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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.

2

Warsaw Ephemeris of the Solar System: DE405/WAW

Sitarski G.

Acta Astronomica

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2002

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Vol. 52, No. 4

471--486

EN

An ephemeris of the Solar System in rectangular coordinates is produced by numerical integration of equations of motion of nine planets, the Moon, and the Sun. Recurrent power series method of 26th order is applied when integrating the equations of motion in barycentric equatorial coordinates. Initial data of the JPL DE405 ephemeris were used to prepare the starting data for our integration. A comparison with the JPL ephemeris shows that our ephemeris gives e.g., the solar coordinates in accordance to within ±5×10-8 a.u. in the interval of thousand years. Starting data for integration of equations of motion of five outer planets were prepared. Initial values of barycentric coordinates and velocity components for Ceres, Pallas, Vesta, and Hygiea were also found on the base of a large number of astrometric observations of those asteroids. A subroutine providing the planetary coordinates and velocity components for any date may be included as a source of necessary planetary data in computer routines for orbital computations. For a convenient practical use, we stored values of coordinates and velocity components of the Solar System in a file for 1001 dates from 1493 January 25.0 to 2150 March 11.0 every 240 days to choose the proper starting data for planetary integration. Examples of the orbital computations using the DE405/WAW ephemeris are presented.

3

On the Small Asteroid 1994 GV

Sitarski G.

Acta Astronomica

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2000

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Vol. 50, No. 3

417--419

EN

Minor planet 1994 GV was observed during only three days but it passed near the Earth and ran over the 28-degree arc on the sky. The orbit of this minor planet crosses the Earth orbit in the descending node. It appeared that according to this nominal orbit determined from the three-day observation arc, a very close approach of the asteroid to the Earth to within 0.000114 a.u. could happen in April 2008. However, our investigations of 600 randomly selected orbits as well as a search of the impact orbit excluded a possibility of a collision of 1994 GV with the Earth.

4

On the Lost Comet 18D/Perrine-Mrkos

Sitarski G.

Acta Astronomica

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1999

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Vol. 49, No. 1

113--116

EN

Periodic comet Perrine-Mrkos was observed during its three consecutive returns during 1955, 1961/62, 1968/69, and since that time it has never been seen. Probably large nongravitational effects quickly changing with time did not allow to make an exact prediction of the future comet's returns. We used 66 astrometric observations of the comet from 1955-1969 to improve the comet's orbits and to determine values of nongravitational parameters. We applied the method based on the rotating and precessing cometary nucleus with a forced precession to determine seven nongravitational parameters. Thus, we found satisfactory solution linking well the last three apparitions with the rms residual equal to 1''.90. The solution gives a very good fit to the earlier observed perihelion passages of the comet in 1909 and in 1896. Therefore, we were able to make prediction of the comet's return in 2002 and we calculated the search ephemeris.

5

Motion of the Dangerous Asteroid 1997 XF11

Sitarski G.

Acta Astronomica

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1999

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Vol. 49, No. 1

103--112

EN

Minor planet 1997 XF11, discovered in December 1997, is moving in the heliocentric orbit which almost intersects the Earth orbit. In October 2028 the asteroid will approach the Earth to within 0.006 a.u. We improved the asteroid's orbit on the base of 151 astrometric observations from 1990-1998. To investigate the long-term motion of the asteroid we randomly selected 500 sets of orbital elements and numerically integrated the equations of motion by the recurrent power series. We followed evolution of minimum distances between orbits of 1997 XF11 and the Earth for moments of consecutive passages of the asteroid through its descending node. We found that only in 2042 the minimum distance between orbits of both planets could be smaller than the Earth radius. However, the asteroid will pass through its descending node in July 2042 while a collision with the Earth could happen during the October encounter. After 2042 the minimum distance between orbits of both planets will be permanently growing up, and hence we estimate that during the next several thousand years collision with the Earth will be impossible. We also investigated the asteroid's motion before 1990. We found that past close approaches of 1997 XF11 to the Earth occurred in 1971 to within 0.032 a.u. and in 1957 to within 0.015 a.u. We calculated ephemerides of the asteroid for those past approaches aiming at finding some old observations of the minor planet. We have also studied accuracy of prediction of the future motion of 1997 XF11 based, however, on 142 observations of the asteroid from 1997/98 only. We found that in that case possibility of collision during 2030-2050, although possible, is completely unpredictable.

6

How to Find an Impact Orbit for the Earth-Asteroid Collision

Sitarski G.

Acta Astronomica

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1999

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Vol. 49, No. 3

421--431

EN

The Earth-crossing asteroids can approach the Earth at dangerously small distances. If the observation arc of a single apparition orbit is short the determined orbital elements and hence a prediction of the future encounter with the Earth are uncertain. We presented the method of finding an impact orbit by the least squares correction with the "forced" equality constraints. As a solution we obtain: (i) initial values of rectangular coordinates and velocity components (and hence the orbital elements) allowing the asteroid to collide with the Earth, and (ii) the minimum value of the \it rms residual resulting from the impact orbit. The latter value is very important since it may serve as a kind of measure of probability of the collision, and in any case it allows us to exclude a possibility of the expected catastrophe. We performed computations of the impact orbits for two asteroids: 1997 XF11 and 1999 AN10. We found two impact orbits for hypothetical collisions of 1997 XF11 with the Earth in 2028 and 2033, and four orbits for 1999 AN10 colliding with the Earth in 2027, 2034, 2036, and 2039. Based on the 101 observations of 1999 AN10 from 1999 Jan. 13 - May 16, we show that its collision with the Earth in 2027 is impossible, but in 2039 it would be more probable. We made also a numerical simulation of the fictitious asteroid which would certainly collide with the Earth. We show that in this case we can easily find the impact orbit, and the rms value evidently does not allow us to exclude a possibility of the collision.

7

Motion of the Minor Planet 4179 Toutatis: Can We Predict Its Collision with the Earth?

Sitarski G.

Acta Astronomica

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1998

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Vol. 48, No. 3

547--561

EN

Minor planet 4179 Toutatis is an Apollo type object with a very small orbit inclination (i=0°.47), hence it has a possibility to approach closely the Earth (an encounter to within 0.01 a.u. is expected in 2004) and might be a good candidate for a future collision with the Earth. We collected 640 astrometric observations of Toutatis from the period 1934-1997 to improve the orbit. We had to include a nongravitational term into equations of motion expressed by a secular change a of the semi-major axis a of the Toutatis orbit to obtain a fully satisfactory solution of the orbit determination. A value a=-0.16×10-10 is two orders smaller than that determined in the case of short-period comets with known nongravitational effects. To investigate the long-term motion of Toutatis we numerically integrated the equations of motion by recurrent power series taking into account perturbations caused by the eight planets from Mercury to Neptun, treating the Earth and Moon as separate bodies, and also by the four biggest asteroids. We randomly varied the orbital elements to examine the Toutatis' motion for a number of different orbits. We present a new method of the random orbit selection which allows us to find a set of different orbits but representing well all the observations used for the orbit correction. Our results confirm a conclusion found by other authors that Toutatis orbit is exceptionally chaotic. Therefore, we are not able to predict the motion of Toutatis further than for 300 years. However, our integrations spanning 1500 years showed that the evolution of position of the descending node of Toutatis' orbit might go also in such a direction that the orbits of Toutatis and of the Earth would intersect in the future. Hence a possibility of the Toutatis-Earth collision is not excluded but it is completely unpredictable. To investigate conditions of a hypothetical collision of a minor planet with the Earth we made the following numerical simulation. Based on the Toutatis' orbit we deduced such orbital elements for a fictitious minor planet "Fatum" that a shape of the orbit was very similar to that of Toutatis, but we knew in advance that "Fatum" would certainly collide with the Earth in September 2004 and we calculated values of the impact parameters. We created a set of 638 artificial observations of "Fatum" in 1988-1997 for the same dates and with the same random observational errors like those of Toutatis. Then we corrected the "Fatum's" orbit for different observational intervals to examine the exactness of the impact prediction in 2004. We found that in 1993 we would be sure that the collision is inevitable, and in 1997 we could determine an impact area on the Earth's surface in range of a square of 100×100 km. We show that if we knew the impact date so early we could undertake an action to avoid the collision by trying to change the "Fatum's" heliocentric velocity only by one cm/sec.

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Forced Precession Model for Three Periodic Comets: 30P/Reinmuth 1, 37P/Forbes, and 43P/Wolf-Harrington

Królikowska M., Sitarski G., Szutowicz S.

Acta Astronomica

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1998

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Vol. 48, No. 1

91--102

EN

The nongravitational motion of three short-period comets - discovered in the twenties and running on similar heliocentric orbits - has been investigated. We used the Sekanina's forced precession model of the rotating cometary nucleus to include the nongravitational terms into equations of the comet's motion. Values of six precessional parameters: A, η, I, φ, fp and s have been determined along with corrections to orbital elements from astrometric observations of the comets. We were able to link successfully all the observations of each comet over interval of time spanning about seventy years. According to our solutions, the nucleus of comet Reinmuth 1 is oblate whereas those of comets Forbes and Wolf-Harrington are prolate along the spin-axis.