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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
Determination of preliminary orbits including
perturbations
V. A. Shefer
Research Institute of Applied Mathematics and Mechanics, Tomsk State
University, Tomsk, Russia
On the basis of the theory of intermediate orbits developed earlier by the au­
thor [1], a new approach to the solution of the problem of preliminary orbit
determination is suggested. This approach enables to take into account the main
perturbations in celestial body motion. The motion under consideration is repre­
sented as a combination of two motions and the corresponding orbit is construct­
ed. The first motion is the uniform rectilinear motion of the fictitious attracting
centre, the mass of which varies in accordance with the first Meshchersky law.
In this case we suppose that the mass of the fictitious centre can take not only
positive values but negative ones as well. The second motion is the motion around
the fictitious centre. It is described by the equations of the Gylden­Meshchersky
problem [2]. The parameters of the required orbit are determined from the bound­
ary conditions of the problem solved and from a number of additional conditions
making it possible to choose the most optimal solution. These parameters are
such that their limiting values at the reference epoch determine the superoscu­
lating intermediate orbit with the third--order tangency. The construction of the
orbit sought is not related with any restrictions in the choice of forces acting on a
body. The only requirement to them consists in the fact that the expressions for
the total vector of acceleration must be a non--zero vector at the reference epoch.
The orbit constructed approximates the real perturbed motion in the neighbour­
hood of the reference epochs better than the Keplerian orbit of the two--body
problem and analogous orbits of other authors.
By the approach suggested the classical problems of orbit determination from
two positions and three complete observations are solved. The algorithm for solv­
ing the second problem includes the solution of the first problem and can be
considered as a generalization of the well--known Lagrange--Gauss method [3, 4].
By the example of orbital motion of the minor planet 1566 Icarus the comparison
of the results of the classical Lagrange--Gauss method and the generalized method
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has been performed. The comparison shows that the new method is a highly ef­
ficient tool for studying the perturbed motion. Its advantage over the classical
method is especially significant in case when we have high accurate observations,
which span short orbit arcs.
Aside from the application to the determination of an unknown orbit from
astronomical positional observations, the method proposed can be very useful in
solving a set of problems of space flight dynamics.
References
1. Shefer V. A. Osculating and Superosculating Intermediate Orbits and Their
Applications. Cel. Mech. & Dyn. Astron., 2002, 82, 19--59.
2. Mestschersky I. W. Works in Mechanics of Bodies with Variable Masses. M.:
GITTL, 1952 (in Russian).
3. Subbotin M. F. Introduction to Theoretical Astronomy. M.: Nauka, 1968,
(in Russian).
4. Escobal P. R. Methods of Orbit Determination. N. Y.--London--Sydney: John
Wiley and Sons, Inc., 1965.
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