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The Zelima asteroid family: Resonant configuration and rotational fission clusters
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The Zelima asteroid family: Resonant configuration and rotational fission clusters V. Carruba *, J.V. Ribeiro S~ ao Paulo State University (UNESP), School of Natural Sciences and Engineering, Guaratinguet a, SP, 12516-410, Brazil
A R T I C L E I N F O
A B S T R A C T
Keywords: Minor planets Asteroids: general Minor planets Asteroids: individual: (633) Zelima Celestial mechanics
Asteroid families are groups of asteroids that are identifiable in domains of proper elements. They may be the outcome of a collision, or the product of rotational fission. Among collisional families, young asteroid families are of interest because they preserve information on the original ejection velocity field with which the asteroid fragments were expelled, and because they may host a significant population of clusters produced by rotational fission among their members. The recently identified Zelima asteroid family is unique because, apart from its very young age of less than 3 Myr, is also interacting with the z1 secular resonance, making it the third asteroid family in this peculiar configuration, after the Agnia and Padua families, and the only one for which all members are in z1 librating states. Because of its extremely young age, the orbits of its members are still very clustered in the domain ðσ ;dσ =dtÞ, with σ ¼ ϖ ϖ 6 þ Ω Ω6 the resonant argument of the z1 resonance. This allow for precisely dating the family with a method not available for any other main belt group. The Zelima family, based on this argument, should be 2:5 1:5 Myr old, and conserved quantities of the z1 secular resonance show that its ejection velocity field is in a range of 1:1þ0:6 0:2 of the escape velocity from the parent body, in agreement with general results for cratering collisional families. Finally, 41.7% of the members of the Zelima family are also members of possible fission clusters, confirming the trend observed for other very young families, like the Jones and Lorre groups.
1. Introduction Asteroid families are group of asteroids characterized by a close proximity in domains of proper elements. Among asteroid families, the group that formed most recently are of particular interest. Families younger than ’ 7 Myr did not evolve significantly because of the effect of non-gravitational forces such as the Yarkovsky effect. Contrary to some of the more evolved asteroid groups, it is therefore easier to obtain information on the original ejection velocity field that formed them (Carruba et al., 2018b). Recently (Tsirvoulis, 2019), identified the (633) Zelima group, a sub-family of the larger and more evolved Eos family, and used the backward integration method, or BIM, to estimate its age to be 2.9 0.2 Myr, so being the product of an extremely recent collisional event. In the BIM the differences of longitudes of nodes and pericenters between two asteroids that are thought to have a common origin are integrated into the past. At the time of the pair formation these differences should both
tend towards values near zero. The Zelima family is remarkable in several ways. All its members are in librating states of the z1 secular resonance, Families interacting with secular resonances are important because conserved quantities of secular dynamics allow to obtain information on the group initial ejection velocity field and age that are not easily available for other asteroid groups (Carruba et al. (2018a)). Until recently, only two families were known to have a majority of their members in resonant states of the z1 secular resonance,1 the Agnia (Vokrouhlický et al., 2006) and Padua (former Lydia) (Carruba, 2009) asteroid families. Only for the Zelima family, however, all the members are in librating z1 states, making it the first known asteroid family for which this phenomenon is observed. Also, in view of the extremely young age of this group, it is reasonable to expect that the resonant angles σ ¼ ϖ ϖ 6 þ Ω Ω6 and frequency ddtσ ¼ g g6 þ s s6 did not had enough time to disperse, unlike the cases of the more evolved Agnia and Padua families. This allows to apply an unique age dating method based on dispersion in a
* Corresponding author. E-mail address: [email protected] (V. Carruba). 1 Secular resonances occur when there is a commensurability between the precession frequency of the longitude of pericenter g or of the node s of an asteroid and a planet. The proper frequencies of planets are identified by a suffix, according to the planet distance with respect to the sun. For Saturn, the suffix is equal to 6. See also Carruba et al. (2018a) for a more in depth definition of secular resonances. https://doi.org/10.1016/j.pss.2019.104810 Received 18 June 2019; Received in revised form 21 November 2019; Accepted 25 November 2019 Available online xxxx 0032-0633/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: Carruba, V., Ribeiro, J.V., The Zelima asteroid family: Resonant configuration and rotational fission clusters, Planetary and Space Science, https://doi.org/10.1016/j.pss.2019.104810
V. Carruba, J.V. Ribeiro
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presence of possible new members with respect to the analysis of (Tsirvoulis, 2019). For this purpose, we looked for new members of the Zelima family in the Asteroid Families Portal (AFP (Radovic et al., 2017), accessed on September 30th, 2019) using the online HCM applicative of the site. In the Hierarchical Clustering Method (HCM), we first define a distance metrics in proper element domain and check if there are objects closer to the body of reference than a given distance. If new objects are found, the process is repeated using the new object as a reference, until no new members are encountered Zappala et al. (1990). Fig. 1 displays the number (left panel) and differential number (right panel, by differential number we mean the number of new family members for each increase of the distance cutoff) for members of the Zelima asteroid family, as found from the AFP. There are two plateaux in the number of family members encountered, at ’ 15 and ’ 23 m/s, which could be associated with possible choices of the optimal distance cutoff for this family. For a distance cutoff of 28 m/s the Zelima sub-family merges with the larger Eos family and it is no longer distinguishable as a separate entity. Using a too large value of distance cutoff may lead to overestimating the numbers of family members, including background Eos family asteroids not related with the Zelima sub-family. On the other hand, using a conservative value may lead to underestimating the real population of family members. To solve this problem, we took advantage of the young age of this family, and we tried to distinguish between background and family members by using BIM. Our approach was the following: we selected the numbered asteroids belonging to the Zelima family at the maximum possible cutoff value of 26 m/s. We then also checked for multiple-opposition possible members by searching for asteroids in 3σ level from the median values of proper ða; e; sinðiÞÞ of currently known Zelima members, using the approach of (Carruba and Nesvorný, 2016). This yielded a population of 51 potential members of the Zelima asteroid dynamical family. We then integrated these asteroids in the past and used the BIM to check for which objects the secular angles converged with respect to the parent body, as also done in the past for other young asteroid families by (Novakovic et al., 2012), which called this method the Selective Backward Integration
plane, that, so far, was not useable for any other family inter-
acting with secular resonances. Another way in which the Zelima family is important is because of the possible existence of fission clusters among its members. Recently, it has also been found that young families may produce a higher fraction of fission clusters than older families. Fission clusters are groups of asteroids that form because of the rotational failure of a parent body. Current models predict that fission clusters may form either because i) of the break-up of a rapidly rotating asteroid into multiple components (Bottke et al., 2002), ii) repeated accelerations up to the point of break-up of a fast spinning object caused by the YORP effect, i.e., the scattering of solar radiation off the surface of an irregularly shaped asteroid and the emission of its own thermal radiation, (Walsh et al., 2008), iii) the fission of a secondary body caused by spin-orbit coupling (Jacobson and Scheeres, 2011), or iv) a low-energy collision on a fast rotating primary (Vokrouhlický, 2017). (Carruba et al., 2019) recently found that the proportion of fission clusters in extremely young asteroid families is much higher than that observed in older families, suggesting that collisions may trigger a subsequent cascade of formations of fission clusters among family members. Again, in view of the extremely young age of this group, it may be reasonable to expect that a significant fraction of its members may be also part of fission clusters. Using the approaches used by (Carruba et al., 2019) for other young asteroid families, in this work we will investigate this hypothesis. This work is so divided: in the second section we will briefly revised the taxonomic and dynamical properties of the Zelima asteroid family. In the third section we will investigate the resonant behavior of this family, and see what information this may yield on the ejection velocity field of the group. Section four will be dedicated to the search for fission clusters inside the Zelima family. Finally, in section five, we will present our conclusions. 1.1. The Zelima family: taxonomic and dynamical properties As a first step in our analysis of the Zelima family, we checked for the
Fig. 1. The number (left panel) and differential number (right panel) of members in the Zelima dynamical family as a function of the velocity cutoff. 2
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120 in the ða; eÞ domain, in a range from 3.0082 to 3.0218 au in a and from 0.074 up to 0.1216 in e. The range in i was from 9:01∘ up to 13:77∘ . All the other initial osculating elements were those of (633) Zelima itself at 58400 MJD. The results of our maps in the two plans are shown in Fig. (3). Black dots displays the proper elements of test particles. Blue full circles identify members of the Zelima family, as originally identified by (Tsirvoulis, 2019), while red full circles displays the orbits of the newly identified possible family members. With the exception of (4077) Asuka, most new members are on orbits quite similar to those of the previously known Zelima asteroids. Mean-motion resonances, such as the 8J:-3S-3A three body resonance known to affect four members of the Zelima family (Tsirvoulis, 2019), are identified as a red vertical line. The main dynamical feature affecting the Zelima asteroid family is, as observed by (Tsirvoulis, 2019), the z1 ¼ g g6 þ s s6 secular resonance. Particles whose frequencies g þ s is in the range g6 þ s6 0:1 arcsec/yr, i.e., those most likely to be in z1 secular resonance librating states, are shown as yellow circles in Fig. (3, panel (a) and (b)). Indeed, most members of the Zelima family are observed to be in the affected region. The implications of this fact will be discussed in the next section.
Method, and by (Carruba et al., 2016). Fig. (2) displays values of the differences in longitudes of the nodes between a possible member and the alleged parent body (633) Zelima (ΔΩ ¼ Ω ΩPB ). Panel (a) shows this differences for the population of 23 asteroids originally identified as members by (Tsirvoulis, 2019). Panel (b) shows the same, but for all 51 asteroids studied in this work. Among these asteroids, we selected those with values of both ΔΩ and Δϖ to within 30∘ from the mean values of ΔΩ and Δϖ of the previously known 23 members of the Zelima asteroid family at the estimated time of family formation, 2.9 Myr ago. 10 new asteroids satisfied this selection criteria, values of their ΔΩ are shown in red in Fig. (2), panel B. For the sake of brevity, we do not display values of Δϖ. The new possible members of the Zelima family are (4077) Asuka, (243212) (2007 UQ48), (325541) (2009 SG58), (489627) (2007 TJ334), (2005 YM141), (2013 YF84), (2015 BL90), (2015 FB407), (2016 BB100), and (2016 CJ291). Among the multi-opposition asteroids, only (2005 YM141) has a Lyapunov exponent of less than 50 106 Myr1 , with all the other multi-opposition asteroids on more chaotic orbits. While the first asteroid in the new sample satisfies the selection criteria and has also an albedo compatible with those of the rest of the family (see discussion below), it is a relative large asteroid (H ¼ 11:00, while (633) Zelima itself has H ¼ 9:73), somewhat more distant from the rest of the family in both proper a and i. It is possible that this could be a false positive identification, but it will be kept in our sample for the time being. We then revised the physical properties of family members. As discussed by (Tsirvoulis, 2019), taxonomic information is only available for (633) Zelima itself, which is a S-type object. No information for any other family member is available in the Sloan Digital Sky Survey-Moving Object Catalog data (SDSS-MOC4 (Ivezic, 2001)). Four objects had albedo data in the WISE and NEOWISE, AKARI, or IRAS databases ((Masiero et al., 2012), (Ishihara, 2010), (Ryan and Woodward, 2010)) and satisfy the good signal-to-noise ratio criteria of (Spoto et al., 2015). These four asteroids have albedo values higher than 0.12, compatible with a S-complex taxonomy. The median value of the albedo is 0.15 0.3. To study the dynamics of the Zelima family we obtained dynamical maps in domains of synthetic proper elements using the approach of (Carruba, 2010). For both the maps in the ða; eÞ and ða; sinðiÞÞ domains we integrated 4200 test particles under the influence of all planets with SWIFT-MVSF, the integrator from the SWIFT package (Levison and Duncan, 1994), modified to account for online filtering of the osculating orbital elements (Broz, 1999). The test particles covered a grid of 35
2. The Zelima family resonant nature Tsirvoulis (2019) recently identified the resonant nature of the Zelima asteroid family and its interaction with the z1 ¼ g g6 þ s s6 secular resonance. This behavior is confirmed in this work, all of the numbered members of the Zelima asteroid families are currently in librating states of the z1 secular resonance. Fig. (4) displays the resonant argument of the asteroid with the lowest librating amplitude around 180∘ , (421572) (2014 OP193), and of an asteroid on a circulating orbit very close to the resonance separatrix, (1861) Komensky, that was already identified as an interloper by Tsirvoulis (2019). All the other 26 numbered members of the Zelima group are all showing libration of the resonant argument of the z1 , with varying oscillation amplitudes. However, none of the six multi-opposition asteroids potential members of the Zelima family identified in the previous section is currently librating. Based on this consideration and on their chaotic orbits, from now on, none of them will be further considered in our analysis. Overall, the Zelima asteroid family is the first family in the main belt for which all known confirmed members are in z1 librating states. This fact, and its extremely young age, allow for a unique age determination approach that will be discussed in the next sub-section.
Fig. 2. Values of ΔΩ for the original population of 23 family members, as identified by (Tsirvoulis, 2019) (panel a), and for a sample of 51 potential members of the family, identified with the criteria discussed in this work (panel b). The vertical dashed line shows the family age, as estimated by (Tsirvoulis, 2019). Horizontal dashed lines displays the 60∘ levels in ΔΩ. Red curves in panel (b) are associated with the ΔΩ values of 10 new potential family members, according to the criteria discussed in the text. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.) 3
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Fig. 3. Dynamical maps in the proper ða; eÞ (panel a) and ða; sinðiÞÞ (panel b) domains for the orbital region of the Zelima family. Black dots show values of synthetic proper elements of test particles in our maps. Yellow full circles display the location of particles likely to be in z1 resonant states, according to the criterion discussed in the text. Blue full circles show the orbital location in the proper element domain of members of the Zelima family as identified by (Tsirvoulis, 2019), while red full circles displays the orbits of the possible new members identified in this work. The vertical red line displays the position of the 8J:-3S-3A three-body resonance. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
2.1. Age estimate from dispersion in the
σ ; ddtσ
frequencies and phases of the asteroid and Saturn were obtained every 1.23 Myr with the procedure discussed in Vokrouhlický et al. (2006), Carruba (2009), that involved a numerical simulation of the asteroid over 20 Myr, under the gravitational influence of all planets. Among the numbered potential new members of the Zelima family identified in the previous section, only (489627) (2007 TJ334) has values of ðσ ; dσ =dtÞ compatible with those of the rest of the family (displayed as a red dot in Fig. (5)) and will be considered as a proper family member hereafter. Contrary to the cases of the Padua and Agnia families, the Zelima family is still quite compact in the ðσ ; dσ =dtÞ domain. For other resonant groups, this information is, however, lost in timescales of the order of ’ 10 Myr. To quantify how tight the family is in this plane, and to try to obtain a family age estimate, we follow the procedure first introduced by Vokrouhlický et al. (2006) for the Agnia family. We use a polar angle Φ in the ðσ ; dσ =dtÞ, obtained using a scaling that maps a ð0∘ ; 360∘ Þ interval of σ and (1,1) “/yr interval in dσ =dt ¼ g þ s g6 s6 into common intervals (1,1). At each time step, the dispersion DΦ can be computed by the
plane
An useful tool when studying asteroid families characterized by their interaction with the z1 secular resonance is the distribution of the members in the ðσ ; dσ =dtÞ plane of the resonance, where σ ¼ ϖ ϖ 6 þ Ω Ω6 is the resonant argument of the z1 resonance, and dσ = dt ¼ g g6 þ s s6 . Most families are thought to have started with a compact distribution of values in the ðσ ;dσ =dtÞ. Points in this planes will describe circular orbits around σ ¼ 180∘ in timescales of the order of 1–2 Myr. Regrettably, after a few orbit cycles the alignments of family members in this plane is lost, and asteroids tend to reach a uniform distributions in less than 10 Myr. This is indeed the case for the already discussed cases of the Agnia and Padua families, both characterized by a uniform distribution of values in this plane. The Zelima family, however, is unique because of both its resonant nature and young age. Fig. (5) displays projections of family members as identified by (Tsirvoulis, 2019) at T ¼ 1:23 Myr (panel a) and T ¼ 20:89 Myr (panel b). The values of proper
Fig. 4. Librating argument ϖ ϖ 6 þ Ω Ω6 of the z1 secular resonance for the (421572) (2014 OP193) (panel a), and for (1861) Komensky (panel b). 4
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Fig. 5. The dispersion in the ðσ ; dσ =dtÞ plane of 23 Zelima family members numerically simulated at T ¼ 1:23 Myr (panel a) and T ¼ 20:89 Myr (panel b), shown as black full circles. The red full triangle identify the position of (489627) (2007 TJ334), the only asteroids among the potential candidates whose position in the ðσ ; dσ =dtÞ plane is in the same region as the remaining members of the Zelima family. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
integrated 225 particles, which is the value of N in Eq. (1). We simulated the orbital evolution of particles in the future and in the past. For the past integrations, we used negative velocities for both planets and particles, and re-scaled the longitudes of nodes and pericenters according to the relationships (Eqs. 2 and 3):
equation: < ΔΦ>2 ¼
X 2 1 Φi Φj ; NðN 1Þ i6¼j
(1)
where N is the number of integrated bodies. For each family members we integrated nine sets of clones. We considered the central value of eccentricity and inclination, plus or minus their orbital uncertainty, obtained from the AstDyS (https://newton.spacedys.com/astdys/, accessed on May 28th, 2019, Knezevic and Milani (2003)). Each value of eccentricity (minimum, central, maximum) is associated to the three possible values of inclination, yielding nine possible combinations. Overall, we
Ω ¼ Ω þ π
(2)
ϖ ¼ Ω ω:
(3)
An uniform distribution of bodies in this polar angle corresponds to < ΔΦ >¼ 103∘ (Vokrouhlický et al., 2006), while the current observed value of < ΔΦ > for the Zelima family is 19:6∘ . If we use the current value of < ΔΦ > as a limit, the estimated age of the Zelima family according to this method would be 2:5 1:5 Myr, in agreement with (Tsirvoulis, 2019) estimate of 2:9 0:2 Myr (see Fig. 6). The distribution in < ΔΦ > would become uniform in the interval 2:5þ6:5 4:3 Myr, which is the upper limit on the age estimate that this method can provide. Overall, it is the first time that a method based on the dispersion of orbits in the ðσ ; dσ =dtÞ domain can provide a rather precise age estimate for a family interacting with the z1 secular resonance. While this age estimate is not as precise as results obtained with the BIM approach,2 we believe that successfully using this approach is an important result for the field of secular dynamics. Founding new extremely young asteroid families interacting with secular resonances other than the z1 could be very valuable to confirm, or not, the validity of this approach. In the next subsection we will investigate the initial ejection velocity
2 We should also point out that, since this method is based on the convergence of values of σ and dσ =dt, and since σ ¼ ϖ ϖ 6 þ Ω Ω6 for the z1 secular resonance, the convergence of σ values is not completely independent from the convergence of the secular angles < ΔΩ and Δϖ required by BIM. Since the ðσ ; dσ =dtÞ method also require convergences of frequency values, that generally do not happen for non-resonant cases, the two methods are not completely equivalent. Yet, it would be interesting if such method could be tested for families in secular resonances other than the z1 , so as to fully test the validity of this approach. Regrettably, currently the Zelima family is the only asteroid group inside a secular resonance with an age of less than 4.0 Myr, as far as we know.
Fig. 6. The time behavior of the dispersion in the Φ polar angles for 207 clones of Zelima family members. The horizontal lines display the current value of the dispersion for real Zelima members, and the < ΔΦ >¼ 103∘ level corresponding to the case of an uniform distribution of bodies. 5
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field of the Zelima asteroid family, and the information that secular dynamics can provide on this subject.
optimal VEJ parameter. For this purpose, we generate several synthetic asteroid families with different values of the VEJ parameter, compute their values of the K ’2 quantity, and assess which of the simulated family is most compatible with the currently observed K ’2 distribution. To quantitatively test this, we can use a χ 2 -like variable defined as (Eq. 6):
2.2. Conserved quantities of the z1 resonance and implications for the initial ejection velocity field (Vokrouhlický et al., 2006) showed that for asteroids in librating 0 states of the z1 secular resonance the quantity K 2 , defined as (Eq. 4): pffiffiffiffiffiffiffiffiffiffiffiffiffi K ’2 ¼ 1 e2 ð2 cosðiÞÞ;
χ2 ¼
(6)
where Nint ¼ 10 is the number of interval used for the values of K ‘2 , qi is the number of real objects in the i th interval in K ‘2 , and pi is the number of simulated family members in the same i th interval. We can then compute the Δχ 2 ¼ χ 2 χ 2min , which is the difference between values of χ 2 and their minimum, and use the χ 2 probability distribution function, given by (Eq. 7):
(4)
is preserved, even when the Yarkosvky force is considered (Carruba, 2009). confirmed this for members of the Padua family in z1 librating 0 states. Fig. (7) displays a histogram of the current distribution of K 2 for Zelima family members (panel a). As a consequence of the preservation of this quantity, proper e and i are anti-correlated. Panel (b) of the same figure show the range of values covered by these quantities for a simulation over 20 Myr. Proper elements were obtained with the method described in (Carruba, 2010). Green full dots show the current positions in the ðe; sinðiÞÞ domain of the Zelima asteroids. The preservation of the original values of the K ’2 quantity allowed to obtain estimates of the initial ejection velocity field that are not available for families not interacting with secular resonances. Here we will attempt to obtain such estimates using the (Vokrouhlický et al., 2006; Carruba, 2009) approach for the case of the Zelima asteroid family. As a first step, we define a simple model for the ejection velocity field of asteroids just after the formation of the family. In this approach we assume that asteroids are ejected with an isotropic ejection velocity field whose standard deviations VSD depends on the asteroid size through a parameter VEJ according to the relationship (Eq. 5): VSD ¼ VEJ ð5km = DÞ;
Nint X ðqi pi Þ2 ; qi i¼1
F x; Ndf ¼
γ Γ
Ndf x ; 2 2
;
(7)
Ndf 2
where x ¼ 0:975 stands for the chosen confidence level, γ N2df ; 2x is the lower incomplete gamma function, and Γ N2df is the gamma function. Ndf ¼ 10 1 ¼ 9 is the number of degrees of freedom, which is given by the number of intervals used to estimate the χ 2 variable (10, in our case), minus the number of parameters estimated, which is one, VEJ , in this application of this method. The value of Δχ 2 consistent with a 97.5% confidence level for nine degrees of freedom is 2.700. Our results are shown in Fig. (8), where in panel (a) we display a histogram of the current distribution of K ’2 values in red, and of the simulated ones that best-fitted the real data, in blue. Panel (b) shows the dependence of Δχ 2 as a function of the VEJ parameter. This analysis yields a value of VEj of 20:0þ3:3 10:4 m/s, which is consistent with the escape velocity from (633) Zelima, that is equal to 17.9 m/s. According to (Carruba and Nesvorný, 2016), most asteroid families have values of VEj ¼ β Vesc that are less than 1.5 the escape velocity from the parent body Vesc , and our value of β ¼ 1:1þ0:2 0:6 , where the errors on β are
(5)
where VEJ is the value of the ejection velocity that we used for the whole family, and D is the asteroid diameter. This method may not provide a realistic model of the actual ejection velocity field of the Zelima family, since (Tsirvoulis, 2019) showed that the actual ejection velocity field is rather asymmetric, as observed for several other cratering asteroid families. Nevertheless, it may provide useful constraints on the magnitude of the original ejection velocity field though an estimate of the
Fig. 7. A histogram of the distribution of the K ’2 conserved quantity of the z1 secular resonance for Zelima family members (panel a). Panel (b) displays the time evolution of proper sinðiÞ versus proper e for the same family members, as blue lines. The green full dots show the current values of e and sinðiÞ of Zelima family members, as identified by (Tsirvoulis, 2019). The red full triangle displays the location of the new possible family member identified in this work. 6
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Fig. 8. Panel (a) show a histogram of current values of the K ’2 quantity (in red) and of the simulated family that best-fitted the data. Panel (b) displays the dependence of Δχ 2 as a function of the VEJ parameter. The horizontal dashed line displays the 97.% confidence level limit. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
identification of fission clusters inside four very young asteroid families, those of (3152) Jones, (7353) Kazuya, (5438) Lorre and (108138) (2001 GB11), so that interested readers can find more details on these methods in that paper. While BIM as already been discussed in the introduction, in CEM, several clones of both the parent body and the other pair object, accounting for the orbital uncertainties and for non-gravitational effect such as the Yarkovsky force are integrated into the past and the time of occurrence of a close encounter between clones of the parent body and the other asteroid that occur at relative speeds and distances smaller than given parameters are registered. The median value of this time provide and estimate of the possible time of separation, while the 5th and 95th percentile of the distribution allow to assess the errors on this age. For illustrative purposes, Fig. (9) displays results of BIM and CEM for asteroid pairs in the Zelima family that were not likely to have formed after the formation of the asteroid family (left panels (a) and (c)), and that are strong candidates to belong to fission clusters that formed afterward (right panels (b) and (d)). We first determined all the fission clusters candidates that satisfy our dmean and q criteria. We then studied them with BIM, without considering any non-gravitational forces, in a purely conservative frame. Since the actual convergence of angles should also account for the Yarkovsky effect (see for instance Carruba et al. (2016) for an application of this method to members of the Karin cluster), and since key parameters of this effect such as the asteroid density and thermal conductivity are not generally known for Zelima family asteroids, we require at this stage of the investigation that the convergence of the angles in the conservative case yields an age estimate less or equal to the Zelima estimated family age of 2.9 Myr, but not superior, like the case of Fig. (9, panel c). For the objects that passed these two phases of our study, we then applied CEM, which provides the most robust results for identifying possible members of rotational fission clusters. A possible pair is identified by CEM if it has a nominal age less than the estimated primary family age. For Zelima family members we could not find any fission cluster between (633) Zelima itself and other family members that satisfied these selection criteria. We found, however, 16 possible pairs among the other family members, that could be members of what (Carruba et al., 2019) defines as tertiary families, i.e., groups that formed after the main family formation event and that do not involve the primary body. Table (1) displays our results for the pairs that were confirmed by both BIM and CEM. The pairs (1861) Komensky and (456020) (2005 YU83),
evaluated considering the upper and lower limit on VEj and dividing these values by the escape velocity, fits in this range. In the next section we will start analyzing the population of fission clusters inside the Zelima asteroid family. 3. Zelima fission clusters population Fission clusters are groups of asteroids extremely close in proper elements domains that may originate because of several proposed mechanisms of asteroid rotational failure, as discussed in the article introduction. Recent studies have shown that young collisional families may produce a higher proportion of young fission clusters than more evolved collisional groups (Carruba et al., 2019). In view of the extreme young estimated age of the Zelima family, in this section we will explore the occurrence of fission clusters among their members. For this purpose, we use the same criteria discussed in (Carruba et al., 2019). We use a distance dmean between a pair of asteroids according to the metric (Eq. 8): rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 δa2 þ 2ðδeÞ2 þ 2ðδ sin iÞ2 ; 4 a
dmean ¼ na
(8)
where ðδa; δe; δ sin iÞ is the separation vector in proper elements, a is the mean value of the pair semi-major axis, and dmean is the standard distance metric as defined in (Zappala et al., 1990). Following (Pravec, 2010), we also use a mass ratio q using the relationship (Eq. 9): 3 q ¼ 10ðδH=5Þ ;
(9)
where δH is the difference in absolute magnitude between the pairs, and where the geometric albedo of the pairs is assumed to be equal. Asteroids are considered good candidates to be part of a cluster if their difference in dmean is less than 5 m/s and if their mass ratio q is less than 0.3, the maximum theoretical and practical limit observed for asteroid pairs by (Pravec, 2010). Once the asteroids more likely to belong to a cluster have been selected, they are then studied with methods based on time-reversal numerical integration, like the backward integration method (BIM, Nesvorný et al. (2002)), and the close encounters method (CEM, Pravec et al. (Pravec, 2010)), to verify if they really belong to a fission cluster, or just share similar orbits. Both BIM and CEM were recently used in (Carruba et al., 2019) for the
7
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Fig. 9. Panel (a) displays the time behavior of ΔΩ and Δϖ for the pair of asteroid (6733) (1992 EF) and (449004) (2012 BY59), for which there is no convergence of the angles before the estimated family formation event, and for the pair (119786) (2002 AE65) and (368472) (2003 SQ191) (panel b), that are most likely to have formed in a subsequent fission pair formation event. Panels (c) and (d) displays the results of CEM for the same asteroid pairs, respectively.
Table 1 The table report the cluster parent body, its members identification, their absolute magnitude, the mass ratio q, the distance in m s1 with respect to the parent body, the number of encounters experienced over the 7 Myr long integration, and the estimated time of separation, with its uncertainty, for the case of the (633) Zelima asteroid family. Cluster
Cluster
H
q
dmean 1
[m s
Number of
P. body
member
enc.
[Myr]
6733
324175
15.9
0.01
2.16
2027
2:855þ3:830 2:547
6733
364004
15.8
0.01
1.82
9071
89714
250011
15.5
0.21
4.38
7505
89714
292616
15.5
0.24
3.33
3748
89714
402372
16.0
0.09
3.88
2727
89714
412371
16.3
0.07
3.76
3101
89714
456020
16.4
0.04
4.67
5352
119786
368472
15.6
0.24
3.58
16352
119786
449004
16.8
0.06
3.46
12786
119786
487314
16.4
0.10
1.82
2885
8
]
Tsep
0:143þ4:964 0:099 0:104þ5:000 0:065 0:192þ5:604 0:162 1:531þ4:935 1:451 1:431þ5:032 1:349 0:188þ5:636 0:152 0:037þ3:402 0:023 0:151þ5:694 0:148 1:613þ4:918 1:565
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Fig. 10. A ða; eÞ (panel a) and ða; sinðiÞÞ (panel b) projection of members of the Zelima family (black circles) and of possible members of fission clusters (yellow full circles). The size of the circles is inversely proportional to the asteroid absolute magnitude according to the figure legend. Other symbols are the same as in Fig. (3). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
and (364004) (2005 UH450) and (449004) (2012 BY59) were not confirmed by BIM and were discarded for an analysis with CEM. (1861) Komensky could also be an interloper, according to Tsirvoulis (2019). Overall, 10 asteroid pairs could be viable candidates to be formed as rotational failures of Zelima family members (or their satellites) after the family formation event. Our results are shown in Fig. (10), where we display an ða; eÞ (panel a) and an ða sinðiÞÞ (panel b) projections of the members of the possible identified tertiary fission clusters (full yellow circles). This analysis, therefore, suggests that up to 54.2% of Zelima members may also be members of rotational fission clusters. Uncertainties on the ages of these possible clusters are quite large, and an origin from the main collisional event that formed the family cannot be completely excluded for any of these asteroids, at this stage of the investigation. High-quality astrometric Gaia data (Spoto et al., 2018) could, in the future, be used to reduce these uncertainties (Carruba et al., 2019), but, unfortunately, at the moment such data is not available for any of the asteroids listed in Table (1). Our results are compatible with the possible presence of three fission clusters, but their existence cannot be yet positively confirmed. For two asteroids, (633) Zelima and (6733) (1992 EF), information is available on their rotation periods. Such information can be used to assess if the groups to which these asteroids belong can be classified as collisional or fission clusters. In the next section we will attempt to do that.
Fig. 11. The position in a ðlog10 ðσ a Þ; ageÞ diagram of the Zelima family (red lines) and its fission clusters (black lines) studied in this work, with their errors. The yellow region displays the area where most collisional families are found. The green area shows the boundaries of the region where fission clusters are more common, as defined in (Carruba et al., 2018b). The dashed horizontal line displays the maximum nominal age found for most rotational fission cluster so far. Data for the fission clusters here identified are available in Table (1). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
4. Cluster classification First, we checked how the clusters identified in the previous section are located in the ðlog10 ðσ a Þ; ageÞ diagram that was introduced by (Carruba et al., 2018b) to classify and distinguish between collisional families and fission clusters. Collisional families tend to form with larger velocities at infinity than rotational fission clusters, and, therefore, tend to have larger values of σ a (Carruba et al., 2018b). identified two regions in ðlog10 ðσ a Þ; ageÞ plane shown in green and yellow, were at the time, most fission clusters and most collisional families were found. Fig. (11) displays our results in this domain. The three clusters all have values of log10 ðσ a Þ compatible with those of other fission clusters, so confirming their origin. Interestingly enough, so are the results for the whole Zelima family, identified in red in the figure, that are more compatible with dispersions associated with fission clusters than with values observed for young collisional families (Carruba et al., 2018b). Is the whole Zelima family actually a fission cluster? To answer this question, we turn our attention to the theory devel-
oped by (Pravec, 2010) to classify fission clusters based on their mass ratio and parent body rotation period. Fig. (12) shows the positions in the ðΔH; P1 Þ diagram of the (633) Zelima family and of the (6733) (1992 EF) cluster, the only groups for which information on the rotation periods of the largest body is available.3 The black dashed line is associated with the optimal values of the parameters for which the theory described in (Pravec, 2010) predicts the formation of a fission cluster: a system scaled angular momentum αL ¼ 1:0, a primary axis ratio a1 =b1 ¼ 1:4 and an initial orbit’s normalized
3 The period of (633) Zelima is 11.730.01 h, while the period of (6733) (1992 EF) is 11.360.01 h (Warner et al., 2009).
9
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Planetary and Space Science xxx (xxxx) xxx
on the dispersion of a polar angle φ in the ðσ ; dσ =dtÞ domain, a method that, so far, only precisely worked for this family. The age predicted by this method, 2:5 1:5 Myr, is in agreement with independent estimates obtained by other authors (Tsirvoulis, 2019). 0 A study of the conserved quantity of the z1 secular resonance, K 2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 e2 ð2 cosðiÞÞ, allowed to estimate that the VEj parameter describing the ejection velocity field of the Zelima family is of the order 1:1þ0:6 0:2 of the estimated escape velocity from the Zelima family parent body, (633) Zelima itself, in agreement with results for other asteroid families that show that most families have VEj values lower than 1.5 the parent body escape velocity (Carruba et al., 2019). Finally, using methods based on backward integrations of asteroid orbits, we identified three possible clusters inside the Zelima asteroid family that could be compatible with a subsequent origin caused by rotational fission: the cluster of (6733) (1992 EF) with 3 members, that of (89714) (2001 YA114) with 6 members, and that of (119786) (2002 AE65) with 4 members. Information on the rotation period of (633) Zelima suggests that the family, according to the theory presented in (Pravec, 2010), should have originated from a collision, despite its unusual very compact distribution in proper a. The rotational period of (6733) (1992 EF) appear to be incompatible with an origin as fission cluster for this group. Obtaining more precise astrometric data on the orbits of the members of the remaining possible clusters identified in this work, as well as information on the orbital periods of (89714) (2001 YA114) and (119786) (2002 AE65) could help clarify if these two clusters indeed formed as results of a rotational failure after the collision that formed family 2.90.2 Myr ago.
Fig. 12. The positions in the ðΔH; P1 Þ diagram of the (633) Zelima family (red lines) and of the (6733) (1992 EF) cluster (black lines), with their uncertainties. The blue and red lines display the lower and upper limiting cases for clusters formed by a fission event, while the dashed line shows the optimal case (see discussion in the text). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
semi-major axis Aini =b1 ¼ 3. The red and blue curves relate to the upper and lower limiting cases, respectively: αL ¼ 1:2, a1 =b1 ¼ 1:2, and Aini = b1 ¼ 2 for the upper curve, while αL ¼ 0:7, a1 =b1 ¼ 1:5, and Aini = b1 ¼ 2 for the lower curve. More information on the rationale for the choice of these parameters is available in (Pravec, 2010). Neither group is in a region comparable with an origin from rotational fission. This suggest that the Zelima family may indeed be the outcome of a collision, regardless of its unusually small value of σ a and that the (6733) (1992 EF) cluster could either not be a real fission cluster,4 or be the outcome of a more recent collision. Considering that i) CEM results for the pair (6733) (1992 EF) and (324175) (2006 AU14) are more compatible with an origin at the same time of the family formation event, and that ii) only the pairs (6733) (1992 EF) and (364004) (2005 UH450) is more robust in terms of CEM results, the more likely explanation appears to be that this could not be a real cluster, but just asteroids on very similar orbits. For the time being, we will not consider this group as a confirmed fission cluster. Finally, obtaining information on the periods of two largest members of the two other clusters identified in this work, (89714) (2001 YA114) and (119786) (2002 AE65) could help understanding the nature of the two remaining possible fission clusters in the Zelima asteroid family.
Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement V. Carruba: Conceptualization, Methodology, Software, Formal analysis, Supervision, Funding acquisition, Visualization, Writing original draft, Writing - review & editing. J.V. Ribeiro: Validation, Visualization, Software, Investigation, Writing - review & editing. Acknowledgments We are grateful to the reviewers of this paper, Dr. Bojan Novakovic and Dr. Georgios Tsirvoulis, for useful comments and suggestions. We would like to thank the S~ao Paulo State Science Foundation (FAPESP) that supported this work via the grant 18/20999-6, and the Brazilian National Research Council (CNPq, grant 301577/2017-0). We are grateful for the use of data from the Asteroid Dynamics Site (AstDys) (http://hamilton.dm.unipi.it/astdys, Knezevic and Milani (2003)). This publication also makes use of data products from the Wide-field Infrared Survey Explorer (WISE) and Near-Earth Objects (NEOWISE), which are a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration.
5. Final considerations In this work we studied the very peculiarly case of the Zelima asteroid family. As far as we know, this is the only family for which all members are in librating states of the z1 secular resonance. Because of its extremely young age, the family members orbits are still clustered in the domain ðσ ; dσ =dtÞ, where σ ¼ ϖ ϖ 6 þ Ω Ω6 is the resonant argument of the z1 resonance. This fact permitted to determine the age of the family based
Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.pss.2019.104810.
4
Alternatively, other formation scenarios than those analyzed by (Pravec, 2010) could be at play. While the (Jacobson and Scheeres, 2011) theory used by (Pravec, 2010) is not compatible with the data of the (6733) (1992 EF) cluster, models like those of (Bottke et al., 2002), (Walsh et al., 2008), or (Vokrouhlický, 2017) could potentially be compatible with a longer rotation period of the primary after the cluster formation. Further research is needed to clarify this issue.
References Bottke W. F. Jr, Vokrouhlický D., Rubincam D. P., Broz M., 2002, Asteroid III, W. F. Bottke Jr, A. Cellino, P. Paolicchi, and R. P. Binzel (eds), University of Arizona Press, Tucson, 395. 10
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Planetary and Space Science xxx (xxxx) xxx Masiero, J.R., Mainzer, A.K., Grav, T., Bauer, J.M., Jedicke, R., 2012. Astron. J. 759, 14. Nesvorný, D., Bottke Jr., W.F., Dones, L., Levison, H.F., 2002. Nature 417, 720. Novakovic, B., Hsieh, H.H., Cellino, A., 2012. Mon. Not. R. Astron. Soc. 424, 1432. Pravec, P., and 25 colleagues, 2010. Nature 466, 1085. Radovic, V., Novakovic, B., Carruba, V., Marceta, D., 2017. Mon. Not. R. Astron. Soc. 470, 576. Ryan, E.L., Woodward, C.E., 2010. Astron. J. 140, 943. Spoto, F., Milani, A., Knezevic, Z., 2015. Icarus 257, 275. Gaia Collaboration: Spoto, F., Tanga, P., Mignard F. and other 448 colleagues, 2018, Gaia data release 2. Astron. Astrophys. 616, A13. Tsirvoulis, G., 2019. Mon. Not. R. Astron. Soc. 482, 2612. Vokrouhlický, D., Broz, M., Bottke, W.F., Nesvorný, D., Morbidelli, A., 2006. Icarus 183, 349. Vokrouhlický, D., and 12 colleagues, 2017. Astron. Astrophys. 598, A91. Walsh, K.J., Richardson, D.C., Michel, P., 2008. Nature 454, 188. Warner, B.D., Harris, A.W., Pravec, P., 2009. Icarus 202, 134. Zappala, V., Cellino, A., Farinella, P., Knezevic, Z., 1990. Astron. J. 100, 2030.
Broz, M., 1999. Diploma Thesis. In: Orbital Evolution of the Asteroid Fragments Due to Planetary Perturbations and Yarkovsky Effects. Charles University, Prague, Czech Republic. Carruba, V., 2009. Mon. Not. R. Astron. Soc. 395, 358. Carruba, V., 2010. Mon. Not. R. Astron. Soc. 408, 580. Carruba, V., Nesvorný, D., 2016. Mon. Not. R. Astron. Soc. 457, 1332. Carruba, V., Nesvorný, D., Vokrouhlický, D., 2016. Astron. J. 151, 164. Carruba, V., Vokrouhlický, D., Novakovic, B., 2018. Planet. Space Sci. 157C, 72. Carruba, V., De Oliveira, E.R., Rodrigues, B., Requena, I., 2018. Mon. Not. R. Astron. Soc. 479, 4815. Carruba, V., Spoto, F., Barletta, W., Aljbaae, S., Fazenda, A., Martins, B., 2019. Nat. Astron. (in press). Ishihara, D., and 32 colleagues, 2010. Astron. Astrophys. 514, A1. and 32 colleagues, 2001. Astron. J. 122, 2749. Ivezic, Z., Jacobson, S.A., Scheeres, D.J., 2011. Icarus 214, 161. Knezevic, Z., Milani, A., 2003. Astron. Astrophys. 403, 1165. Levison, H.F., Duncan, M.J., 1994. Icarus 108, 18.
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The Zelima asteroid family: Resonant configuration and rotational fission clusters V. Carruba *, J.V. Ribeiro S~ ao Paulo State University (UNESP), School of Natural Sciences and Engineering, Guaratinguet a, SP, 12516-410, Brazil
A R T I C L E I N F O
A B S T R A C T
Keywords: Minor planets Asteroids: general Minor planets Asteroids: individual: (633) Zelima Celestial mechanics
Asteroid families are groups of asteroids that are identifiable in domains of proper elements. They may be the outcome of a collision, or the product of rotational fission. Among collisional families, young asteroid families are of interest because they preserve information on the original ejection velocity field with which the asteroid fragments were expelled, and because they may host a significant population of clusters produced by rotational fission among their members. The recently identified Zelima asteroid family is unique because, apart from its very young age of less than 3 Myr, is also interacting with the z1 secular resonance, making it the third asteroid family in this peculiar configuration, after the Agnia and Padua families, and the only one for which all members are in z1 librating states. Because of its extremely young age, the orbits of its members are still very clustered in the domain ðσ ;dσ =dtÞ, with σ ¼ ϖ ϖ 6 þ Ω Ω6 the resonant argument of the z1 resonance. This allow for precisely dating the family with a method not available for any other main belt group. The Zelima family, based on this argument, should be 2:5 1:5 Myr old, and conserved quantities of the z1 secular resonance show that its ejection velocity field is in a range of 1:1þ0:6 0:2 of the escape velocity from the parent body, in agreement with general results for cratering collisional families. Finally, 41.7% of the members of the Zelima family are also members of possible fission clusters, confirming the trend observed for other very young families, like the Jones and Lorre groups.
1. Introduction Asteroid families are group of asteroids characterized by a close proximity in domains of proper elements. Among asteroid families, the group that formed most recently are of particular interest. Families younger than ’ 7 Myr did not evolve significantly because of the effect of non-gravitational forces such as the Yarkovsky effect. Contrary to some of the more evolved asteroid groups, it is therefore easier to obtain information on the original ejection velocity field that formed them (Carruba et al., 2018b). Recently (Tsirvoulis, 2019), identified the (633) Zelima group, a sub-family of the larger and more evolved Eos family, and used the backward integration method, or BIM, to estimate its age to be 2.9 0.2 Myr, so being the product of an extremely recent collisional event. In the BIM the differences of longitudes of nodes and pericenters between two asteroids that are thought to have a common origin are integrated into the past. At the time of the pair formation these differences should both
tend towards values near zero. The Zelima family is remarkable in several ways. All its members are in librating states of the z1 secular resonance, Families interacting with secular resonances are important because conserved quantities of secular dynamics allow to obtain information on the group initial ejection velocity field and age that are not easily available for other asteroid groups (Carruba et al. (2018a)). Until recently, only two families were known to have a majority of their members in resonant states of the z1 secular resonance,1 the Agnia (Vokrouhlický et al., 2006) and Padua (former Lydia) (Carruba, 2009) asteroid families. Only for the Zelima family, however, all the members are in librating z1 states, making it the first known asteroid family for which this phenomenon is observed. Also, in view of the extremely young age of this group, it is reasonable to expect that the resonant angles σ ¼ ϖ ϖ 6 þ Ω Ω6 and frequency ddtσ ¼ g g6 þ s s6 did not had enough time to disperse, unlike the cases of the more evolved Agnia and Padua families. This allows to apply an unique age dating method based on dispersion in a
* Corresponding author. E-mail address: [email protected] (V. Carruba). 1 Secular resonances occur when there is a commensurability between the precession frequency of the longitude of pericenter g or of the node s of an asteroid and a planet. The proper frequencies of planets are identified by a suffix, according to the planet distance with respect to the sun. For Saturn, the suffix is equal to 6. See also Carruba et al. (2018a) for a more in depth definition of secular resonances. https://doi.org/10.1016/j.pss.2019.104810 Received 18 June 2019; Received in revised form 21 November 2019; Accepted 25 November 2019 Available online xxxx 0032-0633/© 2019 Elsevier Ltd. All rights reserved.
Please cite this article as: Carruba, V., Ribeiro, J.V., The Zelima asteroid family: Resonant configuration and rotational fission clusters, Planetary and Space Science, https://doi.org/10.1016/j.pss.2019.104810
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σ ; ddtσ
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presence of possible new members with respect to the analysis of (Tsirvoulis, 2019). For this purpose, we looked for new members of the Zelima family in the Asteroid Families Portal (AFP (Radovic et al., 2017), accessed on September 30th, 2019) using the online HCM applicative of the site. In the Hierarchical Clustering Method (HCM), we first define a distance metrics in proper element domain and check if there are objects closer to the body of reference than a given distance. If new objects are found, the process is repeated using the new object as a reference, until no new members are encountered Zappala et al. (1990). Fig. 1 displays the number (left panel) and differential number (right panel, by differential number we mean the number of new family members for each increase of the distance cutoff) for members of the Zelima asteroid family, as found from the AFP. There are two plateaux in the number of family members encountered, at ’ 15 and ’ 23 m/s, which could be associated with possible choices of the optimal distance cutoff for this family. For a distance cutoff of 28 m/s the Zelima sub-family merges with the larger Eos family and it is no longer distinguishable as a separate entity. Using a too large value of distance cutoff may lead to overestimating the numbers of family members, including background Eos family asteroids not related with the Zelima sub-family. On the other hand, using a conservative value may lead to underestimating the real population of family members. To solve this problem, we took advantage of the young age of this family, and we tried to distinguish between background and family members by using BIM. Our approach was the following: we selected the numbered asteroids belonging to the Zelima family at the maximum possible cutoff value of 26 m/s. We then also checked for multiple-opposition possible members by searching for asteroids in 3σ level from the median values of proper ða; e; sinðiÞÞ of currently known Zelima members, using the approach of (Carruba and Nesvorný, 2016). This yielded a population of 51 potential members of the Zelima asteroid dynamical family. We then integrated these asteroids in the past and used the BIM to check for which objects the secular angles converged with respect to the parent body, as also done in the past for other young asteroid families by (Novakovic et al., 2012), which called this method the Selective Backward Integration
plane, that, so far, was not useable for any other family inter-
acting with secular resonances. Another way in which the Zelima family is important is because of the possible existence of fission clusters among its members. Recently, it has also been found that young families may produce a higher fraction of fission clusters than older families. Fission clusters are groups of asteroids that form because of the rotational failure of a parent body. Current models predict that fission clusters may form either because i) of the break-up of a rapidly rotating asteroid into multiple components (Bottke et al., 2002), ii) repeated accelerations up to the point of break-up of a fast spinning object caused by the YORP effect, i.e., the scattering of solar radiation off the surface of an irregularly shaped asteroid and the emission of its own thermal radiation, (Walsh et al., 2008), iii) the fission of a secondary body caused by spin-orbit coupling (Jacobson and Scheeres, 2011), or iv) a low-energy collision on a fast rotating primary (Vokrouhlický, 2017). (Carruba et al., 2019) recently found that the proportion of fission clusters in extremely young asteroid families is much higher than that observed in older families, suggesting that collisions may trigger a subsequent cascade of formations of fission clusters among family members. Again, in view of the extremely young age of this group, it may be reasonable to expect that a significant fraction of its members may be also part of fission clusters. Using the approaches used by (Carruba et al., 2019) for other young asteroid families, in this work we will investigate this hypothesis. This work is so divided: in the second section we will briefly revised the taxonomic and dynamical properties of the Zelima asteroid family. In the third section we will investigate the resonant behavior of this family, and see what information this may yield on the ejection velocity field of the group. Section four will be dedicated to the search for fission clusters inside the Zelima family. Finally, in section five, we will present our conclusions. 1.1. The Zelima family: taxonomic and dynamical properties As a first step in our analysis of the Zelima family, we checked for the
Fig. 1. The number (left panel) and differential number (right panel) of members in the Zelima dynamical family as a function of the velocity cutoff. 2
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120 in the ða; eÞ domain, in a range from 3.0082 to 3.0218 au in a and from 0.074 up to 0.1216 in e. The range in i was from 9:01∘ up to 13:77∘ . All the other initial osculating elements were those of (633) Zelima itself at 58400 MJD. The results of our maps in the two plans are shown in Fig. (3). Black dots displays the proper elements of test particles. Blue full circles identify members of the Zelima family, as originally identified by (Tsirvoulis, 2019), while red full circles displays the orbits of the newly identified possible family members. With the exception of (4077) Asuka, most new members are on orbits quite similar to those of the previously known Zelima asteroids. Mean-motion resonances, such as the 8J:-3S-3A three body resonance known to affect four members of the Zelima family (Tsirvoulis, 2019), are identified as a red vertical line. The main dynamical feature affecting the Zelima asteroid family is, as observed by (Tsirvoulis, 2019), the z1 ¼ g g6 þ s s6 secular resonance. Particles whose frequencies g þ s is in the range g6 þ s6 0:1 arcsec/yr, i.e., those most likely to be in z1 secular resonance librating states, are shown as yellow circles in Fig. (3, panel (a) and (b)). Indeed, most members of the Zelima family are observed to be in the affected region. The implications of this fact will be discussed in the next section.
Method, and by (Carruba et al., 2016). Fig. (2) displays values of the differences in longitudes of the nodes between a possible member and the alleged parent body (633) Zelima (ΔΩ ¼ Ω ΩPB ). Panel (a) shows this differences for the population of 23 asteroids originally identified as members by (Tsirvoulis, 2019). Panel (b) shows the same, but for all 51 asteroids studied in this work. Among these asteroids, we selected those with values of both ΔΩ and Δϖ to within 30∘ from the mean values of ΔΩ and Δϖ of the previously known 23 members of the Zelima asteroid family at the estimated time of family formation, 2.9 Myr ago. 10 new asteroids satisfied this selection criteria, values of their ΔΩ are shown in red in Fig. (2), panel B. For the sake of brevity, we do not display values of Δϖ. The new possible members of the Zelima family are (4077) Asuka, (243212) (2007 UQ48), (325541) (2009 SG58), (489627) (2007 TJ334), (2005 YM141), (2013 YF84), (2015 BL90), (2015 FB407), (2016 BB100), and (2016 CJ291). Among the multi-opposition asteroids, only (2005 YM141) has a Lyapunov exponent of less than 50 106 Myr1 , with all the other multi-opposition asteroids on more chaotic orbits. While the first asteroid in the new sample satisfies the selection criteria and has also an albedo compatible with those of the rest of the family (see discussion below), it is a relative large asteroid (H ¼ 11:00, while (633) Zelima itself has H ¼ 9:73), somewhat more distant from the rest of the family in both proper a and i. It is possible that this could be a false positive identification, but it will be kept in our sample for the time being. We then revised the physical properties of family members. As discussed by (Tsirvoulis, 2019), taxonomic information is only available for (633) Zelima itself, which is a S-type object. No information for any other family member is available in the Sloan Digital Sky Survey-Moving Object Catalog data (SDSS-MOC4 (Ivezic, 2001)). Four objects had albedo data in the WISE and NEOWISE, AKARI, or IRAS databases ((Masiero et al., 2012), (Ishihara, 2010), (Ryan and Woodward, 2010)) and satisfy the good signal-to-noise ratio criteria of (Spoto et al., 2015). These four asteroids have albedo values higher than 0.12, compatible with a S-complex taxonomy. The median value of the albedo is 0.15 0.3. To study the dynamics of the Zelima family we obtained dynamical maps in domains of synthetic proper elements using the approach of (Carruba, 2010). For both the maps in the ða; eÞ and ða; sinðiÞÞ domains we integrated 4200 test particles under the influence of all planets with SWIFT-MVSF, the integrator from the SWIFT package (Levison and Duncan, 1994), modified to account for online filtering of the osculating orbital elements (Broz, 1999). The test particles covered a grid of 35
2. The Zelima family resonant nature Tsirvoulis (2019) recently identified the resonant nature of the Zelima asteroid family and its interaction with the z1 ¼ g g6 þ s s6 secular resonance. This behavior is confirmed in this work, all of the numbered members of the Zelima asteroid families are currently in librating states of the z1 secular resonance. Fig. (4) displays the resonant argument of the asteroid with the lowest librating amplitude around 180∘ , (421572) (2014 OP193), and of an asteroid on a circulating orbit very close to the resonance separatrix, (1861) Komensky, that was already identified as an interloper by Tsirvoulis (2019). All the other 26 numbered members of the Zelima group are all showing libration of the resonant argument of the z1 , with varying oscillation amplitudes. However, none of the six multi-opposition asteroids potential members of the Zelima family identified in the previous section is currently librating. Based on this consideration and on their chaotic orbits, from now on, none of them will be further considered in our analysis. Overall, the Zelima asteroid family is the first family in the main belt for which all known confirmed members are in z1 librating states. This fact, and its extremely young age, allow for a unique age determination approach that will be discussed in the next sub-section.
Fig. 2. Values of ΔΩ for the original population of 23 family members, as identified by (Tsirvoulis, 2019) (panel a), and for a sample of 51 potential members of the family, identified with the criteria discussed in this work (panel b). The vertical dashed line shows the family age, as estimated by (Tsirvoulis, 2019). Horizontal dashed lines displays the 60∘ levels in ΔΩ. Red curves in panel (b) are associated with the ΔΩ values of 10 new potential family members, according to the criteria discussed in the text. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.) 3
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Fig. 3. Dynamical maps in the proper ða; eÞ (panel a) and ða; sinðiÞÞ (panel b) domains for the orbital region of the Zelima family. Black dots show values of synthetic proper elements of test particles in our maps. Yellow full circles display the location of particles likely to be in z1 resonant states, according to the criterion discussed in the text. Blue full circles show the orbital location in the proper element domain of members of the Zelima family as identified by (Tsirvoulis, 2019), while red full circles displays the orbits of the possible new members identified in this work. The vertical red line displays the position of the 8J:-3S-3A three-body resonance. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
2.1. Age estimate from dispersion in the
σ ; ddtσ
frequencies and phases of the asteroid and Saturn were obtained every 1.23 Myr with the procedure discussed in Vokrouhlický et al. (2006), Carruba (2009), that involved a numerical simulation of the asteroid over 20 Myr, under the gravitational influence of all planets. Among the numbered potential new members of the Zelima family identified in the previous section, only (489627) (2007 TJ334) has values of ðσ ; dσ =dtÞ compatible with those of the rest of the family (displayed as a red dot in Fig. (5)) and will be considered as a proper family member hereafter. Contrary to the cases of the Padua and Agnia families, the Zelima family is still quite compact in the ðσ ; dσ =dtÞ domain. For other resonant groups, this information is, however, lost in timescales of the order of ’ 10 Myr. To quantify how tight the family is in this plane, and to try to obtain a family age estimate, we follow the procedure first introduced by Vokrouhlický et al. (2006) for the Agnia family. We use a polar angle Φ in the ðσ ; dσ =dtÞ, obtained using a scaling that maps a ð0∘ ; 360∘ Þ interval of σ and (1,1) “/yr interval in dσ =dt ¼ g þ s g6 s6 into common intervals (1,1). At each time step, the dispersion DΦ can be computed by the
plane
An useful tool when studying asteroid families characterized by their interaction with the z1 secular resonance is the distribution of the members in the ðσ ; dσ =dtÞ plane of the resonance, where σ ¼ ϖ ϖ 6 þ Ω Ω6 is the resonant argument of the z1 resonance, and dσ = dt ¼ g g6 þ s s6 . Most families are thought to have started with a compact distribution of values in the ðσ ;dσ =dtÞ. Points in this planes will describe circular orbits around σ ¼ 180∘ in timescales of the order of 1–2 Myr. Regrettably, after a few orbit cycles the alignments of family members in this plane is lost, and asteroids tend to reach a uniform distributions in less than 10 Myr. This is indeed the case for the already discussed cases of the Agnia and Padua families, both characterized by a uniform distribution of values in this plane. The Zelima family, however, is unique because of both its resonant nature and young age. Fig. (5) displays projections of family members as identified by (Tsirvoulis, 2019) at T ¼ 1:23 Myr (panel a) and T ¼ 20:89 Myr (panel b). The values of proper
Fig. 4. Librating argument ϖ ϖ 6 þ Ω Ω6 of the z1 secular resonance for the (421572) (2014 OP193) (panel a), and for (1861) Komensky (panel b). 4
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Fig. 5. The dispersion in the ðσ ; dσ =dtÞ plane of 23 Zelima family members numerically simulated at T ¼ 1:23 Myr (panel a) and T ¼ 20:89 Myr (panel b), shown as black full circles. The red full triangle identify the position of (489627) (2007 TJ334), the only asteroids among the potential candidates whose position in the ðσ ; dσ =dtÞ plane is in the same region as the remaining members of the Zelima family. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
integrated 225 particles, which is the value of N in Eq. (1). We simulated the orbital evolution of particles in the future and in the past. For the past integrations, we used negative velocities for both planets and particles, and re-scaled the longitudes of nodes and pericenters according to the relationships (Eqs. 2 and 3):
equation: < ΔΦ>2 ¼
X 2 1 Φi Φj ; NðN 1Þ i6¼j
(1)
where N is the number of integrated bodies. For each family members we integrated nine sets of clones. We considered the central value of eccentricity and inclination, plus or minus their orbital uncertainty, obtained from the AstDyS (https://newton.spacedys.com/astdys/, accessed on May 28th, 2019, Knezevic and Milani (2003)). Each value of eccentricity (minimum, central, maximum) is associated to the three possible values of inclination, yielding nine possible combinations. Overall, we
Ω ¼ Ω þ π
(2)
ϖ ¼ Ω ω:
(3)
An uniform distribution of bodies in this polar angle corresponds to < ΔΦ >¼ 103∘ (Vokrouhlický et al., 2006), while the current observed value of < ΔΦ > for the Zelima family is 19:6∘ . If we use the current value of < ΔΦ > as a limit, the estimated age of the Zelima family according to this method would be 2:5 1:5 Myr, in agreement with (Tsirvoulis, 2019) estimate of 2:9 0:2 Myr (see Fig. 6). The distribution in < ΔΦ > would become uniform in the interval 2:5þ6:5 4:3 Myr, which is the upper limit on the age estimate that this method can provide. Overall, it is the first time that a method based on the dispersion of orbits in the ðσ ; dσ =dtÞ domain can provide a rather precise age estimate for a family interacting with the z1 secular resonance. While this age estimate is not as precise as results obtained with the BIM approach,2 we believe that successfully using this approach is an important result for the field of secular dynamics. Founding new extremely young asteroid families interacting with secular resonances other than the z1 could be very valuable to confirm, or not, the validity of this approach. In the next subsection we will investigate the initial ejection velocity
2 We should also point out that, since this method is based on the convergence of values of σ and dσ =dt, and since σ ¼ ϖ ϖ 6 þ Ω Ω6 for the z1 secular resonance, the convergence of σ values is not completely independent from the convergence of the secular angles < ΔΩ and Δϖ required by BIM. Since the ðσ ; dσ =dtÞ method also require convergences of frequency values, that generally do not happen for non-resonant cases, the two methods are not completely equivalent. Yet, it would be interesting if such method could be tested for families in secular resonances other than the z1 , so as to fully test the validity of this approach. Regrettably, currently the Zelima family is the only asteroid group inside a secular resonance with an age of less than 4.0 Myr, as far as we know.
Fig. 6. The time behavior of the dispersion in the Φ polar angles for 207 clones of Zelima family members. The horizontal lines display the current value of the dispersion for real Zelima members, and the < ΔΦ >¼ 103∘ level corresponding to the case of an uniform distribution of bodies. 5
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field of the Zelima asteroid family, and the information that secular dynamics can provide on this subject.
optimal VEJ parameter. For this purpose, we generate several synthetic asteroid families with different values of the VEJ parameter, compute their values of the K ’2 quantity, and assess which of the simulated family is most compatible with the currently observed K ’2 distribution. To quantitatively test this, we can use a χ 2 -like variable defined as (Eq. 6):
2.2. Conserved quantities of the z1 resonance and implications for the initial ejection velocity field (Vokrouhlický et al., 2006) showed that for asteroids in librating 0 states of the z1 secular resonance the quantity K 2 , defined as (Eq. 4): pffiffiffiffiffiffiffiffiffiffiffiffiffi K ’2 ¼ 1 e2 ð2 cosðiÞÞ;
χ2 ¼
(6)
where Nint ¼ 10 is the number of interval used for the values of K ‘2 , qi is the number of real objects in the i th interval in K ‘2 , and pi is the number of simulated family members in the same i th interval. We can then compute the Δχ 2 ¼ χ 2 χ 2min , which is the difference between values of χ 2 and their minimum, and use the χ 2 probability distribution function, given by (Eq. 7):
(4)
is preserved, even when the Yarkosvky force is considered (Carruba, 2009). confirmed this for members of the Padua family in z1 librating 0 states. Fig. (7) displays a histogram of the current distribution of K 2 for Zelima family members (panel a). As a consequence of the preservation of this quantity, proper e and i are anti-correlated. Panel (b) of the same figure show the range of values covered by these quantities for a simulation over 20 Myr. Proper elements were obtained with the method described in (Carruba, 2010). Green full dots show the current positions in the ðe; sinðiÞÞ domain of the Zelima asteroids. The preservation of the original values of the K ’2 quantity allowed to obtain estimates of the initial ejection velocity field that are not available for families not interacting with secular resonances. Here we will attempt to obtain such estimates using the (Vokrouhlický et al., 2006; Carruba, 2009) approach for the case of the Zelima asteroid family. As a first step, we define a simple model for the ejection velocity field of asteroids just after the formation of the family. In this approach we assume that asteroids are ejected with an isotropic ejection velocity field whose standard deviations VSD depends on the asteroid size through a parameter VEJ according to the relationship (Eq. 5): VSD ¼ VEJ ð5km = DÞ;
Nint X ðqi pi Þ2 ; qi i¼1
F x; Ndf ¼
γ Γ
Ndf x ; 2 2
;
(7)
Ndf 2
where x ¼ 0:975 stands for the chosen confidence level, γ N2df ; 2x is the lower incomplete gamma function, and Γ N2df is the gamma function. Ndf ¼ 10 1 ¼ 9 is the number of degrees of freedom, which is given by the number of intervals used to estimate the χ 2 variable (10, in our case), minus the number of parameters estimated, which is one, VEJ , in this application of this method. The value of Δχ 2 consistent with a 97.5% confidence level for nine degrees of freedom is 2.700. Our results are shown in Fig. (8), where in panel (a) we display a histogram of the current distribution of K ’2 values in red, and of the simulated ones that best-fitted the real data, in blue. Panel (b) shows the dependence of Δχ 2 as a function of the VEJ parameter. This analysis yields a value of VEj of 20:0þ3:3 10:4 m/s, which is consistent with the escape velocity from (633) Zelima, that is equal to 17.9 m/s. According to (Carruba and Nesvorný, 2016), most asteroid families have values of VEj ¼ β Vesc that are less than 1.5 the escape velocity from the parent body Vesc , and our value of β ¼ 1:1þ0:2 0:6 , where the errors on β are
(5)
where VEJ is the value of the ejection velocity that we used for the whole family, and D is the asteroid diameter. This method may not provide a realistic model of the actual ejection velocity field of the Zelima family, since (Tsirvoulis, 2019) showed that the actual ejection velocity field is rather asymmetric, as observed for several other cratering asteroid families. Nevertheless, it may provide useful constraints on the magnitude of the original ejection velocity field though an estimate of the
Fig. 7. A histogram of the distribution of the K ’2 conserved quantity of the z1 secular resonance for Zelima family members (panel a). Panel (b) displays the time evolution of proper sinðiÞ versus proper e for the same family members, as blue lines. The green full dots show the current values of e and sinðiÞ of Zelima family members, as identified by (Tsirvoulis, 2019). The red full triangle displays the location of the new possible family member identified in this work. 6
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Fig. 8. Panel (a) show a histogram of current values of the K ’2 quantity (in red) and of the simulated family that best-fitted the data. Panel (b) displays the dependence of Δχ 2 as a function of the VEJ parameter. The horizontal dashed line displays the 97.% confidence level limit. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
identification of fission clusters inside four very young asteroid families, those of (3152) Jones, (7353) Kazuya, (5438) Lorre and (108138) (2001 GB11), so that interested readers can find more details on these methods in that paper. While BIM as already been discussed in the introduction, in CEM, several clones of both the parent body and the other pair object, accounting for the orbital uncertainties and for non-gravitational effect such as the Yarkovsky force are integrated into the past and the time of occurrence of a close encounter between clones of the parent body and the other asteroid that occur at relative speeds and distances smaller than given parameters are registered. The median value of this time provide and estimate of the possible time of separation, while the 5th and 95th percentile of the distribution allow to assess the errors on this age. For illustrative purposes, Fig. (9) displays results of BIM and CEM for asteroid pairs in the Zelima family that were not likely to have formed after the formation of the asteroid family (left panels (a) and (c)), and that are strong candidates to belong to fission clusters that formed afterward (right panels (b) and (d)). We first determined all the fission clusters candidates that satisfy our dmean and q criteria. We then studied them with BIM, without considering any non-gravitational forces, in a purely conservative frame. Since the actual convergence of angles should also account for the Yarkovsky effect (see for instance Carruba et al. (2016) for an application of this method to members of the Karin cluster), and since key parameters of this effect such as the asteroid density and thermal conductivity are not generally known for Zelima family asteroids, we require at this stage of the investigation that the convergence of the angles in the conservative case yields an age estimate less or equal to the Zelima estimated family age of 2.9 Myr, but not superior, like the case of Fig. (9, panel c). For the objects that passed these two phases of our study, we then applied CEM, which provides the most robust results for identifying possible members of rotational fission clusters. A possible pair is identified by CEM if it has a nominal age less than the estimated primary family age. For Zelima family members we could not find any fission cluster between (633) Zelima itself and other family members that satisfied these selection criteria. We found, however, 16 possible pairs among the other family members, that could be members of what (Carruba et al., 2019) defines as tertiary families, i.e., groups that formed after the main family formation event and that do not involve the primary body. Table (1) displays our results for the pairs that were confirmed by both BIM and CEM. The pairs (1861) Komensky and (456020) (2005 YU83),
evaluated considering the upper and lower limit on VEj and dividing these values by the escape velocity, fits in this range. In the next section we will start analyzing the population of fission clusters inside the Zelima asteroid family. 3. Zelima fission clusters population Fission clusters are groups of asteroids extremely close in proper elements domains that may originate because of several proposed mechanisms of asteroid rotational failure, as discussed in the article introduction. Recent studies have shown that young collisional families may produce a higher proportion of young fission clusters than more evolved collisional groups (Carruba et al., 2019). In view of the extreme young estimated age of the Zelima family, in this section we will explore the occurrence of fission clusters among their members. For this purpose, we use the same criteria discussed in (Carruba et al., 2019). We use a distance dmean between a pair of asteroids according to the metric (Eq. 8): rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 δa2 þ 2ðδeÞ2 þ 2ðδ sin iÞ2 ; 4 a
dmean ¼ na
(8)
where ðδa; δe; δ sin iÞ is the separation vector in proper elements, a is the mean value of the pair semi-major axis, and dmean is the standard distance metric as defined in (Zappala et al., 1990). Following (Pravec, 2010), we also use a mass ratio q using the relationship (Eq. 9): 3 q ¼ 10ðδH=5Þ ;
(9)
where δH is the difference in absolute magnitude between the pairs, and where the geometric albedo of the pairs is assumed to be equal. Asteroids are considered good candidates to be part of a cluster if their difference in dmean is less than 5 m/s and if their mass ratio q is less than 0.3, the maximum theoretical and practical limit observed for asteroid pairs by (Pravec, 2010). Once the asteroids more likely to belong to a cluster have been selected, they are then studied with methods based on time-reversal numerical integration, like the backward integration method (BIM, Nesvorný et al. (2002)), and the close encounters method (CEM, Pravec et al. (Pravec, 2010)), to verify if they really belong to a fission cluster, or just share similar orbits. Both BIM and CEM were recently used in (Carruba et al., 2019) for the
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Fig. 9. Panel (a) displays the time behavior of ΔΩ and Δϖ for the pair of asteroid (6733) (1992 EF) and (449004) (2012 BY59), for which there is no convergence of the angles before the estimated family formation event, and for the pair (119786) (2002 AE65) and (368472) (2003 SQ191) (panel b), that are most likely to have formed in a subsequent fission pair formation event. Panels (c) and (d) displays the results of CEM for the same asteroid pairs, respectively.
Table 1 The table report the cluster parent body, its members identification, their absolute magnitude, the mass ratio q, the distance in m s1 with respect to the parent body, the number of encounters experienced over the 7 Myr long integration, and the estimated time of separation, with its uncertainty, for the case of the (633) Zelima asteroid family. Cluster
Cluster
H
q
dmean 1
[m s
Number of
P. body
member
enc.
[Myr]
6733
324175
15.9
0.01
2.16
2027
2:855þ3:830 2:547
6733
364004
15.8
0.01
1.82
9071
89714
250011
15.5
0.21
4.38
7505
89714
292616
15.5
0.24
3.33
3748
89714
402372
16.0
0.09
3.88
2727
89714
412371
16.3
0.07
3.76
3101
89714
456020
16.4
0.04
4.67
5352
119786
368472
15.6
0.24
3.58
16352
119786
449004
16.8
0.06
3.46
12786
119786
487314
16.4
0.10
1.82
2885
8
]
Tsep
0:143þ4:964 0:099 0:104þ5:000 0:065 0:192þ5:604 0:162 1:531þ4:935 1:451 1:431þ5:032 1:349 0:188þ5:636 0:152 0:037þ3:402 0:023 0:151þ5:694 0:148 1:613þ4:918 1:565
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Fig. 10. A ða; eÞ (panel a) and ða; sinðiÞÞ (panel b) projection of members of the Zelima family (black circles) and of possible members of fission clusters (yellow full circles). The size of the circles is inversely proportional to the asteroid absolute magnitude according to the figure legend. Other symbols are the same as in Fig. (3). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
and (364004) (2005 UH450) and (449004) (2012 BY59) were not confirmed by BIM and were discarded for an analysis with CEM. (1861) Komensky could also be an interloper, according to Tsirvoulis (2019). Overall, 10 asteroid pairs could be viable candidates to be formed as rotational failures of Zelima family members (or their satellites) after the family formation event. Our results are shown in Fig. (10), where we display an ða; eÞ (panel a) and an ða sinðiÞÞ (panel b) projections of the members of the possible identified tertiary fission clusters (full yellow circles). This analysis, therefore, suggests that up to 54.2% of Zelima members may also be members of rotational fission clusters. Uncertainties on the ages of these possible clusters are quite large, and an origin from the main collisional event that formed the family cannot be completely excluded for any of these asteroids, at this stage of the investigation. High-quality astrometric Gaia data (Spoto et al., 2018) could, in the future, be used to reduce these uncertainties (Carruba et al., 2019), but, unfortunately, at the moment such data is not available for any of the asteroids listed in Table (1). Our results are compatible with the possible presence of three fission clusters, but their existence cannot be yet positively confirmed. For two asteroids, (633) Zelima and (6733) (1992 EF), information is available on their rotation periods. Such information can be used to assess if the groups to which these asteroids belong can be classified as collisional or fission clusters. In the next section we will attempt to do that.
Fig. 11. The position in a ðlog10 ðσ a Þ; ageÞ diagram of the Zelima family (red lines) and its fission clusters (black lines) studied in this work, with their errors. The yellow region displays the area where most collisional families are found. The green area shows the boundaries of the region where fission clusters are more common, as defined in (Carruba et al., 2018b). The dashed horizontal line displays the maximum nominal age found for most rotational fission cluster so far. Data for the fission clusters here identified are available in Table (1). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
4. Cluster classification First, we checked how the clusters identified in the previous section are located in the ðlog10 ðσ a Þ; ageÞ diagram that was introduced by (Carruba et al., 2018b) to classify and distinguish between collisional families and fission clusters. Collisional families tend to form with larger velocities at infinity than rotational fission clusters, and, therefore, tend to have larger values of σ a (Carruba et al., 2018b). identified two regions in ðlog10 ðσ a Þ; ageÞ plane shown in green and yellow, were at the time, most fission clusters and most collisional families were found. Fig. (11) displays our results in this domain. The three clusters all have values of log10 ðσ a Þ compatible with those of other fission clusters, so confirming their origin. Interestingly enough, so are the results for the whole Zelima family, identified in red in the figure, that are more compatible with dispersions associated with fission clusters than with values observed for young collisional families (Carruba et al., 2018b). Is the whole Zelima family actually a fission cluster? To answer this question, we turn our attention to the theory devel-
oped by (Pravec, 2010) to classify fission clusters based on their mass ratio and parent body rotation period. Fig. (12) shows the positions in the ðΔH; P1 Þ diagram of the (633) Zelima family and of the (6733) (1992 EF) cluster, the only groups for which information on the rotation periods of the largest body is available.3 The black dashed line is associated with the optimal values of the parameters for which the theory described in (Pravec, 2010) predicts the formation of a fission cluster: a system scaled angular momentum αL ¼ 1:0, a primary axis ratio a1 =b1 ¼ 1:4 and an initial orbit’s normalized
3 The period of (633) Zelima is 11.730.01 h, while the period of (6733) (1992 EF) is 11.360.01 h (Warner et al., 2009).
9
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on the dispersion of a polar angle φ in the ðσ ; dσ =dtÞ domain, a method that, so far, only precisely worked for this family. The age predicted by this method, 2:5 1:5 Myr, is in agreement with independent estimates obtained by other authors (Tsirvoulis, 2019). 0 A study of the conserved quantity of the z1 secular resonance, K 2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 e2 ð2 cosðiÞÞ, allowed to estimate that the VEj parameter describing the ejection velocity field of the Zelima family is of the order 1:1þ0:6 0:2 of the estimated escape velocity from the Zelima family parent body, (633) Zelima itself, in agreement with results for other asteroid families that show that most families have VEj values lower than 1.5 the parent body escape velocity (Carruba et al., 2019). Finally, using methods based on backward integrations of asteroid orbits, we identified three possible clusters inside the Zelima asteroid family that could be compatible with a subsequent origin caused by rotational fission: the cluster of (6733) (1992 EF) with 3 members, that of (89714) (2001 YA114) with 6 members, and that of (119786) (2002 AE65) with 4 members. Information on the rotation period of (633) Zelima suggests that the family, according to the theory presented in (Pravec, 2010), should have originated from a collision, despite its unusual very compact distribution in proper a. The rotational period of (6733) (1992 EF) appear to be incompatible with an origin as fission cluster for this group. Obtaining more precise astrometric data on the orbits of the members of the remaining possible clusters identified in this work, as well as information on the orbital periods of (89714) (2001 YA114) and (119786) (2002 AE65) could help clarify if these two clusters indeed formed as results of a rotational failure after the collision that formed family 2.90.2 Myr ago.
Fig. 12. The positions in the ðΔH; P1 Þ diagram of the (633) Zelima family (red lines) and of the (6733) (1992 EF) cluster (black lines), with their uncertainties. The blue and red lines display the lower and upper limiting cases for clusters formed by a fission event, while the dashed line shows the optimal case (see discussion in the text). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)
semi-major axis Aini =b1 ¼ 3. The red and blue curves relate to the upper and lower limiting cases, respectively: αL ¼ 1:2, a1 =b1 ¼ 1:2, and Aini = b1 ¼ 2 for the upper curve, while αL ¼ 0:7, a1 =b1 ¼ 1:5, and Aini = b1 ¼ 2 for the lower curve. More information on the rationale for the choice of these parameters is available in (Pravec, 2010). Neither group is in a region comparable with an origin from rotational fission. This suggest that the Zelima family may indeed be the outcome of a collision, regardless of its unusually small value of σ a and that the (6733) (1992 EF) cluster could either not be a real fission cluster,4 or be the outcome of a more recent collision. Considering that i) CEM results for the pair (6733) (1992 EF) and (324175) (2006 AU14) are more compatible with an origin at the same time of the family formation event, and that ii) only the pairs (6733) (1992 EF) and (364004) (2005 UH450) is more robust in terms of CEM results, the more likely explanation appears to be that this could not be a real cluster, but just asteroids on very similar orbits. For the time being, we will not consider this group as a confirmed fission cluster. Finally, obtaining information on the periods of two largest members of the two other clusters identified in this work, (89714) (2001 YA114) and (119786) (2002 AE65) could help understanding the nature of the two remaining possible fission clusters in the Zelima asteroid family.
Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. CRediT authorship contribution statement V. Carruba: Conceptualization, Methodology, Software, Formal analysis, Supervision, Funding acquisition, Visualization, Writing original draft, Writing - review & editing. J.V. Ribeiro: Validation, Visualization, Software, Investigation, Writing - review & editing. Acknowledgments We are grateful to the reviewers of this paper, Dr. Bojan Novakovic and Dr. Georgios Tsirvoulis, for useful comments and suggestions. We would like to thank the S~ao Paulo State Science Foundation (FAPESP) that supported this work via the grant 18/20999-6, and the Brazilian National Research Council (CNPq, grant 301577/2017-0). We are grateful for the use of data from the Asteroid Dynamics Site (AstDys) (http://hamilton.dm.unipi.it/astdys, Knezevic and Milani (2003)). This publication also makes use of data products from the Wide-field Infrared Survey Explorer (WISE) and Near-Earth Objects (NEOWISE), which are a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration.
5. Final considerations In this work we studied the very peculiarly case of the Zelima asteroid family. As far as we know, this is the only family for which all members are in librating states of the z1 secular resonance. Because of its extremely young age, the family members orbits are still clustered in the domain ðσ ; dσ =dtÞ, where σ ¼ ϖ ϖ 6 þ Ω Ω6 is the resonant argument of the z1 resonance. This fact permitted to determine the age of the family based
Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.pss.2019.104810.
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Alternatively, other formation scenarios than those analyzed by (Pravec, 2010) could be at play. While the (Jacobson and Scheeres, 2011) theory used by (Pravec, 2010) is not compatible with the data of the (6733) (1992 EF) cluster, models like those of (Bottke et al., 2002), (Walsh et al., 2008), or (Vokrouhlický, 2017) could potentially be compatible with a longer rotation period of the primary after the cluster formation. Further research is needed to clarify this issue.
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