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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
An approximate projection of confidence ellipsoids of
space objects positions onto the celestial sphere
A. P. Baturin
Research Institute of Applied Mathematics and Mechanics, Tomsk, Russia
The approximate method of calculation of a visible border of probable motion
regions (PMR) for space objects (SO) has been considered. The PMR has been
defined as a confidence ellipsoid in the phase space. A confidence ellipsoid is
determined by LSM--estimations of motion--parameter vector “
q and its covariance
matrix “
D (LSM is the least squares method). The considered method takes into
account real dynamics of evolution of PMR, namely, the fact of expansion of the
region mainly along the reference trajectory which is determined by the vector
“
q. Besides, the method includes several acceptable simplifications facilitating the
construction of PMR visible border. As a result, the simple analytical expressions
for border points of PMR projection onto the celestial sphere have been obtained.
The method has been tested by comparison of the obtained border with a
projection of the set of points filling a confidence ellipsoid according to six--
dimensional Gaussian distribution determined by the matrix “
D. The algorithm
for obtaining such set of points may be found in [1]. This technique gives the exact
projection of PMR, so it has been used as a standard one. The asteroids Albert
and Icarus have been chosen as models. For the calculation of LSM­estimations
“
q and “
D the various samples of observations of these objects have been used.
The comparison of the proposed method with the standard one has shown
that obtained border is of sufficiently good accuracy in all considered cases.
The work has been supported by RFBR Grant 01­02­17266.
References
1. Pollak Y. G. Probability simulation on computers. M.: Soviet Radio, 1971,
400 (in Russian).
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