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1

On the Expansion of the Direct Part of the Disturbing Planetary Function

Kamel O. M., Soliman A. S.

Mechanics and Mechanical Engineering

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2016

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Vol. 20, nr 4

371--375

EN

We generalize the expansion of Murray–Dermott for the direct part of the disturbing function using Taylor‘s theorem. We present the values of Δ-s for s = 1, 3, 5, . . . which is essential for high order planetary theories. Murray – Dermott executed the expansion for s = 1 which is necessary for only first order theories.

2

The Potentially Hazardous Asteroid (410777) 2009 FD

Wlodarczyk I.

Acta Astronomica

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2015

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Vol. 65, No. 2

215--231

EN

The asteroid (410777) 2009 FD is placed at the top of the JPL NASA Sentry Risk Table. We show that the predicted probability of the potential impact of the asteroid (410777) 2009 FD in fact depends on (i): used methods of selection and weighing of observational data, (ii): adopted dynamical model with included non-gravitational effects in the motion of asteroid based on cometary approach, and considerably weaker depends on (iii): the used model of the Solar System with different number of massive asteroids as potential perturbers. We computed impact solutions of the asteroid (410777) 2009 FD based on its 296 optical observations from February 24, 2009 to April 02, 2014 and one radar observation from April 07, 2014. We used the freely available ORBFIT Software Package and studied the future evolution of the orbit of (410777) 2009 FD searching for close approaches with the Earth and for the possible impacts up to the year 2200. According to our study the impacts are possible in the years: 2185, 2186, 2190, 2191, 2192, 2194, 2196, and 2198, provided the non-gravitational parameter A2 in the range of (-46.0,+3.0)×10-15 a.u./d2, with the gap between (-25.0,-11.0)×10-15 a.u./d2.

3

On the Dynamical Evolution of Scattered Disk Objects Outside the Planetary System

Gabryszewski R., Rickman H.

Acta Astronomica

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2010

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Vol. 60, No. 4

373--385

EN

We report the results of dynamical simulations, covering Gyr timescales, of fictitious Scattered Disk Objects as a follow-up to an earlier study. Our dynamical model is similar in that it does not include external agents like passing stars or the Galactic tide. Only the four giant planets are explicitly treated as perturbers. We analyze the random-walk behavior of the inverse semi-major axis by means of a simplified circular restricted 3-body problem as an approximate analogue. Our results concerning the role of resonant effects and the transfer efficiency into the orbital energy domain of the inner Oort Cloud are in broad agreement with the earlier papers, and we confirm the important role of external objects (with perihelia beyond Neptune's orbit) in feeding the Oort Cloud. We estimate the efficiency of this transfer to be even somewhat higher than previously found.

4

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.

5

Generating of "Clones" of an Impact Orbit for the Earth-Asteroid Collision

Sitarski G.

Acta Astronomica

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2006

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Vol. 56, No. 3

283--292

EN

If we find an impact orbit of the Earth-crossing asteroid we can determine the impact point location on the Earth surface. If we want to find other orbits, very similar to the impact one, we have to select randomly a number of such "clones" and to integrate equations of motion many times from the osculation epoch to the collision date. Then we can determine a path of hypothetical impact points on a map of the Earth. We elaborated a method allowing us to avoid the repeating of long-term integration. The method is based on a special feature of the cracovian least squares correction applied to the random orbit selection. After finding the impact orbit we randomly select an arbitrary number of "clones", perform only one time-consuming integration, and find quickly many similar impact orbits for the collision date. We applied our method for four chosen asteroids: 2004 VD17, 1950 DA, Apophis (2004 MN4), and Hathor. We show that we are able to "clone" the impact orbit in a very difficult case and when it is impossible to do this in another way.