IS THE CHAOTIC CLOCK TICKING CORRECTLY?saj.matf.bg.ac.rs/156/pdf/047-069.pdf · Bull. Astron. Belgrade 156 (l997), 47 – 69 UDC 523.44–32 Original scientiﬁc paper IS THE CHAOTIC

Bull. Astron. Belgrade � 156 (l997), 47 – 69 UDC 523.44–32Original scientific paper

IS THE CHAOTIC CLOCK TICKING CORRECTLY?

Z. Knezevic and B. Jovanovic

Astronomical Observatory, Volgina 7, 11000 Belgrade, Yugoslavia

(Received: August 22, 1997)

SUMMARY: We performed an extensive analysis of the dynamics of all presentlyknown members of the Veritas asteroid family, in order to check the estimate of itsage given originally by Milani and Farinella (1994), who used the so-called ”chaoticchronology” method for this purpose. We started from a much larger sample offamily members and carried out many more numerical integrations of the entirefamily and, in particular, of the chaotic bodies; integrations of these latter embracedtypically 200 Myr per body, covering a total of about 2.1 Gyr. We show that thedynamics in the region of the phase space occupied by the family is more complexthan previously believed, and that there are bodies with motions ranging fromremarkable stability to strong chaos, due primarily to the effects of the 21/10 Jovianmean motion resonance and its harmonics involving slow angular variables. Sevenstrongly chaotic bodies exhibit a qualitatively similar behavior, in agreement withthe predictions of Milani et al. (1996), but also a number of previously unknownfeatures, the most important one pertaining to the fact that some of the chaoticasteroids, including the largest member of the family 490 Veritas itself, can becaptured into the resonance and stay there for a very long time in a quasi–stablestate. Although, in general, our results do not contradict the conclusions of Milaniand Farinella and fit reasonably well with their estimate of the family age, there arealso some interesting results of this analysis, which open new questions and requirea more thorough and dedicated investigation.

1. INTRODUCTION

The ages of the asteroid families represent oneof the most important pieces of information still mis-sing and badly needed in order to calibrate and con-strain the current models of the collisional evolutionof the asteroid belt, and of the Solar System as awhole. At the same time, this is one of the most dif-ficult problems to deal with in asteroid research, asno ”direct” evidence is available yet (probably until asample–return mission to a family member will havebeen carried out). Several attempts have been madeso far in this direction (c.f. Farinella et al. 1989,Farinella 1994), but with no real success. Recently,

however, Milani and Farinella (1994) found a way to”indirectly” estimate at least an upper bound to theage of a family (more precisely, of a particular fam-ily – that of 490 Veritas), by means of the so–calledchaotic chronology method, based on purely dynami-cal arguments. This result has attracted a great dealof attention, and, as a consequence, a number of pa-pers have appeared in the past few years devoted justto the study of this family.

The Veritas family is a small, compact one,located in the outer part of the main belt, in a re-gion interesting and important from both the dy-namical and the physical points of view. The relia-bility of its membership is rated as very high, andit is also considered to be well defined in the proper

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Z. KNEZEVIC and B. JOVANOVIC

element phase space (Zappala et al. 1995). The dy-namical peculiarity of the Veritas family that hasbeen recognized and used by Milani and Farinella(1994) pertains to the fact that the proper elementsof a couple of family members (including the largestone, 490 Veritas itself) chaotically wander in thephase space of proper elements, instead of remainingnearly constant in time. By integrating the orbitsof the seven numbered family members (plus fourbackground objects) for 72 Myr in the past, theydemonstrated that the rate of chaotic diffusion un-dergone by the two member asteroids might be largeenough that they exit from the region occupied bythe family in the proper element space, after a char-acteristic time much shorter than the lifetime of theSolar System (some 50 Myr). As the physical char-acteristics of asteroid 490 Veritas itself are very sim-ilar to those of the rest of the family, thus undoubt-edly establishing it as family member (unfortunately,no data are available for the other wandering object3542 1964TN2, although, according to Migliorini etal. 1995, it is highly unlikely that any of the pro-posed family members can be a chance interloper),the characteristic time found from the dynamical dif-fusion effect should provide an upper bound estimatefor the age of this family.

A detailed analysis of the dynamics of the twoVeritas family asteroids having chaotic orbits hasbeen performed by Milani et al. (1996), in the frame-work of a study of the phenomenon of stable chaosin the asteroid belt. The Veritas family members arefound to be a paradigm for this particular type ofchaotic motion, and therefore especially suitable forits investigation, as no other causes of chaotic behav-ior are relevant in the region occupied by the family.Milani et al. found that chaos in this case is due tohigh order mean motion resonances with Jupiter, incombination with secular perturbations on the peri-helia and the nodes of the asteroids. These perturba-tions push a number of critical arguments in and outof the resonance and shift the orbit from one reso-nance to another, in a typical irregular manner. Theproper elements are therefore less stable and a slowdiffusion is observed in eccentricity and inclination,which may eventually bring chaotic bodies out of thefamily.

Finally, Di Martino et al. (1997) reported thefirst optical reflectance spectra for seven members ofthe Veritas asteroid family and another body whichjoins the family when a slightly more relaxed crite-rion for membership is adopted. They found a slopevariation in these spectra, spanning a range that in-cludes slopes typical of all the low–albedo, primitivebodies (from C to D type). The possible explanationsof this slope gradient, according to the authors, aretwo: it is due either to a space weathering processsimilar to those affecting S–type asteroids, or to analteration of the parent body’s interior due to ex-tended thermal episodes prior to the family–formingbreakup event. In addition, they reconstructed theoriginal velocity field of the family, deriving it underthe tentative assumption that the present positionof the largest body 490 Veritas is representative ofits true position at the instant of breakup. Evenif this latter assumption may be unrealistic (due to

the chaotic motion of Veritas), they found that thevelocity field can be well represented by a ”jetlike”planar structure, which, in a qualitative sense, doesnot change even when the position of the barycenterof the family is moved.

In view of the above results regarding the Ver-itas family, and of the current understanding of cha-otic phenomena in the motion of asteroids in general,the rationale for the present paper can be summa-rized as follows:

• the conclusions of Milani and Farinella (1994),though quite sound and convincing in the fra-mework of the then available data, were drawnon the basis of a rather limited set of long–term integrations, and a less complete knowl-edge of the membership of the family, i.e. ofits size in the space of proper elements (which,in turn, might significantly affect the estimateof the family age). With a computing powerenhanced dramatically in the last few years,we are now able to perform many more exper-iments, covering much longer time spans thanMilani and Farinella originally did, thus ensur-ing that any conclusion based on the statis-tics of chaotic orbits will become more reli-able; moreover, the membership of the fam-ily is meanwhile increased so that it currentlyconsists of 11 numbered and 11 multi–opposi-tion and single–opposition bodies with reason-ably well determined orbits (i.e., with a qual-ity code for the osculating orbital elementsQCO < 20; see Milani et al. 1994);

• an improved understanding of the dynamicsinvolved with the chaotic Veritas family mem-bers may provide a capability of predictingthe stable chaos type behavior for bodies inthe same narrow semimajor axis region, anda number of newly recognized members of thefamily are found to be located just there. Hav-ing more bodies which exhibit a similar cha-otic behavior would give a more comprehen-sive insight in the complex dynamics involved,possibly strengthen and/or refine the existingresults and conclusions, and even perhaps of-fer some entirely new explanations and ideas.

2. PROCEDURES AND RESULTS

In order to produce comparable results, wehave in general closely followed the procedures ofnumerical integration and output analysis used byMilani and Farinella (1994; hereinafter referred toas M&F). Thus, we have performed numerical inte-grations by means of the same software (ORBIT9a,kindly provided by A. Milani), have analyzed theoutput by using the same routines and procedures(GIFFV software, again provided by A. Milani), andhave used the same software to compute proper el-ements (Milani and Knezevic 1994). Only in mi-nor details, which could not affect the comparisons,we have changed the original approach, and thesechanges will always be mentioned and briefly ex-plained, if necessary, in the following.

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Table I. Results from the 10 Myr integrations. The columns contain the asteroid identification, the mean(a) and RMS (σa) values of the filtered semimajor axis over the integration time span, the Lyapunovcharacteristic exponents LCE and Lyapunov times (T

L). Only the objects with shortest Lyapunov times

entered the second phase of our analysis.

Asteroid a [AU ] σa [AU ] LCE [yr−1] TL

[yr]

490 Veritas 3.17397 0.00125 9.775 × 10−5 10 250844 Leontina 3.19575 0.00008 5.494 × 10−7 1 500 000

1086 Nata 3.16536 0.00003 2.249 × 10−7 > 1 500 0002147 Kharadze 3.17098 0.00003 7.822 × 10−6 127 8002428 Kamenyar 3.16997 0.00004 2.662 × 10−6 375 7002934 Aristophanes 3.16659 0.00005 4.195 × 10−6 238 4003090 Tjossem 3.16906 0.00004 3.971 × 10−7 > 1 500 0003542 1964 TN2 3.17410 0.00122 1.115 × 10−4 8 9715592 Oshima 3.16844 0.00005 8.776 × 10−8 > 1 500 0005594 1991 NK1 3.16765 0.00032 3.387 × 10−5 29 5207231 1985 TQ1 3.16580 0.00004 3.160 × 10−8 > 1 500 000

1976 QL2 3.17038 0.00008 1.083 × 10−5 92 3401981 ES9 3.16474 0.00003 3.013 × 10−7 > 1 500 0001981 EE4 3.17405 0.00124 1.075 × 10−4 9 3021981 EM10 3.17178 0.00006 8.808 × 10−6 113 5001981 ER34 3.16538 0.00003 2.785 × 10−7 > 1 500 0001991 PW9 3.16425 0.00003 3.141 × 10−7 > 1 500 0002123 PL 3.17409 0.00125 1.122 × 10−4 8 9134107 PL 3.16364 0.00003 4.194 × 10−7 > 1 500 0004573 PL 3.17420 0.00112 1.218 × 10−4 8 2101118 T 3 3.17410 0.00119 1.048 × 10−4 9 5421122 T 3 3.18258 0.00004 2.248 × 10−7 > 1 500 000

As already mentioned, at present the Veritasfamily consists of a total of 22 objects. For all ofthem, we have first performed a preliminary numeri-cal integration, lovering 10 Myr, which served solelyfor the purpose of establishing their dynamical state;we also computed the mean/proper semimajor axesand their RMS deviations applying an on–line filter-ing procedure (see Carpino et al. 1987), as well asderived the Lyapunov characteristic exponents andthe corresponding Lyapunov times (the inverse of theslope). The initial conditions for these integrationswere taken from the asteroid osculating orbital ele-ments data base by E. Bowell (Bowell et al. 1994);the results are summarized in Table I.

Some simple conclusions can be drawn by in-specting the data contained in Table I. It is clearthat, from the point of view of the stability of theirmotion, there are three distinct groups of objectsin the family. Stable orbits are characterized by avery small LCE and a Lyapunov time on the orderof millions of years (note that T

Lvalues in excess

of 1.5 Myr have no meaning in our case, because ofthe short time span covered by the integrations). Agroup of objects with T

Lon the order of 105 yr can be

considered as being in an intermediate state betweenstability and chaos (they are only marginally chaotic,and no macroscopic instability due to chaos was ob-served for these objects). Finally, there are 7 stronglychaotic family members, and these are the only ob-

jects which we are actually interested in. Their Lya-punov times are on the order of 104 yr, and they alsoexhibit comparatively large variations (although nolong term evolution) of the semimajor axes, thus rep-resenting typical examples of stable chaos.

All but one of the chaotic objects have theiraverage proper semimajor axes clustered within anarrow strip, located at about 3.174 AU; the onlyobject that is out of this main chaotic strip (5594),is located at a slightly lower semimajor axis, but itis also somewhat less chaotic. All this is in goodagreement with the results of Milani et al. (1996),who found that chaos in this region is due to thehigh order mean motion resonance 21/10, and itssecular harmonics (with critical arguments contain-ing different combinations of perihelia and nodes).As explained in that paper, the region of influence ofa high order mean motion resonance is small, but itsmain effect shows up at some distance from the res-onant value of the fast variables (mean anomalies),depending on the particular combination of slow an-gles involved in the critical argument. Even if thechaotic region is presumably as wide as � 3 × 10−3

AU (judging from the values of σa for chaotic bodies;see Table I), the averaged semimajor axes of chaoticorbits cluster, as expected provided the time spancovered with integration was long enough, near thecenter of the chaotic region.

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A comment is in order here: in view of thelimited size of the chaotic region we are interestedin, the accuracy of the osculating orbital elementsof the bodies in question (especially their semimajoraxes) is of critical importance, in particular to decidewhether all the bodies we see inside the chaotic stripreally belong there, or they just by chance might onlyappear to be there. Out of the seven chaotic mem-bers of the extended Veritas family, only three arenumbered asteroids (that is, with their orbits knownto an accuracy better than 10−4 AU in semimajoraxis). The other four bodies are all single oppositionones, two of which (1981 EE4 and 2123 PL) have or-bits that, according to Bowell’s 1997 data base, canbe rated as reasonably reliable (error in osculatingsemimajor axis on the order of 10−3 AU, and anywaypresumably smaller than the width of the chaoticregion), whereas the remaining two ((1118 T 3 and4573 PL) have rather poorly determined orbits, witherrors exceeding the chaotic region size. The latterbodies should therefore be considered as analogousto ”fictitious” bodies, and we applied for them theusual trick of producing ”clone orbits”, with initialsemimajor axes changed by an amount correspond-

ing to the standard deviations of the nominal ones;the clone integrations have then been analyzed alongwith the nominal ones (unfortunately, with little suc-cess).

The next thing we have done was to computethe time series of proper elements for all the inte-grated bodies, filtering the results again to removeall the remaining perturbations up to 40,000 yr ofperiod. We have then plotted the 10 Myr time evo-lution of the proper eccentricity and proper sine ofinclination for all the 22 members of the Veritas fam-ily (see Fig. 1, to be compared with Fig. 1a of M&F),in order to reveal any change in the size and bound-aries of the extended family with respect to the pre-vious situation (see also a similar plot in Di Martinoet al. 1997, which contains as an additional infor-mation the diameters of the family members). Dueto the remarkable stability of the proper elementsfor the majority of family members the general pic-ture did not change very much in a qualitative sense.Quantitatively, however, there are some changes, butthey are definitely not so large to affect significantlythe overall compactness of the family, or to cause adramatic increase of its size.

Fig. 1. Time evolution of the proper eccentricity and proper sine of inclination for all the Veritas familymembers. Due to the remarkable stability of proper elements the entire variation is confined within a dot onthe plot for the majority of the bodies; a short line near the center of the family corresponds to 2147 Kharadze,which exhibits a residual long term oscillation of the proper eccentricity. Proper elements are plotted onceevery 40 000 yr; for the chaotic bodies crosses indicate the corresponding averages over the integration timespan.

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Perhaps the most important change with re-spect to the previous situation is due to the inclu-sion of asteroid 844 Leontina in the family. Thisis the second largest body in the family, and it hasbeen included in the family only recently (note thatin the M&F paper it was still considered as a back-ground object). It is rather far away from the fam-ily in proper semimajor axis, and at the very edgeof the family in proper eccentricity (note the rela-tively large difference between the proper eccentric-ities found by M&F and those computed in this pa-per; the latter are in good agreement with the cur-rent values provided by Milani and Knezevic 1994—version 6.8.5.— and used by Zappala et al. 1995to define families). If, in addition, one takes into ac-count that this body is found to be difficult to fitinto the possible collisional model and velocity fieldof the family, and that its spectrum has the largestslope of all the observed family members (Di Mar-tino et al. 1997), one wonders whether its familymembership may be only an artifact of the statisti-cal clustering technique (according to Zappala et al.1995, 844 separates from the other family members

at a ”distance” level not very much below the QRLfor the outer–belt region). Anyway, from the pointof view of the size of the family, the issue is not socritical, since in the proper (e, sin i) plane 844 Leon-tina is not very far from the rest of the family, andthere are some other newly added members whichappear to be even farther away.

In Fig. 2 a typical example is shown of the10 Myr evolution of proper elements for one of thechaotic objects (to be compared with Fig. 1b ofM&F), giving rise to ”clouds” in the phase space.Again one point is plotted every 40,000 yr. We haveselected on purpose an object located currently (butalso on the average) near the geometrical center ofthe family, to show that even in that ”unfavourable”case chaotic evolution can readily bring the body atthe very edge of the family, and probably out of thefamily in an experiment of longer duration. Similarplots can be shown for all the chaotic members of thefamily, those close to the family borders already cov-ering an area in the proper (e, sin i) plane extendingwell out of the family.

Fig. 2. Time evolution of proper eccentricity and proper sine of inclination for one of the chaotic bodies,namely 3542 1964 TN2. This asteroid is, on the average, located near the center of the family, but its properelements change in such a way to bring it close to the border of the family in the relatively short interval oftime covered by the integration. It is therefore plausible to expect that a longer integration would show thisbody to exit the family region. Similar plots can be produced for all the chaotic members of the family.

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Fig. 3a. Time evolution of the proper semimajor axis, proper eccentricity and proper sine of inclinationfor 490 Veritas (100 Myr backward integration). A ”capture” into a high order mean motion resonance(a harmonic of the 21/10 resonance) occurred and lasted almost for the entire integration time span. Thisprocess may affect the interpretation of the results in terms of the age of the family.

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Fig. 3b. Time evolution of the proper semimajor axis, proper eccentricity and proper sine of inclinationfor 490 Veritas (forward integration). Only very brief episodes of ”capture” were observed in this case.

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Having thus identified the chaotic objects, wenext performed for all seven of them numerical in-tegrations covering 200 Myr (100 Myr in the past,and the same in the future), repeating essentially thesame procedure we have already described for theprevious step: on–line filtering of the short periodicperturbations, analytical computation of proper ele-ments, and additional filtering to remove remainingperturbations of longer period. For this integration,however, the initial conditions were taken from the1997 release of Bowell’s orbital element data base.

In general, the results were as expected andconfirmed the findings of the preliminary analysis,but they also revealed some new interesting featuresand phenomena, which shed a new light on the com-plex dynamics in this region. In the following weshall present our results on a body–by–body basis,trying at the same time to grasp some common be-haviors and understand their causes.

Let us start from the major family member,that is 490 Veritas. In Fig. 3a we have plotted thetime series of proper semimajor axis, proper eccen-tricity and proper sine of inclination over the 100 Myrbackward integration of the asteroid’s orbit, and inFig. 3b the same has been done for the integra-tion in the future. While the output of the forwardintegration exhibits only some well–known featuresof the motion of Veritas (chaotic wandering of theproper semimajor axis, combined with short episodesof temporary captures in high order mean motionresonances, and large variations of proper eccentric-ity and inclination; see Milani et al. 1996), the back-ward integration reveals a completely different be-havior. After a brief period (a few million years)of instability, the body apparently gets trapped intothe resonance (one of the harmonics involving mostlyperihelia and few, if any, nodes), and stays there forthe rest of the time. The semimajor axis variationsbecome very small, and the proper e and sin i re-main quite stable for the entire integration time span(apart from a secular drift of small amplitude).

This latter result was certainly a kind of sur-prise, not so much because we found 490 Veritastrapped into the resonance, but because the trap-ping could last for so long. This is also an impor-tant result, as it might affect the estimate of theage of the family, based on the probabilistic argu-ments on a ”fast” escape of the major body fromthe family region. In order to check the robustnessof this result, we have performed a couple of inte-grations of clone orbits, obtained by changing theinitial semimajor axis of the nominal orbit by plusor minus 10−7 AU. In both cases we found no trap-ping of the kind observed with the nominal orbit;as a matter of fact, we did observe temporary trap-pings, but lasting up to several millions of years, andnot in the same resonance: one clone remained tem-porarily locked into another harmonic of the 21/10resonance at a slightly larger semimajor axis, whilethe other stayed into the 44/21 resonance, at about3.176 AU, for some 10 Myr (see Figs. 4a and 4b,where a similar behavior, but for another asteroid, isshown). It is therefore clear that the peculiar resultwe found with the nominal orbit is not necessarilycommon to a set of nearby orbits, but that similar,

though not quite the same, phenomena do take placein its neighborhood; a tentative conclusion may bethat the observed long–term resonant trapping mightbe just due to chance and to the essentially unpre-dictable nature of chaotic orbits in general. Lateron, however, we found other examples of the samebehavior, so we had to revise this conjecture.

The consequences for the problem estimatingthe family age are obvious: if family members whichin some experiments drift away from the family zoneafter a given characteristic time, in other experi-ments stay inside the family due to a long term cap-ture into the resonance, the age estimate obtained byM&F must be reconsidered. Any new estimate willbe certainly valid only in probabilistic terms, and inany case more experiments are needed to derive suchan estimate.

Figs. 4a and 4b are analogous to Figs. 3a and3b, but show the time series of the proper elementsfor asteroid 3542 1964 TN2. In this case it is the orbitintegrated into the past that exhibits large chaoticvariations of all the three elements, with only a sin-gle very short capture into the resonance at about−6.7 × 107 yr; the corresponding stabilizing effectagainst the chaotic changes can barely be seen inthe plots of the proper eccentricity and proper sineof inclination. However, captures occurring in theforward integration lasted much longer, even up to10 Myr, and the stabilization of the e, sin i elementsin this case was much more effective. Also worth not-ing in this forward integration is the steady and fastincrease of the proper eccentricity and proper sine ofinclination in the entire integration time span, whichbrings this asteroid out of the family very quickly.This result might really serve as a paradigm for thewell–known problems with orbital predictions basedon the integrations of chaotic bodies. Even if qual-itatively the integrations in the past and the futureare supposed to give statistically similar results, inview of the time reversibility of the N–body prob-lem, quantitatively these results can be quite differ-ent. Therefore, all conclusions based on these inte-grations are subject to a high degree of uncertainty,and only a large number of experiments with similaroutcomes can be given enough credit.

The next couple of figures, 5a and 5b, showsthe outcomes of our integrations for asteroid5594 1991 NK1, which is processed in a slightly dif-ferent manner with respect to the other ones. In thiscase, the plots for the eccentricity and inclinationshow their proper values as obtained by eliminatingthe perturbations up to 100,000 yr of period; this be-cause in this case perturbations with shorter periodscompletely masked the underlying chaotic behavior,which could be revealed only by applying another,more appropriate filter.

The results confirmed what was expected onthe basis of the previous results, as discussed above.Again captures of different durations and into differ-ent resonant harmonics were observed in the propersemimajor axis, associated with episodes of quasi–stable behavior in proper e, sin i. In agreement withthe prediction of Milani et al. (1996), this asteroid,which is located in the zone of the 21/10 mean mo-tion resonance populated with harmonics involvingboth perihelia and nodes, exhibits a typical stable

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Fig. 4a. Proper semimajor axis, proper eccentricity and proper sine of inclination for 3542 1964 TN2

(past). The large variations of the proper elements are a primary feature of this experiment.

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Fig. 4b. Proper semimajor axis, proper eccentricity and proper sine of inclination for 3542 1964 TN2

(future). Capture into different resonances occurs, lasting up to 10 Myr.

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Fig. 5a. Proper semimajor axis, proper eccentricity and proper sine of inclination for 5594 1991 NK1

(past). The jumps in inclination are associated with captures into a resonance located at 3.167 AU

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Fig. 5b. Proper semimajor axis, proper eccentricity and proper sine of inclination for 5594 1991 NK1

(future).

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Fig. 6a. Proper semimajor axis, proper eccentricity and proper sine of inclination for the single–oppositionbody 1981 EE4 (past). The proper e and proper sin i exhibit a long periodic trend.

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Fig. 6b. Proper semimajor axis, proper eccentricity and proper sine of inclination for 1981 EE4 (future).

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Fig. 7a. Proper semimajor axis, proper eccentricity and proper sine of inclination for the single–oppositionbody 2123 PL (past).

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Fig. 7b. Proper semimajor axis, proper eccentricity and proper sine of inclination for 2123 PL (future).Note the capture in the same resonance harmonic in which 490 Veritas remained also locked for almost100 Myr.

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chaos behavior, but with variations of smaller ampli-tude and with longer Lyapunov time. However, it ispuzzling that this body appears to be, in a way, ”toochaotic”, (more chaotic than 2147, for example, whilethe opposite should be expected according to Milaniet al. 1996), again pointing out that the dynamicsin the region of Veritas must be more complex thanit might have been expected. It can be easily veri-fied that the jumps in the proper sine of inclinationvisible in Figs. 5a and 5b are associated with cap-tures into a resonance located at 3.167 AU, which isexactly where the 40/19 mean motion resonance oc-curs, so that the amplified chaos might be due to anoverlapping of the harmonics of the two resonances.

There is not much to say about 1981 EE4. Theresults are shown in Figs. 6a and 6b, and they arequalitatively the same as those for 3542, for exam-ple. One notices a kind of decreasing trend of the

proper eccentricity and proper sine of inclination inthe backward integration, large enough to bring thisasteroid out of the family. Again there are temporarycaptures into resonances, and so on.

The results for 2123 PL are more interesting,in particular those for the forward integration. Whilethe backward integration (Fig. 7a) reveals again thetypical behavior observed in other cases, the forwardone (Fig. 7b) shows another example of a long–termcapture, lasting more than 60 Myr, into the same res-onance in which 490 Veritas has been found to staytrapped. Thus, these long trappings do not seem tobe so exceptional, and the above cautionary conclu-sions on the estimate of the family age are confirmed(see also later).

Another interesting point about these resultspertains to a shorter (� 8 Myr) locking, into a differ-ent resonant harmonic at a slightly higher semimajor

Fig. 8a. Variations of the distance from the ”center of the family” of 490 Veritas for the 100 Myr backwardintegration (one point every 2,000 yr); for a comparison, lines representing the current distances of the twobackground objects 92 and 1351, as well as the 10 Myr variations of the distance of 844 Leontina, are alsogiven. The small square in the lower right corner of the figure shows the 10 Myr variations of the distancefor the two stable family members (2147, upper curve, and 1086, lower one) located near the family center.Apart from the very brief excursion at the beginning of the integration, Veritas remains within the familyboundaries for all the time.

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axis (see Fig. 7b); this locking is also associated witha stabilization of the chaotic oscillations of the properinclination, but at a somewhat lower value than forthe above mentioned long–term locking (proper ec-centricities, on the contrary, in both cases remainedabout the same on the average). Note that the har-monic responsible for the long–term capture, beinglocated at a lower semimajor axis, involves morenodes than the one responsible for the shorter cap-ture (see Milani et al. 1996), and that this might bethe reason for the observed difference.

The results for 4573 PL and its clones did notreveal anything new, while 1118 T3 proved to beanother case of long–term capture in the resonance,lasting this time for some 90 Myr, while its clonesbehaved more regularly.

Note that for the majority of the consideredcases, the evolution of proper elements in the (e, sin i)plane (see Fig. 2) sooner or later brings a given as-teroid out of the family boundaries. Exceptions tothis are cases in which long–term captures occurred,and the single less chaotic body (5594); even theseorbits, on the other hand, wander around creating

characteristic ”clouds” in the phase space, but theydid not exit from the family zone in the covered timespan.

The final result to be considered here per-tains to the variation in the proper element space ofthe distances between chaotic family members in thephase space of proper elements, or between a givenmember and the ”center of the family”. In derivingthese distances in terms of the d1 metrics of Zappalaet al. (1995), we used as origin an arithmetical meanof the 10 Myr averages of proper elements of the fivemost stable members of the family (1086, 2147, 2428,2934, 3090). Note that this is a slightly different ori-gin with respect to that used by M&F (the positionof 1086 itself), but this discrepancy has no practi-cal importance. In Fig. 8a we show the variations ofthe distance of 490 Veritas for the 100 Myr backwardintegration; for a comparison, the lines representingthe current distances of two background objects (as-teroids 92 and 1351), as well as the 10 Myr variationsof the distances of the newly included member of thefamily 844 Leontina, are plotted too. These threelines are there to provide an idea of the limiting dis-

Fig. 8b. Variations of the central distance of 490 Veritas for the 100 Myr integration in the future. Thebody escapes from the family zone already after some 10 Myr.

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Fig. 9. A composite figure showing the variations of the distance from the ”center of the family” for threechaotic members of the Veritas family (3542 1964 TN2, 1984 EE4 and 2123 PL). On the left–hand side,the results coming from the 100 Myr backward integration are shown, and on the right–hand side the resultsfrom the forward integrations (note the different vertical scales in the plots).

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Fig. 10. Variations of the central distance of the two clones of 490 Veritas, for the 100 Myr integration inthe past.

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tance for the family members: obviously the familyborder should be somewhere between 0.014 and 0.015in the arbitrary units of the metrics used in definingdistances. In addition, a small square in the lowerright corner of the figure shows again the 10 Myr vari-ations of the same distance, but for the two stablefamily members (2147, upper curve, and 1086, lowerone) located near the family center. One can readilyconclude from the plot that, apart from the very briefexcursion at the beginning of the integration, Veritas(due, of course, to the capture into the resonance) re-mains within the family boundaries for all the timespan. On the contrary, Fig. 8b, in which the futurevariations of Veritas’ distance are presented, shows acompletely different behavior, that is, a fast increaseof the distance and the escape from the family zonein as little as � 107 yr. In both cases, these resultsare very different from those obtained by M&F, andagain point out that the previous results should beconsidered as preliminary and the conclusions as un-certain, at least until more data are produced andanalyzed.

The results for three other chaotic bodies withreliable orbits are assembled in Fig. 9, where for eachof them the variation of the distance is shown for theforward and the backward integrations. A remark-able feature is the significant past vs. future asym-metry of the results. Moreover, the three future in-tegrations produced quite different outcomes: a fastincrease of the distance and the escape from the fam-ily for asteroid 3542, a wandering across the familyboundary for 1981 EE4, and a similar behavior, butaffected by the resonant capture, for 2123 PL; thislatter body, due to the capture effect, spends mostof the time inside the family borders.

On the contrary, the integrations in the pastshow for 3542 a typical stable–chaos random wan-dering of the distance across the family border andepisodes of temporary escapes and returns, while forthe other two asteroids the behavior is very much thesame as that found by M&F in their study: a chaoticwandering similar to that just described for 3542, fol-lowed by a sudden and fast increase of the distance,occurring between 50 and 60 Myr from the present,and eventually the escape from the family (but seean example of a return after a long ”absence” in theplot for 2123 PL at about −9.7×107 yr). This latterresult is confirmed by the results for the two clonesof Veritas. As seen from Fig. 10, the variations ofthe distances for these two bodies behave in a verysimilar way.

3. DISCUSSION AND CONCLUSIONS

The results presented in this paper are inter-esting from several points of view and represent animportant extension of the results of M&F. First ofall, there are many more data for a detailed analy-sis of the dynamics in the chaotic region, since thesedata are produced by integrations covering a muchlonger total time span and a larger number of chaoticorbits have been integrated. All this provides a newinsight into the complex phenomena taking place in

the Veritas family and reveals some new features ofthe family itself, as well as of a number of its mem-bers.

One of the novel findings of the present studyis the discovery of a number of previously unknownchaotic members of the family. As we pointed outearlier, these bodies were recognized as members ofthe family only recently, together with some othernewly added, but more stable, objects. Their or-bits are not always accurate, but in most cases theyare accurate enough for the purpose of a qualitativestudy of their dynamical evolution. All the stronglychaotic asteroids are located close to each other insemimajor axis, on the right–hand side of the 21/10mean motion Jovian resonance, and all of them haveLyapunov times on the order of 104 yr, in agreementwith predictions of Milani et al. (1996).

The most important result of this analysis per-tains to the estimate of an upper limit for the age ofthe Veritas family. On one hand, we have found thatfor the majority of the chaotic bodies the integrationsin the past reveal a behavior similar to that foundand used by M&F to establish the ≈ 50 Myr upperbound for the family age; all the bodies exhibit acharacteristic stable chaos behavior for several tensof Myr, then suddenly begin to put distance betweenthemselves and the rest of the family, and eventuallyleave the region in the phase space occupied by thefamily. This behavior strengthens the conclusion ofM&F on the relatively young age of the family, andin particular their probabilistic arguments based onthe ergodic principle seem to be fully applicable inthis situation. Moreover, the probability that all theescaping members could have found themselves to-gether and near the other members at some remoteepoch(s) in the past (and that the family was formedduring one of these collective chance returns) appearsto be negligible.

On the other hand, the integrations in the fu-ture revealed a somewhat different picture. Morespecifically, one can say that the qualitative pictureis roughly the same as that for the backward inte-grations, but that in the forward integrations thedistances with respect to the family core increasedin some cases much faster than for the experimentsin the past, bringing these bodies out of the family inless than few tens of Myr. This is in agreement withwhat we already stated above on the time reversibil-ity of the N–body problem, and the essentially un-predictable nature of chaotic motions, and obviouslybrings in a bit of uncertainty when the age of the fam-ily is in question. This uncertainty is enhanced bythe long term resonant captures which we observedin a number of experiments, including those for thelargest family member 490 Veritas: were this phe-nomenon very common, it would be conceivable thefamily to be much older than inferrred from the prob-abilistic argument of M&F. However, this is probablynot the case: considering only the four bodies whichhave accurate osculating orbits and might exit the re-gion occupied by the family, our integrations covereda total of some 1.9 Gyr of chaotic orbital evolutionand we found long term captures for three bodies,lasting some 250 Myr overall; adding also the shortterm captures in different resonances,

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we estimate that there is a probability of some 15 −20 % for any body in this chaotic region to be at anytime locked in a resonance, and stay temporarily in aquasi–stable state within the family borders. If thisestimate is correct, it would be rather unlikely (say,a (0.2)4 ≈ 1.6×10−3 probability) that all the chaoticfamily members at any given time (e.g., now) werestaying in one of these quasi–stable states. Thereforewe conclude that it is also unlikely (though not im-possible) that the family age is longer than 50–100Myr.

Various types of dynamics are found to coex-ist in the Veritas family region. These diverse be-haviors range from a remarkable stability to a strongchaos. Typically, chaotic bodies jump between res-onant harmonics and the resulting variations of the(proper) orbital elements are large and entirely un-predictable; there are also regions in the phase spacewhere the dynamical evolution has all the character-istics of chaotic motion, but the elements are never-theless stable for very long times. This just high-lights the extraordinary complexity of this regionfrom the dynamical point of view, once again stressesthe importance of the high order mean motion res-onances and their effects, and emphasizes the needto investigate these interesting phenomena in moredetail. We plan to work in this direction in the nearfuture.

Acknowledgments –We are indebted to V. Zappala,A. Milani and P. Farinella for discussions and sugges-tions which helped us in clarifying our ideas aboutthis work and in bringing it to its final form. Thiswork is a part of the project ”Astrometrical, Astro-

dynamical and Astrophysical Studies”, supported bythe Ministry of Science and Technology of Serbia.

REFERENCES

Bowell, E., Muinonen, K., and Wasserman, L.H:1994. In: Asteroids, Comets, Meteors 1993(A. Milani, M. Di Martino and A. Cellino,Eds.), Kluwer Acad. Publ., Dordrecht, pp.477.

Carpino, M., Milani, A. and Nobili, A.M.: 1987, As-tron. Astrophys. 181, 182.

Di Martino, M., Migliorini, F., Zappala, V. and Bar-bieri, C.: 1997, Icarus, 127, 112.

Farinella, P., Carpino, M., Froeschle, Ch., Froeschle,Cl., Gonczi, R., Knezevic, Z. and Zappala, V.:1989, Astron. Astrophys. 217, 298.

Farinella, P.: 1994, In: Seventy-Five Years of Hi-rayama Asteroid Families: The Role of Colli-sions in the Solar System History, (Y. Kozai,R.P. Binzel and T. Hirayama, Eds.), ASPConf. Ser. 63, 76.

Migliorini, F., Zappala, V., Vio, R. and Cellino, A.:1995, Icarus 118, 271.

Milani, A. and Farinella, P.: 1994, Nature, 370, 40.Milani, A. and Knezevic, Z.: 1994, Icarus, 107, 219.Milani., A., Bowell, E., Knezevic, Z., Lemaitre, A.,

Morbidelli, A. and Muinonen, K.: 1994, In:Asteroids, Comets, Meteors 1993 (A. Milani,M. Di Martino and A. Cellino, Eds.), KluwerAcad. Publ., Dordrecht, 467.

Milani, A., Nobili, A. and Knezevic, Z.: 1997, Icarus125, 13.

Zappala, V., Bendjoya, Ph., Cellino, A., Farinella, P.and Froeschle, C.: 1995, Icarus, 116, 291.

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DA LI HAOTIQNI QASOVNIK TAQNO OTKUCAVA?

Z. Kne�evi� i B. Jovanovi�

Astronomska opservatorija, Volgina 7, 11000 Beograd, Jugoslavija

UDK 523.44-32Originalni nauqni rad

U radu je prikazana obimna analiza di-namike svih poznatih qlanova asteroidne fa-milije Veritas. Analiza je ura�ena u ciǉuprovere procene starosti ove familije, kojusu dali Milani i Farinella (1994) koriste�i me-todu tzv. ”haotiqne hronologije”. Mi smokrenuli od znatno ve�eg uzorka qlanova fa-milije i izvrxili daleko ve�i broj integra-cija putaǌa svih ǌenih qlanova, a naroqitotela sa haotiqnim kretaǌem; integracije zaove posledǌe su obuhvatile po 200 milionagodina i pokrile ukupno oko 2.1 milijardu go-dina. Pokazalo se da je dinamika u oblastifaznog prostora elemenata kretaǌa koju zauz-ima familija Veritas kompleksnija nego xtose to ranije verovalo i da tu postoje telasa kretaǌem u rasponu od izrazite stabil-nosti do potpunog haosa; haos je uzrokovanuticajem rezonance u sredǌem kretaǌu 21/10

sa Jupiterom i ǌenim harmonicima koji uk-ǉuquju spore uglovne promenǉive. Na�eno jeukupno sedam izrazito haotiqnih tela, kojapokazuju sliqno ponaxaǌe u kvalitativnomsmislu, a u skladu sa predvi�aǌima Milani-jai dr. (1996), ali tako�e i izvesne do sadanepoznate osobine. Od novootkrivenih osobi-na najva�nija se odnosi na qiǌenicu da nekiod haotiqnih asteroida, ukǉuquju�i i najve-�e telo u familiji, sam asteroid 490 Veri-tas, mogu biti uhva�eni u rezonancu i ostatitu vrlo dugo u kvazi-stabilnom staǌu. Iakonaxi rezultati uglavnom nisu u suprotnostisa zakǉuqcima Milani-ja i Farinella-e i dobro sesla�u sa ǌihovom procenom starosti famili-je, postoje i neki interesantni rezultati oveanalize koji otvaraju nova pitaǌa i zahtevajusveobuhvatniju i detaǉniju analizu.

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IS THE CHAOTIC CLOCK TICKING CORRECTLY?saj.matf.bg.ac.rs/156/pdf/047-069.pdf · Bull. Astron. Belgrade 156 (l997), 47 – 69 UDC 523.44–32 Original scientiﬁc paper IS THE CHAOTIC