The heart of the swarm: K2 photometry and rotational characteristics of 56 Jovian Trojan asteroids
Gy. M. Szabó, A. Pál, Cs. Kiss, L. L. Kiss, L. Molnár, O. Hanyecz, E. Plachy, K. Sárneczky, R. Szabó
aa r X i v : . [ a s t r o - ph . E P ] S e p Astronomy & Astrophysics manuscript no. aa˙troj˙rs1 c (cid:13)
ESO 2018September 14, 2018
The heart of the swarm: K2 photometry and rotationalcharacteristics of 56 Jovian Trojan asteroids
Gy. M. Szab´o , , A. P´al , , Cs. Kiss , L. L. Kiss , , L. Moln´ar , O. Hanyecz , , E. Plachy , K. S´arneczky ,and R. Szab´o ELTE Gothard Astrophysical Observatory, 9704 Szombathely, Szent Imre herceg ´utja 112, Hungary Konkoly Observatory, MTA Research Centre for Astronomy and Earth Sciences, Konkoly-Thege Mikl´os ´ut 15-17,1121 Budapest, Hungary; e-mail: [email protected] Department of Astronomy, Lor´and E¨otv¨os University, P´azm´any P´eter s´et´any 1/A, 1117 Budapest, Hungary Sydney Institute for Astronomy, School of Physics A28, University of Sydney, NSW 2006, AustraliaReceived September 14, 2018; accepted September 14, 2018
ABSTRACT
We present fully covered phased light curves for 56 Jovian Trojan asteroids as acquired by the K2 mission of the
Kepler space telescope. This set of objects has been monitored during Campaign 6 and represents a nearly unbiasedsubsample of the population of small Solar System bodies. We derived precise periods and amplitudes for all Trojans,and found their distributions to be compatible with the previous statistics. We point out, however, that ground-basedrotation periods are often unreliable above 20 h, and we find an overabundance of rotation periods above 60 h comparedwith other minor planet populations. From amplitude analysis we derive a rate of binarity of 20 ± ∼ ∼ − cometary-likedensity limit, also suggesting a high internal porosity for Jovian Trojans. One of our targets, asteroid 65227 exhibits adouble rotation period, which can either be due to binarity or the outcome of a recent collision. Key words.
Techniques: photometric
1. Introduction
Trojan asteroids are located at a heliocentric distance of ∼ . populations are suggested, with different origin and different evolution,instead of simply different taxonomical types which weremixed together (Wong et al. 2014, 2015). However, objectsin these populations are still significantly bluer than thetypical “red” objects in the Centaur and trans-Neptunianpopulations (Peixinho et al., 2012; Lacerda et al., 2014). Acommon, outer Solar system origin of Jovian Trojans andtrans-Neptunian objects was recently proposed by Wong &Brown (2016), also suggesting that the retention or loss ofH S in the early Solar system was the likely reason behindthe colour differences we observe today.In addition to the photometric characteristics discussedabove, light curves available for Jovian Trojans can pro-vide shape and rotational frequency distributions, and alsoinformation on the binary fraction. These statistics can becompared with the prediction of various formation and evo-lution models. In this sense, binary or multiple systems es-pecially important, as their observations provide reliablemasses and densities, a key to composition and internalstructure. There are several formation mechanisms pro-posed for multiple systems (see e.g. Merline et al., 2002;
Fig. 1.
Upper panel: The upper view of Field 6 Trojans(large black dots) superimposed to the SDSS MOC3Trojans in the L4 swarm (smaller grey dots) projected tothe Jupiter’s orbital plane. Earth is in origin. Note thatCampaign 6 pointed exactly into the core of the L4 swarm.Lower panel: The field-of-view of K2 Campaign 6 super-imposed with the apparent trajectories of the 56 JupiterTrojans discussed in this paper. On each trajectory, thedots show the motion in 5-days long steps.Noll et al., 2008), therefore their properties give an impor-tant clue on accretional, collisional and radiative processesas well, and may lead to identify differences between thered and the less red groups in the case of Jovian Trojans. AWISE survey found that 20% of Jovian Trojan asteroids areeither extremely elongated objects, or are binaries (Sonnettet al., 2015).In the K2 mission of the Kepler space telescope (Howellet al. 2014) 56 Jovian Trojan asteroids have been observedin Campaign 6, and long, uninterrupted light curves havebeen taken that are free from aliases, giving a more compre-hensive description of these populations then the sparselysampled WISE or ground based data. In this paper wepresent the light curves and photometric properties of these56 Jovian Trojan asteroids, all orbiting at the L4 Lagrangepoint of the Sun-Jupiter system. In Sec. 2 we summarizethe observations and the data reduction schemes used toobtain the light curves of these Trojan asteroids. In Sec. 3we present the statistical properties of this sample. Our re-sults are summarized in Sec. 4. Tabulated data and the lightcurves of individual Trojans are shown in the Appendix. Asimilar study about Main Belt asteroids with K2 will bepublished in a related paper (Szab´o et al., 2016).
Fig. 2.
Typical stamps taken from the image series re-lated to (12974) Halitherses, which was the object with thefastest apparent speed (exceeding 2 pixels per long-cadenceframe). The upper row of image stamps shows the originalframes between the cadences × . ′ × . ′ on the sky.
2. Observations and data reduction
Kepler has observed parts of the apparent trajectories of56 Trojans during K2 Campaign 6, between 2015 July13.95 and 2015 September 30.87 (UTC). In total, Campaign6 data are formed from 3686 long cadence frames andthe stamps allocated for these 56 Trojan asteroids havealso been retrieved in long cadence mode, correpsondingto 29.41 minute sampling frequency). To date, all of themoving objects observed by K2 were trans-Neptunian ob-jects (P´al et al., 2015a; Kiss et al., 2016; P´al et al., 2016)whose apparent speed in the K2 CCD frames were so smallthat during the photometric analysis, these objects can betreated as point sources on the individual long cadenceframes. However, Trojans of Jupiter are located signifi-cantly closer to the inner Solar System, i.e. to K2 itself.Hence, the apparent speed can exceed 0 . ′′ / min, yieldingtrail-like corresponding PSFs on the long cadence frames.As we will describe later on (Sec. 2.2), the photometry of N u m be r o f ob j e c t s Average trail length (pixel/frame) 0 5 10 15 20 25 0 5 10 15 20 25 N u m be r o f ob j e c t s Light curve time coverage (days) 100 200 300 400 500 600 700 800 900 1000 5 10 15 20 N u m be r o f li gh t c u r v e po i n t s Light curve time coverage (days)
Fig. 3.
Statistics related to the light curve acquisition characteristics and the photometric data coverage. See text forfurther explanation.these trails require non-circular apertures. The main pa-rameters of the 56 observed Trojans are summarized inTable A.1. The apparent trajectories of these objects w.r.tthe K2 CCDs are shown in Fig. 1.
In principle, the reduction of K2 long cadence data corre-sponding to Jupiter Trojans follows the similar steps thathave been conducted in former analysis of moving objectsobserved with K2 (P´al et al., 2015a; Kiss et al., 2016). Inorder to correct for the positioning jitter of the spacecraftwe retrieved nearly a dozen of additional stellar sources foreach Trojan and included these in further processing (seee.g. Fig. 1 of P´al et al., 2016). While all of these 56 ob-jects have been observed in the same campaign of K2, thedetermination of the positioning jitter have been done sep-arately for each object. This is essential due to the largefield-of-view: namely, the same pitch, roll and yaw offsetsapplied during the attitude control of the space telescopeyields different apparent centroid shift and field rotation onthe various CCD modules.After the derivation of the positioning jitter of the sub-sequent frames, we registered the frames to the same ref-erence system in order to perform differential image anal-ysis. A series of frames have been pre-selected to form amaster median-combined image which was then subtractedfrom the subsequent frames. Differential aperture photom-etry were then performed using apertures that follow theexpected shape of the apparent elongated PSFs of these ob-jects. The apparent speed of . . ′′ / min mentioned above isequivalent to . K epler CCD pixels per long cadence sam-pling. Therefore, it is essential to take into account theseshapes during the photometry. In Fig. 2 we displayed ashort series of subsequent image stamps and the relatedapertures corresponding to (12974) Halitherses. This objectshowed the the largest apparent speed, namely 1 . . . . . Fig. 4.
Comparison of periods in the literature and ourdeterminations.during the telemetry of K2 data. On average, ∼
10% of thedata points were excluded from the light curves due to theaforementioned reasons. In Fig. 3 we plotted some corre-sponding statistical properties while quantities related tothe light curve statistics are also found in Table A.1.For all of the previously discussed photometric proce-dures, we applied the various tasks of the FITSH package(P´al, 2012). We note here that the features of aperture pho-tometry on non-circular (in fact, arbitrary shaped) aper-tures are included only in the upcoming, version ofthe package. This new version is currently under prepara- A m p li t ude [ m ag ] Fig. 5.
Upper panel: The period–amplitude distribution ofTrojan asteroids. Our data are plotted with large blackdots, data from the literature is plotted with small greendots. See the text for references and discussion. Lowerpanel: The periodogram of (65227) 2002 ES with mul-tiple periods.tion for public release and expected to be published on thewebsite of the software soon.The statistics of light curve acquisition is plotted in Fig.3. The left panel shows the distribution of the apparent traillengths of the Trojans on the long-cadence frames. It caneasily be seen that one-third of the objects have an averagetrail length less than 0 . http://fitsh.szofi.net/ of ≈ / . . We searched for significant periodicities using the Fouriermethod as implemented in the
Period04 program pack-age (Lenz & Breger, 2005), and also the Lomb-Scargle pe-riodogram in the gatspy
Python package . We got verysimilar results in several test cases, therefore we decided tostick to the Lomb-Scargle periods. We note that the errorsof the individual photometric points are taken into account.Only those signals were considered that were significant onthe 3 σ -level compared to the background local noise peri-odogram. We phase-folded the light curves with the bestperiod and its double value, then decided which gives abetter fit based on a visual inspection.In our parallel paper about Main Belt asteroid detec-tions we show that the period determination is usually solidif the coverage exceeds 5 days and duty cycle is above 60%(Szab´o et al., 2016). Since the conditions are well fulfilledfor K2 Trojan asteroids, we could derive reliable solutionsfor most asteroids. In addition to the automatic processingof period determination all phased light curves were in-spected visually, too. We suspect that the occurence of pe-riod biases due to unfortunate data distribution is very low.For some objects the light curve we obtained suggested acomplex brightness variation. In these cases we tested man-ually for periods other than the primary one, but finallyomitted solutions different from the bimodal light curves.The phased light curves were also binned by 0.02 rotationalphase steps, and the full amplitude was accepted as the dif-ference between the minimum and maximum of this binnedlight curve.
3. Results
We present the determined periods and amplitudes in TableA.2. Both parameters span over a wide range (which we dis-cuss later in details), and indicatively, the median value ofthe measured periods is around 13 h, and the median am-plitude is 0 . m
32. The number of high amplitude asteroidsis surprisingly high; all asteroids in our sample exceed 0 . m . m
75 amplitude,belonging to more than 2 : 1 asphericity in sky projec-tion. These findings, as well as the overall distribution of https://github.com/astroML/gatspy/4y.M. Szab´o et al.: K2 photometry of Jupiter Trojan asteroids our data points, are consistent with the results in previouspublications.Our period determinations are plotted against datafrom the literature (Molnar et al. 2008, Mottola et al. 2011,French et al. 2015, Waszczak et al. 2015) in Fig. 4. For peri-ods less than 10 hours our periods are in perfect coincidencewith the previous determinations, while above 15–18 h pe-riod, we could confirm only a fraction of previous periods.Clearly, the power of the uninterrupted K2 light curves isin its accuracy in the long period range, partly because themeasurements are free from daily aliases, and also becausethe stable observation circumstances do not lead to dis-tracting systematics that mimic rotational light variations.In Fig. 5 we plot the amplitude–period diagram forTrojans from the literature, (Molnar et al. 2008, Mottolaet al. 2011, French et al. 2015, Waszczak et al. 2015) com-pared to our results. It is known that Main Belt familiesare characterized by specific brightness variation distribu-tions (Szab´o and Kiss, 2008). For the Trojan asteroids, wecannot confirm such parameter dependencies.Our sample shows a significant overabundance above60 hour periods. Possibly this is due to the limited com-pleteness of previous surveys in the range of very slow ro-tation rate. Due to the unbiased K2 observations that lastfor many days in the most cases, we believe that in thisrange our sample is still balanced, and reflects the commonoccurrence of very slow rotators among Trojan asteroids.This result would worth comparing to a similar sample inthe Main Belt. Interestingly, this is not possible now, sincethe already known distributions suffer the same incomplete-ness as the Trojan asteroids in the very long period range,so we cannot compare our results to the known distribu-tions. Even K2 asteroid surveys are more limited in thevery long period range than in the Trojan swarms, simplybecause Main Belt asteroids move faster and spend muchless time on silicon than Trojan asteroids do. Three light curve features are invoked as alerting signs fora binary asteroid. Leone et al. (1984) defined a light curveamplitude > . m a/b asphericity > .
3. This elongation cannot be ex-plained by a rubble pile body in equilibrium, but instead, aJacobi ellipsoid stretched along the semi major axis of tworubble piles orbiting each other. Another signal of binarityis the slow rotation, possibly reflecting a tidal synchroniza-tion with a modestly far companion. Following the recipeof Sonnett et al. to put the alerting limit at 3 × the averageTrojan period, here we consider a Trojan to rotate unusu-ally slow if the period exceeds 3 × the average, roughly 40hours. The third signal of possible binarity is the presence oftwo periods, reflecting the light variation of the main bodyand the companion which have not been synchronized, orthe forced perturbation of the main body because of thepresence of the companion.In our sample, several asteroids exhibited the describeddiagnostics of binarity. Asteroids 21593, 22056, and 39289were detected to exceed 0 . m . Sonnett et al. (2015) de-rived a WISE magnitude range of 0 . m ± . m
15, for this as-teroid, while here we present a period of 11 . ± .
22 hoursand 0 . m
75 amplitude. Since the amplitude is based on 475K2 photometric points, the photometric error is in the orderof 0 . m
01 in the amplitude. 6 years were passing between theWISE and K2 measurements, therefore the two amplitudesrefer to very different aspect geometry at roughly opposingpositions on the asteroid’s orbit. Since both amplitudes areunusually high, this body should be really elongated and apole position roughly perpendicular to the plane of Ecliptic.Thus, (16152) 1999 YN passed the amplitude criterion ofbinarity, and remains a good candidate for further investi-gations.Putting these detections together, we get 11 K2 Trojanswith signs on binarity. This means ≈ ±
5% rate of bi-naries among L4 Trojans, or at least asteroids with strongsigns of binarity. This determination is unbiased becausethe full light curve coverage and the long observation runs,and is fully compatible to the debiased rate of 14–23% ofSonnett et al. from WISE data, and is consistent or slightlyexceeds a previous estimate of 6–10% (Mann et al., 2007)from Earth-based light curves. with double periods We detected multiple periodicities in the case of (65227)2002 ES , the “one-hump” periods are 49.7 h and 3.53 h,with 0 . m
065 and 0 . m
08 full amplitudes, respectively. The de-tection of these periods is secure above the the noise level of ≈ . m . m
04. These periods are not resonant to each other,and since the period ratio is ≈ . m
16 and 0 . m A = B and C will be in a state of free precession if the rotation axis isinclined to the total angular momentum vector, and theratio of ω rotation rate and the Ω precession rate will beΩ /ω = ( A − C ) /C , independently of the inclination angle of the rotation vector. If we assume a free precession in thecase of (65227) 2002 ES , an ellipsoidal shape and a homo-geneous internal composition, we can derive the axis ratiosto be A/C = 1 . . m We compared the red and less red populations by meansof period and amplitude distributions of the member as-teroids. For this task, we cross-correlated the K2 asteroidsto the SDSS Moving Object Catalog 4 (MOC hereafter),to derive the t color which separates the red and less redmembers the most (Szab´o et al. 2007). In case of multiplemeasurements of the same asteroid, the measurements wereaveraged to get more precise colors.Eleven Trojans with K2 observations were found withan entry in SDSS MOC. Four of them were identified torepresent the less red population – (8241) Agrius, 13331,16152, 23939 – and seven that belong to red population ofTrojans – (1749) Telamon, (5028) Halaesus, 21599, 22056,24357, 59049, and 129602. We added the population mem-bership as a population flag ( r for red, lr for less red) inTable A.2.No specific pattern has been found for red and/or lessred population members. Both groups contain normal rota-tors around the median value of all K2 Trojans ( ≈
13 h pe-riod, ≈ . m
32 amplitude) and also, both exhibit very slow ro-tators, as well (e.g. 180 and 358 hours for 13331 and 22056,respectively) and elongated asteroids (e.g. 0 . m
75 and 0.98for 16152 and 22056, respectively). Because of low counts,we cannot get to more detailed conclusions, but we couldconfidently observe that high amplitude asteroids and slowrotators are not specific to one population only: they arequite common in both populations.
In Fig. 6 we present the size vs. spin rate distribution ofsmall bodies, including near-Earth and main belt aster-oids, Jovian Trojans, Centaurs, and trans-Neptunian ob-jects. The fast rotation of minor planets is limited by theso called spin barrier: there is a critical rotation period atwhich a rubble pile asteroid would fly apart due to its cen-tripetal acceleration. For a specific body, this critical periodcan be estimated as P c ≈ . · p (1 + A ) /ρ , where A is thelight curve amplitude, ρ is the density (in [g cm − ]), and P c is obtained in hours (Pravec & Harris, 2000). Using thisformula the knowledge of the rotation period and the lightcurve amplitude can provide a lower limit estimate of the Fig. 6.
Diameter vs. spin rate of asteroids. Black dots:main belt, red dots: trans-Neptunian objects, green dots:near-Earth asteroids, small blue dots: Jovian Trojans.Data is obtained from the Asteroid Light Curve Database(Warner et al., 2009). The Jovian Trojan asteroids withknown sizes – as determined by NEOWISE (Grav et al.,2012) – are marked by large, filled blue circles.
Fig. 7.
Distribution of normalized spin rates in our sample(blue bars). We used < f > = 1.34 cycle day − for normaliza-tion. Black and blue curves represent the scaled normalizedspin rate distribution of main belt and Jovian Trojan as-teroids, respectively, as obtained from the Asteroid LightCurve Database (Warner et al., 2009).body’s density. This spin barrier is well established for mainbelt asteroids, the critical rotation period is ∼ ∼ − . For main belt asteroids this limit is set by as-teroids with diamters of 1-10 km. However, with the JovianTrojans, we are in the size range of ∼ ∼ − using the fastest rotation periods of ∼ Fig. 8.
Critical densities obtained for our sample, as a func-tion of light curve amplitude (upper panel) and rotationperiod (lower panel).value is different from the critical density obtained for mainbelt asteroids ( ∼ − ) but consistent with the densityof cometary nuclei (A’Hearn, 2011) as well as that of ob-jects from the trans-Neptunian populations (Brown, 2013;Vilenius et al., 2014). Applying the critical density calcu-lation above for our sample, the highest density values weobtain are ∼ − (see Fig.8), lower than that of mainbelt asteroids in the same size range, but the same as ob-tained by French et al. (2015). Even in our unbiased sam-ple, despite that rotation periods longer than ∼ ∼ P >
50 h, ∼
20% of the objects observed by K2in this work. Very long rotation period may be an indicationof binarity, as it is e.g. the case for Jovian Trojan (617)Patroclus where the >
100 h rotation period is explainedby tidal breaking (Mueller et al., 2010). Slow rotation ofsmall ( <
30 km) and low density ( < − ) objects mayhave also been set by the YORP effect (French et al., 2015).Among our slow rotators there are objects that fall intothis susceptible size range according to the sizes derived byGrav et al. (2012) based on NEOWISE observations. In Fig. 9 we plotted the amplitude distribution of ourTrojan asteroid sample (black bars) and compared it withthe amplitude distribution obtained by Binzel & Sauter(2011) for a somewhat smaller sample. The magnitude limitof the detectability of periodic light curve variation for oursample is estimated to be ∆ m min ≈ . m
02 that correspondsto a a/b axis ratio of ( a/b ) > . · ∆ m min = 1 .
02, if onlyshape effects are taken into account. We note again thatlight curve variations were detected for all asteroids in oursample, therefore our sample can be considered to be an un-biased sample, rather than the sample by Binzel & Sauter(2011), where the targets were mostly Jovian Trojan as-teroids for which the existence of detectable brightnessvariations were previously known. A selection bias for theBinzel & Sauter (2011) sample can also be inferred fromthe fact that a Kolmogorov-Smirnov test that comparesthe two cases gives a probability of only ∼
7% that the twosamples are drawn from the same distribution.Our light curves are certainly affected by the spin axisorientations. To test the impact of the geometry on theamplitude statistics, we have corrected the original ampli-tudes following the method given by Binzel, i.e. choosingthe largest value if multiple amplitudes are available in theliterature Warner et al. (using data from the Asteroid LightCurve Database 2009), and applying a correction assuming ϑ = 60 ◦ aspect angle in the case of single amplitudes.The higher frequency of large amplitudes in our samplewith respect to main belt asteroids is even more pronouncedthan in the sample of Binzel & Sauter (2011) (see fig. 20 intheir paper). Binzel & Sauter (2011) interpreted this as ahigher number of elongated objects in the Trojan popula-tion, however, it is still unknown how the collisional historyor other evolutionary effects can explain this deviation.
4. Summary
In this paper we investigated the Trojan L4 asteroids de-tections by K2 and got the following conclusions: – The K2 sample shows a significant fraction ( ≈ P >
50 h) rotation periods. The K2 sam-ple is still unbiased in this period range, therefore thisobservation reflects the actual occurrence of very slowrotators. – The K2 sample shows a significant overabundance oflarge amplitude asteroids. 3 of 56 asteroids exceeded the
Fig. 9.
Light curve amplitude distribution of JovianTrojan asteroids. Black bars represent our original, un-corrected sample. The red and blue bars correspondthe bias-corrected amplitudes of our sample and that inBinzel & Sauter (2011).0 . m ≈ . m a/c > – In the case of (65227) 2002 ES we detected doubleperiodicity, 49.7 and 3.53 hours and 0 . m . m – The excess of large amplitude asteroids, the very slowrotators among Trojans, and the presence of double pe-riods can all be explained by a high rate of binary as-teroids in the L4 cloud. We estimated the occurrenceof binarity between 20-25%, in agreement with previousestimates. – Red and less red populations were found to be identi-cal for light variation properties. The similarity appliesfor both amplitude and period distributions, and also,the presence of very slow rotators and high amplitudeasteroids. – We detected a notable lack of fast rotators amongTrojans. We interpreted it as the effect of a densitybarrier, and estimated the upper limit of the densityof 0.5 g/cm in the K2 sample, in agreement with pre-vious estimates. – We derived the amplitude distribution of K2 Trojan as-teroids and debiased assuming random spin orientation.Both the observed and unbiased distribution differ fromthe results of Binzel & Sauter (2011) since K2 observedsignificantly more asteroids in the high amplitude wingof the distribution. How the collisional history or otherevolutionary effects led to the excess of elongated bodiesin the Trojan cloud still needs further investigations.
Acknowledgements.
Funding for the
Kepler and K2 missions is pro-vided by the NASA Science Mission directorate. The authors ac-knowledge the Kepler team for the extra efforts to allocate specialpixel masks to track moving targets. This work has been supportedby the Lend¨ulet Programme of the Hungarian Academy of Sciences(LP2012-31, LP2014-17), by the OTKA K-109276 and K-104607, theGINOP-2.3.2-15-2016-00003 grant and NKFIH K-115709 and PD-116175 grants of the Hungarian National Research, Development andInnovation Office, a T´eT-14FR-1-2015-0012 grant, and by the Cityof Szombathely under agreement No. 67/177-21/2016. The researchleading to these results has received funding from the EuropeanCommunitys Seventh Framework Programme (FP7/2007-2013) under grant agreement No. 312844 (SPACEINN), the ESA PECS ContractNos. 4000110889/14/NL/NDe and 4000109997/13/NL/KML, theEuropean Unions Horizon 2020 Research and Innovation Programme,Grant Agreement no 687378. L.M. was supported by the J´anos BolyaiResearch Scholarship of the Hungarian Academy of Sciences.All of the data presented in this paper were obtained from theMikulski Archive for Space Telescopes (MAST). STScI is operated bythe Association of Universities for Research in Astronomy, Inc., un-der NASA contract NAS5-26555. Support for MAST for non-HSTdata is provided by the NASA Office of Space Science via grantNNX13AC07G and by other grants and contracts.
References
A’Hearn, M.F., 2011, ARA&A, 49, 281Binzel, R.,P. & Sauter, L.M., 1992, Icarus, 95, 222Borucki, W. J., Koch, D., Basri, G., et al. 2010, Science, 327, 977Brown, M.E., 2013, ApJ, 778, L34Dahlgren, M. 1998, A&A, 336, 1056Dell’Oro, A.; Marzari, F.; Paolicchi, P. & Vanzani, V. 2001, A&A,366, 1053Emery, J. et al., 2016, The Complex History of Trojan Asteroids, inAsteroids IV, Univ. of Arizona, Tucson.Franklin, F. A.; Lewis, N. K.; Soper, P. R. & Holman, M. J. 2004, AJ,128, 1391French L. M., Stephens R. D., Coley D., Wasserman L. H., Sieben J.,2015, Icar, 254, 1Grav T., et al., 2011, ApJ, 742, 40Grav, T., Mainzer, A.K., Bauer, J.M., et al., 2012, ApJ, 759Gomes R., Levison H. F., Tsiganis K., Morbidelli A., 2005, Natur,435, 466Howell, S. B., Sobeck, C., Haas, M., et al. 2014, PASP, 126, 398Kiss, Cs., P´al, A., Farkas-Tak´acs, A. I. et al., 2016, MNRAS, 457,2908Lacerda, P., 2005, The Shapes and Spins of Kuiper Belt Objects, PhDThesis, Sterrewacht - Universiteit LeidenLacerda, P., Fornasier, S., Lellouch, E. et al., 2014, ApJ. 793, L2Lenz, P., Breger, M., CoAst, 146, 53Leone, G., Paolicchi, P., Farinella, P., Zappal´a, V. 1984, A&A, 140,265Mann, R.K., Jewitt, D., Lacerda, P., 2007, AJ, 134, 1133Levison H. F., Morbidelli A., Tsiganis K., Nesvorn´y D., Gomes R.,2011, AJ, 142, 152Merline, W. J.; Weidenschilling, S. J.; Durda, D. D.; Margot, J. L.;Pravec, P. & Storrs, A. D. 2002, Asteroids do have satellites, inAsteroids III, W. F. Bottke Jr., A. Cellino, P. Paolicchi, and R. P.Binzel (eds), University of Arizona Press, Tucson, p.289-312Molnar L. A., Haegert M., J., Hoogeboom K. M., 2008, MPBu, 35,82Moln´ar, L.; P´al, A.; Plachy, E.; Ripepi, V.; Moretti, M. I.; Szab´o, R.& Kiss, L. L. 2015, ApJ, 812, 2Mottola S., et al., 2011, AJ, 141, 170Morbidelli A., Levison H. F., Tsiganis K., Gomes R., 2005, Natur,435, 462Mueller, M., Marchis, F., Emery, J.P., et al., 2010, Icarus, 205, 505Nesvorn´y D., Vokrouhlick´y D., Morbidelli A., 2013, ApJ, 768, 45Noll, K. S., Grundy, W. M., Chiang, E. I., Margot, J.-L. & Kern,S. D. In The Solar System Beyond Neptune, Eds. A.Barucci, H.Boehnhardt, D. Cruikshank, and A. Morbidelli. Univ. of ArizonaPress, Tucson, pp. 345-363.P´al, A. 2012, MNRAS, 421, 1825P´al, A.; Szab´o, R.; Szab´o, Gy. M.; Kiss, L. L.; Moln´ar, L.; S´arneczky,K. & Kiss, Cs. 2015, ApJL, 804, 45P´al, A., Kiss, Cs.; M¨uller, Th. G., Moln´ar, L.; Szab´o, R.; Szab´o, Gy.M., S´arneczky, K. & Kiss, L. L. 2016, AJ, 151, 117Peixinho, N., Delsanti, A., Guilbert-Lepoutre, A., Gafeira, R.,Lacerda, P., 2012, A&A, 546, A86Pravec, P. & Harris, A.W., 2000, Icarus, 148, 12Roig F., Ribeiro A. O., Gil-Hutton R., 2008, A&A, 483, 911Sonnett, S.; Mainzer, A.; Grav, T.; Masiero, J. & Bauer, J. 2015, ApJ,799, 191Szab´o G. M., Ivezi´c ˇZ., Juri´c M., Lupton R., 2007, MNRAS, 377, 1393Szab´o, R., S´arneczky, K., Szab´o, Gy. M., et al. 2015, AJ, 149, 112Szab´o, R., et al. 2016, A&A, resubmittedTsiganis K., Gomes R., Morbidelli A., Levison H. F., 2005, Natur,435, 459Vilenius, E., Kiss, Cs., M¨uller, Th.G., et al., 2014, A&A, 564, A35
Appendix A: Figures and tables
Table A.1.
Summary of the observational characteristicsof the 56 Trojans observed by K2 during Campaign 6. Thecolumns are the following: a) MPC designation of the as-teroid; b) the average apparent trail length (in pixels) ofthe object on the K2 CCD frames; c) the variations of theapparent trail length throughout the observations; d) thenumber of light curve points used in the further analysis;and e) the total time coverage of the light curves in days. a Object b Trail c Trail d LC e Timelength var. points cov.(MPC number) (pixel) (pixel) (days)(1143) Odysseus 2.05 0.11 375 9.52(1749) Telamon 0.36 0.11 629 20.09(3801) Thrasymedes 1.74 0.11 367 10.40(4035) 1986 WD 0.84 0.10 442 10.42(4057) Demophon 0.26 0.14 871 20.25(4138) Kalchas 0.66 0.15 489 10.46(5028) Halaesus 0.84 0.09 466 10.48(5123) 1989 BL 1.11 0.13 461 10.46(5244) Amphilochos 0.32 0.10 808 18.12(5436) Eumelos 0.79 0.12 444 10.40(5652) Amphimachus 0.31 0.12 747 19.96(8241) Agrius 1.84 0.13 461 10.38(9807) 1997 SJ Table A.2. a Object Period Period Amplitude PopulationerrorNumber (h) (h) (mag)1143 10.079 0.194 0.2141749 22.662 0.193 0.059 r3801 20.270 0.672 0.1384035 13.475 0.156 0.1814057 29.925 0.765 0.2094138 29.411 2.001 0.1025028 25.052 1.091 0.225 r5123 19.800 0.140 0.445244 19.566 0.088 0.7315436 21.276 0.315 0.4425652 8.374 0.112 0.2228241 17.902 0.230 0.291 lr9807 331.034 117.56 0.42810989 26.101 0.286 0.32411251 10.448 0.068 0.31312238 7.281 0.041 0.29412974 6.971 0.030 0.51513184 11.934 0.119 0.14913185 11.453 0.124 0.03313331 180.451 31.938 0.444 lr13366 400 105.26 0.22813372 20.176 0.327 0.36213379 13.698 0.138 0.1114690 8.519 0.030 0.20814791 19.615 0.259 0.37115529 375 91.019 0.59216152 11.477 0.223 0.747 lr21593 10.747 0.196 0.99121599 12.651 0.121 0.489 r22056 358.208 16.791 0.976 r23939 12.868 0.104 0.552 lr23947 15.335 0.506 0.06623958 1142.85 154.44 0.35824357 131.868 0 0.385 r24534 19.417 0.368 0.14724537 10.489 0.064 0.3435363 18.779 0.374 0.32838574 7.085 0.084 0.21939270 82.191 4.450 0.14439286 8.421 0.065 0.539289 19.062 0.160 1.0457041 8.562 0.027 0.51658480 32.967 1.815 0.14359049 6.172 0.012 0.686 r63239 60.075 1.940 0.53765210 35.087 0.626 0.40865223 252.631 67.368 0.4465227 7.066 0.025 0.07465240 230.769 54.945 0.0965257 17.863 0.235 0.583984 5.752 0.037 0.08588227 6.355 0.018 0.25888241 5.734 0.013 0.099353363 5.403 0.020 0.356129602 43.010 1.188 0.362 r228102 7.259 0.039 0.24810y.M. Szab´o et al.: K2 photometry of Jupiter Trojan asteroids1143 1749 3801
16 16.5 0 0.5 1 1.5 2 B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m )
17 17.5 18 0 0.5 1 1.5 2 B r i gh t ne ss ( U S N O - B R sys t e m )
17 17.5 0 0.5 1 1.5 2 B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m )
17 17.5 18 0 0.5 1 1.5 2 B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m )
18 18.5 19 0 0.5 1 1.5 2 B r i gh t ne ss ( U S N O - B R sys t e m )
19 19.5 20 0 0.5 1 1.5 2 B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m ) Fig. A.1.
The phased light curves of the Trojan asteroids in K2 Field 6.
18 18.5 19 0 0.5 1 1.5 2 B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m )
19 19.5 20 0 0.5 1 1.5 2 B r i gh t ne ss ( U S N O - B R sys t e m )
18 18.5 19 0 0.5 1 1.5 2 B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m )
18 18.5 19 0 0.5 1 1.5 2 B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m )
19 19.5 20 0 0.5 1 1.5 2 B r i gh t ne ss ( U S N O - B R sys t e m )
18 18.5 19 19.5 20 0 0.5 1 1.5 2 B r i gh t ne ss ( U S N O - B R sys t e m )
19 19.5 20 20.5 21 0 0.5 1 1.5 2 B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m )
19 19.5 20 20.5 0 0.5 1 1.5 2 B r i gh t ne ss ( U S N O - B R sys t e m ) Fig. A.2.
The phased light curves of the Trojan asteroids in K2 Field 6. B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m )
19 19.5 20 0 0.5 1 1.5 2 B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m )
19 19.5 20 0 0.5 1 1.5 2 B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m )
19 19.5 20 0 0.5 1 1.5 2 B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m )
19 19.5 20 20.5 0 0.5 1 1.5 2 B r i gh t ne ss ( U S N O - B R sys t e m )
19 19.5 0 0.5 1 1.5 2 B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m ) Fig. A.3.
The phased light curves of the Trojan asteroids in K2 Field 6. B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m )
19 19.5 20 0 0.5 1 1.5 2 B r i gh t ne ss ( U S N O - B R sys t e m )
19 19.5 20 0 0.5 1 1.5 2 B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m )
20 20.5 21 21.5 22 22.5 0 0.5 1 1.5 2 B r i gh t ne ss ( U S N O - B R sys t e m )
19 19.5 20 0 0.5 1 1.5 2 B r i gh t ne ss ( U S N O - B R sys t e m )
19 19.5 20 0 0.5 1 1.5 2 B r i gh t ne ss ( U S N O - B R sys t e m )
20 20.5 21 0 0.5 1 1.5 2 B r i gh t ne ss ( U S N O - B R sys t e m ) B r i gh t ne ss ( U S N O - B R sys t e m ) Fig. A.4.