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About some properties of jacobian for polynomial mappings of two complex variables

Gawroński B., Jankowska J., Moltzan R., Włodarczyk I., Biernat G.

Journal of Applied Mathematics and Computational Mechanics

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2013

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Vol. 12, nr 4

41--46

EN

In our article we consider jacobian Jac(f,h) of polynomial mapping f = Xk Yk +…+ f1, h = Xk–1 Yk–1 +…+ h1. We give conditions for coordinate h in which constant jacobian Jac(f,h) = Jac(f1,h1) vanishes.

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

Włodarczyk I.

Acta Astronomica

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2007

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Vol. 57, No. 1

103--121

EN

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.

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Stability of the Most Hazardous Mars-Crossers

Kankiewicz P., Włodarczyk I.

Acta Astronomica

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2006

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Vol. 56, No. 4

413--425

EN

The equations of motion of 4190 Mars Crossers (MCs) were numerically integrated to analyze their all possible close approaches to planets in the next 104 years. A sample of asteroids potentially hazardous to Mars was selected and properties of their chaotic motion on larger time scales were determined. For samples of MCs closely approaching Mars, their mean frequency of close encounters was computed. We also analyzed the presence of mean motion and secular resonances. The population of asteroids hazardous to Mars was found and the influence of frequent close approaches and resonances on the stability of their trajectories was estimated. We also estimated the correlation between the frequency of close approaches to Mars and the Lyapunov Time (LT) of these asteroids. Some results concerning the correlation between mean motion/secular resonances and LT were also presented as well as three selected examples of dynamically interesting MCs.

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Prediction of the Motion of Asteroids and Comets Over Long Intervals of Time

Włodarczyk I.

Acta Astronomica

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2001

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Vol. 51, No. 4

357--376

EN

Difference of the mean anomalies of two starting orbits of a minor planet or a comet which only differ by an error of calculating of one of the orbital elements grows rapidly with time. This means that it is almost impossible to predict behavior of minor planets or comets on the orbit outside the period of time called the time of stability in our work. The time of stability for some selected minor planets and comets are given. For some minor planets and comets the time of stability is surprisingly short, about several hundreds years only.