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Evolution of near-Earth asteroids close to mean motion resonances
Planetary and Space Science 49 (2001) 811–815
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Evolution of near-Earth asteroids close to mean motion resonances L.E. Bykova ∗ , T.Yu. Galushina Research Institute of Applied Mathematics and Mechanics, Tomsk State University, 634050, Tomsk, Russia Received 1 September 2000; received in revised form 6 March 2001; accepted 6 March 2001
Abstract The dynamics of asteroids near mean motion resonances is considered in the paper. Results of study of orbital evolution of NEAs 2608 Seneca, 3103 Eger, 3838 Epona, 1994 RB, 1994 CB are presented. Equations of the motion of asteroids have been integrated numerically. In the process of numerical integration perturbations from planets and the Moon have been taken into account. The interval of time is ranging from −3000 to 3000 yr. Evolution of ensembles of 100 test particles with orbital elements nearby their nominal orbits has been c 2001 Elsevier Science Ltd. All rights reserved. considered for each object. Keywords: Asteroid; Mean motion resonance; Numerical simulation
1. Introduction In the paper, the orbital evolution of some near-Earth asteroids (NEAs) near mean motion resonances with the Earth and other planets is considered. Investigation of orbital resonances in the motion of NEAs is very important because stable resonances make it possible to preserve a certain relative con8gurations of an asteroid and planets (Murray and Dermott, 1999) including minimal distances between orbits. Depending on the initial parameters of orbits resonances either protect asteroids from close encounters and collisions with planets prolonging the time of their life or promote “sweeping up” of asteroids from the neighborhood of a corresponding planet orbit. Many papers were devoted to the investigation of population of asteroids approaching the Earth and to the problem of “asteroid hazard” (e.g., Milani et al., 1989; Muinonen, 1999; Tancredi, 1998a; Wiegert et al., 1998). The goal of this paper is to study a medium-term (for several thousand years) orbital evolution of real NEAs: 2608 Seneca, 3838 Epona, 3103 Eger, 1994 CB, 1994 RB. Numerical experiments were carried out by us to solve the following problems: to reveal close encounters and low - order mean motion commensurabilities of these asteroids with the Earth and other planets; to determine stability of revealed resonances and their part in protecting asteroids from close encounters. ∗ Corresponding author. Tel.: +7-3822-410-576; fax: +7-3822-410347. E-mail address: [email protected] (L.E. Bykova).
For prediction of motion of a real NEA on the long time interval it is insuDcient to investigate the evolution of some nominal orbit because the accuracy of initial parameters of the orbit is limited by the precision of the available observational data. Therefore, evolution of ensemble of 100 test particles with initial parameters nearby the nominal orbit has been considered for each NEA.
2. Techniques of investigation We applied the following technique of the investigation of orbital evolution of the real NEAs. Equations of the motion of asteroids were integrated numerically by Everhart 19th order method (Everhart, 1974). In the process of numerical integration perturbations from planets (except Pluto) and the Moon were taken into account using the positions of the planets given by DE406. In order to achieve a higher accuracy of numerical integration, new forms of regularizing and stabilizing equations of motion (Bordovitsyna et al., 1998) obtained by KS-transformation (Stiefel and Scheifele, 1971) were used. For every asteroid, the problem was solved beginning from the analysis of observations and re8ning the initial parameters of the orbit. The evaluation of parameters was carried out by the least-squares method. By comparing the results of toward-and-backward integration of the equations of motion of each asteroid, an admissible interval of prediction from the point of view of preserving the admissible accuracy was determined. Then for each investigated
c 2001 Elsevier Science Ltd. All rights reserved. 0032-0633/01/$ - see front matter PII: S 0 0 3 2 - 0 6 3 3 ( 0 1 ) 0 0 0 3 8 - 1
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L.E. Bykova, T.Yu. Galushina / Planetary and Space Science 49 (2001) 811–815
Table 1 Extremal values of elements of the asteroid nominal orbits. Deviations of extreme and isochronous solutions for a set of 100 orbits from the nominal orbits in the interval (−3000, +3000 yr) Element El
Elextrem max
GElextrem
GElissochr:
Elextrem
min
max
mean
max
mean
2.4403 0.5194 7.782
1×10−2 5×10−2 2
9×10−3 3×10−2 4×10−2
2×10−1 1×10−1 12
1×10−1 5×10−2 9
1.4031 0.3256 20.934
3×10−6
1×10−6
6×10−5
2×10−5
8×10−5
2×10−5
5×10−4
2×10−4
2608 Seneca a (AU) e i (deg)
2.5688 0.6672 19.189
min
GElissochr:
max
mean
max
mean
1.5032 0.5593 29.273
5×10−4 9×10−7 9×10−2
1×10−4 3×10−7 2×10−2
2×10−3 2×10−3 4×10−1
5×10−4 2×10−4 6×10−2
2.4732 0.3834 26.605
2×10−2 3×10−2 8
4×10−3 2×10−3 1
6×10−2 3×10−1 16
3×10−2 8×10−2 4
3838 Epona
3103 Eger a (AU) e i (deg)
max
GElextrem
1.5060 0.7020 40.649 1994 RB
1.4083 0.3550 22.118
8×10−7
3×10−7
2×10−5
7×10−6
2.5271 0.6387 40.899
Fig. 1. Close approaches of 2608 Seneca (a1), 3838 Epona (b1), 3103 Eger (c1), 1994 RB (d1) with the Earth (“•”) and with Mars (“?”), the resonance band (in /day) for these asteroids correspondingly (a2), (b2), (c2), (d2). Sets from 100 test particles are shown by a grey background, nominal orbits are marked out in black color.
object an ensemble of particles with initial parameters close to the nominal orbit was constructed and its evolution was retraced on the permissible interval of time. The initial set of orbits has been generated for 100 test particles in relation to the selected center (initial epoch) with the help of a random number generator on the basis of the normal law of dispersing and a corresponding covariation matrix. 3. Numerical simulation The described approach was applied to the construction of possible motion domains of numbered asteroids 2608 Seneca, 3838 Epona, 3103 Eger, and unnumbered 1994 CB, 1994 RB. All close approaches and commensurabilities of low order with the Earth and other planets were revealed in
the process of numerical simulation of the motion of these objects. Commensurabilities were determined by means of estimation of the resonance band value = k1 na –k2 np , where na is a mean motion of an asteroid, np is a mean motion of a planet, k1 ; k2 are integers. The initial parameters of orbits were taken from E. Bowell catalog on January 22, 1999 (ftp://ftp.lowell.edu/pub/elgb/astorb.dat). For the evaluation of probable errors, we used the following observational data: 3838 Epona–99, 3103 Eger–190 and 2608 Seneca–53 observations in several oppositions, 1994 RB–21 observations on the interval of 2 days, 1994 CB-124 observations on the interval of 211 days. In Table 1 and Figs. 1–5 the results of numerical simulation of orbital evolution for all the enumerated NEAs are shown. January 22, 1999 was taken as an initial epoch. In
L.E. Bykova, T.Yu. Galushina / Planetary and Space Science 49 (2001) 811–815
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Fig. 4. Projection of the orbits of asteroid 2608 Seneca, the Earth, Mars and Jupiter onto the ecliptic plane, in a heliocentric frame.
Fig. 2. Close approaches of 1994 CB with the Earth (a), the resonance band = 2na –nf( /day) (b), osculating heliocentric orbital elements a; e; i (c) for 100 test particles. A set from 100 test particles is shown by a grey background, a nominal orbit is marked out in black color. Initial set of orbits was generated on the basis of the probable errors Gr0 = 4×10−5 AU, Gv0 = 3×10−7 AU/day.
Fig. 3. The average rate of separation of 25 test particles close to Seneca from the nominal orbit (t) (year −1 ).
Table 1, for each asteroid extreme values of osculating elements of the nominal orbit and maximum and mean values of deviations of solutions for 100 test particles from the nominal orbit are given in the intervals of time pointed out
Fig. 5. Critical argument for the asteroids: 1994 RB (a), 1994 CB (b), 3838 Epona (c), 3103 Eger (d).
above. Here orbital elements are denoted by El; Elextrem are extreme (maximum and minimum) values of these elements for the nominal orbit; GElextrem are deviations of maximum (at considered interval) values of orbital elements of the test particles from corresponding values of the nominal orbit, GElisochr: are deviations of isochronous values of elements, GElmax and GElmean are correspondingly the greatest and mean values of deviations among various orbits of the set; Gr0 ; Gv0 are probable errors of initial parameters of orbits, obtained from the analysis of observations by the leastsquares method. In Figs. 1–2 for each asteroid close approaches with the Earth and other planets and the resonance band are shown for the nominal orbits and for the sets of 100 test particles (d is the planetocentric distance of NEA in astronomical units, T is the time in thousands years, = k1 na – k2 np ( =day)). Moreover, in Fig. 2 (c) for 1994 CB the evolution of the osculating orbital elements (semimajor axes a, eccentricity e, inclination i) is shown for the nominal orbit and the set of 100 test particles. Initial sets of orbits were generated on the basis of the probable errors Gr0 = 6×10−6 AU, Gv0 = 3×10−8 AU/day for 2608 Seneca, Gr0 = 1×10−6 AU, Gv0 = 6×10−9 AU/day for 3838 Epona, Gr0 = 1×10−7 AU, Gv0 = 4×10−9 AU/day
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L.E. Bykova, T.Yu. Galushina / Planetary and Space Science 49 (2001) 811–815
for 3103 Eger, Gr0 = 4×10−5 AU, Gv0 = 3×10−7 AU/day for 1994 CB, Gr0 = 1×10−4 AU, Gv0 = 3×10−5 AU/day for 1994 RB. The average rate of separation of test particles from nominal orbits has been determined as (Wisdom, 1983) ln(d(t)=d(t0 )) ; (t) = t − t0 where d is the ordinary Euclidean distance between two close trajectories, t is the time, the index “0” refers to initial values. As an example in Fig. 3 (t) is presented for a set of 25 test particles close to asteroid 2608 Seneca. 4. Results The investigated asteroids have the following commensurabilities of mean motions: 1 : 3 for 3838 Epona and Venus, 3 : 5 for 3103 Eger and the Earth, 1 : 2 for 1994 CB and Venus, 3 : 1 for 2608 Seneca and Jupiter, 3 : 1 for 1994 RB and Jupiter. The evolution of the resonance band (Fig. 1 (b2), (c2), (d2); Fig. 2(b)) shows that all the particles of ensembles of these asteroids except Seneca regularly pass through the value of the precise commensurability = 0, moving away periodically from it at ±11 /day for 3838 Epona and 1994 CB, at ±22 /day for 1994 RB, at ±45 /day for 3103 Eger. It is well seen (Fig. 1(b2)) how a group of close encounters of the asteroid 3838 Epona with Mars during 1390 –1780 yr (0:008¡d¡0:03 AU) inMuences the regular behaviour , however, it does not destroy resonance. Another picture is observed for asteroid 2608 Seneca having a very close approach to the Earth (d ≈ 0:0012 AU). This approach has inMuence on the regular behaviour for the nominal orbit, however, it does not destroy resonance, but some of the test particles near 2608 Seneca move in a rather large neighbourhood of resonance (−30 =day¡¡90 =day). The results presented in Fig. 1(a1) and Table 1 show that for Seneca small variations of the initial orbital elements lead to large deviations of trajectories of test particles from the nominal orbit. Such instability of motion is explained by the relative geometric con8gurations of Seneca and Jupiter and of Seneca and the Earth. The resonant relationship 3 : 1 leads to repeated close approaches between Jupiter and this asteroid every three periods of the asteroid at the aphelion of its orbit. The distance of asteroid 2608 from Jupiter in the aphelion can achieve 1.1 AU. The critical argument of the nominal orbit = 3l − la − 2!a − 2a ◦ librates near the aphelion ( = 180 ) within the range ◦ ◦ (−150 ; +150 ) till the closest encounter with the Earth in 441 year after which its amplitude increases up to ◦ a maximum value ±180 (l ; la ; !a ; a are the mean ecliptic longitude of Jupiter and asteroid, the argument of perihelion and the longitude of the ascending node of asteroid correspondingly). The relative geometry of the
asteroid and the Earth orbits is such that the perihelion of the asteroid orbit is located near the Earth orbit (q ≈ 1 AU) (Fig. 4). That is, the asteroid and the Earth follow the same route for some time. As a result the probability of close approaches to the Earth increases (Fig. 1(a1) and Fig. 4). Thus for asteroid 2608 Seneca the resonance mechanism does not become a protection from collision with the Earth under the given initial conditions. Asteroid 1994 RB also moves in the neighbourhood of resonance 3 : 1 with Jupiter. Its orbit crosses (in projection) the orbits of the Earth and Mars. The critical argument just as for Seneca librates near the aphelion within the range ◦ ◦ (−120 ; +120 ) (Fig. 5a). The minimal distances between Seneca and planets achieve the following values: between the asteroid and Jupiter dmin = 1:85 AU, between the asteroid and the Earth for nominal orbit dmin = 0:04 AU, for the test particles dmin = 0:018 AU, between the asteroid and Mars for nominal orbit dmin = 0:03 AU and for the test particles dmin = 0:001 AU over the range (−3000; 3000 yr). Osculating orbital elements e, i are changed considerably in time (see Table 1). Asteroid 1994 CB moving in the neighbourhood of resonance 1 : 2 with Venus crosses (in projection) only the orbit of the Earth. The elements of its orbit are changed little within the considered interval of time. The critical argument = 2la − lf − !a − a sometimes circulates, sometimes li◦ brates also with large amplitude ±150 (Fig. 5b). The minimal distances between the asteroid and the Earth for the nominal orbit and all the test particles remain within the range of 0:046 6 dmin 6 0:052 AU. Asteroid 3838 Epona moves near resonance 1 : 3 with Venus. Its orbit crosses (in projection) orbits of Mars, the Earth, Venus and approaches the orbit of Mercury. The critical argument librates near a slowly displacing centre lo◦ cated at 310 from the pericenter. The libration amplitude ◦ ◦ is equal to ≈ ±35 , it increases by 2000 yr up to ±55 and ◦ further from 2500 yr it decreases up to ±35 (Fig. 5c). Approaches to the Earth within the range 0:05 6 d 6 0:06 AU take place only on the interval where the amplitude of libration increases. Asteroid 3103 Eger moves in the neighbourhood of the resonance 3 : 5 with the Earth and its orbit crosses (in projection) the orbits of the Earth and Mars. The critical argument ◦ ◦ librates with large amplitude ±150 − 160 approximately (Fig. 5d). The minimal distances between the asteroid and the Earth for the nominal orbit and all the test particles remain within the range of 0:0980 6 dmin 6 0:0981 AU. We constructed also the second ensemble of 50 test particles on the basis of the probable errors given above ampli8ed by 100 times (Gr0 = 1×10−5 AU, Gv0 = 4×10−7 AU/day). The trajectories of these test particles remain suDciently close to the nominal orbit of Eger over 6000 years. In particular, the minimal distances between these test particles and the Earth remain within the range of 0:093 6 dmin 6 0:098 AU. Thus the asteroid 3103 Eger has the stable regular behaviour over the considered time span.
L.E. Bykova, T.Yu. Galushina / Planetary and Space Science 49 (2001) 811–815
5. Conclusions The numerical simulating allows to make the following conclusions. All considered real NEAs and test particles close to them have approaches with the Earth up to the distance d¡ 0:1 AU. These asteroids move near the following loworder mean motion resonances: 1 : 3 for 3838 Epona and Venus, 3 : 5 for 3103 Eger and the Earth, 1 : 2 for 1994 CB and Venus, 3 : 1 for 2608 Seneca and Jupiter, 3 : 1 for 1994 RB and Jupiter. For NEAs 3838 Epona, 3103 Eger, 1994 CB, 1994 RB and all 8ctitious particles close to them, resonance relations are preserved over the whole considered interval (−3000; 3000 yr). However, for all the objects except Epona the resonance arguments librate near 8xed values with a large amplitude or circulate as in the case of 1994 CB. Some test particles near 2608 Seneca leave the area of resonance. The orbital osculating elements and minimal geocentric distances for all test particles close to the real NEAs: 3838 Epona, 3103 Eger, 1994 CB, diOer little from those for nominal orbits. On the contrary the orbital elements of asteroids 2608 Seneca and 1994 RB are changed considerably in time and test particles close to them have rather large deviations from the nominal orbits. Thus the orbital behaviour of NEAs 2608 Seneca and 1994 RB is very sensitive to small variations of initial conditions. The orbits of 3838 Epona, 3103 Eger and 1994 CB are not so sensitive to small variations of initial conditions. The investigations of some authors (e.g., Tancredi, 1998b) show that the Lyapunov times TL for asteroids crossing the orbits of other planets are not large and their orbits are chaotic. According to our estimations for all considered asteroids TL ¡10 000 yr, so over longer timescales the motion of all these asteroids is probably unstable.
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The present paper is the continuation of our earlier papers (Bykova and Timoshenko, 1998, Bykova et al., 1998) on the study of the NEAs dynamics. References Bordovitsyna, T.V., Bykova, L.E., Avdyushev, V.A., 1998. Problems in applications of regularizing and stabilizing KS-transformation to tasks of dynamics of planets’ natural satellites and asteroids. Astron. Geodesy 16, 33–57 (in Russian). Bykova, L.E., Galushina, T.Yu., Timoshenko, L.V., 1998. Investigation of motion of asteroids close to low-order resonances with Earth. Proceedings of conference Fundamental and Applied Problems of Modern Mechanics. Tomsk: Tomsk State University, pp. 163–164 (in Russian). Bykova, L.E., Timoshenko, L.V., 1998. The near-Earth asteroids: encounters with the planets, transformation of the orbital elements. Astron. Geodesy 16, 183–238 (in Russian). Everhart, E., 1974. Implicit single sequence method for integrating orbit. Celest. Mech. 10, 35–55. Milani, A., Carpino, M., Hahn, G., Nobili, A.M., 1989. Dynamics of planet-crossing asteroids classes of orbital behavior. Project SPACEGUARD. Icarus 78, 212–269. Muinonen, K., 1999. Asteroid and Comet Encounters with the Earth. The Dynamics of Small Bodies in the Solar System, A Major Key to Solar System Studies, Vol. 522, NATO ASI Series C: Mathematical Physical Sciences Kluver Academic Publishers, Dordrecht, 127–158. Murray, C.D., Dermott, S.F., 1999. Solar System Dynamics. Cambridge University press, Cambridge. Tancredi, G., 1998a. An asteroid in a Earth-like orbit. Celes. Mech. Dyn. Astron. 69, 119–132. Tancredi, G., 1998b. Chaotic dynamics of planet-encountering bodies. Celes. Mech. Dyn. Astron. 70, 181–200. Stiefel, E.L., Scheifele, G., 1971. Linear and Regular Celestial Mechanics. Springer, Berlin. Wiegert, P., Innanen, K.A., Mikkola, S., 1998. The orbital evolution of near-Earth asteroid 3753. Astron. J. 115, 2604–2613. Wisdom, J., 1983. Chaotic behavior and origin of the 3/1 Kirkwood gap. Icarus 56, 51–74.
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Evolution of near-Earth asteroids close to mean motion resonances L.E. Bykova ∗ , T.Yu. Galushina Research Institute of Applied Mathematics and Mechanics, Tomsk State University, 634050, Tomsk, Russia Received 1 September 2000; received in revised form 6 March 2001; accepted 6 March 2001
Abstract The dynamics of asteroids near mean motion resonances is considered in the paper. Results of study of orbital evolution of NEAs 2608 Seneca, 3103 Eger, 3838 Epona, 1994 RB, 1994 CB are presented. Equations of the motion of asteroids have been integrated numerically. In the process of numerical integration perturbations from planets and the Moon have been taken into account. The interval of time is ranging from −3000 to 3000 yr. Evolution of ensembles of 100 test particles with orbital elements nearby their nominal orbits has been c 2001 Elsevier Science Ltd. All rights reserved. considered for each object. Keywords: Asteroid; Mean motion resonance; Numerical simulation
1. Introduction In the paper, the orbital evolution of some near-Earth asteroids (NEAs) near mean motion resonances with the Earth and other planets is considered. Investigation of orbital resonances in the motion of NEAs is very important because stable resonances make it possible to preserve a certain relative con8gurations of an asteroid and planets (Murray and Dermott, 1999) including minimal distances between orbits. Depending on the initial parameters of orbits resonances either protect asteroids from close encounters and collisions with planets prolonging the time of their life or promote “sweeping up” of asteroids from the neighborhood of a corresponding planet orbit. Many papers were devoted to the investigation of population of asteroids approaching the Earth and to the problem of “asteroid hazard” (e.g., Milani et al., 1989; Muinonen, 1999; Tancredi, 1998a; Wiegert et al., 1998). The goal of this paper is to study a medium-term (for several thousand years) orbital evolution of real NEAs: 2608 Seneca, 3838 Epona, 3103 Eger, 1994 CB, 1994 RB. Numerical experiments were carried out by us to solve the following problems: to reveal close encounters and low - order mean motion commensurabilities of these asteroids with the Earth and other planets; to determine stability of revealed resonances and their part in protecting asteroids from close encounters. ∗ Corresponding author. Tel.: +7-3822-410-576; fax: +7-3822-410347. E-mail address: [email protected] (L.E. Bykova).
For prediction of motion of a real NEA on the long time interval it is insuDcient to investigate the evolution of some nominal orbit because the accuracy of initial parameters of the orbit is limited by the precision of the available observational data. Therefore, evolution of ensemble of 100 test particles with initial parameters nearby the nominal orbit has been considered for each NEA.
2. Techniques of investigation We applied the following technique of the investigation of orbital evolution of the real NEAs. Equations of the motion of asteroids were integrated numerically by Everhart 19th order method (Everhart, 1974). In the process of numerical integration perturbations from planets (except Pluto) and the Moon were taken into account using the positions of the planets given by DE406. In order to achieve a higher accuracy of numerical integration, new forms of regularizing and stabilizing equations of motion (Bordovitsyna et al., 1998) obtained by KS-transformation (Stiefel and Scheifele, 1971) were used. For every asteroid, the problem was solved beginning from the analysis of observations and re8ning the initial parameters of the orbit. The evaluation of parameters was carried out by the least-squares method. By comparing the results of toward-and-backward integration of the equations of motion of each asteroid, an admissible interval of prediction from the point of view of preserving the admissible accuracy was determined. Then for each investigated
c 2001 Elsevier Science Ltd. All rights reserved. 0032-0633/01/$ - see front matter PII: S 0 0 3 2 - 0 6 3 3 ( 0 1 ) 0 0 0 3 8 - 1
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L.E. Bykova, T.Yu. Galushina / Planetary and Space Science 49 (2001) 811–815
Table 1 Extremal values of elements of the asteroid nominal orbits. Deviations of extreme and isochronous solutions for a set of 100 orbits from the nominal orbits in the interval (−3000, +3000 yr) Element El
Elextrem max
GElextrem
GElissochr:
Elextrem
min
max
mean
max
mean
2.4403 0.5194 7.782
1×10−2 5×10−2 2
9×10−3 3×10−2 4×10−2
2×10−1 1×10−1 12
1×10−1 5×10−2 9
1.4031 0.3256 20.934
3×10−6
1×10−6
6×10−5
2×10−5
8×10−5
2×10−5
5×10−4
2×10−4
2608 Seneca a (AU) e i (deg)
2.5688 0.6672 19.189
min
GElissochr:
max
mean
max
mean
1.5032 0.5593 29.273
5×10−4 9×10−7 9×10−2
1×10−4 3×10−7 2×10−2
2×10−3 2×10−3 4×10−1
5×10−4 2×10−4 6×10−2
2.4732 0.3834 26.605
2×10−2 3×10−2 8
4×10−3 2×10−3 1
6×10−2 3×10−1 16
3×10−2 8×10−2 4
3838 Epona
3103 Eger a (AU) e i (deg)
max
GElextrem
1.5060 0.7020 40.649 1994 RB
1.4083 0.3550 22.118
8×10−7
3×10−7
2×10−5
7×10−6
2.5271 0.6387 40.899
Fig. 1. Close approaches of 2608 Seneca (a1), 3838 Epona (b1), 3103 Eger (c1), 1994 RB (d1) with the Earth (“•”) and with Mars (“?”), the resonance band (in /day) for these asteroids correspondingly (a2), (b2), (c2), (d2). Sets from 100 test particles are shown by a grey background, nominal orbits are marked out in black color.
object an ensemble of particles with initial parameters close to the nominal orbit was constructed and its evolution was retraced on the permissible interval of time. The initial set of orbits has been generated for 100 test particles in relation to the selected center (initial epoch) with the help of a random number generator on the basis of the normal law of dispersing and a corresponding covariation matrix. 3. Numerical simulation The described approach was applied to the construction of possible motion domains of numbered asteroids 2608 Seneca, 3838 Epona, 3103 Eger, and unnumbered 1994 CB, 1994 RB. All close approaches and commensurabilities of low order with the Earth and other planets were revealed in
the process of numerical simulation of the motion of these objects. Commensurabilities were determined by means of estimation of the resonance band value = k1 na –k2 np , where na is a mean motion of an asteroid, np is a mean motion of a planet, k1 ; k2 are integers. The initial parameters of orbits were taken from E. Bowell catalog on January 22, 1999 (ftp://ftp.lowell.edu/pub/elgb/astorb.dat). For the evaluation of probable errors, we used the following observational data: 3838 Epona–99, 3103 Eger–190 and 2608 Seneca–53 observations in several oppositions, 1994 RB–21 observations on the interval of 2 days, 1994 CB-124 observations on the interval of 211 days. In Table 1 and Figs. 1–5 the results of numerical simulation of orbital evolution for all the enumerated NEAs are shown. January 22, 1999 was taken as an initial epoch. In
L.E. Bykova, T.Yu. Galushina / Planetary and Space Science 49 (2001) 811–815
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Fig. 4. Projection of the orbits of asteroid 2608 Seneca, the Earth, Mars and Jupiter onto the ecliptic plane, in a heliocentric frame.
Fig. 2. Close approaches of 1994 CB with the Earth (a), the resonance band = 2na –nf( /day) (b), osculating heliocentric orbital elements a; e; i (c) for 100 test particles. A set from 100 test particles is shown by a grey background, a nominal orbit is marked out in black color. Initial set of orbits was generated on the basis of the probable errors Gr0 = 4×10−5 AU, Gv0 = 3×10−7 AU/day.
Fig. 3. The average rate of separation of 25 test particles close to Seneca from the nominal orbit (t) (year −1 ).
Table 1, for each asteroid extreme values of osculating elements of the nominal orbit and maximum and mean values of deviations of solutions for 100 test particles from the nominal orbit are given in the intervals of time pointed out
Fig. 5. Critical argument for the asteroids: 1994 RB (a), 1994 CB (b), 3838 Epona (c), 3103 Eger (d).
above. Here orbital elements are denoted by El; Elextrem are extreme (maximum and minimum) values of these elements for the nominal orbit; GElextrem are deviations of maximum (at considered interval) values of orbital elements of the test particles from corresponding values of the nominal orbit, GElisochr: are deviations of isochronous values of elements, GElmax and GElmean are correspondingly the greatest and mean values of deviations among various orbits of the set; Gr0 ; Gv0 are probable errors of initial parameters of orbits, obtained from the analysis of observations by the leastsquares method. In Figs. 1–2 for each asteroid close approaches with the Earth and other planets and the resonance band are shown for the nominal orbits and for the sets of 100 test particles (d is the planetocentric distance of NEA in astronomical units, T is the time in thousands years, = k1 na – k2 np ( =day)). Moreover, in Fig. 2 (c) for 1994 CB the evolution of the osculating orbital elements (semimajor axes a, eccentricity e, inclination i) is shown for the nominal orbit and the set of 100 test particles. Initial sets of orbits were generated on the basis of the probable errors Gr0 = 6×10−6 AU, Gv0 = 3×10−8 AU/day for 2608 Seneca, Gr0 = 1×10−6 AU, Gv0 = 6×10−9 AU/day for 3838 Epona, Gr0 = 1×10−7 AU, Gv0 = 4×10−9 AU/day
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L.E. Bykova, T.Yu. Galushina / Planetary and Space Science 49 (2001) 811–815
for 3103 Eger, Gr0 = 4×10−5 AU, Gv0 = 3×10−7 AU/day for 1994 CB, Gr0 = 1×10−4 AU, Gv0 = 3×10−5 AU/day for 1994 RB. The average rate of separation of test particles from nominal orbits has been determined as (Wisdom, 1983) ln(d(t)=d(t0 )) ; (t) = t − t0 where d is the ordinary Euclidean distance between two close trajectories, t is the time, the index “0” refers to initial values. As an example in Fig. 3 (t) is presented for a set of 25 test particles close to asteroid 2608 Seneca. 4. Results The investigated asteroids have the following commensurabilities of mean motions: 1 : 3 for 3838 Epona and Venus, 3 : 5 for 3103 Eger and the Earth, 1 : 2 for 1994 CB and Venus, 3 : 1 for 2608 Seneca and Jupiter, 3 : 1 for 1994 RB and Jupiter. The evolution of the resonance band (Fig. 1 (b2), (c2), (d2); Fig. 2(b)) shows that all the particles of ensembles of these asteroids except Seneca regularly pass through the value of the precise commensurability = 0, moving away periodically from it at ±11 /day for 3838 Epona and 1994 CB, at ±22 /day for 1994 RB, at ±45 /day for 3103 Eger. It is well seen (Fig. 1(b2)) how a group of close encounters of the asteroid 3838 Epona with Mars during 1390 –1780 yr (0:008¡d¡0:03 AU) inMuences the regular behaviour , however, it does not destroy resonance. Another picture is observed for asteroid 2608 Seneca having a very close approach to the Earth (d ≈ 0:0012 AU). This approach has inMuence on the regular behaviour for the nominal orbit, however, it does not destroy resonance, but some of the test particles near 2608 Seneca move in a rather large neighbourhood of resonance (−30 =day¡¡90 =day). The results presented in Fig. 1(a1) and Table 1 show that for Seneca small variations of the initial orbital elements lead to large deviations of trajectories of test particles from the nominal orbit. Such instability of motion is explained by the relative geometric con8gurations of Seneca and Jupiter and of Seneca and the Earth. The resonant relationship 3 : 1 leads to repeated close approaches between Jupiter and this asteroid every three periods of the asteroid at the aphelion of its orbit. The distance of asteroid 2608 from Jupiter in the aphelion can achieve 1.1 AU. The critical argument of the nominal orbit = 3l − la − 2!a − 2a ◦ librates near the aphelion ( = 180 ) within the range ◦ ◦ (−150 ; +150 ) till the closest encounter with the Earth in 441 year after which its amplitude increases up to ◦ a maximum value ±180 (l ; la ; !a ; a are the mean ecliptic longitude of Jupiter and asteroid, the argument of perihelion and the longitude of the ascending node of asteroid correspondingly). The relative geometry of the
asteroid and the Earth orbits is such that the perihelion of the asteroid orbit is located near the Earth orbit (q ≈ 1 AU) (Fig. 4). That is, the asteroid and the Earth follow the same route for some time. As a result the probability of close approaches to the Earth increases (Fig. 1(a1) and Fig. 4). Thus for asteroid 2608 Seneca the resonance mechanism does not become a protection from collision with the Earth under the given initial conditions. Asteroid 1994 RB also moves in the neighbourhood of resonance 3 : 1 with Jupiter. Its orbit crosses (in projection) the orbits of the Earth and Mars. The critical argument just as for Seneca librates near the aphelion within the range ◦ ◦ (−120 ; +120 ) (Fig. 5a). The minimal distances between Seneca and planets achieve the following values: between the asteroid and Jupiter dmin = 1:85 AU, between the asteroid and the Earth for nominal orbit dmin = 0:04 AU, for the test particles dmin = 0:018 AU, between the asteroid and Mars for nominal orbit dmin = 0:03 AU and for the test particles dmin = 0:001 AU over the range (−3000; 3000 yr). Osculating orbital elements e, i are changed considerably in time (see Table 1). Asteroid 1994 CB moving in the neighbourhood of resonance 1 : 2 with Venus crosses (in projection) only the orbit of the Earth. The elements of its orbit are changed little within the considered interval of time. The critical argument = 2la − lf − !a − a sometimes circulates, sometimes li◦ brates also with large amplitude ±150 (Fig. 5b). The minimal distances between the asteroid and the Earth for the nominal orbit and all the test particles remain within the range of 0:046 6 dmin 6 0:052 AU. Asteroid 3838 Epona moves near resonance 1 : 3 with Venus. Its orbit crosses (in projection) orbits of Mars, the Earth, Venus and approaches the orbit of Mercury. The critical argument librates near a slowly displacing centre lo◦ cated at 310 from the pericenter. The libration amplitude ◦ ◦ is equal to ≈ ±35 , it increases by 2000 yr up to ±55 and ◦ further from 2500 yr it decreases up to ±35 (Fig. 5c). Approaches to the Earth within the range 0:05 6 d 6 0:06 AU take place only on the interval where the amplitude of libration increases. Asteroid 3103 Eger moves in the neighbourhood of the resonance 3 : 5 with the Earth and its orbit crosses (in projection) the orbits of the Earth and Mars. The critical argument ◦ ◦ librates with large amplitude ±150 − 160 approximately (Fig. 5d). The minimal distances between the asteroid and the Earth for the nominal orbit and all the test particles remain within the range of 0:0980 6 dmin 6 0:0981 AU. We constructed also the second ensemble of 50 test particles on the basis of the probable errors given above ampli8ed by 100 times (Gr0 = 1×10−5 AU, Gv0 = 4×10−7 AU/day). The trajectories of these test particles remain suDciently close to the nominal orbit of Eger over 6000 years. In particular, the minimal distances between these test particles and the Earth remain within the range of 0:093 6 dmin 6 0:098 AU. Thus the asteroid 3103 Eger has the stable regular behaviour over the considered time span.
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5. Conclusions The numerical simulating allows to make the following conclusions. All considered real NEAs and test particles close to them have approaches with the Earth up to the distance d¡ 0:1 AU. These asteroids move near the following loworder mean motion resonances: 1 : 3 for 3838 Epona and Venus, 3 : 5 for 3103 Eger and the Earth, 1 : 2 for 1994 CB and Venus, 3 : 1 for 2608 Seneca and Jupiter, 3 : 1 for 1994 RB and Jupiter. For NEAs 3838 Epona, 3103 Eger, 1994 CB, 1994 RB and all 8ctitious particles close to them, resonance relations are preserved over the whole considered interval (−3000; 3000 yr). However, for all the objects except Epona the resonance arguments librate near 8xed values with a large amplitude or circulate as in the case of 1994 CB. Some test particles near 2608 Seneca leave the area of resonance. The orbital osculating elements and minimal geocentric distances for all test particles close to the real NEAs: 3838 Epona, 3103 Eger, 1994 CB, diOer little from those for nominal orbits. On the contrary the orbital elements of asteroids 2608 Seneca and 1994 RB are changed considerably in time and test particles close to them have rather large deviations from the nominal orbits. Thus the orbital behaviour of NEAs 2608 Seneca and 1994 RB is very sensitive to small variations of initial conditions. The orbits of 3838 Epona, 3103 Eger and 1994 CB are not so sensitive to small variations of initial conditions. The investigations of some authors (e.g., Tancredi, 1998b) show that the Lyapunov times TL for asteroids crossing the orbits of other planets are not large and their orbits are chaotic. According to our estimations for all considered asteroids TL ¡10 000 yr, so over longer timescales the motion of all these asteroids is probably unstable.
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The present paper is the continuation of our earlier papers (Bykova and Timoshenko, 1998, Bykova et al., 1998) on the study of the NEAs dynamics. References Bordovitsyna, T.V., Bykova, L.E., Avdyushev, V.A., 1998. Problems in applications of regularizing and stabilizing KS-transformation to tasks of dynamics of planets’ natural satellites and asteroids. Astron. Geodesy 16, 33–57 (in Russian). Bykova, L.E., Galushina, T.Yu., Timoshenko, L.V., 1998. Investigation of motion of asteroids close to low-order resonances with Earth. Proceedings of conference Fundamental and Applied Problems of Modern Mechanics. Tomsk: Tomsk State University, pp. 163–164 (in Russian). Bykova, L.E., Timoshenko, L.V., 1998. The near-Earth asteroids: encounters with the planets, transformation of the orbital elements. Astron. Geodesy 16, 183–238 (in Russian). Everhart, E., 1974. Implicit single sequence method for integrating orbit. Celest. Mech. 10, 35–55. Milani, A., Carpino, M., Hahn, G., Nobili, A.M., 1989. Dynamics of planet-crossing asteroids classes of orbital behavior. Project SPACEGUARD. Icarus 78, 212–269. Muinonen, K., 1999. Asteroid and Comet Encounters with the Earth. The Dynamics of Small Bodies in the Solar System, A Major Key to Solar System Studies, Vol. 522, NATO ASI Series C: Mathematical Physical Sciences Kluver Academic Publishers, Dordrecht, 127–158. Murray, C.D., Dermott, S.F., 1999. Solar System Dynamics. Cambridge University press, Cambridge. Tancredi, G., 1998a. An asteroid in a Earth-like orbit. Celes. Mech. Dyn. Astron. 69, 119–132. Tancredi, G., 1998b. Chaotic dynamics of planet-encountering bodies. Celes. Mech. Dyn. Astron. 70, 181–200. Stiefel, E.L., Scheifele, G., 1971. Linear and Regular Celestial Mechanics. Springer, Berlin. Wiegert, P., Innanen, K.A., Mikkola, S., 1998. The orbital evolution of near-Earth asteroid 3753. Astron. J. 115, 2604–2613. Wisdom, J., 1983. Chaotic behavior and origin of the 3/1 Kirkwood gap. Icarus 56, 51–74.