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 ob- servations 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 veloci- ties 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 inte- grating 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 grav- itational influence on Chiron’s orbit. The influence of Uranus and Neptune is sev- eral 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 integra- tion processes. Figs. 1a–1d present a differences between minimum (lower curves) and maximum (upper curves) values of the semi-major axes, eccentricities, inclina- tions 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 (val- ues 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.