arXiv:1905.06056v1 [astro-ph.EP] 15 May 2019 rfrnefroeo h w osbemro ouin.W d We solutions. 3 mirror between possible ranged two which the of one for preference Results. WISE. and AKARI, IRAS, from Conclusions. cuttos dpieotc mgs rrdrehe r a are echoes radar ( or able images, s multi-chord optics unless adaptive determine, occultations, to hard otherwise th are Ast of magnitudes. albedo sizes absolute and visual d size with coupled Thermal the when determine compactness. objects, precisely their to us and allow surface also size, inertia, grain thermal regolith like ness, properties on regolith information surface carries asteroids from flux infrared Thermal Introduction 1. ie nsdslysgaue fcasrgan ntesurfa 2013). the Blum in & (Gundlach grains inertia coarser thermal larger of in signatures resulting display fine-grained while ones very inertia), thermal with sized smaller covered displaying usually (thus large are regolith bodies diameter regolith Also, in insulating values. km o of inertia 100 thermal of layer small of surfaces thick in amount the with sulting while small covered surface, are the the targets of on because surfac regolith inertia, fresh fine-grained Young, thermal composition. high and display age their 201 surf to asteroid al. covering connected et regolith (Vokrouhlický the force of properties recoil Physical thermal member under family asteroid persion or 2012), (Carry al densities et asteroid (Bottke o distribution based size-frequency evolution observed collisional the rently example for of studies for de to and method, s inversion in lightcurve adding the worldwide, used stations we multiple struction from hours, 12 wi than increases inertia thermal whether Methods. verify to and TPM, with e words. not Key does which widely, vary inertias thermal sample current htmtyue ooti hp oesrqie o thermop for required models Aims. shape obtain to used photometry rtsdee notesbsrae oee,temlinert thermal However, sub-surface. the into deeper trates Context. .Marciniak A. srnm Astrophysics & Astronomy a 6 2019 16, May .Ditteon R. .Kudak V. .Skrzypek J. .Antonini P. uehe l 05,ytte r seta characteristics essential are they yet 2015), al. et Durech u isaet iiaeteeslcinefcsb producin by effects selection these mitigate to are aims Our ˇ .Hirsch R. .Pilcher F. epeetnwmdl f1 lwrttr htpoieago fi good a provide that rotators slow 11 of models new present We ale oksget htsol oaigatrisshoul asteroids rotating slowly that suggests work Earlier odces h isaantso oaos ecnutdaph a conducted we rotators, slow against bias the decrease To 23 io lnt:atris–tcnqe:pooerc–radia – photometric techniques: – asteroids planets: minor 14 oehrwt u rvoswr,w aeaaye 6so rot slow 16 analysed have we work, previous our with Together , 24 1 .Feuerbach M. , hra rpriso lwyrttn asteroids: rotating slowly of properties Thermal 1 .Alí-Lagoa V. , 1 .Kulczak P. , 28 .Sobkowiak K. , 8 .Horbowicz J. , .Barbotin E. , .Perig V. , − +33 3 n 45 and aucitn.35129corr no. manuscript 24 1 .L Maestre L. J. , 10 − +60 2 .Polakis T. , 9 .G Müller G. T. , 30 1 eut rmatree survey targeted a from Results .Fauvaud S. , .Behrend R. , 1 .Kami K. , .Sonbas E. , m J − 2 s eevd2 aur 09/Acpe pi 2019 April 9 Accepted / 2019 January 25 Received − 1 29 Afiitoscnb on fe h references) the after found be can (Affiliations / nski 2 ´ .Poli M. , 25 K 15 32 10 2 − .Manzini F. , .Szakáts R. , .Garlitz J. , .Thizy O. , 1 1 .Bernasconi L. , . a aebe eemndmil o atrttr u osele to due rotators fast for mainly determined been have ias .K Kami K. M. , ofimteerirsgeto htsoe oaoshv hig have rotators slower that suggestion earlier the confirm ietemlieta eapidatempyia oe ofi to model thermophysical a applied we inertias thermal rive htertto period. rotation the th m ae aafo IEadKpe pc eecps o spi For telescopes. space Kepler and WISE from data cases ome rvdtedaeesadabdso u agt nadto t addition in targets our of albedos and diameters the erived yia oeln (TPM). modelling hysical csare aces nska 2005), . ´ smaller rough- than r cur- n dis- s tellar eroid hp oeso lwrttr,t cl hmadcmuethe compute and them scale to rotators, slow of models shape g ABSTRACT their vail- re- , aehge hra nrista atrrttr because rotators faster than inertias thermal higher have d ese ata 1 ce, 10 5). 16 es ld 3 12 .Roy R. , .Molnár L. , .Trela P. , .Geier S. , .Marks S. , inmcaim:thermal mechanisms: tion tmti bevn apino anbl seod ihp with asteroids main-belt of campaign observing otometric nska ´ pt fa bnac ftemldt vial anyfrom (mean- mainly AKARI available Explorer), data i Survey thermal Also, Infrared TPM. of (Wide-field by abundance WISE studies an asteroid of for spite factor limiting main the d angles. at phase and taken aspects t measurements viewing explain thermal ferent by to separate fails represented or often be lightcurves shape can mal simple it a 2014) such approximation although al. first et sphere, especiall at (Delbo’ when shape, cracking the cycles, thermal for in Such c result cooling. alternating can the and intense, of incli heating intensity axis surface the and spin of of duration and shape the period and dictate rotation spin tion the The study. of knowledge under t the objects input is essential models The these 2015). (fo al. ply used et being Delbo’ see are overview their TPM) (henceforth: models o thermophysical rotators orbit. slow the like to area, inclination axis surface spin same low epis the with long targets heating with Sun targets the studying by of or waveleng deeper, longer using penetrate by de that surface study immediate to possible the be under also layers plane might It created system. som that solar the asteroids, planetesimals shaped leftover of intact history are thermal which and collisional namical, 11 otetemldt.I w ae,teTMaayi hwdac a showed analysis TPM the cases, two In data. thermal the to t .Btiwc ˛ak B - Butkiewicz M. , tr rmordnesre ihszsbten3 n 5 m T km. 150 and 30 between sizes with survey dense our from ators oa h viaiiyo pnadsaemdle agt is targets modelled shape and spin of availability the Today vario properties, thermal asteroid of studies detailed For dy- the into insights valuable provide can properties These 1 30 .H Kim D.-H. , 1 .J Sanabria J. J. , 17 .Urakawa S. , 10 3 , 18 , 4 .Monteiro F. , .Pál A. , .Goncalves R. , 21 3 , , 22 33 5 17 .Pdesa-Gaca - Podlewska E. , .J Kim M.-J. , M. , .Santana-Ros T. , 26 1 .Ogłoza W. , .Crippa R. , 19 Zejmo ˙ .Grice J. , 34 22 n K. and , .Konstanciak I. , to fet nteavailable the in effects ction 12 20 e hra inertias. thermal her 27 .Duffard R. , .Grze I. , vial nrrddata infrared available t hi hra inertias, thermal their o 1 .Oszkiewicz D. , .Skiff B. , h etwv pene- wave heat the n hp recon- shape and n 1 Zukowski , ˙ 6 rtemlinertia thermal ir

.Parley N. , c skowiak ´ roslonger eriods S 2019 ESO 31 13 1 , , , sand ts 1 ycles ap- o odes eper As . of e her- 1 1 lear na- ths , 7 if- , us he , n y 1 a r r Marciniak et al.: Thermal properties of slowly rotating asteroids ing “light” in Japanese), IRAS (The Infrared Astronomical having profound effects on the results from any survey using Satellite), or Herschel space observatories, they also possess asparse-in-time observing cadence. their limitations: for unique solutions from thermophysical mod- Some light on the subject of biases in asteroid models was elling, thermal data have to come from a range of aspect and shed by recent results based on data for asteroids from Gaia phase angles, probing the target rotation with sufficient reso- Data Release 2 (DR2), which facilitated unique determinationof lution at the same time. As a result, so far detailed TPM have spin parameters for around 200 targets (Durechˇ & Hanuš 2018). been applied to less than 200 asteroids (see the compilations by This first, largely unbiased sample of targets (with some excep- Delbo’ et al. 2015; Hanuš et al. 2018b), and only a small frac- tions, see Santana-Ros et al. 2015) modelled using very precise tion of them rotate slowly, with periods exceeding 12 hours. absolute brightnesses revealed interesting results. Half of the Slow rotators are especially interesting in this context, as a trend asteroids in this sample turned out to rotate slowly, with pe- of increasing thermal inertia with period has been found by riods longer than 12 hours, unlike in the sets of models from Harris & Drube (2016), based on estimated beaming parameters the majority of previous surveys, based on ground-based data, (η) determined on data from the WISE space telescope. where shorter periods dominated (e.g. Hanuš et al. 2011, 2013). The distribution of thermal inertia amongst the members of Results from Gaia DR2 confirm the findings of Szabó et al. asteroid families would be crucial for example in the search (2016) and Molnár et al. (2018) based on data from the Kepler for evidence for asteroid differentiation, separating iron rich Space Telescope of the substantial contribution of slow rotators from iron poor family members (Matter et al. 2013). This quan- in the asteroid population. tity is also essential for estimates of the orbital drift caused It should also be noted that models based on sparse data pro- by the Yarkovsky effect, and for studies of regolith properties. vide reliable spin parameters, but only low-resolution, coarse However, the relatively small number of targets with known spin shape representations (flag ’1’ or ’2’ in the shape quality sys- and shape parameters prevents detailed thermophysical studies tem proposed by Hanuš et al. 2018a), limiting their use in other on large groups of asteroids, limiting these to using simple ther- applications (Hanuš et al. 2016). For example, detailed thermo- mal models with spherical shape approximation, often insuffi- physical modelling needs rather high-resolution (flag ’3’) shape cient to explain thermal data taken in various bands and viewing models as input, only possible with large datasets of dense geometries. It is especially vital in the light of high precision lightcurves obtained at various viewing geometries. Such mod- thermal infrared data from the WISE and Spitzer space tele- els usually provide a much better fit to thermal data than shapes scopes now available, with uncertainties comparable to average approximated by a sphere with the same spin axis and period uncertainties of the shape models based on lightcurve inversion (Marciniak et al. 2018). (Delbo’ et al. 2015; Hanuš et al. 2015). In the next section we refer to our targeted survey, then in Nowadays asteroid spin and shape models are created mainly Sect. 3 we describe our modelling methods. Results from both from sparse-in-time visual data (of the order of one to two points lightcurve inversion and thermophysicalmodelling are presented per day) from large surveys like those compiled in the Lowell in Sect. 4, and discussion and conclusionsin Sect. 5. Appendix A photometric database (Hanuš et al. 2011). The low photometric contains details of the observing runs and composite lightcurves, precision of such data (average σ ∼ 0.15 - 0.2 mag), originally while Appendix B presents O-C plots from thermophysicalanal- gathered for astrometric purposes, results in the favouring of ysis. some targets in the modelling, for example mainly those display- ing large amplitudes in each apparition (Durechˇ et al. 2016). A 2. Targeted survey partial solution to the problem is joining sparse data with dense lightcurves (like in e.g. Hanuš et al. 2013), an approach limited Taking all the above-mentioned facts into account, it is clear though by the availability of the latter, which are also strongly that there is a continuing need for targeted, dense-in-time ob- biased towards large amplitudes and short periods. On the other servations of objects omitted by previous and ongoing sur- hand, targeted surveys gathering dense lightcurves favour certain veys. Such targets are mainly slow rotators, and also bodies types of objects, like near Earth asteroids (NEAs) or membersof that either constantly or temporarily display lightcurves of small specific groups or families like Flora, Eos, or Hungariaasteroids. amplitudes. To counteract both selection effects, our observ- The resulting spin and shape determinations, even if taken alto- ing campaign is targeted at main-belt slow rotators (P > 12 gether, do not provide unbiased information on for example the h), which at the same time have small maximum amplitudes spin axis distributions of the whole asteroid population. Instead, (amax ≤0.25 mag). Neither sparse data from large surveys nor they are biased by spin clusters within families (Slivan 2002; available datasets of dense lightcurves allow for their precise Kryszczynska´ 2013) and by preferential retrograde rotation of spin and shape modelling, so they require a dedicated observ- NEAs (Vokrouhlický et al. 2015). Taking all this together can ing campaign. It is worth noting that it is exactly these features – distort the results for the whole population, missing certain spe- small and slow flux variations – that make them challenging for cific cases, so that many elements essential for properly under- spin and shape reconstructions, which make these targets per- standing asteroid dynamics, physics, and evolution are lacking fect calibration sources for a range of infrared observatories (like (Warner & Harris 2011). For example, Jupiter Trojans on aver- ALMA – Atacama Large Millimeter Array, APEX – Atacama age rotate much more slowly than main belt asteroids, so there Pathfinder Experiment, or IRAM – Institut de Radioastronomie might be a gradation of rotation periods with growing heliocen- Millimétrique, Müller & Lagerros 2002), on the condition that tric distance (Marzari et al. 2011). Also, as has been stressed by these variations can be exactly predicted. To this purpose their Warner & Harris (2011), as much as 40% of lightcurves consid- reliable spin and shape model are necessary. ered in the Lightcurve Database (LCDB)1 as reliable, have small The details of our campaign conducted at over 20 stations from 12 countries worldwide, the target selection procedure, maximum amplitude: amax ≤ 0.2 mag, posing additional chal- lenges in period determination due to potential ambiguities, and and first results are described in Marciniak et al. (2015) and Marciniak et al. (2016). In a nutshell, our survey revealed a sub- stantial number of slow rotators which, in spite of having good 1 http://www.MinorPlanet.info/lightcurvedatabase.html. data from dense observations, had wrongly determined rotation

2 Marciniak et al.: Thermal properties of slowly rotating asteroids periods, summarised in LCDB (Warner et al. 2009), a database as relative, and not absolutely calibrated, as from our campaign used in a variety of further studies on asteroid spins and dy- we obtain mostly relative brightness measurements. Only the namics. We corrected those determinations with the new values models with a unique solution, clearly the best in terms of χ2 fit for the periods, and confirmed them in consecutive apparitions. of observed to modelled lightcurves, have been accepted. Each We also constructed spin and shape models for the first sam- of the acceptable shape model solutions has been visually in- ple our targets, using dense lightcurves from multiple appari- spected to fulfil the criterion of rotation around the axis of great- tions, and scaled these models by stellar occultations fitting and est inertia. From the range of accepted solutions the uncertainty thermophysical modelling, with consistent size determinations range of spin axis position and period has been evaluated. As (Marciniak et al. 2018). In the latter work we also presented shown by Kaasalainen et al. (2001) this is the only practical way simultaneous fits to data from three different infrared missions of obtaining realistic uncertainty for spin axis position because (IRAS, AKARI, and WISE), and in spite of potential calibration formal errors like those obtained from the covariance matrix tend problems of each of the missions alone and cross-calibrations to be strongly underestimated. The effects of random noise are between them, we obtained good fits. This gave us confidence in much smaller than systematic errors not uncommon in photo- our models and methods used. metric data, or the combined effects of model shape and spin The most important of our previous results presented in axis uncertainties. The uncertainty of period is also determined Marciniak et al. (2018) were large thermal inertia values ob- from the range of best-fit solutions, and is dictated by the time tained for most of our slow rotators, which followed the trend span of the whole photometric dataset and the period duration found by Harris & Drube (2016). This seemed to support the itself. If the minimum in the periodogram is substantially lower idea that for slow rotators we observe thermal emission from than the others, the period uncertainty can be assumed to be one deeper, more compact layers of the surface, a fact that might hundreth of this minimum width, which is usually of the same open new paths for studies of asteroid regolith. However, due order as the one determined from the best-fit solutions. to the small number of models for slowly rotating asteroids, As is usually the case when models are based exclusively on until recently thermal inertia values from detailed TPM have dense lightcurves, the uncertainty of the spin axis position (pole) been determined for only two such asteroids (227 Elvira and was of the order of a few degrees (see Table 1), and the shape 956 Elisa, Delbo & Tanga 2009, Lim et al. 2011, see fig 5. in models had a smooth appearance. Formally large uncertainty in Harris & Drube (2016), based on compilation by Delbo’ et al. some values of pole longitude (λp) is due to high inclination of (2015), reproduced here in Fig. 12) The above-mentioned trend the pole (βp), translating actually to a relatively small distance was found on the values derived from estimated beaming param- on the celestial sphere. eters using a simplified approach. Recent results from TPM by In thermophysical modelling, the diameter (D) and the ther- Hanuš et al. (2018b) showed a rather large diversity of thermal mal inertia (Γ) are fitted and different surface roughness are tried inertia values for slow rotators within the size range of 10-100 by varying the opening angle of hemispherical craters covering km. Here we further investigate this issue by studying more slow 0.6 of the area of the facets (following Lagerros 1996), cov- rotators in both visible light and thermal infrared radiation. ering rms values between 0.2 and ∼1. Heat diffusion is 1-D Our photometric campaign is ongoing, and new observing and we use the Lagerros approximation (Lagerros 1996, 1998; stations have joined. Our data sources now also include observa- Müller & Lagerros 1998; Müller 2002). The TPM implementa- tions from the Kepler Space Telescope in its extended mission, tion is the one used in Alí-Lagoa et al. (2014), based on that of K2 (Howell et al. 2014). Kepler could not track moving targets Delbo’ & Harris (2002), however the colour correction is treated during a campaign, so solar system objects could be observed ei- differently: instead of colour correcting each facet’s flux based ther by using previously allocated, curved, or boomerang-shaped on its effective temperature, we now use the H and G values tab- masks in accordance with the expected positions of the target ulated in the JPL (Jet Propulsion Laboratory) Horizons database minor bodies (see e.g. Pál et al. 2015) or larger continuous pixel to compute an effective temperature based on the heliocentric masks (so-called supermasks) where such objects also appeared distance at which each observation was taken (following the ap- for several days, but not on purpose(see e.g. Molnár et al. 2018). proach in Usui et al. 2011). Another simplifying assumption is Data processing has been safely established for these observa- that spectral emissivity is constant and equal to 0.9 (see e.g. tions, regardless of whether these were targeted or not. This pro- Delbo’ et al. 2015), which does not seem unreasonable given the cessing is based on the registration of the images (to correct for small spectral contrast of 1–3% found by Licandro et al. (2012) spacecraft jitter) followed by a differential-image photometry by in the 10-µm features of 50–60 km Themis family members involving oblong-shaped apertures (see Pál et al. 2015, Szabó et (the 20-µ emission plateau is expected to be flatter). Once we al. 2017,and Molnáret al. 2018for furtherdetails). The modelof obtain D, we compute the visible geometric albedo using the (100) Hekate presented here has been partially based on Kepler H-G12 values from Oszkiewicz et al. (2011). The Bond albedo data. The asteroid (100) Hekate was observed with a dedicated that would be derived from these values is sometimes differ- set of masks for 5.5 and 4.5 days continuously in Campaigns ent from the Bond albedo value used as input for the TPM 16 and 18, respectively. Table A.1 in the Appendix gives details (Ai). This value was obtained by first averaging the tabulated of each observing run obtained within the present work. Scarce diameter from IRAS, AKARI, and WISE (Tedesco et al. 2005; literature data had to be complemented with from ∼100 up to Usui et al. 2011; Alí-Lagoa et al. 2018; Mainzer et al. 2016) and ∼500 hours of new observations for each target before unique occultations (compiled by Dunham et al. 2016). Then, we used models were feasible. this size to compute the Bond albedo from all available H-G, H-G12, and/or H-G1-G2 values from the Minor Planet Center 3. Lightcurve inversion and thermophysical (Oszkiewicz et al. (2011) or Vereš et al. (2015)), and took again modelling the average value. This approach is somewhat arbitrary but the TPM results are not very sensitive to the value of the Bond For spin and shape reconstructions we use lightcurve inversion albedo. Justification for this, further details, and discussion, in- by Kaasalainen & Torppa (2001), which represents the shape cluding informationon how we estimate our error bars, are given model by a convex polyhedron. All the lightcurves were treated in Appendix B.

3 Marciniak et al.: Thermal properties of slowly rotating asteroids Table 1: Spin parameters of asteroid models obtained in this work, with their uncertainty values. The first column gives the sidereal period of rotation, next there are two sets of pole J2000.0 longitude and latitude. The sixth column gives the rms deviations of the model lightcurves from the data, and next follow the photometric dataset parameters (observing span, number of apparitions, and number of individual lightcurve fragments). Pole solutions preferred by TPM are marked in bold.

Sidereal Pole 1 Pole 2 rmsd Observing span Napp Nlc period [hours] λp βp λp βp [mag] (years)

(100) Hekate 27.07027 104◦ +51◦ 306◦ +52◦ 0.012 1977–2018 8 62 ±0.00006 ±7◦ ±2◦ ±8◦ ±3◦

(109) Felicitas 13.190550 77◦ −26◦ 252◦ −49◦ 0.015 1980–2018 8 62 ±0.000004 ±1◦ ±5◦ ±4◦ ±7◦

(195) Eurykleia 16.52178 101◦ +71◦ 352◦ +83◦ 0.015 2001–2017 7 51 ±0.00001 ±60◦ ±15◦ ±60◦ ±15◦

(301) Bavaria 12.24090 46◦ +61◦ 226◦ +70◦ 0.014 2004–2018 5 30 ±0.00001 ±5◦ ±6◦ ±14◦ ±6◦

(335) Roberta 12.02713 105◦ +48◦ 297◦ +54◦ 0.012 1981–2017 9 52 ±0.00003 ±7◦ ±2◦ ±9◦ ±6◦

(380) Fiducia 13.71723 21◦ +34◦ 202◦ +44◦ 0.016 2008–2018 6 37 ±0.00002 ±2◦ ±3◦ ±1◦ ±2◦

(468) Lina 16.47838 74◦ +68◦ 255◦ +68◦ 0.013 1977–2018 7 40 ±0.00003 ±18◦ ±6◦ ±17◦ ±8◦

(538) Friederike 46.739 168◦ −58◦ 328◦ −59◦ 0.015 2003–2018 8 98 ±0.001 ±30◦ ±5◦ ±25◦ ±10◦

(653) Berenike 12.48357 147◦ +18◦ 335◦ +8◦ 0.009 1984–2018 6 39 ±0.00003 ±3◦ ±2◦ ±2◦ ±1◦

(673) Edda 22.33411 66◦ +64◦ 236◦ +63◦ 0.022 2005–2017 7 49 ±0.00004 ±15◦ ±6◦ ±13◦ ±6◦

(834) Burnhamia 13.87594 77◦ +60◦ 256◦ +69◦ 0.018 2005–2017 6 32 ±0.00002 ±10◦ ±6◦ ±8◦ ±7◦

4. Results models applied are summarised in Table 2, presenting the diam- eter, albedo, and thermal inertia for the best-fitting model solu- In this section for each target separately we refer to previous tion. The values obtained here for diameter are ”scaling values” works containing lightcurves, briefly describe our data, the char- of the given spin and shape solutions listed in Table 1 that would acter of brightness variations, and its implications for the spin otherwise be scale-free. For comparison we also refer there to and shape. Later, the thermal data availability and range are de- the diameters obtained previously from the AKARI (Usui et al. scribed, followed by the results of applying these models in TPM 2011), IRAS (Tedesco et al. 2005), and WISE (Mainzer et al. analysis. In plots like Fig. 1, the fit of model infrared flux vari- 2011; Masiero et al. 2011) surveys. We also added their tax- ations to thermal lightcurves is presented, and plots like Fig. onomic class following Bus & Binzel (2002a,b) and Tholen B.9 in the Appendix show the observation-to-model ratios ver- (1989), so that the albedos could be verified for agreement with sus wavelength, helicentric distance, rotational angle, and phase taxonomy. angle. Table 1 presents spin parameters obtained within this work For practical reasons, the projections of the shape models (sidereal period, spin axis position, and used lightcurve data). and the fit to all the visible lightcurvesare not presented here, but Results from thermophysical modelling with our spin and shape will be available from the DAMIT (Database of Astroid Models

4 Marciniak et al.: Thermal properties of slowly rotating asteroids

Table 2: Asteroid diameters from AKARI, IRAS, and WISE (DA,DI ,DW ) (Tedesco et al. 2005; Usui et al. 2011; Mainzer et al. 2016), compared to our values from TPM (sixth column). We also provide the corresponding visible geometric albedo (pV ) as described in Sect. 3. Finally, we tabulate the nominal thermal inertia (Γ) and its value normalised at 1 AU using the mid-value (Rh) between the maximum and minimum heliocentric distances spanned by each object’s IR data set. To convert Γ values to 1 AU we assumed that Γ is proportional to Rh−3/4 (see the Discussion section). Error bars are 3-σ. Radiometric solution for combined data Thermal inertia Target DA DI DW Taxonomic Diameter pV Thermal inertia Rh at 1 AU [km] [km] [km] type [km] [SI units] [AU] [SI units]

+5 +0.03 +66 +154 100 Hekate 88.52 88.66 91.421 S 87−4 0.22 −0.03 4−2 3.10 9−5

+7 +0.008 +100 +236 109 Felicitas 80.81 89.44 89.000 Ch 85−5 0.065−0.01 40−40 3.15 95−95 +11 ± +55 +121 195 Eurykleia 89.38 85.71 80.330 Ch 87−9 0.06 0.02 15−15 2.85 33−33

+2 +0.004 +60 +125 301 Bavaria 51.90 54.32 55.490 C 55−2 0.047−0.003 45−30 2.65 94−62

+10 +0.014 335 Roberta 92.12 89.07 89.703 B 98−11 0.046−0.008 unconstrained 2.80 unc.

+9 +0.009 +140 +278 380 Fiducia 75.72 73.19 69.249 C 72−5 0.057−0.012 10−10 2.50 20−20

+11 +0.006 +280 +638 468 Lina 59.80 69.34 64.592 CPF 69−4 0.052−0.014 20−20 3.00 46−46 +5 ± +25 +115 538 Friederike 72.86 72.49 79.469 C 76−2 0.06 0.01 20−20 3.50 50−50

+4 +0.02 +120 +273 653 Berenike 46.91 39.22 56.894 K 46−2 0.18 −0.03 40−40 3.00 91−91

+6 +0.03 +33 +72 673 Edda 39.38 37.53 41.676 S 38−2 0.13 −0.05 3−3 2.82 7−7

+8 +0.014 +30 +64 834 Burnhamia 61.44 66.65 66.151 GS: 67−6 0.074−0.016 22−20 2.75 47−43 from Inversion Techniques)2 created by Durechˇ et al. (2010), for Hainaut-Rouelleet al. (1995), and Galad et al. (2009), and models viewing and download, to be used in other applications. complemented them with data from the SuperWASP survey In Appendix A (Figs. from A.1 to A.53), we present all the pre- (Wide Angle Search for Planets, Grice et al. 2017), and our viously unpublished photometric data in the form of composite own data from three more apparitions (see Figs. A.1 to A.4 lightcurves, as they testify the good quality of our spin and shape in the Appendix). SuperWASP cameras are known to suffer solutions: photometric accuracy here is mostly at the level of a from the detector having a temperature dependence, so the few millimagnitudes, which is one to two orders of magnitude absolute lightcurves can be systematically shifted in magnitude better than in sparse data standardly used for asteroid modelling. depending on the temperature of the detector. However, as we Also, with such slow rotation, separate lightcurves would not treat them as relative lightcurves this should not be an issue. show the whole character of the brightness variability, but only a In the last apparition, Hekate was observed for our project by small fraction of the full lightcurve, which is visible only in the the Kepler Space Telescope, resulting in two continuous, four- folded plots. Moreover, such plots enable us to see lightcurve day-long lightcurves of great quality. The K2 data have been evolution caused by phase angle effect, which due to various reduced with the fitsh package, using the same methods that levels of shadowing highlight various shape features. Since we were already applied to targeted observations of Trojan asteroids applied no photometric phase correction, their signatures are and chance observations of main-belt asteroids (MBAs) in the visible on the overlapping fragments that cover the same rota- mission (Pál 2012; Szabó et al. 2016, 2017; Molnár et al. 2018). tional phase, but are spaced by more than a month in time like Overall, Hekate displayed interesting, asymmetric lightcurves for example in Fig. A.18. Apparitions with only one lightcurve of amplitudes ranging from 0.11 to 0.23 magnitude and a long, fragment or those with data covering only a small fraction of 27.07 hours period. full rotation, are not presented here, because creating compos- The spin parameters of our model are presented in Table 1. ite lightcurves would be either impossible or would not present To construct it, initial scanning of parameter space needed an much information on the period or the lightcurve appearance. increased number of trial poles and iteration steps. Overall both However, such data have been used in the modelling, helping to spin solutions fit the visual lightcurves at a very good level of constrain spin and shape. The full list of observing runs is sum- 0.012 mag. marised in Table A.1 in Appendix A. TPM analysis 4.1. (100) Hekate The two mirror pole solutions of Hekate are named AM 1 and We compiled archival lightcurves from four appari- AM 2. We used 52 infrared observations, 32 from IRAS (8 x tions, published by Tedesco (1979), Gil-Hutton (1990), 12 µm,8x25 µm,8x60 µm, 8 x 100 µm), five from AKARI (2 x S9W, 3 x L18W), and 15 from WISE (W4). We assumed 2 http://astro.troja.mff.cuni.cz/projects/asteroids3D. A =0.090 for the Bond albedo.

5 Marciniak et al.: Thermal properties of slowly rotating asteroids

10.5 2.2 AM 2 model AM 1 model Abs.calib. error bar Abs.calib. error bar 10 2 9.5

9 1.8

8.5

Flux (Jy) Flux (Jy) 1.6 8

7.5 1.4 7

6.5 1.2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Rotational phase Rotational phase 5 Fig.1: (100) Hekate W4 data and model thermal lightcurves for AM 1 model Abs.calib. error bar shape model 2. Data error bars are 1-σ. Table B.1 summarises 4.8 the TPM analysis. 4.6

4.4

4.2

Flux (Jy) 4

Both models provide formally acceptable fits, as do the cor- 3.8 2 responding spheres (see Table B.1 and the χ plots in Fig. B.9 3.6 in Appendix B). The fit to the WISE “lightcurve” seems rea- 3.4 sonable for both models, although not around the zero rotational phases shown in Figs. 1, B.1, and B.9 (third panel from the top). 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Rotational phase The IRAS data present a slope in the observation-to-model ra- tios (OMR) versus wavelength plot, which is shown in the top Fig.2: Asteroid (109) Felicitas W3 (top) and W4 (bottom) data panel of Fig. B.9. Model 2 provides a better fit than 1 and the and model thermal light curves for shape model AM 1 (see spheres so we select it as our provisionally favoured solution. Table B.2). Nonetheless, these results could be biased by our neglecting the dependence of thermal conductivity – and therefore thermal in- ertia – with temperature (see Rozitis et al. 2018, and the review TPM analysis in Delbo et al. 2015, for instance), over the wide range of he- The two mirror solutions for Felicitas from Table 1 are denoted liocentric distances sampled (2.6 to 3.6 au). However, there is AM 1 and AM 2. In the thermal approach we used 38 observa- not enough data taken at long heliocentric distances to make a tions, 15 from IRAS (5 x 12 µm,7 x 25 µm,3 x 60 µm), five conclusive statement. fromAKARI (3x S9W, 2 x L18W),and18 fromWISE (9 x W3, To summarise the TPM analysis: the surface roughness is 9 x W4). We assumed A =0.025 for the Bond albedo. +5 not constrained, the diameter is 87−4 km, and geometric albedo, Unlike the AM 2 model, AM 1 provides a formally accept- +0.03 2 pV = 0.22−0.03. We would benefit from additional thermal light able fit (Table B.2, and the χ plot in Fig. B.10 in Appendix B), curves at negative phase angles (i.e. after opposition) and higher although the WISE bands are not fitted equally well: the model heliocentric distances (to study the conductivity dependence overestimates the W4 data and underestimates the W3 (top panel with temperatures). Low thermal inertias of approximately 5 SI in Fig B.10). Even though the model thermal lightcurves miss units3 and low to medium roughness fit the data better. some W3 and W4 fluxes (Fig. 2), we consider it as a preliminary approximated solution given that it fits significantly better than the sphere. Additional thermal curves could help confirm or re- 4.2. (109) Felicitas ject this model and data at negative phase angles could improve the constraints for the thermal inertia. Available photometric data for Felicitas came from the works As usual, the surface roughness is not constrained. From the of Zappala et al. (1983), Harris & Young (1989), and from the +7 +0.008 diameter (85−5 km), we obtain the albedo pV = 0.065−0.01 . SuperWASP archive (Grice et al. 2017). We added to these two apparitions data from five more obtained within our cam- paign, and also data from the WISE satellite obtained in W1 4.3. (195) Eurykleia band, which is dominated by reflected light (thermal contribu- There was no prior publication presenting lightcurves of tion in W1 for this target was estimated at 8% - 30%). The latter Eurykleia, but plots from three apparitions are available on the dataset greatly helped to constrain the model, in an approach observers’ web pages4, indicating a long, 16.52 hours period. first proposed by Durechˇ et al. (2018). Felicitas lightcurves have Here we publish all of these data, adding more recent observa- been changing substantially between the apparitions, reaching tions from four more apparitions. During all the seven appari- peak-to-peak amplitudes from 0.06 to 0.22 mag, with the syn- tions, Eurykleia displayed similarly shaped lightcurves with one odic period around 13.194 hours (Figs. A.6, A.7, and A.8 in the minimum sharper than the other, which was accompanied by an Appendix). additionalbump or shelf (see Figs. A.9 to A.14 in the Appendix).

4 http://obswww.unige.ch/∼behrend/page_cou.html, 3 [Jm−2s−0.5K−1] http://eoni.com/∼garlitzj/Period.htm.

6 Marciniak et al.: Thermal properties of slowly rotating asteroids

9.5 W4 data thermal lightcurve (Fig. 3), although the flux level of the W4 9 Best fit AM 1 (all data used) Best fit AM 1 (W4 not used) data is systematically below our fitted model’s fluxes. We do not Best fit AM 1 (only W4 used) 8.5 have an explanation for such a systematic mismatch, but per- haps additional thermal IR data could correct the TPM diameter 8 of (195) Eurykleia to a higher value closer to 90 km, or a lower 7.5 value closer to 80 km. Thus, with the current dataset, the pos- 7 Flux (Jy) sible diameters range from 78 to 98 km depending on which 6.5 subsets of data we fit (Table B.3). As a compromise, we take the +11 6 mid-value in this range, 87−9 km (3σ level error bars, ∼13% relative error). 5.5 On the other hand, the large 3σ range of possible thermal 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 inertias does not change when we fit different subsets of data, Rotational phase but the error bars are widely asymmetric. We can only provide Fig.3: W4 data and model thermal lightcurves for (195) an upper limit of 70 SIu, with a best-fitting value of Γ=15 SIu. Eurykleia’s shape model AM 1. The different models resulted Surface roughness is not constrained at the 3σ level, but high from fitting different subsets of data. Table B.3 contains the cor- roughness models fit the data better, as is usually the case for responding thermo-physical parameters. main-belt asteroids. Finally, the geometric albedos derived from our radiometric diameters vary slightly depending on the photometric phase cor- The amplitude was stable, being always around 0.24 mag, which rection applied and the source of the absolute magnitudes and indicates a stable aspect angle due to the high inclination of the slope parameters. To compute our final value, we use the H-G12 spin axis. As expected, the resulting model has a high value of values from the Oszkiewicz et al. (2011) table and calculate the |β|, and a formally large range of possible values of λ (see Table maximum and minimum possible pV s using all the three sources 1). listed above. We find pV =0.06 ± 0.02.

TPM analysis 4.4. (301) Bavaria The two mirror solutions from lightcurve inversion are labelled The lightcurve from only one previous apparition of asteroid AM 1 and AM 2. We used a total of 57 infrared observations Bavaria has been reported in the literature (Warner 2004). Our from IRAS (nine epochs x four filters), AKARI (4 S9W + 5 campaign built an extensive lightcurve dataset for this target, L18W) and WISE W4 (12). The WISE W3 data were saturated covering four viewing aspects and a wide range of phase an- for this object and the W4 data, with reported magnitudes be- gles. Our data covered sometimes four to five months within one tween 0.00 and 0.35, are near the -0.6 mag identified as the on- apparition, and less than a year passed between consecutive ob- set of saturation for individual W4 images (Cutri et al. 2012). served apparitions. Such an observing strategy was defined as We assumed A =0.020 for the Bond albedo. the most optimal for spin and shape reconstructions by Slivan Both shape models provide statistically similarly good fits (2012), and is confirmed by our experience. Bavaria during its to all the data, so the TPM cannot reject any of the mirror so- 12.24-hours-long rotation displayed wide minima and ampli- lutions in this case. Using all the data to optimise the χ2 (i.e. tudes from 0.25 to 0.36 magnitudes (Figs. A.15 - A.18 in the considering ν = 57 − 2 = 55 degrees of freedom), the mini- Appendix), exceeding our initial target selection criteria in the 2 mum reduced chi-squared (χ¯m) were 0.51 and 0.60 for the AM course of the campaign; nonetheless it was retained in the target 1 and AM 2 models, respectively. These are significantly bet- list. Although the solution for the sidereal period and two pos- ter than the values obtained for spheres with the same respective sible spin axis positions are very well defined, due to the high spin pole orientation (see Table B.3). value of pole latitude (see Table 1) the shape model vertical ex- Although the thermal data seems to be fitted very well based tent is poorly constrained. 2 on our small χ¯m, the OMR of many IRAS data are slightly but systematically above 1, the WISE OMRs slightly below 1, and the AKARI ones are not fully horizontally aligned (Fig. B.11 in TPM analysis Appendix B, upper panels). Figures 3 and B.2 in Appendix B show that both shape models fit the variation of the W4 thermal The two mirror solutions from Table 1 are AM 1 and AM 2. lightcurve reasonably well. However, the fluxes predicted by the We used 36 thermal observations, 18 from IRAS (9 x 25 µm, TPM solution that best fits all the data (yellow open circles) sys- 9 x 60 µm), six from AKARI (3 x S9W, 3 x L18W), and 12 tematically overestimate most of the W4 observations. This is from WISE (W4). We rejected IRAS 12- and 100-micron data worse for the solution that best fits the AKARI and the IRAS because they contain clear outliers (by a factor of several). We data (open triangles), because the fitted diameter is larger in this assumed A =0.020 for the Bond albedo. case (third line in Table B.3, first and third plot from the top in Both models and the corresponding spheres provide a good 2 Fig. B.12). However, the OMR values for the IRAS and AKARI fit (see Table B.4, and the χ plots in Fig. B.13 in Appendix B), data do align horizontally (Fig. B.11 in Appendix B, lower pan- although with slightly offset diameters and different roughness els). Finally, we can fit the lightcurve much better if we optimise (the spheres require low roughness rather than medium). The the W4 data alone (filled squares),but the thermalinertia is lower AM 1 OMR plots are shown in Fig. B.13, and the W4 model by a factor of three (fourth entry in Table B.3, second and fourth thermal lightcurves with the data in Figs. 4 and B.3. plot in Fig. B.12). As usual, the roughness is not constrained at the 3σ level. +2 +0.004 Conclusions from TPM analysis are the following: both With a diameter 55−2 km, we get pV = 0.047−0.003 and a ther- shapes seem to fit the data well, including the shape of the W4 mal inertia between 10 and 100 SI units.

7 Marciniak et al.: Thermal properties of slowly rotating asteroids

4.2 AM 1 model 10 Abs.calib. error bar W4 data Best fit AM 2 4 9

3.8 8

3.6 Flux (Jy) Flux (Jy) 7

3.4

6 3.2

5 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Rotational phase Rotational phase Fig.4: Asteroid (301) Bavaria AM 1 model thermal lightcurves Fig.5: Asteroid (335) Roberta’s WISE data and model thermal and W4 data (see Table B.4). lightcurves for shape model 2 and the corresponding best-fitting solutions (very low thermal inertia of 15 SIu).

4.5. (335) Roberta We compiled a large dataset covering as many as nine appari- varied substantially from one apparition to another, reaching tions of asteroid Roberta, including data from Binzel (1987), 0.32 mag amplitude in one, and then decreasing down to 0.04 Harriset al. (1992), Lagerkvistet al. (1995), Warner et al. in the other (see Figs. A.22 - A.28 in the Appendix). In the (2007), Pilcher & Martinez (2015), SuperWASP archive (Grice year 2014 Fiducia was observed exclusively on the 0.8m TJO et al. 2017), and our own data (presented in Figs. A.19, A.20, (Telescopi Joan Oró) telescope in the Montsec Observatory and A.21 in the Appendix). This target showed rather rare, in a mode similar to ’dense-sparse cadence’ described by monomodal lightcurves of amplitudes from to mag, 0.13 0.19 Warner & Harris (2011). In spite of the relative sparseness of depending on the aspect. Its rotation period, 12.027 hours, com- datapoints, the lightcurve’s general character is clearly outlined, mensurate with an Earth day, required observations from sites and the synodic period from other apparitions is confirmed (Fig. well spaced in longitude for full coverage. Thanks to such A.24). an extensive dataset both spin and shape solutions are well constrained (Table 1), however different from those found by In the apparition on the verge of the years 2015 and 2016, Blanco et al. (2000), who reported four pole solutions, all much Fiducia was observed under an exceptionally large range of lower in |β| than ours, and a sidereal period longer by 0.027 phase angles (Table A.1), which resulted in distinctive differ- hours, which is a substantial difference. ences in the lightcurve character. Thus we present data from that apparition on two separate plots, one for small, and the other for large phase angles (Appendix, Figs. A.26 and A.25, respec- TPM analysis tively). The two mirror solutions are AM 1 and AM 2. Thermal data The unique solution for the sidereal period could only be consisted of 44 observations, 22 from IRAS (6 x 12 µm,6x25 found in this case during denser scanning of parameter space. In µm,6x60 µm,4x100 µm), nine from AKARI (4 x S9W and 5 spite of clear signatures of low inclination of the spin axis, the x L18W), and 13 from WISE (W4). We assumed A =0.020 for resulting pole solution is located at rather moderate latitudes (see the Bond albedo. Table 1). This might be a symptom of the lightcurve inversion The best solutions for both models are comparably good, method bias against low poles found by Cibulková et al. (2016), and even the spheres provide a formally good fit with com- as small shape modifications can compensate for the shifted spin patible results (see Table B.5, and the χ2, and OMR plots in axis position of the model. Fig. B.14 in Appendix B). The diameter is constrained at the 3σ-level but with a relatively large relative error of ∼10%. The TPM analysis fit to the WISE lightcurve (Figs. 5 and B.4) shows that the fit at phases between 0.0 and 0.10 might be improved with addi- The two obtained mirror solutions are AM 1 and AM 2. We had tional visible data for the inversion (phase 0 corresponds to JD 54 thermal observations at our disposal, 38 from IRAS (10 x 12 2442489.835789). µm,10x25 µm,10x60 µm,8x100 µm), seven from AKARI (5 2 The problem with this model is that the χ¯min versus Γ curve x S9W and 2 x L18W), and nine from WISE (W4). We assumed is very shallow at high Γs, so the thermal inertia is basically A =0.020 for the Bond albedo. unconstrained. The aspect angles sampled by the thermal data The best solutions for both models have comparable 2 2 are concentrated around equatorial values, so additional data at χ¯minand very shallow minima in the χ¯min versus Γ plots. other sub-observer latitudes and a dense thermal lightcurve at Thermal inertias between 0 and 150 SIu fit the data (3σ lim- pre-opposition (positive phase angle) would help. The diameter its), but the best fit is Γ=10 SI units (see Table B.6, the χ2 plot, +10 +0.014 constraint of 98−11 km leads to an albedo of pV = 0.046−0.008. and OMR plot in Fig. B.15 in Appendix B). The corresponding spheres give similar diameters but fit the data better with higher 4.6. (380) Fiducia thermal inertia (Γ ≈200 SIu). Only the most extreme rough- ness values produce fits outside the 3σ range, and low roughness A previous lightcurve of Fiducia was published by Warner solutions (rms<0.30) fit the data better. The shapes reasonably (2004). With this work we add six apparitions, confirming the reproduce the W4 lightcurve (Fig. 6, and B.5), but the fits could synodic period around 13.72 hours. Fiducia’s lightcurve shape probably be improved with “less boxy” shape models.

8 Marciniak et al.: Thermal properties of slowly rotating asteroids

9 8 W4 data W4 data Best fit AM 2 Best fit AM 1 8 7 7

6 6

Flux (Jy) 5 Flux (Jy) 5

4 4 3

2 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Rotational phase Rotational phase Fig.6: Asteroid (380) Fiducia’s WISE data and model thermal Fig.7: Asteroid (468) Lina’s WISE data and model thermal lightcurves for shape model 2 and the corresponding best-fitting lightcurves for shape model 1’s corresponding best-fitting so- solutions (very low thermal inertia of 10 SIu; see Table B.6)). lutions (see Table B.7).

From TPM analysis one can conclude that the roughness is unconstrained, but thermal inertias are lower than 150 at the3σ level (the best fit is for Γ = 10 SIu). The diameter range of the Appendix. Lightcurves of Friederike had minima of differ- +9 +0.009 ent widths and amplitudes from 0.20 to 0.25 mag. 72−5 km leads to an albedo pV =0.057−0.012. In spite of a large and varied dataset, only with the addition of WISE W1 data could the unique solution for period, spin axis, 4.7. (468) Lina and shape be found. These data proved to be a necessary, con- Lightcurves from two previous apparitions of Lina have been tinuous basis lasting a few tens of hours, while other lightcurves published by Tedesco (1979) and Buchheim (2007). Our visual covered only at most 20% of the full rotation. Such extremely observations of this target spanned six apparitions, and we found long period targets are especially challenging for ground-based it displaying complex lightcurves, with wide, wavy maxima and studies, being better targets for wide-field, space-borne obser- narrow minima (Figs. A.29 to A.33 in the Appendix). Peak-to- vatories. Due to this long period and relatively short time span peak amplitudes ranged between 0.13 and 0.18 mag within its of the observational dataset, the precision of the sidereal period 16.48 hours period. Model spin parameters are given in Table 1. determination is lower than in the case of other targets (Table 1). Shape model stretch along the spin axis is somewhat uncertain. TPM analysis TPM analysis The two mirror spin-shape solutions are AM 1 (168◦,-58◦, The two mirror solutions are named AM 1 and AM 2. We used 46.73928 h) and AM 2 (328◦,-59◦, 46.73985 h). Friederike is 33 observations, seven from IRAS (2 x 12 µm,2x25 µm,2x60 well observed at thermal wavelengths: two four-bandIRAS mea- µm,1x100 µm), eight from AKARI (2 x S9W and 6 x L18W), surements, ten measurements by AKARI (4 x S9W, 6 x L18W), and 20 from WISE (W4), and assumed A =0.020 for the Bond taken at different epochs before and after opposition, and 23 albedo. WISE data points (at W3 and W4), also at two epochs before The comparatively fewer data points are available for this and after opposition. object. The best-fitting roughness and thermal inertias are low 2 Solution AM 1 provides a borderline acceptable fit to all (rms ∼ 0.3 and Γ=20 SIu; see Table B.7, and the χ plots in Fig. thermal data simultaneously, as do the corresponding spheres B.16 in Appendix B), even for the spherical models. The fit to (see Table B.8). However, the AM 2 solution seems to match the the WISE lightcurve is reasonably good (Fig. 7, and B.6), but 2 WISE W3 and W4 lightcurves a bit better. The residuals in the the χ¯minis 1.20, which is not optimal. The data sample widely observation-divided-by-model plots (see Fig. B.17) indicate that different heliocentric distances, from 2.5 to 3.5 au (Fig. B.16), the spin-shape solutions are not perfect or – alternatively – that but the residuals do not present a strong trend. there are surface variegations that influence the thermal fluxes. The analysis points to thermal inertias lower than 300 SIu The WISE W3 and W4 lightcurvesfavour AM 2, but do not help and low roughness, but this object requires more thermal data to to settle the spin ambiguity completely. Overall, in this case high constrain these properties better. Even the diameter has a rela- +11 and extremely high surface roughness worked very well and all tively large error bar by TPM standards: 69−4 km. The corre- the radiometric solutions point to low values for the thermal in- +0.006 2 sponding albedo is 0.0520.014 . ertia well below Γ = 20 SIunits (see χ plots in Fig. B.17), with a trend to higher inertias at shorter heliocentric distance (2.6 au) 4.8. (538) Friederike and lower values at the largest heliocentric distance (> 3.5 au). The heliocentric influence on thermal inertia is also visible in the Data previously published came from one apparition and clearly radiometric solutions for the individual datasets (see Table B.8): displayed a very long,46.728 hoursperiod for this target (Pilcher all WISE data are taken at rhelio > 3.4 au, while the AKARI and 2013). In addition to it we used previously unpublished data IRAS data are taken well below 3.0 au. The overall best radio- from the years 2003 and 2006, with only partial coverage though metric solution for AM 2 and extremely high surface roughness +4 (Table A.1), and own data from five more apparitions. Those produces Γ = 20 SIunits, a diameter of 76−2 , and a geometric with good phase coverage are shown in Figs. A.34 - A.37 in albedo of 0.06±0.01.

9 Marciniak et al.: Thermal properties of slowly rotating asteroids

4.9. (653) Berenike The best solutions for both models have comparableand very 2 low χ¯m with very low thermal inertia of about 3 SI units (see Berenike displayed small amplitudes in the range of 0.05 - 0.16 Table B.10 and the χ2 plots in Fig. B.19 in the Appendix B). (Figs. A.38 - A.41 in the Appendix), also in the archival obser- 2 The low χ¯ms suggest the error bars might be overestimated, so vations by Binzel (1987) and Galad & Kornos (2008). However, we normalise the χ¯2 curves to have the minimum at 1 in order in the last apparition observed in this work the amplitude unex- to compute the uncertainties of the parameters (see discussion in pectedly rose to 0.38 mag, being over two times larger than ever, Hanuš et al. 2015). The shapes fit the WISE data well (Fig. 10 which is confirmed by two independent observing runs (Fig. and B.7), and unlike (195) Eurykleia’s case, the best-fitting solu- A.42). That apparition must have provided the only viewing ge- tion also fits the AKARI data with a reasonably flat OMR versus ometry when signatures of full elongation of this target could be wavelength plot (Fig. B.19). visible. This case shows the importance of probing a wide range To conclude, although high thermal inertias of about 70 of geometries for correct reproduction of the spin axis position SI units are still allowed, the best solutions seem to point to and shape elongation. The synodic period was around 12.485 very low thermal inertias of around 3 SI units. The diameters hours, and the full dataset spanned seven apparitions. +6 are constrained to be 38−2 km (3σ error bars), which lead to The shape model is indeed elongated in the equatorial di- +0.03 mensions, and this time the pole position is low, as expected a visible geometric albedo of pV = 0.13−0.05. This value is (see Table 1). Still, the shape model extent along the spin axis somewhat low for a typical S-type (classified based on a visi- is poorly constrained, probably due to the lack of data from ge- ble SMASS spectrum, Small Main-Belt Asteroid Spectroscopic ometries where intermediate amplitudes could be observed. Survey Bus & Binzel 2002a,b).

TPM analysis 4.11. (834) Burnhamia We denotethe two mirror solutions as AM 1 and AM 2. We used Previously observed by Buchheim (2007) and in our campaign, 49 thermal observations, 24 from IRAS (8 x 12 µm,8x25 µm, Burnhamia displayed lightcurves of extrema at unequal levels, 8 x60 µm), eight from AKARI (4 x S9W, 4 x L18W), and 17 13.87 hour period, and amplitudes from 0.15 to 0.25 mag. We from WISE (8 x W3, 9 x W4). The Bond albedo was assumed to accumulated data from six apparitions well spread in longitude, be A =0.070. and present them in Figs. A.48 to A.53 in the Appendix. In the Model AM 1 is significantly better than AM 2 and the lightcurve inversion we obtained two well-defined pole solutions spheres in this case, although the reduced χ2 is 1.1, slightly over (Table 1) with somewhat unconstrained shape models both in the unity (Table B.9, and the χ2 plot in Fig. B.18 in Appendix B). vertical dimension and in the level of smoothness of the shape. The IRAS data residuals show some scatter and the WISE data are reasonably well fitted but still present some waviness in the TPM analysis OMR versus rotational phase plot (Figs. 9 and B.18). The latter could indicate that there is room for improvement of the shape The two mirror solutions are denoted AM 1 and AM 2. We used model. Very high roughness solutions fit the data better, and an 70 thermal observations, 42 from IRAS (11 x 12 µm, 11 x 25 rms lower than 0.3 can be rejected at the 3σ level. µm, 11 x 60 µm, 9 x 100 µm), one from AKARI (L18W), and We conclude that additional densely sampled thermal 26 from WISE (13 x W3, 13 x W4). We assumed A =0.035 for lightcurves and an improved shape model could improve the the Bond albedo. constraints on the thermal inertia and surface roughness. With +4 +0.02 It turned out to be yet another slow rotator with seemingly a diameter 46−2 km, we get pV = 0.18−0.03, and the best fit is low thermal inertia. The three-sigma upper limit on 50 SI is quite obtained for medium values of thermal inertia. 2 2 clear. The χ¯minis ∼ 0.8 for both models (Table B.11, and the χ plots in Fig. B.20 in Appendix B), so the fit is good but there 4.10. (673) Edda is room for improvement. The WISE data are reasonably well fitted (Fig.11, and B.8), but the 100-micron IRAS data are not The first published lightcurve of Edda was obtained within (these data carry less weigh for the fit, however). our project (Marciniak et al. 2016), and was highly asymmet- As usual, the roughness is not constrained, the diameter ric, with maxima unequally spaced in time. Here we present +8 +0.014 is 67−6 km and pV = 0.074−0.016. We could benefit from data from five more apparitions, confirming the period of 22.34 additional thermal light curves at positive phase angles (pre- hours, and non-typical, asymmetric lightcurve behaviour (see opposition) and lower heliocentric distance although there is no Figs. A.43 to A.47 in the Appendix). This dataset was comple- apparent trend in the lowest panel of Fig. B.20. mented by one more apparition with partial coverage by data from the SuperWASP archive (Grice et al. 2017). lightcurve am- plitudes ranged from 0.13 to 0.23 mag. The shape model from 5. Discussion and conclusions lightcurve inversion is somewhat angular, and the fit to some of the lightcurves of smallest amplitude is imperfect. Both spin so- We present here 11 new spin and shape models of slowly rotating lutions are well constrained and are given in Table 1. asteroids from lightcurve inversion on dense lightcurves, with determined sizes, albedos, and thermal parameters. Together TPM analysis with five models from our previous study (Marciniaket al. 2018), we have 16 shape models within our sample of slow- The two mirror solutions are named AM 1 and AM 2. At our rotators with applied TPM. This number substantially enlarges disposal there were 54 infrared observations, 20 from IRAS (7x the available pool of well-studied slow rotators. Models obtained 12 µm,7x25 µm,6x60 µm), six from AKARI (3 x S9W and in this work provide good resolution, “smooth” shape represen- 3 x L18W), and 28 WISE (14 x W3 and 14 x W4). We assumed tations for a multitude of future applications, including calibra- A =0.047 for the Bond albedo. tion in the infrared.

10 Marciniak et al.: Thermal properties of slowly rotating asteroids

One of our aims was to verify whether the thermal inertia in- shape solutions for targets not presented here. The TESS mission deed increases with the rotation period (Harris & Drube 2016). is observing in a similar cadence to the Kepler Space Telescope. Our initial sample (Marciniak et al. 2018) seemed to be in line Moreover, unlike in the case of Kepler, full frames are going with this finding, while our current sample is dominated by tar- to be downlinked, so TESS data are going to be perfect for ex- gets with lower thermal inertias. To ensure this different result is tensive studies of slow rotators (Pál et al. 2018). It is also pos- notrelatedto the fact thatwe used a differentcodeforthis work5, sible that the shape models that best fit visual lightcurves were we modelled one of the targets from Marciniak et al. (2018) not the best possible ones from the thermal point of view, so and reproduced the results. This is expected given that we use the varied-shape TPM method (Hanuš et al. 2015), or simulta- the same model approach and approximations. We also cross- neous fitting of visual and thermal data using a convex inversion checked our results with those estimated by Harris & Drube TPM (Durechˇ et al. 2017) could help to resolve issues with un- (2016) and found one overlapping target, 487 Venetia, with con- constrained solutions. To overcome problems with limited ther- sistent thermal inertia values: 96 SIu in Harris & Drube (2016), mal data we are planning proposals to VLT/VISIR (Very Large and 100 ± 75 SIu in Marciniak et al. (2018). In Fig. 12 we su- Telescope with the VLT spectrometer and imager for the mid- perimposed our results on the plot from Harris & Drube (2016). infrared) and SOFIA (Stratospheric Observatory For Infrared Astronomy) infrared facilities, carefully choosing targets and Taking both of our samples together (from this work and observing geometries to best complement the existing datasets from Marciniak et al. 2018), we find diverse values of thermal in terms of aspect, pre- and post-opposition geometry, and he- inertia, rangingfrom 3 up to 125 SI units. To make a quantitative liocentric distance. comparison, we collected the thermal inertias of all targets with sizes 30 < D < 200 km from Hanuš et al. (2018b), Delbo’ et al. Acknowledgements. This work was supported by the National Science Centre, (2015), and our values and split them into two groups: slow Poland, through grant no. 2014/13/D/ST9/01818. The research leading to these results has received funding from the European Union’s Horizon 2020 Research (P > 12 hrs) and fast (P < 12 hrs) rotators. Neither the two- and Innovation Programme, under Grant Agreement no 687378 (SBNAF). sample Kolmogorov-Smirnov test nor the k-sample Anderson- The research of VK was supported by a grant from the Slovak Research and Darling test rule out the null hypothesis that both samples (with Development Agency, number APVV-15-0458. R. Duffard acknowledges finan- 33 and 36 values, respectively) are drawn from the same dis- cial support from the State Agency for Research of the Spanish MCIU through the ”Center of Excellence Severo Ochoa” award for the Instituto de Astrofísica tribution. The p-values are higher than 0.3, far even from a lax de Andalucía(SEV-2017-0709). The Joan Oró Telescope (TJO) of the Montsec threshold of statistical significance of 5%. Perhaps a larger sam- Astronomical Observatory (OAdM) is owned by the Catalan Government and ple could lead to a firmer conclusion in the future. operated by the Institute for Space Studies of Catalonia (IEEC). This article is This does not necessarily deny the hypothesis of Γ growth based on observations made in the Observatorios de Canarias del IAC with the 0.82m IAC80 telescope operated on the island of Tenerife by the Instituto de with period; instead it might indicate various levels of regolith Astrofísica de Canarias (IAC) in the Observatorio del Teide. This article is based development on the surface, connected to age and/or collisional on observations made with the SARA telescopes (Southeastern Association history, and space weathering. This could be further tested if we for Research in Astronomy), whose nodes are located at the Observatorios de had sufficiently large samples of objects belonging to young and Canarias del IAC on the island of La Palma in the Observatorio del Roque old collisional families, since younger surfaces are expected to de los Muchachos; Kitt Peak, AZ under the auspices of the National Optical Astronomy Observatory (NOAO); and Cerro Tololo Inter-American Observatory have less developed regoliths (e.g. Delbo’ et al. 2014). The inter- (CTIO) in La Serena, Chile. This project uses data from the SuperWASP archive. pretation of our results might also be complicated by the temper- The WASP project is currently funded and operated by Warwick University ature dependence of the thermal conductivity, which we account and Keele University, and was originally set up by Queen’s University Belfast, for by assuming the scaling relation between Γ and heliocen- the Universities of Keele, St. Andrews, and Leicester, the Open University, the Isaac Newton Group, the Instituto de Astrofisica de Canarias, the South African tric distance given for example in Eq. 13 of Delbo’ et al. (2015) Astronomical Observatory, and by STFC. to normalise the values of Γ to 1 au (these values are given in Funding for the Kepler and K2 missions is provided by the NASA Science 3 the last column of Table 2). This scaling is related to the T de- Mission Directorate. The data presented in this paper were obtained from the pendence of the thermal conductivity (e.g. Vasavada et al. 1999), Mikulski Archive for Space Telescopes (MAST). STScI is operated by the 3/2 Association of Universities for Research in Astronomy, Inc., under NASA con- and translates into a thermal inertia dependence of Γ ∝ T . tract NAS5- 26555. Support for MAST for non-HST data is provided by the However, we note that Centaurs and TNOs (trans-Neptunian NASA Office of Space Science via grant NNX09AF08G and by other grants and 2 objects) follow a different behaviour (Γ ∝ T ; Lellouch et al. contracts. 2013), and Rozitis & et al. (2018) have found more extreme de- pendencies in two near-Earth asteroids (Γ ∝ T 4.4 and Γ ∝ T 2.92), although the fitted exponents have large error bars. More References work is certainly needed in this direction. Alí-Lagoa, V., Lionni, L., Delbo, M., et al. 2014, A&A, 561, A45 Our large observing campaign, targeted at about 100 aster- Alí-Lagoa, V., Müller, T. G., Usui, F., & Hasegawa, S. 2018, A&A, 612, A85 oids with slow rotation and small amplitudes, already resulted in Binzel, R. P. 1987, Icarus, 72, 135 the gathering of around 10,000 hours of photometric data, where Blanco, C., Cigna, M., & Riccioli, D. 2000, Planetary and Space Science, 48, 973 hundreds of hours are needed for a unique lightcurve inversion Bottke, W. F., Durda, D. D., Nesvorný, D., et al. 2005, Icarus, 175, 111 solution for each of such targets. There are a few factors further Buchheim, R. K. 2007, Minor Planet Bulletin, 34, 68 limiting the number of targets with all the parameters uniquely Bus, S. J. & Binzel, R. P. 2002a, Icarus, 158, 146 determined, like shape model imperfections,or thermal data lim- Bus, S. J. & Binzel, R. P. 2002b, Icarus, 158, 106 Carry, B. 2012, Planet. Space Sci., 73, 98 ited geometries. In the future we are planning to add more pho- Cibulková, H., Durech,ˇ J., Vokrouhlický, D., Kaasalainen, M., & Oszkiewicz, tometric data to our datasets accumulated already including pho- D. A. 2016, A&A, 596, A57 tometry from the TESS (Transiting Exoplanet Survey Satellite) Cutri et al., R. M. 2012, VizieR Online Data Catalog, 2311, 0 mission, which should improve the uniqueness of many spin and Delbo’, M. & Harris, A. W. 2002, Meteoritics and Planetary Science, 37, 1929 Delbo’, M., Libourel, G., Wilkerson, J., et al. 2014, Nature, 508, 233 5 Delbo’, M., Mueller, M., Emery, J. P., Rozitis, B., & Capria, M. T. 2015, We used the TPM code of Müller (2002) in Marciniak et al. (2018), Asteroids IV, ed. P. Michel, F. E. DeMeo, & W. F. Bottke, 107–128 whereas here we used the TPM code of Delbo’ & Harris (2002) modi- Dunham, D. W., Herald, D., Frappa, E., et al. 2016, NASA Planetary Data fied in Alí-Lagoa et al. (2014) mentioned in Sect. 3. System, 243, EAR

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12 Marciniak et al.: Thermal properties of slowly rotating asteroids

0.8 0.80 AM 1 model AM 2 model Abs.calib. error bar Abs.calib. error bar 0.78 0.75 0.76

0.70 0.74

0.65 0.72

0.7 Flux (Jy)

Flux (Jy) 0.60 0.68

0.55 0.66

0.64 0.50

0.62 0.45 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Rotational phase Rotational phase 2.05 1.15 AM 1 model AM 2 model Abs.calib. error bar 1.10 Abs.calib. error bar 2

1.05 1.95

1.9 1.00 1.85 0.95 1.8 0.90

Flux (Jy) 1.75 Flux (Jy) 0.85 1.7 0.80 1.65 0.75 1.6 0.70 1.55 0.65 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Rotational phase Rotational phase 2.40 AM 2 model Fig.9: Asteroid (653) Berenike WISE data (W3 top, W4 bottom) Abs.calib. error bar and the AM 1 model’s thermal lightcurves (see Table B.9). 2.20

0.75 W3 data 2.00 Best fit AM 1 0.7

Flux (Jy) 1.80 0.65

1.60 0.6 Flux (Jy)

1.40 0.55 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Rotational phase 0.5

3.00 AM 2 model Abs.calib. error bar 0.45 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2.80 Rotational phase 1.7 W4 data 2.60 Best fit AM 1 1.6

Flux (Jy) 2.40 1.5

1.4 2.20

Flux (Jy) 1.3 2.00 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.2 Rotational phase 1.1 Fig. 8: Asteroid (538) Friederike’s WISE W3 (first two plots 1 from the top) and W4 data (last two plots) taken at two epochs, 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 and thermal lightcurves for shape model AM 2 and the cor- Rotational phase responding best-fitting solutions obtained based on all thermal Fig.10: Asteroid (673) Edda’s WISE data and model thermal data. lightcurves for shape model 1’s best-fitting solution (very low thermal inertia of 3 SI units).

13 Marciniak et al.: Thermal properties of slowly rotating asteroids

W3 data 1.4 Best fit AM 1

1.2

1 Flux (Jy)

0.8

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Rotational phase

W4 data 3.4 Best fit AM 1

3.2

3

2.8

Flux (Jy) 2.6

2.4

2.2

2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Rotational phase Fig.11: Asteroid (834) Burnhamia’s WISE data and model ther- mal lightcurves for shape model 1’s best-fitting solution (see Table B.11).

600

500

400

300

Thermal inertia [SI units] 200

100

0 10 100 Rotation period [h] Fig.12: Updated plot from Harris & Drube (2016) of thermal in- ertia dependence on rotation period. Black dots are thermal in- ertias estimated from η values by Harris & Drube (2016), red dots from detailed TPM compiled by Delbo’ et al. (2015), or- ange are slow rotators from“Varied-ShapeTPM” by Hanuš et al. (2018b), blue are from TPM results of Marciniak et al. (2018), and green dots from this work. Our new results do not confirm the growing trend.

14 Marciniak et al.: Thermal properties of slowly rotating asteroids

Appendix A: Visible photometry Details of the observing runs (Table A.1) and composite lightcurves of asteroids with spin and shape models presented here (Figs. A.1 - A.53).

Date λ Phase angle Duration σ Observer Site [deg] [deg] [hours] [mag]

(100) Hekate

2006 Nov 27.2 118.8 13.1 5.3 0.030 - SuperWASP 2006 Nov 29.2 118.7 12.7 5.5 0.026 - SuperWASP 2006 Nov 30.2 118.6 12.5 5.3 0.023 - SuperWASP 2006 Dec 14.2 117.2 9.3 5.7 0.021 - SuperWASP 2006 Dec 15.2 117.1 9.0 4.9 0.019 - SuperWASP 2006 Dec 16.2 117.0 8.7 5.8 0.018 - SuperWASP 2006 Dec 23.1 115.9 6.7 4.8 0.016 - SuperWASP 2006 Dec 30.0 114.7 4.6 4.1 0.032 - SuperWASP

2015 Jul 11.0 314.5 10.0 0.5 0.004 S. Fauvaud Bardon, France 2015 Jul 12.1 314.3 9.6 3.3 0.004 S. Fauvaud Bardon, France 2015 Jul 13.1 314.2 9.2 2.9 0.005 S. Fauvaud Bardon, France 2015 Jul 14.1 314.0 8.8 3.9 0.003 S. Fauvaud Bardon, France 2015 Jul 15.1 313.9 8.4 3.4 0.003 S. Fauvaud Bardon, France 2015 Jul 16.0 313.7 8.0 2.9 0.010 W. Ogłoza Suhora, Poland 2015 Jul 16.0 313.7 8.0 0.4 0.006 J. J. Sanabria Torreaguila, Spain 2015 Jul 16.1 313.7 8.0 3.5 0.004 S. Fauvaud Bardon, France 2015 Jul 21.0 312.9 5.9 3.3 0.006 W. Ogłoza Suhora, Poland 2015 Aug 7.1 309.6 1.9 3.7 0.004 S. Fauvaud Bardon, France 2015 Sep 1.9 305.6 12.6 4.3 0.005 W. Ogłoza Suhora, Poland 2015 Sep 14.9 304.8 16.6 1.9 0.011 - Montsec, CAT, Spain 2015 Sep 19.9 304.8 17.8 3.5 0.008 - Montsec, CAT, Spain 2015 Sep 20.9 304.8 18.1 3.6 0.012 - Montsec, CAT, Spain

2016 Oct 3.1 63.1 15.2 4.5 0.003 S. Fauvaud Bardon, France 2016 Oct 7.0 62.9 14.3 4.7 0.004 S. Fauvaud Bardon, France 2016 Oct 8.0 62.8 14.1 2.2 0.004 S. Fauvaud Bardon, France 2016 Oct 28.1 60.3 8.3 7.9 0.004 S. Fauvaud Bardon, France 2016 Oct 30.2 60.0 7.6 2.5 0.004 S. Fauvaud Bardon, France 2016 Oct 31.0 59.8 7.4 3.5 0.005 S. Fauvaud Bardon, France 2016 Nov 2.3 59.4 6.5 9.3 0.006 T. Polakis, B. Skiff Tempe, AZ, USA 2016 Nov 5.4 58.8 5.5 8.6 0.005 T. Polakis, B. Skiff Tempe, AZ, USA 2016 Nov 6.3 58.6 5.2 9.0 0.005 T. Polakis, B. Skiff Tempe, AZ, USA 2016 Nov 7.3 58.4 4.9 9.0 0.006 T. Polakis, B. Skiff Tempe, AZ, USA 2016 Nov 8.3 58.2 4.6 7.5 0.006 T. Polakis, B. Skiff Tempe, AZ, USA 2016 Nov 9.3 58.0 4.3 8.7 0.005 T. Polakis, B. Skiff Tempe, AZ, USA 2016 Dec 4.2 52.9 6.5 8.0 0.005 T. Polakis, B. Skiff Tempe, AZ, USA 2016 Dec 5.2 52.7 6.8 7.9 0.007 T. Polakis, B. Skiff Tempe, AZ, USA 2017 Jan 10.8 49.7 15.5 4.5 0.013 A. Marciniak Borowiec, Poland

2018 Feb 1.1 135.2 14.5 18.1 0.0004 - Kepler Space Telescope 2018 Feb 2.1 135.2 14.4 27.5 0.0004 - Kepler Space Telescope 2018 Feb 3.3 135.2 14.2 8.8 0.0004 - Kepler Space Telescope 2018 Feb 4.0 135.2 14.1 23.5 0.0004 - Kepler Space Telescope 2018 Feb 5.0 135.2 14.0 23.5 0.0004 - Kepler Space Telescope 2018 Feb 5.9 135.2 13.9 17.6 0.0004 - Kepler Space Telescope 2018 May 19.1 121.2 13.4 24.0 0.0010 - Kepler Space Telescope 2018 May 19.8 121.2 13.6 23.5 0.0011 - Kepler Space Telescope 2018 May 20.8 121.2 13.7 23.5 0.0012 - Kepler Space Telescope 2018 May 21.8 121.1 13.9 23.5 0.0011 - Kepler Space Telescope 2018 May 22.5 121.1 14.0 10.8 0.0013 - Kepler Space Telescope 404.1 hours total Table A.1: Observation details: Mid-time observing date, eclip- tic longitude of the target, sun-target-observer phase angle, duration of the observing run, photometric scatter, observer and site name. See Marciniak et al. (2018) for telescope and site details. The remaining part of the table for the rest of the targets, together with the photometric data from all individual nights are available through the CDS archive, http://cdsarc.u-strasbg.fr.

15 Marciniak et al.: Thermal properties of slowly rotating asteroids

100 Hekate 100 Hekate P=27.069 h P=27.080 h 8,2 -3,3

8,25 -3,25

8,3 -3,2

8,35 2015 Jul 11.0 Bardon -3,15 Jul 12.1 Bardon Jul 13.1 Bardon 8,4 2006 Jul 14.1 Bardon Jul 15.1 Bardon Nov 27.2 SuperWASP Jul 16.0 Suhora Relative C magnitude Nov 29.2 SuperWASP Relative R magnitude -3,1 Jul 16.0 Terre. 8,45 Nov 30.2 SuperWASP Jul 16.1 Bardon Dec 14.2 SuperWASP Jul 21.0 Suhora Dec 15.2 SuperWASP Aug 7.1 Bardon Dec 16.2 SuperWASP -3,05 Sep 1.9 Suhora 8,5 Dec 23.1 SuperWASP Sep 14.9 OAdM Dec 30.0 SuperWASP Sep 19.9 OAdM Zero time at: 2006 Nov 27.0467 UTC, LT corr. Zero time at: 2015 Jul 11.9942 UTC, LT corr. Sep 20.9 OAdM 8,55 -3 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Phase Fig.A.1: Composite lightcurve of (100) Hekate from the year Fig.A.2: Composite lightcurve of (100) Hekate from the year 2006. 2015.

100 Hekate 100 Hekate P=27.072 h P=27.068 h 7,75 8,35 2018 Feb 1.1 Kepler Feb 2.1 Kepler Feb 3.3 Kepler 7,8 8,4 Feb 4.0 Kepler Feb 5.0 Kepler Feb 5.9 Kepler May 19.1 Kepler 7,85 8,45 May19.8 Kepler May 20.8 Kepler May 21.8 Kepler May 22.5 Kepler 2016/2017 7,9 8,5 Oct 3.1 Bardon Oct 7.0 Bardon Oct 8.0 Bardon Oct 28.1 Bardon

Relative R magnitude 7,95 Oct 30.2 Bardon Relative R magnitude 8,55 Oct 31.0 Bardon Nov 2.3 Tempe Nov 5.4 Tempe Nov 6.3 Tempe 8,6 8 Nov 7.3 Tempe Nov 8.3 Tempe Zero time at: 2016 Nov 2.1408 UTC, LT corr. Nov 9.3 Tempe Zero time at: 2018 Jan 31.7150 UTC, LT corr. Dec 4.2 Tempe 8,05 8,65 0 0,1 0,2 0,3 0,4 0,5 0,6 Dec0,7 5.2 Tempe 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Jan 10.8 Bor. Phase Phase Fig.A.3: Composite lightcurve of (100) Hekate from the years Fig.A.4: Composite lightcurve of (100) Hekate based on 2016-2017. Kepler K2 data from the year 2018.

109 Felicitas 109 Felicitas P = 13.191 h P = 13.194 h -2,1

9,4 -2,05

-2

9,5 -1,95

2015 -1,9 Oct 26.9 Bor. Oct 31.7 Bor. Nov 22.8 Bor. Relative C magnitude 9,6 Dec 20.8 Bor. Relative R and C magnitude -1,85 Dec 21.8 OAdM 2006 Dec 22.8 OAdM Jul 1.1 SuperWASP Dec 23.8 OAdM Aug 30.8 SuperWASP -1,8 Dec 24.8 OAdM Dec 25.8 OAdM Sep 14.9 SuperWASP Dec 26.8 OAdM 9,7 Zero time at 2006 Aug 30.7462 UTC, LT corr. Sep 20.8 SuperWASP Zero time at 2015 Oct 31.6875, LT corr. Dec 27.8 OAdM -1,75 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Phase Fig.A.5: Composite lightcurve of (109) Felicitas from the year Fig.A.6: Composite lightcurve of (109) Felicitas from the year 2006. 2015.

16 Marciniak et al.: Thermal properties of slowly rotating asteroids

109 Felicitas 109 Felicitas P = 13.188 h P = 13.198 h -4,5 -5 2017 Jan 22.3 CTIO Jan 27.1 Bor. Jan 29.1 Bor. Feb 16.1 Bor. Feb 22.4 Tempe -4,4 Feb 23.4 Tempe -4,9 Feb 24.4 Tempe Feb 25.4 Tempe Feb 27.4 Tempe Mar 1.4 Tempe Mar 16.5 Winer Mar 17.5 Winer 2018 May 4.3 Lowell May 5.3 Lowell May 6.3 Lowell -4,3 -4,8 May 9.3 Lowell May 10.3 Lowell May 14.3 Lowell Relative R magnitude May 15.3 Lowell May 19.3 Lowell Relative C and R magnitude May 19.3 Winer May 20.3 Lowell May 21.3 Winer May 24.1 Winer May 25.3 Lowell -4,2 -4,7 May 26.2 Lowell May 31.2 Lowell Jun 3.2 Lowell Zero time at 2017 Jan 28.9650 UTC, LT corr. Zero time at 2018 May 21.1700 UTC, LT corr. Jun 13.2 Lowell

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Phase Fig.A.7: Composite lightcurve of (109) Felicitas from the year Fig.A.8: Composite lightcurve of (109) Felicitas from the year 2017. 2018.

195 Eurykleia 195 Eurykleia

P=16.522 h P=16.524 h 2008/2009

28 Oct 2008 30 Oct 2008 -3,9 -3,9 15 Nov 2008 23 Nov 2008 24 Nov 2008 17 Dec 2008 17 Jan 2009 19 Jan 2009 -3,8 -3,8

2004/2005 Nov 24.1 Sozzago Dec 4.1 Engar. Feb 1.9 Linhac. -3,7 Feb 2.8 Sozzago Feb 2.9 Linhac. -3,7 Feb 3.9 Linhac.

Feb 14.9 Linhac. Relative C magnitude Feb 17.8 Sozzago

Relative R and C magnitude Feb 23.8 Hauts P. Feb 25.0 Blauvac Feb 25.9 Blauvac Feb 27.9 Blauvac Feb 28.9 Blauvac Mar 1.9 Hauts P. -3,6 Mar 8.9 Sozzago Mar 9.8 Tradate -3,6 Mar 9.9 Sozzago Zero time at: 2008 Oct 28.1912 UTC, LT corr. Mar 15.9 Linhac. Zero time at: 2008 Oct 28.1912 UTC, LT corr. 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 1,1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Phase Fig.A.9: Composite lightcurve of (195) Eurykleia from the Fig.A.10: Composite lightcurve of (195) Eurykleia from the years 2004-2005. year 2008.

195 Eurykleia 195 Eurykleia

P=16.518 h 2013 P=16.527 h 2015 -4,5 29 Sep, Bor. Sep 24.1 Bor. 30 Sep, Bor. -3,4 Nov 4.1 Bor. 13 Oct, Bor. Nov 5.1 Bor. 3 Nov, Bor. Jan 5.0 Bor. 2 Dec, Bor. Feb 14.9 Bor. Feb 17.9 Bor. 17 Dec, Bor. -4,4 Feb 18.0 Der. Mar 10.0 Bor. -3,3

-4,3

-3,2 Relative C magnitude Relative C and R magnitude -4,2

-3,1

Zero time at: 2013 Sep 29.9142 UTC, LT corr. Zero time at: 2015 Feb 14.7050 UTC, LT corr. -4,1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Phase Fig.A.11: Composite lightcurve of (195) Eurykleia from the Fig.A.12: Composite lightcurve of (195) Eurykleia from the year 2013. year 2015.

17 Marciniak et al.: Thermal properties of slowly rotating asteroids

195 Eurykleia 195 Eurykleia

P=16.52 h 2016 P=16.52 h -2,7 -4,2 Apr 27.1 JKT May 11.1 IAC80 May 25.0 IAC80 Jun 8.2 Winer Jun 8.3 Winer -2,6 Jun 18.2 Winer -4,1

-2,5 -4 Relative R magnitude Relative R magnitude

-2,4 -3,9 2017 Aug 4.0 OAdM Aug 5.0 OAdM Aug 28.0 IAC80, Teide Zero time at: 2016 Apr 26.9767 UTC, LT corr. Zero time at: 2017 Aug 31.8633 UTC, LT corr. Aug 31.9 IAC80, Teide -2,3 -3,8 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Phase Fig.A.13: Composite lightcurve of (195) Eurykleia from the Fig.A.14: Composite lightcurve of (195) Eurykleia from the year 2016. year 2017.

301 Bavaria 301 Bavaria P=12.243 h 2014 P=12.240 h 2015 -2,3 24 Feb, Bor. May 24.4 Winer 2 Mar, Bor. Jun 4.4 Winer 11 Mar, Bor. -2,1 Jun 18.4 Winer 13 Mar, Bor. Sep 5.9 OAdM 23 Mar, Organ M. Sep 6.8 Adi -2,2 1 Apr, Organ M. Sep 6.9 OAdM 22 May, Bor. Sep 10.9 OAdM Oct 2.8 OAdM Oct 3.8 OAdM -2

-2,1

-1,9 Relative C magnitude Relative C and R magnitude -2

-1,8 Zero time at: 2014 Feb 24.0821, LT corr. Zero time at: 2015 Sep 6.6933 UTC, LT corr. -1,9 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Phase Fig.A.15: Composite lightcurve of (301) Bavaria from the year Fig.A.16: Composite lightcurve of (301) Bavaria from the year 2014. 2015.

301 Bavaria 301 Bavaria P=12.244 h P=12.242 h -3,1 -3,1 2016/2017 2018 Oct 12.3 CTIO Jan 23.4 Winer Oct 30.3 CTIO Mar 4.9 Bor. Nov 22.0 Bor Mar 19.1 OASI Jan 18.1 CTIO Mar 20.1 OASI -3 Feb 12.5 BOAO -3

-2,9 -2,9 Relative R and C magnitude Relative R and C magnitude -2,8 -2,8

Zero time at: 2016 Nov 21.7929 UTC, LT corr. Zero time at: 2018 Jan 23.1642 UTC, LT corr. -2,7 -2,7 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Phase Fig.A.17: Composite lightcurve of (301) Bavaria from the Fig.A.18: Composite lightcurve of (301) Bavaria from the year years 2016-2017. 2018.

18 Marciniak et al.: Thermal properties of slowly rotating asteroids

335 Roberta 335 Roberta P=12.027 h 2013 P=12.027 h -3,75 29 Sep, Bor. 18 Oct, Organ M. 19 Oct, Bor. -3,3 19 Oct, Organ M. -3,7 25 Oct, Bor. 21 Nov, Organ M. 18 Dec, Organ M. 23 Dec, Bor. -3,65

-3,2 -3,6 2015 Nov 18.5 Winer Dec 9.0 Bor Feb 14.7 Bisei

Relative C magnitude -3,55 Mar 9.9 Der Mar 17.0 Der Relative C and R magnitude

-3,1 -3,5

Zero time at: 2013 Sep 28.9158 UTC, LT corr. Zero time at: 2015 Feb 14.4592 UTC, LT corr. -3,45 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Phase Fig.A.19: Composite lightcurveof (335) Roberta from the year Fig.A.20: Composite lightcurve of (335) Roberta from the year 2013. 2015.

335 Roberta 380 Fiducia P=12.024 h P=13.71 h -3,65 9,5 2017 Oct 16.0 Bor. Oct 30.9 Bor. -3,6 Nov 16.0 Der. 9,6

-3,55

9,7 -3,5

-3,45 Relative C magnitude 9,8 2008

Relative C and R magnitude Feb 3.0 Villefagnan Feb 7.0 Villefagnan Feb 7.9 Villefagnan -3,4 Feb 9.0 Villefagnan 9,9 Feb 10.0 Villefagnan Zero time at: 2017 Oct 15.8788 UTC, LT corr. Zero time at: 2008 Feb 2.9179 UTC, LT corr. -3,35 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Phase Fig.A.21: Composite lightcurveof (335) Roberta from the year Fig.A.22: Composite lightcurve of (380) Fiducia from the year 2017. 2008.

380 Fiducia 380 Fiducia P=13.75 h P=13.702 h 2014 9,3 -2,8 15 Jun, OAdM 19 Jun, OAdM 20 Jun, OAdM 21 Jun, OAdM 27 Jun, OAdM 9,4 -2,7 29 Jun, OAdM 4 Jul, OAdM

9,5 -2,6

Relative C magnitude 9,6 Relative R magnitude -2,5

2009 Apr 5.0 Blauvac Apr 14.0 Blauvac 9,7 -2,4 Zero time at: 2009 Apr 5.8204 UTC, LT corr. Zero time at: 2014 Jun 19.8678 UTC, LT corr. 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Phase Fig.A.23: Composite lightcurve of (380) Fiducia from the year Fig.A.24: Composite lightcurve of (380) Fiducia from the year 2009. 2014.

19 Marciniak et al.: Thermal properties of slowly rotating asteroids

380 Fiducia 380 Fiducia P=13.716 h P=13.719 h -3,5 -3,1 2015/2016 2015 Aug 31.1 Bor. Oct 11.0 Bor. Sep 1.1 Bor. Oct 31.0 Bor. Sep. 28.0 Bor. -3,4 Dec 30.8 Bor. -3 Jan 22.3 Bor. Feb 25.2 Winer

-3,3 -2,9

Relative C magnitude -3,2 Relative C magnitude -2,8

-3,1 -2,7 Zero time at: 2015 Aug 31.0021 UTC, LT corr. Zero time at: 2015 Oct 31.7696 UTC, LT corr. 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Phase Fig.A.25: Composite lightcurve of (380) Fiducia from the Fig.A.26: Composite lightcurve of (380) Fiducia from the year years 2015-2016 under large phase angle. 2015 under small phase angle.

380 Fiducia 380 Fiducia P=13.717 h P=13.728 h -3,4 -3,8

-3,3 -3,7

-3,2 -3,6

2016/2017 Nov 29.1 Bor. -3,1 Dec 13.0 Bor. -3,5 Dec 28.1 Bor. 2018 Relative C and R magnitude Relative C and R magnitude Jan 8.9 Bor. Mar 4.1 Suhora Jan 26.9 Bor. Mar 26.1 Bor. Feb 8.2 CTIO Apr 10.0 Bor. Feb 15.0 Bor. Apr 13.0 Bor. -3 Mar 11.8 Bor. -3,4 Mar 24.8 Bor. Apr 15.0 Bor. Zero time at: 2016 Dec 12.9042 UTC, LT corr. Zero time at: 2018 Mar 4.0333 UTC, LT corr. May 29.9 Bor. 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Phase Fig.A.27: Composite lightcurve of (380) Fiducia from the Fig.A.28: Composite lightcurve of (380) Fiducia from the year years 2016-2017. 2018.

468 Lina 468 Lina P = 16.48 h P=16.48 h 10 8,25

10,05 8,3

10,1 8,35

10,15 8,4

Relative C magnitude 10,2 Relative C magnitude 8,45 2007 Feb 11.1 SM 2006 Feb 15.2 SM Feb 17.2 SM Jan 3.9 Hauts Patys 10,25 8,5 Feb 20.1 SM Jan 4.9 Hauts Patys Feb 20.3 SM Jan 13.9 Hauts Patys Feb 21.2 SM Jan 14.9 Hauts Patys Zero time at: 2006 Jan 14.7246 UTC, LT corr. Zero time at: 2007 Feb 11.0392 UTC, LT corr. Feb 21.3 SM 10,3 8,55 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Phase Fig.A.29: Composite lightcurve of (468) Lina from the year Fig.A.30: Composite lightcurve of (468) Lina from the year 2006. 2007.

20 Marciniak et al.: Thermal properties of slowly rotating asteroids

468 Lina 468 Lina P=16.56 h 2014 P = 16.480 h -3,8 -3,3 24 Mar, OAdM 23 May, OAdM 26 May, OAdM -3,75 12 Jun, OAdM -3,25

-3,7 -3,2

-3,65 -3,15 2015 May 13.4 Winer May 17.5 Winer May 26.4 Winer May 31.4 Winer Relative R magnitude -3,6 -3,1 Jun 15.4 Winer Relative L and R magnitude Jun 16.4 Winer Jun 26.1 Teide Jul 1.1 Teide -3,55 -3,05 Jul 18.0 OAdM Jul 23.0 OAdM Sep 9.9 OAdM Zero time at: 2014 May 23.8908 UTC, LT corr. Zero time at: 2015 May 13.3612 UTC, LT corr. Sep 14.9 OAdM -3,5 -3 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Phase Fig.A.31: Composite lightcurve of (468) Lina from the year Fig.A.32: Composite lightcurve of (468) Lina from the year 2014. 2015.

468 Lina 538 Friederike P = 16.485 h P=46.724 h -3,4 -4,1 Feb 9, OAdM 2014 Feb 17, OAdM Feb 21, OAdM Feb 22, OAdM -3,35 Feb 23, OAdM Feb 27, OAdM -4 Mar 8, Bor. Mar 16, OAdM -3,3 Mar 17, OAdM Mar 23, OAdM Mar 27, OAdM Apr 17, OAdM -3,25 -3,9 Apr 18, OAdM 2016/2017 Apr 23, OAdM Sep 7.1 Bor. Apr 25, OAdM Nov 2.3 Tempe -3,2 Nov 5.3 Tempe Nov 6.4 Tempe

Relative R and C magnitude Nov 7.3 Tempe Relative R and C magnitude Nov 8.3 Tempe -3,8 Nov 9.3 Tempe -3,15 Dec 2.6 SOAO Dec 4.3 Tempe Dec 5.3 Tempe Zero time at: 2016 Sep 7.0217 UTC, LT corr. Jan 26.8 Bor. Zero time at: 2014 Mar 17.8204 UTC, LT corr. -3,1 -3,7 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Phase Fig.A.33: Composite lightcurve of (468) Lina from the years Fig.A.34: Composite lightcurve of (538) Friederike from the 2016-2017. year 2014.

538 Friederike 538 Friederike P=46.745 h 2017 P=46.73 h -3,4 -3,4 Aug 30.0 Bor. Sep 1.1 OAdM Sep 2.0 OAdM Sep 3.0 OAdM Sep 19.0 OAdM -3,3 -3,3 Sep 20.0 OAdM Sep 25.9 OAdM Sep 27.9 OAdM Oct 7.0 OAdM 2015 Oct 8.0 OAdM Mar 16.1 Bor Oct 21.0 OAdM Mar 17.0 Bor Oct 25.9 OAdM -3,2 Apr 16.4 Winer -3,2 Nov 21.5 SOAO Dec 2.5 SOAO Apr 27.4 Winer Apr 28.3 Winer May 7.3 Winer May 8.3 Winer May 15.8 Adi Relative R and C magnitude Relative R and C magnitude -3,1 May 16.8 Adi -3,1 May 17.2 Winer May 20.2 Winer May 21.2 Winer

Zero time at: 2015 Apr 27.1171 UTC, LT corr. Zero time at: 2017 Aug 30.9121 UTC, LT corr. -3 -3 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Phase Fig.A.35: Composite lightcurve of (538) Friederike from the Fig.A.36: Composite lightcurve of (538) Friederike from the year 2015. year 2017.

21 Marciniak et al.: Thermal properties of slowly rotating asteroids

538 Friederike 653 Berenike P=46.74 h P=12.487 h -3,5 8,5

8,6 -3,4

8,7

2018 -3,3 Oct 18.1 Bor. Oct 19.0 Bor. Nov 1.1 Bor. 8,8 Nov 4.1 OAdM

Nov 6.1 Bor. Relative C magnitude Nov 7.1 Bor. 2005 Relative R and C magnitude Nov 10.4 Organ M. -3,2 Jul 3.1 Engar. Nov 17.4 Organ M. Jul 9.0 Engar. Nov 18.0 Bor. 8,9 Nov 29.3 Organ M. Aug 7.0 Engar. Dec 5.9 Bor. Aug 9.0 Engar. Dec 11.2 OAdM Zero time at: 2005 Aug 8.8438 UTC, LT corr. Zero time at: 2018 Oct 17.9492 UTC, LT corr. Dec 14.3 Organ M. -3,1 9 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Phase Fig.A.37: Composite lightcurve of (538) Friederike from the Fig.A.38: Composite lightcurve of (653) Berenike from the year 2018. year 2005.

653 Berenike 653 Berenike P=12.481 h P=12.488 h -3,4 -3,5

2014 26 Mar, Bor. -3,3 29 Mar, Bor. -3,4 2 Apr, Bor. 16 Apr, Bor. 20 May, Bor. -3,2 -3,3

-3,1 -3,2

Relative C magnitude Relative R magnitude 2015 Jun 18.0 OAdM Jun 19.0 OAdM -3 -3,1 Jun 21.0 OAdM Jun 29.1 Teide Jul 5.0 Teide Aug 10.9 OAdM Zero time at: 2014 Mar 26.8125 UTC, LT corr. Zero time at: 2015 Jun 17.9025 UTC, LT corr. Aug 11.9 OAdM -2,9 -3 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Phase Fig.A.39: Composite lightcurve of (653) Berenike from the Fig.A.40: Composite lightcurve of (653) Berenike from the year 2014. year 2015.

653 Berenike 653 Berenike P=12.487 h P=12.485 h -4 -4,5

-3,9 -4,4

-3,8 -4,3

-3,7 -4,2 2016 Relative R magnitude Relative C magnitude Aug 13.2 ORM 2017/2018 Sep 6.0 OAdM Oct 31.2 CTIO Sep 7.0 OAdM -3,6 -4,1 Nov 18.1 Bor Sep 12.0 OAdM Jan 15.8 Bor. Oct 10.0 OAdM Jan 15.6 SOAO Oct 11.8 OAdM Feb 26.8 Bor. Oct 15.9 OAdM Zero time at: 2016 Aug 23.0958 UTC, LT corr. Zero time at: 2018 Feb 26.7421 UTC, LT corr. -3,5 -4 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Phase Fig.A.41: Composite lightcurve of (653) Berenike from the Fig.A.42: Composite lightcurve of (653) Berenike from the year 2016. years 2017-2018.

22 Marciniak et al.: Thermal properties of slowly rotating asteroids

673 Edda 673 Edda P=22.32 h P=22.34 h 10,9 10,05

10,95 10,1

11 10,15

11,05 10,2

Relative R magnitude 11,1 Relative R magnitude 10,25

2005 2007 Mar 10.9 Engar. 11,15 10,3 Mar 12.0 Engar. Sep 16.9 Blauvac Mar 12.9 Engar. Sep 20.0 Blauvac Zero time at: 2005 Mar 10.7754 UTC, LT corr. Zero time at: 2007 Sep 16.8104 UTC, LT corr. 11,2 10,35 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Phase Fig.A.43: Composite lightcurve of (673) Edda from the year Fig.A.44: Composite lightcurve of (673) Edda from the year 2005. 2007.

673 Edda 673 Edda P=22.338 h P=22.340 h -3,05 -3,95 2013/2014 2016 Nov 25.1 Bor. Jun 8.4 Winer Jan 23.8 Bor. Jun 19.4 Lowell -3 Feb 2.9 Bor. -3,9 Jul 10.0 OAdM Feb 22.8 Bor. Jul 11.0 OAdM Feb 23.0 Bor. Jul 13.0 OAdM Feb 25.8 Bor. Jul 15.0 OAdM Jul 16.0 OAdM -2,95 -3,85 Jul 17.0 OAdM Jul 28.1 Teide Aug 3.0 OAdM Sep 22.9 ORM -2,9 -3,8

Relative C magnitude -2,85 Relative R magnitude -3,75

-2,8 -3,7

Zero time at: 2013 Nov 25.0275 UTC, LT corr. Zero time at: 2016 Sep 22.8104 UTC, LT corr. -2,75 -3,65 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Phase Fig.A.45: Composite lightcurve of (673) Edda from the years Fig.A.46: Composite lightcurve of (673) Edda from the year 2013-2014. 2016.

673 Edda 834 Burnhamia P=22.338 h P = 13.89 h 9,75 8,9 2017 Sep 30.1 Bor. Oct 2.3 Kitt Peak 9,8 Oct 9.9 Bor. Oct 17.0 Der. Nov 23.9 Bor. 9

9,85

9,9 9,1

9,95 Relative R magnitude Relative C and R magnitude 9,2 2005 May 5.0 Engar. 10 May 6.0 Engar. May 8.0 Engar. Zero time at: 2017 Oct 16.8379 UTC, LT corr. Zero time at: 2005 May 5.8846 UTC, LT corr. 10,05 9,3 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Phase Fig.A.47: Composite lightcurve of (673) Edda from the year Fig.A.48: Composite lightcurve of (834) Burnhamia from the 2017. year 2005.

23 Marciniak et al.: Thermal properties of slowly rotating asteroids

834 Burnhamia 834 Burnhamia P = 13.88 h P=13.866 h 9,4 -4,3

9,5 -4,2

9,6 -4,1 Relative R magnitude Relative C magnitude

9,7 2006 -4 Oct 8.0 Engar. 2013/2014 Oct 9.9 Engar. Dec 3.1 Bor. Oct 10.9 Engar. Jan 8.2 Winer Zero time at: 2006 Oct 7.8088 UTC, LT corr. Zero time at: 2013 Dec 3.0000 UTC, LT corr. Mar 24.2 Winer 9,8 -3,9 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Phase Fig.A.49: Composite lightcurve of (834) Burnhamia from the Fig.A.50: Composite lightcurve of (834) Burnhamia from the year 2006. years 2013-2014.

834 Burnhamia 834 Burnhamia P = 13.876 h P = 13.879 h -5,7 2014/2015 Nov 25.4 Winer Nov 26.4 Winer Dec 17.1 OAdM Dec 20.1 OAdM -4 Dec 21.1 OAdM Dec 22.1 OAdM -5,6 Jan 24.0 OAdM Jan 25.0 OAdM Jan 26.0 OAdM Apr 19.2 Winer -3,9

-5,5

-3,8

Relative R magnitude 2016

Relative R and C magnitude Jan 26.4 Winer -5,4 Jan 27.5 Winer Feb 18.4 Winer Feb 23.5 Winer -3,7 Mar 7.5 Winer Mar 17.1 Bor. Zero time at: 2014 Nov 25.3242 UTC, LT corr. Zero time at: 2016 Jan 26.3400 UTC, LT corr. May 7.9 Bor. -5,3 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Phase Fig.A.51: Composite lightcurve of (834) Burnhamia from the Fig.A.52: Composite lightcurve of (834) Burnhamia from the years 2014-2015. year 2016.

834 Burnhamia P = 13.88 h -3

-2,9

-2,8

Relative R magnitude 2017 Jul 29.0 OAdM -2,7 Jul 30.0 OAdM Jul 31.0 Suhora Aug 1.0 Suhora Aug 2.0 OAdM Aug 3.0 OAdM Zero time at: 2017 Jul 31.8262 UTC, LT corr. -2,6 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Phase Fig.A.53: Composite lightcurve of (834) Burnhamia from the year 2017.

24 Marciniak et al.: Thermal properties of slowly rotating asteroids

Appendix B: Thermophysical model details In Sect. 3 in the main text, we providedan overviewof our mod- elling approach. Below, we give a few more details about how we determine the Bond albedo (A) for the TPM and how we es- timate our error bars. We also include all the tables (Tables B.1 to B.11), observations-to-model-ratio plots (Figs. B.9 to B.20), and χ2 versus Γ plots that we used to give full information on the TPM analysis. We fix the value of the Bond albedo to run the model (let us call it Ai). Our TPM diameters (D) do not always lead to the same value of Bond albedo (say, Ao) when we use the H- G12 values. However, the differences are not large enough to be meaningful in practice as long as A ∼ 0.1 or lower. This is because flux is proportionalto T 4, and that in turn is proportional to (1 − A), so for example wrongly assuming A =0.05 instead of A = 0.10 would lead to a ∼5% offset in the flux, which is still comparable to the absolute calibration uncertainties. On the other hand, the effect is indeed strong for objects with higher A (e.g. Vesta, with A ∼ 0.2). To give numbers in our case, all our targets – except one – have As in the range 0.02–0.07, and the differences between Ai and Ao never lead to systematic model flux differences >0.5%, so the effect is negligible. For our highest albedo target, (100) Hekate, we used Ai = 0.10 but got Ao=0.12. This 20% offset leads nonetheless to a ∼2% systematic difference in the model fluxes, which is not ideal but it is well within the absolute cal- ibration flux uncertainties. More quantitatively, we re-ran the TPM for model 2 with Ai=0.12 and got virtually the same size and thermal inertia but a higher surface roughness (rms = 0.45 instead of rms = 0.4). In this case, the slightly lower temperature due to the higher albedo seems to be compensated by a small increase in the roughness. To estimate D and Γ error bars we have followed stan- dard procedures (more details and discussion can be found in Alí-Lagoa et al. 2014; Hanuš et al. 2015). Namely, if we have 2 χ¯min≈1, the 3σ errorbars are given by the range of Γs and Dsof 2 those models with χ¯min< 3p2/ν, where ν is the effective num- 2 ber of degrees of freedom (e.g. Press et al. 1986). When χ¯minis much lower than 1 or up to 1.5, we assume we can scale the χ2 curves to have the minimum at 1 and apply the same for- mula. This is not rigourous mathematically but it is a working assumption to have some estimate of the error bars (Hanuš et al. 2 2015). After all, a very low χ¯mincould be due to some error bars being overestimated (the IRAS data have the largest ones). 2 If χ¯minis about 1.5, we assume it is still a reasonable fit given the model simplification (constant parameters over the surface, wavelength-independent emissivity) or other sources of errors. For instance, we do not consider the uncertainties in shape (no shape modelling scheme so far provides shape uncertainties), ro- tation period, or spin orientation explicitly. These are coupled parameters themselves in the shape modelling so we cannot sim- ply vary them separately within their uncertainties to explore the effects in TPM. Hanus et al. (2015) proposed bootstrapping the visible photometry to obtain ∼30 shape and spin models that were subsequently input to the TPM (the so-called varied-shape TPM). The values of acceptable Γs spanned were significantly larger than the classical approach only in some cases, not sys- tematically.

25 Marciniak et al.: Thermal properties of slowly rotating asteroids Table B.1: Summary of TPM results for (100) Hekate. 2 ± ± Shape model IR data subset χ¯m D 3σ (km) Γ 3σ (SIu) Roughness (rms) Comments

+6 +114 AM 1 All data 0.8 90−4 6−4 Low/Medium (0.30) rms unconstrained at the 3σ level

+6 +160 AM 1 sphere All data 0.8 86−5 20−20 Medium/high (0.6) rms unconstrained at the 3σ level

+5 +66 AM 2 All data 0.65 87−4 4−2 Medium (0.40) rms unconstrained at the 3σ level +8 +144 ∼ AM 2 sphere All data 0.8 86−8 16−16 Medium/high ( 0.6) rms unconstrained at the 3σ level

Table B.2: Summary of TPM results for (109) Felicitas. 2 ± ± Shape model IR data subset χ¯m D 3σ (km) Γ 3σ (SIu) Roughness (rms) Comments

+7 +100 ∼ AM 1 All data 1.1 85−5 40−40 Ext. high ( 1.0) rms unconstrained at the 3σ level

AM 1 sphere All data 2.2 97 200 Ext. high (∼1.0) Bad fit

AM 2 Alldata 2.0 81 4 Ext.high(∼1.0) Bad fit

AM 2 sphere All data 3.4 101 250 Ext. high (∼1.0) Bad fit

Table B.3: Summary of TPM results for (195) Eurykleia fitting different subsets of data. For example, the “Excluding W4” label refers to the fact that the W4 data were not used to optimise the χ2 in that case. 2 ± ± Shape model IR data subset χ¯m D 3σ (km) Γ 3σ (SIu) Roughness (rms) Comments

+5 +37 AM 1 All data 0.51 82−4 15−15 High (0.65) rms>0.34 at the 1σ level, but otherwise unconstrained +7 +65 AM 1 sphere All data 1.23 83−4 25−20 Extr. high (>1.0) Roughness unconstrained

+5 +35 AM 1 Excluding W4 0.50 87−6 15−15 Medium (0.45) Roughness unconstrained

2 AM 1 OnlyW4 0.23 81 5 Low(0.29) Artificiallylow χ¯m. Single-epoch data are insufficient to constrain params.

+5 +40 ∼ AM 2 All data 0.60 86−4 20−20 Extr. high ( 1.0) Roughness rms>0.34 at the 3σ level +8 +60 AM 2 sphere All data 1.17 84−3 30−25 Extr. high (>1.0) Roughness unconstrained

+7 +50 AM 2 Excluding W4 0.58 91−6 15−15 Med.-high (0.5) Fitted with rel. calibration error bars

2 AM 2 Only W4 0.47 80 5 Extr. high (>1.0) Artificially low χ¯m. Single-epoch data are insufficient to constrain params.

Table B.4: Summary of TPM results for (301) Bavaria. 2 ± ± Shape model IR data subset χ¯m D 3σ (km) Γ 3σ (SIu) Roughness (rms) Comments

+2 +60 AM 1 All data 0.34 55−2 40−30 Med.-high (0.5) rms unconstrained at the 3σ level

+3 +90 AM 1 sphere All data 0.7 57−3 30−25 Low (0.20) rms unconstrained at the 3σ level

+2 +50 AM 2 All data 0.36 55−2 50−35 Medium/high (0.60) rms unconstrained at the 3σ level

+3 +70 AM 2 sphere All data 0.7 57−3 30−25 Low (0.20) rms unconstrained at the 3σ level

26 Marciniak et al.: Thermal properties of slowly rotating asteroids Table B.5: Summary of TPM results for (335) Roberta. 2 ± ± Shape model IR data subset χ¯m D 3σ (km) Γ 3σ (SIu) Roughness (rms) Comments

+10 +1500 AM 1 All data 0.70 100−11 70−55 Very low (0.21) Roughness and Γ unconstrained at the 3σ level +8 +1350 ∼ AM 1 sphere All data 0.71 97−11 150−125 Extr. high ( 1.0) Compatible solution but requires extremely high roughness

+10 +1410 AM 2 All data 0.60 98−8 90−85 Low (0.34) Roughness and Γ unconstrained at the 3σ level +8 +1350 ∼ AM 2 sphere All data 0.68 97−11 150−125 Extr. high ( 1.0) Very similar to “sphere AM 1” fit

Table B.6: Summary of TPM results for (380) Fiducia. 2 ± ± Shape model IR data subset χ¯m D 3σ (km) Γ 3σ (SIu) Roughness (rms) Comments

+8 +135 AM 1 All data 0.91 73−5 15−15 Low (0.25) rms<0.90 at the 3σ level ∼ 2 AM 1 sphere All data 1.33 78 200 Extr. high ( 1.0) Borderline acceptable χ¯m but unconstrained thermal props.

+9 +140 AM 2 All data 0.59 72−5 10−10 Low (0.25) rms<0.90 at the 3σ level ∼ 2 AM 2 sphere All data 1.41 72 200 Extr. high ( 1.0) Borderline acceptable χ¯m but unconstrained thermal props.

Table B.7: Summary of TPM results for (468) Lina. 2 ± ± Shape model IR data subset χ¯m D 3σ (km) Γ 3σ (SIu) Roughness (rms) Comments

+11 +280 AM 1 All data 1.20 69−4 20−20 Very low (0.20) rms unconstrained at the 3σ level ∼ 2 AM 1 sphere All data 2.0 65. 35 Low ( 0.30) Bad χ¯m but similar thermal properties

+11 +280 AM 2 All data 1.22 70−5 20−20 Very low (0.2) rms not constrained at the 3σ level ∼ 2 AM 2 sphere All data 2.0 65 30 Low ( 0.30) Bad χ¯m but similar thermal properties

Table B.8: Summary of TPM results for (538) Friederike. 2 ± ± Shape model IR data subset χ¯m D 3σ (km) Γ 3σ (SIu) Roughness (rms) Comments

+2 AM 1 all data 1.65 74−1 < 15 Extr. high (0.9) Bad fit +3 ∼ AM 1 sphere all data 1.52 71−1 < 28 Extr. high 1 Badfit +3 +34 ∼ AM 1 WISE only 0.99 73−2 10−6 Extr. high 1 rms > 0.40

+2 AM 2 all data 2.04 75−1 < 14 High (0.75) Bad fit +3 ∼ AM 2 sphere all data 1.52 71−1 < 28 Extr. high 1 Badfit +5 +25 ∼ AM 2 WISE only 0.72 76−2 20−20 Extr. high 1 rms > 0.45

27 Marciniak et al.: Thermal properties of slowly rotating asteroids Table B.9: Summary of TPM results for (653) Berenike. 2 ± ± Shape model IR data subset χ¯m D 3σ (km) Γ 3σ (SIu) Roughness (rms) Comments

+4 +120 AM 1 All data 1.1 46−2 40−40 Med.-high (0.5) rms<0.3 rejected at the 3σ level

AM 1 sphere All data 2.0 52 100 Extr. high (1.00) Bad fit

AM 2 All data 2.0 52 120 Extr. high (1.00) Bad fit

AM 2 sphere All data 2.6 52 100 Extr. high(1.00) Bad fit

Table B.10: Summary of TPM results for (673) Edda. 2 ± ± Shape model IR data subset χ¯m D 3σ (km) Γ 3σ (SIu) Roughness (rms) Comments

+6 +33 AM 1 All data 0.47 38−2 3−3 Med.-high (0.5) rms>0.34 at the 3σ level

AM 1 sphere All data 1.83 38 5 Medium (0.4) Bad fit

+2 +37 ∼ AM 2 All data 0.59 38−2 3−3 Extr. high ( 1.0) rms>0.34 at the 3σ level

AM 2 sphere All data 1.76 38 10. Medium (0.45) Bad fit

Table B.11: Summary of TPM results for (834) Burnhamia. 2 ± ± Shape model IR data subset χ¯m D 3σ (km) Γ 3σ (SIu) Roughness (rms) Comments

+8 +23 AM 1 All data 0.78 67−6 22−22 Extr. high (0.9) rms unconstrained at the 3σ level +6 +45 ∼ 2 AM 1 sphere All data 1.12 65−5 40−40 Extr. high ( 1.0) Acceptable χ¯m and similar thermal properties

+5 +30 AM 2 All data 0.80 66−4 20−20 High (0.6) rms not constrained at the 3σ level +6 +45 ∼ 2 AM 2 sphere All data 1.12 65−5 40−40 Extr. high ( 1.0) Acceptable χ¯m but similar thermal properties

28 Marciniak et al.: Thermal properties of slowly rotating asteroids

10.5 10 AM 1 model W4 data Abs.calib. error bar Best fit AM 1 10 9 9.5

9 8 8.5 Flux (Jy) Flux (Jy) 8 7

7.5 6 7

6.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Rotational phase Rotational phase Fig.B.1: Asteroid (100) Hekate W4 data and model thermal Fig.B.4: Asteroid (335) Roberta’s WISE data and model thermal lightcurves for shape model 1. Data error bars are 1-σ. Table B.1 lightcurves for shape model 1 and the corresponding best-fitting summarises the TPM analysis. solutions (very low thermal inertia of 15 SIu).

9.5 9 W4 data W4 data 9 Best fit AM 2 (all data used) Best fit AM 1 Best fit AM 2 (W4 not used) 8 Best fit AM 2 (only W4 used) 8.5 7 8

7.5 6

7

Flux (Jy) Flux (Jy) 5 6.5 4 6 3 5.5

5 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Rotational phase Rotational phase Fig.B.2: W4 data and model thermal lightcurves for (195) Fig.B.5: Asteroid (380) Fiducia’s WISE data and model thermal Eurykleia’s shape model AM 2. The different models resulted lightcurves for shape model 1 and the corresponding best-fitting from fitting different subsets of data. Table B.3 contains the cor- solutions (very low thermal inertia of 15 SIu; see Table B.6). responding thermo-physical parameters.

8 W4 data 4.2 AM 2 model Best fit AM 2 Abs.calib. error bar 7 4

6 3.8

3.6 Flux (Jy) 5 Flux (Jy)

3.4 4

3.2 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

3 Rotational phase 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Rotational phase Fig.B.6: Asteroid (468) Lina’s WISE data and model thermal Fig.B.3: Asteroid (301) Bavaria AM 2 model thermal lightcurves for shape model 2’s corresponding best-fitting solu- lightcurves and W4 data (see Table B.4). tions (see Table B.7).

29 Marciniak et al.: Thermal properties of slowly rotating asteroids

0.75 W3 data Best fit AM 2 0.7

0.65

0.6 Flux (Jy)

0.55

0.5

0.45 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Rotational phase

1.7 W4 data Best fit AM 2 1.6

1.5

1.4

Flux (Jy) 1.3

1.2

1.1

1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Rotational phase Fig.B.7: Asteroid (673) Edda’s WISE data and model thermal lightcurves for shape model 2’s best-fitting solution (very low thermal inertia of 3 SI units).

W3 data 1.4 Best fit AM 2

1.2

1 Flux (Jy)

0.8

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Rotational phase

W4 data 3.4 Best fit AM 2

3.2

3

2.8

Flux (Jy) 2.6

2.4

2.2

2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Rotational phase Fig.B.8: Asteroid (834) Burnhamia’s WISE data and model thermal lightcurves for shape model 2’s best-fitting solution (see Table B.11).

30 Marciniak et al.: Thermal properties of slowly rotating asteroids

2 180 10 110 AM 2 (all data) ρ=0.21 ρ=0.34 ρ=0.50 ρ=0.75 ρ=0.25 ρ=0.39 ρ=0.57 ρ=0.88 160 ρ=0.29 ρ=0.44 ρ=0.65 ρ=1.08 105 140 1.5 100 120 2 100 χ 95 1 80

Obs/Mod 90 Reduced

60 Aspect angle (deg.) 85 0.5 Optimised diameter (km) 40 1 80 20

0 0 75 10 100 10 100 Wavelength (micron) Thermal Inertia (SIu)

2 100 10 105 AM 2 (all data) ρ=0.21 ρ=0.34 ρ=0.50 ρ=0.75 ρ=0.25 ρ=0.39 ρ=0.57 ρ=0.88 90 ρ=0.29 ρ=0.44 ρ=0.65 ρ=1.08 100 80 1.5 70 95 2 60 χ 1 1 90 50 Obs/Mod 40 Reduced 85 Wavelength (micron)

0.5 30 Optimised diameter (km) 20 80

10 0 0.1 75 2.6 2.8 3 3.2 3.4 3.6 10 100 Heliocentric distance (au) Thermal Inertia (SIu)

2 180 AM 2 (all data) 160

140 1.5 120

100 1 80 Obs/Mod

60 Aspect angle (deg.) 0.5 40

20

0 0 0 0.2 0.4 0.6 0.8 1 Rotational phase

2 100 AM 2 (all data) 90

80 1.5 70

60 1 50 Obs/Mod 40 Wavelength (micron) 0.5 30 20

10 0 -40 -30 -20 -10 0 10 20 30 40 Phase angle (degree) Fig.B.9: Asteroid (100) Hekate, left from top to bottom: obser- vation to model ratios versus wavelength, heliocentric distance, rotational phase, and phase angle. Depending on the plot, the colour indicates the aspect angle or wavelength at which the observations were taken (see the legend on the right). The plots for AM 1 looked similar. Right: χ2 versus thermal inertia curves for (100) Hekate model 1 (top) and model 2 (bottom). In all the subsequent figures of this kind, the colours denote optimised diameter, while various symbols show various levels 31 of surface roughness. Marciniak et al.: Thermal properties of slowly rotating asteroids

2 180 10 110 AM 1 (all data) ρ=0.21 ρ=0.34 ρ=0.50 ρ=0.75 ρ=0.25 ρ=0.39 ρ=0.57 ρ=0.88 160 ρ=0.29 ρ=0.44 ρ=0.65 ρ=1.08 105 140 1.5 120 100 2 100 χ 1 95 80 Obs/Mod Reduced

60 Aspect angle (deg.) 90

0.5 Optimised diameter (km) 40 1 85 20

0 0 80 10 100 10 100 Wavelength (micron) Thermal Inertia (SIu)

2 100 AM 1 (all data) 90

80 1.5 70

60 1 50 Obs/Mod 40 Wavelength (micron) 0.5 30 20

10 0 2.6 2.8 3 3.2 3.4 3.6 Heliocentric distance (au)

2 180 AM 1 (all data) 160

140 1.5 120

100 1 80 Obs/Mod

60 Aspect angle (deg.) 0.5 40

20

0 0 0 0.2 0.4 0.6 0.8 1 Rotational phase

2 100 AM 1 (all data) 90

80 1.5 70

60 1 50 Obs/Mod 40 Wavelength (micron) 0.5 30 20

10 0 -40 -30 -20 -10 0 10 20 30 40 Phase angle (degree) Fig. B.10: Asteroid (109) Felicitas, left, from top to bottom: ob- servation to model ratios versus wavelength, heliocentric dis- tance, rotational phase, and phase angle for model AM 1 (AM 2 was rejected). Right: χ2 versus thermal inertia curves for model AM1.

32 Marciniak et al.: Thermal properties of slowly rotating asteroids

2 180 10 105 θ=08.0 θ=14.6 θ=21.5 θ=33.3 AM 1 (all data used) θ=11.1 θ=16.6 θ=24.6 θ=40.3 160 θ=12.8 θ=18.9 θ=28.4 θ=52.0 100 140 1.5 120 95 2 100 χ 1 90 80 Obs/Mod Reduced 1

60 Aspect angle (deg.) 85

0.5 3−σ Optimised diameter (km) 40 1−σ 80 20

0 0 75 10 100 10 100 Wavelength (micron) Thermal Inertia (SIu)

2 180 10 115 θ=08.0 θ=14.6 θ=21.5 θ=33.3 AM 2 (All data used) θ=11.1 θ=16.6 θ=24.6 θ=40.3 θ=12.8 θ=18.9 θ=28.4 θ=52.0 160 110 140 1.5 105 120 2 100 χ 100 1 80

Obs/Mod 95 Reduced

60 Aspect angle (deg.) 90 0.5 Optimised diameter (km) 40 1 85 20

0 0 80 10 100 10 100 Wavelength (micron) Thermal Inertia (SIu)

2 180 AM 1 (W4 data not used) 160

140 1.5 120

100 1 80 Obs/Mod

60 Aspect angle (deg.) 0.5 40

20

0 0 10 100 Wavelength (micron)

2 180 AM 2 (W4 data not used) 160

140 1.5 120

100 1 80 Obs/Mod

60 Aspect angle (deg.) 0.5 40

20

0 0 10 100 Wavelength (micron) Fig.B.11: Asteroid (195) Eurykleia, left: observation to model ratios versus wavelength for models AM 1 (first and third plot) and AM 2 (second and fourth) using all data to minimise the χ2 (top) and excluding the W4 data (bottom). The colour indicates the aspect angle at which the observations were taken. The AM 1 model’s pole orientation is such that the WISE data would have been taken at a sub-observer latitude of ∼25◦ south, and the rest of the data at ∼25◦ north. On the other hand, if pole 2 is correct, then all data would have been taken at similar 33 “equator-on” aspect angles between 80◦ and 100◦. Right: χ2 versus thermal inertia curves for model AM1 (top) and AM2 (bottom) using all thermal data. Marciniak et al.: Thermal properties of slowly rotating asteroids

115 θ=08.0 θ=14.6 θ=21.5 θ=33.3 θ=11.1 θ=16.6 θ=24.6 θ=40.3 θ=12.8 θ=18.9 θ=28.4 θ=52.0 110

105 1 100 2

χ 95 3−σ

Reduced 1−σ 90

85 Optimised diameter (km)

80

0.1 75 10 100 Thermal Inertia (SIu) 100 θ=08.0 θ=14.6 θ=21.5 θ=33.3 θ=11.1 θ=16.6 θ=24.6 θ=40.3 θ=12.8 θ=18.9 θ=28.4 θ=52.0 95

1

2 90 χ

3−σ

Reduced 85

1−σ Optimised diameter (km) 80

0.1 75 10 100 Thermal Inertia (SIu) 125 θ=08.0 θ=14.6 θ=21.5 θ=33.3 θ=11.1 θ=16.6 θ=24.6 θ=40.3 θ=12.8 θ=18.9 θ=28.4 θ=52.0 120

115

1 110 2 χ 105

100 Reduced 95 Optimised diameter (km) 90

85

0.1 80 10 100 Thermal Inertia (SIu) 110 θ=08.0 θ=14.6 θ=21.5 θ=33.3 θ=11.1 θ=16.6 θ=24.6 θ=40.3 θ=12.8 θ=18.9 θ=28.4 θ=52.0 105

1 100 2

χ 95 Reduced 90 Optimised diameter (km)

85

0.1 80 10 100 Thermal Inertia (SIu) Fig. B.12: Additional χ2 versus thermal inertia plots for (195) Eurykleia: Top to bottom: Model AM1 fitted to data excluding WISE W4 data, AM1 with W4 data only, model AM2 fitted to data excluding WISE W4 data, AM2 with W4 data only.

34 Marciniak et al.: Thermal properties of slowly rotating asteroids

2 180 10 66 AM 1 (all data) ρ=0.21 ρ=0.34 ρ=0.50 ρ=0.75 ρ=0.25 ρ=0.39 ρ=0.57 ρ=0.88 160 ρ=0.29 ρ=0.44 ρ=0.65 ρ=1.08 64

140 62 1.5 60 120

2 58 100 χ 1 1 56 80 Obs/Mod

Reduced 54

60 Aspect angle (deg.) 52 0.5 Optimised diameter (km) 40 50

20 48

0 0 0.1 46 10 100 10 100 Wavelength (micron) Thermal Inertia (SIu)

2 60 10 66 AM 1 (all data) ρ=0.21 ρ=0.34 ρ=0.50 ρ=0.75 ρ=0.25 ρ=0.39 ρ=0.57 ρ=0.88 ρ=0.29 ρ=0.44 ρ=0.65 ρ=1.08 64 50 62 1.5 60 40 2 58 χ 1 1 56 30 Obs/Mod

Reduced 54

Wavelength (micron) 52 0.5 20 Optimised diameter (km) 50

10 48 0 0.1 46 2.5 2.55 2.6 2.65 2.7 2.75 2.8 2.85 10 100 Heliocentric distance (au) Thermal Inertia (SIu)

2 180 AM 1 (all data) 160

140 1.5 120

100 1 80 Obs/Mod

60 Aspect angle (deg.) 0.5 40

20

0 0 0 0.2 0.4 0.6 0.8 1 Rotational phase

2 60 AM 1 (all data)

50 1.5

40

1 30 Obs/Mod Wavelength (micron) 0.5 20

10 0 -40 -30 -20 -10 0 10 20 30 40 Phase angle (degree) Fig.B.13: Asteroid (301) Bavaria, left, from top to bottom: ob- servation to model ratios versus wavelength, heliocentric dis- tance, rotational phase, and phase angle for model AM 1 (the same plots for AM 2 looked similar). Right: χ2 versus thermal inertia curves for model AM1 (top), and AM2 (bottom).

35 Marciniak et al.: Thermal properties of slowly rotating asteroids

2 180 10 115 AM 2 θ=08.0 θ=14.6 θ=21.5 θ=33.3 θ=11.1 θ=16.6 θ=24.6 θ=40.3 160 θ=12.8 θ=18.9 θ=28.4 θ=52.0 110 140 1.5 105 120 2 100 χ 100 1 80

Obs/Mod 95 Reduced 1 60 Aspect angle (deg.) 90 0.5 Optimised diameter (km) 40 85 20

0 0 80 10 100 10 100 1000 Wavelength (micron) Thermal Inertia (SIu)

2 100 10 115 AM 2 θ=08.0 θ=14.6 θ=21.5 θ=33.3 θ=11.1 θ=16.6 θ=24.6 θ=40.3 90 θ=12.8 θ=18.9 θ=28.4 θ=52.0 110 80 1.5 105 70 2 60 χ 100 1 50

Obs/Mod 95 40 Reduced 1 Wavelength (micron) 90 0.5 30 Optimised diameter (km)

20 85 10 0 80 -40 -30 -20 -10 0 10 20 30 40 10 100 1000 Phase angle (degree) Thermal Inertia (SIu)

2 180 AM 2 160

140 1.5 120

100 1 80 Obs/Mod

60 Aspect angle (deg.) 0.5 40

20

0 0 0 0.2 0.4 0.6 0.8 1 Rotational phase

2 100 AM 2 90

80 1.5 70

60 1 50 Obs/Mod 40 Wavelength (micron) 0.5 30 20

10 0 2.65 2.7 2.75 2.8 2.85 2.9 2.95 Heliocentric distance (au) Fig.B.14: Asteroid (335) Roberta, left: observation to model ratios versus wavelength, phase angle, rotational phase, and he- liocentric distance for model AM 2. The plots for AM 1 looked similar. Right: χ2 versus thermal inertia curves for model AM1 (top), and AM2 (bottom).

36 Marciniak et al.: Thermal properties of slowly rotating asteroids

2 180 10 84 θ=08.0 θ=14.6 θ=21.5 θ=33.3 AM 1 θ=11.1 θ=16.6 θ=24.6 θ=40.3 160 θ=12.8 θ=18.9 θ=28.4 θ=52.0 82

140 80 1.5 120 78 2 100 χ 76 1 1 80 74 Obs/Mod Reduced

60 Aspect angle (deg.) 72

0.5 Optimised diameter (km) 40 70

20 68

0 0 0.1 66 10 100 10 100 Wavelength (micron) Thermal Inertia (SIu)

2 100 10 84 θ=08.0 θ=14.6 θ=21.5 θ=33.3 AM 1 θ=11.1 θ=16.6 θ=24.6 θ=40.3 90 θ=12.8 θ=18.9 θ=28.4 θ=52.0 82

80 80 1.5 70 78 2 60 χ 76 1 1 50 74 Obs/Mod Reduced 40 72 Wavelength (micron) 0.5 30 Optimised diameter (km) 70 20 68 10 0 0.1 66 -40 -30 -20 -10 0 10 20 30 40 10 100 Phase angle (degree) Thermal Inertia (SIu)

2 180 AM 1 160

140 1.5 120

100 1 80 Obs/Mod

60 Aspect angle (deg.) 0.5 40

20

0 0 0 0.2 0.4 0.6 0.8 1 Rotational phase

2 100 AM 1 90

80 1.5 70

60 1 50 Obs/Mod 40 Wavelength (micron) 0.5 30 20

10 0 2.35 2.4 2.45 2.5 2.55 2.6 2.65 Heliocentric distance (au) Fig.B.15: Asteroid (380) Fiducia, left: observation to model ratios versus wavelength, phase angle, rotational phase, and he- liocentric distance for model AM 1. The plots for AM 2 looked similar. Right: χ2 versus thermal inertia curves for model AM1 (top), and AM2 (bottom).

37 Marciniak et al.: Thermal properties of slowly rotating asteroids

2 180 10 90 AM 1 θ=08.0 θ=14.6 θ=21.5 θ=33.3 θ=11.1 θ=16.6 θ=24.6 θ=40.3 160 θ=12.8 θ=18.9 θ=28.4 θ=52.0 85 140 1.5 120 80 2 100 χ 1 75 80 Obs/Mod Reduced

60 Aspect angle (deg.) 70

0.5 Optimised diameter (km) 40 1 65 20

0 0 60 10 100 10 100 Wavelength (micron) Thermal Inertia (SIu)

2 60 10 90 AM 1 θ=08.0 θ=14.6 θ=21.5 θ=33.3 θ=11.1 θ=16.6 θ=24.6 θ=40.3 θ=12.8 θ=18.9 θ=28.4 θ=52.0 50 85 1.5 80 40 2

χ 1 75 30 Obs/Mod Reduced 70 Wavelength (micron)

20 Optimised diameter (km) 0.5 1 65 10 0 60 -40 -30 -20 -10 0 10 20 30 40 10 100 Phase angle (degree) Thermal Inertia (SIu)

2 180 AM 1 160

140 1.5 120

100 1 80 Obs/Mod

60 Aspect angle (deg.) 0.5 40

20

0 0 0 0.2 0.4 0.6 0.8 1 Rotational phase

2 60 AM 1

50 1.5

40

1 30 Obs/Mod Wavelength (micron) 0.5 20

10 0 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 Heliocentric distance (au) Fig.B.16: Asteroid (468) Lina, left: observation to model ra- tios versus wavelength, phase angle, rotational phase, and he- liocentric distance for model AM 1. The plots for AM 2 looked similar. Right: χ2 versus thermal inertia curves for model AM1 (top), and AM2 (bottom).

38 Marciniak et al.: Thermal properties of slowly rotating asteroids

2 180 10 110 AM 1 (all data) ρ=0.21 ρ=0.34 ρ=0.50 ρ=0.75 ρ=0.25 ρ=0.39 ρ=0.57 ρ=0.88 160 ρ=0.29 ρ=0.44 ρ=0.65 ρ=1.08 105

140 1.5 100 120 95 2 100 χ 1 90 80 Obs/Mod Reduced 85

60 Aspect angle (deg.)

0.5 80 Optimised diameter (km) 40 1

20 75

0 0 70 10 100 10 100 Wavelength (micron) Thermal Inertia (SIu)

2 100 10 115 AM 1 (all data) ρ=0.21 ρ=0.34 ρ=0.50 ρ=0.75 ρ=0.25 ρ=0.39 ρ=0.57 ρ=0.88 90 ρ=0.29 ρ=0.44 ρ=0.65 ρ=1.08 110

80 105 1.5 70 100 2 60 χ 95 1 50 90 Obs/Mod 40 Reduced 85 Wavelength (micron)

0.5 30 Optimised diameter (km) 1 80 20 75 10 0 70 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 10 100 Heliocentric distance (au) Thermal Inertia (SIu)

2 180 AM 1 (all data) 160

140 1.5 120

100 1 80 Obs/Mod

60 Aspect angle (deg.) 0.5 40

20

0 0 0 0.2 0.4 0.6 0.8 1 Rotational phase

2 100 AM 1 (all data) 90

80 1.5 70

60 1 50 Obs/Mod 40 Wavelength (micron) 0.5 30 20

10 0 -40 -30 -20 -10 0 10 20 30 40 Phase angle (degree) Fig.B.17: Asteroid (538) Friederike, from top to bottom: ob- servation to model ratios versus wavelength, heliocentric dis- tance, rotational phase, and phase angle for model AM 1 (the AM 2 model on all data did not provide an acceptable fit and was rejected). Right: χ2 versus thermal inertia curves for model AM1 (top), and AM2 (bottom) using WISE W3 and W4 data only.

39 Marciniak et al.: Thermal properties of slowly rotating asteroids

2 180 10 58 AM 1 (all data) ρ=0.21 ρ=0.34 ρ=0.50 ρ=0.75 ρ=0.25 ρ=0.39 ρ=0.57 ρ=0.88 160 ρ=0.29 ρ=0.44 ρ=0.65 ρ=1.08 56

140 1.5 54 120 52 2 100 χ 1 1 50 80 Obs/Mod Reduced 48

60 Aspect angle (deg.)

0.5 46 Optimised diameter (km) 40

20 44

0 0 0.1 42 10 100 10 100 Wavelength (micron) Thermal Inertia (SIu)

2 60 AM 1 (all data)

50 1.5

40

1 30 Obs/Mod Wavelength (micron) 0.5 20

10 0 2.9 2.95 3 3.05 3.1 3.15 Heliocentric distance (au)

2 180 AM 1 (all data) 160

140 1.5 120

100 1 80 Obs/Mod

60 Aspect angle (deg.) 0.5 40

20

0 0 0 0.2 0.4 0.6 0.8 1 Rotational phase

2 60 AM 1 (all data)

50 1.5

40

1 30 Obs/Mod Wavelength (micron) 0.5 20

10 0 -40 -30 -20 -10 0 10 20 30 40 Phase angle (degree) Fig.B.18: Asteroid (653) Berenike, left, from top to bottom: observation to model ratios versus wavelength, heliocentric distance, rotational phase, and phase angle for model AM 1 (the AM 2 model did not provide an acceptable fit and was rejected). Right: χ2 versus thermal inertia curves for model AM1.

40 Marciniak et al.: Thermal properties of slowly rotating asteroids

2 180 10 56 θ=08.0 θ=14.6 θ=21.5 θ=33.3 AM 1 θ θ θ θ =11.1 =16.6 =24.6 =40.3 54 160 θ=12.8 θ=18.9 θ=28.4 θ=52.0 52 140 1.5 50 120 48 2

χ 100 46 1 80 44 Obs/Mod Reduced 1 42

60 Aspect angle (deg.)

40 Optimised diameter (km) 0.5 40 38 20 36 0 0 34 10 100 10 100 Wavelength (micron) Thermal Inertia (SIu)

2 60 10 56 θ=08.0 θ=14.6 θ=21.5 θ=33.3 AM 1 θ=11.1 θ=16.6 θ=24.6 θ=40.3 θ=12.8 θ=18.9 θ=28.4 θ=52.0 54 50 52 1.5 50

40 48 2

χ 46 1 30 44 Obs/Mod Reduced 1 42 Wavelength (micron)

40 Optimised diameter (km) 0.5 20 38 10 36 0 34 -40 -30 -20 -10 0 10 20 30 40 10 100 Phase angle (degree) Thermal Inertia (SIu)

2 180 AM 1 160

140 1.5 120

100 1 80 Obs/Mod

60 Aspect angle (deg.) 0.5 40

20

0 0 0 0.2 0.4 0.6 0.8 1 Rotational phase

2 60 AM 1

50 1.5

40

1 30 Obs/Mod Wavelength (micron) 0.5 20

10 0 2.805 2.81 2.815 2.82 2.825 2.83 2.835 2.84 2.845 Heliocentric distance (au) Fig.B.19: Asteroid (673) Edda, left: observation to model ra- tios versus wavelength, phase angle, rotational phase, and he- liocentric distance for model AM 1. The plots for AM 2 looked similar. Right: χ2 versus thermal inertia curves for model AM1 (top) and AM2 (bottom).

41 Marciniak et al.: Thermal properties of slowly rotating asteroids

2 180 10 95 AM 1 θ=08.0 θ=14.6 θ=21.5 θ=33.3 θ=11.1 θ=16.6 θ=24.6 θ=40.3 160 θ=12.8 θ=18.9 θ=28.4 θ=52.0 90 140 1.5 85 120 2 100 χ 80 1 80

Obs/Mod 75 Reduced

60 Aspect angle (deg.) 70 0.5 Optimised diameter (km) 40 1 65 20

0 0 60 10 100 10 100 Wavelength (micron) Thermal Inertia (SIu)

2 100 10 95 AM 1 θ=08.0 θ=14.6 θ=21.5 θ=33.3 θ=11.1 θ=16.6 θ=24.6 θ=40.3 90 θ=12.8 θ=18.9 θ=28.4 θ=52.0 90 80 1.5 85 70 2 60 χ 80 1 50

Obs/Mod 75 40 Reduced

Wavelength (micron) 70 30 Optimised diameter (km) 0.5 1 20 65 10 0 60 -40 -30 -20 -10 0 10 20 30 40 10 100 Phase angle (degree) Thermal Inertia (SIu)

2 180 AM 1 160

140 1.5 120

100 1 80 Obs/Mod

60 Aspect angle (deg.) 0.5 40

20

0 0 0 0.2 0.4 0.6 0.8 1 Rotational phase

2 100 AM 1 90

80 1.5 70

60 1 50 Obs/Mod 40 Wavelength (micron) 0.5 30 20

10 0 2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 Heliocentric distance (au) Fig.B.20: Asteroid (834) Burnhamia, left: observation to model ratios versus wavelength, phase angle, rotational phase, and heliocentric distance for model AM 1. The plots for AM 2 looked similar. Right: χ2 versus thermal inertia curves for model AM1 (top) and AM2 (bottom).

42 Marciniak et al.: Thermal properties of slowly rotating asteroids

1 Astronomical Observatory Institute, Faculty of Physics, A. Mickiewicz University, Słoneczna 36, 60-286 Poznan,´ Poland. E- mail: [email protected] 2 Max-Planck-Institut für Extraterrestrische Physik, Giessenbachstrasse 1, 85748 Garching, Germany 3 Konkoly Observatory, Research Centre for Astronomy and Earth Sciences, Hungarian Academy of Sciences, H-1121 Budapest, Konkoly Thege Miklós út 15-17, Hungary 4 MTA CSFK Lendület Near-Field Cosmology Research Group 5 Astronomy Department, Eötvös Loránd University, Pázmány P. s. 1/A, H-1171 Budapest, Hungary 6 Institute of Physics, University of Szczecin, Wielkopolska 15, 70- 453 Szczecin, Poland 7 The IEA, University of Reading, Philip Lyle Building, Whiteknights Campus, Reading, RG6 6BX 8 Observatoire des Hauts Patys, F-84410 Bédoin, France 9 Villefagnan Observatory, France 10 Geneva Observatory, CH-1290 Sauverny, Switzerland 11 Les Engarouines Observatory, F-84570 Mallemort-du-Comtat, France 12 Stazione Astronomica di Sozzago, I-28060 Sozzago, Italy 13 Departamento de Sistema Solar, Instituto de Astrofísica de Andalucía (CSIC), Glorieta de la Astronomía s/n, 18008 Granada, Spain 14 Rose-Hulman Institute of Technology, CM 171 5500 Wabash Ave., Terre Haute, IN 47803, USA 15 Observatoíre du Bois de Bardon, 16110 Taponnat, France 16 1155 Hartford St; Elgin, OR USA 17 Instituto de Astrofísica de Canarias, C/ Vía Lactea, s/n, 38205 La Laguna, Tenerife, Spain 18 Gran Telescopio Canarias (GRANTECAN), Cuesta de San José s/n, E-38712, Breña Baja, La Palma, Spain 19 Lincaheira Observatory, Instituto Politécnico de Tomar, 2300-313 Tomar Portugal 20 School of Physical Sciences, The Open University, MK7 6AA, UK 21 Chungbuk National University, 1, Chungdae-ro, Seowon-gu, Cheongju-si, Chungcheongbuk-do, Republic of Korea 22 Korea Astronomy and Space Science Institute, 776 Daedeokdae-ro, Yuseong-gu, 305-348 Daejeon, Korea 23 Institute of Physics, Faculty of Natural Sciences, University of P. J. Šafárik, Park Angelinum 9, 040 01 Košice, Slovakia 24 Laboratory of Space Researches, Uzhhorod National University, Daleka st. 2a, 88000, Uzhhorod, Ukraine 25 Albox, Spain 26 Observatório Nacional Rua General José Cristino, 77, 20921-400 Bairro Imperial de São Cristóvão Rio de Janeiro, RJ, Brasil 27 Mt. Suhora Observatory, Pedagogical University, Podchora ˛zych˙ 2, 30-084, Cracow, Poland 28 4438 Organ Mesa Loop, Las Cruces, New Mexico 88011 USA 29 Command Module Observatory, 121 W. Alameda Dr., Tempe, AZ 85282 USA 30 Observatoire de Blauvac, 293 chemin de St Guillaume, F-84570 St- Estève, France 31 Lowell Observatory, 1400 West Mars Hill Road, Flagstaff, Arizona, 86001 USA 32 University of Adiyaman, Department of Physics, 02040 Adiyaman, Turkey 33 Bisei Spaceguard Center, Japan Spaceguard Association, 1716-3, Okura, Bisei, Ibara, Okayama 714-1411, Japan 34 Kepler Institute of Astronomy, University of Zielona Góra, Lubuska 2, 65-265 Zielona Góra, Poland

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