den jumps from one mean motion resonance to another, has poorly dened H. We further nd that these
ss of smers lieof the
taurs are also removed from the Solar System by ejection on hyper-bolic orbits or by collisions with a planet. Numerical analysis oftheir orbital evolution shows that these objects typically suffer fre-quent close encounters with the giant planets and their orbits arestrongly chaotic.
law behavior; instead, the uctuations increase slowly at shorttimescales and more rapidly at larger timescales, suggesting thatmultiple processes are at work. These two types of behavior arestrongly correlated with Centaur lifetime: with few exceptions,Centaurs exhibiting the diffusion-like behavior have dynamicallifetimes shorter than 22 Myr, whereas the second type havelonger dynamical lifetimes. We also nd that the latter group ofCentaurs are strongly correlated with the resonance-stickingbehavior noted qualitatively in previous studies, in which Centaursbecome temporarily trapped in mean motion resonances with the
* Corresponding author.E-mail addresses: email@example.com (B.L. Bailey), firstname.lastname@example.org
Icarus 203 (2009) 155163
Contents lists availab
.e l(R. Malhotra).years (Levison and Duncan, 1997; Dones et al., 1996; Tiscareno andMalhotra, 2003; Horner et al., 2004; di Sisto et al., 2007). Thesedynamical lifetimes are very short compared to the age of the SolarSystem, implying that the Centaurs are a transitional populationwith a source elsewhere in the system. Likely source populationsare the several dynamical subclasses of the Kuiper belt beyondNeptune (Levison and Duncan, 1997; Volk and Malhotra, 2008).Possible sinks of the Centaur population include the Kuiper beltsscattered disk, the Jupiter-family comets, and the Oort cloud; Cen-
turbing inuence of the four giant planets. (The length of thisintegration is more than 10 times the median dynamical lifetimesof the observed Centaurs found in previous studies.) We then ana-lyzed the Centaurs uctuations in semimajor axis to determinehow the root mean square uctuations evolve over time. We foundtwo distinct types of evolution of the semimajor axis uctuations.The rst is characterized by diffusion-like evolution, wherein themean square uctuations of the semimajor axis increase as apower law of time. The second type does not show this power1. Introduction
The Centaurs are a dynamical claSolar System whose orbital parametbetween those of the Kuiper belt andNumerical simulations have found tremoved from the Solar System on ti0019-1035/$ - see front matter 2009 Elsevier Inc. Adoi:10.1016/j.icarus.2009.03.044two types of behavior are correlated with Centaur dynamical lifetime: most Centaurs whose dynamicallifetime is less than 22 Myr exhibit generalized diffusion, whereas most Centaurs of longer dynamicallifetimes exhibit intermittent resonance sticking. We also nd that Centaurs in the diffusing class arelikely to evolve into Jupiter-family comets during their dynamical lifetimes, while those in the reso-nance-hopping class do not.
2009 Elsevier Inc. All rights reserved.
all bodies in the outerin a range intermediateJupiter-family comets.Centaurs are typicallyes of only a few million
Previous studies have provided detailed qualitative descriptionsof the different types of chaotic behavior of Centaurs. The presentstudy aims for a quantitative analysis of Centaur chaotic dynamicsby using a generalized diffusion approach, which is a relativelynew tool in Solar System dynamics. Towards this end, we startedwith the known sample of Centaurs, and we carried out a 100 mil-lion year (Myr) numerical integration of their orbits under the per-CentaursComets, Dynamics
in which the rms deviation of the semimajor axis grows with time t as t , with Hurst exponent H in therange 0.220.95, however (2) orbital evolution dominated by intermittent resonance sticking, with sud-Two dynamical classes of Centaurs
Brenae L. Bailey a, Renu Malhotra b,*a Program in Applied Mathematics, 617 N. Santa Rita, The University of Arizona, Tucsonb Lunar and Planetary Laboratory, 1629 E. University Blvd., The University of Arizona, T
a r t i c l e i n f o
Article history:Received 19 November 2008Revised 27 March 2009Accepted 31 March 2009Available online 13 May 2009
a b s t r a c t
The Centaurs are a transistrongly chaotic. These obof a few thousand years, atively as random walk andtaurs. Our analysis hasdistinguishable: (1) the ra
journal homepage: wwwll rights reserved.85721, USAn, AZ 85721, USA
population of small bodies in the outer Solar System whose orbits ares typically suffer signicant changes of orbital parameters on timescalestheir orbital evolution exhibits two types of behaviors described qualita-nance-sticking. We have analyzed the chaotic behavior of the known Cen-ealed that the two types of chaotic evolution are quantitativelym walk type behavior is well described by so-called generalized diffusion
le at ScienceDirect
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156 B.L. Bailey, R. Malhotra / Icargiant planets; these Centaurs typically hop from one resonance toanother for much of their dynamical lifetimes. Our analysis showsthat the two types of behavior can be objectively and quantita-tively distinguished, and suggests that the Centaurs may be com-prised of two distinct dynamical classes.
This paper is organized as follows. We summarize previousstudies of Centaur dynamics in Section 2. In Section 3, we describethe numerical simulations we have carried out to explore the orbi-tal evolution of Centaurs. Section 4 describes the analyses that wehave applied to the results of our simulations. In Section 5 we pro-vide a summary and conclusions.
2. Previous work
The Minor Planet Center (MPC) denes Centaurs as objects[with] perihelia beyond the orbit of Jupiter and semimajor axes in-side the orbit of Neptune.1 The MPC provides a combined list ofCentaurs and scattered disk objects. (The latter are not dened onthe MPCs webpages but are generally identied as objects with peri-helia near or slightly beyond Neptunes orbit and semimajor axesgreater than 50 AU.) While there is a general consensus on the lowerbound for Centaur orbits (although Gladman et al. (2008) proposed ahigher cutoff of q > 7:35 AU to exclude objects whose dynamics arecontrolled by Jupiter), the upper bound is open for interpretation.The samples of Centaurs studied by different authors thus varyslightly based on the authors chosen criterion, with some authorsconstraining semimajor axis only and others using constraints basedalso on perihelion and aphelion distance.
The rst Centaur, Chiron, was discovered in 1977 (Kowal, 1989),but only in the past decade or so has there been sufcient comput-ing power to explore the dynamical behavior of Centaurs usinglarge-scale integrations of particle orbits. Dones et al. (1996) sim-ulated the orbital evolution of 800 particles cloned from the sixCentaurs known at that time that had values of semimajor axis abetween 6 and 25 AU. The initial orbital elements of each particlewere identical to those of one of the Centaurs, save for the additionof a random variation in a of order 105 AU. They focused on thedynamical lifetimes of the particles and found median lifetimesof 0.55 Myr for their six ensembles of clones. Lifetime was mostsensitive to perihelion distance: at smaller perihelion, particlesare more likely to encounter one of the more massive planetsand receive large gravitational kicks, and thereby be removedfrom the Centaur population. They also reported that the numberof surviving particles in each ensemble decreased at rst roughlyexponentially with time, then more slowly as a power law.
In work published in 1997, Levison and Duncan followed theevolution of a sample of hypothetical Kuiper belt objects as theyevolved their way inward toward the Sun to become Jupiter-familycomets. A subset of their sample thus spent time as Centaurs.These authors argued that a dynamical classication based on Tiss-erand parameter Tp is more appropriate for objects on planet-encountering orbits than the traditional divisions based solely onsemimajor axis (or equivalently, orbital period). The Tisserandparameter is dened in the context of the circular restricted 3-body problem, and is given by
Tp apa 2aap1 e2
rcos i; 1
where ap is the semimajor axis of the planet, a and e are the semi-major axis and eccentricity of the small body in the heliocentricframe, and i is the inclination of the small body relative to the orbitof the planet. The Tisserand parameter is nearly constant for a givenparticle before and after an encounter with a planet. Note that when1 http://www.cfa.harvard.edu/iau/lists/Unusual.htmli 0, e 0, and a ap, then Tp 3, so values of Tp near 3 indicatethat the orbit of the particle is similar to the orbit of that planet(although this is not guaranteed), and the planet can strongly inu-ence the orbit of the particle. In particular, Levison and Duncanidentify the Centaurs with what they call Chiron-type comets, de-ned by TJ > 3 and a > aJ , where TJ is the Tisserand parameter withrespect to Jupiter and aJ 5:2 AU is the semimajor axis of Jupiter.Jupiter-family comets are dened as objects with 2 < TJ < 3.
Levison and Duncans results suggested that Centaurs can be-come Jupiter-family comets by being handed inward from oneplanet to the next through a series of close encounters. Based onthe approximate conservation of Tp and their assumed initial val-ues of a, e, and i for their hypothetical source population in the Kui-per belt, they calculated that, starting with Neptune, each planetcould scatter a small body just far enough inward to reduce itsperihelion distance so that the body could cross the orbit of thenext planet in. About 30% of the particles in their integrationsdid become Jupiter-family comets.
Tiscareno and Malhotra (2003) carried out a study of thedynamics of all of the known Centaurs as of 2002. Their numericalsimulation included the four giant planets and 53 Centaurs, the lat-ter treated as massless test particles, and they followed the orbitsfor 100 Myr. They chose their sample of Centaurs based on perihe-lion distance q alone, using the criterion 5:2 < q < 30 AU. Particleswere removed from the simulation when they reached eitherr > 20;000 or r < 2:5 AU where r is the heliocentric distance. Fromthis simulation, they found a median dynamical lifetime of 9 Myr,longer than but on the same order as the lifetimes found by Doneset al. (1996). Only 7 of the 53 particles (13%) survived the full100 Myr integration.
Tiscareno and Malhotra also observed qualitatively differenttypes of behavior over time, as indicated by time series plots ofsemimajor axis for different particles. They described these as res-onance hopping and random uctuations, with some particlesexhibiting a combination of both. This work provides the motiva-tion and starting point for our own work, described in Section 3.
Horner et al. (2004) have also investigated the dynamical evolu-tion of Centaurs. They integrated the orbits of 23,328 particlescloned from 32 Centaurs for 3 Myr both forward and backwardin time. From their results, they extrapolate half-lives of 0.532 Myr for each Centaur, dened as the time when half of theensemble of clones of that Centaur have been removed from thesimulation by either reaching a heliocentric distance of 1000 AUor colliding with a massive body. They also estimate a total popu-lation of 44,300 Centaurs with diameters greater than 1 km,based on the fraction of particles in their simulation that becomeshort-period comets and an assumed ux of one new short-periodcomet every 200 years. For this calculation they consider Centaursto be objects with perihelion q > 4 AU and aphelion Q < 60 AU.The authors propose a classication scheme for Centaurs basedon both perihelion and aphelion, suggesting that whichever planetis nearest at those parts of the orbit controls the dynamics of theCentaur. This results in 18 dynamical classes for objects with peri-helia between 4 and 33.5 AU. While the dening assumption is rea-sonable for a very detailed description, the number of categories isprobably too large for the purpose of describing the big picture ofCentaur dynamics.
di Sisto et al. (2007) modeled the origin of the Centaurs from apresumed source in the trans-Neptunian scattered disk population.They generated initial conditions of the source population by debi-asing the orbital element distribution of 95 observed scattered diskobjects, and then performed a numerical integration of 1000 testparticles for 100 Myr. From this simulation, they tracked particles
us 203 (2009) 155163that evolved into the Centaur zone and they estimated the intrinsicpopulation and the orbital distribution of Centaurs. They gave acomprehensive description of the qualitatively different types of
B.L. Bailey, R. Malhotra / Icarevolution found in their simulation. Furthermore, they determineda mean lifetime of 72 Myr for Centaurs, and a very strong andsmooth dependence of lifetime on perihelion distance. The authorsexplained qualitatively that their large value of the mean lifetime(compared with previous studies) was owed to a larger fractionof their sample exhibiting either resonance hopping or pseudo-sta-ble states with near-conservation of perihelion distance in therange between Saturns and Neptunes orbits.
In the present work, we do not attempt to trace the origins ofCentaurs or to determine the distribution of their dynamical life-times. Rather, our goal is to make a step towards a quantitativeanalysis of the qualitative descriptions of Centaurs chaotic orbitalevolution reported in previous studies. We make use of the tools ofgeneralized diffusion (Section 4.1), which is a relatively new appli-cation in Solar System dynamics. Although we analyzed only asmall sample (63 observed Centaurs), our results show that thequalitative behaviors found in previous studies are objectivelyquantiable by means of a Hurst exponent.
Our orbital integrations were done using the RA15 integrator(Everhart, 1985), a 15th-order variable step size method for ordin-ary differential equations, which is part of the public-domain soft-ware package Mercury (Chambers, 1999), designed for N-bodyintegrations in planetary dynamics applications. Everharts orbitintegrator has often been used for cometary orbits and is very sta-ble for large eccentricities and close planetary encounters; theprice for its high accuracy is its larger computational time require-ment, compared to the lower accuracy hybrid symplectic methodalso offered in the Mercury package. As numerical accuracy was apriority for the analysis of the strongly chaotic orbit evolution,and our simulation involved only a small number of known Cen-taurs, the penalty in computational time was not prohibitive. Weused a relative position and relative velocity error tolerance of1012.
Our primary simulation included the Sun,...