2060 Chiron – Chaotic Dynamical Evolution and Its Implications

Home , 2060 Chiron, Oort cloud

ACTA ASTRONOMICA Vol. 52 (2002) pp. 305–315

2060 Chiron – Chaotic Dynamical Evolution and its Implications

by Ryszard Gabryszewski

Space Research Centre, ul. Bartycka 18A, 00-716 Warszawa, Poland e-mail: [email protected]

Received May 24, 2002

ABSTRACT

2060 Chiron – one of the Centaurs orbiting chaotically among the giant planets – is treated as an asteroid and a comet (95P/Chiron) as well. Since the day of the discovery many papers have discussed its past and future fate. In this paper a possibility of Chiron’s dynamical evolution to different cometary orbital types is studied. An ensemble of orbital elements was used to describe Chiron’s dynamics in terms of probability. The ensemble was generated using a unique scheme of elements creation. Dispersion of elements obtained by this method is much smaller compared to ranges obtained by varying the original elements in the ellipsoid of their mean errors. The chaos in Chiron’s dynamical evolution can be seen in 5 to 9 kyrs, although the dispersion of orbital elements is small. Halley type orbits are the rarest noticed orbital types but the number of these objects is three times greater than the number of apparent Halley type comets. The variations of probability of different cometary orbits as a function of time is also presented. The rate of HTC orbit production is only four times lower than the production rate of JFCs after the ﬁrst 50 kyrs of integration. Remarks on the small body transportation mechanisms are also included. Key words: Minor planets, asteroids – Comets: general – Celestial mechanics

1. Introduction

2060 Chiron (95P/Chiron) was discovered by C.T. Kowal in October 1977 (Kowal, Liller and Marsden 1979). It was the ﬁrst of the group of several known objects of this type. Chiron is the only Centaur which exhibits a cometary activity. Previous papers (Hahn and Bailey 1990, Nakamura and Yoshikawa 1993, Oikawa and Everhart 1979) clearly showed that Chiron is on a highly unstable orbit due to planetary perturbations and its dynamical lifetime among the giant planets is short. Kuiper Belt or Oort Cloud are considered as a possible place of its origin (Nakamura and Yoshikawa 1993). 306 A. A.

Investigation of the dynamical evolution of Centaurs group is of importance not only for determination of possible past and future paths of particular objects but also helps to deﬁne the origin and the dynamical evolution of small bodies in the outer region of the Solar System. New scheme for orbital elements creation adopted from Sitarski (1998) was used in computations.

2. Numerical Modeling

2.1. Observational Material and Selection of Orbits Observations of Chiron were taken from the ﬁles available at the Minor Planet Center in Cambridge, USA. The set of apparitions covers time interval since April 24th, 1895 till June 1st, 1998. Observations were selected according to mathemati- cally objective criteria, elaborated by Bielicki and Sitarski (1991), assuming normal distribution of random errors of observations. After the selection process, 623 observations were used for the orbit improvement using the least square method. Or- bital calculations were executed using Sitarski’s orbital program package (Sitarski 1971, 1979, 1984). Table 1 gives the obtained elements of Chiron’s orbit.

Table1 Chiron’s orbital elements derived from the observational data. All elements are given in J2000.0 reference frame

epoch = 1998 July 6.0 (JD 2451000.5) ¦ T = 1996 Feb. 11.25430 ¦ 0.00102 q = 8.45177389 0.00000225

ω ¦ = 209.38546 ¦ 0.00010 e = 0.3800584 0.0000002

Ω ¦ = 339.40743 ¦ 0.00011 i = 6.938870 0.00002 P = 50.34 [yrs] RMS = 0.97 [arcsec]

Values of orbital elements are uncertain due to random errors of observations. That is why a single set of orbital elements has a limited validity for predictions of object’s long time dynamical behavior. An ensemble of possible orbital elements is often used in such cases and the object’s dynamical evolution is described in terms of probability. Usually sets of elements are acquired by varying initial orbital elements within a reasonably small range, in most cases in the range of their mean errors. This procedure forms different orbital element sets but they cannot be treated as element sets of one celestial body. In fact they represent similar orbits of quite different bodies because these orbits do not ﬁt well the observations. The sets of elements, used in modeling and described in this paper, were generated in a different way. A method was devised (Sitarski 1998) for creation of orbits which well represent the observations. This method allows to produce any number of orbital element sets, Vol. 52 307 all of them ﬁtting the observations. Therefore these orbits can be treated as orbits of one celestial body. Chiron’s dynamics was modeled with a use of 900 orbital element sets created by the method proposed by Sitarski. Table 2 shows the ranges of all six elements.

Table2 Dispersion of Chiron’s orbital elements. The upper section shows boundary values for element sets of 900 orbits, separately for each single element. The lower section shows the mean orbit which ﬁts all the observations and the maximum and minimum differences for each element – B1950.0 reference frame

epoch M a e ω Ω i 1998 07 06 17.14702 13.63318528 0.38005919 339.43772 208.65749 6.94435 1998 07 06 17.14689 13.63318083 0.38005808 339.43695 208.65676 6.94421

1998 07 06 17.14695 13.63318306 0.38005859 339.43737 208.65712 6.94430

: ·: ·: ·: ·: ·:

· 00007 00000222 00000060 00035 00037 00005

: : : : : : 00006 00000223 00000051 00042 00036 00009

2.2. Numerical Integration Chiron’s dynamical evolution was investigated by integrating six-body problem including giant planets from Jupiter to Neptune. Initial conditions for the planets were taken from the Bretagnon planetary theory. All starting positions and velocities were reduced to the barycentre of the Sun and inner planets. Each of 901 sets of orbital elements (900 orbits generated using Sitarski’s method and 1 orbit acquired using the least square method) consisted of coordinates and velocities for giant planets (identical values in all cases) and for a massless particle (Chiron’s orbit).

The sets were integrated separately forwards and backwards for ¦150 000 years using 15-th order RADAU integrator (Everhart 1974, Everhart 1975), with error tolerance set to 10 13 . This integrator adjusts the step size to keep the accuracy for all objects taking part in the process. In a process of numerical integration, an error always affects the final solution. There is a possibility that numerical error dominates natural solution while integrating n-body problem over long time scales (Gabryszewski 1998). Therefore the integration process has been fully tested. First, the method of integration was checked by comparing positions of the outer planets published in American Ephemeris and Nautical Almanac (Eckert, Brouwer and Clemence 1951) to those obtained by RADAU method with initial conditions taken from the same source. They agreed in 7–10 decimal places in the time span covered by the Almanac (which is about 400 years). This seems to be a very good outcome – initial conditions, obtained with the use of Cowell method of 308 A. A. integration and the fixed step size of 40 days, are published in the Almanac with accuracy of 11 significant places. The original Chiron orbit was integrated on a 5000 year time scale using three different methods: RADAU, Bulirsch and Stoer (1966) and recurrent power series (Sitarski 1989). The results were comparable up to 7–9 significant places. Moments of time of Chiron’s close encounters to Saturn and Uranus were also compared to the results published by Oikawa and Everhart (1979) to double-check the integration routine. They are consistent till four close encounters. The accuracy of computations were also studied by checking the variations of the Jacobi integral in time. Accuracy of calculations were specified by an ampli- tude of Jacobi integral value variations for two three-body problems: Sun–Saturn– Chiron and Sun–Jupiter–Chiron because these two planets have the strongest gravitational influence on Chiron’s orbit. The influence of Uranus and Neptune is several times weaker. The integral values conserve 12–11 significant places in the 1.5 Myrs of integration time and 13–12 significant places for the time interval of about 200 kyrs.

3. Dynamical Evolution of Chiron

Orbital elements of each generated particle are the direct outcomes of integration processes. Figs. 1a–1d present a differences between minimum (lower curves) and maximum (upper curves) values of the semi-major axes, eccentricities, inclinations to ecliptic plane and perihelion distances of all 901 evaluated orbits, recorded every 150 years. Chiron’s motion is very chaotic. The maximum values of semi-major axes grow fast (Fig. 1a). Close approaches which move object to a different orbits (values of semi-major axes change from 150 a.u. to over 300 a.u.) can be observed after 20 kyrs of integration backwards and forwards. Differences of maximum and minimum values for eccentricities are much larger. We can easily see that the eccentricity values of all orbits are nearly similar in the time span of a few thou- sands years and then increase rapidly to 0.7–0.8. Such huge changes are caused by numerous close encounters with Saturn and Jupiter. A vast range of the possible eccentricities is occupied after 30–40 kyrs of dynamical evolution. Chiron’s orbit is stable in a very short time interval – about 9 kyrs (over 5 kyrs forwards and less than 4 kyrs backwards). There is also no past/future symmetry of the evolution (see Figs. 1a and 1b). This is caused mainly by less frequent close approaches to Saturn at the beginning of the forward integrations. Variations between minimum and maximum values of inclinations (Fig. 1c) and perihelion distances (Fig. 1d) look different then variations of semi-major axes and eccentricities. After a short period of quasi-stability, differences of inclinations and perihelion distances grow very rapidly and then the changes of values are small and slow. Vol. 52 309

Fig. 1. Variations of minimum and maximum values of 4 elements of all 901 evaluated orbits as a function of time: a – semi-major axes, b – eccentricities, c – inclinations to the ecliptic plane, d – perihelion distances. Shadowed area marks the difference between boundary values of the particular element.

Figs. 1a and 1b show that Chiron may have been a short period comet in its earlier history and probably will be such an object in the future. This also means that Chiron is probably not a new object, it had been evolving into different types of cometary orbits and its future dynamical evolution will proceed similarly. Figs. 1c and 1d indicate that Chiron originated rather in the Kuiper Belt than in the Oort Cloud. Long time integrations of a large data sample give good statistical answer to the question about the past and future fate of Chiron. Table 3 presents types of possible Chiron’s orbits. They were classified using standard, currently accepted taxonomy (illustrated in Fig. 2) with a few additional modifications to distinguish Centaurs and Encke type orbits. Adopted definitions of different orbital types are as follows: LPCs (long pe-

riod comets) + KBOs (Kuiper Belt objects): P > 200 years, HTCs (Halley Type

< : <

comets): 20 < P 200 years and q 3 5 a.u., Centaurs: 20 P 200 years and

: q 3 5 a.u., JFCs (Jupiter family comets): P 20 years. Encke type comets are

defined by aphelion distance Q < 4 a.u. which means that the number of objects in this class should be overestimated due to lack of a condition for an orbit’s shape. The probability of occupation of particular orbital types was estimated from τ – the modified mean lifetime of a small object. This parameter has a specific definition. Usually mean lifetime is defined by a sum of lifetimes of objects on a particular orbital type divided by the number of these objects. However such a 310 A. A.

Fig. 2. Standard classification of the cometary objects. P is the period and a is the semi-major axis of a small body. The illustration is taken from Levison (1996). By the kind permission of The Astronomical Society of the Pacific. standard definition can be useless if we want to treat the mean lifetime as a measure of probability because there can be a few objects having huge values of orbital lifetimes. The calculated mean lifetime cannot be treated as a probability of orbital type occupation by numerous objects in such a case. To avoid this problem a modification was made to the standard definition. The modified mean lifetime τ is also defined by a sum of lifetimes of objects on a particular orbital type but the sum is divided by the number of all objects taking part in the integration process (which is 901 here). Due to this specific definition, higher modified mean lifetime of object on a particular orbital type means higher number of objects on this orbital type which gives higher probability of this orbital type occupation. Table3 Number of objects and the modified mean lifetimes τ on respective orbital types

backward integration forward integration

Orbital type deﬁnition number of objects τ (yrs) number of objects τ (yrs)

Centaurs 20 < P 200 901 118929 901 122244 :

and q 3 5

HTCs 20 < P 200 48 6404 33 5778 : and q < 3 5

JFCs P 20 410 30630 330 24023

Encke Q < 4 76 57129 51 49736

KBOs+LPCs P > 200 385 50176 395 50543 Vol. 52 311

More than one-third of objects were and will be Jupiter family comets and only about 3 to 5% were and will become a Halley type comets. About 43% of objects were and will be a Kuiper Belt object or a long period comet, their τ is the one-third of the whole time of investigated dynamical evolution. About 0.5% of objects collided with planets. Some hyperbolic orbits were also observed. Looking only at the Table 3, we can say that the probability that Centaurs will pass to the HTC orbits is small. But this representation will change if we look at Fig. 3. It presents τ of Chiron objects on a cometary orbital types in a 20 kyr sections.

HTCs JFCs Encke 15,000

10,000 TAU [yrs]

5,000

0 -100,000 -50,000 0 50,000 100,000 Time [yrs] τ Fig. 3. Cometary orbits with P 200. Variations of for HTCs, JFCs and Encke type objects in 20 000 year sections.

The mean lifetime of JFC objects grows rapidly in the first 20–40 kyrs and later the increase is much slower. Similar tendency is observed for the HTCs. τ of these objects is growing fast in the first 50 kyrs and then the increase is much slower. The fluctuations of τ are so big in forward integrations that longer time of evolution is needed to distinguish between very slow increase and a quasi-stable state. If we treat τ as a measure of probability of occupying particular orbital types in the past and in the future, we can see that after 50 kyrs of evolution the probability that the Centaur was or will be a HTC is about 4 or 5 times lower than for a JFC object. Chiron can also pass to Encke type, Kuiper Belt and long period cometary orbits. Variations of the mean lifetime for these orbital types have another structure comparing to JFC and HTC cases. The grow of the τ is quasi constant. 312 A. A.

4. Conclusions

Figs. 1a–1d present highly unstable and chaotic evolution of Chiron’s dynamics. As the numerical integrations show, there is a high probability that Chiron was and will be a short period comet so it cannot be a new object. The modeling showed that Centaurs can evolve to the prograde, low inclined HTC objects. Estimates of the HTCs number found at Levison (1997) are low (up to 3%), higher values are presented in Table 3 (about 3 to 5% of all objects). This can be explained as follows: Levison studied the dynamics of different Kuiper Belt objects while this research concerns possible evolution of only one body located in the giant planet region. These estimations refer to the mean number of objects in these populations. The time distribution indicates the fast growth of the HTC and JFC objects in the ﬁrst 50 and 30 kyrs, respectively. If we treat the modiﬁed mean lifetime of objects as a measure of probability of occupation of different orbital types, we can notice that the probability that Chiron was and will be a HTC is only 4 to 5 times lower than JFC object. This means that the population of the HTC objects can reach 20 to 25% of the population of JFCs after 50 kyrs of dynamical evolution. Halley type orbits are the rarest noticed orbital types but the number of these objects is three times greater than the number of apparent Halley type comets. This

leads to a conclusion that the present inner Solar System contains only a small fraction of cometary objects with 20 < P 200 years which could have originated in the region among giant planets. The huge and growing with time probability of Chiron’s evolution to orbits with aphelion distance less than 4 a.u. indicates that the outer planetary region can enlarge cometary and planetoid-type orbits with

P < 9 years. The Encke type orbits seem to be overestimated with the definition accepted in this paper. The Encke type is usually an orbit having an elongated shape and the value of aphelion distance smaller than the value of the semi-major of Jupiter’s orbit. Only this second condition was accepted as a definition of Encke type orbits in this paper. There is also one more remark on Encke type orbits. They were created although the perturbations from Earth and Venus had not been included to the calculations. That means the cumulation of giant planet perturbations is sufficient to push orbits into the inner Solar System and put them away (these orbits were not stable, lots of transitions among different orbital types were observed). One of the most important questions concerning the dynamical evolution of small bodies in the Solar System is how they move from the outer planet region towards Sun. Do they diffuse from one planet to another or are they thrown to the center by the close approaches? Both mechanisms of transportation were observed during the dynamical evolution but the calculations did not allow to state which of them is more efficient. Further studies are needed to analyze these processes. Vol. 52 313 150,000 q Q 100,000 Time [yrs] 50,000 Variations of the perihelion and aphelion distances in time Close encounters to Jupiter Close encounters to Uranus Close encounters to Saturn 0 4 0 20 12

2.5

10.5 q, Q [AU] Q q,

Fig. 4. Particle no 46. Variations of the perihelion and aphelion distances as a function of time. Sections mark close encounters to Uranus, Saturn and Jupiter and their dispersion in time. The length of the sections indicates a small body’s distance to a planet during the close encounter in a.u. units. The maximum distance of close encounter marked by the sections is 1.5 a.u. 314 A. A.

Fig. 4 presents an example of such diffusion process. The perihelion distance of the orbit decreases when the small body suffers numerous close approaches to Saturn after 50 kyrs of the evolution and then to Saturn and Jupiter after 64 kyrs. When the body is not exposed to strong planetary gravitational perturbations its dynamical evolution is very slow – the orbit is quasi-stable. And numerous close encounters to Jupiter diminish the perihelion and aphelion distances (the orbit of the body) very rapidly in about 10 kyrs. After that time the particle seems to be in a resonance relation with the planet. Fig. 4 shows that efficiency of the diffusion process is a result of close encounters. Seldom approaches and weak gravitational perturbations from Uranus and Saturn in the first 50 kyrs had an influence on the orbital evolution (seen as changes of the aphelion distance in time) but were not able to change the perihelion distance effectively. The process of diffusion was initiated by numerous and shallow close approaches to massive bodies. The dynamical investigations of chaotic orbits were done by numerical integrating the motion of a large number of massless bodies. The question is how many bodies should be used in such modeling. The results for the huge sample presented in this paper are comparable only with the results for samples of 100 and more objects. This value should be treated as the lowest limit of the object number in a sample. It seems that samples having less massless bodies do not provide good statistics of a dynamical orbit evolution.

Acknowledgements. The author would like to thank Dr Małgorzata Kró- likowska and prof. Grzegorz Sitarski for their valuable help. Special thanks should also be given to Dr Krzysztof Ziołkowski and Dr Sławomira Szutowicz for helpful discussions. This work was partially ﬁnanced by the KBN grant 2 P03D 012 15. Computa- tions were accomplished in the Networks and Supercomputing Center of Poznan,´ Poland, and in the Interdisciplinary Centre of Mathematical Modelling of Warsaw University.

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