ACTA ASTRONOMICA Vol. 51 (2001) pp. 357–376 Prediction of the Motion of Asteroids and Comets Over Long Intervals of Time by I. Włodarczyk Astronomical Observatory of the Chorzów Planetarium, WPKiW, 41-501 Chorzów, Poland e-mail: [email protected]; [email protected] Received May 18, 2001 ABSTRACT 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. Key words: Minor planets, asteroids – Comets: general 1. Introduction Investigations of the motion of asteroids and comets over long intervals of time gain increasing popularity (Milani et al. 1989, Lecar et al. 1997, Laskar 1996, Lecar and Franklin 1997, Tabachnik and Evans 2000). Some of these works refer to close encounters with the Earth (Michael et al. 1996, Sitarski 1998, 1999, 2000). In all studies the problem of reliability of obtained results appears. First of all errors of determination of orbital elements and of the exponential divergence of nearby orbits limit the precise prediction of the motion of asteroids and comets. For instance the prediction of the close approaches of 4179 Toutatis to the Earth is uncertain beyond 100 years from the present time (Whipple and Shelus 1993) or further than 300 years (Sitarski 1998). Chaotic motion of asteroids is a result of close approaches to planets. Maxi- mum Lyapunov characteristic exponent only reflects the exponential growth of two initially nearby trajectories (Wisdom 1983, Whipple 1995). 358 A. A. The errors of determination of orbital elements from observations, the inevitable accumulation of errors of numerical integration of equations of motion of minor planets or comets and the chaos make it almost impossible to draw definite con- clusions from the obtained computations. For instance, we verified the results of Milani et al. (1989): their suggestion concerning the permanently increasing ec- centricity of the orbit of Geographos is not certain because repeated integration with other values of starting orbital elements gave completely different findings (see Section 4). Therefore we present in this paper the maximal intervals of time beyond which numerical computation of motion of minor planets and comets are not reliable. 2. Method of Computation and Numerical Testing To compute the influence of errors of determination of orbital elements on the prediction of motion of minor planets and comets we constructed, for each starting orbital element of these bodies, a set of neighboring orbits which differ only by adding or subtracting from these starting orbital elements the errors of determina- tion of corresponding orbital elements. Thus, we obtain one unchanged orbit and six changed orbits. Orbital elements of 86 minor planets according to the classifica- tion given by Zellner et al. (1985) were chosen. Equations of motion of these aster- oids and of seven short periodic comets were integrated 10 000 years forward and 10 000 years backward. The equations of motion were integrated using the RA 15 Everhart method (Everhart 1974). As the starting point the elements of the asteroids published in Ephemerides of Minor Planets for 1997 were used and those of comets were taken from The Minor Planet Circulars/Minor Planets and Comets edited by Minor Planet Center, Smithsonian Astrophysical Observatory, Cambridge, USA. Four models of the Solar System were used: Osterwinter model with 9 planets and the Moon treated separately, Osterwinter model with 8 planet and the Earth-Moon barycenter (Osterwinter and Cohen 1972), DE 102 model with 9 planets and the Moon separately (Newhall et al. 1983) and ALPL-1 model (Dybczynski´ and Jopek 1986). Since almost the same times of stability were obtained using different mod- els of the Solar System only the Osterwinter model with 9 planets and the Moon treated separately was used for further computations. First of all it was necessary to test the method of integrations for internal and external consistence. We compared our results of integration of equations of motion of minor planets and comets with those obtained by Sitarski (1968). For example we tested results of motion of the Kearns-Kwee comet which has a close, about 0.03 a.u., approach to Jupiter. The results were almost the same although Sitarski (1968) used another method of integration and another model of the Solar System. These tests show that our method of computation and the used Solar Model are correct. Vol. 51 359 2.1. Minor Planet 1943 Anteros – Example of Computations We chose the minor planet 1943 Anteros as an example for our investigations. This asteroid was discovered on March 13, 1973 by J. Gibson at El Leoncito. An- teros is one of 607 Amor-type asteroids known as of 26 March 2001 (see http://cfa-www.harvard.edu/cfa/ps/lists/Amors.html) These asteroids cross the orbit of Mars and approach the orbit of the Earth. To compute the motion of Anteros the following starting orbital elements were taken: Epoch 1997 December 18.0 ET M a [a.u.] e ω2000 Ω2000 i2000 Æ Æ Æ 143.Æ 49027 1.43007440 0.2558456 338. 15937 246. 45223 8. 70222 Æ Æ Æ Æ ¦ : ¦ : ¦ ¦ ¦ ¦0. 00024 0 00000004 0 0000007 0. 00057 0. 00051 0. 00004 where M – mean anomaly, a – semimajor axis, e – eccentricity, ω2000 – argument of perihelion, Ω2000 – longitude of the ascending node, i2000 – inclination to the orbit. These orbital elements are referred to the J2000 equator and equinox. The orbital elements were computed using observations of Anteros from 1893–1993. Equations of motion of Anteros were integrated 3200 years forward and back- ward using the RA 15 Everhart method (Everhart 1974). The differences in the mean anomaly between unchanged and changed orbits were calculated. The chang- ed orbits were constructed by adding or subtracting from the starting orbital ele- ments the errors of determination of corresponding orbital elements. When the differences in the mean anomaly were greater than 360 Æ then the computations were terminated. The results of calculations are shown in Figs. 1 and 2. Fig. 1 presents the results of integration of the equation of motion for Anteros 3200 years backward and forward after constructing the changed orbit by adding one error of determination of orbital elements to corresponding orbital elements. Fig. 1 shows, that in almost all cases after about 1000 years forward and about 2500 years backward the differences in the mean anomaly between neighboring orbits grow rapidly. Almost the same situation is observed in Fig. 2 when we subtract the errors of determination of orbital elements from orbital elements. This example indicates that it is impossible to predict the behavior of minor planets on the orbit beyond the time called further time of stability. Thus the time of stability for Anteros equals about 1000 years for forward integration and about 2500 years for backward integrations. Figs. 1 and 2 show that the change of any orbital element leads to almost the same results. 3. Problem of (433) Eros and (1943) Anteros Figs. 1 and 2 show the behavior of differences in the mean anomaly of Anteros on the unchanged and changed orbit. Even small changes in the starting orbital elements of this Amor-type asteroid lead to different orbits. It was shown that it 360 A. A. 360 360 (a) (b) 180 180 0 0 et M(deg) delta et M(deg) delta -180 -180 -360 -360 -4000 -2000 0 2000 4000 -4000 -2000 0 2000 4000 time(yrs) time (yrs) 360 360 (c) (d) 180 180 0 0 et M(deg) delta M(deg) delta -180 -180 -360 -360 -4000 -2000 0 2000 4000 -4000 -2000 0 2000 4000 time (yrs) time (yrs) 360 360 (e) (f) 180 180 0 0 et M(deg) delta M(deg) delta -180 -180 -360 -360 -4000 -2000 0 2000 4000 -4000 -2000 0 2000 4000 time (yrs) time (yrs) Fig. 1. Anteros. Time evolution of differences in mean anomaly, δM in degrees, between asteroid on unchanged and changed orbit during forward and backward integration with only one orbital element · · changed: a) – mean anomaly by ·dM , b) – semimajor axis by da, c) – eccentricity by de, ω · Ω d) – argument of perihelion by ·d , e) – longitude of ascending node by d , f) – inclination to ecliptic by ·di. is difficult to predict the behavior of Anteros for about 1000 years forward and about 2500 years backward in time. Eros, another minor planet from the same Amor-type group behaves quite differently. We performed additional 100 000 year integrations and deduced from them that the time of stability of Eros is about 60 000 years forward and backward in time. It is several dozen times greater than that of Anteros. What is the cause of this difference in times of stability of Eros and Anteros? They belong to the same Near Earth Asteroid subgroup of Amors. Their starting orbital elements are very similar: Vol. 51 361 360 360 (a) (b) 180 180 0 0 et M(deg) delta M(deg) delta -180 -180 -360 -360 -4000 -2000 0 2000 4000 -4000 -2000 0 2000 4000 time (yrs) time (yrs) 360 360 (c) (d) 180 180 0 0 et M(deg) delta M(deg) delta -180 -180 -360 -360 -4000 -2000 0 2000 4000 -4000 -2000 0 2000 4000 time (yrs) time (yrs) 360 360 (e) (f) 180 180 0 0 et M(deg) delta M(deg) delta -180 -180 -360 -360 -4000 -2000 0 2000 4000 -4000 -2000 0 2000 4000 time (yrs) time (yrs) Fig.