showed by
Standish and Fienga (2002), the masses of the asteroids do not make it possible
to envisage the position of Mars to better than 1 kilometer over 10 years.
The constants of integration are in the course of determination (Fienga, 1999)
by comparison with the observations.
The planetary solutions are expressed as function of TCB (Barycentric Coor­
dinate Time) (Brumberg, 1991) and are used to calculate the relations between
the time scales TCB and TCG (Geocentric Coordinate Time).
TCG = TCB (1 \Gamma LC ) + periodic terms:
The Table 1 gives a comparison de LC with the determination by Fukushima
(1995) and by Irwin and Fukushima (1999). In this latter solution the uncertainty
is 2 \Theta 10 \Gamma17 .
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Table 1. Secular term LC of TCB\GammaTCG.
Solution LC \Theta 10 8
Fukushima (1995) 1:480 826 845 7
Irwin and Fukushima (1999) 1:480 826 867 41
Our result 1:480 826 866 70
2. Reference systems
Until now, the analytical solutions were built in the ecliptic and the equinox
J2000. But, such a reference frame depends on the solution considered, analytical
or numerical.
Thus, in the current state of the development of the analytical solutions of
the motion of planets, the mean plane of the ecliptic is obtained from the ICRF
by two rotations:
1) a rotation about the z axis of ' with
' = \Gamma 0:053 727 00 = \Gamma 0:000 000 260 476 radian
2) a rotation about the x axis of '' with
'' = 23 ffi 26 0 21:408 800 00 = 0:409 092 614 174 radian:
Following the adoption of the ICRF by IAU in 1997, we propose to define
a reference frame close to the ecliptic, the ``ecliptic ICRF'', obtained from the
ICRF by a rotation about the x axis of '' ICRF where
'' ICRF = 0:409 092 614 radian exactly:
It is in this reference frame that will be built henceforth the analytical plan­
etary solutions.
3. Earth rotation
The analytical solutions VSOP of the motion of planets as well as the solution
ELP2000/82 of the Moon (Chapront­Touz'e and Chapront, 1983) are used to build
the solution SMART97 of the rotation of the rigid Earth (Bretagnon et al, 1998).
We determine the three angles of Euler /, ! and ' with the accuracy of 2 ¯as
for / and ' and of 0.6 ¯as for ! on the interval 1970--2020.
It is not desirable to separate precession and nutation today, also the solutions
are built globally for the polynomial parts and the periodic and of Poisson parts
by taking into account simultaneously the perturbations by the Moon, the Sun
and all the planets.
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Let us note that variables such as `, z and i are singular (they are not defined
in J2000) and do not take place any more to be considered.
The MHB2000 model (Mathews et al, 2002), developed for nutations with
period higher than two days, was adopted by IAU in 2000 and was applied to
the solution of Souchay et al (1999). In 2002, IERS recommended to build the
polynomials of precession which do not appear in the solution of Souchay et
al in order to imprve the solution of Lieske et al (1977). From the solution of
the rotation of the rigid Earth SMART97 more precise and which contains the
polynomials of precession, we build a solution of the rotation of the nonrigid Earth
using the MHB2000 model as well as the diurnal and subdiurnal part starting
from a model under development by Mathews (2002).
Lastly, starting from the dynamical quantities /, !, ' and of the geodetic
precession­nutation of Brumberg (1997), we calculate the kinematical quantities
/K , !K , 'K .
The current solutions of the rotation of the Earth are reckoned from the
ecliptic and the equinox J2000. They will have to be calculated again using better
solutions of the disturbing bodies and by referring them to the `ecliptic ICRF'.
References
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VSOP82. Astron. Astrophys., 1982, 114, 278--288.
2. Bretagnon P., Francou G. Planetary theories in rectangular and spherical
variables. VSOP87 solutions. Astron. Astrophys., 1988, 202, 309--315.
3. Bretagnon P., Francou G, Rocher P., Simon J.­L. SMART97: a new solution
for the rotation of the rigid Earth. Astron. Astrophys., 1998, 329, 329--338.
4. Brumberg V. A. Essential Relativistic Celestial Mechanics. Bristol: Adam
Hilger, 1991.
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th'eories analytiques de leur mouvement. Th`ese, Observatoire de Paris, 1999.
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9. Irwin A. W., Fukushima T. A numerical time ephemeris of the Earth. Astron.
Astrophys., 1999, 348, 642--652.
10. Lieske J. H., Lederle T., Fricke W., Morando B. Expressions for the Preces­
sion Quantities Based upon the IAU (1976) System of Astronomical Con­
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11. Mathews P. M., Herring T. A., Buffet B. A. New nonrigid Earth nutation
series and Earth's interior. J. Geophys. Res., 2002 (in press).
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crossed­nutation and spin­orbit coupling effects. Astron. Astrophys. Supp.
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328.
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