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Volume 507 / No 1 (November III 2009)

A&A, 507 1 (2009) 495-504

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Issue		A&A Volume507, Number1, November III 2009


Page(s)		495 - 504
Section		Planets and planetary systems
DOI		https://doi.org/10.1051/0004-6361/200912696
Published online		03 September 2009

A&A 507, 495-504 (2009)

Datura family: the 2009 update

D. Vokrouhlický¹- J. Durech¹ - T. Micha $\l$ owski² -Yu. N. Krugly³ -N. M. Gaftonyuk⁴ -A. Kryszczynska² - F. Colas⁵- J. Lecacheux⁶ - I. Molotov⁷- I. Slyusarev³ - M. Polinska²- D. Nesvorný⁸ - E. Beshore⁹

1 - Institute of Astronomy, Faculty of Mathematics and Physics, CharlesUniversity, V Holesovickách 2, 18000 Prague 8, Czech Republic
2 - Astronomical Observatory, Adam Mickiewicz University, S $\l$ oneczna 36, 60-286 Poznan,Poland
3 - Institute of Astronomy, Karazin Kharkiv National University, Sumska35, Kharkiv 61022, Ukraine
4 - Crimean Astrophysical Observatory, Simeiz Department, Simeiz 98680,Ukraine
5 - IMCCE-CNRS-Observatoire de Paris, 77 avenue Denfert Rochereau,75014 Paris, France
6 - LESIA-Observatoire de Paris, 77 avenue Denfert Rochereau, 75014Paris, France
7 - Keldysh Institute of Applied Mathematics, RAS, Miusskaya 4, Moscow125047, Russia
8 - Southwest Research Institute, 1050 Walnut St, Suite 300, Boulder,CO 80302, USA
9 - Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ85719, USA

Received 15 June 2009 / Accepted 15 July2009

Abstract
Context. Research of asteroid families has beenrecentlyrefreshed by the discovery of very young ones. These families are ofgreat interest because they represent the product of their parent bodyfragmentation before orbital and physical evolutionary processes canchange them. A cluster of seven objects around the largest body (1270)Datura is of particular interest because it has enough known membersand resides in the inner part of the main asteroid belt, facilitatingobservations.
Aims. We carried out photometric observations of thetwo largestmembers of the Datura family - asteroids (1270) Datura and (90265)2003 CL5 - with the goal of inferring their physicalparameters.We also used numerous astrometric observations of Datura-family membersin the past few years to revisit the age of this cluster.
Methods. We performed numerous photometricobservations of(1270) Datura over several oppositions. We then used thelightcurve inversion method to determine the spin state and shape ofthis asteroid. In the case of (90265) 2003 CL5, for which onlylimited lightcurve data have been acquired so far, we used Fourieranalysis to determine the synodic rotation period during the 2008apparition. We also used backward numerical integration of the improvedorbits of Datura family members to reduce uncertainty in its age.
Results. We determined the rotation state of(1270) Datura,the largest member of its own family. Its major properties are a shortrotation period of $\sim$ 3.36 hand small obliquity, which, however, exhibits $\sim$ $\pm$ $15^\circ$ excursions because of a forced Cassini state of the proper nodalfrequency. Any possible initial non-principal rotation state hasprobably been damped and the asteroid rotates about the shortest axisof the inertia tensor. Its global shape, although convex in ourrepresentation, may reflect regions related to the excavation of thefamily members from the parent body surface. Interestingly, the secondlargest member of the Datura family - (90265) 2003 CL5 -appearsto be very slow rotator with the rotation period $\sim$ 24 h.The large amplitude of its rotation curve suggests that its shape isextremely elongated, possibly bi-lobed. Improved orbits of the familymembers allow us to re-determine the possible age of this family. Wefind an age that is slightly older than previously reported. Using aconservative approach, we obtain an age in the 450 to 600 kyrrange. With strengthened, but plausible, conditions, we find that thecurrent data may support an age of $530\pm 20$ kyr.Further astrometric and photometric observations of the Datura clustermembers are needed to determine its age more accurately.

Key words:minor planets, asteroids - techniques:photometric

1 Introduction

The discovery of very young asteroid clusters (e.g., Nesvornýet al.2006b;Nesvorný & Vokrouhlický 2006;Pravec & Vokrouhlický 2009)started a new phase in the analysis of the asteroid families. This isbecause we have become aware of many dynamical and physical processesthat modify theconfiguration and the observable parameters of their individual memberswith time.These processes prevent information about the initial state of theasteroidfamilies being derived from current observations. In very youngfamilies, of ages less than one million years, most of these processeshad insufficient timeto affect the physical properties of both their members and theiroverall orbital configuration. From their study, we might be able todirectly infer properties of their parent object fragmentation. Thevery youngasteroid families may also have some novel features in terms of theinterplanetary dust distribution (e.g., Vokrouhlický et al.2008;Espy et al.2009).

Table1: Equinoctial orbital elements of the Datura family members as ofMJD 55000.0.

Spectroscopic studies with the goal of calibrating space weatheringprocesses and spectral uniformity as a function of the rotational cycleare ongoing(Mothé-Diniz & Nesvorný 2008; Takato 2008;Chapman et al.2009;Vernazzaet al.2009).Here we begin attempts to characterize the rotational state of theyoung families' members. We focus on the Datura family for two reasons:(i) this cluster has a wealth of currently known members(Table 1);and (ii) residing in the inner part of theasteroid belt, its largest members are assessable forphysical studies to even limited-size instrumentation. Our ideal goalwouldbe to determine the spin state and shape of as many members aspossible, andrelate them to their initial values immediately after the familyformation. Weshall see, however, that already in the (1270) Datura case(Sect. 3)the analysis may not be straightforward, and sometimes we may only beable to infer lower and upperbounds to the initial spin state. So far we have succeeded in acquiringphotometric observations of (1270) Datura and the second largest memberin its family (90265) 2003 CL5. While in the first case, wehave already sufficient data to reconstruct the current spin state andshape (Sect. 3),in the case of asteroid 2003 CL5, we present onlysingle-opposition data that poorly provides a constraint on itsrotation period. Nevertheless, we find that this incomplete informationis veryinteresting because of the large disparity in the rotation rate of thetwo largestmembers in this family and the likely bi-lobed shape of2003 CL5(Sect. 4).In the second part of the paper, we revise the age ofthis family using new astrometric observations of the Datura familymembers acquired since January 2006, and especially recoveryof the asteroid (215619) 2003 SQ168 in December 2007(Sect. 5).With six members on reasonablywell-constrained orbits (Table 1),we are able to reconstruct the past configuration of this asteroidcluster to higher accuracythan before. Surprisingly, we find possible age solutions that areolder than the value of $\sim$ 450 kyrpreviously reported by Nesvorný et al. (2006b).

2 Observations

With its near 10 km size, Datura is a relatively easy targetfor photometric studies. It was observed by Wisniewski et al. (1997)during the 1990/1991 opposition, and Székely et al. (2005)during the 2000 opposition. Analyses ofboth datasets indicated relatively short rotation period of $\sim$ 3.2-3.4 h,but they were limited enough to allow linkage across the 10 yrinterval of time between them.

The discovery of the Datura family in late 2005 provided strongmotivation forphotometric and spectroscopic observations to infer the physicalpropertiesof (1270) Datura and other members of its family. We started ourphotometriccampaign during the 2006 opposition and continued during subsequentobserving timeopportunities. Our main efforts were concentrated on the 2007/2008opposition whenobservations covering a period of nearly six months, and phase anglevalues of up to $26^\circ$ ,were obtained (we did not use the independent observations ofTakato (2008)from mid February 2008 because they roughly coincided with ourdata from the beginning of February 2008). These observations were veryimportantbecause they allowed easy linkage of data from all oppositions andhelped us to resolve the asteroid shape.The synodic period determined by combining lightcurves from thisopposition,3.3583 h, also served as a good starting value of the siderealrotation periodin the global lightcurve inversion procedure. We note however, that thedata in 2006 and 2009 are also important, because they offer novelviewing geometry to the asteroid, thus helping us to resolve its poleand shape. Thecalibrated observations of November 20/21, 2007 allowed us to determinethecolor indexes $B-V=0.81\pm 0.07$ , $V-R = 0.44\pm 0.03$ ,and $R-I = 0.36\pm0.05$ ,which are typical of S type asteroids (e.g., Shevchenko &Lupishko 1998).Additional calibrated observations at different phase angles inRband allowed us to fit $H_0(R)=12.03\pm 0.02$ ,at maximumlightcurve, and $G(R)=0.29\pm 0.01$ using theH-G system (seeFig. A.4 in theAppendix). Using our derivedV-Rvalue, we then obtained $H_0=12.47\pm 0.05$ in visible band, which infers an equivalent diameter of $\sim$ 9.8 km(for an albedo of 0.19). Details of observing geometries forall 20 new lightcurves, including the 3 old ones, are given inTable 2.

Table2: Aspect data for observations of (1270) Datura.

The $\sim$ 3 kmsize asteroid (90265) 2003 CL5, the second largest in theDatura family, is a significantly more difficult target for small-scaletelescopes.Fortunately, the asteroid was close to the pericenter of itsheliocentric orbit duringopposition in 2008 enabling our observations to take place. Weperformed lightcurveobservations of (90265) 2003 CL5 in five nights during thisfavorable apparition with the date and observing geometries indicatedin Table 3.All observations were performed with the 1 mtelescope at the Crimean Astrophysical Observatory at Simeiz. Threeobservations of (90265) 2003 CL5 were calibrated and performedin the standardR band. We also obtained acolor index $V-R = 0.43\pm 0.02$ that fits well the Datura value given above and the spectralclassification S for thisobject (e.g., Mothé-Diniz & Nesvorný2008).

Table3: Aspect data for new observations of (90265) 2003 CL5.

$\begin{figure}\par\includegraphics[width=16.2cm,clip]{12696fig1.eps}\end{figure}$	Figure 1: The shape model of (1270) Datura shown from equatorial level (left and center, $90^\circ$ apart) and pole-on ( right).
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$\begin{figure}\par\includegraphics[width=16.2cm,clip]{12696fig2.eps}\end{figure}$	Figure 2: Examples of Datura's lightcurves (symbols) fitted with our model (solidcurve). The viewing and illumination geometry is given by the aspectangle $\theta$ ,the solar aspect angle $\theta _0$ ,and the solar phase angle $\alpha$ .
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Assuming a phase/slope parameterG=0.2, ourcalibrated observationsallow us to determine absolute magnitude $H_0=15.79\pm 0.05$ in visible band at the maximum of the lightcurve (in this case, thephase-angle analysis waspartly affected by the slow rotation of the asteroid and thus differentobservations were obtained at different phases of the rotation curve).Assuming an albedo of 0.19, this translates into a 2.1 km sizeequivalent to the maximal projection of the system. This is a slightlysmaller value than reported so far from photometry accompanying theastrometric observations.

3 Spin state and shape of (1270) Datura

Our new observations were combined with the older data of Wisniewskiet al.(1997)and Székely et al. (2005), andanalysed using the lightcurve inversion method of Kaasalainen andcollaborators (e.g., Kaasalainen et al.2001,2003).This procedure provides a simultaneous solution of thefundamental rotation parameters, rotation periodP,and pole position $(\lambda,\beta)$ ,in addition to parameters describing the irregular shape of theasteroid. Low-to-moderate phase angle lightcurves are insensitive toconcavities in the asteroid shape, so we are generally only able tosolve for aconvex realization of the body (e.g., Durech & Kaasalainen 2003).Thus, we are unable to directly study the potential concavities of theshapethat might indicate details of the fragmentation of the parent body ofthe family. Nevertheless, the flat planar areas in the derived figureareindicative of the presence of these concavities.The lightcurve inversion procedure also did not require a non-principalaxis (NPA) component indicating that Datura rotates close enough to theshortest axis of its inertia tensor. Moderate initial NPA rotationwouldin all cases be damped with a characteristic timescale of $\sim$ 75 kyrifwe adopted the parameters in Harris (1994),short enough compared to theestimated family age. So the present principal axis rotation does notpreclude initial NPA.

As usual for main-belt targets, the lightcurve inversion method isaffected bya $\sim$ $180^\circ$ ambiguity in the pole longitude. We thus obtained twopossible pole positions: (i) $\lambda = 60^\circ$ , $\beta = 76^\circ$ (hereafter P1);and (ii) $\lambda = 264^\circ$ , $\beta = 77^\circ$ (hereafter P2) with siderealrotation period $P = 3.358100 \pm 0.000003$ h.In each of the cases, the uncertainty in the pole direction is roughly $5^\circ$ .We note that P1 provides a slightlybetter fit to the lightcurves, but P2 cannot be ruled out on the baseof the photometric data. Interestingly, well-plannedinfrared observations could not only improve the size/albedo values forDatura but also help to discriminate between P1 and P2 (see, e.g.,Delbó& Tanga 2009).

Our solution supersedes an attempt of Durech et al. (2009), whousedthe 1990/1991 lightcurves of Wisniewski et al. (1997)and complemented themwith 79 sparse photometry data from the US Naval Observatory catalog.The latitude solution in this reference differs significantly from ourresult, probably because of the limited quality of the dense-photometrydataand the small amount of sparse-photometry data. This exampleillustrates thecontinuing necessity to acquire dense photometry data, to supplementpossibly for more numerous amount of sparse data from the sky-surveyprograms inthe future.

The convex shape model of (1270) Datura associated with the P1solutionis shown in Fig. 1and six representative lightcurves and the corresponding fits are shownin Fig. 2.The planarregions in the northern and southern hemispheres are probably not real,butcaused by the apriori convex representation of the model. It isinterestingto hypothesize that one, or both, may correspond to the impact featurerelated to the origin of the Datura family.

3.1 What was the initial obliquity of Datura?

Given the youth of the Datura family, one is tempted to interpret thecurrent rotation state as the initial one, right after the familyformationevent. We assume that this is true for the short rotation period,because theonly sizeable phenomenon, the Yarkovsky-O'Keefe-Radzievskii-Paddack(YORP)effect, could only have changed rotation period by a few per-milfractionally(see, e.g., Capek & Vokrouhlický 2004). Thesituation is, however,different for the obliquity value.

The pole solutions P1 and P2 have current obliquities $\sim$ $11.2^\circ$ and $\sim$ $15.3^\circ$ ,respectively, measured with respect to the orbital plane. While theYORP effect may have changed theobliquity by only a fraction of a degree (e.g., Capek &Vokrouhlický2004;Bottke et al.2006), itsinitial value canhardly be inferred because of the parameter-dependent interplay of thegravitational torque due to the Sun and inertial torques due to movingorbital plane (known as the Cassini dynamics; e.g. Henrard &Murigande 1987;Vokrouhlický et al.2006).To verify this conclusion, we used the symplecticnumerical scheme of Breiter et al. (2005) topropagate the current rotation state(pole P1 and P2) for 1 Myr into the future. The orbit ofDatura waspropagated using theSWIFT_MVS integrator with atimestep of 5 d andoutput sampling of 50 yr (the asteroid initial data fromTable 1and planetary ephemeris from JPL DE405),and then used by the Breiter et al. scheme to propagate thespin state. The only unknown parameter is the asteroid's dynamicalflattening $\Delta=(C-0.5(A+B))/C$ ,where (A,B,C)are the principal moments of inertia of the body. We used our two shapemodels for Datura associated with the P1 andP2 solutions, assuming a homogeneous distribution of density, to inferthat $\Delta$ ranges from $\sim$ 0.15to $\sim$ 0.3(notethat the lightcurve inversion method provides a reasonable shapeof the body but only poorly constrains its $\Delta$ value).

Figure 3shows the future obliquity evolution of the currentpole positions P1 and P2 for the two assumptions of $\Delta$ :0.2 solidcurve and 0.3 dashed curve. In either of these two cases, the obliquityundergoes large amplitude oscillations up to $\sim$ $30^\circ$ with $\Delta$ -dependentperiodicity. For instance, in the $\Delta =0.2$ casethe principal period of the obliquity change is $\sim$ 54 kyr, significantly largerthan the principal nodal precession period of $\sim$ 35 kyr. This difference,and the overall large amplitude of the obliquity oscillation, arerelated tosignificantly high obliquity value of the forced Cassinistate 2 (e.g., Henrard& Murigande1987;Vokrouhlický et al.2006).For instance in the $\Delta =0.2$ case, the forced Cassini state 2 is at $\sim 8.3^\circ$ and becomes larger than $\sim$ $11^\circ$ for $\Delta =0.3$ .

Figure 3shows that the period, and thus also a phase,of obliquity oscillations critically depends on the unknown $\Delta$ value.Because the lightcurve inversion only poorly constrains $\Delta$ ,theinitial, post-breakup obliquity of Datura cannot be determined. It canbe any value between zero and $\sim$ $30^\circ$ .

$\begin{figure}\par\includegraphics[width=8cm,clip]{12696fig3.eps}\end{figure}$	Figure 3: Obliquity of P1 ( top) and P2 ( bottom)solutions during the next 1 Myr time interval. Numerical modelincludes gravitational torque from the Sun and evolution of Datura'sorbit due to planetary perturbations. The solid lines are for $\Delta =0.2$ value, the dashed lines are for $\Delta =0.3$ value, of the asteroid's dynamical flattening.
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$\begin{figure}\par\includegraphics[width=8cm,clip]{12696fig4.eps}\end{figure}$

Figure 4:

Composite lightcurve of (90265) 2003 CL5 from the 2008opposition. Different symbols for observations during 5 nights inSeptember and October as indicated in the labels. The ordinate is thecalibrated magnitude in standardR band;the calibrated observations were related using the H-G fitted phasecurve assumingG=0.2 value. We used $23.42\pm 0.02$ hrotation period to transform all data into the phase of the rotationcycle (abscissa; zero phase atJD 2 454 710.87034).

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4 Rotation period for (90265) 2003 CL5

From the first night of observations, we were already able to inferthat(90265) 2003 CL5is a slow rotator. We then followed the object over fivenights in September and October 2008. The composite lightcurve(Fig. 4)has a very large amplitude of 1.5 magor more (because of the rotation period close to one day, ourobservations unfortunately did not cover any of the lightcurve minima).The insufficient coverage of the lightcurve prevents us from accuratelydetermining the apparent rotationalperiod during this opposition (we additionally note that the observinggeometry somewhat changed in-between our first and last observations).One of the possible solutions, $23.42\pm 0.02$ h,has been used inFig. 4to fold all observations into a single phase plot. There are, however,other possiblesolutions near this period.

Given the slow rotation rate and small size of (90265)2003 CL5, it isinteresting to explore the extent to which the YORP effect could havechanged therotation period within the estimated age of the Datura family. Usingthe statistical data in Capek & Vokrouhlický (2004;Figs. 6-8),we estimate that the initial rotation period of this asteroid could nothave been shorter than $\sim$ 17.5 h,while it could have been aslong as $\sim$ 32 h.The shorter limit is perhaps more interesting,indicating that the initial rotation period of (90265)2003 CL5 shouldhave been fairly long in any case. We should also mention that theprincipal axis rotation of this asteroid, if confirmed by futureobservations, may provide an important constraint on the formationprocess after parent body fragmentation. We note that the canonicalestimate for the damping timescale of an excited rotation state (e.g.,Harris1994)would give several hundreds of Myr, about a factor100-1000 longer than the family age.

The very large amplitude of the lightcurve suggests an interestingpossibility that this object is an extremely elongated, possiblybi-lobedobject or even a close synchronous binary system.We note that the hydrocode simulations of asteroid fragmentation dopredict large fraction of binary fragments, the so-called escapingejecta binaries (e.g., Durda et al.2004;Nesvorný et al.2006b;Richardson et al.2009).However, it seems that (90265) 2003 CL5 being a bi-lobedobject (perhaps a failed binary) would challenge severalcurrent models. As a result, (90265) 2003 CL5 may provideinteresting information about the fragmentation process of largeasteroids in general, and our ability to model this particular asteroidnumerically. This stronglymotivates further photometric observations of this system in thefuture. In particular, it is important to obtain calibratedobservations that would allow us to unambiguously link all availabledataand cover lightcurve minima that were not seen in our 2008-oppositiondata.

Unfortunately, (90265) 2003 CL5 will be quite faint during the2010 opposition such that small- and medium-scale telescopes will notallow us to continue its photometry. The next truly favorable observingconditions for this asteroid will occur in June and July 2011.Interestingly, at this opposition the ecliptic longitude of theasteroid will span values between $260^\circ$ and $270^\circ$ ,suitably complementary to those in 2008 (Table 3).As a result, observations from this opposition may themselveshelp to exclude some pole positions, even though not yet ableto determine the pole accurately.

5 Re-estimation of Datura family age

In a first attempt to reconstruct the Datura cluster pastconfiguration, and thus infer its age, Nesvorný et al. (2006b)usedorbits of four members of the family. Out of these, only (1270) Daturahad very precisely determined orbital elements. Orbits of the otherthree asteroids - (60151) 1999 UZ6, (90265) 2003 CL5and (203370)2001 WY35 - were significantly less accurate, reflecting thesmall numberof available astrometric observations over short time intervalsof only 6 to 8 years. Since then, the situation has significantlyimproved.

First, Sergio Foglia and colleagues made a significant effortattempting to find young-families' members on archival plates andsucceeded in several cases. For instance, discovery of (90265)2003 CL5on the NEAT program archival frames taken in January 1996 allowed us toextend the known orbital arc of this asteroid by several years into thepast. Second, the discovery of the young families allowed us to prompttargeted astrometry observations to ensure that most of the objectsare observed on every subsequent opposition. In this respect, weshould point out in particular observations made with the1.5 mtelescope at Mt. Lemmon Observatory at the Catalina Sky Survey(CSS).Importantly, both lines of effort led to precovery and recovery ofthe Datura member (215619) 2003 SQ168 nearly simultaneously inDecember 2007.Since then, CSS also acquired astrometry of this object in April andMay 2009 during itsnext opposition. As a result, we can now use this asteroid for thepast reconstruction of the Datura cluster. The information about thecurrently available accuracy of the orbits of Datura members isgiven in Table 1(compare with Table 1 inNesvorný & Vokrouhlický2006).

We used the same approach as Nesvorný et al. (2006b)andNesvorný & Vokrouhlický (2006) todetermine the past orbitalconfiguration of the Datura members. We refer interested readersto these references for details, limiting ourselves here to a minimaldescription of the technique. We considered a largenumber of statistically identical orbital clones for the asteroids.This was necessary because the past orbital history of each ofthe members depends on: (i) the current orbital uncertainty(Table 2);and (ii) unknown physical parameters.The former simply means that we have to take a number of initialorbitaldata for the backward integration uniformly filling the uncertaintyellipsoid in the orbital element space (we call them geometricclones). The second refers to the thermal (Yarkovsky) forces that maysignificantly affect the orbital evolution over timescales of tens tohundreds of thousands of years and that depends on the spin state andsurface properties of the body (e.g., Bottke et al.2006).Therefore, we need to uniformly cover a range of possible pastevolutionsof the orbits with Yarkovsky forces applied. We call these variantsthe Yarkovsky clones.

Performing backward integrationsof the clone versions for each of theasteroids, we can now at each time choose at random theiridentification to construct one possible cluster configuration. Giventhe large numberof integrated clones, many of these identifications are possible andwe can only characterize information about their proximity on astatistical basis. To work quantitatively, we define a targetfunction(e.g., Nesvorný & Vokrouhlický2006)

$\begin{displaymath}\Delta V = na~\sqrt{k_1~\left(\sin i ~\Delta\Omega\right)^2 +k_2~\left(e~ \Delta\varpi\right)^2} ,\end{displaymath}$

(1)

where (a,e,i)are the orbital elements of the largest asteroidin the family,n its heliocentric mean motion,k₁=1andk₂=1/2 and the node andpericenter dispersions are definedby $\left(\Delta\Omega\right)^2=\sum_{ij}(\delta\Omega)_{ij}^2/N_{ij}$ andsimilarly for the pericenter. We sum over all possibleindependent pairs of the asteroids in the particular identificationof the family. For instance, using 6 members of the family, thei andj indexes range from 1-6andN_ij=15.The targetfunction in Eq. (1)reflects that the Datura members aresufficiently close in terms of (a,e,i)elements and basically probessufficient convergence of the secular angles $\Omega$ and $\varpi$ .Its structure also follows from the analysis of theGauss equations of the perturbation calculus (e.g., Bertottiet al.2003;Nesvorný & Vokrouhlický2006)and can be interpreted in terms of the characteristicfragment dispersal velocity. As a rule of thumb, we expect theacceptable solutions to imply/infer a value of $\Delta V$ of the orderof the escape velocity from the estimated parent body of the family,some $\sim$ 5-10 m/s.

The integration of thousands of clones with dense enough time outputbecomes computationally difficult, both as far as CPU time anddisk space are concerned. For that reason, we adopted the followingapproximation:

the past orbital evolution of (1270) Datura is representedby 30 geometric clones, each of which has additionally15 Yarkovsky clones with da/dtvalue negative (this is because we integrate into the past and < $90^\circ$ obliquity for Datura guarantees the sense of its migration due to thethermal forces), implying that altogether we have 450 geometric andYarkovsky clones;
the past orbital evolution of (60151) 1999 UZ6,(90265) 2003 CL5, and (203370) 2001 WY35 isrepresented by 30 geometric clones, each of which has additionally30 Yarkovsky variants, so altogether there are 900 geometricand Yarkovsky clones;
the past orbital evolution of (89309) 2001 VN36and (215619) 2003 SQ168 is represented by 40 geometric clones,each of which has in addition 40 Yarkovsky variants, andaltogether there are 1600 geometric and Yarkovsky clones.

A few comments are in order. First, it appears unnecessary to considerso many clone variants for the largest asteroid (1270)Datura. For instance, the estimated nodal variation caused bymismodeled Yarkovsky forces $\Delta\Omega \sim 0.5~(\partial s/\partial a)~(\Delta a) ~T$ (and similarly for thelongitude of pericenter) is $\sim 0.02^\circ$ or less. Heres is the proper frequency of nodeprecession for which we have $(\partials/\partial a)\simeq -38$ arcsec/yr/AUin the Datura zone (e.g., Vokrouhlický et al.2008), $\Delta a\sim 10^{-5}$ AUis an estimate of the maximum accumulated Yarkovsky change in thesemimajor axisa in the timespan $T\sim 500$ kyr(see, e.g., Bottke et al.2006, forcharacteristic rates of change ina for asteroidsin the main belt).This small influence of the Yarkovsky clones is due toDatura's large size; the similar effect may be an order of magnitudelarger for smaller members in the family. However, by trackinga limited number of geometric clones we found that the underlyingchaoticity of the Datura region may still cause the pure geometricclones disperse in node and pericenter by up to $\sim$ $0.1^\circ$ ,morethan the above estimated Yarkovsky contribution. This test showed thenecessity of including some limited number of geometric and Yarkovskyclones even for (1270) Datura. Second, the larger number of clonesfor (89309) 2001 VN36 and (215619) 2003 SQ168obviously compensates for the largeruncertainty of their past orbital configuration. In the case of(89309) 2001 VN36, this is because of its interaction with the9/16exterior resonance with Mars (e.g., Nesvorný & Vokrouhlický2006,Fig. 1). Indeed, tracking the possible past evolution ofthe geometric clones for this body, we noted that they can disperse onaverage by $\sim$ $2^\circ$ in $\sim$ 500 kyr,with the most distantclones being up to $\sim$ $5^\circ$ away in node and pericenter. The large number of clones for (215619)2003 SQ168 is obviously due to having the most poorlyconstrainedorbit of all Datura members (except for the unused orbit of2003 UD112) and its small size.

When all six asteroids are included, the past Datura clusterreconstructionis in our integration represented by as many as $450\times 900^3\times1600^2=8.4\times 10^{17}$ clone identifications. This numbersurpasses by far CPU possibilities to evaluate target functions $\Delta V$ and $\Delta V'$ for each one of these identificationsat each output from our integrations (20 yr in our case). Wethusonly compute the target functions for randomly chosen 50 milliondifferent identifications at each output step. This is much lessthan the maximum possible number of clone combinations and may implythatwe significantly undersample the results. Fortunately, the situation isimproved by several factors: (i) clones of (1270) Datura and(90265) 2003 CL5 have far more similar past evolutions thanotherbodies; (ii) the close nodal/pericenter configuration, for whichthe target functions $\Delta V$ and $\Delta V'$ remain smaller than somethreshold, may last up to 20 kyr, a factor 10³longer than our outputsampling of 20 yr; and (iii) not all clones have widelydifferentpast orbital evolutions. Taking these factors into consideration,we estimate that the 50 million trial identifications at each timeoutput should be sufficiently representative.

Table 1also indicates that orbits of (1270) Datura and (215619)2003 SQ168 are very close to each other (including the meanlongitudein orbit). For that reason, the two objects were assumed to be pairedobjects. Vokrouhlický & Nesvorný (2008)argued against this possibility, proposing instead that an anomalouslysmall initial separation of (215619) 2003 SQ168 from (1270)Datura caused their current proximity(see Fig. 6 in Vokrouhlický & Nesvorný2008).If well constrained, the convergence of the two orbits in themean anomaly value might then be used as an additional criterionfor the age determination of the Datura family. Unfortunately, thepresent orbital uncertainty of (215619) 2003 SQ168, and lackof knowledgeof its physical parameters, prevents this kind of approach. TheYarkovsky clones of this small asteroid diverge in the meanlongitude in orbit by $\delta \lambda/ 2\pi \simeq (3/4)~(T/P_{\rm orb})~ (\Delta a/ a)$ after timeT; here $P_{\rm orb}\simeq 3.34$ yris the heliocentric revolution periodand $\Delta a$ is the accumulated change in semimajor axisa dueto the Yarkovsky effect in timeT. For $T\simeq 500$ kyr,and theestimated size of (215619) 2003 SQ168, we have $\Delta a\simeq 10^{-4}$ AU.Wethus note that after only $\sim$ 100 kyrthe Yarkovsky clones extendalong the entire elliptic orbit and prevent any deterministic workusing the mean anomaly value.

$\begin{figure}\par\includegraphics[width=8cm,clip]{12696fig5.eps}\end{figure}$

Figure 5:

Number of clone combinations that provided target functions smallerthan 5 m/s: (i) $\Delta V$ from Eq. (1)(part a), top), and(ii) $\Delta V'$ (part b), bottom).Data were collected into 5 kyr wide bins. The Gaussian curvesin grey are for illustration purposes only:(i) 530 kyr mean value and 18 kyr standarddeviation at the top panel; and (ii)532 kyr mean value and 17 kyr standard deviation atthe bottom panel. At each 20 yr spacedoutput, we run 50 million trial configurations of all available Daturaasteroids, except for the single-opposition case of 2003 UD112(Table 1).

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$\begin{figure}\par\includegraphics[width=8cm,clip]{12696fig6.eps}\end{figure}$

Figure 6:

An example of a good convergence of secular angles for the Daturafamily members. Upper and lower panels show pastdifference of longitude of node and longitude of pericenter withrespect to (1270) Datura, the largest asteroid in the family,for: (90265) 2003 CL5 [red], (60151) 1999 UZ6 [green], (203370) 2001WY35 [blue], (89309) 2001 VN36 [cyan] and (215619) 2003 SQ168[magenta]. The vertical dashed line denotes the best convergence time536 kyr for which $\Delta V = 1.4$ m/sin this case.

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5.1 Results

Figure 5shows distribution of age values determined by a number of the clonecombinations corresponding to target functions $\Delta V$ fromEq. (1)(upper panel) and $\Delta V'$ (lower panel)smaller than 5 m/s, rather restrictive value. All Daturaasteroids,except obviously the single-opposition case of 2003 UD112, and50 milliontrial identifications of different clones at each time have been usedinthis analysis. The age distribution is matched satisfactorily by aGaussian(grey curves) with mean values of 530 kyr and532 kyr, respectively, and standarddeviations both of $\sim$ 18 kyr.Figure 6illustratespossible past convergence of the secular angles as referenced to the(1270) Datura value for a good solution that has $\Delta V = 1.4$ m/ssome 536 kyr ago. This result comes as a surprise, because itdeviates significantlyfrom the solution of Nesvorný et al. (2006b)and increments the previously determined age of the Datura family bynearly 100 kyr. This disagreement in the two solutionsrequires some explanation and/oranalysis and that prompted us to perform several further tests.

$\begin{figure}\par\includegraphics[width=8cm,clip]{12696fig7.eps}\end{figure}$	Figure 7: Number of clone combinations that provided target functions smallerthan 13 m/s (light grey histogram) and 10 m/s (darkgrey histogram): (i) $\Delta V$ from Eq. (1)(part a), top),and (ii) $\Delta V'$ (part b), bottom).Data were collected into 5 kyr wide bins. In this case onlyhalf a million trials of the clone identifications was used at eachoutput from the simulation.
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First, we tested how the solution depends on the chosen thresholdfor the target function values, increasing it to 10 m/s and13 m/s.Resulting age distributions are shown in Fig. 7.Because there are many more possible solutions now, we performedonly half a million trials of the clone identifications inthis case. Indeed, we observe that the possible age solutions nowextend to smaller values more compatible with the result ofNesvorný et al. (2006b).However, we would consider the 13 m/scutoff not to provide a strong constraint and this is also reflected inthe numerous solutions found: even if we restricted ourselvesto 50 thousand trials (out of the theoretical maximum value $8.4\times10^{17}$ ), we wouldobtain many solutions. The 13 m/s value translatesindeed into a mean quadratic nodal or pericenter distance of nearly $\Delta\Omega \simeq 0.2^\circ$ .Past orbital evolutions inFig. 6hint that this is rather easily matched over a long interval of time.

$\begin{figure}\par\includegraphics[width=8cm,clip]{12696fig8.eps}\end{figure}$

Figure 8:

Number of clone combinations that provided target functions smallerthan 13 m/s (light grey histogram) and 10 m/s (darkgrey histogram): (i) $\Delta V$ from Eq. (1)(part a), top), and(ii) $\Delta V'$ (part b), bottom).Data were collected into 5 kyr wide bins. Past orbital historyof (89309) 2001 VN36 was not included in this analysis andhalf a million trials of the clone identifications were used at eachoutput from the simulation.

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To determine more exactly the sources of difference between the twosolutions, we neglected the orbital history of (89309)2001 VN36 from oursimulation. This approach was adopted by Nesvorný et al. (2006b)becausethe orbit of this asteroid was assumed to be too uncertain because ofresonance-related chaoticity. Performing again half a million randomidentifications at each time instant in the past, as given by ournumerical propagation, we obtained results shown in Fig. 8.While the bulk of the distributionshifted toward the $450\pm 50$ kyrsolution interval determined by Nesvorný et al. (2006b),there is an extension beyond its 500 kyr age. Choosing alimited amount of trials and effective high-cutoff values for thetarget functions, one may preferentially find the optimal age solutionsin the $450\pm 50$ kyrinterval and we believe that this is what happened.Decreasing the cutoff limit again to 5 m/s for the targetfunctions,the Datura age indeed moves beyond half a million value even inthis simulation where the highly-chaotic orbit of (89309)2001 VN36was excluded.

To summarize, our main conclusions are that: (i) if we are not overlyoptimistic, our analysis suggests an age in between 450 and600 kyrfor the Datura cluster; and (ii) if we were to take our mostcomplete and restricted solution for granted, the cluster age woulddecreasewithin the $530\pm 20$ kyrinterval. Thisis where, effectively, the best-convergence solutions reside. More workappears to be required to resolve the issue of the Datura cluster age.This includes both new astrometric and physical observations (thatwouldconstrain the possible diverse Yarkovsky histories for small Daturamembers),but perhaps also more sophisticated theoretical methods than used sofar.

6 Conclusions

This paper has presented our state-of-art knowledge of the Daturaasteroidfamily at the dawn of the upcoming new-generation surveys such asPanSTARRS or LSST (e.g., Jedicke et al.2007;Ivezic et al.2007). Itwill be interesting to see how many more Datura-family members will bediscovered in the next few years. Input from these new datasets seemscrucial, because with the current instrumentationno additional Datura members have been found since thediscovery of the family in late 2005 (the last three memberswere discovered in 2003). On the other hand, we argue thatit is equally important to continue with further astrometric andphotometric observations of the fainter, currently known members.This is because the newly discovered asteroids will likely bevery small, $\leq$ 2 km,and the lack of spin state information will make their past orbitsuncertain primarily due to unconstrainedYarkovsky forces. As a result, continuing photometric observations ofthe knownmembers may help us to constrain at least the sense of their rotationand thus eliminate many of the clones used in the age determinationsimulations in Sect. 5.

Acknowledgements

The work of D.V. and J.D. was supported by grantsGACR 205/08/0064 and GACR 205/07/P070 of the Czech grant agency and bythe Research Program MSM0021620860 of the Ministry of Education. Thework of T.M. and A.K. was supported by grants N N203 302535 and N N203382136 of the Polish Ministry of Science and Higher Education. Thispaper uses observations made in part at the South African AstronomicalObservatory (SAAO). We thank Richard Kowalski from the Catalina SkySurvey for important help in obtaining astrometry of faint objects inyoung asteroid families including Datura, Alan Harris (SSI) for hiscomments and suggestions that improved the original version of thispaper and Vasilij Shevchenko for help with phase curve analysis of(1270) Datura.

Appendix A: Composite lightcurves and phase function for (1270) Datura

In this Appendix, we present composite lightcurves of (1270) Daturacontaining all our new observations starting from the

$\begin{figure}\par\includegraphics[width=8cm,clip]{12696figA1.eps}\end{figure}$	Figure A.1: Composite lightcurve of (1270) Datura during the 2006 apparition.
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$\begin{figure}\par\includegraphics[width=8cm,clip]{12696figA2.eps}\end{figure}$	Figure A.2: Composite lightcurve of (1270) Datura during the 2007/2008 apparition.Lightcurves obtained in January-February 2008 have smaller amplitudessince the phase angles were smaller (see Table 2). In the sameway, the largest spread corresponds to the observations from beginningand end of the opposition because they were taken with slightlydifferent viewing geometry.
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$\begin{figure}\par\includegraphics[width=8cm,clip]{12696figA3.eps}\end{figure}$	Figure A.3: Composite lightcurve of (1270) Datura during the 2009 apparition.
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2006 apparition(Figs. A.1-A.3). We alsoshow the phase curve for this asteroid using our calibrated RandV band observations and the corresponding fitin theH-G system (Fig. A.4). Ourbest-fit model values are $H_0(R)=12.03\pm 0.02$ ,at the maximum of the lightcurve, and $G(R)=0.29\pm 0.01$ .Weused data from 1991 (Wisniewski et al.1997;with the derivedV-Rvalue),2007/2008 (Simeiz, Kharkiv) and 2009 (Simeiz), see Table 2.

$\begin{figure}\par\includegraphics[width=8cm,clip]{12696figA4.eps}\end{figure}$	Figure A.4: Solar phase angle behavior for (1270) Datura derived from ourR bandcalibrated observations (symbols). The dashed curve is the best-fitmodel using $H_0=12.03\pm 0.02$ and $G=0.29\pm 0.01$ .
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Footnotes

... update: Photometric data is only available in electronic form atthe CDS via anonymous ftp tocdsarc.u-strasbg.fr(130.79.128.5) or viahttp://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/507/495
... NPA: The derived shape model together with the completelightcurve data set is availableatastro.troja.mff.cuni.cz/projects/asteroids3D.
... system: The long rotation period may support the model of abi-lobed shape rather than a synchronous detached system. For example,if the binary system consisted of two spherical bodies with diameters1 km and the bulk density 2 g/cm³,the observed orbital period $\sim$ 24 hrwould imply $\sim$ 3 kmfor the semimajor axis of the mutual orbit. Apparently, such a largerelative separation is not consistent with the shape of the observedlightcurves. Assuming spherical components, the lightcurve amplitudealso cannot be larger than $\sim$ 0.8 mag.Triaxiality of the components may however slightly change theseconclusions.
... integrations: We useSWIFT_MVSY symplecticintegrator (e.g., Broz2006)with all planets included and timestep of 5 days. The initialstate vectors for planets were obtained from JPL DE405 ephemerides atthe same time as the elements of the Datura-cluster asteroids(Table 1).We use rather dense output sampling of 20 yr and integratebackward in time till 700 kyr.
...function: To see if our results are robust, we also evaluated anothertarget function $\Delta V'$ ,defined as in Eq. (1),where $\Delta\Omega$ and $\Delta\varpi$ are performed with respect to the orbit of (1270) Datura only and thesummation is performed over the remaining members in the cluster. Assuch, the $\Delta V'$ quantity measures dispersal velocity field of the Datura-familyfragments with respect to the largest fragment. Note the $\Delta V$ target function may become unrealistically small for a cluster offragments that reside on nearby orbits, yet distant from the largestasteroid in the family.
... proximity: In this paper, we take this standpoint and do notinvestigate the less likely possibility that (215619)2003 SQ168 separated from (1270) Datura in a more recentepoch.
... tests: Obviously, the two age determinations used different orbitsets and this might have influenced the results. However, if the newobservations caused simple shrinking of the uncertainty ellipsoid inthe space of initial orbital elements, we would expect to obtainequally nested solutions: the one with more accurate orbits would beentirely within the one with less accurate orbits. This idea assumesuniform mixing of results from the uncertainty ellipsoid, which islikely, but not entirely guaranteed.

All Tables

Table1: Equinoctial orbital elements of the Datura family membersas of MJD 55000.0.

Table2: Aspect data for observations of (1270) Datura.

Table3: Aspect data for new observations of (90265)2003 CL5.

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