The Dynamical History of Chariklo and its Rings
Jeremy Wood, Jonti Horner, Tobias C. Hinse, Stephen C. Marsden
DDraft version October 12, 2018
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THE DYNAMICAL HISTORY OF CHARIKLO AND ITS RINGS
Jeremy Wood,
1, 2
Jonti Horner,
2, 3
Tobias C. Hinse,
4, 5 and Stephen C. Marsden Hazard Community and Technical College, Community College Drive Hazard, KY USA 41701 Computational Engineering and Science Research Centre, University of Southern Queensland West St, Toowoomba, QLD 4350, Australia Australian Centre for Astrobiology, UNSW Australia, Sydney, NSW 2052, Australia Korea Astronomy and Space Science Institute, 776 Daedukdae-ro, Yuseong-gu, Daejeon 305-348, Republic of Korea Armagh Observatory, College Hill, Armagh BT61 9DG, UK (Accepted March 27, 2017)
Submitted to AJABSTRACTChariklo is the only small Solar system body confirmed to have rings. Given the instability of its orbit, the presenceof rings is surprising, and their origin remains poorly understood. In this work, we study the dynamical history of theChariklo system by integrating almost 36,000 Chariklo clones backwards in time for one Gyr under the influence ofthe Sun and the four giant planets. By recording all close encounters between the clones and planets, we investigatethe likelihood that Chariklo’s rings could have survived since its capture to the Centaur population. Our results revealthat Chariklo’s orbit occupies a region of stable chaos, resulting in its orbit being marginally more stable than those ofthe other Centaurs. Despite this, we find that it was most likely captured to the Centaur population within the last 20Myr, and that its orbital evolution has been continually punctuated by regular close encounters with the giant planets.The great majority ( > Keywords: minor planets, asteroids: individual: 10199 Chariklo, planets and satellites: dynamicalevolution and stability, planets and satellites: rings
Corresponding author: Jeremy [email protected] a r X i v : . [ a s t r o - ph . E P ] M a y Wood et al. INTRODUCTIONThe Centaurs are a dynamically unstable populationof small bodies in the outer Solar system. The firstCentaur to be discovered, Chiron, was discovered in1977. After the discovery, astronomers searched througharchival images, revealing the presence of Chiron on oldphotographic plates, which allowed the object’s orbit tobe precisely determined. It was soon realised that Ch-iron followed an unusual path around the Sun, spend-ing the vast majority of its time between the orbits ofSaturn and Uranus (Kowai et al. 1979). In the decadessince Chiron’s discovery, many other Centaurs have beenfound, all following unstable orbits in the outer Solarsystem. Though the definition of Centaur varies withinthe astronomical community, we will use the definitionadopted by the Minor Planet Center that Centaurs moveon orbits with semi-major axes between those of Jupiterand Neptune, and have perihelia beyond Jupiter’s orbit (e.g. Jewitt 2009; Sheppard et al. 2000). They exhibitextreme dynamical instability (e.g. Horner et al. 2004a;Bailey & Malhotra 2009), being scattered chaotically asa result of regular close encounters with the giant plan-ets.As a result of their extreme dynamical instability, theobserved Centaurs can not simply be the last remainingmembers of a once larger, primordial population. In-stead, the must be continually replenished. Over theyears, a number of other Solar system small body pop-ulations have been suggested as potential sources forthe Centaurs, including captured Oort Cloud comets(Emel’yanenko et al. 2005) (e.g. Brasser et al. 2012;Fouchard et al. 2014), the Jovian Trojans (e.g. Horner &Evans 2006; Horner et al. 2012b), and the Neptune Tro-jans (Horner & Lykawka 2010a,b; Horner et al. 2012a).The primary source population, however, seems likelyto be the trans-Neptunian objects - principally the Scat-tered Disk (e.g. Di Sisto & Brunini 2007; Volk, & Mal-hotra 2008), with a small contribution from the classicalEdgeworth-Kuiper belt (e.g. Levison & Duncan 1997).In turn, the Centaurs are thought to be the primary par-ent population for the short-period comets - with up toa third of Centaurs likely to be captured to that popu-lation at some point during their chaotic evolution (e.g.Horner et al. 2004a).The largest known Centaur is Chariklo, with an esti-mated diameter of approximately 250 km(Fornasier etal. 2014). It moves on a moderately eccentric orbit be-tween the orbits of Saturn and Uranus, with a semi- major axis of 15.8 au. Early dynamical studies showedthat Chariklo moves on a relatively stable orbit for aCentaur, with an estimated dynamical half-life of 10.3Myr (Horner et al. 2004a).In 2013, observations of a chance stellar occultationby Chariklo revealed the unexpected presence of twonarrow rings with radii 391 km and 405 km, respectively- making it the only small body in the Solar systemconfirmed to possess rings (Braga-Ribas et al. 2014).The discovery of Chariklo’s rings was a great surpriseand has prompted significant discussion on their natureand origin, as well as opening up the possibility thatother small bodies such as Chiron could also possessrings (e.g. Ortiz et al. 2015; Pan & Wu 2016).A variety of mechanisms have been proposed to ex-plain the rings, including leftover debris from a collisionwith another small body, debris from the tidal disrup-tion of another small body (El Moutamid et al. 2014),partial tidal disruption of Chariklo itself (Hyodo et al.2016) and dust particles sent into orbit due to an outflowof CO and/or N from Chariklo as a result of cometaryactivity (Pan & Wu 2016).El Moutamid et al. (2014) suggest the possibility thatshepherd satellites could exist around Chariklo mak-ing the rings more stable. Such satellites are known tosculpt the rings of the giant planets - with several exam-ples found in the Saturnian system alone (e.g. Colwellet al. 2009).Whilst such shepherding satellites have not yet beenfound in orbit around Chariklo, their presence wouldpotentially ensure the long term survival of the ring sys-tem.The presence of rings around Chariklo is perhaps par-ticularly surprising when one considers that the orbitsof Centaurs are highly chaotic, as a result of the gravita-tional influence of the giant planets (Tiscareno & Malho-tra 2003; Bailey & Malhotra 2009). On average, a Cen-taur remains just 10 Myr in the Centaur region (Levison& Duncan 1994; Dones et al. 1996; Tiscareno & Malho-tra 2003; Horner et al. 2004a) which is far less than theage of the Solar system (4.6 Gyr).During their lifetime, Centaurs cross the orbits of thegiant planets and most likely experience multiple closeencounters within one Hill radius of those planets duringtheir stay in the Centaur region (Tiscareno & Malhotra2003; Bailey & Malhotra 2009; Araujo et al. 2016). Thisopens up the possibility that Chariklo has had a closeencounter with a giant planet at some time in its pastwhich was so close that the rings as they exist todaywould not have survived.The goal of this work is to determine the dynamicalhistory of Chariklo and its rings; and to examine the hariklo and its Rings CHARIKLO PROPERTIES AND THEORYChariklo was discovered in 1997 by the Spacewatchprogram , moving on an orbit that lies between those ofSaturn and Uranus, within 0.09 au of the location of the4:3 mean motion resonance with Uranus. Its physicalproperties and those of its rings are presented in Table 1.Orbital elements of Chariklo are shown in Table 2.Since its discovery, a number of groups have carriedout observations of Chariklo at a variety of wavelengths,with the goal of disentangling its surface composition.Despite the work that has been carried out, there re-mains significant disagreement on the Centaur’s surfacecomposition. Groussin et al. (2004) report that the re-flectance spectrum of Chariklo is consistent with a sur-face composed of 80% refractory material and 20% waterice.Guilbert et al. (2009) reported water ice in the com-bined spectrum of Chariklo+rings and Duffard et al.(2014) showed that the water ice feature comes onlyfrom the rings, and not from Chariklo. The rings arebelieved to be composed of water ice, silicates, tholinsand some amorphous carbon (Duffard et al. 2014).To date, no cometary activity has been detected forChariklo, despite it passing through perihelion in thelast decade. However, this does not rule out the pos-sibility that it may have displayed cometary activity inthe past (Guilbert et al. 2009).Backwards integrations show that Chariklo has abackward half-life of 9.38 Myr, some 1.6 Myr longer thanthe next largest Centaur Chiron (Horner et al. 2004b).2.1. The Stability of Rings Through Close Encounters:The “Ring Limit” Criterion
The severity of a close encounter between a small body(such as Chariklo) and one of the giant planets has beenshown to depend on the closest approach distance of theencounter, and the velocity of the small body at infinity.(Araujo et al. 2008; Hyodo et al. 2016). In order to de-termine the dynamical history of Chariklo and its rings,we neglect velocity effects following Araujo et al. (2016) http://spacewatch.lpl.arizona.edudiscovery.html accessed29th October, 2016 and compare the minimum close encounter distance be-tween Chariklo and a planet to three different criticaldistances within the Hill sphere of the planet. The firstof these is the distance between Chariklo and a planetat which tidal forces can disrupt a Chariklo-ring par-ticle binary pair instantaneously. This tidal disruptiondistance, R td , for a binary consisting of a massless, out-ermost ring particle in a circular orbit and Chariklo isgiven by: R td ≈ r (cid:16) M p m ch (cid:17) (1)where M p is the mass of the planet, m ch is the massof Chariklo and r is the orbital radius of a ring par-ticle (Agnor & Hamilton 2006; Philpott et al. 2010).When Chariklo is just within the tidal disruption dis-tance to a planet, an outermost ring particle is justoutside Chariklo’s Hill sphere. According to Araujoet al. (2016) the minimum distance obtained betweenChariklo and a planet during a close encounter must be ≤ R td in order for the encounter to have a significanteffect on the rings. We will refer to this distance as the’ring limit’, R . They considered the effect ‘noticeable’ ifthe maximum change in eccentricity of any orbiting ringparticle was at least 0.01.But there is one more critical distance to consider. Atan even closer distance to a planet is the Roche Limit- the distance within which a small body like Chariklocan be torn apart by tidal forces. For a small, sphericalsatellite of a planet, the equation for the Roche limit is(Murray & Dermott 1999): R roche ≈ R ch (cid:16) M p m ch (cid:17) (2)where R ch is the physical radius of Chariklo.Since closer approaches have a larger effect than moredistant ones, the minimum distance, d min , obtained be-tween Chariklo and a planet during a close encountercan be used to assess severity.We now present in Table 3 a severity scale based on d min relative to the distances R H , R, R td and R roche . METHOD3.1.
Chariklo
In order to determine whether Chariklo has experi-enced sufficiently close encounters with the giant planetsto disrupt its rings during its life, we need to be able todetermine its historical dynamical evolution.To do this, we follow the same methodology as thatused in previous studies of dynamically unstable objects(e.g. Horner et al. 2004a; Horner & Lykawka 2010a; Kisset al. 2013; P´al et al. 2015) and follow the evolution of
Wood et al.
Table 1.
Properties of Chariklo and its rings. [1] Altenhoff et al. (2001) [2] Jewitt & Kalas (1998) [3] Groussin et al. (2004) [4]Braga-Ribas et al. (2014) [5] El Moutamid et al. (2014) [6] Fornasier et al. (2014) [7] Campins & Fern´andez (2000) [8] Duffardet al. (2014) [9] Brown & Koresko (1998).Property Value UncertaintyRadius (km) 137
3% water ice Inner Ring Width (km) 7 Inner Ring Radius (km) 391 Outer Ring Width (km) 3 Outer Ring Radius (km) 405 Ring Composition 20% water ice40-70% silicates10-30% tholinssmall quantities of amorphous carbon Table 2.
Orbital elements of Chariklo taken from the Aster-oids Dynamic WWW b site for epoch MJD 2,457,600.0 basedon an observational arc of 9,684.35 days.Element Value ± Uncertainty (1-sigma)eccentricity 0.172265 ± ± ± ± ± ± a http://hamilton.dm.unipi.it/astdys/; accessed 31st Dec., 2015 b http://hamilton.dm.unipi.it/astdys/; accessed 31st Dec., 2015 a suite of clones of Chariklo backwards in time for aperiod of 1 Gyr. By following the evolution of a large Minimum Distance Range Severity d min ≥ R H Very Low10 R td ≤ d min < R H Low R td ≤ d min < R td Moderate R roche ≤ d min < R td Severe d min < R roche Extreme
Table 3.
A scale ranking the close encounter severity be-tween a ringed small body and a planet based on the min-imum distance obtained between the small body and theplanet, d min , during the close encounter. R H ,10 R td , R td and R roche are the Hill radius of the planet, R = 10 × tidal disruption distance, tidal disruption distance and Rochelimit respectively (see text for details). population of Chariklo clones, we can obtain a statisticaloverview of the object’s potential past history. hariklo and its Rings a , eccen-tricity, e , and inclination, i , of the test particles in evensteps through the full ± σ uncertainty ranges in thoseelements. We held the three rotational orbital elements,argument of perihelion, longitude of ascending node andMean Anomaly constant across our population of clones.33 massless test particles per orbital parameter werecreated for parameters a, e and i to yield a total of 33 =35,937 test particles. Test particles were evenly spacedacross the full uncertainty range of the orbital parame-ter.The initial orbital elements of the four giant planetswere found using the NASA JPL HORIZON ephemeris for epoch Jan 1, 2000 at UT 00:00. Inclinations and lon-gitudes for both Chariklo and the planets were relativeto the ecliptic plane.The planets were then integrated (within the heliocen-tric frame) to the epoch MJD 2,457,600.0 - the epoch ofthe Chariklo clones using the Hybrid integrator withinthe
Mercury
N-body dynamics package (Chambers1999). Test particles and planets were then integratedbackwards in time for 1 Gyr in the 6-body problem (Sun,four giant planets and test particle) subject only to thegravitational forces of the Sun and giant planets. Thisintegration time is 100 times longer than the typical life-time of a Centaur ( ∼
10 Myr). Therefore, the conclu-sions presented in this study are limited to within thistime span.For the symplectic integration we chose a time stepof 40 days (Horner et al. 2004a,b) corresponding to ap-proximately 1% of the orbital period of Jupiter, the in-nermost planet at the start of our integrations ensuringan accurate orbit calculation for the giant planets andthe particle during non-close encounter epochs (e.g. Tis-careno & Malhotra 2003).We set the accuracy tolerance parameter for theswitch-over integration algorithm to be 10 − . Thisensured an accurate integration of the test particle dur-ing epochs of high eccentricity excursions as a result ofclose encounters. A close encounter was said to haveoccurred when the distance between a test particle anda planet was ≤ a − e − i elements and the minimum separationobtained between the test particle and planet, d min . http://ssd.jpl.nasa.gov/horizons.cgi?s body=1 Test particles were removed from the simulation bycolliding with a planet, upon reaching a barycentric dis-tance of 1,000 au, achieving e ≥ N = N o e − λt (3)Here, N o is the initial number of test particles, N isthe number of test particles remaining in the Centaurregion at a time t and λ is the decay constant. Thedecay constant can be found from the slope of the best-fit line of a graph of ln( NN o ) vs time. Then the half-life, τ , is given by: τ = -ln(0 . λ (4)The data for the number of test particles remaining inthe Centaur region at a time t was fit to Equation 3 toobtain the decay constant. Then Equation 4 was usedto find the half-life.The half-life gives a best first estimate to Chariklo’sage as a Centaur - with 50% of the clones of Chariklobeing ejected within that time period. We also used thehalf-life in Equation 3 to determine the time at which99% of all Chariklo-like objects would have left the Cen-taur region.3.2. The Severity of Close Encounters and the Mass ofChariklo
In order to gain an understanding of whetherChariklo’s rings existed prior to its capture to the Cen-taur region, or are a more recent addition, we can inves-tigate the times at which test particles had encounterswith the planets that were sufficiently close to disruptthe rings.If the great majority of clones were to experience evena few severe or greater disruptive encounters or a largenumber of low to moderate encounters, this would sug-gest that Chariklo’s rings most likely formed in the rel-atively recent past.On the other hand, if relatively few clones have en-counters deep enough to disturb the rings, then it isclearly feasible that the rings could be primordial (and,equally, such infrequent close encounters might in turnsuggest that any origin for the rings involving the tidaldisruption of Chariklo or an ancient satellite seems un-likely).
Wood et al.
We therefore examined the depths and timings of theclose encounters between test particles and planets andranked the severity of each encounter using the scale inTable 3.As Table 3 along with equations 1, and 2 show,the severity of a close encounter depends in part onChariklo’s mass. This mass was estimated using the av-erage density of Chariklo from Braga-Ribas et al. (2014)of 1,000 kgm and the radius value of 125 km from ElMoutamid et al. (2014). A mass of 8.18 × kg wasobtained. This calculation assumed that the shape ofChariklo was a perfect sphere, as it is nearly sphericalwith the major to minor axis ratio of 1.1 (Fornasier etal. 2014). 3.3. MEGNO and Lifetime maps
In addition to our N-body integrations of Chariklo’sorbital evolution, a complementary suite of calculationswere performed to examine the wider dynamical contextof Chariklo’s orbit.Since sampling very large regions of phase space is im-practical with full-scale N-body integrations, we insteadgenerated a MEGNO (Mean Exponential Growth factorof Nearby Orbits; Cincotta et al. 2003) map for the re-gion of phase space bound by 14 au ≤ a ≤
19 au and e ≤ . ×
800 pixels. Themap was constructed by integrating one test particleper pixel or 30,000 test particles total using the Gragg-Bulirsh-Stoer (Hairer et al. 1993) method.The initial values of a and e for each test particle weredetermined by each location of a pixel on the map. Theintegration algorithm makes use of a variable step-sizedetermined by a relative and absolute tolerance param-eter which both were set to be close to the machineprecision. The total integration time for each particle inthe a-e grid was 1 Myr.MEGNO maps show the chaoticity of a region of a − e space by calculating a parameter (cid:104) Y (cid:105) which is propor-tional to the Lyapunov characteristic exponent at eachpoint. The reader is referred to Cincotta & Sim´o (2000),Go´zdziewski et al. (2001), Cincotta et al. (2003), Gior-dano & Cincotta (2004) and Hinse et al. (2010) for moredetails on MEGNO maps. For an explanation of Lya-punov characteristic exponents the reader is referred toWhipple (1995). (cid:104) Y (cid:105) will asymptotically converge towards 2.0 for quasi-periodic orbits and diverge from 2.0 for chaotic orbits asthe system is allowed to evolve in time. For this work, quasi-periodic orbits were color codedblue and highly chaotic orbits yellow. Test particles wereremoved by following the same criteria as for the long-term integration described earlier in this work. In addi-tion, we terminated a given integration when (cid:104) Y (cid:105) > (cid:104) Y (cid:105) value were recorded. If a test particle sur-vived the entire simulation then its removal time wasrecorded as 1 Myr. A lifetime map was then gener-ated in conjunction with the MEGNO map covering thesame a-e grid space. In the life-time map shortest re-moval times were color coded black and the longest withyellow. RESULTS4.1.
The Dynamical History of Chariklo
Over 70 million close encounters within 3 Hill radiiwere recorded, with roughly 7.1 million of these being atdistances less than one Hill radius. The close encounterswere analysed using eight different subsets of the entireencounter dataset. Five of those subsets examined closeencounters whilst the clone in question was a member ofone of the Solar system’s various small body populations(as detailed below), with the other three described asfollows:1. The set of first close encounters - a first closeencounter is the earliest time chronologically atwhich a close encounter occurred. Each test par-ticle had one and only one of these.2. The set of close encounters at any time at whichthe test particle was classified as a Centaur. Eachtest particle had more than one of these.3. The set of earliest close encounters chronologically(not necessarily a first close encounter) at whicheach test particle was classified as a Centaur. Eachtest particle had one and only one of these.The subsets of close encounters based on the member-ship of the clone in a small body population when theclose encounter occurred are described as follows:1. Inner SS - a ≤ a J
2. Comet - a > a J and q < a J
3. Centaur - a J < a < a N and q > a J
4. TNO - a ≥ a N
5. Ejection - the test particle was being ejected fromthe Solar system at the time of the close encounter hariklo and its Rings Table 4.
The percentage of close encounters as a func-tion of membership of the different small body populationsChariklo’s clones occupied through the course of the integra-tions. Region PercentInner SS 7Comet 9Centaur 53TNO 31Ejection 0.4
Where a is the semi-major axis of the test particle atthe time of the close encounter, a J the semi-major axisof Jupiter, a N the semi-major axis of Neptune and q theperihelion distance of the test particle.The percentage of close encounters which occurredwhen the test particle was in each of the five populationsubsets is shown in Table 4. The Centaur and TNOsubsets dominate with 53% and 31% of the close en-counters respectively. This implies but does not provethat Chariklo entered the Centaur region from beyondNeptune.To determine the dynamical history of Chariklo andif Chariklo did enter the Centaur region from beyondNeptune, three of the subsets were investigated.First, analysis of the Centaur subset showed thatthe average time between consecutive close encounterswithin 3 Hill radii of a planet in the Centaur region was8 kyr. Therefore, on Myr timescales, the earliest timechronologically of a close encounter of a test particlewhile classified as a Centaur was taken to be the ap-proximate time of insertion of the test particle into theCentaur region.The set of earliest close encounters chronologically inwhich each test particle was classified as a Centaur wasused to determine the number of test particles in theCentaur region as function of time. This data was an-alyzed by fitting it to Equation 3 with time measuredfrom the start date backwards in time.Figure 1 shows the decay of the number of test par-ticles in the Centaur region moving backwards in time.Note the reverse ‘s’ shape of the graph. It took ∼ Figure 1.
The decay of test particles from the Centaurregion moving backwards in time. The decay is exponentialfrom 1.0 Myr to 14.1 Myr ago. The straight line is the lineof best fit over this time interval. It has a slope of -0.2346Myr − and linear regression coefficient of -0.998. The slopewas used to find the half-life of ∼ best-fit line. This occurred because by that time manyremaining test particles had evolved onto more stableorbits, which in turn took longer to decay.From the slope of the best-fit line of -0.2346 Myr − and Equation 4, the half-life with respect to removalfrom the Centaur region was calculated to be ∼ Wood et al.
Figure 2.
A histogram of the number of first close encoun-ters over the last 100 Myr. The bin size is 2.5 Myr. counters was studied to determine from what region ofthe Solar system Chariklo entered the Centaur region.Figure 2 shows a histogram of the number of first closeencounters over the last 100 Myr.Table 5 shows statistics on the first close encountersby region of the Solar system. From the table it seemsmost likely that Chariklo entered the Centaur regionfrom an orbit outside that of Neptune, perhaps fromthe Edgeworth-Kuiper Belt (Horner et al. 2004b) orScattered Disk (Duncan et al. 2004; Di Sisto & Brunini2007). Two factors point to this conclusion:1. The small percentage (2%) of the subset of firstclose encounters which were also members of theinner Solar system subset makes it statistically un-likely that Chariklo was captured directly to theinner Solar system from elsewhere (such as a long-period comet orbit, or the main asteroid belt) andthen migrated outwards to the Centaur popula-tion.2. The much larger percentage (63%) and earliestchronological mean time of the subset of first closeencounters which were also members of the TNO
Table 5.
Statistics on the set of first close encounters bysmall body population of the Solar system.Subset Percent Mean Time Ago (Myr)Inner SS 2 15.7Comet 21 12.1Centaur 6 15.5TNO 63 32.6Ejection 8 31 subset makes it statistically likely that Chariklowas a TNO before becoming a Centaur.These dynamical results potentially complement theobserved physical properties of Chariklo, which also sug-gest both an origin beyond the orbit of Neptune, andthat the object has not spent a protracted period in theinner Solar system.First, the presence of volatiles on Chariklo’s surfacesuggests that it has not spent lengthy periods interiorto the Solar system’s ice-line where most sublimation ofvolatile material occurs (Whipple & Sekanina 1979; DiSisto et al. 2009; Brown et al. 2011).Indeed, Levison & Duncan (1997) suggest that just25 kyr in the inner Solar system is enough to entirelydevolatilise comets. However, it should be noted thatChariklo is significantly larger than the nuclei of short-period comets (Weissman & Lowry 2008) - and so couldhave potentially contained far more volatile material,and would therefore been able to survive a longer periodof devolatilisation. Still, the presence of volatiles doessuggest an origin beyond the ice-line - and most likely,beyond the orbit of Neptune.Nevertheless, though the percentages of close encoun-ters occurring in the inner Solar system and Comet sub-sets are relatively small they are not negligible.This allows for the possibility that Chariklo could havebeen active for brief periods in its past, and its ringsreplenished by cometary activity. The only caveat isthat Chariklo would have needed to migrate outwardfrom such an orbit before its volatiles were extinguished.However, such inward and outward migrations are dy-namically feasible (Horner et al. 2004a).It should also be noted that for Chariklo to exhibitcomet-like activity it would not be necessary for the or-bit to be in the inner Solar system as Centaurs beyondJupiter are known to be active (Jewitt 2009).To determine which planet dominated the close en-counters in each population, the close encounters of eachof the five population subsets were subdivided by planet.The results are shown in Table 6. hariklo and its Rings Table 6.
The number of close encounters within one Hillradius by planet and population. Most close encounters oc-curred between the Centaur population and Uranus.Population J S U NInner SS 492255 1531 0 0Comet 452415 139289 11879 6652Centaur 56142 991571 2033886 717860TNO 9026 224005 475861 1465906Ejection 18687 9671 479 571
Uranus dominated the number of close encounters ofthe Centaur subset followed by Saturn, Neptune andJupiter. In the TNO subset, Neptune dominated fol-lowed by Uranus, Saturn, and Jupiter. Thus, statisti-cally, Neptune is most likely responsible for perturbingChariklo into the Centaur region over time.Jupiter dominated the number of close encounters ofthe other three subsets - Inner SS, Comet and Ejection.4.2.
The Dynamical History of the Rings of Chariklo
The values of all ring limits, tidal disruption distancesand Roche limits are shown by planet in Table 7.Jupiter had the largest value of R at 0.02400 au andUranus the smallest at 0.008584 au. All values of R werewell within 1 Hill radius of each planet by an order ofmagnitude or larger.Thus, to have a close encounter of at least moder-ate severity, it must be far closer than the size of theplanet in question’s Hill sphere - sufficiently close, infact, that it would be placed within the domain of theregular satellites of that planet.For example, to have a moderate close encounter withJupiter, Chariklo would have to approach the giantplanet at a distance similar to the orbital radius ofThemisto, or roughly a factor of four times more dis-tant from the planet than Callisto.In other words - disruptive encounters require veryclose encounters, and hence might be expected to berelatively infrequent.This hypothesis is well supported by our data ascan be seen in Tables 8 and 9. Every single clone ofChariklo experienced multiple close encounters, how-ever, the great majority of these approaches were rel-atively distant. Over 99% of all close encounters wereoutside the ring limit R = 10 R td for the planet in ques-tion. Therefore, we conclude that planetary close en-counters have not played a major role in the disruptionof rings. Table 7.
The Hill radii, ring limits, tidal disruption dis-tances and Roche limits for each giant planet, see text fordetails.Planet R H (au) R (au) R td (au) R roche (au)J 0.3387 0.02400 0.002400 7.408 × − S 0.4128 0.01606 0.001606 4.956 × − U 0.4473 0.008584 0.0008584 2.649 × − N 0.7704 0.009069 0.0009069 2.799 × − Table 8.
The number and percent of close encounters byseverity which occurred within one Hill radius of any planet.Severity Number PercentLow 7084469 100.0Moderate 21953 0.0Severe 1025 0.0Extreme 239 0.0
Table 9.
Severity of close encounters by planet which oc-curred within one Hill radius of any planet. 99%, of closeencounters were of low severity. Only 0.0034% of close en-counters were of extreme severity.Severity J S U NLow 1012998 1360893 2520339 2190239Moderate 14685 4884 1675 709Severe 707 219 71 28Extreme 135 71 20 13
Just 35% of the clones experienced at least one en-counter within 10 R td . Thus, over half of the clones neverexperienced even at least a moderate close encounter.Furthermore, since only 0.0034% of the close encoun-ters were extreme, it is unlikely (but still possible) thatthe rings were created by gaseous outflow during a closeencounter (Hyodo et al. 2016) because this would re-quire Chariklo to be closer to the planet than its Rochelimit.This theory of ring formation may be further put indoubt if the purported rings around Chiron (Ortiz etal. 2015) and the Saturnian satellites Rhea and Iape-tus(Sicardy et al. 2016) are confirmed because it wouldsuggest that rings around small bodies are more com-mon and are not formed by a very rare extreme closeencounter.0 Wood et al.
It should be noted that no age of the rings of Chariklocan be stated with absolute certainty since the totaleffects of gaseous outflow, shepherd satellites (if any),ring replenishment and non-gravitational forces are un-known. 4.3.
MEGNO and Lifetime maps
The lifetime map in Figure 3 shows that the longestlifetimes for Chariklo-like orbits in the a − e region boundby 14 au ≤ a ≤
19 au and 0 ≤ e ≤ . ≤ a ≤ . ≤ e < . . ≤ e ≤ .
55 virtually no lifetimesof 1 Myr can be seen for any value of a . Orbits with e > ∼ .
55 have noticeably shorter lifetimes for nearlyall a values to as low as 0.01 Myr. We attribute thisdrop in lifetime to the crossing of Saturn’s orbit.The MEGNO map in Figure 3 shows that the entireregion is dominated by highly chaotic orbits ( (cid:104) Y (cid:105) ≥ a . Instead, rel-atively small islands of low chaoticity (quasi-periodic or-bits) can be seen scattered about the region. The extentof their sizes might well depend on the initial phase an-gle of Chariklo. It is noteworthy that these islands lie inthe same rectangle which contains nearly all of the mostlong-lived orbits seen in the lifetime map.The orbits with relatively longer lifetimes and lowerchaos are said to display stable chaos, and Figure 3shows that Chariklo has one of these orbits. CONCLUSIONSThe dynamical history of Chariklo and its rings wasdetermined using the technique of numerical integrationof massless clones backwards in time for 1 Gyr and byrecording close encounters between test particles and gi-ant planets.We find that Chariklo most likely originated in anorbit beyond Neptune and was likely captured intothe Centaur population via perturbations from Neptunesometime within the last 20 Myr. The backwards half-life with respect to removal of clones from the Centaurregion is ∼ Figure 3.
The lifetime map (top panel) and the MEGNOmap (bottom panel) of Cariklo-like orbits. Chariklo is lo-cated at a = 15 . e = 0 .
172 and is marked by thestar. For the top panel the longest lifetimes are shown inyellow and the shortest in black, while for the bottom panelhighly chaotic orbits are shown in yellow and the least chaoticare blue. rience multiple periods of cometary behaviour through-out their lifetimes.The critical distances of the Hill radius, tidal disrup-tion distance, ‘ring limit‘ (defined as ten times the tidaldisruption distance) and Roche limit were used to createa severity scale for close encounters based solely on theminimum distance obtained between the test particleand planet during the encounter.More than 99% of all close encounters over the courseof our simulations were sufficiently distant that no im-pact on the structure of Chariklo’s rings would be ex-pected. Indeed, just 35% of all clones experience an en-counter within ring limit with one or other of the giantplanets. In other words, 65% of clones never experiencea sufficiently close encounter to significantly disrupt the hariklo and its Rings a and e ) are are found to be dependent on theeccentricity of the orbit with a general trend that orbitswith higher eccentricities have shorter lifetimes. Thecrossing of Saturn’s orbit plays a strong role in reducingthe lifetime of an orbit.Nearly all Chariklo-like orbits in the region boundedby 14 au ≤ a ≤
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