Paper the Maximal Intervals of Time Beyond Which Numerical Computation of Motion of Minor Planets and Comets Are Not Reliable

Home , 1000 Piazzia, 1010 Marlene, 1139 Atami, 1221 Amor, 1373 Cincinnati, 153 Hilda

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 eccentricity 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 determination 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 asteroids 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 models 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 backward using the RA 15 Everhart method (Everhart 1974). The differences in the mean anomaly between unchanged and changed orbits were calculated. The changed orbits were constructed by adding or subtracting from the starting orbital elements 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

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

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. 2. Same as in Fig. 1, but with subtracting error of determination of orbital elements.

Epoch 1997 December 18.0 ET

No. M a [a.u.] e ω2000 Ω2000 i2000

Æ Æ Æ Æ

1 .Æ 98665 1.4583366 0. 2230310 178. 64204 304. 43523 10. 83087

Æ Æ Æ Æ 2 143.Æ 49027 1.4300744 0. 2558456 338. 15937 246. 45223 8. 70222

No. 1 is the orbit of Eros, No. 2 is the orbit of Anteros. The orbital elements are referred to the J2000 equator and equinox. Fig. 3 shows that the orbits of Eros and Anteros and their evolution over 3200 years are similar. The dotted curves denote parts of the orbits placed below the ecliptic plane, Ω and Π are the positions of the ascending node and perihelion of asteroid, respectively, and γ is the direction to the vernal equinox. Orbits of Eros and Anteros approach close to the Earth and cross the orbit of Mars. In these 362 A. A.

Fig. 3. Time evolution of orbit in ecliptic plane for 3200 years forward for Eros (a) and Anteros (b).

3200 year integrations the longitudes of ascending nodes of Eros and of Anteros

Æ Æ Æ decreased from about 304 Æ to 285 and from about 246 to 224 , respectively. In

Figs. 3a and 3b both ascending nodes rotate clockwise. Values of arguments of

Æ Æ perihelion increased in this time from about 178 Æ to 211 and from about 338 to

23 Æ for Eros and Anteros, respectively. Lines of apsis of the orbits of Eros and Anteros move counter-clockwise. The orbits of Anteros oscillate to a greater extent on the ecliptic plane than those of Eros and hence may lead to more frequent close approaches of Anteros to Mars. Smaller separations between Anteros and the Earth near the ascending node and the more tangential orbits of Anteros to the orbit of the Earth than those for Eros are the cause of the different behavior of these asteroids. Fig. 4 shows the evolution of gravitational potential from the Earth acting on Eros and Anteros for the unchanged orbit (a and c) and for the changed orbit obtained by adding 6 km to the starting semimajor axis (b and d). We obtained Fig. 4 by numerical integration of equations of motion for all planets from Mercury to Pluto and for Eros and Anteros over the time span of 3200 years forward. If we take into account the changes of gravitational potential coming from the Earth only then we can see that the gravitational potential minima which belong to Anteros are about twice as deep as for Eros, causing stronger gravitational interac- tions between the Earth and Anteros than between the Earth and Eros. It is worth pointing out the fact that changes of gravitational potential in the case of Anteros are very irregular (Fig. 4c,d) unlike for Eros (Fig. 4a,b) where we can see very regular, symmetrical, clear and limpid structures. The irregular gravitational potential affecting Anteros generates irregularity of its motion. Also after the change of orbit the character of changes of the gravitational potential is quite similar, regular remains regular and irregular remains irregular. Furthermore the potential on Anteros at the end of the time of integration is different for the changed and unchanged orbit contrary to the case of Eros. Vol. 51 363

0 a (a) 0 a+da (b)

-5 -5

-10 -10

-15 -15

-20 -20 rvttoa oeta abtayunits) (arbitrary potential Gravitational rvttoa oeta abtayunits) (arbitrary potential Gravitational -25 -25 0 1000 2000 3000 0 1000 2000 3000 Time(years) Time(years)

0 a (c) 0 a+da (d)

-20 -20

-40 -40 rvttoa oeta abtayunits) (arbitrary potential Gravitational rvttoa oeta abtayunits) (arbitrary potential Gravitational -60 -60 0 1000 2000 3000 0 1000 2000 3000 Time(years) Time(years) Fig. 4. Time evolution of the gravitational potential from the Earth acting on Eros (upper) and An-

teros (lower) with unchanging orbit (left) and changing orbit by adding da = 6 km to the starting semimajor axis (right).

Regular and clear structure of the changes of gravitational potential from the Earth acting on the motion of Eros and the stability of this structure even after a slight starting change of the semimajor axis of the orbit of Eros are the main causes of its regular motion. In the case of Anteros the opposite situation occurs. The changes of the gravitational potential of the Earth acting on Anteros are irregular, unclear and unstable even after a slight starting change in the semimajor axis of Anteros. The above differences in time evolution of the gravitational potential of the Earth acting on these asteroids are the main causes of the shorter time of stability of Anteros. Additionally Eros is locked in 4:7 mean motion resonance with the Earth which is of lower order than that for Anteros, 7:12.

3.1. Time Evolution of the Orbital Elements of Eros and Anteros

The orbits of Eros and Anteros over time spans longer than 100 000 years have been computed by numerical integration. Our investigations referred particularly to the time evolution of semimajor axes and eccentricities of Eros and Anteros. 364 A. A.

1.490 1.490

a (a) a+da (b) 1.480 1.480

1.470 1.470

eiao axis (au) Semimajor 1.460 axis (au) Semimajor 1.460

1.450 1.450 -200000 -100000 0 100000 200000 -200000 -100000 0 100000 200000 Time(years) Time(years)

1.600 1.600 (c) (d) a a+da 1.500 1.500

1.400 1.400

eiao axis (au) Semimajor 1.300 axis (au) Semimajor 1.300

1.200 1.200 -200000 -100000 0 100000 200000 -200000 -100000 0 100000 200000 Time(years) Time(years) Fig. 5. Time evolution of semimajor axis for unchanging orbits (a, c) and for changing orbits by

adding da = 6 km to the starting semimajor axis (b, d). Eros – upper panels, and Anteros – lower panels.

Fig. 5 shows time evolution of the semimajor axis of the orbit of Eros during forward and backward integrations over 100 000 years for unchanged starting elements of the orbit (a), and for starting semimajor axis changed by adding

da = 6 km (b). Similar computations for the orbit of Anteros are presented in Fig. 5c and Fig. 5d, respectively. Note, that the scale of the semimajor axis for Anteros is 10 times larger than for Eros. It is clear from Fig. 5 that it is practically impossible to calculate correctly evolution of the orbit of Eros and Anteros over long intervals of time, if a small change of the starting semimajor axis of the orbit leads to a different evolution of the semimajor axis. Similar evolution of the semimajor axis of Eros and Anteros was given by Milani et al. (1989) where their results of the SPACEGUARD numerical integration of 410 planet-crossing asteroids spanning 200 000 years were also presented. In their paper time evolution of semimajor axes of the orbits of Eros and Anteros are quite different from ours in Fig. 5. Their starting orbital elements of Eros and Anteros were taken from the 1986 version of Marsden’s catalog of asteroid orbits (Marsden 1986). Vol. 51 365

Fig. 5. shows that it is practically impossible to compute evolution of the semimajor axes of Eros and Anteros in time-spans greater than the computed in this paper time of stability – 60 000 years for Eros and 1000 for Anteros, respectively.

4. Time Evolution of Orbital Elements of Geographos

Rationality of investigations of the evolution of orbits over long intervals of time is clearly visible in the case of Geographos. Table 1 lists orbital elements of Geographos taken from various sources. There are the following orbits: Mi- lani, 1986 – orbital elements are from Marsden (1986), they are the same as in Ephemerides of Minor Planets for the year 1986 (EMP 1986); CBK, 1997 – orbital elements were computed in Space Research Center of Polish Academy of Science in Warsaw, Poland from observations of Geographos until 1997; CBK, 1995 – as for CBK 1997 from observations until 1995; EMP, 1998 – orbital elements from Ephemerides of Minor Planets for 1998. Table1 Orbits of Geographos

No. M a [a.u.] e ω1950 Ω1950 i1950 case Orbit: Milani, EMP 1986 1 80.69976 1.24469560 0.33538480 276.56409 336.73378 13.32094 Orbit: CBK, 1997 2 80.70034 1.24469571 0.33538484 276.56421 336.73376 13.32085 Orbit: EMP, 1998 3 80.69969 1.24469551 0.33538534 276.56439 336.73345 13.32081 Orbit: CBK, 1995 – random orbit selection 4 80.70002 1.24469564 0.33538521 276.56425 336.73382 13.32085 unchanged 5 80.70002 1.24469564 0.33538519 276.56422 336.73384 13.32086 Orbit.001 6 80.70003 1.24469564 0.33538518 276.56423 336.73382 13.32085 Orbit.011 7 80.70003 1.24469564 0.33538521 276.56425 336.73382 13.32086 Orbit.111 Epoch: 1986 June 19. The reference system is the mean ecliptic and equinox of 1950.0.

Fig. 6 shows time evolution of the semimajor axis orbit of Geographos in forward and backward integration for time spans of 100 000, 10 000 and 1000 years. Starting orbital elements are taken from CBK, 1997. It is clear, that one could ra- tionally study the evolution of semimajor axis of Geographos’s single orbit only for about 400 years forward and backward. In a longer time span we cannot draw any conclusion about the evolution of the semimajor axis, since a small change of the starting semimajor axis, only 6 km, leads to a remarkable difference in evolution. In Fig. 6a some mean motion resonances of Geographos with the Earth are marked. Milani et al. (1989) published a similar evolution of the semimajor axis of Geographos’s orbit. Their evolution differs not only in the course of variability but also in the existence of some mean motion resonances. 366 A. A. 1.40 1.40

(a) a a+da (b)

1.30 1.30 9:13 9:13 7:10 5:7

1.20 1.20 eiao axis (au) Semimajor eiao axis (au) Semimajor

1.10 1.10 -200000 -100000 0 100000 200000 -200000 -100000 0 100000 200000 Time(years) Time(years)

1.26 1.26 (c) a a+da (d) 1.25 1.25

1.24 1.24 eiao axis (au) Semimajor 1.23 axis (au) Semimajor 1.23

1.22 1.22 -10000 -5000 0 5000 10000 -10000 -5000 0 5000 10000 Time(years) Time(years)

1.255 1.255 a (e) a+da (f) 1.250 1.250

1.245 1.245

1.240 1.240 eiao axis (au) Semimajor eiao axis (au) Semimajor 1.235 1.235

1.230 1.230 -1000 -500 0 500 1000 -1000 -500 0 500 1000 Time(years) Time(years)

Fig. 6. Geographos. Time evolution of the semimajor axis of the orbit of Geographos in forward and backward integration for time spans of 100 000, 10 000 and 1000 years for unchanged orbit (a, c, e)

and for semimajor axis changed by adding da = 6 km (b, dand f).

Time evolution of eccentricity of Geographos’s orbit is similarly unpredictable. Fig. 7 shows time evolution of eccentricity of the orbit of Geographos in forward integration over 100 000 years. Starting elements of the orbit of Geographos are according to a) – Milani (EMP 1986), c) – CBK 1997, e) – EMP 1998. Using the same starting elements as Milani et al. (1989) we obtained the same results (compare our Fig. 7a and Fig. 4b in the paper by Milani et al. 1989). Vol. 51 367

0.40 0.40

a a+da (a) (b) 0.38 0.38

0.36 0.36

0.34 0.34 Eccentricity Eccentricity

0.32 0.32

EMP1986-Milani 0.30 0.30 0 40000 80000 120000 0 40000 80000 120000 Time(years) Time(years)

0.40 0.40

a a+da (c) (d) 0.38 0.38

0.36 0.36

0.34 0.34 Eccentricity Eccentricity

0.32 0.32

CBK1997 0.30 0.30 0 40000 80000 120000 0 40000 80000 120000 Time(years) Time(years)

0.40 0.40

a a+da (e) (f) 0.38 0.38

0.36 0.36

0.34 0.34 Eccentricity Eccentricity

0.32 0.32

EMP1998 0.30 0.30 0 40000 80000 120000 0 40000 80000 120000 Time(years) Time(years) Fig. 7. Geographos. Time evolution of eccentricity of the orbit of Geographos in forward integration for 100 000 years, for unchanged starting elements (a, c, e) and for semimajor axis changed by adding

da = 6 km (b, d,f).

As Fig. 7 shows, time evolution of eccentricity of Geographos’s orbit strongly depends on the starting orbital elements. It is difﬁcult to predict the behavior of Geographos for such a long intervals of time. Also a small change in the starting semimajor axis of each orbit (EMP 1986, CBK 1997 i EMP 1998) leads to different time evolutions of eccentricity. 368 A. A.

Note that while evolution of eccentricities is different in all cases, there exist quasi sinusoidal changes of eccentricities. However, from the presented evolution of rapidly growing eccentricity (Fig. 7a) Milani et al. (1989) draw the conclusion that the Geographos type object transit to higher orbits. This is, however, not visible on other plots in Figs. 7. Untill now the inﬂuence of the error of determination of orbital elements on the behavior of an asteroid or comet were investigated by adding or subtracting the error of determination of a given orbital element to or from the value of the orbital element. Then the equations of motion with the obtained starting orbital elements were integrated. However, there exists a new method which permits the generation of a series of orbits computed from observations by the least squares corrections which well rep- resent all the observations used for orbit computation. This random orbit selection method was adopted to minor planet Toutatis by Sitarski (1998). Similar calculation were made for Geographos using 500 random selected orbits. Table 1 lists elements of some of these orbits. The equations of motion of planets and Geographos were subsequently integrated using these starting elements. It became evident that even random orbit selection does not lead to correct prediction of the orbit behavior of Geographos for a time span longer than our computed time of stability. It is worth noticing that the error of determination of the semimajor axis of Geographos orbit CBK, 1995 was 6 km – the same value as was used in the investigation of the time of stability for all minor planets and comets in this paper.

5. Discussion

Presented results on the long-term motion of Eros, Anteros and Geographos indicate the necessity of careful investigation and the introduction of the concept of the time of stability. In Section 2, using Anteros as an example, we deﬁned the time of stability. As Figs. 1 and 2 show, changing any orbital element of Anteros gives almost the same results. However, the determination of the time of stability of Anteros indicates that it is enough to compute the inﬂuence of the error of determination of the semimajor axis only. What is more, to simplify the process of constructing changed orbits the same error of determination of semimajor axis of Anteros for all minor planets and comets was used. The summary of the computations of times of stability for some minor planets are listed in Table 3. In detail the results are given in Table 2. Table 2 contains: name of selected

asteroid or name of selected comet, time of stability for forward and backward · integration provided we change the semimajor axis only by adding ·da or 10da respectively. Vol. 51 369

Table2

Times of stability

Times of stability [years] Object da 10 da forward backward forward backward

Amors

> > > 433 Eros >10000 10000 10000 10000 887 Alinda 2400 7000 2200 5000 1036 Ganymed 7000 8500 7000 3000 1221 Amor 1200 6000 1000 500 1580 Betulia 5800 4000 5200 4000 1916 Boreas 5500 2500 5000 2500

1943 Anteros 1000 2500 900 2500

> > > 1980 Tezcatlipoca >10000 10000 10000 10000

3122 Florence 2000 1000 2000 1000 > 3199 Nefertiti 2500 >10000 2000 10000 3691 1982 FT 3700 9000 3300 5000

3988 1986 LA 2600 >10000 2500 10000

4015 Wilson-Harrington 1200 1500 1000 1000

> > 4487 Pocahontas >10000 10000 7500 10000

4596 1981 QB 2300 >10000 2200 2000

4947 Ninkasi 3200 8000 3200 500 > 4957 Brucemurray 3300 >10000 2600 10000 5324 Lapunov 8500 10000 7500 8000

5587 1990 SB 1200 6200 1200 6200

> > > 6569 1993 MO >10000 10000 10000 10000

7474 1992 TC 1200 >10000 1200 3000 7480 1994 PC 2400 3000 2400 2500

1993 QA 800 3000 500 500

> > > 1994 QC >10000 10000 10000 10000 1996 FO3 5700 7000 5200 7000 Apollos 1566 Icarus 2500 2500 2400 800 1620 Geographos 400 400 500 400 1685 Toro 8000 2000 8000 1500 1862 Apollo 500 120 280 120 1981 Midas 1200 500 1000 350

2063 Bacchus 500 500 500 500 > 2102 Tantalus 800 >10000 700 10000

2201 Oljato 500 400 500 320 >

3103 Eger 3500 >10000 2200 10000 > 3752 Camillo >10000 1500 10000 1200 4179 Toutatis 250 200 250 200 5189 1990 UQ 500 2100 500 2100 5496 1973 NA 5700 1300 6000 700 1992 CC1 3300 10000 3500 5200 1994 PC1 500 500 300 150

Atens > 2062 Atena >10000 800 10000 600 2100 Ra-Shalom 8100 8000 7100 7000 2340 Hathor 100 100 400 200

Trojans of Mars

> > > 5261 Eureka >10000 10000 10000 10000

Mars crossers

> > >

699 Hela >10000 10000 10000 10000

> > >

1139 Atami >10000 10000 10000 10000

> > >

1474 Beira >10000 10000 10000 10000

> > > 1508 Kemi >10000 10000 10000 10000 370 A. A.

Table2

Times of stability

Times of stability [years] Object da 10 da forward backward forward backward

Mars crossers

> > > 1951 Lick >10000 10000 10000 10000

2629 Rudra 6200 2200 6200 2000

> > >

3496 Arieso >10000 10000 10000 10000

> > > 3581 Alvarez >10000 10000 10000 10000 3800 Karayusuf 2500 400 2400 400 3833 1971 S.C. 3700 1500 2400 1200 4775 Hansen 3000 2600 3000 2600

5201 1983 XF 500 1000 500 800

> > > 5349 Paulharris >10000 10000 10000 10000

Phocaea Group

> > > 25 Phocaea >10000 10000 10000 10000

Main Belt

> > >

1 Ceres >10000 10000 10000 10000

> > >

2 Pallas >10000 10000 10000 10000

> > >

3 Juno >10000 10000 10000 10000

> > >

4 Westa >10000 10000 10000 10000

> > >

253 Mathilde >10000 10000 10000 10000

> > >

1000 Piazzia >10000 10000 10000 10000

> > > 1010 Marlene >10000 10000 10000 10000

Griqua Group

> > >

1362 Griqua (2/1) >10000 10000 10000 10000

> > > 2938 Hopi >10000 10000 10000 10000

Minor planets in mean motion resonances with Jupiter

> > >

588 Achilles (1/1) >10000 10000 10000 10000

> > >

624 Hektor (1/1) >10000 10000 10000 10000

> > >

659 Nestor (1/1) >10000 10000 10000 10000

> > >

153 Hilda (3/2) >10000 10000 10000 10000

> > >

279 Thule (4/3) >10000 10000 10000 10000

> > >

329 Svea (3/1) >10000 10000 10000 10000

> > >

385 Ilmatar (5/2) >10000 10000 10000 10000

> > >

403 Cyane (5/2) >10000 10000 10000 10000

> > >

522 Helga (12/7) >10000 10000 10000 10000

> > >

1362 Griqua (2/1) >10000 10000 10000 10000

> > >

1373 Cincinnati (2/1) >10000 10000 10000 10000

> > >

1921 Pala (2/1) >10000 10000 10000 10000

> > > 1922 Zulu (2/1) >10000 10000 10000 10000 Centaurs 2060 Chiron 4000 2000 3200 1800 5145 Pholus 7700 7000 7000 6000

Kuiper Belt

> > >

1992 QB1 >10000 10000 10000 10000

> > >

1993 FW >10000 10000 10000 10000

> > >

1993 RO >10000 10000 10000 10000

> > > 1993 SB >10000 10000 10000 10000 Comets 2P/Encke 500 1000 400 800 21P/Giacob.-Zinner 600 400 400 300 22P/Kopff 500 400 400 300 26P/Grigg-Skjeller. 400 300 500 500 29P/Schw.-Wach.1 350 300 300 300 39P/Oterma 350 300 300 200 56P/Slaughter-Bur. 250 150 1200 1000 Vol. 51 371

Amors: Amors belong to the ﬁrst group of selected minor planets. They cross the orbit of Mars and approach the orbit of the Earth. More precisely, their semimajor axes a are greater than 1.0 a.u. and distances of perihelion q are between 1.1017 a.u. and 1.3 a.u. The computations of the time of stability of 25 Amors from 607 known as of 26 March 2001 were done. (see http://cfa-www.harvard.edu/cfa/ps/lists/Unusual.html) Only four of them have regular motion for time-spans of 10 000 years. The others show a sudden increase of differences of mean anomaly between minor planets both on unchanged and on changed orbit in this interval of time. Only about 30% of the

investigated Amors have time of stability greater than 10 000 years (see Table 3). :

Apollos: They cross the orbit of the Earth and fulﬁll conditions a>1 0 a.u. and : q<1 017 a.u. The times of stability of 15 Apollos from 601 known (data from the same WWW site as for Amors) were computed. Their times of stability are shorter on average than those for Amors. Only 10% have the time of stability greater than 10 000 years.

Table3 Summary results of times of stability for minor planets

Minor Planets Times of Stability, t [years]

´ > µ

Atens t > 100 15% 10000

´ > µ

Apollos t > 120 10% 10000

´ > µ Amors t > 800 30% 10000

Trojans of Mars t > 10000

´ > µ Mars Crossers t > 500 60% 10000

Main Belt t > 10000

Trojans of Jupiter t > 10000 Minor Planets in Mean Motion Resonances 2:1, 3:1, 3:2, 4:3, 5:2

with Jupiter t > 10000 < Centaurs 2000 < t 7700

Kuiper Belt t > 10000 :

Atens: They orbit most of their time inside the orbit of the Earth. Exactly a < 1 0 : a.u., Q > 0 983 a.u., where Q is the aphelion distance. For 3 studied Atens from 104 known about 15% of their times of stability is greater than 10 000 years. Trojans of Mars: Only 3 Martian Trojans are now known (January 2001): 5261 Eureka discovered in 1990, 1998 VF31 and 2001 DH47 (see http://cfa-www.harvard.edu/cfa/ps/lists/MarsTrojans.html) The studied motion of Eureka has the time of stability greater than 10 000 years. Mars Crossers: They cross the orbit of the Mars and are neither Atens, Amors nor Apollos. Their average semimajor axes are about 2.285 a.u. and distances of perihelion are smaller than 1.666 a.u. About 60% of them have the time of stability 372 A. A. greater than 10 000 years. Phocaea Group: Their average semimajor axes are about 2.368 a.u., eccentricities

are less than 0.1 and inclinations of their orbits to the ecliptic plane are between Æ 18 Æ and 32 . The only representative 25 Phocaea has the time of stability greater than 10 000 years. Main Belt: The computations of the times of stability were performed for 7 from about 100 000 minor planets of the Main Belt. All of them have the times of stability greater than 10 000 years. Griqua Group: They have semimajor axes of about 3.243 a.u., on average, and eccentricities less than 0.35. Their times of stability for two representatives are greater than 10 000 years. Minor planets in mean motion resonances with Jupiter: Among them 3 Trojans out of 964 known in March 2001 were investigated. (see http://cfa-www.harvard.edu/cfa/ps/lists/JupiterTrojans.html) All studied Trojans and other minor planets from this group have the time of stability greater than 10 000 years. Centaurs: In majority their perihelion distances are outside the orbit of Jupiter and the semimajor axes are inside the orbit of Neptune. The computations were made for two of about 67 known (March 2001) Centaurs and Scattered-Disk Objects. Their times of stability are of the order of several thousand years. Kuiper Belt: Their semimajor axes run beyond the orbit of Neptune and the distances of perihelion of some of them are inside the orbit of Neptune. Four objects of 366 known (March 2001) transneptunians objects were examined (see WWW site like for Amors). They all have the time of stability greater than 10 000 years. Comets: All investigated 7 short periodic comets have the time of stability below 1200 years. Our computations of the time of stability of 4179 Toutatis from Apollos (300 years, cf. Table 2) are consistent with the results of Sitarski (1998), who investigated the differences in angular and linear distances in space between positions of this minor planet moving in the standard orbit and in randomly selected orbits. In Sitarski’s paper the times of rapid growth of these differences are between 225 years and 300 years. In his computations Sitarski used different method of integration and different model of the Solar System. Tables 2 and 3 show that minor planets even from the same group may have surprisingly different lengths of times of stability. An attempt at the explanation of the causes of these differences was presented with the use of two minor planets Eros and Anteros in Section 3. It is very difﬁcult to deﬁne mathematically the time of stability. The best way to compute its value is by comparing graphically the behavior of differences in the mean anomaly between unchanged and displaced orbits. However, the time of stability can also be computed automatically by comparing these differences, for example, every 100 years. If the adopted differences become, for instance, Vol. 51 373

greater than 10 Æ then computations are terminated and the time with rapid growth is deﬁned as the time of stability. Recently Włodarczyk (2001) computed the times of stability for 930 Atens, Apollos and Amors (AAA) in time-spans 0.3 Myr.

5.1. Time Evolution of the Orbital Elements of Comets The motion of seven short periodic comets were also studied by numerical integration of equations of motion. As can be seen from Table 2, the times of stability of these comets are very short, about several hundred years only. For four of these comets we computed theirs long-term motion. Table 4 lists the orbital elements of these comets.

Table4 Orbits of comets

2P/Encke 21P/Giac.-Zin. 22P/Kopff 26P/Grigg-Skj. q [a.u.] 0.34 1.03 1.58 1.00 Q [a.u.] 4.10 6.00 5.35 4.93 M 0.99624 1.01585 2.11043 358.01091 a [a.u.] 2.217724 3.515609 3.464497 2.964984 e 0.846899 0.707517 0.544074 0.663806

ω1950 186.49432 172.48554 162.76689 359.34291 Ω1950 333.89087 194.70603 120.27945 212.60029 i1950 11.74951 31.87845 4.72458 21.09195 Epoch 2000 Sep 13 1985 Sep 12 1996 Jul 16 1997 Aug 20

The reference system is the mean ecliptic and equinox of 1950.0.

The orbits of short periodic comets are similar to the orbits of some asteroids. However, in the case of comets a possibility of close encounters with Jupiter exists which can cause a sudden change of their orbits. In this work comets were treated as asteroids. The nongravitational effects were not included. Comets given above have a time of stability of about 400 years but their orbits are different – one can note the differences of the perihelion and aphelion distances, q and Q in Table 4, respectively. In perihelion the orbit of comet Encke crosses the orbit of Mercury, orbits of comets Giacobini-Zinner and Grigg-Skjellerup approach the Earth and the orbit of comet Kopff approaches Mars. In aphelion comet Encke and comet Grigg-Skjellerup do not cross the orbit of Jupiter. Comet Encke is situated about 1.1 a.u. from Jupiter, comet Grigg-Skjellerup is about 0.3 a.u. from Jupiter. The other two comets cross the orbit of Jupiter. Time evolution of the orbits of comets Encke and Kopff for the time-spans of 10 000 years is shown in Figs. 8a and 8b, respectively. The dotted curves denote 374 A. A. the parts of the orbits placed below the ecliptic plane. Time evolution of orbits of comets runs on many different ways. The orbit of comet Encke is precessing and does not change its shape (Fig. 8a). The orbit of comet Kopff evolves strongly and it even crosses the orbit of Neptune (Fig. 8b). The main differences in evolution of the investigated comets comes from the difference of starting orbital elements, particularly perihelion and aphelion.

Also the question of dependence of orbital evolution of comets on the accuracy of their orbital elements was studied. The orbit of comet Encke is not very sensitive to the disturbance of the semimajor axis. The unchanged and changed orbits of this comet are almost similar. Nevertheless, the time of stability of this comet is short, i.e., the position of the comet on the changed orbit is removed far in time from the position of the comet on the unchanged orbit (cf. Table 2). In the case of comet Kopff the addition of a slight disturbance to the starting semimajor axis leads to a different evolution of the orbit. It is worth noting that the orbit of the comet does not cross the orbit of Neptune.

From the obtained computations it results that the precision of determination of the orbital elements of comets strongly inﬂuences the study of their orbital evolution. Even a small disturbance in the starting semimajor axis leads to different behavior of the orbit in time. It is difﬁcult to predict in long time-spans not only the position of the comet on the orbit but also the shape and the position of the orbit in space.

Fig. 8. Time evolution of orbit on ecliptic plane for 10 000 years forward for comet Encke (a) and comet Kopff (b). Vol. 51 375

6. Summary

It is impossible to predict the behavior of minor planets and comets over a time span longer than the time of stability as suggested in our present work. In each case it is necessary to carry out detailed procedure which we propose. We point out that the calculated times of stability for many minor planets and short periodic comets are very short, from several hundred to several thousand years. The principal cause of different times of stability of minor planets and comets are their close approaches to planets. As noticed before it is difﬁcult to deﬁne how to precisely compute the duration of the time of stability. We can study it graphically by comparing the behavior of the differences in the mean anomaly between disturbed and undisturbed orbits and estimate the time when these differences start growing suddenly. We can also do it automatically by recording, for example every 100 years, the mean anomalies of those two neighboring orbits. If the two succeeding differences of the mean anomaly are for instance, greater than 10 Æ then the computations are terminated and the time obtained is the searched time of stability. Table 2 lists in detail times of stability of some selected minor planets and comets in the time span of 10 000 years. In order to investigate the behavior of minor planets and comets in a longer time span when the times of stability are greater than 10 000 years one should carry out further computations including the time span of over 10 000 years.

Acknowledgements. I would like to thank Prof. Grzegorz Sitarski for profound and fruitful help. Also I thank Drs. K. Ziołkowski, M. Królikowska-Sołtan, S. Szutowicz and to R. Gabryszewski from Space Research Center, Polish Academy of Sciences in Warsaw for many helpful discussions.

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