A thermophysical and dynamical study of the Hildas (1162) Larissa and (1911) Schubart
C. F. Chavez, T. G. Müller, J. P. Marshall, J. Horner, H. Drass, B. Carter
MMNRAS , 1–13 (2020) Preprint 3 February 2021 Compiled using MNRAS L A TEX style file v3.0
A thermophysical and dynamical study of the Hildas (1162) Larissa and(1911) Schubart
Cristian F. Chavez , , ★ T. G. Müller , J. P. Marshall , , J. Horner , H. Drass , , B. Carter Centre for Astrophysics, University of Southern Queensland, West St, Darling Heights, Toowoomba, QLD 4350, Australia Centre for Astro-Engineering, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Santiago, Chile Max-Planck-Institut für Extraterrestrische Physik, Giessenbachstraße 1, 85748 Garching, Germany Academia Sinica Institute of Astronomy and Astrophysics, AS/NTU Astronomy-Mathematics Building, No.1, Sect. 4, Roosevelt Rd,Taipei 10617, Taiwan
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
The Hilda asteroids are among the least studied populations in the asteroid belt, despite their potential importance as markers ofJupiter’s migration in the early Solar system. We present new mid-infrared observations of two notable Hildas, (1162) Larissaand (1911) Schubart, obtained using the Faint Object infraRed CAmera for the SOFIA Telescope (FORCAST), and use these tocharacterise their thermal inertia and physical properties. For (1162) Larissa, we obtain an effective diameter of 46.5 + . − . km,an albedo of 0.12 ± + − Jm − s / K − . In addition, our Larissa thermal measurements are wellmatched with an ellipsoidal shape with an axis ratio a/b=1.2 for the most-likely spin properties. Our modelling of (1911) Schubartis not as refined, but the thermal data point towards a high-obliquity spin-pole, with a best-fit a/b=1.3 ellipsoidal shape. Thisspin-shape solution is yielding a diameter of 72 + − km, an albedo of 0.039 ± − s / K − (or 10 + − Jm − s / K − ). As with (1162) Larissa, our results suggest that (1911) Schubart is aspherical, and likely elongatedin shape. Detailed dynamical simulations of the two Hildas reveal that both exhibit strong dynamical stability, behaviourthat suggests that they are primordial, rather than captured objects. The differences in their albedos, along with their diver-gent taxonomical classification, suggests that despite their common origin, the two have experienced markedly different histories. Key words: minor planets, asteroids: individual: (1162) Larissa, (1911) Schubart; radiation mechanisms: thermal; minor planets,asteroids: general; infrared: general; planets and satellites: formation
Over the past few decades, there has been a growing consensusamongst astronomers and planetary scientists that the Solar system’syouth was a chaotic place. Rather than being the result of a sedate, in-situ formation, leading models now suggest that the terrestrial planetswere shaped by cataclysmic collisions (e.g. Cameron & Benz 1991;Canup 2004; Benz et al. 2007; Canup 2012; Asphaug & Reufer 2014),whilst the giant planets underwent dramatic migration, potentiallyspanning hundreds of millions or even billions of kilometres (e.g.Tsiganis et al. 2005; Gomes et al. 2005; Levison et al. 2008; Lykawkaet al. 2009, 2010, 2011; Walsh et al. 2011; Nesvorný 2018).Much of the evidence for the chaotic evolution of the early Solarsystem has come, not from studies of the planets themselves, butfrom analysis of the vast numbers of small bodies littered throughoutthe system . The Solar system’s asteroidal bodies are of particularinterest in determining the true narrative of the system’s past. The ★ E-mail: [email protected] For a detailed recent review of the various theories of Solar system’smigration history, and the abundant evidence that supports them, we directthe interested reader to Horner et al. (2020), and references therein. overwhelming majority of those objects move on orbits betweenthose of Mars and Jupiter, within the Asteroid belt. Those asteroidsare of great scientific value as natural records of the formation andevolution of our Solar system. The study of their orbits and physicalproperties are key research topics to understand the mechanics andevolution of the Solar system as a whole, and particularly the origin,hydration, and collisional history of the Earth (e.g. Morbidelli et al.2000; Strom et al. 2005; Bottke et al. 2005; Gomes et al. 2005;Horner & Jones 2008; Horner et al. 2009; Minton & Malhotra 2011;Nesvorný 2018).In recent years, studies of the Jovian Trojan population havebrought fresh insights to the migration history of the giant planets(e.g. Morbidelli et al. 2005; Nesvorný et al. 2013; Roig & Nesvorný2015; Nesvorný 2018; Nesvorný et al. 2018; Pirani et al. 2019). Thedegree to which the orbits of Jupiter’s Trojans are excited suggeststhat they must have been captured during the migration of the gi-ant planets, rather than having formed in-situ – and as a result, thepopulation has been used as a means to estimate the scale, speed, di-rection, and chaoticity of that migration (e.g. Morbidelli et al. 2005;Nesvorný et al. 2013; Pirani et al. 2019).The Jovian Trojans are not, however, the only population of aster-oidal bodies trapped in mean-motion resonance with Jupiter. Located © a r X i v : . [ a s t r o - ph . E P ] F e b C. F. Chavez et al. between the outer edge of the Asteroid belt and Jupiter’s orbit lie theHildas – a swarm of objects trapped in 3:2 Mean Motion Resonancewith Jupiter, at a semi-major axis of around 4 au (Spratt 1989; Gil-Hutton & Brunini 2008; Milani et al. 2017). It is natural, therefore,to wonder whether this population, too, holds clues that may help usunveil the full narrative of the Solar system’s early days.Following that logic, both Roig & Nesvorný (2015) and Pirani et al.(2019) considered the effect of Jupiter’s migration on the planet’s twomain resonant populations. Roig & Nesvorný (2015) examined the“Jumping-Jupiter” model of Jovian migration, and find that “neitherprimordial Hildas, nor Trojans, survive the instability, confirmingthe idea that such populations must have been implanted from othersources” . Similarly, Pirani et al. (2019) investigated how a rapidinward migration of Jupiter from a more distant initial orbit (beyond ∼
15 au) would have affected the Jovian Trojans and the Hildas. Theyfound that such migration would result in the capture of a massive andexcited Hilda population – something that is strongly at odds with themodern observed distribution of Hildas, which move on remarkablydynamically cold orbits. Their model does, however, successfullyreplicate the observed asymmetry between the observed populationsof the leading and trailing clouds of Jovian Trojans. These recentstudies reveal the importance of the Hildas as a key test for modelsof planet formation, and as such, it is fair to consider the Hildasas a whole an interesting but underrated link to the Solar system’sformation and evolution.The first Hilda asteroid to be discovered was that after which thefamily is named – (153) Hilda, which was found in November 1875by Johann Palisa , some 30 years before the discovery of the firstJovian Trojans (e.g. Wolf 1907; Heinrich 1907; Strömgren 1908).Although the Hildas orbit markedly closer to the Sun than the JovianTrojans, and their population is comparable to that of their morefamous cousins, they remain relatively poorly studied.Recent studies that have focused on the physical properties ofHildas have suggested that these asteroids may have had a commonorigin with Jovian Trojans. Studies of their near infrared spectra(Wong et al. 2017), size distributions (Terai & Yoshida 2018), sizefrequency distributions (Yoshida et al. 2019), and their optical andinfrared colours (from the Sloan Digital Sky Survey; Gil-Hutton &Brunini 2008, and NEOWISE; De Prá et al. 2018), reveal that theoverall properties of the two resonant populations are broadly similar.Such studies have revealed the presence of at least two colli-sional/dynamical families in the Hilda population, both of whichare named for their largest members (e.g. Brož & Vokrouhlický2008). To date, 111 members of the (1911) Schubart family havebeen identified, whilst the (153) Hilda family has at least 216 con-firmed members (Romanishin & Tegler 2018). It seems likely thatfuture surveys will reveal both more members of these families, andnew collisional families with different parent bodies.Taxonomically, the detailed mapping of the asteroid belt carriedout by DeMeo & Carry (2013, 2014) found the Hildas and JovianTrojans to be distinct groups of objects, separate from their clas-sification of the main belt asteroids. The Hildas and Trojans weretypically labelled as P and D-types in contrast to the S and V-typesthat are common throughout the main belt, a result that has beenverified by later studies (Terai & Yoshida 2018; Yoshida et al. 2019).In addition to their similar taxonomic classifications, the Hildas andJovian Trojans both exhibit a clear bimodality in their spectral distri-bution (Terai & Yoshida 2018; De Prá et al. 2018), a fact that could Discovery details taken from the NASA/JPL Small-Body Database Browserat https://ssd.jpl.nasa.gov/sbdb.cgi , accessed on 2020 March 11. be suggestive of a variety of source regions having contributed to themodern Hilda and Trojan populations.To date, amongst the Hildas that have been classified on the basisof their spectral data, the majority of were classified under the low-albedo asteroid classifications C, P and D-type (Dymock 2010; Wonget al. 2017; Szabó et al. 2020) in the Tholen taxonomy (Kaasalainen& Torppa 2001; Warner et al. 2009; de Pater & Lissauer 2015).Of these, the D-type asteroids are undoubtedly the most numerous,followed by P-types and finally C-types (Grav et al. 2012).Some studies state that the general outline for the surface com-position of the Hildas is a mix of organics, anhydrous silicates,opaque material and ice (Dahlgren et al. 1998; Dahlgren et al. 1999;Gil-Hutton & Brunini 2008; De Prá et al. 2018). The surfaces ofthe Hildas have likely experienced significant space weathering, al-though it should be noted that they may well be considered to be morepristine than similar objects in the main asteroid belt, as a result oftheir larger heliocentric distances (Gil-Hutton & Brunini 2008; DePrá et al. 2018).Moreover, De Prá et al. (2018) raised an interesting point on cur-rent knowledge of Hildas concluding that there is a population inthe Jovian Trojans not present in the Hildas, supporting the idea thatTrojans and Hildas may be celestial objects of different origins. In-deed, the stark contrast in mean albedo values amongst sub-groupsin the same Hildas suggests that even within them we can appreciatea kind of intruder of another source (Romanishin & Tegler 2018). Inthe coming years, the Transiting Exoplanet Survey Satellite (
TESS ;Ricker et al. 2014) will aid the identification of Hildas with differentorigins, by providing a wealth of high cadence light curves for theseasteroids. As described by Szabó et al. (2020), that detailed photo-metric data will facilitate a direct comparison between the main beltasteroids, the Hildas, and the Jovian Trojans, allowing the similaritiesand differences between the populations to be studied in depth.In this work, we focus on two of the largest and brightest mem-bers of the Hilda population – (1162) Larissa, and (1911) Schubart.(1162) Larissa was discovered in 1930 by German astronomer KarlReinmuth , and is amongst the brightest Hildas due to its relativelyhigh albedo (from 0.11 (Alí-Lagoa & Delbo 2017) to 0.18 (Grav et al.2012) in the literature), which is markedly higher than the average of0.055 for this group of asteroids (De Prá et al. 2018). Its taxonomy hasbeen variously defined as M-type (Grav et al. 2012), P-type (Warneret al. 2009), and most recently as X-type (De Prá et al. 2018). It doesnot belong to either of the currently identified collisional familieswithin the Hilda population (e.g. Romanishin & Tegler 2018), andas a result, it is plausible that it might be a primordial, undisruptedbody.(1911) Schubart was discovered in 1973 by Swiss astronomerPaul Wild , and is the largest member of a populous collisionalfamily (the Schubart family), containing at least 111 members (Brož& Vokrouhlický 2008). The extremely low albedo calculated for it,under 0.05 in several studies (Grav et al. 2012; Warner et al. 2009;Romanishin & Tegler 2018), makes it one of the darkest asteroids inthe Hilda group and even in the Solar system.We describe the observations taken by the “Faint Object infraRedCAmera for the SOFIA Telescope” (FORCAST, Adams et al. 2010)instrument on the Stratospheric Observatory For Infrared Astronomy(SOFIA, Herter et al. 2012) used in this work in Section 2. Theprocess by which flux values are extracted from the observationsis then described in Section 3. In Section 4, we perform a detailedthermophysical analysis of (1162) Larissa and (1911) Schubart,before investigating the orbital stability of the two asteroids inSection 5. In Section 6, we discuss our results and present our MNRAS , 1–13 (2020) thermophysical and dynamical study of the Hildas (1162) Larissa and (1911) Schubart conclusions. We obtained SOFIA/FORCAST observations of the Hildas (1162)Larissa and (1911) Schubart, which we supplement with archivalmid- and far-infrared data taken by
IRAS (Tedesco 1986),
AKARI (Usui et al. 2011) and
WISE (Mainzer et al. 2011) projects. TheSOFIA images collected for this research were part of a series ofstudies about small bodies and Solar system formation that were per-formed as part of the so-called “Basic Science” observation flights,under the proposal ID 82_0004, mission ID FO64, PI Dr. ThomasMüller, carried out during 2011 June.Our data comprise a series of multi-wavelength mid-infrared imag-ing observations taken on 2011 June 4, for (1911) Schubart, and 2011June 8 for (1162) Larissa. FORCAST is a dual-channel infrared cam-era and spectrograph, able to produce images with a field of view ofapproximately 3.4 (cid:48) × (cid:48) , with a plate scale of 0.768 (cid:48)(cid:48) . One channelis the short-wavelength camera (SWC), which operates at 5 to 26 𝜇 m, whilst the other, the long-wavelength camera (LWC), operatesat 26 to 40 𝜇 m; both channels can be used either individually orsimultaneously (Herter et al. 2013).In order to take advantage of the dual channel nature of FORCAST,two observations were taken per target, in couplet using first the11.1 and 34.8 𝜇 m and then the 19.7 and 31.5 𝜇 m filters, under thesymmetrical two-position chop-nod mode (C2N), to minimise theimpact of a variety of sources of noise, including time variable skybackgrounds and the thermal emission from the telescope itself. Thatmode is recommended for observations of point-like sources such as(1162) Larissa and (1911) Schubart, as described in Herter et al.(2012).The infrared images used in this work are level 3 coadded dataproducts, which means that they were telluric and flux corrected bySOFIA team in units of Jy/pix (Herter et al. 2013). The files werestored and then downloaded from the SOFIA Airborne Observa-tory Public Archive .To complement our new data, we searched for archival observa-tions of our two targets. Fortunately, both (1162) Larissa and (1911)Schubart have been observed multiple times at thermal infrared wave-lengths by other surveys, so that in addition to the 12 SOFIA FOR-CAST images of (1162) Larissa, for the thermophysical analysis weused one observation from IRAS (at 25 𝜇 m), four from AKARI (at18 𝜇 m), and 42 from WISE (21 images at 11.1 𝜇 m and 21 at 22.64 𝜇 m).Moreover, for (1911) Schubart, in addition to the 23 SOFIA FOR-CAST images, we used 14 observations from IRAS (four at 12 𝜇 m,four at 25 𝜇 m, four at 60 𝜇 m and two at 100 𝜇 m), eight from AKARI (four at 9 𝜇 m and four at 18 𝜇 m) and also 39 from WISE (20 at 11.1 𝜇 m and 19 at 22.64 𝜇 m). All the data for both asteroids were takenfrom the SBNAF Infrared Database (Szakáts et al. 2020) at VizieROnline Data Catalog . Full details on the archival images used aregiven in Tables 1 and 2. Accessed on 2019 October 10 VizieR Online Data Catalog: SBNAF Infrared Database Web-site at http://vizier.u-strasbg.fr/viz-bin/VizieR?-source=J/A+A/635/A54 , accessed most recently on 2020 July 14. Earlier in our anal-ysis, we obtained data from the Konkoly Observatory Website at https://konkoly.hu/database.shtml , accessed on 2019 December 17, which
In order to perform a thorough thermophysical analysis of the Hildas(1162) Larissa and (1911) Schubart, we first used the images takenof the asteroids at 11.1, 19.7, 31.5 and 34.8 𝜇 m to determine theflux from the asteroids in each waveband. A visual inspection of theimages was carried out using SAOImageDS9 (Joye & Mandel 2003)to assess their quality. This revealed that in 12 of the 23 imagesof (1911) Schubart there was no clear presence of the asteroid, sothose images were discarded after completing the process of checkingtheir signal-to-noise values, and they were not analysed further. Theimages of (1162) Larissa were all of excellent quality and, as a result,all were used in our research.The selected images were then analysed using a bespoke code thatapplied implementations of standard Python-based image analysistools including the packages astropy (Greenfield et al. 2013) andphotutils (Bradley et al. 2016), yielding the results shown in Tables3 and 4.Although the FORCAST Photometry Recipe reference document recommends aperture photometry radius and annulus dimensions of12 pixels and 15 to 25 pixels, the same document suggests the use ofsmaller apertures to reduce the overall uncertainty in measurement oflow S/N sources. In this sense, given the quite irregular backgroundsof several of the images and the diversity amongst them, we optedto individually optimise the values for the aperture photometry radiiand annuli for each image.We used the Aperture Photometry Tool (APT) (Laher et al. 2012)and DS9 (Joye & Mandel 2003) as complementary tools to decideupon the best values for making the photometry measurements, mak-ing particular use of APT functions Aperture Slice, Radial Profileand Curve of Growth, which were combined with the DS9 horizontaland vertical graphs in order to set the correct values for the apertureand annulus extent as required.We applied appropriate colour correction factors to the measuredflux densities, assuming blackbody input spectra for our targets witha temperature close to 150 K, according to the Hildas’ distance fromthe Sun (Hinrichs et al. 1999). The magnitude of these corrections pereach waveband filter, however, are values quite close to 1 (between1.0005 and 1.009), implying a small correction for the raw fluxescalculated, and were taken from Table 3 of Herter et al. (2013).The resulting aperture photometry measurements obtained from theSOFIA images at 11.1, 19.7, 31.5 and 34.8 𝜇 m are presented inFigures 1 and 2.To determine the uncertainties in our results, we followed theFORCAST Photometry Recipe . The error propagation was done by was used for our preliminary work, but has since been superseded by theupdated VizieR catalog. SOFIA FORCAST Photometry Recipe at , accessed on 2020 April 17. SOFIA FORCAST Photometry Recipe at , accessed on 2020 April 17.MNRAS , 1–13 (2020)
C. F. Chavez et al.
Date [JD] Wavelength [ 𝜇 m] Flux density [Jy] 𝑟 Helio [au] Δ [au] 𝛼 [ ◦ ] Telescope/Instrument ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Table 1.
Summary of the supplementary thermal observations of (1162) Larissa used in this work. The table shows the time at midpoint of the observation (inJulian Date), the reference wavelength (in 𝜇 m), the monochromatic flux density (colour corrected in-band flux density) with its absolute error, the heliocentricdistance ( 𝑟 Helio ), the observatory-centric distance ( Δ ; in au), the phase angle ( 𝛼 ), and the spacecraft telescope used for the observation. adding the photometry, the calibration with the standard star and thecalibration factor in quadrature. In that process, we used values fromTable 5 of Herter et al. (2013) to provide the appropiate calibrationfactor, which was 0.055 for (1162) Larissa, and 0.102 for (1911)Schubart. In order to better understand the physical properties of (1162) Larissaand (1911) Schubart, we here present a detailed thermophysical anal-ysis for both objects, based on the observational data we obtained using SOFIA, coupled with archival data taken by
IRAS (Tedesco1986),
AKARI (Usui et al. 2011) and
WISE (Mainzer et al. 2011), asdetailed in Section 2.The thermophysical analysis performed in this research is basedon the thermophysical model (TPM) detailed in Lagerros (1996);Lagerros (1997, 1998) and Müller & Lagerros (1998, 2002). In theTPM, the surface temperature is calculated from the energy balancebetween absorbed Solar radiation, the thermal emission, and heatconduction into the surface material (here, 1-d heat conduction isconsidered). For a direct comparison with the observed fluxes, weuse a given spin-shape solution to calculate the temperature of each
MNRAS , 1–13 (2020) thermophysical and dynamical study of the Hildas (1162) Larissa and (1911) Schubart Date [JD] Wavelength [ 𝜇 m] Flux density [Jy] 𝑟 Helio [au] Δ [au] 𝛼 [ ◦ ] Telescope/Instrument ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Table 2.
Summary of the supplementary thermal observations of (1911) Schubart used in this work. The table shows the time at the midpoint of the observation(in Julian Date), the reference wavelength (in 𝜇 m), the monochromatic flux density (colour corrected in-band flux density) with its absolute error, the heliocentricdistance ( 𝑟 Helio ), the observatory-centric distance ( Δ , in au), the phase angle ( 𝛼 ), and the spacecraft telescope used for the observation.MNRAS000
Summary of the supplementary thermal observations of (1911) Schubart used in this work. The table shows the time at the midpoint of the observation(in Julian Date), the reference wavelength (in 𝜇 m), the monochromatic flux density (colour corrected in-band flux density) with its absolute error, the heliocentricdistance ( 𝑟 Helio ), the observatory-centric distance ( Δ , in au), the phase angle ( 𝛼 ), and the spacecraft telescope used for the observation.MNRAS000 , 1–13 (2020) C. F. Chavez et al.
Date [JD] Length of the exposure [s] Wavelength [ 𝜇 m] Flux density [Jy] 𝑟 Helio [au] Δ [au] ± ± ± ± ± ± ± ± ± ± ± ± Table 3.
Details of the observations of (1162) Larissa taken using the FORCAST instrument on the SOFIA airborne observatory on 2011 June 8, (JD 2455720).The table shows the time at which the observations were started (in Julian Date), the length of the exposure (in seconds), the reference wavelength (in 𝜇 m), thecolour-corrected monochromatic flux densities with errors, the heliocentric distance of the asteroid ( 𝑟 Helio ) and the observatory-centric distance ( Δ , in au). Dueto the small span of time (0.05 JD), the phase angle value shows no significant change between the first and last observation, and was estimated to be 9.08 ◦ . Date [JD] Length of the exposure [s] Wavelength [ 𝜇 m] Flux density [Jy] 𝑟 Helio [au] Δ [au] ± ± ± ± ± ± - - - - - - - - - - - - - - - - - - - - - - - -2455716.9500 63.41 31.50 1.112 ± ± ± - - - - - - - - -2455716.9752 61.01 34.80 0.879 ± ± - - - Table 4.
Details of the observations of (1911) Schubart taken using the FORCAST instrument on the SOFIA airborne observatory on 2011 June 4, (JD 2455716).The table shows the time at which the observations were started (in Julian Date), the length of the exposure (in seconds), the reference wavelength (in 𝜇 m), thecolour-corrected monochromatic flux densities with errors, the heliocentric distance of the asteroid ( 𝑟 Helio ) and the observatory-centric distance ( Δ , in au). Theimages whose values are given in italics were of too poor quality to be used further in our analysis – the asteroid was not clearly visible in those images, andany analysis therefore yielded results with very low signal-to-noise. Due to the small span of time (0.03 JD), the phase angle value shows no significant changebetween the first and last observation, and was estimated to be 12.12 ◦ . surface facet. The integral over all projected surface elements towardsthe observer then leads to a TPM flux prediction. In principle, wetune the effective size, albedo, and thermal properties, to calculatesimultaneously the reflected light (described by the object’s absolutemagnitude 𝐻 and the phase slope parameter 𝐺 ) and the thermalemission (in direct comparison with the observed thermal emission).This is done for all measurements to find the best-possible solutionin size, albedo, thermal inertia, and, if possible, to constrain theobject’s surface roughness. The results of our analysis are presentedin Figure 3, which shows that, in general, the observational data can be relatively well fit by our final thermophysical model solutions, despitethe lack of high-quality spin-shape information for both targets.For (1162) Larissa, our radiometric tests were performed usingspherical and ellipsoidal spin-shape solutions, with a rotation periodof 6.519148 hours (Slyusarev et al. 2013; Warner & Stephens 2017,J. Durech, private communications). The recent, most likely, polesolutions are (294 ◦ , -28 ◦ ) or (114 ◦ , -23 ◦ ) in ecliptic longitude andlatitude (J. Durech, private communications). In addition (for spher-ical shapes only), we considered spin-axis orientations of (110 ◦ ,75 ◦ MNRAS , 1–13 (2020) thermophysical and dynamical study of the Hildas (1162) Larissa and (1911) Schubart Figure 1. (1162) Larissa mid-infrared photometric data. The graphic showsthe flux densities derived from the four wavelength bands of FORCAST, as afunction of the rotational phase of the asteroid, together with the given errorbars. The obtained mid-infrared photometric light curves follow the trend ofincrements and decrements along the different wavelengths used in each chopnod shot.
Figure 2. (1911) Schubart mid-infrared photometric data. The graphic showsthe flux densities derived from the four wavelength bands of FORCAST, as afunction of the rotational phase of the asteroid, together with the given errorbars. The obtained mid-infrared photometric light curves follow the trend ofincrements and decrements along the different wavelengths used in each chopnod shot.
Cibulková et al. 2018) , equator-on prograde, equator-on retrograde,pole-on during the SOFIA observations. We assumed that the as-teroid had a typical level of surface roughness (Müller & Lagerros1998), and solved for its size, albedo and thermal inertia using allSOFIA-FORCAST data (12 points) plus all of the detailed archivaldata described in section 2.Following this methodology, the radiometric range solution for aspherical object, independent of the spin-axis orientation, and as- Using entries from the database at , accessed November 2020. suming H V =9.59 and G=0.49 (Oszkiewicz et al. 2011) is given by: • Effective diameter: 41 to 48 km • Geometric V-band albedo 0.10 to 0.15 • Thermal inertia below 30 Jm − s − . K − The reduced 𝜒 of the spherical solutions are poor (>4.0). Givenan observed (visible) lightcurve amplitude up to 0.22 mag, the poorfit is likely related to our assumption of a simple spherical shape. Thisis confirmed by deriving radiometric sizes separately for the differentdata sets. Due to the different viewing geometries for IRAS , Akari , WISE , and SOFIA data, the corresponding dataset-specific sizes de-viate substantially and can only be explained by a non-sphericalbody.If we take the two recent spin-pole solutions, we can implementellipsoidal shape solutions with varying a/b ratios from 1.0 to 1.5 toimprove the radiometric analysis. For an a/b ratio of 1.2 we obtainedlower 𝜒 values (around 2.0 for the (114 ◦ , -23 ◦ ) and 2.5 for the (294 ◦ ,-28 ◦ ) pole solution) with the following values: • Effective diameter: 46.5 + . − . km • Geometric V-band albedo: 0.12 ± • Thermal inertia: 15 + − Jm − s − . K − • Preference for the spin-pole solution with (114 ◦ , -23 ◦ ),P_sidereal = 6.519148 h, ellipsoidal shape with a/b=1.2These two spin-poles with the specified ellipsoidal shape have thebig advantage that they can reproduce reasonably well the observedthermal lightcurve (22 data points taking in Jan 2010, and another20 data points taken in July 2010, each time within about one day),and the overall flux changes with wavelengths and phase angles.But based on the thermal data alone, it is not possible to entirelyexclude other spin-shape solutions. However, future observations ofthe asteroid will prove critical in order to mitigate the remaining spin-shape uncertainties and to improve the radiometric solution further.In the case of (1911) Schubart, using the multiple thermal datadescribed in section 2, in combination with a spherical shape, a ro-tation period of 11.9213 h (Stephens 2016), and pole solution of(320 ◦ , 20 ◦ ) (J. Durech, private communication) leads to acceptable 𝜒 values below 2.0 for all spin-axis orientations. The data are rea-sonably well fitted, only the IRAS 𝜇 m observations seem to beproblematic. The derived radiometric size range is between about64 km up to 78 km, the corresponding albedos are between 0.033and 0.42, respectively. However, the analysis process gave a slightpreference for a spin-axis with high-obliquity like the pole-on duringSOFIA observations or one of the recently estimated solutions fromlightcurve inversion techniques.For completeness, we performed additional tests considering thealternative rotation period of 7.9121 hours rotation period proposedin Warell (2017). Changing the rotation period in this manner had noimpact on the results, showing no influence on the spin-size-albedo-TI solution. However, we find that the 11.9 h rotation period seemsto explain the WISE data in a more consistent way.We also investigated if ellipsoidal shape solutions with varying a/bratios (from 1.0 to 1.5) lead to better fits of all data. An improvementis seen only in the high-obliquity case (with the asteroid seen pole-onduring SOFIA observations). Here, an axis ratio a/b=1.3 pushes the 𝜒 close to 1.0. However, the WISE lightcurve data set (36 data pointstaken within 3.6 days in April 2010) is still not very well matched.This points to a more complex shape solution. But if we accept thisbest-fit pole-on spin-shape solution as baseline, then our radiometricsolution provides the following values:
MNRAS , 1–13 (2020)
C. F. Chavez et al. • Effective diameter: 72 + − km • Geometric V-band albedo: 0.039 ± • Thermal inertia: 10 + − Jm − s − . K − In Figure 4, we show (in the upper panels) the degree to which thethermophysical model agrees with our observations, as a function ofwavelength. Aside from the 100 𝜇 m observations of (1911) Schubartmade by IRAS , it is clear that, in general, the model and data are ingood agreement. The observation-to-model ratios are well balancedover a wide range of wavelengths, phase angles, and rotational phases.In the lower panels of Figure 4, we plot the agreement between ourthermophysical model and the gathered observational data against ro-tational phase. For both targets, there appears to be a remaining smallvariation in goodness of fit between our model and the data as a func-tion of rotational phase. In general, there is still a significant scatterin the observational-to-model ratios, which lends further support tothe idea that the two asteroids may well have more complex shapes.
To complement our study of the physical properties of (1162) Larissaand (1911) Schubart, we performed two detailed suites of 𝑛 -bodysimulations to examine the long-term dynamical evolution of the twoasteroids. For each asteroid, we used the Hybrid integrator within the 𝑛 -body dynamics package Mercury (Chambers 1999) to follow theevolution of a swarm of 61 875 test particles for a period of 10 years,under the gravitational influence of the Sun, Jupiter, Saturn, Uranusand Neptune. An integration time step of 60 days was used, and eachindividual particle was followed for the full 10 year duration of thesimulations, unless it collided with one of the four giant planets, theSun, or was ejected to a heliocentric distance of 1000 au.The test particles themselves were generated in a manner similar tothat used in our earlier work (e.g. Horner & Lykawka 2010; Horneret al. 2012a; Horner et al. 2012b), with “clones” generated in a regular4-dimensional grid spanning the full ± 𝜎 uncertainties in the objectssemi-major axis, 𝑎 , eccentricity, 𝑒 , inclination, 𝑖 , and mean anomaly, 𝑀 . In total, 25 unique values of semi-major axis were tested. For eachof these values, 15 unique values of eccentricity were examined,creating a 25 ×
15 grid in 𝑎 − 𝑒 space. For each 𝑎 − 𝑒 ordinate,15 unique inclination values were tested - with 11 unique meananomalies being considered for each individual 𝑎 − 𝑒 − 𝑖 ordinate. Inthis way, we generated a four-dimensional hypercube of test particles,distributed on an even lattice, with dimensions ( × × × ) in ( 𝑎, 𝑒, 𝑖, 𝑀 ) space. The initial orbital elements for the two Hildasused as the basis for our integrations, along with their associateduncertainties, are detailed in Table 6.In stark contrast to our previous studies of resonant Solar systemsmall bodies (Horner & Lykawka 2010; Horner et al. 2012a; Horneret al. 2012b), both (1162) Larissa and (1911) Schubart proved toexhibit staggering dynamical stability. Over the course of our sim-ulations, not a single clone of either object escaped from the Hildapopulation. Over time, the test particles did diffuse to fill the entiretyof the “Hildan Triangle” – the striking feature in plan view of theSolar system occupied by the Hildas (as shown in Figure 5) – butnone became sufficiently excited to escape that population.The extreme stability of the two Hildas argues towards them beingprimordial, rather than captured objects. It stands as an interestingcontrast to the dynamical evolution of objects studied in a similarmanner in the Jovian and Neptunian Trojan populations - all of whichexhibit some degree of instability (e.g. Levison et al. 1997; Marzari & Scholl 2002; Horner & Lykawka 2010; Horner et al. 2012a; Horneret al. 2012b; Di Sisto et al. 2019; Holt et al. 2020). Given the extremestability of these two objects, it seems reasonable to suggest thatfuture studies of the migration of the giant planets and their impacton the Solar system’s young small body populations should examinewhether those models can replicate such tightly captured objectsmoving on dynamically cold orbits in the Hilda region. In this work, we have presented a detailed thermophysical and dy-namical analysis of the Hilda asteroids (1162) Larissa and (1911)Schubart, using observations made using the FORCAST instrumenton the SOFIA airborne observatory on 2011 June 4 and June 8, sup-plemented by archival data from the
IRAS , AKARI and
WISE spaceobservatories. Our dynamical simulations reveal both Hildas to ex-hibit extreme dynamical stability, with not a single one of the 61 875clones generated of each object escaping from the Hilda region in the10 years of our integrations. This finding is strongly suggestive thatthe two objects are primordial in nature, rather than having been cap-tured, since objects in populations that are widely accepted to havebeen captured during the Solar system’s youth tend to exhibit dynam-ical instability to some degree (e.g. Levison et al. 1997; Marzari &Scholl 2002; Horner & Lykawka 2010; Horner et al. 2012a; Horneret al. 2012b; Di Sisto et al. 2019; Holt et al. 2020).Our thermophysical analysis of the two Hildas reveals that bothasteroids have non-spherical shapes, with the physical properties wederive on the basis of our analysis being in broad agreement with pre-viously published values, albeit with markedly reduced uncertainties(See Table 5).For (1162) Larissa, the geometric V-band albedo we obtained(0.12 ± ± ± ± 𝜇 m, which spans the wavelength range of peak thermal emissionfrom asteroids (Harris & Lagerros 2002; Emery et al. 2006). Ourresults yield a refined value for the effective diameter of (1162)Larissa of 46 . + . − . km, a measurement that sits close to the centreof the range of previously published diameters, spanning 40.38 to48.59 km (Tedesco et al. 2004; Usui et al. 2011; Nugent et al. 2015).Our thermophysical analysis of (1911) Schubart proved more chal-lenging, as a result of having to discard a number of our images ofthe asteroid. When our observations are taken in concert with thoseobtained by WISE , we find better agreement with the longer rotationperiod of 11.915 hours proposed in (Warell 2017), rather than theshort 7.9 hour period also presented in that work. By incorporatingthat longer period into our analysis, we obtain a geometric V-bandalbedo for (1911) Schubart of 0.039 ± MNRAS , 1–13 (2020) thermophysical and dynamical study of the Hildas (1162) Larissa and (1911) Schubart Figure 3.
Plot showing the both the observations of (1162) Larissa (top) and (1911) Schubart (bottom) and the best fit thermophysical model obtained fromthose observations. The left plots show the data in linear space, with those on the right showing the same data in a log-log space. The observations are colouredto denote their source - with SOFIA observations shown as red squares,
IRAS as green stars,
Akari as blue plus marks, and
WISE as black circles. For eachobservation, the vertical lines denote the 1- 𝜎 uncertainties. The thermophysical model is plotted in grey. The spectral energy distribution was calculated for theSOFIA/FORCAST epoch at lightcurve mid-point, with all the other fluxes scaled to these SOFIA/FORCAST epochs.(1162) Larissa (1911) SchubartVariables Published data Current research Published data Current research
Effective diameter [km] 40.38-52.16 46 . + . − . + − Geometric albedo in V-band 0.102-0.186 0.12 ± ± − s − . K − ] No value 15 + − No value 10 + − Table 5.
Results of thermophysical modelling for (1162) Larissa and for (1911) Schubart. This Table collates data from Tedesco et al. (2004), Usui et al. (2011),Grav et al. (2012), Nugent et al. (2015), (Alí-Lagoa & Delbo 2017) and De Prá et al. (2018) dynamical family as a whole is 30 per cent darker than the bulk of theHilda population outside that family. Our results yield an effectivediameter for (1911) Schubart of 72 + − km. This result is once again inbroad agreement with those published in the literature, which rangebetween 64.66 and 92.37 km (Tedesco et al. 2004; Usui et al. 2011;Nugent et al. 2015).Our thermophysical analysis also enabled us to determine the ther-mal inertias of the two Hilda asteroids we studied. That analysisyielded values of 15 + − Jm − s − . K − for (1162) Larissa and 10 + − Jm − s − . K − for (1911) Schubart. These results lie within the broad range of values given by Müller & Lagerros (1998) in their study ofa number of the largest main-belt asteroids, whose thermal inertiaswere found to range between 5 and 25 Jm − s − . K − , and stand instark contrast to the outliers found in previous studies of inner mainbelt asteroids, such as (306) Unitas, which yielded values as highas 260 Jm − s − . K − , and (277) Elvira, with values reaching 400Jm − s − . K − (Delbo et al. 2015).Flux variations in the thermal measurements provide some evi-dence that (1911) Schubart has an elongated shape since the spher-ical one produces significant variations in the observation-divided-by-model plot which seem to correlate with the most-likely rotation MNRAS000
Results of thermophysical modelling for (1162) Larissa and for (1911) Schubart. This Table collates data from Tedesco et al. (2004), Usui et al. (2011),Grav et al. (2012), Nugent et al. (2015), (Alí-Lagoa & Delbo 2017) and De Prá et al. (2018) dynamical family as a whole is 30 per cent darker than the bulk of theHilda population outside that family. Our results yield an effectivediameter for (1911) Schubart of 72 + − km. This result is once again inbroad agreement with those published in the literature, which rangebetween 64.66 and 92.37 km (Tedesco et al. 2004; Usui et al. 2011;Nugent et al. 2015).Our thermophysical analysis also enabled us to determine the ther-mal inertias of the two Hilda asteroids we studied. That analysisyielded values of 15 + − Jm − s − . K − for (1162) Larissa and 10 + − Jm − s − . K − for (1911) Schubart. These results lie within the broad range of values given by Müller & Lagerros (1998) in their study ofa number of the largest main-belt asteroids, whose thermal inertiaswere found to range between 5 and 25 Jm − s − . K − , and stand instark contrast to the outliers found in previous studies of inner mainbelt asteroids, such as (306) Unitas, which yielded values as highas 260 Jm − s − . K − , and (277) Elvira, with values reaching 400Jm − s − . K − (Delbo et al. 2015).Flux variations in the thermal measurements provide some evi-dence that (1911) Schubart has an elongated shape since the spher-ical one produces significant variations in the observation-divided-by-model plot which seem to correlate with the most-likely rotation MNRAS000 , 1–13 (2020) C. F. Chavez et al.
Figure 4.
The degree to which the thermophysical models agree with the observations can be demonstrated by taking the observed flux and dividing it bythe predicted values. Here, we show the degree to which the observations and model agree for (1162) Larissa (left) and (1911) Schubart (right). The upperpanels show the agreement between the model and observations as a function of wavelength, whilst the lower panel plots the agreement instead as a function ofrotational phase for the two asteroids. As in Figure 3, the observations are coloured to denote which instrument obtained them. In general, the observations andmodel are in good agreement, albeit with noticeable scatter at any given wavelength.1162 Larissa 1911 SchubartVariables
Best fit value 1- 𝜎 uncertainty Best fit value 1- 𝜎 uncertainty Semi-major axis [au] 3.9234 5.092 × − × − Eccentricity 0.110524 6.291 × − × − Inclination [ ◦ ] 1.888 6.972 × − × − 𝜔 [ ◦ ] 39.791 1.825 × − × − Ω [ ◦ ] 212.222 1.847 × − × − Mean Anomaly [ ◦ ] 59.024 3.447 × − × − Table 6.
Orbital elements, and their associated uncertainties, for (1162) Larissa and (1911) Schubart, as taken from the AstDys database on January 12, 2012. period. Future improved spin-shape solution might resolve this issueand will allow to refine the radiometric solutions for both Hildas.It is interesting that the two asteroids studied, (1162) Larissa and(1911) Schubart, are similar in size and thermal inertia, but so dif-ferent in terms of albedo. Indeed, they seem to mark extremes inthe observed properties of the Hildas - with (1162) Larissa beingunusually reflective, and (1911) Schubart particularly dark.These albedo values motivate us to review the taxonomy of bothasteroids; the P-type assigned to (1162) Larissa in the (LCDB) Asteroid Light Curve Data Base at , accessed on 20th March 2020. and JPL is not in tune with its low thermal inertia of 15 + − Jm − s − . K − , suggesting a clear difference in physical nature com-pared to the P-type low albedo (1911) Schubart and its 10 + − Jm − s − . K − .Romanishin & Tegler (2018) suggest that the (1911) Schubart fam-ily are interlopers to the Hilda region. The low albedo we obtain fromour observations supports the idea that (1911) Schubart is physicallysomewhat different to the other Hildas – but the question remains of NASA/JPL Small-Body Database at https://ssd.jpl.nasa.gov/sbdb.cgi , accessed on 20th March 2020.MNRAS , 1–13 (2020) thermophysical and dynamical study of the Hildas (1162) Larissa and (1911) Schubart Figure 5.
The dispersion of a suite of 61 875 clones of (1162) Larissa (left two columns) and (1911) Schubart (right two columns). The top row shows theclones after 10 Myr of integration time have elapsed, the central row shows them after 100 Myr, and the lower row shows them after 1 Gyr, at the end of oursimulations. For each object, the left-hand column shows the clones in semi-major axis-eccentricity space, whilst the right shows them in plan view, lookingdown at the Solar system from above. In the plan view plots, the yellow point at the centre denotes the location of the Sun, whilst the red dot shows the locationof Jupiter at the instant the clones are plotted. Whilst the clones of both (1162) Larissa and (1911) Schubart take some time to disperse, it is noticeable that theclones of (1911) Schubart do so more slowly, and over a smaller distance in orbital element space. whether those differences are the result of the collision that formedthe (1911) Schubart family, or are instead indicative that the parentof the family is an interloper in the Hilda region. Our dynamicalresults reveal that (1911) Schubart is extremely firmly embedded inthe Hilda population, trapped securely in 3:2 mean-motion resonancewith Jupiter – a result that seems hard to reconcile with its havingbeen captured to the population after the bulk had already formed.This result is potentially supported by the similarity between thethermal inertias of the two objects – suggesting that the divergentalbedos are the result of processes that have occurred through thelifetime of the objects, rather than being indicative of their havingmarkedly different origins.It is clear that further study is needed before we can conclusivelydetermine whether any or all of the Hildas are captured objects,rather than having formed in-situ. The origin of the Hildas is still anopen and crucial question, and it seems likely that the answer to thatquestion will shed new light on the formation and evolution of theSolar system as a whole.
ACKNOWLEDGEMENTS
This research is mainly based on observations taken by theNASA/DLR SOFIA airborne telescope. SOFIA is jointly operatedby the Universities Space Research Association, Inc. (USRA, NASAcontract NAS2-97001), and the Deutsches SOFIA Institut (DSI)under DLR contract 50 OK 0901, to the University of Stuttgart.This work has made use of the NASA/JPL Solar System Dynamicspage, at https://ssd.jpl.nasa.gov/ . It has also made use of theAstDyS database, at https://newton.spacedys.com/astdys/index.php?pc=3.0 , and NASA’s Astrophysics Data System, ADS.Also, this research made use of Astropy, a community-developedcore Python package for Astronomy, Photutils, an affiliated packageof Astropy for detecting and performing photometry of astronomicalsources, and Aperture Photometry Tool (APT), a software created toperform simple and effective aperture photometry analyses.TM has received funding from the European Union’s Horizon 2020Research and Innovation Programme, under Grant Agreement No.687378, as part of the project "Small Bodies Near and Far" (SBNAF).JPM acknowledges research support by the Ministry of Science and
MNRAS000
MNRAS000 , 1–13 (2020) C. F. Chavez et al.
Technology of Taiwan under grants MOST104-2628-M-001-004-MY3, MOST107-2119-M-001-031-MY3, and MOST109-2112-M-001-036-MY3, and Academia Sinica under grant AS-IA-106-M03.CCh wishes to acknowledge the fruitful discussions in the field ofInfrared Astronomy with Dr. Felipe Barrientos.The authors wish to thank the anonymous referee, whose com-ments helped to improve the flow and clarity of this work.
Facilities:
SOFIA (FORCAST)
Software:
Astropy (Astropy Collaboration et al. 2013, 2018),Numpy (Oliphant 2006; van der Walt et al. 2011; Harris et al. 2020),Matplotlib (Hunter 2007), Mercury (Chambers 1999).
DATA AVAILABILITY
The data underlying this article will be shared on reasonable requestto the corresponding author.
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C., Varoquaux G., 2011, Computing in Scienceand Engineering, 13, 22This paper has been typeset from a TEX/L A TEX file prepared by the author. MNRAS000