Mathematical modeling of the asteroids' motion belonging to~the~Apollo and Aten groups

A. F. Zausaev ; M. A. Romanyuk ; A. A. Zausaev

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 24 (2020) no. 4, pp. 692-717.

Résumé

This article evaluates the accuracy of solutions to differential equations of motion, taking into account relativistic effects obtained on the basis of a new principle of interaction, using the example of studies of the evolution of the orbits of five asteroids. A numerical integration of equations of the asteroids' motion with the initial data referred to different points in time is carried out. Based on a comparison of the results of the study, certain patterns are revealed. At time intervals in the absence of rapprochement of the asteroid with the Earth less than 0.1 au it is possible to apply with equal efficiency the differential equations given in the paper. The loss of accuracy of numerical integration is directly dependent on the magnitude of the rapprochement of the asteroid width the Earth. Due to the fact that in the right sides of the equations of motion we have differences of the coordinates of the asteroid and the planet, with sufficient proximity, the relative accuracy of the coordinates is many times greater than the relative accuracy of the difference. For the studied asteroids, when they approach the Earth, the relative error of the difference in the coordinates of the asteroid and the Earth is approximately 227 to 44900 times higher than the limiting relative error of the coordinates of the asteroid itself. Predicting the motion of Apophis after its close approach to the Earth based on the solution of the equations of motion by modern methods leads to large errors, the reduction of which is possible only by improving the initial data of the elements of the orbits of the asteroid. About the possibility of close approach of Apophis with the Earth on a time interval from April 14, 2029 to January 1, 2100 it can be argued with a certain degree of probability. The results of the research can be generalized to all asteroids of Apollo and Aten groups.

Détail

Keywords: numerical integration, differential motion equations, asteroids, 1999 AN$_{10}$, 2001 WN$_5$.
Mots-clés : (99942) Apofis, (367943) Duende, 2012 UE$_{34}$

Bibliographie

[1] Asteroidno-kometnaia opasnost': vchera, segodnia, zavtra [Asteroid-Comet Hazard: Yesterday, Today, and Tomorrow], ed. B. M. Shustov, L. V. Rykhlova, Fizmatlit, Moscow, 2010, 384 pp. (In Russian)

[2] Krasinsky G. A., Pitjeva E. V., Vasilyev M. V., Yagudina E. I., “Hidden mass in the asteroid belt”, Icarus, 158:1 (2002), 98–105 | DOI

[3] Adushkin V. V., Basiev E. U., Zykov Yu. N., et al., Katastroficheskie vozdeistviia kosmicheskikh tel [Catastrophic Impacts of Cosmic Bodies], Akademkniga, Moscow, 2005, 310 pp. (In Russian)

[4] Chebotarev G. A., Analytical and Numerical Methods of Celestial Mechanics, Modern Analytic and Computational Methods in Science and Mathematics, 9, American Elsevier Publishing Co., Inc., 1967, xviii+331 pp. | MR | MR | Zbl

[5] Subbotin M. F., Vvedenie v teoreticheskuiu astronomiiu [Introduction to Theoretical Astronomy], Nauka, Moscow, 1968, 800 pp. (In Russian)

[6] Brumberg V. A., Reliativistskaia nebesnaia mekhanika [Relativistic Celestial Mechanics], Nauka, Moscow, 1972, 384 pp. (In Russian) | MR

[7] Zausaev A. F., “The Investigation of the Motion of Planets, the Moon, and the Sun Based on a New Principle of Interaction”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2014, no. 3(36), 118–131 (In Russian) | DOI | Zbl

[8] Zausaev A. F., Romanyuk M. A., “Comparison of various mathematical models on the example of solving the equations of the movement of large planets and the Moon”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 23:1 (2019), 152–185 (In Russian) | DOI | Zbl

[9] Standish E. M., JPL Planetary and Lunar Ephemerides, DE405/LE405. Interoffice memorandum: JPL IOM 312. F–98-048, 1998, August 26, 18 pp. ftp://ssd.jpl.nasa.gov/pub/eph/planets/ioms/de405.iom.pdf

[10] Zausaev A. F., Zausaev A. A., Matematicheskoe modelirovanie orbital'noi evoliutsii malykh tel Solnechnoi sistemy [Mathematical Modelling of Orbital Evolution of Small Bodies of the Solar System], Mashinostroenie-1, Moscow, 2008, 250 pp. (In Russian)

[11] Zausaev A. F., Romanyuk M. A., Chislennye metody v zadachakh matematicheskogo modelirovaniia dvizheniia nebesnykh tel v Solnechnoi sisteme [Numerical Methods in the Problems of Mathematical Modeling of the Motion of Celestial Bodies in the Solar System], Samara State Technical Univ., Samara, 2017, 265 pp. (In Russian)

[12] Newhall X. X., Standish E M., Williams J. G., “DE 102: A numerically integrated ephemeris of the Moon and planets spanning forty-four centuries”, Astron. Astrophys., 125:1 (1983), 150–167 | Zbl

[13] Zausaev A. F., Abramov V. V., Denisov S. S., Katalog orbital'noi evoliutsii asteroidov, sblizhaiushchikhsia s Zemlei s 1800 po 2204 gg. [Catalog of Orbital Evolution for Asteroid Approaching the Earth since 1800 until 2204], Mashinostroenie, Moscow, 2007, 608 pp. (In Russian)

[14] Everhart E., “Implicit single-sequence methods for integrating orbits”, Celestial Mech., 10:1 (1974), 35–55 | DOI | MR | Zbl

[15] Bordovitsyna T. V., Sovremennye chislennye metody v zadachakh nebesnoi mekhaniki [Modern Numerical Methods in Celestial Mechanics Problems], Nauka, Moscow, 1984, 136 pp. (In Russian)

[16] Zausaev A. F., Zausaev A. A., Ol'khin A. G., “The numerical integration of the equations of motion for large planets (Mercury and Pluto) and the Moon with the radar observations”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2004, no. 26, 43–47 (In Russian) | DOI

[17] Sokolov L. L., Bashakov A. A., Pitjev N. P., “Peculiarities of the motion of asteroid 99942 Apophis”, Sol. Syst. Res., 42:1 (2008), 18–27 | DOI

[18] Zausaev A. F., Derevyanka A. E., “Comparative analysis of mathematical models for estimating the impact probability of asteroid Apophis”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2012, no. 2(27), 192–196 (In Russian) | DOI

[19] Shor V. A., Kochetova O. M., Sokolov L. L., “Dangerous asteroid (99942) Apophis”, Asteroidno-kometnaia opasnost': vchera, segodnia, zavtra [Asteroid-Comet Hazard: Yesterday, Today, and Tomorrow], ed. B. M. Shustov, L. V. Rykhlova, Fizmatlit, Moscow, 2010, 206–223 (In Russian)

[20] Hamming R. W., Numerical Methods for Scientists and Engineers, Dover Publ., New York, NY, 1986, xii+721 pp. | Zbl

[21] Demidovich B. P., Maron I. A., Computational Mathematics, Mir Publ., Moscow, 1981, 688 pp.

[22] Zausaev A. F., Denisov S. S., Derevyanka A. E., “Research of the orbital evolution of asteroid 2012 DA14”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2012, no. 3(28), 211–214 (In Russian) | DOI