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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
Periodic orbits in photogravitational three--body
problem
A. S. Zimovschikov 1 , N. N. Titova 2 , V. N. Tkhai 1
1 Moscow State Academy of Instrument Making and Computer Science,
Moscow, Russia
2 A. A. Dorodnitsyn Computing Center of Russian Academy of Sciences,
Moscow, Russia
In 1619 Ioghann Kepler was the first to propose the hypothesis of light pressure
in order to explain the reason of comet--tail's deviation.
Indeed, in a number of cases, when studying the motion of celestial objects one
should take into account both gravity and repulsive force of light pressure. This
latter force, being qualitatively close to gravity, may also considerably surpass it.
Both repulsive and gravitational forces form the so--called photogravitational
force field. A number of researches has been devoted to a motion of celestial
objects in such a field.
For the first time it was V. V. Radziewsky who stated and solved some prob­
lems in dynamics of particle's of the photogravitational problem. In the pho­
togravitational three--body problem, formulated by him, in contrast to the clas­
sical problem, one of primaries or both of them are the sources of light repulsion.
In this respect the papers by Radziewsky are of most significance for celestial
mechanics.
The restricted photogravitational problem corresponding to a system ``star--
planet--particle'' is under consideration here. The passively gravitating particle is
exposed to the influence of gravity from two primaries. Besides, it is sunjected to
the repulsive force of light pressure from the star. The gravity F g and the light
pressure force F p of the star are collinear and opposite in sign. The effect of the
repulsive force, which pushes away the particle, is reduced to a ``mass reduction''
of radiating object involving the effective mass Qm. The coefficient of reduction
Q for a given particle is a constant. It determines the total effect both of gravity
and of light pressure. This coefficient is defined by formula Q = 1 \Gamma F p =F g .
By analogy to the classical three--body problem in the photogravitational
problem there exist seven relative equilibria, i.e. collinear (L 1 ; L 2 ; L 3 ), triangular
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(L 4 ; L 5 ) and coplanar (L 6 ; L 7 ) libration points. The positions of these points
are mostly depend on the coefficient of reduction.
The present paper deals with construction and global study of families of
symmetric periodic orbits around all collinear libration points. The evolution of
orbits from one kind to another one as well as the reverse of stability property are
under studying. The method proposed here allows to construct and investigate the
stability of all symmetric periodic trajectories of reversible system. The localizing
of such a trajectory is reduced to its analysis in the half--period time interval. The
stability investigation is performed by solving the joint system which contains the
initial equations and the system in variations in the small neighborhood of the
orbit under consideration.
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