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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
Dynamics of artificial satellite orbits with the effects
of luni­solar perturbations
Y. A. Abdel­Aziz 1;2 , E. Wnuk 2
1 National Research Institute of Astronomy and Geophysics (Helwan
Observatory), Helwan, Cairo, Egypt
2 Astronomical Observatory of Adam Mickiewicz University, Sloneczna 36,
60­286 Poznan, Poland
In this paper we propose an analytical solution of some problems of Earth's
artificial satellite moving on high orbits (at altitude above 20 000 km). Perturba­
tions due to geopotential (zonal and tesseral harmonics) and luni­solar attractions
are considered up to the second order. The solution will avoid the singularities
from small eccentricities and small inclinations. We derive the theory of perturba­
tion using the Hori­Lie algorithm and the Hamiltonian canonical transformaion.
We discuss different classes of orbits, in particular geostationary, close to geosta­
tionary and geosynchronous as well as 1:1 and 1:2 resonance cases. We present
also comparision of our analytical solution with numerical integration of motion
for chosen artificial satellites.
High Earth's artificial satellites were set on geostationary or 12­hr orbits (for
example GPS, GLONASS). The region of geostationary ring (GEO, GTO) is now
very crowdy so, there is a very real danger of collision among objects moving on
these orbits. This is the reason for searching for other types of satellite orbits. One
of them might be geosynchronous orbit, but in contrast to geostationary ones its
inclination to the equator is considerably different from zero. Other possibilites
are orbits with perigee close to Earth and apogee being at the distance of 40 000
km or even farther away.
To determine and calculate the position of satellites on these types of orbits
methods of numerical intergation of equations of motion were used until now.
Existing analytical methods can solve this problem only with low accuracy. Dif­
ficulties are caused mainly by the lack of satisfactory analytical solution of the
resonance problem for geosynchronous orbits as well as by the lack of efficient an­
alytical theory combining luni­solar perturbations with geopotential attraction.
Numerical integration is time consuming in some cases, and then for qualitative
analysis of satellite's motion it is necessary to apply analytical solution. This is
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the main reason for developing analytical theory of the motion of high artificial
satellites.
To achieve this purpose, we construct an analytical theory of motion of high
Earth's artificial satellites and cosider the following items by
(1) including the combined effects of geopotential (zonal and tesseral harmon­
ics) and luni­solar attractions;
(2) describing the motion of high geosynchronous satellites;
(3) applying the theory for some satellite missions.
The theory of satellite's motion is derived by applying the Hori­Lie algorithm
and the Hamiltonian canonical transformation method. The initial Hamiltonian
of the problem includes the influence of the gravitation fields of the Earth, the
Sun and the Moon. Single canonical transformation eliminating the periodic terms
(long and short period terms) is applied for the generating functions. With the
use of these functions the final formula describing different types of perturbations
is obtained. Solving the resonance problem needs special treatment.
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