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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
Families of periodic solutions of the planar restricted
three--body problem and their application at
designing the orbit for the space radio telescope
B. B. Kreisman
Astro Space Center of Lebedev Physical Institute of Russian Academy of
Sciences, Moscow, Russia
1. The international project ``RadioAstron'' [1] provides to launch the space
radio telescope with 10--meter antenna and to create together with the global
ground VLBI network a unified system of the ground--space interferometer with
capabilities, approximately in 10 times exceeding capabilities of ground interfer­
ometers. For implementation of this purpose it is required to design regularly
evolved (under the action of the Moon) orbit with radius of apogee ё 400 thou­
sand km and with the greatest possible rotation of its plane and apsides line.
2. The periodic solutions of autonomous Hamiltonian systems [2, 3] belong
to families of the periodic solutions with variable period and index of stability.
For the stable solutions the transformation of the neighbourhood of the solution
for period is reduced to turning through some angle '. If ' rationally expresses
in terms of 2ъ, ' = 2ъ \Delta p=q, p and q are integers, q ? 1, the initial family
with period T is intersected with the family of the periodic solutions with period
qT . Poincare [2] has defined such solutions as the periodic solutions of the
second kind. The monodromy matrix allows to define the directions of
prolongation both of initial family, and of a family, induced by it as
the solutions of second kind. If there is no second integral at q ? 4 in the
neighbourhood of this point of the family of the solutions of second kind there
must exist [3] stable solutions generating their own families of the periodic
solutions of second kind and so ad infinitum.
3. Practically all known periodic solutions of the restricted three--body prob­
lem are symmetric with respect to axis x 1 . It involves that the equations of
motion admit an invariant change of variables. Therefore, the monodromy matrix
has a special form; it is possible to show, that at the absence of the resonance
1:1 the prolongation of symmetric periodic solutions of this problem
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gives only symmetric periodic solutions. It is essential that this state­
ment is valid not only for prolongation of the initial family of periodic
solutions, but for a family, generated by it as solutions of second kind.
In case of a resonance 1:1 it is necessary to take into account the struc­
ture of elementary divisors of the monodromy matrix. In correspondence with
the divisor (ae \Gamma 1) 2 the monodromy matrix has two eigenvectors. One gives the
direction of prolongation of the family of symmetric solutions with variable en­
ergy and duration of period. Second vector gives the direction of prolongation
of new family of the solutions of second kind, for which energy and duration of
period are constant (in the first approximation); these solutions always are
asymmetric. At two other structures of elementary divisors the solutions remain
symmetric.
4. In the restricted three--body problem there exist families of periodic (in
rotating coordinate system) solutions around collinear equilibrium points. For
motion in these orbits Keplerian elements change with a very high speed. If we
select orbital parameters so that it would get in the neighbourhood (in four--
dimensional space of coordinates and velocities) of orbits of one of these families,
the motion will coincide with one around the equilibrium points for some time
and Keplerian elements also will strongly change. The implementation of this idea
has allowed to construct families containing periodic solutions with any rotational
displacement (for one year) of apsides line.
5. The analysis of orbits obtained with the above described algorithms, has
allowed to select a direction of searching real orbits which are met the require­
ments of the project ``RadioAstron''. One of such orbits is described in detail
in [1].
References
1. Kardashev N. S., Kreisman B. B., Ponomarev Yu. N. New orbit and new
capabilities of the project ``RadioAstron''. In: Radioastronomical engineering
and methods. M.: LPhI, 2000, 228, 3--12.
2. Poincare H. Les methodes nouvelles de la Mecanique celeste, t. 1--3. Paris,
Gauthier--Villars, 1892, 385 pp.; 1893, 479 pp.; 1899, 414 pp.
3. Bruno A. D. Theory of orbits. M.: Nauka, 1990, 296 pp. (in Russian).
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