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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
Investigation of stability of NEAs orbital resonance
motions by numerical methods
L. E. Bykova
Tomsk State University, Tomsk, Russia
The problem of orbital stability of resonance Near­Earth asteroids (NEAs)
is considered. Most of NEAs have multiple close approaches to the inner ma­
jor planets. The investigation of motion of such objects presents a complicated
problem of celestial mechanics related with the instability of solutions of differ­
ential equations and motions in the vicinity of close approaches. Investigation of
stability of orbital resonances in the motion of NEAs is very important because
the stable resonances make it possible to preserve certain relative geometric con­
figurations of an asteroid and a planet [1] including minimal distances between
orbits. Depending on the initial parameters of orbits the resonances either protect
asteroids from close encounters and collisions with planets prolonging the time
of their life or promote `sweeping up' of asteroids from the neighborhood of the
orbit of the corresponding planet.
The technique of the investigation of NEAs orbital stability has been devel­
oped by author on the basis of the construction of possible motion domains (R)
of asteroids.
By assuming that the distribution law for errors of observations is close to
the normal one, the initial domains R 0 of possible motion for the object under
investigation have been constructed on the basis of the estimation of the vector
“
q 0 of initial dynamical parameters and the covariance matrix “
D 0 of their errors
obtained from analyzing the observations by the least­squares method (LSM)
R 0 : R 0 (“q 0 ; k 2 “
D 0 ); k = 1; 2; 3; : : : : (1)
Here k is the gain factor of LSM­estimations of the covariance matrix “
D 0 (k = 3
corresponds to the `three sigma' rule).
First of all, the initial domain R 0 has been determined at k = 1 and the
evolution of the domain has been considered over a certain time interval. The
initial set of orbits has been obtained for n test particles with respect to a given
center (an initial epoch) with the help of the random number generator according
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to the normal distribution law and the covariance matrix “
D 0 of errors. Further,
several initial domains R 0 (1) have been constructed at various values of k . If
the scattering of the trajectories of the test particles is not extensive over the
considered time interval then new initial domains are constructed at k ? 1 (e. g.
k = 10; 20; : : :). Otherwise it is done for k ! 1 (e.g., k = 0:1; 0:01; : : :). Varying
the coefficient k by this way and considering the evolution of the initial sets of the
test orbits for each k, we have determined the domain of the initial parameters
“
q 0 in which the resonant relations of mean motions (if they are present) remain
to be stable over a time interval under consideration.
This approach has been used to investigate the stability of the orbital reso­
nances of some single NEAs [2] and of NEAs assembly close to the 3:1 resonance
with Jupiter. For each asteroid the problem has been solved beginning from the
analysis of observations and refining the initial parameters of the orbit. The dif­
ferential equations of motion [3] have been integrated numerically by Everhart
19th order method. Perturbations of the major planets and the Moon have been
taken into account using the positions of the planets given by DE200/LE200 and
DE406. The initial osculating elements of orbits have been taken from Bowell
catalog.
Domains of possible motions have been constructed over the time interval
6000 years. The evolution of osculating orbital elements, resonance band ff (ff =
k 1 n a \Gamma k 2 n p , where n a is the mean motion of the asteroid, n p is the mean motion
of a planet and k 1 , k 2 are integers) and critical arguments has been considered
for each investigated object and for corresponding ensembles of 100 clones.
References
1. Murray C. D. and Dermott S. F. Solar System Dynamics. Cambridge: Cam­
bridge University press, 1999.
2. Bykova L. E. and Galushina T. Yu. Orbital evolution of near­Earth asteroids
close to mean motion resonances. Cel. Mech. & Dyn. Astron., 2002, 82, 265--
284.
3. Bordovitsyna T. V, Avdyushev V. A. and Titarenko V. P. Numerical inte­
gration in the general three­bodies problem. Research in Ballistics and Con­
tiguous Problems of Dynamics, Tomsk: Tomsk State University Publishers,
1998, 2, 164--168 (in Russian).
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