The color-magnitude distribution of Hilda asteroids: Comparison with Jupiter Trojans

0.47 MB·

by  Ian Wong

Draft version January 3, 2017
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
Keywords: minor planets, asteroids: general
THE COLOR-MAGNITUDE DISTRIBUTION OF HILDA ASTEROIDS: COMPARISON WITH JUPITER
TROJANS
Ian Wong and Michael E. Brown
DivisionofGeologicalandPlanetarySciences,CaliforniaInstituteofTechnology,Pasadena,CA91125,USA;[email protected]
Draft version January 3, 2017
ABSTRACT
Current models of Solar System evolution posit that the asteroid populations in resonance with
Jupiter are comprised of objects scattered inward from the outer Solar System during a period of
dynamical instability. In this paper, we present a new analysis of the absolute magnitude and optical
7
color distribution of Hilda asteroids, which lie in the 3:2 mean motion resonance with Jupiter, with
1
the goal of comparing the bulk properties with previously published results from an analogous study
0
of Jupiter Trojans. We report an updated power law fit of the Hilda magnitude distribution through
2
H =14. UsingphotometricdatalistedintheSloanMovingObjectCatalog,weconfirmthepreviously-
n
reportedstrongbimodalityinvisiblespectralslopedistribution,indicativeoftwosub-populationswith
a
differingsurfacecompositions. Whenconsideringcollisionalfamiliesseparately,wefindthatcollisional
J
fragments follow a unimodal color distribution with spectral slope values consistent with the bluer of
2 thetwosub-populations. ThecolordistributionsofHildasandTrojansarecomparableandconsistent

with a scenario in which the color bimodality in both populations developed prior to emplacement
]
P into their present-day locations. We propose that the shallower magnitude distribution of the Hildas
is a result of an initially much larger Hilda population, which was subsequently depleted as smaller
E
bodies were preferentially ejected from the narrow 3:2 resonance via collisions. Altogether, these
.
h observations provide a strong case supporting a common origin for Hildas and Trojans as predicted
p by current dynamical instability theories of Solar System evolution.
-
o
r
t 1. INTRODUCTION isticsofflat,featurelessopticalandnear-infraredspectra
s with reddish colors (e.g., Dahlgren & Lagerkvist 1995;
a Over the past few decades, the classical picture of So-
Dotto et al. 2006; Fornasier et al. 2007; Marsset et al.
[ lar System formation and evolution, in which planets
2014)andsimilar,verylowalbedos(e.g.,Fern´andezetal.
formedandsmoothlymigratedtotheirpresent-dayloca-
1 2003, 2009; Ryan & Woodward 2011). In addition, both
tions within the protoplanetary disk, has been beset by
v HildasandTrojansarenotableinhavingabimodalcolor
significant challenges. The unexpectedly high eccentric-
7 distribution. Analyses of spectral slopes derived from
ities and inclinations of the giant planets, the dynami-
6 photometry contained in the Sloan Digital Sky Survey
cally excited orbital distribution of the Kuiper Belt, and
3 Moving Object Catalog (SDSS-MOC) for both Hildas
the irregular satellites of Jupiter and Saturn are among
0 (Gil-Hutton & Brunini 2008) and Trojans (e.g., Roig
an increasing body of observations that point toward a
0 et al. 2008; Wong et al. 2014) demonstrate a clear bifur-
chaotic restructuring of the Solar System orbital archi-
.
1 tecture after the dispersal of the protoplanetary disk. cation in the optical color distribution and indicate the
0 presence of two classes of objects within the Hildas and
Current theories of Solar System evolution describe a
7 Trojans. Thebimodalityinopticalcolorissupportedby
scenario in which Jupiter and Saturn crossed a mean-
1 bimodality in the infrared reflectivity distribution mea-
motion resonance, setting off a period of dynamical in-
: sured by WISE and NEOWISE for both Hildas (Grav
v stability throughout the middle and outer Solar System
et al. 2012b) and Trojans (Grav et al. 2012a).
i (e.g., Morbidelli et al. 2005). Simulations have shown
X The most direct way of comparing two populations is
thattheprimordialminorbodypopulationsinresonance
by studying their bulk properties, namely, the absolute
r withJupiter(HildasandJupiterTrojans)werefirstemp-
a magnitude distribution and the color distribution. For
tied during this turbulent episode and then replaced
a population with a narrow range of albedos, such as
almost exclusively with planetesimals scattered inward
the Hildas and Trojans, the magnitude distribution is a
fromtheregionbeyondtheicegiants(Gomesetal.2005;
good proxy for the size distribution and contains infor-
Roig & Nesvorny´ 2015). The major implication of these
mation about both the formation environment and the
models is that Kuiper Belt objects, Trojans, and Hildas
subsequent collisional evolution of the population. The
alloriginatedwithinasingleprogenitorpopulationinthe
color distribution reveals the diversity of surface types
outer Solar System and should therefore be largely iden-
and also provides constraints on models of the composi-
tical. By comparing the observable properties of Hildas
tion and origin of objects within the population.
and Trojans, one can evaluate their similarities and/or
In Wong et al. (2014), we carried out an in-depth
differences and thereby empirically test current dynami-
study of the color-magnitude distribution of Trojans. In
cal instability models.
this paper, we present an analogous study for Hildas,
Recent progress in our understanding of Hildas and
in order to obtain a point of reference for comparing
Trojanshasalreadyuncoveredmanynotablesimilarities.
the two populations. We report fits to the total Hilda
Objectsinbothpopulationssharethegeneralcharacter-
magnitude distribution and provide a detailed analysis
2
104
4.05 Total
Hilda family
AU)3.95 103 Schubart family
(a3.85 er
b
m
3.75 nu
20 ve 102
ati
ul
15 m
u
C
()i◦10 101
5
0 100
0.00 0.05 0.10 0.15 0.20 0.25 0.303.75 3.85 3.95 4.05 6 8 10 12 14 16
e a (AU) Absolute magnitude
Fig. 1.— Distribution of the 3801 objects in our Hilda dataset, Fig. 2.— Cumulative absolute magnitude distributions of the
plotted in the space of semi-major axis (a), eccentricity (e), and total Hilda population (white), as well as the Hilda and Schubart
inclination (i). Objects belonging to the Hilda and Schubart col- collisionalfamiliesindividually(magentaandyellow,respectively).
lisional families are denoted by magenta and yellow dots, respec- Thebest-fitpowerlawcurvesdescribingthedistributionsareover-
tively;backgroundHildasaredenotedbybluedots. plotted(dashedgreenlines).
2.1. Magnitude distributions
of the updated color distribution, as derived from the
newest fourth release of the Sloan Moving Object Cat- For each object, we set the absolute magnitude to the
alog (SDSS-MOC4). Special attention is given to ex- value listed in the Asteroid Orbital Elements Databse.
ploring the properties of individual collisional families. The cumulative absolute magnitude distribution of the
The results of our Hilda analysis are compared with our total Hilda population is shown in Figure 2. The mag-
previously published Trojan results and discussed in re- nitude distribution has the characteristic shape seen in
lation to collisional evolution, surface composition, and many minor body populations, with a steeper slope at
dynamicalconsiderationswithintheframeworkofrecent large sizes transitioning to a shallower slope at interme-
dynamical instability models of Solar System evolution diate sizes.
The gentle rollover at H ∼ 16 reflects the onset of in-
2. DATAANDANALYSIS completenessintheHildadataset. InouranalysisofTro-
jans, we determined the completeness limit of the MPC
In this section, we present our analysis of the absolute
Trojan dataset to be H ∼ 11.3 and were able to cor-
magnitudeandopticalcolordistributionsoftheHildaas-
rect the shape of the magnitude distribution for incom-
teroids. The methods used are mostly identical to those
pleteness at fainter magnitudes by utilizing the deeper
described in detail in our previously published analysis
SDSS-MOC4 dataset, which we calculated to be com-
of Jupiter Trojans (see Wong et al. 2014, and references
pleteforTrojansthroughH =12.3. InthecaseofHildas,
therein).
however,theonsetofincompletenessintheSDSS-MOC4
In selecting for Hilda asteroids, we have applied the
dataset occurs at a brighter magnitude (H ∼ 14) than
following constraints in orbital parameter space, which
the MPC dataset, and as such, we are unable to correct
areusedbytheIAUMinorPlanetCenter(MPC)intheir
for incompleteness in the total magnitude distribution.
definition of Hildas: 3.7 ≤ a ≤ 4.1 AU, e ≤ 0.3, and
i≤20◦. Querying theMPC database with these criteria In this paper, we have chosen a conservative upper limit
forouranalysisatH =14. Varyingthislimitby0.5mag
results in a total count of 3801 Hildas (as of October
ineitherdirectiondoesnotsignificantlyaffectthedistri-
2016). Using a less stringent criterion (e.g., extending
the maximum values of (a,e,i) to (4.2 AU,0.4,30◦), as bution fits.
We fit the total differential magnitude distribution,
in Grav et al. 2012b) does not appreciably increase the
Σ(H), to a single power law of the form
overallnumberofHildasanddoesnotsignificantlyaffect
the results of our analysis. Σ(α ,H |H)=10α(H−H0), (1)
Studyingthecolor-magnitudedistributionofHildacol- 1 0
lisional families is of particular relevance in our under- where α is the slope of the distribution, and H is the
0
standingofthecompositionandevolutionofthepopula- threshold magnitude used to properly normalize the dis-
tion as a whole. There exist two major collisional fami- tribution to fit the data.
lieswithintheHildas—theHildaandSchubartfamilies. The best-fit parameter values and 1σ uncertainties
We have created lists of family members using the tab- were computed using a Markov Chain Monte Carlo
ulated results in Nesvorny´ et al. (2015), which identify (MCMC)ensemblesampler. Forthetotalmagnitudedis-
385membersoftheHildafamilyand350membersofthe tribution,thebest-fitparametervaluesareα=0.34+0.02
−0.01
Schubart family. The distribution of Hildas in (a,e,i) andH =6.42±0.29. Wealsoexperimentedwithfitting
0
space is illustrated in Figure 1, with the location of the the total magnitude distribution with a four-parameter
two collisional families highlighted. brokenpowerlaw(e.g.,Wongetal.2014;Wong&Brown
3
2015);however,theadditionofasecondpowerlawslope
is strongly disfavored by the Bayesian Information Cri-
45
terion(∆BIC=8.4;BIC≡−2log(L)+klog(n),whereL 40
is the likelihood of the best-fit solution, k is the number 35
of free parameters, and n is the number of data points). 30
nt25
We also fit the magnitude distributions of the Hilda u
o20
and Schubart collisional families, which are plotted in C15
Figure 2. The best-fit parameters are α=0.40+0.04 and 10
−0.03 5
H = 8.77+0.36 for the Hilda family, and α = 0.43+0.07
0 −0.37 −0.03 0
andH =10.23+0.39 fortheSchubartfamily. Theslopes 30 Background
of the0Hilda an−d0S.5c0hubart collisional family magnitude 25 Hilda family
Schubart family
distributions are steeper than the overall population (at nt20
u
the1.7σand2.5σlevels,respectively),andareconsistent o15
C
with the range of power law slopes derived from numeri- 10
cal simulations of asteroid fragmentation (e.g., α=0.44– 5
0.54 in Jutzi et al. 2010).
0
For each power law fit, we sampled the best-fit distri- 4 2 0 2 4 6 8 10 12 14
bution to create a model magnitude distribution of the Spectral slope (10−5Å−1)
same size as the respective population within the mag-
nitude range under consideration (H < 14). We carried Fig. 3.— Top panel: the overall spectral slope distribution of
Hildas, as derived from SDSS-MOC4 photometry, demonstrating
outatwo-sampleAnderson-Darlingtest,whichevaluates
arobustcolorbimodalitythatdividesthepopulationintoless-red
the null hypothesis that the model distribution and the andredobjects. Bottompanel: thespectralslopedistributionsfor
data are drawn from the same underlying distribution. Hilda and Schubart family members, as well as background non-
In all cases, we could not reject the null hypothesis at family members. Note that the background color distribution is
bimodal,whiletheindividualcollisionalfamilycolordistributions
a confidence level greater than 50%, demonstrating that
arebothunimodal.
themodeldistributionfitsareastatisticallygoodmatch
to the data.
theHildapopulationiscomprisedoftwotypesofobjects,
withcharacteristicallydifferentsurfacecolors. Following
2.2. Color distribution and sub-populations
theterminologyinWongetal.(2014),werefertotheseas
The SDSS-MOC4 lists photometric measurements of the less-red (LR) and red (R) Hildas. We did not detect
minor bodies in the u, g, r, i, z bands. We queried the any significant correlations between spectral slope and
database for Hildas and identified 275 objects that were any orbital parameter, which demonstrates that LR and
observed by the Sloan survey. Following the methods R Hildas are well-mixed within the overall population.
of Roig et al. (2008) and Wong et al. (2014), we cor- We also studied the color distribution of Hilda and
rectedthelistedapparentmagnitudesforsolarcolorsand Schubart family members. The bottom panel of Fig-
derived relative reflectance fluxes (normalized to 1 in r ure 3 shows the color distribution of the two families
band), discarding observations in which any of the band along with the color distribution of non-family Hildas
fluxes had a relative error greater than 10%. For each (i.e., background objects). The key observation here is
observation, the spectral slope S was computed from an thatthecolordistributionoffamilymembersisunimodal
error-weightedlinearleast-squaresfittothefluxesinthe and centered at relatively low spectral slope values con-
g, r, i, z bands. The u band flux was not used in fitting sistent with the LR sub-population, whereas the back-
sincethefluxatthosewavelengthstypicallydeviatessig- ground population (and the Hilda population overall) is
nificantlyfromthelineartrendinthespectrumatlonger bimodal in color. Examining the distribution of back-
wavelengths(Roigetal.2008). Forobjectswithmultiple ground objects with low spectral slope values in orbital
observations,wecalculatedtheweightedaveragespectral parameter space, we do not find any notable correlation
slope. with the location of known family members; therefore,
The spectral slope distribution of Hildas is shown in we do not expect significant contamination of collisional
Figure 3, where a clear bimodality is evident, as was familymemberswithinthebackgroundpopulation. Con-
first reported in Gil-Hutton & Brunini (2008). The ear- versely, the handful of high spectral slope family mem-
lier study used the previous, third release of the Moving bers are likely interlopers and not formally collisional
Object Catalog (SDSS-MOC3) and identified 122 Hildas fragments.
inthedataset. Thelatestreleasemorethandoubledthe These results may indicate that the progenitor bodies
number of Hildas with photometric measurements. To of the Hilda and Schubart families were LR objects. Al-
quantitatively assess the significance of the bimodality, ternatively, the unimodal color distribution of collisional
wefitsingleanddoubleGaussianmodelstothecolordis- fragmentsmaydemonstratethepristineinteriormaterial
tribution and found that the two-peaked model is very of Hilda asteroids, upon irradiation and space weather-
stronglyfavored(∆BIC=45.1). Notably,wefoundthat ing, evolve to take on a less-red color, regardless of the
the bimodality in the color distribution is discernible color of the progenitor body. The latter possibility has
throughout the entire magnitude range covered by the important implications for the our understanding of the
Sloan observations, which counters the observation in origin of the color bimodality, as we discuss in the next
Gil-Hutton & Brunini (2008) of an apparent lack of low section.
spectral slope objects in the range 10<H <12. In order to classify individual objects as LR or R
Thebimodalityinthecolordistributionindicatesthat Hildas, we fit the spectral slope distribution of back-
4
40
Background Hildas
102 35 Background Trojans
30
er
mb 25
u
ative n101 Count20
ul
m 15
u
C
10
5
100 0
9 10 11 12 13 14 15 16 5 0 5 10 15
Absolute magnitude Spectral slope (10−5Å−1)
Fig. 4.—ThecumulativemagnitudedistributionsoftheLRand Fig. 5.—ComparisonoftheHildaandTrojancolordistributions,
Rsub-populations,whereobjects(includingfamilymembers)have with family members removed. For Hildas, all objects brighter
been categorized into the sub-populations by spectral slope. The than H = 14 are shown, while for Trojans, all objects brighter
distributions are statistically distinct from each other at the 98% than H =12.3 are shown; these are the established completeness
confidencelevel. Bothdistributionshaveacharacteristicallywavy limitsofthecorrespondinganalyses(seeWongetal.2014,forthe
shapethatisnotconsistentwithasingleordoublepowerlawcurve. discussionofTrojans). Bothdistributionsshowaclearbifurcation
in color, corresponding to the LR and R sub-populations present
ground Hildas with a double Gaussian and obtained in both populations, with comparable mean colors. The R-to-LR
the mean colors of the LR and R sub-populations — number ratio in both Hilda and Trojan background populations
arealsosimilar.
4.0×10−5 ˚A−1 and 9.3×10−5 ˚A−1, respectively. We
chose to remove family members in our calculation of two populations.
mean colors since the surface composition of fragments The most salient similarity between Hildas and Tro-
may be systematically different than the surfaces of un- jans is their bimodal color distributions. Figure 5 shows
collided Hildas and would therefore not accurately re- the spectral slope distributions for Hildas and Trojans.
flect the initial color distribution. Using an analogous Since fragments from a major collision introduce a sig-
methodology to the one described in Wong et al. (2014), nificant number bias in the color distribution of a pop-
wecategorizedallHildas(includingcollisionalfragments) ulation relative to the initial pre-collision state, we have
with S ≤ 4.0 × 10−5 ˚A−1 as LR and all Hildas with removed the Hilda and Schubart family members in or-
S ≥ 9.3×10−5 ˚A−1 as R, resulting in a sample of 107 der to compare the background populations only. Run-
ning the normal mixture model test on the background
LR and 63 R Hildas. The cumulative absolute magni-
Hilda and Trojan color distributions (Section 2.2), we
tude distributions of the LR and R sub-populations are
find that a two-peaked model is very strongly favored
shown in Figure 4.
over a single-peaked model in both cases (∆BIC values
Both distributions are characterized by wavy shapes
of 16.7 and 39.1, respectively). From the figure, we can
that are not well-described by a single or double power
see that the characteristic mean colors of the LR and R
law; we do not present distribution fits for the indi-
Hildas and Trojans are comparable. The mean colors of
vidual color sub-population magnitude distributions in
this paper. Nevertheless, we compared the LR and R theTrojanLRandRsub-populationsare5.3×10−5 ˚A−1
magnitudedistributionsusingthetwo-sampleAnderson- and 9.6 × 10−5 ˚A−1, respectively, as compared to the
Darling test. We reject the null hypothesis that the LR somewhat bluer mean colorsof the Hilda LR andR sub-
andRmagnitudedistributionsaresampledfromasingle populations(4.0×10−5 ˚A−1 and9.3×10−5 ˚A−1,respec-
underlying distribution at the 0.8% significance level. In tively).
other words, the two color magnitude distributions are In addition, the number ratio of R-to-LR objects is
statistically distinct at the 99.2% confidence level. similarfortheHildasandTrojans. InWongetal.(2014),
we categorized 47 background Trojans as LR and 104 as
3. DISCUSSION R, while in the present work, we obtained a categorized
Having carried out an analysis of the color-magnitude sampleof28LRand56Robjectsoutofthebackground
distribution of Hildas analogous to the one presented for Hilda population. For both non-family Hildas and Tro-
TrojansinWongetal.(2014),wearenowinapositionto jans, the R-to-LR number ratio is roughly 2-to-1.
comparethetwopopulations. Recentdynamicalinstabil- The origin of the color bimodality in the Trojans and
ity models of Solar System evolution describe a common Hildas has long remained unexplained. Earlier expla-
progenitor population of minor bodies in the primordial nations concerning the color bimodality in Trojans sug-
trans-NeptunianregionfromwhichbothHildasandTro- gested that the LR and R populations may have been
jansaresourced. Itfollowsthat, ifthecurrentparadigm sourced from different regions of the solar nebula, with
of Solar System evolution is correct, there should be no- one population originating in the middle Solar System
tablesimilaritiesbetweentheobservablepropertiesofthe and the other scattered in from the outer Solar System.
5
However, within the framework of current dynamical in-
stability models, such a scenario is not supported; in- 103
stead, both LR and R Hildas and Trojans are predicted Hildas
to have been emplaced from the same primordial collec- Trojans
tion of planetesimals in the outer Solar System.
In a hypothesis first proposed in Wong et al. (2014)
and subsequently developed in full in Wong & Brown er102
b
(2016a), we posited that the color bimodality arose m
u
withintheprimordialtrans-Neptunianplanetesimaldisk, n
e
i.e., the purported progenitor population of Trojans and ativ
Hildas. In short, objects in this region accumulated out ul
m
of a mix of rocky material and ices of roughly cometary u101
C
composition, including a significant volume of volatile
ices such as ammonia and methanol. Under the action
of solar irradiation, location-dependent volatile loss led
to differential surface depletion of the various volatile
ices: objects closer in experienced higher surface tem- 100
6 7 8 9 10 11 12 13 14
peratures and faster rates of sublimation, leading to the Absolute magnitude
depletion of the more volatile species from the surface
layers, while objects farther out were colder and thereby
Fig. 6.—Comparisonofthetotalcumulativemagnitudedistribu-
retained some of the more volatile species. tionsofHildas(blackdots)andTrojans(bluesquares). TheTrojan
In the volatile loss model we developed, it was shown distribution has been corrected for incompleteness, following the
thatH Swouldhavebeenakeydistinguishingfactor,di- methodsofWongetal.(2014). TheHildamagnitudedistribution
2 isnotablyshallowerthroughouttheentiremagnituderangeofthe
vidingthetrans-Neptunianplanetesimalpopulationinto
data.
two groups, with the closer objects depleted in H S on
2
their surfaces and farther objects retaining H2S. Irradi- H2S), resulting in LR surface colors.
ation of the volatile ice rich surfaces would have red- Movingontothemagnitudedistributions,wecompare
dened and darkened the surfaces of all objects in the the total cumulative magnitude distributions for Hildas
region, as has been demonstrated in various laboratory andTrojansinFigure6. Thegeneralshapeofthedistri-
experiments(e.g.,Brunettoetal.2006); however, irradi- butions is the same. In Wong et al. (2014), we modeled
ationofobjectsthatretainedH Sontheirsurfaceswould thecollisionalevolutionofTrojansusingtheintrinsiccol-
2
haveproducedsulfur-bearingmoleculesintheirradiated lisional probabilities and impact velocities derived from
mantle, which is expected to provide a significant ad- previously published numerical simulations. We found
ditional reddening (e.g., as in the polar deposits on Io; that current level of collisional activity is insufficient to
Carlson et al. 2007). As a result, we posited that the have produced the observed break in the magnitude dis-
H S-retaining objects would have attained characteristi- tributionatH ∼9startingfromasinglepowerlawinitial
2
callyreddersurfacecolorsthantheH S-depletedobjects. magnitudedistribution(seealsoMarzarietal.1997); in-
2
The subsequent scattering of the trans-Neptunian plan- stead,thebreakislikelytobeaconsequenceofthemuch
etesimaldiskandtheemplacementofHildasandTrojans more intense collisional environment in the early trans-
intotheirpresent-daylocationswouldhavepreservedthis Neptunian planetesimal region from which the Trojans
primordial color bifurcation. and Hildas originated. From our modeling, we showed
This hypothesis for the observed color bimodality has that the collisional evolution of the Trojans, assuming
an important implication that explains another point of current rates, would only have resulted in a slight flat-
similaritybetweenHildasandTrojans—theobservation tening of the power law slope at intermediate sizes.
that all collisional family members are LR. As shown in AmajordifferencebetweentheHildaandTrojanmag-
Section 2.2, both the Hilda and Schubart families are nitude distributions is that the former is significantly
comprised of exclusively LR objects; analyzed spectra of shalloweratallsizes. Inparticular, thepowerlawslopes
objects from the only robustly attested major collisional at intermediate sizes — 0.46±0.01 (Wong et al. 2014)
family in the Trojans — the Eurybates family (Broˇz & and 0.34+0.02 (Section 2.1) for the Trojans and Hildas,
−0.01
Rozehnal 2011) — reveal a similar pattern in which the respectively — are discrepant at the 5.4σ level. In the
family members have a unimodal color distribution cen- context of collisional evolution and assuming that both
tered at relatively low spectral slopes (Fornasier et al. Hildas and Trojans were derived from the same progeni-
2007). Regardless of the original surface color of the tor population and therefore were emplaced with similar
parent bodies, our color bimodality hypothesis offers a initial size distributions, the shallower Hilda magnitude
naturalexplanationfortheobservedtrend. Uponashat- distribution would be indicative of a more active colli-
teringimpact,thefragmentsarecomposedofthepristine sional environment. However, estimates of both the cur-
interiormaterialoftheparentbodies,namely,rockyma- rentintrinsiccollisionalprobabilityandimpactvelocities
terial, water ice, and any remaining volatile ices trapped aresignificantlylowerfortheHildasthanfortheTrojans
in the subsurface. At the much higher temperatures of (Davis et al. 2002, and references therein). This incon-
the Hilda and Trojan regions, the volatile ices sublimate sistency presents a challenge to the idea of a common
instantaneouslyfromthesurfacesofthefamilymembers. originfortheHildasandTrojans asproposed bycurrent
Consequently, irradiation of these volatile-depleted sur- dynamical instability models.
faces would not lead to reddening of the same extent as One possible explanation is apparent when consider-
inthecasewherevolatileicesareretained(inparticular, ingthenumberofmajorcollisionalfamiliesintheHildas
6
and Trojans. Despite its lower current rate of colli- lative LR magnitude distribution was shown in Wong
sional activity, the Hilda population contains two ma- & Brown (2015) to overtake the R magnitude distri-
jor collisional families, with the Hilda family contain- bution at H ∼ 15. While the initial R-to-LR ratio of
ing the largest object (153 Hilda, H = 7.48) in the en- the Hildas was similar to that of the Trojans, as demon-
tire population. Meanwhile, the Trojan population has stratedbyourearliercomparisonofthebackground,un-
only one major family, the Eurybates family, with its collided color distribution (Figure 5), we see that the
largest object (3548 Eurybates, H = 9.7) being signifi- LR sub-population becomes more numerous than the R
cantly smaller than 153 Hilda. All else being equal, the sub-population at a larger size (H ∼13), due to the sig-
frequency of shattering collisions decreases sharply with nificant enrichment of the LR sub-population by Hilda
increasing target size, due to the decrease in the num- and Schubart family members.
berofimpactorscapableoffragmentingthetargetbody. All in all, the comparison of the magnitude and color
Assumingthatthecharacteristicimpactvelocityofares- distributions of Hildas and Trojans reveals several no-
onantpopulationdoesnotchangeappreciablywithtime, table similarities, with the discrepancies in the present-
one way of increasing the collisional probability for large day magnitude distribution shapes addressed by a dy-
targets is by increasing the number of impactors. namically plausible explanation. In turn, the body of
Therefore, the presence of two major collisional fami- observational data analyzed in this work presents a con-
liesintheHildassuggeststhatperhapsthenumberofob- vincingcasethattheHildasandTrojansoriginatedfrom
jects emplaced into the 3:2 resonance initially was much the same progenitor population prior to being emplaced
higher, creating a significantly more active early colli- in their current locations, as is predicted by current dy-
sional environment, but was gradually depleted as col- namical instability models of Solar System evolution.
lisional activity pushed fragments out of the resonance
4. CONCLUSION
and out of the Hilda population. The 3:2 mean motion
resonance with Jupiter has a narrow 0.1 AU-wide stable Inthispaper,weanalyzedtheabsolutemagnitudeand
zone centered at 3.96 AU, surrounded on both sides by optical color distributions of the Hilda asteroids. We
a dynamically chaotic boundary region with very short computed a power law fit to the magnitude distribution
characteristic diffusion times (Ferraz-Mello et al. 1998). through H = 14 and found a slope of α = 0.34+0.02.
−0.01
In Figure 1, the location of the stable zone is evident in Using photometric measurements contained in SDSS-
thesharpdecreaseinobjectdensityoutsideofthecentral MOC4, we calculated the spectral slope of 275 Hildas
region. andconfirmedtherobustbimodalityincolorreportedin
Following a scenario that has been described by sev- Gil-Hutton & Brunini (2008). This bimodality demon-
eral earlier works (e.g., Gil-Hutton & Brunini 2008), if strates that the Hilda population is comprised of two
a collisional fragment is ejected from the central stable groups of objects — less-red and red Hildas. We classi-
region, it is removed from the resonance on a relatively fiedindividualobjectsintothetwocolorsub-populations
short timescale, thereby depleting the magnitude distri- and presented the individual color magnitude distribu-
bution. Therelativeejectionvelocityrequiredforafrag- tions, which were shown to be highly distinct from each
menttopassoutofthecenteroftheresonanceisaround other. We also analyzed the Hilda and Schubart colli-
∆V ∼ 0.16 km/s (Davis et al. 2002). Based on the re- sionalfamiliesseparatelyandfoundthatthefamiliesare
sults of numerical models simulating the fragmentation comprised of LR objects only.
of asteroidal bodies and given the characteristic impact OurcomparisonoftheHildaandTrojancolordistribu-
velocity in the Hilda population (Davis et al. 2002), a tions revealed that both are bimodal, with similar char-
significantfractionofcollisionalfragmentsisexpectedto acteristic LR and R colors. Furthermore, the R-to-LR
have a sufficient ejection velocity to exit the resonance number ratios among non-family Hildas and Trojans are
(Jutzi et al. 2010). These same simulations demonstrate consistent with each other; likewise, both populations
thatthesmallestfragmentstendtobeimpartedthehigh- display the same trend in which collisional family mem-
est ejection velocity. bersareexclusivelyLR.Withintheframeworkofdynam-
Since the smaller bodies experience more frequent col- icalinstabilitymodels,ouranalysisoftheHildaandTro-
lisionsandarealsomorelikelytobeejectedfromthesta- jancolordistributionssupportsourpreviouslypublished
ble zone, the initial Hilda magnitude distribution would hypothesis that the color bimodality seen in both popu-
havebecomedepletedmostseverelyatfaintmagnitudes, lations developed prior to emplacement in their current-
consistent with the relatively shallow Hilda distribution day locations, with the difference in color primarily aris-
when compared with the Trojan distribution. Eventu- ing due to the retention vs. depletion of H S ice on
2
ally, as the total number of Hildas fell due to the re- thesurfacesofplanetesimalswithintheprimordialtrans-
moval of objects from resonance, the intrinsic collisional Neptunian disk.
probability decreased to the present-day value. Comparing the Hilda and Trojan total magnitude dis-
Relatingbacktotheimportantimplicationofourcolor tributions, we showed that the Hilda distribution is sig-
bimodality hypothesis that collisional fragments are LR, nificantlyshallowerthantheTrojandistribution,despite
a higher initial level of collisional activity also explains being much less collisionally active at the present time.
the distinct shapes of the LR and R Hilda magnitude Upon consideration of the number of major collisional
distributions (Figure 4). Collisions enrich the LR pop- families in each population, we proposed an explanation
ulation exclusively and lead to a relative steepening in for the discrepancy in magnitude distributions by posit-
the shape of the LR magnitude distribution with time. ing that the Hilda population upon emplacement was
The overall R-to-LR number should decrease with de- significantlylargerthanthecurrentpopulation. Thishy-
creasing size, since collisions are much more frequent for pothesis naturally explains the higher apparent level of
smaller targets. In the Trojan population, the cumu- collisionalevolutionintheHildamagnitudedistributions
7
(evidenced by the shallower power law slope at interme- twocolorclassesamongthesmallKBOshavemagnitude
diate sizes), since small collisional fragments are readily distributionsthatarestatisticallyindistinguishablefrom
ejected from the narrow stable zone of the 3:2 resonance themagnitudedistributionsofLRandRTrojans. Taken
and removed from the population. together, these studies provide the first body of observa-
We conclude that the bulk properties of Hildas and tional evidence linking the properties of KBOs, Hildas,
Trojans lend strong support to the idea of a shared pro- and Trojans.
genitor population — a major step in validating one The question of the composition of HIldas, Trojans,
of the main predictions of current dynamical instability and similarly sized KBOs remains unresolved and could
models of Solar System evolution. In further validation provide a complementary probe into the similarities
and/or differences between the respective asteroid pop-
of these models, our photometric survey observations of
ulations. Intensive spectroscopic observations of Hildas
small dynamically excited Kuiper Belt objects (KBOs)
and Trojans using current and near-future instruments
in the same size range as Hilda and Trojan asteroids
promise to provide improved constraints on the surface
reveal that these KBOs are likewise bimodal in optical
composition of these bodies, which will help solidify our
color (Wong & Brown 2016b). We also show that the understanding of their origin.
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