101 Trojans: a tale of period bimodality, binaries, and extremely slow rotators from K2 photometry
Csilla E. Kalup, László Molnár, Csaba Kiss, Gyula M. Szabó, András Pál, Róbert Szakáts, Krisztián Sárneczky, József Vinkó, Róbert Szabó, Viktória Kecskeméthy, László L. Kiss
DDraft version February 19, 2021
Typeset using L A TEX twocolumn style in AASTeX62
101 Trojans: a tale of period bimodality, binaries, and extremely slow rotators from K2 photometry
Csilla E. Kalup,
1, 2, 3
L´aszl´o Moln´ar,
1, 4, 3
Csaba Kiss,
1, 4
Gyula M. Szab´o,
5, 6
Andr´as P´al,
1, 2
R´obert Szak´ats, Kriszti´an S´arneczky, J´ozsef Vink´o, R´obert Szab´o,
1, 3, 4
Vikt´oria Kecskem´ethy,
1, 2 and L´aszl´o L. Kiss
1, 3 Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, Konkoly Thege 15-17, H-1121 Budapest, Hungary Department of Astronomy, E¨otv¨os Lor´and University, P´azm´any P´eter s´et´any 1/A, H-1171 Budapest, Hungary MTA CSFK Lend¨ulet Near-Field Cosmology Research Group ELTE E¨otv¨os Lor´and University, Institute of Physics, 1117, P´azm´any P´eter s´et´any 1/A, Budapest, Hungary ELTE E¨otv¨os Lor´and University, Gothard Astrophysical Observatory, 9700 Szombathely, Szent Imre h. u. 112, Hungary MTA-ELTE Exoplanet Research Group, 9700 Szombathely, Szent Imre h. u. 112, Hungary (Accepted to ApJS on February 17, 2021)
ABSTRACTVarious properties of Jovian trojan asteroids such as composition, rotation periods, and photometricamplitudes, or the rate of binarity in the population can provide information and constraints on theevolution of the group and of the Solar System itself. Here we present new photometric properties of45 Jovian trojans from the K2 mission of the
Kepler space telescope, and present phase-folded lightcurves for 44 targets, including (11351) Leucus, one of the targets of the
Lucy mission. We extendour sample to 101 asteroids with previous K2 Trojan measurements, then compare their combinedamplitude- and frequency distributions to other ground-based and space data. We show that there isa dichotomy in the periods of Trojans with a separation at ∼
100 hr. We find that 25% of the sampleare slow rotators (P ≥
30 hr), which excess can be attributed to binary objects. We also show that32 systems can be classified as potential detached binary systems. Finally, we calculate density androtation constraints for the asteroids. Both the spin barrier and fits to strengthless ellipsoid modelsindicate low densities and thus compositions similar to cometary and TNO populations throughoutthe sample. This supports the scenario of outer Solar System origin for Jovian trojans.
Keywords: photometry, Trojan asteroids, Jupiter trojans INTRODUCTIONJovian trojan asteroids are located around the L4and L5 Lagrange points of the Sun-Jupiter system, in1:1 mean-motion resonance. According to the MinorPlanet Center , at the time of writing we know 8190 Jo-vian trojans, but of those in the Asteroid Light CurveDatabase , only ∼
5% have rotational period and/oramplitude information (LCDB, Warner et al. 2009).However, it is possible that there are more than halfa million asteroids out there with diameter of > Corresponding author: Csilla E. [email protected] https://minorplanetcenter.net/iau/lists/JupiterTrojans.html known asteroids is complete down to H <
13 mag abso-lute brightness (i.e. having d (cid:38)
15 km diameter) (Vino-gradova & Chernetenko 2015), and based on SDSS mea-surements, asteroids larger than 10 km are all identified(Szab´o et al. 2007).The surface characteristics of Jovian trojans show sim-ilarities to those of trans-Neptunian objects (Fraser et al.2014; Emery et al. 2015; Wong & Brown 2016), anddynamical models also indicate that they had likelyformed in the outer planetesimal disk, then were im-planted to their current orbits after planetary encoun-ters (Nesvorn´y et al. 2013; Pirani et al. 2019). Rota-tional characteristics, obtained via time-series photome-try, also indicate that the Trojans share more similaritieswith objects in the outer Solar System than with mainbelt asteroids (see,e.g., French et al. 2015).Recently, time-resolved photometry of asteroids havebecome possible with space-based instruments, too: a r X i v : . [ a s t r o - ph . E P ] F e b Kalup et al. both the
Kepler and TESS missions have encounterednumerous objects within the Solar System while observ-ing stellar targets. TESS can observe mostly objectswithin the main belt, but the larger aperture of
Kepler allowed the measurement of fainter and farther objects,such as Hildas, Trojans, centaurs and TNOs during itsK2 mission. The K2 mission consisted of 3 months longCampaigns around the ecliptic plane, which provideda unique opportunity to observe asteroids from space(Howell et al. 2014). These high quality, several-week-long, uninterrupted and continuous measurements arefree from aliases, thus, very suitable for studying aster-oids with rotational periods longer than 10 or even 100hours. In contrast, rotational statistics from previousground-based measurements could be strongly biased,favouring the detection of short (P (cid:46)
12 hr) rotationperiods. More recently, a large number of slow rota-tors were identified in various small body populations,including Centaurs (Marton et al. 2020), Jovian tro-jans (Ryan et al. 2017; Szab´o et al. 2017) and Hildas(Szab´o et al. 2020) using K2 data, and also among mainbelt asteroids, based on TESS measurements (P´al et al.2020).One of the possible explanations of an excess of slowrotators can be binarity. The importance of determiningthe binary fraction of asteroids in a certain populationis that it can provide constraints on the dynamical evo-lution of the Solar System. Using mid-infrared WISEdata, Sonnett et al. (2015) derived that 13%–23% ofTrojans are extremely elongated or binaries. Based onthe Ryan et al. (2017) and Szab´o et al. (2017) Joviantrojan sample Nesvorn´y et al. (2020) suggested that thelarge fraction of ∼
15% of very slow rotators in that sam-ple with periods over 100 hr can be explained by tidallysynchronized binaries, originally formed as equal-sizedbinaries in the massive outer disk at 20-30 au in the earlySolar system.Szab´o et al. (2017) analyzed 56 Jovian trojans fromCampaign 6 (C6). In this work, we expand that K2Trojan sample to 101 asteroids. There are two notabletargets in this paper. The first one is (11351) Leucus,which belongs to the extremely slow rotators, and it waschosen as one of the flyby targets of the
Lucy mission(Levison & Lucy Science Team 2016). The spacecraft isplanned to visit the asteroid in April 2028. The secondnotable target is (13062) Podarkes, which is the princi-pal body of the proposed Podarkes family that belongsto the larger Menelaus clan (Roig et al. 2008).In the framework of the K2 mission our group al-ready published results concerning various objects, frommain belt asteroids (Szab´o et al. 2015, 2016; Moln´aret al. 2018) to trans-Neptunian objects (P´al et al. 2015, 2016), Centaurs (Marton et al. 2020) and irregularmoons (Kiss et al. 2016; Farkas-Tak´acs et al. 2017). K2light curves were also used to model main belt asteroidsby Marciniak et al. (2019) and Podlewska-Gaca et al.(2020). In this work we continue to explore the K2 datafrom the Solar System. In Sections 2 and 3 we describethe observations made by
Kepler and the data reductionsteps. In Section 4 and 5 we present the results obtainedfrom the photometry and the analysis of the rotationalperiods and light curve amplitudes of the asteroids. Fi-nally, a summary is provided in Sect. 6. OBSERVATIONSIn this paper, we present light curves and photometricproperties of 45 Jovian trojan asteroids, observed by
Ke-pler during K2 Campaigns 11-19, between 2016 October21.27 and 2018 September 23.10, in long cadence mode(29.41 min sampling frequency). The apparent trajec-tories allocated for these targets were observed via K2 Guest Observer proposals. The target pixel time seriesfiles have been retrieved from the Mikulsi Archive forSpace Telescopes (MAST). The main parameters of the45 observed Jovian trojans, including the duty cycles foreach object, are summarized in Table 3 in the Appendix.Unlike C6 that looked into the dense regions of the L4group, later campaigns captured only the edges of theL4 and L5 clouds, as shown in Fig. 1. Therefore, espe-cially after C11, only a handful of objects were observedin each run.Two Campaigns required special treatment. C11 wasaimed at the Galactic bulge area, thus, the Trojans werefollowed in front of a very dense stellar background.Therefore, even though 38 targets were observed in C11,many of them were too faint to be detectable againstthe stellar background, hence we were not able to col-lect meaningful photometry. Due to this circumstance,we prioritized targets that were either bright with re-spect to the background stars or moved in front of lessdense stellar fields. Our results from C11 generally agreewith the light curves produced by the K2 Moving Bod-ies Project , which, at the time of writing this paper,included Solar System data processed up to C13.The other Campaign that needed special attentionwas the last one, C19. Kepler operated right untilthe spacecraft ran out of fuel. Due to the drop in fuelpressure, C19 suffered from degraded pointing accuracy,then ended prematurely. Only a roughly two-week longsegment contains useful data that was taken with nomi-nal pointing. Serendipitously, these two week coincided http://archive.stsci.edu/ https://christinahedges.github.io/asteriks/index.html
01 Trojans with K2 cloud0 clouddistance (AU)0 Figure 1.
Positions of the Trojan swarms and the observed targets in the L4 (top row) and L5 (bottom row) clouds in the SolarSystem relative to the Sun and Jupiter, in each Campaign. The position of
Kepler is also marked. Sizes of dots indicate apparentbrightness as seem from
Kepler . Black dots indicate observed targets; black circles in C11 are targets that were observed butwere too faint to produce usable light curves; the object marked with red cross in C16/C18 is 2001 SC101, observed in bothCampaigns. Data obtained from the JPL Horizons service. with the time when the Trojans selected for observationin C19 traversed the pixels placed along their tracks,making it possible to measure their brightness variationsprecisely.The data we use in this paper were proposed throughK2 GO programs GO11051, GO14085, GO15085,GO17028, GO18028 and GO19028 . DATA REDUCTIONThe data reduction steps of K2 long cadence datafor moving objects have been analogous to other earlieranalysis of asteroid targets that already have been dis-cussed by our group in previous works (see, e.g., Szab´oet al. 2017; Moln´ar et al. 2018; Marton et al. 2020).To process Kepler observations, we have utilized theFITSH software package and our external scripts. Inbrief, we assembled mosaic images from the individualTarget Pixel Files of the target and some nearby stars,and derived the astrometric solution to register theminto the same reference system. Then, a master imagewas created from a few dozens of frames that did not https://keplerscience.arc.nasa.gov/k2-approved-programs.html contain the target, which was then subtracted from allthe frames. After that step only signals belonging tomoving and variable features remained on each frame.We generated the coordinates of asteroids from JPLHorizons ephemerides . We converted fluxes to magni-tude based on USNO R magnitudes of the nearby stars(Monet et al. 2003). We note that we compute the samescaling law between Kepler fluxes and R magnitudesthat was determined for Kp magnitudes by Lund et al.(2015). 3.1. Campaigns 11 and 19
C11 was split into two parts due to an initial point-ing error. Many of the Trojan targets were observedduring the first half (C111) for which we did not havean astrometric solution readily available. For those weused the nominal coordinates of the neighboring TPFsand generated the astrometric matches manually. Wethen enlarged the mosaic images of all C11 targets by ∼ https://ssd.jpl.nasa.gov/?horizons Kalup et al. by the sub-pixel position differences between the masterimage and the registered individual frames. Moreover,we are able to use tighter apertures just as we did for theCentaur light curves in Marton et al. (2020). These tightapertures did not cover the edges of the PSFs, which ledto slight flux loss in some cases. We corrected the lightcurves when necessary by matching them to the averagebrightness and amplitude measured on the non-enlargedimages. The differences in average brightness were be-tween 0.05–0.3 mag for most targets. In three cases werestricted the aperture to the center of the PSF whichrequired corrections between 0.5–0.75 mag. Overall, wewere able to derive rotation periods for 14 objects outof 38 observed in this campaign.For C19 we generated the astrometry manually again,based on the nominal TPF coordinates. We tested en-larged pixels for this C19 too but the method did notoffer significant improvements. We decided to use non-enlarged pixels in this campaign.3.2.
Post–processing
Obvious outliers were filtered out from the light curvesafter finishing the photometry. We then compared eachlight curve to the differential images and also filteredout data points that were affected by excess noise fromstellar residuals, CCD crosstalk patterns or partial cov-erage of the PSF. We calculated the duty cycles for eachtarget to quantify how many data points were used inour analysis from each observing window. Values, alongwith Campaign numbers, length, start and end times arelisted in Table 3 in the Appendix. A sample of the datafile containing measurements of all asteroids is shown inTable 1.Finally, we created rectified versions of each lightcurve for the period search. We subtracted a linear fitfrom all light curves except for three targets in order totake into account the slow changes in brightness causedby the changing phase angle and distances to the tar-gets. In two cases we applied no corrections, and in onecase a quadratic fit was required.3.3.
Period search
The final light curves were analyzed with two differentmethods. First, we used our own residual minimizationalgorithm (see P´al et al. 2016; Moln´ar et al. 2018; Mar-ton et al. 2020, for previous studies). In this case, we fitthe data m (∆ t ) with a function m (∆ t ) = A + B cos(2 πf ∆ t ) + C sin(2 πf ∆ t ) (1)where f is a trial frequency, and ∆ t = T − t , where T isthe approximate center of the time series. The dispersion of the light curve residual as a func-tion of the trial frequency, S ( f ), is computed by scan-ning the interval of the physically meaningful frequen-cies with a step size ∆ f = 0 .
002 d − to determine theparameters A , B and C via Eq. 1. Then we searched forthe minimum of the S ( f ) dispersion curves, and the fre-quency corresponding to the minimum was considered asthe best-fit one ( f b ). Note that the best-fit frequenciesobtained with this method give the same result as theLomb-Scargle periodogram or Fast Fourier Transformmethods (see Moln´ar et al. 2018). After the automaticprocessing of the period search, we phase-folded the lightcurves with the best period, P b = f − b and 2 P b , and in-spected all the phased light curves visually. It turnedout that folding with the double period gave the betterfit for every Trojan in this paper. We also binned thephased light curves to see the shape of light curves bet-ter, and obtained the full amplitudes as the differencebetween the maximum and minimum of the binned lightcurve. The main frequencies were also determined withthe FAMIAS code (Zima 2008). The resulting frequen-cies from FAMIAS were identical to our ones obtainedabove. The frequency uncertainties were calculated withFAMIAS. RESULTS4.1.
Rotation periods and amplitudes
We present the photometric properties of our samplein Table 4. The period and amplitude values span awide range. The shortest period is 5.124 hr ((151883)2003 WQ25), while the longest one is 445.73 hr ((11351)Leucus). The smallest amplitude is 0.065 mag ((116567)2004 BV84), and the largest one is 0.9644 mag ((76820)2007 RW105). The median values of the periods andamplitudes are 8.931 hr and 0.3087 mag, respectively.We were able to determine rotation periods and am-plitudes for all but one target. For (60421) 2000 CZ31we did not detect any significant coherent frequencycomponents in the frequency spectrum. We determinedthe average upper limit for the rotation amplitude as (cid:46) .
05 mag at an expected period of 12 hours. Sincelow-frequency noise is present in the spectrum, this up-per limit increases to 0.08 mag for periods longer thanone day.We present the period, frequency and amplitude dis-tributions of the overall sample of Jovian trojans fromall K2 Campaigns (i.e., our sample complemented withTrojans from C6). In Szab´o et al. (2017), asteroid 65227has two periods in the residual spectrum, and we usedtheir double peak solutions (7.06 hr and 49.7 hr) in thelater analysis. We note that the presence of two peri-ods in the spectrum can be a signal of possible binarity,
01 Trojans with K2 Table 1.
Sample table of the photometry of Trojan asteroids observed in Campaigns 11 to 19 K2 mission. The table includes theidentification numbers of the asteroids, the measured Kp brightness and uncertainty from each frame, plus the corresponding equatorialcoordinates (RA and Dec), heliocentric ecliptic longitudes and latitudes ( λ and β ), observer–target range (∆), solar elongation (SOT),and phase angle (STO) values. The entire table is available online.ID JD UTC [day] Kp [mag] σ Kp [mag] RA [deg] DEC [deg] λ [deg] β [deg] ∆ [AU] SOT [deg] STO [deg]1871 2458268.868974 18.9660 0.0131 132.99263 16.51030 139.94735 -0.87773 4.801031 122.60730 9.105941871 2458268.889407 18.8972 0.0127 132.99208 16.51075 139.94890 -0.87750 4.801292 122.58714 9.108121871 2458268.909841 18.9003 0.0113 132.99154 16.51120 139.95045 -0.87727 4.801554 122.56699 9.110301871 2458268.930275 18.8511 0.0133 132.99099 16.51166 139.95201 -0.87703 4.801816 122.54684 9.112481871 2458268.950708 18.8886 0.0150 132.99045 16.51211 139.95356 -0.87680 4.802077 122.52669 9.11466. . . period [h]0510152025303540455055 c o un t a) c o un t b)240.0 24.0 12.0 8.0 6.0 4.8 4.0period [h] 0.0 0.2 0.4 0.6 0.8 1.0 1.2amplitude [mag]01020304050607080 c o un t c) Figure 2. (a) and (b): Period and frequency distributions of K2 Jovian trojan asteroids (red bars), compared to the periodand frequency distributions of Jovian trojans from LCDB (blue bars). (c): K2 and LCDB amplitude distributions, using thesame color coding. Where the LCDB also included a minimum amplitude we used the average of the maximum and minimumobserved amplitudes. therefore we consider these periods as the rotational pe-riods of the two components: the longer period is theorbit of the secondary (which is often tidally locked)and the shorter period is the rotation of the primary(when not tidally locked). We also note that in the caseof 23958, there was a factor–of–2 error in Table A.2,the good period value is the half of the presented value(571.425 hr).Panel (a) and (b) of Fig. 2 show the period and fre-quency distributions, respectively. Red bars representthe overall K2 Jovian trojan sample, blue bars representthe Jovian trojan asteroids obtained from the LCDB.Fig. 2a shows that 38% of the K2 Trojan periods areshorter than 10 hr, while 12% of the sample has verylong rotational periods, between 100–600 hr. The me-dian of the overall sample (12.65 hr) is the same as themedian of Trojans from LCDB (12.75 hr). In panel (b)of Fig. 2 we present the frequency distributions of Joviantrojans. It shows a similar cut off at short periods as theLCDB sample. In Fig. 2c, the histogram shows the dis-tribution of the observed amplitudes. We do not applyany de-bias procedure to the amplitudes as spin axes of Trojans appear to prefer perpendicular orientations tothe plane of the Solar System instead of a uniform distri-bution (Bowell et al. 2014). Where both minimum andmaximum values were listed in the LCDB, we used theaverage value for this figure. Most of the asteroids haveamplitudes similar to the median value, but multiple as-teroids fall into the 0.6–1.0 mag range too. Asteroids inthis range could be either strongly elongated, or binaryobjects.In Fig. 3, we plot the cumulative distribution of Tro-jan rotation periods obtained from the LCDB (blue) andfrom the K2 observations (red). Above ∼
15 hr the slopeof the cumulative distribution of the LCDB sample re-mains steeper than that of the K2 sample, and the differ-ence becomes most pronounced at the long-period wingof the distribution. This means Trojans with longer pe-riods are less represented in the LCDB than in the K2sample.We also compared the distributions of the swarms inFig. 4 and conclude that there is no significant differ-
Kalup et al. C u m u l a t i v e p r o b a b ili t y Figure 3.
Cumulative distribution of Trojan periods fromLCDB (blue) and from K2 (red).MB MB
TESS
JT HildaN 13072 7874 101 112f m [cycle d − ] 3.53 1.49 1.90 1.26P m [h] 6.79 16.06 12.65 19.02N f f [%] 14.8 7.4 0 1.8N s s [%] 11.1 34.9 25.7 38.9N vs
488 1390 13 20r vs [%] 3.7 14.0 12.9 17.85 Table 2.
Summary table of median rotation rates (f m , andthe corresponding period P m ), and the number of slow andfast rotating asteroids in the main belt, as obtained fromdata in the LCDB (MB), main belt asteroid data from theTESS DR1 (MB TESS ) (P´al et al. 2020), K2 Jovian tro-jans (JT, this work), and K2 Hilda asteroids (Szab´o et al.2020). We defined fast rotators (subscript ’f’) as f ≥ − (P ≤ ≤ − (P ≥
30 h) andvery slow rotators (’vs’) as f ≤ − (P ≥
100 h), followingPravec & Harris (2007). ence between them, which is consistent with previousstatistical results (Slyusarev et al. 2018).If we compare our K2 Jovian trojan sample to theproperties of other types of asteroids derived from
Ke-pler (Szab´o et al. 2020) and TESS (P´al et al. 2020) data,we find that slow rotators exist elsewhere, too. It is alsoseen that ground-based observations underestimated thenumber of slow rotators in the Main Belt. In Fig. 5 weplot the rotation rate distributions of these populations.In the cases of Trojans and Hildas, we can see a possibledichotomy in rotational periods. We fitted a Maxwelliandistribution, which describes a collisionally evolved sam- c o un t L4 0.0 0.5 1.0amplitude [mag]0246810121416182022 c o un t L51 10 100 1000period [hr]024681012141618202224262830 c o un t c o un t Figure 4.
Comparison of the distributions of the L4 and L5nodes of Jovian Trojans. ple, to the Trojan sample. While the short-period groupcan be fitted reasonably well with this profile, the long-period group clearly falls outside the distribution. Wealso computed the ratio of fast, slow and very slow ro-tators among the four samples mentioned above, in Ta-ble 2. This, again, shows that there are essentially nofast rotators among the Trojans and Hildas, whereas alarge portion of the sample rotates slowly or very slowly.Moreover, with the TESS observations at hand, the rateof slow and very slow rotators among the main-belt as-teroids is comparable to that of the farther groups ofHildas and Trojans.We compare our period determinations with existingliterature values in Fig. 6. It shows that above 10 hrthe previous rotational periods are significantly differentfrom ours, except for the largest value, which belongs to(11351) Leucus. Leucus is an important asteroid as itwill be one of the targets of the
Lucy mission, hence pre-vious studies measured its period quite accurately (Buieet al. 2018). In another three long-period cases (5638,11397, 10247) we have fully covered phased light curveswith precise periods, which shows the advantage of theuninterrupted K2 measurements compared to ground-based observations, where the changing circumstancesand daily aliases make it more difficult to achieve highaccuracy.In the case of (5638) Deikoon, the current default ro-tation period value in the LCDB is 19 . ± .
011 hr,based on the observations of Molnar et al. (2008) whoconstructed a complex light curve with multiple maxima
01 Trojans with K2 Figure 5.
Rotational frequency distribution of Jovian tro-jan asteroids from the K2 mission (orange bars) comparedwith those of other asteroid samples (black: main belt as-teroids from the LCDB; gray: main belt asteroids form theTESS DR1 sample, P´al et al. 2020; magenta: K2 Hilda as-teroids, Szab´o et al. 2020). The black-orange dashed curverepresent the Maxwellian fit to the K2 Jovian trojan sample.
10 100Periods in this paper [h]10100 P e r i o d s i n li t e r a t u r e [ h ] (11397) 1998 XX93(10247) Amphiaraos Figure 6.
Comparison of periods in the literature and thispaper. Red plus signs represent (5638) Deikoon, that havetwo different rotational period in the literature. and minima. Later, Mottola et al. (2011) also detecteda complex, four-peaked light curve, but with a rotationperiod of 9 . ± .
003 hr. This latter value is very closeto the 9 . ± .
029 hr period we calculated from theK2 data, and the shape of the light curve is also similarto our photometry, indicating that this shorter value isthe correct rotation period for the asteroid.4.2.
Period-amplitude diagram
Panels of Fig. 7 show the period-amplitude distribu-tions of Jovian trojan asteroids. In panel (b) and (c)we used bias-corrected amplitudes. Fig. 7c shows Tro-jans from LCDB (blue dots) with Trojans from K2 (redsquares). We can notice the same period dichotomy that period [h] a m p li t u d e [ m a g ] a) period [h] a m p li t u d e [ m a g ] b) Figure 7.
Period-amplitude distributions of Jovian trojanasteroids. Panel a) shows the observed K2 amplitudes, wheregrey dots are Trojans from C6 and red squares from oursample. In panel b), red squares are the overall K2 Trojansample, and blue dots are Trojans from LCDB, using averageamplitudes if minimum and maximum amplitudes were bothlisted in the database. was evident in Fig. 2 too, with a smaller group cluster-ing at periods above 100 hr. Although the de-biasedamplitudes group at lower values, there still seems tobe gap in large amplitudes ( > . Kalup et al.
Figure 8.
Spin rate versus V -band absolute magnitude ofmain belt, Hilda, and Jovian trojan asteroids. The horizon-tal dashed and dash-dotted lines represent the spin rates off = 10 and 5 d − or the rotation periods of P = 2.4 and 4.8 h.These correspond to critical densities of ∼ ∼ − .5. ANALYSIS5.1.
Spin barrier and critical density
Minor planets have a rotational limit called the spinbarrier, which value is the smallest possible rotationalperiod that a given type of asteroid can have without fly-ing apart due to its centrifugal acceleration. The criticalperiod can be calculated approximately as P c ≈ . · (cid:115) (1 + A ) ρ c (2)where P c is expressed in hours, A is the amplitude ofthe light curve, and ρ c is the critical density in g cm − ,which is a lower limit estimate of the asteroid’s bulkdensity (Pravec & Harris 2000).In Fig. 8 we present the distribution of rotation rateversus the absolute magnitude of the asteroids for ourJovian trojan sample, as well as for main belt asteroidsand Hildas. For main belt asteroids the ’spin barrier’ at ∼
10 d − (dashed horizontal line) corresponds a criticaldensity of ∼ − calculated from Eq. 2. Althoughthere are a few Jovian trojans in the LCDB that haverelatively short rotation periods, the bulk of Jovian tro-jans – most targets in the LCDB (blue dots), and alltargets in the K2 sample (orange symbols) – have a spinrate f (cid:46) − or rotation period P (cid:38) − , significantlylower than in the main belt.Using Eq. 2 we calculated the critical densities eachof the asteroids in our sample. In Fig. 9 we presentthese critical densities complemented with Jovian tro- jans from C6 as a function of rotation periods. It showsthat there are no points below ∼ ∼ − .If our sample is unbiased these values estimate thelower limit of rotational period and the upper limit ofthe critical density of Jovian trojans. We can rule outmost observational biases safely: we detect variationsin all but one of the 101 Trojans. There is a smallchance that some of the long variations are caused bystroboscopic sampling of a rotation period close to thesampling frequency of Kepler . However, as K2 data formain-belt asteroids show, we would be able to detectrotation rates up to twice as short as the fastest Trojanin our sample before reaching the Nyquist limit (Moln´aret al. 2018). There are no Trojans with a measured pe-riod in the LCDB that would rotate fast enough to sufferfrom stroboscopic sampling issues if observed with the
Kepler cadence either. Therefore we can conclude thatwe observe the true rotation limit of the sample.It is also possible that only Trojans below the faintlimit of the K2 observations ( H V (cid:38)
13 mag, or under10–20 km diameter) rotate faster. At this distance theYORP effect may not be sufficient to spin large objectsup (or down). However, as our timescale calculationsin the Appendix indicate, collisions can spin up Trojansmuch more effectively than the YORP effect. Indeed,as Fig. 5 indicates, the fast-rotating group of Trojansappear to follow a Maxwellian distribution, suggestingcollisional evolution.Based on the Maxwellian fit of the short-period groupin Fig. 5 we calculated the expected number of aster-oids with periods between 3–5 hr. In a sample with 100elements, we predict 10–30 asteroids with periods be-low 5 hr, while below 3 hr the expected number dropsto only 0–4 objects. We also calculated the probabilitythat zero objects have periods below a given period inthat sample. We find that at 3 hr there is a 38% chanceof this. The probability increases steeply below 3 hr, itis more than 50% at 2.9 hr already. We conclude thatin the collisionally evolving model, we would expect tofind multiple objects between 3 and 5 hr in a sample of100 asteroids. Neither the Maxwellian distribution, northe YORP effect indicate that we would not detect as-teroids with rotational period below 5 hr, which impliesthere should be another physical reason to explain thecutoff the short period end of the distribution. We inter-pret the observed cutoff at ≈
01 Trojans with K2 c r i t i c a l d e n s i t y [ g c m − ] Figure 9.
Critical densities derived from the overall sampleof K2 Jovian trojans as a function of rotational period. Greypoints are from C6, red squares are from our sample. significantly different from that of the main belt, butvery similar to the TNO population or the nucleus ofcomets. This value is expected from icy bodies withnotable porosity, and it is similar to the bulk densitiesfound for trans-Neptunian objects in the D ≤
500 km sizerange (see e.g. Grundy et al. 2019). It is also in agree-ment with color and albedo measurements (Szab´o et al.2007). 5.2.
Rate of binarity
We estimated the rate of binarity of our sample intwo different ways. First, we used two features that canindicate a binary asteroid, depending on the photomet-ric properties of the asteroids: a.) at least one phaseangle, the amplitude should exceed 0.9 m (Leone et al.1984), and b.) the rotational period should be greaterthan the triple of the median value (Sonnett et al. 2015).Here we approximate the limit of slow rotation to be 30hr (Pravec & Harris 2007). In our sample there are1 (200037) and 11 (11351, 13062, 18071, 26486, 31819,96337, 99306, 111113, 175471, 200037, 296787) asteroidsthat fulfill these amplitude and period requirements, re-spectively. This leads to an estimated ∼
25% rate ofbinaries among our sample, which is consistent with pre-vious calculations (Szab´o et al. 2017). However, this isestimate is different from constraining the true binaryrate, because some objects may produce the expectedfeatures through other mechanisms, and some binarieswill fall outside these selection bounds.To get some hints on the reason behind their lightcurve variations, minor bodies are often classified basedon their spin frequencies and light curve amplitudes:Leone et al. (1984) and Sheppard & Jewitt (2004) identi-fied three main zones on the light curve amplitude versus rotational frequency plane (see Fig. 10a), re-evaluatedby Thirouin et al. (2010) and Benecchi & Sheppard(2013). Light curve variations of objects with smallamplitudes (∆ m ≤ ϑ = π/ − , region 2) are likely rotating singlebodies, if their rotational speed is below the breakuplimit. The rotation of the objects to the left (region 3)is too slow to cause elongation and a corresponding ro-tational light curve. For these objects the light curvesare often explained by binarity (e.g. Leone et al. 1984;Sheppard & Jewitt 2004).As panel (a) of Fig. 10 indicates, densities do notexceed 1.5 g cm − when assuming a strenghtless bodymodel. This is in agreement with the spin barrier re-sult presented before in Sect. 5.1, indicating that theJovian trojans generally have low densities, suggestingicy and/or porous composition.We can also characterise the possibly binary natureof a specific system following Marton et al. (2020). Weuse the estimated ’separation’, a bin , the semi-major axisof the orbit of the potential binary, assuming that thebinary has two equally sized, spherical (of diameter D )and equal-mass components, and a uniform density of 1g cm − . Effective diameters are obtained from the NE-OWISE Derived Diameters and Albedos of Solar Sys-tem Small Bodies database (Mainzer et al. 2019). Ifno geometric albedo was available then we used the me-dian value of the known geometric albedos, p V = 0.079,and calculated the effective diameter from the absolutemagnitude (see Marton et al. 2020, for details). The a bin /D = 1 case is a classical contact binary. Whileformally the minimum requirement for a binary in thisscheme is a bin /D ≥ a bin /D ) (seee.g. Lacerda & Jewitt 2007). Therefore we consider a bin /D ≥ https://sbn.psi.edu/pds/resource/neowisediam.html,https://irsa.ipac.caltech.edu/cgi-bin/Gator/nph-dd Kalup et al.
Figure 10. (a) Light curve amplitude versus frequency of Jovian trojans in this paper and in Szab´o et al. (2017). Blue dash-dotted curves represent the rotational frequencies and light curve amplitudes corresponding to the rotation of a strengthless body(Jacobi ellipsoid), of a specific density (shown at the top in g cm − units). In the blue and purple shaded areas (below ∆m ≤ ρ = 1.0 g cm − curve (or any other constant density chosen), could be elongated due torotation. Objects in region 3 should have densities notably below ρ = 1.0 g cm − in order to be elongated from rotation. (b) Ratioof binary orbit to binary diameter ( a bin /D ) versus the maximum light curve amplitude ( A max ). (c) Rotation period versusdeduced diameter. The black boxes mark the regions A, B and L identified by Pravec & Harris (2007). Colour symbols in allsubfigures correspond to the following: light blue – 1 ≤ a bin /D ≤ ≤ A max ≤ ≤ a bin /D ≤ A max ≥ a bin /D ≥ ≤ A max ≤ a bin /D ≥ A max ≥ a bin /D ≥ A max ≥ a min /D ratio and light curve characteristics—and assuming spin locking—are in region B, where doublesynchronous systems can be found in the main belt.The yellow and light blue points in Fig. 10 are close tothe maximum photometric amplitude a strengthless el-lipsoid can achieve, and only differ by the separation pa-rameter. These are considered edge cases whose classi-fication will need further observations at different phaseangles and orbital geometries.Dark blue points represent asteroids where thestrengthless model cannot account for the photometricamplitude but the estimated separation is too smallfor a detached binary. The objects in this regioncould either be single objects with large surface fea-tures, elongated, non-axisymmetric objects, or couldalso be contact/collapsed binary asteroids. Examplesfor these have been observed directly by various spacemissions. Photographs taken by the NEAR-Shoemakerprobe showed that the shape of the rubble pile asteroid(253) Mathilde is dominated by five deep giant craters(Thomas et al. 1999). The probe also confirmed thatthe near-Earth asteroid (433) Eros rotates about one ofits minor axes, as indicated earlier radar and photomet-ric results (Ostro et al. 1990; Miller et al. 2002). Theseproperties can increase the photometric amplitudes.Finally, multiple contact binary objects, includingnear-Earth asteroid (4179) Toutatis, 67P/Churyumov-Gerasimenko and (486958) Arrokoth are thought to haveformed through low-speed impacts of a binary pair, or—in the case of 67P—through low-energy sub-catastrophiccollisions that leave behind a bi-lobate remnant (Huet al. 2018; Jutzi & Benz 2017; Grishin et al. 2020). If
01 Trojans with K2
The incidence of very slow rotators
Nesvorn´y et al. (2020) estimated a rate ∼
15% of veryslow rotators (P ≥
100 h) in the C6 sample. In our sam-ple we identified four new slow rotators: (11351) Leu-cus, (13062) Podarkes, (26486) 2000 AQ231 and (96337)1997 LG2. With those in the C6 sample this means al-together 13 very slowly rotating targets, ∼ τ YORP for all ofthe very slow rotators, as described in the Appendix.For all these targets—which are D ≥
10 km—we find τ YORP ≥ yr. On the other hand, the timescale to no-tably change the rotational state of an asteroid via colli-sions is 2-3 orders of magnitudes shorter than τ YORP forJovian trojans ( ∼ a bin /D (cid:46) Notable targets
We inspected the light curves of the targets individu-ally. Three asteroids from the sample were notable forvarious reasons. In this section we discuss their photo-metric results in more detail.5.4.1. (11351) Leucus
Perhaps the most famous asteroid among the ones pre-sented in this study is (11351) Leucus, which was chosenas one of the flyby targets of the
Lucy mission (Levison& Lucy Science Team 2016; Levison et al. 2019). Leucusbelongs to the extremely slow rotators, as first reportedby French et al. (2013). A more detailed study based onmore extensive ground-based observations led to a re-fined period of P = 445 .
732 hr (Buie et al. 2018). Leu-cus was observed less than half a year later by the K2mission, during the second half of Campaign 11 (in C112or C11b). Unfortunately, the allocated pixels only pro-vided approximately 300 hr of coverage, i.e., less than afull rotation. Nevertheless, we were able to confirm the445.73 hr rotation period of Leucus, even from the par-tial light curve. Preliminary results from stellar occulta- (11351) Leucus - K2 C11 B r i gh t ne ss ( K p m ag ) (11351) Leucus - LCOGT H r ( m ag ) Rotation phase (P = h) Figure 11.
Phase-folded light curves of (11351) Leucus.Top: the K2 data; bottom: ground-based photometry col-lected by Buie et al. (2018). Blue points are individual mea-surements, black points are phase-binned values. tions indicate that Leucus has a non-ellipsoidal shapebut it is not a binary (although the observations do notrule out a distant companion). The light curve shapeand the long period raises the possibility that this aster-oid may be a dissociated binary proposed by Nesvorn´yet al. (2020).We compared the K2 data to the photometry pub-lished by Buie et al. (2018) in Fig. 11. We found virtu-ally no difference in the light curve amplitude, but thiswas not unexpected since a time span of about 7 rota-tions separated the two measurements and the eclipticlongitude of Leucus changed by only about 10 deg duringthat time. We did find a small but significant phase shiftof ≈ .
02 between the two phase curves. This differenceis most likely caused by the different viewing geometriesbetween the Earth and
Kepler .5.4.2. (13062) Podarkes (13062) Podarkes is the principal body of the proposedPodarkes family that belongs to the larger Menelaus clan(Roig et al. 2008). However, the existence of this family,and other proposed, smaller ones, is not universally ac-cepted, and they do not appear in the general study ofasteroid families by Nesvorn´y et al. (2015). Not muchhas been known about Podarkes itself before, and no \ -citizen-science-unistellar-evscope Kalup et al. B r i gh t ne ss ( U S N O - B R sys t e m ) Rotation phase (P = 245 h)(13062) Podarkes
Figure 12.
Phased-folded light curve of (13062) Podarkes,from Campaign 11. B r i gh t ne ss [ K p m ag ] Rotational phase (P = 89.989 h)2001 SC101 19.5 20 20.5 21 0 0.5 1 1.5 2 B r i gh t ne ss [ K p m ag ] Rotational phase (P = 89.989 h)2001 SC101
Figure 13.
Phased-folded light curves of 2001 SC101 as seenby Kepler in C16 ( upper panel ) and C18 ( lower panel ). Therotational period which was used to fold these light curvesis the average of the two independent periods obtained fromeach Campaigns. The two light curves are in the same phase. light curve or rotation data can be found in the LCDB.We determined its rotational period to be 245 hr, whichmeans it is another slow rotator (Fig. 12). The lightcurve has a fairly large amplitude of 0.645 mag, withvery pronounced, V-shaped minima. 5.4.3. (99306) 2001 SC101
This is the only Trojan asteroid that has been ob-served in two different Campaigns during the mission.The fields-of-view of Campaigns 16 and 18 largely over-lapped but C16 was observed in forward-facing modeand then C18 in the normal, backward-facing mode.During the 105-day gap between the measurements of(99306) 2001 SC101, the telescope traveled to the otherside of the Sun, changing the phase angle it saw the as-teroid at from − ◦ to +10 ◦ . The solar elongation ofthe object also changed by 10 . ◦ . The shape of the lightcurve changed quite noticeably between the two obser-vations. The amplitudes also differ slightly, but closeto the level of uncertainties. Unfortunately, these twoepochs are too close in elongation to meaningfully con-strain the shape and rotational axis of (99306), and noother time-resolved photometry has been published. SUMMARYIn this paper, we determined photometric propertiesof 45 Jovian trojans observed in the K2 mission, andpresent phase-folded light curves for 44 targets. We in-creased our sample to 101 asteroids with previous K2Trojan measurements from C6 (Szab´o et al. 2017), andderived their amplitude- and frequency distributions.These statistics can provide information on the binaryfraction and the composition, which then can be com-pared to predictions of several formation models. Wecompared the rotation frequency distribution of the Tro-jan sample to that of other types of asteroids, like mainbelt asteroids or Hildas, which we previously determinedbased on
Kepler (Szab´o et al. 2017, 2020) and TESSdata (P´al et al. 2020). We arrived to the following con-clusions: • While 38% of the periods of K2 Trojans are shorterthan 10 hr, 25% of the sample are slow rotators(P ≥
30 hr), 12% are very slow rotators with peri-ods between 100–600 hr. • Compared to other types of asteroids observed by
Kepler and TESS, we find slow rotators amongHildas (39%) and main belt asteroids (35%) too.However, there is no Trojan with rotational periodless than 5 hr. • There is a possible dichotomy among Trojans andHildas in the rotational periods. In the case ofTrojans, the short-period group can be fitted witha Maxwellian distribution, which describes a col-lisionally evolved sample, while the long-periodgroup clearly falls outside the distribution.
01 Trojans with K2 • The lack of fast rotators can be interpreted as Tro-jans generally having low densities, suggesting icyand/or porous composition. Rotation rate statis-tics indicate that collisional evolution itself wouldbe able to spin Trojans up to ∼ − , whichis similar to the TNO population or the nucleusof comets, supporting the scenario of outer SolarSystem origin. • Based on period (P ≥
30 hr) and amplitude limits(A ≥ ∼
25% rate of bina-ries among our sample, consistent with previousmeasurements (Szab´o et al. 2017). • Assuming equator-on viewing geometry and max-imum light curve amplitude, a strengthless bodymodel indicates that densities do not exceed 1.5g cm − , in agreement with the low densities fromthe spin barrier. We also characterised the possi-bly binary nature of the K2 Trojan sample, as-suming equally sized, spherical, equal-mass anduniform density (1 g cm − ) components. Wefind 32 systems for which both the separation, a bin /D ≥ A max ≥ • (11351) Leucus was chosen as one of the flyby tar-gets of the Lucy mission. We confirm the 445.73 hr rotation period, which was based on extensiveground-based observations by Buie et al. (2018). • We found that (13062) Podarkes – which is theprincipal body of the proposed but not universallyaccepted Podarkes family (Roig et al. 2008) – alsohas a very slow rotational period (245 hr).Our results once again show that the advantages ofcontinuous and precise photometry collected by spacetelescopes extend to Solar System objects and plane-tary science too. With the TESS primary mission al-ready finished and the first extension under way, hugeamounts of new photometric data can be expected: how-ever, the small aperture of TESS will limit its ability tostudy fainter objects (unless temporal resolution is notrequired, see Holman et al. 2019). This puts most of theTrojans out of its reach, and the results obtained by theK2 mission even more valuable.We wish to thank the anonymous referees for theircomments that helped to improve this paper. Our re-sults includes data collected by the K2 mission. Fund-ing for the K2 mission is provided by the NASA Sci-ence Mission directorate. Cs.E.K. was supported bythe ´UNKP-19-2 New National Excellence Program ofthe Ministry of Human Capacities. L.M. was supportedby the Pr´emium Postdoctoral Research Program of theHungarian Academy of Sciences. The research leadingto these results has been supported by the LP2018–7/2020 grant of the Hungarian Academy of Sciencesand by the K–125015 and GINOP–2.3.2–15–2016–00003grant of the Research, Development and Innovation Of-fice (NKFIH), Hungary. This research has made use ofNASA’s Astrophysics Data System Bibliographic Ser-vices.
Facilities:
K2 (Howell et al. 2014), JPL HORIZONS(Giorgini et al. 1996)
Software:
FITSH (P´al 2012), numpy (Oliphant 2006),matplotlib (Hunter 2007), FAMIAS (Zima 2008), gnuplotREFERENCES
Benecchi, S. D., & Sheppard, S. S. 2013, AJ, 145, 124Bottke, W. F., Broˇz, M., O’Brien, D. P., et al. 2015, TheCollisional Evolution of the Main Asteroid Belt, 701–724Bowell, E., Oszkiewicz, D. A., Wasserman, L. H., et al.2014, Meteoritics and Planetary Science, 49, 95Buie, M. W., Zangari, A. M., Marchi, S., Levison, H. F., &Mottola, S. 2018, AJ, 155, 245 Emery, J. P., Marzari, F., Morbidelli, A., French, L. M., &Grav, T. 2015, The Complex History of Trojan Asteroids(University of Arizona Press), 203–220Farinella, P., Vokrouhlick´y, D., & Hartmann, W. K. 1998,Icarus, 132, 378Farkas-Tak´acs, A., Kiss, C., P´al, A., et al. 2017, AJ, 154,119 Kalup et al.
Fraser, W. C., Brown, M. E., Morbidelli, A. r., Parker, A.,& Batygin, K. 2014, ApJ, 782, 100French, L. M., Stephens, Robert, D., Coley, D. R., et al.2013, Minor Planet Bulletin, 40, 198French, L. M., Stephens, R. D., Coley, D., Wasserman,L. H., & Sieben, J. 2015, Icarus, 254, 1Giorgini, J. D., Yeomans, D. K., Chamberlin, A. B., et al.1996, in Bulletin of the American Astronomical Society,Vol. 28, AAS/Division for Planetary Sciences MeetingAbstracts
01 Trojans with K2 Thomas, P. C., Veverka, J., Bell, J. F., et al. 1999, Icarus,140, 17ˇCapek, D., & Vokrouhlick´y, D. 2004, Icarus, 172, 526Vinogradova, T. A., & Chernetenko, Y. A. 2015, SolarSystem Research, 49, 391Vokrouhlick´y, D., & ˇCapek, D. 2002, Icarus, 159, 449Warner, B. D., Harris, A. W., & Pravec, P. 2009, Icarus,202, 134 Wong, I., & Brown, M. E. 2016, AJ, 152, 90Yoshida, F., & Nakamura, T. 2005, AJ, 130, 2900Zima, W. 2008, Communications in Asteroseismology, 157,387 Kalup et al.
APPENDIXYORP AND COLLISIONAL TIMESCALESThe timescale to significantly change the rotation state of the asteroid by the YORP effect can be estimated, followingVokrouhlick´y & ˇCapek (2002) and ˇCapek & Vokrouhlick´y (2004), as: τ Y ORP = (cid:12)(cid:12)(cid:12)(cid:12) dfdt (cid:12)(cid:12)(cid:12)(cid:12) f = 180 M yr (cid:18) r h . au (cid:19) (cid:18) D . km (cid:19) (cid:18) ρ . g cm − (cid:19) (1)To estimate the YORP timescale for Jovian Trojans we used r h = 5.2 au, D = the values from the actual sizeestimates, and a bulk density of ρ = 1 g cm − . There is a considerable uncertainty in the density of the Jovian Trojans,but there are indications that their densities is lower than that of main belt asteroids (e.g. Marchis et al. 2006).Farinella et al. (1998) estimated the timescale to change a rotation of an asteroid by collisions, τ coll , in the main belt.Using this scheme, we estimated the τ coll for the Jovian Trojans considering the different number density of objectsin the main belt and in the Jovian Trojan swarms. Nakamura & Yoshida (2008) obtained for the L swarm that thenumber of Jovian Trojans larger than 2 km is N(D > L ≈ · , while this is N(D > MB ≈ · in the mainbelt (Bottke et al. 2015). Considering the larger space the main belt asteroids occupy this leads to surface densitiesof σ (D > L ≈ · − km − and σ (D > MB ≈ · − km − . We rescaled the Farinella et al. (1998) τ coll estimate by using the ratio of these surface density values, assuming the same velocity dispersion (∆v ≈ − ) inthe two populations and that the targets and impactors have the same densities. With these assumptions the collisionaltimescale is: τ coll = 28 M yr · (cid:18) R km (cid:19) (2)The τ Y ORP and τ coll timescales of our Jovian Trojan sample are very different: τ coll = 20–66 Myr, while τ Y ORP = 1–100 Gyr, i.e. the YORP timescales are 2-3 orders of magnitudes longer. This suggests that YORP does not playa significant role in the spin rate evolution of Jovian Trojans in our sample (D >
10 km), and it cannot slow downasteroids to very slow rotation before their rotational state would be overwritten by collisions.SUMMARY TABLES AND PLOTSThe appendix contains the summary tables and plots. Table 3 lists the start and end dates of each light curve, thenumber of data points we used in our analysis, and the duty cycle values. The latter is the ratio of the number of datapoints we use over the number of data points that would fill the length of the observation. Dates are light receivaltimes at the spacecraft. Table 4 summarizes the results of our analysis. We provide rotation periods and amplitudesplus associated uncertainties, as well as the node designations and campaign numbers for each target. For (99306)3001 SC101 we list the values obtained from the Campaign 16 and 18 data separately. Finally, in Figures 14–19 wedisplay the light curves, phase curves and residual dispersion spectra of each asteroid.
Table 3 . Observation data and duty cycles. Dates are JD–2450000 (d).Name Start date End date Length (d) No. points Duty cycle(1) (2) (3) (4) (5) (6)(1871) Astyanax 8268.869 8289.936 21.07 961 0.929(4836) Medon 7669.184 7678.788 9.60 453 0.961(5144) Achates 8274.754 8295.841 21.09 1022 0.987(5209) 1989 CW1 7670.267 7679.339 9.07 364 0.817(5638) Deikoon 8296.986 8302.360 5.37 251 0.951
Table 3 continued
01 Trojans with K2 Table 3 (continued)
Name Start date End date Length (d) No. points Duty cycle(1) (2) (3) (4) (5) (6)(9030) 1989 UX5 8199.272 8215.333 16.06 741 0.940(10247) Amphiaraos 8369.484 8384.605 15.12 607 0.818(10664) Phemios 7664.116 7674.701 10.58 218 0.420(11273) 1988 RN11 8258.693 8279.842 21.15 977 0.941(11351) Leucus 7714.935 7723.925 8.99 315 0.714(11397) 1998 XX93 7687.124 7696.258 9.13 333 0.743(12658) Peiraios 7663.646 7674.333 10.69 361 0.688(13062) Podarkes 7669.327 7679.012 9.69 374 0.787(13229) Echion 7666.487 7675.559 9.07 284 0.638(15651) Tlepolemos 7664.300 7674.987 10.69 335 0.639(18071) 2000 BA27 7668.060 7678.726 10.67 392 0.749(18263) Anchialos 7682.772 7689.495 6.72 231 0.700(23126) 2000 AK95 8370.301 8384.380 14.08 523 0.757(23135) 2000 AN146 7666.180 7676.744 10.56 468 0.902(24444) 2000 OP32 8266.560 8287.852 21.29 903 0.864(26486) 2000 AQ231 7659.539 7669.776 10.24 281 0.559(31819) 1999 RS150 7926.014 7936.353 10.34 484 0.954(34835) 2001 SZ249 8269.094 8289.691 20.60 919 0.909(39463) Phyleus 7682.772 7686.941 4.17 151 0.738(60421) 2000 CZ31 8369.484 8383.992 14.51 474 0.666(76820) 2000 RW105 7930.877 7943.076 12.20 534 0.892(76835) 2000 SH255 7930.550 7942.851 12.30 594 0.984(77891) 2001 SM232 8206.077 8227.184 21.11 988 0.953(96337) 1997 LG2 8267.643 8288.669 21.03 896 0.868(99306) 2001 SC101 (C16) 8141.690 8152.377 10.69 501 0.955(99306) 2001 SC101 (C18) 8257.692 8278.841 21.15 895 0.862(100475) 1996 TZ36 8369.484 8376.431 6.95 277 0.812(111113) 2001 VK85 7932.777 7943.750 10.97 463 0.860(111231) 2001 WM60 7938.519 7941.053 2.53 122 0.981(116567) 2004 BV84 7931.674 7941.870 10.20 458 0.915(131635) 2001 XW71 7926.381 7938.478 12.10 513 0.864(134419) Hippothous 8268.930 8289.916 20.99 1014 0.984(151883) 2003 WQ25 8263.740 8284.787 21.05 839 0.812(163731) 2003 KD 8369.484 8379.292 9.81 400 0.831(175471) 2006 QA138 8369.484 8375.839 6.35 289 0.926(200037) 2007 RW105 8369.464 8384.973 15.51 583 0.766(246550) 2008 SO47 8010.609 8023.666 13.06 446 0.696(247019) 1999 XJ55 8023.135 8031.247 8.11 368 0.924(296787) 2009 UR154 8011.508 8025.668 14.16 453 0.652(301013) 2008 JJ18 7925.748 7936.700 10.95 225 0.418(316484) 2010 VM61 8026.935 8040.033 13.10 437 0.680 Kalup et al.
Table 4 . Rotation periods and amplitudes. For (99306) 2001 SC101, we list results from each campaigns individually(starred values).Name Period (h) ∆P (h) Amplitude (mag) ∆A (mag) Lagrange Node Campaign(1) (2) (3) (4) (5) (6) (7)(1871) Astyanax 6.517 0.001 0.140 0.005 L5 C18(4836) Medon 9.844 0.004 0.199 0.007 L4 C11(5144) Achates 5.955 0.0004 0.176 0.003 L5 C18(5209) 1989 CW1 11.59 0.005 0.347 0.032 L4 C11(5638) Deikoon 9.146 0.029 0.077 0.006 L5 C18(9030) 1989 UX5 6.299 0.001 0.488 0.011 L5 C17(10247) Amphiaraos 26.24 0.017 0.264 0.011 L4 C19(10664) Phemios 7.874 0.009 0.260 0.018 L4 C11(11273) 1988 RN11 8.290 0.002 0.160 0.008 L5 C18(11351) Leucus 445.73 15.8 0.77 0.23 L4 C11(11397) 1998 XX93 11.374 0.011 0.279 0.015 L4 C11(12658) Peiraios 8.151 0.003 0.562 0.033 L4 C11(13062) Podarkes 245 1.3 0.645 0.015 L4 C11(13229) Echion 8.43 0.010 0.259 0.034 L4 C11(15651) Tlepolemos 5.819 0.002 0.230 0.015 L4 C11(18071) 2000 BA27 36.0 0.054 0.368 0.022 L4 C11(18263) Anchialos 10.345 0.034 0.426 0.050 L4 C11(23126) 2000 AK95 5.526 0.001 0.186 0.012 L4 C19(23135) 2000 AN146 8.716 0.003 0.308 0.011 L4 C11(24444) 2000 OP32 7.083 0.001 0.220 0.006 L5 C18(26486) 2000 AQ231 440 5.1 0.470 0.072 L4 C11(31819) 1999 RS150 34.828 0.016 0.595 0.005 L5 C14(34835) 2001 SZ249 7.742 0.001 0.306 0.017 L5 C18(39463) Phyleus 10.46 0.041 0.408 0.049 L4 C11(60421) 2000 CZ31 — — < .
05 — L4 C19(76820) 2000 RW105 7.060 0.003 0.388 0.021 L5 C14(76835) 2000 SH255 5.740 0.002 0.202 0.017 L5 C14(77891) 2001 SM232 12.945 0.008 0.229 0.024 L5 C17(96337) 1997 LG2 366.412 1.6 0.646 0.018 L5 C18(99306) 2001 SC101* 90.056 0.537 0.226 0.024 L5 C16(99306) 2001 SC101* 89.787 0.317 0.255 0.017 L5 C18(100475) 1996 TZ36 6.636 0.013 0.231 0.041 L4 C19(111113) 2001 VK85 41.631 0.171 0.297 0.028 L5 C14(111231) 2001 WM60 12.83 0.202 0.317 0.033 L5 C14(116567) 2004 BV84 6.778 0.013 0.065 0.015 L5 C14(131635) 2001 XW71 5.739 0.002 0.309 0.027 L5 C14(134419) Hippothous 7.559 0.002 0.539 0.016 L5 C18
Table 4 continued
01 Trojans with K2 Table 4 (continued)
Name Period (h) ∆P (h) Amplitude (mag) ∆A (mag) Lagrange Node Campaign(1) (2) (3) (4) (5) (6) (7)(151883) 2003 WQ25 5.1243 0.0005 0.083 0.028 L5 C18(163731) 2003 KD 8.103 0.003 0.317 0.015 L4 C19(175471) 2006 QA138 63.66 0.994 0.323 0.036 L4 C19(200037) 2007 RW105 36.47 0.046 0.964 0.060 L4 C19(246550) 2008 SO47 7.787 0.002 0.742 0.047 L4 C15(247019) 1999 XJ55 8.602 0.019 0.070 0.017 L4 C15(296787) 2009 UR154 72.948 0.521 0.397 0.025 L4 C15(301013) 2008 JJ18 12.542 0.011 0.890 0.074 L5 C14(316484) 2010 VM61 18.857 0.045 0.650 0.112 L4 C15 Kalup et al. P ha s e d i s pe r s i on Frequency [c/d](1871) Astyanax 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](4836) Medon 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](5144) Achates 0.125 0.13 0.135 0.14 0.145 0.15 0.155 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](5209) 1989 CW1 0.0325 0.033 0.0335 0.034 0.0345 0.035 0.0355 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](5638) Deikoon 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](9030) 1989 UX5 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1 0.105 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](10247) Amphiaraos 0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](10664) Phemios 19 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=6.517 hr)(1871) Astyanax 17.5 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=9.844 hr)(4836) Medon 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=5.955 hr)(5144) Achates 17.5 18 18.5 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=11.594 hr)(5209) 1989 CW1 18.5 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=9.146 hr)(5638) Deikoon 18.5 19 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=6.299 hr)(9030) 1989 UX5 19 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=26.238 hr)(10247) Amphiaraos 19 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=7.873 hr)(10664) Phemios 19 -10 -8 -6 -4 -2 0 2 4 6 8 10 K ep l e r m ag JD-2458279.065341(1871) Astyanax 17.5 -4 -2 0 2 4 K ep l e r m ag JD-2457673.883478(4836) Medon 17.5 -10 -8 -6 -4 -2 0 2 4 6 8 10 K ep l e r m ag JD-2458285.297590(5144) Achates 18 18.5 19 -4 -2 0 2 4 K ep l e r m ag JD-2457674.414751(5209) 1989 CW1 18.5 -2 0 2 K ep l e r m ag JD-2458299.744148(5638) Deikoon 18.5 19 -8 -6 -4 -2 0 2 4 6 8 K ep l e r m ag JD-2458207.282114(9030) 1989 UX5 19 -6 -4 -2 0 2 4 6 8 K ep l e r m ag JD-2458375.757123(10247) Amphiaraos 19 -8 -6 -4 -2 0 2 K ep l e r m ag JD-2457672.187489(10664) Phemios
Figure 14.
Jovian trojan asteroids observed by the K2 mission between Campaign 11-19. Left: Raw light curves with errors.Middle: Rectified, phase-folded and binned phase curves. Right: Residual dispersion frequency spectra. To fold the raw lightcurves, we used the half of the main frequency (double-peak solution) in all cases.
01 Trojans with K2 P ha s e d i s pe r s i on Frequency [c/d](11273) 1988 RN11 0.464 0.466 0.468 0.47 0.472 0.474 0.476 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](11351) Leucus 0.075 0.08 0.085 0.09 0.095 0.1 0.105 0.11 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](11397) 1998 XX93 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](12658) Peiraios 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](13062) Podarkes 0.136 0.138 0.14 0.142 0.144 0.146 0.148 0.15 0.152 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](13229) Echion 0.08 0.085 0.09 0.095 0.1 0.105 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](15651) Tlepolemos 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](18071) 2000 BA27 19 19.5 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=8.290 hr)(11273) 1988 RN11 18 18.5 19 19.5 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=445.765 hr)(11351) Leucus 18 18.5 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=11.374 hr)(11397) 1998 XX93 18.5 19 19.5 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=8.151 hr)(12658) Peiraios 18.5 19 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=244.898 hr)(13062) Podarkes 19 19.5 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=8.430 hr)(13229) Echion 18.5 19 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=5.819 hr)(15651) Tlepolemos 19.5 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=35.977 hr)(18071) 2000 BA27 19 19.5 -10 -8 -6 -4 -2 0 2 4 6 8 10 K ep l e r m ag JD-2458269.727185(11273) 1988 RN11 18.5 19 19.5 -4 -2 0 2 4 K ep l e r m ag JD-2457719.348247(11351) Leucus 18 18.5 -4 -2 0 2 4 K ep l e r m ag JD-2457691.517678(11397) 1998 XX93 18.5 19 19.5 -4 -2 0 2 4 6 K ep l e r m ag JD-2457668.366405(12658) Peiraios 18.5 19 -4 -2 0 2 4 K ep l e r m ag JD-2457674.333017(13062) Podarkes 19 19.5 -4 -2 0 2 4 K ep l e r m ag JD-2457670.981906(13229) Echion 19 -4 -2 0 2 4 6 K ep l e r m ag JD-2457668.407272(15651) Tlepolemos 19.5 -4 -2 0 2 4 6 K ep l e r m ag JD-2457672.923098(18071) 2000 BA27
Figure 15.
Jovian trojan asteroids observed by the K2 mission between Campaign 11-19, continued. Columns are same as inFig. 14. Kalup et al. P ha s e d i s pe r s i on Frequency [c/d](18263) Anchialos 0.062 0.064 0.066 0.068 0.07 0.072 0.074 0.076 0.078 0.08 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](23126) 2000 AK95 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](23135) 2000 AN146 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](24444) 2000 OP32 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](26486) 2000 AQ231 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](31819) 1999 RS150 0.11 0.115 0.12 0.125 0.13 0.135 0.14 0.145 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](34835) 2001 SZ249 0.19 0.195 0.2 0.205 0.21 0.215 0.22 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](39463) Phyleus 19 19.5 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=10.345 hr)(18263) Anchialos 19.5 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=5.526 hr)(23126) 2000 AK95 17.5 18 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=8.716 hr)(23135) 2000 AN146 19 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=7.083 hr)(24444) 2000 OP32 20 20.5 21 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=440.044 hr)(26486) 2000 AQ231 19 19.5 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=34.828 hr)(31819) 1999 RS150 20 20.5 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=7.742 hr)(34835) 2001 SZ249 20.5 21 21.5 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=10.461 hr)(39463) Phyleus 19 19.5 -2 0 2 4 K ep l e r m ag JD-2457685.673667(18263) Anchialos 19.5 -6 -4 -2 0 2 4 6 8 K ep l e r m ag JD-2458375.981893(23126) 2000 AK95 17.5 18 -6 -4 -2 0 2 4 K ep l e r m ag JD-2457671.737950(23135) 2000 AN146 19 19.5 -10 -8 -6 -4 -2 0 2 4 6 8 10 K ep l e r m ag JD-2458277.205884(24444) 2000 OP32 20 20.5 -6 -4 -2 0 2 4 K ep l e r m ag JD-2457665.260497(26486) 2000 AQ231 19 19.5 -4 -2 0 2 4 K ep l e r m ag JD-2457931.224202(31819) 1999 RS150 20 20.5 -10 -8 -6 -4 -2 0 2 4 6 8 10 K ep l e r m ag JD-2458278.963173(34835) 2001 SZ249 21 21.5 -2 0 2 K ep l e r m ag JD-2457684.733721(39463) Phyleus
Figure 16.
Jovian trojan asteroids observed by the K2 mission between Campaign 11-19, continued. Columns are same as inFig. 14.
01 Trojans with K2 P ha s e d i s pe r s i on Frequency [c/d](60421) 2000 CZ31 0.135 0.14 0.145 0.15 0.155 0.16 0.165 0.17 0.175 0.18 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](76820) 2000 RW105 0.092 0.094 0.096 0.098 0.1 0.102 0.104 0.106 0.108 0.11 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](76835) 2000 SH255 0.194 0.196 0.198 0.2 0.202 0.204 0.206 0.208 0.21 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](77981) 2001 SM232 0.09 0.095 0.1 0.105 0.11 0.115 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](99306) 2001 SC101 0.13 0.132 0.134 0.136 0.138 0.14 0.142 0.144 0.146 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](99306) 2001 SC101 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](96337) 1997 LG2 0.147 0.148 0.149 0.15 0.151 0.152 0.153 0.154 0.155 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](100475) 1996 TZ36 20 20.5 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=7.060 hr)(76820) 2000 RW105 20 20.5 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=5.740 hr)(76835) 2000 SH255 20 20.5 21 21.5 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=12.941 hr)(77981) 2001 SM232 20 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=90.124 hr)(99306) 2001 SC101 20 20.5 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=89.820 hr)(99306) 2001 SC101 20 20.5 21 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=366.412 hr)(96337) 1997 LG2 20 20.5 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=11.594 hr)(100475) 1996 TZ36 20 -8 -6 -4 -2 0 2 4 6 K ep l e r m ag JD-2458377.432679(60421) 2000 CZ31 20 20.5 -6 -4 -2 0 2 4 6 K ep l e r m ag JD-2457936.925177(76820) 2000 RW105 20 20.5 -6 -4 -2 0 2 4 6 K ep l e r m ag JD-2457936.700407(76835) 2000 SH255 20 20.5 21 21.5 -10 -8 -6 -4 -2 0 2 4 6 8 10 K ep l e r m ag JD-2458216.272899(77981) 2001 SM232 20 -4 -2 0 2 4 K ep l e r m ag JD-2458146.982579(99306) 2001 SC101 20 20.5 -10 -8 -6 -4 -2 0 2 4 6 8 10 K ep l e r m ag JD-2458268.153798(99306) 2001 SC101 20 20.5 21 -10 -8 -6 -4 -2 0 2 4 6 8 10 K ep l e r m ag JD-2458277.900626(96337) 1997 LG2 20.5 21-4 -2 0 2 K ep l e r m ag JD-2458372.978153(100475) 1996 TZ36
Figure 17.
Jovian trojan asteroids observed by the K2 mission between Campaign 11-19, continued. Columns are same as inFig. 14. Kalup et al. P ha s e d i s pe r s i on Frequency [c/d](111113) 2001 VK85 0.164 0.165 0.166 0.167 0.168 0.169 0.17 0.171 0.172 0.173 0.174 0.175 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](111231) 2001 WM60 0.0935 0.094 0.0945 0.095 0.0955 0.096 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](116567) 2004 BV84 0.15 0.155 0.16 0.165 0.17 0.175 0.18 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](131635) 2001 XW71 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](134419) Hippothous 0.1376 0.1378 0.138 0.1382 0.1384 0.1386 0.1388 0.139 0.1392 0.1394 0.1396 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](151883) 2003 WQ25 0.095 0.1 0.105 0.11 0.115 0.12 0.125 0.13 0.135 0.14 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](163731) 2003 KD 0.152 0.154 0.156 0.158 0.16 0.162 0.164 0.166 0.168 0.17 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](175471) 2006 QA138 20.5 21 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=41.631 hr)(111113) 2001 VK85 20.5 21 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=12.831 hr)(111231) 2001 WM60 21 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=6.778 hr)(116567) 2004 BV84 20.5 21 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=5.739 hr)(131635) 2001 XW71 20 20.5 21 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=7.559 hr)(134419) Hippothous 20.5 21 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=5.124 hr)(151883) 2003 WQ25 20 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=8.103 hr)(163731) 2003 KD 20 20.5 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=63.660 hr)(175471) 2006 QA138 20.5 21 -6 -4 -2 0 2 4 K ep l e r m ag JD-2457938.907236(111113) 2001 VK85 20.5 21 0 K ep l e r m ag JD-2457939.765448(111231) 2001 WM60 20.5 21 -4 -2 0 2 4 K ep l e r m ag JD-2457936.884310(116567) 2004 BV84 20.5 21 -4 -2 0 2 4 6 K ep l e r m ag JD-2457930.529459(131635) 2001 XW71 20 20.5 21 -10 -8 -6 -4 -2 0 2 4 6 8 10 K ep l e r m ag JD-2458279.433146(134419) Hippothous 20.5 21 -10 -8 -6 -4 -2 0 2 4 6 8 10 K ep l e r m ag JD-2458274.038675(151883) 2003 WQ25 20 20.5 -4 -2 0 2 4 K ep l e r m ag JD-2458374.183736(163731) 2003 KD 20.5 21 -2 0 2 K ep l e r m ag JD-2458372.773817(175471) 2006 QA138
Figure 18.
Jovian trojan asteroids observed by the K2 mission between Campaign 11-19, continued. Columns are same as inFig. 14.
01 Trojans with K2 P ha s e d i s pe r s i on Frequency [c/d](200037) 2007 RW105 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](246550) 2008 SO47 0.057 0.0575 0.058 0.0585 0.059 0.0595 0.06 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](247019) 1999 XJ55 0.176 0.178 0.18 0.182 0.184 0.186 0.188 0.19 0.192 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](296787) 2009 UR154 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](301013) 2008 JJ18 0.445 0.45 0.455 0.46 0.465 0.47 0 2 4 6 8 10 12 14 16 18 20 P ha s e d i s pe r s i on Frequency [c/d](316484) 2010 VM61 20 20.5 21 21.5 22 22.5 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=36.474 hr)(200037) 2007 RW105 20 20.5 21 21.5 22 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=7.787 hr)(246550) 2008 SO47 19.5 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=8.602 hr)(247019) 1999 XJ55 20.5 21 21.5 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=72.948 hr)(296787) 2009 UR154 20 20.5 21 21.5 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=12.542 hr)(301013) 2008 JJ18 20 20.5 21 21.5 22 22.5 0 0.5 1 1.5 2 K ep l e r m ag Phase (P=18.857 hr)(316484) 2010 VM61 20 20.5 21 21.5 22 22.5 -6 -4 -2 0 2 4 6 8 K ep l e r m ag JD-2458376.390565(200037) 2007 RW105 20 20.5 21 21.5 22 -8 -6 -4 -2 0 2 4 K ep l e r m ag JD-2458019.027374(246550) 2008 SO47 19.5 -4 -2 0 2 4 K ep l e r m ag JD-2458027.384717(247019) 1999 XJ55 20.5 21 21.5 -8 -6 -4 -2 0 2 4 K ep l e r m ag JD-2458020.968566(296787) 2009 UR154 20 20.5 21 21.5 22 -6 -4 -2 0 2 4 K ep l e r m ag JD-2457931.632874(301013) 2008 JJ18 20 20.5 21 21.5 22 22.5 -6 -4 -2 0 2 4 6 K ep l e r m ag JD-2458033.453497(316484) 2010 VM61