Error Propagation of the Computed Orbital Elements of Selected Near-Earth Asteroids

Computed orbital elements of asteroids contain errors depending on the errors of observations. In accordance with the procedure described by Sitarski (1998) we can find randomly selected sets of orbital elements which reasonably represent all observations with fixed mean rms residual. In this way we can obtain the error ellipse of the initial orbital elements, and that of the predicted ones. By integrating equations of motion of these computed clones we can obtain a time evolution of changes of the shape of the torus, inside which all the orbits of the clones exist. The time evolution of the configuration of the torus and its size are connected with the asteroid position inside this torus. The larger is the torus the more difficult it is to find the position of the asteroid. The shape of the torus and its time evolution depend mainly on the kind of the asteroid's orbit. If the orbit is more chaotic, then changes of the torus shape are more rapid and the size of the torus is larger. Close approaches of asteroids to planets are the main source of the chaotic motion. This is particularly important in computing their close approaches to Earth. The distances between the minor planet on the nominal orbit and the virtual minor planets around the nominal orbit can attain considerable values. In this work we computed the time necessary for the values of the mean distances of the clones to achieve the dimensions of the Earth radius. In this respect, we investigated the motion of the known earlier asteroids 433 Eros and 1943 Anteros, and the recently discovered minor planets 99942 Apophis (2004 MN4) and 2004 VD17 - the most dangerous to the Earth, according to the Impact Risk Page of NASA (http://neo.jpl.nasa.gov/risk/). It appears that time-span after which dimensions of the torus attain well defined values are strongly correlated with the stability time and they are also connected with frequent and close approaches to the planets. Furthermore, it was investigated whether the computed orbital elements of the asteroids for the epoch of the beginning, middle or end of the observation, influence the behavior of the asteroids. Also the propagation of the region of uncertainty of asteroid position was computed. This can simplify the computing of close approaches of these asteroids to the Earth and the impact risk assessment.