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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
Solution stabilization algorithms for the estimation of
small bodies orbital parameters
L. E. Bykova, V. P. Titarenko
Tomsk State University, Tomsk, Russia
Sometimes there is not enough information for the correct improvement of
the orbits of the Solar system small bodies. As a rule it is caused by lack of
observations or by too short arc of orbit covered by observations. Therefore one
can obtain only a set X of acceptable solutions satisfying the given observations
with the adequate accuracy.
In this work some stabilization algorithms for solution of such problems have
been considered. Algorithms have been constructed on the basis of a linear least
squares method using additional a priori information about the solution. This
additional information is given as restrictions on the solution or its components
[1].
This approach has been developed for the estimation of the initial orbital
parameters of asteroids moving in the vicinity of a resonance with one of large
planets. A condition of keeping the smallness of the resonance bound ff = k 1 n a \Gamma
k 2 n p , where n a is mean daily motion of an asteroid, n p is mean daily motion
of planet, k 1 ; k 2 are integer numbers, is taken as the restriction on the solution.
Based on this condition two algorithms have been constructed.
In the first algorithm additional observations are simulated. Then this ob­
servations are used together with real ones in orbital parameters improvement
by least squares method. The simulated observations are constructed on the test
orbit taken from the region of possible motion of the object. This orbit satisfies
the condition jffj џ ae, where ae ? 0 is a given small quantity. The technique of
construction of possible motion regions is described in [2].
In the second algorithm the least square problem with linear restrictions in the
form of the inequalities for the major axis of asteroid is solved. These restricting
inequalities correspond to a resonance of the object under consideration.
For investigation of the efficiency of such approach the numerical experiment
for the example of some asteroids moving in the vicinity of 3/1 resonance with
Jupiter and having close encounters with inner planets has been made. Equations
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of motion have been integrated numerically by the Everhart method [3]. Pertur­
bations from all large planets and the Moon which coordinates were taken from
ephemerides DE200/LE200 have been taken into account. Equations of motion
in rectangular coordinates have been used in the first algorithm. In the second
one Keplerian elements have been used.
To avoid additional computation errors a singular decomposition algorithm
for the matrix of the conditional equations have been used in the least squares
problem solution. It is well known that this algorithm is stable to the errors of
initial data.
Numerical experiments have shown that both methods allow to obtain ac­
ceptable solutions. However the use of the first algorithm for finding a test orbit
that satisfies condition jffj џ ae, where ae ? 0 is a given small quantity, is not
possible for all asteroids.
References
1. Lawson Ch. E., Hanson R. J. Solving Least Squares Problems. Prentice­Hall,
Inc., Englewood Cliffs, New Jersey, 1974.
2. Bykova L. E., Galushina T. Yu. Orbital evolution of near­Earth asteroids
close to mean motion resonances. Cel. Mech. & Dyn. Astron. 2002, 82, 265--
284.
3. Everhart E. An efficient integrator that uses Gauss­Radau spacing. In: Dy­
namics of comets: their origin and evolution (A. Carusi and G. B. Valsecchi,
Eds.), Dordrecht: Reidel, 1985, 185--202.
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